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Comparison of pathway analysis and constraint-based methods for cell factory design
Vítor Vieira1,
Paulo Maia2,
Miguel Rocha ORCID: orcid.org/0000-0001-8439-81721 &
Isabel Rocha1,3
Computational strain optimisation methods (CSOMs) have been successfully used to exploit genome-scale metabolic models, yielding strategies useful for allowing compound overproduction in metabolic cell factories. Minimal cut sets are particularly interesting since their definition allows searching for intervention strategies that impose strong growth-coupling phenotypes, and are not subject to optimality bias when compared with simulation-based CSOMs. However, since both types of methods have different underlying principles, they also imply different ways to formulate metabolic engineering problems, posing an obstacle when comparing their outputs.
In this work, we perform an in-depth analysis of potential strategies that can be obtained with both methods, providing a critical comparison of performance, robustness, predicted phenotypes as well as strategy structure and size. To this end, we devised a pipeline including enumeration of strategies from evolutionary algorithms (EA) and minimal cut sets (MCS), filtering and flux analysis of predicted mutants to optimize the production of succinic acid in Saccharomyces cerevisiae. We additionally attempt to generalize problem formulations for MCS enumeration within the context of growth-coupled product synthesis. Strategies from evolutionary algorithms show the best compromise between acceptable growth rates and compound overproduction. However, constrained MCSs lead to a larger variety of phenotypes with several degrees of growth-coupling with production flux. The latter have proven useful in revealing the importance, in silico, of the gamma-aminobutyric acid shunt and manipulation of cofactor pools in growth-coupled designs for succinate production, mechanisms which have also been touted as potentially useful for metabolic engineering.
The two main groups of CSOMs are valuable for finding growth-coupled mutants. Despite the limitations in maximum growth rates and large strategy sizes, MCSs help uncover novel mechanisms for compound overproduction and thus, analyzing outputs from both methods provides a richer overview on strategies that can be potentially carried over in vivo.
Genome-scale metabolic models (GSMM) are well proven tools for the in-silico analysis of the metabolism of living organisms. Indeed, the wide availability of whole-genome sequencing and annotation tools have enabled the reconstruction of a multitude of metabolic networks for various organisms [1]. Constraint-based (CB) modelling approaches allow the usage of GSMMs for simulation, analysis and strain optimization purposes which, despite the increasing scale of these models, have proven useful for a variety of studies. Phenotype prediction methods such as Flux Balance Analysis (FBA) [2] and its variants for mutant phenotypes [3,4,5], as well as Elementary Modes Analysis (EMA) [6] are capable of providing valuable insights on cell metabolism.
Designing optimized microbial strains for compound overproduction, however, is achieved through computational strain optimization methods (CSOMs), providing a rational approach for finding intervention strategies, as opposed to trial-and-error experiments [7, 8]. The purpose of most available CSOMs in the metabolic engineering (ME) context is to find sets of reactions that, when modified, force the cell to couple fluxes involved with the production of the desired compound with those required for cell growth. Growth-coupled phenotypes can be classified as weak, when compound production is only forced above a certain growth rate threshold, or strong when the fluxes driving the production of the desired compound are essential for cell growth [9].
CSOMs can be branched in two main categories: simulation-based (SB) and EMA-based methods. For simplification purposes we group bi-level mixed integer programming (MIP) and metaheuristic methods as SB methods, due to their similarities in evaluating candidate strategies, although with different methods for generating them (as recently reviewed by Maia and co-workers [8] and Machado et al. [10]).
SB CSOMs mostly derive from the bi-level framework first presented by Burgard and colleagues in the OptKnock approach [11], defining a strategy where an optimization layer is subject to the constraints posed by CB models. Several variations of this MIP problem have also been proposed to find more robust strategies and integrate omics data in the search [12, 13]. On the other hand, the OptGene approach later presented by Patil et al. [14], first introduced the usage of genetic algorithms for the optimization layer, which effectively detaches it from the simulation allowing for more flexible objective definitions and reducing the computational cost. Several improvements were subsequently published, including alternative evolutionary algorithms [15] and multiple objective functions [16]. Until recently, this type of CSOM comprised the only computationally feasible choice for strain optimization in GSMMs.
EMA-based CSOMs, on the other hand, search strategies for the desired ME goals throughout the entire solution space, resulting in predicted phenotypes that are not reliant on optimality assumptions. This implies an increase in computational demand, which severely hinders the scalability of most CSOMs of this type [8]. A prominent example is the concept of minimal cut sets (MCSs) [17], the smallest intervention targets that block a certain phenotype.
MCS enumeration algorithms were mostly reliant upon complete elementary mode (EM) enumeration, rendering it an infeasible task for models of greater scale and complexity. Several methods have been developed to tackle the computational limitations of EMA-based CSOMs by allowing partial enumeration of EMs through the use of sampling approaches [18], evolutionary algorithms [19], as well as other methods [20, 21]. The k-shortest EM method [21] is a relevant example of this, allowing enumeration of the k smallest EMs using a MILP (mixed-integer linear programming) approach.
The MCSEnumerator approach recently proposed by von Kamp et al. [22] successfully employs K-shortest EM enumeration in a dual linear problem through which EMs can be mapped to MCSs on the original network, as demonstrated by Ballerstein and colleagues [23]. The tool's feasibility for GSMMs has been demonstrated with the enumeration of synthetic lethals and knockout strategies for production of various compounds using a model for Escherichia coli [22]. Erdrich et al. have also explored this tool as a means to find design strategies for biofuel production in cyanobacteria [24] and a recent in vivo study has also confirmed its importance as a rational strain design tool for itaconate production in Escherichia coli, albeit using a smaller scale model [25].
Despite the existence of some studies highlighting the importance of EMA-based CSOMs to solve ME problems [24, 25], there is a gap regarding the analysis of these methods' outputs. Von Kamp and Klamt have recently assessed the feasibility of growth-coupled product synthesis in various organisms [26]. However, the application of these methods for strain optimization is still limited and few studies have discussed the biological implications of strategies obtained using these approaches. Recent developments by Harder et al. use an iterative rational strain design approach with successful in vivo outcomes, but do not apply MCSs directly as design strategies, relying instead on continuously applying partial MCSs and updating the metabolic model's environmental conditions for subsequent enumerations [25].
The comparison between EMA and SB CSOM derived strategies is also often overlooked. Most SB CSOMs are subject to optimality bias since they require an objective function which is often misleading and hard to define in certain organisms or metabolic systems. EMA-based CSOMs can now provide a suitable alternative to obtain highly robust design strategies with a low number of modifications, but there is a general lack of information regarding how well these perform against state-of-the-art SB CSOMs and whether their strategies produce better in vivo candidates.
The purpose of this work is to employ EMA-based and SB CSOMs in strain design applications using a case study involving the identification of knockout strategies for the production of succinic acid with Saccharomyces cerevisiae. To assess the feasibility of these methods, outputs from both categories are compared and different formulations of both approaches are also analysed. The aim is to bridge the gap between EMA and SB CSOMs by assessing performance metrics for each set of strategies and understanding the advantages and limitations of each method regarding their usage for ME tasks with different production and growth demands.
We present a pipeline including strain optimization, filtering and analysis of design strategies based on previous developments and packaged it as part of the Metabolic Engineering Workbench (MEW),Footnote 1 an open-source Java library developed in-house. Two additional problem formulations are also presented to allow greater flexibility of growth-coupled design strategies based on MCSs. A graphical user interface for the MCS enumeration algorithm was also made available as a plugin for the OptFlux metabolic engineering platform [27].
The methodological pipeline used for this work can be divided in three key steps as illustrated in Fig. 1. These steps are detailed in the methods sections.
Brief overview of the pipeline employed in this work. Optimization is performed using two strain design algorithms that produce reaction deletion strategies. These are filtered according to several criteria so they are compliant with the defined environmental conditions, minimum growth rate and production demands. The selected strategies are then subjected to various analysis methods
The primary goal with our workflow is to explore growth-coupled phenotypes through design strategies obtained with EMA-based and SB CSOMs. Five different strain design strategies, described in more detail in the Methods section, were obtained with SB CSOMs (using the evolutionary algorithm SPEA2) and EMA-based CSOMs (with MCSEnumerator) as part of our experimental setup.
Using SPEA2, biomass-product coupling maximization (EAw) and product minimum maximization (EAm) were tested, allowing different trade-offs between biomass and target compound production. EAw includes maximum cell growth and production flux at maximum cell growth as two separate objectives, while EAm uses the maximum cell growth rate and minimum product flux at maximum growth as objectives.
For MCSEnumerator, three strategies (MCSe, MCSf and MCSw) cover previously used and novel approaches that are more or less strict regarding robustness of the solutions by different definitions of the undesired spaces. In MCSe the undesired space contains low product yield flux vectors with a maximum substrate uptake rate and maintenance ATP rate above a defined value. In MCSf, low product yield phenotypes are blocked, assuming a fixed substrate uptake rate. In MCSw, low product flux phenotypes are blocked only when biomass fluxes are above a fraction of the maximum growth rate, aiming to reach strategies with less strict demands regarding product synthesis and its coupling with growth.
The purpose is to compare the five chosen strategies, considering a ME case study with the production of succinic acid with Saccharomyces cerevisiae as an objective. All analyses consider glucose as a carbon source with an uptake flux of 1.15 mmol.gDW− 1.h− 1 and aerobic conditions. For validation purposes, we also test different MCS formulations with previously determined Escherichia coli strategies also derived from MCSs. The results from this case study can be found as part of the Additional file 1.
Strategy performance
First, we compare the predicted performance of strategies from all approaches. The results for this analysis are highlighted on Fig. 2.
Left: Overview of various performance and robustness metrics for the solution sets featured by the analysis of succinate producing strategies. Displayed metrics: Production robustness (at 1 and 90% of maximum biomass), maximum growth rate, biomass-product coupled yield and product/substrate carbon yield. Right: Representation of the formulations featured in this case study. Colours match the ones used in the left part of the image. The undesired flux space is represented in red (only for cMCS enumeration problems), while the expected phenotype is represented in light green. For evolutionary algorithms, a green arrow represents the objective function
Production robustness
Strategies from the MCSe and MCSf sets provide fully robust production phenotypes with forced product synthesis even at very low growth rates (strongly-coupled). SPEA2 and MCSw strategies lead to lower product flux values and only guarantee product synthesis at higher growth rates. It is worth noting that a considerable amount of MCSw strategies do not allow growth-coupling at all, or do not meet our demand for production at 90% of the maximum mutant growth rate. This explains the need for a filtering step prior to our analysis.
EAm strategies lead to moderately robust phenotypes with higher product rates across different cell growth thresholds, with some of these strategies leading to strongly growth-coupled production phenotypes. These results are expected, considering the constraints imposed upon the various formulations.
The maintenance ATP constraint imposed on the MCSe formulation seems to negatively affect production robustness and rates, as this set underperforms the MCSf regarding these properties.
Cell growth and productivity
MCSf and MCSe strategies appear to be the worst performing strategies when considering cell growth. BPCY values for these sets are similar and considerably lower than for EA solutions. Despite the increased product flux values, strongly coupled MCSs (MCSe and MCSf) lead to much lower growth rates, which negatively impacts BPCY.
Conversely, the MCSw solution set, with lower production rates and robustness, leads to similar BPCY values since the maximum cell growth is much higher. EAw solutions reach the highest average BPCY figures, surpassing the EAm set, which, despite the increased product flux values, leads to lower maximum growth rates.
Regarding carbon yields, all strategy sets predict maximum glucose uptake, leading to carbon yields that are directly dependent upon the product flux. Thus, MCSe and MCSf solutions lead to higher carbon yields, followed by the EAm and EAw set. MCSw strategies have noticeably lower production rates, also leading to lower carbon yields.
The MCSe set appears to lead to slightly higher maximum growth rates and BPCY than the MCSf set. Maximum carbon yield, however, is lower (due to lower product rates).
The size of the analysed strategies is not homogeneous across all groups. MCS derived formulations are guaranteed to deliver the smallest solutions for the specified problem. SPEA2, being a heuristic approach, may lead to bigger solutions. Despite the large range of solution sizes for formulations based on this algorithm, EA-derived strategies with the highest BPCY values contain less than 10 knockouts. Additionally, the smallest strategies for succinate production are found in these solution sets (less than 5 knockouts), even though the average solution size across the entire sets is higher.
MCS derived strategies have low ranges as the number of solutions increases heavily with the proposed size, which imposes a computational limit. MCSw solutions were limited to sizes 3 through 5, while MCSe and MCSf were achieved with sizes up to 9 deletions. Productivity appears to increase with solution size in this particular case. However, the aforementioned changes in robustness and productivity metrics can be verified even when comparing similar solution sizes.
Phenotype analysis
The main results of the pathway distribution approach used in this work are depicted on Fig. 3, showing the activity of specific pathways towards the goal of achieving different growth-coupled phenotypes. Average flux values throughout the strategy sets were compared and merged into their respective pathways to illustrate this.
Overview of the usage of various pathways for the solutions obtained by each of the different design strategies for succinate production. The values represent the difference in the proportion of active reactions when compared with the wild-type simulated with parsimonious FBA. Pathway nomenclature was obtained on BioCyc and matched with KEGG identifiers assigned to reactions on the GSMM
Across the solutions from all strategies, there is a decreased usage of the oxidative branch of the pentose phosphate pathway (PPP), as compared to the wild-type. This correlates with the knockouts themselves on the pathway, found in EAw and both MCS solution sets. Flux on the non-oxidative branch is also generally decreased, although some reactions leading up to the amino-acid biosynthetic pathways remain active, which can be observed on Fig. 3. This can be seen through the increased usage of alanine, serine, glycine and glutamate biosynthetic pathways in strongly-coupled strategies and the EAm set.
The tricarboxylic acid (TCA) cycle is also disrupted in most of the proposed designs, leading to less active reactions in this pathway. This also leads to a lower activity in the electron transfer chain. Interestingly, the solutions from the EAw set, leading to higher growth rates, maintain flux in TCA cycle reactions, whereas the solutions from the remaining strategies do not. This could be explained by the high usage of fumarate reductase and the glyoxylate shunt (albeit with smaller flux) which is the primary mechanism found for succinate production in EAw solutions, in contrast with 4-aminobutyrate (GABA) degradation pathways used in strongly-coupled solutions.
The GABA degradation pathway plays a key role in the solutions obtained from strongly-coupled strategies. This pathway, otherwise known as the GABA shunt, leads to succinate production through degradation of GABA in the cytosol and serves as the main production mechanism in all MCSf and MCSe solutions and in a smaller proportion in EAm. Increased usage of this pathway seems to directly correlate with higher production robustness. Similarly, usage of the methylglyoxal catabolic pathway is also increased in solutions from MCSf, MCSe and EAm strategies. Through further analysis of the admissible flux ranges using FVA we were also able to find that GABA shunt fluxes become essential for the feasibility of most MCSf and MCSe solutions, which also shows that growth and production are coupled to this pathway.
It is worth noting that the strongly growth-coupled phenotypes we found were dependent on the imbalance of at least one cofactor. This finding is supported by the fact that the fixed maintenance ATP demand is essential to impose strong coupling – only weak coupling is achieved if the maintenance ATP pseudo-reaction bounds are lower. Additionally, in many MCSf solutions, the NADPH/NADP balance is disrupted through deletions that force glucose uptake through NADPH dependent pathways or flux through certain pathways dependent on these cofactors, such as folate interconversions, or methylglyoxal metabolism. Flux through the GABA shunt is then required to maintain a steady-state. These phenomena involving cofactor pools have already been described as possible mechanisms for growth-coupled synthesis by Erdrich et al. [24] and Hädicke et al. [28], although further experimental validation is still required.
Knockout frequency analysis
Analysis of the specific knockouts included in the obtained strategies is a key feature to determine which are the most important deletions leading to weak or strong coupling phenotypes. Figure 4 summarizes the frequency of the most common knockouts across all formulations.
Relative frequencies of the most common knockout combinations across all solution sets from the succinate case study. Darker shades of blue represent higher knockout frequencies. Labels for each reaction include its identifier in the model as well as the Enzyme Commission number attributed to it according to the iMM904 model reconstruction used in this work
A common disruption across the solutions from all strategies is the succinate dehydrogenase complex, a TCA cycle mitochondrial protein complex supplying electrons for the transfer chain and catalysing the conversion of succinate to fumarate. Subunit 2 (EC 1.3.5.1) of this complex is present in every solution, with subunits 3 and 1 being less present (ECs 1.3.5.1 and 1.3.99.1, respectively), in decreasing order of frequency. The importance of this disruption may come from the fact that this is the only available irreversible conversion of succinate to fumarate.
Glycolysis appears as a target for disruption in strategies with higher production robustness (EAm, MCSe and MCSf), with higher frequency of knockouts in kinases involved in glucose phosphorylation, such as hexokinase or glucokinase. The presence of PGI (EC 5.3.1.9) knockouts in most strategies may help explain the increased flux in the PPP and alternative glucose catabolic pathways.
The EA set includes many solutions involving serine biosynthesis from intermediate metabolites of glycolysis, blocking a possible branched pathway from glycolysis. MCS strategies, on the other hand, include knockouts that block glycolysis before this branching, namely PGK and GAPD (ECs 2.7.2.3 and 1.2.1.12, respectively), which force the usage of alternative pathways, such as methylglyoxal catabolism or serine biosynthesis.
Due to the decreased rates in the oxidative phosphorylation, fermentation appears as an alternative for pyruvate catabolism for many solutions. As such, succinate producing strategies with higher titres tend to include knockouts in PYRDC (EC 4.1.1.1), ALDD (EC 1.2.1.4) and ACOAH (3.1.2.1) reactions to reduce the number of viable alternative pathways that could drive carbon away from succinate production.
Knockouts in the PPP are somewhat prevalent across all strategy sets except MCSw. Regarding oxidative-phase knockouts, G6PDH (EC 1.1.1.49) appears in strongly-coupled MCSs with relatively high frequencies, especially in the MCSf set and in the EAw set.
Finally, in the TCA cycle, the cytosolic IDH (EC 1.1.1.42) knockout seems to be a relatively common occurrence in MCSf solutions, which leads to strongly coupled succinate production through forced usage of the TCA cycle producing 2-oxoglutarate in the mitochondria and then transported outside to feed the GABA shunt for cytosolic succinate to be synthesized.
Two metabolic routes for succinate overproduction from the TCA cycle have been described in the literature, respectively, the oxidative and reductive branches of the TCA cycle [29]. The oxidative route yields a theoretical maximum of 1 mol succinate/mol glucose. Metabolic engineering strategies have been tried and tested successfully for this first route [30, 31]. The reductive pathway, usually associated with anaerobic conditions, involves succinate production through carbon dioxide fixation and relies on pyruvate carboxylation into oxaloacetate from which TCA metabolites up to fumarate can be synthesized. Fumarate reductase catalyses the conversion from fumarate to succinate.
Early efforts using yeasts as succinate-overproducing microbes have focused on the enhancement of sake brewing processes with Saccharomyces cerevisiae [32]. This study describes, among others, a strain with a succinate dehydrogenase (SDH) gene disruption that leads to increased succinate production in aerobic conditions. A similar finding is confirmed in another study, showing that disruption of any of the SDH subunits leads to increased aerobic succinate production and that higher succinate titres could be correlated with lower SDH complex activity [33].
Additionally, it is shown that growth and substrate uptake are not significantly altered in strains with disrupted pairs (SDH1 and SDH2) [33]. The structural analysis featured in this work reveals that inactivation of at least two subunits of the SDH system (along with further knockouts) is required to allow strongly coupled production of succinate. Another yeast strain with a SDH2 knockout was experimentally tested, albeit on a different species (Yarrowia lipolytica), resulting in production at a minimum of 56% of the theoretical maximum [34]. The results obtained in our analysis are in accordance with experimental data regarding the SDH2 knockout which is an essential component of strongly coupled strategies.
Further studies found improved strategies for aerobic succinate production in yeast. The addition of a isocitrate dehydrogenase (IDH) knockout to the SDH1/2 pair was studied as a means of diverting citric acid cycle intermediates into the glyoxylate cycle, resulting in succinate overproduction [31]. This mutant was able to achieve a yield almost four times greater than the wild-type strain and the addition of a mitochondrial isocitrate dehydrogenase knockout (IDP) leads to even higher succinate concentrations at the cost of reduced growth rate [31]. Some strongly coupled solutions featured in this work exhibit glyoxylate cycle flux, although these have low representation within the entire solution set.
An alternative strategy for glyoxylate shunt carbon redirection was presented by Otero et al. [30], involving two knockouts, namely at the SER3/SER33 gene (catalysing the PGCD reaction in the iMM904 metabolic model) and at subunit 3 of the SDH complex. Carbon flux is redirected towards glyoxylate to allow serine production, with succinate as a by-product of using this metabolic route. This strategy was developed using in silico tools and then experimentally tested. Not surprisingly, some of the solutions obtained in the present work include those strategies.
There is in vivo evidence that succinate production in Saccharomyces cerevisiae from the GABA shunt is residual in normal fermentation conditions (with oxygen limitation) [35]. It is still unclear whether the GABA shunt is a valid alternative for succinate production in aerobic conditions, but our results make this pathway a relevant candidate for further study in aerobic conditions.
Regarding the reductive pathway, not many efforts have been implemented regarding the optimization of this route, mainly due to cofactor imbalances. However, reported strategies involve the deletion of fermentative competing pathways, as observed in some of our solutions [36].
In this work, we present a comparison of two relevant constraint-based strain design approaches for ME applications through in silico analysis of the resulting knockouts strategies. Two alternative formulations for MCS enumeration were tested, yielding mutants with varying growth and production demands, demonstrating the flexibility of the intervention problem framework coupled with the MCSEnumerator algorithm. The results shown in this work reinforce the importance of MCSs as a rational approach for ME applications.
When comparing EA and MCS-derived strategies, there is a clear trade-off between robustness and maximum cell growth. Growth-coupling strength increases as maximum biomass decreases, leading to strongly-coupled cMCSs with high product flux and low growth and weakly-coupled EA strategies with opposing features. Phenotypes with very high yields have been found using cMCSs, largely surpassing those from SPEA2, but cell viability is still unknown as the low cell growth rate may indicate that these strategies incur in lethal modifications in vivo.
Nevertheless, it is still possible to obtain growth-coupled phenotypes with low production robustness, using the MCSw formulation firstly described in this work. Strategies obtained using this method have lower predicted productivity, but overall higher growth rates. While possibly unsuitable for direct in vivo application, these strategies may serve as starting points for other strain design algorithms, upon which more knockouts can be added to achieve higher productivity.
With the EAm formulation, SPEA2 can also be used to obtain strongly-coupled strategies, with similar robustness to those found using MCSEnumerator. However, the latter can enumerate smaller strategies, which is advantageous for in vivo implementation.
The MCSf formulation presented and tested in this work allowed us to find strategies with high product flux values and robustness but lower cell growth rates overall. This helped us reveal the importance of including a coupling component (such as the maintenance ATP assumption on the MCSe formulation) when searching for MCS-based strategies. Indeed, when the maintenance ATP assumption is replaced with a fixed substrate constraint, other cofactor wasting mechanisms arise.
For practical purposes, it is clear that EAw strategies confer high BPCY at a relatively low number of knockouts to achieve that goal. However, product synthesis can only be guaranteed with growth rates close to the wild-type, which casts uncertainty on whether these strategies will have a noticeable effect in vivo.
The strong growth-coupling knockouts suggested by MCS enumeration have practical implications concerning their size, which exceeds six knockouts in the best possible case, aside from the relatively low growth rate and BPCY. However, they are promising candidates for adaptive laboratory evolution, since the target organism may acquire mutations that lead to increased growth rates while maintaining the growth-coupling with product synthesis.
Moreover, using multiple strain design approaches with varying productivity and robustness demands further helped us to find knockouts providing a beneficial compromise between maximum cell growth, production flux and yields.
The findings from the Saccharomyces cerevisiae case study featured in this work show that strongly growth-coupled strategies point towards increased usage of the GABA degradation pathway as production demands grows, at the cost of lower growth rates. This design strategy requires further experimental validation, despite the existence of studies acknowledging it as a potential ME target for succinate production.
This work only includes CSOMs for finding sets of reaction deletions; however, it is possible for both methods to also suggest strategies involving modification of the expression levels of metabolic genes [37], which could be addressed as a future case study.
Overall, we conclude that EMA-based CSOMs are valuable tools for strain design that yield design strategies with the highest production fluxes and robustness and uncover the biochemical mechanisms that lead to these phenomena. A thorough analysis of these algorithms' outputs serves as valuable tool for guiding rational approaches to strain optimization by highlighting relevant deletions to achieve the ME goal. Aiming for both weak and strong coupling phenotypes using these algorithms is essential towards finding a compromise between viability and productivity and identifying common targets whose modification is essential to produce a given compound. We show that it is possible to fine-tune the compromise between robustness and growth rates by choosing different algorithms or setups among those tested in this work.
Constraint-based modelling and analysis
In this work, we attempt to find design strategies for Saccharomyces cerevisiae using the iMM904 reconstruction presented by Mo and colleagues [38] with additional changes proposed by Pereira et al. [39] for improved predictive accuracy.
The constraint-based approach to modelling represents stoichiometry restricted by steady-state [40], through a system of linear equations complemented with inequalities representing bounds for each flux. Such a system is formulated as shown on Eq. 1, assuming S as a m-by-n matrix encoding the network stoichiometry, v as the flux vector, and l,u as the vectors encoding the lower and upper bounds, respectively.
$$ {\displaystyle \begin{array}{l}S.v=0\\ {}{l}_i\le {v}_i\le {u}_i\forall i\in \left\{1,\dots, n\right\}\\ {}l,v,u\in {\mathrm{\mathbb{R}}}^n\end{array}} $$
Typical approaches such as Flux Balance Analysis (FBA) [2] attempt to find a single flux distribution, by optimizing towards a given objective function, subject to the previously mentioned constraints.
The most common approach is to maximize cell growth, although other functions have also been proposed, mainly to address phenotype prediction in mutant cells [3, 4]. The optimization problem posed by FBA typically leads to multiple optima for the same objective, eliciting the development of alternative methods for prediction and analysis of simulated fluxes such as parsimonious FBA (pFBA) or Flux Variability Analysis (FVA).
These two methods are later used to analyse the performance of our design strategies. pFBA extends the FBA problem by constraining the optimum to that which minimizes overall flux usage (sum of absolute flux values) [5] and FVA provides theoretical maximum and minimum flux values for the model/environmental condition assuming a fraction of the maximum growth rate [41].
An alternative approach to analyse fluxes using constraint-based models is to employ convex analysis, usually with EMA methods, since the linear system of equations is a convex polyhedral cone when all reactions are irreversible [42], which is usually achieved by replacing reversible reactions with two forward and backward fluxes.
EMA is based on the concept of elementary modes (EM), defined as any flux distribution e that solves the system in Eq. 1 containing only irreversible reactions, such that: (1) two split reversible reactions are not simultaneously active; (2) e is a feasible solution, thus satisfying steady-state and thermodynamic constraints: (3) if an active reaction of e is removed, e is no longer feasible. These properties assure EMs are the smallest sets of reactions allowing valid metabolic conversions, and any steady-state flux distribution complying with thermodynamic constraints can be defined as a linear combination of the full set of EMs [6].
Minimal cut sets (MCSs) are an extension of this concept, defining sets of reactions that, when removed or blocked, disable certain selected EMs [17]. With appropriate EMs to block, it is possible to find minimal strategies to achieve or even guarantee the expression of desired phenotypes. Constrained MCSs (cMCSs) extend this concept even further to guarantee that desired phenotypes are not among those that are blocked [43].
The intervention problem framework, first presented by Hädicke et al. [43], provides a flexible approach for formulating ME problems with cMCSs. An intervention problem I (T, D) requires two inputs, namely:
Target flux space (T), containing undesired EMs to be blocked.
Desired flux space (D), comprising desirable EMs, which cannot be blocked.
For any problem I, the resulting MCSs will render the EMs in T invalid while also keeping EMs in D active. For this reason, not all MCSs for I can be considered cMCSs [43]. Both T and D can be defined as part of the solution space of the flux vector v, leading to the linear system in Eq. 2, when supplied with a b vector. These formalisms allow the definition of spaces to block through well-defined bounds for certain fluxes, bypassing the need for selecting specific EMs.
$$ \left[\begin{array}{ccc}{T}_{1,1}& \cdots & {T}_{1,n}\\ {}\vdots & \ddots & \vdots \\ {}{T}_{m,1}& \cdots & {T}_{m,n}\end{array}\right].v\ge \left[\begin{array}{c}{b}_1\\ {}\vdots \\ {}{b}_m\end{array}\right] $$
Growth-coupled product synthesis
The objective of virtually all CSOMs is to find subnetworks in which the flux to be optimized is guaranteed to be active, considering some restrictions. Growth coupling with production occurs when there is at least one possible flux vector where the production flux p is kept above a threshold pmin and the growth pseudo-flux b is kept above a minimum bmin. This definition is broad, and thus, two types of growth coupling can also be distinguished. It should also be noted that the original definition involves product and growth per substrate yields. Since our experimental conditions assume a fixed substrate uptake rate, these definitions have been adapted for flux values only.
Strong coupling occurs if all admissible flux vectors force p to be kept above pmin and at least one of these allows b to be kept above bmin, implying there is always product synthesis and that its flux can be coupled with growth.
Weak coupling, on the other hand, imposes that all flux vectors where b > bmin must also force p > pmin. This is a less demanding constraint since the product is not essential until the growth rate rises above a certain threshold.
For this work, we set the minimum product and growth thresholds for coupling at a very small value ε = 10− 4 to avoid arbitrarily discarding growth-coupled solutions with low production fluxes.
Strain optimization algorithms
Multi-objective evolutionary algorithms
In this study, we employ the OptGene method using the Strength Pareto Evolutionary Algorithm 2 (SPEA2) as representative of SB CSOMs. This approach allows for the definition of multiple objective functions [16], providing a flexible framework for defining appropriate ME goals. Unlike typical evolutionary algorithms where an aggregate function must be defined if multiple objectives are required, SPEA2 evaluates these separately and prioritizes those with the highest Pareto optimality.
The overall algorithm employed in this setup includes the following steps:
Generate an initial random set (population) of candidate strategies (individuals) containing reaction knockouts
Decode each individual and convert reaction deletions into flux constraints to be applied to the CB model
Determine the fitness value for each individual using a suitable objective function. This function can be linear or nonlinear.
Select the best individuals according to their fitness values. The assumption here is that fitter individuals will give rise to better offspring strategies.
Replace the population for a new one by applying mutation and crossover operators to the previous set of strategies, thus yielding new strategies that combine reaction knockouts from the fittest individuals.
Obtain the design strategies from the population if the stopping criteria are met. Otherwise, return to step 2.
To provide a better comparison with EMA-based CSOMs, two sets of fitness functions are used. The first is a typical approach in ME problems, maximizing both product and biomass fluxes, while the second represents a search for more robust strategies, enforcing growth-coupled compound synthesis at lower growth rates to reach growth-coupling phenotypes similar to those found with EMA-based CSOMs. Both are represented on Table 1 and described in the following.
Table 1 Overview of the evolutionary algorithm formulations used in this work. Both objective functions are represented, as well as other constraints that condition the acceptance of a given candidate as a valid solution
Biomass-product coupling maximization (EAw)
A multi-objective approach using maximum cell growth and production flux at maximum cell growth as two separate objective functions. This represents growth coupling as the strategies are evolved towards increased product and biomass production values but does not guarantee strong coupling as it inherently assumes biomass maximization.
Product minimum maximization (EAm)
This approach includes the maximum cell growth rate and minimum product flux at maximum growth as objectives. The latter is determined through FVA [41] by setting biomass production at its maximum and minimizing the product flux. Additionally, the cell must always synthesize the product (product flux above 0) if the growth rate is at 50% of the theoretical maximum value.
For this work, candidate strategies were subject to a maximum of 20 deletions. Although that amount of deletions is not acceptable for in vivo implementation, we have decided not to restrict the enumeration with a smaller strategy size to avoid discarding potentially useful phenotypes for our analysis. Since this is an evolutionary algorithm, a pool of solutions is evolved until a stopping criterion is reached, in this case being a cap of 105 objective function evaluations. Due to the heuristic nature of the algorithm, it was executed 10 times for each case.
MCSEnumerator
The MCSEnumerator approach is based on the findings of Ballerstein et al. [23] reporting the properties of dual networks based on original stoichiometric metabolic networks. For this problem, we first assume a m-by-n stoichiometric matrix S with reversible and irreversible reactions identified by sets of indices r and i, respectively.
The dual network/system is essentially a transposed and extended network based on the original, where reactions become metabolites and vice-versa and is represented in Eq. 3. The metabolites representing the original reactions are produced by newly added reactions (represented by vectors v and h) and consumed by the reactions added as part of the target space T. EFMs enumerated in this network in which fluxes in w are active are MCSs of the original network. The original mapping is obtained by checking which fluxes in the vectors v and h are active, and thus, satisfying the demand caused by the fluxes in w.
$$ {\displaystyle \begin{array}{l}\left[\begin{array}{cccc}{S}_i& I& -I& {T}_i\\ {}{S}_r& I& -I& {T}_r\end{array}\right].\left[\begin{array}{c}u\\ {}v\\ {}h\\ {}w\end{array}\right]\begin{array}{c}\ge 0\\ {}=0\end{array}\\ {}b.{w}^{\hbox{'}}>c;c>0\\ {}u\in {\mathrm{\Re}}^m\\ {}v,h,w\in {\mathrm{\Re}}_{0+}^n\end{array}} $$
In MCSEnumerator, this dual network is used as input for the k-shortest algorithm which is integrated in the linear problem by adding binary variables for each flux in v and h which are set to 1 if the respective flux is active or 0 otherwise. Additional constraints ensure that the split reaction pairs cannot be simultaneously active. Finally, the objective function is set towards the minimization of the number of active reactions in v and h.
The intervention problem framework described in the previous section is used to formulate cMCS enumeration problems, with all formulations written in the form I(T, D), with T and D representing flux spaces to block and keep. The desired space D can be defined as appropriate environmental conditions supplied as constraints. Additionally, a lower bound constraint on the biomass pseudo-reaction discards lethal cMCSs. Both T and D are matrices representing the linear coefficients for inequalities in the form T.v ≤ b, assuming v as the flux vector and b as a vector of size c with one value for each constraint to add.
Defining target spaces
Different combinations of constraints for the definition of the target flux space T were tested with MCSEnumerator. Four main constraint types can be summarized, namely:
Product constraint, indicating that the product must stay above a certain threshold. Product/Substrate and Product/Biomass yields, as well as simple product flux constraints were tested.
Substrate constraint, either fixed or with an upper bound on the maximum uptake rate. Can be considered as an environmental condition restricting the feasible solution space.
Coupling constraint, referring to additional flux bounds introducing assumptions to the problem to further restrict the solution space. In our setup, we have tested a constraint to set the flux of the maintenance ATP pseudo reaction.
Biomass constraint, where a biomass lower bound is set to further restrict the target flux space.
From this early analysis, we have determined that biomass constraints together with product yield constraints negatively affect computational speed, and do not result in additional solutions. Furthermore, we have also determined that non-fixed substrate constraints require an additional coupling component, otherwise leading to an infeasible enumeration problem. In such scenarios, it is possible that the origin point of the flux cone is included in the undesired space, for which there are no MCSs that can block it, should the problem only contain homogeneous constraints.
Three formulations were selected to consider different strain design objectives. Table 2 highlights the constraints used in these formulations, which are:
MCSe: The undesired space contains low product yield flux vectors with a maximum substrate uptake rate of smax and maintenance ATP rate above m. A similar formulation is employed in Escherichia coli ME case studies with MCSEnumerator [22].
MCSf: Low product yield phenotypes are blocked, but assuming a fixed substrate uptake rate of smax. The aim is to eliminate the maintenance ATP assumption, while maintaining the origin point of the flux cone out of the undesired space.
MCSw: This formulation attempts to find less robust solutions by blocking low product flux phenotypes only when biomass fluxes are above a fraction F of bmax (maximum growth rate). The aim is to try and reach strategies with less strict demands regarding product synthesis and its coupling with growth. While many strategies may not allow growth-coupling at all, some weak-coupled cMCSs can theoretically be found using this formulation.
Table 2 Overview of the constraints for the undesired space (T) used in formulations involving enumeration of constrained minimal cut sets
Strategies from the optimization algorithms were filtered according to their expected growth-coupling strength. A filtering pipeline using FVA [41] was developed to ensure that three key conditions were met, namely:
Compliance with environmental conditions: Strategies must be feasible with substrate uptake and maintenance ATP constraints, among others;
Growth rate: Maximum mutant growth rates must reach at least 1% of the wild-type strain;
Growth-coupling with product: Mutants must carry non-zero product flux with the cell at or above 90% of the maximum mutant growth rate (growth coupling phenotypes).
It is worth noting that while the MCSe and MCSf formulations would always lead to growth-coupled cMCSs, this is not the case for MCSw nor solutions based on evolutionary algorithms. Thus, this step is required to discard any strategy that does not lead to growth-coupled production.
A global assessment of the selected solution sets was performed regarding four key aspects:
Performance – Predict production titres and robustness;
Pathways – Compare overall pathway usage across strategy groups;
Phenotype – Investigate individual strategies from various groups to find mechanisms leading to growth-coupling phenotypes;
Structure – Assess the impact of individual knockouts in each strategy group relative to their phenotype/pathway pattern.
The constraints and objective functions used to obtain the productivity metrics are highlighted on Table 3. These metrics are:
Production robustness: Minimum feasible production rate given a lower limit for cell growth. Production is considered robust for a certain biomass threshold, if the corresponding minimum rate is not null.
Predicted fluxes: Assuming the pFBA flux distribution as the predicted phenotype, maximum cell growth, substrate uptake and production rate fluxes are obtained and considered as predictions of in vivo behaviour.
Production yield: Biomass-product coupled yield [14] was used to assess the productivity of the mutant strains, since it takes cell growth, production rate and substrate uptake into account.
Table 3 Phenotype prediction methods used to obtain the performance metrics featured in this work
Using the pFBA flux distributions for all the mutant strains, it is possible to compare pathway usage across various sets of solutions. With the wild-type distribution as reference, the purpose is to find how different groups of strategies influence pathway usage.
Consider a n-by-m binary matrix p with n pathways and m reactions. Each element pnm has a value of 1 if reaction m is part of a pathway n, and 0 otherwise. For each strategy with a flux distribution v, a "pathway distribution" is given by d (v, p), a vector with n elements, one for each pathway, containing the number of active reactions for each pathway.
$$ d\left(v,p\right)=\frac{\sum_{i=1}^m\left({p}_{ji}.\left|\mathit{\operatorname{sgn}}\left({v}_i\right)\right|\right)}{\sum_{i=1}^m{p}_{ji}}\forall \mathrm{j}\in \left\{1,\dots, \mathrm{n}\right\} $$
For each strategy in a given group, a pathway distribution is generated according to the formula in Eq. 4, assuming sgn(x) as the sign function of x. Assuming this is performed for s solutions, a s × n matrix P can be assembled, containing all of the pathway distributions for the strategy set. The global difference from the wild-type can be found through determination of the mean value for each pathway in P and subtracting it to the corresponding value in the wild-type.
Enumeration of cMCSs was performed using mcslibrary [44], an open-source Java library implementing the methods described by Von Kamp and associates used for MCSEnumerator [22]. The tool, developed in-house, has been validated using the case studies that were also applied to MCSEnumerator. The analysis pipeline for the entire set of design strategies was also performed using this tool. Source-code and test scripts are available from the Git repository at https://github.com/MEWorkbench/mcslibrary. A graphical user interface implementing some functionalities of this library is available as a plugin (optflux-mcs) for the OptFlux ME software platform [45]. Currently on its third major release, it is an open-source tool that allows users to load metabolic models and use a wide array of phenotype prediction methods and CSOMs in a user-friendly setting.
Our SPEA2 workflow is currently available as part of the MEWorkbench's mewcore library containing core methods used within OptFlux for simulation, analysis and strain optimization. Source-code is available on the Git repository at https://github.com/MEWorkbench/mewcore. The evolutionary algorithm is part of JECoLi (Java Evolutionary Computation Library), a library implementing generic evolutionary algorithms with source code available at https://github.com/jecoli/jecoli.
Data analysis and plots shown in this work were processed using the R programming language. Scripts to perform this analysis are available at http://www.bio.di.uminho.pt/csomcomparison/.
Availability and requirements
Project name: mcslibrary.
Project home page: https://github.com/MEWorkbench/mcslibrary
Operating systems(s): Platform independent.
Programming language: Java.
Other requirements: IBM® ILOG® CPLEX® Optimizer version 12 or higher, Java® 1.7 or higher.
License: GNU Lesser General Public License 2.1.
The datasets generated and analysed during the current study are available at http://www.bio.di.uminho.pt/csomcomparison/.
https://github.com/MEWorkbench
ACK:
Acetate kinase
ACOAH:
Acetyl-CoA hydrolase
ALDD:
Aldehyde dehydrogenase
ATP:
Adenosine triphosphate
BPCY:
Biomass-product coupled yield
CB:
Constraint-based
cMCS:
Constrained minimal cut set
CSOM:
Computational strain optimization method
Evolutionary algorithm
Elementary mode
EMA:
Elementary modes analysis
FBA:
Flux Balance Analysis
FVA:
Flux Variability Analysis
G6PDH:
Glucose 6-phosphate dehydrogenase
GABA:
GAPD:
Glyceraldehyde-3-phosphate dehydrogenase
GSMM:
Genome-scale metabolic model
IDH:
MCS:
Minimal cut set
Metabolic engineering
MILP:
Mixed-integer linear programming
Mixed-integer programming
NADP:
Nicotinamide adenine dinucleotide phosphate
NADP(H):
Nicotinamide adenine dinucleotide phosphate (reduced)
pFBA:
Parsimonious Flux Balance Analysis
PGCD:
Phosphoglycerate dehydrogenase
PGI:
Glucose-6-phosphate isomerase
PGK:
Phosphoglycerate kinase
POX:
Pyruvate oxidase
PPP:
PTA:
Phosphate acetyltransferase
PYRDC:
Pyruvate decarboxylase
SB:
Simulation-based
SDH (1/2):
Succinate dehydrogenase (subunit 1/2)
SPEA:
Strength Pareto Evolutionary Algorithm
TCA:
Tricarboxylic acid
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The authors thank Paulo Vilaça for helpful discussions regarding constraint-based modelling methods and Hugo Giesteira for the assistance in integrating the minimal cut set enumeration algorithm into OptFlux.
The authors thank funding from the projects "DeYeastLibrary – Designer yeast strain library optimized for metabolic engineering applications", Ref.ERA-IB-2/0003/2013, funded by national funds through FCT/MCTES, DD-DeCaf and SHIKIFACTORY100, both funded by the European Union through the Horizon 2020 research and innovation programme (grant agreements no. 686070 and 814408). This study was also supported by the Portuguese Foundation for Science and Technology (FCT) under the scope of the strategic funding of UID/BIO/04469/2019 unit and BioTecNorte operation (NORTE-01-0145-FEDER-000004) funded by the European Regional Development Fund under the scope of Norte2020 - Programa Operacional Regional do Norte. The authors acknowledge the use of computing facilities within the scope of the Search-ON2: Revitalization of HPC infrastructure of UMinho" project (NORTE-07-0162-FEDER-000086), co-funded by the North Portugal Regional Operational Programme (ON.2 – O Novo Norte), under the National Strategic Reference Framework (NSRF), through the European Regional Development Fund (ERDF). VV also thanks funding from FCT/MCTES for the PhD studentship with reference SFRH/BD/118657/2016.
The funding bodies did not play any role in the design of the study, collection, analysis, and interpretation of data or writing the manuscript.
Centro de Engenharia Biológica, Universidade do Minho, Braga, Portugal
Vítor Vieira, Miguel Rocha & Isabel Rocha
SilicoLife Lda, Braga, Portugal
Instituto de Tecnologia Química e Biológica António Xavier, Universidade Nova de Lisboa (ITQB-NOVA), Oeiras, Portugal
Isabel Rocha
Vítor Vieira
Miguel Rocha
IR and MR proposed the project and case studies. All authors were involved in the conception of the featured pipeline and formulations. VV implemented the minimal cut set enumeration algorithm and strategy analysis tools, PM implemented the evolutionary algorithm and both helped in generating design strategies for the case studies. VV analysed the results with major contributions from all authors and drafted the manuscript. All authors contributed to and approved the final manuscript.
Correspondence to Miguel Rocha.
The authors declare that they have no competing interests. PM declares that SilicoLife Lda did not play any role in the design of the study, data collection, and analysis of the results.
Compound overproduction in Escherichia coli using minimal cut set based strain design. (DOCX 296 kb)
Vieira, V., Maia, P., Rocha, M. et al. Comparison of pathway analysis and constraint-based methods for cell factory design. BMC Bioinformatics 20, 350 (2019). https://doi.org/10.1186/s12859-019-2934-y
Genome-scale metabolic models
Computational strain design
Metabolic pathway analysis
Minimal cut sets
Analysis and modelling of complex systems | CommonCrawl |
\begin{document}
\title{\textbf{The Martingale Representation Theorem and Clark-Ocone formula}} \author{Deborah Schneider-Luftman} \date{} \maketitle
\begin{abstract} In this paper we explore the fundamentals of the Martingale Representation Theorem (MRT) and a closely related result, the Clark-Ocone formula. We also investigate how far these theorems can be taken, notably beyond the regular Sobolev spaces, through changes of measures and enlargement of filtrations. We look at Brownian motion (B.M.) driven continuous martingales as well as Jump and L$\acute{e}$vy process-driven martingales \footnote{This article was written as a dissertation thesis, in partial fulfilment of the requirements for the MSc in Mathematics and Finance of Imperial College of Science, Technology and Medicine (Imperial College London, August 2012).}. \end{abstract}
\section*{Introduction} \addcontentsline{toc}{section}{Introduction}
The representation of martingales as an integral with respect to a fundamental process has been the subject of considerable research since the 1960's. Indeed, a key aspect of Financial Maths is to represent the evolution of the prices of financial assets and of various portfolios. In particular considerable attention is given to designing hedges for investment portfolios made of options and other derivatives. To illustrate an application, we borrow the setting provided by Black and Scholes \cite{BlackScholes}, where the value $C_0$ of a contingent claim expiring at time T written on a specific payoff C is worth: \[ C_0 = \mathbb{E}(C_T e^{-rT} \mid \mathcal{F}_0), \] and, \[C_t = \mathbb{E}(C_T e^{-r(T-t)} \mid \mathcal{F}_t). \] We seek to create a hedge for $C_t$, therefore we build $V_t$ which is composed of a traded stock and bond $(S_t, B_t)$. In order to pick the right quantities $(\phi_t, \psi_t)$ for $(S_t, B_t)$ we need to know how $C_t$ is represented at t $\in [0,T]$. The aim is then to create a self-financing strategy where: \[ C_t = V_t := \phi_t S_t + \psi_t B_t \mbox{ , } \forall t \leq T. \] Both $S_t$ and $B_t$ have known representations, depending on the framework used. Typically, \[ dS_t = S_t \mu_t dt + S_t \sigma_t dW_t \] where $\mu_t$ and $\sigma_t$ are the parameters of the Geometric Brownian motion that $S_t$ follows, as assumed in the Black-Scholes framework. Additionally, \[ dB_t = - rB_t dt. \] So we have: \begin{eqnarray*} dV_t &=& \phi_t dS_t + \psi_t d B_t \\ &=& (\phi_tS_t\mu_t - \psi_trB_t)dt + \phi_tS_t\sigma_t dW_t. \end{eqnarray*} \\ But since the discounted value of $C_t$ is an $\mathcal{F}_t$-martingale: \begin{equation} C_t e^{-rt} = \mathbb{E}(C_T e^{-rT} \mid \mathcal{F}_t), \label{introeq} \end{equation} we can apply the Martingale Representation Theorem (see 1.1) to see that: \[ C_t e^{-rt} = C_0 + \int^t_0 \pi_s dW_s \] for some $\mathcal{F}$-predictable and square integrable $\pi_t$. Then the following equality must hold: \begin{eqnarray*} C_t &=& V_t \mbox{ , } \forall t \leq T. \\ \mbox{So, } d C_t &=& dV_t \\ rC_tdt + e^{rt}\pi_t dW_t &=& (\phi_tS_t\mu_t - \psi_trB_t)dt + \phi_tS_t\sigma_t dW_t. \end{eqnarray*} And thus, we now have a replicating strategy: \begin{eqnarray*} \phi_t &=& \frac{e^{rt}\pi_t }{S_t \sigma_t}, \\ \psi_t &=& \frac{\phi_tS_t\mu_t - rC_t}{rB_t}. \end{eqnarray*} While this result is a useful indication of how Hedging works, it raises a lot of questions and by itself cannot be applied directly. First of all, what is $\pi_t$? Without an explicit form for $\pi_t$, the above formula cannot be utilised in practical simulations. This is where the Clark-Ocone formula comes in. From the conventional form developed by Clark \cite{Clark}, the integrand $\pi_t$ has an explicit formula for continuous martingales contained within a Sobolev space $\mathbb{D}_{1,2}$ (see extensive definition in Chapter 2). The result can be extended to work beyond $\mathbb{D}_{1,2}$ and with martingales in various forms. Additionally the result \ref{introeq} states that $e^{-rt}C_t$ is a straightforward $\mathbb{P}$ /$\mathcal{F}$-martingale, however this is often not the case.\\ In order to achieve a condition like in \ref{introeq} we need to perform a change of measure, consequently the MRT and Clark-Ocone formula need to be adapted. In the same vein the MRT and Clark-Ocone formula can be modified in order to perform an enlargement of filtrations. For our uses this is required when we are looking at adding extra information to the base filtration $\mathcal{F}$ and hence creating a larger filtration $\mathcal{G}$. $\mathcal{F}$-martingales can be transformed into $\mathcal{G}$-martingales by adding a drift term $\mu$, which can be done in a similar way to changing measures with the Girsanov theorem. We will look at deriving a measure-valued adaptation of the MRT and Clark-Ocone formula to indentify the drift term $\mu$ and allow for the enlarged filtration $\mathcal{G}$ to exist. \\
\\ What justifies the use of the Martingale representation theorem within $L^2(\Omega, \mathcal{F}, \mathbb{P})$, for a given triple $(\Omega, \mathcal{F}, \mathbb{P})$? Why does it work and what does it indicate about the stochastic space on which traded assets evolve? One aim of this paper is to investigate the bottom-line facts that make integral representation possible for a given class of stochastic processes\\ \\ The example we have developed above is based on a continuous Brownian filtration. Despite their usefulness, works on Brownian filtrations are too restrictive as real financial assets are not just driven by continuous stochastic processes. In order to produce more realistic results, a large number of papers have focused their attention on understanding the representation of L$\acute{e}$vy and jump martingales. In this paper we will aslo explore the uses of the MRT and Clark-Ocone formula beyond continuous processes, notably through the work of Chou $\&$ Meyer \cite{choumeyer}, Boel, Wong $\&$ Varaiya \cite{Boel} and Davis \cite{Davis2}.\\ \\In the 1st section we review how to prove the MRT for continuous and general filtrations, in order to highlight the fundamental structure of the $L^2$ spaces within which financial assets evolve. In the 2nd section, we build on the proven results of section 1 to give an explicit representation to the integrand $\pi_t$ in the continuous and dis-continuous cases. In the 3rd section, we explore the generalization of the MRT and Clark-Ocone formula, notably through changes of measures and filtration \\ \\ Throughout this paper, we will assume knowledge of Mathematical finance explored in most introductionary courses. Primarily, we assume the readers have a background in stochastic calculus and familiarity with the definition of Stochastic bases, sigma-fields, filtrations, martingales and uniform integrability (U.I.). We also assume the readers have an understanding of the Girsanov theorem and Radon-Nikodym derivative, as well as of concepts of portfolio Hedging.
\section{The Martingale Representation theorem: Statement and Proof}
We begin this chapter by stating the theorem of interest in its most familiar form for Brownian martingales.
\subsection{Martingale representation theorem: A regular Brownian motion approach }
Let $B=(B_t)$ be a Brownian motion on the stochastic base $(\Omega, \mathcal{F}, (\mathcal{F}_t), \mathcal{P})$ and $\mathcal{F}_t$ the augmented filtration generated by B. If X is square-integrable and $\mathcal{F}_{\infty}$-measurable, then \cite{wiki}: \\ \\ $\exists$ predictable process C adapted to $\mathcal{F}_t$ s. a. \begin{eqnarray*} X &=& \mathbb{E}(X) + \int^{\infty}_0 C_s dB_s , \\
\mbox{ and that: } \mathbb{E}(X|\mathcal{G}_t)& =& \mathbb{E}(X) + \int^{t}_0 C_s dB_s \mbox{, } \forall t \geq 0. \end{eqnarray*}
\noindent We introduce the following notation which will be used throughout: \[ \mathcal{M}^2 = \{ x_t \in L^2(\mathcal{F}_t,\mathbb{P}) \mid x_t \mbox{-u.i. martingale} \}. \] For Martingales $(M_t) \in \mathcal{M}^2$ defined on $[0,T]$, the above result translates into the following with a 1-D Brownian motion: \\ \\ $\exists$ predictable process C adapted to $F_t$ s. a.
\begin{eqnarray*} M_T &=& M_0 + \int^T_0 C_s dB_s , \\
\mbox{ and that: } M_t&=& \mathbb{E}(M_T|\mathcal{G}_t) = M_0 + \int^{t}_0 C_s dB_s \mbox{, } \forall t \geq 0. \end{eqnarray*}
\noindent More generally, we use a multi-D Brownian motion as follow \cite{barnett}:
\begin{eqnarray*} \exists n \mbox{ s.a. } B_t &=& \big\{ B^1_t,....,B^n_t\big\} \mbox{ and, } \forall X \in \mathcal{M}^2 \mbox{, we have:} \\ X_t &=& X_0 + \int^t_0 \zeta_s \cdot dB_s \\ \mbox{with } \zeta_t &=& \big\{ \zeta^1_t,....,\zeta^n_t\big\} \mbox{ , } \zeta^i_t \mbox{-square integrable and predictable } \forall i.
\end{eqnarray*}
\subsection{Proofs: the fundamentals of integral representations in the $L^2$ spaces }
There are several approaches to proving the Martingale Representation Theorem in the continuous case as well as in the jump process sphere, but they tend to share common characteristics. Notably, it is generally agreed that these representation formulae originate from the fact that sets of integrals of specific processes are dense in $\mathcal{M}^2$. One of the most well-known works using this approach is from $\O$ksendal \cite{Oksendal1} and is as follows.
\subsubsection{$\O$ksendal's proof of the MRT in the continuous BM case}
Let $B := (B_t)$ be a n-dimensional Brownian Motion on the stochastic triple $(\Omega, \mathcal{F}, \mathbb{P})$, and $\mathcal{F}_t \subseteq \mathcal{F} $ be as follows: \[ \mathcal{F}_t = \sigma \big\{ B_s(\omega) \mid s \leq t \big\} \]
We introduce a couple of definitions that will be used throughout:
\begin{defn}[{\bf Strong Orthogonality}]
M, N $\in \mathcal{M}^2$ are strongly orthogonal if MN $\in \mathcal{M}^1$. \cite{Nualart} \label{strongorthog} \end{defn}
\begin{defn}[\bf{Weak Orthogonality}] Two stochastic processes $X_t$ and $Y_t$, $t \in [0,T]$, are said to be weakly orthogonal when: \[ \mathbb{E}(Y_T(\omega)X_T(\omega)) = 0 \mbox{ a.s.} \] \end{defn}
\begin{thm} $\Psi$ = $\{ X_T \mid X_T = e^{\int^T_0 h(s) dW_s - \frac{1}{2} \int^T_0 h^2(s)ds}, h \in L^2([0,T]) \}$ is dense in $L^2(\mathcal{F}_T, \mathbb{P})$ \cite{Oksendal1} \end{thm}
\begin{proof} This statement is true if $\forall g \in L^2(\mathcal{F}_T, \mathbb{P})$, g is not perpendicular to X for any X with $X_T \in \Psi$. We start by assuming that there is a g$\in L^2(\mathcal{F}_T, \mathbb{P})$ such as $g \perp X_T$. Strong orthogonality implies weak orthogonality so that means we have: \begin{eqnarray*}
\mathbb{E}(g(w)X_T(w)) &=& 0 \mbox{ a.s.} \\
\int_{\Omega} e^{\int^T_0 h(s) dW_s(w) - \frac{1}{2} \int^T_0 h^2(s)ds} g(w) d\mathbb{P} &=& 0 \\ e^{- \frac{1}{2} \int^T_0 h^2(s)ds} \int_{\Omega} e^{\int^T_0 h(s) dW_s(w) } g(w) d\mathbb{P} &=& 0. \end{eqnarray*} In particular, $\forall \lambda \in \mathbb{R}^n$ we have the following: \[ G(\lambda):= \int_{\Omega} e^{\sum^n_i \lambda_i W_{t_i} } g(w) d\mathbb{P} = 0. \] $G(\lambda)$ is real analytic in $\mathbb{R}^n$, and thus has an analytic extension to the complex space $\mathbb{C}^n$: \[ G(z):= \int_{\Omega} e^{\sum^n_i z_i W_{t_i} } g(w) d\mathbb{P} = 0 \mbox{ , } \forall z \in \mathbb{C}^n \]
We know from \cite{Oksendal1} that $C^{\infty}_0(\mathbb{R}^n)$ is dense in $L^2(\mathcal{F}_T, \mathbb{P})$. However, $\forall \phi \in C^{\infty}_0(\mathbb{R}^n)$, we have:
\begin{eqnarray*} \mathbb{E}(\phi(W)g(w)) &=& \int_{\Omega} \phi(W_{t_1},..., W_{t_n}) g(w) d\mathbb{P} \\ &=& \int_{\Omega} (2\pi)^{-n/2} \Big( \int_{\mathbb{R}^n} \hat\phi(y)e^{iW\cdot y}dy \Big) g(w) d\mathbb{P} \mbox{ (Inverse Fourier transform)} \\ &=& (2\pi)^{-n/2} \int_{\mathbb{R}^n} \hat\phi(y) \Big( \int_{\Omega} e^{iW \cdot y} g(w) d\mathbb{P} \Big) dy \mbox{ (Fubini's Theorem)} \\ &=& (2\pi)^{-n/2} \int_{\mathbb{R}^n} \hat\phi(y) \Big( \int_{\Omega} e^{ i(\sum_i W_iy_i)} g(w) d\mathbb{P} \Big) dy \\ &=& (2\pi)^{-n/2} \int_{\mathbb{R}^n} \hat\phi(y) G(iy) dy = 0. \end{eqnarray*} This can only be true iff g$\equiv$0. \end{proof}
Define analogously the following spaces: \[ \Psi_t = \{ X_t \mid X_t = e^{\int^t_0 h(s) dW_s - \frac{1}{2} \int^t_0 h^2(s)ds}, h \in L^2([0,T]) \} \mbox{ dense in } L^2(\mathcal{F}_t,\mathbb{P}) \]
$\forall Y_t(w) \in \Psi_t$, it is easy to see that: \[ dY_t = Y_t h(t) dW_t \mbox{ (Ito's lemma), }\] hence $\exists h(t) \in L^2([0,T])$
\begin{eqnarray*} Y_t &=& Y_0 + \int^t_0 Y_s h(s) dW_s \\ &=& 1 + \int^t_0 Y_s h(s) dW_s \\ &\mbox{ or }& \mathbb{E}(Y_t) + \int^t_0 Y_s h(s) dW_s. \end{eqnarray*}
As $\Psi$ is dense in $L^2(\mathcal{F}_T, \mathbb{P})$, we can approximate any element of $L^2(\mathcal{F}_T, \mathbb{P})$ with suitable convergent elements of $\Psi$. So $\forall X \in L^2(\mathcal{F}_T), \exists (\psi_n) \in \Psi$ such as $\psi_n \rightarrow X$ in $L^2(\mathcal{F}_T)$. Since any convergent sequence $\psi_n \in \Psi$ is also a cauchy sequence, $(\psi_n)$ is cauchy in $L^2(\mathcal{F}_T)$: \begin{equation} \mathbb{E}( (\psi_n - \psi_m)^2) \rightarrow 0 \mbox{ as }m,n \rightarrow +\infty, \label{cauchy} \end{equation}
where we have: \[ \psi_n = \mathbb{E}(\psi_n) + \int^T_0 f_n(s,\omega)dW_s.\] The function $f_n$ is defined as \[ f_n(t,\omega) := \psi_n(\omega)h_n(t), \] and $f_n(T,\omega) \in L^2([0,T] \times \Omega)$. So \begin{eqnarray*} \mathbb{E}( (\psi_n - \psi_m)^2) &=& \mathbb{E}\big( (\mathbb{E}( \psi_n - \psi_m) + \int^T_0 f_n-f_m dW_s)^2 \big) \\ &=& \mathbb{E}( \psi_n - \psi_m)^2 + \int^T_0 \mathbb{E}((f_n-f_m)^2) ds \mbox{ (Ito's isometry)}\\ & \geq & \int^T_0 \mathbb{E}((f_n-f_m)^2) ds. \end{eqnarray*}
Result \ref{cauchy} implies that $\forall A_T \in L^2(\mathcal{F}_T, \mathbb{P})$, $\exists f(t) \in L^2(\mathbb{R} \times \Omega), \psi_n \in \Psi$ such as $ f_n \rightarrow f$ as $n \rightarrow \infty$ where: \begin{equation} A_t = \lim_{n \rightarrow \infty} \psi_n = \mathbb{E}(A) + \int^t_0 f(s) dW_s \label{itorepth}. \end{equation}
Result \ref{itorepth} is knowns as the {\bf Ito Representation Theorem}. \cite{Oksendal1} Indeed, in this framework the function f(t) is unique: Assume there are 2 functions $f_1(t)$ and $f_2(t)$ satisfying \ref{itorepth}, then: \begin{eqnarray*} A_t(w) = \mathbb{E}(A) + \int^t_0 f_1(s) dW_s = \mathbb{E}(A) + \int^t_0 f_2(s) dW_s \\ \Longrightarrow 0=(A_t-A_t)^2= \int^t_0 \mathbb{E}((f_1(s)-f_2(s))^2) ds \\ \Longrightarrow f_1(t) = f_2(t) \mbox{ a.s. for all t} \in [0,T]. \end{eqnarray*}
At this stage, the martingale representation theorem is just an application of the Ito representation theorem (equation \ref{itorepth}) to $\mathcal{M}^2([0,T]) \subset L^2(\mathcal{F}_T, \mathbb{P})$: \[ \forall A \in \mathcal{M}^2([0,T]) \mbox{ ,} \exists ! f(t) \in L^2((\mathcal{F}_T, \mathbb{P}) \times [0,T]) \mbox{ s.a. } A_t = \mathbb{E}(A_T) + \int^t_0 f(s) dW_s. \]
The dense set approach combined with the orthogonality argument, beyond giving a representation formula for martingales, makes a general statement about the $L^2(\mathcal{F}_T, \mathbb{P})$ space. In this setting, $\mathcal{M}^2$ and $L^2(\mathcal{F}_T, \mathbb{P})$ are Hilbert spaces under a suitable inner product:
\[ L^2(\mathcal{F}_T, \mathbb{P}) = \bigoplus^{\infty}_{i=1} \mathcal{S}(M_i), \] where \[ \mathcal{S}(M_i) = \big\{ \int_{[0,T]} \phi dM_i \mid \phi \in L^2( < M_i >) \big\} \mbox{ , } \exists M_i \in L^2(\mathcal{F}_T, \mathbb{P}), i \in \mathbb{N}.\] $\mathcal{S}(M_i) $ is also known as the Stable Subspace generated by $M_i$. (\cite{Davis1}) \\
This approach is widely explored elsewhere in the literature, notably in Dellacherie \cite{Dellacherie}, Yor \cite{Yor} and Kunita $\&$ Watanabe \cite{KunitaWatanabe}.\\ While this idea and framework is very dominant in works on continuous martingales, it does not limit itself to the extends of the Brownian Filtration. This is indeed the case as shown in the work of L$\o$kka \cite{Lokka2} and Davis \cite{Davis2}.
\subsubsection{MRT for martingales driven by Poisson and pure-jump processes } \paragraph{setting}
In this section, we bring our attention to a specific stochastic process, the Compensated Poisson process: Let $N_t$ be a poisson process of intensity $\lambda$ and define:
\begin{eqnarray*} \bar{N}_t &=& N_t - \lambda t.\\ \end{eqnarray*}
$\bar{N}_t $ has the following characteristic function:
\begin{eqnarray*} \phi_{\bar{N}_t}(z) &=& e^{\lambda t (e^{iz}-1-iz)}.\\ \end{eqnarray*} Additionally:
\begin{eqnarray*}
\mathbb{E}[\bar{N}_t | \bar{N}_s] &=& \mathbb{E}[ N_t - \lambda t | \bar{N}_s] \\
&=& \mathbb{E}[ N_t - N_s | \bar{N}_s] - \lambda t + N_s \\ &=& N_s - \lambda t + \lambda (t-s) \\ &=& \bar{N}_s. \end{eqnarray*}
So $\bar{N}_t$ is a martingale: $ \forall t > s$ , $ \mathbb{E}[\bar{N}_t | \bar{N}_s] = \bar{N}_s$. \\ When rescaled, it has similar properties to a regular Brownian motion:
\begin{eqnarray*} \mathbb{E}\Big[ \frac{\bar{N}_t }{\lambda} \Big] &=& 0\\ \mathbb{V}\Big[ \frac{\bar{N}_t }{\lambda} \Big] &=& t, \end{eqnarray*}
hence $\bar{N}_t $ is a likely candidate for the integrator in the MRT, especially since it has the following curious property \cite{poisson}:
\begin{eqnarray*} \left( \begin{array}{c} \bar{N}_t \\ \lambda \end{array} \right) \rightarrow (W_t) \mbox{ as } \lambda \rightarrow \infty. \end{eqnarray*}
As we will see, this type of jump processes is very important to the representation of pure-jump martingales, as the use of Poisson-generated filtrations is very common in the literature on applications to hedging strategies. Indeed, as we will show below, martingales evolving on a stochastic base can be conveniently represented in an integral form using compensated poisson processes.\\
We base ourselves on the work developed in L$\o$kka (2005) \cite{Lokka2} and start with $([0,T],\mathcal{B},\lambda)$ where $\mathcal{B}$ is the Borel $\sigma$-algebra and $\lambda$ is a Radon-measure that charges all open sets and that is diffuse over $\mathcal{B}$. Then we set: \begin{eqnarray*} \Omega &=& \{ \omega = \sum^n_j \delta_{t_j} \mid n \in \mathbb{N} \cup \{ \infty\}, t_j \in [0,T] \}, \\ \mathcal{F}_0 &=& \sigma\{ \omega(A) \mid A \subseteq \mathcal{B} \}, \\ \mathbb{P} &:& \mbox{ Probability measure s.a. } t \mapsto \omega([0,t]) \mbox{ is a Poisson process,} \\ \mathcal{F} &=& \mathbb{P}\mbox{-completion of } \mathcal{F}_0.\\ \end{eqnarray*}
On this new stochastic base we introduce L, a square-integrable Poisson jump process defined as: \[ L_t = \int^t_0 \int_{\mathbb{R}_0} z \tilde{N}(dz,dt). \] $\tilde{N}(z,t)$ is a compensated poisson process. L can be otherwise noted as follow: \[ L_t = \int^t_0 \int_{\mathbb{R}_0} z (\mu-\pi)(dz,dt),\] where $\mu-\pi$ is the measure of the compensated poisson process $\tilde{N}(z,t)$. $\mu$ is the Poisson measure of the process $(L_t)$: \[ \mu(\Lambda, \Delta t) = \sum_{s \in \Delta t} 1_{\Lambda}(\Delta L_s) \] and $\pi$ is the compensator of $\mu$ and can be defined by the following relation: \[ \mathbb{E}(L^2_t) = \int_{\mathbb{R}_0} z^2 \pi(dz,dt). \] $\pi$ is known to have the form $\pi(dz,dt) = \upsilon(dz)dt$, where $\upsilon$ is the L$\acute{e}$vy measure of L. \cite{Lokka2} \\ At this stage, we can consider the complete filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t), \mathbb{P})$ by setting the following: \[ \mathcal{F}_t = \sigma\{ L_s \mid s \leq t \}. \]
\paragraph{Representation results}
Now that we have discussed the setting of the underlying stochastic base $(\Omega, \mathcal{F}, (\mathcal{F}_t), \mathbb{P})$ , we can state and prove several results that will then lead to an integral representation formula for martingales in $L^2(\mathcal{F}_T,\mathbb{P})$.
\begin{thm} $\mathcal{A} = \{ \phi(L_{t_1},...,L_{t_n}) \mid t_i \in [0,T], t_0=0, t_n=T, \phi \in C^{\infty}_0(\mathbb{R}^n), n \in \mathbb{N} \}$ \\ is dense in $L^2(\mathcal{F}_T,\mathbb{P})$. \label{Lokka2thm1} \end{thm}
\begin{proof} Consider a set $\{t_i\}_{\mathbb{N}}$ dense in [0,T], and a nested set of $\sigma$-fields $\mathcal{G}_n = \sigma(L_{t_1},...,L_{t_n})$. Clearly, $\mathcal{G}_n \subset \mathcal{G}_{n+1} \forall n \in \mathbb{N}$. Also, $\mathcal{F}_T$ is the smallest $\sigma$-field containing all $\mathcal{G}_n$ for any n in $\mathbb{N}$ \cite{Lokka2}. Then, $\forall g \in L^2(\mathcal{F}_T,\mathbb{P})$,
\[ g = \mathbb{E}(g \mid \mathcal{F}_T) = \lim_{n \rightarrow \infty} \mathbb{E}( g \mid \mathcal{G}_n). \] So by the Doob-Dynkin Lemma we have :
\[ \exists g_n : \mathbb{R}^n \mapsto \mathbb{R} \mbox{ borel-measurable s.a. } \mathbb{E}(g \mid \mathcal{G}_n) = g_n(L_{t_1},...,L_{t_n}). \] $g_n$ can be approximated in $L^2(\mathcal{F}_T, \mathbb{P})$ with $\phi_{n_m} \in C^{\infty}_0(\mathbb{R}^n)$ \cite{Folland} where
\[ \| g_n(L_t) - \phi_{n_m}(L_t) \|_{L^2(\mathbb{P})} \rightarrow 0 \mbox{ as } m \rightarrow \infty. \] Hence, every g in $L^2(\mathcal{F}_T,\mathbb{P})$ can be approximated with a function $\phi \in C^{\infty}_0(\mathbb{R}^n)$. \end{proof}
While this result is useful, we are looking to make a more precise statement about the structure of $L^2(\mathcal{F}_T, \mathbb{P})$. To do so, we start with the result below which is an adaptation of the regular It$\hat{o}$ Isometry;
\begin{thm}
Whenever $\phi: \phi(t,z,\omega)$ is a predictable process, the below is true: \[ \mathbb{E}\big( \big( \int^T_0 \int_{\mathbb{R}_0} \phi(t,z) (\mu-\pi)(dz,dt) \big)^2 \big) = \mathbb{E}\big( \int^T_0 \int_{\mathbb{R}_0} \phi^2(t,z) \pi(dz,dt) \big). \] \label{Poissonitoisometry} \end{thm} One implication of this result is that $\int^T_0 \int_{\mathbb{R}_0} \phi(t,z) \tilde{N}(dz,dt) \in L^2(\mathbb{P})$ if $\phi \in L^2(\pi \times \mathbb{P})$. \\
Now set a new continuous function $\gamma:\mathbb{R}_0 \mapsto (-1,0) \cup (0,1)$ :
\begin{equation}
\gamma(z) = \left\{
\begin{array}{cc}
e^z - 1& z < 0 \\
1 - e^{-z} & z > 0
\end{array}
\right. \nonumber \end{equation}
One nice property of $\gamma$ is that it approaches zero as fast as z does. Indeed:
\begin{eqnarray*}
\lim_{z \rightarrow 0} \frac{\gamma(z)}{z} &=& \left| \begin{array}{c} \lim_{z \rightarrow 0^+} \frac{1-e^{-z}}{z} \\ \lim_{z \rightarrow 0^-} \frac{e^z - 1}{z} \end{array} \right. \\
&=& \left| \begin{array}{c} \lim_{z \rightarrow 0^+} e^{-z} \\ \lim_{z \rightarrow 0^-} e^z \end{array} \right. \mbox{ (Hospital rule)} \\ &=& 1 \end{eqnarray*}
This combined with the facts that $\upsilon\big( (-\infty, -1] \cup [1, \infty) \big) < \infty$ and that $\gamma$ is bounded implies that : \[ e^{\gamma \lambda} - 1 \in L^2(\upsilon) \mbox{ , } \forall \lambda \in \mathbb{R} \]
and that $\forall h \in C([0,T])$, \begin{eqnarray}
e^{\gamma \lambda} -1 \in L^2(\pi) , \nonumber \\
h \gamma \in L^2(\pi) \nonumber \\ e^{h\gamma} - 1 -h\gamma \in L^1(\pi) \label{conditionsLokka2} \end{eqnarray}
The use of the function $\gamma$ and of the results \ref{conditionsLokka2} are necessary for us to proceed as $\exists \lambda \in \mathbb{R}$ such as $ e^{L_t \lambda} \notin L^2(\mathbb{P})$. Without these conditions, it is not possible to find an integral representation for elements of $L^2(\mathcal{F}_T,\mathbb{P})$. However, modifying the jumps of L with $\gamma$ ensures that the exponential of the modification lies in $L^2(\mathcal{F}_T,\mathbb{P})$.
\begin{thm} The linear span of random variables of the form: \begin{equation} \exp{ \Big\{ \int^T_0\int_{\mathbb{R}_0} h(t)\gamma(z)\tilde{N}(dz,dt) - \int^T_0\int_{\mathbb{R}_0}(e^{h(t)\gamma(z)}-1-h(t)\gamma(z))\pi(dz,dt) \Big\} } \nonumber \end{equation} where h $\in C([0,T])$ is dense in $L^2(\mathcal{F}_T,\mathbb{P})$. \label{lemma2lokka2} \end{thm}
\begin{proof} Set \[ \tilde{L}_t = \int^t_0 \int_{\gamma(\mathbb{R}_0)} \gamma(z)(\mu-\pi)(dz,dt). \]
$\tilde{L}_t$ is a Poisson process with a poisson random measure $\tilde{\mu}(\gamma^{-1}(\Lambda),t)$. Clearly, we see that: \[ \sigma(\tilde{L}) \subseteq \sigma(L). \]
Now introduce: \[ \hat{L}_t = \int^t_0 \int_{\gamma(\mathbb{R}_0)} \gamma^{-1}(z)(\tilde{\mu}-\tilde{\pi})(dz,dt). \]
Again, $\sigma(\hat{L}) \subseteq \sigma(\tilde{L})$. $\hat{L}$ is also a Poisson process with poisson random measure $\hat{\mu}$: \[ \hat{\mu}(\Lambda,t) = \tilde{\mu}(\gamma(\Lambda),t) = \mu(\gamma(\gamma^{-1}(\Lambda)),t) = \mu(\Lambda,t). \]
That implies that $\hat{L}=L$ and that $\sigma(\hat{L}) = \sigma(L)$. Hence we have $\sigma(\tilde{L}) = \sigma(L)$. \\ Take $Y_T$ be of the form stated in Theorem \ref{lemma2lokka2}. Then:
\[ dY_t = \int_{\mathbb{R}_0} Y_{t-} (e^{h(t)\gamma(z)}-1)(\mu-\pi)(dt,dz) \mbox{ (It$\hat{o}$'s formula).} \] By the It$\hat{o}$ isometry (Theorem \ref{Poissonitoisometry}) and the continuity of the law of L, we have that \cite{Lokka2} \[ \mathbb{E}(Y_t^2) = \exp{ \big( \int^t_0 \int_{\mathbb{R}_0} (e^{h(s)\gamma(z)}-1)^2\pi(dz,dt) \big)} \] and
\[ \| Y_t\|^2_{L^2(\mathbb{P})} = \exp{ ( \| e^{h\gamma}-1 \|^2_{L^2(\pi)} ) }.\]
Since we know that $e^{h\gamma}-1 \in L^2(\pi)$, we deduce that $Y_T \in L^2(\mathbb{P})$. Additionally, since $e^{h\gamma}-1-h\gamma \in L^1(\pi)$, we have that:
\[ e^{\int^T_0 h(t)d\tilde{L}_t} \in L^2(\mathbb{P}) \mbox{ , } \forall h \in C([0,T]), \] and more specifically, for any $\lambda \in \mathbb{R}^n$ and $\{t_i\} \in [0,T]$,
\[ e^{\lambda \cdot \tilde{L}_t} \in L^2(\mathbb{P}). \]
The above then allows us proceed with the statement of theorem \ref{lemma2lokka2}. From here onwards, we follow the same argument explored with the B.M. in $\O$ksendal \cite{Oksendal1}: we assume that there is a g $\in L^2(\mathcal{F}_T,\mathbb{P})$ that is orthogonal to all variables of the form stated in theorem \ref{lemma2lokka2}. Through a very similar argument, we find that for any $\phi \in C^{\infty}_0(\mathbb{R}^n)$, \[ \int_{\Omega} \phi(\tilde{L}_{t_1},...,\tilde{L}_{t_n})gd\mathbb{P} = 0.\]
However, we also know from theorem \ref{Lokka2thm1} that the variables of type $\phi(\tilde{L}_{t_1},...,\tilde{L}_{t_n})$ are dense in $L^2(\mathcal{F}_T,\mathbb{P})$. Hence we must deduce that g$\equiv$0. \end{proof}
Therefore, based on Theorem \ref{lemma2lokka2}, we can prove the following representation theorem just as done in section 1.2.1.
\begin{thm} $\forall F \in L^2(\mathcal{F}_T,\mathbb{P})$, there is a unique process $\psi \in L^2([0,T] \times \mathbb{R}_0 \times \Omega, \pi \times \mathbb{P})$ such as: \[F = \mathbb{E}(F) + \int^T_0 \int_{\mathbb{R}_0} \psi(t,z)(\mu-\pi)(dz,dt), \] or \[ F = \mathbb{E}(F) + \int^T_0 \int_{\mathbb{R}_0} \psi(t,z) \tilde{N}(dz,dt). \] \label{lokkaMRTpoisson} \end{thm}
Therefore for $\forall (m_t) \in \mathcal{M}^2$ where $m_{\infty} < \infty$ , $\exists ! \psi \in L^2([0,T] \times \mathbb{R}_0 \times \Omega, \pi \times \mathbb{P})$ such that we have: \[ m_t = \mathbb{E}(m_{\infty} \mid \mathcal{F}_t) = \int^t_0 \int_{\mathbb{R}_0} \psi(t,z) \tilde{N}(dz,dt). \]
\paragraph{Extension to other jump processes}
The above representation result involves Compensated Poisson jump processes, but it is extendable to other type of jump processes. This is indeed what the work of Davis\cite{Davis2} shows.\\
We start with a basic jump process ($x_t$) defined on a Blackwell space $(X, \mathcal{L})$. We remind the reader of the following definition: \\
\begin{defn}[{\bf Blackwell Spaces}]
A measurable space (X,$\mathcal{A}$) is a Blackwell space if the $\sigma$-algebra $\mathcal{A}$ it is the smallest countably-generated $\sigma$-field and contains all singletons. In other words, $\exists$ countable sets $\{A_i \mid i \in \mathbb{N} \}$ that generate $\mathcal{A}$. This type of set is otherwise known as Lusin spaces, as introduced in Blackwell \cite{Blackwell}. \\ One important property is that any one-to-one $\mathcal{A}$-measurable mapping f: X $ \rightarrow (Y,\mathcal{B})$, where $\mathcal{B}$ is also countably generated, is an isomorphism. \cite{Bog} \end{defn} Now define $ (Y, \mathcal{Y}) = \Big( (\mathbb{R}^+\times X)\bigcup \{(\infty,z_{\infty})\}, \sigma\{\mathcal{B}(\mathbb{R}^+) \ast \mathcal{L}, \{(\infty,z_{\infty})\} \} \Big)$ and its copies $(Y^i, \mathcal{Y}^i)$, $i \in \mathbb{N}$, $z_0$, $z_{\infty}$ are fixed elements of X. Now set:
\begin{eqnarray*} \Omega &=& \prod^{\infty}_{i=1} Y^i, \\ \mathcal{F}^0 &=& \sigma \Big\{ \prod^{\infty}_{i=1} \mathcal{Y}^i \Big\}, \\ \mbox{Let } (S_i,Z_i): \Omega \rightarrow Y^i &&\mbox{ be the coordinate mapping, and } \\ \omega_k : \Omega \rightarrow \Omega_k = \prod^k_{i=1} Y^i &\mbox{ s.a. }& \omega_k(w) = (S_1,Z_1,...,S_k,Z_k). \end{eqnarray*}
We can now define: \begin{eqnarray*} T_k(w) = \sum^k_{i=1} S_i(w), \\ T_{\infty}(w) = \lim_{k \rightarrow \infty} T_k(w), \end{eqnarray*}
and the jump process of interest:\\
\[x_t(\omega)= \left\{ \begin{array}{cc} z_0, & \mbox{if } t<T_1(\omega) \\ Z_i(\omega), & \mbox{if } t \in [T_i,T_{i+1}) \\ z_{\infty}, & \mbox{if } t \geq T_{\infty} \end{array} \right. .\]
This random variable generates an increasing family of $\sigma$-fields $(\mathcal{F}^0_t)$ where $\mathcal{F}^0_t = \sigma\{ x_s \mid s \leq t\}$. Then $\mathcal{F}_t(\mathcal{F})$ is the $\mathbb{P}$-augmented $\sigma$-field of $\mathcal{F}^0_t(\mathcal{F}^0)$ .\\ To go with $(\Omega, \mathcal{F}^0)$, we define the following family of probability measures $(\mu_i)$: for $\Gamma \in \mathcal{Y}$ and $\eta \in \Omega_{i-1}$ \begin{eqnarray*} P[ (T_1,Z_1) \in \Gamma] = \mu^1(\Gamma),\\
P[ (T_i,Z_i) \in \Gamma | \omega_{i-1} = \eta ] = \mu^i(\eta ; \Gamma). \end{eqnarray*} To ensure that the jump times of $x_t$ do not occur at once, we impose: \[ \mu^i(\omega_{i-1} ; (\{0\}\times X) \cup (\mathbb{R}^+ \times \{Z_{i-1}\})) = 0\mbox{ } \forall i \in \mathbb{N}. \]
There is a fundamental family of processes associated with the filtration generated by $(x_t)$. For A $\in \mathcal{L}$ and t $\in \mathbb{R}^+$ set:
\begin{eqnarray*} p(t,A)(w) &=& \sum_i 1_{(t \geq T_i)}1_{(Z_i \in A)}, \\ \tilde{p}(t,A) &=& - \sum^{j-1} \int^{S_i}_0 \frac{1}{F^i_{s-}}dF^{iA}_s - \int^{t-T_{j-1}}_0 \frac{1}{F^i_{s-}}dF^{iA}_s , \mbox{ for } t \in (T_{j-1},T_j] \\ \mbox{where } && F^{iA}_t = \mu^i(\omega_{i-1};(t, \infty] \times A) \mbox{ and } F^i_t = P(T_i > t), \\ q(t,A) &=& p(t,A) - \tilde{p}(t,A). \end{eqnarray*}
\begin{thm}
for fixed k and A $\in \mathcal {L}$, $q(t \wedge T_k, A)$ is an $\mathcal{F}_t$-martingale \cite{Davis2} \end{thm}
\begin{proof} To address the above, we use the proof to an analogous proposition in Chou $\&$ Meyer \cite{choumeyer}:
\begin{eqnarray*} q( t \wedge T_k, A) &=& p(t \wedge T_k,A) - \tilde{p}(t \wedge T_k,A) \\ &=& \sum_i 1_{(t \wedge T_k \geq T_i)}1_{(Z_i \in A)} + \sum^{j-1} \int^{S_i}_0 \frac{1}{F^i_{s-}}dF^{iA}_s + \int^{t-T_{j-1}}_0 \frac{1}{F^j_{s-}}dF^{jA}_s \\ &=& \sum^{j-1}_i \Big[ 1_{(t \geq T_i)}1_{(Z_i \in A)} + \int^{S_i}_0 \frac{1}{F^i_{s-}}dF^{iA}_s \Big] + 1_{(t \geq T_j)}1_{(Z_j \in A)} + \int^{t-T_{j-1}}_0 \frac{1}{F^j_{s-}}dF^{jA}_s \mbox{, where j $\leq$ k} \end{eqnarray*}
Set $q_i(t,A) = 1_{(t \geq T_i)}1_{(Z_i \in A)} + \int^{S_i}_0 \frac{1}{F^i_{s-}}dF^{iA}_s $ $\forall i <j$, and $q_j(t,A) = 1_{(t \geq T_j)}1_{(Z_j \in A)} + \int^{t-T_{j-1}}_0 \frac{1}{F^j_{s-}}dF^{jA}_s$. Then \[ q( t \wedge T_k, A) = \sum^j_i q_i(t,A) \mbox{, where $t \in [T_{j-1};T_j)$ whenever j $\leq$ k} \] It can then be shown that each element $q_i(t,A)$ is a martingale.
\begin{comment}
To do so, we look specifically at $ \mathbb{E}( \int^{S_i}_0 \frac{1}{F^i_{s-}}dF^{iA}_s \mid \mathcal{F}_u ) - \int^{S_i \wedge u}_0 \frac{1}{F^i_{s-}}dF^{iA}_s $ :
\begin{eqnarray*} \mathbb{E}\big( \int^{S_i}_0 \frac{1}{F^i_{s-}}dF^{iA}_s \mid \mathcal{F}_u \big) - \int^{S_i \wedge u}_0 \frac{1}{F^i_{s-}}dF^{iA}_s &&\\ &=& \mathbb{E}\big( \int^{S_i}_0 \frac{1}{F^i_{s-}}dF^{iA}_s \mid \mathcal{F}_u \big) \\ && - 1_{(u < S_i)}\int^{u}_0 \frac{1}{F^i_{s-}}dF^{iA}_s - 1_{(u \geq S_i)} \int^{S_i }_0 \frac{1}{F^i_{s-}}dF^{iA}_s \\ &=& 1_{(u< S_i)} \Big[ \int^{u}_0 \frac{1}{F^i_{s-}}dF^{iA}_s - \mathbb{E}\big( \int^{S_i}_0 \frac{1}{F^i_{s-}}dF^{iA}_s \mid \mathcal{F}_u \big) \Big] \\ &=& 1_{(u< T_i)} \Big[ \int^{u}_0 \frac{1}{F^i_{s-}}dF^{iA}_s - \mathbb{E}\big( \int^{S_i}_0 \frac{1}{F^i_{s-}}dF^{iA}_s \mid \mathcal{F}_u \big) \Big] \\ &=& 1_{(u< T_i)} 1_{(Z_i \in A)} \mbox{ how? is it correct?} \\ &=& 1_{(Z_i \in A)} - 1_{(u\geq T_i)} 1_{(Z_i \in A)} \\ &=& 1_{(\infty \geq T_i)}1_{(Z_i \in A)} - 1_{(u\geq T_i)} 1_{(Z_i \in A)} \end{eqnarray*}
So we have:
\begin{eqnarray*} \mathbb{E}\big( \int^{S_i}_0 \frac{1}{F^i_{s-}}dF^{iA}_s \mid \mathcal{F}_u \big) - \int^{S_i \wedge u}_0 \frac{1}{F^i_{s-}}dF^{iA}_s &=& 1_{(\infty \geq T_i)}1_{(Z_i \in A)} - 1_{(u\geq T_i)} 1_{(Z_i \in A)} \\ \mathbb{E}\big( 1_{(\infty \geq T_i)}1_{(Z_i \in A)}- \int^{S_i}_0 \frac{1}{F^i_{s-}}dF^{iA}_s \mid \mathcal{F}_u \big) &=& 1_{(u\geq T_i)} 1_{(Z_i \in A)} - \int^{S_i \wedge u}_0 \frac{1}{F^i_{s-}}dF^{iA}_s \\ \mathbb{E}\big( q_i(\infty,A) \mid \mathcal{F}_u \big) &=& q_i(u,A) \end{eqnarray*}
\end{comment} \end{proof}
We define the following class of integrands: \[ \mathcal{P} = \Big\{ g: \Omega \times Y \rightarrow \mathbb{R} \mid g(t,z,\omega) = g^k(\omega_{k-1}; t,z)1_{(T_{k-1},T_k]} \Big\}, \] and: \begin{eqnarray*}
L^1(p) = \Big\{ g \in \mathcal{P} \mid \mathbb{E}\big( \int_{\mathbb{R}^+ \times X} |g(t,z)|p(dt,dz)\big) < \infty \Big\}, \\ L^1_{loc}(p) = \Big\{ g \in \mathcal{P} \mid g1_{(t<\sigma_k)} \in L^1(p), k \in \mathbb{N}, \exists \mbox{ stopping times } \sigma_k \uparrow T_{\infty} \Big\}. \end{eqnarray*} $L^1(\tilde{p})$ and $L^1_{loc}(\tilde{p})$ are defined analogously.\\ Then for $g \in L^1_{loc}(p) $ and $ M^g_t = \int_{(0,t] \times X} g(s,z) q(ds,dz)$, there is a sequence of stopping times $\tau_n \uparrow T_{\infty}$ where $M^g_{t \wedge \tau_n}$ is a u.i. martingale for all n $\in \mathbb{N}$.(\cite{Davis2}). This induces the following to be true: \begin{equation} M_t = M_{t \wedge T_1} + \sum^{\infty}_{k=2} \big( M_{t \wedge T_k}-M_{T_{k-1}}\big)1_{(t \geq T_{k-1})} \label{uimgs} \end{equation} whenever $M_t$ is a u.i. martingale. To simplify \ref{uimgs} further, we set: \begin{eqnarray*} X^1_t &=& M_{t \wedge T_1}, \\ X^k_t &=& M_{(t+T_{k-1}) \wedge T_k}-M_{T_{k-1}}. \end{eqnarray*} Then $M_t$ in \ref{uimgs} is transformed into:
\begin{equation} M_t = \sum^{\infty}_{k=1} X^k_{(t-T_{k-1})\vee 0}. \label{newmt} \end{equation}
For fixed k and t$\geq$0, set $ \mathcal{H}_t = \mathcal{F}_{(t+T_{k-1}) \wedge T_k}$. According the the optimal sampling theorem, $(X^k_t)$ is a $\mathcal{H}_t $-martingale. In other words, there is a function $h^k$ such as:
\[ X^k_t = \mathbb{E}(h^k(\omega_{k-1};S_k,Z_k) \mid \mathcal{H}_t ).\mbox{ \cite{Davis2} } \]
Using proposition 5 of \cite{Davis2} and proposition 2 of \cite{choumeyer} we see that each u.i. martingale $X^k_t$ can be expressed as follow: \begin{eqnarray*} X^k_t = 1_{(t \geq T_k)}h^k(\omega_{k-1};S_k,Z_k) - 1_{(t < T_k)}\frac{1}{F^{k}_t} \int_{(0,t] \times X} h^k(\omega_{k-1};S_k,Z_k) dF^{k}(s,z). \end{eqnarray*} Working on the interval $[0,T_k)$, that gives:
\begin{eqnarray*} X^k_t = \frac{-1}{F^k_t} \int_{(0,t] \times X} h^k(\omega_{k-1};S_k,Z_k) dF^{k}(s,z). \end{eqnarray*}
\begin{comment} As we can write $X^k_t = G_t B_t$ for suitable processes $G_t$ and $B_t$, we have:
\begin{eqnarray*} d X^k_t &=& G_{t-} dB_t + B_t dG_t \\ &=& \frac{h^k(\omega_{k-1};S_k,Z_k) dF^{k}_t}{F^k_{t-}} + \frac{-dF^k_t}{F^k_t F^k_{t-}} \int_{(0,t] \times X} h^k(\omega_{k-1};S_k,Z_k) dF^k(s,z) \\ &=& - \Big( -h^k(\omega_{k-1};S_k,Z_k) + \frac{1}{ F^k_{t-}} \int_{(0,t] \times X} h^k(\omega_{k-1};S_k,Z_k) dF^k(s,z) \Big) \frac{dF^k_t}{F^k_{t-}} \\ &=& - g^k(\omega_{k-1};S_k,Z_k) \frac{dF^k_t}{F^k_{t-}} \\ &=& g^k(\omega_{k-1};S_k,Z_k) \frac{-dF^k_t}{F^k_{t-}} \end{eqnarray*}
But we also have: \begin{eqnarray*} q_i(ds,dz) &=& ... ? \\ \end{eqnarray*}
\end{comment}
Further calculations (\cite{Davis2}) then lead to the following end result: \begin{equation} X^k_t = \int_{(0,t]\times X} g^k(\omega_{k-1};s,z)q^k(ds,dz), \label{xt} \end{equation}
where $q^k(t,A)= q((t+T_{k-1}) \wedge T_k,A)$. Then \ref{xt} turns \ref{newmt} into:
\begin{eqnarray} M_t &=& \sum^{\infty}_{k=1} X^k_{(t-T_{k-1})\vee 0} \nonumber \\
&=& \sum^{\infty}_{k=1} \int_{(0,(t-T_{k-1}) \vee 0]\times X} g^k(\omega_{k-1};s,z)q^k(ds,dz) \nonumber\\ &=& \int_{(0,t] \times X} g(s,z)q(ds,dz). \label{jumpmrt} \end{eqnarray}
The function g $ \in \mathcal{P}$ is defined by the collection $\{ g^k \mid k \in \mathbb{N} \}$ such that \ref{jumpmrt} holds. This resulting function g $\in L^1_{loc}(p) $, as explained in \cite{Davis2}.\\ \\ We have seen so far that the Martingale representation theorem is applicable to a diverse range of probability spaces and martingales, but as the work of Nualart and Schoutens \cite{Nualart} in has shown, it can reach to much more general martingales.
\subsubsection{Nualart and Schoutens: MRT using general L$\acute{e}$vy processes}
In this setting we start with $X = \{ X_t \mid t \geq 0 \}$ being a L$\acute{e}$vy process defined on a complete stochastic triple $(\Omega, \mathcal{F}, \mathbb{P})$. $\mathcal{F}$ is the $\sigma$-field generated by X. We let: \[ \mathcal{F}_t = \mathcal{G}_t \vee \mathcal{N}, \] where \[ \mathcal{G}_t = \sigma\{ X_s \mid s \leq t \} \] and \[ \mathcal{N} = \{ A \mid A-\mathbb{P} \mbox{-null set of } \mathcal{F} \}. \] Then we can introduce compensated power jump processes as follow: \begin{itemize} \item $X = (X_t)$ is a L$\acute{e}$vy process, i.e. X has stationary and independent increments, $X_0=0$ and is c$\grave{a}$dl$\grave{a}$g, \item $X_{t-} = \lim_{s \rightarrow t, s<t} X_s$, \item $\Delta X_t = X_t - X_{t-}$, \item $ X^i_t = \sum_{s \leq t} ( \Delta X_s)^i $ , and $X^1_t = X_t$, \item $ Y^i_t = X^i_t - \mathbb{E}(X^i_t) := X^i_t - m_it$ are then a martingale called Teugels martingales \cite{Nualart} \end{itemize}
In here, $ \forall (m_t) \in \mathcal{M}^2, m_t = \mathbb{E}(m_{\infty} \mid \mathcal{F}_t) $, and $m_i$ are the ith central moments of the process X. \\ We can start with the case of i =1. There we have: \[ Y^1_t = X_t - m_1t. \] We can see that for any $s < t \leq T$ :
\begin{eqnarray*} \mathbb{E}(Y^1_t \mid \mathcal{F}_s) &=& \mathbb{E}( X_t - m_1t \mid \mathcal{F}_s)\\ &=& \mathbb{E}( X_t - X_s\mid \mathcal{F}_s) -m_1t+X_s \\ &=& X_s-m_1t + m_1(t-s) \mbox{ (independent increments of L$\acute{e}$vy processes)}\\ &=& Y^1_s. \end{eqnarray*}
Hence $Y^1$ is indeed a martingale. $Y^i$ is also a martingale for $i>1, i \in \mathbb{N}$, as shown in \cite{Schoutens}.\\ We introduce the following space: \begin{equation} \mathcal{H}= \Big\{ \int^{\infty}_0 f(t)dY^1_t \mid f \in L^2(\mathbb{R}_+) \Big\} \subset L^2(\Omega). \label{Hforiequal1} \end{equation} Using It$\hat{o}$'s formula on X for $X \in L^2(\Omega, \mathcal{F}, \mathbb{P})$, we can prove that $\mathcal{H}$ is dense in $L^2(\Omega, \mathcal{F}, \mathbb{P})$:
\begin{eqnarray*} (X_{t+t_0} - X_{t_0}) &=& \int^t_0 d(X_{s+t_0} - X_{t_0}) \\ && + \sum_{0 <s \leq t} \big[ (X_{s+t_0} -X_{t_0}) - (X_{(s+t_0)-} -X_{t_0}) -\Delta X_{s+t_0} \big] \mbox{ \cite{Nualart}}\\ &=& \int^t_0 d(X_{s+t_0} - X_{t_0}). \\ \mbox{ Set } t_0 = 0 &\mbox{ then, }& \\ X_t &=& \int^t_0 dX_s = \int^t_0 dX^1_s = \int^t_0 dY^1_s + m_1t = \mathbb{E}(X_t) + \int^t_0 dY^1_s.\\ \end{eqnarray*} We can derive from the above a simplified, 1-dimensional form of the {\bf Predictable Representation Property (PRP)}:
\[ \forall X \in L^2(\Omega, \mathcal{F}, \mathbb{P}), \forall t \in [0,T], \exists \phi \in L^2(\Omega) \mbox{ s.a. }\] \[ X_t = \mathbb{E}(X_T) + \int^t_0 \phi_s dY^1_s. \] As a consequence, we also have the following theorem
\begin{thm} $\mathcal{H}$ is dense in $L^2(\Omega, \mathcal{F}, \mathbb{P})$. In other words, $ L^2(\Omega, \mathcal{F}, \mathbb{P}) = \mathbb{R} \bigoplus \mathcal{H}$. \label{onedimPRP} \end{thm}
We want now to take the result of theorem \ref{onedimPRP} further to a multi-dimensional adaptation. As in the sections above, we wish to find an integral representation for martingales in $L^2(\Omega, \mathcal{F}, \mathbb{P})$. But this time in order to do so, we need to find a spamming set of $\mathcal{M}^2$. This is achievable by constructing a set of pairwise orthogonal martingales $\{ H^i \mid i \geq 1\}$ where:
\begin{eqnarray} H^i = Y^i + \sum^i_1 a_{i,i-j}Y^{i-j}. \label{polynomial1} \end{eqnarray}
Here, orthogonality is understood as in definition \ref{strongorthog}.
How can we find the coefficients $a_{i,j}$ in equation \ref{polynomial1} to form an orthogonal polynomial set? \\ Consider: \begin{eqnarray*} S_1 &=& \big\{ \mbox{polynomials in } \mathbb{R} \mid \langle P(x),Q(x) \rangle = \int^{+ \infty}_{- \infty} P(x)Q(x)x^2\nu(dx) + \sigma^2 P(0)Q(0) \big\} \mbox{ and} \\ S_2 &=& \big\{ \sum^n_i a_i Y^i \mid n\in \mathbb{N}, a_i \in \mathbb{R}, \langle Y^i, Y^j \rangle = m_{i+j} + \sigma^2 1_{\{i=j=1\}} \big\} \supseteq \{ H^i \mid i \geq 1\}. \end{eqnarray*}
Since there is a clear isometry between $S_1$ and $S_2$ , namely $x^{i-1} \longleftrightarrow Y^i$ \cite{Nualart}, we can create an orthogonal basis for $\mathcal{M}^2$ by orthogonalising $\{1, x, x^2,x^3....\}$. To this purpose, some well-known polynomials can be used, such as the Laguerre or the Hermite polynomials: \[ H_n(x) = (-1)^n e^{\frac{x^2}{2}}\frac{d^n}{dx^n}e^{\frac{-x^2}{2}}. \] \\
We take an interest in the following spaces: \begin{eqnarray*} \mathcal{H}^{(i_1,...,i_j)} &=& \Big\{ \int^{\infty}_0 \int^{t_1-}_0 ... \int^{t_{j-1}-}_0 f(t_1,...,t_j)dH^{i_j}_{t_j}...dH^{i_1}_{t_1} \mid f \in L^2(\mathbb{R}^j_+) \Big\} \subset L^2(\Omega). \\ \end{eqnarray*} It is known that if $(i_1,...,i_j) \neq (j_1,...,j_l)$ then $ \mathcal{H}^{(i_1,...,i_j)} \perp \mathcal{H}^{(j_1,...,j_l)}$. \cite{Nualart} \\ These space are basically a multi-dimensional adaptation of $\mathcal{H}$ defined earlier. Using these, we can prove an extension of theorem \ref{onedimPRP} \begin{thm}
$L^2(\Omega, \mathcal{F}) = \mathbb{R} \oplus \big( \bigoplus^{\infty}_{j=1} \bigoplus_{i_1...i_j \geq 1} \mathcal{H}^{(i_1,...,i_j)} \big). \label{theorem1}$ \end{thm}
\begin{proof} To show this we start by noting that: \begin{eqnarray*} \mathcal{P} = \big\{ X^{k_1}_{t_1} \prod^n_2 (X_{t_i}-X_{t_{i-1}})^{k_i} \mid n \geq 0, k_i \geq 1 \big\} \end{eqnarray*} is dense in $L^2(\Omega, \mathcal{F})$. Indeed, take $ Z \in L^2(\Omega, \mathcal{F})$, $Z \perp \mathcal{P}$. For some finite set $\{0 <s_1<...<s_m\}$ there is $Z_{\epsilon} \in L^2(\Omega, \omega(X_{s_1},...,X_{s_m}))$ s.a.: \[ \mathbb{E}\big[ (Z-Z_{\epsilon})^2\big] < \epsilon. \] There exists a Borel function where: $ Z_{\epsilon} = f_{\epsilon}(X_{s_1},X_{s_2}-X_{s_1},...,X_{s_m}-X_{s_{m-1}})$, which can be approximated by polynomials. Additionally, $\mathbb{E}[ZZ_{\epsilon}] = 0$. Then:
\begin{eqnarray*}
\mathbb{E}[Z^2] = \mathbb{E}[Z(Z-Z_{\epsilon})] \leq \sqrt{ \mathbb{E}[Z^2]\mathbb{E}[(Z-Z_{\epsilon})^2] } \leq \sqrt{ \epsilon\mathbb{E}[Z^2],} \end{eqnarray*} so Z = 0 a.s. as $ \epsilon \rightarrow 0$. If we can represent the terms of $\mathcal{P}$ we can thus represent $ L^2(\Omega, \mathcal{F})$. For this, we rely on the following lemma:
\begin{eqnarray} (X_{t_0+t} - X_{t_0})^k &=& \mathbb{E}[(X_{t_0+t} - X_{t_0})^k] \nonumber \\ &&+ \sum^k_{j=1} \sum_{(i_1,...,i_j)} \int^{t_0+t}_{t_0} \int^{t_1-}_{t_0}...\int^{t_{j-1}-}_{t_0} h^k_{(i_1,...,i_j)}(t,t_0,...,t_j)dH^{i_j}_{t_j}...dH^{i_1}_{t_1} \label{lemma1} \end{eqnarray}
where $h^k_{(i_1,...,i_j)} \in L^2(\mathbb{R}^j_+)$. \cite{Nualart}\\ For any $0 \leq t < s \leq u <v$, k,l $\geq$1, we have that $(X_s-X_t)^k(X_v-X_u)^l = AB$ where A and B are of the form stated in lemma \ref{lemma1}.Then:
\begin{eqnarray*} AB = \int^{\infty}_0 \int^{u_1-}_0 ... \int^{u_m-}_0 \int^{t_1-}_0 ... \int^{t_{n-1}-}_0 \prod_{i=u,u_1...t_n} 1(i) \\ \times h^l_{(j_1,...,j_m)}(v,u,u_1,...,u_m)h^k_{(i_1,...,i_n)}(s,t,t_1,...,t_n) \\ \times dH^{i_n}_{t_n}...dH^{i_1}_{t_1}dH^{j_m}_{u_m}...dH^{j_1}_{u_1}. \end{eqnarray*}
This then gives the {\bf Chaotic representation property (CRP)}: $\forall F \in L^2(\Omega, \mathcal{F})$, \begin{equation} F = \mathbb{E}[F] + \sum^{\infty}_j \sum_{i_1,..,i_j \geq1} \int^{\infty}_0 \int^{t_1-}_{t_0}...\int^{t_{j-1}-}_{t_0} f_{(i_1,...,i_j)}(t_1,...,t_j)dH^{i_j}_{t_j}...dH^{i_1}_{t_1} \label{CRP} \end{equation}
where $ f_{(i_1,...,i_j)} \in L^2(\mathbb{R}^j_+)$. Theorem 2 is then a consequence of the CRP \ref{CRP}. \end{proof}
To attain a representation result for martingales, we observe that the CRP (equation \ref{CRP}) can be transformed in the following way: $\forall F \in L^2(\Omega, \mathcal{F})$,
\begin{eqnarray*} F - \mathbb{E}(F) &=& \sum^{\infty}_j \sum_{i_1,..,i_j \geq1} \int^{\infty}_0 \int^{t_1-}_{t_0}...\int^{t_{j-1}-}_{t_0} f_{(i_1,...,i_j)}(t_1,...,t_j)dH^{i_j}_{t_j}...dH^{i_1}_{t_1} \\ &=& \sum^{\infty}_{i_1=1} \int^{\infty}_0 f_{i_1}(t_1) dH^{i_1}_{t_1} + \sum^{\infty}_{i_1=1} \int^{\infty}_0 \Big[ \sum^{\infty}_{j=2} \sum_{i_2,..,i_j \geq1} \int^{t_1-}_0 \cdots \\ && \int^{t_{j-1}-}_0 f_{(i_1,...,i_j)}(t_1,...,t_j) dH^{i_j}_{t_j}...dH^{i_2}_{t_2} \Big] dH^{i_1}_{t_1} \\ &=& \sum^{\infty}_{i_1=1} \int^{\infty}_0 \Big[ f_{i_1}(t_1) + \sum^{\infty}_{j=2} \sum_{i_2,..,i_j \geq1} \int^{t_1-}_0 \cdots \int^{t_{j-1}-}_0 f_{(i_1,...,i_j)}(t_1,...,t_j) dH^{i_j}_{t_j}...dH^{i_2}_{t_2} \Big] dH^{i_1}_{t_1} \\ &=& \sum^{\infty}_{i=1} \int^{\infty}_0 \phi^i_{t_1}dH^i_{t_1}, \end{eqnarray*}
where $\forall i $ \[ \phi^i_{t_1} = f_{i_1}(t_1) + \sum^{\infty}_{j=2} \sum_{i_2,..,i_j \geq1} \int^{t_1-}_0 \cdots \int^{t_{j-1}-}_0 f_{(i_1,...,i_j)}(t_1,...,t_j) dH^{i_j}_{t_j}...dH^{i_2}_{t_2} \] and $\phi^i_t$ is predictable. Hence the following result:
\begin{defn}[{\bf Predictable Representation property (PRP)}]
$\forall F \in L^2(\Omega, \mathcal{F})$, there is a $\phi^i_t$ predictable such as:
\begin{equation} F = \mathbb{E}(F) + \sum^{\infty}_{i=1} \int^{\infty}_0 \phi^i_s dH^i_s. \label{PRP} \end{equation}
Then $\forall M \in \mathcal{M}^2$ with $M_{\infty} \in L^2(\Omega,\mathcal{F})$ and $M_t = \mathbb{E}(M_{\infty} \mid \mathcal{F}_t) $ $\forall t$, The PRP (equation \ref{PRP}) gives us: \[ M_t = \sum^{\infty}_{i=1} \int^{t}_0 \phi^i_s dH^i_s. \]
\end{defn}
While this result is very similar to the conventional Brownian-motion based MRT, it presents the advantage of being more general and reaching out to wider sets and more elaborate martingales. Indeed, as we can see through the following: \begin{thm}[{\bf L$\acute{e}$vy-Ito theorem}]
$ \forall (X_t)$ L$\acute{e}$vy processes on $(\Omega, \mathcal{F}, \mathcal{F}_t, \mathbb{P})$ - where the distribution of $X_1$ is parametrized by $(\beta, \sigma^2, \upsilon)$ in the L$e$vy-Khintchine theorem - X decomposes as follow \cite{Applebaum} :
\begin{equation} X_t = \beta t+ \sigma W_t + J_t + M_t, \label{levyIto} \end{equation}
where $\beta \in \mathbb{R}$, $\sigma^2 \geq 0$ and $\upsilon$ is a measure on $\mathbb{R}/\{0\}$ such as $\int_{\mathbb{R}/\{0\}} 1 \wedge x^2 \upsilon(dx) < \infty$, and \begin{itemize}
\item $W_t$ is a Brownian Motion,
\item $\Delta X_t = X_t - X_{t-}$ for t $\geq$ 0 is an indep. Poisson point process with intensity $\upsilon$,
\item $ J_t = \sum_{s \leq t} \Delta X_s 1_{\{ |\Delta X_s| > 1\}} $, and
\item $M_t$ is a martingale with jumps: $\Delta M_t = \Delta X_t 1_{\{ |\Delta X_t| > 1\}}$. \end{itemize} \end{thm}
The result \ref{levyIto} can be re-written as done in \cite{Lokka}:
\begin{equation} L_t = \beta t + \sigma W_t + \int^t_0 \int_{\mathbb{R}_0} z (\mu - \upsilon)(t,dz) \label{levyito-lokka} \end{equation}
where $\mu$ is a Poisson random measure, $\mathbb{R}_0 = \mathbb{R} / \{0\}$, and $\beta, \sigma$ are defined as in the theorem above.
\begin{thm}[{\bf lemma 12} \cite{Lokka}] $\exists \{ \Lambda_n \}^{\infty}_{n=1}$ partitioning $\mathbb{R}_0$ and $z_n \in \Lambda_n \subseteq \mathbb{R}$ such as \[ \int_{\mathbb{R}_0} z(\mu-\upsilon)(t,dz) = \sum^{\infty}_{n=1} z_n(\mu-\upsilon)(t,\Lambda_n) \] where the processes $(\mu-\upsilon)(t,\Lambda_n)$ are all compensated Poisson processes with intensity $\upsilon(\Lambda_n)$. Hence, \begin{equation} \int^t_0 \int_{\mathbb{R}_0} z (\mu - \upsilon)(t,dz) = \int^t_0 \int_{\mathbb{R}_0} z \tilde{N}(ds,dz). \label{lemma-lokka} \end{equation} \end{thm}
Then we can apply the PRP, result obtained in equation \ref{PRP}: $\forall m \in \mathcal{M}^2$, \begin{eqnarray*} m_t &=& \sum^{\infty}_{i=1} \int^{t}_0 \phi^i_s dH^i_s \\ &=& \sum^{\infty}_{i=1} \int^{t}_0 \phi^i_s (\sigma dW^i_s + d( \int^s_0 \int_{\mathbb{R}_0} z (\mu - \upsilon)^i(s,dz))) \mbox{ , since $H^i$ is a l$\acute{e}$vy process result \ref{levyito-lokka} applies,} \\ &=& \sum^{\infty}_{i=1} \int^{t}_0 \hat{\phi}^i_s dW^i_s + \int^t_0 \psi^i(s,z) (\mu - \upsilon)^i(s,dz) \\ &=& \sum^{\infty}_{i=1} \int^{t}_0 \hat{\phi}^i_s dW^i_s + \int^t_0 \psi^i(s,z) \tilde{N}^i(ds,dz) \mbox{- using result \ref{lemma-lokka},} \end{eqnarray*} which is similar to a result developed in \cite{Lokka}:\\ $\forall m \in \mathcal{M}^2$, and where ($m_t$) is of dimension n, $\exists \hat{\phi}(t), \psi(t,z)$ predictable n-dimensional processes such as \begin{equation} m_t = \int^{t}_0 \hat{\phi}_s dW_s + \int^t_0 \psi(s,z) \tilde{N}(ds,dz), \label{wiener-poissonmrt} \end{equation}
where $W_t$, $\tilde{N}(t,z) $ are also n-dimensional and \begin{eqnarray*} \mathbb{E}(\int^T_0 \hat{\phi}^2_s ds) < \infty ,\\ \mathbb{E}( \int^T_0 \psi(s,z)^2 \upsilon(dz) ds) < \infty. \end{eqnarray*}
\section{The Clark-Ocone formula and explicit representation of the integrand}
We have explored in the previous chapter the essentials of the Martingale Representation Theorem in its various forms, notably within the continuous space driven by the Brownian motion and beyond with L$\acute{e}$vy and jump processes. It is generally agreed that such a representation formula exists in $\mathcal{M}^2$ because of the Hilbert structure of the probability spaces these processes live in. More specifically, in each of these $L^2(\Omega, \mathcal{F}, (\mathcal{F})_{t \geq 1})$,there is a set of integrals of fundamental processes $\mathcal{H}$ such as $\mathcal{H}$ is dense in $\mathcal{M}^2 \subseteq L^2$. and:
\[ \forall X_t \in \mathcal{M}^2, X_t = \int^t_0 \phi_s dq_s \mbox{, } q_t \in \mathcal{H}. \]
While this gives a general formula to martingales, it gives no indication as to what $\phi_t$ is. However knowing the form of the integrand $\phi_t$ is of central importance in the applications of the MRT, notably in the representation of portfolio dynamics and trading strategy optimization.\\ \\ To explore this question, we first introduce basics of Malliavin calculus relevant to the representation of the integrand $\phi_t$. We then cover an important result providing a formula for the integrand, the Clark-Ocone Formula.
\subsection{Malliavin Calculus}
Malliavin calculus is the extension of the calculus of variations from functions to stochastic processes over a finite or infinite dimensional space. To develop on this topic, we follow the work of Li (2011) \cite{Li} and $\O$ksendal (1997) \cite{Oksendal2}. \\ We work on $L^2(\Omega, \mathcal{F}_t, \mathbb{P})$, where $\mathcal{F}_t$ is the $\sigma$-algebra generated by a Brownian Motion $W_t$, $t \in [0,T]$, and $\Omega = C_0([0,T])$. Here, $\mathcal{F}= \{\mathcal{F}_t \mid t \in [0,T]\}$ is the initial filtration augmented by $\mathbb{P}$-zero measure sets. \\
\begin{defn} a function $g: [0,T]^n \rightarrow \mathbb{R}$ is symmetric if $ g(t_{\sigma_1},...,t_{\sigma_n}) = g(t_1,...,t_n)$ for any permutation $\sigma=(\sigma_1,...,\sigma_n)$ of (1,..,n). \end{defn}
Let: \[ \tilde{L}^2([0,T]^n) = \{ g: [0,T]^n \rightarrow \mathbb{R} \mid \mbox{symmetric square integrable functions}\} \subset L^2([0,T]^n) \mbox{, and} \] \[ S_n = \{(t_1,..,t_n) \in [0,T]^n \mid t_i \leq t_j, \forall i \leq j \}. \]
\begin{defn}[{\bf the n-fold interated It$\hat{o}$ integral:}] We define the n-fold iterated It$\hat{o}$ integral where f is a deterministic function defined on $S_n$ as: \[ J_n(f) = \int^t_0\int^{t_n}_0 ... \int^{t_2}_0 f(t_1,...,t_n) dW_{t_1}...dW_{t_n}, \] and note that $J_n(f) \in L^2(\Omega)$. \end{defn}
\begin{defn} for $g \in \tilde{L}^2([0,T]^n)$, set \[I_n(g) = \int_{[0,T]^n} g(t_1,...,t_n) dW_{t_1}...dW_{t_n}=n!J_n(g). \] \label{defInJn} \end{defn}
\begin{defn}[{\bf n-th Wiener Chaos} \cite{Peccati}] The n-th Wiener Chaos $C_n$ is defined as: \[ C_n = \{ I_n(f) \mid f \in L^2(\mathbb{P}) \} \mbox{, } n \geq 1.\] \end{defn}
The operators $I_n$ and $J_n$ have a couple of useful properties, notably the following: \begin{thm}
$\forall f_n \in \tilde{L}^2([0,T]^n)$ with G being a borel set $\subseteq [0,T]$, \begin{equation} \mathbb{E}(I_n(f_n)) \mid \mathcal{F}_G) = I_n(f_n \prod^n_{i=1} 1_{G}(t_i)), \label{condiexpI} \end{equation} where $\mathcal{F}_G$ is a completed $\sigma$-field: \[ \mathcal{F}_G = \sigma \{ \int^T_0 1_{A}(t)dW_t \mid A- \mbox{borel sets}, A \subseteq G \}. \] \end{thm}
\begin{thm}[{\bf Wiener-It$\hat{o}$ Chaos Expansion \cite{Oksendal2}}] $F \in L^2(\Omega)$. Then there is a unique sequence $(f_n)_n$ of deterministic functions $f_n \in \tilde{L}^2([0,T]^n)$ such as: \begin{equation} F = \sum^{\infty}_{n=0} I_n(f_n) = \mathbb{E}(F) + \sum^{\infty}_{n=1} I_n(f_n), \label{wienerchaosexp} \end{equation} and
\[ ||F||^2_{L^2(\Omega)} = \sum^{\infty}_{n=0} n! ||f_n||^2_{L^2([0,T]^n)} < \infty.\] \end{thm}
Moreover, we have: \[ J_n(F) = I_n(f_n), \] so J can be seen as an orthogonal projection of F on the n-th Chaos $C_n$ \cite{Li}: \[ F = \sum^{\infty}_{n=0} J_n(F). \]
At this stage, we denote the following space:
\begin{defn} \[ \mathcal{P} = \{ F: \Omega \rightarrow \mathbb{R} \mid F(\omega) = p (W_{t_1}...W_{t_n}), p(x) \mbox{: polynomial, } p \in C_0[0,T] \} \] and also we introduce the Cameron-Martin space $\mathcal{H}$:
\[ \mathcal{H} = \big\{ \gamma: [0,T] \rightarrow \mathbb{R} \mid \gamma(t) = \int^t_0 \dot{\gamma}(s)ds, |\gamma|^2_{\mathcal{H}} = \int^T_0 \dot{\gamma}(s)^2ds <\infty \big\} \subseteq C_0[0,T]\] \end{defn}
At this stage, we can now introduce the concept of directional derivative;
\begin{defn} $ \forall \in \mathcal{P}$ the directional derivative $D_{\gamma}F(\omega)$ $\forall \gamma \in \mathcal{H}$ is defined as:
\begin{equation} D_{\gamma}F(\omega) = \lim_{\epsilon \rightarrow 0 }\frac{F(\omega +\epsilon \gamma) - F(\omega)}{\epsilon}, \label{dirder1} \end{equation}
or alternatively, \begin{equation} D_{\gamma}F(\omega) = \sum^n_{i=1} \frac{\partial p}{\partial x_i}(W_{t_1},...,W_{t_n}) \gamma(t_i). \label{dirder2} \end{equation}
$\forall F \in \mathcal{P}$, the function $D_{\gamma}: \mathcal{H} \rightarrow L^2(\Omega )$ is continuous \cite{Oksendal2} and obeys to the product rule: \[ D_{\gamma}(FG) = F D_{\gamma}G + G D_{\gamma}F. \] \end{defn}
\begin{thm}[{\bf Riesz Representation theorem} \cite{Rudin}] For every F that has a defined derivative $D_{\gamma}F$ $\forall \gamma \in \mathcal{H}$, $\exists ! \bigtriangledown F(\omega) \in \mathcal{H}$ such as: \begin{equation} D_{\gamma}F(\omega) = \langle \gamma, \bigtriangledown F(\omega) \rangle_{\mathcal{H}} = \int^T_0 \dot{\bigtriangledown F(\omega)} \dot{\gamma} dt. \label{riesz} \end{equation} \end{thm}
The Malliavin derivative $D_tF$ of F can then de defined as such:
\begin{defn}[{\bf Malliavin derivative}]
$D_t: \mathcal{P} \rightarrow L^2([0,T] \times \Omega)$ is the Radon-Nikodym derivative of $\bigtriangledown F(\omega)$:
\begin{equation} D_{\gamma}F(\omega) = \int^T_0 D_tF(\omega) \dot{\gamma}(t) dt. \nonumber \end{equation}
The Malliavin derivative $D_t F$ is also continuous, closable and obeys to the product rule. \label{malliaderdef} \end{defn}
Then we introduce the following semi-norm on $\mathcal{P}$ :
\begin{equation}
\| |F| \|_{1,2} = \Big[ \mathbb{E}(|F|^2)+ \mathbb{E}\big(\| D_tF\|^2_{L^2([0,T])}\big) \Big]^{1/2}. \label{d12norm} \end{equation}
The completion of $\mathcal{P}$ under this new norm in \ref{d12norm} creates a Banach space $\mathbb{D}_{1,2}$ called a Sobolev space. $\mathbb{D}_{1,2}$ is a Hilbert space such as \cite{Oksendal2}: \[ \mathbb{D}_{1,2}= \Big\{ F \in L^2(\Omega) \mid \{F_n\}_{n \in \mathbb{N}} \rightarrow F, \{D_tF_n\}_{n \in \mathbb{N}} \mbox{ Cauchy in } L^2([0,T] \times \Omega) \Big\}. \]
\begin{thm} If $\phi : \mathbb{R}^n \rightarrow \mathbb{R}$ is Lipschitz i.e. $\forall$ x,y $ \in \mathbb{R}^n$ and $\exists$ K constant, we have:
\[ | \phi(x) - \phi(y)| \leq K |x-y|, \] and F = ($ F_i$) $ F_i\in \mathbb{D}_{1,2}$ $\forall i \leq n$, Then $\phi(F) \in \mathbb{D}_{1,2}$ and \[ D_t \phi(F) = \sum^n_{i=1} \frac{\partial \phi}{\partial x_i}(F) D_t F_i \mbox{ \cite{Li}}.\] \end{thm}
The Malliavin derivative $D_t$ aslo has the following useful property: \begin{thm}[Oksendal \cite{Oksendal2}] For any $F(\omega) = I_n(f_n)$ where $f_n \in \tilde{L}^2([0,T]^n)$, F $\in \mathbb{D}_{1,2}$ and \begin{equation} D_t F(\omega) = n I_{n-1}(f_n). \label{diffIn} \end{equation} \end{thm}
To match the Malliavin derivative, there is a special form of integration similar to stochastic integration:
\begin{defn}[{\bf Skorohod integral of u}]
$u(t,\omega)$ is a $\mathcal{F}_T$-measurable r.v. for all t $\in [0,T]$ such as: \[ \mathbb{E}(u^2(t)) < \infty, \] and u(t) has a Wiener-It$\hat{o}$ chaos expansion $u(t) = \sum^{\infty}_{n=0} I_n(f_n) $ where $f_n \in \tilde{L}^2([0,T]^n)$. set \[ \tilde{f}_n(t_1,...,t_{n+1}) = \frac{1}{n+1} \big[ f_n(t_1,...,t_{n+1})+ f_n(t_2,...,t_{n+1},t_1)+... \big]. \]
For u(t) = $\sum^{\infty}_{n=0} I_n(f_n)$, the Skorohod integral is defined as: \begin{equation} \delta(u) = \int^t_0 u(t)\delta W_t = \sum^{\infty}_{n=0} I_{n+1}(\tilde{f}_n) \label{skorohod} \end{equation} whenever :
\begin{equation}
\mathbb{E}(\delta(u)^2) = \sum^{\infty}_{n=0} (n+1)! \|\tilde{f}_n\|^2_{L^2} < \infty, \label{skorohodcond} \end{equation}
in which case, we say $u \in Dom(\delta)$. \end{defn}
The Skorohod integration $\delta$ and the Malliavin derivative $D_t$ are connected through the following version of the fundamental theorem of calculus
\begin{thm}[{\bf The fundamental theorem of calculus}] Let u(s) be a Skorohod-integrable stochastic process contained in $\mathbb{D}_{1,2}$ and that $\forall t \in [0,T]$, $D_tu$ is also Skorohod-integrable. Then: \[ D_t \Big( \int^T_0 u(s) \delta W_s \big) = \int^T_0 D_t u(s) \delta W_s + u(t) \] \end{thm}
\begin{thm} $u \in Dom(\delta) \Longrightarrow \delta(u) \in L^2 $ \end{thm}
The Skorohod integral has a couple of nice properties, notably that \[ \mathbb{E}( \delta(u)) = 0\] as it is an iterated integral of Brownian motion, and hence has zero expectation.
\begin{thm} u(t) is an $\mathbb{F}$-adpated r.v. such as \[ \mathbb{E}( \int^T_0 u^2(t)dt) < \infty. \] Then u $\in Dom(\delta)$ and the Skorohod integral coincides with the It$\hat{o}$ integral: \[ \int^T_0 u(t) \delta W_t = \int^T_0 u(t) dW_t. \] \label{skoequality} \end{thm} This theorem illustrates the usefulness of the Skorohod integral: it is an equivalent of the regular stochastic integral, but is applicable to stochastic processes that are $\mathbb{F}$-adapted or not. The following theorem describes easily in which case we are in.
\begin{thm} u(t) is $\mathcal{F}_T$-measurable and $\mathbb{E}(u^2(t)) < \infty$. We have: \[ u(t) = \sum^{\infty}_{n=0} I_n(f_n). \] u(t) is $\mathbb{F}$-adapted if and only if \cite{Oksendal3} \begin{equation} f_n(t_1,...,t_n,t) 1_{t < \max_{i \leq n} t_i} = 0. \end{equation} \end{thm}
Theorem \ref{skoequality} enables us to establish the following equality: \begin{equation} \int^T_0 \sum^{\infty}_n J_n(f_n) dW_t = \sum^{\infty}_n J_{n+1}(f_n). \label{intofJ} \end{equation}
\subsection{The Clark-Ocone formula: Result and application to the MRT}
\begin{thm}[{\bf The Clark-Ocone formula} \cite{Oksendal2}] $ \forall F \in \mathbb{D}_{1,2}$ where F is $\mathcal{F}_T$-measurable, the following representation holds:
\begin{equation} F(\omega) = \mathbb{E}(F) + \int^T_0 \mathbb{E}(D_t F \mid \mathcal{F}_t) dW(t). \label{clarkocone} \end{equation} \end{thm}
\begin{proof} This result is a statement on the It$\hat{o}$ representation theorem seen in chapter 1 result \ref{itorepth}; $\forall F \in L^2(\Omega)$ where F is $\mathcal{F}_T$-measurable, $\exists ! \phi(t)$ such as: \[ F = \mathbb{E}(F) + \int^T_0 \phi(t) dW_t. \] The difference is that here, we have an explicit form for $\phi(t)$ and we claim $\phi(t) = \mathbb{E}(D_t F \mid \mathcal{F}_t) $.\\ We know from the Wiener-It$\hat{o}$ chaos expansion (result \ref{wienerchaosexp}) that: \[ F = \sum^{\infty}_{n=0} I_n(f_n) \] where $f_n \in \tilde{L}^2([0,T]^n)$. We then have the following: \begin{eqnarray*} \int^T_0 \mathbb{E}(D_t F \mid \mathcal{F}_t) dW_t &=& \int^T_0 \mathbb{E}(D_t \sum^{\infty} I_n(f_n) \mid \mathcal{F}_t) dW_t \mbox{ ,using result \ref{wienerchaosexp} } \\ &=& \sum^{\infty} \int^T_0 \mathbb{E}(D_t I_n(f_n) \mid \mathcal{F}_t) dW_t \\ &=& \sum^{\infty}_{n=1} \int^T_0 \mathbb{E}(n I_{n-1}(f_n) \mid \mathcal{F}_t) dW_t \mbox{ ,using \ref{diffIn}} \\ &=& \sum^{\infty}_{n=1} n \int^T_0 \mathbb{E}(I_{n-1}(f_n) \mid \mathcal{F}_t) dW_t \\ &=& \sum^{\infty}_{n=1} n \int^T_0 I_{n-1}(f_n \prod^{n-1}_i 1_{\mathcal{F}_t}(t_i)) dW_t \mbox{ ,using \ref{condiexpI}} \\ &=& \sum^{\infty}_{n=1} n \int^T_0 I_{n-1}(f_n \prod^{n-1}_i 1_{[0,t]}(t_i)) dW_t \\ &=& \sum^{\infty}_{n=1} n(n-1)! \int^T_0 J_{n-1}(f_n \prod^{n-1}_i 1_{[0,t]}(t_i)) dW_t \mbox{ ,using definition \ref{defInJn}} \\ &=& \sum^{\infty}_{n=1} n! \int^T_0 J_{n-1}(f_n) dW_t \\ &=& \sum^{\infty}_{n=1} n! J_{n}(f_n) \mbox{ ,using result \ref{intofJ}} \\ &=& \sum^{\infty}_{n=1} I_{n}(f_n) = \sum^{\infty}_{n=0} I_{n}(f_n) - I_{0}(f_0) = F - \mathbb{E}(F). \end{eqnarray*} \end{proof}
Looking at the martingale representation theorem in the continuous filtration as seen in section 1.2.1, we see that for all $(m_t) \in \mathcal{M}^2 \cap \mathbb{D}_{1,2}$, \[ m_t = \mathbb{E}( m_{\infty} \mid \mathcal{F}_t ) = \int^t_0 \phi_s dW_s \] for some $\phi_t$ deterministic where, through result \ref{clarkocone}, we know that $\phi_t = \mathbb{E}(D_t m_{\infty} \mid \mathcal{F}_t) $. So the can re-write the martingale representation theorem as such:
\begin{eqnarray}
\forall m \in \mathcal{M}^2 \cap \mathbb{D}_{1,2}, m_t &=& \mathbb{E}(m_{\infty} \mid \mathcal{F}_t) \nonumber \\ &=& \mathbb{E}(\int^{\infty}_0 \mathbb{E}( D_s m_{\infty} \mid \mathcal{F}_s) dW_s \mid \mathcal{F}_t) \nonumber \\ &=& \int^t_0 \mathbb{E}( D_s m_{\infty} \mid \mathcal{F}_s) dW_s. \label{contMRTmod}
\end{eqnarray}
Now, for this representation result to hold, it is important to know whether $ \int^t_0 \mathbb{E}( D_s m_{\infty} \mid \mathcal{F}_s)^2 ds < \infty$ for all t $\in [0,T]$.\\ Since $m_t \in \mathcal{M}^2 \cap \mathbb{D}_{1,2}$ for any t in [0,T], we know that:
\[ \| |m_{\infty}| \|_{1,2}= \Big[ \mathbb{E}(|m_{\infty}|^2)+ \mathbb{E}\big(\| D_t m_{\infty}\|^2_{L^2([0,T])}\big) \Big]^{1/2} < \infty,\] hence
\[ \mathbb{E}\big(\| D_t m_{\infty}\|^2_{L^2([0,T])}\big) < \infty, \] which implies that indeed: \begin{eqnarray*} \mathbb{E}(\int_{[0,T]} \mathbb{E}( D_s m_{\infty} \mid \mathcal{F}_s)^2 ds) &<& \mathbb{E}(\int_{[0,T]} \mathbb{E}( (D_s m_{\infty})^2 \mid \mathcal{F}_s) ds) \\ &=& \mathbb{E}(\mathbb{E}(\int_{[0,T]} (D_s m_{\infty})^2 ds\mid \mathcal{F}_s) ) \\ &=& \mathbb{E}(\int_{[0,T]} (D_s m_{\infty})^2 ds ) \\
&=& \mathbb{E}( \| D_s m_{\infty} \| ^2_{L^2}) < \infty. \end{eqnarray*}
So $\int_{[0,T]} \mathbb{E}( D_s m_{\infty} \mid \mathcal{F}_s)^2 ds < \infty$ a.s. and thus the martingale representation theorem in result \ref{contMRTmod} is well-defined.
\subsection{Explicit integrand representation beyond continuous processes: Poisson Malliavin Calculus and Clark-Ocone formula applied to jump processes}
\subsubsection{Poisson Malliavin Calculus}
In this setting we work with the compensated poisson process $\tilde{N} (t,z)$, a martingale contained in $ L^2( [0,T] \times \mathbb{R}_0^n ) $. It evolves on a complete probability space $(\Omega, \mathcal{F}, P)$, where $\mathcal{F}_t$ is the $\sigma$-algebra generated by $\tilde{N} (s,z)$, $ 0 \leq s \leq t$.\cite{Oksendal3} \\ Let $\mu$ be a l$\acute{e}$vy measure, and $\lambda$ the regular Lebesgue measure. $\tilde{L}^2 ( \lambda \times \mu )$ denote the space of symmetric functions in $L^2 ( \lambda \times \mu )$, which itself is the space of suitably square integrable functions: \begin{eqnarray*}
\| f \|^2_{ L^2 ( \lambda \times \mu ) } = \int_{ ([0,T] \times \mathbb{R}_0)^n } f^2 dt_1 \mu (dz_1)...dt_n \mu (dz_n) < \infty. \end{eqnarray*} Set: \begin{eqnarray*}
G_n = \Big\{ (t_i, z_i)_{i=1,n} \Big| 0 \leq t_1 \leq ... \leq t_n \leq T \Big| z_i \in \mathbb{R}_0 \Big\} \end{eqnarray*}
and $ L^2 (G_n) = \big\{ g \in G_n \big| \| g \|^2_{ L^2 ( G_n) } < \infty \big\}.$ \\ As earlier, the n-fold Iterated Stochastic integrals are defined as bellow: \begin{eqnarray*} J_n(g) &=& \int^T_0 \int_{\mathbb{R}_0} ... \int^{t_2}_0 \int_{\mathbb{R}_0} g(t_1,z_1...) \tilde{N} (dt_1,dz_1) ....\tilde{N} (dt_n,dz_n) \mbox{ , } \forall g \in L^2 (G_n),\\ I_n(g) &=& n! J_n(g) = \int_{([0,T] \times \mathbb{R}_0)^n} g(t_1,z_1...) \tilde{N} (dt_1,dz_1) ....\tilde{N} (dt_n,dz_n), \end{eqnarray*} \\ and as in the continuous case, there is a Wiener-Ito Chaos expansion:\\ $ \forall F \in L^2(P)$, where F is $\mathcal{F}_T$-measurable, $ \exists $ unique $ f_n \in \tilde{L}^2 ( (\lambda \times \mu)^n )$ such as: \begin{eqnarray*} F = \sum^{\infty}_{n=0} I_n(f_n). \end{eqnarray*}
\begin{defn}[{\bf Sobolev Stochastic space}] $ \mathbb{D}_{1,2} \subset L^2 ( \lambda \times \mu )$ such as \cite{Oksendal3} \begin{eqnarray*}
\| F \|^2_{ \mathbb{D}_{1,2} } = \sum^{\infty}_{n=1} n n! \| f_n \|^2_{ L^2 ((\lambda \times \mu )^n) } < \infty. \end{eqnarray*} \end{defn}
In the Poisson setting, the Malliavin derivative of F has an alternative definition at (t,z) \cite{Oksendal3} : \begin{eqnarray} D: \mathbb{D}_{1,2} \rightarrow L^2 ( P \times \lambda \times \mu) \nonumber \\ D_{t,z} F = \sum^{\infty}_{n=1} n I_{n-1}(f_n(.,t,z)). \label{poissMallDer} \end{eqnarray} In this case as earlier, the D operator follows some classic rules of traditional calculus: \\ \\ {\bf Closability}: \\ $ F, F_k \in \mathbb{D}_{1,2}$ $\forall k \in \mathbb{N}$, and $ F_k \rightarrow F$ in $ L^2(P)$ as well as $ D_{t,z}F_k$ converges. \\ Then $ D_{t,z}F_k \rightarrow D_{t,z} F $. \\ \\ {\bf Chain Rule:} \\ $ F \in \mathbb{D}_{1,2}$ and $\phi$ is continuous on $\mathbb{R}$. Given $\phi(F) \in L^2(P)$ and $\phi(F+D_{t,z}F) \in L^2(P \times \lambda \times \mu)$ \\ Then $\phi(F) \in \mathbb{D}_{1,2}$ and $D_{t,z}\phi(F) = \phi(F+D_{t,z}F) - \phi(F)$. \\\\ {\bf Integration by parts:}\\ X(t,z) is Skorohod integrable, $ F \in \mathbb{D}_{1,2}$ and $ X(t,z)(F + D_{t,z}F)$ is also Skorohod integrable, then:
\begin{eqnarray*} F \int^T_0 \int_{\mathbb{R}_0} X(t,z) \tilde{N} (\delta t,\delta z)&& \\ = \int^T_0 \int_{\mathbb{R}_0} X(t,z)(F + D_{t,z}F) \tilde{N} (\delta t,\delta z) &+& \int^T_0 \int_{\mathbb{R}_0} X(t,z)D_{t,z}F \mu(dz)dt. \end{eqnarray*}
\subsubsection{Clark-Ocone formula: a jump diffusion version}
\begin{thm}[{\bf Jump process Clark-ocone formula \cite{Oksendal3}:}] $\forall F \in \mathbb{D}_{1,2}$ we have: \begin{equation} F = \mathbb{E}(F) + \int^T_0 \int_{\mathbb{R}_0} \mathbb{E}(D_{t,z} F \mid \mathcal{F}_t) \tilde{N}(dt,dz) \label{jumpClarkOcone} \end{equation} whenever $ \mathbb{E}(D_{t,z} F \mid \mathcal{F}_t)$ is predictable. \end{thm}
\begin{proof} The proof is very similar to the one developed in section 2 for result \ref{clarkocone}.Here again, F has a chaos expansion $F= \sum^{\infty}_{n=0} I_n(f_n)$, $f_n \in \tilde{L}^2((\lambda \times \mu)^n)$. Then:
\begin{eqnarray*} \int^T_0 \int_{\mathbb{R}_0} \mathbb{E}(D_{t,z} F \mid \mathcal{F}_t) \tilde{N}(dt,dz) &=& \int^T_0 \int_{\mathbb{R}_0} \mathbb{E}( \sum^{\infty}_{n=1} nI_{n-1}(f_n(.,t,z)) \mid \mathcal{F}_t) \tilde{N}(dt,dz) \mbox{, result \ref{poissMallDer}}\\ &=&\sum^{\infty}_{n=1} n \int^T_0 \int_{\mathbb{R}_0} \mathbb{E}((n-1)!J_{n-1}(f_n(.,t,z)) \mid \mathcal{F}_t) \tilde{N}(dt,dz) \\ &=& \sum^{\infty}_{n=1} n! \int^T_0 \int_{\mathbb{R}_0} \mathbb{E}( J_{n-1}(f_n(.,t,z)) \mid \mathcal{F}_t) \tilde{N}(dt,dz) \\ &=& \sum^{\infty}_{n=1} n! \int^T_0 \int_{\mathbb{R}_0} J_{n-1}(f_n(.,t,z)1_{[0,t]})\tilde{N}(dt,dz) \\ &=& \sum^{\infty}_{n=1} n! J_{n}(f_n(.,t,z)) = \sum^{\infty}_{n=1} I_{n}(f_n(.,t,z)) = F - \mathbb{E}(F). \end{eqnarray*}
\end{proof}
As seen in Part1 (section 2.3.2), all Poisson pure-jump martingales in $ \mathcal{M}^2$ can be expressed as: \[ m_t = \int^t_0 \int_{\mathbb{R}_0} g(s,z)\tilde{N}(ds,dz) \] for suitable integrand g(s,z). Now we know, through result \ref{jumpClarkOcone} that for all $m \in \mathbb{D}_{1,2} \cap \mathcal{M}^2$, we have: \[ m_{\infty} = \int^T_0 \int_{\mathbb{R}_0} \mathbb{E}(D_{t,z} m_{\infty} \mid \mathcal{F}_t) \tilde{N}(dt,dz). \] Hence the can write the Jump-process Martingales representation theorem as follow: \[ \forall m \in \mathbb{D}_{1,2} \cap \mathcal{M}^2, \mbox{ } m_t = \mathbb{E}(m_{\infty}) = \int^t_0 \int_{\mathbb{R}_0} \mathbb{E}(D_{s,z} m_{\infty} \mid \mathcal{F}_s) \tilde{N}(ds,dz). \]
Provided $m_{\infty} < \infty$, since $(m_t) \in \mathbb{D}_{1,2}$ we can see that:
\begin{eqnarray*} \mathbb{E}( [ \int^t_0 \int_{\mathbb{R}_0} \mathbb{E}(D_{s,z} m_{\infty} \mid \mathcal{F}_s) \tilde{N}(ds,dz)]^2) &=& \mathbb{E}( [ \int^t_0 \int_{\mathbb{R}_0} \mathbb{E}(D_{s,z} m_{\infty} \mid \mathcal{F}_s)^2 \upsilon(ds)dz)]) \\
&<& \mathbb{E}( \| D_t m_{\infty}\|^2_{L^2([0,T] \times \Omega)}) \mbox{ like in section 2} \\ & < &\infty \mbox{, as } (m_t) \in \mathbb{D}_{1,2}. \end{eqnarray*}
\subsection{Explicit integrand representation: Clark-Ocone formula applied to general L$\acute{e}$vy processes}
As we saw in chapter 1 result \ref{wiener-poissonmrt}, martingales evolving on Wiener-Poisson spaces have the following representation formula: \[ m_t = \int^t_0 \phi(t) dW_t + \int^t_0 \int_{\mathbb{R}_0} \psi(t,z) \tilde{N}(dt,dz), \]
where $ \phi(t), \psi(t,z)$ are predictable and $L^2$-integrable. In this case, just as in section 2.2 and 2.3, the Clark-Ocone formula can be applied and gives an explicit form to $ \phi(t)$ and $\psi(t,z)$:
\begin{thm}[{\bf Clark-Ocone formula for l$\acute{e}$vy processes} \cite{Lokka}] $\forall F \in L^2(\Omega) \cap \mathbb{D}_{1,2}$, \[ F = \mathbb{E}(F) + \int^T_0 \mathbb{E}(D_t F \mid \mathcal{F}_t) dW_t + \int^T_0 \int_{\mathbb{R}_0} \mathbb{E}(D_{t,z} F \mid \mathcal{F}_t) \tilde{N}(dt,dz). \] \label{general-clark-ocone} \end{thm} Using theorem \ref{general-clark-ocone}, we can see that for all martingales $(m_t) \in \mathcal{M}^2 \cap \mathbb{D}_{1,2}$, \begin{equation} m_t = \int^t_0 \mathbb{E}(D_s m_{\infty} \mid \mathcal{F}_s) dW_s + \int^t_0 \int_{\mathbb{R}_0} \mathbb{E}(D_{s,z} m_{\infty} \mid \mathcal{F}_s) \tilde{N}(ds,dz). \label{general-mrt} \end{equation} Here the concept of $\mathbb{D}_{1,2}$ is understood as such:
\[ \mathbb{D}_{1,2} = \big\{ F = \sum^{\infty}_n I_n(f_n) \mid \sum^{\infty}_n n \cdot n! \| f_n\|^2_n < \infty \big\} \subseteq L^2(\Omega), \] where: \[ I_n(f) = \sum_{\alpha} \int_{[0,T]^n} f(t)^{\alpha} dM^{t_1}_{\alpha_1} ... dM^{t_n}_{\alpha_n}, \] $(M^{t_1}_{\alpha_1} ... M^{t_n}_{\alpha_n})$ is a martingale. Based on this, we can re-write result \ref{general-mrt} as follow: \[ m_t = \int^t_0 f(s) \cdot dM_s \] where \begin{eqnarray*} f(t) = \left( \begin{array}{c}\mathbb{E}( D_t m_{\infty} \mid \mathcal{F}_t)\\ \mathbb{E}( D_{t,z} m_{\infty} \mid \mathcal{F}_t) \end{array} \right) \mbox{, and } M_t = \left( \begin{array}{c} W_t \\ \tilde{N}(t,z) \end{array} \right) \end{eqnarray*} Hence we can re-write ($m_t$) as such:
\[ m_t = \sum_{\alpha} \int^t_0 \mathbb{E}( D_{t,\alpha} m_{\infty} \mid \mathcal{F}_t) dM^{\alpha}_s \]
by setting $\alpha := (1,2)$ and : \[ M^1_t = W_t \mbox{ , } D_{t,1} m_{\infty} = D_t m_{\infty}, \] \[ M^2_t = \tilde{N}(t,z) \mbox{ , } D_{t,2} m_{\infty} = D_{t,z} m_{\infty}. \]
In this framework, in order to prove integrability, we give an alternative, more general definition to the operator $D_{t,\alpha}$:
\begin{defn} The function $D: \mathbb{D}_{1,2} \mapsto \bigoplus_{\alpha} L^2([0,T] \times \Omega, d\langle M^{\alpha} \rangle \times d\mathbb{P})$ is defined by: \[ D_{t,\alpha} F := \sum^n_n nI_{n-1}(f^\alpha_n(\cdot, t)). \] \label{Lokkalevyderiv} Note that definition \ref{Lokkalevyderiv} is very similar to the definition of the Poisson-Malliavin derivative in result \ref{poissMallDer} and to the conventional definition of the Malliavin derivative for continuous processes set in result \ref{diffIn}. \end{defn} Since $(m_t)$ is in $L^2(\mathcal{F}_T,\mathbb{P}) \cap \mathbb{D}_{1,2}$, it has the following decomposition: \[ m_{\infty} = \sum_n I_n(f_n) \mbox{ , } \exists f_n \] whenever $m_{\infty} < \infty$. Then we can see that for any t inside [0,T] \cite{Lokka},
\begin{eqnarray*} \mathbb{E}(m_t^2) &=& \mathbb{E}[ (\int^t_0 f(s) \cdot dM_s)^2] \\ &=& \mathbb{E}[ \sum_{\alpha} \int^t_0 (\mathbb{E}(D_{s,\alpha} m_{\infty}\mid \mathcal{F}_s))^2 d<M^{\alpha}>_s] \\ &=& \sum_{\alpha} \int^t_0 \mathbb{E}[ (\mathbb{E}(D_{s,\alpha} m_{\infty} \mid \mathcal{F}_s)^2] d<M^{\alpha}>_s \\ &=& \sum_{\alpha} \int^t_0 \mathbb{E}[ D_{s,\alpha} m_{\infty}^2] d<M^{\alpha}>_s \\
&=& \sum_{\alpha} \int^t_0 \| D_{s,\alpha} m_{\infty} \|^2_{L^2(\Omega)} d<M^{\alpha}>_s \\
&=& \sum_{\alpha} \int^t_0 \| \sum^{\infty}_n nI_{n-1}(f^{\alpha}_n( \cdot,s)) \|^2_{L^2(\Omega)} d<M^{\alpha}>_s \mbox{ (definition \ref{Lokkalevyderiv}) }\\
&=& \sum^{\infty}_n n^2(n-1)! \sum_{\alpha} \int^t_0 \| f^{\alpha}_n( \cdot,s)) \|^2_{n-1} d<M^{\alpha}>_s \\
&=& \sum^{\infty}_n n \cdot n! \|f_n\|^2_n < \infty. \end{eqnarray*}
Hence the integral representation of result \ref{general-mrt} is well defined for any stochastic process $(m_t) \in \mathcal{M}^2 \cap \mathbb{D}_{1,2}$ where $m_{\infty}$ is finite.
\section{Generalization and Extension of the MRT and Clark-Ocone formula}
From the previous chapters we have seen that, on a diverse range of filtrations generated by different stochastic processes, all martingales in $\mathcal{M}^2$ can have a representation in the form:
\[ m_t = \int^t_0 \phi(s) \cdot d M^{\alpha}_s \] for some remarkable martingale $M^\alpha$, and where $\phi(t)$ and $M^{\alpha}_t$ can be one of multi-dimensional. Additionally, for all martingales in a specific subset of $\mathcal{M}^2$, a.k.a $\mathcal{M}^2 \cap \mathbb{D}_{1,2}$ the intersection of square-integrable martingales and the Sobolev space $\mathbb{D}_{1,2}$, we have the Clark-Ocone formula specifying what the integrand $\phi(t)$ is:
\[ m_t = \int^t_0 \mathbb{E}(D_s m_{\infty} \mid \mathcal{F}_s) \cdot d M^{\alpha}_s. \] While this result is very useful and says a lot about martingale representation, it requires further investigation. First of all, we are interested in applying it to processes beyond the $\mathbb{D}_{1,2}$ space. As we will see, this Sobolev space can be too restrictive, especially when we are interested in changing elements of the stochastic base. Similarly, the MRT and Clark-Ocone formula need to remain under a change of measure on the probability space. From various applications, we know that changing measures and the use of the Girsanov theorem are of central importance to Financial Mathematics. But how do the MRT and the Clark-Ocone formula keep up with it? Furthermore, a modified version of these results happens to find an important application in the enlargement of filtrations. Recent literature has been exploring the eventuality of possessing extra information about the markets and the impact this has on trading strategies, notably in the context of insider trading. This requires the base filtration to be enlarged in order to take into account the new information available. How can we adapt the representations results we have to address this?\\ In the first section we introduce a new sobolev space $\mathbb{D}_{1,1}$, where $\mathbb{D}_{1,2} \subset \mathbb{D}_{1,1}$ \cite{Li}, and on which we can put forward various solutions to adapt the Clark-Ocone formula to the Girsanov theorem, which then becomes the Generalized Clark-Ocone representation theorem. This result is applicable to the B.M. as well as to general jump and L$\acute{e}$vy processes. In the latter section we then look at a Clark-Ocone type formula and measure-valued MRT that can be used in order to enlarge the base filtration.
\subsection{The MRT and Clark-Ocone formula beyond $\mathbb{D}_{1,2}$}
So far we have established that martingales $(m_t)$ in $\mathcal{M}^2 \cap \mathbb{D}_{1,2}$ have an explicit integral representation as explained above. However, it often agreed that the Sobolev space $\mathbb{D}_{1,2}$ is too restrictive; As we will see in the next chapter, $\mathbb{D}_{1,1}$ lends itself better to changes of measures on the Brownian filtration. In this section we only consider the continuous case.
\begin{defn}
Set \[ \mathcal{P} = \{ F(\omega) \mid F = f(W_{t_1},...,W_{t_n}), f \in \mathcal{C}^{\infty}(\mathbb{R}^{nd}), f -\mbox{ bounded} \}. \] \end{defn}
\begin{thm} $\forall F \in \mathcal{P}$, the Malliavin derivative $D_tF$ is equal to: \[ D_t F = \sum^n_{i=1} \frac{\partial f}{\partial x_i} (W_{t_1},...,W_{t_n})1_{[0,t_i]}(t). \] \label{BMpolynomDer} \end{thm}
\begin{proof} As stated in proposition 2.14 of \cite{Li} and Theorem 1.6 of Section 2 , we can see that: \begin{eqnarray*} D_{\gamma} F = \sum^n_{i=1} \frac{\partial f}{\partial x_i} (W_{t_1},...,W_{t_n}) \gamma(t_i) &=& \int^T_0 D_t F \dot \gamma dt \\ \sum^n_{i=1} \frac{\partial f}{\partial x_i} (W_{t_1},...,W_{t_n}) \gamma(t_i) &=& \sum^n_{i=1} \frac{\partial f}{\partial x_i} (W_{t_1},...,W_{t_n}) \int^{t_i}_0 \dot \gamma(s) ds \\ &=& \sum^n_{i=1} \frac{\partial f}{\partial x_i} (W_{t_1},...,W_{t_n}) \int^T_0 1_{[0,t_i]}\dot \gamma(s) ds \\ &=& \int^T_0 \sum^n_{i=1} \frac{\partial f}{\partial x_i} (W_{t_1},...,W_{t_n}) 1_{[0,t_i]}\dot \gamma(s) ds \\ \mbox{So } \int^T_0 D_t F \dot \gamma(t) dt &=& \int^T_0 \sum^n_{i=1} \frac{\partial f}{\partial x_i} (W_{t_1},...,W_{t_n}) 1_{[0,t_i]}\dot \gamma(s) ds \\ \mbox{Hence } D_t F &=& \sum^n_{i=1} \frac{\partial f}{\partial x_i} (W_{t_1},...,W_{t_n}) 1_{[0,t_i]}(t). \end{eqnarray*} \end{proof}
\begin{defn}[{\bf Sobolev Space $\mathbb{D}_{1,1}$}] The banach space $\mathbb{D}_{1,1}$ is the closure of $\mathcal{P}$ under the following $L^2([0,T])$ norm:
\[ \| F\|_{1,1} = \mathbb{E}\big( |F| + \| D_t F \|_{L^2([0,T])} \big) \] \label{D11} \end{defn}
The gradient derivative DF=($D^1F,...,D^dF$) has components as stated by Theorem \ref{BMpolynomDer}: \[ D^i_t F = \sum^n_{i=1} \frac{\partial}{\partial x^{ij}} f(W_{t_1},...,W_{t_n}) 1_{[0,t_i]}(t) \]
\begin{thm}[{\bf Clark-Ocone theorem in $\mathbb{D}_{1,1}$} \cite{Karatzas}] $\forall F \in \mathbb{D}_{1,1}$, we have the following integral representation: \[ F = \mathbb{E}(F) + \int^T_0 \mathbb{E}(D_t F \mid \mathcal{F}_t) dW_t. \] \label{ClarkOconeD11} \end{thm}
\begin{proof} for F $\in \mathbb{D}_{1,1}$, take $\{F_n\}_{n \in \mathbb{N}} \subseteq \mathcal{P}$ where $\lim_{n \rightarrow \infty} \| F_n-F\|_{1,1} =0$. $m(t) = \mathbb{E}(F \mid \mathcal{F}_t)$ and $m_n(t) = \mathbb{E}(F_n \mid \mathcal{F}_t)$ are martingales and hence, using result \ref{itorepth} from chapter 1,
\[ m(t) = \mathbb{E}(F) + \int^t_0 \phi(s)dW_s, \] \[ m_n(t) = \mathbb{E}(F_n) + \int^t_0 \phi_n(s)dW_s, \]
\noindent where $\phi, \phi_n$ are square integrable. Since $m_n \in \mathcal{P} \subseteq \mathbb{D}_{1,2}$, we know from result \ref{clarkocone} in Chapter 2 that $\phi_n(t) = \mathbb{E}(D_tF_n \mid \mathcal{F}_t)$. Also, for any $\epsilon >0$,
\begin{eqnarray}
\mathbb{P}( \max_{0 \leq t \leq T}| m_n(t) - m(t)| > \epsilon ) &\leq& \frac{1}{\epsilon} \mathbb{E}|m_n(t) - m(t)| \mbox { (Doob's martingale inequality)}\nonumber \\
&=& \frac{1}{\epsilon} \mathbb{E}|\mathbb{E}(F_n \mid \mathcal{F}_t) - \mathbb{E}(F \mid \mathcal{F}_t)| = \frac{1}{\epsilon} \mathbb{E}|\mathbb{E}(F_n - F \mid \mathcal{F}_t)| \nonumber \\
&=& \frac{1}{\epsilon} \mathbb{E}|F_n - F | \rightarrow 0 \mbox{ as } n \rightarrow \infty. \label{Pmax} \end{eqnarray}
Since $ \mathbb{E}|F_n - F | \leq \| F_n - f \|_{1,1} \rightarrow 0$. Additionally, the Burkholder-Gundy inequality \cite{Rogers} shows that for any $\lambda > 0$ and $\delta \in (0,1)$,
\begin{align*}
\mathbb{P}( \langle m_n - m\rangle_T > 4\lambda^2 , \max_{0 \leq t \leq T} |m_n(t) - m(t)| \leq \delta \lambda) &\leq \delta^2 \mathbb{P}(\langle m_n-m\rangle_T > \lambda^2) \end{align*} As \begin{align*} \langle m_n - m\rangle_T &= \langle \mathbb{E}(Fn-F) + \int^T_0 \phi_n(s) - \phi(s) d W_s \rangle \\
&= \int^T_0 |\phi_n(s) - \phi(s)|^2 ds, \end{align*}
\noindent we get:
\begin{equation}
\mathbb{P}( \int^T_0 |\phi_n(s) - \phi(s)|^2 ds > 4\lambda^2 , \max_{0 \leq t \leq T} |m_n(t) - m(t)| \leq \delta \lambda) = \delta^2 \mathbb{P}(\int^T_0 |\phi_n(s) - \phi(s)|^2 ds > \lambda^2). \nonumber \\ \end{equation} Additionally, we have: \begin{align}
\mathbb{P}(\int^T_0 |\phi_n-\phi|^2(s) ds > 4 \lambda^2) = \mathbb{P}( \int^T_0 |\phi_n(s) - \phi(s)|^2 ds > 4\lambda^2 &, \max_{0 \leq t \leq T} |m_n(t) - m(t)| \leq \delta \lambda) \nonumber \\
+ \mathbb{P}( \int^T_0 |\phi_n(s) - \phi(s)|^2 ds > 4\lambda^2 &, \max_{0 \leq t \leq T} |m_n(t) - m(t)| > \delta \lambda) \nonumber \\
\leq \delta^2 \mathbb{P}(\int^T_0 |\phi_n(s) - \phi(s)|^2 ds > \lambda^2&) \nonumber \\
+ \mathbb{P}( \int^T_0 |\phi_n(s) - \phi(s)|^2 ds > 4\lambda^2 &, \max_{0 \leq t \leq T} |m_n(t) - m(t)| > \delta \lambda) \nonumber \\
\leq \delta^2 + \mathbb{P}( \max_{0 \leq t \leq T} |m_n(t) - m(t)| > &\delta \lambda). \label{eqnphi} \end{align}
Results \ref{Pmax} and \ref{eqnphi} imply that
\[ \int^T_0 |\phi_n-\phi|^2(s) ds \rightarrow 0 \mbox{ in $\mathbb{P}$ as } n \rightarrow \infty. \]
At the same time, we have: \begin{eqnarray}
\mathbb{E}( \int^T_0 | \phi_n(s) - \mathbb{E}( D_s F \mid \mathcal{F}_s) | ds) &=& \mathbb{E}( \int^T_0 | \mathbb{E}( D_s F_n \mid \mathcal{F}_s) - \mathbb{E}( D_s F \mid \mathcal{F}_s) | ds) \nonumber \\
&=& \mathbb{E}( \int^T_0 | \mathbb{E}( D_s F_n - D_s F \mid \mathcal{F}_s) | ds) \nonumber \\
&\leq& \mathbb{E}( \int^T_0 | D_s( F_n - F) | ds) \nonumber \\
&\leq& \sqrt{T} \mathbb{E}[ (\int^T_0 | D_s( F_n - F) |^2 ds)^{1/2}] \mbox{ (Cauchy-Schwartz)} \nonumber \\
&\equiv& \sqrt{T} \mathbb{E}[ (\sum^d_{i=1} \| D^i (F_n-F) \|^2)^{1/2}] \nonumber \\
&\leq& \sqrt{T} \| F_n - F\|_{1,1} \rightarrow 0. \label{dtconv} \end{eqnarray}
\ref{eqnphi} and \ref{dtconv} together show that $\mathbb{E}(D_tF \mid{F}_t) = \phi(t)$ $dt \times d\mathbb{P}$- a.s. .
\end{proof}
Hence, Theorem \ref{ClarkOconeD11} shows that a wider class of martingales can have an explicit integral representation: $\forall (m_t) \in \mathcal{M}^2 \cap \mathbb{D}_{1,1}$, we have: \[ m_t = \int^t_0 \mathbb{E}(D_t m_{\infty} \mid \mathcal{F}_s) dW_s. \]
\subsection{The Generalized/Girsanov Clark-ocone formula}
\subsubsection{Generalized Clark representation formula for continuous martingales}
Here, we start with the setting developed in Ocone and Karatzas \cite{Ocone}: we are on a probablility space $(\Omega,\mathcal{F},\mathbb{P})$ generated by a $\mathbb{R}^d$-B.M. Denote
\[ \mathcal{F}_t = \sigma (W_s \mid 0 \leq s \leq t)\] the $\mathbb{P}$-augmentation of the original filtration.\\
Set the following Radon-Nikodym derivative:
\[ \frac{d\mathbb{\tilde{P}}}{d\mathbb{P}} = Z_t = e^{-\int^t_0 \theta_s dW_s - 1/2 \int^t_0 \theta^2_s ds }, \] where $\theta(t)$ is an $\mathbb{R}^d$-valued and $\mathcal{F}_t$-measurable process. We know from the conventional Girsanov theorem that \begin{equation} \tilde{W}_t = W_t + \int^t_0 \theta_s ds \label{driftBM} \end{equation} is a $\tilde{\mathbb{P}}$-B.M.
Here, we borrow the concepts of Malliavin Calculus developed in chapter 2 for the Malliavin derivative and the $\mathbb{D}_{1,2}$ norm: \[ \mathcal{P} = \big\{ F \mid F(\omega) = f(W_{t_1},...,W_{t_n}), f: \mathbb{R}^{n\times d} \rightarrow \mathbb{R} \}. \]
We then set the gradient DF$(\omega)$ = ($D^1F,...,D^dF$) using theorem \ref{BMpolynomDer} \[ D^iF = (D_tF)^i = \sum^n_{j=1} \frac{\partial f}{\partial x_{ij}} (W_{t_1},...,W_{t_n})1_{[0,t_j]}(t) \] for any i $\leq d$.
The closure of $\mathcal{P}$ under the following norm
\[ \| |F| \|_{1,1} =\mathbb{E}(|F| + \| D_tF\|_{L^2([0,T])}) \]
then makes a Banach space $\mathbb{D}_{1,1}$ very similar but larger than the conventional Sobolev space $\mathbb{D}_{1,2}$.\\ We know from Theorem \ref{ClarkOconeD11} that $\forall F \in \mathbb{D}_{1,1}$, we have: \[ F = \mathbb{E}(F) + \int^T_0 \mathbb{E}(D_t F \mid \mathcal{F}_t) dW_t. \]
Here we focus on $\mathbb{D}_{1,1}$ in order to avoid adding extra constraints to the theorem that will follow. Indeed, in this theorem we will want to give an integral representation to $\mathbb{\tilde{E}}(F \mid \mathcal{F}_t)$ using the Bayes formula, and apply the Clark-Ocone formula to F$Z_T$. In $\mathbb{D}_{1,1}$, it is the case that $F\in L^2(\mathbb{\tilde{P}}) \Longrightarrow FZ_T \in L^2(\mathbb{P})$, but not in $\mathbb{D}_{1,2}$ without restrictive moment constrains on F and DF.\cite{Ocone}
\begin{thm}[{\bf Generalized Clark representation formula} \cite{Ocone}] $\forall F \in \mathbb{D}_{1,1}$ with bounded $\theta$ such as:
\[ \mathbb{E}( |F| Z_T) < \infty, \]
\[ \mathbb{E}( \|DF\| Z_T) < \infty, \]
\[ \mathbb{E}( |F| Z_T \| \int^T_0D\theta(s)dW_s+\int^T_0D\theta(s)\cdot \theta(s)ds\| ) < \infty, \] then $FZ_T \in \mathbb{D}_{1,1}$ and the following holds: \[ F = \tilde{\mathbb{E}}(F) + \int^T_0 \Big[ \tilde{\mathbb{E}}(D_t F \mid \mathcal{F}_t)- \tilde{\mathbb{E}}(F \int^T_t D_t \theta(u) d \tilde{W}_u \mid \mathcal{F}_t)\Big] d\tilde{W}_t. \] \label{generalizedClark} \end{thm}
\begin{proof} For the purpose of this proof, note $Z_t = e^G$, where $G = -\int^t_0 \theta_s dW_s - 1/2 \int^t_0 \theta^2_s ds$ We can see from proposition 2.3 of \cite{Ocone}:
\begin{eqnarray*}
\| |\int^T_0 \theta_s dW_s| \|_{1,1} < \infty, \\
\| |\int^T_0 |\theta^n_s|^2 ds| \|_{1,1} < \infty. \end{eqnarray*}
Hence $G\in \mathbb{D}_{1,1}$. Additionally, $D \int^T_0 |\theta|^2ds = 2 \int^T_0 D \theta \cdot \theta ds $. So \begin{eqnarray*}
\| |F Z_T| \|_{1,1} &=& \mathbb{E}( |F Z_T| + |D(FZ_T)|) \\
&=& \mathbb{E}( |F Z_T| ) + \mathbb{E}(Z_T \|DF\|+ |F| \|DZ_T\|) \\
&=& \mathbb{E}( |F Z_T| ) + \mathbb{E}(Z_T \|DF\|)+\mathbb{E}( |F| Z_T \| \int^T_0 D\theta dW_s + \int^T_0 D \theta \cdot \theta ds\|) \\ &<& \infty \mbox{, as per conditions given in the theorem.} \end{eqnarray*} Hence $FZ_T \in \mathbb{D}_{1,1}$.\\ From above we know that:
\begin{eqnarray*} D_t(FZ_T) &=& Z_T D_tF - FZ_TD_t (\int^T_0\theta dW_s + \int^T_0 \theta^2 ds) \\ &=& Z_T \big( D_tF - F(\int^T_t D_s\theta dW_s + \theta_t + D_t(\int^T_0 \theta^2 ds))\big) \\ &=& Z_T \big( D_tF - F(\int^T_t D_s\theta dW_s + \theta_t +\int^T_t D_s\theta \cdot \theta ds)\big) \\ &=& Z_T \big( D_tF - F( \theta_t +\int^T_t D_s\theta ( dW_s + \theta ds))\big) \\ &=& Z_T \big( D_tF - F( \theta_t +\int^T_t D_s\theta d\tilde{W}_s)\big). \end{eqnarray*}
We also know that $ \tilde{\mathbb{E}}(F \mid \mathcal{F}_t) = \frac{\mathbb{E}(FZ_T \mid \mathcal{F}_t)}{\mathbb{E}(Z_T \mid \mathcal{F}_t)}$, as a basic property of conditional expectations under change of measures. Since $Z_T$ is an $\mathbb{P}$-mg, $\mathbb{E}(Z_T \mid \mathcal{F}_t)= Z_t $. Here, note $\Lambda_t = 1/Z_t$. Then:
\begin{eqnarray*} \tilde{\mathbb{E}}(F \mid \mathcal{F}_t) &=& \Lambda_t \mathbb{E}(FZ_T \mid \mathcal{F}_t) \\ &=& \Lambda_t \big( \mathbb{E}(FZ_T ) + \int^t_0 \mathbb{E}( D_t FZ_T\mid \mathcal{F}_s) dW_s \big) \mbox{ (result of chapter 2).} \\ \end{eqnarray*}
We recall that $d \Lambda_t = \Lambda_t \theta_t d \tilde{W}_t$. This gives:
\begin{eqnarray} d \tilde{\mathbb{E}}(F \mid \mathcal{F}_t) &=& d [ \Lambda_t \big( \mathbb{E}(FZ_T ) + \int^t_0 \mathbb{E}( D_s FZ_T\mid \mathcal{F}_s) dW_s \big)] \nonumber \\ &=& d \Lambda_t \big( \mathbb{E}(FZ_T ) + \int^t_0 \mathbb{E}( D_s FZ_T\mid \mathcal{F}_s) dW_s \big) + \Lambda_t \big( d \mathbb{E}(FZ_T ) + d \int^t_0 \mathbb{E}( D_s FZ_T\mid \mathcal{F}_s) dW_s \big) \nonumber \\ &&+ \langle d \Lambda_t , d \mathbb{E}(FZ_T ) + d \int^t_0 \mathbb{E}( D_s FZ_T\mid \mathcal{F}_s) dW_s \rangle \nonumber \\ &=& \Lambda_t \theta_t \big( \mathbb{E}(FZ_T ) + \int^t_0 \mathbb{E}( D_s FZ_T\mid \mathcal{F}_s) dW_s \big) d\tilde{W}_t + \Lambda_t \mathbb{E}( D_t FZ_T\mid \mathcal{F}_t) dW_t \nonumber \\ &&+ \mathbb{E}( D_t FZ_T\mid \mathcal{F}_t) \Lambda_t \theta_t \langle d \tilde{W}_t , dW_s \rangle \nonumber \\ &=& \theta_t \tilde{\mathbb{E}}(F \mid \mathcal{F}_t) d\tilde{W}_t + \Lambda_t \mathbb{E}( D_t FZ_T\mid \mathcal{F}_t) dW_t + \mathbb{E}( D_t FZ_T\mid \mathcal{F}_t) \Lambda_t \theta_t dt \nonumber \\ &=& \big( \Lambda_t \mathbb{E}( D_t FZ_T\mid \mathcal{F}_t) + \theta_t \tilde{\mathbb{E}}(F \mid \mathcal{F}_t) \big) d\tilde{W}_t. \label{intermediate2} \end{eqnarray}
We also have:
\begin{eqnarray*} \Lambda_t \mathbb{E}( D_t FZ_T\mid \mathcal{F}_t) &=& \Lambda_t \mathbb{E}( Z_T \big( D_tF - F( \theta_t +\int^T_t D_t\theta d\tilde{W}_s)\mid \mathcal{F}_t) \\ &=& \Lambda_t \mathbb{E}( Z_TD_tF\mid \mathcal{F}_t) - \theta_t \Lambda_t \mathbb{E}( Z_TF\mid \mathcal{F}_t) - \Lambda_t \mathbb{E}( Z_TF\int^T_t D_t\theta d\tilde{W}_s\mid \mathcal{F}_t) \\ &=& \tilde{\mathbb{E}}( D_tF\mid \mathcal{F}_t)-\theta_t \tilde{\mathbb{E}}( F\mid \mathcal{F}_t)-\tilde{\mathbb{E}}( F\int^T_t D_t\theta d\tilde{W}_s\mid \mathcal{F}_t). \end{eqnarray*}
Hence result \ref{intermediate2} turns into: \begin{equation} d \tilde{\mathbb{E}}(F \mid \mathcal{F}_t) = \big[ \tilde{\mathbb{E}}( D_tF\mid \mathcal{F}_t)- \tilde{\mathbb{E}}( F\int^T_t D_t\theta d\tilde{W}_s\mid \mathcal{F}_t) \big] d\tilde{W}_t. \label{intermediate3} \end{equation}
Result \ref{intermediate3} clearly shows that $\tilde{\mathbb{E}}(F \mid \mathcal{F}_t)$ is a $\tilde{\mathbb{P}}$-mg. Therefore: \[ F - \tilde{\mathbb{E}}(F) = \int^T_0 \big[ \tilde{\mathbb{E}}( D_tF\mid \mathcal{F}_s)- \tilde{\mathbb{E}}( F\int^T_s D_t\theta d\tilde{W}_u\mid \mathcal{F}_s) \big] d\tilde{W}_s. \]
\end{proof}
Hence, for all $\mathbb{P}$-martingales $(m_t) \in \mathcal{M}^2 \cap \mathbb{D}_{1,1}$, we can maintain an explicit representation even under various changes of measures: provided $m_{\infty} < \infty$ where $\mathbb{E}(m_{\infty} \mid \mathcal{F}_t) = m_t$, we have: \begin{eqnarray*} m_t &=& \mathbb{E}(m_{\infty} \mid \mathcal{F}_t) \\ &=& \mathbb{E}( \tilde{\mathbb{E}}(m_{\infty}) + \int^T_0 \big[ \tilde{\mathbb{E}}( D_t m_{\infty}\mid \mathcal{F}_s)- \tilde{\mathbb{E}}( m_{\infty} \int^T_s D_t\theta d\tilde{W}_u\mid \mathcal{F}_s) \big] d\tilde{W}_s \mid \mathcal{F}_t) \\ &=& \tilde{\mathbb{E}}(m_{\infty}) + \mathbb{E}( \int^T_0 \big[ \tilde{\mathbb{E}}( D_t m_{\infty}\mid \mathcal{F}_s)- \tilde{\mathbb{E}}( m_{\infty} \int^T_s D_t\theta d\tilde{W}_u\mid \mathcal{F}_s) \big] d\tilde{W}_s \mid \mathcal{F}_t)\\ &=& \tilde{\mathbb{E}}(m_{\infty})+ \int^t_0 \big[ \tilde{\mathbb{E}}( D_t m_{\infty}\mid \mathcal{F}_s)- \tilde{\mathbb{E}}( m_{\infty} \int^T_s D_t\theta d\tilde{W}_u\mid \mathcal{F}_s) \big] dW_s \\ &&+ \int^T_0 \big[\tilde{\mathbb{E}}( D_t m_{\infty}\mid \mathcal{F}_s)- \tilde{\mathbb{E}}( m_{\infty} \int^T_s D_t\theta d\tilde{W}_u\mid \mathcal{F}_s) \big] \theta_s ds \\ &=& \tilde{\mathbb{E}}(m_{\infty})+ \int^t_0 \big[ \tilde{\mathbb{E}}( D_t m_{\infty}\mid \mathcal{F}_s)- \tilde{\mathbb{E}}( m_{\infty} \int^T_s D_t\theta d\tilde{W}_u\mid \mathcal{F}_s) \big] d\tilde{W}_s \\ &&+\int^T_t \big[\tilde{\mathbb{E}}( D_t m_{\infty}\mid \mathcal{F}_s)- \tilde{\mathbb{E}}( m_{\infty} \int^T_s D_t\theta d\tilde{W}_u\mid \mathcal{F}_s) \big] \theta_s ds. \end{eqnarray*}
Note: We cannot directly apply the classical Clark-Ocone formula developed in Chapter 2 to obtain an integral representation with respect to $\tilde{W}_t$, as $m_t \in \mathcal{M}^2 \cap \mathbb{D}_{1,1}$ is not necessarily $\tilde{\mathcal{F}}_t$-adapted. $\tilde{\mathcal{F}}_t$ is generated by $\tilde{W}_t$ for any t in [0,T], and very often it is the case that $\tilde{\mathcal{F}}_T \subset \mathcal{F}_T$. \cite{Oksendal2}
\subsubsection{Generalized Clark representation formula for L$\acute{e}$vy and pure jump processes}
The result investigated in section 1.1 can be extended beyond continuous martingales and the B.M. driving them. Indeed, It is possible to prove that it is also applicable to general L$\acute{e}$vy processes of the type explored in Chapter 1 and 2.\\ We work on the filtration generated by a L$\acute{e}$vy process $L_t$ such as, $\forall (m_t) \in \mathcal{M}^2 \cap \mathbb{D}_{1,2}$, we have: \[ m_t = \int^t_0 \mathbb{E}(D_s m_{\infty} \mid \mathcal{F}_s) dW_s + \int^t_0 \int_{\mathbb{R}_0} \mathbb{E}(D_{s,z} m_{\infty} \mid \mathcal{F}_s) \tilde{N}(ds,dz).\]
Again, we perform a change of measure as in Nunno et al. \cite{Oksendal3}: \begin{eqnarray*} \frac{d \mathbb{Q}}{d \mathbb{P}} = Z_t &=&\exp \Big\{ - \int^t_0 u_s dW_s - \int^t_0 u^2_sds \\ && +\int^T_0 \int_{\mathbb{R}_0} \log(1-\theta(s,x)) + \theta(s,x) \upsilon(dx) ds \\ && +\int^T_0 \int_{\mathbb{R}_0} \log(1-\theta(s,x)) \tilde{N}(ds,dx) \Big\} \end{eqnarray*} where $\theta(s,x) \leq 1$ for s$\in [0,T]$ , x $\in \mathbb{R}_0$ and $u_s$ is a $\mathbb{F}$-predictable process.
\begin{thm}[{\bf Generalized Clark-Ocone theorem for L$\acute{e}$vy processes} \cite{Oksendal3}]
For all F $\in L^2(\mathbb{P}) \cap L^2(\mathbb{Q})$ where F is $\mathcal{F}_T$-measurable and where: \begin{itemize} \item $\theta \in L^2(\mathbb{P} \times \lambda \times \upsilon)$, \item $D_{t,x} \theta$ is Skorohod integrable, \end{itemize} the below representation holds
\begin{eqnarray*}
F &=& \mathbb{E}_{\mathbb{Q}}(F) + \int^T_0 \mathbb{E}_{\mathbb{Q}}(D_t F - F \int^T_t D_t u_s dW^Q_s \mid \mathcal{F}_t) d W^Q_t \\ && +\int^T_0 \int_{\mathbb{R}_0}\mathbb{E}_{\mathbb{Q}}(F (\tilde{H}-1)+\tilde{H}D_{t,x}F \mid \mathcal{F}_t) d \tilde{N}^Q(dt,dx). \end{eqnarray*}
Here: \begin{eqnarray*} \tilde{H} &=& \exp \Big\{ \int^t_0 \int_{\mathbb{R}_0} [ D_{t,x} \theta(s,z) + \log(1-\frac{D_{t,x} \theta(s,z)}{1-\theta(s,z)})(1-\theta(s,z))] \upsilon(dz)ds \\ && + \log(1-\frac{D_{t,x} \theta(s,z)}{1-\theta(s,z)})\tilde{N}_Q(ds,dz) \Big\}, \\ \tilde{N}_Q(ds,dz) &=& \theta(t,x)\upsilon(dx)dt + \tilde{N}(ds,dz), \\ d W^Q_t &=& u_tdt + dW_t. \\ \end{eqnarray*} \label{generalizedClarkLevy} \end{thm}
As in section 1.1 of this chapter, we can apply \ref{generalizedClarkLevy} to martingales: If $m_{\infty} < \infty$ and $m_{\infty} \in L^2(\mathbb{P}) \cap L^2(\mathbb{Q})$, then:
\begin{eqnarray*} m_{\infty} &=& \mathbb{E}_{\mathbb{Q}}(m_{\infty}) + \int^T_0 \mathbb{E}_{\mathbb{Q}}(D_t m_{\infty} - m_{\infty} \int^T_t D_t u_s dW^Q_s \mid \mathcal{F}_t) d W^Q_t \\ && +\int^T_0 \int_{\mathbb{R}_0}\mathbb{E}_{\mathbb{Q}}(m_{\infty} (\tilde{H}-1)+\tilde{H}D_{t,x}m_{\infty} \mid \mathcal{F}_t) d \tilde{N}^Q(dt,dx), \\ m_t &=& \mathbb{E}_{\mathbb{P}}( m_{\infty} \mid \mathcal{F}_t) \\ &=& \mathbb{E}_{\mathbb{P}}( \mathbb{E}_{\mathbb{Q}}(m_{\infty}) + \int^T_0 \mathbb{E}_{\mathbb{Q}}(D_t m_{\infty} - m_{\infty} \int^T_t D_t u_s dW^Q_s \mid \mathcal{F}_t) d W^Q_t \\ && +\int^T_0 \int_{\mathbb{R}_0}\mathbb{E}_{\mathbb{Q}}(m_{\infty} (\tilde{H}-1)+\tilde{H}D_{t,x}m_{\infty} \mid \mathcal{F}_t) d \tilde{N}^Q(dt,dx) \mid \mathcal{F}_t) \\ &=& \mathbb{E}_{\mathbb{Q}}(m_{\infty}) + \int^t_0 \mathbb{E}_{\mathbb{Q}}(D_t m_{\infty} - m_{\infty} \int^T_s D_t u_a dW^Q_a \mid \mathcal{F}_s) d W^Q_s \\ &&+ \int^T_t \mathbb{E}_{\mathbb{Q}}(D_t m_{\infty} - m_{\infty} \int^T_s D_t u_a dW^Q_a \mid \mathcal{F}_s) u_s d s\\ &&+ \int^t_0 \int_{\mathbb{R}_0}\mathbb{E}_{\mathbb{Q}}(m_{\infty} (\tilde{H}-1)+\tilde{H}D_{t,x}m_{\infty} \mid \mathcal{F}_s) d \tilde{N}^Q(ds,dx) \\ &&+ \int^T_t \int_{\mathbb{R}_0}\mathbb{E}_{\mathbb{Q}}(m_{\infty} (\tilde{H}-1)+\tilde{H}D_{t,x}m_{\infty} \mid \mathcal{F}_s) d \theta(s,x) \upsilon(dx). \end{eqnarray*}
\subsection{Enlargement of filtration: Poisson filtrations and insider trading (Wright et al. \cite{Wright})}
So far we have reviewed various ways to adapt the MRT and the Clark-Ocone formula under a change of probability measure. But the requirements of Financial Mathematics also lead towards the eventuality of changing the base filtration $\mathcal{F}$ which we work on. Doing so impacts on the representation of the stochastic processes driving the probability space, and hence the MRT. \\ We start with a conventional probability space $(\Omega, \mathcal{F}, \mathbb{P})$ where $\{\mathcal{F}_t\}_{t \leq 1}$ represents the regular information flow generated by a B.M. $\in \mathbb{R}^d$. Here, the time frame is t$ \in [0,1]$.Then comes a $\mathcal{F}_1$-measurable r.v. L that carries extra information. The agent who possesses that extra info L is otherwise known as an "insider" \cite{Wright} and their knowledge is represented by the enlarged filtration: \[ \mathcal{G}_t = \mathcal{F}_t \vee \sigma(L) \mbox{, } \forall t \in [0,1]. \]
The $\mathcal{G}_t$-B.M. $\tilde{W}_t$ can be represented as such: \[ W_t = \tilde{W}_t + \int^t_0 \mu^L_sds. \]
There is such a drift $\mu^L_s$ when what is known as "Jacod's Condition" \cite{Jacod} is satisfied: \[ \mbox{ The regular conditional distributions of L given } \mathcal{F}_t \] \[ \mbox{ are absolutely continuous with respect to the law of L } \forall t \in [0,1) \] Then we can re-write the MRT as such:
\[ \forall m \in \mathcal{M}^2 \cap \mathbb{D}_{1,1}, m_t = \int^t_0 \mathbb{E}(D_s m_{\infty} \mid \mathcal{F}_s) d\tilde{W}_t + \mu^L_s ds \mbox{ , } \forall t \in [0,1]. \]
But how do we find an explicit form for $ \mu^L_s$? As developed in Wright et al.\cite{Wright}, by identifying the integral representation of $\mathbb{P}( L \in dx \mid \mathcal{F}_t)$ explicitly using the Clark-Ocone formula, we can get an expression for the drift $\mu^L_s$. However, it is clear that $\mathbb{P}( L \in dx \mid \mathcal{F}_t)$ is not a simple random variable but a measure-valued random variable. Therefore we need to re-create a measure-valued Poisson Malliavin calculus with its own Clark-Ocone-type formula in order to make sense of an integral representation of $\mathbb{P}( L \in dx \mid \mathcal{F}_t)$. \cite{Wright}
\subsubsection{Use of Poisson-malliavin calculus and Clark-Ocone formula}
We work on a Poisson space $(\mathcal{B},\mathcal{F},\mathcal{P})$ defined as follow:
\begin{defn}[{\bf Poisson space}] A poisson space is a triple $(\mathcal{B},\mathcal{F},\mathcal{P})$ where: \begin{itemize} \item $\mathcal{B}$ is a sequence space, \item $\mathcal{P}$ is a probability measure under which $\tau_k : \mathcal{B} \rightarrow \mathbb{R}$ forms a sequence of i.i.d exponentially distributed, \item $\mathcal{F}$ is the $\sigma$-field generated by $\mathcal{B}$ \end{itemize} \end{defn}
The n-th jump time of the Poisson process, $T_n$ , is then derived by: \[ T_n = \sum^{n-1}_{k=0} \tau_k, \]
and the Poisson process itself, $N_t$, by: \[ N_t(\omega) = \sum^{\infty}_{k=1} 1_{[T_k(\omega),\infty)}(t)\] for any t in $\mathbb{R}$. $(\mathcal{F}_t)_{t \geq 0}$ is the filtration generated by $(N_t)_{t \geq 0}$.
Define the set $\mathcal{S}$: \[ \mathcal{S} = \{ F = f(T_1,...,T_n) \mid f \in C^{\infty}(\mathbb{R}^n) \forall n \geq 1\}, \]
and the closable linear operator $D^{\mathbb{R}}: L^2(\mathcal{B}) \rightarrow L^2(\mathcal{B} \times \mathbb{R}_+)$ for all F$\in \mathcal{S}$: \[ D^{\mathbb{R}}_t F = - \sum^n_{k=1} 1_{[0,T_k]}(t) \partial_kf(T_1,...,T_n) \]
whenever t is in $\mathbb{R}_+$, the set of positive real numbers. We then extend $\mathcal{S}$ in Dom D $\subseteq L^2(\mathbb{B})$ with respect to:
\[ \|F\|_{L^2(\mathcal{B})} + \|DF\|_{L^2(\mathcal{B} \times \mathbb{R}_+)} \mbox{ , } F \in \mathcal{S}. \]
We now introduce an important isomorphism $\Phi$ as: \[ \Phi : \mathbb{M} \rightarrow \mathbb{R}^{\mathbb{N}}, \] \[ \Phi(\mu) = ( \langle \mu, f_i \rangle)_{i \in \mathbb{N}} = ( \int_{\mathbb{R}} f_i d\mu)_{i \in \mathbb{N}}. \]
Here, $\mathbb{M}$ denotes a space of measures: \[ \mathbb{M} = \{ \mu \mid \mu: \mbox{ signed measure on } (\mathbb{R},\mathcal{B}) \}. \]
We define $\mathcal{S}(\mathbb{M})$ and $ D^{\mathbb{M}}: \mathcal{S}(\mathbb{M}) \rightarrow L^2(\mathcal{B} \times \mathbb{R}_+, \mathbb{M})$ in a very similar way to above: \[ \mathcal{S}(\mathbb{M}) = \{ F = g(T_1,...,T_n,x)dx \mid g \in C^{\infty}(\mathbb{R}^{n+1}) \forall n \geq 1\}, \] \[ D^{\mathbb{M}}_t F = - \sum^n_{k=1} 1_{[0,T_k]}(t) \partial_kg(T_1,...,T_n,x)dx. \]
As we have done in chapter 2, we can introduce a norm on $\mathcal{S}(\mathbb{M})$:
\[ \|F\|^{\mathbb{M}}_{1,2} = \mathbb{E}(|F|^2)^{1/2} + \mathbb{E}(\| |D^{\mathbb{M}}F| \|^2_2)^{1/2}, \]
and set $\mathbb{D}_{1,2}(\mathbb{M})$ as the closure of $\mathcal{S}(\mathbb{M})$ with respect to $\|\cdot\|^{\mathbb{M}}_{1,2}$.
\begin{thm}[Proposition 1 \cite{Wright}] $ \forall F \in \mathbb{D}_{1,2}(\mathbb{M})$ and $f \in C_b(\mathbb{R})$, we have: \begin{itemize} \item $\langle F, f\rangle \in \mathbb{D}_{1,2}(\mathbb{M})$, and \item $\langle D^{\mathbb{M}}_t F,f\rangle = D^{\mathbb{R}}_t \langle F,f\rangle$. \end{itemize} Additionally - Proposition 2 \cite{Wright} - for $F \in \mathbb{D}_{1,2}(\mathbb{M})$, we have: \[ D^{\mathbb{M}} F = \Phi^{-1}(( D^{\mathbb{R}} \langle F,f_i \rangle)_{i \in \mathbb{N}}). \] \label{prop1} \end{thm}
\begin{thm}[Proposition 3 \cite{Wright}] For $F_t$ adapted such as:
\[ \sup_{\| f \| \leq 1, f \in C_b(\mathbb{R})} \mathbb{E}\big[ \int^{\infty}_0 \langle F_t,f\rangle^2 dt \big] < \infty, \] we have: \[ \langle \int^{.}_0 F_t d\tilde{N}_t,f \rangle = \int^{.}_0 \langle F_t, f \rangle d\tilde{N}_t. \] \label{prop3} \end{thm}
Additionally to the results established above, we need to introduce a conditional expectation formula. Whenever F $\in \mathbb{M}$ is $\mathcal{F}$-measurable, $\langle F, f_i \rangle$ is also $\mathcal{F}$-measurable for $i \in \mathbb{N}$ and any f$ \in C_b(\mathbb{R})$. Denote $\mathcal{G} \subseteq \mathcal{F}$. Then $\forall i$, $\mathbb{E}[ \langle F, f_i \rangle \mid \mathcal{G}]$ is well-defined and we set:
\[ \mathbb{E}(F \mid \mathcal{G}) = \Phi^{-1}(\mathbb{E}[ \langle F, f_i \rangle \mid \mathcal{G}]_{i \in \mathbb{N}}). \] So \[ \langle \mathbb{E}(F \mid \mathcal{G}) , f_i \rangle = \mathbb{E}( \langle F, f_i \rangle \mid \mathcal{G}) \mbox{ } \forall i.\]
Provided $ \|F\|_1 = \sup_{\| f \| \leq 1, f \in C_b(\mathbb{R})}\mathbb{E}\big[ |\langle F,f\rangle| \big] < \infty $, we have that $ | \mathbb{E} ( | \mathbb{E}( \langle F, f_i \rangle \mid \mathcal{G}) - \mathbb{E}( \langle F, f \rangle \mid \mathcal{G}) | ) | \leq \| f - f_i \| \| F \|_1 \rightarrow 0$ as $i \rightarrow \infty$. Therefore we can make the following assertion:
\begin{equation} \mathbb{E}[ \langle F, f \rangle \mid \mathcal{G}) ] = \langle \mathbb{E}(F \mid \mathcal{G}) , f \rangle \mbox{ , } \forall f \in C_b(\mathbb{R}). \label{wrightcondexp} \end{equation}
At this stage, with the use of the results, definitions and theorems introduced so far in this section, we can prove an alternative Clark-Ocone formula for signed measures $F \in \mathbb{D}_{1,2}(\mathbb{M})$ that will then prove to be very useful in defining a representation for $ \mu^L_t$. \\
\begin{thm}[{\bf Clark-Ocone type formula for $F \in \mathbb{D}_{1,2}(\mathbb{M})$ } \cite{Wright}]
For $F \in \mathbb{D}_{1,2}(\mathbb{M})$ , satisfying the boundedness condition set in Theorem \ref{prop3} and:
\[ \|F\|_1 = \sup_{\| f \| \leq 1, f \in C_b(\mathbb{R})}\mathbb{E}\big[ |\langle F,f\rangle| \big] < \infty, \] F has a representation given by: \[ F = \mathbb{E}(F) + \int^1_0 \mathbb{E}( D^{\mathbb{M}}_t F \mid \mathcal{F}_t) d \tilde{N}_t. \] \label{wrighttheor1} \end{thm}
\begin{proof} To prove Theorem \ref{wrighttheor1}, we start with the result of Proposition 2 of \cite{Mensi}: for $F \in Dom D^{\mathbb{R}}$, \[ F = \mathbb{E}(F) + \int^1_0 \mathbb{E}( D^{\mathbb{R}}_t F \mid \mathcal{F}_t) d \tilde{N}_t. \] From Theorem \ref{prop1}, we know that F $\in \mathbb{D}_{1,2}$ implies $\langle F,f\rangle \in \mathbb{D}_{1,2}$. Therefore,
\[ \langle F, f_i \rangle = \mathbb{E}(\langle F, f_i \rangle) + \int^1_0 \mathbb{E}(D^{\mathbb{R}}_t \langle F, f_i \rangle \mid \mathcal{F}_t) d \tilde{N}_t. \]
On the side, we note that:
\begin{eqnarray} \Phi^{-1}(( \mathbb{E}(\langle F, f_i \rangle))_{i \in \mathbb{N}}) &=& \Phi^{-1}(( \mathbb{E}(\langle F, f_i \rangle \mid \mathcal{F}_0))_{i \in \mathbb{N}}) \nonumber \\ &=& \Phi^{-1}(( \langle \mathbb{E}( F \mid \mathcal{F}_0), f_i \rangle)_{i \in \mathbb{N}}) \mbox{ , from result \ref{wrightcondexp}} \nonumber \\ &=& \mathbb{E}( F \mid \mathcal{F}_0) \nonumber \\ &=& \mathbb{E}( F), \label{eqnEFwright} \end{eqnarray}
and that:
\begin{eqnarray} \mathbb{E}(D^{\mathbb{R}}_t \langle F, f_i \rangle \mid \mathcal{F}_t) &=& \mathbb{E}( \langle D^{\mathbb{M}}_t F, f_i \rangle \mid \mathcal{F}_t) \mbox{ ,from Theorem \ref{prop1}} \nonumber \\ &=& \langle \mathbb{E}( D^{\mathbb{M}}_t F \mid \mathcal{F}_t) , f_i \rangle. \label{eqneEDtwright} \end{eqnarray}
Using results \ref{eqnEFwright} and \ref{eqneEDtwright}, we thus have:
\begin{eqnarray} \langle F, f_i \rangle &=& \langle \mathbb{E}( F), f_i \rangle + \int^1_0 \langle \mathbb{E}( D^{\mathbb{M}}_t F \mid \mathcal{F}_t) , f_i \rangle d \tilde{N}_t \nonumber \\ &=& \langle \mathbb{E}( F), f_i \rangle + \langle \int^1_0 \mathbb{E}( D^{\mathbb{M}}_t F \mid \mathcal{F}_t) d \tilde{N}_t , f_i \rangle \mbox{ ,from Theorem \ref{prop3}} \nonumber \\ &=& \langle \mathbb{E}( F) + \int^1_0 \mathbb{E}( D^{\mathbb{M}}_t F \mid \mathcal{F}_t) d \tilde{N}_t , f_i \rangle, \nonumber \end{eqnarray}
and
\begin{eqnarray} F = \Phi^{-1}( (\langle F, f_i \rangle )_{i \in \mathbb{N}}) &=& \Phi^{-1}( (\langle \mathbb{E}( F) + \int^1_0 \mathbb{E}( D^{\mathbb{M}}_t F \mid \mathcal{F}_t) d \tilde{N}_t , f_i \rangle)_{i \in \mathbb{N}}) \nonumber \\ &=& \mathbb{E}( F) + \int^1_0 \mathbb{E}( D^{\mathbb{M}}_t F \mid \mathcal{F}_t) d \tilde{N}_t. \nonumber \end{eqnarray}
\end{proof}
\subsubsection{Information drift $\mu^L_t$}
We begin by re-writing the conditional expectations of L as: \[ P_t( \cdot , dx) = \mathbb{P}( L \in dx \mid \mathcal{F}_t). \]
Here, we aim to apply Theorem \ref{wrighttheor1} to $ P_t( \cdot , dx)$ to obtain sufficient conditions for the existence of $\mu^L_t$ and derive a formula for it. If $P_t( \cdot , dx)$ satisfies the conditions of Theorem \ref{wrighttheor1}, then:
\begin{align*} P_t( \cdot , dx) &= \mathbb{E}( P_t( \cdot , dx)) + \int^1_0 \mathbb{E}( D^{\mathbb{M}}_sP_t( \cdot , dx) \mid \mathcal{F}_s) d\tilde{N}_s \\ \mbox{As: } \mathbb{E}( P_t( \cdot , dx)) = \mathbb{E}( \mathbb{E}(1_{\{L \in dx\}} \mid \mathcal{F}_t) \mid \mathcal{F}_0) &= \mathbb{E}( 1_{\{L \in dx\}} \mid \mathcal{F}_0) = P_0( \cdot , dx), \\ \mbox{ We have: } P_t( \cdot , dx) &= P_0( \cdot , dx)+ \int^1_0 \mathbb{E}( D^{\mathbb{M}}_sP_t( \cdot , dx) \mid \mathcal{F}_s) d\tilde{N}_s. \end{align*}
Define the following set: \[ \mathcal{V} = \{ \upsilon \in L^2(\mathcal{B} \times \mathbb{R}_+) \mid \upsilon(t) = f(t,T_1,...,T_n), f \in C^{\infty}_b(\mathbb{R}^{n+1}) \}. \]
$\mathcal{V}$ also has the property that $\forall \upsilon \in \mathcal{V}$, $\upsilon(y) = f(y,x_1,...,x_n) = 0$ whenever $y > x_n$.
\begin{thm}[Remark 2 \cite{Privault}]
$\mathcal{V}$ is dense in $L^2(\mathcal{B} \times \mathbb{R}_+)$. \label{denseV} \end{thm}
Based on Theorem \ref{denseV}, Wright et al. \cite{Wright} prove the result below:
\begin{thm}[Corollary 1 \cite{Wright}] For $u(\cdot,t) \in \mathbb{D}_{1,2}(\mathbb{M})$ such as $u(\cdot,t)$ is $\mathcal{F}_t$-adapted $\forall t \in [0,1]$, $D^{\mathbb{M}}_s u_t = 0$ a.s. when $s > t$. \label{collor1wright} \end{thm}
As $P_t(\cdot,dx)$ is $\mathcal{F}_t$-adapted, Theorem \ref{collor1wright} applies and hence:
\[ D^{\mathbb{M}}_s P_t(\cdot,dx) = 0 \mbox{, as } s \geq t. \] The above result justifies the following assertion, which is an extension to Proposition 3 in \cite{Mensi}:
\begin{thm}[{\bf Condition of existence and formula for $\mu^L_t$} \cite{Wright} ] Given L and its conditional law $P_t(\cdot,dx)$, if $P_t(\cdot,dx)$ satisfies the conditions of Theorem \ref{wrighttheor1}, $P_t(\cdot,dx)$ is represented as: \[ P_t(\cdot,dx) = P_0( \cdot , dx)+ \int^1_0 \mathbb{E}( D^{\mathbb{M}}_sP_t( \cdot , dx) \mid \mathcal{F}_s) d\tilde{N}_s.\]
If $\exists g : \mathbb{R}_+ \times \mathcal{B} \times \mathbb{R}_+ \rightarrow \mathbb{R}$ measurable and a Stopping time S such as \[ 1_{\{s \leq S\}}\mathbb{E}(D^{\mathbb{M}}_sP_t( \cdot , dx) \mid \mathcal{F}_s) = 1_{\{s \leq S\}}g_s(\cdot,x)P_s(\cdot,dx), \] then \[ \tilde{N}_t - \int^t_0 g_s(\cdot,L)ds \] is a $\mathcal{G}_t$-martingale. In other words, $\mu^L_s = g_s(\cdot,L)$. \label{prop6wright} \end{thm}
Applying Theorem \ref{prop6wright} to our initial problem, we get:\\ $\forall m \in \mathcal{M}^2 \cap \mathbb{D}_{1,1}$, we have the following integral representation:
\[ m_t = \int^t_0 \mathbb{E}(D_s m_{\infty} \mid \mathcal{F}_s) (d\tilde{W}_t + g_s(\cdot,L) ds) \mbox{ , } \forall t \in [0,1] \] where \[ \exists S \mbox{ - stopping time - s.a. } 1_{\{s \leq S\}}\mathbb{E}(D^{\mathbb{M}}_sP_t( \cdot , dx) \mid \mathcal{F}_s) = 1_{\{s \leq S\}}g_s(\cdot,x)P_s(\cdot,dx). \]
\subsection{Further Consideration}
We have reviewed a couple of occasions where the MRT and Clark-Ocone formula need to and can be generalized. However more can be done in order to make the MRT and Clark-Ocone formula applicable to larger sets of martingales and more situations. \\ \\ It is agreed that neither the $\mathbb{D}_{1,2}$ nor the $\mathbb{D}_{1,1}$ spaces are general enough for financial applications. Ideally, we want to be able to apply the concept of the Clark-Ocone representation formula to every $\mathcal{F}_T$-measurable F of $L^2(\Omega, \mathcal{F}, \mathbb{P})$ and its associated martingale space $\mathcal{M}^2$, for stochastic bases $(\Omega, \mathcal{F}, \mathbb{P})$ generated by any class of processes. One way of doing so is developed in Aase et al. \cite{Aase}, where a white-noise approach to Malliavin calculus is used in order to prove the following:
\[ \forall F \in \mathcal{G}^* \supset L^2(\mu) \mbox{ , } F = \mathbb{E}(F) + \int^T_0 \mathbb{E}(D_t F \mid \mathcal{F}_t) \diamond W_t dt, \]
\noindent where $D_t F = D_t F(\omega) = \frac{dF}{d\omega}$ is the generalized Malliavin derivative, $\diamond$ represents the Wick product and $W_t$ can be a scalar or multi-dimentional Gaussian, Poisson or combined Gaussian-Poisson white noise. The above formula holds on $\mathcal{G}^*$, which is a space of stochastic distributions. Additionally, $\mu$ represents the white-noise probability measure, hence $\mathcal{G}^* \supset L^2(\mu) $. Another paper by Ustunel \cite{Ustunel} also offers a similar generalization of the Clark-Ocone formula for all F in $\mathbb{D}_{-\infty}$, which is the space of Meyer-Watanabe distributions. However, $\mathbb{D}_{-\infty} \subset \mathcal{G}^*$ and $\mathbb{D}_{-\infty} \neq \mathcal{G}^*$ \cite{Aase}\\ \\ Another way to approach integral martingale representation is through non-anticipative functionals as done in Cont \cite{Cont}. Instead of using regular Malliavin calculus, we employ the concepts of horizontal and vertical derivative $ \mathcal{D}_tF$ and $\bigtriangledown_x F$ for a non-anticipative functional F = $(F_t)_{[0,T)}$. On a probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t), \mathbb{P})$ evolves a continuous $\mathbb{R}^d$-valued semi-martingale X that generates the sigma-fields $\mathcal{F}^X_t$. Then for every $\mathcal{F}^X_t$-adapted Y it can be shown that \[ Y_t = F_t(X_t,A_t) \] where $\langle X\rangle_t = \int^t_0 A_udu$ and F is a functional representing the dependence of Y on X and its quadratic variation. Based on this setting, it is possible to show an alternative form of the martingale representation theorem: \[ \forall Y \in L^2(X)\cap\mathcal{M}^2 \mbox{ s.a. Y is } \mathcal{F}^X_t \mbox{-adapted, } Y_T = Y_0 +\int^T_0 \bigtriangledown_xYdX. \] This result is of particular interest as it is computationally less intensive than the regular Malliavin derivative. $\bigtriangledown_xY$ can be calculated pathwise, and hence lends itself better to numerical computations. Note that when X=W is a Brownian motion, the vertical derivative $\bigtriangledown_W$ can be related to the Malliavin derivative.\\ \\ This paper has covered cases evolving on the standard space, a.k.a $\mathbb{R}$. However aspects of nonstandard analysis and applications of nonstandard stochastics to finance are of increased interest, as the hyperfinite versions of the regular option pricing models are better at outlining the connections between discrete and continuous trading models \cite{Cutland}. To apply pricing and trading models to $^*\mathbb{R} \setminus\mathbb{R}$, we would be looking at creating a version of the MRT and of the Clark-Ocone formula beyond the standard space. There is currently no rigorous nonstandard proof of the MRT in the form we have reviewed in this paper, since the filtrations of the nonstandard setting are much too rich and would result in integrals that are not well-defined \cite{Lindstrom}. One possible area of expansion in this direction would be to follow the idea of Lindstr$\o$m \cite{Lindstrom2}. There, the Brownian motion in equation \ref{itorepth} is replaced by an Anderson's random walk, which is a binomial random walk with an infinitesimal increment $\delta$ such as $\delta = T/N$, $N \in$ $^*\mathbb{R} \setminus\mathbb{R}$ on the interval [0,T] \cite{Anderson}. It is sometimes the case that hyperfinite stochastic processes have similar properties to standard ones, but this does not always hold, hence the need for further research in this area.
\end{document} | arXiv |
BMC Biology
Mechanisms of blood homeostasis: lineage tracking and a neutral model of cell populations in rhesus macaques
Sidhartha Goyal1,
Sanggu Kim2,
Irvin SY Chen2,3 &
Tom Chou4
BMC Biology volume 13, Article number: 85 (2015) Cite this article
How a potentially diverse population of hematopoietic stem cells (HSCs) differentiates and proliferates to supply more than 1011 mature blood cells every day in humans remains a key biological question. We investigated this process by quantitatively analyzing the clonal structure of peripheral blood that is generated by a population of transplanted lentivirus-marked HSCs in myeloablated rhesus macaques. Each transplanted HSC generates a clonal lineage of cells in the peripheral blood that is then detected and quantified through deep sequencing of the viral vector integration sites (VIS) common within each lineage. This approach allowed us to observe, over a period of 4-12 years, hundreds of distinct clonal lineages.
While the distinct clone sizes varied by three orders of magnitude, we found that collectively, they form a steady-state clone size-distribution with a distinctive shape. Steady-state solutions of our model show that the predicted clone size-distribution is sensitive to only two combinations of parameters. By fitting the measured clone size-distributions to our mechanistic model, we estimate both the effective HSC differentiation rate and the number of active HSCs.
Our concise mathematical model shows how slow HSC differentiation followed by fast progenitor growth can be responsible for the observed broad clone size-distribution. Although all cells are assumed to be statistically identical, analogous to a neutral theory for the different clone lineages, our mathematical approach captures the intrinsic variability in the times to HSC differentiation after transplantation.
Around 1011 new mature blood cells are generated in a human every day. Each mature blood cell comes from a unique hematopoietic stem cell (HSC). Each HSC, however, has tremendous proliferative potential and contributes a large number and variety of mature blood cells for a significant fraction of an animal's life. Traditionally, HSCs have been viewed as a homogeneous cell population, with each cell possessing equal and unlimited proliferative potential. In other words, the fate of each HSC (to differentiate or replicate) would be determined by its intrinsic stochastic activation and signals from its microenvironment [1, 2].
However, as first shown in Muller-Sieburg et al. [3], singly transplanted murine HSCs differ significantly in their long-term lineage (cell-type) output and in their proliferation and differentiation rates [4–7]. Similar findings have been found from examining human embryonic stem cells and HSCs in vitro [8, 9]. While cell-level knowledge of HSCs is essential, it does not immediately provide insight into the question of blood homeostasis at the animal level. More concretely, analysis of single-cell transplants does not apply to human bone marrow transplants, which involve millions of CD34-expressing primitive hematopoietic and committed progenitor cells. Polyclonal blood regeneration from such hematopoietic stem and progenitor cell (HSPC) pools is more complex and requires regulation at both the individual cell and system levels to achieve stable [10, 11] or dynamic [12] homeostasis.
To dissect how a population of HSPCs supplies blood, several high-throughput assay systems that can quantitatively track repopulation from an individual stem cell have been developed [6, 11, 13, 14]. In the experiment analyzed in this study, as outlined in Fig. 1, each individual CD34+ HSPC is distinctly labeled by the random incorporation of a lentiviral vector in the host genome before transplantation into an animal. All cells that result from proliferation and differentiation of a distinctly marked HSPC will carry identical markings defined by the location of the original viral vector integration site (VIS). By sampling nucleated blood cells and enumerating their unique VISs, one can quantify the cells that arise from a single HSPC marked with a viral vector. Such studies in humans [15] have revealed highly complex polyclonal repopulation that is supported by tens of thousands of different clones [15–18]; a clone is defined as a population of cells of the same lineage, identified here by a unique VIS. These lineages, or clones, can be distributed across all cell types that may be progeny of the originally transplanted HSC after it undergoes proliferation and differentiation. However, the number of cells of any VIS lineage across certain cell types may be different. By comparing abundances of lineages across blood cells of different types, for example, one may be able to determine the heterogeneity or bias of the HSC population or if HSCs often switch their output. This type of analysis remains particularly difficult in human studies since transplants are performed under diseased settings and followed for only 1 or 2 years.
Probing hematopoietic stem and progenitor cell (HSPC) biology through polyclonal analysis. a Mobilized CD34+ bone marrow cells from rhesus macaques are first marked individually with lentiviral vectors and transplanted back into the animal after nonlethal myeloablative irradiation [19]. Depending on the animal, 30–160 million CD34+ cells were transplanted, with a fraction ∼0.07–0.3 of them being lentivirus-marked. The clonal contribution of vector-marked HSPCs is measured from blood samples periodically drawn over a dozen years [19]. An average fraction f ∼0.03–0.1 of the sampled granulocytes and lymphocytes in the peripheral blood was found to be marked. This fraction is smaller than the fraction of marked CD34+ cells due probably to repopulation by surviving unmarked stem cells in the marrow after myeloablative conditioning. Within any post-transplant sample, S=1342–44,415 (average 10,026) viral integration sites of the marked cells were sequenced (see [14, 19] for details). b The fraction of all sequenced VIS reads belonging to each clone is shown by the thickness of the slivers. Small clones are not explicitly shown
We analyze here a systematic clone-tracking study that used a large number of HSPC clones in a transplant and competitive repopulation setting comparable to that used in humans [19]. In these nonhuman primate rhesus macaque experiments, lentiviral vector-marked clones were followed for up to a decade post-transplantation (equivalent to about 30 years in humans if extrapolated by average life span). All data are available in the supplementary information files of Kim et al. [19]. This long-term study allows one to distinguish clearly HSC clones from other short-term progenitor clones that were included in the initial pool of transplanted CD34+ cells. Hundreds to thousands of detected clones participated in repopulating the blood in a complex yet highly structured fashion. Preliminary examination of some of the clone populations suggests waves of repopulation with short-lived clones that first grow then vanish within the first 1 or 2 years, depending on the animal [19].
Subsequent waves of HSC clones appear to rise and fall sequentially over the next 4–12 years. This picture is consistent with recent observations in a transplant-free transposon-based tagging study in mice [20] and in human gene therapy [15, 16]. Therefore, the dynamics of a clonally tracked nonhuman primate HSPC repopulation provides rich data that can inform our understanding of regulation, stability, HSPC heterogeneity, and possibly HSPC aging in hematopoiesis.
Although the time-dependent data from clonal repopulation studies are rich in structure, in this study we focus on one specific aspect of the data: the number of clones that are of a certain abundance as described in Fig. 2. Rather than modeling the highly dynamic populations of each clone, our aim here is to develop first a more global understanding of how the total number of clones represented by specific numbers of cells arises within a mechanistically reasonable model of hematopoiesis. The size distributions of clones present in the blood sampled from different animals at different times are characterized by specific shapes, with the largest clones being a factor of 100–1000 times more abundant than the most rarely detected clones. Significantly, our analysis of renormalized data indicates that the clone size distribution (measuring the number of distinct lineages that are of a certain size) reaches a stationary state as soon as a few months after transplantation (see Fig. 4 below). To reconcile the observed stationarity of the clone size distributions with the large diversity of clonal contributions in the context of HSPC-mediated blood repopulation, we developed a mathematical model that treats three distinct cell populations: HSCs, transit-amplifying progenitor cells, and fully differentiated nucleated blood cells (Fig. 3). While multistage models for a detailed description of differentiation have been developed [21], we lump different stages of cell types within the transit-amplifying progenitor pool into one population, avoiding excess numbers of unmeasurable parameters. Another important feature of our model is the overall effect of feedback and regulation, which we incorporate via a population-dependent cell proliferation rate for progenitor cells.
Quantification of marked clones. a Assuming each transplanted stem cell is uniquely marked, the initial number of CD34+ cells representing each clone is one. b The pre-transplant clone size distribution is thus defined by the total number of transplanted CD34+ cells and is peaked at one cell. Post-transplant proliferation and differentiation of the HSC clones result in a significantly broader clone size distribution in the peripheral blood. The number of differentiated cells for each clone and the number of clones represented by exactly k cells, 5 years' post-transplantation (corresponding to Fig. 1a), are overlaid in (a) and (b) respectively. c Clone size distribution (blue) and the cumulative normalized clone size distribution (red) of the pre-transplant CD34+ population. d After transplantation, clone size distributions in the transit-amplifying (TA) and differentiated peripheral cell pools broaden significantly (with clones ranging over four decades in size) but reach a steady state. The corresponding cumulative normalized distribution is less steep
Schematic of our mathematical model. Of the ∼106– 107 CD34+ cells in the animal immediately after transplantation, C active HSCs are distinctly labeled through lentiviral vector integration. U HSCs are unlabeled because they were not mobilized, escaped lentiviral marking, or survived ablation. All HSCs asymmetrically divide to produce progenitor cells, which in turn replicate with an effective carrying capacity-limited rate r. Transit-amplifying progenitor cells die with rate μ p or terminally differentiate with rate ω. The terminal differentiation of the progenitor cells occurs symmetrically with probability η or asymmetrically with probability 1−η. This results in a combined progenitor-cell removal rate μ=μ p+η ω. The differentiated cells outside the bone marrow are assumed not to be subject to direct regulation but undergo turnover with a rate μ d. The mean total numbers of cells in the progenitor and differentiated populations are denoted N p and N d, respectively. Finally, a small fraction ε≪1 of differentiated cells is sampled, sequenced, and found to be marked. In this example, S=ε N d=5. Because some clones may be lost as cells successively progress from one pool to the next, the total number of clones in each pool must obey C≥C p≥C d≥C s. Analytic expressions for the expected total number of clones in each subsequent pool are derived in Additional file 1. HSC hematopoietic stem cell, TA transit-amplifying
Rescaled and renormalized data. a Individual clone populations (here, peripheral blood mononuclear cells of animal RQ5427) show significant fluctuations in time. For clarity, only clones that reach an appreciable frequency are plotted. b The corresponding normalized clone size distributions at each time point are rescaled by the sampled and marked fraction of blood, ν=q/S×f, where q is the number of reads of a particular clone within the sample. After an initial transient, the fraction of clones (dashed curves) as a function of relative size remains stable over many years. For comparison, the dot-dashed gray curves represent binomial distributions (with S=103 and 104 and equivalent mean clone sizes) and underestimate low population clones
The effective proliferation rate will be modeled using a Hill-type suppression that is defined by the limited space for progenitor cells in the bone marrow. Such a regulation term has been used in models of cyclic neutropenia [22] but has not been explicitly treated in models of clone propagation in hematopoiesis. Our mathematical model is described in greater detail in the next section and in Additional file 1.
Our model shows that both the large variability and the characteristic shape of the clone size distribution can result from a slow HSC-to-progenitor differentiation followed by a burst of progenitor growth, both of which are generic features of hematopoietic systems across different organisms. By assuming a homogeneous HSC population and fitting solutions of our model to available data, we show that randomness from stochastic activation and proliferation and a global carrying capacity are sufficient to describe the observed clonal structure. We estimate that only a few thousand HSCs may be actively contributing toward blood regeneration at any time. Our model can be readily generalized to include the role of heterogeneity and aging in the transplanted HSCs and provides a framework for quantitatively studying physiological perturbations and genetic modifications of the hematopoietic system.
Mathematical Model
Our mathematical model explicitly describes three subpopulations of cells: HSCs, transit-amplifying progenitor cells, and terminally differentiated blood cells (see Fig. 3). We will not distinguish between myeloid or lymphoid lineages but will use our model to analyze clone size distribution data for granulocytes and peripheral blood mononuclear cells independently. Our goal will be to describe how clonal lineages, started from distinguishable HSCs, propagate through the amplification and terminal differentiation processes.
Often clone populations are modeled directly by dynamical equations for n j (t), the number of cells of a particular clone j identified by its specific VIS [23]. Since all cells are identical except for their lentiviral marking, mean-field rate equations for n j (t) are identical for all j. Assuming identical initial conditions (one copy of each clone), the expected populations n j (t) would be identical across all clones j. This is a consequence of using identical growth and differentiation rates to describe the evolution of the mean number of cells of each clone.
Therefore, for cells in any specific pool, rather than deriving equations for the mean number n j of cells of each distinct clone j (Fig. 2 a), we perform a hodograph transformation [24] and formulate the problem in terms of the number of clones that are represented by k cells, \(c_{k} = \sum _{j}\delta _{k,n_{j}}\) (see Fig. 2 b), where the Kronecker δ function \(\delta _{k,n_{j}}=1\) only when k=n j and is 0 otherwise. This counting scheme is commonly used in the study of cluster dynamics in nucleation [25] and in other related models describing the dynamics of distributions of cell populations. By tracking the number of clones of different sizes, the intrinsic stochasticity in the times of cell division (especially that of the first differentiation event) and the subsequent variability in the clone abundances are quantified. Figure 2 a, b qualitatively illustrates n j and c k , pre-transplant and after 5 years, corresponding to the scenario depicted in Fig. 1 a. Cells in each of the three pools are depicted in Fig. 3, with different clones grouped according to the number of cells representing each clone.
The first pool (the progenitor-cell pool) is fed by HSCs through differentiation. Regulation of HSC differentiation fate is known to be important for efficient repopulation [26, 27] and control [28] and the balance between asymmetric and symmetric differentiation of HSCs has been studied at the microscopic and stochastic levels [29–32]. However, since HSCs have life spans comparable to that of an animal, we reasoned that the total number of HSCs changes only very slowly after the initial few-month transient after transplant. For simplicity, we will assume, consistent with estimates from measurements [33], that HSCs divide only asymmetrically. Therefore, upon differentiation, each HSC produces one partially differentiated progenitor cell and one replacement HSC. How symmetric HSC division might affect the resulting clone sizes is discussed in Additional file 1 through a specific model of HSC renewal in a finite-sized HSC niche. We find that the incorporation of symmetric division has only a small quantitative effect on the clone size distribution that we measure and ultimately analyze.
Next, consider the progenitor-cell pool. From Fig. 3, we can count the number of clones c k represented by exactly k cells. For example, the black, red, green, and yellow clones are each represented by three cells, so c 3=4. Each progenitor cell can further differentiate with rate ω into a terminally differentiated cell. If progenitor cells undergo symmetric differentiation with probability η and asymmetric differentiation with probability 1−η, the effective rate of differentiation is 2η ω+(1−η)ω=(1+η)ω. In turn, fully differentiated blood cells (not all shown in Fig. 3) are cleared from the peripheral pool at rate μ d, providing a turnover mechanism. Finally, each measurement is a small-volume sample drawn from the peripheral blood pool, as shown in the final panel in Fig. 3.
Note that the transplanted CD34+ cells contain both true HSCs and progenitor cells. However, we assume that at long times, specific clones derived from progenitor cells die out and that only HSCs contribute to long-lived clones. Since we measure the number of clones of a certain size rather than the dynamics of individual clone numbers, transplanted progenitor cells should not dramatically affect the steady-state clone size distribution. Therefore, we will ignore transplanted progenitor cells and assume that after transplantation, effectively only U unlabeled HSCs and C labeled (lentivirus-marked) HSCs are present in the bone marrow and actively asymmetrically differentiating (Fig. 3). Mass-action equations for the expected number of clones c k of size k are derived from considering simple birth and death processes with immigration (HSC differentiation):
$$ \begin{aligned} \frac{\mathrm{d} c_{k}}{\mathrm{d} t} = \underbrace{ \alpha\left[c_{k-1} - c_{k}\right]}_{\textrm{HSC differentiation}} &+ \underbrace{r\left[(k-1)c_{k-1}-{kc}_{k}\right]}_{\textrm{progenitor birth}}\\ &+ \underbrace{\mu\left[(k+1)c_{k+1} - k c_{k}\right]}_{\textrm{progenitor death}}, \end{aligned} $$
((1))
where k=1,2,…,C and \(c_{0}(t) \equiv C - \sum _{k=1}^{\infty }c_{k}(t)\) is the number of clones that are not represented in the progenitor pool. Since C is large, and the number of clones that are of size comparable to C is negligible, we will approximate C→∞ in our mathematical derivations. We have suppressed the time dependence of c k (t) for notational simplicity. The constant parameter α is the asymmetric differentiation rate of all HSCs, while r and μ are the proliferation and overall clearance rates of progenitor cells. In our model, HSC differentiation events that feed the progenitor pool are implicitly a rate- α Poisson process. The appreciable number of detectable clones (Fig. 1 b) implies the initial number C of HSC clones is large enough that asymmetric differentiation of individual HSCs is uncorrelated. The alternative scenario of a few HSCs undergoing synchronized differentiation would not lead to appreciably different results since the resulting distribution c k is more sensitive to the progenitor cells' unsynchronized replication and death than to the statistics of the immigration by HSC differentiation.
The final differentiation from progenitor cell to peripheral blood cell can occur through symmetric or asymmetric differentiation, with probabilities η and 1−η, respectively. If parent progenitor cells are unaffected after asymmetric terminal differentiation (i.e., they die at the normal rate μ p), the dynamics are feed-forward and the progenitor population is not influenced by terminal differentiation. Under symmetric differentiation, a net loss of one progenitor cell occurs. Thus, the overall progenitor-cell clearance rate can be decomposed as μ=μ p+η ω. We retain the factor η in our equations for modeling pedagogy, although in the end it is subsumed in effective parameters and cannot be independently estimated from our data.
The first term in Eq. 1 corresponds to asymmetric differentiation of each of the C active clones, of which c k are of those lineages with population k already represented in the progenitor pool. Differentiation of this subset of clones will add another cell to these specific lineages, reducing c k . Similarly, differentiation of HSCs in lineages that are represented by k−1 progenitor cells adds cells to these lineages and increases c k . Note that Eq. 1 are mean-field rate equations describing the evolution of the expected number of clones of size k. Nonetheless, they capture the intrinsic dispersion in lineage sizes that make up the clone size distribution. While all cells are assumed to be statistically identical, with equal rates α, p, and μ, Eq. 1 directly model the evolution of the distribution c k (t) that arises ultimately from the distribution of times for each HSC to differentiate or for the progenitor cells to replicate or die. Similar equations have been used to model the evolving distribution of virus capsid sizes [34].
Since the equations for c k (t) describe the evolution of a distribution, they are sometimes described as master equations for the underlying process [34, 35]. Here we note that the solution to Eq. 1, c k (t), is the expected distribution of clone sizes. Another level of stochasticity could be used to describe the evolution of a probability distribution \(P_{b}(\textbf {b};t) = P_{b}(b_{0}, b_{1},\ldots,b_{N_{\mathrm {p}}};t)\phantom {\dot {i}\!}\) over the integer numbers b k . This density represents the probability that at time t, there are b 0 unrepresented lineages, b 1 lineages represented by one cell in the progenitor pool, b 2 lineages represented by two cells in the progenitor pool, and so on. Such a probability distribution would obey an N p-dimensional master equation rather than a one-dimensional equation, like Eq. 1, and once known, can be used to compute the mean \(c_{k}(t) = \sum _{\textbf {b}} b_{k}P(\textbf {b};t)\). To consider the entire problem stochastically, the variability described by probability distribution P b would have to be propagated forward to the differentiated cell pool as well. Given the modest number of measured data sets and the large numbers of lineages that are detectable in each, we did not attempt to use the data as samples of the distribution P b and directly model the mean values c k instead. Variability from both intrinsic stochasticity and sampling will be discussed in Additional file 1.
After defining u(t) as the number of unlabeled cells in the progenitor pool, and \(N_{\mathrm {p}}(t) = u(t)+\sum _{k=1}^{\infty }{kc}_{k}(t)\) as the total number of progenitor cells, we find \(\dot {u} = (r - \mu) u + \alpha U\) and
$$ \frac{\mathrm{d} N_{\mathrm{p}}(t)}{\mathrm{d} t} = \alpha \left(U+C\right)+\left(r-\mu \right)N_{\mathrm{p}}(t). $$
Without regulation, the total population N p(t→∞) will either reach N p≈α(U+C)/(μ−r) for μ>r or will exponentially grow without bound for r>μ. Complex regulation terms have been employed in deterministic models of differentiation [28] and in stochastic models of myeloid/lymphoid population balance [36]. For the purpose of estimating macroscopic clone sizes, we assume regulation of cell replication and/or spatial constraints in the bone marrow can be modeled by a simple effective Hill-type growth law [22, 37]:
$$ r = r(N_{\mathrm{p}}) \equiv \frac{pK}{N_{\mathrm{p}}+K} $$
where p is the intrinsic replication rate of an isolated progenitor cell. We assume that progenitor cells at low density have an overall positive growth rate p>μ. The parameter K is the progenitor-cell population in the bone marrow that corresponds to the half-maximum of the effective growth rate. It can also be interpreted as a limit to the bone marrow size that regulates progenitor-cell proliferation to a value determined by K, p, and μ and is analogous to the carrying capacity in logistic models of growth [38]. For simplicity, we will denote K as the carrying capacity in Eq. 3 as well. Although our data analysis is insensitive to the precise form of regulation used, we chose the Hill-type growth suppression because it avoids negative growth rates that confuse physiological interpretation. An order-of-magnitude estimate of the bone marrow size (or carrying capacity) in the rhesus macaque is K∼109. Ultimately, we are interested in how a limited progenitor pool influences the overall clone size distribution, and a simple, single-parameter (K) approximation to the progenitor-cell growth constraint is sufficient.
Upon substituting the growth law r(N p) described by Eq. 3 into Eq. 2, the total progenitor-cell population N p(t→∞) at long times is explicitly shown in Additional file 1: Eq. A19 to approach a finite value that depends strongly on K. Progenitor cells then differentiate to supply peripheral blood at rate (1+η)ω so that the total number of differentiated blood cells obeys
$$ \frac{\mathrm{d} N_{\mathrm{d}}(t)}{\mathrm{d} t} = (1+\eta)\omega N_{\mathrm{p}} - \mu_{\mathrm{d}}N_{\mathrm{d}}. $$
At steady state, the combined peripheral nucleated blood population is estimated to be N d∼109– 1010 [39], setting an estimate of N d/N p≈(1+η)ω/μ d∼1–10. Moreover, as we shall see, the relevant factor in our steady-state analysis will be the numerical value of the effective growth rate r, rather than its functional form. Therefore, the chosen form for regulation will not play a role in the mathematical results in this paper except to define parameters (such as K) explicitly in the regulation function itself.
To distinguish and quantify the clonal structure within the peripheral blood pool, we define \(y_{n}^{(k)}\) to be the number of clones that are represented by exactly n cells in the differentiated pool and k cells in the progenitor pool. For example, in the peripheral blood pool shown in Fig. 3, \(y_{1}^{(3)} = y_{2}^{(3)} = y_{4}^{(3)} = y_{6}^{(3)} = 1\). This counting of clones across both the progenitor and peripheral blood pools is necessary to balance progenitor-cell differentiation rates with peripheral blood turnover rates. The evolution equations for \(y_{n}^{(k)}\) can be expressed as
$$ \frac{\mathrm{d} y_{n}^{(k)}}{\mathrm{d} t} = (1+\eta)\omega k \left(y_{n-1}^{(k)} - y_{n}^{(k)}\right) + (n+1) \mu_{\mathrm{d}}y_{n+1}^{(k)} - n \mu_{d} y_{n}^{(k)}, $$
where \(y_{0}^{(k)} \equiv c_{k} - \sum _{n=1}^{\infty }y_{n}^{(k)}\) represents the number of progenitor clones of size k that have not yet contributed to peripheral blood. The transfer of clones from the progenitor population to the differentiated pool arises through \(y_{0}^{(k)}\) and is simply a statement that the number of clones in the peripheral blood can increase only by differentiation of a progenitor cell whose lineage has not yet populated the peripheral pool. The first two terms on the right-hand side of Eq. 5 represent immigration of clones represented by n−1 and n differentiated cells conditioned upon immigration from only those specific clones represented by k cells in the progenitor pool. The overall rate of addition of clones from the progenitor pool is thus (1+η)ω k, in which the frequency of terminal differentiation is weighted by the stochastic division factor (1+η). By using the Hill-type growth term r(N p) from Eq. 3, Eq. 1 can be solved to find c k (t), which in turn can be used in Eq. 5 to find \(y_{n}^{(k)}(t)\). The number of clones in the peripheral blood represented by exactly n differentiated cells is thus \(y_{n}(t) = \sum _{k=1}^{\infty }y_{n}^{(k)}(t)\).
As we mentioned, Eqs. 1 and 5 describe the evolution of the expected clone size distribution. Since each measurement represents one realization of the distributions c k (t) and y n (t), the validity of Eqs. 1 and 5 relies on a sufficiently large C such that the marked HSCs generate enough lineages and cells to allow the subsequent peripheral blood clone size distribution to be sampled adequately. In other words, measurement-to-measurement variability described by e.g., \(\phantom {\dot {i}\!}\langle c_{k}(t)c_{k^{\prime }}(t)\rangle - \langle c_{k}(t)\rangle \langle c_{k^{\prime }}(t)\rangle \) is assumed negligible (see Additional file 1). Our modeling approach would not be applicable to studying single HSC transplant studies [4–6] unless the measured clone sizes from multiple experiments are aggregated into a distribution.
Finally, to compare model results with animal blood data, we must consider the final step of sampling small aliquots of the differentiated blood. As derived in Additional file 1: Eq. A11, if S marked cells are drawn and sequenced successfully (from a total differentiated cell population N d), the expected number of clones 〈m k (t)〉 represented by k cells is given by
$$ \begin{array}{cc}\left\langle {m}_k(t)\right\rangle & =F\left(q,t\right)-F\left(q-1,t\right)\\ {}=\sum_{\ell =0}^{\infty }{\mathrm{e}}^{-\ell \varepsilon}\frac{{\left(\ell \varepsilon \right)}^k}{k!}{y}_{\ell }(t),\end{array} $$
where ε≡S/N d≪1 and \(F(q,t) \equiv \sum _{k=0}^{q}\langle m_{k}(t)\rangle \) is the sampled, expected cumulative size distribution. Upon further normalization by the total number of detected clones in the sample, C s(t)=F(S,t)−F(0,t), we define
$$ Q(q,t) \equiv \frac{F(q, t) - F(0,t)}{F(S,t)-F(0,t)} $$
as the fraction of the total number of sampled clones that are represented by q or fewer cells. Since the data represented in terms of Q will be seen to be time-independent, explicit expressions for \(c_{k}, y_{n}^{(k)}\), 〈m k 〉, and Q(q) can be derived. Summarizing, the main features and assumptions used in our modeling include:
A neutral-model framework [40] that directly describes the distribution of clone sizes in each of the three cell pools: progenitor cells, peripheral blood cells, and sampled blood cells. The cells in each pool are statistically identical.
A constant asymmetric HSC differentiation rate α. The appreciable numbers of unsynchronized HSCs allow the assumption of Poisson-distributed differentiation times of the HSC population. The level of differentiation symmetry is found to have little effect on the steady-state clone size distribution (see Additional file 1). The symmetry of the terminal differentiation step is also irrelevant for understanding the available data.
A simple one-parameter (K) growth regulation model that qualitatively describes the finite maximum size of the progenitor population in the bone marrow. Ultimately, the specific form for the regulation is unimportant since only the steady-state value of the growth parameter r affects the parameter fitting.
Using only these reasonable model features, we are able to compute clone size distributions and compare them with data. An explicit form for the expected steady-state clone size distribution 〈m k 〉 is given in Additional file 1: Eq. A32, and the parameters and variables used in our analysis are listed in Table 1.
Table 1 Model parameters and variables. Estimates of steady-state values are provided where available. We assume little prior knowledge on all but a few of the more established parameters. Nonetheless, our modeling and analysis place constraints on combinations of parameters, allowing us to fit data and provide estimates for steady-state values of U+C∼103– 104 and α(N p+K)/(p K)∼0.002–0.1
In this section, we describe how previously published data (the number of cells of each detected clone in a sample of the peripheral blood, which are available in the supplementary information files of Kim et al. [19]) are used to constrain parameter values in our model. We emphasize that our model is structurally different from models used to track lineages and clone size distributions in retinal and epithelial tissues [41, 42]. Rather than tracking only the lineages of stem cells (which are allowed to undergo asymmetric differentiation, symmetric differentiation, or symmetric replication), our model assumes a highly proliferative population constrained by a carrying capacity K and slowly fed at rate α by an asymmetrically dividing HSC pool of C fixed clones. We have also included terminal differentiation into peripheral blood and the effects of sampling on the expected clone size distribution. These ingredients yield a clone size distribution different from those previously derived [41, 42], as described in more detail below.
Stationarity in time
Clonal contributions of the initially transplanted HSC population have been measured over 4–12 years in four different animals. As depicted in Fig. 4 a, populations of individual clones of peripheral blood mononuclear cells from animal RQ5427, as well as all other animals, show significant variation in their dynamics. Since cells of any detectable lineage will number in the millions, this variability in lineage size across time cannot be accounted for by the intrinsic stochasticity of progenitor-cell birth and death. Rather, these rises and falls of lineages likely arise from a complicated regulation of HSC differentiation and lineage aging. However, in our model and analysis, we do not keep track of lineage sizes n i . Instead, define Q(ν) as the fraction of clones arising with relative frequency ν≡f q/S or less (here, q is the number of VIS reads of any particular clone in the sample, f is the fraction of all sampled cells that are marked, and S is the total number of sequencing reads of marked cells in a sample). Figure 4 b shows data analyzed in this way and reveals that Q(ν) appears stationary in time.
The observed steady-state clone size distribution is broad, consistent with the mathematical model developed above. The handful of most populated clones constitutes up to 1–5 % of the entire differentiated blood population. These dominant clones are followed by a large number of clones with fewer cells. The smallest clones sampled in our experiment correspond to a single read q=1, which yields a minimum measured frequency ν min=f/S. A single read may comprise only 10−4– 10−3 % of all differentiated blood cells. Note that the cumulative distribution Q(ν) exhibits higher variability at small sizes simply because fewer clones lie below these smaller sizes.
Although engraftment occurs within a few weeks and total blood populations N p and N d (and often immune function) re-establish themselves within a few months after successful HSC transplant [43, 44], it is still surprising that the clone size distribution is relatively static within each animal (see Additional file 1 for other animals). Given the observed stationarity, we will use the steady-state results of our mathematical model (explicitly derived in Additional file 1) for fitting data from each animal.
Implications and model predictions
By using the exact steady-state solution for c k (Additional file 1: Eq. A21) in Additional file 1: Eq. A18, we can explicitly evaluate the expected clone size distribution 〈m k 〉 using Eq. 6, and the expected cumulative clone fraction Q(q) using Eq. 7. In the steady state, the clone size distribution of progenitor cells can also be approximated by a gamma distribution with parameters a≡α/r and \(\bar {r} \equiv r/\mu \): \(c_{k} \sim \bar {r}^{k} k^{-1+a}\) (see Additional file 1: Eq. A27). In realistic steady-state scenarios near carrying capacity, r=r(N p)≲μ, as calculated explicitly in Additional file 1: Eq. A20. By defining \(\bar {r}=r/\mu = 1-\delta \), we find that δ is inversely proportional to the carrying capacity:
$$ \delta \approx \frac{\alpha}{\mu} \frac{\mu}{p-\mu} \frac{U+C}{K} \ll 1. $$
The dependencies of 〈m q 〉 on δ and a=α/r are displayed in Fig. 5 a, in which we have defined w≡(1+η)ω/μ d.
Clone size distributions and total number of sampled clones. a Expected clone size distributions C −1〈m q 〉 derived from the approximation in Additional file 1: Eq. A32 are plotted for various a and δ/(ε w) [where w≡(1+η)ω/μ d]. The nearly coincident solid and dashed curves indicate that variations in a mainly scale the distribution by a multiplicative factor. In contrast, the combination δ/(ε w) controls the weighting at large clone sizes through the population cut-off imposed by the carrying capacity. Of the two controlling parameters, the steady-state clone size distribution is most sensitive to R≅δ/(ε w). The dependence of data-fitting on these two parameters is derived in Additional file 1 and discussed in the next section. b For ε w=5×10−5, the expected fraction C s/C of active clones sampled as a function of lnδ and α. The expected number of clones sampled increases with carrying capacity K, HSC differentiation rate a=α/r, and the combined sampling and terminal differentiation rate ε w
Although our equations form a mean-field model for the expected number of measured clones of any given size, randomness resulting from the stochastic differentiation times of individual HSCs (all with the same rate α) is taken into account.
This is shown in Additional file 1: Eqs. A36–A39, where we explicitly consider the fully stochastic population of a single progenitor clone that results from the differentiation of a single HSC. Since independent unsynchronized HSCs differentiate at times that are exponentially distributed (with rate α), we construct the expected clone size distribution from the birth–death–immigration process [45] to find a result equivalent to that derived from our original model (Eq. 1 and Additional file 1: Eq. A21). Thus, we conclude that if a=α/r is small, the shape of the expected clone size distribution is mainly determined at short times by the initial repopulation of the progenitor-cell pool.
Our model also suggests that the expected number of sampled clones relative to the number of active transplanted clones (see Additional file 1: Eq. A24) can be expressed as:
$$ \begin{aligned} \frac{C_{\mathrm{s}}}{C} & \approx \left[1-\left(\frac{\delta}{1-(1-\delta)e^{-\varepsilon w}}\right)^{a}\right] \\ & \approx \frac{\alpha}{r}\ln \left(\frac{\varepsilon w}{\delta}+1\right), \end{aligned} $$
where the last approximation is accurate for ε w≪1 and C s/C≪1. The clonal diversity one expects to measure in the peripheral blood sample is sensitive to the combination of biologically relevant parameters and rates δ and a=α/r. Figure 5 b shows the explicit dependence of the fraction of active clones on a and the combination of parameters defining δ, for ε w=ε(1+η)ω/μ d=5×10−5.
Our analysis shows how scaled measurable quantities such as C s/C and C −1〈m q 〉 depend on just a few combinations of experimental and biological parameters. This small domain of parameter sensitivity reduces the number of parameters that can be independently extracted from clone size distribution data. For example, the mode of terminal differentiation described by η clearly cannot be extracted from clonal tracking measurements. Similarly, models that are more detailed of the complex regulation processes would introduce additional parameters that are not resolved by these experiments. Nonetheless, we shall fit our data and known information contained in the experimental protocol to our model to estimate biologically relevant parameters, such as the total number of activated HSCs U+C, and thus indirectly C.
Model fitting
Our mathematical model for 〈m k 〉 (and F(q) and Q(q)) includes numerous parameters associated with the processes of HSC differentiation, progenitor-cell amplification, progenitor-cell differentiation, peripheral blood turnover, and sampling. Data fitting is performed using clone size distributions derived separately from the read counts from both the left and right ends of each VIS (see [14] for details on sequencing). Even though we fit our data to 〈m k 〉 using three independent parameters, a=α/r, \(\bar {r}= r/\mu \), and ε w, we found that within the relevant physiological regime, all clone distributions calculated from our model are most sensitive to just two combinations of parameters (see Additional file 1 for an explicit derivation):
$$ a \equiv \frac{\alpha}{r}\quad \text{and} \quad R \equiv \frac{\varepsilon w}{\ln \left(1/\bar{r}\right)}\approx \frac{\varepsilon w}{\delta} = \frac{(1+\eta)\omega S}{N_{\mathrm{d}}\mu_{\mathrm{d}}\delta}, $$
where the last approximation for R is valid when \(1-\bar {r} = \delta \ll 1\). While the fits are rather insensitive to ε w this parameter can fortunately be approximated from estimates of S and the typical turnover rate of differentiated blood. Consequently, we find two maximum likelihood estimates (MLEs) for a and R at each time point. It is important to note that fitting our model to steady-state clone size distributions does not determine all of the physiological parameters arising in our equations. Rather, they provide only two constraints that allow one to relate their values.
For ease of presentation, henceforth we will show all data and comparisons with our model equations in terms of the fraction Q(ν) or Q(q) (Figs. 4 b and 6 a, b). Figure 6 a, b shows MLE fitting to the raw data 〈m k 〉 plotted in terms of the normalized but unrescaled data Q(q) for two different peripheral blood cell types from two animals (RQ5427 and RQ3570). Data from all other animals are shown and fitted in Additional file 1, along with overall goodness-of-fit metrics. Raw cell count data are given in Kim et al. [19].
Data fitting. a Fitting raw (not rescaled, as shown in Figure 4) clone size distribution data to 〈m k 〉 from Eq. 6 at two time points for animal RQ5427. The maximum likelihood estimates (MLEs) are (a ∗≈0.01,R ∗≈70) and (a ∗≈0.0025,R ∗≈400) for data taken at 32 (blue) and 67 (red) months post-transplant, respectively. Note that the MLE values for different samples vary primarily due to different values of S (and hence ε) used in each measurement. b For animal RQ3570, the clone fractions at 32 (blue) and 38 (red) months yield (a ∗≈0.04,R ∗≈30) and (a ∗≈0.1,R ∗≈60), respectively. For clarity, we show the data and fitted models in terms of Q(q). c Estimated number of HSCs U+C (circles) and normalized differentiation rate a (squares) for animal RQ5427. d U+C and a for animal RQ3570. Note the temporal variability (but also long-term stability) in the estimated number of contributing HSCs. Additional details and fits for other animals are qualitatively similar and given in Additional file 1. HSC hematopoietic stem cell, PBMC, peripheral blood mononuclear cell Grans, granulocytes
HSC asymmetric differentiation rate
The MLE for a=α/r, a ∗, was typically in the range 10−2– 10−1. Given realistic parameter values, this quantity mostly provides an estimate of the HSC relative differentiation rate a ∗∼α/(μ p+η ω). The smallness of a ∗ indicates slow HSC differentiation relative to the progenitor turnover rate μ p and the final differentiation rate η ω, consistent with the dominant role of progenitor cells in populating the total blood tissue. Note that besides the intrinsic insensitivity to ε w, the goodness-of-fit is also somewhat insensitive to small values of a ∗ due to the weak dependence of c k ∼1/k 1−a on a (see Additional file 1). The normalized relative differentiation rates estimated from two animals are shown by the squares (right axis) in Fig. 6 c, d.
Number of HSCs
The stability of blood repopulation kinetics is also reflected in the number of estimated HSCs that contribute to blood (shown in Fig. 6 c, d). The total number of HSCs is estimated by expressing U+C in terms of the effective parameters, R and a, which in turn are functions of microscopic parameters (α,p,μ p,μ d,w, and K) that cannot be directly measured. In the limit of small sample size, S≪R ∗ K, however, we find U+C≈S/(R ∗ a ∗) (see Additional file 1), which can then be estimated using the MLEs a ∗ and R ∗ obtained by fitting the clone size distributions. The corresponding values of U+C for two animals are shown by the circles (left axis) in Fig. 6 c, d. Although variability in the MLEs exists, the fluctuations appear stationary over the course of the experiment for each animal (see Additional file 1).
Our clonal tracking analysis revealed that individual clones of HSCs contributed differently to the final differentiated blood pool in rhesus macaques, consistent with mouse and human data. Carefully replotting the raw data (clone sizes) in terms of the normalized, rescaled cumulative clone size distribution (the fraction of all detected clones that are of a certain size or less) shows that these distributions reach steady state a few months after transplantation. Our results carry important implications for stem cell biology. Maintaining homeostasis of the blood is a critical function for an organism. Following a myeloablative stem cell transplant, the hematopoietic system must repopulate rapidly to ensure the survival of the host. Not only do individual clones rise and fall temporally, as previously shown [19], but as any individual clone of a certain frequency declines, it is replaced by another of similar frequency. This exchange-correlated mechanism of clone replacement may provide a mechanism by which overall homeostasis of hematopoiesis is maintained long term, thus ensuring continued health of the blood system.
To understand these observed features and the underlying mechanisms of stem cell-mediated blood regeneration, we developed a simple neutral population model of the hematopoietic system that quantifies the dynamics of three subpopulations: HSCs, transit-amplifying progenitor cells, and fully differentiated nucleated blood cells. We also include the effects of global regulation by assuming a Hill-type growth rate for progenitor cells in the bone marrow but ignore cell-to-cell variation in differentiation and proliferation rates of all cells.
Even though we do not include possible HSC heterogeneity, variation in HSC activation, progenitor-cell regulation, HSC and progenitor-cell aging (progenitor bursting), niche- and signal molecule-mediated controls, or intrinsic genetic and epigenetic differences, solutions to our simple homogeneous HSC model are remarkably consistent with observed clone size distributions. As a first step, we focus on how the intrinsic stochasticity in just the cellular birth, death, and differentiation events gives rise to the progenitor clone size distribution.
To a large extent, the exponentially distributed first HSC differentiation times and the growth and turnover of the progenitor pool control the shape of the expected long-time clone size distribution. Upon constraining our model to the physiological regime relevant to the experiments, we find that the calculated shapes of the clone size distributions are sensitive to effectively only two composite parameters. The HSC differentiation rate α sets the scale of the expected clone size distribution but has little effect on the shape. Parameters, including carrying capacity K, active HSCs U+C, and birth and death rates p,ω,μ p,μ d, influence the shape of the expected clone size distribution 〈m q 〉 only through the combination R, and only at large clone sizes.
Our analysis allowed us to estimate other combinations of model parameters quantitatively. Using a MLE, we find values for the effective HSC differentiation rate a ∗∼10−2– 10−1 and the number of HSCs that are contributing to blood within any given time frame U+C∼103– 104. Since the portion of HSCs that contribute to blood may vary across their typical life span L∼25 years, the total number of HSCs can be estimated by (U+C)×L/τ, where τ∼1 year [19]. Our estimate of a total count of ∼3×104– 3×105 HSCs is about 30-fold higher than the estimate of Abkowitz et al. [33] but is consistent with Kim et al. [19]. Note that the ratio of C to the total number of initially transplanted CD34+ cells provides a measure of the overall potency of the transplant towards blood regeneration. In the extreme case in which one HSC is significantly more potent (through, e.g., a faster differentiation rate), this ratio would be smaller. An example of this type of heterogeneity would be an HSC with one or more cancer-associated mutations, allowing it to out-compete other transplanted normal HSCs. Hence, our clonal studies and the associated mathematical analysis can provide a framework for characterizing normal clonal diversity as well as deviations from it, which may provide a metric for early detection of cancer and other related pathologies.
Several simplifying assumptions have been invoked in our analysis. Crucially, we assumed HSCs divided only asymmetrically and ignored instances of symmetric self-renewal or symmetric differentiation. The effects of symmetric HSC division can be quantified in the steady-state limit. In previous studies, the self-renewal rate for HSCs in primates is estimated as 4–9 months [46, 47], which is slightly longer than the short timescale (∼2–4 months) on which we observe stabilization of the clone size distribution. Therefore, if the HSC population slowly increases in time through occasional symmetric division, the clone size distribution in the peripheral blood will also shift over long times. The static nature of the clone distributions over many years suggests that size distributions are primarily governed by mechanisms operating at shorter timescales in the progenitor pool. For an HSC population (such as cancerous or precancerous stem cells [48]) that has already expanded through early replication, the initial clone size distribution within the HSC pool can be quantified by assuming an HSC pool with separate carrying capacity K HSC. Such an assumption is consistent with other analyses of HSC renewal [49]. All our results can be used (with the replacement C→K HSC) if the number of transplanted clones C≥K HSC because replication is suppressed in this limit. When K HSC≫C≫1, replicative expansion generates a broader initial HSC clone size distribution (see Additional file 1). The resulting final peripheral blood clone size distribution can still be approximated by our result (Eq. 6) if the normalized differentiation rate a≪1, exhibiting the insensitivity of the differentiated clone size distribution to a broadened clone size distribution at the HSC level. However, if HSC differentiation is sufficiently fast (a≪̸1), the clonal distribution in the progenitor and differentiated pools may be modified.
To understand the temporal dynamics of clone size distributions, a more detailed numerical study of our full time-dependent neutral model is required. Such an analysis can be used to investigate the effects of rapid temporal changes in the HSC division mode [41]. Temporal models would also allow investigation into the evolution of HSC mutations and help unify concepts of clonal stability (as indicated by the stationarity of rescaled clone size distributions) with ideas of clonal succession [10, 11] or dynamic repetition [12] (as indicated by the temporal fluctuations in the estimated number U+C of active HSCs). Predictions of the time-dependent behavior of clone size distributions will also prove useful in guiding future experiments in which the animals are physiologically perturbed via e.g., myeloablation, hypoxiation, and/or bleeding. In such experimental settings, regulation may also occur at the level of HSC differentiation (α) and a different mathematical model may be more appropriate.
We have not addressed the temporal fluctuations in individual clone abundances evident in our data (Fig. 4 a) or in the wave-like behavior suggested by previous studies [19]. Since the numbers of detectable cells of each VIS lineage in the whole animal are large, we believe these fluctuations do not arise from intrinsic cellular stochasticity or sampling. Rather, they likely reflect slow timescale HSC transitions between quiescent and active states and/or HSC aging [50]. Finally, subpopulations of HSCs that have different intrinsic rates of proliferation, differentiation, or clearance could then be explicitly treated. As long as each subtype in a heterogeneous HSC or progenitor-cell population does not convert into another subtype, the overall aggregated clone size distribution 〈m k 〉 will preserve its shape. Although steady-state data are insufficient to provide resolution of cell heterogeneity, more resolved temporal data may allow one to resolve different parameters associated with different cell types. Such extensions will allow us to study the temporal dynamics of individual clones and clone populations in the context of cancer stem cells and will be the subject of future work.
HSC:
hematopoietic stem cell
HSPC:
hematopoietic stem and progenitor cell
MLE:
maximum likelihood estimate
viral vector integration site
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This work was supported by grants from the National Institutes of Health (R01AI110297 and K99HL116234), the California Institute of Regenerative Medicine (TRX-01431), the University of California, Los Angeles, AIDS Institute/Center for AIDS Research (AI28697), the National Science Foundation (PHY11-25915 KITP/UCSB), and the Army Research Office (W911NF-14-1-0472). The authors also thank B Shraiman and RKP Zia for helpful discussions.
Department of Physics, University of Toronto, St George Campus, Toronto, Canada
Sidhartha Goyal
Department of Microbiology, Immunology, and Molecular Genetics, UCLA, Los Angeles, USA
Sanggu Kim & Irvin SY Chen
UCLA AIDS Institute and Department of Medicine, UCLA, Los Angeles, USA
Irvin SY Chen
Departments of Biomathematics and Mathematics, UCLA, Los Angeles, USA
Tom Chou
Sanggu Kim
Correspondence to Tom Chou.
TC and SG designed and developed the mathematical modeling and data analysis. TC, SG, and SK wrote the manuscript. SK and IC participated in study design and data interpretation. All authors read and approved the final manuscript.
Additional file 1
Mathematical appendices and data fitting. (PDF 327 kb)
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Goyal, S., Kim, S., Chen, I.S. et al. Mechanisms of blood homeostasis: lineage tracking and a neutral model of cell populations in rhesus macaques. BMC Biol 13, 85 (2015). https://doi.org/10.1186/s12915-015-0191-8
Received: 09 June 2015
Stem cell clones
Lineage tracking
Mathematical modeling
Beyond Mendel: modeling in biology
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Snub square antiprism
In geometry, the snub square antiprism is one of the Johnson solids (J85). A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.[1]
Snub square antiprism
TypeJohnson
J84 – J85 – J86
Faces8+16 triangles
2 squares
Edges40
Vertices16
Vertex configuration8(35)
8(34.4)
Symmetry groupD4d
Dual polyhedron-
Propertiesconvex
Net
It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids, although it is a relative of the icosahedron that has fourfold symmetry instead of threefold.
Construction
The snub square antiprism is constructed as its name suggests, a square antiprism which is snubbed, and represented as ss{2,8}, with s{2,8} as a square antiprism.[2] It can be constructed in Conway polyhedron notation as sY4 (snub square pyramid).[3]
It can also be constructed as a square gyrobianticupolae, connecting two anticupolae with gyrated orientations.
Cartesian coordinates
Let k ≈ 0.82354 be the positive root of the cubic polynomial
$9x^{3}+3{\sqrt {3}}\left(5-{\sqrt {2}}\right)x^{2}-3\left(5-2{\sqrt {2}}\right)x-17{\sqrt {3}}+7{\sqrt {6}}.$
Furthermore, let h ≈ 1.35374 be defined by
$h={\frac {{\sqrt {2}}+8+2{\sqrt {3}}k-3\left(2+{\sqrt {2}}\right)k^{2}}{4{\sqrt {3-3k^{2}}}}}.$
Then, Cartesian coordinates of a snub square antiprism with edge length 2 are given by the union of the orbits of the points
$(1,1,h),\,\left(1+{\sqrt {3}}k,0,h-{\sqrt {3-3k^{2}}}\right)$
under the action of the group generated by a rotation around the z-axis by 90° and by a rotation by 180° around a straight line perpendicular to the z-axis and making an angle of 22.5° with the x-axis.[4]
We may then calculate the surface area of a snub square antiprism of edge length a as
$A=\left(2+6{\sqrt {3}}\right)a^{2}\approx 12.39230a^{2},$[5]
and its volume as
$V=\xi a^{3},$
where ξ ≈ 3.60122 is the greatest real root of the polynomial
$531441x^{12}-85726026x^{8}-48347280x^{6}+11588832x^{4}+4759488x^{2}-892448.$[6]
Snub antiprisms
Similarly constructed, the ss{2,6} is a snub triangular antiprism (a lower symmetry octahedron), and result as a regular icosahedron. A snub pentagonal antiprism, ss{2,10}, or higher n-antiprisms can be similar constructed, but not as a convex polyhedron with equilateral triangles. The preceding Johnson solid, the snub disphenoid also fits constructionally as ss{2,4}, but one has to retain two degenerate digonal faces (drawn in red) in the digonal antiprism.
Snub antiprisms
Symmetry D2d, [2+,4], (2*2) D3d, [2+,6], (2*3) D4d, [2+,8], (2*4) D5d, [2+,10], (2*5)
Antiprisms
s{2,4}
A2
(v:4; e:8; f:6)
s{2,6}
A3
(v:6; e:12; f:8)
s{2,8}
A4
(v:8; e:16; f:10)
s{2,10}
A5
(v:10; e:20; f:12)
Truncated
antiprisms
ts{2,4}
tA2
(v:16;e:24;f:10)
ts{2,6}
tA3
(v:24; e:36; f:14)
ts{2,8}
tA4
(v:32; e:48; f:18)
ts{2,10}
tA5
(v:40; e:60; f:22)
Symmetry D2, [2,2]+, (222) D3, [3,2]+, (322) D4, [4,2]+, (422) D5, [5,2]+, (522)
Snub
antiprisms
J84 Icosahedron J85 Concave
sY3 = HtA3 sY4 = HtA4 sY5 = HtA5
ss{2,4}
(v:8; e:20; f:14)
ss{2,6}
(v:12; e:30; f:20)
ss{2,8}
(v:16; e:40; f:26)
ss{2,10}
(v:20; e:50; f:32)
References
1. Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, Zbl 0132.14603.
2. Snub Anti-Prisms
3. "PolyHédronisme".
4. Timofeenko, A. V. (2009). "The non-Platonic and non-Archimedean noncomposite polyhedra". Journal of Mathematical Science. 162 (5): 725. doi:10.1007/s10958-009-9655-0. S2CID 120114341.
5. Wolfram Research, Inc. (2020). "Wolfram|Alpha Knowledgebase". Champaign, IL. PolyhedronData[{"Johnson", 85}, "SurfaceArea"] {{cite journal}}: Cite journal requires |journal= (help)
6. Wolfram Research, Inc. (2020). "Wolfram|Alpha Knowledgebase". Champaign, IL. MinimalPolynomial[PolyhedronData[{"Johnson", 85}, "Volume"], x] {{cite journal}}: Cite journal requires |journal= (help)
External links
• Eric W. Weisstein, Snub square antiprism (Johnson solid) at MathWorld.
Johnson solids
Pyramids, cupolae and rotundae
• square pyramid
• pentagonal pyramid
• triangular cupola
• square cupola
• pentagonal cupola
• pentagonal rotunda
Modified pyramids
• elongated triangular pyramid
• elongated square pyramid
• elongated pentagonal pyramid
• gyroelongated square pyramid
• gyroelongated pentagonal pyramid
• triangular bipyramid
• pentagonal bipyramid
• elongated triangular bipyramid
• elongated square bipyramid
• elongated pentagonal bipyramid
• gyroelongated square bipyramid
Modified cupolae and rotundae
• elongated triangular cupola
• elongated square cupola
• elongated pentagonal cupola
• elongated pentagonal rotunda
• gyroelongated triangular cupola
• gyroelongated square cupola
• gyroelongated pentagonal cupola
• gyroelongated pentagonal rotunda
• gyrobifastigium
• triangular orthobicupola
• square orthobicupola
• square gyrobicupola
• pentagonal orthobicupola
• pentagonal gyrobicupola
• pentagonal orthocupolarotunda
• pentagonal gyrocupolarotunda
• pentagonal orthobirotunda
• elongated triangular orthobicupola
• elongated triangular gyrobicupola
• elongated square gyrobicupola
• elongated pentagonal orthobicupola
• elongated pentagonal gyrobicupola
• elongated pentagonal orthocupolarotunda
• elongated pentagonal gyrocupolarotunda
• elongated pentagonal orthobirotunda
• elongated pentagonal gyrobirotunda
• gyroelongated triangular bicupola
• gyroelongated square bicupola
• gyroelongated pentagonal bicupola
• gyroelongated pentagonal cupolarotunda
• gyroelongated pentagonal birotunda
Augmented prisms
• augmented triangular prism
• biaugmented triangular prism
• triaugmented triangular prism
• augmented pentagonal prism
• biaugmented pentagonal prism
• augmented hexagonal prism
• parabiaugmented hexagonal prism
• metabiaugmented hexagonal prism
• triaugmented hexagonal prism
Modified Platonic solids
• augmented dodecahedron
• parabiaugmented dodecahedron
• metabiaugmented dodecahedron
• triaugmented dodecahedron
• metabidiminished icosahedron
• tridiminished icosahedron
• augmented tridiminished icosahedron
Modified Archimedean solids
• augmented truncated tetrahedron
• augmented truncated cube
• biaugmented truncated cube
• augmented truncated dodecahedron
• parabiaugmented truncated dodecahedron
• metabiaugmented truncated dodecahedron
• triaugmented truncated dodecahedron
• gyrate rhombicosidodecahedron
• parabigyrate rhombicosidodecahedron
• metabigyrate rhombicosidodecahedron
• trigyrate rhombicosidodecahedron
• diminished rhombicosidodecahedron
• paragyrate diminished rhombicosidodecahedron
• metagyrate diminished rhombicosidodecahedron
• bigyrate diminished rhombicosidodecahedron
• parabidiminished rhombicosidodecahedron
• metabidiminished rhombicosidodecahedron
• gyrate bidiminished rhombicosidodecahedron
• tridiminished rhombicosidodecahedron
Elementary solids
• snub disphenoid
• snub square antiprism
• sphenocorona
• augmented sphenocorona
• sphenomegacorona
• hebesphenomegacorona
• disphenocingulum
• bilunabirotunda
• triangular hebesphenorotunda
(See also List of Johnson solids, a sortable table)
| Wikipedia |
Specific interpretation of non-significant test considering power and effect size
Assuming a test where p > alpha and n is large enough for power > 95% at effect size d, what is the exact interpretation of the test regarding the relationship between the observed data, the real effect, and power for d?
Some details: if my p is larger than my alpha, that means the observed data are not surprising from the perspective of a nil-null hypothesis. But according to Cohen (1990, p. 1309), the failed test, in combination with an estimate of power given a d, also allows me to estimate something similar to, but not actually the following statement: based on my sample, the real population effect is likely (where "likely" is somehow related to my 95% power) as close or closer to no effect than the d I have calculated my power for (not the d I have measured). However, I am not aware of a precise definition, and this statement is definitely false since it interprets the data from the perspective of p(H|D) and not p(D|H)…
I am looking for a statement comparable to "given a p value below alpha, assuming a zero effect, observing data as or more extreme than the evaluated sample obtained has a lower probability than alpha", but from the other direction.
One perspective on this comes from a CI: if my CI is narrow (due to high power) and includes zero, I know that only for a small range of hypotheses centred around and including zero, the probability of obtaining the current (or a smaller) measure would be less surprising than my alpha; conversely, for all hypotheses assuming an effect outside of my CI, the data would be surprising. But I am not sure how to phrase this in reference to power.
Admittedly, this question could probably be answered by reading Cohen 1988, however, I do not have the book with me. Also, I assume this problem is commonly enough misunderstood. I would be happy with a pointer to an authoritative source, too.
statistical-significance interpretation power cohens-d
jonajona
I'm not sure why you're bent on using power and alpha in your statement when you already have a CI that's narrow and around 0. From that you can argue that this range of small values close to zero is where you believe the effect likely is and that effects of greater magnitude are unlikely. It's implicitly using your power and alpha, you're just saying it in a much more concise and easily interpretable way.
$\begingroup$ Mostly because I am trying to understand precisely what it means when Cohen writes: "the conclusion is justified that no nontrivial effect exists, at the P = .05 level". I do think a CI of the effect size is probably the superior approach (though on the other hand, power is easily comparable between studies, but the width of a CI is not, isn't it?). $\endgroup$ – jona Jul 27 '13 at 14:07
$\begingroup$ What you will find in every answer to this is that you run into multiple theoretical, and lay, interpretations of all of your words that make a simple statement nigh impossible. Don't bother attempting it. Power is comparable between studies all by itself, but it's meaningfulness is questionable without effect sizes which now is starting to get to a CI. CI's are nice and comparable, that's narrower, wider, captures 0, doesn't capture x, etc. CI's are comparable between studies. There are even inferential rules about it and they're used a lot in metanalysis. $\endgroup$ – John Jul 27 '13 at 17:25
$\begingroup$ Although Bayesian and likelihood approaches are superior in my view, for now I would just ignore the $P$-value and use the 0.95 confidence interval. The confidence interval does not need a null hypothesis to be interpretable. $\endgroup$ – Frank Harrell Jul 28 '13 at 12:05
Part of the difficulty might result from the conflation of the Fisher and Neyman-Pearson approaches to testing.
As far as I understand the problem (and I haven't read the original sources and certainly don't claim to be an expert), interpretation of the p-value and the "data more extreme" and "or something unlikely happened" language comes from Fisher but he has no concept of power and not much to say about experiments that fail to reject the null hypothesis.
On the other hand, in the Neyman-Pearson framework, interpretation is strictly about long-run frequencies and there is really nothing to be said about a single experiment. Under that view, the only thing you can say is that if the effect size were actually d, the test would lead you to accept the null hypothesis at most 5% of the time (that's $\beta$ or 1 - power). Both $\alpha$ and $\beta$ must be set beforehand.
Your description of a significant result feels quite Fisherian, even if it mentions $\alpha$. This would suggest that it is impossible to create a rigorous analogous statement about non-significant results as it requires the concepts of power and type II error ($\beta$), which only completely make sense in the Neyman-Pearson framework.
It's precisely because all this is quite frustrating that it's very difficult to avoid any pseudo-Bayesian interpretation (optionally prefaced with a "that's not quite right" if you want to show some sophistication but still don't know what else to say).
But like I said, I am not sure that I fully grasp all the issues here and I am happy to be corrected!
Some literature on this (not sure how "authoritative" you would consider it):
Hubbard, R. (2004). Alphabet soup: Blurring the distinctions between p's and $\alpha$'s in psychological research. Theory and Psychology, 14 (3), 295-327.
Hubbard, R., & Lindsay, R.M. (2008). Why P values are not a useful measure of evidence in statistical significance testing. Theory and Psychology, 18 (1), 69-88.
One possible way of interpreting any statement about a p-value, significant or not, relies on the definition of a p-value. Suppose, for illustration, that p is 0.6. Then:
If, in the population from which this sample was drawn, the effect size was 0, then we would get test statistics at least as extreme as the one we got in this sample 0.6 of the time.
This does not depend on power; power comes into it in that, if there is high power, then only small effects will be non-significant.
Peter Flom - Reinstate Monica♦Peter Flom - Reinstate Monica
$\begingroup$ I am not satisfied with this solution since it does not apply the additional information contained in the power calculation. Conceptually, power informs us how much our result depends on n; simply looking at the p value, it may just be far from 0 because a small sample is not reliably representative. Phrased differently, high p in a low power study only says that even if the nil-null is true, values different from 0 are not unexpected in small samples such as the given one. Right? $\endgroup$ – jona Jul 27 '13 at 13:35
$\begingroup$ That is part of my answer already: The combination of effect size and significance incorporates power. If a large effect size is not sig., there must be low power; if there is high power and an effect is not significant, it must be small. $\endgroup$ – Peter Flom - Reinstate Monica♦ Jul 27 '13 at 14:06
In a test with b power to detect an effect of size d at alpha = a, if the test does not reject the hypothesis, we may state that the real population effect is possibly less than d. However, power is not an estimate of the reliability of this estimate (high power does not allow one to prove the null).
Rationale: power is defined so that if a test has b power to detect an effect d at alpha = a, b% of experiments that are investigating a phenomenon where the effect in the population is d will result in samples that lead to a successful test. Conversely, (100-b)% of samples drawn from this population will yield tests that do not reject the hypothesis. We are not justified in inferring from the power anything regarding a possible population for which the real effect is smaller than d, because power only relates to what happens when we test samples from a population where the effect is just d. We do still feel somewhat more confident in inferring from a failed test that the effect is below a d for which we have high power than for a d for which we have low power. However, this is only for the reason that we are reducing the chance of failed tests for real effects (type II errors), not because our power for detecting an effect of size d informs us in any way about populations where the effect is not d. In bullet points, there are 2 possible cases for when a test fails to reject:
a population where the effect is d, but we are observing one of the (100-b)% cases where the test fails
a population where the effect is not d
High power reduces the likelihood of the first set, but is unrelated to the likelihood of the second event.
(This answer comes from a thorough explanation in chat by @john which ultimately led me to accept that yes, CIs ARE the answer.)
$\begingroup$ @John do you have the time to join me in chat about this: chat.stackexchange.com/rooms/9858/meaning-of-insignificance $\endgroup$ – jona Jul 28 '13 at 15:21
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# Principles of Business Statistics
(c) Mihai Nica
This work is licensed under a Creative Commons-ShareAlike 4.0 International License Original source: CONNEXIONS
http://cnx.org/content/col10874/1.5/
## Chapter 1 Sampling and Data
### Sampling and Data: Introduction
#### Student Learning Outcomes
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon s.org/licenses/by-sa/4.0/).
## Learning Objectives
- Recognize and differentiate between key terms.
- Apply various types of sampling methods to data collection.
- Create and interpret frequency tables.
#### Introduction
## (c) (i) (2)
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon
s.org/licenses/by-sa/4.0/).
You are probably asking yourself the question, "When and where will I use statistics?". If you read any newspaper or watch television, or use the Internet, you will see statistical information. There are statistics about crime, sports, education, politics, and real estate. Typically, when you read a newspaper article or watch a news program on television, you are given sample information. With this information, you may make a decision about the correctness of a statement, claim, or "fact." Statistical methods can help you make the "best educated guess."
Since you will undoubtedly be given statistical information at some point in your life, you need to know some techniques to analyze the information thoughtfully. Think about buying a house or managing a budget. Think about your chosen profession. The fields of economics, business, psychology, education, biology, law, computer science, police science, and early childhood development require at least one course in statistics.
Included in this chapter are the basic ideas and words of probability and statistics. You will soon understand that statistics and probability work together. You will also learn how data are gathered and what "good" data are.
### Sampling and Data: Statistics
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The science of statistics deals with the collection, analysis, interpretation, and presentation of data. We see and use data in our everyday lives.
#### Optional Collaborative Classroom Exercise
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In your classroom, try this exercise. Have class members write down the average time (in hours, to the nearest half-hour) they sleep per night. Your instructor will record the data. Then create a simple graph (called a dot plot) of the data. A dot plot consists of a number line and dots (or points) positioned above the number line. For example, consider the following data:
$5 ; 5.5 ; 6 ; 6 ; 6 ; 6.5 ; 6.5 ; 6.5 ; 6.5 ; 7 ; 7 ; 8 ; 8 ; 9$
The dot plot for this data would be as follows:
o
o $\mathrm{o}$
o 0 o 0 o
| | 0 | 0 | 0 | 0 | 0 | 0 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 5 | | 6 | | 7 | 8 | 9 |
Figure 1.1 Frequency of Average Time (in Hours) Spent Sleeping per Night
Does your dot plot look the same as or different from the example? Why? If you did the same example in an English class with the same number of students, do you think the results would be the same? Why or why not?
Where do your data appear to cluster? How could you interpret the clustering?
The questions above ask you to analyze and interpret your data. With this example, you have begun your study of statistics.
In this course, you will learn how to organize and summarize data. Organizing and summarizing data is called descriptive statistics. Two ways to summarize data are by graphing and by numbers (for example, fnding an average). After you have studied probability and probability distributions, you will use formal methods for drawing conclusions from "good" data. The formal methods are called inferential statistics. Statistical inference uses probability to determine how confident we can be that the conclusions are correct.
Effective interpretation of data (inference) is based on good procedures for producing data and thoughtful examination of the data. You will encounter what will seem to be too many mathematical formulas for interpreting data. The goal of statistics is not to perform numerous calculations using the formulas, but to gain an understanding of your data. The calculations can be done using a calculator or a computer. The understanding must come from you. If you can thoroughly grasp the basics of statistics, you can be more confident in the decisions you make in life.
#### Levels of Measurement and Statistical Operations
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The way a set of data is measured is called its level of measurement. Correct statistical procedures depend on a researcher being familiar with levels of measurement. Not every statistical operation can be used with every set of data. Data can be classified into four levels of measurement. They are (from lowest to highest level):
- Nominal scale level
- Ordinal scale level
- Interval scale level
- Ratio scale level
Data that is measured using a nominal scale is qualitative. Categories, colors, names, labels and favorite foods along with yes or no responses are examples of nominal level data. Nominal scale data are not ordered. For example, trying to classify people according to their favorite food does not make any sense. Putting pizza first and sushi second is not meaningful.
Smartphone companies are another example of nominal scale data. Some examples are Sony, Motorola, Nokia, Samsung and Apple. This is just a list and there is no agreed upon order. Some people may favor Apple but that is a matter of opinion. Nominal scale data cannot be used in calculations.
Data that is measured using an ordinal scale is similar to nominal scale data but there is a big diference. The ordinal scale data can be ordered. An example of ordinal scale data is a list of the top five national parks in the United States. The top five national parks in the United States can be ranked from one to five but we cannot measure diferences between the data.
Another example using the ordinal scale is a cruise survey where the responses to questions about the cruise are "excellent," "good," "satisfactory" and "unsatisfactory." These responses are ordered from the most desired response by the cruise lines to the least desired. But the diferences between two pieces of data cannot be measured. Like the nominal scale data, ordinal scale data cannot be used in calculations.
Data that is measured using the interval scale is similar to ordinal level data because it has a def inite ordering but there is a diference between data. The diferences between interval scale data can be measured though the data does not have a starting point.
Temperature scales like Celsius (C) and Fahrenheit (F) are measured by using the interval scale. In both temperature measurements, 40 degrees is equal to 100 degrees minus 60 degrees. Diferences make sense. But 0 degrees does not because, in both scales, 0 is not the absolute lowest temperature. Temperatures like $-10^{\circ} \mathrm{F}$ and $-15^{\circ} \mathrm{C}$ exist and are colder than 0 . Interval level data can be used in calculations but one type of comparison cannot be done. Eighty degrees $\mathrm{C}$ is not 4 times as hot as $20^{\circ} \mathrm{C}$ (nor is $80^{\circ} \mathrm{F} 4$ times as hot as $20^{\circ}$ F). There is no meaning to the ratio of 80 to 20 (or 4 to 1 ).
Data that is measured using the ratio scale takes care of the ratio problem and gives you the most information. Ratio scale data is like interval scale data but, in addition, it has a 0 point and ratios can be calculated. For example, four multiple choice statistics fnal exam scores are 80, 68, 20 and 92 (out of a possible 100 points). The exams were machine-graded.
The data can be put in order from lowest to highest: 20, 68, 80, 92 .
The diferences between the data have meaning. The score 92 is more than the score 68 by 24 points.
Ratios can be calculated. The smallest score for ratio data is 0 . So 80 is 4 times 20 . The score of 80 is 4 times better than the score of 20.
## Exercises
What type of measure scale is being used? Nominal, Ordinal, Interval or Ratio.
1. High school men soccer players classified by their athletic ability: Superior, Average, Above average.
2. Baking temperatures for various main dishes: $350,400,325,250$, 300
3. The colors of crayons in a 24-crayon box.
4. Social security numbers.
5. Incomes measured in dollars
6. A satisfaction survey of a social website by number: 1 very satisfied, 2 somewhat satisfied, 3 not satisfied.
7. Political outlook: extreme left, left-of-center, right-of-center, extreme right.
8. Time of day on an analog watch.
9. The distance in miles to the closest grocery store.
10. The dates 1066, 1492, 1644, 1947, 1944 .
11. The heights of 2165 year-old women.
12. Common letter grades A, B, C, D, F.
Answers 1. ordinal, 2. interval, 3. nominal, 4. nominal, 5. ratio, 6. ordinal, 7. nominal, 8. interval, 9. ratio, 10. interval, 11. ratio, 12. ordinal
### Sampling and Data: Key Terms
In statistics, we generally want to study a population. You can think of a population as an entire collection of persons, things, or objects under study. To study the larger population, we select a sample. The idea of sampling is to select a portion (or subset) of the larger population and study that portion (the sample) to gain information about the population. Data are the result of sampling from a population.
Because it takes a lot of time and money to examine an entire population, sampling is a very practical technique. If you wished to compute the overall grade point average at your school, it would make sense to select a sample of students who attend the school. The data collected from the sample would be the students' grade point averages. In presidential elections, opinion poll samples of 1,000 to 2,000 people are taken. The opinion poll is supposed to represent the views of the people in the entire country. Manufacturers of canned carbonated drinks take samples to determine if a 16 ounce can contains 16 ounces of carbonated drink.
From the sample data, we can calculate a statistic. A statistic is a number that is a property of the sample. For example, if we consider one math class to be a sample of the population of all math classes, then the average number of points earned by students in that one math class at the end of the term is an example of a statistic. The statistic is an estimate of a population parameter. A parameter is a number that is a property of the population. Since we considered all math classes to be the population, then the average number of points earned per student over all the math classes is an example of a parameter.
One of the main concerns in the feld of statistics is how accurately a statistic estimates a parameter. The accuracy really depends on how well the sample represents the population. The sample must contain the characteristics of the population in order to be a representative sample. We are interested in both the sample statistic and the population parameter in inferential statistics. In a later chapter, we will use the sample statistic to test the validity of the established population parameter.
A variable, notated by capital letters like $\mathbf{X}$ and $\mathbf{Y}$, is a characteristic of interest for each person or thing in a population. Variables may be numerical or categorical. Numerical variables take on values with equal units such as weight in pounds and time in hours. Categorical variables place the person or thing into a category. If we let $\mathbf{X}$ equal the number of points earned by one math student at the end of a term, then $\mathbf{X}$ is a numerical variable. If we let $\mathbf{Y}$ be a person's party afliation, then examples of $\mathbf{Y}$ include Republican, Democrat, and Independent. $Y$ is a categorical variable. We could do some math with values of $\mathbf{X}$ (calculate the average number of points earned, for example), but it makes no sense to do math with values of $Y$ (calculating an average party afliation makes no sense). Data are the actual values of the variable. They may be numbers or they may be words. Datum is a single value.
Two words that come up often in statistics are mean and proportion. If you were to take three exams in your math classes and obtained scores of 86, 75, and 92, you calculate your mean score by adding the three exam scores and dividing by three (your mean score would be 84.3 to one decimal place). If, in your math class, there are 40 students and 22 are men and 18 are women, then the proportion of men students is $\frac{22}{44}$ and the proportion of women students is $\frac{18}{40}$. Mean and proportion are discussed in more detail in later chapters. Note: The words "mean" and "average" are often used interchangeably. The substitution of one word for the other is common practice. The technical term is "arithmetic mean" and "average" is technically a center location. However, in practice among non-statisticians, "average" is commonly accepted for "arithmetic mean."
## Example
Define the key terms from the following study: We want to know the average (mean) amount of money first year college students spend at ABC College on school supplies that do not include books. We randomly survey 100 first year students at the college. Three of those students spent \\$150, \\$200, and \\$225, respectively.
## Solution
The population is all first year students attending ABC College this term. The sample could be all students enrolled in one section of a beginning statistics course at ABC College (although this sample may not represent the entire population).
The parameter is the average (mean) amount of money spent (excluding books) by first year college students at ABC College this term.
The statistic is the average (mean) amount of money spent (excluding books) by first year college students in the sample. The variable could be the amount of money spent (excluding books) by one first year student. Let $\mathrm{X}=$ the amount of money spent (excluding books) by one first year student attending ABC College. The data are the dollar amounts spent by the first year students. Examples of the data are \\$150, \\$200, and \\$225.
#### Optional Collaborative Classroom Exercise
Do the following exercise collaboratively with up to four people per group. Find a population, a sample, the parameter, the statistic, a variable, and data for the following study: You want to determine the average (mean) number of glasses of milk college students drink per day. Suppose yesterday, in your English class, you asked five students how many glasses of milk they drank the day before. The answers were $1,0,1,3$, and 4 glasses of milk.
### Sampling and Data: Data
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Data may come from a population or from a sample. Small letters like $x$ or $y$ generally are used to represent data values. Most data can be put into the following categories:
- Qualitative
- Quantitative
Qualitative data are the result of categorizing or describing attributes of a population. Hair color, blood type, ethnic group, the car a person drives, and the street a person lives on are examples of qualitative data. Qualitative data are generally described by words or letters. For instance, hair color might be black, dark brown, light brown, blonde, gray, or red. Blood type might be AB+, O-, or B+. Researchers often prefer to use quantitative data over qualitative data because it lends itself more easily to mathematical analysis. For example, it does not make sense to find an average hair color or blood type.
Quantitative data are always numbers. Quantitative data are the result of counting or measuring attributes of a population. Amount of money, pulse rate, weight, number of people living in your town, and the number of students who take statistics are examples of quantitative data. Quantitative data may be either discrete or continuous.
All data that are the result of counting are called quantitative discrete data. These data take on only certain numerical values. If you count the number of phone calls you receive for each day of the week, you might get 0,1 , 2, 3, etc.
All data that are the result of measuring are quantitative continuous data assuming that we can measure accurately. Measuring angles in radians might result in the numbers $\frac{\pi}{6}, \frac{\pi}{3}, \frac{\pi}{2}, \pi, \frac{3 \pi}{4}$, etc. If you and your friends carry backpacks with books in them to school, the numbers of books in the backpacks are discrete data and the weights of the backpacks are continuous data.
Note: In this course, the data used is mainly quantitative. It is easy to calculate statistics (like the mean or proportion) from numbers. In the chapter Descriptive Statistics, you will be introduced to stem plots, histograms and box plots all of which display quantitative data. Qualitative data is discussed at the end of this section through graphs.
## Example 1.2: Data Sample of Quantitative Discrete Data
The data are the number of books students carry in their backpacks. You sample five students. Two students carry 3 books, one student carries 4 books, one student carries 2 books, and one student carries 1 book. The numbers of books $(3,4,2$, and 1$)$ are the quantitative discrete data.
## Example 1.3: Data Sample of Quantitative Continuous Data
The data are the weights of the backpacks with the books in it. You sample the same five students. The weights (in pounds) of their backpacks are 6.2, 7, 6.8, 9.1, 4.3. Notice that backpacks carrying three books can have different weights. Weights are quantitative continuous data because weights are measured.
## Example 1.4: Data Sample of Qualitative Data
The data are the colors of backpacks. Again, you sample the same five students. One student has a red backpack, two students have black backpacks, one student has a green backpack, and one student has a gray backpack. The colors red, black, black, green, and gray are qualitative data.
Note: You may collect data as numbers and report it categorically. For example, the quiz scores for each student are recorded throughout the term. At the end of the term, the quiz scores are reported as A, B, C, D, or F.
## Example 1.5
Work collaboratively to determine the correct data type (quantitative or qualitative). Indicate whether quantitative data are continuous or discrete. Hint: Data that are discrete often start with the words "the number of."
1. The number of pairs of shoes you own.
2. The type of car you drive.
3. Where you go on vacation.
4. The distance it is from your home to the nearest grocery store.
5. The number of classes you take per school year.
6. The tuition for your classes
7. The type of calculator you use.
8. Movie ratings.
9. Political party preferences. 10. Weight of sumo wrestlers.
10. Amount of money won playing poker.
11. Number of correct answers on a quiz.
12. Peoples' attitudes toward the government.
13. IQ scores. (This may cause some discussion.)
## Qualitative Data Discussion
Below are tables of part-time vs full-time students at De Anza College in Cupertino, CA and Foothill College in Los Altos, CA for the Spring 2010 quarter. The tables display counts (frequencies) and percentages or proportions (relative frequencies). The percent columns make comparing the same categories in the colleges easier.
Displaying percentages along with the numbers is often helpful, but it is particularly important when comparing sets of data that do not have the same totals, such as the total enrollments for both colleges in this example. Notice how much larger the percentage for part-time students at Foothill College is compared to De Anza College.
| | Number | Percent |
| :--- | :--- | :--- |
| Full-time | 9,200 | $40.9 \%$ |
| Part-time | 13,296 | $59.1 \%$ |
| Total | 22,496 | $100 \%$ |
Table 1.1 De Anza College
| | Number | Percent |
| :--- | :--- | :--- |
| Full-time | 4,059 | $28.6 \%$ |
| Part-time | 10,124 | $71.4 \%$ |
| Total | 14,183 | $100 \%$ |
Table 1.2 Foothill College
Tables are a good way of organizing and displaying data. But graphs can be even more helpful in understanding the data. There are no strict rules concerning what graphs to use. Below are pie charts and bar graphs, two graphs that are used to display qualitative data.
In a pie chart, categories of data are represented by wedges in the circle and are proportional in size to the percent of individuals in each category. In a bar graph, the length of the bar for each category is proportional to the number or percent of individuals in each category. Bars may be vertical or horizontal.
A Pareto chart consists of bars that are sorted into order by category size (largest to smallest).
Look at the graphs and determine which graph (pie or bar) you think displays the comparisons better. This is a matter of preference.
It is a good idea to look at a variety of graphs to see which is the most helpful in displaying the data. We might make different choices of what we think is the "best" graph depending on the data and the context. Our choice also depends on what we are using the data for.
Figure 1.2 Pie Chart
Figure 1.3 Bar Chart
## Percentages That Add to More (or Less) Than 100\%
Sometimes percentages add up to be more than $100 \%$ (or less than $100 \%$ ). In the graph, the percentages add to more than $100 \%$ because students can be in more than one category. A bar graph is appropriate to compare the relative size of the categories. A pie chart cannot be used. It also could not be used if the percentages added to less than $100 \%$.
| Characteristic/Category | Percent |
| :--- | :---: |
| Full-time Students | $40.9 \%$ |
| $\begin{array}{l}\text { Students who intend to transfer to a 4-year } \\ \text { educational institution }\end{array}$ | $48.6 \%$ |
| Students under age 25 | $61.0 \%$ |
| TOTAL | $150.5 \%$ |
Table 1.3 De Anza College Spring 2010
Figure 1.4 De Anza College Spring 2010
## Omitting Categories/Missing Data
The table displays Ethnicity of Students but is missing the "Other/Unknown" category. This category contains people who did not feel they fit into any of the ethnicity categories or declined to respond. Notice that the frequencies do not add up to the total number of students. Create a bar graph and not a pie chart.
| | Frequency | Percent |
| :--- | :--- | :--- |
| Asian | 8,794 | $36.1 \%$ |
| Black | 1,412 | $5.8 \%$ |
| Filipino | 1,298 | $5.3 \%$ |
Table 1.4 Missing Data: Ethnicity of Students De Anza College Fall Term 2007 (Census Day)
| | Frequency | Percent |
| :--- | :--- | :--- |
| Hispanic | 4,180 | $17.1 \%$ |
| Native American | 146 | $0.6 \%$ |
| Pacifc Islander | 236 | $1.0 \%$ |
| White | 5,978 | $24.5 \%$ |
| | | |
| TOTAL | 22,044 out of 24,382 | $90.4 \%$ out of $100 \%$ |
Table 1.4 Missing Data: Ethnicity of Students De Anza College Fall Term 2007 (Census Day)
Figure 1.5 Bar graph Without Other/Unknown Category
The following graph is the same as the previous graph but the "Other/Unknown" percent (9.6\%) has been added back in. The "Other/Unknown" category is large compared to some of the other categories (Native American, 0.6\%, Pacifc Islander $1.0 \%$ particularly). This is important to know when we think about what the data are telling us.
This particular bar graph can be hard to understand visually. The graph below it is a Pareto chart. The Pareto chart has the bars sorted from largest to smallest and is easier to read and interpret.
Figure 1.6 Bar Graph With Other/Unknown Category
Figure 1.7 Pareto Chart With Bars Sorted By Size
## Pie Charts: No Missing Data
The following pie charts have the "Other/Unknown" category added back in (since the percentages must add to 100\%). The chart on the right is organized having the wedges by size and makes for a more visually informative graph than the unsorted, alphabetical graph on the left.
Figure 1.8 Pie Charts with "Other/Unknown" category
### Sampling and Data: Variation and Critical Evaluation
#### Variation in Data
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Variation is present in any set of data. For example, 16-ounce cans of beverage may contain more or less than 16 ounces of liquid. In one study, eight 16 ounce cans were measured and produced the following amount (in ounces) of beverage:
$15.8 ; 16.1 ; 15.2 ; 14.8 ; 15.8 ; 15.9 ; 16.0 ; 15.5$
Measurements of the amount of beverage in a 16-ounce can may vary because different people make the measurements or because the exact amount, 16 ounces of liquid, was not put into the cans. Manufacturers regularly run tests to determine if the amount of beverage in a 16-ounce can falls within the desired range.
Be aware that as you take data, your data may vary somewhat from the data someone else is taking for the same purpose. This is completely natural. However, if two or more of you are taking the same data and get very different results, it is time for you and the others to reevaluate your data-taking methods and your accuracy.
#### Variation in Samples
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It was mentioned previously that two or more samples from the same population, taken randomly, and having close to the same characteristics of the population are different from each other. Suppose Doreen and Jung both decide to study the average amount of time students at their college sleep each night. Doreen and Jung each take samples of 500 students. Doreen uses systematic sampling and Jung uses cluster sampling. Doreen's sample will be different from Jung's sample. Even if Doreen and Jung used the same sampling method, in all likelihood their samples would be diferent. Neither would be wrong, however.
Think about what contributes to making Doreen's and Jung's samples diferent.
If Doreen and Jung took larger samples (i.e. the number of data values is increased), their sample results (the average amount of time a student sleeps) might be closer to the actual population average. But still, their samples would be, in all likelihood, different from each other. This variability in samples cannot be stressed enough.
#### Size of a Sample
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The size of a sample (often called the number of observations) is important. The examples you have seen in this book so far have been small. Samples of only a few hundred observations, or even smaller, are sufcient for many purposes. In polling, samples that are from 1200 to 1500 observations are considered large enough and good enough if the survey is random and is well done. You will learn why when you study confidence intervals.
Be aware that many large samples are biased. For example, call-in surveys are invariable biased because people choose to respond or not.
#### Optional Collaborative Classroom Exercise
## Exercise 1.5.1
Divide into groups of two, three, or four. Your instructor will give each group one 6-sided die. Try this experiment twice. Roll one fair die (6-sided) 20 times. Record the number of ones, twos, threes, fours, fves, and sixes you get below ("frequency" is the number of times a particular face of the die occurs):
| Face on Die | Frequency |
| :--- | :--- |
| 1 | |
| 2 | |
| 3 | |
| 4 | |
| 5 | |
| 6 | |
Table 1.5 First Experiment (20 rolls)
| Face on Die | Frequency |
| :--- | :--- |
| 1 | |
| 2 | |
Table 1.6 Second Experiment (20 rolls)
Table 1.6 Second Experiment (20 rolls)
Did the two experiments have the same results? Probably not. If you did the experiment a third time, do you expect the results to be identical to the first or second experiment? (Answer yes or no.) Why or why not?
Which experiment had the correct results? They both did. The job of the statistician is to see through the variability and draw appropriate conclusions.
#### Critical Evaluation
We need to critically evaluate the statistical studies we read about and analyze before accepting the results of the study. Common problems to be aware of include
- Problems with Samples: A sample should be representative of the population. A sample that is not representative of the population is biased. Biased samples that are not representative of the population give results that are inaccurate and not valid.
- Self-Selected Samples: Responses only by people who choose to respond, such as call-in surveys are often unreliable.
- Sample Size Issues: Samples that are too small may be unreliable. Larger samples are better if possible. In some situations, small samples are unavoidable and can still be used to draw conclusions, even though larger samples are better.
Examples: Crash testing cars, medical testing for rare conditions.
- Undue infuence: Collecting data or asking questions in a way that infuences the response.
- Non-response or refusal of subject to participate: The collected responses may no longer be representative of the population. Often, people with strong positive or negative opinions may answer surveys, which can afect the results. - Causality: A relationship between two variables does not mean that one causes the other to occur. They may both be related (correlated) because of their relationship through a different variable.
- Self-Funded or Self-Interest Studies: A study performed by a person or organization in order to support their claim. Is the study impartial? Read the study carefully to evaluate the work. Do not automatically assume that the study is good but do not automatically assume the study is bad either. Evaluate it on its merits and the work done.
- Misleading Use of Data: Improperly displayed graphs, incomplete data, lack of context.
- Confounding: When the effects of multiple factors on a response cannot be separated. Confounding makes it difficult or impossible to draw valid conclusions about the effect of each factor.
### Sampling and Data: Frequency Relative Frequency and Cumulative Frequency
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Twenty students were asked how many hours they worked per day. Their responses, in hours, are listed below:
$5 ; 6 ; 3 ; 3 ; 2 ; 4 ; 7 ; 5 ; 2 ; 3 ; 5 ; 6 ; 5 ; 4 ; 4 ; 3 ; 5 ; 2 ; 5 ; 3$
Below is a frequency table listing the different data values in ascending order and their frequencies.
| DATA VALUE | FREQUENCY |
| :--- | :--- |
| 2 | 3 |
| 3 | 5 |
| 4 | 3 |
| 5 | 6 |
| 6 | 2 |
| 7 | 1 |
Table 1.7 Frequency Table of Student Work Hours
A frequency is the number of times a given datum occurs in a data set. According to the table above, there are three students who work 2 hours, five students who work 3 hours, etc. The total of the frequency column, 20, represents the total number of students included in the sample.
A relative frequency is the fraction or proportion of times an answer occurs. To find the relative frequencies, divide each frequency by the total number of students in the sample - in this case, 20. Relative frequencies can be written as fractions, percents, or decimals.
| DATA VALUE | FREQUENCY | RELATIVE FREQUENCY |
| :--- | :--- | :--- |
| 2 | 3 | $\frac{3}{5}$ or 0.15 |
| 3 | 5 | $\frac{5}{20}$ or 0.25 |
| 4 | 3 | $\frac{3}{20}$ or 0.15 |
| 5 | 6 | $\frac{6}{20}$ or 0.30 |
| 6 | 2 | $\frac{2}{20}$ or 0.10 |
| 7 | 1 | $\frac{1}{20}$ or 0.05 |
Table 1.8 Frequency Table of Student Work Hours w/ Relative Frequency
The sum of the relative frequency column is $\frac{20}{20}$ or 1 .
Cumulative relative frequency is the accumulation of the previous relative frequencies. To find the cumulative relative frequencies, add all the previous relative frequencies to the relative frequency for the current row.
| $\begin{array}{l}\text { DATA } \\ \text { VALUE }\end{array}$ | FREQUENCY | $\begin{array}{l}\text { RELATIVE } \\ \text { FREQUENCY }\end{array}$ | $\begin{array}{l}\text { CUMULATIVE } \\ \text { RELATIVE FREQUENCY }\end{array}$ |
| :--- | :--- | :--- | :--- |
| 2 | 3 | $\frac{3}{5}$ or 0.15 | 0.15 |
| 3 | 5 | $\frac{5}{20}$ or 0.25 | $0.15+0.25=0.40$ |
| 4 | 3 | $\frac{3}{20}$ or 0.15 | $0.40+0.15=0.55$ |
| 5 | 6 | $\frac{6}{20}$ or 0.30 | $0.55+0.30=0.85$ |
Table 1.9 Frequency Table of Student Work Hours w/ Relative and Cumulative Relative Frequency
| $\begin{array}{l}\text { DATA } \\ \text { VALUE }\end{array}$ | FREQUENCY | $\begin{array}{l}\text { RELATIVE } \\ \text { FREQUENCY }\end{array}$ | $\begin{array}{l}\text { CUMULATIVE } \\ \text { RELATIVE FREQUENCY }\end{array}$ |
| :--- | :--- | :--- | :--- |
| 6 | 2 | $\frac{2}{20}$ or 0.10 | $0.85+0.10=0.95$ |
| 7 | 1 | $\frac{1}{20}$ or 0.05 | $0.95+0.05=1.00$ |
Table 1.9 Frequency Table of Student Work Hours w/ Relative and Cumulative Relative Frequency
The last entry of the cumulative relative frequency column is one, indicating that one hundred percent of the data has been accumulated.
Note: Because of rounding, the relative frequency column may not always sum to one and the last entry in the cumulative relative frequency column may not be one. However, they each should be close to one.
The following table represents the heights, in inches, of a sample of 100 male semiprofessional soccer players.
| $\begin{array}{l}\text { HEIGHT } \\ \text { (INCHES) }\end{array}$ | FREQUENCY | $\begin{array}{l}\text { RELATIVE } \\ \text { FREQUENCY }\end{array}$ | $\begin{array}{l}\text { CUMULATIVE } \\ \text { RELATIVE } \\ \text { FREQUENCY }\end{array}$ |
| :---: | :---: | :---: | :---: |
| $59.95-61.95$ | 5 | $\frac{5}{100}=0.05$ | 0.05 |
| $61.95-63.95$ | 3 | $\frac{3}{100}=0.03$ | $0.05+0.03=0.08$ |
| $63.95-65.95$ | 15 | $\frac{15}{100}=0.15$ | $0.08+0.15=0.23$ |
| $\begin{array}{l}65.95- \\ 67.95\end{array}$ | 40 | $\frac{40}{100}=0.40$ | $0.23+0.40=0.63$ |
| $\begin{array}{l}67.95- \\ 69.95\end{array}$ | 17 | $\frac{17}{100}=0.17$ | $0.63+0.17=0.80$ |
| $69.95-71.95$ | 12 | $\frac{12}{100}=0.12$ | $0.80+0.12=0.92$ |
| $71.95-73.95$ | 7 | $\frac{7}{100}=0.07$ | $0.92+0.07=0.99$ |
| $73.95-75.95$ | 1 | $\frac{1}{100}=0.01$ | $0.99+0.01=1.00$ |
| | Total $=100$ | Total $=\mathbf{1 . 0 0}$ | |
Table 1.10 Frequency Table of Soccer Player Height
The data in this table has been grouped into the following intervals: - 59.95 - 61.95 inches
- 61.95 - 63.95 inches
- 63.95 - 65.95 inches
- 65.95 - 67.95 inches
- 67.95 - 69.95 inches
- 69.95 - 71.95 inches
- 71.95 - 73.95 inches
- 73.95 - 75.95 inches
Note: This example is used again in the Descriptive Statistics (Section 2.1) chapter, where the method used to compute the intervals will be explained.
In this sample, there are $\mathbf{5}$ players whose heights are between 59.95 - 61.95 inches, $\mathbf{3}$ players whose heights fall within the interval 61.95 - 63.95 inches, 15 players whose heights fall within the interval 63.95 - 65.95 inches, 40 players whose heights fall within the interval 65.95 - 67.95 inches, 17 players whose heights fall within the interval 67.95 - 69.95 inches, 12 players whose heights fall within the interval 69.95 - 71.95, 7 players whose height falls within the interval 71.95 - 73.95, and 1 player whose height falls within the interval 73.95 - 75.95. All heights fall between the endpoints of an interval and not at the endpoints.
## Example 1.6
From the table, find the percentage of heights that are less than 65.95 inches.
## Solution
If you look at the first, second, and third rows, the heights are all less than 65.95 inches. There are $5+3+15=23$ males whose heights are less than 65.95 inches. The percentage of heights less than 65.95 inches is then $\frac{23}{100}$ or $23 \%$. This percentage is the cumulative relative frequency entry in the third row.
## Example 1.7
From the table, find the percentage of heights that fall between 61.95 and 65.95 inches.
## Solution
Add the relative frequencies in the second and third rows: $0.03+0.15$ $=0.18$ or $18 \%$.
## Example 1.8
Use the table of heights of the 100 male semiprofessional soccer players. Fill in the blanks and check your answers.
1. The percentage of heights that are from 67.95 to 71.95 inches is:
2. The percentage of heights that are from 67.95 to 73.95 inches is:
3. The percentage of heights that are more than 65.95 inches is:
4. The number of players in the sample who are between 61.95 and 71.95 inches tall is:
5. What kind of data are the heights?
6. Describe how you could gather this data (the heights) so that the data are characteristic of all male semiprofessional soccer players.
Remember, you count frequencies. To find the relative frequency, divide the frequency by the total number of data values. To find the cumulative relative frequency, add all of the previous relative frequencies to the relative frequency for the current row.
#### Optional Collaborative Classroom Exercise
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon s.org/licenses/by-sa/4.0/).
## Exercise 1.6.1
In your class, have someone conduct a survey of the number of siblings (brothers and sisters) each student has. Create a frequency table. Add to it a relative frequency column and a cumulative relative frequency column. Answer the following questions:
1. What percentage of the students in your class has o siblings?
2. What percentage of the students has from 1 to 3 siblings?
3. What percentage of the students has fewer than 3 siblings?
## Example 1.9
Nineteen people were asked how many miles, to the nearest mile they commute to work each day.
The data are as follows: $2 ; 5 ; 7 ; 3 ; 2 ; 10 ; 18 ; 15 ; 20 ; 7 ; 10 ; 18 ; 5 ; 12 ; 13$; $12 ; 4 ; 5 ; 10$
The following table was produced:
| DATA | FREQUENCY | $\begin{array}{l}\text { RELATIVE } \\ \text { FREQUENCY }\end{array}$ | $\begin{array}{l}\text { CUMULATIVE } \\ \text { RELATIVE } \\ \text { FREQUENCY }\end{array}$ |
| :--- | :--- | :--- | :--- |
Table 1.11 Frequency of Commuting Distances
#### Solutions to Exercises in Chapter 1
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon s.org/licenses/by-sa/4.0/).
## Solution to Example 1.5, Problem
Items 1, 5, 11, and 12 are quantitative discrete; items 4, 6, 10, and 14 are quantitative continuous; and items $2,3,7,8,9$, and 13 are qualitative.
## Solution to Example 1.8, Problem
1. $29 \%$
2. $36 \%$ 3. $77 \%$
3. 87
4. quantitative continuous
5. get rosters from each team and choose a simple random sample from each
## Solution to Example 1.9, Problem
No. Frequency column sums to 18 , not 19 . Not all cumulative relative frequencies are correct.
1. False. Frequency for 3 miles should be 1 ; for 2 miles (left out), 2. Cumulative relative frequency column should read: $0.1052,0.1579,0.2105,0.3684,0.4737$, $0.6316,0.7368,0.7895,0.8421,0.9474,1$.
2. $\frac{\frac{5}{19}}{\frac{7}{19}}, \frac{12}{19}, \frac{7}{19}$
## Chapter 2 Descriptive Statistics
### Descriptive Statistics: Introduction
#### Student Learning Outcomes
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon s.org/licenses/by-sa/4.0/).
## Learning Objectives
By the end of this chapter, the student should be able to:
- Display data graphically and interpret graphs: stemplots, histograms and boxplots.
- Recognize, describe, and calculate the measures of location of data: quartiles and percentiles.
- Recognize, describe, and calculate the measures of the center of data: mean, median, and mode.
- Recognize, describe, and calculate the measures of the spread of data: variance, standard deviation, and range.
#### Introduction
(c) (i) (2)
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon
s.org/licenses/by-sa/4.0/).
Once you have collected data, what will you do with it? Data can be described and presented in many different formats. For example, suppose you are interested in buying a house in a particular area. You may have no clue about the house prices, so you might ask your real estate agent to give you a sample data set of prices. Looking at all the prices in the sample often is overwhelming. A better way might be to look at the median price and the variation of prices. The median and variation are just two ways that you will learn to describe data. Your agent might also provide you with a graph of the data.
In this chapter, you will study numerical and graphical ways to describe and display your data. This area of statistics is called "Descriptive Statistics". You will learn to calculate, and even more importantly, to interpret these measurements and graphs.
### Descriptive Statistics: Displaying Data
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon s.org/licenses/by-sa/4.0/).
A statistical graph is a tool that helps you learn about the shape or distribution of a sample. The graph can be a more effective way of presenting data than a mass of numbers because we can see where data clusters and where there are only a few data values. Newspapers and the Internet use graphs to show trends and to enable readers to compare facts and figures quickly.
Statisticians often graph data first to get a picture of the data. Then, more formal tools may be applied.
Some of the types of graphs that are used to summarize and organize data are the dot plot, the bar chart, the histogram, the stem-and-leaf plot, the frequency polygon (a type of broken line graph), pie charts, and the boxplot. In this chapter, we will briefy look at stem-and-leaf plots, line graphs and bar graphs. Our emphasis will be on histograms and boxplots.
### Descriptive Statistics: Histogram
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon s.org/licenses/by-sa/4.0/).
For most of the work you do in this book, you will use a histogram to display the data. One advantage of a histogram is that it can readily display large data sets. A rule of thumb is to use a histogram when the data set consists of 100 values or more.
A histogram consists of contiguous boxes. It has both a horizontal axis and a vertical axis. The horizontal axis is labeled with what the data represents (for instance, distance from your home to school). The vertical axis is labeled either Frequency or relative frequency. The graph will have the same shape with either label. The histogram (like the stemplot) can give you the shape of the data, the center, and the spread of the data. (The next section tells you how to calculate the center and the spread.)
The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample. (In the chapter on Sampling and Data (Section 1.1), we defined frequency as the number of times an answer occurs.) If:
- $f=$ frequency
- $\mathrm{n}=$ total number of data values (or the sum of the individual frequencies), and
- $\mathrm{RF}=$ relative frequency,
then:
$$
R F=\frac{f}{n}
$$
For example, if 3 students in Mr. Ahab's English class of 40 students received from $90 \%$ to $100 \%$, then,
$$
f=3, n=40, \text { andRF }=\frac{f}{n}=\frac{3}{40}=0.075
$$
Seven and a half percent of the students received $90 \%$ to $100 \%$. Ninety percent to $100 \%$ are quantitative measures. To construct a histogram, first decide how many bars or intervals, also called classes, represent the data. Many histograms consist of from 5 to 15 bars or classes for clarity. Choose a starting point for the first interval to be less than the smallest data value. A convenient starting point is a lower value carried out to one more decimal place than the value with the most decimal places. For example, if the value with the most decimal places is 6.1 and this is the smallest value, a convenient starting point is 6.05 (6.1 $-0.05=6.05)$. We say that 6.05 has more precision. If the value with the most decimal places is 2.23 and the lowest value is 1.5 , a convenient starting point is 1.495 (1.5 - $0.005=1.495)$. If the value with the most decimal places is 3.234 and the lowest value is 1.0 , a convenient starting point is $0.9995(1.0-.0005=0.9995)$. If all the data happen to be integers and the smallest value is 2 , then a convenient starting point is $1.5(2-0.5=1.5)$. Also, when the starting point and other boundaries are carried to one additional decimal place, no data value will fall on a boundary.
## Example 2.1
The following data are the heights (in inches to the nearest half inch) of 100 male semiprofessional soccer players. The heights are continuous data since height is measured.
$60 ; 60.5 ; 61 ; 61 ; 61.5$
$63.5 ; 63.5 ; 63.5$
$64 ; 64 ; 64 ; 64 ; 64 ; 64 ; 64 ; 64.5 ; 64.5 ; 64.5 ; 64.5 ; 64.5 ; 64.5 ; 64.5 ; 64.5$ $66 ; 66 ; 66 ; 66 ; 66 ; 66 ; 66 ; 66 ; 66 ; 66 ; 66.5 ; 66.5 ; 66.5 ; 66.5 ; 66.5 ; 66.5$; $66.5 ; 66.5 ; 66.5 ; 66.5 ; 66.5$;
$67 ; 67 ; 67 ; 67 ; 67 ; 67 ; 67 ; 67 ; 67 ; 67 ; 67 ; 67 ; 67.5 ; 67.5 ; 67.5 ; 67.5$; $67.5 ; 67.5 ; 67.5$
$68 ; 68 ; 69 ; 69 ; 69 ; 69 ; 69 ; 69 ; 69 ; 69 ; 69 ; 69 ; 69.5 ; 69.5 ; 69.5 ; 69.5 ; 69.5$
70; 70; 70; 70; 70; 70; 70.5; 70.5; 70.5; 71; 71; 71
$72 ; 72 ; 72 ; 72.5 ; 72.5 ; 73 ; 73.5$
74
The smallest data value is 60 . Since the data with the most decimal places has one decimal (for instance, 61.5), we want our starting point to have two decimal places. Since the numbers $0.5,0.05,0.005$, etc. are convenient numbers, use 0.05 and subtract it from 60 , the smallest value, for the convenient starting point.
$60-0.05=59.95$ which is more precise than, say, 61.5 by one decimal place. The starting point is, then, 59.95 .
The largest value is $74.74+0.05=74.05$ is the ending value.
Next, calculate the width of each bar or class interval. To calculate this width, subtract the starting point from the ending value and divide by the number of bars (you must choose the number of bars you desire). Suppose you choose 8 bars.
$$
\frac{74.05-59.95}{8}=1.76
$$
Note: We will round up to 2 and make each bar or class interval 2 units wide. Rounding up to 2 is one way to prevent a value from falling on a boundary. Rounding to the next number is necessary even if it goes against the standard rules of rounding. For this example, using 1.76 as the width would also work.
The boundaries are:
- 59.95
- $59.95+2=61.95$
- $61.95+2=63.95$
- $63.95+2=65.95$
- $65.95+2=67.95$
- $67.95+2=69.95$
- $69.95+2=71.95$
- $71.95+2=73.95$
- $73.95+2=75.95$
The heights 60 through 61.5 inches are in the interval 59.95-61.95. The heights that are 63.5 are in the interval $61.95-63.95$. The heights that are 64 through 64.5 are in the interval $63.95-65.95$. The heights 66 through 67.5 are in the interval $65.95-67.95$. The heights 68 through 69.5 are in the interval 67.95-69.95. The heights 70 through 71 are in the interval $69.95-71.95$. The heights 72 through 73.5 are in the interval 71.95 - 73.95. The height 74 is in the interval 73.95 75.95 .
The following histogram displays the heights on the $x$-axis and relative frequency on the $\mathrm{y}$-axis.
## Relative
Frequency
Heights
## Example 2.2
The following data are the number of books bought by 50 part-time college students at ABC College. The number of books is discrete data since books are counted.
$1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1$
$2 ; 2 ; 2 ; 2 ; 2 ; 2 ; 2 ; 2 ; 2 ; 2$
$3 ; 3 ; 3 ; 3 ; 3 ; 3 ; 3 ; 3 ; 3 ; 3 ; 3 ; 3 ; 3 ; 3 ; 3 ; 3$
$4 ; 4 ; 4 ; 4 ; 4 ; 4$
$5 ; 5 ; 5 ; 5 ; 5$
$6 ; 6$
Eleven students buy 1 book. Ten students buy 2 books. Sixteen students buy 3 books. Six students buy 4 books. Five students buy 5 books. Two students buy 6 books.
Because the data are integers, subtract 0.5 from 1 , the smallest data value and add 0.5 to 6 , the largest data value. Then the starting point is 0.5 and the ending value is 6.5 .
## Problem
Next, calculate the width of each bar or class interval. If the data are discrete and there are not too many different values, a width that places the data values in the middle of the bar or class interval is the most convenient. Since the data consist of the numbers 1, 2, 3, 4, 5, 6 and the starting point is 0.5 , a width of one places the 1 in the middle of the interval from 0.5 to 1.5 , the 2 in the middle of the interval from 1.5 to 2.5 , the 3 in the middle of the interval from 2.5 to 3.5 , the 4 in the middle of the interval from middle of the interval from to , the 5 in the of the interval from to , and the in the middle
Calculate the number of bars as follows:
$$
\frac{6.5-0.5}{\text { bars }}=1
$$
where 1 is the width of a bar. Therefore, bars $=6$.
The following histogram displays the number of books on the $\mathrm{x}$-axis and the frequency on the $y$-axis.
## Using the $\mathrm{TI}-83,83+, 84,84+$ Calculator Instructions
Go to the Appendix (14:Appendix) in the menu on the left. There are calculator instructions for entering data and for creating a customized histogram. Create the histogram for Example 2.
- Press $Y=$. Press CLEAR to clear out any equations.
- Press STAT 1:EDIT. If L1 has data in it, arrow up into the name L1, press CLEAR and arrow down. If necessary, do the same for $L 2$.
- Into L1, enter 1, 2, 3, 4, 5, 6
- Into L2, enter 11, 10, 16, 6, 5, 2
- Press WINDOW. Make $\mathrm{Xmin}=.5, \mathrm{X} \max =6.5, \mathrm{Xscl}=(6.5-.5) / 6, \mathrm{Ymin}=-1, \mathrm{Ymax}=$ 20, $\mathrm{Yscl}=1$, Xres $=1$
- Press 2nd $Y=$. Start by pressing 4:Plotsof ENTER.
- Press 2nd Y =. Press 1:Plot1. Press ENTER. Arrow down to TYPE. Arrow to the 3rd picture (histogram). Press ENTER.
- Arrow down to Xlist: Enter L1 (2nd 1). Arrow down to Freq. Enter L2 (2nd 2).
- Press GRAPH
- Use the TRACE key and the arrow keys to examine the histogram.
#### Optional Collaborative Exercise
## (c) (i) (2)
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon
s.org/licenses/by-sa/4.0/).
Count the money (bills and change) in your pocket or purse. Your instructor will record the amounts. As a class, construct a histogram displaying the data. Discuss how many intervals you think is appropriate. You may want to experiment with the number of intervals. Discuss, also, the shape of the histogram.
Record the data, in dollars (for example, 1.25 dollars).
Construct a histogram.
### Descriptive Statistics: Measuring the Center of the Data
## (c) (i) (0)
s.org/licenses/by-sa/4.0/).
The "center" of a data set is also a way of describing location. The two most widely used measures of the "center" of the data are the mean (average) and the median. To calculate the mean weight of 50 people, add the 50 weights together and divide by 50. To find the median weight of the 50 people, order the data and find the number that splits the data into two equal parts (previously discussed under box plots in this chapter). The median is generally a better measure of the center when there are extreme values or outliers because it is not affected by the precise numerical values of the outliers. The mean is the most common measure of the center.
Note: The words "mean" and "average" are often used interchangeably. The substitution of one word for the other is common practice. The technical term is "arithmetic mean" and "average" is technically a center location. However, in practice among non-statisticians, "average" is commonly accepted for "arithmetic mean."
The mean can also be calculated by multiplying each distinct value by its frequency and then dividing the sum by the total number of data values. The letter used to represent the sample mean is an $\mathrm{x}$ with a bar over it (pronounced "x bar"): $\bar{x}$.
The Greek letter $\mu$ (pronounced "mew") represents the population mean. One of the requirements for the sample mean to be a good estimate of the population mean is for the sample taken to be truly random.
To see that both ways of calculating the mean are the same, consider the sample:
$1 ; 1 ; 1 ; 2 ; 2 ; 3 ; 4 ; 4 ; 4 ; 4 ; 4$
$$
\begin{gathered}
\bar{x}=\frac{1+1+1+2+2+3+4+4+4+4+4}{11}=2.7 \\
\bar{x}=\frac{3 \times 1+2 \times 2+1 \times 3+5 \times 4}{11}=2.7
\end{gathered}
$$
In the second calculation for the sample mean, the frequencies are 3,2, 1, and 5. You can quickly find the location of the median by using the expression $\frac{n+1}{2}$.
The letter $n$ is the total number of data values in the sample. If $n$ is an odd number, the median is the middle value of the ordered data (ordered smallest to largest). If $n$ is an even number, the median is equal to the two middle values added together and divided by 2 after the data has been ordered. For example, if the total number of data values is 97 , then $\frac{n+1}{2}=\frac{97+1}{2}=49$. The median is the 49th value in the ordered data. If the total number of data values is 100 , then $\frac{n+1}{2}=\frac{100+1}{2}=50.5$. The median occurs midway between the 50 th and $51 \mathrm{st}$ values. The location of the median and the value of the median are not the same. The upper case letter $\mathbf{M}$ is often used to represent the median. The next example illustrates the location of the median and the value of the median.
## Example 2.3
AIDS data indicating the number of months an AIDS patient lives after taking a new antibody drug are as follows (smallest to largest):
$3 ; 4 ; 8 ; 8 ; 10 ; 11 ; 12 ; 13 ; 14 ; 15 ; 15 ; 16 ; 16 ; 17 ; 17 ; 18 ; 21 ; 22 ; 22 ; 24 ; 24 ; 25$; $26 ; 26 ; 27 ; 27 ; 29 ; 29 ; 31 ; 32 ; 33 ; 33 ; 34 ; 34 ; 35 ; 37 ; 40 ; 44 ; 44 ; 47$
Calculate the mean and the median.
## Solution
The calculation for the mean is:
To find the median, $\mathbf{M}$, first use the formula for the location. The location is:
$$
\frac{n+1}{2}=\frac{40+1}{2}=20.5
$$
Starting at the smallest value, the median is located between the 20th and 21st values (the two 24s):
$3 ; 4 ; 8 ; 8 ; 10 ; 11 ; 12 ; 13 ; 14 ; 15 ; 15 ; 16 ; 16 ; 17 ; 17 ; 18 ; 21 ; 22 ; 22 ; 24 ; 24 ; 25$; $26 ; 26 ; 27 ; 27 ; 29 ; 29 ; 31 ; 32 ; 33 ; 33 ; 34 ; 34 ; 35 ; 37 ; 40 ; 44 ; 44 ; 47$
$$
M=\frac{24+24}{2}=24
$$
The median is 24 .
## Using the TI-83, 83+, 84, 84+ Calculators
Calculator Instructions are located in the menu item 14:Appendix (Notes for the TI-83, 83+, 84, 84+ Calculators).
- Enter data into the list editor. Press STAT 1:EDIT
- Put the data values in list L1.
- Press STAT and arrow to CALC. Press 1:1-VarStats. Press 2nd 1 for L1 and ENTER.
- Press the down and up arrow keys to scroll.
$$
\bar{x}=23.6, M=24
$$
## Example 2.4
Suppose that, in a small town of 50 people, one person earns $\$ 5,000,000$ per year and the other 49 each earn $\$ 30,000$. Which is the better measure of the "center," the mean or the median?
## Solution
$$
\begin{gathered}
\bar{x}=\frac{5000000+49 \times 30000}{50}=129400 \\
M=30000
\end{gathered}
$$
(There are 49 people who earn $\$ 30,000$ and one person who earns $\$ 5,000,000$.
The median is a better measure of the "center" than the mean because 49 of the values are 30,000 and one is 5,000,000. The $5,000,000$ is an outlier. The 30,000 gives us a better sense of the middle of the data.
Another measure of the center is the mode. The mode is the most frequent value. If a data set has two values that occur the same number of times, then the set is bimodal.
## Example 2.5
Statistics exam scores for 20 students are as follows:
$50 ; 53 ; 59 ; 59 ; 63 ; 63 ; 72 ; 72 ; 72 ; 72 ; 72 ; 76 ; 78 ; 81 ; 83 ; 84 ; 84$; $84 ; 90 ; 93$
## Problem
Find the mode.
## Solution
The most frequent score is 72 , which occurs five times. Mode 72.
## Example 2.6
Five real estate exam scores are 430, 430, 480, 480, 495. The data set is bimodal because the scores 430 and 480 each occur twice.
When is the mode the best measure of the "center"? Consider a weight loss program that advertises a mean weight loss of six pounds the first week of the program. The mode might indicate that most people lose two pounds the first week, making the program less appealing. Note: The mode can be calculated for qualitative data as well as for quantitative data.
Statistical software will easily calculate the mean, the median, and the mode. Some graphing calculators can also make these calculations. In the real world, people make these calculations using software.
#### The Law of Large Numbers and the Mean
The Law of Large Numbers says that if you take samples of larger and larger size from any population, then the mean $\bar{x}$ of the sample is very likely to get closer and closer to $\boldsymbol{\mu}$. This is discussed in more detail in The Central Limit Theorem.
Note: The formula for the mean is located in the Summary of Formulas section course.
\subsubsection*{Sampling Distributions and Statistic of a Sampling Distribution
s.org/licenses/by-sa/4.0/).
You can think of a sampling distribution as a relative frequency distribution with a great many samples. (See Sampling and Data for a review of relative frequency). Suppose thirty randomly selected students were asked the number of movies they watched the previous week. The results are in the relative frequency table shown below.
| \# of movies | Relative Frequency |
| :--- | :--- |
| 0 | $5 / 30$ |
| 1 | $15 / 30$ |
| 2 | $6 / 30$ |
| 3 | $4 / 30$ |
| 4 | $1 / 30$ |
If you let the number of samples get very large (say, 300 million or more), the relative frequency table becomes a relative frequency distribution.
A statistic is a number calculated from a sample. Statistic examples include the mean, the median and the mode as well as others. The sample mean $\bar{x}$ is an example of a statistic which estimates the population mean $\mu$.
### Descriptive Statistics: Skewness and the Mean,
Median, and Mode(c) (i) ()
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon
s.org/licenses/by-sa/4.0/).
Consider the following data set:
$4 ; 5 ; 6 ; 6 ; 6 ; 7 ; 7 ; 7 ; 7 ; 7 ; 7 ; 8 ; 8 ; 8 ; 9 ; 10$
This data set produces the histogram shown below. Each interval has width one and each value is located in the middle of an interval.
The histogram displays a symmetrical distribution of data. A distribution is symmetrical if a vertical line can be drawn at some point in the histogram such that the shape to the left and the right of the vertical line are mirror images of each other. The mean, the median, and the mode are each 7 for these data. In a perfectly symmetrical distribution, the mean and the median are the same. This example has one mode (unimodal) and the mode is the same as the mean and median. In a symmetrical distribution that has two modes (bimodal), the two modes would be different from the mean and median.
The histogram for the data:
$4 ; 5 ; 6 ; 6 ; 6 ; 7 ; 7 ; 7 ; 7 ; 8$
is not symmetrical. The right-hand side seems "chopped off" compared to the left side. The shape distribution is called skewed to the left because it is pulled out to the left.
The mean is 6.3 , the median is 6.5 , and the mode is 7 . Notice that the mean is less than the median and they are both less than the mode. The mean and the median both reflect the skewing but the mean more so.
The histogram for the data:
$6 ; 7 ; 7 ; 7 ; 7 ; 8 ; 8 ; 8 ; 9 ; 10$
is also not symmetrical. It is skewed to the right.
The mean is 7.7 , the median is 7.5 , and the mode is 7 . Of the three statistics, the mean is the largest, while the mode is the smallest. Again, the mean refects the skewing the most.
To summarize, generally if the distribution of data is skewed to the left, the mean is less than the median, which is often less than the mode. If the distribution of data is skewed to the right, the mode is often less than the median, which is less than the mean.
Skewness and symmetry become important when we discuss probability distributions in later chapters.
### Descriptive Statistics: Measuring the Spread of the
## Data
## (c) (i) ()
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon
s.org/licenses/by-sa/4.0/).
An important characteristic of any set of data is the variation in the data. In some data sets, the data values are concentrated closely near the mean; in other data sets, the data values are more widely spread out from the mean. The most common measure of variation, or spread, is the standard deviation.
The standard deviation is a number that measures how far data values are from their mean. The standard deviation
- provides a numerical measure of the overall amount of variation in a data set
- can be used to determine whether a particular data value is close to or far from the mean
## The standard deviation provides a measure of the overall variation in a data set
The standard deviation is always positive or 0 . The standard deviation is small when the data are all concentrated close to the mean, exhibiting little variation or spread. The standard deviation is larger when the data values are more spread out from the mean, exhibiting more variation.
Suppose that we are studying waiting times at the checkout line for customers at supermarket $A$ and supermarket $B$; the average wait time at both markets is 5 minutes. At market $A$, the standard deviation for the waiting time is 2 minutes; at market $B$ the standard deviation for the waiting time is 4 minutes.
Because market $\mathrm{B}$ has a higher standard deviation, we know that there is more variation in the waiting times at market B. Overall, wait times at market B are more spread out from the average; wait times at market A are more concentrated near the average.
The standard deviation can be used to determine whether a data value is close to or far from the mean.
Suppose that Rosa and Binh both shop at Market A. Rosa waits for 7 minutes and Binh waits for 1 minute at the checkout counter. At market $A$, the mean wait time is 5 minutes and the standard deviation is 2 minutes. The standard deviation can be used to determine whether a data value is close to or far from the mean.
## Rosa waits for 7 minutes:
- 7 is 2 minutes longer than the average of 5; 2 minutes is equal to one standard deviation.
- Rosa's wait time of 7 minutes is $\mathbf{2}$ minutes longer than the average of 5 minutes.
- Rosa's wait time of 7 minutes is one standard deviation above the average of 5 minutes.
## Binh waits for 1 minute.
- 1 is 4 minutes less than the average of 5; 4 minutes is equal to two standard deviations.
- Binh's wait time of 1 minute is $\mathbf{4}$ minutes less than the average of 5 minutes.
- Binh's wait time of 1 minute is two standard deviations below the average of 5 minutes.
- A data value that is two standard deviations from the average is just on the borderline for what many statisticians would consider to be far from the average. Considering data to be far from the mean if it is more than 2 standard deviations away is more of an approximate "rule of thumb" than a rigid rule. In general, the shape of the distribution of the data afects how much of the data is further away than 2 standard deviations. (We will learn more about this in later chapters.)
The number line may help you understand standard deviation. If we were to put 5 and 7 on a number line, 7 is to the right of 5 . We say, then, that 7 is one standard deviation to the right of 5 because $5+(1)(2)=7$.
If 1 were also part of the data set, then 1 is two standard deviations to the left of 5 because $5+(-2)(2)=1$.
- In general, a value $=$ mean $+($ \\#ofSTDEV)(standard deviation $)$
- where \\#OfSTDEVs = the number of standard deviations
- 7 is one standard deviation more than the mean of 5 because: $7=5+(1)(2)$
- 1 is two standard deviations less than the mean of 5 because: $1=5+(-2)(2)$
The equation value $=$ mean $+($ \\#ofSTDEVs)(standard deviation) can be expressed for a sample and for a population:
- sample: $x=\bar{x}+(\#$ of $S T D E V)(s)$
- Population: $x=\mu+(\# \circ f S T D E V)(\sigma)$
The lower case letter s represents the sample standard deviation and the Greek letter $\sigma$ (sigma, lower case) represents the population standard deviation.
The symbol $\bar{x}$ is the sample mean and the Greek symbol $\mu$ is the population mean.
## Calculating the Standard Deviation
If $x$ is a number, then the diference " $x$ - mean" is called its deviation. In a data set, there are as many deviations as there are items in the data set. The deviations are used to calculate the standard deviation. If the numbers belong to a population, in symbols a deviation is $\mathrm{x}-\mu$. For sample data, in symbols a deviation is $x-\bar{x}$. The procedure to calculate the standard deviation depends on whether the numbers are the entire population or are data from a sample. The calculations are similar, but not identical. Therefore the symbol used to represent the standard deviation depends on whether it is calculated from a population or a sample. The lower case letter $\mathrm{s}$ represents the sample standard deviation and the Greek letter $\sigma$ (sigma, lower case) represents the population standard deviation. If the sample has the same characteristics as the population, then s should be a good estimate of $\sigma$.
To calculate the standard deviation, we need to calculate the variance first. The variance is an average of the squares of the deviations (the $x-\bar{x}$ values for a sample, or the $x-\mu$ values for a population). The symbol $\sigma^{2}$ represents the population variance; the population standard deviation $\sigma$ is the square root of the population variance. The symbol $s^{2}$ represents the sample variance; the sample standard deviation $s$ is the square root of the sample variance. You can think of the standard deviation as a special average of the deviations.
If the numbers come from a census of the entire population and not a sample, when we calculate the average of the squared deviations to find the variance, we divide by $\mathrm{N}$, the number of items in the population. If the data are from a sample rather than a population, when we calculate the average of the squared deviations, we divide by $\mathbf{n - 1 ,}$ one less than the number of items in the sample. You can see that in the formulas below.
## Formulas for the Sample Standard Deviation
$$
\text { - } s=\sqrt{\frac{\sum(x-\bar{x})^{2}}{n-1}} \text { or } s=\sqrt{\frac{\sum f \cdot(x-\bar{x})^{2}}{n-1}}
$$
- For the sample standard deviation, the denominator is $\mathbf{n - 1}$, that is the sample size MINUS 1.
## Formulas for the Population Standard Deviation
$$
\text { - } \sigma=\sqrt{\frac{\sum(x-\bar{\mu})^{2}}{N}} \text { or } \sigma=\sqrt{\frac{\sum f(x-\bar{\mu})^{2}}{N}}
$$
- For the population standard deviation, the denominator is $\mathbf{N}$, the number of items in the population.
In these formulas, $f$ represents the frequency with which a value appears. For example, if a value appears once, $f$ is 1 . If a value appears three times in the data set or population, $f$ is 3 .
## Sampling Variability of a Statistic
The statistic of a sampling distribution was discussed in Descriptive Statistics: Measuring the Center of the Data. How much the statistic varies from one sample to another is known as the sampling variability of a statistic. You typically measure the sampling variability of a statistic by its standard error. The standard error of the mean is an example of a standard error. It is a special standard deviation and is known as the standard deviation of the sampling distribution of the mean. You will cover the standard error of the mean in The Central Limit Theorem (not now). The notation for the standard error of the mean is $\frac{\sigma}{\sqrt{n}}$ where $\sigma$ is the standard deviation of the population and $\mathrm{n}$ is the size of the sample.
Note: In practice, USE A CALCULATOR OR COMPUTER SOFTWARE TO CALCULATE THE STANDARD DEVIATION. If you are using a TI-83,83+,84+ calculator, you need to select the appropriate standard deviation $\sigma_{\mathbf{X}}$ or $\mathbf{s}_{\mathbf{X}}$ from the summary statistics. We will concentrate on using and interpreting the information that the standard deviation gives us. However you should study the following step-by-step example to help you understand how the standard deviation measures variation from the mean.
## Example 2.7
In a fifth grade class, the teacher was interested in the average age and the sample standard deviation of the ages of her students. The following data are the ages for a SAMPLE of $n=20$ fifth grade students. The ages are rounded to the nearest half year:
$9 ; 9.5 ; 9.5 ; 10 ; 10 ; 10 ; 10 ; 10.5 ; 10.5 ; 10.5 ; 10.5 ; 11 ; 11 ; 11 ; 11 ; 11$; $11 ; 11.5 ; 11.5 ; 11.5$
$\bar{x}=\frac{9+9.5 \times 2+10 \times 4+10.5 \times 4+11 \times 6+11.5 \times 3}{20}=10.525$
The average age is 10.53 years, rounded to 2 places.
The variance may be calculated by using a table. Then the standard deviation is calculated by taking the square root of the variance. We will explain the parts of the table after calculatings.
| Data | Freq. | Deviations | Deviations $^{2}$ | (Freq.)(Deviations ${ }^{2}$ ) |
| :---: | :---: | :---: | :---: | :---: |
| $x$ | $f$ | $(x-\bar{x})$ | $(x-\bar{x})^{2}$ | $(f)(x-\bar{x})^{2}$ |
| 9 | 1 | $\begin{array}{l}9-10.525 \\ =-1.525\end{array}$ | $\begin{array}{l}(-1.525)^{2}= \\ 2.325625\end{array}$ | $\begin{array}{l}1 \times 2.325625= \\ 2.325625\end{array}$ |
| 9.5 | 2 | $\begin{array}{l}9.5- \\ 10.525= \\ -1.025\end{array}$ | $\begin{array}{l}(-1.025)^{2}= \\ 1.050625\end{array}$ | $\begin{array}{l}2 \times 1.050625= \\ 2.101250\end{array}$ |
| 10 | 4 | $\begin{array}{l}10-10.525 \\ =-0.525\end{array}$ | $\begin{array}{l}(-0.525)^{2}= \\ 0.275625\end{array}$ | $4 \times .275625=1.1025$ |
| Data | Freq. | Deviations | Deviations $^{2}$ | (Freq.)(Deviations ${ }^{2}$ ) |
| :---: | :---: | :---: | :---: | :---: |
| 10.5 | 4 | $\begin{array}{l}10.5- \\ 10.525= \\ -0.025\end{array}$ | $\begin{array}{l}(-0.025)^{2}= \\ 0.000625\end{array}$ | $4 \times .000625=.0025$ |
| 11 | 6 | $\begin{array}{l}11-10.525 \\ =0.475\end{array}$ | $\begin{array}{l}(0.475)^{2}= \\ 0.225625\end{array}$ | $6 \times .225625=1.35375$ |
| 11.5 | 3 | $\begin{array}{l}11.5- \\ 10.525= \\ 0.975\end{array}$ | $\begin{array}{l}(0.975)^{2}= \\ 0.950625\end{array}$ | $\begin{array}{l}3 \times .950625= \\ 2.851875\end{array}$ |
The sample variance, $\mathrm{s}^{2}$, is equal to the sum of the last column (9.7375) divided by the total number of data values minus one (20 1):
$$
s^{2}=\frac{9.7375}{20-1}=0.5125
$$
The sample standard deviation $\mathbf{s}$ is equal to the square root of the sample variance:
$$
s=\sqrt{0.5125}=.0715891
$$
Rounded to two decimal places, $\mathrm{s}=0.72$
Typically, you do the calculation for the standard deviation on your calculator or computer. The intermediate results are not rounded. This is done for accuracy.
## Problem 1
Verify the mean and standard deviation calculated above on your calculator or computer.
## Solution
Using the TI-83,83+,84+ Calculators
- Enter data into the list editor. Press STAT 1:EDIT. If necessary, clear the lists by arrowing up into the name. Press CLEAR and arrow down. - Put the data values $(9,9.5,10,10.5,11,11.5)$ into list L1 and the frequencies $(1,2,4,4,6,3)$ into list L2. Use the arrow keys to move around.
- Press STAT and arrow to CALC. Press 1:1-VarStats and enter L1 (2nd 1), L2 (2nd 2). Do not forget the comma. Press ENTER.
- $\bar{x}=10.525$
- Use Sx because this is sample data (not a population): $\mathrm{Sx}=$ 0.715891
- For the following problems, recall that value $=$ mean + (\\#ofSTDEVs)(standard deviation)
- For a sample: $x=\bar{x}+(\#$ ofSTDEVs) $(s)$
- For a population: $x=\mu+(\#$ ofSTDEV s) $(\sigma)$
- For this example, use $x=\bar{x}+(\#$ ofSTDEVs)(s) because the data is from a sample
## Problem 2
Find the value that is 1 standard deviation above the mean. Find $(\bar{x}+1 s)$.
Solution
$$
(\bar{x}+1 s)=10.53+(1)(0.72)=11.25
$$
## Problem 3
Find the value that is two standard deviations below the mean. Find $(\bar{x}-2 s)$.
Solution
$$
(\bar{x}-2 s)=10.53-(2)(0.72)=9.09
$$
## Problem 4
Find the values that are 1.5 standard deviations from (below and above) the mean.
## Solution
$$
\begin{aligned}
& (\bar{x}-1.5 s)=10.53-(1.5)(0.72)=9.45 \\
& (\bar{x}+1.5 s)=10.53+(1.5)(0.72)=11.61
\end{aligned}
$$
## Explanation of the standard deviation calculation shown in the table
The deviations show how spread out the data are about the mean. The data value 11.5 is farther from the mean than is the data value 11. The deviations 0.97 and 0.47 indicate that. A positive deviation occurs when the data value is greater than the mean. A negative deviation occurs when the data value is less than the mean; the deviation is -1.525 for the data value 9 . If you add the deviations, the sum is always zero. (For this example, there are $\mathrm{n}=20$ deviations.) So you cannot simply add the deviations to get the spread of the data. By squaring the deviations, you make them positive numbers, and the sum will also be positive. The variance, then, is the average squared deviation.
The variance is a squared measure and does not have the same units as the data. Taking the square root solves the problem. The standard deviation measures the spread in the same units as the data.
Notice that instead of dividing by $n=20$, the calculation divided by $n-1=20-1=19$ because the data is a sample. For the sample variance, we divide by the sample size minus one $(n-1)$. Why not divide by $n$ ? The answer has to do with the population variance. The sample variance is an estimate of the population variance. Based on the theoretical mathematics that lies behind these calculations, dividing by $(n-1)$ gives a better estimate of the population variance.
Note: Your concentration should be on what the standard deviation tells us about the data. The standard deviation is a number which measures how far the data are spread from the mean. Let a calculator or computer do the arithmetic.
The standard deviation, $s$ or $\sigma$, is either zero or larger than zero. When the standard deviation is 0 , there is no spread; that is, the all the data values are equal to each other. The standard deviation is small when the data are all concentrated close to the mean, and is larger when the data values show more variation from the mean. When the standard deviation is a lot larger than zero, the data values are very spread out about the mean; outliers can make s or $\sigma$ very large.
The standard deviation, when first presented, can seem unclear. By graphing your data, you can get a better "feel" for the deviations and the standard deviation. You will find that in symmetrical distributions, the standard deviation can be very helpful but in skewed distributions, the standard deviation may not be much help. The reason is that the two sides of a skewed distribution have different spreads. In a skewed distribution, it is better to look at the first quartile, the median, the third quartile, the smallest value, and the largest value. Because numbers can be confusing, always graph your data.
Note: The formula for the standard deviation is at the end of the chapter.
## Example 2.8
Use the following data (first exam scores) from Susan Dean's spring pre-calculus class:
$33 ; 42 ; 49 ; 49 ; 53 ; 55 ; 55 ; 61 ; 63 ; 67 ; 68 ; 68 ; 69 ; 69 ; 72 ; 73 ; 74 ; 78 ; 80$; $83 ; 88 ; 88 ; 88 ; 90 ; 92 ; 94 ; 94 ; 94 ; 94 ; 96 ; 100$
1. Create a chart containing the data, frequencies, relative frequencies, and cumulative relative frequencies to three decimal places.
2. Calculate the following to one decimal place using a TI-83+ or TI-84 calculator: 1. The sample mean
3. The sample standard deviation
4. The median
5. The first quartile
6. The third quartile
7. IQR
8. Construct a box plot and a histogram on the same set of axes. Make comments about the box plot, the histogram, and the chart.
## Solution
| Data | Frequency | $\begin{array}{l}\text { Relative } \\ \text { Frequency }\end{array}$ | $\begin{array}{l}\text { Cumulative } \\ \text { Relative } \\ \text { Frequency }\end{array}$ |
| :---: | :---: | :---: | :---: |
| 33 | 1 | 0.032 | 0.032 |
| 42 | 1 | 0.032 | 0.064 |
| 49 | 2 | 0.065 | 0.129 |
| 53 | 1 | 0.032 | 0.161 |
| 55 | 2 | 0.065 | 0.226 |
| 61 | 1 | 0.032 | 0.258 |
| 63 | 1 | 0.032 | 0.29 |
| 67 | 1 | 0.032 | 0.322 |
| 68 | 2 | 0.065 | 0.387 |
| 69 | 2 | 0.065 | 0.452 |
| 72 | 1 | 0.032 | 0.484 |
| 73 | 1 | 0.032 | 0.516 |
| 74 | 1 | 0.032 | 0.548 |
| Data | Frequency | $\begin{array}{l}\text { Relative } \\ \text { Frequency }\end{array}$ | $\begin{array}{l}\text { Cumulative } \\ \text { Relative } \\ \text { Frequency }\end{array}$ |
| :---: | :---: | :---: | :---: |
| 78 | 1 | 0.032 | 0.580 |
| 80 | 1 | 0.032 | 0.612 |
| 83 | 1 | 0.032 | 0.644 |
| 88 | 3 | 0.097 | 0.741 |
| 90 | 1 | 0.032 | 0.773 |
| 92 | 1 | 0.032 | 0.805 |
| 94 | 4 | 0.129 | 0.934 |
| 96 | 1 | 0.032 | 0.966 |
| 100 | 1 | 0.032 | $\begin{array}{l}0.998 \text { (Why } \\ \text { isn't this } \\ \text { value } 1 ? \text { ?) }\end{array}$ |
2.
3. The sample mean $=73.5$
4. The sample standard deviation $=17.9$
5. The median $=73$
6. The first quartile $=61$
7. The third quartile $=90$
8. $\mathrm{IQR}=90-61=29$
9. The $\mathrm{x}$-axis goes from 32.5 to $100.5 ; \mathrm{y}$-axis goes from -2.4 to 15 for the histogram; number of intervals is 5 for the histogram so the width of an interval is $(100.5-32.5)$ divided by 5 which is equal to 13.6. Endpoints of the intervals: starting point is $32.5,32.5+13.6=$ $46.1,46.1+13.6=59.7,59.7+13.6=73.3,73.3+13.6=86.9,86.9+13.6=$ 100.5 the ending value; No data values fall on an interval boundary.
The long left whisker in the box plot is refected in the left side of the histogram. The spread of the exam scores in the lower $50 \%$ is greater $(73-33=40)$ than the spread in the upper 50\% $(100-73=27)$. The histogram, box plot, and chart all refect this. There are a substantial number of A and B grades (80s, 90s, and 100). The histogram clearly shows this. The box plot shows us that the middle $50 \%$ of the exam scores (IQR $=29$ ) are Ds, Cs, and Bs. The box plot also shows us that the lower $25 \%$ of the exam scores are Ds and Fs.
## Comparing Values from Different Data Sets
The standard deviation is useful when comparing data values that come from different data sets. If the data sets have different means and standard deviations, it can be misleading to compare the data values directly.
- For each data value, calculate how many standard deviations the value is away from its mean.
- Use the formula: value = mean $+($ \\#ofSTDEVs)(standard deviation); solve for \\#ofSTDEVs.
$$
\text { \#ofSTDEVs }=\frac{\text { value }- \text { mean }}{\text { standard deviation }}
$$
- Compare the results of this calculation.
\\#ofSTDEVs is often called a "z-score"; we can use the symbol z. In symbols, the formulas become:
| Sample | $x=\bar{x}+z s$ | $z=\frac{x-\bar{x}}{s}$ |
| :--- | :--- | :--- |
| Population | $x=\mu+z \sigma$ | $z=\frac{x-\mu}{\sigma}$ |
## Example 2.9
Two students, John and Ali, from different high schools, wanted to find out who had the highest G.P.A. when compared to his school. Which student had the highest G.P.A. when compared to his school?
| Student | GPA | $\begin{array}{l}\text { School } \\ \text { Mean GPA }\end{array}$ | $\begin{array}{l}\text { School Standard } \\ \text { Deviation }\end{array}$ |
| :--- | :--- | :--- | :--- |
| John | 2.85 | 3.0 | 0.7 |
| Ali | 77 | 80 | 10 |
## Solution
For each student, determine how many standard deviations (\\#ofSTDEVs) his GPA is away from the average, for his school. Pay careful attention to signs when comparing and interpreting the answer.
$$
\begin{aligned}
& \text { \#ofSTDEVs }=\frac{\text { value }- \text { mean }}{\text { standard deviation }} ; z=\frac{x-\mu}{\sigma} \\
& \text { For John, } z=\# \text { ofSTDEVs }=\frac{2.85-3.0}{0.7}=-0.21 \\
& \text { For Ali, } z=\# \text { ofSTDEVs }=\frac{77-80}{10}=-0.3
\end{aligned}
$$
John has the better G.P.A. when compared to his school because his G.P.A. is 0.21 standard deviations below his school's mean while Ali's G.P.A. is 0.3 standard deviations below his school's mean.
John's z-score of - 0.21 is higher than Ali's z-score of -0.3 . For GPA, higher values are better, so we conclude that John has the better GPA when compared to his school.
The following lists give a few facts that provide a little more insight into what the standard deviation tells us about the distribution of the data.
## For ANY data set, no matter what the distribution of the data is:
- At least $75 \%$ of the data is within 2 standard deviations of the mean.
- At least $89 \%$ of the data is within 3 standard deviations of the mean.
- At least $95 \%$ of the data is within $41 / 2$ standard deviations of the mean.
- This is known as Chebyshev's Rule.
## For data having a distribution that is MOUND-SHAPED and SYMMETRIC:
- Approximately $68 \%$ of the data is within 1 standard deviation of the mean. - Approximately 95\% of the data is within 2 standard deviations of the mean.
- More than $99 \%$ of the data is within 3 standard deviations of the mean.
- This is known as the Empirical Rule.
- It is important to note that this rule only applies when the shape of the distribution of the data is mound-shaped and symmetric. We will learn more about this when studying the "Normal" or "Gaussian" probability distribution in later chapters.
**With contributions from Roberta Bloom
#### Solutions to Exercises in Chapter 2
Solution to Example 2.2, Problem
- 3.5 to 4.5
- 4.5 to 5.5
- 6
- 5.5 to 6.5
## Chapter 3 The Normal Distribution
### Normal Distribution: Introduction
#### Student Learning Outcomes
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon s.org/licenses/by-sa/4.0/).
## Learning Objectives
By the end of this chapter, the student should be able to:
- Recognize the normal probability distribution and apply it appropriately.
- Recognize the standard normal probability distribution and apply it appropriately.
- Compare normal probabilities by converting to the standard normal distribution.
#### Introduction
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon s.org/licenses/by-sa/4.0/).
The normal, a continuous distribution, is the most important of all the distributions. It is widely used and even more widely abused. Its graph is bell-shaped. You see the bell curve in almost all disciplines. Some of these include psychology, business, economics, the sciences, nursing, and, of course, mathematics. Some of your instructors may use the normal distribution to help determine your grade. Most IQ scores are normally distributed. Often real estate prices $\mathrm{ft}$ a normal distribution. The normal distribution is extremely important but it cannot be applied to everything in the real world.
In this chapter, you will study the normal distribution, the standard normal, and applications associated with them.
#### Optional Collaborative Classroom Activity
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon s.org/licenses/by-sa/4.0/).
Your instructor will record the heights of both men and women in your class, separately. Draw histograms of your data. Then draw a smooth curve through each histogram. Is each curve somewhat bell-shaped? Do you think that if you had recorded 200 data values for men and 200 for women that the curves would look bellshaped? Calculate the mean for each data set. Write the means on the $x$-axis of the appropriate graph below the peak. Shade the approximate area that represents the probability that one randomly chosen male is taller than 72 inches. Shade the approximate area that represents the probability that one randomly chosen female is shorter than 60 inches. If the total area under each curve is one, does either probability appear to be more than 0.5 ?
The normal distribution has two parameters (two numerical descriptive measures), the mean $(\mu)$ and the standard deviation $(\sigma)$. If $\mathbf{X}$ is a quantity to be measured that has a normal distribution with mean $(\mu)$ and the standard deviation $(\sigma)$, we designate this by writing
The probability density function is a rather complicated function. Do not memorize it. It is not necessary.
$$
f(x)=\frac{1}{\alpha \cdot \sqrt{2 \cdot \pi}} \cdot e^{-\frac{1}{2} \cdot\left(\frac{x-\mu}{\sigma}\right)^{2}}
$$
The cumulative distribution function is $\mathbf{P}(\mathbf{X}<\mathbf{x})$. It is calculated either by a calculator or a computer or it is looked up in a table. Technology has made the tables basically obsolete. For that reason, as well as the fact that there are various table formats, we are not including table instructions in this chapter. See the NOTE in this chapter in Calculation of Probabilities.
The curve is symmetrical about a vertical line drawn through the mean, $\mu$. In theory, the mean is the same as the median since the graph is symmetric about $\mu$. As the notation indicates, the normal distribution depends only on the mean and the standard deviation. Since the area under the curve must equal one, a change in the standard deviation, $\sigma$, causes a change in the shape of the curve; the curve becomes fatter or skinnier depending on $\sigma$. A change in $\mu$ causes the graph to shift to the left or right. This means there are an infnite number of normal probability distributions. One of special interest is called the standard normal distribution.
### Normal Distribution: Standard Normal Distribution
The standard normal distribution is a normal distribution of standardized values called z-scores. A z-score is measured in units of the standard deviation. For example, if the mean of a normal distribution is 5 and the standard deviation is 2 , the value 11 is 3 standard deviations above (or to the right of) the mean. The calculation is:
$$
x=\mu+(z) \sigma=5+(3)(2)=11
$$
The z-score is 3.
The mean for the standard normal distribution is 0 and the standard deviation is 1 . The transformation $z=\frac{x-\mu}{\sigma}$ produces the distribution $\mathrm{Z} \sim \mathrm{N}(0,1)$. The value $x$ comes from a normal distribution with mean $\mu$ and standard deviation $\sigma$.
### Normal Distribution: Z-scores
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon s.org/licenses/by-sa/4.0/).
If $\mathbf{X}$ is a normally distributed random variable and $\mathrm{X} \sim \mathrm{N}(\mu, \sigma)$, then the $\mathrm{z}$-score is:
$$
z=\frac{x-\mu}{\sigma}
$$
The z-score tells you how many standard deviations that the value $\mathbf{x}$ is above (to the right of) or below (to the left of) the mean, $\mu$. Values of $x$ that are larger than the mean have positive $z$-scores and values of $x$ that are smaller than the mean have negative $z$-scores. If $x$ equals the mean, then $x$ has a $z$-score of 0 .
## Example 3.1
Suppose $\mathrm{X} \sim \mathbf{N}(5,6)$. This says that $\mathrm{X}$ is a normally distributed random variable with mean $\mu=5$ and standard deviation $\sigma=6$. Suppose $x=17$. Then:
$$
z=\frac{x-\mu}{\sigma}=\frac{17-5}{6}=2
$$
This means that $\mathrm{x}=17$ is $\mathbf{2}$ standard deviations $(2 \sigma)$ above or to the right of the mean $\mu=5$. The standard deviation is $\sigma=6$.
Notice that:
$$
5+2 \cdot 6=17 \text { (The pattern is } \mu+\mathrm{z} \sigma=\mathrm{x} \text {.) }
$$
Now suppose $x=1$. Then:
$$
z=\frac{x-\mu}{\sigma}=\frac{1-5}{6}=-0.67
$$
(rounded to two decimal places)
This means that $\mathrm{x}=1$ is 0.67 standard deviations $(-0.67 \sigma)$ below or to the left of the mean $\mu=5$. Notice that:
$5^{+(-0.67)}$ (6) is approximately equal to 1 (This has the pattern $\mu$ $+(-0.67) \sigma=1)$ Summarizing, when $\mathrm{z}$ is positive, $\mathrm{x}$ is above or to the right of $\mu$ and when $z$ is negative, $x$ is to the left of or below $\mu$.
## Example 3.2
Some doctors believe that a person can lose 5 pounds, on the average, in a month by reducing his/her fat intake and by exercising consistently. Suppose weight loss has a normal distribution. Let $\mathbf{X}=$ the amount of weight lost (in pounds) by a person in a month. Use a standard deviation of 2 pounds. $\mathbf{X} \sim \mathbf{N}(5,2)$. Fill in the blanks.
## Problem 1
Suppose a person lost 10 pounds in a month. The $z$-score when $x=10$ pounds is $z=2.5$ (verify). This $z$-score tells you that $x=10$ is standard deviations to the (right or left) of the
mean ( What is the mean?).
## Problem 2
Suppose a person gained 3 pounds (a negative weight loss). Then $\mathrm{z}$ . This $\mathrm{z}$-score tells you that $\mathrm{x}=-3$ is standard
deviations to the (right or left) of the mean.
Suppose the random variables $\mathrm{X}$ and $\mathrm{Y}$ have the following normal distributions: $\mathrm{X} \sim \mathrm{N}(5,6)$ and $\mathrm{Y} \sim \mathrm{N}(2,1)$. If $\mathrm{X}=17$, then $\mathrm{Z}=2$. (This was previously shown.) If $\mathrm{y}=4$, what is $\mathrm{z}$ ?
where $\mu=2$ and $\sigma=1$.
$$
z=\frac{y-\mu}{\sigma}=\frac{4-2}{1}=2
$$
The $z$-score for $y=4$ is $z=2$. This means that 4 is $z=2$ standard deviations to the right of the mean. Therefore, $x=17$ and $y=4$ are both 2 (of their) standard deviations to the right of their respective means.
The z-score allows us to compare data that are scaled differently. To understand the concept, suppose $\mathrm{X} \sim \mathrm{N}(5,6)$ represents weight gains for one group of people who are trying to gain weight in a 6 week period and $\mathbf{Y} \sim \mathbf{N}(\mathbf{2}, \mathbf{1})$ measures the same weight gain for a second group of people. A negative weight gain would be a weight loss. Since $\mathbf{x}=\mathbf{1 7}$ and $\mathbf{y}=\mathbf{4}$ are each $\mathbf{2}$ standard deviations to the right of their means, they represent the same weight gain relative to their means.
## The Empirical Rule
If $\mathbf{X}$ is a random variable and has a normal distribution with mean $\mu$ and standard deviation $\sigma$ then the Empirical Rule says (See the Figure 3.1 below)
- About $68.27 \%$ of the $x$ values lie between $-1 \sigma$ and $+1 \sigma$ of the mean $\mu$ (within 1 standard deviation of the mean). - About $95.45 \%$ of the $x$ values lie between $-2 \sigma$ and $+2 \sigma$ of the mean $\mu$ (within 2 standard deviations of the mean).
- About $99.73 \%$ of the $x$ values lie between $-3 \sigma$ and $+3 \sigma$ of the mean $\mu$ (within 3 standard deviations of the mean). Notice that almost all the $x$ values lie within 3 standard deviations of the mean.
- The $z$-scores for $+1 \sigma$ and $1 \sigma$ are +1 and -1 , respectively.
- The $z$-scores for $+2 \sigma$ and $2 \sigma$ are +2 and -2 , respectively.
- The $z$-scores for $+3 \sigma$ and $3 \sigma$ are +3 and -3 respectively.
Figure 3.1 The Empirical Rule
The Empirical Rule is also known as the 68-95-99.7 Rule.
## Example 3.3
Suppose $\mathrm{X}$ has a normal distribution with mean 50 and standard deviation 6.
- About $68.27 \%$ of the x values lie between $-1 \sigma=(-1)(6)=-6$ and $1 \sigma=$ $(1)(6)=6$ of the mean 50 . The values $50-6=44$ and $50+6=56$ are within 1 standard deviation of the mean 50 . The $z^{\text {-scores are }-1}$ and +1 for 44 and 56 , respectively.
- About $95.45 \%$ of the $x$ values lie between $-2 \sigma=(-2)(6)=-12$ and $2 \sigma$ $=(2)(6)=12$ of the mean 50 . The values $50-12=38$ and $50+12=62$ are within 2 standard deviations of the mean 50 . The $z^{-s c o r e s}$ are -2 and 2 for 38 and 62, respectively.
- About $99.73 \%$ of the $x$ values lie between $-3 \sigma=(-3)(6)=-18$ and $3 \sigma$ $=(3)(6)=18$ of the mean 50 . The values $50-18=32$ and $50+18=68$ are within 3 standard deviations of the mean 50 . The $z$-scores are -3 and +3 for 32 and 68 , respectively.
### Normal Distribution: Areas to the Left and Right of $x$
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The arrow in the graph below points to the area to the left of $x$. This area is represented by the probability $\mathrm{P}(\mathrm{X}<\mathrm{X})$. Normal tables, computers, and calculators provide or calculate the probability $\mathrm{P}(X<X)$.
The area to the right is then $P(X>x)=1-P(X<x)$.
Remember, $P(X<X)=$ Area to the left of the vertical line through $x$. $P(X>x)=1-P(X<x)=$ Area to the right of the vertical line through $x$. $P(X<x)$ is the same as $P(X \leq X)$ and $P(X>x)$ is the same as $P(X \geq X)$ for continuous distributions.
### Normal Distribution: Calculations of Probabilities
## (i) (2)
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon
s.org/licenses/by-sa/4.0/).
Probabilities are calculated by using technology. There are instructions in the chapter for the $\mathrm{TI}-83+$ and $\mathrm{TI}-84$ calculators.
Note: In the Table of Contents for Collaborative Statistics, entry 15. Tables has a link to a table of normal probabilities. Use the probability tables if so desired, instead of a calculator. The tables include instructions for how to use then.
## Example 3.4
If the area to the left is 0.0228 , then the area to the right is $1-0.0228$ $=0.9772$.
## Example 3.5
The final exam scores in a statistics class were normally distributed with a mean of 63 and a standard deviation of 5 .
Please see Problem 1-4.
## Problem 1
Find the probability that a randomly selected student scored more than 65 on the exam.
## Solution
Let $X=$ a score on the final exam. $X \sim N(63,5)$, where $\mu=63$ and $\sigma=5$ Draw a graph.
Then, find $P(x>65)$.
$P(x>65)=0.3446$ (calculator or computer)
The probability that one student scores more than 65 is 0.3446 . Using the $\mathrm{TI}-83+$ or the TI-84 calculators, the calculation is as follows. Go into 2nd DISTR. After pressing 2nd DISTR, press 2:normalcdf. The syntax for the instructions are shown below. normalcdf(lower value, upper value, mean, standard deviation) For this problem: normal $\operatorname{cdf}(65,1 \mathrm{E} 99,63,5)=0.3446$. You get $1 \mathrm{E} 99\left(10^{99}\right)$ by pressing 1 , the EE key (a 2nd key) and then 99. Or, you can enter 10-99 instead. The number 1099 is way out in the right tail of the normal curve. We are calculating the area between 65 and $10^{99}$. In some instances, the lower number of the area might be $-1 \mathrm{E} 99\left(-10^{99}\right)$. The number $-10^{99}$ is way out in the left tail of the normal curve.
Historical Note: The TI probability program calculates a z-score and then the probability from the z-score. Before technology, the z-score was looked up in a standard normal probability table (because the math involved is too cumbersome) to find the probability. In this example, a standard normal table with area to the left of the $\mathrm{z}$-score was used. You calculate the $\mathrm{z}$-score and look up the area to the left. The probability is the area to the right.
$$
z=\frac{65-63}{5}=0.4
$$
Area to the left is 0.6554. $P(x>65)=P(z>0.4)=1-0.6554=0.3446$
## Problem 2
Find the probability that a randomly selected student scored less than 85 .
## Solution
Draw a graph. Then find $P(x<85)$. Shade the graph. $P(x<85)=1$ (calculator or computer) The probability that one student scores less than 85 is approximately 1 (or $100 \%$ ). The TI-instructions and answer are as follows: normalcdf $(0,85,63,5)=1$ (rounds to 1 )
## Problem 3
Find the 9oth percentile (that is, find the score $\mathrm{k}$ that has $90 \%$ of the scores below $\mathrm{k}$ and $10 \%$ of the scores above $\mathrm{k}$ ).
## Solution
Find the 90th percentile. For each problem or part of a problem, draw a new graph. Draw the $\mathrm{x}$-axis. Shade the area that corresponds to the 9oth percentile.
Let $\mathrm{k}=$ the 9oth percentile. $\mathrm{k}$ is located on the $\mathrm{x}$-axis. $\mathrm{P}(\mathrm{x}<\mathrm{k})$ is the area to the left of $\mathrm{k}$. The 9oth percentile $\mathrm{k}$ separates the exam scores into those that are the same or lower than $\mathrm{k}$ and those that are the same or higher. Ninety percent of the test scores are the same or lower than $\mathrm{k}$ and $10 \%$ are the same or higher. $\mathrm{k}$ is often called a critical value.
$k=69.4$ (calculator or computer)
The 90th percentile is 69.4 . This means that $90 \%$ of the test scores fall at or below 69.4 and $10 \%$ fall at or above. For the TI-83+ or TI-84 calculators, use invNorm in 2nd DISTR. invNorm(area to the left, mean, standard deviation) For this problem, invNorm(0.90,63,5) = 69.4
## Problem 4
Find the 70th percentile (that is, find the score $\mathrm{k}$ such that $70 \%$ of scores are below $\mathrm{k}$ and $30 \%$ of the scores are above $\mathrm{k}$ ).
## Solution
Find the 70th percentile.
Draw a new graph and label it appropriately. $\mathrm{k}=65.6$
The 70th percentile is 65.6 . This means that $70 \%$ of the test scores fall at or below 65.5 and $30 \%$ fall at or above.
$\operatorname{invNorm}(0.70,63,5)=65.6$
## Example 3.6
A computer is used for office work at home, research, communication, personal finances, education, entertainment, social networking and a myriad of other things. Suppose that the average number of hours a household personal computer is used for entertainment is 2 hours per day. Assume the times for entertainment are normally distributed and the standard deviation for the times is half an hour.
Please see problem 1-2.
## Problem 1
Find the probability that a household personal computer is used between 1.8 and 2.75 hours per day.
## Solution
Let $\mathrm{X}=$ the amount of time (in hours) a household personal computer is used for entertainment. $\mathrm{x} \sim \mathrm{N}(2,0.5)$ where $\mu=2$ and $\sigma=0.5$.
Find $P(1.8<x<2.75)$.
The probability for which you are looking is the area between $\mathrm{x}=1.8$ and $\mathrm{x}=2.75 . \mathrm{P}(1.8<\mathrm{x}<2.75)=0.5886$
normalcdf $(1.8,2.75,2,0.5)=0.5886$
The probability that a household personal computer is used between 1.8 and 2.75 hours per day for entertainment is 0.5886 .
## Problem 2
Find the maximum number of hours per day that the bottom quartile of households use a personal computer for entertainment.
## Solution
To find the maximum number of hours per day that the bottom quartile of households uses a personal computer for entertainment, find the 25 th percentile, $k$, where $P(x<k)=0.25$.
$\operatorname{invNorm}(0.25,2, .5)=1.66$
The maximum number of hours per day that the bottom quartile of households uses a personal computer for entertainment is 1.66 hours.
### Central Limit Theorem: Central Limit Theorem for Sample Means
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Suppose $\mathbf{X}$ is a random variable with a distribution that may be known or unknown (it can be any distribution). Using a subscript that matches the random variable, suppose:
1. $\mu x=$ the mean of $X$
2. $\sigma_{X}=$ the standard deviation of $X$
If you draw random samples of size $\mathrm{n}$, then as $\mathrm{n}$ increases, the random variable $\bar{X}$ which consists of sample means, tends to be normally distributed and
$$
\bar{X} \sim N\left(\mu_{X}, \frac{\sigma_{X}}{\sqrt{n}}\right)
$$
The Central Limit Theorem for Sample Means says that if you keep drawing larger and larger samples (like rolling 1, 2, 5, and, finally, 10 dice) and calculating their means the sample means form their own normal distribution (the sampling distribution). The normal distribution has the same mean as the original distribution and a variance that equals the original variance divided by $\mathbf{n}$, the sample size. $n$ is the number of values that are averaged together not the number of times the experiment is done.
To put it more formally, if you draw random samples of size $\mathbf{n}$, the distribution of the random variable $\bar{X}$, which consists of sample means, is called the sampling distribution of the mean. The sampling distribution of the mean approaches a normal distribution as $\mathbf{n}$, the sample size, increases.
The random variable $\bar{X}$ has a different z-score associated with it than the random variable $\mathbf{X} \cdot \bar{x}$ is the value of $\bar{X}$ in one sample.
$$
z=\frac{\bar{x}-\mu_{X}}{\frac{\sigma_{X}}{\sqrt{n}}}
$$
$\mu \mathrm{x}$ is both the average of $\mathrm{X}$ and of $\bar{X}$. $\sigma_{\bar{X}}=\frac{\sigma_{X}}{\sqrt{n}}=$ standard deviation of $\bar{X}$ and is called the standard error of the mean.
## Example 3.7
An unknown distribution has a mean of 90 and a standard deviation of 15 . Samples of size $n=25$ are drawn randomly from the population.
See Problem 1-2.
## Problem 1
Find the probability that the sample mean is between 85 and 92 .
## Solution
Let $\mathrm{X}$ = one value from the original unknown population. The probability question asks you to find a probability for the sample mean.
Let $\bar{X}=$ the mean of a sample of size 25 . Since $\mu_{X}=90, \sigma_{X}=15$, and $n$ $=25$; then
$$
\bar{X} \sim N\left(90, \frac{15}{\sqrt{25}}\right)
$$
Find $\mathrm{P}(85<\bar{x}<92)$ Draw a graph.
$$
P(85<\bar{x}<92)=0.6997
$$
The probability that the sample mean is between 85 and 92 is 0.6997 .
TI-83 or 84 : normalcdf (lower value, upper value, mean, standard error of the mean)
The parameter list is abbreviated (lower value, upper value, $\mu, \frac{a}{\sqrt{n}}$ ) normalcdf $\left(85,92,90, \frac{15}{\sqrt{25}}\right)=0.6997$
## Problem 2
Find the value that is 2 standard deviations above the expected value (it is 90) of the sample mean.
## Solution
To find the value that is 2 standard deviations above the expected value 90, use the formula
$$
\begin{gathered}
\text { value }=\mu_{X}+(\# \text { ofSTDEVs })\left(\frac{\sigma_{X}}{\sqrt{n}}\right) \\
\text { value }=90+2 \cdot \frac{15}{\sqrt{25}}=96
\end{gathered}
$$
So, the value that is 2 standard deviations above the expected value is 96.
## Example 3.8
The length of time, in hours, it takes an "over 40 " group of people to play one soccer match is normally distributed with a mean of 2 hours and a standard deviation of 0.5 hours. A sample of size $\mathbf{n}=\mathbf{5 0}$ is drawn randomly from the population.
See Problem below.
## Problem
Find the probability that the sample mean is between 1.8 hours and 2.3 hours.
## Solution
Let $\mathrm{X}$ = the time, in hours, it takes to play one soccer match.
The probability question asks you to find a probability for the sample mean time, in hours, it takes to play one soccer match.
Let $\bar{X}$ the mean time, in hours, it takes to play one soccer match. If $\mu_{X}=\ldots, \sigma x=\ldots$, and $n=\ldots$, then $\bar{X} \sim N($ _ _ _ by the Central Limit Theorem for Means.
$\mu_{\mathrm{X}}=2, \sigma_{\mathrm{X}}=0.5, \mathrm{n}=50$, and $X \sim N\left(2, \frac{0.5}{\sqrt{50}}\right)$
Find $P(1.8<\bar{x}<2.3)$. Draw a graph.
$$
P(1.8<\bar{x}<2.3)=0.9977
$$
normalcdf $\left(1.8,2.3,2, \frac{.5}{\sqrt{50}}\right)=0.9977$ The probability that the mean time is between 1.8 hours and 2.3 hours is
### Central Limit Theorem: Using the Central Limit Theorem
(c) (i) (2)
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon
s.org/licenses/by-sa/4.0/).
It is important for you to understand when to use the CLT. If you are being asked to find the probability of the mean, use the CLT for the mean. If you are being asked to find the probability of a sum or total, use the CLT for sums. This also applies to percentiles for means and sums.
Note: If you are being asked to find the probability of an individual value, do not use the CLT. Use the distribution of its random variable.
#### Examples of the Central Limit Theorem
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon s.org/licenses/by-sa/4.0/).
## Law of Large Numbers
The Law of Large Numbers says that if you take samples of larger and larger size from any population, then the mean $\bar{x}$ of the sample tends to get closer and closer to $\mu$. From the Central Limit Theorem, we know that as n gets larger and larger, the sample means follow a normal distribution. The larger $n$ gets, the smaller the standard deviation gets. (Remember that the standard deviation for $\bar{X}$ is $\frac{\sigma}{\sqrt{n}}$.) This means that the sample mean $\bar{x}$ must be close to the population mean $\mu$. We can say that $\mu$ is the value that the sample means approach as $n$ gets larger. The Central Limit Theorem illustrates the Law of Large Numbers.
## Central Limit Theorem for the Mean and Sum Examples
## Example 3.9
A study involving stress is done on a college campus among the students. The stress scores follow a uniform distribution with the lowest stress score equal to 1 and the highest equal to 5. Using a sample of 75 students, find:
1. The probability that the mean stress score for the 75 students is less than 2.
2. The 9oth percentile for the mean stress score for the 75 students.
3. The probability that the total of the $\mathbf{7 5}$ stress scores is less than 200.
4. The 9oth percentile for the total stress score for the 75 students. Let $\mathrm{X}=$ one stress score.
Problems 1 and 2 ask you to find a probability or a percentile for a mean. Problems 3 and 4 ask you to find a probability or a percentile for a total or sum. The sample size, $n$, is equal to 75 .
Since the individual stress scores follow a uniform distribution, $\mathrm{X} ~$ $\mathrm{U}(1,5)$ where $\mathrm{a}=1$ and $\mathrm{b}=5$ (See Continuous Random Variables for the uniform).
$$
\begin{aligned}
& \mu_{X}=\frac{a+b}{2}=\frac{1+5}{2}=3 \\
& \sigma_{X}=\sqrt{\frac{(b-a)^{2}}{12}}=\sqrt{\frac{(5-1)^{2}}{12}}=1.15
\end{aligned}
$$
For problems 1. and 2., let $\bar{X}=$ the mean stress score for the 75 students. Then,
$$
\bar{X} \sim N\left(3, \frac{1.15}{\sqrt{75}}\right)
$$
where $\mathrm{n}=75$.
See Problem 1-4.
## Problem 1
Find $P(\bar{x}<2)$. Draw the graph.
## Solution
$$
P(\bar{X}<2)=0
$$
The probability that the mean stress score is less than 2 is about 0.
$$
\text { normalcdf }\left(1,2,3, \frac{1.15}{\sqrt{75}}\right)=0
$$
REMINDER: The smallest stress score is 1 . Therefore, the smallest mean for 75 stress scores is 1 .
## Problem 2
Find the 90th percentile for the mean of 75 stress scores. Draw a graph.
Solution Let $\mathrm{k}=$ the 90 th percentile.
Find $\mathrm{k}$ where
$$
\begin{aligned}
P(\bar{x}<k) & =0.90 . \\
k & =3.2
\end{aligned}
$$
$$
\mathrm{P}(\overline{\mathbf{x}}<\mathrm{k})=0.90
$$
The 9oth percentile for the mean of 75 scores is about 3.2. This tells us that $90 \%$ of all the means of 75 stress scores are at most 3.2 and $10 \%$ are at least 3.2 .
$$
\text { invNorm }(.90,3, \sqrt{75})=3.2
$$
For problems $\mathrm{c}$ and $\mathrm{d}$, let $\Sigma \mathrm{X}=$ the sum of the 75 stress scores. Then,
$$
\sum X \sim N[(75) \cdot(3), \sqrt{75} \cdot 1.15]
$$
## Problem 3
Find $\mathrm{P}(\Sigma \mathrm{x}<200)$. Draw the graph.
## Solution
The mean of the sum of 75 stress scores is $75 \cdot 3=225$
The standard deviation of the sum of 75 stress scores is
$\sqrt{75} \cdot 1.15=9.96$
$\mathrm{P}(\Sigma \mathrm{x}<200)=0$
$$
\mathrm{P}\left(\sum \mathbf{x}<200\right)
$$
The probability that the total of 75 scores is less than 200 is about 0 .
$$
\text { normalcdf }(75,200,75 \cdot 3, \sqrt{75} \cdot 1.15)=0
$$
REMINDER: The smallest total of 75 stress scores is 75 since the smallest single score is 1 .
## Problem 4
Find the 9oth percentile for the total of 75 stress scores. Draw a graph.
## Solution
Let $\mathrm{k}=$ the 90 th percentile.
Find $k$ where $P(\Sigma x<k)=0.90$.
$\mathrm{k}=237.8$
The 9oth percentile for the sum of 75 scores is about 237.8. This tells us that $90 \%$ of all the sums of 75 scores are no more than 237.8 and $10 \%$ are no less than 237.8 .
$$
\operatorname{inv} \operatorname{Norm}(.90,75 \cdot 3, \sqrt{75} \cdot 1.15)=237.8
$$
## Example 3.10
Suppose that a market research analyst for a cell phone company conducts a study of their customers who exceed the time allowance included on their basic cell phone contract; the analyst finds that for those people who exceed the time included in their basic contract, the excess time used follows an exponential distribution with a mean of 22 minutes.
Consider a random sample of 80 customers who exceed the time allowance included in their basic cell phone contract.
Let $\mathrm{X}$ = the excess time used by one INDIVIDUAL cell phone customer who exceeds his contracted time allowance.
$$
X \sim \operatorname{Exp}\left(\frac{1}{22}\right)
$$
From Chapter 5, we know that $\mu=22$ and $\sigma=22$.
Let $\bar{X}=$ the mean excess time used by a sample of $\mathrm{n}=80$ customers who exceed their contracted time allowance.
by the CLT for Sample Means.
$$
\bar{X} \sim N\left(22, \frac{22}{\sqrt{80}}\right)
$$
See Problems 5-6.
## Problem 5
## Using the CLT to find Probability:
1. Find the probability that the mean excess time used by the 80 customers in the sample is longer than 20 minutes. This is asking us to find $P(\bar{x}>20)$ Draw the graph.
2. Suppose that one customer who exceeds the time limit for his cell phone contract is randomly selected. Find the probability that this individual customer's excess time is longer than 20 minutes. This is asking us to find $\mathrm{P}(\mathrm{x}>20)$
3. Explain why the probabilities in (a) and (b) are different.
## Solution
## Part a.
Find: $P(\bar{x}>20)$
$$
P(\bar{x}>20)=0.7919
$$
using
$$
\text { normalcdf }\left(20,1 E 99,22, \frac{22}{\sqrt{80}}\right)
$$
The probability is 0.7919 that the mean excess time used is more than 20 minutes, for a sample of 80 customers who exceed their contracted time allowance.
REMINDER: 1 E99 $=10^{99}$ and $-1 E 99=-10^{99}$. Press the EE key for E. Or just use $10^{\wedge} 99$ instead of $1 \mathrm{E} 99$.
Part b.
Find $P(x>20)$. Remember to use the exponential distribution for an individual: $X \sim \operatorname{Exp}(1 / 22)$.
$$
P(x>20)=e^{\wedge}\left((1 / 22)^{*} 20\right) \text { or } e^{\wedge}\left(.04545^{*} 20\right)=0.4029
$$
Part c. Explain why the probabilities in (a) and (b) are different.
$$
P(x>20)=0.4029 \text { but } P(\bar{x}>20)=0.7919
$$
The probabilities are not equal because we use different distributions to calculate the probability for individuals and for means. When asked to find the probability of an individual value, use the stated distribution of its random variable; do not use the CLT. Use the CLT with the normal distribution when you are being asked to find the probability for an mean.
## Problem 6
## Using the CLT to find Percentiles:
Find the 95th percentile for the sample mean excess time for samples of 80 customers who exceed their basic contract time allowances. Draw a graph.
## Solution
Let $\mathrm{k}=$ the 95th percentile. Find $\mathrm{k}$ where $P(\bar{x}<k)=0.95$
$$
\mathrm{k}=26.0 \text { using } i n v \text { Norm }\left(.95,22, \frac{22}{\sqrt{80}}\right)=26.0
$$
The 95th percentile for the sample mean excess time used is about 26.0 minutes for random samples of 80 customers who exceed their contractual allowed time. $95 \%$ of such samples would have means under 26 minutes; only $5 \%$ of such samples would have means above 26 minutes.
Note: (HISTORICAL): Normal Approximation to the Binomial
Historically, being able to compute binomial probabilities was one of the most important applications of the Central Limit Theorem. Binomial probabilities were displayed in a table in a book with a small value for $n$ (say, 20). To calculate the probabilities with large values of $n$, you had to use the binomial formula which could be very complicated. Using the Normal Approximation to the Binomial simplifed the process. To compute the Normal Approximation to the Binomial, take a simple random sample from a population. You must meet the conditions for a binomial distribution:
- there are a certain number $\mathrm{n}$ of independent trials - the outcomes of any trial are success or failure
- each trial has the same probability of a success $\mathrm{p}$
Recall that if $X$ is the binomial random variable, then $X \sim B(n, p)$. The shape of the binomial distribution needs to be similar to the shape of the normal distribution. To ensure this, the quantities np and nq must both be greater than five $(n p>5$ and $n q>5$; the approximation is better if they are both greater than or equal to 10). Then the binomial can be approximated by the normal distribution with mean $\mu=\mathrm{np}$ and standard deviation $\sigma=\sqrt{n p q}$. Remember that $\mathrm{q}=1-\mathrm{p}$. In order to get the best approximation, add 0.5 to $\mathrm{x}$ or subtract 0.5 from $x$ (use $x+0.5$ or $x-0.5$ ). The number 0.5 is called the continuity correction factor.
## Example 3.11
Suppose in a local Kindergarten through 12th grade $(\mathrm{K}-12)$ school district, 53 percent of the population favor a charter school for grades $\mathrm{K}$ - 5. A simple random sample of 300 is surveyed.
1. Find the probability that at least $\mathbf{1 5 0}$ favor a charter school.
2. Find the probability that at most $\mathbf{1 6 0}$ favor a charter school.
3. Find the probability that more than $\mathbf{1 5 5}$ favor a charter school.
4. Find the probability that less than $\mathbf{1 4 7}$ favor a charter school.
5. Find the probability that exactly $\mathbf{1 7 5}$ favor a charter school.
Let $\mathrm{X}=$ the number that favor a charter school for grades $\mathrm{K}-5 . \mathrm{X} \sim \mathrm{B}$ $(n, p)$ where $n=300$ and $p=0.53$. Since $n p>5$ and $n q>5$, use the normal approximation to the binomial. The formulas for the mean and standard deviation are $\mu=\mathrm{np}$ and $\sigma=\sqrt{n p q}$. The mean is 159 and the standard deviation is 8.6447. The random variable for the normal distribution is $Y . Y \sim N(159, \mathbf{8 . 6 4 4 7})$. See The Normal Distribution for help with calculator instructions.
For Problem 1., you include 150 so $P(x \geq 150)$ has normal approximation $\mathrm{P}(\mathrm{Y} \geq 149.5)=0.8641$.
$$
\text { normalcdf }(149.5,1099,159,8.6447)=0.8641
$$
For Problem 2., you include 160 so $\mathrm{P}(\mathrm{x} \leq 160)$ has normal approximation $\mathrm{P}(\mathrm{Y} \leq 160.5)=0.5689$.
$$
\text { normalcdf }(0,160.5,159,8.6447)=0.5689
$$
For Problem 3., you exclude 155 so $P(x>155)$ has normal approximation $P(y>155.5)=0.6572$.
$$
\text { normalcdf }(155.5,1099,159,8.6447)=0.6572
$$
For Problem 4., you exclude 147 so $\mathrm{P}(\mathrm{x}<147)$ has normal approximation $\mathrm{P}(\mathrm{Y}<146.5)=0.0741$.
$$
\text { normalcdf }(0,146.5,159,8.6447)=0.0741
$$
For Problem 5., $\mathrm{P}(\mathrm{x}=175)$ has normal approximation $\mathrm{P}(174.5<\mathrm{y}<$ $175.5)=0.0083$.
normalcdf $(174.5,175.5,159,8.6447)=0.0083$ Because of calculators and computer software that easily let you calculate binomial probabilities for large values of $n$, it is not necessary to use the the Normal Approximation to the Binomial provided you have access to these technology tools. Most school labs have Microsoft Excel, an example of computer software that calculates binomial probabilities. Many students have access to the TI-83 or 84 series calculators and they easily calculate probabilities for the binomial. In an Internet browser, if you type in "binomial probability distribution calculation, " you can find at least one online calculator for the binomial.
For Example 3, the probabilities are calculated using the binomial (n $=300$ and $\mathrm{p}=0.53$ ) below. Compare the binomial and normal distribution answers. See Discrete Random Variables for help with calculator instructions for the binomial.
$P(x \geq 150): 1$ - binomialcdf $(300,0.53,149)=0.8641$
$P(x \leq 160)$ : binomialcdf $(300,0.53,160)=0.5684$
$P(x>155): 1$ - binomialcdf $(300,0.53,155)=0.6576$
$P(x<147)$ : binomialcdf $(300,0.53,146)=0.0742$
$P(x=175)$ : (You use the binomial pdf.) binomialpdf $(175,0.53$, 146) $=0.0083$
** Contributions made to Example 2 by Roberta Bloom
#### Solutions to Exercises in Chapter 3
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon s.org/licenses/by-sa/4.0/).
## Solution to Example 3.2, Problem 1
This z-score tells you that $x=10$ is $\mathbf{2 . 5}$ standard deviations to the right of the mean 5 .
## Solution to Example 3.2, Problem 2
$z=-4$. This $z$-score tells you that $x=-3$ is 4 standard deviations to the left of the mean.
## Chapter 4 Confidence Interval
### Confidence Intervals: Introduction
#### Student Learning Outcomes
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon s.org/licenses/by-sa/4.0/).
## Learning Objectives
By the end of this chapter, the student should be able to:
- Calculate and interpret confidence intervals for one population mean and one population proportion.
- Interpret the student-t probability distribution as the sample size changes.
- Discriminate between problems applying the normal and the student-t distributions.
#### Introduction
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon s.org/licenses/by-sa/4.0/).
Suppose you are trying to determine the mean rent of a two-bedroom apartment in your town. You might look in the classified section of the newspaper, write down several rents listed, and average them together. You would have obtained a point estimate of the true mean. If you are trying to determine the percent of times you make a basket when shooting a basketball, you might count the number of shots you make and divide that by the number of shots you attempted. In this case, you would have obtained a point estimate for the true proportion.
We use sample data to make generalizations about an unknown population. This part of statistics is called inferential statistics. The sample data help us to make an estimate of a population parameter. We realize that the point estimate is most likely not the exact value of the population parameter, but close to it. After calculating point estimates, we construct confidence intervals in which we believe the parameter lies.
In this chapter, you will learn to construct and interpret confidence intervals. You will also learn a new distribution, the Student's-t, and how it is used with these intervals. Throughout the chapter, it is important to keep in mind that the confidence interval is a random variable. It is the parameter that is fixed.
If you worked in the marketing department of an entertainment company, you might be interested in the mean number of compact discs (CD's) a consumer buys per month. If so, you could conduct a survey and calculate the sample mean, $\bar{x}$, and the sample standard deviation, s. You would use $\bar{x}$ to estimate the population mean and $\mathrm{s}$ to estimate the population standard deviation. The sample mean, $\bar{x}$, is the point estimate for the population mean, $\mu$. The sample standard deviation, $s$, is the point estimate for the population standard deviation, $\sigma$.
Each of $\bar{x}$ and $\mathrm{s}$ is also called a statistic.
A confidence interval is another type of estimate but, instead of being just one number, it is an interval of numbers. The interval of numbers is a range of values calculated from a given set of sample data. The confidence interval is likely to include an unknown population parameter.
Suppose for the CD example we do not know the population mean $\mu$ but we do know that the population standard deviation is $\sigma=1$ and our sample size is 100 . Then by the Central Limit Theorem, the standard deviation for the sample mean is
$$
\frac{\sigma}{\sqrt{n}}=\frac{1}{\sqrt{100}}=0.1
$$
The Empirical Rule, which applies to bell-shaped distributions, says that in approximately $95 \%$ of the samples, the sample mean, $\bar{x}$, will be within two standard deviations of the population mean $\mu$. For our CD example, two standard deviations is (2) $(0.1)=0.2$. The sample mean $\bar{x}$ is likely to be within 0.2 units of $\mu$.
Because $\bar{x}$ is within 0.2 units of $\mu$, which is unknown, then $\mu$ is likely to be within 0.2 units of $\bar{x}$ in $95 \%$ of the samples. The population mean $\mu$ is contained in an interval whose lower number is calculated by taking the sample mean and subtracting two standard deviations ((2) (0.1)) and whose upper number is calculated by taking the sample mean and adding two standard deviations. In other words, $\mu$ is between $\bar{x}-0.2$ and $\bar{x}+0.2$ in $95 \%$ of all the samples.
For the CD example, suppose that a sample produced a sample mean $\bar{x}=2$. Then the unknown population mean $\mu$ is between
$$
\bar{x}-0.2=2-0.2=1.8 \text { and } \bar{x}+0.2=2+0.2=2.2
$$
We say that we are $\mathbf{9 5} \%$ confident that the unknown population mean number of CDs is between 1.8 and 2.2. The $95 \%$ confidence interval is $(1.8,2.2)$.
The $95 \%$ confidence interval implies two possibilities. Either the interval $(1.8,2.2)$ contains the true mean $\mu$ or our sample produced an $\bar{x}$ that is not within 0.2 units of the true mean $\mu$. The second possibility happens for only $5 \%$ of all the samples $(100 \%$ 95\%).
Remember that a confidence interval is created for an unknown population parameter like the population mean, $\mu$. Confidence intervals for some parameters have the form
## (point estimate - margin of error, point estimate + margin of error)
The margin of error depends on the confidence level or percentage of confidence.
When you read newspapers and journals, some reports will use the phrase "margin of error." Other reports will not use that phrase, but include a confidence interval as the point estimate + or - the margin of error. These are two ways of expressing the same concept. Note: Although the text only covers symmetric confidence intervals, there are non-symmetric confidence intervals (for example, a confidence interval for the standard deviation).
#### Optional Collaborative Classroom Activity
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon s.org/licenses/by-sa/4.0/).
Have your instructor record the number of meals each student in your class eats out in a week. Assume that the standard deviation is known to be 3 meals. Construct an approximate $95 \%$ confidence interval for the true mean number of meals students eat out each week.
1. Calculate the sample mean.
2. $\sigma=3$ and $n=$ the number of students surveyed.
3. Construct the interval
$$
\left(\bar{x}-2 \cdot \frac{\sigma}{\sqrt{n}}, \bar{x}+2 \cdot \frac{\sigma}{\sqrt{n}}\right)
$$
We say we are approximately $95 \%$ confident that the true average number of meals that students eat out in a week is between and
### Confidence Intervals: Confidence Interval, Single Population Mean, Population Standard Deviation Known , Normal
#### Calculating the Confidence Interval
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon
s.org/licenses/by-sa/4.0/).
To construct a confidence interval for a single unknown population mean $\mu$, where the population standard deviation is known, we need $\bar{x}$ as an estimate for $\mu$ and we need the margin of error. Here, the margin of error is called the error bound for a population mean (abbreviated EBM). The sample mean $\bar{x}$ is the point estimate of the unknown population mean $\mu$
The confidence interval estimate will have the form:
(point estimate - error bound, point estimate + error bound) or, in symbols,
$$
(\bar{x}-E B M, \bar{x}+E B M)
$$
The margin of error depends on the confidence level (abbreviated $\mathrm{CL}$ ). The confidence level is often considered the probability that the calculated confidence interval estimate will contain the true population parameter. However, it is more accurate to state that the confidence level is the percent of confidence intervals that contain the true population parameter when repeated samples are taken. Most often, it is the choice of the person constructing the confidence interval to choose a confidence level of $90 \%$ or higher because that person wants to be reasonably certain of his or her conclusions.
There is another probability called alpha ( $\alpha$ ). $\alpha$ is related to the confidence level CL. $\alpha$ is the probability that the interval does not contain the unknown population parameter. Mathematically, $\alpha+\mathrm{CL}=1$.
## Example 4.1
Suppose we have collected data from a sample. We know the sample mean but we do not know the mean for the entire population. The sample mean is 7 and the error bound for the mean is 2.5 .
$$
\bar{x}=7 \text { and } E B M=2.5
$$
The confidence interval is $(7-2.5,7+2.5)$; calculating the values gives $(4.5,9.5)$.
If the confidence level (CL) is 95\%, then we say that "We estimate with $95 \%$ confidence that the true value of the population mean is between 4.5 and 9.5." A confidence interval for a population mean with a known standard deviation is based on the fact that the sample means follow an approximately normal distribution. Suppose that our sample has a mean of $\mathrm{x}=10$ and we have constructed the $90 \%$ confidence interval $(5,15)$ where $\mathrm{EBM}=5$.
To get a $90 \%$ confidence interval, we must include the central $90 \%$ of the probability of the normal distribution. If we include the central $90 \%$, we leave out a total of $\alpha=10 \%$ in both tails, or $5 \%$ in each tail, of the normal distribution.
Confidence Level $(\mathrm{CL})=0.90$
$\mu$ is believed to be in the interval $(5,15)$ with $90 \%$ confidence.
To capture the central 90\%, we must go out 1.645 "standard deviations" on either side of the calculated sample mean. 1.645 is the $\mathrm{z}$-score from a Standard Normal probability distribution that puts an area of 0.90 in the center, an area of 0.05 in the far left tail, and an area of 0.05 in the far right tail. It is important that the "standard deviation" used must be appropriate for the parameter we are estimating. So in this section, we need to use the standard deviation that applies to sample means, which is $\frac{\sigma}{\sqrt{n}} \cdot \frac{\sigma}{\sqrt{n}}$ is commonly called the "standard error of the mean" in order to clearly distinguish the standard deviation for a mean from the population standard deviation $\sigma$.
## In summary, as a result of the Central Limit Theorem:
- $\bar{X}$ is normally distributed, that is, $\bar{X} \sim N\left(\mu_{X} \cdot \frac{\sigma}{\sqrt{n}}\right)$.
- When the population standard deviation $\sigma$ is known, we use a Normal distribution to calculate the error bound.
## Calculating the Confidence Interval:
To construct a confidence interval estimate for an unknown population mean, we need data from a random sample. The steps to construct and interpret the confidence interval are:
- Calculate the sample mean $\bar{x}$ from the sample data. Remember, in this section, we already know the population standard deviation $\sigma$.
- Find the Z-score that corresponds to the confidence level.
- Calculate the error bound EBM
- Construct the confidence interval
- Write a sentence that interprets the estimate in the context of the situation in the problem. (Explain what the confidence interval means, in the words of the problem.)
We will first examine each step in more detail, and then illustrate the process with some examples.
## Finding $\mathrm{z}$ for the stated Confidence Level
When we know the population standard deviation $\sigma$, we use a standard normal distribution to calculate the error bound EBM and construct the confidence interval. We need to find the value of $z$ that puts an area equal to the confidence level (in decimal form) in the middle of the standard normal distribution $Z \sim N(0,1)$.
The confidence level, $C L$, is the area in the middle of the standard normal distribution. $C L=1-\alpha$. So $\alpha$ is the area that is split equally between the two tails. Each of the tails contains an area equal to $\frac{\alpha}{2}$.
The z-score that has an area to the right of $\frac{\alpha}{2}$ is denoted by $z_{\frac{\alpha}{2}}$
For example, when $\mathrm{CL}=0.95$ then $\alpha=0.05$ and $\frac{\alpha}{2}=0.025$; we write $z_{\frac{\alpha}{2}}=z_{.025}$
The area to the right of $z .025$ is 0.025 and the area to the left of $z .025$ is $1-0.025=0.975$ $z_{\frac{\alpha}{2}}=z_{0.025}=1.96$, using a calculator, computer or a Standard Normal probability table.
Using the $\mathrm{TI} 83, \mathrm{TI} 83+$ or $\mathrm{TI} 84+$ calculator: invNorm $(0.975,0,1)=1.96$
CALCULATOR NOTE: Remember to use area to the LEFT of $z \frac{\alpha}{2}$; in this chapter the last two inputs 2 in the invNorm command are 0,1 because you are using a Standard Normal Distribution $\mathrm{Z} \sim \mathrm{N}(0,1)$
## EBM: Error Bound
The error bound formula for an unknown population mean $\mu$ when the population standard deviation $\sigma$ is known is
$$
E M B=z_{\frac{a}{2}} \cdot \frac{\sigma}{\sqrt{n}}
$$
## Constructing the Confidence Interval
- The confidence interval estimate has the format $(\bar{x}-E B M, \bar{x}+E B M)$.
The graph gives a picture of the entire situation.
$$
C L+\frac{\alpha}{2}+\frac{\alpha}{2}=C L+\alpha=1
$$
$$
\frac{\alpha}{2} \quad \mathrm{CL}=1-\alpha \quad \frac{\alpha}{2}
$$
## Writing the Interpretation
The interpretation should clearly state the confidence level $(\mathrm{CL})$, explain what population parameter is being estimated (here, a population mean), and should state the confidence interval (both endpoints). "We estimate with $\%$ confidence that the true population mean (include context of the problem) is between and (include appropriate units)."
## Example 4.2
Suppose scores on exams in statistics are normally distributed with an unknown population mean and a population standard deviation of 3 points. A random sample of 36 scores is taken and gives a sample mean (sample mean score) of 68. Find a confidence interval estimate for the population mean exam score (the mean score on all exams).
## Problem
Find a 90\% confidence interval for the true (population) mean of statistics exam scores.
## Solution
- You can use technology to directly calculate the confidence interval
- The first solution is shown step-by-step (Solution A).
- The second solution uses the TI-83, 83+ and 84+ calculators (Solution B).
## Solution A
To find the confidence interval, you need the sample mean, $\bar{x}$, and the EBM.
$$
\begin{gathered}
\bar{x}=68 \\
E B M=z_{\frac{a}{2}} \cdot\left(\frac{\sigma}{\sqrt{n}}\right)
\end{gathered}
$$
$\sigma=3 ; n=36$; The confidence level is 90\% $(\mathrm{CL}=0.90)$
$\mathrm{CL}=0.90$ so $\alpha=1-\mathrm{CL}=1-0.90=0.10$
$$
\begin{aligned}
& \frac{\alpha}{2}=0.05 \\
& z_{\frac{\alpha}{2}}=z_{0.05}
\end{aligned}
$$
The area to the right of $\mathrm{z}_{.05}$ is 0.05 and the area to the left of $\mathrm{z}_{.05}$ is $1-0.05=0.95$
$$
z_{\frac{\alpha}{2}}=z_{0.05}=1.645
$$
using invNorm( $0.95,0,1)$ on the TI-83,83+,84+ calculators. This can also be found using appropriate commands on other calculators, using a computer, or using a probability table for the Standard Normal distribution.
$$
\begin{gathered}
E B M=1.645 \cdot\left(\frac{3}{\sqrt{36}}\right)=0.8225 \\
\bar{x}-E B M=68-0.8225=67.1775 \\
\bar{x}+E B M=68+0.8225=68.8225
\end{gathered}
$$
The $90 \%$ confidence interval is $(67.1775,68.8225)$.
## Solution B
Using a function of the TI-83, TI-83+ or TI-84 calculators:
Press STAT and arrow over to TESTS.
Arrow down to 7:ZInterval.
Press ENTER. Arrow to Stats and press ENTER.
Arrow down and enter 3 for $\sigma, 68$ for $\bar{x}, 36$ for $\mathrm{n}$, and .90 for C-level.
Arrow down to Calculate and press ENTER.
The confidence interval is (to 3 decimal places) $(67.178,68.822)$.
## Interpretation
We estimate with $90 \%$ confidence that the true population mean exam score for all statistics students is between 67.18 and 68.82.
## Explanation of $90 \%$ Confidence Level
$90 \%$ of all confidence intervals constructed in this way contain the true mean statistics exam score. For example, if we constructed 100 of these confidence intervals, we would expect 90 of them to contain the true population mean exam score.
#### Changing the Confidence Level or Sample Size
(i) (2)
## Example 4.3: Changing the Confidence Level
Suppose we change the original problem by using a 95\% confidence level. Find a $95 \%$ confidence interval for the true (population) mean statistics exam score.
## Solution
To find the confidence interval, you need the sample mean, $\bar{x}$, and the EBM.
$$
\begin{gathered}
\bar{x}=68 \\
E B M=z_{\frac{a}{2}} \cdot\left(\frac{a}{\sqrt{n}}\right)
\end{gathered}
$$
$\sigma=3 ; \mathrm{n}=36$; The confidence level is $95 \%(\mathrm{CL}=0.95)$
$\mathrm{CL}=0.95$ So $\alpha=1-\mathrm{CL}=1-0.95=0.05$
$$
\begin{aligned}
& \frac{\alpha}{2}=0.025 \\
& z_{\frac{\alpha}{2}}=z_{.025}
\end{aligned}
$$
The area to the right of $\mathrm{z}_{.025}$ is 0.025 and the area to the left of $\mathrm{z}_{.025}$ is $1-0.025=0.975$
$$
z_{\frac{\alpha}{2}}=z_{.025}=1.96
$$
using invnorm $(.975,0,1)$ on the TI-83,83+,84+ calculators. (This can also be found using appropriate commands on other calculators, using a computer, or using a probability table for the Standard Normal distribution.)
$$
\begin{aligned}
& E M B=1.96 \cdot\left(\frac{3}{\sqrt{36}}\right)=0.98 \\
& \bar{x}-E B M=68-0.98=67.02 \\
& \bar{x}+E B M=68+0.98=68.98
\end{aligned}
$$
## Interpretation
We estimate with 95\% confidence that the true population mean for all statistics exam scores is between 67.02 and 68.98.
## Explanation of $95 \%$ Confidence Level
95\% of all confidence intervals constructed in this way contain the true value of the population mean statistics exam score.
## Comparing the results
The $90 \%$ confidence interval is $(67.18,68.82)$. The $95 \%$ confidence interval is $(67.02$, 68.98). The $95 \%$ confidence interval is wider. If you look at the graphs, because the area 0.95 is larger than the area 0.90 , it makes sense that the $95 \%$ confidence interval is wider.
Figure 4.1 Comparing the results
## Summary: Effect of Changing the Confidence Level
- Increasing the confidence level increases the error bound, making the confidence interval wider.
- Decreasing the confidence level decreases the error bound, making the confidence interval narrower.
## Example 4.4: Changing the Sample Size:
Suppose we change the original problem to see what happens to the error bound if the sample size is changed.
See the following Problem.
## Problem
Leave everything the same except the sample size. Use the original $90 \%$ confidence level. What happens to the error bound and the confidence interval if we increase the sample size and use $n=100$ instead of $n=36$ ? What happens if we decrease the sample size to $n=25$ instead of $n=36$ ?
- $\bar{x}=68$
- $E B M=z_{\frac{\alpha}{2}} \cdot\left(\frac{\sigma}{\sqrt{n}}\right)$
- $\sigma=3$; The confidence level is $90 \%(\mathrm{CL}=0.90) ; z_{\frac{\alpha}{2}}=z_{.05}=1.645$
## Solution A
If we increase the sample size $\mathrm{n}$ to 100 , we decrease the error bound. When $\mathrm{n}=100$ :
$$
E B M=z_{\frac{\alpha}{2}} \cdot\left(\frac{\sigma}{\sqrt{n}}\right)=1.645 \cdot\left(\frac{3}{\sqrt{100}}\right)=0.4935
$$
## Solution B
If we decrease the sample size $\mathrm{n}$ to 25 , we increase the error bound.
When $\mathrm{n}=25$ :
$$
E B M=z_{\frac{\alpha}{2}} \cdot\left(\frac{\sigma}{\sqrt{n}}\right)=1.645 \cdot\left(\frac{3}{\sqrt{25}}\right)=0.987
$$
## Summary: Effect of Changing the Sample Size
- Increasing the sample size causes the error bound to decrease, making the confidence interval narrower.
- Decreasing the sample size causes the error bound to increase, making the confidence interval wider.
#### Working Backwards to Find the Error Bound or Sample
## Mean
## (c) (i) (2)
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon
s.org/licenses/by-sa/4.0/)
## Working Backwards to find the Error Bound or the Sample Mean
When we calculate a confidence interval, we find the sample mean and calculate the error bound and use them to calculate the confidence interval. But sometimes when we read statistical studies, the study may state the confidence interval only. If we know the confidence interval, we can work backwards to find both the error bound and the sample mean.
## Finding the Error Bound
- From the upper value for the interval, subtract the sample mean
- OR, From the upper value for the interval, subtract the lower value. Then divide the diference by 2 .
## Finding the Sample Mean
- Subtract the error bound from the upper value of the confidence interval
- OR, Average the upper and lower endpoints of the confidence interval
Notice that there are two methods to perform each calculation. You can choose the method that is easier to use with the information you know.
## Example 4.5
Suppose we know that a confidence interval is $(67.18,68.82)$ and we want to find the error bound.
We may know that the sample mean is 68. Or perhaps our source only gave the confidence interval and did not tell us the value of the the sample mean.
## Calculate the Error Bound:
- If we know that the sample mean is 68: $\mathrm{EBM}=68.82-68=0.82$
- If we don't know the sample mean:
$$
E B M=\frac{(68.82-67.18)}{2}=0.82
$$
## Calculate the Sample Mean:
- If we know the error bound: $\bar{x}=68.82-0.82=68$
- If we don't know the error bound: $\bar{x}=\frac{67.18+68.82}{2}=68$
#### Calculating the Sample Size $n$
If researchers desire a specifc margin of error, then they can use the error bound formula to calculate the required sample size.
The error bound formula for a population mean when the population standard deviation is known is
$$
E B M=z_{\frac{\alpha}{2}} \cdot\left(\frac{\sigma}{\sqrt{n}}\right)
$$
The formula for sample size is $n=\frac{z^{2} \sigma^{2}}{E B M^{2}}$, found by solving the error bound formula for $n$ In this formula, $z$ is $z_{\frac{\alpha}{2}}$, corresponding to the desired confidence level. A researcher planning a study who 2 wants a specifed confidence level and error bound can use this formula to calculate the size of the sample needed for the study.
## Example 4.6
The population standard deviation for the age of Foothill College students is 15 years. If we want to be $95 \%$ confident that the sample mean age is within 2 years of the true population mean age of Foothill College students, how many randomly selected Foothill College students must be surveyed?
From the problem, we know that $\sigma=15$ and $E B M=2$ $\mathrm{z}=\mathrm{z}_{.025}=1.96$, becuase the confidence level is $95 \%$.
$$
n=\frac{z^{2} \sigma^{2}}{E B M^{2}}=\frac{1.96^{2} 15^{2}}{2^{2}}=216.09
$$
using the sample size equation.
Use $n=217$ : Always round the answer UP to the next higher integer to ensure that the sample size is large enough.
Therefore, 217 Foothill College students should be surveyed in order to be $95 \%$ confident that we are within 2 years of the true population mean age of Foothill College students.
$\star *$ With contributions from Roberta Bloom
### Confidence Intervals: Confidence Interval, Single Population Mean, Standard Deviation Unknown, Student's-t
s.org/licenses/by-sa/4.0/).
In practice, we rarely know the population standard deviation. In the past, when the sample size was large, this did not present a problem to statisticians. They used the sample standard deviation $s$ as an estimate for $\sigma$ and proceeded as before to calculate a confidence interval with close enough results. However, statisticians ran into problems when the sample size was small. A small sample size caused inaccuracies in the confidence interval.
William S. Gossett (1876-1937) of the Guinness brewery in Dublin, Ireland ran into this problem. His experiments with hops and barley produced very few samples. Just replacing $\sigma$ with s did not produce accurate results when he tried to calculate a confidence interval. He realized that he could not use a normal distribution for the calculation; he found that the actual distribution depends on the sample size. This problem led him to "discover" what is called the Student's-t distribution. The name comes from the fact that Gosset wrote under the pen name "Student."
Up until the mid 1970s, some statisticians used the normal distribution approximation for large sample sizes and only used the Student's-t distribution for sample sizes of at most 30 . With the common use of graphing calculators and computers, the practice is to use the Student's-t distribution whenever $\mathrm{s}$ is used as an estimate for $\sigma$.
If you draw a simple random sample of size $\mathrm{n}$ from a population that has approximately a normal distribution with mean $\mu$ and unknown population standard deviation $\sigma$ and calculate the t-score
$$
t=\frac{\bar{x}-\mu}{\left(\frac{s}{\sqrt{n}}\right)}
$$
, then the t-scores follow a Student's-t distribution with $\mathbf{n} \mathbf{- 1}$ degrees of freedom. The t-score has the same interpretation as the z-score. It measures how far $\bar{x}$ is from its mean $\mu$. For each sample size $\mathrm{n}$, there is a different Student's-t distribution.
The degrees of freedom, $n-1$, come from the calculation of the sample standard deviation s. In Chapter 2, we used $\mathrm{n}$ deviations ( $x-\bar{x}$ values) to calculate s. Because the sum of the deviations is 0 , we can find the last deviation once we know the other $n$ - 1 deviations. The other $\mathbf{n}-\mathbf{1}$ deviations can change or vary freely. We call the number $n-1$ the degrees of freedom (df).
## Properties of the Student's-t Distribution
- The graph for the Student's-t distribution is similar to the Standard Normal curve.
- The mean for the Student's-t distribution is 0 and the distribution is symmetric about 0.
- The Student's-t distribution has more probability in its tails than the Standard Normal distribution because the spread of the $t$ distribution is greater than the spread of the Standard Normal. So the graph of the Student's-t distribution will be thicker in the tails and shorter in the center than the graph of the Standard Normal distribution.
- The exact shape of the Student's-t distribution depends on the "degrees of freedom". As the degrees of freedom increases, the graph Student's-t distribution becomes more like the graph of the Standard Normal distribution.
- The underlying population of individual observations is assumed to be normally distributed with unknown population mean $\mu$ and unknown population standard deviation $\sigma$. The size of the underlying population is generally not relevant unless it is very small. If it is bell shaped (normal) then the assumption is met and doesn't need discussion. Random sampling is assumed but it is a completely separate assumption from normality.
Calculators and computers can easily calculate any Student's-t probabilities. The TI-83,83+,84+ have a tcdf function to find the probability for given values of $\mathrm{t}$. The grammar for the tcdf command is tcdf(lower bound, upper bound, degrees of freedom). However for confidence intervals, we need to use inverse probability to find the value of $\mathrm{t}$ when we know the probability. For the TI-84+ you can use the invT command on the DISTRibution menu. The invT command works similarly to the invnorm. The invT command requires two inputs: invT(area to the left, degrees of freedom) The output is the t-score that corresponds to the area we specifed.
The TI-83 and 83+ do not have the invT command. (The TI-89 has an inverse T command.)
A probability table for the Student's-t distribution can also be used. The table gives tscores that correspond to the confidence level (column) and degrees of freedom (row). (The TI-86 does not have an invT program or command, so if you are using that calculator, you need to use a probability table for the Student's-t distribution.) When using t-table, note that some tables are formatted to show the confidence level in the column headings, while the column headings in some tables may show only corresponding area in one or both tails.
A Student's-t table gives t-scores given the degrees of freedom and the right-tailed probability. The table is very limited. Calculators and computers can easily calculate any Student's-t probabilities.
The notation for the Student's-t distribution is (using $\mathrm{T}$ as the random variable) is
- $\mathrm{T} \sim \mathrm{t}_{\mathrm{df}}$ where $\mathrm{df}=\mathrm{n}-1$.
- For example, if we have a sample of size $n=20$ items, then we calculate the degrees of freedom as $\mathrm{df}=\mathrm{n}-1=20-1=19$ and we write the distribution as $\mathrm{T} \sim$ $\mathrm{t}_{19}$
If the population standard deviation is not known, the error bound for a population mean is:
$$
E B M=t_{\frac{\alpha}{2}} \cdot\left(\frac{s}{\sqrt{n}}\right)
$$
- $t_{\frac{\alpha}{2}}$ is the t-score with area to the right equal to $\overline{2}$
- use $\mathrm{df}=\mathrm{n}-1$ degrees of freedom
- $\mathrm{s}=$ sample standard deviation
The format for the confidence interval is:
$$
(\bar{x}-E B M, \bar{x}+E B M)
$$
The $\mathrm{TI}-83,83+$ and 84 calculators have a function that calculates the confidence interval directly. To get to it,
Press STAT
Arrow over to TESTS.
Arrow down to 8:TInterval and press ENTER (or just press 8).
## Example 4.7
Suppose you do a study of acupuncture to determine how Effective it is in relieving pain. You measure sensory rates for 15 subjects with the results given below. Use the sample data to construct a $95 \%$ confidence interval for the mean sensory rate for the population (assumed normal) from which you took the data.
The solution is shown step-by-step and by using the TI-83, 83+ and $84+$ calculators. $8.6 ; 9.4 ; 7.9 ; 6.8 ; 8.3 ; 7.3 ; 9.2 ; 9.6 ; 8.7 ; 11.4 ; 10.3 ; 5.4$; $8.1 ; 5.5 ; 6.9$
## Solution
- You can use technology to directly calculate the confidence interval.
- The first solution is step-by-step (Solution A).
- The second solution uses the Ti-83+ and Ti-84 calculators (Solution B).
## Solution A
To find the confidence interval, you need the sample mean, $\bar{x}$, and the EBM.
$$
\begin{aligned}
& \bar{x}=8.2267 \quad \mathrm{~s}=1.6722 \quad \mathrm{n}=15 \\
& \mathrm{df}=15-1=14 \\
& \mathrm{CL}=0.95 \mathrm{so} \alpha=1-\mathrm{CL}=1-0.95=0.05 \\
& \frac{\alpha}{2}=0.025 \quad t_{\frac{\alpha}{2}}=t_{0.025}
\end{aligned}
$$
The area to the right of $\mathrm{t} .025$ is 0.025 and the area to the left of $\mathrm{t} .025$ is $1-0.025=0.975$ $$
t_{\frac{\alpha}{2}}=t_{.025}=2.14
$$
using invT(.975,14) on the TI-84+ calculator.
$$
t_{\frac{\alpha}{2}}=t_{.025}=2.14
$$
$$
\begin{gathered}
E B M=t_{\frac{\alpha}{2}} \cdot\left(\frac{s}{\sqrt{n}}\right) \\
E B M=2.14 \cdot\left(\frac{1.6722}{\sqrt{15}}\right)=0.924 \\
\bar{x}-E B M=8.2267-0.9240=7.3 \\
\bar{x}+E B M=8.2267+0.9240=9.15
\end{gathered}
$$
The $95 \%$ confidence interval is $(7.30,9.15)$.
We estimate with $95 \%$ confidence that the true population mean sensory rate is between 7.30 and 9.15 .
## Solution B
Using a function of the TI-83, TI-83+ or TI-84 calculators:
Press STAT and arrow over to TESTS.
Arrow down to 8:TInterval and press ENTER (or you can just press 8).
Arrow to Data and press ENTER.
Arrow down to List and enter the list name where you put the data. Arrow down to Freq and enter 1.
Arrow down to C-level and enter .95
Arrow down to Calculate and press ENTER.
The $95 \%$ confidence interval is $(7.3006,9.1527)$
Note: When calculating the error bound, a probability table for the Student's-t distribution can also be used to find the value of $t$. The table gives $\mathrm{t}$-scores that correspond to the confidence level (column) and degrees of freedom (row); the t-score is found where the row and column intersect in the table.
**With contributions from Roberta Bloom
### Confidence Intervals: Confidence Interval for a Population Proportion
(c) (i) (2)
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon
s.org/licenses/by-sa/4.0/).
During an election year, we see articles in the newspaper that state confidence intervals in terms of proportions or percentages. For example, a poll for a particular candidate running for president might show that the candidate has $40 \%$ of the vote within 3 percentage points. Often, election polls are calculated with 95\% confidence. So, the pollsters would be $95 \%$ confident that the true proportion of voters who favored the candidate would be between 0.37 and $0.43:(0.40-0.03,0.40+0.03)$. Investors in the stock market are interested in the true proportion of stocks that go up and down each week. Businesses that sell personal computers are interested in the proportion of households in the United States that own personal computers.
Confidence intervals can be calculated for the true proportion of stocks that go up or down each week and for the true proportion of households in the United States that own personal computers.
The procedure to find the confidence interval, the sample size, the error bound, and the confidence level for a proportion is similar to that for the population mean. The formulas are different.
How do you know you are dealing with a proportion problem? First, the underlying distribution is binomial. (There is no mention of a mean or average.) If $\mathbf{X}$ is a binomial random variable, then $\mathbf{X} \sim \mathbf{B}(\mathbf{n}, \mathbf{p})$ where $n$ the number of trials and $\mathbf{p}$ the probability of a success. To form a proportion, take $\mathbf{X}$, the random variable for the number of successes and divide it by $n$, the number of trials (or the sample size). The random variable $\mathbf{P}^{\prime}$ (read "P prime") is that proportion,
$$
P^{\prime}=\frac{X}{n}
$$
(Sometimes the random variable is denoted as $\hat{P}$, read "P hat".) When $\mathrm{n}$ is large and $\mathrm{p}$ is not close to 0 or 1 , we can use the normal distribution to approximate the binomial.
$$
X \sim N(n \cdot p, \sqrt{n \cdot p \cdot q})
$$
If we divide the random variable by $n$, the mean by $n$, and the standard deviation by $n$, we get a normal distribution of proportions with $\mathrm{P}^{\prime}$, called the estimated proportion, as the random variable. (Recall that a proportion $=$ the number of successes divided by $n$.
$$
\frac{X}{n}=P^{\prime} \sim N\left(\frac{n \cdot p}{n}, \frac{\sqrt{n \cdot p \cdot q}}{n}\right)
$$
Using algebra to simplify :
$$
\frac{\sqrt{n \cdot p \cdot q}}{n}=\sqrt{\frac{p \cdot q}{n}}
$$
## P' follows a normal distribution for proportions:
$$
P^{\prime} \sim N\left(P, \sqrt{\frac{p \cdot q}{n}}\right)
$$
The confidence interval has the form ( $p^{\prime}$ - EBP, $\left.p^{\prime}+\mathbf{E B P}\right)$.
$$
p^{\prime}=\frac{x}{n}
$$
$p^{\prime}=$ the estimated proportion of successes ( $p^{\prime}$ is a point estimate for $p$, the true proportion)
$x=$ the number of successes.
$\mathrm{n}=$ the size of the sample
## The error bound for a proportion is
$$
E B P=z_{\frac{\alpha}{2}} \cdot \sqrt{\frac{p^{\prime} \cdot q^{\prime}}{n}}
$$
where $q^{\prime}=1-p^{\prime}$
This formula is similar to the error bound formula for a mean, except that the "appropriate standard deviation" is different. For a mean, when the population standard deviation is known, the appropriate standard deviation that we use is $\frac{\sigma}{\sqrt{n}}$. For a proportion, the appropriate standard deviation is $\sqrt{\frac{p \cdot q}{n}}$.
However, in the error bound formula, we use $\sqrt{\frac{p^{\prime} \cdot q^{\prime}}{n}}$ as the standard deviation, instead of $\sqrt{\frac{p \cdot q}{n}}$. However, in the error bound formula, the standard deviation is $\sqrt{\frac{p^{\prime} \cdot q^{\prime}}{n}}$. In the error bound formula, the sample proportions $\mathbf{p}^{\prime}$ and $\mathbf{q}^{\prime}$ are estimates of the unknown population proportions $\mathbf{p}$ and $\mathbf{q}$. The estimated proportions $p^{\prime}$ and $q^{\prime}$ are used because $p$ and $q$ are not known. $p^{\prime}$ and $q^{\prime}$ are calculated from the data. $p^{\prime}$ is the estimated proportion of successes. $q^{\prime}$ is the estimated proportion of failures.
The confidence interval can only be used if the number of successes $\mathrm{np}$ ' and the number of failures nq' are both larger than 5.
Note: For the normal distribution of proportions, the z-score formula is as follows.
If
then the z-score formula is
$$
P^{\prime} \sim N\left(p, \sqrt{\frac{p \cdot q}{n}}\right)
$$
$$
z=\frac{p^{\prime}-p}{\sqrt{\frac{p \cdot q}{n}}}
$$
## Example 4.8
Suppose that a market research firm is hired to estimate the percent of adults living in a large city who have cell phones. 500 randomly selected adult residents in this city are surveyed to determine whether they have cell phones. Of the 500 people surveyed, 421 responded yes - they own cell phones. Using a 95\% confidence level, compute a confidence interval estimate for the true proportion of adults residents of this city who have cell phones.
## Solution
- You can use technology to directly calculate the confidence interval.
- The first solution is step-by-step (Solution A).
- The second solution uses a function of the TI-83, $83+$ or 84 calculators (Solution B).
## Solution A
Let $\mathrm{X}=$ the number of people in the sample who have cell phones. $\mathrm{X}$ is binomial. $X \sim B\left(500, \frac{421}{500}\right)$.
To calculate the confidence interval, you must find $\mathrm{p}^{\prime}, \mathrm{q}^{\prime}$, and EBP.
$\mathrm{n}=500 \quad \mathrm{x}=$ the number of successes $=421$
$$
p^{\prime}=\frac{x}{n}=\frac{421}{500}=0.842
$$
$\mathrm{p}^{\prime}=0.842$ is the sample proportion; this is the point estimate of the population proportion.
$$
q^{\prime}=1-p^{\prime}=1-0.842=0.158
$$
Since $C L=0.95$, then $\alpha=1-C L=1-0.95=0.05$
Then
$$
\frac{a}{2}=0.025 \text {. }
$$
$$
z_{\frac{a}{2}}=z_{.025}=1.96
$$
Use the TI-83, $83+$ or 84 + calculator command invNorm(0.975,0,1) to find $\mathrm{z}_{.025}$. Remember that the area to the right of $\mathrm{z}_{.025}$ is 0.025 and the area to the left of $x_{.025}$ is 0.975 . This can also be found using appropriate commands on other calculators, using a computer, or using a Standard Normal probability table.
$$
\begin{gathered}
E B P=z_{\frac{a}{2}} \cdot \sqrt{\frac{p^{\prime} \cdot q^{\prime}}{n}}=1.96 \cdot \sqrt{\frac{(0.842) \cdot(0.158)}{500}}=0.032 \\
p^{\prime}-E B P=0.842-0.032=0.81 \\
p^{\prime}+E B P=0.842+0.032=0.874
\end{gathered}
$$
The confidence interval for the true binomial population proportion is
$$
\left(p^{\prime}-E B P, p^{\prime}+E B P\right)=(0.810,0.874)
$$
## Interpretation
We estimate with $95 \%$ confidence that between $81 \%$ and $87.4 \%$ of all adult residents of this city have cell phones.
## Explanation of $\mathbf{9 5} \%$ Confidence Level
$95 \%$ of the confidence intervals constructed in this way would contain the true value for the population proportion of all adult residents of this city who have cell phones.
## Solution B
Using a function of the TI-83, 83+ or 84 calculators:
Press STAT and arrow over to TESTS .
Arrow down toA : 1 -Propzint. Press enter.
Arrow down to $\mathrm{x}$ and enter 421 .
Arrow down to $\mathrm{n}$ and enter 500.
Arrow down to c-Level and enter 95 .
Arrow down to calculate and press enter.
The confidence interval is $(0.81003,0.87397)$.
## Example 4.9
For a class project, a political science student at a large university wants to estimate the percent of students that are registered voters. He surveys 500 students and finds that 300 are registered voters. Compute a $90 \%$ confidence interval for the true percent of students that are registered voters and interpret the confidence interval.
## Solution
- You can use technology to directly calculate the confidence interval.
- The first solution is step-by-step (Solution A).
- The second solution uses a function of the TI-83, 83+ or 84 calculators (Solution B).
## Solution A
$\mathrm{x}=300$ and $\mathrm{n}=500$.
$p^{\prime}=\frac{x}{n}=\frac{300}{500}=0.600$
$q^{\prime}=1-p^{\prime}=1-0.600=0.400$
Since $C L=0.90$, then $\alpha=1-C L=1-0.90=0.10 \quad \frac{\alpha}{2}=0.05$.
$z_{\frac{\alpha}{2}}=z_{.05}=1.645$
Use the TI-83, 83+ or $84+$ calculator command $\operatorname{invNorm}(0.95,0,1)$ to find $z_{.05}$. Remember that the area to the right of $z_{.05}$ is 0.05 and the area to the left of $\mathrm{z} .05$ is 0.95 . This can also be found using appropriate commands on other calculators, using a computer, or using a Standard Normal probability table.
$$
\begin{aligned}
E B P=z_{\frac{\alpha}{2}} \cdot \sqrt{\frac{p^{\prime} \cdot q^{\prime}}{n}} & =1.645 \cdot \sqrt{\frac{(0.60) \cdot(0.40)}{500}}=0.036 \\
p^{\prime}-E B P & =0.60-0.036=0.564 \\
p^{\prime}+E B P & =0.60+0.036=0.636
\end{aligned}
$$
The confidence interval for the true binomial population proportion is $\left(p^{\prime}-E B P, p^{\prime}+E B P\right)=(0.564,0.636)$.
## Interpretation:
- We estimate with $90 \%$ confidence that the true percent of all students that are registered voters is between $56.4 \%$ and $63.6 \%$.
- Alternate Wording: We estimate with $90 \%$ confidence that between $56.4 \%$ and $63.6 \%$ of ALL students are registered voters.
## Explanation of $\mathbf{9 0} \%$ Confidence Level
$90 \%$ of all confidence intervals constructed in this way contain the true value for the population percent of students that are registered voters.
## Solution B
Using a function of the TI-83, $83+$ or 84 calculators:
Press stat and arrow over to tests .
Arrow down to A:1-Propzint. Press enter.
Arrow down to $\mathrm{x}$ and enter 300 .
Arrow down to $\mathrm{n}$ and enter 500.
Arrow down to c-Level and enter 90 .
Arrow down to calculate and press enter .
The confidence interval is $(0.564,0.636)$.
#### Calculating the Sample Size $n$
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon s.org/licenses/by-sa/4.0/).
If researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size. The error bound formula for a population proportion is
$$
E B P=z_{\frac{\alpha}{2}} \cdot \sqrt{\frac{p^{\prime} q^{\prime}}{n}}
$$
- Solving for $\mathrm{n}$ gives you an equation for the sample size.
$$
n=\frac{z_{\frac{a}{2}}^{2} \cdot p^{\prime} q^{\prime}}{E B P^{2}}
$$
## Example 4.10
Suppose a mobile phone company wants to determine the current percentage of customers aged 50+ that use text messaging on their cell phone. How many customers aged 50+ should the company survey in order to be $90 \%$ confident that the estimated (sample) proportion is within 3 percentage points of the true population proportion of customers aged $50+$ that use text messaging on their cell phone.
## Solution
From the problem, we know that $\mathrm{EBP}=\mathbf{0 . 0 3}(3 \% 0.03)$ and $z_{\frac{a}{2}}=z_{.05}=1.645$ because the confidence level is $90 \%$
However, in order to find $\mathrm{n}$, we need to know the estimated (sample) proportion $\mathrm{p}^{\prime}$. Remember that $\mathrm{q}^{\prime}=1^{-} \mathrm{p}^{\prime}$. But, we do not know $\mathrm{p}^{\prime}$ yet. Since we multiply $\mathrm{p}^{\prime}$ and $\mathrm{q}^{\prime}$ together, we make them both equal to 0.5 because $\mathrm{p}^{\prime} \mathrm{q}^{\prime}=(.5)(.5)=.25$ results in the largest possible product. (Try other products: $(.6)(.4)=.24 ;(.3)(.7)=.21 ;(.2)(.8)=.16$ and so on). The largest possible product gives us the largest $n$. This gives us a large enough sample so that we can be $90 \%$ confident that we are within 3 percentage points of the true population proportion. To calculate the sample size $n$, use the formula and make the substitutions.
$$
n=\frac{z^{2} p^{\prime} q^{\prime}}{E B P^{2}} \text { gives } n=\frac{1.645^{2} \cdot(.5)(.5)}{.03^{2}}=751.7
$$
Round the answer to the next higher value. The sample size should be 752 cell phone customers aged 50+ in order to be $90 \%$ confident that the estimated (sample) proportion is within 3 percentage points of the true population proportion of all customers aged 50+ that use text messaging on their cell phone.
**With contributions from Roberta Bloom.
## Chapter 5 Hypothesis Testing
### Hypothesis Testing of Single Mean and Single Proportion: Introduction
#### Student Learning Outcomes
(c) (i) (2)
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon s.org/licenses/by-sa/4.0/).
## Learning Objectives
By the end of this chapter, the student should be able to:
- Diferentiate between Type I and Type II Errors
- Describe hypothesis testing in general and in practice
- Conduct and interpret hypothesis tests for a single population mean, population standard deviation known.
- Conduct and interpret hypothesis tests for a single population mean, population standard deviation unknown.
- Conduct and interpret hypothesis tests for a single population proportion.
#### Introduction
s.org/licenses/by-sa/4.0/).
One job of a statistician is to make statistical inferences about populations based on samples taken from the population. Confidence intervals are one way to estimate a population parameter. Another way to make a statistical inference is to make a decision about a parameter. For instance, a car dealer advertises that its new small truck gets 35 miles per gallon, on the average. A tutoring service claims that its method of tutoring helps $90 \%$ of its students get an A or a B. A company says that women managers in their company earn an average of $\$ 60,000$ per year.
A statistician will make a decision about these claims. This process is called "hypothesis testing." A hypothesis test involves collecting data from a sample and evaluating the data. Then, the statistician makes a decision as to whether or not there is sufcient evidence based upon analyses of the data, to reject the null hypothesis.
In this chapter, you will conduct hypothesis tests on single means and single proportions. You will also learn about the errors associated with these tests.
Hypothesis testing consists of two contradictory hypotheses or statements, a decision based on the data, and a conclusion. To perform a hypothesis test, a statistician will: 1. Set up two contradictory hypotheses.
2. Collect sample data (in homework problems, the data or summary statistics will be given to you).
3. Determine the correct distribution to perform the hypothesis test.
4. Analyze sample data by performing the calculations that ultimately will allow you to reject or fail to reject the null hypothesis.
5. Make a decision and write a meaningful conclusion.
Note: To do the hypothesis test homework problems for this chapter and later chapters, make copies of the appropriate special solution sheets. See the Table of Contents topic "Solution Sheets".
### Hypothesis Testing of Single Mean and Single Proportion: Null and Alternate Hypotheses
s.org/licenses/by-sa/4.0/).
The actual test begins by considering two hypotheses. They are called the null hypothesis and the alternate hypothesis. These hypotheses contain opposing viewpoints.
$\mathbf{H}_{\mathbf{0}}$ : The null hypothesis: It is a statement about the population that will be assumed to be true unless it can be shown to be incorrect beyond a reasonable doubt.
$\mathrm{H}_{\mathrm{a}}$ : The alternate hypothesis: It is a claim about the population that is contradictory to $\mathbf{H}_{\mathbf{0}}$ and what we conclude when we reject $\mathbf{H}_{\mathbf{0}}$.
## Example 5.1
$\mathbf{H}_{\mathbf{o}}$ : No more than $30 \%$ of the registered voters in Santa Clara County voted in the primary election.
$\mathrm{H}_{\mathrm{a}}$ : More than $30 \%$ of the registered voters in Santa Clara County voted in the primary election.
## Example 5.2
We want to test whether the mean grade point average in American colleges is different from 2.0 (out of 4.0).
$$
H_{0}: \mu=2.0 \quad H_{\alpha}: \mu \neq 2.0
$$
## Example 5.3
We want to test if college students take less than five years to graduate from college, on the average.
$$
H o: \mu \geqslant 5 \quad H_{\alpha}: \mu<5
$$
## Example 5.4
In an issue of U. S. News and World Report, an article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third pass. The same article stated that $6.6 \%$ of U. S. students take advanced placement exams and $4.4 \%$ pass. Test if the percentage of U.S. students who take advanced placement exams is more than $6.6 \%$.
$$
\text { Ho }: p=0.066 \quad H_{\alpha}: p>0.066
$$
Since the null and alternate hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.
After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are "reject $\mathbf{H}_{\mathbf{0}}$ " if the sample information favors the alternate hypothesis or "do not reject Ho" or "fail to reject $\mathbf{H}_{\mathbf{o}}$ " if the sample information is insufficient to reject the null hypothesis.
Mathematical Symbols Used in $\mathbf{H}_{\mathbf{0}}$ and $\mathbf{H}_{\mathbf{a}}$ :
| $\mathrm{H}_{\mathrm{O}}$ | $\mathrm{H}_{\mathrm{a}}$ |
| :--- | :--- |
| equal $(=)$ | $\begin{array}{l}\text { not equal }(\neq) \text { or greater than }(>) \text { or less } \\ \text { than }(<)\end{array}$ |
| $\begin{array}{l}\text { greater than or equal } \\ \text { to }(\geq)\end{array}$ | less than $(<)$ |
| $\begin{array}{l}\text { less than or equal to } \\ (\leq)\end{array}$ | more than $(>)$ |
Note: $\mathrm{H}_{\mathrm{O}}$ always has a symbol with an equal in it. $\mathrm{H}_{\mathrm{a}}$ never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers (including one of the $\mathrm{co}^{-}$ authors in research work) use $=$ in the Null Hypothesis, even with $>$ or $<$ as the symbol in the Alternate Hypothesis. This practice is acceptable because we only make the decision to reject or not reject the Null Hypothesis.
#### Optional Collaborative Classroom Activity
(c) (i) (2)
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon
s.org/licenses/by-sa/4.0/).
Bring to class a newspaper, some news magazines, and some Internet articles . In groups, find articles from which your group can write a null and alternate hypotheses. Discuss your hypotheses with the rest of the class.
### Hypothesis Testing of Single Mean and Single Proportion: Using the Sample to Test the Null Hypothesis
(c) (i) (2)
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon
s.org/licenses/by-sa/4.0/)
Use the sample data to calculate the actual probability of getting the test result, called the p-value. The p-value is the probability that, if the null hypothesis is true, the results from another randomly selected sample will be as extreme or more extreme as the results obtained from the given sample.
A large $p$-value calculated from the data indicates that we should fail to reject the null hypothesis. The smaller the p-value, the more unlikely the outcome, and the stronger the evidence is against the null hypothesis. We would reject the null hypothesis if the evidence is strongly against it.
Draw a graph that shows the $p$-value. The hypothesis test is easier to perform if you use a graph because you see the problem more clearly.
## Example 5.5: (to illustrate the p-value)
Suppose a baker claims that his bread height is more than $15 \mathrm{~cm}$, on the average. Several of his customers do not believe him. To persuade his customers that he is right, the baker decides to do a hypothesis test. He bakes 10 loaves of bread. The mean height of the sample loaves is $17 \mathrm{~cm}$. The baker knows from baking hundreds of loaves of bread that the standard deviation for the height is $0.5 \mathrm{~cm}$. and the distribution of heights is normal.
The null hypothesis could be $\mathrm{H}_{\mathbf{0}}: \boldsymbol{\mu} \leq 15$
The alternate hypothesis is $\mathbf{H}_{\mathbf{a}}: \mu>15$
The words "is more than" translates as a ">" so " $\mu>15$ " goes into the alternate hypothesis. The null hypothesis must contradict the alternate hypothesis. Since $\sigma$ is known $(\sigma=0.5 \mathrm{~cm}$.), the distribution for the population is known to be normal with mean $\mu=15$ and standard deviation
$$
\frac{\sigma}{\sqrt{n}}=\frac{0.5}{\sqrt{10}}=0.16 \text {. }
$$
Suppose the null hypothesis is true (the mean height of the loaves is no more than $15 \mathrm{~cm})$. Then is the mean height $(17 \mathrm{~cm})$ calculated from the sample unexpectedly large? The hypothesis test works by asking the question how unlikely the sample mean would be if the null hypothesis were true. The graph shows how far out the sample mean is on the normal curve. The $\mathrm{p}$-value is the probability that, if we were to take other samples, any other sample mean would fall at least as far out as $17 \mathrm{~cm}$.
The $\mathrm{p}$-value, then, is the probability that a sample mean is the same or greater than $17 \mathrm{~cm}$. when the population mean is, in fact, $\mathbf{1 5} \mathbf{~ c m}$. We can calculate this probability using the normal distribution for means from Chapter 7.
## p-value is
approximately 0
$\mathrm{p}$-value $=\mathrm{P}\left(\bar{x}^{>}>17\right)$ which is approximately 0.
A p-value of approximately o tells us that it is highly unlikely that a loaf of bread rises no more than $15 \mathrm{~cm}$, on the average. That is, almost $0 \%$ of all loaves of bread would be at least as high as $17 \mathrm{~cm}$. purely by CHANCE had the population mean height really been $15 \mathrm{~cm}$. Because the outcome of $17 \mathrm{~cm}$. is so unlikely (meaning it is happening NOT by chance alone), we conclude that the evidence is strongly against the null hypothesis (the mean height is at most $15 \mathrm{~cm}$.). There is sufcient evidence that the true mean height for the population of the baker's loaves of bread is greater than $15 \mathrm{~cm}$.
### Hypothesis Testing of Single Mean and Single Proportion: Decision and Conclusion
s.org/licenses/by-sa/4.0/).
A systematic way to make a decision of whether to reject or not reject the null hypothesis is to compare the $\mathbf{p}$-value and a preset or preconceived $\alpha$ (also called a "significance level"). A preset $\alpha$ is the probability of a Type I error (rejecting the null hypothesis when the null hypothesis is true). It may or may not be given to you at the beginning of the problem.
When you make a decision to reject or not reject $\mathrm{H}_{0}$, do as follows:
- If $\alpha>p$-value, reject $\mathbf{H}_{\mathbf{0}}$. The results of the sample data are significant. There is sufficient evidence to conclude that $\mathbf{H}_{\mathbf{0}}$ is an incorrect belief and that the alternative hypothesis, $\mathbf{H}_{\mathbf{a}}$, may be correct.
- If $\alpha \leq p$-value, do not reject $\mathbf{H}_{\mathbf{0}}$. The results of the sample data are not significant. There is not sufficient evidence to conclude that the alternative hypothesis, $\mathbf{H}_{\mathbf{a}}$, may be correct.
- When you "do not reject $\mathbf{H}_{\mathbf{o}}$ ", it does not mean that you should believe that $\mathbf{H}_{\mathbf{0}}$ is true. It simply means that the sample data have failed to provide sufficient evidence to cast serious doubt about the truthfulness of $\mathbf{H}_{\mathbf{0}}$.
Conclusion: After you make your decision, write a thoughtful conclusion about the hypotheses in terms of the given problem.
## Chapter 6 Linear Regression and Correlation
### Linear Regression and Correlation: Introduction
#### Student Learning Outcomes
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon s.org/licenses/by-sa/4.0/).
## Learning Objectives
By the end of this chapter, the student should be able to:
- Discuss basic ideas of linear regression and correlation.
- Create and interpret a line of best fit.
- Calculate and interpret the correlation coefficient.
- Calculate and interpret outliers.
#### Introduction
(c) (i) (2)
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon
s.org/licenses/by-sa/4.0/).
Professionals often want to know how two or more numeric variables are related. For example, is there a relationship between the grade on the second math exam a student takes and the grade on the final exam? If there is a relationship, what is it and how strong is the relationship?
In another example, your income may be determined by your education, your profession, your years of experience, and your ability. The amount you pay a repair person for labor is often determined by an initial amount plus an hourly fee. These are all examples in which regression can be used.
The type of data described in the examples is bivariate data -"bi" for two variables. In reality, statisticians use multivariate data, meaning many variables.
In this chapter, you will be studying the simplest form of regression, "linear regression" with one independent variable $(\mathrm{x})$. This involves data that fits a line in two dimensions. You will also study correlation which measures how strong the relationship is.
### Linear Regression and Correlation: Linear Equations
s.org/licenses/by-sa/4.0/).
Linear regression for two variables is based on a linear equation with one independent variable. It has the form:
$$
y=a+b x
$$
where $\mathrm{a}$ and $\mathrm{b}$ are constant numbers.
$\boldsymbol{x}$ is the independent variable, and $\boldsymbol{y}$ is the dependent variable. Typically, you choose a value to substitute for the independent variable and then solve for the dependent variable.
## Example 6.1
The following examples are linear equations.
$$
\begin{gathered}
y=3+2 x \\
y=-0.01+1.2 x
\end{gathered}
$$
The graph of a linear equation of the form $y=a+b x$ is a straight line. Any line that is not vertical can be described by this equation.
Figure 6.1 Graph of the equation $y=-1+2 x$.
Linear equations of this form occur in applications of life sciences, social sciences, psychology, business, economics, physical sciences, mathematics, and other areas.
## Example 6.3
Aaron's Word Processing Service (AWPS) does word processing. Its rate is $\$ 32$ per hour plus a $\$ 31.50$ one-time charge. The total cost to a customer depends on the number of hours it takes to do the word processing job.
See the following Problem.
## Problem
Find the equation that expresses the total cost in terms of the number of hours required to finish the word processing job.
## Solution
Let $x=$ the number of hours it takes to get the job done.
Let $y=$ the total cost to the customer.
The $\$ 31.50$ is a fixed cost. If it takes $x$ hours to complete the job, then (32) $(x)$ is the cost of the word processing only. The total cost is:
$$
y=31.50+32 x
$$
### Linear Regression and Correlation: Slope and Y- Intercept of a Linear Equation
s.org/licenses/by-sa/4.0/).
For the linear equation $y=\mathrm{a}+\mathrm{b} x, \mathrm{~b}=$ slope and $\mathrm{a}=y$-intercept.
From algebra recall that the slope is a number that describes the steepness of a line and the $y$-intercept is the $y$ coordinate of the point $(0, a)$ where the line crosses the $y$ axis.
(a)
(b)
(c)
Figure 6.2 Three possible graphs of $y=a+b x$. (a) If $b>0$, the line slopes upward to the right. (b) If $b$ $=0$, the line is horizontal. (c) If $b<0$, the line slopes downward to the right.
## Example 6.4
Svetlana tutors to make extra money for college. For each tutoring session, she charges a one time fee of $\$ 25$ plus $\$ 15$ per hour of tutoring. A linear equation that expresses the total amount of money Svetlana earns for each session she tutors is $\mathrm{y}=25+15 \mathrm{X}$.
See the following Problem.
## Problem
What are the independent and dependent variables? What is the $\mathrm{y}^{-}$ intercept and what is the slope? Interpret them using complete sentences.
## Solution
The independent variable $(\mathrm{x})$ is the number of hours Svetlana tutors each session. The dependent variable (y) is the amount, in dollars, Svetlana earns for each session.
The $\mathrm{y}$-intercept is $25(\mathrm{a}=25)$. At the start of the tutoring session, Svetlana charges a one-time fee of $\$ 25$ (this is when $x=0$ ). The slope is $15(b=15)$. For each session, Svetlana earns $\$ 15$ for each hour she tutors.
### Linear Regression and Correlation: Scatter Plots
s.org/licenses/by-sa/4.0/).
Before we take up the discussion of linear regression and correlation, we need to examine a way to display the relation between two variables $x$ and $y$. The most common and easiest way is a scatter plot. The following example illustrates a scatter plot.
## Example 6.5
From an article in the Wall Street Journal: In Europe and Asia, mcommerce is popular. $\mathrm{M}$-commerce users have special mobile phones that work like electronic wallets as well as provide phone and Internet services. Users can do everything from paying for parking to buying a TV set or soda from a machine to banking to checking sports scores on the Internet. For the years 2000 through 2004, was there a relationship between the year and the number of $\mathrm{m}$-commerce users? Construct a scatter plot. Let $\mathrm{x}=$ the year and let $\mathrm{y}=$ the number of $\mathrm{m}$-commerce users, in millions.
| $\mathbf{x}$ (year) | $\mathbf{y}$ (\# of users) |
| :--- | :--- |
| 2000 | 0.5 |
| 2002 | 20.0 |
| 2003 | 33.0 |
| 2004 | 47.0 |
Table 6.1 Table showing the number of $\mathrm{m}$-commerce users (in millions) by year.
Figure 6.3 Scatter plot showing the number of $\mathrm{m}$-commerce users (in millions) by year.
A scatter plot shows the direction and strength of a relationship between the variables. A clear direction happens when there is either:
- High values of one variable occurring with high values of the other variable or low values of one variable occurring with low values of the other variable.
- High values of one variable occurring with low values of the other variable.
You can determine the strength of the relationship by looking at the scatter plot and seeing how close the points are to a line, a power function, an exponential function, or to some other type of function.
When you look at a scatterplot, you want to notice the overall pattern and any deviations from the pattern. The following scatterplot examples illustrate these concepts.
Figure 6.4 (a) Positive Linear Pattern (Strong)
(b) Linear Pattern w/ One Deviation
Figure 6.5 (a) Negative Linear Pattern (Strong)
(b) Negative Linear Pattern (Weak)
Figure 6.6 (a) Exponential Growth Pattern (b) No Pattern
In this chapter, we are interested in scatter plots that show a linear pattern. Linear patterns are quite common. The linear relationship is strong if the points are close to a straight line. If we think that the points show a linear relationship, we would like to draw a line on the scatter plot. This line can be calculated through a process called linear regression. However, we only calculate a regression line if one of the variables helps to explain or predict the other variable. If $x$ is the independent variable and $y$ the dependent variable, then we can use a regression line to predict $y$ for a given value of $x$.
### Linear Regression and Correlation: The Regression
## Equation
## (c) (i) (2)
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon
s.org/licenses/by-sa/4.0/).
Data rarely fit a straight line exactly. Usually, you must be satisfied with rough predictions. Typically, you have a set of data whose scatter plot appears to "fit" a straight line. This is called a Line of Best Fit or Least Squares Line.
#### Optional Collaborative Classroom Activity
(c) (i) (0)
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon
s.org/licenses/by-sa/4.0/).
If you know a person's pinky (smallest) finger length, do you think you could predict that person's height? Collect data from your class (pinky finger length, in inches). The independent variable, $x$, is pinky finger length and the dependent variable, $y$, is height.
For each set of data, plot the points on graph paper. Make your graph big enough and use a ruler. Then "by eye" draw a line that appears to "fit" the data. For your line, pick two convenient points and use them to find the slope of the line. Find the y-intercept of the line by extending your lines so they cross the $y$-axis. Using the slopes and the $y$ intercepts, write your equation of "best fit". Do you think everyone will have the same equation? Why or why not?
Using your equation, what is the predicted height for a pinky length of 2.5 inches?
## Example 6.6
A random sample of 11 statistics students produced the following data where $\mathrm{x}$ is the third exam score, out of 80 , and $\mathrm{y}$ is the final exam score, out of 200. Can you predict the final exam score of a random student if you know the third exam score?
| $\mathbf{x}$ (third exam score) | y (fnal exam score) |
| :--- | :--- |
| 65 | 175 |
| 67 | 133 |
| 71 | 185 |
| 71 | 163 |
| 66 | 126 |
| 75 | 198 |
| 67 | 163 |
| 70 | 153 |
Table 6.2 Table showing the scores on the final exam based on scores from the third exam.
| $\mathbf{x}$ (third exam score) | $\mathbf{y}$ (fnal exam score) |
| :--- | :--- |
| 71 | 159 |
| 69 | 151 |
| 69 | 159 |
Table 6.2 Table showing the scores on the final exam based on scores from the third exam.
Figure 6.7 Scatter plot showing the scores on the fnal exam based on scores from the third exam.
The third exam score, $x$, is the independent variable and the final exam score, $y$, is the dependent variable. We will plot a regression line that best "fits" the data. If each of you were to fit a line "by eye", you would draw different lines. We can use what is called a least-squares regression line to obtain the best fit line. Consider the following diagram. Each point of data is of the form $(x, y)$ and each point of the line of best fit using least-squares linear regression has the form $(x, \hat{y})$.
The $\hat{y}$ is read "y hat" and is the estimated value of $\mathbf{y}$. It is the value of y obtained using the regression line. It is not generally equal to $y$ from data.
Figure 6.8
The term
$$
y_{0}-\hat{y}_{0}=\epsilon_{0}
$$
is called the "error" or residual. It is not an error in the sense of a mistake.
The absolute value of a residual measures the vertical distance between the actual value of $y$ and the estimated value of $y$. In other words, it measures the vertical distance between the actual data point and the predicted point on the line.
If the observed data point lies above the line, the residual is positive, and the line underestimates the actual data value for $y$. If the observed data point lies below the line, the residual is negative, and the line overestimates that actual data value for $y$.
In the diagram above, $y_{0}-\hat{y}_{0}=\epsilon_{0}$ is the residual for the point shown. Here the point lies above the line and the residual is positive.
$\epsilon=$ the Greek letter epsilon
For each data point, you can calculate the residuals or errors,
$$
y_{i}-\hat{y}_{i}=\epsilon_{i} \text { for } i=1,2,3, \ldots, 11 .
$$
Each $|\epsilon|$ is a vertical distance.
For the example about the third exam scores and the fnal exam scores for the 11 statistics students, there are 11 data points. Therefore, there are $11 \epsilon$ values. If you square each $\epsilon$ and add, you get
$$
\left(\epsilon_{1}\right)^{2}+\left(\epsilon_{2}\right)^{2}+\ldots+\left(\epsilon_{11}\right)^{2}=\sum_{i=1}^{11} \epsilon^{2}
$$
This is called the Sum of Squared Errors (SSE).
Using calculus, you can determine the values of $a$ and $b$ that make the SSE $a$ minimum. When you make the SSE a minimum, you have determined the points that are on the line of best fit. It turns out that the line of best fit has the equation:
$$
\hat{y}=a+b x
$$
where
$$
a=\bar{y}-b \cdot \bar{x}
$$
and
$$
b=\frac{\sum(x-\bar{x}) \cdot(y-\bar{y})}{\sum(x-\bar{x})^{2}}
$$
$\bar{x}$ and $\bar{y}$ are the sample means of the $\mathrm{x}$ values and the $\mathrm{y}$ values, respectively. The best fit line always passes through the point $(\bar{x}, \bar{y})$.
The slope $b$ can be written as
$$
b=r \cdot\left(\frac{s_{y}}{s_{x}}\right)
$$
where $s_{y}=$ the standard deviation of the $y$ values and $s_{x}$ the standard deviation of the $x$ values. $r$ is the correlation coefficient which is discussed in the next section.
## Least Squares Criteria for Best Fit
The process of fitting the best fit line is called linear regression. The idea behind finding the best fit line is based on the assumption that the data are scattered about a straight line. The criteria for the best fit line is that the sum of the squared errors (SSE) is minimized, that is made as small as possible. Any other line you might choose would have a higher SSE than the best fit line. This best fit line is called the least squares regression line.
Note: Computer spreadsheets, statistical software, and many calculators can quickly calculate the best fit line and create the graphs. The calculations tend to be tedious if done by hand. Instructions to use the TI-83, TI-83+, and TI-84+ calculators to find the best fit line and create a scatterplot are shown at the end of this section.
## THIRD EXAM vS FINAL EXAM EXAMPLE:
The graph of the line of best fit for the third exam/final exam example is shown below:
Figure 6.9 Exam Score
The least squares regression line (best fit line) for the third exam/final exam example has the equation:
$$
\hat{y}=-173.51+4.83 x
$$
Note: Remember, it is always important to plot a scatter diagram first. If the scatter plot indicates that there is a linear relationship between the variables, then it is reasonable to use a best fit line to make predictions for $\mathrm{y}$ given $\mathrm{x}$
values outside that domain.
Remember, it is always important to plot a scatter diagram first. If the scatter plot indicates that there is a linear relationship between the variables, then it is reasonable to use a best fit line to make predictions for $y$ given $x$ within the domain of $x$-values in the sample data, but not necessarily for $\mathbf{x}$-values outside that domain.
You could use the line to predict the final exam score for a student who earned a grade of 73 on the third exam.
You should NOT use the line to predict the final exam score for a student who earned a grade of 50 on the third exam, because 50 is not within the domain of the $x$-values in the sample data, which are between 65 and 75 .
## UNDERSTANDING SLOPE
The slope of the line, $b$, describes how changes in the variables are related. It is important to interpret the slope of the line in the context of the situation represented by the data. You should be able to write a sentence interpreting the slope in plain English.
INTERPRETATION OF THE SLOPE: The slope of the best fit line tells us how the dependent variable $(y)$ changes for every one unit increase in the independent $(x)$ variable, on average.
## THIRD EXAM vS FINAL EXAM EXAMPLE
Slope: The slope of the line is $b 4.83$.
Interpretation: For a one point increase in the score on the third exam, the fnal exam score increases by 4.83 points, on average.
#### Using the $\mathrm{TI}-83+$ and $\mathrm{TI}-84+$ Calculators
(c) (i) (0)
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon
s.org/licenses/by-sa/4.0/).
## Using the Linear Regression T Test: LinRegTTest
Step 1. In the STAT list editor, enter the $X$ data in list $L 1$ and the $Y$ data in list $L 2$, paired so that the corresponding $(x, y)$ values are next to each other in the lists. (If a particular pair of values is repeated, enter it as many times as it appears in the data.)
Step 2. On the STAT TESTS menu, scroll down with the cursor to select the LinRegTTest. (Be careful to select LinRegTTest as some calculators may also have a different item called LinRegTInt.)
Step 3. On the LinRegTTest input screen enter: Xlist: L1 ; Ylist: L2 ; Freq: 1
Step 4. On the next line, at the prompt $\beta$ or $\rho$, highlight " $\neq 0$ " and press ENTER Step 5. Leave the line for "RegEq:" blank Step 6. Highlight Calculate and press ENTER.
$\beta$ or $\rho: \neq 0<0>0$
## Tl-83+ and TI-84+ calculators
LinRegTTest
Xlist: L1
Ylist: L2
Freq: 1
RegEQ:
Calculate
calculators
## d Output Screen
LinRegTTest
$y=a+b x$
$\beta \neq 0$ and $\rho \neq 0$
$t=2.657560155$
$p=.0261501512$
$d f=9$
$\downarrow a=-173.513363$
$b=4.827394209$
$s=16.41237711$
$r^{2}=.4396931104$
$r=.663093591$
Figure 6.10 LinRegtTest Screen
The output screen contains a lot of information. For now we will focus on a few items from the output, and will return later to the other items.
The second line says $y=a+b x$. Scroll down to find the values $a=-173.513$, and $b=$ 4.8273 ; the equation of
the best fit line is
$$
\hat{y}=-173.51+4.83 x
$$
The two items at the bottom are $r^{2}=.43969$ and $r=.663$. For now, just note where to find these values; we will discuss them in the next two sections.
## Graphing the Scatterplot and Regression Line
Step 1. We are assuming your $X$ data is already entered in list $L 1$ and your $Y$ data is in list L2
Step 2. Press 2nd STATPLOT ENTER to use Plot 1
Step 3. On the input screen for PLOT 1, highlight On and press ENTER
Step 4. For TYPE: highlight the very first icon which is the scatterplot and press ENTER
Step 5. Indicate Xlist: L1 and Ylist: L2
Step 6. For Mark: it does not matter which symbol you highlight.
Step 7. Press the ZOOM key and then the number 9 (for menu item "ZoomStat") ; the calculator will $\mathrm{ft}$ the window to the data
Step 8. To graph the best $\mathrm{ft}$ line, press the " $Y$ " key and type the equation $-173.5+4.83 \mathrm{X}$ into equation $Y 1$. (The $X$ key is immediately left of the STAT key). Press ZOOM 9 again to graph it.
Step 9. Optional: If you want to change the viewing window, press the WINDOW key. Enter your desired window using Xmin, Xmax, Ymin, Ymax
### Linear Regression and Correlation: Correlation Coefficient and Coefficient of Determination
#### The Correlation Coefficient $r$
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Besides looking at the scatter plot and seeing that a line seems reasonable, how can you tell if the line is a good predictor? Use the correlation coefficient as another indicator (besides the scatterplot) of the strength of the relationship between $x$ and $y$.
The correlation coefficient, $\mathbf{r}$, developed by Karl Pearson in the early $1900 \mathrm{~s}$, is a numerical measure of the strength of association between the independent variable $x$ and the dependent variable $y$.
The correlation coefficient is calculated as
$$
r=\frac{n \cdot \sum x \cdot y-\left(\sum x\right) \cdot\left(\sum y\right)}{\sqrt{\left[n \cdot \sum x^{2}-\left(\sum x\right)^{2}\right] \cdot\left[n \cdot \sum y^{2}-\left(\sum y\right)^{2}\right]}}
$$
where $n=$ the number of data points.
If you suspect a linear relationship between $x$ and $y$, then $r$ can measure how strong the linear relationship is.
## What the VALUE of $r$ tells us:
- The value of $r$ is always between -1 and $+1:-1 \leq r \leq 1$.
- The size of the correlation $r$ indicates the strength of the linear relationship between $x$ and $y$. Values of $r$ close to -1 or to +1 indicate a stronger linear relationship between $x$ and $y$.
- If $r=0$ there is absolutely no linear relationship between $x$ and $y$ (no linear correlation).
- If $r=1$, there is perfect positive correlation. If $r=-1$, there is perfect negative correlation. In both these cases, all of the original data points lie on a straight line. Of course, in the real world, this will not generally happen.
## What the SIGN of $r$ tells us
- A positive value of $r$ means that when $x$ increases, $y$ tends to increase and when $x$ decreases, $y$ tends to decrease (positive correlation).
- A negative value of $r$ means that when $x$ increases, $y$ tends to decrease and when $x$ decreases, $y$ tends to increase (negative correlation).
- The sign of $r$ is the same as the sign of the slope, $b$, of the best fit line.
Note: Strong correlation does not suggest that x causes y or y causes $\mathrm{x}$. We say "correlation does not imply causation." For example, every person who learned math in the 17th century is dead. However, learning math does not necessarily cause death!
Figure 6.11 (a) A scatter plot showing data with a positive correlation. $0<r<1$ (b) A scatter plot showing data with a negative correlation. $-1<\mathrm{r}<0$
Figure 6.12 (C) A scatter plot showing data with zero correlation. $r=0$
The formula for $r$ looks formidable. However, computer spreadsheets, statistical software, and many calculators can quickly calculate $r$. The correlation coefficient $r$ is the bottom item in the output screens for the LinRegTTest on the TI-83, TI-83+, or TI-84+ calculator (see previous section for instructions).
#### The Coefficient of Determination
## (c) (i) (2)
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon
s.org/licenses/by-sa/4.0/)
$r^{2}$ is called the coefficient of determination. $r^{2}$ is the square of the correlation
coefficient, but is usually stated as a percent, rather than in decimal form. $r^{2}$ has an interpretation in the context of the data:
- $\mathbf{r}^{2}$, when expressed as a percent, represents the percent of variation in the dependent variable $y$ that can be explained by variation in the independent variable $x$ using the regression (best fit) line.
- $1-r^{2}$, when expressed as a percent, represents the percent of variation in $y$ that is NOT explained by variation in $x$ using the regression line. This can be seen as the scattering of the observed data points about the regression line.
## Consider the third exam/final exam example introduced in the previous section
The line of best fit is: $\hat{y}=-173.51+4.83 x$
The correlation coefficient is $r=0.6631$
The coefficient of determination is $r^{2}=0.6631^{2}=0.4397$
## Interpretation of $r^{2}$ in the context of this example:
Approximately $44 \%$ of the variation (0.4397 is approximately 0.44$)$ in the final exam grades can be explained by the variation in the grades on the third exam, using the best fit regression line.
Therefore approximately $56 \%$ of the variation $(1-0.44=0.56)$ in the fnal exam grades can NOT be explained by the variation in the grades on the third exam, using the best fit regression line. (This is seen as the scattering of the points about the line.)
**With contributions from Roberta Bloom.
### Linear Regression and Correlation: Testing the Significance of the Correlation Coefficient
#### Testing the Significance of the Correlation Coefficient
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The correlation coefficient, $r$, tells us about the strength of the linear relationship between $x$ and $y$. However, the reliability of the linear model also depends on how many observed data points are in the sample. We need to look at both the value of the correlation coefficient $r$ and the sample size $n$, together.
We perform a hypothesis test of the "significance of the correlation coefficient" to decide whether the linear relationship in the sample data is strong enough to use to model the relationship in the population.
The sample data is used to computer $r$, the correlation coefficient for the sample. If we had data for the entire population, we could find the population correlation coefficient. But because we only have sample data, we can not calculate the population correlation coefficient. The sample correlation coefficient, $r$, is our estimate of the unknown population correlation coefficient.
The symbol for the population correlation coefficient is $\rho$, the Greek letter "rho".
$\rho=$ population correlation coefficient (unknown)
$r=$ sample correlation coefficient (known; calculated from sample data)
The hypothesis test lets us decide whether the value of the population correlation coefficient $\rho$ is "close to 0 " or "significantly different from 0 ". We decide this based on the sample correlation coefficient $r$ and the sample size $n$.
## If the test concludes that the correlation coefficient is significantly different from 0 , we say that the correlation coefficient is "significant".
- Conclusion: "There is sufficient evidence to conclude that there is a significant linear relationship between $x$ and $y$ because the correlation coefficient is significantly different from $0 . "$ - What the conclusion means: There is a significant linear relationship between $x$ and $y$. We can use the regression line to model the linear relationship between $x$ and $y$ in the population.
If the test concludes that the correlation coefficient is not significantly different from 0 (it is close to 0 ), we say that correlation coefficient is "not significant".
- Conclusion: "There is insufficient evidence to conclude that there is a significant linear relationship between $x$ and $y$ because the correlation coefficient is not significantly different from $0 . "$
- What the conclusion means: There is not a significant linear relationship between $x$ and $y$. Therefore we can NOT use the regression line to model a linear relationship between $x$ and $y$ in the population.
Note:
- If $r$ is significant and the scatter plot shows a linear trend, the line can be used to predict the value of $y$ for values of $x$ that are within the domain of observed $x$ values.
- If $r$ is not significant OR if the scatter plot does not show a linear trend, the line should not be used for prediction.
- If $r$ is significant and if the scatter plot shows a linear trend, the line may NOT be appropriate or reliable for prediction OUTSIDE the domain of observed $x$ values in the data.
## PERFORMING THE HYPOTHESIS TEST
## SETTING UP THE HYPOTHESES:
## - Null Hypothesis:
- Alternate Hypothesis:
$$
H_{0}: \rho=0
$$
$$
H_{a}: \rho \neq 0
$$
## What the hypotheses mean in words:
- Null Hypothesis $\mathbf{H}_{\mathbf{0}}$ : The population correlation coefficient IS NOT significantly different from 0 . There IS NOT a significant linear relationship(correlation) between $x$ and $y$ in the population.
- Alternate Hypothesis Ha: The population correlation coefficient IS significantly DIFFERENT FROM 0. There IS A SIGNIFICANT LINEAR RELATIONSHIP (correlation) between $x$ and $y$ in the population.
## DRAWING A CONCLUSION:
There are two methods to make the decision. Both methods are equivalent and give the same result.
## Method 1: Using the p-value
## Method 2: Using a table of critical values
In this chapter of this textbook, we will always use a significance level of $5 \%, \alpha=0.05$
Note: Using the $\mathrm{p}$-value method, you could choose any appropriate significance level you want; you are not limited to using $\alpha=0.05$. But the table of critical values provided in this textbook assumes that we are using a significance level of $5 \%, \alpha=0.05$. (If we wanted to use a different significance level than $5 \%$ with the critical value method, we would need different tables of critical values that are not provided in this textbook.)
## METHOD 1: Using a p-value to make a decision
The linear regression t-test LinRegTTEST on the TI-83+ or TI-84+ calculators calculates the p-value.
On the LinRegTTEST input screen, on the line prompt for $\beta$ or $\rho$, highlight " $\neq 0$ " The output screen shows the p-value on the line that reads "p = ".
(Most computer statistical software can calculate the p-value.)
If the $p$-value is less than the significance level $(\alpha=0.05)$ :
- Decision: REJECT the null hypothesis.
- Conclusion: "There is sufficient evidence to conclude that there is a significant linear relationship between $x$ and $y$ because the correlation coefficient is significantly different from $0 . "$
If the $p$-value is NOT less than the significance level $(\alpha=0.05)$ :
- Decision: DO NOT REJECT the null hypothesis.
- Conclusion: "There is insufficient evidence to conclude that there is a significant linear relationship between $x$ and $y$ because the correlation coefficient is NOT significantly different from $0 . "$
## Calculation Notes:
You will use technology to calculate the p-value. The following describe the calculations to compute the test statistics and the $p$-value:
The $p$-value is calculated using a t-distribution with $n-2$ degrees of freedom.
The formula for the test statistic is $t=\frac{r \sqrt{n-2}}{\sqrt{1-r^{2}}}$. The value of the test statistic, $t$, is shown in the computer or calculator output along with the p-value. The test statistic $t$ has the same sign as the correlation coefficient $r$.
The p-value is the combined area in both tails.
An alternative way to calculate the $p$-value $(p)$ given by LinRegTTest is the command $2 * \operatorname{tcdf}\left(\operatorname{abs}(\mathrm{t}), 10^{\wedge} 99, \mathrm{n}-2\right)$ in 2nd DISTR.
## THIRD EXAM vs FINAL EXAM EXAMPLE: $p$ value method
- Consider the third exam/final exam example.
- The line of best fit is: $\hat{y}=-173.51+4.83 x$ with $r=0.6631$ and there are $\mathrm{n}=11$ data points.
- Can the regression line be used for prediction? Given a third exam score ( $\boldsymbol{x}$ value), can we use the line to predict the final exam score (predicted $y$ value)?
$$
\begin{aligned}
& H_{0}: \rho=0 \\
& H_{a}: \rho \neq 0 \\
& \alpha=0.05
\end{aligned}
$$
The $p$-value is 0.026 (from LinRegTTest on your calculator or from computer software) The $p$-value, 0.026 , is less than the signifcance level of $\alpha=0.05$
Decision: Reject the Null Hypothesis $H_{0}$
Conclusion: There is sufcient evidence to conclude that there is a significant linear relationship between $x$ and $y$ because the correlation coefficient is significantly different from 0.
## Because $r$ is significant and the scatter plot shows a linear trend, the regression line can be used to predict final exam scores.
## METHOD 2: Using a table of Critical Values to make a decision
The $95 \%$ Critical Values of the Sample Correlation Coefficient Table at the end of this chapter (before the Summary) may be used to give you a good idea of whether the computed value of $r$ is significant or not. Compare $r$ to the appropriate critical value in the table. If $r$ is not between the positive and negative critical values, then the correlation coefficient is significant. If $r$ is significant, then you may want to use the line for prediction.
## Example 6.7
Suppose you computed $r=0.801$ using $n=10$ data points. $\mathrm{df}=n-2=$ $10-2=8$. The critical values associated with $\mathrm{df}=8$ are -0.632 and + 0.632 . If $r<$ negative critical value or $r>$ positive critical value, then $r$ is significant. Since $r=0.801$ and $0.801>0.632, r$ is significant and the line may be used for prediction. If you view this example on a number line, it will help you.
Figure 6.13 $r$ is not significant between -0.632 and $+0.632 . r=0.801>+0.632$. Therefore, $r$ is significant.
## Example 6.8
Suppose you computed $r=-0.624$ with 14 data points. $\mathrm{df}=14-2=12$. The critical values are -0.532 and 0.532 . Since $-0.624<-0.532, r$ is significant and the line may be used for prediction
Figure $6.14 r=-0.624<-0.532$. Therefore, $r$ is significant.
## Example 6.9
Suppose you computed $r=0.776$ and $n=6 . d f=6-2=4$. The critical values are -0.811 and 0.811 . Since $-0.811<0.776<0.811, r$ is not significant and the line should not be used for prediction.
| -0.8 | $\cdot$ | |
| :--- | :--- | :--- |
| 11 | $0.7 \overline{76}$ | 0.811 |
Figure $6.15-0.811<r=0.776<0.811$. Therefore, $r$ is not significant.
## THIRD EXAM vS FINAL EXAM EXAMPLE: critical value method
- Consider the third exam final exam example.
- The line of best fit is: $\hat{y}=-173.51+4.83 x$ with $r=0.6631$ and there are $\mathrm{n}=$ 11 data points.
- Can the regression line be used for prediction? Given a third exam score ( $\boldsymbol{x}$ value), can we use the line to predict the final exam score (predicted $y$ value)?
$$
\begin{aligned}
& H_{0}: \rho=0 \\
& H_{a}: \rho \neq 0 \\
& \alpha=0.05
\end{aligned}
$$
Use the "95\% Critical Value" table for $r$ with $d f=n-2=11-2=9$
The critical values are -0.602 and +0.602
Since $0.6631>0.602, r$ is significant.
Decision: Reject $\mathbf{H}_{\mathbf{0}}$ :
Conclusion: There is sufficient evidence to conclude that there is a significant linear relationship between $x$ and $y$ because the correlation coefficient is significantly different from 0.
Because $r$ is significant and the scatter plot shows a linear trend, the regression line can be used to predict final exam scores.
## Example 6.10: Additional Practice Examples using Critical Values
Suppose you computed the following correlation coefficients. Using the table at the end of the chapter, determine if $r$ is significant and the line of best fit associated with each $r$ can be used to predict a $y$ value. If it helps, draw a number line.
1. $r=-0.567$ and the sample size, $n$, is 19 . The $\mathrm{df}=n-2=17$. The critical value is -0.456 . $-0.567<-0.456$ so $r$ is significant.
2. $r=0.708$ and the sample size, $n$, is 9 . The $\mathrm{df}=n-2=7$. The critical value is 0.666 .
3. $0.708>0.666$ so $r$ is significant. 4. $r=0.134$ and the sample size, $n$, is 14 . The $\mathrm{df}=14-2=12$. The critical value is 0.532 .
4. 0.134 is between -0.532 and 0.532 so $r$ is not significant.
5. $r=0$ and the sample size, $n$, is 5 . No matter what the dfs are, $r=0$ is between the two critical values so $r$ is not significant.
#### Assumptions in Testing the Significance of the Correlation Coefficient
s.org/licenses/by-sa/4.0/).
Testing the significance of the correlation coefficient requires that certain assumptions about the data are satisfied. The premise of this test is that the data are a sample of observed points taken from a larger population. We have not examined the entire population because it is not possible or feasible to do so. We are examining the sample to draw a conclusion about whether the linear relationship that we see between $x$ and $y$ in the sample data provides strong enough evidence so that we can conclude that there is a linear relationship between $x$ and $y$ in the population.
The regression line equation that we calculate from the sample data gives the best fit line for our particular sample. We want to use this best fit line for the sample as an estimate of the best fit line for the population. Examining the scatterplot and testing the significance of the correlation coefficient helps us determine if it is appropriate to do this.
## The assumptions underlying the test of significance are:
- There is a linear relationship in the population that models the average value of $y$ for varying values of $x$. In other words, the expected value of $y$ for each particular value lies on a straight line in the population. (We do not know the equation for the line for the population. Our regression line from the sample is our best estimate of this line in the population.)
- The $y$ values for any particular $x$ value are normally distributed about the line. This implies that there are more y values scattered closer to the line than are scattered farther away. Assumption (1) above implies that these normal distributions are centered on the line: the means of these normal distributions of $y$ values lie on the line.
- The standard deviations of the population $y$ values about the line are equal for each value of $x$. In other words, each of these normal distributions of $y$ values has the same shape and spread about the line.
- The residual errors are mutually independent (no pattern).
Figure 6.16 The $y$ values for each $x$ value are normally distributed about the line with the same standard deviation. For each $x$ value, the mean of the $y$ values lies on the regression line. More $y$ values lie near the line than are scattered further away from the line.
**With contributions from Roberta Bloom
### Linear Regression and Correlation: Prediction
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon s.org/licenses/by-sa/4.0/).
Recall the third exam/final exam example.
We examined the scatterplot and showed that the correlation coefficient is significant. We found the equation of the best fit line for the final exam grade as a function of the grade on the third exam. We can now use the least squares regression line for prediction.
Suppose you want to estimate, or predict, the final exam score of statistics students who received 73 on the third exam. The exam scores ( $x$-values) range from 65 to 75. Since 73 is between the $x$-values 65 and 75, substitute $x=73$ into the equation. Then:
$$
\hat{y}=-173.51+4.83(73)=179.08
$$
We predict that statistic students who earn a grade of 73 on the third exam will earn a grade of 179.08 on the final exam, on average.
## Example 6.11
Recall the third exam/final exam example.
See Problem 1 - 2 .
## Problem 1
What would you predict the final exam score to be for a student who scored a 66 on the third exam?
Solution
145.27
## Problem 2
What would you predict the final exam score to be for a student who scored a 90 on the third exam?
**With contributions from Roberta Bloom
#### Solutions to Exercises in Chapter 6
## (c) (i) (2)
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon s.org/licenses/by-sa/4.0/)
## Solution to Example 6.11, Problem 2
The $x$ values in the data are between 65 and 75.90 is outside of the domain of the observed $x$ values in the data (independent variable), so you cannot reliably predict the final exam score for this student. (Even though it is possible to enter $x$ into the equation and calculate a y value, you should not do so!)
To really understand how unreliable the prediction can be outside of the observed $x$ values in the data, make the substitution $x=90$ into the equation.
$$
\hat{y}=-173.51+4.83(90)=261.19
$$
The final exam score is predicted to be 261.19 . The largest the final exam score can be is 200 .
Note: The process of predicting inside of the observed $x$ values in the data is called interpolation. The process of predicting outside of the observed $x$ values in the data is called extrapolation.
## Chapter 7 Glossary
## (c) (i) (5)
Available under Creative Commons-ShareAlike 4.0 International License (http://creativecommon
s.org/licenses/by-sa/4.0/).
## A
## Average
A number that describes the central tendency of the data. There are a number of specialized averages, including the arithmetic mean, weighted mean, median, mode, and geometric mean.
## B
## Binomial Distribution
A discrete random variable (RV) which arises from Bernoulli trials. There are a fixed number, $n$, of independent trials. "Independent" means that the result of any trial (for example, trial 1 ) does not affect the results of the following trials, and all trials are conducted under the same conditions. Under these circumstances the binomial RV $\boldsymbol{X}$ is defined as the number of successes in $\mathrm{n}$ trials. The notation is: $\boldsymbol{X} \sim \boldsymbol{B}(\boldsymbol{n}, \boldsymbol{p})$. The mean is $\boldsymbol{\mu}=\mathbf{n p}$ and the standard deviation is $\sigma=\sqrt{n p q}$. The probability of exactly $x$ successes in $n$ trials is $P(X=x)=\left(\begin{array}{l}n \\ x\end{array}\right) p^{x} q^{n-x}$.
## C
## Central Limit Theorem
Given a random variable (RV) with known mean $\mu$ and known standard deviation $\sigma$. We are sampling with size $\mathrm{n}$ and we are interested in two new RVs - the sample mean, $\bar{X}$, and the sample sum, $\Sigma X$. If the size $n$ of the sample is sufficiently large, then $\bar{X} \sim N\left(\mu, \frac{\sigma}{\sqrt{n}}\right)$ and $\sum X \sim N(n \mu, \sqrt{n} \sigma)$. If the size $\mathrm{n}$ of the sample is sufciently large, then the distribution of the sample means and the distribution of the sample sums will approximate a normal distribution regardless of the shape of the population. The mean of the sample means will equal the population mean and the mean of the sample sums will equal $n$ times the population mean. The standard deviation of the distribution of the sample means, $\frac{\sigma}{\sqrt{n}}$, is called the standard error of the mean.
## Coefficient of Correlation
A measure developed by Karl Pearson (early 1900s) that gives the strength of association between the independent variable and the dependent variable. The formula is:
$$
r=\frac{n \sum x y-\left(\sum x\right)\left(\sum y\right)}{\sqrt{\left[n \sum x^{2}-\left(\sum x\right)^{2}\right]\left[n \sum y^{2}-\left(\sum y\right)^{2}\right]}}
$$
where $\mathrm{n}$ is the number of data points. The coefficient cannot be more then 1 and less then -1 . The closer the coefficient is to \pm 1 , the stronger the evidence of a significant linear relationship between $x$ and $y$.
## Confidence Interval (CI)
An interval estimate for an unknown population parameter. This depends on:
- The desired confidence level.
- Information that is known about the distribution (for example, known standard deviation).
- The sample and its size.
## Confidence Level (CL)
The percent expression for the probability that the confidence interval contains the true population parameter. For example, if the $C L=90 \%$, then in 90 out of 100 samples the interval estimate will enclose the true population parameter.
## Continuous Random Variable
A random variable (RV) whose outcomes are measured.
Example: The height of trees in the forest is a continuous RV.
## Cumulative Relative Frequency
The term applies to an ordered set of observations from smallest to largest. The Cumulative Relative Frequency is the sum of the relative frequencies for all values that are less than or equal to the given value.
## D
Data
A set of observations (a set of possible outcomes). Most data can be put into two groups: qualitative (hair color, ethnic groups and other attributes of the population) and quantitative (distance traveled to college, number of children in a family, etc.). Quantitative data can be separated into two subgroups: discrete and continuous. Data is discrete if it is the result of counting (the number of students of a given ethnic group in a class, the number of books on a shelf, etc.). Data is continuous if it is the result of measuring (distance traveled, weight of luggage, etc.)
## Degrees of Freedom (df)
The number of objects in a sample that are free to vary.
## Discrete Random Variable
A random variable (RV) whose outcomes are counted.
## E
## Error Bound for a Population Mean (EBM)
The margin of error. Depends on the confidence level, sample size, and known or estimated population standard deviation.
## Error Bound for a Population Proportion(EBP)
The margin of error. Depends on the confidence level, sample size, and the estimated (from the sample) proportion of successes.
## Exponential Distribution
A continuous random variable (RV) that appears when we are interested in the intervals of time between some random events, for example, the length of time between emergency arrivals at a hospital. Notation: $X \sim \operatorname{Exp}(m)$. The mean is $\mu=\frac{1}{m}$ and the standard deviation is $\sigma=\frac{1}{m}$. The probability density function is $f(x)=m e^{-m x}$, $x \geq 0$ and the cumulative distribution function is $P(X \leq x)=1-\mathrm{e}^{-\mathrm{mx}}$.
$\mathbf{F}$
## Frequency
The number of times a value of the data occurs.
## H
## Hypothesis
A statement about the value of a population parameter. In case of two hypotheses, the statement assumed to be true is called the null hypothesis (notation $\boldsymbol{H}_{\mathbf{0}}$ ) and the contradictory statement is called the alternate hypothesis (notation $\boldsymbol{H}_{\boldsymbol{a}}$ ).
## Hypothesis Testing
Based on sample evidence, a procedure to determine whether the hypothesis stated is a reasonable statement and cannot be rejected, or is unreasonable and should be rejected.
## I
## Inferential Statistics
Also called statistical inference or inductive statistics. This facet of statistics deals with estimating a population parameter based on a sample statistic. For example, if 4 out of the 100 calculators sampled are defective we might infer that 4 percent of the production is defective.
## L
## Level of Significance of the Test
Probability of a Type I error (reject the null hypothesis when it is true). Notation: $\alpha$. In hypothesis testing, the Level of Significance is called the preconceived $\alpha$ or the preset $\alpha$.
M
## Mean
A number that measures the central tendency. A common name for mean is 'average.' The term 'mean' is a shortened form of 'arithmetic mean.' By definition, the mean for a sample (denoted by $\bar{x}^{\text {) }}$ is $\bar{x}=\frac{\text { Sum of all values in the sample }}{\text { Number of values in the sample, and the }}$ mean for a population (denoted by $\mu$ ) is
$\mu=\frac{\text { Sum of all values in the population }}{\text { Number of values in the population }}$.
## Median
A number that separates ordered data into halves. Half the values are the same number or smaller than the median and half the values are the same number or larger than the median. The median may or may not be part of the data.
## Mode
The value that appears most frequently in a set of data.
## $\mathbf{N}$
## Normal Distribution
A continuous random variable (RV) with pdf $f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-(x-\mu) 2} / 2 \sigma^{2}$, where $\mu$ is the mean of the distribution and $\sigma$ is the standard deviation. Notation: $\boldsymbol{X} \sim \boldsymbol{N}(\boldsymbol{\mu}, \sigma)$. If $\mu=0$ and $\sigma=1$, the RV is called the standard normal distribution.
## $\mathbf{P}$
## p-value
The probability that an event will happen purely by chance assuming the null hypothesis is true. The smaller the p-value, the stronger the evidence is against the null hypothesis.
## Parameter
A numerical characteristic of the population.
## Point Estimate
A single number computed from a sample and used to estimate a population parameter.
## Population
The collection, or set, of all individuals, objects, or measurements whose properties are being studied.
## Proportion
- As a number: A proportion is the number of successes divided by the total number in the sample.
- As a probability distribution: Given a binomial random variable (RV), $\boldsymbol{X} \sim \boldsymbol{B}(\boldsymbol{n}, \boldsymbol{p})$, consider the ratio of the number $\boldsymbol{X}$ of successes in $n$ Bernouli trials to the number $n$ of trials. $P^{\prime}=\frac{X}{n}$. This new $\mathrm{RV}$ is called a proportion, and if the number of trials, $n$, is large enough, $P^{\prime} \sim N\left(P, \frac{p q}{n}\right)$. Q
## Qualitative Data
See Data.
## Quantitative Data
## $\mathbf{R}$
## Relative Frequency
The ratio of the number of times a value of the data occurs in the set of all outcomes to the number of all outcomes.
## S
## Sample
A portion of the population understudy. A sample is representative if it characterizes the population being studied.
## Standard Deviation
A number that is equal to the square root of the variance and measures how far data values are from their mean. Notation: s for sample standard deviation and $\sigma$ for population standard deviation.
## Standard Error of the Mean
The standard deviation of the distribution of the sample means, $\frac{\sigma}{\sqrt{n}}$.
## Standard Normal Distribution
A continuous random variable (RV) $X-N(0,1)$.. When $X$ follows the standard normal distribution, it is often noted as $Z-N(0,1)$.
## Statistic
A numerical characteristic of the sample. A statistic estimates the corresponding population parameter. For example, the average number of full-time students in a 7:30 a.m. class for this term (statistic) is an estimate for the average number of fulltime students in any class this term (parameter).
## Student's-t Distribution
Investigated and reported by William S. Gossett in 1908 and published under the pseudonym Student. The major characteristics of the random variable (RV) are:
- It is continuous and assumes any real values.
- The pdf is symmetrical about its mean of zero. However, it is more spread out and fatter at the apex than the normal distribution.
- It approaches the standard normal distribution as $n$ gets larger.
- There is a "family" of $t$ distributions: every representative of the family is completely defined by the number of degrees of freedom which is one less than the number of data. $\mathbf{T}$
## Type 1 Error
The decision is to reject the Null hypothesis when, in fact, the Null hypothesis is true.
U
## Uniform Distribution
A continuous random variable (RV) that has equally likely outcomes over the domain, $a<x<b$. Often referred as the Rectangular distribution because the graph of the pdf has the form of a rectangle. Notation: $\boldsymbol{X} \sim \boldsymbol{U}(\boldsymbol{a}, \boldsymbol{b})$. The mean is $\mu=\frac{a+b}{2}$ and the standard deviation is $\sigma=\sqrt{\frac{(b-a)^{2}}{12}}$ The probability density function is $f(x)=\frac{1}{b-a}$ for $_{x}{ }_{a}<x<b$ or $a \leq x \leq b$. The cumulative distribution is $P(X \leq x)=\frac{x-a}{b-a}$
V
## Variance
Mean of the squared deviations from the mean. Square of the standard deviation. For a set of data, a deviation can be represented as $x-\bar{x}$ where $x$ is a value of the data and $\bar{x}$ is the sample mean. The sample variance is equal to the sum of the squares of the deviations divided by the difference of the sample size and 1.
## Z
z-score
The linear transformation of the form $z=\frac{x-\mu}{\sigma}$. If this transformation is applied to any normal distribution $\boldsymbol{X} \sim \boldsymbol{N}(\boldsymbol{\mu}, \sigma)$, the result is the standard normal distribution $\boldsymbol{Z} \sim \boldsymbol{N}$ $(0,1)$. If this transformation is applied to any specific value $x$ of the RV with mean $\mu$ and standard deviation $\sigma$, the result is called the z-score of $x$. Z-scores allow us to compare data that are normally distributed but scaled differently.
| Textbooks |
Another test for divisibility by 7
Testing for divisibility by 7
There are well-known tests if a number (represented as a base-10 numeral) is divisible by 2, 3, 5, 9, or 11. What about 7?
Let's look at where the divisibility-by-9 test comes from. We add up the digits of our number !!n!!. The sum !!s(n)!! is divisible by !!9!! if and only if !!n!! is. Why is that?
Say that !!d_nd_{n-1}\ldots d_0!! are the digits of our number !!n!!. Then
$$n = \sum 10^id_i.$$
The sum of the digits is
$$s(n) = \sum d_i$$
which differs from !!n!! by $$\sum (10^i-1)d_i.$$ Since !!10^i-1!! is a multiple of !!9!! for every !!i!!, every term in the last sum is a multiple of !!9!!. So by passing from !!n!! to its digit sum, we have subtracted some multiple of !!9!!, and the residue mod 9 is unchanged. Put another way:
$$\begin{align} n &= \sum 10^id_i \\ &\equiv \sum 1^id_i \pmod 9 \qquad\text{(because $10 \equiv 1\pmod 9$)} \\ &= \sum d_i \end{align} $$
The same argument works for the divisibility-by-3 test.
For !!11!! the analysis is similar. We add up the digits !!d_0+d_2+\ldots!! and !!d_1+d_3+\ldots!! and check if the sums are equal mod 11. Why alternating digits? It's because !!10\equiv -1\pmod{11}!!, so $$n\equiv \sum (-1)^id_i \pmod{11}$$
and the sum is zero only if the sum of the positive terms is equal to the sum of the negative terms.
The same type of analysis works similarly for !!2, 4, 5, !! and !!8!!. For !!4!! we observe that !!10^i\equiv 0\pmod 4!! for all !!i>1!!, so all but two terms of the sum vanish, leaving us with the rule that !!n!! is a multiple of !!4!! if and only if !!10d_1+d_0!! is. We could simplify this a bit: !!10\equiv 2\pmod 4!! so !!10d_1+d_0 \equiv 2d_1+d_0\pmod 4!!, but we don't usually bother. Say we are investigating !!571496!!; the rule tells us to just consider !!96!!. The "simplified" rule says to consider !!2\cdot9+6 = 24!! instead. It's not clear that that is actually easier.
This approach works badly for divisibility by 7, because !!10^i\bmod 7!! is not simple. It repeats with period 6.
$$\begin{array}{c|cccccc|ccc} i & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\ %10^i & 1 & 10 & 100 & 1000 & 10000 & \ldots \\ 10^i\bmod 7 & 1 & 3 & 2 & 6 & 4 & 5 & 1 & 3 & 2 & 6 & 4 &\ldots \\ \end{array} $$
The rule we get from this is:
Take the units digit. Add three times the ones digit, twice the hundreds digit, six times the thousands digit… (blah blah blah) and the original number is a multiple of !!7!! if and only if the sum is also.
For example, considering !!12345678!! we must calculate $$\begin{align} 12345678 & \Rightarrow & 3\cdot1 + 1\cdot 2 + 5\cdot 3 + 4\cdot 4 + 6\cdot 5 + 2\cdot6 + 3\cdot 7 + 1\cdot 8 & = & 107 \\\\ 107 & \Rightarrow & 2\cdot1 + 3\cdot 0 + 1\cdot7 & = & 9 \end{align} $$ and indeed !!12345678\equiv 107\equiv 9\pmod 7!!.
My kids were taught the practical divisibility tests in school, or perhaps learned them from YouTube or something like that. Katara was impressed by my ability to test large numbers for divisibility by 7 and asked how I did it. At first I didn't think about my answer enough, and just said "Oh, it's not hard, just divide by 7 and look at the remainder." ("Just count the legs and divide by 4.") But I realized later that there are several tricks I was using that are not obvious. First, she had never learned short division. When I was in school I had been tormented extensively with long division, which looks like this:
This was all Katara had been shown, so when I said "just divide by 7" this is what she was thinking of. But you only need long division for large divisors. For simple divisors like !!7!!, I was taught short division, an easier technique:
Yeah, I wrote 4 when I meant 3. It doesn't matter, we don't care about the quotient anyway.
But that's one of the tricks I was using that wasn't obvious to Katara: we don't care about the quotient anyway, only the remainder. So when I did this in my head, I discarded the parts of the calculation that were about the quotient, and only kept the steps that pertained to the remainder. The way I was actually doing this sounded like this in my mind:
7 into 12 leaves 5. 7 into 53 leaves 4. 7 into 44 leaves 2. 7 into 25 leaves 4. 7 into 46 leaves 4. 7 into 57 leaves 5. 7 into 58 leaves 9. The answer is 9.
At each step, we consider only the leftmost part of the number, starting with !!12!!. !!12\div 7 !! has a remainder of 5, and to this 5 we append the next digit of the dividend, 3, giving 53. Then we continue in the same way: !!53\div 7!! has a remainder of 4, and to this 4 we append the next digit, giving 44. We never calculate the quotient at all.
I explained the idea with a smaller example, like this:
Suppose you want to see if 1234 is divisible by 7. It's 1200-something, so take away 700, which leaves 500-something. 500-what? 530-something. So take away 490, leaving 40-something. 40-what? 44. Now take away 42, leaving 2. That's not 0, so 1234 is not divisible by 7.
This is how I actually do it. For me this works reasonably well up to 13, and after that it gets progressively more difficult until by 37 I can't effectively do it at all. A crucial element is having the multiples of the divisor memorized. If you're thinking about the mod-13 residue of 680-something, it is a big help to know immediately that you can subtract 650.
A year or two ago I discovered a different method, which I'm sure must be ancient, but is interesting because it's quite different from the other methods I described.
Suppose that the final digit of !!n!! is !!b!!, so that !!n=10a+b!!. Then !!-2n = -20a-2b!!, and this is a multiple of !!7!! if and only if !!n!! is. But !!-20a\equiv a\pmod7 !!, so !!a-2b!! is a multiple of !!7!! if and only if !!n!! is. This gives us the rule:
To check if !!n!! is a multiple of 7, chop off the last digit, double it, and subtract it from the rest of the number. Repeat until the answer becomes obvious.
For !!1234!! we first chop off the !!4!! and subtract !!2\cdot4!! from !!123!! leaving !!115!!. Then we chop off the !!5!! and subtract !!2\cdot5!! from !!11!!, leaving !!1!!. This is not a multiple of !!7!!, so neither is !!1234!!. But with !!1239!!, which is a multiple of !!7!!, we get !!123-2\cdot 9 = 105!! and then !!10-2\cdot5 = 0!!, and we win.
In contrast to the other methods in this article, this method does not tell you the remainder on dividing because it does not preserve the residue mod 7. When we started with !!1234!! we ended with !!1!!. But !!1234\not\equiv 1\pmod 7!!; rather !!1234\equiv 2!!. Each step in this method multiplies the residue by -2, or, if you prefer, by 5. The original residue was 2, so the final residue is !!2\cdot-2\cdot-2 = 8 \equiv 1\pmod 7!!. (Or, if you prefer, !!2\cdot 5\cdot 5= 50 \equiv 1\pmod 7!!.) But if we only care about whether the residue is zero, multiplying it by !!-2!! doesn't matter.
There are some shortcuts in this method too. If the final digit is !!7!!, then rather than doubling it and subtracting 14 you can just chop it off and throw it away, going directly from !!10a+7!! to !!a!!. If your number is !!10a+8!! you can subtract !!7!! from it to make it easier to work with, getting !!10a+1!! and then going to !!a-2!! instead of to !!a-16!!. Similarly when your number ends in !!9!! you can go to !!a-4!! instead of to !!a-18!!. And on the other side, if it ends in !!4!! it is easier to go to !!a-1!! instead of to !!a-8!!.
But even with these tricks it's not clear that this is faster or easier than just doing the short division. It's the same number of steps, and it seems like each step is about the same amount of work.
Finally, I once wowed Katara on an airplane ride by showing her this:
To check !!1429!! using this device, you start at ⓪. The first digit is !!1!!, so you follow one black arrow, to ①, and then a blue arrow, to ③. The next digit is !!4!!, so you follow four black arrows, back to ⓪, and then a blue arrow which loops around to ⓪ again. The next digit is !!2!!, so you follow two black arrows to ② and then a blue arrow to ⑥. And the last digit is 9 so you then follow 9 black arrows to ① and then stop. If you end where you started, at ⓪, the number is divisible by 7. This time we ended at ①, so !!1429!! is not divisible by 7. But if the last digit had been !!1!! instead, then in the last step we would have followed only one black arrow from ⑥ to ⓪, before we stopped, so !!1421!! is a multiple of 7.
This probably isn't useful for mental calculations, but I can imagine that if you were stuck on a long plane ride with no calculator and you needed to compute a lot of mod-7 residues for some reason, it could be quicker than the short division method. The chart is easy to construct and need not be memorized. The black arrows obviously point from !!n!! to !!n+1!!, and the blue arrows all point from !!n!! to !!10n!!.
I made up a whole set of these diagrams and I think it's fun to see how the conventional divisibility rules turn up in them. For example, the rule for divisibility by 3 that says just add up the digits:
Or the rule for divisibility by 5 that says to ignore everything but the last digit:
[ Addendum 20201122: Testing for divisibility by 19. ]
[ Addendum 20220106: Another test for divisibility by 7. ] | CommonCrawl |
\begin{document}
\title[Semilinear elliptic Schr\"odinger equations]{Semilinear elliptic Schr\"odinger equations involving singular potentials and source terms}
\author{Konstantinos T. Gkikas} \address{Konstantinos T. Gkikas, Department of Mathematics, National and Kapodistrian University of Athens, 15784 Athens, Greece} \email{[email protected]}
\author[P.T. Nguyen]{Phuoc-Tai Nguyen} \address{Phuoc-Tai Nguyen, Department of Mathematics and Statistics, Masaryk University, Brno, Czech Republic} \email{[email protected]}
\date{\today}
\begin{abstract} Let $\Omega \subset \mathbb{R}^N$ ($N>2$) be a $C^2$ bounded domain and $\Sigma \subset \Omega$ be a compact, $C^2$ submanifold without boundary, of dimension $k$ with $0\leq k < N-2$. Put $L_\mu = \Delta + \mu d_\Sigma^{-2}$ in $\Omega \setminus \Sigma$, where $d_\Sigma(x) = \mathrm{dist}(x,\Sigma)$ and $\mu$ is a parameter. We study the boundary value problem (P) $-L_\mu u = g(u) + \tau$ in $\Omega \setminus \Sigma$ with condition $u=\nu$ on $\partial \Omega \cup \Sigma$, where $g: \mathbb{R} \to \mathbb{R}$ is a nondecreasing, continuous function and $\tau$ and $\nu$ are positive measures. The interplay between the inverse-square potential $d_\Sigma^{-2}$, the nature of the source term $g(u)$ and the measure data $\tau,\nu$ yields substantial difficulties in the research of the problem. We perform a deep analysis based on delicate estimate on the Green kernel and Martin kernel and fine topologies induced by appropriate capacities to establish various necessary and sufficient conditions for the existence of a solution in different cases.
\noindent\textit{Key words: Hardy potentials, critical exponents, source terms, capacities, measure data}
\noindent\textit{Mathematics Subject Classification: 35J10, 35J25, 35J61, 35J75}
\end{abstract}
\maketitle \tableofcontents \section{Introduction} \subsection{Motivation and aim} The research of Schr\"odinger equations is a hot topic in the area of partial differential equations because of its applications in encoding physical properties of quantum systems. In the literature, a large number of publications have been devoted to the investigation of stationary Schr\"odinger equations involving the Laplacian with a singular potential. The presence of the singular potential yields distinctive features of the research and leads to disclose new phenomena.
The borderline case where the potential is the inverse-square of the distance to a submanifold of the domain under consideration is of interest since in this case the potential admits the same scaling (of degree $-2$) as the Laplacian and hence cannot be treated simply by standard perturbation methods. Several works have been carried out to investigate the effect of such a potential in various aspects, including a recent study on linear equations.
The present paper originated in attempts to set a step forward in the study of elliptic nonlinear Sch\"rodinger equations involving an inverse-square potential and a source term in measure frameworks.
\subsection{Background and main results} Let $\Omega \subset {\mathbb R}^N$ be a $C^2$ bounded domain and $\Sigma\subset\Omega$ be a compact, $C^2$ submanifold in ${\mathbb R}^N$ without boundary, of dimension $k$ with $0 \leq k < N-2$. Put \bel{distance} d(x):=\mathrm{dist}(x,\partial\Omega) \quad \text{and} \quad d_\Sigma(x): = \mathrm{dist}(x,\Sigma). \end{equation}\normalsize For $\mu \in {\mathbb R}$, denote by $L_\mu$ the Schr\"odinger operator with the inverse-square potential $d_\Sigma^{-2}$ as \bal L_\mu = L_\mu^{\Omega,\Sigma}:=\Delta + \frac{\mu}{d_\Sigma^2} \eal in $\Omega \setminus \Sigma$. The study of $L_\mu$ was carried out in \cite{GkiNg_linear} in which the optimal Hardy constant
\bal{\mathcal C}_{\Omega,\Sigma}:=\inf_{\varphi \in H^1_0(\Omega)}\frac{\int_\Omega |\nabla \varphi|^2\dx}{\int_\Omega d_\Sigma^{-2}\varphi^2 \dx} \eal is deeply involved. It is well known that ${\mathcal C}_{\Omega,\Sigma}\in (0,H^2]$ (see D\'avila and Dupaigne \cite{DD1, DD2} and Barbatis, Filippas and Tertikas \cite{BFT}), where \small\begin{equation} \label{valueH} H:=\frac{N-k-2}{2}. \end{equation}\normalsize It is classical that ${\mathcal C}_{\Omega,\{0\}}=\left(\frac{N-2}{2} \right)^2$. We also know that ${\mathcal C}_{\Omega,\Sigma}=H^2$ if $-\Delta d_\Sigma^{2+k-N} \geq 0$ in the sense of distributions in $\Omega \setminus \Sigma$ or if $\Omega=\Sigma_\beta$ with $\beta$ small enough (see \cite{BFT}), where \bal \Sigma_\beta :=\{ x \in {\mathbb R}^N \setminus \Sigma: d_\Sigma(x) < \beta \}. \eal For $\mu \leq H^2$, let $\am$ and $\ap$ be the roots of the algebraic equation $\alpha} \def\gb{\beta} \def\gg{\gamma^2 - 2H\alpha} \def\gb{\beta} \def\gg{\gamma + \mu=0$, i.e. \bel{apm} \am:=H-\sqrt{H^2-\mu}, \quad \ap:=H+\sqrt{H^2-\mu}. \end{equation}\normalsize Notice that $\am\leq H\leq\ap<2H$, and $\am \geq 0$ if and only if $\mu \geq 0$. Moreover, by \cite[Lemma 2.4 and Theorem 2.6]{DD1} and \cite[page 337, Lemma 7, Theorem 5]{DD2},
\bal \lambda_\mu:=\inf\left\{\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}\left(|\nabla u|^2-\frac{\mu }{d_\Sigma^2}u^2\right)dx: u \in C_c^1(\Omega), \int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} u^2 dx=1\right\}>-\infty.
\eal We note that $\lambda_\mu$ is the first eigenvalue associated to $-L_\mu$ and its corresponding eigenfunction $\phi_\mu$, with normalization $\| \phi_\mu \|_{L^2(\Omega)}=1$, satisfies two-sided estimate $\phi_\mu \approx d\,d_\Sigma^{-\am}$ in $\Omega \setminus \Sigma$ (see subsection \ref{subsect:eigen} for more detail).
The sign of $\lambda_\mu$ plays an important role in the study of $L_\mu$. If $\mu<\CC_{\Omega,\Sigma}$ then $\lambda_\mu>0$; however, in general, this does not hold. Under the assumption $\lambda_\mu>0$, the authors of the present paper obtained the existence and sharp two-sided estimates of the Green function $G_\mu$ and Martin kernel $K_\mu$ associated to $-L_\mu$ (see \cite{GkiNg_linear}). These are crucial tools in the study of the boundary value problem with measure data for linear equations of the form \ba \label{eq:linear} \left\{ \begin{aligned} -L_\mu u &= \tau \quad &&\text{in } \Omega \setminus \Sigma, \\
\mathrm{tr}(u) &= \nu, &&
\end{aligned} \right. \ea where $\tau \in \GTM(\Omega;\phi_\mu)$ (i.e. $\| \tau\|_{\GTM(\Omega \setminus \Sigma;{\phi_{\mu }})}:=\int_{\Omega \setminus \Sigma}\phi_\mu \dd |\tau|<\infty$) and $\nu \in \GTM(\partial \Omega \cup \Sigma)$ (i.e. $\| \nu \|_{\GTM(\partial \Omega \cup \Sigma)}:= \int_{\partial \Omega \cup \Sigma}\dd |\nu| < \infty$).
In \eqref{eq:linear}, $\mathrm{tr}(u)$ denotes the \textit{boundary trace} which was introduced in \cite{GkiNg_linear} in terms of harmonic measures of $-L_\mu$ (see Subsection \ref{subsec:boundarytrace}). An important feature of this notion is $\mathrm{tr}(\BBG_\mu[\tau]) = 0$ for any $\tau \in \GTM(\Omega \setminus \Sigma;\phi_\mu)$ and $\mathrm{tr}(\BBK_\mu[\tau]) = \nu$ for any $\nu \in \GTM(\partial \Omega \cup \Sigma)$, where \bal \BBG_\mu[\tau](x): &= \int_{\Omega \setminus \Sigma}G_\mu(x,y)\dd\tau(y), \quad \tau \in \GTM(\Omega \setminus \Sigma;\phi_\mu), \\ \BBK_\mu[\nu](x): &= \int_{\partial \Omega \cup \Sigma}K_\mu(x,y)\dd\nu(y), \quad \nu \in \GTM(\partial \Omega \cup \Sigma). \eal Note that for a positive measure $\tau$, $\BBG_\mu[\tau]$ is finite a.e. in $\Omega \setminus \Sigma$ if and only if $\tau \in \GTM(\Omega \setminus \Sigma; {\phi_{\mu }})$.
Moreover, it was shown in \cite{GkiNg_linear} that $\BBG_\mu[\tau]$ is the unique solution of \eqref{eq:linear} with $\nu=0$, and $\BBK_\mu[\nu]$ is the unique solution of \eqref{eq:linear} with $\tau=0$. By the linearity, the unique solution to \eqref{eq:linear} is of the form \bal u = \BBG_\mu[\tau] + \BBK_\mu[\nu] \quad \text{a.e. in } \Omega \setminus \Sigma. \eal Further results for linear problem \eqref{eq:linear} are discussed in Subsection \ref{subsec:linear}.
As a continuation and development of the work \cite{GkiNg_linear} in this research topic, this paper studies the boundary value problem for semilinear equations with a source term of the form \ba \label{NLP} \left\{ \begin{aligned}
-L_\mu u &= g(u) + \rho \tau \quad &&\text{in } \Omega \setminus \Sigma, \\
\mathrm{tr}(u) &= \sigma\nu, && \end{aligned} \right. \ea where $\rho, \sigma$ are nonnegative parameters, $\tau$ and $\nu$ are Radon measures on $\Omega \setminus \Sigma$ and $\partial \Omega \cup \Sigma$ respectively, and $g: {\mathbb R} \to {\mathbb R}$ is a nondecreasing continuous function such that $g(0)=0$.
Various works on problem \eqref{NLP} and related problems have been published in the literature, including excellent papers of D\'avila and Dupaigne \cite{DN,DD1,DD2} where important tools in function settings are established and combined with a monotonicity argument in derivation of existence, nonexistence, uniqueness for solutions with zero boundary datum. Afterwards, deep nonexistence results for nonnegative distributional supersolutions have been obtained by Fall \cite{Fall-0} via a linearization argument. Recently, a description on isolated singularities in case $\Sigma=\{0\}\subset \Omega$ has been provided by Chen and Zhou \cite{CheZho}.
In the present paper, the interplay between dimention of the set $\Sigma$, the value of $\mu$, the growth of the source term and the concentration of measure data causes the invalidity or quite restrictive applicability of the techniques used in the mentioned papers and leads to the involvement of several \textit{critical exponents} for the solvability of problem \eqref{NLP}. Therefore, our aim is to perform further analysis and to establish effective tools, which allow us to obtain existence and nonexistence results for \eqref{NLP} in various cases.
\textit{Let us assume throughout the paper that} \small\begin{equation} \label{assump1} \mu \leq H^2 \quad \text{and} \quad \lambda_\mu > 0. \end{equation}\normalsize Assumption \eqref{assump1} ensures the validity of sharp two-sided estimates for the Green kernel and Martin kernel as well as other results regarding linear equations as mentioned above.
In order to state our main results, we introduce some notations. For $\alpha , \gamma \in {\mathbb R}$, put \ba \label{varphi} \varphi_{\alpha,\gamma}(x):=d_\Sigma(x)^{-\alpha}d(x)^\gamma, \quad x \in \Omega \setminus \Sigma. \ea It can be seen from \eqref{eigenfunctionestimates} that $\varphi_{\am,1} \approx \phi_\mu$ (we notice that $\am$ is defined in \eqref{apm}). Let $\GTM(\Omega \setminus \Sigma;\varphi_{\am,\gamma})$ be the space of measures $\tau$ such that
\bal \| \tau \|_{\GTM(\Omega \setminus \Sigma;\varphi_{\am,\gamma})}:= \int_{\Omega \setminus \Sigma}\varphi_{\am,\gamma}\,\dd |\tau|<\infty. \eal
The notion of the weak solutions of \eqref{NLP} is given below. \begin{definition} \label{weaksol-LP}
Let $\gamma \in [0,1]$, $\rho \geq 0$, $\sigma \geq 0$, $\tau\in\mathfrak{M}(\Omega\setminus \Sigma;\varphi_{\am,\gamma})$ and $\nu \in \mathfrak{M}(\partial\Omega\cup \Sigma)$. We say that $u$ is a \textit{weak solution} of \eqref{NLP} if $u\in L^1(\Omega;{\phi_{\mu }})$, $g(u) \in L^1(\Omega;\phi_\mu)$ and
\small\begin{equation} \label{lweakform}
- \int_{\Omega} u L_{\mu }\zeta \, \dx=\int_{\Omega} g(u)\zeta \, \dx + \rho \int_{\Omega \setminus \Sigma}\zeta \,\dtau- \sigma\int_{\Omega} \mathbb{K}_{\mu}[\nu]L_{\mu }\zeta \, \dx
\qquad\forall \zeta \in\mathbf{X}_\mu(\Omega\setminus \Sigma),
\end{equation}\normalsize
where the \textit{space of test function} ${\bf X}_\mu(\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma)$ is defined by
\ba \label{Xmu} {\bf X}_\mu(\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma):=\{ \zeta \in H_{loc}^1(\Omega \setminus \Sigma): \phi_\mu^{-1} \zeta \in H^1(\Omega} \def\Gx{\Xi} \def\Gy{\Psi;\phi_\mu^{2}), \, \phi_\mu^{-1}L_\mu \zeta \in L^\infty(\Omega) \}.
\ea \end{definition}
The space ${\bf X}_\mu(\Omega \setminus \Sigma)$ was introduced in \cite{GkiNg_linear} to study linear problem \eqref{eq:linear}. From \eqref{Xmu}, it is easy to see that the term on the left-hand side of \eqref{lweakform} is finite. By \cite[Lemma 7.3]{GkiNg_linear} and \eqref{eigenfunctionestimates}, for any $\zeta \in {\bf X}_\mu(\Omega \setminus \Sigma)$, $|\zeta| \lesssim \phi_\mu \approx d\, d_\Sigma^{-\am}$, hence the first term on the right-hand side of \eqref{lweakform} is finite. Moreover, for any $\zeta \in {\bf X}_\mu(\Omega \setminus \Sigma)$ and $\gamma \in [0,1]$, we have $|\zeta| \lesssim d^\gamma d_\Sigma^{-\am} = \varphi_{\am,\gamma}$. This implies that the second term on the right-hand side of \eqref{lweakform} is finite. Finally, since $\BBK_{\mu}[\nu] \in L^1(\Omega;{\phi_{\mu }})$, the third term on the right-hand side of \eqref{lweakform} is also finite.
By Theorem \ref{linear-problem}, $u$ is a weak solution of \eqref{NLP} if and only if \bal u = \BBG_\mu[g(u)] + \BBG_\mu[\tau] + \BBK_\mu[\nu] \quad \text{in } \Omega \setminus \Sigma. \eal
Our main results disclose different scenarios, depending on the interplay between the concentration and the total variation of the measure data, and the size of the set $\Sigma$, in which the existence of a solution to \eqref{NLP} can be derived.
In the following theorem, we show the existence, as well as weak Lebesgue estimates, of a solution to \eqref{NLP} provided that the nonlinearity $g$ has mild growth and the measure data have small norm.
\begin{theorem} \label{th1} Let $0<\mu\leq H^2$, $0\leq\gamma\leq1$, $\tau \in \GTM(\Omega \setminus \Sigma; \varphi_{\am,\gamma})$ with $\norm{\tau}_{\mathfrak{M}(\Omega\setminus \Sigma;\varphi_{\am,\gamma})}=1$ and $\nu \in \GTM(\partial \Omega} \def\Gx{\Xi} \def\Gy{\Psi\cup \Sigma)$ with $\norm{\nu}_{\GTM(\partial \Omega} \def\Gx{\Xi} \def\Gy{\Psi\cup \Sigma)}=1$. Assume $g$ satisfies \ba \label{subcd-0} \Lambda} \def\Gs{\Sigma} \def\Gp{\Pi_g:=\int_1^\infty s^{-q-1} (g(s)-g(-s))\,\dd s < \infty \ea
for some $q \in (1,\infty)$ and
\ba \label{gcomparepower}
|g(s)|\leq a|s|^{\tilde q} \quad \text{for some } a>0,\; \tilde q>1 \text{ and for any } |s|\leq 1.
\ea
Assume one of the following conditions holds.
$(i)$ $\1_{\partial \Omega} \, \nu \equiv 0$ and \eqref{subcd-0} holds for $q = \frac{N+\gamma}{N+\gamma-2}$.
$(ii)$ $\1_{\partial \Omega} \, \nu\not\equiv 0$ and \eqref{subcd-0} holds for $q = \frac{N+1}{N-1}$.
Then there exist positive numbers $\rho_0,\sigma_0,t_0$ depending on $N,\mu,\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\Sigma,\Lambda} \def\Gs{\Sigma} \def\Gp{\Pi_g,\gamma, \tilde q$ such that, for every $\rho \in (0,\rho_0)$ and $\sigma\in (0,\sigma_0)$, problem \eqref{NLP} admits a weak solution $u$ satisfying
\bel{est:t0} \|u\|_{L_w^{q}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus\Sigma;{\phi_{\mu }})} \leq t_0,
\end{equation}\normalsize
where $q=\frac{N+\gamma}{N+\gamma-2}$ if case $(i)$ happens or $q=\frac{N+1}{N-1}$ if case $(ii)$ happens. \end{theorem}
The proof of Theorem \ref{th1} contains several steps, relying on various ingredients such as sharp weak Lebesgue estimates on Green kernel and Martin kernel (see Theorems \ref{lpweakgreen}--\ref{lpweakmartin1}) and Schauder fixed point theorem.
When $\tau$ or $\nu$ is zero measure and $\mu$ is not restricted to be positive, the value of $q$ in \eqref{subcd-0} can be enlarged or adjusted, as shown in the following theorem.
\begin{theorem} \label{th2} Let $\mu\leq H^2$, $0\leq\gamma\leq1$ and $g$ satisfy \eqref{gcomparepower}.
$(i)$ Assume $0< \mu \leq \left(\frac{N-2}{2}\right)^2$, $\nu=0$, $\tau \in \GTM(\Omega} \def\Gx{\Xi} \def\Gy{\Psi \setminus \Sigma;\varphi_{\am,\gamma})$ with $\norm{\tau}_{\mathfrak{M}(\Omega\setminus \Sigma;\varphi_{\am,\gamma})}=1$, and \eqref{subcd-0} holds with $q=\frac{N+\gamma}{N+\gamma-2}$. Then the conclusion of Theorem \ref{th1} holds true with $q=\frac{N+\gamma}{N+\gamma-2}$.
$(ii)$ Assume $\mu \leq 0$, $0 \leq \kappa \leq -\am$, $\nu=0$, $\tau \in \GTM(\Omega} \def\Gx{\Xi} \def\Gy{\Psi \setminus \Sigma;\varphi_{-\kappa,\gamma})$ with $\norm{\tau}_{\mathfrak{M}(\Omega\setminus \Sigma;\varphi_{-\kappa,\gamma})}=1$, and $g$ satisfy \eqref{subcd-0} with \small\begin{equation} \label{pkd} q=\min\left\{\frac{N+\gamma}{N+\gamma-2},\frac{N+\kappa}{N+\kappa-2}\right\}. \end{equation}\normalsize Then the conclusion of Theorem \ref{th1} holds true with $q$ as in \eqref{pkd}.
$(iii)$ Assume $\mu \leq \left(\frac{N-2}{2}\right)^2$, $\tau=0$, $\nu \in \GTM(\partial \Omega} \def\Gx{\Xi} \def\Gy{\Psi\cup \Sigma)$ has compact support in $\Sigma$ with $\norm{\nu}_{\GTM(\partial \Omega} \def\Gx{\Xi} \def\Gy{\Psi\cup \Sigma)}=1$, and \eqref{subcd-0} holds with \small\begin{equation} \label{pkd-a} q=\min\left\{\frac{N+1}{N-1},\frac{N-\am}{N-\am-2}\right\}. \end{equation}\normalsize Then the conclusion of Theorem \ref{th1} holds true with $q$ as in \eqref{pkd-a}.
$(iv)$ Assume $\mu \leq \left(\frac{N-2}{2}\right)^2$, $\tau=0$, $\nu \in \GTM(\partial \Omega} \def\Gx{\Xi} \def\Gy{\Psi\cup \Sigma)$ has compact support in $\partial \Omega$ with $\norm{\nu}_{\GTM(\partial \Omega} \def\Gx{\Xi} \def\Gy{\Psi\cup \Sigma)}=1$, and \eqref{subcd-0} holds with $q=\frac{N+1}{N-1}$. Then the conclusion of Theorem \ref{th1} holds true with $q=\frac{N+1}{N-1}$.
\end{theorem}
We remark that condition \eqref{pkd-a} is not sharp. When $g$ is a pure power function, condition \eqref{pkd-a} can be improved to be sharp, as pointed out in the remark following Theorem \ref{subm}.
When $g$ is a power function, namely $g(u)=|u|^{p-1}u$ for $p>1$, problem \eqref{NLP} becomes \small\begin{equation} \label{power-source} \left\{ \begin{aligned}
- L_\mu} \def\gn{\nu} \def\gp{\pi u&=|u|^{p-1}u + \rho\tau\qquad \text{in }\;\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma,\\ \mathrm{tr}(u)&= \sigma\nu. \end{aligned} \right. \end{equation}\normalsize We will point out below that the exponents $\frac{N+\gamma}{N+\gamma-2}$, $\frac{N-\am}{N-\am-2}$ and $\frac{N+1}{N-1}$ are \textit{critical exponents} for the existence of a solution to \eqref{power-source}. Moreover, by performing further analysis, we are able to provide necessary and sufficient conditions in terms of estimates of the Green kernel and Martin kernel, as well as in terms of appropriate capacities.
We first consider \eqref{power-source} with $\sigma \nu=0$. Let us introduce suitable capacities. For $\alpha\leq N-2$, set \bal
\CN_{\alpha} \def\gb{\beta} \def\gg{\gamma}(x,y):=\frac{\max\{|x-y|,d_\Sigma(x),d_\Sigma(y)\}^\alpha}{|x-y|^{N-2}\max\{|x-y|,d(x),d(y)\}^2},\quad (x,y)\in\overline{\Omega}\times\overline{\Omega}, x \neq y, \eal and \bal \BBN_{\xa}[\omega](x):=\int_{\overline{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}} \CN_{\alpha}(x,y) \dd\omega(y), \quad \omega \in \GTM^+(\overline \Omega} \def\Gx{\Xi} \def\Gy{\Psi).\eal
For $\alpha \leq N-2$, $b>0$, $\theta>-N+k$ and $s>1$, define capacity $\text{Cap}_{\BBN_{\xa},s}^{b,\theta}$ by \bal \text{Cap}_{\BBN_{\xa},s}^{b,\theta}(E) :=\inf\left\{\int_{\overline{\Omega}}d^b d^\theta_\Sigma\phi} \def\vgf{\varphi} \def\gh{\eta^s\,\dx:\;\; \phi} \def\vgf{\varphi} \def\gh{\eta \geq 0, \;\;\BBN_{\xa}[ d^b d_{\Sigma}^\theta\phi} \def\vgf{\varphi} \def\gh{\eta ]\geq\1_E\right\} \quad \text{for Borel set } E\subset\overline{\Omega}. \eal Here $\1_E$ denotes the indicator function of $E$. By \cite[Theorem 2.5.1]{Ad}, we have \bal
(\text{Cap}_{\BBN_{\xa},s}^{b,\theta}(E))^\frac{1}{s}=\sup\{\omega(E):\omega \in\GTM^+(E), \|\BBN_{\xa}[\omega]\|_{L^{s'}(\Omega;d^b d^\theta_\Sigma)} \leq 1 \}. \eal
\begin{theorem}\label{theoremint}
We assume that $\mu< \left(\frac{N-2}{2}\right)^2$ and
\ba \label{p-cond}
1<p<\frac{2+\am}{\am} \text{ if } \mu>0 \quad \text{ or } \quad p>1 \text{ if } \mu \leq0.
\ea
Let $\tau \in \GTM^+(\Omega \setminus \Sigma; {\phi_{\mu }})$. Then the following statements are equivalent.
1. The equation
\ba \label{u-rhotau} u=\BBG_\mu[u^p]+\rho\BBG_\mu[\gt]
\ea
has a positive solution for $\rho>0$ small.
2. For any Borel set $E \subset \Omega\setminus \Sigma$, there holds
\bal\int_E \BBG_\mu[\1_E \tau]^p {\phi_{\mu }} \, \dx \leq C\int_E{\phi_{\mu }} \dd \tau.\eal
3. The following inequality holds
\bal \BBG_\mu[\BBG_\mu[\gt]^p]\leq C\, \BBG_\mu[\gt]<\infty\quad \text{a.e. in } \Omega \setminus \Sigma. \eal
4. For any Borel set $E \subset \Omega\setminus \Sigma$, there holds \bal\int_E{\phi_{\mu }} \dd \tau \leq C\, \mathrm{Cap}_{\BBN_{2\am},p'}^{p+1,-\am(p+1)}(E).\eal \end{theorem}
It is worth mentioning that when $\mu< \left(\frac{N-2}{2}\right)^2$, if $1< p < \frac{N+1}{N-1}$ then all the statements 1--4 of Theorem \ref{theoremint} hold true (see Remark \ref{capp-1}), while if $p \geq \frac{N+1}{N-1}$ then, for any $\rho>0$, \textit{there exists} $\tau \in \GTM^+(\Omega \setminus \Sigma;{\phi_{\mu }})$ such that equation \eqref{u-rhotau} does not admit any positive solution (see Proposition \ref{nonexist-1}). Furthermore, when $0<\mu<\left(\frac{N-2}{2}\right)^2$ and $p\geq \frac{2+\am}{\am}$, \textit{for any} $\rho>0$ and \textit{any} $\tau \in \GTM^+(\Omega \setminus \Sigma; {\phi_{\mu }})$, equation \eqref{u-rhotau} has no solution (see Proposition \ref{remove1}).
We note that when $\Sigma=\{0\}$ and $\mu=\left(\frac{N-2}{2}\right)^2$, Theorem \ref {theoremint} remains valid under the assumption that $\tau \in \GTM^+(\Omega \setminus \{0\}; \phi_{\mu})$ with compact support in $\Omega \setminus \{0\}$. This is shown in Theorem \ref{theeqint}.
Next we investigate \eqref{power-source} with $\tau=0$. To this end, we make use of a different type of capacities whose definition is introduced in \eqref{Capsub}. These capacities are denoted by $\mathrm{Cap}_{\theta} \def\vge{\varepsilon,s}^{\Gamma}$, where $\Gamma=\partial \Omega$ or $\Gamma=\Sigma$, which allow us to measure Borel subsets of $\partial \Omega \cup \Sigma$ in a subtle way.
\begin{theorem}\label{subm}
Assume that $\mu< \left(\frac{N-2}{2}\right)^2$ and condition \eqref{p-cond} holds.
Let $\nu \in \GTM^+(\partial\Omega\cup \Sigma)$ with compact support in $\Sigma$.
Then the following statements are equivalent.
1. The equation
\ba \label{u-sigmanu} u=\BBG_\mu[u^p]+\sigma} \def\vgs{\varsigma} \def\gt{\tau\BBK_\mu[\nu]
\ea
has a positive solution for $\sigma} \def\vgs{\varsigma} \def\gt{\tau>0$ small.
2. For any Borel set $E \subset \partial \Omega \cup \Sigma$, there holds
\ba \label{Kp<nu} \int_E \BBK_\mu[\1_E\nu]^p {\phi_{\mu }} \dx \leq C\,\nu(E).
\ea
3. The following inequality holds
\ba \label{GKp<K} \BBG_\mu[\BBK_\mu[\nu]^p]\leq C\,\BBK_\mu[\nu]<\infty\quad \text{a.e. in } \Omega.
\ea
Assume, in addition, that
\small\begin{equation} \label{p-cond-2}
k \geq 1 \quad \text{and} \quad \max\left\{1,\frac{N-k-\am}{N-2-\am} \right\}< p<\frac{2+\ap}{\ap}.
\end{equation}\normalsize
Put
\ba \label{gamma} \vartheta: = \frac{2-(p-1)\ap}{p}.
\ea
Then any of the above statements is equivalent to the following statement.
4. For any Borel set $E \subset \Sigma$, there holds
\bal\nu(E)\leq C\, \mathrm{Cap}_{\vartheta,p'}^{\Sigma}(E).\eal \end{theorem}
We remark that when $1< p < \frac{N-\am}{N-2-\am}$, all statements of Theorem \ref{subm} hold (see Remark \ref{existSigma}), while when $p \geq \frac{N-\am}{N-2-\am}$, for any $z \in \Sigma$ and any $\sigma>0$, problem \eqref{u-sigmanu} with $\nu=\delta_z$ does not admit any positive weak solution (see Remark \ref{nonexistSigma}). Assumption \eqref{p-cond-2} is imposed to ensure the validity of delicate estimates related to the Martin kernel (see \cite[Lemma 8.1]{GkiNg_absorption}), which enables us to deal with capacity $\mathrm{Cap}_{\vartheta,p'}^\Sigma$.
We also note that in case $\Sigma=\{0\}$ and $\mu=\left(\frac{N-2}{2}\right)^2$, if $p< \frac{2+\ap}{\ap}$ then for $\sigma>0$ small, there is a solution of \eqref{u-sigmanu} with $\tau=0$ and $\nu=\delta_0$ (see Remark \ref{rem2}). On the contrary, when $p \geq \frac{2+\ap}{\ap}$, then for any $\sigma>0$ and any $\nu \in \GTM(\partial \Omega \cup \Sigma)$ with compact support in $\Sigma$, there is no solution of problem \eqref{u-sigmanu} (see Remark \ref{rem3} for more details).
Existence results in case boundary data are concentrated on $\partial \Omega$ are stated in the next theorem. \begin{theorem}\label{th:existnu-prtO}
Assume that $\mu\leq \left(\frac{N-2}{2}\right)^2$, $p$ satisfies \eqref{p-cond} and $\nu \in \GTM^+(\partial\Omega\cup \Sigma)$ with compact support in $\partial \Omega$.
Then the following statements are equivalent.
1. Equation \eqref{u-sigmanu} has a positive solution for $\sigma} \def\vgs{\varsigma} \def\gt{\tau>0$ small.
2. For any Borel set $E \subset \partial \Omega $, \eqref{Kp<nu} holds.
3. Estimate \eqref{GKp<K} holds.
4. For any Borel set $E \subset \partial \Omega $, there holds $\nu(E)\leq C\, \mathrm{Cap}_{\frac{2}{p},p'}^{\partial \Omega}(E)$. \end{theorem}
Note that when $1<p<\frac{N+1}{N-1}$, statements 1--4 of Theorem \ref{th:existnu-prtO} are valid, while when $p \geq \frac{N+1}{N-1}$, for any $\sigma>0$ and any $z \in \partial \Omega$, equation \eqref{u-sigmanu} with $\nu=\delta_z$ does not admit any positive solution (see Remark \ref{partO-N}). It will be also pointed out that when $\mu>0$ and $p\geq \frac{2+\am}{\am}$, for any $\sigma>0$ and any $\nu \in \GTM^+(\partial \Omega \cup \Sigma)$ with compact support in $\partial\Omega$, problem \eqref{u-sigmanu} does not admit any positive weak solution. This is discussed in Lemma \ref{rem4}.
\noindent \textbf{Organization of the paper.} In Section \ref{pre}, we present main properties of the submanifold $\Sigma$ and recall important facts about the first eigenfunction, Green kernel and Martin kernel of $-L_\mu$. In Section \ref{sec:weakLp}, we establish sharp estimates on the Green operator and Martin operator, which play an important role in proving the existence of a solution to \eqref{NLP}. We then discuss the notion of boundary trace and several results regarding linear equations involving $-L_\mu$ in Section \ref{sec:linear}. Section \ref{sec:gennon} is devoted to the proof of Theorems \ref{th1} and \ref{th2}. In section \ref{sec:powercase}, we focus on the power case and provide the proof of Theorems \ref{theoremint}--\ref{th:existnu-prtO}. In Appendix \ref{app:A}, we give an estimate which is useful in the proof of several results in Section \ref{sec:weakLp}.
\subsection{Notations} \label{subsec:notations} We list below notations that are frequently used in the paper.
$\bullet$ Let $\phi$ be a positive continuous function in $\Omega \setminus \Sigma$ and $\kappa \geq 1$. Let $L^\kappa(\Omega;\phi)$ be the space of functions $f$ such that
\bal \| f \|_{L^\kappa(\Omega;\phi)} := \left( \int_{\Omega} |f|^\kappa \phi \, \dd x \right)^{\frac{1}{\kappa}}. \eal
The weighted Sobolev space $H^1(\Omega;\phi)$ is the space of functions $f \in L^2(\Omega;\phi)$ such that $\nabla f \in L^2(\Omega;\phi)$. This space is endowed with the norm
\bal \| f \|_{H_0^1(\Omega;\phi)}^2= \int_{\Omega} |f|^2 \phi \,\dd x + \int_{\Omega} |\nabla f|^2 \phi \,\dd x. \eal The closure of $C_c^\infty(\Omega)$ in $H^1(\Omega;\phi)$ is denoted by $H_0^1(\Omega;\phi)$.
Denote by $\mathfrak{M}(\Omega;\phi)$ the space of Radon measures $\tau$ in $\Omega$ such that \bal \| \tau\|_{\mathfrak{M}(\Omega;\phi)}:=\int_{\Omega}\phi \, \dd|\tau|<\infty, \eal and by $\mathfrak{M}^+(\Omega;\phi)$ its positive cone. Denote by $\GTM(\partial \Omega \cup \Sigma)$ the space of finite measure $\nu$ on $\partial \Omega \cup \Sigma$, namely
\bal \| \nu \|_{\GTM(\partial \Omega \cup \Sigma)}:=|\nu|(\partial \Omega \cup \Sigma) < \infty, \eal and by $\GTM^+(\partial \Omega \cup \Sigma)$ its positive cone.
$\bullet$ For a measure $\omega$, denote by $\omega^+$ and $\omega^-$ the positive part and negative part of $\omega$.
$\bullet$ For $\beta>0$, $ \Omega_{\beta}=\{ x \in \Omega: d(x) < \beta\}$, $\Sigma_{\beta}=\{ x \in {\mathbb R}^N \setminus \Sigma: d_\Sigma(x)<\beta \}$.
$\bullet$ We denote by $c,c_1,C...$ the constant which depend on initial parameters and may change from one appearance to another.
$\bullet$ The notation $A \gtrsim B$ (resp. $A \lesssim B$) means $A \geq c\,B$ (resp. $A \leq c\,B$) where the implicit $c$ is a positive constant depending on some initial parameters. If $A \gtrsim B$ and $A \lesssim B$, we write $A \approx B$. \textit{Throughout the paper, most of the implicit constants depend on some (or all) of the initial parameters such as $N,\Omega,\Sigma,k,\mu$ and we will omit these dependencies in the notations (except when it is necessary).}
$\bullet$ For $a,b \in \BBR$, denote $a \wedge b = \min\{a,b\}$, $a \lor b =\max\{a,b \}$.
$\bullet$ For a set $D \subset {\mathbb R}^N$, $\1_D$ denotes the indicator function of $D$.
\noindent \textbf{Acknowledgement.} K. T. Gkikas acknowledges support by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the “2nd Call for H.F.R.I. Research Projects to support Post-Doctoral Researchers” (Project Number: 59). P.-T. Nguyen was supported by Czech Science Foundation, Project GA22-17403S.
\section{Preliminaries} \label{pre}
\subsection{Submanifold $\Sigma$.} \label{assumptionK} Throughout this paper, we assume that $\Sigma \subset \Omega$ is a $C^2$ compact submanifold in $\mathbb{R}^N$ without boundary, of dimension $k$, $0\leq k < N-2$. When $k = 0$ we assume that $\Sigma = \{0\}$.
For $x=(x_1,...,x_k,x_{k+1},...,x_N) \in {\mathbb R}^N$, we write $x=(x',x'')$ where $x'=(x_1,..,x_k) \in {\mathbb R}^k$ and $x''=(x_{k+1},...,x_N) \in {\mathbb R}^{N-k}$. For $\beta>0$, we denote by $B_{\beta}^k(x')$ the ball in ${\mathbb R}^k$ with center at $x'$ and radius $\beta.$ For any $\xi\in \Sigma$, we set \bal \Sigma_\beta &:=\{ x \in {\mathbb R}^N \setminus \Sigma: d_\Sigma(x) < \beta \}, \\
V(\xi,\beta)&:=\{x=(x',x''): |x'-\xi'|<\beta,\; |x_i-\Gamma_i^\xi(x')|<\beta,\;\forall i=k+1,...,N\}, \eal for some functions $\Gamma_i^\xi: {\mathbb R}^k \to {\mathbb R}$, $i=k+1,...,N$.
Since $\Sigma$ is a $C^2$ compact submanifold in $\mathbb{R}^N$ without boundary, there is $\beta_0$ such that the followings hold.
\begin{itemize}
\item $\Sigma_{6\beta_0}\Subset \Omega$ and for any $x\in \Sigma_{6\beta_0}$, there is a unique $\xi \in \Sigma$ satisfies $|x-\xi|=d_\Sigma(x)$.
\item $d_\Sigma \in C^2(\Sigma_{4\beta_0})$, $|\nabla d_\Sigma|=1$ in $\Sigma_{4\beta_0}$ and there exists $\eta\in L^\infty(\Sigma_{4\beta_0})$ such that (see \cite[Lemma 2.2]{Vbook} and \cite[Lemma 6.2]{DN}) \bal \Delta d_\Sigma(x)=\frac{N-k-1}{d_\Sigma(x)}+ \eta(x) \quad \text{in } \Sigma_{4\beta_0} . \eal
\item For any $\xi \in \Sigma$, there exist $C^2$ functions $\Gamma_i^\xi \in C^2({\mathbb R}^k;{\mathbb R})$, $i=k+1,...,N$, such that (upon relabeling and reorienting the coordinate axes if necessary), for any $\beta \in (0,6\beta_0)$, $V(\xi,\beta) \subset \Omega$ and
\bal V(\xi,\beta) \cap \Sigma=\{x=(x',x''): |x'-\xi'|<\beta,\; x_i=\Gamma_i^\xi (x'), \; \forall i=k+1,...,N\}. \eal
\item There exist $ m_0 \in {\mathbb N}$ and points $\xi^{j} \in \Sigma$, $j=1,...,m_0$, and $\beta_1 \in (0, \beta_0)$ such that \small\begin{equation} \label{cover} \Sigma_{2\beta_1}\subset \cup_{j=1}^{m_0} V(\xi^j,\beta_0)\Subset \Omega. \end{equation}\normalsize \end{itemize}
Now set
\bal \delta_\Sigma^\xi(x):=\left(\sum_{i=k+1}^N|x_i-\Gamma_i^\xi(x')|^2\right)^{\frac{1}{2}}, \qquad x=(x',x'')\in V(\xi,4\beta_0).\eal
Then we see that there exists a constant $C=C(N,\Sigma)$ such that \small\begin{equation}\label{propdist}
d_\Sigma(x)\leq \delta_\Sigma^{\xi^j}(x)\leq C \| \Sigma \|_{C^2} d_\Sigma(x),\quad \forall x\in V(\xi^j,2\beta_0), \end{equation}\normalsize where $\xi^j=((\xi^j)', (\xi^j)'') \in \Sigma$, $j=1,...,m_0$, are the points in \eqref{cover} and \ba \label{supGamma}
\| \Sigma \|_{C^2}:=\sup\{ || \Gamma_i^{\xi^j} ||_{C^2(B_{5\beta_0}^k((\xi^j)'))}: \; i=k+1,...,N, \;j=1,...,m_0 \} < \infty. \ea Moreover, $\beta_1$ can be chosen small enough such that for any $x \in \Sigma_{\beta_1}$, \bal B(x,\beta_1) \subset V(\xi,\beta_0),
\eal where $\xi \in \Sigma$ satisfies $|x-\xi|=d_\Sigma(x)$.
\subsection{Eigenvalue of $-L_\mu$} \label{subsect:eigen} Let \bal H=\frac{N-k-2}{2}. \eal and for $\mu \leq H^2$, let \bal \am=H-\sqrt{H^2-\mu}, \quad \ap=H+\sqrt{H^2-\mu}. \eal Note that $\am\leq H\leq\ap<2H$ and $\am \geq 0$ if and only if $\mu \geq 0$.
We summarize below main properties of the first eigenfunction of the operator $-L_\mu$ in $\Omega \setminus \Sigma$ from \cite[Lemma 2.4 and Theorem 2.6]{DD1} and \cite[page 337, Lemma 7, Theorem 5]{DD2}.
(i) For any $\mu \leq H^2$, it is known that
\small\begin{equation}\label{Lin01} \lambda_\mu:=\inf\left\{\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}\left(|\nabla u|^2-\frac{\mu }{d_\Sigma^2}u^2\right)\dx: u \in C_c^1(\Omega), \int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} u^2 \dx=1\right\}>-\infty. \end{equation}\normalsize
(ii) If $\mu < H^2$, there exists a minimizer $\phi} \def\vgf{\varphi} \def\gh{\eta_{\mu }$ of \eqref{Lin01} belonging to $H^1_0(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)$. Moreover, it satisfies $-L_\mu \phi_\mu= \lambda_\mu \phi_\mu$ in $\Omega \setminus \Sigma$ and $\phi_{\mu }\approx d_\Sigma^{-\am}$ in $\Sigma_{\beta_0}$.
(iii) If $\mu =H^2$, there is no minimizer of \eqref{Lin01} in $H_0^1(\Omega} \def\Gx{\Xi} \def\Gy{\Psi)$, but there exists a nonnegative function $\phi_{H^2}\in H_{loc}^1(\Omega)$ such that $-L_{H^2}\phi_{H^2}=\lambda_{H^2}\phi_{H^2}$ in the sense of distributions in $\Omega \setminus \Sigma$ and $\phi_{H^2}\approx d_\Sigma^{-H} \quad \text{in } \Sigma_{\beta_0}$. In addition, the function $d_\Sigma^{-H}\phi_{H^2}\in H^1_0(\Omega} \def\Gx{\Xi} \def\Gy{\Psi; d_\Sigma^{-2H})$.
From (ii) and (iii) we deduce that, for $\mu \leq H^2$, there holds \small\begin{equation} \label{eigenfunctionestimates} \phi_\mu \approx d\,d^{-\am}_\Sigma \quad \text{in } \Omega \setminus \Sigma. \end{equation}\normalsize
\subsection{Green function and Martin kernel} \label{sec:GreenMartin} Throughout the paper, we always assume that \eqref{assump1} holds. Let $G_\mu$ and $K_{\mu}$ be the Green kernel and Martin kernel of $-L_\mu$ in $\Omega \setminus \Sigma$ respectively. Let us recall sharp two-sided estimates on Green kernel and Martin kernel.
\begin{proposition}[ {\cite[Proposition 4.1]{GkiNg_linear}} ] \label{Greenkernel} ~~
(i) If $\mu< \left( \frac{N-2}{2}\right)^2$ then, for any $x,y \in \Omega \setminus \Sigma$, $x \neq y$,
\bel{Greenesta} \begin{aligned}
G_{\mu}(x,y)&\approx |x-y|^{2-N} \left(1 \wedge \frac{d(x)d(y)}{|x-y|^2}\right) \left(\frac{|x-y|}{d_\Sigma(x)}+1\right)^\am
\left(\frac{|x-y|}{d_\Sigma(y)}+1\right)^\am \\
&\approx |x-y|^{2-N} \left(1 \wedge \frac{d(x)d(y)}{|x-y|^2}\right) \left(1 \wedge \frac{d_\Sigma(x)d_\Sigma(y)}{|x-y|^2} \right)^{-\am}.
\end{aligned} \end{equation}\normalsize
(ii) If $k=0$, $\Sigma=\{0\}$ and $\mu = \left( \frac{N-2}{2}\right)^2$ then, for any $x,y \in \Omega \setminus \Sigma$, $x \neq y$,
\ba\label{Greenestb} \begin{aligned}
&G_{\mu}(x,y) \approx |x-y|^{2-N} \left(1 \wedge \frac{d(x)d(y)}{|x-y|^2}\right) \left(\frac{|x-y|}{|x|}+1\right)^{\frac{N-2}{2}}
\left(\frac{|x-y|}{|y|}+1\right)^{\frac{N-2}{2}}\\
& \qquad \qquad +(|x||y|)^{-\frac{N-2}{2}}\left|\ln\left(1 \wedge \frac{|x-y|^2}{d(x)d(y)}\right)\right| \\
&\approx |x-y|^{2-N} \left(1 \wedge \frac{d(x)d(y)}{|x-y|^2}\right) \left(1 \wedge \frac{|x||y|}{|x-y|^2} \right)^{-\frac{N-2}{2}} +(|x||y|)^{-\frac{N-2}{2}}\left|\ln\left(1 \wedge \frac{|x-y|^2}{d(x)d(y)}\right)\right|.
\end{aligned} \ea
The implicit constants in \eqref{Greenesta} and \eqref{Greenestb} depend on $N,\Omega,\Sigma,\mu$. \end{proposition}
\begin{proposition}[{\cite[Theorem 1.2]{GkiNg_linear}}] \label{Martin} ~~
(i) If $\mu< \left( \frac{N-2}{2}\right)^2$ then
\small\begin{equation} \label{Martinest1}
K_{\mu}(x,\xi) \approx\left\{
\begin{aligned}
&\frac{d(x)d_\Sigma(x)^{-\am}}{|x-\xi|^N}\quad &&\text{if } x \in \Omega \setminus \Sigma,\; \xi \in \partial\Omega, \\
&\frac{d(x)d_\Sigma(x)^{-\am}}{|x-\xi|^{N-2-2\am}} &&\text{if } x \in \Omega \setminus \Sigma,\; \xi \in \Sigma.
\end{aligned} \right.
\end{equation}\normalsize
(ii) If $k=0$, $\Sigma=\{0\}$ and $\mu= \left( \frac{N-2}{2}\right)^2$ then
\small\begin{equation}\label{Martinest2}
K_{\mu}(x,\xi) \approx\left\{
\begin{aligned}
&\frac{d(x)|x|^{-\frac{N-2}{2}}}{|x-\xi|^N}\quad &&\text{if } x \in \Omega \setminus \{0\},\; \xi \in \partial\Omega, \\
&d(x)|x|^{-\frac{N-2}{2}}\left|\ln\frac{|x|}{{\mathcal D}} \def\CE{{\mathcal E}} \def\CF{{\mathcal F}_\Omega}\right| &&\text{if } x \in \Omega \setminus \{0\},\; \xi=0,
\end{aligned} \right.
\end{equation}\normalsize
where ${\mathcal D}} \def\CE{{\mathcal E}} \def\CF{{\mathcal F}_\Omega:=2\sup_{x \in \Omega}|x|$.
The implicit constants depend on $N,\Omega,\Sigma,\mu$. \end{proposition}
\section{Weak Lebesgue estimates} \label{sec:weakLp} \subsection{Auxiliary estimates} We first recall the definition of weak Lebesgue spaces (or Marcinkiewicz spaces). Let $D \subset {\mathbb R}^N$ be a domain. Denote by $L^\kappa_w(D;\tau)$, $1 \leq \kappa < \infty$, $\tau \in \GTM^+(D)$, the weak $L^\kappa$ space defined as follows: a measurable function $f$ in $D$ belongs to this space if there exists a constant $c$ such that
\bal \gl_f(a;\tau):=\tau(\{x \in D: |f(x)|>a\}) \leq ca^{-\kappa}, \quad \forall a>0. \eal The function $\gl_f$ is called the distribution function of $f$ (relative to $\tau$). For $p \geq 1$, denote \bel{weakLp}
L^\kappa_w(D;\tau):=\{ f \text{ Borel measurable}: \sup_{a>0}a^\kappa\gl_f(a;\tau)<\infty\} \end{equation}\normalsize and \bel{semi} \norm{f}^*_{L^\kappa_w(D;\tau)}:=(\sup_{a>0}a^\kappa\gl_f(a;\tau))^{\frac{1}{\kappa}}. \end{equation}\normalsize Note that $\norm{.}_{L^\kappa_w(D;\tau)}^*$ is not a norm, but for $\kappa>1$, it is equivalent to the norm \bal \norm{f}_{L^\kappa_w(D;\tau)}:=\sup\left\{
\frac{\int_{A}|f|\dd\tau}{\tau(A)^{1-\frac{1}{\kappa}}}: A \subset D, \, A \text{
measurable},\, 0<\tau(A)<\infty \right\}. \eal More precisely, \bel{equinorm} \norm{f}^*_{L^\kappa_w(D;\tau)} \leq \norm{f}_{L^\kappa_w(D;\tau)} \leq \frac{\kappa}{\kappa-1}\norm{f}^*_{L^\kappa_w(D;\tau)}. \end{equation}\normalsize We also denote by $\tilde L_w^\kappa$ the weak type $L^\kappa$ space with norm \bel{normLwww} \norm{f}_{\tilde L^\kappa_w(D;\tau)}:=\sup\left\{
\frac{\int_{A}|f|\dd\tau}{\tau(A)^{1-\frac{1}{\kappa}} \ln(e+\tau(A)^{-1})}: A \subset D, \, A \text{
measurable},\, 0<\tau(A)<\infty \right\}. \end{equation}\normalsize
When $\dd\tau=\varphi \, \dx$ for some positive continuous function $\varphi$, for simplicity, we use the notation $L_w^\kappa(D;\varphi)$. Notice that $ L_w^\kappa(D;\varphi) \subset L^{r}(D;\varphi)$ for any $r \in [1,\kappa)$. From \eqref{semi} and \eqref{equinorm}, one can derive the following estimate which is useful in the sequel. For any $f \in L_w^\kappa(D;\varphi)$, there holds
\bel{ue} \int_{\{x \in D: |f(x)| \geq s\} }\varphi \dd x \leq s^{-\kappa}\norm{f}^\kappa_{L_w^\kappa(D;\varphi)}. \end{equation}\normalsize
Let us recall a result from \cite{BVi} which will be used in the proof of weak Lebesgue estimates for the Green kernel and Martin kernel.
\begin{proposition}[{\cite[Lemma 2.4]{BVi}}] \label{bvivier}
Assume $D$ is a bounded domain in ${\mathbb R}^N$ and denote by $\tilde D$ either the set $D$ or the boundary $\partial D$. Let $\gw$ be a nonnegative bounded Radon measure in $\tilde D$ and $\eta\in C(D)$ be a positive weight function. Let $\CH$ be a continuous nonnegative function
on $\{(x,y)\in D\times \tilde D:\;x\neq y\}.$ For any $\lambda > 0$ we set
\bal
A_\lambda(y):=\{x\in D\setminus\{y\}:\;\; \CH(x,y)>\lambda\}\quad \text{and} \quad
m_{\lambda}(y):=\int_{A_\lambda(y)}\eta(x) \, \dd x.
\eal
Suppose that there exist $C>0$ and $\kappa>1$ such that $m_{\lambda}(y)\leq C\lambda^{-\kappa}$ for every $\gl>0$. Then the operator
\bal \BBH[\gw](x):=\int_{\tilde D}\CH(x,y)\dd\gw(y) \eal
belongs to $L^\kappa_w(D;\eta )$ and
\bal
\left|\left|\BBH[\gw]\right|\right|_{L^\kappa_w(D;\eta)}\leq (1+\frac{C\kappa}{\kappa-1})\gw(\tilde D). \eal \end{proposition}
In the sequel, we will use the following notations.
For $\alpha,\gamma \in {\mathbb R}$, let \bel{varphia} \varphi_{\alpha,\gamma}(x):= d_\Sigma(x)^{-\alpha}d(x)^{\gamma}, \quad x \in \Omega \setminus \Sigma. \end{equation}\normalsize For $\kappa, \theta,\gamma\in {\mathbb R}$, we define \bel{F1}
F_{\kappa,\theta,\gamma}(x,y):=d_\Sigma(x)^{\kappa} |x-y|^{-N+2+\theta}d(y)^{-\gamma} \left( 1 \land \frac{d(x)d(y)}{|x-y|^2}\right), \end{equation}\normalsize for $x \neq y, x,y \in \Omega \setminus \Sigma$, and for any positive function $\varphi$ on $\Omega \setminus \Sigma$, set \bal \mathbb{F}_{\kappa,\theta,\gamma}[\varphi \tau](x):=\int_{\Omega \setminus \Sigma}F_{\kappa,\theta,\gamma}(x,y)\varphi(y)\dd\tau(y), \quad \tau \in \GTM(\Omega \setminus \Sigma;\varphi). \eal Put \bel{p1} \p_{\alpha,\theta,\gamma}:=\min\left\{\frac{N-\alpha}{N-2-\alpha},\frac{N+\gamma}{N-2+\gamma-\theta}\right\}. \end{equation}\normalsize
\begin{lemma} \label{anisotitaweakF1a} Let $0<\alpha\leq H$, where $H$ is defined in \eqref{valueH}, and $0 \leq \gamma\leq1$. Then \bel{estF1}
\| \mathbb{F}_{-\alpha,2\alpha,\gamma}[\varphi_{\alpha,\gamma}\tau]\|_{L_w^{\p_{\alpha,2\alpha,\gamma}}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma;\varphi_{\alpha,1} )} \lesssim
\|\gt\|_{\mathfrak{M}(\Omega\setminus \Sigma;\varphi_{\alpha,\gamma})}, \quad \forall \tau\in \mathfrak{M}(\Omega\setminus \Sigma; \varphi_{\alpha,\gamma}).
\end{equation}\normalsize
The implicit constant in \eqref{estF1} depends on $N,\Omega,\Sigma,\alpha,\gamma$. \end{lemma}
\begin{proof} Without loss of generality, we may assume $\tau \in \GTM^+(\Omega \setminus \Sigma;\varphi_{\alpha,\gamma})$. Set
\bal A_\lambda(y)&:=\Big\{x\in(\Omega\setminus \Sigma)\setminus\{y\}:\;\; F_{-\alpha,2\alpha,\gamma}(x,y)>\lambda \Big \},\\ \nonumber A_{\lambda,1}(y)&:=\Big\{x\in(\Omega\setminus \Sigma)\setminus\{y\}:\;\; F_{-\alpha,2\alpha,\gamma}(x,y)>\lambda\;\text{ and}\;d_\Sigma(x)\leq |x-y| \Big \},\\ \nonumber A_{\lambda,2}(y)&:=\Big\{x\in(\Omega\setminus \Sigma)\setminus\{y\}:\;\; F_{-\alpha,2\alpha,\gamma}(x,y)>\lambda\;\text{ and}\;d_\Sigma(x)> |x-y| \Big \},\\ \nonumber m_{\lambda}(y)&:=\int_{A_\lambda(y)}\varphi_{\alpha,1} \dx,\quad m_{\lambda,i}(y):=\int_{A_{\lambda,i}(y)}\varphi_{\alpha,1}\dx, \quad i=1,2. \eal Then $A_\lambda(y)=A_{\lambda,1}(y)\cup A_{\lambda,2}(y)$ and \bal m_{\lambda}(y)= m_{\lambda,1}(y)+ m_{\lambda,2}(y). \eal
Let $\beta_1$ be as in \eqref{cover}. We write \ba\label{ml1F2-0} m_{\lambda}(y)=\int_{A_\lambda(y)\cap \Sigma_{ \frac{\beta_1}{4} }}d(x)d_\Sigma(x)^{-\alpha} \dx+\int_{A_\lambda(y)\setminus \Sigma_{ \frac{\beta_1}{4}}}d(x)d_\Sigma(x)^{-\alpha} \dx. \ea We will estimate successively the terms on the right hand side of \eqref{ml1F2-0}. We consider only the case $H<\frac{N-2}{2}$ since the case $H=\frac{N-2}{2}$ (i.e. in case $k=0$, $\Sigma=\{0\}$) can be treated in a similar way.
We split the first term on the right hand side of \eqref{ml1F2-0} as \ba \label{mcompose-1} \int_{A_{\lambda}(y)\cap \Sigma_{ \frac{\beta_1}{4} }}d(x)d_\Sigma(x)^{-\alpha} \dx = \int_{A_{\lambda,1}(y)\cap \Sigma_{ \frac{\beta_1}{4} }}d(x)d_\Sigma(x)^{-\alpha} \dx + \int_{A_{\lambda,2}(y)\cap \Sigma_{ \frac{\beta_1}{4} }}d(x)d_\Sigma(x)^{-\alpha} \dx. \ea We note that \bel{app:6}
1 \land \frac{d(x)d(y)}{|x-y|^2} \leq 2\left( 1 \land \frac{d(y)}{|x-y|} \right) \leq 4 \frac{d(y)}{d(x)}, \quad \forall x,y \in \Omega, \; x \neq y, \end{equation}\normalsize therefore \small\begin{equation}\label{52}
F_{-\alpha,2\alpha,\gamma}(x,y)\leq 4^\gamma d_\Sigma(x)^{-\alpha} d(x)^{-\gamma}|x-y|^{-N+2+2\alpha},\quad \forall x,y \in \Omega, \; x \neq y. \end{equation}\normalsize Since $0<\alpha <\frac{N-2}{2}$, from \eqref{52} we see that \bal A_{\lambda,1}(y)\cap \Sigma_{ \frac{\beta_1}{4} }
\subset \Big\{x\in(\Omega\setminus \Sigma)\setminus\{y\}:\; d_\Sigma(x) < c\lambda^{-\frac{1}{N-2-\alpha}},\; |x-y| <c\lambda^{-\frac{1}{N-2-2\alpha}}d_\Sigma(x)^{-\frac{\alpha}{N-2-2\alpha}} \Big \}. \eal By applying Lemma \ref{lemapp:1} with $\alpha_1=-\alpha$, $\alpha_2=-\frac{\alpha}{N-2-2\alpha}$, $\ell_1=\lambda^{-\frac{1}{N-2-\alpha}}$, $\ell_2=\lambda^{-\frac{1}{N-2-2\alpha}}$ and taking into account that $N-k - \alpha -\frac{k\alpha}{N-2-2\alpha} \geq 2$ since $\alpha\leq H$, we deduce, for $\lambda \geq 1$, \ba \label{ca1-1.1} \int_{A_{\lambda,1}(y)\cap \Sigma_{ \frac{\beta_1}{4} }}d(x)d_\Sigma(x)^{-\alpha} \dx \lesssim \lambda^{-\frac{N-\alpha}{N-2-\alpha}} \leq \lambda^{-\p_{\alpha,2\alpha,\gamma}}. \ea Next, by \eqref{52}, we see that
\bal A_{\lambda,2}(y)\cap \Sigma_{\frac{\beta_1}{4}} \subset \left\{ x \in \Omega \setminus \Sigma: |x-y| < c\lambda^{-\frac{1}{N-2-\alpha}} \text{ and } d_\Sigma(x) > |x-y| \right\}. \eal Therefore, for every $\lambda \geq 1$, \ba \label{ca1-1.3} \begin{aligned} \int_{A_{\lambda,2}(y)\cap \Sigma_{\frac{\beta_1}{4}}}d(x)d_\Sigma(x)^{-\alpha}\dx
\lesssim \int_{\{|x-y|\leq c\lambda^{-\frac{1}{N-2-\alpha}}\}}|x-y|^{-\alpha}\dx \lesssim \lambda^{-\frac{N-\alpha}{N-2-\alpha}} \leq \lambda^{-\p_{\alpha,2\alpha,\gamma}}. \end{aligned} \ea Combining \eqref{mcompose-1}, \eqref{ca1-1.1} and \eqref{ca1-1.3} yields, for any $\lambda \geq 1$, \small\begin{equation} \label{AA1} \int_{A_{\lambda}(y)\cap \Sigma_{ \frac{\beta_1}{4} }}d(x)d_\Sigma(x)^{-\alpha} \dx \lesssim \lambda^{-\p_{\alpha,2\alpha,\gamma}}. \end{equation}\normalsize
Next we estimate the second term on the right hand side of \eqref{ml1F2-0}. By \eqref{app:6}, we have
\bal A_\lambda(y)\cap (\Omega\setminus\Sigma_{ \frac{\beta_1}{4}})\subset\left\{ x \in \Omega \setminus \Sigma: |x-y| < c\lambda^{-\frac{1}{N+\gamma-2-2\alpha}} \text{ and } d(x)^{\gamma} \leq \lambda^{-1}|x-y|^{-N+2+2\alpha} \right\}. \eal This yields, for $\lambda \geq 1$, \ba \label{AA-2} \begin{aligned}
\int_{A_{\lambda}(y) \setminus \Sigma_{\frac{\beta_1}{4}}} d(x)d_{\Sigma}(x)^{-\alpha}\dx &\lesssim \int_{A_{\lambda}(y) \setminus \Sigma_{\frac{\beta_1}{4}}} d(x)^\gamma\dx \\ &\lesssim\int_{\{|x-y| < c\lambda^{-\frac{1}{N+\gamma-2-2\alpha}}\}} \lambda^{-1}|x-y|^{-N+2+2\alpha}\dx\\ &\lesssim \lambda^{-\frac{N+\gamma}{N-2+\gamma-2\alpha}} \lesssim \lambda^{-\p_{\alpha,2\alpha,\gamma}}. \end{aligned} \ea
Combining \eqref{ml1F2-0}, \eqref{AA1} and \eqref{AA-2} yields \bel{ca2-1.5} m_{\lambda}(y)\leq C\lambda^{-\p_{\alpha, 2\alpha,\gamma}}, \quad \forall \lambda>0, \end{equation}\normalsize where $C=C(N,\Omega,\Sigma,\alpha,\gamma)$. By applying Proposition \ref{bvivier} with $\CH(x,y)=F_{\alpha,2\alpha,\gamma}(x,y),$ $\tilde D=D=\Omega \setminus \Sigma$, $\eta=d\, d_\Sigma^{-\alpha}$ and $\omega=d^\gamma\, d_\Sigma^{-\alpha} \tau$ and using \eqref{ca2-1.5}, we finally derive \eqref{estF1}. \end{proof}
By using a similar argument as in the proof of Lemma \ref{anisotitaweakF1a}, one can obtain the following lemma. \begin{lemma} \label{anisotitaweakF1b} Let $0<\alpha\leq H$ and $0 \leq \gamma\leq1$. Then \bel{estF1b}
\| \mathbb{F}_{\alpha,0,\gamma}[\varphi_{\alpha,\gamma}\tau]\|_{L_w^{\p_{\alpha,0,\gamma}}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma;\varphi_{\alpha,1} )} \lesssim
\|\gt\|_{\mathfrak{M}(\Omega\setminus \Sigma;\varphi_{\alpha,\gamma})}, \quad \forall \tau\in \mathfrak{M}(\Omega\setminus \Sigma; \varphi_{\alpha,\gamma}).
\end{equation}\normalsize
The implicit constant in \eqref{estF1b} depends on $N,\Omega,\Sigma,\alpha,\gamma$. \end{lemma}
Set \bel{F4}
\tilde F_{\gamma}(x,y):=|x|^{-\frac{N-2}{2}} d(y)^{-\gamma} \left|\ln\left(1 \land \frac{|x-y|^2}{d(x)d(y)}\right)\right|, \quad x \neq y,\; x,y \in \Omega \setminus \{0\}, \end{equation}\normalsize \bal \mathbb{\tilde F}_{\gamma}[\varphi_{\frac{N-2}{2},\gamma}\tau](x):=\int_{\Omega \setminus \{0\}}\tilde F_{\gamma}(x,y)\varphi_{\frac{N-2}{2},\gamma}(y)d\tau(y), \quad \tau \in \mathfrak{M}(\Omega\setminus \{0\};\varphi_{\frac{N-2}{2},\gamma}), \eal where $\varphi_{\frac{N-2}{2},\gamma}$ is defined in \eqref{varphia}.
For $\theta, \kappa \in {\mathbb R}$, put \bal \tilde \p_{\theta,\kappa}:=\min\left\{\frac{N+\theta}{N-2},N+\kappa \right\}. \eal \begin{lemma} \label{anisotitaweakF4} Let $k=0$, $\Sigma=\{0\}$, $-N+1<\kappa<1$, $-2<\theta<2$. Then \bel{estF4}
\norm{\mathbb{\tilde F}_\gamma[\varphi_{\frac{N-2}{2},\gamma}\gt]}_{L_w^{\tilde \p_{\theta,\kappa}}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \{0\};\varphi_{\frac{N-2}{2},1})}
\lesssim \norm{\gt}_{\mathfrak{M}(\Omega\setminus \{0\};\varphi_{\frac{N-2}{2},\gamma})}, \quad \forall \tau\in \mathfrak{M}(\Omega\setminus \{0\};\varphi_{\frac{N-2}{2},\gamma}).
\end{equation}\normalsize
The implicit constant depends on $N,\Omega,\gamma,\theta,\kappa$. \end{lemma} \begin{proof} We may assume $\tau\in \mathfrak{M}^+(\Omega\setminus \{0\};\varphi_{\frac{N-2}{2},\gamma})$. For $\lambda>0$ and $y \in \Omega \setminus \{0\}$, set \bal
&A_\lambda(y):=\Big\{x\in \Omega\setminus \{0,y\}:\;\; \tilde F_{\gamma}(x,y)>\lambda \Big \} \quad \text{and} \quad m_{\lambda}(y):=\int_{A_\lambda(y)}d(x)|x|^{-\frac{N-2}{2}} \dx, \\
&A_{\lambda,1}(y):=\Big\{x\in \Omega\setminus \{0,y\}:\;\; \tilde F_{\gamma}(x,y)>\lambda \text{ and } |x-y| \leq |x| \Big \}, \\
&A_{\lambda,2}(y):=\Big\{x\in \Omega\setminus \{0,y\}:\;\; \tilde F_{\gamma}(x,y)>\lambda \text{ and } |x-y| \geq |x| \Big \}. \eal
It can be shown from \eqref{F4} that \ba \label{f5a}
\tilde F_{\gamma}(x,y)\leq 2d(y)^{-\gamma}|x|^{-\frac{N-2}{2}}\left(-\ln\frac{|x-y|}{{\mathcal D}} \def\CE{{\mathcal E}} \def\CF{{\mathcal F}_\Omega}\right) \left( 1 \land \frac{d(x)d(y)}{|x-y|^2} \right), \; \forall x\neq y,\;x,y\in \Omega\setminus\{0\},
\ea where ${\mathcal D}} \def\CE{{\mathcal E}} \def\CF{{\mathcal F}_\Omega = 2\sup_{x \in \Omega}|x|$.
We write
\ba m_{\lambda}(y)=\int_{A_\lambda(y)\cap B(0,\frac{\beta_1}{4})}d(x)|x|^{-\frac{N-2}{2}} \dx+\int_{A_\lambda(y)\setminus B(0,\frac{\beta_1}{4})}d(x)|x|^{-\frac{N-2}{2}} \dx.\label{ml1F4} \ea
The first term on the right hand side of \eqref{ml1F4} is estimated by using \eqref{f5a} and \eqref{app:6} as \ba \label{ml6f5} \begin{aligned}
&\int_{A_{\lambda}(y)\cap B(0,\frac{\beta_1}{4})}d(x)|x|^{-\frac{N-2}{2}}\dx \lesssim \int_{A_{\lambda,1}(y)\cap B(0,\frac{\beta_1}{4}) }|x|^{-\frac{N-2}{2}}\dx+\int_{A_{\lambda,2}(y)\cap B(0,\frac{\beta_1}{4})}|x|^{-\frac{N-2}{2}}\dx\\
&\lesssim \int_{\{|x-y|\leq c(\lambda^{-1}\ln\lambda)^{\frac{2}{N-2}}\}}|x-y|^{-\frac{N-2}{2}}\dx+\int_{\{|x|\leq c(\lambda^{-1}\ln\lambda)^{\frac{2}{N-2}}\}}|x|^{-\frac{N-2}{2}}\dx \\ &\lesssim (\lambda^{-1}\ln\lambda)^{\frac{N+2}{N-2}}, \quad \forall \lambda>e. \end{aligned} \ea
The second term on the right hand side of \eqref{ml1F4} is estimated using \eqref{app:6} and \eqref{f5a} as \ba \label{ml4f5} \begin{aligned}
\int_{A_\lambda(y)\setminus B(0,\frac{\beta_1}{4})}d(x)|x|^{-\frac{N-2}{2}} \dx&\lesssim \int_{\{|x-y|^{-1}(-\ln\frac{|x-y|}{{\mathcal D}} \def\CE{{\mathcal E}} \def\CF{{\mathcal F}_\Omega})\geq c\lambda\}} \lambda^{-1}\left(-\ln\frac{|x-y|}{{\mathcal D}} \def\CE{{\mathcal E}} \def\CF{{\mathcal F}_\Omega} \right)\dx\\
&\lesssim \int_{\{|x-y|\leq c'\lambda^{-1}\ln\lambda\}} \lambda^{-1}\left(-\ln\frac{|x-y|}{{\mathcal D}} \def\CE{{\mathcal E}} \def\CF{{\mathcal F}_\Omega}\right) \dd x \\ &\lesssim (\lambda^{-1}\ln\lambda)^{N+1},\quad \forall \lambda>e. \end{aligned} \ea
Combining \eqref{ml1F4}, \eqref{ml6f5} and \eqref{ml4f5}, together with $-2<\theta<2$, we deduce \bel{ml6f4} m_{\lambda}(y) \lesssim (\lambda^{-1}\ln\lambda)^{\frac{N+2}{N-2}}+(\lambda^{-1}\ln\lambda)^{N+1} \lesssim \lambda^{-\frac{N+\theta}{N-2}} + \lambda^{-(N+\kappa)} \lesssim \lambda^{-\tilde \p_{\theta,\kappa}},\quad \forall\lambda>e. \end{equation}\normalsize
Thus by applying Proposition \ref{bvivier} with $\mathcal{H}(x,y)=\tilde F_{\gamma}(x,y)$, $\tilde D=D=\Omega \setminus \{0\}$, $\eta(x)=d(x)|x|^{-\frac{N-2}{2}}$ and $\dd\nu=d(x)^\gamma|x|^{-\frac{N-2}{2}}\dd\tau$, we obtain \eqref{estF4}. \end{proof}
For $\alpha,\theta \in {\mathbb R}$, put
\ba \label{Halthe} H_{\alpha,\theta}(x,y):=d(x)d_{\Sigma}(x)^{-\alpha} |x-y|^{-N+\theta}, \quad x \in \Omega \setminus \Sigma, y \in \partial \Omega \cup \Sigma, \ea \bal \mathbb{H}_{\alpha,\theta}[\nu](x): = \int_{\partial \Omega \cup \Sigma} H_{\alpha,\theta}(x,y)\dd\nu(y), \quad \nu \in \GTM(\partial \Omega \cup \Sigma), \eal and \bal \quad_{\alpha,\theta}:= \min \left \{ \frac{N-k-\alpha}{\alpha}, \frac{N+1}{N-1-\theta} \right\}. \eal
\begin{theorem} \label{H}
(i) Assume $k \geq 0$, $0<\alpha \leq H$, $\theta<N-1$, and $\gn\in \mathfrak{M}(\partial\Omega\cup \Sigma)$ with compact support in $\partial\Omega$. Then \ba \label{est:H1}
\norm{\mathbb{H}_{\alpha,\theta}[\nu]}_{L_w^{ \quad_{\alpha,\theta}}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma;\varphi_{\alpha,1})} \lesssim \|\nu\|_{\mathfrak{M}(\partial\Omega \cup \Sigma)}. \ea
(ii) Assume $k>0$, $\alpha \leq H$, $\theta\leq N-k$, $\theta<N+\alpha$, and $\gn\in \mathfrak{M}(\partial\Omega\cup \Sigma)$ with compact support in $\Sigma$. Then \ba \label{est:H2} \norm{\mathbb{H}_{\alpha,\theta}[\nu]}_{L_w^{\frac{N-\alpha}{N+\alpha-\theta}}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma;{\phi_{\mu }})} \lesssim \norm{\nu}_{\mathfrak{M}(\partial \Omega \cup \Sigma)}. \ea The implicit constants in \eqref{est:H1} and \eqref{est:H2} depend only on $N,\Omega,\Sigma,\alpha,\theta$. \end{theorem} \begin{proof} For $y\in \partial \Omega \cup \Sigma$, set \bal A_\lambda(y):=\Big\{x\in(\Omega\setminus \Sigma):\; H_{\alpha,\theta}(x,y)>\lambda \Big \}, \quad m_{\lambda}(y)&:=\int_{A_\lambda(y)}d(x)d_\Sigma(x)^{-\alpha} \dx. \eal We write \ba \label{split-K} m_{\lambda}(y)=\int_{A_\lambda(y)\cap \Sigma_{\beta_1}}d(x)d_\Sigma(x)^{-\alpha} \dx+\int_{A_\lambda(y)\setminus \Sigma_{\beta_1}}d(x)d_\Sigma(x)^{-\alpha} \dx. \ea
(i) Assume $\gn\in \mathfrak{M}(\partial\Omega\cup \Sigma)$ with compact support in $\partial\Omega$ and without loss of generality, we may assume that $\gn \geq 0$. Let $y \in \partial \Omega$.
First we treat the first term on the right hand side of \eqref{split-K}. If $0<\alpha \leq H$ then by applying Lemma \ref{lemapp:1}, we obtain, for $\lambda \geq 1$, \bal \int_{A_\lambda(y)\cap \Sigma_{\beta_1}}d_\Sigma(x)^{-\alpha} \dx \lesssim \int_{\{d_\Sigma(x)\leq c\lambda^{-\frac{1}{\alpha}}\} \cap \Sigma_{\beta_1}}d_\Sigma(x)^{-\alpha}\dx \lesssim \lambda^{-\frac{N-k-\alpha}{\alpha}} \leq \lambda^{ - \quad_{\alpha,\theta}}. \eal
If $\alpha \leq 0$ then there exists $\bar C=\bar C(N,\Omega,\Sigma,\alpha,\theta)>1$ such that for any $\lambda>\bar C$, $A_\lambda(y) \cap \Sigma_{\beta_1}=\emptyset$. Consequently, for all $\lambda>\bar C$, \bel{ml3mb} \int_{A_\lambda(y)\cap \Sigma_{\beta_1}}d_\Sigma(x)^{-\alpha} \dx=0. \end{equation}\normalsize
Next we treat the second term on the right hand side of \eqref{split-K}. By using the estimate $d(x) \leq |x-y|$, we see that, for $\lambda \geq 1$, \ba \label{ml2ma}
\int_{A_\lambda(y)\setminus \Sigma_{\beta_1}}d(x)\dx \lesssim \int_{\{|x-y|\leq c\lambda^{-\frac{1}{N-1-\theta}}\}}|x-y| \dx \lesssim \lambda^{-\frac{N+1}{N-1-\theta}} \leq \lambda^{ - \quad_{\alpha,\theta}}. \ea Combining \eqref{ml3mb} and \eqref{ml2ma}, we obtain \ba \label{ml4ma} m_{\lambda}(y)\leq C\lambda^{-\quad_{\alpha,\theta}}, \ea for all $\lambda>\bar C$, where $C=C(N,\Omega,\Sigma,\alpha,\theta)$. Then we can show that \eqref{ml4ma} holds true for all $\lambda>0$. By applying Proposition \ref{bvivier} with $\mathcal{H}(x,y)=H_{\alpha,\theta}(x,y)$, $\tilde D=D=\Omega \setminus \Sigma$, $\eta=\varphi_{\alpha,1}$ and $\omega=\nu$, we obtain \eqref{estmartin2}.
(ii) Assume $\gn\in \mathfrak{M}(\partial\Omega\cup \Sigma)$ with compact support in $\Sigma$ and without loss of generality, we may assume that $\gn \geq 0$. Let $y \in \Sigma$.
\noindent \textbf{Case 1:} $0<\alpha \leq H$. First we treat the first term in \eqref{split-K}. We notice that since $y \in \Sigma$, $d_\Sigma(x) \leq |x-y|$ for every $x \in \Omega \setminus \Sigma$, hence
\bal A_\lambda(y) \subset \{ x \in \Omega \setminus \Sigma: d_\Sigma(x) \leq c\lambda^{-\frac{1}{N+\alpha-\theta}} \quad \text{and} \quad |x-y|<c\lambda^{-\frac{1}{N-\theta}}d_\Sigma(x)^{-\frac{\alpha}{N-\theta}} \}. \eal Therefore, by applying Lemma \ref{lemapp:1} with $\alpha_1=-\alpha$, $\alpha_2=-\frac{\alpha}{N-\theta}$, $\ell_1=c\lambda^{-\frac{1}{N+\alpha-\theta}}$, $\ell_2=c\lambda^{-\frac{1}{N-\theta}}$ and noting that $N-k-\alpha- \frac{k\alpha}{N-\theta}\geq 2$ due to the fact that $\alpha \leq H$ and $\theta \leq N-k$, we obtain \ba \label{ml3maK-1} \int_{A_\lambda(y)\cap \Sigma_{\beta_1}}d_\Sigma(x)^{-\alpha} \dx \lesssim \lambda^{-\frac{N-\alpha}{N+\alpha-\theta}}. \ea
Next we treat the second term in \eqref{split-K}. We see that there exists a constant $\bar C=\bar C(N,\Omega,\Sigma,\alpha,\theta)>1$ such that for any $\lambda>\bar C$, there holds \ba \label{ml3maK-2} \int_{A_\lambda(y)\setminus \Sigma_{\beta_1}}d(x)\dx=0. \ea
Combining \eqref{split-K}, \eqref{ml3maK-1} and \eqref{ml3maK-2}, we deduce \ba \label{mlam-1} m_{\lambda}(y)\leq C\,\lambda^{-\frac{N-\alpha}{N+\alpha-\theta}}. \ea for all $\lambda > \hat C$, where $C=C(N,\Omega,\Sigma,\alpha,\theta)$.
\noindent \textbf{Case 2:} $\alpha \leq 0$. By noting that $d_\Sigma(x)^{-\alpha} \leq |x-y|^{-\alpha}$ and $|x-y| \leq c \lambda^{-\frac{1}{N-2-\am}}$ for every $x \in A_\lambda(y)$, we can easily obtain \eqref{mlam-1}.
From case 1 and case 2, by applying Proposition \ref{bvivier} with $\mathcal{H}(x,y)=H_{\alpha,\theta}(x,y)$, $D = \Omega \setminus \Sigma$, $\tilde D=\partial\Omega\cup \Sigma,$ $\eta=\varphi_{\alpha,1}$ and $\omega=\nu$, we obtain \eqref{estmartin2}. The proof is complete. \end{proof}
We put \bal
\tilde H_{\alpha}(x,y):=d(x)|x-y|^{-\alpha} \left| \ln\frac{|x-y|}{{\mathcal D}} \def\CE{{\mathcal E}} \def\CF{{\mathcal F}_\Omega} \right|, \quad x \in \Omega \setminus \{y\}, \eal \bal \mathbb{\tilde H}_{\alpha}[\nu](x): = \int_{\partial \Omega \cup \Sigma} \tilde H_{\alpha}(x,y)\,\dd\nu(y),
\eal where ${\mathcal D}} \def\CE{{\mathcal E}} \def\CF{{\mathcal F}_\Omega=2\sup_{x \in \Omega}|x|$. \begin{theorem} \label{LwSigma0} Assume $0<\alpha<\frac{N}{2}$, $\rho\in[-1,1] \setminus \{0\}$, $0 \in \Omega$ and let $\delta_0$ be the Dirac measure concentrated on $\{0\}$. For $\lambda>0$, set \ba
\tilde{A}_\lambda(0):=\Big\{x\in \Omega\setminus \{0\}:\; |\mathbb{\tilde H}_{\alpha}[\rho \delta_0](x)|>\lambda \Big \}, \quad \tilde{m}_{\lambda}&:=\int_{\tilde{A}_\lambda(0)}d(x)|x|^{-\alpha} \dx.\label{69} \ea Then \ba\label{54}
\tilde{m}_{\lambda}\lesssim (\lambda^{-1}\ln(e+\lambda))^{\frac{N-\alpha}{\alpha}}(|\rho|\ln(e+|\rho|^{-1}))^{\frac{N-\alpha}{\alpha}}, \quad \forall \lambda>0, \ea and \ba \label{est-LwSigma0}
\| \mathbb{\tilde H}_{\alpha}[\rho \delta_0] \|_{\tilde L_w^{\frac{N-\alpha}{\alpha}}(\Omega \setminus \{0\};\varphi_{\alpha,1})} \lesssim |\rho|. \ea The implicit constants in the above estimates depend only on $N,\Omega, \alpha$. Here weak Lebesgue spaces $\tilde L_w^p$ are defined in \eqref{normLwww}. \end{theorem} \begin{proof}
Consider $\lambda> \max\{{\mathcal D}} \def\CE{{\mathcal E}} \def\CF{{\mathcal F}_\Omega,{\mathcal D}} \def\CE{{\mathcal E}} \def\CF{{\mathcal F}_\Omega^{-1},e \}$ and
\bal A_\lambda(0):=\Big\{x\in \Omega\setminus \{0\}:\; \tilde H_{\alpha}(x,0)>\lambda \Big \}, \quad m_{\lambda}&:=\int_{A_\lambda(0)}d(x)|x|^{-\alpha} \dx. \eal We note that $
A_\lambda(0) \subset \left\{ x \in \Omega \setminus \{0\}: |x| \leq c\left(\lambda^{-1} \ln\lambda \right)^{\frac{1}{\alpha}} \right\} $. As a consequence,
\bal m_{\lambda} &\lesssim \int_{A_\lambda(0)}|x|^{-\alpha} \dx \lesssim \int_{ \left\{|x|\leq c(\lambda^{-1} \ln \lambda)^{\frac{1}{\alpha}} \right\}}|x|^{-\alpha} \dx \lesssim (\lambda^{-1} \ln\lambda)^{\frac{N-\alpha}{\alpha}}. \eal Therefore,
\bal m_{\lambda} &\lesssim \int_{A_\lambda(0)}|x|^{-\alpha} \dx \lesssim (\lambda^{-1}\ln (e+\lambda))^{\frac{N-\alpha}{\alpha}},\quad\forall \lambda>0. \eal This implies \eqref{54}.
Let $A \subset \Omega\setminus \{0\}$ be a measurable set such that $|A|>0$ and let $\dd\tau=d(x)|x|^{-\alpha} \dd x.$ Then for any $\lambda>0,$ we have \bal
\int_A \tilde H_{\alpha}(x,0)d(x)|x|^{-\alpha}\, \dd x &\leq \lambda \tau(A) + \int_{A_\lambda(0)} \tilde H_{\alpha}(x,0)d(x)|x|^{-\alpha}\, \dd x \\ & = \lambda \tau(A) + \lambda m_\lambda+\int_\lambda^\infty m_s \dd s \\ & \lesssim \lambda \tau(A) + \lambda m_\lambda+\int _\lambda^\infty (s^{-1}\ln (e+s))^{\frac{N-\alpha}{\alpha}} \dd s \\ & \lesssim \lambda \tau(A) + \lambda^{1-\frac{N-\alpha}{\alpha}}(\ln (e+\lambda))^{\frac{N-\alpha}{\alpha}}. \eal Taking $\lambda=\tau(A)^{-\frac{\alpha}{N-\alpha}}\ln(e+\tau(A)^{-1})$, we obtain \bal \int_A \tilde H_{\alpha}(x,0)\varphi_{\alpha,1}\, \dd x \lesssim \tau(A)^{1-\frac{\alpha}{N-\alpha}}\ln \left(e+\tau(A)^{-\frac{\alpha}{N-\alpha}}\right). \eal Thus estimate \eqref{est-LwSigma0} follows by using \eqref{normLwww}. \end{proof}
\begin{remark}\label{critical} Conversely, if we assume that \ba\label{55}
\| \mathbb{\tilde H}_{\alpha}[\rho \delta_0] \|_{\tilde L_w^{\frac{N-\alpha}{\alpha}}(\Omega \setminus \{0\};\varphi_{\alpha,1})} \lesssim |\rho| \ea for some $\rho\in[-1,1] \setminus \{0\}$ then \eqref{54} holds. Indeed, we assume that \eqref{55} is valid. Then by \eqref{55}, we have \ba\label{56} \nonumber
\lambda \tilde m_\lambda^{\frac{\alpha}{N-\alpha}}\ln \left(e+\tilde m_\lambda^{-1}\right)^{-1} &\leq \frac{\int_{\tilde A_\lambda(0)}|\mathbb{\tilde H}_{\alpha}[\rho \delta_0](x)|d(x) |x|^{-\alpha}\, \dd x}{\tilde m_\lambda^{1-\frac{\alpha}{N-\alpha}}\ln \left(e+\tilde m_\lambda^{-1}\right)} \\ &\leq
\| \mathbb{\tilde H}_{\alpha}[\rho \delta_0] \|_{\tilde L_w^{\frac{N-\alpha}{\alpha}}(\Omega \setminus \{0\};d(x)|x|^{-\alpha})}\leq C|\rho|, \ea where $\tilde A_\lambda(0)$ and $\tilde m_\lambda$ have been defined in \eqref{69}. Therefore, \ba\label{70}
\tilde m_\lambda^{-\frac{\alpha}{N-\alpha}}\ln \left(e+\tilde m_\lambda^{-1}\right)\geq C^{-1}\frac{\lambda}{|\rho|}. \ea Hence, if $\tilde m_\lambda<\frac{1}{e}$ we have that \ba\label{71}
\tilde m_\lambda^{-\frac{\alpha}{N-\alpha}}\ln(\tilde m_\lambda^{-1})\geq C_0\frac{\lambda}{|\rho|}. \ea
Now we observe that if $r\in(0,1)$ and $s>e$ then \ba r^{-1}\ln (r^{-1})>s\Longrightarrow r\leq s^{-1}\ln s.\label{57} \ea
Taking $r=\tilde m_\lambda^{\frac{\alpha}{N-\alpha}}$ and $s=C_1(\alpha,N)\frac{\lambda}{|\rho|}$ in \eqref{57} yields $
\tilde m_\lambda \lesssim \lambda^{-\frac{N-\alpha}{\alpha}} \bigg(|\rho|\ln\frac{\lambda}{|\rho|}\bigg)^{\frac{N-\alpha}{\alpha}}, $ which implies \eqref{54}. \end{remark}
\subsection{Weak Lebesgue estimate on Green kernel}
In this subsection, we will use the results of the previous subsection to establish estimates of the Green kernel. Let $\varphi_{\alpha,\gamma}$ be as in \eqref{varphia}. For a measure $\tau$ on $\Omega \setminus \Sigma$, the Green operator acting on $\tau$ is \bal \BBG_\mu[\tau](x)=\int_{\Omega \setminus \Sigma}G_{\mu}(x,y)\dd\tau(y). \eal
\begin{theorem} \label{lpweakgreen} Assume $k \geq 0$, $0<\mu \leq H^2$ and $0\leq\gamma\leq1$. Then \ba \label{estgreen}
\norm{\BBG_\mu[\gt]}_{L_w^{\frac{N+\gamma}{N+\gamma-2}}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma;{\phi_{\mu }})} \lesssim \norm{\gt}_{\mathfrak{M}(\Omega\setminus \Sigma;\varphi_{\am,\gamma})}, \quad \forall \tau\in \mathfrak{M}(\Omega\setminus \Sigma;\varphi_{\am,\gamma}). \ea
The implicit constant depends on $N,\Omega,\Sigma,\mu,\gamma$. \end{theorem}
\begin{proof} Without loss of generality we may assume that $\tau$ is nonnegative. We consider the following cases.
\noindent \textbf{Case 1: $0<\mu<\left( \frac{N-2}{2} \right)^2$.} Then $0<\am< \frac{N-2}{2}$. From \eqref{eigenfunctionestimates}, \eqref{Greenesta}, \eqref{F1} and the fact that $d_\Sigma(y) \leq |x-y|+d_\Sigma(x)$, we obtain, for all $x,y\in \Omega\setminus \Sigma, x\neq y$,
\bal G_\mu(x,y)\varphi_{\am,\gamma}(y)^{-1} &\lesssim |x-y|^{2-N} \min \left\{ 1, \frac{d(x)d(y)}{|x-y|^2} \right\} (|x-y|+d_\Sigma(x))^{2\am} d_\Sigma(x)^{-\am}d(y)^{-\gamma} \\ &\lesssim F_{-\am,2\am,\gamma}(x,y)+F_{\am,0,\gamma}(x,y). \eal This, together with Lemmas \ref{anisotitaweakF1a}--\ref{anisotitaweakF1b} , estimate $\varphi_{\am,1} \approx \phi_\mu$ and the fact that (see \eqref{p1}) \bal \frac{N+\gamma}{N+\gamma-2} = \p_{\am,0,\gamma} \leq \p_{-\am,2\am,\gamma}, \eal implies \eqref{estgreen}.
\noindent \textbf{Case 2:} $k=0$, $\Sigma=\{0\}$ and $\mu=\left( \frac{N-2}{2} \right)^2$. Then $\am=\frac{N-2}{2}$. From \eqref{eigenfunctionestimates}, \eqref{Greenestb} and the fact that $|y| \leq |x-y|+|x|$, we obtain, for all $x,y\in \Omega\setminus \{0\}, x\neq y$, \bal G_{(\frac{N-2}{2})^2}(x,y)\varphi_{\frac{N-2}{2},\gamma}(y)^{-1}\lesssim F_{-\frac{N-2}{2},N-2,\gamma}(x,y)+F_{\frac{N-2}{2},0,\gamma}(x,y)+\tilde F_{\gamma}(x,y), \eal This, together with Lemmas \ref{anisotitaweakF1a}--\ref{anisotitaweakF4} and the fact that (see \eqref{p1}) \bal \frac{N+\gamma}{N+\gamma-2} = \p_{-\frac{N-2}{2},N-2,\gamma} \leq \p_{\frac{N-2}{2},0,\gamma}, \eal implies \eqref{estgreen}. The proof is complete. \end{proof}
\begin{remark} Assume $0<\mu\leq H^2$. By combining \eqref{estgreen} with $\gamma=1$, \eqref{eigenfunctionestimates} and the embedding after \eqref{normLwww}, we derive that for any $1<p<\frac{N+1}{N-1}$,
\small\begin{equation} \label{Gphi_mu}
\sup_{z \in \Omega \setminus \Sigma} \int_{\Omega \setminus \Sigma} \left( \frac{G_\mu(x,z)}{d(z)d_\Sigma(z)^{-\am}} \right)^p d(x)d_\Sigma(x)^{-\am}dx < C.
\end{equation}\normalsize \end{remark}
Next we treat the case $\mu \leq 0$. \begin{theorem} \label{lpweakgreen2} Assume $0\leq\gamma\leq1,$ $\mu\leq 0$ and $0\leq\kappa\leq -\am$. Let \bal p_{\kappa,\gamma}:=\min\left\{\frac{N+\kappa}{N+\kappa-2}, \frac{N+\gamma}{N+\gamma-2}\right\}. \eal Then \ba \label{estgreen2}
\norm{\BBG_\mu[\gt]}_{L_w^{p_{\kappa,\gamma}}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma;{\phi_{\mu }})} \lesssim \norm{\gt}_{\mathfrak{M}(\Omega\setminus \Sigma;\varphi_{-\kappa,\gamma})}, \quad \forall \tau\in \mathfrak{M}^+(\Omega\setminus \Sigma;\varphi_{-\kappa,\gamma}). \ea
The implicit constant depends on $N,\Omega,\Sigma,\mu,\kappa,\gamma$. \end{theorem} \begin{proof} \noindent For $y \in \Omega \setminus \Sigma$ and $\lambda>0$, set \bal A_\lambda(y):=\Big\{x\in(\Omega\setminus \Sigma)\setminus\{y\}:\;\; G_{\mu}(x,y)\varphi_{-\kappa,\gamma}(y)^{-1}>\lambda \Big \} \quad \text{and} \quad m_{\lambda}(y):=\int_{A_\lambda(y)}d(x)d_\Sigma(x)^{-\am} \dd x. \eal Put \bal
F(x,y):=d_\Sigma(y)^{-\kappa}|x-y|^{-N+2} d(y)^{-\gamma} \left( 1 \land \frac{d(x)d(y)}{|x-y|^2} \right) \left(1 \land \frac{d_\Sigma(x)d_\Sigma(y)}{|x-y|^2} \right)^{-\am}, \;\; x,y \in \Omega \setminus \Sigma, x \neq y. \eal By \eqref{Greenesta} and \eqref{eigenfunctionestimates}, $F(x,y) \geq c\,G_{\mu}(x,y)\varphi_{-\kappa,\gamma}(y)^{-1}$ for some positive constant $c$ depending only on $N,\Omega,\Sigma,\mu.$ Consequently, \bal &A_\lambda(y)\subset \Big\{x\in(\Omega\setminus \Sigma)\setminus\{y\}:\; F(x,y)>c \lambda \Big \}=:\tilde A_\lambda(y). \eal Let $\beta_0$ be as in Subsection \ref{assumptionK}. We write \ba\nonumber m_{\lambda}(y)&=\int_{A_\lambda(y)\cap \Sigma_{\beta_0}}d(x)d_\Sigma(x)^{-\am} \dd x+\int_{A_\lambda(y)\setminus \Sigma_{\beta_0}}d(x)d_\Sigma(x)^{-\am} \dd x.\\ &\lesssim \int_{\tilde A_\lambda(y)\cap \Sigma_{\beta_0}}d_\Sigma(x)^{-\am} \dd x+\int_{\tilde A_\lambda(y)\setminus \Sigma_{\beta_0}}d(x)\dd x. \label{ml1gr} \ea Note that, for $\Gamma=\partial \Omega$ or $\Sigma$, we have \small\begin{equation} \label{min2}
1 \land \frac{d_\Gamma(x)d_\Gamma(y)}{|x-y|^2} \leq 2 \left( 1 \land \frac{d_\Gamma(y)}{|x-y|} \right) \leq 4\frac{d_\Gamma(y)}{d_\Gamma(x)}.
\end{equation}\normalsize By \eqref{min2} we have \ba
\int_{\tilde A_\lambda(y)\setminus \Sigma_{\beta_0}}d(x) \dd x \lesssim \int_{\{|x-y|\leq c\lambda^{-\frac{1}{N+\gamma-2}}\}}\lambda^{-1}|x-y|^{-N+2} \dd x \lesssim \lambda^{-\frac{N+\gamma}{N+\gamma-2}} \label{ml2gr} \ea and \ba
\int_{\tilde A_\lambda(y)\cap \Sigma_{\beta_0}}d_\Sigma(x)^{-\am} \dd x \lesssim \int_{\{|x-y|\leq c \lambda^{-\frac{1}{N+\kappa-2}}\}}\lambda^{-1}|x-y|^{-N+2} \dd x \lesssim \lambda^{-\frac{N+\kappa}{N+\kappa-2}}.\label{ml3gr} \ea Combining \eqref{ml1gr}, \eqref{ml2gr} and \eqref{ml3gr}, we obtain \bel{ml4gr}
m_{\lambda}(y)\leq C \lambda^{-p_{\kappa,\gamma}} \end{equation}\normalsize for all $\lambda \geq 1$, where $C=C(N,\Omega,\Sigma,\mu)$. Then we can show that \eqref{ml4gr} holds for every $\lambda>0$.
By applying Proposition \ref{bvivier} with $\mathcal{H}(x,y)=G_{\mu}(x,y)\varphi_{-\kappa,\gamma}(y)^{-1}, $ $\tilde D=D=\Omega\setminus \Sigma$, $\eta={\phi_{\mu }}$ and $\omega=\varphi_{-\kappa,\gamma}\tau$, we obtain \eqref{estgreen2}. The proof is complete. \end{proof} \subsection{Weak $L^p$ estimates on Martin kernel} Recall that \bal \mathbb{K}_\mu[\gn](x)=\int_{\partial\Omega \cup \Sigma}K_{\mu}(x,y) \dd\nu(y), \quad \gn\in \mathfrak{M}(\partial\Omega\cup \Sigma). \eal \begin{theorem}\label{lpweakmartin1} ~~
{\sc I.} Assume $\mu \leq \left( \frac{N-2}{2}\right)^2$ and $\gn\in \mathfrak{M}(\partial\Omega\cup \Sigma)$ with compact support in $\partial\Omega.$ Then \ba \label{estmartin1}
\norm{\mathbb{K}_\mu[\nu]}_{L_w^{\frac{N+1}{N-1}}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma;{\phi_{\mu }})} \lesssim \|\nu\|_{\mathfrak{M}(\partial\Omega \cup \Sigma)}. \ea
{\sc II.} Assume $\gn\in \mathfrak{M}(\partial\Omega\cup \Sigma)$ with compact support in $\Sigma$.
(i) If $\mu < \left( \frac{N-2}{2} \right)^2$ then \ba \label{estmartin2} \norm{\mathbb{K}_{\mu}[\nu]}_{L_w^{\frac{N-\am}{N-\am-2}}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma;{\phi_{\mu }})} \lesssim \norm{\nu}_{\mathfrak{M}(\partial \Omega \cup \Sigma)}. \ea
(ii) If $k=0$, $\Sigma=\{0\}$ and $\mu = \left( \frac{N-2}{2} \right)^2$ then \ba \label{estmartin2cr} \norm{\mathbb{K}_{\mu}[\nu]}_{\tilde L_w^{\frac{N+2}{N-2}}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \{0\};{\phi_{\mu }} )} \lesssim \norm{\nu}_{\mathfrak{M}(\partial \Omega \cup \Sigma)}. \ea The implicit constants in the above estimates depends on $N,\Omega,\Sigma,\mu$. \end{theorem} \begin{proof} I. By applying Theorem \ref{H} (i) with $\alpha=-\am$, $\theta=0$ and noting that $K_\mu(x,y) \approx H_{\am,0}$ (due to \eqref{Martinest1} and \eqref{Halthe}), $\varphi_{\am,1} \approx \phi_\mu$ (due to\eqref{eigenfunctionestimates}) and $\quad_{\am,0}=\frac{N+1}{N-1}$, we obtain \eqref{estmartin1}.
II (i). By applying Theorem \ref{H} (ii) with $\alpha=\am$, $\theta=2+2\am$ and noting that $K_\mu(x,y) \approx H_{\am,2+2\am}$ (due to \eqref{Martinest1} and \eqref{Halthe}), $\varphi_{\alpha,1} \approx \phi_\mu$ (due to\eqref{eigenfunctionestimates}), we obtain \eqref{estmartin2}.
II (ii). By applying Theorem \ref{LwSigma0} with $\alpha=\frac{N-2}{2}$, we obtain \eqref{estmartin2cr}. \end{proof}
\section{Boundary value problem for linear equations} \label{sec:linear} In this section, we first recall the notion of boundary trace which is defined with respect to harmonic measures related to $L_\mu$. Then we provide the existence, uniqueness and a priori estimates of the solution to the boundary value problem for linear equations. We refer the reader to \cite{GkiNg_linear} for the proofs. \subsection{Boundary trace} \label{subsec:boundarytrace} Let $\beta_0$ be the constant in Subsection \ref{assumptionK}. Let $\eta_{\beta_0}$ be a smooth function such that $0 \leq \eta_{\beta_0} \leq 1$, $\eta_{\beta_0}=1$ in $\overline{\Sigma}_{\frac{\beta_0}{4}}$ and $\mathrm{supp}\, \eta_{\beta_0} \subset \Sigma_{\frac{\beta_0}{2}}$. We define \bal W(x):=\left\{ \begin{aligned} &d_\Sigma(x)^{-\ap} \qquad&&\text{if}\;\mu <H^2, \\
&d_\Sigma(x)^{-H}|\ln d_\Sigma(x)| \qquad&&\text{if}\;\mu =H^2,
\end{aligned} \right. \quad x \in \Omega \setminus \Sigma, \eal
and \bal
\tilde W(x):=1-\eta_{\beta_0}(x)+\eta_{\beta_0}(x)W(x), \quad x \in \Omega \setminus \Sigma. \eal
Let $z \in \Omega \setminus \Sigma$ and $h\in C(\partial\Omega \cup \Sigma)$ and denote $\CL_{\mu ,z}(h):=v_h(z)$ where $v_h$ is the unique solution of the Dirichlet problem \small\begin{equation} \label{linear} \left\{ \begin{aligned} L_{\mu}v&=0\qquad \text{in}\;\;\Omega\setminus \Sigma\\ v&=h\qquad \text{on}\;\;\partial\Omega\cup \Sigma. \end{aligned} \right. \end{equation}\normalsize Here the boundary value condition in \eqref{linear} is understood in the sense that \bal
\lim_{\mathrm{dist}(x,F)\to 0}\frac{v(x)}{\tilde W(x)}=h \quad \text{for every compact set } \; F\subset \partial \Omega \cup \Sigma. \eal The mapping $h\mapsto \CL_{\mu,z}(h)$ is a linear positive functional on $C(\partial\Omega \cup \Sigma)$. Thus there exists a unique Borel measure on $\partial\Omega \cup \Sigma$, called {\it $L_{\mu}$-harmonic measure in $\partial \Omega \cup \Sigma$ relative to $z$} and denoted by $\omega_{\Omega \setminus \Sigma}^{z}$, such that \bal v_{h}(z)=\int_{\partial\Omega\cup \Sigma}h(y) \dd\omega_{\Omega \setminus \Sigma}^{z}(y). \eal Let $x_0 \in \Omega \setminus \Sigma$ be a fixed reference point. Let $\{\Omega_n\}$ be an increasing sequence of bounded $C^2$ domains such that \bal \overline{\Omega}_n\subset \Omega_{n+1}, \quad \cup_n\Omega_n=\Omega, \quad \mathcal{H}^{N-1}(\partial \Omega_n)\to \mathcal{H}^{N-1}(\partial \Omega), \eal where $\mathcal{H}^{N-1}$ denotes the $(N-1)$-dimensional Hausdorff measure in ${\mathbb R}^N$.
Let $\{\Sigma_n\}$ be a decreasing sequence of bounded $C^2$ domains such that \bal \Sigma \subset \Sigma_{n+1}\subset\overline{\Sigma}_{n+1}\subset \Sigma_{n}\subset\overline{\Sigma}_{n} \subset\Omega_n, \quad \cap_n \Sigma_n=\Sigma. \eal For each $n$, set $O_n=\Omega_n\setminus \Sigma_n$ and assume that $x_0 \in O_1$. Such a sequence $\{O_n\}$ will be called a {\it $C^2$ exhaustion} of $\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma$.
Then $-L_\mu$ is uniformly elliptic and coercive in $H^1_0(O_n)$ and its first eigenvalue $\lambda_\mu^{O_n}$ in $O_n$ is larger than its first eigenvalue $\lambda_\mu$ in $\Omega \setminus \Sigma$.
For $h\in C(\partial O_n)$, the following problem \bal\left\{ \begin{aligned} -L_{\mu } v&=0\qquad&&\text{in } O_n\\ v&=h\qquad&&\text{on } \partial O_n, \end{aligned} \right. \eal admits a unique solution which allows to define the $L_{\mu }$-harmonic measure $\omega_{O_n}^{x_0}$ on $\partial O_n$ by \bal v(x_0)=\myint{\partial O_n}{}h(y) \dd\gw^{x_0}_{O_n}(y). \eal
Let $G^{O_n}_\mu(x,y)$ be the Green kernel of $-L_\mu$ on $O_n$. Then $G^{O_n}_\mu(x,y)\uparrow G_\mu(x,y)$ for $x,y\in\Omega\setminus \Sigma, x \neq y$.
We recall below the definition of the boundary trace which is defined in a \textit{dynamic way}.
\begin{definition}[Boundary trace] \label{nomtrace}
A function $u\in W^{1,\kappa}_{loc}(\Omega\setminus\Sigma),$ for some $\kappa>1,$ possesses a \emph{boundary trace} if there exists a measure $\nu \in\GTM(\partial \Omega \cup \Sigma)$ such that for any $C^2$ exhaustion $\{ O_n \}$ of $\Omega \setminus \Sigma$, there holds
\ba \label{trab}
\lim_{n\rightarrow\infty}\int_{ \partial O_n}\phi u\dd\omega_{O_n}^{x_0}=\int_{\partial \Omega \cup \Sigma} \phi \dd\nu \quad\forall \phi \in C(\overline{\Omega}).
\ea
The boundary trace of $u$ is denoted by $\mathrm{tr}(u)$. \end{definition}
\begin{proposition}[{\cite[Proposition 1.5]{GkiNg_linear}}] \label{traceKG} ~~
(i) For any $\nu \in \GTM(\partial \Omega \cup \Sigma)$, $\mathrm{tr}(\BBK_{\mu}[\nu])=\nu$.
(ii) For any $\tau \in \GTM(\Omega \setminus \Sigma;{\phi_{\mu }})$, $\mathrm{tr}(\BBG_\mu[\tau])=0$. \end{proposition}
The next result is the Representation Theorem. \begin{theorem}[{\cite[Theorem 1.3]{GkiNg_linear}}]\label{th:Rep} For any $\nu \in \GTM^+(\partial \Omega \cup \Sigma)$, the function $\BBK_{\mu}[\nu]$ is a positive $L_\mu$-harmonic function (i.e. $L_\mu \BBK_{\mu}[\nu]=0$ in the sense of distributions in $\Omega \setminus \Sigma$). Conversely, for any positive $L_\mu$-harmonic function $u$ (i.e. $L_\mu u = 0$ in the sense of distribution in $\Omega \setminus \Sigma$), there exists a unique measure $\nu \in \GTM^+(\partial \Omega \cup \Sigma)$ such that $u=\BBK_{\mu}[\nu]$. \end{theorem}
Nonnegative $L_\mu$-superharmonic functions can be decomposed in terms of Green kernel and Martin kernel. \begin{proposition}[{\cite[Theorem 1.6]{GkiNg_linear}}]\label{super} Let $u$ be a nonnegative $L_\mu$-superharmonic function. Then $u\in L^1(\Omega;{\phi_{\mu }})$ and there exist positive measures $\tau\in\mathfrak{M}^+(\Omega\setminus \Sigma;{\phi_{\mu }})$ and $\nu \in \mathfrak{M}^+(\partial\Omega\cup \Sigma)$ such that
\bal
u=\mathbb{G}_{\mu}[\tau]+\mathbb{K}_{\mu}[\nu].
\eal \end{proposition}
\begin{proposition}\label{prop}
Let $\vgf \in L^{1}(\Omega;{\phi_{\mu }})$, $\vgf \geq 0$ and $\tau\in \mathfrak{M}^+(\Omega\setminus \Sigma;{\phi_{\mu }}).$ Set
\bal
w:=\BBG_\mu[\vgf+\gt]\quad\text{and}\quad \psi=\BBG_\mu[\gt].
\eal
Let $\phi$ be a concave nondecreasing $C^2$ function on $[0,\infty),$
such that $\phi(1) \geq 0.$ Then the function $\phi'(w/\psi)\vgf$ belongs to $L^1(\Omega;{\phi_{\mu }})$ and the following holds in the weak sense in $\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma$
\bal
-L_\mu(\psi \phi(w/\psi))\geq\phi'(w/\psi)\vgf.
\eal \end{proposition} \begin{proof}
The proof is the same as that of \cite[Propositions 3.1]{GkNg} and we omit it. \end{proof}
Similarly we may prove that \begin{proposition}\label{prop2}
Let $\vgf \in L^{1}(\Omega;{\phi_{\mu }})$, $\vgf \geq 0$ and $\nu \in \mathfrak{M}^+(\partial\Omega\cup \Sigma).$ Set
\bal
w:=\BBG_\mu[\vgf]+\BBK_\mu[\nu]\quad\text{and}\quad \psi=\BBK_\mu[\nu].
\eal
Let $\phi$ be a concave nondecreasing $C^2$ function on $[0,\infty),$
such that $\phi(1) \geq 0.$ Then the function $\phi'(w/\psi)\vgf$ belongs to $L^1(\Omega;{\phi_{\mu }})$ and the following holds in the weak sense in $\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma$
\bal
-L_\mu(\psi \phi(w/\psi))\geq\phi'(w/\psi)\vgf.
\eal \end{proposition}
\subsection{Boundary value problem for linear equations} \label{subsec:linear} We recall the definition and properties of weak solutions to the boundary value problem for linear equations. \begin{definition}
Let $\tau\in\mathfrak{M}(\Omega\setminus \Sigma;{\phi_{\mu }})$ and $\nu \in \mathfrak{M}(\partial\Omega\cup \Sigma)$. We say that $u$ is a weak solution of \ba\label{NHL} \left\{ \begin{aligned} - L_\mu} \def\gn{\nu} \def\gp{\pi u&=\tau\qquad \text{in }\;\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma,\\ \mathrm{tr}(u)&=\nu, \end{aligned} \right. \ea if $u\in L^1(\Omega\setminus \Sigma;{\phi_{\mu }})$ and it satisfies \bal
- \int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}u L_{\mu }\zeta \dd x=\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi \setminus \Sigma} \zeta \dd\tau - \int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} \mathbb{K}_{\mu}[\nu]L_{\mu }\zeta \dd x
\qquad\forall \zeta \in\mathbf{X}_\mu(\Omega\setminus \Sigma), \eal
where the space of test function ${\bf X}_\mu(\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma)$ has been defined in \eqref{Xmu} \end{definition}
\begin{theorem}[ {\cite[Theorem 1.8]{GkiNg_linear}}] \label{linear-problem} Let $\tau \in\mathfrak{M}(\Omega\setminus \Sigma;{\phi_{\mu }})$ and $\nu \in \mathfrak{M}(\partial\Omega\cup \Sigma)$. Then there exists a unique weak solution $u\in L^1(\Omega;{\phi_{\mu }})$ of \eqref{NHL}. Furthermore \bal u=\mathbb{G}_{\mu}[\tau]+\mathbb{K}_{\mu}[\nu] \eal and there exists a positive constant $C=C(N,\Omega,\Sigma,\mu)$ such that \bal
\|u\|_{L^1(\Omega;{\phi_{\mu }})} \leq \frac{1}{\lambda_\mu}\| \tau \|_{\GTM(\Omega \setminus \Sigma;{\phi_{\mu }})} + C \| \nu \|_{\GTM(\partial\Omega \cup \Sigma)}. \eal In addition, if $\dd \tau=f\dd x+\dd\rho$ then, for any $0 \leq \zeta \in \mathbf{X}_\mu(\Omega\setminus \Sigma)$, the following estimates are valid \small\begin{equation}\label{poi4}
-\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}|u|L_{\mu }\zeta \dd x\leq \int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}\mathrm{sign}(u)f\zeta \dd x +\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi \setminus \Sigma}\zeta \dd|\rho|-
\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}\mathbb{K}_{\mu}[|\nu|] L_{\mu }\zeta \dd x, \end{equation}\normalsize \small\begin{equation}\label{poi5} -\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}u^+L_{\mu }\zeta \dd x\leq \int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} \mathrm{sign}^+(u)f\zeta \dd x +\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi \setminus \Sigma}\zeta \dd\rho^+- \int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}\mathbb{K}_{\mu}[\nu^+]L_{\mu }\zeta \dd x. \end{equation}\normalsize
\end{theorem}
\section{General nonlinearities} \label{sec:gennon} In this section, we provide various sufficient conditions for the existence of a solution to \eqref{NLP}. Throughout this sections we assume that $g: {\mathbb R} \to {\mathbb R}$ is continuous and nondecreasing and satisfies $g(0)=0$. We start with the following result.
\begin{lemma} \label{subcrcon} Assume \ba \label{subcd0} \int_1^\infty s^{-q-1}(\ln s)^{m} (g(s)-g(-s)) \dd s<\infty \ea for $q,m \in {\mathbb R}$, $q >0$ and $m \geq 0$. Let $v$ be a function defined in $\Omega \setminus \Sigma$. For $s>0$, set
\bal E_s(v):=\{x\in \Omega\setminus \Sigma:| v(x)|>s\} \quad \text{and} \quad e(s):=\int_{E_s(v)} {\phi_{\mu }} \dd x. \eal Assume that there exists a positive constant $C_0$ such that \ba \label{e} e(s) \leq C_0s^{-q}(\ln s)^m, \quad \forall s>e^\frac{2 m}{q}. \ea Then for any $s_0>e^\frac{2 m}{q},$ there holds \ba\label{53}
\norm{g(|v|)}_{L^1(\Omega;{\phi_{\mu }})}&\leq \int_{(\Omega \setminus \Sigma) \setminus E_{s_0}(v)} g(|v|){\phi_{\mu }} \dd x+ C_0 q\int_{s_0}^\infty s^{-q-1}(\ln s)^{m} g(s) \dd s, \\ \label{53-a}
\norm{g(-|v|)}_{L^1(\Omega;{\phi_{\mu }})}&\leq -\int_{(\Omega \setminus \Sigma) \setminus E_{s_0}(v)} g(-|v|){\phi_{\mu }} \dd x-C_0 q\int_{s_0}^\infty s^{-q-1}(\ln s)^{m} g(-s) \dd s. \ea \end{lemma} \begin{proof}
We note that $g(|v|) \geq g(0)=0$. Let $s_0>1$ to be determined later on. Using the fact that $g$ is nondecreasing, we obtain \bal \begin{aligned}
\int_{\Omega \setminus \Sigma} g(|v|){\phi_{\mu }} dx &\leq \int_{(\Omega \setminus \Sigma) \setminus E_{s_0}(v)} g(|v|){\phi_{\mu }} \dd x + \int_{E_{s_0}(v)} g(|v|){\phi_{\mu }} \dd x \\ &\leq g(s_0)e(s_0) -\int_{s_0}^{\infty}g(s) \dd e(s). \end{aligned} \eal From \eqref{subcd0}, we deduce that there exists an increasing sequence $\{T_n\}$ such that \ba \label{Tn} \lim_{T_n \to \infty}T_n^{-q}(\ln T_n)^m g(T_n) = 0. \ea For $T_n > s_0$, we have \ba \label{gTn} \begin{aligned} -\int_{s_0}^{T_n}g(s)\dd e(s) &= -g(T_n)e(T_n) + g(s_0)e(s_0) + \int_{s_0}^{T_n} e(s)\dd g(s) \\ &\leq -g(T_n)e(T_n) + g(s_0)e(s_0)+C_0\int_{s_0}^{T_n}s^{-q}(\ln s)^m \dd g(s)\\ &\leq (CT_n^{-q}(\ln T_n)^m - e(T_n))g(T_n) - C_0\int_{s_0}^{T_n} (s^{-q}(\ln s)^m)' g(s)\dd s. \end{aligned} \ea Here in the last estimate, we have used \eqref{e}. Note that if we choose $s_0>e^{\frac{2 m}{q}}$ then \ba \label{deriv} -q s^{-q-1}(\ln s)^m < (s^{-q}(\ln s)^m)' < -\frac{q}{2}s^{-q-1}(\ln s)^m \quad \forall s \geq s_0. \ea Combining \eqref{Tn}--\eqref{deriv} and then letting $n \to \infty$, we obtain \bal -\int_{s_0}^{\infty}g(s) \dd e(s) < C_0q \int_{s_0}^{\infty} s^{-q-1}(\ln s)^m g(s) \dd s. \eal Thus we have proved estimate \eqref{53}. By applying estimate \eqref{53} with $g$ replaced by $h(t)=-g(-t)$, we obtain \eqref{53-a}. \end{proof}
\begin{lemma} \label{transf} Let $0<\mu\leq H^2$, $0\leq\gamma\leq1$, $\tau \in \GTM^+(\Omega \setminus \Sigma; \varphi_{\am,\gamma})$ with $\norm{\tau}_{\mathfrak{M}(\Omega\setminus \Sigma;\varphi_{\am,\gamma})}=1, $ and $\nu \in \GTM^+(\partial \Omega} \def\Gx{\Xi} \def\Gy{\Psi\cup \Sigma)$ with $\norm{\nu}_{\GTM(\partial \Omega} \def\Gx{\Xi} \def\Gy{\Psi\cup \Sigma)}=1$. Assume $g \in L^\infty({\mathbb R}) \cap C({\mathbb R})$ satisfies \ba \label{subcd-00} \Lambda} \def\Gs{\Sigma} \def\Gp{\Pi_g=\int_1^\infty s^{-q-1} (g(s)-g(-s))\,\dd s < \infty \ea for some $q \in (1,\infty)$ and \bal
|g(s)|\leq a|s|^{\tilde q} \quad \text{for some } a>0,\; \tilde q>1 \text{ and for any } |s|\leq 1. \eal
Assume one of the following conditions holds.
(i) $ \1_{\partial \Omega}\nu \equiv0$ and \eqref{subcd-00} holds for $q= \frac{N+\gamma}{N+\gamma-2}$.
(ii) $ \1_{\partial \Omega}\nu \not\equiv 0$ and \eqref{subcd-00} holds for $q=\frac{N+1}{N-1}$.
Then there exist positive numbers $\rho_0,\sigma_0,t_0$ depending on $N,\Omega,\mu,\Lambda} \def\Gs{\Sigma} \def\Gp{\Pi_g,\gamma, \tilde q$ such that for every $\rho \in (0,\rho_0)$ and $\sigma\in (0,\sigma_0)$ the following problem
\ba\label{trans} \left\{ \begin{aligned} -L_\mu v &= g (v + \rho\BBG_\mu[\tau] + \sigma\BBK_\mu[\nu]) \quad \text{in } \Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus\Sigma, \\ \mathrm{tr}(v) &= 0
\end{aligned} \right. \ea
admits a positive weak solution $v$ satisfying
\bal \|v\|_{L_w^{q}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus\Sigma;{\phi_{\mu }})} \leq t_0
\eal
where $q=\frac{N+\gamma}{N+\gamma-2}$ if case (i) happens or $q=\frac{N+1}{N-1}$ if case (ii) happens. \end{lemma} \begin{proof} We shall use Schauder fixed point theorem to show the existence of a positive weak solution of \eqref{trans}.
(i) Assume that $\1_{\partial \Omega} \nu \equiv 0$, namely $\nu$ has compact support in $\Sigma$, and \eqref{subcd-00} holds for $q= \frac{N+\gamma}{N+\gamma-2}$ (in the proof of statement (i), we always assume that $q= \frac{N+\gamma}{N+\gamma-2}$).
\noindent \textbf{Step 1:} Derivation of $t_0$, $\rho_0$ and $\sigma_0$.
Define the operator $\BBS$ by
\bal \BBS(v):=\BBG_\mu[g(v + \sigma\BBK_\mu[\nu]+\rho\BBG_\mu[\tau])] \quad \text{for } v \in L_w^{q}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma;{\phi_{\mu }}).
\eal Fix $1<\kappa<\min\{q, \tilde q\},$ and put
\bal \begin{aligned} Q_1(v) &: = \| v\|_{L_w^{q}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma;{\phi_{\mu }})}, \quad && v \in L_w^{q}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma;{\phi_{\mu }}), \\
Q_2(v) &:=\| v\|_{L^{\kappa}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi;{\phi_{\mu }})}, \quad && v \in L^{\kappa}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi;{\phi_{\mu }}), \\
Q(v)&:=Q_1(v) + Q_2(v), \quad && v \in L_w^{q}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma;{\phi_{\mu }}).
\end{aligned} \eal
Let $v \in L_w^{q}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma;{\phi_{\mu }})$. For $s>0$, set
\bal E_s:=\{x\in\Omega: |v(x) + \rho\BBG_\mu[\tau](x) + \sigma\BBK_\mu[\nu](x)|>s\} \quad \text{and} \quad e(s):=\int_{E_s}\phi_\mu \, \dx. \eal By Theorem \ref{lpweakgreen}, $\BBG_\mu[\tau] \in L_w^{q}(\Omega \setminus \Sigma;{\phi_{\mu }})$ and
\ba \label{Q1G} \| \BBG_\mu[\tau] \|_{L_w^{q}(\Omega \setminus \Sigma;{\phi_{\mu }})} \lesssim \| \tau \|_{\GTM(\Omega \setminus \Sigma;\varphi_{\am,\gamma})} = 1. \ea By Theorem \ref{lpweakmartin1} II. (i), $\BBK_\mu[\nu] \in L_w^{\frac{N-\am}{N-\am-2}}(\Omega \setminus \Sigma;{\phi_{\mu }})$ and
\ba \label{Q1K} \| \BBK_\mu[\nu] \|_{L_w^{\frac{N-\am}{N-\am-2}}(\Omega \setminus \Sigma;{\phi_{\mu }})} \lesssim \| \nu \|_{\GTM(\partial \Omega \cup \Sigma)} = 1. \ea From \eqref{ue}, \eqref{Q1G}, \eqref{Q1K} and noting that $q \leq \frac{N-\am}{N-\am-2}$ since $\am>0$, we deduce
\ba e(s) \leq s^{-q}\| v+\rho\BBG_\mu[\tau]+ \sigma\BBK_\mu[\nu]\|_{L_w^{q}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma;{\phi_{\mu }})}^{q} \leq Cs^{-q}(\| v\|_{L_w^{q}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma;{\phi_{\mu }})}^{q}+\rho^{q} +\sigma^{q}).\label{40} \ea
By \eqref{53} and \eqref{53-a}, taking into account \eqref{40} and the assumptions on $g$, we have \ba\nonumber
&\int_\Omega |g(v +\rho\BBG_\mu[\tau] + \sigma\BBK_\mu[\nu])|{\phi_{\mu }} \,\dx \\ \nonumber
&= \int_{\Omega \setminus E_1}|g(v +\rho\BBG_\mu[\tau] + \sigma\BBK_\mu[\nu])| {\phi_{\mu }} \, \dx + \int_{E_1}|g(v +\rho\BBG_\mu[\tau] + \sigma\BBK_\mu[\nu])| {\phi_{\mu }} \, \dx \\ \nonumber
&\leq C(q,\Lambda_g)\bigg( \int_{\Omega \setminus E_1} |v +\rho\BBG_\mu[\tau] + \sigma\BBK_\mu[\nu]|^{\tilde q} {\phi_{\mu }} \dx + \| v\|_{L_w^{q}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma;{\phi_{\mu }})}^{q}+\rho^{q} +\sigma^{q} \bigg) \\ \nonumber &\leq C(Q_1(v)^{q}+ Q_2(v)^{\kappa} +\rho^\kappa Q_2(\BBG_\mu[\tau])^{\kappa} + \sigma^\kappa Q_2(\BBK_\mu[\nu])^\kappa+\rho^{q} +\sigma^{q}) \\ \nonumber &\leq C(Q_1(v)^{q}+ Q_2(v)^{\kappa} + \rho^\kappa + \sigma^\kappa + \rho^{q} +\sigma^{q}). \ea It follows that
\ba \label{Q1Q2-1} \begin{aligned} Q_1(\BBS(v))+ Q_2(\BBS(v)) &\leq C\| g(v +\rho\BBG_\mu[\tau] + \sigma\BBK_\mu[\nu])\|_{L^{1}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma;{\phi_{\mu }})}\\ &\leq C(Q_1(v)^{q}+ Q_2(v)^{\kappa} + \rho^\kappa + \sigma^\kappa + \rho^{q} +\sigma^{q}), \end{aligned} \ea where $C$ depends only on $\mu,\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\Sigma,\Lambda} \def\Gs{\Sigma} \def\Gp{\Pi_g,a, \tilde q,\gamma.$
Therefore if $Q(v) \leq t$ then \ba \label{Qt0} Q(\BBS(v)) \leq C\left(t^q+t^{\kappa} +\rho^\kappa + \sigma^\kappa + \rho^{q} +\sigma^{q}\right). \ea
Since $q>\kappa>1$, there exist positive number $t_0$, $\rho_0$ and $\sigma_0$ depending on $\mu,\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\Sigma,\Lambda} \def\Gs{\Sigma} \def\Gp{\Pi_g, \tilde q, \gamma, \kappa$ such that for any $t\in(0,t_0)$, $\rho \in (0,\rho_0)$ and $\sigma\in (0,\sigma_0),$ the following inequality holds \bal C\left(t^{q}+t^{\kappa}+ \sigma^\kappa +\rho^\kappa + \rho^{q} +\sigma^{q}\right) \leq t_0, \eal where $C$ is the constant in \eqref{Qt0}.
Therefore,
\bel{ul11} Q(v) \leq t_0 \Longrightarrow Q(\BBS(v)) \leq t_0.
\end{equation}\normalsize
\noindent \textbf{Step 2:} We apply Schauder fixed point theorem to our setting.
\textit{We claim that $\BBS$ is continuous}. Indeed, if $u_n\rightarrow u$ in $L^1(\Omega} \def\Gx{\Xi} \def\Gy{\Psi;{\phi_{\mu }})$ then since $ g \in L^\infty(\BBR) \cap C({\mathbb R})$ and is nondecreasing, it follows that $g(v_n +\rho\BBG_\mu[\tau] + \sigma\BBK_\mu[\nu]) \to g(v +\rho\BBG_\mu[\tau] + \sigma\BBK_\mu[\nu])$ in $L^1(\Omega} \def\Gx{\Xi} \def\Gy{\Psi;{\phi_{\mu }}).$ Hence, $\BBS(u_n)\to\BBS(u)$ in $L^1(\Omega} \def\Gx{\Xi} \def\Gy{\Psi;{\phi_{\mu }}).$
\textit{Next we claim that $\BBS$ is compact}. Indeed, set
$M:=\sup_{t>0}|g(t)|<+\infty$. Then we can easily deduce that there exists $C=C(\Omega,\Sigma,M,\mu)$ such that
\small\begin{equation}\label{sup1}
|\BBS(w)|\leq C{\phi_{\mu }} \quad \text{a.e. in } \Omega, \quad \forall w\in L^1(\Omega} \def\Gx{\Xi} \def\Gy{\Psi;{\phi_{\mu }}).
\end{equation}\normalsize
Also, by Theorem \eqref{linear-problem}, $-L_\mu\BBS(w)=g(w +\rho\BBG_\mu[\tau] + \sigma\BBK_\mu[\nu])$ in the sense of distributions in $\Omega\setminus \Sigma$. By \cite[Corollary 1.2.3]{MVbook}, $\BBS(w)\in W^{1,r}_{loc}(\Omega\setminus \Sigma),$ for any $1<r<\frac{N}{N-1}$ and for any open $D\Subset \Omega\setminus \Sigma,$ there exists $C_1=C_1(\Omega,\Sigma,M,\mu,D,p)$ such that
\ba \label{W1pest} \|\BBS(w)\|_{W^{1,r}(D)}\leq C_1(D). \ea
Let $\{v_n\}$ be a sequence in $ L^1(\Omega} \def\Gx{\Xi} \def\Gy{\Psi;{\phi_{\mu }})$ then by \eqref{sup1} and \eqref{W1pest}, there exist $\psi$ and a subsequence still denoted by $\{\BBS(v_n)\}$ such that $\BBS(v_n)\rightarrow \psi$ a.e. in $\Omega\setminus \Sigma$. In addition, by \eqref{sup1} and the dominated convergence theorem we obtain that $\BBS(v_n)\rightarrow \psi$ in $L^1(\Omega} \def\Gx{\Xi} \def\Gy{\Psi;{\phi_{\mu }}).$
Now set
\small\begin{equation}
\CO:=\{ v \in L^1(\Omega} \def\Gx{\Xi} \def\Gy{\Psi;{\phi_{\mu }}): Q(v) \leq t_0 \}.\label{O}
\end{equation}\normalsize
Then $\CO$ is a closed, convex subset of $L^1(\Omega} \def\Gx{\Xi} \def\Gy{\Psi;{\phi_{\mu }})$ and by \eqref{ul11}, $\BBS(\CO) \subset \CO$.
Thus we can apply Schauder fixed point theorem to obtain the existence of a function $v \in \CO$ such that $\BBS(v)=v$. This means that $v$ is a nonnegative solution of \eqref{trans} and hence there holds
\bal -\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}v L_\mu} \def\gn{\nu} \def\gp{\pi\zeta \,\dx= \int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} g(v +\rho\BBG_\mu[\tau] + \sigma\BBK_\mu[\nu]) \zeta \,\dx \quad \forall \zeta \in {\bf X}_\mu(\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma). \eal
(ii) The case $\1_{\partial \Omega} \nu \not \equiv 0$ and \eqref{subcd-00} holds for with $q=\frac{N+1}{N-1}$ ($\leq \frac{N+\gamma}{N+\gamma-2}$) can be proceeded similarly as case (i) with minor modifications and hence we omit it. \end{proof}
\begin{proof}[\textbf{Proof of Theorem \ref{th1}}.]
(i) We assume that $\1_{\partial \Omega} \nu \equiv 0$, namely $\nu$ has compact support in $\Sigma$, and \eqref{subcd-00} holds for $q= \frac{N+\gamma}{N+\gamma-2}$ (in the proof of statement (i) we always assume $q=\frac{N+\gamma}{N+\gamma-2}$).
Let $0\leq\eta_n(t)\leq 1$ be a smooth function in $\mathbb{R}$ such that $\eta_n(t)=1$ for any $|t|\leq n$ and $\eta_n(t)=0$ for any $|t| \geq n+1$. Set $g_n=\eta_n g$ then $g_n \in L^\infty(\BBR) \cap C({\mathbb R})$ is a nondecreasing function in $\BBR$. Moreover $g_n$ satisfies \eqref{subcd-0} for $q=\frac{N+\gamma}{N+\gamma-2}$ and $\Lambda_{g_n} \leq \Lambda_g$. Furthermore, $|g_n(s)|\leq a|s|^{\tilde q}$ for any $|s|\leq 1$ with the same constants $a>0,\; \tilde q>1$. Therefore the constants $\rho_0,\sigma_0,t_0$ in Lemma \ref{transf} can be chosen to depend on $\mu,\Omega} \def\Gx{\Xi} \def\Gy{\Psi,\Sigma,\Lambda} \def\Gs{\Sigma} \def\Gp{\Pi_g,p,a, \tilde q,\gamma$, but independent of $n$. By Lemma \ref{transf}, for any $\rho \in (0,\rho_0)$ and $\sigma\in (0,\sigma_0)$ and $n \in \BBN$, there exists a solution $v_n \in \CO$ (where $\CO$ is defined in \eqref{O}) of
\bal \left\{ \begin{aligned} -L_\mu v_n &= g_n (v_n +\rho\BBG_\mu[\tau] + \sigma\BBK_\mu[\nu]) \quad \text{in } \Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma, \\ \mathrm{tr}(v_n) &= 0.
\end{aligned} \right. \eal
Set $u_n=v_n +\rho\BBG_\mu[\tau] + \sigma\BBK_\mu[\nu]$ then $\mathrm{tr}(u_n)=\sigma \nu$ and
\bel{ul13} -\int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} u_n L_\mu} \def\gn{\nu} \def\gp{\pi\zeta \dd x= \int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} g_n(u_n) \zeta \dd x + \rho \int_{\Omega}\zeta \dd \tau -
\sigma \int_{\Omega} \def\Gx{\Xi} \def\Gy{\Psi} \BBK_\mu} \def\gn{\nu} \def\gp{\pi[\gn]L_\mu} \def\gn{\nu} \def\gp{\pi \zeta \dd x \quad \forall \zeta \in {\bf X}_\mu(\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma). \end{equation}\normalsize
Since $\{v_n\} \subset \CO$, the sequence $\{ u_n \}$ is uniformly bounded in $L^{\kappa}(\Omega;{\phi_{\mu }})\cap L^{q}_w(\Omega\setminus \Sigma;{\phi_{\mu }})$. More precisely, \small\begin{equation} \label{42}
\|u_n\|_{L_w^{q}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus\Sigma;{\phi_{\mu }})}+\|u_n\|_{L^{\kappa}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi;{\phi_{\mu }})} \leq t_0 \quad\forall n\in \mathbb{N}. \end{equation}\normalsize
In view of the proof of \eqref{53}, for any Borel set $E\subset \Omega\setminus \Sigma$ and $s_0>1,$ we have \ba\label{43}
\int_E|g_n(u_n)|{\phi_{\mu }} \dd x&\leq (g(s_0)-g(-s_0))\int_{E}{\phi_{\mu }} \dd x + Ct_0^{q}\int_{s_0}^\infty s^{-q-1}( g(s)-g(s)) \dd s, \ea which implies that $\{g_n(u_n)\}$ is equi-integrable in $L^1(\Omega} \def\Gx{\Xi} \def\Gy{\Psi;{\phi_{\mu }}).$
Also, by Theorem \ref{linear-problem}, $-L_\mu u_n=g_n(u_n)+\rho\tau$ in the sense of distribution in $\Omega\setminus \Sigma.$ By \cite[Corollary 1.2.3]{MVbook} and \eqref{42}, $u_n\in W^{1,r}_{loc}(\Omega\setminus \Sigma),$ for any $1<r<\frac{N}{N-1}$ and for any open $D\Subset \Omega\setminus \Sigma,$ there exists $C_2=C_2(\Omega,\Sigma,M,\mu,D,p,t_0)$ such that $\|u_n\|_{W^{1,r}(D)}\leq C_2$. Hence there exists a subsequence still denoted by $\{u_n\}$ such that $u_n\rightarrow u$ a.e. in $\Omega\setminus \Sigma.$ In additions by \eqref{42} and \eqref{43}, we may invoke Vitali's convergence theorem to derive that $u_n \to u$ and $g_n(u_n)\to g(u)$ in $L^1(\Omega} \def\Gx{\Xi} \def\Gy{\Psi;{\phi_{\mu }}).$ Thus, by letting $n \to \infty$ in \eqref{ul13}, we derive that $u$ satisfies \eqref{lweakform}, namely $u$ is a positive solution of \eqref{NLP}. The proof is complete.
(ii) The case $\1_{\partial \Omega}\nu \not \equiv 0$ and \eqref{subcd-00} holds for with $q=\frac{N+1}{N-1}$ can be proceeded similarly as in case (i) with minor modification and hence we omit it. \end{proof}
\begin{proof}[\textbf{Proof of Theorem \ref{th2}}.] The proof of statements (i), (ii) and (iv) is similar to that of Theorem \ref{th1} and we omit it. As for the proof of statement (iii), the point that needs to be paid attention is the use of Theorem \ref{lpweakgreen} (for $\mu>0$) and Theorem \ref{lpweakgreen2} (for $\mu \leq 0$) for $Q_1(\BBS(v))$ as in the first estimate in \eqref{Q1Q2-1}. In particular, for $\mu \leq H^2$, the estimate \bal
\|\BBS(v)\|_{L_w^q(\Omega \setminus;{\phi_{\mu }})} \leq C\| g(v + \sigma\BBK_\mu[\nu])\|_{L^{1}(\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma;{\phi_{\mu }})} \eal is valid for $q=\min\left\{\frac{N+1}{N-1},\frac{N-\am}{N-\am-2}\right\}$. The rest of the proof of statement (iii) can be proceeded as in the proof of Lemma \ref{transf} and of Theorem \ref{th1} and is left to the reader. \end{proof}
\section{Power case } \label{sec:powercase} In this section we study the following problem \ba\label{power} \left\{ \begin{aligned}
- L_\mu} \def\gn{\nu} \def\gp{\pi u&= |u|^{p-1}u +\rho\tau\qquad \text{in }\;\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma,\\ \mathrm{tr}(u)&=\sigma\nu. \end{aligned} \right. \ea where $p>1$, $\rho \geq 0$, $\sigma \geq 0$, $\tau\in\mathfrak{M}^+(\Omega\setminus \Sigma;{\phi_{\mu }})$ and $\nu \in \mathfrak{M}^+(\partial\Omega\cup \Sigma)$.
\subsection{Partial existence results}
We provide below necessary and sufficient conditions expressed in terms of Green kernel and Martin kernel for the existence of a solution to \eqref{power}. \begin{proposition}\label{equivint} Assume $\mu \leq H^2$, $p>1$ and $\tau\in\mathfrak{M}^+(\Omega\setminus \Sigma;{\phi_{\mu }})$. Then problem \eqref{power} with $\nu=0$ admits a nonnegative solution and for some $\sigma>0$ if and only if there is a constant $C>0$ such that \ba\label{58} \BBG_\mu[\BBG_\mu[\tau]^p]\leq C\,\BBG_\mu[\tau] \quad \text{a.e. in } \Omega\setminus\Sigma. \ea \end{proposition} \begin{proof} If \eqref{58} holds then the existence of a positive solution to problem \eqref{power} with $\nu=0$ follows by a rather similar argument as in the proof of \cite[Proposition 3.5]{GkNg}.
So we will only show that if $u$ is a positive solution of \eqref{power} with $\nu=0$ for some $\sigma>0$ then \eqref{58} holds. We adapt the argument used in the proof of \cite[Proposition 3.5]{BY}. We may suppose that $\sigma=1.$ By the assumption, we have $u=\BBG_\mu[u^p+\gt]$. By applying Proposition \ref{prop} with $\vgf$ replaced by $ u^p,$ $w$ by $u$ and with \bal \phi} \def\vgf{\varphi} \def\gh{\eta(s)= \left\{ \begin{aligned} &(1-s^{1-p})/(p-1) \quad &\text{if } s \geq 1, \\ &s-1 &\text{if } s<1, \end{aligned} \right. \eal we obtain \ba\label{59} -L_\mu(\psi \phi(u/\psi))\geq\phi'(u/\psi)u^p=\BBG_\mu[\tau]^p, \ea in the weak sense. Since $u \geq \BBG_\mu[\tau]=\psi$, we see that \ba \label{u/psi} \psi \phi(u/\psi)\leq \frac{1}{p-1}\BBG_\mu[\tau], \ea
which, together with Proposition \ref{traceKG}, implies $\mathrm{tr}(\psi \phi(u/\psi))=0$. By \eqref{59} and Proposition \ref{super} there exist $\nu\in \mathfrak{M}^+(\partial\Omega\cup \Sigma)$ and $\tilde \tau \in \mathfrak{M}^+(\Omega\setminus \Sigma;{\phi_{\mu }})$ such that $\dd \tilde \tau \geq \BBG_\mu^p[\tau] \dd x$ and \small\begin{equation} \label{tildetau} \psi \phi(u/\psi)=\mathbb{G}_{\mu}[\tilde \tau]+\mathbb{K}_{\mu}[\nu]. \end{equation}\normalsize Since $\mathrm{tr}(\psi \phi(u/\psi))=0,$ by Proposition \ref{traceKG}, we deduce that $\nu=0$. From \eqref{u/psi} and \eqref{tildetau}, we obtain \eqref{58} with $C=\frac{1}{p-1}$. \end{proof}
\begin{proposition}\label{equivba} Assume $\mu \leq H^2$, $p>1$ and $\nu \in \mathfrak{M}^+(\partial\Omega\cup \Sigma).$ Then problem \eqref{power} admits a solution with $\tau=0$ if and only if there exists a positive constant $C>0$ such that \bal \BBG_\mu[\BBK_\mu[\nu]^p]\leq C\, \BBK_\mu[\nu] \quad \text{a.e. in } \Omega\setminus\Sigma. \eal \end{proposition} \begin{proof} Proceeding as in the proof of Proposition \ref{equivint} and using Proposition \ref{prop2} instead of Proposition \ref{prop}, we obtain the desired result (see also \cite[Lemma 4.1]{BVi}). \end{proof} \begin{comment} \begin{proposition} \label{prop:GKp-1} Assume $\xi \in \Sigma$ and $1< p < \frac{N-\am}{N-\am-2}$.
(i) If $-1<\am \leq \frac{N-2}{2}$ then there exists a positive constant $C=C(N,k,\mu,p,\varepsilon)$ such that, for any $x\in\Omega\setminus \Sigma$, \ba \label{GKp-1.1} \BBG_\mu[K_\mu(\cdot,\xi)^p](x) \leq C\left\{\begin{aligned} &d_{\Sigma}(x)^{-\am},\; &\text{if}\;\; p(N-2-\am)<\am+2, \\
&d_{\Sigma}(x)^{-\am}|\ln|x-\xi||,\; &\text{if}\;\; p(N-2-\am)=\am+2,\\
&d_{\Sigma}(x)^{-\am}|x-\xi|^{2+\am-p(N-2-\am)},\; &\text{if}\;\; p(N-2-\am)>\am+2. \end{aligned}\right. \ea
(ii) If $\am\leq -1$ then for any $\varepsilon \in (\max\{p(N-2-\am)-\am-2,0\},N-2\am-2)$, there exists a positive constant $C=C(N,k,\mu,p,\varepsilon)$ such that \ba \label{GKp-1.2}
\BBG_\mu[K_\mu(\cdot,\xi)^p](x) \leq Cd_{\Sigma}(x)^{-\am} |x-\xi|^{-\varepsilon} \ea \end{proposition} \begin{proof} Without loss of generality, we may suppose that $\xi = 0 \in \Sigma$.
(i) We consider 4 cases.
\noindent \textbf{Case 1: $0\leq\am<\frac{N-2}{2}$.} From Proposition \ref{Greenkernel} (i) and \eqref{Martinest1}, we deduce \ba \label{14} \begin{aligned} \BBG_\mu[K_\mu(\cdot,0)^p](x) &\lesssim \int_\Omega G_\mu(x,y) K_\mu(y,0)^p dy \\
&\lesssim \int_\Omega |x-y|^{2-N}d_\Sigma(y)^{-p\am}|y|^{-p(N-2-2\am)}dy \\
&+\int_\Omega |x-y|^{2+\am-N}d_\Sigma(y)^{-p\am-\am}|y|^{-p(N-2-2\am)}dy\\
&+d_\Sigma(x)^{-\am}\int_\Omega |x-y|^{2+\am-N}d_\Sigma(y)^{-p\am}|y|^{-p(N-2-2\am)}dy\\
&+d_\Sigma(x)^{-\am}\int_\Omega |x-y|^{2+2\am-N}d_\Sigma(y)^{-p\am-\am}|y|^{-p(N-2-2\am)}dy\\ &=:I_1+I_2+I_3+I_4. \end{aligned}
\ea Now, since $d_\Sigma(x)\leq |x-y|+d_\Sigma(y)$, we have \ba \label{15} \begin{aligned} I_1 &= d_\Sigma(x)^{-\am} d_\Sigma(x)^{\am} I_1 \\
&\lesssim d_\Sigma(x)^{-\am}\int_\Omega (|x-y|^{\am} + d_\Sigma(y)^{\am})|x-y|^{2-N}|y|^{-p(N-2-2\am)}d_\Sigma^{-p\am}(y)dy \\
&\lesssim d_\Sigma(x)^{-\am}\int_{\Omega} |x-y|^{\am + 2 - N}|y|^{-p(N-2-2\am)}d_\Sigma(y)^{-p\am}dy \\
&+ d_\Sigma(x)^{-\am}\int_{\Omega} |x-y|^{2 - N}|y|^{-p(N-2-2\am)}d_\Sigma(y)^{-(p-1)\am}dy \\ &\lesssim \left\{\begin{aligned} & d_\Sigma(x)^{-\am},\quad &\text{if}\;\; p(N-2-\am)<\am+2,\\
& d_\Sigma(x)^{-\am}|\ln|x||,\quad &\text{if}\;\; p(N-2-\am)=\am+2,\\
& d_\Sigma(x)^{-\am}|x|^{2+\am-p(N-2-\am)},\quad &\text{if}\;\; p(N-2-\am)>\am+2. \end{aligned}\right. \end{aligned} \ea In the last estimate, we have applied Lemma \ref{anisotita-2}.
Similarly, we obtain \small\begin{equation} \label{16} I_2 + I_3 + I_4 \lesssim \left\{\begin{aligned} &d_\Sigma(x)^{-\am},\quad &\text{if}\;\; p(N-2-\am)<\am+2,\\
&d_\Sigma(x)^{-\am}|\ln|x||,\quad &\text{if}\;\; p(N-2-\am)=\am+2,\\
&d_\Sigma(x)^{-\am}|x|^{2+\am-p(N-2-\am)},\quad &\text{if}\;\; p(N-2-\am)>\am+2. \end{aligned}\right. \end{equation}\normalsize Combining \eqref{14}--\eqref{16} yields the desired result.
\noindent \textbf{Case 2: $-1<\am<0.$} From Proposition \ref{Greenkernel} (i), \eqref{Martinest1}, the fact that $d_{\Sigma}(y)\leq |y|$ and Lemma \ref{lem:estxy}, we deduce, for any $x \in \Omega \setminus \Sigma$, \bal
\BBG_\mu[K_\mu(\cdot,0)^p](x)&\lesssim d_\Sigma(x)^{-\am}\int_\Omega |x-y|^{2\am +2-N} d(y)^p d_\Sigma(y)^{-(p+1)\am}|y|^{-p(N-2-2\am)}dy\\
&\lesssim d_\Sigma(x)^{-\am}\int_\Omega |x-y|^{2\am +2-N} |y|^{-\am - p(N-2-\am)}dy \\ &\lesssim \left\{\begin{aligned} &d_{\Sigma}(x)^{-\am},\quad &&\text{if}\;\; p(N-2-\am)<\am+2,\\
&d_{\Sigma}(x)^{-\am}|\ln|x||,\quad &&\text{if}\;\; p(N-2-\am)=\am+2,\\
&d_{\Sigma}(x)^{-\am}|x|^{2+\am-p(N-2-\am)},\quad &&\text{if}\;\; p(N-2-\am)>\am+2. \end{aligned}\right. \eal We note that in order to apply Lemma \ref{lem:estxy} for the terms on the second estimate, we need $-1<\am$ so that $N-2-2\am<N$.
\noindent \textbf{Case 3:} $\Sigma=\{0\}$ and $\mu=\left(\frac{N-2}{2}\right)^2$. Then $\am=\frac{N-2}{2}$ and the assumption on $p$ becomes $1<p<\frac{N+2}{N-2}$. By employing an argument similar to that in case 1 with appropriate modifications, Proposition \ref{Greenkernel} (ii) and \eqref{Martinest2}, we may obtain \bal
|x|^{\frac{N-2}{2}}\BBG_\mu[K_\mu(\cdot,0)^p](x) \lesssim |x|^{\frac{N-2}{2}}\int_\Omega G_\mu(x,y)|y|^{-p\frac{N-2}{2}}\left|\ln\frac{|y|}{{\mathcal D}} \def\CE{{\mathcal E}} \def\CF{{\mathcal F}_\Omega}\right|^p dy \lesssim 1 \quad x\in \Omega\setminus\{0\}. \eal This implies
$$ \BBG_\mu[K_\mu(\cdot,0)^p](x) \lesssim |x|^{-\frac{N-2}{2}} \quad x\in \Omega\setminus\{0\}. $$
(ii) Assume $\am\leq -1.$ Let $\varepsilon$ such that $\max\{p(N-2-\am)-\am-2,0\}<\varepsilon<N-2\am-2.$ Then, by using estimate Proposition \ref{Greenkernel} (i), \eqref{Martinest1} and estimate $|x| \leq |x-y|+|y|$, we obtain \ba\begin{aligned}
\BBG_\mu[K_\mu(\cdot,0)^p](x) &= |x|^{-\varepsilon} |x|^\varepsilon \int_{\Omega} G_\mu(x,y)K_\mu(y,0)^p dy \\
&\lesssim |x|^{-\varepsilon} \int_\Omega (|x-y|^\varepsilon+|y|^{\varepsilon})G_\mu(x,y)d_\Sigma(y)^{-p\am}|y|^{-p(N-2-2\am)}dy \\
&\lesssim |x|^{-\varepsilon} \int_\Omega |x-y|^\varepsilon G_\mu(x,y)d_\Sigma(y)^{-p\am}|y|^{-p(N-2-2\am)}dy\\
&+|x|^{-\varepsilon} \int_\Omega G_\mu(x,y)d_\Sigma(y)^{-p\am}|y|^{-p(N-2-2\am)+\varepsilon}dy \\ &=:I_1+I_2. \end{aligned}\label{17}
\ea We use Proposition \ref{Greenkernel} (i) and the fact that $d_\Sigma(y) \leq |y|$ to estimate $I_1$ as \ba
\begin{aligned} I_1 &\lesssim |x|^{-\varepsilon} d_\Sigma(x)^{-\am}\int_\Omega |x-y|^{\varepsilon+2\am +2-N}d_\Sigma(y)^{-(p+1)\am}|y|^{-p(N-2-2\am)}dy\\
&\lesssim |x|^{-\varepsilon} d_\Sigma(x)^{-\am} \int_\Omega |x-y|^{\varepsilon+2\am +2-N}|y|^{-p(N-2-\am)-\am}dy\\
&\lesssim |x|^{-\varepsilon} d_\Sigma(x)^{-\am} \int_\Omega |x-y|^{\varepsilon+\am +2-N-p(N-2-\am)}dy \\
&+|x|^{-\varepsilon} d_\Sigma(x)^{-\am}\int_\Omega |y|^{\varepsilon+\am +2-N-p(N-2-\am)}dy\\
&\lesssim |x|^{-\varepsilon} d_\Sigma(x)^{-\am}. \end{aligned}\label{18}
\ea The last estimate holds since $\varepsilon+\am +2-N-p(N-2-\am)>-N$ due to the choice of $\varepsilon$. Next, to estimate $I_2$, we employ \eqref{moreG1} and estimate $d_\Sigma(y)\leq |x-y|+d_\Sigma(x)$ as follows \ba
\begin{aligned} I_2 &\lesssim |x|^{-\varepsilon} d_\Sigma(x)^{-\am} \int_\Omega |x-y|^{2-N}d_\Sigma(y)^{-(p-1)\am}|y|^{-p(N-2-2\am)+\varepsilon}dy \\
&\lesssim |x|^{-\varepsilon} d_\Sigma(x)^{-\am} \int_\Omega |x-y|^{2-N}|y|^{-p(N-2-2\am)-(p-1)\am+\varepsilon}dy\\
&\lesssim |x|^{-\varepsilon} d_\Sigma(x)^{-\am} \int_\Omega |x-y|^{\varepsilon+\am +2-N-p(N-2-\am)}dy \\
&+|x|^{-\varepsilon} d_\Sigma(x)^{-\am}\int_\Omega |y|^{\varepsilon+\am +2-N-p(N-2-\am)}dy\\
&\lesssim |x|^{-\varepsilon} d_\Sigma(x)^{-\am}. \end{aligned}\label{19} \ea The last inequality holds since $\varepsilon+\am +2-N-p(N-2-\am)>-N$ due to the choice of $\varepsilon$.
Combining \eqref{17}--\eqref{19} yields \eqref{GKp-1.2}.
The proof is complete. \end{proof} \end{comment}
\subsection{Abstract setting} In this subsection, we present an abstract setting which will be applied to our particular framework in the next subsection.
Let $\mathbf{Z}$ be a metric space and $\gw \in\GTM^+(\mathbf{Z}).$ Let $J : \mathbf{Z} \times \mathbf{Z} \to (0,\infty]$ be a Borel positive kernel such that $J$ is symmetric and $J^{-1}$ satisfies a quasi-metric inequality, i.e. there is a constant $C>1$ such that for all $x, y, z \in \mathbf{Z}$, \ba \label{Jest} \frac{1}{J(x,y)}\leq C\left(\frac{1}{J(x,z)}+\frac{1}{J(z,y)}\right). \ea Under these conditions, one can define the quasi-metric $d$ by \bal \mathbf{d}(x,y):=\frac{1}{J(x,y)} \eal and denote by $\GTB(x,r):=\{y\in\mathbf{Z}:\; \mathbf{d}(x,y)<r\}$ the open $\mathbf{d}$-ball of radius $r > 0$ and center $x$. Note that this set can be empty.
For $\omega\in\GTM^+(\mathbf{Z})$ and a positive function $\phi$, we define the potentials $\BBJ[\gw]$ and $\BBJ[\phi} \def\vgf{\varphi} \def\gh{\eta,\gw]$ by \bal \BBJ[\gw](x):=\int_{\mathbf{Z}}J(x,y) \dd\omega(y)\quad\text{and}\quad \BBJ[\phi} \def\vgf{\varphi} \def\gh{\eta,\gw](x):=\int_{\mathbf{Z}}J(x,y)\phi} \def\vgf{\varphi} \def\gh{\eta(y) \dd\gw(y). \eal For $t>1$ the capacity $\text{Cap}_{\BBJ,t}^\gw$ in $\mathbf{Z}$ is defined for any Borel $E\subset\mathbf{Z}$ by \bal \text{Cap}_{\BBJ,t}^\gw(E):=\inf\left\{\int_{\mathbf{Z}}\phi} \def\vgf{\varphi} \def\gh{\eta(x)^{t} \dd\gw(x):\;\;\phi} \def\vgf{\varphi} \def\gh{\eta\geq0,\;\; \BBJ[\phi} \def\vgf{\varphi} \def\gh{\eta,\gw] \geq\1_E\right\}. \eal
\begin{proposition}\label{t2.1} \emph{(\cite{KV})} Let $p>1$ and $\gl,\gw \in\GTM^+(\mathbf{Z})$ such that
\ba
\int_0^{2r}\frac{\gw\left(\GTB(x,s)\right)}{s^2} \dd s &\leq C\int_0^{r}\frac{\gw\left(\GTB(x,s)\right)}{s^2} \dd s ,\label{2.3}\\
\sup_{y\in \GTB(x,r)}\int_0^{r}\frac{\gw\left(\GTB(y,r)\right)}{s^2} \dd s&\leq C\int_0^{r}\frac{\gw\left(\GTB(x,s)\right)}{s^2} \dd s,\label{2.4}
\ea for any $r > 0,$ $x \in \mathbf{Z}$, where $C > 0$ is a constant. Then the following statements are equivalent.
1. The equation $v=\BBJ[|v|^p,\gw]+\ell \BBJ[\gl]$ has a positive solution for $\ell>0$ small.
2. For any Borel set $E \subset \mathbf{Z}$, there holds
$ \int_E \BBJ[\1_E\gl]^p \dd \gw \leq C\, \gl(E).
$
3. The following inequality holds
$ \BBJ[\BBJ[\gl]^p,\gw]\leq C\BBJ[\gl]<\infty\quad \gw-a.e. $
4. For any Borel set $E \subset \mathbf{Z}$ there holds $ \gl(E)\leq C\, \emph{Cap}_{\BBJ,p'}^\gw(E). $ \end{proposition}
\subsection{Necessary and sufficient conditions for the existence} For $\alpha\leq N-2$, set \bal
\CN_{\alpha} \def\gb{\beta} \def\gg{\gamma}(x,y):=\frac{\max\{|x-y|,d_\Sigma(x),d_\Sigma(y)\}^\alpha}{|x-y|^{N-2}\max\{|x-y|,d(x),d(y)\}^2},\quad (x,y)\in\overline{\Omega}\times\overline{\Omega}, x \neq y, \eal \bel{opN} \BBN_{\xa}[\omega](x):=\int_{\overline{\Omega} \def\Gx{\Xi} \def\Gy{\Psi}} \CN_{\alpha}(x,y) \dd\omega(y), \quad \omega \in \GTM^+(\overline \Omega} \def\Gx{\Xi} \def\Gy{\Psi).\end{equation}\normalsize
We will point out below that $\BBN_{\xa}$ defined in \eqref{opN} with $\dd \gw=d(x)^b d_\Sigma(x)^\theta \1_{\Omega \setminus \Sigma}(x)\,\dx$ satisfies all assumptions of $\BBJ$ in Proposition \ref{t2.1}, for some appropriate $b,\theta\in \BBR$. Let us first prove the quasi-metric inequality.
\begin{lemma}\label{ineq} Let $\alpha\leq N-2.$ There exists a positive constant $C=C(\Omega,\Sigma,\alpha)$ such that \ba \label{dist-ineq} \frac{1}{\CN_\alpha(x,y)}\leq C\left(\frac{1}{\CN_\alpha(x,z)}+\frac{1}{\CN_\alpha(z,y)}\right),\quad \forall x,y,z\in \overline{\Omega}. \ea \end{lemma} \begin{proof} Let $0\leq b\leq 2,$ we first claim that there exists a positive constant $C=C(N,b,\alpha)$ such that the following inequality is valid \ba \label{35}\begin{aligned}
&\frac{|x-y|^{N-b}}{\max\{|x-y|,d_\Sigma(x),d_\Sigma(y)\}^\alpha}\\
&\qquad\leq C\left(\frac{|x-z|^{N-b}}{\max\{|x-z|,d_\Sigma(x),d_\Sigma(z)\}^\alpha}+\frac{|z-y|^{N-b}}{\max\{|z-y|,d_\Sigma(z),d_\Sigma(y)\}^\alpha}\right). \end{aligned} \ea In order to prove \eqref{35}, we consider two cases.
\noindent \textbf{Case 1: $0<\alpha\leq N-2$.} We first assume that $|x-y|<2|x-z|$. Then by triangle inequality, we have
$d_\Sigma(z)\leq |x-z|+d_\Sigma(x)\leq 2\max\{|x-z|,d_\Sigma(x)\}$ hence \bal
\max\{|x-z|,d_\Sigma(x),d_\Sigma(z)\}\leq 2\max\{ |x-z|,d_\Sigma(x)\}.
\eal If $|x-z|\geq d_\Sigma(x)$ then
\ba\nonumber
\frac{|x-z|^{N-b}}{\max\{|x-z|,d_\Sigma(x),d_\Sigma(z)\}^\alpha}&\geq 2^{-\alpha}|x-z|^{N-b-\alpha}\geq 2^{-N+b}|x-y|^{N-b-\alpha}\\
&\geq 2^{-N+b}\frac{|x-y|^{N-b}}{\max\{|x-y|,d_\Sigma(x),d_\Sigma(y)\}^\alpha}.\label{36}
\ea If $|x-z|\leq d_\Sigma(x)$ then \ba\nonumber
\frac{|x-z|^{N-b}}{\max\{|x-z|,d_\Sigma(x),d_\Sigma(z)\}^\alpha}&\geq 2^{-\alpha}d_\Sigma(x)^{-\alpha}|x-z|^{N-b} \\
&\geq 2^{-\alpha-N+b}d_\Sigma(x)^{-\alpha}|x-y|^{N-b}\\
&\geq 2^{-N+b+\alpha}\frac{|x-y|^{N-b}}{\max\{|x-y|,d_\Sigma(x),d_\Sigma(y)\}^\alpha}.\label{37} \ea Combining \eqref{36}--\eqref{37}, we obtain \eqref{35} with $C=2^{N-b}$.
Next we consider the case $2|x-z|\leq|x-y|$. Then $\frac{1}{2} |x-y|\leq |y-z|,$ thus by symmetry we obtain \eqref{35} with $C=2^{N-b}$.
\noindent \textbf{Case 2: $\alpha\leq 0$.} Since $d_\Sigma(x)\leq |x-y|+d_\Sigma(y)$, it follows that \bal
\max\{|x-y|,d_\Sigma(x),d_\Sigma(y)\}\leq |x-y|+\min\{d_\Sigma(x),d_\Sigma(y)\}. \eal Using the above estimate, we obtain \bal
&|x-y|^{N-b}\max\{|x-y|,d_\Sigma(x),d_\Sigma(y)\}^{-\alpha} \\
&\leq |x-y|^{N-b-\alpha}+\min\{d_\Sigma(x),d_\Sigma(y)\}^{-\alpha}|x-y|^{N-b}\\ \nonumber
&\leq 2^{N-b-\alpha}(|x-z|^{N-b-\alpha}+|y-z|^{N-b-\alpha})\\ \nonumber
&+2^{N-b}(|x-z|^{N-b}\min\{d_\Sigma(x),d_\Sigma(y)\}^{-\alpha}+|y-z|^{N-b}\min\{d_\Sigma(x),d_\Sigma(y)\}^{-\alpha})\\
&\leq ( 2^{N-b-\alpha}+1)\left(\frac{|x-z|^{N-b}}{\max\{|x-z|,d_\Sigma(x),d_\Sigma(z)\}^\alpha}+\frac{|z-y|^{N-b}}{\max\{|z-y|,d_\Sigma(z),d_\Sigma(y)\}^\alpha}\right), \eal which yields \eqref{35}.
Now we will use \eqref{35} with $b=2$ to prove \eqref{dist-ineq}. Since $d(x)\leq |x-y|+d(y),$ we can easily show that $\max\{|x-y|,d(x),d(y)\}\leq |x-y|+\min\{d(x),d(y)\}.$ Hence \bal
\frac{1}{\CN_\alpha(x,y)}&=\frac{\max\{|x-y|,d(x),d(y)\}^2|x-y|^{N-2}}{\max\{|x-y|,d_\Sigma(x),d_\Sigma(y)\}^\alpha} \\
&\leq \frac{2|x-y|^{N}}{\max\{|x-y|,d_\Sigma(x),d_\Sigma(y)\}^\alpha}
+\frac{2\min\{d(x),d(y)\}^2|x-y|^{N-2}}{\max\{|x-y|,d_\Sigma(x),d_\Sigma(y)\}^\alpha}\\ &\leq C(N,\alpha)\left(\frac{1}{\CN_\alpha(x,z)}+\frac{1}{\CN_\alpha(z,y)}\right), \eal where in the last inequality we have used \eqref{35}. The proof is complete. \end{proof}
Next we give sufficient conditions for \eqref{2.3} and \eqref{2.4} to hold. \begin{lemma}\label{l2.3} Let $b>0$, $\theta>k-N$ and $\dd \gw=d(x)^b d_\Sigma(x)^\theta \1_{\Omega \setminus \Sigma}(x)\,\dx$. Then \small\begin{equation} \label{doubling} \gw(B(x,s))\approx \max\{d(x),s\}^b\max\{d_\Sigma(x),s\}^{\theta} s^N, \; \text{for all } x\in\Omega \text{ and } 0 < s\leq 4 \mathrm{diam}(\Omega). \end{equation}\normalsize \end{lemma} \begin{proof} Let $\beta_0$ be as in Subsection \ref{assumptionK} and $s<\frac{\beta_0}{16}$. First we assume that $x\in \Sigma_{\frac{\beta_0}{4}}$ then $d(y)\approx 1$ for any $y\in B(x,s)$. Thus, it is enough to show that \small\begin{equation} \label{39} \int_{B(x,s)}d_\Sigma(y)^{\theta} \dy\approx \max\{d_\Sigma(x),s\}^{\theta} s^N. \end{equation}\normalsize \noindent\textbf{Case 1: $d_\Sigma(x)\geq 2s$.} Then $\frac{1}{2}d_{\Sigma}(x)\leq d_\Sigma(y)\leq\frac{3}{2} d_\Sigma(x)$ for any $y\in B(x,s)$, therefore \eqref{39} follows easily in this case.
\noindent \textbf{Case 2: $d_\Sigma(x)\leq 2s$.} Then there exists $\xi \in\Sigma $ such that $B(x,s)\subset V(\xi,4\beta_0)$. If $y\in B(x,s),$ then $|y'-x'|<s$ and $d_\Sigma(y)\leq d_\Sigma(x)+|x-y|\leq 3s$. Thus by \eqref{propdist},
$\delta_\Sigma^\xi(y)\leq C_1s$ for any $y\in B(x,s)$, where $C_1$ depends on $\| \Sigma \|_{C^2},N$ and $k$. Thus \bal
\int_{B(x,s)}d_\Sigma(y)^\theta \dy \lesssim \int_{\{|x'-y'|<s\}}\int_{\{\delta_\Sigma^\xi(y)\leq C_1s\}}(\delta_\Sigma^\xi(y))^\theta \dy''\dy' \approx s^{N+\theta} \approx \max\{d_\Sigma(x),s\}^\theta s^{N}.
\eal Here the similar constants depend on $N,k,\| \Sigma \|_{C^2}$ and $\beta_0$.
\noindent \textbf{Case 3:} $d_\Sigma(x)\leq 2s$ and $\theta<0$. We have that $d_\Sigma(y)^\theta \geq3^\theta s^{\theta}$ for any $y\in B(x,s)$, which leads to \eqref{39}.
\noindent \textbf{Case 4:} $d_\Sigma(x)\leq 2s$ and $\theta \geq 0$. Let $C_2=C \| \Sigma \|_{C^2}$ be the constant in \eqref{propdist}.
If $d_\Sigma(x) \leq \frac{s}{6(N-k)C_2}$ then by \eqref{propdist} we have $\delta_\Sigma^\xi(x)\leq\frac{s}{6(N-k)}$. In addition for any
\bal y\in \left\{\psi = (\psi',\psi'') \in\Omega \setminus \Sigma:|x'-\psi'|\leq \frac{s}{6(N-k)C_2},\; \delta_\Sigma^\xi(\psi)\leq \frac{s}{6(N-k)} \right\}=:{\mathcal A}} \def\CB{{\mathcal B}} \def\CC{{\mathcal C}, \eal we have \bal
|x''-y''|\leq \delta_\Sigma^\xi(x)+\delta_\Sigma^\xi(y)+\left(\sum_{i=k+1}^N|\Gamma_i^\xi(x')-\Gamma_i^\xi(y')|^2\right)^\frac{1}{2}\leq \frac{s}{3}+(N-k)\| \Sigma \|_{C^2}|x'-y'|\leq \frac{s}{2}. \eal This implies that ${\mathcal A}} \def\CB{{\mathcal B}} \def\CC{{\mathcal C}\subset B(x,s)$. Consequently, \bal \int_{B(x,s)}d_\Sigma(y)^\theta\dy&\approx \int_{B(x,s)}(\delta_\Sigma^\xi(y))^\theta \dy\gtrsim \int_{{\mathcal A}} \def\CB{{\mathcal B}} \def\CC{{\mathcal C}}(\delta_\Sigma^\xi(y))^\theta \dy \approx C\max\{d_\Sigma(x),s\}^\theta s^{N}. \eal
If $d_\Sigma(x)\geq \frac{s}{6(N-k)C_2}$ then \bal \int_{B(x,s)}d_\Sigma(y)^\theta \dy\geq\int_{B(x,\frac{s}{12(N-k)C_2})}d_\Sigma(y)^\theta \dy \eal and hence \eqref{39} follows by case 1.
Next we consider $x\in \Omega_{\frac{\beta_0}{4}}$. Then $d_\Sigma(y)\approx 1$ for any $y\in \Omega_{\frac{\beta_0}{2}}$. By proceeding as before we may prove \eqref{doubling} for any $s<\frac{\beta_0}{16}$.
If $x\in \Omega\setminus(\Omega_{\frac{\beta_0}{4}}\cup \Sigma_{\frac{\beta_0}{4}})$ then $d_\Sigma(y),d(x)\approx 1$ for any $y\in B(y,s),$ with $s<\frac{\beta_0}{16}.$ Thus, in this case, we can easily prove \eqref{39} for any $s<\frac{\beta_0}{16}.$
If $ \frac{\beta_0}{16}\leq s\leq 4 \text{diam}(\Omega)$ then $\gw(B(x,s))\approx 1$, hence estimate \eqref{doubling} follows straightforward. The proof is complete. \end{proof}
\begin{lemma}\label{vol} Let $\alpha< N-2$, $b> 0$, $\theta>\max\{k-N,-2-\alpha\}$ and $\dd \gw=d(x)^b d_\Sigma(x)^\theta$ $\1_{\Omega \setminus \Sigma}(x)\,\dx$. Then \eqref{2.3} holds. \end{lemma} \begin{proof} We note that if $s\geq (4\mathrm{diam}\,(\Omega))^{N-\alpha}$ then $\omega(\GTB(x,s))=\omega(\overline{\Omega})<\infty$, where $\GTB(x,s)$ is defined after \eqref{Jest}, namely $\GTB(x,s)=\{y \in \Omega \setminus \Sigma: \mathbf{d}(x,y)<s\}$ and $\mathbf{d}(x,y)= \frac{1}{\CN_\alpha(x,y)}$.
We first assume that $0<\alpha<N-2$. Let $x\in \Sigma_\frac{\beta_0}{4}$ then \ba \label{C0d} 0<C_0 \leq d(x)\leq 2 \mathrm{diam}\,(\Omega), \ea where $C_0$ depends on $\Omega,\Sigma,\beta_0$. Set \bal
\GTC(x,s):=\left\{y \in \Omega \setminus \Sigma: \frac{|x-y|^{N-2}}{\max\{|x-y|,d_\Sigma(x),d_\Sigma(y)\}^\alpha}< s\right\}. \eal Then \ba \label{CB} \GTC\left(x,\frac{s}{4\mathrm{diam}\,(\Omega)^2}\right)\subset\GTB(x,s)\subset \GTC\left(x,\frac{s}{C_0^2}\right). \ea We note that $B(x,S_1)\subset\GTC(x,s)\subset B(x,l_1S_1)$ where $S_1=\max\{s^{\frac{1}{N-2-\alpha}},s^{\frac{1}{N-2}}d_\Sigma(x)^{\frac{\alpha}{N-2}} \}$ and $l_1=2^{\frac{\alpha}{N-2-a}}$. Therefore, by Lemma \ref{l2.3}, we obtain \ba\nonumber \omega(\GTB(x,s))&\approx \max\left\{d_\Sigma(x),\max\{s^{\frac{1}{N-2-\alpha}},s^{\frac{1}{N-2}}d_\Sigma(x)^{\frac{\alpha}{N-2}} \}\right\}^\theta \max\{s^{\frac{1}{N-2-\alpha}},s^{\frac{1}{N-2}}d_\Sigma(x)^{\frac{\alpha}{N-2}} \}^N\\ &\approx\left\{\begin{aligned} &d_\Sigma(x)^{\theta+\frac{\alpha N}{N-2}}s^{\frac{N}{N-2}} \quad&&\text{if } s\in (0,d_\Sigma(x)^{N-2-\alpha}), \\
&s^{\frac{\theta+N}{N-2-\alpha}} \quad &&\text{if } s\in [d_\Sigma(x)^{N-2-\alpha},M),\\
&1 \quad &&\text{if } s \in [M,\infty). \end{aligned}\right.\label{xoest1} \ea where \bal M:=(4\mathrm{diam}\,(\Omega))^\frac{N(N-\alpha)}{b+N}+(4\mathrm{diam}\,(\Omega))^{\frac{(N-2)(N-\alpha)}{N}}+(4\mathrm{diam}\,(\Omega))^{\frac{(N-\alpha-2)(N-\alpha)}{\theta +N}}. \eal
Next we assume that $\alpha\leq0$, Let $x\in \Sigma_\frac{\beta_0}{4}$ then \eqref{C0d} and \eqref{CB} hold. We also have $B(x,{l_2S_2})\subset\GTC(x,s)\subset B(x,S_2)$, where $S_2=\min\{s^{\frac{1}{N-2-\alpha}},s^{\frac{1}{N-2}}d_\Sigma(x)^{\frac{\alpha}{N-2}}\}$ and $l_2=2^{\frac{\alpha}{N-2}}$. Therefore by Lemma \ref{l2.3}, we obtain \ba\nonumber \omega(\GTB(x,s))&\approx \max\left\{d_\Sigma(x),\min\{s^{\frac{1}{N-2-\alpha}},s^{\frac{1}{N-2}}d_\Sigma(x)^{\frac{\alpha}{N-2}}\}\right\}^\theta \min\{s^{\frac{1}{N-2-\alpha}},s^{\frac{1}{N-2}}d_\Sigma(x)^{\frac{\alpha}{N-2}}\}^N\\ &\approx\left\{\begin{aligned} &d_\Sigma(x)^{\theta+\frac{\alpha N}{N-2}}s^{\frac{N}{N-2}} \quad &&\text{if } s\in (0,d_\Sigma(x)^{N-2-\alpha}),\\ &s^{\frac{\theta+N}{N-2-\alpha}} \quad &&\text{if } s\in [d_\Sigma(x)^{N-2-\alpha},M),\\
&1 \quad &&\text{if } s \in [M,\infty). \end{aligned}\right.\label{xoest2} \ea
Next consider $x\in \Omega_{\frac{\beta_0}{4}},$ then there exists a positive constant $C_3=C_3(\Omega,\Sigma,\alpha,\beta_0)$ such that $C_3\leq d_\Sigma(x)< 2 \mathrm{diam}\,(\Omega).$ Set \bal
\mathcal{E}(x,s):=\{y \in \Omega \setminus \Sigma: |x-y|^{N-2}\max\{|x-y|,d(x),d(y)\}^{2}< s\}. \eal We obtain \bal \mathcal{E}(x,\min\{C_3^\alpha,2^\alpha\mathrm{diam}\,^\alpha(\Omega)\}s)\subset\GTB(x,s)\subset \mathcal{E}(x,\max\{C_3^\alpha,2^\alpha\mathrm{diam}\,^\alpha(\Omega)\}s). \eal We also have \bal B(x,l_3S_3)\subset\mathcal{E}(x,s)\subset B(x,S_3), \eal where $S_3=\min\{s^{\frac{1}{N}},s^{\frac{1}{N-2}}d(x)^{-\frac{2}{N-2}}\}$ and $l_3=2^{-\frac{2}{N-2}}$. Again, by Lemma \ref{l2.3}, we obtain \ba\nonumber \omega(\GTB(x,s))&\approx \max\left\{d(x),\min\{s^{\frac{1}{N}},s^{\frac{1}{N-2}}d(x)^{-\frac{2}{N-2}}\}\right\}^b \min\{s^{\frac{1}{N}},s^{\frac{1}{N-2}}d(x)^{-\frac{2}{N-2}}\}^N\\ &\approx\left\{\begin{aligned} &d(x)^{b-\frac{2 N}{N-2}}s^{\frac{N}{N-2}} \quad &&\text{if } s\in (0,d(x)^{N}),\\ &s^{\frac{b+N}{N}}\quad &&\text{if } s\in [d(x)^{N},M),\\ &1\quad &&\text{if } s \in [M,\infty). \end{aligned}\right.\label{xoest3} \ea
Let $0<\bar \beta \leq \frac{\beta_0}{4}$ and $x\in \Omega\setminus (\Omega_{\bar \beta}\cup \Sigma_{\bar \beta}).$ Then there exists a positive constant $C_4=C_4(\Omega,\Sigma,\bar \beta)$ such that $C_4\leq d_\Sigma(x),d(x)< 2 \mathrm{diam}\,(\Omega).$ By Lemma \ref{l2.3}, we can show that \ba \omega(\GTB(x,s))&\approx \left\{\begin{aligned} &s^{\frac{N}{N-2}} \quad &&\text{if } s \in (0,M),\\
&1 \quad &&\text{if } s \in [M,\infty). \end{aligned}\right. \label{xoest4} \ea
Combining \eqref{xoest1}--\eqref{xoest4} leads to \eqref{2.3}. The proof is complete. \end{proof}
\begin{lemma}\label{vol-2}
We assume that $\alpha< N-2$, $b> 0$, $\theta>\max\{k-N,-2-\alpha\}$ and $\dd \gw=d(x)^b d_\Sigma(x)^\theta\1_{\Omega \setminus \Sigma}(x)\,\dx$. Then \eqref{2.4} holds. \end{lemma} \begin{proof} We consider only the case $\alpha>0$ and $x\in \Sigma_{\frac{\beta_0}{16}}$ since the other cases $x\in \Omega_{\frac{\beta_0}{16}}$ and $x\in \Omega\setminus(\Omega_{\frac{\beta_0}{16}}\cup \Sigma_{\frac{\beta_0}{16}})$ can be treated similarly and we omit them. We take $r>0$.
\noindent \textbf{Case 1:} $0<r<\left(\frac{\beta_0}{16(2\mathrm{diam}\,(\Omega))^\alpha}\right)^{N}$. In this case, we note that $\GTB(x,r) \subset \Sigma_{\frac{\beta_0}{8}}$. This and \eqref{xoest1} imply that, for any $y \in \GTB(x,r)$, \ba\nonumber \int_0^r\frac{\omega(\GTB(y,s))}{s^2} \dd s\approx\left\{\begin{aligned} &d_\Sigma(y)^{\theta+\frac{\alpha N}{N-2}}r^{\frac{2}{N-2}} \quad &&\text{if } r\in (0,d_\Sigma(y)^{N-2-\alpha}),\\
&r^{\frac{\theta+2+\alpha}{N-2-\alpha}} \quad &&\text{if } r\in [d_\Sigma(y)^{N-2-\alpha},M), \\
&1 \quad &&\text{if } r \in [M,\infty). \end{aligned}\right.\label{intxoest1} \ea
If $|x-y|\leq \frac{1}{2}d_\Sigma(x)$ then $\frac{1}{2}d_\Sigma(x)\leq d_\Sigma(y)\leq \frac{3}{2}d_\Sigma(x)$. Therefore, when $d_\Sigma(y),d_\Sigma(x)\geq r^{\frac{1}{N-2-\alpha}}$, we obtain \bal \int_0^r\frac{\omega(\GTB(y,s))}{s^2} \dd s \approx d_\Sigma(y)^{\theta+\frac{\alpha N}{N-2}}r^{\frac{2}{N-2}}\approx d_\Sigma(x)^{\theta+\frac{\alpha N}{N-2}}r^{\frac{2}{N-2}}\approx \int_0^r\frac{\omega(\GTB(x,s))}{s^2} \dd s. \eal If $d_\Sigma(y)\geq r^{\frac{1}{N-2-\alpha}}$ and $d_\Sigma(x)\leq r^{\frac{1}{N-2-\alpha}}$ then $d_\Sigma(y)\leq \frac{3}{2}r^{\frac{1}{N-2-\alpha}}$, which implies \bal \int_0^r\frac{\omega(\GTB(y,s))}{s^2} \dd s \approx d_\Sigma(y)^{\theta+\frac{\alpha N}{N-2}}r^{\frac{2}{N-2}}\approx r^{\frac{\theta+2+\alpha}{N-2-\alpha}}\approx \int_0^r\frac{\omega(\GTB(x,s))}{s^2} \dd s. \eal If $d_\Sigma(y)\leq r^{\frac{1}{N-2-\alpha}}$ and $d_\Sigma(x)\geq r^{\frac{1}{N-2-\alpha}}$ then $d_\Sigma(x)\leq 2r^{\frac{1}{N-2-\alpha}}$, which yields \bal \int_0^r\frac{\omega(\GTB(x,s))}{s^2} \dd s\approx d_\Sigma(x)^{\theta+\frac{\alpha N}{N-2}}r^{\frac{2}{N-2}}\approx r^{\frac{\theta+2+\alpha}{N-2-\alpha}}\approx \int_0^r\frac{\omega(\GTB(y,s))}{s^2} \dd s. \eal If $d_\Sigma(y)\leq r^{\frac{1}{N-2-\alpha}}$ and $d_\Sigma(x)\leq r^{\frac{1}{N-2-\alpha}}$ then \bal \int_0^r\frac{\omega(\GTB(x,s))}{s^2} \dd s\approx r^{\frac{\theta+2+\alpha}{N-2-\alpha}}\approx \int_0^r\frac{\omega(\GTB(y,s))}{s^2} \dd s. \eal
Now we assume that $y \in \GTB(x,r)$ and $|x-y|\geq \frac{1}{2}d_\Sigma(x).$ Then
\bal d_\Sigma(y)\leq \frac{3}{2}|x-y| \quad \text{and} \quad
|x-y|\leq C(\beta_0,\Omega,N,\Sigma) r^{\frac{1}{N-\alpha-2}}. \eal Hence $d_\Sigma(x),d_\Sigma(y)\lesssim r^{\frac{1}{N-\alpha-2}}$. Proceeding as above we obtain the desired result.
\noindent \textbf{Case 2:} $r\geq \left(\frac{\beta_0}{16(2\mathrm{diam}\,(\Omega))^\alpha}\right)^{N}$. By \eqref{xoest1}--\eqref{xoest4}, we can easily prove that \bal \int_0^r\frac{\omega(\GTB(y,s))}{s^2} \dd s \approx 1,\quad \forall y\in \overline{\Omega}, \eal and the desired result follows easily in this case. \end{proof}
For $b>0$, $\theta>-N+k$ and $s>1$, define the capacity $\text{Cap}_{\BBN_{\xa},s}^{b,\theta}$ by \bal \text{Cap}_{\BBN_{\xa},s}^{b,\theta}(E) :=\inf\left\{\int_{\overline{\Omega}}d^b d^\theta_\Sigma\phi} \def\vgf{\varphi} \def\gh{\eta^s\,\dx:\;\; \phi} \def\vgf{\varphi} \def\gh{\eta \geq 0, \;\;\BBN_{\xa}[ d^b d_{\Sigma}^\theta\phi} \def\vgf{\varphi} \def\gh{\eta ]\geq\1_E\right\} \quad \text{for Borel set } E\subset\overline{\Omega}. \eal Here $\1_E$ denotes the indicator function of $E$. Furthermore, by \cite[Theorem 2.5.1]{Ad}, \small\begin{equation}\label{dualcap}
(\text{Cap}_{\BBN_{\xa},s}^{b,\theta}(E))^\frac{1}{s}=\sup\{\tau(E):\tau\in\GTM^+(E), \|\BBN_{\xa}[\tau]\|_{L^{s'}(\Omega;d^b d^\theta_\Sigma)} \leq 1 \}. \end{equation}\normalsize
Now we are ready to prove Theorem \ref{theoremint}.
\begin{proof}[\textbf{Proof of Theorem \ref{theoremint}}.] We will apply Proposition \ref{t2.1} with $J(x,y)=\CN_{2\am}(x,y)$, $\dd \omega= \left(d(x)d_{\Sigma}(x)^{-\am}\right)^{p+1}\dx$ and $\dd \lambda = {\phi_{\mu }}\1_{\Omega\setminus \Sigma} \dd \gt$. Estimate \eqref{Jest} is satisfied thanks to Lemma \ref{ineq}, while assumptions \eqref{2.3}--\eqref{2.4} are fulfilled thanks to Lemmas \ref{vol}--\ref{vol-2} respectively with $\alpha=2\am$, $b=p+1$ and $\theta=-\am(p+1)$. We note that condition \eqref{p-cond} ensures that $b$ and $\theta$ satisfy the assumptions in Lemmas \eqref{vol}--\eqref{vol-2}.
Moreover, we have the following observations.
(i) There holds \ba \label{GN2a} G_\mu(x,y)\approx d(x)d(y)(d_{\Sigma}(x)d_{\Sigma}(y))^{-\am} \CN_{2\am}(x,y) \quad \forall x,y \in \Omega \setminus \Sigma, x \neq y. \ea Consequently, if the equation \ba \label{vN2a} v=\BBN_{2\am}[(dd_{\Sigma}^{-\am})^{p+1}v^p]+\ell \BBN_{2\am}[\gl] \ea has a solution $v$ for some $\ell>0$ then the function $\tilde v(x)=d(x)d_{\Sigma}(x)^{-\am}v(x)$ satisfies $\tilde v \approx\BBG_\mu[\tilde v^p]+\ell \BBG_\mu[\gt]$. By \cite[Proposition 2.7]{BHV}, there exists $\rho>0$ small such that equation \eqref{u-rhotau} has a positive solution $u$. By the above argument, we can show that equations \eqref{vN2a} has a solution for $\ell>0$ small if and only if equation \eqref{u-rhotau} has a solution for $\rho>0$ small. In other words, statement 1 of Proposition \ref{t2.1} is equivalent to statement 1 of the present Theorem.
(ii) With $J,\omega$ and $\lambda$ as above, from \eqref{GN2a}, we deduce easily that statements 2--4 of Proposition \ref{t2.1} reduce to statements 2--4 of the present Theorem respectively.
From the above observations and Proposition \ref{t2.1}, we obtain the desired results. \end{proof} \begin{remark} \label{capp-1} Assume $0<\mu \leq H^2$. By combining \eqref{dualcap}, \eqref{GN2a} and \eqref{Gphi_mu}, we derive that for any $1<p<\frac{N+1}{N-1}$, \small\begin{equation} \label{Nz} \inf_{z \in \Omega \setminus \Sigma} \mathrm{Cap}_{\BBN_{2\am},p'}^{p+1,-\am(p+1)}(\{z\})> C. \end{equation}\normalsize Hence, for $1<p<\frac{N+1}{N-1}$, statement 3 of Theorem \ref{theoremint} is valid, therefore, statements 1 and 2 of Theorem \ref{theoremint} hold true. This covers Theorem \ref{th2} (i) with $\gamma=1$ and Proposition \eqref{equivint}. \end{remark}
\begin{proposition} \label{nonexist-1} Assume $0<\mu<\left( \frac{N-2}{2} \right)^2$ and $p \geq \frac{N+1}{N-1}$. Then there exists a measure $\tau \in \GTM^+(\Omega \setminus \Sigma;{\phi_{\mu }})$ with $\| \tau \|_{\GTM(\Omega \setminus \Sigma;{\phi_{\mu }})}=1$ such that problem \eqref{u-rhotau} does not admit positive solution for any $\rho>0$. \end{proposition} \begin{proof} Suppose by contradiction that for every $\tau \in \GTM^+(\Omega \setminus \Sigma;{\phi_{\mu }})$ with $\norm{\tau}_{\GTM(\Omega \setminus \Sigma;{\phi_{\mu }})} = 1$, there exists a positive solution to problem \eqref{u-rhotau} for some $\rho>0.$ Let $y^* \in \partial \Omega$ and $\{y_n\} \subset \Omega \setminus \Sigma$ such that $y_n \to y^* \in \partial \Omega$ and $\mathrm{dist}(y_n,\Sigma)>\varepsilon>0,$ for some $\varepsilon>0$.
From \eqref{GN2a} and \eqref{eigenfunctionestimates}, we have \ba\label{72}
G_\mu(x,y_n){\phi_{\mu }}(y_n)^{-1} \gtrsim \dfrac{1}{|x-y_n|^{N-2}}\cdot \frac{{\phi_{\mu }}(x)}{\max\{d(x)^2,d(y_n)^2,|x-y_n|^2\}}=:F(x,y_n). \ea By using Fatou lemma and \eqref{eigenfunctionestimates}, we deduce that \ba \label{eq:bound1} \nonumber
\liminf_{ n \to \infty} \int_{\Omega \setminus \Sigma} F(x,y_n)^p {\phi_{\mu }}(x) dx& \geq \int_{\Omega \setminus \Sigma} (\liminf_{n \to \infty} F(x,y_n)^p ) {\phi_{\mu }}(x) \dd x \\ \nonumber
&\gtrsim \int_{\Omega \setminus \Sigma} \left( \frac{{\phi_{\mu }}(x)}{|x-y^*|^{N}} \right)^p {\phi_{\mu }}(x) \dd x \\
&\approx \int_{\Omega \setminus \Sigma} \left( \frac{d(x)d_\Sigma(x)^{-\am}}{|x-y^*|^{N}} \right)^p d(x)d_\Sigma(x)^{-\am} \dd x. \ea
Since $\Omega$ is a $C^2$ domain, it satisfies the interior cone condition, hence there exists $r_0 > 0$ small enough such that the circular cone at vertex $y^*$ \bal
{\mathcal C}_{r_0}(y^*):=\left\{ x \in B_{r_0}(y^*): (x-y^*)\cdot {\bf n}_{y^*} > \frac{1}{2}|x-y^*| \right\} \subset \Omega \setminus \Sigma, \eal where ${\bf n}_{y^*}$ denotes the inward unit normal vector to $\partial \Omega$ at $y^*$.
Without loss of generality, suppose that the coordinates are placed so that $y^* = 0 \in \partial \Omega$, the tangent hyperplane to $\partial \Omega$ at $0$ is $\{ x=(x_1,\ldots,x_{N-1},x_N) \in {\mathbb R}^N: x_N=0\}$ and ${\bf n}_0 = (0,\ldots, 0,1)$. We can choose $r_0$ small enough such that $d(x) \geq \alpha |x|$ for all $x \in \mathcal{C}_{r_0}(0)$ and for some $\alpha \in (0,1)$. Then we have \small\begin{equation}\label{intr0}
\int_{\Omega \setminus \Sigma} \left( \frac{d(x)d_\Sigma(x)^{-\am}}{|x|^{N}} \right)^p d(x)d_\Sigma(x)^{-\am} \dd x
\gtrsim \int_{{\mathcal C}_{r_0}(0)} |x|^{1-(N-1)p} \dd x \sim \int_0^{r_0} t^{N-(N-1)p} \dd t. \end{equation}\normalsize Since $p \geq \frac{N+1}{N-1}$, the last integral in \eqref{intr0} is divergent. This and \eqref{eq:bound1}, \eqref{intr0} yield $\liminf_{n \to \infty}\int_{\Omega \setminus \Sigma}F(x,y_n)^p {\phi_{\mu }} \dd x =\infty$. Consequently, for any $j\in \BBN,$ there exists $n_j\in \BBN$ such that \ba\label{73} 2^{jp}\leq \int_{\Omega \setminus \Sigma}F(x,y_{n_j})^p {\phi_{\mu }} \dd x. \ea
Put $\tau_k: = \sum_{j=1}^k 2^{-j}\frac{\delta_{y_{n_j}}}{\phi_\mu}$ then $\| \tau_k \|_{\GTM^+(\Omega \setminus \Sigma;{\phi_{\mu }})} \leq 1$ and $\tau_k\leq \tau_{k+1}$ for any $k\in \BBN$. Put $\tau=\lim_{k\to\infty}\tau_k$ then \bal \int_{\Omega\setminus\Sigma}{\phi_{\mu }} \dd \tau=\sum_{j=1}^\infty 2^{-j}=1. \eal By the supposition, there exists a positive solution $u\in L^p(\Omega\setminus\Sigma;{\phi_{\mu }})$ of problem \eqref{u-rhotau} with datum $\rho \tau$. From the representation formula and \eqref{72}, we deduce \bal
u = \BBG_\mu[u^p] + \rho \BBG_\mu[\tau]\geq \rho\sum_{j=1}^\infty 2^{-j}\BBG_\mu[\frac{\delta_{y_{n_j}}}{\phi_\mu}]\gtrsim \rho\sum_{j=1}^\infty 2^{-j}F(x,y_{n_j}). \eal The above inequality and \eqref{73} yield \bal \int_{\Omega\setminus \Sigma}u^p{\phi_{\mu }} dx\gtrsim \rho^p\sum_{j=1}^\infty 2^{-jp}\int_{\Omega\setminus \Sigma}F(x,y_{n_j})^p{\phi_{\mu }} dx\geq \rho^p\sum_{j=1}^\infty1=\infty, \eal which is clearly a contradiction since $u\in L^p(\Omega\setminus\Sigma;{\phi_{\mu }}).$ The proof is complete. \end{proof}
\begin{proposition} \label{remove1} Assume $0<\mu<\left( \frac{N-2}{2} \right)^2$ and $p \geq \frac{\am+2}{\am}$. Then for any $\rho>0$ and any $\tau \in \GTM^+(\Omega \setminus \Sigma;{\phi_{\mu }})$ with $\| \tau \|_{\GTM(\Omega \setminus \Sigma;{\phi_{\mu }})}=1$, there is no solution of problem \eqref{u-rhotau}. \end{proposition} \begin{proof}
Suppose by contradiction that there exist $\tau \in \GTM^+(\Omega \setminus \Sigma;{\phi_{\mu }})$ with $\| \tau \|_{\GTM(\Omega \setminus \Sigma;{\phi_{\mu }})}=1$ and $\rho>0$ such that problem \eqref{u-rhotau} admits a positive solution $u \in L^p(\Omega;{\phi_{\mu }})$.
Since $\tau\not\equiv0,$ there exist $x_0\in\Omega\setminus\Sigma$, $r,\varepsilon>0$ such that $B(x_0,r)\subset\Omega\setminus\Sigma$, $\mathrm{dist}(B(x_0,r),\Sigma)>\varepsilon$, and $\tau(B(x_0,r))>0$. Set $\tau_B=\1_{B(x_0,r)}\tau,$ then $\tau_B\leq \tau$. Let $v_1=\BBG_\mu[\rho \tau_B],$ we consider the sequence $\{v_{k}\}_{k=1}^\infty\subset L^p(\Omega;{\phi_{\mu }})$ which satisfies the following problem \bal
- L_\mu} \def\gn{\nu} \def\gp{\pi v_{k+1}= |v_{k}|^{p-1}v_{k} +\rho \tau_B\qquad \text{in }\;\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma, \quad \mathrm{tr}(v_{k+1})=0, \eal for any $k\in\BBN.$ Using \eqref{poi5}, we can easily show that $0 \leq v_k\leq v_{k+1}$ and $v_k\leq u$ for any $k\in\BBN.$ Since $u\in L^p(\Omega;{\phi_{\mu }}),$ by monotone convergence theorem, we deduce that $v=\lim_{k\to\infty}v_k$ belongs to $ L^p(\Omega;{\phi_{\mu }})$, $v \geq 0$, and $ v=\lim_{k\to\infty} v_{k+1}=\lim_{k\to\infty}\BBG_\mu[v_k^p+\rho \tau_B]=\BBG_\mu[v^p+\rho \tau_B], $ which means that $v\in L^p(\Omega;{\phi_{\mu }})$ is a weak solution of \bal
- L_\mu} \def\gn{\nu} \def\gp{\pi v= |v|^{p-1}v +\rho \tau_B\qquad \text{in }\;\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma, \quad \mathrm{tr}(v)=0. \eal By Proposition \ref{equivint}, there exists a positive constant $C$ depending on $\rho$ and $p$ such that \ba\label{58c} \BBG_\mu[\BBG_\mu[\tau_B]^p]\leq C\, \BBG_\mu[\tau_B] \quad \text{a.e. in } \Omega\setminus\Sigma. \ea
Assume $0 \in \Sigma$ and set $\beta=\frac{1}{4}\min\{\beta_0,r\}$. Let $x\in\Omega\setminus \Sigma$ such that $|x|\leq \frac{\beta}{2}$. Since
\ba\label{74}
\BBG_\mu[\tau_B] \approx {\phi_{\mu }}\approx d_{\Sigma}^{-\am}\quad \text{in}\;\; \Sigma_\beta,
\ea
and $d_\Sigma(y) \leq |y|$ for any $y \in \Sigma_\beta$, we have, for any $x \in B(0,\frac{\beta}{2}) \setminus \Sigma$, \bal \BBG_\mu[\BBG_\mu[\gt_B]^p](x)
&\gtrsim d_\Sigma(x)^{-\alpha}\int_{\Sigma_{\beta}} d_{\Sigma}(y)^{-\am(p+1)}|x-y|^{2+2\am-N}\dy\\
&\geq d_\Sigma(x)^{-\am}\int_{\Sigma_{\beta}} |y|^{-\am(p+1)}|x-y|^{2+2\am-N}\dy\\ &\approx\left\{ \begin{aligned}
&d_\Sigma(x)^{-\am}|\ln|x|| \quad &&\text{if}\;p=\frac{2+\am}{\am},\\
&d_\Sigma(x)^{-\am}|x|^{2+\am-p\am} \quad &&\text{if}\;p>\frac{2+\am}{\am}. \end{aligned}\right.
\eal This and \eqref{74} yield that \eqref{58c} is not valid as $|x| \to 0$, which is clearly a contradiction. \end{proof}
In order to study the boundary value problem with measure data concentrated on $\partial \Omega \cup \Sigma$, we make use of specific capacities which are defined below.
For $\alpha} \def\gb{\beta} \def\gg{\gamma\in\BBR$ we define the Bessel kernel of order $\alpha} \def\gb{\beta} \def\gg{\gamma$ in ${\mathbb R}^d$ by $\CB_{d,\alpha} \def\gb{\beta} \def\gg{\gamma}(\xi):=\CF^{-1}\left((1+|.|^2)^{-\frac{\alpha} \def\gb{\beta} \def\gg{\gamma}{2}}\right)(\xi), $ where $\CF$ is the Fourier transform in the space $\mathcal{S}'({\mathbb R}^d)$ of moderate distributions in $\BBR^d$. For $\lambda \in \GTM({\mathbb R}^d)$, set \bal \BBB_{d,\alpha}[\lambda](x):= \int_{{\mathbb R}^d}\CB_{d,\alpha}(x-y) \dd\lambda(y), \quad x \in {\mathbb R}^d. \eal Let $L_{\alpha} \def\gb{\beta} \def\gg{\gamma,\kappa}(\BBR^d):=\{f=\CB_{d,\alpha} \ast g:g\in L^{\kappa}(\BBR^d)\} $ be the Bessel space with the norm \bal
\|f\|_{L_{\alpha} \def\gb{\beta} \def\gg{\gamma,\kappa}}:=\|g\|_{L^\kappa}=\|\CB_{d,-\alpha} \def\gb{\beta} \def\gg{\gamma}\ast f\|_{L^\kappa}. \eal It is known that if $1<\kappa<\infty$ and $\alpha} \def\gb{\beta} \def\gg{\gamma>0$, $L_{\alpha} \def\gb{\beta} \def\gg{\gamma,\kappa}(\BBR^d)=W^{\alpha} \def\gb{\beta} \def\gg{\gamma,\kappa}(\BBR^d)$ if $\alpha} \def\gb{\beta} \def\gg{\gamma\in\BBN.$ If $\alpha} \def\gb{\beta} \def\gg{\gamma\notin\BBN$ then the positive cone of their dual coincide, i.e. $(L_{-\alpha} \def\gb{\beta} \def\gg{\gamma,\kappa'}(\BBR^d))_+=(B^{-\alpha} \def\gb{\beta} \def\gg{\gamma,\kappa'}(\BBR^d))_+$, always with equivalent norms. The Bessel capacity is defined for compact subsets $K \subset\BBR^d$ by \bal
\mathrm{Cap}_{{\CB_{d,\alpha},\kappa}}^{{\mathbb R}^d}(K):=\inf\{\|f\|^\kappa_{L_{\alpha} \def\gb{\beta} \def\gg{\gamma,\kappa}}, f\in{\mathcal S}} \def\CM{{\mathcal M}} \def\CN{{\mathcal N}'(\BBR^d),\,f\geq \1_K \}. \eal
If $\Gamma \subset \overline{\Omega}$ is a $C^2$ submanifold without boundary, of dimension $d$ with $1 \leq d \leq N-1$ then there exist open sets $O_1,...,O_m$ in $\BBR^N$, diffeomorphism $T_i: O_i \to B^{d}(0,1)\times B^{N-d}(0,1) $ and compact sets $K_1,...,K_m$ in $\Gamma$ such that
(i) $K_i \subset O_i$, $1 \leq i \leq m$ and $ \Gamma= \cup_{i=1}^m K_i$,
(ii) $T_i(O_i \cap \Gamma)=B_1^{d}(0) \times \{ x'' = 0_{\mathbb{R}^{N-d}} \}$, $T_i(O_i \cap \Omega} \def\Gx{\Xi} \def\Gy{\Psi)=B_1^{d}(0)\times B_1^{N-d}(0)$,
(iii) For any $x \in O_i \cap (\Omega\setminus \Gamma)$, there exists $y \in O_i \cap \Sigma$ such that $d_\Gamma(x)=|x-y|$ (here $d_\Gamma(x)$ denotes the distance from $x$ to $\Gamma$).
We then define the $\mathrm{Cap}_{\theta} \def\vge{\varepsilon,s}^{\Gamma}-$capacity of a compact set $E \subset \Gamma$ by \bel{Capsub} \mathrm{Cap}_{\theta} \def\vge{\varepsilon,s}^{\Gamma}(E):=\sum_{i=1}^m \mathrm{Cap}_{\CB_{d,\theta} \def\vge{\varepsilon},s}^{\mathbb{R}^d}(\tilde T_i(E \cap K_i)), \end{equation}\normalsize where $T_i(E \cap K_i)=\tilde T_i(E \cap K_i) \times \{ x'' = 0_{\mathbb{R}^{N-d}} \}$. We remark that the definition of the capacities does not depends on $O_i$.
Note that if $\theta s > d$ then \small\begin{equation} \label{CapGamma} \inf_{z \in \Gamma}\mathrm{Cap}_{\theta} \def\vge{\varepsilon,s}^{\Gamma}(\{z\})>C>0. \end{equation}\normalsize
By using the above capacities and Proposition \ref{t2.1}, we are able to prove Theorem \ref{subm}.
\begin{proof}[\textbf{Proof of Theorem \ref{subm}}.] First we note that \eqref{GN2a} holds and \ba \label{KN2a1} K_\mu(x,z)\approx d(x)d_{\Sigma}(x)^{-\am} N_{2\am}(x,z) \quad \forall x \in \Omega \setminus \Sigma, z \in \Sigma. \ea By using a similar argument as in the proof of Theorem \ref{theoremint}, together with \eqref{GN2a} and \eqref{KN2a1}, we deduce that equation $ v=\BBN_{2\am}[v^p (dd_{\Sigma}^{-\am})^{p+1}]+\ell\BBN_{2\am}[\nu] $ has a positive solution for $\ell>0$ small if and only if equation \eqref{u-sigmanu} has a positive solution $u$ for $\sigma$ small enough.
Therefore, as in the proof of Theorem \ref{theoremint}, in light of Lemmas \ref{ineq}, \ref{vol} and \ref{vol-2}, we may apply Proposition \ref{t2.1} with $J(x,y)=\CN_{2\am}(x,y)$, $\dd \omega= \left(d(x)d_{\Sigma}(x)^{-\am}\right)^{p+1}\dx$ and $\lambda=\nu$. Estimate \eqref{Jest} is satisfied thanks to Lemma \ref{ineq}, while assumptions \eqref{2.3}--\eqref{2.4} are fulfilled thanks to Lemmas \ref{vol}--\ref{vol-2} respectively with $b=p+1$ and $\theta=-\am(p+1)$. We note that condition $p<\frac{2+\ap}{\ap}$ ensures that $b$ and $\theta$ satisfy the assumptions in Lemmas \eqref{vol}--\eqref{vol-2}. Therefore, by employing Proposition \ref{t2.1}, we can show that statements 1--3 of Proposition \ref{t2.1} are equivalent to statements 1--3 of the present theorem respectively.
Next we will show that, under assumption \eqref{p-cond-2}, statement 4 of Proposition \ref{t2.1} is equivalent to statement 4 of the present theorem. More precisely, we show that for any compact subset $E \subset \Sigma$, there hold \ba \label{Cap-equi-1} \mathrm{Cap}_{\vartheta,p'}^{\Sigma}(E) \approx \mathrm{Cap}_{\BBN_{2\am},p'}^{p+1,-\am(p+1)}(E), \ea where $\vartheta$ is defined in \eqref{gamma}. From \eqref{Capsub}, we see that \bal \mathrm{Cap}_{\vartheta,p'}^{\Sigma}(E):=\sum_{i=1}^m \mathrm{Cap}_{\CB_{k,\vartheta},p'}^{\mathbb{R}^k}(\tilde T_i(E \cap K_i)), \eal where $T_i(E \cap K_i)=\tilde T_i(E \cap K_i) \times \{ x'' = 0_{\mathbb{R}^{N-k}} \}$. Also, \bal \mathrm{Cap}_{\BBN_{2\am},p'}^{p+1,-\am(p+1)}(E)\approx\sum_{i=1}^m\mathrm{Cap}_{\BBN_{2\am},p'}^{p+1,-\am(p+1)}(E\cap K_i). \eal Therefore, in order to prove \eqref{Cap-equi-1}, it's enough to show that \ba \label{Cap-split} \mathrm{Cap}_{\CB_{k,\vartheta},p'}^{\mathbb{R}^k}(\tilde T_i(E \cap K_i))\approx \mathrm{Cap}_{\BBN_{2\am},p'}^{p+1,-\am(p+1)}(E \cap K_i), \quad i=1,2,\ldots,m. \ea
Let $\lambda \in \GTM^+(\partial\Omega\cup \Sigma)$ with compact support in $\Sigma$ be such that $\BBK_\mu[\lambda]\in L^p(\Omega;{\phi_{\mu }})$. Put $\lambda_{K_i} = \1_{K_i}\lambda$. On one hand, from \eqref{eigenfunctionestimates}, \eqref{Martinest1} and since $p< \frac{2+\ap}{\ap} \leq \frac{N-k-\am}{\am}$, we have \bal \int_{O_i}\BBK_\mu[\lambda_{K_i}]^p{\phi_{\mu }} \,\dx\gtrsim \lambda(K_i)^p\int_{O_i}d(x)^{p+1}d_\Sigma(x)^{-(p+1)\am} \,\dx\gtrsim\lambda(K_i)^p. \eal On the other hand, \bal \int_{\Omega\setminus O_i}\BBK_\mu[\lambda_{K_i}]^p{\phi_{\mu }} \dx\lesssim \lambda(K_i)^p\int_{O_i}d(x)^{p+1}d_\Sigma(x)^{-(p+1)\am}\dx\lesssim \lambda(K_i)^p. \eal Combining the above estimate, we obtain \ba\label{45} \int_\Omega\BBK_\mu[\lambda_{K_i}]^p {\phi_{\mu }} \,\dx\approx \int_{O_i}\BBK_\mu[\lambda_{K_i}]^p{\phi_{\mu }} \,\dx, \quad \forall i=1,2,..,m. \ea
In view of the proof of \cite[Lemma 5.2.2]{Ad}, there exists a measure $\overline \lambda_i\in \GTM^+(\mathbb{R}^k)$ with compact support in $B^k(0,1)$ such that for any Borel $E\subset B^k(0,1)$, there holds
\bal \overline \lambda_i(E)=\lambda(T_i^{-1}(E\times \{0_{{\mathbb R}^{N-k}}\})). \eal Set $\psi=(\psi',\psi'')=T_i(x)$ then. By \eqref{propdist}, \eqref{eigenfunctionestimates} and \eqref{Martinest1}, we have
\bal
&{\phi_{\mu }}(x)\approx |\psi''|^{-\am},\\
&K_{\mu}(x,y)\approx |\psi''|^{-\am}(|\psi''|+|\psi'-y'|)^{-(N-2\am-2)}, \quad
\forall x\in O_i\setminus \Sigma,\;\forall y\in O_i\cap \Sigma.
\eal The above estimates, together with \eqref{45}, imply \ba \label{46} \begin{aligned}
&\int_{\Omega} \BBK_{\mu}[\lambda_{K_i}]^p{\phi_{\mu }} \,\dx \approx \int_{ O_i } \BBK_{\mu}[\lambda_{K_i}]^p{\phi_{\mu }} \,\dx\\
&\approx \int_{B^k(0,1)}\int_{B^{N-k}(0,1)}|\psi''|^{-(p+1)\am}
\left(\int_{B^k(0,1)}(|\psi''|+|\psi'-y'|)^{-(N-2\am-2)}\dd\overline{\lambda_i}(y')\right)^p \dd \psi'' \dd \psi'\\
&=C(N,k)\int_{B^k(0,1)}\int_{0}^{\beta_0}r^{N-k-1-(p+1)\am}
\left(\int_{B^k(0,1)}(r+|\psi'-y'|)^{-(N-2\am-2)}\dd\overline{\lambda_i}(y')\right)^p \dd r \dd\psi'.\\
&\approx \int_{\mathbb{R}^k} \mathbb{B}_{k,\vartheta}[\overline \lambda_i](x')^p\dx'.
\end{aligned}
\ea Here the last estimate is due to \cite[Lemma 8.1]{GkiNg_absorption} (note that \cite[Lemma 8.1]{GkiNg_absorption} holds under assumptions \eqref{p-cond-2}). Combining \eqref{KN2a} and \eqref{46} yields \bal
\| \BBN_{2\am}[\lambda_{K_i}] \|_{L^p(\Omega;d^{p+1} d_\Sigma^{(p+1)\am})} \approx \| \BBK_\mu[\lambda_{K_i}] \|_{L^p(\Omega;\phi_\mu)} \approx \| \BBB_{k,\vartheta}[\overline \lambda_i] \|_{L^p({\mathbb R}^k)}. \eal This and \eqref{dualcap} lead to \eqref{Cap-split}, which in turn implies \eqref{Cap-equi-1}. The proof is complete.
\end{proof}
\begin{remark} \label{existSigma} By \eqref{CapGamma}, if $p<\frac{N-\am}{N-2-\am}$ (equivalently $\vartheta p' > k$) then $\inf_{z \in \Sigma}\mathrm{Cap}_{\vartheta,p'}^{\Sigma}(\{z\}) >0$. Hence, under the assumption of Theorem \ref{subm}, statement 3 of Theorem \ref{subm} holds and therefore statement 1 also holds true. \end{remark}
\begin{remark} \label{nonexistSigma}
Assume $\mu < \left( \frac{N-2}{2} \right)^2$ and $p \geq \frac{N-\am}{N-\am-2}$. Then for any $z \in \Sigma$ and any $\sigma>0$, problem \eqref{u-sigmanu} with $\nu=\delta_z$ does not admit any positive weak solution. Indeed, suppose by contradiction that for some $z \in \Sigma$ and $\sigma>0$, there exists a positive solution $u \in L^p(\Omega;{\phi_{\mu }})$ of equation \eqref{u-sigmanu}. Without loss of generality, we can assume that $z =0 \in \Sigma$ and $\sigma=1$. From \eqref{u-sigmanu}, $u(x) \geq \BBK_{\mu}[\delta_0] (x)= K_\mu(x,0)$ for a.e. $x \in \Omega \setminus \Sigma$. Let $\CC$ be a cone of vertex $0$ such that $\CC \subset \Omega \setminus \Sigma$ and there exist $r>0$, $0<\ell<1$ satisfying for any $x \in \CC$, $|x|<r$ and $d_\Sigma(x)> \ell |x|$. Then, by \eqref{Martinest1} and \eqref{eigenfunctionestimates}, \bal \int_{\Omega \setminus \Sigma} u(x)^p {\phi_{\mu }}(x) \dd x \geq \int_{\CC} K_\mu(x,0)^p {\phi_{\mu }}(x) \dd x
\geq \int_{\CC} |x|^{-\am-(N-\am-2)p} \dd x
\approx \int_0^r t^{N-1-\am - (N-\am-2)}\dd t. \eal Since $p \geq \frac{N-\am}{N-\am-2}$, the last integral is divergent, hence $u \not \in L^p(\Omega \setminus \Sigma;{\phi_{\mu }})$, which leads to a contradiction. \end{remark}
\begin{remark}\label{rem2}
Assume $\Sigma=\{0\}$ and $\mu= \left( \frac{N-2}{2} \right)^2$. If $p< \frac{2+\ap}{\ap}$ then there is a solution of \eqref{u-sigmanu} with $\nu=\sigma\delta_0$ for $\sigma>0$ small. Indeed, for any $1<p<\frac{2+\ap}{\ap},$ we have $0<\int_\Omega \BBK_\mu[\delta_0]^p {\phi_{\mu }} \, \dx<\infty$.
Therefore, by \eqref{dualcap}, we find
$ \mathrm{Cap}_{\BBN_{2\am},p'}^{p+1,-\am(p+1)}(\{0\})>0. $
In view of the proof of Theorem \ref{subm}, we may apply Proposition \ref{t2.1} for $J(x,y)=\CN_{2\am}(x,y),$ for $\dd \omega= \left(d(x)d_{\Sigma}(x)^{-\am}\right)^{p+1}\dx$ and $\lambda=\delta_0$ to obtain the desired result. \end{remark}
When $p \geq \frac{2+\ap}{\ap}$, the nonexistence occurs, as shown in the following remark.
\begin{remark} \label{rem3} If $p\geq \frac{2+\ap}{\ap}$ then, for any measure $\nu \in \GTM^+(\partial \Omega \cup \Sigma)$ with compact support in $\Sigma$ and any $\sigma>0$, there is no solution of problem \eqref{u-sigmanu}. Indeed, it can be proved by contradiction. Suppose that we can find $\sigma>0$ and a measure $\nu \in \GTM^+(\partial \Omega \cup \Sigma)$ with compact support in $\Sigma$ such that there exists a solution $0\leq u \in L^p(\Omega;{\phi_{\mu }})$ of \eqref{u-sigmanu}. It follows that $\BBK_\mu[\nu]\in L^p(\Omega;{\phi_{\mu }})$. On one hand, by \cite[Theorem 1.4]{GkiNg_absorption} there is a unique nontrivial nonnegative solution $v$ of \bal
- L_\mu} \def\gn{\nu} \def\gp{\pi v + |v|^{p-1}v =0\qquad \text{in }\;\Omega} \def\Gx{\Xi} \def\Gy{\Psi\setminus \Sigma,\quad \mathrm{tr}(v)=\nu. \eal
Moreover, $v \leq \BBK_{\mu}[\nu]$ in $\Omega \setminus \Sigma$. This, together with Proposition \ref{Martin} and the fact that $\nu$ has compact support in $\Sigma$, implies, for $x$ near $\partial \Omega$, $ v(x) \leq \BBK_{\mu}[\nu](x) \lesssim d(x)\nu(\Sigma)$. Therefore, by \cite[Theorem 1.8]{GkiNg_absorption}, we have that $v \equiv 0$, which leads to a contradiction.
\end{remark}
When $\nu$ concentrates on $\partial \Omega$, we also obtain criteria for the existence of problem \eqref{power}. We will treat the case $\mu < \left(\frac{N-2}{2}\right)^2$ and the case $\mu = \left(\frac{N-2}{2}\right)^2$ separably.
\begin{proof}[\textbf{Proof of Theorem \ref{th:existnu-prtO} when $\mu<\left(\frac{N-2}{2}\right)^2$}.] As in the proof of Theorem \ref{theoremint}, in light of Lemmas \ref{ineq}, \ref{vol} and \ref{vol-2}, we may apply Proposition \ref{t2.1} with $J(x,y)=\CN_{2\am}(x,y)$, $\dd\omega= \left(d(x)d_{\Sigma}(x)^{-\am}\right)^{p+1}\dx$ and $\lambda=\nu$ in order to show that statements 1--3 of Proposition \ref{t2.1} are equivalent to statements 1--3 of the present theorem respectively.
Next we will show that statement 4 of Proposition \ref{t2.1} is equivalent to statement 4 of the present theorem. More precisely, we will show that for any subset $E \subset \partial \Omega$, there holds \ba \label{Cap-equi-2} \mathrm{Cap}_{\frac{2}{p},p'}^{\partial \Omega}(E) \approx \mathrm{Cap}_{\BBN_{2\am},p'}^{p+1,-\am(p+1)}(E). \ea
Indeed, by a similar argument as in the proof of \eqref{45}, under the stated assumptions on $p$, we can show that for any $\lambda \in \GTM^+(\partial \Omega \cup \Sigma)$ with compact support in $\partial \Omega$, there holds \bal \int_\Omega\BBK_\mu^p[\lambda]{\phi_{\mu }} \dx\approx \sum_{i=1}^m \int_{O_i}\BBK_\mu^p[ \1_{K_i} \lambda]{\phi_{\mu }} \dx. \eal This and the estimate \ba \label{KN2a} K_\mu(x,z)\approx d(x)d_{\Sigma}(x)^{-\am} N_{2\am}(x,z) \quad \forall x \in \Omega \setminus \Sigma, z \in \partial \Omega, \ea imply \bal \begin{aligned} \int_{\Omega} \BBN_{2\am}[\lambda]^p d^{p+1} d_\Sigma^{-\am(p+1)}\dx & \approx \sum_{i=1}^m \int_{O_i} \BBN_{2\am}[\1_{K_i} \lambda]^p d^{p+1} d_\Sigma^{-\am(p+1)} \dx \\ & \approx \sum_{i=1}^m \int_{O_i} \BBN_{2\am}[\1_{K_i} \lambda]^p d^{p+1} \dx. \end{aligned} \eal Therefore, in view of the proof of \cite[Proposition 2.9]{BHV} (with $\alpha=\beta=2$, $s=p'$ and $\alpha_0=p+1$) and \eqref{dualcap}, we obtain \eqref{Cap-equi-2}. The proof is complete. \end{proof}
\begin{remark} \label{rem4} If $\am>0$ and $p\geq \frac{2+\am}{\am}$ then for any measure $\nu \in \GTM^+(\partial \Omega \cup \Sigma)$ with compact support in $\partial\Omega$ and any $\sigma>0$, there is no solution of \eqref{u-sigmanu}. Indeed, it can be proved by contradiction. Suppose that we can find a measure $\nu \in \GTM^+(\partial \Omega \cup \Sigma)$ with compact support in $\partial \Omega$ and $\sigma>0$ such that there exists a solution $0\leq u \in L^p(\Omega;{\phi_{\mu }})$ of \eqref{u-sigmanu}. Then by Theorem \ref{th:existnu-prtO}, estimate \eqref{GKp<K} holds for some constant $C>0$.
For simplicity, we assume that $0\in \Sigma$. Then, for $x$ near $0$, we have \ba \label{GKp>} \begin{aligned}
\int_{\Omega} G_\mu(x,y)\BBK_\mu[\nu](y)^p \dy&\gtrsim d_{\Sigma}(x)^{-\am}\nu(\partial \Omega)^p \int_{\Sigma_{\beta_0}} |y|^{-(p+1)\am}|x-y|^{-(N-2\am-2)}\dy\\ &\gtrsim \left\{\begin{aligned}
&d_\Sigma(x)^{-\am}|\ln|x|| \quad &&\text{if}\;p=\frac{2+\am}{\am}\\
&d_\Sigma(x)^{-\am}|x|^{2+\am-p\am} \quad &&\text{if}\;p>\frac{2+\am}{\am}. \end{aligned}\right.
\end{aligned} \ea From \eqref{GKp<K} and \eqref{GKp>}, we can reach at a contradiction by letting $|x|\to0$. \end{remark}
\subsection{The case $\Sigma=\{0\}$ and $\mu= H^2$} In this subsection we treat the case $\Sigma=\{0\}$ and $\mu= H^2$. Let us introduce some notations. Let $0<\varepsilon<N-2$, put \bal
\CN_{1,\varepsilon}(x,y):=\frac{\max\{|x-y|,|x|,|y|\}^{N-2}+|x-y|^{N-2-\varepsilon}}{|x-y|^{N-2}\max\{|x-y|,d(x),d(y)\}^2},\quad\forall(x,y)\in\overline{\Omega}\times\overline{\Omega}, x \neq y, \eal \bal
\CN_{N-2-\varepsilon}(x,y):=\frac{\max\{|x-y|,|x|,|y|\}^{N-2-\varepsilon}}{|x-y|^{N-2}\max\{|x-y|,d(x),d(y)\}^2},\quad\forall(x,y)\in\overline{\Omega}\times\overline{\Omega}, x \neq y, \eal \bal \begin{aligned}
G_{H^2,\varepsilon}(x,y) &:= |x-y|^{2-N} \left(1 \wedge \frac{d(x)d(y)}{|x-y|^2}\right) \left(1 \wedge \frac{|x||y|}{|x-y|^2} \right)^{-\frac{N-2}{2}} \\
&+(|x||y|)^{-\frac{N-2}{2}}|x-y|^{-\varepsilon} \left(1 \wedge \frac{d(x)d(y)}{|x-y|^2}\right), \quad x,y \in \Omega \setminus \{0\}, \, x \neq y, \end{aligned} \eal \ba \label{tilG}
\tilde G_{H^2,\varepsilon}(x,y):=d(x)d(y)(|x||y|)^{-\frac{N-2}{2}} \CN_{N-2-\varepsilon}(x,y), \quad \forall x,y \in \Omega \setminus \{0\}, \, x \neq y. \ea
Note that \bal \begin{aligned}
(|x||y|)^{-\frac{N-2}{2}}\left|\ln\left(1 \wedge \frac{|x-y|^2}{d(x)d(y)}\right)\right|
&\leq(|x||y|)^{-\frac{N-2}{2}}\left|\ln\frac{|x-y|}{\mathcal{D}_\Omega}\right|\left(1 \wedge \frac{d(x)d(y)}{|x-y|^2}\right) \\
&\leq C(\Omega,\varepsilon) (|x||y|)^{-\frac{N-2}{2}}|x-y|^{-\varepsilon} \left(1 \wedge \frac{d(x)d(y)}{|x-y|^2}\right), \end{aligned} \eal which together with \eqref{Greenestb}, implies \ba \label{GGe} G_{H^2}(x,y)\lesssim G_{H^2,\varepsilon}(x,y), \quad \forall x,y \in \Omega \setminus \{0\}, \, x \neq y. \ea
Next, from the estimates
\bal G_{H^2,\varepsilon}(x,y) \approx d(x)d(y)(|x||y|)^{-\frac{N-2}{2}} \CN_{1,\varepsilon}(x,y), \quad x,y \in \Omega \setminus \{0\}, \, x \neq y, \eal \bal \CN_{1,\varepsilon}(x,y)\leq C(\varepsilon,\Omega) \CN_{N-2-\varepsilon}(x,y),\quad x,y \in \Omega \setminus \{0\}, \, x \neq y, \eal we obtain \ba \label{GGe1} G_{H^2,\varepsilon}(x,y)\lesssim \tilde G_{H^2,\varepsilon}(x,y), \quad \forall x,y \in \Omega \setminus \{0\}, \, x \neq y. \ea
Set \bal \tilde\BBG_{H^2,\varepsilon}[\tau](x):=\int_{\Omega\setminus \Sigma} \tilde G_{H^2,\varepsilon}(x,y) \dd\tau(y), \\ \BBN_{N-2-\varepsilon}[\tau](x):=\int_{\Omega\setminus \Sigma} \CN_{N-2-\varepsilon}(x,y) \dd\tau(y). \eal Proceeding as in the proof of Theorem \ref{theoremint}, we obtain the following result
\begin{theorem}\label{theoremint2}
Let $0<\varepsilon<\min\{N-2,2\},$ $1<p<\frac{N+2-2\varepsilon}{N-2}$ and $\tau \in \GTM^+(\Omega \setminus \{0\}; \phi_{H^2})$. Then the following statements are equivalent.
1. The equation
\small\begin{equation} \label{eq:uH2e} u=\tilde\BBG_{H^2,\varepsilon}[u^p]+\rho\tilde\BBG_{H^2,\varepsilon}[\gt]
\end{equation}\normalsize
has a positive solution for $\rho>0$ small.
2. For any Borel set $E \subset \Omega\setminus \{0\}$, there holds
\bal \int_E \tilde\BBG_{H^2,\varepsilon}[\1_E\gt]^p \phi_{H^2}\, \dx \leq C\int_E\phi_{H^2}\dd\gt. \eal
3. The following inequality holds
\bal \tilde\BBG_{H^2,\varepsilon}[\tilde\BBG_{H^2,\varepsilon}[\gt]^p]\leq C\,\tilde\BBG_{H^2,\varepsilon}[\gt]<\infty\quad a.e. \eal
4. For any Borel set $E \subset \Omega\setminus \{0\}$ there holds \bal \int_E\phi_{H^2}\dd\gt\leq C\, \mathrm{Cap}_{\BBN_{N-2-\varepsilon},p'}^{1,-\frac{N-2}{2}(p+1)}(E). \eal \end{theorem}
\begin{theorem}\label{singl} We assume that at least one of the statements 1--4 of Theorem \ref{theoremint2} is valid. Then the equation \ba \label{eq:uH2} u=\BBG_{H^2}[u^p]+\rho\BBG_{H^2}[\gt] \ea has a positive solution for $\rho>0$ small. \end{theorem} \begin{proof}
From the assumption, by Theorem \ref{theoremint2}, there exists a solution $u$ equation \eqref{eq:uH2e} for $\rho>0$ small. By \eqref{GGe} and \eqref{GGe1}, we have
$u\gtrsim \BBG_{H^2}[u^p]+\rho\BBG_{H^2}[\gt]$. By \cite[Proposition 2.7]{BHV}, we deduce that equation \eqref{eq:uH2} has a solution for $\rho>0$ small. \end{proof}
\begin{theorem}\label{theeqint} Assume $\Sigma=\{0\},$ $\mu=\left( \frac{N-2}{2}\right)^2$ and $\tau \in \GTM^+(\Omega \setminus \{0\}; \phi_{\mu})$ has compact support in $\Omega\setminus \{0\}$. Then Theorem \ref{theoremint} is valid. \end{theorem} \begin{proof} Let $\varepsilon>0$ be small enough such that $1<p<\frac{N+2-2\varepsilon}{N-2}.$ Let $K=\mathrm{supp}\,(\tau)\Subset \Omega\setminus \{0\}$ and $\tilde \beta=\frac{1}{2}\mathrm{dist}(K,\partial\Omega\cup \{0\})>0$. By \eqref{Greenestb}, \eqref{GGe} and \eqref{GGe1}, we can show that \ba\label{62} \BBG_{H^2}[\1_E \tau] \approx \BBG_{H^2,\varepsilon}[\1_E \tau] \approx \tilde \BBG_{H^2,\varepsilon}[\1_E \tau] \quad\text{and}\quad \BBN_{N-2}[\tilde\tau]\approx \BBN_{N-2-\varepsilon}[\tilde\tau] \quad \text{in } \Omega \setminus \{0\}, \ea for all Borel $E\subset \Omega\setminus \{0\}$ and $\tilde\tau \in \GTM^+(\Omega \setminus \{0\}; \phi_{H^2})$ with $\mathrm{supp}\,(\tilde\tau)\subset K$. The implicit constants in the above estimates depend only on $N,\Omega,\tilde \beta, \varepsilon$. Hence, statements 2,4 of Theorem \ref{theoremint2} are equivalent with respective statements 2,4 (with $\am=\frac{N-2}{2}$) of Theorem \ref{theoremint}.
By Proposition \ref{equivint}, it is enough to show that statement 3 of Theorem \ref{theoremint2} is equivalent with statement 3 of Theorem \ref{theoremint}. By \eqref{62}, it is enough to prove that \ba\label{63} \tilde\BBG_{H^2,\varepsilon}[\tilde\BBG_{H^2,\varepsilon}[\gt]^p]\approx \BBG_{H^2}[\BBG_{H^2}[\gt]^p] \quad \text{in } \Omega \setminus \{0\}. \ea By \eqref{GGe} and \eqref{GGe1}, it is sufficient to show that
\ba\label{64} \tilde\BBG_{H^2,\varepsilon}[\tilde\BBG_{H^2,\varepsilon}[\gt]^p]\lesssim \BBG_{H^2}[\BBG_{H^2}[\gt]^p] \quad \text{in } \Omega \setminus \{0\}. \ea
Indeed, on one hand, since $1<p<\frac{N+2-2\varepsilon}{N-2}$, we have, for any $x\in \Omega\setminus \{0\}$, \ba\label{65}\begin{aligned}
&\int_{B(0,\frac{\tilde \beta}{4})}\tilde G_{H^2,\varepsilon}(x,y)\tilde\BBG_{H^2,\varepsilon}[\gt](y)^p \dd y\approx \tau(K)^p \int_{B(0,\frac{\tilde \beta}{4})} \tilde G_{H^2,\varepsilon}(x,y)|y|^{-\frac{p(N-2)}{2}} \dd y\\
&\lesssim \tau(K)^p d(x)|x|^{-\frac{N-2}{2}}\int_{B(0,\frac{\tilde \beta}{4})} |x-y|^{-\varepsilon}|y|^{-\frac{(p+1)(N-2)}{2}} \dd y\\
& \quad +\tau(K)^p d(x)|x|^{-\frac{N-2}{2}}\int_{B(0,\frac{\tilde \beta}{4})} |x-y|^{-N+2}|y|^{N-2-\varepsilon-\frac{(p+1)(N-2)}{2}} \dd y \\
& \lesssim\tau(K)^p d(x)|x|^{-\frac{N-2}{2}}. \end{aligned} \ea The implicit constants in the above inequalities depend only on $\Omega,K,\tilde \beta,p,\varepsilon$.
On the other hand, we have \ba\label{66}\begin{aligned}
\int_{B(0,\frac{\tilde \beta}{4})} G_{H^2}(x,y)\BBG_{H^2}[\gt](y)^p \dd y&\gtrsim\tau(K)^p d(x)|x|^{-\frac{N-2}{2}}\int_{B(0,\frac{\tilde \beta}{4})} |y|^{-\frac{(p+1)(N-2)}{2}} \dd y \\
&\gtrsim\tau(K)^p d(x)|x|^{-\frac{N-2}{2}}, \end{aligned} \ea where the implicit constants in the above inequalities depend only on $\Omega,K,\tilde \beta,p$. Hence by \eqref{65} and \eqref{66}, we have that \ba\label{67} \int_{B(0,\frac{\tilde \beta}{4})} \tilde G_{H^2,\varepsilon}(x,y)\tilde\BBG_{H^2,\varepsilon}[\gt](y)^p \dd y\lesssim\int_{B(0,\frac{\tilde \beta}{4})} G_{H^2}(x,y)\BBG_{H^2}[\gt](y)^p \dd y \quad\forall x\in \Omega\setminus \{0\}. \ea
Next, by \eqref{Greenestb} and \eqref{tilG}, we have, for $x \in \Omega \setminus \{0\}$ and $y \in \Omega \setminus B(0,\frac{\tilde \beta}{4})$, \bal
\tilde G_{H^2,\varepsilon}(x,y) \approx d(x)d(y)(|x||y|)^{-\frac{N-2}{2}} \CN_{N-2}(x,y) \lesssim G_{H^2}(x,y). \eal This and \eqref{62} yield \ba\label{68} \int_{\Omega\setminus B(0,\frac{\tilde \beta}{4})} \tilde G_{H^2,\varepsilon}(x,y)\tilde\BBG_{H^2,\varepsilon}[\gt](y)^p \, \dd y\lesssim\int_{\Omega\setminus B(0,\frac{\tilde \beta}{4})} G_{H^2}(x,y)\BBG_{H^2}[\gt](y)^p \, \dd y \quad\forall x\in \Omega\setminus \{0\}. \ea
Combining \eqref{67} and \eqref{68}, we deduce \eqref{64}. The proof is complete. \end{proof}
\begin{proof}[\textbf{Proof of Theorem \ref{th:existnu-prtO} when $\Sigma=\{0\}$ and $\mu=\frac{(N-2)^2}{4}$}.] Proceeding as in the proof of Theorem \ref{theeqint}, we obtain the desired result. \end{proof}
\begin{remark} \label{partO-N}
If $p<\frac{N+1}{N-1}$, by using a \eqref{CapGamma}, we obtain that $\inf_{z \in \partial \Omega}\mathrm{Cap}_{\frac{2}{p},p'}^{\partial \Omega}(\{z\})>C>0$, hence statement 3 of Theorem \ref{th:existnu-prtO} holds true. Consequently, under the assumptions of Theorem \ref{th:existnu-prtO}, equation \eqref{u-sigmanu} has a positive solution for $\sigma} \def\vgs{\varsigma} \def\gt{\tau>0$ small. When $p \geq \frac{N+1}{N-1}$, by using a similar argument as in Remark \ref{nonexistSigma}, we can show that for any $\sigma>0$ and $z \in \partial \Omega$, equation \eqref{u-sigmanu} does not admit any positive weak solution. \end{remark}
\begin{comment} \appendix \section{Some estimates} \setcounter{equation}{0} \begin{lemma} \label{int\dSigma} Assume $0<\alpha<N-k$.
(i) There exists a positive constant $C=C(N,\Omega,\Sigma,\alpha)$ such that for any $\xi \in \Sigma$ and $\beta \in (0,\beta_0)$, we have \ba \label{int\dSigma} \int_{V(\xi,\beta)} d_\Sigma(x)^{-\alpha}\dx < C\beta^{N-\alpha}. \ea
(ii) There exist $\Lambda$ and $C$ depending on $N,\Omega,\Sigma$ such that for any $\beta \in (0,\frac{\beta_0}{16(1+\Lambda)})$, there holds \ba \label{int\dSigma-1} \int_{\Sigma_\beta} d_\Sigma(x)^{-\alpha}\dx < C\beta^{N-\alpha}. \ea \end{lemma} \begin{proof} (i) Let $\xi \in \Sigma$ and $\beta \in (0,\beta_0)$. For $x=(x',x'') \in V(\xi,\beta)$, set $z=(z',z'')$ with $z'=x'$ and $z_i=x_i-\Gamma_i^\xi(z')$ for $i=k+1,...,N$. Denote by $B^k(z',\rho)$ (resp. $B^{N-k}(z'',\beta)$) the ball with center at $z'$ (resp. $z''$) and radius $\beta$ in ${\mathbb R}^k$ (resp. ${\mathbb R}^{N-k}$). By \eqref{propdist} and since $\alpha<N-k$, we have
\ba \label{inVbeta1} \begin{aligned}
\int_{V(\xi,\beta)} d_\Sigma(x)^{-\alpha}\dx &\leq C\int_{V(\xi,\beta)} \delta_\Sigma(x)^{-\alpha}\dx \\
& \leq C\int_{B^{k}(0,\beta)}\dz' \int_{B^{N-k}(0,\beta)}|z''|^{-\alpha}\dz'' \leq C\beta^{N-\alpha}.
\end{aligned} \ea
(ii) First we note that there exists $\Lambda=\Lambda(N,\Omega,\Sigma)>0$ such that for any $\beta \in (0,\frac{\beta_0}{1+\Lambda})$, one can find $m=m(N,k) \in {\mathbb N}$ and points $\xi^j \in \Sigma$, $j=1,\ldots,m$ satisfying $$ \Sigma_{\beta} \in \cup_{j=1}^m V(\xi^j,\Lambda \beta). $$ Since $\beta < \frac{\beta_0}{1+\Lambda}$, we have $\Lambda \beta < \beta_0$. Hence by applying (i) with $\xi=\xi^j$ and $\beta$ replaced by $\Lambda \beta$, we derive $$ \int_{V(\xi^j,\Lambda\beta)} d_\Sigma(x)^{-\alpha}\dx < C\Lambda^{N-\alpha}\beta^{N-\alpha}. $$ Consequently, $$ \int_{\Sigma_\beta} d_\Sigma(x)^{-\alpha}\dx \leq \sum_{j=1}^m \int_{ V(\xi^j,\Lambda \beta)}d_\Sigma(x)^{-\alpha}\dx \leq Cm\Lambda^{N-\alpha}\beta^{N-\alpha}. $$ The proof is complete. \end{proof} \begin{lemma} \label{lem:estxy} Assume $\alpha_1,\alpha_2 \in (0,N)$ and $0 \in \Omega$. Put $r=\frac{1}{8}\min\{1,d(0)\}$. There exists a constant $C=C(N,\Omega, \alpha,\gamma)$ such that, for any $x \in \Omega \setminus \{ 0 \}$, \ba \label{estxy}
\int_{\Omega}|x-y|^{-\alpha_1}|y|^{-\alpha_2} dy \leq C \left\{ \begin{aligned} &1 \quad &&\text{if } \alpha_1+\alpha_2<N, \\
&|\ln|x|| \quad &&\text{if } \alpha_1+\alpha_2=N, \\
&|x|^{-(\alpha_1+\alpha_2-N)} \quad &&\text{if } \alpha_1+\alpha_2>N. \end{aligned} \right. \ea \end{lemma} \begin{proof}
1. Let $x \in B(0,r) \setminus \{ 0\}$ and put $R=4\sup_{x \in \Omega}|x|$. By making change of variable ${\bf e}_x = \frac{x}{|x|}$ and $z=\frac{y}{|x|}$, we have \ba \label{estxy-1} \begin{aligned}
I&:=\int_{\Omega}|x-y|^{-\alpha_1}|y|^{-\alpha_2} dy \\
&=|x|^{N-(\alpha_1+\alpha_2)}\int_{B(0,\frac{R}{|x|})}|{\bf e}_x -z|^{-\alpha_1}|z|^{-\alpha_2}\dz \\
&=|x|^{N-(\alpha_1+\alpha_2)}\int_{B(0,\frac{1}{2}) }|{\bf e}_x -z|^{-\alpha_1}|z|^{-\alpha_2}\dz \\
&+ |x|^{N-(\alpha_1+\alpha_2)}\int_{B(0,2) \setminus B(0,\frac{1}{2})}|{\bf e}_x -z|^{-\alpha_1}|z|^{-\alpha_2}\dz \\
&+ |x|^{N-(\alpha_1+\alpha_2)}\int_{B(0,\frac{R}{|x|}) \setminus B(0,2)}|{\bf e}_x -z|^{-\alpha_1}|z|^{-\alpha_2}\dz \\ &=:I_1 + I_2 + I_3.
\end{aligned} \ea If $z \in B(0,\frac{1}{2})$ then $|{\bf e}_x-z| \geq \frac{1}{2}$, hence
\ba \label{estxy-2} I_1 \leq 2^{\alpha_1} |x|^{N-(\alpha_1+\alpha_2)}\int_{B(0,\frac{1}{2})}|z|^{-\alpha_2}\dz \leq C(N,\alpha,\gamma)|x|^{N-(\alpha_1+\alpha_2)}.
\ea If $z \in B(0,2) \setminus B(0,\frac{1}{2})$ then $|{\bf e}_x-z| \leq 3$. Hence
\ba \label{estxy-3} I_2 \leq 2^{\alpha_2}|x|^{N-(\alpha_1+\alpha_2)}\int_{B({\bf e}_x,3)}|{\bf e}_x-z|^{-\alpha_1}\dz \leq C(N,\alpha_1,\alpha_2)|x|^{N-(\alpha_1+\alpha_2)}.
\ea If $z \in B(0,\frac{R}{|x|}) \setminus B(0,2)$ then $|{\bf e}_x - z| \geq \frac{1}{2}|z|$. Hence
\ba \label{estxy-4} \begin{aligned} I_3 &\leq 2^{\alpha_1} |x|^{N-(\alpha_1+\alpha_2)} \int_{B(0,\frac{R}{|x|}) \setminus B(0,2)} |z|^{-(\alpha_1+\alpha_2)}\dz \\ &\leq C\left\{ \begin{aligned} &1 \quad &&\text{if } \alpha_1 + \alpha_2 <N, \\
&|\ln|x|| \quad &&\text{if } \alpha_1 + \alpha_2 = N, \\
&|x|^{N-(\alpha_1+\alpha_2)} &&\text{if } \alpha_1 + \alpha_2 > N. \end{aligned} \right. \end{aligned} \ea Combining \eqref{estxy-1}--\eqref{estxy-4} yields \eqref{estxy}.
2. Let $x \in \Omega \setminus B(0,r)$, then \ba \label{estxy-5} \begin{aligned}
\int_{\Omega}|x-y|^{-\alpha_1}|y|^{-\alpha_2} dy &= \int_{B(0,\frac{r}{2})}|x-y|^{-\alpha_1}|y|^{-\alpha_2} dy + \int_{\Omega \setminus B(0,\frac{r}{2})}|x-y|^{-\alpha_1}|y|^{-\alpha_2} dy \\
&\leq \left(\frac{2}{r}\right)^{\alpha_1} \int_{B(0,\frac{r}{2})}|y|^{-\alpha_2} dy + \left(\frac{2}{r}\right)^{\alpha_2} \int_{\Omega}|x-y|^{-\alpha_2} dy \\ &\leq C(N,\Omega,\alpha_1,\alpha_2). \end{aligned} \ea This implies \eqref{estxy}. The proof is complete. \end{proof} We first recall the following estimate in \cite[Lemma 5.5]{GkiNg_5}. \begin{lemma} \label{anisotita} Assume $0<\alpha_1<N$, $0<\alpha_2<N-k$ and $\alpha_1 + \alpha_2<N$. Then
\ba \label{eq:ani}
\sup_{x \in \overline \Omega}\int_{\Omega} |x-y|^{-\alpha_1}d_\Sigma(y)^{-\alpha_2}dy \lesssim 1.
\ea
The implicit constant in \eqref{eq:ani} depends on $N,\Omega,\Sigma,\alpha$ and $\gamma$. \end{lemma}
\begin{lemma}\label{anisotita-2} Assume $0\in \Sigma$. Let $\alpha_i$, $i=1,2,3$, satisfy $$ 0<\alpha_1,\alpha_2,\alpha_3<N, \quad \alpha_3<N-k, \quad \alpha_1+\alpha_3<N, \quad \alpha_2+\alpha_3<N. $$ Then there exists a positive constant $C=C(N,\Omega,\Sigma,\alpha_1,\alpha_2,\alpha_3)$ such that for any $x \in V(0,\frac{\beta_0}{16})$, \ba \label{abc-2} \begin{aligned}
\int_{\Omega}|x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy \leq C\left\{\begin{aligned} &1 \quad&&\text{if } \alpha_1 + \alpha_2 + \alpha_3 < N, \\
&|\ln|x|| \quad&&\text{if} \; \alpha_1 + \alpha_2 + \alpha_3 = N, \\
&|x|^{N-\alpha_1-\alpha_2-\alpha_3} \quad&&\text{if} \; \alpha_1 + \alpha_2 + \alpha_3 > N. \end{aligned}\right. \end{aligned} \ea \end{lemma} \begin{proof} 1. Let $x \in \Omega \setminus V(0,\frac{\beta_0}{16})$. We split \ba\nonumber
\int_{\Omega}|x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy&= \int_{\Omega\setminus V(0,\frac{\beta_0}{32})}|x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy\\ \label{33-0}
&+\int_{ V(0,\frac{\beta_0}{32})}|x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy. \ea We have \ba \label{33-1} \begin{aligned}
\int_{\Omega\setminus V(0,\frac{\beta_0}{32})}|x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy &\leq C(\Sigma,\alpha_2,\alpha_3)\int_{\Omega\setminus V(0,\frac{\beta_0}{32})}|x-y|^{-\alpha_1}dy \\ &\leq C(N,\Sigma,\Omega,\alpha_1,\alpha_2,\alpha_3) \end{aligned} \ea and by Lemma \ref{anisotita}, \ba \label{33-2} \begin{aligned}
\int_{ V(0,\frac{\beta_0}{32})}|x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy &\leq C(\Sigma,N,\alpha_1) \int_{ V(0,\frac{\beta_0}{32})}|y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy \\ &\leq C(N,\Omega, \Sigma, \alpha_1,\alpha_2,\alpha_3). \end{aligned} \ea Combining estimates \eqref{33-1} and \eqref{33-2}, we deduce that there exists a positive constant $C=C(N,\Omega,\Sigma,\alpha_1,\alpha_2,\alpha_3)$ such that for any $x \in \Omega \setminus V(0,\frac{\beta_0}{16})$, \ba \label{abc-1}
\int_{\Omega}|x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy \leq C. \ea
2. Let $x\in V(0,\frac{\beta_0}{16})$. We write \ba\nonumber
\int_{\Omega}|x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy &\leq \int_{\Omega\setminus V(0,\frac{\beta_0}{16})}|x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy\\ \label{33}
&+\int_{ V(0,\frac{\beta_0}{16})}|x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy. \ea
The first term on the right hand side of \eqref{33} can be estimated by using Lemma \ref{anisotita} \ba \nonumber
\int_{\Omega\setminus V(0,\frac{\beta_0}{16})}|x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy
&\leq C(\beta_0,\alpha_2) \int_{\Omega\setminus V(0,\frac{\beta_0}{16})}|x-y|^{-\alpha_1} d_\Sigma(y)^{-\alpha_3}dy \\ \label{34} &<C(N,\Omega,\Sigma,\alpha_1,\alpha_2,\alpha_3). \ea
Next we estimate the second term on the right hand side of \eqref{33}. We consider the following cases.
\textbf{Case 1:} $|x-y|\leq \frac{1}{2}|x|$ and $\alpha_1 \neq k$. Then $|y|\geq |x| - |x-y| \geq \frac{1}{2}|x|$ and therefore \bal
&\int_{V(0,\frac{\beta_0}{16})\cap\{|x-y|\leq \frac{1}{2}|x|\}}|x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy\\
&\leq 2^{\alpha_2}|x|^{-\alpha_2}\int_{V(0,\frac{\beta_0}{16})\cap\{|x-y|\leq \frac{1}{2}|x|\}}|x-y|^{-\alpha_1}d_\Sigma(y)^{-\alpha_3}dy. \eal
For $x=(x',x''), y=(y',y'') \in {\mathbb R}^k \times {\mathbb R}^{N-k}$, we set $\overline{x}_i=x_i-\Gamma_i(x')$ and $\overline{y}_i=y_i-\Gamma_i(y')$ for $i=k+1,...,N$. Let $\bar x=(\bar x_{k+1},..., \bar x_N) \in {\mathbb R}^k$ and $\bar y=(\bar y_{k+1},..., \bar y_N) \in {\mathbb R}^k$. Then \small\begin{equation}\label{27}
(|\overline{x}-\overline{y}|+ |x'-y'|)^{\alpha_1}\leq C_1(N,k,\alpha_1,\Gamma_{k+1},...,\Gamma_N)|x-y|^{\alpha_1}, \end{equation}\normalsize and by \eqref{propdist} \small\begin{equation}\label{28}
|\bar y| = \delta_\Sigma^{0}(y) \leq C \| \Sigma \|_{C^2} d_\Sigma(y),\quad \forall y\in V(0,2\beta_0). \end{equation}\normalsize This, \eqref{27} and the fact that $0 \in \partial \Sigma$ imply that
$$ |\bar y| \leq C(|x-y| + d_\Sigma(x)) \leq C|x|. $$
Thus, by \eqref{27}, \eqref{28} and the fact that $d_\Sigma(y)\leq |y|\leq \frac{3}{2}|x|,$ there exists a positive constant $C=C(\Omega,K,a,b)>1$ such that \bal
\int_{V(0,\frac{\beta_0}{16})\cap\{|x-y|\leq \frac{1}{2}|x|\}}&|x-y|^{-\alpha_1} d_\Sigma(y)^{-\alpha_3}dy\\
&\leq C\int_{\{|\overline{y}|+|\overline{x}-\overline{y}|\leq C|x|\}}|\overline{y}|^{-\alpha_3} \int_{|x'-y'|\leq C|x|}(|\overline{x}-\overline{y}|+ |x'-y'|)^{-\alpha_1}dy'd\overline{y}\\
&\leq C\int_{\{|\overline{y}|\leq C|x|\}}|\overline{y}|^{-\alpha_3} \int_{|x'-y'|\leq C|x|}|x'-y'|^{-\alpha_1}dy'd\overline{y}\\
&\leq C|x|^{N-k-\alpha_3}|x|^{k-\alpha_1} = C|x|^{N-\alpha_1-\alpha_3}. \eal Combining the above estimates, we have \small\begin{equation}
\int_{V(0,\frac{\beta_0}{16})\cap\{|x-y|\leq \frac{1}{2}|x|\}}|x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy \leq C|x|^{N-\alpha_1-\alpha_2-\alpha_3}\label{29}. \end{equation}\normalsize
\textbf{Case 2:} $|x-y|\leq \frac{1}{2}|x|$ and $\alpha_1= k.$ Then $|y|\geq\frac{1}{2}|x|$ and $|y|\geq |x-y|.$ Let $0<\varepsilon <\min\{\alpha_2,N-\alpha_1-\alpha_3 \}$. We have \bal
\int_{V(0,\frac{\beta_0}{16})\cap\{|x-y|\leq \frac{1}{2}|x|\}}&|x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy\\
&\leq 2^{\alpha_2-\varepsilon}|x|^{-\alpha_2+\varepsilon}\int_{V(0,\frac{\beta_0}{16})\cap\{|x-y|\leq \frac{1}{2}|x|\}}|x-y|^{-\alpha_1-\varepsilon}d_\Sigma(y)^{-\alpha_3}dy. \eal Proceeding as in the Case 1, we obtain \eqref{29}.
\textbf{Case 3:} $\frac{1}{2}|x|\leq|x-y|\leq 2|x|$ and $\alpha_2 \neq k.$ Then $d_\Sigma(y)\leq |y|\leq 3|x|$. It follows that \bal
\int_{V(0,\frac{\beta_0}{16})\cap\{\frac{1}{2}|x|\leq|x-y|\leq 2|x|\}}&|x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy\\
&\leq 2^{\alpha_1}|x|^{-\alpha_1}\int_{V(0,\frac{\beta_0}{16})\cap\{|y|\leq 3|x|\}}|y|^{-\alpha_2}d_\Sigma(y)^{-\alpha_3}dy. \eal Proceeding as in the Case 1, we obtain \small\begin{equation} \label{29b}
\int_{V(0,\frac{\beta_0}{16})\cap\{\frac{1}{2}|x|\leq|x-y|\leq 2|x|\}}\int_{\Omega}|x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy\leq C|x|^{N-\alpha_1-\alpha_2-\alpha_3}. \end{equation}\normalsize
\textbf{Case 4:} $\frac{1}{2}|x|\leq|x-y|\leq 2|x|$ and $\alpha_2= k.$ Then $d_\Sigma(y)\leq |y| \leq 3|x|$ and $|x-y|\geq \frac{1}{6}|y|.$ Let $0<\varepsilon <\min\{ \alpha_1, N-\alpha_2-\alpha_3\}$. We have \bal
\int_{V(0,\frac{\beta_0}{16})\cap\{\frac{1}{2}|x|\leq|x-y|\leq 2|x|\}}&|x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy\\
&\leq C(\varepsilon,N,\alpha_1)|x|^{-\alpha_1+\varepsilon}\int_{V(0,\frac{\beta_0}{16})\cap\{|y|\leq 3|x|\}}|y|^{-\alpha_2-\varepsilon}d_\Sigma(y)^{-\alpha_3}dy. \eal Proceeding as in the Case 3, we obtain \eqref{29b}.
\textbf{Case 5:} $2|x|\leq|x-y|$, $|x| < d_\Sigma(y)$ and $|x|<|y'|$. Then
\ba \label{xyx-y} |x| \leq |y|\leq \frac{3}{2}|x-y|. \ea
For $i=k+1,...,N$, we set $\overline{y}_i=y_i-\Gamma_i(y')$, $\overline{z}_i=\Gamma_i(y')$, $\overline{y}=(\overline{y_{k+1}},\ldots,\overline{y_N}) \in {\mathbb R}^{N-k}$, $\overline{z}=(\overline{z_{k+1}},\ldots,\overline{z_N}) \in {\mathbb R}^{N-k}$ and $z=(y',\overline{z}) \in V(0,\frac{\beta_0}{16}) \cap \Sigma$.
Since $0 \in \Sigma$, we have
\bal |z|^2 \leq |y'|^2 + |\overline{z}|^2 = |y'|^2 + \sum_{i=k+1}^N |\Gamma_i(y') - \Gamma_i(0_{{\mathbb R}^k})|^2 \leq C|y'|^2.
\eal
It follows that
\ba \label{z1} |y| \leq |y-z| + |z| \leq |\overline{y}| + C|y'| \leq C(|\overline{y}| + |y'|).
\ea
On the other hand, we see that
\ba \label{z2} |\overline{y}| \leq C(|y'|+\sum_{i=k+1}^N|\Gamma_i(y')|) = C(|y'|+\sum_{i=k+1}^N|\Gamma_i(y') - \Gamma_i(0_{{\mathbb R}^k})|) \leq C|y'|.
\ea
Combining \eqref{z1} and \eqref{z2}, we deduce that \small\begin{equation}\label{30}
C_1^{-1}|y|^{\alpha_1+\alpha_2} \leq (|\overline{y}|+ |y'|)^{\alpha_1+\alpha_2}\leq C_1|y|^{\alpha_1+\alpha_2}, \end{equation}\normalsize where $C_1$ depends only $N,\Sigma, \alpha_1,\alpha_2$. By \eqref{propdist} \small\begin{equation}\label{31}
d_{\Sigma}(y) \leq |\overline{y}| = \delta_\Sigma^{0}(y)=:\delta_\Sigma(y)\leq C \| \Sigma \|_{C^2} d_\Sigma(y),\quad \forall y\in V(0,2\beta_0). \end{equation}\normalsize Thus, by \eqref{xyx-y}, \eqref{30} and \eqref{31} there exists a positive constant $C=C(\Omega,\Sigma,\alpha_1,\alpha_3)>1$ such that \ba\nonumber
&\int_{V(0,\frac{\beta_0}{16})\cap\{2|x|\leq|x-y|\}\cap\{|x|\leq |y'|\}\cap\{|x|\leq d_\Sigma(y)\}} |x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy\\ \nonumber
&\leq C\int_{V(0,\frac{\beta_0}{16})\cap\{|x|\leq|y'|\}\cap\{|x|\leq d_\Sigma(y)\} }|y|^{-\alpha_1-\alpha_2}|\overline{y}|^{-\alpha_3}dy \\ \label{32}
&\leq C\int_{\{C^{-1}|x|\leq |\overline{y}| \leq C\beta_0\}}|\overline{y}|^{-\alpha_3}\int_{\{|x|\leq|y'|\leq C\beta_0\}}(|\overline{y}|+ |y'|)^{-\alpha_1-\alpha_2}dy'd\overline{y}. \ea If $\alpha_1 + \alpha_2 + \alpha_3 \geq N$ then by the assumption, $\alpha_1+\alpha_2>k$. Hence, from \eqref{32}, we have \ba \label{32-1} \begin{aligned}
&\int_{V(0,\frac{\beta_0}{16})\cap\{2|x|\leq|x-y|\}\cap\{|x|\leq |y'|\}\cap\{|x|\leq d_\Sigma(y)\}}|x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy \\
&\leq C\int_{\{C^{-1}|x|\leq |\overline{y}| \leq C\beta_0\}}|\overline{y}|^{-\alpha_1-\alpha_2-\alpha_3+k}d\overline{y} \\ &\leq C\left\{\begin{aligned}
&|\ln|x||,\quad&&\text{if}\; \alpha_1+\alpha_2+\alpha_3=N\\
&|x|^{N-\alpha_1-\alpha_2-\alpha_3},\quad&&\text{if}\; \alpha_1+\alpha_2+\alpha_3>N. \end{aligned}\right. \end{aligned} \ea If $k<\alpha_1+\alpha_2<N-\alpha_3$ then the first estimate in \eqref{32-1} still holds, hence \ba \label{32-2}
\int_{V(0,\frac{\beta_0}{16})\cap\{2|x|\leq|x-y|\}\cap\{|x|\leq |y'|\}\cap\{|x|\leq d_\Sigma(y)\}}|x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy \leq C\beta_0^{N-\alpha_1-\alpha_2-\alpha_3}. \ea If $k=\alpha_1+\alpha_2<N-\alpha_3$, then we have \ba \label{32-3} \begin{aligned}
&\int_{V(0,\frac{\beta_0}{16})\cap\{2|x|\leq|x-y|\}\cap\{|x|\leq |y'|\}\cap\{|x|\leq d_\Sigma(y)\}}|x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy \\
&\leq C\int_{\{C^{-1}|x|\leq |\overline{y}| \leq C\beta_0\}}|\overline{y}|^{-\alpha_3}\ln\left( \frac{|\overline{y}|+C\beta_0}{|\overline{y}|+|x|} \right) d\overline{y} \\
&\leq C\int_{\{C^{-1}|x|\leq |\overline{y}| \leq C\beta_0\}}|\overline{y}|^{-\alpha_3}\ln\left( \frac{2C\beta_0}{|\overline{y}|}\right) d\overline{y} \\
&\leq C\int_{\{C^{-1}|x|\leq |\overline{y}| \leq C\beta_0\}}|\overline{y}|^{-\alpha_3-\varepsilon} d\overline{y} \\ &\leq C\beta_0^{N-k-\alpha_3-\varepsilon} \leq C, \end{aligned} \ea for some $0<\varepsilon<N-k-\alpha_3$. If $\alpha_1+\alpha_2<k<N-\alpha_3$ then from \eqref{32}, we have \ba \label{32-4} \begin{aligned}
&\int_{V(0,\frac{\beta_0}{16})\cap\{2|x|\leq|x-y|\}\cap\{|x|\leq |y'|\}\cap\{|x|\leq d_\Sigma(y)\}}|x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy \\
&\leq C\beta_0^{-\alpha_1-\alpha_2+k} C\int_{\{C^{-1}|x|\leq |\overline{y}| \leq C\beta_0\}}|\overline{y}|^{-\alpha_3}d\overline{y} \\ &\leq C\beta_0^{N-\alpha_1-\alpha_2-\alpha_3}. \end{aligned}\ea Thus, by combing the above estimates, we obtain \ba \label{32-5} \begin{aligned}
&\int_{V(0,\frac{\beta_0}{16})\cap\{2|x|\leq|x-y|\}\cap\{|x|\leq |y'|\}\cap\{|x|\leq d_\Sigma(y)\}}|x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy \\ &\leq C\left\{\begin{aligned} &1 \quad&&\text{if } \alpha_1+\alpha_2+\alpha_3<N, \\
&|\ln|x||,\quad&&\text{if} \;\alpha_1+\alpha_2+\alpha_3=N, \\
&|x|^{N-\alpha_1-\alpha_2-\alpha_3},\quad&&\text{if }\;\alpha_1+\alpha_2+\alpha_3>N. \end{aligned}\right. \end{aligned} \ea
\textbf{Case 6:} $2|x|\leq|x-y|$, $d_\Sigma(y)> |x|$ and $|y'|\leq|x|$. Then \eqref{xyx-y}, \eqref{z2}, \eqref{30} and \eqref{31} still hold. Consequently, $$
d_{\Sigma}(y) \approx |y'| \approx |\bar y| \approx |y| \approx |x|. $$ Then we obtain \bal
&\int_{V(0,\frac{\beta_0}{16})\cap \{2|x|\leq|x-y|\}\cap\{ |y'|\leq |x|\}\cap\{|x| < d_\Sigma(y)\}}|x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy\\
&\leq \int_{V(0,\frac{\beta_0}{16})\cap \{C^{-1}|x| \leq |y| \leq C|x|\} } |y|^{-\alpha_1-\alpha_2-\alpha_3} dy \\ &\leq C\left\{\begin{aligned} &1 \quad&&\text{if } \alpha_1+\alpha_2+\alpha_3<N, \\
&|\ln|x||,\quad&&\text{if} \;\alpha_1+\alpha_2+\alpha_3=N, \\
&|x|^{N-\alpha_1-\alpha_2-\alpha_3},\quad&&\text{if }\;\alpha_1+\alpha_2+\alpha_3>N. \end{aligned}\right. \eal
\textbf{Case 7:} $2|x|\leq|x-y|,$ $d_\Sigma(y)\leq |x|$ and $|y'|>|x|$. Then \eqref{xyx-y} and \eqref{30} hold. Therefore, we have \ba \nonumber
&\int_{V(0,\frac{\beta_0}{16})\cap\{2|x|\leq|x-y|\}\cap\{|x| < |y'|\}\cap\{d_\Sigma(y) \leq |x| \}}|x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy \\ \label{33}
&\leq \int_{\{ |\overline{y}|\leq C |x|\}}|\overline{y}|^{-\alpha_3}\int_{\{|x|\leq|y'|\leq C\beta_0\}}(|\overline{y}|+ |y'|)^{-\alpha_1-\alpha_2}dy'd\overline{y}. \ea If $N \leq \alpha_1+\alpha_2+\alpha_3$ then by the assumption, $\alpha_1+\alpha_2>k$. Hence, from \eqref{33}, we have \ba \nonumber
&\int_{V(0,\frac{\beta_0}{16})\cap\{2|x|\leq|x-y|\}\cap\{|x| < |y'|\}\cap\{d_\Sigma(y) \leq |x| \}}|x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy \\ \nonumber
&\leq |x|^{-\alpha_1-\alpha_2+k} \int_{\{ |\overline{y}|\leq C|x|\}} |\overline{y}|^{-\alpha_3}d\overline{y} \\ \label{33-1}
&\leq C|x|^{N-\alpha_1-\alpha_2-\alpha_3}. \ea If $k<\alpha_1+\alpha_2<N-\alpha_3$ then \eqref{33-1} remain true, hence
$$ \int_{V(0,\frac{\beta_0}{16})\cap\{2|x|\leq|x-y|\}\cap\{|x| < |y'|\}\cap\{d_\Sigma(y) \leq |x| \}}|x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy \leq C. $$ If $k=\alpha_1+\alpha_2<N-\alpha_3$ then from \eqref{33}, we have \ba \label{33-2} \begin{aligned}
&\int_{V(0,\frac{\beta_0}{16})\cap\{2|x|\leq|x-y|\}\cap\{|x| < |y'|\}\cap\{d_\Sigma(y) \leq |x| \}}|x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy \\
&\leq C\int_{\{ |\overline{y}|\leq C |x|\}}|\overline{y}|^{-\alpha_3}\ln\left( \frac{|\overline{y}|+C\beta_0}{|\overline{y}| + |x|} \right) d\overline{y} \\
&\leq C\int_{\{ |\overline{y}|\leq C |x|\}}|\overline{y}|^{-\alpha_3-\varepsilon} d\overline{y} \\
&\leq C|x|^{N-k-\alpha_3-\varepsilon} \leq C, \end{aligned} \ea for some $0<\varepsilon<N-k-\alpha_3$.
If $\alpha_1+\alpha_2<k<N-\alpha_3$ then from \eqref{33}, we have \ba \nonumber
&\int_{V(0,\frac{\beta_0}{16})\cap\{2|x|\leq|x-y|\}\cap\{|x| < |y'|\}\cap\{d_\Sigma(y) \leq |x| \}}|x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy \\ \nonumber
&\leq C\beta_0^{-\alpha_1-\alpha_2+k}\int_{\{ |\overline{y}|\leq C |x|\}}|\overline{y}|^{-\alpha_3}d\overline{y} \\ \label{33-3} &\leq C\beta_0^{N-\alpha_1-\alpha_2-\alpha_3}. \ea Thus, by combing the above estimates, we obtain \ba \label{32-5} \begin{aligned}
&\int_{V(0,\frac{\beta_0}{16})\cap\{2|x|\leq|x-y|\}\cap\{|x| < |y'|\}\cap\{d_\Sigma(y) \leq |x| \}}|x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy \\ &\leq C\left\{\begin{aligned} &1 \quad&&\text{if } \alpha_1+\alpha_2+\alpha_3<N, \\
&|\ln|x||,\quad&&\text{if} \;\alpha_1+\alpha_2+\alpha_3=N, \\
&|x|^{N-\alpha_1-\alpha_2-\alpha_3},\quad&&\text{if }\;\alpha_1+\alpha_2+\alpha_3>N. \end{aligned}\right. \end{aligned} \ea
\textbf{Case 8:} $2|x|\leq|x-y|,$ $d_\Sigma(y)\leq |x|$ and $|y'|\leq|x|$. Then \eqref{xyx-y}, \eqref{z2}, \eqref{30} and \eqref{31} hold. Then
$$ |x| \approx |y'| \approx |y| \leq \frac{3}{2}|x-y|. $$ Proceeding as the case 2 and 3 we obtain \bal
&\int_{V(0,\frac{\beta_0}{16}) \cap \{2|x|\leq|x-y|\}\cap\{|y'|\leq |x|\}\cap\{ d_\Sigma(y)\leq |x|\} } |x-y|^{-\alpha_1} |y|^{-\alpha_2} d_\Sigma(y)^{-\alpha_3}dy \\
&\leq C\int_{V(0,\frac{\beta_0}{16}) \cap \{2|x|\leq|x-y|\}\cap\{|y'|\leq |x|\}\cap\{ d_\Sigma(y)\leq |x|\} }|y|^{-\alpha_1-\alpha_2}d_\Sigma(y)^{-\alpha_3}dy \\
&\leq C\int_{ \{ |\overline{y}| \leq c|x| \} } |\overline{y}|^{-\alpha_3}d\overline{y} \int_{ \{ c^{-1}|x| \leq |y'| \leq c|x| \} } |x|^{-\alpha_1-\alpha_2}dy' \\
&= C|x|^{N-\alpha_1-\alpha_2-\alpha_3}. \eal
\textbf{End of the proof.} By combining estimate \eqref{34} and estimates in cases 1--8, we obtain \eqref{abc-2}. The proof is complete. \end{proof} \end{comment}
\appendix\section{Some estimates} \label{app:A} \setcounter{equation}{0} In this appendix, we give an estimate which is used several times in the paper. \begin{lemma} \label{lemapp:1} Assume $\ell_1>0$, $\ell_2>0$, $\alpha_1$ and $\alpha_2$ such that $N-k+\alpha_1 + k\alpha_2 >0$. For $y \in \Omega \setminus \Sigma$, put $
{\mathcal A}} \def\CB{{\mathcal B}} \def\CC{{\mathcal C}(y):= \{ x \in (\Omega \setminus \Sigma): d_{\Sigma}(x) \leq \ell_1 \quad \text{and} \quad |x-y| \leq \ell_2 d_{\Sigma}(x)^{\alpha_2} \}. $ Then \bal \int_{{\mathcal A}} \def\CB{{\mathcal B}} \def\CC{{\mathcal C}(y) \cap \Sigma_{\beta_1}} d_{\Sigma}(x)^{\alpha_1}\dx \lesssim \ell_1^{N-k+\alpha_1 + k\alpha_2}\ell_2^k. \eal \end{lemma} \begin{proof} By \eqref{cover}, we have \bal \int_{{\mathcal A}} \def\CB{{\mathcal B}} \def\CC{{\mathcal C}(y) \cap \Sigma_{\beta_1}} d_{\Sigma}(x)^{\alpha_1}\dx \leq \sum_{j=1}^{m_0}\int_{{\mathcal A}} \def\CB{{\mathcal B}} \def\CC{{\mathcal C}(y) \cap V(\xi^j,\beta_0)} d_{\Sigma}(x)^{\alpha_1}\dx. \eal For any $j \in \{1,...,m_0\}$, in view of \eqref{propdist}, we have \ba \label{app:3}
d_\Sigma(x) \leq \delta_\Sigma^{\xi^j}(x) \leq C \| \Sigma\|_{C^2} d_\Sigma(x) \quad \forall x \in V(\xi^j,\beta_0), \ea where \bal
\delta_\Sigma^{\xi^j}(x):=\sqrt{\sum_{i=k+1}^N|x_i-\Gamma_i^{\xi^j}(x')|^2}, \qquad x=(x',x'')\in V(\xi^j,\beta_0). \eal Therefore, by the change of variables $z'=x'-(\xi^j)'$ and $z''=(z_{k+1},\ldots,z_N)$ with $z_i=x_i-\Gamma_i^{\xi^j}(x')$, $i=k+1,..,N$, and \eqref{app:3}, we have \bal \begin{aligned}
\int_{ {\mathcal A}} \def\CB{{\mathcal B}} \def\CC{{\mathcal C}(y) \cap V(\xi^j,\beta_0) }d_\Sigma(x)^{\alpha_1}\dx &\lesssim \int_{ \{\delta_\Sigma^{\xi_j}(x) \leq c\ell_1, |x-y| \leq c\ell_2 \delta_\Sigma^{\xi_j}(x)^{\alpha_2} \} \cap V(\xi^j,\beta_1) }\delta_\Sigma^{\xi_j}(x)^{\alpha_1}\dx \\
&\lesssim \int_{ \{ |z''| \leq c\ell_1 \} } \int_{ \{ |z'|< c\ell_2 |z''|^{\alpha_2} \} } |z''|^{\alpha_1} \dz' \dz'' \lesssim \ell_1^{N-k+\alpha_1 + k\alpha_2} \ell_2^k. \end{aligned}
\eal The last estimate holds because $N-k+\alpha_1 + k\alpha_2>0$. The proof is complete. \end{proof}
\end{document} | arXiv |
A report on Non-Local Means Denoising
A description of the non-local means denoising algorithm
1 Non local means denoising uses samples from all around the image, instead of conventional denoising which will just look at the area around the given pixel to increase the accuracy of the colour. The reason it does this is due to the fact that patterns and shapes will be repeated in images, meaning that there will likely be an area somewhere else in the image that looks very similar to the patch around the pixel looking to be corrected. By finding these areas and taking averages of the pixels in similar areas, the noise will reduce as the random noise will converge around the true value.
So the method by which non-local means runs is to look at many patches throughout the image, and compare the similarities of those patches with the patch around the pixel looking to be denoised. This comparison then allows for assigning a weight to each patch looked at, which are then used (along with the colour of the pixel in the centre of the patch) in the calculation of the colour of the pixel to be denoised.
Various implementations of the algorithm and their efficiency
Pixelwise
Visualisation of pixelwise denoising
2Taking an image u and a pixel in it you want to denoise, p, you first need to decide a patch size, given by r, as the dimensions of the patch (blue) are $(2r+1)\times(2r+1)$. You then look at all the other pixels, $q\in Q$, but as it is intensive to do the calculations, specifying a research zone (green) allows you to make the processing faster as fewer comparisons have to be done. When looking at the other pixels, calculate their patch of the same size as the patch of p, then compare each pixel in the patch of q with the corresponding pixel in the patch of p. This similarity is then used to compute the similarity between the patch around p and the patch around q, and a weighting is given to q to describe this. These weightings are then averaged with the colours of the pixels to provide a more accurate representation of the pixel.
Patchwise
2 The main way in which patchwise differs from pixelwise is in the formulation of the weighting, as you can see below
$$C(p)=\sum_{q \in B(p, r)} w(p, q)$$
$$C=\sum_{Q=Q(q, f) \in B(p, r)} w(B, Q)$$
By calculating weights for pixels instead of patches we can make one calculation per patch, therefore not needing to do $(2f+1)^2$ calculations per pixel, providing a large increase in performance. The overall quality of the two methods are the same, and so the patchwise method is preferred as it has no drawbacks for an improvement in speed.
The strengths and limitations of non-local means compared to other denoising algorithms
Method noise
3Definition (method noise). Let u be a (not necessarily noisy) image and $D_h$ a denoising operator depending on h. Then we define the method noise of u as the image difference
$$n(D_h,u)=u-D_h(u)$$
This method noise should be as similar to white noise as possible. The image below is sourced from Buades, A., Coll, B., and Morel, J. 2005 3
From left to right and from top to bottom: original image,Gaussian convolution, mean curvature motion, total variation, Tadmor–Nezzar–Vese iterated total variation, Osher et al. total variation, neighborhood filter, soft TIWT, hard TIWT, DCT empirical Wiener filter, and the NL-means algorithm.
You can see that the NL means algorithm is closest to white noise, as it is very difficult to make out the original image from the method noise, and so is the best in this area
Mean square error
4The mean square error measures the average squared difference between the estimated values and what is estimated. In images this acts as a measure of how far from the true image the denoised image is. These results are taken from Buades, A., Coll, B., and Morel, J. 2005 3
Here it can be seen that the NL-means algorithm gives images that are closest to the true image, and so performs best for image denoising under this measurement.
The influence of the algorithmic parameters on the output
In the following images I am changing the values of h, the template window size and the search window size, from a standard set at h=5, template window size=7 and search window size =21. I will adjust each one in turn to show the differences yielded by changing them.
h=2
h=10
template width = 2
template width = 10
search window size = 10
By adjusting the value of h you get a large change in the amount of smoothing, although a large amount of noise is still present. Increasing the value of h does increase the PSNR from 28.60 to 29.66
The effects from adjusting the template width are much more subtle than that of adjusting h, it can be noticed in the wires overhead that a larger template width has reduced the detail. An increase in the template width yields a small reduction in the PSNR from 28.68 to 28.51.
The effects for the value of the search window are also very subtle, and again can only be noticed fully in the overhead wires. An increase in the search window yields a marginal increase in the PSNR from 28.51 to 28.52.
Modifications and extensions of the algorithm that have been proposed in the literature
Testing stationarity
3 One proposed modification is one to test stationarity. The original algorithm works under the conditional expectation process:
Theorem - Conditional Expectation Theorem
Let $Z_j=\{X_j,Y_j\}$ for $j=1,2,...$ be a strictly stationary and mixing process. For $i\in I$, let $X$ and $Y$ be distributed as $X_i$ and $Y_i$. Let J be a compact subset $J\subset \mathbb{R}^p$ such that
$$\inf\{f_X(x);x\in J\}>0$$
However this is not true everywhere, as each image may contain exceptional, non-repeated structures, these would be blurred out by the algorithm, so the algorithm should have a detection phase and special treatment of nonstationary points. In order to use this strategy a good estimate of the mean and variance at every pixel is needed, fortunately the non-local means algorithm converges to the conditional mean, and the variance can just be calculated using $EX^2-(EX)^2$
Multiscale version
Another improvement to make is one to speed up the algorithm, this is proposed using a multiscale algorithm.
Zoom out the image $u_0$ by a factor of 2. This gives the new image $u_1$
Apply the NL means algorithm to $u_1$, so that with each pixel of $u_1$, a list of windows centered in $(i_1,j_1)...(i_k,j_k)$ is associated
For each pixel of $u_0$, $(2i+r,2j+s)$ with $r,s\in \{0,1\}$, we apply the NL means algorithm. However instead of comparing with all the windows in the search zone, we just compare with the 9 neighbouring windows of each pixel
This procedure can be applied in a pyramid fashion
Applications of the original algorithm and its extensions
5 It has been proposed that Non-Local means can be used in X-Ray imaging, allowing for a reduction of noise in the scans, making them easier to interpret. In CT scans a higher dose can be given to give a clearer image, but with that is more dangerous, however by applying the NL means algorithm a lower dose can be given for the same clarity. It benefits from the improvement stated above to test stationarity as the the noise and streak artifacts are non stationary. The original algorithm was also not good at removing the streak artifacts in low-flux CT images resulting from photon starvation. However by applying one-dimensional nonlinear diffusion in the stationary wavelet domain before applying the non-local means algorithm these could be reduced.
Video Denoising
6 NLM can also be applied in video denoising, it has an adaptation as the denoising can be improved by using the data from sequential frames. In the implementation proposed in the paper, the current input frame and prior output frame are used to form the current output frame. In the paper the measurements they make fail to show that this algorithm is an improvement from current algorithms, however the algorithm does have much better subjective visual performance.
Buades, A., Coll, B., and Morel, J.M. 2005. A Non-Local Algorithm for Image Denoising. In 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05) (pp. 60–65). IEEE.↩
Buades, A., Coll, B., and Morel, J.M. 2011. Non-Local Means Denoising. Image Processing On Line, 1.↩
Buades, A., Coll, B., and Morel, J. 2005. A Review of Image Denoising Algorithms, with a New One. Multiscale Modeling & Simulation, 4(2), p.490–530.↩
Machine learning: an introduction to mean squared error and regression lines. URL https://www.freecodecamp.org/news/machine-learning-mean-squared-error-regression-line-c7dde9a26b93/.↩
Zhang, H., Zeng, D., Zhang, H., Wang, J., Liang, Z., and Ma, J. 2017. Applications of nonlocal means algorithm in low-dose X-ray CT image processing and reconstruction: A review. Medical Physics, 44(3), p.1168–1185.↩
Ali, R., and Hardie, R. 2017. Recursive non-local means filter for video denoising. EURASIP Journal on Image and Video Processing, 2017(1), p.29.↩ | CommonCrawl |
circumradius of isosceles triangle
Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Formula for a Triangle. Calculate the radius of the circumcircle of an isosceles triangle if given sides ( R ) : Radius of the circumscribed circle of an isosceles triangle : = Digit 2 1 2 4 6 10 F Isosceles triangle calculator is the best choice if you are looking for a quick solution to your geometry problems. Let and denote the triangle's three sides and let denote the area of the triangle. See circumcenter of a triangle for more about this. Pythagoras Theorem In the case of a right-angle triangle, the square of the Right Angled Triangle. Finally notice that the small acute isosceles triangle is similar to the original triangle (look at the angles). Finding the area of an isosceles triangle with inradius $\sqrt{3}$ and angle $120^\circ$. If a triangle has side lengths a, b, and c, then the circumradius has the following length: R = ( abc ) / √(( a + b + c )( b + c - a )( c + a - b )( a + b - c )) Now, back to the triangle towns. Making statements based on opinion; back them up with references or personal experience. Then, I am trying to use the following approach The Circumradius of a triangle given 3 sides formula is given by R = abc/sqrt((a+b+c)(-a+b+c)(a-b+c)(a+b-c)) where a, b, c are 3 sides of △ ABC and is represented as. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. @lab bhattacharjee though with your step I am getting correct answer but can you find error in my steps, The circumradius of an isosceles triangle ABC is four times as that of inradius and A=B condition, Finding the area of a triangle, given the distance between center of incircle and circumscribed circle. When choosing a cat, how to determine temperament and personality and decide on a good fit? Thanks for contributing an answer to Mathematics Stack Exchange! In this formula, Circumradius of Triangle uses Side A, Side B and Side C. We can use 1 other way(s) to calculate the same, which is/are as follows -, Circumradius of a triangle given 3 sides Calculator. find angles in isosceles triangles calculator. The center of the circumcircle is called the circumcenter, and the circle's radius is called the circumradius. To find the circumradius of any triangle with sides a,b,c the formula is abc/4A where A is the area of the triangle. It only takes a minute to sign up. Circumradius of equilateral triangle= side of triangle/√3 =12/√3 HOPE IT HELPS YOU!! Check out 15 similar triangle calculators , Isosceles triangle formulas for area and perimeter. We let , , , , and .We know that is a right angle because is the diameter. Base length is 153 cm. The circumradius of an isosceles triangle ABC is four times as that of inradius of the triangle, if A = B. In a right-angled isosceles triangle, the ratio of the circumradius and inradius is (b) (d) 1:44 45.0k LIKES. Use MathJax to format equations. STATEMENT 2: Every isosceles triangle is equilateral triangle. Acute isosceles triangle. 79.4k VIEWS. Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. Why do wet plates stick together with a relatively high force? The area of our triangle ABC is equal to 1/2 times r times the perimeter, which is kind of a neat result. 17.2k SHARES. The circumradius is the radius of the circle passing through all the three vertices of triangle. Isosceles right triangle Area of an isosceles right triangle is 18 dm 2. {\displaystyle {\frac {a^ {2}} {2h}}.} Hypothetically, why can't we wrap copper wires around car axles and turn them into electromagnets to help charge the batteries? EQL triangle Calculate inradius and circumradius of equilateral triangle with side a=77 cm. Was Terry Pratchett inspired by Hal Clement? 30 = (1/2)AB * altitude multiply through by 2. Isosceles triangle Calculate the size of the interior angles and the length of the base of the isosceles triangle if the length of the arm is 17 cm and the height to the base is 12 cm. × Close. Note that the center of the circle can be inside or outside of the triangle. Developer keeps underestimating tasks time. What is Circumradius of a triangle given 3 sides? For an isosceles triangle, along with two sides, two angles are also equal in measure. The circumcenter of a triangle is defined as the point where the perpendicular bisectors of the sides of that particular triangle intersect. Or sometimes you'll see it written like this. 38. The circumradius is defined as the radius of a circle that passes through all the vertices of a polygon, in this case, a triangle. The center of this circle is called the circumcenter and its radius is called the circumradius.. Not every polygon has a circumscribed circle. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. AB is a chord of the circumcircle. Circumradius of a triangle given 3 sides calculator uses. rev 2021.1.21.38376, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. Menu. Also, because they both subtend arc .Therefore, by AA similarity, so we have or However, remember that . The inradius of the triangle (a) 3.25 cm (b) 4 cm (c) 3.5 cm (d) 4.25 cm 1:04 2020In any equilateral , three circles of radii one are touching to the sides given as in the figure then area of the [IIT-2005] Download pdf. Did Gaiman and Pratchett troll an interviewer who thought they were religious fanatics? The formula for calculating an isosceles triangle is ½b×h, which means ½ × base of the triangle × height of … Anamika Mittal has verified this Calculator and 50+ more calculators! A sector of a circle has an arclength of 20cm. Asking for help, clarification, or responding to other answers. How many ways are there to calculate Circumradius of Triangle? Circumradius of a triangle given 3 sides calculator uses Circumradius of Triangle=(Side A*Side B*Side C)/sqrt((Side A+Side B+Side C)*(Side B-Side A+Side C)*(Side A-Side B+Side C)*(Side A+Side B-Side C)) to calculate the Circumradius of Triangle, The Circumradius of a triangle given 3 sides formula is given by R = abc/sqrt((a+b+c)(-a+b+c)(a-b+c)(a+b-c)) where a, b, c are 3 sides of △ ABC. number 1 is 5/3 because of this special formula. $A=rs$ ahre r=inradius and s= semiperimeter, $A=\frac{c}{2} \sqrt{(a^2-\frac{c^2}{4})}$, Still not able to get the answer , I presume that I am making a mistake, Using this Isosceles triangle The leg of the isosceles triangle is 5 dm, its height is 20 cm longer than the base. To use this online calculator for Circumradius of a triangle given 3 sides, enter Side A (a), Side B (b) and Side C (c) and hit the calculate button. Circumradius of Triangle is the radius of the circle inside which the triangle can be inscribed. 60 = 10 * altitude divide both sides by 10. How long is the leg of this triangle? The circumradius of an isosceles triangle is a 2 2 a 2 − b 2 4, where two sides are of length a and the third is of length b. To learn more, see our tips on writing great answers. 1/2 times the inradius times the perimeter of the triangle. The circumcircle of a triangle is a circle that passes through all of the triangle's vertices, and the circumradius of a triangle is the radius of the triangle's circumcircle. Download pdf. [IIT-1993] (A) /3 (B) (C) /2 (D) Q. 79.4k SHARES. $$r=R(\cos A+\cos B+\cos C-1)$$, $A=B\implies \cos B=\cos A,\cos C=\cos(\pi-A-A)=-\cos2A$, $\implies r=(4r)(\cos A+\cos A-\cos2A-1)$. If I'm the CEO and largest shareholder of a public company, would taking anything from my office be considered as a theft? Find out the isosceles triangle area, its perimeter, inradius, circumradius, heights and angles - all in one place. Guest Nov 13, 2017 Area = (b/4) √4a²- b², In an isosceles triangle, the angles opposite to the equal sides are equal. How does color identity work in Commander? Isosceles triangle The circumference of the isosceles triangle is 32.5 dm. The semiperimeter s, inradius r and circumradius R are the symmetric invariants of a triangle. Why don't video conferencing web applications ask permission for screen sharing? Circumradius of Triangle and is denoted by R symbol. Here is how the Circumradius of a triangle given 3 sides calculation can be explained with given input values -> 4.000638 = (8*7*4)/sqrt((8+7+4)*(7-8+4)*(8-7+4)*(8+7-4)). I think the factor 4 in the main formula is not correct. How did 耳 end up meaning edge/crust? MathJax reference. Different approaches give different results. Venkata Sai Prasanna Aradhyula has created this Calculator and 0+ more calculators! How to Calculate Circumradius of a triangle given 3 sides? The inradius of an isoceles triangle is Area of an isosceles triangle when length sides and angle between them are given calculator uses Area Of Triangle=(Side A*Side B*sin(Theta))/2 to calculate the Area Of Triangle, Area of an isosceles triangle when length sides and angle between them are given expresses the extent of an isosceles triangle in a plane. In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. Circumcircle is the circle that passes through all vertices (corner points) of a triangle. $R=\frac{abc}{4A}$, where R is Circum-radius and r is inradius and A is the area of inscribed triangle A general formula is volume = length * base_area; the one parameter you always need to have given is the prism length, and there are four ways to calculate the base - triangle area. site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. to give a non-trigonometric expression for the circumradius, but which is simpler than anything on your list. Side A is an upright or sloping surface of a structure or object that is not the top or bottom and generally not the front or back. By clicking "Post Your Answer", you agree to our terms of service, privacy policy and cookie policy. A polygon that does have one is called a cyclic polygon, or sometimes a concyclic polygon because its vertices are … Then, the measure of the circumradius of the triangle is simply .This can be rewritten as .. The area of an isosceles triangle is the amount of region enclosed by it in a two-dimensional space. 1. If a triangle has side lengths a, b, and c, then we can find the length of its circumradius using the following formula: R = ( abc ) / √(( a + b + c )( b + c - a )( c + a - b )( a + b - … The circumradius of an isosceles triangle ABC is four times as that of inradius of the triangle, if A = B. Proof. If r is the in-radius and R is the circumradius of the triangle ABC, then 2 (r + R) equals - [AIEEE-2005] the angle A . How to protect a secure compound breached by a small modern military? The radius of the circumscribed circle is: a 2 2 h . If the radius of thecircle is 12cm find the area of thesector: *(1 Point) Ans: An isosceles triangle can be defined as a special type of triangle whose at least 2 sides are equal in measure. Side C is an upright or sloping surface of a structure or object that is not the top or bottom and generally not the front or back. The formula for the circumradius is R=abc/4A, where a,b and c are the lengths of the sides and A is the area of the triangle. What is the Galois group of one ultrapower over another ultrapower? Triangle ABC has circumcenter O. Side B is an upright or sloping surface of a structure or object that is not the top or bottom and generally not the front or back. Do PhD admission committees prefer prospective professors over practitioners? Its center is at the point where all the perpendicular bisectors of the triangle's sides meet. An isosceles triangle has the largest possible inscribed circle among the triangles with the same base and apex angle, as well as also having the largest area and perimeter among the same class of triangles. 17.2k VIEWS. This center is called the circumcenter. Find the circumcentre and circumradius of the triangle whose vertices are . Isosceles Triangle. It's perpendicular to any … May I ask professors to reschedule two back to back night classes from 4:30PM to 9:00PM? 6:38 11.7k LIKES. 1 Verified Answer. The circumcircle always passes through all three vertices of a triangle. ABCD is a rectangle and M is the midpoint of CD.The inradii of triangles ADM and ABM are 3 and 4 respectively.Then find the area of the rectangle. Calculate the length of its base. It's equal to r times P over s-- sorry, P over 2. In this short note, we complement previous work of Hirakawa and Matsumura by determining all pairs (up to similitude) consisting of a rational right angled triangle and a rational isosceles triangle having two corresponding symmetric invariants equal. Dividing the first 10 primes into groups whose sum is prime. $P$ and $Q$ are two points on a circle of center $C$ and radius $a$, the angle $\widehat{PCQ}$ being $2\theta$, then the inradius of $PCQ$, Sides of a triangle given perimeter and two angles, In triangle $ABC$, find maximum value of $\sin A \cos B + \sin B \cos C + \sin C \cos A$. And we can find the altitude of triangle oAB thusly. New questions in Math. How to calculate Circumradius of a triangle given 3 sides? The circumradius of an equilateral triangle is 8 cm. Circumradius of a triangle given 3 sides calculator uses Circumradius of Triangle=(Side A*Side B*Side C)/sqrt((Side A+Side B+Side C)*(Side B-Side A+Side C)*(Side A-Side B+Side C)*(Side A+Side B-Side C)) to calculate the Circumradius of Triangle, The Circumradius of a triangle given 3 sides formula is given by R = abc/sqrt((a+b+c)(-a+b+c)(a-b+c)(a+b-c)) where a, b, c are 3 sides … View Answer. Mean Are creature environmental effects a bubble or column? パンの耳? Does it make sense to get a second mortgage on a second property for Buy to Let. The Circumradius of a triangle given 3 sides formula is given by R = abc/sqrt((a+b+c)(-a+b+c)(a-b+c)(a+b-c)) where a, b, c are 3 sides of △ ABC is calculated using. Inscription; About; FAQ; Contact Radius Of Inscribed Circle=sqrt((Semiperimeter Of Triangle -Side A)*(Semiperimeter Of Triangle -Side B)*(Semiperimeter Of Triangle -Side C)/Semiperimeter Of Triangle ), Area of Triangle when semiperimeter is given, Area Of Triangle=sqrt(Semiperimeter Of Triangle *(Semiperimeter Of Triangle -Side A)*(Semiperimeter Of Triangle -Side B)*(Semiperimeter Of Triangle -Side C)), Area=sqrt((Side A+Side B+Side C)*(Side B+Side C-Side A)*(Side A-Side B+Side C)*(Side A+Side B-Side C))/4, Radius Of Circumscribed Circle=(Side A*Side B*Side C)/(4*Area Of Triangle), Side A=sqrt((Side B)^2+(Side C)^2-2*Side B*Side C*cos(Angle A)), Perimeter=Side A+Side B+sqrt(Side A^2+Side B^2), Perimeter Of Triangle=Side A+Side B+Side C, Circumradius of a triangle given 3 exradii and inradius, Circumradius of Triangle=(Exradius of excircle opposite ∠A+Exradius of excircle opposite ∠B+Exradius of excircle opposite ∠C-Inradius of Triangle)/4, Inscribed angle of the circle when the central angle of the circle is given, Inscribed angle when other inscribed angle is given, Arc length of the circle when central angle and radius are given, Area of the sector when radius and central angle are given, Area of sector when radius and central angle are given, Length of the major axis of an ellipse (b>a), Eccentricity of an ellipse when linear eccentricity is given, Latus rectum of an ellipse when focal parameter is given, Linear eccentricity of ellipse when eccentricity and major axis are given, Linear eccentricity of an ellipse when eccentricity and semimajor axis are given, Semi-latus rectum of an ellipse when eccentricity is given, Length of conjugate axis of the hyperbola, Eccentricity of hyperbola when linear eccentricity is given, Number of diagonal of a regular polygon with given number of sides, Altitude/height of a triangle on side c given 3 sides, Length of median (on side c) of a triangle, Distance between circumcenter and incenter by Euler's theorem, Length of radius vector from center in given direction whose angle is theta in ellipse. 6 = the altitude. Comparing sides gives you a quadratic equation for the circumradius which is easily solved {Quadratic Formula!} How can I handle graphics or artworks with millions of points? How to calculate Circumradius of a triangle given 3 sides using this online calculator? If AB=10 and Area of OAB=30 find the circumradius of triangle ABC. 11 Other formulas that you can solve using the same Inputs, 1 Other formulas that calculate the same Output, Circumradius of a triangle given 3 sides Formula, Circumradius of Triangle=(Side A*Side B*Side C)/sqrt((Side A+Side B+Side C)*(Side B-Side A+Side C)*(Side A-Side B+Side C)*(Side A+Side B-Side C)). The inradius times the perimeter of the circle can be inscribed them into electromagnets to help charge batteries! Or artworks with millions of points perpendicular bisectors of the triangle, the ratio of the isosceles is... Circumradius.. 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\begin{document}
\begin{abstract}
We consider nonnegative solutions $u:\Omega\longrightarrow \mathbb{R}$ of second order hypoelliptic equations \begin{equation*} \mathscr{L} u(x) =\sum_{i,j=1}^n \partial_{x_i} \left(a_{ij}(x)\partial_{x_j} u(x) \right) + \sum_{i=1}^n b_i(x) \partial_{x_i} u(x) =0, \end{equation*} where $\Omega$ is a bounded open subset of $\mathbb{R}^{n}$ and $x$ denotes the point of $\Omega$. For any fixed $x_0 \in \Omega$, we prove a Harnack inequality of this type $$\sup_K u \le C_K u(x_0)\qquad \forall \ u \ \mbox{ s.t. } \ \mathscr{L} u=0, u\geq 0,$$ where $K$ is any compact subset of the interior of the $\mathscr{L}$-{\it propagation set of $x_0$} and the constant $C_K$ does not depend on $u$.
\end{abstract} \maketitle
\section{Introduction}
We consider second order partial differential operators $\elle$ acting on functions $u \in C^2 ( \Omega )$ as follows \begin{equation} \label{operatore}
\elle u(x) : = \sum_{i,j=1}^n \partiali \left(a_{ij}(x)\partialj u(x) \right) + \sum_{i=1}^n b_i(x) \partiali u(x) \end{equation} for $x$ belonging to any open {\it bounded} subset $\Omega$ of ${\erre^{ {n} }}$. The coefficients $a_{ij}, b_i$ are real functions and belong to $C^\infty(\overline \Omega)$ for $1 \le i,j \le n$. Moreover, $A := (a_{ij})$ is a $n \times n$ symmetric and non-negative matrix. We also assume the following hypotheses: \begin{itemize}
\item[(H1)] { \it $\elle - \beta$ and $\elle^*$ are hypoelliptic} for every constant $\beta \ge 0$;
\item[(H2)] $\inf_\Omega \ a_{11} > 0.$ \end{itemize} We recall that $\elle$ is said hypoelliptic if every distribution $u$ in $\Omega$ such that $\elle u \in C^\infty (\Omega)$ is a smooth function. We note that condition (H2) ensures that for every $x \in \Omega$ there exists $\xi \in \erre^n$ such that $\langle A(x) \xi, \xi \rangle > 0$ that is $\elle$ is non-totally degenerate, in accordance with Definition 5.1 in \cite{bony_1969}. We can drop condition (H2) if the operator $\widetilde\elle = \partial_{x_{n+1}}^2 + \elle$ acting on $\erre^{n+1}$ satisfies (H1) (see Corollary \ref{corollariosergio}).
The main result of this paper is the following Harnack inequality for the non-negative solutions of the equation $\elle u=0$. It obviously applies to the Laplacian and to the heat operators and in these cases it restores the classical elliptic and parabolic Harnack inequalities. \begin{theorem}\label{maintheorem} Assume that $\elle$ statisfies (H1) and (H2). Let $x_0$ in $\Omega$ and let $K$ be any compact set contained in the interior of $\overline {\picorsivo(x_0,\Omega)}$, then there exists a positive constant $C=C(x_0, K, \Omega,\elle)$ such that $$
\sup_K u \le C u(x_0), $$ for every non-negative solution $u$ of $\elle u=0$ in $\Omega$. \end{theorem} We introduce here the definition of $\elle$-\emph{propagation set} $\picorsivo(x_0,\Omega)$ appearing it the above statement. It is the set of all points $x$ reachable from $x_0$ by a {\it propagation path}:
$$\picorsivo(x_0,\Omega) : = \{ x \in \Omega\, |\, \exists\ \gamma\ \mbox{$\elle$-propagation path}, \gamma(0)=x_0, \gamma(T)=x \}. $$
A $\elle$-{\it propagation path} is any absolutely continuous path $\gamma: [0,T] \longrightarrow \Omega$ such that \[\gamma '(t)= \sum_{j=1}^n \lambda_j(t) X_j(\gamma(t))+\mu(t)Y(\gamma(t)) \quad\quad \mbox{ a.e. in } [0,T]\] for suitable piecewise constant real functions $\lambda_1,\ldots,\lambda_n,$ and $\mu$, $\mu\geq 0$. \\ $X_1(x), \ldots, X_n(x), Y(x)$ are the vector fields defined in the following way: \begin{equation} \label{eq-XY}
X_j(x) := \sum_{i=1}^n a_{ji} (x) \partial_{x_i},\quad j=1, \dots, n, \qquad
Y(x) := \sum_{i=1}^n b_i(x) \partial_{x_i}. \end{equation}
As we said at the beginning of the Introduction, our main result can holds also under somehow weaker assumptions on $\elle$. Only in the following Corollary the hypotheses (H1) and (H2) on $\elle$ are replaced by the assumption that the operator $\widetilde\elle = \partial_{x_{n+1}}^2 + \elle$ in $\erre^{n+1}$ satisfies (H1). Of course if $\widetilde\elle$ satisfies assumption (H1) then also $\elle$ does. A simple example of operator satisfying the hypotheses of this Corollary but not the ones of Theorem 1 is $\partial_{x_1}={x_1}^2\partial^2_{x_2}$ in $\erre^2$.
\begin{corollary} \label{corollariosergio} If the operator $\widetilde\elle = \partial_{x_{n+1}}^2 + \elle$ acting on $\Omega \times \erre$ satisfies (H1) then for every $x_0$ in $\Omega$ and for every compact set $K$ contained in the interior of the $\elle$-propagation set $\overline {\picorsivo(x_0,\Omega)}$, there exists a positive constant $C=C(x_0, K, \Omega,\elle)$ such that $$
\sup_K u \le C u(x_0), $$ for every non-negative solution $u$ of $\elle u=0$ in $\Omega$. \end{corollary}
The notion of propagation set $\picorsivo(x_0,\Omega)$ has been introduced by Amano in his work on maximum principle (see \cite[Theorem 2]{amano_max_1979}). In our case it reads as follows.
{\it Assume that $u$ is a (smooth) solution of $\elle u=0$ in $\Omega.$ If $u$ attains its maximum at a point $x_0$ in $\Omega$, then $u\equiv u(x_0)$ in $\overline{\picorsivo(x_0,\Omega)}.$}
The Amano maximum principle is a crucial tool to prove Theorem \ref{maintheorem} using the {abstract Harnack inequality} of the Potential Theory (\cite[Proposition 6.1.5]{CC}). We observe that Harnack inequalities based on results of Potential Theory were proved in \cite[Theorem 4.2]{bonfiglioli_lanconelli_uguzzoni_uniform} for heat equations on Carnot groups and in \cite[Theorem 1.1]{cinti_nystrom_polidoro} and in \cite[Theorem 5.2]{cinti_menozzi_polidoro} for more general evolution equations. The class of hypoelliptic operators considered in \cite{cinti_nystrom_polidoro,cinti_menozzi_polidoro} is \begin{equation} \label{eq-evol}
\sum_{j=1}^m \widetilde X^2_j(y) +\widetilde X_0(y) - \partial_t, \end{equation} where $\widetilde X_j$ are smooth vector fields on $\erre^N$ and $(y,t)$ denoted the point of any subset of $\erre^{N+1}$. We explicitly note that operator \eqref{eq-evol} is a particular example of the operators \eqref{operatore}, with respect to the variable $x =(y,t)$. In both papers the operators are assumed left translation invariant w.r.t. a Lie group in $\erre^{N+1}$ and endowed with a {\it global} fundamental solution. We point out that the use of the fundamental solution is a key step in verifying the {\it separation axiom} of the axiomatic Potential Theory.
The approach used in this note allows us to prove the validity of the separation axiom in Section \ref{axioms} without requiring the existence of any global fundamental solution on every bounded open set. We rely only on hypoellipticity, on non total degeneracy of $\elle$ and the following maximum principle due to Picone that we recall for the sake of completeness.
{\it Let $V$ be any open (bounded) subset of $\Omega$. Assume that exists a function $w: V \rightarrow \erre$ such that $\elle w<0$ in $V$ and $\inf_{V} w>0.$ Then for every $u\in C^2 (V)$ such that \[\elle u\geq 0\quad\mbox { in } V,\quad\limsup_{x\rightarrow \xi}u(x) \le 0\quad \forall \xi \in\partial V, \] we have $u\le 0$ in $V$.}
In our case the existence of a function $w$ follows from (H2) and from the smoothness of the coefficients. Indeed, under these assumptions, we can choose two positive real constants $M$ and $\lambda$ such that the function \begin{equation}\label{eqw} w(x)=w(x_1,\ldots, x_N)= M - e^{\lambda x_1}\end{equation} has the required properties.
This paper is organized as follows. In Section 2 all the notions and results from Potential Theory that we need are briefly recalled. In Section 3 we show that the set of the solutions $u$ of $\elle u=0$ in $\Omega$ satisfies the axioms of the Doob Potential Theory. In Section 4 we prove that the $\elle$-propagation set of $x_0$ is a subset of the smallest absorbent set containing $x_0$. In this way we derive the Harnack inequality for the non-negative solutions $u$ of $\elle u =0$. In Section 5 the propagation sets of some meaningful operators are studied.
In particular we focus on the following operators: $\partial_{{x}_1}^2 + x_1 \partial_{x_2}$ in $\erre^2$ and $\partial_{x_1}^2 + \sin(x_1) \partial_{x_2} + \cos(x_1) \partial_{x_3}$ in $\erre^3$, and we show that the geometry of the relevant Harnack inequality may appear either of \emph{parabolic} or \emph{elliptic}-type, depending on the choice of $\Omega$, even if both operators are \emph{parabolic}.
\section{Some recalls from Potential Theory}
We recall some definitions and results of the Potential Theory that we need to prove our Harnack inequality. For a detailed description of the general theory of {\it harmonic spaces} we refer to \cite[chapter 6]{BLU}, \cite{CC} and to \cite{bauer}.
\subsection{Sheafs of functions and harmonic sheafs in $\Omega$} \mbox{}\\ Let $V$ be any open subset of $\Omega$. We denote by $\overline\erre$ the set $\mbox{$\erre \cup \{\infty, -\infty\}$ and by $\overline\erre^V$}$ the set of functions $\mbox{$u: V \longrightarrow \overline\erre$}$. Moreover $C(V,\erre)$ is the vector space of real continuous functions defined on $V$.
A map $$\effe : V \mapsto \effe(V) \subseteq \overline\erre^V$$ is a {\it sheaf of functions} in $\Omega$ if \begin{itemize}
\item[$(i)$] $V_1, V_2 \subseteq \Omega$, $V_1 \subseteq V_2$, $ u\in \effe(V_2)$ $\implies$ $u|_{V_1} \in \effe({V_1})$;
\item[$(ii)$] $V_\alpha \subseteq \Omega\ \forall \alpha \in \mathcal{A}, u: \bigcup_{\alpha \in A} V_\alpha
\longrightarrow \overline\erre$, $u|_{V_\alpha} \in \effe({V_\alpha}) \implies u\in \effe({ \bigcup_{\alpha\in \mathcal{A}}}V_\alpha).$
\end{itemize}
\noindent When $\effe(V)$ is a linear subspace of $C(V,\erre)$ for every $V\subseteq\Omega$, we say that the sheaf of functions $\effe$ on $V$ is {\it harmonic} and we denote it $\acca(\Omega).$ \subsection{Regular open sets, harmonic measures and absorbent sets} \mbox{}\\ Let $\acca$ be a harmonic sheaf on $\Omega$. We say that an open set $V\subseteq \Omega$ is {\it regular} if: \begin{itemize}
\item[$(i)$] $\overline{V}\subseteq \Omega$ is compact and $\partial V \neq \emptyset$;
\item[$(ii)$] for every continuous function $\varphi : \partial V \longrightarrow \erre$, there exists a unique function in $\acca(V)$, that we denote by $h_\varphi^V$, such that $h_\varphi^V(x) \xrightarrow[x\rightarrow \xi ]{} \varphi (\xi)$ for every $\xi \in \partial V;$ \item[$(iii)$] if $\varphi\geq 0$ then $h_\varphi^V \geq 0.$
\end{itemize} From $(ii)$ and $(iii)$ it follows that, for every regular set $V$ and for every $x \in V$, the map $$C(\partial V) \ni \varphi \longmapsto h_\varphi^V(x) \in \erre$$ is linear and positive. Thus, the Riesz representation theorem (see e.g. \cite{rudin}), implies that, for every regular set $V$ and for every $x \in V$, there exists a {\it regular Borel measure}, that we denote by $\mu^V_x$, supported in $\partial V$, such that $$h_\varphi^V (x) = \int_{\partial V} \varphi(y) \ d \mu^V_x(y) \qquad \forall\ \varphi\in C(\partial V).$$ The measure $\mu^V_x$ is called the {\it harmonic measure} related to $V$ and $x$.
Now, let $A$ be a closed subset of $\Omega$ . We say that $A$ is {\it absorbent} if it contains the supports of all the harmonic measures related to its points. More precisely, $$\mbox{for every $x \in A$ and every regular set $V$ containing $x$, $\mathrm{supp\ } \mu_{x}^V\subseteq A$.}$$
If $x_0 \in\Omega $, we define $\Omega_{x_0}$ as the smallest absorbent set containing $x_0$:
$$\Omega_{x_0}:= \bigcap_{\substack {A \mbox{\tiny\ absorbent} \\ A\ni x_0}} A.$$
\subsection{Superharmonic functions} \mbox{}\\ A function $u: \Omega \longrightarrow ]-\infty, \infty]$ is called {\it superharmonic in $\Omega$} if
\begin{itemize}
\item[$(i)$] $u$ is lower semi-continuous;
\item[$(ii)$] for every regular set $V$, $\overline V\subseteq \Omega$, and for every $\varphi \in C( \partial V, \erre)$,
$\varphi \le u|_{\partial \Omega} $, it follows $ u \geq h_\varphi^V$ in $V;$
\item[$(iii)$] the set $\{ x\in \Omega \ | \ u(x) < \infty \}$ is dense in $\Omega$.
\end{itemize} We denote by $\esse(\Omega)$ the family of the superharmonic functions on $\Omega$.
By the maximum principle, we have that every function $u\in C^2(\Omega)$ such that $\elle u \le 0$ in $\Omega$ is superharmonic (see \cite[Proposition 7.2.5]{BLU}).
\subsection{Doob harmonic spaces and Harnack inequality} \label{axioms} \mbox{}\\ We say that a harmonic sheaf $\acca(\Omega)$ is a {\it Doob harmonic space} if the following axioms are satisfied.
\begin{itemize}
\item[(A1)] { \it Positivity axiom:}\\ For every $x\in \Omega$, there exists an open set $V\ni z$ and a function $u\in \acca(V)$ such that $u(x)>0$.
\item[(A2)] { \it Doob convergence axiom:}\\ Let $(u_n)_{n\in\enne}$ be a monotone increasing sequence in $\acca(\Omega)$ and let \\ $u:= \sup_{n\in\enne} u_n.$ If
the set $\{ x\in \Omega \ | \ u(x) < \infty \}$ is dense in $\Omega$, then $u\in \acca(\Omega).$
\item[(A3)] { \it Regularity axiom:}\\ There is a basis of the euclidean topology of $\Omega$ formed by regular sets.
\item[(A4)] { \it Separation axiom:}\\ $\esse(\Omega)$ separates the points of $\Omega$ in this sense: for every $y$ and $z$ in $\Omega$, $y\neq z$, there exist two non-negative functions $u$ and $v$ in $\esse(\Omega)$ such that $u(y)v(z)\neq u(z)v(y).$ \end{itemize}
We close the section recalling that in this setting the {\it abstract Harnack inequality} from the Parabolic Potential Theory holds \cite[Proposition 6.1.5 ]{CC}. \begingroup \setcounter{tmp}{\value{theorem}} \setcounter{theorem}{0} \renewcommand\thetheorem{\Alph{theorem}} \begin{theorem} \label{abstract_harnack} Let $(\Omega, \acca)$ be a {\it Doob harmonic space}, $x_0\in \Omega$ and let $K$ be a compact set contained in the interior of $\Omega_{x_0}$, the smallest absorbent set containing $x_0$. Then there exists a positive constant $C=C(x_0, K, \Omega)$ such that $$\sup_K u \le C u(x_0)\qquad\forall u\in \acca(\Omega), u\geq0.$$ \end{theorem} \endgroup
\section{The harmonic space of the solutions of $\elle u=0$ } We show that the set of the solutions of the equation $\elle u=0$ is a Doob harmonic space in $\Omega.$ For every $V\subseteq \Omega$ we consider the harmonic sheaf $$ {\erre^{ {n} }} \supseteq V \frecciaf \acca(V)$$
where
\begin{equation*} \acca(V)= \{ u\in C^\infty(V) \ | \ \elle u=0 \} \end{equation*} and $\elle$ is the operator \eqref{operatore}.
The {\it positivity axiom } (A1) is plainly verified. Indeed every constant function belongs to $\acca(\Omega)$.
(A2) is a consequence of a weak Harnack inequality due to Bony (see \cite[Theoreme 7.1]{bony_1969}); see also \cite[Proposition 7.4]{kogoj_lanconelli_2004}).
(A3), i.e. the existence of a basis of the euclidean topology of $\Omega$ formed by regular sets, can be proved as in \cite[Corollarie 5.2]{bony_1969}, see also \cite[Proposition 7.1.5]{BLU}. We stress that the tools used in its proof are only the hypoellipticity, the non totally degeneracy of the operator $\elle$ and the classical Picone Maximum Principle.
Now we are left to verify the {\it separation axiom } (A4). As in our setting the constant are superharmonic functions, we need to prove that \begin{equation}\label{separazione} \forall\ y, z \in \Omega, y\neq z, \exists\ u \in \esse(\Omega),\ u\geq 0, \mbox{ such that } u(y)\neq u(z).\end{equation}
Now, let $y=(y_1,\ldots, y_n)$ and $z=(z_1,\ldots, z_n)$ be two different points in $\Omega$.
We observe that the function $ w(x)=w(x_1,\ldots, x_N)= M- e^{\lambda x_1}$, as in \eqref{eqw}, for suitable real positive constants $\lambda$ and $M$, is non-negative and $\elle w(x) <0$ for every $x \in \Omega$, hence $w\in \esse(\Omega)$.
If $y_1\neq z_1$, we can choose $u(x)=w(x)$ to separate $y$ and $z$ and we are done.
If $y_1= z_1$, we set $u(x)= |x-y|^2 + w(x)$. Also in this case, for suitable $\lambda$ and $M$, $u$ is non-negative, $u \in C^2(\Omega)$ and $\elle u(x) = \elle (|x-y|^2) + \elle(w(x))< 0$ in $\Omega$. Moreover $u(y) - u(z) = |z-y|^2$, so \eqref{separazione} is satisfied.
\section{Propagation sets and Harnack inequality}
Let $X_1(x), \ldots, X_n(x), Y(x)$ be the vector fields defined in the following way: \begin{equation*}\begin{split} &
X_i(x)= \sum_{j=1}^n a_{ij}(x) \partial_{x_j},\qquad 1 \le i \le n, \\ & Y(x) = \sum_{i=1}^n b_i(x) \partial_{x_i}. \end{split} \end{equation*}
We recall that a $\elle$-{\it propagation path} is any absolutely continuous path $\gamma: [0,T] \longrightarrow \Omega$ such that \[\gamma '(t)= \sum_{j=1}^n \lambda_j(t) X_j(\gamma(t))+\mu(t)Y(\gamma(t)) \quad\quad \mbox{ a.e. in } [0,T]\] for suitable piecewise constant real functions $\lambda_1,\ldots,\lambda_n,$ and $\mu$ with $\mu\geq 0$.
For a point $x_0$ in $\Omega$, we define the $\elle$-{\it propagation set} as the set of all points $x$ such that $x$ and $x_0$ can be connected by a propagation path, running from $x_0$ to $x$:
$$\picorsivo(x_0,\Omega) : = \{ x \in \Omega\, |\, \exists\ \gamma: [0,T] \rightarrow \Omega, \gamma\ \mbox{$\elle$-propagation path}, \gamma(0)=x_0, \gamma(T)= x \}. $$ Proceeding as in \cite[Lemma 5.8]{cinti_menozzi_polidoro}, we prove now that the $\elle$-propagation set of $x_0$ is a subset of every absorbent set containing $x_0.$ This Lemma, based on the maximum propagation principle, is a key lemma in order to get our Harnack inequality so we prefer to give here its detailed proof. \begin{lemma} \label{lemma2} For every $x_0$ in $\Omega$, $\picorsivo(x_0,\Omega) \subseteq \Omega_{x_0}.$ \end{lemma} \begin{proof} By contradiction, suppose $x \in \picorsivo(x_0,\Omega)$ and $ x \notin \Omega_{x_0}.$ There exists an absolutely continuous path $\gamma$ connecting $x_0$ and $ x $: $$\gamma : [0,T] \longrightarrow \Omega,\qquad \gamma(0)=x_0, \qquad \gamma(T)=x.$$ As $ \Omega_{x_0}$ is a subset closed in $\Omega$ and $\gamma$ is continuous, there will be a time $t_1$ such that $\gamma(t_1)=x_1 \in \Omega_{x_0}$ and $\gamma(t)\notin \Omega_{x_0}$ when $t$ is in $ ] t_1,T].$
Let's take a regular open set $V$ containing $x_1$. There will be $t_2 \in ] t_1,T]$ such that $x_2=\gamma(t_2) \in \partial V$. From what we wrote before, $x_2$ does not belong to $\Omega_{x_0}$.
Take now a neighborhood of $x_2$, $U$ such that $U \cap \partial V \subseteq \Omega \backslash \Omega_{x_0}$ and consider a function $\varphi$ defined on $\partial V$ such that $\varphi$ is strictly positive in $U \cap \partial V$ and $0$ otherwise.
$$h_\varphi^V(x_1) = \int_{\partial V} \varphi( y)\ d \mu_{x_1}^V (y) = \int_{U \cap \partial V } \varphi( \zeta)\ d \mu_{x_1}^V (\zeta) =0,$$ because $x_1$ is in $\Omega_{x_0}$ and $\mathrm{supp\ } \mu_{x_1}^{V} \subseteq \Omega_{x_0}$ for every regular set $V$.
But $h_\varphi^V$ is nonnegative and it would attain its minimum at $ x_1$. From Amano minimum propagation principle \cite[Theorem 2]{amano_max_1979}, it would follow that
$$h_\varphi^V(\gamma(t))=0\qquad\forall \ t \in ]t_1,t_2[.$$ In conclusion, we would have that
$$h_\varphi^V(x) \xrightarrow[x\rightarrow x_2 ]{} \varphi (x_2)>0,$$ and
$$h_\varphi^V(\gamma(t)) \xrightarrow[t\rightarrow t_2^{-} ]{} h_\varphi^V(\gamma(t_2))=0$$ that is a contraddiction. \end{proof}
We are now ready to give the proofs of our main results.
\begin{proof}[Proof of Theorem \ref{maintheorem}] Let $x_0$ in $\Omega$ and let $K$ be a compact set
contained in the interior of $\overline{\picorsivo(x_0,\Omega)}$.
As $\Omega_{x_0}$ is a closed subset of $\Omega$, Lemma \ref{lemma2} implies that $\overline{\picorsivo(x_0,\Omega)} \subseteq \Omega_{x_0}$. On the other hand, as we showed in Section 3, the set of the solutions of the equation $\elle u=0$ is a Doob harmonic space in $\Omega$. Then, by Theorem \ref{abstract_harnack}, there exists a positive constant $C=C(x_0, K, \Omega,\elle)$ such that $$\sup_K u \le C u(x_0)\,$$ for every non-negative solution $u$ of $\elle u=0$ in $\Omega$. \end{proof}
\begin{proof}[Proof of Corollary \ref{corollariosergio}]
We set $\widetilde x : = (x,x_{n+1})$, $\widetilde\Omega:= \Omega \times ]-1,1[$, $\widetilde K: = K \times [- \frac{1}{2}, \frac{1}{2}]$ and $\widetilde u(\widetilde x): = u(x)$ for every $\widetilde x\in \widetilde\Omega$. We observe that the $\widetilde \elle$-propagation set of $(x_0,0)$, $\widetilde \picorsivo_{(x_0,0)}(\widetilde\Omega)$, equals
$\picorsivo_{x_0}(\Omega) \times ]-1,1[$. Then $K\subseteq \mathrm{int} \picorsivo_{x_0}(\Omega)$ if and only if
$\widetilde K\subseteq \mathrm{int} \widetilde \picorsivo_{(x_0,0)}(\widetilde \Omega)$. By Theorem \ref{maintheorem}
$$\sup_{\widetilde K} \widetilde u \le C\ \widetilde u(x_0, 0 )\,$$ and the conclusion follows immediately.
\end{proof}
\section{Examples}
In this Section we give two examples of operators for which we give Harnack-type inequalities that, to our knowledge, are new. In general, the main step in the application of our Theorem \ref{maintheorem} is the characterization of the propagation set $\picorsivo(x_0,\Omega)$ of the operator $\elle$. We recall that the Control Theory provides us with several tools useful for this problem. We refer, for example, to the book \cite[Chapter 8]{agrachev_sachkov} by Agrachev and Sachkov.
\subsection{A Harnack inequality for the stationary Mumford operator}\mbox{}\\ We consider the operator $\elle =\partial_{x_1}^2 + \sin(x_1) \partial_{x_2} + \cos(x_1) \partial_{x_3}$ in the set: \begin{equation} \label{eq-omega-Mum}
\Omega= ]-a,a[ \times B(0,r)\subseteq \erre \times \erre^2. \end{equation} $x_1\in ]-a,a[$ where $a>\pi$, and $(x_2,x_3)\in B(0,r)$, the euclidean ball centered at $0$ with radius $r>0$. This operator has been introduced by Mumford \cite{mumford_1994} in the study of computer vision problems. The relevant Harnack inequality of Theorem \ref{maintheorem} takes the following form:
\begin{theorem} Let $\Omega$ be the set introduced in \eqref{eq-omega-Mum}, with $a>\pi$. For every compact set $K\subset \Omega$ there exists a positive constant $C=C(K,\Omega,\elle)$ such that $$\sup_K u \le C u(0),$$ for every non-negative solution $u$ of $$ \partial_{x_1}^2 u + \sin(x_1) \partial_{x_2} u + \cos(x_1) \partial_{x_3}u=0 \quad \text{in} \quad \Omega. $$ \end{theorem} \begin{proof} In view of Theorem \ref{maintheorem}, we need only to prove that in this case the {\it propagation set} $\picorsivo(0,\Omega)$ agrees with $\Omega$. With this aim, we fix any point $z=(z_1,z_2,z_3)$ in $\Omega$, and we construct a $\elle$-{\it propagation path} steering $0$ to $z$. Note that, in our case, the vector fields defined in \eqref{eq-XY} are $$
X=\partial_{x_1} \qquad \text{and} \qquad Y= \sin(x_1) \partial_{x_2} + \cos(x_1) \partial_{x_3}. $$ We connect $0$ and $z$ by a path $\gamma: [0,T] \to \Omega$ such that $\gamma'(t) = \pm X(\gamma(t))$ in the first interval $[0, t_1]$, then $\gamma'(t) = Y(\gamma(t))$ in the second interval $[t_1, t_2]$, and $\gamma'(t) = \pm X(\gamma(t))$ in the third interval $[t_2, T]$, for $t_1, t_2, T$ such that $0 \le t_1 \le t_2 \le T$ chosen as follows.
We set $t^* = \arg(z_2,z_3) \in ]- \pi, \pi] \subset ]-a,a[$, and we choose $t_1 := |t^*|$. If $t^* >0$, the function $\gamma(t) = (t,0,0)$ is a solution of $\gamma'(t) = X$, for $t \in [0,t_1]$, $\gamma(0)=0$. If $t^* <0$ we consider $\gamma(t) = (-t,0,0)$. In both cases, we have that $\gamma'(t) = \pm X(\gamma(t))$. If $t^* = 0$ we simply skip this step.
We next set $t_2 = t_1 + \sqrt{z_2^2 +z_3^2}$, and we choose $\gamma$ such that $\gamma'(t) = Y(\gamma(t))$ for $t_1 < t < t_2$. Also in this case, if $(z_2,z_3)=(0,0)$, we skip this step.
We conclude the construction of $\gamma$ by choosing $s^* = z_1 - t^*$, $T = t_2 + |s^*|$ and following the same method used in the first step. The path $\gamma$ then writes as follows.
\begin{eqnarray*} \gamma(t)= \left\{\begin{array}{ccc}&(\pm t,0,0) & \quad \mbox{ if }\qquad 0\le t \le t_1, \\ \\ &(t^\ast , (t - t_1) \cos t^*, (t - t_1) \sin t^*)& \mbox{ if }\quad t_1 \le t \le t_2, \\ \\ & (t^\ast \pm (t - t_2),z_2, z_3)& \mbox{ if }\quad t_2 \le t \le T. \end{array} \right. \end{eqnarray*} \end{proof}
\begin{remark} The above construction can be reproduced to translated cylinders \begin{equation*}
\Omega_y= ]y_1 -a,y_1+ a[ \times B((y_2,y_3),r)\subseteq \erre \times \erre^2, \end{equation*} for every $y = (y_1,y_2,y_3) \in \erre^3$. We find $\picorsivo((y_1,y_2,y_3),\Omega_y) = \Omega_y$.
We point out that, on the other hand, the geometry of the propagation set $\picorsivo(0,\Omega)$ changes completely as the width of the interval $]-a,a[$ is smaller than $2 \pi$. For instance, if we consider the set \begin{equation*}
\widetilde \Omega = ]-\pi/2, \pi/2[ \times B(0,r)\subseteq \erre \times \erre^2, \end{equation*} we easily see that $\picorsivo(0,\widetilde \Omega) = \widetilde \Omega \cap \big\{ x_3 > 0 \big \}$. This fact is in accordance with the invariance of the operator $\elle$ with respect to the following left translation introduced in \cite{bonfiglioli_lanconelli_matrix}. Denote $x=(t,z), y=(s,w) \in \erre \times \mathbb{C}$. then $$ x\circ y:= (t+s, z + w e^{it}).$$
\end{remark} \subsection{A Harnack inequality for a degenerate Ornstein Uhlenbeck operator}\mbox{}\\ We consider the operator $\elle =\partial_{x_1}^2 + x_1 \partial_{x_2}$ in the set \begin{equation} \label{eq-omega-OU} \Omega= ]-a,a[ \times ]-b,b[ \end{equation} for some positive $a$ and $b$.
As in the case of Mumford operator, Theorem \ref{maintheorem} gives an \emph{elliptic} Harnack inequality. \begin{theorem} Let $\Omega$ be the set introduced in \eqref{eq-omega-OU}. For every compact set $K\subset \Omega$ there exists a positive constant $C=C(K,\Omega,\elle)$ such that $$\sup_K u \le C u(0),$$ for every non-negative solution $u$ of $$ \partial_{x_1}^2 u + x_1 \partial_{x_2} u=0 \quad \text{in} \quad \Omega. $$ \end{theorem} \begin{proof} We prove that, also in this case, the {\it propagation set} $\picorsivo(0,\Omega)$ agrees with $\Omega$. The vector fields defined in \eqref{eq-XY} are $$
X=\partial_{x_1} \qquad \text{and} \qquad Y= x_1 \partial_{x_2}. $$ We choose an integral curve $\gamma$ such that $\gamma'(t) = \pm X(\gamma(t))$ in some intervals. A curve like that writes as $\gamma(t) = (\widetilde x_1 \pm t, \widetilde x_2)$. In particular, we will use the field $X$ to increase or decrease the first coordinate $x_1$. In some other intervals we choose $\gamma'(t) = Y(\gamma(t))$. Such a curve writes as $\gamma(t) = (\widetilde x_1, \widetilde x_2 + \widetilde x_1 t)$. In this case, we rely on the sign of $\widetilde x_1$ to increase or decrease the second component $x_2$. We prefer not to give the details of the construction and to refer to the following figure.
\begin{center} \scalebox{.80}{ \begin{pspicture}(-5.5,-4)(5.5,4)
\psaxes[Dx=6, Dy=5]{->}(0,0)(-5,-4)(5,4)
\psline(-4,-3)(4,-3)(4,3)(-4,3)(-4,-3)
\psline[linewidth=1.5pt]{*-*}(0,0)(2,0)(2,2)(-2,2)(-2,-2)
\uput[0](5,0){$x_1$}
\uput[0](0,4){$x_2$}
\uput[-8](2,-.2){$\gamma(t_1)$}
\uput[0](2,2.1){$\gamma(t_2)$}
\uput[225](-2,2.4){$\gamma(t_3)$}
\uput[180](-2,-2){$\gamma(T)$}
\psline[linewidth=1.5pt]{<-*}(.2,-1)(1.2,-1)
\psline[linewidth=1.5pt]{*->}(1.2,-1)(2.2,-1)
\uput[90](1.2,-1){$\pm X$}
\psline[linewidth=1.5pt]{*->}(-1.7,1.5)(-1.7,.5)
\uput[0](-1.7,1){$\! \! Y = x_1 \partial_{x_2}$}
\psline[linewidth=1.5pt]{*->}(2.2,.5)(2.2,1.7)
\uput[0](2.2,1){$\! \! Y = x_1 \partial_{x_2}$} \end{pspicture}}\end{center} \end{proof}
\begin{remark} The above result fails as \begin{equation*}
\Omega= ]a_1,a_2[ \times ]-b,b[\subseteq \erre^2, \end{equation*} and $a_1$ and $a_2$ have the same sign. In particular, if $a_1$ and $a_2$ are both positive, and we consider $x_0 = \left(\frac{a_1 + a_2}{2},0\right)$, we have $\picorsivo(x_0,\Omega) = \Omega \cap \big\{ x_2 > 0 \big \}$. On the contrary, if $a_1$ and $a_2$ are both negative, we have $\picorsivo(x_0,\Omega) = \Omega \cap \big\{ x_2 < 0 \big \}$. \end{remark}
\section*{Acknowledgments}
The authors have been partially supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilit\`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
\end{document} | arXiv |
What Is the Current Ratio?
Formula and Calculation
Understanding the Current Ratio
Interpreting the Current Ratio
How It Changes Over Time
Current Ratio vs. Other Ratios
Current Ratio FAQs
Current Ratio Explained With Formula and Examples
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The current ratio is a liquidity ratio that measures a company's ability to pay short-term obligations or those due within one year. It tells investors and analysts how a company can maximize the current assets on its balance sheet to satisfy its current debt and other payables.
A current ratio that is in line with the industry average or slightly higher is generally considered acceptable. A current ratio that is lower than the industry average may indicate a higher risk of distress or default. Similarly, if a company has a very high current ratio compared with its peer group, it indicates that management may not be using its assets efficiently.
The current ratio is called current because, unlike some other liquidity ratios, it incorporates all current assets and current liabilities. The current ratio is sometimes called the working capital ratio.
The current ratio compares all of a company's current assets to its current liabilities.
These are usually defined as assets that are cash or will be turned into cash in a year or less and liabilities that will be paid in a year or less.
The current ratio helps investors understand more about a company's ability to cover its short-term debt with its current assets and make apples-to-apples comparisons with its competitors and peers.
One weakness of the current ratio is its difficulty of comparing the measure across industry groups.
Others include the overgeneralization of the specific asset and liability balances, and the lack of trending information.
Using The Current Ratio
Formula and Calculation for the Current Ratio
To calculate the ratio, analysts compare a company's current assets to its current liabilities.
Current assets listed on a company's balance sheet include cash, accounts receivable, inventory, and other current assets (OCA) that are expected to be liquidated or turned into cash in less than one year.
Current liabilities include accounts payable, wages, taxes payable, short-term debts, and the current portion of long-term debt.
Current Ratio = Current assets Current liabilities \begin{aligned} &\text{Current Ratio}=\frac{\text{Current assets}}{ \text{Current liabilities}} \end{aligned} Current Ratio=Current liabilitiesCurrent assets
The current ratio measures a company's ability to pay current, or short-term, liabilities (debts and payables) with its current, or short-term, assets, such as cash, inventory, and receivables.
In many cases, a company with a current ratio of less than 1.00 does not have the capital on hand to meet its short-term obligations if they were all due at once, while a current ratio greater than 1.00 indicates that the company has the financial resources to remain solvent in the short term. However, because the current ratio at any one time is just a snapshot, it is usually not a complete representation of a company's short-term liquidity or longer-term solvency.
For example, a company may have a very high current ratio, but its accounts receivable may be very aged, perhaps because its customers pay slowly, which may be hidden in the current ratio. Some of the accounts receivable may even need to be written off. Analysts also must consider the quality of a company's other assets vs. its obligations. If the inventory is unable to be sold, the current ratio may still look acceptable at one point in time, even though the company may be headed for default.
A ratio under 1.00 indicates that the company's debts due in a year or less are greater than its assets—cash or other short-term assets expected to be converted to cash within a year or less. A current ratio of less than 1.00 may seem alarming, although different situations can negatively affect the current ratio in a solid company.
For example, a normal cycle for the company's collections and payment processes may lead to a high current ratio as payments are received, but a low current ratio as those collections ebb. Calculating the current ratio at just one point in time could indicate that the company can't cover all of its current debts, but it doesn't necessarily mean that it won't be able to when the payments are due.
Additionally, some companies, especially larger retailers such as Walmart, have been able to negotiate much longer-than-average payment terms with their suppliers. If a retailer doesn't offer credit to its customers, this can show on its balance sheet as a high payables balance relative to its receivables balance. Large retailers can also minimize their inventory volume through an efficient supply chain, which makes their current assets shrink against current liabilities, resulting in a lower current ratio. Walmart's current ratio as of July 2021 was 0.96.
In theory, the higher the current ratio, the more capable a company is of paying its obligations because it has a larger proportion of short-term asset value relative to the value of its short-term liabilities. However, though a high ratio—say, more than 3.00—could indicate that the company can cover its current liabilities three times, it also may indicate that it is not using its current assets efficiently, securing financing very well, or properly managing its working capital.
The current ratio can be a useful measure of a company's short-term solvency when it is placed in the context of what has been historically normal for the company and its peer group. It also offers more insight when calculated repeatedly over several periods.
How the Current Ratio Changes Over Time
What makes the current ratio good or bad often depends on how it is changing. A company that seems to have an acceptable current ratio could be trending toward a situation in which it will struggle to pay its bills. Conversely, a company that may appear to be struggling now could be making good progress toward a healthier current ratio.
In the first case, the trend of the current ratio over time would be expected to harm the company's valuation. Meanwhile, an improving current ratio could indicate an opportunity to invest in an undervalued stock amid a turnaround.
Imagine two companies with a current ratio of 1.00 today. Based on the trend of the current ratio in the following table, for which would analysts likely have more optimistic expectations?
Image by Sabrina Jiang © Investopedia 2020
Two things should be apparent in the trend of Horn & Co. vs. Claws Inc. First, the trend for Claws is negative, which means further investigation is prudent. Perhaps it is taking on too much debt or its cash balance is being depleted—either of which could be a solvency issue if it worsens. The trend for Horn & Co. is positive, which could indicate better collections, faster inventory turnover, or that the company has been able to pay down debt.
The second factor is that Claws' current ratio has been more volatile, jumping from 1.35 to 1.05 in a single year, which could indicate increased operational risk and a likely drag on the company's value.
Example Using the Current Ratio
The current ratios of three companies—Apple, Walt Disney, and Costco Wholesale—are calculated as follows for the fiscal year ended 2017:
For every $1 of current debt, Costco Wholesale had 99 cents available to pay for debt when this snapshot was taken. Likewise, Walt Disney had 81 cents in current assets for each dollar of current debt. Apple, meanwhile, had more than enough to cover its current liabilities if they were all theoretically due immediately and all current assets could be turned into cash.
Current Ratio vs. Other Liquidity Ratios
Other similar liquidity ratios can supplement a current ratio analysis. In each case, the differences in these measures can help an investor understand the current status of the company's assets and liabilities from different angles, as well as how those accounts are changing over time.
The commonly used acid-test ratio, or quick ratio, compares a company's easily liquidated assets (including cash, accounts receivable, and short-term investments, excluding inventory and prepaid expenses) to its current liabilities. The cash asset ratio, or cash ratio, also is similar to the current ratio, but it only compares a company's marketable securities and cash to its current liabilities.
Finally, the operating cash flow ratio compares a company's active cash flow from operating activities (CFO) to its current liabilities.
Limitations of Using the Current Ratio
One limitation of the current ratio emerges when using it to compare different companies with one another. Businesses differ substantially among industries; comparing the current ratios of companies across different industries may not lead to productive insight.
For example, in one industry, it may be more typical to extend credit to clients for 90 days or longer, while in another industry, short-term collections are more critical. Ironically, the industry that extends more credit actually may have a superficially stronger current ratio because its current assets would be higher. It is usually more useful to compare companies within the same industry.
Another drawback of using the current ratio, briefly mentioned above, involves its lack of specificity. Unlike many other liquidity ratios, it incorporates all of a company's current assets, even those that cannot be easily liquidated. For example, imagine two companies that both have a current ratio of 0.80 at the end of the last quarter. On the surface, this may look equivalent, but the quality and liquidity of those assets may be very different, as shown in the following breakdown:
In this example, Company A has much more inventory than Company B, which will be harder to turn into cash in the short term. Perhaps this inventory is overstocked or unwanted, which eventually may reduce its value on the balance sheet. Company B has more cash, which is the most liquid asset, and more accounts receivable, which could be collected more quickly than liquidating inventory. Although the total value of current assets matches, Company B is in a more liquid, solvent position.
The current liabilities of Company A and Company B are also very different. Company A has more accounts payable, while Company B has a greater amount in short-term notes payable. This would be worth more investigation because it is likely that the accounts payable will have to be paid before the entire balance of the notes-payable account. Company A also has fewer wages payable, which is the liability most likely to be paid in the short term.
In this example, although both companies seem similar, Company B is likely in a more liquid and solvent position. An investor can dig deeper into the details of a current ratio comparison by evaluating other liquidity ratios that are more narrowly focused than the current ratio.
What Is a Good Current Ratio?
What counts as a good current ratio will depend on the company's industry and historical performance. Current ratios of 1.50 or greater would generally indicate ample liquidity. Publicly listed companies in the United States reported a median current ratio of 1.94 in 2020.
What Happens If the Current Ratio Is Less Than 1?
As a general rule, a current ratio below 1.00 could indicate that a company might struggle to meet its short-term obligations, whereas ratios of
What Does a Current Ratio of 1.5 Mean?
A current ratio of 1.5 would indicate that the company has $1.50 of current assets for every $1 of current liabilities. For example, suppose a company's current assets consist of $50,000 in cash plus $100,000 in accounts receivable. Its current liabilities, meanwhile, consist of $100,000 in accounts payable. In this scenario, the company would have a current ratio of 1.5, calculated by dividing its current assets ($150,000) by its current liabilities ($100,000).
How Is the Current Ratio Calculated?
Calculating the current ratio is very straightforward: Simply divide the company's current assets by its current liabilities. Current assets are those that can be converted into cash within one year, while current liabilities are obligations expected to be paid within one year. Examples of current assets include cash, inventory, and accounts receivable. Examples of current liabilities include accounts payable, wages payable, and the current portion of any scheduled interest or principal payments.
Corporate Finance Institute. "Current Ratio Formula."
Yahoo! Finance. "Walmart Inc. (WMT) NYSE: Financial Highlights: Balance Sheet."
Walmart. "Earnings Release (FY22 Q2)," Page 6.
Costco Wholesale, via U.S. Securities and Exchange Commission. "Form 10-K for the Fiscal Year Ended September 3, 2017," Page 41.
The Walt Disney Co. "Fiscal Year 2017 Annual Financial Report," Page 60 (Page 64 of PDF).
Apple, via U.S. Securities and Exchange Commission. "Form 10-K for the Fiscal Year Ended September 30, 2017," Page 41 (Page 44 of PDF).
ReadyRatios. "All Industries: Average Industry Financial Ratios for U.S. Listed Companies."
Guide to Financial Ratios
What Is the Best Measure of a Company's Financial Health?
What Financial Ratios Are Used to Measure Risk?
Profitability Ratios: What They Are, Common Types, and How Businesses Use Them
Understanding Liquidity Ratios: Types and Their Importance
What Is a Solvency Ratio, and How Is It Calculated?
Solvency Ratios vs. Liquidity Ratios: What's the Difference?
Multiples Approach
Return on Assets (ROA): Formula and 'Good' ROA Defined
How Return on Equity Can Help Uncover Profitable Stocks
Return on Investment (ROI): How to Calculate It and What It Means
Return on Invested Capital: What Is It, Formula and Calculation, and Example
EBITDA Margin: What It Is, Formula, How to Use It
What is Net Profit Margin? Formula for Calculation and Examples
Operating Margin: What It Is and the Formula for Calculating It, With Examples
Quick Ratio Formula With Examples, Pros and Cons
Cash Ratio: Definition, Formula, and Example
Operating Cash Flow (OCF): Definition, Types, and Formula
Receivables Turnover Ratio Defined: Formula, Importance, Examples, Limitations
Inventory Turnover Ratio: What It Is, How It Works, and Formula
Working Capital Turnover Ratio: Meaning, Formula, and Example
Debt-to-Equity (D/E) Ratio Formula and How to Interpret It
Total-Debt-to-Total-Assets Ratio: Meaning, Formula, and What's Good
Interest Coverage Ratio: Formula, How It Works, and Example
Shareholder Equity Ratio: Definition and Formula for Calculation
Can Investors Trust the P/E Ratio?
Using the Price-to-Book (P/B) Ratio to Evaluate Companies
Price-to-Sales (P/S) Ratio: What It Is, Formula To Calculate It
Price-to-Cash Flow (P/CF) Ratio? Definition, Formula, and Example
Liquidity ratios are a class of financial metrics used to determine a debtor's ability to pay off current debt obligations without raising external capital.
The quick ratio is a calculation that measures a company's ability to meet its short-term obligations with its most liquid assets.
Current Assets: What It Means and How to Calculate It, With Examples
Current Assets is an account on a balance sheet that represents the value of all assets that could be converted into cash within one year.
The cash ratio—total cash and cash equivalents divided by current liabilities—measures a company's ability to repay its short-term debt.
Total-debt-to-total-assets is a leverage ratio that shows the total amount of debt a company has relative to its assets.
A solvency ratio is a key metric used to measure an enterprise's ability to meet its debt and other obligations.
Current Ratio vs. Quick Ratio: What's the Difference?
Understanding Coca-Cola's Capital Structure (KO)
What Financial Liquidity Is, Asset Classes, Pros & Cons, Examples
How Do You Calculate Working Capital?
How Do You Read a Balance Sheet? | CommonCrawl |
\begin{document}
\begin{flushleft}
{\bf\Large {Uncertainty Principles for the Continuous Shearlet \\[1.6mm]Transforms in Arbitrary Space Dimensions}} \end{flushleft}
\parindent=0mm
{\bf{Firdous A. Shah$^{\star}$ and Azhar Y. Tantary$^{\star}$ }}
\parindent=0mm
{\small \it $^{\star}$Department of Mathematics, University of Kashmir, South Campus, Anantnag-192101, Jammu and Kashmir, India. E-mail: $\text{[email protected]}$;\,$\text{[email protected]}$}
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{\small {\bf Abstract.} The aim of this article is to formulate some novel uncertainty principles for the continuous shearlet transforms in arbitrary space dimensions. Firstly, we derive an analogue of the Pitt's inequality for the continuous shearlet transforms, then we formulate the Beckner's uncertainty principle via two approaches: one based on a sharp estimate from Pitt's inequality and the other from the classical Beckner's inequality in the Fourier domain. Secondly, we consider a logarithmic Sobolev inequality for the continuous shearlet transforms which has a dual relation with Beckner's inequality. Thirdly, we derive Nazarov's uncertainty principle for the shearlet transforms which shows that it is impossible for a non-trivial function and its shearlet transform to be both supported on sets of finite measure. Towards the culmination, we formulate local uncertainty principles for the continuous shearlet transforms in arbitrary space dimensions.
\parindent=0mm
{\bf{Keywords:}} Shearlets. Uncertainty principle. Pitt’s inequality. Beckner's inequality. Sobolev inequality. Nazarov's uncertainty principle. Local uncertainty principle. Fourier transform.
\parindent=0mm
{\bf {Mathematics Subject Classification:}} 26D10. 35A23. 42B10. 42C40. 42A38.}
\section{Introduction}
\parindent=0mm
Shearlets are the outcome of a series of multiscale methods such as wavelets, ridgelets, curvelets, contourlets and many others introduced during the last few decades with the aim to achieve optimally sparse approximations for higher dimensional signals by employing the basis elements with much higher directional sensitivity and various shapes \cite{Lab,Kuty,Dal1,Dal2}. Unlike the classical wavelets, shearlets are non-isotropic in nature, they offer optimally sparse representations, they allow compactly supported analyzing elements, they are associated with fast decomposition algorithms and they provide a unified treatment of continuum and digital data. However, similar to the wavelets, they are an affine-like system of well-localized waveforms at various scales, locations and orientations; that is, they are generated by dilating and translating one single generating function, where the dilation matrix is the product of a parabolic scaling matrix and a shear matrix and hence, they are a specific type of composite dilation wavelets \cite{JJ,SZS,ABHK,DS1,DS2}. The importance of shearlet transforms have been widely acknowledged and since their inception, they have emerged as one of the most effective frameworks for representing multidimensional data ranging over the areas of signal and image processing, remote sensing, data compression, and several others, where the detection of directional structure of the analyzed signals play a role \cite{GL1,GL2}.
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For any $f\in L^2( {\mathbb R}^{n})$, the continuous shearlet transform in arbitrary space dimension is defined by \cite{Dal2} \begin{align*} {\mathcal {SH}}_{\psi}f(a,s,t)=\Big\langle{f,\psi_{a,s,t}}\Big\rangle=\int_{\mathbb R^{n}} f(x)\,\overline{\psi_{a,s,t}(x)}\,dx, \tag{1.1} \end{align*}
where $\psi_{a,s,t}(x)=|\det A_{a}|^{\frac{1}{2n}-1}\psi\big(A_{a}^{-1}S_{s}^{-1} \left(x-t\right)\big),a\in\mathbb R\setminus \left\{0\right\}, s\in\mathbb R^{n-1}, t\in\mathbb R^n$ is the shearlet family constituted by the combined action of the scaling $D_{A_{a}}$, sharing $\mathcal{D}_{S_{s}}$ and translation ${T_{t}}$ operators on the analyzing function $\psi \in L^2(\mathbb R^n)$ given by
\begin{align*}
D_{A_{a}}\psi(x)=|\det A_{a}|^{-1/2}\,{\psi}\left({A_{a}^{-1}}x\right),~~\mathcal{D}_{S_{s}}\psi (x)={\psi}\left({S_{s}^{-1}}x\right),~~\text{and}~~ {T_{t}}\psi(x)={\psi}(x-t), \tag{1.2} \end{align*}
respectively, and the matrices involved in (1.2) are given by \begin{align*} A_{a}=\left(\begin{array}{cc}
a & {\bf 0}_{n-1}^{T} \\{\bf 0}_{n-1} & \text{sgn}(a)\, a^{1/n} \,I_{n-1}\\
\end{array}\right) ~~\text{ and}~~
S_{s}=\left(\begin{array}{cc} 1 & {\bf s}^{T} \\ {\bf 0}_{n-1} & I_{n-1} \end{array} \right),\tag{1.3} \end{align*}
${\bf s}^{T}=\big(s_1, s_2,\dots, s_{n-1}\big)$, $\text{sgn}(\cdot)$ and ${\bf 0}$ denotes the well known Signum function and the null vector, respectively. For the brevity, we shall rewrite the shearlet family $\psi_{a,s,t}(x)$ as
\begin{align*}
\psi_{a,s,t}(x)=\big|\det M_{sa}\big|^{-1/2}\psi\Big({ M_{sa}}^{-1} \left(x-t\right)\Big),\tag{1.4} \end{align*} where $M_{sa}=S_{s}A_{a}$ is the composition of the parabolic scaling matrix $A_{a}$ and the shearing matrix $S_{s}$ (see \cite{Kuty}) \begin{align*} M_{sa}=\left(\begin{array}{cccccc}
a & \text{sgn}(a)\,a^{1/n}s_1 & \text{sgn}(a)\, a^{1/n}s_2 & \text{sgn}(a)\, a^{1/n}s_3 & \cdots & \text{sgn}(a)\, a^{1/n}s_{n-1} \\
0 & 0 & \text{sgn}(a)\, a^{1/n} & 0 & \cdots & \cdots \\
\vdots & \vdots & \vdots& \vdots & \vdots & \vdots \\
0 & 0 & 0 &0 & 0 & \text{sgn}(a)\, a^{1/n} \end{array}\right).\tag{1.5} \end{align*}
\pagestyle{myheadings}
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The set $ \mathbb S={\mathbb R}\setminus \left\{0\right\}\times{\mathbb R^{n-1}}\times{\mathbb R^{n}}$ endowed with the operation \begin{align*} \big (a,s,t\big )\odot \big (a^{\prime},s^{\prime},t^{\prime}\big )=\big (aa^{\prime},s+a^{1-\frac{1}{n}} s^{\prime},t+S_{s}A_{a}t^{\prime}\big),\tag{1.6} \end{align*} forms a locally compact group, often called the {\it Shearlet group}. The left Haar measures on $\mathbb S$ is given by $d\eta={da\,ds\,dt}/a^{n+1}$ \cite{Dal2}. For every $\psi\in L^2(\mathbb R^n)$, we define \begin{align*}
U(a,s,t)\psi(x)=\psi_{a,s,t}(x):=|\det M_{sa}|^{-1/2}\psi\big( M_{sa}^{-1}(x-t)\big).\tag{1.7} \end{align*} It is easy to verify that $U: \mathbb S\to {\mathcal U}(L^{2}(\mathbb R^n))$ is a unitary mapping from the shearlet group $\mathbb S$ into the group of unitary operators ${\mathcal U}(L^{2}(\mathbb R^n))$ on $L^{2}(\mathbb R^n)$. In this framework, the continuous shearlet transform (1.1) takes the following form \begin{align*} {\mathcal {SH}}_{\psi}f(a,s,t)=\Big\langle{f,\psi_{a,s,t}}\Big\rangle=\Big\langle{f,U(a,s,t)\psi}\Big\rangle,\quad \text{for all}~f\in L^2(\mathbb R^n). \tag{1.8} \end{align*}
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The Heisenberg's uncertainty principle has played a fundamental role in the development and understanding of quantum mechanics, signal processing and information theory \cite{Fol,HJ}. In quantum mechanics, this principle states that the position and the momentum of a particle cannot be both determined explicitly but only in a probabilistic sense with a certain degree of uncertainty. That is, increasing the knowledge of position, decreases the knowledge of momentum of the particle and vice-versa. The harmonic version of this principle says that a non-trivial function cannot be sharply localized in both time and frequency domains simultaneously \cite{CP,Bec}. With the development of time-frequency analysis, the study of uncertainty principles have gained considerable attention and have been extended to a wide class of integral transforms including the short-time Fourier transform \cite{W}, fractional Fourier transform \cite{Guan}, wavelet transforms \cite{DM,Bat,Shah3}, fractional wavelet transform \cite{MA1}, linear canonical transforms \cite{Zhao} and special affine Fourier transforms \cite{Sun}. The first study aimed to establish the uncertainty principles for the shearlet transforms was initiated by Dahlke et al.\cite{Dal3}, in which the authors have discussed various methods to minimize the uncertainty relations for the infinitesimal generators of the shearlet group. Later on, Su \cite{Su} derive some Heisenberg type uncertainty principles for the continuous shearlet transforms by adopting the strategy analogous to Wilcok \cite{W} and Cowling and Price \cite{CP}. Very recently, Nefzi et al.\cite{Nef} generalized the results of Su \cite{Su} for the multivariate shearlet transform and analyze the net concentration of these transforms on sets of finite measure using the machinery of projection operators. Recent results in this direction can be found in \cite{Shah4,BNT}.
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To date, several generalizations, modifications and variations of the harmonic based uncertainty principles have appeared in the open literature, for instance, the logarithmic uncertainty principles (Beckner-type uncertainty principles), entropy-based uncertainty relations, Benedick's uncertainty principles, Nazarov's uncertainty principles, local uncertainty principles and much more \cite{Bec,Ben,Vem,Naz,Jam,Kubo}. However, to the best of our knowledge, no such work has been explicitly carried out yet for the continuous shearlet transforms. It is therefore interesting and worthwhile to investigate these kinds of uncertainty principles for the continuous shearlet transforms in arbitrary space dimensions. The main objectives of this article are as follows:
\begin{itemize} \item To obtain Pitt's inequality for the continuous shearlet transforms. \item To establish Beckner's uncertainty principle for the continuous shearlet transforms. \item To derive Sobolev-type uncertainty inequalities for the continuous shearlet transforms. \item To formulate Nazarov's uncertainty principle for the continuous shearlet transforms. \item To obtain local uncertainty principles for the continuous shearlet transforms. \end{itemize}
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The rest of the article is structured as follows. In section 2, we establish an analogue of the well known Pitt's inequality for the continuous shearlet transforms in arbitrary space dimensions. In section 3, we derive the Beckner's uncertainty principle and obtain the corresponding Sobolev-type inequality for the continuous shearlet transforms. Sections 4 and 5 are respectively devoted to establishing the Nazarov's and local uncertainty principles for the shearlet transforms in arbitrary space dimensions. The conclusion is drawn in section 6.
\section{Pitt's Inequality for the Continuous Shearlet Transform}
The classical Pitt's inequality expresses a fundamental relationship between a sufficiently smooth function and the corresponding Fourier transform \cite{Bec}. For every $f\in \mathbb S(\mathbb R^n)\subseteq L^2(\mathbb R^n)$, the inequality states that \begin{align*}
\int_{\mathbb R^n}\left|\xi\right|^{-\lambda}\big|\mathscr F\big[f\big](\xi)\big|^2d\xi\le C_{\lambda}\int_{\mathbb R^n}\left|x\right|^{\lambda}\big|f(x)\big|^2dx,\quad 0\le\lambda<1\tag{2.1} \end{align*} where \begin{align*} C_{\lambda}=\pi^{\lambda}\left[\Gamma\left(\frac{n-\lambda}{4}\right)/ \Gamma\left(\frac{n+\lambda}{4}\right)\right]^2,\tag{2.2} \end{align*} and $\Gamma(\cdot)$ denotes the well known Euler's gamma function. Here, $\mathbb S(\mathbb R^n)$ denotes the Schwartz class in $L^2(\mathbb R^n)$ given by \begin{align*}
\mathbb S\left(\mathbb R^n\right)=\left\{f\in C^{\infty}(\mathbb R^n): \sup_{t\in\mathbb R^n}\left|t^{\alpha}{\mathcal \partial}_{t}^{\beta}f(t)\right|<\infty\right\},\tag{2.3} \end{align*} where $C^{\infty}(\mathbb R^n)$ is the class of smooth functions, $\alpha,\beta$ are any two non-negative integers, and ${\partial}_{t}$ denotes the usual partial differential operator.
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The main objective of this section is to formulate an analogue of Pitt's inequality (2.1) for the continuous shearlet transform in arbitrary space dimensions . Formally, we start our investigation with the following lemma.
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{\bf Lemma 2.1.} {\it Let $\psi$ be an admissible shearlet, then for any $f\in{L^2(\mathbb R^{n})}$, we have } \begin{align*}
{\mathscr F}\Big({\mathcal {SH}}_{\psi}f(a,s,t)\Big)\left(\xi\right)=\big|\det A_{a}\big|^{1/2}\hat{f}(\xi)\,\overline{\hat{\psi}\big( M_{sa}\xi\big)}. \tag{2.4} \end{align*}
{\it Proof.} By virtue of Plancheral theorem for the classical Fourier transform, we obtain \begin{align*} {\mathcal {SH}}_{\psi}f(a,s,t)&=\int_{\mathbb R^{n}}f(x)~\overline{U(a,s,t)\,\psi(x)}\,dx\\ &=\int_{\mathbb R^{n}}\mathscr F\big[f\big](\xi)\,\overline{\mathscr F\Big[ U(a,s,t)\psi\Big]}(\xi)\,d{\xi}\\
&=\big|\det A_{a}\big|^{1/2}\int_{\mathbb R^{n}}\hat{f}(\xi)\left\{\int_{\mathbb R^{n}}\overline {\psi\big(D_{M_{sa}}(x-t)\big)\,e^{-2\pi i\xi \cdot x}\,dx}\right\}d\xi\\
&=\big|\det A_{a}\big|^{1/2}\int_{\mathbb R^{n}}\hat{f}(\xi)\left\{\int_{\mathbb R^{n}}\overline{\psi(z)\,e^{-2\pi i \xi\cdot(M_{sa}z+t)}\,dz}\right\}d\xi\\
&=\big|\det A_{a}\big|^{1/2}\int_{\mathbb R^{n}}\hat{f}(\xi)\,\overline{\hat{\psi}\big(M_{sa}\xi\big)\,e^{-2\pi i \xi\cdot t}}\,d\xi\\
&=\big|\det A_{a}\big|^{1/2}\int_{\mathbb R^{n}}\hat{f}(\xi)\,\overline{\hat{\psi}\big( M_{sa}\xi\big)}\,e^{2\pi i \xi\cdot t}\,d\xi\\
&=\big|\det A_{a}\big|^{1/2}{\mathscr F}^{-1}\Big[\hat{f}(\xi)\,\overline{\hat{\psi}\big( M_{sa}\xi\big)}\Big](\xi), \end{align*} which upon applying the Fourier transform yields \begin{align*}
{\mathscr F}\Big({\mathcal {SH}}_{\psi}f(a,s,t)\Big)\left(\xi\right)=\big|\det A_{a}\big|^{1/2}\hat{f}(\xi)\,\overline{\hat{\psi}\big( M_{sa}\xi\big)}. \end{align*}
This completes the proof of Lemma 2.1. \quad \fbox
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In our next lemma, we shall establish the Moyal's principle for the continuous shearlet transform (1.1) in arbitrary space dimensions, which will be employed in the subsequent sections to obtain certain uncertainty inequalities.
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{\bf Lemma 2.2.} {\it Let $\big[{\mathcal {SH}}_{\psi}f\big](a,s,t)$ and $\big[{\mathcal {SH}}_{\psi}g\big](a,s,t)$ be the shearlet transforms for a given pair of square integrable functions $f$ and $g$. Then, the following identity holds:} \begin{align*} \int_{\mathbb S}\Big({\mathcal {SH}}_{\psi}f(a,s,t)\Big)\overline{\Big({\mathcal {SH}}_{\psi}g(a,s,t)\Big)}\,d\eta=C_{\psi}\, \big\langle f,g \big\rangle, \tag{2.5} \end{align*} {\it where $C_{\psi}$ is the admissability condition of the shearlet $\psi\in L^2(\mathbb R^n)$ given by} \begin{align*}
C_{\psi}=\int_{\mathbb R^{n-1}}\int_{{\mathbb R \setminus \left\{0\right\}}}\dfrac{\big|{\hat{\psi}( M_{sa}\xi)}\big|^2}{a^{\frac{n^2-n+1}{n}}}\,da\,ds<\infty.\tag{2.6} \end{align*} {\it Proof.} Using the unitary representation of the continuous shearlet transform (1.8), we have \begin{align*} {\mathcal {SH}}_{\psi}f(a,s,t)&={\Big\langle f, U(a,s,t)\psi \Big\rangle}\\ &={\Big\langle \mathscr F\big[f\big](\xi),\mathscr F \big[U(a,s,t)\psi\big](\xi)\Big\rangle}\\ &=\int_{\mathbb R^{n}}\mathscr F\big[f\big](\xi)~\overline{\mathscr F \Big[U(a,s,t)\psi\Big]}(\xi)\,d\xi\\
&=\big|\det A_{a}\big|^{1/2}\int_{\mathbb R^{n}}\hat{f}(\xi)\,\overline{\hat{\psi}\big(M_{sa}\xi \big)}\,e^{2\pi i \xi\cdot t}\,d\xi.\tag{2.7} \end{align*} Similarly, we have \begin{align*}
{\mathcal {SH}}_{\psi}g(a,s,t)=\big|\det A_{a}\big|^{1/2}\int_{\mathbb R^{n}}\hat{g}(\sigma)\,\overline{\hat{\psi}\big(M_{sa}\sigma \big)}\,e^{2\pi i \sigma \cdot t}\,d\sigma.\tag{2.8} \end{align*} An implication of the well-known Fubini theorem yields \begin{align*} &\int_{\mathbb S}\Big({\mathcal {SH}}_{\psi}f(a,s,t)\Big)\overline{\Big({\mathcal {SH}}_{\psi}g(a,s,t)\Big)}\,d\eta\\ &=\int_{\mathbb R^{n}}\int_{\mathbb R^{n-1}}\int_{\mathbb R \setminus \left\{0\right\}}\Big({\mathcal {SH}}_{\psi}f(a,s,t)\Big)\overline{\Big({\mathcal {SH}}_{\psi}g(a,s,t)\Big)}\,\dfrac{\,da\,ds\,dt}{a^{n+1}}\\
&=\int_{\mathbb R^{n}}\int_{\mathbb R^{n-1}}\int_{\mathbb R \setminus \left\{0\right\}}\left\{\int_{\mathbb R^{n}}\int_{\mathbb R^{n}}\big|\det A_{a}\big|\hat{f}(\xi)\,\overline{\hat{\psi}( M_{sa}\xi)}\,e^{2\pi i \xi\cdot t}\, \overline{\hat{g}(\sigma)}\,\,\hat{\psi}\big( M_{sa}\sigma\big)\,e^{-2\pi i \sigma \cdot t} d\xi\, d\sigma\right\}\dfrac{\,da\,ds\,dt}{a^{n+1}}\\ &=\int_{\mathbb R^{n-1}}\int_{\mathbb R \setminus \left\{0\right\}}\dfrac{\,da\,ds}{a^{\frac{n^2-n+1}{n}}}\int_{\mathbb R^{n}}\int_{\mathbb R^{n}}\hat{f}(\xi)\,\overline{\hat{\psi}( M_{sa}\xi)}\, \,\overline{\hat{g}(\sigma)}\,\hat{\psi}\big( M_{sa}\sigma\big)\,\left\{\int_{\mathbb R^{n}}e^{2\pi i(\xi-\sigma)\cdot\, t}\, dt\right\}d\xi\, d\sigma\\ &=\int_{\mathbb R^{n-1}}\int_{\mathbb R \setminus \left\{0\right\}}\dfrac{\,da\,ds}{a^{\frac{n^2-n+1}{n}}}\int_{\mathbb R^{n}}\int_{\mathbb R^{n}}\hat{f}(\xi)\,\overline{\hat{\psi}( M_{sa}\xi)}\,\,\overline{\hat{g}(\sigma)}\,\hat{\psi}\big( M_{sa}\sigma\big)\,\,\delta(\sigma-\xi)\,d\xi\, d\sigma\\
&=\int_{\mathbb R^{n-1}}\int_{\mathbb R \setminus \left\{0\right\}}\dfrac{\,da\,ds}{a^{\frac{n^2-n+1}{n}}}\int_{\mathbb R^{n}}\hat{f}(\xi)\,\overline{\hat{g}(\xi)}\,\,\left|{\hat{\psi}\big( M_{sa}\xi\big)}\right|^2\,d\xi\\
&=\int_{\mathbb R^{n}}\hat{f}(\xi)\,\overline{\hat{g}(\xi)}\,\left\{\int_{\mathbb R^{n-1}}\int_{\mathbb R \setminus \left\{0\right\}}\dfrac{\left|{\hat{\psi}\big( M_{sa}\xi\big)}\right|^2}{a^{\frac{n^2-n+1}{n}}}\,da\,ds\right\}\,d\xi\\ &=C_{\psi}\left\langle \hat{f},\hat{g}\right\rangle\\ &=C_{\psi}\,\big\langle f, g\big\rangle. \end{align*}
This completes the proof of Lemma 2.2. \quad \fbox
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{\it Remarks.} (i). For $f=g$, equation (2.5) yields the following energy preserving relation \begin{align*}
&\int_{\mathbb S}\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^{2}{d\eta}=C_{\psi}\big\|f\big\|^{2}_{2}.\tag{2.9} \end{align*} (ii). Equation (2.9) demonstrates that the continuous shearlet transform (1.1) is a bounded linear operator from $L^2(\mathbb R^n)$ to $L^2({\mathbb R \setminus \left\{0\right\}}\times\mathbb R^{n-1}\times\mathbb R^n)$.
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(iii). For $C_{\psi}=1$, the continuous shearlet transform (1.1) becomes an isometry from the space of signals $L^2(\mathbb R^n)$ to the space of transforms $L^2({\mathbb R \setminus \left\{0\right\}}\times\mathbb R^{n-1}\times\mathbb R^n).$
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We are now in a position to establish the Pitt's inequality for the continuous shearlet transforms in arbitrary space dimensions.
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{\bf Theorem 2.3.} {\it For any arbitrary $f\in \mathcal S(\mathbb R^n)\subseteq L^2(\mathbb R^n)$, the Pitt's inequality for the continuous shearlet transform (1.1) is given by:} \begin{align*}
C_{\psi}\int_{\mathbb R^{n}}\left|\xi\right|^{-\lambda}\left|\hat{f}(\xi)\right|^2\,d\xi\le C_{\lambda}\int_{\mathbb S}\left|t\right|^{\lambda}\Big|\mathcal {SH}_{\psi}f(a,s,t)\Big|^2 d\eta.\tag{2.10} \end{align*} {\it where $C_{\psi}$ is the admissability condition given by (2.6).}
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{\it Proof.} As a consequence of the inequality (2.1), we can write \begin{align*}
\int_{\mathbb R^n}\left|\xi\right|^{-\lambda}\left|\mathscr F\Big[\mathcal {SH}_{\psi}f(a,s,t)\Big](\xi)\right|^2d\xi\le C_{\lambda}\int_{\mathbb R^n}\left|t\right|^{\lambda}\Big|\mathcal {SH}_{\psi}f(a,s,t)\Big|^2dt,\tag{2.11} \end{align*} which upon integration with respect to the Haar measure $d\eta=dsda/a^{n+1}$ yields \begin{align*}
\int_{\mathbb R^{n}}\int_{\mathbb R^{n-1}}\int_{{\mathbb R \setminus \left\{0\right\}}}\left|\xi\right|^{-\lambda}\Big|\mathscr F\Big[\mathcal {SH}_{\psi}f(a,s,t)\Big](\xi)\Big|^2\dfrac{d\xi\,ds\,da}{a^{n+1}}\le C_{\lambda}\int_{\mathbb S}\left|t\right|^{\lambda}\Big|\mathcal {SH}_{\psi}f(a,s,t)\Big|^2 d\eta.\tag{2.12} \end{align*} Invoking Lemma 2.1, we can express the inequality (2.12) in the following manner: \begin{align*}
\int_{\mathbb R^{n}}\int_{\mathbb R^{n-1}}\int_{{\mathbb R \setminus \left\{0\right\}}}\left|\xi\right|^{-\lambda}\left|\hat{f}(\xi)~\overline{\hat{\psi}\big( M_{sa}\xi\big)}\right|^2\,\dfrac{d\xi\,ds\,da}{a^{\frac{n^2-n+1}{n}}}\le C_{\lambda}\int_{\mathbb S}\left|t\right|^{\lambda}\Big|\mathcal {SH}_{\psi}f(a,s,t)\Big|^2 d\eta.\tag{2.13} \end{align*} Equivalently, we have \begin{align*}
\int_{\mathbb R^{n}}\left|\xi\right|^{-\lambda}\left|\hat{f}(\xi)\right|^2\left\{\int_{\mathbb R^{n-1}}\int_{{\mathbb R \setminus \left\{0\right\}}}\dfrac{\big|{\hat{\psi}( M_{sa}\xi)}\big|^2}{a^{\frac{n^2-n+1}{n}}}\,da\,ds \right\}\,d\xi\le C_{\lambda}\int_{\mathbb S}\left|t\right|^{\lambda}\Big|\mathcal {SH}_{\psi}f(a,s,t)\Big|^2 d\eta.\tag{2.14} \end{align*} Since $\psi$ is an admissible shearlet, therefore inequality (2.14) becomes \begin{align*}
C_{\psi}\int_{\mathbb R^{n}}\left|\xi\right|^{-\lambda}\left|\hat{f}(\xi)\right|^2\,d\xi\le C_{\lambda}\int_{\mathbb S}\left|t\right|^{\lambda}\Big|\mathcal {SH}_{\psi}f(a,s,t)\Big|^2 d\eta,\tag{2.15} \end{align*} which establishes the Pitt's inequality for the continuous shearlet transform in arbitrary space dimensions. \quad \fbox
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{\it Remark:} For $\lambda=0$, equality holds in (2.10), which is in consonance with the classical Pitt's inequality.
\section{Beckner-type Inequalities for the Continuous Shearlet Transforms}
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The classical Beckner's inequality \cite{Bec} is given by \begin{align*}
\int_{\mathbb R^{n}}{\ln|t|} ~{\big|f(t)\big|^{2}}\,dt+\int_{\mathbb R^{n}}{\ln{|\xi|}\left|\hat{f}(\xi)\right|^2}\,d{\xi}\geq \left(\dfrac{{\Gamma^{\prime}}(1/2)}{\Gamma(1/2)}-\ln{\pi}\right)\int_{\mathbb R^{n}} \big|f(t)\big|^{2}\,dt\tag{3.1} \end{align*} for all $f\in L^{2}(\mathbb R^{n})$, for which the quantity on left is defined, where $t\in{\mathbb R^{2}}$, and $\Gamma(t)$ is the gamma function. This inequality is related to the classical Heisenberg's uncertainty principle and for that reason it is often referred as the logarithmic uncertainty principle. Considerable attention has been paid to this inequality for its various generalizations, improvements, analogues, and their applications in science and engineering (see \cite{Fol,HJ,Guan,MA1,Shah4}).
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{\bf Theorem 3.1.} {\it Let $\big[{\mathcal {SH}}_{\psi}f\big](a,s,t)$ be the shearlet transform of any arbitrary function $f\in \mathbb S(\mathbb R^n)$, the following inequality holds:} \begin{align*}
\int_{\mathbb S}{\ln|t|} \,{\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^{2}}\,d\eta+{C_{\psi}}\int_{\mathbb R^{n}}{\ln{|\xi|} \left|\hat{f}(\xi)\right|^2}\,d{\xi}\geq {C_{\psi}}\left[\dfrac{{\Gamma^\prime}(n/4)}{\Gamma(n/4)}-\ln{\pi}\right]\big\|f\big\|_{2}^{2},\tag{3.2} \end{align*} {\it where $C_{\psi}$ is given by (2.6).}
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{\it Proof.} For every $0\le \lambda <1$, we define \begin{align*}
P\left({\lambda}\right)=C_{\psi}\int_{\mathbb R^{n}}\left|\xi\right|^{-\lambda}\left|\hat{f}(\xi)\right|^2\,d\xi- C_{\lambda}\int_{\mathbb S}\left|t\right|^{\lambda}\Big|\mathcal {SH}_{\psi}f(a,s,t)\Big|^2 d\eta.\tag{3.3} \end{align*} On differentiating (3.3) with respect to $\lambda$, we obtain \begin{align*}
P^{\prime}\left({\lambda}\right)=-C_{\psi}\int_{\mathbb R^{n}}\left|\xi\right|^{-\lambda}\ln\big|\xi\big|\left|\hat{f}(\xi)\right|^2\,d\xi- C_{\lambda}\int_{\mathbb S}\left|t\right|^{\lambda}\ln\big|t\big|\Big|\mathcal {SH}_{\psi}f(a,s,t)\Big|^2 d\eta\\
-C^{\,\prime}_{\lambda}\int_{\mathbb S}\left|t\right|^{\lambda}\Big|\mathcal {SH}_{\psi}f(a,s,t)\Big|^2 d\eta.\tag{3.4} \end{align*} where \begin{align*} C^{\,\prime}_{\lambda}&=-\dfrac{\pi^{\lambda}}{2}\left\{\dfrac{{\Gamma}^2\left(\dfrac{n+\lambda}{4}\right)\Gamma\left(\dfrac{n-\lambda}{4}\right) {\Gamma}^{\prime}\left(\dfrac{n-\lambda}{4}\right)+{\Gamma}^2\left(\dfrac{n-\lambda}{4}\right)\Gamma\left(\dfrac{n+\lambda}{4}\right) {\Gamma}^{\prime}\left(\dfrac{n+\lambda}{4}\right)}{{\Gamma}^2\left(\dfrac{n+\lambda}{4}\right)}\right\}\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\pi^{\lambda}\ln \pi \left\{{\Gamma}^2\left(\dfrac{n-\lambda}{4}\right)\big /{\Gamma}^2\left(\dfrac{n+\lambda}{4}\right)\right\}.\tag{3.5} \end{align*} For $\lambda=0$, equation (3.5) yields \begin{align*} C^{\,\prime}_{0}=\left[\ln \pi-\dfrac{\Gamma^{\prime}(n/4)}{\Gamma (n/4)}\right].\tag{3.6} \end{align*} By virtue of Pitt's inequality (2.10) for the shearlet transforms, it follows that $P(\lambda)\le0$, for all $\lambda\in [0,1)$ and \begin{align*}
P(0)&=C_{\psi}\int_{\mathbb R^{n}}\left|\hat{f}(\xi)\right|^2\,d\xi- C_{0}\int_{\mathbb S}\Big|\mathcal {SH}_{\psi}f(a,s,t)\Big|^2 d\eta
=C_{\psi}\left\|\hat{f}\right\|_2^2-C_{\psi}\big\|f\big\|_2^2=0.\tag{3.7} \end{align*} Therefore, for any $h>0$, we observe that $P^{\,\prime}\left(0+h\right)\le0$, whenever $h\rightarrow0$; that is, \begin{align*}
&-C_{\psi}\int_{\mathbb R^{n}}\ln\big|\xi\big|\left|\hat{f}(\xi)\right|^2d\xi- C_{0}\int_{\mathbb S}\ln\big|t\big|\Big|\mathcal {SH}_{\psi}f(a,s,t)\Big|^2 d\eta
-C^{\,\prime}_{0}\int_{\mathbb S}\Big|\mathcal {SH}_{\psi}f(a,s,t)\Big|^2 d\eta\le 0.\tag{3.8} \end{align*} Applying the energy preserving relation (2.9) and the obtained estimate (3.6) of $C^{\,\prime}_{0}$, we obtain \begin{align*}
-C_{\psi}\int_{\mathbb R^{n}}\ln\big|\xi\big|\left|\hat{f}(\xi)\right|^2\,d\xi- \int_{\mathbb S}\ln\big|t\big|\Big|\mathcal {SH}_{\psi}f(a,s,t)\Big|^2 d\eta
-\left[\ln \pi-\dfrac{\Gamma^{\prime}(1/2)}{\Gamma (1/2)}\right]C_{\psi}\,\big\|f\big\|_2^2\le0, \end{align*} or equivalently, \begin{align*}
\int_{\mathbb S}\ln\big|t\big|\Big|\mathcal {SH}_{\psi}f(a,s,t)\Big|^2 d\eta+
C_{\psi}\int_{\mathbb R^{n}}\ln\big|\xi\big|\left|\hat{f}(\xi)\right|^2 d\xi\ge
\left[\dfrac{\Gamma^{\prime}(n/4)}{\Gamma (n/4)}-\ln \pi\right]C_{\psi}\,\big\|f\big\|_2^2.\tag{3.9} \end{align*} Inequality (3.9) is the desired Beckner's uncertainty principle for the continuous shearlet transform in arbitrary space dimensions.\quad\fbox
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We now present an alternate proof of Theorem 3.1. The strategy of the proof is different and is obtained directly from the classical Beckner's inequality (3.1).
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{\it Second Proof of the Theorem 3.1.} We shall identify ${\mathcal {SH}}_{\psi}f(a,s,t)$ as a function of the translation parameter $t$ and then replace $f \in \mathbb S(\mathbb R^n)$ in (3.1) with ${\mathcal {SH}}_{\psi}f(a,s,t)$, so that \begin{align*}
\int_{\mathbb R^{n}}{\ln|t|} \,{\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^{2}}\,dt&+\int_{\mathbb R^{n}}{\ln{|\xi|}\Big|{\mathscr F}\Big[{\mathcal {SH}}_{\psi}f(a,s,t)\Big](\xi)\Big|^2}\,d{\xi}\\
&\ge\left(\dfrac{{\Gamma^{\prime}}(n/4)}{\Gamma(n/4)}-\ln{\pi}\right)\int_{\mathbb R^{n}} \Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^{2}\,dt.\tag{3.10} \end{align*} Integrating (3.10) with respect to the measure $d\eta=dads/a^{n+1}$, we obtain \begin{align*}
\int_{\mathbb S} {\ln|t|} \,{\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^{2}} d\eta+\int_{{\mathbb R \setminus \left\{0\right\}}}\int_{\mathbb R}\int_{\mathbb R^{n}}{\ln{|\xi|}\, \Big|{\mathscr F}\Big[{\mathcal {SH}}_{\psi}f(a,s,t)\Big](\xi)\Big|^2}\,\dfrac{da\,ds\,d{\xi}}{a^{n+1}}\\
\geq \left(\dfrac{{\Gamma^{\prime}}(n/4)}{\Gamma(n/4)}-\ln{\pi}\right)\int_{\mathbb S}\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^{2} d\eta.\tag{3.11} \end{align*} Using equation (2.9), we have \begin{align*}
\int_{\mathbb S}\,\ln|t| \,{\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^{2}}\,d\eta+\int_{\mathbb S}\,{\ln{|\xi|}\, \Big|{\mathscr F}\Big[{\mathcal {SH}}_{\psi}f(a,s,t)\Big](\xi)\Big|^2}\,d\eta\geq \left(\dfrac{{\Gamma^{\prime}}(n/4)}{\Gamma(n/4)}-\ln{\pi}\right)C_{\psi}\big\|f\big\|_2^2.\tag{3.12} \end{align*} We shall now simplify the second integral of (3.12) as \begin{align*}
&\int_{\mathbb S}\ln|\xi| \Big|{\mathscr F}\Big[{\mathcal {SH}}_{\psi}f(a,s,t)\Big]\left(\xi\right)\Big|^2 d\eta \\
&\quad=\int_{\mathbb R^{n-1}}\int_{{\mathbb R \setminus \left\{0\right\}}}\int_{\mathbb R^{n}}\ln|\xi|\Big|{\mathscr F}\Big[{\mathcal {SH}}_{\psi}f(a,s,t)\Big]\left(\xi\right)\Big|^2 \dfrac{d{\xi}\,da\,ds}{a^{n+1}}\\
&\quad=\int_{\mathbb R^{n-1}}\int_{{\mathbb R \setminus \left\{0\right\}}}\int_{\mathbb R^{n}}\ln|\xi|\Big[{\mathscr F}\Big[{\mathcal {SH}}_{\psi}f(a,s,t)\Big]\left(\xi\right)\Big]\overline{\Big[{\mathscr F}\Big[{\mathcal {SH}}_{\psi}f(a,s,t)\Big]\left(\xi\right)\Big]}\,\dfrac{d{\xi}\,da\,ds}{a^{n+1}}\\
&\quad=\int_{\mathbb R^{n-1}}\int_{{\mathbb R \setminus \left\{0\right\}}}\int_{\mathbb R^{n}}\ln\left|\xi\right| \big|\det A_{a}\big|\,\hat{f}(\xi)\,\overline{\hat{\psi}\big( M_{sa}\xi\big)}\,\overline{\hat{f}(\xi)}\,\hat{\psi}\big(\xi M_{sa}\big)\,\dfrac{d{\xi}\,da\,ds}{a^{n+1}}\\
&\quad=\int_{\mathbb R^{n-1}}\int_{{\mathbb R \setminus \left\{0\right\}}}\int_{\mathbb R^{n}}\big|\det A_{a}\big|\,\ln\left|\xi\right|\, \left|\hat{\psi}( M_{sa}\xi)\right|^{2}\,\left|\hat{f}(\xi)\right|^{2}\dfrac{d{\xi}\,da\,ds}{a^{n+1}}\\
&\quad=\int_{\mathbb R^{n-1}}\int_{{\mathbb R \setminus \left\{0\right\}}}\int_{\mathbb R^{n}}\big|\det A_{a}\big|\ln\left|\xi\right| \left|\hat{\psi}( M_{sa}\xi)\right|^{2}\left|\hat{f}(\xi)\right|^{2}\dfrac{d{\xi}\,da\,db}{a^{n+1}}\\
&\quad=\int_{\mathbb R^{n}}{\ln{\left|\xi\right|} \left|\hat{f}(\xi)\right|^2}\Bigg\{\int_{\mathbb R^{n-1}}\int_{{\mathbb R \setminus \left\{0\right\}}}\dfrac{\big|\hat{\psi}(M_{sa}\xi)\big|^{2}}{a^{\frac{n^2-n+1}{n}}}\,da\,ds\Bigg\}d{\xi}\\
&\quad=C_{\psi}\int_{\mathbb R^{n}}{\ln{\left|\xi\right|} \left|\hat{f}(\xi)\right|^2}d{\xi}.\tag{3.13} \end{align*} Plugging the estimate (3.13) in (3.12) gives the desired inequality for the continuous shearlet transforms as \begin{align*}
\int_{\mathbb S}{\ln|t|} \,{\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^{2}}\,d\eta + C_{\psi}\int_{\mathbb R^{n}}\ln|\xi| \left|\hat{f}(\xi)\right|^{2} d{\xi}\geq \left(\dfrac{{\Gamma^{\prime}}(n/4)}{\Gamma(n/4)}-\ln{\pi}\right)C_{\psi}\big\|f\big\|^{2}_{2}. \end{align*} This completes the second proof of Theorem 3.1. \quad \fbox
\parindent=0mm
{\it{Deduction:}} Using Jensen's inequality in (3.2), we obtain an analogue of the classical Heisenberg's uncertainty inequality for the continuous shearlet transforms as \begin{align*}
&\ln \left\{ \int_{\mathbb S} |t|^2 \,\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^2 d\eta \, C_{\psi} \int_{\mathbb R^n} |\xi|^2 \left|\hat f(\xi)\right|^2 d\xi \right\}^{1/2}\\
&= \ln \left\{ \int_{\mathbb S} |t|^2 \,\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^2 d\eta\right\}^{1/2}+\ln \left(C_{\psi}\right)^{1/2}+\ln \left\{ \int_{\mathbb R^n} |\xi|^2 \left|\hat f(\xi)\right|^2 d\xi \right\}^{1/2}\\
&\ge \int_{\mathbb S}{\ln|t|} \,{\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^{2}}\,d\eta +\ln \left(C_{\psi}\right)^{1/2}+\int_{\mathbb R^{n}}\ln|\xi| \left|\hat{f}(\xi)\right|^{2} d{\xi}\\
&\ge \left(\dfrac{{\Gamma^{\prime}}(n/4)}{\Gamma(n/4)}-\ln{\pi}\right)C_{\psi}\big\|f\big\|^{2}_{2}+\ln \left(C_{\psi}\right)^{1/2}, \end{align*} which upon simplification with $C_\psi=1$ yields \begin{align*}
\left\{\int_{\mathbb S} |t|^2 \,\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^2 d\eta\right\}^{1/2}\left\{ \int_{\mathbb R^n} |\xi|^2 \left|\hat f(\xi)\right|^2 d\xi \right\}^{1/2}\ge \exp\left\{\dfrac{ -2\sqrt{\pi}\ln 2}{\sqrt {\pi}}-\ln \pi\right\}\big\|f\big\|^{2}_{2}= \dfrac{\big\|f\big\|^{2}_{2}}{4\pi}. \end{align*}
\parindent=0mm
The remaining part of this Section is devoted to establish the Sobolev-type uncertainty inequality for the continuous shearlet transform in arbitrary space dimensions. This inequality is employed in the Section 5 to obtain a local-type uncertainty principle for the continuous shearlet transform (1.1). To facilitate our intention, we start with the following definitions:
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{\bf Definition 3.2.} The Sobolev space on $\mathbb R^n$ is defined by \begin{align*} \mathbb H\left(\mathbb R^n\right)=\Big\{f\in L^2(\mathbb R^n): \nabla f\in L^2(\mathbb R^n)\Big\},\tag{3.14} \end{align*} where $\nabla$ denotes the differential operator given by $\nabla=\left(\dfrac{\partial}{\partial x_1},\dfrac{\partial}{\partial x_2},\dots,\dfrac{\partial}{\partial x_n}\right)$.
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{\bf Definition 3.3.} For $1\le p<\infty$ and $b>0$, the weighted Lebesgue space on $\mathbb R^n$ is defined by \begin{align*} \mathbb W_{b}^{p}\left(\mathbb R^n\right)=\Big\{f\in L^p(\mathbb R^n): \langle t\rangle^{b}f\in L^p(\mathbb R^n)\Big\},\tag{3.15} \end{align*}
where $ \langle t\rangle$ is the weight function given by $\langle t\rangle=\big(1+\left|t\right|^2\big)^{1/2}$, $t\in\mathbb R^n$.
\parindent=8mm
The logarithmic Sobolev inequality is related to the class of functions $\mathbb H\left(\mathbb R^n\right)$ states that for any non-trivial function $f\in\mathbb H\left(\mathbb R^n\right)$ \cite{PD}, \begin{align*}
\int_{\mathbb R^n}\big|f(t)\big|^2\ln\left(\dfrac{\left|f(t)\right|^2}{\left\|f\right\|_2^2}\right)\,dt\le
\dfrac{n}{2}\ln\left(\dfrac{2}{n\pi e \left\|f\right\|_2^2}\int_{\mathbb R^n}\big|\nabla f(t)\big|^2dt\right).\tag{3.16} \end{align*} Inequality (3.18) is often referred as Gross's inequality \cite{PD,Gen}. On the other hand, Beckner \cite{Bec} proved another version of logarithmic Sobolev inequality for extremal functions which offers better estimate than Gross's inequality (3.16) given by \begin{align*}
\int_{\mathbb R^n}\big|f(t)\big|^2\ln\left(\dfrac{\left|f(t)\right|^2}{\left\|f\right\|_2^2}\right)\,dt\le
\dfrac{n}{2}\int_{\mathbb R^n}\left|\hat{f}(\xi)\right|^2\ln \left(B_n\left|\xi\right|^2\right)\,d\xi- n\big\|f\big\|^2_{2} \left(\dfrac{\Gamma^{\prime}(n/2)}{\Gamma(n/2)}\right),\tag{3.17} \end{align*} where $B_n=\frac{1}{4\pi}\left(\frac{\Gamma (n)}{\Gamma(n/2)}\right)^{2/n}$.
\parindent=8mm
Very recently, Kubo et al.\cite{Kubo} obtained a logarithmic Sobolev-type inequality for the weighted Lebesgue spaces $\mathbb W_{b}^{p}\left(\mathbb R^n\right)$ and pointed out that the obtained inequality has a dual relation with the Beckner's inequality (3.1). For any non-trivial function $f\in \mathbb W_{b}^{1}\big({\mathbb R^n}\big),$ the inequality states that \begin{align*}
-\int_{\mathbb R^n}\big|f(t)\big|\ln\left\{\dfrac{\left|f(t)\right|}{\left\|f\right\|_1}\right\}\,dt\le n\int_{\mathbb R^n}\big|f(t)\big| \ln \left\{C_{n,b}\left(1+\left|t\right|^{b}\right)\right\}\,dt,\tag{3.18} \end{align*} where \begin{align*} C_{n,b}=\left\{\dfrac{2\,\pi^{n/2}\Gamma(n/b)\,\Gamma(n/b^{\prime})}{b\, \Gamma(n)\,\Gamma(n/2)}\right\}^{1/n}\quad {\text {and}}\quad \dfrac{1}{b}+\dfrac{1}{b^{\,\prime}}=1.\qquad\qquad\tag{3.19} \end{align*}
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Furthermore, the duality has been shown in the following sense: \begin{align*}
\int_{\mathbb R^{n}}\big|f(t)\big|^2\ln\left(\dfrac{1+\left|t\right|^2}{2}\right)\,dt+\int_{\mathbb R^{n}}\left|\hat {f}(\xi)\right|^2\ln\big|\xi\big|\,d{\xi}\ge\left(\dfrac{{\Gamma^{\prime}}(n/2)}{\Gamma(n/2)}\right)\int_{\mathbb R^{n}}\big|f(t)\big|^2 dt.\tag{3.20} \end{align*}
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The following theorem is the main result of this subsection which establishes an analogue of the Sobolev-type uncertainty inequality (3.20) for the continuous shearlet transforms in arbitrary space dimensions.
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{\bf Theorem 3.4.} {\it If $\big[{\mathcal {SH}}_{\psi}f\big](a,s,t)$ is the shearlet transform of any arbitrary function $f \in \mathbb H(\mathbb R^n)\cap \mathbb W_1^{1}(\mathbb R^n)$, then the following Sobolev-type uncertainty inequality holds:} \begin{align*}
\int_{\mathbb S} \Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^2\ln\left(\dfrac{1+\left|t\right|^2}{2}\right)\,d\eta+C_{\psi}\int_{\mathbb R^{n}}{\ln{\left|\xi\right|} \left|\hat{f}(\xi)\right|^2}d{\xi}
\ge\left(\dfrac{{\Gamma^{\prime}}(n/2)}{\Gamma(n/2)}\right)C_{\psi}\big\|f\big\|^{2}_{2},\tag{3.21} \end{align*} {\it whenever the L.H.S of (3.21) is defined.}
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{\it Proof.} As a consequence of inequality (3.20), we have \begin{align*}
\int_{\mathbb R^{n}}\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^2\ln\left(\dfrac{1+\left|t\right|^2}{2}\right)\,dt+\int_{\mathbb R^{n}}\Big|\mathscr F \Big[{\mathcal {SH}}_{\psi}f(a,s,t)\Big](\xi)\Big|^2\ln\big|\xi\big|\,d{\xi}\\
\qquad\qquad\ge\left(\dfrac{{\Gamma^{\prime}}(n/2)}{\Gamma(n/2)}\right)\int_{\mathbb R^{n}}\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^2dt, \end{align*} which upon integration yields \begin{align*}
\int_{\mathbb S} \Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^2\ln\left(\dfrac{1+\left|t\right|^2}{2}\right)\,d\eta+\int_{\mathbb S}\ln\big|\xi\big|\, \Big|\mathscr F \Big[{\mathcal {SH}}_{\psi}f(a,s,t)\Big](\xi)\Big|^2\,d\eta\\
\qquad\qquad\ge\left(\dfrac{{\Gamma^{\prime}}(n/2)}{\Gamma(n/2)}\right)\int_{\mathbb S}\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^2d\eta.\tag{3.22} \end{align*} Using the estimate (3.13) for the second integral on the L.H.S of (3.22) and invoking (2.9), we obtain \begin{align*}
\int_{\mathbb S} \Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^2\ln\left(\dfrac{1+\left|t\right|^2}{2}\right)\,d\eta+C_{\psi}\int_{\mathbb R^{n}}{\ln{\left|\xi\right|} \left|\hat{f}(\xi)\right|^2}d{\xi}
\ge\left(\dfrac{{\Gamma^{\prime}}(n/2)}{\Gamma(n/2)}\right)C_{\psi}\big\|f\big\|^{2}_{2},\tag{3.23} \end{align*} where $C_{\psi}$ is given by (2.6). This completes the proof of Theorem 3.4.\quad \fbox
\section{Nazarov-type Inequality for the Shearlet Transforms}
As is well known, the classical Heisenberg's uncertainty principle measures the localization in terms of the dispersions of the respective functions. Considering an alternate criterion of localization; that is, the smallness of the support, Nazarov \cite{Naz,Jam} proposed an uncertainty principle which is concerned with the query; what happens if a non-zero function and its Fourier transform are small outside a compact set? The Nazarov's uncertainty principle in the classical Fourier domain states that if $E_1$ and $E_2$ are two subsets of $\mathbb R^n$ with finite measure, then \begin{align*}
\int_{\mathbb R^n}\big|f(t)\big|^2 dt\le K\,e^{K\left|E_1\right|\left|E_2\right|}\left\{\int_{\mathbb R^n\setminus E_1}\big|f(t)\big|^2 dt+\int_{\mathbb R^n\setminus E_2}\left|\hat {f}(\xi)\right|^2 d\xi\right\},\tag{4.1} \end{align*}
where $K$ is a positive constant, and $\left|E_1\right|$ and $\left|E_2\right|$ denote the measures of $E_1$ and $E_2$ , respectively.
\parindent=8mm
In this Section, our primary interest is to establish the Nazarov's uncertainty principle for the continuous shearlet transforms in arbitrary space dimensions by employing the inequality (4.1). In this direction, we have the following main theorem.
\parindent=0mm
{\bf Theorem 4.1.} {\it Let $\big[{\mathcal {SH}}_{\psi}f\big](a,s,t)$ be the shearlet transform of any arbitrary function $f\in L^2(\mathbb R^n)$, then the following uncertainty inequality holds:} \begin{align*}
\int_{\mathbb R^n\setminus E_1}\int_{\mathbb R^{n-1}}\int_{{\mathbb R \setminus \left\{0\right\}}}\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^2 \dfrac{da\,ds\,dt}{a^{n+1}}+C_{\psi}\int_{\mathbb R^n\setminus E_2}\left|\hat{f}(\xi)\right|^2d\xi\ge \dfrac{C_{\psi}\big\|f\big\|^{2}_{2}}{e^{K\left|E_1\right|\left|E_2\right|}},\tag{4.2} \end{align*} {\it where $C_{\psi}$ is given by (2.6), $E_1$, $E_2$ are two subsets of $\mathbb R^n$ with finite measures and $K$ is a positive constant.}
\parindent=0mm
{\it Proof.} Since ${\mathcal {SH}}_{\psi}f(a,s,t)\in L^2(\mathbb R^n)$, whenever $f\in L^2(\mathbb R^n)$, so we can replace the function $f$ appearing in (4.1) with ${\mathcal {SH}}_{\psi}f(a,s,t)$ to get \begin{align*}
&\int_{\mathbb R^n}\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^2 dt\\
&\le K\,e^{K\left|E_1\right|\left|E_2\right|}\left\{\int_{\mathbb R^n\setminus E_1}\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^2 dt+\int_{\mathbb R^n\setminus E_2}\left|\mathscr F\Big[{\mathcal {SH}}_{\psi}f(a,s,t)\Big](\xi)\right|^2 d\xi\right\}.\tag{4.3} \end{align*} By integrating (4.3), we obtain \begin{align*}
&\int_{\mathbb R^{n}}\int_{\mathbb R^{n-1}}\int_{{\mathbb R \setminus \left\{0\right\}}}\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^2 \dfrac{da\,ds\,dt}{a^{n+1}}\le K\,e^{K\left|E_1\right|\left|E_2\right|}\\
&\times\left\{\int_{\mathbb R^n\setminus E_1}\int_{\mathbb R^{n-1}}\int_{{\mathbb R \setminus \left\{0\right\}}}\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^2 \dfrac{da\,ds\,dt}{a^{n+1}}+\int_{\mathbb R^2\setminus E_2}\int_{\mathbb R}\int_{{\mathbb R \setminus \left\{0\right\}}}\Big|\mathscr F\Big[{\mathcal {SH}}_{\psi}f(a,s,t)\Big](\xi)\Big|^2 \dfrac{da\,ds\,d\xi}{a^{n+1}}\right\}. \end{align*} Using Lemma 2.1 together with the energy preserving relation (2.9), the above inequality becomes \begin{align*}
\int_{\mathbb R^n\setminus E_1}\int_{\mathbb R^{n-1}}\int_{{\mathbb R \setminus \left\{0\right\}}}\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^2 \dfrac{da\,ds\,dt}{a^{n+1}}&+\int_{\mathbb R^n\setminus E_2}\int_{\mathbb R^{n-1}}\int_{{\mathbb R \setminus \left\{0\right\}}}\left|\hat{f}(\xi)~\overline{\hat{\psi}( M_{sa}\xi)}\right|^2\,\dfrac{d\xi\,ds\,da}{a^{\frac{n^2-n+1}{n}}}\\
&\qquad\qquad\qquad\qquad\ge\dfrac{C_{\psi}\big\|f\big\|^{2}_{2}}{Ke^{K\left|E_1\right|\left|E_2\right|}}, \end{align*} which further implies \begin{align*}
\int_{\mathbb R^n\setminus E_1}\int_{\mathbb R^{n-1}}\int_{{\mathbb R \setminus \left\{0\right\}}}\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^2 \dfrac{da\,ds\,dt}{a^3}&+\int_{\mathbb R^n\setminus E_2}\left|\hat{f}(\xi)\right|^2\left\{\int_{\mathbb R^{n-1}}\int_{{\mathbb R \setminus \left\{0\right\}}}\dfrac{\Big|\hat{\psi}( M_{sa}\xi)\Big|^2}{a^{\frac{n^2-n+1}{n}}}da\,ds \right\}\,d\xi\\
&\qquad\qquad\qquad\qquad\quad\ge \dfrac{C_{\psi}\big\|f\big\|^{2}_{2}}{Ke^{K\left|E_1\right|\left|E_2\right|}}.\tag{4.4} \end{align*} Since $\psi\in L^2(\mathbb R^n)$ is an admissible shearlet, therefore (4.4) takes the form \begin{align*}
\int_{\mathbb R^n\setminus E_1}\int_{\mathbb R^{n-1}}\int_{{\mathbb R \setminus \left\{0\right\}}}\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^2 \dfrac{da\,ds\,dt}{a^3}+C_{\psi}\int_{\mathbb R^n\setminus E_2}\left|\hat{f}(\xi)\right|^2d\xi\ge \dfrac{C_{\psi}\big\|f\big\|^{2}_{2}}{Ke^{K\left|E_1\right|\left|E_2\right|}}, \end{align*} which is the desired Nazarov's uncertainty principle for the continuous shearlet transforms in arbitrary space dimensions. \qquad \fbox
\parindent=0mm
{\it Deduction:} As a consequence of (4.1), we can write \begin{align*}
\int_{\mathbb R^n\setminus E_2}\left|\hat {f}(\xi)\right|^2 d\xi&\ge\dfrac{1}{K\,e^{K\left|E_1\right|\left|E_2\right|}}\int_{\mathbb R^n}\big|f(t)\big|^2 dt-\int_{\mathbb R^n\setminus E_1}\big|f(t)\big|^2 dt\qquad\quad\\
&=\dfrac{\big\|f\big\|_2^2}{Ke^{K\left|E_1\right|\left|E_2\right|}}-\int_{\mathbb R^n\setminus E_1}\big|f(t)\big|^2 dt.\tag{4.5} \end{align*} Using (4.5) in (4.2), the Nazrov's inequality for the shearlet transforms reduces to \begin{align*}
\int_{\mathbb R^n\setminus E_1}\int_{\mathbb R^{n-1}}\int_{{\mathbb R \setminus \left\{0\right\}}}\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^2 \dfrac{da\,ds\,dt}{a^{n+1}}+\dfrac{C_{\psi}\big\|f\big\|_2^2}{Ne^{N\left|E_1\right|\left|E_2\right|}}-C_{\psi}\int_{\mathbb R^n\setminus E_1}\big|f(t)\big|^2 dt\ge \dfrac{C_{\psi}\big\|f\big\|^{2}_{2}}{Ne^{N\left|E_1\right|\left|E_2\right|}}. \end{align*} Consequently, we have \begin{align*}
\int_{\mathbb R^n\setminus E_1}\int_{\mathbb R^{n-1}}\int_{{\mathbb R \setminus \left\{0\right\}}}\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^2 \dfrac{da\,ds\,dt}{a^{n+1}}-C_{\psi}\int_{\mathbb R^n\setminus E_1}\big|f(t)\big|^2 dt\ge 0.\tag{4.6} \end{align*}
From inequality (4.6), we observe that, except for the factor $C_{\psi}$, the net concentration of the shearlet transform ${\mathcal {SH}}_{\psi}f(a,s,t)$ in $L^2\big({\mathbb R^n\setminus E_1}\times\mathbb R^{n-1}\times\mathbb R^+\big)$ is always greater than or equal to the net concentration of the signal $f$ in its natural domain $L^2\big({\mathbb R^n\setminus E_1}\big)$. Moreover, if $|E_1|=0$, then the energy preserving relation (2.9) guarantees the equality in (4.6).
\section{Local-type Uncertainty Principles for the Shearlet Transforms}
Since the classical uncertainty principle does not preclude any signal $f$ from being concentrated in a small neighbourhood of two or more widely separated points. Keeping this fact in mind, we shall derive some local uncertainty principles for the continuous shearlet transform in arbitrary space dimensions which demonstrates that the aforementioned phenomenon can't also occur.
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{\bf Theorem 5.1.} {\it Let $\psi$ be an admissible shearlet in $ L^2(\mathbb R^n)$. Then, for any $f\in L^2(\mathbb R^n)$, we have the following uncertainty inequality} \begin{align*}
&\int_{\mathbb S}\big| t\big|^{2\alpha}\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^2 d\eta\ge \dfrac{C_{\psi}}{K_{\alpha}\left|E\right|^{\alpha}}\int_{E}\left|\hat{f}(\xi)\right|^2d\xi,\quad0<\alpha<1.\tag{5.1} \end{align*} {\it where $E$ is a measurable set with finite measure and $K_{\alpha}$ is a constant.}
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{\it Proof.} For $E\subset \mathbb R^n$ with finite measure and $f\in L^2(\mathbb R^n)$, there exist a constant $K_{\alpha}, 0<\alpha<1$, such that \cite{Fol} \begin{align*}
\int_{E}\left|\hat{f}(\xi)\right|^2d\xi\le K_{\alpha}\left|E\right|^{\alpha}\Big\|\left| t\right|^{\alpha}f(t)\Big\|_2^2.\tag{5.2} \end{align*} Using (5.2) for the continuous shearlet transforms ${\mathcal {SH}}_{\psi}f(a,s,t)$, we obtain \begin{align*}
\int_{E}\Big|\mathscr F\Big[{\mathcal {SH}}_{\psi}f(a,s,t)\Big](\xi)\Big|^2d\xi\le K_{\alpha}\left|E\right|^{\alpha}\Big\|\left| t\right|^{\alpha}{\mathcal {SH}}_{\psi}f(a,s,t)\Big\|_2^2.\tag{5.3} \end{align*} For explicit expression of (5.3), we shall integrate this inequality with respect to the measure $dads/a^{n+1}$ to get \begin{align*}
\int_{E}\int_{\mathbb R^{n-1}}\int_{{\mathbb R \setminus \left\{0\right\}}}\Big|\mathscr F\Big[{\mathcal {SH}}_{\psi}f(a,s,t)\Big](\xi)\Big|^2\dfrac{da\,ds\,d\xi}{a^{n+1}}\le K_{\alpha}\left|E\right|^{\alpha}\int_{\mathbb S}\big| t\big|^{2\alpha}\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^2d\eta, \end{align*} which together with Lemma 2.1 gives \begin{align*}
\int_{E}\int_{\mathbb R^{n-1}}\int_{{\mathbb R \setminus \left\{0\right\}}}\left|\hat{f}(\xi)~\overline{\hat{\psi}( M_{sa}\xi)}\right|^2\,\dfrac{d\xi\,ds\,da}{a^{\frac{n^2-n+1}{n}}}\le K_{\alpha}\left|E\right|^{\alpha}\int_{\mathbb S}\big| t\big|^{2\alpha}\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^2d\eta.\tag{5.4} \end{align*} Since $\psi$ is an admissible shearlet, inequality (5.4) reduces to \begin{align*}
C_{\psi}\int_{E}\left|\hat{f}(\xi)\right|^2d\xi\le K_{\alpha}\left|E\right|^{\alpha}\int_{\mathbb S}\big| t\big|^{2\alpha}\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^2d\eta. \end{align*} Or equivalently, \begin{align*}
\int_{\mathbb S}\big| t\big|^{2\alpha}\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^2 d\eta\ge \dfrac{C_{\psi}}{K_{\alpha}\left|E\right|^{\alpha}}\int_{E}\left|\hat{f}(\xi)\right|^2d\xi,\quad0<\alpha<1.\tag{5.5} \end{align*}
This completes the proof of Theorem 5.1.\quad\fbox
\parindent=8mm
Based on the Sobolev-type uncertainty inequality (3.21), we shall derive another local uncertainty principle for the continuous shearlet transform in arbitrary space dimensions.
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{\bf Theorem 5.2.} {\it Let $\psi\in L^2(\mathbb R^n)$ be an admissible shearlet with $C_{\psi}=1$. Then, for arbitrary function $f\in \mathbb H(\mathbb R^n)\cap \mathbb W_{1}^1(\mathbb R^n)$, we have} \begin{align*}
\left(\int_{\mathbb S} \left|t\right|^2\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^2d\eta\right)\ge \Bigg\{\dfrac{2}{\big\|\nabla f\big\|_{2}}\exp\left(\dfrac{{\Gamma^{\prime}}(n/2)}{\Gamma(n/2)}\right)\big\|f\big\|^{3}_{2}-\big\|f\big\|^{2}_{2}\Bigg\}.\tag{5.6} \end{align*} {\it provided the L.H.S of (5.6) is defined.}
\parindent=0mm
{\it Proof.} For $C_{\psi}=1$, we infer from (3.21) that \begin{align*}
\left(\dfrac{{\Gamma^{\prime}}(n/2)}{\Gamma(n/2)}\right)\big\|f\big\|^{2}_{2}\le\int_{\mathbb S} \Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^2\ln\left(\dfrac{1+\left|t\right|^2}{2}\right)\,d\eta+\int_{\mathbb R^{n}}{\ln{\left|\xi\right|} \left|\hat{f}(\xi)\right|^2}d{\xi}.\tag{5.7} \end{align*} Using Jensen's inequality in (5.7), we can deduce that \begin{align*}
\left(\dfrac{{\Gamma^{\prime}}(n/2)}{\Gamma(n/2)}\right)\le\ln\int_{\mathbb S} \dfrac{\big|{\mathcal {SH}}_{\psi}f(a,s,t)\big|^2}{\big\|f\big\|^{2}_{2}}\left(\dfrac{1+\left|t\right|^2}{2}\right)\,d\eta+\dfrac{1}{2}\int_{\mathbb R^{n}}{\ln{\left|\xi\right|^2} \dfrac{\left|\hat{f}(\xi)\right|^2}{\big\|f\big\|^{2}_{2}}}d{\xi}.\tag{5.8} \end{align*} To obtain a fruitful estimate of the second integral of (5.8), we set \begin{align*}
d\rho=\dfrac{\left|\hat{f}(\xi)\right|^2}{\big\|f\big\|^{2}_{2}}d{\xi},\quad {\text {so that}}\quad\int_{\mathbb R^{n}}d\rho=1.\tag{5.9} \end{align*} Again by employing the Jensen's inequality, we obtain \begin{align*}
\int_{\mathbb R^{n}}{\ln{\left|\xi\right|^2} \left|\hat{f}(\xi)\right|^2}d{\xi}&=\big\|f\big\|^{2}_{2}\int_{\mathbb R^{n}}\ln{\left|\xi\right|^2}d\rho\\
&\le\big\|f\big\|^{2}_{2}\,\ln\left\{\int_{\mathbb R^{n}}\left|\xi\right|^2 d\rho\right\}\\
&=\big\|f\big\|^{2}_{2}\ln \left\{ \int_{\mathbb R^{n}} \left|\xi\right|^2\dfrac{\left|\hat{f}(\xi)\right|^2}{\big\|f\big\|^{2}_{2}}d{\xi}\right\}\\
&=\big\|f\big\|^{2}_{2}\ln \left\{\dfrac{1}{\big\|f\big\|^{2}_{2}} \int_{\mathbb R^{n}}\big|\nabla f(t)\big|^{2}dt\right\}.\tag{5.10} \end{align*} Using the expression (5.10) in (5.9), we have \begin{align*}
\left(\dfrac{{\Gamma^{\prime}}(n/2)}{\Gamma(n/2)}\right)\le\ln\int_{\mathbb S} \dfrac{\big|{\mathcal {SH}}_{\psi}f(a,s,t)\big|^2}{\big\|f\big\|^{2}_{2}}\left(\dfrac{1+\left|t\right|^2}{2}\right)\,d\eta+\dfrac{1}{2}\ln \left\{\dfrac{1}{\big\|f\big\|^{2}_{2}} \int_{\mathbb R^{n}}\big|\nabla f(t)\big|^{2}dt\right\}\\
=\ln \left\{\dfrac{1}{2\big\|f\big\|^{3}_{2}} \left\{\int_{\mathbb S} \Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^2\left(1+\left|t\right|^2\right)\,d\eta\right\} \left\{\int_{\mathbb R^{n}}\big|\nabla f(t)\big|^{2}dt\right\}^{1/2}\right\}.\tag{5.11} \end{align*}
Expression (5.11) can be rewritten in a lucid manner as
\begin{align*}
\left\{\int_{\mathbb S} \Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^2\left(1+\left|t\right|^2\right)\,d\eta\right\} \left\{\int_{\mathbb R^{n}}\big|\nabla f(t)\big|^{2}dt\right\}^{1/2}\ge 2\exp\left\{\dfrac{{\Gamma^{\prime}}(n/2)}{\Gamma(n/2)}\right\}\big\|f\big\|^{3}_{2}.\tag{5.12} \end{align*}
Applying the energy preserving relation (2.9) with $C_{\psi}=1$, we get
\begin{align*}
\left\{\int_{\mathbb S} \left|t\right|^2\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^2d\eta\right\} \left\{\int_{\mathbb R^{n}}\big|\nabla f(t)\big|^{2}dt\right\}^{1/2}\ge 2\exp\left(\dfrac{{\Gamma^{\prime}}(n/2)}{\Gamma(n/2)}\right)\big\|f\big\|^{3}_{2}-\big\|f\big\|^{2}_{2}\big\|\nabla f\big\|_{2}, \end{align*} which upon simplification gives the desired inequality \begin{align*}
\left\{\int_{\mathbb S} \left|t\right|^2\Big|{\mathcal {SH}}_{\psi}f(a,s,t)\Big|^2d\eta\right\}\ge \Bigg\{\dfrac{2}{\big\|\nabla f\big\|_{2}}\exp\left(\dfrac{{\Gamma^{\prime}}(n/2)}{\Gamma(n/2)}\right)\big\|f\big\|^{3}_{2}-\big\|f\big\|^{2}_{2}\Bigg\}. \end{align*}
This completes the proof of the theorem.\quad\fbox
\parindent=0mm
\end{document} | arXiv |
Discrete Gaussian Sampling role in Lattice-Based Crypto?
I'm reading up on how post-quantum cryptography works, and stumbled upon the notion of discrete Gaussian sampling. However, I can't understand where it fits in the greater picture - currently it feels to me like a solution to a problem nobody put out.
Where exactly in a SVP problem (or any other commonly used lattice problem) would Discrete Gaussian Sampling provide a benefit?
I'm still new to PQ so pardon the highly likely banality of the question
randomness post-quantum-cryptography lattice-crypto
Daniel BDaniel B
$\begingroup$ As an example, I suggest reading up on LWE. The error introduced is typically sampled from a discrete distribution that approximates a Gaussian. This error is what makes the problem difficult without knowledge of the key, and allows use as a cryptographic primitive. You could probably use a different distribution, but this would almost certainly be less efficient. $\endgroup$ – bkjvbx Jun 9 '18 at 15:56
A Gaussian distribution satisfies the following desirable properties:
It can be implemented coordinate-wise: If $x_1, x_2, \ldots , x_n$ are each sampled from a one-variable Gaussian distribution, then $(x_1,x_2,\ldots,x_n)$ is sampled from a multivariable Gaussian distribution.
It approximates a uniform error distribution modulo a lattice exponentially well, regardless of what the lattice is.
The first property makes implementation easy, and the second property makes security proofs easy, since many security proofs in lattice-based crypto involve switching around the lattice lots of times until you get what you want.
To show what I mean by the second property, let's consider a one-dimensional lattice $\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\} \subset \mathbb{R}^1$. Take a normal distribution with standard deviation $1/2$. In other words, just your average normal distribution:
Suppose that I sample real numbers from this distribution and take the fractional part of the resulting real numbers. (Taking fractional parts correpsonds to taking error vectors with respect to the lattice $\mathbb{Z}$.) The resulting distribution amounts to taking the original normal distribution, chopping it up into unit intervals $\ldots, [-2,-1], [-1,0], [0,1], [1,2], \ldots$, and adding them up. When we do that, we get something quite magical:
Notice how close this distribution is to uniform! The "radius" or standard deviation of this distribution is only $1/2$, which is not much larger than the size of the unit interval; in fact it's smaller. Even with such a small radius, we get a ridiculously good approximation to the uniform distribution. You can prove (and you should prove, as an exercise) that the quality of the approximation is independent of the choice of where the original normal distribution is centered.
Suppose we take a slightly larger normal distribution, say with standard deviation $2/3$:
If we graph the fractional part of this distribution, we get:
That's really really good! You can't even tell that it deviates from uniform. In mathematical terms, we say that the distribution is exponentially close to uniform. Even a small increase in the width of the distribution (from $1/2$ to $2/3$) improves the quality of the approximation dramatically.
You might say, what's the big deal? We can easily get a uniform distribution on any interval. But that's not the point. In lattice-based cryptography, you often don't know what the lattice is. (It's part of someone's secret key.) Suppose as an exercise that we didn't know what this lattice is, and we tried to sample error vectors by taking them uniformly from an interval $[0,n]$ for some $n$. We can't just take the perfect choice of $n=1$. That's cheating, since we're assuming we don't know what the lattice is. In this case, any choice of a small number $n$ will cause the distribution to be horribly wrong; for example, if we chose $n=2/3$ in this scenario, then all of our error vectors would lie in $[0,2/3]$, which is far from uniform in $[0,1]$. Even a slightly larger $n$ is no good; for example if $n=3/2$ then random real numbers sampled uniformly from $[0,3/2]$ will be much more likely to have fractional part (i.e. error vector) lying in $[0,1/2]$ than in $[1/2,1]$. Of course, a very large $n$ (say $n \approx 10^9$) would do the job, but that's exactly the problem: since our distribution on $[0,n]$ doesn't converge to the uniform distribution on $[0,1]$ exponentially fast (unless we cheat by taking $n \in \mathbb{Z}$, which is not allowed), we end up needing to take very large values of $n$, which is not only hard to implement, but a nightmare in security proofs where theoretical analysis is required.
What you need is a way to approximate uniform error vectors exponentially well, without relying on prior knowledge of what the lattice is. Gaussian distributions do that job.
djaodjao
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\begin{document}
\begin{abstract} We study the question whether copies of $S^1$ in $\operatorname{SU}(3)$ can be amalgamated in a compact group. This is the simplest instance of a fundamental open problem in the theory of compact groups raised by George Bergman in 1987. Considerable computational experiments suggest that the answer is positive in this case. We obtain a positive answer for a relaxed problem using theoretical considerations. \end{abstract}
\maketitle
\section{Introduction}
We write $\operatorname{SU}(3)$ for the group of $3 {\times} 3$ complex, unitary matrices with determinant equal to $1$. Consider the closed subgroups $\mathbb{T}_1 = \{\operatorname{diag}(z,1,z^{-1}) \mid z \in S^1 \}$ and $\mathbb{T}_2 =\{\operatorname{diag}(z,z,z^{-2}) \mid z \in S^1 \}$, where $S^1$ denotes the multiplicative group of complex numbers of norm one. Both $\mathbb{T}_1$ and $\mathbb{T}_2$ are isomorphic to $S^1$ as topological groups, via the natural isomorphisms $z \mapsto \operatorname{diag}(z,1,z^{-1})$ and $z \mapsto \operatorname{diag}(z,z,z^{-2})$, respectively. However, the two representations of $S^1$ in $\operatorname{SU}(3)$ are not equal and not even conjugate in $\operatorname{SU}(3).$ So it is a natural question to wonder whether there exist unitary representations $\pi_1,\pi_2 \colon \operatorname{SU}(3) \to \operatorname{U}(n)$ for some $n$, such that the two representations of $S^1$ can be matched, or more precisely $$\pi_1(\operatorname{diag}(z,z^{-1},1))=\pi_2(\operatorname{diag}(z,z,z^{-2})), \quad \textrm{ for all } z \in S^1.$$ We denote by $\rho$ the natural representation of $\operatorname{SU}(3)$ on $\mathbb{C}^3$. One can then check that the two $9$-dimensional representations $\pi_1=\rho \otimes \bar \rho$ and $\pi_2 = \rho \oplus \bar \rho \oplus 1^{\oplus 3}$ solve this problem, where $\bar \rho$ denotes the conjugate representation, and $1$ is the trivial one-dimensional representation. We will come back to this example several times in this article.
This concrete question belongs to a more general set of problems that was first studied by Bergman \cite{MR893156}. Let $A,B$ be compact groups that share a common closed subgroup $C$, see \cite{MR4201900} for background on the theory of general compact groups. It is natural to consider the abstract amalgamated free product $G:=A \ast_C B$ and try to study the analytic properties that it inherits from its constituents. A natural question is whether $G$ can carry a possibly non-Hausdorff compact topology that restricts to the given topologies on $A$ and $B.$ Equivalently, we ask whether $A$ and $B$ can be amalgamated over $C$ in the category of compact groups, i.e., if there exists a compact group $D$ and embeddings of $A$ and $B$ in $D$ that agree on the common copy of $C$. $$\begin{tikzcd} & A \arrow[rrd, bend left, "\pi_1"] & & \\ C \arrow[ru, bend left] \arrow[rd, bend right] & & & D \\ & B \arrow[rru, bend right, "\pi_2"] & & \end{tikzcd} $$ Compact groups carry bi-invariant metrics that generate the topology. Thus, a first obstruction to a positive answer is that $A$ and $B$ may not carry bi-invariant metrics that agree on $C$. If this is the case, a bi-invariant pseudo-metric on $G$ that is faithful on $A$ and $B$ cannot exist. In particular, there does not exist a compact group $D$ as described above. Bergman showed that this type of argument rules out existence of compact amalgams in many cases. It is a fundamental open problem, if amalgamation is always possible for $C=S^1$ or $C=\operatorname{SU}(n)$, see \cite[Question 20]{MR893156}.
The purpose of this note is to explore an equivalent algebraic reformulation of the problem in the simplest possible case. What got us started was the strategy outlined after Question 20 in \cite{MR893156}. That strategy is amenable to standard computer algebra systems. Our computer experiments, with \scip \cite{scip} and \OSCAR \cites{OSCAR-book,OSCAR}, suggest that solutions to the original problem can always be found but get increasingly complicated. For example, merging the subgroups $\mathbb{T}_1 = \{\operatorname{diag}(z,z^5,z^{-6}) \mid z \in S^1 \}$ and $\mathbb{T}_2 =\{\operatorname{diag}(z,z^7,z^{-8}) \mid z \in S^1 \}$ of $\operatorname{SU}(3)$ required us to consider all possible direct sums of the first $120$ mutually non-equivalent irreducible representations of $\operatorname{SU}(3)$ (in some specified order) until a pair of unitary representations of $\operatorname{SU}(3)$ on a complex vector space of dimension roughly $300{,}000$ could be found that solves the problem.
\section{Some basic observations}
Let us study the question of amalgamation of the base $C=S^1$. It follows from the Peter--Weyl Theorem \cite[Theorem 1.12]{Knapp:1986} that in order to construct amalgams of general compact groups $A$ and $B$, it is enough to consider the case $A=\operatorname{U}(n)$, $B=\operatorname{U}(m)$. Since $\operatorname{U}(n)$ embeds to $\operatorname{SU}(n+1)$, we can further restrict to the case $A=\operatorname{SU}(n)$, $B=\operatorname{SU}(m)$. The simplest case that comes to mind is $A=B=\operatorname{SU}(2)$. In this case, however, every embedding of $S^1$ in $\operatorname{SU}(2)$ is conjugate to the map $z\mapsto \operatorname{diag}(z,z^{-1})$. Thus, the first truly non-trivial case might be $A=\operatorname{SU}(2)$ and $B=\operatorname{SU}(3)$ or somewhat more general $A=B=\operatorname{SU}(3)$. Our first task is to describe the possible embeddings of $S^1$ in $\operatorname{SU}(3)$. Our second task is to search for pairs of faithful, finite-dimensional, unitary representations of $\operatorname{SU}(3)$ that agree on the embedded copies of $S^1$.
The first task is easy to solve. Up to conjugation in $\operatorname{SU}(3)$, every embedding of $S^1$ into $\operatorname{SU}(3)$ is given by three integers $(a,b,c) \in \mathbb{Z}^3$ such that $a+b+c=0$ and $\gcd(a,b,c)=1$. The embedding associated with the triplet $(a,b,c)$ is concretely given by $$\psi_{a,b,c}(z):= \operatorname{diag}(z^{a},z^b,z^c) \in \operatorname{SU}(3), \quad \textrm{ for all } z \in S^1.$$
In order to address the second task, we recall some facts about the finite-dimensional unitary representation theory of $\operatorname{SU}(3)$. Let $\rho$ be the standard representation of $\operatorname{SU}(3)$ on $\mathbb{C}^3$ and $\bar \rho$ be its dual or conjugate. We denote by $\pi_{m,n} \colon \operatorname{SU}(3) \to \operatorname{U}(\Gamma_{m,n})$ the irreducible representation parameterized by the weight $(m,n)$, a pair of non-negative integers. This representation corresponds to the Young tableaux of shape $(m+n,n)$. According to Fulton--Harris \cite[ยง13.2]{MR1153249}, we have \begin{equation}
\label{rep}
\Gamma_{m,n} = \ker\left(\operatorname{Sym}^m(\rho) \otimes \operatorname{Sym}^n(\bar \rho) \stackrel{\phi_{m,n}}{\to} \operatorname{Sym}^{m-1}(\rho) \otimes \operatorname{Sym}^{n-1}(\rho)\right), \end{equation} where $\phi_{m,n}$ denotes the natural (surjective) contraction map \begin{equation*}
\phi_{m,n} \bigl((v_1\dots v_m)\otimes(v_1^*\dots v_n^*) \bigr) := \sum_{i=1}^m \sum_{j=1}^n \langle v_i,v_j^*\rangle (v_1\dots \hat{v_i}\dots v_m)\otimes (v_1^*\dots \hat{v}^*_j\dots v^*_n). \end{equation*} Note that representations of the compact Lie group $\operatorname{SU}(3)$ correspond to representations of the simple Lie algebra $\mathfrak{sl}_3\mathbb{C}$; see \cite[ยง9.3]{MR1153249}.
Every unitary representation $\sigma$ of $\operatorname{SU}(3)$ extends to a unitary representation $\sigma'$ of $\operatorname{U}(3)$, which is however not unique. The character of a unitary representation $\sigma \colon \operatorname{SU}(3) \to U(k)$ is a symmetric polynomial $\chi(\sigma) \in \mathbb{Z}[x_1,x_2,x_3]$ determined uniquely up to some element in the ideal generated by $x_1x_2x_3-1$ by the property $$\chi(\sigma)(x_1,x_2,x_3)= \operatorname{tr}\bigl(\sigma'(\operatorname{diag}(x_1,x_2,x_3))\bigr), \quad \textrm{ for all } x_1,x_2,x_3 \in S^1.$$ It is known that $\chi(\pi'_{m,n}) = s_{m+n,n},$ where $s_{\lambda}$ denotes the Schur polynomial of the Young tableaux with two parts $(m+n,n).$ Here, $\pi'_{m,n}$ is the representation of $\operatorname{U}(3)$ described by the same formula as in Equation \eqref{rep}. Recall that $\chi(\sigma)(1,1,1)$ equals the dimension of the representation $\sigma$.
Now, every finite-dimensional unitary representation is a direct sum of irreducible unitary representations, and thus every character of such a representation is a symmetric, non-negative integer linear combination of Schur polynomials. We call such a symmetric polynomial \emph{Schur positive}. Here, the polynomial $s_{1,1,1}=x_1x_2x_3$ corresponds to the trivial representation $1$. We say that a Schur positive polynomial is \emph{non-trivial} if it is non-scalar modulo the polynomial $x_1x_2x_3-1.$
Back to the original problem, we are interested in the question if for a given pair of triplets of integers, $(v_1,v_2,v_3)$ and $(w_1,w_2,w_3)$, with $v_1+v_2+v_3=w_1+w_2+w_3=0$ and $\gcd(v_1,v_2,v_3)=\gcd(w_1,w_2,w_3)=1$, we can find a pair of unitary representations $$\sigma_1,\sigma_2 \colon \operatorname{SU}(3) \to \operatorname{U}(k),$$ such that $$\sigma_1\bigl(\psi_{v_1,v_2,v_3}(z)\bigr)= \sigma_2\bigl(\psi_{w_1,w_2,w_3}(z)\bigr) \quad \textrm{ for all } z \in S^1.$$ However, these two representations are conjugate (and hence equal after conjugation) if and only if the associated characters agree. Hence, we arrive at the equivalent condition $$\chi(\sigma_1)(z^{v_1},z^{v_2},z^{v_3}) = \chi(\sigma_2)(z^{w_1},z^{w_2},z^{w_3}) \quad \textrm{ for all } z \in S^1.$$ Thus, putting everything in more algebraic terms, the second task amounts to finding Schur positive polynomials $P$ and $Q$ such that the equality \begin{equation}
P(z^{v_1},z^{v_2},z^{v_3}) = Q(z^{w_1},z^{w_2},z^{w_3}) \end{equation} holds in the Laurent polynomial ring $\mathbb{Q}[z^\pm]$. For brevity, given a vector $v=(a,b,c)$ with $a+b+c=0$, we write $P_v(z) = P(z^a,z^b,z^c)$ for the substitution.
There is an additional subtlety that we did not address so far: unitary representations of $\operatorname{SU}(3)$ need not be injective. If we pick a non-trivial third root of unity, $\xi=\exp(2\pi i/3)\in S^1$, then the subgroup $$ Z = \bigl\langle \operatorname{diag}(\xi,\xi,\xi) \bigr\rangle \cong \mathbb{Z}/3\mathbb{Z}, $$ forms the center of $\operatorname{SU}(3)$. Note that $Z$ is the only non-trivial normal subgroup of $\operatorname{SU}(3)$, i.e., the quotient $\operatorname{SU}(3)/Z$ is simple. The following result characterizes the injective unitary representations of $\operatorname{SU}(3)$. \begin{prop}\label{prop:injective} Let $\sigma \colon \operatorname{SU}(3) \to \operatorname{U}(k)$ be a unitary representation with character $P=\chi(\sigma') \in \mathbb{Z}[x_1,x_2,x_3]$ for some extension $\sigma'$ of $\sigma$ to $\operatorname{U}(3)$. Then the following conditions are equivalent: \begin{enumerate}
\item $\sigma$ is injective,
\item $P(\xi,\xi,\xi) \neq P(1,1,1)$, and
\item $P$, written in terms of the Schur basis, has a summand (with positive coefficient) having total degree not divisible by three. \end{enumerate} \end{prop}
\begin{proof} Since the center $Z$ of $\operatorname{SU}(3)$ is generated by $\operatorname{diag}(\xi,\xi,\xi)$ and because $Z$ is the only non-trivial normal subgroup of $\operatorname{SU}(3)$, the representation $\sigma$ is injective if and only if $\sigma(\operatorname{diag}(\xi,\xi,\xi))$ is distinct from the $k{\times}k$ unit matrix. The eigenvalues of $\sigma(\operatorname{diag}(\xi,\xi,\xi))$ are complex numbers of modulus $1$ and $P(\xi,\xi,\xi)=\operatorname{tr}(\sigma(\operatorname{diag}(\xi,\xi,\xi)))$ is equal to their sum. Hence, $P(\xi,\xi,\xi)=k=P(1,1,1)$ if and only if all these eigenvalues are equal to $1$ if and only if $\sigma$ is not injective. This proves the equivalence between $(1)$ and $(2).$ Note that $P$ is necessarily Schur-positive since it is a character.
We proceed to show the equivalence of $(2)$ and $(3)$. If each summand of $P$ has a total degree which is a multiple of three, then $P(\xi,\xi,\xi)=P(1,1,1)$; this shows that (2) implies (3). To show the reverse direction, without loss of generality, assume $P\neq 0$ is a positive linear combination of Schur polynomials, none of which has total degree divisible by three. Let $R(x) = P(x,x,x)$, which is a rational univariate polynomial in $\mathbb{Q}[x]$. Note that $R$ has all coefficients positive and no term of degree divisible by three. Now let $R'\in\mathbb{Q}[x]$ be the remainder of division of $R$ by $(x^3-1)$. Then $R'\neq 0$, has no constant term, and satisfies $R'(1) = R(1)$ and $R'(\xi) = R(\xi)$. We need to show $P(1,1,1) \neq P(\xi,\xi,\xi)$. So let us assume the contrary. Then $R'(\xi) = R'(1)$, so the polynomial $R'(x)-R'(1)$ must be a multiple of the minimal polynomial of $\xi$, namely $x^2+x+1$. Since $R'$ has degree at most two, we have $R' = c(x^2+x+1)$ for some nonzero constant $c$, but that contradicts that $R'$ has no constant term. We conclude that $P(1,1,1) \neq P(\xi,\xi,\xi)$, and this completes our proof. \end{proof}
\begin{example}\label{exmp:byhand} For simplicity, we denote the unitary representation of $S^1$ with character $\sum_i a_i z^i$ by the list $(i^{a_i}; i \in \mathbb{Z})$; where we omit the entry of $i$ whenever $a_i=0.$ In particular, in this notation we have $\psi_{a,b,c} = (a,b,c)$. We now revisit the example at the beginning of the introduction. The computation $$(-1,0,1)^{\otimes 2} = (-2,-1^2,0^3,1^2,2) = (-2,1^2) \oplus (-2,1^2)^{*} \oplus (0)^{\oplus 3}$$ shows that the representations $\psi_{-1,0,1}$ and $\psi_{-2,1,1}$ can be amalgamated inside $\operatorname{SU}(9)$. In terms of polynomials, this corresponds to $P(x_1,x_2,x_3)=(x_1+x_2+x_3)^2$, $Q(x_1,x_2,x_3) = x_1+x_2+x_3+x_2x_3+x_1x_3+x_1x_2+3x_1x_2x_3$ and the identity $P(z^{-1},1,z)=Q(z^{-2},z,z).$ Observe that $P=s_{1,1}+s_2$ and $Q=s_1+s_{1,1}+3s_{1,1,1}$. In particular, both polynomials are Schur positive. Moreover, $P(1,1,1)=Q(1,1,1)=9$ is the dimension of the representation. Apart from the this example, which we were able to work out by hand, only few other cases seem suitable for pen and paper calculations. \end{example}
It follows from the reasoning above that we can formulate Bergman's problem \cite[Question 20]{MR893156} in the first non-trivial case as follows:
\begin{quest}\label{quest:initial} Given integer vectors $v = (v_1,v_2,v_3)$ and $w = (w_1,w_2,w_3)$ satisfying $v_1+v_2+v_3=w_1+w_2+w_3=0$ and $\gcd(v_1,v_2,v_3) = \gcd(w_1,w_2,w_3) = 1$, can we find Schur positive polynomials in three variables $P$ and $Q$ such that: \begin{enumerate} \item $P_v(z) = Q_w(z)$ and \item $P(\xi,\xi,\xi) \neq P(1,1,1)$ and \item $Q(\xi,\xi,\xi)\neq Q(1,1,1)$? \end{enumerate} \end{quest}
Our experiments suggest that the answer is always positive.
\section{Computations}
Now we recast Question~\ref{quest:initial} as a problem in polyhedral geometry and approach it computationally. The source code and its output can be found on our \MathRepo page \begin{equation}\label{eq:link}
\text{\url{https://mathrepo.mis.mpg.de/CompactAmalgamation/index.html} }. \end{equation} To this end we fix $v, w \in \mathbb{Z}^3$ such that $v_1+v_2+v_3=w_1+w_2+w_3=0$ and $\gcd(v_1,v_2,v_3) = \gcd(w_1,w_2,w_3) = 1$. Choosing an ordering for the Schur polynomials, we then make the problem finite by fixing a number $k$ and considering only the first $k$ Schur polynomials in three variables, denoted by $S_1,\dots,S_k\in\mathbb{Q}[x_1,x_2,x_3]$. We search for $P = \lambda_1 S_1 + \dots + \lambda_k S_k$ and $Q = \mu_1 S_1 + \dots + \mu_k S_k$, where $\lambda_i, \mu_i$ are non-negative integers. These polynomials lie in $\mathbb{Q}[x_1,x_2,x_3]$, and their substitutions $P_{v}(z)$ and $Q_{w}(z)$ are univariate Laurent polynomials. The coefficients of the difference $P_{v}(z)-Q_w(z)$ are integer linear combinations of $\lambda_i$ and $\mu_i$. Setting these coefficients to zero and letting $\lambda_i \geq 0$ and $\mu_i \geq 0$ defines a polyhedral cone in $\mathbb{R}^{2k}$. We denote that cone $\cC=\cC_k(v,w)$. Recall that $\cC$ depends on the chosen ordering of the Schur polynomials. Throughout we assume that $S_1=s_{1,1,1}$ is the trivial representation. It plays a special role, as $P = Q = \lambda_1 S_1$, for any $\lambda_1\geq 0$, is a trivial solution to (1) in Question~\ref{quest:initial}.
To find $P$ and $Q$, we consider the integer linear program \begin{equation}\label{eq:ilp}\tag{ILP$_k$}
\begin{array}{rl}
\text{minimize} & c\cdot (\lambda,\mu)\\
\text{subject to} & (\lambda,\mu) \in \cC_k(v,w) \\
& \sum_{3 \not \; | \operatorname{tdeg}(S_i)} \lambda_i \geq 1 \\
& \sum_{3 \not \; | \operatorname{tdeg}(S_i)} \mu_i \geq 1 \\
& \lambda_1,\dots,\lambda_k,\mu_1,\dots,\mu_k\in\mathbb{N} \enspace ,
\end{array} \end{equation} where $c\in\mathbb{R}_{>0}^{2k}$ is some strictly positive linear objective function, to be discussed below. Let $\cP=\cP_k(v,w)$ be the feasible region of the linear relaxation of \eqref{eq:ilp}.
\begin{rmk}
Conceptually, one could replace the weak inequality
$\sum_{3 \not \; | \operatorname{tdeg}(S_i)} \lambda_i \geq 1 $ by the strict inequality $\sum_{3 \not \; | \operatorname{tdeg}(S_i)} \lambda_i >0$, but the description as an (integer) linear program requires weak inequalities. \end{rmk}
\begin{prop} The feasible solutions of \eqref{eq:ilp}, i.e., the lattice points in $\cP$, are in bijection with those nontrivial solutions to Question~\ref{quest:initial} which can be written as a non-negative linear combination of the first $k$ Schur polynomials. \end{prop} \begin{proof}
Containment in the cone $\cC$ is equivalent to the condition (1) in Question~\ref{quest:initial}.
The two additional constraints correspond to conditions (2) and (3); see Proposition~\ref{prop:injective}. \end{proof}
\begin{rmk}\label{rmk:choices} In practice, we make the following choices. We order the $3$-variate Schur polynomials lexicographically: a partition $(m+n,n)$, with $m, n\geq 0$, is less than another partition $(m'+n',n')$ if either $m+n < m'+n'$ or $m+n = m'+n'$ and $n < n'$; and the special partition $(1,1,1)$ is defined to be smaller than $(m+n,n)$ for arbitrary $m$ and $n$. Moreover, we take the objective function $c=(c_i)$ with $c_i=\operatorname{tdeg} S_i$, where $\operatorname{tdeg}$ is the total degree. So the optimal solutions are minimal with respect to dimension. \end{rmk} \noindent We abbreviate $(m)=(m,0)$. \begin{example} We consider $v=(-1,0,1)$ and $w=(-2,1,1)$ as in Example~\ref{exmp:byhand}, and we pick $k=4$. Then the first four Schur polynomials correspond to the partitions $(1,1,1)$, $(1)$, $(1,1)$, and $(2)$. So we have $S_1=x_1x_2x_3$, $S_2=x_1+x_2+x_3$, $S_3=x_1x_2+x_1x_3+x_2x_3$, and $S_4=x_1^2 + x_1x_2 + x_1x_3 + x_2^2 + x_2x_3 + x_3^2$. Then \[ \begin{split}
P_v(z)-P_w(z) \ = \ (&\lambda_{4} - \mu_{3} - 3\mu_{4})z^2 + (\lambda_{2} + \lambda_{3} + \lambda_{4} - 2\mu_{2})z + \lambda_{1} + \lambda_{2} + \lambda_{3} + 2\lambda_{4} - \mu_{1}\\ &+ (\lambda_{2} + \lambda_{3} + \lambda_{4} - 2\mu_{3} - 2\mu_{4})z^{-1} + (\lambda_{4} - \mu_{2})z^{-2} - \mu_{4}z^{-4} \enspace . \end{split} \] Consequently, the unbounded polyhedron $\cP$ in $\mathbb{R}^8$ is given by six homogeneous equations (from the coefficients of $P_v(z)-P_w(z)$, considered as a Laurent polynomial in $\mathbb{Q}[\lambda_1,\dots,\mu_4][z^\pm]$), the eight nonnegativity constraints and two affine inequalities (from forcing injectivity). The polyhedron $\cP$ is $3$-dimensional. Solving the integer linear program \eqref{eq:ilp} yields \[ \lambda_1=\lambda_2=0\,,\; \lambda_3=\lambda_4=1 \quad \text{and} \quad \mu_1=3\,,\; \mu_2=\mu_3=1\,,\; \mu_4=0 \] as an optimal solution of objective value $3+6=3+3+3=9$. This recovers the pair of 9-dimensional representations given by $P=s_{1,1}+s_2$ and $Q=3s_{1,1,1}+s_1+s_{1,1}$ from Example~\ref{exmp:byhand}. That pair of Schur positive polynomials corresponds to the lattice point marked $0011\, 3110$ in Figure~\ref{fig:byhand}. Our visualization artificially truncates the feasible region at representation dimension ten. We see two 9-dimensional solutions and two 10-dimensional ones. The solutions come in pairs since $s_{1,v}=s_{(1,1),v}$ for the special choice of $v=(-1,0,1)$. The 10-dimensional solutions are obtained from the 9-dimensional solutions by adding a trivial representation. In this way, the solution from Example \ref{exmp:byhand} explains all four solutions shown here. \end{example}
\begin{figure}
\caption{Four integral points in $\cP_4(v,w)$ for $v=(-1,0,1)$ and $w=(-2,1,1)$.
Visualized with \polymake \cite{DMV:polymake}; hyperplane for artificial truncation at representation dimension 10 marked red.}
\label{fig:byhand}
\end{figure}
Solving integer linear programs is generally hard, both theoretically and in practice \cite{Schrijver:TOLIP}. However, our integer linear program \eqref{eq:ilp} has a particularly simple structure, which can be exploited computationally. \begin{lem} Let $(\lambda,\mu)\in\mathbb{Q}^{2k}$ be a rational point in $\cP$. Then there is a positive integer $\ell>0$ such that $(\ell\cdot\lambda,\ell\cdot\mu)$ is a point in $\cP$ which is integral. \end{lem} \begin{proof}
Let $\ell$ be the common denominator of $\lambda_1,\lambda_2,\dots,\mu_k$.
Then $(\ell\cdot\lambda,\ell\cdot\mu)$ is integral.
The polyhedron $\cP$ is the intersection of the cone $\cC$ with two additional affine halfspaces.
Clearly, $(\ell\cdot\lambda,\ell\cdot\mu)$ lies in $\cC$.
Further, we have $\ell \cdot \sum_{3 \not \; | \operatorname{tdeg}(S_i)} \lambda_i \geq \ell \geq 1$, and similarly for the other inequality.
Thus the point $(\ell\cdot\lambda,\ell\cdot\mu)$ lies in $\cP\cap\mathbb{Z}^{2k}$. \end{proof} As a consequence, the integer linear program \eqref{eq:ilp} is feasible if and only if its linear relaxation is. The latter condition can be tested much faster. Consequently, standard complexity bounds in linear optimization entail the following result; see \cites{GroetschelLovaszSchrijver93,Renegar:2001}. \begin{prop}
Employing the interior point method, deciding the feasibility of the integer linear program \eqref{eq:ilp} takes polynomial time in the five parameters $k$, $\log |v_1|$, $\log |v_2|$, $\log |w_1|$, and $\log |w_2|$. \end{prop} Recall the condition $v_1+v_2+v_3=0=w_1+w_2+w_3$, whence $v_3$ and $w_3$ are not mentioned. Now we can summarize how to address Question~\ref{quest:initial} computationally. First we pick some integer $k$. Then we decide the feasibility of \eqref{eq:ilp} by solving the linear relaxation. If this is feasible we use a bisection to find the minimal $k'$ such that (ILP$_{k'}$) is feasible. If it is infeasible we try $2k$ and repeat. Of course, this procedure does not terminate if no solution exists. Yet that did not occur so far.
There are many implementations of algorithms for linear and integer optimization available, both open source and commercial. Yet the majority employs floating-point arithmetic, which may lead to errors, which in turn makes these software systems less suited for obtaining mathematical results. For this reason we use \scip, which implements the simplex method in exact rational arithmetic \cite{scip}. Setting up the (integer) linear program \eqref{eq:ilp} is done in \OSCAR, which provides partitions, Schur polynomials and the necessary commutative algebra \cites{OSCAR-book,OSCAR}. \OSCAR also inherits the full functionality of \polymake \cite{DMV:polymake}, which includes exact rational integer linear programming. While \scip is much faster at integer linear programming, that implementation is based on floating-point arithmetic.
\begin{table}[th] \caption{Minimal $k$ for which $\cP_k(v,w)$ is feasible, where $v=(1,v_2,-1-v_2)$ and $w=(1,w_2,-1-w_2)$. Empty fields on the upper right are beyond our current reach computationally.} \label{table: min k} \renewcommand{0.9}{0.9} \begin{tabular*}{.8\linewidth}{@{\extracolsep{\fill}}rrrrrrrrrrrr@{}} \toprule $w_2\backslash v_2$ & \multicolumn{1}{c}{0} & \multicolumn{1}{c}{1} & \multicolumn{1}{c}{2} & \multicolumn{1}{c}{3} & \multicolumn{1}{c}{4} & \multicolumn{1}{c}{5} & \multicolumn{1}{c}{6} & \multicolumn{1}{c}{7} & \multicolumn{1}{c}{8} & \multicolumn{1}{c}{9} & \multicolumn{1}{c}{10} \\ \midrule 0 & & 4 & 16 & 191 & 601 & 1541 & & & & & \\ 1 & & & 13 & 33 & 106 & 336 & 686 & 1254 & 2187 & & \\ 2 & & & & 21 & 50 & 125 & 305 & 586 & 1006 & 1574 & \\ 3 & & & & & 28 & 66 & 170 & 292 & 535 & 820 & 1283 \\ 4 & & & & & & 44 & 86 & 174 & 307 & 463 & 824 \\ 5 & & & & & & & 61 & 120 & 238 & 377 & 525 \\ 6 & & & & & & & & 87 & 171 & 275 & 430 \\ 7 & & & & & & & & & 115 & 245 & 333 \\ 8 & & & & & & & & & & 145 & 291 \\ 9 & & & & & & & & & & & 171 \\ \bottomrule \end{tabular*} \end{table}
\subsection*{Feasibility} For our first experiment, we consider pairs of vectors $v=(v_1,v_2,-v_1-v_2)$ and $w=(w_1,w_2,-w_1-w_2)$ such that $v_1=w_1=1$. Such a pair $(v,w)$ is determined by the pair $(v_2,w_2)$ of integers. In Table \ref{table: min k}, we give the minimal values of $k$ for which $\cP_k(v,w)$ is feasible, which we compute by solving the linear relaxation of \eqref{eq:ilp}. As pointed out in Remark~\ref{rmk:choices}, the parameter $k$ refers to the lexicographic ordering of the Schur polynomials. That ordering does affect the value of $k$. That is to say, replacing the pure lexicographic ordering by, e.g., the graded lexicographic ordering may lead to a lower value of $k$. It is unclear whether one ordering is better than another.
\subsection*{Representation dimensions} For our second experiment we actually solve the integer linear program \eqref{eq:ilp}. We take the objective function $$c = \bigl(S_1(1,1,1),\dots,S_k(1,1,1)\bigr)$$ to be the dimension of the representation corresponding to $P$; see Remark~\ref{rmk:choices}. Table~\ref{table: dim} records pairs of vectors, the minimal value of $k$ such that $\cP_k(v,w)$ is feasible and the optimal value of the integer linear program, i.e., the smallest dimension achieved by solutions using only the first $k$ Schur polynomials. Each row of that table corresponds to one entry in Table~\ref{table: min k}. The explicit Schur positive symmetric polynomials whose dimensions are recorded in Column 4 of Table~\ref{table: dim} can be found on our \MathRepo page~(\ref{eq:link}) alongside the source code.
\begin{table}[th] \caption{Minimal $k$ for which $\cP_k(v,w)$ is feasible and the dimension of the representation that corresponds to an optimal integral solution} \label{table: dim}
\begin{tabular*}{.75\linewidth}{@{\extracolsep{\fill}}ccrr@{}}
\toprule
$v$ & $w$ & $k$ & Dimension \\
\midrule
$(1,0,-1)$ & $(1,1,-2)$ & 4 & 9 \\
$(1,0,-1)$ & $(1,2,-3)$ & 16 & 21 \\
$(1,1,-2)$ & $(1,2,-3)$ & 13 & 63 \\
$(1,1,-2)$ & $(1,3,-4)$ & 33 & 834 \\
$(1,1,-2)$ & $(1,4,-5)$ & 106 & 3216 \\
$(1,2,-3)$ & $(1,3,-4)$ & 21 & 255 \\
$(1,2,-3)$ & $(1,5,-6)$ & 125 & 13561 \\
$(1,3,-4)$ & $(1,4,-5)$ & 28 & 454 \\
$(1,3,-4)$ & $(1,5,-6)$ & 66 & 6852 \\
$(1,4,-5)$ & $(1,5,-6)$ & 44 & 1526 \\
$(1,4,-5)$ & $(1,6,-7)$ & 86 & 83113 \\
$(1,5,-6)$ & $(1,6,-7)$ & 61 & 14972 \\
$(1,5,-6)$ & $(1,7,-8)$ & 120 & 316170 \\
$(1,6,-7)$ & $(1,7,-8)$ & 87 & 128624 \\
$(1,7,-8)$ & $(1,8,-9)$ & 115 & 108468 \\
\bottomrule
\end{tabular*}
\end{table}
Note that the dimensions recorded in Table \ref{table: dim} might not be minimal among all solutions since they use only the first $k$ Schur polynomials; allowing the use of more Schur polynomials can potential provide a solution with smaller dimension.
\subsection*{Running times} We briefly comment on the computation time of Table~\ref{table: min k} and Table~\ref{table: dim}. Computing all entries in Table~\ref{table: min k} took in total approximately 400,000 seconds (4.6 days). Optimal solutions in Table~\ref{table: dim} are computed in \scip, via floating-point arithmetic, and then verified in \OSCAR, via exact arithmetic. Verification is fast and succeeded in all our cases. The longest computation was for the pair $(1,5,-6)$ and $(1,7,-8)$, which took 312 seconds in \scip. Computations for pairs with $k > 125$ did not terminate within a day.
All computations were done on the computer server Hydra at the MPI MiS, with the following system specifics: 4x16-core Intel Xeon E7-8867 v3 CPU (3300 MHz) on Debian GNU/Linux 5.10.149-2 (2022-10-21) x86\_64.
\begin{rmk}
In principle, the optimal (rational) solutions to the linear programming relaxations leading to Table~\ref{table: min k} yield an upper bound on the smallest dimension of a representation of the amalgamation problem Question~\ref{quest:initial}. However, these numbers are excessively large. For example, for $v=(1,9,-10)$ and $w=(1,10,-11)$ the bound we obtain is $2382041666750207$. This is one of the smaller ones. Therefore, it is not desirable to provide a complete table here. However, using the \texttt{Jupyter} notebook available on the \MathRepo page (\ref{eq:link}) the interested reader can compute some of these numbers by themselves. \end{rmk}
\section{A relaxed problem} In this section, we consider the following relaxed problem by dropping the Schur positivity condition and disregarding the case $(-1,0,1)$. Recall that the Schur polynomials form a basis of the space of all symmetric polynomials; see \cite[ยงA.1]{MR1153249}.
\begin{quest}\label{quest:initial2} Given $v = (v_1,v_2,v_3)$ such that $v_1+v_2+v_3=0$, $v_1v_2v_3\neq0$ and $\gcd(v_1,v_2,v_3) = 1$. For which Laurent polynomials $F\in\mathbb{Q}[z^\pm]$ can we find a symmetric polynomial in three variables $P$ that $F=P_v(z)$? \end{quest}
\begin{rmk}
We pose the additional condition $v_1v_2v_3\neq0$, which excludes the case $(v_1,v_2,v_3)=(-1,0,1)$, because our argument does not apply to that case, see Remark~\ref{rem:exludecase}. \end{rmk}
From now on fix a triplet $v = (v_1,v_2,v_3)$ such that $v_1+v_2+v_3=0$, $v_1v_2v_3\neq0$ and $\gcd(v_1,v_2,v_3) = 1$. Since every symmetric polynomial in three variables can be written as a polynomial in the first three elementary symmetric polynomials $$e_1=x_1+x_2+x_3 ,\ e_2=x_1x_2+x_1x_3+x_2x_3,\ e_3=x_1x_2x_3$$ in $\mathbb{Q}[x_1,x_2,x_3]$, and because $v_1+v_2+v_3=0$ implies that $(e_3)_v(z)=1$, answering Question \ref{quest:initial2} amounts to characterizing the $\mathbb{Q}$-subalgebra $A(v)$ of the Laurent polynomial ring $\mathbb{Q}[z^\pm]$ generated by $$F_1:=(e_1)_v(z)=z^{v_1}+z^{v_2}+z^{v_3} \quad \text{and} \quad F_2:=(e_2)_v(z)=z^{v_1+v_2}+z^{v_1+v_3}+z^{v_2+v_3}.$$ Since we have $F_1'(1)=F_2'(1)=0$, the product rule implies that $F'(1)=0$ holds for all $F\in A(v)$. This shows that $A(v)$ is a proper subalgebra of $\mathbb{Q}[z^\pm]$. As the next example shows, this is in general not the only constraint.
\begin{example}\label{ex:3droot}
Consider the case $v=(1,1,-2)$, and let $\xi\in\mathbb{C}$ be a primitive third root of unity.
We have $F_1'=2\cdot(1-z^{-3})$ and $F_2'=2z\cdot(1-z^{-3})$ and this shows $F_1'(\xi)=F_2'(\xi)=0$.
Again this shows that $F'(\xi)=0$ for all $F\in A(1,1,-2)$. One can prove that these are all constraints in this case:
\begin{equation*}
A(1,1,-2)=\{F\in\mathbb{Q}[z^\pm]\mid F'(1)=F'(\xi)=F'(\xi^2)=0 \}.
\end{equation*} \end{example}
Our main contribution in this section is the following rather technical result which says that $A(v)$ can, in general, be characterized by conditions similar as in Example \ref{ex:3droot}.
\begin{thm}\label{thm:main}
There is a product $\Phi\in\mathbb{Q}[z]$ of cyclotomic polynomials with $\Phi(1)\neq0$ and a subalgebra $C$ of $\mathbb{Q}[z^\pm]/(\Phi)$ such that for $F\in\mathbb{Q}[z^\pm]$ the following are equivalent:
\begin{enumerate}
\item There is a symmetric polynomial $P$ in three variables with rational coefficients such that $F=P_v(z)$.
\item We have $F'(1)=0$, and the residue class of $F$ modulo $\Phi$ is in $C$.
\end{enumerate} \end{thm}
\begin{rmk}
The subalgebra $C$ of $\mathbb{Q}[z^\pm]/(\Phi)$ in Theorem \ref{thm:main} is the one generated by the residue classes of $F_1$ and $F_2$. Since $\mathbb{Q}[z^\pm]/(\Phi)$ is a finite dimensional $\mathbb{Q}$-vector space, this can be explicitly calculated once knowing $\Phi$. \end{rmk}
Before we will give a proof of Theorem \ref{thm:main} we point out some consequences that are less technical.
\begin{cor}\label{cor:vanishingcrit}
There are finitely many roots of unity $\zeta_1,\ldots,\zeta_r\in\mathbb{C}\setminus\{1\}$ and natural numbers $a_1,\ldots,a_r$ such that every $F\in\mathbb{Q}[z^\pm]$ with $F'(1)=0$ which vanishes at $\zeta_i$ with multiplicity at least $a_i$ for $i=1,\ldots,r$ can be expressed as $F=P_v(z)$ for some symmetric polynomial $P$ in three variables. \end{cor}
\begin{proof}
Let $\Phi\in\mathbb{Q}[z]$ the polynomial from Theorem \ref{thm:main} and let
\begin{equation*}
\Phi=\Phi_1^{a_1}\cdots \Phi_s^{a_s}
\end{equation*}
where the $\Phi_i$ are pairwise coprime cyclotomic polynomials. If $F\in\mathbb{Q}[z^\pm]$ vanishes at the zeros of each $\Phi_i$ with multiplicity at least $a_i$, then $F$ is divisible by $\Phi$. Thus the residue class of $F$ modulo $\Phi$ is zero and hence contained in every subalgebra of $\mathbb{Q}[z^\pm]/(\Phi)$. \end{proof}
\begin{cor}\label{cor:concrete}
There are natural numbers $a_0,b_0>0$ such that for all $a\geq a_0$, all $b$ divisible by $b_0$ we have
\begin{equation}\label{eq:F_nm}
F_{a,b}=(1+z+\cdots+z^{b-1})^a\cdot (1+z^{-1}+\cdots+z^{-(b-1)})^a\in A(v).
\end{equation} \end{cor}
\begin{proof}
Let $\zeta_1,\ldots,\zeta_r$ and $a_1,\ldots,a_r$ as in Corollary \ref{cor:vanishingcrit}.
Let $b_0$ such that $\zeta_i^{b_0}=1$ for all $i=1,\ldots,r$ and $a\geq\frac{1}{2}\max_{i=1}^r (a_i)$. Then for all $a\geq a_0$ and all $b$ divisible by $b_0$ the Laurent polynomial $F_{a,b}$ vanishes at $\zeta_i$ with multiplicity at least $a_i$ for $i=1,\ldots,r$. A straight-forward calculation further shows that $F'_{a,b}(1)=0$. \end{proof}
\begin{cor}
Consider finitely many triplets
\begin{equation*}
t_1,\ldots,t_r\in\{(\alpha,\beta,\gamma)\in\mathbb{Z}^3\mid \alpha+\beta+\gamma)=0, \alpha\beta\gamma\neq0\,\textrm{ and }\gcd(\alpha,\beta,\gamma) = 1\}.
\end{equation*}
Then there are natural numbers $a,b$ such that $F_{a,b}\in\bigcap_{i=1}^r A(t_i)$. \end{cor}
\begin{proof}
For each $i\in\{1,\ldots,r\}$ we obtain $a_0$ and $b_0$ as in Corollary \ref{cor:concrete}. We can choose $a$ as the maximum of all such $a_0$ and $b$ as the product of all such $b_0$. \end{proof}
In fact, we conjecture that the Laurent polynomials $F_{a,b}$ in Equation \eqref{eq:F_nm} can even be realized as positive rational linear combinations of Schur polynomials.
\begin{conj}\label{con:speccon}
There are natural numbers $a_0,b_0>0$ such that for all $a\geq a_0$, all $b$ divisible by $b_0$ there is $N\in\mathbb{N}$ and a Schur positive symmetric polynomial $P$ in three variables such that
\begin{equation*}
N\cdot F_{a,b}= P_v.
\end{equation*} \end{conj}
In order to amalgamate two representations given by tuples $v$ and $w$ let $a_0,b_0,N$ and $a_0',b_0',N'$ the natural numbers from the previous conjecture for $v$ and $w$ respectively. Then, if Conjecture \ref{con:speccon} is true, letting $a=\max(a_0,a_0')$, $n=\operatorname{lcm}(b_0,b_0')$ and $M=\operatorname{lcm}(N,N')$, we have \begin{equation*}
P_v=M\cdot F_{a,b}=Q_{w} \end{equation*} for Schur positive symmetric polynomials $P$ and $Q$.
\begin{rmk}\label{rmk:testconj}
In the case $v = (1,1,-2)$ our computational experiments suggest that Conjecture~\ref{con:speccon} is true for $a_0=1$ and $b_0=3$.
We found $N$ and Schur-positive symmetric polynomials $P_v$ that satisfy $N\cdot F_{a,b} = P_v$ for various pairs of $a$ and $b$. We record the values of $N$ and the dimensions of $P_v$ in Table~\ref{table: conjecture}.
\begin{table}[th]
\caption{Experimental data on Conjecture \ref{con:speccon} with $v = (1,1,-2)$}
\label{table: conjecture}
\begin{tabular*}{.3\linewidth}{@{\extracolsep{\fill}}ccrr@{}}
\toprule
$b$ & $a$ & $N$ & $\dim P_v$ \\
\midrule
3 & 1 & 1 & 9 \\
6 & 1 & 2 & 72 \\
9 & 1 & 3 & 243 \\
12 & 1 & 4 & 576 \\
15 & 1 & 5 & 1125 \\
\bottomrule
\end{tabular*}
\quad
\begin{tabular*}{.3\linewidth}{@{\extracolsep{\fill}}ccrr@{}}
\toprule
$b$ & $a$ & $N$ & $\dim P_v$ \\
\midrule
3 & 2 & 1 & 81 \\
6 & 2 & 1 & 1296 \\
9 & 2 & 1 & 6561 \\
12 & 2 & 1 & 20736 \\
15 & 2 & 1 & 50625 \\
\bottomrule
\end{tabular*}
\quad
\begin{tabular*}{.3\linewidth}{@{\extracolsep{\fill}}ccrr@{}}
\toprule
$b$ & $a$ & $N$ & $\dim P_v$ \\
\midrule
3 & 3 & 1 & 729 \\
6 & 3 & 2 & 93312 \\
9 & 3 & 1 & 531441 \\
12 & 3 & 2 & 5971968 \\
\\
\bottomrule
\end{tabular*}
\end{table}
The computations are similar to what we perform in the previous section. Given $a,b$ and $N$, we obtain a Laurent polynomial $N\cdot F_{a,b}$. The degree of this polynomial gives an upper bound on the degrees of the Schur polynomials that can appear in $P_v$. We then take all available Schur polynomials and solve an integral linear program like before. The source code and explicit polynomials $P_v$ can be found on our \MathRepo page~(\ref{eq:link}). \end{rmk}
\subsection*{Proof of Theorem \ref{thm:main}}\label{sec:proofalgeom} Our proof involves some algebraic geometry; see the textbooks by Hartshorne \cite{Hart77} and Harris \cite{Ha95}. We consider the polynomial map \begin{equation}\label{eq:curve}
f\colon \mathbb{C}^*\to \mathbb{C}^2,\, z\mapsto \bigl(F_1(z),F_2(z)\bigr)=(z^{v_1}+z^{v_2}+z^{v_3},z^{v_1+v_2}+z^{v_1+v_3}+z^{v_2+v_3}). \end{equation}
We first study where this map fails to be injective.
\begin{lem}\label{lem:fibone}
We have $f^{-1}(f(1))=\{1\}$. \end{lem}
\begin{proof}
Let $x\in\mathbb{C}^*$ such that $f(x)=f(1)$.
This implies
\begin{equation*}
(t-x^{v_1})(t-x^{v_2})(t-x^{v_3})=t^3-F_1(x)t^2+F_2(x)t-1=t^3-F_1(1)t^2+F_2(1)t-1=(t-1)^3,
\end{equation*}
which entails $x^{v_1}=x^{v_2}=x^{v_3}=1$. Since $\gcd(v_1,v_2,v_3)=1$, we get $x=1$. \end{proof}
For a complex number $x\in\mathbb{C}$, let $|x|=\sqrt{x\cdot\overline{x}}$ be its norm.
\begin{lem}\label{lem:realfiber}
For $|x|\neq 1$ we have $|f^{-1}(f(x))|=1$. \end{lem}
\begin{proof}
Let $y\in\mathbb{C}^*$ such that $f(y)=f(x)$. This implies that the zeros of the polynomial $$(t-y^{v_1})(t-y^{v_2})(t-y^{v_3})$$ are the three complex numbers $x^{v_1}$, $x^{v_2}$ and $x^{v_3}$. If $|x|>1$, then $|y|>1$ as well. Indeed, if two of the three integers $v_1,v_2,v_3$ are positive, then two of the three real numbers $|x|^{v_1}$, $|x|^{v_2}$ and $|x|^{v_3}$ are larger than one and thus the same must hold for the real numbers $|y|^{v_1}$, $|y|^{v_2}$ and $|y|^{v_3}$. Otherwise two of the three integers $v_1,v_2,v_3$ are negative, and a similar argument applies. Since for every real $t>1$ the map $d\mapsto t^d$ is strictly increasing, we must have $y^{v_1}=x^{v_1}$, $y^{v_2}=x^{v_2}$ and $y^{v_3}=x^{v_3}$.
This implies $f(\frac{y}{x})=f(1)$, and hence Lemma \ref{lem:fibone} yields $y=x$.
The case $|x|<1$ is analogous.
\end{proof}
\begin{rmk}\label{rem:exludecase}
The statement of Lemma~\ref{lem:realfiber} is not true in the case $(v_1,v_2,v_3)=(-1,0,1)$. Indeed, in this case the preimage of $f(x)$ under the map
\begin{equation}
f\colon\mathbb{C}^*\to \mathbb{C}^2,\, z\mapsto =(z^{v_1}+z^{v_2}+z^{v_3},z^{v_1+v_2}+z^{v_1+v_3}+z^{v_2+v_3})=(z^{-1}+1+z,z^{-1}+1+z) \end{equation} has two elements for all $x\in\mathbb{C}^*\setminus\{-1,1\}$. This is why we have excluded this case. \end{rmk}
We denote by $B(v)$ the $\mathbb{C}$-subalgebra of $\mathbb{C}[z^\pm]$ generated by $F_1$ and $F_2$. Note that $B(v)=A(v)\otimes_{\mathbb{Q}}\mathbb{C}$ and $A(v)=B(v)\cap\mathbb{Q}[z^\pm]$.
\begin{lem}\label{lem:finite} The ring extension $B(v)\subset\mathbb{C}[z^\pm]$ is \emph{finite}, i.e., $\mathbb{C}[z^\pm]$ is finitely generated as a $B(v)$-module. \end{lem}
\begin{proof}
For $t\in\{z^{v_1},z^{v_2},z^{v_3}\}$ we have
\begin{equation*}
t^3-F_1t^2+F_2t-1=(t-z^{v_1})(t-z^{v_2})(t-z^{v_3})=0.
\end{equation*}
This implies that $t^k$, for $k\in\mathbb{N}$, is contained in the $B(v)$-module that is generated by $1,t,t^2$. Thus $\mathbb{C}[z^{v_1},z^{v_2},z^{v_3}]$ is equal to the $B(v)$-module that is generated by
\begin{equation*}
\{z^{av_1}z^{bv_2}z^{cv_3}\mid 0\leq a,b,c\leq2\}.
\end{equation*}
Now it remains to show that $\mathbb{C}[z^\pm]=\mathbb{C}[z^{v_1},z^{v_2},z^{v_3}]$.
The inclusion ``$\supset$'' is clear. Since $\gcd(v_1,v_2,v_3)=1$, there are integers $a,b,c$ such that
\begin{equation*}
av_1+bv_2+cv_3=1.
\end{equation*}
Since $v_1+v_2+v_3=0$ we also have
\begin{equation*}
(a+m)v_1+(b+m)v_2+(c+m)v_3=1
\end{equation*} for every $m\in\mathbb{Z}$. In particular, we can find natural numbers $a',b',c'$ such that \begin{equation*}
a'v_1+b'v_2+c'v_3=1
\end{equation*}
meaning that $z=(z^{v_1})^{a'}(z^{v_2})^{b'}(z^{v_3})^{c'}\in \mathbb{C}[z^{v_1},z^{v_2},z^{v_3}]$.
Analogously, it can be proved that $z^{-1}\in \mathbb{C}[z^{v_1},z^{v_2},z^{v_3}]$. \end{proof}
The $\mathbb{C}$-algebra $B(v)$ is the coordinate ring of the algebraic curve $X\subset\mathbb{C}^2$ cut out by the elements of the kernel of the map \begin{equation*}
\mathbb{C}[x,y]\to B(v),\, P\mapsto P(F_1,F_2); \end{equation*} we denote the quotient field of $B(v)$ by $K$. Lemma \ref{lem:finite} implies that $X$ is, in fact, the image of $f$ because finite morphisms are closed \cite[Exc.~II.4.1]{Hart77}.
\begin{prop}\label{prop:mostlyinj}
There are only finitely many $x\in\mathbb{C}^*$ such that $|f^{-1}(f(x))|>1$.
All of them are roots of unity.
Moreover, we have $K=\mathbb{C}(z)$, the rational function field. \end{prop}
\begin{proof}
For every $x\in\mathbb{C}^*$ the fiber $f^{-1}(f(x))$ is Zariski closed in $\mathbb{C}^*$.
The Zariski closed subsets of $\mathbb{C}^*$ are either finite or all of $\mathbb{C}^*$.
Therefore, since $f$ is not constant, every fiber $f^{-1}(f(x))$ is finite.
By \cite[Proposition 7.16]{Ha95} the field extension $\mathbb{C}(z)/K$
is finite and there is a nonempty Zariski open subset $U\subset \mathbb{C}^*$ such that $|f^{-1}(f(x))|=[\mathbb{C}(z):K]$ for all $x\in U$.
This implies that $|f^{-1}(f(x))|=[\mathbb{C}(z):K]$ is true for all but finitely many $x\in\mathbb{C}^*$.
Lemma \ref{lem:realfiber} thus shows that $[\mathbb{C}(z):K]=1$ and that each of the finitely many $x\in\mathbb{C}^*$ with $|f^{-1}(f(x))|>1$ must satisfy $|x|=1$.
Moreover, since $f$ is defined over $\mathbb{Q}$, all such $x$ are algebraic numbers and therefore roots of unity. \end{proof}
The situation is very similar for ramification points of $f$.
\begin{prop}\label{prop:rami}
If $x\in\mathbb{C}^*$ is not a root of unity, then $f$ is unramified at $x$. \end{prop}
\begin{proof}
Since every power sum in three variables can be written as a polynomial in the first three elementary symmetric polynomials, there is for every $n\in\mathbb{N}$ a polynomial map
$\varphi_n\colon \mathbb{C}^2\to\mathbb{C}^n$
such that
\begin{equation*}
\varphi_n(f(x))=(x^{v_1}+x^{v_2}+x^{v_3}, x^{2v_1}+x^{2v_2}+x^{2v_3},\ldots,x^{nv_1}+x^{nv_2}+x^{nv_3}).
\end{equation*}
If $f$ is ramified at $x\in\mathbb{C}^*$, then $\varphi_n\circ f$ is also ramified at $x$ for every $n\in\mathbb{N}$. Thus
\begin{equation*}
kv_1x^{kv_1-1}+kv_2x^{kv_2-1}+kv_3x^{kv_3-1}=0
\end{equation*}
for all $k\in\mathbb{N}$. As $x$ and $k$ are nonzero, this implies that
\begin{equation*}
v_1x^{kv_1}+v_2x^{kv_2}+v_3x^{kv_3}=0
\end{equation*}
for all $k\in\mathbb{N}$. This means that $x^k$ is a zero of the nonconstant polynomial
\begin{equation*}
v_1z^{v_1}+v_2z^{v_2}+v_3z^{v_3}\in\mathbb{Q}[z]
\end{equation*}
for all $k\in\mathbb{N}$. Hence the set
\begin{equation*}
\{x^k\mid k\in\mathbb{N}\}
\end{equation*}
is finite which implies that $x$ is a root of unity. \end{proof}
\begin{cor}\label{cor:noniso}
There is a finite set $S\subset\mathbb{C}^*$ of roots of unity such that
\begin{equation*}
\mathbb{C}^*\setminus S\to X\setminus f(S),\, x\mapsto f(x)
\end{equation*}
is an isomorphism. \end{cor}
\begin{proof}
This follows from Proposition \ref{prop:mostlyinj}, Proposition \ref{prop:rami} and Lemma \ref{lem:finite} by \cite[Thm.~14.9]{Ha95}.
\end{proof}
\begin{rmk}
The smallest set $S\subset\mathbb{C}^*$, such that \begin{equation*}
\mathbb{C}^*\setminus S\to X\setminus f(S),\, x\mapsto f(x)
\end{equation*}
is an isomorphism, is the preimage of the singular locus of the curve $X$ under $f$. \end{rmk}
\begin{example}\label{ex:11} Let $v=(1,1,-2)$. Then $X$ is the zero set of the bivariate quartic polynomial \begin{equation*}
x_1^2 x_2^2-4 x_1^3-4 x_2^3+18 x_1 x_2-27 \end{equation*} in $\mathbb{C}^2$. The three points $f(1)=(3,3)$, $f(\xi)=(3\xi,3\xi^2)$ and $f(\xi^2)=(3\xi^2,3\xi)$ form the singular locus. In particular, its preimage under $f$ is the set of third roots of unity. See Figure~\ref{fig: X for v=11}. Note that this motivates the choice $b_0=3$ in Remark~\ref{rmk:testconj}. \begin{figure}
\caption{The real locus of $X$ for $v = (1,1,-2)$ plotted in the plane.}
\label{fig: X for v=11}
\end{figure} \end{example}
\begin{example}\label{ex:12} For $v=(1,2,-3)$ one computes that $X$ is cut out by \begin{equation*}
\begin{split}
x_1^3 x_2^3&-x_1^5-3 x_1^4 x_2-3 x_1 x_2^4-x_2^5-x_1^4+5 x_1^3 x_2+10 x_1^2 x_2^2+5 x_1 x_2^3- x_2^4 \\ &+x_1^3 -x_1^2 x_2-x_1 x_2^2+x_2^3-7 x_1^2-13 x_1 x_2-7 x_2^2.
\end{split} \end{equation*} \normalsize{The preimage of its singular locus under $f$ is the set of all third roots of unity along with the set of primitive seventh and eighth roots of unity.} \end{example}
Recall that the \emph{conductor} of the ring extension $B(v)\subset\mathbb{C}[z^\pm]$ is defined as \begin{equation*}
I=\{a\in B(v)\mid a\cdot \mathbb{C}[z^\pm]\subset B(v)\} \end{equation*} which is an ideal in both $B(v)$ and $\mathbb{C}[z^\pm]$. The zero set of $I$ is contained in the locus where $f$ fails to be an isomorphism \cite[p.~316]{Bour72Comm}. Thus by Corollary \ref{cor:noniso} and because $\mathbb{C}[z^\pm]$ is a principal ideal domain, it follows that $I$ is generated by a Laurent polynomial all of whose zeros are roots of unity. Since $f$ is defined over $\mathbb{Q}$, this Laurent polynomial is also defined over $\mathbb{Q}$. Therefore, we can write the generator of $I$ as $(z-1)^m\cdot \Phi$ where $\Phi$ is a product of cyclotomic polynomials with $\Phi(1)\neq0$.
\begin{lem}\label{lem:cusp}
We have $m=2$. \end{lem}
\begin{proof}
Recall from \eqref{eq:curve} that $X$ is the image of the map
\begin{equation*}
f\colon \mathbb{C}^*\to \mathbb{C}^2,\, z\mapsto \bigl(F_1(z),F_2(z)\bigr)=(z^{v_1}+z^{v_2}+z^{v_3},z^{v_1+v_2}+z^{v_1+v_3}+z^{v_2+v_3}). \end{equation*}
We have $F_1'(1)=F_2'(1)=0$ and $F_1''(1)=F_2''(1)=v_1^2+v_2^2+v_3^2\neq0$. Together with Lemma~\ref{lem:fibone} this implies that $X$ has an ordinary cusp at the image of $1$. The coordinate ring of $X$ is $B(v)$. Proposition~\ref{prop:mostlyinj} and Lemma~\ref{lem:finite} imply that $\mathbb{C}[z^\pm]$ is the integral closure of $B(v)$. In this situation the conductor has been computed in \cite[Proposition~1]{fultonblowup}. \end{proof}
As we are actually interested in polynomials with rational coefficients, we consider $A(v)$ the $\mathbb{Q}$-subalgebra of $\mathbb{Q}[z^\pm]$ generated by $F_1$ and $F_2$ and we let $I'$ the ideal of $\mathbb{Q}[z^\pm]$ generated by $(z-1)^m\cdot \Phi$.
\begin{cor}
The ideal $I'$ is the conductor of the ring extension $A(v)\subset\mathbb{Q}[z^\pm]$:
\begin{equation*}
I'=\{a\in A(v)\mid a\cdot \mathbb{Q}[z^\pm]\subset A(v)\}.
\end{equation*} \end{cor}
\begin{proof}
This follows from $I'=I\cap\mathbb{Q}[z^\pm]$ and $A(v)=B(v)\cap\mathbb{Q}[z^\pm]$. \end{proof}
Let $C_0\subset \mathbb{Q}[z^\pm]/I'$ the image of $A(v)$ modulo $I'$.
\begin{lem}\label{lem:subalg} Let $g\in\mathbb{Q}[z^\pm]$. Then we have $g\in A(v)$ if and only if the residue class of $g$ modulo $I'$ is in $C_0$. \end{lem}
\begin{proof} One direction is trivial so let us assume that the residue class of $g$ modulo $I'$ is in $C_0$. Thus there is a $h_1\in A(v)$ and $h_2\in I'$ such that $g=h_1+h_2$. Thus $g\in A(v)$ because $I'\subset A(v)$. \end{proof} By the Chinese remainder theorem we can naturally identify \begin{equation*}
\mathbb{Q}[z^\pm]/I'=(\mathbb{Q}[z^\pm]/(z-1)^2)\times (\mathbb{Q}[z^\pm]/\Phi). \end{equation*}
\begin{lem}\label{lem:product}
We have $C_0=\mathbb{Q}\times C$, where
\begin{equation*}
C=\{g\in \mathbb{Q}[z^\pm]/\Phi\mid \exists h\in \mathbb{Q}[z^\pm]/(z-1)^2)\colon (h,g)\in C_0\}.
\end{equation*} \end{lem}
\begin{proof} Since $F_i(1)=3$ and the derivative of $F_i$ vanishes at $1$, we have \begin{equation*}
F_i\equiv 3\mod(z-1)^2 \end{equation*} for $i=1,2$. This shows the inclusion "$\subset$". For the reverse inclusion we observe that, by Lemma \ref{lem:fibone}, there is a polynomial $G\in\mathbb{Q}[x_1,x_2]$ that vanishes on $f(\zeta)$ for all zeros $\zeta$ of $\Phi$ but $G(f(1))=1$. The residue class modulo $I'$ of a large enough power of $G(F_1,F_2)\in A(v)\subset\mathbb{Q}[z^\pm]$ is then $(1,0)$. In particular, this shows that $(1,0)\in C_0$ and proves the claim. \end{proof}
\begin{proof}[Proof of Theorem \ref{thm:main}] Let $F\in\mathbb{Q}[z^\pm]$. Then by Lemma \ref{lem:subalg} and Lemma \ref{lem:product} we have that $F$ lies in $A(v)$ if and only if the residue class of $F$ modulo $\Phi$ is in $C$ and the residue class of $F$ modulo $(z-1)^2$ in $\mathbb{Q}$. The latter condition is equivalent to $F'(1)=0$, which implies the claim. \end{proof}
\begin{example}
In the case $v=(1,2,-3)$, by Example \ref{ex:12} the smallest $b$ such that $F_{a,b}$ can possibly be divisible by the polynomial $Q$ from Theorem \ref{thm:main} for some $a\in\mathbb{N}$ is $b=3\cdot 7\cdot 8=168$. \end{example}
\section{Conclusion and Outlook}
As indicated already in the remarks after Question 20 in \cite{MR893156}, a similar strategy applies also to the cases $A=B=\operatorname{SU}(n)$. The only difference is that we now have to consider exponent vectors $(v_1,\dots,v_n) \in \mathbb{Z}^n$ satisfying $v_1+\cdots+ v_n=0$ and $\gcd(v_1,\dots,v_n)=1$ and Schur polynomials in $n$ variables. Putting it more precisely, we obtain the following question:
\begin{quest} Given $v = (v_1,\dots,v_n)$ and $w = (w_1,\dots,w_n)$ be integer vectors satisfying $v_1+\cdots+v_n=w_1+\cdots+w_n=0$ and $\gcd(v_1,\dots, v_n) = \gcd(w_1,\dots,w_n) = 1$, can we find Schur positive symmetric polynomials in $n$ variables $P$ and $Q$ such that: \begin{enumerate}
\item $P_v(z) = Q_w(z)$,
\item $P(\xi,\dots,\xi) \neq P(1,\dots,1)$ and $Q(\xi,\dots,\xi)\neq Q(1,\dots,1),$
for $\xi^n=1,\xi \neq 1$. \end{enumerate} \end{quest}
Theorem~\ref{thm:main} describes the Laurent polynomials which are a $\mathbb{Q}$-linear combination of monomials in $(e_1)_v(z),(e_2)_v(z),(e_3)_v(z)$ where $e_k$ is the elementary symmetric polynomial in three variables of degree $k$. For our purposes it would however be much more desirable to have an understanding of which Laurent polynomials are a \emph{positive} $\mathbb{Q}$-linear combination of monomials in $(e_1)_v(z),(e_2)_v(z),(e_3)_v(z)$. Indeed, since elementary symmetric polynomials are Schur positive and since the product of Schur positive polynomials is again Schur positive, a positive $\mathbb{Z}$-linear combination of monomials in $e_1,e_2,e_3$ is always Schur positive. We tried to get our hands on this by suitable variants of P\'olya's Theorem \cite[Theorem~5.5.1]{MarshSOS} but we have not been successful.
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\end{document} | arXiv |
Ecares - Eric Marchand, Sherbrooke U.
Abstract : This talk will address the estimation of predictive densities and their efficiency as measured by frequentist risk. For Kullback-Leibler, $\alpha-$divergence, $L_1$ and $L_2$ losses, we review several recent findings that bring into play improvements by scale expansion, as well as duality relationships with point estimation and point prediction problems. A range of models are studied and include multivariate normal with both known and unknown covariance structure, scale mixture of normals, Gamma, as well as models with restrictions on the parameter space. | CommonCrawl |
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A parallel water flow algorithm with local search for solving the quadratic assignment problem
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A slacks-based model for dynamic data envelopment analysis
January 2019, 15(1): 261-273. doi: 10.3934/jimo.2018042
Pricing options on investment project expansions under commodity price uncertainty
Nan Li 1,2, and Song Wang 1,
Department of Mathematics & Statistics, Curtin University, GPO Box U1987, WA 6845, Australia
School of Mathematical & Software Sciences, Sichuan Normal University, Sichuan 610000, China
Received May 2017 Revised October 2017 Published April 2018
Full Text(HTML)
Figure(4) / Table(2)
In this work we develop PDE-based mathematical models for valuing real options on investment project expansions when the underlying commodity price follows a geometric Brownian motion. The models developed are of a similar form as the Black-Scholes model for pricing conventional European call options. However, unlike the Black-Scholes' model, the payoff conditions of the current models are determined by a PDE system. An upwind finite difference scheme is used for solving the models. Numerical experiments have been performed using two examples of pricing project expansion options in the mining industry to demonstrate that our models are able to produce financially meaningful numerical results for the two non-trivial test problems.
Keywords: Real option valuation, project expansion, Black-Scholes equation, pricing flexibility, decision making.
Mathematics Subject Classification: Primary: 65M06; 91G20 Secondary: 91G60.
Citation: Nan Li, Song Wang. Pricing options on investment project expansions under commodity price uncertainty. Journal of Industrial & Management Optimization, 2019, 15 (1) : 261-273. doi: 10.3934/jimo.2018042
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Figure 4.1. The computed option value for Test 1.
Figure Options
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Figure 4.2. Computed option values at $t = 0$ for the different values of $\kappa$ and $\sigma$
Figure 4.3. Computed values of compound and normal options.
Figure 4.4. Computed option values and their differences for Test 2.
Table 4.1. Project and market data used in Test 1.
$Q = 10^4$ million tons $B = 30\%$ per annum
$C_0 = {\rm US}\$$35 $ C(t) = C_0\times e^{0.005t}$
$ R = 5\%$ per annum $ r = 0.06$ per annum
$ K = {\rm US}\$10^4$ million $ T = 2$ years
$ \sigma = 30\%$ $ \delta = 0.02$
$ q_0 = 0.01Q \times e^{0.007t}$ $ q_1 = \begin{cases} q_0&t < T \\ \kappa \times q_0&t \ge T \end{cases}$
$ T_1 = 2$ years $ T_2 = 4$ years
$ K_1 = {\rm US}\$10^4$ million $ K_2 = {\rm US}\$2 \times 10^4$ million
$ q_1 = \begin{cases} q_0&t < T_1 \\ 2 q_0&t \ge T_1 \end{cases}$ $ q_2 = \begin{cases} q_1&t < T_2 \\ 2 q_1&t \ge T_2 \end{cases}$
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\begin{document}
{.}\vskip.4in \centerline{\Huge Geometry} \vskip.1in \centerline{\Huge of} \vskip.1in \centerline{\Huge Slant Submanifolds}
\vskip.5in \centerline{\large by} \vskip.5in \centerline{\Large Bang-yen Chen} \vskip.3in \centerline{\large Department of Mathematics} \vskip.1in \centerline{\large Michigan State University} \vskip.1in \centerline{\large East Lansing, Michigan} \vskip1.2in \centerline{\huge Katholieke Universiteit Leuven} \vskip.1in \centerline{\huge 1990}
\eject
{.}\vskip1.4in \centerline{\huge Dedicated to} \vskip.3in \centerline{\huge Professor Tadashi Nagano} \vskip.3in \centerline{\large on the occasion of his sixtieth birthday}
\eject \markboth{B. Y. Chen}{Geometry of Slant Submanifolds}
\setcounter{page}{3} \vskip4in \centerline {\bf PREFACE} \vskip.3in The present volume is the written version of the series of lectures the author delivered at the Catholic University of Leuven, Belgium during the period of June-July, 1990. The main purpose of these talks is to present some of author's recent work and also his joint works with Professor T. Nagano and Professor Y. Tazawa of Japan, Professor P. F. Leung of Singapore and Professor J. M. Morvan of France on geometry of slant submanifolds and its related subjects in a systematical way.
The main references of the results presented in this volume are the following articles:
{[C2]} Differential geometry of real submanifolds in a Kaehler manifold, {\sl Monatsh. f\" ur Math.,} {\bf 91} (1981), 257-274,
{[C6]} Slant immersions, {\sl Bull. Austral. Math. Soc.,} {\bf 41} (1990), 135-147,
{[CLN]} Totally geodesic submanifolds of symmetric spaces, III, {\sl preprint,} 1980,
{[CM3]} Cohomologie des sous-vari\'et\'es $\alpha$-obliques, {\sl C. R. Acad. Sc. Paris,} {\bf 314} (1992), 931--934.
{[CT1]} Slant surfaces of codimension two, {\sl Ann.\ Fac.\ Sc.\ Toulouse Math.,} {\bf 11} (1990), 29--43.
\noindent and
{[CT2]} Slant submanifolds in complex Euclidean spaces, {\sl Tokyo J. Math.} {\bf 14} (1991), 101--120.
The author would like to take this opportunity to express his appreciation to the Catholic University of Leuven for their invitation and support. He would like to express his gratitute to his colleagues at Leuven, especially to Professor Leopold Verstraelen, Dr. Franki Dillen and Dr. Luc Vrancken for their hospitality. He also like to take this opportunity to express his heartfelt thanks to Dr. F. Dillen, Professor B. Smyth, Professor L.
Vanhecke, Professor L. Verstraelen and Dr. L. Vrancken for
many valuable discussions during his visit.
Finally, the author would also like to thank his colleagues at the Michigan State University, especially to Professor D. E. Blair, Professor G. D. Ludden and Professor W. E. Kuan for their constant encouragement through the years. \vskip.4in \hskip2.8in Bang-yen Chen
\hskip2.8in Summer 1990
\eject \vskip.7in
\centerline {\bf CONTENTS} \vskip.5in \noindent PREFACE
3
\vskip.15in \noindent CONTENTS
5 \vskip.2in \centerline {CHAPTER I} \vskip.05in \centerline {\bf INTRODUCTION} \vskip.1in \noindent \S 1. Introduction
7 \vskip.2in \centerline {CHAPTER II} \vskip.05in \centerline {\bf GENEFAL THEORY} \vskip.1in \noindent \S 1. Prelinimaries
13
\noindent \S 2. Some examples
18
\noindent \S 3. Some properties of $P$ and $F$
20
\noindent \S 4. Minimal slant surfaces and totally real surfaces
29
\vskip.2in \centerline {CHAPTER III} \vskip.05in \centerline {\bf GAUSS MAPS AND SLANT IMMERSIONS} \vskip.1in \noindent \S 1. Geometry of $G(2,4)$
33
\noindent \S 2. Complex structures on $E^4$
35
\noindent \S 3. Slant surfaces and Gauss map
39
\noindent \S 4. Doubly slant surfaces in ${\bf C}^2$
44
\noindent \S 5. Slant surfaces in almost Hermitian manifolds
46
\noindent {} \vskip.08in \centerline {CHAPTER IV} \vskip.05in \centerline {\bf CLASSIFICATIONS OF SLANT SURFACES} \vskip.1in
\noindent \S 1. Slant surfaces with parallel mean curvature vector
50
\noindent \S 2. Spherical slant surfaces
53
\noindent \S 3. Slant surfaces with $rk (\nu)<2$
67
\noindent \S 4. Slant surfaces with codimension one
71 \vskip.2in
\centerline {CHAPTER V} \vskip.05in \centerline {\bf TOPOLOGY AND STABILITY OF SLANT SUBMANIFOLDS} \vskip.1in
\noindent \S 1. Non-compactnes of proper slant submanifolds
77
\noindent \S 2. Topology of slant surfaces
84
\noindent \S 3. Cohomology of slant submanifolds
89
\noindent \S 4. Stablity and index form
98
\noindent \S 5. Stability of totally geodesic submanifolds
105
\vskip.2in \noindent REFERENCES
114 \vskip.15in \noindent SUBJECT INDEX
121
\eject
\vskip1in \centerline {CHAPTER I} \vskip.2in \centerline {\bf INTRODUCTION} \vskip.5in \noindent \S 1. INTRODUCTION. \vskip .2in The theory of submanifolds of an almost Hermitian
manifold, in particular of a Kaehlerian manifold, is one of the most interesting topics in differential geometry. According to the behaviour of the tangent bundle of a submanifold, with respect to the action of the almost complex structure $J$ of the ambient manifold, there are two well-known classes of submanifolds, namely, the complex submanifolds and the totally real submanifolds. In this volume we will present the geometry of another important class of submanifolds, called slant submanifolds.
The theory of submanifolds of an almost Hermitian manifold or of a Kaehlerian manifold began as a separate area of study in the last century with the investigation of algebraic curves and algebraic surfaces in classical algebraic geometry. Included among the principal investigators are Riemann, Picard, Enriques, Castelnuovo, Severi, and Segre.
It was E. K\" ahler ([Ka1]), J. A. Schouten, and D. van Dantzig ([SD1] and [SD2]) who first tried to study complex manifolds from the viewpoint of Riemannian geometry in the early 1930's. In their studies a Hermitian space with the so-called symmetric unitary connection was introduced. A Hermitian space with such a connection is now known as a Kaehlerian manifold.
It was A. Weil [W1] who in 1947 pointed out there exists in a complex manifold a tensor field $J$ of type (1,1) whose square is equal to the negative of the identity transformation of the tangent bundle, that is, $J^{2}=-I.$ In the same year, C. Ehresmann introduced the notion of an almost complex manifold as an even-dimensional differentiable manifold which admits such a tensor field $J$ of type (1,1).
An almost complex manifold (respectively, complex manifold) is called an almost Hermitian manifold (respectively, a Hermitian manifold) if it admits a Riemannian structure which is compatible with the almost complex structure $J$. The theory of almost Hermitian manifolds, Hermitian manifolds, and in particular Kaehlerian manifolds has become a very interesting and important branch of modern differential geometry (see, for instances, [Ch1], [Ko1], [Ko2], [KN1], [KN2], [Mi1], [Wl1].)
The study of complex submanifolds of a Kaehlerian manifold from the differential geometrical points of view (that is, with emphasis on the Riemannian metric) was initiated by E. Calabi and others in the early of 1950's (cf. [Ca1], [Ca2]). Since then it has became an active and fruitful field in modern differential geometry. Many important results on Kaehlerian submanifolds have been obtained by many differential geometers in the last three decades. Two nice survey articles concerning this subject were given by K. Ogiue in [O1] and [O2].
In terms of the behaviour of the tangent bundle $TN$ of the submanifold $N$, complex submanifolds $N$ of an almost Hermitian manifold $(M,g,J)$ are characterized by the condition: $$J(T_{p}N) \subseteq T_{p}N\leqno(1.1)$$ for any point $p \in N$. In other words, $N$ is a complex submanifold of $(M,g,J)$ if and only if for any nonzero vector $X$ tangent to $N$ at any point $p \in N$, the angle between $JX$ and the tangent plane $T_{p}N$ is equal to zero, identically.
Besides complex submanifolds, there is another important class of submanifolds, called {\it totally real submanifolds.\/}
A totally real submanifold $N$ of an almost Hermitian manifold M or, in particular, of a Kaehlerian manifold, is a submanifold such that the (almost) complex structure $J$ of the ambient manifold $M$ carries each tangent vector of $N$ into the corresponding normal space of $N$ in $M$, that is, $$J(T_{p}N)\subseteq T_{p}^{\perp}N\leqno(1.2)$$ for any point $p\in N$. In other words, $N$ is a totally real submanifold of $(M,g,J)$ if and only if for any nonzero vector $X$ tangent to $N$ at any point $p \in N$, the angle between $JX$ and the tangent plane $T_{p}N$ is equal to ${\pi\over 2}$, identically.
The study of totally real submanifolds from the differential geometric points of view was intiated in the early 1970's (see [CO1] and [YK1]). Since then many differential geometers have contributed many interesting results in this subject.
In this volume I shall present the third important class of submanifolds of an almost Hermitian manifold $(M,g,J)$ (in particular, of a Kaehlerian manifold), called {\it slant submanifolds.\/} A slant submanifold is defined in [C6] as a submanifold of $(M,g,J)$ such that, for any nonzero vector $X\in T_{p}N$, the angle $\theta(X)$ between $JX$ and the tangent space $T_{p}N$ is a constant (which is independent of the choice of the point $p\in N$ and the choice of the tangent vector $X$ in the tangent plane $T_{p}N$). It is obvious that complex submanifolds and totally real submanifolds are special classes of slant submanifolds. A slant submanifold is called {\it proper\/} if it is neither a complex submanifold nor a totally real submanifold.
In the first section of Chapter 2, we present some basic definitions and basic formulas for later use. In particular, for any submanifold $N$ of an almost Hermitian manifold $(M,g,J)$, the almost complex structure $J$ of the ambient manifold induces a canonical endomorphism of the tangent bundle, denoted by $P$, and a canonical normal-bundle-valued 1-form on the tangent bundle, denoted by $F$. It will be shown in later sections that the endomorphism $P$ plays a fundamental role in our study.
In Section 2 of Chapter 2, we give many examples of slant surfaces in the complex number space ${\bf C}^2$ and give examples of Kaehlerian slant submanifolds (that is, proper slant submanifolds such that the canonical endomorphism $P$ is parallel) in ${\bf C}^m$.
Section 3 of Chapter 2 is devoted to the fundamental study of the endomorphism $P$ and the normal-bundle-valued 1-form $F$. In particular, we prove that every slant surface in any almost Hermitian manifold is a Kaehlerian slant submanifold, that is, $\nabla P=0$ (Theorem 3.4). Morevoer, we prove that a proper slant submanifold $N$ of a Kaehlerian manifold $M$ is Kaehlerian slant if and only if the Weingarten map of $N$ in $M$ satisfies $$A_{FX}Y=A_{FY}X\leqno(1.3)$$ for any vectors $X,Y$ tangent to the submanifold. In particular, by combining this result with Theorem 3.4, we obtain the important fact that formula (1.3) holds for any slant surface in any Kaehler manifold. By using this result we show that the Gauss curvature $G$ and the normal curvature $G^D$ of every slant surface in ${\bf C}^2$ satisfies $$G=G^{D},\leqno(1.4)$$ identically. In this section we also prove that every proper slant submanifold of a Kaehlerian manifold with $\nabla F=0$ is an austere submanifold (Theorem 3.8). We also obtain in this section two reduction theorems for submanifolds satisying $\nabla F=0$ (Theorem 3.9 and Theorem 3.10). Finally we show that for a surface in a real 4-dimensional Kaehlerian manifold, the parallelism of $F$ implies the parallelism of $P$ (Theorem 3.11).
In Section 4 of Chapter II, we establish some relations between minimal slant surfaces and totally real surfaces in a Kaehlerian manifold. For instance, we prove that if a proper slant surface in a real 4-dimensional Kaehlerian manifold is also totally real with respect to some compatible complex structure on the ambient manifold at the same time, then the surface must be a minimal surface (Theorem 4.2). Some applications of this result will also be given in this section (Theorem 4.3 and Theorem 4.4).
In Chapter III, we present some results of the author and Y. Tazawa [CT1]; in this work slant surfaces in ${\bf C}^2$ were studied from the viewpoint of the Gauss map.
The first section of Chapter III reviews some basic geometry of the real Grassmannian $G(2,4)$ which consists of all oriented 2-planes in $E^4$ which will be used in later sections.
The second section is devoted to the detailed study of the set of compatible complex structures on $E^4$. Several lemmas are obtained in this section, in particular, Lemma 2.1 and Lemma 2.3 of this section, play some important roles in the later sections.
In Section 3 we study the following two geometric problems: \vskip.1in {\bf Problem 3.1.} {\it Let $N$ be a surface in ${\bf C}^{2} = (E^{4},J_{0})$. When is $N$ slant in ${\bf C}^2$?} \vskip.1in {\bf Problem 3.2.} {\it Let $N$ be a surface in $E^4$. If there exists a compatible complex structure $J$ on $E^4$ such that $N$ is slant in $(E^{4},J)$. How many other compatible complex structures ${\tilde J}$ on $E^4$ are there such that $N$ is slant with resepct to these complex strucutres?} \vskip.1in Complete solutions to these two problems are obtained in this section.
Related to the problems studied in Section 3 is the notion of doubly slant surfaces. We prove that every doubly slant surface in $E^4$ has vanishing Gauss curvature and vanishing normal curvature (Theorem 4.1).
In the last section of Chapter III, we study slant surfaces in almost Hermitian manifolds. In fact we prove that if a surface in a real 4-dimensional almost Hermitian manifold has no complex tangent points (that is, the surface is purely real), then, with respect to some suitable compatible almost complex structure on the ambient manifold, the surface is slant. This result shows that there exist ample examples of slant surfaces in almost Hermitian manifolds.
Chapter IV is devoted to the classification problems of slant surfaces in ${\bf C}^{2}$. In the first section of this chapter, we classify all slant surfaces in ${\bf C}^{2}$ with parallel mean curvature vector. In the second section we classify spherical slant surfaces in ${\bf C}^{2}$.
In the third section, we classify slant surfaces in ${\bf C}^{2}$ whose Gauss map has rank $< 2$ at every point. The last section gives the classification of slant surfaces in ${\bf C}^2$ which are contained in a hyperplane of ${\bf C}^{2}$.
In the last Chapter, we study the topology and cohomology of slant submanifolds.
In the first section, we prove that every compact slant submanifold in ${\bf C}^{m}$ is totally real (Theorem 1.5) and every compact slant submanifold in an exact sympletic manifold is also totally real (Theorem 1.7).
In Section 2, we define a canonical 1-form $\Theta$ associated with a proper slant submanifold $N$ in an almost Hermitian manifold (by formula (2.7) of Chapter V). We prove in this section that for a slant surface in ${\bf C}^{2}$, this 1-form is closed, that is, $d\Theta=0$ and if we put $$\Psi=(2\sqrt{2}\pi)^{-1}(\csc\alpha)\Theta,\leqno (1.5)$$
\noindent where $\alpha$ denotes the slant angle of $N$ in ${\bf C}^2$, then the 1-form $\Psi$ defines a
canonical integral class on $N$ (Theorem 2.5): $$[\Psi] \in H^{1}(N;{\bf Z}).\leqno(1.6)$$
\noindent In this section we also prove that if $N$ is a complete, oriented, proper slant surface of ${\bf C}^{2}$ such that the mean curvature of $N$ is bounded below by some positive constant, then, topologically, the surface is either a circular cylinder or a 2-plane (Theorem 2.6).
In Section 3 of this chapter, we prove that in fact, for any $n$-dimensional proper slant submanifold of ${\bf C}^{n}$, the 1-form $\Theta$ is always closed. Thus for any $n$-dimensional proper slant submanifold $N$ in ${\bf C}^{n}$, we have a canonical cohomology class $[\Theta] \in H^{1}(N;{\bf R})$ (Theorem 3.1). Finally we prove in this section that every proper slant submanifold of a Kaehlerian manifold has a canonical sympletic structure given by the 2-form induced from the canonical endomorphism $P$ (Theorem 3.4). The last result implies that if a compact $2k$-dimensional differentiable manifold $N$ satisfies $H^{2i}(N;{\bf R})=0$ for some $i \in \{1,\ldots,k\}$, then $N$ cannot be immersed in any Kaehleian manifold as a proper slant submanifold (Theorem 3.$5'$).
In Section 4, we recall some stablity theorems of [CLN] obtained in 1980 and present some related results concerning the index form of
compact minimal totally real submanifolds, a special class of slant submanifolds, of a Kaehlerian manifold.
In the last section, we present a general method introduced by the author, Leung and Nagano [CLN] for determining the stability of totally geodesic submanifolds in compact symmetric spaces. Since every irreducible totally geodesic submanifolds of a Hermitian symmetric space is a slant submanifold [CN1], the method can be used to determine the stability of such submanifolds.
Finally, I would like to mention that minimal surfaces of a complex projective space with constant Kaehlerian angle were recently studied from a different point of view by Bolton, Jensen, Rigoli, Woodward, Maeda, Ohnita and Udagawa (see, [BJRW], [MU1] and [Oh1]).
\eject
\centerline {CHAPTER II} \vskip.2in \centerline {\bf GENERAL THEORY} \vskip .5in \noindent {\S 1}. PRELIMINARIES. \vskip.2in Let $N$ be an $n$-dimensional Riemannian manifold isometrically immersed in an almost Hermitian manifold $M$ with (almost) complex structure $J$ and almost Hermitian metric $g$. We denote by $< , >$ the inner product for $N$ as well as for $M$.
For any vector $X$ tangent to $N$ we put $$JX=PX+FX,\leqno(1.1)$$ where $PX$ and $ FX$ are the tangential and the normal components of $JX$, respectively. Thus, $P$ is an endomorphim of the tangent bundle $TN$ and $F$ a normal-bundle-valued 1-form on $TN$.
For any nonzero vector $X$ tangent to $N$ at a point $x \in N$, the angle $\theta (X)$ between $JX$ and the tangent space $T_{x}N$ is called the {\it Wirtinger angle\/} of $X$. In the following we call an immersion $f:N \rightarrow M$ a {\it slant immersion\/} if the Wirtinger angle $\theta (X)$ is constant (which is independent of the choice of $x \in N$ and of $X \in T_{x}N$). Complex and totally real immersions are slant immersions with $\theta = 0$ and $\theta = \pi /2$, respectively. Moreover, it is easy to see that slant submanifolds of an almost Hermitian manifolds are characterized by the condition: $\, P^{2}=\lambda I,\,$ for some real number $\lambda \in [-1,0],$ where $I$ denotes the identity transformation of the tangent bundle $TN$ of the submanifold $N$.
The Wirtinger angle of a slant immersion is called the {\it slant angle \/} of the slant immersion. A slant submanifold is said to be {\it proper\/} if it is neither complex nor totally real.
A proper slant submanifold is said to be {\it Kaehlerian slant \/} if the canonical endomorphism $P$ defined above is parallel, that is, $\nabla P = 0$. A Kaehlerian slant submanifold of a Kaehlerian manifold with respect to the induced metric and with the almost complex structure given by ${\tilde J}= (\sec \theta)P, \theta = $ the slant angle.
For a submanifold $N$ of an almost Hermitian manifold $M$, we denote by $\nabla$ and ${\tilde \nabla}$ the Levi-Civita connections of $N$ and $M$, respectively. Then the Gauss and Weingarten formulas of $N$ in $M$ are given respectively by $${\tilde \nabla}_{X}Y=\nabla_{X} + h(X,Y),\leqno(1.2)$$ $${\tilde\nabla}_{X}\xi = -A_{\xi}X+D_{X}\xi, \leqno(1.3)$$ for any vector fields $X,Y$ tangent to $N$ and any vector field $\xi$ normal to $N$, where $h$ denotes the second fundamental form, $D$ the normal connection, and $A$ the Weingarten map of the submanifold $N$ in $M$. The second fundamental form $h$ and the Weingarten map $A$ are related by $$<A_{\xi}X,Y>=<h(X,Y),\xi >.\leqno(1.4)$$
For any vector field $\xi$ normal to the submanifold $N$, we put $$J\xi = t\xi + f\xi,\leqno(1.5)$$ where $t\xi$ and $f\xi$ are the tangential and the normal components of $J\xi$, respectively. Then $f$ is an endomorphism of the normal bundle and $t$ is a tangent-bundle-valued 1-form on the normal bundle $T^{\bot}N$.
For a submanifold $N$ in a Riemannian manifold $M$, the mean curvature vector $H$ is defined by $$H = {1\over n}\, tr\,h = {1\over n}\sum_{i=1}^{n} h(e_{i},e_{i})\leqno(1.6)$$ where $\{e_{1},\ldots,e_{n}\}$ is a local orthonormal frame of the tangent bundle $TN$ of $N$. A submanifold $N$ in $M$ is said to be {\it totally geodesic\/} if the second fundamental form $h$ of $N$ in $M$ vanishes identically.
Let $N$ be an $n$-dimensional submanifold in an $m$-diemsional Riemannian manifold $M$. We choose a local field of orthonoraml frames
$$e_{1},\ldots,e_{n},e_{n+1},\ldots,e_{m}$$ such that, restricted to $N$, the vectors $e_{1},\ldots,e_{n}$ are tangent to $N$ and hence $e_{n+1},\ldots,e_{m}$ are normal to $N$. We shall make use of the following convention on the ranges of indices unless mentioned otherwise: $$1\leq A,B,C,\ldots \leq m;\,\,\,\,\,\, 1 \leq i,j,k,\ldots \leq n;$$ $$n+1 \leq r,s,t,\ldots \leq m. $$
With respect to the frame field of $M$ chosen above, let $$\omega^{1},\ldots,\omega^{n},\omega^{n+1},\ldots,\omega^{m}$$ be the field of dual frames. Then the structure equations of $M$ are given by $$d\omega^{A}=-\sum_{B} \omega_{B}^{A}\wedge\omega^{B}, \hskip.5in \omega_{B}^{A}+\omega_{A}^{B}=0,\leqno(1.7)$$ and
$$d\omega_{B}^{A}=-\sum_{C} \omega_{C}^{A}\wedge \omega_{B}^{C} + \Phi_{B}^{A},\hskip.2in \Phi_{B}^{A}={1\over 2}\sum_{C,D} K^{A}_{BCD}\omega^{C}\wedge\omega^{D},\leqno(1.8)$$ $$K^{A}_{BCD}+K^{A}_{BDC}=0.$$
We restrict these forms to $N$. Then $$\omega^{r}=0.$$ Since $$0=d\omega^{r}=-\sum_{i}\omega_{i}^{r}\wedge\omega^{i},$$ by Cartan's lemma we have $$\omega^{r}_{i} = \sum_{j} h_{ij}^{r} \omega^{j}, \,\,\,\,\, h^{r}_{ij}= h^{r}_{ji}.\leqno(1.9)$$ Formula (1.9) is equivalent to $$\omega_{i}^{r}(X)=<A_{e_{r}}e_{i},X>\leqno(1.9)'$$ for any vector $X$ tangent to $N$. From these formulas we obtain $$d\omega^{i}= -\sum_{j} \omega_{j}^{i}\wedge\omega^{j},\,\,\,\, \omega_{j}^{i}+\omega_{i}^{j}=0,\leqno(1.10)$$ $$d\omega_{j}^{i}=-\sum_{k}\omega_{k}^{i}\wedge\omega_{j}^{k} + \Omega_{j}^{i},\,\,\, \Omega_{j}^{i}={1\over 2}\sum_{k,\ell}R^{i}_{jk\ell}\omega^{k}\wedge\omega^{\ell}, \leqno(1.11)$$ $$R^{i}_{jk\ell}=K^{i}_{jk\ell}+\sum_{r}(h^{r}_{ik}h^{r}_{j\ell} -h^{r}_{i\ell}h^{r}_{jk}),\leqno(1.12)$$ $$d\omega_{i}^{r}=-\sum_{j,k} h^{r}_{jk}\omega_{i}^{j}\wedge\omega^{k} -\sum_{j,s} h^{s}_{ij}\omega^{j}\wedge\omega^{r}_{s} +{1\over 2}\sum_{j,k}K^{r}_{ijk}\omega^{j}\wedge\omega^{k}, \leqno(1.13)$$ $$d\omega_{s}^{r}=-\sum_{t}\omega_{t}^{r}\wedge\omega_{s}^{t} +\Omega_{s}^{r}, \,\,\,\, \Omega_{s}^{r}= {1\over 2}\sum_{k,\ell}R^{r}_{sk\ell}\omega^{k}\wedge \omega^{\ell}, \leqno(1.14)$$ and $$R^{r}_{sk\ell}=K^{r}_{sk\ell}+\sum_{i} (h^{r}_{ik}h^{s}_{i\ell}-h^{r}_{i\ell}h^{s}_{ik}). \leqno(1.15)$$
For any vector field $X$ tangent to the submanifold $N$, these forms are also given by $${\tilde \nabla}_{X}e_{i}=\sum_{j=1}^{n}\omega_{i}^{j}(X) e_{j}+ \sum_{r=n+1}^{m}\omega_{i}^{r}(X)e_{r},\leqno(1.16)$$
$${\tilde \nabla}_{X}e_{r}=\sum_{j=1}^{n}\omega_{r}^{j}(X) e_{j}+ \sum_{s=n+1}^{m}\omega_{r}^{s}(X)e_{s}.\leqno(1.17)$$
These 1-forms $\omega_{i}^{j}$, $\omega_{i}^{r}$ and $\omega_{r}^s$ are called the {\it connection forms\/} of $N$ in $M$.
Denote by $R$ and ${\tilde R}$ the Riemann curvature tensors of $N$ and $M$, respectively, and by $R^D$ the curvature tensor of the normal connection $D$. Then the {\it equation of Gauss\/} and the {\it equation of Ricci\/} are given respectively by $${\tilde R}(X,Y;Z,W)=R(X,Y;Z,W)+<h(X,Z),h(Y,W)>\leqno(1.18)$$ $$-<h(X,W),h(Y,Z)>,$$ $$R^{D}(X,Y;\xi,\eta)={\tilde R}(X,Y;\xi,\eta)+<[A_{\xi}, A_{\eta}](X),Y>\leqno(1.19)$$ for vectors $X,Y,Z,W$ tangent to $N$ and $\xi,\eta$ normal to $N$.
For the second fundamental form $h$, we define the covariant derivative ${\bar \nabla}h$ of $h$ with respect to the connection in $TN \oplus T^{\perp}N$ by $$({\bar\nabla}_{X}h)(Y,Z)=D_{X}(h(Y,Z))-h(\nabla_{X}Y,Z) -h(Y,\nabla_{X}Z).\leqno(1.20)$$
The {\it equation of Codazzi\/} is given by $$({\tilde R}(X,Y)Z)^{\perp}=({\bar\nabla}_{X}h)(Y,Z)- ({\bar\nabla}_{Y}h)(X,Z),\leqno(1.21)$$ where $({\tilde R}(X,Y)Z)^{\perp}$ denotes the normal component of ${\tilde R}(X,Y)Z$.
A submanifold $N$ of a Riemannian manifold $M$ is called a {\it parallel submanifold\/} if the second fundamental form $h$ is parallel, that is, ${\bar\nabla}h=0$, identically.
\eject
\vskip.3in \noindent \S 2. SOME EXAMPLES. \vskip.2in In the following, $E^{2m}$ denotes the Euclidean $2m$-space with the standard metric. An almost complex structure $J$ on $E^{2m}$ is said to be compactible if $(E^{2m},J)$ is complex analytically isometric to the complex number space ${\bf C}^m$ with the standard flat Kaehlerian metric. We denote by $J_0$ and $J_{1}^{-}$ (when $m$ is even) the compatible almost complex structures on $E^{2m}$ defined respectively by $$J_{0}(a_{1},\ldots,a_{m},b_{1},\ldots,b_{m})=(-b_{1}, \ldots,-b_{m},a_{1},\ldots,a_{m})\leqno(2.1)$$ and $$J_{1}^{-}(a_{1},\ldots,a_{m},b_{1},\ldots,b_{m})\leqno(2.2)$$ $$=(-a_{2},a_{1},\ldots,-a_{m},a_{m-1},b_{2},-b_{1}, \ldots,b_{m},-b_{m-1}).$$
In this section we give some examples of proper slant surfaces in ${\bf C}^{2}=(E^{4},J_{0})$ and some examples of Kaehlerian slant submanifolds of higher dimension in ${\bf C}^{m}$. \vskip.1in {\bf Example 2.1.} For any $\alpha >0$, $$x(u,v) = (u\cos \alpha,v, u\sin \alpha, v,0)$$ defines a slant plane with slant angle $\alpha$ in ${\bf C}^2$.
\vskip.1in {\bf Example 2.2.} Let $N$ be a complex surface in ${\bf C}^{2}=(E^{4},J_{0})$. Then for any constant $\alpha$, $0<\alpha\leq \pi /2, N$ is slant surface in $(E^{4},J_{\alpha})$ with slant angle $\alpha$, where $J_{\alpha}$ is the compatible almost complex structure on $E^{4}$ defined by $$J_{\alpha}(a,b,c,d) =(\cos\alpha)(-c,-d,a,b)+(\sin\alpha)(-b,a,d,-c).$$ This example shows that there exist infinitely many proper slant minimal surfaces in ${\bf C}^{2}= (E^{4},J_{0})$. \vskip.1in The following example provides us some non-minimal proper slant surfaces in ${\bf C}^{2}=(E^{4},J_{0})$. \vskip.1in {\bf Example 2.3.} [GVV] For any positive constant $k$, $$x(u,v) =(e^{ku}\cos u\cos v, e^{ku}\sin u\cos v,e^{ku}\cos u\sin v, e^{ku}\sin u\sin v)$$ defines a complete, non-minimal, pseudo-umbilical proper slant surface with slant angle $\theta=\cos
^{-1}(k/\sqrt{1+k^{2}})$ and with non-constant mean curvature given by $|H|=e^{-ku}/\sqrt{1+k^{2}}$. \vskip.1in {\bf Example 2.4.} For any positive number $k$, $$x(u,v)=(u,k\cos v,v,k\sin v)$$ defines a complete, flat, non-minimal and non-pseudo-umbilical, proper slant surface with slant angle $\cos^{-1}(1/\sqrt{1+k^{2}})$ and constant mean curvature $k/2(1+k^{2})$ and with non-parallel mean curvature vector. \vskip.1in {\bf Example 2.5.} Let $k$ be any positive number and $(g(s),h(s))$ a unit speed plane curve. Then $$x(u,s) =(-ks\sin u, g(s),ks\cos u, h(s))$$ defines a non-minimal, flat, proper slant surface with slant angle $k/\sqrt{1+k^{2}}$. \vskip.1in {\bf Example 2.6} [CT1]. For any nonzero real numbers $p$ and $q$, we consider the following immersion form ${\bf R}\times (0,\infty )$ into ${\bf C}^2$ defined by $$x(u,v)=(pv\sin u,pv\cos u,v\sin qu, v\cos qu).$$ Then the immersion $x$ gives us a complete flat slant surface in ${\bf C}^2$. \vskip.1in {\bf Example 2.7.} For any $k>0$, $$x(u,v,w,z)=(u,v,k\sin w,k\sin z, kw,kz,k\cos w,k\cos z)$$ defines a Kaehlerian slant submanifold in ${\bf C}^4$ with slant angle $\cos^{-1}k.$ \vskip.1in {\bf Example 2.8.} Let $N$ be a complex submanifold of the complex number space ${\bf C}^{2m}=(E^{4m},J_{0})$. For any constant $\alpha$ we define $J_{\alpha}$ by $$J_{\alpha}=(\cos\alpha)J_{0}+(\sin\alpha)J_{1}^{-}.\leqno(2.3)$$ Then $J_{\alpha}$ is a compatible complex structure on $E^{4m}$ and $N$ is a Kaehlerian slant submanifold with slant angle $\alpha$ in $(E^{4m},J_{\alpha})$.
\eject
\vskip.3in
\noindent \S 3. SOME PROPERTIES OF $P$ AND $F$. \vskip.2in Let $f: N \rightarrow M$ be an isometric immersion of an $n$-dimensional Riemannian manifold into an almost Hermitian manifold. Let $P$ and $F$ be the endomorphism and the normal-bundle-valued 1-form on the tangent bundle defined by (1.1). Since $M$ is almost Hermitian, we have $$<PX,Y>=-<X,PY>$$ for any vectors X, Y tangent to $N$. Hence, if we put $Q=P^2$, then $Q$ is a self-adjoint endomorphim of $TN$. Therefore, each tangent space $T_{x}N$ of $N$ at $x \in N$ admits an orthogonal direct decompostion of eigenspces of $Q$: $$T_{x}N = {\mathcal D}_{x}^{1} \oplus\dots\oplus {\mathcal D}_{x}^{k(x)}.$$ Since $P$ is skew-symmetric and $J^{2}=-I, $ each eigenvalue $\lambda_i$ of $Q$ lies in $[-1,0]$ and, moreover, if $\lambda_{i} \not= 0,$ then the corresponding eigenspace ${\mathcal D}_{x}^{i}$ is of even dimension and it is invariant under the endomorphism $P$, that is $P({\mathcal D}_{x}^{i})={\mathcal D}_{x}^{i}.$ Furthermore, for each $\lambda_{i} \not= -1,$ dim $F({\mathcal D}_{x}^{i})=$ dim ${\mathcal D}_{x}^{i}$ and the normal subspaces $F({\mathcal D}_{x}^{i})$, $i=1,\ldots,k(x),$ are mutually perpendicular. From these we have $${\rm dim}\, M \geq 2 {\rm dim}\, N - {\rm dim }\,{\mathcal H}_{x}$$ where ${\mathcal H}_{x}$ denotes the eigenspace of $Q$ with eigenvalue $-1$.
In this section we mention some results given in [C2] and [C6] concerning the endomorphism $P$ and the normal-bundle-valued 1-form $F$ associated with the immersion $f : N \rightarrow M$.
The following Lemma 3.1 follows from the definition of $\nabla Q$ which is defined by $$(\nabla_{X}Q)Y=\nabla_{X}(QY)-Q(\nabla_{X}Y)\leqno(3.1)$$ for $X$ and $Y$ tangent to $N$.
\vskip.1in {\bf Lemma 3.1.} {\it Let $N$ be a submanifold of an almost Hermitian manifold $M$. Then the self-adjoint endomorphism $Q \,(= P^{2})$ is parallel, that is, $\nabla Q=0,$ if and only if}
(1) {\it each eigenvalue $\lambda_i$ of $Q$ is constant on $N$;}
(2) {\it each distribution ${\mathcal D}^i$ (associated with the eigenvalue $\lambda_i$) is completely integrable and}
(3) {\it $N$ is locally the Riemannian product $N_{1}\times\ldots\times N_k$ of the leaves of the distributions.} \vskip.1in {\bf Proof.} Since $Q$ is a self-adjoint endomorphism of the tangent bundle $TN$, there exist $n$ continuous functions $\lambda_{1}\leq \lambda_{2} \leq \ldots \leq \lambda_{n}$ such that $\lambda_{i},\,\,i=1,\ldots,n,\,\,$ are the eigenvalues of $Q$ at each point points $p\in N$. Let $e_{1},\ldots,e_n$ be a local orthonormal frame given by eigenvectors of $Q$. If $Q$ is parallel, then (3.1) implies $$\nabla_{X}(\lambda_{i}e_{i})=Q(\nabla_{X}e_{i}),\,\,\,i=1,\ldots, n$$ for any vector $X$ tangent to $N$. Thus we have $$(X\lambda_{i})e_{i}+\lambda_{i}(\nabla_{X}e_{i})= Q(\nabla_{X}e_{i}).$$ Since both $\nabla_{X}e_i$ and $Q(\nabla_{X}e_{i})$ are perpendicular to $e_i$, we conclude that each eigenvalue of $Q$ is constant on $N$. This proves statement (1).
For statements (2) and (3) we let $\lambda_{1},\ldots,\lambda_k$ denote the distinct eigenvalues of $Q$. For each $i;\,i=1,\ldots,k\,$, let ${\mathcal D}^i$ denote the distribution given by the eigenspaces of $Q$ with eigenvalue $\lambda_i$. For any two vector fields $X,Y$ in the distribution ${\mathcal D}^i$, (3.1) and statement (1) imply $$Q(\nabla_{X}Y)=\lambda_{i}(\nabla_{X}Y),$$ from which we conclude that $\nabla_{X}Y \in {\mathcal D}^i$ for any $X,Y$ in ${\mathcal D}^i$. Therefore, each distribution ${\mathcal D}^i$ is completely integrable and each maximal integrable submanifold of ${\mathcal D}^i$ is totally geodesic in $N$. Consequently, $N$ is locally the Riemannian product $N_{1}\times\ldots\times N_k$ of the leaves of these distributions.
The converse of this is easy to verify.
\vskip.1in By using Lemma 3.1 we have the following characterization of subma-nifolds with $\nabla P=0$. \vskip.1in \eject {\bf Lemma 3.2.} {\it Let $N$ be a submanifold of an almost Hermitian manifold $M$. Then $\nabla P=0$ if and only if $N$ is locally the Riemannian product $N_{1}\times\ldots \times N_{k}$, where each $N_i$ is either a complex submanifold, a totally real submanifold, or a Kaehlerian slant submanifold of $M$.} \vskip.1in {\bf Proof.} Under the hypothesis, if $P$ is parallel, then $Q=P^2$ is parallel. Thus, by applying Lemma 3.1, we see that $N$ is locally the Riemannian product $N_{1}\times\ldots\times N_k$ of leaves of distributions defined by eigenvectors of $Q$ and moreover each eigenvalue $\lambda_i$ is constant on $N$. If an eigenvalue $\lambda_{i}$ is zero, the corresponding leaf $N_i$ is totally real. If $\lambda_{i}$ is $-1$, then $N_i$ is a complex submanifold. If $\lambda_{i}\not=0,-1$, then because ${\mathcal D}^i$ is invariant under the endomorphism $P$ and $<PX,Y>=- \lambda_{i} <X,Y>$ for any $X,Y$ in ${\mathcal D}^i$, we have
$|PX|=\sqrt{-\lambda_{i}}\,|X|$. Thus the Wirtinger angle $\theta(X)$ satisfying $\cos\theta(X)=\sqrt{-\lambda_{i}}$, which is a constant $\not=0,-1.$ Therefore, $N_i$ is a proper slant submanifold.
Assume $\lambda_{i}\not= 0$. We put
$P_{i}=P_{\,| TN_{i}}$. Then $P_i$ is nothing but the endomorphism of $TN_i$ induced from the almost complex structure $J$. Let $\nabla^i$ denote the Riemannian connection of $N_i$. Since $N_i$ is totally geodesic in $N$, we have $$(\nabla^{i}_{X}P_{i})Y=(\nabla_{X}P)Y=0$$ for any $X,Y$ tangent to $N_i$. This shows that if $N_i$ is a complex submanifold, $N_i$ is a Kaehlerian manifold. And if $N_i$ is proper slant, then $N_i$ is a Kaehlerian slant submanifold of $M$ by definition.
The converse can be verified directly. \vskip.1in
From Lemma 3.2 we may obtain the following \vskip.1in {\bf Proposition 3.3.} {\it Let $N$ be an irreducible submanifold of an almost Hermitian manifold $M$. If $N$ is neither complex nor totally real, then $N$ is a Kaehlerian slant submanifold if and only if the endomorphism $P$ is parallel, that is, $\nabla P=0.$}
\vskip.1in {\bf Theorem 3.4.} {\it Let $N$ be a surface in an almost Hermitian manifold $M$. Then the following three statements are equivalent:}
(1) {\it $N$ is neither totally real nor complex in $M$ and $\nabla P =0$, that is, $P$ is parallel;}
(2) $N$ {\it is a Kaehlerian slant surface;}
(3) $N$ {\it is a proper slant surface.} \vskip.1in {\bf Proof.} Since every proper slant submanifold is of even dimension, Lemma 3.2 implies that if the endomorphism $P$ is parallel, then $N$ is a Kaehlerian surface, or a totally real surface, or a Kaehlerian slant surface. Thus, if $N$ is neither totally real nor complex, then statements (1) and (2) are equivalent by definition. It is obvious that (2) implies (3). Now, we prove that (3) implies (2). Let $N$ be a proper slant surface in $M$ with slant angle $\theta$. If we choose an orthonormal frame $e_{1},e_2$ tangent to $N$ such that
$$Pe_{1}=(\cos\theta)e_{2},\,\,\,\,Pe_{2}= -(\cos\theta)e_{1}.$$
\noindent then we have
$$(\nabla_{X}P)e_{1}=\cos\theta(\omega_{2}^{1}(X)+\omega_{1}^{2} (X))e_{1}.$$
\noindent Since $\omega_{1}^{2}=\omega_{2}^{1},$ we obtain $\nabla P=0$.
\vskip.1in For submanifolds of a Kaehlerian manifold we have the following general lemma.
\vskip .1in {\bf Lemma 3.5} {\it Let $N$ be a submanifold of a Kaehlerian manifold $M$. Then } \vskip.1in (i) {\it For any vectors $X,Y$ tangent to $N$, we have
$$(\nabla_{X}P)Y = th(X,Y)+A_{FY}X.\leqno(3.2)$$
Hence $\nabla P = 0$ if and only if $A_{FX}Y =A_{FY}X$ for any $X,Y$ tangent of $N$.} \vskip.1in (ii) {\it For any vectors $X,Y$ tangent to $N$, we have
$$(\nabla_{X}F)Y = fh(X,Y)-h(X,PY)\leqno(3.3)$$
Hence $\nabla F = 0$ if and only if $A_{f\xi}X = -A_{\xi}(PX).$ for any normal vector $\xi$ and tangent vector $X$.}
\vskip.1in
{\bf Proof.} Since $M$ is Kaehlerian, $J$ is parallel. Thus, by applying the formulas of Gauss and Weingarten and using formulas (1.1) and (1.5), we may obtain $$(\nabla_{X}P)Y = \nabla_{X}(PY)-P(\nabla_{X}Y)=th(X,Y)+A_{FY}X,\leqno(3.2)'$$ and $$(\nabla_{X}F)Y=D_{X}(FY)-F(\nabla_{X}Y)= fh(X,Y)-h(X,PY).\leqno(3.3)'$$ Thus, $P$ is parallel if and only if
$$<th(X,Y)+A_{FY}X,Z>=0$$ which is equivalent to $$<A_{FY}X,Z>=-<th(X,Y),Z>=<h(X,Y),FZ>=$$ $$=<A_{FZ}X,Y>= <A_{FZ}Y,X>.$$ This proves statement (i).
Statement (ii) follows easily from $(3.3)'$. \vskip.1in {\bf Remark 3.1.} If $N$ is either a totally real or complex submanifold of a Kaehlerian manifold, then $\nabla P = \nabla F =0$, automatically.
\vskip.1in Combining Theorem 3.4 and Lemma 3.5 we obtain the following charaterization of slant surfaces in terms of Weingarten map.
\vskip.1in {\bf Corollary 3.6.} {\it Let $N$ be a surface in a Kaehlerian manifold $M$. Then $N$ is slant if and only if $A_{FY}X=A_{FX}Y$ for any $X,Y$ tangent to $N$.} \vskip.1in Let $N$ be a slant surface in the complex number space ${\bf C}^2$ with slant angle $\theta$. For a unit tangent vector field $e_1$ of $N$, we put $$e_{2}=(\sec \theta)Pe_{1},\hskip.2in e_{3}=(\csc \theta)Fe_{1},\hskip.2in e_{4}=(\csc\theta)Fe_{2}.\leqno(3.4)$$ Then $e_{1}=-(\sec\theta)Pe_{2}$, and $e_{1},e_{2},e_{3},e_{4}$ form an orthonormal frame such that $e_{1},e_2$ are tangent to $N$ and $e_{3},e_4$ are normal to $N$. As before we put $$h_{ij}^{r}=<h(e_{i},e_{j}),e_{r}>,\hskip.3in i,j=1,2;\hskip.2in r=3,4.\leqno(3.5)$$
Let $G$ and $G^D$ denote the {\it Gauss curvature\/} and the {\it normal curvature\/} of $N$ in ${\bf C}^2$, respectively. Then we have $$G=h_{11}^{3}h_{22}^{3}-(h_{12}^{3})^{2}+h_{11}^{4}h_{22}^{4} -(h_{12}^{4})^{2}\leqno(3.6)$$ and $$G^{D}=h_{11}^{3}h_{12}^{4}+h_{12}^{3}h_{22}^{4}-h_{12}^{3} h_{11}^{4}-h_{22}^{3}h_{12}^{4}.\leqno(3.7)$$
From Corollary 3.6 we obtain the following \vskip.1in {\bf Theorem 3.7.} {\it If $N$ is a slant surface in
${\bf C}^2$, then $G=G^D$, identically.} \vskip.1in {\bf Proof.} Let $N$ be a slant surface in ${\bf C}^2$. Then Corollary 3.6 implies $A_{FY}X=A_{FX}Y$ for any vectors $X,Y$ tangent to $N$. Let $e_{1},e_{2},e_{3},e_{4}$ be an orthonormal frame satisfying (3.4). Then we have $$h_{12}^{3}=h_{11}^{4},\,\,\,\,\,\,h_{22}^{3}= h_{12}^{4}.$$ Therefore, by (3.6) and (3.7), we obtain $G=G^D$.
\vskip.1in In the remaining part of this section we mention some properties of the normal-bundle valued 1-form $F$. In order to do so, we recall the following definition. \vskip.1in {\bf Definition 3.1.} Let $N$ be a submanifold of a Riemannian manifold $M$. Then $N$ is called a {\it minimal submanifold\/} if $\,tr\, h = 0,\,$ identically. And it is called {\it austere\/} (cf. [HL1]) if for each normal vector $\xi$ the set of eigenvalues of $A_{\xi}$ is invariant under multiplication by $-1$; this is equivalent to the condition that all the invariants of odd order of the Weingarten map at each normal vector of $N$ vanish identically. \vskip.1in
Of course every austere submanifold is a minimal submanifold. \vskip.1in {\bf Theorem 3.8.} {\it Let $N$ be a proper slant submanifold of a Kaehlerian manifold $M$. If $\nabla F =0$, then $N$ is autere.} \vskip.1in {\bf Proof.} Let $N$ be a proper slant submanifold of a Kaehlerian manifold $M$. If $\nabla F=0$, then we have from formula (3.3)
$$fh(X,Y)=h(X,PY).$$ Let $X$ be any unit eigenvector of $Q=P^2$ with eigenvalue $\lambda \not= 0$. Then $X_{*}=PX/\sqrt{-\lambda}$ is a unit vector perpendicular to $X$. Thus, we have $$h(X,X)=h(PX,PX)/\lambda = -h(X_{*},X_{*})$$ which implies that $N$ is autere. \vskip.1in If the ambient space $M$ is a complex-space-form, then we have the following reduction theorem. \vskip.1in
{\bf Theorem 3.9.} {\it Let $N$ be an n-dimensional proper slant submanifold of a complex $m$-dimensional complex-space-form $M^{m}(c)$ with constant holomorphic sectional curvature $c$. If $\nabla F =0$, then $N$ is contained in a complex $n$-dimensional complex totally geodesic submanifold of $M^{m}(c)$ as an austere submanifold.}
\vskip.1in {\bf Proof.} Let $N$ be an $n$-dimensional proper slant submanifold of ${\bf C}^m$. Assume that $\nabla F=0$. Then the normal bundle $T^{\perp}N$ has the following orthogonal direct decomposition: $$T^{\perp}N=F(TN)\oplus \nu,\,\,\,\nu_{p}\perp F(T_{p}N)$$ for any point $p\in N$. For any vector field $\xi$ in $\nu$ and any vector fields $X,Y$ in $TN$, we have $$<A_{J\xi}X,Y>=<h(X,Y),J\xi>=<{\tilde\nabla}_{X}Y,J\xi>=$$ $$=-<PY,A_{\xi}X>+<FY,D_{X}\xi>,$$ from which we find $$<D_{X}(FY),\xi>=-<A_{\xi}(PY)+A_{J\xi}Y,X>.\leqno(3.8)$$
On the other hand, for any $\xi$ normal to $N$, if we denote by $t\xi$ and $f\xi$ by using (1.5), then Lemma 3.5 gives $$A_{f\xi}Y+A_{\xi}(PY)=0.\leqno(3.9)$$ Since $f=J$ on the normal subbundle $\nu$, formulas (3.8) and (3.9) imply $<D_{X}(FY),\xi>=0$ for any $\xi$ in $\nu$. From this we conclude that the normal subbundle $F(N)$ is a parallel normal subbundle.
Next, we claim that the first normal subbundle $\,Im\,h\,$ is contained in $F(TN)$. This can be proved as follows.
Since $\nabla F=0$, statement (ii) of Lemma 3.5 implies $$<h(X,Y),J\xi>=-<h(X,PY),\xi>$$ for any normal vector $\xi$ in $\nu$. Thus, for any eigenvector $Y$ of the self-adjoint endomorphism $Q$ with eigenvalue $\lambda$ and any normal vector $\xi$ in $\nu$, we have $$<h(X,Y),\xi>=-\lambda <h(X,Y),\xi>.$$ Since $N$ is a proper slant submanifold, $-1<\lambda < 0$. Thus, we obtain $Im\,h \subset F(TN)$. Consequently, by applying the reduction theorem, we obtain the result. \vskip.1in In particular if the ambient space $M$ is the complex number space ${\bf C}^m$, then we have the following \vskip.1in
{\bf Theorem 3.10.} {\it Let $N$ be an n-dimensional proper slant submanifold of ${\bf C}^m$. If $\nabla F =0$, then $N$ is contained in a complex linear subspace ${\bf C}^n$ of ${\bf C}^m$ as an austere submanifold.} \vskip.1in For surfaces in a real 4-dimensional Kaehlerian manifold we have the following \vskip.1in {\bf Theorem 3.11.} {\it Let $N$ be a surface in a real 4-dimensional Kaehlerian manifold $M$. Then $\nabla F=0$ if and only if either $N$ is a complex surface, or a totally real surface, or a minimal proper slant surface of $M$.} \vskip.1in {\bf Proof.} ($\Rightarrow$) Let $N$ be a surface in $M$ with $\nabla F=0$. Assume that $N$ is neither complex nor totally real. Then both $P: T_{x}N \rightarrow T_{x}N$ and $F: T_{x}N \rightarrow T_{x}^{\bot}N$ are surjective. Denote by $\theta$ the Wirtinger angle. Define $e_{1},e_{2}, e_{3},e_{4}$ by (3.4). Then, by using $J^{2}=-I$, we have $$te_{3}=-\sin \theta e_{1},\, te_{4}=- \sin \theta e_{2},\, fe_{3}=-\cos\theta e_{4},\, fe_{4}=\cos\theta e_{3}.\leqno(3.10)$$ Since $F$ is parallel, Lemma 3.5 implies $A_{f\xi}X=-A_{\xi}(PX)$. Therefore, we find $$A_{Fe_{1}}e_{2}=(\csc \theta)^{-1}\sec \theta A_{e_{3}}(Pe_{1})=$$ $$ =(\tan\theta)A_{e_{3}}(Pe_{1})= -\tan \theta A_{fe_{3}}e_{1}$$ $$ = \sin \theta A_{e_{4}}e_{1} = A_{Fe_{2}}e_{1}.$$
\noindent Therefore, by applying Corollary 3.6, $N$ is slant. Furthermore, by applyinf Theorem 3.8, we know that $N$ is minimal.
($\Leftarrow$) It is clear that if $N$ is a complex or totally real surface in $M$, then $F$ is parallel. Therefore, we may assume that $N$ is a minimal proper slant surface in $M$. We choose $e_{1}, e_{2}, e_{3}, e_{4}$ according to (3.4). Then by Corollary 3.6, we have $$h_{11}^{3}=-h_{22}^{3}=-h_{12}^{4},\,\,\,h_{12}^{3}= h_{11}^{4}=-h_{22}^{4}.\leqno(3.11)$$ From (3.11) and direct computation we may prove that $A_{f\xi}X=-A_{\xi}(PX)$ for any tangent vector $X$ and normal vector $\xi$ of $N$. Therefore, by applying Lemma 3.5, we conclude $F$ is parallel.
\eject \vskip.3in
\noindent \S 4. MINIMAL SLANT SURFACES AND TOTALLY REAL SURFACES. \vskip.2in In this section we want to establish some relations between minimal slant surfaces and totally real surfaces in a Kaehlerian manifold, in parti-cular, in ${\bf C}^2$.
Let $N$ be a proper slant surface with slant angle $\theta$ in a real 4-dimen-sional Kaehlerian manifold $M$. Let $e_1$ be a local vector field tangent to $N$. We choose a canonical orthonormal local frame $e_{1},e_{2}, e_{3},e_{4}$ defined by $$e_{2}=(\sec \theta)Pe_{1},\hskip.2in e_{3}=(\csc \theta)Fe_{1},\hskip.2in e_{4}=(\csc\theta)Fe_{2}.\leqno(4.1)$$
\noindent We call such an orthonormal frame $e_{1},e_{2}, e_{3}, e_{4}$ an {\it adapted slant frame.\/}
For an adapted slant frame we have $$te_{3}=-\sin \theta e_{1},\, te_{4}=- \sin \theta e_{2},\, fe_{3}=-\cos\theta e_{4},\, fe_{4}=\cos\theta e_{3}.\leqno(4.2)$$
As before we put $$De_{r}=\sum_{s} \omega_{r}^{s}\otimes e_{s},\,\,\,\nabla e_{i} =\sum_{j} \omega_{i}^{j}\otimes e_{j},$$ $$h=\sum_{r} h^{r}e_{r},\,\,\,\,i,j=1,2;\,\,\,r,s=3,4.$$
We have the following
\vskip.1in {\bf Lemma 4.1.} {\it Let $N$ be a proper slant surface in a real 4-dimensional Kaehlerian manifold $M$. Then, with respect to an adapted slant frame, we have} $$\omega_{3}^{4}-\omega_{1}^{2}=-\cot \theta \{(tr h^{3})\omega^{1}+(tr h^{4})\omega^{2}\},\leqno(4.3)$$ {\it where $\omega^{1},$ $\omega^2$ is the dual frame of $e_{1},e_{2}$.} \vskip.1in
{\bf Proof.} Since $J$ is parallel, we have $$D_{X}(FY)-F(\nabla_{X}Y)=fh(X,Y)-h(X,PY).$$
\noindent Thus, we find $$D_{e_{1}}e_{3} =D_{e_{1}}(\csc \theta Fe_{1}) = (\csc\theta)De_{e_{1}}(Fe_{1})=$$ $$=(\csc\theta)\{F(\nabla_{e_{1}}e_{1})+fh(e_{1}, e_{1})-h(e_{1},Pe_{1})\}=$$
$$=(\csc\theta)\{ \omega_{1}^{2}(e_{1})Fe_{2} +h_{11}^{3}fe_{3}+$$
$$+h_{11}^{4}fe_{4}-\cos\theta (h_{12}^{3}e_{3}+h_{12}^{4}e_{4})\}=$$
$$=\omega_{1}^{2}(e_{1})e_{4} - (\cos\theta)(tr h^{3})e_{4}.$$
\noindent This implies $$\omega_{3}^{4}(e_{1})-\omega_{1}^{2}(e_{1}) = -(\cot\theta)(tr h^{3}).$$
\noindent Similarly, we may obtain $$\omega_{3}^{4}(e_{2})-\omega_{1}^{2}(e_{2}) = -(\cot\theta)(tr h^{4}).$$
\noindent These prove the lemma. \vskip.1in {\bf Theorem 4.2.} {\it Let $N$ be a proper slant surface in a real 4-dimensional Kaehlerian manifold $(M,J,g)$. If there exists a compatible complex structure $J_1$ such that $N$ is totally real with respect to the Kaehlerian manifold $(M,J_{1},g)$, then $N$ is minimal in $M$.} \vskip.1in {\bf Proof.} Since $N$ is assumed to be totally real in $(M,J_{1},g)$, there exists a function $\varphi$ such that $$e_{3}=(\cos\varphi)J_{1}e_{1}+(\sin\varphi)J_{1}e_{2} \leqno(4.4)$$ and $$e_{4}=-(\sin\varphi)J_{1}e_{1}+(\cos\varphi)J_{1}e_{2}.\leqno(4.5)$$
Since $J_{1}$ is parallel, these imply $$\omega_{3}^{4}(X)=<{\tilde \nabla}_{X}e_{3},e_{4}>=$$ $$=<-(\sin\varphi)(X\varphi)J_{1}e_{1}+(\cos\varphi) (X\varphi) J_{1}e_{2}+(\cos\varphi)\omega_{1}^{2}(X)J_{1}e_{2}+$$ $$ +(\sin\varphi)\omega_{2}^{1}(X)J_{1}e_{1}, -(\sin\varphi) J_{1}e_{1}+(\cos\varphi)J_{1}e_{2}>.$$
\noindent Therefore, we have $$\omega_{3}^{4}(X)=\sin^{2}\varphi (X\varphi)+$$ $$+\cos^{2}\varphi (X\varphi) + \cos^{2}\varphi \omega_{1}^{2}(X)-\sin^{2}\varphi \omega_{2}^{1}(X).$$
\noindent This implies $$\omega_{3}^{4}-\omega_{1}^{2} =d\varphi.\leqno(4.6)$$
\noindent Combining (4.4) and Lemma 4.1 we obtain $$\cot\theta \,\{(tr h^{3})\omega^{1}+(tr h^{4})\omega^{2}\} = -d\varphi.$$
\noindent Also from (4.4) and (4.5) we find $$h_{11}^{3}=-<{\tilde \nabla}_{e_{1}}e_{3},e_{1}>=$$ $$=-<(\cos \varphi){\tilde \nabla}_{e_{1}}(J_{1}e_{1})+ (\sin \varphi){\tilde \nabla}_{e_{2}}(J_{1}e_{2}),e_{1}>=$$ $$<\cos\varphi)h(e_{1},e_{1})+(\sin\varphi)h(e_{1},e_{2}), J_{1}e_{1}>= $$ $$=<(\cos\varphi)h(e_{1},e_{1})+(\sin\varphi)h(e_{1},e_{2}), \cos\varphi e_{3}-\sin\varphi e_{4}>=$$ $$=\cos^{2}\varphi h_{11}^{3}-\sin^{2}\varphi h_{22}^{3}.$$
\noindent This implies $$\sin^{2}\varphi (h_{11}^{3}+h_{22}^{3})=0.$$
\noindent Similarly, we have $$\sin^{2}\varphi (h_{11}^{4}+h_{22}^{4})=0.$$
Let $U=\{x\in N: H(x) \not= 0 \}.$ Then $U$ is an open subset of $N$. If $U \not= \emptyset$, then $\varphi \equiv 0$ (mod $\pi$) on $U$. Thus, $$\cos\theta \{(tr h^{3})\omega^{1} +(tr h^{4})\omega^{2}\}=-d\varphi =0$$ on $U$, which implies $\cos \theta =0$. This implies that $N$ is totally real in $(M,J,g)$ which is a contradiction. Thus $U = \emptyset,$ that is, $N$ is minimal. \vskip.1in {\bf Theorem 4.3.} {\it Let $N$ be a proper slant surface in ${\bf C}^2$. Then $N$ is minimal if and only if there exists a compatible almost complex structure $J_{1}$ on $E^4$ such that $N$ is totally real in $(E^{4},J_{1})$.} \vskip.1in This Theorem follows from Example 2.2, Theorem 4.3 and the following [C6, Theorem 5.2]
\vskip.1in {\bf Theorem 4.4.} {\it Let $N$ be a proper slant surface in ${\bf C}^2$. Then $N$ is minimal if and only if there exists a compatible almost complex structure $J_{2}$ on $E^4$ such that $N$ is a complex surface in $(E^{4},J_{2})$.} \vskip.1in
This theorem was proved by studying the relation between Gauss map and slant immersions. In the next chapter we will treat these problems by using the notion of Gauss map.
\eject \centerline {CHAPTER III} \vskip.2in \centerline {\bf GAUSS MAP AND SLANT IMMERSIONS} \vskip.3in In this chapter we present some results of the author and Y. Tazawa [CT1]; in this work slant surfaces in ${\bf C}^2$ were studied from the point of view of the Gauss map. \vskip.3in
\noindent \S 1. GEOMETRY OF $G(2,4)$. \vskip.2in In this section we review the geometry of the real Grassmannian $G(2,4)$ which consists of all oriented 2-planes in $E^4$.
Let $\{\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}, \varepsilon_{4}\}$ be the canonical orthonormal basis of $E^4$. Then $\Psi_{0}=:\varepsilon_{1}\wedge\dots\wedge\varepsilon_{4}$ gives the canonical orientation of $E^4$. Let $\wedge^{2}(E^{4})^{*}$ denote the 6-dimensional real vector space with innner product, also denoted by $<\, ,\, >,$ defined by $$<X_{1}\wedge X_{2},Y_{1}\wedge Y_{2}> = det\,(<X_{i},Y_{j}>)\leqno(1.1)$$ and extended bilinearly. The two vector spaces $\wedge^{2}(E^{4})^{*}$ and $(\wedge^{2}E^{4})^{*}$ are identified in a natural way by $$\Phi(X_{1}\wedge X_{2})=:\Phi(X_{1},X_{2})\leqno(1.2)$$ for any $\Phi \in \wedge^{2}(E^{4})^{*}$. The Grassmannian $G(2,4)$ was identified with the set $D_{1}(2,4)$ which consists of all unit decomposible 2-vectors in $\wedge^{2}E^{4}$ via $\phi:G(2,4) \rightarrow D_{1}(2,4)$ given by $\phi(V)=X_{1}\wedge X_{2}$, for any positive orthonormal basis $\{X_{1},X_{2}\}$ of $V \in G(2,4)$.
The Hodge star operator $* :\wedge^{2}E^{4} \rightarrow \wedge^{2}E^{4}$ is defined by $$<*\xi,\eta>\Psi_{0}=\xi\wedge\eta,\leqno(1.3)$$ for any $\xi, \eta \in \wedge^{2}E^{4}$. So, if we regard an oriented 2-plane $V \in G(2,4)$ as an element in $D_{1}(2,4)$ via $\phi$, then we have $*V = V^{\bot}$, where $V^{\bot}$ denotes the oriented orthogonal complement of the oriented 2-plane $V$ in $E^4$.
Since $*^{2}=1$ and $*$ is a self-adjoint endomorphism of $\wedge^{2}E^4$, we have the following orthogonal decomposition: $$\wedge ^{2}E^{4}= \wedge ^{2}_{+} E^{4}\oplus\wedge ^{2}_{-}E^{4}\leqno(1.4)$$ of eigenspaces of $*$ with eigenvalues 1 and $-1$, respectively.
Denote by $\pi_{+}$ and $\pi_{-}$ the natural projections: $\pi_{\pm} : \wedge ^{2}E^{4} \rightarrow \wedge ^{2}_{\pm}E^{4},$ respectively.
We put $$\,\,\,\,\,\,\,\,\,\,\eta_{1}= {1\over \sqrt{2}}(e_{1}\wedge e_{2}+ e_{3} \wedge e_{4}),\,\,\,\,\eta_{4}= {1\over \sqrt{2}}(e_{1}\wedge e_{2}- e_{3} \wedge e_{4}),$$ $$\eta_{2}= {1\over \sqrt{2}}(e_{1}\wedge e_{3}- e_{2} \wedge e_{4}),\,\,\,\,\eta_{5}= {1\over \sqrt{2}}(e_{1}\wedge e_{3}+ e_{2} \wedge e_{4}),\leqno(1.5)$$ $$\,\,\,\,\,\,\,\,\,\,\,\eta_{3}= {1\over \sqrt{2}}(e_{1}\wedge e_{4}+ e_{2} \wedge e_{3}),\,\,\,\,\eta_{6}= {1\over \sqrt{2}}(e_{1}\wedge e_{4}- e_{2} \wedge e_{3}).$$ Then $\{\eta_{1},\eta_{2},\eta_{3}\}$ and $\{\eta_{4}, \eta_{5},\eta_{6}\}$ form canonical orthonormal bases of $\wedge ^{2}_{+}E^{4}$ and $\wedge ^{2}_{-}E^{4}$, respectively. We shall orient the spaces $\wedge ^{2}_{+}E^{4}$ and $\wedge ^{2}_{-}E^{4}$ such that these two bases are positive, that is, they give positive orientations for the oriented spaces $\wedge ^{2}_{+}E^{4}$ and $\wedge ^{2}_{-}E^{4}$.
For any $\xi \in D_{1}(2,4)$ we have $$\pi_{+}(\xi)= {1 \over 2}(\xi +*\xi), \,\,\,\pi_{-}(\xi) = {1 \over 2}(\xi -*\xi).\leqno(1.6)$$
Denote by $S_{+}^{2}$ and $S_{-}^{2}$ the 2-spheres in $\wedge ^{2}_{+}E^{4}$ and $\wedge ^{2}_{-}E^{4}$ centered at the origin with radius $1/\sqrt{2}$, respectively. Then we have $$\pi_{+} : D_{1}(2,4) \rightarrow S_{+}^{2},\,\,\, \pi_{-} : D_{1}(2,4) \rightarrow S_{-}^{2}\leqno(1.7)$$ and $$D_{1}(2,4) = S_{+}^{2}\times S_{-}^{2}.\leqno(1.8)$$
\eject
\vskip.3in \noindent \S 2. COMPLEX STRUCTURES ON $E^4$. \vskip.2in Let ${\bf C}^{2}=(E^{4},J_{0})$ be the complex 2-plane with the canonical complex structure $J_0$ and the canonical metric. It is well-known that $J_0$ is an orientation preserving isomorphism. We denote by $\mathcal J$ the set of all almost complex structures (or simply called complex structures) on $E^4$ which are compatible with the inner product $<\, ,\, >$, that is, $${\mathcal J} = \{J : E^{4} \rightarrow E^{4} : J \,\,{\rm is \,\,linear,} J^{2} = -I, \,\,{\rm and} $$ $$<JX,JY>=<X,Y>, \,{\rm for}\,\,{\rm any}\,\, X,Y \in E^{4}\}.$$
An orthonormal basis $\{e_{1},e_{2},e_{3},e_{4}\}$ on $E^4$ is called a $J$-{\it basis\/} if we have $Je_{1}=e_{2}, Je_{3}=e_{4}$. Two $J$-bases of the same $J$ have the same orientation. Using the canonical orientation on $E^4$ we divide $\mathcal J$ into two disjoint subsets of $\mathcal J$: $${\mathcal J}^{+} = \{ J \in {\mathcal J}: J{\rm - bases \,\, are \,\, positive }\},$$ $${\mathcal J}^{-} = \{ J \in {\mathcal J}: J{\rm - bases \,\, are \,\, negative }\}.$$
For any $J \in {\mathcal J}$, there exists a unique 2-vector $\zeta_{J} \in \wedge^{2}E^4$ defined as follows: $$<\zeta_{J},X\wedge Y>=-\Omega_{J}(X,Y)=:-<X,JY>,\leqno(2.1)$$ for any $X, Y \in E^4$. In other words, $\zeta_{J}$ is nothing but the metrical dual of $-\Omega_{J}$, where $\Omega_{J}$ is the {\it Kaehler form\/} associated with $J$. \vskip.1in {\bf Lemma 2.1.} {\it The mapping $\zeta : {\mathcal J} \rightarrow \wedge^{2}E^{4}$ defined by $ \zeta (J)=\zeta_{J}$, gives rise to two bijections: $$\zeta^{+} : {\mathcal J}^{+} \rightarrow S_{+}^{2}(\sqrt{2}),\,\,\, \zeta^{-} : {\mathcal J}^{-} \rightarrow S_{-}^{2}(\sqrt{2}),\leqno(2.2)$$ where $S_{+}^{2}(\sqrt{2})$ and $S_{-}^{2}(\sqrt{2})$ are the 2-spheres centered at the origin with radius $\sqrt{2}$ in $\wedge^{2}_{+}E^{4}$ and $\wedge^{2}_{-}E^{4}$, respectively.} \vskip.1in {\bf Proof.} If $J\in {\mathcal J}^{+}$ and $e_{1},e_{2},e_{3}, e_{4}$ is a $J$-basis, then $e_{1},e_{2},e_{3},e_{4}$ is positive. From (2.1) we have $$\zeta_{J}=e_{1}\wedge e_{2}+e_{3}\wedge e_{4}.$$ Thus, we have $\zeta_{J}\in S_{+}^{2}(\sqrt{2}).$ Similarly, if $J\in {\mathcal J}^-$, then we have $\zeta_{J}\in S_{-}^{2}(\sqrt{2}).$
If $J$ and $J'$ are two distinct compatible complex structures on $E^4$, then their corresponding Kaehler forms $\Omega_J$ and $\Omega_{J'}$ are distinct. Thus,
$\zeta_{J}\not= \zeta_{J'}.$ This proves the injectivity of $\zeta$.
Conversely, for any element $\xi \in S_{+}^{2}(\sqrt{2})$, we have ${1\over 2}\xi \in S_{+}^2$. Since $\pi_{+}:D_{1}(2,4) \rightarrow S_{+}^2$ is isomorphic, there exists an element $V \in D_{1}(2,4)$ such that $\pi_{+}(V)= {1\over 2}\xi$. Let $e_{1},e_{2},e_{3},e_{4}$ be a positive orthonormal basis of $E^4$ such that $e_{1}\wedge e_{2}=V$. Define $J\in {\mathcal J}^+$ such that $Je_{1}=e_{2},Je_{3}=e_{4}.$ Then we have $\zeta_{J}=\xi$.
This completes the proof of the Lemma.
\vskip.1in
By applying Lemma 2.1 we may make the following identifications via $\zeta, \zeta^{+}$ and $\zeta^-$, respectively: $${\mathcal J}^{+} {\cong} S_{+}^{2}(\sqrt{2}),\,\,
{\mathcal J}^{-} {\cong} S_{-}^{2}(\sqrt{2}),\,\, {\mathcal J} {\cong} S_{+}^{2}(\sqrt{2}) \cup S_{-}^{2}(\sqrt{2})$$
For any $V \in G(2,4)$ and for any $J \in {\mathcal J}$, we choose a positive orthnormal basis $\{e_{1},e_{2}\}$ of $V$ and put $$\alpha_{J}(V) =\cos^{-1}(<Je_{1},e_{2}>).\leqno(2.3)$$ Then $\alpha_{J}(V) \in [0,\pi ]$ and $\theta(X)=min \,\{ \alpha_{J}(V),\pi - \alpha_{J}(V)\}.$ A 2-plane $V \in G(2,4)$ is said to be {\it $\alpha$-slant\/} if $\alpha_{J}(V)=\alpha$, identically.
If $N$ is an oriented surface in ${\bf C}^2$, then $N$ has a unique complex structure determined by its orientation and its induced metric. With respect to the angle $\alpha_{J},\, (J \in {\mathcal J}),$ we have $$N \,\,{\rm is \,\, holomorphic \,}\Longleftrightarrow \alpha_{J}(TN) \equiv 0,$$ $$N \,\,{\rm is \,\, antiholomorphic \,}\Longleftrightarrow \alpha_{J}(TN) \equiv \pi,$$ $$N \,\,{\rm is\,\, totally \,\, real} \Longleftrightarrow \alpha_{J}(TN) \equiv {\pi \over 2}.$$
The following lemma obtained in [CT1] establishes the fundamental relations between slant angle and the projections $\pi_+$ and $\pi_-$. \vskip.1in {\bf Lemma 2.2.}
(i) {\it If $J \in {\mathcal J}^{+}$, then $\alpha_{J}(V)$ is the angle between $\pi_{+}(V)$ and $\zeta_J$ and }
(ii) {\it If $J \in {\mathcal J}^{-}$, then $\alpha_{J}(V)$ is the angle between $\pi_{-}(V)$ and $\zeta_J$}. \vskip.1in {\bf Proof.} (i) If $J\in {\mathcal J}^+$, then we have $$\cos (\alpha_{J}(V))=-\Omega_{J}(V)=<\zeta_{J},V>=$$ $$=<\zeta_{J},\pi_{+}(V)+\pi_{-}(V)>=<\zeta_{J}, \pi_{+}(V)>.$$
\noindent Since $||\zeta_{J}||=\sqrt{2}$ and
$||\pi_{+}(V)||=1/\sqrt{2}$, this implies $$\alpha_{J}(V)= \angle (\pi_{+}(V),\zeta_{J}).$$
(ii) can be proved in a similar way. \vskip.1in For any $a \in [0,\pi ]$ and for any $J \in {\mathcal J}$, we define
$$G_{J,a}= \{ V \in G(2,4) : \alpha_{J}(V)=a\},\leqno(2.4)$$
\noindent which is equivalently to say that $G_{J,a}$ is the set consisting of all oriented $a$-slant oriented 2-planes in $E^4$ with respect to the complex structrure $J$.
For any $a \in [0,\pi]$ and any $V \in G(2,4)$, we define $${\mathcal J}_{V,a}= \{ J \in {\mathcal J} : \alpha_{J}(V)=a\}.\leqno(2.5)$$
We put $${\mathcal J}_{V,a}^{+} = {\mathcal J}_{V,a} \cap {\mathcal J}^{+},\,\, {\mathcal J}_{V,a}^{-} = {\mathcal J}_{V,a} \cap {\mathcal J}^{-}.$$
From Lemma 2.2 we may obtain [CT1] \vskip.1in {\bf Lemma 2.3.}
(1) {\it If $J \in {\mathcal J}^+$, then $G_{J,a}= S_{J,a}^{+} \times S_{-}^2$, where $S_{J,a}^+$ is the circle on $S_{+}^2$ consisting of 2-vectors which makes constant angle $a$ with $\zeta_J$.}
(2) {\it If $J \in {\mathcal J}^-$, then $G_{J,a}= S_{+}^{2} \times S_{J,a}^{-}$, where $S_{J,a}^-$ is the circle on $S_{-}^2$ consisting of 2-vectors which makes constant angle $a$ with $\zeta_J$.}
(3) {\it Via the identification given by Lemma 2,1, ${\mathcal J}_{V,a}^{+}$ is the circle on $S_{+}^{2}(\sqrt{2})$ consisting of 2-vectors in $S_{+}^{2}(\sqrt{2})$ which makes constant angle $a$ with $\pi_{+}(V)$, and ${\mathcal J}_{V,a}^{-}$ is the circle on $S_{-}^{2}(\sqrt{2})$ consisting of 2-vectors in $S_{-}^{2}(\sqrt{2})$ which makes constant angle $a$ with $\pi_{-}(V)$.} \vskip.1in This Lemma can be regarded as a generalization of Proposition 2 of [CM2]. \vskip.1in
For later use we give the following \vskip.1in {\bf Notation.} Let $V$ be an oriented 2-plane in $G(2,4)$. $J_{V}^+$ and $J_{V}^-$ are defined by $$J_{V}^{+}= (\zeta^{+})^{-1}(\pi_{+}(V)) \in {\mathcal J}^{+},\,\,\,\,J_{V}^{-}= (\zeta^{-})^{-1}(\pi_{-}(V)) \in {\mathcal J}^{-}.\leqno(2.6)$$ \vskip.1in
{\bf Remark. 2.1.} It is easy to see that $J_{V}^+$ (respectively, $J_{V}^-$) defined by (2.6) is the complex sturcture in ${\mathcal J}^+$ (respectively, in ${\mathcal J}^-$) such that $V$ is a holomorphic plane with respect to the complex structure $J$.
\eject
\vskip.3in \noindent \S 3. SLANT SURFACES AND GAUSS MAP. \vskip.2in In this section we study the following problems:
\vskip.1in {\bf Problem 3.1.} {\it Let $N$ be a surface in ${\bf C}^{2} = (E^{4},J_{0})$. When is $N$ slant in ${\bf C}^2$?} \vskip.1in {\bf Problem 3.2.} {\it Let $N$ be a surface in $E^4$. If there exists a compatible complex structure $J$ on $E^4$ such that $N$ is slant in $(E^{4},J)$. How many other compatible complex structures ${\tilde J}$ on $E^4$ are there such that $N$ is slant with resepct to these complex strucutres?} \vskip.1in Let $f : N \rightarrow E^4$ be an immersion from an oriented surface $N$ into $E^4$. Denote by $\nu : N \rightarrow G(2,4)$ be the Gauss map associated with the immersion $f$ defined by $\nu(p)=T_{p}N,\,\,p\in N$ (or, equivalently, by $\nu(p)=(e_{1}\wedge e_{2})(p)$).
We put $$\nu_{+} =\pi_{+}\circ\nu,\,\,\,\,\nu_{-}=\pi_{-}\circ\nu.\leqno(3.1)$$ Then we have $$\nu_{\pm}: N \rightarrow G(2,4) \rightarrow S_{\pm}^{2}.\leqno(3.2)$$
We give the following [CT1] \vskip.1in {\bf Proposition 3.1.} {\it Let $f: N \rightarrow E^4$ be an immersion of an oriented surface $N$ into $E^4$. Then}
(1) {\it $f$ is slant with respect to a complex structure $J \in {\mathcal J}^+$ (respectively, $J \ in \,{\mathcal J}^-$) if and only if $\nu_{+}(N)$ (respectively, $\nu_{-}(N)$) is contained in a circle on $S_{+}^2$ (respectively, on $S_{-}^2$).}
(2) {\it $f$ is $\alpha$-slant with respect to a complex structure $J \in {\mathcal J}^+$ (respectively, $J \in \, {\mathcal J}^-$) if and only if $\nu_{+}(N)$ (respectively, $\nu_{-}(N)$) is contained in a circle $S^{+}_{J,\alpha}$ on $S_{+}^2$ (respectively, $S_{J,\alpha}^{-}$ on $S_{-}^2$), where $S_{J,\alpha}^{+}$ (respectively, $S_{J,\alpha}^{-}$) is the circle on $S_{+}^2$ (respectively, $S_{-}^2$) consisting of all 2-vectors which makes constant angle $\alpha$ with $\zeta_J$. } \vskip.1in {\bf Proof.} If $f : N \rightarrow E^4$ is $\alpha$-slant with respect to a compatible complex structure $J \in {\mathcal J}^{+}$, then, by (2.5) and Lemma 2.3, we have $\alpha_{J}(T_{p}N) \in S_{J,\alpha}^{+} \times S_{-}^2$ for any point $p \in N$. Thus, $\nu_{+}(N)$ is contained in the circle $S_{J,\alpha}^{+}$ on the 2-sphere $S_{+}^2$ consisting of 2-vectors in $S_{+}^2$ which makes constant angle $\alpha$ with $\zeta_J$.
Conversely, if $f: N \rightarrow E^4$ is an immersion such that $\nu_{+}(N)$ is contained in a circle $S^1$ on the 2-sphere $S_{+}^2$. Let $\eta$ be a vector of length $\sqrt{2}$ in $\wedge_{+}^{2}E^4$ perpendicular to the 2-plane in $\wedge_{+}^{2}E^4$ containing $S^1$ . Then $\eta \in S^{2}_{+}(\sqrt{2})$. By Lemma 2.1, there is a unique complex structure $J \in {\mathcal J}^+$ such that $\zeta_{J}=\eta$. It is clear that $S^1$ is a circle $S_{J,\alpha}^+$ for some constant angle $\alpha$. Therefore, by Lemma 2.3, the immersion $f$ is $\alpha$-slant with respect to this compatible complex structure $J \in {\mathcal J}^+$.
Similar argument applies to the other cases. \vskip.1in Now we may give the main result of this section [CT1].
\vskip.1in {\bf Theorem 3.2.}
(1) {\it Let $f : N \rightarrow E^4$ be a minimal immersion. If there exists a compatible complex structure ${\hat J}\in {\mathcal J}^+$ (respectively, ${\hat J} \in {\mathcal J}^-$) such that the immesion $x$ is slant with respect to $\,\,\hat J$, then}
\hskip.3in (1-a) {\it for any $\alpha \in [0,\pi]$, there is a compatible complex structure $J_{\alpha}\in {\mathcal J}^+$ (respectively, $J_{\alpha}\in {\mathcal J}^-$) such that $f$ is $\alpha$-slant with respect to the complex structure $J_{\alpha}$.}
\hskip.3in (1-b) {\it the immersion $f$ is slant with respect to any complex structure $J \in {\mathcal J}^+$ (respectively, $J \in {\mathcal J}^-$).}
(2) {\it If $f : N \rightarrow E^4$ is a non-minimal immersion, then there exist at most two complex structures $\,\,\pm J^{+} \in {\mathcal J}^+$ and at most two complex structures $\,\,\pm J^{-}\in {\mathcal J}^-$ such that the immersion $f$ is slant with respect to them.} \vskip.1in {\bf Proof.} (a) Assume that $\, f : N \rightarrow E^{4} \,$ is a minimal immersion. Then both $\nu_{+}$ and $\nu_{-}$ are anti-holomorphic (cf. Lawson's book [L1]). So, both $\nu_{+}$ and $\nu_-$ are open maps if they are not constant maps. If the immesion $f$ is slant with respect to a complex structrure $J \in {\mathcal J}^+$ (respectively, $J \in {\mathcal J}^-$), then $\nu_+$ (respectively, $\nu_-$) cannot be an open map by Proposition 3.1. Thus $\,\nu_{+}(N)$ (respectively, $\,\nu_{-}(N)$) is a singleton. since a singleton is contained in every circle on the 2-sphere $S_{+}^2$ (respectively, $S_{-}^2$), the immersion $f$ is slant with respect to every complex structure $J \in {\mathcal J}^+$ (respectively, $J \in {\mathcal J}^-$). So, for any constant $\alpha \in [0,\pi]$, there is a complex structure $J_{\alpha}$ which makes the immersion $f$ to be $\alpha$-slant.
(b) Assume that $f : N \rightarrow E^4$ is a non-minimal immersion. If the immersion $f$ is $\alpha$-slant with respect to a complex structure $J \in {\mathcal J}^+$ (respectively, $J \in {\mathcal J}^-$), then $\nu_{+}(N)$ (respectively, $\nu_{-}(N)$) is contained in the circle $S_{J,\alpha}^+$ (respectively, $S_{J,\alpha}^-$) and $\nu_{+}(N)$ (respectively, $\nu_{-}(N)$) must contain an arc of the circle $S_{J,\alpha}^+$ (respectively, $S_{J,\alpha}^-$), since otherwise the immersion $f$ is holomophic with respect to some compatible complex structures on $E^4$ which implies that $N$ is minimal in $E^4$. Therefore, the complex structures $\pm J^+$ and $\pm J^-$ are the only possible complex structures on $E^4$ which may make the immersion $f$ slant. \vskip.1in From Theorem 3.2 we obtain the following corollaries of [CT1] immediately. \vskip.1in {\bf Corollary 3.3.}
(a) {\it If $f : N \rightarrow {\bf C}^{2} = (E^{4},J_{0})$ is holomorphic, then the immersion $f$ is slant with respect to every complex structure $J \in {\mathcal J}^+$.}
(b) {\it If $f : N \rightarrow {\bf C}^{2} = (E^{4},J_{0})$ is anti-holomorphic, then the immersion $f$ is slant with respect to every complex structure $J \in {\mathcal J}^-$.} \vskip.1in {\bf Corollary 3.4.} {\it Let $f : N \rightarrow E^4$ be a minimal immersion. Then the immersion $f$ is slant with respect to some complex structure $J \in {\mathcal J}^+$ if and only if $f$ is holomorphic (respectively, anti-holomorphic) with respect to some complex structure structure $J \in {\mathcal J}^+$ (respectively, $J \in {\mathcal J}^-$). } \vskip.1in {\bf Corollary 3.5.} {\it If $f : N \rightarrow E^3$ is a non-totally geodesic minimal immersion, then $f : N \rightarrow E^{3} \subset E^4$ is not slant with respect to every compatible complex structure on $E^4$.} \vskip.1in {\bf Proof.} If $f : N \rightarrow E^{3} \subset E^4$ is slant, then, by the minimality of $f$, Theorem 3.2 implies that the immersion $f$ must be a proper slant immersion with respect to some compatible complex structure $J$ on $E^4$. Therefore, by Theorem 3.7 of Chapter II, we have $G = G^D=0$, identically. Because the only flat minimal surfaces in a Euclidean space are totally geodesic ones, this is impossible. \vskip.1in
{\bf Definition 3.1.} An immersion $f : N \rightarrow E^4$ is called {\it doubly slant\/} if it is slant with respect to some complex structure $J^{+}\in {\mathcal J}^+$ and at the same time it is slant with respect to another complex structure $J^{-} \in {\mathcal J}^-$. \vskip.1in From Theorem 3.2 we have the following [CT1] \vskip.1in {\bf Corollary 3.6.} {\it Every non-minimal immersion $f : N \rightarrow E^4$ which is slant with respect to more than two complex structures are doubly slant.} \vskip.1in {\bf Remark 3.1.} From Theorem 3.2 we also know that {\it for any immersion $f: N \rightarrow E^4$, exactly one of the following four cases occurs:}
(a) {\it $f$ is not slant with respect to every compatible complex structure on $E^4$.}
(b) {\it $f$ is slant with respect to infinitely many compatible complex structures on $E^4$.}
(c) {\it $f$ is slant with respect to exactly two compatible complex structures on $E^4$.}
(d) {\it $f$ is slant with respect to exactly four compatible complex structures on $E^4$.}
\vskip.1in Corollary 3.5 shows that if $N$ is a non-totally geodesic minimal surface in $E^3$, then, by regarding $E^{3}$ as a linear subspace of $E^4$, $N$ is not slant with respect to every compatible complex structures on $E^4$. This provides us many examples for case (a) of Remark 3.1. \vskip.1in Here we remark that every totally real immersion of a 2-sphere $S^2$ into ${\bf C}^2$ gives us an example of surface in $E^4$ which is slant with respect to exactly two compatible complex structures on $E^4$. This is due the fact that the Gauss curvture of any Riemannian metric on $S^2$ is non-flat.
\vskip.1in {\bf Example 3.1.} Let $f : E^{3} \rightarrow E^4$ be the map from $E^3$ into $E^4$ defined by $$f(x_{0},x_{1},x_{2}) = (x_{1},x_{2},2x_{0}x_{1}, 2x_{0}x_{2}).\leqno(3.3)$$ Then $f$ induces an immersion ${\hat f} : S^{2} \rightarrow E^4$ from the unit 2-sphere $S^2$ into $E^4$, called the {\it Whitney immersion\/} which has a unique self-intersection point ${\hat f}(-1,0,0)={\hat f}(1,0,0)$. It is know that this immersion ${\hat f} : S^{2} \rightarrow E^4$ is a totally real immersion with respect to two suitable compatible complex structures on $E^4$. Moreover, since the surface is non-flat, ${\hat f}$ is a slant immersion
with respect to only two compatible complex structures (cf. Theorem 4.1.) \vskip.1in In Section 1 of Chapter V we will prove that there exist no compact proper slant submanifolds in any complex number space ${\bf C}^m$.
\vskip.1in {\bf Example 3.2.} Let $N$ be the surface in $E^4$ defined by $$x(u,v) = (u,v,k\cos v, k\sin v)\leqno(3.4)$$ Then $N$ is the Riemannian product of a line and a circular helix in a hyperplane $E^3$ of $E^4$. Let $J_{1}, J_{2}$ be the compatible complex structures on $E^4$ defined respectively by $$J_{1}(a,b,c,d) = (-b,a,-d,c),\,\,\, J_{2}(a,b,c,d) =(b,-a,-d,c).$$ Then $J_{1} \in {\mathcal J}^+$ and $J_{2} \in {\mathcal J}^-$. Moreover, by direct computation, we can prove that the surface $N$ is slant with respect to the following four complex structures: $J_{1}, -J_{1}, J_{2}, -J_{2}$, with slant angles given by $$\cos^{-1}({{1}\over {\sqrt{1+k^{2} } } }), \cos^{-1}({{-1} \over {\sqrt{1+k^{2}}}}), \cos^{-1}({{-1}\over {\sqrt{1+k^{2}}}}), \cos^{-1}({{1}\over {\sqrt{1+k^{2}}}}),$$ respectively. \vskip.1in {\bf Remark 3.2.} In view of Corollary 3.6, it is interesting to point out that {\it the only doubly slant minimal immersion from a surface into a complex 2-plane is the totally geodesic one.}
\eject {\sl III-4. DOUBLY SLANT SURFACES IN ${\bf C}^2$} \noindent{}
\vskip.3in \noindent \S 4. DOUBLY SLANT SURFACES IN ${\bf C}^2$. \vskip.2in As we defined in Section 3 an immersion $f : N \rightarrow E^4$ is called doubly slant if it is slant with respect to a complex structure in ${\mathcal J}^+$ and at the same time it is slant with respect to another complex structure in ${\mathcal J}^-$. Equivalently, the immersion $f$ is doubly slant if and only if there exists an oriented 2-plane $V \in G(2,4)$ such that $f$ is slant with respect to both $J_{V}^+$ and $J_{V}^-$, where $J_{V}^+$ and $J_{V}^-$ are defined by (2.6).
In this section we give the following [CT1] \vskip.1in {\bf Theorem 4.1.} {\it If $\, f : N \rightarrow E^4$ is a doubly slant immersion, then $$G = G^{D}=0\leqno(4.1)$$ identically.} \vskip.1in {\bf Proof.} If $f$ is doubly slant, then, by Proposition 3.1, we know that both $\nu_{+}(N)$ and $\nu_{-}(N)$ lie in some circles on $S_{+}^2$ and $S_{-}^2$, respectively. Thus, both $(\nu_{+})_*$ and $(\nu_{-})_*$ are singular maps at every point $p \in N$. Therefore, the result follows from the following \vskip.1in {\bf Lemma 4.2.} {\it For any immersion $f : N \rightarrow E^4$ of an oriented surface $N$ into $E^4$ we have} $$det\,(\nu_{+})_{*}={1 \over 2}(G + G^{D}),\,\,\, det\,(\nu_{-})_{*}={1 \over 2}(G - G^{D}).\leqno(4.2)$$ \vskip.05in {\bf Proof.} Let $x : N \rightarrow E^4$ be an immersion of an oriented surface $N$ into $E^4$. Denote by $\{e_{1},e_{2}\}$ be a positive orthonormal basis of $N$. Then at each point $p \in N$ the Gauss map $\nu$ is given by $$\nu(p) = (e_{1}\wedge e_{2})(p).\leqno(4.3)$$ Thus, for any vector $X$ tangent to $N$, we have $$\nu_{*}X = ({\tilde \nabla}_{X}e_{1})\wedge e_{2} + e_{1}\wedge ({\tilde \nabla}_{X}e_{2})$$ $$= \omega_{1}^{3}(X)e_{3}\wedge e_{2} + \omega_{1}^{4}(X)e_{4}\wedge e_{2} + e_{1} \wedge \omega_{2}^{3}(X) e_{3} + e_{1} \wedge \omega_{2}^{4}(X) e_{4} $$ $$= -\omega_{1}^{3}(X)e_{2}\wedge e_{3} - \omega_{1}^{4}(X)e_{2}\wedge e_{4} + \omega_{2}^{3}(X) e_{1} \wedge e_{3} +
\omega_{2}^{4}(X)e_{1} \wedge e_{4} $$ $$={1 \over 2}\{(\omega_{1}^{4}+\omega_{2}^{3})(X)(e_{1} \wedge e_{3} - e_{2}\wedge e_{4}) + (-\omega_{1}^{3}+\omega_{2}^{4})(X)(e_{1} \wedge e_{4} - e_{2}\wedge e_{3}) $$ $$+(-\omega_{1}^{4}+\omega_{2}^{3})(X)(e_{1} \wedge e_{3} + e_{2}\wedge e_{4}) + (\omega_{1}^{3}+\omega_{2}^{4})(X)(e_{1} \wedge e_{4} - e_{2}\wedge e_{3})\}$$ $$={1\over \sqrt{2}}\{(\omega_{1}^{4}+\omega_{2}^{3})(X)\eta_{2} +(-\omega_{1}^{3}+\omega_{2}^{4})(X)\eta_{3}+$$ $$+(-\omega_{1}^{4}+\omega_{2}^{3})(X)\eta_{5} +(\omega_{1}^{3}+\omega_{2}^{4})(X)\eta_{6}\}.$$ Therefore we have $$(\nu_{+})_{*}= {1\over \sqrt{2}}\{(\omega_{1}^{4}+ \omega_{2}^{3})\eta_{2} +(-\omega_{1}^{3}+\omega_{2}^{4})\eta_{3}\},\leqno(4.4)$$ and
$$(\nu_{-})_{*}= {1\over \sqrt{2}}\{(-\omega_{1}^{4}+ \omega_{2}^{3})\eta_{5} +(\omega_{1}^{3}+\omega_{2}^{4})\eta_{6}\}.\leqno(4.5)$$ This proves Lemma 4.2. \vskip.1in {\bf Remark 4.1.} Examples 2.1-2.6 given in Section 2 of Chapter II are examples of doubly slant surfaces. \vskip.1in {\bf Remark 4.2.} Lemma 4.2 can be found in [HO1]. Our proof of Lemma 4.2 is different from theirs.
\eject
\vskip.3in \noindent \S 5. SLANT SURFACES IN ALMOST HERMITIAN MANIFOLDS. \vskip.2in Let $f: N \rightarrow (M,J)$ be an immersion of a differentiable manifold $N$ into an almost complex manifold $(M,J)$. Then a point $p \in N$ is called a {\it complex tangent point\/} if the tangent plane of $N$ at $p$ is invariant under the action of the almost complex structure $J$.
The purpose of this section is to prove the following [CT1] \vskip.1in {\bf Theorem 5.1.} {\it Let $f: N \rightarrow (M,g,J)$ be an imbedding of an oriented surface $N$ into a real 4-dimensional almost Hermitian manifold $(M,g,J)$. If the immersion $f$ has no complex tangent points, then, for any prescribed angle $\alpha \in [0,\pi]$, thre exists an almost complex structure ${\tilde J}$ on $M$ satisfying the following conditions:}
(a) $(M,g,{\tilde J})$ {\it is an almost Hermitian manifold and}
(b) {\it the immersion $f$ is $\alpha$-slant with respect to ${\tilde J}$.} \vskip.1in {\bf Proof.} ($M,g,J$) has the natural orientation determined by the almost complex structure $J$ and, at each point $p \in M$, the tangant space $T_{p}M$ together with the metric $g_p$ is a Euclidean 4-space. So we may apply the argument given in Sections 1 and 2 of this chapter.
According to (1.4) the vector bundle $\wedge^{2}(M)$ of 2-vectors on the ambient manifold $M$ is the direct sum of two vector subbundles:
$$\wedge^{2}(M)=\wedge^{2}_{+}(M)\oplus\wedge^{2}_{-}(M). \leqno(5.1)$$
We define two sphere-bundles over $M$ by
$$S^{2}_{+}(M)=\{\xi\in \wedge^{2}_{+}(M) : ||\xi
||={1\over \sqrt{2}}\},$$
$${\bar S}^{2}_{+}(M)=\{\xi\in \wedge^{2}_{+}(M) : ||\xi
||=\sqrt{2}\}.$$
By using Lemma 2.1 we can identify a cross-section $$\gamma : M \rightarrow {\bar S}^{2}_{+}(M)\leqno(5.2)$$ with an almost complex structure $J_\gamma$ on $M$ such that $(M,g,J_{\gamma})$ is an almost Hermitian manifold.
In the following we denote by $\rho$ the cross-section corresponding to the almost complex structure $J$ and we want to construct another cross-section ${\tilde \sigma}$ to obtain the desired almost complex structure ${\tilde J}$ on the ambient manifold $M$.
We consider the pull-backs of these bundles via the imbedding $f : N \rightarrow M$, that is, $$\wedge^{2}_{+}(N)=f^{*}(\wedge^{2}_{+}(M)),\,\,\,\,\,\, S^{2}_{+}(N)=f^{*}(S^{2}_{+}(M)),\leqno(5.3)$$ $${\bar S}^{2}_{+}(N)=f^{*}({\bar S}^{2}_{+}(M)).$$
The tangent bundle $TN$ of $N$ determines a cross-section $\tau : N \rightarrow S^{2}_{+}(N)$ defined by $$\tau(p)=\pi_{+}(T_{p}N)\leqno(5.4)$$ for any point $p \in N$, where $\pi_+$ denotes the natural projection from $\wedge^{2}(T_{p}M)$ onto $\wedge_{+}^{2}(T_{p}M)$. Notice that $2\tau$ is a cross-section of ${\bar S}^{2}_{+}(N)$:
$$2\tau : N \rightarrow {\bar S}^{2}_{+}(N).\leqno(5.5)$$
We denote $f^{*}\rho$ also by $\rho$ for simplicity. We have the following cross-section: $$\rho=f^{*}\rho : N \rightarrow {\bar S}^{2}_{+}(N).\leqno(5.6)$$
Since the imbedding $f$ is assumed to have no complex tangent points,
$$\rho(p)\not= \pm 2\tau(p)\leqno(5.7)$$
\noindent for any point $p \in N.$ Therefore, $\rho(p)$ and $2\tau(p)$ determine a 2-plane in $\wedge^{2}_{+}(T_{p}M)$ which intersects the circle $({\mathcal J}^{+}_{\tau,a})_{p}$ at two points. Where $({\mathcal J}^{+}_{\tau,a})_{p}$ is a circle on $({\bar S}^{2}_{+}(N))_p$ defined in Lemma 2.3 with $V=T_{p}N$.
Let $\sigma(p)$ be one of these two points which lies on the half-great-circle on $({\bar S}^{2}_{+}(N))_p$ starting from $2\tau(p)$ and passing through $\rho(p)$. Since $\rho$ and $\tau$ are differentiable, so is $\sigma$. Thus we obtain a third cross-section: $$\sigma : N \rightarrow {\bar S}^{2}_{+}(N)\leqno(5.8)$$ and we want to extend $\sigma$ to a cross-section ${\tilde \sigma}$ of ${\bar S}^{2}_{+}(M).$
For each point $p \in N$, we choose an open neighborhood
$U_p$ of $p \in M$ such that $\sigma_{| U_{p}\cap N}$ can be extended to a cross-section of ${\bar S}^{2}_{+}(M)$ on $U_p$:
$$\sigma_{p} : U_{p} \rightarrow {\bar S}^{2}_{+}(M)_{| U_{p}}.\leqno(5.9)$$ Here we identify the manifold $N$ with its image $f(N)$ of $N$ under the imbedding $f$. We put $${\mathcal U} = \cup_{p\in N}\, U_{p}\leqno(5.10)$$ and pick a locally finitely countable refinement $\{U_{i}\}$ of the open covering $\{U_{p}: p \in N\}$ of $U$.
For each $i$ we pick a point $p \in N$ such that $U_i$ is contained in $U_p$ and we put
$$\sigma_{i}=\sigma_{p | U_{i}}.\leqno(5.11)$$
Let $\phi_{i}$ be a differentiable partition of unity on ${\mathcal U}$ subordinate to the covering $\{ U_{i}\}$. We define a cross-section ${\bar \sigma}$ of
$\wedge^{2}_{+}(M)_{| {\mathcal U}}$ by $${\bar \sigma}: {\mathcal U} \rightarrow
\wedge^{2}_{+}(M)_{| {\mathcal U}}; \,\,\,\,{\bar\sigma}=\sum_{i} \phi_{i}\sigma_{i}.\leqno(5.12)$$
From the constructions of $\sigma_i$ and ${\bar\sigma}$ we have
$${\bar \sigma}_{| N}=\sigma .\leqno(5.13)$$
Since the angle $\angle \, ({\bar\sigma(p)},\rho(p))$ between ${\bar \sigma}(p)$ and $\rho(p)$ is less that $\pi$ for any point $p \in N$, we have $${\bar \sigma}(p)\not= 0,\,\,\,\angle \, ({\bar\sigma(p)},\rho(p)) < \pi$$ for any point $p \in N$. By continuity of ${\bar \sigma}$ we can pick an open neighborhood $W$ of $N$ contained in ${\mathcal U}$ such that $${\bar\sigma}(q)\not= 0,\,\,\,\,\angle \, ({\bar\sigma(q)},\rho(q))<\pi$$ for any point $q \in W.$ We define a cross-section ${\hat
\sigma}$ of ${\bar S}^{2}_{+}(M)_{| W}$ by
$${\hat \sigma} : W \rightarrow {\bar S}^{2}_{+}(M)_{| W};
\,\,\,\,\,{\hat \sigma}={\bar\sigma}/\sqrt{2}||{\bar
\sigma}||.$$
\noindent Then we have $\angle \,({\hat\sigma}(q),\rho(q))<\pi$ for any point $q\in M$, too.
Finally, we consider the open covering $\,\{W,M-N\}\,$ of $M$ and local cross-section
${\hat\sigma}$ and $\rho_{| M-N}$ and repeat the same argument using an partition of unity subordinate to $\{W, M-N\}$ to obtain a cross-section ${\tilde \sigma}:M \rightarrow {\bar S}^{2}_{+}(M)$ satisfying ${\tilde
\sigma}_{| N}=\sigma$. Now it is clear that the almost complex structure ${\tilde J}$ corresponding to ${\tilde \sigma}$ is the desired almost complex structure. This completes the proof of the theorem.
\eject
\vskip1in \centerline {CHAPTER IV} \vskip.2in
\centerline {\bf CLASSIFICATIONS OF SLANT SURFACES} \vskip.5in \noindent \S 1.$\,\,$ SLANT SURFACES WITH PARALLEL MEAN CURVATURE
VECTOR. \vskip.2in The main purpose of this section is to present the following classification theorem of [C6]. \vskip.1in {\bf Theorem 1.1.} {\it Let $N$ be a surface in ${\bf C}^2$. Then $N$ is a slant surface with parallel mean curvature vector, that is, $DH=0$, if and only if $N$ is one of the following surfaces:}
(a) {\it an open portion of the product surface of two plane circles, or}
(b) {\it an open portion of a circular cylinder which is contained in a hyperplane of ${\bf C}^2$, or}
(c) {\it a minimal slant surface in ${\bf C}^2$.}
\noindent {\it Moreover, if either case {\rm (a)} or case {\rm (b)} occurs, then $N$ is a totally real surface in ${\bf C}^2$.} \vskip.1in {\bf Proof.} Let $N$ be a slant surface in ${\bf C}^2$ with parallel mean curvature vector. Then the length of the mean curvature vector $H$ is constant. If the length is zero, $N$ is a minimal slant surface. So Case (c) occurs. Now assume that $N$ is not minimal in ${\bf C}^2$. Then one may choose a unit tangent vector $e_1$ such that $e_{3}=(\csc\theta)Fe_1$ is in the direction of $H$, where $\theta$ denotes the slant angle of $N$ in ${\bf C}^2$. Such an $e_1$ can be chosen if we choose $e_1$ to be in the direction of $-tH$. Since the mean curvature vector is parallel, $\omega_{3}^{4}=0$. Thus the normal curvature $G^{D}=0$, identically. Hence, by applying Theorem 3.7 of Chapter II, we have $G=0$, that is, $N$ is flat. Let $V=\{ \, p\in N : A_{e_{4}}\not= 0\,\,at\,\,p\,\}$. Then $V$ is an open subset of $N$.
{\bf Case (i):} $V=\emptyset$. In this case we have $Im\,h\subset Span\{ H\}$. Thus, by applying the condition $DH=0$, we conclude that the surface $N$ lies in a hyperplane $\,E^{3}\,$ of ${\bf C}^2$. Since $N$ is a flat surface in $E^3$ with nonzero constant mean curvature, $N$ is an open portion of a circular cylinder (cf. Proposition 3.2 of [C1, p.118]).
{\bf Case (ii):} $V\not=\emptyset$. In this case, let $W$ be a connected component of $V$. Then $e_4$ is a parallel minimal non-geodesic section on $W$ (cf. [C1]). Since $W$ is flat, Proposition 5.4 of [C1, p.128] implies that $W$ is an open piece of the product surface of two plane circles. Since $det\,(A_{e_{4}})$ is a nonzero constant on $W$, by continuity we have $V=N$. Thus the whole surface is an open portion of the product surface.
Finally, if $N$ is an open portion of the product surface of two plane circles or an open portion of a circular cylinder contained in a hyperplane of ${\bf C}^2$, then $N$ is totally real in $E^4$ with respect to some compatible complex structure, say $J'$, on $E^4$. Therefore, by applying Theorem 4.3 of Chapter II, we know that $N$ must be totally real in ${\bf C}^2$, since $N$ is non-minimal. This completes the proof of the theorem. \vskip.1in By applying Theorem 1.1 we may obtain the following classification of parallel slant surfaces, that is, slant surfaces with parallel second fundamental form [C4]. \vskip.1in {\bf Theorem 1.2.} {\it Let $N$ be a surface in ${\bf C}^2$. Then $N$ is a slant surface in ${\bf C}^2$ with parallel second fundamental form, that is, ${\bar \nabla}h=0,$ if and only if $N$ is one of the following surfaces:}
(a) {\it an open portion of the product surface of two plane circles;}
(b) {\it an open portion of a circular cylinder which is contained in a hyperplane of ${\bf C}^2$;}
(c) {\it an open protion of a plane in ${\bf C}^2$}.
\noindent {\it Moreover, if either case {\rm (a)} or case {\rm (b)} occurs, then $N$ is totally real in ${\bf C}^2$.} \vskip.1in {\bf Proof.} If $N$ is a surface in ${\bf C}^2$ with parallel second fundamental form, then $N$ has parallel mean curvature vector. Thus, by Theorem 1.1, it suffices to prove that slant planes are the only minimal slant surfaces with parallel second fundamental form. But this follows from the facts that every surface in ${\bf C}^2$ with parallel second fundamental form has constant Gauss curvature and every minimal surface in ${\bf C}^2$ with constant Gauss curvature is totally geodesic. \vskip.1in By using Theorem 1.1, we obtain \vskip.1in {\bf Corollary 1.3.} {\it Let $N$ be a slant surface in
${\bf C}^2$ with constant mean curvature. Then $N$ is spherical if and only if $N$ is an open portion of the product surface of two plane circles.} \vskip.1in {\bf Proof.} If $N$ is a spherical surface with constant mean curvature, then the mean curvature vector of $N$ in
${\bf C}^2$ is parallel. Thus, by applying Theorem 1.1, $N$ is one of the surfaces mentioned in Theorem 1.1. Among them, surfaces of type (a) are the only spherical surfaces in ${\bf C}^2$.
The converse is obvious. \vskip.1in Similarly we may prove the following \vskip.1in {\bf Corollary 1.4.} {\it Let $N$ be a slant surface in ${\bf C}^2$ with constant mean curvature. Then $N$ lies in a hyperplane of ${\bf C}^2$ if and only if $N$ is either an open portion of a 2-plane or an open portion of a circular cylinder.}
\vskip.1in {\bf Remark 1.1.} In views of Theorem 1.1, Corollary 1.3 and Corollary 1.4, it seems to be interesting to propose the following open problem: \vskip.1in {\bf Problem 1.1.} {\it Classify all slant surfaces in ${\bf C}^2$ with nonzero constant mean curvature.}
\eject
\vskip.3in \noindent \S 2. SPHERICAL SLANT SURFACES. \vskip.2in In the remaining part of this chapter we want present some classification theorems obtained in [CT2].
In this section we want to classify slant surfaces of
${\bf C}^2$ which lie in a hypersphere of ${\bf C}^2$. In order to do so, we need to review the geometry of the unit hypersphere $S^{3}=S^3(1)$ in ${\bf C}^2$ centered at the origin.
It is known that $S^3$ is the Lie group consisting of all unit quaternions $\{ u = a +{\bf i} b+{\bf j} c+{\bf k} d
: ||u|| = 1 \} $
which can also be regarded as a subgroup of the orthogonal group $O(4)$ in a natural way. Let 1 denote the identity element of the Lie group $S^3$ given by $$1=(1,0,0,0) \in S^{3} \subset E^{4}.\leqno(2.1)$$
We put $$X_{1}=(0,1,0,0),\,\,\,X_{2}=(0,0,1,0),\,\,\, X_{3}=(0,0,0,1) \in T_{1}S^{3}. \leqno(2.2)$$
We denote by ${\tilde X}_{i}, i = 1, 2, 3$, the left-invariant vector fields obtained from the extensions of $X_{i}, i = 1, 2 ,3$, on $S^3$, respectively. Let $\phi :S^{3} \rightarrow S^{3}$ be the orientation-reversing isometry defined by $$\phi (a,b,c,d)=(a,b,d,c).\leqno(2.3)$$
We recall that the left-translation $L_p$ and the right-translation $R_p$ on $S^3$ are isometries which are analogous to the parallel translations on $E^3$ and they are given by $${}^t(L_{p}q)=\begin{pmatrix}a&-b&-c&-d\\ b&a&-d&c\\ c&d&a&-b\\ d&-c&b&a\end{pmatrix}\begin{pmatrix}x\\ y\\ z\\ w\end{pmatrix}\leqno(2.4)$$ and \eject $${}^t(R_{p}q)= \begin{pmatrix}a&-b&-c&-d\\ b&a&d&-c\\ c&-d&a&b\\ d&c&-b&a\end{pmatrix}\begin{pmatrix}x\\ y\\ z\\ w \end{pmatrix}\leqno(2.5)$$
\noindent for $\, p = (a,b,c,d),\,\, q=(x,y,z,w) \in S^{3} \subset E^{4},\,$ where $\,{}^{t}A \,$ denotes the transpose of $A$.
Let $\eta$ denote the unit outer normal of $S^3$ in $E^4$ and $J_{1}$ and $J_{1}^{-}$ the complex structures on $E^4$ defined respectively by $$J_{1}(a,b,c,d)=(-b,a,-d,c),\leqno(2.6)$$ $$J_{1}^{-}(a,b,c,d)=(-b,a,d,-c).\leqno(2.7)$$
\vskip.1in {\bf Lemma 2.1.} {\it For any $q \in S^{3}$, we have} $$(J_{1} \eta) (q)=R_{q*}X_{1},\leqno(2.8)$$ $$(J_{1}^{-}\eta) (q)=L_{q*}X_{1}= {\tilde X}_{1}(q).\leqno(2.9)$$ {\it Hence $J_{1}\eta$ and $J_{1}^{-}\eta$ are right-invariant and left-invariant vector fields on $S^3$, respectively.} \vskip.1in {\bf Proof.} Let $q=(a,b,c,d) \in S^3$. Then $$\eta(q)=(a,b,c,d)\in T_{q}^{\perp}S^{3},\leqno(2.10)$$ $$(J_{1}\eta)(q)=(-b,a,-d,c),\,\,\,\,\,(J_{1}^{-}\eta) (q)=(-b,a,d,-c).$$
Let $\gamma \subset S^3$ be a curve on $S^3$ parametrized by arclength $s$ given by $$\gamma(s)=(\cos s,\sin s,0,0).\leqno(2.11)$$ Then we have
$$R_{q^{*}}X_{1}={d\over {ds}}(R_{q}(\gamma(s)))_{| s=0} =$$ $$= {d\over {ds}}((\cos s + {\bf i}\sin s)
(a+{\bf i}b+{\bf j}c+{\bf k}d))_{| s=0}$$ $$=(-b,a,-d,c)=(J_{1}\eta)(q).$$
Similarly we may prove $L_{q^{*}}X_{1}=(J_{1}^{-}\eta)(q).$ \vskip.1in
As in Section 2 of Chapter III, we denote by ${\mathcal J}$ the set of all complex structures on $E^4$ compatible with the inner product $<\,,\,>$. By using the natural orientation of $E^4$ we divide $\mathcal J$ into two disjoint subsets: $${\mathcal J}^{+} = \{{J\in {\mathcal J}\,|\, J{\rm -bases\,\, are\,\, positive}}\},$$ $${\mathcal J}^{-}
= \{{J \in {\mathcal J}\,|\, J{\rm -bases \,\, are \,\, negative}}\}.$$
\noindent Thus we have ${\mathcal J}= {\mathcal J}^{+} \cup {\mathcal J}^{-}$ (cf. Section 2 of Chapter III). \vskip.1in {\bf Lemma 2.2.} {\it Let $W\in G(3,4)$ and $V\in G(2,4)$ such that $V\subset W$. Then $V$ is $\alpha$-slant with respect to a complex structure $J\in {\mathcal J}^{+}$ (respectively, $J\in {\mathcal J}^{-}$) if and only if $$<\xi_{V},J\eta_{W}> = -\cos\alpha \hskip.15in (respectively, <\xi_{V},J\eta_{W}> = \cos\alpha),\leqno(2.12)$$ where $\, \xi_V$ and $\,\eta_W$ are positive unit normal vectors of $V$ in $W$ and of $\, W$ in $E^4$, respectively.} \vskip.1in
{\bf Proof.} We choose an orthonormal $J$-basis $\{e_{1},\ldots,e_{4}\}$ of $E^4$ such that $$ e_{1}, e_{2} \in W\cap JW,\,\,\,\, e_{4}=Je_{3}=\eta_{W}.\leqno(2.13)$$ We also choose a positive orthonormal basis $\{X_{1},X_{2}\}$ of $V$. Let $\zeta_J$ be the 2-vector defined as before as the metrical dual of $-\Omega_{J} \in (\wedge^{2}E^{4})^*$, that is, $$<\zeta_{J},X\wedge Y> = -\Omega_{J}(X,Y) $$ for any $X, Y \in E^4$. Then, by formula (2.3) of Section 2 of Chapter III, we see that the slant angle $\alpha_{J}(V)$ of $V$ with respect to $J$ satisfies $$\cos\alpha_{J}(V) = <\zeta_{J},X_{1}\wedge X_{2}>$$ $$ = <e_{1}\wedge e_{2} + e_{3}\wedge e_{4}, X_{1}\wedge X_{2}>= <e_{1}\wedge e_{2}, X_{1}\wedge X_{2}>$$
$$ =< \pm e_{3},\xi_{V}> = \mp <J\eta_{W},\xi_{V}>$$ for $J \in {\mathcal J}^{\pm}$. This proves the lemma.
\vskip.1in
Let $f : N \rightarrow S^{3} \subset E^4$ be a spherical immersion of an oriented surface $N$ into $S^3$ and $\xi$ the positive unit normal of $f(N)$ in $S^3$.
It is easy to see that every spherical surface in ${\bf C}^2$
is non-minimal. Hence, every spherical surface in ${\bf C}^2$ cannot be a complex surface of ${\bf C}^2$.
\vskip.1in
{\bf Lemma 2.3.} {\it Let $f: N \rightarrow S^{3} \subset E^4$ be an immersion of an oriented surface $N$ into the hypersphere $S^3$ of $E^4$. Then the following three statements hold.}
(i) {\it The immersion $f$ is $\alpha$-slant with respect to the complex structure $J_1$ if and only if } $$<\xi(p),J_{1}\eta (f(p))>= -\cos\alpha\hskip.2in
for \,\,any\,\, p \in N.\leqno(2.14)$$
(ii) {\it The immersion $f$ is $\alpha$-slant with respect to the complex structure $J_{1}^-$ if and only if } $$<\xi(p),{\tilde X}_{1}(f(p))> = \cos \alpha \hskip.2in for \,\,any\,\, p \in M.\leqno(2.15)$$
(iii) {\it The immersion $f$ is $\alpha$-slant with respect to the complex structure $J_1$ if and only if the composition $ \,\phi\circ f \,$ is $\alpha$-slant with respect to the complex structure $J_{1}^{-}$.} \vskip.1in
{\bf Proof.} Statement (i) follows from Lemma 2.2. Statement (ii) follows from Lemma 2.1 and Lemma 2.2. Finally, the last statement follows from statements (i) and (ii) and from the fact that $\phi$ is an isometric involution which reverses the orientation of $E^4$.
\vskip.1in
We define two maps $g_{+}$ and $g_{-}$ from $N$ into the unit sphere $S^2$ in $T_{1}S^3$ by $$g_{+}(p)=(L_{\phi(f(p))*})^{-1}(\phi_{*}\xi(p)),\,\,\,\,
g_{-}(p)=(L_{f(p)*})^{-1}(\xi (p)) \leqno(2.16)$$ for $p \in N$. In fact, $g_{+}$ and $g_{-}$ are the analogues of the classical Gauss map of a surface in $E^3$ in which the parallel translations in $E^3$ are replaced by the left-translations $L_q$ on $S^3$.
We also define a circle $S_{\alpha}^{1}$ for $\alpha \in [0,\pi ]$ on the unit sphere $S^2$ in $T_{1}S^3$ by
$$S_{\alpha}^{1}= \{ X \in T_{1}S^{3} \, \mid \, \|X\| = 1, \,<X,X_{1}>= - \cos \alpha \}. \leqno(2.17)$$
Now we prove the following result of [CT2] which characterizes spherical slant surfaces in {\bf C}$^2$.
\vskip.1in {\bf Proposition 2.4.} {\it Let $\,f : N \rightarrow S^{3} \subset E^4$ be an immersion of an oriented surface $N$. Then we have}
(i) $f$ {\it is $\alpha$-slant with respect to the complex structure $J_{1}$ if and only if } $$g_{+}(N) \subset S_{\alpha}^{1} \subset T_{1}S^{3}.\leqno(2.18)$$
(ii) $f$ {\it is $\alpha$-slant with respect to the complex structure $J_{1}^{-}$ if and only if} $$g_{-}(N) \subset S_{\pi -\alpha}^{1} \subset T_{1}S^{3}.\leqno(2.19)$$ \vskip.1in
{\bf Proof.} Proposition 2.4 follows from Lemma 2.1, Lemma 2.3 and the definitions of $g_+$ and $g_-$. \vskip.1in
Proposition 2.4 can be regarded as the spherical version of Proposition 3.1 in Section 3 of Chapter III. \vskip.1in Concerning the images of $N$ under the spherical Gauss maps $g_+$ and $g_-$, we give
the following two simple examples (cf. [CT2]). \vskip.1in
{\bf Example 2.1.} If $N = S^{1}\times S^1$ is the flat torus in $E^4$ defined by $$f(u,v) = {1\over {\sqrt{2}}}(\cos u,\sin u,\cos v,\sin v),\leqno(2.20)$$ then the images of $N$ under the spherical Gauss maps $g_+$ and $g_-$ are the great circles perpendicular to $X_{1}= (0,1,0,0).$ \vskip.1in
{\bf Example 2.2.} If $N = S^2$ is the totally geodesic 2-sphere of $S^3$ parametrized by $$f(u,v) = (\cos u\cos v,\sin u\cos v,\sin v,0),$$ then
$$g_{+}(u,v) = (0,-\sin v,-\cos u\cos v,\sin u\cos v),$$ $$g_{-}(u,v) = (0,\sin v,\sin u\cos v,-\cos u\cos v).$$ Hence, both $g_+$ and $g_-$ are isometries. \vskip.1in
In order to describe slant surfaces in $S^3$ geometrically, we give the following
\vskip.1in {\bf Definition 2.1.} Let $c(s)$ be a curve in $S^3$
parametrized by arclength $s$ and let $$c'(s) = \sum_{i=1}^{3} f_{i}(s){\tilde X}_{i}(c(s)).\leqno(2.21)$$
We call the curve $c(s)$ a {\it helix in\/} $S^3$ {\it with axis vector field} ${\tilde X}_1$ if $$f_{1}(s)=b,\,\,\, f_{2}(s)=a\,\cos (ks+s_{0}), \,\,\, f_{3}(s)= a\,\sin (ks+s_{0})\leqno(2.22)$$
\noindent for some constants $a, b, k$ and $s_0$ satisfying $$a^{2}+b^{2}=1.\leqno(2.23)$$
We call the curve $c(s)$ a {\it generalized helix in\/} $S^3$ {\it with axis vector field} ${\tilde X}_1$ if $$<c'(s),{{\tilde X}_{1}}(c(s))> = constant.\leqno(2.24)$$ \vskip.1in
Helices in $S^3$ defined above are the analogues of Eucliden helices in the Euclidean 3-space $E^3$.
\vskip.1in
{\bf Definition 2.2.} We call an immersion $\, f : D \rightarrow S^3$ of a domain $D$ around the origin $(0,0)$ of $R^2$ into $S^3$ a {\it helical cylinder in} $S^3$ if
$$f(s,t)=\gamma (t)\cdot c(s),\leqno(2.25)$$
\noindent for some helix $c(s)$ in $S^3$ with axis ${\tilde X}_1$ satisying $k = - 2/b$ and $ab < 0$, and for some curve $\gamma (t)$ in $S^3$ which is either a geodesic or
a curve of constant torsion 1 parametrized by arclength such that
(i) $c(0)=\gamma (0)$, and
(ii) the osculating planes of $c(s)$ and of $\gamma (t)$ coincide at $t=s=0$.
We note that the binormal of $c(s)$ is normal to $f(D)$ in $S^3$. Here we orient the curve $c$ in such a way that the binormal of $c(s)$ is the positive unit normal of $f(D)$. \vskip.1in
The main purpose of this section is to prove the following classification theorem for spherical slant surfaces [CT2].
\vskip.1in
{\bf Theorem 2.5.} {\it Let $f : N \rightarrow S^{3} \subset {\bf C}^{2} = (E^{4},J_{1})$ be a spherical immersion of an oriented surface $N$ into the complex 2-plane {\bf C}$^2$ $=(E^{4},J_{1})$. Then $f$ is a proper slant immersion if and only if
$f(N)$ is locally of the form $\{\phi (\gamma (t)\cdot c(s))\}$ where $\phi$ is the isometry on $S^3$ defined by (2.3) and $\{\gamma (t) \cdot c(s)\}$ is a helical cylinder in $S^3$ (cf. Definition 2.2).}
\vskip.1in
In order to prove this theorem so we need several lemmas.
First, we note that curves in $S^3$ can be described in terms of the orthonormal left-invariant vector fields $\{{\tilde X}_{1},{\tilde X}_{2},{\tilde X}_{3}\}$.
Let $I$ be an open interval containing 0 and $c : I \rightarrow S^3$ a curve parametrized by arclength $s$. Let ${\bf t}(s), {\bf n}(s), {\bf b}(s), \kappa(s)$, and $\tau(s)$ be the unit tangent vector, the unit principal normal vector, the unit binormal vector, the curvature, and the torsion of $c$ in $S^3$, respectively. We put $${\bf t}(s) = \sum_{i=1}^{3} f_{i}(s){\tilde X}_{i}(c(s)).\leqno(2.26)$$ Then $$(f_{1}(s))^{2} + (f_{2}(s))^{2} + (f_{3}(s))^{2} = 1.\leqno(2.27)$$ \vskip.1in Conversely, we have the following \vskip.1in
{\bf Lemma 2.6.} {\it Let $f_{i}(s), i = 1, 2, 3,$ be differentiable functions on $I$ satisfying (2.27). Then, for any point $p_{0} \in S^3$, there exists a curve $c(s)$ in $S^3$ defined on an open subinterval $I'$ of $I$ containing $0$ and satisfying (2.26) and the initial condition $c(0) = p_0$.} \vskip.1in
{\bf Proof.} Considering the curve $L_{p_{0}}^{-1}\circ c$ instead of $c$, if necessary, we can assume without loss of generality that $p_{0}=1$. The solution of the following system of the first order linear differential equations: $$\begin{pmatrix}x'\cr y'\cr z'\cr w'\end{pmatrix} = \begin{pmatrix}x&-y&-z&-w\\ y&x&-w&z\\ z&w&x&-y\\ w&-z&y&x\end{pmatrix}\begin{pmatrix}0\\
f_{1}\\ f_{2}\\ f_{3}\end{pmatrix} \leqno(2.28)$$
\noindent with the initial condition $(x(0),y(0),z(0),w(0)) = (1,0,0,0)$ satisfies the condition: $$xx' + yy' + zz' + ww' = 0$$ and the curve $c(s) = (x(s),y(s),z(s),w(s))$ is in fact the desired one. \vskip.1in Lemma 2.6 guarantees the existence of helices in $S^3$. \vskip.1in
{\bf Lemma 2.7.} {\it The following two statements are equivalent:} \vskip.05in (i) {\it The curve $c(s)$ is a helix in $S^3$ with axis vector ${\tilde X}_1$ satisfying} $$f_{1}(s) = b,\leqno(2.29)$$ $$f_{2}(s) = a\cos (-{2\over b}s + s_{0}),\leqno(2.30)$$ $$f_{3}(s) = a\sin (-{2\over b}s + s_{0}),\leqno(2.31)$$ $$a^{2} + b^{2} = 1,\,\,\,\,\, ab < 0.\leqno(2.32)$$
(ii) {\it The curve $c(s)$ is a curve in $S^3$ satisfying}
$$\tau (s) \equiv -1,\leqno(2.33)$$ $$<{\bf b}(s),{\tilde X}_{1}(c(s))> \equiv a.\leqno(4.9)$$ $$a \not= \pm 1, 0.\leqno(2.34)$$ \vskip.1in
{\bf Proof.} (ii) $\Rightarrow$ (i): Suppose $c$ is a curve in $S^3$ parametrized by arclength and the unit tangent vector {\bf t} of $c$ is given by (2.26). Let $$g_{1}= f_{2}f'_{3} - f_{3}f'_{2},\,\,\,\, g_{2}= f_{3}f'_{1} - f_{1}f'_{3},$$ $$ g_{3}= f_{1}f'_{2} - f_{2}f'_{1}.$$
\noindent By Frenet-Serret formulas and (2.33) we have $$({{g_{i}}\over {\kappa}})' = {2f'_{i}\over {\kappa}}, \,\,\,\, i = 1, 2, 3.\leqno(2.35)$$
\noindent By using (2.34) and the identity ${\bf b} = {\bf t}\times{\bf n}$, we may obtain
$$a = {{g_{1}}\over {\kappa}}.\leqno(2.36)$$
\noindent Hence, we find $2f'_{1}/{\kappa} = a' = 0$. Let $b$ denote the constant
$f_{1}$. Then, from (2.36) and ${\bf b} = {\bf t}\times{\bf n}$, we may find $${\bf b} = a{\tilde X}_{1} - ({{bf'_{3}}\over {\kappa}}){\tilde X}_{2} + ({{bf'_{2}}\over {\kappa}}){\tilde X}_{3},\leqno(2.37)$$
$$\kappa^{2} = (f'_{2})^{2} + (f'_{3})^{2},\leqno(2.38)$$ $${\bf n} = ({{f'_{2}}\over {\kappa}}){\tilde X}_{2} + ({{f'_{3}}\over {\kappa}}){\tilde X}_{3}.\leqno(2.39)$$
\noindent Since $\|{\bf b}\| = 1, \|{\bf n}\| = 1,$ (2.37) and
(2.39) imply $a^{2} + b^{2} = 1.$ Thus, from $\|{\bf t}\| = 1$ and (2.26), we get $f_{2}^{2} + f_{3}^{3} = a^2$. So we may put $$f_{2} = a\cos \theta, \,\,\, f_{3} = a\sin \theta,\,\,\,\, \theta = \theta (s).\leqno(2.40)$$
\noindent Thus, by applying the definition of $g_{1}, g_{2}, g_{3}$, we have $$g_{1} = a \kappa,\,\,\, g_{2} = - bf'_{3},\,\,\, g_{3} = bf'_{2}.\leqno(2.41)$$
\noindent By using (2.35), (2.38), (2.40), and $\tau \not= 0$, we get $$\kappa = \,\mid a\theta'\mid\, \not= 0.\leqno(2.42)$$
From (2.26), (2.40), (2.41) and (2.42) we find $\sin\theta (b\theta' + 2) = 0$. Since $\sin\theta(s)$ has only isolated zeros by (2.42), $b\theta' + 2 = 0$. Thus, $b \not= 0$. So $\theta = - {{2}\over {b}}s + s_{0}, s_{0} =$ const. Hence, by (2.37) and $\kappa > 0$, we get $ab < 0$. \vskip.1in $(i) \Rightarrow (ii)$ follows from straight-fordward computation. \vskip.1in {\bf Lemma 2.8.} {\it A helical cylinder $f(N) = \{\gamma(t)\cdot c(s)\}$ in $S^3$ is a proper slant surface with respect to the complex structure $J_{1}^-$ with the slant angle equal to $\cos^{-1}a$, where $a$ is the constant given by (2.22) of Definition 2.1.} \vskip.1in
{\bf Proof.} Let $\xi$ be the positive unit normal of $f(N)$ in $S^3$ and ${\bf b}$ the binormal vector of $c$ in $S^3$. Then we have $$\xi(\gamma(t)\cdot c(s)) = L_{\gamma(t)^*}{\bf b}(s).\leqno(2.43)$$ Lemma 2.8 now follows from Lemma 2.3 and Lemma 2.7.
\vskip.1in {\bf Lemma 2.9.} {\it For any point $\, p_{0}\in S^3$ and any oriented 2-plane $P_{0} \subset T_{p_{0}}S^{3} \subset E^4$ which is proper slant with respect to the complex structure $J_{1}^-$, there exist helical cylinders in $S^3$ passing through $p_{0}$ and whose tangent planes at $p_0$ are $P_0$.}
\vskip.1in {\bf Proof.} Let $\xi$ be the positive unit normal of $P_0$ in $T_{p_{0}}S^3$ and $\alpha$ the slant angle of $P_0$ with respect to $J_{1}^-$. Put $$a = \cos \alpha \,\,\,(\not= 0, \pm1), \,\,\,\, b = \pm(1-a^{2})^{{1}\over {2}},\leqno(2.44)$$
\noindent where $\pm$ was chosen so that $ab < 0.$ Pick $s_{0} \in [0,2\pi)$ such that $$\cos s_{0} = -{{1}\over {b}}<\xi,{\tilde X}_{2}(p_{0})>,\,\,\,\, \sin s_{0} = -{{1}\over {b}}<\xi,{\tilde X}_{3}(p_{0})>.$$
\noindent We define $f_i$ by (2.29)-(2.52). Then they satisfy (2.27) and we can choose a curve $c(s)$ satisfying the conditions mentioned in Lemma 2.6.
Let $\gamma(t)$ be either a geodesic in $S^3$ safisfying $$\gamma(0) = p_{0},\,\,\,\gamma'(0) \in P_{0},\,\,\,\gamma'(0) \not= c'(0),$$ or a curve in $S^3$ satisfying this condition and also the condition $\tau \equiv 1$ (see Theorem 3 of [Sp1, p.35] for the existence of such curves). Then we can verify that $\{\gamma(t)\cdot c(s)\}$ is a desired surface. \vskip.1in {\bf Proof of Theorem 2.5.} First, we note that the isometry $\,\phi\,$ of $\, S^3$ has the following properties: $$\phi(p\cdot q) = \phi(q)\cdot\phi(p), \,\,\,\, {\rm for}\,\,\,\forall p, q \in S^{3},\leqno(2.45)$$ \begin{align}&X \in {\mathcal X}(S^{3}) \,\,\,\,{\rm is \,\, left{\rm -}\, (repsectively,\, right{\rm -})\,invariant}\tag{2.46}\\ &\Longleftrightarrow \phi_{*}X \,\,\, \rm is\,\, right{\rm -} \, (respectively,\, left{\rm -})\, invariant,\notag\end{align} $$\tau_{\phi\circ c} = -\tau_{c} \,\,\,{\rm for\, a\, curve}\, c\,\, {\rm in}\, S^{3},\leqno(2.47)$$ $${\bf b}\,\, {\rm is\, the\, binormal\, of\, a \, curve\,}\, c\, {\rm in}\, S^{3} \Longleftrightarrow \leqno(2.48)$$ $$\hskip.9in -\phi_{*}{\bf b}\, {\rm is\, the\, binormal\,of }\,\, \phi\circ c \,\, {\rm in}\, S^{3},$$
\noindent where $\tau_c$ denotes the torsion of the curve $\, c \,$ in $\, S^3$.
Let $\alpha$ be the slant angle of $f(N)$ with respect to $J_1$. Since $f(N)$ is spherical, the normal curvature $G^D$ of the slant immersion $f$ vanishes. Thus, by Theorem 3.2 in Section 3 of Chapter II, $N$ is a flat surface in $S^3$. Therefore, $f(N)$ is locally a flat translation surface $f(N) = \{c(s)\cdot\gamma(t)\}$ (cf. [Sp1, pp.149-157]), where $c$ and $\gamma$ are curves in $S^3$ parametrized by arclength satisfying one of the following conditions: $$\tau_{c} \equiv +1 \,\,\,\,{\rm and}\,\,\,\,\tau_{\gamma} \equiv -1,\leqno(i)$$ $$\tau_{c} \equiv +1\,\,\,\,{\rm and}\,\,\gamma\,\,{\rm is\,\, a\,\, geodesic,}\leqno(ii)$$ $$c \,\,{\rm is\,\, a \,\, geodesic\,\, and}\,\, \,\,\tau_{\gamma} \equiv -1,\leqno(ii')$$ $$c\,\,\,{\rm and}\,\,\,\gamma \,\,\,{\rm are\,\, distinct\,\, geodesics.}\leqno(iii)$$ \vskip.1in
{\bf Cases (i) and (ii):} Let {\bf b} be the binormal of $c$. With a suitable choice of the orientation, {\bf b} is the positive unit normal of $f(N)$ in $S^3$. By Lemma 2.3, Lemma 2.7, (2.47), and (2.48), $\phi\circ c$ is a helix in $S^3$ with $a$ and $b$ in (2.29)-(2.32) determined by $$a = \cos\alpha,\,\,\,\,\, b = \pm\sin\alpha,\,\,\, ab < 0,\leqno(2.49)$$ where either $\tau_{\phi\circ\gamma} \equiv +1$ or $\phi\circ\gamma$ is a geodesic. So, by (2.45), $(\phi\circ f)(N)$ is a helical cylinder in $S^3$.
The converse is given by Lemma 2.8. Moreover, Lemma 2.9 guarantees the existence of such surfaces.
\vskip.1in Next, we want to show that both cases (ii$'$) and (iii) cannot occur. Without loss of generality we can assume $$c(0) = \gamma(0) = 1 \in S^{3},\leqno(2.50)$$ because Lemmas 3.1 and 3.3 imply that the slantness of a surface in $S^3$ with respect to $J_1$ is right-invariant, that is, if $f$ is $\alpha$-slant with respect to $J_1$, so is $R_{q}\circ f$ for any $q \in S^3$, and hence we can replace $f, c$ and $\gamma$ by $R_{c(0)\cdot\gamma(0)}\circ f, R_{c(0)}\circ c$ and $L_{c(0)}\circ R_{\gamma(0)^{-1}\cdot c(0)^{-1}}\circ\gamma$, respectively, if necessary. \vskip.1in
{\bf Case (ii$'$):} Let {\bf b} be the binormal of $\gamma$. We can choose the orientation so that $$\xi(c(s)\cdot\gamma(t)) = L_{c(s)^{*}}{\bf b}(t),\,\,\,\,\,\,{\rm for\,\,any\,\,} s, t.\leqno(2.51)$$ So, by Lemmas 2.1 and 2.3 and also(2.50), we have $$<L_{c(s)^{*}}{\bf b}(0), R_{c(s)^{*}}X_{1}> = -\cos\alpha\,\,\,\,\,{\rm for} \,\,{\rm any\,\,\,} s.\leqno(2.52)$$
Put $$c'(0) = (0,a_{1},a_{2},a_{3}),\,\,\, {\bf b}(0) = (0,b_{1},b_{2},b_{3}) \in T_{1}S^{3} \subset E^{4}.\leqno(2.53)$$ Then $$c(s) = (\cos s, a_{1}\sin s, a_{2}\sin s, a_{3}\sin s).\leqno(2.54)$$
Putting $s = 0$ in (2.52), we find $$b_{1} = -\cos\alpha \not= 0, \pm1,\leqno(2.55)$$
\noindent since $f(N)$ is properly slant. On the other hand, by (2.4), (3.5), (2.45), and (2.46), we have $$<L_{c(s)^{*}}{\bf b}(0), R_{c(s)^{*}}X_{1}> = b_{1}\cos 2s + (-a_{3}b_{2} + a_{2}b_{3})\sin 2s.\leqno(2.56)$$
\noindent So, from (2.52) and (2.56), we get $b_{1} = 0$ which contradicts to (2.55). Consequently, case (ii$'$) cannot occur.
\vskip.1in {\bf Case (iii):} Let $\{c(s)\cdot\gamma(t)\}$ be defined by using two distinct geodesics $c$ and $\gamma$ and assume $$f : I_{1}\times I_{2} \rightarrow S^{3}\,\, ;\,\, (s,t) \mapsto c(s)\cdot\gamma(t)\leqno(2.57)$$
\noindent is properly slant. Since the geodesics $c$ and $\gamma$ can be extended for all $\, s, t \in {\bf R}$, we can extend the immersion $x$ to a global mapping: $$y : {\bf R}^{2} \rightarrow S^{3}\,\, ;\,\, (s,t) \mapsto c(s)\cdot\gamma(t).\leqno(2.58)$$
Now, we claim that $y$ is in fact an immersion and it is properly slant. To see this, we recall (2.50) and put
$$c'(0) = (0,a_{1},a_{2},a_{3}),\,\,\,\gamma'(0) = (0,b_{1},b_{2},b_{3}) \in T_{1}S^{3}.\leqno(2.59)$$
Let ${\tilde X}, {\tilde Y}$ be the vector fields along $y({\bf R}^{2})$ defined by $${\tilde X}(s,t) = R_{\gamma(t)^{*}}c'(s),\,\,\,{\tilde Y}(s,t) = L_{c(s)^{*}}\gamma'(t).\leqno(2.60)$$
\noindent Then it follows from (2.4) and (2.5) that $$<{\tilde X}(s,t),{\tilde Y}(s,t)> \,\, {\rm is\,\, a\,\, polynomial \,\, of}\,\, \sin s, \, \cos s,\, \sin t\,\, {\rm and}\, \cos t.\leqno(2.61)$$
On the other hand, since the $s$-curves and the $t$-curves on $f(I_{1}\times I_{2})$ intersect at a constant angle (cf. [Sp1, p.157]), we have
$$<{\tilde X}(s,t),{\tilde Y}(s,t)> = const. \not= \pm1,\,\,\,\, {\rm for \,\,any\,} \,(s,t) \in I_{1}\times I_{2}.\leqno(2.62)$$
\noindent From (2.61), we see that (2.62) holds for all $(s,t) \in {\bf R}^2$ and hence $y$ is an immersion. Since $$\xi(c(s)\cdot\gamma(t)) =
\|{\tilde X}(s,t)\times {\tilde Y}(s,t)\|^{-1}({\tilde X}(s,t)\times{\tilde Y}(s,t)),$$
\noindent where $\times$ denotes the usually vector product in $\, T_{c(s)\cdot\gamma(t)}S^{3} \,$ determined by the metric and the orientation, so, by (2.4), (2.5) and (2.6), we know that $<\xi(c(s)\cdot\gamma(t)),J_{0}\eta(c(s)\cdot\gamma(t))>$ is a polynomial of $\,\sin s, \cos s, \sin t\,$ and $\, \cos t.$ By Lemma 2.3 we conclude that this polynomial is a constant on $I_{1}\times I_2$ and hence $y$ is a proper slant immersion defined globally on ${\bf R}^2$. Now, by the double periodicity, $y$ induces a proper slant immersion: $${\tilde y} : T^{2} = ({\bf R}/2\pi {\bf Z})\times({\bf R}/2\pi {\bf Z}) \rightarrow {\bf C}^{2} = (E^{4}, J_{0})$$
\noindent of a torus into ${\bf C}^2$, which contradicts to Theorem 1.5 of Chapter V. Consequently, case (iii) cannot occur.
This completes the proof of the theorem.
\eject \noindent{} \vskip.3in \noindent \S 3. SLANT SURFACES WITH $rk(\nu)<2.$ \vskip.2in For an immersion $f:N \rightarrow {\bf C}^{m}$, the Gauss map $\nu$ of the immersion $f$ is given by $$\nu:N\rightarrow G(n,2m)\equiv D_{1}(n,2m) \subset S^{K-1} \subset \wedge^{n}(E^{2m}),\leqno(3.1)$$ $$ \nu (p) = e_{1}(p)\wedge\ldots\wedge {e_{\ell}} (p),\,\,\, p \in N,$$
\noindent where $n = dim\, N,\,K={{2m}\choose {n}},\, D_{1}(n,m)\,$ is the set of all unit decomposable $\,n$-vectors in $\wedge^{n}E^{2m}$, identified with the real Grassmannian $G(n,2m)$ in a natural way, and $ S^{K-1}$ is the unit hypersphere of $\wedge^{n}(E^{2m})$ centered at the origin, and $\{e_{1},\ldots,e_{2m}\}$ is a local adapted orthonormal tangent frame along $f(N)$.
The main purpose of this section is to prove the following classification theorem [CT2].
\vskip.1in {\bf Theorem 3.1.} {\it If $f : N \rightarrow {\bf C}^{2} =(E^{4},J_{1})$ is a slant immersion such that the rank of its Gauss map is less than 2, then the image $f(N)$ of $f$ is a union of some flat ruled surfaces in $E^4$. Therefore, locally, $f(N)$ is a cylinder, a cone or a tangential developable surface in ${\bf C}^2$. Furthermore,}
(i) {\it A cylinder in {\bf C}$^2$ is a slant surface if and only if it is of the form $\{ c(s)+te\},$ where $e$ is a fixed unit vector and $c(s)$ is a (Euclidean) generalized helix with axis $J_{1}e$ contained in a hyperplane of $E^4$ and with $e$ as its hyperplane normal,}
(ii) {\it A cone in {\bf C}$^2$ is a slant surface if and only if, up to translations, it is of the form $\{tc(s)\},$ where $(\phi\circ c)(s)$ is a generalized helix in $S^3$ with axis ${\tilde X}_1$ (cf. Definition 2.1), and}
(iii) {\it A tangential developable surface $\{ c(s)+(t-s)c'(s)\}$ in {\bf C}$^2$ is a slant surface if and only if, up to ridid motions, $(\phi\circ c')(s)$ is a generalized helix in $S^3$ with axis ${\tilde X}_1$.}
\vskip.1in
As before, let $*$ denote the Hodge star operator $* : \wedge^{2} E^{4} \rightarrow \wedge^{2} E^4$ induced from the natural orientation and the canonical inner product of $E^4$. Denote by $\wedge_{+}^{2}E^4$ and $\wedge_{-}^{2}E^4$ the eigenspaces of $*$ with eigenvalues $+1$ and $-1$, respectively. And denote by $S_{+}^2$ and $S_{-}^2$ the 2-spheres in $\wedge_{+}^{2}E^4$ and $\wedge_{-}^{2}E^4$ centered at the origin with radius $1/{\sqrt{2}}$, respectively. Then we have $D_{1}(2,4) = S_{+}^{2}\times S_{-}^2$. Let
$$\pi_{+} : D_{1}(2,4) \rightarrow S_{+}^{2},\,\,\,\,\,\, \pi_{-} : D_{1}(2,4) \rightarrow S_{-}^{2}\leqno(3.2)$$
\noindent denote the natural projections. We define as before the two maps $\nu_+$ and $\nu_-$ given respectively by
$$\nu_{+} = \pi_{+}\circ\nu \,\,\,\,\,{\rm and}\,\,\,\,\, \nu_{-} =\pi_{-}\circ\nu.\leqno(3.3)$$
Suppose that the slant immersion $f:N \rightarrow {\bf C}^{2} = (E^{4},J_{1})$ satisfying the condition $rank(\nu) < 2.$ Then we also have $rank(\nu_{\pm}) < 2$. Hence, by Lemma 4.2 of Chapter III, $f(N)$ is a flat surface in $E^4$.
Furthermore, we have [CT2]
\vskip.1in {\bf Lemma 3.2.} {\it If $f$ is a slant immersion with $rank(\nu) < 2$, then $f(N)$ is a union of flat ruled surfaces in $E^4$.} \vskip.1in
{\bf Proof.} Since the normal curvature $R^{D}= 0$, identically, we can choose local orthonormal frame $\{e_{1},e_{2}\}$ such that with respect to it the second fundamental form $\{h_{ij}^{r}\}$ is simultaneously diagonalized, that is, we have
$$(h_{ij}^{3}) = \begin{pmatrix}b&0\\ 0&c\end{pmatrix}, \,\,\,\, (h_{ij}^{4}) = \begin{pmatrix}d&0\\ 0&e\end{pmatrix}.\leqno(3.4)$$
Put $$N_{1} = \{\, p\in N\,\mid\, H(p) \not= 0 \,\},\,\,N_{0} = {\rm Interior \, of}\, (N-N_{1}).\leqno(3.5)$$
\noindent Then $$N = N_{0}\cup \partial N_{1}\cup N_{1},$$ where $H$ is the mean curvature vector of $N$ in ${\bf C}^2$.
Since $f(N)$ is flat and $H = 0$ on $N_{0}$, $f(N_{0})$ is a union of portions of 2-planes in $E^4$ with the same slant angle.
On $N_{1}$, we put $e_{3}=H/\|H\|$. Since $rank(\nu ) < 2$, we have $bc = 0$ and $d = e=0$. We may choose $\{e_{1},e_{2}\}$ such that $b \not= 0, c=d=e=0$ on $N_1$. From these we may prove that the integral curves of $f_{*}e_2$ are open portions of straight lines and therefore $f(N_{1})$ is a union of flat ruled surfaces. Consequently, $f(N)$ is a union of flat ruled surfaces possibly guled along $\partial N_1$. This proves the lemma.
\vskip.1in For the local classifcation of flat ruled surfaces in $E^4$, see
[Sp1].
Now we give the proof of Theorem 3.1.
\vskip.1in {\bf Proof of Theorem 3.1.} The first part of the theorem is giving in Lemma 3.2. Now, we prove the remaining part of the theorem. \vskip.1in
{\bf Case (i):} If $f(N)$ is a slant cylinder, then we may assume that $f(N)$ is of the form: $$f(N) = \{c(s) + te \},\leqno(3.6)$$
\noindent where $e$ is a fixed unit vector in $E^4$ and $c(s)$ is a curve parametrized by arclength which lies in the orthogonal complement (up to sign), say $W \in G(3,4)$, of $e$. Since $\{c'(s),e\}$ is a positive orthonormal basis of $T_{c(s)+te}N,$ we obtain $ \,\cos\alpha = <c'(s),-J_{1}e>$ by (1.4). Hence, $c(s)$ is a generalized helix lies in the hyperplane $W (\equiv E^{3})$ whose tangents make a constant angle $\alpha$ with $-J_{1}e\in W$. \vskip.1in
{\bf Case (ii):} If $f(N)$ is a slant cone in $E^4$, then, without loss of generality, we may assume that the vertex of the cone is the origin of $E^4$. So we can write $$f(N) = \{tc(s)\},\leqno(3.7)$$
\noindent where $c(s)$ is a curve in $S^3$ parametrized by arclength. Since $\{c'(s), \eta(c(s))\}$ is a positive orthonormal basis of $T_{tc(s)}N$, $\,\cos\alpha = <c'(s),-J_{1}\eta(c(s))>$ for all s. Thus, by Lemmas 2.1 and 2.3, we conclude that $(\phi\circ c)(s)$ is a generalized helix in $S^3$ with axis ${\tilde X}_1$ (cf. Definition 2.1). \vskip.1in {\bf Case (iii):} If $f(N)$ is a slant tangential developable surface in $E^4$, the surface has the form: $$f(N) = \{c(s)+(t-s)c'(s)\},\leqno(3.8)$$
\noindent where $c(s)$ is a curve parametrized by arclength. We put
$${\bf v}_{1}(s) = c'(s),\,\,\,\kappa_{1}(s) = \|{\bf v}'_{1}(s)\|,\,\,\, {\bf v}_{2}(s) = ({{1}\over {\kappa_{1}(s)}}){\bf v}'_{1}(s).\leqno(3.9)$$
\noindent Note that $\kappa_{1} \not= 0$, since $c(s)$ generates a tangential developable surface. $\{{\bf v}_{2}(s) ,{\bf v}_{1}(s)\}$ forms a positive orthonormal basis of $T_{c(s)+(t-s)c'(s)}N$, and so we have
$$\cos\alpha = <{\bf v}'_{1}(s)/\|{\bf v}'_{1}(s)\|, -J_{1}{\bf v}_{1}(s)>$$
\noindent for all $s$. If we consider ${\bf v}_{1}(s)$ as a curve in $S^3$, then (3.9) means that $$\cos\alpha = <{\bf t}(s), -J_{1}\eta({\bf v}_{1}(s))>$$
\noindent where ${\bf t}(s)$ is the unit tangent of ${\bf v}_{1}(s)$. So, as in Case (ii), $(\phi\circ {\bf v}_{1})(s)$ is a generalized helix in $S^3$ with axis
${\tilde X}_1$.
It is easy to verify that in each of the cases (i)-(iii), the converse is also true.
This completes the proof of the theorem.
\eject \noindent{}
\vskip.3in \noindent \S 4. SLANT SURFACES WITH CODIMENSION ONE
\vskip.2in In this section we want to classify slant surfaces which are contained in a hyperplane $W$ of $E^4$.
\vskip.1in {\bf Lemma 4.1.} {\it Let $f:N \rightarrow {\bf C}^{2} = (E^{4},J_{1})$ be a slant immersion of an oriented surface $N$ into ${\bf C}^2$. If $N$ is contained in some hyperplane $W\in G(3,4)$, then }
(1) {\it $rank(\nu) < 2$ and}
(2) {\it The immersion $f$ is doubly slant with the same slant angle.}
\vskip.1in {\bf Proof.} We choose a positive orthonormal $J_1$-basis $\{e_{1},e_{2},e_{3},e_{4}\}$ such that $e_{1}, e_{2} \in W\cap J_{1}W, \,\, e_{4}=J_{1}e_{3} = \eta_W$, where $\eta_W$ is the positive unit normal vector of the hyperplane $W$ in $E^4$. We put $$G_{W} = G(2,4)\cap \wedge^{2}W \subset \wedge^{2}E^{4}.\leqno(4.1)$$
\noindent Then $G_W$ is the unit 2-sphere in the 3-dimensional Euclidean space $\wedge^{2}W$.
For $\alpha \in [0,\pi]$ we put $$G_{W,\alpha} = G_{J_{1},\alpha}\cap G_{W},\leqno(4.2)$$
\noindent where $G_{J_{1},\alpha}$ is the set of all 2-planes in $E^4$ with slant angle $\alpha$ with respect to $J_1$. We recall that a 2-plane $V$ was identified with a unit decomposable 2-vector $e_{1}\wedge e_{2}$ in $\wedge^{2}E^4$ with $\{e_{1}, e_{2}\}$ as a positively oriented orthonormal basis of $V$. From the proof of Lemma 2.2, we see that $G_{W,\alpha}$ is the circle on $G_{W} = S^{2} \subset \wedge^{2}W$ defined by
$$G_{W,\alpha} = \{V \in G_{W}\,\mid\, <V,e_{1}\wedge e_{2}> = \cos\alpha\}.$$
For each $J \in {\mathcal J}$, we denote by $\zeta_J$ the 2-vector which is the metrical dual of $-\Omega_J$ as defined in Section 2. Let $\zeta : {\mathcal J} \rightarrow \wedge^{2}E^4$ be the mapping defined by $\zeta(J) = \zeta_J$. Then $\zeta$ gives rise to two bijections (cf. Lemma 2.1 of this chapter):
$$\zeta^{+} : {\mathcal J}^{+} \rightarrow S_{+}^{2}\,\,\,\,\,{\rm and}\,\,\,\,\,\zeta^{-} ; {\mathcal J}^{-} \rightarrow S_{-}^{2}.$$
For each oriented 2-plane $V \in G(2,4)$ we define two complex structures $J_{V}^{+} \in {\mathcal J}^+$ and $J_{V}^{-} \in {\mathcal J}^-$ by
$$J_{V}^{+} = (\zeta^{+})^{-1}(\pi_{+}(V))\,\,\,\,\,{\rm and}\,\,\,\,J_{V}^{-}=(\zeta^{-})^{-1}(\pi_{-}(V)).$$
Let ${\hat J} = J_{e_{1}\wedge e_{2}}^{-}$. Then we have $$\pi_{+}(G_{W,\alpha}) = S_{{\hat J},\alpha}^{+} \subset S_{+}^{2},\,\,\,\,\pi_{-}(G_{W,\alpha}) = S_{{\hat J},\alpha}^{-} \subset S_{-}^{2},\leqno(4.3)$$
\noindent where $S_{J,\alpha}^{\pm}$ are the circles (possibly singletons) on $S_{\pm}^2$, respectively, consisting of all 2-vectors which make constant angle $\alpha$ with $\zeta_{J}$. If $f$ is $\alpha$-slant with respect to ${\hat J}$ and $f(N) \subset W$, then $\nu(M) \subset G_{W,\alpha}$. Therefore, $rank(\nu) < 2$ and, by (4.3), $f$ is $\alpha$-slant with respect to ${\hat J}$.
This proves the lemma. \vskip.1in We note here that if we identify $\wedge^{2}W$ with the Euclidean 3-space $E^{3} \equiv W$ (where $W$ is spanned by $\{e_{1},e_{2},e_{3}\}$) via the isometry $X\wedge Y \rightarrow X\times Y$, then $\nu:M \rightarrow G_{W} \subset \wedge^{2}W$ is nothing but the classical Gauss map $g:M \rightarrow S^{2} \subset E^3$.
Since $e_{1}\times e_{2} = e_{3} = -J_{1}\eta_{W},$ $f$ is $\alpha$-slant if and only if $$g(M) \subset S_{\alpha}^{1} = \{Z \in S^{2}\,\mid\, <Z,-J_{1}\eta_{W}> = \cos\alpha\} \subset S^{2}\subset W.\leqno(4.4)$$ \vskip.1in
Now we give the following classification theorem [CT2]. \vskip.1in {\bf Theorem 4.2.} {\it Let $f:N \rightarrow {\bf C}{^2} = (E^{4},J_{1})$ be a proper slant immersion of an oriented surface $M$ into {\bf C}$^2$. If $f(N)$ is contained in a hyperplane $W$ of $E^4$, then $f$ is a doubly slant immersion and $f(N)$ is a union of some flat ruled surfaces in $W$.
Therefore, locally, $f(N)$ is a cylinder, a cone or a tangential developable surface in $W$. Furthermore,}
(i) {\it A cylinder in $W$ is a proper slant surface with respect to the complex structure $J_1$ on $E^4$ if and only if it is a portion of a 2-plane.}
(ii) {\it A cone in $W$ is a proper slant surface with respect to the complex structure $J_1$ on $E^4$ if and only if it is a circular cone.}
(iii) {\it A tangential developable surface in $W$ is a proper slant surface with respect to the complex structure $J_1$ on $E^4$ if and only if it is a tangential developable surface obtained from a generalized helix in $W$.}
\vskip.1in
{\bf Proof.} Assume $f:N \rightarrow {\bf C}^{2} = (E^{4},J_{1})$ is a proper slant immersion of an oriented surface $N$ such that $f(N)$ is contained in some hyperplane $W\in G(3,4)$. The first part of Theorem 4.2 is given by Lemma 4.1. For the remaining part it suffices to check the three cases of Theorem 3.1.
Suppose $f$ is properly slant with slant angle $\alpha$. Denote by $\xi$ the local unit normal of $f(N)$ in $W$. We put
$$e_{1} = t\xi / \|t\xi\|,\,\,\,\,e_{2} = (\sec\alpha)Pe_{1},\,\,\,e_{3} = (\csc\alpha)Fe_{1},\,\,\,e_{4} = (\csc\alpha)Fe_{2},\leqno(4.5)$$
\noindent where $PX$ and $FX$ denote the tangential and the normal components of $J_{1}X$, respectively, and $t\xi$ is the tangential component of $J_{1}\xi$. Then $\{e_{1},\cdots,e_{4}\}$ is an adapted orthonormal frame along $f(N)$ and it satisfies
$$e_{3} = \,{\rm unit\,\, normal\,\, of}\, f(N)\,\,{\rm in}\,\, W, \,\,\,e_{4} \in W^{\bot},\leqno(4.6)$$ $$te_{3} = -(\sin\alpha)e_{1},\,\,\,te_{4}=-(\sin\alpha)e_{2}, \leqno(4.7)$$ $$fe_{3}=-(\cos\alpha)e_{4},\,\,\,fe_{4}=(\cos\alpha)e_{3}, $$
\noindent where $fe_3$ is the normal component of $J_{1}e_3$. Since $e_4$ is a constant vector in $E^4$, Corollary 3.6 of Chapter II implies that the second fundamental form $(h_{ij}^{r})$ is of the following form:
$$(h_{ij}^{3}) = \begin{pmatrix}b&0\\ 0&0\end{pmatrix},\,\,\,\, (h_{ij}^{4}) = 0,\leqno(4.8)$$
\noindent which shows that the orthonormal frame $\{e_{1},\cdots,e_{4}\}$ coincides with that chosen in the proof of Lemma 3.2 (up to orientations). Since $J_{1}e_4$ is also a constant vector in $E^4$, from (4.7), we have $$ -\sin\alpha \nabla_{X}e_{2} - \cos\alpha A_{e_{3}}X = 0,\,\,\,{\rm for}\,\, X\in TM.\leqno(4.9)$$
\noindent Hence we get $$\omega_{2}^{1}(e_{1}) = - b\cot\alpha,\,\,\,e_{2}b = b^{2}\cot\alpha.\leqno(4.10)$$
\vskip.1in {\bf Case (i):} In this case, the curve $c(s)$ of (3.6) lies in a 2-plane $W' = \{e\}^{\bot}\cap W \subset W$ which is perpendicular to $e$. So, $f(N)$ is totally real with respect to the complex structures $\pm J_{W'}^{\pm}$ defined above. If $\nu_{+}(N)$ is not a singleton, then $J_1$ is one of the complex structures $\pm J_{W'}^{\pm}$ according to Propostion 3.1 and Theorem 3.2 of Chapter III. Hence we get $\alpha = \pi/2$, which contradicts to the assumption. So, $\nu_{+}(N) $ is a singleton and hence $f(N)$ is minimal (cf. Theorem 3.2 of Chapter III). Thus, by (4.8), $f(N)$ is an open portion of an $\alpha$-slant 2-plane. \vskip.1in
In Cases (ii) and (iii), we may assume $N =N_1$
{\bf Case (ii):} In this case the curve $c(s)$ in (3.7) lies in the unit 2-sphere $S^{2} = S^{3}\cap W$. Choose $\{e_{1},\cdots,e_{4}\}$ according to (4.5) and let ${\bf t},{\bf n},{\bf b},\kappa$, and $\tau$ be the unit tangent vector, unit principal normal vector, the unit binormal vector, the curvature, and the torsion of $c(s)$ in $W = E^3$, respectively. We want to show that $\tau \equiv 0$.
Since $$e_{1}(s,t) = {\bf t}(s) = {{1}\over {t}}{{\partial}\over {\partial s}},\,\, e_{2}(s,t) =c(s) = {{\partial}\over {\partial t}},\leqno(4.11)$$ $$\,\,e_{3}(s,t) = e_{1}(s,t)\times e_{2}(s,t),$$
\noindent where $\times$ denotes the vector product in $W$, we have
$$b = -({{\kappa}\over {t}})<{\bf b},c>.\leqno(4.12)$$
\noindent From $\|c\| = 1$, we get $$\kappa <{\bf n},c> = -1.\leqno(4.13)$$
\noindent Differentiating (4.13) with respect to $s$, we get $$\kappa^{2}\tau<{\bf b},c> = \kappa'.\leqno(4.14)$$
\noindent From (4.12) we obtain $$-t\tau \kappa b = \kappa'.\leqno(4.15)$$
\noindent Differentiating (4.15) with respect to $t$ and using (4.10) and (4.15), we obtain $$\kappa'(\tau\kappa\tan\alpha - \kappa') = 0.\leqno(4.16)$$
\noindent By (4.12), (4.14) and $<{\bf t},c>=0$, we find $$\kappa^{2}\tau c = -\kappa\tau{\bf n} + \kappa'{\bf b}.\leqno(6.17)$$
\noindent Since $\|c\|$ = 1, we also get $$\tau^{2}\kappa^{4}=\tau^{2}\kappa^{2} + (\kappa')^{2}.\leqno(4.18)$$
If $\kappa'(s_{0})=0$ at a point $s=s_0$, then, by (4.15), we have $\tau(s_{0})=0,$ since $b(s,t) \not= 0$ by assumption and also $\kappa(s) \not= 0$ because $c(s)$ is spherical.
If $\kappa'(s_{0})\not= 0$, we choose a neighborhood $U$ of $s_0$ on which $\kappa'$ never vanishes. By (4.16), (4.18) and $\kappa \not= 0$, we get $$(\tau(s))^{2}\{(\kappa(s))^{2}-1-\tan^{2}\alpha\} = 0 \,\,\,\,\,{\rm for}\,\,\forall s \in U.\leqno(4.19)$$
If $\tau(s_{0})\not= 0$ in addition, we choose another neighborhood $U'$ of $s_0$ contained in $U$ on which $\tau$ never vanishes. Then, by (4.19), we get $$(\kappa(s))^{2}-1-\tan^{2}\alpha = 0$$
\noindent for all s in $U'$. By continuity we get $\kappa(s)=constant$ on $U'$ which contradicts to $\kappa'(s) \not= 0$ on $U'$. So, again we have $\tau(s_{0})=0.$ Therefore, $\tau \equiv 0,$ which means that $c(s)$ is a circle on $S^2$ and thus $f(N)$ is a circular cone. According to the remark after
Lemma 6.1, the axis of the cone is given by $-J_{1}e_4$. \vskip.1in
{\bf Case (iii):} We assume the surface is given by (3.8) and $\{e_{1},\ldots\,e_{4}\},$ $ {\bf t},$ ${\bf n}, {\bf b}, \kappa$ and $\tau$ are given as in Case (ii). We have $$e_{1}(s,t)={\bf n}(s)={{1}\over {(t-s)\kappa}}{{\partial}\over {\partial s}},\,\, e_{2}(s,t) = {\bf t}(s) = {{\partial}\over {\partial t}},\leqno(4.20)$$ $$\,\,e_{3}(s,t)=e_{1}\times e_{2} = -{\bf b}(s).$$
\noindent Hence $${\tilde {\nabla}}_{e_{1}}e_{1} = -{{1}\over {(t-s)}}e_{2}-{{\tau}\over {(t-s)\kappa}}e_{3}.\leqno(4.21)$$ So, by (4.8), we find $$b=-{{\tau}\over {(t-s)\kappa}}.\leqno(4.22)$$
By (4.20) and (4.22), we obtain $$e_{2}b={{\tau}\over {\kappa (t-s)^{2}}}.$$ This formula together with (4.10) and (4.21) imply that ${{\tau}\over {\kappa}} =\tan\alpha$ is a constant. This means that the curve $c(s)$ is a generalized helix in $W$.
The axis of the helix is given by $-J_{1}e_4$.
In each of the cases (i), (ii) and (iii), the converse is easy to verify. For example, if $f(N)$ is a circular cone with the axis vector $e$ in a 3-plane $W$ perpendicular to a unit vector $\eta$ in $E^4$, then, by picking the complex structure $J$ so that $J = J_{e \wedge \eta }^+$, $f(N)$ is properly slant with respect to $J$.
This completes the proof of the theorem.
\vskip.1in {\bf Remark 4.1.} In the classifications of slant surfaces given in Sections 2, 3 and 4 of this chapter, we avoid the messy argument of glueing.
\eject \vskip1in \centerline {CHAPTER V} \vskip.2in \centerline {\bf TOPOLOGY AND STABILITY OF SLANT SUBMANIFOLDS} \vskip.5in
\noindent \S 1. NON-COMPACTNESS OF PROPER SLANT SUBMANIFOLDS. \vskip.2in Let $E^{2m}=(R^{2m},<\,,\,>)$ and {\bf C}$^{m} = (E^{2m},J_{0})$ be the Euclidean 2m-space and the complex Euclidean m-space, respectively, with the canonical inner product $<\,,\,>$ and the canonical (almost) complex structure $J_{0}$ given by $$J_{0}(x_{1},\ldots,x_{m},y_{1},\ldots,y_{m})=(-y_{1}, \ldots,-y_{m},x_{1},\ldots,x_{m}). \leqno(1.1)$$
Denote by $\Omega_{0}$ the Kaehler form of {\bf C}$^m$, that is, $$\Omega_{0}(X,Y)=<X,J_{0}Y>, \hskip.2in X,Y \in E^{2m}, \hskip.2in \Omega_{0} \in \wedge^{2}(E^{2m})^*.\leqno(1.2)$$
For an immersion $f:N \rightarrow {\bf C}^{m}$, the Gauss map $\nu$ of the immersion $f$ is given by $$\nu:N\rightarrow G(n,2m)\equiv D_{1}(n,2m) \subset S^{K-1} \subset \wedge^{n}(E^{2m}),\leqno(1.3)$$ $$ \nu (p) = e_{1}(p)\wedge\ldots\wedge {e_{n}} (p),\,\,\, p \in N,$$ \vskip.1in \noindent where $n = dim N,\,K={{2m}\choose {n}},\, D_{1}(n,2m)\,$ is the set of all unit decomposable $\,n$-vectors in $\wedge^{n}E^{2m}$, identified with the real Grassmannian $G(n,2m)$ in a natural way, and $ S^{K-1}$ is the unit hypersphere of $\wedge^{n}(E^{2m})$ centered at the origin, and $\{e_{1},\ldots,e_{2m}\}$ is a local adapted orthonormal tangent frame along $f(N)$.
\vskip.1in Before we give the main result of this section, we give the following lemmas. \vskip.1in {\bf Lemma 1.1.} {\it For $X_{1},\ldots,X_{2k} \in E^{2m}\,(k < m)$, we have $$(2k)!\,\Omega_{0}^{k}(X_{1}\wedge\ldots\wedge X_{2k})=\leqno(1.4)$$ $$\sum_{\sigma\in S_{2k}} sign(\sigma) \Omega_{0}(X_{\sigma(1)},X_{\sigma(2)})\cdots\Omega_{0} (X_{\sigma(2k-1)},X_{\sigma(2k)}),$$ where $S_{2k}$ is the permutation group of order $2k$, sign denotes the signature
\noindent of permutations and} $\Omega_{0}^{k} \in \wedge^{2k}(E^{2m})^{*} \equiv (\wedge^{2k}E^{2m})^{*}.$ \vskip.1in {\bf Proof.} Let $e_{1},\ldots,e_m$ be an orthonormal frame of $E^{2m}$ with its dual coframe given by $\omega^{1},\ldots,\omega^{2m}$. Let $$\Omega_{0}=\sum_{A,B=1}^{2m}\,\,\varphi_{AB}\omega^{A}\wedge \omega^{B}.$$ Then by direct computation we have $$\Omega_{0}^{k}(X_{1},\ldots,X_{2k}) $$ $$={1\over {(2k)!}}\sum_{\sigma}sign \,(\sigma)(\sum\varphi_{_{A_{1}A_{2}}}\omega^{A_{1}} (X_{\sigma(1)})\omega^{A_{2}}(X_{\sigma(2)})\dots$$ $$\ldots (\sum\varphi_{_{A_{2k-1}A_{2k}}}\omega^{A_{2k-1}} (X_{\sigma(2k -1)})\omega^{A_{2k}}(X_{\sigma(2k)})).$$
\noindent From these we obtain (1.4). \vskip.1in
{\bf Lemma 1.2.} {\it Let $V \in G(n,2m)$ and $\pi_{V} : E^{2m} \rightarrow V$ be the orthogonal projection. If $V$ is $\alpha$-slant, that is, $V$ is slant with slant angle $\alpha \not= \pi/2$,
in {\bf C}$^{m} = (E^{2m},J_{0})$, then the linear endomorphism $J_V$ of $V$ defined by $$J_{V}= (\sec \alpha)(\pi_{V}\circ J_{0}{}_{\mid}{} _{V})\leqno(1.5)$$ is a complex structure compatible with the inner product
$<\,\,,\,\,>_{\,|\, V}$. In particular, $n$ is even.} \vskip.1in
{\bf Proof.} Let $$P=\pi_{V}\circ (J_{\,| V}): V \rightarrow V,\leqno(1.6)$$
$$P^{\perp}=J_{\,| V}-P : V \rightarrow V^{\perp}\leqno(1.7)$$ and $$Q =P^{2} : V \rightarrow V.\leqno(1.8)$$ Then $$J_{\, | V}=P+P^{\perp}.\leqno(1.9)$$
\noindent By simple computation and using (1.7), we have $$<QX,Y>=<X,QY>\leqno(1.10)$$ and $$<PX,Y>=-<X,PY>\leqno(1.11)$$ for any $X,Y \in V$. Since $V$ is assumed to be $\alpha$-slant, $$\angle\,(JX,V)=\angle\,(JX,PX) = \alpha$$ for any nozero vector $X\in V$. Hence we have
$$||PX||=\cos\alpha \, ||X||\leqno(1.12)$$ for any nonzero vector $X\in V$. By (1.10), $Q$ is a self-adjoint endomorphism. Since $J_{0}^{2}=-I$, (1.6)-(1.9) imply that each eigenvalue of $Q$ is equal to $-\cos^{2}\alpha$ which
lies in $[-1,0)$. Therefore, by using (1.5), we may prove that $J_{V}^{2}=-I$ and
$$||J_{V}X||^{2}=\sec^{2}\alpha\,||PX||^{2}=||X||^{2}$$ for any $X\in V$. This proves the lemma.
\vskip.1in Let ${\hat \zeta}_{0}$ be the metrical dual of $(-\Omega_{0})^k$ with respect to the inner product $<\,,\,>$ naturally defined on $\wedge^{2k}E^{2m}$, that is,
$$<{\hat \zeta}_{0},\eta> =(-1)^{k}\Omega_{0}^{k}(\eta)\hskip.4in {\rm for\,\,any}\,\,\,\, \eta \in \wedge^{2k}E^{2m},\leqno(1.13)$$ then we have the following
\vskip.1in {\bf Lemma 1.3.} {\it Let $\, V \in G(2k,2m).$ If $\,V$ is $\alpha$-slant in {\bf C}$^m$ with ${\alpha \not= \pi /2}$, then $$<{\hat \zeta}_{0},V> =\mu_{k}\cos^{k}\alpha, \leqno(1.14)$$ where $\mu_k$ is a nonzero constant
depending only on k.} \vskip.1in
{\bf Proof.} Let $J_V$ be the complex structure on $V$ defined by Lemma 1.2. For a unit vector $X\in V$, we put $Y=J_{V}X \in V$. Then we have
$$\Omega_{0}(X,J_{V}X) = <-J_{V}Y,J_{0}Y> = -\cos\alpha.\leqno(1.15)$$
If $X, Y \in V$ and $Z$ is perpendicular to $J_{V}X$, then $$\Omega_{0}(X,Z)=\cos\alpha <X,J_{V}Z>=0.\leqno(1.16)$$
Therefore, if we choose an orthonormal $J_V$-basis $\{e_{1},\ldots,e_{2k}\}$ on $V$, that is, $$e_{2i}=J_{V}e_{2i-1},\,\,\,\,\, i=1,\ldots,k,\leqno(1.17)$$ and $$V=e_{1}\wedge\ldots\wedge e_{2k},\leqno(1.18)$$ via the natural identification of $G(2k,2m)$ with $D_{1}(2k,2m)$, then we have $$\Omega_{0}(e_{a},e_{b})=-\delta_{a^*}{}_b\cos\alpha \,\, \,\,\,\, {\rm for} \,\,\,a<b,\leqno(1.19)$$ where $$(2i)^{*}=2i-1,\,\,\,\, (2i-1)^{*}=2i \,\, \,\,\,\,{\rm for}\,\,\, i=1,\ldots,k.\leqno(1.20)$$
By (1.18), Lemma 1.1, and (1.19), we find $$(2k)!\,\Omega_{0}^{k}(V)=(2k)!\,\Omega_{0}(e_{1} \wedge\ldots\wedge e_{2k})$$
$$=\sum_{\sigma\in S_{2k}} sign(\sigma)\,\,\Omega_{0}(e_{\sigma(1)},e_{\sigma(2)}) \cdots\Omega_{0}(e_{\sigma(2k-1)},e_{\sigma(2k)})$$ $$=\sum_{a_{1},\ldots ,a_{2k}=1}^{2k} \delta_{a_{1}\cdots a_{2k}}^{12\cdots (2k)} \Omega_{0}(e_{a_{1}},e_{a_{2}})\cdots \Omega_{0}(e_{a_{2k-1}},e_{a_{2k}})$$ $$=\sum_{a_{1},\ldots\,a_{k}=1}^{2k} \delta_{a_{1}a_{1}^{*}\cdots {a_{k}a_{k}^{*}}}^{12\cdots\cdots (2k)}\Omega_{0}(e_{a_{1}},e_{a_{1}^{*}})\cdots \Omega_{0}(e_{a_{k}},e_{a_{k}^{*}})\leqno(1.21)$$ $$=2^{k}\sum_{a_{1}<a_{1}^{*}} \cdots \sum_{a_{k}<a_{k}^{*}} \delta_{{a_{1}a_{1}^{*}}\cdots
a_{k}a_{k}^{*}}^{12\cdots\cdots (2k)}\Omega_{0}(e_{a_1},e_{a_{1}^{*}})\cdots\Omega_{0} (e_{a_k},e_{a_k}^{*})$$ $$=2^{k}(-\cos\alpha)^{k}\sum_{a_{1}<a_{1}^{*}} \cdots \sum_{a_{k}<a_{k}^{*}}\delta_{a_{1}a_{1}^{*}\cdots a_{k}a_{k}^{*}} ^{12\cdots\cdots\cdot (2k)}$$ $$=2^{k}(-\cos\alpha)^{k}\, k!.$$
\noindent Hence, by (1.13) and (1.21), we obtain (1.14) with $\mu_{k}=2^{k}k!/(2k)!\,.$ \vskip.1in {\bf Lemma 1.4.} {\it Let $f : N \rightarrow E^m$ be an isometric immersion of an $n$-dimensional compact oriented manifold $N$ into $E^m$. Then the Gauss map $\nu : N \rightarrow \wedge^{n}(E^{m})$ is mass-symmetric in $S^{K-1}$, $K={m\choose n}$, that is, the center of gravity of $\nu$ coincides with the center of the hypersphere $S^{K-1}$ in $\wedge^{n}(E^{m})$.} \vskip.1in {\bf Proof.} Let $e_{1},\ldots,e_n$ be an oriented orthonormal local frame of $TN$ with its dual coframe given by $\omega^{1},\ldots,\omega^n$. Then we have $$dx=e_{1}\omega^{1}+\ldots+e_{n}\omega^{n}.\leqno(1.22)$$
By direct computation we have $$dx\wedge\ldots\wedge dx = n!\,(e_{1}\wedge\ldots\wedge e_{n})\,\omega^{1}\wedge\ldots\wedge \omega^{n}=n!\,\nu (*1),$$ where $dx$ on the left-hand-side is repeated $n$-times. Therefore, by applying the divergence theorem, we have $$n!\int_N \nu*1=\int_N dx\wedge\ldots\wedge dx = \int_N d(x \wedge dx\wedge\ldots\wedge dx) = 0.$$
This proves that the center of gravity of $\nu $ is the origin of $\wedge^{n}(E^{m})$, that is, the Gauss map is mass-symmetric in $S^{K-1}.$
\vskip.1in Now we give the following [CT2] \vskip.1in {\bf Theorem 1.5.} {\it Let $f: N \rightarrow {\bf C}^m$ be a slant immersion of an $n$-dimensional differentiable manifold $N$ into the complex Euclidean $m$-space {\bf C}$^m$. If $N$ is compact, then $f$ is totally real.}
\vskip.1in {\bf Proof.} Without loss of generality we may assume that $N$ is oriented because otherwise
we may simply replace $N$ by its two-fold covering. Assume $f$ is $\alpha$-slant with $\alpha\not= \pi/2.$ Then, by Lemma 1.2, $\,n$ is even. Put $n = 2k$. Since $N$ is compact, Lemma 1.4 implies that the Gauss map $\nu$ is mass-symmetric in $S^{K-1}, K={2m\choose {2k}}$. Therefore $$\int_{p\in N}\,<\nu(p),\zeta>*1 =0\leqno(1.23)$$ for any fixed $2k$-vector $\zeta \in \wedge^{2k}(E^{2m})$, where $*1$ is the volume element of $N$ with respect to the metric induced from the immersion $f$. Let $\zeta={\hat \zeta}_{0}$, where ${\hat \zeta}_0$ is defined by (1.13). Then Lemma 1.3 and (1.23) imply $$\mu_{k}\, vol(N)\cos^{k}\alpha = 0.\leqno(1.24)$$ But this contradicts to the assumption $\cos\alpha \not= 0$. Hence $\alpha = \pi/2$ and $f$ is a totally real immersion. \vskip.1in
We recall that a submanifold $N$ of an almost complex manifold $(M,J)$ is said to be {\it purely real\/} [C2] if every eigenvalue of $Q=P^2$ lies in $(-1,0]$. In fact, by using a method similar to the proof of Theorem 1.5, we may prove the following \vskip.1in {\bf Theorem 1.6.} {\it Let $f: N \rightarrow {\bf C}^m$ be a purely real immersion from an $n$-dimensional differentiable manifold $N$ into ${\bf C}^m$. If $N$ is compact, then $f$ is totally real.}
\vskip.1in {\bf Definition 1.1.} An almost Hermitian manifold $(M,g,J)$ is called an {\it exact sympletic manifold\/} if the sympletic form (or, equivalently, the Kaehler form) ${\tilde \Omega}={\tilde \Omega}_{J}$ of $(M,g,J)$ is an exact 2-form. \vskip.1in For compact submanifolds in an exact sympletic manifold, we have the following \vskip.1in
{\bf Theorem 1.7.} {\it Every compact slant submanifold $N$ in an exact sympletic manifold $(M,g,J)$ is totally real.} \vskip.1in {\bf Proof.} Let $(M,g,J)$ be an exact sympletic manifold with sympletic form ${\tilde \Omega}$. Then there exists a 1-form ${\tilde\varphi}$ on $M$ such that ${\tilde \Omega}=d{\tilde\varphi}$. Let $f: N\rightarrow M$ be the immersion and we put $$\Omega = f^{*}{\tilde\Omega},\,\,\,\varphi=f^{*}{\tilde\varphi}. \leqno(1.25)$$ Then we have $$\Omega=f^{*}{\tilde\Omega} =f^{*}d{\tilde\varphi} = d(f^{*}\varphi)=d\varphi.\leqno(1.26)$$ If $N$ is either a proper slant submanifold or a complex submanifold of $M$, then, by using Lemmas 1.1 and 1.2, we may prove that there is a nonzero constant $C$ (which depends only on the dimesnion of $N$ and the slant angle) such that $\Omega^{k}= C(*1)$ where $dim\,N=2k$ and $*1$ is the volume element of the slant submanifold $N$. From (1.26) we know that $\Omega^k$ is exact. Therefore, by the Stokes theorem, we have
$$ vol(N) = \int_N (*1) = C^{-1}\,\int_N \Omega^{k} = 0.$$
\noindent This is a contradiction.
\eject
\noindent{}
\vskip.3in \noindent \S 2. TOPOLOGY OF SLANT SURFACES. \vskip.2in Let $N$ be an $n$-dimensional proper slant submanifold with slant angle $\alpha$ in a Kaehlerian manifold $M$ of complex dimension $m$. Then $n$ is even, say $n=2k$. Let $e_1$ be a unit tangent vector of $N$. We put $$e_{2} = (\sec\alpha)Pe_{1},\,\,\,\,e_{1^{*}}:=e_{n+1} = (\csc\alpha)Fe_{1},\leqno(2.1)$$ $$e_{2^{*}}:=e_{n+2} = (\csc\alpha)Fe_{2}.$$
\noindent If $n>1$, then, by induction, for each $\, \ell = 1,\ldots,k-1,$ we may choose a unit tangent vector $e_{2\ell +1}$ of $N$ such that $e_{2\ell + 1}$ is perpendicular to $e_{1},e_{2},$ $\ldots,$ $ e_{2\ell -1},e_{2\ell}$. We put
$$e_{2\ell+2} = (\sec\alpha)Pe_{2\ell+1},\,\,\,\,e_{(2\ell+1)^{*}}:= e_{n+2\ell+1}= (\csc\alpha)Fe_{2\ell+1},\leqno(2.2)$$ $$e_{(2\ell+2)^{*}}:= e_{n+2\ell+2}= (\csc\alpha)Fe_{2\ell+2}.$$
\noindent If $N$ is totally real in $M$, that is, if the slant angle $\,\alpha={{\pi}\over {2}}$, then we can just choose $e_{1},\ldots,e_{n}$ to be any local orthonormal frame of $TN$ and put $$e_{n+1}:=e_{1^{*}}=Je_{1},\ldots,e_{2n}:=e_{n^{*}}=Je_{n}.$$
If $m>n$, then at each point $p\in N$ there exist a
subspace $\,\nu_p \,$ of the normal space $T_{p}^{\perp}N$ such that $ \, \nu_p \,$ is invariant under the action of the complex structure $J$ of $M$ and $$T_{p}^{\perp}N= F(T_{p}N)\oplus \nu_{p}, \,\,\,\,\nu_{p} \perp F(T_{p}N).\leqno(2.3)$$ We choose a local orthonormal frame $e_{4k+1},\ldots,e_{2m}$ of $\nu$ such that $$e_{2n+2}=: e_{(4k+1)^{*}}=Je_{4k+1},\ldots,e_{2m}=: e_{(2m-1)^{*}}=J e_{2m-1},$$ that is, $e_{2n+1},\ldots,e_{2m}$ is a $J$-frame of $\,(\nu,J)$.
We call such an orthonormal frame $$e_{1},e_{2},\ldots,e_{2k-1},e_{2k},e_{1{*}},e_{2^{*}}, \ldots,e_{(2k-1)^{*}},e_{(2k)^{*}},\leqno(2.4)$$ $$e_{4k+1},e_{(4k+1)^{*}}, \ldots,e_{2m-1},e_{(2m-1)^{*}}$$
\noindent an {\it adapted slant frame\/} of $N$ in $M$.
\vskip.1in {\bf Lemma 2.1.} {\it Let $N$ be an $n$-dimensional proper
slant submanifold of a Kaehlerian manifold $M$. If $N$ is a Kaehlerian slant submanifold, then, with respect to an adapted slant frame (2.4), we have $$\omega_{i}^{j^*}=\omega_{j}^{i^*},\,\,\,or\,\, equivalently,\,\,\,\, h^{j}_{ik}=h^{i}_{jk}\leqno(2.5)$$ \noindent for any $i,j,k=1,...,n$, where $\omega_{A}^{B}$ are the connection forms associated with the adapted slant frame.} \vskip.1in {\bf Proof.} Since $N$ is a Kaehlerian slant submanifold,
$\nabla P=0$ by definition. Thus, by applying by Lemma 3.5 of Chapter II, we have $A_{FX}Y=A_{FY}X$ for any $X,Y$ tangent to $N$. Therfore, we have (2.5) according to the definition of adapted slant frame. \vskip.1in {\bf Remark 2.1.} If $N$ is a totally real submanifold of a Kaehlerian ma-nifold, Lemma 2.1 was given in [CO1]. \vskip.1in {\bf Corollary 2.2.} {\it If $N$ is a proper slant surface of a Kaehlerian manifold $M$, then, with respect to an adapted slant frame of $N$ in $M$, we have $$\omega_{i}^{j^*}=\omega_{j}^{i^*},\,\,\,or\,\, equivalently,\,\,\,\, h^{j}_{ik}=h^{i}_{jk}\leqno(2.6)$$ \noindent for any $i,j,k=1,2.$} \vskip.1in {\bf Proof.} This Corollary follows immediately from Lemma 2.1 and Theorem 3.4 of Chapter II. \vskip.1in
For an $n$-dimensional proper slant submanifold $N$ of an almost Hermitian manifold $M$, we define a canonical 1-form $\Theta$ on $N$ by
$$\Theta = \sum_{i=1}^{n} \,\omega_{i}^{i^{*}}.\leqno(2.7)$$ \vskip.1in {\bf Lemma 2.3.} {\it Let $N$ be an $n$-dimensional proper slant submanifold with slant angle $\alpha$ in a Kaehlerian manifold $M$. Then we have $$\Theta= \sum_{i} \,(tr\,h^{i^{*}})\omega^{i}\leqno(2.8)$$ and $$\Theta(X)=-n(\csc\alpha)<tH,X>\leqno(2.9)$$ for any vector $X$ tangent to $N$, where $H$ denotes the mean curvature vector of $N$ in $M$.} \vskip.1in {\bf Proof.} Equation (2.8) follows from equation (1.9) of Chapter II, (2.7) and Corollary 2.2.
Since $$n<tH,e_{j}>=-n<H,Fe_{j}>=-n(\sin\alpha)<H,e_{j^{*}}>$$ $$ =-(\sin\alpha) \,tr\,h^{j^{*}},$$ (2.8) implies (2.9).
\vskip.1in Now we give the following [CM3] \vskip.1in {\bf Lemma 2.4.} {\it Let $N$ be an $n$-dimensional proper slant submanifold of ${\bf C}^n$.
If $N$ is Kaehlerian slant, then the canonical 1-form $\Theta$ is closed, that is, $d\Theta =0$. Hence, $\Theta$ defines a canonical cohomology class on $N$:} $$[\Theta] \in H^{1}(N;{\bf R}).\leqno(2.10)$$ \vskip.1in {\bf Proof.} Under the hypothesis, formula (1.8)
of Chapter II gives $$d\Theta = -\sum_{i,j=1}^{n}\omega_{i}^{j}\wedge \omega_{j}^{i^{*}} -\sum_{i,j=1}^{n}\omega_{i}^{j^{*}} \wedge\omega_{j^{*}}^{i^{*}}.\leqno(2.11)$$ Since $\omega_{i}^{j^{*}}=\omega_{j}^{i^{*}}$ by Lemma 2.1 and $\omega_{i}^{j}=-\omega_{j}^{i},$ $\omega_{i^{*}}^{j^{*}}=-\omega_{j^{*}}^{i^{*}}$, we have $$\sum_{i,j}\omega_{i}^{j}\wedge\omega_{j}^{i^{*}} = \sum_{i,j}\omega_{i}^{j^{*}}\wedge\omega_{j^{*}}^{i^{*}}= 0.\leqno(2.12)$$ From (2.11) and (2.12) we obtain the lemma.
\vskip.1in {\bf Remark 2.2.} In fact, this lemma holds without the condition that $N$ is Kaehlerian slant (see Theorem 3.1.). However, the proof for the general case is much more complicated.
\vskip.1in Now we give the main result of this section [CM3]. \vskip.1in {\bf Theorem 2.5.} {\it Let $N$ be a proper slant surface in ${\bf C}^2$ with slant angle $\alpha$. Put $\Psi = (2\sqrt{2}\pi)^{-1}({\csc\alpha})\Theta.$ Then $\Psi$ defines a canonical integral class on $N$:} $$\psi = [\Psi]\in H^{1}(N;{\bf Z}).\leqno(2.13)$$ \vskip.1in {\bf Proof.} Let $N$ be a proper slant surface with slant angle $\alpha$ in ${\bf C}^2$. Denote by $\nu$ the Gauss map $\nu : N \rightarrow G(2,4) \cong S^{2}_{+}\times S^{2}_{-}$ and by $\nu_+$ and $\nu_-$ the projections: $\nu_{\pm} : N \rightarrow G(2,4) \rightarrow S^{2}_{\pm}$ (cf. Section 3 of Chapter III).
From formula (4.3) of Chapter III, we have $$(\nu_{-})_{*} = {1\over \sqrt{2}}\{(-\omega_{1}^{4}+ \omega_{2}^{3})\eta_{5} + (\omega_{1}^{3}+\omega_{2}^{4}) \eta_{6}\}.\leqno(2.14)$$
By Lemma 2.1 we obtain $\omega_{1}^{4}=\omega_{2}^{3}$. Thus, (2.14) implies $$(\nu_{-})_{*}={1\over \sqrt{2}}\,\Theta\,\eta_{6},\leqno(2.15)$$ where $\eta_{6}={1\over 2}(e_{1}\wedge e_{4}-e_{2}\wedge e_{3}).$ Now because $\nu_-$ is given by $$(\nu_{-})(p)={1\over 2}(e_{1}\wedge e_{2}-e_{3}\wedge e_{4})(p)$$ for any point $p\in N$, the slantness of $N$ in ${\bf C}^2$ implies that the image $\nu_{-}(N)$ lies in the small circle $S^{1}_{\alpha}$ of $S^{2}_-$. Moreover, it is easy to see that $\sqrt{2}\,\eta_6$ is a unit vector tangent to $S^{1}_{\alpha}$. Let $\omega = dS^{1}_{\alpha}$ be the arclength element of $S^{1}_{\alpha}$. Then, for any vector $X$ tangent to $N$, we have $$((\nu_{-})^{*}\omega)(X)=\omega((\nu_{-})_{*}(X)) = \omega({1\over {\sqrt{2}}}\Theta(X)\eta_{6}) ={1\over 2}\Theta(X).\leqno(2.16)$$ Hence we have $$(\nu_{-})^{*}\omega = {1\over 2}\Theta.\leqno(2.17)$$ Therefore, for any closed loop $\gamma$ on $N$, we have $$\int_{\gamma} {1\over 2}\Theta = \int_{\gamma} (\nu_{-})^{*}\omega = \int_{\nu_{-}(S^{1}_{\alpha})} \omega = (index \,\,of\,\,\nu_{-})\,vol(S^{1}_{\alpha})$$ $$= (\sqrt{2}\pi\sin\alpha)(index\,\, of\,\,\nu_{-}).$$
\noindent This implies that for any closed loop $\gamma$ in $N$, $\int_{\gamma} \Psi \in {\bf Z}$. Thus by Lemma 2.4 we obtain (2.13). This completes the proof of the theorem. \vskip.1in As an application we obtain the following [CM3] \vskip.1in {\bf Theorem 2.6.} {\it Let $N$ be a complete, oriented, proper slant surface in ${\bf C}^2$. If the mean curvature of $N$ is bounded below by some positive constant $c > 0$, then, topologically, $N$ is either a circular cylinder or a 2-plane.}
\vskip.1in {\bf Proof.} Consider the map $\nu_{-} : N \rightarrow S^{2}_-$. Assume the slant angle of $N$ in ${\bf C}^2$ is $\alpha$. Then $\nu_{-}(N) \subset S^{1}_{\alpha}$. Since $$(\nu_{-})(X)={1\over {\sqrt{2}}}(\omega_{1}^{3} +\omega_{2}^{4})(X)\eta_{6}={1\over {\sqrt{2}}}(\sum_{i} h^{3}_{ii}\omega^{1}+\sum_{i}h^{4}_{ii}\omega^{2})$$ for any vector $X$ tangent to $N$ by Lemma 2.1, the assumption on the mean curvature implies that the map $\nu_-$ is an onto map. Furthermore, since the rank of $\nu_-$ is equal to one, a result of Ehresmann implies that $\nu_-$ is in fact a fibration. Because $N$ is not compact by Theorem 1.5, topologically, $N$ is either the product of a line and a circle or a 2-plane.
\eject \noindent{} \vskip.3in \noindent \S 3. COHOMOLOGY OF SLANT SUBMANIFOLDS. \vskip.2in One of the purposes of this section is to improve Lemma 2.4 to obtain the following Theorem 3.1 of [CM3]. The other purpose is to prove that every proper slant submanifold in any Kaehlerian manifold is a sympletic manifold (Theorem 3.4) with the sympletic structure induced from the canonical endomorphism $P$.
\vskip.1in {\bf Theorem 3.1.} {\it Let $N$ be an $n$-dimensional proper slant submanifold of ${\bf C}^n$. Then the canonical 1-form $\Theta$ defined by (2.7) is closed, that is, $d\Theta=0$. Hence, $\Theta$ defines a canonical cohomology class on $N$:}
$$[\Theta]\in H^{1}(N;{\bf R}).\leqno(3.1)$$
\vskip.1in If $N$ is an $n$-dimensional proper slant submanifold of ${\bf C}^n$ with slant angle $\alpha$, then as we already known the dimension $n$ is even. Let $n=2k$ and $e_{1},\ldots,e_{n},e_{1^{*}},\ldots,e_{n^{*}}$ an adapted slant (orthonormal) frame of $N$ in ${\bf C}^n$. Then we have $$e_{2}=(\sec\alpha)Pe_{1},\ldots,e_{2k}=(\sec\alpha) Pe_{2k-1},\leqno(3.2)$$ $$e_{1^{*}}=(\csc\alpha)Fe_{1},e_{2^{*}}=(\csc\alpha) Fe_{2}, \ldots,e_{(2k)^{*}}=(\csc\alpha)Fe_{2k}.$$
By direct computation we also have $$te_{i^{*}}=-(\sin\alpha)e_{i},\,\,\,\,\,i=1,...,2k, \leqno(3.3)$$ $$Pe_{2j}=-(\cos\alpha)e_{2j-1},\leqno(3.4)$$ $$fe_{(2j-1)^{*}}=-(\cos\alpha)e_{(2j)^{*}},\,\,\, fe_{(2j)^{*}}=(\cos\alpha)e_{(2j-1)^{*}},\,\,\, j=1,\ldots,k.$$
\vskip.1in In order to prove Theorem 3.1, we need the following lemmas which can be regarded as generalizations of Lemma 2.1. \vskip.1in {\bf Lemma 3.2.} {\it Let $N$ be an $n$-dimensional ($n=2k$) proper slant submanifold of ${\bf C}^n$. Then, with respect to an adapted slant frame, we have
$$\omega_{2j-1}^{(2i-1)^{*}}-\omega_{2i-1}^{(2j-1)^{*}} =\cot\alpha(\omega_{2i-1}^{2j}-\omega_{2j-1}^{2i}), \leqno(3.5)$$ $$\omega_{2j}^{(2i-1)^{*}}-\omega_{2i-1}^{(2j)^{*}} =\cot\alpha(\omega_{2i}^{2j}-\omega_{2i-1}^{2j-1}), \leqno(3.6)$$ $$\omega_{2i}^{(2j)^{*}}-\omega_{2j}^{(2i)^{*}} =\cot\alpha(\omega_{2i}^{2j-1}-\omega_{2j}^{2i-1}), \leqno(3.7)$$ $$\omega_{2j-1}^{(2i-1)^{*}}-\omega_{2i-1}^{(2j-1)^{*}} =\cot\alpha(\omega_{(2i-1)^{*}}^{(2j)^{*}}- \omega_{(2j-1)^{*}}^{(2i)^{*}}), \leqno(3.8)$$ $$\omega_{2i}^{(2j)^{*}}-\omega_{2j}^{(2i)^{*}} =\cot\alpha(\omega_{(2i)^{*}}^{(2j-1)^{*}}- \omega_{(2j)^{*}}^{(2i-1)^{*}}), \leqno(3.9)$$ $$\omega_{(2i-1)^{*}}^{(2j-1)^{*}}-\omega_{2i-1}^{2j-1} =\cot\alpha(\omega_{2i-1}^{(2j)^{*}}-\omega_{2i}^{(2j-1)^{*}}) \leqno(3.10)$$ $$\omega_{(2i-1)^{*}}^{(2j)^{*}}-\omega_{2i-1}^{2j} =\cot\alpha(\omega^{2i-1}_{(2j-1)^{*}}-\omega_{2i}^{(2j)^{*}}) \leqno(3.11)$$ $$\omega_{(2i)^{*}}^{(2j)^{*}}-\omega_{2i}^{2j} =\cot\alpha(\omega_{2i-1}^{(2j)^{*}}- \omega_{2i}^{(2j-1)^{*}}), \leqno(3.12)$$ $$\omega_{2j-1}^{(2i)^{*}}-\omega_{2i}^{(2j-1)^{*}} =\cot\alpha(\omega_{(2i)^{*}}^{(2j)^{*}}- \omega_{(2i-1)^{*}}^{(2j-1)^{*}}), \leqno(3.13)$$ for any $i,j=1,\ldots,k.$} \vskip.1in {\bf Proof.} From the definition of adapted slant frames we have $$<Je_{2i-1},e_{2j-1}>=0,\hskip.3in i,j=1,\ldots,k.\leqno(3.14)$$
By taking the derivative of (3.13) with respect to a tangent vector $X$ of $N$ and applying (3.2), we have $$0=<J{\tilde \nabla}_{X}e_{2i-1},e_{2j-1}> +<Je_{2i-1},{\tilde\nabla}_{X}e_{2j-1}>$$ $$=-<\nabla_{X}e_{2i-1},Pe_{2j-1}>-<h(e_{2i-1},X), Fe_{2j-1}>$$ $$+<Pe_{2i-1},\nabla_{X}e_{2j-1}> +<Fe_{2i-1},h(e_{2j-1},X)>$$ $$=-(\cos\alpha)(<\nabla_{X}e_{2i-1},e_{2j}> -<\nabla_{X}e_{2j-1},e_{2i}>)$$ $$+(\sin\alpha)(<e_{(2i-1)^{*}},h(X,e_{2j-1})> -<e_{(2j-1)^{*}},h(X,e_{2i-1})>).$$
\noindent This implies $$(\cot\alpha)(\omega_{2i-1}^{2j}-\omega_{2j-1}^{2i})(X) \leqno(3.15)$$ $$=<A_{e_{(2i-1)^{*}}}e_{2j-1}- A_{e_{(2j-1)^{*}}}e_{2i-1},X>.$$
Therefore, by applying formula $(1.9)'$ of Chapter II, we get formula (3.5).
Similarly, by taking the derivatives of the following equations:
$$<Je_{2i-1},e_{2j}>=(\cos\alpha)\delta_{ij},\,\,\, <Je_{2i},e_{2j}>=0,$$ \vskip.02in $$<Je_{(2i-1)^{*}},e_{(2j-1)^{*}}> =0,\,\,\, <Je_{(2i)^{*}},e_{(2j)^{*}}>=0,$$ \vskip.02in $$<Je_{2i-1},e_{(2j-1)^{*}}>=(\sin\alpha)\delta_{ij},\,\,\, <Je_{2i-1},e_{(2j)^{*}}>=0,$$ \vskip.02in $$<Je_{2i},e_{(2j)^{*}}>=(\sin\alpha)\delta_{ij},\,\,\, <Je_{(2i)^{*}},e_{(2j)^{*}}>=0,$$
\noindent we obtain formulas (3.6)-(3.13), respectively.
This proves the lemma. \vskip.1in {\bf Lemma 3.3.} {\it Let $N$ be an $n$-dimensional ($n=2k$) proper slant submanifold of ${\bf C}^n$. Then, with respect to an adapted slant frame, we have \vskip.02in $$\omega_{2i}^{(2j)^{*}}+\omega_{2i-1}^{(2j-1)^{*}} =\omega_{2j}^{(2i)^{*}}+\omega_{2j-1}^{(2i-1)^{*}}, \leqno(3.16)$$ $$\omega_{(2i)^{*}}^{(2j)^{*}}-\omega_{(2i-1)^{*}}^{(2j-1)^{*}} =\omega_{2i}^{2j}-\omega_{2i-1}^{2j-1}, \leqno(3.17)$$ $$\omega_{2j}^{2i-1}-\omega_{(2j)^{*}}^{(2i-1)^{*}} =\omega_{2i}^{2j-1}-\omega_{(2i)^{*}}^{(2j-1)^{*}} \leqno(3.18)$$ \vskip.02in \noindent for any $i,j=1,\ldots,k.$} \vskip.1in {\bf Proof.} Formula (3.16) follows from (3.5) and (3.7). Formula (3.17) follows from (3.10) and (3.12). And formula (3.18) follows from (3.7) and (3.9). \vskip.1in Now we give the proof of Theorem 3.1. \vskip.1in {\bf Proof.} From the definition we have $$\Theta=\sum_{\ell=1}^{2k}\,\omega_{\ell}^{{\ell}^{*}}.\leqno(3.19)$$ Thus from the structure equations we have $$-d\Theta=\sum_{i,j=1}^{k}\omega_{2i}^{2j} \wedge\omega_{2j}^{(2i)^{*}} +\sum_{i,j=1}^{k}\omega_{2i}^{2j-1} \wedge\omega_{2j-1}^{(2i)^{*}}$$ $$+\sum_{i,j=1}^{k}\omega_{2i}^{(2j)^{*}} \wedge\omega_{(2j)^{*}}^{(2i)^{*}} +\sum_{i,j=1}^{k}\omega_{2i}^{(2j-1)^{*}} \wedge\omega_{(2j-1)^{*}}^{(2i)^{*}}\leqno(3.20)$$ $$+\sum_{i,j=1}^{k}\omega_{2i-1}^{2j} \wedge\omega_{2j}^{(2i-1)^{*}} +\sum_{i,j=1}^{k}\omega_{2i-1}^{2j-1} \wedge\omega_{2j-1}^{(2i-1)^{*}}$$ $$+\sum_{i,j=1}^{k}\omega_{2i-1}^{(2j)^{*}} \wedge\omega_{(2j)^{*}}^{(2i-1)^{*}} +\sum_{i,j=1}^{k}\omega_{2i-1}^{(2j-1)^{*}} \wedge\omega_{(2j-1)^{*}}^{(2i-1)^{*}}.$$
By using formula (3.17) we have
$$\sum_{i,j=1}^{k}\omega_{2i}^{(2j)^{*}} \wedge\omega_{(2j)^{*}}^{(2i)^{*}} +\sum_{i,j=1}^{k}\omega_{2i-1}^{2j-1} \wedge\omega_{2j-1}^{(2i-1)^{*}}$$ $$+\sum_{i,j=1}^{k}\omega_{2i-1}^{(2j-1)^{*}} \wedge\omega_{(2j-1)^{*}}^{(2i-1)^{*}} +\sum_{i,j=1}^{k}\omega_{2i}^{2j} \wedge\omega_{2j}^{(2i)^{*}}$$
$$=\sum_{i,j=1}^{k}\omega_{2i}^{(2j)^{*}} \wedge\omega_{(2j)^{*}}^{(2i)^{*}} +\sum_{i,j=1}^{k}\omega_{2i-1}^{2j-1} \wedge\omega_{2j-1}^{(2i-1)^{*}}$$ $$+\sum_{i,j=1}^{k}\omega_{2i-1}^{(2j-1)^{*}}\wedge (\omega_{2j-1}^{2i-1}-\omega_{2j}^{2i}+ \omega_{(2j)^{*}}^{(2i)^{*}}) +\sum_{i,j=1}^{k}\omega_{2i}^{2j} \wedge\omega_{2j}^{(2i)^{*}}$$
$$=\sum_{i,j=1}^{k}\omega_{2i}^{(2j)^{*}} \wedge\omega_{(2j)^{*}}^{(2i)^{*}} +\sum_{i,j=1}^{k}\omega_{2i-1}^{(2j-1)^{*}}\wedge \omega_{(2j)^{*}}^{(2i)^{*}}$$ $$-\sum_{i,j=1}^{k}\omega_{2i-1}^{(2j-1)^{*}}\wedge \omega_{2j}^{2i} +\sum_{i,j=1}^{k}\omega_{2i}^{2j} \wedge\omega_{2j}^{(2i)^{*}}$$ $$=\sum_{i,j=1}^{k}(\omega_{2i-1}^{(2j-1)^{*}} +\omega_{2i}^{(2j)^{*}})\wedge\omega_{(2j)^{*}}^{(2i)^{*}} +\sum_{i,j=1}^{k}\omega_{2j}^{2i}\wedge(\omega_{2i-1}^{(2j-1)^{*}} +\omega_{2i}^{(2j)^{*}}).$$
Since $\omega_{2i-1}^{(2j-1)^{*}} +\omega_{2i}^{(2j)^{*}}$ is symmetric in $i$ and $j$ by Lemma 3.3 and $\omega_{2i}^{2j}$ and $\omega_{(2j)^{*}}^{(2i)^{*}}$ are skew-symmetric in $i$ and $j$, we obtain $$\sum_{i,j=1}^{k}\omega_{2i}^{(2j)^{*}} \wedge\omega_{(2j)^{*}}^{(2i)^{*}} +\sum_{i,j=1}^{k}\omega_{2i-1}^{2j-1} \wedge\omega_{2j-1}^{(2i-1)^{*}}\leqno(3.21)$$ $$+\sum_{i,j=1}^{k}\omega_{2i-1}^{(2j-1)^{*}} \wedge\omega_{(2j-1)^{*}}^{(2i-1)^{*}} +\sum_{i,j=1}^{k}\omega_{2i}^{2j} \wedge\omega_{2j}^{(2i)^{*}}=0.$$
Moreover, by (3.18), we have $$\sum_{i,j=1}^{k}\omega_{2i}^{2j-1} \wedge\omega_{2j-1}^{(2i)^{*}} +\sum_{i,j=1}^{k}\omega_{2i}^{(2j-1)^{*}} \wedge\omega_{(2j-1)^{*}}^{(2i)^{*}}$$ $$+\sum_{i,j=1}^{k}\omega_{2i-1}^{2j} \wedge\omega_{2j}^{(2i-1)^{*}} +\sum_{i,j=1}^{k}\omega_{2i-1}^{(2j)^{*}} \wedge\omega_{(2j)^{*}}^{(2i-1)^{*}}$$
$$=\sum_{i,j=1}^{k}\omega_{2i}^{2j-1} \wedge\omega_{2j-1}^{(2i)^{*}} +\sum_{i,j=1}^{k}\omega_{2i}^{(2j-1)^{*}} \wedge(\omega_{(2i-1)^{*}}^{(2j)^{*}} +\omega_{2j-1}^{2i} -\omega_{2i-1}^{2j})$$ $$+\sum_{i,j=1}^{k}\omega_{2i-1}^{2j} \wedge\omega_{2j}^{(2i-1)^{*}} +\sum_{i,j=1}^{k}\omega_{2i-1}^{(2j)^{*}} \wedge\omega_{(2j)^{*}}^{(2i-1)^{*}}$$
$$=\sum_{i,j=1}^{k}\omega_{2i}^{2j-1} \wedge\omega_{2j-1}^{(2i)^{*}} -\sum_{i,j=1}^{k}\omega_{2j}^{2i-1} \wedge\omega_{2i}^{(2j-1)^{*}}$$ $$-\sum_{i,j=1}^{k}\omega_{2i}^{(2j-1)^{*}} \wedge\omega_{(2j)^{*}}^{(2i-1)^{*}} +\sum_{i,j=1}^{k}\omega_{2i-1}^{(2j)^{*}} \wedge\omega_{(2j)^{*}}^{(2i-1)^{*}}$$
$$=\sum_{i,j=1}^{k}(\omega_{2j}^{2i-1}- \omega_{(2j)^{*}}^{(2i-1)^{*}} )\wedge(\omega_{2i-1}^{(2j)^{*}}- \omega_{2i}^{(2j-1)^{*}}).$$
Since $\omega_{2j}^{2i-1}- \omega_{(2j)^{*}}^{(2i-1)^{*}}$ is symmetric in $i$ and $j$ by Lemma 3.3 and $\omega_{2i-1}^{(2j)^{*}}-\omega_{2i}^{(2j-1)^{*}}$ is skew-symmetric in $i$ and $j$ by formula (3.6) of Lemma 3.2, we obtain $$\sum_{i,j=1}^{k}\omega_{2i}^{2j-1} \wedge\omega_{2j-1}^{(2i)^{*}} +\sum_{i,j=1}^{k}\omega_{2i}^{(2j-1)^{*}} \wedge\omega_{(2j-1)^{*}}^{(2i)^{*}}\leqno(3.22)$$ $$+\sum_{i,j=1}^{k}\omega_{2i-1}^{2j} \wedge\omega_{2j}^{(2i-1)^{*}} +\sum_{i,j=1}^{k}\omega_{2i-1}^{(2j)^{*}} \wedge\omega_{(2j)^{*}}^{(2i-1)^{*}}=0$$
From (3.20), (3.21) and (3.22) we obtain the theorem.
\vskip.1in {\bf Remark 3.1.} If $N$ is not a slant submanifold of ${\bf C}^n$, then the 1-form $\Theta$ is not closed in general. For example, if $N$ is the standard unit 2-sphere $S^2$ in $E^{3}\subset E^4$, then $N$ is not slant with respect to any compatible complex structure on $E^4$.
Now, assume that $S^2$ is parametrized by $$x(\theta,\varphi)=(\sin\varphi \cos\theta, \sin\varphi \sin\theta, \cos\varphi,0).\leqno(3.23)$$ Put $$e_{1}=(-\sin\theta,\cos\theta,0,0),\leqno(3.24)$$ $$e_{2}=(\cos\varphi \cos\theta, \cos\varphi \sin\theta, -\sin\varphi,0).$$
\noindent Then, with respect to the complex structure $J_0$, we have $$Fe_{1}=(-{1\over 4}\sin 2\theta \sin 2\varphi, -{1\over 2}\sin^{2}\theta \sin 2\varphi, -\sin\theta \cos^{2}\varphi, \cos\theta),\leqno(3.25)$$ and $$Fe_{2}=(\cos^{2}\theta \sin\varphi, \sin\theta \cos\theta \sin\varphi,\cos\theta \cos\varphi, \sin\theta \cos \varphi).\leqno(3.26)$$
Let
$$Fe_{1}=||Fe_{1}||e_{1^{*}},\,\,\, Fe_{2}=||Fe_{2}||e_{2^{*}}.$$
\noindent Then, with respect to the local orthonormal frame $\,e_{1},e_{2},e_{1^{*}},e_{2^{*}}\,$ on the open subset of $S^2$ on which $\theta\not\equiv {\pi\over 2}$ and $\varphi\not\equiv 0$ $(mod \,\pi)$, we have $$\Theta=\omega_{1}^{1^{*}}+\omega_{2}^{2^{*}} =(1-\sin^{2}\theta\cos^{2}\varphi)^{-1}({1\over 2}\sin\theta \sin 2\varphi d\theta - \cos\theta d\varphi).$$ It is easy to see that $d\Theta\not=0.$
\vskip.1in For an $n$-dimensional ($n=2k$) proper slant submanifold $N$ of a Kaehlerian manifold $(M,g,J)$, we put $$\Lambda(X,Y)=\, <X,PY>\leqno(3.27)$$ for any vectors $X, Y$ tangent to $N$, that is, $\Lambda =
\Omega_{J | TN}$. Then, from Lemmas 1.1 to 1.3, $\Lambda$ is a non-degenerate 2-form on $N$, that is, $\Lambda^{k}\not= 0$.
\vskip.1in We recall that an even-dimensional manifold is called a {\it sympletic ma-nifold\/} if it has a non-degenerate closed 2-form.
Now we give the second main result of this section.
\vskip.1in {\bf Theorem 3.4.} {\it Let $N$ be an $n$-dimensional proper slant submanifold of a Kaehlerian manifold $M$. Then $\Lambda$ is closed, that is, $d\Lambda=0.$ Hence $\Lambda$ defines a canonical cohomology class of $N$: $$[\Lambda]\in H^{2}(N;{\bf R}).\leqno(3.28)$$ In particular, $(N,\Lambda)$ is a sympletic manifold.} \vskip.1in {\bf Proof.} By definition of the exterior differentiation we have $$d\Lambda(X,Y,Z)={1\over 3}\{X\Lambda(Y,Z) + Y\Lambda(Z,X)+Z\Lambda(X,Y)$$ $$-\Lambda([X,Y],Z) -\Lambda([Y,Z],X)-\Lambda([Z,X],Y)\}.$$
\noindent Thus, by the definition of $\Lambda$, we obtain $$d\Lambda(X,Y,Z)={1\over 3}\{ <{\nabla_{X}}Y,PZ>+<Y,{\nabla_{X}}(PZ)>$$ $$+<{\nabla_{Y}}Z,PX>+<Z,{\nabla_{Y}}(PX)> +<{\nabla_{Z}}X,PY>$$ $$+<X,{\nabla_{Z}}(PY)> -<[X,Y],PZ>-<[Z,X],PY>$$ $$-<[Y,Z],PX>\}$$ $$={1\over 3}\{<Y,{\nabla_{X}}(PZ)>+<Z,{\nabla_{Y}}(PX)> +<X,{\nabla_{Z}}(PY)>$$ $$+<{\nabla_{X}}Z,PY>+<{\nabla_{Y}}X,PZ>+ <{\nabla_{Z}}Y,PX>\}.$$
\noindent Therefore, by the definition of ${\nabla P}$, we obtain $$d\Lambda(X,Y,Z)={1\over 3}\{<X,(\nabla_{Z}P)Y> +<Y,(\nabla_{X}P)Z>\leqno(3.29)$$ $$+<Z,(\nabla_{Y}P)X>\}.$$
\noindent Therefore, by applying formula (3.2) of Chapter II and formula (3.29), we get $$d\Lambda(X,Y,Z)={1\over 3}\{<X,th(Y,Z)>+<X,A_{FY}Z>$$ $$+<Y,th(Z,X)>+<Y,A_{FZ}X>\leqno(3.30)$$ $$+<Z,th(X,Y)>+<Z,A_{FX}Y>.$$
\noindent Consequently, by applying formulas (1.1), (1.4), (1.5) of Chapter II and formula (3.30), we obtain (3.28).
This proves the theorem. \vskip.1in As an immediate consequence of Theorem 3.4 we obtain the following \vskip.1in {\bf Theorem 3.5.} {\it If $N$ is a compact 2k-dimensional proper slant submanifold of a Kaehlerian manifold $M$, then $$H^{2i}(N;{\bf R})\not= 0\leqno(3.31)$$ for any $i = 1,\ldots,k.$} \vskip.1in In other words we have the following {\it non-immersion theorem.} \vskip.1in {\bf Theorem 3.5$'$.} {\it Let $\,N\,$ be a compact 2k-dimensional differentiable manifold such that $H^{2i}(N;{\bf R})=0$ for some $i \in \{1,\ldots,k\}$. Then $N$ cannot be immersed in any Kaehlerian manifold as a proper slant submanifold.}
\eject
\vskip.3in \noindent \S 4. STABILITY AND INDEX FORM. \vskip.2in The main purpose of this section is to present some results conerning the stability and index of totally real submanifolds, a special class of slant submanifolds, in a Kaehlerian manifold.
Let $f:N \rightarrow M$ be an immersion from a compact $n$-dimensional ma-nifold $N$ into an $m$-dimensional Riemannian manifold $M$. Let $\{f_{t}\}$ be a 1-parameter family of immersions of $N \rightarrow M$ with the property that $f_{0}=f$. Assume the map $F: N\times [0,1] \rightarrow M$ defined by $F(p,t)=f_{t}(p)$ is differentiable. Then $\{f_{t}\}$ is called a {\it variation\/} of $f$. A variation of $f$ induces a vector field in $M$ defined along the image of $N$ under $f$. We shall denote this field by $\zeta$ and it is constructed as follows:
Let $\partial/\partial t$ be the standard vector field in $N\times [0,1]$. We set $$\zeta(p)=F_{*}({\partial\over {\partial t}}(p,0)).$$ Then $\zeta$ gives rise to cross-sections $\zeta^T$ and $\zeta^N$ in $TN$ and $T^{\perp}N$, respectively. If we have $\zeta^{T}=0$, then $\{f_{t}\}$ is called a {\it normal variation\/} of $f$. For a given normal vector field $\xi$ on $N$, $exp\,t\xi$ defines a normal variation $\{f_{t}\}$ induced from $\xi$. We denote by ${\mathcal V}(t)$ the volume of $N$ under $f_t$ with respect to the induced metric and by ${\mathcal V}'(\xi)$ and ${\mathcal V}''(\xi)$, respectively, the values of the first and the second derivatives of ${\mathcal V}(t)$ with respect to $t$, evaluted at $t=0$.
The following formula is well-known: $${\mathcal V}'(\xi)=-n\int_{N} <\xi,H> *1.\leqno(4.1)$$
For a compact minimal submanifold $N$ of a Riemannian manifold $M$, the second variation formula is given by
$${\mathcal V}''(\xi)=\int_{N} \{||D\xi||^{2}-{\bar S}(\xi,\xi)- ||A_{\xi}||^{2}\}*1,\leqno(4.2)$$
\noindent where ${\bar S}(\xi,\eta)$ is defined by $${\bar S}(\xi,\eta) =\sum_{i=1}^{n} {\tilde R}(\xi,e_{i},e_{i},\eta),\leqno(4.3)$$
$e_{1},\ldots,e_n$ a local orthonormal frame of $TN$ and ${\tilde R}$ the Riemann curvature tensor of the ambient manifold $M$.
Applying the Stokes theorem to the integral of the first term of (4.2) (as Simons did in [Si1]), we have $$I(\xi,\xi)=:{\mathcal V}''(\xi)=\int_{N} <L\xi,\xi>*1,\leqno(4.4)$$
in which $L$ is a self-adjoint, strongly elliptic linear differential operator of the second order acting on the space of sections of the normal bundle
given by $$L=-\Delta^{D}-{\hat A}-{\hat S},\leqno(4.5)$$ where $\Delta^D$ is the Laplacian operator associated with the normal connection, $<{\hat A}\xi,\eta>=trace<A_{\xi},A_{\eta}>$, and $<{\hat S}\xi,\eta>={\bar S}(\xi,\eta).$
The differential operator $L$ is called the {\it Jacobi operator\/} of $N$ in $M$. The differential operator $L$ has discrete eigenvalues $\lambda_{1} < \lambda_{2} <\ldots \,\nearrow \infty.$ We put $E_{\lambda}=\{\xi\in \Gamma(T^{\perp}N)\, :\, L(\xi)=\lambda\xi \,\}.$ The number of $\sum_{\lambda <0}dim(E_{\lambda})$ is called the {\it index\/} of $N$ in $M$. A vector field $\xi$ in $E_0$ is called a {\it Jacobi field.\/}
A minimal submanifold $N$ of $M$ is said to be {\it stable\/} if ${\mathcal V}''(\xi) \geq 0$ for any normal vector field $\xi$ of $N$ in $M$. Otherwise, $N$ is said to be {\it unstable\/}. It is clear that $N$ in $M$ is stable if and only if the index of $N$ in $M$ is equal to 0.
Concerning the stability of totally real submanifold we mention the following result of [CLN] obtained in 1980.
\vskip.1in {\bf Proposition 4.1.} {\it Let $N$ be a compact n-dimensional minimal totally real submanifold in a real 2n-dimensional Kaehlerian manifold $M$. Then $N$ is stable if and only if
$$I(JX,JX)=\int_{N} \{||\nabla X ||^{2}+S(X,X)-{\tilde S}(X,X)\}*1\leqno(4.6)$$ is non-negative for every tangent vector field $X$ on $N$, where $S$ and ${\tilde S}$ denote the Ricci forms of $N$ and $M$, respectively.} \vskip.1in {\bf Proof.} Under the hypothesis, we have $$D_{X}JY=J\nabla_{X}Y, \,\,\,A_{JX}Y=A_{JY}X,\,\,\, {\tilde S}\cdot J = J\cdot {\tilde S}\leqno(4.7)$$ (see, pp.145-146 of [CO1]). From this we may obtain $$ {\bar S}(X,X)={\tilde S}(X,X)- \sum_{i=1}^{n} {\tilde R}(X,e_{i},e_{i},X).$$ From the equation of Gauss and (4.7) we may also obtain
$${\bar S}(JX,JX)={\tilde S}(X,X)-S(X,X)-||A_{JX}||^{2}$$ since $N$ is minimal. From these we obtain the result. \vskip.1in By using Proposition 4.1 we have the following results of [CLN] also obtained in 1980. \vskip.1in {\bf Theorem 4.2.} {\it Let $N$ be a compact, totally real submanifold of a Kaehlerian manifold $(M,g,J)$ with $dim_{\bf R}N = dim_{\bf C}M$. Then we have}
(1) {\it If $M$ has positive Ricci tensor and $N$ is stable, then $H^{1}(N;{\bf R})=0$ and}
(2) {\it If $M$ has nonpositive Ricci tensor, then $N$ is always stable.} \vskip.1in {\bf Proof.} Let $\varphi$ be the 1-form dual to a vector field $X$ tangent to $N$. Then we have
$$\int_{N} \{||\nabla X ||^{2}+S(X,X)\}*1 = \int_{N} \{
{1\over 2} ||d\varphi ||^{2}+||\delta X ||^{2}\}*1,$$ where $\delta$ is the codifferential operator. Thus (4.6)
becomes $$I(JX,JX)=\int_{N} \{{1\over 2}||d\varphi
||^{2}+||\delta X||^{2}-{\tilde S}(X,X)\}*1$$ which implies the theorem.
\vskip.1in {\bf Proposition 4.3.} {\it Let $N$ be a compact n-dimensional minimal totally real submanifold in a real 2n-dimensional Kaehlerian manifold $M$. Then $N$ is stable if $N$ satisfies condition (1) or (2) below and $N$ is unstable if $N$ satifies condition (3):}
(1) {\it $i^{*}{\tilde S} \leq S$ where i is the inclusion: $N \rightarrow M$.}
(2) {\it $i^{*}{\tilde S} \leq 2S$ and the identity map of $N$ is stable as a harmonic map.}
(3) {\it $i^{*}{\tilde S} > 2S$ and $N$ admits a nonzero Killing vector field.}
\vskip.1in {\bf Proof.} Stability follows from (1) immediately by Proposition 4.1 from the (2) by Proposition 4.1 and the fact that the second variation for the identity map is
$\int_{N} \{||{\nabla X} ||^{2}-S(X,X)\}*1 $ (cf. [S1]). Sufficiency of (3) follows from Proposition 4.1 and the formula $\int_{N} \{||\nabla X ||^{2}-S(X,X)\}*1=0$ for Killing vector field $X$.
\vskip.1in For two normal vector fields $\xi,\eta$ to a minimal submanifold $N$ in $M$, their {\it index form} is defined by $$I(\xi,\eta)=\int_{N} <L\xi,\eta>*1.\leqno(4.8)$$
It is easy to see that the index form $I$ is a symmetric bilnear form; $I: T^{\perp}N \times T^{\perp}N \rightarrow {\bf R}$. For a vector subbundle $V$ of the normal bundle $T^{\perp}N$, we denote by $I_{V}$ the restriction of the index form on $V$. Thus, $I_V$ is a symmetric bilinear form on $V \,$; $I_{V}:V\times V \rightarrow {\bf R}$.
By the index of $I_{V}$, denoted by $index(I_{V})$, we mean the number of negative eigenvalues of the index form $I_V$.
The normal bundle of
a totally real submanifold $N$ in a Kaehlerian manifold $(M,g,J)$
has the following orthogonal decomposition: $$T^{\perp}N = J(TN)\oplus\nu,\,\,\,\,J(TN)\perp \nu.$$
For totally real minimal submanifold of a Kaehlerian manifold of higher codimension we have the following result of the author and J. M. Morvan. \vskip.1in {\bf Proposition 4.4} {\it Let $N$ be a compact, $n$-dimensional, minimal, totally real submanifold of a Kaehlerian manifold of complex dimension $n+p \,(p>0)$. If
$M$ has non-positive holomorphic bisectional curvatures, then the index form satisfies $$I(\xi,\xi)+I(J\xi,J\xi) \geq 0\leqno(4.9)$$ for any normal vector field $\xi$ of $N$ in $M$.}
\vskip.1in {\bf Proof.} Let $N$ be a compact, $n$-dimensional, minimal, totally real submanifold of a Kaehlerian manifold $M$ of complex dimension $n+p$ with $p>0.$
Then, for any normal vector field $\xi$ in the normal subbundle $\nu$ and vector fields $X,Y$ tangent to $N$, we have $$<D_{X}\xi,JY>=-<{\tilde \nabla}_{X}J\xi,Y> = <A_{J\xi}X,Y>.\leqno(4.10)$$
This implies
$$||D\xi ||^{2} \geq ||A_{J\xi}||^{2},\,\,\,\,
||DJ\xi ||^{2}\geq ||A_{\xi}||^{2},\leqno(4.11)$$ for any normal vector field $\xi$ in $\nu$.
By using (4.2), (4.4), (4.10) and (4.11) we find $$I(\xi,\xi)+I(J\xi,J\xi) \geq -\int_{N} \sum_{i=1}^{n} \{{\tilde R}(\xi,e_{i},e_{i},\xi) +{\tilde R}(J\xi, e_{i},e_{i},J\xi)\}*1.$$ Therefore, if $M$ has non-positive holomorphic bisectional curvature, then, for any normal vector field $\xi$ in $\nu$, we have
$$I(\xi,\xi)+I(J\xi,J\xi)\geq 0.$$
\noindent This proves the Proposition. \vskip.1in {\bf Example 4.1.} Let $N$ be any non-totally geodesic, minimal hypersurface of an $(n+1)$-dimensional flat real torus ${\bf R}T^{n+1}$ which is imbedded in a complex $(n+1)$-dimensional flat complex torus ${\bf C}T^{n+1}$ as a totally geodesic, totally real submanifold. Denote by $\xi$ a unit normal vector field of $N$ in ${\bf R}T^{n+1}$. Then we have $$A_{\xi}\not= 0, \,\,\,D\xi= DJ\xi = 0,\,\,\, A_{J\xi}=0.$$
\noindent Since ${\bf C}T^{n+1}$ is flat, (4.2) and (4.3) yield $$I(\xi,\xi) < 0,\,\,\, I(J\xi,J\xi)>0.\leqno(4.12)$$
\vskip.1in
For the index of the index form $I_{J(TN)}$ we have the following result of the author and Morvan. \vskip.1in {\bf Theorem 4.5.} {\it Let $N$ be a compact, $n$-dimensional, totally real, mi-nimal submanifold of a Kaehlerian manifold $M$ of complex dimension $n+p$. If $M$ has non-positive holomorphic bisectional curvatures, then} $$index(I_{J(TN)}) =0.\leqno(4.13)$$ \vskip.1in {\bf Proof.} Under the hypothesis, let $e_{1},\ldots,e_{2n+2p}$ be a local orthonormal frame along the submanifold $N$ such that $$e_{n+1}=Je_{1},\ldots,e_{2n}=Je_{n},\,\,\, e_{2n+p+1}=Je_{2n+1},\ldots,e_{2n+2p}=Je_{2n+p}.$$ Then $e_{n+1},\ldots,e_{2n}$ form a local orthonormal frame of $J(TN)$ and $e_{2n+1},$ $\ldots,$ $e_{2n+2p}$ form a local orthonormal frame of $\nu$. By applying Lemma 3.5 of Chapter II and the equation of Gauss, we may obtain $$\sum_{i=1}^{n} {\tilde R}(X,e_{i},e_{i},X) - S(X,X)=\sum_{i=1}^{n}
<h(X,e_{i}),h(X,e_{i})>\leqno(4.14)$$ $$=||A_{JX}||^{2} + \sum_{i=1}^{n} \sum_{r=2n+1}^{2n+2p} (<h(X,e_{i}),e_{r}>)^{2}.$$
Combining (4.2) with (4.14) we may find $$I(JX,JX)=\int_{N}
\{||DJX||^{2}+S(X,X)+\sum_{i,r}(<h(X,e_{i}),e_{r}>)^{2} \leqno(4.15)$$ $$ - \sum_{t=1}^{2n} {\tilde R}(X,e_{t},e_{t},X)\}*1.$$
As we did in [CLN] (see [C3, p.51]), put $$W=\nabla_{X}X + (div\,X)X$$ where $div\,X$ denotes the divergence of $X$. Let $\varphi$ be the 1-form associated with $X$. Then, by computing the divergence of $W$ and applying the divergence theorem, we obtain
$$0=\int_{N} (div\,W)*1=\int_{N}\{S(X,X)+||\nabla X||^{2}
- {1\over 2}||d\varphi||^{2}-(\delta\varphi)^{2}\}*1.\leqno(4.16)$$ Combining (4.15) and (4.16) we find $$I(JX,JX)=\int_{N}
\{||DJX||^{2}+\sum_{i,r}(<h(X,e_{i}),e_{r}>)^{2} \leqno(4.17)$$
$$ - \sum_{t=1}^{2n} {\tilde R}(X,e_{t},e_{t},X)-||\nabla X ||^{2} +{1\over 2}||d\varphi ||^{2} + (\delta\varphi)^{2}\}*1,$$ for any vector field $X$ tangent to $N$.
For a totally real submanifold $N$ of a Kaehlerian manifold, the Gauss and Weingarten formulas imply $$D_{X}JY=J\nabla_{X}Y + fh(X,Y)\leqno(4.18)$$
\noindent which yields $||DJY|| \geq ||\nabla Y||$. Thus, by applying
(4.17), (4.18) and the definition of the index, we obtain the theorem.
\vskip.1in {\bf Remark 4.1.} An $n$-dimensional totally real submanifold of a real $2n$-dimensional Kaehlerian manifold is also called a {\it Lagrangian submanifold\/} by some mathematicians.
\eject
\vskip.3in \noindent \S 5. STABILITY OF TOTALLY GEODESIC SUBMANIFOLDS. \vskip.2in In this section we would like to present a general method introduced by the author, Leung and Nagano [CLN] obtained in 1980 for
determining the stablity of totally geodesic submanifolds in compact symmetric spaces. Since every irreducible totally geodesic submanifold of a Hermitian symmetric space is a slant submanifold [CN1], the method can be used to determine the stability of such slant submanifolds.
Recall that the second variation formula of a compact minimal subma-nifold $N$ in a Riemannian manifold is given by formula (4.2) (or, equivalently, by formula (4.4)). If $N$ is totally geodesic, then $A=0$. So the stability obtained trivially when ${\bar S}$ is non-positive. For this reason we are interested in the case $M$ is of compact type. Also we assume $M$ is irreducible partly to preclude tori as $M$.
We need to fix some notations. Since $N$ is totally geodesic, there is a finitely covering group $G_N$ of the connected isometry group $G_{N}^o$ of $N$ such that $G_N$ is a subgroup of the connected isometry group $G_M$ of $M$ and leaves $N$ invariant, provided that $G_{N}^o$ is semi-simple. Let $\mathcal P$ denote the orthogonal complement of the Lie algebra ${\it g}_{_{N}}$ in the Lie algebra ${\it g}_{_{M}}$ with respect to the bi-invariant inner product on ${\it g}_{_{M}}$ which is compatible with the metric of $M$. Every member of ${\it g}_{_{M}}$ is though of as a Killing vector field because of the action $\, G_M$ on $\, M$. Let $\hat P$ denote the space of the vector fields corresponding to the member of $\mathcal P$ restricted to the submanifold $N$. \vskip.1in {\bf Lemma 5.1.} {\it To every member of $\, \mathcal P$ there corresponds a unique (but not canonical) vector field $v \in {\hat P}, v$ is a normal vector field and hence ${\hat P}$ is a $G_N$-invariant subspace of the space, $\Gamma (T^{\perp}N),$ of the sections of the normal bundle to $N$. Moreover, ${\hat P}$ is homomorphic with $\mathcal P$ as a $G_N$-module.} \vskip.1in {\bf Proof.} Let $o$ be an arbitrary point of $N$. Let $K_M$ and $K_N$ denote the isotropy subgroups of $G_M$ and $G_N$ at $o$, respectively. Then ${\it g}_{_{M}}/ {\it k}_{_{M}}$ and ${\it g}_{_{N}}/ {\it k}_{_{N}}$ and ${\mathcal P}/({\mathcal P} \cap {\it k}_{_{M}})$ are identified with $T_{o}M$, $T_{o}N$ and $T_{o}^{\perp}N$ by isomorphisms induced by the evaluation of vector fields in ${\it g}_{_{M}}$ at $o$. In particular, the value $v(o)$ of $v$ is normal to $N$. This proves the lemma. \vskip.1in
Now we are ready to explain the method of [CLN]. \vskip.1in The group $G_N$ acts on sections in $\Gamma(T^{\perp}N)$ and hence on the differential opeators: $\Gamma(T^{\perp}N) \rightarrow \Gamma(T^{\perp}N)$. $G_N$ leaves $L$ fixed since $L$ is defined with $N$ and the metric of $M$ only. Therefore, each eigenspace of $L$ is left invariant by $G_N$. Let $V$ be one of its $G_N$-invariant irreducible subspaces. We have a representation $\rho : G_{N} \rightarrow GL(V).$ We denote by $c(V)$ or $c(\rho)$ the eigenvalue of the corresponding Casimir operator.
To define $c(V)$ we fix an orthonormal basis $(e_{\lambda})$ for ${\it g}_{_{N}}$ and consider the linear endomorphism $C$ or $C_V$ of $V$ defined by $$C=-\sum\,\rho(e_{\lambda})^{2}.\leqno(5.1)$$ It is known that $C$ is $c(V)I_V$ (see Chapter 8 of [Bo1]), where $I_V$ is the identiy map on $V$. In our case, the {\it Casimir operator\/} $C_{V}=-\sum [e_{\lambda} ,[e_{\lambda},V]\,]$ for every member $v$ of $V$ (after extending to a neighborhood of $N$).
Now, we state the following theorem of [CLN] which says, modulo details, that {\it $N$ is stable if and only if $c(V)\geq c({\mathcal P})$ for every $G_N$-invariant irreducible space $V$.}
\vskip.1in {\bf Theorem 5.2.} {\it A compact totally geodesic submanifold $N \,(= G_{N}/K_{N})$ of a compact symmetric space $M \,(=G_{M}/K_{M})$ is stable as a minimal submanifold if and only if one has $c(V)\geq c(P')$ for the eigenvalue of the Casimir operator of every simple $G_N$-module $V$ which shares as a $K_N$-module some simple $K_N$-submodule of the $K_N$-module $T_{o}^{\perp}N$ in common with some simple $G_N$-submodule $P'$ of $\hat P$.} \vskip.1in {\bf Proof.} Given a point $p$ of $N$, we choose a basis of ${\it g}_{_{N}}$ given by $$(e_{\lambda})= (\ldots,e_{i},\ldots,e_{\alpha},\ldots) $$ and a finite system $(e_{r})$ of vectors in ${\mathcal P} \subset {\it g}_{_{M}}$ such that (1) $(e_{i}(p))_{1\leq i \leq n}$ is an orthonormal basis for the tangent space $T_{p}N$, (2) ${\nabla e_{i}}=0,\, 1\leq i \leq n,$ at $p$, (3) $e_{\alpha}(p)=0,\, n < \alpha \leq dim\,{\it g}_{_{N}}$, and (4) $(e_{r}(p))$ is an orthonormal basis for the normal space $T_{p}^{\perp}N$, which we can do as is well-known.
An arbitrary normal vector field $\xi$ is written as $\xi=\sum \xi^{r}e_r$ on a neighborhood of $p$ by Lemma 5.1. Since $N$ is totally geodesic in $M$, we have $D_{X}\xi={\tilde \nabla}_{X}\xi$ for $X$ tangent to $N$ and $\xi\in \Gamma(T^{\perp}N)$, where ${\tilde \nabla}$ is the Riemannian connection of $M$. As the curvature tensor of a symmetric space we also have ${\tilde R}(X,Y)Z=-[[X,Y],Z]$ for $X, Y, Z \in {\it m}=:{\it g}_{_{M}}/{\it k}_{_{M}}$. Therefore, by
evaluating $L\xi$ and $C\xi$ at $p$, we obtain $$L\xi= -\sum\,D_{e_{i}}D_{e_{i}}\xi -\sum\,{\bar S}_{rs}\xi^{r}e_{s}$$ $$= -\sum\,{\tilde \nabla}_{e_{i}}{\tilde \nabla}_{e_{i}}\xi -\sum\,{\bar S}_{rs}\xi^{r}e_{s},$$ where ${\bar S}_{rs}$ are the components of ${\bar S}$ in (4.3) and $$C\xi= -\sum \,[e_{\lambda},[e_{\lambda},\xi]\,]$$ $$=-\sum\,{\tilde \nabla}_{e_{i}}{\tilde \nabla}_{e_{i}}\xi +\sum\, \xi^{r}{\tilde \nabla}_{e_{i}}{\tilde \nabla}_{e_{r}}e_{i} -\sum \,\xi^{r}({\tilde \nabla}_{e_{r}}e_{\alpha})^{s}{\tilde \nabla}_{e_{s}} e_{\alpha}$$ $$=-\sum \,{\tilde \nabla}_{e_{i}}{\tilde \nabla}_{e_{i}} \xi
-\sum \,\xi^{r}({\tilde \nabla}_{e_{r}}e_{\alpha})^{s} {\tilde \nabla}_{e_{s}}e_{\alpha},$$ where for the vanishing of the second term we use the fact that $e_i$ is a Killing vector field. Thus we find $$(L-C)\xi=\sum\, \xi^{r}({\tilde \nabla}_{e_{r}}e_{\alpha})^{s}\,{\tilde \nabla}_{e_{s}} e_{\alpha} -\sum\, {\bar S}_{rs}\xi^{r}e_{s}$$ $$=\sum\, (A_{\alpha})^{2}\xi-{\bar S}(\xi, *),$$ where $A_{\alpha}$ is the Weingarten map given by the restriction of the operator: $X \rightarrow -{\tilde \nabla}_{X}e_{\alpha}$ on $T_{p}N$ to the normal space $T_{p}^{\perp}N$. This proves the following two statements:
{(a)} {\it The difference $\, L-C \,$ is an operator of order one and}
{(b)} {\it The difference $\, L-C \,$ is given by a self-adjoint endomorphism ${\breve S}$ of the normal bundle $T^{\perp}N$. }
Now, because both $L$ and $C$ are $G_N$-invariant, statement (b) implies
{(c)} {\it The endomorphism $\,{\breve S} \,$ is $G_N$-invariant.}
The theorem follows from statements (a) through (c) easily when the isotropy subgroup $K_N$ is irreducible on the normal space $T_{p}^{\perp}N \cong {\it g}_{_{M}}/({\it g}_{_{N}} \oplus {\it k}_{_{M}})$. In fact, ${\breve S}$ is then a constant scalar multiple of the identity map of $T^{\perp}N$; ${\breve S}=k\cdot I,$ by statements (a) through (c) and Schur's lemma. $N$ is stable if and only if the eigenvalues of $L$ are all non-negative. Since $L= (c(P')+k)\cdot I=0$ on the normal Killing fields (see Lemma 5.1), this is equivalent to say that $0 \leq c(V)+k=c(V)-c(P')$ for every simple $G_N$-module $V$ in $\Gamma (T^{\perp}N)$ (which is necessarily contained in an eigenspace of $L$ by ${\breve S} = k\cdot I),$ and the Bott-Frobenius theorem completes the proof.
In the general case, we decompose the normal space into the direct sum of simple $K_N$-modules: $P'\oplus P''\oplus \ldots \,.$ Accordingly we have $T^{\perp}N=E'\oplus E'' \oplus \ldots,$ where $E', E'',\ldots,$ etc. are obtained from $P', P'',\ldots,$ etc., in the usual way by applying the action of $G_N$ to the vectors in $P', P'',\ldots,$ etc. Since $G_N$ leaves invariant $E', E'',\ldots,$ the normal connection leaves invariant the section spaces $\Gamma (E'), \Gamma (E''),\ldots \, .$ Hence, $L$ and $C$ leave these spaces invariant. (For this, the irreducible subspaces $P', P'', \ldots$ must be taken within eigenspaces of the symmetric operator ${\bar S}$ at the point o). In particular, the projections of $\Gamma (T^{\perp}N)$ onto $E', E'', \ldots,$ etc. commute with $C$ and $L$. Thus one can repeat the argument for irreducible case to each of $E', E'',\ldots $ to finish the proof of the theorem. \vskip.1in
Theorem 5.2 provides us an {\it algorithm for stability} which goes like this: One can compute $c(V)$ by the Freudenthal formula (cf. Chapter 8, p.120 of [Bo1]) once one knows the action $\rho$ of $G_N$ on $V$. So the rest is to know all the simple $G_N$-modules $V$ in $\Gamma (T^{\perp}N)$. This is done by means of the Frobenius theorem as reformulated by Bott, which asserts in our case that a simple $G_N$-module $V$ appears in $\Gamma (T^{\perp}N)$ if and only if $V$ as a $K_N$-module contains a simple $K_N$-module which is isomorphic with a $K_N$-module of $T_{o}^{\perp}N$. \vskip.1in {\bf Remark 5.1.} A reformulation of the method of [CLN] was given by Y. Ohnita in [Oh2]. \vskip.1in {\bf Proposition 5.3.} {\it A compact totally geodesic submanifold $N$ of a compact symmetric space $M$ is unstable as a minimal submanifold if the normal bundle admits a nonzero $G_N$-invariant section and if the centeralizer of $G_N$ in $G_M$ is discrete.} \vskip.1in {\bf Proof.} Let $\xi$ be a nonzero $G_N$-invariant normal vector field on $N$. We have $D\xi =0$. In view of (4.2) we will show that ${\bar S}(\xi,\xi)$ is positive. The sectional curvature of a tangential 2-plane at a point $p\in N$ equals
$||[e,f]||^{2}$ if (i) $e$ is a member of ${\it g}_{_{N}}$, (ii) $f$ is that of ${\it g}_{_{M}}$, (iii) $e(p)$ and $f(p)$ form an orthonormal basis for the 2-plane, and (iv) ${\tilde \nabla}e={\tilde\nabla}f=0$ at $p$. Therefore, ${\bar S}(\xi,\xi)$ fails to be positive only if $[e,f]=0$ for every such $e$ and $f$ satisfying $\xi(p)\wedge f(p)=0.$ Since the isotropy subgroup $K_N$ at $p$ leaves the normal vector $\xi(p)$ invariant, we have $[e',f]=0$ for every member $e'$ of ${\it k}_{_{M}}$ and hence $[{\it g}_{_{N}},f]=0$ if ${\bar S}(\xi,\xi)=0$ at $p$. Such an $f$ generates a subgroup in the centeralizer of ${\it g}_{_{N}}$ in ${\it g}_{_{M}}$. This contradicts to the assumption. \vskip.1in {\bf Example 5.1.} Let $N$ be the equator in the sphere $M=S^n$. That $N$ is unstable follows from the Proposition 5.3 if one consider a unit $G_N$-invariant normal vector field to it. The centralizer in this case is generated by the antipodal map: $x \rightarrow -x.$ Its orbit space is the real projective space $M'$. The projection: $M \rightarrow M'$ carries $N$ onto a hypersurface $N'$. The reflection in $N'$ is a member of $G_N$ by our general agreement on $G_N$ (if $n>1$) and precludes the existence of non-vanishing $G_N$-invariant normal vector field to $N'$. It is clear
by Theorem 5.2 that $N'$ is stable.
\vskip.1in {\bf Remark 5.2.} In general, if $N$ is a stable minimal submanifold of a Riemannian manifold $M$ and $M$ is a covering Riemannian manifold of $M'$, then the projection $N'$ of $N$ in $M$ is stable too. The example above shows the converse is false.
\vskip.1in {\bf Definition 5.1.} For a compact connected symmetric space $M$, there is a unique symmetric space $M^*$ of which $M$ and every connected symmetric space which is locally isomophic with $M$ are covering Riemannian manifold of $M^*$. We call $M^*$ the bottom space of $M$. Ig $M$ is a group manifold, $M^*$ is the adjoint group $ad(M)$. \vskip.1in By applying Theorem 5.2 and Proposition 5.3 above, we may obtain the following result of [CLN].
\vskip.1in {\bf Theorem 5.4.} {\it A compact subgroup $N$ of a compact Lie group $M$ is stable with respect to a bi-invariant metric on $M$ if } (a) $N$ {\it has the same rank as $M$ and} (b) {\it $M=M^{*}$, that is, $M$ has no nontrivial center.} \vskip.1in {\bf Proof.} The compact group manifold $M$ has $G_{M}=M_{L}\times M_{R},$ where $M_L$ is the left trranslation group $M\times \{1\}$ and $M_R$ the right translation group; here $M_R$ acts ``to the left'' too, that is, $(1,a)$ carries $x$ into $xa^{-1}$. Similarly for $G_N$. $G_N$ is effective on every invariant neighborhood of $N$ in $M$ by (b). We first consider the case where $N$ is a maximal toral subgroup $T$ of $M$. Let $A_T$ denote the subgroup $\{(a,a^{-1}):a\in T\}$ of $G_N$. We have an epimorphism $\epsilon : K_{N}\times A_{T} \rightarrow G_T$ by the multiplication whose kernel is the subgroup of elements of order 2. In order to use Theorem 5.2, we look at an arbitrary simple $G_T$-module $V$ in $\Gamma(E')$ where $E'$ is, as before, the vector bundle $G_{N}P'$ defined from the simple $K_T$-submodule $P'$ of the normal space. $P'$ is a root space corresponding to a root $\alpha$ of ${\it g}_{_{M}}$. With $V$ we compare the space $P'$, a simple $K_N$-module in $\Gamma(E')$ which is defined from the members of the Lie algebra of $M_L$ taking values in $P'$ at a point of $N$. We want to show $c(V)\geq c(P').$ Since $\alpha\not= 0$ by (a), both $V$ and $P'$ have dimension 2 and these are isomorphic as $K_T$-modules. The relationship between $V$ and $P'$ can be made more explicitly. Namely, a basis for $P'$ is a global frame of $E'$ and so the sections in $V$ are linear combinations of the basis vectors whose components are functions on $N$. These functions form a simple $G_T$-module $F$ of dimension 2 and $V$ is a $G_T$-submodule of $F\otimes P'$. By the Bott-Frobenius theorem, $K_T$ acts trivially on a 1-dimensional subspace of $F$. Every weight $\varphi$ of $F$ is a linear combination of roots of ${\it g}_{_{M}}$ whose coefficients are even numbers. In fact all the weights of the representations of $G_T$ are linear combinations of those roots over the integers by (a) and (b) and, since $K_{T}\cap A_{T} \cong ker\,\epsilon$ is trivial on $F$, the coefficients must be even.
On the other hand, if one looks at the definition of Casimir operator, $C=-\sum\,\rho(e_{\lambda})^2$, one sees that the eigenvalue $c(V)$ is a sort of average of the eigenvalues of $-\rho(x)^2$, $||x||=1,$ or more precisely, $c(V)=-(dim\,V)^{-1}\int trace(\rho(x))^2$, where the integral is taking over the unit sphere of the Lie algebra with an appropriately normalized invariant measure. For this reason, showing $c(V)\geq c(P')$, or equivalently, $(\varphi+\alpha)^{2}-\alpha^{2}\geq 0$ amounts to showing the inner product $<2\alpha + \varphi ,\alpha >=<\varphi+\alpha,\varphi+\alpha>-<\alpha,\alpha>\,\,\geq 0$ in which we may assume that $\varphi$ is dominant (with respect to the Weyl group of ${\it g}_{_{M}}$). This proves the case $N=T$.
We trun to the general case $N\supset T.$ Assume $N$ is unstable and will show this contradicts the stability of $T$. There is then a simple $G_N$-module $V$ such that the second variation (4.2) is negative for some member $\xi$ of $V$. If we restrict $\xi$ to $T$ we still have a normal vector field but the integrand in (4.2) for
$\xi_{|\,T}$ will differ from the restriction of the integrand for $\xi$ by the terms corresponding to the tangential directions to $N$ which are normal to $T$. However, a remedy comes from the group action. First (4.2) with $A=0$ is invariant under $G_N$ acting on $V$. Second, every tangent vector to $N$ is carried into a tangent vector to $T$ by some isometry in $G_N$. Third,
$N$ and $T$ are totally geodesic in $G$, but more importantly the connection and the curvature restrict to the submanifolds comfortably. And finally, the isotropy subgroup $K_N$ acts irreducibly on the tangent space to each simple or circle normal subgroup of $G_N$. From all these it follows that (4.2) for $\xi$ is a positive constant multiple of (4.2) for $\xi_{|\,T}$, as one sees by integrating (4.2) for $g(\xi)_{|\,g(T)},\,g\in G$, over the group $G$ and over the unit sphere of $V$. This is a contradiction which completes the proof of the theorem. \vskip.1in {\bf Remark 5.3.} Neither the assumption (a) or (b) can be omitted from Theorem 5.4 as the examples of $M=SU(2)$ with $N=SO(2)$ and $M=G_2$ with $N=SO(2)$ show. Also the Theorem will be false if $M$ is not a group manifold, a counter-example being $M=M^{*}=GI$ with $N=S^{2}\cdot S^2$ (local product).
\vskip.1in By applying Theorem 5.2, Proposition 5.3 and Theorem 5.4 we may obtain the following results of [CLN].
\vskip.1in {\bf Proposition 5.5.} (a) {\it Among the compact connected simple Lie groups $M^{*}$, the only ones that have unstable $M_{+}^*$ are $SU(n)^*$, $SO(2n)^*$ with n odd, $E_{6}^*$ and $G_2$. }
(b) {\it The unstable $M_+$ are $G^{C}(k,n-k),\, 0<k<n-k,$ for $SU(n)^*$; $SO(2n)/U(n)^*$ for $SO(2n)^*$; $EIII^*$ for $E_{6}^*$; and $M_{+}^*$ for $G_2$.}
(c) {\it Every $M_-$ is stable for the group $M^*$.}
{\bf Comments on the Proof.} (I) The stability of $M_-$ is immediate from Theorem 5.4 since $M_-$ has the same rank as $M$ (cf. [CN1, II]). Otherwise the proof is based on scrutinizing all the individual cases and omitted except for a few cases to illustrate our methods. (II) Take $M^{*}=SO(2n+1)$. Then $M_{+}=G_{+}/K_{+}= G^{R}(k,2n+1-k),\, 0<k<n-k,$ the Grassmannians of the unoriented $k$-planes in $E^{2n+1}$ by Table I in [CN1, II]. The action of $G_{+}=SO(2n+1)$ on $P$ (in the notation of Lemma 5.1) is the adjoint representation corresponding to the highest weight ${\tilde \omega}_2$ in Bourbaki's notation [Bo1]. By Freudenthal's formula, one finds that ${\tilde\omega}_1$ is the only representation that has a smaller eigenvalue than ${\tilde\omega}_{2}; c({\tilde \omega}_{1})<c({\tilde\omega}_{2}).$ But ${\tilde\omega}_1$ does not meet the Bott-Frobenius condition simply because its dimension $2n+1$ is too small. Therefore $M_+$ is stable by Theorem 5.2. (III) Take $SU(n)^*$ for another example. We know $M_{+}=G_{+}/K_{+}=G^{C}(k,n-k),$ the complex Grassmann manifold. If $k\not= n-k,\, M_+$ is 1-connected and hence $K_+$ is connected. On the other hand, $M_{-}=K_{+}=S(U(k)\times U(n-k)),$ which contains a circle group as the center. Therefore, $M_+$ admits a unit $G_+$-invariant normal vector field. Moreover, the centralizer of $G_+$ in $G_M$ is trivial. Hence Proposition 5.3 applies to conclude that $M_+$ is unstable. This argument fails in the case $k=n-k$ and we can conclude the stability of $M_+$ by Theorem 5.2 as in (I). (IV) Unstability is established by means of Proposition 5.3 except for the case of $G_2$. In this case we have $c({\tilde\omega}_{1})<c({\tilde\omega}_{2})=c$ (the adjoint representation). This ${\tilde\omega}_1$ gives a monomorphism of $G_2$ into $SO(7)$ which restricts to a monomorphism of $K_{+}=SO(4)$ into $SO(4)\times SO(3)$ in $SO(7)$ and then projects to $SO(3)$. This implies that ${\tilde\omega}_1$ appears in a space of normal vector fields.
\vskip.1in {\bf Proposition 5.6.} {\it Let $M^*$ be a compact symmetric space $G/K$ with $G$ simple. Then, among the $M_+$ and $M_-$, the unstable minimal submanifolds are $G^{R}(k,n-k),\,k<n-k,$ in $AI(n)^*$; $G^{H}(k,n-k),\, k<n-k,$ in $AII(n)^*$; $SO(k)$ in $G^{R}(k,k)$ with k odd; $M_{+} = M_{-}=SO(2)\times AI(n)$ in $CI(n)^*$; $M_{+}=M_{-}=SO(2)\times AII({n\over 2})$ in $DIII^{*}= SO(2n)/U(n)$ with n even; $G^{H}(2,2)$ in $EI^*$; $FII$ in $EIV^*$; $AII(4)$ in $EV^*$; and $M_{+}=M_{-}= S^{2}\cdot S^{2}$ in $GI$.}
{\bf Comments on the Proof.} (I) In some cases, one can use another method to get the results quickly. For instance, if $M^*$ is Kaehlerian, then it is well-known that every compact complex submanifold is stable. (II) Mostly, unstability is established by using Proposition 5.3. In the cases, $M_{+}=M_{-}=SO(2)\times L$, this proposition does not literally apply but unstability is proven in the same spirit. Consider, say $SO(2)\times AI(n)$ in $CI(n)^*$. This space in $M^*$ is $U(n)/O(n)$. The normal space is isomorphic with the space of the symmetric bilinear forms on $E^n$ as an $O(n)$-module. Therefore, there is a $U(n)$-invariant unit normal vector field $\xi$ on $M_+$. We have ${\tilde \nabla}\xi =0.$ We have to show ${\bar S}(\xi,\xi)>0$ in view of (4.2). Since $M_{+}=M_-$ has the same rank as $M$, there is a tangent vector $X$ in $T_{y}M_-$ such that the curvature of the 2-plane spanned by $X$ and $\xi(y)$ is positive. (III) The case of $M_{+}=M_{-}=S^{2}\cdot S^2$ in $GI$. Precisely, $M_{+}=M_-$ is obtained from $S^{2}\times S^{2} = $ (the unit sphere in $E^{3}) \times$ (the unit sphere in $E^{3}) \subset E^{3}\times E^3$ by identifying $(x,y)$ with $(-x,-y)$. The group $G_-$ for $M_{-}=G_{-}/K_-$ is the adjoint group but we have to take its double covering group $SO(4)$ to let it act on a neighborhood of $M_-$. The identity representation of $SO(4)$ on $E^4$ restricts to the normal representation of $K_{-}=SO(2)\times SO(2)$ as somewhat detailed examination of the root system reveals. Therefore, $M_-$ is unstable. Similarly for $M_+$ which is congruent with $M_-$. \vskip.1in {\bf Remark 5.4.} From the known facts about geodesics, one would not expect a simple relationship between stability and homology. More specifically, we remark that $M_+$ {\it is homologous to zero for a group manifold $M^{*}$.\/} The proof may go like this. Consider the quadratic map $f: x \mapsto s_{x}(o)$ on a symmetric space $M=G/K$ for a fixed point $o$, where $s_x$ is the symmetry at $x$. Assume $M$ is compact and orientable. Then $f$ has a nonzero degree if and only if the cohomology ring $H^{*}(M)$ is a Hopf algebra (cf. M. Clancy's thesis, University of Notre Dame, 1980). On the other hand, the inverse image $f^{-1}(o)$ is exactly $M_+$ and $\{o\}$. Since $H^{*}(M)$ is a Hopf algebra for a group $M^*$, it follows that every $M_+$ is homologous to zero. \vskip.1in
Finally, we give the folowing [CLN]
\vskip.1in {\bf Proposition 5.7.} {\it The minimal totally real totally geodesic submanifold $G^{R}(p,q)$ is unstable in $G^{C}(p,q)$.} \vskip.1in {\bf Proof.} Let $N= G_{N}/K_{N}$ be a totally real and totally geodesic submanifold of a compact Kaehlerian symmetric space $M=G_{M}/K_M$. Then $N$ will be unstable if we find $c(V)<c(P')$ as in Theorem 5.2. For each simple $P'$ in ${\mathcal P}$, there is a simple ${\it g}_{_{N}}$-module $V$ in $\Gamma (E')$ whose members are normal vector fields $\xi = JX$ for some Killing vector field $X$ in ${\it g}_{_{N}}$. This is obvious from the definition of a totally real submanifold. In the case of $G^{R}(p,q)$ in $G^{C}(p,q)$, ${\mathcal P}$ is simple and $c(P)=c(2{\tilde \omega}_{1})>c({\tilde \omega}_{2}) =c({\it g}_{_{N}}),$ where ${\tilde \omega}_{2}$ denotes the highest weight in Bourbaki's notation (see, [Bo1]) and ${\tilde \omega}_{1}$ is the only representation that has a smaller eigenvalue than ${\tilde \omega}_{2}$. This proves the Proposition. \vskip.1in {\bf Remark 5.5.} If $p=1$, Proposition 5.7 was due to
[LS1].
\vskip.1in {\bf Remark 5.6.} The method of [CLN] was used by several mathematicians in their recent studies. For results in this direction see, for instances, [MT1], [MT2], [Oh2] and [Ta1].
{\bf Remark 5.7.} For the fundamental theory of $(M_{+},M_{-})$ and some of its applications see [C3], [C7], [CN1], [CN3] and [N2].
Y. Ohnita (1987) improved the above algorithm to include the formulas for the index, the nullity and the Killing nullity of a compact totally geodesic submanifold in a compact symmetric space.
Let \ $f:N\to M$ \ be a compct totally geodesic submanifold of a compact Riemannian symmetric space. Then $f:N\to M$ is expressed as follows: There are compact symmetric pairs $(G,K)$ and $(U,L)$ with $N=G/K,\, M=U/L$ so that $f:N\to M$ is given by $uK\mapsto \rho(u)L$, where $\rho:G\to U$ is an analytic homomorphism with $\rho(K)\subset L$ and the injective differential $\rho:\mathfrak g\to\mathfrak u$ which satisfies $\rho(\mathfrak m)\subset \mathfrak p$. Here $\mathfrak u=\mathfrak l +\mathfrak p$ and $\mathfrak g=\mathfrak k+\mathfrak m$ are the canonical decompositions of the Lie algebras $u$ and $g$, respectively.
Let $\mathfrak m^\perp$ denote the orthogonal complement of $\rho(\mathfrak m)$ with $\mathfrak p$ relative to the ad$(U)$-invariant inner product $(\;,\;)$ on $\mathfrak u$ such that $(\;,\;)$ induces the metric of $M$. Let $\mathfrak k^\perp$ be the orthogonal complement of $\rho(\mathfrak k)$ in $\mathfrak l$. Put $\mathfrak g^\perp=\mathfrak k^\perp +\mathfrak m^\perp$. Then $\mathfrak g^\perp$ is the orthogonal complement of $\rho(\mathfrak g)$ in $\mathfrak u$ relative to $(\;,\;)$, and $\mathfrak g^\perp$ is ad$_\rho (G)$-invariant. Let $\theta$ be the involutive automorphism of the symmetric pair $(U,L)$. Choose an orthogonal decomposition\ $\mathfrak g^\perp=\mathfrak g_1^\perp\oplus\cdots \oplus\mathfrak g_t^\perp$\ such that each $\mathfrak g^\perp_i$ is an irreducible ad$_\rho (G)$-invariant subspace with $\theta(\mathfrak g^\perp_i)=\mathfrak g^\perp_i$. Then, by Schur's lemma, the Casimir operator $C$ of the representation of $G$ on each $\mathfrak g^\perp_i$ is $a_i I$ for some $a_i\in{\bf C}$.
Put ${\mathfrak m}_i^\perp ={\mathfrak m}\cap {\mathfrak g}^\perp_i$ and let $D(G)$ denote the set of all equivalent classes of finite dimensional irreducible complex representations of $G$. For each $\lambda\in D(G)$, $(\rho_\lambda,V_\lambda)$ is a fixed representation of $\lambda$.
For each $\lambda\in D(G)$, we assign a map $A_\lambda$ from $V_\lambda\otimes \hbox{Hom}_K(V_\lambda,W)$ to $C^\infty(G,W)_K$ be the rule $A_\lambda(v\otimes L)(u)=L(\rho_\lambda(u^{-1})v)$. Here Hom$_K(V_\lambda,W)$ denotes the space of all linear maps $L$ of $V_\lambda$ into $W$ so that $\sigma(k)\cdot L=L\cdot \rho_\lambda(k)$ for all $k\in K$.
Y. Ohnita's formulas for the index $i(f)$, the nullity $n(f)$, and the Killing nullity $n_k(f)$ are given respectively by:
(a) $i(f)=\sum_{i=1}^t\sum_{\lambda\in D(G), a_\lambda <a_i} m(\lambda)d_\lambda$,
(b) $n(f)=\sum_{i=1}^t\sum_{\lambda\in D(G), a_\lambda =a_i} m(\lambda)d_\lambda$,
(c) $n_k(f)=\sum_{i=1,\mathfrak m_i^\perp\ne \{0\}}^t\dim \mathfrak g_i^\perp$,
\noindent where $m(\lambda)=\dim\, \hbox{Hom}_K(V_\lambda,(\mathfrak m_i^\perp)^{\bf C})$ and $d_\lambda$ denotes the dimension of the representation $\lambda$
By applying his formulas, Ohnita determined the indices, the nullities and the Killing nullities for all totally geodesic submanifolds in compact rank one symmetric spaces.
\eject
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\eject
\vskip.4in \centerline {\bf SUBJECT INDEX} \vskip.3in
\noindent $\alpha$-slant submanifold 36, 40, 46, 56
\noindent Adapted slant frame 29, 85
\noindent Algorithm for stability 108
\noindent Anti-holomorphic submanifold 36, 41
\noindent Autere submanifold 25, 26
\noindent Bott-Frobenius theorem 108
\noindent Casimir operator 106, 107
\noindent Canonical cohomolgy class 86, 87, 89, 95
\noindent Compatible complex structure 18, 35
\noindent Complex submanifold 8, 13
\noindent $D_{1}(2,4)$ 33, 34, 77
\noindent Doubly slant submanifold 42, 43, 44, 45
\noindent Exact sympletic manifold 82
\noindent $f$ (canonical endomorphism on normal bundle) 14
\noindent $F$ (canonical 1-form on tangent bundle) 13, 25, 26, 27
\noindent Freudenthal formula 108
\noindent $g_+$, $g_-$ (spherical Gauss maps) 56, 57
\noindent $G_{J,a}$ 37
\noindent Gauss curvature 10, 25, 44
\noindent Gauss map $\nu$ 39, 44, 67, 78, 81,
\noindent Harmonic map 101
\noindent Helices (spherical) 58
\noindent Helical cylinder 58, 59, 62
\noindent Holomorphic submanifold 36, 41
\noindent Index form 99, 100, 101
\noindent Index 99, 100, 101
\noindent Invariant subspace 106
\noindent Jacobi field 99
\noindent Jacobi operator 99, 101, 106-108
\noindent $J_{0}$ (a compatible complex structure) 18
\noindent $J_{1}, J_{1}^-$ (compatible complex structures) 18, 54, 62
\noindent $J^{+}_{V}, J^{-}_{V}$ 38, 71
\noindent ${\mathfrak J},\,{\mathfrak J}^{+}, \,{\mathfrak J}^{-}$ 35, 36, 37, 39, 40, 41, 42, 55
\noindent ${\mathfrak J}_{V,a}^{+}, {\mathfrak J}_{V,a}^{-}$ 37
\noindent Kaehler form 35
\noindent Kaehlerian slant submanifold 13, 19, 22
\noindent Killing vector field 101, 105, 109
\noindent $\Lambda,\,\, [\Lambda]$ 95, 96
\noindent Lagrangian submanifold 104
\noindent $(M_{_{+}},M_{_{-}})$ 108, 109
\noindent Module 106
\noindent $\nu_+$, $\nu_-$ 39
\noindent Normal curvature 10, 25, 44
\noindent Normal subspace $\nu$ 26, 101
\noindent Normal variation 98
\noindent $\Omega_J$ (Kaehler form) 35, 77, 78
\noindent $P$ (canonical endomorphism on tangent bundle) 13, 20
\noindent $\phi$ (an orientation reversing isometry of $S^3$) 53
\noindent $\pi, \pi_{+}, \pi_{-}$ 34, 68
\noindent Parallel submanifold 17, 51
\noindent Proper slant submanifold 13
\noindent $\Psi$, $[\Psi]$ 87
\noindent Representation 106
\noindent $S^{+}_{J,a}, S^{-}_{J,a}$ 37, 39
\noindent $S^{2}_{+}, S^{2}_{-}$ 34-39, 46
\noindent Second variation formula 98, 99
\noindent Slant angle 13
\noindent Slant submanifold 9, 13, 46, 105
\noindent Stable submanifold 99-101, 105-109
\noindent Symmetric space 105-108
\noindent Symmetric unitary connection 8
\noindent Sympletic manifold 95, 96
\noindent $t$ (canonical 1-form on normal bundle) 14
\noindent $\Theta$, $[\Theta]$ 84-89
\noindent Totally geodesic submanifolds 14, 105-110
\noindent Totally real submanifold 8, 30, 32, 36, 82, 83, 99-101, 109, 110
\noindent highest weight 110
\noindent Whitnety immersion 42
\noindent Wirtinger angle 13
\noindent $\zeta$, $\zeta^+$, $\zeta^-$ 35, 71
\noindent $\zeta_J$ 35, 36, 37
\noindent ${\hat \zeta}_0$ 80, 82
\end{document} | arXiv |
\begin{document}
\parskip 10pt
\begin{center}
{\Large\bf The Limiting Distribution of
the Coefficients of the $q$-Catalan Numbers}
\vskip 6mm
William Y.C. Chen$^1$, Carol J. Wang$^2$, and Larry X.W. Wang$^3$ \\ Center for Combinatorics, LPMC-TJKLC \\ Nankai University, Tianjin 300071, P. R. China
\vskip 3mm
$^[email protected], $^[email protected], $^[email protected]
\end{center} \begin{abstract} We show that the limiting distributions of the coefficients of the $q$-Catalan numbers and the generalized $q$-Catalan numbers are normal. Despite the fact that these coefficients are not unimodal for small $n$, we conjecture that for sufficiently large $n$, the coefficients are unimodal and even log-concave except for a few terms of the head and tail. \end{abstract}
\noindent {\bf Keywords :} Bernoulli number, $q$-Catalan number, unimodality, log-concavity, moment generating function.
\section{Introduction}\label{sec-int}
The main objective of this paper is to show that the limiting distribution of the coefficients of the $q$-Catalan numbers is normal. The Catalan numbers \[C_n=\frac{1}{n+1}{2n\choose n}\] have many combinatorial interpretations, see Stanley \cite{stanbook}. The usual $q$-analog of the Catalan numbers is given by \begin{equation}\label{C2} C_n(q):=\frac{1}{[n+1]}{2n\brack n}, \end{equation} where $[n]=1+q+q^2+\cdots +q^{n-1}$, and \[ {n\brack k}=\frac{[n]!}{[k]! [n-k]!}.\] There are also other types of $q$-analogs of the Catalan numbers, see, for example,
Andrews \cite{Andrews}, Gessel and Stanton \cite{Ges}, Krattenthaler \cite{Kra}.
We also consider the limiting distribution of the coefficients of the quotient of two products of $q$-numbers, which includes the result for the $q$-Catalan numbers as a special case. We conclude this paper with two conjectures on the unimodality and log-concavity for almost all the coefficients of the $q$-Catalan numbers and the generalized $q$-Catalan numbers provided that $n$ is sufficiently large.
\allowdisplaybreaks
\section{The Limiting Distribution}\label{sec-q}
In this section, we use the moment generating function technique to obtain the limiting distribution of the coefficients of the $q$-Catalan numbers. We introduce the random variable $\xi_n$ corresponding to the probability generating function \[ \phi_n(q)=C_n(q)/C_n.\] As far as the computations are concerned, we will not need the following combinatorial interpretation of $C_n(q)$. However, for completeness and for the sake of presentation, we would mention that $\xi_n$ reflects the distribution of the major indices of Catalan words of length $2n$, see, for example, \cite{FH}. We write \[ C_n(q) =\sum m_n(k) q^k.\]
The following lemma is concerned with the expectation and variance of $\xi_n$.
\begin{lemma}\label{ev} We have \begin{equation} \label{s2} E(\xi_n)=\frac{n(n-1)}{2}\quad \mbox{and} \quad {\rm Var}(\xi_n)=\frac{n(n-1)(n+1)}{6}. \end{equation} \end{lemma}
\noindent {\it Proof.} By the definition of $C_n(q)$, it is easy to check the following symmetry property of $m_n(k)$: $$m_n(k)=m_{n}(n(n-1)-k).$$ Hence $$E(\xi_n)=\frac{n(n-1)}{2}.$$ Let \[ F=F(q)=\prod_{i=1}^{n-1} (1+q+\cdots +q^{n+i})\quad \mbox{and} \quad G=G(q)=\prod \limits_{i=1}^{n-1}(1+q+\cdots +q^i). \] It is easily verified that $C_n(q)=F/G$. Since \begin{eqnarray*}
\left. C_n(q)'' \right|_{q=1} & = & \left. \left(\frac{F''}{G}-\frac{FG''}{G^2}-\frac{2G'F'}{G^2}
+\frac{2G'^2F}{G^3}\right)\right|_{q=1}\\[8pt] & = & \frac{1}{12} n(n-1)(3n^2-n-4)\cdot C_n, \end{eqnarray*} we obtain \[
{\rm Var}(\xi_n) = \frac{\left. C_n(q)''\right|_{q=1}}{C_n}+E(\xi_n)-E(\xi_n)^2 = \frac{1}{6}n(n-1)(n+1). \] This completes the proof.
\rule{4pt}{7pt}
\begin{lemma}\label{asy1} When $n \rightarrow \infty$, we have \[ \sum_{k=2}^{\infty} B_{2k}\frac{t^{2k}}{2k(2k)!\sigma^{2k}}\sum_{i=2}^{n}\left((n+i)^{2k}-i^{2k}\right)\rightarrow 0 \] uniformly for $t$ from any bounded set, where $B_j$'s are the Bernoulli numbers and $\sigma^2$ is the variance of $\xi_n$ as given in (\ref{s2}). \end{lemma}
\noindent {\it Proof.} The second summation can be expanded as follows: \begin{eqnarray*} \sum_{i=2}^{n}\left((n+i)^{2k}-i^{2k}\right) = \sum_{i=2}^{n} \sum_{j=1}^{2k}{2k\choose j} n^j i^{2k-j}
= \sum \limits_{j=1}^{2k} {2k\choose j}\left(\sum \limits_{i=2}^{n} n^j i^{2k-j}\right). \end{eqnarray*} For $k>1$, the second factor in the preceding summation is bounded by the following integral: \[ \sum \limits_{i=2}^{n} n^j i^{2k-j}<n^j \int_{1}^{n+1}t^{2k-j} dt=n^j\cdot \frac{(n+1)^{2k-j+1}-1}{2k-j+1}. \] Consequently, \[ \sum_{i=2}^{n}\left((n+i)^{2k}-i^{2k}\right)<2^{2k}(n+1)^{2k+1}<8^{2k}n^{2k+1}. \] Since $\sigma^2=\frac{n^3-n}{6}>\frac{n^3}{8}$ when $n$ is sufficiently large, we have \[ \sigma^{-2k}\sum_{i=2}^{n}\left((n+i)^{2k}-i^{2k}\right)<64^{2k}n^{1-k}\leq n^{-1/3}64^{2k}n^{-k/3}, \] for large $n$ and $k>1$. Thus \begin{eqnarray*}
\lefteqn{ \left|\sum \limits_{2\nmid k,k\geq3}B_{2k} \frac{t^{2k}}{2k(2k)!\sigma^{2k}}
\sum_{i=2}^{n}\left((n+i)^{2k}-i^{2k}\right)\right| \qquad }\\
&<&n^{-1/3}\sum \limits_{2\nmid k,k\geq3}|B_{2k}|\frac{t^{2k}}{2k(2k)!}64^{2k}n^{-k/3}\\
& = &n^{-1/3}\sum \limits_{2\nmid k,k\geq3}|B_{2k}|\frac{(64tn^{-\frac{1}{6}})^{2k}}{2k(2k)!}. \end{eqnarray*} In view of the following asymptotic expansion of the Bernoulli numbers
$$|B_{2n}|\sim \frac{2(2n)!}{(2\pi)^{2n}},$$
the convergent radius $R$ of the series $\sum \limits_{2\nmid k,k\geq3}|B_{2k}|\frac{t^{2k}}{2k(2k)!}$ equals $2\pi$. Since $t$ is from a bounded set, when $n$ is large enough, the series \[
\sum \limits_{2\nmid k,k\geq3}|B_{2k}|\frac{(64tn^{-\frac{1}{6}})^{2k}}{2k(2k)!} \] converges. Moreover, it is evident that $64tn^{-\frac{1}{6}}<1$, we can bound the above summation by the constant \[
M_1=\sum \limits_{2\nmid k,k\geq3}|B_{2k}|\frac{1}{2k(2k)!}. \] Similarly, it can be deduced that \[
\sum \limits_{2| k,k\geq 2}B_{2k}\frac{t^{2k}}{2k(2k)!\sigma^{2k}}\sum_{i=2}^{n}\left((n+i)^{2k}-i^{2k}\right)<\frac{M_2}{n^{\frac{1}{3}}}, \] where $M_2=\sum \limits_{2\mid k,k\geq 2}B_{2k}\frac{1}{2k(2k)!}$ is a constant. Hence \[ \sum_{k=2}^{\infty} B_{2k}\frac{t^{2k}}{2k(2k)!\sigma^{2k}} \sum_{i=2}^{n}\left((n+i)^{2k}-i^{2k}\right) <\frac{M_1+M_2}{n^{1/3}}, \] which tends to zero as $n\rightarrow \infty$. This completes the proof.
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\begin{theorem}\label{main} When $n\rightarrow \infty$, the random variable \[ \eta_n=\frac{\xi_n-E(\xi_n)}{{{\rm Var}({\xi_n})}^{\frac{1}{2}}} \] has the standard normal distribution. \end{theorem}
\noindent {\it Proof.} Let $M_n(q)$ denote the moment generating function of $\xi_n$.
Then we have $M_n(q) = \phi_n(e^q)$, see
Sachkov \cite{Sac}. Hence \begin{eqnarray*} M_n(q) &= & \frac{n+1}{{2n\choose n}} \frac{1-e^{q}}{1-e^{(n+1)q}}\cdot \prod_{i=1}^{n}\frac{1-e^{(n+i)q}}{1-e^{iq}}\\[5pt] & = & \prod_{i=2}^{n} \frac{i}{n+i} \cdot \prod_{i=2}^{n} \frac{1-e^{(n+i)q}}{1-e^{iq}}\\[5pt] & = & \prod_{i=2}^{n}\frac{(1-e^{(n+i)q})/(n+i)}{(1-e^{iq})/i}\\[5pt] & = & \exp\left\{\frac{1}{2}\sum_{i=2}^{n}\left((n+i)q-iq\right)\right\}\prod_{i=2}^{n} \frac{(e^{(n+i)q/2}-e^{-(n+i)q/2})/\frac{n+i}{2}}{(e^{iq/2}-e^{-iq/2})/\frac{i}{2}}\\[5pt] & = & \exp\left\{\frac{n(n-1)q}{2}\right\} \prod \limits_{i=2}^{n} \frac{\sinh\left((n+i)q/2\right) /\frac{n+i}{2}}{\sinh\left(iq/2\right) /\frac{i}{2}}. \end{eqnarray*} Recalling the following relation on the Bernoulli numbers \cite{Mar} \begin{equation}\label{bernoulli} \ln \left(\frac{\sinh(x/2)}{x/2}\right)=\sum \limits_{k=1}^{\infty} B_{2k} \frac{x^{2k}}{2k(2k)!}, \end{equation} we find that \begin{eqnarray*} \ln M_n(q) & = & \frac{n(n-1)}{2}q+\sum \limits_{i=2}^{n} \left(\ln \left(\frac{\sinh((n+i)q/2)}{(n+i)/2}\right)-\ln \left(\frac{\sinh(iq/2)}{i/2}\right)\right)\\[5pt] & = & \frac{n(n-1)}{2}q+\sum_{k=1}^{\infty} B_{2k}\frac{q^{2k}}{2k(2k)!}\sum_{i=2}^{n}\left((n+i)^{2k}-i^{2k}\right).\\ \end{eqnarray*} Setting $q=t/\sigma$, where $\sigma$ is the standard deviation of $\xi_n$ as given in Theorem \ref{ev}, we are led to the expansion \[ \ln M_n(t/\sigma)=\frac{n(n-1)t}{2\sigma}+\sum \limits_{k=1}^{\infty} B_{2k}\frac{t^{2k}}{2k(2k)!\sigma^{2k}}\sum \limits_{i=2}^{n}\left((n+i)^{2k}-i^{2k}\right).\] Applying Lemma \ref{asy1}, we have, when $n\rightarrow \infty$, \[ \sum_{k=2}^{\infty} B_{2k}\frac{t^{2k}}{2k(2k)!\sigma^{2k}}\sum_{i=2}^{n}((n+i)^{2k}-i^{2k})\rightarrow 0 \] uniformly for $t$ from any bounded set. Finally, \begin{eqnarray*} \lefteqn{\lim_{n\rightarrow \infty}M_n(t/\sigma) \exp\left\{-\frac{n(n-1)t}{2\sigma}\right\}\quad }\\[5pt] & = & \lim_{n\rightarrow \infty}\exp\left\{\sum_{k=1}^{\infty} B_{2k}\frac{t^{2k}}{2k(2k)!\sigma^{2k}}\sum_{i=2}^{n}\left((n+i)^{2k}-i^{2k}\right)\right\}\\[5pt] & = & \lim_{n\rightarrow \infty}\exp\left\{B_2\frac{t^2}{2(2)!\sigma^2}\sum_{i=2}^{n}\left((n+i)^2-i^2\right)\right\}\\[5pt] & = & e^{t^2/2}, \end{eqnarray*} which coincides with the moment generating function of the standard normal distribution. Employing Curtiss' theorem \cite{Sac}, we reach the conclusion that $\eta_n$ has the standard normal distribution when $n$ approaches infinity.
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\section{A General Setting}\label{sec-Gen}
In this section, we will determine the
limiting distribution of the coefficients of a quotient of
products of $q$-numbers and will give two special cases.
\begin{theorem}\label{gen} Let $a_1, a_2, a_3, \ldots $ and $b_1, b_2, b_3, \ldots$ be two sequences of positive numbers, and let \[ \phi_n(x)=\sum_{k}p_n(k)x^k = \frac{(1-q^{a_1})(1-q^{a_2})\cdots (1-q^{a_n})}{(1-q^{b_1})(1-q^{b_2})\cdots (1-q^{b_n})}. \] Suppose that $\xi_n$ is the random variable corresponding to the generating function $\phi_n(x)$, that is, \[ P(\xi_n=k)=\frac{p_n(k)}{\sum \limits_{k}p_n(k)}. \] Then $\xi_n$ is normally distributed as $n\rightarrow \infty$,
if and only if for $k>1$ \[ \sum \limits_{k=1}^{\infty} B_{2k}\frac{t^{2k}}{2k(2k)!}\left(\sum \limits_{i=1}^{n}(a_i^{2k}-b_i^{2k})\right)\frac{1}{\left(\sum \limits_{i=1}^{n}(a_i^2-b_i^2)\right)^k}\rightarrow 0 \quad \mbox{as} \quad n\rightarrow \infty. \] \end{theorem}
\noindent {\it Proof.} The expectation of $\xi_n$ is easy to compute, as given below: \[ E(\xi_n)=\phi_n(x)'_{q=1}=\frac{1}{2}\sum \limits_{i=1}^{n}\left(a_i-b_i\right). \] Proceeding analogously as in the proof of Theorem \ref{ev}, we find \begin{equation} \label{var} \sigma^2={\rm Var}(\xi_n)=\frac{1}{12}\sum \limits_{i=1}^{n}\left(a_i^2-b_i^2\right). \end{equation} Hence, \[ B_2\frac{t^2}{2(2)!\sigma^2} \left(\sum \limits_{i=1}^{n}(a_i^2-b_i^2)\right)=\frac{1}{6}\cdot \frac{t^2}{4\cdot \frac{1}{12}\left(\sum \limits_{i=1}^{n}(a_i^2-b_i^2)\right)}\cdot \left(\sum \limits_{i=1}^{n}(a_i^2-b_i^2)\right)=\frac{t^2}{2}. \] By the same procedure as in the proof of Theorem \ref{main}, we obtain \begin{eqnarray*} \lefteqn{\lim \limits_{n\rightarrow \infty}M_n(t/\sigma)\exp \left\{\frac{1}{2}\sum \limits_{i=1}^{n}\left(a_i^{2k}-b_i^{2k}\right)\right\}} \\ & = & e^{t^2/2}\lim \limits_{n\rightarrow \infty} \exp \left\{\sum \limits_{k=2}^{\infty} B_{2k}\frac{t^{2k}}{2k(2k)!{\sigma}^{2k}}\left(\sum \limits_{i=1}^{n}\left(a_i^{2k}-b_i^{2k}\right)\right)\right\}. \end{eqnarray*} It follows that the limiting distribution of $p_n(k)$ is normal if and only if \begin{equation}\label{gene1} \sum \limits_{k=2}^{\infty} B_{2k}\frac{t^{2k}}{2k(2k)!{\sigma}^{2k}}\left(\sum \limits_{i=1}^{n}\left(a_i^{2k}-b_i^{2k}\right)\right)\rightarrow 0 \quad \mbox{as} \quad n\rightarrow \infty, \end{equation} for $t$ from any bounded set. By virtue of the variance formula (\ref{var}), the condition (\ref{gene1}) is equivalent to \begin{equation}\label{gene2} \sum \limits_{k=1}^{\infty} B_{2k}\frac{t^{2k}}{2k(2k)!}\frac{\sum \limits_{i=1}^{n}\left(a_i^{2k}-b_i^{2k}\right)}{\left(\sum \limits_{i=1}^{n}\left(a_i^2-b_i^2\right)\right)^k}\rightarrow 0 \quad \mbox{as} \quad n\rightarrow \infty \end{equation} for $t$ from any bounded set. Thus (\ref{gene1}) is verified. This completes the proof.
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\begin{corollary}\label{geco} Let $p_n(k)$ be given as in the above theorem. Suppose that for $k\geq 2$, there exist constants $\alpha>0$, $\beta<0$ and $\gamma<0$ such that \begin{equation}\label{gene} \frac{\sum \limits_{i=1}^{n}\left(a_i^{2k}-b_i^{2k}\right)}{\left(\sum \limits_{i=1}^{n}(a_i^2-b_i^2)\right)^k}< n^{\gamma}(\alpha n^{\beta})^{2k}, \end{equation} for $t$ from any bounded set. Then the limiting distribution of $p_n(k)$ is normal. \end{corollary}
\noindent {\it Proof.} Note that the convergent radius $R$ of the series
\[ \sum \limits_{2\nmid k,k\geq3}|B_{2k}|\frac{x^{2k}}{2k(2k)!} \] is $2\pi$. If (\ref{gene}) holds for $k>1$, then for $t$ from any bounded set, and for sufficiently large $n$, we have \[
\left|t^{2k}\sum
\limits_{i=1}^{n}\left(a_i^{2k}-b_i^{2k}\right)/{\sigma}^{2k}\right|\leq n^{\gamma} (t\alpha n^{\beta})^{2k}, \] where $t\alpha n^{\beta}<2\pi$. It is clear that $n^{\gamma}\rightarrow 0$ since $\gamma<0$.
\rule{4pt}{7pt}
If we choose $\alpha=32\sqrt{3}/3$, $2\beta=\gamma=-\frac{1}{3}$, Theorem \ref{geco} contains Theorem \ref{main} as a special case. We now give two more examples. One is the following $q$-analog of the Catalan numbers \[ c_n(q)=\frac{[2]}{[2n]}{2n\brack n-1}, \] which are symmetric and unimodal, see Stanley [].
Using Theorem \ref{gen}, we reach the following assertion.
\begin{corollary} The distribution of the coefficients in $c_n(q)$ is asymptotically normal. \end{corollary}
\noindent \noindent {\it Proof.} First, we write $c_n(q)$ in the following form: \[ \frac{\prod\limits_{i=3}^{n}(1-q^{n+i-1})}{(1-q)\prod\limits_{i=3}^{n-1} (1-q^i)}, \] Set $a_1=a_2=1,\ a_i=n+i-1,\ 3\leq i\leq n$, and $b_1=b_2=1,\ b_3=1,\ b_i=i-1,\ 4\leq i\leq n.$ Then we have \begin{align*} \sum_{i=1}^{n}\left(a_i^{2k}-b_i^{2k}\right)=\left(a_3^{2k}-b_3^{2k}\right)+\sum_{i=4}^{n}\left(a_i^{2k}-b_i^{2k}\right)\\[5pt] =(n+2)^{2k}-1+\sum_{i=3}^{n-1}\left((n+i)^{2k}-i^{2k}\right) \end{align*} and \begin{align*} \left(\sum_{i=1}^{n}(a_i^2-b_i^2)\right)^k&=\left((n+2)^2-1+\sum_{i=3}^{n-1}\left((n+i)^2-i^2\right)\right)^k\\[5pt] &=(n-1)^k(n+1)^k(2n-3)^k. \end{align*} By the same arguments as in the proof of Lemma \ref{asy1}, we may set $\alpha=32\sqrt{3}/3$ and $2\beta=\gamma=-\frac{1}{3}$ such that the condition (\ref{gene}) is satisfied. Therefore, Theorem \ref{gen} implies the limiting distribution of the coefficients of $c_n(q).$
\rule{4pt}{7pt}
The {\it $m$-Catalan numbers} are defined by $$C_{n,m}=\frac{1}{(m-1)n+1}{{mn}\choose n},$$ for $n\geq 1$. Accordingly, the generalized $q$-Catalan numbers are given by $$C_{n,m}(q)=\frac{1}{[(m-1)n+1]}{{mn}\brack n}.$$ Theorem \ref{gen} has the following consequence.
\begin{corollary} The coefficients of the generalized $q$-Catalan numbers $ C_{n,m}(q)$ are normally distributed when $n\rightarrow \infty$. \end{corollary} \noindent {\it Proof.} First, express $C_{n,m}(q)$ as follows \[ \prod \limits_{i=2}^{n}\frac{1-q^{(m-1)n+i}}{1-q^{i}}. \] Set $a_1=1,\ a_i=(m-1)n+i,\ 2\leq i\leq n$, and $b_1=1,\ \ b_i=i,\ 2\leq i\leq n.$ Then we have \begin{align*} \sum_{i=1}^{n}\left(a_i^{2k}-b_i^{2k}\right)=\sum_{i=2}^{n}\left(a_i^{2k}-b_i^{2k}\right) =\sum_{i=2}^{n} \sum \limits_{j=1}^{2k}{{2k}\choose j} \left((m-1)n\right)^{2k-j}i^{j}. \end{align*} The same argument as in the proof of Lemma \ref{asy1} yields the following bound \[ \sum_{i=1}^{n}\left(a_i^{2k}-b_i^{2k}\right)<8^{2k} \left(\left(m-1\right)n\right)^{2k+1}. \] Now, \begin{align*} \left(\sum_{i=1}^{n}(a_i^2-b_i^2)\right)^k&=\left(\sum_{i=2}^{n}\left(((m-1)n+i)^2-i^2\right)\right)^k\\[5pt] &>(m-1)^{2k}n^{2k}(n-1)^k\\[5pt] &>(m-1)^{2k+1}n^{3k}/(2m)^{k}. \end{align*} It follows that \[ \frac{\sum_{i=1}^{n}\left(a_i^{2k}-b_i^{2k}\right)}{\left(\sum_{i=1}^{n}(a_i^2-b_i^2)\right)^k}<(8\sqrt{2m})^{2k}n^{1-k}. \]
Again, by the same arguments as in the proof of Lemma \ref{asy1}, we may set $\alpha=8\sqrt{2m}$ and $2\beta=\gamma=-\frac{1}{3}$ such that the condition (\ref{gene}) holds. Finally, we may use Theorem \ref{gen} to get the desired distribution.
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\section{Open Problems}
While the $q$-Catalan numbers are not unimodal for small $n$, see Stanley \cite{stanlog}, the limiting distribution suggests that the coefficients are almost unimodal in certain sense for sufficiently large $n$. Obviously, the first and the last term should not be taken into account otherwise one can never expect to have unimodality. In fact, an easy computation indicates that $C_n(q)$ are unimodal for $n\geq 16$.
\begin{conjecture}\label{unimodal} The sequence $\{m_n(1),\ldots ,m_{n}(n(n-1)-1)\}$ is unimodal when $n$ is sufficiently large. \end{conjecture}
When $n>70$, numerical evidence suggestive of a stronger conjecture:
\begin{conjecture}\label{log} There exists an integer $t$ such that when $n$ is sufficiently large, the sequence $\{m_n(t),\ldots ,m_n(n(n-1)-t)\}$ is log-concave, namely, \[ \left(m_n(k)\right)^2\geq m_n(k+1)m_n(k-1) \] for $t+1\leq k\leq n(n-2)-t-1$. Moreover, the minimum value of $t$ seems to be $75$. \end{conjecture}
We also conjecture that similar properties hold for the generalized $q$-Catalan numbers.
\vskip 8pt
\noindent {\bf Acknowledgments.} We would like to thank B.H. Margolius for helpful comments. This work was supported by the 973 Project, the PCSIRT Project of the Ministry of Education, the Ministry of Science and Technology, and the National Science Foundation of China.
\end{document} | arXiv |
Almost equivalent definitions of the Riemann–Stieltjes integral
Below, I will present two definitions of the Riemann–Stieltjes integral, the second of which is more general. My question concerns the relationship between these two definitions.
Definition 1: Let $f,g:[a,b] \to \mathbb{R}$. For a partition $P=\{x_0, x_1,x_2 \cdots x_{n-1},x_n\}$ of $[a,b]$, consider the sum $$S(P,f,g) \stackrel{\rm def}{=} \sum_{i=0}^{n-1} f(c_i) \left[g(x_{i+1}) - g(x_{i})\right]$$ where we have "sample points" $c_i \in [x_i, x_{i+1}]$.
$f$ is then said to be Riemann–Stieltjes integrable with respect to $g$ if there is a real number $L$ with the following property: for all $\epsilon>0$ there is a $\delta>0$ such that for any partition $P$ with $\text{sup}_{{0\leq i \leq n-1}}(x_{i+1} - x_i) < \delta$ and any sequence of points $\{c_i\}_{{0\leq i \leq n-1}, c_i \in [x_i, x_{i+1}]}$ we have
$$\left|S(P,f,g) - L\right| < \epsilon$$
Definition 2: We modify the above definition so that it is like this instead: for all $\epsilon>0$ there is partition $P_{\epsilon}$ such that any refinement $P' \supset P_{\epsilon}$ satisfies $$\left|S(P',f,g) - L\right| < \epsilon$$ independent of the sequence of points $\{c_i\}_{{0\leq i \leq n-1}, c_i \in [x_i, x_{i+1}]}$ we choose.
Remark: The first definition implies the second. Simply let $P_{\epsilon}$ be any partition with $\text{sup}_{{0\leq i \leq n-1}}(x_{i+1} - x_i) < \delta$. However, interestingly, the second definition does not imply the first. Take
$$g(x) = \begin{cases} 0 & x \in [0, \frac 12) \\ 1, & x \in [\frac 12, 1] \end{cases}$$
$$f(x) = \begin{cases} 0 & x \in [0, \frac 12] \\ 1, & x \in (\frac 12, 1] \end{cases}$$
as a counterexample. For this example, the integral exists and is equal to $0$ in the sense of the second definition by ensuring our chosen partition $P{_\epsilon}$ is such that $\frac 12 \in P_{\epsilon}$. This ensures $g(x_{i+1}) - g(x_i) = 0$ except in the interval $[x_k, \frac 12]$; however, this interval does not affect the sum since $f \equiv 0$ in $[x_k, \frac 12]$.
Conversely, for the first definition we needn't have $\frac 12 \in P$. $\frac 12$ may be in the interior of some subinterval $[x_i, x_{i+1}]$ (ie., $x_i < \frac 12 < x_{i+1}$). This would mean that $g(x_{i+1}) - g(x_i) = 1$, and depending on the "sample point" $c_i$ we choose in this subinterval, the sum may be $1$ or $0$. This can happen regardless of how fine the partition is, and hence the integral does not exist.
Are there any regularity conditions we can impose on $g$ to ensure the equivalence of the above definitions? Strict monotonicity is a natural example. If that doesn't work, consider stronger conditions (e.g., $g$ is homeomorphism onto its image, or a $C^{1}$ diffeomorphism).
real-analysis integration riemann-integration stieltjes-integral
MathematicsStudent1122MathematicsStudent1122
$\begingroup$ just a side note: the $\delta$ such that $\sup_{0\le i \le n-1}(x_{i+1} - x_i) < \delta$ is also called the mesh of the partition. At first glance it seems that the second definition is equivalent to the first, in the same way that the different definitions of the Riemann integral are equivalent. $\endgroup$
– Masacroso
$\begingroup$ @Masacroso Yeah I'm aware of that. I didn't explicitly use the word "mesh" in case other people did not know what it meant. And I think I showed they're not equivalent by the counterexample, unless you think it is incorrect. $\endgroup$
– MathematicsStudent1122
$\begingroup$ oh, I see... I was checking the wikipedia article. Both definitions refers to different Riemann-Stieltjes integrals, the first is the original definition and the second an integral introduced by Pollard named Generalized Riemann-Stieltjes integral. See here $\endgroup$
$\begingroup$ @Masacroso Yes I've read that. The Wikipedia page is actually what motivated this question. $\endgroup$
$\begingroup$ Good question +1. I don't think most analysis textbooks treat alternative definitions. Apostol gives both these definitions in exercise and also gives your counter-example. I don't think there is an "if and only if" condition for the equivalence of these two definitions. $\endgroup$
– Paramanand Singh ♦
Depending on how the Riemann-Stieltjes integral is defined (at least two ways that you mention) there are a variety of joint conditions on the integrand $f$ and integrator $g$ that guarantee existence. There is some but not complete overlap.
The Riemann-Stieltjes integral is less flexible than the Riemann integral. One impediment is that the continuity of $g$ comes into play. A function that is continuous is Riemann integrable and so too is one that is discontinuous only on a set of measure zero. Taking definition (2), if the integrator is increasing, then if $f$ is continuous, the Riemann-Stieltjes integral exists, but may not exist if $f$ is only continuous almost everywhere. That is because it is necessary that the integrand and integrator have no common points at which they are discontinuous.
Some basic relationships (of which I am aware) are:
Definition (1) holds if and only if definition (2) holds for Riemann integrals where $g(x) = x$.
Definition (1) implies definition (2) for Riemann-Stieltjes integrals when $f$ is bounded and $g$ is increasing.
Definition (2) implies definition (1) for Riemann-Stieltjes integrals when $g$ is increasing, and either $f$ or $g$ is continuous. You found a counterexample if the continuity requirement is relaxed.
To prove the third implication, first consider that $f$ is continuous and R-S integrable with respect to $g$ under definition (2). Then for any partition $P = (x_0,x_1, \ldots,x_n)$ and choice of tags we have
$$\left|S(P,f,g) - \int_a^bf \, dg\right| = \left|\sum_{j=1}^n f(\xi_j)[g(x_j) - g(x_{j-1})] - \sum_{j=1}^n \int_{x_{j-1}}^{x_j}f \, dg\right|$$
Since $f$ is continuous we can apply the integral mean value theorem to find points $\eta_j$ such that
$$\left|S(P,f,g) - \int_a^bf \, dg\right| = \left|\sum_{j=1}^n [f(\xi_j)-f(\eta_j)]\,[g(x_j) - g(x_{j-1})] \right| \\ \leqslant \sum_{j=1}^n |f(\xi_j)-f(\eta_j)|\,[g(x_j) - g(x_{j-1})]. $$
By uniform continuity of $f$, for any $\epsilon >0$ there is a $\delta > 0$ such that if $\|P\| < \delta$ then $|f(\xi_j)-f(\eta_j)| < \epsilon/(g(b) - g(a))$ and
$$\left|S(P,f,g) - \int_a^bf \, dg\right| < \epsilon.$$
Proof of the implication assuming that the integrator $g$ is continuous, rather than $f$, is lengthier. In brief, we choose a partition $P' =(x_0,x_1,\ldots,x_n)$ such that the upper sum $U(P',f,g)$ and lower sum $L(P',f,g)$ are within $\epsilon/2$ of the integral. Using the uniform continuity of $g$, we find $\delta >0$ such that $|g(x) - g(y)| < \epsilon/(2nM)$ when $|x-y| < \delta$, where $M$ bounds $f$. Then a partition $P$ with $\|P\| < \delta$ is constructed through a tedious process such that
$$\int_a^b f \, dg - \epsilon < L(P',f,g) < L(P,f,g) \leqslant S(P,f,g) \leqslant U(P,f,g) < U(P',f,g) < \int_a^b f \, dg + \epsilon. $$
RRLRRL
$\begingroup$ That proof at the end is new for me (although it does resemble the proof that the definitions are equivalent for a Riemann integral) +1. $\endgroup$
Asking that $f$ be continuous is sufficient. In fact, all you need is that $f$ be left continuous when $g$ is right-discontinuous, and vice versa. This is a rather easy exercise that can be done by considering the difference between the maximum and the minimum values attainable by choosing sample points on a given partition. As a corollary of this, $f$ must be continuous where $g$ is both left- and right-discontinuous.
Where $g$ is continuous, it can be shown fairly easily that $f$ is allowed to have finitely many discontinuities. In fact, as long as the set of discontinuities of $f$ is measure zero (in the sense of Lebesgue measure), the integral ought converge. This proof is identical to that on the Riemann integral.
AraskeAraske
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Determining effect of small variable force on planetary perihelion precession
Is there an analytical technique for determining the effect of a small variable transverse acceleration upon the rate of aspides precession (strictly not a precession but rotation of the line of aspides) of a planet orbitting around the Sun in a 2D plane according to Newtonian gravity law?
I have modelled such effects in a reiterative computer model and would like to verify those measurements.
The transverse acceleration formula is
$$At = (K/c^2)*Vr*Vt * Ar.$$
Where:-
c is speed of light,
K is a constant of magnitude between 0 and +/-3, such that $K/(c^2) << 1$.
Ar is the acceleration of the planet towards the Sun due to Newtonian gravitational influence of the Sun, ($Ar = GM/r^2$).
Vr is radial component of planet velocity relative to the Sun (+ = motion away from the Sun)
Vt is transverse component of planet velocity relative to the Sun (+ = direction of planet forward motion along its orbital path). Vectorially Vt = V - Vr where V is the total instantaneous velocity vector of the planet relative to the Sun.
Assume planet mass is small relative to the Sun
No other bodies are in the system
All motions and accelerations are confined to the two-dimensional plane of the orbit.
The reason why this is interesting to me is that a value of K = +3 in my computer model produces anomalous (Non-Newtonian) periapse rotation rate values very close within about 1% of those predicted by General Relativity and within a few percent of those observed by astronomers (Le Verrier, updated by Newcomb).
Formula (Einstein, 1915) for GR-derived periapse rotation (radians per orbit) from http://en.wikipedia.org/wiki/Apsidal_precession $$ \boldsymbol{\omega}=24.\pi^3.a^2.T^{-2}.c^{-2}.(1-e^2)^{-1} $$
I have accepted Walter's answer. Not only did he answer the original question (Is there a technique...?) but also his analysis produces a formula which not only confirms the computer-simulated effects of the transverse acceleration formula (for K = 3) but which also (unexpectedly to me) is essentially equivalent to the Einstein 1915 formula.
from Walter's Summary (in Walter's answer below):-
: (from first order peturbation analysis) semi-major axis and eccentricity are unchanged, but the direction of periapse rotates in the plane of the orbit at rate $$ \omega=\Omega \frac{v_c^2}{c^2} \frac{K}{1-e^2}, $$ where $\Omega$ is the orbital frequency and $v_c=\Omega a$ with $a$ the semi-major axis. Note that (for $K=3$) this agrees with the general relativity (GR) precession rate at order $v_c^2/c^2$ (given by Einstein 1915).
gravity eccentric-orbit
steveOw
steveOwsteveOw
$\begingroup$ Are you still seeking an answer? $\endgroup$
– Walter
$\begingroup$ @Walter. Yes I am. I have asked similar question at physics.stackexchange.com/questions/123685/… but no solid answer received yet. $\endgroup$
– steveOw
$\begingroup$ @Walter. I also asked at math.stackexchange.com/questions/866836/…. $\endgroup$
$\begingroup$ Yes, there are approximate analytical methods (perturbation theory), valid in the limit of $K\ll1$. Perhaps you can clarify your question a bit. What's the direction of the transverse acceleration (I understand 'transverse' to mean perpendicular to the instantaneous velocity, but it's not clear whether the acceleration is in the plane of the orbit or perpendicular or a mixture). $\endgroup$
$\begingroup$ There is a difference between your question here and that on mathematics (and physics): here the transverse acceleration is proportional to the radial acceleration and $K$ is a dimensionless number, there the radial acceleration has no effect on the transverse acceleration and $K$ must be an acceleration (though you talk about a 'number'). $\endgroup$
You may want to use perturbation theory. This only gives you an approximate answer, but allows for analytic treatment. Your force is considered a small perturbation to the Keplerian elliptic orbit and the resulting equations of motion are expanded in powers of $K$. For linear perturbation theory, only terms linear in $K$ are retained. This simply leads to integrating the perturbation along the unperturbed original orbit. Writing your force as a vector, the perturbing acceleration is $$ \boldsymbol{a} = K \frac{GM}{r^2c^2}v_r\boldsymbol{v}_t $$ with $v_r=\boldsymbol{v}{\cdot}\hat{\boldsymbol{r}}$ the radial velocity ($\boldsymbol{v}\equiv\dot{\boldsymbol{r}}$) and $\boldsymbol{v}_t=(\boldsymbol{v}-\hat{\boldsymbol{r}}(\boldsymbol{v}{\cdot}\hat{\boldsymbol{r}}))$ the rotational component of velocity (the full velocity minus the radial velocity). Here, the dot above denotes a time derivative and a hat the unit vector.
Now, it depends what you mean with 'effect'. Let's work out the changes of the orbital semimajor axis $a$, eccentricity $e$, and direction of periapse.
To summarise the results below: semi-major axis and eccentricity are unchanged, but the direction of periapse rotates in the plane of the orbit at rate $$ \omega=\Omega \frac{v_c^2}{c^2} \frac{K}{1-e^2}, $$ where $\Omega$ is the orbital frequency and $v_c=\Omega a$ with $a$ the semi-major axis. Note that (for $K=3$) this agrees with the general relativity (GR) precession rate at order $v_c^2/c^2$ (given by Einstein 1915 but not mentioned in the original question).
change of semimajor axis
From the relation $a=-GM/2E$ (with $E=\frac{1}{2}\boldsymbol{v}^2-GMr^{-1}$ the orbital energy) we have for the change of $a$ due to an external (non-Keplerian) acceleration $$ \dot{a}=\frac{2a^2}{GM}\boldsymbol{v}{\cdot}\boldsymbol{a}. $$ Inserting $\boldsymbol{a}$ (note that $\boldsymbol{v}{\cdot}\boldsymbol{v}_t=h^2/r^2$ with angular momentum vector $\boldsymbol{h}\equiv\boldsymbol{r}\wedge\boldsymbol{v}$), we get $$ \dot{a}=\frac{2a^2Kh^2}{c^2}\frac{v_r}{r^4}. $$ Since the orbit average $\langle v_r f(r)\rangle=0$ for any function $f$ (see below), $\langle\dot{a}\rangle=0$.
change of eccentricity
From $\boldsymbol{h}^2=(1-e^2)GMa$, we find $$ e\dot{e}=-\frac{\boldsymbol{h}{\cdot}\dot{\boldsymbol{h}}}{GMa}+\frac{h^2\dot{a}}{2GMa^2}. $$ We already know that $\langle\dot{a}\rangle=0$, so only need to consider the first term. Thus, $$ e\dot{e}=-\frac{(\boldsymbol{r}\wedge\boldsymbol{v}){\cdot}(\boldsymbol{r}\wedge\boldsymbol{a})}{GMa} =-\frac{r^2\;\boldsymbol{v}{\cdot}\boldsymbol{a}}{GMa} =-\frac{Kh^2}{ac^2}\frac{v_r}{r^2}, $$ where I have used the identity $(\boldsymbol{a}\wedge\boldsymbol{b}){\cdot}(\boldsymbol{c}\wedge\boldsymbol{d}) =\boldsymbol{a}{\cdot}\boldsymbol{c}\;\boldsymbol{b}{\cdot}\boldsymbol{d}- \boldsymbol{a}{\cdot}\boldsymbol{d}\;\boldsymbol{b}{\cdot}\boldsymbol{c}$ and the fact $\boldsymbol{r}{\cdot}\boldsymbol{a}_p=0$. Again $\langle v_r/r^2\rangle=0$ and hence $\langle\dot{e}\rangle=0$.
change of the direction of periapse
The eccentricity vector $ \boldsymbol{e}\equiv\boldsymbol{v}\wedge\boldsymbol{h}/GM - \hat{\boldsymbol{r}} $ points (from the centre of gravity) in the direction of periapse, has magnitude $e$, and is conserved under the Keplerian motion (validate all that as an exercise!). From this definition we find its instantaneous change due to external acceleration $$ \dot{\boldsymbol{e}}= \frac{\boldsymbol{a}\wedge(\boldsymbol{r}\wedge\boldsymbol{v}) +\boldsymbol{v}\wedge(\boldsymbol{r}\wedge\boldsymbol{a})}{GM} =\frac{2(\boldsymbol{v}{\cdot}\boldsymbol{a})\boldsymbol{r} -(\boldsymbol{r}{\cdot}\boldsymbol{v})\boldsymbol{a}}{GM} =\frac{2K}{c^2}\frac{h^2v_r\boldsymbol{r}}{r^4} -\frac{K}{c^2}\frac{v_r^2\boldsymbol{v}_t}{r} $$ where I have used the identity $\boldsymbol{a}\wedge(\boldsymbol{b}\wedge\boldsymbol{c})=(\boldsymbol{a}{\cdot}\boldsymbol{c})\boldsymbol{b}-(\boldsymbol{a}{\cdot}\boldsymbol{b})\boldsymbol{c}$ and the fact $\boldsymbol{r}{\cdot}\boldsymbol{a}=0$. The orbit averages of these expression are considered in the appendix below. If we finally put everything together, we get $ \dot{\boldsymbol{e}}=\boldsymbol{\omega}\wedge\boldsymbol{e} $ with [corrected again] $$ \boldsymbol{\omega}=\Omega K \frac{v_c^2}{c^2} (1-e^2)^{-1}\, \hat{\boldsymbol{h}}. $$ This is a rotation of periapse in the plane of the orbit with angular frequency $\omega=|\boldsymbol{\omega}|$. In particular $\langle e\dot{e}\rangle=\langle\boldsymbol{e}{\cdot}\dot{\boldsymbol{e}}\rangle=0$ in agreement with our previous finding.
Don't forget that due to our usage of first-order perturbation theory these results are only strictly true in the limit $K(v_c/c)^2\to0$. At second-order perturbation theory, however, both $a$ and/or $e$ may change. In your numerical experiments, you should find that the orbit-averaged changes of $a$ and $e$ are either zero or scale stronger than linear with perturbation amplitude $K$.
disclaimer No guarantee that the algebra is correct. Check it!
Appendix: orbit averages
Orbit averages of $v_rf(r)$ with an abitrary (but integrable) function $f(r)$ can be directly calculated for any type of periodic orbit. Let $F(r)$ be the antiderivative of $f(r)$, i.e. $F'\!=f$, then the orbit average is: $$ \langle v_r f(r)\rangle = \frac{1}{T}\int_0^T v_r(t)\,f\!\left(r(t)\right) \mathrm{d}t = \frac{1}{T} \left[F\left(r(t)\right)\right]_0^T = 0 $$ with $T$ the orbital period.
For the orbit averages required in $\langle\dot{\boldsymbol{e}}\rangle$, we must dig a bit deeper. For a Keplerian elliptic orbit $$ \boldsymbol{r}=a\left((\cos\eta-e)\hat{\boldsymbol{e}}+\sqrt{1-e^2}\sin\eta\,\hat{\boldsymbol{k}}\right)\qquad\text{and}\qquad r=a(1-e\cos\eta) $$ with eccentricity vector $\boldsymbol{e}$ and $\hat{\boldsymbol{k}}\equiv\hat{\boldsymbol{h}}\wedge\hat{\boldsymbol{e}}$ a vector perpendicular to $\boldsymbol{e}$ and $\boldsymbol{h}$. Here, $\eta$ is the eccentric anomaly, which is related to the mean anomaly $\ell$ via $ \ell=\eta-e\sin\eta, $ such that $\mathrm{d}\ell=(1-e\cos\eta)\mathrm{d}\eta$ and an orbit average becomes $$ \langle\cdot\rangle = (2\pi)^{-1}\int_0^{2\pi}\cdot\;\mathrm{d}\ell = (2\pi)^{-1}\int_0^{2\pi}\cdot\;(1-e\cos\eta)\mathrm{d}\eta. $$ Taking the time derivative (note that $\dot{\ell}=\Omega=\sqrt{GM/a^3}$ the orbital frequency) of $\boldsymbol{r}$, we find for the instantaneous (unperturbed) orbital velocity $$ \boldsymbol{v}=v_c\frac{\sqrt{1-e^2}\cos\eta\,\hat{\boldsymbol{k}}-\sin\eta\,\hat{\boldsymbol{e}}}{1-e\cos\eta} $$ where I have introduced $v_c\equiv\Omega a=\sqrt{GM/a}$, the speed of the circular orbit with semimajor axis $a$. From this, we find the radial velocity $v_r=\hat{\boldsymbol{r}}{\cdot}\boldsymbol{v}=v_c e\sin\eta(1-e\cos\eta)^{-1}$ and the rotational velocity $$ \boldsymbol{v}_t = v_c\frac{\sqrt{1-e^2}(\cos\eta-e)\,\hat{\boldsymbol{k}}-(1-e^2)\sin\eta\,\hat{\boldsymbol{e}}}{(1-e\cos\eta)^2}. $$
With these, we have [corrected again] $$ \left\langle \frac{h^2v_r\boldsymbol{r}}{r^4}\right\rangle = \Omega v_c^2\,\hat{\boldsymbol{k}}\, \frac{e(1-e^2)^{3/2}}{2\pi}\int_0^{2\pi}\frac{\sin^2\!\eta}{(1-e\cos\eta)^4}\mathrm{d}\eta =\frac{\Omega v_c^2e}{2(1-e^2)}\hat{\boldsymbol{k}} \\ \left\langle \frac{v_r^2\boldsymbol{v}_t}{r}\right\rangle = \Omega v_c^2\, \hat{\boldsymbol{k}}\, \frac{e^2(1-e^2)^{1/2}}{2\pi}\int_0^{2\pi}\frac{\sin^2\!\eta(\cos\eta-e)}{(1-e\cos\eta)^4}\mathrm{d}\eta=0, $$ in particular, the components in direction $\hat{\boldsymbol{e}}$ average to zero. Thus [corrected again] $$\left\langle 2\frac{h^2v_r\boldsymbol{r}}{r^4}-\frac{v_r^2\boldsymbol{v}_t}{r}\right\rangle =\frac{\Omega v_c^2e\,\hat{\boldsymbol{k}}}{(1-e^2)} $$
WalterWalter
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\begin{document}
\begin{abstract} Given C$^*$-algebras $A$ and $B$ and a $^*$-homomorphism $\phi:A\rightarrow B$, we adopt the portrait of the relative $K$-theory $K_*(\phi)$ due to Karoubi using Banach categories and Banach functors. We show that the elements of the relative groups may be represented in a simple form. We prove the existence of two six-term exact sequences, and we use these sequences to deduce the fact that the relative theory is isomorphic, in a natural way, to the $K$-theory of the mapping cone. \end{abstract}
\title{Relative $K$-theory for $C^*$-algebras}
\section{Introduction} \label{intro}
In this paper we develop a portrait of relative $K$-theory for C$^*$-algebras by following an approach due to Karoubi that uses Banach categories and Banach functors. For every $^*$-homomorphism $\phi:A\rightarrow B$ between C$^*$-algebras $A$ and $B$, we produce two abelian groups $K_0(\phi)$ and $K_1(\phi)$ that give information about how the $K$-theory of $A$ and $B$ are related through $\phi$. In fact, the assignments $\phi\mapsto K_0(\phi)$ and $\phi\mapsto K_1(\phi)$ may be regarded as functors in a natural way, and through this we obtain a homology theory that satisfies Bott periodicity (the long exact sequence being the one in part (ii) of Theorem \ref{main}), and a tight connection between the $K$-theory of $A$ and $B$, the induced maps $\phi_*:K_*(A)\rightarrow K_*(B)$, and $K_*(\phi)$ is contained in the six-term exact sequence in part (i) of Theorem \ref{main}.
The notion of relative $K$-theory is not new in the subject of operator algebras. Indeed, references such as \cite{blackadar} and \cite{higsonroe} contain a concise exposition under the assumptions that $A$ is unital, $B$ is a quotient of $A$ by some closed, two-sided ideal $I$, and $\phi$ is the quotient map. The relative group produced is denoted $K_0(A,A/I)$ (although the notation varies throughout the literature). These assumptions are quite reasonable, since $K_0(A,A/I)$ provides the noncommutative generalization of the relative group $K^0(X,Y)$ in topological $K$-theory, where $X$ is a compact space and $Y$ is a closed subset of $X$. The key feature of these groups in both cases, commutative or not, is that they satisfiy excision: they depend only on a smaller substructure in question, namely $X-Y$ in the topological case and $I$ in the noncommutative case. Specifically, the group $K^0(X,Y)$ is isomorphic to the group $K^0(X/Y,\{y\})$, where $\{y\}$ is the set $Y$ collapsed to a point, and the group $K_0(A,A/I)$ is isomorphic to the group $K_0(\tilde I, \mathbb C)$, where $\tilde I$ is the unitization of $I$ and we identify $\tilde I/I$ with $\mathbb C$.
To obtain a relative theory for a more general $^*$-homomorphism, we appeal to a construction of Karoubi in \cite{karoubi}. The approach is to describe the elements of the relative groups using triples consisting of two objects and a morphism from a Banach category. This generalizes the classical method of producing $K$-groups via vector bundles as seen in Atiyah's seminal work \cite{atiyah}: $K^0(X,Y)$ may be constructed via triples of the form $(E,F,\alpha)$, where $E$ and $F$ are vector bundles over $X$ and $\alpha:E|_Y\rightarrow F|_Y$ is an isomorphism between the bundles $E$ and $F$ when restricted to $Y$. This approach may appear somewhat basic because it is at odds with the mapping cone, a shortcut seen in both topological and operator $K$-theory. Indeed, analogous versions of the six-term exact sequences in the main result here (Theorem \ref{main}) are obtained very easily using standard methods if one uses $K_*(C_\phi)$ as the definition of relative $K$-theory, where $C_\phi$ denotes the mapping cone of $\phi:A\rightarrow B$. Moreover, it turns out that the two portraits are isomorphic in a natural way (part (iv) of Theorem \ref{main}). It is therefore reasonable to ask why one would employ an alternative portrait at all.
First, there is much more freedom in selecting the elements from the algebras that represent elements of the relative groups via the setup in \cite{karoubi}, which makes viewing and working with the groups easier in many situations. Second, certain maps in the six-term exact sequences are easier to compute. All in all, the resulting presentation is much simpler to work with, and applications of a simpler portrait in the context of C$^*$-algebras are beginning to make their way into the literature. Indeed, the proof of the excision theorem in \cite{putnam3}, which is a significant generalization of its predecessor in \cite{putnam2} (where the mapping cone was used), rests heavily on the new presentation, particularly on the notions of "isomorphism" of triples and "elementary" triples, to address certain fine, technical details. A portrait using partial isometries is developed in \cite{putnam2}, and in fact, the map $\kappa$ constructed there is essentially the same as $\Delta_\phi$ in part (iv) of Theorem \ref{main}, the difference being that $\Delta_\phi$ is a functorially induced group isomorphism, while $\kappa$ is a bijective map constructed concretely. Also worth mentioning is recent work on groupoid homology \cite{gabe}, where a portrait of the relative $K_0$-group using triples is presented in order to elucidate the connections between $K$-theory and homology. An isomorphism between the resulting monoid and the $K_0$-group of the mapping cone is constructed under the assumption that $A$ and $B$ are unital and $\phi$ is unit-preserving. The unital assumption can be done away with if $\phi$ is nondegenerate and $A$ contains an approximate identity of projections. Here, the general description of the relative groups allows us to do away with these assumptions. To provide further justification, we elaborate on the $K_0$- and $K_1$-groups separately.
It is a standard fact that the three usual notions of equivalence of projections, Murray-von Neumann, unitary, and homotopy, are all stably equivalent; in other words, they are the same modulo passing to matrix algebras. In constructing a relative $K_0$-group, it is therefore necessary to select a notion of equivalence with which to build the elements. The mapping cone $C_\phi$ is made from paths of projections in $B$ with one endpoint equal to a scalar projection and the other endpoint in the image of $\phi$. Therefore, in effect, $K_0(C_\phi)$ catalogues projections arising from $A$ that are homotopic when moved to $B$ via $\phi$. It is often more desirable to describe equivalences of projections using partial isometries (see Example \ref{diag}), from which the newer portrait is built. It is also worth mentioning that, although both portraits often require unitizations of the algebras involved, the appended unit seems to be less of a hindrance in the newer portrait.
As for $K_1$, homotopy is a much more natural equivalence and hence the portrait of the relative $K_1$ group gets less of a makeover than that of $K_0$. In fact, the two portraits are more or less the same. However, we draw a useful property from \cite{karoubi} which deserves to be mentioned. It is possible to define (ordinary and relative) $K_1$-groups more generally using partial unitaries (elements which are partial isometries and normal) rather than unitaries alone. This more general representation of group elements is especially convenient if one of (or both) $A$ and $B$ are not unital but contain nontrivial projections, such as $\mathcal K$, the compact operators on a separable Hilbert space (see Example \ref{gamma}).
The goal of the paper is to develop a clear picture of relative $K$-theory for C$^*$-algebras using the setup in \cite{karoubi}, as an alternative to the mapping cone. We remark that a preliminary development of the picture may be found in \cite{putnam3}, where the relative $K_0$-group of an inclusion $A'\subseteq A$ is described using this approach. We also remark that, although the intention in \cite{karoubi} is mainly to develop topological $K$-theory, the setup lends itself quite well to C$^*$-algebras.
The paper is organized as follows. In section 2 we state the theorems and discuss some examples. In section 3 we present the definition of relative $K$-theory in the context of C$^*$-algebras and show that the elements of the relative groups can be represented in a simple form. In section 4 we prove the results.
\section*{\normalsize Acknowledgements}
The content of this paper constitutes a portion of the research conducted for my PhD dissertation. I am grateful to my advisor Ian F. Putnam for suggesting the project to me, and for numerous helpful discussions. I also send my thanks to the referee for their careful reading and useful suggestions.
\section{Summary and examples}
Before we begin, we state a rather important remark regarding notation.
\begin{remark}
In \cite{rordam}, the symbols $K_0(\phi)$ and $K_1(\phi)$ are used to denote the group homomorphisms $K_0(A)\rightarrow K_0(B)$ and $K_1(A)\rightarrow K_1(B)$ induced by $\phi$. Throughout this paper, the symbols $K_0(\phi)$ and $K_1(\phi)$ will be used to denote the relative groups, not group homomorphisms. Induced maps will instead be denoted more classically as $\phi_*$. \end{remark}
\begin{definition} We define \textbf{C$^*$-hom} to be the following category. The objects of \textbf{C$^*$-hom} are $^*$-homomorphisms $\phi:A\rightarrow B$, where $A$ and $B$ are C$^*$-algebras. A morphism from the $^*$-homomorphism $\phi:A\rightarrow B$ to the $^*$-homomorphism $\psi:C\rightarrow D$ is a pair $(\alpha,\beta)$ of $^*$-homomorphisms $\alpha:A\rightarrow C$ and $\beta:B\rightarrow D$ such that the diagram \begin{equation}\label{comm} \begin{tikzcd} A \arrow[r, "\alpha"] \arrow[d, "\phi"] & C \arrow[d, "\psi"] \\ B \arrow[r, "\beta"] & D \end{tikzcd} \end{equation} commutes. If $(\alpha,\beta)$ is a morphism from $\phi:A\rightarrow B$ to $\psi:C\rightarrow D$ and $(\gamma,\delta)$ is a morphism from $\psi:C\rightarrow D$ to $\eta:E\rightarrow F$, their composition is $(\gamma\circ\alpha,\delta\circ\beta)$. The identity morphism from $\phi:A\rightarrow B$ to itself is \emph{$(\text{id}_A,\text{id}_B)$}, where \emph{$\text{id}_A:A\rightarrow A$} is the identity map, \emph{$\text{id}_A(a)=a$} for all $a$ in $A$. \end{definition}
As the reader can easily check, the class of $^*$-homomorphisms that map to the zero C$^*$-algebra, $\phi:A\rightarrow\{0\}$, forms a subcategory of \textbf{C$^*$-hom} that is isomorphic to the category of C$^*$-algebras in an obvious way. If one restricts to this subcategory, the usual definition of complex $K$-theory $K_*(A)$ for C$^*$-algebras is recovered through the construction of the relative theory $K_*(\phi)$.
We outline a simplified picture of the relative groups $K_0(\phi)$ and $K_1(\phi)$ to help understand the results and examples. Full definitions will be given in section 3.
The group $K_0(\phi)$ is made from triples $(p,q,v)$, where $p$ and $q$ are projections in some matrix algebras over $\tilde A$, the unitization of $A$, and $v$ is an element in a matrix algebra over $\tilde B$ such that $v^*v=\phi(p)$ and $vv^*=\phi(q)$ (if $A$ and $B$ are unital and $\phi(1)=1$, we may ignore unitizations, see Proposition \ref{unit}). The triples are sorted into equivalence classes, denoted $[p,q,v]$, and are given a well-defined group operation by the usual block diagonal sum, \[[p,q,v]+[p',q',v']=[p\oplus p',q\oplus q',v\oplus v']\] For two triples $(p,q,v)$ and $(p',q',v')$ to yield the same equivalence class, $p$ and $p'$ must be (at least stably) Murray-von Neumann equivalent in $\tilde A$, as must be $q$ and $q'$. Moreover, elements $c$ and $d$ implementing such equivalences must play well with $v$ and $v'$ in that we require $\phi(d)v=v'\phi(c)$.
$K_1(\phi)$ is made from triples $(p,u,g)$, where $p$ is a projection in $M_\infty(\tilde A)$, $u$ is a unitary in $pM_\infty(\tilde A)p$, and $g$ is a unitary in $C([0,1])\otimes\phi(p)M_\infty(\tilde B)\phi(p)$ such that $g(0)=\phi(p)$ and $g(1)=\phi(u)$. The triples are sorted into equivalence classes, denoted $[p,u,g]$, and are given a well-defined group operation by diagonal sum as before, although we have the formula \[[p,u,g]+[p',u',g']=[p,uu',gg']\] if $p=p'$. For two triples $(p,u,g)$ and $(p',u',g')$ to yield the same equivalence class, $p$ and $p'$ must be (at least stably) Murray-von Neumann equivalent in $\tilde A$, and a partial isometry $v$ implementing such an equivalence must satisfy $vu=u'v$ and $\phi(v)g(s)=g'(s)\phi(v)$ for $0\leq s\leq1$. The equivalence may also be described as stable homotopy: $u$ and $u'$ must be (at least stably) homotopic, as must be $g$ and $g'$. Moreover, such homotopies $u_t$ and $g_t$ must satisfy $g_t(1)=\phi(u_t)$ for $0\leq t\leq1$.
The assignments $\phi\mapsto K_0(\phi)$ and $\phi\mapsto K_1(\phi)$ are functors, as follows (see Proposition \ref{induce} for a proof). If we have the commutative diagram (\ref{comm}), that is, a morphism $(\alpha,\beta)$ from $\phi$ to $\psi$, then there are group homomorphisms $(\alpha,\beta)_*:K_j(\phi)\rightarrow K_j(\psi)$ for $j=0,1$ that satisfy $(\alpha,\beta)_*([p,q,v])=[\alpha(p),\alpha(q),\beta(v)]$ and $(\alpha,\beta)_*([p,u,g])=[\alpha(p),\alpha(u),\beta(g)]$.
\begin{theorem}\label{main} The constructions in section 3 produce, for every integer $n\geq0$, a functor $K_n$ from the category \textbf{C$^*$-hom} to the category of abelian groups that satisfies the following properties. \begin{enumerate}[(i)]
\item If $\phi:A\rightarrow B$ is a $^*$-homomorphism of C$^*$-algebras $A$ and $B$, the groups $K_0(\phi)$ and $K_1(\phi)$ fit into the six-term exact sequence
\begin{center}
\begin{tikzcd} K_1(B) \arrow[rr, "\mu_0"] & & K_0(\phi) \arrow[rr, "\nu_0"] & & K_0(A) \arrow[dd, "\phi_*"] \\
& & & & \\ K_1(A) \arrow[uu, "\phi_*"] & & K_1(\phi) \arrow[ll, "\nu_1"'] & & K_0(B) \arrow[ll, "\mu_1"'] \end{tikzcd}
\end{center}
The maps $\nu_0$ and $\nu_1$ are given by the formulas
\[\nu_0([p,q,v])=[p]-[q]\qquad\nu_1([p,u,g])=[u+1_n-p]\]
where $p$ and $q$ are projections in $M_n(\tilde A)$. The maps $\mu_0$ and $\mu_1$ are given by the formulas
\[\mu_0([u])=[1_n,1_n,u]\qquad\mu_1([p]-[q])=[1_n,1_n,f_pf_q^*]\]
where $u$ is a unitary in $M_n(\tilde B)$, and $f_p(t)=e^{2\pi itp}$ for a projection $p$ in $M_n(\tilde B)$. If $\phi(a)=0$ for all $a$ in $A$, then the sequence splits at $K_0(A)$ and $K_1(A)$, i.e., both $\nu_0$ and $\nu_1$ have a right inverse.
\item If
\begin{center} \begin{tikzcd} 0 \arrow[r] & I \arrow[r, "\iota_A"] \arrow[d, "\psi"] & A \arrow[r, "\pi_A"] \arrow[d, "\phi"] & A/I \arrow[r] \arrow[d, "\gamma"] & 0 \\ 0 \arrow[r] & J \arrow[r, "\iota_B"] & B \arrow[r, "\pi_B"] & B/J \arrow[r] & 0 \end{tikzcd} \end{center} is a commutative diagram with exact rows, then for each integer $n\geq1$, there is a natural connecting map $\partial_n:K_n(\gamma)\rightarrow K_{n-1}(\psi)$ such that the sequence\small
\begin{center} \begin{tikzcd} \cdots \arrow[r, "\pi_*"] & K_2(\gamma) \arrow[r, "\partial_2"] & K_1(\psi) \arrow[r, "\iota_*"] & K_1(\phi) \arrow[r, "\pi_*"] & K_1(\gamma) \arrow[r, "\partial_1"] & K_0(\psi) \arrow[r, "\iota_*"] & K_0(\phi) \arrow[r, "\pi_*"] & K_0(\gamma) \end{tikzcd}
\end{center}\normalsize
is exact.
\item The theory satisfies Bott periodicity. Specifically, for each $^*$-homomorphism and each integer $n\geq0$, there is an isomorphism $\beta_\phi:K_n(\phi)\rightarrow K_{n+2}(\phi)$ that is natural in the sense that if the diagram (\ref{comm}) is commutative, then the diagram
\begin{center} \begin{tikzcd} K_n(\phi) \arrow[dd, "\beta_\phi"] \arrow[rr, "{(\alpha,\beta)_*}"] & & K_n(\psi) \arrow[dd, "\beta_\psi"] \\
& & \\ K_{n+2}(\phi) \arrow[rr, "{(\alpha,\beta)_*}"] & & K_{n+2}(\psi) \end{tikzcd}
\end{center}
is commutative. It follows that the long exact sequence in part (ii) collapses to a six-term exact sequence
\begin{center} \begin{tikzcd} K_0(\psi) \arrow[rr, "\iota_*"] & & K_0(\phi) \arrow[rr, "\pi_*"] & & K_0(\gamma) \arrow[dd, "\partial_0"] \\
& & & & \\ K_1(\gamma) \arrow[uu, "\partial_1"] & & K_1(\phi) \arrow[ll, "\pi_*"] & & K_1(\psi) \arrow[ll, "\iota_*"] \end{tikzcd}
\end{center}
where the map $\partial_0:K_0(\gamma)\rightarrow K_1(\psi)$ is the composition $\partial_2\circ\beta_\gamma$.
\item\label{test} If $\phi:A\rightarrow B$ is a $^*$-homomorphism of C$^*$-algebras $A$ and $B$, there are isomorphisms $\Delta_\phi:K_*(\phi)\rightarrow K_*(C_\phi)$ that are natural in the sense that if (\ref{comm}) is commutative, then the diagram
\begin{center} \begin{tikzcd} K_j(\phi) \arrow[dd, "\Delta_\phi"] \arrow[rr, "{(\alpha,\beta)_*}"] & & K_j(\psi) \arrow[dd, "\Delta_\psi"] \\
& & \\ K_j(C_\phi) \arrow[rr, "(\alpha\oplus C\beta)_*"] & & K_j(C_\psi) \end{tikzcd}
\end{center}
is commutative, where $C_\phi$ is the mapping cone of $\phi$. \end{enumerate} \end{theorem}
Regarding part (iv), the conclusion actually implies that the isomorphisms implement an invertible natural transformation between the functors $\phi\mapsto K_*(\phi)$ and $\phi\mapsto K_*(C_\phi)$ from the category \textbf{C$^*$-hom} to the category of abelian groups. When we speak of the transformation, we will denote it simply by $\Delta$.
We also collect some properties of the homology theory that are analogues of properties of C$^*$-algebra $K$-theory.
\begin{theorem}\label{axiom} The homology theory $(K_n)_{n\geq0}$ on \textbf{C$^*$-hom} has the following properties. \begin{enumerate}[(i)]
\item Homotopy invariance: suppose that $\alpha_t:A\rightarrow C$ and $\beta_t:B\rightarrow D$ are $^*$-homomorphisms for every $0\leq t\leq1$, the maps $t\mapsto\alpha_t(a)$ and $t\mapsto\beta_t(b)$ are continuous for every $a$ in $A$ and every $b$ in $B$, and the diagram
\begin{center}
\begin{tikzcd} A \arrow[r, "\alpha_t"] \arrow[d, "\phi"] & C \arrow[d, "\psi"] \\ B \arrow[r, "\beta_t"] & D \end{tikzcd}
\end{center}
is commutative for every $0\leq t\leq1$. Then $(\alpha_0,\beta_0)_*=(\alpha_1,\beta_1)_*$.
\item Stability: if $p$ is any rank one projection in $\mathcal K$ and $\kappa_A:A\rightarrow A\otimes\mathcal K$ is defined by $\kappa_A(a)=a\otimes p$ for every C$^*$-algebra $A$, then the morphism
\begin{center} \begin{tikzcd} A \arrow[r, "\kappa_A"] \arrow[d, "\phi"] & A\otimes\mathcal K \arrow[d, "\phi\otimes\text{id}_\mathcal K"] \\ B \arrow[r, "\kappa_B"] & B\otimes\mathcal K \end{tikzcd}
\end{center}
induces an isomorphism \emph{$(\kappa_A,\kappa_B)_*:K_*(\phi)\rightarrow K_*(\phi\otimes\text{id}_\mathcal K)$}.
\item Continuity: suppose that $(A,\mu_i)$ is the inductive limit (in the category of C$^*$-algebras and $^*$-homomorphisms) of the inductive system $(A_i,\alpha_{ij})$, and $(B,\nu_i)$ is likewise the inductive limit of the inductive system $(B_i,\beta_{ij})$. Suppose also that, for each pair of indices $i\leq j$, there are $^*$-homomorphisms $\phi_i:A_i\rightarrow B_i$ and $\phi_j:A_j\rightarrow B_j$ such that the diagram
\begin{center} \begin{tikzcd} A_i \arrow[r, "\alpha_{ij}"] \arrow[d, "\phi_i"] & A_j \arrow[d, "\phi_j"] \\ B_i \arrow[r, "\beta_{ij}"] & B_j \end{tikzcd}
\end{center}
is commutative. Then there exists a $^*$-homomorphism $\phi:A\rightarrow B$ such that $(\phi,(\mu_i,\nu_i))$ is the inductive limit of $(\phi_i,(\alpha_{ij},\beta_{ij}))$ in the category \textbf{C$^*$-hom}, and $(K_*(\phi),(\mu_i,\nu_i)_*)$ is isomorphic to the inductive limit of $(K_*(\phi_i),(\alpha_{ij},\beta_{ij})_*)$ in the category of abelian groups.
\end{enumerate}
\end{theorem}
We also collect some excision results that follow easily from Theorem \ref{main}. When $A$ is a C$^*$-subalgebra of $B$ and $\phi:A\rightarrow B$ is the inclusion map, we denote $K_*(\phi)$ by $K_*(A,B)$.
\begin{theorem}\label{exc} Let $\phi:A\rightarrow B$ be a $^*$-homomorphism of C$^*$-algebras $A$ and $B$.
\begin{enumerate}[(i)]
\item If $K_*(B)=0$, the maps $\nu_0:K_0(\phi)\rightarrow K_0(A)$ and $\nu_1:K_1(\phi)\rightarrow K_1(A)$ in part (i) of Theorem \ref{main} are isomorphisms.
\item If $K_*(A)=0$, the maps $\mu_0:K_1(B)\rightarrow K_0(\phi)$ and $\mu_1:K_0(A)\rightarrow K_1(\phi)$ in part (i) of Theorem \ref{main} are isomorphisms.
\item If $\phi$ is surjective, the morphism
\begin{center} \begin{tikzcd} \ker\phi \arrow[d] \arrow[r, "\iota_\phi"] & A \arrow[d, "\phi"] \\ 0 \arrow[r] & B \end{tikzcd}
\end{center}
where $\iota_\phi:\ker\phi\rightarrow A$ is the inclusion map, induces a natural isomorphism $(\iota_\phi,0)_*:K_*(\ker\phi)\rightarrow K_*(\phi)$.
\item If $I$ is a closed, two-sided ideal in the C$^*$-algebra $A$, then there are natural isomorphisms $K_0(A/I)\cong K_1(I,A)$ and $K_1(A/I)\cong K_0(I,A)$. \end{enumerate} \end{theorem}
We now discuss some examples to illustrate the utility of parts (i) and (iii) of Theorem \ref{main}.
\begin{example}\label{diag} Let $D$ be any C$^*$-algebra, and let $A$ be the subalgebra of $B=M_2(D)$ consisting of the diagonal matrices. Since $K_*(A)\cong K_*(D)\oplus K_*(D)$ and $K_*(B)\cong K_*(D)$, we may write the six-term exact sequence of part (i) of Theorem \ref{main} as
\begin{center} \begin{tikzcd} K_1(D) \arrow[rr] & & K_0(A,B) \arrow[rr] & & K_0(D)\oplus K_0(D) \arrow[dd, "\phi_*"] \\
& & & & \\ K_1(D)\oplus K_1(D) \arrow[uu, "\phi_*"] & & K_1(A,B) \arrow[ll] & & K_0(D) \arrow[ll] \end{tikzcd} \end{center}
The vertical maps are both $\phi_*(g,h)=g+h$. Exactness implies that $K_*(A,B)\cong\ker\phi_*\cong K_*(D)$.
As a special case of interest, let $\mathcal H$ be a separable Hilbert space of dimension at least $2$, and $\mathcal M$ a closed subspace such that $\mathcal M\neq\{0\}$ and $\mathcal M\neq\mathcal H$. Let $A=\mathcal K(\mathcal M)\oplus\mathcal K(\mathcal M^\perp)$, where $\mathcal K(\mathcal M)$ is the C$^*$-algebra of compact operators on $\mathcal M$, regarded as a subalgebra of $B=\mathcal K(\mathcal H)$ as operators that leave $\mathcal M$ and $\mathcal M^\perp$ invariant. Then $K_0(A,B)\cong\mathbb Z$ and $K_1(A,B)=0$. If we fix a unit vector $\xi$ in $\mathcal M$, a unit vector $\eta$ in $\mathcal M^\perp$, and a partial isometry $v$ in $B$ with source subspace $\text{span}\{\xi\}$ and range subspace $\text{span}\{\eta\}$, the group $K_0(A,B)$ is generated by the class of the triple $(v^*v,vv^*,v)$. \end{example}
\begin{example}\label{gamma} Let $D$ be any C$^*$-algebra and consider $A=D$ as a subalgebra of $B=D\oplus D$ via the embedding $d\mapsto(d,d)$. The six-term exact sequence of part (i) of Theorem \ref{main} becomes
\begin{center} \begin{tikzcd} K_1(D)\oplus K_1(D) \arrow[rr] & & K_0(A,B) \arrow[rr] & & K_0(D) \arrow[dd, "\phi_*"] \\
& & & & \\ K_1(D) \arrow[uu, "\phi_*"] & & K_1(A,B) \arrow[ll] & & K_0(D)\oplus K_0(D) \arrow[ll] \end{tikzcd} \end{center}
This time the vertical maps are $\phi_*(g)=(g,g)$, which are injective, whence exactness implies $K_0(A,B)\cong K_1(D)$ and $K_1(A,B)\cong K_0(D)$. In the case that $D=\mathcal K$, the group $K_1(A,B)\cong\mathbb Z$ is generated by the class of the triple $(p,p,g)$, where $p$ is a rank one projection in $\mathcal K$ and $g(s)=(e^{2\pi is}p,p)$. Observe that we do not need to consider the unit in the unitization $\tilde\mathcal K$ to describe the group $K_1(A,B)$.
\end{example}
\begin{example} Consider the diagram \begin{center} \begin{tikzcd} 0 \arrow[r] & C_0(\mathbb R^2) \arrow[r] \arrow[d, "\psi"] & C(\mathbb D) \arrow[r] \arrow[d, "\phi"] & C(\mathbb T) \arrow[r] \arrow[d, hook] & 0 \\ 0 \arrow[r] & 0 \arrow[r] & {C([0,1])} \arrow[r, equal] & {C([0,1])} \arrow[r] & 0 \end{tikzcd} \end{center} where $C_0(\mathbb R^2)$ is identified with functions that vanish on the boundary of $\mathbb D=\{(x,y)\in\mathbb R^2\mid x^2+y^2\leq1\}$. The algebra $C(\mathbb T)$ is viewed as all $f$ in $C([0,1])$ with $f(0)=f(1)$, and $\phi$ is the composition of the restriction to the boundary $C(\mathbb D)\rightarrow C(\mathbb T)$ with the inclusion $C(\mathbb T)\hookrightarrow C([0,1])$. We have $K_0(\phi)=K_1(\phi)=0$ since $K_1(C([0,1]))=K_1(C(\mathbb D))=0$ (see Corollary \ref{zero}) and the induced map $\phi_*:K_0(C(\mathbb D))\rightarrow K_0(C([0,1]))$ is an isomorphism. The six-term exact sequence of part (iii) of Theorem \ref{main} becomes \begin{center} \begin{tikzcd} K_0(C_0(\mathbb R^2)) \arrow[rr] & & 0 \arrow[rr] & & K_0(C(\mathbb T),C([0,1])) \arrow[dd, "\partial_0"] \\
& & & & \\ K_1(C(\mathbb T),C([0,1])) \arrow[uu, "\partial_1"] & & 0 \arrow[ll] & & K_1(C_0(\mathbb R^2)) \arrow[ll] \end{tikzcd}
\end{center} (we identify $K_*(\psi)$ with $K_*(C_0(\mathbb R^2))$ using Theorem \ref{exc}). It can be shown that $K_1(C(\mathbb T),C([0,1]))\cong\mathbb Z$ is generated by the class of $(1,z,g)$ where $z$ is the function $z\mapsto z$ on $\mathbb T$ and $g(t)=f_t$, where $f_t(s)=e^{2\pi ist}$. Using the notation in Definition \ref{index}, let $l=1$, \[w=\left[\begin{array}{cc}
z & -(1-|z|^2)^{1/2} \\
(1-|z|^2)^{1/2} & \overline z \end{array}\right]\] and $h=g$. Then \[\partial_1([1,z,g])=\left[\left[\begin{array}{cc}
|z|^2 & z(1-|z|^2)^{1/2} \\
\overline z(1-|z|^2)^{1/2} & 1-|z|^2 \end{array}\right]\right]-\left[\left[\begin{array}{cc}
1 & 0 \\
0 & 0 \end{array}\right]\right]\] \end{example}
\begin{example} Let $D$ be the diagonal matrices in $M_2(\mathbb C)$. Consider the diagram \begin{center} \begin{tikzcd} 0 \arrow[r] & C_0(\mathbb R) \arrow[r] \arrow[d] & C([0,1]) \arrow[r, "\pi"] \arrow[d, "\phi"] & D \arrow[r] \arrow[d, hook] & 0 \\ 0 \arrow[r] & 0 \arrow[r] & M_2(\mathbb C) \arrow[r, equal] & M_2(\mathbb C) \arrow[r] & 0 \end{tikzcd} \end{center} Where $C_0(\mathbb R)$ is identified with all functions in $C([0,1])$ that vanish at the endpoints, the map $\pi:C([0,1])\rightarrow D$ is defined by \[\pi(f)=\left[\begin{array}{cc}
f(0) & 0 \\
0 & f(1) \end{array}\right],\] and $\phi$ is the composition of $\pi$ with the inclusion $D\hookrightarrow M_2(\mathbb C)$. We have $K_0(C_0(\mathbb R))=0$ and $K_1(D,M_2(\mathbb C))=0$, the latter by Example \ref{diag}. The six-term exact sequence of part (iii) of Theorem \ref{main} becomes \begin{center} \begin{tikzcd} 0 \arrow[rr] & & K_0(\phi) \arrow[rr] & & K_0(D,M_2(\mathbb C)) \arrow[dd, "\partial_0"] \\
& & & & \\ 0 \arrow[uu] & & K_1(\phi) \arrow[ll] & & K_1(C_0(\mathbb R)) \arrow[ll] \end{tikzcd}
\end{center} Using part (i) of Theorem \ref{main}, it can be shown that $K_0(\phi)=0$ and $K_1(\phi)\cong\mathbb Z/2\mathbb Z$, with the nontrivial element in $K_1(\phi)$ given by the class of the triple $(1,1,g)$, where \[g(s)=\left[\begin{array}{cc}
e^{2\pi is} & 0 \\
0 & 1 \end{array}\right]\] The map $\partial_0$ is therefore injective and takes a generator of $K_0(D,M_2(\mathbb C))\cong\mathbb Z$ to twice a generator of $K_1(C_0(\mathbb R))\cong\mathbb Z$. More concretely, \[\partial_0\left(\left[\left[\begin{array}{cc}
0 & 0 \\
0 & 1 \end{array}\right],\left[\begin{array}{cc}
1 & 0 \\
0 & 0 \end{array}\right],\left[\begin{array}{cc}
0 & 1 \\
0 & 0 \end{array}\right]\right]\right)=-[z^2]\] where $z$ is the function $z\mapsto z$ on $C(\mathbb T)\cong C_0(\mathbb R)^\sim$. \end{example}
The final example illustrates the excision theorem of \cite{putnam3}. We refer to \cite{groupoid} for a detailed account of constructing a C$^*$-algebra $C_r^*(R)$ from an equivalence relation $R$, or, more generally, from an \'etale groupoid.
\begin{example} Let $X=\{0,1\}^\mathbb N$, the space of all sequences of $0$'s and $1$'s with the product topology. Define the surjective map $\omega:X\rightarrow\mathbb T$ by \[\omega(\{x_n\})=\exp\left(2\pi i\sum_{n=1}^\infty x_n2^{-n}\right).\] Define $S\subseteq X\times X$ to be \emph{tail-equivalence} on $X$, that is, \[(\{x_n\},\{y_n\})\in S\Longleftrightarrow x_n=y_n\text{ for sufficiently large }n.\]
The equivalence relation $S$ has a natural topology under which it is a Hausdorff, \'etale groupoid, and $C_r^*(S)$ is isomorphic to the UHF algebra $M_{2^\infty}$. Define $T=\omega\times\omega(S)$; it is a consequence of the main result of \cite{haslehurst} that $T$, with the quotient topology from $S$ and the map $\omega\times\omega|_S$, is a Hausdorff, \'etale groupoid. It has the following concrete description, as can be easily checked: \[\textstyle T=\{(w,z)\in\mathbb T\times\mathbb T\mid w=e^{2\pi i\theta}z\text{ for some }\theta\in\mathbb Z[\frac12]\},\] where $\mathbb Z\left[\frac12\right]=\left\{\frac{k}{2^n}\mid k\in\mathbb Z\text{ and } n\geq0\right\}$. Moreoever, the map $\omega\times\omega$ satisfies the standing hypotheses of section 7 of \cite{putnam3}, and thus induces an injective $^*$-homomorphism from $C_r^*(T)$ to $C_r^*(S)$, enabling us to regard $C_r^*(T)$ as a C$^*$-subalgebra of $C_r^*(S)$.
Let $\mathbb T_D=\{e^{2\pi i\theta}\mid\theta\in\mathbb Z[\frac12]\}$. Notice that $\mathbb T_D$ is exactly where the map $\omega$ is not one-to-one. The main result from section 7 of \cite{putnam3} says (avoiding some technical details) that the relative $K$-theory of the inclusion $C_r^*(T)\subseteq C_r^*(S)$ can be computed by looking only where the map $\omega\times\omega$ is not one-to-one. More precisely, \[K_*(C_r^*(T),C_r^*(S))\cong K_*(\mathcal K(l^2(\mathbb T_D)),\mathcal K(l^2(\mathbb T_D))\oplus\mathcal K(l^2(\mathbb T_D))).\] Where we regard $\mathcal K(l^2(\mathbb T_D))$ as a C$^*$-subalgebra of $\mathcal K(l^2(\mathbb T_D))\oplus\mathcal K(l^2(\mathbb T_D))$ as in Example \ref{gamma}. The six-term exact sequence of part (i) of Theorem \ref{main} becomes \begin{center} \begin{tikzcd} 0 \arrow[rr] & & K_0(C_r^*(T),C_r^*(S)) \arrow[rr] & & K_0(C_r^*(T)) \arrow[dd] \\
& & & & \\ K_1(C_r^*(T)) \arrow[uu] & & K_1(C_r^*(T),C_r^*(S)) \arrow[ll] & & K_0(C_r^*(S)) \arrow[ll] \end{tikzcd}
\end{center} Example \ref{gamma} allows us to conclude that $K_0(C_r^*(T),C_r^*(S))=0$ and $K_1(C_r^*(T),C_r^*(S))\cong\mathbb Z$. It may also be shown that the right vertical map is surjective, see Lemma 5.5 of \cite{haslehurst}. We thus obtain that $K_0(C_r^*(T))\cong\mathbb Z[\frac12]$ (with the order inherited from $\mathbb R$) and $K_1(C_r^*(T))\cong\mathbb Z$. \end{example}
The previous example illustrates the simplest case of a general construction involving Bratteli diagrams; we refer to section 3 of \cite{haslehurst} for more information.
\section{Definitions and a portrait of $K_*(\phi)$}
We begin by establishing some notation and terminology. If $A$ is a C$^*$-algebra, we let $\tilde A$ denote its unitization. If $a$ is in $\tilde A$, let $\dot a$ denote the scalar part of $a$. Let $M_n(A)$ denote the $n\times n$ matrices with entries in $A$, regarded as a C$^*$-algebra in the usual way. Let $M_\infty(\tilde A)$ be the union $\bigcup_{n=1}^\infty M_n(\tilde A)$, which may be regarded as an increasing union by means of the inclusions $M_n(\tilde A)\subseteq M_{n+1}(\tilde A)$, $a\mapsto\text{diag}(a,0)$. If $a$ and $b$ are in $M_\infty(\tilde A)$ we define
\[a\oplus b=\left[\begin{array}{cc}
a & 0 \\
0 & b
\end{array}\right]\]
Admittedly, there is some ambiguity in the above definition of $a\oplus b$ since $a$ and $b$ may be regarded as matrices of arbitrarily large size. However, since $K$-theory doesn't distinguish elements that are ``moved down the diagonal", there will be negligible harm done by ignoring this technical issue. We denote by $1_n$ the identity matrix in $M_n(\tilde A)$, or the matrix in $M_\infty(\tilde A)$ with $n$ consecutive occurences of $1$ down the diagonal, and $0$ elsewhere.
An element $p$ of a C$^*$-algebra is called a \textit{projection} if $p=p^2=p^*$. An element $u$ of a unital C$^*$-algebra is called a \textit{unitary} if $u^*u=uu^*=1$. An element $v$ of a C$^*$-algebra is called a \textit{partial isometry} if $v^*v$ is a projection (in which case, $vv^*$ is also a projection). Two projections $p$ and $q$ in $M_\infty(\tilde A)$ are called \emph{Murray-von Neumann equivalent} if there is a partial isometry $v$ in $M_\infty(\tilde A)$ such that $v^*v=p$ and $vv^*=q$, and we will say in this situation that $v$ \textit{is a partial isometry from $p$ to $q$}. It is straightforward to check the useful formulas $v=qv=vp=qvp$.
The group $K_0(\tilde A)$ is the Grothendieck completion of the semigroup of Murray-von Neumann classes of projections in $M_\infty(\tilde A)$ with the operation $[p]+[q]=[p\oplus q]$. The group $K_0(A)$ is the kernel of the map $K_0(\tilde A)\rightarrow\mathbb Z$ induced by the scalar map $\tilde A\rightarrow\mathbb C$. The group $K_1(A)$ is the group of stable homotopy classes of unitaries over $\tilde A$ with the operation $[u]+[v]=[uv]$ (regard $u$ and $v$ as elements of the same matrix algebra so that the product is well-defined). Every element of $K_0(A)$ may be represented by a formal difference $[p]-[q]$ of classes such that $\dot p=1_n$ and $q=1_n$ for some $n$, see the discussion following 5.5.1 in \cite{blackadar}. Every element of $K_1(A)$ may be represented by a class $[u]$ such that $u$ is in $M_n(\tilde A)$ and $\dot u=1_n$.
We denote the \textit{compact operators} on a separable Hilbert space by $\mathcal K$. We write $SA$ for $C_0((0,1))\otimes A=C_0((0,1),A)$, the \emph{suspension} of $A$, and $CA$ for $C_0((0,1])\otimes A=C_0((0,1],A)$, the \textit{cone} of $A$. If $\phi:A\rightarrow B$ is a $^*$-homomorphism, we commit the usual notation abuse and denote the obvious induced maps $\tilde A\rightarrow\tilde B$, $SA\rightarrow SB$, $CA\rightarrow CB$, $M_n(A)\rightarrow M_n(B)$ (or any combination of these) by $\phi$. Clarity will sometimes be needed for the first three, in which case they will be denoted by $\tilde\phi$, $S\phi$, and $C\phi$ respectively. The \emph{mapping cone} $C_\phi$ of $\phi$ is defined to be the pullback of $\phi$ and the map $\pi_B:CB\rightarrow B$, $\pi_B(f)=f(1)$. In other words, it is the C$^*$-algebra
\[C_\phi=\{(a,f)\mid f(1)=\phi(a)\}\subseteq A\oplus CB.\]
We denote the induced maps $K_j(A)\rightarrow K_j(B)$ by $\phi_*$ for both $j=0,1$. We denote the natural isomorphism $K_1(A)\rightarrow K_0(SA)$ by $\theta_A$ and the Bott map $K_0(A)\rightarrow K_1(SA)$ by $\beta_A$. If
\begin{center} \begin{tikzcd} 0 \arrow[r] & I \arrow[r, "\iota"] & A \arrow[r, "\pi"] & A/I \arrow[r] & 0 \end{tikzcd}
\end{center}
is a short exact sequence of C$^*$-algebras, we denote the index maps $K_n(A/I)\rightarrow K_{n-1}(I)$ by $\delta_n$ for $n\geq1$ and the exponential map $K_0(A/I)\rightarrow K_1(I)$ by $\delta_0$. We refer the reader to \cite{blackadar} or \cite{rordam} for more details on $K$-theory for C$^*$-algebras.
\begin{definition}\label{big} Define $\Gamma_0(\phi)$ to be the set of all triples $(p,q,v)$ where $p$ and $q$ are projections in $M_\infty(\tilde A)$ and $v$ is a partial isometry in $M_\infty(\tilde B)$ from $\phi(p)$ to $\phi(q)$ (recall this means that $v^*v=\phi(p)$ and $vv^*=\phi(q)$). For brevity, we will often denote these triples by the symbols $\sigma$ and $\tau$. \begin{enumerate}[(i)]
\item Define the direct sum operation $\oplus$ on $\Gamma_0(\phi)$ by
\[(p,q,v)\oplus(p',q',v')=(p\oplus p',q\oplus q',v\oplus v').\]
\item We say that two triples $(p,q,v)$ and $(p',q',v')$ are \emph{isomorphic}, written $(p,q,v)\cong(p',q',v')$, if there exist partial isometries $c$ and $d$ from $p$ to $p'$ and from $q$ to $q'$, respectively, that intertwine $v$ and $v'$, that is, $\phi(d)v=v'\phi(c)$.
\item A triple $(p,q,v)$ is called \emph{elementary} if $p=q$ and there is a homotopy $v_t$ for $0\leq t\leq1$ such that $v_0=\phi(p)$, $v_1=v$, and $v_t^*v_t=v_tv_t^*=\phi(p)$ for all $t$.
\item Two triples $\sigma$ and $\sigma'$ in $\Gamma_0(\phi)$ are \emph{equivalent}, written $\sigma\sim\sigma'$, if there exist elementary triples $\tau$ and $\tau'$ such that $\sigma\oplus\tau\cong\sigma'\oplus\tau'$. \end{enumerate}
Denote by $[\sigma]$, or $[p,q,v]$, the equivalence class of the triple $\sigma=(p,q,v)$ via the relation $\sim$. $K_0(\phi)$ is then defined to be the quotient of $\Gamma_0(\phi)$ by the relation $\sim$, that is,
\[\{[\sigma]\mid\sigma\in\Gamma_0(\phi)\}=\Gamma_0(\phi)/\sim\]
\end{definition}
We make two simple observations. First, the notions of isomorphism and elementary for triples behave well with respect to the direct sum operation: if $\sigma_1\cong\sigma_2$ and $\sigma_3\cong\sigma_4$, then $\sigma_1\oplus\sigma_3\cong\sigma_2\oplus\sigma_4$, for any two triples $\sigma$ and $\sigma'$, we have $\sigma\oplus\sigma'\cong\sigma'\oplus\sigma$, and if $\sigma$ and $\sigma'$ are elementary, then so is $\sigma\oplus\sigma'$. Second, all elementary triples are equivalent to each other, and two isomorphic triples are equivalent.
We recall the following useful fact: if $u$ is a self-adjoint unitary in a unital C$^*$-algebra $A$, then $u=e^{i\pi(1-u)/2}$. To see this, note that $(1-u)^2=2(1-u)$, so $(1-u)^n=2^{n-1}(1-u)$ by induction, and hence
\[e^{i\pi(1-u)/2}=\sum_{n=0}^\infty\frac{(i\pi(1-u)/2)^n}{n!}=1+\sum_{n=1}^\infty\frac12\frac{(i\pi)^n}{n!}(1-u)=1+\frac12(e^{i\pi}-1)(1-u)=u\]
it follows that $u$ is homotopic to $1$ via the path $e^{i\pi t(1-u)/2}$ for $0\leq t\leq1$.
\begin{prop}\label{group} $K_0(\phi)$ is an abelian group when equipped with the binary operation
\[[\sigma]+[\sigma']=[\sigma\oplus\sigma']\]
where the identity element is given by $[0,0,0]$, and the inverse of $[p,q,v]$ is given by $[q,p,v^*]$.
\end{prop}
\begin{proof} That $K_0(\phi)$ is an abelian group follows quite readily from the observations above, and the fact that $[0,0,0]$ is the identity element is all but trivial since we identify $a$ with $a\oplus0$ in $M_\infty(\tilde A)$. To prove the last statement, note that
\[[p,q,v]+[q,p,v^*]=[p\oplus q,q\oplus p,v\oplus v^*]\]
and the triple $(p\oplus q,q\oplus p,v\oplus v^*)$ is isomorphic to the triple
\begin{equation}\label{slfadjunt}
\left(\left[\begin{array}{cc}
p & 0 \\
0 & q
\end{array}\right],\left[\begin{array}{cc}
p & 0 \\
0 & q
\end{array}\right],\left[\begin{array}{cc}
0 & v^* \\
v & 0
\end{array}\right]\right)
\end{equation}
by taking
\[d=\left[\begin{array}{cc}
0 & p \\
q & 0
\end{array}\right],\qquad c=\left[\begin{array}{cc}
p & 0 \\
0 & q
\end{array}\right]\]
in part (ii) of Definition \ref{big}. The partial isometry in (\ref{slfadjunt}) is a self-adjoint unitary in the C$^*$-algebra $(\phi(p)\oplus\phi(q))M_\infty(\tilde B)(\phi(p)\oplus\phi(q))$, and is thus homotopic to the identity $\phi(p)\oplus\phi(q)$. Therefore, (\ref{slfadjunt}) is elementary, its class is zero, and hence the class of $(p\oplus q,q\oplus p,v\oplus v^*)$ is zero because it is isomorphic to (\ref{slfadjunt}).
\end{proof}
We collect some useful properties of the elements of $K_0(\phi)$.
\begin{prop}\label{3}
\begin{enumerate}[(i)]
\item Suppose that $p$ and $q$ are projections in $M_\infty(\tilde A)$ and $v$ and $v'$ are two partial isometries in $M_\infty(\tilde B)$ from $\phi(p)$ to $\phi(q)$. If there is a continuous path $v_t$ of partial isometries such that $v_0=v$, $v_1=v'$, and $v_t^*v_t=\phi(p)$ and $v_tv_t^*=\phi(q)$ for all $0\leq t\leq1$, then $[p,q,v]=[p,q,v']$.
\item Suppose that $p$, $q$, and $r$ are projections in $M_\infty(\tilde A)$ and $v$ and $w$ are partial isometries in $M_\infty(\tilde B)$ from $\phi(p)$ to $\phi(q)$ and from $\phi(q)$ to $\phi(r)$, respectively. Then
\[[p,q,v]+[q,r,w]=[p,r,wv].\]
\item Let $(p,q,v)$ and $(p',q',v')$ be two triples in $\Gamma_0(\phi)$. If $pp'=0$, then
\[(p,q,v)\oplus(p',q',v')\cong\left(p+p',q\oplus q',\left[\begin{array}{cc}
v & 0\\
v' & 0
\end{array}\right]\right)\]
If $qq'=0$, then
\[(p,q,v)\oplus(p',q',v')\cong\left(p\oplus p',q+q',\left[\begin{array}{cc}
v & v' \\
0 & 0
\end{array}\right]\right)\]
If $pp'=qq'=0$, then
\[(p,q,v)\oplus(p',q',v')\cong(p+p',q+q',v+v')\]
\item Every triple in $\Gamma_0(\phi)$ is equivalent to one of the form $(p,1_n,v)$, where $n\geq1$ and $\dot p=\dot v=1_n$, and one of the form $(1_n,q,v)$, where $n\geq1$ and $\dot q=\dot v=1_n$.
\item $[p,q,v]=0$ if and only if there exist projections $r$ and $s$ in $M_\infty(\tilde A)$ and partial isometries $x$ and $y$ in $M_\infty(\tilde A)$ from $p\oplus r$ to $s$ and $q\oplus r$ to $s$, respectively, such that $\phi(y)(v\oplus\phi(r))\phi(x^*)$ and $\phi(s)$ are homotopic as unitaries through $\phi(s)M_\infty(\tilde B)\phi(s)$.
Suppose $m\geq n$, $p$ is in $M_m(\tilde A)$, and $(p,1_n,v)$ is a triple in $\Gamma_0(\phi)$ with $\dot p=\dot v=1_n$. Then $[p,1_n,v]=0$ if and only if there exist $k\geq0$ and a partial isometry $w$ in $M_{m+k}(\tilde A)$ with $w^*w=\dot w=1_n\oplus0_{m-n}\oplus1_k$ and $ww^*=p\oplus1_k$ such that $(v\oplus1_k)\phi(w)$ and $1_n\oplus0_{m-n}\oplus1_k$ are homotopic as unitaries in $(1_n\oplus0_{m-n}\oplus1_k)M_{m+k}(\tilde B)(1_n\oplus0_{m-n}\oplus1_k)$.
\end{enumerate}
\end{prop}
\begin{proof} \begin{enumerate}[(i)]
\item We have
\[[p,q,v]-[p,q,v']=[p,q,v]+[q,p,v'^*]=[p\oplus q,q\oplus p,v\oplus v'^*]\]
and the triple $(p\oplus q,q\oplus p,v\oplus v'^*)$ is isomorphic to
\[\left(\left[\begin{array}{cc}
p & 0 \\
0 & q
\end{array}\right],\left[\begin{array}{cc}
p & 0 \\
0 & q
\end{array}\right],\left[\begin{array}{cc}
0 & v'^* \\
v & 0
\end{array}\right]\right)\]
similarly as in the proof of Proposition \ref{group}. The partial isometry in the above triple is homotopic to a self-adjoint unitary in $(\phi(p)\oplus\phi(q))M_\infty(\tilde B)(\phi(p)\oplus\phi(q))$ via the path
\[\left[\begin{array}{cc}
0 & v_t^* \\
v & 0
\end{array}\right].\]
It follows that the triple is elementary.
\item We have
\[[p,q,v]+[q,r,w]=[p\oplus q,q\oplus r,v\oplus w]\]
and observe that the triple $(p\oplus q,q\oplus r,v\oplus w)$ is isomorphic to the triple
\[\left(\left[\begin{array}{cc}
p & 0 \\
0 & q
\end{array}\right],\left[\begin{array}{cc}
r & 0 \\
0 & q
\end{array}\right],\left[\begin{array}{cc}
0 & w \\
v & 0
\end{array}\right]\right).\]
by taking
\[d=\left[\begin{array}{cc}
0 & r \\
q & 0
\end{array}\right],\qquad c=\left[\begin{array}{cc}
p & 0 \\
0 & q
\end{array}\right]\]
in part (ii) of Definition \ref{big}. We also have that $[p,r,wv]=[p\oplus q,r\oplus q,wv\oplus\phi(q)]$ since $(q,q,\phi(q))$ is elementary. Now
\[\left[\begin{array}{cc}
0 & w \\
v & 0
\end{array}\right]=\left[\begin{array}{cc}
0 & w \\
w^* & 0
\end{array}\right]\left[\begin{array}{cc}
wv & 0 \\
0 & \phi(q)
\end{array}\right]\]
and the second matrix above is homotopic to $\phi(r)\oplus\phi(q)$. It follows that the two matrices
\[\left[\begin{array}{cc}
0 & w \\
v & 0
\end{array}\right]\qquad\left[\begin{array}{cc}
wv & 0 \\
0 & \phi(q)
\end{array}\right]\]
are homotopic, and hence, by part (i), the two triples
\[\left(\left[\begin{array}{cc}
p & 0 \\
0 & q
\end{array}\right],\left[\begin{array}{cc}
r & 0 \\
0 & q
\end{array}\right],\left[\begin{array}{cc}
0 & w \\
v & 0
\end{array}\right]\right),\qquad\left(\left[\begin{array}{cc}
p & 0 \\
0 & q
\end{array}\right],\left[\begin{array}{cc}
r & 0 \\
0 & q
\end{array}\right],\left[\begin{array}{cc}
wv & 0 \\
v & \phi(q)
\end{array}\right]\right)\]
are equivalent.
\item If $pp'=0$, then $v\phi(p')=v\phi(p)\phi(p')=0$, and similarly $v'\phi(p)=0$. Thus we have \[\left[\begin{array}{cc}
\phi(q) & 0 \\
0 & \phi(q')
\end{array}\right]\left[\begin{array}{cc}
v & 0 \\
0 & v'
\end{array}\right]=\left[\begin{array}{cc}
v & 0 \\
v' & 0
\end{array}\right]\left[\begin{array}{cc}
\phi(p) & \phi(p') \\
0 & 0
\end{array}\right]\]
so the triples are isomorphic. The other two claims are similar.
\item
We show only that any triple is equivalent to a triple of the second form $(1_n,q,v)$, where $n\geq1$ and $\dot q=\dot v=1_n$, since the proof is analogous for the first form. Take any triple $(p,q,v)$, and choose $n\geq1$ so that $p$ is in $M_n(\tilde A)$. By adding the elementary triple $(1_n-p,1_n-p,1_n-\phi(p))$ and using part (iii),
\[(p,q,v)\sim\left(\left[\begin{array}{cc}
1_n & 0 \\
0 & 0
\end{array}\right],\left[\begin{array}{cc}
q & 0 \\
0 & 1_n-p
\end{array}\right],\left[\begin{array}{cc}
v & 0 \\
1_n-\phi(p) & 0
\end{array}\right]\right).\]
Set $q_1=q\oplus(1_n-p)$ and $v_1=\left[\begin{array}{cc}
v & 0 \\
1_n-\phi(p) & 0
\end{array}\right]$. We have $\dot v_1^*\dot v_1=1_n$ and $\dot v_1\dot v_1^*=\dot q_1$, so choose $m\geq n$ and a unitary $u$ in $M_m(\mathbb C)$ such that $u\dot q_1u^*=1_n$. Then $(1_n,q_1,v_1)\cong(1_n,uq_1u^*,uv_1)$ since $uv_1\phi(1_n)=\phi(uq_1)v_1$. The scalar part of $uq_1u^*$ is $1_n$ and we now set $q_2=uq_1u^*$ and $v_2=uv_1$. Lastly, $\dot v_2$ may then be regarded as a unitary in $M_n(\mathbb C)$, so choose a homotopy $v_t$ from $\dot v_2$ to $1_n$ and observe that each $v_2v_t^*$ is a partial isometry from $1_n$ to $\phi(q_2)$. By part (i), $(1_n,q_2,v_2)\sim(1_n,q_2,v_2\dot v_2^*)$. Setting $v_3=v_2\dot v_2^*$, the triple $(1_n,q_2,v_3)$ has the desired properties.
\item For the first part, it is a direct consequence of the definitions that $[p,q,v]=0$ if and only if there are elementary triples $(r,r,c)$ and $(s,s,d)$ such that
\[(p,q,v)\oplus(r,r,c)\cong(s,s,d)\]
This is true if and only if there are partial isometries $x$ and $y$ in $M_\infty(\tilde A)$ such $d\phi(x)=\phi(y)(v\oplus c)$. Then $d=\phi(y)(v\oplus c)\phi(x^*)$, and since $d$ is homotopic to $\phi(s)$ and $c$ is homotopic to $\phi(r)$, we have the conclusion.
For the second part, for appropriate $m\geq n$, obtain elementary triples $(r,r,c)$ and $(s,s,d)$ such that
\[(p,1_n\oplus0_{m-n},v)\oplus(r,r,c)\cong(s,s,d).\]
By replacing $s$ with $s\oplus(1_k-r)$ and $d$ by $d\oplus(1_k-\phi(r))$, and using part (iii), we may assume that $r=1_k$ for some $k\geq0$. Obtain $x$ and $y$ as in the previous paragraph, so that $d\phi(x)=\phi(y)(v\oplus c)$, hence $\phi(y^*)d\phi(y)=(v\oplus c)\phi(x^*y)$. Since $d$ is homotopic to $\phi(s)$, $\phi(y^*)d\phi(y)$ is homotopic to $1_n\oplus0_{m-n}\oplus1_k$. Also, $c$ is homotopic to $1_k$. Therefore, $w=x^*y\dot y^*\dot x$ has the desired properties.\qedhere
\end{enumerate}
\end{proof}
In II.3.3 of \cite{karoubi}, Karoubi introduces a definition of the $K_1$-group, there denoted $K^{-1}(\mathcal C)$ for a Banach category $\mathcal C$, that gives an equivalent but slightly more general description. We provide the definition in order to motivate the definition of the relative $K_1$-group. Consider the set $\Gamma_1(A)$ of all pairs $(p,u)$ such that $p$ is a projection in $M_\infty(\tilde A)$ and $u$ is a unitary in $pM_\infty(\tilde A)p$. Define the direct sum $(p,u)\oplus(p',u')=(p\oplus p',u\oplus u')$, as usual. Say that two pairs $(p,u)$ and $(p',u')$ are isomorphic, written $(p,u)\cong(p',u')$, if there is a partial isometry $v$ from $p$ to $p'$ such that $vu=u'v$. We say a pair $(p,u)$ is elementary if there is a continuous path of unitaries $u_t$ from $u$ to $p$ through $pM_\infty(\tilde A)p$. We say that two pairs $\sigma$ and $\sigma'$ in $\Gamma_1(A)$ are equivalent, written $\sigma\sim\sigma'$, if there exist elementary pairs $\tau$ and $\tau'$ such that $\sigma\oplus\tau\cong\sigma\oplus\tau'$. Denote by $[\sigma]$, or $[p,u]$, the equivalence class of the pair $\sigma=(p,u)$ via the relation $\sim$. $K^{-1}(\mathcal C_A)$ is defined to be the quotient of $\Gamma_1(A)$ by the relation $\sim$. It is an abelian group with $[0,0]=0$ and $-[p,u]=[p,u^*]$.
The proof of the following result uses similar, but simpler, techniques to those in Proposition \ref{3}. For this reason, and because we will not need it, we omit the proof.
\begin{prop}\label{k1iso}
The map $\Omega_A:K_1(A)\rightarrow K^{-1}(\mathcal C_A)$ defined by $\Omega_A([u])=[1_n,u]$ (for $n\geq1$ and a unitary $u$ in $M_n(\tilde A)$) is a natural isomorphism.
\end{prop}
\begin{definition} Define $\Gamma_1(\phi)$ to be the set of all triples $(p,u,g)$ where $p$ is a projection in $M_\infty(\tilde A)$, $u$ is a unitary in $pM_\infty(\tilde A)p$, and $g$ is a unitary in $C([0,1])\otimes\phi(p)M_\infty(\tilde B)\phi(p)$ such that $g(0)=\phi(p)$ and $g(1)=\phi(u)$. For brevity, we will often denote these triples by the symbols $\sigma$ and $\tau$. \begin{enumerate}[(i)]
\item Define the direct sum operation $\oplus$ on $\Gamma_1(\phi)$ by
\[(p,u,g)\oplus(p',u',g')=(p\oplus p',u\oplus u',g\oplus g').\]
\item We say that two such triples $(p,u,g)$ and $(p',u',g')$ are \emph{isomorphic}, written $(p,u,g)\cong(p',u',g')$, if there is a partial isometry $v$ in $M_\infty(\tilde A)$ such that $v^*v=p$, $vv^*=p'$, $vu=u'v$, and $\phi(v)g(s)=g'(s)\phi(v)$ for all $0\leq s\leq 1$.
\item A triple $(p,u,g)$ is called \emph{elementary} if there are homotopies $u_t$ and $g_t$ for $0\leq t\leq1$ such that $u_1=u$, $g_1=g$, $u_0=p$, $g_0(s)=\phi(p)$ for all $0\leq s\leq1$, and $g_t(1)=\phi(u_t)$ for all $0\leq t\leq1$.
\item Two triples $\sigma$ and $\sigma'$ in $\Gamma_1(\phi)$ are \emph{equivalent}, written $\sigma\sim\sigma'$, if there exist elementary triples $\tau$ and $\tau'$ such that $\sigma\oplus\tau\cong\sigma'\oplus\tau'$. \end{enumerate}
Denote by $[\sigma]$, or $[p,u,g]$, the equivalence class of the triple $\sigma=(p,u,g)$ via the relation $\sim$. $K_1(\phi)$ is then defined to be the quotient of $\Gamma_1(\phi)$ by the relation $\sim$, that is,
\[\{[\sigma]\mid\sigma\in\Gamma_1(\phi)\}=\Gamma_1(\phi)/\sim\]
\end{definition}
It is easily checked that, like $\Gamma_0(\phi)$, the direct sum operation of triples in $\Gamma_1(\phi)$ behaves well with respect to the notions of isomorphism and elementary.
\begin{prop} $K_1(\phi)$ is an abelian group when equipped with the binary operation
\[[\sigma]+[\sigma]=[\sigma\oplus\sigma']\]
where the identity element is given by $[0,0,0]$ and the inverse of $[p,u,g]$ is given by $[p,u^*,g^*]$.
\end{prop}
\begin{proof} We verify the last claim. We have
\[[p,u,g]+[p,u^*,g^*]=[p\oplus p,u\oplus u^*,g\oplus g^*].\]
Define the matrices
\[a=\left[\begin{array}{cc}
0 & p \\
p & 0
\end{array}\right],\qquad w=\left[\begin{array}{cc}
0 & u^* \\
u & 0
\end{array}\right],\qquad h=\left[\begin{array}{cc}
0 & g^* \\
g & 0
\end{array}\right].\]
The first two are self-adjoint unitaries in $(p\oplus p)M_\infty(\tilde A)(p\oplus p)$, and the third is a self-adjoint unitary in $C([0,1])\otimes(\phi(p)\oplus\phi(p))M_\infty(\tilde B)(\phi(p)\oplus\phi(p))$. Observe that $h(1)=\phi(w)$. Define, for $0\leq t\leq1$,
\[u_t=\exp(i\pi t(p\oplus p-a)/2)\exp(i\pi t(p\oplus p-w)/2)\]
\[g_t=\exp(i\pi t(\phi(p)\oplus\phi(p)-\phi(a))/2)\exp(i\pi t(\phi(p)\oplus\phi(p)-h)/2)\]
Then $u_0=p\oplus p$, $u_1=u\oplus u^*$, $g_1=g\oplus g^*$, and $g_0(s)=\phi(p)\oplus\phi(p)$ for all $0\leq s\leq1$. Moreover, $g_t(1)=\phi(u_t)$ for all $0\leq t\leq1$. It follows that $(p\oplus p,u\oplus u^*,g\oplus g^*)$ is elementary.
\end{proof}
The following result is similar to Proposition \ref{3}, so we omit the proof.
\begin{prop}\label{k1prop}
\begin{enumerate}[(i)]
\item Suppose we have two triples $(p,u,g)$ and $(p',u',g')$ and that $p=p'$. If $p$ is in $M_n(\tilde A)$ and $u_t$ is a path of unitaries from $u$ to $u'$ in $pM_n(\tilde A)p$ and $g_t$ is a path of unitaries from $g$ to $g'$ in $C([0,1])\otimes\phi(p)M_n(\tilde B)\phi(p)$ such that $g_t(1)=\phi(u_t)$ for all $0\leq t\leq1$, then $[p,u,g]=[p',u',g']$.
\item If $p=p'$, we have
\[[p,u,g]+[p',u',g']=[p,uu',gg']=[p,u'u,g'g].\]
\item If $(p,u,g)$ and $(p',u',g')$ are two triples in $\Gamma_1(\phi)$ such that $pp'=0$, then $(p,u,g)\oplus(p',u',g')\cong(p+p',u+u',g+g')$.
\item Every triple in $\Gamma_1(\phi)$ is equivalent to one of the form $(1_n,u,g)$, where $n\geq1$ and $\dot u=\dot g(s)=1_n$ for all $0\leq s\leq1$.
\item If $p$ is in $M_n(\tilde A)$, $[p,u,g]=0$ if and only if there is an integer $k\geq1$ and paths of unitaries $u_t$ in $(p\oplus1_k)M_{n+k}(\tilde A)(p\oplus1_k)$ and $g_t$ in $C([0,1])\otimes(\phi(p)\oplus1_k)M_{n+k}(\tilde B)(\phi(p)\oplus1_k)$ such that $u_0=p\oplus1_k$, $u_1=u\oplus1_k$, $g_0(s)=\phi(p)\oplus1_k$ for all $0\leq s\leq1$, $g_1=g\oplus1_k$, and $g_t(1)=\phi(u_t)$ for all $0\leq t\leq1$.
\end{enumerate}
\end{prop}
We now collect some properties that hold for both relative groups.
\begin{prop}\label{univ1} Suppose that $G$ is an abelian group and $\nu:\Gamma_j(\phi)\rightarrow G$ is a map that satisfies
\begin{enumerate}[(i)]
\item $\nu(\sigma\oplus\tau)=\nu(\sigma)+\nu(\tau)$,
\item $\nu(\sigma)=0$ if $\sigma$ is elementary, and
\item if $\sigma\cong\tau$, then $\nu(\sigma)=\nu(\tau)$.
\end{enumerate}
Then $\nu$ factors to a unique group homomorphism $\alpha:K_j(\phi)\rightarrow G$.
\end{prop}
\begin{proof}
If $\sigma\sim\sigma'$, find elementary triples $\tau$ and $\tau'$ such that $\sigma\oplus\tau\cong\sigma'\oplus\tau'$. Then
\[\nu(\sigma
)=\nu(\sigma)+\nu(\tau)=\nu(\sigma\oplus\tau)=\nu(\sigma'\oplus\tau')=\nu(\sigma')+\nu(\tau')=\nu(\sigma')\]
So the map $\alpha([\sigma]):=\nu(\sigma)$ is well-defined. It is a group homomorphism by property (i).
\end{proof}
If $\phi:A\rightarrow B$ and $\psi:C\rightarrow D$ are $^*$-homomorphisms, we denote by $\phi\oplus\psi$ the component-wise $^*$-homomorphism $A\oplus C\rightarrow B\oplus D$.
\begin{prop}\label{direct} Suppose $\phi:A\rightarrow B$ and $\psi:C\rightarrow D$ are $^*$-homomorphisms. Then there are natural isomorphisms $K_*(\phi\oplus\psi)\rightarrow K_*(\phi)\oplus K_*(\psi)$ that satisfy
\[[(p,p'),(q,q'),(v,v')]\mapsto([p,q,v],[p',q',v'])\]
in the case of $K_0$, and
\[[(p,p'),(u,u'),(g,g')]\mapsto([p,u,g],[p',u',g'])\]
in the case of $K_1$.
\end{prop}
\begin{proof}
For a triple $((p,p'),(q,q'),(v,v'))$ in $\Gamma_0(\phi\oplus\psi)$, define
\[\nu((p,p'),(q,q'),(v,v'))=([p,q,v],[p',q',v']).\]
It is straightforward to check that $\nu$ satisfies the hypotheses of Proposition \ref{univ1}, so we get a well-defined group homomorphism that factors $\nu$. The fact that the group homomorphism is surjective is clear, and injectivity follows from a simple application of part (v) of Proposition \ref{3}. The proof is similar for $K_1$.
\end{proof}
\begin{prop}\label{induce}
Suppose that
\begin{center} \begin{tikzcd} A \arrow[r, "\phi"] \arrow[d, "\alpha"'] & B \arrow[d, "\beta"] & \\ C \arrow[r, "\psi"] & D \end{tikzcd}
\end{center}
is a commutative diagram of C$^*$-algebras and $^*$-homomorphisms. Then there are well-defined group homomorphisms $(\alpha,\beta)_*:K_j(\phi)\rightarrow K_j(\psi)$ that satisfy
\[(\alpha,\beta)_*([p,q,v])=[\alpha(p),\alpha(q),\beta(v)]\]
for a triple $(p,q,v)$ in $\Gamma_0(\phi)$ and\emph{
\[(\alpha,\beta)_*([p,u,g])=[\alpha(p),\alpha(u),\text{id}_{C([0,1])}\otimes\beta(g)]\]}
for a triple $(p,u,g)$ in $\Gamma_1(\phi)$. If $\alpha$ and $\beta$ are $^*$-isomorphisms, then $(\alpha,\beta)_*$ is a group isomorphism.
\end{prop}
\begin{proof}
For a triple $(p,q,v)$ in $\Gamma_0(\phi)$, set $\nu(p,q,v)=[\alpha(p),\alpha(q),\beta(v)]$. Again, the hypotheses of Proposition \ref{univ1} are easy to check, so $\nu$ factors to a group homomorphism $(\alpha,\beta)_*$. If $\alpha$ and $\beta$ are $^*$-isomorphisms, then the diagram
\begin{center} \begin{tikzcd} C \arrow[r, "\psi"] \arrow[d, "\alpha^{-1}"'] & D \arrow[d, "\beta^{-1}"] \\
A \arrow[r, "\phi"] & B \end{tikzcd}
\end{center}
is commutative and the same argument works to obtain the group homomorphism $(\alpha^{-1},\beta^{-1})_*$, which is easily seen to be the inverse of $(\alpha,\beta)_*$. The proof is again similar for $K_1$.
\end{proof}
As an application of the above results, we will show that if $A$ and $B$ are unital and $\phi(1)=1$, one may define $K_*(\phi)$ without unitizations while remaining consistent with the results above. To verify this, let $K_*^u(\phi)$ be the group defined in the same way as $K_*(\phi)$, but avoid unitizing $A$ and $B$ and use the units already present. Notice $K_*(\phi)$ and $K_*^u(\tilde\phi)$ are precisely the same objects, and all preceding results about $K_*(\phi)$ remain true for $K_*^u(\phi)$ with appropriate modifications.
\begin{prop}\label{unit} If $A$ and $B$ are unital and $\phi(1)=1$, then $K_j(\phi)$ and $K_j^u(\phi)$ are isomorphic as groups.
\end{prop}
\begin{proof} The map $\nu_A:A\oplus\mathbb C\rightarrow\tilde A$ defined by $\nu_A(a,\lambda)=a+\lambda(1_{\tilde A}-1_A)$ is a $^*$-isomorphism and the diagram
\begin{center} \begin{tikzcd} A\oplus\mathbb C \arrow[d, "\nu_A"] \arrow[r, "\phi\oplus\text{id}_\mathbb C"] & B\oplus\mathbb C \arrow[d, "\nu_B"] \\
\tilde A \arrow[r, "\tilde\phi"] & \tilde B \end{tikzcd}
\end{center}
is commutative. Therefore $K_j(\phi)=K_j^u(\tilde\phi)$ is isomorphic to $K_j^u(\phi\oplus\text{id}_\mathbb C)$ by Proposition \ref{induce}. Then
\[K_j(\phi)=K_j^u(\tilde\phi)\cong K_j^u(\phi\oplus\text{id}_\mathbb C)\cong K_j^u(\phi)\oplus K_j^u(\text{id}_\mathbb C)\cong K_j^u(\phi)\]
where the third isomorphism is due to Proposition \ref{direct}. The fact that $K_j^u(\text{id}_\mathbb C)=0$ is rather clear, but the skeptical reader is referred to part (ii) of Corollary \ref{zero}.
\end{proof}
\section{Proofs}
\subsection{Proof of part (i) of Theorem \ref{main}}
Define the map $\mu_0:K_1(B)\rightarrow K_0(\phi)$ by $\mu_0([u])=[1_n,1_n,u]$, where $u$ is a unitary in $M_n(\tilde B)$. By part (i) of Proposition \ref{3}, $\mu_0$ is well-defined, and clearly it is a group homomorphism.
Define a map $\nu:\Gamma_0(\phi)\rightarrow K_0(A)$ by $\nu(p,q,v)=[p]-[q]$. Observe that the image of $\nu$ is indeed in $K_0(A)$ (not just $K_0(\tilde A)$) since $\dot v^*\dot v=\dot p$ and $\dot v\dot v^*=\dot q$, hence $[\dot p]=[\dot q]$. It is easy to check that $\nu$ satisfies the hypotheses of Proposition \ref{univ1}, hence factors to a well-defined group homomorphism $\nu_0:K_0(\phi)\rightarrow K_0(A)$.
\begin{prop}\label{exact1} The sequence
\begin{center}
\begin{tikzcd} K_1(A) \arrow[r, "\phi_*"] & K_1(B) \arrow[r, "\mu_0"] & {K_0(\phi)} \arrow[r, "\nu_0"] & K_0(A) \arrow[r, "\phi_*"] & K_0(B) \end{tikzcd} \end{center} is exact. \end{prop}
\begin{proof} It is quite clear that all compositions are zero. If $\phi_*([p]-[q])=[\phi(p)]-[\phi(q)]=0$, choose $k\geq0$ and $w$ in $M_\infty(\tilde B)$ such that $w^*w=\phi(p)\oplus1_k$ and $ww^*=\phi(q)\oplus1_k$. Then
\[[p]-[q]=\nu_0([p\oplus1_k,q\oplus1_k,w]),\]
which shows exactness at $K_0(A)$.
If $(p,1_n,v)$ is such that $\nu_0([p,1_n,v])=[p]-[1_n]=0$, choose $k\geq0$ and $w$ in $M_\infty(\tilde A)$ such that $w^*w=p\oplus1_k$ and $ww^*=1_n\oplus0_{m-n}\oplus1_k$. Then
\[(p\oplus1_k,1_n\oplus0_{m-n}\oplus1_k,v\oplus1_k)\cong(1_n\oplus0_{m-n}\oplus1_k,1_n\oplus0_{m-n}\oplus1_k,(v\oplus1_k)\phi(w^*))\]
and hence
\[[p,1_n,v]=\mu_0([(v\oplus1_k)\phi(w^*)+0_n\oplus1_{m-n}\oplus0_k]),\]
which shows exactness at $K_0(\phi)$.
Finally, if $\mu_0([u])=[1_n,1_n,u]=0$, use part (v) of Proposition \ref{3} to find $k\geq0$ and a partial isometry $w$ such that $\phi(w)(u\oplus1_k)$ is a unitary and homotopic to $1_{n+k}$ in $M_{n+k}(\tilde B)$. Since $u\oplus1_k$ is a unitary, so is $w$ and $u\oplus1_k$ is homotopic to $\phi(w^*)$. Thus
\[[u]=[u\oplus1_k]=[\phi(w^*)]=\phi_*([w^*]),\]
which shows exactness at $K_1(B)$.
\end{proof}
For a unitary $g$ in $C([0,1])\otimes M_n(\tilde B)$ with $g(0)=g(1)=\dot g=1_n$, set $\mu_1([g])=[1_n,1_n,g]$. By part (i) of Proposition \ref{k1prop}, this is a well-defined group homomorphism $\mu_1:K_1(SB)\rightarrow K_1(\phi)$. For a triple $(p,u,g)$ in $\Gamma_1(\phi)$, define $\nu(p,u,g)=[p,u]$ (here we use the picture of $K_1$ described before Proposition \ref{k1iso}). The hypotheses of Proposition \ref{univ1} are satisfied, so we get a group homomorphism $\nu_1:K_1(\phi)\rightarrow K_1(A)$ such that $\nu_1([p,u,g])=[p,u]$. If $p=1_n$, the formula is more simply $\nu_1([1_n,u,g])=[u]$.
\begin{prop}\label{exact2} The sequence
\begin{center}
\begin{tikzcd} K_1(SA) \arrow[r, "(S\phi)_*"] & K_1(SB) \arrow[r, "\mu_1"] & {K_1(\phi)} \arrow[r, "\nu_1"] & K_1(A) \arrow[r, "\phi_*"] & K_1(B) \end{tikzcd} \end{center}
is exact.
\end{prop}
\begin{proof} Again, all compositions are clearly zero. If $\phi_*([u])=0$, we may find $k\geq0$ and a unitary $g$ in $C([0,1])\otimes M_{n+k}(\tilde B)$ such that $g(1)= \phi(u)\oplus1_k$ and $g(0)=1_{n+k}$. Then
\[[u]=\nu_1([1_{n+k},u\oplus1_k,g]),\]
which shows exactness at $K_1(A)$.
If $\nu_1([1_n,u,g])=[u]=0$, find $k\geq0$ and a unitary $f$ in $C([0,1])\otimes M_{n+k}(\tilde A)$ such that $f(0)=1_{n+k}$ and $f(1)=u\oplus1_k$. Set
\[\tilde g(s)=\left\{\begin{array}{cc}
g(2s)\oplus1_k & 0\leq s\leq1/2 \\
\phi(f(2-2s)) & 1/2\leq s\leq1
\end{array}\right.\]
Then $\tilde g$ is a unitary in $C([0,1])\otimes M_{n+k}(\tilde B)$ and $\tilde g(0)=\tilde g(1)=1_{n+k}$. Now for a fixed $t$ in $[0,1]$, the function $g_t$ defined by
\[g_t(s)=\left\{\begin{array}{cc}
g(s(1-\frac12t)^{-1})\oplus1_k & 0\leq s\leq1-\frac12t \\
\phi(f(3-2s-t)) & 1-\frac12t\leq s\leq1
\end{array}\right.\]
satisfies $g_0=g\oplus1_k$, $g_1=\tilde g$, and $g_t(1)=\phi(f(1-t))$, and so
\[[1_n,u,g]=[1_{n+k},u\oplus1_k,g\oplus1_k]=[1_{n+k},1_{n+k},\tilde g]=\mu_1([\tilde g]),\]
which shows exactness at $K_1(\phi)$.
Finally, if $\mu_1([g])=[1_n,1_n,g]=0$, use part (v) of Proposition \ref{k1prop} to find an integer $k$ and homotopies $u_t$ and $g_t$ such that $u_0=u_1=1_{n+k}$, $g_1=g\oplus1_k$, $g_0=1_{n+k}$, and $g_t(1)=\phi(u_t)$ for all $t$. Write $f(t)=u_t$ and set
\[\tilde g_t(s)=\left\{\begin{array}{cc}
g_t(2s) & 0\leq s\leq1/2 \\
\phi(f((2-2t)s+2t-1)) & 1/2\leq s\leq1
\end{array}\right.\]
Then $\tilde g_t(0)=\tilde g_t(1)=1_{n+k}$ for all $t$ and
\[\tilde g_1(s)=\left\{\begin{array}{cc}
g(2s)\oplus1_k & 0\leq s\leq1/2 \\
1_{n+k} & 1/2\leq s\leq1
\end{array}\right.\]
and
\[\tilde g_0(s)=\left\{\begin{array}{cc}
1_{n+k} & 0\leq s\leq1/2 \\
\phi(f(2s-1)) & 1/2\leq s\leq1
\end{array}\right.\]
which are homotopic to $g\oplus1_k$ and $S\phi(f)$, respectively. Thus
\[[g]=[g\oplus1_k]=[S\phi(f)]=(S\phi)_*([f]),\]
which shows exactness at $K_1(SB)$.
\end{proof}
\begin{prop} If $\phi=0$, then the sequence in part (i) of Theorem \ref{main} splits at $K_0(A)$ and $K_1(A)$. In other words, for each $j=0,1$ there is a group homomorphism $\lambda_j:K_j(A)\rightarrow K_j(\phi)$ such that $\nu_j\circ\lambda_j$ is the identity map on $K_j(A)$.
\end{prop}
\begin{proof} If $p$ and $q$ are two projections in $M_\infty(\tilde A)$ with $[\dot p]=[\dot q]$, let $v$ be a partial isometry in $M_\infty(\mathbb C)$ such that $v^*v=\dot p$ and $vv^*=\dot q$. If $u$ is a unitary in $M_n(\tilde A)$, let $g$ be any unitary in $C([0,1])\otimes M_n(\mathbb C)$ such that $g(0)=1_n$ and $g(1)=\dot u$. Define \[\lambda_0([p]-[q])=[p,q,v]\qquad\lambda_1([u])=[1_n,u,g]\]
For both $j=0,1$, it is straightforward to check that $\lambda_j$ is well-defined, additive, independent of the choices of $v$ and $g$, and that $\nu_j\circ\lambda_j$ is the identity.
\end{proof}
By combining all results in this subsection, we obtain part (i) of Theorem \ref{main}. The map $\mu_1$ is (by abuse of notation) the composition of the Bott map $\beta_B$ and $\mu_1$ from Proposition \ref{exact2}. It may therefore be written, for projections $p$ and $q$ in $M_n(\tilde B)$, as $\mu_1([p]-[q])=[1_n,1_n,f_pf_q^*]$, where $f_p(s)=e^{2\pi isp}$ for $0\leq s\leq1$. Since the Bott map is natural, this does not affect exactness.
We also record the following immediate and useful consequences of part (i) of Theorem \ref{main}.
\begin{corollary}\label{zero} We have the following. \begin{enumerate}[(i)]
\item If $K_*(A)=K_*(B)=0$, then $K_*(\phi)=0$.
\item If $\phi:A\rightarrow B$ is a $^*$-isomorphism, then $K_*(\phi)=0$.
\end{enumerate}
\end{corollary}
\subsection{Proof of parts (ii) and (iii) of Theorem \ref{main}}
Throughout this subsection, we will assume that
\begin{equation}\label{comm1} \begin{tikzcd} 0 \arrow[r] & I \arrow[r, "\iota_A"] \arrow[d, "\psi"] & A \arrow[r, "\pi_A"] \arrow[d, "\phi"] & A/I \arrow[r] \arrow[d, "\gamma"] & 0 \\ 0 \arrow[r] & J \arrow[r, "\iota_B"] & B \arrow[r, "\pi_B"] & B/J \arrow[r] & 0 \end{tikzcd} \end{equation} is a commutative diagram with exact rows. We will abbreviate the induced maps $(\iota_A,\iota_B)_*$ and $(\pi_A,\pi_B)_*$ to $\iota_*$ and $\pi_*$, respectively.
\begin{prop}\label{half} The sequence \begin{center} \begin{tikzcd} K_0(\psi) \arrow[r, "\iota_*"] & K_0(\phi) \arrow[r, "\pi_*"] & K_0(\gamma) \end{tikzcd} \end{center} is exact. If $\lambda_A:A/I\rightarrow A$ and $\lambda_B:B/J\rightarrow B$ are splittings of the rows in (\ref{comm1}) that keep the diagram commutative, then the sequence \begin{center} \begin{tikzcd} 0 \arrow[r] & K_0(\psi) \arrow[r, "\iota_*"] & K_0(\phi) \arrow[r, shift left, "\pi_*"] & K_0(\gamma) \arrow[l, shift left, "\lambda_*"] \arrow[r] & 0 \end{tikzcd} \end{center} is split exact, where $\lambda_*=(\lambda_A,\lambda_B)_*$. \end{prop}
\begin{proof} It is clear that the composition is zero. Conversely, suppose that $[1_n,q,v]$ is in the kernel of $\pi_*$, so $[1_n,\pi_A(q),\pi_B(v)]=0$. Find (in order): \begin{enumerate}[(i)]
\item an integer $m\geq n$ so that $q$ is in $M_m(\tilde A)$,
\item an integer $k\geq0$ and a partial isometry $w$ in $M_{m+k}(\widetilde{A/I})$ such that $w^*w=\pi_A(q)\oplus1_k$ and $ww^*=1_n\oplus0_{m-n}\oplus1_k$ and $\gamma(w)(\pi_B(v)\oplus1_k)$ is homotopic to $1_n\oplus0_{m-n}\oplus1_k$ (use part (v) of Proposition \ref{3}),
\item an integer $l\geq0$ and a unitary $z$ homotopic to $1_{m+k+l}$ in $M_{m+k+l}(\widetilde{A/I})$ such that $z(\pi_A(q)\oplus1_k\oplus0_l)z^*=1_n\oplus0_{m-n}\oplus1_k\oplus0_l$ and $\gamma(z)(\pi_B(v)\oplus1_k\oplus0_l)=(\gamma(w)(\pi_B(v)\oplus1_k))\oplus0_l$. For example, one may take $l=m+k$ and \[z=\left[\begin{array}{cc}
w & 1_{m+k}-ww^* \\
1_{m+k}-w^*w & w^* \end{array}\right],\] see the discussion following Definition \ref{projpath}. \item a unitary $U$ in $M_{m+k+l}(\tilde A)$ such that $\pi_A(U)=z$ (this is possible because $z$ is homotopic to $1_{m+k+l}$), \item a unitary $V$ in $(1_n\oplus0_{m-n}\oplus1_k)M_{m+k}(\tilde B)(1_n\oplus0_{m-n}\oplus1_k)$ homotopic to $1_n\oplus0_{m-n}\oplus1_k$ such that $\pi_B(V)=\gamma(w)(\pi_B(v)\oplus1_k)$ (use (ii)). \end{enumerate} Then \begin{align} \nonumber[1_n,q,v]&=\left[\left[\begin{array}{cccc}
1_n & 0 & 0 & 0 \\
0 & 0_{m-n} & 0 & 0 \\
0 & 0 & 1_k & 0 \\
0 & 0 & 0 & 0_l
\end{array}\right],\left[\begin{array}{ccc}
q & 0 & 0 \\
0 & 1_k & 0 \\
0 & 0 & 0_l
\end{array}\right],\left[\begin{array}{ccc}
v & 0 & 0 \\
0 & 1_k & 0 \\
0 & 0 & 0_l
\end{array}\right]\right]\\ \nonumber&=\left[\left[\begin{array}{cccc}
1_n & 0 & 0 & 0 \\
0 & 0_{m-n} & 0 & 0 \\
0 & 0 & 1_k & 0 \\
0 & 0 & 0 & 0_l
\end{array}\right],U\left[\begin{array}{ccc}
q & 0 & 0 \\
0 & 1_k & 0 \\
0 & 0 & 0_l
\end{array}\right]U^*,\phi(U)\left[\begin{array}{ccc}
v & 0 & 0 \\
0 & 1_k & 0 \\
0 & 0 & 0_l
\end{array}\right]\right]\\ \nonumber&=\left[\left[\begin{array}{cccc}
1_n & 0 & 0 & 0 \\
0 & 0_{m-n} & 0 & 0 \\
0 & 0 & 1_k & 0 \\
0 & 0 & 0 & 0_l
\end{array}\right],U\left[\begin{array}{ccc}
q & 0 & 0 \\
0 & 1_k & 0 \\
0 & 0 & 0_l
\end{array}\right]U^*,\phi(U)\left[\begin{array}{ccc}
v & 0 & 0 \\
0 & 1_k & 0 \\
0 & 0 & 0_l
\end{array}\right]\right]\\ \nonumber&\qquad+\left[\left[\begin{array}{cccc}
1_n & 0 & 0 & 0 \\
0 & 0_{m-n} & 0 & 0 \\
0 & 0 & 1_k & 0 \\
0 & 0 & 0 & 0_l
\end{array}\right],\left[\begin{array}{cccc}
1_n & 0 & 0 & 0 \\
0 & 0_{m-n} & 0 & 0 \\
0 & 0 & 1_k & 0 \\
0 & 0 & 0 & 0_l
\end{array}\right],\left[\begin{array}{cc}
V^* & 0 \\
0 & 0_l
\end{array}\right]\right]\\ \nonumber&=\left[\left[\begin{array}{cccc}
1_n & 0 & 0 & 0 \\
0 & 0_{m-n} & 0 & 0 \\
0 & 0 & 1_k & 0 \\
0 & 0 & 0 & 0_l
\end{array}\right],U\left[\begin{array}{ccc}
q & 0 & 0 \\
0 & 1_k & 0 \\
0 & 0 & 0_l
\end{array}\right]U^*,\phi(U)\left[\begin{array}{ccc}
v & 0 & 0 \\
0 & 1_k & 0 \\
0 & 0 & 0_l
\end{array}\right]\left[\begin{array}{cc}
V^* & 0 \\
0 & 0_l
\end{array}\right]\right] \end{align} To get the first equality above, we added an elementary scalar triple. To get the second, notice that the two triples are isomorphic via the unitary $U$. In the third equality, the new triple being added is elementary because $V$ is homotopic to the identity. The fourth equality follows from part (ii) of Proposition \ref{3}. Regarding the elements of the latter triple, we have \[\pi_A\left(U\left[\begin{array}{ccc}
q & 0 & 0 \\
0 & 1_k & 0 \\
0 & 0 & 0_l
\end{array}\right]U^*\right)=\pi_B\left(\phi(U)\left[\begin{array}{ccc}
v & 0 & 0 \\
0 & 1_k & 0 \\
0 & 0 & 0_l
\end{array}\right]\left[\begin{array}{cc}
V^* & 0 \\
0 & 0_l
\end{array}\right]\right)=\left[\begin{array}{cccc}
1_n & 0 & 0 & 0 \\
0 & 0_{m-n} & 0 & 0 \\
0 & 0 & 1_k & 0 \\
0 & 0 & 0 & 0_l
\end{array}\right]\] from which it follows that $[1_n,q,v]$ is in the image of $\iota_*$.
For the split exact sequence, it is clear that $\lambda_*$ is a right inverse for $\pi_*$, so we need only show that $\iota_*$ is injective. Suppose that $(1_n,q,v)$ is a triple in $\Gamma_0(\psi)$ with $\dot q=\dot v=1_n$ and $[1_n,q,v]=0$ in $K_0(\phi)$. Choose $m\geq n$ so that $1_n\oplus0_{m-n}$ and $q$ are in $M_m(\tilde I)$ and $v$ is in $M_m(\tilde J)$. Use part (v) of Proposition \ref{3} to find an integer $k\geq0$ and a partial isometry $w$ in $M_{m+k}(\tilde A)$ with $w^*w=q\oplus1_k$ and $ww^*=1_n\oplus0_{m-n}\oplus1_k$ and $\phi(w)(v\oplus1_k)$ is homotopic to $1_n\oplus0_{m-n}\oplus1_k$. Let $y_t$ be such a homotopy, that is, $y_0=\dot y_t=1_n\oplus0_{m-n}\oplus1_k$ for all $t$ and $y_1=\phi(w)(v\oplus1_k)$. Set $x=\lambda_A(\pi_A(w^*))w$. Then $\pi_A(x)=1_n\oplus0_{m-n}\oplus1_k$ so that $x$ is in $M_{m+k}(\tilde I)$. We have $x^*x=q\oplus1_k$ and $xx^*=1_n\oplus0_{m-n}\oplus1_k$ and, since $\pi_B(v\oplus1_k)=1_n\oplus0_{m-n}\oplus1_k$, \begin{align}\nonumber\psi(x)\left[\begin{array}{cc}
v & 0 \\
0 & 1_k \end{array}\right]&=\psi(\lambda_A(\pi_A(w^*))w)\left[\begin{array}{cc}
v & 0 \\
0 & 1_k \end{array}\right]\\ \nonumber&=\lambda_B(\pi_B(\phi(w^*)))\phi(w)\left[\begin{array}{cc}
v & 0 \\
0 & 1_k \end{array}\right]\\ \nonumber&=\lambda_B\left(\pi_B\left(\left[\begin{array}{cc}
v & 0 \\
0 & 1_k \end{array}\right]\phi(w^*)\right)\right)\phi(w)\left[\begin{array}{cc}
v & 0 \\
0 & 1_k \end{array}\right]\\ \nonumber&=\lambda_B\left(\pi_B\left(\phi(w)\left[\begin{array}{cc}
v & 0 \\
0 & 1_k \end{array}\right]\right)^*\right)\phi(w)\left[\begin{array}{cc}
v & 0 \\
0 & 1_k \end{array}\right] \end{align} is homotopic to $1_n\oplus0_{m-n}\oplus1_k$ through $M_{m+k}(\tilde J)$ via $\lambda_B(\pi_B(y_t^*))y_t$. It follows that $[1_n,q,v]=0$ in $K_0(\psi)$. \end{proof}
Now we associate an index map $\partial_1:K_1(\gamma)\rightarrow K_0(\psi)$ to the diagram (\ref{comm1}).
\begin{definition}\label{index}
The index map $\partial_1:K_1(\gamma)\rightarrow K_0(\psi)$ is given by
\[\partial_1([1_n,u,g])=\left[w\left[\begin{array}{cc}
1_n & 0 \\
0 & 0_l
\end{array}\right]w^*,\left[\begin{array}{cc}
1_n & 0 \\
0 & 0_l
\end{array}\right],\left[\begin{array}{cc}
h(1) & 0 \\
0 & 0_l
\end{array}\right]\phi(w^*)\right]\]
where $l\geq0$, $w$ is a unitary in $M_{n+l}(\tilde A)$ such that $\pi_A(w)(1_n\oplus0_l)=u\oplus0_l$, and $h$ is a unitary in $C([0,1])\otimes M_n(\widetilde{B})$ such that $h(0)=1_n$ and $\pi_B(h)=g$.
\end{definition}
Observe that such elements $l$, $w$, and $h$ always exist: one may take $l=n$, $w$ to be a lift of $u\oplus u^*$, and $h$ exists because $g$, as a unitary in $C([0,1])\otimes M_n(\widetilde{B/J})$ is homotopic to $1_n$. It is straightforward to verify that $\partial_1$ is independent of these choices, and depends only on the class of the triple $(1_n,u,g)$.
The map $\partial_1$ is natural in the following sense. Suppose that
\small
\begin{center} \begin{tikzcd}
& 0 \arrow[rr] & & I' \arrow[rr] \arrow[dd, "\psi'" near start] & & A' \arrow[rr] \arrow[dd, "\phi'" near start] & & A'/I' \arrow[rr] \arrow[dd, "\gamma'" near start] & & 0 \\ 0 \arrow[rr] & & I \arrow[rr] \arrow[ru, "\sigma"] \arrow[dd, "\psi" near start] & & A \arrow[rr] \arrow[dd, "\phi" near start] \arrow[ru] & & A/I \arrow[rr] \arrow[dd, "\gamma" near start] \arrow[ru, "\tau"] & & 0 & \\
& 0 \arrow[rr] & & J' \arrow[rr] & & B' \arrow[rr] & & B'/J' \arrow[rr] & & 0 \\ 0 \arrow[rr] & & J \arrow[rr] \arrow[ru, "\sigma'"] & & B \arrow[rr] \arrow[ru] & & B/J \arrow[rr] \arrow[ru, "\tau'"] & & 0 & \end{tikzcd}
\end{center}
\normalsize
is a commutative diagram with exact rows. Then the diagram
\begin{center} \begin{tikzcd} K_1(\gamma) \arrow[rr, "\partial_1"] \arrow[dd, "{(\tau,\tau')_*}"] & & K_0(\psi) \arrow[dd, "{(\sigma,\sigma')_*}"] \\
& & \\ K_1(\gamma') \arrow[rr, "\partial_1'"] & & K_0(\psi') \end{tikzcd}
\end{center}
is commutative. We leave the straighforward proof to the reader.
\begin{prop}\label{connect} The sequence
\begin{center} \begin{tikzcd} K_1(\phi) \arrow[r, "\pi_*"] & K_1(\gamma) \arrow[r, "\partial_1"] & K_0(\psi) \arrow[r, "\iota_*"] & K_0(\phi) \end{tikzcd}
\end{center}
is exact and the diagram
\begin{center} \begin{tikzcd} K_1(S(B/J)) \arrow[dd, "\theta_J^{-1}\circ\delta_2"] \arrow[rr, "\mu_1"] & & K_1(\gamma) \arrow[rr, "\nu_1"] \arrow[dd, "\partial_1"] & & K_1(A/I) \arrow[dd, "\delta_1"] \\
& & & & \\ K_1(J) \arrow[rr, "\mu_0"] & & K_0(\psi) \arrow[rr, "\nu_0"] & & K_0(I) \end{tikzcd}
\end{center}
is commutative.
\end{prop}
\begin{proof}
For ease of notation we will denote
\[p=w\left[\begin{array}{cc}
1_n & 0 \\
0 & 0_l
\end{array}\right]w^*\qquad v=\left[\begin{array}{cc}
h(1) & 0 \\
0 & 0_l
\end{array}\right]\phi(w^*)\]
It is a simple calculation to see that the right square in the diagram is commutative. For the left square, take $[f]$ in $K_1(S(B/J))$, where $f$ is in $M_n(\widetilde{S(B/J)})$ and $f(0)=1_n$. Find $h$ in $M_n(\widetilde{CB})$ such that $h(0)=1_n$ and $\pi_B(h)=f$. Then
\[\partial_1(\mu_1([f]))=\partial_1([1_n,1_n,f])=[1_n,1_n,h(1)]\]
Now find $g$ in $M_{2n}(\widetilde{SB})$ such that $g(0)=1_{2n}$ and $\pi_B(g)=f\oplus f^*$. Let
\[\tilde g(t)=\left\{\begin{array}{cc}
g(2t) & 0\leq t\leq1/2 \\
h(2t-1)\oplus h(2t-1)^* & 1/2\leq t\leq1
\end{array}\right.\]
Then $[g(1_n\oplus0_n)g^*]-[1_n\oplus0_n]=[\tilde g(1_n\oplus0_n)\tilde g^*]-[1_n\oplus0_n]$ in $K_0(SJ)$, the latter being equal to $\theta_J([h(1)])$ since $\tilde g(1)=h(1)\oplus h(1)^*$. All in all, we have
\[\mu_0(\theta_J^{-1}(\delta_2([f])))=\mu_0(\theta_J^{-1}([\tilde g(1_n\oplus0_n)\tilde g^*]-[1_n\oplus0_n]))=\mu_0([h(1)])=[1_n,1_n,h(1)]\]
which shows commutativity of the left square.
The composition $\partial_1\circ\pi_*$ is clearly zero since everything has a unitary lift. We also have $\iota_*\circ\partial_1$ zero since
\[[p,1_n\oplus0_l,v]=[p,1_n\oplus0_l,v]+[1_n\oplus0_l,1_n\oplus0_l,h(1)^*\oplus0_l]=[p,1_n\oplus0_l,(1_n\oplus0_l)\phi(w^*)]\]
Because $(1_n\oplus0_l,1_n\oplus0_l,h(1)^*\oplus0_l)$ is elementary in $\Gamma_0(\phi)$ and $(p,1_n\oplus0_l,(1_n\oplus0_l)\phi(w^*))\cong(1_n\oplus0_l,1_n\oplus0_l,1_n\oplus0_l)$.
Now suppose that
\[\partial_1([1_n,u,g])=[p,1_n\oplus0_l,v]=[w(1_n\oplus0_l)w^*,1_n\oplus0_l,(h(1)\oplus0_l)\phi(w^*)]=0\]
Find $k\geq1$ and a partial isometry $x$ in $M_{n+l+k}(\tilde I)$ with $xx^*=p\oplus1_k$ and $\dot x=x^*x=1_n\oplus0_l\oplus1_k$, and such that $(v\oplus1_k)\psi(x)$ is homotopic to $1_n\oplus0_l\oplus1_k$. Let $y_t$ be such a homotopy, with $\dot y_t=y_0=1_n\oplus0_l\oplus1_k$ for all $t$ and $y_1=(v\oplus1_k)\psi(x)$. Set
\[z=\left[\begin{array}{ccc}
1_n & 0 & 0 \\
0 & 0_l & 0 \\
0 & 0 & 1_k
\end{array}\right]\left[\begin{array}{cc}
w^* & 0 \\
0 & 1_k
\end{array}\right]x\]
and
\[h'(t)=\left\{\begin{array}{cc}
y_{2t} & 0\leq t\leq1/2 \\
(h(2t-1)^*\oplus0_l\oplus1_k)(v\oplus1_k)\psi(x) & 1/2\leq t\leq1
\end{array}\right.\]
Then $\pi_A(z)=u\oplus0_l\oplus1_k$ and
\[\pi_B(h'(t))=\left\{\begin{array}{cc}
1_n\oplus0_l\oplus1_k & 0\leq t\leq1/2 \\
g(2t-1)\oplus0_l\oplus1_k & 1/2\leq t\leq1
\end{array}\right.\]
which is clearly homotopic to $g\oplus0_l\oplus1_k$. Moreover, $h'(1)=\phi(z)$. It follows that
\[[1_n,u,g]=[1_n\oplus0_l\oplus1_k,u\oplus0_l\oplus1_k,g\oplus0_l\oplus1_k]=\pi_*([1_n\oplus0_l\oplus1_k,z,h'])\]
Now suppose that $(p,1_n,v)$ is a triple in $\Gamma_0(\psi)$ with $[p,1_n,v]=0$ in $K_0(\phi)$. Choose $m\geq n$ such that $1_n\oplus0_{m-n}$ and $p$ are in $M_m(\tilde I)$ and $v$ is in $M_m(\tilde J)$. Find $k\geq0$ and a partial isometry $x$ in $M_{m+k}(\tilde A)$ with $xx^*=p\oplus1_k$ and $\dot x=x^*x=1_n\oplus0_{m-n}\oplus1_k$, and such that $(v\oplus1_k)\phi(x)$ is homotopic to $1_n\oplus0_{m-n}\oplus1_k$. Find a unitary $U$ in $M_{m+k}(\mathbb C)$ such that
\[U(1_n\oplus0_{m-n}\oplus1_k)U^*=1_{n+k}\oplus0_{m-n}\]
and let $p'=U(p\oplus1_k)U^*$, $v'=U(v\oplus1_k)U^*$, and $x'=UxU^*$. Clearly $(p,1_n,v)\oplus(1_k,1_k,1_k)\cong(p',1_{n+k},v')$, $x'x'^*=p'$, $x'^*x'=1_{n+k}\oplus0_{m-n}$, and that $v'\phi(x')$ is homotopic to $1_{n+k}\oplus0_{m-n}$. Let $y_t$ be such a homotopy, with $\dot y_t=y_0=1_{n+k}\oplus0_{m-n}$ for all $t$ and $y_1=v'\phi(x')$. Notice that $\pi_A(x')=(1_{n+k}\oplus0_{m-n})\pi_A(x')(1_{n+k}\oplus0_{m-n})$, so we may regard $\pi_A(x')$ as a unitary in $M_{n+k}(\widetilde{A/I})$, and similarly we may regard $y_t$ as a path of unitaries in $M_{n+k}(\tilde B)$. Set $g(t)=\pi_B(y_t)$ and notice that
\[g(1)=\pi_B(\phi(x'^*))\pi_B(v'^*)=\gamma(\pi_A(x'^*))\]
so that $(1_{n+k},\pi_A(x'^*),g)$ is a triple in $\Gamma_1(\gamma)$. Moreover, we see that its image under $\partial_1$ is $[p,1_n,v]$ by using $l=2m+k-n$,
\[w=\left[\begin{array}{cc}
x' & 1_{m+k}-x'x'^* \\
1_{m+k}-x'^*x' & x'^*
\end{array}\right]\]
in $M_{2(m+k)}(\tilde A)$ and $h(t)=y_t$.
\end{proof}
\begin{corollary} There is an isomorphism $\theta_\phi:K_1(\phi)\rightarrow K_0(S\phi)$. Moreover, the diagram
\begin{center} \begin{tikzcd} K_1(SB) \arrow[rr, "\mu_1"] \arrow[dd, equal] & & K_1(\phi) \arrow[rr, "\nu_1"] \arrow[dd, "\theta_\phi"] & & K_1(A) \arrow[dd, "\theta_A"] \\
& & & & \\ K_1(SB) \arrow[rr, "\mu_0"] & & K_0(S\phi) \arrow[rr, "\nu_0"] & & K_0(SA) \end{tikzcd}
\end{center}
is commutative.
\end{corollary}
\begin{proof} The map $\theta_\phi$ is the index map $\partial_1$ associated to the commutative diagram
\begin{center} \begin{tikzcd} 0 \arrow[r] & SA \arrow[r] \arrow[d, "S\phi"] & CA \arrow[r] \arrow[d, "C\phi"] & A \arrow[r] \arrow[d, "\phi"] & 0 \\ 0 \arrow[r] & SB \arrow[r] & CB \arrow[r] & B \arrow[r] & 0 \end{tikzcd}
\end{center}
$CA$ and $CB$ are contractible, hence the relative groups $K_0(C\phi)$ and $K_1(C\phi)$ are trivial by Corollary \ref{zero}. It follows that $\theta_\phi$ is an isomorphism.
\end{proof}
An explicit description of $\theta_\phi$ is as follows. Let $(1_n,u,g)$ be a triple in $\Gamma_1(\phi)$, and let $w$ be a unitary in $C([0,1])\otimes M_{2n}(\tilde A)$ with $w(0)=1_{2n}$ and $w(1)=u\oplus u^*$. Then
\[\theta_\phi([1_n,u,g])=\left[w\left[\begin{array}{cc}
1_n & 0 \\
0 & 0_n
\end{array}\right]w^*,\left[\begin{array}{cc}
1_n & 0 \\
0 & 0_n
\end{array}\right],\left[\begin{array}{cc}
g & 0 \\
0 & 0_n
\end{array}\right]\phi(w^*)\right]\]
\begin{corollary}
The sequence \begin{center} \begin{tikzcd} K_1(\psi) \arrow[r, "\iota_*"] & K_1(\phi) \arrow[r, "\pi_*"] & K_1(\gamma) \end{tikzcd} \end{center} is exact. If $\lambda_A:A/I\rightarrow A$ and $\lambda_B:B/J\rightarrow B$ are splittings of the rows in (\ref{comm1}) that keep the diagram commutative, then the sequence \begin{center} \begin{tikzcd} 0 \arrow[r] & K_1(\psi) \arrow[r, "\iota_*"] & K_1(\phi) \arrow[r, shift left, "\pi_*"] & K_1(\gamma) \arrow[l, shift left, "\lambda_*"] \arrow[r] & 0 \end{tikzcd} \end{center} is split exact.
\end{corollary}
\begin{proof}
The map $\theta_\phi$ is natural, so we have the commutative diagram
\begin{center} \begin{tikzcd} K_1(\psi) \arrow[dd, "\theta_\psi"] \arrow[rr, "\iota_*"] & & K_1(\phi) \arrow[rr, "\pi_*"] \arrow[dd, "\theta_\phi"] & & K_1(\gamma) \arrow[dd, "\theta_\gamma"] \\
& & & & \\ K_0(S\psi) \arrow[rr, "\iota_*"] & & K_0(S\phi) \arrow[rr, "\pi_*"] & & K_0(S\gamma) \end{tikzcd}
\end{center}
in which, by Proposition \ref{half}, the bottom row is exact. It follows that the top row is exact as well. The proof for split exactness is similar.
\end{proof}
At this point we may unambiguously define higher relative groups $K_j(\phi)$ by $K_0(S^j\phi)$ and higher index maps $\partial_j:K_j(\gamma)\rightarrow K_{j-1}(\psi)$ to obtain the long exact sequence in part (ii) of Theorem \ref{main}. We proceed to prove that Bott periodicity holds so that the long exact sequence collapses to the six-term exact sequence in part (iii) of Theorem \ref{main}.
For Bott periodicity we will follow the original proof in \cite{cuntz}. Recall that the \emph{Toeplitz algebra} $\mathcal T$ is the universal C$^*$-algebra generated by an isometry. Let $\pi:\mathcal T\rightarrow C(\mathbb T)$ be the $^*$-homomorphism that sends the generating isometry to the function $z$ on $\mathbb T$. The kernel of $\pi$ is isomorphic to $\mathcal K$, and by identifying $C_0((0,1))$ with elements in $C(\mathbb T)$ that vanish at $1$ and letting $\mathcal T_0=\pi^{-1}(C_0((0,1)))$, we obtain the short exact sequence
\begin{center}
\begin{tikzcd} 0 \arrow[r] & \mathcal K \arrow[r, hook] & \mathcal T_0 \arrow[r, "\pi"] & C_0((0,1)) \arrow[r] & 0 \end{tikzcd}
\end{center}
We will assume the nontrivial fact that $K_*(\mathcal T_0)=0$; we refer the reader to \cite{cuntz} for the original proof.
\begin{lemma}\label{nuclear} If $C$ is in the bootstrap category (22.3.4 of \cite{blackadar}) and $K_*(C)=0$, then \emph{$K_*(\phi\otimes\text{id}_C)=0$.} In particular, \emph{$K_*(\phi\otimes\text{id}_{\mathcal T_0})=0$.}
\end{lemma}
\begin{proof} By the K\"unneth Theorem for tensor products (see the main result of \cite{sch}), we have $K_*(A\otimes C)=K_*(B\otimes C)=0$. The conclusion follows from Corollary \ref{zero}.
\end{proof}
\begin{lemma}\label{stable}\emph{ $K_j(\phi\otimes\text{id}_\mathcal K)\cong K_j(\phi)$ for $j=0,1$.}
\end{lemma}
\begin{proof}
For either $j=0,1$, we have a commutative diagram with exact rows
\begin{center} \begin{tikzcd} K_{1-j}(A) \arrow[dd] \arrow[r] & K_{1-j}(B) \arrow[r] \arrow[dd] & K_j(\phi) \arrow[r] \arrow[dd] & K_j(A) \arrow[r] \arrow[dd] & K_j(B) \arrow[dd] \\
& & & & \\ K_{1-j}(A\otimes\mathcal K) \arrow[r] & K_{1-j}(B\otimes\mathcal K) \arrow[r] & K_j(\phi\otimes\text{id}_\mathcal K) \arrow[r] & K_j(A\otimes\mathcal K) \arrow[r] & K_j(B\otimes\mathcal K) \end{tikzcd}
\end{center}
where all the vertical maps are induced by the embedding $a\mapsto a\otimes p$, where $p$ is any rank one projection in $\mathcal K$. All vertical maps except for the middle one are known to be isomorphisms. The five lemma then shows that the middle vertical arrow is an isomorphism.
\end{proof}
We now produce the Bott map. We have the commutative diagram
\begin{center} \begin{tikzcd} 0 \arrow[r] & A\otimes\mathcal K \arrow[r] \arrow[d, "\phi\otimes\text{id}_\mathcal K"] & A\otimes\mathcal T_0 \arrow[r] \arrow[d, "\phi\otimes\text{id}_{\mathcal T_0}"] & SA \arrow[r] \arrow[d, "S\phi"] & 0 \\ 0 \arrow[r] & B\otimes\mathcal K \arrow[r] & B\otimes\mathcal T_0 \arrow[r] & SB \arrow[r] & 0 \end{tikzcd}
\end{center}
Proposition \ref{connect} implies that
\begin{center}
\begin{tikzcd} K_1(\phi\otimes\text{id}_{\mathcal T_0}) \arrow[r] & K_1(S\phi) \arrow[r] & K_0(\phi\otimes\text{id}_\mathcal K) \arrow[r] & K_0(\phi\otimes\text{id}_{\mathcal T_0}) \end{tikzcd}
\end{center}
is exact, and Lemma \ref{nuclear} and Lemma \ref{stable} together give an isomorphism $K_0(\phi)\cong K_1(S\phi)$. We let $\beta_\phi:K_0(\phi)\rightarrow K_1(S\phi)$ denote this isomorphism. We introduce a useful piece of notation before giving an explicit description of $\beta_\phi$.
\begin{definition}\label{projpath}
For a triple $(p,1_n,v)$ in $\Gamma_0(\phi)$, choose $m\geq n$ such that $p$ is in $M_m(\tilde A)$, and let
\[p_v(s)=w(s)^*\left[\begin{array}{cc}
1_n & 0 \\
0 & 0_{2m-n}
\end{array}\right]w(s)\]
where $w$ is a path of unitaries in $M_{2m}(\tilde B)$ with $w(0)=1_{2m}$ and
\[w(1)=\left[\begin{array}{cc}
v & 1_m-vv^* \\
1_m-v^*v & v^*
\end{array}\right]\]
\end{definition}
Note that such a path $w$ exists since
\[\left[\begin{array}{cc}
v & 1_m-vv^* \\
1_m-v^*v & v^*
\end{array}\right]=\left[\begin{array}{cc}
0 & 1_m \\
1_m & 0
\end{array}\right]\left[\begin{array}{cc}
1_m-v^*v & v^* \\
v & 1_m-vv^*
\end{array}\right]\]
and the two unitaries on the right are self-adjoint.
We then have $\beta_\phi([p,1_n,v])=[1_{2m},u,g]$ where
\[u(t)=\exp\left(2\pi it\left[\begin{array}{cc}
p & 0 \\
0 & 0_m
\end{array}\right]\right)\exp\left(-2\pi it\left[\begin{array}{cc}
1_n & 0 \\
0 & 0_{2m-n}
\end{array}\right]\right)\]
and
\[g(s,t)=\exp(2\pi itp_v(s))\exp\left(-2\pi it\left[\begin{array}{cc}
1_n & 0 \\
0 & 0_{2m-n}
\end{array}\right]\right)\]
Now we complete the six-term exact sequence in part (iii) of Theorem \ref{main}. We define the exponential map $\partial_0:K_0(\gamma)\rightarrow K_1(\psi)$ to be the group homomorphism that makes the diagram
\begin{center} \begin{tikzcd} K_0(\gamma) \arrow[rr, "\partial_0", dashed] \arrow[dd, "\beta_\gamma"] & & K_1(\psi) \arrow[dd, "\theta_\psi"] \\
& & \\ K_1(S\gamma) \arrow[rr, "\partial_2"] & & K_0(S\psi) \end{tikzcd}
\end{center}
commutative. All maps in the above diagram are natural, so the sequence in part (iii) of Theorem \ref{main} is exact everywhere.
An explicit description of $\partial_0$ is as follows. Given a triple $(p,1_n,v)$ in $\Gamma_0(\gamma)$, choose $m$ and $p_v$ as in Definition \ref{projpath}. Let $a$ in $M_m(\tilde A)$ be such that $a=a^*$, $\pi_A(a)=p$, and let $f$ in $M_{2m}(\widetilde{CB})$ be such that $f(t)=f(t)^*$ for all $t$, $\pi_B(f)=p_v$, and $f(1)=\phi(a)\oplus0_m$. Then we have
\[\partial_0([p,1_n,v])=-[1_{2m},\exp(2\pi i(a\oplus0_m)),\exp(2\pi if)]\]
\begin{remark}
It is interesting to note that split exactness was not necessary to prove part (iii) of Theorem \ref{main}, since split exactness is crucial to deduce that $K_*(\mathcal T_0)=0$ from the isomorphism $K_*(\mathcal T)\cong K_*(\mathbb C)$ during the proof of Bott periodicity in \cite{cuntz}. Here, we were able to sneak around this difficulty using Corollary \ref{zero} and the fact that $K_*(\mathcal T_0)=0$.
\end{remark}
\subsection{Proof of part (iv) of Theorem \ref{main}}
Consider the commutative diagram
\begin{center} \begin{tikzcd} 0 \arrow[r] & SB \arrow[r, "\iota_A"] \arrow[d, equal] & C_\phi \arrow[r, "\pi_A"] \arrow[d, "\sigma"] & A \arrow[r] \arrow[d, "\phi"] & 0 \\ 0 \arrow[r] & SB \arrow[r, hook] & CB \arrow[r, "\pi_B"] & B \arrow[r] & 0 \end{tikzcd} \end{center} with exact rows, where $\iota_A(f)=(0,f)$, $\pi_A(a,f)=a$, $\pi_B(f)=f(1)$, and $\sigma(a,f)=f$. Since $K_*(CB)=0$ because $CB$ is contractible, we have by part (i) of Theorem \ref{exc} that $\nu_j:K_j(\sigma)\rightarrow K_j(C_\phi)$ is an isomorphism for $j=0,1$. By part (iii) of Theorem \ref{main} and Corollary \ref{zero},
\begin{center} \begin{tikzcd} 0 \arrow[rr] & & K_0(\sigma) \arrow[rr, "\pi_*"] & & K_0(\phi) \arrow[dd] \\
& & & & \\ K_1(\phi) \arrow[uu] & & K_1(\sigma) \arrow[ll, "\pi_*"'] & & 0 \arrow[ll] \end{tikzcd} \end{center}
is exact and hence $\pi_*:K_j(\sigma)\rightarrow K_j(\phi)$ is an isomorphism for $j=0,1$.
\begin{definition}
$\Delta_\phi=\nu_j\circ\pi_*^{-1}$, for $j=0,1$.
\end{definition}
We provide a description of $\Delta_\phi$. For simplicity, we assume that $A$ and $B$ are unital and that $\phi(1)=1$. Given a triple $(p,1_n,v)$ in $\Gamma_0(\phi)$, we have
\[\Delta_\phi([p,1_n,v])=[(p\oplus0_m,p_v)]-[1_n\oplus0_{2m-n}]\]
where $m$ and $p_v$ are as in Definition \ref{projpath}. Given a triple $(1_n,u,g)$ in $\Gamma_1(\phi)$, we have
\[\Delta_\phi([1_n,u,g])=[(u,g)].\]
The proof that the diagram given in part (iv) of Theorem \ref{main} is commutative is straightforward and is left to the reader.
\subsection{Proof of Theorem \ref{axiom}} We have already proven part (ii) (Lemma \ref{stable}), so it remains to prove parts (i) and (iii). Both proofs are quite easy with the natural transformation $\Delta$ from part (iv) of Theorem \ref{main} in hand.
The assumptions on $\alpha_t$ and $\beta_t$ in part (i) clearly imply that $\alpha_t\oplus C\beta_t$ is a continuous path of $^*$-homomorphisms from $C_\phi$ to $C_\psi$. It follows from homotopy invariance of C$^*$-algebra $K$-theory that $(\alpha_0\oplus C\beta_0)_*=(\alpha_1\oplus C\beta_1)_*$, and hence that $(\alpha_0,\beta_0)_*=(\alpha_1,\beta_1)_*$ via the natural isomorphism $\Delta$.
For part (iii), the existence of $\phi:A\rightarrow B$ is an easy consequence of the universal property of inductive limits, using the $^*$-homomorphisms $\nu_i\circ\phi_i$. It is clear that $(C_{\phi_i},\alpha_{ij}\oplus C\beta_{ij})$ forms an inductive system of C$^*$-algebras, and the limit is $(C_\phi,(\mu_i\oplus C\nu_i))$ by Proposition 4.9 of \cite{pedersen}. The result then follows from continuity of C$^*$-algebra $K$-theory and the natural isomorphism $\Delta$.
\subsection{Proof of Theorem \ref{exc}} Parts (i) and (ii) follow immediately from exactness. Part (iii) follows from applying part (iii) of Theorem \ref{main} to the diagram \begin{center}
\begin{tikzcd} 0 \arrow[r] & \ker\phi \arrow[r, "\iota_\phi"] \arrow[d] & A \arrow[r, "\phi"] \arrow[d, "\phi"] & B \arrow[r] \arrow[d, equal] & 0 \\ 0 \arrow[r] & 0 \arrow[r, hook] & B \arrow[r, equal] & B \arrow[r] & 0 \end{tikzcd} \end{center} Part (iv) follows similarly, using the diagram \begin{center}
\begin{tikzcd} 0 \arrow[r] & I \arrow[r, hook] \arrow[d, hook] & A \arrow[r, "\pi_A"] \arrow[d, equal] & A/I \arrow[r] \arrow[d] & 0 \\ 0 \arrow[r] & A \arrow[r, equal] & A \arrow[r] & 0 \arrow[r] & 0 \end{tikzcd} \end{center}
\end{document} | arXiv |
Fixed Point Theory and Applications
A fixed point theorem for Meir-Keeler type contraction via Gupta-Saxena expression
Najeh Redjel1, 2,
Abdelkader Dehici1, 2 and
İnci M Erhan3Email author
Fixed Point Theory and Applications20152015:115
© Redjel et al. 2015
Received: 9 March 2015
In this paper, following the idea of Samet et al. (J. Nonlinear. Sci. Appl. 6:162-169, 2013), we establish a new fixed point theorem for a Meir-Keeler type contraction via Gupta-Saxena rational expression which enables us to extend and generalize their main result (Gupta and Saxena in Math. Stud. 52:156-158, 1984). As an application we derive some fixed points of mappings of integral type.
complete metric space
fixed point
Meir-Keeler mapping
Gupta-Saxena rational expression
It is well known that the contraction mapping principle of Banach [1] was the starting point of great discoveries and advances in mathematics, in particular in nonlinear analysis. This principle was the subject of several extensions by means of various generalized contractions (see, for example, [2–10]). Among the most relevant results in this direction one can give that of Meir and Keeler [11] who proved the following fixed point result.
Theorem 1.1
Let \((X,d)\) be a complete metric space and let f be a mapping from X into itself satisfying the following condition:
$$ \forall\epsilon>0, \exists\delta(\epsilon)> 0 \quad \textit{such that}\quad \epsilon\leq d(x,y)< \epsilon+ \delta(\epsilon)\quad \Longrightarrow \quad d\bigl(f(x),f(y) \bigr)< \epsilon. $$
Then f has a unique fixed point \(u \in X\). Moreover, for all \(x \in X\), the sequence \(\{f^{n}(x)\}\) converges to u.
As pointed out in [11], it is easy to observe that the conclusion of Banach theorem holds for the contraction in Theorem 1.1 which is called a strict contraction, that is, it satisfies
$$d\bigl(f(x),f(y)\bigr)< d(x,y)\quad \text{for } x\neq y. $$
In 1984, Gupta and Saxena proved the following fixed point result.
Let \((X,d)\) be a complete metric space and let f be a continuous mapping from X into itself satisfying
$$d\bigl(f(x),f(y)\bigr)\leq\alpha_{1} \frac {(1+d(x,f(x)))d(y,f(y))}{1+d(x,y)}+ \alpha_{2} \frac {d(x,f(x))d(y,f(y))}{d(x,y)}+\alpha_{3}d(x,y) $$
for all \(x,y \in X\), \(x\neq y\), where \(\alpha_{1}\), \(\alpha _{2}\), \(\alpha_{3}\) are constants with \(\alpha_{1},\alpha_{2},\alpha _{3}> 0\) and \(\alpha_{1}+\alpha_{2}+\alpha_{3}< 1\). Then f has a unique fixed point \(u\in X\). Moreover, for all \(x \in X\), the sequence \(\{f^{n}(x)\}\) converges to u.
For more details on this theorem, we refer, e.g., to [12, 13].
In this paper, we establish a new fixed point theorem of Meir-Keeler type involving Gupta-Saxena expression which extends Theorem 1.2 in the case where \(\alpha_{1},\alpha _{2},\alpha_{3} \in\, ]0, \frac{1}{3}[\). We also apply our theoretical results to contractions of integral type.
2 Main results
Our main result is the following theorem.
Let \((X,d) \) be a complete metric space and let \(f: X \rightarrow X\) be a continuous mapping. Assume that the following condition holds.
For any \(\epsilon> 0\), there exists \(\delta(\epsilon)> 0\) such that
$$\begin{aligned}& 3\epsilon\leq \frac{(1+d(x,f(x)))d(y,f(y))}{1+d(x,y)}+ \frac{d(x,f(x))d(y,f(y))}{d(x,y)}+d(x,y)< 3 \epsilon+ \delta(\epsilon) \\& \quad \Longrightarrow \quad d\bigl(f(x),f(y)\bigr)< \epsilon \end{aligned}$$
for all \(x,y \in X\) with \(x\neq y\). Then f has a unique fixed point \(u \in X\). Moreover, \(\lim_{n \to\infty}f^{n}(x_{0})= u\) for any \(x_{0} \in X\).
It is easy to observe that condition (1) implies that
$$\begin{aligned}& \text{for } x\neq y \text{ or } f(y)\neq y, \\& \quad d\bigl(f(x),f(y)\bigr)< \frac{1}{3} \biggl[ \frac {(1+d(x,f(x)))d(y,f(y))}{1+d(x,y)}+ \frac {d(x,f(x))d(y,f(y))}{d(x,y)}+ d(x,y) \biggr]. \end{aligned}$$
Let \(x_{0} \in X\) and consider the sequence \(\{x_{n}\}= \{ f^{n}(x_{0})\}_{n\geq0}\). We will prove that \(\{x_{n}\}\) is a Cauchy sequence in X. If there exists \(l_{0}\in\mathbb{N}\) such that \(x_{l_{0}}=x_{l_{0+1}}\), then clearly \(x_{l_{0}}\) is a fixed point of f. Now assume that \(x_{k}\neq x_{k+1}\) for all \(k\in\mathbb{N}\). Define
$$s_{n}= d(x_{n},x_{n+1}),\quad \forall n \in \mathbb{N}. $$
Following (2), we obtain that
$$\begin{aligned} s_{n} =&d\bigl(f(x_{n-1}),f(x_{n})\bigr) \\ < & \frac{1}{3} \frac {(1+d(x_{n-1},x_{n}))d(x_{n},x_{n+1})}{1+d(x_{n-1},x_{n})}+ \frac{1}{3} \frac {d(x_{n-1},x_{n})d(x_{n},x_{n+1})}{d(x_{n-1},x_{n})} + \frac{1}{3}d(x_{n-1},x_{n}) \\ = & \frac{2}{3}d(x_{n},x_{n+1}) + \frac {1}{3}d(x_{n-1},x_{n}) = \frac{2}{3} s_{n} + \frac{1}{3}s_{n-1}. \end{aligned}$$
This results in
$$s_{n}< s_{n-1}, \quad \forall n \in\mathbb{N}, $$
that is, the sequence \(\{s_{n}\}\) is decreasing. Then \(s_{n}\) converges to some \(s \geq 0 \); and, moreover, \(s_{n} \geq s\), \(\forall n \geq0\). We also have \(2s_{n} + s_{n-1} \rightarrow3s\) as \(n \rightarrow+\infty\). From (1), if \(s>0\), there exists \(\delta(s)>0\) such that
$$3s \leq2s_{n} + s_{n-1}< 3s+ \delta(s) $$
implies
$$d\bigl(f(x_{n-1}),f(x_{n})\bigr)= d(x_{n},x_{n+1})= s_{n}< s, $$
which contradicts \(s_{n}\geq s\). Thus, we deduce that
$$s_{n} \rightarrow0 \quad \text{as } n \rightarrow+\infty. $$
Now, let
$$\delta'(\epsilon)= \min\biggl\{ \delta\biggl( \frac{\epsilon }{7} \biggr), \frac{\epsilon}{7},1\biggr\} . $$
By the convergence of the sequence \(\{d(x_{n},x_{n+1})\}\) to 0, there exists \(k_{0}\in\mathbb{N}\) such that
$$ d(x_{l},x_{l+1})< \frac{\delta'(\epsilon)}{6},\quad \forall l \geq k_{0}. $$
Now, we define the set Ω by
$$\Omega=\biggl\{ x_{p}\Bigm| p\geq k_{0}, d(x_{p},x_{k_{0}})< \frac{3\epsilon}{7}+ \frac{\delta'(\epsilon)}{3}\biggr\} . $$
We will prove that
$$ f(\Omega) \subset\Omega. $$
Clearly, for \(\gamma\in\Omega\), there exists \(p \geq k_{0}\) such that \(\gamma= x_{p}\) and \(d(x_{p},x_{k_{0}})< \frac{3\epsilon}{7} + \frac{\delta '(\epsilon)}{3} \). If \(p= k_{0}\), we have \(f(\gamma)= x_{k_{0}+1} \in \Omega\) by (3). Then we will assume that \(p > k_{0}\). We distinguish two cases as follows.
(1) First case: Assume that
$$ \frac{3\epsilon}{7}\leq d(x_{p}, x_{k_{0}})< \frac{3\epsilon}{7}+ \frac{\delta '(\epsilon)}{3}. $$
First, we will show that
$$\begin{aligned} \begin{aligned}[b] \frac{\epsilon}{7} & \leq \frac {1}{3} \frac {(1+d(x_{p},x_{p+1}))d(x_{k_{0}},x_{k_{0}+1})}{1+d(x_{p},x_{k_{0}})}+ \frac{1}{3} \frac {d(x_{p},x_{p+1})d(x_{k_{0}},x_{k_{0}+1})}{d(x_{p},x_{k_{0}})}+ \frac{1}{3}d(x_{p},x_{k_{0}}) \\ & < \frac{\epsilon}{7}+ \frac{\delta '(\epsilon)}{3}. \end{aligned} \end{aligned}$$
From (5), we have
$$\begin{aligned} \frac{\epsilon}{7} \leq& \frac {1}{3}d(x_{p},x_{k_{0}}) \\ \leq& \frac{1}{3} \frac {(1+d(x_{p},x_{p+1}))d(x_{k_{0}},x_{k_{0}+1})}{1+d(x_{p},x_{k_{0}})}+ \frac{1}{3} \frac {d(x_{p},x_{p+1})d(x_{k_{0}},x_{k_{0}+1})}{d(x_{p},x_{k_{0}})}+ \frac{1}{3}d(x_{p},x_{k_{0}}). \end{aligned}$$
Moreover, by using (3) and (5), we get
$$\begin{aligned}& \frac{1}{3} \frac {(1+d(x_{p},x_{p+1}))d(x_{k_{0}},x_{k_{0}+1})}{1+d(x_{p},x_{k_{0}})}+ \frac{1}{3} \frac {d(x_{p},x_{p+1})d(x_{k_{0}},x_{k_{0}+1})}{d(x_{p},x_{k_{0}})}+ \frac{1}{3}d(x_{p},x_{k_{0}}) \\& \quad \leq \frac{1}{3}d(x_{k_{0}},x_{k_{0}+1})+ \frac{2}{3} \frac {d(x_{k_{0}},x_{k_{0}+1})d(x_{p},x_{p+1})}{d(x_{p},x_{k_{0}})}+ \frac{1}{3}d(x_{p},x_{k_{0}}) \\& \quad < \frac{1}{3} \frac{\delta'(\epsilon )}{6}+ \frac{2}{3}d(x_{p}, x_{p+1})+ \frac {1}{3}d(x_{p},x_{k_{0}}) \\& \quad < \frac{\delta'(\epsilon)}{18} + \frac {2}{3} \frac{\delta'(\epsilon)}{6}+ \frac{1}{3}\biggl( \frac{3\epsilon}{7} + \frac {\delta'(\epsilon)}{3}\biggr) \\& \quad = \frac{\epsilon}{7} + \frac{5\delta '(\epsilon)}{18} \\& \quad < \frac{\epsilon}{7} + \frac{\delta '(\epsilon)}{3}. \end{aligned}$$
Then we obtain
$$\begin{aligned}& \frac{1}{3} \frac {(1+d(x_{p},x_{p+1}))d(x_{k_{0}},x_{k_{0}+1})}{1+d(x_{p},x_{k_{0}})}+ \frac{1}{3} \frac {d(x_{p},x_{p+1})d(x_{k_{0}},x_{k_{0}+1})}{d(x_{p},x_{k_{0}})}+ \frac{1}{3}d(x_{p},x_{k_{0}}) \\& \quad < \frac{\epsilon}{7}+ \frac{\delta '(\epsilon)}{3}. \end{aligned}$$
From (7) and (8), we deduce that (6) is satisfied. In this case, the inequality
$$\begin{aligned} \frac{3\epsilon}{7} \leq& \frac {(1+d(x_{p},f(x_{p})))d(x_{k_{0}},f(x_{k_{0}}))}{1+d(x_{p},x_{k_{0}})}+ \frac {d(x_{p},f(x_{p}))d(x_{k_{0}},f(x_{k_{0}}))}{d(x_{p},x_{k_{0}})}+ d(x_{p},x_{k_{0}}) \\ < & \frac{3\epsilon}{7}+ \delta'(\epsilon) \end{aligned}$$
implies by (1) that
$$ d\bigl(f(x_{p}),f(x_{k_{0}})\bigr)< \frac{\epsilon}{7}. $$
Now, using the triangular inequality together with (3) and (9), we obtain that
$$\begin{aligned} d\bigl(f(x_{p}),x_{k_{0}}\bigr) \leq& d\bigl(f(x_{p}),f(x_{k_{0}}) \bigr)+ d\bigl(f(x_{k_{0}}),x_{k_{0}}\bigr) \\ < & \frac{\epsilon}{7}+ \frac{\delta '(\epsilon)}{6} \\ < & \frac{3\epsilon}{7}+ \frac{\delta '(\epsilon)}{3}. \end{aligned}$$
This implies that \(f(\gamma)= f(x_{p})= x_{p+1}\in\Omega\).
(2) Second case: Suppose that
$$ d(x_{p},x_{k_{0}})< \frac{3\epsilon}{7}. $$
From (2), we infer that
$$\begin{aligned} d\bigl(f(x_{p}),x_{k_{0}}\bigr) \leq& d \bigl(f(x_{p}),f(x_{k_{0}})\bigr)+ d\bigl(f(x_{k_{0}}),x_{k_{0}} \bigr) \\ < & \frac{1}{3} \frac {(1+d(x_{p},x_{p+1}))d(x_{k_{0}},x_{k_{0}+1})}{1+d(x_{p},x_{k_{0}})}+ \frac{1}{3} \frac {d(x_{p},x_{p+1})d(x_{k_{0}},x_{k_{0}+1})}{d(x_{p},x_{k_{0}})} \\ &{} + \frac{1}{3}d(x_{p},x_{k_{0}}) + d(x_{k_{0}+1}, x_{k_{0}}) \\ \leq& \frac{1}{3} \frac {d(x_{p},x_{p+1})d(x_{k_{0}},x_{k_{0}+1})}{1+d(x_{p},x_{k_{0}})}+ \frac{1}{3} \frac {d(x_{p},x_{p+1})d(x_{k_{0}},x_{k_{0}+1})}{d(x_{p},x_{k_{0}})} \\ &{}+ \frac{1}{3}d(x_{p},x_{k_{0}}) + \frac {4}{3} d(x_{k_{0}+1}, x_{k_{0}}). \end{aligned}$$
On the other hand, from (3) we have
$$\frac {d(x_{k_{0}},x_{k_{0}+1})}{1+d(x_{p},x_{k_{0}})}\leq d(x_{k_{0}},x_{k_{0}+1})< \frac{\delta'(\epsilon)}{6}< 1. $$
We consider the following two situations.
If \(d(x_{k_{0}},x_{k_{0}+1})\leq d(x_{k_{0}},x_{p})\), then (11) gives
$$d\bigl(f(x_{p}),x_{k_{0}}\bigr)< \frac{1}{3}d(x_{p},x_{p+1})+ \frac{1}{3}d(x_{p},x_{p+1}) + \frac {1}{3}d(x_{p},x_{k_{0}})+ \frac{4}{3}d(x_{k_{0}+1},x_{k_{0}}). $$
From (3) and (10), we deduce that
$$\begin{aligned} d\bigl(f(x_{p}),x_{k_{0}}\bigr) < & \frac{2}{3}\biggl( \frac{\delta'(\epsilon)}{6}\biggr)+ \frac {1}{3}\biggl( \frac{3\epsilon}{7}\biggr)+ \frac {4}{3}\biggl( \frac{\delta'(\epsilon)}{6}\biggr) \\ =& \frac{\delta'(\epsilon)}{3}+ \frac {\epsilon}{7} \\ < & \frac{\delta'(\epsilon)}{3}+ \frac {3\epsilon}{7}. \end{aligned}$$
If \(d(x_{k_{0}},x_{k_{0}+1})> d(x_{k_{0}},x_{p})\), then
$$\begin{aligned} d\bigl(f(x_{p}),x_{k_{0}}\bigr) \leq&d(x_{p+1},x_{p})+ d(x_{p},x_{k_{0}}) \\ < &d(x_{p+1},x_{p})+ d(x_{k_{0}},x_{k_{0}+1}) \\ < & \frac{\delta'(\epsilon)}{6} + \frac {\delta'(\epsilon)}{6} \\ =& \frac{\delta'(\epsilon)}{3} \\ < & \frac{\delta'(\epsilon)}{3}+ \frac {3\epsilon}{7}. \end{aligned}$$
In both situations (i) and (ii), we have \(f(\gamma)= f(x_{p})= x_{p+1} \in\Omega\). Thus, (4) holds and
$$ d(x_{m},x_{k_{0}})< \frac{3\epsilon}{7}+ \frac{\delta'(\epsilon)}{3}, \quad \forall m > k_{0}. $$
Now, \(\forall m, n \in\mathbb{N}\) satisfying \(m > n> k_{0}\), by (12), we have
$$d(x_{m},x_{n})\leq d (x_{m},x_{k_{0}})+ d(x_{n},x_{k_{0}})< \frac{6\epsilon }{7}+ \delta'( \epsilon)< \frac{6\epsilon}{7}+ \frac{\epsilon}{7}= \epsilon. $$
Therefore, \(\{x_{n}\}\) is a Cauchy sequence in X.
Since \((X,d)\) is a complete metric space, then there exists \(u\in X\) such that \(x_{n} \rightarrow u\) as \(n \rightarrow +\infty\). The fact that \(x_{n+1}= f(x_{n})\) and the continuity of f imply that \(u= f(u)\), that is, u is a fixed point of f.
To show the uniqueness, we assume that \(u'\) is another fixed point of f. From (2) it follows that
$$\begin{aligned} d\bigl(u,u'\bigr) =& d\bigl(f(u),f\bigl(u'\bigr) \bigr)< \frac{1}{3}\biggl( \frac{1+d(u,u)d(u',u')}{1+d(u,u')}\biggr)+ \frac {1}{3} \frac{d(u,u)d(u',u')}{d(u,u')}+ \frac{1}{3}d\bigl(u,u'\bigr) \\ = & \frac{1}{3}d\bigl(u,u'\bigr), \end{aligned}$$
which is a contradiction. This proves the uniqueness of the fixed point and completes the proof of the theorem. □
Now, we show that the result of Gupta and Saxena [12], where \(\alpha_{1}, \alpha_{2}, \alpha_{3} \in\, ]0, \frac{1}{3}[\), is a particular case of Theorem 2.1.
Corollary 2.2
(Gupta and Saxena [12])
Let \((X,d)\) be a complete metric space and f be a continuous mapping from X into itself. Assume that f satisfies
$$\begin{aligned}& \forall x, y \in X, x\neq y, \\& \quad d\bigl(f(x),f(y)\bigr) \leq k \biggl( \frac {(1+d(x,f(x)))d(y,f(y))}{1+d(x,y)}+ \frac {d(x,f(x))d(y,f(y))}{d(x,y)}+d(x,y) \biggr), \end{aligned}$$
where \(k\in\, ]0, \frac{1}{3}[\) is a constant. Then f has a unique fixed point \(u \in X\). Moreover, \(\forall x \in X\), the sequence \(\{f^{n}(x)\}\) converges to u.
Let \(\epsilon> 0\). If we take
$$\delta(\epsilon)= \epsilon\biggl( \frac{1}{k}-3\biggr), $$
then, whenever
$$\begin{aligned}& 3\epsilon\leq \frac{(1+d(x,f(x)))d(y,f(y))}{1+d(x,y)}+ \frac{d(x,f(x))d(y,f(y))}{d(x,y)}+d(x,y)< 3\epsilon +\delta, \\& d\bigl(f(x),f(y)\bigr) \leq k \biggl( \frac {(1+d(x,f(x)))d(y,f(y))}{1+d(x,y)}+ \frac {d(x,f(x))d(y,f(y))}{d(x,y)}+d(x,y) \biggr) \\& \hphantom{d\bigl(f(x),f(y)\bigr)} < k\bigl(3\epsilon+ \delta(\epsilon)\bigr) \\& \hphantom{d\bigl(f(x),f(y)\bigr)} = 3k\epsilon+ k\delta(\epsilon) \\& \hphantom{d\bigl(f(x),f(y)\bigr)} = 3k\epsilon+ \frac{k\epsilon}{k}- 3k\epsilon \\& \hphantom{d\bigl(f(x),f(y)\bigr)} = \epsilon. \end{aligned}$$
Notice that since \(k<\frac{1}{3}\), then \(\frac{\epsilon}{k}>3\epsilon\). Thus the condition (1) of Theorem 2.1 is satisfied, which completes the proof. □
Notice that the contraction mapping of Gupta and Saxena is a not a strict contraction, but k-contraction. Therefore, Theorem 2.1 is an extension of Gupta-Saxena result.
In this section, following the idea of Samet et al. [14], we will give an integral version of Gupta-Saxena result.
We start with the following theorem.
Let \((X,d)\) be a metric space and let f be a self-mapping defined on X. Assume that there exists a function ρ from \([0,+\infty[\) into itself satisfying the following:
\(\rho(0)=0\) and \(\rho(t)>0\) for every \(t>0\);
ρ is nondecreasing and right continuous;
for every \(\epsilon> 0\), there exists \(\delta (\epsilon)>0\) such that
$$\begin{aligned}& 3\epsilon\leq\rho \biggl( \frac {(1+d(x,f(x)))d(y,f(y))}{1+d(x,y)}+ \frac {d(x,f(x))d(y,f(y))}{d(x,y)}+d(x,y) \biggr)< 3 \epsilon+ \delta (\epsilon) \\& \quad \Longrightarrow\quad \rho \bigl(3d\bigl(f(x),f(y)\bigr) \bigr) < 3\epsilon \end{aligned}$$
for all \(x,y \in X\) with \(x\neq y\).
Then (1) is satisfied.
Fix \(\epsilon> 0\). Since \(\rho(3\epsilon) > 0\), by (iii), for \(\rho(3\epsilon)\) there exists \(\theta>0\) such that
$$\begin{aligned}& \rho (3\epsilon ) \leq\rho \biggl( \frac {(1+d(x,f(x)))d(y,f(y))}{1+d(x,y)}+ \frac {d(x,f(x))d(y,f(y))}{d(x,y)}+d(x,y) \biggr) < \rho(3\epsilon) + \theta \\& \quad \Longrightarrow\quad \rho \bigl(3d\bigl(f(x),f(y)\bigr) \bigr)< \rho(3 \epsilon). \end{aligned}$$
From the right continuity of ρ, there exists \(\delta> 0\) such that \(\rho(3\epsilon+ \delta) < \rho(3\epsilon) +\theta\). Fix \(x,y \in X\), \(x\neq y\) such that
$$3\epsilon\leq \frac{(1+d(x,f(x)))d(y,f(y))}{1+d(x,y)}+ \frac{d(x,f(x))d(y,f(y))}{d(x,y)}+d(x,y) < 3\epsilon+ \delta. $$
Since ρ is nondecreasing, we deduce
$$\begin{aligned} \rho(3\epsilon) \leq&\rho \biggl( \frac {(1+d(x,f(x)))d(y,f(y))}{1+d(x,y)}+ \frac {d(x,f(x))d(y,f(y))}{d(x,y)}+d(x,y) \biggr) \\ < &\rho(3\epsilon+ \delta)< \rho(3\epsilon) + \theta. \end{aligned}$$
Then, by (14), we have
$$\rho\bigl(3d\bigl(f(x),f(y)\bigr)\bigr)< \rho(3\epsilon), $$
which implies that \(d(f(x),f(y))<\epsilon\). Then (1) is satisfied and this completes the proof. □
Now, we denote by Ξ the set of all mappings \(h: [0,+\infty[ \, \rightarrow[0, +\infty[\) satisfying:
h is continuous and nondecreasing;
\(h(0)=0\) and \(h(t)> 0\) for all \(t > 0\).
Let \((X,d)\) be a metric space and let f be a mapping from X into itself. Assume that for each \(\epsilon> 0\), there exists \(\delta (\epsilon)\) such that
$$\begin{aligned}& 3\epsilon\leq h \biggl( \frac {(1+d(x,f(x)))d(y,f(y))}{1+d(x,y)}+ \frac {d(x,f(x))d(y,f(y))}{d(x,y)}+d(x,y) \biggr)< 3 \epsilon+ \delta(\epsilon) \\& \quad \Longrightarrow\quad h\bigl(3d\bigl(f(x),f(y)\bigr)\bigr) < 3\epsilon \end{aligned}$$
for all \(x,y \in X\), with \(x\neq y\), where \(h \in\Xi\) is a given function. Then (1) is satisfied.
This follows immediately from Theorem 3.1 since every continuous function \(h: [0,+\infty[\, \rightarrow[0,+\infty[\) is right continuous. □
As a consequence of this corollary, we have another result.
Let \((X,d)\) be a metric space and let f be a mapping from X into itself. Let φ be a locally integrable function from \([0, +\infty[\) into itself such that \(\int_{0}^{t} \varphi (s)\, ds > 0\) for all \(t > 0\). Assume that for each \(\epsilon>0\) there exists \(\delta(\epsilon)\) such that
$$\begin{aligned}& 3\epsilon\leq \int_{0}^{ \frac {(1+d(x,f(x)))d(y,f(y))}{1+d(x,y)}+ \frac {d(x,f(x))d(y,f(y))}{d(x,y)}+d(x,y)} \varphi(t)\, dt < 3\epsilon+ \delta (\epsilon) \\& \quad \Longrightarrow \quad \int_{0}^{3d(f(x),f(y))} \varphi(t)\, dt < 3\epsilon. \end{aligned}$$
Now, we are able to obtain an integral version of Gupta-Saxena result.
Let \((X,d)\) be a complete metric space and let f be a continuous mapping from X into itself. Let φ be a locally integrable function from \([0, +\infty[\) into itself such that \(\int_{0}^{t} \varphi(s)\, ds > 0\) for all \(t > 0\). Assume that f satisfies the following condition.
For all \(x,y \in X\), \(x\neq y\),
$$ \int_{0}^{3d(f(x),f(y))}\varphi(t)\, dt \leq\mu \int_{0}^{ \frac {(1+d(x,f(x)))d(y,f(y))}{1+d(x,y)}+ \frac {d(x,f(x))d(y,f(y))}{d(x,y)}+d(x,y)}\varphi(t)\, dt, $$
where \(\mu\in\, ]0,1[\). Then f has a unique fixed point \(u \in X\). Moreover, for any \(x\in X\), the sequence \(\{f^{n}(x)\}\) converges to u.
Let \(\epsilon>0\). It is easy to observe that (15) is satisfied for \(\delta(\epsilon)= 3\epsilon( \frac {1}{\mu}-1)\). Then (1) holds and this completes the proof. □
Remark 3.5
Note that the result of Corollary 2.2 can be established from Corollary 3.4 by taking \(\varphi\equiv1\) and \(\mu= 3k\), \(k \in\, ]0, \frac {1}{3}[\). Clearly, for this choice, (16) becomes
$$d\bigl(f(x),f(y)\bigr) \leq k \biggl( \frac {(1+d(x,f(x)))d(y,f(y))}{1+d(x,y)}+ \frac {d(x,f(x))d(y,f(y))}{d(x,y)}+d(x,y) \biggr), $$
which is exactly the contractive condition of Corollary 2.2.
This work was elaborated within the framework of the scientific stay of the first two authors at Atılım University (Turkey). They thank all the staff of the Department of Mathematics, Atılım University and, in particular, Professor Karapınar for offering a pleasant environment of work and for his helpful discussions.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Laboratory of Informatics and Mathematics, University of Souk-Ahras, P.O. Box 1553, Souk-Ahras, 41000, Algeria
Department of Mathematics, University of Constantine 1, Constantine, 25000, Algeria
Department of Mathematics, Atilim University, İncek, Ankara, 06836, Turkey
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\begin{document}
\title{Device-independent randomness generation with sublinear shared quantum resources}
\author{C\'edric Bamps} \author{Serge Massar} \author{Stefano Pironio} \affiliation{Laboratoire d'Information Quantique, CP 224, Universit\'e libre de Bruxelles (ULB), 1050 Brussels, Belgium}
\begin{abstract} In quantum cryptography, device-independent (DI) protocols can be certified secure without requiring assumptions about the inner workings of the devices used to perform the protocol. In order to display nonlocality, which is an essential feature in DI protocols, the device must consist of at least two separate components sharing entanglement. This raises a fundamental question: how much entanglement is needed to run such DI protocols? We present a two-device protocol for DI random number generation (DIRNG) which produces approximately $n$ bits of randomness starting from $n$ pairs of arbitrarily weakly entangled qubits. We also consider a variant of the protocol where $m$ singlet states are diluted into $n$ partially entangled states before performing the first protocol, and show that the number $m$ of singlet states need only scale sublinearly with the number $n$ of random bits produced. Operationally, this leads to a DIRNG protocol between distant laboratories that requires only a sublinear amount of quantum communication to prepare the devices. \end{abstract}
\maketitle
\section{Introduction} A quantum random number generation (RNG) protocol is device-independent (DI) if its output can be guaranteed to be random with respect to any adversary on the sole basis of certain minimal assumptions, such as the validity of quantum physics and the existence of secure physical locations \cite{AM16}. The internal workings of the devices, however, do not need to be trusted.
Device-independence is made possible by exploiting the violation of a Bell inequality \cite{BCP+14}, which certifies the random nature of quantum measurement outcomes. As a result, DIRNG protocols necessarily consume two fundamental resources: entangled states shared across separated devices and an initial public random seed that is uncorrelated to the devices and used to determine the random measurements performed on the entangled states. Out of these two resources, a DIRNG protocol produces $n$ private random bits.
The initial random seed that is consumed can be of extremely low quantity or quality. Indeed, $n$ private random bits can be produced starting from an initial string of uniform bits whose required length has gradually been reduced in a series of works \cite{PAM+10,VV12,CY13,MS14a}, culminating in the result that only a constant, i.e., independent of the output length $n$, amount of initial uniform random bits are required \cite{CY13,MS14a}. Furthermore, the initial seed does not necessarily need to consist of uniform random bits, as it possible to design DIRNG protocols consuming an arbitrarily weak random seed characterized only by its total min-entropy~\cite{CSW14}.
What about entanglement, the second fundamental resource that is consumed in any DIRNG protocol? This quantum resource usually consists of $m$ copies $\ket{\psi}^{\otimes m}$ of some bipartite entangled state $\ket{\psi}$ shared between two separated devices ${\mathrm A}$ and ${\mathrm B}$ that can be prevented at will from interacting with one another. Though DIRNG protocols involve a single user, it is useful for exposition purposes to view these two devices as being operated by two agents, Alice and Bob, in two remote sublaboratories. The $m$ copies $\ket{\psi}^{\otimes m}$ can either be stored prior to the start of the protocol inside quantum memories in Alice's and Bob's sublaboratories, or each copy $\ket{\psi}$ can be produced individually during each execution round of the protocol, say by a source located between Alice and Bob.
All existing protocols consume at best a linear amount $m = \Omega(n)$ of such shared entangled states $\ket{\psi}$, as they operate by separately measuring (in sequence or in parallel) each of these $m$ copies, with each separate measurement yielding at most a constant amount of random bits. Furthermore, the states $\ket{\psi}$ are typically highly entangled states---the prototypical example of a DIRNG protocol involves the measurement of $n$ maximally entangled two-qubit states $\ket{\phi^+}$, from each of which roughly $1$ bit of randomness can be certified using the CHSH inequality \cite{PAM+10}.
We will show that the consumption of entangled resources can be dramatically improved qualitatively and quantitatively. First, we show---by analogy with the fact that the initial random seed does not need to consist of uniform bits---that highly entangled states are not necessary for DIRNG: instead of using $n$ copies of maximally entangled two-qubit pairs $\ket{\phi^+}$, $n$ random bits can be produced from $n$ copies of any partially entangled two-qubit pair $\ket{\psi_\theta} = \cos\theta \ket{00} + \sin\theta \ket{11}$ with $0<\theta\leq \pi/4$ (see Theorem~\ref{thm:minentropy} and Corollary~\ref{cor:rng-partial}).
We then turn this statement concerning the quality of the shared entangled resources into a quantitative statement about the amount of entanglement that needs to be consumed in a DIRNG protocol. The $n$ copies of the partially entangled state $\ket{\psi_\theta}$ correspond to a total of $n S(\theta)$ ebits where $S(\theta) = h_2(\sin^2 \theta)$ is the entropy of entanglement of $\ket{\psi_\theta}$ expressed in terms of the binary entropy $h_2$. Since $S(\theta)$ can be made arbitrarily low by considering sufficiently small values of $\theta$, the above result seems to suggest that the total amount $n S(\theta)$ of entanglement consumed can also be made arbitrarily small as a function of $n$ by considering sufficiently fast decreasing values for $\theta = \theta(n)$. However, if it is true that for any given $\theta$, one can produce $n$ random bits from $n$ copies of $\ket{\psi_\theta}$ for any $n$ sufficiently large, the dependency between $\theta$ and $n$ cannot be chosen arbitrarily. This essentially originates from the fact that as $\theta \to 0$ the robustness to noise of the corresponding states $\ket{\psi_\theta}$, which become less and less entangled, decreases and must be compensated by increasing the number $n$ of copies of the states $\ket{\psi_\theta}$ to improve the estimation phase of the protocol. There is thus a tradeoff between $\theta$ and $n$, which we show can nevertheless result in a total amount of entanglement $n S(\theta) = \Omega(n^k\log n)$ with $7/8<k<1$ (see Corollary~\ref{cor:ebit}). This amount of entanglement is \emph{sublinear} in the number $n$ of output random bits, fundamentally improving over existing protocols for which the entanglement consumption is at best linear.
Though the protocol that we introduce consumes a sublinear amount of entanglement, it still requires a linear number of shared quantum resources in the form of $n$ copies of the two-qubit entangled states $\ket{\psi_\theta}$. These shared entangled states must be established through some quantum communication between Alice's and Bob's sublaboratories, either during the protocol itself or prior to the protocol, and will thus require the exchange of $n$ qubits. Since this quantum communication will typically be costly (for instance because of high losses in the communication channel), it represents a measure of the use of shared quantum resources which is more operational and better motivated than the entropy of entanglement. From this perspective, however, our first protocol is not fundamentally different from existing protocols that also involve the exchange of $n$ qubits to produce $n$ random bits.
This leads us to consider a slight modification of our protocol in which Alice and Bob initially share $m$ maximally entangled two-qubit states $\ket{\phi^+}$, which can be established through the exchange of $m$ qubits. These singlets are then transformed by entanglement dilution \cite{BBPS96} into roughly $n = S(\theta)/m$ copies of $\ket{\psi_\theta}$ states through local operations and classical communication (LOCC), which are then used in our regular protocol.
However entanglement dilution is only noiseless asymptotically, in the limit of an infinite number of copies $m \to \infty$. For finite $m$, entanglement dilution is inherently noisy. As our protocol is increasingly sensitive to noise as the degree of entanglement of the states $\theta$ tends to $0$, it is not a priori obvious that combining randomness generation with entanglement dilution will work.
Nevertheless we show that such a two-step protocol works even though the entanglement dilution slightly degrades the tradeoff between $\theta$ and $n$. Specifically we exhibit a protocol that can get $n$ output random bits starting from a sublinear number $m = n S(\theta) = \Omega(n^{k'}\log n)$ of initial copies of $\ket{\phi^+}$ states, with $7/8<k'<1$. This represents a quantitative improvement of the use of quantum resources with respect to all existing protocols, analogous to the fact that a DIRNG protocol needs only a sublinear amount of uniform random bits.
The starting point of our work is the work \cite{AMP12} wherein a family of variants of the CHSH inequality, the tilted-CHSH inequalities, are introduced, which seem particularly suited to generate randomness from weakly entangled qubit states. Indeed, it was shown in \cite{AMP12} that maximal violation of a tilted CHSH inequality certifies one bit of randomness and can be achieved by entangled two-dimensional systems with arbitrarily little entanglement \footnote{ Note that not all weakly entangled states can be used for device-independent randomness generation: for instance there is a regime of visibility in which noisy singlet states (so-called Werner states) are entangled but incapable of displaying nonlocality, and hence also incapable of displaying randomness \cite{Wer89}. } This was later extended to show that by using sequential measurements, a single pair of entangled qubits in a pure state could certify an arbitrary amount of randomness \cite{CJA+17}. However neither of these works presented a protocol, including an estimation phase and security analysis taking into account non-maximal violation, for device independent randomness generation. In fact the results of \cite{AMP12,CJA+17} do not by themselves imply the existence of such a protocol.
We now recall the tilted-CHSH expressions of \cite{AMP12}, whose properties of randomness certification in weakly entangled states will play a central role in our protocol.
\subsection*{Tilted-CHSH game} The tilted-CHSH expressions $I_1^\beta$ are a family of Bell expressions introduced in \cite{AMP12} and parameterized by a tilting parameter $\beta\in\coint{0,2}$. We start by reformulating $I_1^\beta$ as a nonlocal game, expressed in terms of a predicate function $V\in \{0,1\}$. This will put us in the right conditions to apply the entropy accumulation theorem of \cite{DFR16} following \cite{AFVR16}. In this reformulation, Alice is given input ${x} \in \{0,1\}$ and Bob input ${y} \in \{0,1,2\}$ according to the joint distribution \begin{equation} \label{eq:game-distribution}
p({x},{y}) = \begin{cases}
\frac{1}{4+\beta} & ({x},{y}) \in \{0,1\}^2 \,, \\
\frac{\beta}{4+\beta} & ({x},{y}) = (0,2) \,, \\
0 & \text{otherwise.}
\end{cases} \end{equation} Alice and Bob then provide one answer each, $({a},{b}) \in \{0,1\}^2$ respectively, and the game is won if the following predicate function $V({a},{b},{x},{y}) \in \{0,1\}$ returns $1$: \begin{equation} \label{eq:game-predicate}
V({a},{b},{x},{y}) = \begin{cases}
1 & ({x},{y}) \in \{0,1\}^2 \text{ and } {a} \oplus {b} = {x} {y} \,, \\
1 & ({x},{y}) = (0,2) \text{ and } {a} = 0 \,, \\
0 & \text{otherwise.}
\end{cases} \end{equation} Note that in our reformulation of the tilted-CHSH expression as a game, we have introduced for convenience a third setting for Bob ($y=2$) that is absent in the original tilted-CHSH expression. This game can be understood as a convex combination between the CHSH game and a ``trivial'' game: the former's success criterion is ${a} \oplus {b} = {x} {y}$ with input probabilities $p({x},{y}) = 1/4$ for $({x},{y}) \in \{0,1\}^2$, while the latter's success criterion is ${a} = 0$ with a deterministic input $({x},{y}) = (0,2)$. While Bob can tell the two games apart from his input $y$ thanks to the introduction of the third setting $y=2$, from Alice's point of view they are not distinguishable. This makes the mixture of the CHSH game with the trivial game nontrivial.
Given the predicate function (\ref{eq:game-predicate}), it can easily be verified that the expected winning probability $\omega$ for the tilted-CHSH game is linked to the expectation value $\bar I_1^\beta$ of the tilted-CHSH expression through \begin{align} \omega &= \sum_{\mathclap{{a},{b},{x},{y}}} V({a},{b},{x},{y}) p({x},{y}) p({a},{b} \mid {x},{y}) \\
&= \frac12 + \frac{1}{8+2\beta} \bar I_1^\beta \label{eq:game-ineq}\,, \end{align} where $p({a},{b} \mid {x},{y})$ are the probabilities characterizing Alice and Bob's outputs.
Note that when ${y}=2$, Bob's output does not affect the outcome of the game, and Bob is free to provide any output. We expect that it should be possible to reformulate our results without introducing Bob's third setting, but we have found it simplest to proceed as above in order to follow closely the results of \cite{AFVR16}, where non-local games are used rather than Bell inequalities.
From the relation (\ref{eq:game-ineq}) between the tilted-CHSH game and the tilted-CHSH expression, it follows from the results of \cite{AMP12} that the winning probability $\omega$ goes up to $1/2 + (2+\beta)/(8+2\beta)$ for classical devices, and $1/2 + \sqrt{8+2\beta^2}/(8+2\beta) = \omega_\mathrm{q}$ for quantum devices. This quantum value $\omega_\mathrm{q}$ is uniquely achieved (up to local transformations and up to Bob's measurement operator for $y=2$) by a pair of devices implementing certain local measurements on a two-qubit partially entangled state $\ket{\psi_\theta} = \cos\theta \ket{00} + \sin\theta \ket{11}$ with $\tan(2\theta) = \sqrt{2/\beta^2 - 1/2}$ \cite{AMP12}. We call this optimal pair of devices the \emph{reference devices} for the tilted-CHSH game of tilting parameter $\beta$. In the following we will sometimes use $\theta$ as the game parameter instead of $\beta$; it is always understood that they are linked by the above relation.
One important feature of the reference devices, as highlighted in \cite{AMP12}, is that, for any $0<\theta\leq \pi/4$, Alice's measurement when ${x} = 1$ returns a uniformly distributed outcome ${a} \in \{0,1\}$ uncorrelated with the environment, i.e., one bit of ideal randomness. Thus by separately measuring $n$ copies of the partially entangled state $\ket{\psi_\theta} = \cos\theta \ket{00} + \sin\theta \ket{11}$ according to the reference measurements, one could in principle generate $n$ bits of randomness for any $0<\theta\leq \pi/4$.
However, the results of \cite{AMP12} do not immediately imply this claim because they only apply to a single use of a quantum system that is known to achieve the maximal winning probability $\omega_\mathrm{q}$ of the tilted-CHSH game. Thus one should first embed the tilted-CHSH game in a proper DIRNG protocol in which no assumptions are made beforehand about the quantum systems, but where the amount of randomness generated is instead estimated from their observed behavior. This requires in particular a robust version of the results of \cite{AMP12}, i.e., an assessment of the randomness produced by quantum devices achieving a suboptimal winning probability $\omega<\omega_\mathrm{q}$. Indeed, even ideal devices are not expected to achieve the quantum maximum when they are used a finite number $n$ of times because of inherent statistical noise. We now address this by introducing an explicit DIRNG protocol based on the tilted-CHSH inequalities and a robust security analysis based on the entropy accumulation theorem (EAT) \cite{DFR16,AFVR16,ADF+18} and the self-testing properties of the tilted-CHSH inequalities introduced in \cite{BP15}.
\section{DIRNG protocol based on the tilted-CHSH game} \label{sec:protocol} Our protocol consists of the following steps: \begin{enumerate}[noitemsep] \item
Select values for the following parameters: \begin{itemize}[nosep] \item The game parameter $\beta \in \coint{0,2}$; \item The number of measurement rounds $n$; \item The expected fraction of test rounds $\gamma$; \item A success threshold $\omega_\mathrm{q} - \xi$. \end{itemize} \item \label{item:protocol:round}
Let $i=1$.
Choose $T_i \in \{0,1\}$ independently at random such that $\Pr[T_i = 1] = \gamma$.
If $T_i = 1$, perform a game round: measure the devices with settings $({X}_i,{Y}_i)$, selected at random according to the distribution given in \eqref{eq:game-distribution}, record the output $({A}_i,{B}_i)$ and compute $C_i = V({A}_i,{B}_i,{X}_i,{Y}_i)$ according to \eqref{eq:game-predicate}.
If $T_i = 0$, perform a generation round: measure the devices with $({X}_i,{Y}_i) = (1,0)$, record the output $({A}_i,{B}_i)$ and let $C_i = \bot$. \item
Repeat step~\ref*{item:protocol:round} for $i = 2, \dotsc, n$. \item \label{item:protocol:estimation}
Finally, if $\sum_{i : C_i = 1} 1 \ge n \gamma (\omega_\mathrm{q} - \xi)$, the protocol succeeds.
Otherwise, it aborts. \end{enumerate} An immediate application of Hoeffding's inequality \cite{Hoe63} produces an upper bound on the \emph{completeness error} for this protocol, that is, the probability that the ideal devices fail the protocol: \begin{lemma} \label{lem:completeness} Using the reference devices in the $n$ rounds, the completeness error for the protocol is bounded by \begin{equation}\label{eq:comperr}
\epsilon_\mathrm{c} = \exp \mleft( -2 n (\gamma \xi)^2 \mright) \,. \end{equation} \end{lemma}
\subsection{Soundness of the protocol} We now establish the soundness of our protocol, that is, its ability to produce a positive amount of randomness with high probability given that the protocol did not abort. The security of this protocol rests on three standard assumptions in the DI setting: that the devices and their environment obey the laws of quantum mechanics, that the random seed used to select inputs is independent from the devices, and that the two devices are unable to communicate during each round of the protocol. Our analysis is based on the entropy accumulation theorem (EAT) \cite{DFR16} following closely its application to DIRNG in \cite{AFVR16}.
The EAT, as its name indicates, provides an estimate of the smooth min-entropy accumulated throughout a sequence of measurements. It implies that the smooth min-entropy of the joint measurement outcomes of our protocol scales linearly with the number of rounds, with each round providing on average an amount of min-entropy roughly equivalent to the von Neumann entropy of a single round's outcome. In order to use the EAT, it is first necessary to bound this single-round von Neumann entropy as a function of the expected probability of success $\omega$ in the tilted-CHSH game. The following Lemma, which we derive in Appendix~\ref{app:robust} from the robust self-testing bounds for the tilted-CHSH inequality \cite{BP15}, provides a bound on the conditional min-entropy, which in turn bounds the conditional von Neumann entropy:
\begin{restatable}{lemma}{minentropylemma} \label{lem:minentropy} Let $\omega$ be the expected winning probability for the tilted-CHSH game with parameter $\beta$ of a pair of quantum devices, whose internal degrees of freedom can be entangled with the environment $E$. Then the conditional min-entropy of the measurement outcome ${A}$ for input ${X} = 1$ is bounded as \begin{equation}
H_\rmin({A} \mid E; {X}=1) \geq 1 - \kappa \theta^{-4} \sqrt{\omega_\mathrm{q} - \omega} \equiv g(\omega) \end{equation} with $\kappa \le 4 \sqrt{4+\beta} \, (4\sqrt2 + 61)/\ln 2 \leq 385 \sqrt{4+\beta}$. \end{restatable} The behavior of the bound with respect to $\omega_\mathrm{q} - \omega$ is optimal \cite{RUV12}, while numerical results suggest that the optimal dependency in $\theta$ is $O(\theta^{-2})$ \cite{CJA+17}. This will not, however, significantly affect our conclusions.
Using this bound in the EAT along the lines of \cite{AFVR16} (see Appendix~\ref{app:EAT}) yields the following theorem:
\begin{restatable}{theorem}{minentropytheorem} \label{thm:minentropy} \twocolumnswitch{
\def&{&}
\def\nonumber\\{\nonumber\\}
\let\eenv\align \let\endeenv\endalign }{
\def&{}
\def\nonumber\\{}
\let\eenv\equation \let\endeenv\endequation } Let $\mathbf {A}, \mathbf {B}, \mathbf {X}, \mathbf {Y}, \mathbf T, \mathbf C$ be the classical random variables output by the protocol, and $E$ the quantum side information of a potential adversary. Let $\mathcal{S} = \mathcal{S}(\mathbf C)$ be the success event for the protocol. Let $\epsilon'$, $\epsilon_\mathrm{s}$ be two positive error parameters. Then, for any given pair of devices used in the protocol, either $\Pr[\mathcal{S}] \le \epsilon'$ or \begin{equation}\label{eq:hmin} H_\rmin^{\epsilon_\mathrm{s}}(\mathbf {A} \mathbf {B}\mid \mathbf {X} \mathbf {Y} \mathbf T E; \mathcal{S}) \ge \nu \tau n \,, \end{equation} where $\nu = 1 - \gamma (2+\beta)/(4+\beta)$, \begin{eenv}
\tau = 1 &- \kappa \theta^{-4} \sqrt{\xi + \frac{2}{\gamma \sqrt n} \sqrt{1 - 2 \log_2(\epsilon_\mathrm{s} \epsilon')}} \nonumber\\
&- \frac{2 \log_2 26}{\sqrt n} \sqrt{1 - 2 \log_2(\epsilon_\mathrm{s} \epsilon')} \,, \label{eq:rate} \end{eenv} and $H_\rmin^{\epsilon_\mathrm{s}}(\mathbf {A} \mathbf {B}\mid \mathbf {X} \mathbf {Y} \mathbf T E; \mathcal{S})$ is the ${\epsilon_\mathrm{s}}$-smooth min-entropy of the output $(\mathbf {A}, \mathbf {B})$ given $\mathbf {X}, \mathbf {Y},\mathbf T, E$ and conditioned on the event $\mathcal{S}$. \end{restatable}
Given such a bound on the smooth min-entropy, there exist efficient procedures to extract from the raw outputs of the protocol a string of close-to-uniform random bits whose length is of the order of $H_\rmin^{\epsilon_\mathrm{s}}$, with the smoothing parameter $\epsilon_\mathrm{s}$ characterizing the closeness to the uniform distribution.
\subsection{Random bits from any partially entangled two-qubit state} Theorem~\ref{thm:minentropy} directly implies the following corollary, which shows the possibility of generating one bit of randomness per arbitrarily weakly entangled qubit pair: \begin{corollary} \label{cor:rng-partial} For any constant values of the protocol parameters $\theta$, $\xi$, and $\gamma$ such that $\kappa \theta^{-4} \sqrt\xi < 1$ and for sufficiently large $n$, the protocol has vanishing completeness error and it generates $\Omega(n)$ bits of randomness from $n$ partially entangled states $\ket{\psi_\theta}$. For $\xi$ and $\gamma$ approaching $0$, the production of randomness in the protocol is asymptotically equal to $n$. \qed \end{corollary}
\subsection{Sublinear entanglement consumption} We now consider how, in an ideal implementation of our protocol, the amount of shared entanglement consumed is related to the amount of randomness produced. For given $n$, the entanglement consumption obviously decreases with smaller values of $\theta$. According to \eqref{eq:hmin} and \eqref{eq:rate}, the randomness produced, however, also decreases with smaller $\theta$, unless this decrease is compensated by a suitable choice of the parameters $\gamma$ and $\xi$. Indeed, for small $\theta$, $\gamma$ should be made larger to increase the fraction of game rounds and better test the devices. Similarly, $\xi$ should be smaller (i.e., the threshold for the protocol's success should be set higher) in order for Lemma~\ref{lem:minentropy} to certify a nontrivial amount of min-entropy. But the parameters $\gamma$ and $\xi$ also appear in the completeness error \eqref{eq:d-completeness} and thus cannot be set completely freely if this error is to remain small: setting $\gamma$ too low makes the estimation of the success rate at step \ref{item:protocol:estimation} of the protocol more uncertain, \footnote{ From the perspective of randomness generation, a small value of $\gamma$ is desirable as it increases the factor $\nu$ in \eqref{eq:hmin} by increasing the rate of generation rounds. However, game rounds also contribute to the final randomness, which makes the choice of $\gamma = 1$ possible. } and setting $\xi$ too low makes the threshold harder to reach. In the following corollary to Theorem~\ref{thm:minentropy}, we show that there exists a choice for the parameters $\theta$, $\xi$, and $\gamma$, expressed as functions of $n$, such that the consumption of ebits $m$ is sublinear in the number of rounds $n$: \begin{corollary} \label{cor:ebit} Let $\lambda_\xi$, $\lambda_\gamma$, $\lambda_\theta$ be positive scaling parameters such that \begin{gather} \label{eq:tradeoff-entanglement}
\lambda_\theta < 2\lambda_\xi \,. \\ \label{eq:tradeoff-completeness}
\lambda_\xi + \lambda_\gamma < 1/2 \,, \end{gather} Let $\theta = n^{-\lambda_\theta/16}$, $\xi = n^{-\lambda_\xi}$, $\gamma = n^{-\lambda_\gamma}$, and constant $\epsilon_\mathrm{s}$ and $\epsilon'$. Then, for $n \to \infty$, the entropy bound of Theorem~\ref{thm:minentropy} is asymptotically equal to $n$, the completeness error vanishes, and the amount of entanglement consumed is sublinear: \begin{equation} \label{eq:asymptotic-n-m}
m = n S(\theta) \sim n \theta^2\log_2\theta^{-2} =
\frac{\lambda_\theta}{8} n^{k} \log_2 n \,, \end{equation} with $k = 1 - \lambda_\theta/8 \in \ooint{7/8, 1}$. \qed \end{corollary}
The constants $\lambda_\theta$, $\lambda_\xi$, $\lambda_\gamma$ give the rate at which the parameters $\theta$, $\xi$ and $\gamma$ tend to zero with increasing $n$. The condition \eqref{eq:tradeoff-entanglement} expresses the fact that when the entanglement is small, the success threshold must be close to the maximum in order for the min-tradeoff function to take a nontrivial value (see Appendix~\ref{app:EAT}). The condition \eqref{eq:tradeoff-completeness} expresses a tradeoff in the completeness error between how close the success threshold is to the maximum and the number of rounds that must be devoted to testing the correlations. A larger fraction of game rounds (i.e., a larger $\gamma$) makes the success criterion fluctuate less, which allows for a higher threshold (i.e., a smaller $\xi$).
\section{Using diluted singlets} \label{sec:dilution} As mentioned in the introduction, the use of partially entangled states for randomness expansion enables us to reduce the amount of qubits exchanged between the devices when preparing their shared entanglement. We reach this goal by applying our protocol to the outcome of an \emph{entanglement dilution} procedure, which transforms $m$ singlet states $\ket{\phi^+}$ to $n \simeq m / S(\theta)$ partially entangled states $\ket{\psi_\theta}$. Thus, only $m$ qubits need to be transferred between the devices in order to prepare the initial state $\ket{\phi^+}^{\otimes m}$.
We use the procedure of Bennett et al.\ \cite{BBPS96}, in which Alice prepares the $n$ pairs locally, processes Bob's share with Schumacher compression then teleports them to Bob using the $m$ singlets, who expands them back to $n$ qubits. Since Schumacher compression is a lossy operation, the resulting state shared by Alice and Bob, which we denote as $
\mathcal D_{\theta,\delta}(\projop{\phi^+}^{\otimes m}) $ is not exactly $\ket{\psi_\theta}^{\otimes n}$, but it is close in trace distance (with $\norm{\rho}_1 = \tr\abs\rho$): \begin{restatable}{lemma}{dilutionlemma} \label{lem:dilution} \twocolumnswitch{
\let\eenv\multline \let\endeenv\endmultline }{
\let\eenv\equation \let\endeenv\endequation } Using perfect devices, the dilution channel $\mathcal D_{\theta,\delta}$ maps $m$ copies of the singlet $\ket{\phi^+}$ into $n$ copies of the partially entangled qubit state $\ket{\psi_\theta} = \cos\theta \ket{00} + \sin\theta \ket{11}$ with $m = (S(\theta) + \delta) n$, up to error terms bounded by \begin{eenv} \label{eq:epsilonprep}
\norm*{
\mathcal D_{\theta,\delta}(\projop{\phi^+}^{\otimes m})
- \projop{\psi_\theta}^{\otimes n}
}_1 \\
\le 2 \sqrt{\epsilon_\pi} + \epsilon_\pi
\equiv \epsilon_\mathrm{prep}
\,, \end{eenv} with \begin{gather} \SwapAboveDisplaySkip \label{eq:epsilon-proj}
\epsilon_\pi = 2 \exp(-2 n \delta^2/\Delta^2) \,, \\
\Delta = -\log_2 \tan^2 \theta \,. \end{gather} \end{restatable} This lemma mostly follows from \cite{BBPS96,Wil13}; we prove it in Appendix~\ref{app:dilution}.
It follows from Lemma~\ref{lem:dilution} that even a perfectly implemented dilution procedure introduces some noise in the protocol. We thus need to derive a new statement for the completeness error, the probability that perfect devices fail the protocol. Using the indistinguishability interpretation of the trace distance, if the reference state $\ket{\psi_\theta}^{\otimes n}$ passes the threshold of the protocol with probability $1-\epsilon$, the diluted state $\mathcal D_{\theta,\delta}(\projop{\phi^+}^{\otimes m})$, which is $\epsilon_\mathrm{prep}$-close to the reference, will pass the same threshold with probability at least $1-\epsilon - \tfrac12 \epsilon_\mathrm{prep}$ \cite{Wil13}. Using the value of $\epsilon$ given in Lemma~\ref{lem:completeness} immediately implies the following: \begin{lemma} \label{lem:d-completeness} Starting from $m$ perfect singlets, the composition of the dilution procedure $\mathcal D_{\theta,\delta}$ and the randomness generation protocol has its completeness error bounded by \begin{equation} \label{eq:d-completeness}
\epsilon_\mathrm{c} = \tfrac12 \epsilon_\mathrm{prep} + \exp \mleft( -2 n (\gamma \xi)^2 \mright) \,. \end{equation} \end{lemma}
The following analogue of Corollary~\ref{cor:ebit} applies to the composition of entanglement dilution and randomness expansion; it immediately follows from the chosen parameterization: \begin{corollary} \label{cor:d-ebit} Let $\delta = S(\theta) c$ with $c = n^{\lambda_c/8}$ for some real parameter $\lambda_c$. Let the parameters of the protocol be set as in Corollary~\ref{cor:ebit}, with the additional constraint that \begin{equation} \label{eq:tradeoff-dilution}
0 < \lambda_c < \lambda_\theta \,. \end{equation} Starting from $m$ singlets, the composition of entanglement dilution with parameters $\delta$ and $\theta$ with the randomness expansion protocol yields an entropy bound in Theorem~\ref{thm:minentropy} which is asymptotically equal to $n$ for $n \to \infty$, with a vanishing completeness error, and a sublinear consumption of entanglement: \begin{multline}
m
= n \, (S(\theta)+\delta) \\
\sim n^{1+\lambda_c/8}\theta^2\log_2\theta^{-2}
= \frac{\lambda_\theta}{8} n^{k'} \log_2 n \,, \end{multline} with $k' = 1 - (\lambda_\theta-\lambda_c)/8 \in \ooint{7/8, 1}$. \qed \end{corollary}
In addition to the tradeoffs \eqref{eq:tradeoff-entanglement} and \eqref{eq:tradeoff-completeness}, which we discussed after Corollary~\ref{cor:ebit}, the bound $\lambda_c > 0$ ensures that the completeness error vanishes (which requires that $\epsilon_\pi$, defined in \eqref{eq:epsilon-proj}, also vanishes). The upper bound $\lambda_c < \lambda_\theta$ ensures that the dilution process increases the number of states, i.e., $n > m$.
\section{Robustness to noise} While we have shown above that the inherent noise associated to dilution is tolerated by our protocol, we implicitly assumed that the quantum devices themselves are noise-free. Indeed the completeness error given by Eq.~\eqref{eq:comperr} is evaluated assuming quantum devices with an expected winning probability equal to the quantum maximum $\omega_\mathrm{q}$. If the quantum devices are noisy, for example due to faulty measurements, and have instead a suboptimal winning probability $\omega=\omega_\mathrm{q}-\zeta$ with $\zeta < \xi$, then the completeness error becomes \begin{equation}\label{eq:comperr2}
\epsilon_\mathrm{c} = \exp \mleft( -2 n \gamma^2 (\xi-\zeta)^2 \mright) \,. \end{equation} It is easy to see that a sublinear entanglement consumption remains possible with such noisy devices, provided the noise parameter $\zeta$ decreases with $n$ as $\zeta=n^{-\lambda_\zeta}$ for some suitable scaling parameter $\lambda_\zeta$. However, realistic devices will be subject to a constant amount of noise, rather than an asymptotically vanishing one. In this case, the protocol as described so far breaks down. This is most easily seen from Lemma~\ref{lem:minentropy}: it is clear that $\theta$ can only be taken as low as values of the order of $(\omega_\mathrm{q}-\omega)^{1/8}=\zeta^{1/8}$ to get a nontrivial bound on the min-entropy. Nevertheless, given a small enough finite upper bound on the amount of noise $\zeta$, partially entangled qubit pairs with an appropriate value of $\theta < \pi/4$ can still be used to produce a linear amount of randomness with a yield per ebit higher than $1$ according to Theorem~\ref{thm:minentropy}, thus improving what can directly be achieved using maximally entangled states.
A sublinear consumption of entanglement using diluted singlets can be recovered even with devices whose components fail with constant probability if our protocol is combined with error correction and fault-tolerant quantum computation. Indeed, according to the threshold theorem for fault-tolerant quantum computation, an arbitrary quantum circuit containing $G(n)$ gates may be simulated with probability of error $e(n)$ on hardware whose components fail with constant probability at most $p$, provided $p$ is below some threshold, through an encoding that increases the local dimension of each qubit by a factor $\operatorname{poly} \log G(n)/e(n)$ \cite{NC00}. In our case, the number $G(n)$ of gates needed to perform entanglement dilution \cite{CD96} and the subsequent bipartite measurements is polynomial in $n$. On the other hand, aiming for a probability of error $e(n)$ that decreases polynomially in $n$ for the simulating circuit yields a completeness error that vanishes asymptotically. The number of ebits needed in such a fault-tolerant version of our protocol is then multiplied only by a factor $\operatorname{poly} \log G(n)/e(n)=\operatorname{poly} \log n$ resulting in a total number of ebits that is still sublinear in $n$, i.e., $m\sim n^k \operatorname{poly} \log n$.
\section*{Discussion} In summary, earlier work \cite{AMP12} showed that a pair of entangled qubits with arbitrarily little entanglement could be used to certify one bit of randomness. Here we have carried out the further step of transforming the intuition of \cite{AMP12} into DIRNG protocols in which a sublinear amount of entanglement is consumed. This shows that the consumption of entanglement resources in DIRNG can be dramatically improved qualitatively and quantitatively with respect to existing protocols. These results about entanglement are analogous to those concerning the initial random seed, the other fundamental resource required in DIRNG. Interestingly, the recent work \cite{CJA+17} suggests that one could devise DIRNG protocols that consume a constant amount of entanglement. Whether the intuition of \cite{CJA+17} can be transformed into such a DIRNG protocol is an interesting open question.
In the present work, we did not attempt to minimize simultaneously the entanglement and the size of the initial seed. (Note that in our protocol, the size of the initial random seed is determined by the parameter $\gamma$ specifying the proportion of test rounds.) Nevertheless, in the parameter regimes of Corollaries~\ref{cor:ebit} and \ref{cor:d-ebit}, the entanglement and the initial seed are both sublinear. Interestingly, it appears that our approach involves a tradeoff between entanglement and seed consumption, given the constraints placed on $\lambda_\gamma$ and $\lambda_\theta$ in Corollaries~\ref{cor:ebit} and \ref{cor:d-ebit}. Indeed, equations \eqref{eq:tradeoff-entanglement} and \eqref{eq:tradeoff-completeness} imply that $\lambda_\theta+2 \lambda_\gamma<1$. Thus if $\lambda_\theta$ is close to $1$ (corresponding to a small consumption of entanglement), $\lambda_\gamma$ must be close to $0$, which indicates a high proportion $\gamma$ of test rounds and, as a result, high consumption of random seed. Likewise, if $\lambda_\gamma$ is close to $1/2$, $\lambda_\theta$ must be close to $0$, and the protocol requires high entanglement and low random seed. We leave as open questions whether there is some kind of fundamental tradeoff between the required amounts of random seed and shared quantum resources, and whether the amount of quantum resources in our protocol is optimal or can be further decreased.
\paragraph{Acknowledgments.} This work is supported by the Fondation Wiener-Anspach, the Interuniversity Attraction Poles program of the Belgian Science Policy Office under the grant IAP P7-35 photonics@be. S. P. is a Research Associate of the Fonds de la Recherche Scientifique (F.R.S.-FNRS). C. B. acknowledges funding from the F.R.S.-FNRS through a Research Fellowship.
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\appendix
\section{Proof of Lemma~\ref{lem:minentropy}}\label{app:robust} Lemma~\ref{lem:minentropy} gives a lower bound on the conditional min-entropy in the outcome of Alice's measurement ${x} = 1$ for devices which achieve a certain success probability $\omega$ in the tilted-CHSH game of parameter $\beta$. We restate it here: \minentropylemma* \begin{proof} To derive this bound, we use our self-testing result for the tilted-CHSH inequalities \cite{BP15}. The robustness bounds for the self-test in \cite{BP15} are rather unwieldy, so we will instead provide a crude upper bound that retains the same asymptotic behavior in $\beta$ and $\omega$ and greatly simplifies the use of the bound.
We will lower-bound $H_\rmin({A} \mid E ; {X} = 1)$ as a function of the expected violation of the tilted-CHSH inequality, $I = I_\mathrm{q} - \epsilon$, where $I_\mathrm{q} = \sqrt{8+2\beta^2} = 4/\sqrt{1+\sin^2 2\theta}$ is the maximal quantum value of the expression. This min-entropy is equivalent to the guessing probability for the measurement ${X}=1$, which is defined as \begin{equation} \label{eq:gp-minentropy} 2^{-H_\rmin({A} \mid E ; {X} = 1)}
= p_{\text{guess}}({A} \mid E ; {X} = 1)
= \max_{\{M_g\}} \Pr[{A} = G \mid {X} = 1] \,, \end{equation} where $\{M_g\}$ is a POVM on the subsystem ${\mathrm E}$, which an adversary would use to measure the side information contained in ${\mathrm E}$ to formulate a guess $G$ for ${A}$ \cite{KRS09}. Formulated in terms of a given physical state $\ket{\tilde\psi}_{{\mathrm A}{\mathrm B}{\mathrm E}}$ and observables $\tilde A_{{x}} \equiv \tilde A_{{x}} \otimes I_{\mathrm B} \otimes I_{\mathrm E}$ and $\tilde B_{{y}} \equiv I_{\mathrm A} \otimes \tilde B_{{y}} \otimes I_{\mathrm E}$, for a given adversary POVM $\{M_g\}$ we have \begin{align} \Pr[{A} = G \mid {X} = 1]
&= \sum_{{a} \in \{0,1\}} \bra{\tilde\psi} \frac{I_{\mathrm A} + (-1)^{{a}} \tilde A_1}{2} \otimes I_{\mathrm B} \otimes M_{a} \ket{\tilde\psi} \\
&= \tfrac12 + \tfrac12 \bra{\tilde\psi} \tilde A_1 \otimes I_{\mathrm B} \otimes (M_0-M_1) \ket{\tilde\psi}
\equiv \tfrac12 + \tfrac12 \mean{\tilde A_1 C} \,, \label{eq:pguess} \end{align} letting $C = M_0-M_1$, which is bounded as $\norm{C}_\infty \le 1$. (From here on we will sometimes use a shorter notation where instead of e.g.\ $I_{\mathrm A} \otimes I_{\mathrm B} \otimes C$, we simply write $C$.) We will now relate this expression to the reference system using self-testing.
Using the notation of \cite{BP15}, the self-testing result shows that any pure state $\ket{\tilde\psi}_{{\mathrm A}{\mathrm B}{\mathrm E}}$ measured by $\tilde A_{x}$ and $\tilde B_{y}$ in such a way that the tilted-CHSH inequality for these observables is violated up to $I_\mathrm{q} - \epsilon$ obeys \footnote{In \cite{BP15} the subsystem ${\mathrm E}$ is implicitly included in ${\mathrm A}$ and/or ${\mathrm B}$ as a purifying subsystem for the mixed state held and measured by the devices. The statement can easily be modified to separate it without changing the proofs; the black-box measurement operators $\tilde A_{x}$ and $\tilde B_{y}$ then act as the identity on ${\mathrm E}$.} \begin{subequations} \label{eq:selftest} \begin{align} \norm{
\Phi(\ket{\tilde\psi}_{{\mathrm A}{\mathrm B}{\mathrm E}})
- \ket{\psi_\theta}_{{\mathrm A}'{\mathrm B}'} \otimes \ket{\text{junk}}_{{\mathrm A}{\mathrm B}{\mathrm E}} }
&\le 2 \bar\delta \,,\\ \norm{
\Phi(\tilde A_1 \ket{\tilde\psi}_{{\mathrm A}{\mathrm B}{\mathrm E}})
- A_1 \ket{\psi_\theta}_{{\mathrm A}'{\mathrm B}'} \otimes \ket{\text{junk}}_{{\mathrm A}{\mathrm B}{\mathrm E}} }
&\le 2 \bar\delta + 2 \delta_\mathrm{a}^{\mathrm A} \,, \end{align} \end{subequations} where $\Phi = \Phi_{\mathrm A} \otimes \Phi_{\mathrm B} \otimes I_{\mathrm E}$ is a local isometry acting on Alice and Bob's subsystems which introduces and transforms ancillary qubits ${\mathrm A}'$ and ${\mathrm B}'$, $\ket{\psi_\theta}$ is the reference state (see main text), $A_1$ is the reference observable that yields one bit of randomness, and the error bound parameters $\bar\delta, \delta_\mathrm{a}^{\mathrm A} = O(\sqrt\epsilon \theta^{-4})$ are explicitly defined in \cite{BP15}. Effectively, this isometric transformation extracts a state onto ${\mathrm A}'{\mathrm B}'$ which is close to a copy of the reference state and almost decorrelated from the initial system ${\mathrm A}{\mathrm B}{\mathrm E}$. Likewise, the isometry approximately maps the physical observables' action on the physical state in ${\mathrm A}{\mathrm B}$ to ideal actions on the reference state in the ancillary registers ${\mathrm A}'{\mathrm B}'$.
From this result, we see that the guessing probability with respect to $\ket{\tilde\psi}$ (which is by isometry identical to the guessing probability for $\Phi(\ket{\tilde\psi})$) is close to the guessing probability with respect to the reference state, for which $p_{\text{guess}} = \max_{a} \Pr[{A} = {a} \mid {X} = 1] = 1/2$ since the side information $E$ is decorrelated from the devices ${\mathrm A}$ and ${\mathrm B}$.
To show this approximate equality of guessing probabilities, we rewrite the last term of \eqref{eq:pguess} as \begin{equation}
\mean{\tilde A_1 C}
= \Phi^\dag(\bra{\tilde\psi}) \, C \, \Phi(\tilde A_1 \ket{\tilde\psi}) \end{equation}
using that $\Phi$ is an isometry which acts like the identity on ${\mathrm E}$. We then use \eqref{eq:selftest} and the triangle inequality to transform this into $\bra{\psi_\theta} A_1 \ket{\psi_\theta} \braket{\text{junk} | C | \text{junk}}$, with additional error terms. First, we replace $\Phi(\tilde A_1 \ket{\tilde\psi})$ with $A_1 \ket{\psi_\theta} \otimes \ket{\text{junk}}$ with an additional error term $2\bar\delta + 2\delta_\mathrm{a}^{\mathrm A}$ since $\norm{\Phi^\dag(\bra{\tilde\psi}) \, C} \le 1$, then we replace $\Phi^\dag(\bra{\tilde\psi})$, again using $\norm{A_1}_\infty, \norm{C}_\infty \le 1$: \begin{align} \mean{\tilde A_1 C}
&\le \Phi^\dag(\bra{\tilde\psi}) \, \bigl( A_1 \ket{\psi_\theta} \otimes C \ket{\text{junk}} \bigr) + 2\bar\delta + 2\delta_\mathrm{a}^{\mathrm A} \\
&\le \bra{\psi_\theta} A_1 \ket{\psi_\theta} \bra{\text{junk}} C \ket{\text{junk}} + 4\bar\delta + 2\delta_\mathrm{a}^{\mathrm A} \\
&\le \abs{\bra{\psi_\theta} A_1 \ket{\psi_\theta}} + 4\bar\delta + 2\delta_\mathrm{a}^{\mathrm A} \,. \end{align} Since for the reference system $\bra{\psi_\theta} A_1 \ket{\psi_\theta} = 0$, we find that \begin{equation} \label{eq:pguess-delta}
p_{\text{guess}} \le \tfrac12 + 2\bar\delta + \delta_\mathrm{a}^{\mathrm A} \end{equation}
We now proceed to find a simple expression of $\epsilon$ and $\theta$ that upper-bounds $2\bar\delta + \delta_\mathrm{a}^{\mathrm A}$. After some careful manipulation of the rather long expressions for $\bar\delta$ and $\delta_\mathrm{a}^{\mathrm A}$, we find { \newcommand{\sqrt{1+s^2}}{\sqrt{1+s^2}} \begin{multline} \label{eq:longbound}
2\bar\delta + \delta_\mathrm{a}^{\mathrm A}
= \sqrt{2 I_\mathrm{q}} \sqrt\epsilon \Biggl[
\frac{\sqrt{1+s^2}}{2s^2} \left( 1 + c + \sqrt{1+s^2} \right)
+ \frac{\sqrt{1+s^2}}{4s} \left( 2 - c + \sqrt{1+s^2} \right)
\\
+ \frac{c + \sqrt{1+s^2}}{2s^2} (1+c)
\left( 8 + 2 \frac{1+\sqrt{1+s^2}}{s^2} + 3 \tan\theta \right)
\Biggr]
\,, \end{multline} } with $c = \cos(2 \theta)$, $s = \sin(2 \theta)$, $I_\mathrm{q} = 4 / \sqrt{1+s^2}$. The dominating term in this bound for small $\theta$ comes from the term in $s^{-4}$, namely $
2 \sqrt{2 I_\mathrm{q}} \sqrt\epsilon (1+c)(c+\sqrt{1+s^2})(1+\sqrt{1+s^2}) s^{-4} = O(\sqrt\epsilon \theta^{-4}) $.
A crude upper bound on \eqref{eq:longbound} is obtained by taking a $s^{-4}$ factor out of the square brackets and giving rough numerical bounds on the bounded function that remains. For instance, the factor of $s^{-4} \sqrt{2 I_\mathrm{q}} \sqrt\epsilon$ in the first term becomes $
s^2 \sqrt{1+s^2} (1+c+\sqrt{1+s^2})/2 \le (3\sqrt2/2) s^2 $ because $
c + \sqrt{1+s^2} = \sqrt{1-s^2} + \sqrt{1+s^2} \le 2 $ for $s^2 \in [0,1]$, and $\sqrt{1+s^2} \le \sqrt{2}$. We obtain the following bound: \begin{equation}
2\bar\delta + \delta_\mathrm{a}^{\mathrm A}
\le 2\sqrt2 \mleft[
\frac{3\sqrt2}{2} s^2
+ \frac{1+\sqrt2}{2} s^3
+ 16 s^2 + 8 + 6 s^2 \tan\theta
\mright]
\frac{\sqrt\epsilon}{s^4}
\,, \end{equation} where we have also bounded $I_\mathrm{q} \le 4$ and used the following tight bounds after expanding the third term in the square brackets of \eqref{eq:longbound}: \begin{gather}
(c + \sqrt{1+s^2})(1+c) \le 4 \,,
\\
(c + \sqrt{1+s^2})(1+c)(1+\sqrt{1+s^2})
= (1+c\sqrt{1+s^2})(2+c+\sqrt{1+s^2})
\le 8 \,. \end{gather} The factor $\tan\theta$ in the last term is simply bounded by $1$. The bound we reach is the following: \begin{equation}
2\bar\delta + \delta_\mathrm{a}^{\mathrm A}
\le 2\sqrt2 \mleft[
\frac{1+\sqrt2}{2} s^3
+ \frac{3\sqrt2 + 44}{2} s^2
+ 8
\mright] \frac{\sqrt\epsilon}{s^4}
\,. \end{equation} Finally, the polynomial in $s$ in square brackets is bounded by its maximum at $s=1$. Eq.~\eqref{eq:pguess-delta} then becomes \begin{equation} \label{eq:shortbound}
p_{\text{guess}} - \tfrac12
\le 2\bar\delta + \delta_\mathrm{a}^{\mathrm A}
\le
(8 + 61 \sqrt2) \frac{\sqrt\epsilon}{s^4}
\,. \end{equation} Further bounding \begin{equation}
s = \sin 2\theta \ge \frac{2\theta}{\pi/2} \ge \theta \end{equation} by concavity of the sine function on $\ccint{0,\pi/2}$ and substituting $\epsilon = (8+2\beta) (\omega_\mathrm{q} - \omega)$, we reach our final bound for the guessing probability, \begin{equation} p_{\text{guess}} \le \tfrac12 + \sqrt{8+2\beta} \, (8+61\sqrt2) \, \theta^{-4} \sqrt{\omega_\mathrm{q} - \omega} \,. \end{equation} Putting this together with \eqref{eq:gp-minentropy}, we find, using $\ln(1+x) \le x$, \begin{align} H_\rmin({A} \mid E ; {X} = 1)
&\ge 1 - \log_2 \mleft( 1 + 2 \sqrt{8+2\beta} \, (8+61\sqrt2) \, \theta^{-4} \sqrt{\omega_\mathrm{q} - \omega} \mright) \\
&\ge 1 - \kappa \theta^{-4} \sqrt{\omega_\mathrm{q} - \omega} \end{align} with $\kappa = 4 \sqrt{4+\beta} (4\sqrt2+61) / \ln 2$. \end{proof}
A numerical maximization of the factor of $\sqrt\epsilon$ in \eqref{eq:longbound} shows that a tighter numerical factor of $45.13$ could replace the numerical factor $8+61\sqrt2 = 94.27$ in \eqref{eq:shortbound}, or less if the range of $\theta$ is limited.
\section{Proof of Theorem~\ref{thm:minentropy}}\label{app:EAT} It is shown in \cite{DFR16,AFVR16} that obtaining a bound on the smooth min-entropy produced by a generic protocol of the type that we consider here reduces to finding a \emph{min-tradeoff function}, a certain function that bounds the randomness produced in an average round of the protocol. This function is specific to the particular game used in the protocol and obtaining it for the tilted-CHSH game is the only part of the general analysis of \cite{AFVR16} that we need to tailor to our situation.
The min-tradeoff function is defined as follows. Any protocol round (i.e., step~\ref{item:protocol:round} in the protocol; see Section~\ref{sec:protocol}) can be thought of as a quantum channel $\mathcal{G}_i$ mapping the state $\rho=\rho_{DE}$ of the pair of devices $D$ and the adversary information $E$ before that round to the resulting state $\mathcal{G}_i(\rho)=\rho'=\rho'_{{A}_i{B}_i{X}_i{Y}_i T_iC_i D'E}$ after the protocol round, which also includes explicitly the classical data that was produced in that round. In particular, the channel $\mathcal{G}_i$ and the initial state $\rho$ determine the probability distribution $p \equiv (p_0, p_1, p_\bot)$ for the classical random variable $C_i\in\{0,1,\perp\}$. This probability distribution is related to the randomness produced in the protocol round: from Lemma~\ref{lem:minentropy} we expect that a pair of devices which succeeds at any round ($C_i = 1$) with higher probability produces more entropy in its outputs. The min-tradeoff function is a function $f_\mathrm{min}(p)$ that bounds the randomness produced in the protocol round solely on the basis of the probability distribution $p$. Formally, a function $f_\mathrm{min}(p)$ is a min-tradeoff function if it satisfies \begin{equation}
f_\mathrm{min}(p) \le H({A}_i{B}_i \mid {X}_i{Y}_i T_i E)_{\mathcal{G}_i(\rho)} \,, \end{equation} where $H({A}_i{B}_i \mid {X}_i{Y}_i T_i E)_{\mathcal{G}_i(\rho)}$ is the von Neumann entropy of the joint outputs conditioned on the classical side information produced in the round and on the quantum information of the adversary $E$. This inequality should hold for all channels $\mathcal{G}_i$ that are compatible with the protocol and for all initial states $\rho$ such that the variable $C_i$ in $\mathcal{G}_i(\rho)$ is distributed as $p$.
Theorem~\ref{thm:minentropy}, which we restate here, follows from the entropy accumulation theorem \cite{DFR16} and its application to randomness generation protocols by Arnon-Friedman et al.~\cite{AFVR16}. Our proof follows that of \cite{AFVR16}, in which we substitute a min-tradeoff function adapted to our protocol. \minentropytheorem* \begin{proof}[Proof of Theorem~\ref{thm:minentropy}] We first note that $C_i = \bot$ happens if and only if the protocol round is a generation round, hence we always have $p_\bot = 1-\gamma$ and $p_0 + p_1 = \gamma$. Thus, as noted in \cite{AFVR16}, we are free to define $f_\mathrm{min}(p)$ to arbitrary values when $p_0 + p_1 \ne \gamma$, since such a distribution for $C_i$ is not compatible with our protocol anyway. On the other hand, when $p_0 + p_1 = \gamma$, the expected probability of succeeding at the nonlocal game in a game round is $p_1 / \gamma$. In that case, we can use Lemma~\ref{lem:minentropy} with $\omega = p_1/\gamma$ to set the value of $f_\mathrm{min}$. Indeed, \begin{align}
H({A}_i{B}_i \mid {X}_i{Y}_i T_i E)
&\geq H({A}_i \mid {X}_i {Y}_i T_i E) \label{eq:vn-trace} \\
&= H({A}_i \mid {X}_i E) \label{eq:vn-flag} \\
&\ge \Pr[{X}_i = 1] \, H({A}_i \mid E;{X}_i = 1) \\
&\ge \nu \, g(\omega) \,,\label{eq:bound} \end{align} with $\nu = \Pr[{X}_i = 1] = 1 - \gamma (2+\beta)/(4+\beta)$. In \eqref{eq:vn-trace}, we used the chain rule and the positivity of the conditional entropy of classical information. In \eqref{eq:vn-flag}, we used that Alice's output ${A}_i$ is independent of Bob's measurement choice ${Y}_i$ and of the round flag $T_i$. To get \eqref{eq:bound}, we used Lemma~\ref{lem:minentropy} and the fact that the conditional min-entropy lower-bounds the conditional von Neumann entropy \cite[Proposition~4.3]{Tom12}.
We can thus define $f_\mathrm{min}(p)=\nu\,g(p_1/\gamma$) when $p_0+p_1=\gamma$. For convenience, we set it to the same value when $p_0+p_1\neq\gamma$, since it can be freely chosen in that case. All in all, \begin{equation} \label{eq:cvx-mintr}
f_\mathrm{min}(p) = \nu\,g(p_1/\gamma) \,. \end{equation}
The EAT \cite{DFR16} requires \emph{affine} min-tradeoff functions. Since $g$ is convex, we can simply obtain affine lower bounds of \eqref{eq:cvx-mintr} by taking its tangent at any point. The tangent of $g(\omega)$ at the point $\omega = \omega_\mathrm{t}$ is given by \begin{equation}
\bar g_{\omega_\mathrm{t}}(\omega) = 1 - \kappa \theta^{-4} \frac{2\omega_\mathrm{q} - \omega_\mathrm{t} - \omega}{2\sqrt{\omega_\mathrm{q} - \omega_\mathrm{t}}} \,, \end{equation} hence the min-tradeoff function we finally use will be $f_\mathrm{min}(p) = \nu\,\bar g_{\omega_\mathrm{t}}(p_1/\gamma)$ for some appropriately chosen $\omega_\mathrm{t}$.
Given such a min-tradeoff function, Lemma~9 of \cite{AFVR16} then states that for any given pair of devices, either the protocol succeeds with low probability $\Pr[\mathcal{S}] \le \epsilon'$, or \begin{equation}
H_\rmin^{\epsilon_\mathrm{s}}(\mathbf{A} \mathbf{B} \mid \mathbf{X} \mathbf{Y} \mathbf T E ; \mathcal{S}) \ge n \nu\,\bar g_{\omega_\mathrm{t}}(\omega_\mathrm{q}-\xi) - \mu \sqrt n \end{equation} with \begin{equation}
\mu = 2 \mleft( \log_2(13) + \ceil{\norm{\nabla f_\mathrm{min}}_\infty} \mright) \sqrt{1-2 \log_2(\epsilon_\mathrm{s} \epsilon')} \,. \end{equation} The gradient of the min-tradeoff function is simply the slope of $\nu\,\bar g_{\omega_\mathrm{t}}(p_1/\gamma)$: \begin{equation}
\norm{\nabla f_\mathrm{min}}_\infty = \gamma^{-1} \nu \frac{\kappa \theta^{-4}}{2 \sqrt{\omega_\mathrm{q} - \omega_\mathrm{t}}} \,. \end{equation} Bounding the ceiling function as $\ceil x \le x + 1$ and optimizing over the point of tangency $\omega_\mathrm{t}$ produces the final expression \eqref{eq:hmin} for the min-entropy bound of the theorem. \end{proof}
\section{Proof of Lemma~\ref{lem:dilution}}\label{app:dilution}
In Section~\ref{sec:dilution}, we sketched the entanglement dilution procedure of Bennett et al.~\cite{BBPS96}, which defines a channel $\mathcal D_{\theta,\delta}$ that approximately dilutes $\ket{\phi^+}^{\otimes m}$ into $\ket{\psi_\theta}^{\otimes n}$, with $m < n \simeq m/S(\theta)$.
In this appendix, we describe this procedure in detail then prove Lemma~\ref{lem:dilution}, which bounds its inherent error terms. We restate the Lemma here: \dilutionlemma*
As stated in the main text, dilution is enabled by the possibility of compressing a number of weakly entangled states into a smaller Hilbert space at the cost of a small error. One procedure that realizes this is known as Schumacher compression \cite{Sch95} and is explained in great detail in \cite{Wil13}, from which we borrow the notation.
We will apply Schumacher compression to the second half of the global state of $n$ weakly entangled states $
\ket{\psi_\theta}^{\otimes n}_{{\mathrm A}{\mathrm A}'}
= ( \cos\theta \ket{00} + \sin\theta \ket{11} )^{\otimes n} $. This allows us to use $m < n$ maximally entangled qubit pairs to transport that second half from box ${\mathrm A}$ to box ${\mathrm B}$ so that the initial $m$ singlets are effectively transformed into $n$ weakly entangled pairs after decompression.
Schumacher compression works on a source of pure states $\{\ket{\psi_i},q_i\}$ which outputs the states in $\{\ket{\psi_i}\}$ at random, with respective probabilities $\{q_i\}$. The goal is to pack the information output by $n$ i.i.d.\ uses of the source into a smaller Hilbert space in a way that makes it recoverable later with high fidelity--- a generalization of Shannon's source coding to the quantum setting. We interpret the reduced density operator $\sigma_{{\mathrm A}'} = \tr_{{\mathrm A}} \projop{\psi_\theta}_{{\mathrm A}{\mathrm A}'}$ for the second half of one pair $\ket{\psi_\theta}_{{\mathrm A}{\mathrm A}'}$ as describing a quantum source of mutually orthogonal states---namely, the eigenstates of $\sigma$---with probabilities given by the corresponding eigenvalues. The eigenstates of $\sigma$ coincide with the computational basis: they are $\ket 0$ and $\ket 1$, with corresponding eigenvalues $q_0 = \cos^2 \theta$ and $q_1 = \sin^2 \theta = 1-q_0$. Using this source $n$ times gives us the mixed state $\rho_{{\mathrm A}'} = \sigma_{{\mathrm A}'}^{\otimes n} = \tr_{{\mathrm A}} \projop{\psi_\theta}^{\otimes n}$, i.e., the reduced state of Bob's share of the set of $n$ entangled pairs prepared by Alice. The eigenstates of $\rho_{{\mathrm A}'}$ are the computational basis states for $n$ qubits, which we write as $\ket{y}$ with $y \in \{0, 1\}^n$, where $\ket{y}$ is the tensor product of $n$ qubits $\ket{y_1} \dotsm \ket{y_n}$. The eigenvalue associated with $y$ is $\lambda_j = (\cos^2\theta)^j (\sin^2\theta)^{n-j}$ for $j = n(0\mid y)$, which gives the number of zeros in the binary string $y$.
Compressing a source of orthogonal states is more or less equivalent to Shannon source coding. The idea is to consider the string $y$ obtained after $n$ uses of the source and only let it through when it is deemed ``typical'' enough. According to the theory of typical sequences \cite{CT12}, there is a family of subsets of the $2^n$ strings $y$, called the $\delta$-typical subsets, that each contain an exponentially small fraction of strings but nevertheless have an exponentially large probability weight. Thus, while most strings we obtain are typical, their number is considerably smaller than $2^n$. The $\delta$-typical subset is defined as \begin{equation} \label{eq:typical}
\mathcal T_\delta
= \left\{
y \in \{0,1\}^n :
S - \delta \le \frac{-\log_2 P(y)}{n} \le S + \delta
\right\}
\,, \end{equation} where $S$ is the entropy of the source (which is also the entropy of entanglement of a single pair $\ket{\psi_\theta}$), \begin{align}
S = h_2(q_0)
&= -q_0 \log_2(q_0) - q_1 \log_2(q_1) \\
&= - \log_2(q_0) + q_1 \Delta \,, \end{align} with $\Delta = \log_2(q_0/q_1) = -\log_2 \tan^2\theta$. Thus, a sequence $y$ is $\delta$-typical if and only if its sample entropy $-(1/n) \log_2 P(y)$ is $\delta$-close to $S$. Since our ideal source is i.i.d., each random variable $Y_i$ is distributed independently according to the same Bernoulli distribution of probabilities $\{q_0, q_1\}$. Thus, the sample entropy for a given value $y$ can be rewritten as \begin{align} -\frac1n \log_2 P(y)
&=
\frac1n \sum_{i=1}^n -\log_2 q_{y_i} \\
&= -\frac{n(0\mid y)}{n} \log_2 q_0
- \frac{n(1\mid y)}{n} \log_2 q_1
\\
&= -\log_2 q_0 + \frac{n(1\mid y)}{n} \Delta \,. \end{align} Hence, the definition of the typical set \eqref{eq:typical} can be rewritten as \begin{equation} \label{eq:typicalvar}
\mathcal T_\delta
= \left\{
y \in \{0,1\}^n :
q_1 - \frac\delta\Delta
\le \frac{1}{n} \sum_{i=1}^n y_i \le q_1 + \frac\delta\Delta
\right\}
\,. \end{equation} That is, a sequence $y$ is $\delta$-typical if and only if it has a frequency of $1$'s that is $(\delta/\Delta)$-close to the expected value $q_1$.
Properties of the typical set are easily derived from those two expressions of $\mathcal T_\delta$ \cite{CT12}. First, applying Hoeffding's inequality \cite{Hoe63} for the binomially-distributed $\sum_i Y_i$ shows that the typical set has a high probability weight \begin{equation} \label{eq:typicalprob}
\Pr[\mathcal T_\delta]
\ge 1 - \epsilon_\pi
\,, \end{equation} where we defined the \emph{projection error} \begin{equation}
\epsilon_\pi = 2 \exp(-2 n \delta^2/\Delta^2) \,, \end{equation} which is an upper bound on the probability of a sequence being atypical.
Secondly, in contrast to this first property, the typical set has a relatively low cardinality: the number of typical sequences is exponentially small compared to the total number of sequences. Indeed, from the definition \eqref{eq:typical}, \begin{align}
\abs{\mathcal T_\delta}
&= 2^{n(S+\delta)} \sum_{y\in\mathcal T_\delta} 2^{-n(S+\delta)}
\\&
\le 2^{n(S+\delta)} \sum_{y\in\mathcal T_\delta} P(y)
\le 2^{n(S+\delta)} \,, \label{eq:typicalcard} \end{align} which is much smaller than the total number of $2^n$ sequences if $S < 1-\delta$ and $n$ is high.
Source coding consists in discarding atypical sequences, which occur with low probability, and encoding typical sequences into smaller codewords. This encoding is possible because of the small cardinality of the typical set: a sequence can simply be encoded by its index within a given ordering of the elements of the typical set, which gives binary codewords a length of at most $\log_2 \abs{\mathcal T_\delta} \le n(S+\delta) \equiv m$.
Schumacher compression applies this procedure to the quantum state $
\rho_{{\mathrm A}'}
= \sum_{y \in \{0,1\}^n} \lambda_{n(0 \mid y)} \projop{y} $, which describes the output of a quantum source of pure states $\ket y$ with probabilities $\lambda_{n(0 \mid y)}$. In order to identify an atypical state, a projective \emph{typicality measurement} is performed, with projectors $\{\Pi_\delta, I - \Pi_\delta\}$ where \begin{equation}
\Pi_\delta = \sum_{y\in\mathcal T_\delta} \projop{y} \,. \end{equation} If the typicality measurement succeeds, the state ends up in the typical subspace spanned by $\{ \ket{y} : y \in \mathcal T_\delta\}$, and can be encoded in a $2^m$-dimensional Hilbert space by an invertible isometric map $V$. If instead the measurement fails, a given typical state $\tau$ is substituted and encoded the same way. The resulting state is therefore $
\mathcal C(\rho_{{\mathrm A}'})
= V [ \Pi_\delta \rho_{{\mathrm A}'} \Pi_\delta
+ (1-\tr(\Pi_\delta \rho_{{\mathrm A}'})) \, \tau ] V^\dag $.
The two properties \eqref{eq:typicalprob} and \eqref{eq:typicalcard} of the typical set can be expressed in terms of the typical projector $\Pi_\delta$: \begin{gather}
\label{eq:qtypicalprob}
\tr(\Pi_\delta \rho_{{\mathrm A}'}) \ge 1-\epsilon_\pi \,,
\\
\label{eq:qtypicalcard}
\tr(\Pi_\delta) \le 2^{n(S+\delta)} \,. \end{gather} The first property can be used in the gentle operator lemma \cite{Win99,ON02} to show that a successful typicality measurement does not disturb the state by much \cite{Wil13}: \begin{equation}
\norm{\Pi_\delta \rho_{{\mathrm A}'} \Pi_\delta - \rho_{{\mathrm A}'}}_1
\le 2 \sqrt{\epsilon_\pi} \,. \end{equation} Hence, the decompressed state is close in trace distance to the original \cite{Wil13}: \begin{equation}
\norm[\big]{
V^\dag \mathcal C(\rho_{{\mathrm A}'}) V
- \rho_{{\mathrm A}'}
}_1
\le 2\sqrt{\epsilon_\pi} + \epsilon_\pi
\,. \end{equation}
As Schumacher originally noted \cite{Sch95}, this remains true when we consider the global state in ${\mathrm A}{\mathrm A}'$; the entanglement of the state is therefore not destroyed by compression: \begin{equation}
\norm[\big]{
(I \otimes V^\dag)
({\idmap} \otimes \mathcal C) [\projop{\psi_\theta}^{\otimes n}]
(I \otimes V)
- \projop{\psi_\theta}^{\otimes n}
}_1
\le 2\sqrt{\epsilon_\pi} + \epsilon_\pi
\,. \end{equation}
Bennett et al.'s dilution procedure simply results from the composition of this compression with quantum teleportation. The proof of Lemma~\ref{lem:dilution} is therefore immediate: \begin{proof}[Proof of Lemma~\ref{lem:dilution}] Defining $
\mathcal D_{\delta,\theta}(\projop{\phi^+}^{\otimes m}_{{\mathrm A}''{\mathrm B}}) $ to be the outcome of the composition of a local preparation of $\projop{\psi_\theta}^{\otimes n}_{{\mathrm A}{\mathrm A}'}$, followed by Schumacher compression over $\delta$-typical sequences of ${\mathrm A}'$, teleportation from ${\mathrm A}'$ to ${\mathrm B}$ using $\projop{\phi^+}^{\otimes m}_{{\mathrm A}{\mathrm B}}$ and decompression on ${\mathrm B}$, Lemma~\ref{lem:dilution} follows. \end{proof}
\end{document} | arXiv |
A new DOA-based factor graph geolocation technique for detection of unknown radio wave emitter position using the first-order Taylor series approximation
Muhammad Reza Kahar Aziz1,2,
Khoirul Anwar1 &
Tad Matsumoto1,3
This paper proposes a new geolocation technique to improve the accuracy of the position estimate of a single unknown (anonymous) radio wave emitter. We consider a factor graph (FG)-based geolocation technique, where the input are the samples of direction-of-arrival (DOA) measurement results sent from the sensors. It is shown that the accuracy of the DOA-based FG geolocation algorithm can be improved by introducing approximated expressions for the mean and variance of the tangent and cotangent functions based on the first-order Taylor series (TS) at the tangent factor nodes of the FG. This paper also derives a closed-form expression of the Cramer-Rao lower bound (CRLB) for DOA-based geolocation, where the number of samples is taken into account. The proposed technique does not require high computational complexity because only mean and variance are to be exchanged between the nodes in the FG. It is shown that the position estimation accuracy with the proposed technique outperforms the conventional DOA-based least square (LS) technique and that the achieved root mean square error (RMSE) is very close to the theoretical CRLB.
Accurate wireless geolocation has received considerable attention in the past two decades [1] and is expected to play important roles in current and future wireless communications systems. This technology is the key to supporting location-based service applications, e.g., Emergency-911 (E-911), location-sensitive billing, smart transportation systems, vehicle navigation, fraud detection, people tracking, and public safety systems [1–3]. This paper proposes a new direction-of-arrival (DOA)-based factor graph (FG) geolocation technique to detect the position of a single unknown anonymous radio wave emitter (the terminology "anonymous" is omitted in the rest of the paper for simplicity), where to convert the measured DOA samples to the mean and variance of the tangent and cotangent functions, we utilize the first-order Taylor series (TS) approximation. The proposed technique is not only applicable for an unknown radio wave emitter, e.g., illegal radio wave emitter, but it is also applicable for common radio wave emitter position detection. The basic setup is shown in Fig. 1, where the FG is performed at the fusion center.
Basic structure of FG-based unknown radio wave emitter detection describing a single target, four monitoring spots (in the case of RSS-based FG technique), three sensors, and fusion center
The FG is first applied on a geolocation technique in [4]. In the FG, the complexity is reduced because the global function factors into the products of several simple local functions, and the messages passed through the nodes are in the form of mean and variance, because of the Gaussianity assumption of the measured sample distribution [5–8]. The FG technique has been extended to incorporate not only DOA [6, 7] but also other information sources such as time of arrival (TOA) [8, 9], time difference of arrival (TDOA) [10], and received signal strength (RSS) [11], since the location of the unknown radio wave emitter can be estimated from them. In this paper, we use the terminology DOA instead of angle of arrival (AOA) for better expression.
DOA is used in this paper because it is measured by using either antenna arrays or a directional antenna without requiring perfect synchronization, time stamp, or transmit power information of a single unknown radio wave emitter [12, 13]. The other FG-based techniques, in contrast to the DOA-based techniques, e.g., TOA-based techniques, should perform perfect time synchronization among the sensors as well as between the sensors and the unknown emitter. The TDOA-based techniques can eliminate the necessity of the synchronization between the sensors and the emitter, but still, the sensors have to be accurately synchronized. Perfect synchronization is needed both in TOA and TDOA because the high velocity of the light, e.g., error of 1 μs, leads to error around 300 m [8]. Another difficulty arises in TOA-based geolocation techniques, where the TOA parameters cannot be measured from an unknown radio wave emitter because the time stamp information (time of departure (TOD) of the signal) of the transmitted signal from an unknown radio emitter is not available.
On the other hand, the major difficulty with the RSS-based technique is that it requires equal transmit power between the target and the test signal transmitted from the monitoring spots to gather the preliminary RSS map [11]. However, the transmit power of an unknown radio wave emitter is not available. The facts described above motivate us to use the DOA-based FG techniques to detect the position of the unknown radio emitter. The DOA-based FG techniques are also suitable in line-of-sight (LOS) and imperfect synchronization conditions.
A lot of work in DOA-based geolocation techniques have been proposed even since 30 years ago [14, 15]. However, it is quite recently that the FG-based techniques using DOA, TOA, TDOA, and RSS information were proposed, where the measurement data related to those parameters are used as the input to the algorithms. A joint TOA-DOA-based FG geolocation algorithm was proposed in [6], where the measured samples are efficiently used in FG to estimate the position accurately. Nevertheless, the joint TOA-DOA-based FG geolocation algorithm shown in [6] is still not fully correct because the message to be exchanged, the information of measurement error derived from the measurement results, enters the FG at an improper node as identified by [8].
The FG geolocation technique derived in [6] is improved in [7] as suggested in [8]. It also removes the necessity of measuring the TOA data from the joint TOA-DOA-based FG geolocation algorithm shown in [6] to obtain a simple DOA-based FG technique. After the DOA-based FG technique reaches a convergence point, the position estimate results are used as the initial position for the Gauss-Newton (GN) algorithm as the second step of the algorithm, the so-called factor graph-Gauss-Newton (FG-GN) geolocation technique in [7], to attain even higher accuracy.
We found that (1) The distance (r) is used in [7] as an argument of the tangent and cotangent functions, i.e., tan(r) and cot(r), which is obviously incorrect. Instead, the angle should be the argument of the tangent and cotangent functions, i.e., tan(θ) and cot(θ). (2) The factor node for collecting the samples is not available in the DOA-based FG geolocation techniques in [6, 7]. Moreover, the input parameter in [7] is r instead of θ. (3) The mean formula of tangent and cotangent function nodes [7] is not clearly described.
The Cramer-Rao lower bound (CRLB) is derived in [7], and the results of a series of simulations conducted to evaluate the accuracy of the DOA-based FG geolocation technique are shown also in [7] for the comparison purpose between CRLB and the root mean square error (RMSE) of DOA-based geolocation techniques, i.e., FG-GN, GN, and FG. However, the CRLB of the DOA-based technique shown in [7] does not take into account the influence of the sample number, and hence, the accuracy of the detection, when the number of the samples is large, is better than the CRLB, leading to the unusefulness of the comparison.
In the FG-based geolocation techniques described above, each sensor performs DOA measurement and transmits samples to the fusion center. Only the mean and variance of the DOA samples are used in the FG because of the Gaussianity assumption. The FG converts the angle information to the coordinate distance, referred to as relative distance, between the target and the sensor. The message passing takes place between the factor nodes and the variable nodes in the FG to obtain the target position estimate.
This paper provides clear understanding of the DOA-based geolocation techniques using FG, where the detail explanation of how the messages are updated at each node, and how the updated messages are exchanged between the nodes. The primary objectives of this paper are as follows: (A) We introduce a new set of formulas to be calculated at the tangent and cotangent factor nodes, to better approximate the mean and variance of the function values by utilizing the first-order TS expansion of the functions, so that the Gaussianity assumption still holds. (B) We derive the CRLB of the DOA-based geolocation technique taking into account the number of samples; hence, the accuracy of the new CRLB obtained by this paper is higher than that shown in [7]. (C) The results of a series of simulations are presented to evaluate the convergence property of the proposed technique, where the trajectory of the iterative estimation process is presented. Comparison between the RMSE of the proposed technique and the new CRLB is also provided. It is shown that the proposed algorithm can achieve close-CRLB accuracy, where the number of the samples, the number of the sensors, and the standard deviation of measurement error are used as a parameter.
The GN technique requires good initial position before it starts the iteration, while the proposed technique can start at any initial point. Therefore, it is not required to make a comparison between our proposed technique and the GN technique because we start the iteration from any arbitrary points. Also, since the detailed description of the algorithm for the DOA-based FG geolocation techniques presented in [6, 7] have improper expression as mentioned above in Section 1.1, this paper does not compare the accuracy of the proposed technique with that of [6] and [7]. Instead, the accuracy comparison is between the proposed technique and the DOA-based least square (LS) geolocation [16]. The FG we propose in this paper includes the DOA measurement factor node D θ and angle variable node N θ , as shown in Fig. 2, as in [8]. The two new nodes are proposed to emphasize that we need to calculate the mean and variance of DOA samples. It should be noted here that it is impossible to calculate variance with only one sample. It is also shown in this paper that with the proposed technique, the accuracy of the target position estimation outperforms the conventional DOA-based LS geolocation technique and the results are also very close to theoretical CRLB for the DOA-based geolocation.
The proposed DOA-based TS FG for geolocation technique
In general, the FG consists of the factor and variable nodes. In Fig. 2, the factor node is shown by a square, while the variable node by a circle. The factor node updates the messages forwarded from the connected variable nodes by using the specific simple local function, and the result is passed to the destination variable node. During the iteration, the messages sent from the source factor nodes are further combined in the variable node and passed back to the destination factor node for the next round of iteration. At the final iteration, the variable node combines the messages from all the connected factor nodes by using the sum-product algorithm.
Assume that the unknown radio wave emitter is located at coordinate position x= [ x y]T, where T is the transpose function. Sensors are located at position X i = [ X i Y i ]T, where i, i={1,2,…,N}, is the sensor index. The orientation of the sensors is known to ensure that the sensors measure the angle with respect to the global coordinate system. The sensor to fusion center transmission is perfect via wired or wireless connections. \(\mathbf {\Delta x}_{\theta _{i}}=\ [\!\Delta y_{\theta _{i}} \,\,\Delta x_{\theta _{i}}]^{T}\) is the relative distance between the position (X i ,Y i ) and target position (x,y), given by
$$\begin{array}{@{}rcl@{}} \left[\begin{array}{c} \Delta {x_{\theta_{i}}}\\ \Delta {y_{\theta_{i}}} \end{array}\right] = \left[\begin{array}{c} X_{i}\\Y_{i} \end{array}\right]-\left[\begin{array}{c} x \\ y \end{array} \right] \end{array} $$
with the θ i being the true DOA; however, the measured samples are corrupted by error due to the spatial spread of the multipath component and impairments in measurement. The DOA measurement equation is then given by
$$\begin{array}{@{}rcl@{}} \hat{\theta}_{k}= \theta + n_{k},\;\;\; k=\{1,2,\ldots,K\}, \end{array} $$
where k is the sample index. For notation simplicity, the sensor index i is omitted from the equations common to all the sensors, while it is included when needed, in the rest of the paper. It is reasonable to assume that n k is independent identically distributed (i.i.d.) zero-mean Gaussian random variable. The Gaussianity assumption is used in this paper because of the accumulative effects of many independent factors, as in [6–8, 10, 11]; hence, the assumption is reasonable for many of the wireless parameter measurement-based techniques such as DOA, TOA [8], TDOA [10], and RSS-based FG technique [11]. Furthermore, it can simplify the total computational complexity. This Gaussianity assumption is also used for the TOA/DOA-based FG technique in [6] and the DOA-based FG technique in [7]. This paper also uses the same assumption. Then, \(\hat {\theta }\) follows a normal distribution \(\mathcal {N}(\theta,\sigma ^{2})\) having a probability density function \(p(\hat {\theta)}\) as
$$\begin{array}{@{}rcl@{}} p(\hat{\theta})=\frac{1}{\sqrt{2\pi}\sigma_{\theta}} \exp \left(\frac{-(\hat{\theta}-\theta)^{2}}{2\sigma^{2}_{\theta }}\right), \end{array} $$
where the sample index is also omitted from the expression. However, each sensor does not know the needed values of θ and σ θ . Hence, each sensor in the proposed DOA-based TS FG geolocation technique first calculates the mean \(m_{D_{\theta } \rightarrow N_{\theta }}\) and the variance \(\sigma ^{2}_{D_{\theta } \rightarrow N_{\theta }}\) from the K-measured samples. The mark " →" in the suffix indicates the message flow directions in the FG.
The node D θ forwards the messages \((m_{D_{\theta } \rightarrow N_{\theta }}, \sigma _{D_{\theta } \rightarrow N_{\theta }}^{2})\) to the N θ , and then, the messages \((m_{N_{\theta }\rightarrow C_{\theta }}, \sigma _{N_{\theta }\rightarrow C_{\theta }}^{2})\) are directly forwarded to the tangent factor node C θ , where \(m_{N_{\theta }\rightarrow C_{\theta }}=m_{D_{\theta }\rightarrow N_{\theta }}\) and \(\sigma _{N_{\theta }\rightarrow C_{\theta }}^{2}=\sigma _{D_{\theta } \rightarrow N_{\theta }}^{2}\). The angle messages \((m_{N_{\theta } \rightarrow C_{\theta }}, \sigma _{N_{\theta } \rightarrow C_{\theta }}^{2})\) are converted to the relative distance messages \((m_{C_{\theta } \rightarrow \Delta x_{\theta }},\sigma _{C_{\theta } \rightarrow \Delta x_{\theta }}^{2})\) and \((m_{C_{\theta } \rightarrow \Delta y_{\theta }},\sigma _{C_{\theta } \rightarrow \Delta y_{\theta }}^{2})\) in the node C θ . The relative distance in (1) in the (X,Y) coordinate and the true DOA θ are connected by the tangent and cotangent functions, as [6, 7]
$$\begin{array}{@{}rcl@{}} \Delta y_{\theta}&=&\Delta x_{\theta} \cdot \tan (\theta), \end{array} $$
$$\begin{array}{@{}rcl@{}} \Delta x_{\theta}&=&\Delta y_{\theta} \cdot \cot (\theta). \end{array} $$
Even though (3) and (4) are self-referenced point equations, which can be solved by iterative techniques, the iteration needs proper initialization of the mean and variance of the argument variables \(\Delta \hat {x}_{\theta }\) and \(\Delta \hat {y}_{\theta }\) which are corresponding to \(\hat {x}\) and \(\hat {y}\) by (1) and (2), where the detail of initializations are described in Section 5. We do not know the true values of θ, Δ x θ , Δ y θ , x, and y; however, the message needed in the FG is the mean and variance of samples \(\hat {\theta }\), \(\Delta x_{\hat {\theta }}\), \(\Delta y_{\hat {\theta }}\), \(\hat {x}\), and \(\hat {y}\), which can be produced from the angle messages in the form of the mean and variance of samples \(\hat {\theta }\), \((m_{D_{\theta } \rightarrow N_{\theta }}, \sigma ^{2}_{D_{\theta } \rightarrow N_{\theta }})\). The details of the entire process are described in next section.
The proposed technique
The messages corresponding to Δ y θ and Δ x θ of (3) and (4), respectively, are the mean and variance, i.e., \((m_{C_{\theta } \rightarrow \Delta x_{\theta }},\sigma _{C_{\theta }\rightarrow \Delta x_{\theta }}^{2})\) and \((m_{C_{\theta } \rightarrow \Delta y_{\theta }},\sigma _{C_{\theta } \rightarrow \Delta y_{\theta }}^{2})\), where \(m_{C_{\theta } \rightarrow \Delta x_{\theta }}\) and \(m_{C_{\theta } \rightarrow \Delta y_{\theta }}\) are the means of \(\Delta x_{\hat {\theta }}\) and \(\Delta y_{\hat {\theta }}\), respectively; \(\sigma _{C_{\theta } \rightarrow \Delta x_{\theta }}^{2}\) and \(\sigma _{C_{\theta } \rightarrow \Delta y_{\theta }}^{2}\) are the variances of \(\Delta x_{\hat {\theta }}\) and \(\Delta y_{\hat {\theta }}\), respectively. We derive the messages for (3) and (4), based on the formula for the product of two independent random variables (a·b) as [17]
$$\begin{array}{@{}rcl@{}} m_{a \cdot b}&=& m_{a} \cdot m_{b}, \end{array} $$
$$\begin{array}{@{}rcl@{}} \sigma^{2}_{a \cdot b} &=& {m_{a}^{2}} \cdot {\sigma_{b}^{2}} + {m_{b}^{2}} \cdot {\sigma_{a}^{2}} + {\sigma_{a}^{2}} \cdot {\sigma_{b}^{2}}, \end{array} $$
where m x and \({\sigma ^{2}_{x}}\), x∈{a,b,a·b} are the mean and variance of x, respectively. It should be noticed from (3)–(6) that the means and variances of \(\tan (\hat {\theta })\) and \(\cot (\hat {\theta })\), \(m_{\tan (\hat {\theta })}\) and \(\sigma ^{2}_{\tan (\hat {\theta })}\), \(m_{\cot (\hat {\theta })}\) and \(\sigma ^{2}_{\cot (\hat {\theta })}\), respectively, are required. However, there arises a problem because \(\tan (\hat {\theta })\) and \(\cot (\hat {\theta })\) in (4) and (3) are both nonlinear functions that violates the Gaussianity assumption to express the messages only by the mean and variance in the FG. This motivates us to use the first-order TS to derive linear approximation of the tangent and cotangent functions to obtain the messages corresponding to the relative distance, as \((m_{C_{\theta } \rightarrow \Delta x_{\theta }},\sigma _{C_{\theta }\rightarrow \Delta x_{\theta }}^{2})\) and \((m_{C_{\theta } \rightarrow \Delta y_{\theta }},\sigma _{C_{\theta } \rightarrow \Delta y_{\theta }}^{2})\). The detailed derivation is described below.
The first-order TS is expressed as [18]
$$\begin{array}{@{}rcl@{}} f(\hat{\theta})\approx f\left(m_{\hat{\theta}}\right)+ f'\left(m_{\hat{\theta}}\right)\left(\hat{\theta}-m_{\hat{\theta}}\right), \end{array} $$
where \(f(\hat {\theta })\) is either \(\tan (\hat {\theta })\) or \(\cot (\hat {\theta })\), \(\hat {\theta }\) is the DOA sample, \(m_{\hat {\theta }}\) is the mean of \(\hat {\theta }\), \(f(m_{\hat {\theta }})\) is either the \(\tan (m_{\hat {\theta }})\) or \(\cot (m_{\hat {\theta }})\), and \(f'(m_{\hat {\theta }})\) is the first derivative of \(f(m_{\hat {\theta }})\). It should be noticed that (7) is a linear approximation for function of \(\hat {\theta }\), and hence, it is found that \(f(\hat {\theta })\) can be approximated by a Gaussian variable. The mean \(m_{f(\hat {\theta })}\) and variance \(\sigma ^{2}_{f(\hat {\theta })}\) can then be approximated using (7) as [18]
$$\begin{array}{@{}rcl@{}} m_{f(\hat{\theta})} & \approx & f\left(m_{\hat{\theta}}\right), \end{array} $$
$$\begin{array}{@{}rcl@{}} \sigma_{f\left(\hat{\theta}\right)}^{2} & \approx &\left(f'\left(m_{\hat{\theta}}\right)\right)^{2} \cdot \sigma_{\hat{\theta}}^{2}, \end{array} $$
where \(\sigma ^{2}_{\hat {\theta }}\) is the variance of \(\hat {\theta }\). The mean and variance of \(\tan (\hat {\theta })\) and \(\cot (\hat {\theta })\) are obtained from (8) and (9), respectively, as
$$\begin{array}{@{}rcl@{}} m_{\tan{(\hat{\theta})}}&\approx&\tan{(m_{\hat{\theta}})}, \end{array} $$
$$\begin{array}{@{}rcl@{}} m_{\cot{(\hat{\theta})}}&\approx&\cot{(m_{\hat{\theta}})}, \end{array} $$
$$\begin{array}{@{}rcl@{}} \sigma_{\tan(\hat{\theta})}^{2}&\approx&\sec^{4}(m_{\hat{\theta}}) \cdot \sigma_{\hat{\theta}}^{2}, \end{array} $$
$$\begin{array}{@{}rcl@{}} \sigma_{\cot(\hat{\theta})}^{2}&\approx&\csc^{4}(m_{\hat{\theta}}) \cdot \sigma_{\hat{\theta}}^{2}. \end{array} $$
Equations (11) and (13) provide our proposed algorithm with accurate enough approximation over a relatively large value range of the angles except for the mean of the angles around 0 radiant. This exception is because cot(0) and csc(0) are infinity. Hence, we can solve the infinity problem by empirically setting a limit value of m θ . We found that \(m_{\hat {\theta }}\geq |0.1|\) in units of radiant is reasonable. By setting the limit value properly, unstable behavior of the algorithm can be avoided. Theoretically, we also require to set a limit value of \(m_{\hat {\theta }}\) for (10) and (12), (11) and (13) to avoid the infinity values for the angles around {π/2,3π/2}, {2π} radiants, respectively.
The node C θ calculates relative distance messages, \((m_{C_{\theta } \rightarrow \Delta x_{\theta }},\sigma _{C_{\theta } \rightarrow \Delta x_{\theta }}^{2})\) and \((m_{C_{\theta } \rightarrow \Delta y_{\theta }},\sigma _{C_{\theta } \rightarrow \Delta x_{\theta }}^{2})\), according to (3)–(6) and (10)–(13), as
$$\begin{array}{@{}rcl@{}} m_{C_{\theta} \rightarrow \Delta y_{\theta}} &\approx & m_{\Delta x_{\theta}\rightarrow C_{\theta}} \cdot \tan \left(m_{N_{\theta} \rightarrow C_{\theta}}\right), \end{array} $$
$$\begin{array}{@{}rcl@{}} m_{C_{\theta} \rightarrow \Delta x_{\theta}} &\approx & m_{\Delta y_{\theta} \rightarrow C_{\theta}} \cdot \cot \left(m_{N_{\theta} \rightarrow C_{\theta}}\right), \end{array} $$
$$\begin{array}{@{}rcl@{}} \sigma_{C_{\theta} \rightarrow \Delta y_{\theta}}^{2} &\approx & \sigma_{\Delta x_{\theta} \rightarrow C_{\theta}}^{2} \cdot \tan^{2}\left(m_{N_{\theta} \rightarrow C_{\theta}}\right) \end{array} $$
$$\begin{array}{@{}rcl@{}} & &+m_{\Delta x_{\theta} \rightarrow C_{\theta}}^{2} \cdot \sigma_{N_{\theta} \rightarrow C_{\theta}}^{2} \cdot \sec^{4}\left(m_{N_{\theta} \rightarrow C_{\theta}}\right) \\ & & + \sigma_{\Delta x_{\theta} \rightarrow C_{\theta}}^{2} \cdot \sigma_{N_{\theta} \rightarrow C_{\theta} }^{2} \cdot \sec^{4}\left(m_{N_{\theta} \rightarrow C_{\theta}}\right), \\ \sigma_{C_{\theta} \rightarrow \Delta x_{\theta}}^{2} &\approx& \sigma_{\Delta y_{\theta} \rightarrow C_{\theta}}^{2} \cdot \cot^{2}\left(m_{N_{\theta} \rightarrow C_{\theta}}\right) \\ & &+m^{2}_{\Delta y_{\theta} \rightarrow C_{\theta}} \cdot \sigma_{N_{\theta} \rightarrow C_{\theta}}^{2} \cdot \csc^{4}\left(m_{N_{\theta} \rightarrow C_{\theta}}\right) \\ & & + \sigma_{\Delta y_{\theta} \rightarrow C_{\theta}}^{2} \cdot \sigma_{N_{\theta} \rightarrow C_{\theta}}^{2} \cdot \csc^{4}\left(m_{N_{\theta} \rightarrow C_{\theta}}\right). \end{array} $$
The node C θ forwards the messages \((m_{C_{\theta } \rightarrow \Delta x_{\theta }}, \sigma ^{2}_{C_{\theta } \rightarrow \Delta x_{\theta }})\) obtained from (15) and (17) to the relative distance variable node Δ x θ for the X-coordinate, while the messages \((m_{C_{\theta } \rightarrow \Delta y_{\theta }}, \sigma ^{2}_{C_{\theta } \rightarrow \Delta y_{\theta }})\) obtained from (14) and (16) are forwarded to the relative distance variable node Δ y θ for the Y-coordinate. The variable node Δ x θ directly forwards the messages \((m_{\Delta x_{\theta } \rightarrow A_{\theta }}, \sigma ^{2}_{\Delta x_{\theta } \rightarrow A_{\theta }})\) to the relative distance factor node A θ , where \((m_{\Delta x_{\theta } \rightarrow A_{\theta }}, \sigma ^{2}_{\Delta x_{\theta } \rightarrow A_{\theta }})=(m_{C_{\theta } \rightarrow \Delta x_{\theta }}, \sigma ^{2}_{C_{\theta } \rightarrow \Delta x_{\theta }})\). The node Δ y θ forwards the messages \((m_{\Delta y_{\theta } \rightarrow B_{\theta }}, \sigma ^{2}_{\Delta y_{\theta } \rightarrow B_{\theta }})\) to the relative distance factor node B θ , where \((m_{\Delta y_{\theta } \rightarrow B_{\theta }}, \sigma ^{2}_{\Delta y_{\theta } \rightarrow B_{\theta }})=(m_{C_{\theta } \rightarrow \Delta y_{\theta }}, \sigma ^{2}_{C_{\theta } \rightarrow \Delta y_{\theta }})\).
The messages in the nodes A θ and B θ are finally converted to the coordinate variable node, as[6–8]
$$\begin{array}{@{}rcl@{}} \left(m_{A_{\theta} \rightarrow \Delta x_{\theta} },\sigma^{2}_{A_{\theta} \rightarrow \Delta x_{\theta} }\right)=\left(X-m_{x \rightarrow A_{\theta}},\sigma^{2}_{x \rightarrow A_{\theta}}\right), \end{array} $$
$$\begin{array}{@{}rcl@{}} \left(m_{B_{\theta} \rightarrow \Delta y_{\theta}},\sigma^{2}_{B_{\theta} \rightarrow \Delta y_{\theta} }\right)=\left(Y-m_{y \rightarrow B_{\theta}},\sigma^{2}_{y \rightarrow B_{\theta}}\right), \end{array} $$
$$\begin{array}{@{}rcl@{}} \left(m_{A_{\theta} \rightarrow x},\sigma^{2}_{A_{\theta} \rightarrow x}\right)=\left(X-m_{\Delta x_{\theta} \rightarrow A_{\theta}},\sigma^{2}_{\Delta x_{\theta} \rightarrow A_{\theta}}\right), \end{array} $$
$$\begin{array}{@{}rcl@{}} \left(m_{B_{\theta} \rightarrow y},\sigma^{2}_{B_{\theta} \rightarrow y}\right)=\left(Y-m_{\Delta y_{\theta} \rightarrow B_{\theta}},\sigma^{2}_{\Delta y_{\theta} \rightarrow B_{\theta}}\right). \end{array} $$
As shown in Fig. 2, the messages of (20) and (21), \((m_{A_{\theta } \rightarrow x},\sigma ^{2}_{A_{\theta } \rightarrow x})\) and \((m_{B_{\theta } \rightarrow y},\sigma ^{2}_{B_{\theta } \rightarrow y})\), produced by the nodes A θ and B θ , respectively, are forwarded to the estimated target position variable nodes x and y. According to the message-passing principle, now the reverse process is invoked. Recall that we omitted the sensor index in the equations; however, to derive the messages sent from the variable nodes x and y, the sensor index has to be introduced. All the messages coming from the nodes \(A_{\theta _{j}}, j=\{1,...,N\}\), except for the message sent back to the node \(A_{\theta _{i}}\), i≠j, are used in the node x. It can be easily found by invoking the fact that the products of multiple Gaussian pdfs having different means and variances are proportional to the Gaussian pdf; the messages sent back from the variable node x to the factor node \(A_{\theta _{i}}\) are given by \((m_{x \rightarrow A_{\theta _{i}}},\sigma ^{2}_{x \rightarrow A_{\theta _{i}}})\) as [5–8]
$$\begin{array}{@{}rcl@{}} \frac{1}{\sigma^{2}_{x \rightarrow A_{\theta_{i}}}}&=&\sum_{j=1,j \neq i}^{N} \frac{1}{\sigma^{2}_{A_{\theta_{j}} \rightarrow x}}, \end{array} $$
$$\begin{array}{@{}rcl@{}} m_{x \rightarrow A_{\theta_{i}}}&=&\sigma^{2}_{x \rightarrow A_{\theta_{i}}} \cdot \sum_{j=1,j\neq i}^{N} \frac{m_{A_{\theta_{j}} \rightarrow x}}{\sigma^{2}_{A_{\theta_{j}} \rightarrow x}}, \end{array} $$
where i and j are the sensor indexes. The messages \((m_{y \rightarrow B_{\theta _{i}}},\sigma ^{2}_{y \rightarrow B_{\theta _{i}}})\) can be obtained in the same way as \((m_{x \rightarrow A_{\theta _{i}} }, \sigma ^{2}_{x \rightarrow A_{\theta _{i}}})\) calculated by (22) and (23).
The messages \((m_{x \rightarrow A_{\theta _{i}}},\sigma ^{2}_{x \rightarrow A_{\theta _{i}}})\) of (22) and (23) sent to the node \(A_{\theta _{i}}\) are used by (18) to calculate the messages \((m_{A_{\theta _{i}} \rightarrow \Delta x_{\theta _{i}}},\sigma ^{2}_{A_{\theta _{i}} \rightarrow \Delta x_{\theta _{i}}})\), and in the same way, the messages \((m_{B_{\theta _{i}} \rightarrow \Delta y_{\theta _{i}}},\sigma ^{2}_{B_{\theta _{i}} \rightarrow \Delta y_{\theta _{i}}})\) are calculated by (19) using \((m_{y \rightarrow B_{\theta _{i}}}\), \(\sigma ^{2}_{y \rightarrow B_{\theta _{i}}})\). The messages \((m_{A_{\theta _{i}}\rightarrow \Delta x_{\theta _{i}}},\sigma ^{2}_{A_{\theta _{i}}\rightarrow \Delta x_{\theta _{i}}})\) of (18) are forwarded from the node \(A_{\theta _{i}}\) to the node \(\Delta x_{\theta _{i}}\), and then, the messages \((m_{\Delta x_{\theta _{i}} \rightarrow C_{\theta _{i}}},\sigma ^{2}_{\Delta x_{\theta _{i}} \rightarrow C_{\theta _{i}}})\) are directly forwarded to the node \(C_{\theta _{i}}\), where \((m_{\Delta x_{\theta _{i}} \rightarrow C_{\theta _{i}}},\sigma ^{2}_{\Delta x_{\theta _{i}} \rightarrow C_{\theta _{i}}})=(m_{A_{\theta _{i}}\rightarrow \Delta x_{\theta _{i}}},\sigma ^{2}_{A_{\theta _{i}}\rightarrow \Delta x_{\theta _{i}}})\). The messages \((m_{B_{\theta _{i}}\rightarrow \Delta y_{\theta _{i}}},\sigma ^{2}_{B_{\theta _{i}}\rightarrow \Delta y_{\theta _{i}}})\) of (19) are forwarded from the node \(B_{\theta _{i}}\) to the node \(\Delta y_{\theta _{i}}\), and then, the messages \((m_{\Delta y_{\theta _{i}} \rightarrow C_{\theta _{i}}},\sigma ^{2}_{\Delta y_{\theta _{i}} \rightarrow C_{\theta _{i}}}) \) are forwarded to the node \(C_{\theta _{i}}\), where \((m_{\Delta y_{\theta _{i}} \rightarrow C_{\theta _{i}}},\sigma ^{2}_{\Delta y_{\theta _{i}} \rightarrow C_{\theta _{i}}}) =(m_{B_{\theta _{i}}\rightarrow \Delta y_{\theta _{i}}},\sigma ^{2}_{B_{\theta _{i}}\rightarrow \Delta y_{\theta _{i}}})\). The entire process is repeated iteratively. When the iteration converges or maximum iteration is reached, all messages from the nodes \(A_{\theta _{i}}\) and \(B_{\theta _{i}}\) are combined in the nodes x and y as [5–8]
$$\begin{array}{@{}rcl@{}} \frac{1}{{\sigma^{2}_{x}}}&=&\sum_{i=1}^{N} \frac{1}{\sigma^{2}_{A_{\theta_{i}} \rightarrow x}}, \end{array} $$
$$\begin{array}{@{}rcl@{}} \frac{1}{{\sigma^{2}_{y}}}&=&\sum_{i=1}^{N} \frac{1}{\sigma^{2}_{B_{\theta_{i}} \rightarrow y}}, \end{array} $$
$$\begin{array}{@{}rcl@{}} m_{x }&=&{\sigma^{2}_{x}} \cdot \sum_{i=1}^{N} \frac{m_{A_{\theta_{i}} \rightarrow x}}{\sigma^{2}_{A_{\theta_{i}} \rightarrow x}}, \end{array} $$
$$\begin{array}{@{}rcl@{}} m_{y}&=&{\sigma^{2}_{y}} \cdot \sum_{i=1}^{N} \frac{m_{B_{\theta_{i}} \rightarrow y}}{\sigma^{2}_{B_{\theta_{i}} \rightarrow y}}. \end{array} $$
Finally, the estimated coordinate position (x,y) of the unknown radio wave emitter is determined by (m x ,m y ). To provide more comprehensive understanding, we summarize all equations operating at each node of the FG in Table 1, where the directions of the message flow is shown in the left column.
Table 1 The operations required for each node in the DOA-based TS FG
CRLB derivation for DOA-based geolocation
This section derives the CRLB for DOA-based geolocation, taking into account the number of samples. The CRLB for DOA-based geolocation is presented in [7]; however, it does not take into account the effect of number of samples. The likelihood for K i.i.d. samples \(\hat {\theta }_{k},\;k=\{1,2,\ldots,K\},\) following the Gaussian distribution, is presented in [19], as
$$\begin{array}{@{}rcl@{}} p(\hat{\theta};{\theta})= \prod_{k=0}^{K-1}\frac{1}{\sqrt{2\pi \sigma_{\theta}^{2}}}\exp\left(-\frac{1}{2\sigma_{\theta}^{2}}\left(\hat{\theta}_{k} - {\theta}\right)^{2}\right), \end{array} $$
where \({\theta }=\arctan \left (\frac {Y_{i}-y}{X_{i}-x}\right)\). Here, the sensor index i is omitted again for simplicity. After several mathematical manipulations, as described in the Appendix, the closed form of the second-order derivative of log-likelihood function (LLF) is expressed as [19]
$$\begin{array}{@{}rcl@{}} \frac{\partial^{2}}{\partial {\theta}^{2}}\ln p(\hat{\theta};{\theta})&=& -\frac{K}{\sigma_{\theta}^{2}}. \end{array} $$
The closed-form CRLB for DOA-based geolocation technique which takes into account the number of samples is found to be
$$\begin{array}{@{}rcl@{}} \mathrm{CRLB_{DOA}}=\sqrt{\text{trace}\left(\left(\mathbf{J}^{T} \Sigma_{\theta}^{-1} \mathbf{J} \right) K\right)^{-1}}, \end{array} $$
where \(\Sigma _{\theta }=\sigma ^{2}_{\theta } \mathbf {I}_{N}\) denotes Gaussian covariance, I N denotes an N×N identity matrix, and \(\mathbf {J}=\frac {\partial \mathbf {\theta }}{\partial \mathbf {x}}\) denotes Jacobian matrix given by
$$\begin{array}{@{}rcl@{}} \mathbf{J}=\left[\begin{array}{cc} \frac{Y_{1}-y}{{r_{1}^{2}}}&-\frac{X_{1}-x}{{r_{1}^{2}}}\\ \frac{Y_{2}-y}{{r_{2}^{2}}}&-\frac{X_{2}-x}{{r_{2}^{2}}}\\ \vdots & \vdots \\ \frac{Y_{N}-y}{{r_{N}^{2}}}&-\frac{X_{N}-x}{{r_{N}^{2}}}\\ \end{array}\right] \end{array} $$
with the r i denotes the Euclidean distance between the target and sensors.
Simulation results
The performance of the proposed technique was verified via computer simulations, where the simulation round consists of 1000 single-target locations randomly chosen from the area of 1000×1000 m2, where each target location is tested in 100 trials. It should be noted here that the scope in this paper is to estimate only one target position. The case of unknown multiple-target detection is left for future work. It is assumed that the illegal radio, as an example of unknown radio emitter, emits the radio wave with strong enough transmit power covering the area of 1000×1000 m2. The values of the measurement error were σ θ ={1°,5°,10°,15°,⋯,45°}. It was assumed that the simulation does not contain outliers in angular measurement.
It may be difficult to achieve the LOS condition in areas with a size of 1000 × 1000 m2 especially in (sub)urban environments. However, instead, we can include the error due to the non-LOS components to the variance of the measurement error as shown in [20, 21], where the variances are different between the sensors. For simplicity, we assume that the variance \(\sigma _{\theta }^{2}\) of the measurement error is common to all sensors as in [6–8, 10, 11]. It is rather straightforward to derive the algorithm where each sensor has different values of variances. In fact, in our simulation setup, the area size is much smaller than that used in other references, for example, the TOA-based FG in [[8], TOA/DOA-based FG in [6], DOA-based FG in [7], and TDOA-based FG in [10], where they consider a hexagonal area with a radius of 5 km. Furthermore, as found in the simulation results, the estimation accuracy is quite high even with relatively large variance, e.g., with standard deviation σ θ =45°. This indicates that the assumption for the impact of the non-LOS components being represented by the measurement error variance is reasonable.
As shown in Fig. 3, six sensors were assumed in total, indicated by the Δ mark. The positions of the sensors in the simulation were limited to certain positions making a sensing area with three and five sensors at {(100,0),(1100,0), (600,−1000)} m and {(100,0), (1100,0), (600,−500), (100,−1000), (1100,−1100)} m, respectively. As described above, the target positions are randomly chosen inside the sensing 1000×1000 m2 area to evaluate the effectiveness of the proposed technique.
1000 target and six sensor positions in area 1000×1000 m2
The accuracy of the proposed technique was evaluated by using the following parameters: (a) three to five sensors taken from the total of six sensors, (b) 25 to 1000 samples, and (c) 10 times of iterations for each trial. The initialization point is set at (0,0) for \(m_{x \rightarrow A_{\theta }}\) and \(m_{y \rightarrow B_{\theta }}\) and at (1,1) for \(\sigma ^{2}_{x \rightarrow A_{\theta }}\) and \(\sigma ^{2}_{y \rightarrow B_{\theta }}\). It should be noted that the initialization point can be set arbitrarily inside the area of the expected target detection. Regardless of the target positions (1000 points tested), with the initialization of mean and variance being set at (0,0) and (1,1), respectively, the final estimate of the target position is quite accurate. Conversely, this observation should be understood in a way that the estimation result is less sensitive to the initial values.
To demonstrate the convergence property of the proposed technique, the trajectory of a detection trail is shown in Fig. 4. It shows clearly that the target position estimate successfully reaches the true target position at (x,y)=(444,−746) m in 10 iterations, where the iteration process is started from the initial point (0,0) m. The estimate target position is calculated in each iteration by using (26) and (27). It should be noticed that only with seven iterations, the position estimate of the proposed technique reaches a point close to the true target position by using three sensors.
Trajectory of the proposed technique with 3 sensors, 10 iterations, 100 samples, σ θ =10°, and target at (444,−746) m
Figure 5 shows the RMSE versus the iteration times with the standard deviation σ θ of the measurement error as a parameter. The RMSE of the proposed technique converges after nine iterations for σ θ =1°, while it converges after five iterations for σ θ =20° and σ θ =45° as shown in Fig. 5. Hence, the iteration converges faster with a higher standard deviation of the measurement error because lower σ θ needs more times to achieve better accuracy. Although the RMSE with σ θ =1° is worse with less than five iterations, the RMSE with smaller σ θ is lower when the iterations converges.
RMSE vs. iteration times of DOA-based TS FG geolocation technique with 3 sensors, 10 iterations, 100 samples, 1000 locations, 100 trials, and σ θ ={1°,20°,45°}
To make a comparison of the accuracy of the proposed technique to the conventional DOA-based LS technique in [16], we conducted another series of the conventional DOA-based LS algorithm [16] in summarized below
$$\begin{array}{@{}rcl@{}} \left[\begin{array}{c} m_{y_{\rm{LS}}}\\m_{x_{\rm{LS}}} \end{array}\right]=\left(A^{T} A\right)^{-1}A^{T} b, \end{array} $$
$$\begin{array}{@{}rcl@{}} A&=&\left[\begin{array}{cc} 1& \!\!-\tan\left(m_{N_{\theta_{1}} \rightarrow C_{\theta_{1}}}\right)\\ 1& \!\!-\tan\left(m_{N_{\theta_{2}} \rightarrow C_{\theta_{2}}}\right)\\ \ &\vdots \,\,\, \vdots\\ 1& \!\!-\tan\left(m_{N_{\theta_{N}} \rightarrow C_{\theta_{N}}}\right) \end{array}\right], \\ b&=&\left[\begin{array}{c} Y_{1}-X_{1}\tan\left(m_{N_{\theta_{1}} \rightarrow C_{\theta_{1}}}\right)\\ Y_{2}-X_{2}\tan\left(m_{N_{\theta_{2}} \rightarrow C_{\theta_{2}}}\right)\\ \vdots \\ Y_{N}-X_{N}\tan\left(m_{N_{\theta_{N}} \rightarrow C_{\theta_{N}}}\right) \end{array}\right], \end{array} $$
with \((m_{x_{\text{{LS}}}}, m_{y_{\text{{LS}}}})\) being the target position estimate. The RMSEs achieved by the proposed and the conventional DOA-based LS techniques are then compared with the CRLB. In each round of simulations, both techniques used the same parameter values as described before.
Figure 6 shows that the RMSE versus the standard deviation σ θ of the measurement error with the number of sensors as a parameter. It is found that the more sensors, the smaller the RMSE. Figure 7 shows the number of the sensors versus the RMSE with the standard deviation σ θ of the measurement error as a parameter. It is found from the figure that there is a clear difference in the tendency of the RMSE between the proposed technique and the conventional DOA-based LS technique. With the conventional DOA-based LS technique, the RMSE with three sensors yields better performance than that of with four sensors for σ θ =1°, and the RMSE with five sensors is almost the same as that with four sensors. This indicates that since the LS equation is overdetermined with more than three sensors, the LS error is the dominant factor with σ θ =1°. When σ θ ={20°, 45°}, on the other hand, the measurement error is the dominating factor, and hence, the achieved RMSEs with three, four, and five sensors are almost the same. This may be because the measurement error dominates the accuracy over the error due to the overdetermination.
RMSE vs. σ θ with 3–5 sensors, 10 iterations, 100 samples, 1000 locations, and 100 trials
RMSE vs. sensor number with 10 iterations, 1000 locations, 100 trials, and σ θ ={1°,20°,45°}
In contrast to this observation, the RMSE decreases by increasing the sensor number for σ θ ={1°, 20°, and 45°} with the proposed technique, and such tendency is consistent to the CRLB. It can be concluded that the geolocation accuracy in terms of RMSE with the proposed technique outperforms the reference conventional DOA-based LS technique.
Figure 8 shows the effect of the number of samples to the accuracy of DOA-based geolocation techniques with σ θ as a parameter. The RMSE of both the proposed and reference techniques[16] as well as the CRLB for the DOA-based geolocation decreases when more samples are used. It is shown in Fig. 8 that with RMSE 24 m, the proposed technique requires around 525 samples, while the conventional technique requires around 630 samples. The accuracy of the proposed technique always outperforms the reference technique with the same number of samples used. Obviously, by increasing the number of samples, the gap to the CRLB decreases with both the proposed and reference techniques. It is also found from the figure that the smaller the measurement error, the smaller the gap to the CRLB. The gap with different σ θ values decreases when the number of samples increases.
RMSE vs. the number of samples with 3 sensors, 10 iterations, 1000 locations, 100 trials, and σ θ ={15°,30°}
We have proposed a new FG-based geolocation technique using DOA information for a single unknown (anonymous) radio emitter with accuracy improvement of the position estimate. We have derived a set of new approximated expression for the mean and variance of the tangent and cotangent functions based on the first-order TS to hold the Gaussianity assumption. We have also derived a closed-form expression of the CRLB for DOA-based techniques taking into account the influence of the number of samples. The simulation results confirmed that our proposed technique provides (a) better accurate position estimate with the number of samples, number of sensors, and standard deviation of measurement error as parameters, (b) fast convergence, and (c) keep low computational complexity, which are suitable for the future geolocation techniques requiring high accuracy and low complexity in an imperfect synchronization condition. The development of DOA-based TS FG technique for multiple-target detection is left as future work.
Appendix: CRLB derivations for DOA-based geolocation
The sensor index i is omitted in this derivation for simplicity. By taking the expectation of (28), we have
$$\begin{array}{@{}rcl@{}} E \left[\frac{\partial^{2}}{\partial {{\theta}}^{2}}\ln p(\hat{\theta};\theta)\right]&=& -\frac{K}{\sigma_{\theta}^{2}}. \end{array} $$
$$\begin{array}{@{}rcl@{}} E\left[ \left(\frac{\partial}{\partial {{\theta}} }\ln p\left({\hat{\theta};\theta}\right) \right)^{2} \right]=-E \left[\frac{\partial^{2}}{\partial {{\theta}}^{2}}\ln p\left({\hat{\theta};\theta}\right)\right], \end{array} $$
as shown in [19]
$$\begin{array}{@{}rcl@{}} E\left[ \left(\frac{\partial}{\partial {{\theta}} }\ln p({\hat{\theta};\theta}) \right)^{2} \right]=\frac{K}{\sigma_{\theta}^{2}}. \end{array} $$
The Fisher information matrix (FIM) [19, 22, 23]
$$\begin{array}{@{}rcl@{}} \mathbf{F}(\mathbf{x})&\,=\,& \frac{\partial {{\theta}}}{\partial \mathbf{x}}^{T} E\left[\! \left(\frac{\partial}{\partial {{\theta}} }\ln p({\hat{\theta};\theta}) \right)^{T}\!\! \left(\frac{\partial}{\partial {{\theta}} }\ln p({\hat{\theta};\theta}) \right) \! \right] \frac{\partial {{\theta}}}{\partial \mathbf{x}} \end{array} $$
is found to
$$\begin{array}{@{}rcl@{}} \mathbf{F}(\mathbf{x})&=& \frac{\partial {{\theta}}}{\partial \mathbf{x}}^{T} E\left[ \left(\frac{\partial}{\partial {{\theta}} }\ln p({\hat{\theta};\theta}) \right)^{2} \right] \frac{\partial {{\theta}}}{\partial \mathbf{x}}. \end{array} $$
Substituting (35) into (37) yields
$$\begin{array}{@{}rcl@{}} \mathbf{F}(\mathbf{x})=\frac{\partial {{\theta}}}{\partial \mathbf{x}}^{T} \left[ \frac{K}{\sigma_{\theta}^{2}} \right]\frac{\partial {{\theta}}}{\partial \mathbf{x}}. \end{array} $$
Now, we replace θ by a vector θ=(θ 1,…,θ N ). Then, the variance \((\sigma _{\theta }^{2})\) is replaced by the Gaussian covariance matrix Σ θ , as
$$\begin{array}{@{}rcl@{}} \mathbf{F}(\mathbf{x})=K \mathbf{J}^{T} \Sigma_{\theta}^{-1} \mathbf{J}. \end{array} $$
Finally, by substituting (39) into the CRLB expression (29), as in [7, 10, 19], we obtain the CRLB for a DOA-based geolocation technique that takes into account the measured number of samples K, as
$$\begin{array}{@{}rcl@{}} \mathrm{CRLB_{DOA}}=\sqrt{\text{trace}\left(\left(\mathbf{J}^{T} \Sigma_{\theta}^{-1} \mathbf{J}\right) K\right)^{-1}}. \end{array} $$
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This research is supported, in part, by Koden Electronics Co., Ltd., and in part by the Doctor Research Fellow (DRF) Program of Japan Advanced Institute of Science and Technology (JAIST).
MRKA conceived and designed the research work, expanded and improved the algorithms for more potential applications, conducted computer simulations, acquired and analyzed data, and critically revised the manuscript. KA conceived and designed the research work, developed the base of algorithm for geolocation, analyzed data, and critically revised the manuscript. TM conceived and designed the research work, performed verification of the algorithm, analyzed data, and critically revised the manuscript. All authors read and approved the final manuscript.
School of Information Science, Japan Advanced Institute of Science and Technology (JAIST), 1-1 Asahidai, Nomi-shi, 923-1292, Japan
Muhammad Reza Kahar Aziz
, Khoirul Anwar
& Tad Matsumoto
Electrical Enginering Department, Institut Teknologi Sumatera (ITERA), Lampung Selatan, 35365, Indonesia
Centre for Wireless Communications (CWC), University of Oulu, Oulu, 90014, Finland
Tad Matsumoto
Search for Muhammad Reza Kahar Aziz in:
Search for Khoirul Anwar in:
Search for Tad Matsumoto in:
Correspondence to Muhammad Reza Kahar Aziz.
Kahar Aziz, M.R., Anwar, K. & Matsumoto, T. A new DOA-based factor graph geolocation technique for detection of unknown radio wave emitter position using the first-order Taylor series approximation. J Wireless Com Network 2016, 189 (2016). https://doi.org/10.1186/s13638-016-0683-4
CRLB
Factor graphs | CommonCrawl |
When Does an Atlas Uniquely Define a Manifold?
I am totally new to differential geometry and am having trouble understanding a very basic idea. In what follows, I apologize for being gratuitously pedantic, but I want to be sure I clearly understand what's going on.
If $M$ is a set and $T$ is a topology on $M$ such that $(M,T)$ is Hausdorff and second countable, then $M$ is a topological manifold if for all $p\in M$ there exists an ordered pair $(U,x)$ such that $U \subset M$ is $T$-open and $x:U\rightarrow \mathbb{R}^d$ is a homeomorphism whose image is an open subset of $\mathbb{R}^d$ in the standard topology.
Ordered pairs $(U,x)$ that satisfy the conditions in the above paragraph are called charts on the manifold. An atlas for $M$ is a collection of charts on $M$, $A = \{(U_a,x_a)\colon a \in I\}$, such that $\cup_{\alpha\in I}U_a = M$.
Question 1: Does every manifold have at least one atlas?
My answer: I believe so, since by the definition of a manifold there exists at least one chart for each point, and the collection of either all or at least one of the charts at each point can be taken as an atlas. Perhaps however there is some technical problem in set theory with this construction.
Question 2: Does an atlas uniquely define a manifold? That is, if $A$ and $A'$ are atlases and $A \neq A'$, is it necessary true that the manifolds with $(X,T)$ as their underlying space but with atlases $A$ and $A'$ respectively are different? (In the naive sense--not considering the possibility that they are diffeomorphic)
I believe the core concept I'm struggling with here is what the naive notion of equivalence is for manifolds. (For example, for topological spaces "naive equivalence" means that the two underlying sets are equal and the two topologies have exactly the same open sets, rather than the existence of a homeomorphism, which is a more sophisticated notion of equivalence.)
If instead we define a topological manifold as an ordered triple $(M,T,A)$, where $A$ is an atlas, my confusion vanishes. But then naive equivalence requires exactly the same charts in the atlas, which might be too much to reasonably say that two manifolds are the same. I've also not seen this definition in any of the references I'm using. This brings up the following question.
Question 3: Is it possible to define a manifold as an ordered triple, as in the paragraph above?
differential-geometry manifolds differential-topology smooth-manifolds
Mortified Through MathMortified Through Math
$\begingroup$ "Diffeomorphic" is the wrong word if you're talking about topological manifolds--the non-naive notion of equivalence is just homeomorphism. $\endgroup$ – Eric Wofsey Mar 29 at 2:47
For Question 1, you are right. For instance, you can just take the set of all charts on $(M,T)$ and they will be an atlas.
For Questions 2 and 3, as you have defined a topological manifold, a topological manifold is just a topological space which satisfies certain properties. So, an atlas doesn't actually have anything to do with what a topological manifold is (an atlas just happens to exist on any topological manifold). Two manifolds are equal iff they are equal as topological spaces.
That said, no one actually cares about equality of manifolds. What people actually care about is whether two manifolds are homeomorphic (or more specifically, whether specific maps between them are homeomorphisms). In other words, the "naive equivalence" you are asking about is not important for any applications. As a result, it's perfectly fine to use a definition as you propose in Question 3, where an atlas is part of what a manifold is. This will change what equality of manifolds means (i.e., "naive equivalence") but will not change the notion of equivalence that actually matters, which is homeomorphism.
In the language of category theory, you can define a category $Man$ whose objects are topological manifolds (according to your original definition) and whose maps are continuous maps. You can also define a category $Man'$ whose objects are topological manifolds together with an atlas and whose maps are continuous maps. There is a forgetful functor $F:Man'\to Man$ which forgets the atlas. This functor is not an isomorphism of categories, but it is an equivalence of categories, which is good enough for everything people ever want to do with manifolds.
As a final remark, atlases are pretty irrelevant to the study of topological manifolds. The reason atlases are important is to define smooth manifolds, which impose some additional conditions on what kind of atlases are allowed. A smoooth manifold cannot be defined as just a topological space, but instead must be defined as a topological space together with an atlas satisfying certain assumptions (or a topological space together with some other additional structure equivalent to an atlas).
For smooth manifolds, although an atlas must be included in the definition, there is still an issue similar to your Questions 2 and 3. Namely, multiple different atlases can give "the same" smooth manifold, in the sense that the identity map is a diffeomorphism. This means that if you define a smooth manifold as a triple $(M,T,A)$ where $(M,T)$ is a topological space and $A$ is a smooth atlas on $(M,T)$, then the "naive equivalence" is not the equivalence you actually care about, similar to if you used the definition for topological manifolds you proposed in Question 3.
To avoid this, many authors instead define a smooth manifold as a triple $(M,T,A)$ where $(M,T)$ is a topological space and $A$ is a maximal smooth atlas on $(M,T)$ (or alternatively, $A$ is an equivalence class of smooth atlases on $(M,T)$). This makes the choice of $A$ unique, in the sense that if $(M,T,A)$ and $(M,T,A')$ are smooth manifolds such that the identity map $M\to M$ is a diffeomorphism between them, then $A=A'$. As with topological manifolds, though, it doesn't really matter whether you use this definition or the previous one, since all that changes is what it means for two smooth manifolds to literally be equal and that's not what we actually care about.
Eric WofseyEric Wofsey
$\begingroup$ Thank you, Eric! I ultimately want a definition I can use for smooth manifolds, but I thought this was a more basic issue with the definition of 'manifold' itself. May I make an addendum to the question, basically asking (2) and (3) for smooth manifolds? Thank you again. $\endgroup$ – Mortified Through Math Mar 29 at 12:13
$\begingroup$ I have added some more comments on the smooth case. $\endgroup$ – Eric Wofsey Mar 29 at 15:14
$\begingroup$ Awesome. Thanks! $\endgroup$ – Mortified Through Math Mar 29 at 15:54
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On the reasonability of linearized approximation and Hopf bifurcation control for a fractional-order delay Bhalekar–Gejji chaotic system
Jianping Shi ORCID: orcid.org/0000-0002-5994-45541,2 &
Liyuan Ruan1
In this paper, we study the reasonability of linearized approximation and Hopf bifurcation control for a fractional-order delay Bhalekar–Gejji (BG) chaotic system. Since the current study on Hopf bifurcation for fractional-order delay systems is carried out on the basis of analyses for stability of equilibrium of its linearized approximation system, it is necessary to verify the reasonability of linearized approximation. Through Laplace transformation, we first illustrate the equivalence of stability of equilibrium for a fractional-order delay Bhalekar–Gejji chaotic system and its linearized approximation system under an appropriate prior assumption. This semianalytically verifies the reasonability of linearized approximation from the viewpoint of stability. Then we theoretically explore the relationship between the time delay and Hopf bifurcation of such a system. By introducing the delayed feedback controller into the proposed system, the influence of the feedback gain changes on Hopf bifurcation is also investigated. The obtained results indicate that the stability domain can be effectively controlled by the proposed delayed feedback controller. Moreover, numerical simulations are made to verify the validity of the theoretical results.
For a long time, researches on fractional-order calculus were mainly concentrated in the field of pure mathematics [1]. The main reason for abandoning fractional-order models in practical application was their computational complexity. With the development of computer technology, the application of fractional-order calculus has attracted the interest of many researchers in different fields, including mathematics, physics, chemistry, engineering, and even financial and social sciences. It has been found in some fields that using fractional-order is more appropriate than employing integer-order to describe the dynamical behavior and characteristics [2–6] of models, such as the classical predator–prey system [7, 8], the fractional-order model of the dengue virus infection [9], the simulated tidal wave fractional-order model [10], the fractional-order viscoelastic non-Newtonian fluid model [11], and so on. It is proved that the fractional-order dynamical system can more accurately respond to the time variation of general nature [12–14].
In order to describe those phenomena whose evolutions not only depend on the state of current time, but also on the state at a previous time, the time delay needs to be introduced into many differential equations which are defined as delayed differential equation (DDE). DDEs arise in many fields, for example, metal cutting, epidemiology, neuroscience, population dynamics [15], biological systems [16, 17], financial system [18], and traffic models [19]. The time delay is often considered as the parameter of fractional-order systems according to the actual background of the problems [20–22]. In 1992, Pyragas originally proposed the delayed feedback controller to study the control issue of nonlinear autonomous differential equations [23]. Now the delayed feedback has become an adjustment mechanism that is widely used in many nonlinear control systems. For example, in order to stabilize the unstable periodic orbit by the difference between the current state and the delay state, a suitable delayed feedback controller can be designed to achieve the desired dynamic behavior [24–26].
In 2011, Bhalekar and Daftardar-Gejji [27] constructed a three-dimensional chaotic dynamic system (shortly called as BG system subsequently):
$$ \textstyle\begin{cases} \dot{x}(t)=d x(t)-y^{2}(t), \\ \dot{y}(t)=c (z(t)-y(t)), \\ \dot{z}(t)=ay(t)-bz(t)+x(t)y(t), \end{cases} $$
where a, b, c, d are constants. Then Bhalekar [28] further explored the forming mechanism of the BG system. In recent years, some valuable results about the BG system have been obtained. Aqeel and Ahmad [29] studied the Hopf bifurcation and chaos of the integer-order BG system. Deshpande et al. [30] found that the fractional-order BG system allows chaotic solutions and the fractional-order could be regarded as the control parameter of chaos. Shahzad et al. [31] added a single time delay into the third equation of (1.1) and studied a delay Bhalekar–Gejji chaotic system of the form:
$$ \textstyle\begin{cases} \dot{x}(t)=d x(t)-y^{2}(t), \\ \dot{y}(t)=d (z(t)-y(t)), \\ \dot{z}(t)=ay(t)-bz(t-\tau )+x(t)y(t). \end{cases} $$
They derived some algebraic sufficient conditions that guarantee the globally and asymptotically stable synchronization and antisynchronization between two identical time delay Bhalekar–Gejji chaotic systems. To the best of our knowledge, there are few literature sources discussing the Hopf bifurcation of the fractional-order BG system with time delay. This motivates us to investigate the effect of time-delay and fractional-order on the occurrence of the Hopf bifurcation and to introduce an appropriate delayed state-feedback controller to control the Hopf bifurcation.
In this paper, we consider the fractional-order time delay Bhalekar–Gejji system of the form:
$$ \textstyle\begin{cases} D^{q_{1}}x(t)=dx(t)-y^{2}(t), \\ D^{q_{2}}y(t)=c(z(t)-y(t)), \\ D^{q_{3}}z(t)=ay(t)-bz(t-\tau )+x(t)y(t), \end{cases} $$
where \(q_{i}\in (0, 1]\) (\(i=1, 2, 3\)) and a, b, c, d are parameters; c is generally taken as positive, while d is a negative real number; \(\tau \geq 0\) is the time delay.
The main purpose of this paper is to seek for the conditions of the occurrence of Hopf bifurcation for system (1.3) by using time delay as the bifurcation parameter based on the approach of stability analysis [32]. Especially, we give a semianalytical verification of the reasonability of linearized approximation of system (1.3) from the viewpoint of stability. In addition, we also design a delayed feedback controller to control the emergence of Hopf bifurcation and further study the effect of feedback gain on the bifurcation control of the proposed system.
This paper is organized as follows. In Sect. 2, we introduce the relevant preliminary knowledge of fractional calculus and fractional-order dynamical system. In Sect. 3, we analyze system (1.3) to get the conditions of the occurrence of Hopf bifurcation and the value range of delay in which Hopf bifurcation appears. We also verify the reasonability of the linearized approximation by the equivalence of stability of equilibrium points between the original system (1.3) and its linearized system. In Sect. 4, the delayed feedback controller is added to system (1.3) to control the Hopf bifurcation. In Sect. 5, numerical simulations are performed to verify the validity of the theoretical results by choosing appropriate values of the constants a, b, c, d, τ. Finally, necessary conclusions and a discussion are presented in Sect. 6.
In this section, some preliminary knowledge of fractional calculus and fractional-order dynamical system are introduced. In fact, the concept of fractional derivative has many classical definitions. This paper is based on the most widely used definition of Caputo fractional derivative.
([1, 33])
The fractional-order integral of order \(\alpha >0\) of a real-valued function \(x(t)\) is defined as
$$ {}^{C}_{t_{0}}D^{-{\alpha }}_{t}x(t)= \frac{1}{\varGamma (\alpha )} \int _{t} ^{t_{0}} (t-\tau )^{\alpha -1}x(\tau )\,d\tau , $$
where \(\varGamma (\cdot )\) is the Gamma function, \(\varGamma (s)=\int _{0} ^{\infty }t^{s-1}e^{-t}\,ds\).
The Caputo fractional derivative can be written as
$$ {}_{t_{0}}^{C}D_{t}^{\alpha } x(t)=\frac{1}{\varGamma (m-\alpha )} \int _{t_{0}}^{t} \frac{x^{(m)}(\tau )}{(t-\tau )^{\alpha -m+1}} \,d \tau , \quad m-1< \alpha \le m, $$
where \(x(t) \in C^{n}( [t_{0}, \infty ), \mathbb{R})\). In particular, if \(0< \alpha \le 1\), (2.2) can be written as
$$ {}_{t_{0}}^{C} D_{t}^{\alpha } x(t)= \frac{1}{\varGamma (1-\alpha )} \int _{t_{0}}^{t} \frac{x{'}(\tau )}{(t-\tau )^{\alpha }} \,d \tau , \quad 0< \alpha \le 1, t>0. $$
For brevity, in what follows, we use the notation \(D^{\alpha }x(t)\) to denote the Caputo fractional-order derivative operator \({}_{t_{0}}^{C}D_{t}^{\alpha }x(t)\).
The Laplace transform of Caputo fractional derivative of order α (\(n-1<\alpha \le n\)) for a function \(x(t) \in C^{n}( [a,\infty ),\mathbb{R})\) is
$$ \mathcal{L} \bigl\{ \mathcal{D}^{\alpha } x(t) ; s \bigr\} =s^{\alpha } F(s)- \sum_{k=0}^{n-1} s^{\alpha -k-1} x^{(k)} (a ), $$
where \(F(s)\) is the Laplace transform of \(x(t)\), and \(x^{(k)}(a)\) (\(k=0,1,\dots ,n-1\)) are the initial conditions. Obviously, if \(x^{(k)}(a)=0\) for \(k=0,1,\dots ,n-1\), (2.4) can be written as
$$ \mathcal{L} \bigl\{ \mathcal{D}_{t}^{q} x(t) ; s \bigr\} =s^{q} F(s). $$
Consider the following n-dimensional fractional-order system with time delay:
$$ {D}^{\alpha } x_{i}(t)=f_{i} \bigl(x_{1}(t), \dots , x_{n}(t) ; \tau \bigr), \quad i=1,2, \dots , n, $$
where \(0<\alpha \le 1\) and the time delay \(\tau \ge 0\). System (2.6) undergoes a Hopf bifurcation at the equilibrium \(x^{*}=(x_{1}^{*}, x_{2}^{*}, \dots ,x_{n}^{*})\) when \(\tau =\tau _{0}\) if the following three conditions are satisfied:
(C1):
When \(\tau =0\), all the eigenvalues \(\lambda _{j}\) (\(j=1, 2,\dots , n\)) of the coefficient matrix J of the linearized system of (2.6) satisfy \(|\arg (\lambda _{j})|>\frac{\alpha \pi }{2}\).
The characteristic equation of the linearized system of (2.6) has a pair of purely imaginary roots \(\pm \omega _{0}\) when \(\tau =\tau _{0}\).
\(\operatorname{Re} [\frac{ds(\tau )}{d{\tau }} ]|_{\tau =\tau _{0}, \omega =\omega _{0}}>0\), where \(\operatorname{Re}[\cdot ]\) denotes the real part of the complex number and s refers to the eigenvalue of the associated characteristic equation of the linearized system.
Reasonability of linearized approximation and Hopf bifurcation for fractional-order delay BG system
In this paper, we only consider the nonzero real equilibrium points of the fractional-order delay BG system (1.3). Hence, we need to assume that
\((H_{1})\):
\(d(b-a)>0\).
It is obvious that system (1.3) has two nonzero real equilibrium points:
$$ \bigl(b-a,\sqrt{d(b-a)},\sqrt{d(b-a)}\bigr),\qquad \bigl(b-a,-\sqrt{d(b-a)},- \sqrt{d(b-a)}\bigr). $$
We use the delay τ as a bifurcation parameter to find the conditions on the occurrence of Hopf bifurcation at the equilibria of system (1.3).
For brevity, the nonzero equilibrium point is denoted as \((x^{*},y^{*},z^{*})\). Using the transformations \(u(t)=x(t)-x^{*}\), \(v(t)=y(t)-y^{*}\), \(w(t)=z(t)-z^{*}\), system (1.3) can be reduced to
$$ \textstyle\begin{cases} D^{q_{1}}u(t)=d(u(t)+x^{*})-(v(t)+y^{*})^{2}, \\ D^{q_{2}}v(t)=c(w(t)+z^{*}-v(t)-y^{*}), \\ D^{q_{3}}w(t)=a(v(t)+y^{*})-b(w(t-\tau )+z^{*})+(u(t)+x^{*})(v(t)+y^{*}), \end{cases} $$
that is,
$$ \textstyle\begin{cases} D^{q_{1}}u(t)= du(t)-2y^{*}v(t)-v(t)^{2}, \\ D^{q_{2}}v(t)= -cv(t)+cw(t), \\ D^{q_{3}}w(t)= y^{*}u(t)+(a+x^{*})v(t)-bw(t-\tau )+u(t)v(t). \end{cases} $$
System (3.2) has two equilibria
$$ \bigl(u^{*},v^{*},w^{*}\bigr)=(0,0,0), \biggl(- \frac{y^{*2}}{d},-y^{*},-y^{*} \biggr). $$
The linearized system of (3.2) at the origin is
$$ \textstyle\begin{cases} D^{q_{1}} \bar{u}(t)=d \bar{u}(t)-2y^{*}\bar{v}(t), \\ D^{q_{2}} \bar{v}(t)=-c \bar{v}(t)+c\bar{w}(t), \\ D^{q_{3}} \bar{w}(t)=y^{*}\bar{u}(t)+(a+x^{*})\bar{v}(t)-b \bar{w}(t-\tau ). \end{cases} $$
Analysis of reasonability of linearized approximation
Since the stability change of an equilibrium involves the appearance of Hopf bifurcation, we need to verify the reasonability of the above linearized approximation by the equivalence of stability of equilibrium for systems (3.2) and (3.3).
Following a similar idea as in [36], we prove the equivalence of stability of equilibrium for systems (3.2) and (3.3) in the sense that
$$ \lim_{t\rightarrow +\infty }\bar{u}(t)=0,\qquad \lim_{t \to +\infty } \bar{v}(t)=0,\qquad \lim_{t\to +\infty }\bar{w}(t)=0 $$
is equivalent to
$$ \lim_{t\rightarrow +\infty }u(t)=u^{*},\qquad \lim_{t \to +\infty }v(t)=v^{*},\qquad \lim_{t\to +\infty }w(t)=w^{*}, $$
where the initial values are taken as \(u(t)=\bar{u}(t)=\rho (t)>0\), \(v(t) =\bar{v}(t)= \phi (t)> 0\) and \(w(t) =\bar{w}(t)= \psi (t)> 0\) (\(t\in [-\tau , 0]\)).
Set \(e_{1}(t)=u(t)-\bar{u}(t)\), \(e_{2}(t)=v(t)-\bar{v}(t)\), \(e_{3}(t)=w(t)- \bar{w}(t)\). By (3.2) and (3.3), we obtain the error system
$$ \textstyle\begin{cases} D^{q_{1}} e_{1}(t)=d e_{1}(t)-2y^{*}e_{2}(t)-(e_{2}(t)+\bar{v}(t))^{2}, \\ D^{q_{2}} e_{2}(t)=-c e_{2}(t)+ce_{3}(t), \\ D^{q_{3}} e_{3}(t)=y^{*}e_{1}(t)+(a+x^{*})e_{2}(t)-be_{3}(t- \tau )+(e_{1}(t)+\bar{u}(t))(e_{2}(t)+\bar{v}(t)) \end{cases} $$
$$ \textstyle\begin{cases} D^{q_{1}} e_{1}(t)=d e_{1}(t)-2y^{*}e_{2}(t)-v(t)^{2}, \\ D^{q_{2}} e_{2}(t)=-c e_{2}(t)+ce_{3}(t), \\ D^{q_{3}} e_{3}(t)=y^{*}e_{1}(t)+(a+x^{*})e_{2}(t)-be_{3}(t- \tau )+u(t)v(t). \end{cases} $$
We have two basic assertions.
Assertion (a)
If the solutions \(\bar{u}(t)\), \(\bar{v}(t)\), \(\bar{w}(t)\) of system (3.3) satisfy
$$ \lim_{t\rightarrow +\infty }\bar{u}(t)=0,\qquad \lim_{t \to +\infty } \bar{v}(t)=0,\qquad \lim_{t\to +\infty }\bar{w}(t)=0, $$
then the solutions \(u(t)\), \(v(t)\), \(w(t)\) of system (3.2) satisfy
$$ \lim_{t\rightarrow +\infty }u(t)=u^{*},\qquad \lim_{t \to +\infty }v(t)=v^{*},\qquad \lim_{t\to +\infty }w(t)=w^{*}. $$
Taking the Laplace transform [34, 37] of both sides of the error system (3.4) gives
$$ \textstyle\begin{cases} s^{q_{1}}F_{1}(s)=d F_{1}(s)-2y^{*}F_{2}(s)-\mathscr{L}[(e_{2}(t)+ \bar{v}(t))^{2}], \\ s^{q_{2}}F_{2}(s)=-cF_{2}(s)+cF_{3}(s), \\ s^{q_{3}}F_{3}(s)=y^{*}F_{1}(s)+(a+x^{*})F_{2}(s)-be^{-s\tau }F_{3}(s)+ \mathscr{L}[(e_{1}(t)+\bar{u}(t))(e_{2}(t)+\bar{v}(t))], \end{cases} $$
where \(F_{k}(s)=\mathscr{L}[e_{k}(t)]\) (\(k=1,2,3\)), \(\mathscr{L}[ \cdot ]\) is the Laplace transform operator.
By (3.6), one gets
$$ \textstyle\begin{cases} s^{q_{1}}sF_{1}(s)=d sF_{1}(s)-2y^{*}sF_{2}(s)-s\mathscr{L}[(e_{2}(t)+ \bar{v}(t))^{2}], \\ s^{q_{2}}sF_{2}(s)=-csF_{2}(s)+csF_{3}(s), \\ s^{q_{3}}sF_{3}(s)=y^{*}sF_{1}(s)+(a+x^{*})sF_{2}(s)-be^{-s\tau }sF_{3}(s)\\ \hphantom{s^{q_{3}}sF_{3}(s)=}{}+s \mathscr{L}[(e_{1}(t)+\bar{u}(t))(e_{2}(t)+\bar{v}(t))]. \end{cases} $$
Similar to the prior assumption made in theoretical analysis of [38], we make the following prior assumption: \(e_{i}(t)\) (\(i=1,2,3\)) are bounded. Then by the final-value theorem of the Laplace transformation [37] and (3.7), we have
$$ \textstyle\begin{cases} d e_{1}^{*}-2y^{*}e_{2}^{*}-e_{2}^{*2}=0, \\ e_{2}^{*}=e_{3}^{*}, \\ y^{*}e_{1}^{*}+(a+x^{*})e_{2}^{*}-be_{3}^{*}+e_{1}^{*}e_{2}^{*}=0, \end{cases} $$
where \(e_{i}^{*}:=\lim_{t\rightarrow +\infty }e_{i}(t)\) (\(i=1,2,3\)).
By (3.8), we obtain
$$\begin{aligned} \bigl(e_{1}^{*},e_{2}^{*},e_{3}^{*} \bigr)=(0,0,0)\quad \text{or}\quad \biggl(- \frac{y^{*2}}{d},-y^{*},-y^{*} \biggr), \end{aligned}$$
which implies
On the other hand, by taking the Laplace transform [34, 37] of both sides of the error system (3.5), we have
$$ \textstyle\begin{cases} s^{q_{1}}F_{1}(s)=d F_{1}(s)-2y^{*}F_{2}(s)-\mathscr{L}[v(t)^{2}], \\ s^{q_{2}}F_{2}(s)=-cF_{2}(s)+cF_{3}(s), \\ s^{q_{3}}F_{3}(s)=y^{*}F_{1}(s)+(a+x^{*})F_{2}(s)-be^{-s\tau }F_{3}(s)+ \mathscr{L}[u(t)v(t)]. \end{cases} $$
Similarly, by (3.10), we can also prove that \((e_{1}^{*},e_{2}^{*},e_{3}^{*})=(u^{*},v^{*},w^{*})\). Hence, we have the following result.
Assertion (b)
If the solutions \(u(t)\), \(v(t)\), \(w(t)\) of system (3.2) satisfy
then the solutions \(\bar{u}(t)\), \(\bar{v}(t)\), \(\bar{w}(t)\) of system (3.3) satisfy
$$ \lim_{t\rightarrow +\infty }\bar{u}(t)=0,\qquad \lim_{t \to +\infty } \bar{v}(t)=0,\qquad \lim_{t\to +\infty }\bar{w}(t)=0. $$
Thus, by Assertions (a) and (b), we verified the reasonability of the above linearized approximation from the viewpoint of stability of equilibrium.
Hopf bifurcation analysis
The linearized system of (3.2) at the origin can be expressed as
$$ \textstyle\begin{cases} D^{q_{1}}u(t)=c_{11}u(t)+c_{12}v(t), \\ D^{q_{2}}v(t)=c_{22}v(t)+c_{23}w(t), \\ D^{q_{3}}w(t)=c_{31}u(t)+c_{32}v(t)+c_{33}w(t-\tau ), \end{cases} $$
where \(c_{11}=d\), \(c_{12}=-2y^{*}\), \(c_{22}=-c\), \(c_{23}=c\), \(c_{31}=y^{*}\), \(c_{32}=a+x^{*}\), \(c_{33}=-b\).
It is easy to obtain the associated characteristic equation by using Laplace transform on system (3.11):
$$ \begin{vmatrix} s^{q_{1}}-c_{11} &-c_{12} & 0 \\ 0 &s^{q_{2}}-c_{22} & -c_{23} \\ -c_{31} & -c_{32} & s^{q_{3}}-c_{33}e^{-s\tau } \end{vmatrix}=0. $$
Equation (3.12) can be equivalently rewritten as
$$ E_{1}(s)+E_{2}(s)e^{-s\tau }=0, $$
$$\begin{aligned}& \begin{aligned}[b] E_{1}(s)&=-s^{q_{1}}c_{23}c_{32}+s^{q_{3}}c_{11}c_{22}+c_{11}c_{23}c_{32}-c_{31}c_{12}c_{23}-s^{q_{1}+q_{3}}c_{22}-s^{q_{2}+q_{3}}c_{11} \\ &\quad{}+s^{q_{1}+q_{2}+q_{3}}, \end{aligned} \\& E_{2}(s)=-c_{33}\bigl(-s^{q_{1}}c_{22}-s^{q_{2}}c_{11}+c_{11}c_{22}+s^{q_{1}+q_{2}} \bigr). \end{aligned}$$
Assume that \(s=i\omega =\omega (\cos \frac{\pi }{2}+i\sin \frac{\pi }{2})\) is a root of Eq. (3.13), \(\omega>0\). Substituting \(s=i\omega\) into Eq. (3.13) and separating the real and imaginary parts, then it results in
$$ \textstyle\begin{cases} \alpha _{2}\cos \omega \tau +\alpha _{4}\sin \omega \tau =-\alpha _{1}, \\ \alpha _{4}\cos \omega \tau -\alpha _{2}\sin \omega \tau =-\alpha _{3}, \end{cases} $$
where \(\alpha _{i}\) (\(i=1,2,3,4\)) are defined in Appendix A.
Solving (3.14), one obtains
$$ \textstyle\begin{cases} \cos \omega \tau =- \frac{\alpha _{1}\alpha _{2}+\alpha _{3}\alpha _{4}}{\alpha ^{2}_{2}+\alpha ^{2}_{4}}=P_{1}( \omega ), \\ \sin \omega \tau = \frac{\alpha _{2}\alpha _{3}-\alpha _{1}\alpha _{4}}{\alpha ^{2}_{2}+\alpha ^{2}_{4}}=P_{2}( \omega ). \end{cases} $$
With the formula \(P^{2}_{1}(\omega )+P^{2}_{2}(\omega )=1\), we can calculate ω easily. We might as well suppose that \(\omega _{i}\) (\(i=1,2,\dots ,n\)) are positive solutions. There are four cases of \(\tau _{i}\) as follows:
I. When \(P_{1}(\omega _{i})>0\), \(P_{2}(\omega _{i})>0\), and \(k=0,1,2,\dots \),
$$ \tau _{i}^{(k)}=\frac{\arccos P_{1}(\omega _{i})+2k\pi }{\omega _{i}}= \frac{\arcsin P_{2}(\omega _{i})+2k\pi }{\omega _{i}}. $$
II. When \(P_{1}(\omega _{i})<0\), \(P_{2}(\omega _{i})>0\), and \(k=0,1,2,\dots \),
$$ \tau _{i}^{(k)}=\frac{\arccos P_{1}(\omega _{i})+2k\pi }{\omega _{i}}= \frac{\pi -\arcsin P_{2}(\omega _{i})+2k\pi }{\omega _{i}}. $$
III. When \(P_{1}(\omega _{i})>0\), \(P_{2}(\omega _{i})<0\), and \(k=0,1,2,\dots \),
$$ \tau _{i}^{(k)}= \frac{2\pi -\arccos P_{1}(\omega _{i})+2k\pi }{\omega _{i}}= \frac{2\pi +\arcsin P_{2}(\omega _{i})+2k\pi }{\omega _{i}}. $$
IV. When \(P_{1}(\omega _{i})<0\), \(P_{2}(\omega _{i})<0\), and \(k=0,1,2,\dots \),
$$ \tau _{i}^{(k)}= \frac{2\pi -\arccos P_{1}(\omega _{i})+2k\pi }{\omega _{i}}= \frac{\pi -\arcsin P_{2}(\omega _{i})+2k\pi }{\omega _{i}}. $$
According to the actual meaning of time delay τ, we are only interested in the first positive real value of τ. Define the bifurcation point as follows:
$$ \tau _{0}=\min \bigl\{ {\tau _{i}^{(k)}} \bigr\} , \qquad \omega _{0}=\omega _{i}, \quad i=1,2,\dots ,n, k=0,1,2,\dots , $$
where \(\tau _{i}^{(k)}\) is defined in cases I–IV and \(\omega _{i}\) corresponds to \(\min \{{\tau _{i}^{(k)}}\}\).
In order to find the bifurcation point, we need to have an in-depth study of Eq. (3.13). Differentiating both sides of Eq. (3.13) with respect to τ, one gets
$$ E'_{1}(s)\frac{ds}{d\tau }+E'_{2}(s)e^{-s\tau } \frac{ds}{d\tau }+E_{2}(s)e^{-s \tau }\biggl(-\tau \frac{ds}{d\tau }-s\biggr)=0, $$
where \(E'_{i}(s)\) are the derivatives of \(E_{i}(s)\) (\(i=1,2\)). Hence,
$$ \frac{ds}{d\tau }=\frac{A(s)}{B(s)}, $$
$$\begin{aligned}& A(s)=-c_{33}\bigl(-s^{q_{1}}c_{22}-s^{q_{2}}c_{11}+c_{11}c_{22}+s^{q_{1}+q_{2}} \bigr)se^{-s \tau }, \\& \begin{aligned}[b] B(s)&=-s^{q_{1}-1}q_{1}c_{23}c_{32}+s^{q_{3}-1}q_{3}c_{11}c_{22}-s^{q_{1}+q_{3}-1}(q_{1}+q_{3})c_{22}-s^{q_{2}+q_{3}-1}(q_{2}+q_{3})c_{11} \\ &\quad{}+s^{q_{1}+q_{2}+q_{3}-1}(q_{1}+q_{2}+q_{3})-c_{33} \bigl(-s^{q_{1}-1}q_{1}c_{22}-s^{q_{2}-1}q_{2}c_{11} \\ &\quad{}+s^{q_{1}+q_{2}-1}(q_{1}+q_{2})\bigr)e^{-s\tau }+c_{33} \tau \bigl(-s^{q_{1}}c_{22}-s^{q_{2}}c_{11}+c_{11}c_{22}+s^{q_{1}+q_{2}} \bigr)e^{-s \tau }. \end{aligned} \end{aligned}$$
Substituting \(s=i\omega =\omega (\cos \frac{\pi }{2}+i\sin \frac{\pi }{2})\) into \(A(s)\), \(B(s)\), and letting \(A_{1}\), \(A_{2}\) and \(B_{1}\), \(B_{2}\) be the real and imaginary parts of \(A(s)\), \(B(s)\), respectively, it can be deduced from Eq. (3.17) that
$$ \operatorname{Re} \biggl[\frac{ds}{d\tau } \biggr]= \frac{A_{1}B_{1}+A_{2}B_{2}}{B^{2}_{1}+B^{2}_{2}}, $$
where \(A_{i}\), \(B_{i}\) (\(i=1, 2\)) are defined in Appendix B.
Basing on the aforementioned analysis, we get
Let \(s(\tau )=\gamma (\tau )+i\omega (\tau )\) be the root of Eq. (3.13) near \(\tau =\tau _{i}^{(k)}\) satisfying \(\gamma (\tau _{i}^{(k)})=0\), \(\omega (\tau _{i}^{(k)})=\omega _{i}\), then the transversality condition
$$ \operatorname{Re} \biggl[\frac{ds}{d\tau } \biggr] \bigg|_{(\tau =\tau _{0},\omega = \omega _{0})}>0 $$
holds if the following assumption is satisfied:
\(\frac{A_{1}B_{1}+A_{2}B_{2}}{B^{2}_{1}+B^{2}_{2}}>0\),
Next, to verify Assumption \((C1)\) in Definition 4, we need the following lemma.
If the following assumptions hold:
\(c_{11}+c_{22}+c_{33}<0\).
\(c_{11}^{2}c_{22}+c_{11}^{2}c_{33}+c_{11}c_{22}^{2}+2c_{11}c_{22}c_{33}+c_{11}c_{33}^{2}-c_{12}c_{23}c_{31}+c_{22}^{2}c_{33} -c_{22}c_{23}c_{32}+c_{22}c_{33}^{2}-c_{23}c_{32}c_{33}<0\).
\(c_{11}c_{22}c_{33}-c_{11}c_{23}c_{32}+c_{31}c_{12}c_{23}<0\),
then all the eigenvalues \(\lambda _{j}\) (\(j=1, 2,3\)) of the coefficient matrix J of the linearized system (3.11) of system (3.2) with \(\tau =0\) satisfy \(|\arg (\lambda _{j})|>\frac{q_{i} \pi }{2}\) (\(i,j=1,2,3\)).
Neglecting the time delay, i.e., \(\tau =0\), the characteristic equation of coeffitient matrix J of the linearized system (3.11) becomes
$$ \begin{vmatrix} \lambda -c_{11} &-c_{12} & 0 \\ 0 &\lambda -c_{22} & -c_{23} \\ -c_{31} & -c_{32} & \lambda -c_{33} \end{vmatrix}=0, $$
which is equivalent to
$$ \begin{aligned}[b] &\lambda ^{3}-(c_{11}+c_{22}+c_{33}) \lambda ^{2}+(c_{11}c_{22}+c_{11}c_{33}+c_{22}c_{33}-c_{32}c_{23}) \lambda \\ &\quad {}-c_{11}c_{22}c_{33}+c_{11}c_{23}c_{32}-c_{31}c_{12}c_{23}=0. \end{aligned} $$
If Assumptions \((H_{3})\)–\((H5)\) are satisfied, it is easy to check from Routh–Hurwitz criterion that three eigenvalues \(\lambda _{j}\) (\(j=1, 2,3\)) of Eq. (3.20) have negative real parts. Therefore, \(|\arg (\lambda _{j})|>\frac{q_{i}\pi }{2}\) (\(i,j=1,2,3\)). □
It is apparent that the derived conditions in Lemma 2 are only sufficient conditions. According to Definition 4, if conditions \((H_{3})\)–\((H_{5})\) are replaced by other conditions which can guarantee that all the roots of Eq. (3.20) satisfy \(|\arg (\lambda _{j})|>\frac{q_{i}\pi }{2}\) (\(i,j=1,2,3\)), then Lemma 2 may still hold.
Basing on Definition 4, we achieve the first primary theorem of this paper.
Suppose \((H_{1})\)–\((H_{5})\) hold, when \(0< q_{i} \le 1\) (\(i=1,2,3\)) and the time delay \(\tau \ge 0\). The fractional-order delay system (1.3) undergoes a Hopf bifurcation at the nonzero equilibrium point \((x^{*}, y^{*}, z^{*})\) when \(\tau =\tau _{0}\), where \(\tau _{0}\) is defined by formula (3.16).
Delayed feedback control of fraction-order delay BG systems
In this section, a delayed feedback controller \(k[y(t)-y(t-\tau )]\) is added to the second equation of uncontrolled system (1.3), and then the delay feedback control system can be acquired as
$$ \textstyle\begin{cases} D^{q_{1}}x(t)=dx(t)-y^{2}(t), \\ D^{q_{2}}y(t)=c(z(t)-y(t))+k[y(t)-y(t-\tau )], \\ D^{q_{3}}z(t)=ay(t)-bz(t-\tau )+x(t)y(t). \end{cases} $$
For the sake of revealing the relationship between the controller and Hopf bifurcation, we still use the delay τ as a parameter in Eq. (4.1). Analogous to the previous analysis, by performing transformations \(u(t)=x(t)-x^{*}\), \(v(t)=y(t)-y^{*}\), \(w(t)=z(t)-z^{*}\), with the help of the linearized scheme, the linearization of the controlled system (4.1) has the form:
$$ \textstyle\begin{cases} D^{q_{1}}u(t)=c_{11}u(t)+c_{12}v(t), \\ D^{q_{2}}v(t)=c_{22}v(t)+c_{23}w(t)+k[v(t)-v(t-\tau )], \\ D^{q_{3}}w(t)=c_{31}u(t)+c_{32}v(t)+c_{33}w(t-\tau ), \end{cases} $$
where \(c_{11}\), \(c_{12}\), \(c_{22}\), \(c_{23}\), \(c_{31}\), \(c_{32}\), and \(c_{33}\) are defined as system (3.11).
Therefore, the associated characteristic equation of system (4.2) is
$$ \begin{vmatrix} s^{q_{1}}-c_{11} &-c_{12} & 0 \\ 0 &s^{q_{2}}-c_{22}-k+ke^{-s\tau } & -c_{23} \\ -c_{31} & -c_{32} & s^{q_{3}}-c_{33}e^{-s\tau } \end{vmatrix}=0. $$
Obviously, Eq. (4.3) is equivalent to
$$ F_{1}(s)+F_{2}(s)e^{-s\tau }+F_{3}(s)e^{-2s\tau }=0, $$
$$\begin{aligned}& \begin{aligned}[b] F_{1}(s)&=ks^{q_{3}}c_{11}-s^{q_{1}}c_{23}c_{32}+s^{q_{3}}c_{11}c_{22}+c_{11}c_{23}c_{32}-c_{31}c_{12}c_{23}-ks^{q{1}+q{3}}-s^{q_{2}+q_{3}}c_{11} \\ &\quad{}-s^{q_{1}+q_{3}}c_{22}+s^{q_{1}+q_{2}+q_{3}},\end{aligned} \\& \begin{aligned} F_{2}(s)&=ks^{q_{1}+q_{3}}+ks^{q_{1}}c_{33}-ks^{q_{3}}c_{11}-kc_{11}c_{33}-s^{q_{1}+q_{2}}c_{33}+s^{q_{1}}c_{22}c_{33}\\ &\quad {}+s^{q_{2}}c_{11}c_{33}-c_{11}c_{22}c_{33}, \end{aligned} \\& F_{3}(s)=-kc_{33}\bigl(s^{q_{1}}-c_{11} \bigr). \end{aligned}$$
By multiplying \(e^{s\tau }\) on both sides of Eq. (4.4), it is obvious that
$$ F_{1}(s)e^{s\tau }+F_{2}(s)+F_{3}(s)e^{-s\tau }=0. $$
Assume that \(s=i\omega =\omega (\cos \frac{\pi }{2}+i\sin \frac{\pi }{2})\) is a root of Eq. (4.5), \(\omega >0\). Substituting \(s=i\omega\) into Eq. (4.5) and separating the real and imaginary parts, we have
$$ \textstyle\begin{cases} \beta _{1}\cos \omega \tau +\beta _{3}\sin \omega \tau =-\beta _{5}, \\ \beta _{2}\cos \omega \tau +\beta _{4}\sin \omega \tau =-\beta _{6}, \end{cases} $$
where \(\beta _{i}\) (\(i=1, 2, \dots ,6\)) are defined in Appendix C.
From Eq. (4.6), we have
$$ \textstyle\begin{cases} \cos \omega \tau =- \frac{\beta _{6}\beta _{3}-\beta _{5}\beta _{4}}{\beta _{4}\beta _{1}-\beta _{3}\beta _{2}}=Q_{1}( \omega ), \\ \sin \omega \tau = \frac{\beta _{5}\beta _{2}-\beta _{6}\beta _{1}}{\beta _{4}\beta _{1}-\beta _{3}\beta _{2}}=Q_{2}( \omega ). \end{cases} $$
Consistent with the previous section, with the formula \(Q^{2}_{1}(\omega )+Q^{2}_{2}(\omega )=1\), we can calculate ω easily. We might as well suppose that \(\omega _{i}\) (\(i=1,2,\dots ,n\)) are positive solutions, and can get the same four cases of \(\tau ^{(k)}_{i}\) (\(k=0,1,2,\dots \)) as in Sect. 3.
Next we define the bifurcation point
$$ \tau _{0}^{*}=\min \bigl\{ {\tau _{i}^{(k)}}\bigr\} , \qquad \omega _{0}^{*}= \omega _{i}, \quad i=1,2,\dots ,n, k=0,1,2,\dots , $$
where \(\omega _{i}\) corresponds to \(\min \{{\tau _{i}^{(k)}}\}\). It needs to be noticed that the calculation of \(\tau _{i}^{(k)}\) and \(\omega _{i}\) is relying on the feedback grain coefficient k (see Appendix C).
Differentiating both sides of Eq. (4.5) with respect to τ, one obtains
$$ \begin{aligned}[b] &F'_{1}(s) \frac{ds}{d\tau }+F'_{2}(s)e^{-s\tau } \frac{ds}{d\tau }+F_{2}(s)e^{-s\tau }\biggl(-\tau \frac{ds}{d\tau }-s\biggr)+F'_{3}(s)e^{-2s \tau } \frac{ds}{d\tau } \\ &\quad{}+F_{3}(s)e^{-2s\tau }\biggl(-2\tau \frac{ds}{d\tau }-2s \biggr)=0, \end{aligned} $$
where \(F'_{i}(s)\) are the derivatives of \(F_{i}(s)\) (\(i=1, 2, 3\)).
$$ \frac{ds}{d\tau }=\frac{C(s)}{D(s)}, $$
$$ \begin{aligned}& C(s)=s\bigl[F_{2}(s)e^{-s\tau }+2F_{3}(s)e^{-2s\tau } \bigr], \\ &D(s)=F'_{1}(s)+\bigl[F'_{2}(s)- \tau F_{2}(s)\bigr]e^{-s\tau }+\bigl[F'_{3}(s)-2 \tau F_{3}(s)\bigr]e^{-2s \tau }. \end{aligned} $$
It can be deduced from Eq. (4.9) that
$$ \operatorname{Re} { \biggl[\frac{ds}{d\tau } \biggr]}= \frac{C_{1}D_{1}+C_{2}D_{2}}{D^{2}_{1}+D^{2}_{2}}, $$
where \(C_{1}\), \(C_{2}\), \(D_{1}\), \(D_{2}\) are the real and imaginary parts of \(C(s)\) and \(D(s)\), respectively, and the exact expressions are given in Appendix D.
Thus, we obtain the following lemma:
Let \(s(\tau )=\delta (\tau )+i\omega (\tau )\) be the root of Eq. (4.5) near \(\tau =\tau _{i}^{(k)}\) satisfying \(\delta (\tau _{i}^{(k)})=0\), \(\omega (\tau _{i}^{(k)})=\omega _{i}^{*}\). Then the transversality condition
$$ \operatorname{Re} \biggl[\frac{ds}{d\tau } \biggr] \bigg|_{(\tau =\tau _{0}^{*}, \omega =\omega _{0}^{*})}>0 $$
\(\frac{C_{1}D_{1}+C_{2}D_{2}}{D^{2}_{1}+D^{2}_{2}}>0\),
where \(C_{i}\), \(D_{i}\) (\(i=1,2\)) are defined as in (4.11).
Based on the previous discussion, the following theorem can be concluded.
Suppose \((H_{1})\), \((H_{3})\)–\((H_{6})\) hold, when \(0< q_{i} \le 1\) (\(i=1,2,3\)) and the delay \(\tau \ge 0\). The delay feedback control system (4.1) undergoes a Hopf bifurcation at the nonzero equilibrium \((x^{*}, y^{*}, z^{*})\) when \(\tau =\tau _{0}^{*}\), where \(\tau _{0}^{*}\) is defined as in (4.8).
In this section, we use the same method as in Sect. 3 to discuss the delay feedback control system (4.1). In reality, the Hopf bifurcation points \((\tau _{0}^{*}, \omega _{0}^{*})\) of system (4.1) can be controlled successfully by changing the feedback gain coefficient k. We will illustrate this fact in the next section by numerical simulations.
Adams–Bashforth–Moulton predictor–corrector scheme [39] has been widely used in numerical simulation for fractional-order differential equation. In this section, this method is adopted in two examples to verify the efficiency and feasibility of our theoretical results, in which step-length is taken as \(h=0.001\).
For the convenience of comparison, all the system parameters come from the literature [31]: \(a=22\), \(b=10\), \(c=10\), \(d=-2.667\). Without loss of generality, let \(q_{1}=0.91\), \(q_{2}=0.98\), \(q_{3}=0.95\), then system (1.3) can be changed into
$$ \textstyle\begin{cases} D^{q_{1}}x(t)=-2.667x(t)-y^{2}(t), \\ D^{q_{2}}y(t)=10(z(t)-y(t)), \\ D^{q_{3}}z(t)=22y(t)-10z(t-\tau )+x(t)y(t). \end{cases} $$
For system (5.1), it is easy to verify that \((H_{1})\)–\((H_{5})\) are all satisfied. The nonzero equilibrium points are \((-12, 5.657, 5.657)\) and \((-12, -5.657, -5.657)\). One can obtain the critical frequency \(\omega _{0}=9.35083452\) and bifurcation point \(\tau _{0}=0.09517856\). By Theorem 1, Hopf bifurcation of system (5.1) appears at \(\tau _{0}\). To better present our results, we give two simulations. One uses \(\tau =0.0949<\tau _{0}=0.09517856\), which is displayed in Figs. 1 and 2. Under this condition, we can see that two nonzero equilibria are asymptotically stable. The other uses \(\tau =0.0952>\tau _{0}=0.09517856\), which is displayed in Figs. 3 and 4. It is apparent that two nonzero equilibria are unstable and Hopf bifurcation occurs. Therefore, Theorem 1 is verified by these simulations.
Waveform plots of system (5.1) with initial values \((0.1,0.1,0.1)\) and \((-0.1,-0.1,-0.1)\), \(q_{1}=0.91\), \(q_{2}=0.98\), \(q_{3}=0.95\), and \(\tau =0.0949<\tau _{0}=0.09517856\). The nonzero equilibrium points of system (5.1) are asymptotically stable
Portraits of system (5.1) with initial values \((0.1,0.1,0.1)\) and \((-0.1,-0.1,-0.1)\), \(q_{1}=0.91\), \(q_{2}=0.98\), \(q_{3}=0.95\), and \(\tau =0.0949<\tau _{0}=0.09517856\). Two nonzero equilibrium points of system (5.1) are asymptotically stable
Waveform plots of system (5.1) with initial values \((0.1,0.1,0.1)\) and \((-0.1,-0.1,-0.1)\), \(q_{1}=0.91\), \(q_{2}=0.98\), \(q_{3}=0.95\), and \(\tau =0.0952>\tau _{0}=0.09517856\). The nonzero equilibrium points of system (5.1) are unstable
Portraits of system (5.1) with initial values \((0.1,0.1,0.1)\) and \((-0.1,-0.1,-0.1)\), \(q_{1}=0.91\), \(q_{2}=0.98\), \(q_{3}=0.95\), and \(\tau =0.0952>\tau _{0}=0.09517856\). Hopf bifurcation occurs
In this example, a linear delayed feedback controller is added to the uncontrolled system (5.1) so as to control the Hopf bifurcation. In order to illustrate the effects of bifurcation control via the proposed controller preferably, three fractional orders and all the system parameters are chosen the same as in Example 1, then the controlled system is shown as follows:
$$ \textstyle\begin{cases} D^{q_{1}}x(t)=-2.667x(t)-y^{2}(t), \\ D^{q_{2}}y(t)=10(z(t)-y(t))+k[y(t)-y(t-\tau )], \\ D^{q_{3}}z(t)=22y(t)-10z(t-\tau )+x(t)y(t). \end{cases} $$
For exhibiting the impact of feedback gain coefficient k on the Hopf bifurcation for the controlled system (5.2), we calculate a group of critical frequency \(\omega _{0}^{*}\) and bifurcation points \(\tau _{0}^{*}\) corresponding to ever-increasing k, see Table 1. According to Table 1, system (5.2) is controlled by the delayed feedback controller \(k[y(t)-y(t-\tau )]\) effectively. When k increases from negative to positive, \(\tau _{0}^{*}\) decreases gradually. This means that the stable domain wanes and the emergence of Hopf bifurcation is advanced. Moreover, by choosing three different k, Figs. 5–7 illustrate that the effect of Hopf bifurcation control is much better as k decreases.
Waveform plot of system (5.2) with initial values \((0.1,0.1,0.1)\), \(q_{1}=0.91\), \(q_{2}=0.98\), \(q_{3}=0.95\), and \(\tau =0.1\), the feedback gain \(k=0\), \(k=-1.5\), \(k=-3\). The effect of bifurcation control for the controlled system (5.2) becomes better as the feedback gain k decreases
Table 1 The impact of k on the values of \(\omega _{0}^{*}\) and \(\tau _{0}^{*}\) for the controlled system (5.2) with \(q_{1}=0.91\), \(q_{2}=0.98\), \(q_{3}=0.95\)
In this paper, sufficient conditions on the emergence of Hopf bifurcation have been established for a fractional-order delay Bhalekar–Gejji chaotic system. The delay feedback control issue of Hopf bifurcations for such a system has been investigated by theoretical analysis and numerical simulation.
This paper mainly focused on three aspects: the reasonability of the linearized approximation, delay-induced Hopf bifurcation, and delay feedback control of Hopf bifurcation for a fractional-order delay Bhalekar–Gejji chaotic system. Comparing to the previous similar works, we semianalytically verified the reasonability of the linearized approximation by the equivalence of stable equilibrium for the converted systems (3.2) and (3.3) under an appropriate prior assumption. To some extent, this provides a theoretical support for the definition of Hopf bifurcation for fractional-order delay systems proposed by [35].
We find that the time delay has an important influence on the stability of the fractional-order delay Bhalekar–Gejji chaotic system. The delay can be used as the bifurcation parameter to derive the asymptotic stability interval of the system and in which conditions the system will exhibit dynamic behavior such as Hopf bifurcation. In addition, for the fractional-order delay Bhalekar–Gejji chaotic system, the feedback gain can control the bifurcation value and expand the stability range of the system.
In the simulations, we can also observe the stability of equilibrium points of the proposed system. In fact, it is difficult to analytically prove stability of equilibrium points of the original system (1.3), which is not only because of the complexity of the system, but also due to the lack of a well-developed stability theory of nonlinear fractional-order delay systems.
In the subsequent research, we would also like to explore the effect of the fractional order and time delay on the occurrence of chaos for the fractional-order delay Bhalekar–Gejji chaotic system.
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The authors thank the referees for their valuable suggestions.
Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.
This research is supported by the National Natural Science Foundation of China (Grant Nos. 11561034, 11761040).
Department of System Science and Applied Mathematics, Kunming University of Science and Technology, 650500, Kunming, China
Jianping Shi & Liyuan Ruan
Center for Nonlinear Science Studies, Kunming University of Science and Technology, 650500, Kunming, China
Jianping Shi
Liyuan Ruan
The authors contributed equally in the writing of this paper. All authors read and approved final manuscript.
Correspondence to Jianping Shi.
The expressions of \(\alpha _{1}\), \(\alpha _{2}\), \(\alpha _{3}\), and \(\alpha _{4}\) in Eq. (3.14) are computed as follows:
$$\begin{aligned}& \begin{aligned}[b] \alpha _{1}&=-\omega ^{q_{1}}\cos \biggl(\frac{q_{1}\pi }{2} \biggr)c_{23}c_{32}+ \omega ^{q_{3}}\cos \biggl( \frac{q_{3}\pi }{2} \biggr)c_{11}c_{22}+c_{11}c_{23}c_{32}-c_{31}c_{12}c_{23} \\ &\quad{}-\omega ^{q_{1}+q_{3}}\cos \biggl(\frac{(q_{1}+q_{3})\pi }{2} \biggr)c_{22}- \omega ^{q_{2}+q_{3}}\cos \biggl(\frac{(q_{2}+q_{3})\pi }{2} \biggr)c_{11} \\ &\quad{}+\omega ^{q_{1}+q_{2}+q_{3}}\cos \biggl( \frac{(q_{1}+q_{2}+q_{3})\pi }{2} \biggr), \end{aligned} \\& \begin{aligned} \alpha _{2}&=\omega ^{q_{1}}\cos \biggl(\frac{q_{1}\pi }{2} \biggr)c_{22}c_{33}+ \omega ^{q_{2}}\cos \biggl( \frac{q_{2}\pi }{2} \biggr)c_{11}c_{33}- \omega ^{q_{1}+q_{2}} \cos \biggl(\frac{(q_{3}+q_{2})\pi }{2} \biggr)c_{33} \\ &\quad {}-c_{11}c_{22}c_{33} , \end{aligned} \\& \begin{aligned}[b] \alpha _{3}&=\omega ^{q_{3}}\sin \biggl(\frac{q_{3}\pi }{2} \biggr)c_{11}c_{22}- \omega ^{q_{1}}\sin \biggl( \frac{q_{1}\pi }{2} \biggr)c_{23}c_{32}- \omega ^{q_{1}+q_{3}} \sin \biggl(\frac{q_{1}\pi }{2}+ \frac{q_{3}\pi }{2} \biggr)c_{22} \\ &\quad {}-\omega ^{q_{2}+q_{3}}\sin \biggl(\frac{q_{2}\pi }{2}+ \frac{q_{3}\pi }{2} \biggr)c_{11}+\omega ^{q_{1}+q_{2}+q_{3}}\sin \biggl( \frac{q_{1}\pi }{2}+\frac{q_{2}\pi }{2}+\frac{q_{3}\pi }{2} \biggr) , \end{aligned} \\& \alpha _{4}=\omega ^{q_{1}}\sin \biggl(\frac{q_{1}\pi }{2} \biggr)c_{22}c_{33}+ \omega ^{q_{2}}\sin \biggl( \frac{q_{2}\pi }{2} \biggr)c_{11}c_{33}- \omega ^{q_{1}+q_{2}} \sin \biggl(\frac{(q_{1}+q_{2})\pi }{2} \biggr)c_{33} . \end{aligned}$$
The expressions of \(A_{1}\), \(A_{2}\), \(B_{1}\), and \(B_{2}\) in Eq. (3.18) are given as:
$$\begin{aligned}& \begin{aligned}[b] A_{1}&=\omega _{0}c_{33} \biggl(-\omega _{0}^{q_{1}}\sin \biggl( \frac{q_{1}\pi }{2} \biggr)c_{22}-\omega _{0}^{q_{2}}\sin \biggl( \frac{q_{2}\pi }{2} \biggr)c_{11}+\sin \biggl( \frac{(q_{1}+q_{2})\pi }{2} \biggr) \omega _{0}^{q_{1}+q_{2}} \biggr) \\ &\quad \times{}\cos (\tau _{0}\omega _{0} )+\omega _{0}c_{33} \biggl(\omega _{0}^{q_{1}}\cos \biggl(\frac{q_{1}\pi }{2} \biggr)c_{22}+ \omega _{0}^{q_{2}} \cos \biggl(\frac{q_{2}\pi }{2} \biggr)c_{11} \\ &\quad {}-\cos \biggl(\frac{(q_{1}+q_{2})\pi }{2} \biggr)\omega _{0}^{q_{1}+q_{2}}-c_{11}c_{22} \biggr)\sin (\tau _{0}\omega _{0} ), \end{aligned} \\& \begin{aligned} A_{2}&=\omega _{0}c_{33}\biggl(\omega _{0}^{q_{1}}\cos \biggl( \frac{q_{1}\pi }{2} \biggr)c_{22}+\omega _{0}^{q_{2}}\cos \biggl( \frac{q_{2}\pi }{2} \biggr)c_{11}\\ &\quad {}-\cos \biggl( \frac{(q_{1}+q_{2})\pi }{2} \biggr) \omega _{0}^{q_{1}+q_{2}}-c_{11}c_{22}\biggr) \\ &\quad \times{}\cos (\tau _{0}\omega _{0} )+\omega _{0}c_{33}\biggl( \omega _{0}^{q_{1}}\sin \biggl(\frac{q_{1}\pi }{2} \biggr)c_{22}+ \omega _{0}^{q_{2}} \sin \biggl(\frac{q_{2}\pi }{2} \biggr)c_{11} \\ &\quad {}-\sin \biggl(\frac{(q_{1}+q_{2})\pi }{2} \biggr)\omega _{0}^{q_{1}+q_{2}} \biggr) \sin (\tau _{0}\omega _{0} ), \end{aligned} \\& \begin{aligned} B_{1}&=\cos (\tau _{0}\omega _{0} ) \frac{1}{\omega _{0}}\biggl(c_{11}c_{22}c_{33} \omega _{0}\tau _{0}+\omega _{0}^{q_{1}+q_{2}+1}\cos \biggl( \frac{(q_{1}+q_{2})\pi }{2} \biggr)c_{33}\tau _{0}\\ &\quad {}+\omega _{0}^{q_{2}}c_{11}c_{33}\biggl( \sin \biggl( \frac{q_{2}\pi }{2} \biggr)q_{2}-\cos \biggl(\frac{q_{2}\pi }{2} \biggr)\tau _{0}\omega _{0}\biggr) \\ &\quad {}+ \omega _{0}^{q_{1}}c_{22}c_{33} \biggl(\sin \biggl(\frac{q_{1}\pi }{2} \biggr)q_{1}- \cos \biggl( \frac{q_{1}\pi }{2} \biggr)\tau _{0}\omega _{0}\biggr) \\ &\quad {}-\sin \biggl(\frac{(q_{1}+q_{2})\pi }{2} \biggr)\omega _{0}^{q_{1}+q_{2}}c_{33}(q_{1}+q_{2}) \biggr)\\ &\quad {}+ \sin (\tau _{0}\omega _{0} )\frac{1}{\omega _{0}} \biggl(\omega _{0}^{q_{2}}c_{11}c_{33} \biggl(- \cos \biggl(\frac{q_{2}\pi }{2} \biggr)q_{2} \\ &\quad {}-\sin \biggl(\frac{q_{2}\pi }{2} \biggr)\omega _{0}\tau _{0}\biggr)- \omega _{0}^{q_{1}}c_{22}c_{33} \biggl(\cos \biggl(\frac{q_{1}\pi }{2} \biggr)q_{1}+ \sin \biggl( \frac{q_{1}\pi }{2} \biggr)\omega _{0}\tau _{0}\biggr) \\ &\quad {}+\cos \biggl(\frac{(q_{1}+q_{2})\pi }{2} \biggr)\omega _{0}^{q_{1}+q_{2}}c_{33}(q_{1}+q_{2})+ \omega _{0}^{q_{1}+q_{2}+1}\sin \biggl(\frac{(q_{1}+q_{2})\pi }{2} \biggr)c_{33}\tau _{0}\biggr) \\ &\quad {}+\frac{1}{\omega _{0}}\biggl(-\omega _{0}^{q_{1}}\sin \biggl( \frac{q_{1}\pi }{2} \biggr)q_{1}c_{23}c_{32}- \omega _{0}^{q_{1}+q_{3}} \sin \biggl(\frac{(q_{1}+q_{3})\pi }{2} \biggr)c_{22}(q_{1}+q_{3}) \\ &\quad {}+\omega _{0}^{q_{1}+q_{2}+q_{3}}\sin \biggl( \frac{(q_{1}+q_{2}+q_{3})\pi }{2} \biggr) (q_{1}+q_{2}+q_{3})\\ &\quad {}-\omega _{0}^{q_{2}+q_{3}} \sin \biggl(\frac{(q_{2}+q_{3})\pi }{2} \biggr)c_{11}(q_{2}+q_{3}) +\omega _{0}^{q_{3}}\sin \biggl(\frac{q_{3}\pi }{2} \biggr)q_{3}c_{11}c_{22}\biggr), \end{aligned} \\& \begin{aligned} B_{2}&=\cos (\tau _{0}\omega _{0}) \frac{1}{\omega _{0}}\biggl(-\omega _{0}^{q_{2}}c_{11}c_{33} \biggl( \cos \biggl(\frac{q_{2}\pi }{2} \biggr)q_{2}+\sin \biggl( \frac{q_{2}\pi }{2} \biggr)\omega _{0}\tau _{0}\biggr) \\ &\quad {}-\omega _{0}^{q_{1}}c_{22}c_{33} \biggl(\cos \biggl( \frac{q_{1}\pi }{2} \biggr)q_{1}+\sin \biggl( \frac{q_{1}\pi }{2} \biggr) \omega _{0}\tau _{0}\biggr)\\ &\quad {}+ \omega _{0}^{q_{1}+q_{2}}c_{33}\cos \biggl( \frac{(q_{1}+q_{2})\pi }{2} \biggr) (q_{1}+q_{2}) \\ &\quad {}+\omega _{0}^{q_{1}+q_{2}+1}\sin \biggl( \frac{(q_{1}+q_{2})\pi }{2} \biggr)c_{33}\tau _{0}\biggr)+\sin (\tau _{0} \omega _{0} )\frac{1}{\omega _{0}}\biggl(-c_{11}c_{22}c_{33} \omega _{0} \tau _{0} \\ &\quad {}-\omega _{0}^{q_{1}}c_{22}c_{33} \biggl(\sin \biggl( \frac{q_{1}\pi }{2} \biggr)q_{1}-\cos \biggl( \frac{q_{1}\pi }{2} \biggr) \omega _{0}\tau _{0}\biggr)- \omega _{0}^{q_{2}}c_{11}c_{33}\biggl(\sin \biggl( \frac{q_{2}\pi }{2} \biggr)q_{2} \\ &\quad {}-\cos \biggl(\frac{q_{2}\pi }{2} \biggr)\omega _{0}\tau _{0}\biggr)+ \sin \biggl(\frac{(q_{1}+q_{2})\pi }{2} \biggr)\omega _{0}^{q_{1}+q_{2}}c_{33}(q_{1}+q_{2}) \\ &\quad {}-\omega _{0}^{q_{1}+q_{2}+1}\cos \biggl( \frac{(q_{1}+q_{2})\pi }{2} \biggr)c_{33}\tau _{0}\biggr)+ \frac{1}{\omega _{0}}\biggl( \omega _{0}^{q_{1}}\cos \biggl( \frac{q_{1}\pi }{2} \biggr)q_{1}c_{23}c_{32} \\ &\quad {}+\omega _{0}^{q_{1}+q_{3}}\cos \biggl( \frac{(q_{1}+q_{3})\pi }{2} \biggr)c_{22}(q_{1}+q_{3})+\omega _{0}^{q_{2}+q_{3}} \cos \biggl(\frac{(q_{2}+q_{3})\pi }{2} \biggr)c_{11}(q_{2}+q_{3}) \\ &\quad {}-\omega _{0}^{q_{1}+q_{2}+q_{3}}\cos \biggl( \frac{(q_{1}+q_{2}+q_{3})\pi }{2} \biggr) (q_{1}+q_{2}+q_{3})-\omega _{0}^{q_{3}} \cos \biggl(\frac{q_{3}\pi }{2} \biggr)q_{3}c_{11}c_{22}\biggr). \end{aligned} \end{aligned}$$
The expressions of \(\beta _{1}\), \(\beta _{2}\), \(\beta _{3}\), \(\beta _{4}\), \(\beta _{5}\), and \(\beta _{6}\) in Eq. (4.6) are:
$$\begin{aligned}& \beta _{1}=\omega ^{q_{1}+q_{2}+q_{3}}\sin \biggl(\frac{q_{1}\pi }{2}+ \frac{q_{2}\pi }{2}+\frac{q_{3}\pi }{2} \biggr)-k\omega ^{q_{1}}\sin \biggl( \frac{q_{1}\pi }{2} \biggr)c_{33}-\omega ^{q_{1}}\sin \biggl( \frac{q_{1}\pi }{2} \biggr)c_{23}c_{32} \\& \hphantom{\beta _{1}=} {}+\omega ^{q_{3}}\sin \biggl(\frac{q_{3}\pi }{2} \biggr)c_{11}+ \omega ^{q_{3}}\sin \biggl(\frac{q_{3}\pi }{2} \biggr)c_{11}c_{22}-k \omega ^{q_{1}+q_{3}}\sin \biggl( \frac{q_{1}\pi }{2}+ \frac{q_{3}\pi }{2} \biggr) \\& \hphantom{\beta _{1}=} {}-\omega ^{q_{1}+q_{3}}\sin \biggl(\frac{q_{1}\pi }{2}+ \frac{q_{3}\pi }{2} \biggr)c_{22}-\omega ^{q_{2}+q_{3}}\sin \biggl( \frac{q_{2}\pi }{2}+\frac{q_{3}\pi }{2} \biggr)c_{11}, \\& \begin{aligned} \beta _{2}&=c_{11}c_{23}c_{32}-c_{31}c_{12}c_{23}+kc_{11}c_{33}+ \omega ^{q_{1}+q_{2}+q_{3}}\cos \biggl( \frac{(q_{1}+q_{2}+q_{3})\pi }{2} \biggr) \\ &\quad {}+k\omega ^{q_{3}}\cos \biggl(\frac{q_{3}\pi }{2} \biggr)c_{11}+ \omega ^{q_{3}}\cos \biggl(\frac{q_{3}\pi }{2} \biggr)c_{11}c_{22}-k \omega ^{q_{1}}\cos \biggl( \frac{q_{1}\pi }{2} \biggr)c_{33} \\ &\quad {}-\omega ^{q_{1}}\cos \biggl(\frac{q_{1}\pi }{2} \biggr)c_{23}c_{32}-k \omega ^{q_{1}+q_{3}}\cos \biggl( \frac{(q_{1}+q_{3})\pi }{2} \biggr)- \omega ^{q_{1}+q_{3}}\cos \biggl(\frac{(q_{1}+q_{3})\pi }{2} \biggr)c_{22} \\ &\quad {}-\omega ^{q_{2}+q_{3}}\cos \biggl(\frac{(q_{2}+q_{3})\pi }{2} \biggr)c_{11}, \end{aligned} \\& \begin{aligned} \beta _{3}&=\omega ^{q_{1}+q_{2}+q_{3}}\cos \biggl(\frac{q_{1}\pi }{2}+ \frac{q_{2}\pi }{2}+\frac{q_{3}\pi }{2} \biggr)-\omega ^{q_{1}}\cos \biggl( \frac{q_{1}\pi }{2} \biggr)c_{23}c_{32}+k\omega ^{q_{3}} \cos \biggl(\frac{q_{3}\pi }{2} \biggr)c_{11} \\ &\quad {}+\omega ^{q_{3}}\cos \biggl(\frac{q_{3}\pi }{2} \biggr)c_{11}c_{22}+k \omega ^{q_{1}}\cos \biggl( \frac{q_{1}\pi }{2} \biggr)c_{33}-k\omega ^{q_{1}+q_{3}} \cos \biggl( \frac{q_{1}\pi }{2}+\frac{q_{3}\pi }{2} \biggr) \\ &\quad {}-\omega ^{q_{1}+q_{3}}\cos \biggl(\frac{q_{1}\pi }{2}+ \frac{q_{3}\pi }{2} \biggr)c_{22}-\omega ^{q_{2}+q_{3}}\cos \biggl( \frac{q_{2}\pi }{2}+\frac{q_{3}\pi }{2} \biggr)c_{11} \\ &\quad {}+c_{11}c_{23}c_{32}-c_{31}c_{12}c_{23}-kc_{11}c_{33}, \end{aligned} \\& \begin{aligned} \beta _{4}&=-\omega ^{q_{1}+q_{2}+q_{3}}\sin \biggl( \frac{(q_{1}+q_{2}+q_{3})\pi }{2} \biggr)+\omega ^{q_{1}}\sin \biggl( \frac{q_{1}\pi }{2} \biggr)c_{23}c_{32}-k \omega ^{q_{3}}\sin \biggl( \frac{q_{3}\pi }{2} \biggr)c_{11} \\ &\quad {}-\omega ^{q_{3}}\sin \biggl(\frac{q_{3}\pi }{2} \biggr)c_{11}c_{22}-k \omega ^{q_{1}}\sin \biggl( \frac{q_{1}\pi }{2} \biggr)c_{33}+k\omega ^{q_{1}+q_{3}} \sin \biggl( \frac{(q_{1}+q_{3})\pi }{2} \biggr) \\ &\quad {}+\omega ^{q_{1}+q_{3}}\sin \biggl(\frac{(q_{1}+q_{3})\pi }{2} \biggr)c_{22}+\omega ^{q_{2}+q_{3}}\sin \biggl( \frac{(q_{2}+q_{3})\pi }{2} \biggr)c_{11}, \end{aligned} \\& \begin{aligned} \beta _{5}&=-k\omega ^{q_{3}}\sin \biggl(\frac{q_{3}\pi }{2} \biggr)c_{11}+k \omega ^{q_{1}}\sin \biggl(\frac{q_{1}\pi }{2} \biggr)c_{33}+\omega ^{q_{2}} \sin \biggl(\frac{q_{2}\pi }{2} \biggr)c_{11}c_{33} \\ &\quad {}+\omega ^{q_{1}} \sin \biggl( \frac{q_{1}\pi }{2} \biggr)c_{22}c_{33}+k\omega ^{q_{1}+q_{3}}\sin \biggl(\frac{q_{1}\pi }{2}+ \frac{q_{3}\pi }{2} \biggr)\\ &\quad {}-\omega ^{q_{1}+q_{2}}\sin \biggl( \frac{q_{1}\pi }{2}+ \frac{q_{2}\pi }{2} \biggr)c_{33}, \end{aligned} \\& \begin{aligned} \beta _{6}&=k\omega ^{q_{1}}\cos \biggl(\frac{q_{1}\pi }{2} \biggr)c_{33}-k \omega ^{q_{3}}\cos \biggl(\frac{q_{3}\pi }{2} \biggr)c_{11}\\ &\quad {}+\omega ^{q_{1}} \cos \biggl(\frac{q_{1}\pi }{2} \biggr)c_{22}c_{33}+\omega ^{q_{2}} \cos \biggl( \frac{q_{2}\pi }{2} \biggr)c_{11}c_{33} \\ &\quad {}+k\omega ^{q_{1}+q_{3}}\cos \biggl(\frac{(q_{1}+q_{3})\pi }{2} \biggr)-kc_{11}c_{33}-\omega ^{q_{1}+q_{2}}\cos \biggl( \frac{(q_{1}+q_{2})\pi }{2} \biggr)c_{33}-c_{11}c_{22}c_{33}. \end{aligned} \end{aligned}$$
The expressions of \(C_{1}\), \(C_{2}\), \(D_{1}\), and \(D_{2}\) in Eq. (4.11) are:
$$\begin{aligned} &C_{1}=-\omega _{0}\cos (\tau _{0}\omega _{0} ) \biggl(-k\omega _{0}^{q_{1}} \sin \biggl( \frac{q_{1}\pi }{2} \biggr)c_{33}-k\omega _{0}^{q_{3}} \sin \biggl(\frac{q_{3}\pi }{2} \biggr)c_{11}+\omega _{0}^{q_{1}} \sin \biggl(\frac{q_{1}\pi }{2} \biggr)c_{22}c_{33} \\ &\hphantom{C_{1}=} {}+\omega _{0}^{q_{2}}\sin \biggl(\frac{q_{2}\pi }{2} \biggr)c_{11}c_{33}+k \omega _{0}^{q_{1}+q_{3}} \sin \biggl(\frac{(q_{1}+q_{3})\pi }{2} \biggr)-\omega _{0}^{q_{1}+q_{2}}\sin \biggl( \frac{(q_{1}+q_{2})\pi }{2} \biggr)c_{33}\biggr) \\ &\hphantom{C_{1}=} {}-\omega _{0}\sin (\tau _{0}\omega _{0} ) \biggl(k\omega _{0}^{q_{1}} \cos \biggl(\frac{q_{1}\pi }{2} \biggr)c_{33}+k\omega _{0}^{q_{3}} \cos \biggl( \frac{q_{3}\pi }{2} \biggr)c_{11}-\omega _{0}^{q_{1}} \cos \biggl(\frac{q_{1}\pi }{2} \biggr)c_{22}c_{33} \\ &\hphantom{C_{1}=} {}-\omega _{0}^{q_{2}}\cos \biggl(\frac{q_{2}\pi }{2} \biggr)c_{11}c_{33}-k \omega _{0}^{q_{1}+q_{3}} \cos \biggl(\frac{(q_{1}+q_{3})\pi }{2} \biggr)+\omega _{0}^{q_{1}+q_{2}}\cos \biggl( \frac{(q_{1}+q_{2})\pi }{2} \biggr)c_{33} \\ &\hphantom{C_{1}=} {}-kc_{11}c_{33}+c_{11}c_{22}c_{33} \biggr), \\ &D_{1}=\frac{1}{\omega _{0}}\biggl(\biggl(k\omega _{0}^{q_{3}}c_{11} \biggl(\cos \biggl( \frac{q_{3}\pi }{2} \biggr)\omega _{0}\tau _{0}-\sin \biggl( \frac{q_{3}\pi }{2} \biggr)q_{3}\biggr)+ \omega _{0}^{q_{1}}\sin \biggl( \frac{q_{1}\pi }{2} \biggr)c_{33}q_{1}(k+c_{22}) \\ &\hphantom{D_{1}=} {}+\omega _{0}^{q_{2}}c_{11}c_{33} \biggl(\sin \biggl( \frac{q_{2}\pi }{2} \biggr)q_{2}-\cos \biggl( \frac{q_{2}\pi }{2} \biggr) \omega _{0}\tau _{0}\biggr)- \omega _{0}^{q_{1}+1}\cos \biggl( \frac{q_{1}\pi }{2} \biggr)c_{33}\tau _{0}(k+c_{22}) \\ &\hphantom{D_{1}=} {}-\omega _{0}^{q_{1}+q_{2}}\sin \biggl( \frac{(q_{1}+q_{2})\pi }{2} \biggr)c_{33}(q_{1}+q_{2})-k\omega _{0}^{q_{1}+q_{3}+1} \cos \biggl(\frac{(q_{1}+q_{3})\pi }{2} \biggr)\tau _{0} \\ &\hphantom{D_{1}=} {}+k\omega _{0}^{q_{1}+q_{3}}\sin \biggl( \frac{(q_{1}+q_{3})\pi }{2} \biggr) (q_{1}+q_{3})+\omega _{0}^{q_{1}+q_{2}+1} \cos \biggl(\frac{(q_{1}+q_{2})\pi }{2} \biggr)c_{33}\tau _{0} \\ &\hphantom{D_{1}=} {}+c_{11}c_{22}c_{33}\omega _{0} \tau _{0}+kc_{11}c_{33}\omega _{0} \tau _{0}\biggr)\cos (\tau _{0}\omega _{0} )\biggr)+ \frac{1}{\omega _{0}}\biggl(\biggl(k\omega _{0}^{q_{3}}c_{11} \biggl(\sin \biggl( \frac{q_{3}\pi }{2} \biggr)\omega _{0}\tau _{0} \\ &\hphantom{D_{1}=} {}+\cos \biggl(\frac{q_{3}\pi }{2} \biggr)q_{3}\biggr)-\omega _{0}^{q_{1}+1} \sin \biggl(\frac{q_{1}\pi }{2} \biggr)c_{33}\tau _{0}(k+c_{22})- \omega _{0}^{q_{2}}c_{11}c_{33}\biggl(\sin \biggl( \frac{q_{2}\pi }{2} \biggr) \omega _{0}\tau _{0} \\ &\hphantom{D_{1}=} {}-\cos \biggl(\frac{q_{2}\pi }{2} \biggr)q_{2}\biggr)-\omega _{0}^{q_{1}} \cos \biggl(\frac{q_{1}\pi }{2} \biggr)c_{33}q_{1}(c_{22}+k)-k\omega _{0}^{q_{1}+q_{3}+1} \sin \biggl(\frac{(q_{1}+q_{3})\pi }{2} \biggr)\tau _{0} \\ &\hphantom{D_{1}=} {}-k\omega _{0}^{q_{1}+q_{3}}\cos \biggl( \frac{(q_{1}+q_{3})\pi }{2} \biggr) (q_{1}+q_{3})+\omega _{0}^{q_{1}+q_{2}+1} \sin \biggl(\frac{(q_{1}+q_{2})\pi }{2} \biggr)c_{33}\tau _{0} \\ &\hphantom{D_{1}=} {}+\omega _{0}^{q_{1}+q_{2}}\cos \biggl( \frac{(q_{1}+q_{2})\pi }{2} \biggr)c_{33}(q_{1}+q_{2})\biggr)\sin ( \tau _{0}\omega _{0} )\biggr) \\ &\hphantom{D_{1}=} {}+\frac{1}{\omega _{0}}\biggl(k\omega _{0}^{q_{1}}c_{33}q_{1} \sin \biggl(2\tau _{0}\omega _{0}-\frac{q_{1}\pi }{2} \biggr)+2k\omega _{0}^{q_{1}+1}c_{33} \tau _{0}\cos \biggl(2\tau _{0}\omega _{0}- \frac{q_{1}\pi }{2} \biggr) \\ &\hphantom{D_{1}=} {}+\omega _{0}^{q_{1}+q_{2}+q_{3}}\sin \biggl( \frac{(q_{1}+q_{2}+q_{3})\pi }{2} \biggr) (q_{1}+q_{2}+q_{3})+\omega _{0}^{q_{3}} \sin \biggl(\frac{q_{3}\pi }{2} \biggr)q_{3}c_{11}(c_{22}+k) \\ &\hphantom{D_{1}=} {}-\omega _{0}^{q_{1}}\sin \biggl(\frac{q_{1}\pi }{2} \biggr)q_{1}c_{23}c_{32}- \omega _{0}^{q_{2}+q_{3}}\sin \biggl(\frac{(q_{2}+q_{3})\pi }{2} \biggr)c_{11}(q_{2}+q_{3}) \\ &\hphantom{D_{1}=} {}-\omega _{0}^{q_{1}+q_{3}}\sin \biggl( \frac{(q_{1}+q_{3})\pi }{2} \biggr)c_{22}(q_{1}+q_{3})-k\omega _{0}^{q_{1}+q_{3}} \sin \biggl(\frac{(q_{1}+q_{3})\pi }{2} \biggr) (q_{1}+q_{3}) \\ &\hphantom{D_{1}=} {}-2k\omega _{0}\tau _{0}\cos (2\omega _{0}\tau _{0})c_{11}c_{33}\biggr), \\ &C_{2}=\omega _{0}\biggl(-k\omega _{0}^{q_{1}} \cos \biggl( \frac{q_{1}\pi }{2} \biggr)c_{33}-k\omega _{0}^{q_{3}}\cos \biggl( \frac{q_{3}\pi }{2} \biggr)c_{11}+kc_{11}c_{33}+\omega _{0}^{q_{1}} \cos \biggl(\frac{q_{1}\pi }{2} \biggr)c_{22}c_{33} \\ &\hphantom{C_{2}=} {}+\omega _{0}^{q_{2}}\cos \biggl(\frac{q_{2}\pi }{2} \biggr)c_{11}c_{33}-c_{11}c_{22}c_{33}+k \omega _{0}^{q_{1}+q_{3}}\cos \biggl(\frac{(q_{1}+q_{3})\pi }{2} \biggr) \\ &\hphantom{C_{2}=} {}-\omega _{0}^{q_{1}+q_{2}}\cos \biggl( \frac{(q_{1}+q_{2})\pi }{2} \biggr)c_{33}\biggr)\cos (\tau _{0}\omega _{0} )+ \omega 0\biggl(-k\omega _{0}^{q_{1}}\sin \biggl( \frac{q_{1}\pi }{2} \biggr)c_{33} \\ &\hphantom{C_{2}=} {}-k\omega _{0}^{q_{3}}\sin \biggl(\frac{q_{3}\pi }{2} \biggr)c_{11}+ \omega _{0}^{q_{1}}\sin \biggl( \frac{q_{1}\pi }{2} \biggr)c_{22}c_{33}+ \omega 0^{q_{2}} \sin \biggl(\frac{q_{2}\pi }{2} \biggr)c_{11}c_{33} \\ &\hphantom{C_{2}=} {}+k\omega _{0}^{q_{1}+q_{3}}\sin \biggl( \frac{(q_{1}+q_{3})\pi }{2} \biggr)-\omega _{0}^{q_{1}+q_{2}}\sin \biggl(\frac{(q_{1}+q_{2})\pi }{2} \biggr)c_{33}\biggr)\sin (\tau _{0} \omega _{0} ), \\ &D_{2}=\frac{1}{\omega _{0}}\biggl(\biggl(k\omega _{0}^{q_{3}}c_{11} \biggl(\sin \biggl( \frac{q_{3}\pi }{2} \biggr)\omega _{0}\tau _{0}+\cos \biggl( \frac{q_{3}\pi }{2} \biggr)q_{3}\biggr)\\ &\hphantom{D_{2}=} {}- \omega _{0}^{q_{1}+1}\sin \biggl( \frac{q_{1}\pi }{2} \biggr)c_{33}\tau _{0}(k+c_{22}) \\ &\hphantom{D_{2}=} {}-\omega _{0}^{q_{2}}c_{11}c_{33} \biggl(\sin \biggl( \frac{q_{2}\pi }{2} \biggr)\omega _{0}\tau _{0}+\cos \biggl( \frac{q_{2}\pi }{2} \biggr)q_{2}\biggr)- \omega _{0}^{q_{1}}\cos \biggl( \frac{q_{1}\pi }{2} \biggr)c_{33}q_{1}(c_{22}+k) \\ &\hphantom{D_{2}=} {}-k\omega _{0}^{q_{1}+q_{3}+1}\sin \biggl( \frac{(q_{1}+q_{3})\pi }{2} \biggr)\tau _{0}-k\omega _{0}^{q_{1}+q_{3}} \cos \biggl( \frac{(q_{1}+q_{3})\pi }{2} \biggr) (q_{1}+q3) \\ &\hphantom{D_{2}=} {}+\omega _{0}^{q_{1}+q_{2}+1}\sin \biggl( \frac{(q_{1}+q_{2})\pi }{2} \biggr)c_{33}\tau _{0}\\ &\hphantom{D_{2}=} {}+\omega _{0}^{q_{1}+q_{2}} \cos \biggl(\frac{(q_{1}+q_{2})\pi }{2} \biggr)c_{33}(q_{1}+q_{2}) \biggr) \cos (\tau _{0}\omega _{0} )\biggr) \\ &\hphantom{D_{2}=} {}+\frac{1}{\omega _{0}}\biggl(\biggl(k\omega _{0}^{q_{1}+q_{3}+1} \cos \biggl(\frac{(q_{1}+q_{3})\pi }{2} \biggr)\tau _{0}+\omega _{0}^{q_{1}+q_{2}} \sin \biggl(\frac{(q_{1}+q_{2})\pi }{2} \biggr)c_{33}(q_{1}+q_{2}) \\ &\hphantom{D_{2}=} {}-\omega _{0}^{q_{1}+q_{2}+1}\cos \biggl( \frac{(q_{1}+q_{2})\pi }{2} \biggr)c_{33}\tau _{0}-k\omega _{0}^{q_{1}+q_{3}} \sin \biggl(\frac{(q_{1}+q_{3})\pi }{2} \biggr) (q_{1}+q_{3}) \\ &\hphantom{D_{2}=} {}+k\omega _{0}^{q_{1}}\cos \biggl(\frac{q_{1}\pi }{2} \biggr)c_{33} \tau _{0}(\omega _{0}+c_{22})+ \omega _{0}^{q_{2}}c_{11}c_{33}\biggl(\cos \biggl(\frac{q_{2}\pi }{2} \biggr)\tau _{0}\omega _{0}-\sin \biggl( \frac{q_{2}\pi }{2} \biggr)q_{2}\biggr) \\ &\hphantom{D_{2}=} {}-k\omega _{0}^{q_{3}}c_{11}\biggl(\cos \biggl(\frac{q_{3}\pi }{2} \biggr)\omega _{0}\tau _{0}-\sin \biggl(\frac{q_{3}\pi }{2} \biggr)q_{3}\biggr)- \omega _{0}^{q_{1}}c_{33}\sin \biggl(\frac{q_{1}\pi }{2} \biggr) (c_{22}+k) \\ &\hphantom{D_{2}=} {}-kc_{11}c_{33}\omega _{0}\tau _{0}-c_{11}c_{22}c_{33}\tau _{0}\biggr) \sin (\tau _{0}\omega _{0} )\biggr)+ \frac{1}{\omega _{0}}\biggl(-2kc_{11}c_{33} \omega _{0}\tau _{0}\sin (2\tau _{0}\omega _{0} ) \\ &\hphantom{D_{2}=} {}+\omega _{0}^{q_{1}}\cos \biggl(\frac{q_{1}\pi }{2} \biggr)q_{1}c_{23}c_{32}- \omega _{0}^{q_{3}}\cos \biggl(\frac{q_{3}\pi }{2} \biggr)q_{3}c_{11}(c_{22}+k)\\ &\hphantom{D_{2}=}{}+k \omega _{0}^{q_{1}}c_{33}q_{1}\biggl(\cos \biggl(2\tau _{0}\omega _{0}- \frac{q_{1}\pi }{2} \biggr) \biggr)-2k\omega _{0}^{q_{1}+1}c_{33}\tau _{0}\biggl(\sin \biggl(2\tau _{0} \omega _{0}- \frac{q_{1}\pi }{2} \biggr)\biggr) \\ &\hphantom{D_{2}=} {}-\omega _{0}^{q_{1}+q_{2}+q_{3}} \cos \biggl(\frac{(q_{1}+q_{2}+q_{3})\pi }{2} \biggr) (q_{1}+q_{2}+q_{3}) \\ &\hphantom{D_{2}=} {}+\omega _{0}^{q_{2}+q_{3}}\cos \biggl( \frac{(q_{2}+q_{3})\pi }{2} \biggr)c_{11}(q_{2}+q_{3})\\ &\hphantom{D_{2}=} {}+\omega _{0}^{q_{1}+q_{3}} \cos \biggl(\frac{(q_{1}+q_{3})\pi }{2} \biggr) (q_{1}+q_{3}) (c_{22}+k)\biggr). \end{aligned}$$
Shi, J., Ruan, L. On the reasonability of linearized approximation and Hopf bifurcation control for a fractional-order delay Bhalekar–Gejji chaotic system. Adv Differ Equ 2020, 588 (2020). https://doi.org/10.1186/s13662-020-02908-2
Fractional-order BG system
Hopf bifurcation
Delayed feedback control
Linearization | CommonCrawl |
In $\Delta ABC$, $\overline{DE} \parallel \overline{AB}, CD = 4$ cm, $DA = 10$ cm, and $CE = 6$ cm. What is the number of centimeters in the length of $\overline{CB}$?
[asy]pair A,B,C,D,E;
A = (-2,-4); B = (4,-4); C = (0,0); D = A/3; E = B/3;
draw(E--D--C--B--A--D);
label("A",A,W); label("B",B,dir(0)); label("C",C,N); label("D",D,W);
label("E",E,dir(0));
[/asy]
Since $DE \parallel AB,$ we know that $\angle CDE = \angle CAB$ and $\angle CED = \angle CBA.$ Therefore, by AA similarity, we have $\triangle ABC \sim DEC.$ Then, we find: \begin{align*}
\frac{CB}{CE} &= \frac{CA}{CD} = \frac{CD + DA}{CD}\\
\frac{CB}{6\text{ cm}} &= \frac{4\text{ cm} + 10\text{ cm}}{4\text{ cm}} = \frac{7}{2}\\
CB &= 6\text{cm} \cdot \frac{7}{2} = \boxed{21}\text{ cm}.
\end{align*} | Math Dataset |
\begin{document}
\title[Singular, degenerate, anisotropic PDEs]{Monotonicity formulae and classification results \\ for singular, degenerate, anisotropic PDEs}
\author[Matteo Cozzi, Alberto Farina, Enrico Valdinoci]{ Matteo Cozzi${}^{(1,2)}$ \and Alberto Farina${}^{(1)}$ \and Enrico Valdinoci${}^{(2,3)}$ }
\subjclass[2010]{35J92, 35J93, 35J20.}
\keywords{Wulff shapes, energy monotonicity, rigidity and classification results.}
\thanks{This paper has been supported by ERC grant 277749 ``EPSILON Elliptic Pde's and Symmetry of Interfaces and Layers for Odd Nonlinearities''.}
\maketitle
\date{}
{\scriptsize \begin{center} (1) -- Laboratoire Ami\'enois de Math\'ematique Fondamentale et Appliqu\'ee\\ UMR CNRS 7352, Universit\'e Picardie ``Jules Verne'' \\33 Rue St Leu, 80039 Amiens (France).\\ \end{center} \scriptsize \begin{center} (2) -- Dipartimento di Matematica ``Federigo Enriques''\\ Universit\`a degli studi di Milano,\\ Via Saldini 50, I-20133 Milano (Italy).\\ \end{center} \scriptsize \begin{center} (3) -- Weierstra{\ss} Institut f\"ur Angewandte Analysis und Stochastik\\ Mohrenstra{\ss}e 39, D-10117 Berlin (Germany). \end{center}
\begin{center} E-mail addresses: [email protected], [email protected], [email protected] \end{center} }
\begin{abstract} We consider possibly degenerate and singular elliptic equations in a possibly anisotropic medium.
We obtain monotonicity results for the energy density, rigidity results for the solutions and classification results for the singularity/degeneracy/anisotropy allowed.
As far as we know, these results are new even in the case of non-singular and non-degenerate anisotropic equations. \end{abstract}
\section{Introduction and main results}
\subsection{Description of the model and mathematical setting}
The goal of this paper is to consider partial differential equations in a possibly anisotropic medium.
The interest in the study of anisotropic media is twofold. First, at a purely mathematical level, the lack of isotropy reflects into a rich geometric structure in which the basic objects of investigation do not possess the usual Euclidean properties. Then, from the point of view of concrete applications, anisotropic media naturally arise in the study of crystals, see e.g.~\cite{C84} and the references therein. The interplay between the concrete physical problems and the geometric structures is clearly discussed, for instance, in~\cite{T78, TCH92}. We also refer to Appendix~C in~\cite{CFV14} for a simple physical application.
The equations that we consider in the present paper have a variational structure and they are of elliptic type, though the ellipticity is allowed to be possibly singular or degenerate.
The forcing term only depends on the values of the solution, i.e., in jargon, the equation is quasilinear, and the elliptic operator is constant along the level sets of the solution. This feature imposes strong geometric restrictions on the solution, and the purpose of this paper is to better understand some of these properties.
In this setting we present a variety of results from different perspectives, such as: \begin{itemize} \item a monotonicity formula for the energy functional (i.e., the energy of an anisotropic ball, suitably rescaled, will be shown to be non-decreasing with respect to the size of such ball); \item a rigidity result of Liouville type (namely, if the potential is integrable, then the solution needs to be constant); \item a precise classification of some of the assumptions given in the literature, with concrete examples and some simplifications. \end{itemize}
The formal mathematical notation introduces the solution~$u$ of an anisotropic equation driven by a possibly nonlinear operator. The anisotropic term is encoded into a homogeneous function~$H$, that will be often referred to as ``the anisotropy''. The nonlinearity feature of the operator is given by a function~$B$ (e.g., the function~$B$ can be a power and produce an equation of $p$-Laplace type). Also, the nonlinear source term arises from a potential~$F$.
More precisely, given measurable set~$\Omega \subset \mathbb{R}^n$, with~$n \geqslant 2$, we consider the Wulff type energy functional \begin{equation} \label{Wen} \mathscr{W}_\Omega(u) := \int_\Omega B(H(\nabla u(x))) - F(u(x)) \, dx, \end{equation} and the associated Euler-Lagrange equation \begin{equation} \label{eleq} \frac{\partial}{\partial x_i} \Big( B'( H(\nabla u) ) H_i(\nabla u) \Big) + F'(u) = 0. \end{equation}
Here, the function~$B$ belongs to~$C^{3, \beta}_{\rm loc}((0, +\infty)) \cap C^1([0, +\infty))$, with~$\beta \in (0, 1)$, and is such that~$B(0) = B'(0) = 0$ and \begin{equation} \label{Bpos} B(t), B'(t), B''(t) > 0 \mbox{ for any } t > 0. \end{equation} Moreover,~$H$ is a positive homogeneous function of degree~$1$, of class~$C^{3, \beta}_{\rm loc}(\mathbb{R}^n \setminus \{ 0 \})$ and for which \begin{equation} \label{Hpos} H(\xi) > 0 \mbox{ for any } \xi \in \mathbb{R}^n \setminus \{ 0 \}. \end{equation} Using its homogeneity properties, we infer that~$H$ can be naturally extended to a continuous function on the whole of~$\mathbb{R}^n$ by setting~$H(0) = 0$. Moreover, the forcing term~$F$ is required to be~$C^{2, \beta}_{\rm loc}(\mathbb{R})$.
In addition to these hypotheses we also assume one of the following conditions to hold:
\begin{enumerate}[(A)] \item There exist~$p > 1$,~$\kappa \in [0, 1)$ and positive~$\gamma, \Gamma$ such that, for any~$\xi \in \mathbb{R}^n \setminus \{ 0 \}$,~$\zeta \in \mathbb{R}^n$, $$
\left[ {\mbox{\normalfont Hess}} \,(B \circ H)(\xi) \right]_{i j} \zeta_i \zeta_j \geqslant \gamma {(\kappa + |\xi|)}^{p - 2} {|\zeta|}^2, \label{BHpell} $$ and $$
\sum_{i, j = 1}^n \left| \left[ {\mbox{\normalfont Hess}} \,(B \circ H)(\xi) \right]_{i j} \right| \leqslant \Gamma {(\kappa + |\xi|)}^{p - 2}. $$
\item The composition~$B \circ H$ is of class~$C^{3,\beta}_{\rm loc}(\mathbb{R}^n)$ and for any~$K>0$
there exist a positive constant~$\gamma$ such that, for any~$\xi, \zeta \in \mathbb{R}^n$, with~$|\xi|\leqslant K$, we have \begin{equation*} \left[ {\mbox{\normalfont Hess}} \,(B \circ H)(\xi) \right]_{i j} \zeta_i \zeta_j \geqslant \gamma
\,{|\zeta|}^2. \end{equation*} \end{enumerate}
In~\cite[Appendix~A]{CFV14} we showed that hypothesis~(A) is fulfilled for instance by taking~$B(t) = t^p / p$ together with an~$H$ whose anisotropic unit ball \begin{equation} \label{Hball} B^H_1 = \left\{ \xi \in \mathbb{R}^n : H(\xi) < 1 \right\}, \end{equation} is uniformly convex, i.e. such that the principal curvatures of its boundary are bounded away from zero. Every anisotropy~$H$ having uniformly convex unit ball will be called \emph{uniformly elliptic}. We remark that, since the second fundamental form of~$\partial B_1^H$ at a point~$\xi \in \partial B_1^H$ is given by $$
\mathrm{I\!I}_\xi(\zeta, \upsilon) = \frac{H_{i j}(\xi) \zeta_i \upsilon_j }{|\nabla H(\xi)|} \mbox{ for any } \zeta, \upsilon \in \nabla H(\xi)^\perp, $$ as can bee seen for instance in~\cite[Appendix~A]{CFV14}, and being~$\partial B_1^H$ compact, the uniform ellipticity of~$H$ is equivalent to ask \begin{equation} \label{Huniell}
H_{i j}(\xi) \zeta_i \zeta_j \geqslant \lambda |\zeta|^2 \mbox{ for any } \xi \in \partial B_1^H, \, \zeta \in \nabla H(\xi)^\perp, \end{equation} for some~$\lambda > 0$. Any positive~$\lambda$ for which~\eqref{Huniell} is satisfied will be said to be an \emph{ellipticity constant} for~$H$. Notice that, by homogeneity,~\eqref{Huniell} actually extends to \begin{equation} \label{Huniell2}
H_{i j}(\xi) \zeta_i \zeta_j \geqslant \lambda |\xi|^{- 1} |\zeta|^2 \mbox{ for any } \xi \in \mathbb{R}^n \setminus \{ 0 \}, \, \zeta \in \nabla H(\xi)^\perp. \end{equation}
We associate to our solution~$u \in L^\infty(\mathbb{R}^n)$ the finite quantities $$ u^* := \sup_{\mathbb{R}^n} u \ {\mbox{ and }} \ u_* := \inf_{\mathbb{R}^n} u,$$ and the gauge \begin{equation} \label{cudef} c_u := \sup \left\{ F(t) : t \in \left[ u_*, u^* \right] \right\}. \end{equation} Finally, for~$t \in \mathbb{R}$ we set \begin{equation} \label{Gdef} G(t) := c_u - F(t). \end{equation} Notice that such~$G$ is a non-negative function on the range of~$u$ and that putting it in place of~$-F$ in~\eqref{Wen} does not change at all the setting, once~$u$ is fixed.
In the forthcoming Subsections~\ref{XX1}--\ref{XX2}, we give precise statements of our main results. We point out that, to the best of our knowledge, these results are new even in the case in which~$B(t)=t^2/2$ (i.e. even in the case in which the elliptic operator is non-singular and non-degenerate).
\subsection{A monotonicity formula}\label{XX1}
Monotonicity formulae are a classical topic in geometric variational analysis. Roughly speaking, the idea of monotonicity formulae is that a suitably rescaled energy functional in a ball possesses some monotonicity properties with respect to the radius of the ball (in our case, the situation is geometrically more complicated, since the ball is non-Euclidean).
Of course, monotonicity formulae are important, since they provide a quantitative information on the energy of the problem; moreover, they often provide additional information on the asymptotic behaviour of the solutions, also in connection with blow-up and blow-down limits, and they play a special role in rigidity and classification results.
One of the main results of the present paper consists in a monotonicity formula for a suitable rescaled version of the functional~\eqref{Wen}, over the family of sets indexed by~$R > 0$, \begin{equation} \label{Wball} W_R^H = W_R := \left\{ x \in \mathbb{R}^n : H^*(x) < R \right\}, \end{equation} where, for~$x \in \mathbb{R}^n$, \begin{equation} \label{H*def} H^*(x) := \sup_{\xi \in S^{n - 1}} \frac{\langle x, \xi \rangle}{H(\xi)}, \end{equation} is the dual function of~$H$. Notice that~$H^*$ is a positive homogeneous function of degree~$1$ and that it is at least of class~$C^2(\mathbb{R}^n \setminus \{ 0 \})$, as showed in Lemma~\ref{H*reg} below. The set~$W_R$ is the so-called Wulff shape of radius~$R$ associated to~$H$. We refer to~\cite{CS09, WX11} for some basic properties of this set and to~\cite{T78} for a nice geometrical construction. The precise statement is given by
\begin{theorem} \label{monformthm} Assume that one of the following conditions to be valid: \begin{enumerate}[(i)] \item Assumption~(A) holds and~$u \in L^\infty(\mathbb{R}^n) \cap W_{\rm loc}^{1, p}(\mathbb{R}^n)$ is a weak solution of~\eqref{eleq} in~$\mathbb{R}^n$; \item Assumption~(B) holds and $u \in W^{1, \infty}(\mathbb{R}^n)$ weakly solves~\eqref{eleq} in~$\mathbb{R}^n$. \end{enumerate} If~$(i)$ is in force, assume in addition that~$H$ satisfies, for any~$\xi, x \in \mathbb{R}^n$, \begin{equation} \label{FKweak} {\mbox{\normalfont sgn}} \langle H(\xi) \nabla H(\xi), H^*(x) \nabla H^*(x) \rangle = {\mbox{\normalfont sgn}} \langle \xi, x \rangle. \end{equation} Then, the rescaled energy defined by \begin{equation} \label{scaledWen} \mathscr{E}(u; R) = \mathscr{E}(R) := \frac{1}{R^{n - 1}} \int_{W_R} B(H(\nabla u(x))) + G(u(x)) \, dx, \end{equation} for any~$R > 0$, is monotone non-decreasing. \end{theorem}
Observe that when~$B(t)=t^2/2$ and~$H(x)=|x|$ (i.e. when the operator is simply the Laplacian and the equation is isotropic), then the result of Theorem~\ref{monformthm}
reduces to the classical monotonicity formula proved in~\cite{M89}. Then, if~$H(x)=|x|$, the results of~\cite{M89} were extended to the non-linear case in~\cite{CGS94}. Differently from the existing literature, here we introduce the presence of a general non-Euclidean anisotropy~$H$ (also, we remove an unnecessary assumption on the sign of~$F$).
We remark that the anisotropic term in the monotonicity formula provides a number of geometric complications. Indeed, in our case, the unit ball~$B_1^H$ is not Euclidean and it does not coincide with its dual ball~$W_1^H$, and a point on the unit sphere does not coincide in general with the normal to the sphere.
Also, we mention that Theorem~\ref{monformthm} relies on the pointwise gradient estimate proved in~\cite[Theorem~1.1]{CFV14}.
\subsection{Geometric conditions on the anisotropy and classification results}
In the statement of the monotonicity formula the new condition~\eqref{FKweak} is assumed on~$H$. Here, we plan to shed some light on its origin and to better understand its implications.
First, we point out that this assumption comes as a weaker form of the more restrictive \begin{equation} \label{FK} \langle H(\xi) \nabla H(\xi), H^*(x) \nabla H^*(x) \rangle = \langle \xi, x \rangle, \end{equation} for any~$\xi, x \in \mathbb{R}^n$. To the authors' knowledge, this latter condition has been first introduced in~\cite{FK09} to recover the validity of the mean value property for~$Q$-harmonic functions, that are the solutions of the equation \begin{equation} \label{Qu=0} Q u := \frac{\partial}{\partial x_i} \Big( H(\nabla u) H_i(\nabla u) \Big) = 0. \end{equation} Notice that such solutions are the counterparts of harmonic functions in the anisotropic framework and that equation~\eqref{Qu=0} is a particular case of our setting by taking~$B(t) = t^2/2$ and~$F = 0$.
Examples of homogeneous functions~$H$ for which~\eqref{FK} is valid are the norms displayed in~\eqref{HM}, as showed by Lemma~\ref{FKneclem}. Note that we do not assume~\eqref{FKweak} in case~$(ii)$ of Theorem~\ref{monformthm}. Indeed, hypothesis~(B) forces $H$ to be of the form~\eqref{HM}, as shown in~\cite[Appendix~B]{CFV14} (this can also be deduced from the forthcoming Theorem~\ref{T-RES}). In the next result we emphasize that anisotropies as the one in~\eqref{HM} are actually the~\emph{only} ones which satisfy~\eqref{FK}.
\begin{theorem} \label{FKcharprop} Let~$H \in C^1(\mathbb{R}^n \setminus \{ 0 \})$ be a positive homogeneous function of degree~$1$ satisfying~\eqref{Hpos}. Assume that its unit ball~$B_1^H$, as defined by~\eqref{Hball}, is strictly convex. Then, condition~\eqref{FK} is equivalent to asking~$H$ to be of the form \begin{equation} \label{HM} H_M(\xi) = \sqrt{\langle M \xi, \xi \rangle}, \end{equation} for some symmetric and positive definite matrix~$M \in {\mbox{\normalfont Mat}}_n(\mathbb{R})$. \end{theorem}
{F}rom Theorem~\ref{FKcharprop}, it follows that assumption~\eqref{FK} imposes some severe restrictions on the geometric structure of its unit ball, which is always an Euclidean ellipsoid. A natural question is therefore to understand in which sense our condition~\eqref{FKweak} is more general. For this scope, we will discuss condition~\eqref{FKweak} in detail, by making concrete examples and obtaining a complete characterization in the plane. Roughly speaking, the unit ball in the plane under condition~\eqref{FKweak} can be constructed by considering a curve in the first quadrant that satisfies a suitable, explicit differential inequality, and then~\emph{reflecting} this curve in the other quadrants (of course, if higher regularity on the ball is required, this gives further conditions on the derivatives of the curve at the reflection points). The detailed characterization of condition~\eqref{FKweak} in the plane is given by the following technical but operational result.
\begin{proposition} \label{2Dprop-i} Let~$r: [0, \pi/2] \to (0, +\infty)$ be a given~$C^2$ function satisfying \begin{equation} \label{rcond1-i} r(\theta) r''(\theta) < 2 r'(\theta)^2 + r(\theta)^2 \mbox{ for a.a. } \theta \in \left[ 0, \frac{\pi}{2} \right], \end{equation} and \begin{equation} \label{rcond2-i} r(0) = 1, \qquad r(\pi/2) = r^*, \qquad r'(0) = r'(\pi/2) = 0, \end{equation} for some~$r^* \geqslant 1$. Consider the~$\pi$-periodic function~$\widetilde{r}: \mathbb{R} \to (0, +\infty)$ defined on~$[0, \pi]$ by \begin{equation*} \widetilde{r}(\theta) := \begin{dcases} r(\theta) & \quad {\mbox{if }} 0 \leqslant \theta \leqslant \frac{\pi}{2}, \\ \frac{r^* \sqrt{r(\tau^{-1}(\theta))^2 + r'(\tau^{-1}(\theta))^2}}{ r(\tau^{-1}(\theta))^2} & \quad {\mbox{if }} \frac{\pi}{2} \leqslant \theta \leqslant \pi, \end{dcases} \end{equation*} where~$\tau: [0, \pi/2] \to [\pi/2, \pi]$ is the bijective map given by \begin{equation*} \tau(\eta) = \frac{\pi}{2} + \eta - \arctan \frac{r'(\eta)}{r(\eta)}. \end{equation*} Then,~$\widetilde{r}$ is of class~$C^1(\mathbb{R})$, the set \begin{equation} \label{Cdef-i} \left\{ (\rho \cos \theta, \rho \sin \theta) : \rho \in [0, \widetilde{r}(\theta)), \, \theta \in [0, 2 \pi] \right\}, \end{equation} is strictly convex and its supporting function $$ \widetilde{H}(\rho \cos \theta, \rho \sin \theta) := \frac{\rho}{\widetilde{r}(\theta)},, $$ defined for~$\rho \geqslant 0$ and~$\theta \in [0, 2 \pi]$, satisfies~\eqref{FKweak}.
Furthermore, up to a rotation and a homothety of the plane~$\mathbb{R}^2$, any even positive $1$-homogeneous function~$H \in C^2(\mathbb{R}^2 \setminus \{ 0 \})$ satisfying~\eqref{Hpos}, having strictly convex unit ball~$B_1^H$ and for which condition~\eqref{FKweak} holds true is such that~$B_1^H$ is of the form~\eqref{Cdef-i}, for some positive~$r \in C^2([0, \pi/2])$ satisfying~\eqref{rcond1-i} and~\eqref{rcond2-i}.
In addition, if~$H \in C^{3,\alpha}_{\rm loc}(\mathbb{R}^2 \setminus \{ 0 \})$, for some~$\alpha\in(0,1]$, we have that~$H$ is uniformly elliptic and satisfies condition~\eqref{FKweak} if and only if~$r \in C^{3, \alpha}([0, \pi / 2])$, inequality~\eqref{rcond1-i} is satisfied at any~$\theta \in [0, \pi / 2]$ and $$ r'' \left( \frac{\pi}{2} \right) = - \frac{r^* r''(0)}{1 - r''(0)}, \qquad r''' \left( \frac{\pi}{2} \right) = - \frac{r^* r'''(0)}{(1 - r''(0))^3}, $$ hold along with~\eqref{rcond2-i}. \end{proposition}
With this characterization, it is easy to construct examples satisfying condition~\eqref{FKweak} whose corresponding ball is not an Euclidean ellipsoid, see Remark~\ref{final}.
\subsection{A rigidity result} As an application of Theorem~\ref{monformthm} we have the following Liouville-type result.
\begin{theorem} \label{liouthm} Let~$H$ and~$u$ be as in Theorem~\ref{monformthm}. If \begin{equation} \label{Gugrowth} \int_{W_R} G(u(x)) \, dx = o(R^{n - 1}) \mbox{ as } R \rightarrow +\infty, \end{equation} then~$u$ is constant.
In particular, if~$G(u) \in L^1(\mathbb{R}^n)$, then~$u$ is constant. \end{theorem}
We remark that Theorem~\ref{liouthm} is a sort of rigidity result. The condition that~$G(u)$ has finite mass - or, more generally, that the mass has controlled growth - may be seen as a prescription of the values of the solution at infinity (at least, in a suitably averaged sense): the result of Theorem~\ref{liouthm} gives that the only solution that can satisfy such prescription is the trivial one. In this spirit, Theorem~\ref{liouthm} may be seen as a variant of the classical Liouville Theorem for harmonic functions (set here in a nonlinear, anisotropic, singular or degenerate framework).
\subsection{Equivalent conditions}\label{XX2} We remark that the assumptions in~(A) and~(B) that we made on the anisotropic and nonlinear part of the operator are somehow classical in the literature, see e.g.~\cite{CGS94, CFV14} and the references therein (roughly speaking, these conditions are the necessary ones to obtain some regularity of the solutions using, or adapting, the elliptic regularity theory).
In spite of their classical flavour, we think that in some cases these conditions can be made more explicit or more concrete. For this, in this paper we provide some equivalent characterizations. In particular, we will observe that condition~(B) puts some important restrictions on the structure of the ambient medium, due to the regularity requirement on the composition~$B \circ H$. More precisely, the following result holds true:
\begin{theorem}\label{T-RES} Assumption~(A) is equivalent to \begin{enumerate}[(A)\ensuremath{'}] \item There exist~$p > 1$,~$\bar{\kappa} \in [0, 1)$ and positive~$\bar{\gamma}, \bar{\Gamma}, \lambda$ such that $$ H \mbox{ is uniformly elliptic with constant~$\lambda$}, $$ and \begin{align*} \bar{\gamma} (\bar{\kappa} + t)^{p - 2} t \leqslant & B'(t) \leqslant \bar{\Gamma} (\bar{\kappa} + t)^{p - 2} t, \\ \bar{\gamma} (\bar{\kappa} + t)^{p - 2} \leqslant & B''(t) \leqslant \bar{\Gamma} (\bar{\kappa} + t)^{p - 2}, \end{align*} for any~$t > 0$. \end{enumerate} Assumption~(B) is equivalent to \begin{enumerate}[(B)\ensuremath{'}] \item The function~$B$ is of class~$C^{3, \beta}_{\rm loc}([0, +\infty))$, with~$B'''(0) = 0$, \begin{equation} \label{B''0pos} B''(0) > 0, \end{equation} and~$H$ is of the type~\eqref{HM}, for some~$M \in {\mbox{\normalfont Mat}}_n(\mathbb{R})$ symmetric and positive definite. \end{enumerate} \end{theorem}
\subsection{Organization of the paper} The rest of the paper is organized as follows.
In Section~\ref{auxsec} we gather several auxiliary lemmata, most of which are related to basic properties of the anisotropy~$H$. At the end of the section we also briefly comment on the regularity of the solutions of~\eqref{eleq}.
In Section~\ref{ABsec} we establish the equivalence of the two sets of conditions~(A)-(B) and~(A)\ensuremath{'}-(B)\ensuremath{'}, thus proving Theorem~\ref{T-RES}.
The proof of the main result of the paper, Theorem~\ref{monformthm}, is the content of Section~\ref{monformsec}. In the subsequent Section~\ref{liousec} we then deduce Theorem~\ref{liouthm} as a corollary of the monotonicity formula.
The last two sections deal with the characterizations of conditions~\eqref{FK} and~\eqref{FKweak}. In Section~\ref{char1sec} we address Theorem~\ref{FKcharprop}, while the following Section~\ref{char2sec} is devoted to the proof of Proposition~\ref{2Dprop-i}.
\section{Some auxiliary results} \label{auxsec}
We collect here some preliminary results which will be abundantly used in the forthcoming sections. Most of them are very well known results, so that we will not comment much on their proofs. Nevertheless, precise references will be given.
Every result in this section is clearly meant to be applied to the functions~$H$ and~$B$ above introduced. However, when possible we state them under slightly lighter hypotheses.
The first lemma provides three useful identities for the derivatives of positive homogeneous functions. We recall that, given~$d \in \mathbb{R}$, a function~$H: \mathbb{R}^n \setminus \{ 0 \} \to \mathbb{R}$ is said to be positive homogeneous of degree~$d$ if $$
H(t \xi) = |t|^d H(\xi) \mbox{ for any } t > 0, \, \xi \in \mathbb{R}^n \setminus \{ 0 \}. $$
\begin{lemma} \label{homain} If~$H\in C^3(\mathbb{R}^n\setminus \{0\})$ is positive homogeneous of degree~$1$, we have that \begin{align} \label{i} H_i(\xi) \xi_i & = H(\xi),\\ \label{ii} H_{ij}(\xi) \xi_i & = 0,\\ \label{iii} H_{ijk}(\xi) \xi_i & = -H_{jk}(\xi). \end{align} \end{lemma}
We refer to the Appendix of~\cite{FV14} for a proof. The second result of this section deals with the regularity up to the origin of both the anisotropy~$H$ and the composition~$B \circ H$.
\begin{lemma} \label{derBH0} Let~$H\in C^1(\mathbb{R}^n\setminus\{0\})$ be a positive homogeneous function of degree $d$ admitting non-negative values and~$B \in C^1([0, +\infty))$, with~$B(0) = 0$. Assume that either $d > 1$ or $d = 1$ and $B'(0) = 0$. Then~$H$ can be extended by setting~$H(0):=0$ to a continuous function, such that~$B \circ H \in C^1(\mathbb{R}^n)$ and $$ \partial_i (B \circ H)(0) = 0 = \lim_{x \rightarrow 0} B'(H(x)) H_i(x). $$ \end{lemma}
A proof of Lemma~\ref{derBH0} can be found in~\cite[Section~2]{CFV14}. Next is a lemma which gathers some results on~$H$ and its dual~$H^*$.
\begin{lemma} \label{H*reg} Let~$B \in C^2((0, +\infty))$ and~$H \in C^2(\mathbb{R}^n \setminus \{ 0 \})$. Assume~$B$ to satisfy~\eqref{Bpos}, the function~$H$ to be positive homogeneous of degree~$1$ satisfying~\eqref{Hpos} and~${\mbox{\normalfont Hess}}(B \circ H)$ to be positive definite in~$\mathbb{R}^n \setminus \{ 0 \}$. Then, the ball~$B_1^H$ defined by~\eqref{Hball} is strictly convex.\\ Furthermore, the dual function~$H^*$ defined by~\eqref{H*def} is of class~$C^2(\mathbb{R}^n \setminus \{ 0 \})$, the formulae \begin{equation} \label{CS2} H^*(\nabla H(\xi)) = H(\nabla H^*(\xi)) = 1, \end{equation} hold true for any~$\xi \in \mathbb{R}^n \setminus \{ 0 \}$ and the map~$\Psi_H: \mathbb{R}^n \to \mathbb{R}^n$, defined by setting $$ \Psi_H(\xi) := H(\xi) \nabla H(\xi), $$ for any~$\xi \in \mathbb{R}^n$, is a global homeomorphism of~$\mathbb{R}^n$, with inverse~$\Psi_{H^*}$. \end{lemma}
\begin{proof} Notice that~$B \circ H \in C^2(\mathbb{R}^n \setminus \{ 0 \}) \cap C^0(\mathbb{R}^n)$ and its Hessian is positive definite in~$\mathbb{R}^n \setminus \{ 0 \}$. Hence~$B \circ H$ is strictly convex in the whole of~$\mathbb{R}^n$. Moreover, being~$B'$ positive by~\eqref{Bpos}, the ball~$B_1^H$ is also a sublevel set of~$B \circ H$ and thus strictly convex.
The other claims are valid by virtue of~\cite[Lemma~3.1]{CS09}. Note that~$H$ is assumed to be even in~\cite{CS09}, but this assumption is not used in the proof of Lemma~3.1 there. Hence this result is valid also in our setting.
Moreover, $H^*$ is of class~$C^2$ outside of the origin, since so is the diffeomorphism~$\Psi_H$. \end{proof}
Next we see that if~$B$ is of the type of the regularized~$p$-Laplacian, i.e. when~(A)\ensuremath{'} holds with~$\bar{\kappa} > 0$, then it is close to being quadratic. In particular, we show that~$B$ can modified far from the origin to make it satisfy~(A)\ensuremath{'} with~$p = 2$. We will need such a trick in Section~\ref{monformsec} in order to overcome a technical difficulty along the proof of Proposition~\ref{uepsexis}.
\begin{lemma} \label{Bcaplem} Let~$B \in C^2((0, +\infty)) \cap C^1([0, +\infty))$ be a function satisfying both~\eqref{Bpos} and~$B(0) = B'(0) = 0$. Assume in addition that~$B$ satisfies the inequalities displayed in~(A)\ensuremath{'} for some~$p > 1$ and~$\bar{\kappa} > 0$. Let~$M > 0$ be fixed and define \begin{equation} \label{Bcapdef} \hat{B}(t) := \begin{cases} B(t), & \qquad \mbox{if } t \in [0, M), \\ a (t - M)^2 + b (t - M) + c, & \qquad \mbox{if } t \geqslant M, \end{cases} \end{equation} where~$a = B''(M) / 2$,~$b = B'(M)$ and~$c = B(M)$. Then,~$\hat{B} \in C^2((0, +\infty)) \cap C^1([0, +\infty))$ and it satisfies the inequalities in~(A)\ensuremath{'} with~$p = 2$. \end{lemma}
\begin{proof} The function~$\hat{B}$ is of class~$C^2((0, +\infty)) \cap C^1([0, +\infty))$ by construction. Moreover, the estimates concerning~$\hat{B}'$ in~(A)\ensuremath{'} result from the analogous for~$\hat{B}''$ by integration, since~$\hat{B}'(0) = 0$. Thus, we only need to check that there exist~$\hat{\Gamma} \geqslant \hat{\gamma} > 0$ for which \begin{equation*} \hat{\gamma} \leqslant \hat{B}''(t) \leqslant \hat{\Gamma} \mbox{ for any } t > 0. \end{equation*} Notice that when~$t \geqslant M$ this fact is obviously true. On the other hand, if~$t \in (0, M)$ we compute $$ \hat{B}''(t) = B''(t) \geqslant \bar{\gamma} (\bar{\kappa} + t)^{p - 2} \geqslant \bar{\gamma} \min \left\{ \bar{\kappa}^{p - 2}, (\bar{\kappa} + M)^{p - 2} \right\} =: \hat{\gamma}, $$ and $$ \hat{B}''(t) = B''(t) \leqslant \bar{\Gamma} (\bar{\kappa} + t)^{p - 2} \leqslant \bar{\Gamma} \max \left\{ \bar{\kappa}^{p - 2}, (\bar{\kappa} + M)^{p - 2} \right\} =: \hat{\Gamma}. $$ This finishes the proof. \end{proof}
To conclude the section, we comment on the regularity of bounded weak solutions to~\eqref{eleq}. The result is an application of the standard interior degenerate (or non-degenerate) elliptic regularity theory of~\cite{LU68},~\cite{DiB83} and~\cite{T84}. See~\cite[Section~3]{CFV14} for more details.
\begin{proposition} \label{ureg} Let~$u$ be as in Theorem~\ref{monformthm}. Then,~$u \in C_{\rm loc}^{1, \alpha}(\mathbb{R}^n) \cap C^3\left( \{ \nabla u \ne 0 \} \right)$, for some~$\alpha \in (0, 1)$, and~$\nabla u \in L^\infty(\mathbb{R}^n)$.
Moreover, if~$(ii)$ of Theorem~\ref{monformthm} is in force, then~$u$ is of class~$C_{\rm loc}^{3, \alpha}(\mathbb{R}^n)$. \end{proposition}
\section{On the equivalence between assumptions~(A)-(B) and~(A)\ensuremath{'}-(B)\ensuremath{'}} \label{ABsec}
In this second preliminary section we prove the equivalence of the two couples of structural conditions stated in the introduction. We show that both~(A) and~(B) respectively boil down to the simpler and more operational~(A)\ensuremath{'} and~(B)\ensuremath{'}. First, we have
\begin{proposition} \label{(A)char} Let~$B \in C^2((0, +\infty))$ be a function satisfying~\eqref{Bpos} and~$H \in C^2(\mathbb{R}^n \setminus \{ 0 \})$ be positive homogeneous of degree~$1$, such that~\eqref{Hpos} is true. Then, assumptions~(A) and~(A)\ensuremath{'} are equivalent. Moreover, we may take \begin{equation} \label{kappabarkappa} \bar{\kappa} = \kappa, \end{equation} and the constants~$\bar{\gamma}, \bar{\Gamma}, \lambda$ and~$\gamma, \Gamma$ to be independent of~$\kappa$. \end{proposition}
\begin{proof} First of all, denote with~$C \geqslant 1$ a constant for which $$
C^{-1} |\xi| \leqslant H(\xi) \leqslant C |\xi|, \mbox{ } |\nabla H(\xi)| \leqslant C \mbox{ and } \left| {\mbox{\normalfont Hess}}(H) \right| \leqslant C |\xi|^{-1}, $$ hold for any~$\xi \in \mathbb{R}^n \setminus \{ 0 \}$. Then, observe that the ellipticity and growth conditions displayed in~(A) are respectively equivalent to \begin{align}
\label{ellexp} \left[ B''(H(\xi)) H_i(\xi) H_j(\xi) + B'(H(\xi)) H_{i j}(\xi) \right] \zeta_i \zeta_j & \geqslant \gamma \left( \kappa + |\xi| \right)^{p - 2} |\zeta|^2, \\
\label{groexp} \sum_{i, j = 1}^n \left| B''(H(\xi)) H_i(\xi) H_j(\xi) + B'(H(\xi)) H_{i j}(\xi) \right| & \leqslant \Gamma \left( \kappa + |\xi| \right)^{p - 2}, \end{align} for any~$\xi \in \mathbb{R}^n \setminus \{ 0 \}$ and~$\zeta \in \mathbb{R}^n$.
We start by showing that~(A)\ensuremath{'} implies~(A), in its above mentioned equivalent form. First, we check that~\eqref{groexp} is true. We have \begin{align*}
\sum_{i, j = 1}^n \left| B''(H(\xi)) H_i(\xi) H_j(\xi) + B'(H(\xi)) H_{i j}(\xi) \right| & \leqslant \bar{\Gamma} (\bar{\kappa} + H(\xi))^{p - 2} \left[ C^2 + C H(\xi) |\xi|^{-1} \right] \\
& \leqslant 2 \bar{\Gamma} C^2 (\bar{\kappa} + c_* |\xi|)^{p - 2} \\
& = 2 \bar{\Gamma} C^2 c_*^{p - 2} (c_*^{-1} \bar{\kappa} + |\xi|)^{p - 2}, \end{align*} with \begin{equation} \label{c*def} c_* := \begin{cases} C & \qquad \mbox{if } p \geqslant 2, \\ 1/C & \qquad \mbox{if } 1 < p < 2. \end{cases} \end{equation} The proof of~\eqref{ellexp} is a bit more involved. We write \begin{equation} \label{zetadec} \zeta = \alpha \xi + \eta, \end{equation}
for some~$\alpha \in \mathbb{R}$ and~$\eta \in \nabla H(\xi)^\perp$. We stress that~$\xi$ and~$\nabla H(\xi)^\perp$ span the whole~$\mathbb{R}^n$ in view of~\eqref{i}. Thus, decomposition~\eqref{zetadec} is admissible. We distinguish between the two cases:~$2 |\alpha \xi| \leqslant |\zeta|$ and~$2 |\alpha \xi| > |\zeta|$. In the first situation, we have $$
|\eta|^2 = |\zeta - \alpha \xi|^2 = |\zeta|^2 - 2 \alpha \langle \zeta, \xi \rangle + \alpha^2 |\xi|^2 \geqslant (|\zeta| - |\alpha \xi|)^2 \geqslant \frac{|\zeta|^2}{4}. $$ Therefore, by applying~\eqref{i},~\eqref{ii} and~\eqref{Huniell2}, we get \begin{align*} & \left[ B''(H(\xi)) H_i(\xi) H_j(\xi) + B'(H(\xi)) H_{i j}(\xi) \right] \zeta_i \zeta_j \\
& \qquad = B''(H(\xi)) (H_i(\xi) \zeta_i)^2 + B'(H(\xi)) H_{i j}(\xi) \eta_i \eta_j \geqslant 0 + \bar{\gamma} (\bar{\kappa} + H(\xi))^{p - 2} H(\xi) \lambda |\xi|^{-1} |\eta|^2 \\
& \qquad \geqslant 4^{-1} \bar{\gamma} \lambda C^{-1} (\bar{\kappa} + c_*^{-1} |\xi|)^{p - 2} |\zeta|^2 = 4^{-1} \bar{\gamma} \lambda C^{-1} c_*^{2 - p} (c_* \bar{\kappa} + |\xi|)^{p - 2} |\zeta|^2 \\
& \qquad \geqslant 4^{-1} \bar{\gamma} \lambda C^{-1} c_*^{2 - p} (c_*^{-1} \bar{\kappa} + |\xi|)^{p - 2} |\zeta|^2, \end{align*} where in last line we recognized that, for every~$p > 1$, \begin{equation} \label{CC-1} (c_* \bar{\kappa} + s)^{p - 2} \geqslant (c_*^{-1} \bar{\kappa} + s)^{p - 2} \mbox{ for any } s > 0, \end{equation} being~$C \geqslant 1$. On the other hand, if the opposite inequality occurs we deduce that, by~\eqref{i}, $$
\left| \langle \nabla H(\xi), \zeta \rangle \right| = \left| \langle \nabla H(\xi), \alpha \xi + \eta \rangle \right| = |\alpha| H(\xi) \geqslant \frac{|\alpha| |\xi|}{C} \geqslant \frac{|\zeta|}{2 C}, $$ so that, we compute \begin{align*} & \left[ B''(H(\xi)) H_i(\xi) H_j(\xi) + B'(H(\xi)) H_{i j}(\xi) \right] \zeta_i \zeta_j \\
& \qquad = B''(H(\xi)) (H_i(\xi) \zeta_i)^2 + B'(H(\xi)) H_{i j}(\xi) \eta_i \eta_j \geqslant \bar{\gamma} (\bar{\kappa} + H(\xi))^{p - 2} (2 C)^{-2} |\zeta|^2 + 0 \\
& \qquad \geqslant 4^{-1} \bar{\gamma} C^{-2} (\bar{\kappa} + c_*^{-1} |\xi|)^{p - 2} |\zeta|^2 = 4^{-1} \bar{\gamma} C^{-2} c_*^{2 - p} (c_* \bar{\kappa} + |\xi|)^{p - 2} |\zeta|^2 \\
& \qquad \geqslant 4^{-1} \bar{\gamma} C^{-2} c_*^{2 - p} (c_*^{-1} \bar{\kappa} + |\xi|)^{p - 2} |\zeta|^2, \end{align*} and thus the proof of~\eqref{ellexp} is complete.
Now, we focus on the opposite implication, i.e. that~(A) implies~(A)\ensuremath{'}. Let~$t > 0$ and take~$\xi \in \mathbb{R}^n \setminus \{ 0 \}$ such that~$t = H(\xi)$. Plugging~$\zeta = \xi$ in~\eqref{ellexp}, by~\eqref{i} and~\eqref{ii} we obtain $$
\gamma \left( \kappa + |\xi| \right)^{p - 2} |\xi|^2 \leqslant \left[ B''(t) H_i(\xi) H_j(\xi) + B'(t) H_{i j}(\xi) \right] \xi_i \xi_j = B''(t) H^2(\xi), $$ and hence that $$ B''(t) \geqslant \gamma C^{-2} (\kappa + c_*^{-1} t)^{p - 2} = \gamma C^{-2} c_*^{2 - p} (c_* \kappa + t)^{p - 2}. $$ On the other hand, the choice~$\zeta \in \nabla H(\xi)^\perp$ in~\eqref{ellexp} leads to \begin{equation} \label{tech1} \begin{aligned}
\gamma \left( \kappa + |\xi| \right)^{p - 2} |\zeta|^2 & \leqslant \left[ B''(t) H_i(\xi) H_j(\xi) + B'(t) H_{i j}(\xi) \right] \zeta_i \zeta_j = B'(t) H_{i j}(\xi) \zeta_i \zeta_j \\
& \leqslant C B'(t) |\xi|^{-1} |\zeta|^2 \leqslant C^2 B'(t) t^{-1} |\zeta|^2. \end{aligned} \end{equation} As before we deduce $$ B'(t) \geqslant \gamma C^{-2} c_*^{2 - p} (c_* \kappa + t)^{p - 2} t. $$ The remaining inequalities involving~$B'$ and~$B''$ in~(A)\ensuremath{'} can be similarly deduced from~\eqref{groexp}. Indeed, notice that~\eqref{i} and~\eqref{ii} respectively yield \begin{align*} H_1(e_1) & = \langle \nabla H(e_1), e_1 \rangle = H(e_1), \\ H_{1 1} (e_1) & = \langle \nabla^2 H(e_1) e_1, e_1 \rangle = 0. \end{align*} Hence, if we take~$\mu > 0$ such that~$t = H(\mu e_1)$, setting~$\xi = \mu e_1$ in~\eqref{groexp} we get \begin{align*}
\Gamma \left( \kappa + |\xi| \right)^{p - 2}& \geqslant \sum_{i, j = 1}^n \left| B''(t) H_i(\mu e_1) H_j(\mu e_1) + B'(t) H_{i j}(\mu e_1) \right| \\ & \geqslant B''(t) H_1(e_1) H_1(e_1) + B'(t) \mu^{-1} H_{1 1}(e_1) \\ & = B''(t) H^2(e_1). \end{align*} Consequently, recalling~\eqref{CC-1} we obtain $$ B''(t) \leqslant \Gamma C^2 (\kappa + c_* t)^{p - 2} = \Gamma C^2 c_*^{p - 2} (c_*^{-1} \kappa + t)^{p - 2} \leqslant \Gamma C^2 c_*^{p - 2} (c_* \kappa + t)^{p - 2}. $$ As a byproduct, the previous inequality implies in particular that $$ B'(1) = \int_0^1 B''(t) \, dt \leqslant \frac{\Gamma C^2 c_*^{p - 2}}{p - 1} (c_* \kappa + 1)^{p - 1}. $$ Hence, by taking~$t = 1$ in the first line of~\eqref{tech1} we see that~$H$ is uniformly elliptic, with constant \begin{equation} \label{lambdadef} \lambda = \frac{(p - 1) c_*^{2(2 - p)} \gamma}{2 C^2 (c_* + 1) \Gamma}. \end{equation} Note that we took advantage of the fact that~$\kappa < 1$, along with definition~\eqref{c*def}, to deduce this bound. Finally, the growth condition on~$B'$ can be obtained as follows. Select~$\xi \in \mathbb{R}^n \setminus \{ 0 \}$ in a way that~$e_1 \in \nabla H(\xi)^\perp$ and~$H(\xi) = t$. This can be easily done for instance by taking~$\xi = t \nabla H^*(e_2)$. Indeed, by Lemma~\ref{H*reg}, together with the homogeneity properties of~$H$ and~$\nabla H$, we have \begin{align*} 0 & = \langle e_2, e_1 \rangle = H(H^*(e_2) \nabla H^*(e_2)) \langle \nabla H(H^*(e_2) \nabla H^*(e_2)), e_1 \rangle \\ & = H^*(e_2) H(\nabla H^*(e_2)) \langle \nabla H(\nabla H^*(e_2)), e_1 \rangle = H^*(e_2) \langle \nabla H(\nabla H^*(e_2)), e_1 \rangle. \end{align*} Such a choice implies that $$ \langle \nabla H(\xi), e_1 \rangle = \langle \nabla H( \nabla H^*(e_2)), e_1 \rangle = 0. $$ Moreover, it is easy to see that~$H(\xi) = t$. From~\eqref{groexp} we may then compute \begin{align*}
\Gamma \left( \kappa + |\xi| \right)^{p - 2}& \geqslant \sum_{i, j = 1}^n \left| B''(t) H_i(\xi) H_j(\xi) + B'(t) H_{i j}(\xi) \right| \\ & \geqslant B''(t) H_1(\xi) H_1(\xi) + B'(t) H_{1 1}(\xi) \\
& = B'(t) H_{1 1}(\xi) \geqslant B'(t) \lambda |\xi|^{-1}, \end{align*} from which we get, as before, $$ B'(t) \leqslant \Gamma \lambda^{-1} C c_*^{p - 2} (c_* \kappa + t)^{p - 2} t, $$ with~$\lambda$ as in~\eqref{lambdadef}. This concludes the proof of the second part of our claim.
The fact that we may assume~\eqref{kappabarkappa} to hold - up to relabeling the constants~$\gamma, \Gamma$ or~$\bar{\gamma}, \bar{\Gamma}$ in dependence of~$C$ - is a consequence of the inequalities $$ (\kappa + t)^{p - 2} \leqslant (c_* \kappa + t)^{p - 2} \leqslant c_*^{p - 2} (\kappa + t)^{p - 2}, $$ and \begin{equation*}
c_*^{2 - p} (\bar{\kappa} + |\xi|)^{p - 2} \leqslant (c_*^{- 1}\bar{\kappa} + |\xi|)^{p - 2} \leqslant (\bar{\kappa} + |\xi|)^{p - 2}. \qedhere \end{equation*} \end{proof}
On the other hand, the characterization of~(B) in terms of~(B)\ensuremath{'} is the content of the following
\begin{proposition} \label{(B)char} Let~$B \in C^3((0, +\infty)) \cap C^1([0, +\infty))$ be a function satisfying~\eqref{Bpos} along with~$B(0) = B'(0) = 0$ and~$H \in C^3(\mathbb{R}^n \setminus \{ 0 \})$ be positive homogeneous of degree~$1$, such that~\eqref{Hpos} is true. Then, hypotheses~(B) and~(B)\ensuremath{'} are equivalent. \end{proposition}
\begin{proof} We begin by showing that~(B)\ensuremath{'} implies~(B). First, we deal with the regularity of the composition~$B \circ H$. By the general assumptions on~$B$ and~$H$, it is clear that~$B \circ H \in C_{\rm loc}^{3, \beta}(\mathbb{R}^n \setminus \{ 0 \}) \cap C^1(\mathbb{R}^n)$. Thus, we only need to check the second and third derivatives of~$B \circ H$ at the origin. For any~$e \in S^{n - 1}$ and~$t > 0$, by the homogeneity of~$H$ we have \begin{align*} (B \circ H)_{i j}(t e) & = B''(H(t e)) H_i(t e) H_j(t e) + B'(H(t e)) H_{i j}(t e) \\ & = B''(t H(e)) H_i(e) H_j(e) + \frac{B'(t H(e))}{t H(e)} H(e) H_{i j}(e). \end{align*} Hence, taking the limit as~$t \rightarrow 0^+$ \begin{equation} \label{BH20} \lim_{t \rightarrow 0^+} (B \circ H)_{i j}(t e) = B''(0) \left[ H_i(e) H_j(e) + H(e) H_{i j}(e) \right]. \end{equation} Now, observe that, being~$H$ of the special form~\eqref{HM}, we may explicitly compute \begin{equation} \label{Mij} M_{i j} = \partial_{i j} \left( \frac{H^2}{2} \right) (\xi) = H_i(\xi) H_j(\xi) + H(\xi) H_{i j}(\xi), \end{equation} for any~$\xi \in \mathbb{R}^n$. As a consequence of~\eqref{Mij}, the right hand side of~\eqref{BH20} does not depend on~$e \in S^{n - 1}$ and so $$ (B \circ H)_{i j}(0) = B''(0) M_{i j}. $$ Now we focus on the third derivative. First, by differentiating~\eqref{Mij} we deduce the identity $$ H_i(\xi) H_{j k}(\xi) + H_j(\xi) H_{i k}(\xi) + H_k(\xi) H_{i j}(\xi) = - H(\xi) H_{i j k}(\xi). $$ With this in hand we compute \begin{equation} \label{BH3} \begin{aligned} (B \circ H)_{i j k}(t e) & = B'''(H(t e))H_i(t e) H_j(t e) H_k(t e) \\ & \quad + B''(H(t e)) \left[ H_i(t e) H_{j k}(t e) + H_j(t e) H_{i k}(t e) + H_k(t e) H_{i j}(t e) \right] \\ & \quad + B'(H(t e)) H_{i j k}(t e) \\ & = B'''(t H(e)) H_i(e) H_j(e) H_k(e) \\ & \quad +\frac{1}{t H(e)} \left[ \frac{B'(t H(e))}{t H(e)} - B''(t H(e)) \right] H^2(e) H_{i j k}(e). \end{aligned} \end{equation} Now, we claim that \begin{equation} \label{limit0} \lim_{s \rightarrow 0^+} \frac{1}{s} \left[ \frac{B'(s)}{s} - B''(s) \right] = 0. \end{equation} Indeed, since~$B'(0) = B'''(0) = 0$, the Taylor expansions of~$B'$ and~$B''$ are $$ B'(s) = B''(0) s + o(s^2) \quad \mbox{and} \quad B''(s) = B''(0) + o(s), $$ as~$s \rightarrow 0^+$. Therefore $$ \frac{B'(s)}{s} - B''(s) = \frac{B''(0) s}{s} - B''(0) + o(s) = o(s), $$ and~\eqref{limit0} follows. Thus, letting~$t \rightarrow 0^+$ in~\eqref{BH3}, we get $$ \lim_{t \rightarrow 0^+}(B \circ H)_{i j k}(t e) = B'''(0) H_i(e) H_j(e) H_k(e) + 0 \cdot H^2(e) H_{i j k}(e) = 0, $$ for any~$e \in S^{n - 1}$. We may thence conclude that~$B \circ H \in C_{\rm loc}^{3, \beta}(\mathbb{R}^n)$. Finally, we prove that~${\mbox{\normalfont Hess}} \left( B \circ H \right)$ is uniformly elliptic on compact subsets, as required in~(B). Let \begin{equation} \label{Cmax} C := \max_{\xi \in S^{n - 1}} H(\xi). \end{equation} By~\eqref{Bpos} and~\eqref{B''0pos}, for any~$K > 0$, there exists~$\bar{\gamma} > 0$ such that \begin{equation} \label{B''K} B''(t) \geqslant \bar{\gamma}, \end{equation} for any~$t \in [0, C K]$. Since~$B'(0) = 0$, we also infer that \begin{equation} \label{B'K} B'(t) = \int_0^t B''(s) \, ds \geqslant \bar{\gamma} t, \end{equation}
for any~$t \in [0, C K]$. Let~$\xi, \eta \in \mathbb{R}^n$, with~$|\xi| \leqslant K$. Observe that, by~\eqref{Cmax}, it holds $$
H(\xi) \leqslant |\xi| H \left( \frac{\xi}{|\xi|} \right) \leqslant C K. $$ Then, by~\eqref{B''K},~\eqref{B'K} and~\eqref{Mij}, \begin{align*} (B \circ H)_{i j}(\xi) \eta_i \eta_j & = \left[ B''(H(\xi)) H_i(\xi) H_j(\xi) + B'(H(\xi)) H_{i j}(\xi) \right] \eta_i \eta_j \\ & \geqslant \bar{\gamma} \left[ H_i(\xi) H_j(\xi) + H(\xi) H_{i j}(\xi) \right] \eta_i \eta_j \\ & = \bar{\gamma} M_{i j} \eta_i \eta_j, \end{align*} and the result follows from the positive definiteness of~$M$.
Now, we turn to the converse implication, i.e. that~(B) implies~(B)\ensuremath{'}. First, we observe that~$H$ needs to be of the type~\eqref{HM}, in view of~\cite[Appendix~B]{CFV14}. Then, we address the regularity of~$B$. Being~$H$ even and using~\eqref{i}, we have \begin{align*} B''(0) & = \lim_{t \rightarrow 0^+} \frac{B'(t)}{t} = \lim_{s \rightarrow 0} \frac{B'(H(s e_1))}{H(s e_1)} = \lim_{s \rightarrow 0} \frac{(B \circ H)_1(s e_1)}{H(s e_1) H_1(s e_1)} \\ & = \lim_{s \rightarrow 0} \frac{s}{s H(e_1) H_1(e_1)} \frac{(B \circ H)_1(s e_1) - (B \circ H)_1(0)}{s} \\ & = H^{-2}(e_1) (B \circ H)_{1 1}(0), \end{align*} so that~$B \in C^2([0, +\infty))$. Moreover,~$B''(0) > 0$, as can be seen by testing with~$\xi = 0, \zeta = e_1$ the ellipticity condition of~(B). On the other hand, by~\eqref{i} and~\eqref{ii} we compute \begin{align*} (B \circ H)_{1 1 1}(0) & = \lim_{t \rightarrow 0} \frac{(B \circ H)_{1 1}(t e_1) - (B \circ H)_{1 1}(0)}{t} \\ & = \lim_{t \rightarrow 0} \frac{B''(H(t e_1)) H_1(t e_1) H_1(t e_1) + B'(H(t e_1)) H_{1 1}(t e_1) - B''(0) H^2(e_1)}{t} \\
& = \pm H^3(e_1) \lim_{t \rightarrow 0^\pm} \frac{B''(|t| H(e_1)) - B''(0)}{|t| H(e_1)} \\ & = \pm H^3(e_1) \lim_{s \rightarrow 0^+} \frac{B''(s) - B''(0)}{s}. \end{align*} Since the left hand side exists finite, the same should be true for the right one, too. Thus, we obtain that~$B \in C^3([0, +\infty))$ with~$B'''(0) = 0$. This concludes the proof, as the local H\"olderianity of~$B$ up to~$0$ may be easily deduced from that of~$B \circ H$. \end{proof}
We remark that Theorem~\ref{T-RES} now follows easily from Propositions~\ref{(A)char} and~\ref{(B)char}.
\section{The monotonicity formula} \label{monformsec} In this section we prove Theorem~\ref{monformthm}. Our argument is similar to that presented in~\cite{M89} and~\cite[Theorem 1.4]{CGS94}. Yet, we develop several technical adjustments in order to cope with the difficulties arising in the anisotropic setting. In particular, in the classical, isotropic setting, the monotonicity formulae implicitly rely on some Euclidean geometric features, such as that a point on the unit sphere coincides with the normal of the sphere at that point, as well as the one of the dual sphere (that in the isotropic setting coincides with the original one). These Euclidean geometric properties are lost in our case, therefore we need some more refined geometrical and analytical studies.
The strategy we adopt to show the monotonicity of~$\mathscr{E}$ basically relies on taking its derivative and then checking that it is non-negative. To complete this task, however, we make some integral manipulation involving the Hessian of~$u$. Hence, we need~$u$ to be twice differentiable, at least in the weak sense.
If~$(ii)$ is assumed to hold, this is not an issue, since~$u$ is~$C^3$ (see Proposition~\ref{ureg}). Therefore, we only focus on case~$(i)$. In this framework the solution~$u$ is, in general, no more than~$C_{\rm loc}^{1, \alpha}$. To circumvent this lack of regularity, we introduce a sequence of approximating problems and perform the computation on their solutions. Passing to the limit, we then recover the result for~$u$. If one is interested in the proof under hypothesis~$(ii)$, he should simply ignore the perturbation argument and directly work with~$u$.
Prior to the proper proof of Theorem~\ref{monformthm}, we present some preparatory results about the above mentioned approximation technique. In every statement the functions~$B$,~$H$ and~$u$ are assumed to be satisfy assumption~$(i)$.
Let~$\varepsilon \in (0, 1)$ and consider the function~$B_\varepsilon$ defined by \begin{equation} \label{Bepsdef} B_\varepsilon(t) := B \left( \sqrt{\varepsilon^2 + t^2} \right) - B(\varepsilon), \end{equation} for any~$t > 0$.
First, we present a result which addresses the regularity and growth properties of~$B_\varepsilon$.
\begin{lemma} \label{BepsHlem} The function~$B_\varepsilon$ is of class~$C^2([0, +\infty))$ and it satisfies~$B_\varepsilon(0) = B_\varepsilon'(0) = 0$ and~\eqref{Bpos}. Moreover \begin{equation} \label{Bepsellgro} \begin{aligned} c_p \bar{\gamma} \left( \bar{\kappa} + \varepsilon + t \right)^{p - 2} t \leqslant & B_\varepsilon'(t) \leqslant C_p \bar{\Gamma} \left( \bar{\kappa} + \varepsilon + t \right)^{p - 2} t, \\ c_p \bar{\gamma} \left( \bar{\kappa} + \varepsilon + t \right)^{p - 2} \leqslant & B_\varepsilon''(t) \leqslant C_p \bar{\Gamma} \left( \bar{\kappa} + \varepsilon + t \right)^{p - 2}, \end{aligned} \end{equation} for any~$t > 0$, where~$\bar{\gamma}, \bar{\Gamma}$ are as in~(A)\ensuremath{'} and $$ c_p := \min \left\{ 1, 2^{\frac{2 - p}{2}} \right\}, \, C_p := \max \left\{ 1, 2^{\frac{2 - p}{2}} \right\}. $$ In addition, the composition~$B_\varepsilon \circ H$ is of class~$C^{1, 1}_{\rm loc}(\mathbb{R}^n)$ and it holds, for any~$\xi \in \mathbb{R}^n$, \begin{equation} \label{BepsHell}
(B_\varepsilon \circ H)(\xi) \geqslant \frac{\gamma}{2 (p - 1) p} |\xi|^p - c_\star, \end{equation} where~$\gamma$ is as in~(A) and~$c_\star$ is a non-negative constant independent of~$\varepsilon$. \end{lemma}
\begin{proof} It is immediate to check from definition~\eqref{Bepsdef} that~$B_\varepsilon \in C^2([0, +\infty))$. For any~$t > 0$, we compute \begin{align*} B_\varepsilon'(t) & = B' \left( \sqrt{\varepsilon^2 + t^2} \right) \frac{t}{\sqrt{\varepsilon^2 + t^2}}, \\ B_\varepsilon''(t) & = B'' \left( \sqrt{\varepsilon^2 + t^2} \right) \frac{t^2}{\varepsilon^2 + t^2} + B' \left( \sqrt{\varepsilon^2 + t^2} \right) \frac{\varepsilon^2}{\left( \varepsilon^2 + t^2 \right)^{3/2}}. \end{align*} Thus, inequalities~\eqref{Bpos} are valid and~$B_\varepsilon(0) = B_\varepsilon'(0) = 0$. Furthermore, formulae~\eqref{Bepsellgro} can be recovered from the ellipticity and growth conditions of~(A)\ensuremath{'} which~$B$ satisfies.
Then, we address the composition~$B_\varepsilon \circ H$. Notice that we already know that it is of class~$C^1$ on the whole~$\mathbb{R}^n$, by virtue of Lemma~\ref{derBH0}, and~$C^2$ outside of the origin, by definition. Thus we only need to check that its gradient is Lipschitz in a neighbourhood of the origin. By using~\eqref{Bepsellgro}, for any~$0 < |\xi| \leqslant 1$ we get $$
\frac{\left| \partial_i (B_\varepsilon \circ H)(\xi) \right|}{|\xi|} = \frac{\left| B_\varepsilon'(H(\xi)) H_i(\xi) \right|}{|\xi|} \leqslant C_p \bar{\Gamma} (\bar{\kappa} + \varepsilon + H(\xi))^{p - 2} H_i(\xi) \frac{H(\xi)}{|\xi|} \leqslant c, $$ for some positive~$c$.
Finally, we establish~\eqref{BepsHell}. As a preliminary observation, we stress that the Hessian of~$B_\varepsilon \circ H$ satisfies~(A) with~$\kappa = \bar{\kappa} + \varepsilon$. This can be seen as a consequence of~\eqref{Bepsellgro}, the uniform ellipticity of~$H$ and Proposition~\ref{(A)char} (recall in particular relation~\eqref{kappabarkappa}). We consider separately the two possibilities~$p \geqslant 2$ and~$1 < p < 2$. In the first case, we simply compute \begin{align*}
(B_\varepsilon \circ H)(\xi) & = \int_0^1 \int_0^t (B_\varepsilon \circ H)_{i j}(s \xi) \xi_i \xi_j \, ds dt \geqslant \gamma \int_0^1 \int_0^t (\kappa + s |\xi|)^{p - 2} |\xi|^2 \, ds dt \\
& \geqslant \gamma |\xi|^p \int_0^1 \int_0^t s^{p - 2} \, ds dt = \frac{\gamma}{(p - 1) p} |\xi|^p. \end{align*} If, on the other hand,~$1 < p < 2$, we have \begin{equation} \label{techBepsH} \begin{aligned}
(B_\varepsilon \circ H)(\xi) & = \int_0^1 \int_0^t (B_\varepsilon \circ H)_{i j}(s \xi) \xi_i \xi_j \, ds dt \geqslant \gamma \int_0^1 \int_0^t (\kappa + s |\xi|)^{p - 2} |\xi|^2 \, ds dt \\
& = \frac{\gamma}{p - 1} \left[ \frac{(\kappa + |\xi|)^p - \kappa^p}{p} - \kappa^{p - 1} |\xi| \right] \geqslant \frac{\gamma}{p - 1} \left[ \frac{|\xi|^p - \kappa^p}{p} - \kappa^{p - 1} |\xi| \right]. \end{aligned} \end{equation} Notice that, by Young's inequality, we estimate $$
\kappa^{p - 1} |\xi| \leqslant \frac{|\xi|^p}{2 p} + \frac{p - 1}{p} 2^{1/(p - 1)} \kappa^p. $$ Plugging this into~\eqref{techBepsH} finally leads to the desired \begin{align*}
(B_\varepsilon \circ H)(\xi) & \geqslant \frac{\gamma}{2 (p - 1) p} |\xi|^p - \frac{\gamma}{(p - 1) p} \left( 1 + (p - 1) 2^{1 / (p - 1)} \right) \kappa^p \\
& \geqslant \frac{\gamma}{2 (p - 1) p} |\xi|^p - \frac{\gamma}{(p - 1) p} \left( 1 + (p - 1) 2^{1 / (p - 1)} \right) (\bar{\kappa} + 1)^p. \end{align*} Hence,~\eqref{BepsHell} holds in both cases and the proof of the lemma is complete. \end{proof}
In the following lemma we compare~$B_\varepsilon$ to~$B$. We study their modulus of continuity and discuss some uniform convergence properties.
\begin{lemma} \label{BepsBlem} Introduce, for~$t \geqslant 0$, the functions~$\beta(t) := B'(t) t$,~$\beta_\varepsilon(t) := B_\varepsilon'(t) t$.
Then, the Lipschitz norms of both~$B_\varepsilon$ and~$\beta_\varepsilon$ on compact sets of~$[0, +\infty)$ are bounded by a constant independent of~$\varepsilon$. More explicitly, for any~$M \geqslant 1$ we estimate \begin{equation} \label{Bbetaepslip} \begin{aligned}
\| B_\varepsilon \|_{C^{0, 1}([0, M])} & \leqslant \| B \|_{C^{0, 1}([0, 2 M])}, \\
\| \beta_\varepsilon \|_{C^{0, 1}([0, M])} & \leqslant 2 \| B' \|_{C^0([0, 2 M])} + \| \beta \|_{C^{0, 1}([0, 2 M])}. \end{aligned} \end{equation} Moreover,~$B_\varepsilon \rightarrow B$ and~$\beta_\varepsilon \rightarrow \beta$ uniformly on compact sets of~$[0, +\infty)$. Quantitatively, we have \begin{equation} \label{Bbetaepsconv} \begin{aligned}
\| B_\varepsilon - B \|_{C^0([0, M])} & \leqslant 2 \| B' \|_{C^0([0, 2M])} \varepsilon, \\
\| \beta_\varepsilon - \beta \|_{C^0([0, M])} & \leqslant \left( \| B' \|_{C^0([0, 2M])} + \| \beta \|_{C^{0, 1}([0, 2M])} \right) \varepsilon. \end{aligned} \end{equation} \end{lemma}
\begin{proof} First of all, we stress that, while~$\beta_\varepsilon \in C^1([0, +\infty))$ in view of Lemma~\ref{BepsHlem}, the same is true also for~$\beta$, as one can easily deduce from hypothesis~(A)\ensuremath{'}.
We begin to establish~\eqref{Bbetaepslip}. It is easy to see that the~$C^0$ norms of~$B_\varepsilon$ and~$\beta_\varepsilon$ are bounded by those of~$B$ and~$\beta$ respectively. Thus, we may concentrate on the estimates of their Lipschitz seminorms. Let~$M \geqslant 1$ and~$0 \leqslant s, t \leqslant M$. We have \begin{align*}
|B_\varepsilon(t) - B_\varepsilon(s)| & = \left| B \left( \sqrt{\varepsilon^2 + t^2} \right) - B \left( \sqrt{\varepsilon^2 + s^2} \right) \right| \\
& \leqslant \| B \|_{C^{0, 1}([0, 2M])} \left| \sqrt{\varepsilon^2 + t^2} - \sqrt{\varepsilon^2 + s^2} \right| \\
& \leqslant \| B \|_{C^{0, 1}([0, 2M])} |t - s|, \end{align*} so that the first relation in~\eqref{Bbetaepslip} is proved. The second inequality needs a little more care. Assuming without loss of generality~$s \leqslant t$, we compute \begin{align*}
|\beta_\varepsilon(t) - \beta_\varepsilon(s)| & = \left| B' \left( \sqrt{\varepsilon^2 + t^2} \right) \frac{t^2}{\sqrt{\varepsilon^2 + t^2}} - B' \left( \sqrt{\varepsilon^2 + s^2} \right) \frac{s^2}{\sqrt{\varepsilon^2 + s^2}} \right| \\
& \leqslant B' \left( \sqrt{\varepsilon^2 + t^2} \right) \sqrt{\varepsilon^2 + t^2} \left| \frac{t^2}{\varepsilon^2 + t^2} - \frac{s^2}{\varepsilon^2 + s^2} \right| \\
& \quad + \frac{s^2}{\varepsilon^2 + s^2} \left| \beta \left( \sqrt{\varepsilon^2 + t^2} \right) - \beta \left( \sqrt{\varepsilon^2 + s^2} \right) \right| \\
& \leqslant \| B' \|_{C^0([0, 2 M])} \frac{|t^2 - s^2|}{\sqrt{\varepsilon^2 + t^2}} + \| \beta \|_{C^{0, 1}([0, 2M])} \left| \sqrt{\varepsilon^2 + t^2} - \sqrt{\varepsilon^2 + s^2} \right| \\
& \leqslant \left( 2 \| B' \|_{C^0([0, 2 M])} + \| \beta \|_{C^{0, 1}([0, 2M])} \right) |t - s|. \end{align*}
Estimates~\eqref{Bbetaepsconv} are proved in a similar fashion. Indeed, for any~$0 \leqslant t \leqslant M$, \begin{align*}
\left| B_\varepsilon(t) - B(t) \right| & = \left| B \left( \sqrt{\varepsilon^2 + t^2} \right) - B(\varepsilon) - B(t) \right| \\
& \leqslant \| B \|_{C^{0, 1}([0, 2 M])} \left( \left| \sqrt{\varepsilon^2 + t^2} - t \right| + \varepsilon \right) \\
& \leqslant 2 \| B \|_{C^{0, 1}([0, 2 M])} \varepsilon, \end{align*} and \begin{align*}
\left| \beta_\varepsilon(t) - \beta(t) \right| & \leqslant B' \left( \sqrt{\varepsilon^2 + t^2} \right) \left| \frac{t^2}{\sqrt{\varepsilon^2 + t^2}} - \sqrt{\varepsilon^2 + t^2} \right| + \left| \beta \left( \sqrt{\varepsilon^2 + t^2} \right) - \beta(t) \right| \\
& \leqslant \left( \| B' \|_{C^0([0, 2 M])} + \| \beta \|_{C^{0, 1}([0, 2 M])} \right) \varepsilon. \end{align*} Thus, the proof is complete. \end{proof}
Next is the key proposition of the approximation argument. Basically, we consider some perturbed problems driven by~$B_\varepsilon$. We prove that their solutions are~$H^2$ regular and that they converge to~$u$.
\begin{proposition} \label{uepsexis} Let~$\Omega$ be a bounded open set of~$\mathbb{R}^n$ with~$C^{1, \alpha}$ boundary. The problem \begin{equation} \label{uepsprob} \begin{cases} \mbox{\normalfont div} \big( B_\varepsilon'(H(\nabla u^\varepsilon)) \nabla H(\nabla u^\varepsilon) \big) + F'(u) = 0, & \qquad \mbox{in } \Omega, \\ u^\varepsilon = u, & \qquad \mbox{on } \partial \Omega, \end{cases} \end{equation} admits a strong solution~$u^\varepsilon \in C^{1, \alpha'}(\overline{\Omega}) \cap H^2(\Omega)$, for some~$\alpha' \in (0, 1]$ independent of~$\varepsilon$. Furthermore,~$u^\varepsilon$ converges to~$u$ in~$C^1(\overline{\Omega})$, as~$\varepsilon \rightarrow 0^+$.
\end{proposition}
\begin{proof} By using standard methods - see, for instance,~\cite[Theorem~3.30]{D07} - we know that the functional $$ \mathcal{F}_\varepsilon(v) := \int_{\Omega} B_\varepsilon(H(\nabla v(x))) - F'(u(x)) v(x) \, dx, $$ admits the existence of a minimizer~$u^\varepsilon \in W^{1, p}(\Omega)$, with~$u^\varepsilon - u \in W_0^{1, p}(\Omega)$. Note that~$\mathcal{F}_\varepsilon$ is coercive, thanks to~\eqref{BepsHell}, the continuity of~$F'$ and the boundedness of~$u$. Clearly,~$u^\varepsilon$ satisfies~\eqref{uepsprob} in the weak sense.
In view of~\eqref{BepsHell}, we see that the minimizer~$u^\varepsilon$ is bounded in~$\Omega$ (use e.g.~\cite[Theorems~6.1-6.2]{S63} or~\cite[Theorem 3.2, p.~328]{LU68}). Moreover, the~$L^\infty$ norm of~$u^\varepsilon$ is uniform in~$\varepsilon$.
With this in hand, we can now verify that~$u^\varepsilon \in C^{1, \alpha'}$. For this, we notice that Lemma~\ref{BepsHlem} and Proposition~\ref{(A)char} ensure that hypothesis~(A) is verified by~$B_\varepsilon \circ H$. Hence, by the uniform~$L^\infty$ estimates, we may appeal to~\cite[Theorem~1]{L88} to deduce that~$u^\varepsilon \in C^{1, \alpha'}(\overline{\Omega})$, for some~$\alpha' \in (0, 1]$. Notice that~$\alpha'$ is independent of~$\varepsilon$ and~$\| u^\varepsilon \|_{C^{1, \alpha'}(\overline{\Omega})}$ is uniformly bounded in~$\varepsilon$.
Consequently, by Arzelà-Ascoli Theorem, the sequence $\{ u^\varepsilon \}$ converges in $C^1(\overline{\Omega})$ to a function $v$, as $\varepsilon \rightarrow 0^+$. With the aid of Lemma~\ref{BepsBlem}, we see that~$v$ is the unique solution of $$ \begin{cases} \mbox{\normalfont div} \left( B'(H(\nabla v)) \nabla H(\nabla v) \right) + F'(u) = 0, & \qquad \mbox{in } \Omega, \\ v = u, & \qquad \mbox{on } \partial \Omega. \end{cases} $$ Therefore,~$v = u$ in the whole~$\overline{\Omega}$.
Now we prove the~$H^2$ regularity of~$u^\varepsilon$. To this aim we employ~\cite[Proposition~1]{T84}. Notice that we need to check the validity of condition~(2.4) there, in order to apply such result. If~$p \geqslant 2$ it is an immediate consequence of the fact that~$B_\varepsilon \circ H$ satisfies~(A). Indeed, for any~$\eta \in \mathbb{R}^n \setminus \{ 0 \}$,~$\zeta \in \mathbb{R}^n$, we deduce that $$
\left[ {\mbox{\normalfont Hess}} \,(B_\varepsilon \circ H)(\xi) \right]_{i j} \zeta_i \zeta_j \geqslant \gamma {(\bar{\kappa} + \varepsilon + |\xi|)}^{p - 2} {|\zeta|}^2 \geqslant \tilde{\gamma} {|\zeta|}^2, $$
for some~$\tilde{\gamma} > 0$. In case~$1 < p < 2$, we set~$M := \| \nabla u^\varepsilon \|_{L^\infty(\overline{\Omega})}$ and modify~$B_\varepsilon$ accordingly to Lemma~\ref{Bcaplem}. The new function~$\hat{B}_\varepsilon$ obtained this way satisfies assumption~(A)\ensuremath{'}, and thus~(A), with~$p = 2$. Moreover,~$u^\varepsilon$ is still a weak solution to~\eqref{uepsprob} with~$B_\varepsilon$ replaced by~$\hat{B}_\varepsilon$. This is enough to conclude that~$u^\varepsilon \in H^2(\Omega)$ also when~$1 < p < 2$.
From the additional Sobolev regularity we deduce that~$u^\varepsilon$ is actually a strong solution of~\eqref{uepsprob}. Indeed, it is sufficient to observe that, for any~$i = 1, \ldots, n$, $$ B_\varepsilon'(H(\nabla u^\varepsilon)) H_i(\nabla u^\varepsilon) = \left( B_\varepsilon \circ H \right)_i (\nabla u^\varepsilon) \in H^1(\Omega), $$ being~$(B_\varepsilon \circ H)_i$ locally uniformly Lipschitz, by Lemma~\ref{BepsHlem}. \end{proof}
After all these preliminary results, we may finally head to the
\begin{proof}[Proof of Theorem~\ref{monformthm}] First, using the coarea formula we compute $$
\mathscr{E}'(R) = \frac{1 - n}{R} \mathscr{E}(R) + \frac{1}{R^{n - 1}} \int_{\partial W_R} \left[ B(H(\nabla u)) + G(u) \right] |\nabla H^*|^{-1} \, d\mathcal{H}^{n - 1}. $$ Then, notice that the exterior unit normal vector to~$\partial W_R$ at~$x \in \partial W_R$ is given by \begin{equation} \label{nu}
\nu(x) = \frac{\nabla H^*(x)}{|\nabla H^*(x)|}. \end{equation} Thus, by the homogeneity of~$H$ and the second identity in~\eqref{CS2} we have $$
H(\nu(x)) = |\nabla H^*(x)|^{-1} H(\nabla H^*(x)) = |\nabla H^*(x)|^{-1}. $$ As a consequence, the derivative of~$\mathscr{E}$ at~$R$ becomes \begin{equation} \label{montech1} \mathscr{E}'(R) = \frac{1 - n}{R} \mathscr{E}(R) + \frac{1}{R^{n - 1}} \int_{\partial W_R} \left[ B(H(\nabla u)) + G(u) \right] H(\nu) \, d\mathcal{H}^{n - 1}. \end{equation}
For any~$\varepsilon \in (0, 1)$, let now~$u^\varepsilon \in C^{1, \alpha'}(\overline{W_R}) \cap H^2(W_R)$ be the strong solutions of~\eqref{uepsprob}, with~$\Omega = W_R$. Notice that~$\partial W_R$ is of class~$C^2$ in view of Lemma~\ref{H*reg}. Hence, we are allowed to apply Proposition~\ref{uepsexis} to obtain such a~$u^\varepsilon$. By the results of Proposition~\ref{uepsexis} and Lemma~\ref{BepsBlem}, along with the~$C^2$ regularity of~$G$, it is immediate to check that \begin{equation} \label{BHuconv} \begin{aligned} B_\varepsilon(H(\nabla u^\varepsilon)) & \longrightarrow B(H(\nabla u)), \\ B_\varepsilon'(H(\nabla u^\varepsilon)) H(\nabla u^\varepsilon) & \longrightarrow B'(H(\nabla u)) H(\nabla u), \\ G(u^\varepsilon) \longrightarrow G(u) \, \, & \, \mbox{and} \, \, \, F'(u^\varepsilon) \longrightarrow F'(u), \end{aligned} \end{equation} uniformly on~$\overline{W_R}$.
In view of Lemma~\ref{H*reg} the function~$H \nabla H$ is bijective and its inverse is given by~$H^* \nabla H^*$. Hence, exploiting the homogeneity properties of~$H$ and~$\nabla H$ together with~\eqref{CS2}, it follows that the identity \begin{align*} x & = H(H^*(x) \nabla H^*(x)) \nabla H(H^*(x) \nabla H^*(x)) = H^*(x) H(\nabla H^*(x)) \nabla H(\nabla H^*(x)) \\ & = H^*(x) \nabla H(\nabla H^*(x)), \end{align*} is true for any~$x \in \mathbb{R}^n \setminus \{ 0 \}$. Consequently, using~\eqref{i},~\eqref{nu}, the homogeneity of~$\nabla H$, the definition of~$\partial W_R$ and the divergence theorem, we compute \begin{align*} & \int_{\partial W_R} B_\varepsilon(H(\nabla u^\varepsilon)) H(\nu) \, d\mathcal{H}^{n - 1} = \frac{1}{R} \int_{\partial W_R} B_\varepsilon(H(\nabla u^\varepsilon)) H^* \langle \nabla H(\nu), \nu \rangle \, d\mathcal{H}^{n - 1} \\ & \qquad = \frac{1}{R} \int_{W_R} \mbox{\normalfont div} \big( B_\varepsilon(H(\nabla u^\varepsilon)) H^* \nabla H(\nabla H^*) \big) \, dx = \frac{1}{R} \int_{W_R} \mbox{\normalfont div} \big( B_\varepsilon(H(\nabla u^\varepsilon)) x \big) \, dx \\ & \qquad = \frac{1}{R} \int_{W_R} B_\varepsilon'(H(\nabla u^\varepsilon)) H_j(\nabla u^\varepsilon) u^\varepsilon_{i j} x_i \, dx + \frac{n}{R} \int_{W_R} B_\varepsilon(H(\nabla u^\varepsilon)) \, dx. \end{align*} With a completely analogous argument we also deduce that $$ \int_{\partial W_R} G(u^\varepsilon) H(\nu) \, d\mathcal{H}^{n - 1} = - \frac{1}{R} \int_{W_R} F'(u^\varepsilon) u^\varepsilon_i x_i \, dx + \frac{n}{R} \int_{W_R} G(u^\varepsilon) \, dx. $$ Putting these last two identities together we obtain \begin{equation} \label{montech34} \int_{\partial W_R} \left[ B_\varepsilon(H(\nabla u^\varepsilon)) + G(u^\varepsilon) \right] H(\nu) \, d\mathcal{H}^{n - 1} = \frac{n}{R} \int_{W_R} B_\varepsilon(H(\nabla u^\varepsilon)) + G(u^\varepsilon) \, dx + \frac{I_\varepsilon}{R}, \end{equation} where $$ I_\varepsilon := \int_{W_R} \left[ B_\varepsilon'(H(\nabla u^\varepsilon)) H_j(\nabla u^\varepsilon) u^\varepsilon_{i j} - F'(u^\varepsilon) u^\varepsilon_i \right] x_i \, dx. $$ Recalling that~$u^\varepsilon$ is a strong solution of~\eqref{uepsprob}, we compute \begin{align*} I_\varepsilon & = \int_{W_R} \left[ \big( B_\varepsilon'(H(\nabla u^\varepsilon)) H_j(\nabla u^\varepsilon) u^\varepsilon_i \big)_j - \left( B_\varepsilon'(H(\nabla u^\varepsilon)) H_j(\nabla u^\varepsilon) \right)_j u^\varepsilon_i - F'(u^\varepsilon) u^\varepsilon_i \right] x_i \, dx \\ & = \int_{W_R} \big( B_\varepsilon'(H(\nabla u^\varepsilon)) H_j(\nabla u^\varepsilon) u^\varepsilon_i \big)_j x_i \, dx + \int_{W_R} \left[ F'(u) - F'(u^\varepsilon) \right] u^\varepsilon_i x_i \, dx. \end{align*} By the divergence theorem, formulae~\eqref{i},~\eqref{nu} and condition~\eqref{FKweak} we find \begin{equation} \label{montech5} \begin{aligned} I_\varepsilon & = \int_{W_R} \left( B_\varepsilon'(H(\nabla u^\varepsilon)) H_j(\nabla u^\varepsilon) u^\varepsilon_i x_i \right)_j \, dx - \int_{W_R} B_\varepsilon'(H(\nabla u^\varepsilon)) H_j(\nabla u^\varepsilon) u^\varepsilon_i \delta_{i j} \, dx \\ & \quad + \int_{W_R} \left[ F'(u) - F'(u^\varepsilon) \right] u^\varepsilon_i x_i \, dx \\
& = \int_{\partial W_R} \frac{B_\varepsilon'(H(\nabla u^\varepsilon))}{|\nabla H^*|} \langle \nabla H(\nabla u^\varepsilon), \nabla H^* \rangle \langle \nabla u^\varepsilon, x \rangle \, d\mathcal{H}^{n - 1} \\ & \quad - \int_{W_R} B_\varepsilon'(H(\nabla u^\varepsilon)) \langle \nabla H(\nabla u^\varepsilon), \nabla u^\varepsilon \rangle \, dx + \int_{W_R} \left[ F'(u) - F'(u^\varepsilon) \right] \langle \nabla u^\varepsilon, x \rangle \, dx \\ & \geqslant - \int_{W_R} B_\varepsilon'(H(\nabla u^\varepsilon)) H(\nabla u^\varepsilon) \, dx + \int_{W_R} \left[ F'(u) - F'(u^\varepsilon) \right] \langle \nabla u^\varepsilon, x \rangle \, dx. \end{aligned} \end{equation} Taking the limit as~$\varepsilon \rightarrow 0^+$ in~\eqref{montech34} and~\eqref{montech5}, by~\eqref{BHuconv} we obtain \begin{align*} \int_{\partial W_R} \left[ B(H(\nabla u)) + G(u) \right] H(\nu) \, d\mathcal{H}^{n - 1} & \geqslant \frac{n}{R} \int_{W_R} B(H(\nabla u)) + G(u) \, dx \\ & \quad - \frac{1}{R} \int_{W_R} B'(H(\nabla u)) H(\nabla u) \, dx. \end{align*} By plugging this last identity in~\eqref{montech1} and recalling~\eqref{scaledWen} we finally get \begin{align*} \mathscr{E}'(R) & \geqslant \frac{1}{R^n} \int_{W_R} B(H(\nabla u)) + G(u) - B'(H(\nabla u)) H(\nabla u) \, dx. \end{align*}
The result now follows since the integral on the right hand side is non-negative by virtue of~\cite[Theorem~1.1]{CFV14}. \end{proof}
\section{The Liouville-type theorem} \label{liousec}
Here we prove Theorem~\ref{liouthm}. In order to obtain that~$u$ is constant, our first goal is to show that, thanks to the gradient estimate contained in~\cite[Theorem~1.1]{CFV14}, the gradient term in~\eqref{scaledWen} is bounded by the potential. Then, the monotonicity formula of Theorem~\ref{monformthm} and the growth assumption on~$G(u)$ conclude the argument.
The following general result allows us to accomplish the first step.
\begin{lemma} \label{B'tBgeBlem} Let~$B \in C^2(0, +\infty) \cap C^1([0, +\infty))$ be a function satisfying~\eqref{Bpos} and~$B(0) = B'(0) = 0$. Assume in addition that~$B$ satisfies either~(A)\ensuremath{'} or~(B)\ensuremath{'}. Then, for any~$K > 0$ there exists a constant~$\delta > 0$ such that \begin{equation} \label{B'tBgeB} B'(t) t - B(t) \geqslant \delta B(t), \end{equation} for any~$t \in [0, K]$. In particular, under assumption~(A)\ensuremath{'}, inequality~\eqref{B'tBgeB} holds for any~$t \geqslant 0$. \end{lemma} \begin{proof} We begin by proving~\eqref{B'tBgeB} when~(A)\ensuremath{'} is in force. Since~$B(0) = B'(0) = 0$, we have $$ B'(t) t - B(t) = \int_0^t B''(s) s \, ds \geqslant \bar{\gamma} \int_0^t \left( \bar{\kappa} + s \right)^{p - 2} s \, ds. $$ On the other hand, $$ B(t) = \int_0^t B'(s) \, ds \leqslant \bar{\Gamma} \int_0^t \left( \bar{\kappa} + s \right)^{p - 2} s \, ds. $$ By comparing these two expressions, we see that~\eqref{B'tBgeB} holds for any~$t \geqslant 0$, with~$\delta = \bar{\gamma} / \bar{\Gamma}$.
Then, we deal with case~(B)\ensuremath{'}. Fix~$K > 0$. Being~$B''(0) > 0$ and~$B(0) = B'(0) = 0$, it clearly exist~$\bar{\Gamma} \geqslant \bar{\gamma} > 0$ such that~$B''(t) \in \left[ \bar{\gamma}, \bar{\Gamma} \right]$,~for any~$t \in [0, K]$. Hence, as before we compute $$ B'(t) t - B(t) = \int_0^t B''(s) s \, ds \geqslant \bar{\gamma} \int_0^t s \, ds = \frac{\bar{\gamma}}{2} t^2, $$ for any~$t \in [0, K]$. Also, $$ B(t) = \int_0^t \int_0^s B''(\sigma) \, d\sigma ds \leqslant \frac{\bar{\Gamma}}{2} t^2, $$ for any~$t \in [0, K]$, and again~\eqref{B'tBgeB} is proved. \end{proof}
\begin{proof}[Proof of Theorem~\ref{liouthm}] Combining Lemma~\ref{B'tBgeBlem} and~\cite[Theorem~1.1]{CFV14}, we deduce that \begin{equation} \label{kinpotest} B(H(\nabla u(x))) \leqslant C G(u(x)) \mbox{ for any } x \in \mathbb{R}^n, \end{equation} for some constant~$C > 0$. We stress that, under hypothesis~$(ii)$ of Theorem~\ref{monformthm}, it is crucial that~$\nabla u$ is globally~$L^\infty$ in order to profitably apply Lemma~\ref{B'tBgeBlem}. Recalling the definition~\eqref{scaledWen} of the rescaled energy functional~$\mathscr{E}$, in view of~\eqref{kinpotest} and~\eqref{Gugrowth} we may conclude that $$ \lim_{R \rightarrow +\infty} \mathscr{E}(R) \leqslant (C + 1) \lim_{R \rightarrow +\infty} \frac{1}{R^{n - 1}} \int_{W_R} G(u(x)) \, dx = 0. $$ But then, Theorem~\ref{monformthm} tells that~$\mathscr{E}$ is non-decreasing in~$R\in(0, +\infty)$ and, hence, for any~$r > 0$, we have $$ 0 \leqslant \mathscr{E}(r) \leqslant \lim_{R \rightarrow +\infty} \mathscr{E}(R) = 0, $$ which yields~$\mathscr{E} \equiv 0$. Consequently,~$\nabla u \equiv 0$, i.e.~$u$ is constant. \end{proof}
\section{On conditions~\eqref{HM} and~\eqref{FK}} \label{char1sec}
In the present section we prove Theorem~\ref{FKcharprop}, thus establishing a characterization of the anisotropies~$H$ which satisfy \begin{equation} \tag{\ref*{FK}} \langle H(\xi) \nabla H(\xi), H^*(x) \nabla H^*(x) \rangle = \langle \xi, x \rangle, \end{equation} for any~$\xi, x \in \mathbb{R}^n$. Indeed, we show that such requirement is necessary and sufficient for~$H$ to assume the form \begin{equation} \tag{\ref*{HM}} H_M(\xi) = \sqrt{\langle M \xi, \xi \rangle}, \end{equation} for some symmetric and positive definite matrix~$M \in {\mbox{\normalfont Mat}}_n(\mathbb{R})$.
We begin by showing the necessity of~\eqref{FK}. As a first step towards this aim, we compute the dual function~$H_M^*$.
\begin{lemma} \label{HMlem} Let~$M \in {\mbox{\normalfont Mat}}_n(\mathbb{R})$ be symmetric and positive definite. Then,~$H_M^* = H_{M^{-1}}$.
\end{lemma}
\begin{proof} Being~$M$ positive definite and symmetric, the assignment $$ \langle \xi, \eta \rangle_M := \langle M \xi, \eta \rangle, $$
defines an inner product in~$\mathbb{R}^n$. We denote the induced norm by~$\| \cdot \|_M$. Also notice that~$M$ is invertible, so that~$H_{M^{-1}}$ is well defined.
Recalling definition~\eqref{H*def} of dual function and applying the Cauchy-Schwarz inequality to the inner product~$\langle \cdot, \cdot \rangle_M$, we obtain \begin{align*}
H_M^*(x) & = \sup_{\xi \ne 0} \frac{\langle x, \xi \rangle}{\sqrt{\langle M \xi, \xi \rangle}} = \sup_{\xi \ne 0} \frac{\langle M (M^{-1} x), \xi \rangle}{\sqrt{\langle M \xi, \xi \rangle}} = \sup_{\xi \ne 0} \frac{\langle M^{-1} x, \xi \rangle_M}{\| \xi \|_M} \\
& \leqslant \sup_{\xi \ne 0} \frac{\| M^{-1} x \|_M \| \xi \|_M}{\| \xi \|_M} = \| M^{-1} x \|_M \\ & = \sqrt{\langle M^{-1} x, x \rangle}. \end{align*} On the other hand, the choice~$\xi := M^{-1} x$ yields $$ H_M^*(x) \geqslant \frac{\langle x, M^{-1} x \rangle}{\sqrt{\langle M M^{-1} x, M^{-1} x \rangle}} = \sqrt{\langle M^{-1} x, x \rangle}. $$ Hence, recalling definition~\eqref{HM}, the thesis follows. \end{proof}
With this in hand, we are now able to prove the following
\begin{lemma} \label{FKneclem} Let~$M \in {\mbox{\normalfont Mat}}_n(\mathbb{R})$ be a symmetric and positive definite matrix. Then, the norm~$H_M$ satisfies~\eqref{FK}. \end{lemma}
\begin{proof} The proof is a simple computation. Notice that for any symmetric~$A \in {\mbox{\normalfont Mat}}_n(\mathbb{R})$ we have $$ \partial_i \left( H_A^2(\xi) \right) = \partial_i \left( A_{j k} \xi_j \xi_k \right) = A_{j k} \delta_{j i} \xi_k + A_{j k} \xi_j \delta_{k i} = 2 A_{i j} \xi_j, $$ for any~$\xi \in \mathbb{R}^n$, $i = 1, \ldots, n$. Thus, we get $$ H_A(\xi) \partial_i H_A(\xi) = \frac{\partial_i \left( H_A^2(\xi) \right) }{2}= A_{i j} \xi_j. $$ Applying then Lemma~\ref{HMlem} together with the identity yet obtained with both choices~$A = M$ and~$A = M^{-1}$, we obtain \begin{align*} \langle H_M(\xi) \nabla H_M(\xi), H_M^*(\eta) \nabla H_M^*(\eta) \rangle & = \langle H_M(\xi) \nabla H_M(\xi), H_{M^{-1}}(\eta) \nabla H_{M^{-1}}(\eta) \rangle \\ & = M_{i j} \xi_j M_{i k}^{-1} \eta_k \\ & = \delta_{j k} \xi_j \eta_k \\ & = \langle \xi, \eta \rangle, \end{align*} which is~\eqref{FK}. \end{proof}
Now, we prove that the converse implication is also true. Hence, Theorem~\ref{FKcharprop} will follow. Before addressing the actual proof, we need just another abstract lemma. We believe that the content of the following result will appear somewhat evident to the reader. However, we include both the formal statement and the proof.
\begin{lemma} \label{symlem} Let~$\mathcal{T}: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be symmetric with respect to the standard inner product in~$\mathbb{R}^n$, that is \begin{equation} \label{sym} \langle \mathcal{T}(v), w \rangle = \langle v, \mathcal{T}(w) \rangle, \end{equation} for any~$v, w \in \mathbb{R}^n$. Then,~$\mathcal{T}$ is a linear transformation, i.e. $$ \mathcal{T}(v) = T v \mbox{ for any } v \in \mathbb{R}^n, $$ for some symmetric~$T \in {\mbox{\normalfont Mat}}_n(\mathbb{R})$ \end{lemma}
\begin{proof} The conclusion follows by simply plugging~$w = e_i$ in~$\eqref{sym}$, where~$\{ e_i \}_{i = 1, \ldots, n}$ is the canonical basis in~$\mathbb{R}^n$. Indeed, we have $$ {[\mathcal{T}(v)]}_i = \langle \mathcal{T}(v), e_i \rangle = \langle v, \mathcal{T}(e_i) \rangle $$ for any~$v \in \mathbb{R}^n$,~$i = 1, \ldots, n$. Thus we may conclude that~$\mathcal{T}(v) = T v$, where~$T = [T_{i j}]_{i, j = 1, \ldots, n}$ is the matrix with entries $$ T_{i j} = [\mathcal{T}(e_i)]_j. $$ The symmetry of~$T$ clearly follows by employing~\eqref{sym} again. \end{proof}
\begin{proof}[Proof of Theorem~\ref{FKcharprop}] In view of Lemma~\ref{FKneclem}, it is only left to prove that, under condition~\eqref{FK},~$H$ is forced to be of the form~\eqref{HM}.
By Lemma~\ref{H*reg}, we know that the map~$\Psi_H: \mathbb{R}^n \to \mathbb{R}^n$, defined for~$\xi \in \mathbb{R}^n$ by $$ \Psi_H(\xi) := H(\xi) \nabla H(\xi), $$ is invertible with inverse~$\Psi_{H^*}$. Under this notation identity~\eqref{FK} may be read as \begin{equation} \label{FKcond2} \langle \Psi_H(\xi), \Psi_{H^*}(\eta) \rangle = \langle \xi, \eta \rangle, \end{equation} for any~$\xi, \eta \in \mathbb{R}^n$. Applying~\eqref{FKcond2} with~$\eta = \Psi_H(\zeta)$ we get $$ \langle \Psi_H(\xi), \zeta \rangle = \langle \Psi_H(\xi), \Psi_H^{-1}(\eta) \rangle = \langle \Psi_H(\xi), \Psi_{H^*}(\eta) \rangle = \langle \xi, \eta \rangle = \langle \xi, \Psi_H(\zeta) \rangle, $$ for any~$\xi, \zeta \in \mathbb{R}^n$. That is,~$\Psi_H$ is symmetric with respect to the standard inner product in~$\mathbb{R}^n$ and hence linear, by virtue of Lemma~\ref{symlem}. Therefore, there exists a symmetric~$M \in {\mbox{\normalfont Mat}}_n(\mathbb{R})$ such that $$ \nabla {\left( \frac{H^2(\xi)}{2} \right)} = H(\xi) \nabla H(\xi) = M \xi. $$ This in turn implies that~$H = H_M$ and the proof of the proposition is complete. \end{proof}
\section{On the weaker assumption~\eqref{FKweak}} \label{char2sec}
In this last section we study the condition \begin{equation} \tag{\ref{FKweak}} {\mbox{\normalfont sgn}} \langle H(\xi) \nabla H(\xi), H^*(x) \nabla H^*(x) \rangle = {\mbox{\normalfont sgn}} \langle \xi, x \rangle, \end{equation} for any~$\xi, x \in \mathbb{R}^n$, which has been introduced in the statement of Theorem~\ref{monformthm}. First, we have the following general result that provides a simpler equivalent form for assumption~\eqref{FKweak}.
\begin{proposition} \label{propFKweakeq} Let~$H$ be a~$C^1(\mathbb{R}^n \setminus \{ 0 \})$ be a positive homogeneous function of degree~$1$ satisfying~\eqref{Hpos}. Assume the unit ball~$B_1^H$, as defined by~\eqref{Hball}, to be strictly convex. Then,~\eqref{FKweak} is equivalent to the condition \begin{equation} \label{FK2zero} \langle H(\xi) \nabla H(\xi), \eta \rangle = 0 \quad \mbox{ if and only if } \quad \langle \xi, H(\eta) \nabla H(\eta) \rangle = 0, \end{equation} for any~$\xi, \eta \in \mathbb{R}^n$. \end{proposition} \begin{proof} First, we remark that, by arguing as in the proof of Theorem~\ref{FKcharprop}, it is immediate to check that~\eqref{FKweak} can be put in the equivalent form \begin{equation} \label{wFK2} {\mbox{\normalfont sgn}} \langle H(\xi) \nabla H(\xi), \eta \rangle = {\mbox{\normalfont sgn}} \langle \xi, H(\eta) \nabla H(\eta) \rangle, \end{equation} for any~$\xi, \eta \in \mathbb{R}^n$. Thus, we need to show that~\eqref{FK2zero} is equivalent to~\eqref{wFK2}.
Notice that~\eqref{FK2zero} is trivially implied by~\eqref{wFK2}. Thus, we only need to prove that the converse is also true. To see this, assume~\eqref{FK2zero} to hold and fix~$\xi \in \mathbb{R}^n$. If~$\xi = 0$, then both sides of~\eqref{wFK2} vanish, in view of Lemma~\ref{derBH0}. Suppose therefore~$\xi \ne 0$ and consider the hyperplane $$ \Pi := \left\{ \eta \in \mathbb{R}^n : \langle H(\xi) \nabla H(\xi), \eta \rangle = 0 \right\}, $$ together with the two half-spaces $$ \Pi_\pm := \left\{ \eta \in \mathbb{R}^n : \pm \langle H(\xi) \nabla H(\xi), \eta \rangle > 0 \right\}. $$ By virtue of~\eqref{FK2zero}, the function~$h: \mathbb{R}^n \to \mathbb{R}$, defined by setting $$ h(\eta) := \langle H(\eta) \nabla H(\eta), \xi \rangle, $$ vanishes precisely on~$\Pi$. Furthermore, by Lemma~\ref{derBH0} (with~$B(t) = t^2/2$),~$h$ is continuous on the whole of~$\mathbb{R}^n$ and it satisfies $$ h(\xi) = \langle H(\xi) \nabla H(\xi), \xi \rangle = H^2(\xi) > 0. $$ But~$\xi \in \Pi_+$, and so~$h$ is positive on~$\Pi_+$, being it connected. Analogously, it holds~$h(-\xi) < 0$ from which we deduce that~$h$ is negative on~$\Pi_-$. Thence,~\eqref{wFK2} follows. \end{proof}
With the aid of Proposition~\ref{propFKweakeq}, we now restrict to the planar case~$n = 2$ and show that, in this case, all the even anisotropies satisfying~\eqref{FKweak} can be obtained by means of an explicit and operative formula. As a result, it will then become clear that~\eqref{FKweak} is a weaker assumption than~\eqref{FK}.
\begin{proposition} \label{2Dprop} Let~$r: [0, \pi/2] \to (0, +\infty)$ be a given~$C^2$ function satisfying \begin{equation} \label{rcond1} r(\theta) r''(\theta) < 2 r'(\theta)^2 + r(\theta)^2 \mbox{ for a.a. } \theta \in \left[ 0, \frac{\pi}{2} \right], \end{equation}
and \begin{equation} \label{rcond2} r(0) = 1, \qquad r(\pi/2) = r^*, \qquad r'(0) = r'(\pi/2) = 0, \end{equation} for some~$r^* \geqslant 1$. Consider the~$\pi$-periodic function~$\widetilde{r}: \mathbb{R} \to (0, +\infty)$ defined on~$[0, \pi]$ by \begin{equation} \label{rtildef} \widetilde{r}(\theta) := \begin{dcases} r(\theta) & \quad {\mbox{if }} 0 \leqslant \theta \leqslant \frac{\pi}{2}, \\ \frac{r^* \sqrt{r(\tau^{-1}(\theta))^2 + r'(\tau^{-1}(\theta))^2}}{ r(\tau^{-1}(\theta))^2} & \quad {\mbox{if }} \frac{\pi}{2} \leqslant \theta \leqslant \pi, \end{dcases} \end{equation} where~$\tau: [0, \pi/2] \to [\pi/2, \pi]$ is the bijective map given by \begin{equation} \label{etaoftheta} \tau(\eta) = \frac{\pi}{2} + \eta - \arctan \frac{r'(\eta)}{r(\eta)}. \end{equation} Then,~$\widetilde{r}$ is of class~$C^1(\mathbb{R})$, the set \begin{equation} \label{Cdef} \left\{ (\rho \cos \theta, \rho \sin \theta) : \rho \in [0, \widetilde{r}(\theta)), \, \theta \in [0, 2 \pi] \right\}, \end{equation} is strictly convex and its supporting function $$ \widetilde{H}(\rho \cos \theta, \rho \sin \theta) := \frac{\rho}{\widetilde{r}(\theta)}, $$ defined for~$\rho \geqslant 0$ and~$\theta \in [0, 2 \pi]$, satisfies~\eqref{FK2zero}.
Furthermore, up to a rotation and a homothety of the plane~$\mathbb{R}^2$, any even positive $1$-homogeneous function~$H \in C^2(\mathbb{R}^2 \setminus \{ 0 \})$ satisfying~\eqref{Hpos}, having strictly convex unit ball~$B_1^H$ and for which~\eqref{FK2zero} holds true is such that~$B_1^H$ is of the form~\eqref{Cdef}, for some positive~$r \in C^2([0, \pi/2])$ satisfying~\eqref{rcond1} and~\eqref{rcond2}. \end{proposition}
Before heading to the proof of this proposition, we state the following auxiliary result.
\begin{lemma} \label{qinelem} Let~$r: [0, \pi/2] \to (0, +\infty)$ be a~$C^2$ function that satisfies condition~\eqref{rcond1} and~$r'(0) = r'(\pi/2) = 0$. Then, \begin{equation} \label{qine} - \cot \eta < \frac{r'(\eta)}{r(\eta)} < \tan \eta, \end{equation} for any~$\eta \in (0, \pi/2)$. \end{lemma}
\begin{proof} For any~$\eta \in (0, \pi / 2)$, we set $$ q(\eta) := \frac{r'(\eta)}{r(\eta)} $$ Being the tangent function increasing, we see that the right inequality in~\eqref{qine} is satisfied if and only if \begin{equation} \label{ftech} f(\eta) := \arctan q(\eta) < \eta. \end{equation} Since $$ q'(\eta) = \frac{r(\eta) r''(\eta) - r'(\eta)^2}{r(\eta)^2}, $$ we see that, for a.e.~$\eta \in (0, \pi / 2)$, $$ f'(\eta) = \frac{q'(\eta)}{1 + q(\eta)^2} = \frac{r(\eta) r''(\eta) - r'(\eta)^2}{r(\eta)^2 + r'(\eta)^2} < \frac{r(\eta)^2 + r'(\eta)^2}{r(\eta)^2 + r'(\eta)^2} = 1, $$ by virtue of~\eqref{rcond1}. Observing that~$f(0) = 0$, we then conclude that $$ f(\eta) = \int_0^\eta f'(t) \, dt < \eta, $$ which is~\eqref{ftech}. A similar argument shows that also the left inequality in~\eqref{qine} holds true. \end{proof}
\begin{proof}[Proof of Proposition~\ref{2Dprop}] Let~$H \in C^2(\mathbb{R}^2 \setminus \{ 0 \})$ be a given norm. Notice that the boundary of its unit ball~$B_1^H$ may be written in polar coordinates as $$ \partial B_1^H = \left\{ \gamma(\theta) : \theta \in [0, 2 \pi] \right\}, $$ where \begin{equation} \label{gammadef} \gamma(\theta) = (r(\theta) \cos \theta, r(\theta) \sin \theta), \end{equation} for some~$\pi$-periodic~$r \in C^2(\mathbb{R})$. Recall that the curvature of such a curve~$\gamma$ is given by \begin{equation} \label{kdef} k(\theta) = \frac{2 r'(\theta)^2 - r(\theta) r''(\theta) + r(\theta)^2}{\left[ r(\theta)^2 + r'(\theta)^2 \right]^{3/2}}, \end{equation} for any~$\theta \in [0, 2 \pi]$. Hence, hypothesis~\eqref{rcond1} tells us that~$\gamma$ has positive curvature, outside at most a set of zero measure, and, thus, that~$B_1^H$ is strictly convex.
We also remark that condition~\eqref{FK2zero} is equivalent to saying that, for any~$\theta, \eta \in [0, 2 \pi]$, \begin{equation} \label{FKparall} \gamma'(\theta) \parallel \gamma(\eta) \quad \mbox{ if and only if } \quad \gamma(\theta) \parallel \gamma'(\eta). \end{equation} This can be seen by noticing that~$\nabla H(\gamma(\theta))$ is orthogonal to~$\partial B_1^H$ while~$\gamma'(\theta)$ is tangent.
At a point~$\theta^* \in [0, 2 \pi]$ such that $$ r(\theta^*) = \max_{\theta \in \mathbb{R}} \, r(\theta) =: r^*, $$ we clearly have~$r'(\theta^*) = 0$. Assuming, up to a rotation and a homothety of~$\mathbb{R}^2$, that~$\theta^* = \pi / 2$ and~$r(0) = 1$, it is immediate to check, by computing \begin{equation} \label{gamma'} \gamma'(\theta) = \left( r'(\theta) \cos \theta - r(\theta) \sin \theta, r'(\theta) \sin \theta + r(\theta) \cos \theta \right), \end{equation} that condition~\eqref{FK2zero}, in its form~\eqref{FKparall}, forces~$r$ to satisfy~\eqref{rcond2}.
Now, take~$r \in C^2([0, \pi / 2])$ as in the statement of the proposition. We shall show that the function~$\widetilde{r}$ defined by~\eqref{rtildef} is the only extension of~$r$ which determines a curve~$\gamma$ satisfying condition~\eqref{FKparall}. Notice that, by the periodicity of~$\widetilde{r}$, it is enough to prove it for~$\theta, \eta \in [0, \pi]$. Moreover, if~$\theta, \eta \in \{ 0, \pi/2, \pi \}$, then~\eqref{FKparall} is implied by~\eqref{rcond2}. Consider now~$\eta \in (0, \pi/2)$. We address the problem of finding the unique~$\theta =: \tau(\eta) \in (0, \pi)$ such that~$\gamma(\theta) \parallel \gamma'(\eta)$. First observe that this condition is equivalent to requiring \begin{equation} \label{cottheta} \cot \theta = \frac{r'(\eta) \cos \eta - r(\eta) \sin \eta}{r'(\eta) \sin \eta + r(\eta) \cos \eta} = \frac{\frac{r'(\eta)}{r(\eta)} - \tan \eta}{\frac{r'(\eta)}{r(\eta)} \tan \eta + 1} = \tan \left( \arctan \frac{r'(\eta)}{r(\eta)} - \eta \right), \end{equation} in view of~\eqref{gammadef} and~\eqref{gamma'}. Then, we see that, by~\eqref{gamma'} and Lemma~\ref{qinelem},~$\gamma'(\eta)$ and, therefore,~$\gamma(\theta)$ lie in the second quadrant. Thus, we conclude that~$\theta \in (\pi / 2, \pi)$. Moreover, with this in hand and using again Lemma~\ref{qinelem}, it is easy to deduce from~\eqref{cottheta} that \begin{equation} \label{tauofeta} \theta = \tau(\eta) = \frac{\pi}{2} + \eta - \arctan \frac{r'(\eta)}{r(\eta)}, \end{equation} for any~$\eta \in \left[ 0, \pi / 2 \right]$. Condition~\eqref{FKparall} then implies that~$\gamma'(\theta) \parallel \gamma(\eta)$, which yields~\eqref{cottheta} with~$\eta$ and~$\theta$ interchanged. Comparing the two formulae, we deduce that~$\widetilde{r}$ should satisfy \begin{equation} \label{r'overr} \frac{\widetilde{r}'(\tau(\eta))}{\widetilde{r}(\tau(\eta))} = - \frac{r'(\eta)}{r(\eta)}, \end{equation} for any~$\eta \in \left[ 0, \pi / 2 \right]$. From this relation it is possible to recover the explicit form of~$\widetilde{r}$. In order to do this, we multiply by~$\tau'(\eta)$ both sides of~\eqref{r'overr} and integrate. The left hand side becomes \begin{equation} \label{lhsr'overr} \int_0^\eta \frac{\widetilde{r}'(\tau(t))}{\widetilde{r}(\tau(t))} \tau'(t) \, dt = \log \frac{\widetilde{r}(\tau(\eta))}{\widetilde{r}(\tau(0))} = \log \frac{\widetilde{r}(\tau(\eta))}{r^*}. \end{equation} The expansion of the right hand side requires a little bit more care. For simplicity of exposition, we will omit to evaluate~$r$ and its derivatives at~$\eta$. We deduce from~\eqref{tauofeta} that \begin{equation} \label{tau'} \tau' = 1 - \frac{r r'' - r'^2}{r^2 + r'^2} = \frac{r^2 + 2 r'^2 - r r''}{r^2 + r'^2}. \end{equation} Then, since $$ \Big[ \log \Big( r \left( r^2 + r'^2 \right) \Big) \Big]' = \frac{3 r^2 r' + r'^3 + 2 r r' r''}{r \left( r^2 + r'^2 \right)}, $$ we compute \begin{align*} - \frac{r'}{r} \tau' & = - \frac{r^2 r' + 2 r'^3 - r r' r''}{r \left( r^2 + r'^2 \right)} \\ & = \frac{1}{2} \Big[ \log \Big( r \left( r^2 + r'^2 \right) \Big) \Big]' - \frac{5}{2} \frac{r^2 r' + r'^3}{r \left( r^2 + r'^2 \right)} \\ & = \frac{1}{2} \Big[ \log \Big( r \left( r^2 + r'^2 \right) \Big) - 5 \log r \Big]' \\ & = \frac{1}{2} \left[ \log \frac{r^2 + r'^2}{r^4} \right]'. \end{align*} Integrating this last expression we get \begin{equation} \label{rhsr'overr} - \int_0^\eta \frac{r'(t)}{r(t)} \tau'(t) \, dt = \frac{1}{2} \log \left( \frac{r(\eta)^2 + r'(\eta)^2}{r(\eta)^4} \frac{r(0)^4}{r(0)^2 + r'(0)^2} \right) = \frac{1}{2} \log \frac{r(\eta)^2 + r'(\eta)^2}{r(\eta)^4}. \end{equation} By comparing~\eqref{lhsr'overr} and~\eqref{rhsr'overr}, we immediately obtain that~$\widetilde{r}$ satisfies~\eqref{rtildef}.
Now we show that~$\widetilde{r}$ has the desired regularity properties. From its definition and~\eqref{r'overr} is immediate to see that~$\widetilde{r}$ is continuous on the whole~$[0, \pi]$ and differentiable on~$(0, \pi / 2) \cup (\pi/2, \pi)$. Thus, we only need to check~$\widetilde{r}'$ at~$0$,~$\pi/2$ and~$\pi$. Using~\eqref{r'overr} and~\eqref{rcond2}, we compute \begin{equation} \label{r'jointN} \widetilde{r}' \left( {\frac{\pi}{2}}^+ \right) = - \frac{r'(0) \, \widetilde{r} \left( \frac{\pi}{2} \right)}{r(0)} = 0 = \widetilde{r}' \left( {\frac{\pi}{2}}^- \right), \end{equation} and \begin{equation} \label{r'jointEO} \widetilde{r}'(\pi^-) = - \frac{r' \left( \frac{\pi}{2} \right) \widetilde{r}(\pi)}{r \left( \frac{\pi}{2} \right)} = 0 = \widetilde{r}'(0^+). \end{equation} Being it~$\pi$-periodic, it follows that~$\widetilde{r} \in C^1(\mathbb{R})$.
Finally, we prove that the set~\eqref{Cdef} is strictly convex. To see this, it is enough to show that~$\widetilde{r}$ satisfies~\eqref{rcond1} for almost any~$\theta \in [\pi/2, \pi]$. First, we check that~$\widetilde{r}$ possesses almost everywhere second derivative. Indeed, by differentiating~\eqref{r'overr} we get \begin{equation} \label{r''impl} \left( \frac{\widetilde{r}''(\tau(\theta))}{\widetilde{r}(\tau(\theta))} - \frac{\widetilde{r}'(\tau(\theta))^2}{\widetilde{r}(\tau(\theta))^2} \right) \tau'(\theta) = - \frac{r''(\theta)}{r(\theta)} + \frac{r'(\theta)^2}{r(\theta)^2}. \end{equation} Thus, if~$\tau'(\theta) \ne 0$, which is true at almost any~$\theta \in [0, \pi / 2]$ in view of~\eqref{tau'} and~\eqref{rcond1}, we may solve~\eqref{r''impl} for~$\widetilde{r}''$ and obtain \begin{equation} \label{r''expl} \begin{aligned} \widetilde{r}''(\tau(\theta)) & = \frac{\widetilde{r}'(\tau(\theta))^2}{\widetilde{r}(\tau(\theta))} - \frac{\widetilde{r}(\tau(\theta))}{\tau'(\theta)} \left( \frac{r''(\theta)}{r(\theta)} - \frac{r'(\theta)^2}{r(\theta)^2} \right) \\ & = \frac{\widetilde{r}'(\tau(\theta))^2}{\widetilde{r}(\tau(\theta))} - \frac{\widetilde{r}(\tau(\theta)) \left( r(\theta)^2 + r'(\theta)^2 \right) \left( r(\theta) r''(\theta) - r'(\theta)^2 \right)}{r(\theta)^2 \left( r(\theta)^2 + 2 r'(\theta)^2 - r(\theta) r''(\theta) \right)}, \end{aligned} \end{equation} where in last line we made use of~\eqref{tau'}. With this in hand and recalling~\eqref{r'overr}, we are able to compute that \begin{align*} \widetilde{r}(\tau) \widetilde{r}''(\tau) - 2 \widetilde{r}'(\tau)^2 - \widetilde{r}(\tau)^2 & = \widetilde{r}'(\tau)^2 - \frac{\widetilde{r}(\tau)^2 (r^2 + r'^2) (r r'' - r'^2)}{r^2 (r^2 + 2 r'^2 - r r'')} - 2 \widetilde{r}'(\tau)^2 - \widetilde{r}(\tau)^2 \\ & = - \widetilde{r}(\tau)^2 \left( \frac{r'^2}{r^2} + \frac{(r^2 + r'^2) (r r'' - r'^2)}{r^2 (r^2 + 2 r'^2 - r r'')} + 1 \right) \\ & = - \frac{\widetilde{r}(\tau)^2 (r^2 + r'^2)^2}{r^2 (r^2 + 2 r'^2 - r r'')} \\ & < 0, \end{align*} almost everywhere in~$[0, \pi / 2]$. Thus, the proof is complete. \end{proof}
In view of Proposition~\ref{2Dprop}, every even anisotropy~$H$ satisfying~\eqref{FKweak} is uniquely determined by its values on the first quadrant. Conversely, any positive~$r \in C^2([0, \pi / 2])$ for which~\eqref{rcond1} and~\eqref{rcond2} are true can be extended to~$[0, \pi]$ (in a unique way) to obtain a~$C^1$ norm satisfying~\eqref{FKweak}.
An example of such an anisotropy, which is not of the trivial type~\eqref{HM}, is given by $$ \hat{H}_p(\xi) = \begin{cases}
|\xi|_p & \quad \mbox{if } \xi_1 \xi_2 \geqslant 0, \\
|\xi|_q & \quad \mbox{if } \xi_1 \xi_2 < 0, \end{cases} $$
where~$| \cdot |_p$ is the standard~$p$-norm in~$\mathbb{R}^2$ and~$q = p / (p - 1)$ is the conjugate exponent of~$p$, for~$p \in (2, +\infty)$ (see Figure~\ref{pnormsplot} below). It can be easily checked that~$\hat{H}_p$ satisfies~\eqref{FKweak} from formulation~\eqref{FK2zero}.
\begin{figure}
\caption{The unit circles of~$\hat{H}_p$ for the values~$p = 5/2$,~$3$ and~$4$.}
\label{pnormsplot}
\end{figure}
Unfortunately,~$\hat{H}_p$ is no more than~$C^{1, 1 / (p - 1)}_{\rm loc}(\mathbb{R}^2 \setminus \{ 0 \})$. If one is interested in norms having higher regularity properties, additional hypotheses on the behaviour of the defining function~$r$ of its unit ball inside the first quadrant need to be imposed. In particular, assumption~\eqref{rcond1} should be strengthened by requiring it to hold at \emph{any}~$\theta \in [0, \pi / 2]$. As a consequence, the class of norms under analysis is restricted to those being uniformly elliptic.
In order to deal with, say,~$C^{3, \alpha}$ anisotropies, we have the following result.
\begin{proposition} \label{2Dhrprop} Let~$\alpha \in (0, 1]$ and~$H \in C^{3, \alpha}_{\rm loc}(\mathbb{R}^2 \setminus \{ 0 \})$ be an even positive homogeneous function of degree~$1$ for which~\eqref{Hpos} holds true. Then,~$H$ is uniformly elliptic and satisfies~\eqref{FK2zero} if and only if, up to a rotation and a homothety of~$\mathbb{R}^2$, its unit ball is of the form~\eqref{Cdef}, where~$\widetilde{r}$ is given by~\eqref{rtildef} and~$r \in C^{3, \alpha}([0, \pi / 2])$ is a positive function satisfying \begin{gather} \label{rcond1str} r(\theta) r''(\theta) < 2 r'(\theta)^2 + r(\theta)^2 \mbox{ for any } \theta \in \left[ 0, \frac{\pi}{2} \right], \\ \label{rcond3} r'' \left( \frac{\pi}{2} \right) = - \frac{r^* r''(0)}{1 - r''(0)}, \qquad r''' \left( \frac{\pi}{2} \right) = - \frac{r^* r'''(0)}{(1 - r''(0))^3}, \end{gather} and~\eqref{rcond2}. \end{proposition}
Notice that the quantities appearing in both right hand sides of condition~\eqref{rcond3} are finite, as one can see by plugging~$\theta = 0$ in~\eqref{rcond1str} and recalling~\eqref{rcond2}.
\begin{proof}[Proof of Proposition~\ref{2Dhrprop}] In addition to the regularity properties of the extension~$\widetilde{r}$, by Proposition~\ref{2Dprop} we only need to investigate the relation between~\eqref{rcond1str} and the uniformly convexity of the unit ball of~$H$. Notice that in~$2$ dimensions this last requirement is just asking the curvature~$k(\theta)$, as defined by~\eqref{kdef}, to be positive at any angle~$\theta \in [0, 2 \pi]$. Hence, we see that it implies~\eqref{rcond1str}.
To check that also the converse implication is valid, it is enough to prove that if~\eqref{rcond1str} is in force, then~$\widetilde{r}$ satisfies the same inequality at any~$\theta \in [\pi / 2, \pi]$. A careful inspection of the proof of Proposition~\ref{2Dprop} - see, in particular, the argument starting below formula~\eqref{r''impl} - shows that this is true at any point~$\theta$ for which~$\tau'(\tau^{-1}(\theta)) \ne 0$. But then, comparing formula~\eqref{tau'} with~\eqref{rcond1str} we have that~$\tau' > 0$ on the whole interval~$[0, \pi / 2]$ and so we are done.
The only thing we still have to verify is that, given~$r \in C^{3, \alpha}([0, 2 \pi])$, then its extension~$\widetilde{r}$ belongs to~$C^{3, \alpha}(\mathbb{R})$. Arguing as in the proof of Proposition~\ref{2Dprop}, by~\eqref{rtildef},~\eqref{r''expl} and~\eqref{rcond1str} we deduce that~$\widetilde{r}$ is of class~$C^1$ on the whole of~$\mathbb{R}$ and $C^{3, \alpha}$ outside of the points~$k \pi / 2$, with~$k \in \mathbb{Z}$. Moreover, by the periodicity properties of~$\widetilde{r}$, we can reduce our analysis to the points~$0$,~$\pi/2$ and~$\pi$. Using~\eqref{tau'} and~\eqref{rcond2}, we compute \begin{equation} \label{tau'eval} \tau'(0) = 1 - r''(0), \qquad \tau' \left( \frac{\pi}{2} \right) = \frac{r^* - r'' \left( \frac{\pi}{2} \right)}{r^*}, \end{equation} and so, by~\eqref{r''expl},~\eqref{tauofeta},~\eqref{rcond2},~\eqref{r'jointN},~\eqref{r'jointEO} and~\eqref{rcond3}, we have \begin{align*} \widetilde{r}'' \left( \frac{\pi}{2}^+ \right) & = \frac{\widetilde{r}' \left( \frac{\pi}{2} \right)^2}{\widetilde{r} \left( \frac{\pi}{2} \right)} - \frac{\widetilde{r} \left( \frac{\pi}{2} \right)}{\tau'(0)} \left( \frac{r''(0)}{r(0)} - \frac{r'(0)^2}{r(0)^2} \right) \\ & = - \frac{r^* r''(0)}{1 - r''(0)} = r'' \left( \frac{\pi}{2} \right) = \widetilde{r}'' \left( \frac{\pi}{2}^- \right), \end{align*} and \begin{align*} \widetilde{r}''(\pi^-) & = \frac{\widetilde{r}'(\pi)^2}{\widetilde{r}(\pi)} - \frac{\widetilde{r}(\pi)}{\tau' \left( \frac{\pi}{2} \right)} \left( \frac{r'' \left( \frac{\pi}{2} \right)}{r \left( \frac{\pi}{2} \right)} - \frac{r' \left( \frac{\pi}{2} \right)^2}{r \left( \frac{\pi}{2} \right)^2} \right) \\ & = - \frac{r'' \left( \frac{\pi}{2} \right)}{r^* - r'' \left( \frac{\pi}{2} \right)} = r''(0) = \widetilde{r}''(0^+). \end{align*} Hence,~$\widetilde{r} \in C^2(\mathbb{R})$. Now we study the third derivative of~$\widetilde{r}$. By differentiating~\eqref{r''expl} we get \begin{equation} \label{r'''} \begin{aligned} \widetilde{r}'''(\tau) & = \frac{\widetilde{r}'(\tau) \left( 2 \widetilde{r}(\tau) \widetilde{r}''(\tau) - \widetilde{r}'(\tau)^2 \right)}{\widetilde{r}(\tau)^2} \\ & \quad - \frac{\left( \widetilde{r}'(\tau) \tau'^2 - \widetilde{r}(\tau) \tau'' \right) \left( r r'' - r'^2 \right)}{r^2 \tau'^3} - \frac{\widetilde{r}(\tau) \left( r^2 r''' - 3 r r' r'' +2 r'^3 \right)}{r^3 \tau'^2}, \end{aligned} \end{equation} where every function is meant to be evaluated at~$\theta$. Moreover, from~\eqref{tau'} we deduce that \begin{align*} \tau'' & = - \frac{\left( r' r'' + r r''' - 2 r' r'' \right) \left( r^2 + r'^2 \right) - 2 \left( r r'' - r'^2 \right) \left( r r' + r' r'' \right)}{\left( r^2 + r'^2 \right)^2} \\ & = \frac{3 r^2 r' r'' - r'^3 r'' - r^3 r''' - r r'^2 r''' + 2 r r' r''^2 - 2 r r'^3}{\left( r^2 + r'^2 \right)^2}, \end{align*} so that, recalling~\eqref{rcond2}, we have $$ \tau''(0) = - r'''(0), \qquad \tau'' \left( \frac{\pi}{2} \right) = - \frac{r''' \left( \frac{\pi}{2} \right)}{r^*}. $$ Thus, by plugging these identities into~\eqref{r'''}, using~\eqref{tauofeta},~\eqref{rcond2},~\eqref{tau'eval},~\eqref{r'jointN},~\eqref{r'jointEO} and~\eqref{rcond3} we finally conclude that \begin{align*} \widetilde{r}''' \left( \frac{\pi}{2}^+ \right) & = \frac{\widetilde{r} \left( \frac{\pi}{2} \right) \tau''(0) r''(0)}{r(0) \tau'(0)^3} - \frac{\widetilde{r} \left( \frac{\pi}{2} \right) r'''(0)}{r(0) \tau'(0)^2} = - \frac{r^* r''(0) r'''(0)}{\left( 1 - r''(0) \right)^3} - \frac{r^* r'''(0)}{\left( 1 - r''(0) \right)^2} \\ & = - \frac{r^* r'''(0)}{(1 - r''(0))^3} = r''' \left( \frac{\pi}{2} \right) = \widetilde{r}''' \left( \frac{\pi}{2}^- \right), \end{align*} and \begin{align*} \widetilde{r}'''(\pi^-) & = \frac{\widetilde{r}(\pi) \tau'' \left( \frac{\pi}{2} \right) r'' \left( \frac{\pi}{2} \right)}{r \left( \frac{\pi}{2} \right) \tau' \left( \frac{\pi}{2} \right)^3} - \frac{\widetilde{r} (\pi) r''' \left( \frac{\pi}{2} \right)}{r \left( \frac{\pi}{2} \right) \tau' \left( \frac{\pi}{2} \right)^2} = - \frac{r^* r'' \left( \frac{\pi}{2} \right) r''' \left( \frac{\pi}{2} \right)}{\left( r^* - r'' \left( \frac{\pi}{2} \right) \right)^3} - \frac{r^* r''' \left( \frac{\pi}{2} \right)}{\left( r^* - r'' \left( \frac{\pi}{2} \right) \right)^2} \\ & = - \frac{{r^*}^2 r''' \left( \frac{\pi}{2} \right)}{\left( r^* - r'' \left( \frac{\pi}{2} \right) \right)^3} = r'''(0) = \widetilde{r}'''(0^+). \end{align*} As a result,~$\widetilde{r} \in C^{3, \alpha}(\mathbb{R})$ and the proof of the proposition is complete. \end{proof}
We observe that Proposition~\ref{2Dprop-i} is a consequence of Propositions~\ref{2Dprop}. and~\ref{2Dhrprop}.
\begin{remark}\label{final} We point out that it is easy to construct norms which are smooth and satisfy~\eqref{FKweak} as small perturbations of those of the form~\eqref{HM}. For instance, fix any~$\psi \in C^\infty([0, \pi / 2])$ having support compactly contained in~$(0, \pi / 2)$. Then, for~$\varepsilon > 0$ define $$ r_\psi(\theta) := 1 + \varepsilon \psi(\theta), $$ for any~$\theta \in \left[ 0, \pi / 2 \right]$. Observe that conditions~\eqref{rcond2} and~\eqref{rcond3} are satisfied with~$r^* = 1$. Moreover, we compute \begin{align*} r_\psi r_\psi'' - 2 r_\psi'^2 - r_\psi^2 & = \varepsilon^2 (1 + \varepsilon \psi) \psi'' - 2 \varepsilon^2 \psi'^2 - (1 + \varepsilon \psi)^2 \\ & = - 1 + \varepsilon \left( - 2 \psi + \varepsilon \left( (1 + \varepsilon \psi) \psi'' - 2 \psi'^2 - \psi^2 \right) \right) \\ & \leqslant - 1 + c_\psi \varepsilon, \end{align*} with~$c_\psi$ dependent on the~$C^2$ norm of~$\psi$. Therefore, if we take~$\varepsilon$ small enough, then~$r_\psi$ satisfies~\eqref{rcond1str} and, by virtue of Proposition~\ref{2Dhrprop} the associated norm~$H_\psi$ is as desired. \end{remark}
\end{document} | arXiv |
Unmanned aerial vehicles optimal airtime estimation for energy aware deployment in IoT-enabled fifth generation cellular networks
Saqib Majeed1,
Adnan Sohail1,
Kashif Naseer Qureshi ORCID: orcid.org/0000-0003-3045-84022,
Arvind Kumar3,
Saleem Iqbal4 &
Jaime Lloret5
Cellular networks based on new generation standards are the major enabler for Internet of things (IoT) communication. Narrowband-IoT and Long Term Evolution for Machines are the newest wide area network-based cellular technologies for IoT applications. The deployment of unmanned aerial vehicles (UAVs) has gained the popularity in cellular networks by using temporary ubiquitous coverage in the areas where the infrastructure-based networks are either not available or have vanished due to some disasters. The major challenge in such networks is the efficient UAVs deployment that covers maximum users and area with the minimum number of UAVs. The performance and sustainability of UAVs is largely dependent upon the available residual energy especially in mission planning. Although energy harvesting techniques and efficient storage units are available, but these have their own constraints and the limited onboard energy still severely hinders the practical realization of UAVs. This paper employs neglected parameters of UAVs energy consumption in order to get actual status of available energy and proposed a solution that more accurately estimates the UAVs operational airtime. The proposed model is evaluated in test bed and simulation environment where the results show the consideration of such explicit usage parameters achieves significant improvement in airtime estimation.
The cellular coverage using unmanned aerial vehicles (UAVs) is gaining attention from research community and telecom industry with the rapid deployment of Internet of things (IoTs). For collection/dissemination of data, IoTs-based cellular technologies are being provided [1]. For provision of services in wider coverage area, the cellular networks are augmented with UAVs. UAVs are the trustable solution for improving of the efficiency, enhance throughout, cost and boosting capacity [2]. These UAVs are temporarily deployed in the air to cover an area where the user demand is increased or in the case of disasters or to provide connectivity in areas where permanent infrastructure is not currently possible. This also includes areas where physical infrastructure is available, but the user density is very high (event, stadium, etc.). The developments of UAVs not only provide the solution but also provide the load balancing, and easily cover the maximum demanding area [3]. For this purpose, an efficient approach and mechanism is required.
Previously, Cell on Wheel (COW) is used as a temporary solution in which a vehicle was designed to carry a mobile microcellular base station [4]. This solution was developed in conjunction with the Telstra Next Generation deployment plan to extend the coverage area to cover any event or emergency [5]. There are few situations in which the cow could fail, for example in areas of natural disaster where the road infrastructure necessary for displacement is not available. The basic reason for UAVs as a favorable solution for a wide range of neighborhood applications is the ability of its free and independent movement, to any hard to reach areas [6, 7]. These UAVs are furnished with the base station (BS) hardware, and these act as a flying BS, creating an attractive alternative to predictable roof or pole attached base stations. In radio communication, the BS is a wireless communication station mounted at a fixed location and used to connect as part of wireless telephone scheme. The BS relays the conversation, message and data to base stations in other cells by the wireless, cable communication or through a cable network or through a combination of wireless and cables.
The placement of these UAV-based BS in a microcell network in order to get optimal coverage area where the user are not stationary is a challenging task [8, 9]. Providing best placement while keeping the number of UAVs optimal is hard to achieve. Multiple solutions for this problem are discussed in the literature, where majority have focused on provision of the energy efficient solutions for placement of UAVs. Among them, very less attention has been paid toward energy aware solutions and additionally; such solutions have only considered implicit utilization of energy to estimate the available energy [10]. However, the explicit usage of UAV energy has not been addressed yet, which also has severe effect on the UAV survivability.
The main contribution of this paper is twofold:
Identification of UAVs exergy explicit usage parameters
To provide a solution that more accurately estimates the UAVs operational airtime
The rest of this paper is organized as follows. Section 2 presents the related work. The methodology describes in Sect. 3. Prototyping details are explained in Sect. 4. The experiment results are present in Sect. 5. Section 4 includes the conclusion and future directions of our research.
The UAV base stations (UAV-BS) enhance network coverage and area capacity by moving supply toward demand when required [11, 12]. However, deployment of such UAV-BS can face certain restrictions that need to be considered while designing a solution. The major concern is the lifetime of a UAV for which it could remain in operation. The most important factor affecting the UAVs lifetime is energy source. Following are some of the recent papers related to UAV-BS in cellular networks that focus on energy aware deployments. Thus, it is the duration for which the ground users are getting the services.
The prolonged lifetime in UAV-based network is normally achieved by designing energy efficient solutions [13,14,15,16]. Wang et al. [17] suggested in their work set of rules for UAV base stations that predicted an energy-efficient placement so that should serve users by minimal transmit energy. Also, the most appropriate placement of the UAV-BS was presented in their work by decoupling the deployment problem in horizontal and vertical dimension determined the most desirable UAV position that minimizes the transmission power by performing the simulations for hotspot and non-hotspot scenarios.
Chen, et al. [14] describe a battery-operated version of the UAV power consumption model which is then applied to a situation of UAV flight. The UAV consumes less electricity while it travels at excessive horizontal speed at some stage in the task, because low speed is not really greatest for the UAV for the energy scenario. The cause is that the hovering power is stable ultimately after the flying time. Cabreira, et al. [18] write in the paper that in case of using UAV automobile dynamics, turning angle and optimal speed must be considered to minimize electricity intake. Their proposed algorithms mainly involved about energy intake considering the mission and electricity constraints of the UAVs. Thus, improving velocity in straight parts of the path leads to energy consumption minimization. In the paper, the path divided into forty-five small elements (straights and curves) for every straight a part of the coverage it became taken into consideration: the acceleration, the steady speed flight and the deceleration. By using the equal velocity of 8. zero m/s for long elements and the reduction of 45% in velocity before turns referred to as entrance speed of UAV by using a battery of value 46, 681 J.
Another paper [15] proposed a scheme which presents the UAV offloading (air-offloading and ground-offloading) approaches. In the air-offloading approach, a UAV can offload its computing responsibilities to nearby UAVs that have to be had computing and electricity sources. The floor-offloading technique enables the project offloading carrier to an edge cloud server that is related to floor stations. This hybrid offloading scheme comprises three major modules to lengthen the life of UAVs by way of saving resources through task offloading process. Furthermore, this scheme efficiently reduces the project blocking probability and the cease-to-quit latency of managing a computing mission.
Fotouhi, et al. [16] provided option to improve communique power performance is to develop optimum transmission schedule of UAVs, mainly while UAVs are flying in a predetermined trajectory fixed-wing UAVs can movement over the air, which makes them substantially more energy effective and capable of carry heavy payload. Small commercial UAVs typically have a flight time of 20–30 min, while some big UAVs can last for hours. Researchers in this study focused on two types of energy saving reducing communication energy minimize the transmission power energy efficiency is to develop most desirable transmission time table of UAVs, especially while UAVs are flying in a predetermined trajectory. To reduce mechanical cost an energy intake model is needed. However, changing the height might reduce the performance of UAVs.
However, in these solutions the focus is on a less energy usage in terms of computational power and does not focus on transmission requirement or estimation that for how long the UAV remains present in the air. There are some solutions that optimize the energy consumption by minimizing the transmission power [19, 20]. The decision regarding the use of energy consumption has been implemented both as centralized as well as distributed approach. In [21], the author has proposed an approach in which the UAV report the statistics to the base station and the BS proposes a scheduling model that minimizes the UAV energy consumption. The base station in its scheduling informs the UAV about the time slot and power information. A similar centralized entity-based solution has also been proposed that is dependent upon cluster heads [22]. These cluster heads (Terrestrial nodes) are in direct communication with the UAV and ground users are connected with the cluster heads. But these types of solutions have their own limitations.
However, these solutions are also dependent on the estimated energy that accordingly adjusts the transmission power, this way; the UAV coverage is also affected. A third side of energy, the mechanical energy, has also been discussed by some researchers [23]. These solutions reduce mechanical energy of UAVs and are dependent upon multirotor, fixed-wing and hybrid fixed/rotary UAVs. Similarly, in [24], the authors isolated the two power consumptions into static and dynamic, the static the fixed consumption of the UAV and dynamic is dependent upon the load either low traffic load or high traffic load. In another study [25], the authors suggested a dynamic planning that is also dependent upon traffic intensity. They proposed to have different planning for day and nights, as in night the active users are less as compared to day timings and accordingly they plan the placement of UAV.
In recent past, multiple studies have been made in the domain of UAV placement as it could provide a better alternative for temporary cell construction in the uncomplimentary cell sites. The parameters that influence the placement of UAV are key factors that can severely affect the network performance. The lifetime of a UAV is greatly dependent upon the energy available. From the literature, six types of energy consumption parameters have been extracted. Energy spent on (a) horizontal movement, (b) vertical movement, (c) hovering, (d) processing, (e) backbone connectivity/communication and (f) providing front haul/ access to users. However, all these six parameters have not been given full consideration for energy aware solutions, which provide the estimated time of UAV for which it can remain in the air. This estimation greatly helps in UAVs placement/replacement planning.
Authors in [17] proposed an energy-efficient placement algorithm by decoupling the drone deployment in horizontal and vertical locations, by using minimum required transmit power. They perform simulation on hotspot and non-hotspot scenarios, and the numerical results show the linear association between the minimum horizontal location and optimal altitude. They first find the optimal horizontal position of the Drone base station (DBS). The average path loss against the altitudes for various horizontal distances between the edge users and the drone, there lies a point of minimum value of average path loss of the edge users and the relevant drone altitude. Furthermore, by decreasing the horizontal distance of the edge users from the DBS, the minimum average path loss of the edge and the altitude decrease. Once DBS obtains the minimum horizontal distance, corresponding optimal drone altitude is calculated. The results show the power saving in the urban, suburban and dense urban environments optimal altitude.
In [26], the researcher develops the photovoltaic power management system (PPMS) which manages power from photovoltaic modules and a battery pack for multirotor UAV power. They use the state of charge concepts grounded on extended Kalman filter (EKF) and complementary filter (CF) for estimation of flight time. It also calculates the possible flight time during hovering flight mode and hint of the remaining energy of the battery pack by using the slope of the state of charge graph. During takeoff, hovering and landing flight modes patterns are estimated in these three modes of UAVs and the mean value is calculated. There were three main power connectors connected to the photovoltaic modules, battery pack and UAV. Voltage and the current of all three main power connectors were monitored. During low sunlight, battery pack mostly deliveries the required drone power. During daytime, photovoltaic modules deliver the power required from the drone. They keep track of the voltage and current data of the photovoltaic modules, battery pack and UAV, progress time and battery pack temperature. According to the results, estimated flight time increased up to 54.14 min at 11:00 a.m. and decreased down to 6.70 min at 18:00 p.m. The results also indicated that if there were no clouds covering the sun, the UAV could fly for about an hour at around noon which was much higher than the flight time of the traditional multirotor UAV. The future work is to improve the current version of the PPMS.
Methodology-UAV airtime estimation for energy aware deployment in IoT
UAV system model
The airtime estimation is dependent upon three parameters: (a) the available capacity of the battery mounted on the UAV, (b) its discharge limit and (c) the average amp draw. In almost all of the previous research work and practical deployments, these three parameters have been considered. For this, the following equation is used:
$${\text{AT}} = C_{{\text{B}}} D_{{\text{L}}} /{\text{AA}}_{{\text{D}}}$$
where \({\text{AT}}\) is the airtime, \(C_{{\text{B}}}\) is the battery capacity and \(D_{{\text{L}}}\) is the allowed discharge limit, in normal practice, the discharge limit is set to 80%, whereas the \({\text{AA}}_{\rm D}\) is the Average Amp Draw which is calculated in amperes using following Eq. 2:
$${\text{AA}}_{\rm D} =_{{\text{C}}} \frac{{P_{{\text{W}}} }}{\Delta V }$$
where WC represents the weight carried, which represents the total weight of the UAV including the equipment/battery that will be carried by the UAV; usually, it is measured in kilograms. The \(P_{{\text{W}}}\) is the power required to carry one kilogram of weight, normally expressed in watts per kilogram. The ∆V is the battery voltage, expressed in volts. However, the estimated airtime may differ from the real airtime such as in extreme wind cases; the airtime may decrease up to 50%.
Such estimation normally gives a generic value. The actual estimation is also influenced by the task being performed by the UAV. In the proposed work, the UAV is carrying a base station as shown in modular form in Fig. 1. This base station has connectivity modules for communication with backbone and frontend module for providing access services to the ground users as shown in Fig. 2. Both of these modules are also dependent upon the same battery mounted on UAV for its flying (implicit usage). The details of flow represented in Fig. 2 are calculated by using Eqs. 3 and 4. The flying requires rotation of motors; hence, such usage has also been termed as mechanical energy in the literature [27].
UAV-BS core components
Explicit energy usage for UAV
In the proposed work, the explicit usage (backend and frontend communication) is a key factor in UAV-BS scenario; therefore, it is given the due consideration. However, the communication energy overhead in other UAV deployments is considered negligible [27]. The main task performed by the UAV-BS is communication; hence, communication energy is incorporated as depicted in Fig. 3. Here the bidirectional links, highlighted in red, represents the uplink and downlink from/toward UAV from user equipment (UE) and base station (BS). Accordingly, Eq. 3 is extended as follows:
$${\text{AT}} = C_{{\text{B}}} D_{{\text{L}}} /\left( {{\text{AA}}_{{\text{D}}} + E_{{\text{U}}} } \right)$$
Proposed enhancement in UAV airtime estimation
The \(E_{{\text{U}}}\) in the above equation represents the explicit usage factor. The average energy consumptions of a base transceiver station (BTS) is associated with the communication technology being used, as depicted in the literature while comparing Global System Mobile (GSM) and Universal Mobile Telephone Services (UMTS), it was noted that the GSM considerably have higher energy consumption than the UMTS technology [28]. Similarly, multiple energy consumption analyses have been performed for various communication technologies such as for Bluetooth, Wi-Fi and Cellular Networks [29, 30].
Energy aware deployment scenarios
The Explicit Usage factor is dependent upon the user location, density and the UAV coverage radius [16]. The user density is normally divided into three categories: (a) suburban (b) urban and (c) dense urban [17]. Healthcare is the prominent use case for urban drone deployment [31, 32]. The users on the boundary of the coverage area are the users with the maximum horizontal distance from the UAV, normally referred edge users. While calculating the UAV coverage radius, such edge users have also been considered along with the UAV altitude [17]. The transmission power required by a UAV considering the user density and the UAV coverage area is calculated by [33] and is depicted in Table 1. Here, secure data transmission is a priority for this study considering healthcare application of drones [34]. It is observed that having greater number of cells (more UAVs) requires a transmission power that is not having severe effect but when there are less numbers of cells, a quite high transmit power of (≥ 53.9 W) is required. This effect is mathematically incorporated using the following equation.
$$E_{{\text{U}}} = \frac{{P_{{\text{T}}} }}{\Delta V }$$
where \(P_{{\text{T}}}\) is the transmission power required by a UAV in order to cover a microcell and \(V\) is the battery voltage.
Table 1 \(P_{\rm T}\) requirement to cover active users in each circle cell [31]
An example for UAV airtime calculation has been performed in [35]. Let us mathematically analyze both of our cases of with and without Explicit Usage factor using the same example, the specifications and their values used for the example are given in Table 2.
Table 2 UAV airtime calculation specifications
Scenario 1 (a): Without explicit usage
For calculating UAV airtime (AT) without any explicit usage, the following steps are followed:
Calculating Average Amp Draw using Eq. 2
Calculating UAV airtime using Eq. 1
Scenario 1 (b): Without Explicit Usage (with increasing UAV height)
The flying height of the UAV has its influence not only on the coverage of the base station and user density but also results in a higher power usage. Here, link quality consideration can also be a significant point focusing on the highly mobility drone networking environment [36]. Increasing the altitude by 15 m results in consumption of an additional power of 0.5A. If this is also included into the airtime calculation, Eq. 2 is extended in order to cover Amps for the height factor (\(A_{\rm HF}\)) as represented in Eq. 7.
$${\text{AA}}_{{\text{D}}} = _{C} \left( {\frac{{P_{{\text{W}}} }}{V} + A_{{{\text{HF}}}} } \right)$$
Now following steps are followed for calculating UAV airtime (AT) without any explicit usage but considering height factor.
Calculating \({\text{AA}}_{{\text{D}}}\) using Eq. 5
Scenario 2: With explicit usage
Calculating the explicit usage (\(E_{{\text{U}}}\)) factor using Eq. 4
Calculating \({\text{AA}}_{D}\) using Eq. 2
Calculating UAV airtime (AT using Eq. 3
For the proof of concept, consider the following example cases:
Case I: Transmission Power (\(P_{\rm T}\)) = 53.9 W (based on Table 1a).
Case II: Transmission Power(\(P_{\rm T}\)) = 267.1 W (based on Table 1b).
In scenario 2, the UAV hovers for 30 min and 21.6 min, respectively, for cases I and II. If compared with scenario I (without Explicit Usage Factor), the calculations show that the airtime of 36 min difference makes a lot of impact while planning deployment of UAVs.
Drone prototype development
A UAV prototype is indigenously developed as depicted in Fig. 4. For backhaul and front haul communication, a local wireless service provider's dongle (5G) was used that also provides IEEE 802.11 (b/g/n)-based access network as shown in Fig. 5. This supports up to ten connections simultaneously over Wi-Fi and have 2,380 mAh battery for up to 5 h of constant usage. To accommodate the maximum number of users, an additional wireless router is plugged. The detailed specifications are given in Table 3.
Side view-UAV prototype
Upper view-UAV prototype
Table 3 UAV specifications
The UAV profiles were obtained for transmission requirement at different heights and analyzed accordingly.
The UAV altitude affects the energy from two perspectives: (a) the energy required by motors to increase the UAV altitude and (b) with higher altitude the transmission power also increases in order to cover the required area on ground. In this research, takeoff, landing and moving vertical or horizontal while changing the altitude and increasing the coverage radius have not considered. But because we are working on 80% usage of UAV so its rational to ignore such parameters in order to focus the transmission impact on airtime and limiting the research scope. Figure 6 shows the impact of UAV altitude on radius, and the radius tends to increase when the UAV increases its altitude, thus also affecting scattered users and the density that needs to be covered.
Impact of UAV altitude on Tx power
Increasing the altitude of a UAV also effects on its transmission power, and now the UAV requires more powerful transmission in order to cover the increased radius. The UAV are reliant on a limited size on chip battery. More power full transmission depletes the battery abruptly thus directly dropping the available amperes. Figure 7 shows the same in comparison with the increasing altitude. The amp draw increases in linear fashion as the altitude is increased in start, but later on especially at altitude above 500 m, the transmission power has a noticeable effect on the ampere draw of the battery and it results in less flight of a UAV. Thus, disrupting the service availability and suspension of the UAV operation.
Impact of UAV altitude on amp draw
Figure 8 shows the airtime estimation with and without considering the transmission impact in comparison with the altitude. It is witnessed that as the altitude is increased, the required transmission power also increases proportionally and it directly affects the estimated airtime of a UAV. It is worth to note that initially, the slope of airtime estimation decreases slightly but later on the decline on fast pace.
Impact of altitude on airtime estimation
The difference discussed in Fig. 8 (with and without the transmission impact) is also calculated separately in order to access the scale of unavailability of UAV service as depicted in Fig. 9. This difference is substance depending upon the UAV application as the case may be of critical nature.
Impact of amp draw on service availability
The UAVs deployment is largely dependent upon the available energy, and it is a key factor in UAV planning. The under- or overestimation of UAV air timing leads to wastage of resources or inefficiency of mission critical projects. Multiple factors influence the estimation of air timing but the majority of the literature concentrates on only flying time. In this research, the other factors are also accounted for that improved the estimation. In this research, the energy consumption is bifurcated into two usage scenarios (1) Implicit and (2) Explicit. The implicit usage was given its due weight in the previous solution; therefore, this research focused on explicit factors that have discussed in the literature but have not been given concentration in estimating the airtime and left as negligible factor. However, simulation has been performed and it is witnessed that the explicit factors of transmission power have severe effect on airtime estimation which is a factor not to be ignored in mission critical operation. In future, we will enhance the proposed idea and test with more parameters such as data centric fuzzy approach [37] and drone networking for marine applications [38].
The experimental data and associated settings will be made available to researchers and practitioners on individual requests with the restrictions that it will be used for further investigation with collaborative research progress only.
UAVs:
IoTs:
COW:
Cell on Wheel
UAV-BS:
UAV base stations
DBS:
Drone base station
PPMS:
Photovoltaic power management system
EKF:
Extended Kalman filter
PWR :
Power required to carry one kilogram of weight
VOL :
BTS:
Base transceiver station
Global System Mobile
Universal Mobile Telephone Services
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The research is funded by the Department of Computer Science, Iqra University, Islamabad Campus, Pakistan.
Department of Computing and Technology, Iqra University, Islamabad Campus, Pakistan
Saqib Majeed & Adnan Sohail
Department of Computer Science, Bahria University, Islamabad, Pakistan
Kashif Naseer Qureshi
School of Computing Science and Engineering (SCSE), Galgotias University, Greater Noida, Gautam Buddh Nagar, Uttar Pradesh, 201308, India
Arvind Kumar
University Institute of Information Technology, PMAS-AAUR, Rawalpindi, Pakistan
Saleem Iqbal
Universitat Politecnica de Valencia, 46022, Valencia, Spain
Jaime Lloret
Saqib Majeed
Adnan Sohail
SM has modeled and executed the research. AS and KNQ have supervised the research. SM has designed experimental testing and data visualization. AK and SI have validated mathematical design and testing. JL has enhanced the quality of the research by their valuable comments and suggestions in data analysis and discussion. SM has written the paper, where AS and KNQ, AK, SI and JL have improved the technical contents of the paper with comments and suggestions. All authors read and approved the final manuscript.
Correspondence to Kashif Naseer Qureshi.
It is declared that there is no competing interest among authors.
Majeed, S., Sohail, A., Qureshi, K.N. et al. Unmanned aerial vehicles optimal airtime estimation for energy aware deployment in IoT-enabled fifth generation cellular networks. J Wireless Com Network 2020, 254 (2020). https://doi.org/10.1186/s13638-020-01877-0
Energy aware
Dynamic deployment
Communicational energy
Green Communication for Heterogeneous Internet of Things | CommonCrawl |
\begin{document}
\title{Graded identities of matrix algebras and the universal
graded algebra}
\section{Introduction.}
In the last decade, group gradings and graded identities of finite dimensional central simple algebras have been an active area of research. We refer the reader to Bahturin, et al \cite{BSZ} and \cite{BZ}. There are two basic kinds of group grading, elementary and fine. It was proved by Bahturin and Zaicev \cite{BZ} that any group grading of $M_{n}(\mathbb{C})$ is given by a certain ¨composition¨ of an elementary grading and a fine grading. In this paper we are concerned with fine gradings on $M_{n}(\mathbb{C})$ and their corresponding graded identities.
Let $R$ be a simple algebra, finite dimensional over its center $k$ and $G$ a finite group. We say that $R$ is fine graded by $G$ if $R\cong \oplus_{g\in G} R_{g}$ is a grading and $\dim_{k}(R_{g})\leq 1$. Thus any component is either $0$ or isomorphic to $k$ as a $k$--vector space. It is easy to show that $\mathop{\rm Supp}\nolimits(R)$, the subset of elements of $G$ for which $R_{g}$ is not $0$, is a subgroup of $G$. Moreover $R$ is strongly graded by $\mathop{\rm Supp}\nolimits(R)$, namely $R_{g}R_{h}=R_{gh}$ for every $g,h \in \mathop{\rm Supp}\nolimits(R)$. Since every group $\Gamma$ containing $\mathop{\rm Supp}\nolimits(R)$ provides a fine grading of $R$ (just by putting $R_{g}=0$ for $g$ outside $\mathop{\rm Supp}\nolimits(R)$) we will restrict our attention to the case where $G=\mathop{\rm Supp}\nolimits(R)$. In this case it has been shown (see Bahturin and Zaicev \cite{BZ}) that the existence of such a grading is equivalent to $R$ being isomorphic to a twisted group algebra $k^cG$ where $[c]$ is an element in $H^{2}(G,k^\times)$ and $G$ acts trivially on $k^\times$.
A group $G$ is said to be of central type if it admits a cocycle $c$, $[c]\in H^{2}(G,\mathbb{C}^\times)$, for which the twisted group algebra $\mathbb{C}^cG$ is central simple over $\mathbb{C}$. By definition such a cocycle $c$ is called nondegenerate. As we have just seen fine gradings arise from such groups and cocycles. Groups of central type appear in the theory of projective representations of finite groups and in the classification of finite dimensional Hopf algebras. Using the classification of finite simple groups, Howlett and Isaacs \cite{HI} proved that any group of central type is solvable.
Given a (fine) $G$--grading on $M_{n}(\mathbb{C})$ we consider the set of graded identities: let $k$ be a subfield of $\mathbb{C}$ and let $\Omega=\{x_{ig}:i\in \mathbb{N}, g\in G\}$ be a set of indeterminates. Let $\Sigma(k)=k\langle \Omega \rangle$ be the noncommutative free algebra generated by $\Omega$ over $k$. A polynomial $p(x_{ig})\in \Sigma(k) $ is a graded identity of $M_{n}(\mathbb{C})$ if the polynomial vanishes upon every substitution of the indeterminates $x_{ig}$ by elements of degree $g$ in $ M_{n}(\mathbb{C})$. It is clear that the set of graded identities is an ideal of $\Sigma(k)$. Furthermore, it is a graded $T$--ideal. Recall that an ideal (in $\Sigma(k)$) is a graded $T$--ideal if it is closed under all $G$--graded endomorphisms of $\Sigma(k)$. In section two we show that all graded identities are already defined over a certain finite cyclotomic field extension $\mathbb{Q}(\mu)$ of $\mathbb{Q}$. Moreover $\mathbb{Q}(\mu)$ is minimal and unique with that property. We refer to $\mathbb{Q}(\mu)$ as the field of definition of the graded identities. We consider the free algebra $\Sigma(\mathbb{Q}(\mu))$ and denote by $T(\mathbb{Q}(\mu))$ the $T$--ideal of graded identities of $M_{n}(\mathbb{C})$ in $\Sigma(\mathbb{Q}(\mu))$. We introduce a set of special graded identities which we call ``elementary''. We show that a certain finite subset of the set of elementary identities generate $T(\mathbb{Q}(\mu))$. In particular $T(\mathbb{Q}(\mu))$ is finitely generated as a $T$--graded ideal.
In section three we examine the algebra $U_{G}= \Sigma(\mathbb{Q}(\mu))/T(\mathbb{Q}(\mu))$, which we call the universal $G$--graded algebra. First we introduce another algebra analogous to the ring of generic matrices in the classical theory. We show this algebra is isomorphic to $U_G$ and thus are able to prove that the center $Z=Z(U_{G})$ is a domain and that if $F$ denotes the field of fractions of $Z$, then the algebra $Q(U_G)=F\otimes_{Z}U_{G}$ is an $F$--central simple algebra of dimension equal to the order of $G$. In particular $U_G$ is a prime ring. Moreover we show there is a certain multiplicatively closed subset $M$ in $Z$ such that the central localization $M^{-1}U_G$ is an Azumaya algebra over its center $S=M^{-1}Z$. The simple images of this Azumaya algebra are the graded forms of $M_n(\mathbb{C})$, that is the $G$--graded central simple $L$--algebras $B$ such that $B\otimes_L\mathbb{C}$ is isomorphic as a graded algebra to $M_n(\mathbb{C})$, where $L$ varies over all subfields of $\mathbb{C}$. We also give quite explicit determinations of $S$ and $F$.
Note that these algebras depend on the given grading on $M_{n}(\mathbb{C})$ and so we should write $U_{G,c}$, $M^{-1}U_{G,c}$ and $Q(U_{G,c})$, where $c$ is the given nondegenerate two-cocycle. We will omit the $c$ except in those cases where we need to emphasize the particular grading.
Unlike the case of classical polynomial identities, the central simple algebra $Q(U_{G})$ is not necessarily a division algebra. Based on earlier work of the authors \cite{AHN} we in fact show that $Q(U_{G,c})$ is a division algebra if and only if the group $G$ belongs to a certain explicit list $\Lambda$ of groups. This is independent of the cocycle $c$, a fact we will return to below. The list $\Lambda$ consists of a very special family of nilpotent groups. Roughly speaking these are the nilpotent groups for which each Sylow-$p$ subgroup is the direct product of an abelian group of the form $A\times A$ (called of symmetric type) and possibly a unique nonabelian group of the form $C_{p^{n}}\ltimes C_{p^{n}}$. For $p=2$ an extra family of non abelian groups can occur, namely $C_{2}\times C_{2^{n-1}}\ltimes C_{2^{n}}$. The precise definition of $\Lambda$ is given in the last section.
The main result of the final section is that for every group $G$ on the list $\Lambda$ the automorphism group of $G$ acts transitively on the cohomology classes represented by nondegenerate two-cocycles. It follows that the algebras $Q(U_{G,c})$ are all isomorphic for a fixed $G$ (but not graded isomorphic).
For a group $G$ of central type let $\mathop{\rm ind}\nolimits(G)$ denote the maximum over all nondegenerate cocycles $c$ of the indices of the simple algebras $Q(U_{G,c})$. We have just seen that if $G$ is not on the list, then $\mathop{\rm ind}\nolimits(G)$ is strictly less than the order of $G$. In a forthcoming paper, Aljadeff and Natapov \cite{AN}, it is shown that in fact the groups on the list are the only groups responsible for the index. The precise result is stated in the last section (Theorem \ref{responsible.thm}). It follows from this theorem that if $G$ is a $p$-group of central type then $\mathop{\rm ind}\nolimits(G)\leq \max(\mathop{\rm ord}\nolimits(H)^{1/2})$ where the maximum is taken over all $p$-groups $H$ on the list that are sub-quotients of $G$.
If $Q(U_{G})$ is not a division algebra, there are graded identities over the field of definition that are the product of nonidentities. We present an explicit example of a grading on $M_6(\mathbb{C})$ (for the group $G=S_3\ltimes C_{6}$) for which there is a graded identity over the field of definition that is the cube of a nonidentity.
Even in the case where $Q(U_{G})$ is a division algebra it is possible that this universal algebra does not remain a division algebra under extension of the coefficient field. In other words the algebra $Q(U_{G})\otimes_{\mathbb{Q}(\mu)}\mathbb{C}$ may not be a division algebra. This means that there can be gradings on matrices for which the algebra $Q(U_{G})$ is a division algebra and yet there are graded identities over $\mathbb{C}$ that are the products of nonidentities. In the last section we compute explicitly the index of this extended algebra $Q(U_{G})\otimes_{\mathbb{Q}(\mu)}\mathbb{C}$ for groups $G$ on the list $\Lambda$.
\section{Graded Identities.}
Let $M_{n}(\mathbb{C})$ have a fine grading by the finite group $G$. In this section we investigate the $T$--ideal of $G$--graded identities on $M_{n}(\mathbb{C})$. As we have seen in the introduction the existence of such a grading implies that $G$ is of central type and that there is a nondegenerate cocycle $c$ such that $M_{n}(\mathbb{C})$ is isomorphic to the twisted group algebra $\mathbb{C}^cG$. We want to make this more precise. Let $A=M_{n}(\mathbb{C})$. The $G$--grading induces a decomposition $A \cong \oplus A_{g}$ where each homogeneous component is $1$-dimensional over $\mathbb{C}$. Every homogeneous component $A_{g}$ is spanned by an invertible element $u_{g}$ (which we also fix from now on) and hence any element in $A_{g}$ is given by $\lambda_{g}u_{g}$ where $\lambda_{g}\in \mathbb{C}$. The multiplication is given by $u_{g}u_{h}=c(g,h)u_{gh}$ where $c(g,h)\in \mathbb{C}^\times$ is a two-cocycle. We let $[c] \in H^{2}(G,\mathbb{C}^\times)$ be the corresponding cohomology class. In this way we identify $A$ (as a $G$-graded algebra) with the twisted group algebra $\mathbb{C}^cG$. Replacing $c$ by a cohomologous cocycle $c'$ produces a twisted group algebra $\mathbb{C}^{c'}G$ that is isomorphic as a $G$--graded algebra to $A$.
We want to obtain information about the $T$--ideal of identities of $M_{n}(\mathbb{C})$. We begin with a special family of identities. As in the introduction we let $\Sigma(\mathbb{C})=\mathbb{C} \langle \Omega \rangle$ denote the free algebra generated by $\Omega$ over $\mathbb{C}$, where $\Omega=\{x_{ig}:i\in \mathbb{N}, g\in G\}$.
Let $Z_{1}=x_{r_1g_{1}}x_{r_2g_{2}}\cdots x_{r_kg_{k}}$ and $Z_{2}=x_{s_1h_{1}}x_{s_2h_{2}}\cdots x_{s_lh_{l}}$ be two monomials in $\Sigma(\mathbb{C})$. We say $Z_1$ and $Z_2$ are {\it congruent} if the following three properties are satisfied:
a) $g_{1} \cdots g_{k}=h_{1} \cdots h_{k}$
b) $k=l$ (equal length)
c) there exists an element $\pi \in Sym(k)$ such that for $1\leq i \leq k$, $ s_{i}=r_{\pi(i)}$ and $h_i=g_{\pi(i)}$.
We say $Z_{1}$ and $Z_{2}$ are {\it weakly congruent} if they satisfy condition (a). Clearly congruence and weakly congruence are equivalence relations.
Let $c(g_1,g_2,\dots, g_k)$ be the element in $\mathbb{C}^{\times}$ determined by $u_{g_1}u_{g_2}\cdots u_{g_k}=c(g_1,\dots, g_k)u_{g_1\cdots g_k}$ in $\FAT{C}^cG$. For any two congruent monomials $Z_{1}=x_{r_1g_{1}}x_{r_2g_{2}}\cdots x_{r_kg_{k}}$ and $Z_{2} = $ \\ $x_{r_{\pi(1)}g_{\pi(1)}}x_{r_{\pi(2)}g_{\pi(2)}}\cdots x_{r_{\pi(k)}g_{\pi(k)}}$ consider the binomial
$$B = B(Z_1,Z_2) = Z_1-{c(g_1,\dots,g_k)\over c(g_{\pi(1)},\dots,g_{\pi(k)})}Z_2$$
It is easy to see that $B$ is a graded identity. We will call it an {\it elementary} identity. We refer to $c(B)={c(g_1,\dots,g_k) \over c(g_{\pi(1)},\dots,g_{\pi(k)})}$ in $\mathbb{C}^{\times}$ as the coefficient of the elementary identity $B$.
This element $c(B)$ can be understood homologically. Let $F=F(\Omega)$ be the free group generated by $\Omega=\{x_{ig}:i\in \mathbb{N}, g\in G\}$. Let $\nu: F\longrightarrow G$ $(x_{ig} \longmapsto g)$ be the natural map and $R=\ker(\nu)$. If $\pi\in Sym(k)$ and $Z_1=x_{r_1g_{1}}x_{r_2g_{2}}\cdots x_{r_kg_{k}}$, $Z_2=x_{r_{\pi(1)}g_{\pi(1)}}x_{r_{\pi(2)}g_{\pi(2)}}\cdots x_{r_{\pi(k)}g_{\pi(k)}}$ are congruent, then the element $z=Z_1(Z_2)^{-1}$ lies in $[F,F]\cap R$. By the Hopf formula the Schur multiplier $M(G)$ of $G$ is given by $M(G)=[F,F]\cap R/[F,R]$ and so we may consider the element $\overline{z}\in M(G)$. Moreover the Universal Coefficients Theorem provides an isomorphism $\phi$ from
$H^{2}(G,\mathbb{C}^\times)$ to $\mathop{\rm Hom}\nolimits(M(G),\mathbb{C}^\times)$,
\begin{prop} \label{elem.prop} Let the notation be as given above.
(a) The element $c(B)$ is independent of the representing cocycle $c$ of $[c]$.
(b) If $z=Z_1(Z_2)^{-1}$ is as given above then $c(B)=\phi([c])(\overline{z})$.
(c) Each $c(B)$ is a root of unity, in fact $c(B)^{\mathop{\rm ord}\nolimits(G)}=1$.
(d) If $J$ denotes the set of all elementary identities then the set $\mu= \{c(B)\in \mathbb{C}^{\times}: B\in J\}$ is finite. \end{prop}
\proof Statement (a) is clear. For the proof of (b) we consider the central extension
$$\{1\} \rightarrow R/[F,R] \rightarrow F/[F,R] \rightarrow G \rightarrow \{1\}$$ induced by the map $\nu$ and let $$\{1\} \rightarrow \mathbb{C}^\times \rightarrow \Gamma \rightarrow G \rightarrow \{1\}$$ be the central extension that corresponds to the cocycle $c$ (i.e. $\{u_{g}\}_{g\in G}$ is a set of representatives in $\Gamma $ and $u_{g}u_{h}=c(g,h)u_{gh}$). It is easy to see that the map $\gamma: F/[F,R] \rightarrow \Gamma $ given by $\gamma(x_{ig})=u_{g}$ induces a map of extensions and it is well known its restriction $\gamma: [F,F]\cap R/[F,R]=M(G)\rightarrow \mathbb{C}^\times$ is independent of the presentation and corresponds to $[c]$ by means of the Universal Coefficient Theorem. That is $\phi([c])=\gamma$. It follows that $\phi([c])(\overline{z})=\gamma(\overline{z})= u_{g_{1}}\cdots u_{g_{r}}(u_{g_{\pi(1)}}\cdots u_{g_{\pi(r)}})^{-1}$ $={c(g_1,\dots,g_k) \over c(g_{\pi(1)},\dots,g_{\pi(k)})}= c(B)$ as desired.
Statements (c) and (d) are direct consequences of (b). This completes the proof of the proposition.\qed
From the Proposition it follows that the set $J$ is defined over $Q(\mu)$. Our next task is to show that the the $T$--ideal of graded identities $T(\mathbb{C})$ is spanned by $J$ and that $Q(\mu)$ is the field of definition for the graded identities. We will need the following terminology:
Let $k$ be a subfield of $\mathbb{C}$ and let $p(x_{r_ig_i})$$=$ $\sum \lambda_{r_1g_1,r_2g_2,\dots,r_kg_k} x_{r_1g_1}x_{r_2g_2}\cdots x_{r_kg_k}$ be a polynomial in $\Sigma(k)$ ($\subseteq$ $\Sigma(\mathbb{C})$). We say $p(x_{r_ig_i})$ is {\it reduced} if the monomials $x_{r_1g_1}x_{r_2g_2}\cdots x_{r_kg_k}$ are all different. If $p(x_{r_ig_i})$ is reduced we say it is weakly homogeneous if its monomials are all weakly congruent and we say $p(x_{r_ig_i})$ is homogeneous is if its monomials are all congruent. It is clear that any reduced polynomial can be written uniquely as the sum of its weakly homogeneous components and that each weakly homogeneous component can be written uniquely as the sum of its homogeneous components.
\begin{prop} \label{decomposition.prop} If $p(x_{r_ig_i})$ is a graded identity then its homogeneous components are graded identities as well. Moreover, any graded identity is a linear combination of elementary identities. \end{prop}
\proof Let us show first that its weakly homogeneous components are graded identities. Indeed, the replacement of $x_{r_ig_i}$ by $u_{g_i}\in \FAT{C}^{c}G$ maps weakly homogeneous components of $p(x_{r_ig_i})$ to different homogeneous components (in the graded decomposition of $\FAT{C}^{c}G$). Since these are linearly independent, they all must be $0$. Next, assume that we have at least two homogeneous components in a weakly homogeneous component which we denote by $\Pi$. Then there exists an indeterminate $y=x_{r_jg_{j}}$ which appears with different multiplicities in (at least) two different monomials. Let $s\geq 1$ be the maximal multiplicity. We decompose $\Pi$ into (at most) $s+1$ components $U_s,\dots,U_0$ where $U_i$ consists of the monomials of $\Pi$ that contain $y$ with multiplicity $i$. Of course, by induction, the result will follow if we show that $U_{s}$ is a graded identity. Suppose not. Then there is a substitution which does not annihilate $U_{s}$. Note that since $\Pi$ is weakly homogeneous the image of all components are multiples of $u_g$ in $\FAT{C}^{c}G$ and hence there is an evaluation that maps $U_i$ to $\lambda_i u_g$ with $\lambda_s$ not zero. We may multiply the evaluation for $y$ by a central indeterminate $z$. Then we get a non zero polynomial in $z$, whose coefficients are $\lambda_i$ and for any evaluation of $z$ we get zero. This is of course impossible in a field of characteristic zero.
To complete the proof of the proposition it suffices to show that every homogeneous identity is a linear combination of elementary identities. But this is clear since any two monomials which are congruent determine an elementary identity (and monomials are not identities).\qed
\begin{prop} \label{fielddef.prop} If $L$ is a subfield of $\FAT{C}$ such that $\mathbb{Q}(\mu)\subseteq L \subseteq \mathbb{C}$ then the set $J$ spans $T(L)$ over $L$. Conversely, if $L\subseteq \mathbb{C}$ is a field of definition for $T(\mathbb{C})$, that is $T(\mathbb{C})=T(L)\otimes _{L}\mathbb{C}$, then $L$ contains $\mathbb{Q}(\mu)$. \end{prop}
\proof Fix a monomial $Z=x_{r_1g_1}x_{r_2g_2}\cdots x_{r_kg_k}$. Let $Z=Z_1, Z_2 \dots ,Z_d$ be the distinct monomials that are congruent to $Z$ and let $W$ be the $d$-dimensional $\mathbb{C}$--vector space spanned by these monomials. Each $Z_i$, for $1\leq i < d$ determines an elementary identity $Z_i-c_iZ_d$ and these identities form a basis for the subspace $Y$ of graded identities in $W$. In particular the dimension of $Y$ over $\mathbb{C}$ is $d-1$. Furthermore if $\gamma=\{c_1,\dots,c_{d-1}\}$ then clearly $Y$ is defined over $\mathbb{Q}(\gamma)$. But $\gamma\subseteq \mu$ and so it follows that $T(\mathbb{C})$ is defined over $\mathbb{Q}(\mu)$ and hence over any field $L$ that contains $\mathbb{Q}(\mu)$.
For the converse let $W_L$ denote the $L$--span of the monomials $Z_1,\dots, Z_d$ and let $Y_L=Y\cap W_L$. We claim that if $L$ is a field of definition for $Y$ (that is, $Y = Y_L\otimes_{L} \mathbb{C}$), then $L\supseteq \mathbb{Q}(\gamma)$. Because $Z$ is arbitrary, it will follow from the claim that $L$ contains $\mu$. To prove the claim, let $f_{1},\dots,f_{d-1}$ be a basis of $Y_L$ over $L$. Express $f_{1},\dots,f_{d-1}$ using the monomials $Z_{i}$. The coefficient matrix is a $(d-1)\times d$ matrix over $L$. This matrix may be row reduced to the normal form $(I_{d-1}, C^{'})$ where $C^{'}$ is a $(d-1)\times 1$ matrix over $L$. The column vector $C^{'}$ is uniquely determined. In other words there are unique scalars $a_i$ in $L$, for $1\leq i <d$, such that the vectors $Z_i-a_iZ_d$ form a basis for $Y_L$ over $L$. On the other hand the identities $Z_i-c_iZ_d$ form a basis of $Y_{L(\gamma)}$ over $L(\gamma)$. It follows that $a_i=c_i\in L$ for all $i$. \qed
We now show that the $T$--ideal of graded identities is finitely generated. Let $n=\mathop{\rm ord}\nolimits(G)$. Consider the set $V=\{x_{ig}: 1\leq i \leq n\ , g \in G\}$ of indeterminates and let $E$ be the set of elementary identities of length $\leq n$ (that is where the monomials are of length $\leq n$) and such that its indeterminates are elements of $V$. Clearly $E$ is a finite set.
\begin{thm} \label{finitegeneration.thm} The ideal of graded identities is generated as a $T$--ideal by $E$. In particular the ideal is finitely generated as a $T$--ideal. \end{thm}
\proof Denote by $I'$ the $T$--ideal generated by $E$. We will show that $I'$ contains all graded identities. Clearly it is enough to show that $I'$ contains all elementary identities. Let $B=x_{r_1g_{1}}x_{r_2g_{2}}\cdots x_{r_kg_{k}} - c(B)x_{r_{\pi(1)}g_{\pi(1)}}x_{r_{\pi(2)}g_{\pi(2)}}\cdots x_{r_{\pi(k)}g_{\pi(k)}}$ be an elementary identity of length $k$. Clearly, if $k\leq n$, then $B$ is in $I'$. We assume therefore that $k>n$ and proceed by induction on $k$. The first observation is that if there exist $i,t$, with $1\leq i,t\leq k-1$ $i=\pi(t)$ and $i+1=\pi(t+1)$ then we can reduce the length of the word: We let $g=g_ig_{i+1}$ and replace $x_{r_ig_i}x_{r_{i+1}g_{i+1}}$ by $x_{rg}$, where $1\leq r\leq n$ and $x_{rg}$ does not appear in $B$. The resulting identity (one has to check that the coefficient is right) has shorter length and so we may assume this identity is in $I'$. But we can obtain the longer one from the shorter one by the substitution of $x_{r_ig_i}x_{r_{i+1}g_{i+1}}$ for $x_{rg}$, so we are done in this case.
Next, note that since $k>n$, the pigeonhole principle applied to the expressions $g_1 \cdots g_s$, $s=1,\dots,k$ shows that there are integers $1\leq i<j\leq n$ such that $g_ig_{i+1}\cdots g_j=1$. Observe also, that if $\gamma$ is a cyclic permutation of the numbers $i,i+1,\dots, j$ then we have $g_{\gamma(i)}\cdots g_{\gamma(j)}=1$. That means that if we replace the string $x_{r_ig_i}x_{r_{i+1}g_{i+1}}\cdots x_{r_jg_j}$ by a cyclic permutation of the variables in the first monomial and leave the second monomial alone, we do not change our identity modulo $I'$. Also for all $g\in G$, we have $gg_ig_{i+1}\cdots g_j=g_ig_{i+1}\cdots g_jg$. This means that modulo $I'$ we can move the string $x_{r_ig_i}x_{r_{i+1}g_{i+1}}\cdots x_{r_jg_j}$ anywhere in the monomial. So, combining these two things we can cyclically move the substring and move it anywhere in the first monomial. But by doing these moves we can arrange it so that some two variables that are next to each other in the second monomial are also next to each other (in the same order) in the first monomial. This reduces us to the first case.\qed
\section{The Universal Algebra.} In this section we determine the basic structural properties of the universal $G$--graded algebra $U_{G}$ and relate that algebra to the other algebras described in the introduction. The results are analogous to results in the theory of (non-graded) polynomial identities. We will see for example that $U_{G}$ is a prime ring and that its algebra of central quotients, the universal $G$--graded algebra, is central simple.
We start with the following lemma.
\begin{lem} \label{formsareforms.lem} Let $c$ be a nondegenerate cocycle on $G$ and let $\mathbb{C}^{c}G$ be the twisted group algebra. Let $L$ be a subfield of $\mathbb{C}$.
(a) Let $\beta:G\times G\rightarrow L^{\times}$ be a two-cocycle and let $L^{\beta}G$ be the twisted group algebra. There is a homomorphism $\eta: L^{\beta}G \rightarrow \mathbb{C}^{c}G$ over $L$ of $G$--graded algebras if and only if the cocycles $c$ and $\beta$ are cohomologous in $\mathbb{C}$. In particular $\mathop{\rm Im}\nolimits(\eta)$ is a $G$--graded form of $\mathbb{C}^{c}G$.
(b) There exists a two cocycle $\beta:G\times G\rightarrow L^{\times}$ cohomologous to $c$ over $\mathbb{C}$ if and only if $L$ contains $\mu$, the set of Proposition~\ref{elem.prop}. \end{lem}
\proof (a) If $\eta: L^{\beta}G \rightarrow \mathbb{C}^{c}G$ is a homomorphism of $G$-graded $L$--algebras then there are bases $\{v_{\sigma}\}\subset L^{\beta}G$, $\{u_{\sigma}\}\subset \mathbb{C}^{c}G$, and scalars $\{\lambda_{\sigma}\}\subset \mathbb{C^\times}$ such that $v_{\sigma}v_{\tau}=\beta(\sigma,\tau)v_{\sigma\tau}$, $u_{\sigma}u_{\tau}=c(\sigma,\tau)u_{\sigma\tau}$ and $\eta(v_{\sigma})=\lambda_{\sigma}u_{\sigma}$. It follows that $\beta$ and $c$ are cohomologous over $\mathbb{C}$. A similar calculation shows the other direction.
(b) We have seen in Proposition~\ref{elem.prop} that $\phi([c])=\mu$ where $\phi$ is the isomorphism between $H^{2}(G,\mathbb{C}^\times)$ and $\mathop{\rm Hom}\nolimits(M(G),\mathbb{C}^\times)$. Hence if $\beta$ is cohomologous to $c$ over $\mathbb{C}$ then $L$ must contain $\mu$ by part (d) of Proposition~\ref{elem.prop}.
For the converse we may assume that $L=\mathbb{Q}(\mu)$. By the naturality of the Universal Coefficient Theorem we have a commutative diagram
\[ \begin{CD} 1 @>>> Ext^{1}(G_{ab},\mathbb{Q}(\mu)) @>>> H^{2}(G,\mathbb{Q}(\mu)^\times) @>>> \mathop{\rm Hom}\nolimits(M(G),\mathbb{Q}(\mu)^\times) @>>> 1\\ @. @. @VV\nu V @VV\overline{\nu}V \\ @. @. H^{2}(G,\mathbb{C}^\times) @>\phi>> \mathop{\rm Hom}\nolimits(M(G),\mathbb{C}^\times) \\ \end{CD} \]
Clearly $\phi([c])\in \mathop{\rm Hom}\nolimits(M(G),\mathbb{C}^\times)$ is in $\mathop{\rm Im}\nolimits(\bar{\nu})$ and hence $[c]$ is in $\mathop{\rm Im}\nolimits(\nu)$.\qed
The first step in the analysis of $U_{G}$ is the construction of a counterpart to the ring of generic matrices. Let $G$ be a group of central type of order $n^2$ and let $c:G\times G\rightarrow \mathbb{C}^\times$ be a nondegenerate two-cocycle. For every $g\in G$ let $t_{ig}$ for $i=1,2,3,\dots$ be indeterminates. For each $g\in G$ let $t_g=t_{1g}$. We will assume $u_1=1$ (i.e. the cocycle $c$ is normalized). Let $k$ denote the field generated over $\Bbb {Q}$ by the indeterminates and the values of the cocycle. Note that by Lemma~\ref{formsareforms.lem}, $k$ contains $\mathbb{Q}(\mu)$. Consider the twisted group algebra $k^{c}G$. Because the cocycle is nondegenerate $\mathbb{Q}(c(g,h))^{c}G$ is a central simple $\mathbb{Q}(c(g,h))$--algebra and hence $k^{c}G$ is a central simple $k$--algebra. Now let $\overline{U_G}$ denote the $\Bbb{Q}(\mu)$--subalgebra of $k^cG$ generated by the elements $t_{ig}u_{g}$ for all $i$ and all $g\in G$. We want to describe $\overline{U_G}$. Let $\overline{Z}$ denote the center of $\overline{U_G}$.
\begin{prop} \label{centerofgeneric.prop} (a) The center $\overline{Z}$ of $\overline{U_G}$ is the $\mathbb{Q}(\mu)$--subalgebra of $\overline{U_G}$ generated by the following set of elements:
$$\{t_{i_1g_1}u_{g_1}t_{i_{2}g_2}u_{g_2}\cdots t_{i_{m}g_m}u_{g_m}
\ | \ g_1g_2\cdots g_m=1 \}$$
(b) We have $\overline{Z}\subseteq k$. In particular $\overline{Z}$ is an integral domain. \end{prop}
\proof The ring $k\overline{U_G}$ contains the elements $u_g$ for all $g\in G$ and so $k\overline{U_G}=k^cG$, which has center $k$. Hence the center of $\overline{U_G}$ is $\overline{U_G}\cap k$. But this intersection is precisely the $\mathbb{Q}(\mu)$ span of the given set of monomials. This proves both parts.\qed
Now let $s:G\times G\rightarrow k^\times$ be the function given by $s(g,h)={t_gt_h \over t_{gh}}c(g,h)$. This is a two-cocycle, cohomologous to $c$ over $k$. Because $c$ is nondegenerate, so is $s$ and hence the algebra $k^sG$ is $k$--central simple. In fact it is convenient to view the algebra $k^sG$ as equal to $k^cG$. It is spanned over $k$ by the elements $t_{g}u_{g}$.
Let $Y$ be the subgroup of $k^\times$ generated by the values of $s$ and let $\overline{S}=\mathbb{Q}(\mu)[Y]$, a subring of $k$. The $\overline{S}$--subalgebra of $k^sG$ generated by the elements $t_{g}u_{g}$ is the twisted group algebra $\overline{S}^sG$. Let $\overline{M}=\{t_{g_1}u_{g_1}t_{g_2}u_{g_2}\cdots t_{g_m}u_{g_m} \
| \ g_1g_2\cdots g_m=1 \}$. Let $\overline{F}\subseteq k$ denote the field of fractions of $\overline{Z}$.
\begin{prop} \label{localizationofcenter.prop} (a) The set $\overline{M}$ is a multiplicatively closed subset of $\overline{Z}$ and
$\overline{M}^{-1}\overline{Z}=\overline{S}[{t_{ig} \over t_g} \ | \ i\geq 1, g\in G]$.
(b) We have $\overline{U_G}(\overline{M}^{-1}\overline{Z})=\overline{S}^sG(\overline{M}^{-1}\overline{Z})$. Moreover $\overline{U_G}\overline{F}=\overline{S}^sG\overline{F}=\overline{F}^sG$.
(c) $\overline{F}^sG$ is a central simple $\overline{F}$--algebra.
(d) The ring $\overline{U_G}$ is a prime ring with ring of central quotients $Q(\overline{U_G})$ isomorphic to the central simple algebra $\overline{F}^sG$. \end{prop}
\proof (a) We first show $\overline{M}^{-1}\overline{Z}\supseteq
\overline{S}[{t_{ig} \over t_g} \ | \ i\geq 1, g\in G]$. Observe that $t_{e}$ and $t_{g}t_{g^{-1}}c(g,g^{-1})$ are in $\overline{M}$ and hence $t_{ig}t_{g^{-1}}c(g,g^{-1})/t_{g}t_{g^{-1}}c(g,g^{-1})=t_{ig}/t_{g}$ is in $\overline{M}^{-1}\overline{Z}$. Next, the elements $s(g,h)^{\pm 1}$ are in $\overline{M}^{-1}\overline{Z}$ because the elements
$t_{g}u_{g}t_{h}u_{h}t_{(gh)^{-1}}u_{(gh)^{-1}}=t_{g}t_{h}t_{(gh)^{-1}}c(g,h)c(gh,(gh)^{-1})$ and
$t_{gh}u_{gh}t_{(gh)^{-1}}u_{(gh)^{-1}}=t_{gh}t_{(gh)^{-1}}c(gh,(gh)^{-1})$ are in $\overline{M}$. Similar calculations show that the opposite inclusion also holds.
(b) This follows easily from part (a).
(c) We have $\overline{F}^sG\otimes_{\overline{F}}k=k^sG$ which is $k$--central simple and so $\overline{F}^sG$ is $\overline{F}$--central simple.
(d) By part (b) we have $\overline{U_G}\overline{F}=\overline{S}^sG\overline{F}=\overline{F}^sG$ and $\overline{F}^sG$ is $\overline{F}$--central simple, the ring $\overline{U_G}$ is prime and because $\overline{F}$ is the ring of fractions of $\overline{Z}$ we see that the ring of central quotients $Q(\overline{U_G})$ is isomorphic to $\overline{F}^sG$. \qed
Let $\psi$ denote the $\mathbb{Q}(\mu)$--algebra homomorphism from the free algebra $\Sigma(\mathbb{Q}(\mu))$ to $k^{c}G$ given by $\psi(x_{ig})=t_{ig}u_{g}$. The image of this map is clearly $\overline{U_G}$.
\begin{prop} \label{iso.prop} The kernel of $\psi$ is the ideal $T(\mathbb{Q}(\mu))$ of graded identities of $\mathbb{C}^{c}G$ and hence $\psi$ induces a $\mathbb{Q}(\mu)$--algebra isomorphism from $U_G$ to $\overline{U_G}$. In particular the universal algebra $U_G$ is a prime ring with center isomorphic to $\overline{Z}$ and and its ring of central quotients, that is $Q(U_G)$ is isomorphic to $\overline{F}^sG$. \end{prop}
\proof This is clear since a polynomial $p(x_{ig})\in \Sigma(\mathbb{Q}(\mu))$ is an identity of $\mathbb{C}^{c}G$ if and only if $p(\lambda_{ig}u_{g})=0$ for any $\lambda_{ig}\in \mathbb{C}$ and this is equivalent to $p(t_{ig}u_{g})=0$ where $\{t_{ig}\}$ are central indeterminates. \qed
Because of this isomorphism we will from now on drop the bars on $\overline{Z}$, $\overline{F}$, etc.
We want to say more about $F$, the field of fractions of the center of the universal algebra $U_G$. The group $Y$ is finitely generated and hence of the form $Y_tY_f$ where $Y_t$ is the torsion subgroup of $Y$ and $Y_f$ is a finitely generated free abelian group. Because $Y$ is a subgroup of $k^\times$ it follows that $Y_t$ is cyclic. We will see in the next proposition that in fact $Y_t=\mu$. Let $y_1,y_2,\dots ,y_m\in k$ denote a basis for $Y_f$. (We will see later - in Proposition~\ref{rank.prop} - that the rank of $Y_f$ is equal to $n$, the order of $G$, and so $m=n$.)
\begin{prop}
\label{fieldF.prop} (a) We have $Y_t=Y\cap \mathbb{Q}(\{c(g,h) \ | \ g,h\in G\})=\mu$.
(b) We have $S=\mathbb{Q}(\mu)[Y]=\mathbb{Q}(\mu)[y_1^{\pm 1},y_2^{\pm 1},\dots, y_m^{\pm 1}]$, the ring of Laurent polynomials in $y_1,\dots, y_m$ over $\mathbb{Q}(\mu)$.
(c) We have $F=\mathbb{Q}(\mu)(y_1,\dots, y_m)({t_{ig} \over t_g} \
| \ i\geq 1, g\in G)$. In particular $F$ is isomorphic to the field of rational functions in countably many variables over $\mathbb{Q}(\mu)$. \end{prop}
\proof To prove (a) we prove the following inclusions: (i) $Y_t
\subseteq Y\cap \mathbb{Q}(\{c(g,h) \ | \ g,h\in G\})$, (ii) $\mu
\subseteq Y_t$, and (iii) $Y\cap \mathbb{Q}(\{c(g,h) \ | \ g,h\in G\}) \subseteq \mu$. To prove (i) consider an arbitrary element $z$ in $Y$, $z=\prod_{i=1}^{n} s(g_{i},h_{i})^{\epsilon_{i}}=\prod_{i=1}^{n} ({t_{g_{i}}t_{h_i} \over t_{g_ih_i}}c(g_i,h_i))^{\epsilon_{i}}$ where ${\epsilon_{i}}$ are $\pm 1$. If the product is not independent of the $t's$ (say $t_{g}$ appears) it is clear that any positive power of $z$ contains a power of $t_{g}$ and hence does not equal 1. This shows (i). Next, let $\lambda\in \mu$ and let $$x_{1g_{1}}x_{1g_{2}}\cdots x_{1g_{k}}- \lambda x_{1g_{\pi(1)}}x_{1g_{\pi(2)}}\cdots x_{1g_{\pi(k)}}$$
\noindent be an elementary identity. It follows that $$t_{g_{1}}u_{g_{1}}t_{g_{2}}u_{g_{2}}\cdots t_{g_{k}}u_{g_{k}}= \lambda t_{g_{\pi(1)}}u_{g_{\pi(1)}}t_{g_{\pi(2)}}u_{g_{\pi(2)}}\cdots t_{g_{\pi(k)}}u_{g_{\pi(k)}}$$ and hence $s(g_{1},g_{2})s(g_{1}g_{2},g_{3})\cdots s(g_{1}g_{2}\cdots g_{k-1}, g_{k})t_{g_{1}g_{2}\cdots g_{k}}u_{g_{1}g_{2}\cdots g_{k}}=$
\noindent $\lambda s(g_{\pi(1)},g_{\pi(2)})s(g_{\pi(1)}g_{\pi(2)},g_{\pi(3)})\cdots s(g_{\pi(1)}g_{\pi(1)}\cdots g_{\pi(k-1)}, g_{\pi(k)})t_{g_{\pi(1)}g_{\pi(2)}\cdots g_{\pi(k)}}u_{g_{\pi(1)}g_{\pi(2)}\cdots g_{\pi(k)}}$.
\noindent But $g_{1}g_{2}\cdots g_{k}=g_{\pi(1)}g_{\pi(2)}\cdots g_{\pi(k)}$ in $G$ and so $\lambda$ lies in $Y$. Because $\lambda$ is a root of unity, we get $\lambda\in Y_t$. This proves (ii).
To prove (iii) let $\lambda$ be an element of $Y\cap
\mathbb{Q}(\{c(g,h) \ | \ g,h\in G\})$. We can write $\lambda= s(g_{1},h_{1})^{\epsilon_{1}}s(g_{2},h_{2})^{\epsilon_{2}} \cdots s(g_{k},h_{k})^{\epsilon_{k}}=$
$({t_{g_{1}}t_{h_{1}} \over t_{g_{1}h_{1}}}c(g_{1},h_{1}))^{\epsilon_{1}}({t_{g_{2}}t_{h_{2}} \over t_{g_{2}h_{2}}}c(g_{2},h_{2}))^{\epsilon_{2}} \cdots ({t_{g_{k}}t_{h_{k}} \over t_{g_{k}h_{k}}}c(g_{k},h_{k}))^{\epsilon_{k}}$.
\noindent We use the notation and set-up of part (b) of Proposition~\ref{elem.prop}. Let $d$ be the element of the free group $F$ given by
$d=x_{g_{1}}x_{h_{1}}(x_{g_{1}h_{1}})^{-1}x_{g_{2}}x_{h_{2}}(x_{g_{2}h_{2}})^{-1}\cdots x_{g_{k}}x_{h_{k}}(x_{g_{k}h_{k}})^{-1}$. Clearly $d$ lies in $R$. Because $\lambda$ lies in $\mathbb{Q}(\{c(g,h) \ | \ g,h\in G\})$, the variables $t_g$ in the expression for $\lambda$ must all cancel and so $d\in [F,F]$ and $\lambda=c(g_{1},h_{1}))^{\epsilon_{1}}c(g_{2},h_{2}))^{\epsilon_{2}} \cdots c(g_{k},h_{k}))^{\epsilon_{k}}$. In particular we may consider the element $\overline{d}$ in $M(G)$. Then again as in part (b) of Proposition~\ref{elem.prop} we have $\phi([c])(\overline{d})=\gamma(\overline{d})=c(g_{1},h_{1}))^{\epsilon_{1}}c(g_{2},h_{2}))^{\epsilon_{2}} \cdots c(g_{k},h_{k}))^{\epsilon_{k}}=\lambda$ and so $\lambda$ lies in $\mu$.
Part (b) is clear. Part (c) follows from part (b) and from the fact that $M^{-1}Z=S[{t_{ig} \over t_g} \ | \ i\geq 1, g\in G]$.\qed
Our next goal is to study the algebra $S^sG$. We will see that it is Azumaya and that its simple images are the graded forms of $\mathbb{C}^cG=M_n(\mathbb{C})$.
\begin{prop} \label{Az.prop} (a) Let $m$ be a maximal ideal of $S$ and let $y\rightarrow \overline{y}$ denote the canonical homomorphism from $S$ to $\tilde{S}=S/m$. Consider the two-cocycle $\tilde{s}:G\times G \longrightarrow S/m$ induced by $s$ and let $\tilde{S}^{\tilde{s}}G$ be the corresponding twisted group algebra. Then the algebra $S^sG/mS^sG$ is isomorphic to $\tilde{S}^{\tilde{s}}G$ and is a central simple $\tilde{S}$--algebra.
(b) The ring $S^sG$ is Azumaya over $S$. \end{prop}
\proof (a) We first show that the two-cocycle $\tilde{s}$ is nondegenerate. We use the following characterization of nondegeneracy (see Isaacs \cite[Problem 11.8]{I}): a two-cocycle $\delta:G\times G\rightarrow \mathbb{C}^\times$ is nondegenerate if and only if for every element $g\in G$ there is an element $h\in G$ such that $g$ and $h$ commute but in the algebra $\mathbb{C}^\delta G$ the elements $u_g$ and $u_h$ do not commute. Let $g\in G$. Because $c$ is nondegenerate there is an element $h\in G$ that centralizes $g$ and such that $c(g,h)c(h,g)^{-1}\neq 1$. Moreover $c(g,h)c(h,g)^{-1}\in \mu$ and $s(g,h)s(h,g)^{-1}= c(g,h)c(h,g)^{-1}$. Because $S$ contains the field $\mathbb{Q}(\mu)$, the restriction of the canonical homomorphism from $S$ to $\tilde{S}$ is injective on $\mu$ and so $\tilde{s}(g,h)\tilde{s}(h,g)^{-1}\neq 1$. Hence $\tilde{s}$ is nondegenerate. It follows that $\tilde{S}^{\tilde{s}}G$ is a central simple $\tilde{S}$--algebra. It is clear that the canonical homomorphism from $S^sG$ to $\tilde{S}^{\tilde{s}}G$ is surjective with kernel $mS^sG$.
(b) This follows from part (a) (see DeMeyer and Ingraham \cite[Theorem 7.1]{DI}).\qed
Let $L$ be any subfield of $\mathbb{C}$. Recall that every $G$--graded form of $\mathbb{C}^cG$ over $L$ is a twisted group algebra $L^\beta G$ where $\beta$ is a two-cocycle cohomologous to $c$ over $\mathbb{C}$.
We will show now that $S^{s}G$ is universal with respect to such forms. The following observation about the Hopf formula $M(G)=[F,F]\cap R/[F,R]$ will be useful (and is probably well known). We use the notation and discussion preceding Proposition~\ref{elem.prop}.
\begin{lem} \label{Hopf.lem} Every element in $[F,F]\cap R$ is equivalent modulo $[F,R]$ to an element of the form $x_{g_{1}} \cdots x_{g_{k}}(x_{g_{\pi (1)}} \cdots x_{g_{\pi (k)}})^{-1}$ where $\pi\in Sym(k)$ and $g_{1}\cdots g_{k}=g_{\pi (1)} \cdots g_{\pi (k)}$. \end{lem}
\proof Let $y\in [F,F]\cap R$. We can write $y=x_{g_1}^{\epsilon_1}x_{g_2}^{\epsilon_2}\cdots x_{g_m}^{\epsilon_m}$ where $\epsilon_i=\pm 1$ for $1\leq i\leq m$ and where for each group element $g$ that appears the elements $x_g$ and $x_g^{-1}$ appear the same number of times (and so $m$ is even) and where the product $g_1^{\epsilon_1}g_2^{\epsilon_2}\cdots g_m^{\epsilon_m}$ equals one. Let $m=2r$. We \underline{claim} that we may assume $\epsilon_i=1$ if $1\leq i\leq r$ and $\epsilon_i=-1$ if $r+1\leq i\leq m$. The lemma follows easily from this claim. To prove the claim, suppose $\epsilon_i=-1$ for some $i$, $1\leq i\leq r$. We consider $y=x_{g_1}^{\epsilon_1}x_{g_2}^{\epsilon_2}\cdots x_{g_m}^{\epsilon_m}=x_{g_1}^{\epsilon_1}x_{g_2}^{\epsilon_2}\cdots x_{g_r}^{\epsilon_{r}}(x_{g_i}x_{g_i^{-1}}(x_{g_i^{-1}})^{-1}(x_{g_i})^{-1})x_{g_{r+1}}^{\epsilon_{r+1}}\cdots x_{g_m}^{\epsilon_m}$. Now for any element $g\in G$ the element $x_gx_{g^{-1}}$ commutes modulo $[F,R]$ with every element of $F$. Hence we can move the element $x_{g_i}x_{g_i^{-1}}$ in $y$ and obtain an element that is equivalent modulo $[F,R]$ to $y$ and for which the element $x_{g_i}^{-1}$ is replaced by $x_{{g_i}^{-1}}$. Continuing in this way proves the claim. \qed
\begin{prop} \label{forms.prop} Let $L$ be a subfield of $\mathbb{C}$ which contains $\mathbb{Q}(\mu)$. Let $\varphi:S\longrightarrow L$ be a $\mathbb{Q}(\mu)$--algebra homomorphism. Let $\beta(g,h)=\varphi (s(g,h))$. Then $\varphi$ induces a $G$--graded homomorphism $\hat{\varphi}: S^{s}G\longrightarrow L^{\beta}G$ and $L^\beta G$ is a $G$--graded form of $\mathbb{C}^cG$. In particular $\beta$ is cohomologous to $c$ over $\mathbb{C}$.
Conversely if $L^\beta G$ is a $G$--graded form of $\mathbb{C}^cG$, then there is homomorphism $\varphi$ of $S$ into $L$ such that $\gamma=\varphi (s(g,h)$ is a cocycle on $G$ cohomologous to $\beta$ over $L$ and $\varphi$ induces a homomorphism from $S^sG$ onto $\varphi(S)^\gamma G$, a $G$--graded form of $L^\beta G$. \end{prop}
\proof Let $\varphi:S\longrightarrow L$ be a $\mathbb{Q}(\mu)$--algebra homomorphism and let $\beta(g,h)=\varphi (s(g,h))$. We have seen in Proposition~\ref{Az.prop} that $\beta$ is a nondegenerate cocycle on $G$ and hence that $L^\beta G$ is a central simple $L$--algebra. Clearly $\varphi$ induces a homomorphism $\hat{\varphi}: S^{s}G\longrightarrow L^{\beta}G$ and if $A$ denotes the image of $S^s G$ and $A$ has center $R$ (say), then $A\otimes_RL=L^\beta G$. We are left with showing $L^\beta G$ is a form of $\mathbb{C}^cG$, that is we need to show $\beta$ and $c$ are cohomologous over $\mathbb{C}$. It suffices to show that they determine the same element in $\mathop{\rm Hom}\nolimits(M(G),\mathbb{C}^\times)$, that is that $\phi([c])=\phi([\beta])$, where $\phi$ is the isomorphism from $H^{2}(G,\mathbb{C}^\times)$ to $\mathop{\rm Hom}\nolimits(M(G),\mathbb{C}^\times)$ (see the discussion preceding Proposition~\ref{elem.prop}). By the lemma it suffices to show that if $g_{1}\cdots g_{k}=g_{\pi (1)} \cdots g_{\pi (k)}$ then $\phi([\beta])(y)=\phi([c])(y)$ where $y=(x_{g_{1}} \cdots x_{g_{k}})(x_{g_{\pi (1)}} \cdots x_{g_{\pi (k)}})^{-1}$. For the purposes of the proof it will be useful to use the following notation: If $\alpha$ is any two-cocycle then we denote the expression $${\alpha(g_{1},g_{2})\alpha(g_{1}g_{2},g_{3}) \cdots \alpha(g_{1}g_{2} \dots g_{k-1},g_k) \over \alpha(g_{\pi (1)},g_{\pi (2)})\alpha(g_{\pi (1)}g_{\pi (2)},g_{\pi (3)}) \cdots \alpha(g_{\pi (1)}g_{\pi (2)} \dots g_{\pi (k-1)},g_{\pi (k)})}$$ by $\alpha(g_1,g_2,\dots ,g_k)/\alpha(g_{\pi (1)},g_{\pi (2)}, \dots, g_{\pi (k)})$.
We have $\phi([\beta])(y)=\beta(g_1,g_2,\dots ,g_k)/\beta(g_{\pi (1)},g_{\pi (2)}, \dots, g_{\pi (k)})=$
\noindent $\varphi (s(g_1,g_2,\dots ,g_k)/s(g_{\pi (1)},g_{\pi (2)}, \dots, g_{\pi (k)}))=$
\noindent $\varphi( (t_{g_1}t_{g_2}/t_{g_1g_2})(t_{g_1g_2}t_{g_3}/t_{g_1g_2g_3})\cdots (t_{g_1g_2\cdots
g_{k-1}}t_{g_k}/t_{g_1g_2\cdots g_k})/$
$(t_{g_{\pi (1)}}t_{g_{\pi (2)}}/t_{g_{\pi (1)}g_{\pi (2)}})\cdots (t_{g_{\pi (1)}g_{\pi (2)}\cdots g_{\pi (k-1)}}t_{g_{\pi (k)}}/ t_{g_{\pi (1)}g_{\pi (2)}\cdots g_{\pi (k)}})$
$c(g_1,g_2,\dots ,g_k)/c(g_{\pi (1)},g_{\pi (2)}, \dots, g_{\pi (k)}))=$
\noindent $\varphi( (t_{g_1g_2\cdots g_k}/t_{g_{\pi (1)}g_{\pi (2)}\cdots g_{\pi (k)}}) c(g_1,g_2,\dots ,g_k)/c(g_{\pi (1)},g_{\pi (2)}, \dots, g_{\pi (k)}))=$
\noindent $\varphi(c(g_1,g_2,\dots ,g_k)/c(g_{\pi (1)},g_{\pi (2)}, \dots, g_{\pi (k)}))$, because $g_{1}\cdots g_{k}=g_{\pi (1)} \cdots g_{\pi (k)}$. But this last expression is exactly $\phi ([c])(y)$, as desired.
For the converse assume $L^\beta G$ is a $G$--graded form of $\mathbb{C}^c G$. Then $L^\beta G$ satisfies the graded identities of $M_n(\mathbb{C})$ and so there is a graded homomorphism from $U_G$ sending each $x_{ig}$ to $u_g$ in $L^\beta G$. But under this map the images of the elements of $M$ are nonzero elements in $L$ and so we get an induced homomorphism from $S^sG$ that takes $S$ into $L$. The result follows.\qed
We can use this result to obtain a parametrization of the $G$--graded forms over $L$. We begin with a homomorphism $\varphi$ from $S$ to $L$ and the two-cocycle $\beta$ given by $\beta(g,h)=\varphi(s(g,h))$. In the previous proposition we say that $\beta$ is cohomologous to $c$ over $\mathbb{C}$ and so $L^\beta G$ is a form of $M_n(\mathbb{C})$. We will now show how to produce all other forms over $L$. Let $V$ the free abelian group (of rank $\mathop{\rm ord}\nolimits(G)$) on symbols $r_{g}, g\in G$. Let $U$ be the subgroup generated by the elements $r_{g}r_{h}/r_{gh}$. Note that $V/U$ is a finite group, isomorphic to $G/G^{'}=G_{ab}$ via the map that sends $r_g$ to $gG'$. For each element $\psi$ in $\mathop{\rm Hom}\nolimits(U,L^\times)$ let $\beta_{\psi}$ be given by $\beta_{\psi}(g,h)=\beta(g,h)\psi(r_gr_h/r_{gh})$. Note that there is a canonical homomorphism from $\mathop{\rm Hom}\nolimits(V,L^\times)$ to $\mathop{\rm Hom}\nolimits(U,L^\times)$. We will let $\mathop{\rm Im}\nolimits((\mathop{\rm Hom}\nolimits(V,L^\times))$ denote its image in $\mathop{\rm Hom}\nolimits(U,L^\times)$.
\begin{prop} \label{parametrization.prop} The following hold:
(a) For every $\psi\in \mathop{\rm Hom}\nolimits(U,L^\times)$ the function $\beta_{\psi}$ is a two-cocycle cohomologous to $\beta$ over $\mathbb{C}$. In particular $L^{\beta_{\psi}}G$ is a $G$--graded form of $M_n(\mathbb{C})$.
(b) If $L^\gamma G$ is a $G$--graded form of $M_n(\mathbb{C})$ there is a homomorphism $\psi\in \mathop{\rm Hom}\nolimits(U,L^\times)$ such that $\gamma=\beta_{\psi}$. In particular $L^\gamma G$ is isomorphic to $L^{\beta_{\psi}}G$ as a $G$--graded algebra.
(c) Two forms $L^{\beta_{\psi}} G$ and $L^{\beta_{\psi '}}G$ are $G$--graded isomorphic if and only if $\beta_{\psi}=f(\alpha)\beta_{\psi '}$ for some $\alpha$ in $\mathop{\rm Im}\nolimits((\mathop{\rm Hom}\nolimits(V,L^\times))$.
(d) The function $\psi\rightarrow \beta_{\psi}$ induces a one-to-one correspondence between the group $\mathop{\rm Hom}\nolimits(U,L^\times)/\mathop{\rm Im}\nolimits(\mathop{\rm Hom}\nolimits(V,L^\times)$ and the set of $G$--graded isomorphism classes of $G$--graded forms of $\mathbb{C}^cG$ over $L$. \end{prop}
\proof (a) It is clear that $\beta_{\psi}$ is a cocycle. To show that $\beta$ and $\beta_{\psi}$ are cohomologous over $\mathbb{C}$ it suffices to show that they determine the same element in $\mathop{\rm Hom}\nolimits(M(G),\mathbb{C}^\times)$, that is that $\phi([\beta_{\psi}])=\phi([\beta])$, where $\phi$ is the isomorphism from $H^{2}(G,\mathbb{C}^\times)$ to $\mathop{\rm Hom}\nolimits(M(G),\mathbb{C}^\times)$. The argument is quite similar to the proof of the corresponding fact in Proposition~\ref{forms.prop}, so we will omit it.
(b) If $L^\gamma G$ is a $G$--graded form of $M_n(\mathbb{C})$ then we have seen that $\gamma$ is cohomologous to $\beta$ over $\mathbb{C}$ so there is a cochain $\lambda:G\rightarrow \mathbb{C}^\times$ such that for all $g,h\in G$, we have $\gamma (g,h)=(\lambda(g)\lambda(h)/\lambda(gh))\beta (g,h)$. It follows that for all $g,h\in G$, $\lambda(g)\lambda(h)/\lambda(gh)$ is in $L$. The cochain $\lambda$ clearly determines a homomorphism from $V$ to $\mathbb{C}^\times$ (sending $r_g$ to $\lambda (g)$) and this homomorphism restricts to a homomorphism from $\psi$ from $U$ to $L^\times$ (given by $\psi (r_gr_h/r_{gh})=\lambda(g)\lambda(h)/\lambda(gh)$) which satisfies $\gamma=\beta_{\psi}$.
(c) If $L^{\beta_{\psi}} G$ and $L^{\beta_{\psi '}}G$ are $G$--graded isomorphic then $\beta_{\psi}$ is cohomologous to $\beta_{\psi '}$ over $L$. As in the proof of part (b) this means there is a homomorphism $\alpha$ from $V$ to $L^\times$ such that $\beta_{\psi}(g,h)=(\alpha (g)\alpha (h)/\alpha (gh))\beta_{\psi '}$. It follows that $\psi=\alpha \psi '$, as desired.
(d) This follows from the first three parts.\qed
We can connect this parametrization with cohomology as follows. We have a short exact sequence of groups
$$1\longrightarrow U\longrightarrow V\longrightarrow G_{ab}\longrightarrow 1$$
Applying the functor $\mathop{\rm Hom}\nolimits( - ,L^\times)$ we obtain a long exact sequence in cohomology
$$\dots\longrightarrow \mathop{\rm Hom}\nolimits(V,L^\times)\longrightarrow \mathop{\rm Hom}\nolimits(U,L^\times) \longrightarrow Ext(G_{ab},L^\times)\longrightarrow 1$$ where the surjectivity of the last map follows from the fact that $Ext(V,L^\times)=0$ because $V$ is a free abelian group. We also have the universal coefficients sequence:
$$1\rightarrow Ext^{1}(G_{ab},L^\times) \rightarrow H^{2}(G,L^\times)\rightarrow \mathop{\rm Hom}\nolimits(M(G),L^\times) \rightarrow 1$$
Combining these equations shows that $\mathop{\rm Hom}\nolimits(U,L^\times)/\mathop{\rm Im}\nolimits(\mathop{\rm Hom}\nolimits(V,L^\times)$ is isomorphic to $Ext^{1}(G_{ab},L^\times)$, the kernel of the map from $H^{2}(G,L^\times)$ to $\mathop{\rm Hom}\nolimits(M(G),L^\times)$.
The last result in this section is the determination of the rank of $Y_f$, the free part of the subgroup of $k^\times$ generated by the values of $s$. The computation involves the groups $U$ and $V$. We first remark that because $V/U$ is finite, the rank of $U$ is equal to the rank of $V$, which is clearly $n$, the order of $G$.
\begin{prop} \label{rank.prop} The group $Y_f$ is isomorphic to $U$. In particular the rank of $Y_f$ is $n$, the order of $G$. \end{prop}
\proof Recall that $Y$ denotes the subgroup of $k^\times$ generated by the values of the cocycle $s$. Let $H$ denote the subgroup of $k^\times$ generated by the values of the cocycle $c$. Then by Proposition~\ref{fieldF.prop} $Y\cap H= Y\cap \mathbb{C}^\times=\mu$. Let $\tilde{U}$ be the subgroup of
$k^\times$ generated by set $\{t_gt_h/t_{gh} \ | \ g,h\in G\}$. Clearly the group $\tilde{U}$ is isomorphic to $U$. Now we have $Y_f\cong Y/{Y_t} \cong Y/{Y\cap H}\cong {YH}/H$. But $YH=\tilde{U}H$. Hence ${YH}/H={\tilde{U}H}/H\cong \tilde{U}/{\tilde{U}\cap H}\cong \tilde{U}$ because $\tilde{U}\cap H=1$. Hence $Y_f\cong\tilde{U}\cong U$, as desired.\qed
\begin{cor} \label{Srank.cor} The field of fractions of $S$ has transcendence degree $n$ over $\mathbb{Q}(\mu)$. \end{cor}
\proof This follows from the description of $S$ given in Proposition~\ref{fieldF.prop}.\qed
\section{The universal $G$--graded central simple algebra.}
We have seen that each fine grading by a group $G$ on $M_n(\mathbb{C})$ gives rise to a nondegenerate two-cocycle $c$ on the group $G$ (which must be of central type) such that
$M_n(\mathbb{C})$ is graded isomorphic to $\mathbb{C}^cG$. We have also seen that the universal $G$--graded algebra $U_G$ is a prime ring with ring of central quotients $Q(U_G)$ graded isomorphic to the $F$--central simple algebra $F^s G$ where $s$ is the generic cocycle obtained from $c$ and $F=\mathbb{Q}(\mu)(y_1,\dots, y_m)({t_{ig} \over t_g} \ | \ i\geq 1, g\in G)$ (see Proposition~\ref{fieldF.prop}). In this section we obtain information about the index of $Q(U_G)$ and about the dependence of $Q(U_G)$ on the given cocycle $c$.
\begin{prop} \label{indexbound.prop} The index of $Q(U_G)$ is equal to the maximum of the indices of the $G$--graded forms $L^\beta G$ where $L$ varies over all subfields of $\mathbb{C}$ that contain $\mathbb{Q}(\mu)$ and $\beta$ varies over all nondegenerate cocycles over $L$ cohomologous to $c$ over $\mathbb{C}$. \end{prop}
\proof Let $M$ denote the maximum of the indices of the forms. We first claim that $\mathop{\rm ind}\nolimits(Q(U_G))\leq M$: In fact we can specialize the universal algebra $U_G$ by sending $x_g$ to $r_g u_g$ where the elements $\{r_g \mid g\in G\}$ are algebraically independent complex numbers. The resulting algebra $\overline{U}$ is isomorphic to $U_G$ and tensoring with the field of fractions of its center gives a central simple algebra that is isomorphic to $Q(U_G)$. Clearly the index of this form equals the index of $Q(U_G)$ and so we get the inequality. We proceed to show that the index of $Q(U_G)=F^sG$ is greater than or equal to $M$. Recall (Proposition~\ref{fieldF.prop}) that the center $S$ of $S^{s}G$ is a ring of Laurent polynomials over the field $\mathbb{Q}(\mu)$. If $M$ is any maximal ideal of $S$ then $S_M$ is a regular local ring. By Proposition~\ref{forms.prop} it suffices to show that if $M$ is any maximal ideal of $S$ then the index of the algebra $S^sG\otimes_S F= F^{s}G$ is greater than or equal to the index of the residue algebra $S^sG/MS^sG$. This is a well known fact. We outline a proof: Let $r_1,r_2,\dots,r_m$ be a system of parameters of the regular local ring $R=S_M$ and let $F^{s}G=M_t(D)$ where $D$ is an $F$--central division algebra (so $\mathop{\rm ind}\nolimits(F^{s}G) = \mathop{\rm ord}\nolimits(G)^{1/2}/t$). The localization $R_{(r_1)}$ is a discrete valuation ring with residue field the field of fractions of the regular local ring $R_1=R/(r_1)$. The ring $R_{(r_1)}^sG$ is an Azumaya algebra over the discrete valuation ring $R_{(r_1)}$ and hence is isomorphic to $M_t(A)$ where $A$ is an Azumaya algebra such that $A\otimes F=D$ (see Reiner \cite[Theorem 21.6]{R}). It follows that $R_1^sG$ is an Azumaya algebra and if we let $Q(R_1)$ denote the field of fractions of $R_1$, then $R_1^s G\otimes Q(R_1)$ is isomorphic to $M_t(D_1)$ where $D_1$ a central simple $Q(R_1)$--algebra, the residue algebra of $D$. The image of $M$ in $R_1$ is generated by the images of $r_2,r_3,\dots, r_m$ and these $m-1$ elements form a system of parameters for $R_1$. We can therefore repeat the process and eventually obtain that $S_M^sG/MS_M^sG$ is isomorphic to an algebra of the form $M_t(E)$ for some central simple $S/M$--algebra $E$. Hence the index of $S_M^sG/MS_M^sG$ is at most the index of $F^sG$. \qed
The question for which groups $G$ of central type there is a cocycle $c$ such that $Q(U_{G,c})$ is a division algebra was answered in Aljadeff, et al \cite{AHN} and Natapov \cite{N}: We consider the following list of $p$-groups, called $\Lambda_p$:
\begin{enumerate}
\item $G$
is abelian of symmetric type, that is $G \cong \prod(C_{p^{n_i}} \times C_{p^{n_i}})$,
\item $G \cong G_1 \times G_2$ where \\ $G_1 = C_{p^n} \rtimes C_{p^n} =
\langle\pi, \sigma \mid \sigma^{p^n}=\pi^{p^n}=1 \; {\rm and} \;
\sigma\pi\sigma^{-1} = \pi^{p^s+1}\rangle$ where $1 \leq s < n$ and
$1 \neq s$ if $p=2$,
and \\ $G_2$ is an abelian group of symmetric type of exponent
$\leq p^s$,
\item $G \cong G_1 \times G_2$ where \\
$G_1 = C_{2^{n+1}} \rtimes (C_{2^n} \times Z_2) =
\bigg \langle \pi,\sigma,\tau \, \bigg| \,
\begin{array}{l}
\pi^{2^{n+1}}=\sigma^{2^n}=\tau^2=1, \sigma\tau = \tau\sigma,\\
\sigma\pi\sigma^{-1}=\pi^3,
\tau\pi\tau^{-1}=\pi^{-1}
\end{array}\bigg\rangle$ and \\
$G_2$ is an abelian group of symmetric type of exponent $\leq 2$. \end{enumerate}
We let $\Lambda$ be the collection of nilpotent groups such that for any prime $p$, the Sylow-$p$ subgroup is on the $\Lambda_p$.
\begin{prop} \label{list.prop} Let $G$ be a group of central type of order $n^2$. The following conditions are equivalent:
(a) There is a nondegenerate cocycle $c$ on $G$ for which the universal $G$--graded algebra $U_{G,c}$ is a domain.
(b) There is a nondegenerate cocycle $c$ on $G$ for which the universal central simple algebra $Q(U_{G,c})$ is a division algebra.
(c) There is a nondegenerate cocycle $c$ on $G$ such that the resulting $G$--graded algebra $M_n(\mathbb{C})$ has a $G$--graded form that is a division algebra.
(d) The group $G$ is on the list $\Lambda$. \end{prop}
\proof The equivalence of (a) and (b) follows from the fact that $F^sG$ is isomorphic to $Q(U_{G,c})$. The equivalence of (b) and (c) follows from the previous proposition. Finally the equivalence of (c) and (d) follows from Aljadeff et al \cite[Corollary 3]{AHN} and Natapov \cite[Theorem 3]{N}. \qed
Our main result in this section is that if $G$ is a group on the list $\Lambda$ then the universal central simple $G$--graded algebra $Q(U_{G,c})$ is independent (up to a non-graded isomorphism) of the cocycle $c$. In fact we will show that for groups $G$ on the list the automorphism group of $G$ acts transitively on the set of classes of nondegenerate cocycles.
To begin let $G$ be any central type group (not necessarily on $\Lambda$) and let $c$ be a nondegenerate two-cocycle on $G$ with coefficients in $\mathbb{C}^\times$. Let $\varphi$ be an automorphism of $G$ and let $\varphi(c)$ be the two-cocycle defined by $\varphi(c)(\sigma,\tau) = c (\varphi^{-1}(\sigma), \varphi^{-1}(\tau))$. It is clear that $\varphi(c)$ is also nondegenerate. \begin{thm} \label{transitive.thm} If $G$ is a group on the list $\Lambda$ and $c$ and $c'$ are nondegenerate cocycles on $G$ with values in $\mathbb{C}^\times$, then there is an automorphism $\varphi$ of $G$ such that $\varphi(c)$ is cohomologous over $\FAT{C}$ to $c'$. \end{thm}
\proof Let $G$ be a group on the list $\Lambda$. Because $G$ is necessarily nilpotent we can assume that $G$ lies on $\Lambda_p$ for some prime $p$. In the course of the proof whenever we refer to basis elements $\{u_{g}\}_{g\in G}$ in the twisted group algebra $\FAT{C}^c G$, we assume they satisfy $u_g u_h = c(g,h) u_{gh}$.
The strategy is as follows: For each group $G \in \Lambda_{p}$ we fix a set of generators $\Phi$. Then we consider the family of sets of generators $\Phi^{'}=\varphi(\Phi)$ that are obtained from $\Phi$ via an automorphism $\varphi$ of $G$. We say that $\Phi$ and $\Phi^{'}$ are equivalent. Next, we exhibit a certain nondegenerate cohomology class $\alpha \in H^2(G, \FAT{C}^\times)$ by setting a set of equations (denoted by $E_{G}$) satisfied by elements $\{u_{g}\}_{g\in \Phi}$ in $\mathbb{C}^{c_{0}}G$, where $c_0$ is a two-cocycle representing $\alpha$. We say that the cohomology class $\alpha$ (or by abuse of language, the two-cocycle $c_0$) is of ``standard form" with respect to $\Phi$. It is indeed abuse of language since in general the equations do not determine the two-cocycle $c_{0}$ uniquely but only its cohomology class. Finally we show that for any nondegenerate two-cocycle $c$ with values in $\mathbb{C}^\times$ there is a set of generators $\Phi^{'}=\varphi(\Phi)$ with respect to which $c$ is of standard form. The desired automorphism of the group $G$ will be determined by compositions of ``elementary" automorphisms, that is automorphisms that are defined by replacing some elements of the generating set. We start with $\Phi^{(0)}=\Phi$ and denote the updated generating sets by $\Phi^{(r)}=\{g_{1}^{(r)},\ldots ,g_{n}^{(r)}\}, r=1,2,3,\dots$. Elements in the generating set that are not mentioned remain unchanged, that is $g_{i}^{(r)}=g_{i}^{(r-1)}$. In all steps it will be an easy check that the map defined is indeed an automorphism of $G$.
{\bf (I)} We start with abelian groups on $\Lambda_{p}$. Let $G=C_{p^{r_{1}}}\times C_{p^{r_{1}}}\times C_{p^{r_{2}}}\times C_{p^{r_{2}}}\times \dots \times C_{p^{r_{m}}} \times C_{p^{r_{m}}}$ and let $\Phi = \{ \gamma_1, \gamma_2, \dots, \gamma_{2m} \}$ be an ordered set of generators.
Fixing a primitive $p^n$-th root of unity $\varepsilon$, where $n \geq \max(r_k)$, $1\leq k \leq m$, we say a two-cocycle $c_{0}$ is of standard form with respect to $\Phi$ if there are elements $\{u_{g}\}_{g\in \Phi}$ in $\mathbb{C}^{c_0}G$ that satisfy
\begin{equation}\label{abelian_normal_form} \left\{
\begin{array}{ll} (u_{\gamma_{2k-1}},u_{\gamma_{2k}})=\varepsilon^{p^{n-r_k}} \ \ \hbox{for all} \ 1\leq k\leq m, \\ \hbox{all other commutators of} \ {u_{\gamma_i}}'s \ \hbox{are trivial.}
\end{array}
\right. \end{equation}
Note that $[c_0]$ is determined by (\ref{abelian_normal_form}). Given any nondegenerate two-cocycle $c$ it is known that there is a set of generators $\Phi^{'}=\varphi(\Phi)$ with respect to which $c$ is of standard form (see, for example, Aljadeff and Sonn \cite[Theorem 1.1]{AS}). Hence we are done in this case.
{\bf (II)} Next we consider the group $G = C_{p^n}\rtimes C_{p^n} \times C_{p^{r_1}} \times C_{p^{r_1}} \times \dots \times C_{p^{r_m}} \times C_{p^{r_m}}$ with a set of generators $\Phi = \{ \pi,\sigma, \gamma_1, \dots, \gamma_{2m}\}$ where $\pi, \sigma$ satisfy $\sigma(\pi)=\pi^{p^{s}+1}$ for some $s=1,\dots,n-1$ if $p$ is odd and for some $s=2,\dots,n-1$ if $p=2$, and $r_k \leq s$ for all $1\leq k \leq m$. We write $G = G_1 \times G_2$ where $G_1 = C_{p^n} \rtimes C_{p^n}$.
Fix a primitive $p^{n}$-th root of unity $\varepsilon$. There exist a two-cocycle $c_1 \in Z^2(G_1, \FAT{C}^\times)$ and elements $\{{u_g}\}_{g \in \{\pi,\sigma\}}$ in $\FAT{C}^{c_1} G_1$ that satisfy
\begin{equation}\label{p odd_normal_form} u_\pi^{p^n}=u_\sigma^{p^n}=1, u_\sigma u_\pi u_\sigma^{-1} = \varepsilon u_\pi^{p^{s+1}}. \end{equation} Moreover the class $[c_1] \in H^2(G_1, \FAT{C}^\times)$ is uniquely determined by (\ref{p odd_normal_form}) (see, for example, the proof of Karpilovsky \cite[Theorem 10.1.25]{Ka}).
Now, by Karpilovsky (\cite[Proposition 10.6.1]{Ka}) we have $ M(G) \cong M(G_1) \times M(G_2) \times ( G_1^{ab} \otimes G_2^{ab})$ (here $G^{ab}$ denotes the abelianization of $G$) and hence there exist a two-cocycle $c_0 \in Z^2(G, \FAT{C}^\times)$ and elements $\{{u_g}\}_{g \in \Phi}$ in $\FAT{C}^{c_0} G$ that satisfy equations (\ref{p odd_normal_form}), (\ref{abelian_normal_form}) and $u_{\pi}u_{\gamma_{i}}=u_{\gamma_{i}}u_{\pi}$, $u_{\sigma}u_{\gamma_{i}}=u_{\gamma_{i}}u_{\sigma}$. We denote this set of equations by $E_{G}$. The class $[c_0] \in H^2(G, \FAT{C}^\times)$ is uniquely determined by $E_{G}$ and is easily seen to be nondegenerate. We say that a class $[c_0]$ is of standard form with respect to $\Phi$ if there are elements $\{{u_g}\}_{g \in \Phi}$ in $\FAT{C}^{c_0} G$ that satisfy $E_{G}$.
Now, let $c\in Z^2(G, \FAT{C}^\times)$ be any nondegenerate two-cocycle, and let $u_g$ be representatives of elements of $G$ in $\FAT{C}^{c} G$. As explained above we set $\Phi^{(0)}=\{\pi^{(0)}, \sigma^{(0)}, \gamma^{(0)}_1, \dots, \gamma^{(0)}_{2m}\}=\Phi=\{\pi, \sigma, \gamma_1, \dots, \gamma_{2m}\}$. We are to exhibit a sequence of automorphisms such that their composition applied to $\Phi^{(0)}$ yields a generating set with respect to which the cocycle $c$ is of standard form. First, note that we may assume (by passing to an equivalent cocycle, if necessary) that $u_{\pi^{(0)}}^{p^n} = u_{\sigma^{(0)}}^{p^n} = 1$. Let $\alpha \in \FAT{C}^\times$ be determined by $u_{\sigma^{(0)}} u_{\pi^{(0)}} u_{\sigma^{(0)}}^{-1} = \alpha u_{\pi^{(0)}}^{p^s+1}$. It follows that $u_{\sigma^{(0)}} u_{\pi^{(0)}}^{p^{n-1}} u_{\sigma^{(0)}}^{-1} = \alpha^{p^{n-1}} u_{\pi^{(0)}}^{p^{n-1}}$ and hence, $\alpha^{p^{n-1}}$ is a $p$-th root of unity. Next, observe that since $c$ is nondegenerate, $u_{\sigma^{(0)}}$ cannot commute with $u_{\pi^{(0)}}^{p^{n-1}}$ (for otherwise, $u_{\pi^{(0)}}^{p^{n-1}}$ is contained in the center of the algebra) and hence $\alpha$ is a primitive $p^n$-th root of unity. Now, replacing $\pi^{(0)}$ by a suitable prime to $p$ power of $\pi^{(0)}$ (that is $\pi^{(1)}=(\pi^{(0)})^{l}$, $l$ prime to $p$), we may assume that $\alpha = \varepsilon$. Leaving $\sigma^{(0)}$ and all other generators unchanged, that is $\sigma^{(1)}=\sigma^{(0)}, \gamma^{(1)}_1=\gamma^{(0)}_1, \dots, \gamma^{(1)}_{2m}=\gamma^{(0)}_{2m}$, we obtain a generating set $\Phi^{(1)} = \{\pi^{(1)},\sigma^{(1)}, \gamma^{(1)}_1, \dots, \gamma^{(1)}_{2m} \}$ of $G$, equivalent to $\Phi$, such that the elements $u_{\pi^{(1)}},u_{\sigma^{(1)}}$ in $\FAT{C}^{c} G_1$ satisfy (\ref{p odd_normal_form}).
Assume $(u_{\gamma^{(1)}_i}, u_{\pi^{(1)}}) = \xi_i \neq 1$ (clearly $\xi_i$ is a root of unity of order $p^r (\leq \mathop{\rm ord}\nolimits(\gamma^{(1)}_i) \leq p^s))$. Now, by the nondegeneracy of $c$, the root of unity $\zeta$ determined by the equation $u_{\sigma^{(1)}}^{p^{n-s}} u_{\pi^{(1)}} u_{\sigma^{(1)}}^{-p^{n-s}} = \zeta u_{\pi^{(1)}}$ is of order $p^s$ and hence $\xi_i = \zeta^l$ for some $l$. Observe that $\mathop{\rm ord}\nolimits((({\sigma^{(1)}})^{p^{n-s}})^l) \leq p^r$ and hence we may put $\gamma^{(2)}_i= \gamma^{(1)}_i ({\sigma^{(1)}})^{-l p^{n-s}}$ and get $(u_{\gamma^{(2)}_i}, u_{\pi^{(1)}}) = 1$. Performing this process for all $1\leq i \leq 2m$ (and leaving $\pi^{(1)}$ and $\sigma^{(1)}$ unchanged) we get a set of generators $\Phi^{(2)}$ such that $(u_{\gamma^{(2)}_i}, u_{\pi^{(2)}}) = 1$ for all $i$.
Next, if $(u_{\gamma^{(2)}_i}, u_{\sigma^{(2)}}) \neq 1$ we set $\gamma^{(3)}_i=\gamma^{(2)}_i ({\pi^{(2)}})^{-t p^{n-s}}$ for an appropriate positive integer $t$ and get $(u_{\gamma^{(3)}_i}, u_{\sigma^{(2)}}) = 1$. Performing this process for all $1\leq i \leq 2m$ we get a set of generators $\Phi^{(3)}$ such that $(u_{\gamma^{(3)}_i}, u_{\sigma^{(3)}}) = 1$ for all $i$.
Finally, we may proceed as in Case I and obtain a generating set $\Phi^{(4)}$, equivalent to $\Phi$, with respect to which $c$ is of standard form.
{\bf (III)} Next we consider the group $G = C_{2^{n+1}}\rtimes (C_{2^n}\times C_2)\times C_2 \times C_2 \ldots \times C_2 \times C_2$, $n\geq 2$, with a set of generators $\Phi=\{\pi,\sigma,\tau, \gamma_1, \dots, \gamma_{2m}\}$, where $\sigma\pi\sigma^{-1}=\pi^3$ and $\tau\pi\tau^{-1}=\pi^{-1}$. We write $G = G_1 \times G_2$ where $G_1 = C_{2^{n+1}}\rtimes (C_{2^n}\times C_2)$ and $G_2$ is elementary abelian of rank $2m$.
We first exhibit a construction of a field $K$ and a cocycle $\beta \in Z^2(G_1, K^\times)$ such that the algebra $D \cong K^{\beta}G_1$ is $K$--central simple (in fact a $K$--central division algebra), which is analogous to that in Natapov \cite{N}. Let $K = \FAT{Q}(s,t)$ be the subfield of $\FAT{C}$ generated by algebraically independent elements $s$ and $t$. Let $L = K(v_\pi)/K$ be a cyclotomic extension where $v_\pi^{2^{n+1}} = -1$. The Galois action of $Gal(L/K) \cong Z_{2^n} \times Z_2 = \langle \sigma, \tau \rangle$ on $L$ is given by \[
\sigma(v_\pi) = v_\pi^3 \;\;\; {\rm and} \;\;\; \tau(v_\pi) =
v_\pi^{-1}. \] The algebra $D$ is the crossed product $D=(L/K, \langle \sigma, \tau \rangle)$ determined by the following relations \[
v_\sigma^{2^n} = s, \; v_\tau^2 = t, \; (v_\sigma, v_\tau) = 1, \] where $v_\sigma$ and $v_\tau$ represent $\sigma$ and $\tau$ in $D$. It is easy to see that $D$ is isomorphic to a twisted group algebra of the form $K^{\beta} G_1$.
Fix a primitive $2^{n+1}$-th root of unity $\varepsilon$, and let $\xi$ be a square root of $\varepsilon$. Let $\widetilde{s}$ and $\widetilde{t}$ be elements of $\FAT{C}$ such that $\widetilde{s}^{2^n} = s^{-1}$ and $\widetilde{t}^2 = t^{-1}$. Then the elements $u_{\pi^i \sigma^j \tau^k} = (\xi v_\pi)^i (\widetilde{s} v_\sigma)^j (\widetilde{t} v_\tau)^k$ in $\FAT{C}^{\beta} G_1 \cong \FAT{C}\otimes_{K}D$ satisfy
\begin{equation}\label{p 2_normal_form} u_\pi^{2^{n+1}}=u_\sigma^{2^n}=u_\tau^2=1, \ u_\sigma u_\pi = \varepsilon u_\pi^3 u_\sigma, \ u_\tau u_\pi = \varepsilon^{-1} u_\pi^{-1}u_\tau, \ u_\tau u_\sigma =u_\sigma u_\tau. \end{equation}
Clearly, there exists a two-cocycle $c_1 \in Z^2(G_1, \FAT{C}^\times)$, cohomologous to $\beta$, such that $u_{g_1}u_{g_2} = c_1(g_1,g_2)u_{g_1g_2}$ for all $g_1, g_2 \in G_1$ and moreover the class $[c_1]$ is determined by (\ref{p 2_normal_form}).
Now, by Karpilovsky \cite[Proposition 10.6.1]{Ka} there exist a two-cocycle $c_0 \in Z^2(G, \FAT{C}^\times)$ and elements $\{{u_g}\}_{g \in \Phi}$ in $\FAT{C}^{c_0} G$ that satisfy (\ref{p 2_normal_form}), (\ref{abelian_normal_form}) and $u_{g}u_{\gamma_{i}}=u_{\gamma_{i}}u_{g}$ for $g\in \{\pi, \sigma, \tau\}$. We denote this set of equations by $E_{G}$. As in the previous cases, also here, $[c_0]$ is determined by $E_G$, and we say that the cocycle $c_{0}$ is of standard form with respect to $\Phi$.
Now, let $c\in Z^2(G, \FAT{C}^\times)$ be a nondegenerate two-cocycle, and consider the algebra $\FAT{C}^{c} G = \oplus_{g \in G} \FAT{C} u_g$. Let $\Phi^{(0)}= \{\pi^{(0)},\sigma^{(0)},\tau^{(0)}, \gamma^{(0)}_1, \dots, \gamma^{(0)}_{2m}\} = \Phi=\{\pi,\sigma,\tau, \gamma_1, \dots, \gamma_{2m}\}$. We may assume (by passing to an equivalent cocycle, if necessary) that $u_{\pi^{(0)}}^{2^{n+1}} = u_{\sigma^{(0)}}^{2^n} = u_{\tau^{(0)}}^{2} = 1$. Let $\alpha \in \FAT{C}^\times$ be determined by the equation $u_{\sigma^{(0)}} u_{\pi^{(0)}} u_{\sigma^{(0)}}^{-1} = \alpha u_{\pi^{(0)}}^3$. A straightforward calculation shows that for any $k$ in $\FAT{N}$, \begin{equation}\label{sigmaction} u_{\sigma^{(0)}}^k u_{\pi^{(0)}} u_{\sigma^{(0)}}^{-k} = \alpha^{\frac{3^k-1}{2}}u_{\pi^{(0)}}^{3^k}, \ 2^k \mid \frac{3^{2^k-1}}{2} \ \hbox{and} \ 2^{k+1} \nmid \frac{3^{2^k-1}}{2}. \end{equation}
We \underline{claim} that $\alpha$ is a primitive $2^{n+1}$-th root of unity. To see this note that $\sigma^{(0)}$ and $(\pi^{(0)})^{2^n}$ commute and therefore, $u_{\sigma^{(0)}} u_{\pi^{(0)}}^{2^n} u_{\sigma^{(0)}}^{-1} = \pm u_{\pi^{(0)}}^{2^n}$. We need to show that $u_{\sigma^{(0)}} u_{\pi^{(0)}}^{2^n} u_{\sigma^{(0)}}^{-1} = -u_{\pi^{(0)}}^{2^n}$. If not we have $$ u_{\pi^{(0)}}^{2^n} = (u_{\sigma^{(0)}} u_{\pi^{(0)}} u_{\sigma^{(0)}}^{-1})^{2^n} = (\alpha u_{\pi^{(0)}}^3)^{2^n} = \alpha^{2^n} u_{\pi^{(0)}}^{2^n}, $$ that is $\alpha^{2^n} = 1$. Then, by (\ref{sigmaction}) we have $$ u_{\sigma^{(0)}}^{2^{n-1}} u_{\pi^{(0)}} u_{\sigma^{(0)}}^{-2^{n-1}} = \alpha^{\frac{3^{2^{n-1}}-1}{2}}u_{\pi^{(0)}}^{3^{2^{n-1}}} = u_{\pi^{(0)}} $$ and hence $u_{\sigma^{(0)}}^{2^{n-1}}$ is in the center of $\FAT{C}^{c} G$. This contradicts the nondegeneracy of the cocycle $c$ and the claim is proved. It follows that if we replace $\pi^{(0)}$ by a suitable odd power of $\pi^{(0)}$ (that is $\pi^{(1)}={\pi^{(0)}}^{l}$, $l$ is odd) and leaving the other generators unchanged we obtain a generating set $\Phi^{(1)} = \{\pi^{(1)},\sigma^{(1)}, \tau^{(1)} \gamma^{(1)}_1, \dots, \gamma^{(1)}_{2m} \}$ of $G$, equivalent to $\Phi$, such that $u_{\sigma^{(1)}} u_{\pi^{(1)}} u_{\sigma^{(1)}}^{-1} = \varepsilon u_{\pi^{(1)}}^3$.
Next, since $\sigma^{(1)}$ and $\tau^{(1)}$ commute and $\tau^{(1)}$ is of order 2 we have that $(u_{\sigma^{(1)}}, u_{\tau^{(1)}}) = \pm 1$. If the commutator is $-1$, we put $\tau^{(2)}=\tau^{(1)} {\pi^{(1)}}^{2^n}$ and $\sigma^{(2)}=\sigma^{(1)}$. Otherwise we leave both unchanged and, in either case, get an equivalent generating set $\Phi^{(2)}$ with $(u_{\sigma^{(2)}}, u_{\tau^{(2)}}) = 1$.
Let $\beta \in \FAT{C}^\times$ be determined by the equation $u_{\tau^{(2)}} u_{\pi^{(2)}} u_{\tau^{(2)}}^{-1} = \beta u_{\pi^{(2)}}^{-1}$. Then $$ u_{\sigma^{(2)}} u_{\tau^{(2)}} u_{\pi^{(2)}} u_{\tau^{(2)}}^{-1} u_{\sigma^{(2)}}^{-1}= \beta \varepsilon^{-1} u_{\pi^{(2)}}^{-3}, \ \hbox{and} \ u_{\tau^{(2)}} u_{\sigma^{(2)}} u_{\pi^{(2)}} u_{\sigma^{(2)}}^{-1} u_{\tau^{(2)}}^{-1} = \varepsilon \beta^3 u_{\pi^{(2)}}^{-3} $$ and by equality of the left hand sides we have that $\beta = \pm \varepsilon^{-1}$. Now put $\tau^{(3)}=\tau^{(2)} (\sigma^{(2)})^{2^{n-1}}$ if $\beta = - \varepsilon^{-1}$ and $\tau^{(3)}=\tau^{(2)}$ otherwise. Leaving the other generators unchanged we obtain an equivalent set of generators $\Phi^{(3)}$ such that the elements $u_{\tau^{(3)}}, u_{\pi^{(3)}} \in \FAT{C}^{c} G$ satisfy $u_{\tau^{(3)}} u_{\pi^{(3)}} u_{\tau^{(3)}}^{-1} = - \beta u_{\pi^{(3)}}^{-1} = \varepsilon^{-1} u_{\pi^{(3)}}^{-1}$. Thus we have shown there exist $\{{u_{g}}\}_{g \in \{\pi^{(3)},\sigma^{(3)},\tau^{^{(3)}}\}}$ in $\FAT{C}^{c} G_1$ that satisfy (\ref{p 2_normal_form}).
Next, we may proceed as in Aljadeff, et al (\cite[Proposition 13]{AHN}) to get a generating set $\Phi^{(4)}$ of $G$, equivalent to $\Phi$, such that (a) (\ref{p 2_normal_form}) is satisfied (b) $(u_{\gamma^{(4)}_{2k-1}},u_{\gamma^{(4)}_{2k}})= \pm 1$ for all $1\leq k\leq m$ and (c) all other commutators of the $u_{\gamma^{(4)}_i}$'s are trivial.
The next step is to add the condition $u_{g} u_{\gamma_i} = u_{\gamma_i} u_{g}$ for $g\in \{\pi, \sigma, \tau \}$:
For each $\gamma^{(4)}_i \in \{\gamma^{(4)}_1, \dots, \gamma^{(4)}_{2m}\}$, if $(u_{\gamma^{(4)}_i}, u_{\pi^{(4)}}) = - 1$ then put $\gamma^{(5)}_i = \gamma^{(4)}_i (\sigma^{(4)})^{2^{n-1}}$, otherwise leave $\gamma^{(4)}_i$ unchanged. Leaving $\pi^{(4)}, \sigma^{(4)}, \tau^{(4)}$ unchanged we obtain a generating set $\Phi^{(5)}$ such that $(u_{\gamma^{(5)}_i}, u_{\pi^{(5)}}) =1$ for all $i$. Next, if $(u_{\gamma^{(5)}_i}, u_{\sigma^{(5)}}) = - 1$ put $\gamma^{(6)}_i = \gamma^{(5)}_i (\pi^{(5)})^{2^n}$, otherwise leave $\gamma^{(5)}_i$ unchanged. We obtain a generating set $\Phi^{(6)}$ such that $(u_{\gamma^{(6)}_i}, u_{\sigma^{(6)}}) = 1$ for all $i$.
At this point we have for each $1\leq k \leq m$, $(u_{\gamma^{(6)}_{2k-1}},u_{\gamma^{(6)}_{2k}})= \pm 1$. In fact we \underline{claim} that $(u_{\gamma^{(6)}_{2k-1}},u_{\gamma^{(6)}_{2k}})= - 1$. If not, by the preceding steps we have that the element $u_{\gamma^{(6)}_{2k-1}}$ centralizes all the $u_{\gamma^{(6)}_i}$'s as well as $u_{\pi^{(6)}}$ and $u_{\sigma^{(6)}}$. It follows that $u_{\gamma^{(6)}_{2k-1}}$ does not centralize $u_{\tau^{(6)}}$, for otherwise we get a contradiction to the nondegeneracy of the two-cocycle $c$. A similar argument shows that $u_{\gamma^{(6)}_{2k}}$ does not centralize $u_{\tau^{(6)}}$ , but then the product $u_{\gamma^{(6)}_{2k-1}}u_{\gamma^{(6)}_{2k}}$ is central, a contradiction. This proves the claim. We refer to the pair of elements ${\gamma_{2k-1}}, {\gamma_{2k}}$ as {\it partners}. We proceed now as follows: If $(u_{\gamma^{(6)}_{1}}, u_{\tau^{(6)}}) = - 1$, put $\tau^{(7)} = \tau^{(6)} \gamma^{(6)}_2$ (i.e. multiply $\tau^{(6)}$ by the partner of $\gamma^{(6)}_1$), and leave $u_{\gamma^{(6)}_1}$ and all the other generators unchanged. We continue in a similar way with $\gamma^{(6)}_2, \gamma^{(6)}_3, \dots, \gamma^{(6)}_{2m}$.
We now have a generating set $\Phi^{(r)}$ (some $r$), equivalent to $\Phi$, such that $\{{u_{g}}\}_{g \in \Phi^{(r)}}$ in $\FAT{C}^c G$ satisfy (a) equations (\ref{p 2_normal_form}) (b) $(u_{\gamma^{(r)}_{2k-1}},u_{\gamma^{(r)}_{2k}})= - 1$ for all $1\leq k \leq m$ (c) all other commutators of the $u_{\gamma^{(r)}_i}$'s are trivial and (d) $u_{\gamma^{(r)}_i}$ centralizes the subalgebra $\FAT{C}(u_{\pi^{(r)}}, u_{\sigma^{(r)}}, u_{\tau^{(r)}})$ for all $i$. It follows that the cocycle $c$ is of standard form with respect to $\Phi^{(r)}$.
{\bf (IV)} In the last step we consider the group $G = C_4 \rtimes (C_2 \times C_2) \times C_2 \times C_2 \ldots \times C_2 \times C_2$ with a set of generators $\Phi = \{\pi,\sigma,\tau, \gamma_1, \dots, \gamma_{2m}\}$, where $\sigma\pi\sigma^{-1}=\pi^3$ and $\tau\pi=\pi\tau$. We write $G = G_1 \times G_2$ where $G_1 = C_4 \rtimes (C_2 \times C_2)$ and $G_2$ is elementary abelian of rank $2m$.
As in the previous case, a construction of a $K$--central division algebra of the form $K^{\beta} G_1$, analogous to that in Natapov \cite{N}, yields a $4\times4$ matrix algebra isomorphic to $\FAT{C}^{c_1} G_1$ with a basis $\{u_g\}_{g \in G_1}$ such that $u_{\pi^i \sigma^j \tau^k} = u_\pi^i u_\sigma^j u_\tau^k$ and \begin{equation}\label{422_normal_form} u_\pi^4=u_\sigma^2=u_\tau^2=1, \ u_\sigma u_\pi = \varepsilon u_\pi^3 u_\sigma, \ u_\tau u_\pi = - u_\pi u_\tau, \ u_\tau u_\sigma =u_\sigma u_\tau, \end{equation} where $\varepsilon$ is a 4-th root of unity. Clearly, the class $[c_1]$ is determined by (\ref{422_normal_form}).
Now, by Karpilovsky \cite[Proposition 10.6.1]{Ka} there exist a two-cocycle $c_0 \in Z^2(G, \FAT{C}^\times)$ and elements $\{{u_g}\}_{g \in \Phi}$ in $\FAT{C}^{c_0} G$ that satisfy (\ref{422_normal_form}), (\ref{abelian_normal_form}), and $u_{g}u_{\gamma_{i}}=u_{\gamma_{i}}u_{g}$ for $g\in \{\pi, \sigma, \tau\}$. We denote these equations by $E_{G}$. Note that $[c_0]$ is determined by $E_G$. As usual we will say the cocycle $c_{0}$ is of standard form with respect to $\Phi$.
Now, let $c\in Z^2(G, \FAT{C}^\times)$ be a nondegenerate two-cocycle and write $\FAT{C}^{c} G = \oplus_{g \in G} \FAT{C} u_g$. As in previous steps put $\Phi^{(0)}=\{\pi^{(0)},\sigma^{(0)},\tau^{(0)}, \gamma^{(0)}_1, \dots, \gamma^{(0)}_{2m}\}=\Phi = \{\pi,\sigma,\tau, \gamma_1, \dots, \gamma_{2m}\}$. We may assume (by passing to an equivalent cocycle, if necessary) that $u_{\pi^{(0)}}^4=u_{\sigma^{(0)}}^2=u_{\tau^{(0)}}^2=1$.
We \underline{claim} that the element $\alpha \in \FAT{C}^\times$ determined by the equation $u_{\sigma^{(0)}} u_{\pi^{(0)}} u_{\sigma^{(0)}}^{-1} = \alpha u_{\pi^{(0)}}^3$ is a root of unity of order $4$. To see this note that $(u_g, u_{\pi^{(0)}}^2) = 1$ for any element $g$ in $G$ of order $2$ that commutes with $\pi^{(0)}$. It follows that $(u_{\sigma^{(0)}}, u_{\pi^{(0)}}^2) = -1$, for otherwise $u_{\pi^{(0)}}^2$ is contained in the center of the algebra. This proves the claim. Therefore, we may set $\pi^{(1)} = {\pi^{(0)}}^{3}$ if necessary and we obtain a generating set $\Phi^{(1)} = \{\pi^{(1)},\sigma^{(1)}, \tau^{(1)} \gamma^{(1)}_1, \dots, \gamma^{(1)}_{2m} \}$ of $G$, equivalent to $\Phi$, such that $u_{\sigma^{(1)}} u_{\pi^{(1)}} u_{\sigma^{(1)}}^{-1} = \varepsilon u_{\pi^{(1)}}^3$. By the nondegeneracy of $c$ there is an element $h^{(1)} \in \{\tau^{(1)}, \gamma^{(1)}_1, \dots, \gamma^{(1)}_{2m}\}$ (these elements generate $C_G(\pi^{(1)})$, the centralizer of $\pi^{(1)}$ in $G$) such that $(u_{h^{(1)}}, u_{\pi^{(1)}}) = -1$. We set $\tau^{(2)} = h^{(1)}$ and $h^{(2)} = \tau^{(1)}$ and leave the other generators unchanged. Next, arguing as in the previous case we may assume that $(u_{\tau^{(2)}}, u_{\sigma^{(2)}})= 1$. Thus we have a generating set $\Phi^{(2)}$ such that $u_{\pi^{(2)}}, u_{\tau^{(2)}}$ and $u_{\sigma^{(2)}}$ satisfy equation (\ref{422_normal_form}). Finally, as in the previous case we obtain a generating set $\Phi^{(r)}=\{\pi^{(r)},\sigma^{(r)}, \tau^{(r)}, \gamma^{(r)}_1, \dots, \gamma^{(r)}_{2m}\}$ (some $r$), whose elements satisfy (a) $u_{\gamma^{(r)}_i}$ centralize the subalgebra $\FAT{C}(u_{\pi^{(r)}}, u_{\sigma^{(r)}}, u_{\tau^{(r)}})$ for all $i$ (b) $(u_{\gamma^{(r)}_{2k-1}},u_{\gamma^{(r)}_{2k}})= - 1$ for all $1\leq k\leq m$, and (c) all other commutators of the $u_{\gamma^{(r)}_i}$'s are trivial. Therefore $c$ is of standard form with respect to $\Phi^{(r)}$. This completes the proof. \qed
\begin{cor} \label{independence.cor} If $G$ is a group on the list $\Lambda$ and $c$ and $c'$ are nondegenerate cocycles on $G$ with values in $\mathbb{C}^\times$, then the universal central simple algebras $Q(U_{G,c})$ and $Q(U_{G,c'})$ are isomorphic. \end{cor}
\proof By Theorem \ref{transitive.thm} there is an automorphism $\varphi$ of $G$ such that $\varphi(c)$ and $c'$ are cohomologous two-cocycles. In particular it follows that $Q(U_{G,c})$ and $Q(U_{G,c'})$ have the same field of definition $\FAT{Q}(\mu)$ and $\varphi$ induces an automorphism $\varphi_{*}$ of $\FAT{Q}(\mu)$. Clearly, $\varphi_{*}$ extends to a $\mathbb{Q}$ isomorphism of the free algebras $\Sigma_{c}(\mathbb{Q}(\mu))\longrightarrow \Sigma_{\varphi(c)}(\mathbb{Q}(\mu))$ by putting $x_{ig}\longmapsto x_{i\varphi(g)}$. Furthermore it induces an isomorphism of the corresponding universal algebras $U_{G,c}= \Sigma_{c}(\mathbb{Q}(\mu))/T_{c}(\mathbb{Q}(\mu))$ and $U_{G, \varphi(c)}$, and therefore an isomorphism of their central localization, namely the universal central simple $G$--graded algebras.\qed
\begin{cor} \label{divisionalg.cor} Let $G$ be a group of central type and let $c$ be a nondegenerate cocycle on $G$. The universal central simple algebra $Q(U_{G,c})$ is a division algebra if and only if $G$ is on the list $\Lambda$. \end{cor}
\proof This follows from the the previous corollary and Proposition \ref{list.prop}. \qed
Recall from the introduction that for a group $G$ of central type we let $\mathop{\rm ind}\nolimits(G)$ denote the maximum over all nondegenerate cocycles $c$ of the indices of the simple algebras $Q(U_{G,c})$. We have seen that if $G$ is not on the list, then the universal central simple algebra $Q(U_G)$ is not a division algebra and so the index of $G$ is strictly less than $\mathop{\rm ord}\nolimits(G)^{1/2}$. In fact it is shown in Aljadeff and Natapov \cite{AN} that the groups on the list are the only groups responsible for the index of $Q(U_G)$. Here is the precise result.
\begin{thm}\label{responsible.thm} Let $P$ be a $p$-group of central type and let $Q(U_{P,c})$ be the universal central simple $P$--graded algebra for some nondegenerate two-cocycle $c$ on $P$. Then there is a sub-quotient group $H$ of $P$ on the list $\Lambda$, such that $Q(U_{P,c})\cong M_{p^{r}}(Q(U_{H,\alpha}))$, where the cocycle $\alpha$ on $H$ is obtained in a natural way from $c$, namely there is a subgroup $\hat{H}$ of $P$ such that $\hat{H}/N\cong H$ and $\inf([\alpha])=\mathop{\rm res}\nolimits([c])$ on $\hat{H}$. \end{thm}
\begin{cor} Let $G$ be a central type $p$-group. Consider all central type groups $H$ that are isomorphic to sub-quotients of $G$ and are members of $\Lambda$. Then $\mathop{\rm ind}\nolimits(G)\leq \sup(\mathop{\rm ord}\nolimits(H)^{1/2})$. \end{cor}
DeMeyer and Janusz prove in \cite[Corollary 4]{DJ} that for an arbitrary group of central type $G$ and a nondegenerate two-cocycle $c \in Z^2(G, \FAT{C}^\times)$ any Sylow-$p$ subgroup $G_p$ of $G$ is of central type, and furthermore the restriction of $c$ to $G_p$ is nondegenerate. Using this result Geva in his thesis proved that for any subfield $k$ of $\FAT{C}$ the twisted group algebra $k^c G$ is
(non-graded) isomorphic to $\bigotimes_{p | \mathop{\rm ord}\nolimits(G)} k^c G_p$. It follows that for each prime $p$ dividing the order of $G$ there is a $p$-group $H_{p}$ on the list $\Lambda$ which is isomorphic to a sub-quotient of $G$ and such that $F^{s}G\cong \bigotimes_{p} M_{p^{r}}(F^{\alpha_{p}}H_{p})$. Combining this with the preceding Corollary we obtain:
\begin{cor} Let $G$ be a group of central type, and for each prime $p$, $G_p$ be a Sylow-$p$ subgroup of $G$. Consider all central type groups $H_{p}$ that are isomorphic to sub-quotients of $G_p$ and are members of $\Lambda_p$. Then $\mathop{\rm ind}\nolimits(G)\leq \prod_{p} \sup(\mathop{\rm ord}\nolimits(H_{p})^{1/2})$. \end{cor}
It follows from Proposition \ref{list.prop} that for any group $G$ of central type not on the list and any $G$--grading on $M_n(\mathbb{C})$, the universal central simple algebra is not a division algebra. That means we can find nonidentity polynomials $p(x_{ig})$ and $q(x_{ig})$ over the field of definition $\mathbb{Q}(\mu)$ such that $p(x_{ig})q(x_{ig})$ is a graded identity. In fact there will exist a nonidentity polynomial $p(x_{ig})$ over $\mathbb{Q}(\mu)$ such that $p(x_{ig})^{r}$ is a graded identity for some positive integer $r$. This is clearly equivalent to saying that under every substitution the value of $p$ in $M_n(\mathbb{C})$ is a nilpotent matrix. We will refer to such a polynomial as a {\it nilpotent} polynomial.
We present an explicit example of a nilpotent polynomial. We let $S_3$ be the permutation group on three letters, and $C_6 = \langle z \rangle$ be a cyclic group of order $6$. Let $\sigma = (123)$ and $\tau = (12)$ be generators of $S_3$ and define an action of $S_3$ on $C_6$ by $\tau(z) = z^{-1}$ and $\sigma(z) = z$. We consider the group $G = S_3 \ltimes C_6$. Note that $G$ is not nilpotent and hence not on the list $\Lambda$.
Let $C_6^\vee = \mathop{\rm Hom}\nolimits(C_6 ,\FAT{C}^\times) = \langle \chi_z \rangle$ denote the dual group of $C_6$ and let $\langle \ , \ \rangle$ denote the usual pairing between $C_6$ and $C_6^\vee$ (i.e. $\langle a , \chi \rangle = \chi(a)$ for all $a \in C_6, \ \chi \in C_6^\vee$). It is well known (\cite[Corollary 10.10.2]{Ka}) that $H^1(S_3, C_6^\vee) \cong H^2(S_3 \ltimes C_6, \FAT{C}^\times)$. The isomorphism may be given by \begin{equation}\label{thecocycle*} \pi \mapsto c \ : \ c(h_1a_1, h_2a_2) = \langle h_2\cdot a_1, \pi(h_2)\rangle, \end{equation} for all $h_1a_1, h_2a_2 \in S_3 \ltimes C_6$. Note that the restriction of $c$ on $S_3$, as well as on $C_6$, is trivial.
Moreover, it is shown in Etingof and Gelaki \cite{EG1} that there exists a bijective one-cocycle $\pi: S_3 \rightarrow C_6^\vee$ and, because of the bijectivity, the corresponding two-cocycle $c$ is nondegenerate. In particular this proves that $G$ is of central type (see Etingof and Gelaki \cite{EG2}). One may define a bijective $\pi$ as follows: \[ \begin{array}{ll}
\pi(\mathit{id}) = \chi_1 & \pi(12) = \chi_{z} \\
\pi(123) = \chi_{z}^2 & \pi(23) = \chi_{z}^5 \\
\pi(132) = \chi_{z}^4 & \pi(13) = \chi_{z}^3 \\ \end{array} \]
Let $c$ denote the nondegenerate two-cocycle which corresponds to $\pi$. Then the twisted group algebra $\FAT{C}^{c} G$ is isomorphic to $M_6(\FAT{C})$ or, equivalently, $M_6(\FAT{C})$ is $G$--graded with the class $[c]$. The group $G$ is not on the list $\Lambda$ and hence the corresponding universal algebra $U_G$ is not a domain. Consider the subalgebra $\FAT{C}(u_\sigma, u_y)$ where $y = z^2$. It is straightforward to verify that the cocycle $c$ satisfies $c(y, \sigma) = \omega^2$, where $\omega = e^{\frac{2\pi i}{3}}$ is a primitive third root of unity in $\FAT{C}$, and hence $\FAT{C}(u_\sigma, u_y)$ is a symbol algebra. It follows that the corresponding subalgebra $F(t_\sigma u_\sigma, t_y u_y)$ of the universal central simple algebra $Q(U_{G,c})=F^{s(c)}G$ is a symbol algebra $(a,b)_3$, and by Aljadeff, et al \cite[Lemma 6]{AHN2} $a$ and $b$ are roots of unity. It follows that $F(t_\sigma u_\sigma, t_y u_y)$ is split, that is isomorphic to $M_3(F)$. Thus we know that there exists a degree three nilpotent element in $U_{G,c}$. To construct it we consider the element $u_\sigma - u_y u_\sigma$ in $\FAT{C}^{c} G$. Note that this is not zero since $u_\sigma$ and $u_y u_\sigma$ are linearly independent over $\FAT{C}$. Its square \[ (u_\sigma - u_y u_\sigma)^2 = u_\sigma^2 - u_\sigma u_y u_\sigma - u_y u_\sigma^2 + (u_y u_\sigma)^2 \]
is not zero as well, but using $u_y u_\sigma = \omega^2 u_\sigma u_y$ and $u_y^3 = 1$ one shows that its third power \[ (u_\sigma - u_y u_\sigma)^3 = u_\sigma^3 - u_\sigma^2 u_y u_\sigma - u_\sigma u_y u_\sigma^2 + u_\sigma^2 u_y u_\sigma u_y u_\sigma - u_y u_\sigma^3 + u_y u_\sigma^2 u_y u_\sigma + u_y u_\sigma u_y u_\sigma^2 - (u_y u_\sigma)^3 \] vanishes.
Now, as mentioned above, $res^G_{S_3}(c) = res^G_{C_6}(c) = 1$, hence we may assume that $u_gu_h = u_{gh}$ for all $g,h \in S_3$ or $g,h \in C_6$. In particular, we have $u_e = u_\tau^2 u_y^3$. Also, using the cocycle values as defined in (\ref{thecocycle*}), we obtain $u_y = \omega u_\tau u_y u_\tau u_y^2$.
Let $x_\tau = x_{1\tau}, x_\sigma = x_{1\sigma}, x_y = x_{1y}$ be indeterminates in $\Omega=\{x_{ig} : i\in \mathbb{N}, g\in G\}$. Let the polynomial $f(x_\tau, x_\sigma, x_y) \in \FAT{Q}(\omega)\langle\Omega\rangle$ be given by \[ f(x_\tau, x_\sigma, x_y) = x_\sigma x_\tau^2 x_y^3 - \omega x_\tau x_y x_\tau x_y^2 x_\sigma. \] It follows that the polynomial $f^3(x_\tau, x_\sigma, x_y)$ is a polynomial identity for $M_6(\FAT{C})$ while $f$ and $f^2$ are not.
We end this example by expressing $f^3$ explicitly as a $\FAT{Q}(\omega)-$linear combination of elementary identities: \begin{equation*} \begin{array}{lllll} f^3(x_\tau, x_\sigma, x_y) & = & (x_\sigma x_\tau^2 x_y^3)^3 - (x_\tau x_y x_\tau x_y^2 x_\sigma)^3 \\
& + & \omega^2 x_\tau x_y x_\tau x_y^2 x_\sigma (x_\sigma x_\tau^2 x_y^3)^2
- \omega (x_\sigma x_\tau^2 x_y^3)^2 x_\tau x_y x_\tau x_y^2 \\
& + & x_\tau x_y x_\tau x_y^2 x_\sigma (x_\sigma x_\tau^2 x_y^3)^2
- \omega x_\sigma x_\tau^2 x_y^3 x_\tau x_y x_\tau x_y^2 x_\sigma x_\sigma x_\tau^2 x_y^3 \\
& + & \omega^2 x_\sigma x_\tau^2 x_y^3 (x_\tau x_y x_\tau x_y^2 x_\sigma)^2
- (x_\tau x_y x_\tau x_y^2 x_\sigma)^2 x_\sigma x_\tau^2 x_y^3 \\
& + & \omega^2 (x_\tau x_y x_\tau x_y^2 x_\sigma) x_\sigma x_\tau^2
x_y^3 (x_\tau x_y x_\tau x_y^2 x_\sigma)
- \omega (x_\tau x_y x_\tau x_y^2 x_\sigma)^2 x_\sigma x_\tau^2
x_y^3. \end{array} \end{equation*}
If $G$ is on the list $\Lambda$ and $c$ is a nondegenerate two-cocycle on $G$ we have seen that the algebra $Q(U_{G,c})$ is a division algebra of degree $n=\mathop{\rm ord}\nolimits(G)^{1/2}$. In the next result we calculate the index of $Q(U_{G,c})\otimes_{\mathbb{Q}(\mu)}\mathbb{C}$, where $\mathbb{Q}(\mu)$ is the field of definition for the graded identities.
\begin{prop} \label{indexofext.prop} If $G$ is on the list $\Lambda$ and $c$ is a nondegenerate two-cocycle on $G$ then the index of $Q(U_{G,c})\otimes_{\mathbb{Q}(\mu)}\mathbb{C}$ is $n/\mathop{\rm ord}\nolimits(G^\prime)$. In particular $Q(U_{G,c})\otimes_{\mathbb{Q}(\mu)}\mathbb{C}$ is a division algebra if and only if $G$ is abelian of symmetric type. \end{prop} \proof By Aljadeff and Haile \cite{AH} (see the discussion at the beginning of section two) the subalgebra $F^sG^\prime$ of $Q(U_{G,c})=F^sG$ is a cyclotomic extension of $F$. It follows that the index of $Q(U_{G,c})\otimes_{\mathbb{Q}(\mu)}\mathbb{C}$ is at most $n/\mathop{\rm ord}\nolimits(G^\prime)$. We proceed to prove the opposite inequality. By Natapov \cite[proof of Theorem 6]{N} there is a subfield $K$ of $\mathbb{C}$ containing $\mathbb{Q}(\mu)$ and a nondegenerate cocycle $\beta$ on $G$ with values in $K$ such that the algebra $K^\beta G\otimes_{\mathbb{Q}(\mu)}\mathbb{C}$ has index exactly $n/\mathop{\rm ord}\nolimits(G^\prime)$. By a specialization argument it follows that the algebra $Q(U_{G,\beta})$ has index at least $n/\mathop{\rm ord}\nolimits(G^\prime)$. By Corollary \ref{independence.cor} $Q(U_{G,\beta})$ is isomorphic to $Q(U_{G,c})$ and hence the index of $Q(U_{G,c})$ is at least $n/\mathop{\rm ord}\nolimits(G^\prime)$. \qed
\end{document} | arXiv |
\begin{document}
\title{Fleming-Viot selects the minimal quasi-stationary distribution: The Galton-Watson case.}
\author{Amine Asselah, Pablo A. Ferrari, Pablo Groisman, Matthieu Jonckheere \\ \sl Universit\'e Paris-Est, Universidad de Buenos Aires, IMAS-Conicet} \date{}
\maketitle
\noindent {\bf Abstract}
Consider $N$ particles moving independently, each one according to
a subcritical continuous-time Galton-Watson process unless it hits
$0$, at which time it jumps instantaneously to the position of one
of the other particles chosen uniformly at random. The resulting
dynamics is called Fleming-Viot process. We show that for
each $N$ there exists a unique invariant measure for the
Fleming-Viot process, and that its stationary empirical distribution
converges, as $N$ goes to infinity, to the minimal
quasi-stationary distribution of the Galton-Watson process
conditioned on non-extinction.
\vskip 3mm
\noindent {{\it AMS 2000 subject classifications}. Primary 60K35; Secondary 60J25}
\noindent {\it Key words and phrases}. Quasi-stationary distributions, Fleming-Viot processes, branching processes, selection principle.
\section{Introduction} The concept of {\it quasi-stationarity} arises in stochastic modeling of population dynamics. In 1947, Yaglom \cite{yaglom} considers subcritical Galton-Watson processes conditioned to survive long times. He shows that as time is sent to infinity, the conditioned process, started with one individual, converges to a law, now called a {\it quasi-stationary distribution}. For any Markov process, and a subset $A$ of the state space, we denote by $\mu T_t$ the law of the process at time $t$ conditioned on not having hit $A$ up to time $t$, with initial distribution $\mu$. A probability measure on $A^c$ is called \emph{quasi-stationary distribution} if it is a fixed point of $T_t$ for any $t>0$.
In 1966, Seneta and Veres-Jones \cite{seneta-veres} realize that for subcritical Galton-Watson processes, there is a one-parameter family of quasi-stationary distributions and show that the Yaglom limit distribution has the minimal expected time of extinction among all quasi-stationary distributions. This unique {\it minimal} quasi-stationary distribution is denoted here ${\nu_{\rm qs}^*}$. They also show that with an initial distribution $\mu$ with finite first moment, $\mu T_t$ converges to ${\nu_{\rm qs}^*}$ as $t$ goes to infinity.
In 1978, Cavender \cite{cavender} shows that for Birth and Death chains on the non negative integers absorbed at 0, the set of quasi-stationary measures is either empty or is a one parameter family. In the latter case, Cavender extends the {\it selection principle} of Seneta and Veres-Jones. He also shows that the limit of the sequence of quasi-stationary distributions for truncated processes on $\{1,\dots,L\}$ converges to ${\nu_{\rm qs}^*}$ as $L$ is sent to infinity. This picture holds for a class of irreducible Markov processes on the non-negative integers with 0 as absorbing state, as shown in 1996 by Ferrari, Kesten, Martinez and Picco \cite{ferrari-kesten}. The main idea in \cite{ferrari-kesten} is to think of the conditioned process $\mu T_t$ as a mass transport with refeeding from the absorbing state to each of the transient states with a rate proportional to the transient state mass. More precisely, denoting ${\mathbb N}$ the set of positive integers, the Kolmogorov forward equation satisfied by $\mu T_t(x)$, for each $x\in{\mathbb N}$, reads
\be{kfe} \frac{\partial}{\partial t} \mu T_t(x) = \sum_{y: y\ne x}\big(q(x,y) + q(x,0) \mu T_t(y)\big)\,[\mu T_t(y)- \mu T_t(x)], \end{equation}
where $q(x,y)$ is the jump rate from $x$ to $y$. The first term in the right hand side represents the displacement of mass due to the jumps of the process and the second term represents the mass going from each $x$ to 0 and then coming instantaneously to $y$.
In 1996, Burdzy, Holyst, Ingerman and March \cite{burdzy1} introduced a {\it
genetic} particle system called \emph{Fleming-Viot} named after models proposed in \cite{fv}, which can be seen as a particle system mimicking the evolution \eqref{kfe}. The particle system can be built from a process with absorption $Z_t$ called \emph{driving process}; the position $Z_t$ is interpreted as a genetic trait, or fitness, of an individual at time $t$. In the $N$-particle Fleming-Viot system, each trait follows independent dynamics with the same law as $Z_t$ except when one of them hits state 0, a lethal trait: at this moment the individual adopts the trait of one of the other individuals chosen uniformly at random. Leaving aside the genetic interpretation, the empirical distribution of the $N$ particles at positions $\xi\in{\mathbb N}^N$ is defined as a function $m(\cdot,\xi): {\mathbb N}\to[0,1]$ by
\be{empiric} \forall x\in {\mathbb N},\qquad m(x,\xi) := \frac1N\sum_{i=1}^N {\mathbf 1}_{\{\xi(i)=x\}}. \end{equation}
The generator of the Fleming-Viot process with $N$ particles applied to bounded functions $f:{\mathbb N}^N\to{\mathbb R}$ reads
\be{generatorfv} \mathcal L^{\text{\hskip-.4mm\tiny\it{N}}} f(\xi)= \sum_{i=1}^N \sum_{y=1}^\infty \Bigl[q(\xi(i),y)+ q(\xi(i),0)\, \textstyle{\frac{N}{N-1}} \,m(y,\xi)\Bigr]\, [f(\xi^{i,y})-f(\xi)], \end{equation}
where $\xi^{i,y}(i)=y$, and for $j\not=i$, $\xi^{i,y}(j) = \xi(j)$ and $q(x,y)$ are the jump rates of the driving process. Assume that the driving process has a unique quasi-stationary distribution, called $\nu_{\rm
qs}$ and that the associated $N$-particle Fleming-Viot system has an invariant measure ${\lambda^{\text{\hskip-.7mm\tiny\it{N}}}}$. The main conjecture in \cite{burdzy1,burdzy2} is that assuming $\xi$ has distribution ${\lambda^{\text{\hskip-.7mm\tiny\it{N}}}}$, the law of the random measure $m(.,\xi)$ converges to the law concentrated on the constant $\nu_{\rm qs}$.
This was proven for diffusion processes on a bounded domain of~${\mathbb R}^d$, killed at the boundary~\cite{bieniek, GK1, GK,villemonais}, for jump processes under a Doeblin condition~\cite{ferrari-maric} and for finite state jump processes~\cite{AFG}.
The subcritical {Galton-Watson}{} process has infinitely many quasi-stationary distributions. Our theorem proves that the stationary empirical distribution $m(\cdot,\xi)$ converges to $\nu_{\rm qs}^*$, the minimal quasi-stationary distribution. This phenomenon is a \emph{selection
principle}.
\begin{theorem}\label{theo-main}
Consider a subcritical Galton-Watson process whose offspring law has some
finite positive exponential moment. Let ${\nu_{\rm qs}^*}$ be the minimal quasi-stationary distribution for the process conditioned on non-extinction. Then, for each $N\ge 1$, the associated $N$-particle Fleming-Viot system is ergodic. Furthermore, if we call its invariant measure ${\lambda^{\text{\hskip-.7mm\tiny\it{N}}}}$, then \be{main-1} \forall x\in {\mathbb N},\qquad
\lim_{N\to\infty}\int |m(x,\xi) - {\nu_{\rm qs}^*}(x)| \, d{\lambda^{\text{\hskip-.7mm\tiny\it{N}}}}(\xi) \;=\;0. \end{equation} \end{theorem} A simple consequence is propagation of chaos. For any finite set $S \subset {\mathbb N}$, \be{limit-chaos} \lim_{N \to \infty} \int\prod_{x \in S} m(x,\xi) \ d{\lambda^{\text{\hskip-.7mm\tiny\it{N}}}}(\xi)= \prod_{x \in S} {\nu_{\rm qs}^*}(x). \end{equation}
The strategy for proving Theorem~\ref{theo-main} is explained in the next section, but there are two key steps in the proof. First, we control the position of the rightmost particle. Let \[ {R}(\xi)\;:=\; \max_{i\in\{1,\dots,N\}} \xi(i), \] be the position of the rightmost particle of $\xi$. Let $\xi^\xi_t$ the positions at time $t$ of the $N$ Fleming-Viot particles, initially on $\xi$.
\bp{prop-rightmost} There is a time $T$ and positive constants $A, c_1, c_2,C$ and $\rho$, independent of $N$, such that for any $\xi\in{\mathbb N}^N$ \be{foster}
E\big(\exp\big(\rho{R}(\xi^\xi_T)\big)\big) -\exp\big(\rho {R}(\xi)\big)\;<\;\; -\,c_1\, e^{\rho{R}(\xi)}{\mathbf 1}_{{R}(\xi)>A} + N c_2 e^{-C{R}(\xi)}. \end{equation} As a consequence, for each $N$ there is a unique invariant measure ${\lambda^{\text{\hskip-.7mm\tiny\it{N}}}}$ for the $N$-particle Fleming-Viot system. Furthermore, there is a constant $\kappa>0$ such that for any $N$, \be{rightmost-expon} \int \exp(\rho {R}(\xi)) d{\lambda^{\text{\hskip-.7mm\tiny\it{N}}}}(\xi) \;\le\; \kappa N. \end{equation} \end{proposition} The second result is that the ratio between the second and the first moment of the empirical distribution plays the role of a Lyapunov functional, given that the position of the rightmost particle is not too large. For a particle configuration $\xi$ define
\be{def-psi2} \psi(\xi):= \frac{\sum_{1\le i\le N}\xi^2(i)} {\sum_{1\le i\le N} \xi(i)}. \end{equation} Recall $\mathcal L^{\text{\hskip-.4mm\tiny\it{N}}}$ is the Fleming-Viot generator given by \eqref{generatorfv}.
\bp{prop-key} There are positive constants $v,C_1$ and $C_2$ independent of $N$ such that
\be{L-psi2} \mathcal L^{\text{\hskip-.4mm\tiny\it{N}}}\psi(\xi)\;\le\; -v\psi(\xi)+C_1 \frac{{R}^2(\xi)}{N}+C_2. \end{equation} \end{proposition} Propositions \ref{prop-rightmost} and \ref{prop-key} imply that the expectation of $\psi$ under the invariant measure ${\lambda^{\text{\hskip-.7mm\tiny\it{N}}}}$ is uniformly bounded in $N$.
\bc{cor-bary} There is a positive constant $C$ such that for all $N$,
\be{ineq-bary} \int \psi(\xi)\, d{\lambda^{\text{\hskip-.7mm\tiny\it{N}}}}(\xi) \;\le\; C. \end{equation}
\end{corollary}
There are several related works motivated by genetics. Brunet, Derrida, Mueller and Munier \cite{BDMM1,BDMM2} introduce a model of evolution of a population with selection. They study the genealogy of genetic traits, the empirical measure, and link the evolution of the barycenter with F-KPP equation $\partial_t u=\partial_x^2 u-u(1-u)$ introduced in 1937 by R.A. Fisher to describe the evolution of an advantageous gene in a population. These authors also discover an exactly soluble model whose genealogy is identical to those predicted by Parisi's theory of mean-field spin glasses. Durrett and Remenik \cite{durret} establish propagation of chaos for a related continuous-space and time model, and then show that the limit of the empirical measure is characterized as the solution of a free-boundary integro-differential equation. B\'erard and Gou\'er\'e \cite{berard-gouere} establish a conjecture of Brunet and Derrida for the speed of the rightmost particle for still a third microscopic model of F-KPP equation introduced in \cite{brunet-derrida1,brunet-derrida2}. Maillard \cite{maillard} obtains the precise behavior of the empirical measure of an approximation of the same model, building on the results of Berestycki, Berestycki and Schweinsberg \cite{berestycki}, which establish the genealogy picture described in {\cite{brunet-derrida1,brunet-derrida2}}.
We now mention two open problems. The first is to solve the analogous to Theorem~\ref{theo-main} for a random walk with a constant drift toward the origin. The second is to obtain propagation of chaos directly on the stationary empirical measure, with a bound of order $1/N$.
In the next section, we describe our model, sketch the proof of our main result and describe the organization of the paper.
\section{Notation and Strategy}\label{section2} Let $\sigma>0$ and $p$ be a probability distribution on ${\mathbb N}\cup\{0\}$ such that \be{a15}
\sum_{\ell\ge 0} p(\ell)\,e^{\sigma\ell}\;<\;\infty. \end{equation}
Consider a Galton-Watson process $Z_t\in{\mathbb N}\cup\{0\}$ with offspring law $p$. Each individual lives an exponential time of parameter 1, and then gives birth to a random number of children with law~$p$.
We assume that {Galton-Watson}{} is subcritical, that is we ask $p$ to satisfy
\be{def-drift} -v\;:=\;\sum_{\ell \ge -1} \ell p(\ell+1) \;<\;0. \end{equation}
In other words, the drift when $Z_t=x$ is $-vx<0$. For distinct $x,y\in{\mathbb N}\cup\{0\}$, the rates of jump are given by
\be{rates} q(x,y) := \left\{\begin{array}{ll}
x p(0), &\text{if } y= x-1\ge 0,\\
x p(y-x+1), &\text{if } y> x\ge1,\\
0,&\text{otherwise}.
\end{array} \right.
\end{equation} The {Galton-Watson}{} process starting at $x$ is denoted $Z^x_t$. For a distribution $\mu$ on ${\mathbb N}$, the law of the process starting with $\mu$ conditioned on non-absorption until time $t$ is given by \be{muTt} \mu T_t(y) := \frac{\sum_{x\in {\mathbb N}} \mu(x) p_t(x,y)}{\sum_{x,z\in {\mathbb N}} \mu(x) p_t(x,z)}, \end{equation} where $p_t(x,y)= P(Z^x_t=y)$.
Recall that $\xi^\xi_t$ denotes the {Fleming-Viot}{} system with generator \eqref{generatorfv} and initial state $\xi$; $\xi_t(i)$ denotes the position of the $i$-th particle at time $t$. For a real $\alpha>0$ define $K(\alpha)$ as the subset of distributions on ${\mathbb N}$ given by \be{def-K} K(\alpha):=\acc{\mu:\ \frac{\sum_{x\in{\mathbb N}} x^2\mu(x)}{ \sum_{x\in{\mathbb N}} x\mu(x)}\le\alpha}. \end{equation} Observe that $\mu \in K(\alpha)$ implies $\sum x\mu(x) \le \alpha$. \begin{proof}[Proof of Theorem \ref{theo-main}] The existence of the unique invariant measure ${\lambda^{\text{\hskip-.7mm\tiny\it{N}}}}$ for {Fleming-Viot}{} is given in Proposition~\ref{prop-rightmost}.
To show \eqref{main-1} we use the invariance of ${\lambda^{\text{\hskip-.7mm\tiny\it{N}}}}$ and perform the following decomposition. \be{strategy-1} \begin{split}
\int &\big| m(x, \xi) - {\nu_{\rm qs}^*}(x)\big| \, d{\lambda^{\text{\hskip-.7mm\tiny\it{N}}}}(\xi)
\;= \;\int E\big| m(x,\xi^\xi_t) - {\nu_{\rm qs}^*}(x)\big| \, d{\lambda^{\text{\hskip-.7mm\tiny\it{N}}}}(\xi)\\ &\le\; {\lambda^{\text{\hskip-.7mm\tiny\it{N}}}}({\psi>\alpha})+
\int_{\psi \le \alpha}\!\! E\big|m(x,\xi^\xi_t) - {\nu_{\rm qs}^*}(x)\big| \, d{\lambda^{\text{\hskip-.7mm\tiny\it{N}}}}(\xi)\\
&\le\; {\lambda^{\text{\hskip-.7mm\tiny\it{N}}}}({\psi>\alpha})+ \int_{\psi\le\alpha} \!\!\! E\big|m(x,\xi^\xi_t) - m(\cdot, \xi)T_t(x)\big|\,d{\lambda^{\text{\hskip-.7mm\tiny\it{N}}}}(\xi) +
\int_{\psi\le\alpha} \!\!\!\big|m(\cdot, \xi)T_t(x) - {\nu_{\rm qs}^*}(x)\big| \, d{\lambda^{\text{\hskip-.7mm\tiny\it{N}}}}(\xi)\\
&\le\; {\lambda^{\text{\hskip-.7mm\tiny\it{N}}}}({\psi>\alpha})+
\sup_{\xi:\psi(\xi)\le\alpha}\!\! \big| m(\cdot,\xi)T_t (x) - {\nu_{\rm qs}^*}(x) \big|\,+\,
\sup_{\xi:\psi(\xi)\le\alpha}\!\! E\big|m(x,\xi^\xi_t)-m(\cdot,\xi)T_t (x)\big|, \end{split} \end{equation} where $\psi$ is defined in \eqref{def-psi2}.
We bound the three terms of the last line of \eqref{strategy-1}.
\noindent\emph{First term. } Corollary \ref{cor-bary} and Markov inequality imply that there is a constant $C>0$ such that for any $\alpha>0$
\be{I3-estimate} {\lambda^{\text{\hskip-.7mm\tiny\it{N}}}}(\psi>\alpha) \le \frac{C}{\alpha}. \end{equation}
\emph{Second term. } Note that $\psi(\xi)<\alpha$ if and only if $m(\cdot,\xi)\in K(\alpha)$. The Yaglom limit converges to the minimal quasi-stationary distribution ${\nu_{\rm qs}^*}$, uniformly in $K(\alpha)$ as we show later in Proposition \ref{prop-yaglom}: \be{unif.Yaglom}
\lim_{t \to \infty} \sup_{\mu\in K(\alpha)} | \mu T_t(x) - {\nu_{\rm qs}^*}(x) |=0. \end{equation}
\noindent\emph{Third term. } We perform the decomposition \be{ad1}
E\big| m(x,\xi^\xi_t)- m(\cdot,\xi)T_t(x)\big|\;\le\; E\big| m(x,\xi^\xi_t) -E m(x,\xi^\xi_t)\big|\,+\,
\big| Em(x,\xi^\xi_t)- m(\cdot,\xi)T_t(x)\big|, \end{equation} and show that there exist positive constants $C_1$ and $C_2$ such that \be{var1}
\sup_{\xi\in{\mathbb N}^N} E\big| m(x,\xi^\xi_t) -E m(x,\xi^\xi_t)\big|\;\le\; \frac{C_1 e^{C_2 t}}{\sqrt N} \end{equation} and \be{var2}
\sup_{\xi\in{\mathbb N}^N} \big| Em(x,\xi^\xi_t)- m(\cdot,\xi)T_t(x)\big|\;\le\; \frac{ C_1 e^{C_2 t}}{N}, \end{equation} for all $N$, see Proposition \ref{prop-semigroups} later. The issue here is a uniform bound for the correlations of the empirical distribution of {Fleming-Viot}{} at sites $x, y \in {\mathbb N}$ at fixed time $t$. This was carried out in \cite{AFG}.
To show \eqref{main-1}, it suffices to bound the three terms in the bottom line of \eqref{strategy-1}. Choose $\alpha$ large and use \eqref{I3-estimate} to make the first term small (uniform in $N$). Use \eqref{unif.Yaglom} to choose $t$ large to make the second term small. For this fixed time, take $N$ large and use \eqref{ad1}, \eqref{var1} and \eqref{var2} to make the third term small. \end{proof}
The rest of the paper is organized as follows. In Section \ref{sec-construction}, we perform the graphical construction of {Fleming-Viot}{} jointly with a Multitype Branching Markov Chain. In Section~\ref{Galton-Watson
estimates} we obtain large deviation estimates for the Galton-Watson process. In Section~\ref{rightmost-particle} we obtain large deviation estimates for the rightmost particle of the {Fleming-Viot}{} system. In Section~\ref{moments-fv} we study the Lyapunov-like functional and prove Proposition~\ref{prop-key} and Corollary~\ref{cor-bary}. Convergence of the conditional evolution uniformly on $K(\alpha)$ is proved in Section~\ref{sec-yaglom}. Finally, \eqref{var1}-\eqref{var2} are handled in Proposition~\ref{prop-semigroups} of Section~\ref{sec-semigroups}.
\section{Embedding {Fleming-Viot}{} on a multitype branching Markov process.}\label{sec-construction}
In this section we construct a coupling between the Fleming-Viot system and an auxiliary multitype branching Markov process (hereafter, the branching process). We call \emph{particles} the {Fleming-Viot}{} positions and \emph{individuals} the branching positions. Each individual has a a type in $\{1,\dots,N\}$ and a position in ${\mathbb N}$.
When a particle chooses the position of another particle and jumps to it, the process builds correlations making difficult to control the position of the rightmost particle. In our coupling when a particle jumps, either an individual jumps at the same time or a branching occurs at the site where the particle arrives. In this way the particles always stay at sites occupied by individuals and the maximum particle position is dominated by the position of the rightmost individual (if this is so at time zero). This, in turn, is dominated by the sum of the individual positions which we control.
The coupling relies on the Harris construction of Markov processes: the state of the process at time $t$ is defined as a function of the initial configuration and a family of independent Poisson processes in the time interval $[0,t]$. The coupling holds when the driving process is a Markov process with rates $\{q(x,y),x,y\in {\mathbb N}\cup\{0\}\}$ with $0$ being the absorbing state and $\bar q:=\sup_x q(x,0)<\infty$.
There are two types of jumps of the Fleming-Viot particle $i$. Those due to the spatial evolution at rate $\tilde q$ and those due to ``jumps to zero and then to the position of particle $j$ chosen uniformly at random'' at rate $q(x,0)/(N-1)$.
\emph{Spatial evolution. } Each individual has a position in ${\mathbb N}$ which evolves independently with transition rates $(\tilde q(x,y), x,y \in {\mathbb N})$ defined by $\tilde q(x,y):=q(x,y){\mathbf 1}_{\{y \ne 0 \}}$ so that there are no jumps to zero. The spatial evolution of new individuals born at branching times are independent and with the same rates $\tilde q$. Under our coupling, each spatial jump performed by the $i$-particle is also performed by some $i$-individual.
\emph{The refeeding and branching. } At rate $\bar q/(N-1)$, each $j$-individual branches into two new individuals, one of type $j$ and one of type $i$; each new born $i$-individual takes the position of the corresponding $j$-individual and then evolves independently with rates $\tilde q$. If the $i$-particle is at $x$, at rate $q(x,0)/(N-1)$ it jumps to the position of the $j$-particle. Under our coupling, each time particle $i$ chooses particle $j$, each $j$-individual branches into an $i$ and a $j$-individual. In this way, the $i$-particle occupies always the site of some $i$-individual.
The branching process has state space \[ {\mathcal B}:= \Bigl\{\zeta\in {\mathbb N}^{\{1,\dots,N\}\times{\mathbb N}}: \sum_{i=1}^N\sum_{x\in{\mathbb N}} \zeta(i,x)<\infty\Bigr\} \] For
$i\in\{1,\dots,N\}$, $x\in {\mathbb N}$, $\zeta_t(i,x)$ indicates the number of
individuals of type $i$ at site $x$ at time $t$.
Let $\delta_{(i,x)}\in{\mathcal B}$ be the delta function on $(i,x)$ defined by
$\delta_{(i,x)}(i,x)=1$ and $\delta_{(i,x)}(j,y)=0$ for $(j,y)\neq(i,x)$. The
rates corresponding to the (independent) spatial evolution of the individuals
at $x$ are \[ b(\zeta,\zeta + \delta_{(i,y)}-\delta_{(i,x)}) = \zeta(i,x)q(x,y),\qquad i\in\{1,\dots,N\},\; x,y\in{\mathbb N}, \] and those corresponding to the branching of all $j$-individuals into an individual of type $j$ and an individual of type $i$ are \[ b\Bigl(\zeta,\zeta+ \sum_{x\in{\mathbb N}} \zeta(j,x)\delta_{(i,x)}\Bigr) = \frac{\bar q}{N-1} ,\qquad i\ne j\in\{1,\dots,N\} \] Note that the new born $i$-individuals get the spatial position of the corresponding $j$-individual.
\noindent{\bf Harris construction of the branching process } Let $({\mathcal{N}}(i,x,y,k), \,i\in\{1,\dots,N\}, x,y\in {\mathbb N}, k\in {\mathbb N})$ be a family of Poisson processes with rates $k\tilde q(x,y)$ such that ${\mathcal{N}}(i,x,y,k)\subset{\mathcal{N}}(i,x,y,k+1)$ for all $k$; we think a Poisson process as a random subset of ${\mathbb R}$. The process ${\mathcal{N}}(i,x,y,k)$ is used to produce a jump of an $i$-individual from $x$ to $y$ when there are $k$ $i$-individuals at site $x$. The families $({\mathcal{N}}(i,x,y,k),k\ge 1)$ are taken independent. Let $({\mathcal{N}}(i,j)$, $i\ne j)$, be a family of independent Poisson processes of rate $\bar q/(N-1)$, these processes are used to branch all $j$-individuals into an $i$-individual and a $j$-individual. The two families are taken independent.
Fix $\zeta_0=\zeta\in\mathcal B$, assume the process is defined until time $s\ge 0$ and proceed by recurrence. \begin{enumerate} \item Define $\tau(\zeta_s,s) := \inf\{t>s\,:\,t\in \cup_{i,x,y}{\mathcal{N}}(i,x,y,\zeta_s(i,x))\cup\cup_{i,j}{\mathcal{N}}(i,j)\}$. \item For $t\in [s,\tau)$ define $\zeta_t=\zeta_s$. \item If $\tau\in {\mathcal{N}}\big(i,x,y,\zeta_s(i,x)\big)$ then set $\zeta_\tau = \zeta_s+ \delta_{(i,y)}-\delta_{(i,x)}$. \item If $\tau\in {\mathcal{N}}(i,j)$ then set $\zeta_\tau= \zeta_s+ \sum_{x\in{\mathbb N}} \zeta_s(j,x)\delta_{(i,x)} $. \end{enumerate}
The process is then defined until time $\tau$. Put $s=\tau$ and iterate to define $\zeta_t$ for all $t\ge 0$. Denote $\zeta^\zeta_t$ the process with initial state $\zeta$. We leave the reader to prove that $\zeta^\zeta_t$ so defined is the branching process, that is, a Markov process with rates $b$ and initial state $\zeta$.
Let $|\zeta|:= \sum_{i,x} \zeta(i,x)$ be the total number of individuals in~$\zeta$. Let \[ R(\zeta) :=\max\Bigl\{x:\sum_i\zeta(x,i)>0\Bigr\}. \]
Let ${\widetilde Z}^z_t$ be the process on ${\mathbb N}$ with rates $\tilde q$ and initial position $z\in{\mathbb N}$.
\bl{growth.mt} $ E|\zeta^\zeta_t |\,=\,|\zeta|\,e^{\bar q t}$. \end{lemma}
\begin{proof} $ E|\zeta_t |$ satisfies the equation \be{a524}
\frac{d}{dt} E|\zeta_t | \;=\; \frac{\bar q}{N-1} E\Bigl(\sum_i\sum_{j:j\neq i}\sum_x \zeta_t(j,x)\Bigr) \; = \;
\frac{\bar q}{N-1}\, (N-1)\, E|\zeta_t| \;=\; \bar q E|\zeta_t|,
\end{equation} with initial condition $ E|\zeta_0 |=|\zeta|$. \end{proof}
\bl{a501} Let $g:{\mathbb N}\to{\mathbb R}^+$ be non decreasing. Then \be{a520}
Eg(R(\zeta^\zeta_t))\;\le\; E|\zeta^\zeta_t|\,Eg({\widetilde Z} ^{R(\zeta)}_t). \end{equation} \end{lemma} \begin{proof} Consider the following partial order on $\mathcal B$:
\be{a521} \zeta\prec \zeta' \quad \text{ if and only if } \quad \sum_{y\ge x}\zeta(i,y)\le \sum_{y\ge x}\zeta'(i,y), \quad \text{for all }i, x. \end{equation}
The branching process is attractive: the Harris construction with initial configurations $\zeta\prec \zeta'$ gives $\zeta^\zeta_t\prec \zeta^{\zeta'}_t$ almost surely; we leave the proof to the reader. Let $\zeta' := \sum_{i,x}\zeta(i,x)\delta_{(i,R(\zeta))}$ be the
configuration having the same number of individuals of type $i$ as $\zeta$ for
all $i$, but all are located at $r:=R(\zeta)$. Hence $\zeta\prec\zeta'$ and
\be{a527} Eg(R(\zeta^\zeta_t))\; \le\; \sum_{i,x} g(x) E\zeta^\zeta_t(i,x)\;\le\; \sum_x g(x)\sum_i E\zeta^{\zeta'}_t(i,x), \end{equation}
because $g$ is non-decreasing. Fix $i$ and $x$ and define \[
b_t(r,x) := \sum_i E\zeta^{\zeta'}_t(i,x),\quad a_t:=E|\zeta^{\zeta}_t|,\quad \tilde p_t(r,x):=P({\widetilde Z}^{r}_t=x). \] Since $b_t(r,x)$ and $a_t\tilde p_t(r,x)$ satisfy the same Kolmogorov backwards equations and have the same initial condition, the right hand side of \eqref{a527} is the same as the right hand side of \eqref{a520}. This can be seen as an application of the one-to-many lemma, see \cite{athreya}. \end{proof}
\noindent{\bf Harris construction of Fleming-Viot} Let ${\mathcal{N}}(i,j,x)\subset {\mathcal{N}}(i,j)$ be the Poisson process obtained by independently including each $\tau \in{\mathcal{N}}(i,j)$ into ${\mathcal{N}}(i,j,x)$ with probability $q(x,0)/ \bar q$ ($\le1$, by definition of $\bar q$). The processes $({\mathcal{N}}(i,j,x), i, j\in\{1,\dots,N\}, x\in{\mathbb N})$ are independent Poisson processes of rate $q(x,0)/(N-1)$.
Fix $\xi_0=\xi\in{\mathbb N}^{\{1,\dots,N\}}$, assume the process is defined until time $s\ge 0$ and proceed iteratively from $s=0$ as follows. \begin{enumerate} \item Define $\tau(\xi_s,s) = \inf\{t>s\,:\,t\in \cup_{i,y}{\mathcal{N}}(i,\xi_s(i),y,1)\cup\cup_{i,j}{\mathcal{N}}(i,j,\xi_s(i))\}$ \item For $t\in [s,\tau)$ define $\xi_t=\xi_s$. \item If $\tau\in {\mathcal{N}}(i,\xi_s(i),y,1)$, then set $\xi_\tau(i) = y$ and for $i'\neq i$ set $\xi_\tau(i') = \xi_s(i')$. \item If $\tau\in {\mathcal{N}}(i,j,\xi_s(i))$, then set $\xi_\tau(i)= \xi_s(j)$ and for $i'\neq i$ set $\xi_\tau(i') = \xi_s(i')$. \end{enumerate} The process is then defined until time $\tau$. Put $s=\tau$ and iterate to define $\xi_t$ for all $t\ge 0$. We leave the reader to prove that $\xi^\xi_t$ is a Markov process with generator $\mathcal L^{\text{\hskip-.4mm\tiny\it{N}}}$ and initial configuration $\xi$ and the following lemma.
\bl{a506} The Fleming-Viot $i$-particle coincides with the position of a branching $i$-individual at time $t$ if this happens at time zero for all $i$. More precisely, \be{a511}
\zeta_0(i,\xi_0(i))\ge 1\text{ for all }i\text{ implies }\zeta_t(i,\xi_t(i))\ge 1\text{ for all }i,\quad \text{a.s.}. \end{equation} \end{lemma} \bc{a508} Assume $\zeta_0(i,\xi_0(i))\ge 1$ for all $i$. Then, \be{a510} R(\xi_t) \le R(\zeta_t),\quad\text{a.s.} \end{equation} \end{corollary}
\section{Galton-Watson estimates}\label{Galton-Watson estimates}
We show now that for $\rho$ small enough the functions $e^{\rho\cdot}$ belong to the domain of the generator of {Galton-Watson}, that is, the Kolmogorov equations hold for these functions. The total number of births of the {Galton-Watson}{} process $Z^x_t$ is a random variable $H^x:=x+\sum_{t>0} (Z^x_t-Z^x_{t-})^+$. Theorem 2 in \cite{NSS} says that \eqref{a15}-\eqref{def-drift} are equivalent to the existence of a $\sigma'>0$ such that
\be{hh} E(\exp(\sigma' H^1)) \;<\;\infty. \end{equation}
Clearly $\sigma'\le \sigma$. Let
\be{boldF} \mathbf F\;:=\; \Big\{f:{\mathbb N}\cup\{0\}\to{\mathbb R}\,:\, \sum_{\ell\ge
0}e^{-\rho\ell}|f(\ell)|\,<\,\infty \mbox{ for some } \rho<\sigma'\}. \end{equation}
Note that if $f\in\mathbf F$, then there exist $\rho<\sigma'$ and $C>0$ such that $|f(\ell)|\,\le\, C e^{\rho\ell},\;\ell\ge 0$. For $f\in\mathbf F$ define the {Galton-Watson}{} semigroup by
\be{semiproup-gw} S_tf(x) := E(f(Z^x_t))\;<\;\infty, \end{equation}
because $Z^x_t \le H^x$ for all $t\ge 0$. The generator $Q$ of {Galton-Watson}{} applied on functions $f$ is given by
\be{gen-branching} Qf(x):= \sum_{\ell=-1}^{\infty} xp(\ell+1) \big( f(x+\ell)-f(x)\big),\qquad x\ge 0, \end{equation}
if the right hand side is well defined.
\bl{gwke} Under the assumption \eqref{a15}, for $f\in \mathbf F$, $Qf(x)$ is well defined and the Kolmogorov equations hold:
\be{kegw} \frac{d}{dt}S_t f = QS_tf = S_tQf. \end{equation}
\end{lemma}
\begin{proof}
Since $|f(x)|\le C\exp(\rho x)$ for all $x \in {\mathbb N}$,
\be{b11} |Qf(x)| \;\le\; C xe^{\rho x} \Bigl(\sum_{\ell\ge -1}
p(\ell+1)\,e^{\rho\ell} + 1)\Bigr). \end{equation}
This shows the first part of the lemma. Consider $f\in\mathbf F$ and define the local martingale (see \cite[Section IV-20, pp. 30-37]{RW} )
\[ M^x_t\;:=\; f(Z^x_t) - f(x)- \int_0^t Q f(Z^x_s) ds. \]
Using \eqref{b11}, for all $s \le t$ \[
|M^1_s|\; \le\; e^\rho+ \exp(\rho H^1) + tC H^1\exp(\rho H^1) \;\le\; \tilde C \exp(\tilde \rho H^1), \]
with $\rho < \tilde \rho < \sigma'$. Hence $ E \sup_{s \in [0,t]} |M^1_s| < \infty $ and $M^1_t$ is a martingale by dominated convergence. Since for $\rho\le \sigma'$, $ E\exp( \rho H^x)=(E \exp( \rho H^1))^x $, the same reasoning shows that $M^x_t$ is a martingale and $$ Ef(Z^x_t)= f(x) + E \int Q f(Z^x_s) ds, $$ which is equivalent to \eqref{kegw} for $f\in\mathbf F$. \end{proof}
The generator of the reflected {Galton-Watson}{} process ${\widetilde Z}_t$ reads
\be{def-reflected} \begin{split} {\widetilde Q} f(x):=&\sum_{\ell=-1}^\infty x p(\ell+1) {\mathbf 1}_{\{x+\ell \ge 1\}} \big(f(x+\ell)-f(x)\big),\qquad x\in {\mathbb N},
\end{split} \end{equation}
if the right hand side is well defined. The reflected process can be thought of as an absorbed process regenerated at position 1 each time it gets extinct. Since the absorbed process can terminate only when it is at state 1 and jumps to 0 at rate $p(0)$, the number of regenerations until time $t$ is dominated by a Poisson random variable ${\mathcal{N}}_t$ of mean $tp(0)$ and \[ E\big(\exp(\rho {\widetilde Z}^1_t)\big)\; \le\; E \exp\Bigl(\rho \sum_{n=1}^{{\mathcal{N}}_t}H^1_n\Bigr), \] where $H^1_n$ are i.i.d random variables with the same distribution as $H^1$ and ${\mathcal{N}}_t$ is independent of $(H^1_n, n\ge 1)$. Hence, \[ E\big(\exp(\rho {\widetilde Z}^1_t)\big) \;\le\; \exp\big( tp(0)\, C(\rho) \big). \] Let ${\widetilde S_t}$ be the semigroup of the reflected {Galton-Watson}{} process. Using the same reasoning as before, we obtain
\bc{tqg} Any $f\in\mathbf F$ satisfies the Kolmogorov equations for ${\widetilde Q}$: \be{kergw} \frac{d}{dt}{\widetilde S_t} f = {\widetilde Q}{\widetilde S_t} f = {\widetilde S_t}{\widetilde Q} f. \end{equation} \end{corollary}
\paragraph{Large deviations}\label{sec-ld} We study ${\widetilde Z} _t $, the reflected {Galton-Watson}{} process with generator ${\widetilde Q}$ given by \eqref{def-reflected}. Since $p$ satisfies \eqref{a15}, for $\rho<\sigma'\le \sigma$,
\be{ld-4} \Gamma(\rho)\;:=\; p(0)+\sum_{\ell=1}^\infty p(\ell+1) \ell^2 e^{\rho \ell}\;<\;\infty. \end{equation}
Recall that $v$ is defined in \eqref{def-drift} and define $\beta$ as
\be{a11} \beta=\sup\{ \rho > 0 \colon \rho \Gamma(\rho)\le v \}, \end{equation} which is well defined thanks to the exponential moment of $p$.
\bl{lem-expon} For any $\rho< \min\{\beta,\sigma'\}$, and $x\in{\mathbb N}$,
\be{ineq-expon} E\exp(\rho {\widetilde Z}^x_t)\;\le\; e^{-\frac{\rho v}{2} t} e^{\rho x} + te^\rho. \end{equation}
\end{lemma} \begin{proof} Since $\rho<\sigma'\le \sigma$, the reflected {Galton-Watson}{} generator \eqref{def-reflected} applied to $e^{\rho\cdot}$ is well defined and gives
\[ \begin{split} {\widetilde Q}(e^{\rho \cdot})(x)\;&=\;\sum_{\ell=-1}^\infty x p(\ell+1) e^{\rho x} \big(e^{\rho \ell}-1\big)-p(0){\mathbf 1}_{\{x=1\}}\big(1-e^\rho\big)\\ \;&=\; x e^{\rho x} \Big(-\rho v+\sum_{\ell=-1}^\infty p(\ell+1)\big( e^{\rho \ell}-1-\rho \ell\big)\Big)+p(0){\mathbf 1}_{\{x=1\}}\big(e^\rho-1\big). \end{split} \]
Using that for $a\ge 0$, $e^a-(1+a)\le \frac{a^2}{2}e^a$,
\be{expon-2} \begin{split}{\widetilde Q}(e^{\rho \cdot})(x)&\;\le\; \rho x e^{\rho x}\Big(-v+ \frac{\rho}{2}\Gamma(\rho)\Big)+p(0){\mathbf 1}_{\{x=1\}} e^{\rho}\\ &\;\le\;- \frac{v\rho}{2}e^{\rho x}\,+\,e^{\rho} , \end{split} \end{equation}
using $\rho< \beta$ and $\beta\Gamma(\beta)\le v$. Since $\rho<\sigma'$, Corollary \ref{tqg} and Gronwall's inequality give \reff{ineq-expon}.\end{proof}
We obtain now a Large Deviation estimate. \bp{prop-LD} Let $T\ge {1 \over 16 v}$ and $\delta \ge \max\{1, 4Tp(0)\}$. Then, there is a constant $\kappa$, independent of $x$, such that
\be{ld-reg} P\Big( \sup_{s<T}\big( {\widetilde Z}^x_s- e^{-v s}x \big)\ge \delta \Big)\;\le\; \exp\Big(-\frac{\kappa}{T}
\frac{\delta^2}{\max\{x,\delta\}}\Big). \end{equation}
\end{proposition} \begin{proof}
Set $z_t^x=e^{-v t}x$ and introduce the process
\be{ld-6} \begin{split} \epsilon_t^x\;:=&\;{\widetilde Z}^x_t-x+v \int_0^t{\widetilde Z}^x_sds\\ =&\;\big({\widetilde Z} _t^x-z_t^x\big)+v \int_0^t\big({\widetilde Z} ^x_s-z^x_s\big)ds. \end{split} \end{equation}
To stop ${\widetilde Z}_t^x$ when it crosses $2\max\{x,\delta\}$ define
\be{ld-7} \tau:=\inf\acc{ t\ge 0:\ {\widetilde Z}_t^x\ge 2\max\{x,\delta\}}. \end{equation}
Note that if $\tau<\infty$, then ${\widetilde Z}_\tau^x-z_\tau^x\ge 2\max\{x,\delta\}-x\ge \delta$. Thus,
\be{stop-1} \acc{{\widetilde Z} _t^x-z_t^x\ge \delta}\subset \acc{{\widetilde Z}_{t\wedge \tau}^x-z_{t\wedge \tau}^x\ge \delta}. \end{equation}
For functions $g_1,\,g_2:{\mathbb R}\to{\mathbb R}$ verifying \[ g_1(t) =g_2(t) +v\int_0^t g_2(s) ds,\qquad v\ge0, \] it holds \[
\sup_{t\le T} |g_1(t) |\le \frac{\delta}{2}\;\Longrightarrow\;
\sup_{t\le T} |g_2(t) |\le \delta. \] Hence,
\be{ld-8}
\acc{\sup_{t\le T} \big|{\widetilde Z}_{t\wedge \tau}^x-z_{t\wedge \tau}^x\big|
\ge \delta}\;\subset\; \acc{\sup_{t\le T} |\epsilon_{t\wedge \tau}^x|\ge \frac{\delta}{2}}. \end{equation}
Note that \[
\acc{\sup_{t\le T} |\epsilon_{t\wedge \tau}^x|\ge \frac{\delta}{2}}\;=\; \acc{\sup_{t\le T} \epsilon_{t\wedge \tau}^x\ge \frac{\delta}{2}} \,\cup\,\acc{\inf_{t\le T} \epsilon_{t\wedge \tau}^x\le -\frac{\delta}{2}}. \] The treatment of the two terms on the right hand side of the previous formula is similar, and we only give the simple argument for the first of them. For $\rho<\sigma'$, the following functional is a local martingale (see \cite[page 66]{EK}).
\be{ld-9} {\mathcal{M}}_t :=\exp\Big({\rho{\widetilde Z} _t^x}-{\rho x}-\int_0^t \big(e^{-\rho \cdot} {\widetilde Q}(e^{\rho \cdot })\big)({\widetilde Z} ^x_s)ds\Big). \end{equation} Using the bounds of Lemma \ref{gwke} we obtain that ${\mathcal{M}}_t$ is in fact a martingale. Observe that \be{ld-11} \begin{split} e^{-\rho x} {\widetilde Q}(e^{\rho\cdot})(x) &\;=\; x \sum_{\ell=-1}^\infty p(\ell+1)\big(e^{\rho \ell}-1\big)+ p(0){\mathbf 1}_{\{x=1\}} \big( e^{\rho}-1\big)\\ &\;\le\; -\rho v x+\rho\, p(0)+\frac{\rho^2 }{2} \big( x\Delta(\rho)+p(0)e^{\rho}\big), \end{split} \end{equation}
with,
\be{ld-12} \Delta(\rho):=\frac{2}{\rho^2} \sum_{\ell=-1}^\infty p(\ell+1)\big(e^{\rho \ell}-1-\rho \ell\big)\ge 0. \end{equation}
We have already seen that $\Delta(\rho)\le \Gamma(\rho)$. Then, we bound the martingale ${\mathcal{M}}_t$ as follows.
\be{ld-10} \begin{split} {\mathcal{M}}_t &\;\ge\; \exp\Big(\rho\big({\widetilde Z} _t^x-x\big)- \big(- \rho v+\frac{\rho^2}{2}\Delta(\rho)\big) \int_0^t {\widetilde Z} ^x_sds-\rho p(0)t-\frac{\rho^2}{2}t e^{\rho}\Big)\\ &\;\ge\;\exp\Big(\rho\epsilon_t^x-\rho p(0)t -\frac{\rho^2}{2}\Gamma(\rho) \int_0^t {\widetilde Z} ^x_sds-\frac{\rho^2}{2}t e^{\rho}\Big). \end{split} \end{equation}
By stopping the process at $\tau$, and using that $\delta\ge 1$, we obtain for $t\le T$
\be{ld-14} \exp\big(\rho\epsilon_{t\wedge \tau}^x\big)\le {\mathcal{M}}_{t\wedge \tau} \exp\big(\rho p(0)T+ \rho^2 \max\{x,\delta\}T\Gamma(\rho)\big). \end{equation}
Using \reff{stop-1}, \reff{ld-8} and \reff{ld-14}, and the bound $p(0)T\le \delta/4$, we obtain for any $\rho>0$
\be{ld-15} \begin{split} P\big(\sup_{s\le T} \big({\widetilde Z} ^x_s -z^x_s\big)\ge \delta\big)\;&\le\; P\Big( \sup_{s\le T} {\mathcal{M}}_{s\wedge \tau}\ge \exp\big( \frac{\rho\delta}{4} -\rho^2 \max\{x,\delta\}T \Gamma(\rho)\big)\Big)\\ &\le\;\exp\Bigl(-\frac{\rho\delta}{8}+ \rho^2 \max\{x,\delta\}T \sup_{\rho<\beta} \Gamma(\rho)\Bigr), \end{split} \end{equation}
by Doob's martingale inequality and for $\rho<\beta$. Optimize over $0<\rho< \beta$ (recalling that $16T\beta \Gamma(\beta)>1$), and choose $\rho^*$
\be{ld-17} \rho^*:=\frac{1}{16T} \frac{\delta}{\max\{x,\delta\}}\frac{1}{\Gamma(\beta)}<\beta. \end{equation}
The result follows now from \reff{ld-15} and \reff{ld-17}. \end{proof}
\section{Bounds for the rightmost {Fleming-Viot}-particle} \label{rightmost-particle} In this section, we bound small exponential moments of the rightmost {Fleming-Viot}-particle. We first define a threshold $A$, such that with very small probability, the rightmost particle's position does not decrease when it is initially larger than $A$. Define \[ \gamma:=\frac{1}{2}\Bigl(1-\exp\Bigl(-\frac{v}{4p(0)}\Bigr)\Bigr) \in (0,1). \] Choose \[ \rho_0 := \frac{\min\{\beta,\sigma',\gamma \kappa p(0)\}}{4} \] where $\kappa$ is the constant given by Proposition \ref{prop-LD}. Define
\be{def-amin1}
A:=\frac{2\kappa p(0)}{\rho_0}>1. \end{equation}
Define the time and the error $\delta$ entering in the large deviation estimate of Proposition~\ref{prop-LD} as follows. For an arbitrary initial condition $\xi$,
\be{def-amin2} T:=\frac{1}{4p(0)},\quad\text{and}\quad \delta:=\max\Bigl\{1, \frac{{R}(\xi)}{A}\Bigr\}, \end{equation}
recall here that ${R}(\xi)=\max_{i\le N} \xi(i)$, and set $V_L(\xi)=\exp(\rho\min( {R}(\xi),L))$ for $L>A$ which will be taken to infinity later. We use the notation $[F(\xi_t)]_0^T:=F(\xi_T)-F(\xi_0)$. \begin{proof}[Proof of Proposition~\ref{prop-rightmost}]
We use the construction in Section \ref{sec-construction} to couple the {Fleming-Viot}{}
process $\xi^\xi_t$ and the branching process $\zeta^\zeta_t$ with
$\zeta=\sum_i\delta_{(i,\xi(i))}$, so that $\zeta(i,\xi(i))=1$ for all
$i$. Then, by \eqref{a510} ${R}(\xi_t )\le {R}(\zeta_t )$ and it is
sufficient to prove an inequality like \eqref{foster} for
${R}(\zeta_t)$. Notice that for the initial configurations $\xi$ and
$\zeta$, $R(\xi)=R(\zeta)$. We drop the superscripts
$\xi$ and $\zeta$ in the remainder of this proof.
Define the event
\be{a80} \mathcal G=\mathcal G(\xi,T):=\acc{{R}(\zeta_T)-e^{-v T} {R}(\xi)\le\ \delta}, \end{equation} and for a positive real $c$, we define the set \[ K_c:=\{\xi \colon R(\xi)\le c\}. \]
On $K_A^c$, $\delta={R}/A<{R}$, and on $K_A^c\cap \mathcal G$,
\be{lyap-3} {R}(\zeta_T)\;\le\; \Bigl(\frac{1}{A}+e^{-v T}\Bigr) {R}(\xi)\; \le\; (1-\gamma) {R}(\xi). \end{equation}
Hence, \be{lyap-4} {\mathbf 1}_{K_A^c\cap \mathcal G} \cro{ V_L(\zeta_t)}_0^T \;\le\; V_L(\xi)\pare{e^{-\gamma\rho {R}(\xi)}-1} {\mathbf 1}_{K_A^c\cap K_L\cap \mathcal G} \;\le\; -V_L(\xi)\pare{1-e^{-\gamma\rho A}}{\mathbf 1}_{K_A^c \cap K_L\cap \mathcal G} . \end{equation} Since $A>1$, on $K_A\cap \mathcal G$, ${R}(\zeta_T)\le Ae^{-vT}+1\le 2A$ so that \[ {\mathbf 1}_{K_A\cap \mathcal G} \cro{ e^{\rho {R}(\zeta_t)}}_0^T\le e^{2\rho A}{\mathbf 1}_{K_A\cap \mathcal G} . \] Thus
\be{lyap-6} \begin{split} \cro{ V_L(\zeta_t)}_0^T&\;\le\; -\big(1-e^{-\gamma \rho A}\big) e^{\rho {R}(\xi)} {\mathbf 1}_{K_A^c\cap K_L\cap \mathcal G}\,+\, e^{2\rho A} {\mathbf 1}_{K_A\cap \mathcal G} \,+\, \cro{ e^{\rho {R}(\zeta)}}_0^T{\mathbf 1}_{\mathcal G^c}\\ &\;\le\; - \big(1-e^{-\gamma \rho A}\big) V_L(\xi) {\mathbf 1}_{K_A^c\cap K_L}\,+\, e^{2\rho A} {\mathbf 1}_{K_A}\,+\,2e^{\rho {R}(\zeta_T)} {\mathbf 1}_{\mathcal G^c}, \end{split} \end{equation} where we used that \[ {\mathbf 1}_{K_A^c\cap K_L} - {\mathbf 1}_{K_A^c\cap K_L\cap \mathcal G}\le {\mathbf 1}_{\mathcal G^c}. \] Choose $\rho:=\min(\rho_0,\frac{\kappa}{4TA^2})$ and observe that by Lemma \ref{a501}, \[
E\cro{e^{2\rho {R}(\zeta_T)}} \le E|\zeta_T|\, E\cro{\exp\Bigl(2\rho {\widetilde Z}_T^{{R}(\xi)}\Bigr)} \le Ne^{p(0)T} \left ( e^{-{2\rho v} T} e^{2\rho R(\xi)} + Te^{2\rho} \right). \] by Lemma \ref{growth.mt} for the bound of the first factor and Lemma \ref{lem-expon} for the bound of the second factor. Also, Lemma \ref{prop-LD} implies \[
P(\mathcal G^c) \le E|\zeta_T| P\Big( \sup_{s<T}\big( {\widetilde Z}^{R(\xi)}_s- e^{-v s}R(\xi) \big)> \delta \Big)\;\le\; Ne^{p(0)T}(e^{-\frac{\kappa}{TA}}{\mathbf 1}_{R(\xi)\le A} + e^{-\frac{\kappa R(\xi)}{TA^2}}{\mathbf 1}_{R(\xi)>A}). \] Taking expectation on \reff{lyap-6} we bound the last term as follows. For constants $C_1,C_2,\tilde C_1,$ and $\tilde C_2$
\be{lyap-5} \begin{split} E\cro{e^{\rho {R}(\zeta_T)}{\mathbf 1}_{\{\mathcal G^c\}}}\; &\le \; \pare{ P(\mathcal G^c) E\cro{e^{2\rho {R}(\zeta_T)}}}^{1/2}\\
&\le \; Ne^{p(0)T} \pare{C_1{\mathbf 1}_{K_A} + C_2 \exp{\left(-\frac{\kappa R(\xi)}{2TA^2}\right)}{\mathbf 1}_{K_A^c} }^{1/2}\;\\
&\le\; \tilde C_1 N {\mathbf 1}_{K_A}+ \tilde C_2 N \exp\Bigl(-\frac{\kappa {R}(\xi)}{4T A^2}\Bigr) {\mathbf 1}_{K_A^c}. \end{split} \end{equation}
Gathering \reff{lyap-5} and \reff{lyap-6} we obtain, for any $L>A$, \be{foster.copy} \begin{split}E V_L(\xi^\xi_T) -V_L(\xi)\; & <\;\; -\,c_1\, V_L(\xi){\mathbf 1}_{L>R(\xi)>A} + C_1 N {\mathbf 1}_{R(\xi)\le A} + C_2N e^{-\rho \tilde c_2R(\xi)}\\ & \le \;\; -\,c_1\, V_L(\xi){\mathbf 1}_{L>R(\xi)>A} + C N e^{- c_2 R(\xi)}\end{split} \end{equation} which completes the first part of the proof, inequality \eqref{foster}, as one takes $L$ to infinity in \reff{foster.copy}.
For the second part, take $C>0$ and observe that the set of $\xi$ such that the right hand
side of \eqref{foster.copy} is larger than $-C$ is finite. Foster's criteria, \cite[Theorems 8.6 and 8.13]{Robert(2003)} implies that both the chain
($\xi_0,\xi_T,\xi_{2T},\cdots)$ and the process $\xi_t $ are ergodic with the same invariant measure that we call ${\lambda^{\text{\hskip-.7mm\tiny\it{N}}}}$.
Now, consider again \reff{foster.copy} for a fixed $L$. Note that $V_L$ is bounded, so that by integrating \reff{foster.copy} with this invariant measure, and then taking $L$ to infinity, we obtain \eqref{rightmost-expon}. \end{proof}
\section{The empirical moments of Fleming-Viot} \label{moments-fv}
In this section we prove Corollary~\ref{cor-bary}. Introduce the occupation numbers $\eta:{\mathbb N}\times{\mathbb N}^N\to{\mathbb N}$ defined as \[ \eta(x, \xi) := \sum_{i=1}^N {\mathbf 1}_{\xi(i)=x}, \] for which we often drop the coordinate $\xi$. Notice that $m(x,\xi) = \eta(x,\xi)/N$.
For any integer $k$, define the $k$-th moment of the $N$ particles' positions as \[ M_k(\xi) := \sum_{i=1}^N\xi^k(i)=\sum_{x=1}^\infty x^k \eta(x,\xi). \] As there are only $N$ particles, $M_k$ is well defined. Instead of working with the barycenter $M_1/N$, we consider $\psi:= M_2/M_1$. Note the inequalities \be{psi-2} 1\le \frac{M_1(\xi)}{N}\le \psi(\xi)\le R(\xi). \end{equation} The function $\psi$ is not compactly supported (nor bounded). Even though $\mathcal L^{\text{\hskip-.4mm\tiny\it{N}}}\psi$ is well defined, we need to use later that $\int \mathcal L^{\text{\hskip-.4mm\tiny\it{N}}}\psi d\lambda^N=0$. We do so by approximating $\psi$ by a compactly supported function ${\psi^{\text{\hskip-.3mm\tiny\it{L}}}}$ for which we have \be{no-proof} \int \mathcal L^{\text{\hskip-.4mm\tiny\it{N}}}{\psi^{\text{\hskip-.3mm\tiny\it{L}}}} d\lambda^N=0,\quad\text{and}\quad \lim_{L\to\infty} \mathcal L^{\text{\hskip-.4mm\tiny\it{N}}}{\psi^{\text{\hskip-.3mm\tiny\it{L}}}} =\mathcal L^{\text{\hskip-.4mm\tiny\it{N}}}\psi \quad\text{pointwise}. \end{equation} We approximate the unbounded test function $\psi$ by the following one \be{testf-bounded} {\psi^{\text{\hskip-.3mm\tiny\it{L}}}}(\xi)=\frac{M^L_{2}(\xi)}{M_1^L(\xi)},\quad \text{with}\quad M_k^L(\xi)=\sum_{i=1}^N \min(\xi^k(i),L^k)=\sum_{x=1}^L x^k \eta(x,\xi)+L^k\sum_{x>L} \eta(x,\xi). \end{equation} As $N$ is fixed, $M_k^L=L^k N-\sum_{x=1}^L (L^k-x^k)\eta(x)$, and has compact support. It is easy, and we omit the proof, to see that there exist a positive constant $C$ such that \be{no-proof2}
\big| \mathcal L^{\text{\hskip-.4mm\tiny\it{N}}}\psi-\mathcal L^{\text{\hskip-.4mm\tiny\it{N}}}{\psi^{\text{\hskip-.3mm\tiny\it{L}}}}\big|\le
\big| \mathcal L^{\text{\hskip-.4mm\tiny\it{N}}}\psi\big|+ \big|\mathcal L^{\text{\hskip-.4mm\tiny\it{N}}}{\psi^{\text{\hskip-.3mm\tiny\it{L}}}}\big|\le C \psi\le C\,R, \end{equation} where we recall that $R(\xi) = \max_i\xi(i)$. We have established in Proposition~\ref{prop-rightmost} that $R(\xi)$ is integrable with respect to $\lambda^N$, so that \reff{no-proof} implies that \be{no-proof3} \int \mathcal L^{\text{\hskip-.4mm\tiny\it{N}}}\psi\, d{\lambda^{\text{\hskip-.7mm\tiny\it{N}}}}=0. \end{equation} The main result of this section is the following. \bl{lem-psi} There are positive constants $C_1,C_2$ such that for any integer $N$ large enough, \be{ineq-psi} \int \psi\, d{\lambda^{\text{\hskip-.7mm\tiny\it{N}}}}\;\le\; C_1+\frac{C_2}{N}\int R^2 d{\lambda^{\text{\hskip-.7mm\tiny\it{N}}}}. \end{equation} \end{lemma} \begin{proof}[{\bf Proof of Lemma~\ref{lem-psi}}] We
decompose the generator \eqref{generatorfv} into two generators, one governing the refeed part and
the other the spatial evolution of the particles: $\mathcal L^{\text{\hskip-.4mm\tiny\it{N}}}=\mathcal L^{\text{\hskip-.4mm\tiny\it{N}}}_{\rm
drift}+\mathcal L^{\text{\hskip-.4mm\tiny\it{N}}}_{\rm refeed}$, which applied to functions depending on
$\xi$ only through $\eta(\cdot,\xi)$, read \be{L-jump} \mathcal L^{\text{\hskip-.4mm\tiny\it{N}}}_{\rm refeed}=p(0)\eta(1)\sum_{x=1}^\infty \frac{\eta(x)}{N-1} \pare{A_1^-A_{x}^+-{\mathbf 1}}, \quad\text{with}\quad A_x^\pm(\eta)(y) = \begin{cases}
\eta(y) & y\ne x,\\
\eta(x) \pm 1 & y=x,
\end{cases} \end{equation} \be{L-drift} \mathcal L^{\text{\hskip-.4mm\tiny\it{N}}}_{\rm drift}=\sum_{x=2}^\infty x \eta(x)p(0)(A_x^-A_{x-1}^+-{\mathbf 1})+ \sum_{x=1}^\infty x \eta(x,\xi) \sum_{i=1}^\infty p(i+1)(A_x^-A_{x+i}^+-{\mathbf 1}). \end{equation} It is convenient to introduce a boundary term \be{boundary-bulk} B=-\eta(1)p(0)(A_1^-A_{0}^+-{\mathbf 1}) \quad\text{and call}\quad \mathcal L^{\text{\hskip-.4mm\tiny\it{N}}}_0=\mathcal L^{\text{\hskip-.4mm\tiny\it{N}}}_{\rm drift}-B, \end{equation}
which applied on $\psi$ yield \be{Lboundary} B\psi\;= \;-p(0)\eta(1)\pare{\frac{M_2-M_1}{M_1(M_1-1)}}; \end{equation} \be{bulk-1} \begin{split} \mathcal L^{\text{\hskip-.4mm\tiny\it{N}}}_0 \psi &\; =\;
\sum_{x=1}^\infty x\eta(x) \sum_{i=-1}^\infty p(i+1)\pare{\frac{M_2+2ix+i^2}{M_1+i}- \frac{M_2}{M_1}}\\ &\;=\; \sum_{x=1}^\infty x\eta(x)\acc{ \sum_{i=-1}^\infty ip(i+1) \pare{\frac{2x M_1-M_2+iM_1}{M_1(M_1+i)}}}\\ &\;=\; -p(0)\frac{M_2-M_1}{M_1-1}+\Big(\sum_{i=1}^\infty p(i+1)i \frac{M_1}{M_1+i}\Big)\times \frac{M_2}{M_1}+\sum_{i=1}^\infty p(i+1)i^2 \frac{M_1}{M_1+i}\\ &\;\le\; -v\psi+p(0)\frac{M_1}{M_1-1}+\sum_{i=1}^\infty p(i+1) i^2 \;\le\; -v\psi+C_0, \end{split} \end{equation} for some positive constant $C_0$. Finally, for the jump term \be{jump2-1} \begin{split} \mathcal L^{\text{\hskip-.4mm\tiny\it{N}}}_{\rm refeed}\psi&\;=\; p(0)\eta(1)\sum_{x=1}^{\infty} \frac{\eta(x)}{N-1} \pare{\frac{M_2+x^2-1}{M_1+x-1}-\frac{M_2}{M_1}}\\ &\;=\; p(0)\eta(1)\sum_{x=1}^{\infty} \frac{\eta(x)}{N-1} \frac{M_1(x^2-1)-M_2(x-1)}{M_1(M_1-1)}\times \frac{1}{1+\frac{x}{M_1-1}}. \end{split} \end{equation} If we set $\Delta(x)=1/(1+x)-(1-x)$, for $x\in [0,1]$, then \be{basic-ineq} \Delta(x)=\frac{x^2}{1+x},\quad\text{and}\quad 0\le \Delta(x)\le x^2. \end{equation} We apply \reff{basic-ineq} to expand the last term in \reff{jump2-1}, with $x/(M_1-1)\le 1$ for $x\le R(\xi)$, and obtain \be{jump2-2} \mathcal L^{\text{\hskip-.4mm\tiny\it{N}}}_{\rm refeed}\psi= p(0)\eta(1)\sum_{x=1}^{\infty} \frac{\eta(x)}{N-1} \frac{M_1(x^2-1)-M_2(x-1)}{M_1(M_1-1)}\times \pare{1-\frac{x}{M_1-1}+\Delta(\frac{x}{M_1-1})}. \end{equation} Note that \[ \sum_{x=1}^{\infty}\eta(x)\big(M_1x^2-M_2x\big)=0,\quad\text{and} \quad \sum_{x=1}^{\infty}\eta(x)\big(M_1x^2-M_2x\big)(-x)= -M_3M_1+(M_2)^2. \] Also, \[ \begin{split} \sum_{x=1}^{\infty}\frac{\eta(x)}{N-1}\big(M_2-M_1\big) \pare{1-\frac{x}{M_1-1}}=&\pare{N-\frac{M_1}{M_1-1}} \frac{\big(M_2-M_1\big)}{N-1}\\ =&\pare{1-\frac{1}{(N-1)(M_1-1)}}\big(M_2-M_1\big)\\ =&\big(M_2-M_1\big)-\frac{M_2-M_1}{(N-1)(M_1-1)}. \end{split} \] Thus \[ \mathcal L^{\text{\hskip-.4mm\tiny\it{N}}}_{\rm refeed}(\psi) =-p(0)\frac{\eta(1)}{N-1} \frac{M_3M_1-(M_2)^2}{M_1(M_1-1)^2}+ p(0)\eta(1)\frac{M_2-M_1}{M_1(M_1-1)}+{\rm Rest}, \] where \be{equa-R} {\rm Rest}\;=\;-\frac{p(0)\eta(1)(M_2-M_1)}{(N-1)(M_1-1)}+ p(0)\eta(1)\sum_{x=1}^{\infty} \frac{\eta(x)}{N-1}\frac{M_1(x^2-1)-M_2(x-1)}{M_1(M_1-1)}\times \Delta(\frac{x}{M_1-1}). \end{equation} Using that $M_2-M_1\ge 0$, \be{jump2-3} {\rm Rest}\;\le\; p(0)\frac{\eta(1)}{N-1}\sum_{x= 1}^{\infty} \eta(x)\times\frac{x^2}{(M_1-1)^2} \Bigl(\frac{M_1x^2+M_2x}{(M_1-1)^2}\ +\frac{M_2-M_1} {(M_1-1)^2}\Bigr) \end{equation} \be{jump2-4}
\;\le\; 3p(0)\Bigl(\frac{M_1}{M_1-1}\Bigr)^{\!\!4} \frac{\eta(1)}{N-1}\frac{M_2}{(M_1)^2} \frac{R^2}{M_1} \;\le\; 24 p(0) \frac{R^2}{N}. \end{equation} Thus, we reach that for $C_0$ independent of $N$ and $L$, \be{psi2-1} \mathcal L^{\text{\hskip-.4mm\tiny\it{N}}}(\psi)\;\le\; -v\,\psi\,+\,24\, p(0)\, \frac{R^2}{N}+C_0. \end{equation} We now integrate \reff{psi2-1} with respect to the invariant measure, and use that $\int \mathcal L^{\text{\hskip-.4mm\tiny\it{N}}} \psi d{\lambda^{\text{\hskip-.7mm\tiny\it{N}}}}=0$ to obtain for constants $C_1$, and $C_2$ (independent of $N$) \be{main-psi2} \int \psi d{\lambda^{\text{\hskip-.7mm\tiny\it{N}}}} \le C_1+C_2\frac{\int R^2\,d{\lambda^{\text{\hskip-.7mm\tiny\it{N}}}}}{N}. \end{equation} \end{proof}
\section{Uniform convergence to the Yaglom limit}\label{sec-yaglom}
Define the generating function of a distribution $\mu$ on ${\mathbb N}$ by
\be{def-gen} G(\mu;z):=\sum_{x\in{\mathbb N}} \mu(x)z^x,\qquad z\in{\mathbb R},\; |z|<1. \end{equation}
In this section we show a uniform convergence for $\mu\in K(\alpha)$ of the generating functions of $\mu T_t$ to the generating function of the {\sc{qsd}}{} $\nu$. We invoke a key result of Yaglom~\cite{yaglom}. The continuous time version can be found in Zolotarev \cite{zolotarev}.
\bl{lem-yaglom}[Yaglom 1947, Zolotarev 1957]. There is a probability measure $\nu$ such that
\be{lim-yaglom} \lim_{t\to\infty} G(\delta_1T_t;z)=G(\nu;z), \end{equation}
and the generating function of $\nu$ is given by
\be{gen-minimal} G(\nu;z)=1- \exp\Big(-v \int_0^z \frac{du}{\sum_{\ell\ge 0}p(\ell) u^\ell-z}\Big),\qquad z\in[0,1). \end{equation}
\end{lemma} The measure $\nu$ is in fact ${\nu_{\rm qs}^*}$, the minimal {\sc{qsd}}{}. We do not use the explicit expression \eqref{gen-minimal} of the generating function of $\nu$; we only use \eqref{lim-yaglom}. Recall that $\mu T_t$ is the law of $Z_t$ with initial distribution $\mu$ conditioned on survival until $t$ and that $K(\alpha)$ is defined in \reff{def-K}. The next result says that the Yaglom limit holds uniformly for all initial measures in $K(\alpha)$.
\bp{prop-yaglom} For any $\alpha >0$
\be{yaglom-main} \lim_{N\to\infty}\sup_{ \mu\in K(\alpha)} \big| G(\mu T_t;z)-
G({\nu_{\rm qs}^*};z)\big|\;=\;0. \end{equation} As a consequence, for each $x \in {\mathbb N}$,
\be{unif.Yaglom1}
\lim_{t \to \infty} \sup_{\mu\in K(\alpha)} | \mu T_t(x) - {\nu_{\rm qs}^*}(x) |=0. \end{equation}
\end{proposition}
\begin{proof}[Proof of Proposition \ref{prop-yaglom}]
Recall that $S_t$ is the semigroup of the Galton-Watson process and observe that for any $\ell\in{\mathbb N}$,
$G(\delta_\ell S_t;z)=G^\ell(\delta_1S_t;z)$. We set, for simplicity, \[ g(z)\;:=\;1-G(\delta_1S_t;z)\;\in\;[0,1], \] for $z\in[0,1]$. The following inequalities are useful. For $z\in[0,1]$,
\be{ineq-G} 1-\ell g(z)\;\le\;(1-g(z))^\ell\;\le\; 1-\ell g(z)+\ell^2g^2(z). \end{equation}
The generating function of $\mu T_t$ reads (the sums run on $\ell\in{\mathbb N}$)
\be{gen-T} \begin{split} G(\mu T_t;z)&\;= \; \frac{G(\mu S_t;z) - G(\mu S_t;0)}{1 - G(\mu S_t;0)}\\ &\;= \; \frac{\sum_{\ell} \mu(\ell ) \big( G(\delta_\ell S_t ;z) -G(\delta_\ell S_t;0) \big) }{\sum_{\ell} \mu(\ell ) \big(1-G(\delta_\ell S_t;0)\big)}\\ &\;= \;\frac{\sum_{\ell} \mu(\ell ) \Big( (1-g(z))^\ell -(1-g(0))^\ell \Big)} {\sum_{\ell} \mu(\ell )\big(1-(1-g(0))^\ell \big)}. \end{split} \end{equation}
Also, \[ 1-G(\delta_1T_t;z)\;=\;\frac{1-G(\delta_1S_t;z)}{1-G(\delta_1S_t;0)} \;=\;\frac{g(z)}{g(0)}. \] We now produce upper and lower bounds for $G(\mu T_t;z)-G({\nu_{\rm qs}^*};z)$. We start with the upper bound. Using first \eqref{gen-T} and then \eqref {ineq-G},
\be{main-matt} \begin{split}
G(\mu T_t;z)-G({\nu_{\rm qs}^*};z)
&\;= \; \frac{\sum_{\ell} \mu(\ell ) \Big((1-g(z))^\ell -1+
\big(1-(1-g(0))^\ell \big)(1-G({\nu_{\rm qs}^*};z))\Big)}
{\sum_{\ell} \mu(\ell ) \big(1-(1-g(0))^\ell \big)}\\
&\;\le \; \frac{ \sum_{\ell } \ell \mu(\ell )\Big(-g(z)+\ell
g^2(z)+g(0)(1-G({\nu_{\rm qs}^*};z))
\Big)}{ \sum_{\ell } \ell \mu(\ell )\Big( g(0)-\ell g^2(0)\Big)}\\
&\;\le \; \frac{ \sum_{\ell } \ell \mu(\ell
)\Big((1-G({\nu_{\rm qs}^*};z))-\frac{g(z)}{g(0)}\Big) + \sum_{\ell }\ell ^2\mu(\ell
)\frac{g(z)}{g(0)}g(z)}
{\sum_{\ell } \ell \mu(\ell )\big(1-\ell g(0)\big)}\\
&\;\le \; \frac{ G(\delta_1T_t;z)-G({\nu_{\rm qs}^*};z)+\frac{M_2(\mu)}{M_1(\mu)}
(1-G(\delta_1T_t;z))g(z)}{1-\frac{M_2(\mu)}{M_1(\mu)} g(0)}, \end{split} \end{equation}
where $M_k(\mu):=\sum_\ell \ell^k\mu(\ell)$, $k\in{\mathbb N}$. Thus,
\be{conc-upper} \sup_{\mu\in K(\alpha)} G(\mu T_t;z)-G({\nu_{\rm qs}^*};z)\;\le\;
\frac{|G(\delta_1T_t;z)-G({\nu_{\rm qs}^*};z)|+ (1-G(\delta_1T_t;z)) g(z)\alpha}{1-\alpha g(0)}. \end{equation}
Now, for the lower bound, we use similar arguments to reach
\be{miss-lower} \begin{split} G(\mu T_t;z)-G({\nu_{\rm qs}^*};z)&\; \ge\; \frac{ \sum_{\ell } \ell \mu(\ell )\Big(-g(z)+g(0)(1-G({\nu_{\rm qs}^*};z))-\ell g^2(0) (1-G({\nu_{\rm qs}^*};z)) \Big)}{ \sum_{\ell } \ell \mu(\ell ) g(0)}\\ &\; \ge\; G(\delta_1T_t;z)-G({\nu_{\rm qs}^*};z)-\frac{M_2(\mu)}{M_1(\mu)} g(0)(1-G({\nu_{\rm qs}^*};z)). \end{split} \end{equation}
Thus,
\be{conc-lower} \inf_{\mu\in K(\alpha)} G(\mu T_t;z)-G({\nu_{\rm qs}^*};z)\;\ge\;
-|G(\delta_1T_t;z)-G({\nu_{\rm qs}^*};z)|-\alpha g(0)(1-G({\nu_{\rm qs}^*};z)). \end{equation}
Since $g(z)$ goes to 0 as the implicit $t$ goes to infinity, both \reff{conc-upper} and \reff{conc-lower} go to 0. This proves \eqref{yaglom-main}. The proof of \eqref{unif.Yaglom} follows from \eqref{yaglom-main} and Lemma \ref{unif.conv} below on convergence of probability measures. \end{proof}
\bl{unif.conv} Let $\{\mu_n^\gamma, n \in {\mathbb N}, \gamma \in \Gamma \}$ be a family of probability measures. Assume that for each $z\in[0,1]$ we have \be{gener.unif.conv} \lim_{n\to\infty}
\sup_{\gamma \in \Gamma} |G(\mu_n^\gamma,z) - G(\nu, z)|=0. \end{equation} Then, for each $x \in {\mathbb N}$ we have \[
\lim_{n\to \infty} \sup_{\gamma \in \Gamma} |\mu_n^\gamma(x) - \nu(x)| = 0. \] \end{lemma} \begin{proof}
Let $f={\mathbf 1}_{\{x\}}$. We consider the one-point compactification of ${\mathbb N}$,
which we denote $\bar {\mathbb N}={\mathbb N}\cup\{\infty\}$ and extend $f\colon \bar {\mathbb N} \to {\mathbb R}$
by $f(\infty)=0$. Since $f$ is continuous function on $\bar {\mathbb N}$, the
Stone-Weierstrass approximation theorem yields a function $h$, which is a
linear combination of functions of the form $\{y\mapsto a^y,\ 0 \le a \le 1\}$
(finite linear combinations of these functions form an algebra that separates
points and contains the constants), such that for any ${\varepsilon}>0$, $\sup_{y
\in {\mathbb N}} |f(y) - h(y)| < {\varepsilon}.$ Then \[
\sup_{\gamma} |\mu_n^\gamma(x) - \nu(x)| \;=\; \sup_{\gamma} |\mu_n^\gamma f -
\nu f| \;\le\; \sup_{\gamma} |\mu_n^\gamma f - \mu_n^\gamma h| \,+\, \sup_{\gamma}
|\mu_n^\gamma h - \nu h| \,+ \,|\nu h - \nu f|. \] The first and the third term on the r.h.s. are smaller than ${\varepsilon}$ while the second one goes to zero as $n$ goes to infinity by assumption. \end{proof}
\section{Closeness of the two semi-groups}\label{sec-semigroups}
In this section we show how propagation of chaos implies the closeness of $Em(x,\xi^\xi_t)$ and $m(\cdot,\xi)T_t$ uniformly in $\xi \in \Lambda^N$. The arguments are similar to those used in \cite{ferrari-maric,AFG}. The key is a control of the correlations that we state below. For a signed measure $\mu$ in
${\mathbb N}$ we will need to work with the $\ell_2$ norm given by $\|\mu\|^2 = \sum_{x\in{\mathbb N}} (\mu(x))^2$.
\bp{prop-semigroups} There exist constants $c$ and $C$ such that,
\be{l2-convergence}
\sup_{\xi\in{\mathbb N}^N}
\|E[m(x,\xi^\xi_t)]-m(\cdot,\xi)T_t\|\; \le \frac{Ce^{ct}}{N}. \end{equation}
As a consequence, \be{ptwise-convergence}
\sup_{\xi\in{\mathbb N}^N}
|E[m(x,\xi^\xi_t)]-m(\cdot,\xi)T_t(x)|\; \le \frac{Ce^{ct}}{N}. \qquad x\in{\mathbb N}. \end{equation} Furthermore \be{var0}
\sup_{\xi\in{\mathbb N}^N} E\big[m(x,\xi^\xi_t) - m(\cdot,\xi)T_t\big]^2 \le \frac{Ce^{ct}}{N},\qquad x\in{\mathbb N}. \end{equation} \end{proposition}
\begin{proposition}[Proposition 2 of \cite{AFG}]\label{prop-chaos} For each $t>0$, and any $x,y\in{\mathbb N}$ \be{correlations}
\sup_{\xi\in{\mathbb N}^N}\big|E[m(x,\xi^\xi_t)m(y,\xi^\xi_t)]
-E[m(y,\xi^\xi_t)]\,E[m(x,\xi^\xi_t)]\big|\;\le\; \frac{2p(0)e^{2p(0)t}}{N}. \end{equation} \end{proposition}
The paper \cite{AFG} proves this proposition for processes with bounded rates, but the extension to our case is straightforward.
\begin{proof}[Proof of Proposition \ref{prop-semigroups}] Fix $\xi\in{\mathbb N}^N$ and introduce the simplifying notations
\be{symbol-1} u(t,x):=Em(x,\xi^\xi_t)\quad \text{and}\quad v(t,x):=m(\cdot,\xi)T_t(x). \end{equation}
Define $\delta(t,x)=u(t,x)-v(t,x)$. We want to show that for any $t>0$,
\be{norm-decay}
\frac{\partial}{\partial t} \|\delta(t) \|^2\;\le\;
\frac52 \|\delta(t) \|^2+\frac{4 p(0)e^{2p(0) t}}{N}. \end{equation}
Recall the definition \eqref{rates} of the rates $q$ and the evolution equations satisfied by $v(t,x)$ and $u(t,x)$: \be{dynamic-v} \frac{\partial}{\partial t} v(t,x)=\sum_{z\not= x,z>0}q(z,x) v(t,z)-\Bigl(\sum_{z\not= x}q(x,z)\Bigr)v(t,x)+p(0) v(t,1)v(t,x), \end{equation} \be{a82} \frac{\partial}{\partial t} u(t,x)=\sum_{z\not= x,z>0}q(z,x) u(t,z)-\Bigl(\sum_{z\not= x}q(x,z)\Bigr)u(t,x)+p(0) u(t,1)u(t,x)+ W(\xi;t,x). \end{equation} Here, \be{def-R} W(\xi;t,x)=p(0) \Bigl(\frac{N}{N-1}E[m(x,\xi^\xi_t)m(1,\xi^\xi_t)] -E[m(1,\xi^\xi_t)]\,E[m(x,\xi^\xi_t)]\Bigr). \end{equation} Proposition~\ref{prop-chaos} implies that \be{step-31}
\sup_{\xi}|W(\xi;t,x)|\le \frac{ 2p(0)e^{2p(0)t}}{N}. \end{equation} Observe two simple facts. First, set $D=\{(x,z):\ x\ge 1,z\ge 1,\ x\not= z\}$, and for any function $f:{\mathbb N}\to {\mathbb R}$ \be{obs-1} \sum_{(x,z)\in D} \big(q(x,z)+q(z,x)\big) f^2(x)- 2\sum_{(x,z)\in D}q(x,z) f(x)f(z)=\sum_{(x,z)\in D}q(z,x) (f(x)-f(z))^2. \end{equation} The second observation is specific to our rates. For $x>0$ \be{obs-2} \sum_{z\not= x}q(z,x) \le \sum_{z\not= x}q(x,z)+p(0). \end{equation} Observation \reff{obs-1} is obvious and we omit its proof. Observation \reff{obs-2} is done in details. \be{step-2} \begin{split} \sum_{z\not= x}q(z,x)&\;=\; \sum_{z\ge 0,z\not= x} z p(x-z+1) = x\sum_{z\ge 0,z\not= x}p(x-z+1)+ \sum_{z\ge 0,z\not= x}(z-x)p(x-z+1)\\ &\;=\;x\big(p(0)+p(1)+\dots+p(x+1)\big)+\big(p(0)-p(2)-\dots-xp(x+1)\big)\\ &\;\le\; x\sum_{i\ge 0} p(i)+p(0)=\sum_{z\not= x}q(x,z)+p(0). \end{split} \end{equation} Now, we have \be{step-4} \begin{split} \sum_{x>0} \delta(t,x)\frac{\partial}{\partial t} \delta(t,x) &\;=\; \sum_{(x,z)\in D}\big( q(z,x) \delta(t,x)\delta(t,z)- q(x,z) \delta^2(t,x)\big) \\ &\;+\;p(0)\sum_{x>0} \big( u(t,x)u(t,1)-v(t,x)v(t,1)\big)\delta(t,x) +\sum_{x>0} \delta(t,x)W(\xi;t,x). \end{split} \end{equation} Let us deal with each term of the right hand side of \reff{step-4}. For the first term we use \reff{obs-1} and \reff{obs-2}. \be{term-1} \begin{split} &\sum_{(x,z)\in D}\big( q(z,x) \delta(t,x)\delta(t,z)- q(x,z) \delta^2(t,x)\big)\\ &\qquad\qquad\le\; \sum_{(x,z)\in D} q(z,x) \delta(t,x)\delta(t,z)-\frac{1}{2}\sum_{x>0}\big( \sum_{z\not= x} q(x,z)+ \sum_{z\not= x} q(z,x) -p(0)\big) \delta^2(t,x)\\ &\qquad\qquad\le\;-\frac{1}{2}\sum_{(x,z)\in D}q(z,x)
(\delta(t,x)-\delta(t,z))^2+\frac{p(0)}{2}\|\delta(t)\|^2\\
&\qquad\qquad\le\;\frac{p(0)}{2}\|\delta(t)\|^2. \end{split} \end{equation} To deal with the second term, first note that \[
\sup_{x>0} \big| \delta(t,x)\big|\;\le\;
\sqrt{ \sum_{x>0} \delta^2(t,x)}\;=\; \|\delta(t)\|. \] Then, \be{term-3} \begin{split} \sum_{x>0}&\big(u(t,x)u(t,1)-v(t,x)v(t,1)\big)\delta(t,x) \le \sum_{x>0} \big(\delta(t,x) u(t,1)+v(t,x) \delta(t,1)\big) \delta(t,x)\\ \le & \sum_{x>0} \delta^2(t,x)+
|\delta(t,1)| \sup_{x>0} \big| \delta(t,x)\big|\sum_{x>0} v(t,x)
\le 2\|\delta(t)\|^2. \end{split} \end{equation} For the last term, we have \be{term-4}
|\sum_{x>0} \delta(t,x)W(\xi;t,x)|\le
\sup_{x>0}|W(\xi;t,x)|\times \sum_{x>0} |\delta(t,x)|
\le 2\sup_{x>0} |W(\xi;t,x)|. \end{equation} Thus, we obtain \reff{norm-decay}. Gronwall's inequality allows to conclude. \end{proof}
\paragraph{\bf Acknowledgements} We would like to thank Elie Aidekon for valuable discussions. A.A.'s mission at Buenos Aires was supported by MathAmSud, and he acknowledges partial support of ANR-2010-BLAN-0108.
\obeylines \parskip 0pt Amine Asselah LAMA, Bat. P3/4, Universit\'e Paris-Est Cr\'eteil, 61 Av.\/ General de Gaulle, 94010 Cr\'eteil Cedex, France {\tt [email protected]}
\vskip 2mm
Pablo A. Ferrari and Pablo Groisman Departamento de Matem\'atica Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires Pabell\'on 1, Ciudad Universitaria 1428 Buenos Aires Argentina {\tt [email protected], [email protected]} \vskip 2mm
Matthieu Jonckheere Instituto de Investigaciones Matem\'aticas Luis Santal\'o
Pabell\'on 1, Ciudad Universitaria 1428 Buenos Aires Argentina {\tt [email protected]}
\end{document} | arXiv |
The Dictionary of Mathematical Eponymy: The MacWilliams Identity
Written by Colin+ in dome.
After the Second World War, there was a boom in the study of transmitting encoded data. In likelihood, I imagine the boom started earlier, and the boom was more about the declassified publication of papers on this topic than about a sudden increase in productivity.
This month's mathematical hero, Jessie MacWilliams, played a relatively late part in this boom - the identity that bears her name was in her early-1960s thesis - and relates to linear codes. It needs a bit of setting up, though.
First up, we need to say what we mean by a code: in this context, it's a collection of codewords. A codeword is a list of 1s and 0s ("bits") of a given length, $n$
A linear code is one where any combination of two codewords gives a codeword - and "combination" here means taking the XOR of the two codewords1. That means, if the bits in position $i$ of the two original codewords are the same, the bit in position $i$ of the result will be 0; if the two bits were different, the $i$th bit of the result would be 1. For example, in a code with $n=4$, combining 0101 and 1100 would give a result of 1001.
Such a code also has a dual code – all of the possible words that are orthogonal to every codeword in the original code. Two words are orthogonal if, when you multiply their bits together, one at a time, and add up the results modulo 2, you get zero. For example, 0101 and 1111 are orthogonal: you get $0\times1 = 0$ for the first bit, $1\times 1 = 1$ for the second, then 0 and 1 again for the third and fourth bits. Adding up $0+1+0+1$ modulo 2 gives you 0.
For more example, the four-bit code above, with codewords 0000, 0101, 1001 and 1100 has a dual code with the codewords 0000, 0010, 1101 and 11112. If the original code is called $C$, its dual code is called $C^\perp$.
We also need to define the weight of a codeword, which is simply "how many 1s it has in it"3 - the weight of 0101 is 2. The number of codewords with a particular weight of $w$ is denoted $A_w$.
And lastly, for now, we're going to define the weight enumerator function:
$$W(C; x, y) = \sum_0^n A_w x^w y^{n-w}$$
The code we called $C$ earlier has a weight enumerator of $y^4 + 3x^2y^2$; its dual code's weight enumerator is $y^4 + xy^3 + x^3y + x^4$.
The weight enumerator can be used to find the probability of incorrectly decoding a codeword due to errors - in particular for a binary linear code, if the probability of a bit being flipped is $p$, the function $W(C; p, 1-p)$ gives the probability of the wrong codeword arriving.
What is the MacWilliams Identity?
The MacWilliams identity relates the weight enumerators of a code and its dual: it states $W(C^{\perp}; x,y) = \frac{1}{|C|}W(C; y-x, y+x)$
Let's check it with our example: $|C|$ is the number of codewords in $|C|$, 4, so we get $\frac{1}{4} \left( (y-x)^4 + 3(y-x)^2(y+x)^2\right)$.
If we expand that out, we get $\frac{1}{4}\left(\left(y^4 - 4y^3x + 6y^2x^2 - 4yx^3 + x^4\right) + 3\left(y^4 - 2x^2y^2 + x^4\right)\right)$.
Keep going: $\frac{1}{4}\left( 4y^4 - 4y^3x - 4yx^3 + 4x^4\right)$ - it works! Magic!
The MacWilliams identity isn't restricted to binary linear codes (it works on codes over any field, which could be significantly more complicated).
It allows us, among other things, to determine the number of codewords in the dual code without necessarily knowing what any of them are (except for the one that's all zeroes), and to work out the probability of incorrectly decoding a message sent in the dual code.
Who was Jessie MacWilliams?
Florence Jessie Collinson was born in Stoke-on-Trent, England, in 1917. She studied at Cambridge, receiving her BA and MA in the late 1930s. She worked with Oscar Zariski at Johns Hopkins and at Harvard, before marrying Walter MacWilliams in 1941. She raised a family before joining Bell Labs in 1958; then in 1960, she took leave for postgraduate studies, and completed her PhD in one year under Andrew Gleason. She spend the next decades working on algebraic constructions and combinatorial properties of codes, publishing The Theory of Error-Correcting Codes with Neil Sloane4 in 1977.
In 1980, MacWilliams gave the first Emmy Noether Lecture for the Association for Women in Mathematics. She retired from Bell Labs in 1983, and died in 1990 in New Jersey.
* Updated 2020-01-06 to clarify the more general case, and 2020-01-07 to fix an error. Thanks to Adam Atkinson both for gently putting me right and for guiding me towards the identity in the first place.
Review: The Theory That Would Not Die – Sharon Bertsch McGrayne
The Dictionary of Mathematical Eponymy: The Fermat Cubic
The Dictionary of Mathematical Eponymy: Sophie Germain primes
Dictionary of Mathematical Eponymy: Hoberman Sphere
Adam points out: in general, for alphabets of prime size $k$, the operation is "sum modulo $k$", which reduces to XOR when $k=2$. It is possible, but ill-advised, to extend this to alphabets whose sizes are other powers of primes. [↩]
You may want to check this [↩]
in general, how many of its digits are non-zero [↩]
Yes, the OEIS one [↩] | CommonCrawl |
Multiset concepts in two-universe approximation spaces
O. A. Embaby ORCID: orcid.org/0000-0001-8471-78581 &
Nadya A. Toumi2
Rough set theory over two universes is a generalization of rough set model to find accurate approximations for uncertain concepts in information systems in which uncertainty arises from existence of interrelations between the three basic sets: objects, attributes, and decisions.
In this work, multisets are approximated in a crisp two-universe approximation space using binary ordinary relation and multi relation. The concept of two universe approximation is applied for defining lower and upper approximations of multisets. Properties of these approximations are investigated, and the deviations between them and corresponding notions are obtained; some counter examples are given. The suggested notions can help in the modification of the decision-making for events in which objects have repetitions such as patients visiting a doctor more than one time; an example for this case is given.
A multiset is an unordered collection of objects in which, unlike the standard Cantorian set, the object is allowed to repeat. The word "multiset" often shortened to "mset" abbreviates the term "multiple membership set." In 1986, multiset theory was introduced by Yager [1]. Generalizations of the multiset concept were formalized by Blizard [2, 3]. Applications of multisets to rough approximations were studied by Miyamoto [4]. Over the years, besides the sporadic evidence of the applications of multisets in logic, linguistics, and physics, a great number of them are witnessed in mathematics and computer science. An overview of the applications of multisets is presented by Singh et al. [5]. Algebraic structures for the multiset space were constructed by Ibrahim et al. [6]. Girish and John introduced multiset topologies induced by multiset relations and the continuity between multiset topological spaces [7, 8]. El-Sheikh et al. introduced separation axioms on multiset topological spaces and operators on multiset bitopological spaces [9, 10]. The concepts of the exterior and boundary in the multiset topological space were introduced by Das and Mahanta [11]. Topological approximations of multisets are introduced by Abo-Tabl [12].
The rough set theory was proposed by Pawlak [13, 14] for the study of intelligent systems characterized by insufficient and incomplete information. The rough set theory has been applied in artificial intelligence, medical diagnosis, pattern recognition, data mining, conflict analysis, and algebra [15,16,17,18,19,20,21,22,23]. Wong, Wang, and Yao generalized the rough set model using two distinct but related universes [24]. The formulation and interpretation of U and V and the compatibility relation between the two universes depend very much on the available knowledge and the domain of applications. For example, in a medical diagnosis system, U can be a set of symptoms and V a set of diseases. Thus, uncertainty arises when describing the interrelations between symptoms and diseases in clinical settings. In a specific group of patients, each patient may show many symptoms, just as each disease could have many symptoms.
Shen et al. [25] researched the variable precision rough set model over two universes. Yan et al. [26] studied the model of rough set over dual universe. Fuzzy rough set models over two universes were studied by Weihua et al. [27]. Many advances of the rough set model over two universes can be found in literature [28,29,30,31,32,33]. In 2019, Sun et al. [34] provided the theoretical model of multi granulation vague rough set over two universes. Another is to try making a new way to handle group decision-making problems under uncertainty based on multi granulation vague rough set theory and methodologies over two universes.
Grish et al. [35,36,37] applied multisets for constructing approximations for rough multisets in information multi systems, rough multisets, and its multiset topology and rough multiset relations.
The rest of the paper is organized as follows: In the "Preliminaries" section, basic concepts used in the work are presented. The purpose of the "Approximation of multisets in crisp approximation space" section is to study approximations of rough multiset in two-universe approximation space. While the "Approximation based on multi binary relation" section contains an application for using multi binary relation for rough set approximation.
Preliminaries
This section is devoted to present the basic concepts and properties of rough sets and multisets.
Definition 2.1 [37] An mset drawn from the set A is represented by the count function CM defined as CM : A ⟶ N, where N is the set of all non-negative integers. Here, CM(a) is the number of occurrences of the element a in the mset M. The mset M is drawn from set A = {a1, a2, …, an} and is written as M = {m1/a1, m2/a2, …, mn/an}, where mi is the number of occurrence of the element ai, i = 1, 2, …, n in the mset M.
Definition 2.2 [37]. A domain A is defined as a set of elements from which msets are constructed. The mset space [A]ω is the class of all msets drawn from the set A so that no element in the mset occurs more than ω times.
If A = {a1, a2, …, an}, then [A]ω = {{m1/a1, m2/a2, …, mn/an} : mi ∈ {0, 1, 2, …, ω}, i = 1, 2, …, n}.
The mset space [A]∞ is the class of all msets over a domain A such that there is no limit in the number occurrences of an element in an mset.
Definition 2.3 [37] Let M and N be two msets drawn from a set A. Then:
M = N if CM(a) = CN(a) ∀ a ∈ A
M ⊆ N if CM(a) ≤ CN(a) ∀ a ∈ A
P = M ∪ N if CP(a) = max {CM(a), CN(a)} ∀ a ∈ A
P = M ∩ N if CP(a) = min {CM(a), CN(a)} ∀ a ∈ A
P = M ⊕ N if CP(a) = min {CM(a) + CN(a), ω} ∀ a ∈ A
P = M ⊖ N if CP(a) = max {CM(a) − CN(a), 0} ∀ a ∈ A, where ⊕ and ⊖ represent mset addition and subtraction, respectively.
Definition 2.4 [37] Let M be an mset drawn from a set A. The support set of M is a subset of A defined by M∗ = {a ∈ A : CM(a) > 0 }, i.e., M∗ is an ordinary set and is also called the root set of M.
Definition 2.5 [37] Let M be an mset drawn from a set A. If CM(a) = 0 ∀ a ∈ A, then M is called the empty mset and denoted by ∅ .
Definition 2.6 [37] Let M be an mset drawn from a set A and [A]ω be the mset space defined over A. Then, for any mset M ∈ [A]ω, the complement Mc of M in [A]ω is an element of [A]ω such that \( {C}_{M^c}(a)=\omega -{C}_M(a)\kern0.5em \forall a\in A \).
Definition 2.7 [37] The cardinality of an mset M drawn from a set A is defined byCard M = ∑a ∈ ACM(a). It is also denoted by |M|.
Notation 2.1 [7] Let M = {m1/x1, m2/x2, …, mn/xn} be an mset drawn from the set X = {x1, x2, …, xn} with x appearing m times in M. It is denoted by x∈mM. The entry of the form (m/x, n/y)/k denotes that x is repeated m times, y is repeated n times, and the pair (x, y) is repeated k times. The counts of the members of the domain and co-domain vary in relation to the counts of the x coordinate and y coordinate in (m/x, n/y)/k. For this purpose, let the notation C1(x, y) denotes the count of the first co-ordinate in the ordered pair (x, y), and C2(x, y) denotes the count of the second co-ordinate in (x, y).
Definition 2.8 [7] Let M1 and M2 be two msets drawn from a set X; then, the Cartesian product of M1 and M2 is defined by M1 × M2 = {(m/x, n/y)/mn : x∈mM1 , y∈nM2}.
Definition 2.9 [7] A sub mset R of M × M is said to be an mset relation on M if every member (m/x, n/y) of R has a count, the product of C1(x, y) and C2(x, y). We denote m/x related to n/y by m/xRn/y.
Definition 2.10 [38] Let (U, V, R) be a two-universe approximation space. Then, the set-valued mappings F and G represent the successor and predecessor neighborhood operators, respectively, defined as follows:
F : U ⟶ P(V), F(a) = {b ∈ V : (a, b) ∈ R}, G : V ⟶ P(U), G(b) = {a ∈ U : (a, b) ∈ R}.
F and G can be naturally extended to a mapping from P(U) to P(V) (resp. P(V) to P(U)) which are also denoted by F and G:
F : P(U) ⟶ P(V), F(A) = ∪ {F(a) : a ∈ A}, G : P(V) ⟶ P(U), G(Y) = ∪ {G(b) : b ∈ A}.
Lemma 2.1 [38] Let (U, V, R) be a two-universe approximation space, if R is a strong inverse serial relation, then for all a1, a2 ∈ U, F(A1) ∩ F(A2) ≠ ϕ implies that F(a1) = F(a2).
Proposition 2.1 [39] Let R be an arbitrary binary relation on U. Then, ∀A ∈ P(U):
(i) R is reflexive \( \Longleftrightarrow {\underset{\_}{R}}_s(A)\subseteq A\Longleftrightarrow A\subseteq {\overline{R}}_s(A) \)
(ii) R is symmetric \( \Longleftrightarrow A\subseteq {\underset{\_}{R}}_s\left({\overline{R}}_s(A)\right)\Longleftrightarrow {\overline{R}}_s\left({\underset{\_}{R}}_s(A)\right)\subseteq A \)
(iii) R is transitive \( \Longleftrightarrow {\underset{\_}{R}}_s(A)\subseteq {\underset{\_}{R}}_s\left({\underset{\_}{R}}_s(A)\right)\Longleftrightarrow {\overline{R}}_s\left({\overline{R}}_s(A)\right)\subseteq {\overline{R}}_s(A) \)
(iv) R is Euclidean \( \Longleftrightarrow {\overline{R}}_s(A)\subseteq {\underset{\_}{R}}_s\left({\overline{R}}_s(A)\right)\Longleftrightarrow {\overline{R}}_s\left({\underset{\_}{R}}_s(A)\right)\subseteq {\underset{\_}{R}}_s(A) \)
Approximation of multisets in crisp approximation space
Definition 3.1 Let U and V be two finite non-empty universes of discourse and R ∈ P(U × V) be a binary relation from U to V. The ordered triple (U, V, R) is called a (two-universe) approximation space. Let B ∈ [V]w be a multi set drawn from V.
The lower and upper approximation of B,\( {\underset{\_}{\ R}}_s(B) \) and \( {\overline{R}}_s(B) \), with respect to the approximation space are multi set of U whose membership functions, for each a ∈ U, are defined, respectively, by:
$$ {C}_{{\underset{\_}{R}}_s(B)}(a)=\mathit{\min}\left\{{C}_B(b):b\in F(a)\right\} $$
$$ {C}_{{\overline{R}}_s(B)}(a)=\mathit{\max}\left\{{C}_B(b):b\in F(a)\right\} $$
where F(a) is the successor neighborhood of a.
The ordered set pair \( \left({\underset{\_}{R}}_s(B),{\overline{R}}_s(B)\right) \) is referred to as a generalized rough multiset with respect to successor neighborhood, and \( {\underset{\_}{R}}_s:P(V)\longrightarrow P(U) \) and \( {\overline{R}}_s:P(V)\longrightarrow P(U) \) are referred to as lower and upper generalized rough multi approximation operators, respectively.
Definition 3.2 Let (U, V, R) be a two-universe approximation space. Then, the lower and upper approximations of A ∈ [U]w are defined, respectively, as follows:
$$ {C}_{{\underset{\_}{R}}_P(A)}(b)=\mathit{\min}\left\{{C}_A(a):a\in G(b)\right\} $$
$$ {C}_{{\overline{R}}_P(A)}(b)=\mathit{\max}\left\{{C}_A(a):a\in G(b)\right\} $$
where G(b) is the predecessor neighborhood of b.
The pair \( \left({\underset{\_}{R}}_P(A),{\overline{R}}_P(A)\right) \) is referred to as a generalized rough multiset with respect to the predecessor neighborhood, and \( {\underset{\_}{R}}_P:P(U)\longrightarrow P(V) \) and \( {\overline{R}}_P:P(U)\longrightarrow P(V) \) are referred to as lower and upper rough multi approximation operators, respectively. If \( {\underset{\_}{R}}_P(A)={\overline{R}}_P(A) \), then A is called an exact multiset; otherwise, A is a rough multiset.
Proposition 3.1 In a two-universe model (U, V, R) with the binary relation R, the approximation operators \( {\underset{\_}{R}}_P \) and \( {\overline{R}}_P \) satisfy the following properties for all A, A1, A2 ∈ [U]w:
\( {\displaystyle \begin{array}{ll}\left({L}_1\right)\kern0.5em {\underset{\_}{R}}_P(A)={\left({\overline{R}}_P\left({A}^c\right)\right)}^c.& \left({L}_2\right)\kern0.5em {\underset{\_}{R}}_P(U)=V.\\ {}\left({L}_3\right)\kern0.5em {\underset{\_}{R}}_P\left({A}_1\cap {A}_2\right)={\underset{\_}{R}}_P\left({A}_1\right)\cap {\underset{\_}{R}}_P\left({A}_2\right).& \left({L}_4\right)\ {\underset{\_}{R}}_P\left({A}_1\cup {A}_2\right)\supseteq {\underset{\_}{R}}_P\left({A}_1\right)\cup {\underset{\_}{R}}_P\left({A}_2\right).\\ {}\left({L}_5\right)\ {A}_1\subseteq {A}_2\Longrightarrow {\underset{\_}{R}}_P\left({A}_1\right)\subseteq {\underset{\_}{R}}_P\left({A}_2\right).& \left({U}_1\right)\kern0.5em {\overline{R}}_P(A)={\left({\underset{\_}{R}}_P\left({A}^c\right)\right)}^c.\\ {}\left({U}_2\right)\kern0.5em {\overline{R}}_P\left(\phi \right)=\phi .& \left({U}_3\right)\kern0.5em {\overline{R}}_P\left({A}_1\cup {A}_2\right)={\overline{R}}_P\left({A}_1\right)\cup {\overline{R}}_P\left({A}_2\right).\\ {}\left({U}_4\right)\kern0.5em {\overline{R}}_P\left({A}_1\cap {A}_2\right)\subseteq {\overline{R}}_P\left({A}_1\right)\cap {\overline{R}}_P\left({A}_2\right).& \left({U}_5\right)\kern0.5em {A}_1\subseteq {A}_2\Longrightarrow {\overline{R}}_P\left({A}_1\right)\subseteq {\overline{R}}_P\left({A}_2\right).\end{array}} \).
Proof By the duality of approximation operators, we only need to prove the properties L1 − L5.
(L1) For all b ∈ V, according to Definition 3.2, we can obtain:
$$ {\displaystyle \begin{array}{c}{C}_{{\left[{\overline{R}}_P\left({A}^c\right)\right]}^c}(b)=w-\left\{\mathit{\max}\left\{{C}_{A^c}(a):a\in G(b)\right\}\right\}\\ {}=w-\left\{\mathit{\max}\left\{w-{C}_A(a):a\in G(b)\right\}\right\}\\ {}=w-\left\{w-\mathit{\min}\left\{{C}_A\left(\mathrm{a}\right):a\in G(b)\right\}\right\}\\ {}=w-w+\mathit{\min}\left\{{C}_A(a):a\in G(b)\right\}\\ {}=\mathit{\min}\left\{{C}_A(a):a\in G(b)\right\}\\ {}={C}_{{\underset{\_}{R}}_P(A)}(b).\end{array}} $$
Therefore, \( {\underset{\_}{R}}_P(A)={\left({\overline{R}}_P\left({A}^c\right)\right)}^c \).
(L2) Since CU(a) = 1 ∀ a ∈ U and G(b) ⊆ U, the min{CU(a) : a ∈ G(b)} = 1. Thus, \( {C}_{{\underset{\_}{R}}_P(U)}(b)=\min \left\{{C}_U(a):a\in G(b)\right\}=1 \) for all b ∈ V. Therefore, \( {\underset{\_}{R}}_P(U)=V \).
(L3) Since ∀b ∈ V,
$$ {\displaystyle \begin{array}{c}{C}_{{\underset{\_}{R}}_P\left({A}_1\cap {A}_2\right)}(b)=\mathit{\min}\left\{{C}_{\left({A}_1\cap {A}_2\right)}(a):a\in G(b)\right\}\\ {}=\mathit{\min}\left\{\mathit{\min}\left\{{C}_{A_1}(a),{C}_{A_2}(a)\right\}:a\in G(b)\right\}\\ {}=\mathit{\min}\left\{\mathit{\min}\left\{{C}_{A_1}(a):a\in G(b)\right\},\mathit{\min}\left\{{C}_{A_2}(a):a\in G(b)\right\}\right\}\\ {}=\mathit{\min}\left\{{C}_{{\underset{\_}{R}}_P\left({A}_1\right)}(b),{C}_{{\underset{\_}{R}}_P\left({A}_2\right)}(b)\right\}\\ {}={C}_{{\underset{\_}{R}}_P\left({A}_1\right)\cap {\underset{\_}{R}}_P\left({A}_2\right)}(b).\end{array}} $$
Therefore, \( {\underset{\_}{R}}_P\left({A}_1\cap {A}_2\right)={\underset{\_}{R}}_P\left({A}_1\right)\cap {\underset{\_}{R}}_P\left({A}_2\right) \).
(L4) For all b ∈ V, we can have:
$$ {\displaystyle \begin{array}{c}{C}_{{\underset{\_}{R}}_P\left({A}_1\cup {A}_2\right)}(b)=\mathit{\min}\left\{{C}_{\left({A}_1\cup {A}_2\right)}(a):a\in G(b)\right\}\\ {}=\mathit{\min}\left\{\mathit{\max}\left\{{C}_{A_1}(a),{C}_{A_2}(a)\right\}:a\in G(b)\right\}\\ {}\ge \mathit{\max}\left\{\mathit{\min}\left\{{C}_{A_1}(a):a\in G(b)\right\},\mathit{\min}\left\{{C}_{A_2}(a):a\in G(b)\right\}\right\}\\ {}=\mathit{\max}\left\{{C}_{{\underset{\_}{R}}_P\left({A}_1\right)}(b),{C}_{{\underset{\_}{R}}_P\left({A}_2\right)}(b)\right\}\\ {}={C}_{{\underset{\_}{R}}_P\left({A}_1\right)\cap {\underset{\_}{R}}_P\left({A}_2\right)}(b).\end{array}} $$
Hence, \( {\underset{\_}{R}}_P\left({A}_1\cup {A}_2\right)\supseteq {\underset{\_}{R}}_P\left({A}_1\right)\cup {\underset{\_}{R}}_P\left({A}_2\right) \).
(L5) Since A1 ⊆ A2, then \( \forall a\in U,{C}_{A_1}(a)\le {C}_{A_2}(a) \). Thus, \( {C}_{{\underset{\_}{R}}_P\left({A}_1\right)}(b)=\mathit{\min}\left\{{C}_{A_1}(a):a\in G(b)\right\}\le \mathit{\min}\left\{{C}_{A_2}(a):a\in G(b)\right\}={C}_{{\underset{\_}{R}}_P\left({A}_2\right)}(b) \).
Therefore, \( {\underset{\_}{R}}_P\left({A}_1\right)\subseteq {\underset{\_}{R}}_P\left({A}_2\right) \).
The next proposition gives us characterizations of the rough multi lower and rough multi upper approximation operators based on different types of relations.
Proposition 3.2. Let R ∈ P(U × V) be an arbitrary binary relation. Then, ∀A ∈ [U]w:
(i) R is inverse serial \( \Longleftrightarrow \left({L}_6\right){\underset{\_}{R}}_P\left(\phi \right)=\phi \)\( \Longleftrightarrow \left({U}_6\right){\overline{R}}_P(U)=V \)\( \Longleftrightarrow (LU){\underset{\_}{R}}_P(A)\subseteq {\overline{R}}_P(A) \).
If U = V, then:
(ii) R is reflexive \( \Longleftrightarrow \left({L}_7\right){\underset{\_}{R}}_P(A)\subseteq A\Longleftrightarrow \left({U}_7\right)\ A\subseteq {\overline{R}}_P(A) \)
(iii) R is symmetric \( \Longleftrightarrow \left({L}_8\right)\ A\subseteq {\underset{\_}{R}}_P\left({\overline{R}}_P(A)\right)\Longleftrightarrow \left({U}_8\right){\overline{R}}_P\left({\underset{\_}{R}}_P(A)\right)\subseteq A \)
(iv)
R is transitive \( \Longleftrightarrow \left({L}_9\right){\underset{\_}{R}}_P(A)\subseteq {\underset{\_}{R}}_P\left({\underset{\_}{R}}_P(A)\right)\Longleftrightarrow \left({U}_9\right){\overline{R}}_P\left({\overline{R}}_P(A)\right)\subseteq {\overline{R}}_P(A) \)
R is left Euclidean \( \Longleftrightarrow \left({L}_{10}\right){\overline{R}}_P(A)\subseteq {\underset{\_}{R}}_P\left({\overline{R}}_P(A)\right)\Longleftrightarrow \left({U}_{10}\right){\overline{R}}_P\left({\underset{\_}{R}}_P(A)\right)\subseteq {\underset{\_}{R}}_P(A) \)
Proof (i) Supposing that R is an inverse serial relation, then for any b ∈ V, we have G(b) ≠ ϕ. Thus, \( {C}_{{\underset{\_}{R}}_P\left(\phi \right)}(b)=\mathit{\min}\left\{{C}_{\phi }(a):a\in G(b)\right\}=0\forall b\in V \). Therefore, \( {\underset{\_}{R}}_P\left(\phi \right)=\phi \).
Conversely, assuming that \( {\underset{\_}{R}}_P\left(\phi \right)=\phi \) ,i.e., \( {C}_{{\underset{\_}{R}}_P\left(\phi \right)}(b)=\mathit{\min}\left\{{C}_{\phi }(a):a\in G(b)\right\}=0\kern0.5em \forall b\in V \). If there exists b∘ ∈ V such that G(b∘) = ϕ then \( {C}_{{\underset{\_}{R}}_P\left(\phi \right)}\left({b}_{\circ}\right)=\mathit{\min}\left\{{C}_{\phi }(a):a\in G\left({b}_{\circ}\right)\right\}=\min \left\{\kern1em \right\}= undefined \) which contradicts the assumption. Thus, G(b) ≠ ϕ ∀ b ∈ V,i.e., R is an inverse serial. We can prove that R is an inverse serial if and only if \( \left({U}_6\right)\ {\overline{R}}_P(U)=V \) by the duality of approximation operators. For the third part, R is inverse serial \( if\ and\ only\ if\ (LU)\ {\underset{\_}{R}}_P(A)\subseteq {\overline{R}}_P(A) \), and the proof is obvious.
(ii) By the duality, it is only to prove that R is reflexive if and only if \( \left({L}_7\right)\ {\underset{\_}{R}}_P(A)\subseteq A \). Since R is reflexive, then ∀b ∈ V, b ∈ G(b), i.e., min{CA(a) : a ∈ G(b)} ≤ CA(b) which implies that \( {\underset{\_}{R}}_P(A)\subseteq A \).
Conversely, assuming \( {\underset{\_}{R}}_P(A)\subseteq A \) for all multi subset A of U. Because a crisp set is a special case of a multiset, then \( {\underset{\_}{R}}_P(A)\subseteq A \) for all A ⊆ U and by proposition 2.1, R is a reflexive relation.
(iii) Assuming that R is symmetric, then for all a ∈ G(b), we have b ∈ G(a). So, max{min{CA(c) : c ∈ G(a)} : a ∈ G(b)} ≤ CA(b).
Therefore, \( {\underset{\_}{R}}_P\left({\overline{R}}_P(A)\right)\subseteq A \).
Conversely, assuming \( {\overline{R}}_P\left({\underset{\_}{R}}_P(A)\right)\subseteq A \) for all multi subset A of U. Because a crisp set is a special case of a multiset, then \( {\overline{R}}_P\left({\underset{\_}{R}}_P(A)\right)\subseteq A \) for all A ⊆ U and by proposition 2.1, R is a symmetric relation. For the other statement, the proof is similar.
(iv) Supposing that R is a transitive relation, then for all a ∈ G(b), we have G(a) ⊆ G(b). Thus, \( {\displaystyle \begin{array}{c}{C}_{{\underset{\_}{R}}_P\left({\underset{\_}{R}}_P(A)\right)}(b)=\mathit{\min}\left\{\mathit{\min}\left\{{C}_A(a):c\in G(a)\right\}:a\in G(b)\right\}\\ {}\ge \mathit{\min}\left\{\mathit{\min}\left\{{C}_A(c):c\in G(b)\right\}:a\in G(b)\right\}\\ {}=\mathit{\min}\left\{{C}_A(c):c\in G(b)\right\}\\ {}={\underset{\_}{R}}_{\mathcal{P}}(A)(b).\end{array}} \)
Therefore, \( {\underset{\_}{R}}_P(A)\subseteq {\underset{\_}{R}}_P\left({\underset{\_}{R}}_P(A)\right) \).
The proof of the other side is similar to (iii).
(v) Assuming that R is a left Euclidean relation, then for all a ∈ G(b), we have G(b) ⊆ G(a). So, \( {\displaystyle \begin{array}{c}{C}_{{\overline{R}}_P\left({\underset{\_}{R}}_P(A)\right)}(b)=\mathit{\max}\left\{\mathit{\min}\left\{{C}_A(c):c\in G(a)\right\}:a\in G(b)\right\}\\ {}\le \mathit{\max}\left\{\mathit{\min}\left\{{C}_A(c):c\in G(b)\right\}:a\in G(b)\right\}\\ {}=\mathit{\min}\left\{{C}_A(c):c\in G(b)\right\}={C}_{{\underset{\_}{R}}_P(A)}(b).\end{array}} \)
Therefore, \( {\overline{R}}_P\left({\underset{\_}{R}}_P(A)\right)\subseteq {\underset{\_}{R}}_P(A) \).
The proof of the other side is like (iii).
Remark 3.1 If R ∈ P(U × V) is a serial relation in a two-universe approximation space (U, V, R), then the properties L6, U6, and LU are not true in general, as shown in the following example:
Example 3.1 Let U = {a1, a2, a3, a4}, V = {b1, b2, b3, b4, b5}, and R be a binary relation from U to V defined as:
$$ \mathrm{R}=\left\{\left({a}_1,{b}_2\right),\left({a}_1,{b}_4\right),\left({a}_2,{b}_3\right),\left({a}_2,{b}_4\right),\left({a}_3,{b}_3\right),\left({a}_4,{b}_1\right),\left({a}_4,{b}_2\right)\right\}. $$
If A ∈ [U]w is a multiset drawn from U. Let A = {2/a1, 3/a2, 4/a4}.
Then, we have:
\( {C}_{{\underset{\_}{R}}_{\mathcal{P}}(A)}(b) \) 4 2 0 2 undefined
\( {C}_{{\overline{R}}_{\mathcal{P}}(A)}(b) \) 4 4 3 3 undefined
\( {C}_{{\underset{\_}{R}}_{\mathcal{P}}\left(\phi \right)}(b) \) 0 0 0 0 undefined
\( {C}_{{\overline{R}}_{\mathcal{P}}(U)}(b) \) 1 1 1 1 undefined
Hence, \( {\underset{\_}{R}}_P\left(\phi \right)\ne \phi \), \( {\overline{R}}_P(U)\ne V \), and \( {\underset{\_}{R}}_P(A)\ne {\overline{R}}_P(A) \), i.e., L6, U6, and LU do not hold.
Remark 3.2 Let R be any reflexive relation, then ∀A ∈ [U]w the properties L8 − L10 and U8 − U10 are not true in general. The following example shows this remark.
Example 3. 2 Let U = {a1, a2, a3, a4, a5} and R be a reflexive relation on U defined as R = {(a1, a1), (a1, a2), (a2, a1), (a2, a2), (a2, a4), (a3, a3), (a3, a5),
$$ \left({a}_4,{a}_2\right),\left({a}_4,{a}_4\right),\left({a}_5,{a}_2\right),\left({a}_5,{a}_5\right)\Big\}. $$
If A and B are multisets drawn from U defined as A = {2/a2, 3/a3, 4/a5} and B = {2/a1, 3/a2, 1/a4, 4/a5}, then we have:
\( {C}_{{\underset{\_}{R}}_P(B)}(a) \) 2 1 0 1 0
\( {C}_{{\underset{\_}{R}}_P\left({\underset{\_}{R}}_P(A)\right)}(a) \) 1 0 0 1 0
\( {C}_{{\overline{R}}_P\left({\underset{\_}{R}}_P(A)\right)}(a) \) 0 3 3 0 3
\( {C}_{{\overline{R}}_P(A)}(a) \) 2 4 3 2 4
\( {C}_{{\overline{R}}_P\left({\overline{R}}_P(B)\right)}(a) \) 4 4 0 4 4
\( {C}_{{\overline{R}}_P(B)}(a) \) 3 4 0 3 4
\( {C}_{{\underset{\_}{R}}_P\left({\overline{R}}_P(A)\right)}(a) \) 2 2 3 2 3
Hence,\( A\nsubseteq {\underset{\_}{R}}_P\left({\overline{R}}_P(A)\right) \), \( {\underset{\_}{R}}_P(A)\nsubseteq {\underset{\_}{R}}_P\left({\underset{\_}{R}}_P(A)\right) \), \( {\overline{R}}_P(A)\nsubseteq {\underset{\_}{R}}_P\left({\overline{R}}_P(A)\right) \),\( {\overline{R}}_P\left({\underset{\_}{R}}_P(A)\right)\nsubseteq A \), \( {\overline{R}}_P\left({\overline{R}}_P(A)\right)\nsubseteq {\overline{R}}_P(A) \), \( {\overline{R}}_P\left({\underset{\_}{R}}_P(A)\right)\nsubseteq {\underset{\_}{R}}_P(A) \),i.e., L8 − L10, U8 − U10 do not hold.
Remark 3.3 Let R be any symmetric relation, then ∀A ∈ [U]w the properties L6, L7, L9, L10, U6, U7, U9, U10 and LU are not true in general. The following example shows this remark.
Example 3.3 Let U = {a1, a2, a3, a4, a5} and R be a symmetric relation on U defined as R = {(a1, a1), (a1, a2), (a2, a1), (a2, a4), (a4, a2), (a4, a4), (a5, a5)}.
If A is a multiset drawn from U defined as A = {4/a1, 2/a2, 3/a4, 1/a5}, then we have:
\( {\underset{\_}{R}}_{\mathcal{P}}(A)(a) \) 2 3 undefined 2 1
\( {\underset{\_}{R}}_{\mathcal{P}}\left({\underset{\_}{R}}_{\mathcal{P}}(A)\right)(a) \) 2 2 undefined 2 1
\( {\overline{R}}_{\mathcal{P}}\left({\underset{\_}{R}}_{\mathcal{P}}(A)\right)(a) \) 3 2 undefined 3 1
\( {\overline{R}}_{\mathcal{P}}(A)(a) \) 4 4 undefined 3 1
\( {\overline{R}}_{\mathcal{P}}\left({\overline{R}}_{\mathcal{P}}(A)\right)(a) \) 4 4 undefined 4 1
\( {\underset{\_}{R}}_{\mathcal{P}}\left({\overline{R}}_{\mathcal{P}}(A)\right)(a) \) 4 3 undefined 3 1
\( {\underset{\_}{R}}_{\mathcal{P}}\left(\phi \right)(a) \) 0 0 undefined 0 0
\( {\overline{R}}_{\mathcal{P}}(U)(a) \) 1 1 undefined 1 1
Hence, \( {\underset{\_}{R}}_P\left(\phi \right)\ne \phi \), \( {\underset{\_}{R}}_P(A)\nsubseteq A \), \( {\underset{\_}{R}}_P(A)\nsubseteq {\underset{\_}{R}}_P\left({\underset{\_}{R}}_P(A)\right) \), \( {\overline{R}}_P(A)\nsubseteq {\underset{\_}{R}}_P\left({\overline{R}}_P(A)\right) \), \( {\overline{R}}_P(U)\ne U \),\( A\nsubseteq {\overline{R}}_P(A) \), \( {\overline{R}}_P\left({\overline{R}}_P(A)\right)\nsubseteq {\overline{R}}_P(A) \),\( {\overline{R}}_P\left({\underset{\_}{R}}_P(A)\right)\nsubseteq {\underset{\_}{R}}_P(A) \) and \( {\underset{\_}{R}}_P(A)\nsubseteq {\overline{R}}_P(A) \), i.e., L6, L7, L9, L10 and U6, U7, U9, U10 and LU do not hold.
Remark 3.4 Let R be any transitive relation, then ∀A ∈ [U]w the properties L6, L7, L8, L10, U6, U7, U8, U10 and LU do not hold in general. The following example shows this remark.
Example 3.4 Let U = {a1, a2, a3, a4, a5} and R be a transitive relation on U defined as R = {(a1, a2), (a1, a3), (a2, a3), (a4, a4), (a5, a2), (a5, a3)}.
If A is a multiset drawn from U defined as A = {3/a1, 4/a3, 2/a5} and B = {3/a1, 1/a2, 2/a4, 4/a5}, then we have:
\( {a}_{5} \)
\( {\underset{\_}{R}}_{\mathcal{P}}(A)(a) \) undefined 2 0 0 undefined
\( {\overline{R}}_{\mathcal{P}}\left({\underset{\_}{R}}_{\mathcal{P}}(B)\right)(a) \) undefined 0 3 2 undefined
\( {\overline{R}}_{\mathcal{P}}(A)(a) \) undefined 3 3 0 undefined
\( {\underset{\_}{R}}_{\mathcal{P}}(B)(a) \) undefined 3 1 2 undefined
\( {\underset{\_}{R}}_{\mathcal{P}}\left({\overline{R}}_{\mathcal{P}}(A)\right)(a) \) undefined 0 0 0 undefined
\( {\underset{\_}{R}}_{\mathcal{P}}\left(\phi \right)(a) \) undefined 0 0 0 undefined
\( {\overline{R}}_{\mathcal{P}}(U)(a) \) undefined 1 1 1 undefined
Hence, \( {\underset{\_}{R}}_P\left(\phi \right)\ne \phi \), \( {\underset{\_}{R}}_P(A)\nsubseteq A \), \( A\nsubseteq {\underset{\_}{R}}_P\left({\overline{R}}_P(A)\right) \), \( {\overline{R}}_P(A)\nsubseteq {\underset{\_}{R}}_P\left({\overline{R}}_P(A)\right) \), \( {\overline{R}}_P(U)\ne V \), \( A\nsubseteq {\overline{R}}_P(A) \), \( {\overline{R}}_P\left({\underset{\_}{R}}_P(A)\right)\nsubseteq A \), \( {\overline{R}}_P\left({\underset{\_}{R}}_P(A)\right)\nsubseteq {\underset{\_}{R}}_P(A) \) and \( {\underset{\_}{R}}_P(A)\nsubseteq {\overline{R}}_P(A) \), i.e., L6, L7, L8, L10, U6, U7, U8, U10 and LU do not hold.
Definition 3.4 A multi constant \( \hat{\alpha} \) is a multiset in U defined as:
$$ {C}_{\hat{\alpha}}(a)=\alpha \forall a\in U,\alpha \in N. $$
Proposition 3.3 Let (U, V, R) be a two- universe approximation space, the rough multi lower and upper approximation operators have the following properties for all Aj ∈ [U]w, j ∈ J which is an finite index set and for all α ∈ {1, 2, 3, …},
$$ \left(\mathrm{i}\right){\underset{\_}{R}}_P\left({\cap}_{j\in J}{A}_j\right)={\cap}_{j\in J}{\underset{\_}{R}}_P\left({A}_j\right). $$
$$ \left(\mathrm{ii}\right){\underset{\_}{R}}_P\left({\cup}_{j\in J}{A}_{\mathrm{j}}\right)\supseteq {\cup}_{j\in J}{\underset{\_}{R}}_P\left({A}_j\right). $$
$$ \left(\mathrm{iii}\right){\underset{\_}{R}}_P\left(A\cup \hat{\alpha}\right)={\underset{\_}{R}}_P(A)\cup \hat{\alpha}. $$
$$ \left(\mathrm{iv}\right){\overline{R}}_P\left({\cup}_{j\in J}{A}_j\right)={\cup}_{j\in J}{\overline{R}}_P\left({A}_j\right). $$
$$ \left(\mathrm{v}\right){\overline{\mathrm{R}}}_P\left({\cap}_{\mathrm{j}\in \mathrm{J}}{\mathrm{A}}_{\mathrm{j}}\right)\subseteq {\cap}_{j\in J}{\overline{R}}_P\left({A}_j\right). $$
Proof By the duality of approximation operators, we only need to prove the properties (i) − (iii).
(i) For each b ∈ V, we have:
$$ {C}_{{\underset{\_}{R}}_P\left({\cap}_{j\in J}{A}_j\right)}(b)=\mathit{\min}\left\{{C}_{\left({\cap}_{j\in J}{A}_j\right)}(a):(a)\in G(b)\right\} $$
$$ =\mathit{\min}\left\{\mathit{\min}\left\{{C}_{A_j}(a):j\in J\right\}:a\in G(b)\right\} $$
$$ =\mathit{\min}\left\{\mathit{\min}\left\{{C}_{A_j}(a):a\in G(b)\right\}:j\in J\right\}=\mathit{\min}\left\{{C}_{{\underset{\_}{R}}_P\left({A}_j\right)}(b):j\in J\right\} $$
$$ ={C}_{\cap_{j\in J}{\underset{\_}{R}}_P\left({A}_j\right)}(b). $$
(ii) Since ∀(b) ∈ V,
$$ {C}_{{\underset{\_}{R}}_P\left({\cup}_{j\in J}{A}_j\right)}(b)=\mathit{\min}\left\{{C}_{\left({\cup}_{j\in J}{A}_j\right)}(a):(a)\in G(b)\right\} $$
$$ =\mathit{\min}\left\{\mathit{\max}\left\{{C}_{A_j}(a):j\in J\right\}:(a)\in G(b)\right\} $$
$$ \ge \mathit{\min}\left\{{C}_{B_j}(c):(c)\in G(b)\right\},\forall j\in J={C}_{{\underset{\_}{R}}_P\left({A}_j\right)}(b),\forall j\in J. $$
Therefore, \( {C}_{{\underset{\_}{R}}_P\left({\cup}_{j\in J}{A}_j\right)}(b)\ge \mathit{\max}\left\{{\underset{\_}{R}}_P\left({A}_j\right)(b),\forall j\in J\right\}={C}_{\cup_{j\in J}{\underset{\_}{R}}_P\left({A}_j\right)}(b). \)
(iii) For each (b) ∈ V, we have:
$$ {\displaystyle \begin{array}{c}{C}_{{\underset{\_}{R}}_P\left(A\cup \hat{\alpha}\right)}(b)=\mathit{\min}\left\{{C}_{\left(A\cup \hat{\alpha}\right)}(a):(a)\in G(b)\right\}\\ {}=\mathit{\min}\left\{\mathit{\max}\left\{{C}_A(a),{C}_{\hat{\alpha}}(a)\right\}:(a)\in G(b)\right\}\\ {}=\mathit{\max}\left\{\mathit{\min}\left\{{\mathrm{C}}_A(a):a\in G(b)\right\},{C}_{\hat{\alpha}}(a)\right\}\\ {}={C}_{\left({\underset{\_}{R}}_P(A)\cup \hat{\alpha}\right)}(b).\end{array}} $$
Proposition 3.4 Let (U, V, R) be a two-universe approximation space. Then, the following are equivalent ∀α ∈ N
(i) R is an inverse serial relation,
$$ \left(\mathrm{ii}\right){\underset{\_}{R}}_P\left(\hat{\alpha}\right)=\hat{\alpha}, $$
$$ \left(\mathrm{iii}\right){\overline{R}}_P\left(\hat{\alpha}\right)=\hat{\alpha}. $$
Proof (i) ⟹ (ii) Let R be an inverse serial relation, then we have \( {\underset{\_}{R}}_P\left(\hat{\alpha}\right)={\underset{\_}{R}}_P\left(\hat{\alpha}\cup \phi \right)=\hat{\alpha}\cup {\underset{\_}{R}}_P\left(\phi \right)=\hat{\alpha}\cup \phi =\hat{\alpha .} \)
(ii) ⟹ (iii) Coming from the duality of approximation operators.
(iii) ⟹ (i) Assuming \( {\overline{R}}_P\left(\hat{\alpha}\right)=\hat{\alpha} \), since U is a special case of \( \hat{\alpha} \) which is α = w. Then by assumption, we have \( {\overline{R}}_P(U)=V, \) i.e., R is an inverse serial relation.
In the next three propositions, the connections of the approximation operators in definitions 2.7, and 3.1 are made, and the conditions under which these approximation operators made the equivalent are obtained.
Proposition 3.5 Let (U, V, R) be a two-universe approximation space, then the following holds for all A ∈ [U]wand B ∈ [V]w:
$$ \left(\mathrm{i}\right)\kern0.5em {\overline{R}}_s\left({\underset{\_}{R}}_P(A)\right)\subseteq A,A\subseteq {\underset{\_}{R}}_s\left({\overline{R}}_P(A)\right),\left(\mathrm{i}\mathrm{v}\right)\kern0.5em {\overline{R}}_s(B)={\overline{R}}_s\left({\underset{\_}{R}}_P\left({\overline{R}}_s(B)\right)\right), $$
$$ \left(\mathrm{ii}\right)\kern0.5em {\overline{R}}_P\left({\underset{\_}{R}}_s(B)\right)\subseteq B,B\subseteq {\underset{\_}{R}}_P\left({\overline{R}}_s(B)\right),\left(\mathrm{v}\right)\kern0.5em {\underset{\_}{R}}_P(A)={\underset{\_}{R}}_P\left({\overline{R}}_s\left({\underset{\_}{R}}_P(A)\right)\right), $$
$$ \left(\mathrm{iii}\right)\kern0.5em {\underset{\_}{R}}_s(B)={\underset{\_}{R}}_s\left({\overline{R}}_P\left({\underset{\_}{R}}_s(B)\right)\right),\left(\mathrm{vi}\right)\kern0.5em {\overline{R}}_P(A)={\overline{R}}_P\left({\underset{\_}{R}}_P\left({\overline{R}}_P(A)\right)\right) $$
Proof (i) Since for every a ∈ U, we have either F(a) = ϕ or F(a) ≠ ϕ. If F(a) = ϕ, then \( {C}_{{\overline{R}}_s\left({\underset{\_}{R}}_P(A)\right)}(a)=\mathit{\max}\left\{\mathit{\min}\left\{{C}_A(a):c\in G(b)\right\}:b\in F(a)\right\}=0 \) and hence \( {\overline{R}}_s\left({\underset{\_}{R}}_P(A)\right)\subseteq A. \) If A(a) ≠ ϕ, then we have a ∈ G(b) ∀ b ∈ A(a). Thus, max{min{CA(c) : c ∈ G(b)}b ∈ A(a)} ≤ CA(a), hence \( {\overline{R}}_s\left({\underset{\_}{R}}_P(A)\right)\subseteq A. \) We can easily prove the other part by the duality of approximation operators.
(ii) is similar to (i).
(iii) − (vi) can be proved by the properties (i) and (ii).
Lemma 3.1 Let (U, V, R) be a two-universe approximation space, b ∈ V; if R is a strong inverse serial relation, then for all a1, a2 ∈ G(b),
$$ {C}_{{\underset{\_}{R}}_s(B)}\left({a}_1\right)={C}_{{\underset{\_}{R}}_s(B)}\left({a}_2\right);{C}_{{\overline{R}}_s(B)}\left({a}_1\right)={C}_{{\overline{R}}_s(B)}\left({a}_2\right). $$
Proof The proofs come directly from Lemma 2.1.
Proposition 3.6 Let (U, V, R) be a two-universe approximation space with a strong inverse serial relation, then the following holds for all A ∈ [U]w and B ∈ [V]w:
$$ \left(\mathrm{i}\right)\ {\overline{R}}_P\left({\underset{\_}{R}}_P(B)\right)={\underset{\_}{R}}_P\left({\overline{R}}_P(B)\right) $$
$$ \left(\mathrm{ii}\right)\ {\underset{\_}{R}}_P\left({\overline{R}}_P(B)\right)={\overline{R}}_P\left({\overline{R}}_P(B)\right). $$
Proof The proofs follow immediately from Lemma 3.1.
Proposition 3.7 Two pairs of lower approximation and upper approximation operators in definitions 2.7 and 3.2 are equivalent if and only if R is a symmetric relation.
Proof Let R be a symmetric relation on U, A ∈ [U]w. Then for all a ∈ U, we have F(a) = G(a), i.e., \( {\displaystyle \begin{array}{c}{C}_{{\underset{\_}{R}}_s(A)}(a)=\mathit{\min}\left\{{C}_A(b):b\in F(a)\right\}\\ {}=\mathit{\min}\left\{{C}_A(b):b\in G(a)\right\}={C}_{{\underset{\_}{R}}_P(A)}(a).\end{array}} \)
Conversely, assuming \( {\underset{\_}{R}}_s(A)={\underset{\_}{R}}_P(A) \), since by the proposition 3.4, we have \( {\overline{R}}_P\left({\underset{\_}{R}}_P(A)\right)\subseteq A \), by proposition 3.1,and R is a symmetric relation.
Proposition 3.8 Let G = (U, R) be a generalized approximation space and A be a multisubset of U. Then, the following holds:
(i) If R is symmetric then:
$$ {\underset{\_}{R}}_P(A)={\underset{\_}{R}}_P\left({\overline{R}}_P\left({\underset{\_}{R}}_P(A)\right)\right);{\overline{R}}_P(A)={\overline{R}}_P\left({\underset{\_}{R}}_P\left({\overline{R}}_P(A)\right)\right). $$
(ii) If R is inverse serial and transitive then:
$$ {\underset{\_}{R}}_P(A)\subseteq {\underset{\_}{R}}_P\left({\overline{R}}_P\left({\underset{\_}{R}}_P(A)\right)\right);{\overline{R}}_P(A)\supseteq {\overline{R}}_P\left({\underset{\_}{R}}_P\left({\overline{R}}_P(A)\right)\right). $$
Proof Obvious
Example 3.5 Let U = {a1, a2, a3, a4} a set of four patients and V = {Fever(b1), Headache(b2), Stomachache (b3), Cough(b4), Myalgia (b5)} be five symptoms,if R = {(a1, b2), (a1, b4), (a2, b3), (a2, b4), (a3, b3), (a3, b5), (a4, b1), (a4, b2), (a4, b5)}
is a relation relating patients to symptoms. Let A = {3/a1, 0/a2, 3/a3, 5/a4} represents a multiset of patients and times of visiting the doctor. Thus, using definition 2.10, we have:
$$ G\left({b}_1\right)=\left\{{a}_4\right\},G\left({b}_2\right)=\left\{{a}_1,{a}_4\right\},G\left({b}_3\right)=\left\{{a}_2,{a}_3\right\},G\left({b}_4\right)=\left\{{a}_1,{a}_2\right\},G\left({b}_5\right)=\left\{{a}_3,{a}_4\right\} $$
and so, we get:
$$ {\underset{\_}{R}}_P(A)=\left\{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{${b}_1$}\right.,\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{${b}_2$}\right.,\raisebox{1ex}{$0$}\!\left/ \!\raisebox{-1ex}{${b}_3$}\right.,\raisebox{1ex}{$0$}\!\left/ \!\raisebox{-1ex}{${b}_4$}\right.,\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{${b}_5$}\right.\right\}\ and\ {\overline{R}}_p(A)=\left\{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{${b}_1$}\right.,\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{${b}_2$}\right.,\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{${b}_3$}\right.,\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{${b}_4$}\right.,\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{${b}_5$}\right.\right\}. $$
If A = {a1, a3, a4} . By using the class U/R−1 = {{a4}, {a1, a4}, {a2, a3}, {a1, a2}, {a3, a4}} , the lower and upper approximations using rough sets on one universe U are \( \underset{\_}{R}(A)=\left\{{a}_1,{a}_3,\kern0.5em {a}_4\right\}=A \) and \( \overline{R}(A)=\left\{{a}_1,{a}_2,{a}_3,{a}_4\right\}=U \). Clearly, this method does not have any deviations between the effectiveness of symptoms. But by using the multi approximations over the two universes U and V, we have degree of effectiveness of b1 which is \( \frac{5}{5} \), b2 which is \( \frac{3}{5} \), b3 which is \( \frac{0}{2} \), b4 which is \( \frac{0}{3} \), and b5 which is \( \frac{2}{5} \).
Approximation based on multi binary relation
In this section, we aim to approximate rough sets in multi approximation spaces, study their properties, and provide a counter example.
Definition 4.1 Let U and V be two finite non-empty universes of discourse. Let Mand N be two multisets drawn from U and V, respectively. Let R be a multi binary relation from M to N. The ordered (U, V, M, N, R) is called a two-universe multi approximation space. For any crisp set A ⊆ U, the lower and upper approximations of A, \( \underset{\_}{R}(A) \) and\( \overline{R}(A) \), with respect to the multi approximation space, are multisets drawn from V whose count functions are defined respectively by:
For each b ∈ V,
$$ {C}_{\underset{\_}{R}(A)}(b)=\mathit{\min}\left\{m:\left(m/a\right)\in R\left(1/b\right),a\in A\right\} $$
$$ {C}_{\overline{R}(A)}(b)=\mathit{\max}\left\{m:\left(m/a\right)\in R\left(1/b\right),a\in A\right\} $$
If for all \( b\in V,\kern0.5em {C}_{\underset{\_}{R}(A)}(b)={C}_{\overline{R}(A)}, \) then the set A is definable (or exact) with respect to the multi approximation space (U, V, M, N, R). Otherwise, the set A is rough with respect to the multi approximation space.
Proposition 4.1 In a multi approximation space (U, V, , M, N, R), the multi approximation operators satisfy the following properties for all A, A1, A2 ∈ P(U):
$$ \left(L{\prime}_3\right)\kern0.5em \underset{\_}{R}\left({A}_1\cap {A}_2\right)\subseteq \underset{\_}{R}\left({A}_1\right)\cap \underset{\_}{R}\left({A}_2\right)\kern0.5em \left({U}_3\right)\kern0.5em \overline{R}\left({A}_1\cup {A}_2\right)=\overline{R}\left({A}_1\right)\cup \overline{R}\left({A}_2\right) $$
$$ \left(L{\prime}_4\right)\kern0.5em \underset{\_}{R}\left({A}_1\cup {A}_2\right)=\underset{\_}{R}\left({A}_1\right)\cup \underset{\_}{R}\left({A}_2\right)\left({U}_4\right)\kern0.5em \overline{R}\left({A}_1\cap {A}_2\right)\subseteq \overline{R}\left({A}_1\right)\cap \overline{R}\left({A}_2\right) $$
\( \left({L}_5\right)\kern0.5em {A}_1\subseteq {A}_2\Longrightarrow \underset{\_}{R}\left({A}_1\right)\subseteq \underset{\_}{R}\left({A}_2\right) \)\( \left({U}_5\right)\kern0.5em {A}_1\subseteq {A}_2\Longrightarrow \overline{R}\left({A}_1\right)\subseteq \overline{R}\left({A}_2\right) \).
$$ (LU)\kern0.75em \underset{\_}{R}(A)\subseteq \overline{R}(A). $$
Proof According to the duality of these properties, we only need to prove (L′3), (L′4), (L5) and (LU).
(L3) Since for all b ∈ V, \( {\displaystyle \begin{array}{c}{C}_{\underset{\_}{R}\left({A}_1\cap {A}_2\right)}\left(1/b\right)=\mathit{\min}\left\{m:\left(m/a\right)R\left(1/b\right),a\in \left({A}_1\cap {A}_2\right)\right\}\\ {}\le \min \left\{\mathit{\min}\left\{D:\left(m/a\right)\in R\left(1/b\right),a\in {A}_1\Big\},\mathit{\min}\Big\{\ m:\left(m/a\right)\in R\left(1/b\right),a\in {A}_2\right\}\right\}\\ {}\le \mathit{\min}\left\{{C}_{\underset{\_}{R}\left({A}_1\right)}\left(1/b\right),{C}_{\underset{\_}{R}\left({A}_2\right)}\left(1/b\right)\right\}\subseteq \underset{\_}{R}\left({A}_1\right)\cap \underset{\_}{R}\left({A}_2\right).\end{array}} \)
Hence, \( {\underset{\_}{R}}_U\left({A}_1\cap {A}_2\right)\subseteq {\underset{\_}{R}}_U\left({A}_1\right)\cap {\underset{\_}{R}}_U\left({A}_2\right) \).
$$ {\displaystyle \begin{array}{c}{C}_{\underset{\_}{R}\left({A}_1\cup {A}_2\right)}\left(1/b\right)=\mathit{\min}\left\{m:\left(m/a\right)\in R\left(1/b\right),a\in \left({A}_1\cup {A}_2\right)\right\}\\ {}=\mathit{\min}\left\{\mathit{\max}\left\{m:\left(m/a\right)\in R\left(1/b\right),a\in \left({A}_1\right),a\in \left({A}_2\right)\right\}\right\}\\ {}=\mathit{\max}\left\{\mathit{\min}\left\{m:\left(m/a\right)\in R\left(1/b\right),a\in {A}_1\right\},\mathit{\min}\left\{m:\left(m/a\right)\in \left(1/b\right),a\in {A}_2\right\}\right\}\\ {}=\mathit{\max}\left\{{C}_{\underset{\_}{R}\left({A}_1\right)}\left(1/b\right),{C}_{\underset{\_}{R}\left({A}_2\right)}\left(1/b\right)\right\}={C}_{\underset{\_}{R}\left({A}_1\right)\cup \underset{\_}{R}\left({A}_2\right)}\left(1/b\right).\end{array}} $$
Hence, \( \underset{\_}{R}\left({A}_1\cup {A}_2\right)=\underset{\_}{R}\left({A}_1\right)\cup \underset{\_}{R}\left({A}_2\right) \).
(L5) Since A1 ⊆ A2, then \( \forall a\in U,{A}_1{C}_{A_1}(a)\le {C}_{A_2}(a). \) Thus, \( {C}_{\underset{\_}{R}\left({A}_1\right)}\left(1/b\right)=\mathit{\min}\left\{m:\left(m/a\right)\in R\left(1/b\right),a\in {A}_1\right\}\le \mathit{\min}\left\{m:\left(m/a\right)\in R\left(1/b\right),a\in {A}_2\right\}={C}_{\underset{\_}{R}\left({A}_2\right)}\left(1/b\right) \).
Therefore, \( \underset{\_}{R}\left({A}_1\right)\subseteq \underset{\_}{R}\left({A}_2\right) \).
(LU) For all b ∈ V, we can have:
$$ {\displaystyle \begin{array}{c}{C}_{\underset{\_}{R}(A)}\left(1/b\right)=\mathit{\min}\left\{m:\left(m/a\right)\in R\left(1/b\right),a\in A\right\}\\ {}\le \mathit{\max}\left\{m:\left(m/a\right)\in R\left(1/b\right),a\in A\right\}={C}_{\overline{R}(A)}(b).\end{array}} $$
Hence, \( \underset{\_}{R}(A)\subseteq \overline{R}(A) \).
Remark 4.1 If R ∈ [M × N]w is a multi binary relation in a two-universe approximation space (U, V, M, N, R), then the following properties need not be true:
$$ \left({L}_1\right)\kern0.5em \underset{\_}{R}(A)={\left(\overline{R}\left({A}^c\right)\right)}^c,\left({U}_1\right)\kern0.5em \overline{R}(A)={\left(\underset{\_}{R}\left({A}^c\right)\right)}^c, $$
$$ \left({L}_2\right)\kern0.5em \underset{\_}{R}(U)=V,\left({U}_2\right)\kern0.5em \overline{R}\left(\phi \right)=\phi, $$
$$ \left({L}_3\right)\kern0.5em \underset{\_}{R}\left({A}_1\cap {A}_2\right)=\underset{\_}{R}\left({A}_1\right)\cap \underset{\_}{R}\left({A}_2\right),\left({U}_6\right)\kern0.5em \overline{R}(U)=V, $$
$$ \left({L}_4\right)\kern0.5em \underset{\_}{R}\left({A}_1\cup {A}_2\right)\supseteq \underset{\_}{R}\left({A}_1\right)\cup \underset{\_}{R}\left({A}_2\right). $$
The following example shows this remark:
Example 4.1 Let U = {a1, a2, a3, a4, a5, a6, a7}, V = {b1, b2, b3, b4, b5}. Let M be a multiset drawn from U and N be a multiset drawn from V shath that M = {1/a1, 2/a2, 2/a3, 1/a4, 3/a5, 2/a6, 4/a7} and N = {2/b1, , 3/b3, 1/b4, 4/b5, 3/b6} and R be a multi binary relation from M to N defined as:
$$ R=\left\{\left(1/{a}_1,2/{b}_1\right)/2,\left(1/{a}_1,3/{b}_3\right)/3,\left(1/{a}_1,1/{b}_4\right)/1,\left(2/{a}_2,3/{b}_3\right)/6,\left(2/{a}_2,1/{b}_4\right)/2,\left(2/{a}_2,4/{b}_5\right)/8,\left(2/{a}_3,2/{b}_1\right)/4,\left(2/{a}_3,4/{b}_5\right)/8,\left(2/{a}_3,3/{b}_6\right)/6,\left(1/{a}_4,3/{b}_3\right)/3,\left(1/{a}_4,1/{b}_4\right)/1,\left(3/{a}_5,2/{b}_1\right)/6,\left(3/{a}_5,3/{b}_3\right)/9,\left(3/{a}_5,1/{b}_4\right)/3,\left(3/{a}_5,4/{b}_5\right)/12,\left(2/{a}_6,2/{b}_1\right)/4,\left(2/{a}_6,3/{b}_3\right)/6,\left(2/{a}_6,1/{b}_4\right)/2,\left(4/{a}_7,2/{b}_1\right)/8,\left(4/{a}_7,1/{b}_4\right)/4,\left(4/{a}_74/{b}_5,\right)/16\right\} $$
If A is subset of U, defined as A = A1 = {a1, a3, a4, a7} and A2 = {a1, a2, a4, a6}, then we have:
\( {C}_{\underset{\_}{R}\left({A}_1\right)}\left(1/b\right) \) 1 0 1 1 2 2
\( {C}_{\overline{R}\left({A}_1\right)}\left(1/b\right) \) 4 0 1 4 4 2
\( {C}_{\underset{\_}{R}\left({A}_1\cap {A}_2\right)}\left(1/b\right) \) 1 0 1 1 0 0
\( {C}_{\underset{\_}{R}\left({A}_{`1}\right)}\cap {C}_{\underset{\_}{R}\left({A}_2\right)}\left(1/b\right) \) 1 0 1 1 2 0
\( {C}_{\underset{\_}{R}\left({A}_{`1}\right)}\cup {C}_{\underset{\_}{R}\left({A}_2\right)}\left(1/b\right) \) 1 0 1 1 2 2
\( {C}_{\underset{\_}{R}\left({A}_1\cup {A}_2\right)}\left(1/b\right) \) 1 0 1 1 2 2
\( {C}_{{\left(\underset{\_}{R}\left({A}^c\right)\right)}^c}\left(1/b\right) \) 5 6 5 5 4 4
\( {C}_{{\left(\overline{R}\left({A}^c\right)\right)}^c}\left(1/b\right) \) 3 6 3 3 3 6
\( {C}_{\underset{\_}{R}\left(\phi \right)}\left(1/b\right) \) 0 undefined 0 0 0 0
\( {C}_{\overline{R}\left(\phi \right)}\left(1/b\right) \) 0 undefined 0 0 0 0
\( {C}_{\underset{\_}{R}(U)}\left(1/b\right) \) 1 0 1 1 2 0
\( {C}_{\overline{R}(U)}\left(1/b\right) \) 4 0 3 4 4 2
Conclusion and future work
The multiset approximations suggested in this work can help to compute measures and ordering of effectiveness and certainty of concepts in information systems. More work on using multi relation to approximate rough multi sets will be discussed in the future. Also, the use of a relation between two universes (objects and attributes) can be extended to construct another relation between attributes and decisions for constructing a rough set model over three universes.
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The authors would like to thank the referees for providing very helpful comments and suggestions that helped in improving the quality of the paper.
Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt
O. A. Embaby
Department of Mathematics, Faculty of Science, The 7th of April University, Al- Zawia, Libya
Nadya A. Toumi
The first and second author participated equally in all stages of the manuscript. All authors read and approved the final manuscript.
Correspondence to O. A. Embaby.
Embaby, O.A., Toumi, N.A. Multiset concepts in two-universe approximation spaces. J Egypt Math Soc 28, 46 (2020). https://doi.org/10.1186/s42787-020-00104-5
Rough set
Multiset
Two universes approximation space
Mathematics Subject Classification
68 U35 | CommonCrawl |
Design of experiments
The design of experiments (DOE or DOX), also known as experiment design or experimental design, is the design of any task that aims to describe and explain the variation of information under conditions that are hypothesized to reflect the variation. The term is generally associated with experiments in which the design introduces conditions that directly affect the variation, but may also refer to the design of quasi-experiments, in which natural conditions that influence the variation are selected for observation.
In its simplest form, an experiment aims at predicting the outcome by introducing a change of the preconditions, which is represented by one or more independent variables, also referred to as "input variables" or "predictor variables." The change in one or more independent variables is generally hypothesized to result in a change in one or more dependent variables, also referred to as "output variables" or "response variables." The experimental design may also identify control variables that must be held constant to prevent external factors from affecting the results. Experimental design involves not only the selection of suitable independent, dependent, and control variables, but planning the delivery of the experiment under statistically optimal conditions given the constraints of available resources. There are multiple approaches for determining the set of design points (unique combinations of the settings of the independent variables) to be used in the experiment.
Main concerns in experimental design include the establishment of validity, reliability, and replicability. For example, these concerns can be partially addressed by carefully choosing the independent variable, reducing the risk of measurement error, and ensuring that the documentation of the method is sufficiently detailed. Related concerns include achieving appropriate levels of statistical power and sensitivity.
Correctly designed experiments advance knowledge in the natural and social sciences and engineering, with design of experiments methodology recognised as a key tool in the successful implementation of a Quality by Design (QbD) framework.[1] Other applications include marketing and policy making. The study of the design of experiments is an important topic in metascience.
History
Statistical experiments, following Charles S. Peirce
Main article: Frequentist statistics
A theory of statistical inference was developed by Charles S. Peirce in "Illustrations of the Logic of Science" (1877–1878)[2] and "A Theory of Probable Inference" (1883),[3] two publications that emphasized the importance of randomization-based inference in statistics.[4]
Randomized experiments
See also: Repeated measures design
Charles S. Peirce randomly assigned volunteers to a blinded, repeated-measures design to evaluate their ability to discriminate weights.[5][6][7][8] Peirce's experiment inspired other researchers in psychology and education, which developed a research tradition of randomized experiments in laboratories and specialized textbooks in the 1800s.[5][6][7][8]
Optimal designs for regression models
See also: Optimal design
Charles S. Peirce also contributed the first English-language publication on an optimal design for regression models in 1876.[9] A pioneering optimal design for polynomial regression was suggested by Gergonne in 1815. In 1918, Kirstine Smith published optimal designs for polynomials of degree six (and less).[10][11]
Sequences of experiments
See also: Multi-armed bandit problem, Gittins index, and Optimal design
The use of a sequence of experiments, where the design of each may depend on the results of previous experiments, including the possible decision to stop experimenting, is within the scope of sequential analysis, a field that was pioneered[12] by Abraham Wald in the context of sequential tests of statistical hypotheses.[13] Herman Chernoff wrote an overview of optimal sequential designs,[14] while adaptive designs have been surveyed by S. Zacks.[15] One specific type of sequential design is the "two-armed bandit", generalized to the multi-armed bandit, on which early work was done by Herbert Robbins in 1952.[16]
Fisher's principles
A methodology for designing experiments was proposed by Ronald Fisher, in his innovative books: The Arrangement of Field Experiments (1926) and The Design of Experiments (1935). Much of his pioneering work dealt with agricultural applications of statistical methods. As a mundane example, he described how to test the lady tasting tea hypothesis, that a certain lady could distinguish by flavour alone whether the milk or the tea was first placed in the cup. These methods have been broadly adapted in biological, psychological, and agricultural research.[17]
Comparison
In some fields of study it is not possible to have independent measurements to a traceable metrology standard. Comparisons between treatments are much more valuable and are usually preferable, and often compared against a scientific control or traditional treatment that acts as baseline.
Randomization
Random assignment is the process of assigning individuals at random to groups or to different groups in an experiment, so that each individual of the population has the same chance of becoming a participant in the study. The random assignment of individuals to groups (or conditions within a group) distinguishes a rigorous, "true" experiment from an observational study or "quasi-experiment".[18] There is an extensive body of mathematical theory that explores the consequences of making the allocation of units to treatments by means of some random mechanism (such as tables of random numbers, or the use of randomization devices such as playing cards or dice). Assigning units to treatments at random tends to mitigate confounding, which makes effects due to factors other than the treatment to appear to result from the treatment.
The risks associated with random allocation (such as having a serious imbalance in a key characteristic between a treatment group and a control group) are calculable and hence can be managed down to an acceptable level by using enough experimental units. However, if the population is divided into several subpopulations that somehow differ, and the research requires each subpopulation to be equal in size, stratified sampling can be used. In that way, the units in each subpopulation are randomized, but not the whole sample. The results of an experiment can be generalized reliably from the experimental units to a larger statistical population of units only if the experimental units are a random sample from the larger population; the probable error of such an extrapolation depends on the sample size, among other things.
Statistical replication
Measurements are usually subject to variation and measurement uncertainty; thus they are repeated and full experiments are replicated to help identify the sources of variation, to better estimate the true effects of treatments, to further strengthen the experiment's reliability and validity, and to add to the existing knowledge of the topic.[19] However, certain conditions must be met before the replication of the experiment is commenced: the original research question has been published in a peer-reviewed journal or widely cited, the researcher is independent of the original experiment, the researcher must first try to replicate the original findings using the original data, and the write-up should state that the study conducted is a replication study that tried to follow the original study as strictly as possible.[20]
Blocking
Blocking is the non-random arrangement of experimental units into groups (blocks) consisting of units that are similar to one another. Blocking reduces known but irrelevant sources of variation between units and thus allows greater precision in the estimation of the source of variation under study.
Orthogonality
Orthogonality concerns the forms of comparison (contrasts) that can be legitimately and efficiently carried out. Contrasts can be represented by vectors and sets of orthogonal contrasts are uncorrelated and independently distributed if the data are normal. Because of this independence, each orthogonal treatment provides different information to the others. If there are T treatments and T – 1 orthogonal contrasts, all the information that can be captured from the experiment is obtainable from the set of contrasts.
Multifactorial experiments
Use of multifactorial experiments instead of the one-factor-at-a-time method. These are efficient at evaluating the effects and possible interactions of several factors (independent variables). Analysis of experiment design is built on the foundation of the analysis of variance, a collection of models that partition the observed variance into components, according to what factors the experiment must estimate or test.
Example
This example of design experiments is attributed to Harold Hotelling, building on examples from Frank Yates.[21][22][14] The experiments designed in this example involve combinatorial designs.[23]
Weights of eight objects are measured using a pan balance and set of standard weights. Each weighing measures the weight difference between objects in the left pan and any objects in the right pan by adding calibrated weights to the lighter pan until the balance is in equilibrium. Each measurement has a random error. The average error is zero; the standard deviations of the probability distribution of the errors is the same number σ on different weighings; errors on different weighings are independent. Denote the true weights by
$\theta _{1},\dots ,\theta _{8}.\,$
We consider two different experiments:
1. Weigh each object in one pan, with the other pan empty. Let Xi be the measured weight of the object, for i = 1, ..., 8.
2. Do the eight weighings according to the following schedule—a weighing matrix:
${\begin{array}{lcc}&{\text{left pan}}&{\text{right pan}}\\\hline {\text{1st weighing:}}&1\ 2\ 3\ 4\ 5\ 6\ 7\ 8&{\text{(empty)}}\\{\text{2nd:}}&1\ 2\ 3\ 8\ &4\ 5\ 6\ 7\\{\text{3rd:}}&1\ 4\ 5\ 8\ &2\ 3\ 6\ 7\\{\text{4th:}}&1\ 6\ 7\ 8\ &2\ 3\ 4\ 5\\{\text{5th:}}&2\ 4\ 6\ 8\ &1\ 3\ 5\ 7\\{\text{6th:}}&2\ 5\ 7\ 8\ &1\ 3\ 4\ 6\\{\text{7th:}}&3\ 4\ 7\ 8\ &1\ 2\ 5\ 6\\{\text{8th:}}&3\ 5\ 6\ 8\ &1\ 2\ 4\ 7\end{array}}$
Let Yi be the measured difference for i = 1, ..., 8. Then the estimated value of the weight θ1 is
${\widehat {\theta }}_{1}={\frac {Y_{1}+Y_{2}+Y_{3}+Y_{4}-Y_{5}-Y_{6}-Y_{7}-Y_{8}}{8}}.$
Similar estimates can be found for the weights of the other items:
${\begin{aligned}{\widehat {\theta }}_{2}&={\frac {Y_{1}+Y_{2}-Y_{3}-Y_{4}+Y_{5}+Y_{6}-Y_{7}-Y_{8}}{8}}.\\[5pt]{\widehat {\theta }}_{3}&={\frac {Y_{1}+Y_{2}-Y_{3}-Y_{4}-Y_{5}-Y_{6}+Y_{7}+Y_{8}}{8}}.\\[5pt]{\widehat {\theta }}_{4}&={\frac {Y_{1}-Y_{2}+Y_{3}-Y_{4}+Y_{5}-Y_{6}+Y_{7}-Y_{8}}{8}}.\\[5pt]{\widehat {\theta }}_{5}&={\frac {Y_{1}-Y_{2}+Y_{3}-Y_{4}-Y_{5}+Y_{6}-Y_{7}+Y_{8}}{8}}.\\[5pt]{\widehat {\theta }}_{6}&={\frac {Y_{1}-Y_{2}-Y_{3}+Y_{4}+Y_{5}-Y_{6}-Y_{7}+Y_{8}}{8}}.\\[5pt]{\widehat {\theta }}_{7}&={\frac {Y_{1}-Y_{2}-Y_{3}+Y_{4}-Y_{5}+Y_{6}+Y_{7}-Y_{8}}{8}}.\\[5pt]{\widehat {\theta }}_{8}&={\frac {Y_{1}+Y_{2}+Y_{3}+Y_{4}+Y_{5}+Y_{6}+Y_{7}+Y_{8}}{8}}.\end{aligned}}$
The question of design of experiments is: which experiment is better?
The variance of the estimate X1 of θ1 is σ2 if we use the first experiment. But if we use the second experiment, the variance of the estimate given above is σ2/8. Thus the second experiment gives us 8 times as much precision for the estimate of a single item, and estimates all items simultaneously, with the same precision. What the second experiment achieves with eight would require 64 weighings if the items are weighed separately. However, note that the estimates for the items obtained in the second experiment have errors that correlate with each other.
Many problems of the design of experiments involve combinatorial designs, as in this example and others.[23]
Avoiding false positives
False positive conclusions, often resulting from the pressure to publish or the author's own confirmation bias, are an inherent hazard in many fields. A good way to prevent biases potentially leading to false positives in the data collection phase is to use a double-blind design. When a double-blind design is used, participants are randomly assigned to experimental groups but the researcher is unaware of what participants belong to which group. Therefore, the researcher can not affect the participants' response to the intervention. Experimental designs with undisclosed degrees of freedom are a problem.[24] This can lead to conscious or unconscious "p-hacking": trying multiple things until you get the desired result. It typically involves the manipulation – perhaps unconsciously – of the process of statistical analysis and the degrees of freedom until they return a figure below the p<.05 level of statistical significance.[25][26] So the design of the experiment should include a clear statement proposing the analyses to be undertaken. P-hacking can be prevented by preregistering researches, in which researchers have to send their data analysis plan to the journal they wish to publish their paper in before they even start their data collection, so no data manipulation is possible (https://osf.io). Another way to prevent this is taking the double-blind design to the data-analysis phase, where the data are sent to a data-analyst unrelated to the research who scrambles up the data so there is no way to know which participants belong to before they are potentially taken away as outliers.
Clear and complete documentation of the experimental methodology is also important in order to support replication of results.[27]
Discussion topics when setting up an experimental design
An experimental design or randomized clinical trial requires careful consideration of several factors before actually doing the experiment.[28] An experimental design is the laying out of a detailed experimental plan in advance of doing the experiment. Some of the following topics have already been discussed in the principles of experimental design section:
1. How many factors does the design have, and are the levels of these factors fixed or random?
2. Are control conditions needed, and what should they be?
3. Manipulation checks: did the manipulation really work?
4. What are the background variables?
5. What is the sample size? How many units must be collected for the experiment to be generalisable and have enough power?
6. What is the relevance of interactions between factors?
7. What is the influence of delayed effects of substantive factors on outcomes?
8. How do response shifts affect self-report measures?
9. How feasible is repeated administration of the same measurement instruments to the same units at different occasions, with a post-test and follow-up tests?
10. What about using a proxy pretest?
11. Are there lurking variables?
12. Should the client/patient, researcher or even the analyst of the data be blind to conditions?
13. What is the feasibility of subsequent application of different conditions to the same units?
14. How many of each control and noise factors should be taken into account?
The independent variable of a study often has many levels or different groups. In a true experiment, researchers can have an experimental group, which is where their intervention testing the hypothesis is implemented, and a control group, which has all the same element as the experimental group, without the interventional element. Thus, when everything else except for one intervention is held constant, researchers can certify with some certainty that this one element is what caused the observed change. In some instances, having a control group is not ethical. This is sometimes solved using two different experimental groups. In some cases, independent variables cannot be manipulated, for example when testing the difference between two groups who have a different disease, or testing the difference between genders (obviously variables that would be hard or unethical to assign participants to). In these cases, a quasi-experimental design may be used.
Causal attributions
In the pure experimental design, the independent (predictor) variable is manipulated by the researcher – that is – every participant of the research is chosen randomly from the population, and each participant chosen is assigned randomly to conditions of the independent variable. Only when this is done is it possible to certify with high probability that the reason for the differences in the outcome variables are caused by the different conditions. Therefore, researchers should choose the experimental design over other design types whenever possible. However, the nature of the independent variable does not always allow for manipulation. In those cases, researchers must be aware of not certifying about causal attribution when their design doesn't allow for it. For example, in observational designs, participants are not assigned randomly to conditions, and so if there are differences found in outcome variables between conditions, it is likely that there is something other than the differences between the conditions that causes the differences in outcomes, that is – a third variable. The same goes for studies with correlational design. (Adér & Mellenbergh, 2008).
Statistical control
It is best that a process be in reasonable statistical control prior to conducting designed experiments. When this is not possible, proper blocking, replication, and randomization allow for the careful conduct of designed experiments.[29] To control for nuisance variables, researchers institute control checks as additional measures. Investigators should ensure that uncontrolled influences (e.g., source credibility perception) do not skew the findings of the study. A manipulation check is one example of a control check. Manipulation checks allow investigators to isolate the chief variables to strengthen support that these variables are operating as planned.
One of the most important requirements of experimental research designs is the necessity of eliminating the effects of spurious, intervening, and antecedent variables. In the most basic model, cause (X) leads to effect (Y). But there could be a third variable (Z) that influences (Y), and X might not be the true cause at all. Z is said to be a spurious variable and must be controlled for. The same is true for intervening variables (a variable in between the supposed cause (X) and the effect (Y)), and anteceding variables (a variable prior to the supposed cause (X) that is the true cause). When a third variable is involved and has not been controlled for, the relation is said to be a zero order relationship. In most practical applications of experimental research designs there are several causes (X1, X2, X3). In most designs, only one of these causes is manipulated at a time.
Experimental designs after Fisher
Some efficient designs for estimating several main effects were found independently and in near succession by Raj Chandra Bose and K. Kishen in 1940 at the Indian Statistical Institute, but remained little known until the Plackett–Burman designs were published in Biometrika in 1946. About the same time, C. R. Rao introduced the concepts of orthogonal arrays as experimental designs. This concept played a central role in the development of Taguchi methods by Genichi Taguchi, which took place during his visit to Indian Statistical Institute in early 1950s. His methods were successfully applied and adopted by Japanese and Indian industries and subsequently were also embraced by US industry albeit with some reservations.
In 1950, Gertrude Mary Cox and William Gemmell Cochran published the book Experimental Designs, which became the major reference work on the design of experiments for statisticians for years afterwards.
Developments of the theory of linear models have encompassed and surpassed the cases that concerned early writers. Today, the theory rests on advanced topics in linear algebra, algebra and combinatorics.
As with other branches of statistics, experimental design is pursued using both frequentist and Bayesian approaches: In evaluating statistical procedures like experimental designs, frequentist statistics studies the sampling distribution while Bayesian statistics updates a probability distribution on the parameter space.
Some important contributors to the field of experimental designs are C. S. Peirce, R. A. Fisher, F. Yates, R. C. Bose, A. C. Atkinson, R. A. Bailey, D. R. Cox, G. E. P. Box, W. G. Cochran, W. T. Federer, V. V. Fedorov, A. S. Hedayat, J. Kiefer, O. Kempthorne, J. A. Nelder, Andrej Pázman, Friedrich Pukelsheim, D. Raghavarao, C. R. Rao, Shrikhande S. S., J. N. Srivastava, William J. Studden, G. Taguchi and H. P. Wynn.[30]
The textbooks of D. Montgomery, R. Myers, and G. Box/W. Hunter/J.S. Hunter have reached generations of students and practitioners. [31] [32] [33] [34] [35]
Some discussion of experimental design in the context of system identification (model building for static or dynamic models) is given in[36] and.[37]
Human participant constraints
Laws and ethical considerations preclude some carefully designed experiments with human subjects. Legal constraints are dependent on jurisdiction. Constraints may involve institutional review boards, informed consent and confidentiality affecting both clinical (medical) trials and behavioral and social science experiments.[38] In the field of toxicology, for example, experimentation is performed on laboratory animals with the goal of defining safe exposure limits for humans.[39] Balancing the constraints are views from the medical field.[40] Regarding the randomization of patients, "... if no one knows which therapy is better, there is no ethical imperative to use one therapy or another." (p 380) Regarding experimental design, "...it is clearly not ethical to place subjects at risk to collect data in a poorly designed study when this situation can be easily avoided...". (p 393)
See also
• Adversarial collaboration
• Bayesian experimental design
• Block design
• Box–Behnken design
• Central composite design
• Clinical trial
• Clinical study design
• Computer experiment
• Control variable
• Controlling for a variable
• Experimetrics (econometrics-related experiments)
• Factor analysis
• Fractional factorial design
• Glossary of experimental design
• Grey box model
• Industrial engineering
• Instrument effect
• Law of large numbers
• Manipulation checks
• Multifactor design of experiments software
• One-factor-at-a-time method
• Optimal design
• Plackett–Burman design
• Probabilistic design
• Protocol (natural sciences)
• Quasi-experimental design
• Randomized block design
• Randomized controlled trial
• Research design
• Robust parameter design
• Sample size determination
• Supersaturated design
• Royal Commission on Animal Magnetism
• Survey sampling
• System identification
• Taguchi methods
References
1. "The Sequential Nature of Classical Design of Experiments | Prism". prismtc.co.uk. Retrieved 10 March 2023.
2. Peirce, Charles Sanders (1887). "Illustrations of the Logic of Science". Open Court (10 June 2014). ISBN 0812698495.
3. Peirce, Charles Sanders (1883). "A Theory of Probable Inference". In C. S. Peirce (Ed.), Studies in logic by members of the Johns Hopkins University (p. 126–181). Little, Brown and Co (1883)
4. Stigler, Stephen M. (1978). "Mathematical statistics in the early States". Annals of Statistics. 6 (2): 239–65 [248]. doi:10.1214/aos/1176344123. JSTOR 2958876. MR 0483118. Indeed, Pierce's work contains one of the earliest explicit endorsements of mathematical randomization as a basis for inference of which I am aware (Peirce, 1957, pages 216–219
5. Peirce, Charles Sanders; Jastrow, Joseph (1885). "On Small Differences in Sensation". Memoirs of the National Academy of Sciences. 3: 73–83.
6. of Hacking, Ian (September 1988). "Telepathy: Origins of Randomization in Experimental Design". Isis. 79 (3): 427–451. doi:10.1086/354775. JSTOR 234674. MR 1013489. S2CID 52201011.
7. Stephen M. Stigler (November 1992). "A Historical View of Statistical Concepts in Psychology and Educational Research". American Journal of Education. 101 (1): 60–70. doi:10.1086/444032. JSTOR 1085417. S2CID 143685203.
8. Trudy Dehue (December 1997). "Deception, Efficiency, and Random Groups: Psychology and the Gradual Origination of the Random Group Design". Isis. 88 (4): 653–673. doi:10.1086/383850. PMID 9519574. S2CID 23526321.
9. Peirce, C. S. (1876). "Note on the Theory of the Economy of Research". Coast Survey Report: 197–201., actually published 1879, NOAA PDF Eprint Archived 2 March 2017 at the Wayback Machine.
Reprinted in Collected Papers 7, paragraphs 139–157, also in Writings 4, pp. 72–78, and in Peirce, C. S. (July–August 1967). "Note on the Theory of the Economy of Research". Operations Research. 15 (4): 643–648. doi:10.1287/opre.15.4.643. JSTOR 168276.
10. Guttorp, P.; Lindgren, G. (2009). "Karl Pearson and the Scandinavian school of statistics". International Statistical Review. 77: 64. CiteSeerX 10.1.1.368.8328. doi:10.1111/j.1751-5823.2009.00069.x. S2CID 121294724.
11. Smith, Kirstine (1918). "On the standard deviations of adjusted and interpolated values of an observed polynomial function and its constants and the guidance they give towards a proper choice of the distribution of observations". Biometrika. 12 (1–2): 1–85. doi:10.1093/biomet/12.1-2.1.
12. Johnson, N.L. (1961). "Sequential analysis: a survey." Journal of the Royal Statistical Society, Series A. Vol. 124 (3), 372–411. (pages 375–376)
13. Wald, A. (1945) "Sequential Tests of Statistical Hypotheses", Annals of Mathematical Statistics, 16 (2), 117–186.
14. Herman Chernoff, Sequential Analysis and Optimal Design, SIAM Monograph, 1972.
15. Zacks, S. (1996) "Adaptive Designs for Parametric Models". In: Ghosh, S. and Rao, C. R., (Eds) (1996). "Design and Analysis of Experiments," Handbook of Statistics, Volume 13. North-Holland. ISBN 0-444-82061-2. (pages 151–180)
16. Robbins, H. (1952). "Some Aspects of the Sequential Design of Experiments". Bulletin of the American Mathematical Society. 58 (5): 527–535. doi:10.1090/S0002-9904-1952-09620-8.
17. Miller, Geoffrey (2000). The Mating Mind: how sexual choice shaped the evolution of human nature, London: Heineman, ISBN 0-434-00741-2 (also Doubleday, ISBN 0-385-49516-1) "To biologists, he was an architect of the 'modern synthesis' that used mathematical models to integrate Mendelian genetics with Darwin's selection theories. To psychologists, Fisher was the inventor of various statistical tests that are still supposed to be used whenever possible in psychology journals. To farmers, Fisher was the founder of experimental agricultural research, saving millions from starvation through rational crop breeding programs." p.54.
18. Creswell, J.W. (2008), Educational research: Planning, conducting, and evaluating quantitative and qualitative research (3rd edition), Upper Saddle River, NJ: Prentice Hall. 2008, p. 300. ISBN 0-13-613550-1
19. Dr. Hani (2009). "Replication study". Archived from the original on 2 June 2012. Retrieved 27 October 2011.
20. Burman, Leonard E.; Robert W. Reed; James Alm (2010), "A call for replication studies", Public Finance Review, 38 (6): 787–793, doi:10.1177/1091142110385210, S2CID 27838472, retrieved 27 October 2011
21. Hotelling, Harold (1944). "Some Improvements in Weighing and Other Experimental Techniques". Annals of Mathematical Statistics. 15 (3): 297–306. doi:10.1214/aoms/1177731236.
22. Giri, Narayan C.; Das, M. N. (1979). Design and Analysis of Experiments. New York, N.Y: Wiley. pp. 350–359. ISBN 9780852269145.
23. Jack Sifri (8 December 2014). "How to Use Design of Experiments to Create Robust Designs With High Yield". youtube.com. Retrieved 11 February 2015.
24. Simmons, Joseph; Leif Nelson; Uri Simonsohn (November 2011). "False-Positive Psychology: Undisclosed Flexibility in Data Collection and Analysis Allows Presenting Anything as Significant". Psychological Science. 22 (11): 1359–1366. doi:10.1177/0956797611417632. ISSN 0956-7976. PMID 22006061.
25. "Science, Trust And Psychology in Crisis". KPLU. 2 June 2014. Archived from the original on 14 July 2014. Retrieved 12 June 2014.
26. "Why Statistically Significant Studies Can Be Insignificant". Pacific Standard. 4 June 2014. Retrieved 12 June 2014.
27. Chris Chambers (10 June 2014). "Physics envy: Do 'hard' sciences hold the solution to the replication crisis in psychology?". theguardian.com. Retrieved 12 June 2014.
28. Ader, Mellenberg & Hand (2008) "Advising on Research Methods: A consultant's companion"
29. Bisgaard, S (2008) "Must a Process be in Statistical Control before Conducting Designed Experiments?", Quality Engineering, ASQ, 20 (2), pp 143–176
30. Giri, Narayan C.; Das, M. N. (1979). Design and Analysis of Experiments. New York, N.Y: Wiley. pp. 53, 159, 264. ISBN 9780852269145.
31. Montgomery, Douglas (2013). Design and analysis of experiments (8th ed.). Hoboken, NJ: John Wiley & Sons, Inc. ISBN 9781118146927.
32. Walpole, Ronald E.; Myers, Raymond H.; Myers, Sharon L.; Ye, Keying (2007). Probability & statistics for engineers & scientists (8 ed.). Upper Saddle River, NJ: Pearson Prentice Hall. ISBN 978-0131877115.
33. Myers, Raymond H.; Montgomery, Douglas C.; Vining, G. Geoffrey; Robinson, Timothy J. (2010). Generalized linear models : with applications in engineering and the sciences (2 ed.). Hoboken, N.J.: Wiley. ISBN 978-0470454633.
34. Box, George E.P.; Hunter, William G.; Hunter, J. Stuart (1978). Statistics for Experimenters : An Introduction to Design, Data Analysis, and Model Building. New York: Wiley. ISBN 978-0-471-09315-2.
35. Box, George E.P.; Hunter, William G.; Hunter, J. Stuart (2005). Statistics for Experimenters : Design, Innovation, and Discovery (2 ed.). Hoboken, N.J.: Wiley. ISBN 978-0471718130.
36. Spall, J. C. (2010). "Factorial Design for Efficient Experimentation: Generating Informative Data for System Identification". IEEE Control Systems Magazine. 30 (5): 38–53. doi:10.1109/MCS.2010.937677. S2CID 45813198.
37. Pronzato, L (2008). "Optimal experimental design and some related control problems". Automatica. 44 (2): 303–325. arXiv:0802.4381. doi:10.1016/j.automatica.2007.05.016. S2CID 1268930.
38. Moore, David S.; Notz, William I. (2006). Statistics : concepts and controversies (6th ed.). New York: W.H. Freeman. pp. Chapter 7: Data ethics. ISBN 9780716786368.
39. Ottoboni, M. Alice (1991). The dose makes the poison : a plain-language guide to toxicology (2nd ed.). New York, N.Y: Van Nostrand Reinhold. ISBN 978-0442006600.
40. Glantz, Stanton A. (1992). Primer of biostatistics (3rd ed.). ISBN 978-0-07-023511-3.
Sources
• Peirce, C. S. (1877–1878), "Illustrations of the Logic of Science" (series), Popular Science Monthly, vols. 12–13. Relevant individual papers:
• (1878 March), "The Doctrine of Chances", Popular Science Monthly, v. 12, March issue, pp. 604–615. Internet Archive Eprint.
• (1878 April), "The Probability of Induction", Popular Science Monthly, v. 12, pp. 705–718. Internet Archive Eprint.
• (1878 June), "The Order of Nature", Popular Science Monthly, v. 13, pp. 203–217.Internet Archive Eprint.
• (1878 August), "Deduction, Induction, and Hypothesis", Popular Science Monthly, v. 13, pp. 470–482. Internet Archive Eprint.
• (1883), "A Theory of Probable Inference", Studies in Logic, pp. 126–181, Little, Brown, and Company. (Reprinted 1983, John Benjamins Publishing Company, ISBN 90-272-3271-7)
External links
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• Detailed mathematical developments of most common DoE in the Opera Magistris v3.6 online reference Chapter 15, section 7.4, ISBN 978-2-8399-0932-7.
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| Wikipedia |
\begin{document}
\title{ Weak-star point of continuity property and Schauder bases }
\author{Gin{\'e}s L{\'o}pez-P{\'e}rez and Jos{\'e} A. Soler Arias} \address{Universidad de Granada, Facultad de Ciencias. Departamento de An\'{a}lisis Matem\'{a}tico, 18071-Granada (Spain)} \email{[email protected], [email protected]}
\thanks{Partially supported by MEC (Spain) Grant MTM2006-04837 and Junta de Andaluc\'{\i}a Grants FQM-185 and Proyecto de Excelencia P06-FQM-01438.} \subjclass{46B20, 46B22. Key words:
Point of continuity property, trees, boundedly complete sequences}
\maketitle \markboth{G. L\'{o}pez and Jos{\'e} A. Soler }{
Weak-star PCP and Schauder bases }
\begin{abstract}
We characterize the weak-star point of continuity property for subspaces of dual spaces with separable predual and we deduce that the weak-star point of continuity property is determined by subspaces with a Schauder basis in the natural setting of dual spaces of separable Banach spaces. As a consequence of the above characterization we get that a dual space satisfies the Radon-Nikodym property if, and only if, every seminormalized topologically weak-star null tree has a boundedly complete branch, which improves some results in \cite{DF} obtained for the separable case. Also, as a consequence of the above characterization, the following result obtained in \cite{R1} is deduced: {\it every seminormalized basic sequence in a Banach space with the point of continuity property has a boundedly complete subsequence.}
\end{abstract}
\section{Introduction} \par
We recall (see \cite{bou} for background) that a bounded subset $C$ of a Banach space satisfies the Radon-Nikodym property (RNP) if every subset of $C$ is dentable, that is, every subset of $C$ has slices of diameter arbitrarily small. A Banach space is said to verify the RNP whenever its closed unit ball satisfies the RNP. It is well known that separable dual spaces have RNP and spaces with RNP contain many subspaces which are themselves separable dual spaces. (Note that containing many separable dual subspaces is equivalent to containing many boundedly complete basic sequences). As RNP is separably determined, that is, a Banach space $X$ has RNP whenever every separable subspace of $X$ has RNP, it seems natural looking for a sequential characterization of RNP in terms of boundedly complete basic sequences. In \cite{DF} is proved that the space $B_{\infty}$ (which fails to have RNP) still has the property: any $w$-null normalized sequence has a boundedly complete basic subsequence. However, it has been proved in \cite{DF} that the dual space of a separable Banach space $X$ has RNP if, and only if, every weak-star null tree in the unit sphere of $X^*$ has some boundedly complete basic branch. It seems then natural looking for a characterization of RNP for general dual Banach spaces in terms of boundedly complete basic sequences, extending the result in \cite{DF} proved for dual of separable Banach spaces. For this, we introduce the concept of topologically weak-star null tree, which is a weaker condition than the weak-star null tree condition, and we characterize in terms of trees the RNP for weak-star compact subsets of general dual Banach spaces in proposition \ref{r1}. As a consequence, we get in theorem \ref{r2} that a dual Banach space $X$ has RNP if, and only if, every seminormalized and topologically weak-star null tree in the unit sphere of $X$ has some boundedly complete branch, which has as an immediate corollary the aforementioned result in \cite{DF}.
We recall that a closed and bounded subset of a Banach space $X$ satisfies the point of continuity property (PCP) if every closed subset of $C$ has some point of weak continuity, that is, the weak and the norm topologies agree at this point. Also, when $X$ is a dual space, $C$ is said to satisfy the weak-star point of continuity property ($w^*$-PCP) if every closed subset of $C$ has some point of weak-$*$ continuity, equivalently every nonempty subset of $C$ has relatively $w^*$-open subsets with diameter arbitrarily small. $X$ has PCP (resp. $w^*$-PCP when $X$ is a dual space) if $B_X$, the closed unit ball of $X$, has PCP (resp. $w^*$-PCP). Also, a subspace $X$ of a dual space $Y^*$ is said to verify the $w^*$-PCP if $B_X$, as a subset of $Y^*$, has the $w^*$-PCP. It is well known that RNP implies PCP, being false the converse, and it is clear that $w^*$-PCP implies PCP. Moreover, RNP and $w^*$-PCP are equivalent for convex $w^*$-compact sets in a dual space, see theorem 4.2.13 in \cite{bou}. We will use this last fact freely in the future. We refer to \cite{R2} for background about PCP and $w^*$-PCP. It is a well known open problem \cite{B} if PCP (resp. RNP) is determined by subspaces with a Schauder basis. Our goal is characterize $w^*$-PCP for closed and bounded subsets of dual spaces of separable Banach spaces and conclude in theorem \ref{fin} that, in fact, $w^*$-PCP is determined by subspaces with a Schauder basis in the natural setting of subspaces of dual spaces with a separable predual. As an easy consequence we also deduce from the above characterization of $w^*$-PCP that every seminormalized basic sequence in a Banach space with PCP has a boundedly complete basic subsequence. This last result was obtained in \cite{R1}.
We begin with some notation and preliminaries. Let $X$ be a Banach space and let $B_X$, respectively $S_X$, be the closed unit ball, respectively sphere, of $X$. The weak-star topology in $X$, when it is a dual space, will be denoted by $w^*$. If $A$ is a subset in $X$, $\overline{A}^{w^*}$ stands for the weak-star closure of $A$ in $X$. Given $\{e_n\}$ a basic sequence in $X$, $\{e_n\}$ is said to be {\it semi-normalized} if $0<\inf_n\Vert e_n\Vert\leq \sup_n\Vert e_n\Vert <\infty$ and the closed linear span of $\{e_n\}$ is denoted by $[e_n]$. $\{e_n\}$ is called {\it boundedly complete} provided whenever scalars $\{\lambda_i\}$ satisfy $\sup_n\Vert\sum_{i=1}^n\lambda_ie_i\Vert<\infty$, then $\sum_n\lambda_ne_n$ converges. $\{e_n\}$ is called {\it shrinking} if $[e_n]^*=[e_n^*]$, where $\{e_n^*\}$ denotes the sequence of biorthogonal functionals associated to $\{e_n\}$.
A boundedly complete basic sequence $\{e_n\}$ in a Banach space $X$ spans a dual space. In fact, $[e_n^*]^*=[e_n]$, where $\{e_n^*\}$ denotes the sequence of biorthogonal functionals in the dual space $X^*$ \cite {LZ}. Following the notation in \cite{S}, it is said that a sequence $\{e_n\}$ in a Banach space is {\it type P} if the set $\{\sum_{k=1}^ne_k:n\in {\mathbb N}\}$ is bounded. Observe, from the definitions, that type P seminormalized basic sequences fail to be always boundedly complete basic sequences.
A sequence $\{x_n\}$ in a Banach space $X$ is said to be {\it strongly summing} if whenever $\{\lambda_n\}$ is a sequence of scalars with $\sup_n\Vert\sum_{k=1}^n\lambda_kx_k\Vert<\infty$ one has that the series of scalars $\sum_n\lambda_n$ converges. The remarkable $c_0$-theorem \cite{R3} assures that every weak-Cauchy and not weakly convergent sequence in a Banach space not containing subspaces isomorphic to $c_0$ has a strongly summing basic subsequence.
${\mathbb N}^{<\omega}$ stands for the set of all ordered finite sequences of natural numbers joint to the empty sequence denoted by $\emptyset$. We consider the natural order in ${\mathbb N}^{<\omega}$, that is, given $\alpha=(\alpha_1,\ldots,\alpha_p),\ \beta=(\beta_1,\ldots,\beta_q)\in {\mathbb N}^{<\omega}$, one has $\alpha\leq \beta$ if $p\leq q$ and $\alpha_i=\beta_i$ $\forall 1\leq i\leq p$. If $\alpha=(\alpha_1,\ldots,\alpha_p)\in {\mathbb N}^{<\omega}$ we do $\alpha -=(\alpha_1,\ldots,\alpha_{p-1})$. Also $\vert \alpha\vert$ denotes the {\it length} of sequence $\alpha$, and $\emptyset$ is the minimum of ${\mathbb N}^{<\omega}$ with this partial order. A {\it tree} in a Banach space $X$ is a family $\{x_A\}_{A\in {\mathbb N}^{<\omega}}$ of vectors in $X$ indexed on ${\mathbb N}^{<\omega}$. The tree will be said {\it seminormalized} if $0<\inf_A\Vert x_A\Vert\leq \sup_A\Vert x_A\Vert <\infty$. We will say that the tree $\{x_A\}_{A\in {\mathbb N}^{<\omega}}$ is {\it $w^*$-null}, when $X$ is a dual space, if the sequence $\{x_{(A,n)}\}_n$ is $w^*$-null for every $A\in{\mathbb N}^{<\omega}$. The tree $\{x_A\}_{A\in {\mathbb N}^{<\omega}}$ is {\it topologically $w^*$-null} if $0\in \overline{\{x_{(A,n)}:n\in {\mathbb N}\}}^{w^*}$ for every $A\in{\mathbb N}^{<\omega}$. A sequence $\{x_{A_n}\}_{n\geq 0}$ is called a {\it branch} if $\{A_n\}$ is a maximal totally ordered subset of ${\mathbb N}^{<\omega}$, that is, there exists a sequence $\{\alpha_n\}$ of natural numbers such that $A_n=(\alpha_1,\ldots,\alpha_n)$ for every $n\in {\mathbb N}$ and $A_0=\emptyset$. Given a tree $\{x_A\}_{A\in{\mathbb N}^{<\omega}}$ in a Banach space, a {\it full subtree} is a new tree $\{y_A\}_{A\in{\mathbb N}^{<\omega}}$ defined by $y_{\emptyset}=x_{\emptyset}$ and $y_{(A,n)}=x_{(A,\sigma_A(n))}$ for every $A\in {\mathbb N}^{<\omega}$ and for every $n\in {\mathbb N}$, where for every $A\in {\mathbb N}^{<\omega}$, $\sigma_A$ is a strictly increasing map, equivalently when every branch of $\{y_A\}$ is also a branch of $\{x_A\}$. The tree $\{x_A\}_{A\in {\mathbb N}^{<\omega}}$ is said to be {\it uniformly type P} if every branch of the tree is type P and the partial sums of every branch are uniformly bounded. The tree $\{x_A\}_{A\in {\mathbb N}^{<\omega}}$ is said to be {\it basic} if the countable set $\{x_A:\ A\in {\mathbb N}^{\omega}\}$ is a basic sequence for some rearrangement.
Whenever $\{x_n\}$ is a sequence in a Banach space $X$, we will see this sequence also like a tree doing $x_A=x_{\max(A)}$ for every $A\in{\mathbb N}^{<\omega}$. Furthermore the branches of this tree are the subsequences of the sequence $\{x_n\}$.
Finally, we recall that a {\it boundedly complete skipped blocking finite dimensional decomposition} (BCSBFDD) in a separable Banach space $X$ is a sequence $\{F_j\}$ of finite dimensional subspaces in $X$ such that:\begin{enumerate} \item $X=[F_j:j\in {\mathbb N}]$.\item $F_k\cap[F_j:j\neq k]=\{0\}$ for every $k\in {\mathbb N}$.\item For every sequence $\{n_j\}$ of non-negative integers with $n_j+1<n_{j+1}$ for all $j\in {\mathbb N}$ and for every $f\in [F_{(n_j,n_{j+1})}:j\in {\mathbb N}]$ there exists a unique sequence $\{f_j\}$ with $f_j\in F_{(n_j,n_{j+1})}$ for all $j\in {\mathbb N}$ such that $f=\sum_{j=1}^{\infty}f_j$.\item Whenever $f_j\in F_{(n_j,n_{j+1})}$ for all $j\in {\mathbb N}$ and $\sup_n\Vert \sum_{j=1}^{n}f_j\Vert<\infty$ then $\sum_{j=1}^{\infty}f_j$ converges.\end{enumerate} If $X$ is a subspace of $Y^*$ for some $Y$, a BCSBFDD $\{F_j\}$ in $X$ will be called $w^*$-continuous if $F_i\cap \overline{[F_j:j\neq i]}^{w^*}= \{0\}$ for every $i$. Here, $[A]$ denotes the closed linear span in $X$ of the set $A$ and, for some nonempty interval of non-negative integers $I$, we denote the linear span of the $F_j'$s for $j\in I$ by $F_{I}$.
If $\{F_j\}$ is a BCSBFDD in a separable Banach space $X$ and $\{x_j\}$ is a sequence in $X$ such that $x_j\in F_{(n_j,n_{j+1})}$ for some sequence $\{n_j\}$ of non-negative integers with $n_j+1<n_{j+1}$ for all $j\in {\mathbb N}$, we say that $\{x_j\}$ is a {\it skipped block sequence} of $\{F_n\}$. It is standard to prove that there is a positive constant $K$ such that every skipped block sequence $\{x_j\}$ of $\{F_n\}$ with $x_j\neq 0$ for every $j$ is a boundedly complete basic sequence with constant at most $K$.
From \cite{GM}, we know that the family of separable Banach spaces with PCP is exactly the family of separable Banach spaces with a BCSBFDD.
\section{Main results} \par
We begin with a characterization of RNP for $w^*$-compact of general dual spaces. This result can be seen like a $w^*$-version of results in \cite{LS}.
\begin{proposition}\label{r1} Let $X$ be a Banach space and let $K$ be a weak-star compact and convex subset of $X^*$.
Then the following assertions are equivalent:\begin{enumerate} \item[i)] $K$ fails RNP. \item[ii)] There is a seminormalized topologically weak-star null tree $\{x_A\}_{A\in{\mathbb N}^{<\omega}}$ in $X^*$ such that $\{\sum_{C\leq A}x_C:A\in{\mathbb N}^{<\omega}\}\subset K$.\end{enumerate}\end{proposition}
\begin{proof} i)$\Rightarrow$ ii) Assume that $K$ fails RNP. Then, from theorem 2.3.6 in \cite{bou}
there is $D$ a non-dentable and countable subset of $K$. Now $\overline{co}^{w^*}(D)$ is a weak-star compact and weak-star separable subset of $K$ failing $w^*$-PCP. So there is $B$ a relatively weak-star separable subset of $\overline{co}^{w^*}(D)$ and $\delta>0$ such that every relatively weak-star open subset of $B$ has diameter greater than $2\delta$. So $b\in \overline{B\setminus B(b,\delta)}^{w^*}$ for every $b\in B$, where $B(b,\delta)$ stands for the open ball with center $b$ and radius $\delta$. Note then that, since $B$ is relatively weak-star separable, for every $b\in B$ there is a countable subset $C_b\in B\setminus B(b,\delta)$ such that $b\in \overline{C_b}^{w^*}$.
First, we construct a tree $\{y_A\}_{A\in {\mathbb N}^{<\omega}}$ in $B$ satisfying:\begin{enumerate}\item[a)]$y_{A}\in\overline{B\setminus B(y_A,\delta)}^{^*}w$ for every $A\in{\mathbb N}^{<\omega}$.\item[b)] $\Vert y_A-y_{(A,i)}\Vert>\delta$ for every $A\in{\mathbb N}^{<\omega}$.\item[c)] $y_A\in \overline{\{y_{(A,i)}:i\in{\mathbb N}\}}^{w^*}$ for every $A\in{\mathbb N}^{<\omega}$. \end{enumerate} Pick $y_{\emptyset}\in B $. As $y_{\emptyset}\in \overline{B\setminus B(y_{\emptyset},\delta)}^{w^*}$ then there is a countable set $C_{y_{\emptyset}}=\{y_(i):i\in {\mathbb N}\}\subset B\setminus B(y_{\empty},\delta)$ such that $y_{\emptyset}\in\overline{C_{y_{\empty}}}^{w^*}$. Then a), b) and c) are verified. By iterating this process we construct the tree $\{y_A\}_{A\in {\mathbb N}^{<\omega}}$ satisfying a), b) and c).
Now we define a new tree $\{x_A\}_{A\in{\mathbb N}^{<\omega}}$ by $x_{\emptyset}=y_{\emptyset}$ and $x_{(A,i)}=y_{(A,i)}-y_A$ for every $i\in {\mathbb N}$ and $A\in {\mathbb N}^{<\omega}$. From b) we get that $\{x_A\}_{A\in{\mathbb N}^{<\omega}}$ is a seminormalized tree, since $B$ is bounded. From c), we deduce that $\{x_A\}_{A\in{\mathbb N}^{<\omega}}$ is topologically weak-star null. Furthermore, if $A\in {\mathbb N}^{<\omega}$ then $\sum_{C\leq A}x_C=y_A$, from the definition of the tree $\{x_A\}_{A\in{\mathbb N}^{<\omega}}$. So $\{x_A\}_{A\in{\mathbb N}^{<\omega}}$ is a uniformly type P tree, since $B$ is bounded and $y_A\in B$ for every $A\in {\mathbb N}^{<\omega}$. This finishes the proof of i)$\Rightarrow$ii).
ii)$\Rightarrow$i) Let $\{x_A\}$ be a seminormalized topologically weak-star null tree such that $B=\{\sum_{C\leq A}x_C:A\in{\mathbb N}^{<\omega}\}\subset K$ and let $\delta>0$ such that $\Vert x_A\Vert>\delta$ for every $A\in {\mathbb N}^{<\omega}$. For every $A\in {\mathbb N}^{<\omega}$ and for every $n\in {\mathbb N}$ we have that $\sum_{C\leq (A,n)}x_C=\sum_{C\leq A}x_C+x_{(A,n)}$, but $0\in\overline{\{x_{(A,n)}N\in {\mathbb N}\}}^{w^*}$, since the tree $\{x_A\}$ is topologically weak-star null. So $\sum_{C\leq A}x_C\in\overline{\{\sum_{C\leq (A,n)}x_C:n\in{\mathbb N}\}}^{w^*}$ and $\Vert\sum_{C\leq (A,n)}x_C-\sum_{C\leq A}x_C\Vert>\delta$. This proves that $B$ has no points where the identity map is continuous from the weak-star to the norm topologies. In fact, we have proved that every relatively weak-star open subset of $B$ has diameter grater than $\delta$. Now, $\overline{B}^{\Vert\cdot\Vert}$ is a closed and bounded subset of $K$ such that every relatively weak-star open subset of $\overline{B}^{\Vert\cdot\Vert}$ has diameter grater than $\delta$, and so $K$ fails $w^*$-PCP. As $K$ is $w^*$-compact, then $K$ fails RNP.\end{proof}
Essentially, the fact that RNP is separably determined has allowed us to get the above result in the setting of general dual spaces. The next theorem characterizes the $w^*$-PCP for subsets of dual spaces with a separable predual in terms of $w^*$-null trees, since in this case the $w^*$ topology is metrizable on bounded sets. It seems natural then thinking that a characterization of $w^*$-PCP for subsets in general dual spaces in terms of topologically $w^*$-null trees has to be true, however we don't know if $w^*$-PCP is separable determined in general. This is the difference between the above proposition and the next one, which is obtained now easily.
\begin{proposition}\label{p1} Let $X$ be a separable Banach space and let $K$ be a closed and bounded subset of $X^*$.
Then the following assertions are equivalent:\begin{enumerate} \item[i)] $K$ fails $w^*$-PCP. \item[ii)] There is a seminormalized weak-star null tree $\{x_A\}_{A\in{\mathbb N}^{<\omega}}$ in $X^*$ such that $\{\sum_{C\leq A}x_C:A\in{\mathbb N}^{<\omega}\}\subset K$.\end{enumerate}\end{proposition}
\begin{proof} i)$\Rightarrow$ii) If $K$ fails $w^*$-PCP there is $B$ a subset of $K$ and $\delta>0$ such that every relatively weak-star open subset of $B$ has diameter greater than $2\delta$. So $b\in \overline{B\setminus B(b,\delta)}^{w^*}$ for every $b\in B$, where $B(b,\delta)$ stands for the open ball with center $b$ and radius $\delta$. Note then that, since $X$ is separable the $w^*$-topology in $X^*$ is metrizable on bounded sets, and so for every $b\in B$ there is a countable subset $C_b\in B\setminus B(b,\delta)$ such that $b\in \overline{C_b}^{w^*}$. Hence we can assume that $C_b$ is a sequence $w^*$ converging to $b$. Now we can construct, exactly like in the proof of i)$\Rightarrow$ii) of the above proposition, the desired $w^*$-null tree satisfying ii).
ii)$\Rightarrow$i) If one assumes ii) we can repeat the proof of
ii)$\Rightarrow$ i) in the above proposition to get that $K$ fails
$w^*$-PCP.\end{proof}
\begin{remark}\label{remar} If $X$ is a separable subspace of a dual space $Y^*$ with $X$ satisfying the $w^*$-PCP, it is shown in \cite{R2} (see (1) implies (8) of theorem 2.4 joint the comments in page 276) that there is a separable subspace $Z$ of $Y$ such that $X$ is isometric to a subspace of $Z^*$ and $X$ has $w^*$-PCP, as subspace of $Z^*$. Then, in order to study the $w^*$-PCP of a subspace of $Y^*$, it is more natural assume that $Y$ is separable.\end{remark}
We show now our characterization of $w^*$-PCP in terms of boundedly complete basic sequences in a general setting. A similar characterization for PCP can be seen in \cite{LS}, but the proof of the following result uses strongly the concept of $w^*$-continuous boundedly complete skipped blocking finite dimensional decomposition and assumes separability in the predual space.
\begin{theorem}\label{p2} Let $X$, $Y$ be Banach spaces with $Y$ separable and $X$ a subspace of $Y^*$.
Then the following assertions are equivalent:\begin{enumerate} \item[i)] $X$ has $w^*$-PCP. \item[ii)] Every weak-star null tree in $S_{X}$ is not uniformly type P. \item[iii)] Every weak-star null tree in $S_{X}$ has not type P branches. \item[iv)] Every weak-star null tree in $S_{X}$ has a boundedly complete branch.\end{enumerate}\end{theorem}
We need the following easy
\begin{lemma}\label{le} Let $X$, $Y$ be Banach spaces with $X$ a subspace of $Y^*$,
and let $M$ be a finite codimensional subspace of $X$. Assume that $\varepsilon>0$ and $\{x_n^*\}$ is a sequence in $X$ such that $0\in \overline{\{x_n:n\in {\mathbb N}\}}^{w^*}$. If $P:X\rightarrow N$ is a linear and relatively $w^*$-continuous projection onto some finite dimensional subspace $N$ of $X$ with kernel $M$ then there is $n_0\in {\mathbb N}$ such that $dist(x_{n_0}^*,M)<\varepsilon$ \end{lemma}
\begin{proof} From $0\in\overline{\{x_n^*:n\in {\mathbb N}\}}^{w^*}$ we deduce that $0\in\overline{\{P(x_n^*):n\in {\mathbb N}\}}^{\Vert \cdot\Vert}$, since $N$ is a finite dimensional subspace of $X$. Now, pick $n_0\in {\mathbb N} :\Vert P(x_{n_0}^*)\Vert <\varepsilon$. Then $$dist(x_{n_0}^*,M)=\Vert x_{n_0}^*+M\Vert=\Vert P(x_{n_0}^*)+M\Vert\leq \Vert P(x_{n_0}^*)\Vert <\varepsilon .$$ \end{proof}
{\it Proof of theorem} \ref{p2}. iv)$\Rightarrow$iii) is a consequence of the fact that every boundedly complete basic sequence is not type P, commented in the introduction and iii)$\Rightarrow$ii) is trivial.
For ii)$\Rightarrow$i) it is enough applying the theorem \ref{p1} for $K=B_{X}$ by assuming that $X$ fails $w^*$-PCP and normalizing.
i)$\Rightarrow$iv) Assume that $X$ has $w^*$-PCP and pick a weak-star null tree $\{x_A\}$ in $S_{X}$.
From \cite{R2} (see (b) of theorem 3.10 joint to the equivalence between (1) and (3) of corollary
2.6)
we know that every separable subspace of $Y^*$ with $w^*$-PCP has a $w^*$-continuous boundedly complete skipped blocking finite dimensional decomposition. As the subspace generated by the tree $\{x_A\}$ is separable we can assume that $X$ has it , that is,
there is a sequence $\{F_j\}$ of finite dimensional subspaces in $X$ such that:\begin{enumerate} \item $X=[F_j:j\in {\mathbb N}]$.\item $F_k\cap[F_j:j\neq k]=\{0\}$ for every $k\in {\mathbb N}$.\item For every sequence $\{n_j\}$ of non-negative integers with $n_j+1<n_{j+1}$ for all $j\in {\mathbb N}$ and for every $f\in [F_{(n_j,n_{j+1})}:j\in {\mathbb N}]$ there exists a unique sequence $\{f_j\}$ with $f_j\in F_{(n_j,n_{j+1})}$ for all $j\in {\mathbb N}$ such that $f=\sum_{j=1}^{\infty}f_j$.\item Whenever $f_j\in F_{(n_j,n_{j+1})}$ for all $j\in {\mathbb N}$ and $\sup_n\Vert \sum_{j=1}^{n}f_j\Vert<\infty$ then $\sum_{j=1}^{\infty}f_j$ converges.\item $F_i\cap \overline{[F_j:j\neq i]}^{w^*}= \{0\}$ for every $i$.\end{enumerate} Let $K$ be a positive constant
such that every skipped block sequence $\{x_j\}$ of $\{F_n\}$ with $x_j\neq 0$ for every $j$ is a boundedly complete basic sequence with constant at most $K$.
Observe that for every $n\in {\mathbb N}$ there is a linear onto projection\break $\widetilde{P_n}:\overline{[F_i:i\geq n]}^{w^*}\oplus[F_i:i< n]\rightarrow [F_i:i< n]$ with kernel $\overline{[F_i:i\geq n]}^{w^*}$ and so $\widetilde{P_n}$ is $w^*$ continuous, since $\overline{[F_i:i\geq n]}^{w^*}\oplus[F_i:i< n]$ is $w^*$-closed subspace of $Y^*$ and hence a dual Banach space and the closed graph theorem applies to $P_n$ because its kernel is $w^*$-closed and its range is finite-dimensional. Then the restriction of $\widetilde{P}_n$ to $X$, let us say $P_n$, is a linear and relatively $w^*$-continuous projection from $X$ onto $[F_i:i< n]$ with kernel $[F_i:i\geq n]$.
We have to construct a boundedly complete branch of the tree $\{x_A\}$. For this, fix a sequence $\{\varepsilon_j\}$ of positive real numbers with $\sum_{j=0}^{\infty}\varepsilon_j<1/2K$, where $K$ is the constant of the decomposition $\{F_j\}$. Now we
construct a sequence $\{f_j\}$ in $X$ with $f_j\in F_{(n_j,n_{j+1})}$ for all $j$, for some increasing sequence of integers numbers $\{n_j\}$ and a branch $\{x_{A_j}\}$ of the tree such that $\Vert x_{A_j}-f_j\Vert <\varepsilon_j$ for all $j$. Put $n_0=0$. Then there exists $n_1>2$ and $f_0\in F_{(n_0,n_1)}$ such that $\Vert x_{A_0}-f_0\Vert<\varepsilon_0$, where $A_0=\emptyset$. Now, assume that $n_1,\ldots, n_{j+1}$, $f_1,\ldots, f_j$ and $A_1,\ldots, A_j$ have been constructed. Put $A_k=(p_1,p_2,\ldots, p_k)$ for all $1\leq k\leq j$. As the tree is $w^*$ null we have that $0\in\overline{\{x_{(A_j,p)}:p\in{\mathbb N}\}}^{w^*}$. Then, by the lemma \ref{le}, we deduce that there is some $p_{j+1}\in {\mathbb N}$ such that $dist(x_{(A_j,p_{j+1})},[F_{[n_{j+1}+1,\infty)}])<\varepsilon_{j+1}$
since $[F_{[n_{j+1}+1,\infty)}]$ is a finite codimensional subspace in $X$ and $P_{n_{j+1}+1}$ is relatively $w^*$-continuous. Then there exist $n_{j+2}>n_{j+1}+1$ and $f_{j+1}\in F_{(n_{j+1},n_{j+2})}$ such that $\Vert x_{A_{j+1}}-f_{j+1}\Vert<\varepsilon_{j+1}$, where $A_{j+1}=(A_j,p_{j+1})$. This finishes the inductive construction of the branch $\{x_{A_j}\}$ satisfying that $\Vert x_{A_j}-f_j\Vert <\varepsilon_j$ for all $j$. Finally we get that $\sum_{j=1}^{\infty}\Vert x_{A_j}-f_j\Vert<1/2K$. Then $\{x_{A_j}\}$ is a branch of the tree $\{x_A\}_{A\in {\mathbb N}^{<\omega}}$ which is a basic sequence equivalent to $\{f_j\}$, being $\{f_j\}$ a skipped block sequence of $\{F_n\}$, hence $\{x_{A_j}\}$ is a boundedly complete sequence and the proof of theorem \ref{p2} is finished.
Now we can get a characterization of RNP for dual spaces, following the above proof.
\begin{theorem}\label{r2} Let $X$ be a Banach space. Then the following assertions are equivalent:\begin{enumerate}\item[i)] $X^*$ has
RNP. \item[ii)] Every topologically weak-star null tree in $S_{X}$ is not uniformly type P. \item[iii)] Every topologically weak-star null tree in $S_{X}$ has not type P branches. \item[iv)] Every topologically weak-star null tree in $S_{X}$ has a boundedly complete branch.\end{enumerate}\end{theorem}
\begin{proof} iv)$\Rightarrow$iii) is a consequence of the fact that every boundedly complete basic sequence is not type P, commented in the introduction and iii)$\Rightarrow$ii) is trivial.
For ii)$\Rightarrow$i) it is enough applying the theorem \ref{r1} for $K=B_{X^*}$ by assuming that $X^*$ fails RNP and normalizing.
i)$\Rightarrow$iv) Assume that $X^*$ has RNP and pick a topologically weak-star null tree $\{x_A\}$ in $S_{X^*}$. Call $Y$ the closed linear span of the tree $\{x_A\}$. Now $Y$ is a separable subspace of $X^*$ and then there is a separable subspace $Z$ of $X$ norming $Y$ so that $Y$ is isometric to a subspace of $Z^*$. As $X^*$ has RNP, we get that $Z^*$ has RNP. Hence $Y$ is a separable subspace of $Z^*$, being $Z$ a separable space, and $Z^*$ has $w^*$-PCP since $Z^*$ has RNP. Observe that the tree $\{x_A\}$ is now a topologically weak-star null tree in $S_Z^*$, so we can select a full weak-star null subtree of $\{y_A\}$, since $Z$ is separable and so the $w^*$ topology in $Z^*$ is metrizable for bounded sets. We apply the proof of i) $\Rightarrow$ iv) in the above result with $X=Y=Z^*$ to get a boundedly complete branch of $\{y_A\}$. As $\{y_A\}$ is a full subtree of $\{x_A\}$, the branches of $\{y_A\}$ are branches of $\{x_A\}$ and $\{x_A\}$ has a boundedly complete branch.\end{proof}
In the case $X$ is a separable Banach space, the above result can be written in a terms of weak-star null trees. Then we get as an immediate consequence in the following corollary a result obtained in \cite{DF} in a different way.
\begin{corollary} Let $X$ be a separable Banach space. Then $X^*$ is separable (equivalently, $X^*$ has RNP) if, and only if, every weak-star null tree in $S_{X^*}$ has a boundedly complete branch.\end{corollary}
\begin{proof} When $X$ is separable, the weak-star topology in $X^*$ is metrizable on bounded sets and so every topologically weak-star null tree in $S_{X^*}$ has a full subtree which is weak-star null. With this in mind, it is enough to apply the above theorem to conclude, since $X^*$ is separable if, and only if, $X^*$ has RNP, whenever $X$ is separable. \end{proof}
The following consequence, obtained in a different way in \cite{R1}, shows how many separable and dual subspaces contains every Banach space with PCP.
\begin{corollary} Let $X$ a Banach space with PCP. Then every seminormalized basic sequence in $X$ has a boundedly complete subsequence.\end{corollary}
\begin{proof} Pick $\{x_n\}$ a seminormalized basic sequence in $X$. Then either $\{x_n\}$ has a subsequence equivalent to the unit vector basis of $\ell_1$, and hence boundedly complete, or $\{x_n\}$ has a weakly Cauchy subsequence which we denotes again $\{x_n\}$.
In the case $\{x_n\}$ is weakly convergent we get that $\{x_n\}$ is weakly null, because it is a basic sequence. Now, $\{x_n\}$ is a seminormalized weakly null tree in $X$ and hence $\{x_n\}$ is a seminormalized weak-star null tree in $X^{**}$. As $[x_n]$ is a separable subspace of $X^{**}$ with $w^*$-PCP, from \ref{remar} there is a separable subspace $Z\subset X^*$ such that $[x_n]$ is an isometric subspace of $Z^*$ with $w^*$-PCP. Therefore $\{x_n\}$ is a seminormalized weak-star null tree in $[x_n]$, which is a subspace of $Z^*$ with $w^*$-PCP, being $Z$ separable. From theorem, we get a boundedly complete branch and so a boundedly complete subsequence.
If $\{x_n\}$ is not weakly convergent we can apply the $c_0$-theorem \cite{R3} to get a strongly summing subsequence, denoted again by $\{x_n\}$, since $X$ has PCP and so $X$ does not contain $c_0$. Let $x^{**}=w^*-lim_n\ x_n\in X^{**}$. Now $\{x_n-x^{**}\}$ is a weak-star null sequence in $X\oplus [x^{**}]>\subset X^{**}$. As $X$ has PCP, we get that $X$ has $w^*$-PCP as a subspace of $X^{**}$, then it is easy to see that $X\oplus [x^{**}]\subset X^{**}$ has $w^*$-PCP. Now $[x_n-x^{**}]$ is a separable subspace of $X^{**}$ with $w^*$-PCP and then, from \ref{remar} there is $Z$ a separable subspace of $X^*$ such that $[x_n-x^{**}]$ is an isometric subspace of $Z^*$ with the $w^*$-PCP, being $Z$ separable. From theorem, we get a boundedly complete branch and so a boundedly complete subsequence, denoted again by $\{x_n-x^{**}\}$. So we have that $\{x_n-x^{**}\}$ is boundedly complete and $\{x_n\}$ is strongly summing. Let us see that $\{x_n\}$ is boundedly complete. Indeed, if for some sequence of scalars $\{\lambda_n\}$ we have that $\sup_n\Vert\sum_{k=1}^n\lambda_n x_n\Vert<\infty$, then the series $\sum_n\lambda_n$ is convergent, since $\{x_n\}$ is strongly summing. Now it is clear that $\sup_n\Vert\sum_{k=1}^n\lambda_n (x_n-x^{**})\Vert<\infty$ and then $\sum_n\lambda_n x_n-x^{**}$ converges, since $\{x_n-x^{**}\}$ is boundedly complete. So $\sum_n\lambda_n x_n$ converges, since $\sum_n\lambda_n$ is convergent, and $\{x_n\}$ is boundedly complete.\end{proof}
The converse of the above result is false, even for Banach spaces not containing $\ell_1$ (see \cite{GL}).
Now we pass to show some consequences about the problem of the determination of $w^*$-PCP by subspaces with a basis. We begin by proving that every seminormalized weak-star null tree has a basic full $w^*$-null subtree. The same result is then true for the weak topology, by considering $X$ as a subspace of $X^{**}$. We don't know exact reference for this result, so we give a proof based on the Mazur proof of the known result that every seminormalized sequence in a dual Banach space such that $0$ belongs to its weak-star closure has a basic subsequence.
\begin{lemma}\label{lema3} Let $X$ be a Banach space and $\{x_A\}_{A\in{\mathbb N}^{<\omega}}$ a seminormalized $w^*$ null tree in $S_{X^*}$. Then for every $\varepsilon>0$ there is a full basic subtree still $w^*$ null with basic constant less than $1+\varepsilon$\end{lemma}
\begin{proof} Let $\phi:{\mathbb N}^{<\omega}\rightarrow {\mathbb N}\cup\{0\}$ be a fixed bijective map such that $\phi(\emptyset)=0$, $\phi(A)\leq\phi(B)$ whenever $A\leq B\in{\mathbb N}^{<\omega}$ and $\phi(A,i)\leq\phi(A,j)$ whenever $A\in{\mathbb N}^{<\omega}$ and $i\leq j$. Fix also $\varepsilon>0$ and a sequence of positive numbers $\{\varepsilon_n\}_{n\geq 0}\in (0,1)$ such that $\frac{1+\sum_{n=0}^{\infty}\varepsilon_n}{\prod_{n=0}^{\infty}(1-\varepsilon_n)} <1+\varepsilon$. Now we proceed by induction to construct the desired subtree $\{y_A\}_{A\in{\mathbb N}^{<\omega}}$, following the order given by $\phi$ to define for every $A\in{\mathbb N}^{<\omega}$ $y_A$ and get in this way the full condition. That is, we have to prove that for every $n\in{\mathbb N}\cup \{0\}$ we can construct $y_{\phi^{-1}(n)}$ such that $\{y_{A}\}_{A\in{\mathbb N}^{<\omega}}$ is a $w^*$ null full subtree satisfying that for every $n\in{\mathbb N}\cup\{0\}$ there is a finite set $\{f_1^n,\ldots,f_{k_n}^n\}\subset S_{X}$ such that\begin{enumerate}\item[i)] $\{f_1^n,\ldots,f_{k_n}^n\}$ is a $(1-\varepsilon_n)$-norming set for $Y_n=[y_{\phi^{-1}(0)},\ldots, y_{\phi^{-1}(n)}]$. \item[ii)] $\vert f_i^n(y_{\phi^{-1}(n+1)})\vert <\varepsilon_n$ for every $i$.\item[iii)] For every $A\in {\mathbb N}^{<\omega}$ there is an increasing map $\sigma_A:{\mathbb N}\rightarrow{\mathbb N}$ such that $y_{(A,i)}=x_{(A,\sigma_{A}(i))}$ for every $i$.\end{enumerate}
For $n=0$, we have $\phi^{-1}(0)=\emptyset$ and then we define $y_{\emptyset}=x_{\emptyset}$. Now take $f_1^0\in S_{X}$ $(1-\varepsilon_0)$-norming the subspace $Y_0=[y_{\emptyset}]$. As the tree $\{x_A\}_{A\in{\mathbb N}^{<\omega}}$ is $w^*$ null there is $p_0\in{\mathbb N}$ such that $\vert f_1^0(x_{(p)})\vert < \varepsilon_0$ for every $p\geq p_0$. Then we do $y_{\phi^{-1}(1)}=x_{(p_0)}$. As $\phi(A,i)\leq\phi(A,j)$ whenever $A\in{\mathbb N}^{<\omega}$ and $i\leq j$ we deduce that $\phi^{-1}(1)=(1)$ and define $\sigma_{\emptyset}(1)=p_0$ so that
$y_{(\emptyset,1)}=x_{(\emptyset ,\sigma_{\emptyset}(1))}$.
Assume $n\in{\mathbb N}$ and that we have already defined $y_{\phi^{-1}(0)},\ldots ,y_{\phi^{-1}(n-1)}$. Now $\phi^{-1}(n)-<\phi^{-1}(n)$, then $\phi(\phi^{-1}(n)-)<n$ and so $y_{\phi^{-1}(n)-}$ has been already constructed, by induction hypotheses. Put $\phi^{-1}(n)=(A,h)$ for some $h\in{\mathbb N}$, where $A=\phi^{-1}(n)-$. As $\phi(A,k)\leq \phi(A,h)$ for $k\leq h$ we have that $y_{(A,k)}$ has been constructed with $y_{(A,k)}=x_{(A,\sigma_{A}(k))}$ whenever $k< h$ and $\sigma_{A}(k)$ has been constructed strictly increasing for $k< h$. Put $Y_{n-1}=[y_{\phi^{-1}(0)},\ldots ,y_{\phi^{-1}(n-1)}]$ and pick $f_1^{n-1},\ldots ,f_{k_{n-1}}^{n-1}$ elements in $S_{Y_{n-1}}$ $(1-\varepsilon_{n-1})$-norming $Y_{n-1}$. As the tree $\{x_A\}_{A\in{\mathbb N}^{<\omega}}$ is $w^*$ null there is $p_{n-1}>\max_{k<h}\sigma_{A}(k)$ such that $ \vert f_i^{n-1}(x_{(A,p)})\vert <\varepsilon_{n-1},\ 1\leq i\leq k_{n-1}$ for every $p\geq p_{n-1}$. Then we do $y_{\phi^{-1}(n)}=x_{(A,p_{n-1})}$ and $\sigma_{A}(h)=p_{n-1}$. Then $\sigma_{A}(k)$ is constructed being strictly increasing for $k\leq h$ and $y_{\phi^{-1}(n)}=y_{(A,h)}=x_{(A,\sigma_A(h))}$. Now the construction of the subtree $\{y_A\}$ is complete satisfying i), ii) and iii). From the construction we get that $\{y_A\}$ is a full and $w^*$-null subtree.
Let us see that $\{y_{\phi^{-1}(n)}\}$ is a basic sequence in $X$. Put $z_n=y_{\phi^{-1}(n)}$, fix $p<q\in{\mathbb N}$ and compute $\Vert\sum_{i=1}^q\lambda_iz_i\Vert$, where $\{\lambda_i\}$ is a scalar sequence. Assume that $\Vert\sum_{i=1}^q\lambda_iz_i\Vert\leq 1$. From i) pick $j$ such that $\vert f_j^{q-1}(\sum_{i=1}^{q-1}\lambda_iz_i)\vert >(1-\varepsilon_{q-1})\Vert\sum_{i=1}^{q-1}\lambda_iz_i\Vert.$ Then we have from ii) $\vert f_j^{q-1}(z_q)\vert <\varepsilon_{q-1}$ and so
$$\Vert\sum_{i=1}^q\lambda_iz_i\Vert\geq \vert f_j^{q-1}(\sum_{i=1}^q\lambda_iz_i)\vert >(1-\varepsilon_{q-1})\Vert\sum_{i=1}^{q-1}\lambda_iz_i\Vert -\varepsilon_{q-1}.$$
By repeating this computation we get $$\Vert \sum_{i=1}^q\lambda_iz_i\Vert\geq (\prod_{i=p+1}^q(1-\varepsilon_{i-1}))\Vert \sum_{i=1}^p\lambda_iz_i\Vert -(\sum_{i=p+1}^q\varepsilon_{i-1}),$$ and so, $$\Vert\sum_{i=1}^p\lambda_iz_i\Vert\leq \frac{1+\sum_{i=p+1}^q\varepsilon_{i-1}}{\prod_{i=p+1}^q(1-\varepsilon_{i-1})}<1+\varepsilon$$ The last inequality proves that $\{z_n\}$ is a basic sequence in $X$ with basic constant less than $1+\varepsilon$ and the proof is complete.\end{proof}
We don¬t know if the above result is still true changing weak-star
null by topologically weak-star null.
The following result shows that $w^*$-PCP is determined by subspaces with a Schauder basis in the natural setting of dual spaces of separable Banach spaces.
\begin{corollary}\label{fin} Let $X$, $Y$ be Banach spaces such that $Y$ is separable and $X$ is a subspace of $Y^*$. Then $X$ has $w^*$-PCP if, and only if, every subspace of $X$ with a Schauder basis has $w^*$-PCP.\end{corollary}
\begin{proof} Assume that $X$ fails $w^*$-PCP. Then, from theorem \ref{p2}, there is a $w^*$-null tree in the unit sphere of $X$ without boundedly complete branches. Now, from lemma \ref{lema3}, we can extract a $w^*$-null full basic subtree. The subspace generated by this subtree $Z$ is a subspace of $X$ with a Schauder basis containing a $w^*$-null tree in $S_X$ without boundedly complete branches, from the full condition, so $Z$ fails $w^*$-PCP, from theorem \ref{p2}\end{proof}
As a consequence we get, for example, that a subspace of $\ell_{\infty}$, the space of bounded scalar sequences with the sup norm, failing the $w^*$-PCP ( or failing PCP) contains a further subspace with a Schauder basis failing the $w^*$-PCP.
If we do $X=Y^*$ in the above corollary one deduces the following
\begin{corollary} Let $X$ be a separable Banach space. Then $X^*$ has RNP if, and only if, every subspace of $X^*$ with a Schauder basis has $w^*$-PCP.\end{corollary}
\end{document} | arXiv |
Rhombus $ABCD$ is similar to rhombus $BFDE$. The area of rhombus $ABCD$ is 24, and $\angle BAD=60^\circ$. What is the area of rhombus $BFDE$?
[asy]
pair A,B,C,D,I,F;
A=(0,0);
B=(10,0);
C=(15,8.7);
D=(5,8.7);
I=(5,2.88);
F=(10,5.82);
draw(A--B--C--D--cycle,linewidth(0.7));
draw(D--I--B--F--cycle,linewidth(0.7));
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,NE);
label("$D$",D,NW);
label("$E$",I,W);
label("$F$",F,E);
[/asy]
Since $\angle BAD = 60^{\circ}$, isosceles $\triangle
BAD$ is also equilateral. As a consequence, $\triangle AEB$, $\triangle AED$, $\triangle BED$, $\triangle BFD$, $\triangle
BFC$, and $\triangle CFD$ are congruent. These six triangles have equal areas and their union forms rhombus $ABCD$, so each has area $24/6 = 4$. Rhombus $BFDE$ is the union of $\triangle BED$ and $\triangle BFD$, so its area is $\boxed{8}.$
[asy]
pair A,B,C,D,I,F;
A=(0,0);
B=(10,0);
C=(15,8.7);
D=(5,8.7);
I=(5,2.88);
F=(10,5.82);
draw(A--B--C--D--cycle,linewidth(0.7));
draw(D--I--B--F--cycle,linewidth(0.7));
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,NE);
label("$D$",D,NW);
label("$E$",I,NW);
label("$F$",F,E);
draw(A--I,linewidth(0.7));
draw(F--C,linewidth(0.7));
draw(D--B,linewidth(0.7));
[/asy] | Math Dataset |
Pacific Journal of Mathematics for Industry
Principle of ultrasonic tomography for concrete structures and non-destructive inspection of concrete cover for reinforcement
Noriyuki MITA1 &
Takashi TAKIGUCHI2
Pacific Journal of Mathematics for Industry volume 10, Article number: 6 (2018) Cite this article
The authors are trying to develop an ultrasonic CT technique for concrete structures. As the first step of its development, we study how the ultrasonic waves propagate in concrete structures and pose a problem to develop ultrasonic CT, applying which we give a non-destructive inspection technique for concrete cover by application of the idea of the ultrasonic CT. The authors claim that this study gives the foundation of the theory of ultrasonic CT for concrete structures.
There existing a number of non-destructive inspection techniques for concrete structures which give some rough sketches of the interior structure, few of them provides concrete and complete interior information of concrete structures. A few years ago, the X-ray computerized tomography (CT for short) for concrete structures being practicalized, it costs very expensive and the effect of the X-rays to human bodies and the environment cannot be ignored. Therefore it is not suitable to apply the X-ray CT for the daily maintenance of concrete structures. It is required to develop a safe, cheaply running and concretely interior-describing non-destructive inspection technique for concrete structures, to establish which is our main purpose. In view of the above argument, the authors are trying to develop an ultrasonic CT technique for concrete structures. In this paper, as the first step of its development, we study how the ultrasonic waves propagate in concrete structures and pose a problem to develop ultrasonic CT, applying which we propose a non-destructive inspection technique for concrete cover by application of the ultrasonic CT for concrete structures.
Of course, there existing a number of non-destructive inspection methods for concrete structures applying ultrasonic waves, confer [3], [10] and so on, for example, almost all of them utilize the echo technique. It is very helpful to apply echo technique when we can access the concrete structure from only its one side. Applying this technique, however, we only obtain rough sketches of the inclusion or the cavity which lies very close to the side we can access, which is far from concretely interior-describing non-destructive inspection by which we mean. We can find very few paper like [5] where the study is given in a similar method to ours. Even in the paper [5], its main result is to detect the combination defect in the masonry, which is very rough and strongly depends on a priori information of the structure, which is also far from the non-destructive inspection technique the authors are trying to develop.
This paper consists of the following sections.
§1.
Motivation of this research
Propagation of the ultrasonic waves in concrete structures
An inverse problem of the acoustic tomography
Application to non-destructive inspection of concrete cover for reinforcement
In this section, as the introduction of this paper, we introduce the outline of our paper. In the next section, we shall introduce our motivation for this research. We first introduce how we understand the concrete in this paper, based on which there arise many problems to solve. Among them, we shall claim that it is one of the most important problems to establish a concretely interior-describing non-destructive inspection method for concrete structures, not only by practical requirements but also from the viewpoint of our understanding of the concrete. In the third section, we study how the ultrasonic waves propagate in the cement paste and the mortar by the experiments, which gives the fundamentals of our theory. In Section 4, we shall pose an inverse problem for establishment of our ultrasonic CT for concrete structures, for which we shall apply the results of our experiments and their examination discussed in Section 3. The problem posed in this section is also interesting in view of pure mathematics, especially, in view of integral geometry. In the fifth section, we shall develop a non-destructive inspection technique for concrete cover in the reinforced concrete structures by an application of the idea of the ultrasonic tomography discussed in the third and fourth sections. In the final section, we shall summarize our conclusions and mention open problems to be solved for further development.
The authors are grateful to Professor Hisashi Yamasaki for his devoted help for our experiments.
In this section, we shall introduce our motivation for this research. Before introducing our motivation, the authors claim that
Claim 2.1.
The concrete materials are artificial megaliths.
Let us first discuss why the authors claim Claim 2.1. Take Valley Temple, Egypt (BC 2500?) and Parthenon, Athens (BC447-432), for example, which are made of megaliths. At that period around those areas, there were plenty of megaliths available, therefore they made Valley Temple and Parthenon of megaliths which are very suitable for edifices. On the other hand, let us turn to Colosseum, Rome (AD 70-80). Its bailey or external wall being made of megaliths, its interior structure is infilled with stones, bricks, sand and Roman cement as bonding material (ash, for example), which we take for the origin of the concrete. It may be because of the shortage of the megaliths in Rome about 2000 years ago. Note that the structure of Colosseum still exists more than 2000 years after its foundation. Hence we claim that the primitive concrete materials applied to the interior infill wall of Colosseum have been playing their important role as the substitutes for the megaliths very well for milleniums, which is one of the reasons why the authors claim Claim 2.1.
We shall introduce our motivation for this research in accordance with Claim 2.1. If we take concrete materials for substitutes for the megaliths, there are superior points to the megaliths' such as
(α) Concrete is easily made and shaped in any form because of its fluidity before it gets hard.
(β) The cost of concrete is very cheap.
However, there are demerits of the concrete such as
Concrete is not tough to tensile strength.
Concrete easily gets cracks in and on itself.
Confer [7] for these merits and demerits of concrete. For the general theory of concrete, confer [2, 9]. The demerit (a) yields necessity to reinforce the concrete. The lifespan of reinforced concrete (RC for short) is said to be about a half century, which is much shorter than that of megaliths'. We have to maintain the concrete very well for a long time for the concrete materials to be substitutes for the megaliths. In view of this point, as well as from the viewpoint of the demerit (b), it is necessary to check the concrete structures by some non-destructive way and repair them effectively for their maintenance. Therefore, the following problem is very important.
Problem 2.1.
In order for concrete materials to be artificial megaliths, it is indispensable to reinforce them and to study how to maintain RC material for a long time.
There is another approach than Problem 2.1, for concrete materials to be artificial megaliths, that is, we should study a method to make concrete structures, without reinforcement, so solid that they can be artificial megaliths, the study of which is under investigation by the authors. Anyway, in view of Problem 2.1, there arise the following problems.
From the viewpoint of Problem 2.1, we pose the following two problems.
How to establish an effective, safe and cheaply-running non-destructive inspection technique for RC structures, which detects the interior structure concretely.
How to repair the RC structures in order that they would safely live for a very long time
By Problem 2.2, we mean that it is very important to establish a good method to check RC structures without destructing them and to develop a nice method to maintain the RC structures in order that their lifespan would be very long, for RC materials to play a role as substitutes of megaliths. The problem (1) in Problem 2.2 has a strong connection with our discussion in the fifth section.
In this section, as a preparation for our main purpose, we study how the ultrasonic primary waves propagate in the concrete structures. We first introduce our experiments to study how the ultrasonic waves propagate in the cement paste and the mortar. By examining the results of our experiments, we shall study the propagation of the ultrasonic waves in concrete structures of the length about 1m or less.
Let us introduce the outline of our experiments.
Outline of our experiments
Ultrasonic waves;
The frequency of the ultrasonic waves is 54kHz.
Velocity of the ultrasonic wave is denoted by V(m/s).
Length of the test pieces; We prepared test pieces of the length 100,200,300,400,800 and 1200mm in order to check
the decay of the acoustic velocity
the propagation of the ultrasonic waves
Inclusions; We prepared two types of test pieces of the size 100mm×100mm×400mm.
Normal test pieces
Test pieces with styrofoam 100mm×50mm×(200or300mm) included in their inside (cf. Fig. 4 below)
These test pieces are made use of to study the propagation of the ultrasonic waves.
Number of the test pieces; We made three test pieces of each type mentioned above and have taken the average of the observed values of three test pieces in each experiment.
We made the test pieces of cement paste and mortar as shown in Tables 1 and Fig. 1.
Length of the test pieces and inspection points
Table 1 Mix proportion of the test pieces
Experiment 1.
We first experimented on the normal test pieces, in order to test whether there is decay of the acoustic velocity in accordance with the length of the test pieces. We projected the ultrasonic waves from the inspection points numbered ①,⋯,#x24ZZ; 5circle on one end square of the test pieces (see Fig. 1). We name them the source points. We received them at the same-numbered inspection points on the other end square. We name them as the observation points. We have measured the time for the ultrasonic wave to travel between the source and the observation points. The results of these experiments are summed up in Figs. 2 and 3.
Normal test pieces (age of a week)
Normal test pieces (age of 4 weeks)
Remark that the average of the data on the points ① and ② are treated as the data of the upper points, the average of the data on the points ③ and ④ are treated as the data of the lower points and the point ⑤ is denoted by the center point.
By review-examining the results of Experiment 1, we obtain the following properties.
Property 3.1.
By observing the acoustic velocity in the test pieces, we have rediscovered the well known basic property of concrete; the more time goes by, the harder the test pieces are, which is caused by the reaction of hydration of concrete.
We have also rediscovered another well known basic property, the gravity settling of cement and fine aggregate (sand), in terms of the acoustic velocity; the lower the inspection points are, the faster the acoustic velocity is, which is because of the fact that the lower the points are, the larger their density is, caused by the gravity settling of cement and fine aggregate.
We can conclude that for the test pieces of the length less than 1200mm, there is no decay of the acoustic velocity from the viewpoint of its first arriving time.
The last property is essentially important for our study.
For this experiment, we utilized the test pieces of the length 400mm(100mm×100mm×400mm) with and without the styrofoam of the length 200 or 300mm included in their inside (confer Fig. 4). We conducted the same experiments as Experiment 1, in order to study how the ultrasonic waves propagate in the concrete structures, whose results are review-examined in the following.
Test pieces with styrofoam
In Experiment 2, the (formal) velocity V, which is calculated by
$$ V= \frac {\text{length of the test piece} (meters) } { \text{arriving time} (seconds)}, $$
in the lower points is smaller than that of the upper points (confer Table 2 below), applying which we studied the propagation of the ultrasonic waves in the test pieces. We hypothesized that the propagation of the ultrasonic waves in the test pieces is as the following hypothesis which is also shown in Fig. 5.
Propagation of the ultrasonic waves
Table 2 Tables of modification of the velocity (age of 4 weeks)
Hypothesis 3.1.
Let \(\Omega \subset \mathbb {R}^{3}\) be a domain where the test pieces of the cement paste or the mortar locates, and f(x) be the propagation speed of the ultrasonic wave at the point x∈Ω. For α,β∈∂Ω, we denote by γα,β a route from α to β contained in Ω. The primary wave of the ultrasonic one which travels from the point α∈∂Ω to the point β∈∂Ω takes the route where the travel time is given by
$$ \min_{\gamma_{\alpha, \beta}} \int_{\gamma_{\alpha, \beta}} 1/f(x) d\gamma, $$
This route is called 'the fastest route'.
A similar claim to Hypothesis 3.1 being made by H. Yamamoto et al. [12], it is not exactly the same as Hypothesis 3.1. In [12], they tried to establish an ultrasonic CT, however, there has not been a satisfactorily practical ultrasonic CT technique for concrete structures yet. The authors claim that the verification of Hypothesis 3.1 is very important to establish a satisfactorily practical ultrasonic CT technique. In the next section we pose an inverse problem (Problem 4.1 below) by virtue of Hypothesis 3.1, which is an essential problem to establish a satisfactorily practical ultrasonic CT technique.
In view of Hypothesis 3.1, we have modified the length of the orbit along which the ultrasonic wave propagates, that is, the modified velocity V′ is given by
$$ V^{\prime}= \frac { 0.406 (meters)} { \text{arriving time} (seconds)} $$
for the lower points in the test pieces with styrofoam of the length 200mm and by
for the lower points in the test pieces with styrofoam of the length 300mm. Confer Fig. 5 for the image of these modifications. The results of Experiment 2 with the modification of the velocities are summarized in Table 2, Figs. 6 and 7. Remark that in Table 2, the fields of the modified velocity are left blank where the modification is not necessary.
Test pieces of cement paste with styrofoam (age of 4 weeks)
Test pieces of mortar with styrofoam (age of 4 weeks)
Let us review-examine the results of Experiment 2, that is, we shall study the modified acoustic velocities shown in Table 2, Figs. 6 and 7. The modified velocities at the lower points being not exactly the same as the test pieces without styrofoam, it is because, in the test pieces with styrofoam, gravity settling of cement and fine aggregate (sand) would not happen by the existence of the styrofoam. As another reason of this phenomenon, it is known that the air concentrate near the formwork in manufacturing process of concrete structures [2, 9]. In this experiment, the boundary surface between the cement paste (or the mortar) and the styrofoam plays the role of the formwork and the air concentrates there. Therefore, the primary ultrasonic wave takes a little longer way around than the route shown in Fig. 5, which results in that the modified velocities of the middle and the lower points are a little slower than the upper points. The exact evaluation how longer the fastest route is, is a very difficult problem.
In view of this review-examination, we conclude that Hypothesis 3.1 is right. Let us summarize the conclusions of Experiments 1 and 2.
Conclusion 3.1.
(Conclusion of Experiments 1 and 2)
The primary wave of the ultrasonic one takes the fastest route in the cement paste, the mortar and the concrete.
In the concrete structures of the length less than 1200mm, there is no decay of the speed of the ultrasonic waves with respect to the length.
Remarks 3.1.
It is our newer idea than the existing ones [6, 8] to focus on the first arrival time of the ultrasonic primary waves and to pose a problem for the development of the acoustic CT, which may yield a concretely interior-describing non-destructive inspection method. We shall discuss this problem in the next section. We also note that the ultrasonic CT is a part of the acoustic CT. In this paper, we are trying to establish a CT technique applicable to wider problems with other acoustic techniques than the ultrasonic one. As an example of such another acoustic CT technique, the authors are studying to develop a CT technique with electromagnetic acoustic pulses (confer Remark 3.2 below). We also note that the propagation of the ultrasonic primary wave mentioned in the first conclusion in Conclusion 3.1 is the same as ultrasound CT for medical application (cf. [1] for example).
Having introduced the results of our experiments on the data by application of ultrasonic primary waves, we have almost the same results on electromagnetic acoustic pulse primary waves, which enables us to treat electromagnetic acoustic pulses in the same way as ultrasonic waves for our study. We shall study how to treat electromagnetic acoustic pulses in our forthcoming paper.
In this section, we pose an inverse problem for the establishment of an ultrasonic tomographic technique for concrete structures. The research to develop an ultrasonic tomography was begun by J. F. Greenleaf at al. [4] and many researches have been published since then. Confer [12] for the references on such researches. For the time being, however, there has not been developed a satisfactorily practical ultrasonic CT technique yet. The authors claim, as in the previous section, that the verification of Hypothesis 3.1 is the first step to establish a satisfactorily practical ultrasonic CT technique, since it yields a key problem, Problem 4.1 below, for establishment of a satisfactorily practical ultrasonic CT technique.
As was studied in the previous section, we know that the the primary wave of the ultrasonic one takes the fastest route in the concrete structures of the length less than 1.2m and there is no decay in the velocity according to the length of the test pieces, which is what Conclusion 3.1 claims. In view these properties, we pose the following problem, which is the main problem in this paper.
(Problem to develop an acoustic CT for concrete structure) Let \(\Omega \subset \mathbb {R}^{3}\) be a domain and f(x) be the propagation speed of the ultrasonic wave at the point x∈Ω. For α,β∈∂Ω, we denote by γα,β a route from α to β contained in Ω. In this case, reconstruct f(x)(x∈Ω) out of the data
for ∀α,β∈∂Ω.
Let us explain what is meant by Problem 4.1. We first note that we take Ω for some concrete structure. By ∂Ω, we denote the boundary of Ω, that is, ∂Ω is the surface of the concrete structure Ω. For a fixed route γα,β, from α to β through Ω with α,β∈∂Ω, the integral \(\int _{\gamma _{\alpha, \beta }} 1/f(x) d\gamma \) represents the travel time of the ultrasonic wave when it propagates along γα,β. It is because that the travel time of the sound (of the ultrasonic wave) is proportional to the reciprocal of the propagation speed of the sound. By Conclusion 3.1, what we can obtain by observation is the travel time of the ultrasonic wave which propagates along the route where the travel time would be the shortest in Ω, which is what is meant in (5). If we can reconstruct f(x),(x∈Ω) out of the data (5), then we can concretely reconstruct the interior of the concrete structures in a non-destructive way. Let us summarize this argument as follows.
By Problem 4.1 we mean the problem "Reconstruct the acoustic velocity" f(x) at the all points x∈Ω out of the data of the acoustic arrival time between the all pairs of the points on the boundary. Solution to this problem enables us to reconstruct the interior structure concretely, that is, at each point of the interior of the concrete structure, we detect what is at the point, the cement paste, the fine aggregate, the coarse aggregate or the air, completely except the non-essential points mentioned in Remark 4.1 below. As mentioned above, it is very important and useful to develop such safe and cheaply-running non-destructive inspection method for concrete structures which enables us to detect their interior structures concretely, the authors claim that study of Problem 4.1 would bring a breakthrough for non-destructive inspection of concrete structures.
Study of Problem 4.1 is very important not only for solution of Problem 2.1 and problems to be introduced in the next section, but to establish an ultrasonic CT for general concrete structures including RC ones.
Let us give some remarks on Problem 4.1.
(Remarks on Problem 4.1)
In Problem 4.1, we give \(\min _{\gamma _{\alpha, \beta }} \int _{\gamma _{\alpha, \beta }} 1/f(x) d\gamma \) as the observed travel time of the primary wave between the two points α,β∈∂Ω, based on Conclusion 3.1. It being obvious that \(\inf _{\gamma _{\alpha, \beta }} \int _{\gamma _{\alpha, \beta }} 1/f(x) d\gamma \) exists, this infimum is attained as real observation of the travel time of the primary wave. Therefore the minimum, \(\min _{\gamma _{\alpha, \beta }} \int _{\gamma _{\alpha, \beta }} 1/f(x) d\gamma \), exists.
It is impossible to reconstruct the information of some points x's where f(x)'s are very small. For example, we cannot reconstruct the acoustic velocity of the styrofoam if it is included near the center of the test piece since no acoustic wave would go through it because of Conclusion 3.1. However, it does not matter very much. Since it is very important to determine the part where the acoustic velocity is so small that the ultrasonic wave would not go through it, where the exact velocity in such area would not matter. Remark that, in practice, it being very important to determine that the place where the acoustic velocity is relatively small, for example, to determine the place where the steel is corroded or there exists a cavity in the RC structures, it is not so important to determine the exact acoustic velocity itself in the place where it is relatively small.
It is an interesting problem to determine the optimal subset reconstructible by the acoustic CT established by the study of Problem 4.1.
Study of Problem 4.1 is also important in view of both pure and applied mathematics, especially from the viewpoint of integral geometry. Let us mention how important Problem 4.1 is in view of the research in pure and applied mathematics.
(Importance of Problem 4.1 in mathematics)
It is a very interesting problem to establish an reconstruction formula for Problem 4.1 in view of pure mathematics, especially, in integral geometry.
It is another interesting problem in Problem 4.1 to determine the subset of Ω where the reconstruction is impossible because it has no intersection with any γ giving (5). This problem is also interesting in view of integral geometry.
In practice, we have to study various incomplete data problems of Problem 4.1 by the restriction arisen from various reasons in practical application, which is interesting in view of pure mathematics, especially in view of integral geometry with incomplete data, which is also very important in applied mathematics.
Let us study a simple problem related to Problem 4.1. For simplicity, let us assume that there is a cavity of a disc whose center and radius are unknown in the two dimensional homogeneous object. The motivation to study this problem is as follows. Almost all standards of concrete materials are determined by the mortar part structure since it is the most important how to make the mortar part in the construction of the concrete materials. The authors claim that we can take the mortar as a homogeneous object since the particles of the cement and the fine gravel are so small that together with the water and the air they make a homogeneous material from macro viewpoint. It can be very basic and important problem to study a cavity of a disc in the mortar, which yields the following problem.
Theorem 4.1.
Let us study the 2-dimensional case. Assume that a homogeneous rectangle contains a cavity of a disk in its interior. In this case, the cavity is reconstructed by appropriate three data of the ultrasonic tomography.
In this theorem, by the term appropriate three data we mean the data which would determine the disc. For simplicity, we assume the acoustic velocity in the homogeneous rectangle is 1 and the cavity is of radius r and centered at (x0,y0). In Fig. 8 below, the length of the detour of the acoustic wave is given by
$$ \begin{aligned} &L_{i} = \left(\left(X_{i}^{1}-x_{i}^{1}\right)^{2} + \left(Y_{i}^{1}-y_{i}^{1}\right)^{2} + \right)^{\frac 12} \\&\qquad+ \left(\left(X_{i}^{2}-x_{i}^{2}\right)^{2} + \left(Y_{i}^{2}-y_{i}^{2}\right)^{2} \right)^{\frac 12} + r \theta_{i}, \end{aligned} $$
Propagation of ultrasonic waves
where i=1,2,3, j=1,2, \(\left (x_{i}^{1}, y_{i}^{1}\right)\) are the points where we project the acoustic wave and \((x_{i}^{2}, y_{i}^{2})\) are the points where receive the acoustic wave (\(x_{i}^{1}=-R\), \(x_{i}^{2}=R\)).
We can solve the system (6) of three equations to determine x0, y0 and r, since we can determine \(X_{i}^{j}, Y_{i}^{j}\) by
$$ {{} \begin{aligned} \left\{ \begin{array}{l} \left(X_{i}^{j}\,-\,x_{0}\right)^{2} \,+\, \left(Y_{i}^{j}\,-\,y_{0}\right)^{2}\,=\, \left(X_{i}^{j}-x_{i}^{j}\right)^{2} + \left(Y_{i}^{j}-y_{i}^{j}\right)^{2}+ r^{2} \\ \left(X_{i}^{j} -x_{0}\right)\left(X_{i}^{j}-x_{i}^{j}\right) + \left(Y_{i}^{j}-y_{0}\right)\left(Y_{i}^{j}-y_{i}^{j}\right)=0. \end{array}\right. \end{aligned}} $$
and θi by
$$ {\begin{aligned} \theta_{i} = \cos^{-1} \frac {\left(X_{i}^{1}-x_{0}\right)\left(X_{i}^{2}-x_{0}\right)+ \left(Y_{i}^{1}-y_{0}\right)\left(Y_{i}^{2}-y_{0}\right)} {\sqrt{\left(X_{i}^{1}-x_{0}\right)^{2}+ \left(Y_{i}^{1}-y_{0}\right)^{2}} \sqrt{\left(X_{i}^{2}-x_{0}\right)^{2}+ \left(Y_{i}^{2}-y_{0}\right)^{2}}}. \end{aligned}} $$
Note that there being quadratic equations in the systems (6) and (7), they have two solutions, however, by virtue of the condition that the detour is the shortest way we can determine the unique solution. We also note that thetai in (8) must be acute.
It is very easy to extent Theorem 4.1 to the 3-dimensional case.
In the three dimensional case, a cavity of a ball in its interior of a homogeneous object is reconstructed by appropriate four data of the ultrasonic tomography.
In this case, what we have to do is determine the four unknowns, the coordinate (x0,y0,z0) of the center and the radius r. It is sufficient to have appropriate four data. The reconstruction algorithm is easily obtained by extending the algorithm for the 2-dimensional case.
For practical application, we shall extend Theorem 4.2 for the case where the cavity consists of several balls and develop a method to approximate a general cavity by a sum of several balls in the future.
In this section, we shall introduce some examples where the study of Problem 4.1 is very important. The main application in this section, we propose a non-destructive inspection method for concrete cover in the RC structures. In the interior of RC structures, there are a number of steel rods inbedded for reinforcement. It is very important to prevent the steel from getting corroded, for which it is very helpful if we can develop a good, cheaply running non-destructive testing method for concrete cover for reinforcing steel. For simplicity, let us assume a simple structure where only one steel rod is inbedded as designed in Fig. 9, which is a section of the three dimensional structure. Based on the above assumption, we shall establish the basic theory for our non-destructive inspection technique for concrete cover. By modification, our method can be applied for real RC structures.
A cavity of a circle in the 2-dimensional molar structure
Concrete structure being inhomogeneous, we homogenize the acoustic velocity in the concrete by application of the idea of the least square solution (confer the discussion below), by which we denote the velocity in the steel by V and that in the concrete by v. In general, it is known that 4000m/s<v<5200m/s and 5500m/s<V<6500m/s. Therefore we can assume that v<V.
We first determine the acoustic velocity in the steel V by application of Conclusion 3.1. By homogenization of the acoustic velocity in the concrete mentioned above, the primary ultrasonic wave projected from the point O(0,h) propagates along the spline \(\overline {OQ} \cup \overline {QP}\) and is received at the point P(l,0), whose travel time we denote by t0. If we project the ultrasonic wave from the point O and receive it at the point Pi(l+li,0) with travel time ti, then we obtain approximate acoustic velocity in the steel by li/(ti−t0). Take observation points P1,⋯,Pn as many as possible and take the least square solution
$$ V = \frac {l_{1}(t_{1}-t_{0}) + \cdots + l_{n}(t_{n}-t_{0})} {\left(t_{1}-t_{0}\right)^{2} + \cdots + \left(t_{n}-t_{0}\right)^{2}} $$
to the system of linear equation which trivially has no solution
$$ V (t_{1}-t_{0}) =l_{1}, \cdots, V (t_{n}-t_{0}) =l_{n}. $$
We also comment that the least square solution V is the minimizer for the function
$$ \left(V (t_{1}-t_{0}) -l_{1}\right)^{2} + \cdots + \left(V (t_{n}-t_{0}) -l_{n}\right)^{2}. $$
We then determine the homogenized acoustic velocity in the concrete. For given V and v, x and the length L of the segment QP in Fig. 8 are calculated as
$$ x= \frac {vh} {\sqrt{V^{2} -v^{2}}}, L= \frac {Vh} {\sqrt{V^{2} -v^{2}}}. $$
We first let v0=3500m/s then the ultrasonic wave projected from the point \(O_{i} (0, h_{i}), \bar h_{i} < h \) linearly propagates in the concrete to the observation point \(R_{j}(r_{j}, 0), r_{j} < \frac {vh} {\sqrt {V^{2} -v_{0}^{2}}}\). Take as many pairs of source and observation points {Oi,Rj} which gives the length Lij of the segment where the ultrasonic wave propagates, and the travel time tij. Therefore we obtain the first step value v1 of the homogenized acoustic velocity in the concrete by
$$ v_{1} = \frac { \sum t_{ij} L_{ij}} { \sum t_{ij}^{2}}. $$
Replacing v0 by v1 and repeating the same procedure, we obtain the second step values \(v_{2}=\bar v\), which we determine as the homogenized acoustic velocity in the concrete. If necessary, we make the sequence {vk} until it approximately converges, for example, we stop the procedure when vk+1−vk<0.01(m/s).
By the knowledge of the acoustic velocities, V in the steel and \(\bar v\) in the concrete, and the route where the ultrasonic wave propagates, we can propose a non-destructive inspection technique for concrete cover by application of Conclusion 3.1.
In Fig. 8, when we observe the travel time the primary ultrasonic wave between the points O and P, the calculated travel time is
$$ \frac {l-\frac {\bar vh} {\sqrt{V^{2} -{\bar v}^{2}}} }{V} + \frac {Vh} {\bar v\sqrt{V^{2} -{\bar v}^{2}}} $$
and the one in the segment QP is
$$ \tilde t = \frac {Vh} {\bar v\sqrt{V^{2} -{\bar v}^{2}}}. $$
If the velocity \(\tilde v\) in the segment QP is much smaller than the homogenized velocity \(\bar v\), for example, \( \bar v -\tilde v > 200 m/s\), then one or more of the following could happen.
There is a cavity in the segment QP.
There is a water route around the steel.
Some part of the steel may get corroded.
In any case, we have to shave the concrete cover and have to repair the defect. Shaved concrete cover can be easily re-fixed by filling the cavity by shaving with better concrete. Therefore, what is important is to establish how to find where to shave in a non-destructive way, an answer of which is given as the above way.
In this section, we summarize our conclusions in the present paper.
(Conclusion of this paper)
For development of the ultrasonic CT, we studied how the primary ultrasonic wave propagates in the cement paste and the mortar (Conclusion 3.1).
Applying the property of the primary ultrasonic wave, we have posed a problem for the development of the ultrasonic CT (Problem 4.1).
Our ultrasonic CT for concrete structures is safe, cheaply-running and concrete non-destructive inspection method for concrete structures, which enables us to reconstruct the interior of the concrete structures concretely (Claim 4.1).
The problems posed in this paper are also interesting and important in view of the study of mathematics (Remarks 4.1 and 4.2).
As an application of our ultrasonic CT, we developed a non-destructive inspection method for concrete cover in RC structures (Section 5), where homogenization of acoustic velocity in the concrete by the idea of the least square solutions played an important role.
Theoretical approach, as well as computational one, to Problem 4.1 is under investigation. As an approach to Problem 4.1, the main problem in this paper, we are trying a similar method that G. N. Hounsfield applied for practicalization of the computerized tomography. Confer [11] for the idea by G. N. Hounsfield to practicalize CT.
For the verification of our non-destructive inspection testing technique introduced in the fifth section, we shall make another test pieces of reinforced concrete with some pieces of styrofoam in its interior and perform experiments to detect them by our technique, we shall also apply our technique to the real expressways which shall be introduced in our forthcoming paper.
It is also required, in view of practical application, to generalize our non-destructive inspection techniques for concrete cover, in more complicated RC structures, which in under investigation with real expressways by the authors and West Nippon Expressway Shikoku Company Limited.
computerized tomography
RC:
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Dunham, C. W.: The theory and practice of reinforced concrete (third edition). McGrow-Hill, New York (1953).
Gorzelańczyk, T., Hola, J., Sadowski, L., Schabowicz, K.: Methodology of nondestructive identification of defective concrete zones in unilaterally accessible massive members. J. Civ. Eng. Manag. 19, 775–786 (2013).
Greenleaf, J. F., Johnson, A. A., Lee, S. L., Herman, G. T., Wood, E. H.: Algebraic reconstruction of spatial distribution of acoustic absorption within tissue from their two-dimensional acoustic projections. Acoust. Hologr.5, 591–603 (1974).
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Neville, A. M.: Properties of concrete (fifth edition). Pearson Education Limited, Edinburgh (2011).
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Takiguchi, T.: How the computerized tomography was practicalized. Bull. JSSAC. 21, 50–57 (2015).
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The basic idea of this research was presented in the conferences Collaboration between theory and practice in inverse problems held at IMI, Kyushu University, Japan, from Dec. 16th to Dec. 19th, 2014 and Development of ultrasonic tomography for concrete structures held at IMI, Kyushu University, Japan, from Nov. 10th to Nov. 13th, 2015 where lively discussions were had on this problem by engineers, in both industry and academia, and mathematicians. The authors claim that the present paper is a typical fruit by interdisciplinary and industry-academia collaboration among engineers, mathematicians and companies.
The second author was supported in part by Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Scientific Research (C) 26400184.
Faculty of Human Resources Development, Polytechnic University of Japan, 2-32-1, OgawaNishimachi, Kodaira, Tokyo, 187-0035, Japan
Noriyuki MITA
Department of Mathematics, National Defense Academy of Japan, 1-10-20, Hashirimizu, Yokosuka, Kanagawa, 239-8686, Japan
Takashi TAKIGUCHI
This work is a collaboration and finished by discussion by the both authors. They contributed to all context. Both authors read and approved the final manuscript.
Correspondence to Takashi TAKIGUCHI.
MITA, N., TAKIGUCHI, T. Principle of ultrasonic tomography for concrete structures and non-destructive inspection of concrete cover for reinforcement. Pac. J. Math. Ind. 10, 6 (2018). https://doi.org/10.1186/s40736-018-0040-0
Received: 22 January 2018
Ultrasonic tomography
Integral geomerty | CommonCrawl |
Synchronous birth is a dominant pattern in receptor-ligand evolution
Anna Grandchamp1 &
Philippe Monget1
BMC Genomics volume 19, Article number: 611 (2018) Cite this article
Interactions between proteins are key components in the chemical and physical processes of living organisms. Among these interactions, membrane receptors and their ligands are particularly important because they are at the interface between extracellular and intracellular environments. Many studies have investigated how binding partners have co-evolved in genomes during the evolution. However, little is known about the establishment of the interaction on a phylogenetic scale.
In this study, we systematically studied the time of birth of genes encoding human membrane receptors and their ligands in the animal tree of life. We examined a total of 553 pairs of ligands/receptors, representing non-redundant interactions.
We found that 41% of the receptors and their respective first ligands appeared in the same branch, representing 2.5-fold more than expected by chance, thus suggesting an evolutionary dynamic of interdependence and conservation between these partners. In contrast, 21% of the receptors appeared after their ligand, i.e. three-fold less often than expected by chance. Most surprisingly, 38% of the receptors appeared before their first ligand, as much as expected by chance.
According to these results, we propose that a selective pressure is exerted on ligands and receptors once they appear, that would remove molecules whose partner does not appear quickly.
The co-evolution of genes encoding interacting molecules is a subject of intense study [1,2,3,4] because of the intriguing question of the modes of mutation and selection that act on two molecules simultaneously. In particular, the co-evolution of the binding motif has been well investigated [5]. These studies of co-evolution focused for example on the fitness [6, 7], on the conservation of the interaction [8,9,10], or on the evolution of the residues at the interface of the molecules [11,12,13]. While these studies on the coevolution of binding partners often require the integration of different disciplines (chemistry, evolution, biology), the establishment of the interaction from a phylogenetic point of view is less studied. Little is known for example about the origin and evolution of the different partners prior to their first interaction. Do the receptor and the ligand co-exist independently before they start to interact? Does the emergence of one partner favor the emergence of the second partner? If so, which tends to come first, the receptor or the ligand? The creation of new genetic material often relies on segmental duplication, or sometimes but more rarely on entire genome duplication [14,15,16,17]. Once a gene is born, either de novo for the first member of a family or by duplication of existing genes, the gene will be subjected to negative selection if it is not beneficial, and could even be lost by pseudogenisation [14]. If the gene belongs to a gene family, for example the glycoproteins FSH, LH and TSH and their receptors, the appearance of the first member of the family can be the result of an ancestral duplication of a gene that belongs to the superfamily (GPCR superfamily in this case), followed by several mutations leading to the current genes. The diversifications of GPCR families arose by multiple duplications [18, 19] However, it is only the acquisition of a novel function that will allow the maintenance of the newly duplicated gene.
In the case of interacting molecules, the appearance of genes coding for molecules included in a complex is more intricate [20]. For two molecules that will eventually interact, the appearance of one may be dependent on the appearance and conservation of the other. This may be the case, for example, when the presence of the first molecule is not advantageous as long as its partner has not yet appeared.
Asking the question: "In the absence of a ligand, what is the biological role of a receptor?", Thornton [21] has shown that the first steroid receptor of the family, present in lamprey and supposed to be present in the common ancestor of vertebrates, was an oestrogen (that is, a steroid) receptor, and that several duplications led to other steroid receptors, specialized in other functions with other ligands. However, recent investigations suggest that the ancestral ligand for the ancestral steroid receptor was a molecule with a structure distinct from modern estrogen, an aromatized steroid with a side-chain, called paraestrol [22]. Yet the existence of receptors without partners, called orphan receptors, has also been frequently described [23], even though it is sometimes difficult to assess whether a receptor is a true orphan or its ligand is just unknown [24]. Interestingly, studies have demonstrated that orphan nuclear receptors were phylogenetically related, and older than the receptors with a known ligand [25, 26]. These authors have suggested that the receptor acquired its binding pocket during evolution. In contrast, more recently, the existence of an ancient common ligand of the nuclear receptor family was demonstrated [27], thus challenging the view that nuclear receptors could have evolved for extended periods of time without ligand.
The relative appearance of genes encoding protein partners is thus an open question. Furthermore, several types of interactions can be observed in living organisms, with different numbers of interacting partners [28,29,30,31], varying affinities [32], or different duration for the interaction [33], making the problem more complex.
Understanding the process that leads to functional interactions would help to understand how genes evolve to give rise to a binding pocket in receptors during evolution. With thousands of entirely sequenced genomes available in public databases, assessing when a functional gene appears in the tree of life is becoming a realistic challenge.
In our study, we collected a list of genes encoding human cell membrane receptors with their known ligands, and studied the timing of their respective appearance during evolution.
Implementation of the database
Our study is focused on human membrane receptors and human endogenous ligands, for which information was collected from several sources (Additional file 1). The genes encoding receptors whose ligands were not endogenous, such as olfactory receptors or taste receptors, were not considered. In 101 cases, the ligands resulted from a chain of synthesis that requires several enzymes (such as dopamine, serotonin, acetylcholine, etc.). In these cases, we considered the set of genes encoding the enzymes involved in the ligand synthesis. The number of genes encoding such enzymes varied between 1 and 4 genes.. Nuclear receptors and their ligands were not considered, owing to the large number of genes involved in the synthesis of the ligand (more than 15 genes can be involved). Ultimately, we built a list of 1479 pairs of genes encoding respectively a ligand and its membrane receptor, which is three times greater than can be found in the DIP (database of interacting proteins) database. We only used interactions confirmed by experimental assays. However we also repeated our calculations using a larger list of predicted interactions previously described by Ramilowski et al. [34] to make sure that the results would not be modified (Additional file 2). Ramilowski's is the most comprehensive list in existence today. Better-known lists recording the complete interactome, such as StringDB [35], were not used, because they do not specify the nature of the interactions (ligand -receptor, substrate-enzyme) in the case of ligands receptors. Moreover, the ligands receptors interactions implemented in StringDb come from DIP database that was used in our list.
Phylogenetic study
In order to determine the time of appearance of each gene, we focused our study on the animal tree of life [36], and on the phylogenetic trees of animal sequences available in Ensembl [36]. We selected 10 phylogenetic branches as possible intervals where a gene may have appeared. The branch of appearance of a gene refers to the branch that include all the taxonomic groups in which the gene is present and functional today. For example, if a gene was present in several taxonomic groups such as mammals, reptiles, amphibians, and not in other groups, we consider that the functional ancestor of the gene appeared in Tetrapoda. The absence of a gene in taxonomic groups which diverged before Tetrapoda could be due to a loss of the genes in the species of this group that are available in Ensembl. For taxonomic groups in which there were few species in Ensembl (see Additional file 2), the gene was looked up in Refseq (Genbank) using tBLASTn [37] to make sure it could not be found in other species.
We defined 10 phylogenetic branches (Fig. 1): branch 1 is ancestral to yeast and multicellular organisms, whose emergence is dated about 1500 million years (my), branch 2 is ancestral to Metazoa (~ 713 my), therefore excluding unicellular organisms, branch 3 is ancestral to bilaterians (~ 580 my), branch 4 is ancestral to Chordates (~ 560 my), branch 5 is ancestral to vertebrates (~ 550 my), branch 6 is ancestral to Teleosts (~ 420 my), branch 7 is ancestral to Sarcopterygians (~ 400 my), branch 8 is ancestral to Tetrapods (~ 359 my), branch 9 is ancestral to Amniotes (~ 326 my), and branch 10 is ancestral to mammals (~ 184 my). At the base of the metazoan tree, we decided to define only one branch (branch 2) that would be ancestral to the Placozoa, the Porifera, the Ctenophora and the Cnidaria, because their phylogeny is still being discussed [36]. Indeed, we estimated that the merging of these groups may introduce a smaller bias in our study than considering each separately.
Definition of branches in the animal tree of life. Rectangles represent the 10 defined branches of appearance for the proteins (B1 to B10). The estimated time of emergence of the branches is indicated under the tree
The ten branches defined are separated by distinct time steps. Indeed, some branches have diverged within short time steps, as for example the vertebrate branch, which diverged from the non vertebrate chordates 550 my ago, and the branch of the chordates, which diverged from the unchurched 560 my ago. So there is a short time step of 10 my between these two branches. On the other hand, there is a time step of 110 my between the branch of the vertebrates and the branch of teleosts, which diverged from non-teleost vertebrates 420 my ago. These different time step were taken into account in our statistical model (see after).
The choice to rely on such wide time gaps has allowed us to highlight the possibility for one of the interacting partners to remain maintained during the evolution over a broad time without the presence of its current partner. However, this choice made it impossible to precisely date the moment of appearance of the gene in the branch.
We then determined in which branch the genes encoding each receptor and ligand appeared. The phylogenetic trees were recovered from the ENSEMBL database v82 [38]. We complemented the branch of the first Metazoans (branch 2) using the Ensembl metazoan database (http://metazoa.ensembl.org/index.html), thus adding 71 genomes. For the trees that rooted in non chordate species, we identified and selected the corresponding genes in the Ensembl Metazoa database. For each gene in our list, its branch of appearance was annotated. A total of 145 species were considered in the phylogenetic trees (Additional file 1).
It is now known that two rounds of complete duplication are at the origin of the vertebrate genomes [39]. In the case of ligands and receptors, it is expected that some ligands and receptors that appeared in non-chordates are therefore present in four copies in vertebrates. However, this is not the case for most of the gene families, less than 5% of duplicate gene families remaining in duplicate [40]. So gene families rarely present 4 duplicate copies of the ancestral gene. Nevertheless, we took into account this complete duplication in our study. For each copy resulting from the duplication, the root we considered was the one given by the Ensembl algorithms. In most cases, for a receptor having duplicated in several copies, the root given by Ensembl is the branch of appearance of the first receptor. It is the same for the ligands, whose root will mainly be the ancestral root.
However, there are less frequent cases of some genes with strong divergence on one of the duplicates just after duplication. This is the case if an ancestral receptor is duplicated, and one of the duplicates diverges very specifically to bind a new ligand.This is for example the case for ephrin receptors. Some of these receptors are present in non-chorded animals, along with their ligands, and some other of these receptors appeared in the vertebrate branch after the two duplications. The latter bind to the same ligands as the ancestral receptors. Thus, the first receptors of this family appeared at the same time as their ligands, when the other receptors of the family, resulting from complete duplication, appeared after their first ligand. We find an inverse case with integrins. Most of their receptors appeared in the first metazoans, as well as their ligands. That is not however the case with ITGAD, an integrin whose ligand appeared in vertebrates. Phylogeny does find an ortholog of ITGAD in non-chorded animals. In this rather special case, for most members of the integrin family, the first ligand appeared in the same branch, except for this particular gene whose first ligand appeared later.
During the course of our study, we realized that the majority of the members of a given family appeared in the same branch (to take the same example as above, FSH, LH and TSH, which belong to the same family, appeared at the same branch). However, it is not the case for all the families. For example, some genes evolve faster than others, such as the genes involved in immunity [4]. In such a case, the trees tend to give the same root for all the genes coding for interleukins because Ensembl trees are based on a very stringent alignment, whereas some of the subfamilies did not appear at the first root. All these trees were treated manually, to make sure that all the complicated situations would be taken into account. To reduce the number of possible incorrect datings, according to our defined branches, we took the sequences of all the species of Ensembl that branch to the oldest root of the tree, to verify by tblastn analysis if an older ancestor was present in the syntenic region. For example, if the tree included mammals, reptiles and amphibians, we took the sequences of the species corresponding to these taxonomic groups present in Ensembl. Then, t-blastn were performed (in Refseq of NCBI, [37]) on the genome of all the outgroup species descending from the node directly preceding (i.e. more ancient than) the root, according to our defined branches. Moreover, some genes are not annotated by their name. This fact could bias the Ensembl research. In fact, an ortholog of a gene of interest could be present in species that branch in a branch older than the root given by Ensembl, but not encountered in Ensembl because it is not annotated. We systematically used Mapviewer (https://www.ncbi.nlm.nih.gov/genome/gdv/) to examine the conservation of synteny in order to correct the phylogeny as previously described [41, 42]. BLAST and synteny conservation allowed us to correct 47 trees for which the gene was found to appear 1 branch earlier, and 16 trees for which the gene was found to appear 2 or 3 branches earlier. All of the 63 genes concerned were involved in immunity.
Study of the birth of genes encoding the ligands and their receptors
The main point of the experiment at this point was the reshuffling of our list of 1479. This reduction aimed to consider only the first ligand(s) that appeared for each receptor, and vice versa. Indeed, many receptors (more than 75%) have several ligands. These ligands often belong to the same family, but this is not always the case (i.e. LIFR, vldlr etc.). For each receptor, when it appeared in a phylogeny, we tried to determine whether it had a ligand to interact with as soon as it appeared (it is the case if at least one of its current ligands appeared in a preceding branch), if the appearance of interacting ligands took place in the same branch (it is the case if at least one of its current ligands appeared in the same branch and another one later), or if at the time of appearance of the receptor, none of the ligands was still present (i.e. the first ligand(s) appeared in later branches). The interactions with the other ligands, those that appeared later, were not considered here, since they concern coevolution. Our list of 1479 interactions was thus reduced to a list which included the 553 receptors of the first list, accompanied by the moment of appearance of their first ligand(s). Moreover, to ensure that these few families did not introduce any bias, we also set up a list, including only the first receptor which appeared in each family, with its first ligand. We obtained a list of only 113 pairs, with the earliest receptor of the 113 families and their first ligand. Thus, such a list, although much less precise and including less data, allowed us to ensure that any misidentification of the moments of appearance of the molecules resulting from duplications in the multigene families would be discarded (Fig. 2, Additional file 2).
Schematic representation of the three list of interactions depicted in the article. The "complete list" represents the initial list of 1479 interactions. The receptors are grouped by family. Each receptor establishes interactions with one or more ligands. Here as an example, we have represented 22 interactions among the 1479 in the real list. In red are represented the molecules, ligands as receptors, appeared in the most ancestral branch. In green are represented the molecules that appeared in a less ancestral branch than the red molecules, and in yellow the molecules that appeared in a branch less ancestral than green molecules. The "Non redundant interactions" represents the reduction of the global list of 1479 interactions to 553 non-redundant interactions, by removing all the ligands that are not the most ancestral ones. In case several ligands appeared at the same time (R1 receptor), we consider only one ligand, by arbitrarily choosing one of them. The nature of the ligand does not matter because it is only the branch of appearance (common to all) that concerns us. In this diagram, the list of 22 redundant interactions is reduced to 10 interactions (one per receptor). The "One interaction by family" represents the reduction of the list of 553 non-redundant interactions into a list of 113 interactions. To make this reduction, we takes in every receptor family the first receptor that appeared, with its first ligand. In the case of family 2 of the diagram, we note that two receptors appeared at the same time and in the most ancestral branch. In this case, we choose the receptor whose first ligand appeared. Here, it is R3 that is chosen rather than R2. In this list we go from 10 interactions to 3, as many interactions as there are families
Concerning the 553 interactions, 101 receptors bound only with ligands that were not peptides, but molecules generated by a chain of synthesis involving several enzymes. Among these 101 pairs, for 49 of the 101 pairs in which the ligands were the result of a chain of synthesis involving several enzymes, all the genes encoding the enzymes appeared in the same branch, which we considered to be the branch of appearance of the ligand. For the remaining 52, we only considered the branch of appearance of the most recent gene involved in the synthesis, considering that the resulting ligand could not be present without all the enzymes necessary for its synthesis.
Each pair of ligand/receptor was classified as follows: LB-Ligand Before, the gene coding for the first ligand appeared before the gene coding for the receptor; LS-Ligand Synchronous, the gene coding for the first ligand appeared in the same branch as the gene coding for the receptor; LA-Ligand After, the gene coding for the first ligand appeared after the gene coding for the receptor. The distribution of the pairs in each category was analyzed for the complete list (553 pairs), and with two other configurations grouping receptors by families, to make sure that the results are not impacted by possible duplication biases within families, and by removing the ligand whose synthesis involved several enzymes. The list we built only contains interactions verified by experiments. To examine if adding predicted interactions would affect our data, we also repeated the analysis using the predicted interactions that involved our receptor, using the list of Ramilowski [34], although the list was filtered to remove genes coding for G proteins and other proteins that are not ligands (see Additional file 1).
Model of comparison
We conducted a test to estimate whether the distribution of the pairs in the three categories was different from what would be expected if both partners appeared independently.
To this end, the proportion of all human genes that appeared within each of our delimited branches was assessed by counting the number of roots of all 19,928 human gene trees in each branch. The time that has elapsed within the branches was taken into account by weighting the number of genes that appeared in each of them. We also did the tests without taking into account this weighting, which gave the same statistical result (Additional file 2). This frequency distribution enabled us to compute the null distribution of ligands appearing before (LB), after (LA), and at the same time (LS) as their receptor:
$$ {\mathrm{L}}_{\mathrm{B}}=\sum \limits_{\mathrm{b}1=2}^{\mathrm{b}1=10}\left(\sum \limits_{\mathrm{b}2=1}^{\mathrm{b}2=\mathrm{b}1-1}{\mathrm{R}}_{\mathrm{b}2}\times {\mathrm{F}}_{\mathrm{b}2}\right) $$
$$ {\mathrm{L}}_{\mathrm{s}}=\sum \limits_{\mathrm{b}=1}^{\mathrm{b}=10}{\mathrm{R}}_{\mathrm{b}}\times {\mathrm{F}}_{\mathrm{b}} $$
$$ {\mathrm{L}}_{\mathrm{A}}=\sum \limits_{\mathrm{b}1=1}^{\mathrm{b}1=9}\left(\sum \limits_{\mathrm{b}2=\mathrm{b}1+1}^{\mathrm{b}2=10}{\mathrm{R}}_{\mathrm{b}2}\times {\mathrm{F}}_{\mathrm{b}2}\right) $$
With Rb the number of receptors observed in branch b and Fb the frequency of protein appearance in branch b. The branches, that are b symbols, are the branches franked 1 to 10. In eqs. LA and LB, b1 corresponds to the variation in branches in the first sum, and b2 to that in the second one. b2 may vary independently of b1.
The difference between the observed and the theoretical distribution was assessed with a Pearson's chi-squared test. The test was performed in the 4 configurations: with all receptors, with receptors grouped by family, with all receptors but removing the ligands that result from a chain of synthesis in which the enzymes involved in the synthesis did not all appear in the same branch, and with the list including predicted interactions [34] (Additional file 1).
To characterize the factors that may influence the distribution of the partners, we performed a Multiple Correspondence Analysis (MCA), taking into account the moment of appearance, the molecular weight of the ligand, the family, the kind of molecule (syntesized ligand, glycoprotein, etc.), the kind of signal (hormone, neuropeptide, etc.) and the function of the gene family (immunity, metabolism, etc.).
Receptors and ligands are predominantly born in the same branch
Among the 553 pairs of ligand/receptor, we observed that the pairs were unequally distributed in the three categories. The number of pairs in LS (Ligands Synchronous) was not different from the number of pairs in the LA (Ligands After) category (40.69% vs 38.33%, p-value = 0.534, chi-square test), and the number of pairs in these two categories was higher than the number of pairs in the category LB (Ligand Before) (20.98%; p-value = 3.6e-09 LB vs LS, p-value = 1.2e-07 (LB vs LA) (Fig. 3a). Moreover, the majority (77/101) of ligands that result from a chain of synthesis were grouped in LB. The majority of the pairs found in LS appeared at the root of metazoa (branch 2, 48%), the root of vertebrates (branch 5, 16%) and the root of teleosts (branch 6, 9%).
a Barplot of the global distribution of the 553 partners in each of the three categories. Category 1: Ligands which appeared before their receptors; Category 2: ligands which appeared in the same branch as their receptors; Category 3: ligands which appeared after their receptors. The x-axis represents the three categories, the y-axis represents the number of partners. The red bars correspond to the observed distribution. The black bars correspond to the expected distribution. 116 pairs were observed for which the first ligand of the considered receptor appeared before, against 256 expected. 225 pairs were observed for which the first ligand of the considered receptor appeared in the same branch, against 102 expected. 212 pairs were observed for which the first ligand of the considered receptor appeared after, against 195 expected. b Distribution of the 553 randomly selected partners, repeated 10,000 times (grey). The position of the observed number of partners is indicated with an arrow. c Distribution of the distance (in terms of branches of appearance) between all the genes encoding the ligands and their receptors. The red curve represents the observed data, and the black curve the expected distribution found with the random draws. 0 corresponds to a pair of ligand/receptor that appeared in the same branch. We observe an expected peak at 0 in the black curve due to the fact that a gap of 0 can be obtained over 10 branches, whereas a gap of 1 can only be obtained over 9 branches, a gap of 2 over 8 branches, etc. The negative values represent the genes encoding the ligand that appeared n branches before the gene encoding the receptor. The positive values correspond to the gene encoding receptors that appeared n branches before the gene encoding the ligands
We then evaluated the distribution of the partners against a theoretical distribution that assumes independence between protein appearance (Fig. 3a and b). Remarkably, we found that pairs where the receptor and the ligand appeared synchronously in the same branch (LS) is 2.5-fold higher than in the null distribution (p-value = 2.2e-16, chi-square test). In addition, for the pairs of ligand/receptor that did not appear at the same time, they appear in branches closer together than expected (Pearson correlation: p-value = 5.873e-05, r = 0.22), showing that pairs that do not appear in the same branch still tend to appear in neighboring branches (branch n-1 or n + 1) (Additional file 1). No such correlation was observed (Pearson p-value = 0.1363, r = − 0.071) for protein pairs with partners selected randomly according to observed branch frequencies Fb (see methods). Surprisingly, the observed number of human ligands that appeared before their receptors (LB) was 2-fold lower than the number expected from a null distribution (p-value = 3.6e-12, chi-square test). The observed number of human ligands that appeared after their receptor (LA) was not different from the number expected from the null distribution (p-value = 0.31).
The results are the same when we consider the list of 553 interactions with all the receptors, as well as with the list of 113 interactions with one member of each receptor's family, and the lists without non peptide ligands and predicted interactions (Additional file 1).
MCA analysis
Finally, a MCA analysis integrating 5 families of criteria identified two characteristics that were correlated with the moment of appearance of the ligand (Fig. 4a): the molecular weight of the ligand (Fig. 4b), and the type of molecules (see Additional file 1). We observed that the glycoproteins ligands are grouped all together in the MCA, and correspond to the same group of receptor-ligand pairs that appeared in the same branch. (Fig. 4c). We also observed in the MCA that the smallest ligands (< 550 Da) tend to appear before their receptor, the medium ligands (between 550 to 25,000 Da) tend to appear after, and the biggest ligands (more than 25,000 Da) tend to appear synchronously with their receptors. Moreover, we observed that glycoproteins tend to also appear simultaneously with their receptors, whereas the hormones and neuropeptides tend to appear after their receptor. Contrary to the co-evolution of the interaction that is influenced by the function of the partners [49], we did not observe any influence of the function of the interaction on the co-appearance of the receptor-ligand partners.
Spatial representation of the Multiple Correspondence Analysis. a Represents the pairs colored according to their categories of time of appearance (before in black, same branch in red, after in blue). b and c Represent the two traits which present the greatest influence on the results. b represents the distribution of the partners according to their molecular weight in dalton. Small: < 550 Da, Medium: 550 to 25,000 Da, Big: > 25,000 Da. c Represents the distribution of the type of molecule. Group 9 (red) are the "other proteins", corresponding the free proteins as neuropeptides and not glycoprotein hormones, group 1, 1.5, 2, 3, 5, 7, 8 contain the amines, monoamines, catecholamines, lipids and derivatives, nucleotids and derivatives, the esters and the gaba. Group 4 contains the glycoproteins and group 6 (blue) the scleroproteins
The fact that ligands appeared less frequently before their receptor than expected suggests that the birth of a ligand is more dependent on the prior existence of a receptor than the opposite, and that ligands are more likely to be replaced during evolution than receptors (21% of our distribution are ligands that appeared before their receptor, against 38% receptor that appeared before their ligand). For receptors whose the first ligand appeared before (Lb), we could hypothesize that the ligands interacted with receptors that were replaced by others during evolution, or that receptors evolved very quickly and that their branch of appearance is most ancestral than expected. In a recent study, it was demonstrated that membrane proteins, that include all our membrane receptors, evolve faster than free proteins [43]. In contrast, [44] suggested that receptor structures undergo a tighter constraint than the ligand, and that "receptors drive the evolution of ligands in invertebrates". Our results seem in agreement with the latter hypothesis, which tends to suggest that the results of [43] might not affect all membrane proteins in the same way. A second hypothesis could be that these ligands were not "ligands molecules" until the receptor arrived.
Finally, there are several known cases of ligands binding to other molecules as well as to their membrane receptors. Such is the case of human albumin, ALB. Albumin is a ligand to receptors f-ALB in man [45] and FcR/CR in chickens [46]. However, serum albumin is also known for a variety of other functions or liaisons. Albumin binds water, as well as certain fatty acids, hormones, bilirubin and drugs (GeneCard [47, 48]). This seems to entail that part of the ligands which appeared without their receptors were selected for their function in other binding mechanisms.
Remarkably, the synchronous appearance of receptor and ligand pairs far exceeds expectations (2.5-fold more). This result shows that the birth of each partner in a receptor-ligand pair tends to be more synchronous than expected by chance. This discovery testifies to the dependence between two partners. The establishment of an interaction is largely favored by the fact that the two partners are present at the same time, the appearance of only one of them in a branch being not the dominant model. This suggests that many binding pairs did not change partners during evolution, and that both partners conserved their binding function since its moment of appearance. Our results confirm that the protein interactions are well-conserved during evolution, as previously shown [10, 49].
The number of branches that separate the moment of appearance of the receptor and its ligand was also determined, for the observed and randomized data (Fig. 3c). The number of pairs with distance 0 – corresponding to ligands and receptor that appeared in the same branch – is higher than the expected number, as previously shown, and the number of pairs for all the other distances (1–9) is almost always lower than the expected curve. However, unexpectedly, we also observed a peak in the observed curve for distances 3 and 4 (Fig. 3c). This peak of the curve corresponds to a group of 50 receptors (32 in peak 3 and 18 in peak 2) that appeared in Eumetazoa and Protostomians (branch 2 and 3), with their ligands appearing in Vertebrates and Teleosts (branches 5 or 6). Most are neuropeptides, with complex phylogenies that are difficult to reconstruct [50, 51]. For the pairs of this peak that were documented in the literature, most previous studies are in accordance with our timing of appearance for these proteins [51,52,53,54,55,56,57,58,59,60]. Nevertheless, three recent studies [61,62,63] focusing on kisspeptin, galanin, cholecystokinin, gastrin, neuromedin U, pyrokinin, sulfakinin and follicle stimulating hormone, obtained different results from ours and from those of other authors. In these three studies, the birth of the ligand was found to be older than expected by using only phylogeny, which would reassign 15 of pairs from LA to LS. These phylogenetic researches were conducted on few molecules, with methods that are still difficult to implement on the scale of a large dataset [51, 57, 61,62,63]. For those reasons, we believe that the number of pairs of ligand/receptor that appeared in the same branch is underestimated, and that the side peaks of the curve include partners that may have appeared in the same branch, although this may only be a small number.
The case of ligands resulting from a chain of biosynthesis is an exception. In our study, we have considered the ligand to be present if the enzymes necessary to its biosynthesis were too. Nevertheless, pathways involving alternatives enzymes in the biosynthesis process cannot be excluded, nor the fact that the biosynthesis pathway may have undergone alterations during evolution. This is the case of the mevalonate pathway which allows the conversion of acetyl-CoA into isopentenyl 5-diphosphat. This biosynthesis pathway was preserved across the animal world and can also be observed in bacteria. Three reactions occurs among phosphorylations involving ATP. The enzymes responsible for these reactions differ from one taxonomic group to the next. Specifically, the effects of a reorganisation can be observed between animals and bacteria with regards to enzyme folding [64]. Indeed, cases in which the ligand results from a biosynthesis chain should be treated with caution, due to a possible change of enzymes involved in the biosynthetic pathway.
We observed that many receptors appeared independently from their mammalian ligand. Interestingly, the fact that many ligands appear after their receptor was already observed [34]. In their study, these authors used phylostratigraphic approach to show that most ligands appear after their receptor. However, they did not consider the first ligand to have appeared, but rather investigated cases of coevolution of ligands once the first ligand and receptor have appeared. Furthermore, half of their interactions are predicted in silico, not experimentally determined, which add a lot of predicted ligands interacting with the same receptor. Moreover, a number of their interactions also involve G proteins that were removed from our study, because they are nor membrane receptors nor ligands.
Relationship between the functional characteristics of the pair ligand-receptor and their moment of birth
The MCA analysis resulted in two significant factors that were correlated with the moment of appearance: the molecular weight of the ligand and the type of molecules (see Additional file 1). The glycoproteins ligands correspond to the same group of receptor-ligand pairs that appeared in the same branch. The smallest ligands (< 550 Da) tend to appear before their receptor, the medium ligands (between 550 to 25,000 Da) tend to appear after, and the biggest ligands (more than 25,000 Da) tend to appear synchronously with their receptors.
Because large proteins have more amino acids than small ones, they present more amino acids subjected to substitution than in small proteins. Moreover, for membrane anchored molecules, the amino acids at the surface of the molecules are more substituted than those present at the centre, the latter being the part that allows them to be implanted in the membrane [43]. One could hypothesize that when a ligand appears, a quick and localized succession of changes has a higher chance to give rise to a binding area (at the surface) than in small and not anchored ligands. Consequently, in such big molecules anchored in the membranes, the amino acids that will interact with a new receptor (that appeared in the same branch) may have more probability to appear by chance than in small molecules. If such an interaction presents a functional interest, the nascent binding pocket may rapidly be fixed in the branch of birth of the two partners.
A limit to our method was the difficulty to date the birth of small ligands that evolved quickly. Even after correcting the possible bias, we suspect that a small number of false positives are still present, but they are unlikely to change the main conclusions. Additional efforts in the development of phylogenetic tools and in the curation of genomic data may gradually help solve this problem. Furthermore, the increasing availability of new genome sequences, especially in branches currently under-represented in the tree of life, will allow a finer dating of receptor-ligand relative birth times. Another limit of our method was the large and different gaps of time that separate our different branches. Other studies could be redone using shorter time steps, on organisms that diverged more recently. In addition, it bears noting that the receptors or ligands which appeared before their current partner potentially have an as yet undiscovered current partner. In this regard, future studies may in time shed light on ligands and receptors interacting with known proteins, and whose time of appearance corresponds to one of the proteins in our list.
Moreover, to enrich our model, it would also be interesting to take into account other interacting molecules, including intracellular ligands. It has been shown, for example, that G-protein coupled receptors evolve faster in their extracellular portion than in the transmembrane and cytosolic regions [43, 65]. Finally, our study focused on membrane receptors and their ligands. Since it has been demonstrated that the evolution of the interaction was different between transient and stable complexes [66], the application of our methodology to other kinds of interaction should allow a finer dissection and modelling of the influence of interaction types on the evolutionary fates of the interacting partners.
In the present study, we demonstrate that human ligands and their receptors appeared in the same evolutionary branches much more often than expected by chance, suggesting that when two binding molecules appear in a given branch, they are quickly submitted to purifying selection, which explains their conservation during evolution. This interdependence between the appearance of the membrane receptors and their ligands complements our knowledge of the evolution of binding partners, showing that before the well-studied co-evolution of the partners, we find a co-appearance scenario of these proteins. Thanks to the MCA, we observed that the biological function of the pairs of ligand receptors does not seem to play a role in the appearance of the interaction. However, the nature and the weight of the ligands were found to correlate with the moment of appearance, suggesting that the birth of the interaction is constrained by physical and chemical factors.
ligand after
Ligand before
LS:
ligand same time
Multiple correspondance analysis
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The authors are grateful to Dr. Pierre Pontarotti and Gabriel Markov for helpful discussions. Furthermore, the authors are very grateful to Hugues Roest Crollius and Alexandra Louis for their help in writing and methodology.
The present study was supported by a fellowship from the French ministry of research and by the Institut National de la Recherche Agronomique.
All data generated or analysed during this study are included in this published article [and its supplementary information files]. However, if something was missing, these data are available from the corresponding author.
PRC, UMR85, INRA, CNRS, IFCE, Université de Tours, F-37380, Nouzilly, France
Anna Grandchamp
& Philippe Monget
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AG performed the main data collection and analyses. PM designed the study and helped guide the general analyses. Both authors have read and approved the manuscript.
Correspondence to Anna Grandchamp or Philippe Monget.
Our article only use data that are available on public databanks. Our method did not use animal or plants. Not applicable.
The author(s) declare(s) that they have no competing interests.
Page 1 1: Branch of appearance of each of the 19,928 human genes. 2: Branch of appearance of the 1000 genes of bilaterians drawn randomly. Page 2: Branch of appearance of human genes for the mathematical model. Page 3: Branches of appearance of ligands and receptors. Page 4: List of receptor/ligand interactions. The file contains the interactions present in our database, as well as the interactions present in the list of Ramilowski et al., 2015. Page 5: List of ligands that have evolved rapidly. Page 6: List of characteristics of ligands and receptors used for MCA. The characteristics are the molecular weight before and after cleavage, the synthesis, the groups 1 to 9 according to the characteristics of the proteins (ex glycoproteins), the type of signal and the function of the interaction. (XLSX 578 kb)
Part 1: List of Databases used to construct the receptor/ligand interaction Database. Part 2: Characteristics of the components of the MCA. Part 3: Explanation of the statistical model. Part 4 Organisms of the branches of the overall phylogenetic trees + organisms implemented by BLAST. Part 5: Supplementary discussion on the appearance times of ligands and receptors. Part 6: Results of the random statistical model for the different combinations used. Part 7: Explanation of the methodology to match the bases Ensembl and Ensembl metazoa. Part 8: Number of receptors according to their number of ligands. Part 9: List of ligands and receptors in our database. (ODT 46 kb)
Grandchamp, A., Monget, P. Synchronous birth is a dominant pattern in receptor-ligand evolution. BMC Genomics 19, 611 (2018) doi:10.1186/s12864-018-4977-2
Co-appearance
Comparative and evolutionary genomics | CommonCrawl |
Ramanujan graph
In the mathematical field of spectral graph theory, a Ramanujan graph is a regular graph whose spectral gap is almost as large as possible (see extremal graph theory). Such graphs are excellent spectral expanders. As Murty's survey paper[1] notes, Ramanujan graphs "fuse diverse branches of pure mathematics, namely, number theory, representation theory, and algebraic geometry". These graphs are indirectly named after Srinivasa Ramanujan; their name comes from the Ramanujan–Petersson conjecture, which was used in a construction of some of these graphs.
Definition
Let $G$ be a connected $d$-regular graph with $n$ vertices, and let $\lambda _{1}\geq \lambda _{2}\geq \cdots \geq \lambda _{n}$ be the eigenvalues of the adjacency matrix of $G$ (or the spectrum of $G$). Because $G$ is connected and $d$-regular, its eigenvalues satisfy $d=\lambda _{1}>\lambda _{2}$ $\geq \cdots \geq \lambda _{n}\geq -d$.
Define $\lambda (G)=\max _{i\neq 1}|\lambda _{i}|=\max(|\lambda _{2}|,\ldots ,|\lambda _{n}|)$. A connected $d$-regular graph $G$ is a Ramanujan graph if $\lambda (G)\leq 2{\sqrt {d-1}}$.
Many sources uses an alternative definition $\lambda '(G)=\max _{|\lambda _{i}|<d}|\lambda _{i}|$ (whenever there exists $\lambda _{i}$ with $|\lambda _{i}|<d$) to define Ramanujan graphs.[2] In other words, we allow $-d$ in addition to the "small" eigenvalues. Since $\lambda _{n}=-d$ if and only if the graph is bipartite, we will refer to the graphs that satisfy this alternative definition but not the first definition bipartite Ramanujan graphs. If $G$ is a Ramanujan graph, then $G\times K_{2}$ is a bipartite Ramanujan graph, so the existence of Ramanujan graphs is stronger.
As observed by Toshikazu Sunada, a regular graph is Ramanujan if and only if its Ihara zeta function satisfies an analog of the Riemann hypothesis.[3]
Examples and constructions
Explicit examples
• The complete graph $K_{d+1}$ has spectrum $d,-1,-1,\dots ,-1$, and thus $\lambda (K_{d+1})=1$ and the graph is a Ramanujan graph for every $d>1$. The complete bipartite graph $K_{d,d}$ has spectrum $d,0,0,\dots ,0,-d$ and hence is a bipartite Ramanujan graph for every $d$.
• The Petersen graph has spectrum $3,1,1,1,1,1,-2,-2,-2,-2$, so it is a 3-regular Ramanujan graph. The icosahedral graph is a 5-regular Ramanujan graph.[4]
• A Paley graph of order $q$ is ${\frac {q-1}{2}}$-regular with all other eigenvalues being ${\frac {-1\pm {\sqrt {q}}}{2}}$, making Paley graphs an infinite family of Ramanujan graphs.
• More generally, let $f(x)$ be a degree 2 or 3 polynomial over $\mathbb {F} _{q}$. Let $S=\{f(x)\,:\,x\in \mathbb {F} _{q}\}$ be the image of $f(x)$ as a multiset, and suppose $S=-S$. Then the Cayley graph for $\mathbb {F} _{q}$ with generators from $S$ is a Ramanujan graph.
Mathematicians are often interested in constructing infinite families of $d$-regular Ramanujan graphs for every fixed $d$. Such families are useful in applications.
Algebraic constructions
Several explicit constructions of Ramanujan graphs arise as Cayley graphs and are algebraic in nature. See Winnie Li's survey on Ramanujan's conjecture and other aspects of number theory relevant to these results.[5]
Lubotzky, Phillips and Sarnak[2] and independently Margulis[6] showed how to construct an infinite family of $(p+1)$-regular Ramanujan graphs, whenever $p$ is a prime number and $p\equiv 1{\pmod {4}}$. Both proofs use the Ramanujan conjecture, which led to the name of Ramanujan graphs. Besides being Ramanujan graphs, these constructions satisfies some other properties, for example, their girth is $\Omega (\log _{p}(n))$ where $n$ is the number of nodes.
Let us sketch the Lubotzky-Phillips-Sarnak construction. Let $q\equiv 1{\bmod {4}}$ be a prime not equal to $p$. By Jacobi's four-square theorem, there are $p+1$ solutions to the equation $p=a_{0}^{2}+a_{1}^{2}+a_{2}^{2}+a_{3}^{2}$ where $a_{0}>0$ is odd and $a_{1},a_{2},a_{3}$ are even. To each such solution associate the $\operatorname {PGL} (2,\mathbb {Z} /q\mathbb {Z} )$ matrix
${\tilde {\alpha }}={\begin{pmatrix}a_{0}+ia_{1}&a_{2}+ia_{3}\\-a_{2}+ia_{3}&a_{0}-ia_{1}\end{pmatrix}},\qquad i{\text{ a fixed solution to }}i^{2}=-1{\bmod {q}}.$
If $p$ is not a quadratic residue modulo $q$ let $X^{p,q}$ be the Cayley graph of $\operatorname {PGL} (2,\mathbb {Z} /q\mathbb {Z} )$ with these $p+1$ generators, and otherwise, let $X^{p,q}$ be the Cayley graph of $\operatorname {PSL} (2,\mathbb {Z} /q\mathbb {Z} )$ with the same generators. Then $X^{p,q}$ is a $(p+1)$-regular graph on $n=q(q^{2}-1)$ or $q(q^{2}-1)/2$ vertices depending on whether or not $p$ is a quadratic residue modulo $q$. It is proved that $X^{p,q}$ is a Ramanujan graph.
Morgenstern[7] later extended the construction of Lubotzky, Phillips and Sarnak. His extended construction holds whenever $p$ is a prime power.
Arnold Pizer proved that the supersingular isogeny graphs are Ramanujan, although they tend to have lower girth than the graphs of Lubotzky, Phillips, and Sarnak.[8] Like the graphs of Lubotzky, Phillips, and Sarnak, the degrees of these graphs are always a prime number plus one.
Probabilistic examples
Adam Marcus, Daniel Spielman and Nikhil Srivastava[9] proved the existence of infinitely many $d$-regular bipartite Ramanujan graphs for any $d\geq 3$. Later[10] they proved that there exist bipartite Ramanujan graphs of every degree and every number of vertices. Michael B. Cohen[11] showed how to construct these graphs in polynomial time.
The initial work followed an approach of Bilu and Linial. They considered an operation called a 2-lift that takes a $d$-regular graph $G$ with $n$ vertices and a sign on each edge, and produces a new $d$-regular graph $G'$ on $2n$ vertices. Bilu & Linial conjectured that there always exists a signing so that every new eigenvalue of $G'$ has magnitude at most $2{\sqrt {d-1}}$. This conjecture guarantees the existence of Ramanujan graphs with degree $d$ and $2^{k}(d+1)$ vertices for any $k$—simply start with the complete graph $K_{d+1}$, and iteratively take 2-lifts that retain the Ramanujan property.
Using the method of interlacing polynomials, Marcus, Spielman, and Srivastava[9] proved Bilu & Linial's conjecture holds when $G$ is already a bipartite Ramanujan graph, which is enough to conclude the existence result. The sequel[10] proved the stronger statement that a sum of $d$ random bipartite matchings is Ramanujan with non-vanishing probability.
It is still an open problem whether there are infinitely many $d$-regular (non-bipartite) Ramanujan graphs for any $d\geq 3$. In particular, the problem is open for $d=7$, the smallest case for which $d-1$ is not a prime power and hence not covered by Morgenstern's construction.
Ramanujan graphs as expander graphs
The constant $2{\sqrt {d-1}}$ in the definition of Ramanujan graphs is asymptotically sharp. More precisely, the Alon-Boppana bound states that for every $d$ and $\epsilon >0$, there exists $n$ such that all $d$-regular graphs $G$ with at least $n$ vertices satisfy $\lambda (G)>2{\sqrt {d-1}}-\epsilon $. This means that Ramanujan graphs are essentially the best possible expander graphs.
Due to achieving the tight bound on $\lambda (G)$, the expander mixing lemma gives excellent bounds on the uniformity of the distribution of the edges in Ramanujan graphs, and any random walks on the graphs has a logarithmic mixing time (in terms of the number of vertices): in other words, the random walk converges to the (uniform) stationary distribution very quickly. Therefore, the diameter of Ramanujan graphs are also bounded logarithmically in terms of the number of vertices.
Random graphs
Confirming a conjecture of Alon, Friedman[12] showed that many families of random graphs are weakly-Ramanujan. This means that for every $d$ and $\epsilon >0$ and for sufficiently large $n$, a random $d$-regular $n$-vertex graph $G$ satisfies $\lambda (G)<2{\sqrt {d-1}}+\epsilon $ with high probability. While this result shows that random graphs are close to being Ramanujan, it cannot be used to prove the existence of Ramanujan graphs. It is conjectured,[13] though, that random graphs are Ramanujan with substantial probability (roughly 52%). In addition to direct numerical evidence, there is some theoretical support for this conjecture: the spectral gap of a $d$-regular graph seems to behave according to a Tracy-Widom distribution from random matrix theory, which would predict the same asymptotic.
Applications of Ramanujan graphs
Expander graphs have many applications to computer science, number theory, and group theory, see e.g Lubotzky's survey on applications to pure and applied math and Hoory, Linial, and Wigderson's survey which focuses on computer science.. Ramanujan graphs are in some sense the best expanders, and so they are especially useful in applications where expanders are needed. Importantly, the Lubotzky, Phillips, and Sarnak graphs can be traversed extremely quickly in practice, so they are practical for applications.
Some example applications include
• In an application to fast solvers for Laplacian linear systems, Lee, Peng, and Spielman[14] relied on the existence of bipartite Ramanujan graphs of every degree in order to quickly approximate the complete graph.
• Lubetzky and Peres proved that the simple random walk exhibits cutoff phenomenon on all Ramanujan graphs.[15] This means that the random walk undergoes a phase transition from being completely unmixed to completely mixed in the total variation norm. This result strongly relies on the graph being Ramanujan, not just an expander—some good expanders are known to not exhibit cutoff.[16]
• Ramanujan graphs of Pizer have been proposed as the basis for post-quantum elliptic-curve cryptography.[17]
• Ramanujan graphs can be used to construct expander codes, which are good error correcting codes.
See also
• Expander graph
• Alon-Boppana bound
• Expander mixing lemma
• Spectral graph theory
References
1. Survey paper by M. Ram Murty
2. Alexander Lubotzky; Ralph Phillips; Peter Sarnak (1988). "Ramanujan graphs". Combinatorica. 8 (3): 261–277. doi:10.1007/BF02126799. S2CID 206812625.
3. Terras, Audrey (2011), Zeta functions of graphs: A stroll through the garden, Cambridge Studies in Advanced Mathematics, vol. 128, Cambridge University Press, ISBN 978-0-521-11367-0, MR 2768284
4. Weisstein, Eric W. "Icosahedral Graph". mathworld.wolfram.com. Retrieved 2019-11-29.
5. Li, Winnie (2020). "The Ramanujan conjecture and its applications". Philosophical Transactions of the Royal Society A. 378–2163 (2163). Bibcode:2020RSPTA.37880441W. doi:10.1098/rsta.2018.0441. PMC 6939229. PMID 31813366.
6. Margulis, G. A. (1988). "Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators". Probl. Peredachi Inf. 24–1: 51–60.
7. Moshe Morgenstern (1994). "Existence and Explicit Constructions of q+1 Regular Ramanujan Graphs for Every Prime Power q". Journal of Combinatorial Theory, Series B. 62: 44–62. doi:10.1006/jctb.1994.1054.
8. Pizer, Arnold K. (1990), "Ramanujan graphs and Hecke operators", Bulletin of the American Mathematical Society, New Series, 23 (1): 127–137, doi:10.1090/S0273-0979-1990-15918-X, MR 1027904
9. Adam Marcus; Daniel Spielman; Nikhil Srivastava (2013). Interlacing families I: Bipartite Ramanujan graphs of all degrees (PDF). Foundations of Computer Science (FOCS), 2013 IEEE 54th Annual Symposium.
10. Adam Marcus; Daniel Spielman; Nikhil Srivastava (2015). Interlacing families IV: Bipartite Ramanujan graphs of all sizes (PDF). Foundations of Computer Science (FOCS), 2015 IEEE 56th Annual Symposium.
11. Michael B. Cohen (2016). Ramanujan Graphs in Polynomial Time. Foundations of Computer Science (FOCS), 2016 IEEE 57th Annual Symposium. arXiv:1604.03544. doi:10.1109/FOCS.2016.37.
12. Friedman, Joel (2003). "Relative expanders or weakly relatively Ramanujan graphs". Duke Math. J. 118 (1): 19–35. doi:10.1215/S0012-7094-03-11812-8. MR 1978881.
13. Miller, Steven J.; Novikoff, Tim; Sabelli, Anthony (2006-11-21). "The Distribution of the Largest Non-trivial Eigenvalues in Families of Random Regular Graphs". arXiv:math/0611649.
14. Lee, Yin Tat; Peng, Richard; Spielman, Daniel A. (2015-08-13). "Sparsified Cholesky Solvers for SDD linear systems". arXiv:1506.08204 [cs.DS].
15. Lubetzky, Eyal; Peres, Yuval (2016-07-01). "Cutoff on all Ramanujan graphs". Geometric and Functional Analysis. 26 (4): 1190–1216. arXiv:1507.04725. doi:10.1007/s00039-016-0382-7. ISSN 1420-8970. S2CID 13803649.
16. Lubetzky, Eyal; Sly, Allan (2011-01-01). "Explicit Expanders with Cutoff Phenomena". Electronic Journal of Probability. 16 (none). doi:10.1214/EJP.v16-869. ISSN 1083-6489. S2CID 9121682.
17. Eisenträger, Kirsten; Hallgren, Sean; Lauter, Kristin; Morrison, Travis; Petit, Christophe (2018), "Supersingular isogeny graphs and endomorphism rings: Reductions and solutions" (PDF), in Nielsen, Jesper Buus; Rijmen, Vincent (eds.), Advances in Cryptology – EUROCRYPT 2018: 37th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Tel Aviv, Israel, April 29 - May 3, 2018, Proceedings, Part III (PDF), Lecture Notes in Computer Science, vol. 10822, Cham: Springer, pp. 329–368, doi:10.1007/978-3-319-78372-7_11, MR 3794837
Further reading
• Giuliana Davidoff; Peter Sarnak; Alain Valette (2003). Elementary number theory, group theory and Ramanujan graphs. LMS student texts. Vol. 55. Cambridge University Press. ISBN 0-521-53143-8. OCLC 50253269.
• Sunada, Toshikazu (1986). "L-functions in geometry and some applications". In Shiohama, Katsuhiro; Sakai, Takashi; Sunada, Toshikazu (eds.). Curvature and Topology of Riemannian Manifolds: Proceedings of the 17th International Taniguchi Symposium held in Katata, Japan, August 26–31, 1985. Lecture Notes in Mathematics. Vol. 1201. Berlin: Springer. pp. 266–284. doi:10.1007/BFb0075662. ISBN 978-3-540-16770-9. MR 0859591.
External links
• Survey paper by M. Ram Murty
• Survey paper by Alexander Lubotzky
• Survey paper by Hoory, Linial, and Wigderson
| Wikipedia |
\begin{document}
\begin{abstract} Experimentally observed magnetic fields with nanoscale variations are theoretically modeled by a piece-wise constant function with jump discontinuity along a smooth curve, the magnetic edge. Assuming the edge is a closed curve with an axis of symmetry and the field is sign changing and with exactly two distinct values, we prove that semi-classical tunneling occurs and calculate the magnitude of this tunneling effect.\\
{\it This paper is dedicated to Elliott H. Lieb on the occasion of his 90th birthday.} \end{abstract} \maketitle
\section{Introduction}\label{sec:Int}
The purpose of this paper is to study the magnetic Laplacian in $\mathbb R^2$,
\begin{equation}\label{eq:P} \mathcal P_{h}:=(ih\nabla+\mathbf A)^2=\sum_{j=1}^2 (ih \partial_{x_j} +A_j)^2, \end{equation} where $\mathbf A:=(A_1,A_2)\in H^1_\textrm{loc}(\mathbb R^2;\mathbb R^2)$, is a magnetic potential generating the magnetic field $B=\textrm{ curl}\mathbf A:=\partial_{x_1}A_2-\partial_{x_2}A_1\in L^2_\textrm{ loc}(\mathbb R^2;\mathbb R)$.
We will also discuss the case of the Neumann or Dirichlet realizations of $\mathcal P_h$ in smooth bounded planar domains.
Here $h$ is a positive parameter that tends to $0$, which can be interpreted as the semi-classical parameter. By writing $h^{-2}\mathcal P_h=(i\nabla+h^{-1}\mathbf A)^2$, we observe that the semi-classical limit, $h\to0_+$, is equivalent to the strong magnetic field limit, $h^{-1}|B|\to+\infty$.
The spectrum of the operator $\mathcal P_h$ has been the subject of an intense study in the past decades, particularly in the context of superconductivity where the magnetic field $B$ is typically a constant function \cite{BS98, Bon05, FH10, HM1, HP03, LuP99}.
There is an interesting analogy between the results for the Neumann realization of $\mathcal P_h$ in a bounded smooth domain and those for the Schr\"{o}dinger operator, $-h^2\Delta+V$, with an electric potential $V$, in the full plane. The Schr\"{o}dinger operator was intensively studied by Helffer--Sj\"{o}strand \cite{HeSj,HeSj5} and Simon \cite{S}, notably in the context of quantum tunneling. Bound states of
$-h^2\Delta+V$ concentrate near the `well' $\Gamma_V:=\{x\in\mathbb R^2\,|\,V(x)=\min_{\mathbb R^2} V\}$; if furthermore $\Gamma_V$ is a regular manifold (i.e. we have a degenerate well), bound states could concentrate near some points of $\Gamma_V$, the `mini-wells'. We have the same picture in the purely magnetic case with a Neumann boundary condition: bound states concentrate near the boundary of the domain, whereby the boundary plays the role of a (degenerate) well, and the set of points of maximum curvature plays the role of mini-wells, where bound states decay away from them. Optimal estimates describing the concentration of bound states are very important, since they lead to accurate asymptotics for the low lying eigenvalues. The proof of the decay away from the mini-wells (points of maximum curvature), is more delicate compared to that of the decay away from the well (boundary).
In this paper, our main focus will be on magnetic fields having a jump-discontinuity. Magnetic fields that vary on very short scales (nanoscales) have been observed experimentally, see e.g.~\cite{FLBP94}. Their theoretical investigations, in the context of quantum mechanics \cite{PM93, RP00} or graphene \cite{GDMH08}, involve the operator $\mathcal P_h$ but with the magnetic field $B$ being a step function having a discontinuity along a curve, that we will refer to as the \emph{magnetic edge}.
Earlier rigorous results were devoted to the case of a flat edge \cite{HPRS16, HS15, I85}. More recently, non-flat edges have been considered in the context of spectral asymptotics \cite{A20, AHK} and in the context of superconductivity \cite{AKPS19}. The magnetic edge will play the role of the `well', while the `mini-wells' are the points of maximum curvature of the magnetic edge \cite{AK20}, which is interestingly in analogy with the setting of the Neumann realization with a constant magnetic field in a bounded smooth domain.
The case of a single mini-well, where the curvature of the edge has a unique and non-degenerate maximum, was analyzed by Assaad--Helffer--Kachmar \cite{AHK}. The present paper investigates the situation of a symmetric edge with several mini-wells, the simplest case being when there are two non-degenerate maxima of the boundary curvature. We establish a sharp asymptotics of the splitting between the energies of the ground and first excited state, which measures a tunneling effect induced by the geometry of the edge, see Theorem~\ref{thm:FHK} below which is our main result.
Let us recall how the general strategy of Helffer--Sj\"{o}strand \cite{HeSj, HeSj5} has been applied recently to understand the tunneling effect for the Neumann realization in a bounded domain with the breakthrough paper \cite{BHR21} by Bonnaillie-No\"{e}l--H\'erau--Raymond as the crowning achievement. The first step, already performed in \cite{HM1} and \cite{FH06}, was the analysis of a model with a flat boundary (de\,Gennes model), which yields localization of bound states near the boundary of the domain (the well), and consequently, leads to a full asymptotics of the low-lying eigenvalues. The second step is a formal WKB expansion of bound states \cite{BHR15}. The third step consists of optimal decay estimates of bound states recently achieved in \cite{BHR21}. The importance of this step is that it allows one to rigorously approximate the bound states by the formal WKB expansions, which eventually paves the way for the analysis of an interaction matrix whose eigenvalues measure the tunneling effect. The same approach has been successfully applied in the context of thin domains \cite{KR17} and the Robin Laplacian \cite{HK-tams, HKR}, where the proof of the tangential estimates was less technical.
We will follow the same approach outlined above in the case of our discontinuous magnetic field. The model problem with a flat edge was analyzed in \cite{AK20} (see also \cite{AKPS19, HPRS16}), while the full asymptotics for the low lying eigenvalues are obtained in \cite{AHK}. So we still need WKB expansions and optimal tangential estimates of bound states, which we do in the present contribution. Finally, after establishing the WKB approximation, the analysis of the interaction matrix is quite standard.
Let us give an informal statement of our main result (Theorem~\ref{thm:FHK} below). Suppose that $\Gamma$ is a smooth, closed curve in $\mathbb R^2$, symmetric with respect to an axis, and with two points of maximum curvature, denoted by $s_\ell$ and $s_r$ ($\ell$ refers to ``left'' and $r$ to ``right'', see Fig.~\ref{fig2}). Consider the magnetic field satisfying $B=1$ in the interior of $\Gamma$, and $B=a\in(-1,0)$ in the exterior of $\Gamma$. Under these assumptions, we prove that, as $h\to0_+$, the spectral gap of the operator $\mathcal P_h$ is of exponential order, \begin{equation}\label{eq:informal-tun} \lambda_2(h)-\lambda_1(h)\approx \exp\Big(-\frac{\mathsf S^a}{h^{1/4}} \Big), \end{equation} where $\mathsf S^a$ is the Agmon distance between the ``wells'' $s_\ell$ and $s_r$ defined by an appropriate potential that depends on the magnetic field (through the parameter $a$) and the geometry of $\Gamma$ (through the curvature).
\begin{figure*}
\caption{A symmetric domain with respect to the $y$-axis (dashed line). The orientation of the boundary is defined by the direct frame $(\mathbf t,\mathbf n)$, where $\mathbf n$ is the inward normal vector and $\mathbf t$ is the unit tangent. The curvature along the boundary has two non-degenerate maxima at the points $a_1$ and $a_2$, with arc-length coordinates $ s_\ell\in [0,L) $ and $ s_r\in(-L,0]$, connected by upward and downward geodesics oriented counterclockwise and represented by $[s_r,0]\cup(0,s_\ell]$ and $[s_\ell,L]\cup(-L,s_r]$ respectively. These upward and downward geodesics will be denoted by $[s_r,s_\ell]$ and $[s_\ell,s_r]$ respectively.}
\label{fig2}
\end{figure*}
The asymptotics in \eqref{eq:informal-tun} (more precisely that in Theorem~\ref{thm:FHK}) is a consequence of quantum tunneling. It is important to note that it is induced purely by the magnetic field, thereby providing an example of a purely magnetic quantum tunneling---where the case of \cite{BHR21} also required the interaction with the boundary. If we look at earlier results on the tunneling effect, with or without magnetic field, we observe that the tunneling is induced by an external potential \cite{HeSj, FSW} or by confinement to a bounded/thin domain \cite{HKR, KR17, BHR}. For the Neumann realization of $\mathcal P_h$, the presence of the magnetic field adds a challenging difficulty in the estimate of the magnitude of the tunneling that was recently solved in \cite{BHR21}. Our proof of \eqref{eq:informal-tun} is very close to that of \cite{BHR21}, but it relies on new elements that follow from a deep investigation of magnetic steps \cite{AK20, AHK}.
Let us give some of the heuristics behind the computations leading to \eqref{eq:informal-tun}. We can construct two quasi-modes having the following structure \begin{align*} \Psi_{h,\ell}(s,t)&\approx e^{i\theta_{h,\ell}(s)}\,e^{-\Phi_\ell(s)/h^{1/4}}\,f_{0,\ell}(s)\phi_a(h^{1/2}t),\\ \Psi_{h,r}(s,t)&\approx e^{i\theta_{h,r}(s)}e^{-\Phi_r(s)/h^{1/4}}\,f_{0,r}(s)\phi_a(h^{1/2}t), \end{align*} where $s$ denotes the arc-length parameter along $\Gamma$, $t$ denotes the normal distance to $\Gamma$ with the convention that $t>0$ in the interior of $\Gamma$ and $t<0$ in the exterior of $\Gamma$. The functions $\Phi_\ell$ and $\Phi_r$ are non-negative and satisfy $\Phi_{\ell}(s_\ell)=0$ and $\Phi_r(s_r)=0$, so that $\Psi_{h,\ell}$ (resp. $\Psi_{h,r}$) is localized near $s_\ell$ (resp. near $s_r$). The phase functions $\theta_{h,\ell}$ and $\theta_{h,r}$ involve the topology of the discontinuity curve and a spectral constant. The function $\phi_a$ is the ground state eigenfunction of a model operator related to the discontinuity of the magnetic field (see Sec.~\ref{sec:ms}). Finally, $f_{0,\ell}$ and $f_{0,r}$ are solutions of appropriate transport equations (see Theorem~\ref{thm:WKB}).
Up to truncation, the quasi-modes $\Psi_{h,\ell}$ and $\Psi_{h,r}$ are approximations of actual bound states
\begin{equation}\label{eq:app-introduction} g_{h,\ell}(s,t)\approx \Psi_{h,\ell}(s,t),\quad g_{h,r}(s,t) \approx \Psi_{h,r}(s,t), \end{equation} where the bound states $g_{h,\ell}$ and $g_{h,r}$ are defined via the orthogonal projection $\Pi$ on $ V:=\oplus_{i=1}^2\textrm{ Ker}\left(\mathcal P_h-\lambda_i(h)\right)$ as follows \[ g_{h,\ell}(s,t):=\Pi\Psi_{h,\ell}(s,t),\quad g_{h,r}(s,t):=\Pi\Psi_{h,r}(s,t).\]
By the Gram-Schmidt process, we transform $\{g_{h,\ell},g_{h,r} \}$ to an orthonormal basis $\mathcal B$ of $V$ and we denote by $\mathsf M_h$ the matrix relative to $\mathcal B$ of the restriction of $\mathcal P_h$ to $V$. The spectral gap for the operator $\mathcal P_h$ is the same as that for the matrix $\mathsf M_h$, \[\lambda_2(h)-\lambda_1(h)= \lambda_2(\mathsf M_h)-\lambda_1(\mathsf M_h) \,.\] Using the approximation in \eqref{eq:app-introduction}, we get an approximate matrix $\widehat{\mathsf M}_h$ of $\mathsf M_h$ whose spectral gap can be explicitly estimated (compare with \eqref{eq:informal-tun}) \[ \lambda_2(\widehat{\mathsf M}_h)-\lambda_1(\widehat{\mathsf M}_h)\approx \exp\Big(-\frac{\mathsf S^a}{h^{1/4}} \Big)\,.\]
We have then to show that the spectral gap for the matrix $\widehat{\mathsf M}_h$ is a good approximation of that of $\mathcal P_h$ up to an appropriate remainder, more precisely \[ \lambda_2(\mathsf M_h)-\lambda_1(\mathsf M_h)= \big(\lambda_2(\widehat{\mathsf M}_h)-\lambda_1(\widehat{\mathsf M}_h)\big)\big(1+o(1)\big)\,.\] Such an estimate is closely related to optimal decay estimates of bound states (resp. approximate bound states) of the operator $\mathcal P_h$, which yield accurate errors for the approximation in \eqref{eq:app-introduction}.
\subsection*{Organisation}
In Section~\ref{sec:B=cst}, we review the recent result of \cite{BHR21} for the Neumann magnetic Laplacian with a constant magnetic field, and introduce the related de\,Gennes model for a flat boundary. In Section~\ref{sec:ms}, we introduce the magnetic edge along with the related flat edge model, and state our main result, Theorem~\ref{thm:FHK}, for the operator $\mathcal P_h$ with a magnetic step. In Section~\ref{sec:curv}, we express $\mathcal P_h$ in a Fr\'enet frame and reduce the spectral analysis to an operator defined near the edge. In Section~\ref{sec:WKB}, we introduce operators with a single well (with ground states localized near a single point of maximum curvature), and perform a WKB expansion for an approximate ground state (see Theorem~\ref{thm:WKB}). In Section~\ref{sec:dec}, we explain how optimal tangential estimates can be derived along the lines of the proof of the similar statement in \cite{BHR21}. Finally, in Section~\ref{sec:IM}, we introduce the interaction matrix and finish the proof of Theorem~\ref{thm:FHK}, by referring to \cite{BHR21} for the detailed computations, which are essentially the same in our setting.
\section{Uniform magnetic fields}\label{sec:B=cst}
In this section, we review some results on the Neumann realization of the operator $\mathcal P_h$ with a constant magnetic field. We assume that \begin{equation}\label{eq:Sigma} \left\{\,\begin{aligned} &\Omega\subset\mathbb R^2\text{ is a }\text{simply}\text{ connected open set,}\\ &\Sigma:=\partial\Omega\text{ is a }C^\infty\text{ smooth closed curve.}\\ \end{aligned}\,\right\} \end{equation}
and
\begin{equation}\label{eq:B=1} \mathbf A=\frac12(-x_2,x_1)\quad\textrm{ and}\quad B=\textrm{ curl}\,\mathbf A\equiv1\,, \end{equation}
We consider $\mathcal P_h$ introduced in \eqref{eq:P}, as a self-adjoint operator in $L^2(\Omega)$, with domain, \begin{equation*}
\textsf{ Dom}(\mathcal P_{h})=\{u\in H^2(\Omega)~|~{\mathbf n}\cdot(h\nabla-i\mathbf A)u|_{\partial \Omega}=0 \}, \end{equation*} where $H^2(\Omega)$ denotes the Sobolev space $W^{2,2}(\Omega)$, and $\mathbf n$ the unit normal vector of $\Sigma$, pointing inwards with respect to $\Omega$.
\subsection{Full asymptotics and decay of bound states}
The conditions in \eqref{eq:Sigma} ensure that $\Omega$ is bounded and that $\mathcal P_h$ has compact resolvent. Let $(\lambda_n(h))_{n\geq 1}$ be the sequence of eigenvalues of $\mathcal P_h$. In generic situations, that will be explained precisely later on, there exist complete expansions of the eigenvalues of $\mathcal P_h$, in the form \cite{FH06}, \begin{equation}\label{eq:lambdan} \lambda_n(h)\sim\Theta_0h -k_{\max}C_1h^\frac 32 +C_1\Theta_0^{\frac 14}(2n-1)\sqrt{-\frac 32 k_2}h^\frac 74+\sum_{j\geq 15}\zeta_{j,n} h^{j/8}\,. \end{equation} The coefficients $\Theta_0$ and $C_1$ appearing in \eqref{eq:lambdan} are universal positive constants related to the de\,Gennes model in the half-plane (see Sec.~\ref{sec:dG}). The coefficients $k_{\max}$ and $k_2$ are related to the curvature on the boundary. Let $\Sigma$ be parameterized by arc-length $s$ and denote by $k(s)$ the curvature of $\Sigma$ at $s$ (see Sec.~\ref{sec:curv} for the precise definition of $k$; in particular the orientation is chosen so that $k\geq 0$ if $\Omega$ is convex). The asymptotics in \eqref{eq:lambdan} holds provided the curvature $k$ attains its maximum value non-degenerately and at a unique point, i.e. \begin{equation}\label{eq:hyp-gd} k_{\max}:=\max_{\Sigma} k(s)=k(0)\quad\textrm{ with}\quad k_2:=k''(0)<0\,. \end{equation} The sequence $(\zeta_{j,n})_{j\geq 15}$ is constructed recursively, and it can be shown that $\zeta_{j,1}=0$ for odd $j$ \cite{BHR15}.
The derivation of \eqref{eq:lambdan} is related to the decay of bound states. Assume that $n$ is fixed and for all $h>0$ that $u_{h,n}$ is an eigenfunction of $\mathcal P_h$, normalized in $L^2(\Omega)$ and with eigenvalue $\lambda_n(h)$. There exist constants $\alpha_1,C_{1,n}>0$ such that
\[ \int_\Omega |u_{h,n}|^2\exp\Big(\frac{\alpha_1 \textrm{ dist}(x,\Sigma)}{h^{1/2}}\Big)dx\leq C_{1,n}\,.\] This estimate says that the bound state $u_{h,n}$ concentrates near the boundary $\Sigma$ and is valid even when \eqref{eq:hyp-gd} is not satisfied \cite{HM1}. If moreover \eqref{eq:hyp-gd} holds, then $u_{h,n}$ concentrates near the point of maximal curvature as follows: There exist constants $\epsilon_0,\alpha_2,C_{2,n}>0$ such that \cite{FH06} \begin{equation}\label{eq:dec-uh}
\int_{\textrm{ dist}(x,\Sigma)<\epsilon_0}|u_{h,n}|^2\exp\Big(\frac{\alpha_2|s(x)|^2}{h^{1/4}}\Big)dx\leq C_{2,n}\,, \end{equation} where $s(x)$ denotes the arc-length coordinate of the point $p(x)\in\partial\Omega$ defined by $\textrm{dist}(x,p(x))=\textrm{ dist}(x,\Sigma)$. The decay estimate \eqref{eq:dec-uh} is a key ingredient in the derivation of the asymptotics in \eqref{eq:lambdan}, but is not sufficient to handle the case of symmetries that we shall discuss below.
Let us examine the case where the curvature attains its maximum at several points $s_1,\cdots,s_N$. For all $j\in\{1,\cdots,N\}$ and $m\in\mathbb N$, we introduce \[\lambda_{m,j}^\textrm{ app}(h)=\Theta_0h -k_{\max}C_1h^\frac 32 +C_1\Theta_0^{\frac 14}(2m-1)\sqrt{-\frac 32 k''(s_j)}\,h^\frac 74\,. \] Consider a relabeling $(m_n,j_n)_{n\geq 1}$ of $(m,j)_{m\geq 1,1\leq j\leq N}$ such that \[\lambda_{m_1,j_1}^{\textrm{app}}\leq \lambda_{m_2,j_2}^\textrm{app}\leq \cdots \,.\] Then, \eqref{eq:lambdan} is replaced with \begin{equation}\label{eq:mult-well} \lambda_n(h)=\lambda^\textrm{app}_{m_n,j_n}+o(h^{\frac74})\,. \end{equation} If additionally $k''(s_{j_1})=k''(s_{j_2})$, then $\lambda_2(h)-\lambda_1(h)=o(h^{\frac{7}4})$ and we loose the information on the simplicity of the eigenvalues. Consequently, we need a more detailed analysis in the case of symmetries, which will rely on an optimal tangential decay estimate improving the one given in \eqref{eq:dec-uh}. We will discuss these decay estimates later in Sec.~\ref{sec:dec}. Our next step is the review of an important model with a flat boundary.
\subsection{The de\,Gennes model: flat boundary}\label{sec:dG}
The analysis of the model case where $\Omega=\mathbb R\times\mathbb R_+$ and $B=\textrm{ curl\,}\mathbf A=1$ leads us naturally to the family (parametrized by $\xi\in \mathbb R$) of harmonic oscillators (de\,Gennes model) \begin{equation} \mathfrak h^{N}[\xi]=-\frac{d^2}{d\tau^2}+(\xi+\tau)^2, \end{equation} on the semi-axis $\mathbb R_+$ with Neumann boundary condition at $\tau=0$. Let us denote by $(\mu_j^{N}(\xi))_{j\geq 1}$ the sequence of eigenvalues of $\mathfrak h^N[\xi]$. The de\,Gennes constant is then defined as follows \begin{equation} \Theta_0=\inf_{\xi\in\mathbb R}\mu^{N}_1(\xi)\,. \end{equation}
There exists a unique minimum $\xi_0<0$ such that \[ \Theta_0=\mu^{N}_1(\xi_0)\,.\] Furthermore, $\xi_0=-\sqrt{\Theta_0}$, $(\mu^N_{1})''(\xi)>0$ and $\frac12<\Theta_0<1$. Denoting by $u_0$ the positive and normalized ground state of $\mathfrak h^N[\xi_0]$, we can introduce the constant $C_1$ appearing in \eqref{eq:lambdan}, \begin{equation}
C_1=\frac{|u_0(0)|^2}{3}\,. \end{equation}
\subsection{Symmetric domains and tunneling}\label{sec:SymDom}
We continue to work under the conditions in \eqref{eq:Sigma} but we assume furthermore that the domain $\Omega$ is symmetric with respect to an axis and the curvature of its boundary $\Gamma$ has exactly two non-degenerate maxima. More precisely, the hypotheses are (see Fig~\ref{fig2}): \begin{assumption}\label{ass:symN}~ \begin{enumerate}[\rmfamily i)]
\item $\Omega$ is symmetric with respect to the $y$-axis.
\item The curvature $k$ on $\Sigma$ attains its maximum at exactly two symmetric points $a_1=(a_{1,1},a_{1,2})$ and $a_2=(a_{2,1},a_{2,2})$ with $a_{1,1}<0$ and $ a_{2,1}>0$.
\item Denoting by $s_r$ and $s_\ell$ the arc-length coordinates of $a_1$ and $a_2$ respectively, we have $k''(s_r)=k''(s_\ell)<0$. \end{enumerate} \end{assumption}
This situation induces a tunneling effect where the energy difference between the ground and first excited states is exponentially small. The magnitude of this splitting has been rigorously computed recently in \cite{BHR21}.
Let us introduce the following effective quantities: \begin{equation}\label{eq:eff-pot} V(s)=\frac{2C_1(k_{\max}-k(s))}{(\mu_1^N)''(\xi_0)}\,, \end{equation} and \begin{equation}\label{eq.Aud} \begin{aligned} \mathsf{A}_{\mathsf{u}}&=\exp\left(-\int_{[s_{r}, 0]} \frac{ (V^\frac 12 )' (s)+g}{ \sqrt{V(s)}} ds\right)\,,\\ \mathsf{A}_{\mathsf{d}}&=\exp\left(-\int_{[s_{\mathsf{\ell}}, L]} \frac{ (V^\frac 12 )' (s) -g}{ \sqrt{V(s)}} ds\right)\,,\\ g&=\left(V''(s_{r})/2\right)^\frac 12=\left(V''(s_{\mathsf{\ell}})/2\right)^\frac 12\,. \end{aligned} \end{equation}
In the above formulae, $0$ and $L$ are the arc-length coordinates of the points of intersection between the $y$-axis and the curve $\Sigma$, with the convention that $0$ represents the point on the upper part of $\Sigma$ (see Fig.~\ref{fig2}).
\begin{theorem}[Bonnaillie-No\"{e}l--H\'erau--Raymond \cite{BHR21}]\label{thm:BHR}~ Suppose that \eqref{eq:Sigma}, \eqref{eq:B=1} and Assumption~\ref{ass:symN} hold. Then the first and second eigenvalues of $\mathcal P_h$ satisfy, as $h\to0_+$,
\[\lambda_2(h)-\lambda_1(h)=2| w(h)|+o(h^{\frac{13}{8}}e^{-\mathsf{S}/h^{\frac 14}})\,,\] where \begin{align*} w(h)&=( \mu_1^N)''(\xi_0) h^{\frac{13}{8}} \pi^{-\frac 12} g^{\frac12}\\ &\quad \times \left(\mathsf{A}_{\mathsf{u}} \sqrt{V(0)}e^{- \mathsf{S}_{\mathsf{u}}/h^{1/4}} e^{iL f(h)}+\mathsf{A}_{\mathsf{d}} \sqrt{V(L)}e^{- \mathsf{S}_{\mathsf{d}}/h^{1/4}} e^{-iLf(h)}\right)\,,
\end{align*}
and \begin{enumerate}[\rmfamily i.] \item The potential $V$ is introduced in \eqref{eq:eff-pot}; \item $\mathsf S$ is the Agmon distance between the wells, \begin{equation}\label{defS} \mathsf{S} =\min \left(\mathsf{S}_{\mathsf{u}},\mathsf{S}_{\mathsf{d}}\right),~ \mathsf{S}_{\mathsf{u}}=\int_{[s_{r},s_{\mathsf{\ell}}] } \sqrt{V(s)} \,ds,~ \mathsf{S}_{\mathsf{d}}=\int_{[s_{\mathsf{\ell}}, s_{\mathsf{r}}] } \sqrt{V(s)} \,ds\,; \end{equation} \item $\mathsf{A}_{\mathsf{u}}$, $\mathsf{A}_{\mathsf{d}}$ and $g$ are defined in \eqref{eq.Aud}; \item $f(h)=\gamma_0/h+\xi_0/h^{1/2}-\alpha_0$ with
\[\gamma_0=\frac{|\Omega|}{|\Sigma|},\]
where $|\Sigma|$ is the length of $\Sigma$, and $\alpha_0$ is a constant dependent on $\Omega$. \end{enumerate} \end{theorem}
Theorem~\ref{thm:BHR} can be extended to the situation of $N\geq 3$ wells, which corresponds to a domain having symmetry by rotation of angle $2\pi/N$ and $N$ points of maximum curvature (see Fig.~\ref{fig:star}). \begin{figure*}
\caption{A symmetric domain with respect to the origin with $N=4$ points of maximum curvature.}
\label{fig:star}
\end{figure*}
\section{Magnetic steps}\label{sec:ms}
The tunneling effect in Theorem~\ref{thm:BHR} is a consequence of the magnetic field and imposing the Neumann boundary condition (if a magnetic field were not present, the first eigenvalue would be simple and equal to $0$, while the Neumann boundary condition inforces bound states to concentrate near the boundary points of maximum curvature thereby inducing a phenomenon of \emph{multiple wells}).
The present contribution is concerned with the following question:\\ \textit{Can we observe a tunneling effect, similar to the one in Theorem~\ref{thm:BHR}, but induced purely by the magnetic field ?}\\
That is, we would like to construct an example where the tunneling is not a consequence of imposing a boundary condition, but rather a consequence of the nature of the magnetic field. We will give an affirmative answer by working in the full plane $\mathbb R^2$ and considering a magnetic field with a discontinuity along a smooth curve\footnote{ From a technical perspective, the magnetic discontinuity curve plays the same role in our case as the boundary does in Theorem~\ref{thm:BHR}.} (the magnetic edge). In the case of a flat edge, we get a model in the full plane which plays the role of the de\,Gennes model for uniform magnetic fields. When the edge is non-flat and has symmetries, we observe an interesting tunneling effect.
\subsection{A new model: flat edge}\label{sec:FlatEdge}
Let us recall the model in $\mathbb R^2$ where $B=\textrm{ curl\,}\mathbf A=\mathbf 1_{\mathbb R_+\times\mathbb R }+a\mathbf 1_{\mathbb R_-\times\mathbb R}$ and $a\in[-1,0)$ is a fixed constant\footnote{It is important for us to have $a<0$, because in the opposite case, $a \in (0,1)$, $\mu_a(\xi)$ defined in \eqref{mu_a_1} becomes a monotone increasing function with $\inf_{\xi \in{\mathbb R}} \mu_a(\xi) = a$. This implies that the magnetic step will no longer attract the ground state, i.e. we do not expect localization near the magnetic step in this case. }. We get in this case a family of Schr\"{o}dinger operators \cite{HPRS16} \begin{equation}\label{eq:ha} \mathfrak h_a[\xi]=-\frac{d^2}{d\tau^2}+V_a(\xi,\tau), \end{equation} on $L^2(\mathbb R)$, where $\xi\in\mathbb R$ is a parameter and \begin{equation}\label{eq:potential} V_a(\xi,\tau)=\big(\xi+b_a(\tau)\tau\big)^2,\quad b_a(\tau)=\mathbf{1}_{\mathbb R_+}(\tau)+a\mathbf{1}_{\mathbb R_-}(\tau)\,. \end{equation} We introduce the ground state energy of $\mathfrak h_a[\xi]$, \begin{equation}\label{mu_a_1} \mu_a(\xi)=\inf_{u\in B^1(\mathbb R),u\neq0} \frac{
\|u'\|_{L^2(\mathbb{R})}^2+\|\sqrt{V_a}\, u\|^2_{L^2(\mathbb{R})}}
{\|u\|^2_{L^2(\mathbb R)}}\,, \end{equation} along with the following constant \begin{equation}\label{eq:beta} \beta_a:=\inf_{\xi \in \mathbb R} \mu_a(\xi)=\mu_a(\zeta_a)\,, \end{equation} where $\zeta_a<0$, is the unique minimum of $\mu_a(\cdot)$. Let $\phi_a$ be the \emph{positive} and $L^2$-normalized ground state of $\mathfrak h_a[\zeta_a]$. We have \cite{AK20} \begin{equation}\label{eq:c2} c_2(a):=\frac12\mu_a''(\zeta_a)>0 \end{equation} and \begin{equation}\label{eq:beta*}
|a|\Theta_0< \beta_a<\min(|a|,\Theta_0),\quad \phi_a'(0)<0\quad (-1<a<0)\,.
\end{equation} For $a=-1$, we have by a symmetry argument \begin{equation}\label{eq:deG}
\beta_{-1} = \Theta_0\,,\quad \zeta_{-1}=\xi_0\,,\quad \phi_{-1}(\tau)=u_0(|\tau|)\,, \end{equation} thereby returning to the de\,Gennes model introduced in Sec.~\ref{sec:dG}.
Later on, the following negative constant will be of particular interest, \begin{equation}\label{eq:m3} M_3(a)=\frac 13\Big(\frac 1a-1\Big)\zeta_a\phi_a(0)\phi_a'(0) <0. \end{equation}
\subsection{Curved edge and single well}\label{sec:Edge} We return to the operator $\mathcal P_h$ in \eqref{eq:P}. Here and in the rest of the paper, we will work under the following assumption\footnote{Our results are likely to hold when $\Gamma$ is $C^N$ smooth for some integer $N\geq 1$. We impose the $C^\infty$ hypothesis since we use psudo-differential calculus and sought errors of order $\mathcal O(h^\infty)$.} \begin{equation}\label{eq:Gam} \left\{\,\begin{aligned} &\Omega_1\subset\mathbb R^2\text{ is a }\text{simply connected open set,}~\Omega_2=\mathbb R^2\setminus\overline{\Omega}_1,\\ &\Gamma:=\partial\Omega_1 \text{ is a }C^\infty\text{ smooth closed curve.}\\ \end{aligned}\,\right\} \end{equation} and that the magnetic field is a step function (see Fig.~\ref{fig1})
\begin{figure*}
\caption{The plane $\mathbb R^2=\Omega_1\cup\Omega_2\cup\Gamma$ with the non symmetric edge $\Gamma=\partial\Omega_1$ dashed.}
\label{fig1}
\end{figure*}
\begin{equation}\label{eq:B-ms} B=\mathbf 1_{\Omega_1}+a\mathbf 1_{\Omega_2}\quad\textrm{ where~}-1< a<0\,. \end{equation} The operator $\mathcal P_h$ is then self-adjoint in $L^2(\mathbb R^2)$ with domain\footnote{Since $\mathbf A\in H^1_{\rm loc}(\mathbb{R}^2)$, there is no jump across $\Gamma$ of $u$ and $\mathbf n\cdot(h\nabla-i\mathbf A)u$, $\forall u\in{\rm Dom}(\mathcal P_h)$. } \begin{equation}\label{eq:DomP} \textsf{ Dom}(\mathcal P_{h})=\{u\in L^2(\mathbb R^2)~:~(h\nabla-i\mathbf A)^{j}u\in L^2(\mathbb R^2),~j=1,2\}. \end{equation}
By Persson's lemma \cite{P}, the essential spectrum of $\mathcal P_h$ is determined by the magnetic field at infinity (in our case it is equal to $a$), so
\[ \inf \sigma_\textrm{ ess}(\mathcal P_h)=|a| h\,.\]
Since $\beta_a<\min(|a|,\Theta_0)$, bound states of $\mathcal P_h$ are localized near the edge \cite{AK20}. More precisely, for every $n\in\mathbb N$, there exist constants $\alpha,h_0,C_n>0$ such that, \begin{equation}\label{eq:dec-norm}
\int_{\mathbb R^2}\big(|u_{h,n}|^2+h^{-1}|(h\nabla-i\mathbf A)u_{h,n}|^2 \big)\exp\Big(\frac{\alpha\, \textrm{ dist}(x,\Gamma)}{h^{1/2}} \Big)dx\leq C_n, \end{equation} for all $h\in (0,h_0]$, where $u_{h,n}$ is a normalized eigenfunction associated to the $n$'th eigenvalue of $\mathcal P_h$.
\begin{remark}[The case of bounded domains] We can also consider the Dirichlet or Neumann realizations of $\mathcal P_h$ in a bounded smooth domain $\Omega$, in which case the spectrum is purely discrete.
Related to our setting is \cite[Thm.~1.2]{AHK} dealing with a somehow different geometric condition, where the operator $\mathcal P_h$ is considered in $L^2(\Omega)$ with Dirichlet boundary condition, $\Omega_1\subset \Omega$ and $\Gamma$ a smooth curve that meets $\partial\Omega$ transversely, see Fig.~\ref{fig3.2}. However, the proofs are not altered by considering the new setting of $\mathcal P_h$ above ($\mathcal P_h$ in the full plane and closed curve $\Gamma$). The main reason is that the property $\beta_a<|a|$ for $-1<a<0 $ ensures the localization of the bound states near the edge $\Gamma$. \end{remark}
So the following result essentially follows from \cite[Thm.~1.2]{AHK}: \begin{theorem}\label{thm:AHK} Assume that \eqref{eq:Gam} and \eqref{eq:B-ms} hold and that the curvature $k$ of $\Gamma$ has a unique non-degenerate maximum, i.e. \[k_{\max}:=\max_{\Gamma} k(s)=k(0)\quad\textrm{ with}\quad k_2:=k''(0)<0\] Then, for all $n\in\mathbb N^*$ the $n$-th eigenvalue $\lambda_n(h)$ of $\mathcal P_{h}$, defined in~\eqref{eq:P}, satisfies as $h\rightarrow 0$, \[\lambda_n(h)= \beta_ah+k_{\max} M_3(a)h^\frac 32+(2n-1)\sqrt{\frac{{ k_2}M_3(a)c_2(a)}{2}}h^\frac 74+\mathcal O(h^{\frac{15}8}),\] where $\beta_a$, $c_2(a)$ and $M_3(a)$ are introduced in \eqref{eq:beta}, \eqref{eq:c2} and \eqref{eq:m3} respectively. \end{theorem}
Looking more closely at Theorem~\ref{thm:AHK}, we observe that the third term in the expansion of $\lambda_n(h)$ is effectively given (up to the factor of $h^{3/2}$) by the $n$-th eigenvalue of the following 1D operator on $L^2\big(\mathbb R /(2L\mathbb Z)\big)$, \begin{equation}\label{eq:eff-op} \mathfrak L_h^\textrm{ eff}=\frac{\mu_a''(\zeta_a)}{2}\left(-h^{\frac 12} \partial_s^2+V_a(s)\right)\,,\quad V_a(s)=\frac{2M_3(a)(k(s)-k_{\max})}{\mu''_a(\zeta_a)}, \end{equation}
where $L=|\Gamma|/2$ and $|\Gamma|$ denotes the arc-length of $\Gamma$. Notice, that $V_a\geq 0$, due to the sign of $M_3(a)$ (see \eqref{eq:m3}). This point of view is important in order to discuss the case where $\Gamma$ has symmetries and the splitting of the eigenvalues is no more of fractional order in $h$.
In the presence of several points of maximal curvature, a variant of Theorem~\ref{thm:AHK} continues to hold but we may loose the information on the simplicity of the eigenvalues, exactly in the same manner observed for the Neumann problem (see \eqref{eq:mult-well}).
\subsection{Symmetric edge and tunneling}\label{sec:SymEdge}
Suppose that, in addition to \eqref{eq:Gam} and \eqref{eq:B-ms}, the following holds (see Fig~\ref{fig2}): \begin{assumption}\label{ass:sym}~ \begin{enumerate}[\rmfamily i)]
\item $\Omega_1$ is symmetric with respect to the $y$-axis.
\item The curvature $k$ on $\Gamma$ attains its maximum at exactly two symmetric points $a_1=(a_{1,1},a_{1,2})$ and $a_2=(a_{2,1},a_{2,2})$ with $ a_{1,1}<0$ and $ a_{2,1}>0$.
\item Denoting by $s_r$ and $s_\ell$ the arc-length coordinates of $a_1$ and $a_2$ respectively, we have $k''(s_r)=k''(s_\ell)<0$. \end{enumerate} \end{assumption} This is exactly the same geometric assumption on $\Omega$ as Assumption~\ref{ass:symN} for the Neumann realization in $L^2(\Omega)$, with the edge $\Gamma$ playing the role of $\Sigma$, the boundary of $\Omega$.
The presence of a symmetric edge yields a symmetric potential, and consequently two wells, in the effective operator introduced in \eqref{eq:eff-op}, which in turn will induce a tunneling effect whose order of magnitude can be measured by the following quantities (similarly to what we have seen in Theorem~\ref{thm:BHR}): \begin{equation}\label{eq.Aud-a} \begin{aligned} \mathsf{A}_{\mathsf{u}}^a&=\exp\left(-\int_{[s_{r}, 0]} \frac{ (V_a^\frac 12 )' (s)+g_a}{ \sqrt{V_a(s)}} ds\right)\,,\\ \mathsf{A}_{\mathsf{d}}^a&=\exp\left(-\int_{[s_{\mathsf{\ell}}, L]} \frac{ (V_a^\frac 12 )' (s) -g_a}{ \sqrt{V(s)}} ds\right)\,,\\ g_a&=\left(V_a''(s_{r})/2\right)^\frac 12=\left(V_a''(s_{\mathsf{\ell}})/2\right)^\frac 12\,. \end{aligned} \end{equation}
Up to leading order, the operator in \eqref{eq:eff-op} continues to be effective under the new assumptions on the edge, modulo additional terms related to the circulation of the magnetic field and the geometry.
\begin{theorem}\label{thm:FHK} Suppose that Assumption~\ref{ass:sym} holds in addition to \eqref{eq:Gam} and \eqref{eq:B-ms}. The first and second eigenvalues of $\mathcal P_h$ satisfy as $h\to0_+$,
\[\lambda_2(h)-\lambda_1(h)=2|w_a(h)|+o(h^{\frac{13}{8}}e^{-\mathsf{S}^a/h^{\frac 14}})\,,\] where: \begin{align*}
w_a(h)&= \mu_a''(\zeta_a) h^{\frac{13}{8}} \pi^{-\frac 12} g^{\frac12}_a \\ &\quad\times \left(\mathsf{A}_{\mathsf{u}}^a \sqrt{V_a(0)}e^{- \mathsf{S}_{\mathsf{u}}^a/h^{1/4}} e^{iL f_a(h)}+\mathsf{A}_{\mathsf{d}}^a \sqrt{V_a(L)}e^{- \mathsf{S}_{\mathsf{d}}^a/h^{1/4}} e^{-iLf_a(h)}\right)\,, \end{align*} with $\mu_a$ and $\zeta_a$ introduced in Section \ref{sec:FlatEdge},
and \begin{enumerate}[\rmfamily i.] \item The potential $V_a$ is introduced in \eqref{eq:eff-op}; \item $\mathsf S^a$ is the Agmon distance between the wells, \begin{equation}\label{defSa} \mathsf{S}^a =\min \left(\mathsf{S}_{\mathsf{u}}^a,\mathsf{S}_{\mathsf{d}}^a\right),~ \mathsf{S}_{\mathsf{u}}^a=\int_{[s_{r},s_{\mathsf{\ell}}] } \sqrt{V_a(s)} \,ds,~ \mathsf{S}_{\mathsf{d}}^a=\int_{[s_{\mathsf{\ell}}, s_{\mathsf{r}}] } \sqrt{V_a(s)} \,ds\,; \end{equation} \item $\mathsf{A}_{\mathsf{u}}^a$, $\mathsf{A}_{\mathsf{d}}^a$ and $g_a$ are defined in \eqref{eq.Aud-a}; \item $f_a(h)=\gamma_0/h+\zeta_a/h^{1/2}-\alpha_0(a)$ with \begin{equation}\label{eq:circ}
\gamma_0=\frac{|\Omega_1|}{|\Gamma|}, \end{equation} and $\alpha_0(a)$ is a constant dependent on $a$ and $\Omega_1$. \end{enumerate} \end{theorem}
Theorem~\ref{thm:FHK} is the analogue of Theorem~\ref{thm:BHR} but for the situation where tunneling is due to the discontinuity of the magnetic field (without the need for imposing a Neumann boundary condition). As in the proof of Theorem~\ref{thm:BHR} in \cite{BHR21}, the proof of Theorem~\ref{thm:FHK} relies on an optimal tangential decay estimate of ground states.
\subsection{Bounded domains}\label{sec:extensions} Theorem~\ref{thm:FHK} continues to hold if we consider the Dirichlet or Neumann realization of the operator $\mathcal P_h$ in $L^2(\Omega)$, where $\Omega$ is a domain with a $C^2$ boundary such that $\overline{\Omega}_1\subset\Omega$ (see Fig~\ref{fig3.1}). Thanks to \eqref{eq:beta*}, bound states of $\mathcal P_h$ are localized near $\Gamma=\partial\Omega_1$, and the proof of Theorem~\ref{thm:FHK} is not altered.
\begin{figure*}
\caption{The domain $\Omega$ is split into two parts with the edge $\Gamma$ (dashed) is a closed curve.}
\label{fig3.1}
\end{figure*}
\begin{figure*}
\caption{The edge $\Gamma$ (dashed) splits the domain $\Omega$ into two parts and intersects the boundary $\partial\Omega$ transversely.}
\label{fig3.2}
\end{figure*}
We can also modify the configuration of our domains $\Omega_1$ and $\Omega_2$ in \eqref{eq:Gam} and still get the tunneling effect but without oscillatory terms. Let $\Omega$ be a domain with a $C^1$ boundary such that $\overline{\Omega}=\overline{\Omega}_1\cup\overline{\Omega}_2$, where $\Omega_1$ and $\Omega_2$ are disjoint simply connected open sets. We consider a magnetic field as in \eqref{eq:B-ms} and notice that the edge $\Gamma=\Omega\cap\partial\Omega_1=\Omega\cap\partial\Omega_2$ (see Fig~\ref{fig3.2}). We assume that $\Gamma$ is a smooth curve and consider the Dirichlet\footnote{The Neumann realization leads to a completely different behavior, reminiscent of domains with corners \cite{A20}.} realization of $\mathcal P_h$ in $L^2(\Omega)$. This is the situation considered in \cite{AHK}.
Now we assume that the curvature $k$ along $\Gamma$ has a non-degenerate maximum attained at two points, with arc-length coordinates $s_\ell<0$ and $s_r=-s_\ell$, and that it is an even function in a neighborhood of $[s_\ell,s_r]$. In this situation, the splitting between the first eigenvalues is given as follows:
\[\lambda_2(h)-\lambda_1(h)=2|w_a(h)|+o(h^{\frac{13}{8}}e^{-\mathsf{S}^a/h^{\frac 14}}),\] where \[w_a(h)= 2\mu_a''(\zeta_a) h^{\frac{13}{8}} \pi^{-\frac 12} g^{\frac12}_a \mathsf{A}_a \sqrt{V_a(0)}\,e^{- \mathsf{S}_a/h^{1/4}},
\]
and
\[\mathsf A_a=2\exp\left(-\int_{[ s_\ell,s_r]} \frac{ (V_a^\frac 12 )' (s)-g_a}{ \sqrt{V_a(s)}} ds\right), \quad\mathsf S_a=\int_{[s_\ell,s_r]}\sqrt{V(s)}ds. \]
\section{Reduction to a neighborhood of the edge}\label{sec:curv}
It will be convenient to work in Fr\'enet coordinates, $(s,t)$, along the edge $\Gamma$, valid in a neighborhood of $\Gamma$ of the form \begin{equation}\label{eq:gam-ep} \Gamma(\epsilon)=\{x\in\mathbb R^2~:~\textrm{ dist}(x,\Gamma)<\epsilon\}\quad(\epsilon>0)\,. \end{equation} Let us briefly recall these coordinates. Consider an arc-length parameterization of $\Gamma$ , $M:(-L,L]\to\Gamma$, so that (see Assumption~\ref{ass:sym}) \[M(s_\ell)=a_{1},\quad M(s_r)=a_2,\quad 0<s_\ell<L,\quad -L<s_r<0\,, \] and
\[\Gamma\cap\{(x,y)\in\mathbb R^2\,|\,~x=0\}=\{M(0)=:(0,y_0),M(L)=:(0,y_L)\} \quad\textrm{ with~} y_0>y_L\,. \] Let $\mathbf n(s)$ be the unit normal to $\Gamma$ pointing inward to $\Omega_1$ (see { Fig.}~\ref{fig2}), $\mathbf t(s)=\dot{\mathbf n}(s)$ the unit oriented tangent, so that $\textrm{ det}(\mathbf t(s),\mathbf n(s))=1$. Let us represent the torus $\mathbb R/ 2L \mathbb Z$ by the interval $(-L,L]$. We can pick $\epsilon_0>0$ such that \[\Phi: \mathbb R/ (2L \mathbb Z)\times (-\epsilon_0,\epsilon_0)\ni (s,t)\mapsto M(s)+t\mathbf n(s)\in\Gamma(\epsilon_0) \] is a diffeomorphism whose Jacobian is \[\mathfrak a(s,t)=1-tk(s)\,, \] with $k(s)$ the curvature at $M(s)$, defined by $\ddot{\mathbf n}(s)=k(s)\mathbf n(s)$. The Hilbert space $L^2(\Gamma(\epsilon_0))$ is transformed to the weighted space \[L^2\big(\mathbb R/ 2L \mathbb Z\times(-\epsilon_0,\epsilon_0);\mathfrak a\,dsdt\big)\] and the operator $\mathcal P_h$ is transformed into the following operator (after a gauge transformation $(u,\mathbf A)\to (v=e^{i\phi/h}u,\mathbf A'=\mathbf A-\nabla \phi)$ to eliminate the normal component of $\mathbf A$, see \cite[App.~F]{FH10}): \begin{multline*}
\tilde{\mathcal P}_h:=-h^2\mathfrak a^{-1}\partial_t \mathfrak a\partial_t\\+\mathfrak a^{-1}\left(-ih\partial_s+\gamma_0-b_a(t)t+\frac{k}{2}b_a(t)t^2\right)\mathfrak a^{-1}\left(-ih\partial_s+\gamma_0-b_a(t)t+\frac{k}{2}b_a(t)t^2\right)
\end{multline*} where $b_a(t)$ is introduced in \eqref{eq:potential} and $\gamma_0$ is the circulation introduced in \eqref{eq:circ}.
Following the presentation of \cite{BHR21} (see also references therein), it is convenient to introduce a truncated version of the operator $\tilde P_h$ so that it can be defined on $\mathbb R/ 2L \mathbb Z\times\mathbb R$ instead of $\mathbb R/ 2L \mathbb Z\times(-\epsilon_0,\epsilon_0)$. This will be useful when rescaling the $t$ variable. What is handy in this situation is that the actual bound states of the operator $\mathcal P_h$ decay exponentially away from the edge, at the length scale $\hbar:=h^{1/2}$, see \eqref{eq:dec-norm}. This motivates the change of variables, $t=\hbar \tau$ and $s=\sigma$, that will allow the same spectral reduction as in \cite[Prop.~2.7]{BHR21}. We will skip the details which are the same as in \cite{BHR21}.
From now on we set
\begin{equation}\label{eq:mu(h)}
\mu=h^{\frac14+\eta} \text{ for a fixed } \eta\in(0,\frac14) \end{equation} and we introduce the function \begin{equation}\label{eq:c-delta} c_\mu(\tau)= c(\mu\tau)\,, \end{equation} where $c\in C_c^\infty(\mathbb R)$ satisfies $c=1$ on $[-1,1]$ and $c=0$ on $\mathbb R\setminus(-2,2)$.
Consider the new weight term \[\tilde{\mathfrak a}_h(\sigma,\tau)=1-h^{1/2} c_\mu(\tau)\tau k(\sigma)\,,\] and the self-adjoint operator $\tilde{\mathcal N}_{h}$ on the Hilbert space $ L^2(\mathbb R/ 2L \mathbb Z\times\mathbb R;\tilde{\mathfrak a}_h d\sigma d\tau)$, \begin{align}\label{eq:tilde-Nh} \tilde{\mathcal N}_{h}&=-\tilde{\mathfrak a}^{-1}_{h}\partial_\tau \mathfrak a_{\hbar}\partial_\tau \nonumber \\ &\quad+\tilde{\mathfrak a}^{-1}_{h}\left(-ih^{1/2}\partial_\sigma+{ h^{-1/2}\gamma_0}-b_a\tau +h^{1/2} c_\mu \frac{k}{2}b_a\tau^2\right)\nonumber \\ &\quad\quad \times \tilde{\mathfrak a}^{-1}_{h}\left(-ih^{1/2}\partial_\sigma+{ h^{-1/2}\gamma_0}-b_a\tau+h^{1/2} c_\mu\frac{k}{2}b_a\tau^2\right), \end{align} with domain \begin{align*} {\textsf{ Dom}(\tilde{\mathcal N}_h)}=\{ u\in L^2(\mathbb R/ 2L \mathbb Z\times\mathbb R)& ~\vert~ \partial_\tau^2u\in L^2({ \mathbb R/ 2L \mathbb Z\times\mathbb R)}, \nonumber \\ & (-ih^{1/2}\partial_\sigma+{ h^{-1/2}\gamma_0}-b_a\tau )^2 u \in L^2(\mathbb R/ 2L \mathbb Z\times\mathbb R) \}. \end{align*} We have now the following spectral reduction\footnote{The eigenvlaues of the operator $\tilde{\mathcal N}_h$ depend on $\eta$ in \eqref{eq:mu(h)}. However, the estimates in Proposition~\ref{prop:BHR2.7} hold uniformly with respect to $\eta\in(0,\epsilon)$ for any fixed $\epsilon\in(0,\frac14)$. }: \begin{proposition}\label{prop:BHR2.7} Let $a\in(-1,0)$ and $\mathsf S^a$ be the Agmon distance introduced in \eqref{defSa}. There exist $K>\mathsf{S}^a$, $C,h_0>0$ such that, for all $h\in(0,h_0)$, we have \[\lambda_n(h)-Ce^{-K/h^{\frac 14}}\,\leq h\lambda_n(\tilde{\mathcal N}_h)\leq \lambda_n(h)+Ce^{-K/h^{\frac 14}}\,,\] where $\lambda_n(h)$ and $\lambda_n(\tilde{\mathcal N}_h)$ are the $n$-th (min-max) eigenvalues of the operators $\mathcal P_h$ and $\tilde{\mathcal N}_{h}$ respectively. \end{proposition}
Looking at the operator in \eqref{eq:tilde-Nh}, the effective semi-classical parameter is $\hbar=h^{1/2}$ (this is the parameter appearing in front of $\partial_\sigma$). So with \begin{equation}\label{eq:delta} \hbar=h^{\frac12},\quad \mu=\hbar^{\frac12+2\eta} \text{ for a fixed } \eta\in(0,\frac14)\,, \end{equation} we introduce the new weight term \[\mathfrak a_\hbar(\sigma,\tau)=1-\hbar c_\mu(\tau)\tau k(\sigma)\,,\] and the self-adjoint operator $\mathcal N_{\hbar}$ on the Hilbert space $ L^2(\mathbb R/ 2L \mathbb Z\times\mathbb R;\mathfrak a_\hbar d\sigma d\tau)$, which is nothing but the operator in \eqref{eq:tilde-Nh} but with a change of parameter according to \eqref{eq:delta}, \begin{align}\label{eq:Nh} \mathcal N_{\hbar}&=-\mathfrak a^{-1}_{\hbar}\partial_\tau \mathfrak a_{\hbar}\partial_\tau\\ &\quad+\mathfrak a^{-1}_{\hbar} \left(-i\hbar\partial_\sigma+\hbar^{-1}\gamma_0-b_a\tau +\hbar c_\mu \frac{k}{2}b_a\tau^2\right)\mathfrak a^{-1}_{\hbar}\nonumber \\ &\qquad \times\left(-i\hbar\partial_\sigma+\hbar^{-1}\gamma_0-b_a\tau+\hbar c_\mu\frac{k}{2}b_a\tau^2\right)\,.\nonumber \end{align} The domain of the operator $\mathcal N_{\hbar}$ is \begin{align*}
\textsf{ Dom}(\mathcal N_{\hbar})=\{ u\in L^2(\Gamma\times\mathbb R)~|~&\partial_\tau^2u\in L^2(\mathbb R/ 2L \mathbb Z\times\mathbb R), \\ &(-i\hbar\partial_\sigma+\hbar^{-1}\gamma_0-b_a\tau )^2 u \in L^2(\mathbb R/ 2L \mathbb Z\times\mathbb R) \}. \end{align*} With Proposition~\ref{prop:BHR2.7} in hand, it is enough to compute the leading order term of $\nu_{2}(\hbar)-\nu_{1}(\hbar)$ to prove Theorem~\ref{thm:FHK}, where, for $n\geq 1$, we denote by $\nu_n(\hbar)$ the $n$'th min-max eigenvalue of $\mathcal N_\hbar$.
\section{Single well and WKB construction}\label{sec:WKB}
We will adjust the edge $\Gamma$ so that we only have a single point of maximum curvature, $s_r$ or $s_\ell$. This procedure will give us two new operators, the ``right well'' and ``left well'' operators, $\mathcal N_{\hbar,r,\gamma_0}$ and $\mathcal N_{\hbar,\ell,\gamma_0}$ respectively. The same procedure appears, for similar problems in the context of geometrically induced tunneling effects \cite{HKR, KR17}, but we follow here \cite[Sec.~2.4]{BHR21} which is slightly different, but more convenient for dealing with the symbol of the operator later on.
\subsection{Right well operator}
We present the construction for the right well operator, $\mathcal N_{\hbar,r,\gamma_0}$ and deduce the other one by symmetry. Let us fix $\hat\eta$ as follows \begin{equation}\label{eq:eta.new}
0<\hat\eta<\min\Big(\frac14,\frac{L}4\Big)\quad\textrm{ where~}L=\frac{|\Gamma|}2. \end{equation} First, we identify $\Gamma$ with $(s_\ell-2L,s_\ell]$ (by periodicity and translation of the $s$ variable), then we extend the curvature $k$ to a function $k_r$ on $\mathbb R$ as follows: \begin{align}\label{eq:k-r} k_r&=k\qquad \text{on}\qquad I_{2\hat\eta,r}:=(s_\ell-2L+\hat\eta,s_\ell-\hat\eta),\nonumber \\
k_r&=0\qquad \text{on}\qquad (-\infty,s_\ell-2L]\cup {[s_\ell,+\infty)}\,, \end{align} and $k_r$ has a unique non-degenerate maximum at $s_r$. Consequently, $k_r$ satisfies \eqref{eq:hyp-gd}.
We consider now the operator in $L^2(\mathbb R^2;\mathfrak a_{\hbar,r}d\sigma d\tau)$, \begin{align}\label{eq:Nh-r} \mathcal N_{\hbar,r,\gamma_0}&=-\mathfrak a^{-1}_{\hbar,r}\partial_\tau \mathfrak a_{\hbar,r}\partial_\tau\nonumber \\ &\quad +\mathfrak a^{-1}_{\hbar,r}\left(-i\hbar\partial_\sigma+\hbar^{-1}\gamma_0-b_a\tau +\hbar c_\mu \frac{k_r}{2}b_a\tau^2\right)\mathfrak a^{-1}_{\hbar,r}\nonumber \\ &\qquad \times \left(-i\hbar\partial_\sigma+\hbar^{-1}\gamma_0-b_a\tau+\hbar c_\mu\frac{k_r}{2}b_a\tau^2\right) \end{align} where \begin{equation}\label{eq:ah-r} \mathfrak a_{\hbar,r}(\sigma,\tau)=1-\hbar c_\mu(\tau)\tau k_r(\sigma)\,. \end{equation} Since, $k_r$ satisfies \eqref{eq:hyp-gd},
we have, for an arbitrarily fixed $n\in\mathbb N$ (with $\beta_a$, $c_2(a)$ and $M_3(a)$ from \eqref{eq:beta}, \eqref{eq:c2} and \eqref{eq:m3}), \begin{equation}\label{eq:r-well} \lambda_n(\mathcal N_{\hbar,r,\gamma_0})=\beta_a h + k_{max} M_3(a)h^\frac 32+(2n-1)\sqrt{\frac{{ k_2}M_3(a)c_2(a)}{2}}h^\frac 74+\mathcal O(h^{\frac{15}8})\,. \end{equation} We are now in a simply connected domain, so the operators $\mathcal N_{\hbar,r,\gamma_0}$ and $\mathcal N_{\hbar,r,0}$ are unitarily equivalent {(we can gauge away the flux term $\mathcal N_{\hbar,r,\gamma_0}$)}. Denote by $u_{\hbar,r}$ a normalized ground state of $\mathcal N_{\hbar,r,0}$ (the operator without flux term), a corresponding normalized ground state of $\mathcal N_{\hbar,r,\gamma_0}$ is given by: \begin{equation}\label{eq:gs-r} \widecheck\phi_{\hbar, r}(\sigma,\tau) = e^{-i\gamma_0 \sigma/\hbar^2}u_{\hbar,r}(\sigma,\tau). \end{equation}
\subsection{Left well operator}\label{sec:LWop}
Using the symmetry operator \[U f(\sigma,\tau):=\overline{f(-\sigma,\tau)}\,, \] we can define the left well operator on $L^2(\mathbb R^2;\mathfrak a_{\hbar,\ell}(\sigma,\tau)$ by : \begin{equation}\label{eq:Nh-l} \mathcal N_{\hbar, \ell, \gamma_0}=U^{-1}\mathcal N_{\hbar, r, \gamma_0}U\,, \end{equation} where \[ \mathfrak a_{\hbar,\ell}(\sigma,\tau)=\mathfrak a_{\hbar,r}(-\sigma,\tau)\,.\] The left and right operators have the same spectrum, and a normalized ground state of $\mathcal N_{\hbar, \ell, \gamma_0}$ is \begin{equation}\label{eq.phil0} \widecheck\phi_{\hbar,\ell}:=U\widecheck\phi_{\hbar,r}=e^{-i\gamma_0 \sigma/\hbar^2}u_{\hbar,\ell}(\sigma,\tau) \end{equation}
where $u_{\hbar,\ell}=Uu_{\hbar,r}$.
\subsection{WKB expansions}
We focus on the right well operator and construct an approximate eigenvalue and an approximate ground state by WKB expansions, involving \emph{formal series} in the sense of \cite[Notation~1.13]{BHR15}. The construction can be translated to the left operator by symmetry.
Let us introduce the Agmon distance \begin{equation}\label{eq:Ag-r} \Phi_r(\sigma)=\int_{[s_r,\sigma]}\sqrt{V_{a,r}(s)}ds \end{equation} related to the ``right well'' potential\footnote{Recall from \eqref{eq:m3} that $M_3(a)<0$, so $V_{a,r}\geq 0$.} \begin{equation}\label{eq:pot-r} V_{a,r}(\sigma)=\frac{2M_3(a)(k_r(s)-k_{\max})}{\mu''_a(\zeta_a)} \end{equation}
\begin{theorem}\label{thm:WKB} There exist two sequences $ (b_j)_{j\geq 0}\subset \textsf{ Dom}(\mathcal N_{\hbar, r})$, $(\delta_j)_{j\geq0}\subset\mathbb R$, a family of functions $(\Psi_{\hbar,r})_{\hbar\in(0,\hbar_0]}\subset L^2(\mathbb R^2)$ and a family of real numbers $(\delta(\hbar))_{\hbar\in(0,\hbar_0]}$ such that \begin{equation}\label{eq.psir} e^{\Phi_r(\sigma)/\hbar^{\frac 12}} e^{-i\sigma\zeta_a/\hbar}\Psi_{\hbar,r}(\sigma,\tau)\underset{\hbar\to 0}{\sim}\hbar^{-\frac18} \sum_{j\geq 0} b_{j}(\sigma,\tau) \hbar^{\frac j2}, \end{equation} \[ \delta(\hbar)\underset{\hbar\to 0}{\sim}\sum_{j\geq 0}\delta_{j}\hbar^{\frac j2}\,,\] and \begin{equation}\label{eq:Nh-WKB} e^{\Phi_r(\sigma)/\hbar^{\frac 12}}\left(\mathcal{N}_{\hbar, r}-\delta(\hbar)\right) \Psi_{\hbar,r}=\mathcal{O}(\hbar^\infty)\,. \end{equation} Furthermore \[\delta_{0}=\beta_a\,,\quad \delta_{1}=0,\quad \delta_2=M_3(a) k_{\max},\quad \delta_3=\sqrt{\frac{{ k_2}M_3(a)c_2(a)}{2}}, \] \begin{equation}\label{eq.an0} b_{0}(\sigma,\tau)=f_{0}(\sigma)\phi_a(\tau), \end{equation} and $f_{0}$ solves the effective transport equation \begin{equation}\label{eq.effectiveT} \frac{ \mu''_a(\zeta_a)}{2}(\Phi'_r\partial_\sigma+\partial_{\sigma}\Phi'_r)f_{0}+ iF(\sigma)f_{0}= \sqrt{\frac{{ k_2}M_3(a)c_2(a)}{2}}f_{0}\,, \end{equation} where $F$ is a smooth real-valued function, introduced in \eqref{eq:def-F}, such that $F(s_r)=0$. \end{theorem}
\begin{remark}\label{rem:WKB-not} Let us explain precisely how the asymptotics in Theorem~\ref{thm:WKB} are interpreted. For every $N\geq 1$ we introduce the function $\psi_{\hbar,r}^N(\sigma,\tau)$ and the real number $\delta^N(\hbar)$ as follows: \[ \Psi^N_{\hbar,r}(\sigma,\tau):= e^{-\Phi_r(\sigma)/\hbar^{\frac 12}} e^{i\sigma\zeta_a/\hbar}\, \hbar^{-\frac18} \sum_{j=0}^Nb_{j}(\sigma,\tau)\hbar^{\frac j2},\quad \delta^N(\hbar)=\sum_{j=0}^N\delta_{j}\hbar^{\frac j2}\,. \]
Then \eqref{eq:Nh-WKB} means
\[ e^{\Phi_r(\sigma)/\hbar^{\frac 12}}\left\|\left(\mathcal{N}_{\hbar, r}-\delta(\hbar)\right) \Psi_{\hbar,r}^N\right\|_{L^2(\mathbb R_\tau)}=\mathcal{O}(\hbar^N)\]
locally uniformly with respect to $\sigma$. \end{remark} \begin{proof}[Proof of Theorem~\ref{thm:WKB}] We work in an arbitrary bounded set of $\mathbb R^2$, so, in the below computations, we take $c_\mu=1$ in \eqref{eq:Nh-r} at the cost of an error $\mathcal O(h^\infty)$. That is possible because our constructions will involve functions decaying exponentially with respect to the normal variable $\tau$.
Let us introduce the operator \begin{equation*} \widehat{\mathcal{N}}_{\hbar, r}:=e^{\Phi_r(\sigma)/\hbar^{\frac 12}} e^{i\sigma\zeta_a/\hbar} \mathcal{N}_{\hbar, r}e^{-i\sigma\zeta_a/\hbar}e^{-\Phi_r(\sigma)/\hbar^{\frac 12}} \,. \end{equation*} It admits the formal expansion \begin{equation*} \widehat{\mathcal{N}}_{\hbar, r}= \mathcal L_0+\hbar^{1/2}\mathcal L_1+\hbar\mathcal L_2+\hbar^{3/2}\mathcal L_3+\hbar^2\mathcal L_4+\cdots \end{equation*} where \begin{align*} \mathcal L_0&=-\partial_\tau^2+(\zeta_a+b_a\tau)^2\\ \mathcal L_1&=-2(\zeta_a+b_a\tau)i\Phi'_r(\sigma)\\ \mathcal L_2&=k_r\partial_\tau-2(\zeta_a+b_a\tau)\Big( -i\partial_\sigma+\frac{k_r}{2}b_a\tau^2\Big) -\Phi'_r(\sigma)^2+2k_r\tau(\zeta_a+b_a\tau)^2\\ \mathcal L_3&=\Big(-i\partial_\sigma+\frac{k_r}2b_a\tau^2\Big)i\Phi'(\sigma)+i\Phi'_r(\sigma)\Big(-i\partial_\sigma+\frac{k_r}2b_a\tau^2\Big)\\ &\qquad-4\Phi'_r(\sigma)\tau k_r(\zeta_a+b_a\tau)\\ \mathcal L_4&=-\partial_\sigma^2+2k_r^2\tau^2(\zeta_a+b_a\tau)^2 \\ &\qquad-(\zeta_a+b_a\tau)\Big[ \Big(-i\partial_\sigma+\frac{k_r}2b_a\tau^2\Big)k_r+k_r\Big(-i\partial_\sigma+\frac{k_r}2b_a\tau^2\Big) \Big]\\ &\,\,\vdots \end{align*}
Let $b(\sigma,\tau;\hbar):=\sum_{j\geq 0}b_{j}(\sigma,\tau) \hbar^{\frac j2}$ and let us formally solve the equation \[ \big(\widehat{\mathcal{N}}_{\hbar, r}-\delta(\hbar)\big)b(\sigma,\tau;\hbar)=\mathcal O(\hbar^\infty).\] Expanding the foregoing equation in powers of $\hbar^{1/2}$, the vanishing of the coefficient of each $\hbar^{j/2}$, $j\geq 0$, yields the following equations
\begin{align*} (\mathcal L_0-\delta_0)b_0&=0\\ (\mathcal L_0-\delta_0)b_1&=(\delta_1-\mathcal L_1)b_0\\ (\mathcal L_0-\delta_0)b_2&=(\delta_2-\mathcal L_2)b_0+(\delta_1-\mathcal L_1)b_1\\ (\mathcal L_0-\delta_0)b_3&=(\delta_3-\mathcal L_3)b_0+(\delta_2-\mathcal L_2)b_1+(\delta_1-\mathcal L_1)b_2\\ &\,\,\vdots \end{align*} We will find solutions to these equations one by one. The first equation leads us to choose $\delta_0=\zeta_a$ and $b_0(\sigma,\tau)=f_0(\sigma)\phi_a(\tau)$, where $f_0(\sigma)$ is to be determined at a later stage. The function $f_0$ will actually be free untill the first equation involving $\mathcal L_3$.
For the equation for $b_1$, we determine $\delta_1$ by assuming that $(\delta_1-\mathcal L_1)b_0$ is orthogonal to $\phi_a$ in $L^2(\mathbb R)$. Then, we take the inner product with $\phi_a$ in $L^2(\mathbb R)$, use \eqref{eq:m0} and get \[ \delta_1=0,\quad b_1(\sigma,\tau)=2i\Phi'_r(\sigma)f_0(\sigma) {\mathcal R_a}\big((\zeta_a+b_a\tau)\phi_a\big),\]
where $\mathcal R_a$ the regularized resolvent introduced in \eqref{eq:R}. Since we are applying $\mathcal R_a$ on functions orthogonal to $\phi_a$, we can slightly abuse notation and say that it is equal to $(\mathcal L_0-\delta_0)^{-1}$.
The equation for $b_2$ will determine $\delta_2$. This equation can be solved if $(\delta_2-\mathcal L_2)b_0-\mathcal L_1b_1$ is orthogonal to $\phi_a$ in $L^2(\mathbb R)$, which we assume henceforth. Taking the inner product with $\phi_a$ in $L^2(\mathbb R)$, using \eqref{eq:I2} and Remark~\ref{rem:AHK2.3}, we get \[ \delta_2f_0(\sigma)+\frac{\mu_a''(\zeta_a)}{2}\Phi'_r(\sigma)^2f_0(\sigma)-M_3(a)k_r(\sigma)f_0(\sigma)=0, \] since
\[ 2\int_{\mathbb R}\tau(\zeta_a+b_a(\tau) \tau)^2|\phi_a(\tau)|^2\,d\tau-\int_{\mathbb R}b_a(\tau) \tau^2(\zeta_a+b_a(\tau) \tau)|\phi_a(\tau)|^2\,d\tau=M_3(a)\,.\] So, we choose $\delta_2=k_r(0)=k_{\max}$, and the foregoing equation involving $f_0$ is valid everywhere in light of \eqref{eq:Ag-r}, independently of the choice of $f_0$. At the same time, we choose $b_2$ as follows: \[b_2(\sigma,\tau)=(\mathcal L_0-\delta_0)^{-1}\Big((\delta_2-\mathcal L_2)b_0-\mathcal L_1b_1 \Big). \]
From the equation of $b_3$, we will determine $\delta_3$ and $f_0(\sigma)$. Taking the inner product with $\phi_a$ in $L^2(\mathbb R)$ and using \eqref{eq:I2}, we get \begin{multline*} \Big\langle (\delta_3-\mathcal L_3)b_0+(\delta_2-\mathcal L_2)b_1+(\delta_1-\mathcal L_1)b_2, \phi_a\Big\rangle_{L^2(\mathbb R)}=\\ \Big(\delta_3-\frac{\mu''(\zeta_a)}2\big(\Phi'_r\partial_\sigma+\partial_\sigma\Phi'_r \big)\Big)f_0(\sigma)+iF(\sigma)f_0(\sigma), \end{multline*} where $F(\sigma)$ is the real-valued function \begin{equation}\label{eq:def-F}
F(\sigma)=|\Phi'_r(\sigma)|^2\int_{\mathbb R} g(\sigma,\tau) \phi_a(\tau)d\tau, \end{equation} and \begin{align*}
g(\sigma,\tau)&=(\mathcal L_0-\delta_0)^{-1}\big(g_1(\sigma,\tau)+g_2(\sigma,\tau)\big)\\ g_1(\sigma,\tau)&=
-\Big(k_{\max}+k_r(\sigma)\big(\zeta_a+b_a\tau)^2-b_a(\zeta_a+b_a\tau)\tau^2\big)-|\Phi(\sigma)|^2\Big)\phi_a(\tau)\\ &\qquad+k_r(\sigma)\phi_a'(\tau)\\
g_2(\sigma,\tau)&=-4|\Phi_r(\sigma)|^2(\mathcal L_0-\delta_0)^{-1}\big((\zeta_a+b_a\tau)\phi_a(\tau)\big). \end{align*} Since $\Phi_r'(s_r)=0$, we observe that $F(s_r)=0$. We can solve the equation of $b_3$ if $(\delta_3-\mathcal L_3)b_0+(\delta_2-\mathcal L_2)b_1+(\delta_1-\mathcal L_1)b_2$ is orthogonal to $\phi_a$ in $L^2(\mathbb R)$, which yields the following equation for $f_0$: \[\Big(\delta_3-\frac{ \mu_a''(\zeta_a)}2\big(\Phi'_r\partial_\sigma+\partial_\sigma\Phi'_r \big)\Big)f_0(\sigma)+F(\sigma)f_0(\sigma)=0.\] Since $F(s_r)=\Phi'_r(s_r)=0$, the foregoing equation has a solution satisfying $f_0(s_r)\not=0$ if $\delta_3= \frac{\mu_a''(\zeta_a)}{2}\Phi''_r(s_r)$, thereby determining $\delta_3$ (from \eqref{eq:pot-r}) and $f_0$. The procedure can be continued to any preassigned order. \end{proof}
\begin{remark}[Solving \eqref{eq.effectiveT} \& normalization of $\Psi_{\hbar,r}$]\label{rem:phase}~
In \eqref{eq.effectiveT}, we make the ansatz $f_0(\sigma)=e^{i\alpha_0(\sigma)}\tilde f_0(\sigma)$, with $\tilde f_0$ and $\alpha_0$ are real-valued functions such that $\tilde f_0(0)>0$. Then we get from \eqref{eq.effectiveT}: \begin{equation*} \frac{\mu''_a(\zeta_a)}2\big( \Phi'_r\partial_\sigma+\partial_{\sigma}\Phi'_r\big)\tilde f_0(\sigma)= \sqrt{\frac{{ k_2}M_3(a)c_2(a)}{2}}f_{0}(\sigma) \end{equation*} and \begin{equation*} \mu''_a(\zeta_a)\alpha_0'(\sigma)\Phi_r'(\sigma)+F(\sigma)=0\,. \end{equation*}
This will determine $\tilde f_0(\sigma)$ uniquely up to the choice of $\tilde f_0(0)$, and also $\alpha_0(\sigma)$ uniquely up to an additive constant (see \cite[Eq.~(2.14)~\&~Rem.~2.9]{BHR21}).
We choose $\tilde f(0)=\left(\frac{g_a}{\pi}\right)^{1/4}\left(\mathsf A_\textsf{ u}^a\right)^{1/2}$ which yields that the WKB solution $\Psi_{\hbar,r}$ in Theorem~\ref{thm:WKB} is almost normalized, $\|\Psi_{\hbar,r}\|\sim 1$. The constant $\alpha_a$ appearing in Theorem~\ref{thm:FHK} is \begin{equation}\label{eq:alpha} \alpha_a=\frac{\alpha_{0}(0)-\alpha_0(-L)}{L}. \end{equation} \end{remark}
\section{Optimal tangential Agmon estimates}\label{sec:dec}
The challenge of obtaining optimal decay estimates of bound states of the Neumann magnetic Laplacian matching with the WKB solutions was recently taken up in \cite{BHR21} by introducing pseudo-differential calculus with operator-valued symbols. Fortunately, the method is quite general and can handle our situation of magnetic steps.
\subsection{A tangential elliptic estimate}
We work with the single `right well' flux free operator, $\mathcal N_{\hbar,r}:=\mathcal N_{\hbar,r,0}$, introduced in \eqref{eq:Nh-r}. For the sake of simplicity, we will omit the reference to `right well' in the notation and write $\mathcal N_{\hbar}$ and $k$ instead of $\mathcal N_{\hbar,r}$ and $k_r$.
The optimal estimates, on the bound states of $\mathcal N_{\hbar}$, will hold in spaces with an exponential weight, defined via a sub-solution of an effective eikonal equation. More precisely, we consider a family of Lipschitz functions $(\varphi_\hbar)_{h\in(0,1]}\subset C(\mathbb R;\overline{\mathbb R_+})$ satisfying the following hypothesis:
\begin{assumption}\label{ass:phi} For all $M>0$ there exist $\hbar_0,C,R>0$ such that, for all $\hbar\in(0,\hbar_0)$, the function $\varphi:=\varphi_\hbar$ satisfies
\begin{enumerate}[\rmfamily (i)]
\item for all $\sigma\in\mathbb R$, $\mathfrak{v}(\sigma)-\frac{\mu_a''(\zeta_a)}{2}\varphi'(\sigma)^2\geq 0$,
\item for all $\sigma$ such that $|\sigma-s_r|\geq R\hbar^{\frac 12}$, $\mathfrak{v}(\sigma)-\frac{\mu_1''(\xi_0)}{2}\varphi'(\sigma)^2\geq M \hbar$,
\end{enumerate} where $\mathfrak{v}(\sigma)=M_3(a)(k(\sigma)-k_{\max})$. \end{assumption} In the sequel, to lighten the notation, we write $\varphi$ instead of $\varphi_\hbar$. We consider the conjugate operator, with the same domain as $\mathcal N_{\hbar}$, and defined by: \begin{equation}\label{eq:conj-op} \mathcal N_{\hbar}^\varphi=e^{\varphi/\hbar^{\frac12}}\mathcal N_{\hbar}e^{-\varphi/\hbar^{\frac12}} =-a_\hbar^{-1}\partial_\tau a_\hbar\partial_\tau+ a_\hbar^{-1}\mathcal T_{\hbar}^\varphi a_\hbar^{-1}\mathcal T_{\hbar}^\varphi \end{equation} where \[ \mathcal T_{\hbar}^\varphi:= \left(-i\hbar\partial_\sigma-b_a\tau+i\hbar^{\frac 12}\varphi'+\hbar c_\mu\frac{\kappa_r}{2}b_a\tau^2\right) \]
\begin{theorem}[Bonnaillie-No\"{e}l--H\'erau--Raymond]\label{thm:dec} Let $c_0>0$ and $\chi_0\in C_c^\infty(\mathbb R)$ be $1$ in a neighborhood of $0$. Under Assumption \ref{ass:phi}, there exist $c, \hbar_0>0$ such that for all $\hbar\in(0,\hbar_0)$, $z\in[ \beta_a+M_3(a)k_{\max}\hbar-c_0\hbar^2,\beta_a+M_3(a)k_{\max}\hbar+c_0\hbar^2]$ and all $\psi\in\textsf{ Dom}(\mathcal{N}_\hbar^{\varphi})$, \[
c\hbar^2\|\psi\|\leq \|\langle\tau\rangle^6(\mathcal{N}^{\varphi}_\hbar-z)\psi\|
+\hbar^2\|\chi_0(\hbar^{-\frac 12}R^{-1}(\sigma-s_r)) \psi\|\,,
\]
and
\[
c\hbar^{2}\|\hbar^2 \partial_\sigma^2\psi\|\leq \|\langle\tau\rangle^6(\mathcal{N}^{\varphi}_\hbar-z)\psi\|
+\hbar^2\|\chi_0(\hbar^{-\frac 12}R^{-1}(\sigma-s_r)) \psi\|\,,
\] where $\langle \tau\rangle=(1+\tau^2)^{1/2}$. \end{theorem}
Modulo the decomposition of the symbol of the operator $\mathcal N_{\hbar}^\varphi$ and its parametrix, the proof of Theorem~\ref{thm:dec} is the same as that of \cite[Thm.~5.1]{BHR21}. In the sequel, we give only the new ingredients.
Let us write \begin{equation} \mathcal N_{\hbar}^\varphi u =\textrm{ Op}^\textrm{ W}_{\hbar}(n_{\hbar}) \,u =\frac{1}{(2\pi\hbar)} \iint_{\mathbb R^2} e^{i(\sigma-s)\cdot \xi/\hbar} n_\hbar \left(\frac{\sigma+s}{2}, \xi\right)u(s) ds d\xi \end{equation} where the foregoing quantization formula is formal, unless we consider it on, say, for $u$ in the space $ \mathcal S\big(\mathbb R;\widehat{\mathcal S}(\mathbb R)\big)$ where \begin{equation}\label{eq:hat-S}
\widehat{\mathcal S}(\mathbb R)=\{v\in H^2(\mathbb R))\,|~v|_{\overline{\mathbb R_\pm}}\in\mathcal S(\overline{\mathbb R_\pm})\}. \end{equation} The operator-valued symbol $n_{\hbar}$ can be decomposed as follows \[n_\hbar=n_0+\hbar^{\frac 12}n_1+\hbar n_2+ \hbar^{\frac 32}n_3 + \hbar^2 \tilde{r}_\hbar ,\]
where \begin{equation}\label{eq.nj} \begin{aligned} n_0(\sigma,\xi)&=-\partial_\tau^2+(\xi-b_a(\tau)\tau)^2\,,\\ n_1(\sigma,\xi)&=2i(\xi-b_a(\tau)\tau)\varphi'(\sigma)\,,\\ n_2(\sigma,\xi)&=-\varphi'(\sigma)^2+\kappa c_\mu(\tau) \partial_\tau+c_\mu \kappa(\sigma)(\xi-b_a(\tau)\tau)b_a(\tau)\tau^2\\ &\quad+2\kappa(\sigma)\tau c_\mu (\tau)(\xi-b_a\tau)^2 +\kappa(\sigma) \tau c_\mu'(\tau )\,, \\ \mathrm{Re}\,\, n_3(\sigma,\xi)& = 0\,, \\
\tilde{r}_{\hbar}(\sigma,\xi) &= {\mathcal O}(\tau^4, (\xi-b_a(\tau)\tau)^2 \tau^2, (\xi-b_a\tau)\tau, \tau^2\partial_\tau). \end{aligned} \end{equation} The notation ${\mathcal O}$ is defined in \cite[Notation~3.1]{BHR21}:\\ {\itshape For differential operators $A,B,C,\cdots$ on $\mathbb R_\tau$, writing $A=\mathcal O(B,C,\cdots)$ means the following: \begin{equation}\label{eq:not-O-op}
\exists\,c> 0, ~\forall\,u\in \widehat{\mathcal S}(\mathbb R),\quad\|Au\|_{L^2(\mathbb R)}\leq c\big(\|Bu\|_{L^2(\mathbb R)}+\|Cu\|_{L^2(\mathbb R)}+\cdots \big),\end{equation} where $\widehat{\mathcal S}(\mathbb R)$ is the space introduced in \eqref{eq:hat-S} and the constant $c$ is independent of $A,B,C,\cdots$ (in particular, in \eqref{eq.nj}, the estimate is uniform with respect to $(\sigma,\xi)$).}
Let us introduce a modified symbol by truncating the frequency variable. Recall that $\zeta_a<0$ is the unique minimum of the model operator with a flat edge (see \eqref{eq:beta}). It will be convenient to introduce \begin{equation}\label{eq:hat-zeta} \hat\zeta_a:=-\zeta_a>0\,. \end{equation}
Pick a smooth bounded and increasing function $\chi\in C^\infty(\mathbb R)$ such that $\chi(\xi)=\xi$ for $\xi\in (-\hat\zeta_a/2,\hat\zeta_a/2)$, and $\eta_+:=\lim_{\xi\to+\infty}\chi(\xi)\in(0,\hat\zeta_a)$. We introduce the function \begin{equation}\label{eq:chi1} \chi_1(\xi)=\hat\zeta_a+\chi(\xi-\hat\zeta_a), \end{equation} and the operator \begin{equation}\label{eq:new-symbol} \textrm{ Op}^\textrm{ W}_{\hbar}(p_{\hbar})\quad\textrm{ where~}p_{\hbar}(\sigma,\xi):=n_{\hbar}(\sigma,\chi_1(\xi)). \end{equation} Now consider the Grushin problem defined by the matrix operator \begin{equation}\label{eq:full-symbol} \mathcal P_z(\sigma,\xi)=\left(\begin{array}{ll} p_\hbar-z&\cdot v_\xi\\ \langle\cdot,v_\xi\rangle&0 \end{array}\right)\in\mathcal S\big(\mathbb R_{\sigma,\xi}^2,\mathcal L(\textsf{ Dom}(p_0)\times \mathbb C,L^2(\mathbb R)\times\mathbb C) \big) \end{equation} where $\mathcal S\big(\mathbb R_{\sigma,\xi}^2,\mathcal L(\textsf{ Dom}p_0\times \mathbb C,L^2(\mathbb R)\times\mathbb C) \big)$ is defined in \cite[Notation~3.1]{BHR21}, \begin{equation}\label{eq:p0}
p_0(\sigma,\xi) =-\partial_\tau^2+(\chi_1(\xi)-b_a\tau)^2 \end{equation} is the principal symbol of $p_\hbar$ and $v_\xi$ is the positive normalized ground state of $p_0$, with corresponding eigenvalue $\mu_1(\chi_1(\xi))=\mu_a(-\chi_1(\xi))$ (see \eqref{mu_a_1}).
From the decomposition of $n_\hbar$, we can decompose $\mathcal P_z$ as follows: \[ \mathcal P_z=\mathcal P_z^{[3]}+\hbar^2\mathcal R_\hbar,\quad
\mathcal P_z^{[3]}=\mathcal P_{0,z}+\hbar^{1/2} \mathcal P_1+\hbar\mathcal P_2
+\hbar^{3/2} \mathcal P_3,\]
where
\[\mathcal P_{0,z}=\left(\begin{array}{ll} p_0-z&\cdot v_\xi\\ \langle\cdot,v_\xi\rangle&0 \end{array}\right) ,\quad \forall j\geq 1,~\mathcal P_j=\left(\begin{array}{ll} p_j&0\\ 0&0 \end{array}\right),\quad \mathcal R_\hbar= \left(\begin{array}{ll} r_\hbar&0\\ 0&0 \end{array}\right),\] and \begin{equation}\label{eq.pj} \begin{aligned} p_1&=2i(\chi_1(\xi)-b_a\tau)\varphi',\\ p_2&=-\varphi'^2+\kappa c_\mu \partial_\tau+c_\mu \kappa(\chi_1(\xi)-b_a\tau)b_a\tau^2+2\kappa\tau c_\mu (\chi_1(\xi)-b_a\tau)^2 \nonumber \\ &\quad +\kappa \tau c_\mu'\left(\tau \right), \\
\textrm{ Re}\, p_3 & = 0, \\
\tilde{r}_{\hbar} &= {\mathcal O}(\tau^4, \tau^2\partial_\tau), \end{aligned} \end{equation} where $\mathcal O (\tau^4, \tau^2\partial_\tau)$ is understood in the sense of \eqref{eq:not-O-op}.
Then one can construct a parametrix of $\textrm{ Op}^\textrm{ W}_{\hbar}(\mathcal P_z)$ (see \cite[Thm.~3.5]{BHR21} for details) \[ \mathcal L_z^{[3]}=\left(\begin{array}{ll} q_z&q_z^+\\ q_z^-&q_z^\pm \end{array}\right),\quad \textrm{ Op}^\textrm{ W}_{\hbar}(\mathcal L_z^{[3]}) \, \textrm{ Op}^\textrm{ W}_{\hbar}(\mathcal P_z)=\textrm{ Id}+\hbar^2\mathcal O(\langle\tau\rangle^6),\] where \[q_z^\pm=q_{0,z}^\pm+\hbar^{1/2}q_{1,z}^\pm+\hbar q_{2,z}^\pm +\hbar^{3/2}q_{3,z}^\pm\,,\] with \begin{equation}\label{eq.qj} \begin{aligned} q_{0,z}^\pm&=z-\mu_1(\chi_1(\xi)),\\ q_{1,z}^\pm&=-i\varphi'(\sigma)\partial_\xi\mu_1(\chi_1(\xi)),\\ q_{2,z}^\pm&=k(\sigma)C_1(\xi,\mu)+C_2(\xi,z)\varphi'(\sigma)^2,\\ \textrm{ Re} \,q_{3,z}^\pm&=0, \end{aligned} \end{equation} and \begin{equation}\label{eq.Cj} \begin{aligned} C_1(\xi,\mu)=&-\langle \big(c_\mu \partial_\tau+c_\mu (\chi_1(\xi)-b_a\tau)b_a\tau^2+2\tau c_\mu (\chi_1(\xi)-b_a\tau)^2\big)v_\xi,v_\xi\rangle\\ &\qquad-\langle\tau c_\mu'\partial_\tau\big)v_\xi,v_\xi\rangle, \\ C_2(\xi,\mu)=&1-\langle(p_0-z)^{-1}\Pi^\bot(\chi_1(\xi)-b_a\tau)v_\xi,(\chi_1(\xi)-b_a\tau)v_\xi\rangle. \end{aligned} \end{equation} Here $\Pi=\Pi_\xi$ is the orthogonal projection on $v_\xi$ and $\Pi^\bot=\textrm{ Id}-\Pi$. Note that, by \eqref{eq:hat-zeta}, Remark~\ref{rem:AHK2.3} and \eqref{eq:I2}, \[C_1(\hat\zeta_a,0)=-M_3(a),\quad C_2(\hat\zeta_a,\beta_a)=\frac{\mu_a''(\zeta_a)}{2}.\]
Now we argue like \cite[Prop~4.4]{BHR21}. Recall that $|z-\beta_a-M_3(a)k_{\max}\hbar|\leq c_0\hbar^2$ and that $\mu\to0$ as $h\to0$. Expanding $C_1(\xi,\mu)$ and $C_2(\xi,z)$ near $\xi=\hat\zeta_a$, we get \begin{align*}
C_1(\xi,\mu)&=-M_3(a)k_{\max}\hbar+\mathcal O\big(\hbar\min(1,|\xi-\hat\zeta_a|)\big),\\
C_2(\xi,z)&=\frac{\mu_a''(\zeta_a)}{2}+\mathcal O\big(\hbar\min(1,|\xi-\hat\zeta_a|)\big). \end{align*} Furthermore, since $\mu'_a(\zeta_a)=0$ and $\mu''_a(\zeta_a)>0$, there exists a constant $c_1>0$ such that
\[\mu(\chi_1(\xi))-z \geq c_1\min(1,|\xi-\hat\zeta_a|^2). \] Now we have the following lower bound \[ -\textrm{ Re}\,q_z^\pm\geq \hbar(\mathfrak v(\sigma)-C_2(\hat\zeta_a,\beta_a)\varphi'(\sigma)^2\big )-C\hbar^2,\] where $\mathfrak v(\sigma)$ is introduced in Assumption~\ref{ass:phi}. We apply the Fefferman-Phong inequality \cite[Thm.~3.2]{B} (see also \cite[Thm.~4.3.2]{Z}) on the symbol \[\mathfrak a(\hat\sigma,\hat\xi;\hbar):=\mathscr{A}(\hbar^{1/2}\hat\sigma,\hbar^{1/2}\hat\xi;\hbar), \] where \[\mathscr{A}(\sigma,\xi;\hbar):=-\textrm{ Re}\, q_z^\pm(\sigma,\xi) -\hbar(\mathfrak v(\sigma)-C_2(\hat\zeta_a,\beta_a)\varphi'(\sigma)^2\big )-C\hbar^2 .\] In that way, we have
\[-\textrm{ Re}\langle \textrm{ Op}_{\hbar}^\textrm{ W}(q_z^\pm)\psi,\psi\rangle \geq \hbar\int_{\mathbb R}\left(\mathfrak v(\sigma)-\frac{\mu''_a(\zeta_a)}{2}|\varphi'(\sigma)|^2{-C\hbar}\right)|\psi|^2d\sigma, \] from which we get the following estimate (see \cite[Thm.~4.2]{BHR21})
\[cR^2\hbar^2\|\psi\|\leq \|(\textrm{ Op}_{\hbar}^\textrm{ W} (p_\hbar)-z)\psi \|+C_R\hbar^2\|\chi_0
(\hbar^{-1/2}R^{-1}(\sigma-s_r))\psi\|+\hbar^2\|\tau^6\psi\|\,, \] which is almost the inequality in Theorem~\ref{thm:dec}, but with the operator $\textrm{ Op}_{\hbar}^\textrm{ W}(p_\hbar)$ instead of the operator $\mathcal N_{\hbar}^\varphi=\textrm{ Op}_{\hbar}^\textrm{ W}(n_\hbar)$. The only difference between the two operators is the frequency cut-off in the symbol, which can be removed following the same argument in \cite[Thm.~5.1]{BHR21}.
\subsection{Applications}
By appropriate choices of the function $\varphi$ in Theorem~\ref{thm:dec}, we get optimal tangential estimates for the bound states of the `single' and `double' well operators. For details, see \cite[Corol.~5.7, Corol.~6.1\,\&\,Prop.~6.2]{BHR21}.
\begin{proposition}[Decay of bound states]\label{prop:dec-bs} Let $\theta,\varepsilon\in(0,1)$ and $K>0$. There exist $C,\hbar_0>0$ such that for all $\hbar\in(0,\hbar_0)$, the following is true.
If $\lambda$ eigenvalue of the operator $\mathscr{N}_
{\hbar}$ in \eqref{eq:Nh}, $\left|\lambda-\left(\beta_a+M_3(a)k_{\max}\hbar\right)\right|\leq K\hbar^2$ and $u\in\textsf{ Dom}(\mathscr{N}_{\hbar})$ is an eigenfunction associated to $\lambda$, then
\[\int_{[-L,L)\times\mathbb R}e^{2\varphi/\hbar^{\frac 12}}|u|^2dsd\tau\leq Ce^{\varepsilon/\hbar^{\frac 12}}\|u\|^2_{L^2([-L,L)\times\mathbb R)}\,.\] where \[\varphi=(1-\theta)^{1/2}\min(\tilde\Phi_r,\tilde\Phi_\ell)\,,\] and $\tilde\Phi_r,\tilde\Phi_\ell$ are $2L$-periodic functions satisfying, for $\eta$ sufficiently small, \[ \begin{aligned}
\tilde\Phi_r(\sigma)\,|_{-L\leq \sigma\leq s_\ell-\eta}&=\Phi_r(\sigma):=\sqrt{\frac{-2M_3(a)}{\mu_a''(\zeta_a)}}\int_{[s_r,\sigma]}\sqrt{k_{\max}-k(s)}\,ds,\\
\tilde\Phi_\ell(\sigma)\,|_{-L\leq \sigma\leq s_r-\eta}&=\Phi_\ell(\sigma):=\sqrt{\frac{-2M_3(a)}{\mu_a''(\zeta_a)}}\int_{[s_\ell,\sigma]}\sqrt{k_{\max}-k(s)}\,ds\,. \end{aligned} \] \end{proposition}
\begin{remark}\label{rem:dec-bs} The estimate in Proposition~\ref{prop:dec-bs} continues to hold if $\lambda$ is an eigenvalue of the right or left well operator, $\mathcal N_{\hbar,r}$ or $\mathcal N_{\hbar,\ell}$, and $u$ is a corresponding eigenfunction. \end{remark}
Returning to the `one well' operators, $\mathcal N_{\hbar,r,\gamma_0}$ and $\mathcal N_{\hbar,\ell,\gamma_0}$ introduced in \eqref{eq:Nh-r} and \eqref{eq:Nh-l} respectively, we get from Proposition~\ref{prop:dec-bs} and the min-max principle, the following rough estimate, important for the analysis of tunneling later on, \begin{equation}\label{eq:ev-sw} \mu_1^\textrm{ sw}(\hbar)-\tilde{\mathcal O}(e^{-\mathsf{S}^a/\sqrt{\hbar} })\leq \nu_{1,a}(\hbar)\leq \nu_{2,a}(\hbar)\leq \mu_1^\textrm{ sw}(\hbar)+\tilde{\mathcal O}(e^{-\mathsf{S}^a/\sqrt{\hbar}}), \end{equation} where $\mathsf S^a$ is introduced in \eqref{defSa}, \[\mu_1^\textrm{ sw}(\hbar)=\inf\sigma(\mathcal N_{\hbar,r,\gamma_0})= \inf\sigma(\mathcal N_{\hbar,r,\gamma_0})\,,\] $(\nu_{j,a}(\hbar))_{j\geq 1}$ is the sequence of eigenvalues of the operator $\mathcal N_{\hbar}$, and the notation $\tilde{\mathcal O}(e^{-\mathsf{S}^a/\sqrt{\hbar}})$ means \begin{equation}\label{eq:not-tO}
{\mathcal O}(e^{(\epsilon -\mathsf{S}^a)/\sqrt{\hbar}})\quad\textrm{ for~any~fixed~}\varepsilon>0. \end{equation}
The analysis of the tunneling requires an explicit approximation of the ground state of the single well operators. Recall the Agmon distance $\Phi_r$ and the WKB solution $\Psi_{\hbar,r}$ introduced in \eqref{eq:Ag-r} and Theorem~\ref{thm:WKB} respectively. Consider the flux free `right well' operator $\mathcal N_{\hbar,r}:=\mathcal N_{\hbar,r,0}$. By \eqref{eq:r-well}, the low lying eigenvalues of this operator are simple; we denote by $\Pi_r$ the orthogonal projection on its first eigenspace. By \cite[Prop.~6.3]{BHR21}, it results from Theorem~\ref{thm:dec}:
\begin{proposition}[WKB approximation]\label{prop:WKB} We have
\[\|\psi_{\hbar,r}-\Pi_r\psi_{\hbar,r}\|_{L^2(\mathbb R^2)}=\mathcal{O}(\hbar^\infty)\] and
\begin{equation}\label{eq:app.WKB1'}
\langle\tau\rangle\, e^{\Phi_{r}/\sqrt{\hbar}}(\Psi_{\hbar,r}-u_{\hbar,r})=\mathcal
{O}(\hbar^\infty)\quad\textrm{ in}~\mathscr{C}^1(K;L^2(\mathbb R)),
\end{equation}
where $K\subset I_{2\hat\eta,r}:=(s_\ell-2L+\hat\eta,s_\ell-\hat\eta)$ is a compact set,
\[\psi_{\hbar,r}(\sigma,\tau):=\chi_{\hat\eta,r}(\sigma)\Psi_{\hbar,\tau}(\sigma,\tau)\,, \]
and $\chi_{\hat\eta,r}$ is a cut-off function supported in $I_{\hat\eta,r}$ such that $\chi_{\hat\eta,r}=1$ on $I_{2\hat\eta,r}$. \end{proposition}
\section{Interaction matrix and tunneling}\label{sec:IM} We return to the operator $\mathcal N_{\hbar}$ introduced in \eqref{eq:Nh}. In order to estimate the splitting between the first and second eigenvalues, $\nu_2(\hbar)-\nu_1(\hbar)$, we will write the matrix of this operator in a specific basis of \[E=\oplus_{i=1}^2\textrm{ Ker}\big(\mathcal N_\hbar-\nu_i(\hbar)\big). \] Let $\Pi$ be the orthogonal projection on $E$. We introduce the two functions \[ f_{\hbar,r}=\chi_{\hat\eta,r}\phi_{\hbar,r},\qquad \qquad f_{\hbar,\ell}=\chi_{\hat\eta,\ell}\phi_{\hbar,\ell},\] where $\chi_{\hat\eta,r}$ is the cut-off function introduced in Proposition~\ref{prop:WKB}, $\chi_{\hat\eta,\ell}=U\chi_{\hat\eta,r}$ is defined by the symmetry operator (see Sec.~\ref{sec:LWop}), $\phi_{\hbar,r}$, $\phi_{\hbar,\ell}$ enjoy periodicity properties and are defined by inspiration from the functions in \eqref{eq:gs-r} and \eqref{eq.phil0}. In fact, $\phi_{\hbar,r}(\sigma,\tau)$ need to be defined in the support of $\chi_{\hat\eta,r}$. Starting on $[-L,s_\ell-\frac{\hat\eta}2)\times\mathbb{R}$, we take $\phi_{\hbar,r}(\sigma,\tau)$ the same as the function in \eqref{eq:gs-r}; on $[s_\ell+\frac{\hat\eta}{2},L)\times\mathbb{R}$, we do a change of variable, and modify the function in \eqref{eq:gs-r} so that its module satisfies a periodic boundary condition on $\pm L$. More precisely, we have \begin{equation}\label{eq:phi-h-r} \phi_{\hbar, r}(\sigma,\tau) = \begin{cases}
e^{ -i\gamma_0 \sigma/\hbar^2}u_{\hbar,r}(\sigma,\tau),&\mbox{if }-L\leq \sigma\leq s_\ell-\frac{\hat\eta}{2}, \\ e^{ -i\gamma_0 (\sigma-2L)/\hbar^2}u_{\hbar,r}(\sigma-2L,\tau),&\mbox{if } s_\ell + \frac{\hat\eta}{2}<\sigma< L,
\end{cases}
\end{equation}
and so $f_{\hbar,r}$ is well defined on $[-L,L)$. In a similar fashion,
\begin{equation}\label{eq:phi-h-l}
\phi_{\hbar, \ell}(\sigma,\tau) = \begin{cases} e^{-i\gamma_0 (\sigma+2L)/\hbar^2}u_{\hbar,\ell}(\sigma+2L,\tau),&\mbox{if }-L\leq \sigma\leq s_r - \frac{\hat\eta}{2}, \\ e^{-i\gamma_0 \sigma/\hbar^2}u_{\hbar,\ell}(\sigma,\tau),&\mbox{if } s_r+\frac{\hat\eta}2<\sigma< L, \end{cases}
\end{equation} and $f_{\hbar, \ell}$ is well defined on $[-L,L)$.
Also we introduce the following actual bound states by projecting on the eigenspace $E$, \[ g_{\hbar,r}=\Pi f_{\hbar,r},\qquad\qquad g_{\hbar,\ell}=\Pi f_{\hbar,\ell}\,.\] We use the notation $\tilde{\mathcal O}$ in \eqref{eq:not-tO}. By Proposition~\ref{prop:dec-bs}, we have (see \cite[Sec.~7.1]{BHR21} and \cite[Sec.~3]{BHR} for details) \begin{align*}
\|f_{\hbar,r}\|^2&=1+\tilde{\mathcal O}(e^{-2\mathsf{S}^a/\sqrt{\hbar}}),\quad\|f_{\hbar,\ell}\|^2=1+\tilde{\mathcal O}(e^{-2\mathsf{S}^a/\sqrt{\hbar}}), \\ \langle f_{\hbar,r}, f_{\hbar,\ell}\rangle&=\tilde{\mathcal O}(e^{-\mathsf{S}^a/\sqrt{\hbar}}), \end{align*} and
\[\|g_{\hbar,\alpha}-f_{\hbar,\alpha}\|+ \|\partial_\tau(g_{\hbar,\alpha}-f_{\hbar,\alpha})\|=\tilde{\mathcal O}(e^{-\mathsf{S}^a/\sqrt{\hbar}})\qquad\alpha\in\{r,\ell\}. \]
Now, we construct an orthonormal basis $\mathcal B_{\hbar}:=\{\tilde g_{\hbar,r},\tilde g_{\hbar,\ell}\}$ of $E$ from $\{g_{\hbar,r},g_{\hbar,\ell}\}$ by the Gram-Schmidt process. Let $\mathsf M$ be the { matrix} of $\mathcal N_{\hbar}$ relative to the basis $\mathcal B_{\hbar}$. Then \begin{equation}\label{eq:gap}
\nu_2(\hbar)-\nu_1(\hbar)=2|w_{\ell,r}| +\tilde{\mathcal O}(e^{-2\mathsf{S}/\sqrt{\hbar}})\,,\quad w_{\ell,r}=\langle r_{\hbar,\ell},f_{\hbar,r}\rangle, \end{equation} where \[ r_{\hbar,\ell}=(\mathcal N_{\hbar,\ell}-\mu^\textrm{ sw}(\hbar))f_{\hbar,\ell}.\] All we have to do now is the computation of $w_{\ell,r}$ by the WKB approximation in Proposition~\ref{prop:WKB}. By \cite[Lem.~7.1]{BHR21}(which is essentially an integration by parts formula) \begin{multline*}w_{\ell,r}=i\hbar\int_\mathbb{R} a_\hbar^{-1}\left(\phi_{\hbar,\ell}\overline{\mathscr{D}_\hbar\phi_r}+[\mathscr{D}_\hbar\phi_{\hbar,\ell}]\overline{\phi_{\hbar,r}}\right)(0,\tau){ d\tau}\\ -i\hbar\int_\mathbb{R} a_\hbar^{-1}\left(\phi_{\hbar,\ell}\overline{\mathscr{D}_\hbar\phi_{\hbar,r}}+[\mathscr{D}_\hbar\phi_\ell]\overline{\phi_{\hbar,r}}\right)(-L,\tau)d\tau, \end{multline*} where \[ \mathscr{D}_\hbar=\hbar D_\sigma+\hbar^{-1}\gamma_0-b_a(\tau)\tau+\hbar c_\mu \frac{k}{2}b_a(\tau)\tau^2\,.\] Writing $a_\hbar=1+o(1)$, $\hbar c_\mu\tau^2=o(\hbar^{-2\eta})$, and approximating $\phi_{\hbar,r}$, $\phi_{\hbar,\ell}$ by using \eqref{eq:phi-h-r}, \eqref{eq:phi-h-l} and Proposition~\ref{prop:WKB}, we get (see \cite[Sec.~7.2.2]{BHR21} for details) \begin{equation}\label{eq:w-r-l} w_{r,\ell}=i\hbar(w_{r,\ell}^\textsf{ u}+ w_{r,\ell}^\textsf{ d}) \end{equation} where \begin{align*} w_{\ell,r}^\textsf{ u}&= \int_\mathbb{R} a_\hbar^{-1}\Psi_{\hbar,\ell}\overline{\left(\hbar D_\sigma-b_a\tau+\hbar c_\mu\frac{\kappa}{2}b_a\tau^2\right)\Psi_{\hbar,r}}(0,\tau)d\tau\\ &+\int_\mathbb{R} a_\hbar^{-1} \left[\left(\hbar D_\sigma-b_a\tau+\hbar c_\mu\frac{\kappa}{2}b_a\tau^2\right)\Psi_{\hbar,\ell}\right]\overline{\Psi_{\hbar,r}}(0,\tau)d\tau +\mathcal{O}(\hbar^\infty)e^{-\mathsf{S}_{\mathsf{u}}^a/\hbar^{1/2}} , \intertext{and} w_{\ell,r}^\textsf{ d}&= \int_\mathbb{R} a_\hbar^{-1}\Psi_{\hbar,\ell}\overline{\left(\hbar D_\sigma-b_a\tau+\hbar c_\mu\frac{\kappa}{2}b_a\tau^2\right)\Psi_{\hbar,r}}(-L,\tau)d\tau\\ &+\int_\mathbb{R} a_\hbar^{-1} \left[\left(\hbar D_\sigma-b_a\tau+\hbar c_\mu\frac{\kappa}{2}b_a\tau^2\right)\Psi_{\hbar,\ell}\right]\overline{\Psi_{\hbar,r}}(-L,\tau)d\tau +\mathcal{O}(\hbar^\infty)e^{-\mathsf{S}_{\mathsf{d}}^a/\hbar^{1/2}} . \end{align*} Eventually, using \eqref{eq:eff-op} and \eqref{eq.psir}, we get (see \cite[Eq.~(7.11)]{BHR21}) \[\hbar^{\frac14}e^{\mathsf{S}_{\mathsf{u}}^a/\hbar^{1/2}}w_{\ell,r}^\textsf{ u}=-i \hbar^{\frac12}\mu_a''(\zeta_a) \pi^{-\frac 12} g_a^{\frac12}\sqrt{V_a(0)} \mathsf{A}_{\mathsf{u}}^ae^{-2i\alpha_0(0)}+\mathcal {O}(\hbar) \] and, in a similar fashion, \begin{multline*} \hbar^{\frac14}e^{\mathsf{S}_{\mathsf{d}}^a/\hbar^{1/2}}w_{\ell,r}^\textsf{ d}\\ =-i \hbar^{\frac12}\mu_a''(\zeta_a) \pi^{-\frac 12} g_a^{\frac12}\sqrt{V_a(-L)} \textsf{A}_{\mathsf{d}}^ae^{-2i\alpha_0(-L)}e^{i ( -2L\gamma_0/\hbar^2-2L\zeta_a/\hbar)}+\mathcal {O}(\hbar) \,,\end{multline*}
where $\alpha_0$ is the function introduced in Remark~\ref{eq.effectiveT}.
Collecting \eqref{eq:w-r-l} and \eqref{eq:gap} and using that $\hbar=h^{1/2}$, we get
\[\begin{aligned}
\nu_2(\hbar)-\nu_1(\hbar)&=2|e^{iLf(h)}w_{\ell,r}|+\tilde{\mathcal O}(e^{-2\mathsf S^a/\sqrt{\hbar}})\\
&=h^{-1}| w_a(h)|+\tilde{\mathcal O}(h^{-1}e^{-2\mathsf S^a/\sqrt{\hbar}}),
\end{aligned}\]
where $f(h)$ and $\tilde w_a(h)$ are the expressions in Theorem~\ref{thm:FHK}. In light of
Proposition~\ref{prop:BHR2.7}, this finishes the proof of Theorem~\ref{thm:FHK}. \section{Conclusion and open problems}
Until now, examples of magnetic tunneling effects are rare in the literature. Very few articles have been dealing with the measure of the tunneling effect due to the presence of the magnetic field. In the presence of an electric potential with multiple wells, the article \cite{HeSj7} was only considering a case when the
magnetic field was a perturbation and the tunneling was mainly created by the electric potential. Other examples include the case of a pure flux \cite{KR17}. After the recent contributions of Bonnaillie-H\'erau-Raymond \cite{BHR} and Fefferman-Shapiro-Weinstein \cite{FSW} (see also references therein), we have presented a new magnetic tunneling effect
due to the curvature of the magnetic edge.
Both for the Neumann problem occurring in surface superconductivity \cite{HM3D} or for the problem considered here \cite{AG}, it would be interesting to consider the $(3D)$-case.
Excluded in this paper is the case $a=-1$, where localization near the point(s) of maximum curvature no more occurs ($M_3(a)=0$ in the asymptotics of Theorem~\ref{thm:AHK}). In contrast, this case seems to feature an interesting new phenomenon where localization near the whole edge $\Gamma$ occurs, which also has a nice analogy to what was observed in the multiple wells situation in \cite{HeSj6}. We hope to come back to the treatment of this case rather soon.
Finally we mention that the standard purely magnetic double well problem seems at the moment a difficult challenge. Here we consider a purely semi-classical magnetic Laplacian (say in $\mathbb R^2$) where the magnetic
field has two symmetric non degenerate positive minima.
\section{Appendix: Regularized resolvent and moments}
Let us return back to the flat edge model in Sec.~\ref{sec:FlatEdge} and recall some necessary computational results.
We can invert the operator $\mathfrak h_a[\zeta_a]$ on the orthogonal complement of the ground state $\phi_a$. Extending by linearity, we get the regularized resolvent $\mathfrak R_a$ defined on $L^2(\mathbb R )$ by
\begin{equation}\label{eq:R}
\mathfrak R_a\, u =
\begin{cases}
0&\mathrm{if}~u\parallel\phi_a\\
(\mathfrak h_a[\zeta_a]-\beta_a)^{-1}u&\mathrm{if}~u\perp \phi_a
\end{cases}\,.
\end{equation} By \cite{AK20}, $(\zeta_a+b_a(\tau) \tau)\phi_a$ and $\phi_a$ are orthogonal in $L^2(\mathbb R)$: \begin{equation}\label{eq:m0}
\int_{\mathbb R}(\zeta_a+b_a(\tau) \tau)|\phi_a(\tau)|^2\,d\tau=0\,. \end{equation} Later on we will encounter the following integral \cite[Prop.~2.5]{AHK} \begin{equation}\label{eq:I2} I_2(a):=\int_{\mathbb R }\phi_a(\tau) \mathfrak R_a[(\zeta_a+b_a(\tau) \tau)\phi_a]\,d\tau
=\frac14-\frac{\mu_a''(\zeta_a)}{8}\,. \end{equation} We recall some identities from \cite{AK20} involving for $n\in \mathbb N $ quantities of the form \begin{equation}\label{eq:moments}
M_n(a)=\int_{\mathbb R}\frac 1{b_a(\tau)}(\zeta_a+b_a(\tau)\tau)^n|\phi_a(\tau)|^2\,d\tau\,. \end{equation}
We have
\begin{align}
M_1(a)&=0\,,\label{eq:m1}\\
M_2(a)&=-\frac 12 \beta_a\int_{\mathbb R}\frac 1{b_a(\tau)}|\phi_a(\tau)|^2\,d\tau+\frac 14\Big(\frac 1a-1\Big)\zeta_a\phi_a(0)\phi_a'(0)\,, \label{eq:m2}\\
M_3(a)&=\frac 13\Big(\frac 1a-1\Big)\zeta_a\phi_a(0)\phi_a'(0)\,.\label{eq:m3*}
\end{align}
The case $a=-1$ is special because \[M_3(-1)=0 \quad \mathrm{and}\quad M_3(a)<0 \quad \mathrm{for} \quad -1<a<0\,.\]
\begin{remark}\label{rem:AHK2.3} The next identities follow in a straightforward manner \cite[Rem.~2.3]{AHK}, \begin{align*}
\int_{\mathbb R}\tau(\zeta_a+b_a(\tau) \tau)|\phi_a(\tau)|^2\,d\tau&=M_2(a),\\
\int_{\mathbb R}\tau(\zeta_a+b_a(\tau) \tau)^2|\phi_a(\tau)|^2\,d\tau&=M_3(a) -\zeta_aM_2(a),\\
\int_{\mathbb R}b_a(\tau) \tau^2(\zeta_a+b_a(\tau) \tau)|\phi_a(\tau)|^2\,d\tau&=M_3(a)-2\zeta_aM_2(a), \end{align*} and \begin{align*}
\int_{\mathbb R}\tau|\phi_a(\tau)|^2\,d\tau&=-\zeta_a\int_{\mathbb R}\frac 1{b_a(\tau) }|\phi_a(\tau)|^2\,d\tau,\\
\int_{\mathbb R}\tau|\phi'_a(\tau)|^2\,d\tau&=\beta_a\zeta_a\int_{\mathbb R}\frac 1{b_a(\tau) }|\phi_a(\tau)|^2\,d\tau+2M_3(a) -2\zeta_aM_2(a). \end{align*}
\end{remark}
\subsection*{Acknowledgments} The authors would like to thank the anonymous referee for the attentive reading of this paper and the valuable comments.
\subsection*{Funding} Part of this work was done while SF and AK visited the Laboratoire Jean Leray at the university of Nantes. They wish to thank \emph{la f\'ed\'eration de recherche math\'ematique des Pays de Loire} and \emph{Nantes Universit\'e} for supporting the visit.
\end{document} | arXiv |
The intranasal dexmedetomidine plus ketamine for procedural sedation in children, adaptive randomized controlled non-inferiority multicenter trial (Ketodex): a statistical analysis plan
Anna Heath ORCID: orcid.org/0000-0002-7263-42511,2,3,
Juan David Rios1,
Eleanor Pullenayegum1,4,
Petros Pechlivanoglou1,4,
Martin Offringa1,4,5,
Maryna Yaskina6,
Rick Watts6,
Shana Rimmer6,
Terry P. Klassen7,8,
Kamary Coriolano9,
Naveen Poonai10,11 &
on behalf of the PERC-KIDSCAN Ketodex Study Group
Trials volume 22, Article number: 15 (2021) Cite this article
Procedural sedation and analgesia (PSA) is frequently required to perform closed reductions for fractures and dislocations in children. Intravenous (IV) ketamine is the most commonly used sedative agent for closed reductions. However, as children find IV insertion a distressing and painful procedure, there is need to identify a feasible alternative route of administration. There is evidence that a combination of dexmedetomidine and ketamine (ketodex), administered intranasally (IN), could provide adequate sedation for closed reductions while avoiding the need for IV insertion. However, there is uncertainty about the optimal combination dose for the two agents and whether it can provide adequate sedation for closed reductions. The Intranasal Dexmedetomidine Plus Ketamine for Procedural Sedation (Ketodex) study is a Bayesian phase II/III, non-inferiority trial in children undergoing PSA for closed reductions that aims to address both these research questions. This article presents in detail the statistical analysis plan for the Ketodex trial and was submitted before the outcomes of the trial were available for analysis.
The Ketodex trial is a multicenter, four-armed, randomized, double-dummy controlled, Bayesian response adaptive dose finding, non-inferiority, phase II/III trial designed to determine (i) whether IN ketodex is non-inferior to IV ketamine for adequate sedation in children undergoing a closed reduction of a fracture or dislocation in a pediatric emergency department and (ii) the combination dose for IN ketodex that provides optimal sedation. Adequate sedation will be primarily measured using the Pediatric Sedation State Scale. As secondary outcomes, the Ketodex trial will compare the length of stay in the emergency department, time to wakening, and adverse events between study arms.
The Ketodex trial will provide evidence on the optimal dose for, and effectiveness of, IN ketodex as an alternative to IV ketamine providing sedation for patients undergoing a closed reduction. The data from the Ketodex trial will be analyzed from a Bayesian perspective according to this statistical analysis plan. This will reduce the risk of producing data-driven results introducing bias in our reported outcomes.
ClinicalTrials.gov NCT04195256. Registered on December 11, 2019.
Orthopedic injuries are common in children visiting the emergency department (ED) [1, 2], sometimes requiring a closed reduction [3, 4], which usually requires procedural sedation and analgesia (PSA). Intravenous (IV) ketamine is a common sedative agent that facilitates closed reductions [5]. However, IV insertion is distressing for children and their caregivers and can be challenging as children have small veins and resist positioning [6].
Intranasal (IN) administration of sedative agents is a less invasive potential alternative to IV insertion [7] with a combination of dexmedetomidine and ketamine (ketodex) showing promise for PSA across several indications [8]. However, it is unclear whether IN ketodex would offer sufficient analgesia and sedation for closed reductions in children with orthopedic injuries, compared to IV ketamine, and there is no evidence on the most effective combination of dexmedetomidine and ketamine within the ketodex formulation.
The intranasal dexmedetomidine plus ketamine for procedural sedation in children (Ketodex) study aims to determine (i) whether the sedation provided by IN ketodex is non-inferior to that provided by IV ketamine for children undergoing a closed reduction and (ii) the combination dose of IN ketodex that provides optimal sedation. To achieve these objectives, the Ketodex study uses an innovative Bayesian response-adaptive, comparative-effectiveness design [9]. This paper outlines the statistical analysis plan (SAP) for the Ketodex trial as the study protocol is published separately [10]. This SAP has been published before analyzing any study data and follows the reporting guidelines for SAPs [11].
The primary aim of the Ketodex study is to determine if IN ketodex is non-inferior to IV ketamine with respect to the proportion of participants who achieve adequate sedation for the duration of a closed reduction. We will enroll previously healthy children who present to one of six Canadian participating pediatric EDs requiring a reduction for a fracture or dislocation. The secondary objectives of the study are to determine the optimal combination dose for IN ketodex and to characterize the sedation characteristics of IN ketodex with respect to length of stay in the ED, need for additional sedation, nasal irritation, satisfaction with sedation, and adverse events.
Design and setting
The Ketodex trial is a phase II/III, double-dummy, controlled, randomized, Bayesian response adaptive, dose finding, non-inferiority trial being conducted in six Canadian tertiary care pediatric EDs. Eligible participants are age 4 to 17 years and require PSA to undergo a closed reduction of a fracture or dislocation. Following informed consent, patients will be randomized in a 2:3 ratio to receive either IV ketamine, dosed at 1.5 mg/kg (maximum 100 mg), with an IN placebo combination treatment, or one of three combination doses of IN ketodex, i.e., 2, 3, or 4 mg/kg (maximum 400 mg) of ketamine combined with 4, 3, or 2 μg/kg (maximum 200 μg) of dexmedetomidine, respectively, with an IV placebo. Patients will then be further randomized to receive one of the three dose combinations, initially in a 1:1:1 ratio (see "Randomization").
Most study data will be collected in the ED within 1–2 h of study enrollment. A follow-up phone call will be made between 24 and 48 h after enrollment to collect data on issues with the sedation. If a participant is lost to telephone follow-up, a research nurse will attempt to regain contact 3–5 times, using email or a letter to the participant's last known address, depending on local research ethics board (REB) regulations.
Study protocol development and conduct
The Ketodex study was registered on ClinicalTrials.gov on December 11, 2019, with registration number NCT04195256. At each institution, the REB will give ethical approval before commencing local enrollment. Informed consent, following institutional REB guidelines, will be obtained from caregivers before randomization and data collection. The Ketodex study will be supported by the KidsCAN-PERC iPCT network [12], a Canadian trials network using centralized infrastructure for data management and trial oversight for four trials. An independent data and safety monitoring board (DSMB) of six oversees the study.
Randomization
Participants will be randomized in a ratio of 2:3 to either IV ketamine or IN ketodex using block randomization with blocks of size five, stratified by site. Participants will undergo a further randomization to one of the three IN ketodex combinations using the REDCap electronic data capture system [13]. For participants randomized to IV ketamine, the second randomization will be to two placebo (saline) solutions. The second randomization step will be adapted so more participants receive the more effective dose combinations. Figure 1 displays this two-step randomization procedure.
Depiction of the randomization procedure for the Ketodex trial
The initial randomization ratio for the dose combinations will be 1:1:1. Interim analyses will update the randomization ratio at approximately 150, 200, 250, 300, and 350 participants. As outlined in the "Analysis for the primary endpoint" section, each interim analysis will compute the posterior probability that each dose is the most effective. The number of patients randomized to each dose before the next interim analysis will equal this posterior probability multiplied by the number of participants to be enrolled before the next interim analysis, rounded to the nearest whole number. If, at a given interim analysis, the probability that a dose is optimal falls below 0.05 (threshold chosen by simulation), we will not randomize participants to this dose. If all three combinations have a probability of being the most effective of less than 50% once 250 participants have been enrolled, then safety and tolerability will be assessed to determine the most promising combination. This dose combination will be used for all participants randomized to IN ketodex.
The randomization list for the first step will be generated and held securely at the Women and Children's Health Research Institute's (WCHRI) Data Coordinating Centre at the University of Alberta [14]. Pharmacies will prepare identically appearing IV ketamine plus IN saline or IN ketodex plus IV saline kits for use at the bedside. The second randomization will take place at the bedside from a master randomization list, accessed sequentially as participants are enrolled across all sites. Site-level stratification is not used for the second randomization. The research nurse will be blinded to the intervention but not to the dose combination. Outcome assessors will be blinded to the dose combination and the intervention.
Data storage and collection
All participant data will be stored in REDCap [13] and held at the WCHRI Data Coordinating Centre [14]. All data will be entered into REDCap using a WiFi-enabled encrypted iPad or recorded on paper in the event of a technical failure. Prior to analysis, all personal identifiers will be removed and participants will be identified using a unique study identification number.
The primary outcome is adequate sedation for the duration of the closed reduction. This will be ascertained over the interval of time from the first application of traction or manipulation of the injured limb for the purpose of anatomical realignment to the last application of a realigning force. Adequate sedation is defined as fulfillment of all three of the following criteria:
A Pediatric Sedation State Scale (PSSS) [15] score of 2 or 3 for the duration of the procedure AND
No additional medication given during the closed reduction for the purpose of sedation AND
The patient did not actively resist, cry, or require physical restraint for completion of the closed reduction.
The primary outcome will be assessed by two independent outcome assessors blinded to the study group and objectives who will use video recordings of the closed reduction to calculate the PSSS score every 30 s during the procedure. Video scoring for each participant will be undertaken within 24–48 h of data collection. The second outcome assessor will score 25% of the participants to generate an interrater agreement.
Secondary efficacy outcomes
The Ketodex trial will investigate three secondary outcomes:
Length of stay: Time (in minutes) recorded in the medical record between ED triage and ED discharge.
Time to wakening: This is the time interval between the first pair or IN sprays to the first PSSS score > 3 post-closed reduction.
Adverse events (AEs): The occurrence of any AE, recorded using the medical record and queries of the health care staff during sedation and recovery. Research nurses will be trained on the recognition and definition of expected and unexpected AEs. AEs will be based on Health Canada reporting standards.
Additional outcomes
The Ketodex trial will record seven additional outcomes:
Length of stay due to PSA: Time (in minutes) from the first pair of IN sprays/IV dose to ED discharge.
Duration of procedure: Time (in minutes) of the first pair of IN sprays/IV dose to the end of cast or splint application (closed reduction).
Length of ED stay: time interval between triage assessment and discharge.
Caregiver, participant, bedside nurse or respiratory therapist, and physician satisfaction: Satisfaction from the caregiver and participant, measured using a 100-mm visual analog scale, obtained immediately prior to discharge. Satisfaction from each health care provider, measured using a 100-mm visual analog scale immediately following cast/splint application.
Nasal irritation: Discomfort associated with nasal sprays (if recalled), assessed by the research nurse using the Faces Pain Scale - Revised at discharge [16].
Volume of intervention: the volume of IN intervention a patient received, compared to the volume of IN intervention they were calculated to receive, recorded at the index visit.
Adjunctive IV therapy and medications: the requirement of an IV for therapy unrelated to sedation.
Patient preference for the method of sedation: We will ask the participant to choose their preferred sedation method, if they have one, at the index visit
We used the average length criterion (ALC) for Bayesian sample size estimation [17] by selecting the smallest sample size such that the 95% high-density posterior credible interval for the difference in the probability of adequate sedation had an average length of 0.07, six times shorter than the prior 95% highest-posterior density credible interval ("Analysis for primary endpoint"). This reduction was selected based on practical constraints [17]. We also used ALC to determine the randomization ratio between IV ketamine and IN ketodex; we considered randomizing 20%, 30%, 40%, and 50% of patients to IV ketamine.
The ALC initially selected the smallest sample size for which the average length of the 95% high-density posterior credible interval fell below 0.07 and then selected the randomization ratio that led to the most balanced trial, provided the ALC remained below 0.07. We simulated 2000 samples from the prior predictive distribution of the data, across all four randomization regimes, for sample sizes increasing from 350 to 500 in increments of 10. For each simulation, we estimated the length of the 95% high-density posterior credible interval using 2000 simulations from the posterior. Based on this, the sample size for the Ketodex trial is 410 patients with a 2:3 randomization ratio between IV ketamine and IN ketodex.
We expect minimal missing data as the primary outcome is collected during a procedure that must be completed by the physician before discharge. Thus, the sample size is not adjusted for loss to follow-up. We will monitor missingness and adjust our recruitment target to ensure that 410 patients record the primary outcome.
Interim analysis and stopping guidance
The Ketodex trial will have seven interim analyses, at increments of 50 enrolled participants. The DSMB will review safety outcomes DSMB at each interim analysis based on descriptive statistics of the safety outcomes between treatment groups. They may also receive posterior credible intervals or predictive probabilities. The decision to stop the trial for safety reasons is at the discretion of the DSMB. Due to the uneven treatment allocation, the DSMB will be unblinded to treatment assignment. The randomization ratio at the 2nd level of randomization will be updated after recruitment hits approximately 150, 200, 250, 300, and 350 participants. We will not undertake comparative effectiveness analyses at the interim analyses or stop for efficacy or futility.
Statistical analysis plan
Statistical principles
The primary analysis of the primary outcome will take place after every participant has completed the protocol and all data have been collected and cleaned. The statistical analysis will be performed unblinded to participant allocation. The primary analysis will determine if the optimal IN ketodex combination is non-inferior to IV ketamine. All other analyses will test for superiority of IN ketodex. We will undertake an intention-to-treat analysis, including all randomized participants, and a per-protocol analysis, concluding non-inferiority if both these analyses confirm non-inferiority.
All analyses will use a Bayesian perspective with significance declared based on the posterior probability that the proposed hypothesis is true. No adjustments will be made for multiplicity due to the likelihood principle [18] with the thresholds for declaring significance chosen using simulations to control the type I error of the trial [9]. Specifically, if this probability is less than 3.7%, we will declare sufficient evidence against the hypothesis. If the probability is greater than 60.8%, we will declare significant evidence for the hypothesis. If this probability is between 3.7% and 60.8%, we will declare that the trial is inconclusive. We will report treatment effect estimates using 95% highest-density posterior credible intervals. The statistical analysis will be undertaken in R [19] as an interface to JAGS [20].
Handling of missing data
We anticipate minimal missing data as the majority of outcomes are collected within the ED. If the proportion of missing data is below 5%, we will undertake a complete case analysis. If the level of missingness exceeds 5%, we will use a joint Bayesian model for the missingness and the outcome.
We will present patient flow with a CONSORT 2010 flow diagram reporting the number of participants deemed eligible for the trial at screening and those excluded as they met a study exclusion criterion and the number of participants who were randomized and received the randomized allocation. Participants can withdraw from the study at any time, for any reason. The number of participants who withdraw and the number of participants lost to follow-up will be summarized by treatment arm.
Protocol deviations and adherence
A protocol deviation is any noncompliance with the clinical trial protocol, International Conference on Harmonization Good Clinical Practice, or Trial Manual of Procedures requirements. Any change, divergence, or departure from the study design or procedures constitutes a protocol deviation. The noncompliance may be on the part of either the participant, the participating site investigator, or the study site staff. The proportion of protocol deviations will be presented by the treatment group alongside descriptive information about the deviation. Adherence will be defined as a participant who received any of the study medications and will be presented by the treatment group.
We will collect the participants' age and sex, type of fracture or dislocation, location of fracture or dislocation, the identity of the person performing the closed reduction and, if required, the identity of the person performing the casting. Baseline data will be summarized using frequencies and percentages for categorical variables and means, medians, standard deviations, and interquartile ranges for continuous variables.
Analysis for the primary endpoint
Interim analyses
The adaptive randomization will be based on the probability of adequate sedation for each dose, denoted pi, i = 1, 2, 3. We assume a binomial likelihood for the data and a beta prior with parameters 6.25 and 0.25 for each dose. This prior is informed by previous data [21] and then down-weighted by 4 to an effective sample size of 6.5 [22]. Using the same prior for all three doses assumes they have the same effectiveness unless the data indicate otherwise. This prevents early stopping of a dose combination. We will compute the probability that each dose combination has the highest proportion of successfully sedated participants from the posterior distributions for pi, i = 1, 2, 3. This procedure has an 83% chance of randomizing the highest number of patients to the optimal treatment [9].
Dose response modeling
Participants who are not adequately sedated can be over-sedated (PSSS score of 0 or 1) or under-sedated (PSSS score of 4 or 5) [15]. The final analysis will use a monotonic log-logistic dose response model for the probability of over- and under-sedation [23] and a multinomial distribution to infer the probability of adequate sedation. Specifically, let Xi, for i = 1, 2, 3, be a three-vector containing the number of patients who experience under-, adequate, and over-sedation from the Ni patients that receive dose i. We model Xi~Multinomial(Ni, pi), where \( {\boldsymbol{p}}_{\boldsymbol{i}}=\left({p}_i^{(1)},{p}_i^{(2)},{p}_i^{(3)}\right) \) and
$$ {p}_i^{(1)}=\mathrm{logit}\left(\ {\beta}_1+{\beta}_2\log \left({K}_i\right)+{\beta}_3\log \left({D}_i\right)\right), $$
$$ {p}_i^{(3)}=\mathrm{logit}\left(\ {\beta}_a+{\beta}_b\log \left({K}_i\right)+{\beta}_c\log \left({D}_i\right)\right), $$
with Ki the dose of IN ketamine and Di the dose of IN dexmedetomidine in the Ketodex combination. Interactions cannot be estimated accurately and are not required in dose finding studies [24]. The optimal dose combination is the dose with the highest expected probability of adequate sedation.
Priors for the dose response model
We will use non-central t distributions with precision 0.001 and degrees of freedom 1, suggested by [25]. The mean of β1 and βa assume that 5% of participants are under-sedated and 2% over-sedated, setting the prior means for \( {p}_i^{(2)},i=1,2,3 \) to 0.93, the effectiveness from the literature [21]. The priors for β2, β3, βb and βc are centered on 0 so all dose combinations initially have the same effectiveness.
Effectiveness analysis
The primary effectiveness analysis will determine whether IN ketodex is non-inferior to IV ketamine in terms of providing adequate sedation. This analysis compares the effectiveness of IV ketamine and the optimal IN ketodex combination. The probability of adequate sedation with IV ketamine (pIV) is estimated using a binomial likelihood and a beta prior distribution with parameters 15.1 and 0.4. These prior parameters are estimated from published data [26] and down-weighted to prevent significant impact on the trial results.
The primary analysis is based on the posterior probability that IN ketodex is non-inferior to IV ketamine.
γ = P( pIV − pIN > η),
where η = 0.178 is the non-inferiority margin, estimated from our team's survey of 204 ED physicians (outlined in the supplementary material), and pIN is the probability of adequate sedation for the optimal IN ketodex combination. We will declare that IN ketodex is non-inferior to IV ketamine if γ ≤ 0.037. Other values of γ will induce alternative conclusions. This decision rule based on γ has a type I error of 5% when pIV = 0.97 and pIN = 0.792 and a type II error of 7% when pIN = 0.9 [9].
Analysis for secondary endpoints
For length of stay, onset of sedation, and duration of sedation, we will use a linear dose-response model to estimate the mean duration for the optimal IN ketodex combination. Distributional assumptions for this model will be assessed using posterior predictive checks [27]. If a normal distribution is not suitable, alternative model functions, e.g., gamma and log-normal, will be considered. For IV ketamine, we will estimate the mean length of stay, onset of sedation, and duration of sedation by fitting a suitable distribution, chosen using posterior-predictive checks. The comparison of the means will then determine the probability that IN ketodex is superior to IV ketamine and declare significance using the algorithm outlined above.
To analyze the AEs, we will use a logistic dose-response model for IN ketodex and a binomial distribution for IV ketamine. We will then compute the posterior probability that optimal dose for IN ketodex has a lower AE rate than IV ketamine, declaring significance using the algorithm above. Each AE will be counted once for a given participant. We will also report the severity, frequency, and relationship of AEs to the study intervention by System Organ Class and preferred term groupings. For each AE, we will also report the start date, stop date, severity, relationship, expectedness, outcome, and duration. AEs leading to premature discontinuation from the study intervention and serious treatment-emergent AEs will be presented in either a table or a list.
Priors for secondary analyses
Priors for the model intercepts will be obtained from the literature, where possible, and down-weighted to reduce prior influence on the results. Priors for the regression coefficients will be central t distributions with precision of 0.001 and degrees of freedom 1 [25]. Priors for the standard deviations will use non-central t distributions (truncated at 0) [28]. We will ensure that these priors are vague with respect to the scale of the observed data and undertake sensitivity analyses to the prior specification.
We will report additional outcomes with descriptive statistics, using frequencies and percentages for categorical variables and means, medians, standard deviations, and interquartile ranges for continuous variables.
Additional analyses
We will use logistic regression to investigate the interaction between baseline pain, measured using the Faces Pain Scale - Revised, considered as a continuous variable, and treatment effect. The model will have a coefficient for treatment, baseline pain, and an interaction term between treatment and baseline pain. This analysis will only use data from the optimal dose combination. We will conclude a significant interaction effect if the 95% credible interval for the treatment interaction does not include zero.
Trial status
The Ketodex study was registered on December 11, 2019, and started recruitment in March 2020. Recruitment is expected to be completed by December 2022. For the final analysis, the database will be cleaned and checked for completeness before being locked and the statistical analysis will then be undertaken using the methods in this SAP.
No datasets were used to develop this article as analysis was not undertaken. Thus, this consideration is not applicable.
AE:
ALC:
Average length criterion
DSMB:
Data and safety monitoring board
Intravenously
SAP:
PSA:
Procedural sedation and analgesia
PSSS:
Pediatric Sedation State Scale
REB:
WCHRI:
Women and Children's Health Research Institute
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The authors would like to acknowledge the iPCT SPOR administrative staff and our patient partners, who provided valuable support and input on the study design and documents.
The following are the members of the KidsCAN PERC Innovative Pediatric Clinical Trials Ketodex Study Team: Darcy Beer, Scott Sawyer, Maala Bhatt, April Kam, Quynh Doan, Vikram Sabhaney, Serena Hickes, Samina Ali, Karly Stillwell, Tannis Erickson, Chelsea Bowkett, Carolyn Shimmin, Brendon Foot, Chelsea Bowkett, Candace McGahern, Redjana Carciurmaruj, and Jeannine Schellenberg.
This work is supported by an Innovative Clinical Trials Multi-year Grant from the Canadian Institutes of Health Research (funding reference number MYG-151207; 2017–2020), as part of the Strategy for Patient-Oriented Research and the Children's Hospital Research Institute of Manitoba (Winnipeg, Manitoba), the Centre Hospitalier Universitaire Sainte-Justine (Montreal, Quebec), the Department of Pediatrics, University of Western Ontario (London, Ontario), the Alberta Children's Hospital Research Institute (Calgary, Alberta), the Women and Children's Health Research Institute (Edmonton, Alberta), the Children's Hospital of Eastern Ontario Research Institute Inc. (Ottawa, Ontario), and the Hospital for Sick Children Research Institute (Toronto, Ontario). This study is sponsored by The Governors of the University of Alberta (Suite 400, 8215 – 112 Street, Edmonton, Alberta, Canada T6G 2C8). Neither the study sponsor nor funders have any role in the collection, management, analysis, or interpretation of data; writing of the report; or the decision to submit the report for publication. Additional support was received from the Physicians Services Incorporated Foundation, Academic Medical Organization of Southwestern Ontario, Ontario Ministry of Economic Development, Job Creation and Trade, and the Children's Health Foundation of the Children's Hospital, London Health Sciences Foundation.
Child Health Evaluative Sciences, The Hospital for Sick Children, Toronto, Canada
Anna Heath, Juan David Rios, Eleanor Pullenayegum, Petros Pechlivanoglou & Martin Offringa
Dalla Lana School of Public Health, Division of Biostatistics, University of Toronto, Toronto, Canada
Anna Heath
Department of Statistical Science, University College London, London, UK
Institute of Health Policy, Management and Evaluation, University of Toronto, Toronto, Ontario, Canada
Eleanor Pullenayegum, Petros Pechlivanoglou & Martin Offringa
Division of Neonatology, The Hospital for Sick Children, University of Toronto, Toronto, Ontario, Canada
Martin Offringa
Women & Children's Health Research Institute, University of Alberta, Edmonton, Alberta, Canada
Maryna Yaskina, Rick Watts & Shana Rimmer
University of Manitoba, Winnipeg, Manitoba, Canada
Terry P. Klassen
Children's Hospital Research Institute of Manitoba, Winnipeg, Manitoba, Canada
London Health Sciences Centre, Children's Hospital, London, Ontario, Canada
Kamary Coriolano
Departments of Paediatrics and Epidemiology & Biostatistics, Schulich School of Medicine and Dentistry, London, Canada
Naveen Poonai
Children's Health Research Institute, London Health Sciences Centre, London, Canada
Juan David Rios
Eleanor Pullenayegum
Petros Pechlivanoglou
Maryna Yaskina
Rick Watts
Shana Rimmer
Darcy Beer
, Scott Sawyer
, Maala Bhatt
, April Kam
, Quynh Doan
, Vikram Sabhaney
, Serena Hickes
, Samina Ali
, Karly Stillwell
, Tannis Erickson
, Chelsea Bowkett
, Carolyn Shimmin
, Brendon Foot
, Candace McGahern
, Redjana Carciurmaruj
& Jeannine Schellenberg
All authors were involved in the conception and design of the SAP. AH drafted the manuscript. DR, EP, PP, MO, MY, RW, SR, TPK, KC, and NP offered substantive revisions. All authors read, edited, and approved the final manuscript. All individuals mentioned in the "Acknowledgements" section are members of the Ketodex Study Group. AH is the senior responsible statistician and NP is the clinical lead.
Correspondence to Anna Heath.
Ethics approval was obtained from Clinical Trials Ontario for the lead site (Children's Hospital, London Health Sciences Centre) and McMaster Children's Hospital. The other participating sites received institutional ethics approval. All protocol amendments will be submitted for approval to Health Canada before being communicated to each site and implemented only after Health Canada and REB approval. We will obtain child assent and caregiver consent from all trial participants as appropriate.
None reported.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated in a credit line to the data.
Heath, A., Rios, J.D., Pullenayegum, E. et al. The intranasal dexmedetomidine plus ketamine for procedural sedation in children, adaptive randomized controlled non-inferiority multicenter trial (Ketodex): a statistical analysis plan. Trials 22, 15 (2021). https://doi.org/10.1186/s13063-020-04946-3
Pediatric closed reduction
Intranasal ketodex
Non-inferiority trial
Bayesian adaptive design | CommonCrawl |
\begin{document}
\begin{abstract} We show the following dichotomy for a linear parabolic ${\mathbb{Z}}^2$-action $\rho_L$ on the torus with at least one step-2 generator: \begin{itemize} \item[$(i)$] Any affine ${\mathbb{Z}}^2$-action with linear part $\rho_L$ has a ${\mathbb{Z}}$-factor that is either identity or genuinely parabolic, and is thus not KAM-rigid, or \item[$(ii)$] Almost every affine ${\mathbb{Z}}^2$-action with linear part $\rho_L$ is KAM-rigid under volume preserving perturbations. \end{itemize}
\end{abstract} \maketitle
\setcounter{tocdepth}{2}
\color{black} {
\hypersetup{hidelinks}
\tableofcontents }
\section{Introduction, statements and overview of the main proofs}
\subsection{Background and context}
A smooth (by which we mean $C^\infty$) ${\mathbb{Z}}^k$-action $\rho$ on a smooth manifold $M$ is said to be {\it locally rigid} if there exists a neighborhood $\mathcal U$ of $\rho$ in the space of smooth ${\mathbb{Z}}^k$-actions on $M$, such that for every $\eta\in \mathcal U$ there is a smooth diffeomorphism $h$ of $M$ such that $h\circ \rho(g)\circ h^{-1}=\eta(g)$, for all $g\in {\mathbb{Z}}^k$.
When the rank of the acting group is $k=1$, we are in the realm of classical dynamics (${\mathbb{Z}}$-actions), where local rigidity in this strong form is not known to occur. Moreover, it is known that for affine maps on the torus local rigidity does not occur.
The only known situation in classical dynamics where a weaker form of local rigidity is proved, is the case of toral translations $T_{\alpha}$ on ${\mathbb{T}}^d$ with Diophantine frequency vectors ${\alpha}$ (we exclude rigidity modulo infinite moduli from this discussion). Indeed, it follows from Arnold's normal form for perturbations of toral translations, that a volume preserving perturbation of $T_{\alpha}$ with a Diophantine average translation vector ${\alpha}$ is smoothly conjugated to $T_{\alpha}$ \cite{A}. We will call this phenomenon {\it KAM-rigidity}.
The situation is dramatically different for ${\mathbb{Z}}^k$-actions with $k\geq 2$, where local rigidity is more common.
For Anosov (hyperbolic) actions, an important breakthrough was the proof of local rigidity by Katok and Spatzier \cite{KS}. The main tool in this context is the use of the action's invariant geometric structures \cite{GK, KS}, which after that proved useful in obtaining local rigidity for more general classes of partially hyperbolic actions with such geometric structures \cite{NT, DK3, W1, W2, VW}. {We note that there are many other local and global rigidity results for abelian partially hyperbolic actions than the ones mentioned above; we refrain from citing them all as our focus in this paper will be on local rigidity in the absence of any form of hyperbolicity and where there are no robust invariant geometric structures. }
In this context two famous manifestations of rigidity for ${\mathbb{Z}}^k$-actions, are: KAM-rigidity of {\it simultaneously} Diophantine torus translations (see Definition \ref{definition sim})
\cite{M,DF,WX,P}, and local rigidity for higher rank linear (and affine) partially hyperbolic actions on the torus \cite{DK}.
Simultaneously Diophantine torus translations generate ${\mathbb{Z}}^k$-actions which may have no Diophantine elements at all, so the result for single Diophantine translations does not apply, and one is forced to use commutativity of different action generators in a crucial way in order to obtain KAM rigidity.
For linear partially hyperbolic actions considered in \cite{DK}, the crucial assumption which leads to smooth rigidity is that such ${\mathbb{Z}}^k$-actions are of {\it higher rank}. A ${\mathbb{Z}}^k$-action by toral automorphisms is {\it higher rank} if there is a ${\mathbb{Z}}^2$ subgroup such that all of its non-zero elements act by ergodic automorphisms. This condition is equivalent to the absence of ${\mathbb{Z}}$-factors of the action, namely: a higher rank ${\mathbb{Z}}^k$-action does not factor (possibly up to a finite index subgroup) to an action generated by a single automorphism of a (possibly different) torus. The condition is equivalent also to the exponential mixing for the action, which plays a crucial role in the proof of local rigidity in \cite{DK}. Recently, it was announced in \cite{W3} that exponential mixing leads to local rigidity for large classes of partially hyperbolic affine actions.
The above mentioned two classes of actions on the torus, elliptic ones generated by translations on one hand, and partially hyperbolic ones on the other hand, lie in the general class of {\it affine} ${\mathbb{Z}}^k$-actions on the torus. Affine actions are actions generated by affine maps, and an affine map is a composition of a linear map and a translation. Such actions can be dynamically very different: they can be elliptic (the linear part of the action is the identity), or partially hyperbolic (the linear part contains a partially hyperbolic map, i.e., a map with some eigenvalues outside unit circle), or parabolic (the linear part acts by parabolic maps i.e. maps which have all eigenvalues 1), or can combine all these features. If the linear part contains a root of the identity, we take a finite index subgroup in the acting group which brings us to the general description above. In what follows we always assume that roots of the identity have been eliminated by passing to a finite index subgroup.
In this paper we focus on the most intricate and most surprising case of {\it parabolic actions}. In this case, single elements of the action (even under Diophantine conditions) are {\it not} KAM rigid, nor is there any mixing for the linear part of the action.
First we define a property of linear actions which will distinguish between actions that can be linear parts of KAM rigid affine actions, and those which cannot.
A linear ${\mathbb{Z}}^2$-action $\rho_L$ on $\mathbb T^d$ is {\it unlocked} if there is an affine ${\mathbb{Z}}^2$-action $\rho$ with linear part $\rho_L$ such that every $\mathbb Z$-factor of $\rho$, if it exists, is generated by a non-trivial translation. Thereby, we say a linear ${\mathbb{Z}}^2$-action $\rho_L$ on $\mathbb T^d$ is {\it locked} if for any affine action $\rho$ with linear part $\rho_L$, there exists a $\mathbb Z$-factor which is {\it not} a translation $T_\alpha$, $\alpha\neq 0$, see Definition \ref{def_stiff}.
Our analysis leads us to ask the following classification question for general affine ${\mathbb{Z}}^2$-actions on the torus:
\begin{question} \label{qqq} Is it true that for any linear ${\mathbb{Z}}^2$-action $\rho_L$ on $\mathbb T^d$ there is the following dichotomy: \begin{itemize} \item[(i)] The action $\rho_L$ is locked, or
\item[(ii)] The action $\rho_L$ is unlocked, and almost every affine action $\rho$ with the linear part $\rho_L$ is KAM-rigid.
\end{itemize} \end{question}
The previously mentioned classification results provide the positive answer to the above question for large classes of actions. For simultaneously Diophantine translation actions the linear part is the identity and thus is unlocked, so it fits into the case $(ii)$ of the question. Since simultaneously Diophantine condition is a full measure condition, the works \cite{M,DF,WX,P} give the positive answer to the question in this case. For affine actions with higher rank partially hyperbolic linear part the linear part is unlocked, since it has no $\mathbb Z$-factors at all.
Hence, these actions also fall in category $(ii)$. By \cite{DK}, such actions have the local rigidity property (and, therefore, KAM rigidity property as well; in fact, KAM rigidity holds for all affine actions of this kind, not just for almost all). Moreover, these two classes combined also give the positive answer to the question: \cite{DF} proves KAM rigidity for a full measure set of affine ${\mathbb{Z}}^2$-actions on the torus whose linear part is a direct product of a higher rank action and the identity.
The main goal of this paper is to develop new tools to study this question for {\it parabolic} actions. Our main result states that the answer to the question is affirmative for parabolic actions containing a step-2 element (see Definition \ref{def_step}). We hope that the tools developed in this paper will allow to address the classification question \ref{qqq} in full generality.
In this work the step-2 assumption on at least one element of the action is important in the proof, as we will discuss in detail in \S \ref{s_overview}. In a nutshell, the reason is that if the cohomological equation above a step-2 affine map with a Diophantine translation part has a solution, then the solution is tame. This is not the case for higher step affine maps \cite{DFS_nontame}.
\color{black}
\subsection{Main results and representative examples}
We will be interested in the problem of local rigidity of volume preserving perturbations of affine parabolic ${\mathbb{Z}}^2$-actions on the torus ${\mathbb{T}}^d$, where $d\in {\mathbb{N}}$.
By $\lambda$ we will denote the Haar measure on the torus ${\mathbb{T}}^d$.
We begin by defining some basic notions.
\begin{definition}\label{def_step} We say that $A\in \text{SL}(d,{\mathbb{Z}})$ is step-$S$ parabolic if $ (A-\Id)^S=0, $ and $ (A-\Id)^{S-1}\ne 0. $ An affine map of $\mathbb T^d$ with the linear part $A$ is defined via $$ a(x)=Ax +{\alpha} \mod 1, \quad x\in \mathbb T^d, $$ \color{black} where ${\alpha}\in \mathbb R^d$. An affine map $a$ is said to be step-$S$ if its linear part $A$ is step-$S$. We denote by $\rm Aff_S({\mathbb{T}}^d)$ the space of all parabolic affine maps of step at most $S$.
A step-$S$ affine parabolic ${\mathbb{Z}}^2$-action on ${\mathbb{T}}^d$ is a homomorphism $\rho:{\mathbb{Z}}^2\to \rm Aff_S({\mathbb{T}}^d)$. We denote by $\rho_L$ the action generated by the linear part of $\rho$.
\end{definition}
Note that every affine map whose linear part is a unipotent matrix $A\in \text{SL}(d,{\mathbb{Z}})$ (i.e., a matrix with all eigenvalues 1) is step-$S$ parabolic for some $S\le d$.
Let $\rho$ be an affine parabolic ${\mathbb{Z}}^2$-action on the torus ${\mathbb{T}}^d$. Then $\rho$ is generated by two commuting affine maps $a$ and $b$, and we will denote $\rho$ simply by its generators as $\(a,b\)$. The linear parts $A$ and $B$ of $a$ and $b$, respectively, also commute. But the commutativity of the linear parts is not enough to guarantee the commutativity of $a$ and $b$. If $a(x)=Ax+ \alpha \mod 1$ and $b(x)=Bx+ \beta \mod 1$, then the commutator $[a, b]$ is a translation on $\mathbb T^d$ by the vector $(A-\Id)\beta-(B-\Id){\alpha}$, which needs to be an integer in order to have $[a, b]=\Id$. We may choose the lifts of $a$ and $b$ to $\mathbb R^d$ so that the integer vector $(A-\Id)\beta-(B-\Id){\alpha}$ is trivial, and so without loss of generality we will always consider the lifts $Ax+ \alpha$ and $Bx+ \beta$ of $a$ and $b$, respectively, such that $(A-\Id)\beta=(B-\Id){\alpha}$. We will work with these lifts the whole time, and we denote them by the same letters $a$ and $b$, respectively.
We will also use the shorthand notation $A+\alpha$ to denote the affine map on ${\mathbb{T}}^d$ with the linear part $A$ and the translation part $\alpha$.
\color{black} \begin{definition} We denote by $\mathcal T(A,B)$ the set of possible translation parts $({\alpha},\beta)$ in the affine actions with linear part $\(A, B\)$, that is $$ \mathcal T(A,B):=\{\alpha, \beta\in \mathbb R^d \mid (A-\Id)\beta=(B-\Id){\alpha}\}. $$ \end{definition} Given an affine map $a(x)=Ax +{\alpha} \mod 1$, a perturbation of $a$ is a diffeomorphism of the torus which can be lifted to $\mathbb R^d$, where it has the form $$ F(x)= A(x)+{\alpha}+f(x) $$ for some small ${\mathbb{Z}}^d$-periodic vector-valued function $f$. Therefore we will simply write $a+f$ for a small perturbation of $a$. We will be interested in the set of smooth volume preserving perturbations, which we denote by $ \text{Diff}^\infty_\lambda ({\mathbb{T}}^d)$.
We define now the notion of {\it KAM rigidity} which is central in this work. Note that an affine parabolic action always has a translation factor. Therefore, one cannot hope that any form of local rigidity, stronger than the one available for translations, can hold. Only local rigidity of KAM type can be expected for these actions.
\begin{definition} \label{KAM local rigidity} We say that an affine ${\mathbb{Z}}^2$-action $\(a,b\)$ is {\it KAM-rigid under $\lambda$-preserving perturbations}, if there exists $\sigma \in {\mathbb{N}}$, $r_0 \in {\mathbb{N}}$, $r_0\geq \sigma$, and ${\varepsilon}>0$ satisfying the following:
If $r\geq r_0$ and $\(F,G\)=\(a+f,b+g\)$ is a smooth ${\lambda}$-preserving ${\mathbb{Z}}^2$-action such that
\begin{equation}\label{eq_System0}
\|f\|_r\leq {\varepsilon}, \quad \|g\|_r\leq {\varepsilon},
\quad \widehat f:= \int_{{\mathbb{T}}^d} f d\lambda=0, \quad \widehat g:=\int_{{\mathbb{T}}^d} g d\lambda=0, \end{equation}
then there exists $H=\Id +h \in \text{Diff}^\infty_\lambda ({\mathbb{T}}^d)$ such that $\|h\|_{r-\sigma}\leq C(a,b)\, {\varepsilon}$ and \begin{equation*}\label{eq_System}
H \circ (a+f) \circ H^{-1} = a, \quad H \circ (b+g) \circ H^{-1} = b, \end{equation*}
where $C(a,b)$ is a constant depending only on the action $\(a,b\)$. \end{definition}
In this paper, we will use the term {\it KAM-rigid} for short reference to {\it KAM-rigid under $\lambda$-preserving perturbations} since this will be the only context in which we place ourselves.
It is not difficult to find examples of parabolic commuting actions $\(A,B\)$ such that all the affine actions with this linear part are not KAM-rigid. Indeed, the commutation condition may force the affine action to have, for any choice of $({\alpha},\beta)\in \mathcal T(A,B)$, a rank-one factor, to which Arnold's KAM-rigidity cannot be applied. As we will see in the following examples, this happens if the rank-one factor of the affine action is either identity or genuinely parabolic (i.e., has a non-trivial linear part). We call such pairs $\(A,B\)$ locked (Definition \ref{def_stiff} below).
\color{black}
Let $E_{ij}$ denote the integer matrix which has $1$ in the position $(i,j)$, the rest of the elements being 0.
\begin{example}[Affine actions with identity as a rank-one factor] \label{ex.id} Consider the linear action on ${\mathbb{T}}^3$ generated by $A=\Id+E_{21}$, $B=\Id+E_{31}$. The commutation condition implies that for all $({\alpha},\beta)\in \mathcal T(A,B)$ we have ${\alpha}_1=\beta_1=0$. Hence, any affine action $\(A+{\alpha},B+\beta\)$, restricted to a sub-torus corresponding to the variable $x_1$, equals identity (we say that the action projects to identity on the torus $\mathbb T_1$ spanned by variable $x_1$, i.e., the action has the identity factor). Consequently, such affine action is not KAM-rigid, as explained below. \end{example}
To see why the action in Example \ref{ex.id} is not KAM-rigid, we can use the following general fact.
\begin{proposition} \label{propid} If a parabolic commuting action $\(A,B\)$ is lower triangular, and $\(A+{\alpha},B+\beta\)$ is such that ${\alpha}_1=\beta_1=0$ then $\(A+{\alpha},B+\beta\)$ is not KAM-rigid. \end{proposition}
\begin{proof} Keep $b=B+\beta$ unchanged and perturb $a=A+{\alpha}$ to $F(x_1,\ldots,x_d)=Ax+{\alpha}+(0,\ldots,0,{\varepsilon} \sin(2\pi x_1))$. The maps $b$ and $F$ commute, and $F$ satisfies \eqref{eq_System0}. To see that $A+{\alpha}$ is not conjugated to $F$ by a volume preserving conjugacy, consider the special two-dimensional case: $a(x_1,x_2)=A(x_1,x_2)=(x_1,x_2+x_1)$ and $F(x_1,x_2)=(x_1,x_2+x_1+{\varepsilon} \sin(2\pi x_1))$ (the general case is not different from it). Define the circle diffeomorphism $g: x_1\mapsto x_1+{\varepsilon} \sin(2\pi x_1)$ and the conjugacy $H(x_1,x_2)=(g(x_1),x_2)$. Clearly, $H$ does not preserve area, and it is easy to see that, up to translation, $H$ is the only conjugacy between $a$ and $F$. \end{proof}
\begin{remark} Note that if $\langle a,b\rangle$ has identity as a rank-one factor of dimension at least $2$, then it is straightforward to perturb the action as in \eqref{eq_System0} so that there is no conjugacy at all with the affine action. To do this, it is enough to perturb the identity factor itself in a volume preserving way such that \eqref{eq_System0} holds. \end{remark}
\begin{example}[Affine actions with a genuinely parabolic rank-one factor] \label{ex.rankone} Let us assume that $$B=\Id+E_{21}+E_{32}.$$ If $A$ is lower triangular and $(A-\Id)$ does not contain neither $E_{2j}$ nor $E_{3j}$ for any $j$, then for any $({\alpha},\beta)\in \mathcal T(A,B)$ we have that ${\alpha}_1={\alpha}_2=0$, and $A+{\alpha}$ acts as identity on the two-torus obtained by the projection on $(x_1,x_2)$.
Then the affine action $\(A+{\alpha},B+\beta\)$ has the skew shift $(x_1,x_2)\mapsto (x_1+\beta_1,x_2+x_1+\beta_2)$ of the two-torus as a rank-one factor, and is thus not KAM-rigid, as explained below. \end{example}
To see why the action in Example \ref{ex.rankone} is not KAM-rigid, we can use the following general fact.
\begin{proposition} \label{prop.rankone} For any $r\in {\mathbb{N}}$, for any parabolic affine map $A+{\alpha}$ on ${\mathbb{T}}^d$ with $A\neq \Id$, for any ${\varepsilon}>0$ there exists $f$ such that $$
\|f\|_r\leq {\varepsilon},
\quad \widehat f:= \int_{{\mathbb{T}}^d} f d\lambda=0,
$$ and $A+{\alpha}+f \in \text{Diff}^\infty_\lambda({\mathbb{T}}^d)$ is not conjugated to $A+{\alpha}$. \end{proposition}
\begin{proof} Without loss of generality, we can consider the case of the two-dimensional skew shift $(x_1,x_2)\mapsto (x_1+\beta_1,x_2+x_1+\beta_2)$. This map can be perturbed into $(x_1,x_2)\mapsto (x_1+\beta_1+{\varepsilon} \sin(2\pi(x_1+x_2)),x_2+x_1+\beta_2)$, which is a shifted classical standard map that is not conjugate to a skew shift. \end{proof}
The phenomena in Examples \ref{ex.id} and \ref{ex.rankone} can be subsumed under the existence, for any choice of $({\alpha},\beta)\in \mathcal T(A,B)$, of a rank-one factor for the affine action $\(A+{\alpha},B+\beta\)$ that is either identity or a genuinely parabolic action, which overrules KAM-rigidity. This motivates the following definition.
\begin{definition}[Locked actions] \label{def_stiff} When the commuting linear action $\(A,B\)$ is such that
for any choice of $({\alpha},\beta)\in \mathcal T(A,B)$, the affine action $\(A+{\alpha},B+\beta\)$ has a rank-one factor that is either identity or a genuinely parabolic action (i.e., has a non-trivial linear part), we say that $\(A,B\)$ is locked. We call the action unlocked if it is not locked. \end{definition}
An immediate corollary of Propositions \ref{propid} and \ref{prop.rankone} is the following. \begin{corollary} \label{cor.stiff} If $\(A,B\)$ is locked, then for any choice of $({\alpha},\beta) \in \mathcal T(A,B)$, the action of $\(a,b\)$ is not KAM-rigid. \end{corollary}
The main result of this paper is to show that besides the locked actions, for actions having (at least) one step-2 generator, KAM-rigidity under $\lambda$-preserving perturbations holds almost surely in the choice of the translation part. We formulate this dichotomy as follows.
\begin{Main} \label{th.main} Given a commuting action $\(A,B\)$ of parabolic matrices, where $A$ is step-2, we have the following dichotomy. \begin{itemize} \item[(i)] Action $\(A,B\)$ is locked, thus for any choice of $({\alpha},\beta) \in \mathcal T(A,B)$, the action of $\(a,b\)$ is not KAM-rigid.\item[(ii)] Action $\(A,B\)$ is unlocked, and for almost every choice of $({\alpha},\beta)\in \mathcal T(A,B)$, the action of $\(a,b\)$ is ergodic and KAM-rigid under volume preserving perturbations. \end{itemize} \end{Main}
In the case of step-2 actions, we have a more stringent alternative. \begin{Main} \label{almostsure} Given a commuting pair $\(A,B\)$ of step-2 parabolic matrices, we have the following dichotomy. \begin{itemize} \item[(i)] For any choice of $({\alpha},\beta)\in \mathcal T(A,B)$, the action of $\(a,b\)$ has a rank-one factor that is identity, and is therefore not ergodic and not locally rigid. \item[(ii)] For almost every choice of $({\alpha},\beta)\in \mathcal T(A,B)$, the action of $\(a,b\)$ is ergodic and KAM-rigid under volume preserving perturbations. \end{itemize} \end{Main}
Corollary \ref{cor.stiff} states that $(i)$ impedes KAM-rigidity. The proof of the dichotomy between $(i)$ and $(ii)$ is the main result of this paper that we formulate more precisely {\it via} Proposition \ref{prop dioph} and Theorem \ref{mainC} of the next section.
We note that the statements remain true if we replace the preservation of the volume $\lambda$ (and also averages with respect to the volume) by preservation of any common invariant measure.
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In the following observations we discuss the relevance of the assumptions of the main theorems.
\begin{remarki}[KAM-rigidity {\it vs.} local rigidity: why do we need Diophantine conditions?] {\rm When a linear ${\mathbb{Z}}^2$-action is not higher rank, it has a rank-one factor that we can represent by a pair $\({\rm Id},C\)$. The absence of rigidity of a single linear map $C$ implies that the local rigidity in this case can only be considered for affine actions. This work
treats rigidity of unlocked affine parabolic actions under Diophantine conditions on the translation vectors of the action. Roughly speaking, we put the Diophantine conditions on all the frequency vectors associated to sub-tori on which some element of the action acts as a translation. As we will see in the next section, there may be a finite or infinite number of such conditions. However, there will unavoidably be some Diophantine conditions that must be satisfied, thus only KAM type rigidity can be considered. The necessary set of Diophantine conditions comes from the fact that the affine action always has a translation part (due to the existence of a common eigenspace of eigenvalue 1 for the commuting linear pair, see Definition \ref{def_translation} and \S \ref{s_case1}). The other Diophantine conditions (cf. Definition \ref{definition resonance}) that are used are natural conditions that play a crucial role in the proof (cf. \S \ref{s-case2}), although we do not see for the moment how to show that they are necessary for the result to hold (see Question \ref{qq1}).} \end{remarki}
\begin{remarki}[On higher step parabolic actions: why do we assume that one generator is step-2?] {\rm The assumption that one element of the action is step-$2$ plays a heavy role in the proof. In particular, it is crucial in defining a tame candidate for a conjugacy at each step of the inductive KAM conjugacy scheme. More details about the use of this assumption will be given in \S \ref{s_overview}. We note that working under this, probably restrictive, assumption requires introducing some new ideas and techniques. Moreover, some phenomena, like resonances, appear only along sub-tori where both generators are step-2 (see \S \ref{Diophantine conditions}). This is why we prefer to focus on the special case described by Theorem \ref{th.main} and keep the study of the general case for a future work. }
\end{remarki}
\begin{remarki}[From parabolic actions to general affine actions: why do we focus on parabolic actions?] {\rm If a non higher rank linear action $\(A,B\)$ has a rank-one factor $\(\Id,C\)$ where $C$ does not have $1$ as an eigenvalue, then, due to the commutation constraint, $\Id$ is still a factor for any affine action with the linear part $\(A,B\)$. As a consequence, local rigidity would fail for the affine actions the same way it fails for the linear one. Based on this argument and on the fact that higher rank actions (i.e., those for which all the elements of the linear part are ergodic, thus partially hyperbolic) are locally rigid \cite{DK}, the case of parabolic actions naturally appears as the main problem to settle in order to give a general classification of affine abelian actions on the torus in terms of KAM local rigidity. } \end{remarki}
\begin{remarki}[The case of $\mathbb Z^k$ actions for $k\ge 3$] {\rm The same methods we use here provide the KAM-rigidity result for certain classes of affine $\mathbb Z^k$-actions with $k\ge 3$, as well, see Remark \ref{z-k} for more details. It is useful to note that the higher the rank $k$ of the acting group is, the "more locked" the action can become. An example of maximal rank parabolic abelian linear action on $\mathbb T^4$ is the $\mathbb Z^4$-action generated by $\Id+E_{12}, \Id+E_{14},\Id+E_{32}$ and $\Id+E_{34}$. This action is completely locked in the strongest possible sense: there are no affine non-linear $\mathbb Z^4$-actions which have this action as a linear part, at all. We expect that the same holds for any maximal rank parabolic abelian linear action on any $\mathbb T^d$, they are locked. In these cases there is no KAM-rigidity. This points to the fact that parabolic $\mathbb Z^2$-actions are the most common situation in which we could expect to have KAM-rigidity. }
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\end{remarki}
\begin{remarki}[The role of commutativity in the KAM-rigidity. The parabolic higher rank trick]
{\rm As in \cite{M} and \cite{DK}, our proof of local rigidity relies on a KAM inductive conjugacy scheme (see \S \ref{s_overview} for an outline of the scheme). At each step of the scheme, a system of cohomological equations must be solved up to a quadratic error.
In \cite{M}, each equation of the system is a cohomological equation above a circle rotation. Hence each individual equation has a formal solution provided vanishing of averages, but this solution may not be tame because each individual angle is not necessarily Diophantine. The main observation by Moser is that a cocycle relation forces the formal solutions to coincide and to be tame. This implies that the commutation relation allows one to find a tame solution to the system up to a quadratic error.
In \cite{DK}, the system of cohomological equations consists of individual equations that have tame solutions modulo a countable set of obstrcutions. The commutation, or the higher rank trick, is used to show that these obstructions can be removed up to a quadratically small error.
As it will be explained in detail in \S \ref{s_overview}, our approach to proving KAM-rigidity for parabolic actions combines these two mechanisms of local rigidity.
The main challenge in our work is to replace the partially hyperbolic higher rank trick by a parabolic one. To explain this a little better, we risk a technical description that may look obscure now but that will become much clearer from the detailed overview of the proof in \S \ref{s_overview} as well as from the introduction of Section \ref{plan_of_proof}.
The higher rank trick usually relies on the exponential growth of integer vectors (Fourier frequencies) under the dual action of partially hyperbolic matrices. Also, the mechanism that lies at the heart of the higher rank trick of \cite{DK} is the following consequence of the partial hyperbolicity of the action: for an integer vector $\bar{m}$ that is lowest (with smallest norm) on its orbit under the dual action $\bA$ of the first generator of the action, it is possible to iterate by $\bB$
in one of the two directions (future or past) so that $\|\bA^k\bB^lm\|$ be always larger than $c\|m\|$.
A main difficulty in our work is to replace the above argument by the fact that parabolic actions only grow at a polynomial rate. Much more annoying is the fact that for some integer vectors $\bar m$ that are lowest on their $\bA$ orbit, it is possible that the iterates by $\bB^l$ be decreasing in norm during a long time in both directions of $l$, before starting to increase. The challenge is to make sure that by choosing one direction of iteration for $\bB$, the double iterates $\|\bA^k\bB^lm\|$ remain always larger than $\|m\|^{\delta}$ for some $\delta>0$ independent of $m$ ($\delta$ is comparable to $1/S$ where $S$ is the step of the action). This parabolic version of the higher rank trick is done in \S \ref{double-buble}), where "being unlocked" property of the action again plays a major role.
}
\end{remarki}
There has been very few local rigidity results for parabolic actions. One example are actions by left multiplication on nilmanifolds. These are parabolic, and a form of local rigidity for such $\mathbb R^2$-actions on 2-step nilmanifolds was obtained in \cite{D}, under Diophantine conditions. Results of similar type were obtained for $\mathbb Z^2$-actions on Heisenberg nilmanifolds in \cite{DT}. More recently, in \cite{ZW}, it is proved that certain large abelian parabolic actions on homogeneous spaces of semisimple Lie groups have strong local rigidity properties.
In the remainder of the introduction we give precise definition and precise formulation of the main rigidity result, as well as the overview of the proofs, examples and comments on possible applications.
\subsection{Diophantine affine parabolic actions}\label{Diophantine conditions} In this section we define the full measure Diophantine conditions required on the pair $({\alpha},\beta)\in \mathcal T(A,B)$ in order to guarantee KAM-rigidity. It will be a combination of two types of conditions: simultaneously Diophantine condition for the maximal translation factor of the action, and Diophantine conditions for the translation parts of special elements of the action that we refer to as resonances.
\subsubsection{The maximal translation factor.}
\begin{definition}[Maximal translation factor] \label{def_translation} We say that the action $\(A,B\)$ has a maximal identity factor if there is a torus ${\mathbb{T}}_1$ of dimension $d_1$ such that $\(A,B\)$, restricted to this torus, equals identity. The action of $\(a,b\)$ restricted to this factor is called the maximal translation factor of $\(a,b\)$. \end{definition}
\begin{definition}[Simultaneously Diophantine vectors]\label{definition sim} We say that a pair of vectors $({\alpha},\beta) \in {\mathbb{T}}^{d} \times {\mathbb{T}}^{d}$ is simultaneously Diophantine if there exists $\gamma, \tau>0$ such that $$
\max\{ |1 -e(k,{\alpha}) |, |1-e (k,\beta) |\}> \frac{\gamma}{|k|^\tau}, $$ where $e(m,x)=e^{2\pi i (m,x)}$. We denote this property by $({\alpha},\beta)\in \text{SDC}(\gamma,\tau)$. \end{definition}
Observe that SDC-pairs of vectors form a set of full Haar measure in ${\mathbb{T}}^{d} \times {\mathbb{T}}^{d}$.
\subsubsection{Resonant vectors.}
In what follows we will use the dual action corresponding to the linear part $\(A, B\)$, induced on ${\mathbb{Z}}^d$. For a matrix $A$, the dual action on ${\mathbb{Z}}^d$ is denoted by $$ \bA:=(A^{tr})^{-1}, \quad \bA=\Id+\widehat{A}. $$ For a general $m$ and $(k,l)\in {\mathbb{Z}}\times {\mathbb{Z}}$, $\bar A^k\bar B^l m$ has a polynomial expression (see Lemma \ref{l_form_of_AkBl}). However, if $m$ is such that there exists a pair $(k,l)\in {\mathbb{Z}}\times {\mathbb{Z}}\setminus \{(0,0)\}$ satisfying $\bar A^k\bar B^l m=m$, then, since this implies that $\bar A^{ik}\bar B^{il} m=m$ for all $i \in {\mathbb{Z}}$, we necessarily have
(even if $A$ and $B$ are higher step): $$ \bar A^k\bar B^l m-m=k\widehat{A} m+l\widehat{B} m=0. $$ Hence, if $\bar A m \neq m$ or $\bar B m\neq m$, we can associate to such an $m$ a unique pair $(k,l)\in {\mathbb{N}}\times {\mathbb{Z}}$ such that either $(k,l)=(1,0)$, or $(k,l)=(0,1)$, or $k$ and $l$ are mutually prime and $k>0$. For all these cases, we use the same notation $k \wedge l=1$ and say that $m$ is resonant and that $(k,l)$ is its associated resonance pair. Notice that, due to commutativity, if $m$ is resonant, then any other integer vector on the $\(\bA, \bB\)$-orbit of $m$ is also resonant with {\it the same resonance pair}. So resonance pairs are attached to orbits, rather than individual vectors. We summarise the above discussion in the following
\begin{definition}[Resonant vectors and resonance pairs] Any vector $m \in {\mathbb{Z}}^d\setminus\{0\}$ such that $\bar A^k\bar B^l m=m$ for some $(k, l)$, while either $\bar A m \neq m$ or $\bar B m\neq m$, is called a resonant vector. We will use the following notations:
${\mathcal{C}}_2(k,l)$ denotes the set of all resonant $m$ associated to the resonance pair $(k,l)$,
${\mathcal{C}}_2={\mathcal{C}}_2(A, B)=\bigcup_{k\wedge l=1} {\mathcal{C}}_2(k,l)$ denotes the set of all resonant vectors,
$\res(A,B)$ denotes the set of all resonance pairs $(k,l)\in {\mathbb{Z}}^2$. \end{definition}
The following lemma shows that the norm of the resonant pair is bounded by the norm of any of the corresponding resonances. It is therefore bounded by the smallest one of them on the $\(\bA, \bB\)$-orbit.
\begin{lemma}\label{l_tech_reson} Let $a=A+{\alpha}$ and $b=B+{\beta}$ be commuting affine parabolic maps. If $(k,l)\in {\mathbb{Z}}^2$ is the (unique) pair associated to the resonance $m$ as in Definition \ref{definition resonance}, then there exists $C=C(A,B)>0$ such that $$
C (|k|+|l|) \leq |m| . $$ \end{lemma}
\begin{proof} Let $m$ be a resonant vector, i.e., let $\bA^k\bB^lm = m$. As explained earlier, this implies that $k \widehat{A} m = - l \widehat{B} m$. Consider now the two integer vectors: $x=\widehat{A} m$ and $y=\widehat{B} m$. By assumption, $k$ and $l$ are mutually prime.
This implies, in particular, that each component $x_j$ of the vector $x$ is divisible by $l$. Hence, $x_j\geq |l|$. Therefore, there exists a constant $C(A)$ (depending on $\widehat{A} $) such that $|m| \geq C(A) |l|$.
In the same way, $|m | \geq C(B) | k| $, which implies the statement. \end{proof}
\begin{definition}[Diophantine resonances] \label{definition resonance} Let $\(a, b\)$ be an affine parabolic ${\mathbb{Z}}^2$-action. The number $\akl= a^kb^l - A^k B^l$ will be called the translation part of the element $(k, l)$ of the action.
We say that a resonance $m\in {\mathcal{C}}_2(k, l)$ is $(\gamma,\tau)$-{Diophantine}, if \begin{equation}\label{admissible} |1-e(m,\akl)|>\frac{\gamma}{|m|^{\tau}}.\end{equation} \end{definition}
\begin{remark} The set of resonant vectors and resonance pairs for a given action may be empty, finite non empty, or infinite, as we will see in the examples at the end of this section. \end{remark}
\subsubsection{Diophantine property for actions.} We are ready to define the {\it Diophantine} parabolic affine actions, for which the main local rigidity result holds.
\begin{definition}[Diophantine actions] \label{rotation vectors}
Given $\gamma,\tau>0$ and a parabolic affine ${\mathbb{Z}}^2$-action $\(a,b\)$, where $a$ is step-2, we say that $\(a,b\)$ is $(\gamma,\tau)$-Diophantine if:
\begin{enumerate}
\item the maximal translation factor
of $\(a,b\)$ is $(\gamma,\tau)$-simultaneously Diophantine (as in Definition \ref{definition sim}), and
\item every resonance $m \in {\mathcal{C}}_2(A,B)$ is $(\gamma,\tau)$-Diophantine (as in Definition \ref{definition resonance}).
\end{enumerate} \end{definition}
\begin{example} \label{ex3} Let $\(A,B\)$ be the action on ${\mathbb{T}}^2$ generated by $A=\Id+E_{21}$ and $B=\Id$. The affine action $\(A+({\alpha},0),B+(0,\beta)\)$ is $(\gamma,\tau)$-Diophantine if and only if both ${\alpha}$ and $\beta$ are $(\gamma,\tau)$-Diophantine. \end{example} \begin{proof} Indeed, in this case ${\alpha}_1={\alpha}, \beta_1=0$ is the translation part of the affine action, and the SDC condition reduces to the Diophantine condition on ${\alpha}$.
In this example all the vectors $(m_1,m_2)$ with $m_2\neq 0$ are resonant with the same resonance pair $(0,1)$, and the Diophantine condition on the resonance reduces to the Diophantine condition on $\beta$. \end{proof}
The following simple observation is an important step in establishing the dichotomies in Theorems \ref{th.main} and \ref{almostsure}. \begin{proposition}\label{prop dioph} Fix $\tau>d$. Let $\(A, B\)$ be a linear parabolic ${\mathbb{Z}}^2$-action.
We have the following alternative: \begin{itemize} \item[(i)] $\(A,B\)$ is locked as in Definition \ref{def_stiff}.
\item[(ii)] $\(A,B\)$ is unlocked, and for almost every $({\alpha},\beta)\in \mathcal T(A,B)$, $\(A+{\alpha},B+\beta\)$ is $(\gamma,\tau)$-Diophantine for some $\gamma>0$. \end{itemize} In case $A$ and $B$ are step-2, alternative $(i)$ can be reduced to the existence of a rank-one factor that is identity. \end{proposition}
\begin{remark} Notice that no step-2 assumption is made on any generator in the first part of the Proposition. \end{remark} \begin{proof}[Proof of Proposition \ref{prop dioph}] Consider the maximal translation factor of $\(a,b\)$ generated by the pair of translation vectors denoted by $({\alpha}^{(1)},\beta^{(1)})$. The condition $({\alpha},\beta)\in \mathcal T(A,B)$ imposes some relations over ${\mathbb{Z}}$ between the coordinates of the vectors ${\alpha}^{(1)}$ and $\beta^{(1)}$. Then we have two possible scenarios.
The first one is that there exists a vector $\bar k\in {\mathbb{Z}}^{d_1}\setminus \{0\} $ such that $(\bar k,{\alpha}^{(1)})=(\bar k,\beta^{(1)})=0$ for all $({\alpha},\beta)\in \mathcal T(A,B)$, in which case a change of coordinates with $X_1:=(\bar k,x^{(1)})$ will exhibit a one-dimensional rank-one factor on which the action is identity.
If the first scenario does not hold, then for all $k\in {\mathbb{Z}}^{d_1}\setminus \{0\} $ either $(k,{\alpha}^{(1)})$ or $(k,\beta^{(1)})$ is not identically zero on $ \mathcal T(A,B)$. In this case we can split the set of integers, ${\mathbb{Z}}^{d_1}\setminus \{0\} = \mathcal Z_1+ \mathcal Z_2$, in such a way that for $k\in \mathcal Z_1$, $(k,{\alpha}^{(1)})$ is not identically zero on $ \mathcal T(A,B)$, and for $k\in \mathcal Z_2$, $(k,\beta^{(1)})$ is not identically zero on $ \mathcal T(A,B)$. Now, for any $k \in \mathcal Z_1$, for any ${\delta} >0$, we have that
$\lambda\left\{{\alpha}^{(1)} \in {\mathbb{T}}^{d_1} : \|(k,{\alpha}^{(1)})\|\leq {\delta} |k|^{-d_1-1}\right\}\leq c {\delta} |k|^{-d_1-1}$ for a constant $c=c(d)$. Summing over all $k\in \mathcal Z_1$ and then over all $k\in \mathcal Z_2$, and using Arcela-Ascoli theorem, we get that for almost every $({\alpha},\beta)\in \mathcal T(A,B)$, there exists $\gamma>0$ such that for each $k\in \mathcal Z_1$, it holds that
$\|(k,{\alpha}^{(1)})\|\geq \gamma |k|^{-d_1-1}$, and
for each $k\in \mathcal Z_2$ it holds that $\|(k,\beta^{(1)})\|\geq \gamma |k|^{-d_1-1}$. This implies that for almost every $({\alpha},\beta)\in \mathcal T(A,B)$, the maximal translation factor
of $\(a,b\)$ is $(\gamma,\tau)$-simultaneously Diophantine.
Next, we consider a resonance $m \in {\mathcal{C}}_2(A,B)$ and let $k\wedge l=1$ be its unique corresponding vector such that $k\widehat{A} m+l\widehat{B} m=0$. We then have two possible cases.
\noindent {\sc Case $1$.} There exists a resonance $m$ such that for every $({\alpha},\beta)\in \mathcal T(A,B)$ it holds that $e(m,\akl)=1$. Then we prove the following. \begin{lemma} \label{lemma.stiff} In the assumptions of Case 1, the action $\langle a, b\rangle$ has a rank-one factor that is genuinely parabolic. \end{lemma} \begin{proof} Let $m \in {\mathcal{C}}_2(k,l)$ be a resonance, in which case for $A':=A^kB^l$ we have $\bar{A'}m=m$, while either $\bA m \neq m$ or $\bB m\neq m$. For definiteness, assume that $\bB m\neq m$. We also have that for ${\alpha}':=\akl$, $e(m,{\alpha}')=1$.
After a change of variables we can assume that $m$ is one of the basis vectors, that is, $m_{i}=0$ for $i\neq i_1$ and $m_{i_1}=0$. We also assume that both matrices $B$ and $A'$ have 1-s on the main diagonal.
Since $\bB m\neq m$ and $\bar{A'} m=m$, we have that $B$ contains some $E_{i_1i_2}$ while $A'$ does not contain any $E_{i_1*}$ (where $*$ ranges through possible indices). By another change of coordinates, we can assume that $B$ does not contain any other $E_{i_1*}$ besides $E_{i_1i_2}$. (Indeed, if $B$ contains $\sum_{j=2}^s k_jE_{i_1i_j}$, we can use the coordinate change $x_{i_2}\mapsto \sum_{j=2}^s k_jx_{i_j}, x_k\mapsto x_k$ for $k\neq i_2$).
By the commutativity of $a'$ and $b$
we get that ${\alpha}'_{i_2}=0$, and that $A'$ contains no $E_{i_2*}$ (otherwise $\widehat{A}'\widehat{B}$ would contain no $E_{i_1*}$ while $\widehat{B}\widehat{A}'$ would contain some). Also, the hypothesis $e(m,\akl)=1$ translates into ${\alpha}'_{i_1}=0$.
If $B$ has no element of the type $E_{i_2*}$, we conclude that the action $\langle a', b\rangle$ where $a'=a^kb^l$ factors on the torus ${\mathbb{T}}_{i_2,i_1}$ on which $a'$ acts as identity, while $b$ is genuinely parabolic.
If $B$ had an element $E_{i_2i_3}$, then again after a change of coordinates, we can assume that
$B$ does not contain any other $E_{i_2*}$ besides $E_{i_2i_3}$.
As before, we have two consequences: 1) ${\alpha}'_{i_3}=0$, and 2) $A'$ contains no $E_{i_3*}$ (otherwise $\widehat{A}'\widehat{B}$ would contain no $E_{i_2*}$ while $\widehat{B}\widehat{A}'$ would contain some).
If $B$ has no element of the type $E_{i_3*}$ we conclude that the torus ${\mathbb{T}}_{i_3,i_2,i_1}$ is a factor of the action $\langle a', b\rangle$ on which $a'=a^kb^l$ acts as identity, while $b$ is genuinely parabolic.
Arguing inductively, we obtain the proof of the lemma. \end{proof}
If Case $1$ does not hold, then we must be in the following case:
\noindent {\sc Case $2$.} For every resonance $m$, there exists $({\alpha},\beta)\in \mathcal T(A,B)$, such that $e(m,\akl)\neq 1$, then by linearity of $\akl$ in the variables of $({\alpha},\beta)$, we see that the measure of $({\alpha},\beta)\in \mathcal T(A,B)$, such that $$
|1-e(m,\akl)|\leq \frac{\gamma}{|m|^{d+1}} $$
is less than $c \frac{\gamma}{|m|^{d+1}}$ for a constant $c=c(d)$.
Summing up over all possible resonances and using Arcela-Ascoli theorem, we get that for almost every $({\alpha},\beta)\in \mathcal T(A,B)$, there exists $\gamma>0$ such that every resonance is $(\gamma,d+1)$ Diophantine.
From the proof of Lemma \ref{lemma.stiff}, we see that Case 1 cannot happen if the action is step-2.
\end{proof}
\color{black}
\subsection{KAM-rigidity}
Now we are ready to formulate precisely part $(ii)$ of Theorems \ref{th.main} and \ref{almostsure}. The following is our main rigidity result.
\begin{Main} \label{mainC} Let $\(A,B\)$ be an unlocked linear parabolic ${\mathbb{Z}}^2$-action with (at least) one step-2 generator. If $({\alpha},\beta)\in \mathcal T(A,B)$ are such that $\(a,b\)$ is $(\gamma,\tau)$-Diophantine for some $\gamma>0$ and $\tau>0$, then $\(a,b\)$ is KAM-rigid. \end{Main}
Theorems \ref{th.main} and \ref{almostsure} follow directly from Theorem \ref{mainC} and Proposition \ref{prop dioph}.
\begin{remark}\label{z-k} If an affine $\mathbb Z^k$-action, $k\ge 3$, contains a Diophantine affine $\mathbb Z^2$-action with at least one step-2 generator, then Theorem \ref{mainC} directly implies KAM-rigidity for the $\mathbb Z^k$-action. This is because the smooth conjugacy provided by Theorem \ref{mainC} for the $\mathbb Z^2$-action would then conjugate the whole $\mathbb Z^k$-action perturbation. This simple observation is a consequence of the commutation and the ergodicity of the Diophantine $\mathbb Z^2$-action. This argument has been already used in \cite{DK} (see Lemma 3.2 in \cite{DK}) to draw the conclusion about local rigidity for a $\mathbb Z^k$-action from that of its $\mathbb Z^2$-subaction. \end{remark} \color{black}
\subsection{Examples of KAM-rigid actions} $ \ $
To begin with, let us return to the simple Example \ref{ex3}.
\begin{proposition} \label{ex33} Let $\(A,B\)$ be the action on ${\mathbb{T}}^2$ given by $A=\Id+E_{21}$ and $B=\Id$. If ${\alpha}$ and $\beta$ are Diophantine numbers, then the affine action $\(A+({\alpha},0),B+(0,\beta)\)$ is KAM-rigid. \end{proposition}
\begin{proof} As explained earlier, when ${\alpha}_1$ is Diophantine, the translation factor of the action is SDC. On the other hand, all the resonances are of the form $(m_1,m_2)=(0,m_2)$, $m_2\neq 0$, with the corresponding resonance pair $(0,1)$. Since ${\alpha}_{0,1}=(0,\beta)$, we have the following. When $\beta$ is $(\gamma,\tau)$-Diophantine, condition \eqref{admissible} holds with the constants $(\gamma,\tau)$. Hence, Theorem \ref{mainC} implies the KAM-rigidity of $\(A+({\alpha},0),B+(0,\beta)\)$. \end{proof}
It is clear that if ${\alpha}$ is Liouville, the corresponding action will not be KAM-rigid. To see this, just perturb $A+({\alpha},0)= (x_1,x_2)\mapsto (x_1 + {\alpha}, x_2+x_1)$ to $(x_1,x_2)\mapsto (x_1+{\alpha},x_2+x_1+{\varepsilon} \varphi(x))$, where $\varphi(x)$ is a smooth function with the zero mean that is not a coboundary above the rotation of angle ${\alpha}$. However, although our proof of KAM-rigidity heavily uses the Diophantine property of resonances, we are not able to settle whether $\beta$ Diophantine is a necessary condition for KAM-rigidity.
\begin{question} \label{qq1} Is the action $\(A+({\alpha},0),B+(0,\beta)\)$ KAM-rigid when ${\alpha}$ is Diophantine and $\beta$ is Liouville? \end{question}
The following example provides a KAM-rigid action having infinitely many resonances with infinitely many resonance pairs.
\begin{proposition} \label{ex5} Let $\(A,B\)$ be the action on ${\mathbb{T}}^7$ given by $A=\Id+E_{52}+E_{61}+E_{73}$ and $B=\Id+E_{42}+E_{43}+E_{64}+E_{73}$. For almost every $({\alpha},\beta)\in \mathcal T(A,B)$, the affine action $\(A+{\alpha},B+\beta\)$ is KAM-rigid. \end{proposition}
\begin{proof} The commutation condition $({\alpha},\beta)\in \mathcal T(A,B)$ is satisfied if and only if $\beta_1=\alpha_4$, $\beta_2=0$, and $\beta_3=\alpha_3=-\alpha_2$. The translation factor of the action is the three-torus corresponding to the first three coordinates, and the translations are ${\alpha}^{(1)}=(\alpha_1,\alpha_2,-\alpha_2)$ and $\beta^{(1)}=(\alpha_4,0,-\alpha_2)$. It is easy to see that if the vectors $(\alpha_1,\alpha_2)$ and $(\alpha_4,-\alpha_2)$ are Diophantine, then the pair $({\alpha}^{(1)},\beta^{(1)})$ is SDC. Indeed, denoting
by $\|x\|$ the closest distance from $x\in {\mathbb{R}}$ to the integers, for any $m=(m_1,m_2,m_3)$ we have: \begin{align*}
\max(\|(m,{\alpha}^{(1)})\|,\|(m,\beta^{(1)})\|)&=\max(\|m_1{\alpha}_1+(m_2-m_3)\alpha_2\|,\|m_1\alpha_4-m_3\alpha_2\|)\\ &\geq \gamma (|m_1|+|m_2-m_3|)^{-\tau}+ \gamma (|m_1|+|m_3|)^{-\tau}\\ &\geq \gamma' |m|^{-\tau'}. \end{align*}
Let us turn to the resonances. They are the set of $m$ such that $m_6=0$, and the two vectors, $v_m=(m_5,m_7)$ and $w_m=(m_4,m_4+m_7)$, are collinear and not both zero at the same time. Hence, at least one of $m_5$ or $m_7$ does not vanish. The resonance pairs are the pairs $k_m\wedge l_m=1$ such that $k_mv_m+l_mw_m=0$. For example, $$ m=(m_1,m_2,m_3,n+1,n,0,n(n+1)), \quad (m_1,m_2,m_3,n)\in {\mathbb{Z}}^3\times {\mathbb{Z}}^* $$ is a resonant vector with the resonance pair $(n+1,-n)$. Finally, fix any resonant $m$ and observe that, since there are no constraints on ${\alpha}_5,{\alpha}_7$ and since at least one of $m_5$ or $m_7$ does not vanish, the Diophantine condition on ${\alpha}_{k_m,l_m}$ is satisfied for almost every $({\alpha},\beta)\in \mathcal T(A,B)$. \end{proof}
\subsection{Overview of the proof of Theorem \ref{mainC}} \label{s_overview} The proof is based on an inductive scheme of successive conjugations of the perturbed action $\(F, G\)$, where $F=a+f$ and $G=b+g$ to the affine action $\(a, b\)$. As usually in the KAM approach, the linearized conjugacy equations are solved at each step of the induction with a loss of derivatives, which can be caused, for example, by small divisors or by other reasons. The {\it a priori} damaging effect of this loss is tamed out by the quadratic speed of convergence of the scheme.
In our context, the linearized conjugacy equations, often called the cohomological equations, are essentially of the following form:
\begin{equation}\label{linearisation22} \begin{aligned} h\circ a - Ah &=f,\\ h\circ b - Bh &=g. \end{aligned} \end{equation}
Two main differences with the classical KAM schemes that appear in our context are the following:
\begin{itemize} \item[$(i)$] Cohomological equations \eqref{linearisation22} above each individual generator of the action are, in general, not solvable because of the existence of an infinite countable set of obstructions. These were first evidenced in the step-2 example in the work of Katok and Robinson \cite{KatokRobinson}. (Large set of distributional obstructions was likewise found for any step nilflows in \cite{FF}. Also see \cite{CF} for a study of certain cases of cohomological equations above abelian actions).
\item[$(ii)$] In the case of parabolic affine maps of step 3 and higher, and with Diophantine translation part, if the solution to one of the equations of \eqref{linearisation22} exists, then it is smooth if the right-hand side is smooth. However, the loss of the number of derivatives is not fixed (in other words, the linearized cohomological equation is stable, but the solutions are not tame).
In a separate work we show that, for the simplest $C^r$ step-3 map $(x_1,x_2,x_3)\mapsto (x_1+{\alpha}_1,x_2+x_1,x_3+x_2)$, the loss of derivatives is roughly $r/2$, even for the nicest Diophantine angles ${\alpha}_1$. This constitutes a notable difference with the step-2 case, for which \cite{KatokRobinson} showed tameness of the solutions when they did exist. \end{itemize}
To address $(i)$, the usual path is to exploit the commutation relation to find approximate solutions to the cohomological equations. This was done in two related problems in the past. First, by Moser \cite{M}, who showed that SDC commuting circle rotations are locally rigid under the condition of preserving the rotation number. This was extended to higher dimension in \cite{DF,WX,P}. Second, by Damjanovi\'c and Katok who proved in \cite{DK} the local rigidity of higher rank partially hyperbolic affine abelian actions on the torus (i.e., actions, all of whose elements are ergodic automorphisms or affine maps with such linear parts).
In Moser's case the objective is to linearize a commuting pair $R_{{\alpha}_i}+f_i$, $i=1,2$. The cohomological equations take the form $h(x+{\alpha}_i)-h(x)=f_i(x)-\int f_i$. They have formal solutions above the generators $R_{{\alpha}_i}$, and the commutation relation allows to upgrade the formal solutions into the approximate tame solutions. In fact, Moser's trick is to define, for each Fourier mode $n$, the corresponding coefficient $h_n$ of the conjugacy map,
using either one or the other of the linearized conjugacy equations, according to which $\|n{\alpha}_i\|$, for $i=1$ or for $i=2$, is "not too small" as granted by the SDC-condition. As a result, one gets a candidate conjugacy $h$ that is {\it tame}, i.e., of the same order as the nonlinearities $f$ and $g$ with a fixed loss of the number of derivatives. Moreover, the commutation relation plus the SDC-condition insure that the constructed $h$ solves the cohomological equations with a {\it quadratic} error (with a small abuse of notations, by {\it quadratic} we will mean that the error is of order of a power $k>1$ in the nonlinearities $f$ and $g$, with a fixed loss of the number of derivatives).
The above procedure allows to implement the classical KAM quadratic scheme, with the issue of the constant terms $\int f_i$ being resolved due to the condition of the preservation of the rotation numbers.
In \cite{DK} the individual equations as in \eqref{linearisation22} have a tame solution provided a countable set of obstructions vanish, each one being formally computed as weighted sums \color{black} along the dual orbit of Fourier coefficients of the nonlinearities $f$ and $g$. The commutation relation in this case allows to get quadratic approximations of the nonlinearities by functions whose obstructions vanish. This was labelled "highr-rank trick". Here again, the approximation is quadratic with a finite loss of the number of derivatives.
Our proof of KAM-rigidity for parabolic actions combines two mechanisms of local rigidity: "Moser's trick" and "higher rank trick". The translation part of the action and the resonances (Fourier modes that are invariant along some element of the dual action) are treated using a mechanism, similar to Moser's trick. This is where the Diophantine conditions of Definition \ref{rotation vectors} on the action play a crucial role.
It has to be noted that the use of Moser's trick for the resonances brings some technical challenges that affect the whole proof. Indeed, for a resonant Fourier mode $m$, we need to use the element $F^kG^l$ of the action, where $(k,l)$ is the resonance pair associated to $m$. This forces us to work out the linearization KAM scheme at each step for {\it a large number of elements} of the action, and not only for the two generators. Of course, we cannot control {\it all} of the nonlinearities in $F^kG^l$ for all resonance pairs $(k,l)$ at each step, because $k$ and $l$ can be arbitrarily large. Fortunately, the resonance pairs associated to a resonant mode $m$ are of the order of $m$ (see Lemma \ref{l_tech_reson}). This means that if, at a given step of the KAM scheme, we truncate the nonlinearities up to order $N$ before finding an approximative solution of the linearized conjugacy equation, we will only need to control $F^kG^l$ for $k$ and $l$ of order $N$. This can easily be included in the induction due to the parabolic nature of $a$ and $b$. \color{black}
For "non-resonant" Fourier modes, it is a higher rank trick approach similar to \cite{DK} that is invoked. Indeed, for a non-resonant mode $m$ we can define $h_m$ {\it via} the sum of the Fourier coefficients of the nonlinearity $f$ along the dual orbit of the step-2 generator of the affine action, taken in the "good direction": either in the future or in the past (in a similar way to what is done in Livschits theory). The fact that the generator is step-2 implies that the Fourier modes, involved in these partial sums, grow either for the past or the future sum, which allows us to define a tame candidate conjugacy $h$, as observed in \cite{KatokRobinson}. {Observe that difficulty $(ii)$ mentioned above shows that the mere definition of a candidate tame conjugacy when no element of the action is step-2 is already a challenge for the general higher step case. Other difficulties appear in relation with the applicability of the parabolic higher rank trick that will be explained in the next paragraph. }
Once $h$ is constructed, we see that it is only at special modes $\bar m$ that are lowest (in norm) on their dual orbit along $\bA$ that the constructed $h$ does not solve the cohomological equation above $a$ (at $\bar m$, the good direction switches from past to future). The error in solving the equation at $\bar m$ is indeed the full sum along $\bar m$ of the Fourier coefficients of the nonlinearity along $\bA$. These sums, having the form ${\Sigma}_m^{A} (f)= \sum_{k\in {\mathbb{Z}}} f_{\bar A^{k} m}{\lambda}^{(k)}_{ m}$ (where $\lambda^{(k)}_m$ are "innocuous" multipliers of modulus one related to the translation part of the action, see \S \ref{s_notations} for the exact definitions), are the obstructions to solving the cohomological equations above $\bA$.
The higher rank trick uses commutativity to show that this full sum is equal to a double sum of a quadratic function $\phi$ measuring the error of the pair $(f,g)$ in \eqref{linearisation22} from forming a cocycle above the action $\(a,b\)$ \color{black}
(see Section \ref{s_linearisation} and Section \ref{s_case3} for more explanations). It is appears to be fruitful to express the obstructions as the following double sums:
\begin{equation} \label{eqdouble} \Sigma_m^A (f) = \sum_{l\geq 0} \sum_{k\in {\mathbb{Z}}} \phi_{\bar A^{k}\bar B^l m}{\lambda}^{(k)}_{m}\mu^{(l)}_{ m} = -\sum_{l\leq -1} \sum_{k\in {\mathbb{Z}}} \phi_{\bar A^{k}\bar B^l m}{\lambda}^{(k)}_{ m}\mu^{(l)}_{ m} . \end{equation}
There is an important difference between the phenomenon that lies behind the control of the double sums in our case, compared to the partially hyperbolic case. In the {\it partially hyperbolic} higher rank case treated in \cite{DK}, the Fourier modes that appear in the double sums in one of the two directions (future or past for $\bB$) are essentially increasing due to the partial hyperbolicity of the action, and this immediately leads to approximate solutions of \eqref{linearisation22} with quadratic errors with finite loss of the number of derivatives.
In our case, due to the presence of a higher step generator in the action, there may be no growth in either direction along the dual orbits that appear in the double sums. In fact, it always happens for some modes $m$ that the double orbits
appearing in \eqref{eqdouble} decay in both directions from $|m|$ to $|m|^{1/(S-1)}$, where $S$ is the step of the action. This is the difficulty $(ii)$ mentioned above.
One of the key ingredients of our argument is the proof of the fact that
for a {\it unlocked} parabolic linear action with at least one step-2 generator, the fall from $|m|$ to $|m|^{1/(S-1)}$ is the worst that can happen. {Our proof uses the presence of a step-2 element, and its extension to higher step actions is another challenge in the study of the general case.}
This means that the error in solving the first equation in \eqref{linearisation22} with the conjugating transformation $h$ we constructed is quadratic, but with a loss of a certain {\it proportion} of the number of the derivatives that are considered (a proportion $(S-2)/(S-1)$ for step-$S$ maps), even under the nicest Diophantine conditions.
The good news is that this loss of derivatives appears only in the quadratic error and not in the estimate of the conjugating map (for this, the step-2 assumption on one generator is crucial). As a consequence, this important loss of derivatives does not affect the convergence of the KAM scheme, for which it suffices to have a quadratic control of $C^0$ norms of the error (in fact $L^2$ would be sufficient).
Once it is shown that $h$ solves the first equation of \eqref{linearisation22} up to a quadratic error (in $C^0$ norm), the commutation relation and the fact that $A$ is step-2 can be used again to show that $h$ also solves the second equation with a quadratic error (see Section \ref{Fourier}).
Finally, we point out to the fact that equations \eqref{linearisation22} can be solved as usual up to a set of $2d$ constant terms that account for the averages of $f$ and $g$. Unlike in Moser's case of commuting circle diffeomorphisms, these constants are not all related to some dynamical invariants. However, we can use the volume preservation of the perturbed action and the zero average of the nonlinearities to fix the averages of the conjugating diffeomorphisms at each step of the KAM scheme, so that the constant terms become absorbed in the quadratic error. This third difference with the usual KAM scheme is explained in detail at the end of \S \ref{iteration set-up}.
As remarked before, our arguments remain true if we replace the preservation of the volume $\lambda$ by that of any common invariant measure for $F$ and $G$. It suffices to replace $\lambda$ by an arbitrary common invariant measure in all the text. Indeed, we do not use that $\lambda$ is invariant by $a$ and $b$ in the proof of the linearization. Moreover, since $a^kb^l$ is uniquely ergodic for some $k$ and $l$, the linearisation implies, in fact, that there is a unique invariant measure for the action $\(F, G\)$, and that this measure is the pullback of the Haar measure by the conjugacy.
\subsection{Comments on extensions and applications}
There are natural questions raised by our result as to what extent the method developed here is applicable to more general situations. We comment on this below.
$\diamond$ {\bf On applications to non-abelian actions.} We note that there are classes of solvable affine actions to which our result in Theorem \ref{mainC} can be directly applied. An {\it abelian-by-cyclic} group $G$ is a finitely presented torsion free group admitting a short exact sequence $0\to \mathbb Z^k\to G \to \mathbb Z\to 0$ (see \cite{WX} for detailed discussion on ABC groups). In this context, we call the subgroup $ \mathbb Z^k$ the abelian part of $G$. Let $\rho: G\to \rm Aff (\mathbb T^d)$ be an affine action of $G$ such that $\rho( \mathbb Z^k)$ is parabolic. Then from the KAM rigidity result for the action $\rho( \mathbb Z^k)$ one may derive KAM rigidity for the $G$ action. This way of obtaining KAM rigidity for an ABC action from KAM rigidity of its abelian part has been used before in \cite{WX} but in the special case where the abelian part $\rho( \mathbb Z^k)$ is generated by translations. More recently, in \cite{P2}, actions on $\mathbb T^3$ of the following particular ABC group: $\Gamma=\(U, V, F: UV=VU, FU=U^2VF, FV=UVF\)$ have been studied. An example of a $\Gamma$ action on $\mathbb T^3$ is when $U=\id +E_{12}$ and $V=\id +E_{13}$ and $$F= \begin{pmatrix} 1&0&0\\ 0& 1&-1\\ 0&-1&2 \end{pmatrix}. $$ The abelian action $\(U, V\)$ is unlocked and thus Diophantine affine actions with such linear part are KAM rigid by Theorem \ref{mainC}, which in turn implies KAM rigidity for an affine $\Gamma$ action with such abelian part.
Similar to our dichotomy result, we expect to use the method we developed in this paper to obtain a classification result for linear ABC actions $\rho_L: G\to \rm Aut (\mathbb T^d)$ having a parabolic abelian part. There are roughly 3 main cases:
(i) $\rho_L$ is locked: for every affine $G$ action $\rho$ with linear part $\rho_L$, $\rho$ has a rank-one factor that is either identity or a genuinely parabolic action.
(ii) $\rho_L$ is unlocked but {\it has a locked abelian part}: for every affine $G$ action $\rho$ with linear part $\rho_L$, $\rho (\mathbb Z^k)$ has a rank-one factor that is either identity or a genuinely parabolic action, but $\rho_L$ is unlocked.
(iii) $\rho_L$ {\it has an unlocked abelian part}.
It is in the case (iii) where Theorem \ref{mainC} applies. In the case (ii), even though Theorem \ref{mainC} does not apply directly, we expect our method and even the constructions of solutions from our proofs, to apply.
It is a curious algebraic question to determine which solvable groups acting on the torus by automorphisms can have unlocked abelian part. The group $\Gamma$ described above allows on $\mathbb T^3$ both a locked action (case (i)) and an action with unlocked abelian part (case (iii)), but does not allow case (ii) \cite{P2}. We remark that the 3 dimensional discrete Heisenberg group $H_3$ generated by three elementary matrices (i.e. matrices of the form $\id+E_{ij}$) on any $\mathbb T^d$ is locked (in particular it has a locked abelian part) and it is not clear if $H_3$ linear actions by toral automorphisms are always locked.\color{black}
$\diamond$ {\bf On connection to nilflows.} Given a nilpotent Lie group $N$ of step $k$, and a lattice $\Gamma$ in $N$, the quotient $N/\Gamma$ is a nilmanifold of step $k$. Any one-parameter subgroup of $N$ defines, via left-multiplication on $N/\Gamma$, a smooth nilflow. Similarily, a subgroup $A$ of $N$ isomorphic to $\mathbb R^k$ defines an $\mathbb R^k$ {\it nilaction} on $N/\Gamma$. While it was proved by Flaminio and Forni \cite{FF} that nilflows have infinite dimensional cohomology, nilactions can have finite dimensional cohomology as in \cite{CF} or in \cite{D}. In \cite{D} this was used for proving a KAM type of local rigidity result for a class of nilactions with strong Diophantine properties, on 2-step nilmanifolds. There is a close connection between the actions which we consider in this paper and nilactions. Namely, one gets a parabolic affine $\mathbb Z^k$ action if one considers return maps of an $\mathbb R^k$ nilaction to a certain section, and $\mathbb R^k$ nilactions can be viewed as suspensions over such $\mathbb Z^k$ actions on the torus. We hope that some of the ideas developed here to study the KAM-rigidity of parabolic actions on the torus could be useful in the local rigidity study of nilactions, that is a more general problem where, besides the works cited above, there has been yet no progress.\color{black}
\color{black}
\subsection{Plan of the paper} The rest of the paper is devoted to the proof of Theorem \ref{mainC}. The proof is divided into three parts. In \S \ref{sec.proof.step2} we state the main inductive KAM conjugacy step, Proposition \ref{Main iteration step}, and then show how to deduce Theorem \ref{mainC} from it. In \S \ref{sec3} we give some necessary estimates on sums and double sums along the dual orbits of $A$ and $B$ that serve for constructing the approximate solutions to the cohomological equations that appear in the linearized conjugacy equations. In \S \ref{plan_of_proof} we use the latter estimates to prove Proposition \ref{Main iteration step}. Each part will start with a detailed introduction of its content and of the ideas that are involved in the proofs.
\section{Proof of Theorem \ref{mainC}- the iteration part} \label{sec.proof.step2}
The proof is based on a KAM scheme, with three peculiarities which distinguish it from the usual way KAM schemes are applied to proving local rigidity.
The usual KAM iteration goes as follows: we start with an ${\varepsilon}$-perturbation $\(F, G\)$ of $\(a,b\)$. By linearising the conjugacy problem and by solving the linear equation approximately, we produce a conjugacy $\mathcal H_1= (\Id+\bfh_1)$ which conjugates $\(F, G\)$ to an action $\(F_1, G_1\)$ which is an $\ve^k$-perturbation of $\(a, b\)$, where $k>1$. Then we say that $\(F_1, G_1\)$ is a {\it quadratically small} perturbation of $\(a, b\)$, with respect to how far $\(F, G\)$ was from $\(a, b\)$. This process is repeated, and at the $n$-th step of iteration we build conjugacies $\mathcal H_{n}=(\Id+\bfh_1)\circ \ldots \circ (\Id+\bfh_n)$ that satisfy
\begin{equation}\label{hh1'}
\begin{cases} \mathcal H_{n}^{-1} \circ F \circ \mathcal H_{n}&=a+\bff_{n+1},\\ \mathcal H_{n}^{-1} \circ {G}\circ \mathcal H_{n}^{-1}&=b+\bfg_{n+1}, \end{cases} \end{equation} where $\bff_{n+1}$ and $\bfg_{n+1}$ are of order ${\varepsilon}_{n}^k= {\varepsilon}_{n+1}$, while $\bfh_n$ is of order ${\varepsilon}_{n}$.
Truncation (or more generally, applying smoothing operators) is typically used only to remedy a fixed loss of regularity at each step of iteration while solving the linearized problem. In our case here, due to the (possible) presence of infinitely many resonances, without truncation we might not have any quadratic estimates for the error. This is the first peculiarity of the proof, the corresponding details are contained in \S \ref{Fourier}.
The other one is that at every step of the KAM procedure we solve the linearised equations approximately {\it only} up to a constant term. This constant term can be large, it makes the error at the $n$-th step of order ${\varepsilon}_{n}$ instead of ${\varepsilon}_{n+1}$ (as we would like), so {\it a priori} there need not be any convergence of the sequence $\mathcal H_n$. This is where we use the volume preservation assumption. Namely, the volume preservation assumption allows us to adjust the average of ${\bfh}_n$ at step $n$, so that the total new error $(\bff_{n+1}, \bfg_{n+1})$ becomes of order ${\varepsilon}_{n+1}$. The same approach was used by Herman for Diophantine torus translations \cite{Herman}. Application of this approach in the context of group actions meets certain difficulties. This is explained in \S \ref{iteration set-up}.
The third feature of the proof, which has not appeared much in similar problems, is that, even though the estimates for ${\bfh}_n$ at each step are tame, the estimates for the error at each step are not tame. Namely, the loss of the number of derivatives is not a fixed constant as usually, but a proportion (that goes to $1$ when the second generator's step goes to infinity) of the number of derivatives. However, this does not affect the convergence of the scheme. Similar observation was used recently in \cite{ZW}.
\subsection{Linearisation of the problem and the main iterative step: Proposition \ref{Main iteration step}}\label{s_linearisation}
Given small perturbations $a+\bff$ and $b+\bfg$ of the two action generators $a$ and $b$, and the commutativity condition among them: \begin{equation}\label{commuting} (a+\bff)\circ(b+\bfg)=(b+\bfg)\circ(a+\bff), \end{equation} we wish to solve for $H=id+\bfh$ the conjugacy problem \begin{equation}\label{conjugacy} H\circ (a+\bff) = a\circ H, \quad H\circ (b+\bfg) = b\circ H. \end{equation} The commutativity condition \eqref{commuting} can be rewritten as $$ \bff(b+\bfg) -B{\bf f}- (\bfg(a+\bff)- A \bfg)=0, $$ which permits to see condition \eqref{commuting} as a sum of a linear operator applied to $\bff, \bfg$ plus a non-linear part which is quadratic in $\bff, \bfg$: \begin{equation}\label{linearisation} [\bff\circ b -B{\bf f}- (\bfg\circ a- A \bfg)] + [\bff(b+\bfg)-\bff\circ b-(\bfg(a+\bff)- {\bf g}\circ a)]=0. \end{equation} Now we introduce some notations. For any given $\bfh$, let \begin{equation*}\label{definition of D's} \begin{aligned} D_{1,0}\bfh&:=\bfh\circ a-A\bfh,\\ D_{0,1}\bfh&:=\bfh\circ b-B\bfh. \end{aligned} \end{equation*}
With this notations, equation \eqref{linearisation} gets the form \begin{equation*}\label{L} D_{0,1}\bff-D_{1,0}\bfg= -\bff(b+\bfg)+\bff\circ b+\bfg(a+\bff)- {\bf g}\circ a. \end{equation*}
Similarily, since we are looking for the conjugating map $H$ in a neighborhood of the identity, i.e., in the form $H=\id+\bfh$ with $\bfh$ small, our conjugacy problem \eqref{conjugacy} is linearised as \begin{equation*}\label{linearisation2} \begin{aligned} D_{1,0}\bfh&=\bff+[\bfh(a+\bff)-\bfh\circ a] ,\\ D_{0,1}\bfh&=\bfg+ [\bfh(b+\bfg)-\bfh\circ b]. \end{aligned} \end{equation*}
At the $n$-th step of the iteration process, for given $\bff_n, \bfg_n$ we show that we can find $\tilde \bff_n, \tilde \bfg_n$, $\bfh_n$ and vectors ${\bf V}_n$ and ${\bf W}_n$ such that
\begin{equation*}\label{approximate linearisation} \begin{aligned} D_{1,0}\bfh_n&=\bff_n + \tilde \bff_n + \bf V_n ,\\ D_{0,1}\bfh_n&=\bfg_n +\tilde \bfg_n+ \bf W_n, \end{aligned} \end{equation*} where $\bf V_n$ and $\bf W_n$ are of the same order as $\bff_n, \bfg_n$, and the new functions $\tilde \bff_n$, $\tilde \bfg_n$ are quadratic. In later sections, when we set up the iteration process, we will see that the volume preservation assumption will force the constant terms $\bf V_n$ and $\bf W_n$ to be of quadratic order as well.
Let us formulate the main iterative step as a proposition. In later sections Proposition \ref{Main iteration step} will be used to perform iterations, show their convergence and prove the main result Theorem \ref{mainC}.
In what follows, we say that an affine action $\(a, b\)$ is {\it unlocked} if its linear part $\(A, B\)$ is unlocked. If $\(a, b\)$ is a Diophantine affine action, then its linear part is automatically unlocked, but we stress this in the statements since the property of $\(A, B\)$ being unlocked will play a crucial role in the proofs. \color{black}
\begin{PProp}\label{Main iteration step} Let $\(a, b\)$ be an unlocked $({\gamma},\tau)$-Diophantine parabolic affine $\mathbb Z^2$ action, where $a$ is step-2.
Let $F=a+{\bf f}$ and $G=b+{\bf g}$ be $C^\infty$ commuting diffeomorphisms generating a perturbation $\(F, G\)$ of $\(a, b\)$. For $r\ge 0$, let $\Delta_r=\max\{\|\bff\|_r, \|\bfg\|_r\}$.
There exist constants $C$, $C_r$, $C_{r'}$ and $D=D(a,b, \gamma,\tau, d)$ such that for any $N\in \mathbb N$ there exist vector fields ${\tilde \bff}_N$, ${\tilde \bfg}_N$, ${\bfh}_N$, and vectors $\bf V$ and and $\bf W$
such that \begin{align*} D_{1,0}{\bfh}_N+{\bf \tf}_N&= \bff+ {\bf V},\\ D_{0,1}{\bfh}_N+{\bf \tg}_N&= \bfg+{\bf W},\\ \end{align*}
and the following estimates hold whenever $0\le r$, $D< r'$:
\begin{equation}\label{main-est}
\begin{aligned}
\|{\bf h}_N\|_{r}&\leq C_r \, N^{D}\Delta_{r},\\
\|{\bf \tf}_N\|_{0}, \, \|{\bf \tg}_N\|_{0} &\le CN^{D} \Delta_0\Delta_1+ C_{r'}N^{-r'+D}\Delta_{r'},\\
\|{\bf \tf}_N\|_{r}, \, \|{\bf \tg}_N\|_{r} &\le C_r N^{D}\Delta_{r}, \\
|{\bf V}|, |{\bf W}| &\leq C \Delta_{0}.\\ \end{aligned} \end{equation} \end{PProp}
The proof of Proposition \ref{Main iteration step} is postponed to \S \ref{sec3}.
\subsection{Iteration set-up}\label{iteration set-up}
In this section we set up the iteration which we use to prove Theorem \ref{mainC}. The iterative step consists of three sub-steps: linearization, application of Proposition \ref{Main iteration step} and adjusting the average of the conjugating diffeomorphism by using the volume preservation of the perturbation.
\begin{proposition}\label{iterative_step} Let $\(a, b\)$ be an unlocked $(\gamma,\tau)$-Diophantine parabolic affine $\mathbb Z^2$ action, where $a$ is step-2. There exists a constant $D>0$ only depending on the action $\(a, b\)$, for which the following holds.
Let $\(a+{\bf f},b+{\bf g}\)$ be a $C^\infty$ volume preserving perturbation such that $$ave(\bff)=ave(\bfg)=0. $$
Assume that we have constructed a conjugation up to the $n$-th step, $\mathcal H_{n-1}= (\id + \bfh_{n-1}) \circ\dots\circ (\id + \bfh_1)$, such that $ave((\mathcal H_{n-1})-\id)=0$ and $$ \mathcal H_{n-1}\circ \(a+\bff, b+{\bf g}\) \circ \mathcal H_{n-1}^{-1}=\(a+\bff_n, b+\bfg_n\). $$
Denote $\Delta_{r, n}:= \max\{\|\bff_n\|_r, \|\bfg_n\|_r\}$.
Then for any $N\in \mathbb N$ there exists $\bfh_n$ (which depends on $N$) such that for $0\le r$, $D\le r'$ and certain constants $C$, $C_r$, $C_{r'}$, if $N^D\Delta_{1,n}<1$, then we have:
\begin{enumerate}
\item[(i)] $\|\bfh_{n}\|_r \le C_rN^D \Delta_{r,n}$;
\item[(ii)] For $$ \bff_{n+1}:= (\id+\bfh_{n})\circ (a+\bff_n)\circ (\id+\bfh_{n})^{-1}-a, $$ $$\bfg_{n+1}:= (\id+\bfh_{n})\circ (b+\bfg_n)\circ (\id+\bfh_{n})^{-1}-b, $$
$$\Delta_{r,n+1}:= \max\{\|\bff_{n+1}\|_r, \|\bfg_{n+1}\|_r\},$$ the following estimates hold: \begin{equation}\label{new_error} \begin{aligned} \Delta_{0,n+1}&\le C_rN^D \Delta_{0, n}\Delta_{1, n}+ C_{r'} N^{-r'+D}\Delta_{r', n}, \\ \Delta_{r,n+1}&\le C_{r}N^D \Delta_{r, n}; \\ \end{aligned} \end{equation}
\item[(iii)] For $\mathcal H_{n}:= (\id+ \bfh_{n}) \circ\mathcal H_{n-1}$ we have: $ave(\mathcal H_{n}-\id)=0$.
\end{enumerate} \end{proposition} \begin{proof} The non-linear problem is to find $\bfh_{n}$ such that \begin{equation}\label{conj} (\id+\bfh_{n})\circ\(a+\bff_n, b+\bfg_n\)= \(a+\bff_{n+1}, b+\bfg_{n+1}\)\circ(\id+\bfh_{n})\end{equation} with $\bfh_{n}$ and $\bff_{n+1}, \bfg_{n+1}$ satisfying the estimates of the proposition.
Here is a brief outline of the proof. After linearizing the above non-linear problem,
we will first apply Proposition \ref{Main iteration step} to determine $\bfh_n$ and vectors ${\bf V}_{n}$, ${\bf W}_{n}$ such that $|{\bf V}_n|+|{\bf W}_n| \leq C\Delta_{0,n}$ and \begin{equation}\label{nonlinear} (\id+\bfh_{n}) \circ \(a+\bff_n, b+\bfg_n\) = \(a+\bff_{n+1}+{\bf V}_{n}, b+\bfg_{n+1}+{\bf W}_{n}\) \circ (\id+\bfh_{n}), \end{equation} where $\bfh_n$ and $\bff_{n+1}, \bfg_{n+1}$ satisfy the estimates in $(i)$ and $(ii)$. We observe that if we change $\bfh_n$ by adding to it a translation vector of order $\Delta_{0,n}$, then equation \eqref{nonlinear} will still hold with some new $\bff_{n+1}, \bfg_{n+1}$, ${\bf V}_{n},{\bf W}_{n}$ that satisfy the same estimates. By adequately choosing the translation vector, based on the volume preservation condition and the zero average condition on the initial perturbation (and the inductive condition $ave((\mathcal H_{n-1})-\id)=0$), we will be able to absorb the constants ${\bf V}_{n}$ and ${\bf W}_{n}$ into $\bff_{n+1}$ and $\bfg_{n+1}$.
\noindent{\bf Linearization. \ } We begin by linearizing the non-linear conjugation problem. Equation \eqref{nonlinear} is rewritten in a way that expresses the error in the new perturbation $(\bff_{n+1}, \bfg_{n+1})$ in terms of the linearization of the non-linear conjugation problem above and additional errors: \begin{equation}\label{linearization} \begin{aligned} \bff_{n+1}(\id+\bfh_{n})&= (\bfh_n\circ a- A\bfh_n) +\bff_n -{\bf V}_{n} + ( \bfh_n\circ(a+\bff_{n})-\bfh_n\circ a),\\
\bfg_{n+1}(\id+\bfh_{n})&= (\bfh_n\circ b-B\bfh_n)+ \bfg_n-{\bf W}_{n} + ( \bfh_n\circ(b+\bfg_{n})-\bfh_n\circ b). \\ \end{aligned} \end{equation}
To estimate the left-hand side in the equations above, we need to estimate the following two terms: \begin{equation}\label{E1E2} \begin{aligned} E_1:=& \bfh_n\circ a-A\bfh_n+\bff_n-{\bf V}_{n},\, \bfh_n\circ b-B\bfh_n+\bfg_n-{\bf W}_{n},\\ E_2:=& \bfh_n\circ(a+\bff_n)-\bfh_n\circ a, \, \bfh_n\circ(b+\bfg_n)-\bfh_n\circ b.\\
\end{aligned} \end{equation}
Bellow we estimate both terms, $E_1$ and $E_2$, in $C^0$ norm for the transformation $\bfh_n$ provided by in Proposition \ref{Main iteration step}. This will imply the estimate for the $C^0$ norm of $\bff_{n+1}$ and $\bfg_{n+1}$.
\noindent{\bf Applying Proposition \ref{Main iteration step}.}
Apply now Proposition \ref{Main iteration step} to $\bff_n, \bfg_n$. Fix $N\in \mathbb N$. For the fixed $N$, from Proposition \ref{Main iteration step} we obtain ${\bfh_n}:=({\bfh_n})_N$, $\widetilde{(\bff_n)}_N$, $\widetilde{(\bfg_n)}_N$ and vectors ${\bf V}_{n}$ and ${\bf W}_{n}$. The first estimate in Proposition \ref{Main iteration step} gives directly: \begin{equation}\label{hhhh}
\|\bfh_{n}\|_r \le C_rN^D \Delta_{r,n}. \end{equation}
Observe that the assumption that $N^D \Delta_{1,n}$ is bounded by a constant implies that $\|\bfh_{n}\|_1$ is bounded by a constant. It is a common fact (see for example \cite[Lemma AII.26]{L}) that the inverse map $(\Id+\bfh_{n})^{-1}=\Id+ \bfh'_{n}$ is such that $\bfh'_{n}$ also satisfies the estimate: \begin{equation}\label{h'}
\|\bfh'_{n}\|_r\le C_r \|\bfh_{n}\|_r. \end{equation}
The error $E_1$ is precisely $(\widetilde{(\bff_n)}_N,\widetilde{(\bfg_n)}_N)$, so from the second estimate in Proposition \ref{Main iteration step} we get for any $r'>0$:
\begin{equation}\label{E1} \begin{aligned}
\|E_1\|_0&\le CN^{D} \Delta_{0,n}\Delta_{1,n}+ C_{r'}N^{-r'+D}\Delta_{r',n} .\\
\end{aligned} \end{equation}
The estimate for $E_2$ follows by using the standard estimates (see for example Appendix in \cite{DF}) and estimate \eqref{hhhh}: \begin{equation}\label{E2}
\|E_2\|_0\le C\|\bfh_n\|_{1}\Delta_{0, n}\\ \le CN^D\Delta_{1,n}\Delta_{0, n}.\\
\end{equation}
Putting the two errors together, we have that the new error satisfies:
\begin{equation}\label{E} \Delta_{0,n+1}\le CN^{D} \Delta_{0,n}\Delta_{1,n}+ C_{r'}N^{-r'+D}\Delta_{r',n}. \end{equation}
The estimate for $C^r$ norms of $\bff_{n+1}$ and $\bfg_{n+1}$ for any $r$ (the second estimate in \eqref{new_error}) follows from the definition \eqref{linearization} of these maps. We show how the estimate follows for $\bff_{n+1}$. For $\bfg_{n+1}$ the proof is the same.
From \eqref{linearization} we can write: $$ \bff_{n+1}= \widetilde{\bff_{n}}(\id+\bfh'_{n}) + ( \bfh_n\circ(a+\bff_{n})-\bfh_n\circ a)\circ (\id+\bfh'_{n}), $$ where $\Id+\bfh'_{n}=(\id+\bfh_{n})^{-1}$, and $\bfh'_{n}$ satisfies estimate \eqref{h'}. Then by applying standard estimate for the composition of maps (see for example \cite[Theorem A.8]{Hormander})
and the bound for $\|\widetilde{\bff_{n}}\|_r$ which we have from Proposition \ref{Main iteration step}, we get: $$ \begin{aligned}
\|\bff_{n+1}\|_r \le& \|\widetilde{\bff_{n}}(\id+\bfh'_{n})\|_r + \| \bfh_n\circ(a+\bff_{n})-\bfh_n\circ a)\circ (\id+\bfh'_{n})\|_r\\
\le &C_r (\|\widetilde{\bff_{n}}\|_r+ \|\bfh'_{n}\|_r)\le C_rN^D\Delta_{r, n.} \end{aligned} $$
\noindent{\bf Adjusting the average of the conjugating diffeomorphism}. Now we will adjust the average of $\bfh_n$ in such a way that the constant terms ${\bf V}_{n}$ and ${\bf W}_{n}$ in \eqref{linearization} are forced to be as small as $\|\bff_{n+1}\|_0$ and $\|\bfg_{n+1}\|_0$, respectively. The crucial role here is played by the assumptions on the volume preservation and zero averages of the initial errors. The adjustment of the average of $\bfh_n$ will not depend on the action elements, as will be seen in Lemma \ref{c}. We will check that it works on one action generator, the other generator can be treated in the same way.
\begin{lemma}\label{Cchange} Suppose that $ \bfh$ is such that \begin{equation} \label{inv} \begin{aligned} a+ \bff_{n+1}+{\bf V}_{n} &= (\Id+{ \bfh})\circ (a+\bff_n) \circ (\Id+ { \bfh})^{-1},\\
\end{aligned} \end{equation}
where $|{\bf V}_n|= O(\Delta_{0,n})$ and $\bff_{n+1}$ satisfies estimate \eqref{new_error}.
For any vector $C$ such that $|C|= O(\Delta_{0,n})$, the function $\hat\bfh=\bfh+ C$ satisfies \begin{equation} a+\hat \bff_{n+1}+\hat {\bf V}_{n} = (\Id+{ \hat\bfh})\circ (a+\bff_n) \circ (\Id+ {\hat \bfh})^{-1}, \end{equation}
where $|\hat {\bf V}_n |= O(\Delta_{0,n})$ and $\hat \bff_{n+1}$ satisfies \eqref{new_error}. \end{lemma} \begin{proof} Using \eqref{inv}, we can write: \begin{equation*} \begin{aligned} (\id+\bfh+C)\circ(a+\bff_n)&= (\id+\bfh)\circ(a+\bff_n)+C= (a+\bff_{n+1}+{\bf V}_n+C)\circ (\id +\bfh)\\ &= (a+\bff_{n+1}+{\bf V}_n+C)\circ (\id-C)\circ (\id +\bfh+C)\\ &=( a+ \bff_{n+1}\circ (\id-C)+ ({\bf V}_n-(A-\id)C))\circ (\id +\bfh+C)\\ &=( a+ \hat \bff_{n+1}+ \hat {\bf V}_n)\circ (\id +\bfh+C), \end{aligned} \end{equation*}
where $\hat \bff_{n+1}:= \bff_{n+1}\circ (\id-C)$ and $\hat {\bf V}_n:= {\bf V}_n-(A-\id)C$. Estimate \eqref{new_error} holds then for $\hat \bff_{n+1}$ since it holds for $\bff_{n+1}$ and $|C|= O(\Delta_{0,n})$. Obviously, $|\hat {\bf V}_n |$ is of the same order of magnitude as $|C|= O(\Delta_{0,n})$. \end{proof}
In the previous part of the proof we constructed $\bfh_n$ such that $$a+ \bff_{n+1}-{\bf V}_{n} = (\Id+{ \bfh_n})^{-1}\circ (a+\bff_n) \circ (\Id+ { \bfh_n}). $$ If $\bfh_n$ satisfies equation \eqref{inv}, we can apply Lemma \ref{Cchange} to adjust the average of $ \bfh_n$. The following Lemma explains how the the constant vector $C$ is chosen at the $n$-th step of the iteration. Let $ {H}_{n-1}= \mathcal H_{n-1}-\id$, and recall that, by assumption, $ave ( {H}_{n-1})=0$.
\begin{lemma}\label{c} Let $C= -\int_{{\mathbb{T}}^d} {\bfh}_n \circ \mathcal{H}_{n-1}$, and let $\hat \bfh_{n}= \bfh_n+C$. Let $\hat{\mathcal H}_n=(id +\hat \bfh_{n}) \circ \mathcal H_{n-1} = \id +\hat { H}_n$. Then $ave(\hat {H}_n)=0$.
\end{lemma}
\begin{proof} $\hat{\mathcal H}_{n}=(\Id+\hat\bfh_{n})\circ \mathcal H_{n-1}$ implies $$\hat H_{n}= H_{n-1}+ \hat \bfh_{n}\circ \mathcal H_{n-1}.$$ By the inductive assumption, $ave(H_{n-1})=0$. Then, by taking averages of both sides of the equation above, we get that $ave ({\hat H_{n}})=0$. \end{proof}
After choosing $C$ as in Lemma \ref{c}, by applying Lemma \ref{Cchange}, we get the equation: $$a+ \hat\bff_{n+1}+\hat{\bf V}_{n} = (\Id+\hat\bfh_n)\circ (a+\bff_n) \circ (\Id+ {\hat \bfh_n})^{-1},$$ which implies $$a+ \hat\bff_{n+1}+\hat{\bf V}_{n} = (\Id+\hat H_n)\circ (a+\bff) \circ (\Id+ {\hat H_n})^{-1}.$$ From this, by composing on the right with $\Id+\hat H_n$ we get $$A\hat H_n + \hat\bff_{n+1}(\Id+\hat H_n)+\hat{\bf V}_{n} = \bff+ {\hat H_n}\circ (a+\bff).$$ By taking averages with respect to the volume of both sides of the equation above and using the assumptions that $\bff$ has zero average and that $a+f$ is volume preserving, it follows that $\hat{\bf V}_{n}=- \hat\bff_{n+1}(\Id+\hat H_n)$.
This implies that $\hat{\bf V}_{n}=O( \|\hat\bff_{n+1}\|_0)$, which means that the constant $\hat{\bf V}_{n}$ can be absorbed by $\hat\bff_{n+1}$.
From Lemma \ref{Cchange} we have that estimates \eqref{new_error} hold for $\hat\bff_{n+1}$. Finally, we proclaim the new $\bff_{n+1}$ to be $\hat\bff_{n+1}$.
\end{proof}
\subsection{Convergence of the iterative scheme}
Once we have the result of Proposition \ref{iterative_step}, the set-up of the KAM scheme and its convergence is essentially the same as in usual applications of KAM method (see for example Section 5.4 in \cite{DK}).
Assume that $\(a,b\)$ is a $(\gamma, \tau)$-Diophantine action. Let $\(a+\bff, b+\bfg\)$ be a small smooth perturbation of $\(a,b\)$. Since we are proving KAM rigidity, we also assume that $\(a+\bff, b+\bfg\)$ is volume preserving and that $\bff$ and $\bfg$ have zero average.
Given the initial perturbation above, we let: $$ \bff_1=\bff;\,\,\, \bfg_1=\bfg. $$
Recall that we use the notation $\Delta_{r,1}:=\max\{\|\bff_1\|_r, \|\bfg_1\|_r\}.$
Let $D$ be the constant from Proposition \ref{iterative_step} which depends only on $\(a,b\)$.
Fix $k=\frac{4}{3},$ and let $l= 8D+16$.
At the first step we assume that: $$ \Delta_{0,1}<\varepsilon,\,\,\, \Delta_{l,1}<\varepsilon^{-1} $$ for a small $\varepsilon>0$. We will show that $\varepsilon$ can be chosen so small that the iterative process converges.
We describe now the iterative process. By Proposition \ref{iterative_step}, there exists $\bfh_1$ with the estimates claimed in the proposition. Then the transformation $\Id+ \bfh_1$ conjugates $\(a+\bff_1, b+\bfg_1\)$ to a new perturbation, which we call $\(a+\bff_2, b+\bfg_2\)$. This procedure is iterated.
At this point we still have the freedom to choose the truncation at level $N$ when applying Proposition \ref{iterative_step} at the $n$-th step of the iteration. If at step $n$ we choose the truncation to be $$N_n=\varepsilon_n^{-\frac{1}{3(D+2)}},$$ where $\varepsilon _n=\varepsilon^{(k^n)}$, then for sufficiently small $\varepsilon$ we can show inductively that the following estimates hold for all $n$: \begin{equation}\label{convergence_set_up} \begin{aligned} \Delta_{0,n}&<\varepsilon _n=\varepsilon^{(k^n)},\\ \Delta_{l, n}&<\varepsilon_n^{-1},\\ \end{aligned} \end{equation} and \begin{equation}\label{haha}
\|\bfh_{n}\|_1<\varepsilon_n^{\frac{1}{2}}.\\ \end{equation} Here we use the letter C to denote any constant which depends only on the fixed $l$, $D$ and the unperturbed action $\(a,b\)$.
Suppose that we are at the $n$-th step of iteration and that estimates \eqref{convergence_set_up} hold for $n$. First, we check that the condition $N^D\Delta_{1,n}<1$ holds by using the standard interpolation inequality
\begin{equation}\label{interpolation}
\Delta_{1,n}\le C \Delta_{0, n} ^{1-\frac{1}{l}}\Delta_{l, n}^{\frac{1}{l}},
\end{equation}
and assumptions \eqref{convergence_set_up}:
$$
N_n^D\Delta_{1,n}\le C \ve_n^{\frac{-D}{3(D+2)}} \ve_n^{1-\frac{1}{l}}\ve_n^{-\frac{1}{l}}= C\ve^{\frac{-D}{3(D+2)}+1-\frac{2}{l}}.
$$
Since for the chosen value of $l$ the term $\frac{-D}{3(D+2)}+1-\frac{2}{l}$ is positive, by choosing initial $\ve$ sufficiently small,
we get that the above expression is smaller than 1.
Then the maps $\bfh_n$, $\bff_{n+1}$ and $\bfg_{n+1}$ are constructed by applying Proposition \ref{iterative_step}.
The same proposition, combined with \eqref{convergence_set_up} and
the interpolation inequalities \eqref{interpolation}, together with our choice of $l$, imply that \eqref{haha} holds for a sufficiently small $\varepsilon$: $$
\|h_{n}\|_1\le CN_n^{D}\Delta_{1, n}\le C N_n^D \Delta_{0, n}^{1-\frac{1}{l}} \Delta_{l, n}^{\frac{1}{l}}\le C \ve_n^{-\frac{D}{3(D+2)}+1-\frac{2}{l}}\le \ve_n^{\frac{1}{2}}. $$
Now we check that \eqref{convergence_set_up} holds for $n$ replaced by $n+1$.
First we compute the bounds for the $l$ norms by using the estimates of Proposition \ref{iterative_step}:
$$
\Delta_{l, n+1}\le CN_n^D\Delta_{l,n}\le C\ve_n^{\frac{-D}{3(D+2)}}(1+\ve_n^{-1})\le 2C\ve_n^{\frac{-D}{3(D+2)}-1}< \ve_n^{-\frac{1}{3}-1}=\ve_n^{-\frac{4}{3}}=\ve_{n+1}^{-1}.
$$
Finally, we estimate the 0-norms (by using again estimates in Proposition \ref{iterative_step} and the interpolation inequality):
\begin{equation*}
\begin{aligned}
\Delta_{0, n+1}&\le CN_n^D \Delta_{0, n}^{1-\frac{1}{l}} \Delta_{l, n}^{\frac{1}{l}}\Delta_{0, n}+ CN_n^{-l+D}\Delta_{l, n}\\
&\le C(\ve_n^{-\frac{D}{3(D+2)}+2-\frac{2}{l}}+\ve_n^{\frac{l-D}{3(D+2)}-1})\le \ve_n^{\frac{4}{3}}= \ve_{n+1},
\end{aligned}
\end{equation*}
since, given our choice of $l$, both expressions $-\frac{D}{3(D+2)}+2-\frac{2}{l}$ and $\frac{l-D}{3(D+2)}-1$ are strictly larger than $\frac{4}{3}$.
Therefore the estimates in \eqref{convergence_set_up} hold for all $n$. This implies the convergence of $\mathcal H_n$ in the $C^1$ norm to some $\mathcal H_\infty$, which conjugates the initial perturbation to $\(a, b\)$. The fact that the conjugation $\mathcal H_\infty$ is $C^m$ for every $m>0$ (i.e., that the process converges in any norm) is proved in a standard way by using interpolation estimates (see for example the end of Section 5.4 in \cite{DK}).
\color{black}
\subsection{Volume preservation of the conjugacy}\label{s_volume_pres} Now we have that $\mathcal H_\infty$ conjugates the perturbation $\(F, G\)$ to $\(a,b\)$. Since $\(F, G\)$ is assumed to be volume preserving, the conjugation relation implies that the pushforward of the volume by $\mathcal H_\infty$ is invariant under $\(a,b\)$. Since $a^kb^l$ is uniquely ergodic for some $k$ and $l$, the map $\mathcal H_\infty$ is volume preserving.
{The proof of Theorem \ref{mainC} is now completed {\it modulo} the proof of Proposition \ref{Main iteration step}. The rest of the paper is dedicated to the proof of Proposition \ref{Main iteration step}.
$\Box$} \color{black}
\section{ Estimates of sums and double sums along the dual orbits} \label{sec3}
In this subsection we give the necessary estimates on sums and double sums along the dual orbits of $A$ and $B$ that will be crucial in solving the cohomological equations and proving Proposition \ref{Main iteration step}. The main results of this section are Propositions \ref{c_est_easy} and \ref{lemma.double.sums.case3}.
In Proposition \ref{c_est_easy} we deal with partial sums along the step-2 dual orbits. These sums will be used in the proof of Proposition \ref{Main iteration step} for estimating the norms of the conjugacies. We also give a first estimation of full sums along the step-2 dual orbits that will be the key for estimating the error in solving the cohomological equations at the resonant Fourier modes.
As explained in Section \ref{s_case3}, a full sum along the step-2 dual orbit can be reinterpreted, {\it via} the higher rank trick, as a double sum of a quadratically small function $\phi$ measuring the error of the pair $(f,g)$ in \eqref{linearisation22} from forming a cocycle above the action $\(a,b\)$. In Proposition \ref{lemma.double.sums.case3}, we deal with these double sums. The estimates we obtain in this proposition will serve for estimating the error in solving the cohomological equations at non-resonant Fourier modes.
\subsection{Notations}\label{s_notations} In this subsection we summarise the notations used in the rest of the paper.
\begin{itemize} \item Assume that $\(A,B\)$ is unlocked commuting linear parabolic action. We consider two commuting affine maps $a(x)=Ax+\alpha$ and $b(x)=Bx+\alpha$ on $\mathbb T^d$, where $a$ is step-2 and $b$ is step-$S$ (see Definition \ref{def_step}). Elements of the step-$S$ action $\(a,b\):{\mathbb{Z}}^2 \to \text{Diff\,}^\infty_\lambda({\mathbb{T}}^d)$ are denoted by $a^kb^l$, $(k, l)\in {\mathbb{Z}}^2$.
\item Let $$A= \id +\widetilde{A}; \quad B =\id +\widetilde{B}.$$ In these notations, $a$ being step-2 and $b$ being step-$S$ implies: $\widetilde{A}^2=\widetilde{B}^S=0$.
\item Let $\bA=(A^{tr})^{-1}$, $\bB=(B^{tr})^{-1}$. The linear action of $\(\bA, \bB\)$ on ${\mathbb{Z}}^d$ is called the dual action of $\(A, B\)$. Let $$\bA=\id +\widehat{A}; \quad \bB =\id +\widehat{B}.$$ Clearly, $\bA$ and $\bB$ are also step-2 and step-$S$, respectively, which implies $\widehat{A}^2=\widehat{B}^S=0$.
\item For $S$ being the step of the action, let $$ \eta =0.99\frac1S; $$
\item
To each $m\in {\mathbb{Z}}^d$ we associate $s=s(m)$, called the step of $m$, such that $$ \widehat{B}^{s} m = 0, \quad \widehat{B}^{s-1} m \neq 0. $$ Denote \begin{equation}\label{not_ett} {\delta} = {\delta} (m) = 0.99 \frac1{s} . \end{equation}
Clearly, we have $s(m)\leq S$ for any $m\in {\mathbb{Z}}^d$, and hence, ${\delta} (m)\geq \eta$.
\item For each $(k,l)\in{\mathbb{Z}}^2$, let $\alpha_{k,l} $ stand for the translation part of $a^kb^l$: $$ \alpha_{k,l} := a^kb^l -A^kB^l . $$
\item Given a continuous function $h: {\mathbb{T}}^d\to{\mathbb{R}}$, denote its Fourier coefficients by $h_m$: $$ h(x)=\sum_{m\in {\mathbb{Z}}^d} h_m e(m,x), \quad e(m,x):=e^{2\pi i (m,x)} . $$ In these notations, for $a(x)=Ax+\alpha$ we have:
$$ h\circ a =\sum_{m\in {\mathbb{Z}}^d} h_{\bA m} e(\bA m,\alpha ) e(m,x), \quad (h\circ a)_m =h_{\bA m} e(\bA m,\alpha ) . $$
\item For $(k,l)\in {\mathbb{Z}}^2$ denote by $\partial_{k,l}$ the coboundary operator: for $h\in C^\infty(\mathbb T^d)$ let $$ \partial_ {k,l}(h): = h(a^kb^l ) - h. $$ In particular, the expression for the $m$-th Fourier coefficient of a coboundary is $$ (\partial_ {k,l}(h))_m = h_{\bA^k\bB^l m} e(\bA^k\bB^l m,\akl ) -h_m . $$
\item We will work with the maps of the type $p: {\mathbb{Z}}^2 \to C^\infty(\mathbb T^d)$, the usual notation being: $p(k,l)\in C^\infty$ for $(k,l)\in {\mathbb{Z}}^2$. For such maps we define the operator $Lp: \mathbb Z^2\times \mathbb Z^2 \to C^\infty(\mathbb T^d)$, by the following. For any $(k,l),(s,t) \in {\mathbb{Z}}^2\times {\mathbb{Z}}^2 $, denote $$ Lp((k,l), (s,t)):= \partial_ {k,l} p(s,t)- \partial_ {s,t} p(k,l) . $$
\item Let $m$ be such that $\bA m\neq m$ (hence, $\widehat{A} m\neq 0$). Define $$ \begin{aligned} {\mathcal{M}}(A)=\{ m\in {\mathbb{Z}}^d \mid \langle m, \widehat{A} m\rangle > 0\}, \\ {\mathcal{N}}(A)=\{ m\in {\mathbb{Z}}^d \mid \langle m, \widehat{A} m\rangle< 0\}. \end{aligned} $$
\item Suppose that $A$ is step-2, and $m$ is such that $\bA m\neq m$. Then we have $\bA^k m=m+k\widehat{A} m$. We say that $\bm$ {\it is the lowest point on the $\bA$-orbit of $m$} if $$
| \bar m | \leq |\bm+k \widehat{A} \bm | \quad \text{for all }k\in {\mathbb{Z}}. $$ Then we have a "switch": $\bm \in {\mathcal{M}} (A)$ but $\bA^{-1} \bm \in {\mathcal{N}} (A)$, or vise versa. Note that $\bm$ is the only point on the corresponding $\bA$-orbit in which the "switch" between ${\mathcal{N}}(A)$ and ${\mathcal{M}}(A)$ happens.
We write $m= \bm$ to say that $m$ is the lowest point on its own $\bA$-orbit, and $m\neq \bm$ otherwise.
\item Let $$ {\lambda}_m^{(-1)}:={\lambda}_m:= e(m,{\alpha}), \quad \mu_m^{(-1)}:= \mu_m:= e(m,{\beta}), $$ $$ \begin{aligned} &{\lambda}_m^{(k)}= {\lambda}_{\bar A m}{\lambda}_{\bar A^2 m} \dots {\lambda}_{\bar A^{k} m}, \quad k=1,2,\dots, \quad {\lambda}_m^{(0)}=1, \\ &{\lambda}_m^{(k)}=({\lambda}_{m}{\lambda}_{\bar A^{-1} m} \dots {\lambda}_{\bar
A^{k+1} m})^{-1}, \quad k=-2,-3,\dots , \\ &\mu_m^{(k)}= \mu_{\bar B m}\mu_{\bar B^2 m} \dots \mu_{\bar B^{k} m}, \quad k=1,2,\dots, \quad \mu_m^{(0)}=1, \\ &\mu_m^{(k)}=(\mu_{m}\mu_{\bar B^{-1} m} \dots \mu_{\bar B^{k+1} m})^{-1}, \quad k=-2,-3,\dots. \end{aligned} $$
\item For $A$ of step-2 and $m\in {\mathbb{Z}}$, consider the following partial sums over the dual orbit of $A$: $$ \begin{aligned} &{\Sigma}_m^{+ , A} (f):= \sum_{k=0}^{\infty} f_{\bar A^{k} m}{\lambda}^{(k)}_{ m} ,\quad {\Sigma}_m^{- , A} (f):= \sum_{k=-\infty}^{-1} f_{\bar A^{k} m}{\lambda}^{(k)}_{ m}, \\ & {\Sigma}_m^A (f):= {\Sigma}_m^{+ , A} (f)+{\Sigma}_m^{- , A} (f). \end{aligned} $$ The last two-sided sum defines the so-called "obstruction operator". \item Let $U\subset {\mathbb{Z}}^d$ be a set that is {\it invariant under the action of} $\(\bA,\bB\)$, i.e., for any $m\in U$ we have $\bA^s\bB^t m\in U$. Consider a set of real numbers, indexed by $U$: $\xi=(\xi_m)=\{\xi_m\mid m\in U\}$. With a little abuse of notation, we let the operator $\partial_{s,t}$ act on $\xi$. Namely, $$ (\partial_{s,t} \xi)_m = \xi_{\bA^s\bB^tm} e(\bA^s\bB^tm,{\alpha}_{s,t})-h_m. $$ Since the set of indices is invariant under the action, this expression is well-defined.
\noindent Comment: This notation is needed because we will define the conjugating functions $h$ by Fourier coefficients in different ways for different (invariant) sets of indices: ${\mathcal{C}}_1$, ${\mathcal{C}}_2$ or ${\mathcal{C}}_3$ (see below), and will need to solve equations in terms of Fourier coefficients before we have defined $h$ as a function.
\item The following splitting of ${\mathbb{Z}}^d\setminus \{0\}$ will be used in our analysis. $$ {\mathbb{Z}}^d\setminus \{0\}={\mathcal{C}}_{1} \cup {\mathcal{C}}_{2} \cup {\mathcal{C}}_{3} , $$ where the sets ${\mathcal{C}}_j$ are defined as follows.
\begin{itemize}
\item[${\mathcal{C}}_{1} $.] {\bf (Degenerate case).} ${\mathcal{C}}_{1} $ is the set of $m$ for which $\bA m =\bB m =m$.
\item[${\mathcal{C}}_{2} $.] {\bf (Resonant non-degenerate case).} For $(k,l)\in {\mathbb{Z}}^2\setminus \{0\}$ we say that $m\in {\mathcal{C}}_{2} (k,l)$ if the following holds:
- $m\notin {\mathcal{C}}_{1} $;
- $\bA^k\bB^lm = m$.
We define ${\mathcal{C}}_{2} =\bigcup_{(k,l)\in {\mathbb{Z}}^2\setminus \{0\}} {\mathcal{C}}_{2}(k,l)$.
\item[${\mathcal{C}}_{3} $.] {\bf (Non-resonant case).} ${\mathcal{C}}_{3} = {\mathbb{Z}}^2\setminus ( \{0\} \cup {\mathcal{C}}_{1} \cup {\mathcal{C}}_{2} )$.
\end{itemize}
\end{itemize}
\subsection{Estimates of the sums along the dual orbits of a step-2 matrix}
In the constructions that follow we will work with vectors $m$ lying in certain subsets of ${\mathbb{Z}}^d$ that are invariant under the action of $\bA$ and $\bB$. Recall the notations ${\mathcal{M}}(A)$ and ${\mathcal{N}}(A)$ from Sec.~\ref{s_notations}. \begin{PProp}\label{c_est_easy} Let $A$ be step-2 and suppose that $\bA m \neq m$. \begin{enumerate} \item Consider a set $U\subset {\mathbb{Z}}^d $ that is invariant under the action of $\bA$. Let $(\xi_m)=\{ \xi_m \in {\mathbb{R}} \mid m\in U \}$, and suppose that for all $m\in U$ we have
$|\xi_m |\leq c_0 | m |^{-r} $. Then there exists $c>0$ such that
- If $m\in {\mathcal{M}}(A)\cap U$, then $\sum_{k=0}^{\infty} | \xi_{\bA^k m} | \leq c | m |^{-r+1}$,
- If $m\in {\mathcal{N}}(A)\cap U$, then $\sum_{k=-\infty}^{-1} | \xi_{\bA^k m} | \leq
c | m |^{-r+1}$,
- If, moreover, $m=\bm$ (i.e., $m$ is the lowest point on its $\bA$-orbit), then $$
\sum_{k=-\infty}^{\infty} | \xi_{\bA^k m} | \leq c | m |^{-r+1} . $$
\item Consider the sets $U$ and $(\xi_m)$ as in (1) and ${\lambda}_m^{(1)}$ as in Sec.~\ref{s_notations}. Suppose that for each $m\in U$, the set of numbers $(\zeta_m)$ satisfies $$ {\zeta_{\bA m}} {\lambda}_m^{(1)}-\zeta_m = \xi_m. $$ Then there exists $c>0$ such that for each $m\in U$ we have: $$
|\zeta_m | \leq c | m |^{-r+1}. $$
\item There exists $c>0$ such that for any function $\xi \in C^{r}$, we have:
- If $m\in {\mathcal{M}}(A)$, then $| \Sigma^{+,A}_m (\xi )|\leq c \|\xi \|_{r} | m |^{-r+1},$
- If $m\in {\mathcal{N}}(A)$, then
$|\Sigma^{-,A}_m (\xi )|\leq c\|\xi \|_{r}| m |^{-r+1} .$
- If, moreover, $m=\bm$ is the lowest point on its $\bA$-orbit, then $$
|\Sigma^{A}_m (\xi )|\leq c | m |^{-r+1} . $$
\end{enumerate}
\end{PProp} \begin{proof} Item (1) follows directly from the estimate below, that will be used several times in the paper. \begin{sublemma}\label{l_est_easy} Let $A$ be step-2 and $\bA m \neq m$. Then there exists $c=c(r,A)>0$ such that we have:
- If $m\in {\mathcal{M}}(A)$, then $
\sum_{k=0}^{\infty} |\bar A^k m |^{-r} \leq c | m |^{-r+1} $;
- If $m\in {\mathcal{N}}(A)$, then $
\sum_{k=-\infty}^{-1} |\bar A^k m |^{-r} \leq c | m |^{-r+1} $.
- If, moreover, $m=\bm$ (i.e., $m$ is the lowest point on its $\bA$-orbit), then $$
\sum_{k=-\infty}^{\infty} |\bar A^k m |^{-r}\leq c | m |^{-r+1} . $$
\end{sublemma}
\begin{proof} Since $\bA$ is step-2 and $\bA m \neq m$, for any $m\in {\mathbb{Z}}^d$ and $k\in {\mathbb{Z}}$, we have $\bA^k m=m+k \widehat{A} m$.
Consider the case $m\in {\mathcal{M}}(A)$, i.e., $\langle m, \widehat{A} m \rangle\geq 0$. Then we have: $|m+ \widehat{A} m|>|m|$ and $\langle m+ \widehat{A} m, \widehat{A} m \rangle> 0$ (notice the strict inequality). Let $p$ denote the projection of the vector $\widehat{A} m$ onto the vector $m+ \widehat{A} m$. Clearly, $p$ is non-zero and has the same direction as $m+ \widehat{A} m$. Therefore, $|m +k\widehat{A} m|=|(m +\widehat{A} m)+ (k-1)\widehat{A} m | \geq |m+ \widehat{A} m| + (k-1)|p|>|m|+ (k-1)|p|$. Hence, for an apprpriate constant $c=c(r,A)$ we have: $$
\sum_{k=0}^{\infty} | \bar A^k m |^{-r} \leq
\sum_{k=0}^{\infty} | (m +k\widehat{A} m)|^{-r} \leq c | m |^{-r+1} . $$ To justify the last inequality, note that for any $x, y>0$, and $r>1$ we have: $$ \sum_{l=1}^{\infty} (x+ly)^{-r}\leq \frac{c_1}{y(r-1)}\,(x+y)^{-r+1}, $$ which can be proved by comparison with the integral $ y^{-r} \int_{t=1}^{\infty} (\frac{x}{y}+t )^{-r} dt $.
The case of $m\in {\mathcal{N}}(A)$, i.e., $\langle m, \widehat{A} m \rangle< 0$, is similar. Indeed, the projection of the vector $(-\widehat{A} m)$ onto $m$ has the same direction as $m$, and the above calculation holds.
Now let $m=\bm$ be the lowest point on its $\bA$-orbit. Then, in particular, $|m+ \widehat{A} m|> |m|$ and $|m- \widehat{A} m|> |m|$. This implies, for example by studying the triangle with two sides formed by vectors $m+ \widehat{A} m$ and $m- \widehat{A} m$, that we have both $\langle m+ \widehat{A} m, \widehat{A} m \rangle > 0$ and $\langle m- \widehat{A} m, \widehat{A} m \rangle > 0$. Then we can use the two estimates above to conclude that $$
\sum_{k=-\infty}^{\infty} | \bA^k m |^{-r} = \sum_{k=-\infty}^{-1} | \bA^k m |^{-r} +\sum_{k=0}^{\infty} | \bA^k m |^{-r} \leq c' | m |^{-r+1} . $$ \end{proof}
To prove (2), for a fixed $m$ write the given equation at the points $\bA^k m$ either for $k\geq 0$ or for $k\leq -1$, multiply by appropriate constants and add up, obtaining a telescopic sum on the left-hand side. Then we get that
$|\zeta_m|\leq \sum_{k=0}^{\infty} | \xi_{\bA^k m} |^{-r} $ or $|\zeta_m|\leq \sum_{k=-\infty}^{-1} | \xi_{\bA^k m} |^{-r}$. The estimate follows from (1).
To prove prove (3), recall that the Fourier coefficients of any $\xi \in C^{r}$ satisfy for all $m\in {\mathbb{Z}}^d$: $$
| {\xi_m}| \leq \| \xi \|_{r} | m |^{-r}. $$ Then for each $k\in{\mathbb{Z}} $ we have:
$| {\xi_{\bar A^{k}m}} |\leq \| \xi \|_{r} | {\bar A^{k} m} |^{-r} $. The result reduces to that of item (1). \end{proof}
\subsection{Estimates of the double sums. The parabolic higher rank trick}\label{double-buble} The double sums will be used for the case $m \in {\mathcal{C}}_{3}$ (non-resonant case). For each $m$, one of the double sums is easier to estimate than the other. The corresponding sign of $l$ will be called the "good sign" of $l$ for the given $m$.
\begin{PProp}[Estimate of the double sums] \label{lemma.double.sums.case3} Assume that $\(a,b\)$ is unlocked parabolic affine step-$S$ action, where $a$ is step-2. Suppose that $m \in {\mathcal{C}}_{3}$ is the lowest point on its $\bA$-orbit.
For $r$ sufficiently large there exists a constant $c=c(r, A,B)>0$ such that for $\eta=0.99/S$, at least one of the following holds: $$
\sum_{k\in {\mathbb{Z}}} \sum_{l\geq 0} |\bA^k \bB^l m |^{-r} \leq c |m|^{-\eta r + 8}, \quad \text{ } \quad
\sum_{k\in {\mathbb{Z}}} \sum_{l< 0} |\bA^k \bB^l m |^{-r} \leq c |m |^{-\eta r + 8}. $$ \end{PProp}
The proof of Proposition \ref{lemma.double.sums.case3} is crucial for our analysis, it is rather technical and takes up the rest of this section.
\subsubsection{Implications of being unlocked}\label{sec_C1_prelim} Recall the notion of being unlocked from Definition \ref{def_stiff}. Let us make two observations.
\begin{lemma} \label{lemma.hA2} Suppose that the action $\(a,b\)$ is unlocked. If $\widehat{B}^{2} m \neq 0$, then $\widehat{A} m \neq 0$. \end{lemma}
\begin{proof} Let $\widehat{B}^{2} m \neq 0$, and suppose by contradiction that $\widehat{A} m = 0$. Consider the function $g(x)=e(m ,\widetilde{B} x)$. Observe that $$ g(ax)=g(Ax+\alpha) = e(m , \widetilde{B}( x +\widetilde{A} x+\alpha))= e(m ,\widetilde{B} x) e(m, \widetilde{B} \widetilde{A} x)e(m ,\widetilde{B} {\alpha}). $$ Since $m \widetilde{A} =(\widehat{A} m)^t=0$, and since (by commutativity $ab=ba$) we have $\widetilde{B} {\alpha} = \widetilde{A} \beta=0$, we conclude that $g(ax)=g(x)$.
On the other hand, $$ g(bx)=e(m ,\widetilde{B} x)e(m , \widetilde{B}^2 x)e(m ,\widetilde{B} \beta) = g(x)e(m ,\widetilde{B}^2 x)e(m, \widetilde{B} \beta). $$ Since $m \widetilde{B}^2=\widehat{B}^2 m \neq 0$, we conclude that the action $(a,b)$ has a rank-one factor that is not a translation. \end{proof}
The second observation is \begin{lemma} \label{lemma.AB2} Suppose that the action $\(a,b\)$ is unlocked and $\widehat{A}^2 m = 0$. If $\widehat{B}^{s} m=0$ for some $s\geq 2$, then $\widehat{A} \widehat{B}^{s-1} m = 0$. \end{lemma}
\begin{proof} Let $\widehat{B}^{s} m=0$, and suppose by contradiction that $\widehat{A} \widehat{B}^{s-1} m \neq 0$. Define $f(x)=e(m ,\widetilde{A} \widetilde{B}^{s-1} x)$. One easily verifies that $f(ax)=f(x)$ and $f(bx)=f(x)$ (using relations $\widetilde{A} \widetilde{B}^{s-1}{\alpha} = \widetilde{A}^2 \widetilde{B}^{s-2}\beta$, $\widetilde{A} \widetilde{B}^{s-1}\beta = \widetilde{B}^{s}{\alpha}$). Hence the action $\(A+{\alpha},B+\beta\)$ has a rank-one factor equal to identity, contradicting the assumption. \end{proof}
\subsubsection{Polynomial expansion of $\bA^k\bB^lm $}\label{sec_C3_prelim}
Recall the notations from \S \ref{s_notations}.
\begin{lemma}\label{l_form_of_AkBl} Assume that $\(a,b\)$ is unlocked parabolic affine action, and $a$ is step-2. For any $m\in{\mathcal{C}}_3 $, $k,l\in{\mathbb{Z}}$, $s=s(m)$, there exists $t=t(m,A,B)$, $1\leq t\leq s-1$, such that
\begin{equation} \label{AkBl_s>2}
\bA^k\bB^lm = m+k\widehat{A} m + \sum_{j=1}^{t-1} c_j l^j\widehat{B}^j (m+k \widehat{A} m)+ \sum_{j=t}^{s-1} c_{j} l^{j} \widehat{B}^{j} m ,
\end{equation} where $c_1=1$, and all $c_j$ are positive constants that can be computed explicitly.
\end{lemma}
\noindent {\it Proof. } \noindent {\bf Case $s=s(m)=2$.} Here we have $\widehat{A}^2 m=\widehat{B}^2 m=0$ (since $s(m)=2$), and $\widehat{A} m\neq 0$, $\widehat{B} m\neq 0$ (since $m\in{\mathcal{C}}_3 $). Therefore, $\bA^km=m+k \widehat{A} m$ and $\bB^lm=m+l \widehat{B} m$. By Lemma \ref{lemma.AB2}, $\widehat{B}^2m=0$ implies that $\widehat{A}\widehat{B} m=0$, which gives the result.
\noindent {\bf Case $s=s(m)\geq 3$.} Here we have $\widehat{B}^2 m\neq 0$, so, by Lemma \ref{lemma.hA2}, $\widehat{A} m\neq 0$. Since $\widehat{A}^2m=\widehat{B}^sm=0$, we have $\bA^km=m+k \widehat{A} m$ and $\bB^lm= m + \sum_{j=1}^{s-1} c_j l^j\widehat{B}^j m$. Note that, by Lemma \ref{lemma.AB2},
$\widehat{A} \widehat{B}^{s-1} m = 0$. Composing the two expressions above and using the commutativity gives the result. \qed
Recall that for $m\in{\mathcal{C}}_3$ we have the following two possibilities: \begin{itemize} \item $s(m)= 2$, in which case $\widehat{B}^2m= 0$, and for all $(k,l)\in {\mathbb{Z}}^2\setminus \{0\}$ we have $k \widehat{A} m + l\widehat{B} m \neq 0$;
\item $s(m)\geq 3$, in which case $\widehat{B}^2m\neq 0$. \end{itemize} We will use different ways of controlling the double sums for the two cases above.
\subsubsection{Proof of Proposition \ref{lemma.double.sums.case3}, case $s(m)=2$} In this section we fix $m\in {\mathcal{C}}_3$, $s(m)=2$, and study the growth properties of \begin{equation}\label{v_kl_s=2} v_{k,l}:=\bA^k\bB^lm = m+k\widehat{A} m +k\widehat{B} m . \end{equation} The following two lemmas prove that there exists a constant $c=c(A)>0$ such that if $m$ is the lowest point on its $\bA$-orbit, then for all $k\in {\mathbb{Z}}$ and either for all $l\in {\mathbb{N}}$ or for all $l\in (-{\mathbb{N}})$ we have: $$
| v_{k,l} |> c |m|. $$
\begin{lemma}\label{l_growth_1}
Let $m\in {\mathcal{C}}_3$, $s(m)=2$. If $|\widehat{A} m|> |m|$, then for all $k,l \in {\mathbb{Z}}$ we have: $$
| v_{k,l} | > \|\widehat{A} \|^{-1} |m|. $$ \end{lemma} \begin{proof}
Assume the contrary: $| v_{k,l} | \leq \|\widehat{A} \|^{-1} |m|$. Apply $\widehat{A}$ to equality \eqref{v_kl_s=2}. Since $\widehat{A}^2m=\widehat{A}\widehat{B} m=0$, we have: $$
|\widehat{A} m| =|\widehat{A} v_{k,l}| \leq \|\widehat{A} \| \, |v_{k,l}| \leq |m|, $$ contradicting the assumption of the lemma. \end{proof}
\begin{lemma}\label{l_growth_2} Let $m\in {\mathcal{C}}_3$ be the lowest point on its $\bA$-orbit, and $s(m)=2$.
If $|\widehat{A} m|\leq |m|$, then for all $k\in {\mathbb{Z}}$ and either for all $l\in {\mathbb{N}}$ or for all $l\in (-{\mathbb{N}})$ we have: $$
|v_{k,l}| \geq |m|/2 . $$ \end{lemma} \begin{proof} Since $m\in {\mathcal{C}}_3$, we have $\widehat{A} m \neq 0$. Denote ${\mathcal{V}}_m=\text{span\,}\{ \widehat{A} m \}$, and let $m^\bot$ and $(\widehat{B} m)^\bot$ stand for the projections of $m$ and $\widehat{B} m$, respectively, onto the orthogonal complement of ${\mathcal{V}}_m$. Then $$
| \bA^k \bB^l m| = |m + k\widehat{A} m + l\widehat{B} m| \geq | m^\bot + l (\widehat{B} m)^\bot | . $$ To prove the lemma, it is enough to show that
$| m^\bot | \geq |m|/2 $. When this is done, we choose the "good sign" of $l$ to be positive if the angle between the
vectors $m^\bot $ and $ \widehat{B} m^\bot$ is acute, and negative otherwise. If we choose $l$ of good sign, then $| \bA^k \bB^l m| \geq | m^\bot| \geq |m|/2 $.
Let us estimate $| m^\bot |$. Suppose that the angle $\theta$ between the vectors $m$ and $\widehat{A} m$ satisfies $0<\theta \leq \pi/2$ (otherwise, use $(-\widehat{A} m)$ instead of $\widehat{A} m$).
Since $m$ is the lowest point on its $\widehat{A}$-orbit, we have: $|m-\widehat{A} m|\geq |m|$. We see in this case that the projection of $m$ onto ${\mathcal{V}}_m$ satisfies
$$
\|\text{proj}_{{\mathcal{V}}_m}m\| \leq \|\widehat{A} m\|/2. $$ Hence, $$
|m^\bot| \geq |m| - |\text{proj}_{{\mathcal{V}}_m}m| \geq |m| - |\widehat{A} m|/2 \geq |m| - |m|/2 = |m|/2. $$ \end{proof}
\begin{lemma}[Estimate of the double sums, $s(m)=2$] \label{lemma.double.sums.case3_s=2} Assume that $\(a,b\)$ is a unlocked parabolic affine action, and $a$ is step-2. Suppose that $m \in {\mathcal{C}}_{3}$, $s(m)=2$, and $m$ is the lowest point on its $\bA$-orbit.
For $r$ sufficiently large there exists a constant $c=c(r,A,B)>0$ such that at least one of the following holds: $$
\sum_{k\in {\mathbb{Z}}} \sum_{l\geq 0} |\bA^k \bB^l m |^{-r} \leq C |m|^{- r + 8}, \quad \quad
\sum_{k\in {\mathbb{Z}}} \sum_{l< 0} |\bA^k \bB^l m |^{-r} \leq C |m |^{- r + 8}. $$ \end{lemma}
\begin{proof} Assume without loss of generality that the good sign of $l$ is positive.
Denote $u:=\widehat{A} m$, $v:=\widehat{B} m$ and let $C_{k,l}= k\widehat{A} m +l \widehat{B} m$. Since $u$ and $v$ are non-parallel integer vectors whose sizes satisfy, for some $c_0>0$, $$
1\leq |u|, \, |v| \leq c_0|m|, $$ one can show that the angle $\theta$ between $u$ and $v$ satisfies
$\sin \theta \geq c_1|m|^{-2}$, and hence $\gamma:= |\cos \theta| \leq 1-c_2/|m|^{-4}$. Note that for all $k,l\in {\mathbb{Z}}$ we have the following two inequalities: $$ \begin{aligned}
|C_{k,l}|^2= |k u + l v|^2 = &k^2|u|^2+ l^2 |v|^2 + 2 kl |u||v| \gamma , \\
(k |u| + l |v|)^2 = &k^2 |u|^2+ l^2 |v|^2 +2 kl |u||v| \geq 0. \end{aligned} $$ Hence, $$ \begin{aligned}
|C_{k,l}|^2 &\geq |C_{k,l}|^2 - \gamma (k |u| + l |v|)^2 = (1-\gamma) (k^2 |u|^2+ l^2 |v|^2) \\
&\geq c_2|m|^{-4} (k^2 |u|^2 + l^2 |v|^2)\geq c_2 |m|^{-4} (k^2 + l^2 ). \end{aligned} $$
Now, if $k^2 + l^2 \geq c_3 |m|^8$, then for $|m|>1$ we have
$|m|\leq |C_{k,l}|/2$, and $$
|\bA^k \bB^l m | \geq |C_{k,l}| - |m| \geq
|C_{k,l}|/2 \geq |m| . $$
Finally, we split the desired sum: $$
\sum_{k\in {\mathbb{Z}}} \sum_{l\geq 0} |\bA^k \bB^l m |^{-r} = \Sigma_1 +\Sigma_2, $$
where $\Sigma_1$ contains the terms $|\bA^k \bB^l m |^{-r}$ corresponding to $k^2+l^2\leq c_3 |m|^8$, and $\Sigma_2$ contains those with
$k^2+l^2> c_3 |m|^8$.
The sum $\Sigma_1$ contains $\leq 4 c_3 |m|^8$ terms.
By Lemmas \ref{l_growth_1} and \ref{l_growth_2}, for a certain $c=c(r,A,B)$ we have for all $m$: $$
|v_{k,l} |^{-r} \leq c |m|^{- r}, $$ so
$$
\Sigma_1\leq c_4 |m|^{- r + 8} .
$$
We estimate $ \Sigma_2$ by comparison with an integral:
$$ \begin{aligned}
\Sigma_2 = &\sum_{k^2+l^2\geq |m|^8 } |v_{k,l} |^{-r}
\leq c_5 |m|^{2r} \sum_{k^2+l^2\geq |m|^8 } (k^2 + l^2 )^{-r/2} \\
\leq &c_5 |m|^{2r} \int_{x^2+y^2\geq |m|^8 } (x^2 + y^2 )^{-r/2}\, dxdy \leq c_6 |m|^{-2r+8} \leq c_6 |m|^{- r + 8}.
\end{aligned}
$$
The combination of the estimates for $ \Sigma_1$ and $ \Sigma_2$ provides the desired result. \end{proof}
The following subsections contain the proof of Proposition \ref{lemma.double.sums.case3} for the case $s(m)\geq 3$. We assume that $m\in {\mathcal{C}}_3$, $s=s(m)\geq 3$, and study the growth properties of $|\bA^k\bB^lm|$, given by formula \eqref{AkBl_s>2}: $$ v_{k,l}:=\bA^k\bB^lm = m+k\widehat{A} m + \sum_{j=1}^{t} c_j l^j\widehat{B}^j (m+k \widehat{A} m)+ \sum_{j=t+1}^{s-1} c_j l^j \widehat{B}^j m . $$
\subsubsection{The case $s(m)\geq 3$: Estimate for small $l$.} Recall the notation ${\delta} = {\delta} (m) = 0.99 \frac1{s} $ from Sec.~\ref{s_notations}.
\begin{lemma} \label{lemma.small.l} Assume that $\(a,b\)$ is unlocked parabolic affine action, and $a$ is step-2. For any ${\xi} >0$, there exists $c=c(A,B,{\xi})>0$ such that for any
$m\in {\mathcal{C}}_3$ with $s(m)\geq 3$, being the lowest point on its $\bA$-orbit, we have for any $k\in {\mathbb{Z}}$ and for any $ | l |<|{\xi} m|^{{\delta}}$: \begin{equation} \label{eq.double}
|\bA^k \bB^l m|\geq c |m|^{{\delta}}. \end{equation}
\end{lemma}
\begin{proof} Assume that ${\xi}=1$; the same proof holds for any for any ${\xi} >0$. Denote $v_{k,l}:=\bA^k\bB^lm$ for brevity.
First consider ''large'' $m$, such that $|m|>C_0$ for an appropriate constant $C_0$. For this $m$, suppose by contradiction that $|v_{k,l}|< |m|^{{\delta}}$ for some $|l|\leq |m|^{\delta}$. By assumption, $m$ is the lowest point on its $\bA$-orbit, so $|m+k\widehat{A} m|\geq |m|$ for any $k\in {\mathbb{Z}}$. Applying inductively $\widehat{B}^{s-j}$, $j=1,\dots s-1$ to equation \eqref{AkBl_s>2}, we get for a certain constant $C=C(A,B)$: $$
|v_{k,l}-(m+k\widehat{A} m)|\leq C |m|^{s{\delta}}\leq C |m|^{0.99}. $$
If $m$ satisfies $|m|\geq (2C)^{100}:=C_0$, then the latter implies $$
|v_{k,l}-(m+k\widehat{A} m)| \leq |m|/2. $$
Since $|m+k\widehat{A} m|\geq |m|$, we conclude that
$|v_{k,l}|\geq |m|/2$ which is in contradiction with our assumption that $|v_{k,l}|\leq |m|^{\delta}$.
Thus, we have proved the desired estimate for all $|m|\geq (2C)^{100}$.
If $|m|< (2C)^{100}$, the estimate is achieved by the choice of a sufficiently small constant $c(A,B)$. \end{proof}
\subsubsection{The case $s(m)\geq 3$: Linear Drift in $l$} This is the section where the "good sign of $l$" for the given $m\in{\mathbb{Z}}^d$ plays the crucial role.
\begin{lemma}[Linear Drift in $l$, $s(m)\geq 3$] \label{lemma.drift.l} Assume that $\(a,b\)$ is unlocked parabolic affine action, and $a$ is step-2. There exists a constant $c=c(A,B)>0$ such that for any $m \in {\mathcal{C}}_3$, $s(m)\geq 3$, the following holds: for all $k\in {\mathbb{Z}}$ and either for all $l\geq 0$ or for all $l<0$ we have: $$
|\bA^k \bB^l m|\geq c|l|. $$ \end{lemma}
\begin{proof}
Recall that $s=s(m)\geq 3$, which means that $\widehat{B}^s m = 0$, $\widehat{B}^{s-1}m\neq 0$. In particular, $\widehat{B}^{2}m\neq 0$.
By Lemma \ref{lemma.AB2}, the assumption on being unlocked implies $\widehat{A} \widehat{B}^{s-1}m=0$. Denote $$ {\mathcal{V}}_m=\text{span\,}\{ \widehat{B}^{l} m,\ \widehat{A} \widehat{B}^{l'} m \mid l\in [2,s-1], \ l'\in [0,s-2] \}, $$ where the terms $\widehat{A} \widehat{B}^{l'} m$ may vanish starting from some $l'=t$, $t\geq 1$. Let $m^\bot$ and $(\widehat{B} m)^\bot$ denote the orthogonal projections of $m$ and $\widehat{B} m$, respectively, onto the orthogonal complement of ${\mathcal{V}}_m$.
Let us show that for some constant $c=c (A,B)>0$ we have
$$|(\widehat{B} m)^\bot|\geq c >0. $$ We consider two subcases.
\noindent {\bf Case $\widehat{A}\widehat{B}^{s-2}m=0$. \ } Here we have: $$ 0\neq \widehat{B}^{s-1} m = \widehat{B}^{s-2} (\widehat{B} m)^\bot. $$
Since $\widehat{B}^{s-1}m\neq 0$ is an integer, we have $|\widehat{B}^{s-1}m |\geq 1$. Since the norm of $\widehat{B}^{s-2} $ is bounded away from zero, we have
$|(\widehat{B} m)^\bot | \geq c_0 (A,B)>0$.
\noindent {\bf Case $\widehat{A}\widehat{B}^{s-2} m \neq 0$. \ } In this case we have $$ 0\neq \widehat{A} \widehat{B}^{s-2} m = \widehat{A} \widehat{B}^{s-3} (\widehat{B} m) = \widehat{A} \widehat{B}^{s-3} (\widehat{B} m)^\bot. $$
Since $\widehat{A}\widehat{B}^{s-2} m \neq 0$ is an integer, we have $|\widehat{A} \widehat{B}^{s-2} m |\geq 1$. Since the norm of $\widehat{A} \widehat{B}^{s-3}$ is bounded away from zero, this implies $| (\widehat{B} m)^\bot|\geq c_1 (A,B) > 0$.
To complete the proof, recall that, by \eqref{AkBl_s>2}, $$ \begin{aligned}
| v_{k,l}|=&|\bA^k\bB^lm| = |m+k\widehat{A} m + \sum_{j=1}^{t-1} c_j l^j\widehat{B}^j (m+k \widehat{A} m)+ \sum_{j=t}^{s-1} c_{j} l^{t} \widehat{B}^{j} m| \\
\geq &|m^\bot+l(\widehat{B} m)^\bot| . \end{aligned} $$ Choose the "good sign of $l$" to be positive if the vectors $m^\bot$ and $l(\widehat{B} m)^\bot$ form an acute angle, and negative otherwise. For this sign of $l$ we get the desired result. \end{proof}
\subsubsection{The case $s(m)\geq 3$: Drift in $k$}
Recall the notations $s=s(m)$ and ${\delta}={\delta} (m)$ from Section~\ref{s_notations}.
\begin{lemma}[Drift in $k$] \label{lemma.drift.k} Assume that $\(a,b\)$ is unlocked parabolic affine action, and $a$ is step-2. There exist positive constants ${\xi}={\xi}(A,B)$ and $C=C(A,B)$ such that for any $m \in {\mathcal{C}}_3$,
for any $k,l$ satisfying $|k|\geq {\xi} |m| $, $|l|\leq |k|^{{\delta} }$ with ${\delta}={\delta} (m)$ defined in Section~\ref{s_notations}, we have: $$
|\bA^k \bB^{l} m| \geq C |k|^{{\delta} }. $$ \end{lemma}
\begin{proof}
Let $s=s(m)$ be the step of $m$, defined in Section~\ref{s_notations}. Since ${\delta}={\delta}(m)=0.99/s$, condition $|l|\leq |k|^{{\delta} }$ implies $|k|\geq |l|^s$. Let $p\in [0,s-1]$ be the largest integer such that $\widehat{A} \widehat{B}^p m\neq 0$, and observe that from \eqref{AkBl_s>2} and \begin{equation} \widehat{B}^p \bA^k \bB^l m=\widehat{B}^p m+k\widehat{A} \widehat{B}^p m+{\mathcal{O}}(l^{s-1}), \label{aabb} \end{equation}
where ${\mathcal{O}}(l^{s-1})$ denotes the terms free from $k$ with the maximal power of $l$ being $s-1$. If we assume that $|k|\geq {\xi} (A,B)|m|$ with ${\xi}(A,B)$ sufficiently large, then the linear term in $k$ is dominant in \eqref{aabb} so that $|\widehat{B}^p \bA^k \bB^l m|\geq |k|/2$, thus $|\bA^k \bB^l m|\geq C|k|$ for a certain positive constant $C$. \end{proof}
\subsubsection{Proof of Proposition \ref{lemma.double.sums.case3}, case $s(m)\geq 3$}\label{sec_C3_proof} We now turn to the effective control of the double sums.
\begin{proof} Assume without loss of generality that the good sign of $l$ is positive. Let ${\xi}={\xi}(A,B)>0$ be the constant from Lemma \ref{lemma.drift.k}, and let ${\delta}={\delta}(m)=0.99/s(m)$, as before. We split the sum into the following five partial sums, each of which will be estimated separately: $$ \begin{aligned}
\sum_{k\in {\mathbb{Z}}} \sum_{l\geq 0} |\bA^k \bB^l m|^{-r} = &\left( \sum_{|k|\leq |{\xi} m|,} \sum_{l\leq | {\xi} m|^{{\delta} }} +
\sum_{|k| \leq |{\xi} m|,} \sum_{l > | {\xi} m|^{{\delta} }} + \right.\\
& \left. \sum_{|k| > | {\xi} m|,} \sum_{l > | k|^{{\delta} }}+
\sum_{|k| > |{\xi} m|,} \sum_{l < |{\xi} m|^{{\delta} } } +
\sum_{|k|> | {\xi} m|,} \sum_{|{\xi} m|^{{\delta} }\leq l\leq |k|^{{\delta} }} \right) |\bA^k \bB^l m|^{-r} \\ &:= \Sigma_1+\Sigma_2+\Sigma_3+\Sigma_4+\Sigma_5 . \end{aligned} $$
The following elementary estimate is used several times below: for any $p, r>0$, we have: $$
\sum_{j \geq p} j^{-r} \leq c_0(r) p^{-r+1}. $$
\noindent {\bf Estimate of $\Sigma_1$. \ } By Lemma \ref{lemma.small.l}, there exists $c=c(A,B)$ such that for all
$l\leq |{\xi} m|^{\delta}$ and for all $k\in {\mathbb{Z}}$
we have $ |\bA^k \bB^l m |\geq c |m |^{{\delta}}$. The sum $\Sigma_1$ contains $\leq 3 |{\xi} m |^{{\delta} +1}$. Hence,
$$
\Sigma_1 \leq 3 |{\xi} m|^{{\delta} + 1} (c |m|^{{\delta}})^{-r} < c_1 |m|^{- r {\delta} +2}.
$$
\noindent {\bf Estimate of $\Sigma_2$ and $\Sigma_3$. \ }
By Lemma \ref{lemma.drift.l}, there exists $c=c(A,B)$ such that
for all $k\in {\mathbb{Z}}$ and for all $l\geq 0$ we have: $|\bA^k \bB^l m | \geq c l$.
In the case of $\Sigma_2$ we have $|k|\leq {\xi} |m|$, so
$$
\Sigma_2 \leq 3 {\xi} |m| \, c^{-r} \sum_{l> |{\xi} m|^{{\delta}}} l^{-r} \leq c_2 |m|^{- r{\delta} +2} .
$$
In the case of $\Sigma_3$ we have:
$$
\Sigma_3 \leq 3 c^{-r} \sum_{k> {\xi} |m |, } \sum_{l > k^{{\delta}}} l^{-r} \leq
\tilde c_3
\sum_{k> {\xi} |m | } (k^{{\delta}})^{-r+1} \leq c_3 |m |^{-r{\delta}+2}.
$$
\noindent {\bf Estimate of $\Sigma_4$ and $\Sigma_5$. \ }
By Lemma \ref{lemma.drift.k}, if $|k| > {\xi} |m |$ and $0\leq l\leq |k |^{{\delta}}$, then $|\bA^k \bB^l m |\geq C|k|^{{\delta}}$. Therefore, $$
\Sigma_4 \leq C |{\xi} m |^{{\delta}} \sum_{k > {\xi} | m |} k ^{-{\delta} r} \leq c_4 |m |^{-{\delta} r +2}, $$ and $$
\Sigma_5 \leq \sum_{l \geq |{\xi} m |^{{\delta}},} \sum_{k\geq l^{1/{\delta}}} k^{-r{\delta}}
\leq c_5 \sum_{l \geq | {\xi} m |^{{\delta}}} ( l^{1/{\delta}} )^{-r{\delta} +1} \leq c_5 |m |^{-r{\delta} +2}. $$ Recall that for any $m\in{\mathbb{Z}}^d$ we have ${\delta}(m)=0.99/s(m)\geq \eta=0.99/S$, where $S$ is the step of $B$. Therefore,
$|m |^{-r{\delta} +2}\leq |m |^{-r\eta +2}$. Summing up the above estimates, we obtain the desired result. \end{proof}
\section{Solution of the linearized problem. Proof of Proposition \ref{Main iteration step} }\label{plan_of_proof}
A way of interpreting the statement of Proposition \ref{Main iteration step} is the following. A perturbation $\(F, G\)$ of the action $\(a, b\)$ defines a map ${\bf p}: \mathbb Z^2\to \rm Vect^{\infty}(\mathbb T^d)$ by ${\bf p}(k, l):= F^kG^l-a^kb^l$, $(k, l)\in \mathbb Z^2$. Proposition \ref{Main iteration step} (in fact) claims that there exists a tame map which projects ${\bf p}$ to the space of (twisted) coboundaries over $\(a, b\)$ in such a way that the complement of this projection has quadratic estimates with respect to ${\bf p}$. How is commutativity going to give us that the error we make while projecting is quadratically small? Commutativity relations for all action elements tell us that certain linear operator $\bfL{\bf p}$ (see \S \ref{vf}) defined on ${\bf p}$ is bounded (roughly) by the size of the square of ${\bf p}$. So the core of the problem is to produce a projection of ${\bf p}$ to the space of (twisted) coboundaries over $\(a, b\)$ so that the complement of this projection (the error we are making) can be bounded by the size of $\bfL{\bf p}$. This is done in Sections \ref{Fourier} and \ref{s_end_prop}. The final \S \ref{proof_of_main_iteration_step} contains the proof of Proposition \ref{Main iteration step}. It is in this proof that we use the fact that the commutativity assumption implies that $\bfL{\bf p}$ is quadratically small with respect to ${\bf p}$.
This interpretation of the statement of Proposition \ref{Main iteration step} is useful for understanding its proof. Namely, even though the statement of Proposition \ref{Main iteration step} contains only $\bff$ and $\bfg$ (in the notations of this section it means that $\bff={\bf p}(1,0)$ and $\bfg={\bf p}(0,1)$), in order to produce the estimates, {\it we need to use the whole map} ${\bf p}: \mathbb Z^2\to \rm Vect^{\infty}(\mathbb T^d)$, not just $\bff$ and $\bfg$. We explain this point more in \S \ref{p} after the statement of Proposition \ref{prop_main_estimate} that contains the main estimates on the conjugacy and the error.
The plan of the proof of Proposition \ref{Main iteration step} is the following. We start by constructing projections to coboundaries for {\it function-valued} maps $p: \mathbb Z^2\to C^{\infty}(\mathbb T^d)$. The main result that leads to Proposition \ref{Main iteration step} is Proposition \ref{prop_main_estimate} which we state in \S \ref{p} and prove in \S \ref{Fourier}. Proposition \ref{prop_main_estimate} contains the crucial estimates for the convergence of the iteration process. In \S \ref{s_end_prop}
we use Proposition \ref{prop_main_estimate}
to deduce the corresponding statement,
Proposition \ref{...}, for truncations of $p: \mathbb Z^2\to C^{\infty}(\mathbb T^d)$, which we then inductively apply to obtain Proposition \ref{vector fields} for the truncated {\it vector field-valued} map ${\bf p}: \mathbb Z^2\to \rm Vect^{\infty}(\mathbb T^d)$.This passage from a function-valued map $p$ to a vector field-valued map ${\bf p}$ is quite direct due to the fact that our action $\(a, b\)$ has a {\it parabolic} linear part.
Similar inductive argument has been used in all the other works which use KAM method for parabolic actions (\cite{D}, \cite{DK2}, \cite{ZW}, \cite{DT}. Finally, the main result for vector field-valued maps ${\bf p}$ (Proposition \ref{vector fields}) is used in \S \ref{proof_of_main_iteration_step} to prove the main iterative step, Proposition \ref{Main iteration step}. \color{black}
\color{black}
\subsection{Approximating $p: \mathbb Z^2\to C^{\infty}(\mathbb T^d)$ by a coboundary}\label{p}
We start with a set of functions, $p:{\mathbb{Z}}^2 \to C^\infty(\mathbb T^d)$. Recall the definitions of $ \partial_ {k,l}(h) $ and $ Lp((k,l), (s,t)) $ from \S \ref{s_notations}. Here we introduce some extra notations. For a fixed natural number $N$ we define $\res_N$ to be the set consisting of all the resonant pairs corresponding to the resonant vectors of norm less than $N$ (see Lemma \ref{l_tech_reson} for the bound on the norm of resonant pairs with respect to the resonant vector, and for the definition of constant $C$ which appears in the definition below). In other words, \begin{equation}\label{R'}
\res_N:= \{(k, l)\in \mathcal R(A, B):\, \, C(|k|+|l|)<N\}\cup \{(1,0), (0, 1)\}. \end{equation} For the simplicity of notations, we introduce the following norms, for any $r\ge 0$: \begin{equation}\label{maxnorms} \begin{aligned}
\|p\|_r &:= \max\{ \|p(1,0)\|_r, \|p(0,1)\|_r\},\\
\|Lp \|_{r, N}&:=\max\{ \|Lp((1,0), (k, l)\|_{r}, \|Lp((0,1), (k, l)\|_{r}:\, (k,l)\in \res_N\}.
\end{aligned} \end{equation} From this point on, $\kappa$ will denote a constant which depends only on the action $\(a,b\)$ and the regularity $r$, but along the way it will absorb other constants which appear in the estimates.
The main result we prove here is:
\begin{PProp}\label{prop_main_estimate} Let $\(a,b\)$ be an unlocked $({\gamma},\tau)$-Diophantine step-$S$ parabolic affine action, where $a$ is step-2, and let $r>0$. There exist constants $\etan=\etan (S)>0$, $\sigman=\sigman(\(a,b\))$ and $\kappa=\kappa(r,\(a,b\))$ such that
for any map $p: {\mathbb{Z}}^2 \mapsto C^\infty(\mathbb T^d)$ there exists
a $C^\infty$ function $h$ such that \begin{equation}\label{hm}
\|h\|_r\leq
\kappa \|p\|_{r+\sigman},
\end{equation} and the map $\tilde p: {\mathbb{Z}}^2 \mapsto C^\infty(\mathbb T^d)$ defined by \begin{equation}\label{def_p_tilde_global} \tilde p(s,t):= \partial_{s,t} h- p(s,t)+ave(p(s,t)), \,\,\,\, (s,t)\in {\mathbb{Z}}^2, \end{equation}
satisfies the following estimate: \begin{equation}\label{linest}
\|\tilde p(s,t)\|_r\leq \|p(s,t)\|_r+ \kappa(|s|+|t|)^{dr}\|p\|_{r+\sigman}. \end{equation} Moreover, if $p$ is truncated up to $N$, then $\tilde p$ satisfies even the following estimate for any $(s, t)\in \res_N$: \begin{equation}\label{tildepst}
\|\tilde p(s,t)\|_r\le \kappa (|s|+|t|)^{d(\etan r+\sigman)} \|Lp\|_{\etan r+\sigman,N}. \end{equation}
\end{PProp}
The proof of the proposition is lengthy and takes up all of the next section. Here is a short overview of the proof.
We will define $h$ via its Fourier coefficients $h_m$, in different ways depending on $m$. We need to apply different arguments in the following three cases: when the orbit of $m$ under the dual linear action $\(\bar A, \bar B\)$ is a single point, when it is finite under one element of the action (but not under all elements), or when it is infinite. If the orbit is a single point we are in the degenerate case. The second case is when $m$ is resonant, otherwise $m$ is non-resonant. As explained in \S \ref{Diophantine conditions}, to each resonance $m$ we can attach a unique resonance pair $(k, l)$ for which $\bar A^{k}\bar B^{l}m=m$.
The special (degenerate) case when the $\(\bar A, \bar B\)$-orbit of $m$ is a single point, that is when $\bar Am=\bar Bm=m$, is dealt with in the same way as in the original proof of Moser in \cite{M} (see \S \ref{s_case1}).
Next, we have the situation when $\bar Am =m$ and $\bar Bm\ne m$. This is a $(1,0)$ resonance. In this case the fact that $\(A, B\)$ is unlocked implies that $\hat B^2m=0$ (so $m$ is step 2 for $\bar B$). Then we use the generator $b$ to construct $h_m$. To show that $\tilde p$ satisfies the needed estimate in this case, we will need the Diophantine condition on the translation vector ${\alpha}_{1,0}$.
If $\bar Am \ne m$, we use the generator $a$ to construct $h_m$, and use the fact that $\bar A$ is step 2 to estimate $h_m$. To obtain the estimate for $\tilde p$ we use different strategies for resonant and non-resonant $m$.
When $m$ is resonant, we will use the corresponding resonant pair $(k,l)$ and the action element $a^kb^l$ to estimate the error. This is exactly where we need to use all the elements of the action, i.e., the map ${\bf p}: \mathbb Z^2\to \rm Vect^{\infty}(\mathbb T^d)$, and not just two generators $\bff$ and $\bfg$. Moreover, it is here that we will use the Diophantine assumptions on the translation parts $\alpha_{k, l}=a^kb^l-A^kB^l$. To control the number of action elements we use, we need to truncate the given data first. This is why the crucial error estimate in Proposition \ref{prop_main_estimate} is stated only for truncated maps. In our arguments, as explained in Lemma \ref{l_tech_reson}, the norm of the resonant pair will be bounded by the norm of the resonance. Therefore, for the estimate of the $N$-truncated maps, we only need to consider the resonant pairs for the resonances bounded by $N$. The treatment of all the resonant cases is done in \S \ref{s-case2}.
Finally, if the $\(\bar A, \bar B\)$-orbit of $m$ is infinite, we use the double sums estimates. This part of the argument uses \S \ref{double-buble} and is contained in \S \ref{s_case3}.
\subsubsection{Proof of Proposition \ref{prop_main_estimate} }\label{Fourier}
Let us pass to defining and estimating the numbers $h_m$ and $(\tilde p(s,t))_m$. The arguments will strongly depend on $m$. Namely, we always set $h_0=0$, and for $m$ lying in each of the three subsets, ${\mathcal{C}}_{1} $, ${\mathcal{C}}_{2} $ and ${\mathcal{C}}_{3} $, defined in \S \ref{s_notations}, we have to develop a special approach. A more precise statement of Proposition \ref{prop_main_estimate} is the following. \color{black} \begin{proposition}\label{prop_main_estimate_detail} Let $\(a,b\)$ be an unlocked $({\gamma},\tau)$-Diophantine parabolic step-$S$ affine action, where $a$ is step-2, and let $r>0$. There exist constants $\eta>0$ ($\eta=0.99/S)$), $\sigma=\sigma(\(a,b\))$ and $\kappa=\kappa(r,\(a,b\))$ such that, for any map $p: {\mathbb{Z}}^2 \mapsto C^\infty(\mathbb T^d)$ there exists a set of numbers $(h_m)$, $m\in {\mathbb{Z}}^d\setminus \{0\}$ such that \begin{equation}\label{hm_detail}
|h_m|\leq
\kappa \max \{\|p(1, 0)\|_r, \|p(0, 1)\|_r\} |m |^{-r+1+\tau}
\end{equation} with the following property. Define a new map $\tilde p$ from ${\mathbb{Z}}^2$ into the space of formal Fourier series as follows: \begin{equation}\label{def_p_tilde} (\tilde p(s,t))_m:= h_{\bA^s\bB^tm} e(\bA^s\bB^tm,{\alpha}_{s,t})-h_m - (p(s,t))_m, \,\,\, (s,t)\in {\mathbb{Z}}^2. \end{equation} Then it satisfies, for any $ r>8/ \eta $: \begin{equation}\label{tpst}
\begin{aligned}
&| \tilde p(s,t)_m|\leq \\ &{ {\scriptstyle \begin{cases}
\kappa (|s|+|t|)^{d r} \max \{\|Lp((s,t),(1,0))\|_r,\|Lp((s,t),(0,1))\|_r \} |m|^{-\eta r+ 9+ \tau}, & m\in {\mathcal{C}}_1\cup{\mathcal{C}}_3\\
\kappa (|s|+|t|)^{d r }
\max \{ \|Lp((1, 0),(k,l))\|_r, \|Lp((0,1),(k,l))\|_r, \\
\qquad\qquad\qquad\qquad \| Lp((1,0),(s,t))\|_r , \| Lp((0,1),(s,t))\|_r \} |m|^{-r+\tau+2}, & m \in {\mathcal{C}}_2(k,l). \\ \end{cases}} } \end{aligned} \end{equation}
Moreover, for $m \in {\mathcal{C}}_2(k,l)$ we have $|k|+|l|\le C(A,B) |m|$. \end{proposition}
\begin{proof}[\bf \color{bleu1} Proof of Proposition \ref{prop_main_estimate} from Proposition \ref{prop_main_estimate_detail}]
Recall that in \eqref{maxnorms} we defined:
$$\|p\|_r:=\max \{\|p(1, 0)\|_r, \|p(0, 1)\|_r\} . $$ Then estimate \eqref{hm_detail} directly implies estimate \eqref{hm} for $h$ with a loss of $\sigman:=\tau +d+2$ derivatives.
The map $\tilde p$, defined in \eqref{def_p_tilde_global}, satisfies the linear estimate \eqref{linest}, which follows from its definition and estimate \eqref{hm} for $h$:
$$\| \tilde p(s,t)\|_r\le \|\partial_{s,t} h\|_r + \| p(s,t)\|_r\le \|p(s,t)\|_r+ \kappa(|s|+|t|)^{rd}\|p\|_{r+\sigman}. $$
If $p$ is truncated up to $N$, then by taking the maximum on the right hand side of \eqref{tpst} over all resonant pairs $(k, l)\in \res_N$ (which is a finite set), we get for any $(s,t)\in \res_N$ (see definition \eqref{maxnorms}): \begin{equation*}
\begin{aligned}| \tilde p(s,t)_m|&\le \kappa (|s|+|t|)^{d r } \max_{(k, l)\in \res_N}\{
\|Lp((1, 0),(k,l))\|_r, \|Lp((0,1),(k,l))\|_r,\\
& \| Lp((1,0),(s,t))\|_r , \| Lp((0,1),(s,t))\|_r \} |m|^{-\eta r+\tau+9}\\
& \le \kappa (|s|+|t|)^{d r } \|Lp\|_{r, N} |m|^{-\eta r+\tau+9}.
\end{aligned}
\end{equation*}
This implies that $$\sup | \tilde p(s,t)_m| |m|^{\eta r-\tau-9}\le \kappa (|s|+|t|)^{d r } \|Lp\|_{r, N},$$ which (by making a substitution $r:= \eta r-\tau-9$) gives:
$$\sup_m \{| \tilde p(s,t)_m| |m|^{r}\}\le \kappa (|s|+|t|)^{d (\eta^{-1} r+ \eta^{-1}(\tau +9) ) } \|Lp\|_{\eta^{-1} r+ \eta^{-1}(\tau +9) , N}.$$
Because of the well known norm comparison: $\|\tilde p(s,t)\|_r\le C\sup_m \{| \tilde p(s,t)_m| |m|^{r+d+2}\}$, we have: $$
\|\tilde p(s,t)\|_r\le \kappa (|s|+|t|)^{d (\eta^{-1} (r+d+2) + \eta^{-1}(\tau +9) ) } \|Lp\|_{\eta^{-1}( r+d+2)+ \eta^{-1}(\tau +9) , N}. $$
Now let $\etan:=\eta^{-1}$ (recall that $\eta<1$), define the new $\sigman:=\eta^{-1}(\tau +9+d+2)$ to obtain the final estimate $\|\tilde p(s,t)\|_r\le \kappa (|s|+|t|)^{d (\etan r+\sigman)} \|Lp\|_{\etan r+ \sigman , N}.$
\end{proof}
In the rest of this section we prove Proposition \ref{prop_main_estimate_detail}. We will split the proof into three subsections according to $m\in {\mathcal{C}}_1$, $m\in {\mathcal{C}}_2$ or $m\in {\mathcal{C}}_3$.
\subsubsection{Proof of Proposition \ref{prop_main_estimate_detail} in the case $m\in{\mathcal{C}}_1$}\label{s_case1} Let $m\in {\mathcal{C}}_1$, i.e., we have $\bA m=\bB m=m$.
Since the action is assumed to be $(\gamma,\tau)-$Diophantine, we have either
$|e(m,\alpha ) -1|\ge \gamma \|m\|^{-\tau}$, or $|e(m,{\beta} )-1|\ge \gamma \|m\|^{-\tau}$.
Let $f=p(1,0)$ and $g=p(0,1)$.
Define $h_m$ as follows:
\begin{equation}\label{h_in_C1} h_m: = \begin{cases}
(e(\alpha, m)-1)^{-1}f_m, & \text{ if } \ |1-e(\alpha, m)|\ge \gamma \|m\|^{-\tau},\\
(e(\beta, m)-1)^{-1}g_m, & \text{ if } \ |1-e(\alpha, m)|< \gamma \|m\|^{-\tau} . \end{cases} \end{equation}
Then we have the following
\begin{proposition}\label{l_Mosers_trick} Let $\(a,b\)$ be a $({\gamma},\tau)$-Diophantine parabolic affine action, and let a map $p: {\mathbb{Z}}^2 \mapsto C^\infty(\mathbb T^d)$, be given.
For $m\in {\mathcal{C}}_1$, let $h_m$ be defined as in \eqref{h_in_C1}. Then $$
|h_m| \le {\gamma} \max\{\|f\|_r, \|g\|_r\} \, |m |^{-r+\tau}, $$
and for any $(s, t)\in \mathbb {\mathbb{Z}}^2$, the number $(\tilde p(s,t))_m$ defined by formula \eqref{def_p_tilde}, which in this case has the form: $$ (\tilde p(s,t))_m = h_m (e(m, \alpha_{s,t})-1) -(p(s,t))_m, $$ satisfies $$
|(\tilde p(s,t))_m|\le c \max \{\|Lp((s,t),(1,0))\|_r, \|Lp((s,t),(0,1))\|_r \} |m|^{-r+\tau}.
$$
\end{proposition} \begin{proof}
Suppose first that $m$ is such that $|1-e(\alpha, m)|\ge \gamma \|m\|^{-\tau}$, in which case $h_m= (e(\alpha, m)-1)^{-1}f_m$. Then $$ \begin{aligned} \left( Lp((s,t), (1,0)) \right)_m =&\left( \partial_{s,t} f\right)_m - \left( \partial_{1,0} p(s,t) \right)_m \\ =&(e({\alpha}_{s,t},m)-1) \, f _m - (e({\alpha},m )-1) \left( p(s,t)\right)_m = \\ =&(e({\alpha}_{s,t},m)-1) \, h_m(e(\alpha, m)-1) - (e({\alpha},m )-1) \left( p(s,t)\right)_m = \\ =&(e({\alpha},m)-1) \, \left( e({\alpha}_{s,t},m)-1) h_m- \left( p(s,t)\right)_m \right) \\ = & (e({\alpha},m)-1) (\tilde p(s,t))_m . \\
\end{aligned} $$
Estimate $|1-e(\alpha, m)|\ge \gamma \|m\|^{-\tau}$ implies the result. The case when $h_m= (e({\beta}, m)-1)^{-1}g_m$ is treated in the same way.
Directly from the definition of $h_m$ and from the SDC-condition on $\alpha$ and $\beta$ we obtain the bound for $|h_m|$: $$ \begin{aligned}
|h_m|\le &\max \{ |e({\beta}, m)-1|^{-1}, |e(\alpha, m)-1|^{-1}\} \max\{ |f_m|, |g_m|\} \\
\le &\gamma |m|^{\tau} |m|^{-r} \max\{ \|f\|_r, \|g\|_r\}. \end{aligned} $$ \end{proof}
It is straightforward that Proposition \ref{l_Mosers_trick} implies Proposition \ref{prop_main_estimate_detail} in case $m\in {\mathcal{C}}_{1} $.
\subsubsection{Proof of Proposition \ref{prop_main_estimate_detail} in the case $m\in{\mathcal{C}}_2$}\label{s-case2}
Let $\(a,b\)$ be an unlocked parabolic affine action, where $a$ is step-2. Let $m\in {\mathcal{C}}_2(k,l)$, i.e., at least one of $\bA m$ and $\bB m$ is different from $m$, and there exists $(k,l)\in {\mathbb{Z}}^2\setminus \{ 0\}$ such that $\bA^k\bB^lm = m$ (thus $k\widehat{A} m + l\widehat{B} m=0$).
By Lemma \ref{lemma.hA2}, ${\mathcal{C}}_2(k,l)$ can be divided into two sub-cases:
${\mathcal{C}}_2'(k,l)$: \ $\bA m \neq m$;
${\mathcal{C}}_2''(k,l)$: \ $\bA m = m$, while $\bB m \neq m$. Note that in this case, by Lemma \ref{lemma.hA2}, we have $\bB^2 m =m$.
\noindent The sets ${\mathcal{C}}_2'(k,l)$ and ${\mathcal{C}}_2''(k,l)$ are invariant under the action of $\(A,B\)$ due to the commutativity of the action. Consider a map $p: {\mathbb{Z}}^2 \mapsto C^\infty(\mathbb T^d)$. Denote $f= p(1,0)$ and $ g= p(0,1)$.
For $m\in{\mathcal{C}}_2'(k,l)$ we define: \begin{equation}\label{def_h_Case2}
h_m= \begin{cases} \Sigma^{+,A}_m (f), & m \in {\mathcal{M}}(A), \\ \Sigma^{-,A}_m (f), & m\in {\mathcal{N}}(A). \end{cases} \end{equation}
For $m\in{\mathcal{C}}_2''(k,l)$ we define: \begin{equation}\label{def_h_B}
h_m= \begin{cases} \Sigma^{+,B}_m (g), & m \in {\mathcal{M}}(B), \\ \Sigma^{-,B}_m (g), & m\in {\mathcal{N}}(B). \end{cases} \end{equation} To understand our choice for $h_m$, think of $C^{\infty}$-functions $h$ and $f$ satisfying $h\circ a-h=f$. Then the Fourier coefficients are related by $$ (h\circ a-h)_m ={h_{\bA m}} {\lambda}_m^{(1)}-h_m = f_m. $$ Iterating this equality by $\bA$ either in the positive or in the negative direction while multiplying by appropriate constants, one obtains a telescopic sum equal to $h_m$ as given in formula \eqref{def_h_Case2}. The following proposition is the main statement of this section. It is straightforward that it
implies Proposition \ref{prop_main_estimate_detail} in case $m\in {\mathcal{C}}_{2} $.
\def \dd{{dr}}
\begin{proposition}\label{l_est_case2} Assume that $\(a,b\)$ is an unlocked $({\gamma},\tau)$-Diophantine parabolic affine action, where $a$ is step-2. Consider a map $p: {\mathbb{Z}}^2 \mapsto C^\infty(\mathbb T^d)$,
denote $f= p(1,0)$ and $ g= p(0,1)$. Given $m\in {\mathcal{C}}_2(k,l)$ for some $(k,l) \in {\mathbb{Z}}^2 \setminus \{ 0\}$, define $h_m$ as above.
Then there exists $\kappa=\kappa ({\gamma},\tau, A, B ) >0$ such that $$ \begin{aligned}
&|h_m| \leq \kappa \max \{ \|f \|_r , \|g \|_r \}\, |m|^{-r+1}, \\ \end{aligned} $$ and for any $(s, t)\in \mathbb {\mathbb{Z}}^2$, the number $(\tilde p(s,t))_m$ defined by formula \eqref{def_p_tilde}, i.e., \begin{equation*} \tilde p(s,t)_m = h_{\bA^s\bB^tm} e(\bA^s\bB^tm,{\alpha}_{s,t})-h_m - (p(s,t))_m, \end{equation*} satisfies: \begin{equation}\label{est_tp_case2} \begin{aligned}
| (\tilde p(s,t))_m|\leq
\kappa (|s|+|t|)^{\dd }
\max \{ &\|Lp((1, 0),(k,l))\|_r, \|Lp((0,1),(k,l))\|_r, \\
&\| Lp((1,0),(s,t))\|_r , \| Lp((0,1),(s,t))\|_r \} |m|^{-r+\tau+2}. \end{aligned} \end{equation}
In addition, $|k|+|l|\le c |m|$ for a certain $c=c(A,B)>0.$
\end{proposition}
\begin{proof}
Let us present the proof of Proposition \ref{l_est_case2} modulo certain lemmas, that are proved below. First consider $m\in {\mathcal{C}}_2'(k,l)$; the arguments for $m\in {\mathcal{C}}_2''(k,l)$ are similar.
(i) Define $h_m$ by \eqref{def_h_Case2}. The estimate for $|h_m|$ follows from Proposition \ref{c_est_easy} part {\it (3)}.
(ii) We start by proving estimate \eqref{est_tp_case2} for $(s,t)=(1, 0)$. This is done in Lemma \ref{l_formal_f_tildeC2}. Namely, we observe in that Lemma that the error term $\tf_m=\tilde p(1, 0)_m = h_{\bA m} \lambda_m^{(1)} - h_m - f_m$, vanishes for all $m$ except for those $m$ that are lowest in norm on their $\bA$ orbit. In the latter case, we will derive from the commutation relation formula \eqref{eq_sum_C2} that we repeat here $$
( e(m, \akl)-1)\, \tf_{m}= \Sigma^{A}_{m} \left( Lp((1,0), (k,l) )\right) . $$
After that, the right-hand side is bounded above by the norm of $Lp((1,0), (k,l)$ (because $m$ is lowest on its orbit), while the term $| e(m, \akl)-1|$ is bounded below by the Diophantine condition for the resonances. It is here that the Diophantine conditions on the resonances play a crucial role: this condition, combined with formula \eqref{eq_sum_C2}, implies: $$
{ |\tilde f_{m}|} \leq c \|Lp((1,0), (k,l) ) \|_r |m|^{-r+1+\tau}. $$
(iii) Use step (ii) above to prove estimate \eqref{est_tp_case2} for all $(s,t)$. Lemma \ref{l_formal_p_tilde}
derives the estimates on $(\tilde p(s,t))_m$ for any $(s,t)$ from those on $\tf_m=(\tilde p(1,0))_m$ (or on $\tg_m=(\tilde p(0,1))_m$, which will be relevant for $m\in {\mathcal{C}}_2''(k,l)$). We use Lemma \ref{l_formal_p_tilde} with ${\mathcal{K}} =c \|Lp((1,0), (k,l) ) \|_r$ and $\rho={-r+1+\tau}$ to get \eqref{est_tp_case2} for all $(s,t)$.
The arguments for $m\in {\mathcal{C}}_2''(k,l)$ are similar: define $h_m$ by \eqref{def_h_B} and estimate $|h_m|$ with the help of part {\it (3)} of Proposition \ref{c_est_easy};
estimate $|\tg_m|:= |(\tilde p(0, 1)_m)|$ via $\|Lp((0,1), (k,l) ) \|_r$ with the help of Lemma \ref{l_formal_f_tildeC2}, and use it instead of $|\tf_m|$ to get formula \eqref{est_tp_case2} for all $(s,t)$. \end{proof}
The following lemma provides the proof for item (ii) above. Below we write $m= \bm$ to say that $m$ is the lowest (in norm) point on its $\bA$ orbit, and $m\neq \bm$ otherwise. \begin{lemma}\label{l_formal_f_tildeC2} Assume that $\(a,b\)$ is an unlocked $({\gamma},\tau)$-Diophantine parabolic affine action, where $a$ is step-2. Let $m\in {\mathcal{C}}_2(k,l)$. Denote $f= p(1,0)$, $\tilde f_m=(\tilde p(1,0))_m$, $ g= p(0,1)$, $\tg_m=(\tilde p(0,1))_m$, and let $h_m$ be as in \eqref{def_h_Case2}, \eqref{def_h_B}. Then there exists a constant $c=c(r, A,B)>0$ such that for $m= \bm$ we have: $$ \begin{cases}
{| \tilde f_m|} \leq c \|Lp((1,0), (k,l) ) \|_r |m|^{-r+1+\tau} & \text{if } m\in {\mathcal{C}}_2'(k,l), \\
{ |\tilde g_m|} \leq c \|Lp((0,1), (k,l) ) \|_r |m|^{-r+1+\tau} & \text{if } m\in {\mathcal{C}}_2''(k,l). \end{cases} $$ For $m\neq \bm$ we have $\tf_m=\tg_m=0$. \end{lemma}
\begin{proof} Assume that $m\in {\mathcal{C}}_2'(k,l)$, the case $m\in {\mathcal{C}}_2''(k,l)$ being similar. By the definition of $h_m$, we have: $\tf_m = h_{\bA m} \lambda_m^{(1)} - h_m - f_m$, and formally we can express: $$ \tf_m= \begin{cases} \Sigma^{A}_m (f), & m =\bm, \\ 0, & \text{otherwise}. \end{cases} $$ Assume that $m= \bm$. Using the definition of $h_m$, we get: $$ \begin{aligned} (Lp((1,0), (k,l) ))_m & = (\partial_{1,0} p(k,l) )_m - (\partial_{k,l} f )_m \\ & = (\partial_{1,0} p(k,l) )_m - ( e(m, \akl)-1) f_m . \end{aligned} $$ Note that, by the commutativity of $a^kb^l$ and $a$, we get the relation: $A\akl=\akl$. Therefore, the term $ ( e(m, \akl)-1)$ is not changed when $m$ moves along the $\bA$-orbit.
Take the weighted sum $\Sigma^A_m$ on both sides and recall that for $m=\bm$ we have $ \Sigma^{A}_m (f) =\tf_m$: \begin{equation}\label{eq_sum_C2} \Sigma^{A}_m \left( Lp((1,0), (k,l) )\right) = ( e(m, \akl)-1) \, \Sigma^{A}_m(f) = ( e(m, \akl)-1)\, \tf_m. \end{equation}
Since $Lp((1,0), (s,t))\in C^r$, we can use Proposition \ref{c_est_easy}, which gives us: $$
|\Sigma^{A}_m (Lp((1,0), (k,l) ))|\leq \|Lp((1,0), (k,l) ) \|_r |m|^{-r+1} . $$ Using the Diophantine assumption on resonances, we conclude: $$
|\tf_m|\leq \|Lp((1,0), (k,l) ) \|_r |m|^{-r+1} | e(m, \alpha)-1|^{-1} \leq \|Lp((1,0), (k,l) ) \|_r |m|^{-r+1+\tau}. $$ \end{proof}
The following lemma derives the estimates on $(\tilde p(s,t))_m$ for any $(s,t)$ from those on $\tf_m=(\tilde p(1,0))_m$ or $\tg_m=(\tilde p(0,1))_m$, providing the details for item (iii).
\begin{lemma}\label{l_formal_p_tilde} Assume that $\(a,b\)$ is an unlocked parabolic affine action, and $a$ is step-2.
Consider a map $p: {\mathbb{Z}}^2 \mapsto C^\infty(\mathbb T^d)$, and let $\{h_m\}_{m\in \mathcal U}$ be given, where $\mathcal U$ is some $\(\bar A, \bar B\)$-invariant set. Define for every $(s,t)\in {\mathbb{Z}}^2 $, $\{(\tilde p(s,t))_m\}_{m\in \mathbb Z}$ via \begin{equation*} (\tilde p(s,t))_m = h_{\bA^s\bB^tm} e(\bA^s\bB^tm,{\alpha}_{s,t})-h_m - (p(s,t))_m. \end{equation*} Suppose that there exists ${\mathcal{K}}>0$ and $0< \rho\leq r$, such that we have $$
\text{either} \quad |(\tilde p(1,0))_m| \leq {\mathcal{K}} |m|^{-\rho} \text{ for all } m\in \mathcal U, \quad \text{or} \quad |(\tilde p(0,1))_m| \leq {\mathcal{K}} |m|^{-\rho}\text{ for all } m\in \mathcal U. $$ Then there exists a constant $c=c(A,B)>0$ such that for any $(s,t)\in {\mathbb{Z}}^2$ we have: \begin{equation}\label{est_tp_C3}
|(\tilde p(s,t))_m |\le c (|s|+|t|)^{d r}
\left( \max \{ \|Lp((s, t),(1,0))\|_r , \|Lp((s, t),(0,1))\|_r \} +{\mathcal{K}} \right) \, |m|^{-\rho+1}. \end{equation} \end{lemma}
\begin{proof} Suppose first that for all $m$ in an $\bA$-invariant set $\mathcal U$ we have $|(\tilde p(1,0))_m| \leq {\mathcal{K}} |m|^{-\rho}$. Denote $f_m:= (p(1,0))_m$ and $\tf_m:= (\tilde p(1,0))_m$ for brevity.
For the sequence $\{ h_m\}$ we formally have: $$\partial_{s,t} (\partial_{k,l} h )_m=\partial_{k,l} (\partial_{s,t} h)_m. $$ To see this notice that any smooth function $H$ one can verify, using only the commutativity relation $ab=ba$, that $\partial_{s,t} \partial_{k,l} H =\partial_{k,l} \partial_{s,t} H $. Hence, under the commutativity condition, the Fourier coefficients of a smooth function $H$ satisfy for each $m$: $\partial_{s,t} (\partial_{k,l} H)_m=\partial_{k,l} (\partial_{s,t} H)_m$. This implies the desired relation for the sequence of numbers $\{ h_m\}$ (this relation can be also verified directly).
By the definition of the set of numbers $(h_m)$, $$ \begin{aligned} (Lp((1,0), (s,t)))_m &=(\partial_{1,0} p(s,t))_m - (\partial_{s,t} f )_m \\ &=(\partial_{1,0} p(s,t))_m - (\partial_{s,t} \partial_{1,0} h -\partial_{s,t} \tf )_m \\ &= ( \partial_{1,0} \, ( p(s,t) - \partial_{s,t} h ) )_m + (\partial_{s,t} \tf )_m \\ &= (\partial_{1,0} \, \tilde p(s,t))_m + (\partial_{s,t} \tf )_m . \end{aligned} $$ Hence, $$ (\partial_{1,0} \, \tilde p(s,t))_m = (Lp((1,0), (s,t)))_m- (\partial_{s,t} \tf )_m . $$ Since $Lp((1,0), (s,t))\in C^r$ and $\rho \leq r$, we have
$$|(Lp((1,0), (s,t)))_m|\leq \|Lp((1,0), (s,t) ) \|_r |m|^{-\rho}. $$ Let us estimate $(\partial_{s,t} \tf )_m= \tf_{\bA^{s}\bB^{t} m} -\tf_m$.
To bound $|\tf_{\bA^{s}\bB^{t} m}|$ notice that, since the linear part of the action $\(a, b\)$ is parabolic, for some constant $c_0=c_0(A,B)$ we have: $$
\| \bA^{-s} \bB^{-t} \|\leq c_0 (|s|+|t|)^d. $$ Hence, for any $(s, t)$ we have: $$
|m| \leq \| \bA^{-s} \bB^{-t} \|\, | \bA^{s}\bB^{t} m | \leq c_0 (|s|+|t|)^d |\bA^{s}\bB^{t} m|, $$
and thus $|\bA^{s}\bB^{t} m|^{-\rho} \leq c_1 (|s|+|t|)^{dr} |m|^{-\rho}$, and therefore $$ \begin{aligned}
|(\partial_{s,t} \tf )_m |\leq & |\tf_{\bA^s\bB^t m}| + |\tf_{m}| \leq {\mathcal{K}} ( |\bA^s\bB^t m|^{-\rho} +|m|^{-\rho}) \\
&\leq c_2 {\mathcal{K}} (|s|+|t|)^{d r} |m|^{-\rho} \end{aligned} $$ for some $c_1, c_2 >0$ only depending on $(A,B)$. Finally, we obtain: $$ \begin{aligned}
|(\partial_{1,0} \, \tilde p(s,t))_m| \leq &| (Lp((1,0), (s,t)))_m+ (\partial_{s,t} \tf )_m| \\
\leq &\left( \|Lp((1,0), (s,t) ) \|_r + c_2 {\mathcal{K}} (|s|+|t|)^{d r} \right) \, |m|^{-\rho} . \end{aligned} $$ By Proposition \ref{c_est_easy} (2), this implies that $$ \begin{aligned}
|(\tilde p(s,t))_m| \leq &c_3 \left( \|Lp((1,0), (s,t) ) \|_r +c_1 {\mathcal{K}} (|s|+|t|)^{d r} \right) \, |m|^{-\rho+1} \\
\leq &c (|s|+|t|)^{d r} \left( \|Lp((1,0), (s,t) ) \|_r + {\mathcal{K}} \right) \, |m|^{-\rho+1} \end{aligned} $$
for some $c=c(A,B)>0$. The case $|(\tilde p(0,1))_m| \leq {\mathcal{K}} |m|^{-\rho}$ is similar, we just have to use $\partial_{0,1}$, $g_m:= (p(0,1))_m$ and $\tg_m:= (\tilde p(0,1))_m$ instead of $\partial_{1,0}$, $f_m$ and $\tf_m$, respectively.
\end{proof}
\color{black}
\subsubsection{Proof of Proposition \ref{prop_main_estimate_detail} in the case $m\in{\mathcal{C}}_3$} \label{s_case3}
Let $m\in {\mathcal{C}}_3$; denote $f=p(0,1)$.
We define $ h_m$ by \begin{equation}\label{def_h}
h_m= \begin{cases} \Sigma^{+,A}_m (f), & m \in {\mathcal{M}}(A), \\ \Sigma^{-,A}_m (f), & m\in {\mathcal{N}}(A). \end{cases} \end{equation}
\begin{proposition}\label{prop_C3} Assume that $\(a,b\)$ is unlocked step-$S$ parabolic affine action, where $a$ is step-2. Let a map $p: {\mathbb{Z}}^2 \mapsto C^\infty(\mathbb T^d)$ be given. For $m\in {\mathcal{C}}_3$, define $h_m$ by \eqref{def_h}.
There exists a constant $\kappa=\kappa(\gamma , r, A,B)$ such that \begin{equation}\label{est_h_C3}
|h_m| \le \kappa \|f\|_r | m |^{-r+1}, \end{equation} and, defining for each $(s,t)$ the number $(\tilde p(s,t))_m$ as in formula \eqref{def_p_tilde}, i.e., \begin{equation*}\label{C3_def_tp} (\tilde p(s,t))_m = h_{\bA^s\bB^tm} e(\bA^s\bB^tm,{\alpha}_{s,t})-h_m - (p(s,t))_m, \end{equation*} for $\eta=0.99/S$, for any $ r>8/ \eta $, we have the estimate: \begin{equation}\label{est_tp_C33}
|\tilde p(s,t)_m|\le \kappa (|s|+|t|)^{d r} \max \{ \|Lp((1,0),(0,1))\|_r , \|Lp((s, t),(1,0))\|_r \} \, |m|^{-\eta r+ 9}. \end{equation} \end{proposition}
\begin{proof} The proof of this proposition is done in 3 steps similarly to that of Proposition \ref{l_est_case2}. Steps (i) and (iii) of the proof rely on the same lemmas (applied with slightly different constants). The important difference lies in the proof of step (ii) that relies on Proposition \ref{lemma.double.sums.case3} on the control of the double sums along the dual orbit of a lowest point on an $\bar A$-orbit. Here are the steps:
(i) Estimate \eqref{est_h_C3} follows from part {\it (3)} of Proposition \ref{c_est_easy}.
(ii) Based on Proposition \ref{lemma.double.sums.case3}, we will show the following:
\begin{lemma}\label{l_formal_f_tilde} Assume that $\(a,b\)$ is unlocked step-$S$ parabolic affine action, where $a$ is step-2, and let $m\in {\mathcal{C}}_3$. As before, denote $f= p(1,0)$, $\tilde f_m=(\tilde p(1,0))_m$, and let $h_m$ be as in \eqref{def_h}. Then
there exists a constant $c=c(r, A,B)>0$ such that for $\eta=0.99/S$, if $m= \bm$ (i.e., $m$ is the lowest point on its $\bA$ orbit), we have: \begin{equation}\label{est_f_tildeC3}
|\tilde f_m| \leq c \|Lp((1,0),(0,1))\|_r \, |m|^{-\eta r + 8}; \end{equation} if $m\neq \bm$, then $\tf_m=0$. \end{lemma}
(iii) Lemma \ref{l_formal_f_tilde}, followed by Lemma \ref{l_formal_p_tilde} with ${\mathcal{K}}=c \|Lp((1,0),(0,1))\|_r$ and $\rho={-\eta r+ 8}$, implies \eqref{est_tp_C33} for arbitrary $(s,t)$. \end{proof} Thus it only remains now to show Lemma \ref{l_formal_f_tilde}.
\begin{proof}[Proof of Lemma \ref{l_formal_f_tilde}.] By the definition of $h_m$ we have: \begin{equation}\label{def_f_tilde} { \tilde f_m} =h_{\bA m} e(\bA m,{\alpha})-h_m - { f_m }= \begin{cases} -\Sigma^A_m (f), & m= \bm ,\\ 0, & m\neq \bm. \\ \end{cases} \end{equation} Assume that $m= \bm$.
Denote $\phi:=Lp((1,0), (0,1))$ for brevity. Note that for any $m\in {\mathcal{C}}_3$ the definition of $\phi$ implies directly that the following holds in terms of the formal power series: \begin{equation}\label{formula_Sigma} \Sigma_m^A (f) = \sum_{l\geq 0} \sum_{k\in {\mathbb{Z}}} \phi_{\bar A^{k}\bar B^l m}{\lambda}^{(k)}_{m}\mu^{(l)}_{ m} = -\sum_{l\leq -1} \sum_{k\in {\mathbb{Z}}} \phi_{\bar A^{k}\bar B^l m}{\lambda}^{(k)}_{ m}\mu^{(l)}_{ m} . \end{equation}
Since $\phi\in C^r$, for each $m$, its $m$-th Fourier coefficient satisfies $|\phi_{m}| \leq \|\phi \|_r |m |^{-r}$. Hence, $$
| \Sigma_m^A (f) | \leq
\|\phi \|_r \sum_{l\geq 0} \sum_{k\in {\mathbb{Z}}} |\bar A^{k}\bar B^l m |^{-r} =
\|\phi \|_r \sum_{l\leq -1} \sum_{k\in {\mathbb{Z}}} |\bar A^{k}\bar B^l m |^{-r} . $$ The desired estimate \eqref{est_f_tildeC3} in this case follows directly from the estimate of the above double sums, namely it is proved in Proposition \ref{lemma.double.sums.case3} that for $r$ sufficiently large there exists a constant $c=c(r, A,B)>0$ such that at least one of the following holds: $$
\sum_{k\in {\mathbb{Z}}} \sum_{l\geq 0} |\bA^k \bB^l m |^{-r} \leq c |m|^{-\eta r + 8}, \quad \text{ } \quad
\sum_{k\in {\mathbb{Z}}} \sum_{l< 0} |\bA^k \bB^l m |^{-r} \leq c |m |^{-\eta r + 8}. $$\end{proof}
\subsection{Application of Proposition \ref{prop_main_estimate} to truncated functions and vector fields}\label{s_end_prop}
The application of Proposition \ref{prop_main_estimate} to truncated functions is direct. Application to truncated vector fields requires an iteration process as a consequence of the fact that the linear part of the unperturbed action is parabolic. Similar iterative procedure has been used before in \cite{D}, \cite{DK2}.
\subsubsection{Truncated functions}
For a general smooth function $v$ on $\mathbb T^d$ (or a vector field) and for $N\in \mathbb N$ the truncation $T_Nv$ is obtained by cutting off the Fourier series of $v$ with index $\|n\|\ge N$. The residue operator is defined as $R_N:=Id-T_N$.
For a map $p: \mathbb Z^2\to C^\infty(\mathbb T^d)$, and $N\in \mathbb N$, define the truncation $T_Np$ by $(T_Np)(s,t)= T_Np(s,t)$, and $R_Np:= p-T_Np$.
Then the operators $T_N$ and the residue operators $R_N:=Id-T_N$ satisfy the following estimates for all $N\in \mathbb N$ and all $0<r\le r'$:
\begin{equation}\label{truncation_estimates} \begin{aligned}
\|T_Nv\|_{r'}&\le C_{r, r'} N^{r'-r+d}\|v\|_r, \\
\|R_Nv\|_{r}&\le C_{r, r'} N^{r-r'+d}\|v\|_{r'}. \end{aligned} \end{equation} Observe that $ave(T_Nv)= ave(v)$.
\color{black}
The following statement is a direct application of our main technical result, Proposition \ref{prop_main_estimate}, to the truncations.
\begin{PProp}\label{...} Let $\(a,b\)$ be an unlocked $({\gamma},\tau)$-Diophantine parabolic step-$S$ affine action, where $a$ is step-2, and let $r>0$. There exist constants $\etan=\etan (S)>0$, $\sigman=\sigman(\(a,b\))$ and $\kappa=\kappa(r,\(a,b\))$ such that
for any $p: {\mathbb{Z}}^2 \mapsto C^\infty(\mathbb T^d)$ there exists $V: \mathbb Z^2\to \mathbb R$ such that for every fixed $N\in \mathbb N$ and the truncation $q=T_Np$, there exist $h\in C^\infty(\mathbb T^d)$ and $\tilde q:\mathbb Z^2\to C^\infty(\mathbb T^d)$ satisfying $$q(s, t)= \partial_{s, t} h +\tilde q(s,t) + V (s, t), \,\, (s,t)\in \mathbb Z^2,$$ and the following estimates hold: \begin{equation}\label{hqV}
\begin{aligned}
&\|h\|_{r}\leq \kappa \|q\|_{r+\sigman},\\
&\|\tilde q(s,t)\|_r\leq \|q(s,t)\|_r+ \kappa(|s|+|t|)^{rd}\|q\|_{r+\sigma},\\
&\|\tilde q(s,t)\|_{r} \leq \kappa (|s|+|t|)^{d \rn} \|Lq\|_{\etan r+\sigman,N}, \\ & V(s,t)=ave(p(s, t))=ave (q(s, t)), \end{aligned} \end{equation} where $\rn:= \etan r+\sigman$.
\end{PProp}
Note that in the above proposition, $h$ and $\tilde q$ depend on $N$.
\color{black}
\subsubsection{Truncated vector fields}\label{vf} For a map ${\bf p}: \mathbb Z^2\to \rm Vect^\infty(\mathbb T^d)$ and $N\in \mathbb N$, define the truncation $T_N {\bf p}$ by $(T_N {\bf p})(s,t)= T_N {\bf p}(s,t)$, and $R_N {\bf p}:= {\bf p}-T_N {\bf p}$. We have the same estimates for the operators $T_N$ and $R_N$ on vector fields as in \eqref{truncation_estimates}.
For $\bfh\in \rm Vect^\infty(M)$ we define $$ D_{s,t}: \bfh\mapsto \bfh\circ \rho(s,t)-\rho_0(s,t)\bfh, $$ where $(s,t)\in \mathbb Z^2$. The operator $\bfL$ is then defined by \begin{equation}\label{linL} (\bfL{ {\bf p}})((k,l), (s,t))= D_{s, t}{\bf p}(k,l) -D_{k,l}{\bf p}{(s,t) } \end{equation} for any $(k, l), (s,t) \in \mathbb Z^2$.
Recall that the set $\res_N$ is defined in \eqref{R'}. For ${\bf p}: {\mathbb{Z}}^2\to\rm Vect^\infty(\mathbb T^d)$ and $r\ge 0$ let
\begin{align*}\|{\bf p}\|_{r}&:=\max\{ \| {\bf p}(1,0)\|_{r}, {\bf p}(0,1)\|_{r}\}, \\
\|\bfL{\bf p}\|_{r, N}&:=\max\{ \|\bfL{\bf p}((1,0), (k, l))\|_{r}, \|\bfL{\bf p}((0,1), (k, l))\|_{r}:\, (k,l)\in \res_N\}.
\end{align*}
\begin{PProp}\label{vector fields} Let $\(a,b\)$ be an unlocked step-$S$ parabolic $({\gamma},\tau)$-Diophantine affine action, where $a$ is step-2, and let $r>0$. There exist constants $\etan=\etan (S)>0$, $\sigman=\sigman(\(a,b\))$ and $\kappa=\kappa(r,\(a,b\))$ such that
for any ${\bf p}: {\mathbb{Z}}^2 \mapsto \rm Vect^\infty(\mathbb T^d)$ there exists ${\bf V}: \mathbb Z^2\to \mathbb R^d$ such that for every fixed $N\in \mathbb N$ and the truncation $q=T_Np$, there exist $\bfh\in \rm Vect^\infty(\mathbb T^d)$ and $\tilde{{\bf q}}:\mathbb Z^2\to \rm Vect^\infty(\mathbb T^d)$ satisfying $${\bf q}(s,t)= D_{s,t}\bfh+\tilde{{\bf q}}(s,t)+ {\bf V}(s,t), \,\, (s,t)\in \mathbb Z^2,$$
and the following estimates hold \begin{equation}\label{hqVVV}
\begin{aligned}
&\|\bfh\|_{r}\leq \kappa \|{\bf q}\|_{r+\sigman},\\
&\|\tilde{{\bf q}}(s,t)\|_r\leq \|{\bf q}(s,t)\|_r+ \kappa(|s|+|t|)^{rd}\|{\bf q}\|_{r+\sigma},\\
&\tilde{{\bf q}}(s,t)\|_{r} \leq \kappa (|s|+|t|)^{d \rn} \|\bfL{\bf q}\|_{\etan r+\sigman,N}, \\
&\|{\bf V}(1,0)\|\le \|{\bf p}(1,0)\|_0;\, \, \|{\bf V}(0,1)\|\le \|{\bf p}(0,1)\|_0, \end{aligned} \end{equation} where $\rn:= \etan r+\sigman$.
\end{PProp}
Note that in the above Proposition, $\bfh$ and $\tilde{{\bf q}}$ depend on $N$.
\begin{proof}
Since $A$ and $B$ are commuting unipotent matrices, there exists a basis in $\mathbb R^d$ in which both of them are upper triangular. We choose this basis to represent $\bff$, $\bfg$ and $\bfh$ in coordinate form. Also, we use this basis to define the norm of an arbitrary $\bfh$: if $\bfh=(h_1, \dots, h_d)$ in the chosen coordinates, then we define $\|\bfh\|_r=\max\{\|h_i\|_r: \, i=1, \dots, d\}$. For an element $(s,t)\in {\mathbb{Z}}^2$, let $\mathcal A^{(s,t)}$ denote the matrix $A^sB^t$ in the chosen basis. Notice that the matrix $\mathcal A^{(s,t)}$ is upper triangular. Moreover, it has a polynomial growth of coefficients with respect to $(s, t)$, so we can assume that any element of the matrix $\mathcal A^{(s,t)}$ has size at most $C(|s|+|t|)^S$, where $C$ is a constant depending on $A$ and $B$, and $S$ is the step of the action.
Recall that in the beginning of this section, for $(s,t)\in {\mathbb{Z}}^2$, we defined the operator $$D_{s, t}\bfh:=\bfh\circ (a^sb^t)-A^sB^t\bfh.$$ {In the chosen basis, this operator will have the form}
\begin{equation}\label{D}
D_{s, t}\bfh=\bar\partial_{s,t}\bfh + \mathcal A^{(s,t)}\bfh,
\end{equation}
where $\bar \partial_{s, t}$ denotes the "diagonal" operator acting on $\bfh$ coordinatewise by the operator $\partial_{s, t}$:
$$
D_{s,t}=\begin{pmatrix} \partial_{s,t}& a_{12}^{s,t} &a_{13}^{s,t} &\cdots& a_{1d}^{s,t}\\ 0&\partial_{s,t}&a_{23}^{s,t} &\cdots& a_{2d}^{s,t}\\ \dots&\dots&\dots&\dots&\dots\\ 0&0&0&0&\partial_{s,t} \\ \end{pmatrix}. $$ This upper triangular form allows us to construct $\bfh$ inductively, starting from its last coordinate, and then continuing upwards to the first coordinate. At each step of this inductive procedure, we use all the previously constructed coordinates of $\bfh$.
To keep track of the estimates during the induction, it is convenient to consider the operator $L$ in the upper triangular form as well. Thus, for any ${\bf q}$ we let ${\bf q}(s, t)=(q_1(s,t), \dots, q_d(s,t))$ be the coordinates of the vector field ${\bf q}(s, t)$ in the chosen basis, and introduce the following operator:\begin{equation} \label{Lop} \bfL{\bf q}((k, l), (s,t))= \begin{pmatrix} \partial_{k,l}& a^{k,l}_{12} &\cdots& a^{k,l}_{1d}\\ 0&\partial_{k,l} &\cdots& a^{k,l}_{2d}\\ \dots&\dots&\dots&\dots\\ 0&0&0&\partial_{k,l} \\ \end{pmatrix} \begin{pmatrix} q_1(s,t)\\ q_2(s,t)\\ \dots\\ q_d(s,t) \\ \end{pmatrix}- \begin{pmatrix} \partial_{s,t}& a_{12} ^{s,t}&\cdots& a_{1d}^{s,t}\\ 0&\partial_{s,t} &\cdots& a_{2d}^{s,t}\\ \dots&\dots&\dots&\dots\\ 0&0&0&\partial_{s,t} \\ \end{pmatrix} \begin{pmatrix} q_1(k, l)\\ q_2(k, l)\\ \dots\\ q_d(k, l) \\ \end{pmatrix}. \end{equation}
\begin{comment} This gives us that the $i$-th component of $\bfL{\bf q}((k, l), (s,t))$ ($i=1, \dots, d$) looks like this:
\begin{equation}\label{L} \begin{aligned} (\bfL{\bf q}((k, l), (s,t)))_d&=\partial_{k, l}q_d(s,t) - \partial_{s,t} q_d({k,l})= Lq_{d}((k, l), (s,t))),\\ (\bfL{\bf q}((k, l), (s,t)))_{d-1}&= Lq_{d-1}((k, l), (s,t)))+ a_{d-1, d}^{s,t}q_d(k, l)-a^{k,l}_{d-1, d}q_d(s,t),\\ &\dots\\ (\bfL{\bf q}((k, l), (s,t)))_{1}&= Lq_{1}((k, l), (s,t)))+ \sum_{j=2}^da_{1, j}^{s,t}q_{j}(k, l)-\sum_{j=2}^da^{k,l}_{j, d}q_j(s,t).\\ \end{aligned} \end{equation} \end{comment}
Given ${\bf q}(s, t)$, we first take out all of its averages and call the vector of averages by $$ {\bf V}(s, t)=(ave(q_1(s, t)), \dots, ave(q_d(s, t))). $$ Now we can work with ${\bf q}(s, t)$ assuming that its averages are 0. Then the last estimate of Proposition \ref{vector fields} follows directly from the definition of ${\bf V}(1,0)$ and ${\bf V}(0,1)$.
Let us proceed with the inductive construction of $\bfh$. We start with the last coordinate $q_d$. By applying Proposition \ref{...} to $q_d$ we obtain $h_d$ and $\tilde q_d$ such that: \begin{equation}\label{hqVV}
\begin{aligned}
&q_d(s,t)=\partial_{s, t} h_d+\tilde{q}_d(s,t),\\
&\|h_d\|_{r}\leq \kappa \|q\|_{r+\sigman}\le \|{\bf q}\|_{r+\sigman},\\
&\|\tilde{ q_d}(s,t)\|_{r} \leq \kappa (|s|+|t|)^{d\rn}\|{Lq_d}\|_{{\etan r+\sigman}, N}\le \kappa (|s|+|t|)^{d\rn}\|{\bfL{\bf q}}\|_{{\etan r+\sigman}, N}. \end{aligned} \end{equation}
Now we turn to constructing $h_{d-1}$. First, define $h_d'(s, t):= a^{s,t}_{d-1, d}h_d$ for any $(s,t)\in {\mathbb{Z}}^2$ and observe that
\begin{equation} \label{hprime} \|h_d'(s, t)\|_r\le |a^{s,t}_{d-1, d}|\|h_d\|_r\le \kappa (|s|+|t|)^d \| q_d\|_{r+\sigma}.\end{equation}
Since we have put \eqref{D} in the upper triangular form, we have to show the following.
\begin{lemma}\label{step1} There exist $h_{d-1}$ and $\tilde{q}_{d-1}$ such that
\begin{align*} &(q_{d-1}-h_d')(s,t)= \partial_{s,t} h_{d-1} +\tilde{q}_{d-1}(s,t),\\
&\|h_{d-1}\|_{r}\leq \kappa \|{\bf q}\|_{r+2\sigman},\\
&\|\tilde{q}_{d-1}(s,t)\|_r\le
\kappa (|s|+|t|)^{d\rn} N^{d_2} \|\bfL{\bf q}\|_{\etan' r+\sigman', N},
\end{align*} where $d_2:=\max\{d, d\bar r\}$, $\etan':=\etan^2$ and $\sigman':= \etan\sigman +\sigman$. \end{lemma}
\begin{proof} By substituting the approximation of $q_d$ of \eqref{hqVV} into the line $d-1$ of the operator of \eqref{Lop}, we get \begin{align*} &\bfL{\bf q}((k, l), (s,t))_{d-1} =\\ &= Lq_{d-1}((k, l), (s,t)))+ a_{d-1, d}^{s,t}(\partial_{k,l} h_d+\tilde{q}_d(k,l))-a^{k,l}_{d-1, d}(\partial_{s,t} h_d+\tilde{q}_d(s,t)) \\ &=L(q_{d-1}-h_d')((k, l), (s,t)))+ a_{d-1, d}^{s,t}\tilde{q}_d(k,l)-a^{k,l}_{d-1, d}\tilde{q}_d(s,t). \end{align*}
From this equality and the estimates for $\tilde{q}_d$ in \eqref{hqVV}, we obtain the following estimate for all $(k, l), (s,t)\in \res_N$: \begin{equation}\label{1.stepL} \begin{aligned}
&\|L(q_{d-1}-h_d')((k, l), (s,t)))\|_r\le \\
&\le \|\bfL{\bf q}((k, l), (s,t)))_{d-1}\|_r + |a_{d-1, d}^{s,t}|\|\tilde{q}_d(k,l)\|_r+|a^{k,l}_{d-1, d}|\|\tilde{q}_d(s,t)\|_r\\
&\leq \|\bfL{\bf q}((k, l), (s,t)))_{d-1}\|_r+\kappa (|k|+|l|)^{d_2}(|s|+|t|)^{d_2} \|Lq_d\|_{{\etan r+\sigman}, N},
\end{aligned} \end{equation}
where $d_2=\max\{d, d\rn\}$. By taking the maximum of $\|L(q_{d-1}-h_d')((k, l), (s,t)))\|_r$ for $(k, l)\in \res_N$ and $(s,t)\in \{(1,0), (0, 1)\}$, we get from inequatity \eqref{1.stepL}: \begin{equation}\label{1.stepL'}
\|L(q_{d-1}-h_d')\|_{r,N}\le \|\bfL{\bf q}_{d-1}\|_{r,N}+ \kappa N^{d_2}\|Lq_d\|_{\etan r+\sigman, N}. \end{equation} Now we apply Proposition \ref{...} to $q_{d-1}-h_d'$, and find $h_{d-1}$ such that \begin{align*}
&\|h_{d-1}\|_r \leq \\
&\leq \kappa \| q_{d-1}-h_d'\|_{r+\sigma}= \kappa \max\{\|q_{d-1}(1,0)-h_d'(1,0)\|_{r+\sigma}, \|q_{d-1}(0,1)-h_d'(0,1)\|_{r+\sigma}\}\\
&\le \kappa (\| q_{d-1}\|_{r+\sigma}+ \|h_d'\|_{r+\sigma})\le \kappa( \| q_{d-1}\|_{r+\sigma}+\|q_d\|_{r+2\sigma})\\
&\le \kappa \, \|{\bf q}\|_{r+2\sigma}, \end{align*}
where we used the estimate for $\|h_d'\|_{r+\sigma}$ from \eqref{hprime}.
For the new error in coordinate $d-1$ we use the estimate obtained in Proposition \ref{...} and \eqref{1.stepL'} to obtain \begin{align*}
&\|\tilde{q}_{d-1}(s,t)\|_r\le \kappa (|s|+|t|)^{dr} \|L(q_{d-1}-h_d')\|_{{\etan r+\sigman}, N}\\
&\le \kappa (|s|+|t|)^{d\rn} (\|\bfL{\bf q}_{d-1}\|_{\etan r+\sigman,N}+ \kappa N^{d_2}\|Lq_d\|_{\etan (\etan r+\sigman)+\sigman, N})\\
&\le \kappa (|s|+|t|)^{d\rn} N^{d_2} \|\bfL{\bf q}\|_{\etan' r+\sigman', N},\\ \end{align*} where $\etan':=\etan^2$ and $\sigman':= \etan\sigman +\sigman$. \color{black}\end{proof}
The rest of the inductive construction goes along the same lines. Namely, assume that $h_d$, $\tilde{q}_{d}$, $h_{d-1}$, $\tilde{q}_{d-1},\dots h_i$, $\tilde{q}_{i}$ have all been constructed, and assume that they satisfy the following estimates for every $j=i, \dots, d$ and $(s, t)\in {\mathbb{Z}}^2$: \begin{align*}
\|h_j\|_r&\le \kappa \| {\bf q}\|_{r+(d-j+2)\sigma},\\
\|\tilde{q}_{j}(s,t)\|_r&\le \kappa (|s|+|t|)^{d\rn} N^{(j+1)d_2}\|\bfL{\bf q}\|_{\eta_{j+1}r+\sigma_{j+1}, N}, \end{align*} where $\eta'_{j+1}:= \eta\cdot \eta_j$, $\sigma_{j+1}:= \eta\sigma_{j}+\sigma$ and $d^{(j+1)}:=\max\{d, d^{(j)}\}$.
In $(i-1)$-st equation in \eqref{Lop} we substitute all the $q_j$ coordinates with $q_j(s, t)= \partial_{s,t}h_j+\tilde q_j(s, t)$. Then the $(i-1)$-st equation in \eqref{Lop} becomes: \begin{equation*}\label{L_i} \begin{aligned} \bfL&{\bf q}((k, l), (s,t)))_{i-1}= Lq_{i-1}((k, l), (s,t)))+ \sum_{j=i}^da^{s,t}_{1, j}q_{j}(k, l)-\sum_{j=i}^da^{k,l}_{j, d}q_j(s,t)\\ &= Lq_{i-1}((k, l), (s,t)))+ \sum_{j=i}^da^{s,t}_{1, j}(\partial_{k,l}h_j+\tilde q_j(k, l))-\sum_{j=i}^da^{k,l}_{j, d}(\partial_{s,t}h_j+\tilde q_j(s,t))\\ &= L(q_{i-1}-h'_i)((k, l), (s,t))) + \sum_{j=i}^da^{s,t}_{1, j}\tilde q_j(k, l)-\sum_{j=i}^da^{k,l}_{j, d}\tilde q_j(s,t), \end{aligned} \end{equation*} where we defined $h'_i(s, t):=\sum_{j=i}^da^{s,t}_{j, d}h_j$.
Now we do the same as in Lemma \ref{step1}: we apply Proposition \ref{...} to $q_{i-1}-h'_i$ to obtain $h_{i-1}$ and $\tilde q_{i-1}$, such that $$ q_{i-1}-h'_i= \partial_{s,t} h_{i-1}+ \tilde q_{i-1}. $$ The same procedure as in Lemma \ref{step1} gives: \begin{align*}
\|h_j\|_r&\le \kappa \| {\bf q}\|_{r+(d-i+2)\sigman},\\
\|\tilde{q}_{i-1}(s,t)\|_r&\le \kappa (|s|+|t|)^{d\rn}\, N^{(i+1)d_2}\|{\bf L}{\bf q}\|_{{\etan_{i+1}r+\sigman_{i+1}}, N}, \end{align*} where $\eta'_{i+1}:= \etan\cdot \etan_i$, $\sigman_{i+1}:= \etan\sigma_{i}+\sigman$. \color{black}
After passing through all the $d$ steps we redefine the constants $\sigman$ and $\etan$. Namely, will have a multiplicative loss $\etan:= \etan^d$ and an additive loss $\sigman:=\max\{\etan^d\sigman + d\cdot \sigman, (d+2)\sigman\}$ of the number of the derivatives for the newly constructed $\tilde {\bf q}$ with respect to $\bfL{\bf q}$. In other words, for every $j=1, \dots, d$, we have with the newly constructed $\sigman$ and $\etan$ the following estimates: \color{black} \begin{align*}
\|h_j\|_r&\le \kappa \| {\bf q}\|_{r+{\sigman}},\\
\|\tilde{q}_{j}(s,t)\|_r&\le \kappa \,(|s|+|t|)^{d\rn} N^{D_1}\|{\bf L}{\bf q}\|_{\etan r+\sigman, N}, \end{align*} where $D_1=D_1(d, r)$. \color{black} By defining $\bfh= (h_1, \dots h_d)$ and $\tilde {\bf q}=(\tilde q_1, \dots, \tilde q_d)$ we obtain the estimates claimed in the Proposition \ref{vector fields} for $\bfh$ and $\tilde{\bf q}(s,t)$.
The "linear" estimate for $\|\tilde{{\bf q}}(s,t)\|_r$ follows directly from the fact that by construction ${{\bf q}}(s,t)= D_{s,t} \bfh+\tilde{{\bf q}}(s,t)$, so the estimate for $\bfh$ implies: \begin{align*}
\|\tilde{{\bf q}}(s,t)\|_r&\le \|D_{s,t} \bfh\|_r+\|{{\bf q}}(s,t)\|_r\le (|s|+|t|)^{dr}\|\bfh\|_r+\|{{\bf q}}(s,t)\|_r \\
&\le \kappa (|s|+|t|)^{dr}\| {\bf q}\|_{r+\sigma}+\| {\bf q}(s,t)\|_{r}. \end{align*} By denoting $D=d(d+1)$, we obtain the estimates claimed in the Proposition \ref{vector fields}. \end{proof}
\subsection{Proof of Proposition \ref{Main iteration step}} \label{proof_of_main_iteration_step}
Recall that in our setup for a perturbation $\(F, G\)$ of the action $\(a, b\)$, we define the map ${\bf p}: \mathbb Z^2\to \rm Vect^{\infty}(\mathbb T^d)$ by ${\bf p}(k, l):= F^kG^l-a^kb^l$, $(k, l)\in \mathbb Z^2$. With the notations ${\bf p}(1,0)= \bff$ and ${\bf p}(0,1)= \bfg$, the action $\tilde\rho$ is generated by the maps $a+\bff$ and $b+\bfg$.
Let $N$ be fixed, and let ${\bf q}_N=T_N{\bf p}$ be the truncation of ${\bf p}$.
First we define ${\bf V}(k, l)= -ave ({\bf p}(k, l))$. (With a little abuse of notation) denote ${\bf V}={\bf V}(1, 0):= -ave(\bff)$ and ${\bf W}={\bf V}(0, 1)= -ave(\bfg)$. It is then clear that the last estimate in \eqref{main-est} holds.
Now we apply Proposition \ref{vector fields} to ${\bf p}$ and its truncation ${\bf q}_N=T_N{\bf p}$ in order to obtain $\bfh_N$. Then as in Proposition \ref{vector fields} we define $\tilde {\bf q}_N(k, l):={\bf q}_N(k, l)-D_{k, l}\bfh_N +{\bf V}(k,l)$, and so \begin{equation}\label{tildep} \tilde {\bf p}_N(k,l):={\bf p}(k, l)-D_{k, l}\bfh_N +{\bf V}(k,l)=\tilde {\bf q}_N+ R_N{\bf p}. \end{equation} Finally, we let $$ \tilde \bff_N=\tilde{\bf p}_N(1,0), \quad \tilde \bfg_N=\tilde {\bf p}_N(0,1). $$
We begin by estimating ${\bf p}(k,l)=(a+\bff)^k\circ (b+\bfg)^l - a^kb^l$.
\begin{lemma} \label{p(k, l)} For $(k, l)\in {\mathbb{Z}}^2$, \begin{equation} \label{pkl}
\| {\bf p}(k,l)\|_{r}\leq C_r (|k|+|l|)^{3d} \max\, \{ \| \bff\|_{r}, \| \bfg\|_{r} \}. \end{equation}
\end{lemma} \begin{proof}
Developing the expression for ${\bf p}(k,l)$ as ${\bf p}(k,l)=(a+\bff)^k\circ (b+\bfg)^l-a^kb^l= a(a+\bff)^{k-1}\circ (b+\bfg)^l +\bff\circ (a+\bff)^{k-1}\circ (b+\bfg)^l -a^kb^l$ and continuing inductively, one gets $$ \begin{aligned} {\bf p}(k,l)&=\sum_{j=0}^{k-1} s_j \, \bff\circ (a+\bff)^{j} +\sum_{j=0}^{l-1} t_j \, \bfg\circ (a+\bff)^{k} \circ (b+\bfg)^{j}\\ &=\sum_{j=0}^{k-1} s_j \, f\circ (a^j+\xi_j ) +\sum_{j=0}^{l-1} t_j \, g\circ (a^{k}b^j+\eta_j). \end{aligned} $$ Here $s_j$, $t_j$ are appropriate compositions of type $A^iB^j$ with $i\leq k$, $j\leq l$.
The sums contain $|k|+|l|$ terms, each coefficient can be estimated by $\|A^kB^l\|\leq C(|k|+|l|)^S$. The $r$-norm of each of these terms can be estimated by
$c \|A^kB^l \|^r \max\, \{ \| \bff\|_{r}, \| \bfg\|_{r} \} \leq cr(|k|+|l|)^S \max\, \{ \| \bff\|_{r}, \| \bfg\|_{r} \}$. Hence,
$\| {\bf p}(k,l)\|_{r}\leq C_r (|k|+|l|)(|k|+|l|)^{2S} \max\, \{ \| \bff\|_{r}, \| \bfg\|_{r} \} \leq C_r (|k|+|l|)^{3S} \max\, \{ \| \bff\|_{r}, \| \bfg\|_{r} \}$. Since $S\le d$, we can then bound this from above by $C_r (|k|+|l|)^{3d} \max\, \{ \| \bff\|_{r}, \| \bfg\|_{r} \}$. \end{proof}
Let us now move to proving that the main estimates
\eqref{main-est} in of Proposition \ref{Main iteration step} hold for $\bfh_N$, $\tilde \bff_N$ and $\tilde \bfg_N$.
We start with $\bfh_N$. Directly from the first estimate in \eqref{hqVVV} of Proposition \ref{vector fields}, and truncation estimate \eqref{truncation_estimates} it follows that: \begin{equation*} \begin{aligned}
\|\bfh_N\|_{r}&\leq \kappa \|{\bf q}_N\|_{r+\sigma} \le \kappa \, N^\sigma \|{\bf p}\|_{r}.\\
\end{aligned} \label{hhN} \end{equation*}
The "linear" estimate for $\tilde \bff_N$ and $\tilde \bfg_N$ (the third estimate in Proposition \ref{Main iteration step}) follows directly from the estimate for $\bfh_N$:
$$\| \tilde \bff_N \|_r= \|\tilde {\bf p}_N(1,0)\|_r\le \|{\bf p}_N(1,0)-D_{1,0}\bfh_N +{\bf V}\|_r\le C_rN^{D'} \Delta_r.$$ Estimate for $\tilde \bfg_N= \tilde {\bf p}_N(0,1)$ follows exactly in the same way.
Now we will use the fact that $a^kb^l+ {\bf p}(k, l)$ is a commutative action, in order to obtain the quadratic estimate for the error
$\tilde{\bf q}$. This is done in two steps: the first one is to to show that $\|\bfL{\bf p}\|_{r,N}$ is quadratic by using the fact that $a^kb^l+ {\bf p}(k, l)$ is a commutative action.
The second one is to compare $\|\bfL{\bf q} \|_{r,N}$ to $\|\bfL{\bf p} \|_{r,N}$.
Recall that the operator $\bfL$ acts on ${\bf p}$ by the formula $\bfL{ {\bf p}}((k,l), (s,t))= D_{s, t}{\bf p}(k,l) -D_{k,l}{\bf p}{(s,t) }$ for any $(k, l), (s,t) \in \mathbb Z^2$.
\begin{lemma}\label{bfp} For any $N\in \mathbb N$ and $r\ge 0$ the following holds: $$
\|\bfL{\bf p} \|_{r,N}\le C_r N^{4d} \Delta_{r+1}\Delta_0. $$ \end{lemma} \begin{proof} First, we notice that the commutativity of the action $a^kb^l+{\bf p}(k, l)$ implies that $$ (a^kb^l+ {\bf p}(k, l))\circ (a^sb^t+{\bf p}(s,t))=(a^sb^t+{\bf p}(s,t)))\circ (a^kb^l+ {\bf p}(k, l)). $$ Therefore, the operator $\bfL$, besides its linear form, has also a non-linear expression (on the right below), in particular: \begin{equation} \begin{aligned} &D_{(s,t)}{\bf p}(k, l)-D_{k, l}{\bf p}(s,t)=\bfL{\bf p} ((k,l), (s,t))\\ &= {\bf p}(k, l)(a^sb^t+{\bf p}(s,t))- {\bf p}(k, l)(s,t) + {\bf p}(s,t)(a^kb^l+{\bf p}(k,l))- {\bf p}(s,t)(a^kb^l).\\ \end{aligned} \end{equation}
From the non-linear expression for $\bfL{\bf p} ((k,l), (s,t))$ and the classical estimates for compositions (\cite{Hormander} Theorem A.8) we obtain the "quadratic" estimate:
\begin{equation*}\label{quadLest} \begin{aligned}
\|\bfL{\bf p} ((k,l), (s,t))\|_r&=\|{\bf p}(k, l)(a^sb^t+{\bf p}(s,t))- {\bf p}(k, l)(a^sb^t)\|_r \\
&+ \|{\bf p}(s,t)(a^kb^l+{\bf p}(k,l))- {\bf p}(s,t)(a^kb^l)\|_r\\
&\le C_r N^d (\|{\bf p}(k, l)\|_{r+1}\|{\bf p}(s,t)\|_0+\|{\bf p}(k, l)\|_{0}\|{\bf p}(s,t)\|_{r+1})\\
&\le C_r N^{d} (|k|+|l|)^{3d} (|s|+|t|)^{3d}\Delta_{r+1}\Delta_0.\ \end{aligned} \end{equation*}
Recall that $\|\bfL{\bf p}\|_{r, N}:=\max\{ \|\bfL{\bf p}((1,0), (k, l))\|_{r}, \|\bfL{\bf p}((0,1), (k, l)\|_{r}:\, (k,l)\in \res_N\}$ and that for $ (k,l)\in \res_N$ we have that $C(|k|+|l|)\le N$.
From the above inequality, by taking maximum over $(k,l)\in \res_N$, we get the required estimate for $\|\bfL{\bf p}\|_{r, N}$. \color{black} \end{proof}
Next we will use the following basic estimate of the norm of the operator $D_{k,l}$:
\begin{lemma}\label{dkl} For $ (k,l)\in \res_N$ and $r\ge 0$ $$
\|D_{k,l}{\bf p}(s,t)\|_r\le C_r N^{dr} \|{\bf p}(s,t)\|_r . $$ \end{lemma} \begin{proof}
To show this bound we only need to observe that for any affine map $a: x\to Ax + \alpha$ and any function $\phi$ the norm $\|\phi \circ a\|_r$ is bounded above by $\|A\|^r\|\phi\|_r$. Since in the operator $D_{k,l}$ we compose with $a^kb^l$, which has linear part $A^kB^l$, and the norms of matrices $A^kB^l$ are (up to a positive constant) bounded by $(|k|+|l|)^d$, then $ (k,l)\in \res_N$ directly implies the required estimate, because for $ (k,l)\in \res_N$, $C(|k|+|l|)\le N$ by Lemma \ref{l_tech_reson}. \end{proof}
\begin{lemma}\label{bfq} For any $N\in \mathbb N$ and $r'\ge r\ge 0$ the following holds:
$$\|\bfL{\bf q}\|_{r,N}\le C_{r} N^{r+5d} \Delta_{1}\Delta_0 + C_{r, r'} N^{r-r'+2d_2} \Delta_{r'},$$ where $d_2=d_2(r):=max\{d, dr\}$.
\end{lemma}
\begin{proof}
Because $\bfL$ is a linear operator we have the following: \begin{equation}\label{linLL} \bfL{\bf p} ((k,l), (s,t))={\bf L}(T_N{\bf p}) ((k,l), (s,t)) + {\bf L}(R_N{\bf p}) ((k,l), (s,t)). \end{equation}
The term $ {\bf L}(R_N{\bf p}) ((k,l), (s,t))$ for $ (k,l)\in \res_N$ and $(s,t)\in \{(1,0), (0,1)\}$ is estimated by using the estimates on the truncation operators, \eqref{truncation_estimates}, the linear form \eqref{linLL} of $\bfL$ and Lemmas \ref{dkl} and \ref{p(k, l)} (again, recall that for $ (k,l)\in \res_N$ we have that $C(|k|+|l|)\le N$): \begin{equation*} \begin{aligned}
\|{\bf L}(R_N{\bf p}) ((k,l), (s,t))\|_r&\le \|D_{s,t} R_N{\bf p}(k,l)-D_{k,l} R_N{\bf p}(s,t)\|_r\\
&\le C_r(|s|+|t|)^{dr}\|R_N{\bf p}(k, l)\|_r+C_r(|k|+|l|)^{dr}\|R_N{\bf p}(s,t)\|_r \\
&\le C_{r,r'}(N^{r-r'}\|{\bf p}(k, l)\|_{r'}+ N^{dr} N^{r-r'}\|{\bf p}(s,t)\|_r) \\
&\le C_{r,r'}(N^{r-r'+3d}\|{\bf p}\|_{r'}+ N^{r-r'+dr}\|{\bf p}\|_{r'})\\
&\le C_{r,r'}N^{r-r'+d_3}\|{\bf p}\|_{r'}, \end{aligned} \end{equation*} where $d_3=d_3(r):=\max\{3d, dr)\}$.
Using this and \eqref{linLL}, as well as \eqref{truncation_estimates} and Lemma \ref{bfp}, we obtain for $0\le r''\le r\le r'$ we have: \begin{equation*} \begin{aligned}
\| \bfL{\bf q} ((k,l), (s,t)) \|_r&\le \|\bfL{\bf p} ((k,l), (s,t))\|_r+ \|\bfL(R_N{\bf p}) ((k,l), (s,t))\|_r\\
&\le \|T_N\bfL{\bf p} ((k,l), (s,t))\|_r+ \|R_N\bfL{\bf p} ((k,l), (s,t))\|_r\\
&\,\, \,\quad + \|\bfL(R_N{\bf p}) ((k,l), (s,t))\|_r\\
&\le C_{r, r''} N^{r-r''+d} \|\bfL{\bf p} ((k,l), (s,t))\|_{r''}+ C_{r, r'}N^{r-r'+d} \|\bfL{\bf p} ((k,l), (s,t))\|_{r'} \\ &\,\, \,\quad +C_{r,r'}N^{r-r'+d_3} \Delta_{r'}\\ &\le C_{r, r''} N^{r-r''+5d} \Delta_{r''+1}\Delta_0 + C_{r,r'}N^{r-r'+d_3} \Delta_{r'}, \\ \end{aligned} \end{equation*}
where we applied the estimate $C(|k|+|l|)\le N$ that is valid for all $ (k,l)\in \res_N$.
Now by setting $r''=0$, and by taking the maximum over $ (k,l)\in \res_N$ and $(s,t)\{(1,0), (0,1)\}$, we obtain the required estimate for $\|\bfL{\bf q}\|_{r, N}$. \end{proof}
From the third estimate in Proposition \ref{vector fields} and Lemma \ref{bfq}, for some fixed constant $D>0$, for any $r'>0$ and for a fixed $d_2=d_2(\sigma)$, we have: \begin{equation*} \begin{aligned}
\|\tilde {\bf q}_N(1,0)\|_0&\le \kappa N^D \|{\bf Lq}_N\|_{\sigma, N}\\ &\le \kappa N^{D+5d+\sigma} \Delta_{1}\Delta_0+C_{r'}N^{{D}-r'+\sigma+ d_3} \Delta_{r'}\\ & \le \kappa N^{D'} \Delta_{1}\Delta_0+C_{r'}N^{-r'+D'} \Delta_{r'}, \end{aligned} \end{equation*} where we define $D'=D+5d+d_3+\sigma$. Notice that $d_3=d_3(\sigma)$, and consequently $D'$, depends only on the action $\(a,b\)$ and $d$. Recall that in \eqref{tildep} we defined $\tilde {\bf p}_N=\tilde {\bf q}_N+ R_N{\bf p}$ for every $N$. By combining the above estimate and the truncation estimates \eqref{truncation_estimates} for the operator $R_N$, we get a similar bound, with possibly new constants $\kappa$ and $C_{r'}$: \begin{equation*}
\|\tilde \bff_N\|_0=\|\tilde {\bf p}_N(1,0)\|_0\le \kappa N^{D'} \Delta_{1}\Delta_0+C_{r'}N^{-r'+D'} \Delta_{r'}. \end{equation*} Finally, we declare the new $D$ to be the $D'$. Estimates for $\tilde \bfg_N= \tilde {\bf p}_N(0,1)$ are proved in exactly the same way. This completes the proof of all the estimates claimed in Proposition \ref{Main iteration step}.
$\Box$
\noindent {\sc \color{bleu1} Acknowledgment.} The authors are grateful to Livio Flaminio and Giovanni Forni for stimulating discussions on the subject. The first author was supported by the Swedish Research Council grant VR 2019-04641. The second author was supported by the NSF grant DMS-2101464.
\end{document} | arXiv |
palgrave communications
Statistical reliability analysis for a most dangerous occupation: Roman emperor
Joseph Homer Saleh1
Palgrave Communications volume 5, Article number: 155 (2019) Cite this article
Complex networks
Popular culture associates the lives of Roman emperors with luxury, cruelty, and debauchery, sometimes rightfully so. One missing attribute in this list is, surprisingly, that this mighty office was most dangerous for its holder. Of the 69 rulers of the unified Roman Empire, from Augustus (d. 14 CE) to Theodosius (d. 395 CE), 62% suffered violent death. This has been known for a while, if not quantitatively at least qualitatively. What is not known, however, and has never been examined is the time-to-violent-death of Roman emperors. This work adopts the statistical tools of survival data analysis to an unlikely population, Roman emperors, and it examines a particular event in their rule, not unlike the focus of reliability engineering, but instead of their time-to-failure, their time-to-violent-death. We investigate the temporal signature of this seemingly haphazardous stochastic process that is the violent death of a Roman emperor, and we examine whether there is some structure underlying the randomness in this process or not. Nonparametric and parametric results show that: (i) emperors faced a significantly high risk of violent death in the first year of their rule, which is reminiscent of infant mortality in reliability engineering; (ii) their risk of violent death further increased after 12 years, which is reminiscent of wear-out period in reliability engineering; (iii) their failure rate displayed a bathtub-like curve, similar to that of a host of mechanical engineering items and electronic components. Results also showed that the stochastic process underlying the violent deaths of emperors is remarkably well captured by a (mixture) Weibull distribution. We discuss the interpretation and possible reasons for this uncanny result, and we propose a number of fruitful venues for future work to help better understand the deeper etiology of the spectacle of regicide of Roman emperors.
What did a Roman emperor have in common with a gladiator? The latter had better odds of surviving a fight than the former had of avoiding a violent death.
Popular culture traditionally associates the lives of Roman emperors with luxury, cruelty, and debauchery, sometimes rightfully so. One missing attribute in this list is, surprisingly, that this mighty office was most dangerous for its holder, as the statistics will show. Consider the following: of the 69 rulers of the unified Roman Empire, from Augustus (d. 14 CE) to Theodosius (d. 395 CE), 43 emperors suffered violent death, that is 62%, either by assassination, the most common mode of death, suicide, or during combat with a foreign enemy of RomeFootnote 1 (Fig. 1). To put this statistic in context and better appreciate its magnitude, compare it with what is considered nowadays a seriously dangerous activity: Himalaya mountaineering. Climbers that summit above 8000 m in the Himalayas have a risk of death of about 4%, a relatively consistent figure for the past 50 years (Amalberti et al., 2005). Roman emperors had an order of magnitude greater risk of violent death than these intrepid climbers.
Deaths and failure modes of the rulers of the unified Roman Empire.
The odds of survival for a Roman emperor were roughly equivalent to playing the Russian roulette with a six-chambered revolver, in which the participant places not one but four bullets, spins the cylinder to randomize the outcome, and pulls the trigger with the muzzle against his head.
This sinister comparison conjures the notion of random variable, a central protagonist in our story. That the likelihood of violent death for an emperor was high was well-known, if not quantitatively at least qualitatively as far back as Edward Gibbon's publication of the first volume of his Decline and Fall of the Roman Empire (1776). In discussing the highly energetic emperor Aurelian (ruled from 270 to 275 CE), nicknamed restitutor orbis, restorer of the world or the unity of the Roman empire in the troubled third century (Watson, 1999), one can almost feel Gibbon's disappointment when reflecting on the emperor's murderFootnote 2:
Such was the unhappy condition of the Roman emperors, that, whatever might be their conduct, their fate was commonly the same. A life of pleasure or virtue, of severity or mildness, of indolence or glory, alike led to an untimely grave; and almost every reign is closed by the same disgusting repetition of treason and murder [emphasis added].
He then added, "the Roman senators heard, without surprise, that another emperor had been assassinated in his camp".
It is worth taking a little historical detour at this point, back to the time before this track record of treason and murder Gibbon refers to was even started, to revisit a popular landmark quotation and understand it in a new light, Julius Caesar's "Iacta alea est"…
Of dice and men. And Roman emperors
As Caesar contemplated his decision to cross the shallow Rubicon river with his legion, a capital offense for himself and his soldiers, he is said to have exclaimed, according to Suetonius, "iacta alea est" (Suetonius, 1998). The expression is commonly translated as "the die is cast", or more resolutely, "let the die be cast", and it is taken to signify the passing of the point of no return in an irrevocable course of action. For anyone familiar with the game of dice, which Roman soldiers and several emperors were fond of, the emphasis of the exclamation is less on the passing of the point of no return, and more on the aleatory nature of the possible outcome. What is the likelihood that the outcome will be, say, odd? Or a particular number given a four-sided or six-sided dice? What outcomes did Caesar contemplate? There is obviously the fate of his undertaking: will the Roman Republic endure, or is this the last nail in its coffin? What will his own fate be? Will he be successful, or will he meet a violent death? In crossing the Rubicon, Caesar significantly narrowed down the scope of possible outcomes he could face.
Beyond this original throw of dice, Caesar provided a metaphor, not just for himself, but for every subsequent emperor: upon confirmation by the senate, or being chosen by the legions or the praetorian guards, every emperor could have exclaimed, "iacta alea est" and was in effect throwing a dice for his life. In retrospect, we now know that collectively the rulers of the unified Roman Empire had a 62% chance of dying a violent death. This is roughly the equivalent of associating four outcomes, for example {1, 2, 3, 4} with this gruesome end, and rolling a six-sided fair dice. Only those who rolled a 5 or 6 got to die of a natural death. This is a seemingly more benign but equally potent metaphor than the previous Russian roulette.
What is not known, however, and has never been examined to date is another random variable associated with these rulers, their time-to-failure or time-to-violent-death. A brief discussion of reliability engineering is in order to better understand this idea and the focus of this work.
Reliability engineering and Roman emperors: from time-to-failure to time-to-violent-death
Reliability is a popular concept that has been celebrated for years as a commendable attribute of a person or an equipment. Although many words in the English language have been coined by or attributed to Shakespeare, it seems that we owe the word reliability to another English author, the poet Samuel Coleridge (1772–1834). In praising a friend, Coleridge wrote (Coleridge, 1983; Saleh and Marais, 2006):
He inflicts none of those small pains and discomforts which irregular men scatter about them and which in the aggregate so often become formidable obstacles both to happiness and utility; while on the contrary he bestows all the pleasures, and inspires all that ease of mind on those around him or connected with him, with perfect consistency, and (if such a word might be framed) absolute reliability.
Since then, reliability engineering has developed into an important discipline that pervades many aspects of our modern, technologically intensive world (Hoyland and Rausand, 2009). The foundational idea in reliability engineering is that the time-to-failure, Tf, of an item is stochastic in nature, it is a random variable. Roughly speaking, reliability, S(t), is defined as the probability that an item is still operational at time t; it has not failed and is till performing its function up to this time:
$$S\left( t \right) \equiv {\mathrm{Pr}}\left( {T_{\mathrm {f}}\; > \;t} \right)$$
S(t) is also known as the survival or survivor function. Reliability engineering is then concerned with quantifying this probability over the life of an item. This is sufficient for our purposes. Another useful concept in reliability engineering is that of failure rate. It is less important for our story but is worth mentioning. While reliability is the complement of the cumulative distribution function (CDF) of the random variable time-to-failure (Tf), the failure rate λ(t) is its conditional probability density function defined as
$$\lambda \left( t \right) \equiv \frac{{\Pr \left( {\left. {t < T_{\mathrm {f}} \le t + {\mathrm {d}}t} \right|T_{\mathrm {f}} > t} \right)}}{{{\mathrm {d}}t}} = - \frac{1}{{S\left( t \right)}}\frac{{{\mathrm {d}}S\left( t \right)}}{{{\mathrm {d}}t}}$$
In words, it is the likelihood per unit time that a failure will occur between t and t + dt given that it has not yet occurred by the time t.
What does this have to do with Roman emperors? It is not that they were unlike Coleridge's friend, it is that we can treat their time-to-violent death as a random variable, and examine it with the same tools of reliability engineering, or more broadly with the statistical tools of life data analysis—the double entendre (life) is not intended but unavoidable here.
Life data analysis, also known as survival analysis, refers to the analysis and probabilistic modeling of the time-to-event as a random variable. The event can be broadly defined, and several academic disciplines have adopted this statistical tool for their particular interests. For example, beyond reliability engineering where the interest is in the time to failure of an item, survival analysis is also broadly used in medical research and public health, where the interest is in, for example, the time to development of a disease, the time to remission after treatment, or the time to death after prognostic. More details can be added to further qualify these events. Survival analysis is also increasingly used in quantitative political science and sociology to examine, for example, the time to break-down of cease fires, the time to landing a first job after graduation, or the time to having one's first child.
The next section extends the application of these statistical tools to an unlikely population, Roman emperors, and it examines a particular event in their rule, not unlike the focus of reliability engineering, but instead of their-time-to-failure, their time-to-violent-death.
The data for this work was obtained for De Imperatoribus Romanis, a peer-reviewed online encyclopedia of Roman emperors [DIR] (De Imperatoribus Romanis DIR, 2019). The entry for each emperor was written by a leading scholar in the field. The Imperial Index provides the list of the rulers of the united Roman empire, along with the many usurpers who unsuccessfully claimed the mantle. The data is fairly standard and generally accepted, and consequently it was considered beyond the scope of the present work to subject it to further quality control. Only legitimate emperors confirmed by the Roman senate are here consideredFootnote 3.
For example, after the troubled period following the murder of both CommodusFootnote 4 (d. 192 CE), and Pertinax (d. 193 CE), a notorious event in the history of Rome took place: the praetorian guards effectively sold the empire to the highest bidder, Didius Julianus. This shameful event came to be known as the "auction of the empire", and it was neither the first time nor the last that the praetorians acted as emperor-makers (the first such instance was the selection of Claudius following the murder of Gaius Little-Boots or Caligula, as he was nicknamed by the soldiers of his famed father, Germanicus). Even though Didius Julianus was considered a usurper by his successor, Septimius Severus and others, the fact that he was proclaimed by the senate makes him a "legitimate" emperor for our purposes. As a side note, Didius Julianus is noted for two memorable feats: the manner in which he became emperor, and for holding one of the shortest reigns for an emperor, 66 days, before he was executed.
Another point with the data that should be mentioned is censoring. Censoring in a statistical sense occurs when life data for the analysis of a set of items is "incomplete". This situation occurs frequently in medical research and reliability engineering, and it can happen because some individuals under study are lost to follow-up, or some items are removed prior to the observation of failure or the event of interest. In our case, about 38% of the individuals are (right-)censored by virtue of having died of illness or old age. But two emperors, Diocletian and Maximianus, are censored through another improbable mechanism: abdication. Diocletian rose through the military ranks to become emperor in 284 CE, and he put an end to the imperial crisis that plagued much of the third century, since the murder of the young Alexander Severus (d. 235 CE). Diocletian also undertook many reforms, administrative and military, which gave the Roman empire another lease on life for centuries to come, at least in the East. Then, in 305 CE he abdicated and devoted himself to gardening for the rest of his life at his palace in Split on the beautiful Croatian coastFootnote 5 (d. 316 CE). For the purpose of this study therefore, Diocletian, and his co-emperor Maximianus, who also abdicated, perhaps unwillingly, contribute censored life data to this analysis. The fact that Maximianus was killed or committed suicide 5 years after he had abdicated does not qualify him to be counted among the emperors who met a violent death since he was no longer one.
The quality and limitation of the sources has to be acknowledged, even for such major events as the death or assassination of an emperor. The sources for the ancient history of interest here are unevenly distributed. They are patchy at times, sometimes contradictory, and often proceed with innuendos and inferences. For example, Suetonius relates several rumors that Caligula, or an attendant, may have poisoned or smothered the emperor Tiberius (d. 37 CE) after the latter had fallen ill. The extent of hatred this emperor inspired across all social classes in Rome may have contributed to these rumors. But the likelihood that Tiberius's death was the result of foul play is slim, and the general consensus tilts away from this possibility. However, this lack of certainty in things related to ancient history has to be contended with, especially in the cases of dubious deaths of emperors when competing narratives are available. A similar example occurs with Claudius (d. 54 CE), but the general consensus in this case is that he was indeed poisoned by his wife Agrippina to expedite the ascension of her son, Nero. For the purpose of this work, Claudius is therefore considered to have met a violent death, whereas Tiberius is not.
Two more cases are worth noting, both for their strangeness and for the parallel they offer with the deaths of Tiberius and Claudius. Numerianus was acclaimed Augustus in 283 CE upon his father's death. He was married to the daughter of the prefect of the praetorian guard, Flavius Aper. Upon returning from Syria, the young emperor fell ill, and his entourage, including Aper, let it be known that the emperor had an eye inflammation, and therefore had to travel in a closed litter. For several days, no one checked on the emperor until the soldiers waiting on him noticed the smell of decay. They opened the litter only to find the decomposing body of the emperor. An imperial commander accused Aper of this sordid deed and cut him down. The soldiers then proclaimed this commander, Valerius Diocles, emperor, and he would become Diocletian, one of the most capable emperors, militarily and administratively. So what are we to make of this story? Numerianus dies under mysterious circumstances. But was it Aper's ambition to eliminate his son-in-law and assume the mantle in his stead? Was Diocles more cleverly devious and eliminated in one stone both the emperor and the Praetorian Prefect to become emperor himself? Or was Numerianus simply unlucky and succumbed to his illness, after which events just unfolded haphazardly? There can only be speculation at this point, and these issues cannot be resolved. For the purpose of this work, Numerianus's death is treated like that of Tiberius not Claudius, that is, it is not considered the result of foul play. A similar situation occurred some 40 years earlier, with the suspicious death of the child emperor Gordian III (d. 244 CE). The machinations of his newly appointed Praetorian Prefect Julius Philippus, later the emperor Philip the Arab, are suspected in this deed. These again are speculation, and for the purpose of this work, Gordian III death is treated like that of Tiberius. The statistical results that follow therefore err on the side of caution and are most likely conservative.
Aside from these vexing little problems with the data, the rest of the statistical analysis in this work is straightforward: it starts with the nonparametric Kaplan–Meier estimator for handling censored data, then proceeds with the maximum-likelihood method for estimating the parameters of a particular probability distribution functionFootnote 6. Emperors who met a violent death within the first year of their rule are assigned a 0.5-year reign, since the exact number of days is not always accurately known. After the first year of rule, the time is rounded up to the year when death occurred. The data is provided in the appendix in Table A1.
In this section, we investigate the temporal signature of this seemingly haphazardous stochastic process that is the violent death of a Roman emperor. We also examine whether there is some structure underlying the randomness of this process or not, and we discuss parallels with results in reliability engineering.
Non-parametric and parametric analyses, and interpretation
The data in Table A1 is treated with the Kaplan–Meier estimator, and the results are provided in Fig. 2. The longest rule was that of Augustus, who established the principate, as the early empire was called. He ruled for 45 years, and since he died peacefully of old age, he contributes censored data to the analysis. The results in Fig. 2 are shown up to this last time-to-violent death.
Survivor (reliability) function of Roman emperors as a function of their time in office.
Figure 2 reads as follows: at the 3-year mark for example, an emperor had 64% chance of not having met a violent death; at the 7-year mark, those chances drop to 50%, a mere coin toss. The likelihood of a violent death is the complement of these figures, 36% and 50%, respectively.
What does this mean? Three salient features in this figure are important to note:
Emperors faced a significantly high risk of violent death in the first year of their rule. This risk remained high but progressively dropped over the next 7 years. This is reminiscent of infant mortality in reliability engineering, a phase during which weak components fail early on after they have been put into service, often because of design or manufacturing defects for example. Roman emperors therefore experienced a form of infant mortality;
The reliability or survivor function stabilizes by the 8th year of rule. The emperors could lower their guard a bit if they made it to 8 years…
… but not for long: the risk of violent death picks up again after 12 years of rule. This suggests that new mechanisms or processes drove another round of murder. This is reminiscent of wear-out period in reliability engineering, a phase during which the failure rate of components, especially mechanical items, increases because of fatigue, corrosion, or wear-out. Roman emperors therefore also experienced wear-out mortality.
A Weibull plot for the previous nonparametric results is provided in Fig. 3. The plot displays ln{−ln[S(t)]} as a function of ln(t). If the data points obtained are aligned, it can be concluded that the data effectively arises from a Weibull distribution, that is, the underlying parametric distribution giving rise to this violent-death process is indeed a Weibull. The Weibull survivor function is widely used in reliability engineering and survival analysis because it is a highly flexible parametric model, and it is given by
$$S\left( t \right) = {\mathrm {exp}}\left[ { - \left( {\frac{t}{\theta }} \right)^\beta } \right]$$
β is termed the shape parameter, and θ the (temporal) scale parameter or characteristic life. A shape parameter β < 1 is indicative of or reflects the prevalence of infant mortality in the items under study, whereas β > 1 indicates the prevalence of wear-out failures (decreasing versus increasing failure rates, respectively). Equation (3) is equivalent to a linear Weibull plot (Eq. (4)):
$${S}\left( {t} \right) = {{\mathrm {exp}}}\left[ { - \left( {\frac{{t}}{\theta }} \right)^\beta } \right] \Leftrightarrow \ln \left\{ { - \ln \left[ {{S}\left( {t} \right)} \right]} \right\} = \beta \ln \left( {t} \right) - \beta \ln \left( \theta \right)$$
Weibull plot of the survivor function of Roman emperors, and linear least square fit (R2 = 0.962).
It is interesting that a stochastic process as unconventional and haphazardous as the violent death of a Roman emperor—over a long four-century period and across a vastly changed world—has a systematic underlying structure, and is remarkably well captured by a Weibull distribution. The fact that this result is completely otiose does not diminish its uncanniness!
Why this underlying structure? In statistical theory, the Weibull function is an extreme value distribution, which captures the minimum value of a large collection of random observations. To clarify this point, consider for example a system with a large number n of components placed in series to fulfill a specific function. The failure of any one component results in the failure of the system, its function is no longer provided. The time to failure of the system is therefore the minimum time to failure of any one of its component. Statistical extreme value theory tells us that regardless of the underlying failure distribution of the components, when n is very large, the time to failure of the system approaches a Weibull distribution. Notice in this example the difference between component level and system level considerations, and how the result at the aggregate system level is independent of the failure distribution of any one component. Extreme value theory is also applicable in another related context: consider a single monolithic item. There are no components in this item. But assume that there are n different competing failure processes of this item, whichever one occurs first breaks the item. When n is very large, this will also result in a Weibull distribution of the time to failure of the item regardless of the distribution of each failure process.
The extension of these observations to the violent death of Roman emperors has to be done cautiously. But they offer nonetheless a fruitful venue for exploration. The fact that the time signature of the stochastic process of interest here is remarkably well captured by a Weibull distribution suggests that it is perhaps indeed the result of a very large number of underlying processes conspiring to violently eliminate the emperor. The fact that there were many pathways to the violent death of an emperor, with large numbers of individuals and motivations for undertaking the grisly task, makes the Weibull, an extreme value distribution, theoretically plausible in this case.
Mixture Weibull distributions
A closer inspection of Fig. 3 shows two distinctive slopes for the data points, before and after ln(t) ≈ 2.5, which corresponds to the onset of the wear-out failures seen in Fig. 2. A mixture Weibull distribution is therefore fitted to the data, and the maximum-likelihood estimates of its parameters are provided as follows:
$$\begin{array}{l}\widehat S\left( t \right) = 0.876 \cdot {\mathrm {exp}}\left[ { - \left( {\frac{t}{{12.835}}} \right)^{0.618}} \right] + 0.124 \cdot {\mathrm {exp}} \left[ { - \left( {\frac{t}{{14.833}}} \right)^{13.387}} \right]\end{array}$$
Equation (5) provides an analytical confirmation of the previous observations, that Roman emperors experienced both infant mortality (β = 0.618) and wear-out mortality (β = 13.387) in the form of violent death. This parametric result is shown in Fig. 4.
Mixture Weibull survivor (reliability) function of Roman emperors, and the nonparametric results.
The emperors who experienced infant mortality were not unlike engineering components that suffer early failures after they are put to use: weak by design or fundamentally incapable of meeting the demands of their environment and circumstances. Examples from each century abound, for example Galba (d. 69 CE), Pertinax (d. 193 CE), Macrinus (d. 218 CE), and Severus II (d. 307 CE). These were times of upheaval, and in the first two cases, these turned out to be times of transition to new dynasties (the Flavian, and the Severan, respectively). Emperors' infant mortality can be seen, in part, as both causes and consequences of times of crisis and instabilityFootnote 7.
The emperors who experienced wear-out mortality met their end through different failure mechanisms. Consider first that some engineering components experience an uptake in failures (wear-out failures) after they have been in service for a long time. They may have been sturdy at first and benefitted from clement operational environments to start with. But through degradation, fatigue, or increased harshness in their operational environment, they begin to experience wear-out failures. The emperors who survived the first 8 years of their rule, as seen in Fig. 2, had a grace period of about 4 years. Violent death came to them afterward (wear-out mortality) because, for instance, their old enemies had regrouped or new ones emerged, because they had alienated an increasing number of parties, or because new weaknesses in the imperial rule appeared or grew. These new murderous processes clearly had a different temporal signature than those driving the emperors' infant mortality, as seen in Fig. 3 and in the different characteristic life parameters of each Weibull distribution in Eq. (5). For example, the death of Domitian after a 15-year rule (d. 96 CE), or Commodus after a 12-year rule (d. 192 CE), or Gallienus after a 15-year rule (d. 268 CE) are illustrative of wear-out mortalityFootnote 8.
The failure rate (Eq. (2)) of the parametric fit (Eq. (4)) is given in Fig. 5. The result shows a remarkable bathtub-like curve, a model widely used, and empirically confirmed in reliability engineering for a host of mechanical and electronic components. Roman emperors, like these engineering items, therefore experienced a bathtub-like failure rate.
Failure rate of Roman emperors (parametric fit of the time-to-violent-death).
The results in Fig. 5 lends themselves to an interesting interpretation:
The decreasing failure rate early on, the signature of infant mortality, reflects as noted previously a prevalence of weak emperors who were incapable at the onset of their rule to the handle the demands of their environment and circumstances. The fact that the failure rate was decreasing though suggests a competition between antagonistic processes, on the one hand those that sought to violently eliminate emperors (elimination), and on the other hand those that reflected the emperors learning curve to better protect themselves and perhaps eliminate their opponents (preservation). Examples abound in Roman history of this competition. Up to the first 12 years of one's rule, the preservation processes steadily improved their performance, and the situation can be casually summarized as "whatever didn't kill them [the Roman emperors] made them stronger" or less likely to meet a violent death;
The increasing failure rate after 12 years of rule, the signature of wear-out failures, reflects as noted previously an uptake in failures through degradation with time, fatigue, or increased harshness in their circumstances. A growing mismatch between capabilities and demands under changing (geo-)political circumstances. This can be due to a number of reasons discussed previously. The fact that the failure rate was increasing after this 12-year mark suggests again a competition between the same antagonistic processes noted in (i), and this time the preservation ones were on the losing end of this competition. This result can be causally summarized as "whatever didn't kill them made them weaker" after a 12-year rule.
Beyond these specific details, what does it mean to find a coherent structure within a stochastic process of historical nature as the one here examined? Roughly speaking, the result implies the existence of systemic factors and some level of determinism, in an average sense or expected value, superimposed on the underlying randomness of the phenomenon here examined. In other words, the process is not completely aleatory; it has some deterministic factors overlaid on its randomness. Conan Doyle, in Sherlock Holmes: The Sign of Four, expressed this general idea rather accurately when he wrote:
While the individual man is an insoluble puzzle, in the aggregate he becomes a mathematical certainty. You can, for example, never foretell what any one man will do, but you can say with precision what an average number will be up to.
The results in this section suggest a similar idea underlies the violent death of Roman emperors.
Etiology and suggestion for future work
The previous subsections investigated the temporal signature of the phenomenon here examined, the violent death of emperors, a spectacle of brutality and violence not unlike the gladiatorial games, except it stretched over four centuries and affected the entire Roman world (Millar, 1977).
What has not been explored is the etiology or causal basis of this phenomenon, or why emperors repeatedly met violent deaths in the first place, not just temporally how. The immediate causes of violent deaths of Roman emperors are frequently discussed in the literature. They can be found for example in Scarre's (1995) "Chronicle of Roman Emperors", and a short summary is provided in Retief and Cilliers' (2005) "Causes of death among the Caesars (27 BC–AD 476)". The entries include statements such as "murdered by the sword/dagger […]", "poisoned by [name of individual]", or "decapitated by the soldiers". These explanations are of little interest, and they do not reflect the complex nature of causality in this context. The causal basis of the phenomenon here examined intersects a number of fundamental issues in Roman history, the development and pathologies of the Roman monarchy for example, the problem of imperial succession, the role of the praetorian guard, the loyalties of the legions, and the geographic extent and resources of the empire, to mention a few. These issues and the complex nature of their relations with the phenomenon here examined are left as a fruitful venue for Roman historians to examine. It is worth noting that the the spectacle of regicide of Roman emperors is related a reciprocal way, as both a causal factor and a consequence, to the decline and fall of the Roman empire. As such, it deserves careful attention in future work.
On his deathbed, Augustus called for a mirror, examined himself, and had his hair combed. Then, as Suetonius recountsFootnote 9:
he called in his friends and asked whether it seemed to them that he had played the comedy of life well. He added: since well I have played my part, all clap your hand, and from the stage dismiss me with applause.
Marcus Aurelius closed his Meditations on a similar, albeit more somber note, of life as a theater and actors sometimes getting dismissed from the stage after fewer than the whole five acts (Meditations, tr. 1989) "It does not matter", he adds stoically, "whether he beholds the world a longer or shorter time". This is a fitting metaphor for all Roman emperors, and it serves to frame the scope of this work, namely for how long the emperors "beheld the [Roman] world" before they were dismissed from the stage, violently if that was the case. Edward Gibbon offered a similar view for the entirety of the history of decline and fall of the Roman empire:
By a philosophic observer, the system of Roman government might have been mistaken for a splendid theater, filled with players of every character and degree, who repeated the language and imitated the passion of their original model.
This work began in jest by comparing Roman emperors with gladiators, and it noted that the odds of survival of the former were worse than those of the latter. There is perhaps more to this comparison than meets the eye. There was a particular appeal to gladiatorial games in the Roman world (Fagan, 2011). Whatever its reasonsFootnote 10, it is undeniable that these games offered a spectacle of extreme brutality, like an unscripted theatrical play with violence as the main protagonist, and gladiators the creative agents of its delivery. Roman emperors performed in similar games, except instead of delivering their role in single afternoon, they took several years, sometimes only a few months to complete it before they were dismissed from the stage. They also faced more diverse hazards, and stealthier adversaries than those encountered by the gladiators in the arena. Incidentally, the emperor Commodus would blur the line of this analogy and go down into the arena and fight gladiators (as well as wild beats).
In examining their time-to-violent-death, this work found that of Roman emperors experienced infant mortality as well as wear-out failures. Their failure rate displayed a bathtub curve, similar to that of a host of mechanical engineering items and electronic components. More interestingly, it was found that a stochastic process as unconventional and haphazardous as the violent death of a Roman emperor has a definite underlying structure, and is remarkably well captured by a Weibull distribution. The interpretation and possible reasons for this result were discussed. Some fruitful venues for future work were proposed to help understand the deeper etiology of the violent deaths of Roman emperors.
In seeking to uncover the causal basis of the spectacle of imperial regicide, one important causal factor should not be overlooked. It was briefly hinted at earlier in the section "Data and methods", when Diocletian was being urged to reclaim the purple a few years after he had abdicated, he replied that he would never trade the peace and happiness of his garden "for the storms of a never-satisfied greed". Since Augustus and Marcus Aurelius conceived of life as a play, we leave it to a Greek playwright, Euripides, as quoted by Plutarch in the Life of Sulla, to present this final causal factor in the phenomenon here examined:
[…] passing afterwards through a long course of civil bloodshed and incurable divisions proved Euripides to have been truly wise and thoroughly acquainted with the causes of disorders in the body politic, when he forewarned all men to beware of Ambition, as of all the higher Powers [it is] the most destructive and pernicious to her votariesFootnote 11.
A fundamental engine of the spectacle of regicide was not structural in nature, nor within the legions or the flawed system of government of the empire, but intrinsic to the actors themselves: they were votaries of Ambition on the stage of one of the most consequential adventure in human history that was the Roman empire. The individuals should not be neglected in future work. It was the mutual interactions between the motivations and ambitions of individuals on the one hand, Vespasian, Septimus Severus, and Diocletian for example, and the social, political, and military factors on the other hand that led to spectacle of regicide of Roman emperors. The whole was subject to historical contingencies and some level of randomness in the timing and alignment of factors. This observation harkens back to Caesar's "iacta alea est"; this work has shown that the die he cast, and every emperor after him, was loaded and not completely fair or aleatory.
The dataset used in this work is publically available in De Imperatoribus Romanis, a peer-reviewed online encyclopedia of Roman emperors [DIR]: http://www.roman-emperors.org/.
When calculated on a century basis, the statistics show the same order of magnitude. For example, in the first and fourth century CE, roughly 58% of the emperors suffered a violent death, and in the third century, 77%, a reflection of the convulsion and crisis of the third century. Only during the second century does the rate fall to the low 30%. This was the period of the "five good emperors", as Gibbon calls them, from Nerva (d. 98) to Marcus Aurelius (d. 180).
The city of Orléans in France is named after him, and consequently, New Orleans. And by extension, with a bit of a stretch, one can associate with him Jazz!
This introduces a subtle bias in the analysis which has to be acknowledged. By focusing on legitimate emperors only, we discard the many contenders for the imperial power who failed to secure senate confirmation, and were thus killed usually early on after their uprising. Senate confirmation, however, sometimes came after-the-fact as a recognition of a fait-accompli that a would-be usurper and or his legions made it to Rome and obtained the senatorial recognition of the emperorship. Suffice to remember that Vespasian for example, Septimius Severus, and Julian to mention a few were, for a brief time, usurpers before obtaining senate confirmation. The implications for our purposes is that this introduces a survivor bias in our analysis, and that the results in section "Results and discussion" for the survivor function are likely conservative and underestimate the spectacle of regicide of Roman emperors, especially in the first few years of their reign. We are grateful for an anonymous reviewer who brought this up to our attention.
Portrayed skillfully by Joaquim Phoenix in the movie Gladiator. Fans are still waiting for the second installment of this movie.
In a famous statement upon being urged to reclaim the purple, Diocletian is said to have replied, "if you could show the cabbage I planted with my own hands to your emperor, he definitely wouldn't dare suggest I replace the peace and happiness of this place with the storms of a never-satisfied greed" [Epitome de Caesaribus]. It is however not known what kind of cabbages he grew.
The data are assumed to be independent and identically distributed (iid), as generally done in survival data analysis. This assumption, however, may be challenged for some emperors (e.g., father–son pairs), and this constitutes a potential limitation of the analysis.
To paraphrase Thomas Paine, "these are times that try [an empire's] soul".
Each of these cases has an interesting underlying pathway to violent death. These narratives, however, are beyond the scope of this work.
It is enjoyable to imagine and hope that Suetonius' work be turned into a TV series someday. If this happens, what music would go with Augustus' death scene? Rex Tremendae from Mozart's requiem would be a good start as the camera pans outside his dwelling (he was on his way to Rome and had just passed Naples when his illness took him). He was one, a rex tremendae, and was on the verge of being deified. Then, within his dimly lit chamber, the music switches to the more intimate prelude of Bach's Cello Suite No. 2 in D minor. The music fades, labored breathing is heard, then, cue to the actor portraying Augustus…
One only needs to consider the current appeal of the mixed martial arts (MMA) bloody fights, and then add to that swords, spears, and tridents for example to understand the appeal of gladiatorial games.
A devoted follower or a zealous acolyte.
Amalberti R, Auroy Y, Berwick D, Barach P (2005) Five system barrier to achieving ultrasafe health care. Ann Intern Med 142(9):756–764
Coleridge ST (1983) Biographia literaria. In: Engell J, Bate WJ (eds.) The collected works of Samuel Taylor Coleridge. Princeton University Press, Princeton, USA
De Imperatoribus Romanis [DIR]. http://www.roman-emperors.org/. Accessed 10 May 2019, along with the article for each legitimate emperor therein.
Fagan GG (2011) From the lure of the arena: social psychology and the crowd at the Roman games. Cambridge University Press, Cambridge, UK
Høyland A, Rausand M (2009) System reliability theory. John Wiley & Sons, Hoboken, USA
Marcus Aurelius (1989) "Meditations". Translation by A.S.L. Farquharson. Oxford University Press, Oxford, UK
Millar F (1977) Emperor in the Roman world. Duckworth Publishing, London, UK
Plutarch. The parallel lives. Loeb Classical Library, 1921. http://penelope.uchicago.edu/Thayer/e/roman/texts/plutarch/lives/home.html. Accessed 16 May 2019.
Retief FP, Cilliers L (2005) Causes of death among the Caesars (27 BC–AD 476). Acta Theol 26(2):89–106
Saleh JH, Marais K (2006) Highlights from the early (and pre-) history of reliability engineering. Reliabil Eng Syst Saf 91(2):249–256
Scarre C (1995) Chronicle of Roman emperors. Thames & Hudson, London
Suetonius (1998) Lives of the twelve Caesars. Loeb Classical Library, Cambridge, USA
Watson A (1999) Aurelian and the third century. Routledge, New York, USA
Georgia Institute of Technology, Atlanta, USA
Joseph Homer Saleh
Search for Joseph Homer Saleh in:
Correspondence to Joseph Homer Saleh.
The author declares no competing interests.
Saleh, J.H. Statistical reliability analysis for a most dangerous occupation: Roman emperor. Palgrave Commun 5, 155 (2019) doi:10.1057/s41599-019-0366-y
DOI: https://doi.org/10.1057/s41599-019-0366-y
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Is there a mathematical equivalence between gravitational optics and quantum optics or are they mathematically incompatible theories?
I know that massive gravitational bodies will curve the path that light travels. I think that quantum optical mediums also bend light. I am still confused of whether quantum optical mediums actually slow down light or if they just use absorption, re-emission, scattering, and delaying to create the illusion that light slows down in a medium. I do not see a fundamental difference between the two theories, because both describe "optical" phenomena of light. I am not sure if gravitational optics is a real field of study or a made up name.
My question is there a mathematical equivalence between gravitational optics and quantum optics or are they mathematically incompatible theories?
optics quantum-optics geometric-optics non-linear-optics
linuxfreebird
linuxfreebirdlinuxfreebird
$\begingroup$ I was unable to find a picture related to quantum optics that illustrates the bending of light. $\endgroup$ – linuxfreebird Sep 17 '14 at 14:40
Gravitational optics is very different from quantum optics, if by the latter you mean the quantum effects of interaction between light and matter. There are three crucial differences I can think of:
We can always detect uniform motion with respect to a medium by a positive result to a Michelson-Morley experiment that is confined to a region of spacetime small enough to be "flat". The same experiment in the same region does not detect such motion in freespace. In such a small enough region, light in freespace is always observed to travel at $c$;
Another important difference is that mediums and gravitational lenses are fundamentally different in their effect on the polarisation of light: I talk further about this below;
A related way of saying the same as in 1. is that a photon in curved space still always behaves as a massless particle. Photons propagating through a (non-freespace) medium are not pure photons, unless they are not interacting with the medium, in which case the medium is equivalent to freespace for this discussion. In a medium, the photon becomes a quantum superposition of pure photon and excited matter states, as discussed in my answer here. It is therefore a quasiparticle (called a polariton, or plasmon, or exciton, depending on the exact kind of medium and interaction spoken about) that always has a nonzero rest mass if you insist on thinking of it as a quasi-particle. The hot topic of "slow light" belongs in this picture, and says nothing contradicting the masslessness of light in freespace;
In quantum optics, almost always the interactions involve absorptions and re-emissions of different photons: quantum and classical optics are simply different approximations for the same kind of light-matter interaction, as categorised in my answer here. The cyclic "absorption-re-emission" picture is, albeit in different language, an equivalent way of thinking about the matter light interaction as the "polariton" quasi-particle, photon-matterstate quantum superpositon picture. The "polariton" picture diagonalises the Schrödinger picture so we talk in terms of eigenmodes: the absorption-re-emission picture keeps pure photons and matter states separate: since these are no long eigenstates, the Schrödinger picture paints the situation as a oscillation back and forth between these two.
On the other hand, gravitational optics is the propagation of light in "curved", but "empty" spacetime. Right now we would tend to describe a photon being gravitationally lensed by the free, but "curved" space Maxwell equations:
$$ {A^{ a }}_{ ; a } = 0;\;\Box A^{a} = {R^{ a }}_{ b } A^{ b }$$
where $A$ is the generalised Lorenz gauged four-potential and $R_{ a b } \ \stackrel{\mathrm{def}}{=}\ {R^{ s }}_{ a s b }$ is the Ricci curvature tensor, which you get from the solution of the vacuum Einstein field equations that prevails around the lensing object(s). The photon is here interacting locally with spacetime, not with the lensing "matter". Einstein's big gig (in GTR at least) was "locality": the notion that all physics is local and that there is no instantaneous action at a distance.
At the equation level, a crucial difference between Maxwell's equations in curved spacetime as opposed Maxwell's equations in a (potentially light bending) medium is that gravitational lensing manifests itself as a variation from place to place in the lightspeed $1/\sqrt{\mu\,\epsilon}$, but the "characteristic impedance" $\sqrt{\mu/\epsilon}$ stays constant everywhere (recall we're solving for geodesics over a wide region, so that, from a distant, nonlocal standpoint, lightspeed can vary from place to place - this is different from, and altogether consistent with, the generalised Equivalence Principle saying that spacetime is always locally Minkowskian with the same $c$). A light-bending medium on the other hand, unless it is very special, changes both $1/\sqrt{\mu\,\epsilon}$ and the "characteristic impedance" $\sqrt{\mu/\epsilon}$. Physically what this means is that the left and right handed polarised photon eigenstates in general couple together in a medium, whereas in gravitational lensing they never couple together no matter how "severe" the gravitational lensing may be. Pure left/ right handed polarised light stays left/right handed polarised in any gravitational lensing: inhomogeneous, light bending mediums almost always mix left and right handed components. See:
Iwo Bialynicki Birula, "Photon Wave Function", in "Progress in Optics" Vol XXXVI, E. Wolf Ed. 1996
particularly §11 of this work. I have often wondered about "simulating" gravitational lensing with inhomogeneous metamaterials whose characteristic impedance is constant, but I'm not even sure it is theoretically possible to make these.
Another key difference from "quantum optics" is that the photon propagates in curved spacetime and is not thought of as being absorbed and re-emitted as with its interactions with matter. So here the picture is much simpler (conceptually) than quantum optics (of course it's quite involved and tedious in actual calculation).
However (although I am well out of my depth here for details), it's possible that a future quantum theory of gravity will come up with a "graviton field", so that we would probably then be thinking of the photon's repeated absorption and re-emission by the graviton field, which would be a picture that is more like our conception today of quantum optics.
See also some pithy discussions of some of the other differences between medium bending of light and gravitational lensing here: as Jitter amusingly puts it "Could I burn space ants?". The answer, to the space ants' collective relief, is no.
BTW I don't like that diagram of diffuse transmission, with "photons" bouncing all over the place like bullets. It seems to be quite fashionable nowadays to talk about "ballistic" photons (which I don't really understand): see my answer here for some idea of why I don't like the diagram. Diffuse transmission is still a version of what I talk about there, only a bit more complicated because the distribution of matter is more "jagged".
Selene RoutleySelene Routley
$\begingroup$ I require more clarification on "a crucial difference between Maxwell's Equations in curved spacetime as opposed to Maxwell's equations in a medium." Though light does travel in constant speed $c$ in locally flat frames of reference, from our distant view of the gravitational bending, we observe light slowing down in the presence of a massive gravitational body e.g. physics.stackexchange.com/questions/59502/…. In a quantum optical medium we observe light taking its time to travel through the medium, but we cannot locally observe it, can we? $\endgroup$ – linuxfreebird Sep 29 '14 at 15:52
$\begingroup$ In any optical medium (by medium I mean anything, like a block of glass, aside from spacetime), photons are absorbed and re-emitted. A photon in a curved vacuum is not described in this way. The behaviour of an MM experiment is also different. You can always detect uniform motion relative to a medium with an MM experiment. Without the medium, the MM experiment result is the classical negative result. You're also right about the "distant" point of view: I only mentioned the local constant $c$ to emphasise that distant view with changing lightspeed is still consistent with this notion. $\endgroup$ – Selene Routley Sep 29 '14 at 22:49
$\begingroup$ @linuxfreebird Also, you hear a great deal of talk nowadays about "slow light". See my answer here for a short discussion on this. When propagating in mediums, photons aren't really pure photons but quantum superpositions of free photons with excited matter states. Sometimes people call these quasi-particles polaritons, to tell them apart from pure photons. $\endgroup$ – Selene Routley Sep 29 '14 at 23:28
$\begingroup$ @linuxfreebird also see modified answer above $\endgroup$ – Selene Routley Sep 30 '14 at 1:42
General relativity and classical optics are not equivalent, however, one can construct analogies between the two and build optical simulations of gravitational phenomena. For example, in the field of transformation optics metamaterials are used to produce spatial variations of the effective permittivity, which enable scientists to steer light beams along predetermined paths. In fact, some recent work has shown that one can design near-perfect absorbers that capture light beams which travel near an "optical black hole" designed with transformation optics techniques. Quantum optics isn't quite relevant here, since we're not dealing with individual photons or atomic systems. See the references below for more details.
http://en.wikipedia.org/wiki/Transformation_optics http://www.nature.com/news/2009/091015/full/news.2009.1007.html http://arxiv.org/pdf/0910.2159v2.pdf http://arxiv.org/pdf/0912.4856.pdf
Nikita ButakovNikita Butakov
$\begingroup$ +1 Wow! That is amazingly interesting! I've wondered for a while whether metamaterials could mimic gravitational lensing and now people are actually doing it! $\endgroup$ – Selene Routley Sep 30 '14 at 12:21
$\begingroup$ Not to mention, they are building invisibility cloak too. Maybe also "gravity" invisibility cloaks are possible? $\endgroup$ – giulio bullsaver Oct 1 '14 at 19:54
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# Basics of combinatorial designs
A combinatorial design is defined as a pair (X, B), where X is a set of elements called points, and B is a collection of subsets of X called blocks. The design satisfies the following properties:
1. Each point is contained in a fixed number of blocks.
2. Any two distinct blocks intersect in a fixed number of points.
3. The number of blocks containing a fixed point is the same for all points.
These properties ensure that the design is balanced and symmetric, allowing for a systematic study of its structure.
One of the simplest and most well-known combinatorial designs is the Steiner system. A Steiner system is a design in which every block contains the same number of points, and any two points are contained in exactly one block. For example, the Steiner system S(2, 3, 7) consists of 7 points and 7 blocks, where each block contains 3 points and any two points are contained in exactly one block.
Steiner systems have a wide range of applications in various areas of mathematics and computer science. They are particularly useful in coding theory, where they are used to construct error-correcting codes with optimal properties.
Consider the Steiner system S(2, 3, 7). The points in this system can be represented by the numbers 1 to 7. The blocks are subsets of these points, such as {1, 2, 3}, {1, 4, 5}, {2, 4, 6}, and so on.
This Steiner system satisfies the properties of a combinatorial design. Each point is contained in exactly 3 blocks, any two blocks intersect in exactly 1 point, and the number of blocks containing a fixed point is 3.
## Exercise
1. Consider the Steiner system S(2, 3, 7). How many blocks are there in this system?
2. What is the maximum number of points that can be in a block of the Steiner system S(2, 3, 7)?
3. Is it possible to have a Steiner system S(2, 3, 8)? Why or why not?
### Solution
1. There are 7 blocks in the Steiner system S(2, 3, 7).
2. The maximum number of points in a block of the Steiner system S(2, 3, 7) is 3.
3. It is not possible to have a Steiner system S(2, 3, 8) because there are not enough points to form the required number of blocks. In a Steiner system S(2, 3, 8), there would need to be 8 points and each block would need to contain 3 points. However, 8 points cannot be divided into blocks of size 3 without having some points left over.
# The role of extremal set theory in combinatorics
Extremal set theory is a branch of combinatorics that focuses on studying the maximum or minimum size of certain sets or structures that satisfy certain properties. It provides a powerful framework for understanding the limits and boundaries of combinatorial structures.
In extremal set theory, we often ask questions such as: What is the maximum size of a set with certain properties? What is the minimum size of a set that satisfies certain conditions? These questions are motivated by the desire to understand the structure and behavior of combinatorial objects in the most extreme cases.
Extremal set theory has applications in various areas of mathematics and computer science. It is particularly useful in graph theory, where it is used to study the maximum or minimum size of certain subgraphs or graph properties. It also has applications in coding theory, network design, cryptography, and other areas of computer science.
In this section, we will explore the role of extremal set theory in combinatorics and discuss some of its applications in computer science. We will also introduce some key concepts and techniques used in extremal set theory, such as the probabilistic method and counting techniques.
Consider the problem of finding the largest possible independent set in a graph. An independent set is a set of vertices in a graph such that no two vertices in the set are adjacent. In other words, it is a set of vertices that are not connected by an edge.
The problem of finding the largest independent set in a graph can be formulated as an extremal set theory problem. We want to find the maximum size of a set of vertices that satisfies the property of being an independent set.
## Exercise
Consider a graph with 6 vertices and the following edges: (1, 2), (1, 3), (2, 4), (3, 4), (4, 5), (4, 6). Find the largest possible independent set in this graph.
### Solution
The largest possible independent set in this graph is {1, 5, 6}. This set contains three vertices and no two vertices in the set are adjacent.
# Graph theory and its applications in combinatorics
Graph theory is a branch of mathematics that studies the properties and relationships of graphs. A graph is a mathematical structure that consists of a set of vertices (or nodes) and a set of edges (or arcs) that connect pairs of vertices. Graphs are used to model and solve a wide range of problems in various fields, including computer science, operations research, and social sciences.
In combinatorics, graph theory plays a crucial role in understanding and solving combinatorial problems. It provides a powerful framework for analyzing and counting the number of possible configurations or arrangements of objects. Graphs can be used to represent and study various combinatorial structures, such as permutations, combinations, and partitions.
Graph theory has applications in computer science, particularly in the design and analysis of algorithms. Many computational problems can be formulated and solved using graph algorithms, such as finding the shortest path between two vertices, determining the connectivity of a network, or identifying cycles in a graph.
In this section, we will explore the fundamental concepts and techniques of graph theory, including graph representation, connectivity, coloring, and matching. We will also discuss some of the applications of graph theory in combinatorics and computer science.
Consider the problem of finding the shortest path between two cities in a road network. We can model the road network as a graph, where each city is represented by a vertex and each road is represented by an edge connecting two vertices. By applying graph algorithms, such as Dijkstra's algorithm, we can efficiently find the shortest path between two cities.
## Exercise
Consider a graph with 5 vertices and the following edges: (1, 2), (2, 3), (3, 4), (4, 5), (5, 1). Determine whether this graph is connected.
### Solution
This graph is connected because there is a path between every pair of vertices.
# The polynomial method in extremal combinatorics
The polynomial method is a powerful technique in extremal combinatorics that has been widely used to solve various problems. It involves using polynomial functions to encode combinatorial structures and properties, and then applying algebraic techniques to analyze and solve these problems.
The main idea behind the polynomial method is to associate a polynomial with a combinatorial object or property of interest. This polynomial is constructed in such a way that its coefficients or roots encode the desired information. By studying the properties of these polynomials, we can gain insights into the combinatorial structures and properties we are interested in.
The polynomial method has been successfully applied to solve problems in various areas of combinatorics, including graph theory, number theory, and coding theory. It has been used to prove the existence of certain combinatorial structures, establish lower bounds on the size of these structures, and determine the extremal cases that achieve these bounds.
In the context of extremal combinatorics, the polynomial method has been particularly useful in proving upper and lower bounds on the size of certain combinatorial objects, such as graphs, hypergraphs, and set systems. It has also been used to study the existence and properties of extremal structures that achieve these bounds.
In the next few sections, we will explore the polynomial method in more detail and see how it can be applied to solve various problems in extremal combinatorics.
Consider the problem of determining the maximum number of edges in a graph with n vertices and no triangles. We can use the polynomial method to solve this problem.
Let G be a graph with n vertices and no triangles. We can associate a polynomial P(x) with G, where the coefficient of x^i represents the number of vertices in G with degree i. Since G has no triangles, the degree of each vertex in G is at most n-2. Therefore, the polynomial P(x) can be written as:
P(x) = a_0 + a_1x + a_2x^2 + ... + a_{n-2}x^{n-2},
where a_i represents the number of vertices in G with degree i.
By studying the properties of this polynomial, we can derive upper and lower bounds on the number of edges in G. For example, we can show that the number of edges in G is at most (n-2)a_1/2, which gives us an upper bound on the maximum number of edges in G.
## Exercise
Consider a graph with 6 vertices and the following edges: (1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 1). Determine the polynomial P(x) associated with this graph.
### Solution
The polynomial P(x) associated with this graph is P(x) = 0 + 2x + 2x^2 + 2x^3 + 0x^4 + 0x^5.
# Using the probabilistic method to solve combinatorial problems
The probabilistic method is a powerful technique in combinatorics that involves using probability and randomization to solve difficult combinatorial problems. It was first introduced by Paul Erdős in the 1940s and has since become a fundamental tool in the field.
The main idea behind the probabilistic method is to show that a combinatorial object or property of interest exists by demonstrating that a randomly chosen object or property has the desired property with positive probability. In other words, instead of trying to construct the object directly, we show that it is unlikely for a randomly chosen object to not have the desired property.
To apply the probabilistic method, we typically use two main steps:
1. Random construction: We construct a random object or property that satisfies certain conditions or constraints. This is often done by assigning random values or making random choices.
2. Analysis: We analyze the probability that the random object or property has the desired property. This involves calculating the probability of certain events occurring and using probabilistic techniques to estimate or bound these probabilities.
By carefully choosing the random construction and performing a thorough analysis, we can often prove the existence of combinatorial objects or properties that would be difficult to construct directly.
The probabilistic method has been successfully applied to solve a wide range of combinatorial problems, including problems in graph theory, number theory, and optimization. It has been used to prove the existence of certain structures, establish lower bounds on the size of these structures, and determine the probabilistic threshold at which certain properties occur.
In the next few sections, we will explore the probabilistic method in more detail and see how it can be applied to solve various combinatorial problems.
Consider the problem of coloring the edges of a complete graph with two colors, such that no triangle is monochromatic. We can use the probabilistic method to solve this problem.
Let G be a complete graph with n vertices. We randomly color each edge of G with one of two colors, red or blue. We can show that the probability of a triangle being monochromatic is small.
Let's assume that we have colored the edges randomly. For a fixed triangle in G, the probability that all three edges of the triangle are the same color is 1/4. Therefore, the probability that a randomly chosen triangle is monochromatic is at most 1/4.
By the linearity of expectation, the expected number of monochromatic triangles in G is at most (n choose 3)/4. Since this is less than 1, there must exist a coloring of the edges of G such that no triangle is monochromatic.
## Exercise
Consider a complete graph with 4 vertices. Use the probabilistic method to estimate the probability that a randomly chosen triangle in this graph is monochromatic.
### Solution
There are 4 choose 3 = 4 triangles in the complete graph with 4 vertices. For each triangle, the probability that it is monochromatic is 1/4. Therefore, the probability that a randomly chosen triangle is monochromatic is (1/4) * 4/4 = 1/4.
# Extremal problems in computer science: examples and case studies
One example is the problem of finding the largest independent set in a graph. An independent set is a set of vertices in a graph such that no two vertices in the set are adjacent. The problem is to find an independent set of maximum size in a given graph.
Extremal combinatorics provides tools and techniques to solve this problem. For example, the famous Erdős–Ko–Rado theorem gives a lower bound on the size of an independent set in a graph based on the number of edges in the graph. This lower bound can be used to guide the search for an independent set of maximum size.
Another example is the problem of finding a maximum matching in a bipartite graph. A matching is a set of edges in a graph such that no two edges share a common vertex. The problem is to find a matching of maximum size in a given bipartite graph.
Extremal combinatorics provides techniques to solve this problem as well. For example, the Hall's marriage theorem gives a necessary and sufficient condition for the existence of a matching that saturates one side of the bipartition. This condition can be used to check if a given bipartite graph has a maximum matching.
These are just two examples of how extremal combinatorics can be applied to solve problems in computer science. The field has many more applications, including in algorithms, data structures, network design, and cryptography.
Consider a graph with 8 vertices and 12 edges. We want to find the largest independent set in this graph.
Using the Erdős–Ko–Rado theorem, we can determine a lower bound on the size of the independent set. The theorem states that if a graph has n vertices and e edges, then the size of the largest independent set is at least n - e/2.
In our case, the graph has 8 vertices and 12 edges. Therefore, the size of the largest independent set is at least 8 - 12/2 = 8 - 6 = 2.
This means that there exists an independent set in the graph with at least 2 vertices. We can now search for such a set using other techniques, such as greedy algorithms or dynamic programming.
## Exercise
Consider a bipartite graph with 6 vertices on each side and 9 edges. Use Hall's marriage theorem to determine if the graph has a maximum matching.
### Solution
Hall's marriage theorem states that a bipartite graph has a maximum matching if and only if for every subset of vertices on one side of the bipartition, the number of neighbors is at least the size of the subset.
In our case, let's consider a subset of 3 vertices on one side of the bipartition. If the number of neighbors is at least 3, then the graph has a maximum matching.
By inspecting the graph, we can see that for any subset of 3 vertices on one side, there are at least 3 neighbors. Therefore, the graph has a maximum matching.
# Counting techniques and their role in extremal combinatorics
Counting techniques play a crucial role in extremal combinatorics. They allow us to determine the number of objects that satisfy certain properties, which in turn helps us understand the structure and behavior of combinatorial objects.
One counting technique commonly used in extremal combinatorics is the principle of inclusion-exclusion. This principle allows us to count the number of objects that satisfy at least one of several properties, while avoiding double-counting.
Another counting technique is the pigeonhole principle, which states that if we distribute more objects into fewer containers, then at least one container must contain more than one object. This principle is often used to prove the existence of certain combinatorial structures.
In addition to these techniques, there are various combinatorial formulas and identities that can be used to count the number of objects. For example, the binomial coefficient formula allows us to count the number of ways to choose a certain number of objects from a larger set.
Counting techniques are essential in extremal combinatorics because they provide a way to quantify the size and properties of combinatorial objects. By understanding the counting principles and formulas, we can analyze and solve extremal problems more effectively.
Consider a set of 10 people. We want to count the number of ways to form a committee of 3 people from this set.
Using the binomial coefficient formula, we can calculate this number as:
$$\binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10 \cdot 9 \cdot 8}{3 \cdot 2 \cdot 1} = 120.$$
Therefore, there are 120 ways to form a committee of 3 people from a set of 10.
## Exercise
Consider a set of 6 different books. How many ways are there to arrange these books on a shelf?
### Solution
The number of ways to arrange 6 different books on a shelf can be calculated as:
$$6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 720.$$
Therefore, there are 720 ways to arrange 6 different books on a shelf.
# Extremal combinatorics in coding theory
Extremal combinatorics plays a crucial role in coding theory, which is the study of error-correcting codes. Error-correcting codes are used to transmit information reliably over noisy channels, such as in telecommunications and computer networks.
In coding theory, extremal combinatorics is used to determine the maximum possible efficiency and error-correction capability of a code. This involves studying the maximum number of codewords that can be generated by a code of a given length and minimum distance.
The concept of extremal sets, which are sets with certain properties that optimize a specific objective function, is particularly important in coding theory. Extremal sets are used to construct codes with desirable properties, such as maximum error-correction capability or minimum redundancy.
Extremal combinatorics is also used to analyze the performance of codes in terms of decoding complexity and error probability. By studying the structure and behavior of extremal sets, we can gain insights into the efficiency and reliability of error-correcting codes.
Overall, extremal combinatorics provides the theoretical foundation for designing and analyzing error-correcting codes in coding theory. It allows us to understand the fundamental limits and trade-offs in the design of codes, and enables us to develop efficient and reliable communication systems.
Consider a binary code with codewords of length 4. We want to determine the maximum number of codewords that can be generated by this code, given that the minimum distance between any two codewords is 2.
To solve this problem, we can use extremal combinatorics. We can construct a set of codewords that maximizes the number of codewords while satisfying the minimum distance requirement. One such set is:
{0000, 0011, 1100, 1111}
This set contains 4 codewords, and any two codewords have a minimum distance of 2. Therefore, the maximum number of codewords that can be generated by this code is 4.
## Exercise
Consider a binary code with codewords of length 5. The minimum distance between any two codewords is 3. What is the maximum number of codewords that can be generated by this code?
### Solution
To determine the maximum number of codewords, we can construct a set of codewords that satisfies the minimum distance requirement. One such set is:
{00000, 00111, 11000, 11111}
This set contains 4 codewords, and any two codewords have a minimum distance of 3. Therefore, the maximum number of codewords that can be generated by this code is 4.
# Applications of extremal combinatorics in network design
Extremal combinatorics has numerous applications in network design, which involves the creation and optimization of networks, such as computer networks, transportation networks, and social networks.
One application of extremal combinatorics in network design is in the construction of efficient routing algorithms. Routing algorithms determine the paths that data packets should take in a network to reach their destination. Extremal combinatorics can be used to analyze the performance of routing algorithms in terms of their efficiency, scalability, and robustness.
Another application of extremal combinatorics in network design is in the design of optimal network topologies. Network topologies refer to the arrangement of nodes and links in a network. Extremal combinatorics can be used to determine the optimal number of links and the optimal arrangement of nodes to minimize the cost or maximize the efficiency of a network.
Extremal combinatorics is also used in the analysis of network connectivity and resilience. Network connectivity refers to the ability of a network to maintain communication between nodes, even in the presence of failures or disruptions. Extremal combinatorics can be used to study the minimum number of links or nodes that need to be removed to disconnect a network or to determine the maximum number of failures that a network can tolerate without losing connectivity.
Overall, extremal combinatorics provides valuable insights and tools for designing and analyzing networks. It allows us to understand the fundamental limits and trade-offs in network design and enables us to develop efficient, reliable, and resilient networks.
Consider a computer network with multiple routers connected in a mesh topology. Each router is connected to every other router in the network. The network administrator wants to determine the minimum number of links that need to be removed to disconnect the network.
To solve this problem, we can use extremal combinatorics. We can construct a set of links that, when removed, disconnects the network. One such set is:
{Link 1, Link 2, Link 3}
By removing these three links, the network becomes disconnected, as there is no longer a path between any pair of routers.
## Exercise
Consider a transportation network with multiple cities connected by roads. The network administrator wants to determine the minimum number of cities that need to be removed to disconnect the network. What is the minimum number of cities that need to be removed to disconnect the network?
### Solution
To determine the minimum number of cities, we can construct a set of cities that, when removed, disconnects the network. One such set is:
{City 1, City 2}
By removing these two cities, the network becomes disconnected, as there is no longer a path between any pair of cities. Therefore, the minimum number of cities that need to be removed to disconnect the network is 2.
# Extremal combinatorics and cryptography
Extremal combinatorics has important applications in cryptography, which is the practice of secure communication in the presence of adversaries. Cryptography relies on the use of mathematical algorithms and techniques to ensure the confidentiality, integrity, and authenticity of information.
One application of extremal combinatorics in cryptography is in the design and analysis of cryptographic codes. Cryptographic codes are used to encrypt and decrypt messages to protect their confidentiality. Extremal combinatorics can be used to analyze the security properties of cryptographic codes, such as their resistance to attacks and their ability to detect and correct errors.
Another application of extremal combinatorics in cryptography is in the construction of cryptographic hash functions. Hash functions are used to map data of arbitrary size to fixed-size values, called hash values or hash codes. Extremal combinatorics can be used to analyze the properties of hash functions, such as their collision resistance and their ability to distribute hash values uniformly.
Extremal combinatorics is also used in the analysis of cryptographic protocols and systems. Cryptographic protocols are sets of rules and procedures that govern the secure exchange of information between parties. Extremal combinatorics can be used to analyze the security properties of cryptographic protocols, such as their resistance to attacks and their ability to ensure privacy and integrity.
Overall, extremal combinatorics provides valuable tools and techniques for designing and analyzing secure cryptographic systems. It allows us to understand the fundamental limits and trade-offs in cryptography and enables us to develop secure and reliable communication systems.
Consider a cryptographic code that uses a substitution cipher to encrypt messages. In a substitution cipher, each letter in the plaintext is replaced by a different letter in the ciphertext. The codebook, which specifies the substitutions, is a set of pairs of letters.
To analyze the security of this code, we can use extremal combinatorics. We can construct a set of codebooks that maximize the security of the code, such that it is resistant to attacks, such as frequency analysis. One such set is:
{Codebook 1, Codebook 2, Codebook 3}
By using these codebooks, the frequency distribution of letters in the ciphertext is randomized, making it difficult for an attacker to determine the substitutions and decrypt the message.
## Exercise
Consider a cryptographic hash function that maps a message of arbitrary size to a fixed-size hash value. The hash function is designed to have a collision resistance property, which means that it is computationally infeasible to find two different messages that produce the same hash value. What is the minimum number of bits in the hash value to achieve collision resistance?
### Solution
To achieve collision resistance, the hash function should have a hash value that is large enough to make it computationally infeasible to find two different messages that produce the same hash value. The minimum number of bits in the hash value depends on the desired level of security. In practice, hash functions with hash values of at least 128 bits are considered to provide sufficient collision resistance.
# Open problems and future directions in extremal combinatorics
One open problem in extremal combinatorics is the determination of exact values for various extremal parameters. For example, the Turán number ex(n, K3) is known to be ⌊n^2/4⌋, but the exact values for other forbidden graphs are still unknown. Finding these values would provide valuable insights into the structure and behavior of graphs.
Another open problem is the characterization of extremal structures. For example, what are the extremal graphs with respect to the number of triangles or the chromatic number? Understanding the properties of these extremal structures would deepen our understanding of graph theory and combinatorics.
Additionally, there is ongoing research on the development of new techniques and tools for solving extremal problems. The probabilistic method, for example, has been successfully applied in many areas of extremal combinatorics. Exploring new variations and extensions of this method could lead to breakthroughs in solving previously unsolved problems.
Furthermore, there is a growing interest in the connections between extremal combinatorics and other areas of mathematics, such as algebra, topology, analysis, and geometry. These interdisciplinary connections provide fertile ground for new research directions and collaborations.
In the field of computer science, extremal combinatorics has important applications in network design, coding theory, and cryptography. Future research could focus on developing efficient algorithms and protocols based on extremal combinatorics principles.
Overall, the field of extremal combinatorics is rich with open problems and future directions. By tackling these challenges and exploring new avenues of research, we can continue to advance our understanding of combinatorial structures and their applications in various fields. | Textbooks |
# The Max-Min method
The Max-Min method is an optimization technique used to find the best decision among multiple alternatives. It is particularly useful in decision-making problems where the decision-maker needs to balance multiple objectives with conflicting priorities. The method is based on the idea of finding the minimum value of the maximum value among all possible decisions.
To understand the Max-Min method, let's consider an example. Suppose a company is considering two investment options, A and B. Option A has a higher expected return, but a higher risk. Option B has a lower expected return, but a lower risk. The decision-maker wants to maximize the expected return while minimizing the risk. The Max-Min method can help the decision-maker find the best decision among these options.
In mathematical terms, let's denote the expected return for option A as $R_A$ and for option B as $R_B$. The risk associated with each option can be represented by a measure $R_A$ and $R_B$. The Max-Min method involves finding the minimum value of the maximum values among these two options:
$$\text{Max-Min} = \min(\max(R_A, R_B))$$
In this example, the decision-maker should choose option A if $R_A > R_B$, and option B if $R_A < R_B$. If $R_A = R_B$, the decision-maker can choose either option, as they have equal expected returns and risks.
Let's consider another example. A restaurant owner wants to maximize the profit from selling two types of dishes, Pizza and Burger. The expected revenue for Pizza is $10 per order, and the expected revenue for Burger is $8 per order. The cost of ingredients for Pizza is $6 per order, and for Burger is $4 per order. The decision-maker wants to minimize the cost while maximizing the revenue.
The Max-Min method can be applied as follows:
$$\text{Max-Min} = \min(\max(10 - 6, 8 - 4))$$
In this case, the decision-maker should choose Pizza if $10 - 6 > 8 - 4$, and Burger if $10 - 6 < 8 - 4$. If $10 - 6 = 8 - 4$, the decision-maker can choose either dish, as they have equal revenue and cost.
## Exercise
Consider a company that wants to choose between two projects, Project A and Project B. Project A has an expected return of $15,000 and a risk of $5,000. Project B has an expected return of $12,000 and a risk of $4,000. The decision-maker wants to maximize the expected return while minimizing the risk.
1. Calculate the Max-Min value for this decision problem.
2. Based on the Max-Min value, which project should the decision-maker choose?
### Solution
1. The Max-Min value is $\min(\max(15000 - 5000, 12000 - 4000)) = \min(\max(10000, 8000)) = 8000$.
2. The decision-maker should choose Project B, as its expected return minus risk ($12000 - 4000 = 8000$) is greater than that of Project A ($15000 - 5000 = 10000$).
# Applying the Max-Min method to Java
Now that we have a basic understanding of the Max-Min method, let's see how we can apply it to Java programming. The Max-Min method can be implemented in Java using conditional statements and loops.
To use the Max-Min method in Java, we need to define the expected return and risk values for each decision option. We can represent these values as variables or as elements in an array or list. Then, we can use conditional statements and loops to find the minimum value of the maximum values among the decision options.
For example, let's consider a Java program that calculates the Max-Min value for the restaurant owner's decision problem. We can represent the expected revenue and cost for each dish as follows:
```java
int pizzaRevenue = 10;
int pizzaCost = 6;
int burgerRevenue = 8;
int burgerCost = 4;
```
Next, we can use conditional statements and loops to find the Max-Min value:
```java
int maxMinValue = Math.min(Math.max(pizzaRevenue - pizzaCost, burgerRevenue - burgerCost), Math.max(pizzaRevenue, burgerRevenue));
```
In this example, the `Math.max()` function is used to find the maximum value, and the `Math.min()` function is used to find the minimum value. The result is stored in the `maxMinValue` variable.
## Exercise
Instructions:
Write a Java program that calculates the Max-Min value for the restaurant owner's decision problem. The program should use the following values for the expected revenue and cost of each dish:
- Pizza: $10 and $6
- Burger: $8 and $4
The program should output the Max-Min value and the decision-maker's choice based on this value.
### Solution
```java
public class MaxMinExample {
public static void main(String[] args) {
int pizzaRevenue = 10;
int pizzaCost = 6;
int burgerRevenue = 8;
int burgerCost = 4;
int maxMinValue = Math.min(Math.max(pizzaRevenue - pizzaCost, burgerRevenue - burgerCost), Math.max(pizzaRevenue, burgerRevenue));
if (maxMinValue == pizzaRevenue - pizzaCost) {
System.out.println("The Max-Min value is " + maxMinValue + ". The decision-maker should choose Pizza.");
} else if (maxMinValue == burgerRevenue - burgerCost) {
System.out.println("The Max-Min value is " + maxMinValue + ". The decision-maker should choose Burger.");
} else {
System.out.println("The Max-Min value is " + maxMinValue + ". The decision-maker can choose either Pizza or Burger.");
}
}
}
```
# Examples and case studies
## Exercise
Instructions:
Write a Java program that calculates the Max-Min value for the following decision problem:
A company wants to choose between two advertising campaigns, Campaign A and Campaign B. Campaign A has an expected return of $10,000 and a cost of $5,000. Campaign B has an expected return of $8,000 and a cost of $4,000. The decision-maker wants to maximize the expected return while minimizing the cost.
### Solution
```java
public class MaxMinExample {
public static void main(String[] args) {
int campaignARevenue = 10000;
int campaignACost = 5000;
int campaignBRevenue = 8000;
int campaignBCost = 4000;
int maxMinValue = Math.min(Math.max(campaignARevenue - campaignACost, campaignBRevenue - campaignBCost), Math.max(campaignARevenue, campaignBRevenue));
if (maxMinValue == campaignARevenue - campaignACost) {
System.out.println("The Max-Min value is " + maxMinValue + ". The decision-maker should choose Campaign A.");
} else if (maxMinValue == campaignBRevenue - campaignBCost) {
System.out.println("The Max-Min value is " + maxMinValue + ". The decision-maker should choose Campaign B.");
} else {
System.out.println("The Max-Min value is " + maxMinValue + ". The decision-maker can choose either Campaign A or Campaign B.");
}
}
}
```
# Implementing the Max-Min method in Java
## Exercise
Instructions:
Write a Java program that calculates the Max-Min value for the following decision problem:
A company wants to choose between three advertising campaigns, Campaign A, Campaign B, and Campaign C. Campaign A has an expected return of $12,000 and a cost of $6,000. Campaign B has an expected return of $10,000 and a cost of $5,000. Campaign C has an expected return of $8,000 and a cost of $4,000. The decision-maker wants to maximize the expected return while minimizing the cost.
### Solution
```java
public class MaxMinExample {
public static void main(String[] args) {
int[] campaignRevenue = {12000, 10000, 8000};
int[] campaignCost = {6000, 5000, 4000};
int maxMinValue = Integer.MAX_VALUE;
int chosenCampaign = -1;
for (int i = 0; i < campaignRevenue.length; i++) {
int maxValue = Math.max(campaignRevenue[i] - campaignCost[i], campaignRevenue[i]);
if (maxValue < maxMinValue) {
maxMinValue = maxValue;
chosenCampaign = i;
}
}
System.out.println("The Max-Min value is " + maxMinValue + ". The decision-maker should choose Campaign " + (chosenCampaign + 1) + ".");
}
}
```
# Developing a Java program for optimal decision-making
## Exercise
Instructions:
Write a Java program that takes user input for the expected return and cost of each decision option. The program should calculate the Max-Min value and output the decision-maker's choice based on this value.
### Solution
```java
import java.util.Scanner;
public class MaxMinExample {
public static void main(String[] args) {
Scanner scanner = new Scanner(System.in);
System.out.println("Enter the number of decision options:");
int numOptions = scanner.nextInt();
int[] campaignRevenue = new int[numOptions];
int[] campaignCost = new int[numOptions];
for (int i = 0; i < numOptions; i++) {
System.out.println("Enter the expected return for Campaign " + (i + 1) + ":");
campaignRevenue[i] = scanner.nextInt();
System.out.println("Enter the cost for Campaign " + (i + 1) + ":");
campaignCost[i] = scanner.nextInt();
}
int maxMinValue = Integer.MAX_VALUE;
int chosenCampaign = -1;
for (int i = 0; i < campaignRevenue.length; i++) {
int maxValue = Math.max(campaignRevenue[i] - campaignCost[i], campaignRevenue[i]);
if (maxValue < maxMinValue) {
maxMinValue = maxValue;
chosenCampaign = i;
}
}
System.out.println("The Max-Min value is " + maxMinValue + ". The decision-maker should choose Campaign " + (chosenCampaign + 1) + ".");
}
}
```
# Testing and debugging the Java program
## Exercise
Instructions:
Test the Java program you developed in the previous section using the following values for the expected return and cost of each decision option:
- Campaign A: $12,000 and $6,000
- Campaign B: $10,000 and $5,000
- Campaign C: $8,000 and $4,000
### Solution
The Java program should output the following result:
```
The Max-Min value is 8000. The decision-maker should choose Campaign 3.
```
# Conclusion: The role of the Max-Min method in decision making and optimization
In conclusion, the Max-Min method is a powerful optimization technique that can be applied to a wide range of decision-making problems. By using the Max-Min method in Java, decision-makers can find the best decision among multiple alternatives while balancing multiple objectives with conflicting priorities.
The Max-Min method can be implemented using conditional statements, loops, and data structures in Java. This flexibility allows the method to be adapted to various decision-making scenarios and user requirements.
In this textbook, we have covered the fundamentals of the Max-Min method, its application to Java programming, and the implementation of the method in Java programs. We have also discussed the importance of testing and debugging these programs to ensure their accuracy and reliability.
In summary, the Max-Min method is a valuable tool for decision-makers who need to balance multiple objectives with conflicting priorities. By using the Max-Min method in Java, decision-makers can make more informed and optimal choices in their decision-making processes. | Textbooks |
Journal of Global Optimization
Nonmonotone line searches for unconstrained multiobjective optimization problems
Kanako Mita
Ellen H. Fukuda
Nobuo Yamashita
In the last two decades, many descent methods for multiobjective optimization problems were proposed. In particular, the steepest descent and the Newton methods were studied for the unconstrained case. In both methods, the search directions are computed by solving convex subproblems, and the stepsizes are obtained by an Armijo-type line search. As a consequence, the objective function values decrease at each iteration of the algorithms. In this work, we consider nonmonotone line searches, i.e., we allow the increase of objective function values in some iterations. Two well-known types of nonmonotone line searches are considered here: the one that takes the maximum of recent function values, and the one that takes their average. We also propose a new nonmonotone technique specifically for multiobjective problems. Under reasonable assumptions, we prove that every accumulation point of the sequence produced by the nonmonotone version of the steepest descent and Newton methods is Pareto critical. Moreover, we present some numerical experiments, showing that the nonmonotone technique is also efficient in the multiobjective case.
Multiobjective optimization Steepest descent method Newton method Nonmonotone line search Pareto optimality
This work was supported by the Kyoto University Foundation, and the Grant-in-Aid for Scientific Research (C) (17K00032 and 19K11840) from Japan Society for the Promotion of Science.
We would like to thank the anonymous referees for their suggestions, which improved the original version of the paper.
Here, we list the test problems used in Sect. 7. For each problem, we state the original reference, the number of variables n, the number of objective functions m, the convexity property, the objective functions, and the bounds L and U of the box constraints.
Das and Dennis (DD1) [1]: \(n=5\), \(m=2\), nonconvex,1\(^{,}\)2
$$\begin{aligned} F_{1}(x)&=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2},\\ F_{2}(x)&=3x_{1}+2x_{2}-\frac{x_{3}}{3}+0.01(x_{4}-x_{5})^{3}, \end{aligned}$$
\(L=(-20, \ldots , -20)^{\top }\), and \(U=(20, \ldots , 20)^{\top }\).
Fliege, Graña Drummond and Svaiter (FDS) [4]: \(n=10\), \(m=3\), convex,\(^2\)
$$\begin{aligned} F_{1}(x)&=\frac{1}{n^{2}}\sum _{i=1}^{n}i(x_{i}-i)^{4},\\ F_{2}(x)&=\exp \left( \sum _{i=1}^{n}\frac{x_{i}}{n}\right) +\Vert x\Vert _{2}^{2},\\ F_{3}(x)&=\frac{1}{n(n+1)}\sum _{i=1}^{n}i(n-i+1)e^{-x_{i}}, \end{aligned}$$
\(L=(-2, \ldots , -2)^{\top }\), and \(U=(2,\ldots , 2)^{\top }\).
Jin, Olhofer and Sendhoff (JOS1) [8]: \(n=5\), \(m=2\), quadratic convex,\(^2\)
$$\begin{aligned} F_{1}(x)&=\frac{1}{n}\sum _{i=1}^{n}x_{i}^{2},\\ F_{2}(x)&=\frac{1}{n}\sum _{i=1}^{n}(x_{i}-2)^{2}, \end{aligned}$$
\(L=(-2,\ldots , -2)^{\top }\), and \(U=(2,\ldots , 2)^{\top }\).
Kim and Weck (KW2) [9]: \(n=2\), \(m=2\), nonconvex,
$$\begin{aligned} F_{1}(x) =&-3(1-x_{1})^{2} \exp (-x_{1}^{2}-(x_{2}+1)^{2})\\&+ 10\left( \frac{x_{1}}{5}-x_{1}^{3}-x_{2}^{5}\right) \exp (-x_{1}^{2}-x_{2}^{2})\\&+3 \exp (-(x_{1}+2)^{2}-x_{2}^{2})-0.5(2x_{1}+x_{2}),\\ F_{2}(x) =&-3(1+x_{2})^{2} \exp (-x_{2}^{2}-(1-x_{1})^{2}) \\&+ 10\left( -\frac{x_{2}}{5}+x_{2}^{3}+x_{1}^{5}\right) \exp (-x_{1}^{2}-x_{2}^{2})\\&+ 3 \exp (-(2-x_{2})^{2}-x_{1}^{2}), \end{aligned}$$
\(L=(-3,-3)^{\top }\), and \(U=(3,3)^{\top }\).
Stadler and J. Dauer (SD) [18]: \(n=4\), \(m=2\), convex,
$$\begin{aligned} F_{1}(x)&=2x_{1}+\sqrt{2}x_{2}+\sqrt{2}x_{3}+x_{4},\\ F_{2}(x)&=\frac{2}{x_{1}}+\frac{2\sqrt{2}}{x_{2}}+\frac{2\sqrt{2}}{x_{3}}+\frac{2}{x_{4}}, \end{aligned}$$
\(L=\left( 1, \sqrt{2}, \sqrt{2}, 1\right) ^{\top }\), and \(U=(3, 3, 3, 3)^{\top }\).
Zitzler, Deb and Thiele (ZDT1) [21]: \(n=30\), \(m=2\), convex,\(^2\)
$$\begin{aligned} F_{1}(x)&=x_{1},\\ F_{2}(x)&=g(x)\left( 1-\sqrt{\frac{x_{1}}{g(x)}}\right) , \end{aligned}$$
with \(g(x)=1+9\sum _{i=2}^{n} x_{i}/(n-1)\), \(L=(0,\ldots , 0)^{\top }\), and \(U=\left( \frac{1}{100},\ldots , \frac{1}{100}\right) ^{\top }\).
Zitzler, Deb and Thiele (ZDT4) [21]: \(n=10\), \(m=2\), nonconvex,\(^2\)
with \(g(x)=1+10(n-1)+\sum _{i=2}^{n}\left( x_{i}^{2}-10\cos (4\pi x_{i})\right) \), \(L=\left( \frac{1}{100}, -5,\ldots , -5\right) ^{\top }\), and \(U=(1, 5,\ldots , 5)^{\top }\).
Toint (TOI4) [19, Problem 4]: \(n=4\), \(m=2\), convex,3
$$\begin{aligned} F_1(x)&=x_1^2+x_2^2+1,\\ F_2(x)&=0.5\left( (x_1-x_2)^2+(x_3-x_4)^2\right) +1, \end{aligned}$$
TRIDIA [19, Problem 8]: \(n=3\), \(m=3\), convex,\(^{22}\)
$$\begin{aligned} F_1(x)&=(2x_{1}-1)^{2}, \\ F_2(x)&=2(2x_{1}-x_{2})^{2}, \\ F_3(x)&=3(2x_{2}-x_{3})^{2}, \end{aligned}$$
\(L=(-1, -1, -1)^{\top }\), and \(U=(1, 1, 1)^{\top }\).
Shifted TRIDIA [19, Problem 9]: \(n=4\), \(m=4\), nonconvex,\(^{23}\)
$$\begin{aligned} F_{1}(x)&=(2x_{1}-1)^{2}+x_{2}^{2},\\ F_{i}(x)&=i(2x_{i-1}-x_{i})^{2}-(i-1)x_{i-1}^{2}+ix_{i}^{2} \quad i=2,3,\\ F_{4}(x)&=4(2x_{3}-x_{4})^{2}-3x_{3}^{2}, \end{aligned}$$
Rosenbrock [19, Problem 10]: \(n=4\), \(m=3\), nonconvex,\(^{23}\)
$$\begin{aligned} F_{i}(x)&=100(x_{i+1}-x_{i}^{2})^{2}+(x_{i+1}-1)^{2}, \quad i=1,2,3, \end{aligned}$$
Helical valley [14, Problem (7)]: \(n=3\), \(m=3\), nonconvex,\(^3\)
$$\begin{aligned} F_{1}(x)&=\left\{ \begin{array}{ll} \displaystyle { \left[ 10\left( x_3-\frac{5}{\pi }\arctan \left( \frac{x_2}{x_1}\right) \right) \right] ^2,} &{} \text{ if }\;x_1>0,\\ \displaystyle { \left[ 10\left( x_3-\frac{5}{\pi }\arctan \left( \frac{x_2}{x_1}\right) -5\right) \right] ^2,} &{} \text{ if }\;x_1<0 \end{array}\right. \\ F_{2}(x)&=\left( 10\left( (x_1^2+x_2^2)^{1/2}-1\right) \right) ^2,\\ F_{3}(x)&=x_3^2, \end{aligned}$$
Gaussian [14, Problem (9)]: \(n=3\), \(m=15\), nonconvex,\(^3\)
$$\begin{aligned} F_{i}(x)&=x_1\exp \left( \frac{-x_2(t_i-x_3)^2}{2}\right) -y_i, \end{aligned}$$
where \(t_i=(8-i)/2\), \(i=1,\ldots ,m\) and \(y_i\) is given as
\(y_i\)
\(L=(-2, -2, -2)^{\top }\), and \(U=(2, -2, 2)^{\top }\).
Brown and Dennis [14, Problem (16)]: \(n=4\), \(m=5\), nonconvex,\(^3\)
$$\begin{aligned} F_i(x)&=\left( x_1+t_i x_2-e^{t_i}\right) ^2+\left( x_3+x_4\sin (t_i)-\cos (t_i)\right) ^2, \end{aligned}$$
where \(t_i=i/5\), \(L=(-25, -5, -5, -1)^{\top }\), and \(U=(25, 5, 5, 1)^{\top }\).
Trigonometric [14, Problem (26)]: \(n=4\), \(m=4\), nonconvex,\(^{23}\)
$$\begin{aligned} F_i(x)&=\left( n-\sum _{j=1}^{n}\cos x_j+i\left( 1-\cos x_i\right) -\sin x_i\right) ^2, \quad i=1,\ldots ,4, \end{aligned}$$
Linear function – rank 1 [14, Problem (33)]: \(n=10\), \(m=4\), convex,\(^{23}\)
$$\begin{aligned} F_i(x)&=\left( i\left( \sum _{j=1}^{n}jx_j\right) -1\right) ^2, \quad i=1,\ldots ,4, \end{aligned}$$
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1.Department of Applied Mathematics and Physics, Graduate School of InformaticsKyoto UniversityKyotoJapan
Mita, K., Fukuda, E.H. & Yamashita, N. J Glob Optim (2019). https://doi.org/10.1007/s10898-019-00802-0
Accepted 24 June 2019 | CommonCrawl |
Branching quantifier
In logic a branching quantifier,[1] also called a Henkin quantifier, finite partially ordered quantifier or even nonlinear quantifier, is a partial ordering[2]
$\langle Qx_{1}\dots Qx_{n}\rangle $
of quantifiers for Q ∈ {∀,∃}. It is a special case of generalized quantifier. In classical logic, quantifier prefixes are linearly ordered such that the value of a variable ym bound by a quantifier Qm depends on the value of the variables
y1, ..., ym−1
bound by quantifiers
Qy1, ..., Qym−1
preceding Qm. In a logic with (finite) partially ordered quantification this is not in general the case.
Branching quantification first appeared in a 1959 conference paper of Leon Henkin.[3] Systems of partially ordered quantification are intermediate in strength between first-order logic and second-order logic. They are being used as a basis for Hintikka's and Gabriel Sandu's independence-friendly logic.
Definition and properties
The simplest Henkin quantifier $Q_{H}$ is
$(Q_{H}x_{1},x_{2},y_{1},y_{2})\varphi (x_{1},x_{2},y_{1},y_{2})\equiv {\begin{pmatrix}\forall x_{1}\,\exists y_{1}\\\forall x_{2}\,\exists y_{2}\end{pmatrix}}\varphi (x_{1},x_{2},y_{1},y_{2}).$
It (in fact every formula with a Henkin prefix, not just the simplest one) is equivalent to its second-order Skolemization, i.e.
$\exists f\,\exists g\,\forall x_{1}\forall x_{2}\,\varphi (x_{1},x_{2},f(x_{1}),g(x_{2})).$
It is also powerful enough to define the quantifier $Q_{\geq \mathbb {N} }$ (i.e. "there are infinitely many") defined as
$(Q_{\geq \mathbb {N} }x)\varphi (x)\equiv (\exists a)(Q_{H}x_{1},x_{2},y_{1},y_{2})[\varphi (a)\land (x_{1}=x_{2}\leftrightarrow y_{1}=y_{2})\land (\varphi (x_{1})\rightarrow (\varphi (y_{1})\land y_{1}\neq a))].$
Several things follow from this, including the nonaxiomatizability of first-order logic with $Q_{H}$ (first observed by Ehrenfeucht), and its equivalence to the $\Sigma _{1}^{1}$-fragment of second-order logic (existential second-order logic)—the latter result published independently in 1970 by Herbert Enderton[4] and W. Walkoe.[5]
The following quantifiers are also definable by $Q_{H}$.[2]
• Rescher: "The number of φs is less than or equal to the number of ψs"
$(Q_{L}x)(\varphi x,\psi x)\equiv \operatorname {Card} (\{x\colon \varphi x\})\leq \operatorname {Card} (\{x\colon \psi x\})\equiv (Q_{H}x_{1}x_{2}y_{1}y_{2})[(x_{1}=x_{2}\leftrightarrow y_{1}=y_{2})\land (\varphi x_{1}\rightarrow \psi y_{1})]$
• Härtig: "The φs are equinumerous with the ψs"
$(Q_{I}x)(\varphi x,\psi x)\equiv (Q_{L}x)(\varphi x,\psi x)\land (Q_{L}x)(\psi x,\varphi x)$
• Chang: "The number of φs is equinumerous with the domain of the model"
$(Q_{C}x)(\varphi x)\equiv (Q_{L}x)(x=x,\varphi x)$
The Henkin quantifier $Q_{H}$ can itself be expressed as a type (4) Lindström quantifier.[2]
Relation to natural languages
Hintikka in a 1973 paper[6] advanced the hypothesis that some sentences in natural languages are best understood in terms of branching quantifiers, for example: "some relative of each villager and some relative of each townsman hate each other" is supposed to be interpreted, according to Hintikka, as:[7][8]
${\begin{pmatrix}\forall x_{1}\,\exists y_{1}\\\forall x_{2}\,\exists y_{2}\end{pmatrix}}[(V(x_{1})\wedge T(x_{2}))\rightarrow (R(x_{1},y_{1})\wedge R(x_{2},y_{2})\wedge H(y_{1},y_{2})\wedge H(y_{2},y_{1}))].$
which is known to have no first-order logic equivalent.[7]
The idea of branching is not necessarily restricted to using the classical quantifiers as leaves. In a 1979 paper,[9] Jon Barwise proposed variations of Hintikka sentences (as the above is sometimes called) in which the inner quantifiers are themselves generalized quantifiers, for example: "Most villagers and most townsmen hate each other."[7] Observing that $\Sigma _{1}^{1}$ is not closed under negation, Barwise also proposed a practical test to determine whether natural language sentences really involve branching quantifiers, namely to test whether their natural-language negation involves universal quantification over a set variable (a $\Pi _{1}^{1}$ sentence).[10]
Hintikka's proposal was met with skepticism by a number of logicians because some first-order sentences like the one below appear to capture well enough the natural language Hintikka sentence.
$[\forall x_{1}\,\exists y_{1}\,\forall x_{2}\,\exists y_{2}\,\varphi (x_{1},x_{2},y_{1},y_{2})]\wedge [\forall x_{2}\,\exists y_{2}\,\forall x_{1}\,\exists y_{1}\,\varphi (x_{1},x_{2},y_{1},y_{2})]$
where
$\varphi (x_{1},x_{2},y_{1},y_{2})$
denotes
$(V(x_{1})\wedge T(x_{2}))\rightarrow (R(x_{1},y_{1})\wedge R(x_{2},y_{2})\wedge H(y_{1},y_{2})\wedge H(y_{2},y_{1}))$
Although much purely theoretical debate followed, it wasn't until 2009 that some empirical tests with students trained in logic found that they are more likely to assign models matching the "bidirectional" first-order sentence rather than branching-quantifier sentence to several natural-language constructs derived from the Hintikka sentence. For instance students were shown undirected bipartite graphs—with squares and circles as vertices—and asked to say whether sentences like "more than 3 circles and more than 3 squares are connected by lines" were correctly describing the diagrams.[7]
See also
• Game semantics
• Dependence logic
• Independence-friendly logic (IF logic)
• Mostowski quantifier
• Lindström quantifier
• Nonfirstorderizability
References
1. Stanley Peters; Dag Westerståhl (2006). Quantifiers in language and logic. Clarendon Press. pp. 66–72. ISBN 978-0-19-929125-0.
2. Antonio Badia (2009). Quantifiers in Action: Generalized Quantification in Query, Logical and Natural Languages. Springer. p. 74–76. ISBN 978-0-387-09563-9.
3. Henkin, L. "Some Remarks on Infinitely Long Formulas". Infinitistic Methods: Proceedings of the Symposium on Foundations of Mathematics, Warsaw, 2–9 September 1959, Panstwowe Wydawnictwo Naukowe and Pergamon Press, Warsaw, 1961, pp. 167–183. OCLC 2277863
4. Jaakko Hintikka and Gabriel Sandu, "Game-theoretical semantics", in Handbook of logic and language, ed. J. van Benthem and A. ter Meulen, Elsevier 2011 (2nd ed.) citing Enderton, H.B., 1970. Finite partially-ordered quantifiers. Z. Math. Logik Grundlag. Math. 16, 393–397 doi:10.1002/malq.19700160802.
5. Blass, A.; Gurevich, Y. (1986). "Henkin quantifiers and complete problems" (PDF). Annals of Pure and Applied Logic. 32: 1–16. doi:10.1016/0168-0072(86)90040-0. hdl:2027.42/26312. citing W. Walkoe, Finite partially-ordered quantification, Journal of Symbolic Logic 35 (1970) 535–555. JSTOR 2271440
6. Hintikka, J. (1973). "Quantifiers vs. Quantification Theory". Dialectica. 27 (3–4): 329–358. doi:10.1111/j.1746-8361.1973.tb00624.x.
7. Gierasimczuk, N.; Szymanik, J. (2009). "Branching Quantification v. Two-way Quantification" (PDF). Journal of Semantics. 26 (4): 367. doi:10.1093/jos/ffp008.
8. Sher, G. (1990). "Ways of branching quantifers" (PDF). Linguistics and Philosophy. 13 (4): 393–422. doi:10.1007/BF00630749. S2CID 61362436.
9. Barwise, J. (1979). "On branching quantifiers in English". Journal of Philosophical Logic. 8: 47–80. doi:10.1007/BF00258419. S2CID 31950692.
10. Hand, Michael (1998). "Reviewed work: On Branching Quantifiers in English, Jon Barwise; Branching Generalized Quantifiers and Natural Language. Generalized Quantifiers, Linguistic and Logical Approaches, Dag Westerståhl, Peter Gärdenfors; Ways of Branching Quantifiers, Gila Sher". The Journal of Symbolic Logic. 63 (4): 1611–1614. doi:10.2307/2586678. JSTOR 2586678. S2CID 117833401.
External links
• Game-theoretical quantifier at PlanetMath.
| Wikipedia |
Benefits of Prepartum Nest-building Behaviour on Parturition and Lactation in Sows - A Review
Yun, Jinhyeon;Valros, Anna 1519
https://doi.org/10.5713/ajas.15.0174 PDF KSCI
It is well known that prepartum sows have an innate motivation to build a nest before parturition. Under commercial conditions, however, the farrowing crate, which is widely used in modern pig husbandry, inhibits this innate behaviour through the lack of space, materials, or both. Thus, restriction of nest-building behaviour could generate increased stress, resulting in a decrease in maternal endogenous hormones. Hence, it could lead to detrimental effects on farrowing and lactating performance. Here we review interactions between prepartum nest-building behaviour, stress and maternal endogenous hormone levels, and discuss their effects on parturition, lactation, and welfare of sows and offspring.
A Genome-wide Scan for Selective Sweeps in Racing Horses
Moon, Sunjin;Lee, Jin Woo;Shin, Donghyun;Shin, Kwang-Yun;Kim, Jun;Choi, Ik-Young;Kim, Jaemin;Kim, Heebal 1525
Using next-generation sequencing, we conducted a genome-wide scan of selective sweeps associated with selection toward genetic improvement in Thoroughbreds. We investigated potential phenotypic consequence of putative candidate loci by candidate gene association mapping for the finishing time in 240 Thoroughbred horses. We found a significant association with the trait for Ral GApase alpha 2 (RALGAP2) that regulates a variety of cellular processes of signal trafficking. Neighboring genes around RALGAP2 included insulinoma-associated 1 (INSM1), pallid (PLDN), and Ras and Rab interactor 2 (RIN2) genes have similar roles in signal trafficking, suggesting that a co-evolving gene cluster located on the chromosome 22 is under strong artificial selection in racehorses.
Differential Expression of miR-34c and Its Predicted Target Genes in Testicular Tissue at Different Development Stages of Swine
Zhang, Xiaojun;Zhao, Wei;Li, Chuanmin;Yu, Haibin;Qiao, YanYan;Li, Aonan;Lu, Chunyan;Zhao, Zhihui;Sun, Boxing 1532
To verified the target genes of miR-34c, bioinformatics software was used to predict the targets of miR-34c. Three possible target genes of miR-34c related to spermatogenesis and male reproductive development: zinc finger protein 148 (ZNF148), kruppel-like factor 4 (KLF4), and platelet-derived growth factor receptor alpha (PDGFRA) were predicted. Then, the expression of miR-34c and its target genes were detected in swine testicular tissue at different developmental stages by quantitative polymerase chain reaction. The results suggested that the expression of PDGFRA has the highest negative correlation with miR-34c. Then immunohistochemical staining was done to observe the morphology of swine testicular tissue at 2-days and 3, 4, 5-months of age, which indicated that PDGFRA was mainly expressed in the support cells near the basement membrane during the early development stages of testicular tissue, but that the expression of PDGFRA was gradually reduced in later stages. Therefore, western blot analyzed that the highest expression of PDGFRA was generated in 2-days old testicular tissues and the expression levels reduced at 3 and 4-months old, which correlated with the results of immunohistochemical staining. In conclusion, PDGFRA is a target gene of miR-34c.
Genome-wide Association Study to Identify Quantitative Trait Loci for Meat and Carcass Quality Traits in Berkshire
Iqbal, Asif;Kim, You-Sam;Kang, Jun-Mo;Lee, Yun-Mi;Rai, Rajani;Jung, Jong-Hyun;Oh, Dong-Yup;Nam, Ki-Chang;Lee, Hak-Kyo;Kim, Jong-Joo 1537
Meat and carcass quality attributes are of crucial importance influencing consumer preference and profitability in the pork industry. A set of 400 Berkshire pigs were collected from Dasan breeding farm, Namwon, Chonbuk province, Korea that were born between 2012 and 2013. To perform genome wide association studies (GWAS), eleven meat and carcass quality traits were considered, including carcass weight, backfat thickness, pH value after 24 hours (pH24), Commission Internationale de l'Eclairage lightness in meat color (CIE L), redness in meat color (CIE a), yellowness in meat color (CIE b), filtering, drip loss, heat loss, shear force and marbling score. All of the 400 animals were genotyped with the Porcine 62K SNP BeadChips (Illumina Inc., USA). A SAS general linear model procedure (SAS version 9.2) was used to pre-adjust the animal phenotypes before GWAS with sire and sex effects as fixed effects and slaughter age as a covariate. After fitting the fixed and covariate factors in the model, the residuals of the phenotype regressed on additive effects of each single nucleotide polymorphism (SNP) under a linear regression model (PLINK version 1.07). The significant SNPs after permutation testing at a chromosome-wise level were subjected to stepwise regression analysis to determine the best set of SNP markers. A total of 55 significant (p<0.05) SNPs or quantitative trait loci (QTL) were detected on various chromosomes. The QTLs explained from 5.06% to 8.28% of the total phenotypic variation of the traits. Some QTLs with pleiotropic effect were also identified. A pair of significant QTL for pH24 was also found to affect both CIE L and drip loss percentage. The significant QTL after characterization of the functional candidate genes on the QTL or around the QTL region may be effectively and efficiently used in marker assisted selection to achieve enhanced genetic improvement of the trait considered.
Effect of a c-MYC Gene Polymorphism (g.3350G>C) on Meat Quality Traits in Berkshire
Oh, J.D.;Kim, E.S.;Lee, H.K.;Song, K.D. 1545
c-MYC (v-myelocytomatosis viral oncogene homologue) is a transcription factor that plays important role in many biological process including cell growth and differentiation, such as myogenesis and adipogenesis. In this study, we aimed to detect MYC gene polymorphisms, their genotype frequencies and to determine associations between these polymorphisms and meat quality traits in Berkshire pigs. We identified a single nucleotide polymorphism (SNP) in intron 2 of MYC gene by Sanger sequencing, i.e., g.3350G>C (rs321898326), that is only found in Berkshire pigs, but not in other breeds including Duroc, Landrace, and Yorkshire pigs that were used in this study. Genotypes of total 378 Berkshire pigs (138 sows and 240 boars) were determined using Hha I restriction enzyme digestion after polymerase chain reaction. Observed allele frequencies of GG, GC, and CC genotypes were 0.399, 0.508, and 0.093 respectively. Statistical analysis indicated that the g.3350G>C polymorphism was significantly associated with $pH_{45min}$ and cooking loss (p<0.05), suggesting that g.3350G>C SNP can be used for pre-selection of $pH_{45min}$ and cooking loss traits in Berkshire pigs.
Genome-wide Association Study (GWAS) and Its Application for Improving the Genomic Estimated Breeding Values (GEBV) of the Berkshire Pork Quality Traits
Lee, Young-Sup;Jeong, Hyeonsoo;Taye, Mengistie;Kim, Hyeon Jeong;Ka, Sojeong;Ryu, Youn-Chul;Cho, Seoae 1551
The missing heritability has been a major problem in the analysis of best linear unbiased prediction (BLUP). We introduced the traditional genome-wide association study (GWAS) into the BLUP to improve the heritability estimation. We analyzed eight pork quality traits of the Berkshire breeds using GWAS and BLUP. GWAS detects the putative quantitative trait loci regions given traits. The single nucleotide polymorphisms (SNPs) were obtained using GWAS results with p value <0.01. BLUP analyzed with significant SNPs was much more accurate than that with total genotyped SNPs in terms of narrow-sense heritability. It implies that genomic estimated breeding values (GEBVs) of pork quality traits can be calculated by BLUP via GWAS. The GWAS model was the linear regression using PLINK and BLUP model was the G-BLUP and SNP-GBLUP. The SNP-GBLUP uses SNP-SNP relationship matrix. The BLUP analysis using preprocessing of GWAS can be one of the possible alternatives of solving the missing heritability problem and it can provide alternative BLUP method which can find more accurate GEBVs.
Physiological Responses and Lactation to Cutaneous Evaporative Heat Loss in Bos indicus, Bos taurus, and Their Crossbreds
Jian, Wang;Ke, Yang;Cheng, Lu 1558
Cutaneous evaporative heat loss in Bos indicus and Bos taurus has been well documented. Nonetheless, how crossbreds with different fractional genetic proportions respond to such circumstances is of interest. A study to examine the physiological responses to cutaneous evaporative heat loss, also lactation period and milk yield, were conducted in Sahiwal (Bos indicus, n = 10, $444{\pm}64.8kg$, $9{\pm}2.9years$), Holstein Friesian (Bos taurus, HF100% (n = 10, $488{\pm}97.9kg$, $6{\pm}2.8years$)) and the following crossbreds: HF50% (n = 10, $355{\pm}40.7kg$, $2{\pm}0years$) and HF87.5% (n = 10, $489{\pm}76.8kg$, $7{\pm}1.8years$). They were allocated so as to determine the physiological responses of sweating rate (SR), respiration rate (RR), rectal temperature (RT), and skin temperature (ST) with and without hair from 06:00 h am to 15:00 h pm. And milk yield during 180 days were collected at days from 30 to 180. The ambient temperature-humidity-index (THI) increased from less than 80 in the early morning to more than 90 in the late afternoon. The interaction of THI and breed were highly affected on SR, RR, RT, and ST (p<0.01). The SR was highest in Sahiwal ($595g/m^2/h$) compared to HF100% ($227g/m^2/h$), and their crossbreds both HF50% ($335g/m^2/h$) and HF87.5% ($299g/m^2/h$). On the other hand, RR was higher in HF87.5% (54 bpm) and both HF100% (48 bpm) and HF50% (42 bpm) than Sahiwal (25 bpm) (p<0.01). The RT showed no significant differences as a result of breed (p>0.05) but did change over time. The ST with and without hair were similar, and was higher in HF100% ($37.4^{\circ}C$; $38.0^{\circ}C$) and their crossbred HF50% ($35.5^{\circ}C$; $35.5^{\circ}C$) and HF87.5% ($37.1^{\circ}C$; $37.9^{\circ}C$) than Sahiwal ($34.8^{\circ}C$; $34.8^{\circ}C$) (p<0.01). Moreover, the early lactation were higher at HF100% (25 kg) and 87.5% (25 kg) than HF50% (23 kg) which were higher than Sahiwal (18 kg) while the peak period of lactation was higher at HF100% (35 kg) than crossbreds both HF87.5% and HF50% (32 kg) which was higher than Sahiwal (26 kg) (p<0.05). In conclusion, sweating and respiration were the main vehicle for dissipating excess body heat for Sahiwal, HF and crossbreds, respectively. The THI at 76 to 80 were the critical points where the physiological responses to elevated temperature displayed change.
Embryo Aggregation Promotes Derivation Efficiency of Outgrowths from Porcine Blastocysts
Lee, Sang-Goo;Park, Jin-Kyu;Choi, Kwang-Hwan;Son, Hye-Young;Lee, Chang-Kyu 1565
Porcine embryonic stem cells (pESCs) have become an advantageous experimental tool for developing therapeutic applications and producing transgenic animals. However, despite numerous reports of putative pESC lines, deriving validated pESC lines from embryos produced in vitro remains difficult. Here, we report that embryo aggregation was useful for deriving pESCs from in vitro-produced embryos. Blastocysts derived from embryo aggregation formed a larger number of colonies and maintained cell culture stability. Our derived cell lines demonstrated expression of pluripotent markers (alkaline phosphatase, Oct4, Sox2, and Nanog), an ability to form embryoid bodies, and the capacity to differentiate into the three germ layers. A cytogenetic analysis of these cells revealed that all lines derived from aggregated blastocysts had normal female and male karyotypes. These results demonstrate that embryo aggregation could be a useful technique to improve the efficiency of deriving ESCs from in vitro-fertilized pig embryos, studying early development, and deriving pluripotent ESCs in vitro in other mammals.
Effect of Soyabean Isoflavones Exposure on Onset of Puberty, Serum Hormone Concentration and Gene Expression in Hypothalamus, Pituitary Gland and Ovary of Female Bama Miniature Pigs
Fan, Juexin;Zhang, Bin;Li, Lili;Xiao, Chaowu;Oladele, Oso Abimbola;Jiang, Guoli;Ding, Hao;Wang, Shengping;Xing, Yueteng;Xiao, Dingfu;Yin, Yulong 1573
This study was to investigate the effect of soyabean isoflavones (SIF) on onset of puberty, serum hormone concentration, and gene expression in hypothalamus, pituitary and ovary of female Bama miniature pigs. Fifty five, 35-days old pigs were randomly assigned into 5 treatment groups consisting of 11 pigs per treatment. Results showed that dietary supplementation of varying dosage (0, 250, 500, and 1,250 mg/kg) of SIF induced puberty delay of the pigs with the age of puberty of pigs fed basal diet supplemented with 1,250 mg/kg SIF was significantly higher (p<0.05) compared to control. Supplementation of SIF or estradiol valerate (EV) reduced (p<0.05) serum gonadotrophin releasing hormone and luteinizing hormone concentration, but increased follicle-stimulating hormone concentration in pigs at 4 months of age. The expression of KiSS-1 metastasis-suppressor (KISS1), steroidogenic acute regulatory protein (StAR) and 3-beta-hydroxysteroid dehydrogenase/delta-5-delta-4 isomerase ($3{\beta}-HSD$) was reduced (p<0.01) in SIF-supplemented groups. Expression of gonadotropin-releasing hormone receptor in the pituitary of miniature pigs was reduced (p<0.05) compared to the control when exposed to 250, 1,250 mg/kg SIF and EV. Pigs on 250 mg/kg SIF and EV also showed reduced (p<0.05) expression of cytochrome P450 19A1 compared to the control. Our results indicated that dietary supplementation of SIF induced puberty delay, which may be due to down-regulation of key genes that play vital roles in the synthesis of steroid hormones.
Effect of Lipid Sources with Different Fatty Acid Profiles on Intake, Nutrient Digestion and Ruminal Fermentation of Feedlot Nellore Steers
Fiorentini, Giovani;Carvalho, Isabela P.C.;Messana, Juliana D.;Canesin, Roberta C.;Castagnino, Pablo S.;Lage, Josiane F.;Arcuri, Pedro B.;Berchielli, Telma T. 1583
The present study was conducted to determine the effect of lipid sources with different fatty acid profiles on nutrient digestion and ruminal fermentation. Ten rumen and duodenal fistulated Nellore steers (268 body weight${\pm}27kg$) were distributed in a duplicated $5{\times}5$ Latin square. Dietary treatments were as follows: without fat (WF), palm oil (PO), linseed oil (LO), protected fat (PF; Lactoplus), and whole soybeans (WS). The roughage feed was corn silage (600 g/kg on a dry matter [DM] basis) plus concentrate (400 g/kg on a DM basis). The higher intake of DM and organic matter (OM) (p<0.001) was found in animals on the diet with PF and WF (around 4.38 and 4.20 kg/d, respectively). Treatments with PO and LO decreased by around 10% the total digestibility of DM and OM (p<0.05). The addition of LO decreased by around 22.3% the neutral detergent fiber digestibility (p = 0.047) compared with other diets. The higher microbial protein synthesis was found in animals on the diet with LO and WS (33 g N/kg OM apparently digested in the rumen; p = 0.040). The highest C18:0 and linolenic acid intakes occurred in animals fed LO (p<0.001), and the highest intake of oleic (p = 0.002) and C16 acids (p = 0.022) occurred with the diets with LO and PF. Diet with PF decreased biohydrogenation extent (p = 0.05) of C18:1 n9,c, C18:2 n6,c, and total unsaturated fatty acids (UFA; around 20%, 7%, and 13%, respectively). The diet with PF and WF increased the concentration of $NH_3-N$ (p<0.001); however, the diet did not change volatile fatty acids (p>0.05), such as the molar percentage of acetate, propionate, butyrate and the acetate:propionate ratio. Treatments PO, LO and with WS decreased by around 50% the concentration of protozoa (p<0.001). Diets with some type of protection (PF and WS) decreased the effects of lipid on ruminal fermentation and presented similar outflow of benefit UFA as LO.
Effect of Feeding a Mixed Microbial Culture Fortified with Trace Minerals on the Performance and Carcass Characteristics of Late-fattening Hanwoo Steers: A Field Study
Kwak, W.S.;Kim, Y.I.;Lee, S.M.;Lee, Y.H.;Choi, D.Y. 1592
This study was conducted to determine the effects of feeding a trace minerals-fortified microbial culture (TMC) on the performance and carcass characteristics of late-fattening Hanwoo steers. A mixture of microbes (0.6% [v/w] of Enterobacter sp., Bacillus sp., Lactobacillus sp., and Saccharomyces sp.) was cultured with 99% feedstuff for ensiling and 0.4% trace minerals (zinc, selenium, copper, and cobalt). Sixteen late-fattening steers (mean age, 21.8 months) were allocated to two diets: a control diet (concentrate mix and rice straw) and a treated diet (control diet+3.3% TMC). At a mean age of 31.1 months, all the steers were slaughtered. The addition of TMC to the diet did not affect the average daily weight gain of the late fattening steers, compared with that of control steers. Moreover, consuming the TMC-supplemented diet did not affect cold carcass weight, yield traits such as back fat thickness, longissimus muscle area, yield index or yield grade, or quality traits such as meat color, fat color, texture, maturity, marbling score, or quality grade. However, consumption of a TMC-supplemented diet increased the concentrations of zinc, selenium, and sulfur (p<0.05) in the longissimus muscle. With respect to amino acids, animals consuming TMC showed increased (p<0.05) concentrations of lysine, leucine, and valine among essential amino acids and a decreased (p<0.05) concentration of proline among non-essential amino acids. In conclusion, the consumption of a TMC-supplemented diet during the late-fattening period elevated the concentrations of certain trace minerals and essential amino acids in the longissimus muscle, without any deleterious effects on performance and other carcass characteristics of Hanwoo steers.
Effects of Supplemental Mannanoligosaccharides on Growth Performance, Faecal Characteristics and Health in Dairy Calves
Kara, Cagdas;Cihan, Huseyin;Temizel, Mutlu;Catik, Serkan;Meral, Yavuz;Orman, Abdulkadir;Yibar, Artun;Gencoglu, Hidir 1599
Twenty Holstein calves were used to investigate the effects of mannanoligosaccharides (MOS) supplementation in the whole milk on growth performance, faecal score, faecal pH, selected faecal bacterial populations and health during the preweaning period. Healthy calves selected by clinical examination were allocated to one of the two groups (control [CG] and experimental [EG]) at 5 days old. Each group consisted of 5 male and 5 female calves. Each calf in EG was supplemented with 7 g/d of a MOS product (Celmanax) from 5 days to 56 days of age. MOS supplement was mixed with the whole milk once in the morning and administered to the calves in EG via nipple bottle, whereas the calves in CG were fed the whole milk without MOS. Calves were weaned at 56 days of age. The final body weight, average daily weight gain (ADG) and average daily feed intake (ADFI) were statistically similar (p>0.05) but were higher by 3.70%, 6.66%, and 10.97%, respectively, in MOS than in control calves. Feed efficiency (ADG/ADFI) was also similar in two calves group. While faecal scores did not differ on day 5, 7, 14, 21, 28, 42, 49, and 56 between groups, EG had a higher faecal score (p = 0.05) than CG on day 35. Faecal concentration of Lactobacillus was lower (p<0.05) in EG compared with CG. No differences (p>0.05) in faecal concentrations of Bifidobacterium, Clostridium perfringens, and Escherichia coli were found between groups. Although there were no significant differences (p>0.05) in the incidence of diarrhoea, treatment days for diarrhoea and the costs associated with diarrhoea treatments between groups, collectively, the observed reductions in treatment days and the cost of diarrhoea treatments accompanying increases in final body weight, ADG and ADFI for EG may indicate potential benefit of MOS in treatment of diarrhoea.
The Effect of Yerba Mate (Ilex Paraguariensis) Supplementation on Nutrient Degradability in Dairy Cows: An In sacco and In vitro Study
Hartemink, Ellen;Giorgio, Daniela;Kaur, Ravneet;Di Trana, Adriana;Celi, Pietro 1606
This study was carried out to investigate the effects of Yerba Mate (YM) supplementation on nutrients' degradation, in vitro dry matter disappearance, gas production and rumen ammonia concentration. Three rumen-fistulated Holstein Friesian cows were used for the in situ incubations and provided rumen liquor for in vitro incubations. The inclusion of YM in a control diet (pasture+pellets) affected some in sacco degradation parameters. YM supplementation decreased the effective degradability and degradation rate of pasture crude protein (CP), and it seems to slow down the degradation of pasture neutral detergent fiber. A significant increase of degradation of pasture acid detergent fiber (ADF) was detected after YM inclusion in the control diet. YM supplementation reduced in vitro gas production of pasture and ammonia concentration of pellets. The addition of YM in ruminant diet could decrease ammonia production and increase protein availability for productive purposes. The moderate presence of tannins in YM could have affected the degradation kinetics of pasture CP and ADF and the ammonia production of pellets.
Growth Performance of Early Finishing Gilts as Affected by Different Net Energy Concentrations in Diets
Lee, Gang Il;Kim, Kwang-Sik;Kim, Jong Hyuk;Kil, Dong Yong 1614
The objectives of the current experiment were to study the response of the growth performance of early finishing gilts to different net energy (NE) concentrations in diets, and to compare the NE values of diets between calculated NE values and measured NE values using French and Dutch CVB (Centraal Veevoederbureau; Central Bureau for Livestock Feeding) NE systems. In a metabolism trail, the NE concentrations in five diets used for the growth trial were determined based on digestible nutrient concentrations, digestible energy, and metabolizable energy using a replicated $5{\times}5$ Latin square design with 10 barrows (initial body weight [BW], $39.2{\pm}2.2kg$). In a growth trial, a total of 60 early finishing gilts (Landrace${\times}$Yorkshire; initial BW, $47.7{\pm}3.5kg$) were allotted to five dietary treatments of 8.0, 9.0, 10.0, 11.0, and 12.0 MJ NE/kg (calculated, as-is basis) with 12 replicate pens and one pig per pen in a 42-d feeding experiment. The NE and amino acid (AA) concentrations in all diets were calculated based on the values from NRC (2012). Ratios between standardized ileal digestible AA and NE concentrations in all diets were closely maintained. Pigs were allowed ad libitum access to feed and water. Results indicated that calculated NE concentrations in diets (i.e., five dietary treatments) were close to measured NE concentrations using French NE system in diets. The final BW was increased (linear and quadratic, p<0.05) with increasing NE concentrations in diets. Furthermore, average daily gain (ADG) was increased (linear and quadratic, p<0.01) with increasing NE concentrations in diets. There was a quadratic relationship (p<0.01) between average daily feed intake and NE concentrations in diets. Feed efficiency (G:F) was also increased (linear, p<0.01) as NE concentrations in diets were increased. The NE intake per BW gain (kcal NE/kg of BWG) was increased (linear, p<0.01) with increasing NE concentrations in diets that were predicted from both French and Dutch CVB NE systems. Linear regression indicated that predictability of daily NE intake from the BW of pigs was very low for both French ($R^2$, 0.366) and Dutch CVB ($R^2$, 0.374) NE systems. In conclusion, increasing NE concentrations in diets increase BW, ADG, G:F, and NE intake per BW gain of early finishing gilts. The BW of early finishing gilts is not a good sole variable for the prediction of daily NE intake.
Evaluation of Acid Digestion Procedures to Estimate Mineral Contents in Materials from Animal Trials
Palma, M.N.N.;Rocha, G.C.;Valadares Filho, S.C.;Detmann, E. 1624
Rigorously standardized laboratory protocols are essential for meaningful comparison of data from multiple sites. Considering that interactions of minerals with organic matrices may vary depending on the material nature, there could be peculiar demands for each material with respect to digestion procedure. Acid digestion procedures were evaluated using different nitric to perchloric acid ratios and one- or two-step digestion to estimate the concentration of calcium, phosphorus, magnesium, and zinc in samples of carcass, bone, excreta, concentrate, forage, and feces. Six procedures were evaluated: ratio of nitric to perchloric acid at 2:1, 3:1, and 4:1 v/v in a one- or two-step digestion. There were no direct or interaction effects (p>0.01) of nitric to perchloric acid ratio or number of digestion steps on magnesium and zinc contents. Calcium and phosphorus contents presented a significant (p<0.01) interaction between sample type and nitric to perchloric acid ratio. Digestion solution of 2:1 v/v provided greater (p<0.01) recovery of calcium and phosphorus from bone samples than 3:1 and 4:1 v/v ratio. Different acid ratios did not affect (p>0.01) calcium or phosphorus contents in carcass, excreta, concentrate, forage, and feces. Number of digestion steps did not affect mineral content (p>0.01). Estimated concentration of calcium, phosphorus, magnesium, and zinc in carcass, excreta, concentrated, forage, and feces samples can be performed using digestion solution of nitric to perchloric acid 4:1 v/v in a one-step digestion. However, samples of bones demand a stronger digestion solution to analyze the mineral contents, which is represented by an increased proportion of perchloric acid, being recommended a digestion solution of nitric to perchloric acid 2:1 v/v in a one-step digestion.
Comparison of Carcass and Sensory Traits and Free Amino Acid Contents among Quality Grades in Loin and Rump of Korean Cattle Steer
Piao, Min Yu;Jo, Cheorun;Kim, Hyun Joo;Lee, Hyun Jung;Kim, Hyun Jin;Ko, Jong-Youl;Baik, Myunggi 1629
This study was performed to compare carcass traits, sensory characteristics, physiochemical composition, and contents of nucleotides, collagen, and free amino acids among quality grades (QG) and to understand the association between QG and above parameters in loin and rump of Korean cattle steer. Loin and rump samples were obtained from 48 Korean cattle steers with each of four QG (QG 1++, 1+, 1, and 2; average 32 months of age). Carcass weight and marbling score (MS) were highest in QG 1++, whereas texture score measured by a meat grader was highest in QG 2. A correlation analysis revealed that MS (r = 0.98; p<0.01) and fat content (r = 0.73; p<0.01) had strong positive correlations with QG and that texture had a strong negative correlation (r = -0.78) with QG. Fat content in loin was highest but protein and moisture contents were lowest in QG 1++. Our results confirmed that a major determinant of QG is the MS; thus, intramuscular fat content. The International Commission on Illumination $L^*$, $a^*$, and $b^*$ values in loin were highest in QG 1++. Numeric values of shear force in loin were lowest in QG 1++, whereas those of tenderness, juiciness, and overall acceptability tended to be highest in QG 1++ without statistical significance. QG was strongly correlated with juiciness (r = 0.81; p<0.01) and overall acceptability (r = 0.87; p<0.001). All sensory characteristics were higher (p<0.05) in loin than those in rump. Adenosine-5'-monophosphate (AMP) and inosine-5'-monophosphate (IMP) contents in both loin and rump did not differ among QGs. No nucleotide (AMP, IMP, inosine, hypoxanthine) was correlated with any of the sensory traits. Total, soluble, and insoluble collagen contents in loin were higher in QG 1++ than those in QG 1. All three collagens had lower content in loin than that in rump. All three collagens were positively correlated with tenderness, juiciness, and overall acceptability. Glutamic acid content did not significantly differ among the four QGs in either loin or rump. In conclusion, it is confirmed that QG is associated with sensory traits but nucleotide contents in beef may not be a major factor determining meat palatability in the present study.
Dietary Protein Sources Affect Internal Quality of Raw and Cooked Shell Eggs under Refrigerated Conditions
Wang, X.C.;Zhang, H.J.;Wu, S.G.;Yue, H.Y.;Wang, J.;Li, Jie;Qi, Guang-Hai 1641
This study was conducted to evaluate the effects of various protein sources (soybean meal, SBM; cottonseed protein, CSP; double-zero rapeseed meal, DRM) on the internal quality of refrigerated eggs. A total of 360 laying hens (32 wk of age) were randomly allotted to six treatment groups (five replicates per treatment) and fed diets containing SBM, CSP, or DRM individually or in combination with equal crude protein content (SBM-CSP, SBM-DRM, and CSP-DRM) as the protein ingredient(s). A $6{\times}3$ factorial arrangement was employed with dietary types and storage time (0 d, 2 wk, and 4 wk) as the main effects. After 12 wk of diet feeding, a total of 270 eggs were collected for egg quality determination. The egg Haugh unit (HU) in the CSP, SBM-DRM, and DRM groups were significantly lower than those in the SBM and SBM-CSP groups. The hardness and springiness of the cooked yolk in the CSP group were significantly higher than those in the other treatment groups. A lower HU, lower yolk index and higher albumen pH were observed in the DRM group compared to the SBM and SBM-CSP groups when the eggs were stored to 4 wk, and the HU was improved in the CSP-DRM group compared to the DRM group (p<0.05). Higher yolk hardness was observed in the CSP group compared to the other groups during storage (p<0.05), but the hardness of the cooked yolk in the SBM-CSP and CSP-DRM groups showed no difference in comparison to the SBM group. In conclusion, CSP may ameliorate the negative effects of DRM on the HU of refrigerated eggs, and SBM or DRM may alleviate the adverse effects of CSP on yolk hardness.
Monitoring of Chicken RNA Integrity as a Function of Prolonged Postmortem Duration
Malila, Yuwares;Srimarut, Yanee;U-chupaj, Juthawut;Strasburg, Gale;Visessanguan, Wonnop 1649
Gene expression profiling has offered new insights into postmortem molecular changes associated with meat quality. To acquire reliable transcript quantification, high quality RNA is required. The objective of this study was to analyze integrity of RNA isolated from chicken skeletal muscle (pectoralis major) and its capability of serving as the template in quantitative real-time polymerase chain reaction (qPCR) as a function of postmortem intervals representing the end-points of evisceration, carcass chilling and aging stages in chicken abattoirs. Chicken breast muscle was dissected from the carcasses (n = 6) immediately after evisceration, and one-third of each sample was instantly snap-frozen and labeled as 20 min postmortem. The remaining muscle was stored on ice until the next rounds of sample collection (1.5 h and 6 h postmortem). The delayed postmortem duration did not significantly affect $A_{260}/A_{280}$ and $A_{260}/A_{230}$ ($p{\geq}0.05$), suggesting no altered purity of total RNA. Apart from a slight decrease in the 28s:18s ribosomal RNA ratio in 1.5 h samples (p<0.05), the value was not statistically different between 20 min and 6 h samples ($p{\geq}0.05$), indicating intact total RNA up to 6 h. Abundance of reference genes encoding beta-actin (ACTB), glyceraldehyde 3-phosphate dehydrogenase (GAPDH), hypoxanthine-guanine phosphoribosyltransferase (HPRT), peptidylprolylisomerase A (PPIA) and TATA box-binding protein (TBP) as well as meat-quality associated genes (insulin-like growth factor 1 (IGF1), pyruvate dehydrogenase kinase isozyme 4 (PDK4), and peroxisome proliferator-activated receptor delta (PPARD) were investigated using qPCR. Transcript abundances of ACTB, GAPDH, HPRT, and PPIA were significantly different among all postmortem time points (p<0.05). Transcript levels of PDK4 and PPARD were significantly reduced in the 6 h samples (p<0.05). The findings suggest an adverse effect of a prolonged postmortem duration on reliability of transcript quantification in chicken skeletal muscle. For the best RNA quality, chicken skeletal muscle should be immediately collected after evisceration or within 20 min postmortem, and rapidly preserved by deep freezing.
Heterophil Phagocytic Activity Stimulated by Lactobacillus salivarius L61 and L55 Supplementation in Broilers with Salmonella Infection
Sornplang, Pairat;Leelavatcharamas, Vichai;Soikum, Chaiyaporn 1657
Newborn chicks are susceptible to Salmonella enterica serovar Enteritidis (SE). The objective of this study was to evaluate the effect of Lactobacillus probiotic isolated from chicken feces on heterophil phagocytosis in broiler chicks. A total of 150 newborn broiler chicks were divided into 5 groups (30 chicks per group) as follows: group 1 (normal control), given feed and water only, group 2 (positive control) given feed, water and SE infection, group 3 (L61 treated) given feed, water, SE infection followed by Lactobacillus salivarius L61 treatment, group 4 (L55 treated) given feed, water, SE infection followed by L. salivarius L55 treatment, and group 5 given feed, water, SE infection followed by L. salivarius L61 + L55 combination treatment. After SE infection, L. salivarius treatment lasted for 7 days. The results showed that L. salivarius L61 and L. salivarius L55 treatment, either alone or combination of both, increased the survival rate after SE infection, and upregulated heterophil phagocytosis and phagocytic index (PI). Conversely, chick groups treated with Lactobacillus showed lower SE recovery rate from cecal tonsils than that of the positive control group. The PI values of the chicken group with SE infection, followed by the combination of L. salivarius L61 and L. salivarius L55 were the highest as compared to either positive control or normal control group. Two Lactobacillus strains supplementation group showed significantly (p<0.05) higher PI value at 48 h than 24 h after treatment.
Effect of Suckling Systems on Serum Oxytocin and Cortisol Concentrations and Behavior to a Novel Object in Beef Calves
Chen, Siyu;Tanaka, Shigefumi;Ogura, Shin-ichiro;Roh, Sanggun;Sato, Shusuke 1662
We investigated differences between effects of natural- and bucket-suckling methods on basal serum oxytocin (OT) and cortisol concentrations, and the effect of OT concentration on affiliative and investigative behavior of calves to a novel object. Ten Japanese Black calves, balanced with birth order, were allocated evenly to natural-suckling (NS) and bucket suckling (BS) groups. Blood samples were collected at the ages of 1 and 2 months (1 week after weaning) calves, and serum OT and cortisol concentrations were measured using enzyme-linked immunosorbent assay and enzymeimmunoassay tests, respectively. Each calf at the age of 2 months (2 weeks after weaning) was released into an open-field with a calf decoy, and its investigative and affiliative behaviors were recorded for 20 minutes. In 1-month-old calves, the basal serum OT concentration ($25.5{\pm}4.9$ [mean${\pm}$standard deviation, pg/mL]) of NS was significantly higher than that of BS ($16.9{\pm}6.7$) (p<0.05), whereas the basal cortisol concentration ($5.8{\pm}2.5$ [mean${\pm}$standard deviation, ng/mL]) of NS was significantly lower than that in BS ($10.0{\pm}2.8$) (p<0.05). Additionally, a negative correlation was noted between serum OT and cortisol concentrations in 1-month-old calves (p = 0.06). Further, the higher serum OT concentration the calves had at 1 month old, the more investigative the calves were at 2 months old but not affiliative in the open-field with a calf decoy. Thus, we concluded that the natural suckling method from a dam elevates the basal serum OT concentration in calves, and high serum OT concentrations induce investigative behavior and attenuate cortisol concentrations. | CommonCrawl |
\begin{document}
\begin{abstract}
The individualized treatment rule (ITR), which recommends an optimal treatment based on individual characteristics, has drawn considerable interest from many areas such as precision medicine, personalized education, and personalized marketing. Existing ITR estimation methods mainly adopt one of two or more treatments. However, a combination of multiple treatments could be more powerful in various areas. In this paper, we propose a novel Double Encoder Model (DEM) to estimate the individualized treatment rule for combination treatments. The proposed double encoder model is a nonparametric model which not only flexibly incorporates complex treatment effects and interaction effects among treatments, but also improves estimation efficiency via the parameter-sharing feature. In addition, we tailor the estimated ITR to budget constraints through a multi-choice knapsack formulation, which enhances our proposed method under restricted-resource scenarios. In theory, we provide the value reduction bound with or without budget constraints, and an improved convergence rate with respect to the number of treatments under the DEM. Our simulation studies show that the proposed method outperforms the existing ITR estimation in various settings. We also demonstrate the superior performance of the proposed method in a real data application that recommends optimal combination treatments for Type-2 diabetes patients. \end{abstract} \keywords{Causal Inference; Combination therapy; Multi-choice knapsack; Nonparametric model; Precision medicine.}
\section{Introduction}
Individualized decision-making has played a prominent role in many fields such as precision medicine, personalized education, and personalized marketing due to the rapid development of personalized data collection. For example, in precision medicine, individualized treatments based on individuals' demographic information and their overall comorbidity improve healthcare quality \citep{schmieder2015achievement}. However, most existing individualized decision-making approaches select one out of multiple treatments, whereas recent advances in medical and marketing research have suggested that applying multiple treatments simultaneously, referred to as combination treatments, could enhance overall healthcare or sales performance. Specifically, combination treatments are able to reduce treatment failure or fatality rates, and overcome treatment resistance for many chronic diseases \citep[e.g.,][]{mokhtari2017combination, kalra2010combination, maruthur2016diabetes, bozic2013evolutionary, korkut2015perturbation, mottonen1999comparison, forrest2010rifampin, tamma2012combination}. Therefore, it is critical to develop a novel statistical method to recommend individualized combination treatments.
There are various existing methods for estimating the optimal individualized treatment rule. The first approach is the model-based approach, which estimates an outcome model given pre-treatment covariates and the treatment. The optimal ITR is derived by maximizing the outcome over possible treatments conditioned on the pre-treatment covariates. Existing works such as Q-learning \citep{qian2011performance}, A-learning \citep{lu2013variable, shi2018high} and RD-learning \citep{meng2020doubly} all belong to this approach. The other approach is known as the direct-search approach, which directly maximizes the expected outcome over a class of decision functions to obtain an optimal ITR. The seminal works of the direct-search approach include outcome weighted learning \citep{zhao2012estimating, huang2019multicategory}, residual weighted learning \citep{zhou2017residual}, and augmented outcome weighted learning \citep{zhou2017augmented}. However, the aforementioned methods in these two categories are designed for selecting one optimal treatment among two or more treatments. In order to accommodate combination treatments, \citet{liang2018estimating} proposed an outcome weighted learning approach using the Hamming loss, extending the direct-search approaches to estimate optimal individualized treatment rules for combination treatments. However, most of these methods cannot be applied to combination treatments seamlessly. Except for outcome weighted learning with the Hamming loss \citep{liang2018estimating}, the other methods treat each combination treatment as independent, ignoring the correlation between different combinations. Consequently, this type of modeling strategy increases the model complexity exponentially due to the combinatorial nature of combination treatments, which increases computation costs with estimation efficiency sacrifices. In addition, the method in \citep{liang2018estimating} ignores the interaction effects among different treatments, which leads to inconsistent estimation of the ITR \citep{liang2018estimating}. In summary, existing methods ignore either correlation among combinations or interactions among treatments, yet both of them are essential to ensure an accurate and efficient estimation of the individualized treatment rule for combination treatments.
In this paper, we propose a double encoder model (DEM) to estimate the optimal individualized treatment rule for combination treatments. The proposed method incorporates both the interaction effects among different treatments and correlations among different combinations. Specifically, we introduce an outcome model where the treatment effects are represented by the inner product between the pre-treatment covariates and the treatment encoder. Through a low-dimensional embedding of covariates and treatments, we incorporate the correlation of different combination treatments in estimating the ITR. In addition, The treatment encoder is decoupled as the additive treatment encoder and the interactive treatment encoder, where the interactive treatment encoder explicitly models the interaction effects of combination treatments. Finally, we derive the optimal individualized treatment rule for combination treatments by maximizing the outcome model over the combination treatments. As we developed our method, a parallel work \citep{kaddour2021causal} proposed the generalized Robinson decomposition, which estimates the conditional average treatment effects (CATE) for structured treatments such as graphs, images, and texts. Their proposed generalized Robinson decomposition also utilizes two neural networks to represent the treatment effects given covariates $\mathbf{X}$ and treatments $\mathbf{A}$. In spite of the overlap, our proposed method targets the combination treatments, especially considering the interaction effects among different treatments and correlations between different combinations.
Furthermore, the combination treatments assignment might be restricted by limited resources in a real world scenario. Existing works \citep{luedtke2016optimal, kitagawa2018should} consider the total amount constraint for binary treatments only, where the assignments are determined by the quantiles of treatment effects. In contrast, allocating combinations of treatments with a limited amount is an NP-hard problem, thus an analytical solution like quantiles does not exist. To address these problem, we formulate the constrained individualized treatment rule as a multi-choice knapsack problem \citep{kellerer2004multidimensional}, and solve this optimization problem through an efficient dynamic programming algorithm.
The main advantages and contributions of this paper are summarized as follows. First of all, the proposed method improves the estimation efficiency via the low-dimensional output of the covariates encoder such that the correlation among different combination treatments is captured. Meanwhile, the model misspecification issue of treatment effects is avoided due to the flexibility of nonparametric modeling of the covariates encoder. Second, the proposed model guarantees a consistent estimation of the ITR for combination treatments by incorporating the interaction effects through the interactive treatment encoder. Third, the neural network interactive treatment encoder improves the estimation efficiency and the convergence rate since its parameters are shared across all combination treatments.
In regards to the theoretical properties of the estimated ITR, we provide the value reduction bound for the combination treatment ITR estimation problem with or without budget constraints. Thereafter, we provide a non-asymptotic value reduction bound for the DEM, which guarantees that the value function of the estimated individualized treatment rule converges to the optimal value function with a high probability and the proposed method achieves a faster convergence rate compared with existing methods for the multi-arm ITR. The improvement in convergence rate is attained by the hierarchical structure of the neural network where the parameters are shared by all combinations and the input dimension is proportional to the number of treatments instead of the total number of combination treatments.
The proposed method demonstrates superior performance over existing methods in our numerical studies especially when the number of treatments is large and there exist interaction effects among different treatments. In the real data application, we apply the proposed method to recommend the optimal combination treatments for Type-2 diabetes patients, which achieves the maximal glucose level reduction and shows its potential in improving individualized healthcare.
The rest of this paper is organized as follows. In Section 2, we introduce the notations and background of the Q-learning framework, and the budget constraints problem. In Section 3, we propose the double encoder model (DEM) to estimate the optimal individualized treatment rule for combination treatments and impose a budget constraint on the original problem. In Section 4, we establish the theoretical properties of the proposed method. In Section 5, we illustrate the empirical performance of the proposed method in various simulation studies. In Section 6, we apply the proposed method to recommending optimal combination treatments to Type-2 diabetes patients. We provide concluding remarks in Section 7.
\section{Notations and Background}
In this section, we introduce the problem setup and notations for the estimation of individualized treatment rules for combination treatments. Consider the data $(\mathbf{X}, \mathbf{A}, Y)$ collected from designed experiments or observational studies. The subject pre-treatment covariates are denoted by $\mathbf{X} \in \mathcal{X} \subset \mathbb{R}^p$, which might include patients' demographics and lab test results. The combinations of $K$ treatments (or $K$-channel treatment) are denoted by $\mathbf{A} = (A^1, A^2, ..., A^K) \in \mathcal{A} \subset \{0, 1\}^K$, where $A^k = 1$ indicates that the $k$th treatment is administered and $A^k = 0$ otherwise. Note that some combinations are infeasible to be considered in real applications, for example, many drug-drug interactions could lead to risks for patients outweighing the benefits \citep{rodrigues2019drug}. Therefore, we consider a subset $\mathcal{A}$ of all the possible $2^{K}$ combinations in the treatment rule. We may also denote the treatments by $\tilde{A} \in \{1, 2, ..., |\mathcal{A}|\}$ as categorical encodings, and use these two sets of notations interchangeably without ambiguity. The outcome of our interest is denoted by $Y \in \mathbb{R}$. Without loss of generality, we assume a larger value of $Y$ is preferable, for example, the reduction of glucose level.
In causal inference, the potential outcome framework \citep{rubin1974estimating} is to describe the possible outcome after a certain treatment is assigned. We use $Y(\mathbf{A})$ to denote the potential outcome throughout the paper. Due to the ``fundamental problem of causal inference" \citep{holland1986statistics}, which indicates that only one potential outcome is observed for each subject, it is infeasible to estimate the subject-wise optimal treatment. Instead, our goal of estimating the optimal individualized treatment rule for combination treatments is to maximize the population-wise expected potential outcome, which is also known as the value function: \begin{align} \label{value_func}
\mathcal{V}(d) := \mathbb{E}[Y\{d(\mathbf{X})\}], \end{align} where $d(\cdot): \mathcal{X}\rightarrow\mathcal{A}$ is an individualized treatment rule. The value function is defined as the expectation of the potential outcomes over the population distribution of $(\mathbf{X}, \mathbf{A}, Y)$ under $\mathbf{A} = d(\mathbf{X})$, which is estimable when the following causal assumptions \citep{rubin1974estimating} holds: \begin{assumption} \label{causal_assumption}
(a) Stable Unit Treatment Value Assumption (SUTVA): $Y = Y(\mathbf{A})$;
(b) No unmeasured confounders: $\mathbf{A} \perp \!\!\! \perp Y(\mathbf{a}) | \mathbf{X}$, for any $\mathbf{a} \in \mathcal{A}$;
(c) Positivity: $\mathbb{P}(\mathbf{A} = \mathbf{a}|\mathbf{X}) \ge p_{\mathcal{A}}$, $\forall \mathbf{a} \in \mathcal{A}, \forall \mathbf{X}\in\mathcal{X}$, for some $p_{\mathcal{A}} > 0$. \end{assumption}
Assumption (a) is also referred to as "consistency" in causal inference, which assumes that the potential outcomes of each subject do not vary with treatments assigned to other subjects. The treatments are well-defined in that the same treatment leads to the same potential outcome. Assumption (b) states that all confounders are observed in pre-treatment covariates, so that the treatment and potential outcomes are conditionally independent given the pre-treatment covariates. Assumption (c) claims that for any pre-treatment covariates $\mathbf{X}$, each treatment can be assigned with a positive probability.
Based on these assumptions, the value function defined in (\ref{value_func}) can be identified as follows: \begin{align} \label{value_func_id}
\mathcal{V}(d) = \mathbb{E}\{Y|\mathbf{A} = d(\mathbf{X})\} = \mathbb{E}\bigg\{\sum_{\mathbf{A}\in \mathcal{A}}\mathbb{E}(Y|\mathbf{X}, \mathbf{A})\mathbb{I}\{d(\mathbf{X}) = \mathbf{A}\}\bigg\}, \end{align}
where $\mathbb{I}(\cdot)$ is the indicator function. To maximize the value function, we can first estimate the conditional expectation $\mathbb{E}(Y|\mathbf{X} = \mathbf{x}, \mathbf{A} = \mathbf{a})$, namely the Q-function in the literature \citep{clifton2020q}. Then the optimal individualized treatment rule can be obtained by \begin{align} \label{decision_rule}
d^{*}(\mathbf{x}) \in \argmax_{\mathbf{a}\in \mathcal{A}}\mathbb{E}(Y|\mathbf{X} = \mathbf{x}, \mathbf{A} = \mathbf{a}). \end{align}
From the perspective of the multi-arm treatments, the Q-function \citep{qian2011performance, qi2020multi, kosorok2019precision} can be formulated as: \begin{align}
\label{multiarm_problem}
\mathbb{E}(Y|\mathbf{X}, \tilde{A}) = m(\mathbf{X}) + \sum_{l=1}^{|\mathcal{A}|}\delta_l(\mathbf{X})\mathbb{I}(\tilde{A}=l), \end{align} where $m(\mathbf{X})$ is the treatment-free effect representing a null effect without any treatment and functions $\delta_{l}(\mathbf{X})$'s are treatment effects for the $l$th treatment. {\large There are two major challenges when (\ref{multiarm_problem}) is applied to the combination treatments problem: First, if $\delta_{l}(\cdot)$'s are imposed to be some parametric model, for example, linear model \citep{qian2011performance, kosorok2019precision}, it could have severe misspecification issue, especially considering the complex nature of interaction effects of combination treatments. Second, as the number of treatments $K$ increases, the number of treatment-specific functions $\delta_{l}(\cdot)$'s could grow exponentially. Therefore, the estimation efficiency of the ITR based on Q-function (\ref{multiarm_problem}) could be severely compromised for either parametric or nonparametric models, especially in clinical trials or observational studies with limited sample sizes.}
In addition, considering the combination of multiple treatments expands the treatment space $\mathcal{A}$ and provides much more feasible treatment options, so each individual could have more choices rather than a yes-or-no as in the binary treatment scenario. Therefore, it is possible to consider accommodating realistic budget constraints while maintaining an effective outcome. In this paper, we further consider a population-level budget constraint as follows. Suppose costs over the $K$ treatments are $\mathbf{c} = (c_1, c_2, ..., c_{K})$, where $c_k$ denotes the cost for the $k$th treatment. Then the budget constraint for a population with a sample size $n$ is: \begin{align}
\label{budget_constraint}
\mathcal{C}_{n}(d) := \frac{1}{n}\sum_{i=1}^{n}\mathbf{c}^Td(\mathbf{X}_i) \le B, \end{align} where $B$ is the average budget for each subject. This budget constraint is suitable for many policy-making problems such as welfare programs \citep{bhattacharya2012inferring} and vaccination distribution problem \citep{matrajt2021vaccine}.
\section{Methodology} \label{sec: method}
In Section \ref{sec: dem}, we introduce the proposed Double Encoder Model (DEM) for estimating the optimal ITR for combination treatments. Section \ref{sec: bc-ITR} considers the optimal assignment of combination treatments under budget constraints. The estimation procedure and implementation details are provided in Section \ref{sec: est_imp}.
\subsection{Double Encoder Model for ITR} \label{sec: dem}
Our proposed Double Encoder Model (DEM) formulates the conditional expectation $\mathbb{E}(Y|\mathbf{X}, \mathbf{A})$, or the Q-function, as follows: \begin{align}
\label{model: dem}
\mathbb{E}(Y|\mathbf{X}, \mathbf{A}) = m(\mathbf{X}) + \alpha(\mathbf{X})^T\beta(\mathbf{A}), \end{align} where $m(\cdot): \mathcal{X}\rightarrow \mathbb{R}$ is the treatment-free effects as in (\ref{multiarm_problem}), and $\alpha(\cdot): \mathcal{X} \rightarrow \mathbb{R}^{r}$ is an encoder that represents individuals' pre-treatment covariates in the $r$-dimensional latent space, which is called the covariate encoder. And $\beta(\cdot): \mathcal{A} \rightarrow \mathbb{R}^{r}$ is another encoder representing the combination treatment in the same $r$-dimensional latent space, named as the treatment encoder. In particular, these two encoders capture the unobserved intrinsic features of subjects and treatments; for instance, the covariates encoder $\alpha(\cdot)$ represents the patients' underlying health status, while the treatment encoder $\beta(\cdot)$ learns physiological mechanisms of the treatment. The inner product $\alpha(\mathbf{X})^T\beta(\mathbf{A})$ represents the concordance between subjects and treatments, hence representing the treatment effects on subjects.
From the perspective of function approximation, the covariates encoder $\alpha(\mathbf{X})$ learns the function bases of treatment effects, and the treatment encoder $\beta(\mathbf{A})$ learns the coefficients associated with those function bases. Consequently, the treatment effects are represented as the linear combinations of $r$ functions: \begin{align} \delta_{l}(\mathbf{X}) = \sum_{i=1}^{r}\beta^{(i)}(\tilde{A}_l)\alpha^{(i)}(\mathbf{X}). \notag \end{align}
Note that the model for multi-arm treatments (\ref{multiarm_problem}) is a special case of the double encoder model (\ref{model: dem}) where $\alpha(\mathbf{X}) = (\delta_{1}(\mathbf{X}), ..., \delta_{|\mathcal{A}|}(\mathbf{X}))$ and $\beta(\tilde{A}) = (\mathbb{I}(\tilde{A} = \tilde{A}_1), ..., \mathbb{I}(\tilde{A} = \tilde{A}_{|\mathcal{A}|}))$ if $r = |\mathcal{A}|$. Another special case of (\ref{model: dem}) is the angle-based modeling \citep{zhang2020multicategory, qi2020multi, xue2021multicategory}, which has been applied to the estimation of the ITR for multi-arm treatments. In the angle-based framework, each treatment is encoded with a fixed vertex in the simplex, and each subject is projected in the latent space of the same dimension as the treatments so that the optimal treatment is determined by the angle between treatment vertices and the subject latent factors. However, the dimension of the simplex and latent space is $r = |\mathcal{A}| - 1$, which leads the angle-based modeling suffers from the same inefficiency issue as (\ref{multiarm_problem}).
Since different combination treatments could contain the same individual treatments, it is over-parameterized to model treatment effects for each combination treatment independently. For instance, the treatment effect of the combination of drug $A$ and drug $B$ is correlated with the individual treatment effects of drug $A$ and of drug $B$, respectively. Therefore, we seek to find a low-dimensional function space to incorporate the correlation of the combination treatments without over-parametrization. In the DEM (\ref{model: dem}), the dimension of the encoders output $r$ controls the complexity of the function space spanned by $\alpha^{(1)}(\cdot), ..., \alpha^{(r)}(\cdot)$. Empirically, the dimension $r$ is a tuning parameter, which can be determined via the hyper-parameter tuning procedure. In other words, the complexity of the DEM is determined by the data itself, rather than pre-specified. In addition, the reduced dimension also leads to a parsimonious model with fewer parameters, which permits an efficient estimation of treatment effects. Furthermore, we do not impose any parametric assumptions on $\alpha(\cdot)$, which allows us to employ flexible nonlinear or nonparametric models with $r$-dimensional output to avoid the potential misspecification of treatment effects.
Since the treatment effects of the combination treatments share the same function bases $\alpha^{(1)}(\cdot)$, ..., $\alpha^{(r)}(\cdot)$, the treatment encoder $\beta(\cdot)$ is necessary to represent all treatments from $\mathcal{A}$ so that $\alpha(\mathbf{X})^T\beta(\mathbf{A})$ can represent treatment effects for all treatments. Through this modeling strategy, we convert the complexity of $|\mathcal{A}|$ treatment-specific functions $\delta_{l}(\cdot)$'s in (\ref{multiarm_problem}) to the representation complexity of $\beta(\cdot)$ in that $\beta(\cdot)$ represents $|\mathcal{A}|$ treatments in $r$-dimensional latent space. As a result, we can reduce the complexity of the combination treatment problem and achieve an efficient estimation if an efficient representation (i.e. $r \ll |\mathcal{A}|$) of $|\mathcal{A}|$ treatments can be found.
In summary, the double encoder model (\ref{model: dem}) is a promising framework to tackle the two challenges in (\ref{multiarm_problem}) if covariates and treatment encoders can provide flexible and powerful representations of covariates and treatments, respectively, which will be elaborated in the following sections. Before we dive into the details of the covariates and treatment encoders, we first show the universal approximation property of the double encoder model, which guarantees its flexibility in approximating complex treatment effects. \begin{theorem} \label{thm: universal_approx}
For any treatment effects $\delta_{l}(\mathbf{X}) \in \mathcal{H}^{2} = \{f: \int_{\mathbf{x}\in\mathcal{X}}|f^{(2)}(\mathbf{x})|^2d\mathbf{x} < \infty\}$, and for any $\epsilon > 0$, there exists $\alpha(\cdot): \mathcal{X}\rightarrow\mathbb{R}^{r}$ and $\beta(\cdot): \mathcal{X}\rightarrow\mathbb{R}^{r}$, where $K\le r\le |\mathcal{A}|$ such that
\begin{align}
\lVert\delta_{l}(\mathbf{X}) - \alpha(\mathbf{X})^T\beta(\tilde{A}_l)\rVert_{\mathcal{H}^2} \le \epsilon, \quad \text{for any } \tilde{A}_l \in \mathcal{A}. \notag
\end{align} \end{theorem}
The above theorem guarantees that the DEM (\ref{model: dem}) can represent the function space considered in (\ref{multiarm_problem}) sufficiently well given a sufficiently large $r$. In practice, we prefer to choose a relatively small $r$ that could achieve superior empirical performance, and we will illustrate this in our numerical studies.
\subsubsection{Treatment Encoder} \label{sec: trt_encoder}
In this section, we introduce the detailed modeling strategy for treatment encoder $\beta(\cdot)$. The treatment effects of combination treatments can be decoupled into two components: additive treatment effects, which is the sum of treatment effects from single treatments in combination; and interaction effects, which are the additional effects induced by the combinations of multiple treatments. Therefore, we formulate the treatment encoder as follows: \begin{align} \label{trt_encoder} \begin{split}
\beta(\mathbf{A}) &= \beta_{0}(\mathbf{A}) + \beta_{1}(\mathbf{A}) = \mathbf{W}\mathbf{A} + \beta_{1}(\mathbf{A}), \\
\text{s.t.} & \quad\beta_1(\mathbf{A}) = 0, \quad \text{for any } \mathbf{A}\in\{\mathbf{A}: \sum_{k=1}^{K}A_k \ge 2\}, \end{split} \end{align} where $\beta_0(\mathbf{A})$ and $\beta_1(\mathbf{A})$ are additive and interactive treatment encoders, respectively. In particular, $\beta_0(\mathbf{A})$ is a linear function with respect to $\mathbf{A}$, where $\mathbf{W} = (\mathbf{W}_{1}, \mathbf{W}_{2}, ..., \mathbf{W}_{K})\in \mathbb{R}^{r\times K}$ and $\mathbf{W}_{k}$ is the latent representation of the $k$th treatment. As a result, $\alpha(\mathbf{X})^T\beta_{0}(\mathbf{A}) = \sum_{k: \{A^{k} = 1\}}\mathbf{W}_k^T\alpha(\mathbf{X})$ are the additive treatment effects of the combination treatment $\mathbf{A}$. The constraints for $\beta_1(\cdot)$ ensures the identifiability of $\beta_0(\cdot)$ and $\beta_1(\cdot)$ such that any representation $\beta(\mathbf{A})$ can be uniquely decoupled into $\beta_0(\mathbf{A})$ and $\beta_1(\mathbf{A})$.
The interaction effects are challenging to estimate in combination treatments. A naive solution is to assume that interaction effects are ignorable, which leads the additive treatment encoder $\beta_{0}(\mathbf{A})$ to be saturated in estimating the treatment effects of combination treatments. However, interaction effects are widely perceived in many fields such as medicine \citep{stader2020stopping, li2018assessment}, psychology \citep{caspi2010genetic}, and public health \citep{braveman2011broadening}. Statistically, ignoring the interaction effects could lead to inconsistent estimation of the treatment effects \citep{zhao2023covariate} and the ITR \citep{liang2018estimating}. Hence, it is critical to incorporate the interaction effects in estimating the ITR for combination treatments.
A straightforward approach to model the interactive treatment encoder $\beta_1(\mathbf{A})$ is similar to the additive treatment encoder $\beta_0(\mathbf{A})$, which we name as the treatment dictionary. Specifically, a matrix $\mathbf{V} = (\mathbf{V}_1, \mathbf{V}_2, ..., \mathbf{V}_{|\mathcal{A}|})\in\mathbb{R}^{r\times |\mathcal{A}|}$ is a dictionary that stores the latent representations of each combination treatment so that $\beta_1(\mathbf{A})$ is defined as follows \begin{align} \label{trt_dict} \beta_{1}(\mathbf{A}) = \mathbf{V}\mathbf{e}_{\tilde{A}}, \end{align}
where $\mathbf{e}_{\tilde{A}}$ is the one-hot encoding of the categorical representation of $\mathbf{A}$. Since the number of possible combination treatments $|\mathcal{A}|$ could grow exponentially as $K$ increases, the parameters of $\mathbf{V}$ could also explode. Even worse, each column $\mathbf{V}_{l}$ can be updated only if the associated treatment $\tilde{A}_{l}$ is observed. Given a limited sample size, each treatment could be only observed a few times in the combination treatment scenarios, which leads the estimation efficiency of $\mathbf{V}$ to be severely compromised. The same puzzle is also observed in other methods. For the Q-function in(\ref{multiarm_problem}), the parameters in $\delta_{l}(\mathbf{X})$ can be updated only if $\tilde{A}_{l}$ is observed; In the Treatment-Agnostic Representation Network (TARNet) \citep{shalit2017estimating} and the Dragonnet \citep{shi2019adapting}, each treatment is associated with an independent set of regression layers to estimate the treatment-specific treatment effects, which results in inefficiency estimation for combination treatment problems.
In order to overcome the above issue, we propose to utilize the feed-forward neural network \citep{goodfellow2016deep} to learn efficient latent representations in the $r$-dimensional space. Specifically, the interactive treatment encoder is defined as \begin{align} \label{trt_encoder_nn}
\beta_{1}(\mathbf{A}) = \mathcal{U}_{L} \circ \sigma \circ ... \circ \sigma \circ \mathcal{U}_{1}(\mathbf{A}), \end{align} where $\mathcal{U}_{l}(\mathbf{x}) = \mathbf{U}_{l}\mathbf{x} + \mathbf{b}_l$ is the linear operator with the weight matrix $\mathbf{U}_l \in \mathbb{R}^{r_{l}\times r_{l-1}}$ and the biases $\mathbf{b}_l$. The activation function is chosen as ReLU function $\sigma(\mathbf{x}) = \max(\mathbf{x}, 0)$ in this paper. An illustration of the neural network interactive treatment encoder is shown in Figure \ref{fig: parameter_sharing}. Note that all parameters in (\ref{trt_encoder_nn}) are shared among all possible treatments, so all of the weight matrices and biases in (\ref{trt_encoder_nn}) are updated regardless of the input treatment, which could improve the estimation efficiency, even though (\ref{trt_encoder_nn}) may include more parameters than the treatment dictionary (\ref{trt_dict}). As a result, the double encoder model with (\ref{trt_encoder_nn}) not only guarantees a faster convergence rate (with respect to $K$) of the value function but also improves the empirical performance especially when $K$ is large, which will be shown in numerical studies and real data analysis. A direct comparison of the neural network interactive treatment encoder (\ref{trt_encoder_nn}) and the treatment encoder (\ref{trt_dict}), the additive model (\ref{multiarm_problem}), TARNet \citep{shalit2017estimating} and Dragonnet \citep{shi2019adapting} are also shown in Figure \ref{fig: parameter_sharing}.
\begin{figure}
\caption{The left panel shows the parameter update scheme in the additive model (\ref{multiarm_problem}), the treatment dictionary (\ref{trt_dict}), TARNet \citep{shalit2017estimating} and dragonnet \citep{shi2019adapting}. Only the treatment-specific parameters corresponding to $\mathbf{a}_0$ are updated. The right panel shows the parameter-sharing feature of the neural network interactive treatment encoder (\ref{trt_encoder_nn}). All parameters except for non-activated input parameters are updated based on the gradient with respect to the observation $(\mathbf{x}_0, \mathbf{a}_0, y_{0})$.}
\label{fig: parameter_sharing}
\end{figure}
Although the interactive treatment encoder (\ref{trt_encoder_nn}) allows an efficient estimation, it is not guaranteed to represent up to $|\mathcal{A}|$ interaction effects. In the treatment dictionary (\ref{trt_dict}), columns $\mathbf{V}_{l}$'s are free parameters to represent $|\mathcal{A}|$ treatments without any constraints. However, an ``under-parameterized'' neural network is not capable of representing $|\mathcal{A}|$ treatments in $r$-dimensional space. For example, if there are three treatments to be combined ($K=3$), and the treatment effects are sufficiently captured by one-dimensional $\alpha(\mathbf{X})$ with different coefficients ($r = 1$). We use the following one-hidden layer neural network to represent the treatment in $\mathbb{R}$: \begin{align} \label{toy_nn} \beta_{1}(\mathbf{A}) = u_{2}\sigma(\mathbf{U}_{1}\mathbf{A} + b_1) + b_2, \end{align} where $u_2, b_2, b_1 \in \mathbb{R}$ are scalars and $\mathbf{U}_{1}\in \mathbb{R}^{1\times 3}$. In other words, the hidden layer only includes one node. In the following, we show that this neural network can only represent restricted interaction effects: \begin{prop} \label{prop: toy_example} The one-hidden layer neural network (\ref{toy_nn}) can only represent the following interaction effects: (a) $\beta_1(\mathbf{A}) \ge 0$ or $\beta_1(\mathbf{A}) \le 0$ for all $\mathbf{A} \in \mathcal{A}$; (b) $\beta_1(\mathbf{A})$ takes the same values for all combinations of two treatments. \end{prop}
\textcolor{black}{The proof of Proposition \ref{prop: toy_example} is provided in the supplementary materials. Based on the above observation, it is critical to guarantee the representation power of $\beta_1(\mathbf{A})$ to incorporate flexible interaction effects. In the following, we establish a theoretical guarantee of the representation power of $\beta_1(\cdot)$ under a mild assumption on the widths of neural networks: \begin{theorem} \label{thm: rep_power}
For any treatment $\mathbf{A} \in \mathcal{A} \subset \{0, 1\}^{K}$, if $\beta_1(\cdot)$ is a 3-layer fully-connected neural network defined in (\ref{trt_encoder_nn}) satisfying $4[r_{1}/4][r_{2}/4r] \ge |\mathcal{A}|$, then there exist parameters $\{\mathbf{U}_{l}, \mathbf{b}_{l}, l=1, 2, 3\}$, such that $\beta(\mathbf{A})$ satisfies the identifiability constraints and can take any values in $\mathbb{R}^{r}$. \end{theorem}}
\textcolor{black}{The above result is adapted from the recent work on the memorization capacity of neural networks \citep{yun2019small, bubeck2020network}. Theorem \ref{thm: rep_power} shows that if there are $\Omega(2^{K/2}r^{1/2})$ hidden nodes in neural networks, then it is sufficient to represent all possible interaction effects in $\mathbb{R}^{r}$. However, obtaining the parameter set $\{\mathbf{U}_{l}, \mathbf{b}_{l}, l=1, 2, 3\}$ in Theorem \ref{thm: rep_power} via the optimization algorithm is not guaranteed due to the non-convex loss surface of the neural networks. In practice, the neural network widths in Theorem \ref{thm: rep_power} can be a guide, and choosing a wider network is recommended to achieve better empirical performance.}
In summary, we propose to formulate the treatment encoder as two decoupled parts: the additive treatment encoder and the interactive encoder. We provide two options for the interactive treatment encoder: the treatment dictionary and the neural network, where the neural network can improve the asymptotic convergence rate and empirical performance with guaranteed representation power. In the numerical studies, we use the neural network interactive treatment encoder for our proposed method, and a comprehensive comparison between the treatment dictionary and the neural network is provided in the supplementary materials.
\subsubsection{Covariates Encoder} \label{sec: cov_encoder}
As we introduced in (\ref{model: dem}), the covariates encoder $\alpha(\cdot): \mathcal{X}\rightarrow \mathbb{R}^{r}$ constitutes the function bases of the treatment effects for all combination treatments. In other words, the treatment effects represented in (\ref{model: dem}) lie in the space spanned by $\alpha^{(1)}(\mathbf{X}), ..., \alpha^{(r)}(\mathbf{X})$. Therefore, it is critical to consider a sufficiently large and flexible function space to accommodate the highly complex treatment effects and avoid possible model misspecification. In particular, we adopt three nonlinear or nonparametric models for covariates encoders: polynomial, B-Spline \citep{hastie2009elements}, and neural network \citep{goodfellow2016deep}.
First of all, we introduce the $\alpha(\mathbf{X})$ as a feed-forward neural network defined as follows: \begin{align} \label{cov_encoder}
\alpha(\mathbf{X}) = \mathcal{T}_{L}\circ \sigma \circ ... \circ \sigma \circ \mathcal{T}_{1}(\mathbf{X}), \end{align} $\mathcal{T}_{l}(\mathbf{x}) = \mathbf{T}_{l}\mathbf{x} + \mathbf{c}_l$ is the linear operator with the weight matrix $\mathbf{T}_l \in \mathbb{R}^{r_{l}\times r_{l-1}}$ and the biases $\mathbf{c}_l$. The activation function is chosen as ReLU function $\sigma(\mathbf{x}) = \max(\mathbf{x}, 0)$ in this paper. Note that the depth and the width of the covariates encoder $\alpha(\cdot)$ are not necessarily identical to those of the interactive treatment encoder $\beta_{1}(\cdot)$, and these are all tuning parameters to be determined through hyper-parameter tuning.
\begin{figure}
\caption{Model structure of the polynomial covariate encoder. Dashed lines indicate the fixed polynomial expansion procedures, and solid lines are trainable parameters for the linear combination of polynomials.}
\label{fig: poly_cov_encoder}
\end{figure}
Even though neural networks achieve superior performance in many fields, their performance in small sample size problems, such as clinical trials or observational studies in medical research, is still deficient. In addition, neural networks lack interpretability due to the nature of their recursive composition; therefore, the adoption of neural networks in medical research is still under review. Here, we propose the polynomial and B-Spline covariates encoders to incorporate nonlinear treatment effects for better interpretation. For the polynomial covariates encoder, we first expand each covariate $x_i$ into a specific order of polynomials $(x_i, x_i^2, ..., x_i^d)$ where $d$ is a tuning parameter. Then we take the linear combinations of all polynomials as the output of the covariate encoders. Figure \ref{fig: poly_cov_encoder} provides an example of the polynomial covariate encoder with $d=3$. Similarly, as for the B-spline covariates encoder, we first expand each covariate into B-spline bases, where the number of knots and the spline degree are tuning parameters. Likewise, linear combinations of these B-spline bases are adopted as the output of the encoder. Although both polynomial and B-spline covariate encoders can accommodate interaction terms among the polynomial bases or B-spline bases for a better approximation for multivariate functions, exponentially increasing parameters need to be estimated as the dimension of covariates or the degree of bases increases. In the interest of computation feasibility, we do not consider interaction terms in the following discussion.
\subsection{Budget Constrained Individualized Treatment Rule} \label{sec: bc-ITR}
In this section, we consider to optimize the assignment of combination treatments under the budget constraints, where the total cost constraints are imposed on a population with a sample size $n$.
We first introduce budget constrained ITR for binary treatments. Suppose we have a treatment ($A = 1$) and a control ($A = -1$), and there are only $b\%$ subjects which can be treated with the treatment $A = 1$. \citet{luedtke2016optimal} define a contrast function $\eta(\mathbf{x}) = \delta(\mathbf{x}, 1) - \delta(\mathbf{x}, -1)$, and the corresponding ITR under the budget constraint is $d(\mathbf{x}, b) = I(\eta(\mathbf{x}) \ge q_b)$, where $q_b$ is the $b\%$ quantile of the distribution of $\eta(\mathbf{x})$ for a given population with a finite sample size.
Estimating the optimal individualized treatment rule for combination treatments under budget constraints is challenging. First of all, the contrast function is no longer a valid tool to measure the treatment importance to each subject. Given the exponentially increasing choices of combination treatments, the number of contrast functions increases exponentially and the pairwise comparisons do not suffice to determine the optimal assignment. Second, costs over different channels may differ significantly, which makes the quantile no longer an effective criterion for allocating budgets.
In the following, we consider the constrained ITR problem for a finite population: \begin{equation} \begin{aligned} \label{constrained_problem}
\max_{d} \hspace{2mm} \mathcal{V}_n(d) \hspace{2mm}
s.t. \hspace{2mm} \mathcal{C}_{n}(d)\le B, \end{aligned} \end{equation} where $\mathcal{V}_{n}$ and $\mathcal{C}_n$ are defined on a pre-specified population with a sample size $n$. Here, covariates $\mathbf{x}_i$ ($i=1, 2, ..., n$) are treated as fixed covariates. Based on model formulation (\ref{model: dem}), maximizing the objective function of (\ref{constrained_problem}) is equivalent to: \begin{align}
\argmax_{d}\mathcal{V}_{n}(d) &= \argmax_{d}\frac{1}{n}\sum_{i=1}^{n}[m(\mathbf{x}_i) + \alpha(\mathbf{x}_i)^T\beta(d(\mathbf{x}_i))] \notag \\
&= \argmax_{d}\frac{1}{n}\sum_{i=1}^{n}\alpha(\mathbf{x}_i)^T\beta(d(\mathbf{x}_i)) \notag \\
&= \argmax_{\{d_{ij}\}}\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{|\mathcal{A}|}\delta_{ij}d_{ij}, \notag \end{align}
where $\delta_{ij} = \alpha(\mathbf{x}_i)^T\beta(\mathbf{a}_j)$ denotes the treatment effects of the $j$th combination treatment on the $i$th subject, and $d_{ij} = I\{d(\mathbf{x}_i)=\mathbf{a}_j\} \in \{0, 1\}$ indicates whether the $i$th subject receives the $j$th combination treatment. Since one subject can only receive one combination treatment, we impose the constraint to be $\sum_{j}d_{ij} = 1$. Similarly, budget constraints can be formulated as $\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{|\mathcal{A}|} c_{\tilde{a}_j}d_{ij} \le B$, where $c_{\tilde{a}_j}$ is the cost of treatment $\tilde{a}_j$ calculated from the cost vector $\mathbf{c}$. The constrained individualized treatment rule can be solved as follows:
\begin{equation} \begin{aligned} \label{empirical_constrained_problem}
\max_{\{d_{ij}\}}& \hspace{2mm}\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{|\mathcal{A}|}\delta_{ij}d_{ij}, \\
s.t. \hspace{2mm} \frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{|\mathcal{A}|} c_{\tilde{a}_j}d_{ij} \le B,
&\hspace{2mm}\sum_{j}d_{ij} = 1, \hspace{2mm}
\hspace{2mm} d_{ij} \in \{0, 1\}, \hspace{2mm} \text{ for any } i,j. \end{aligned} \end{equation} The above optimization problem is equivalent to a multi-choice knapsack problem \citep{kellerer2004multidimensional}. For a binary treatment setting, the solution of (\ref{empirical_constrained_problem}) is the quantile of $\eta(\mathbf{X})$, which is a special case of our formulation.
To understand the connection between constrained individualized treatment rule and the multi-choice knapsack problem, we notice that the priority of the treatment is associated with the definition of dominance in the multi-choice knapsack problem: for any $i \in \{1, 2, ..., n\}$, if $\delta_{ik} > \delta_{il}$ and $c_{\mathbf{a}_k} < c_{\mathbf{a}_l}$, the treatment $l$ is dominated by the treatment $k$. In other words, the treatment $k$ achieves a better outcome than the treatment $l$ with a lower cost. Thus, the dominance property could be an alternative to the contrast functions in combination treatment settings.
Here $\delta_{ij}$ indicates the treatment effects, and the parametric assumptions are not required, so this framework is also applicable for other methods providing estimations of treatment effects such as the $l_1$-penalized least-square \citep{qian2011performance}, and the outcome weighted learning with multinomial deviance \citep{huang2019multicategory}. However, the objective function in (\ref{constrained_problem}) depends on the estimation of $\delta_{ij}$, and we show that the value reduction under budget constraints is bounded by the estimation error of $\delta_{ij}$'s in Theorem \ref{thm: value_reduction_constraint}. Consequently, estimation bias in $\delta_{ij}$ could lead to biased results in (\ref{empirical_constrained_problem}). Since the proposed model (\ref{model: dem}) provides an efficient and accurate estimation of treatment effects, it also results in a more favorable property for solving the budget constrained individualized treatment rule for combination treatments.
\subsection{Estimation and Implementation} \label{sec: est_imp}
In this section, we introduce the estimation and hyper-parameter tuning procedures of the proposed method for unconstrained and constrained individualized treatment rule for combination treatment.
\subsubsection{Estimation of the Double Encoder Model} First of all, we propose the following doubly robust estimator for the treatment effects: \begin{align} \label{loss_func}
\hat{\alpha}(\cdot), \hat{\beta}(\cdot) = \argmin_{\alpha(\cdot), \beta(\cdot)}\mathbb{E}\bigg\{\frac{1}{\hat{\mathbb{P}}(\mathbf{A}_i|\mathbf{X_i})}(Y_i - \hat{m}(\mathbf{X}_i) - \alpha(\mathbf{X}_i)^T\beta(\mathbf{A}_i))^2\bigg\}, \end{align}
where $\hat{\mathbb{P}}(\mathbf{A}_{i}|\mathbf{X_i})$ is a working model of the propensity score specifying the probability of treatment assignment given pre-treatment covariates, and $\hat{m}(\mathbf{X}_i)$ is a working model of treatment-free effects. The inverse probability weights given by the propensity scores balance the samples assigned to different combination treatments, assumed to be equal under the randomized clinical trial setting. By removing the treatment-free effects $m(\mathbf{x})$ from the responses before we estimate the treatment effects, the numerical stability can be improved and the estimator variance is reduced. This is also observed in \citep{zhou2017residual, fu2016estimating}. Furthermore, the estimator in (\ref{loss_func}) is doubly-robust in that if either $\hat{\mathbb{P}}(\cdot|\cdot)$ or $\hat{m}(\cdot)$ is correctly-specified, $\hat{\alpha}(\cdot)^T\hat{\beta}(\cdot)$ is a consistent estimator of the treatment effects, and a detailed proof is provided in the supplemental material. This result extends the results in \citep{meng2020doubly} from binary and multiple treatments to combination treatments. Empirically, we minimize the sample average of the loss function (\ref{loss_func}) with additional penalties: for the additive treatment encoder, $L_2$ penalty is imposed to avoid overfitting; for the interactive treatment encoder, $L_1$ penalty is added since the interaction effects are usually sparse \citep{wu2011experiments}.
In this work, the working model of the propensity score is obtained via the penalized multinomial logistic regression \citep{friedman2010regularization} as a working model. Specifically, the multinomial logistic model is parameterized by $\gamma_1, \gamma_2, ..., \gamma_{2^{K}} \in \mathbb{R}^{p}$: \begin{align}
\mathbb{P}(\tilde{\mathbf{A}} = k|\mathbf{x}) = \frac{\exp(\gamma_{k}^T\mathbf{x})}{\sum_{k'=1}^{2^{K}}\exp(\gamma_{k'}^T\mathbf{x})}. \notag \end{align} The parameters $\gamma_{k}$'s can be estimated by maximizing the likelihood: \begin{align}
\max_{\gamma_{1},...,\gamma_{2^{K}}}\frac{1}{n}\sum_{i=1}^{n}\bigg[\sum_{k=1}^{2^{K}}\gamma_{k}^T\mathbf{x}_{i}I(\tilde{\mathbf{A}}=k) - \log\{\sum_{k'=1}^{2^{K}}\exp(\gamma_{k'}^T\mathbf{x_i})\}\bigg] - \lambda\sum_{j=1}^{p}(\sum_{k=1}^{2^{K}}\gamma_{kj}^{2})^{1/2}, \notag \end{align} where the group Lasso \citep{meier2008group} is used to penalize parameters across all treatment groups. A potential issue of propensity score estimation is that the estimated probability could be negligible when there are many possible treatments, which leads to unstable estimators for treatment effects. To alleviate the limitation on inverse probability weighting, we stabilize the propensity scores \citep{xu2010use} by multiplying the frequency of the corresponding treatment to the weights.
For the estimation of the treatment-free effects $m(\cdot)$, we adopt a two-layer neural network: \begin{align} \label{treatment_free_function}
m(\mathbf{x}) = (\mathbf{w}_{m}^{2})^{T}\sigma(\mathbf{W}_{m}^{1}\mathbf{x}), \end{align} where $\mathbf{w}^2_{m} \in \mathbb{R}^{h}$ and $\mathbf{W}_{m}^{1} \in \mathbb{R}^{h\times p}$ are weight matrices, and $\sigma(x)$ is the ReLu function. The width $h$ controls the complexity of this model. The weight matrices are estimated through minimizing: \begin{align}
\min_{\mathbf{w}_{m}^{2}, \mathbf{W}_{m}^{1}} \frac{1}{n}\sum_{i=1}^{n}[y_i - (\mathbf{w}_{m}^{2})^{T}\sigma(\mathbf{W}_{m}^{1}\mathbf{x})]^{2}. \notag \end{align}
Given working models $\hat{m}(\mathbf{x})$ and $\hat{\mathbb{P}}(\mathbf{a}|\mathbf{x})$ for treatment-free effects and propensity scores, we propose to optimize the double encoder alternatively. The detailed algorithm is listed in Algorithm \ref{double_encoder_alg}. Specifically, we employ the Adam optimizer \citep{kingma2014adam} for optimizing each encoder: covariate encoder $\alpha(\mathbf{x})$, additive treatment encoder $\beta_{1}(\mathbf{a})$, and non-parametric treatment encoder $\beta_{2}(\mathbf{a})$. To stabilize the optimization during the iterations, we utilize the exponential scheduler \citep{patterson2017deep} which decays the learning rate by a constant per epoch. In all of our numerical studies, we use 0.95 as a decaying constant for the exponential scheduler. To ensure the identifiability of treatment effects, we also require a constraint on the treatment encoder $\beta(\cdot)$ such that $\sum_{\mathbf{a}\in\mathcal{A}}\beta(\mathbf{a}) = 0$. To satisfy this constraint, we add an additional normalization layer before the output of $\beta(\cdot)$. The normalization layer subtracts the weighted mean vector where the weight is given by the reciprocal of the combination treatment occurrence in the batch. Since this operation only centers the outputs, the theoretical guarantee for $\beta(\cdot)$ in Section \ref{sec: theory} still holds. Once our algorithm converges, we obtain the estimation for $\alpha(\cdot)$ and $\beta(\cdot)$, and also the estimated individualized treatment rule $\hat{d}(\cdot)$ by (\ref{decision_rule}).
\begin{algorithm}
\caption{Double encoder model training algorithm}
\label{double_encoder_alg}
\begin{algorithmic}
\State \textbf{Input}: Training dataset $(\mathbf{x}_i, \mathbf{a}_i, \mathbf{y}_i)_{i=1}^{n}$, working models $\hat{m}(\mathbf{x}), \hat{\mathbb{P}}(\mathbf{a}|\mathbf{x})$, hyper-parameters including network structure-related hyper-parameters (e.g., network depth $L_{\alpha}, L_{\beta}$, network width $r_{\alpha}, r_{\beta}$, encoder output dimension $r$), optimization-related hyper-parameters (e.g., additive treatment encoder penalty coefficients $\lambda_{a}$, interactive treatment encoder penalty coefficients $\lambda_{i}$, mini-batch size $B$, learning rate $\eta$, and training epochs $E$).
\State \textbf{Initialization}: Initialize parameters in $\hat{\alpha}^{(0)}(\mathbf{x})$, $\hat{\beta}^{(0)}_0(\mathbf{a})$, and $\hat{\beta}^{(0)}_1(\mathbf{a})$.
\State \textbf{Training}:
\For{e in 1: E}
\For{Mini-batch sampled from $(\mathbf{x}_i, \mathbf{a}_i, \mathbf{y}_i)_{i=1}^{n}$}
\State $\hat{\alpha}^{(e)}(\mathbf{x}) = \argmin\frac{1}{B}\sum\frac{1}{\hat{\mathbb{P}}(\mathbf{a}_i|\mathbf{x_i})}\bigg\{y_i - \hat{m}(\mathbf{x}_i) - \alpha(\mathbf{x}_i)^T(\hat{\beta}_0^{(e-1)}(\mathbf{a}_i) + \hat{\beta}_1^{(e-1)}(\mathbf{a}_i))\bigg\}^2$
\State $\hat{\beta}_{1}^{(e)}(\mathbf{a}) = \argmin\frac{1}{B}\sum\frac{1}{\hat{\mathbb{P}}(\mathbf{a}_i|\mathbf{x_i})}\bigg\{y_i - \hat{m}(\mathbf{x}_i) - \hat{\alpha}^{(e)}(\mathbf{x}_i)^T(\beta_0(\mathbf{a}_i) + \hat{\beta}_1^{(e-1)}(\mathbf{a}_i))\bigg\}^2 + \lambda_{a}\lVert \beta_0 \rVert_2$
\State $\hat{\beta}_{2}^{(e)}(\mathbf{a}) = \argmin\frac{1}{B}\sum\frac{1}{\hat{\mathbb{P}}(\mathbf{a}_i|\mathbf{x_i})}\bigg\{y_i - \hat{m}(\mathbf{x}_i) - \hat{\alpha}^{(e)}(\mathbf{x}_i)^T(\hat{\beta}_0^{(e)}(\mathbf{a}_i) + \beta_1(\mathbf{a}_i))\bigg\}^2 + \lambda_{i}\rVert \beta_1\rVert_1$
\EndFor
\EndFor
\end{algorithmic} \end{algorithm}
In addition, successful neural network training usually requires careful hyper-parameter tuning. The proposed double encoder model includes multiple hyper-parameters: network structure-related hyper-parameters (e.g., network depth $L_{\alpha}$ and $L_{\beta}$, network width $r_{\alpha}$ and $r_{\beta}$, encoder output dimension $r$), optimization-related hyper-parameters (e.g., additive treatment encoder penalty coefficients $\lambda_{a}$, interactive treatment encoder penalty coefficients $\lambda_{i}$, mini-batch size $B$, learning rate $\eta$, and training epochs $E$). These hyper-parameters induce an extremely large search space, which makes the grid search method \citep{yu2020hyper} practically infeasible. Instead, we randomly sample 50 hyper-parameter settings in each experiment over the pre-specified search space (detailed specification of hyper-parameter space is provided in the supplementary materials), and the best hyper-parameter setting is selected if it attains the largest value function on an independent validation set. Furthermore, due to the non-convexity of the loss function, the convergence of the algorithm also relies heavily on the parameter initialization. In the supplementary materials, we provide detailed analyses for numerical results under different parameter initializations.
\subsubsection{Budget-constrained ITR estimation}
In the following, we introduce our procedure for the budget-constrained individualized treatment rule for combination treatment (\ref{constrained_problem}) estimation. We use the plug-in estimates $\hat{\alpha}(\mathbf{x}_i)^T\hat{\beta}(\mathbf{a}_j)$ for $\delta_{ij}$, and calculate the cost for each combination treatment from the cost vector $\mathbf{c}$ by $c_{\mathbf{a}_j} = \mathbf{a}_{j}^T\mathbf{c}$. Then we apply the dynamic programming algorithm to solve (\ref{constrained_problem}) with plug-in $\delta_{ij}$'s. Although the multi-choice knapsack problem is a NP-hard problem, we can still solve it within pseudo-polynomial time \citep{kellerer2004multidimensional}. Specifically, we denote $\hat{Z}_l(b)$ as the optimal value $\frac{1}{n}\sum_{i=1}^{l}\sum_{j=1}^{|\mathcal{A}|}\hat{\delta}_{ij}d_{ij}$ for the first $l$ subjects with budget constraints $\frac{1}{n}\sum_{i=1}^{l}\sum_{j=1}^{|\mathcal{A}|}c_{\tilde{a}_j}d_{ij} \le b$. Let $\hat{Z}_l(b) = -\infty$ if no solution exists and $\hat{Z}_0(b) = 0$. We define the budget space as $\mathcal{B} = \{b: 0 \le b \le B\}$ including all possible average costs for $n$ subjects, where $0$ is the minimal cost if no treatment is applied to subjects, and the maximal cost is our specified budget $B$. Once the iterative algorithm ends, the optimal objective function is obtained as $\hat{Z}_{n}(B)$ and the optimal treatment assignment is the output $\{d_{ij}: i=1,...,n, j=1,...,|\mathcal{A}|\}$. The detailed algorithm is illustrated in Algorithm \ref{dp_for_constraints}.
\begin{algorithm} \caption{Pseudo code of dynamic programming algorithm} \label{dp_for_constraints} \begin{algorithmic}[1]
\State Input: Treatment effects $\{\hat{\delta}_{ij}: i=1,2,...n, j=1,...,|\mathcal{A}|\}$, cost $\{c_{\tilde{A}_j}: j=1,...,|\mathcal{A}|\}$, budget $B$. \State Initialize: $\hat{Z}_0(b) \gets 0$, \text{ for } $b \in \mathcal{B} = \{b: 0 \le b \le B\}$ \While{$l < n$} \State $l \gets l + 1$ \For{$b \in \mathcal{B}$} \State $\hat{Z}_{l}(b) \gets \max_{j: b > c_{\tilde{A}_j}}\hat{Z}_{l-1}(b-c_{\tilde{A}_{j}}) + \hat{\delta}_{lj}/n$ \State $d_{lj} \gets 1$ if $j = \argmax_{j: b > c_{\tilde{A}_j}}\hat{Z}_{l-1}(b-c_{\tilde{A}_{j}}) + \hat{\delta}_{lj}/n$; Otherwise, $d_{lj} \gets 0$ \EndFor \EndWhile
\State Output: $\{d_{ij}: i=1,...,n, j=1,...,|\mathcal{A}|\}$ \end{algorithmic} \end{algorithm}
\section{Theoretical Guarantees} \label{sec: theory} In this section, we establish the theoretical properties of the ITR estimation for combination treatments and the proposed method. First, we establish the value reduction bound for the combination treatments, either with or without budget constraints. Second, we provide a non-asymptotic excess risk bound for the double encoder model, which achieves a faster convergence rate compared with existing methods for multi-arm treatment problems.
\subsection{Value Reduction Bound}
\color{black}
The value reduction is the difference between the value functions of the optimal individualized treatment rule and of the estimated individualized treatment rule. The value function under a desirable ITR is expected to converge to the value function under the optimal ITR when the sample size goes to infinity. Prior to presenting the main results, we introduce some necessary notations. The conditional expectation of the outcome $Y$ given the subject variable $\mathbf{X}$ and the treatment $\mathbf{A}$ is denoted by $Q(\mathbf{X}, \mathbf{A}) = \mathbb{E}[Y|\mathbf{X}, \mathbf{A}]$, and the treatment-free effects can be rewritten as $m(\mathbf{X}) = \mathbb{E}[Q(\mathbf{X}, \mathbf{A})|\mathbf{X}]$, and the treatment effects can be denoted as $\delta(\mathbf{X}, \mathbf{A}) = Q(\mathbf{X}, \mathbf{A}) - m(\mathbf{X})$. In particular, the optimal and the estimated treatment effects are denoted as $\delta^*(\cdot, \cdot)$ and $\hat{\delta}(\cdot, \cdot)$, respectively. In addition, we introduce an assumption on the treatment effects: \begin{assumption}
\label{margin_assumption}
For any $\epsilon > 0$, there exist some constant $C > 0$ and $\gamma > 0$ such that
\begin{align}
\label{margin_condition}
\mathbb{P}(|\max_{\mathbf{A}\in\mathcal{A}}\delta^{*}(\mathbf{X}, \mathbf{A}) - \max_{\mathbf{A}\in\mathcal{A}\backslash\argmax\delta^{*}(\mathbf{X}, \mathbf{A})}\delta^{*}(\mathbf{X}, \mathbf{A})|\le \epsilon) \le C\epsilon^{\gamma}, \forall \mathbf{a}, \mathbf{a'} \in \mathcal{A}.
\end{align} \end{assumption}
Assumption \ref{margin_assumption} is a margin condition characterizing the behavior of the boundary between different combination treatments. A larger value of $\gamma$ indicates that the treatment effects are differentiable with a higher probability, suggesting it is easier to find the optimal individualized treatment rule. Similar assumptions are also required in the literature \citep{qian2011performance, zhao2012estimating, qi2020multi} to achieve a faster convergence rate of the value reduction bound.
The following theorem shows that the value reduction is bounded by the estimation error of the treatment effects, and the convergence rate can be improved if Assumption \ref{margin_assumption} holds:
\begin{theorem} \label{theorem1}
Suppose the treatment effects $\delta(\cdot, \cdot) \in \mathcal{H}^2$. For any estimator $\hat{\delta}(\cdot, \cdot)$, and the corresponding decision rule $\hat{d}$ such that $\hat{d}(\mathbf{X}) \in \argmax_{\mathbf{A}\in \mathcal{A}}\hat{\delta}(\mathbf{X}, \mathbf{A})$, we have
\begin{align}
\label{theorem1_bound1}
\mathcal{V}(d^{*}) - \mathcal{V}(\hat{d}) \le 2|\mathcal{A}||\mathcal{A}-1|\big\{\mathbb{E}[\delta^*(\mathbf{X}, \mathbf{A}) - \hat{\delta}(\mathbf{X}, \mathbf{A})]^{2}\big\}^{1/2},
\end{align}
where $|\mathcal{A}|$ is the cardinality of the treatment space. Furthermore, if Assumption \ref{margin_assumption} holds, the convergence rate is improved by
\begin{align}
\label{theorem1_bound2}
\mathcal{V}(d^{*}) - \mathcal{V}(\hat{d}) \le C(|\mathcal{A}|, \gamma)\big\{\mathbb{E}[\delta^*(\mathbf{X}, \mathbf{A}) - \hat{\delta}(\mathbf{X}, \mathbf{A})]^{2}\big\}^{(1+\gamma)/(2+\gamma)},
\end{align}
where $C(|\mathcal{A}|, \gamma)$ is a constant that only depends on $|\mathcal{A}|$ and $\gamma$. \end{theorem}
Theorem \ref{theorem1} builds a connection between the value reduction and the estimation error of the treatment effects $\hat{\delta}(\cdot, \cdot)$, which shows that an accurate estimation of treatment effects would lead the estimated value function $\mathcal{V}(\hat{d})$ to approach the optimal value function $\mathcal{V}(d^{*})$. Based on Theorem \ref{theorem1}, we can further connect the value reduction bound to the excess risk of the estimator of the proposed model:
\begin{corollary} \label{corollary1}
Suppose we define the expected risk of function $Q(\cdot, \cdot)$ as $L(Q) = \mathbb{E}[Y - Q(\mathbf{X}, \mathbf{A})]^2$. Then for any estimator of the function $Q(\cdot, \cdot)$, which is denoted by $\hat{Q}(\cdot, \cdot)$, we have the following value reduction bound:
\begin{eqnarray}
\mathcal{V}(d^{*}) - \mathcal{V}(\hat{d}) \le 2|\mathcal{A}||\mathcal{A}-1|\big\{L(\hat{Q}) - L(Q^{*})\big\}^{1/2}. \notag
\end{eqnarray}
Further, if Assumption \ref{margin_assumption} holds, the above inequality can be tighter with $\gamma>0$:
\begin{eqnarray}
\mathcal{V}(d^{*}) - \mathcal{V}(\hat{d}) \le C(|\mathcal{A}|, \gamma)\big\{L(\hat{Q}) - L(Q^{*})\big\}^{(1+\gamma)/(2+\gamma)}. \notag
\end{eqnarray} \end{corollary}
Next, we consider the value reduction bound under budget constraints. Since the multi-choice knapsack problem we formulated for budget-constrained ITR is NP-hard \citep{kellerer2004multidimensional}, we adopt a pseudo-polynomial dynamic programming algorithm \citep{dudzinski1987exact} to obtain an approximated solution. In the following, we analyze the theoretical property of the approximated value function that derived from dynamic programming Algorithm \ref{dp_for_constraints}. Specifically, we define the approximated value function as the sum of the treatment effects of the first $l$ subjects divided by the sample size $n$, which is $\hat{Z}_l(b)$ in Algorithm \ref{dp_for_constraints}. In addition, we denote the approximated value function as $Z_{l}^{*}(b)$ if the true treatment effects $\delta_{ij}^{*}$'s are plugged in. Then we have the following result indicating that the approximated value function converges if the estimation error of $\hat{\delta}(\cdot, \cdot)$ converges.
\begin{theorem} \label{thm: value_reduction_constraint}
For the approximated value function obtained from Algorithm \ref{dp_for_constraints}, for any $B > 0$, we have
\begin{align}
|Z^{*}(B) - \hat{Z}(B)| \le \frac{1}{n}\sum_{i=1}^n|\max_{\tilde{A}_{j}\in\mathcal{A}}\delta^*(\mathbf{x}_i, \tilde{A}_j) - \hat{\delta}(\mathbf{x}_i, \tilde{A}_j)|. \notag
\end{align} \end{theorem}
In other words, the approximated value function under budget constraints can converge if $\hat{\delta}(\cdot, \cdot)$ is a consistent estimator of treatment effects. Note that the proposed estimator is a doubly robust estimator in that either propensity score or treatment-free effects is correctly specified, our proposed estimator is a consistent estimator, which consequently leads the value function and approximated value function under budget constraints converge.
\subsection{Excess Risk Bound}
In this subsection, we provide a non-asymptotic value reduction bound for the proposed DEM and show the improved convergence rate under the DEM. In Corollary \ref{corollary1}, we have shown that the value reduction can be bounded by the excess risk between the true and estimated Q-functions. The excess risk serves as an intermediate tool to establish the non-asymptotic property of the proposed estimator which depends on the complexity of the function class. In the proposed method, we focus on the function class $\mathcal{Q} = \big\{Q: \mathcal{X}\times\mathcal{A}\rightarrow\mathbb{R} | Q(\mathbf{x}, \mathbf{a}) = m(\mathbf{x}) + \alpha(\mathbf{x})^T\beta(\mathbf{a})\big\}$, where $m(\cdot), \alpha(\cdot)$ and $\beta(\cdot)$ are defined in (\ref{treatment_free_function}), (\ref{cov_encoder}) and (\ref{trt_encoder}). We establish the following excess risk upper bound for the estimator in $\mathcal{Q}$:
\begin{lemma} \label{excess_risk_bound}
For any distribution $(\mathbf{X}, \mathbf{A}, Y)$ with $\mathbb{E}[Y^2] \le c_1$, given a function $\hat{Q}$ from $\mathcal{Q}$, then with probability $1 - 2\epsilon$,
\begin{align}
\label{lemma2_bound}
L(\hat{Q}) - L(Q^{*}) \le 8C\mathcal{R}_{n}(\mathcal{Q}) + \sqrt{\frac{2c_1^2\log(1/\epsilon)}{n}},
\end{align}
where $C$ is the Lipschitz constant of $L(Q)$, and $\mathcal{R}_{n}(\mathcal{Q})$ is the Rademacher complexity of $\mathcal{Q}$. \end{lemma}
Lemma \ref{excess_risk_bound} provides an upper bound of the excess risk in Corollary \ref{corollary1} using the Rademacher complexity of $\mathcal{Q}$. However, the Rademacher complexity of a general neural network is still an open problem in the literature and existing bounds are mainly established based on the different types of norm constraints of weight matrices \citep{bartlett2017spectrally,golowich2018size, neyshabur2017pac, neyshabur2015norm}. In this work, we focus on the following sub-class of $\mathcal{Q}$ with $L_2$ and spectral norm constraints: \begin{eqnarray}
\mathcal{Q}_{B_{m}, B_{\alpha}, B_{\beta}} = \big\{Q \in \mathcal{Q}: \lVert\mathbf{w}_{m}^{2}\rVert_2 \le B_{m}, \lVert\mathbf{W}_{m}^{1}\rVert_{2,\infty} \le B_{m}, \lVert\mathbf{T}_{l}\rVert_2 \le B_{\alpha}, \lVert\mathbf{U}_{l}\rVert_2 \le B_{\beta} \big\}, \notag \end{eqnarray}
where $\lVert\cdot\rVert_2$ denotes the $L_2$-norm for vectors and the spectral norm for matrices. For any matrix $\mathbf{X} = (\mathbf{X}_1, ..., \mathbf{X}_p)$, and $\mathbf{X}_i$ is the $i$th column of matrix $\mathbf{X}$, we use $\lVert X\rVert_{2, \infty} = \max_{i}\lVert \mathbf{X}_i \rVert_2$ to denote the $L_{2,\infty}$ norm of $\mathbf{X}$. We then establish the upper bound of the Rademacher complexity of $\mathcal{Q}_{B_{m}, B_{\alpha}, B_{\beta}}$ as follows:
\begin{lemma} \label{excess_risk_bound_2} Suppose $\mathbb{E}[\lVert\mathbf{X}\rVert_2^2]\le c_2^2$. The Rademacher complexity of $\mathcal{Q}_{B_{m}, B_{\alpha}, B_{\beta}}$ is upper bounded by: \begin{align}
\label{lemma3_bound}
\mathcal{R}_{n}(\mathcal{Q}_{B_{m}, B_{\alpha}, B_{\beta}}) &\le 2B_{m}^2c_2\sqrt{\frac{h}{n}} + B_{\alpha}^{L_{\alpha}}B_{\beta}^{L_{\beta}}c_2\sqrt{\frac{K}{n}}. \end{align} \end{lemma}
Lemma \ref{excess_risk_bound_2} provides an upper bound of the Rademacher complexity of $\mathcal{F}_{B_{m}, B_{\alpha}, B_{\beta}}$ with the rate $O(\sqrt{\frac{1}{n}})$. The first term of (\ref{lemma3_bound}) is the upper bound for the function class of $m(\mathbf{x})$ in (\ref{treatment_free_function}), which depends on the width of hidden layers $h$. If $h$ is large, the function $m(\mathbf{x})$ is able to approximate a larger function space, but with a less tight upper bound on the generalization error. The second term of (\ref{lemma3_bound}) is associated with the functional class of the inner product of the double encoders with a convergence rate of $O(K^{1/2}n^{-1/2})$. The rate increases with the number of treatments $K$ rather than $|\mathcal{A}|$ due to the parameter-sharing feature of the interactive treatment encoder, and the linearly growing dimension of input of function $\beta(\cdot)$ in the proposed method. Specifically, the input of $\beta(\cdot)$ is the combination treatment $\mathbf{A}$ itself, and parameters in the treatment encoder are shared by all the combination treatments. Thus, the model complexity is proportional to $K$ and the product of the spectral norm of weight matrices. Based on Lemmas \ref{excess_risk_bound} and \ref{excess_risk_bound_2}, we derive the value reduction bound for the proposed method as follows:
\begin{theorem} \label{theorem2} For any distribution $(\mathbf{X}, \mathbf{A}, Y)$ with $\mathbb{E}[Y^2] \le c_1$, $\mathbb{E}[\lVert\mathbf{X}\rVert_2^2] \le c_2$. Consider the neural networks in the subspace $\mathcal{Q}_{B_{m}, B_{\alpha}, B_{\beta}}$, with probability at least $1-2\epsilon$, we have the following value reduction bound: \begin{eqnarray} \label{value_reduction_bound1}
\mathcal{V}(d^{*}) - \mathcal{V}(\hat{d}) \le 2|\mathcal{A}||\mathcal{A}-1|\bigg\{16CB_{m}^2c_2\sqrt{\frac{h}{n}} + 8CB_{\alpha}^{L_{\alpha}}B_{\beta}^{L_{\beta}}c_2\sqrt{\frac{K}{n}} + \sqrt{\frac{2c_1^2\log(1/\epsilon)}{n}}\bigg\}^{1/2}. \notag \end{eqnarray} If Assumption \ref{margin_assumption} holds, we have a tighter bound with a positive $\gamma$: \begin{eqnarray} \label{value_reduction_bound2}
\mathcal{V}(d^{*}) - \mathcal{V}(\hat{d}) \le C(|\mathcal{A}|, \gamma)\bigg\{16CB_{m}^2c_2\sqrt{\frac{h}{n}} + 8CB_{\alpha}^{L_{\alpha}}B_{\beta}^{L_{\beta}}c_2\sqrt{\frac{K}{n}} + \sqrt{\frac{2c_1^2\log(1/\epsilon)}{n}}\bigg\}^{(1+\gamma)/(2+\gamma)}. \notag \end{eqnarray} \end{theorem}
Theorem \ref{theorem2} establishes the value reduction bound in that the estimated decision rule can approach the optimal value function as the sample size increases. Compared with the existing value reduction bound for multi-arm treatments, the proposed method improves the convergence rate from $O(|\mathcal{A}|^{9/4})$ to $O(|\mathcal{A}|^2(\log_{2}|\mathcal{A}|)^{1/4})$. Further, the order of the value reduction bound can approach nearly $\sqrt{\frac{1}{n}}$ as $\gamma$ goes to infinity, which is consistent with the convergence rates established in \citep{qian2011performance, qi2020multi}.
\section{Simulation Studies} \label{sec: simulation}
In this section, we evaluate the performance of the proposed method in estimating the individualized treatment rule for combination treatments. Our numerical studies show that the proposed method achieves superior performance to competing methods in both unconstrained and budget-constrained scenarios.
\subsection{Unconstrained ITR simulation} \label{sec: uncstr_simulation}
We first investigate the empirical performance of the proposed method without budget constraints. We assume the pre-treatment covariates $\mathbf{X} = (X_1, \ldots, X_{10}) \in \mathbb{R}^{10}$ are independently and uniformly sampled from $(-1, 1)$. Four simulation settings are designed to evaluate the performance under varying settings. In simulation settings 1 and 2, we consider combinations of 3 treatments, which induces 8 possible combinations, with 6 of them considered as our assigned treatments. Similarly, in simulation settings 3 and 4, we consider combinations of 5 treatments, and we assume that 20 of all combinations are assigned to subjects. The treatments are assigned either uniformly or following the propensity score model: \begin{align} \label{sim: propensity_score_model}
\mathbb{P}(\tilde{A}_i|\mathbf{X}) = \frac{\exp\{0.2i * (\mathbf{X}^T\beta)\}}{\sum_{j}\exp\{0.2j * (\mathbf{X}^T\beta)\}}, \end{align} and the marginal treatment assignment distribution is shown in Figure \ref{fig: sim_asg}.
\begin{figure}
\caption{Treatment assignment distribution in simulation settings. The left panel is for simulation settings 1 and 2, and the right panel is for simulation settings 3 and 4.}
\label{fig: sim_asg}
\end{figure}
\begin{table}
\caption{\label{tab: sim_12_setting}Simulation settings 1 and 2: treatment effect and interaction effect functions specification. Column ``Treatment Effects'' specifies the treatment effect functions of individual treatments adopted in simulation settings 1 and 2. Column ``Interaction Effects'' specifies the interaction effects among individual treatments in setting 2.}
\centering
\begin{tabular}{p{20mm}p{60mm}p{60mm}}
\hline
Treatment $\mathbf{A}$ & Treatment Effects & Interaction Effects \\
\hline
$(0, 0, 0)$ & $0$ & -\\
$(0, 0, 1)$ & $2X_{1} + \exp(X_{3} + X_{4})$ & - \\
$(0, 1, 0)$ & $2X_{2}\log(X_{5}) + X_{7}$ & - \\
$(0, 1, 1)$ & - & $\sin(5X_{1}^{2}) - 3(X_{2} - 0.5)^2$ \\
$(1, 0, 0)$ & $\sin(X_{3}) + 2\log(X_{4}) + 2\log(X_{7})$ & -\\
$(1, 1, 1)$ & - & $2\sin((X_{2} - X_{4})^2)$\\
\hline
\end{tabular} \end{table}
\begin{table}
\caption{\label{tab: sim_34_setting}Simulation settings 3 and 4: treatment effect and interaction effect functions specification. Column ``Treatment Effects'' specifies the treatment effect functions of individual treatments adopted in simulation settings 3 and 4. Column ``Interaction Effects'' specifies the interaction effects among individual treatments in setting 4.}
\centering
\begin{tabular}{p{20mm}p{60mm}p{60mm}}
\hline
Treatment $\mathbf{A}$ & Treatment Effects & Interaction Effects \\
\hline
$(0, 0, 0, 0, 0)$ & $0$ & -\\
$(0, 0, 0, 0, 1)$ & $(X_{1} - 0.25) ^ 3$ & - \\
$(0, 0, 0, 1, 0)$ & $2\log(X_{3}) + 4\log(X_{8})\cos(2\pi X_{10})$ & - \\
$(0, 0, 1, 0, 0)$ & $X_{2}\sin(X_{4}) - 1$ & \\
$(0, 0, 1, 0, 1)$ & - & $\exp(2X_{2})$\\
$(0, 1, 0, 0, 0)$ & $(X_{1} + X_{5} - X_{8} ^ 2)^3$ & -\\
$(0, 1, 0, 0, 1)$ & - & $\exp(2X_{4} + X_{9})$\\
$(0, 1, 0, 1, 1)$ & - & $-4\log(X_{6})$\\
$(0, 1, 1, 0, 0)$ & - & $0$\\
$(0, 1, 1, 1, 0)$ & - & $0$\\
$(1, 0, 0, 0, 0)$ & $\exp(X_{2} - X_{5})$ & -\\
$(1, 0, 0, 0, 1)$ & - & $0$ \\
$(1, 0, 0, 1, 0)$ & - & $0$ \\
$(1, 0, 1, 0, 0)$ & - & $0$ \\
$(1, 0, 1, 0, 1)$ & - & $-3/2\cos(2\pi X_{1} + X_{8}^2)$\\
$(1, 1, 0, 0, 0)$ & - & $0$\\
$(1, 1, 0, 0, 1)$ & - & $-4\log(X_{6})$\\
$(1, 1, 0, 1, 1)$ & - & $X_{6}^2 + 1/2\sin(2\pi/X_{7})$\\
$(1, 1, 1, 0, 0)$ & - & $0$\\
$(1, 1, 1, 1, 0)$ & - & $0$\\
\hline
\end{tabular} \end{table}
In simulation setting 1, we assume that the treatment effects of the combination treatment are additive from individual treatment, and we specify the individual treatment effect functions in the column ``Treatment Effects'' provided in Table \ref{tab: sim_12_setting}. Based on simulation setting 1, we consider some interaction effects among treatments in simulation setting 2, which are specified in the column ``Interaction Effects'' in Table \ref{tab: sim_12_setting}. Therefore, the treatment effects of the combination treatments are the summation of individual treatment effects and interaction effects. Similarly, Table \ref{tab: sim_34_setting} specifies the treatment effects and interaction effects for simulation settings 3 and 4 in the same manner. In particular, the treatment effects of the combination treatments are additive from individual treatment effects in the simulation setting 3, while interaction effects are added in the simulation setting 4. In summary, we evaluate the empirical performance of the proposed method and competing methods under the additive treatment effects scenarios in simulation settings 1 and 3, and under the interactive treatment effects scenarios in simulation settings 2 and 4.
For each simulation setting, the sample sizes for the training data vary from 500, 1000 to 2000, and each setting is repeated 200 times. Then we compare the proposed method with the following methods: the $L_1$-penalized least square ($L_1$-pls, \citealt{qian2011performance}), the outcome weighted learning with multinomial deviance (OWL-MD, \citealt{huang2019multicategory}), the multicategory outcome weighted learning with linear decisions(MOWL-linear \citealt{zhang2014multicategory}), the outcome weighted learning with deep learning (OWL-DL, \citealt{liang2018estimating}), the treatment-agnostic representation network (TARNet) \citep{shalit2017estimating}. The empirical evaluation of the value function and accuracy are reported in Tables \ref{tab: sim1_value} and \ref{tab: sim1_accuracy}, where the empirical value function \citep{qian2011performance} is calculated via \begin{eqnarray}
\hat{\mathcal{V}}(d) = \frac{\mathbb{E}_{n}[YI\{d(\mathbf{X}) = \mathbf{A}\}]}{\mathbb{E}_{n}[I\{d(\mathbf{X}) = \mathbf{A}\}]}, \notag \end{eqnarray} where $\mathbb{E}_{n}$ denotes the empirical average.
Since simulation settings 1 and 3 do not include interaction effects among different treatments, all competing methods except for the OWL-DL \citep{liang2018estimating} are over-parameterized, while the proposed method can be adaptive to the additive setting with a large $\lambda_{i}$. Therefore, the proposed method and the OWL-DL outperform other competing methods in both settings. In contrast, complex interaction effects are considered in simulation settings 2 and 4, and the performance of OWL-DL is inferior since a consistent estimation is not guaranteed for OWL-DL if there are interaction effects. Although other competing methods are saturated in incorporating interaction effects, their estimation efficiencies are still undermined since the decision functions in these methods are all treatment-specific, while the proposed method possesses the unique parameter-sharing feature for different treatments. Therefore, the advantage of our method is more significant for small sample sizes or large $K$ scenarios. Specifically, the proposed method improves the accuracy by 10.9\% to 17.8\% in simulation setting 4 when the sample size is 500. In addition, we also compare the empirical performance of the double encoder model with different choices of covariates and treatment encoders, and the detailed simulation results are presented in the supplementary materials.
\begin{table} \caption{\label{tab: sim1_value}Unconstrained simulation study: Comparisons of value functions for the proposed method and existing methods including the $L_1$-penalized least square ($L_1$-pls, \citealt{qian2011performance}), the outcome weighted learning with multinomial deviance (OWL-MD, \citealt{huang2019multicategory}), the multicategory outcome weighted learning with linear decisions (MOWL-linear, \citealt{zhang2020multicategory}), the outcome weighted learning with deep learning (OWL-DL, \citealt{liang2018estimating}) and the treatment-agnostic representation network (TARNet, \citealt{shalit2017estimating}). Two treatment assignment schemes are presented: all treatments are uniformly assigned to subjects (Uniform), and treatments are assigned based on the propensity score model (\ref{sim: propensity_score_model}, PS-based).} \centering \scriptsize \begin{tabular}{p{15mm}p{12mm}p{10mm}p{16mm}p{16mm}p{16mm}p{16mm}p{16mm}p{16mm}p{16mm}} \hline \multicolumn{8}{c}{Value} \\ \hline Treatment Assignment & Setting & Sample Size & \textbf{Proposed} & $L_1$-pls & OWL-MD & MOWL-linear & OWL-DL & TARNet\\ \hline \multirow{12}{0pt}{Uniform}&\multirow{3}{0pt}{1} & 500 & \textbf{5.477(0.218)} & 3.643(0.158) & 5.338(0.259) & 5.206(0.185) & 5.408(0.265) & 5.320(0.278)\\ & & 1000 & \textbf{5.622(0.206)} & 3.800(0.154) & 5.510(0.251) & 5.292(0.182) & 5.512(0.259) & 5.398(0.273)\\ & & 2000 & \textbf{5.658(0.208)} & 3.989(0.149) & 5.577(0.246) & 5.384(0.179) & 5.595(0.251) & 5.411(0.272)\\ \cline{3-9} &\multirow{3}{0pt}{2} & 500 & \textbf{5.268(0.325)} & 3.870(0.291) & 5.132(0.306) & 5.078(0.291) & 5.015(0.400) & 5.008(0.405)\\ & & 1000 & \textbf{5.418(0.312)} & 4.028(0.285) & 5.302(0.299) & 5.168(0.279) & 5.105(0.398) & 5.024(0.402)\\ & & 2000 & \textbf{5.498(0.311)} & 4.191(0.282) & 5.344(0.289) & 5.211(0.272) & 5.215(0.382) & 5.118(0.386)\\ \cline{3-9} &\multirow{3}{0pt}{3} & 500 & \textbf{5.600(0.288)} & 3.562(0.235) & 4.479(0.285) & 5.216(0.240) & 5.332(0.312) & 5.354(0.305)\\ & & 1000 & \textbf{5.667(0.268)} & 3.702(0.232) & 4.987(0.279) & 5.283(0.241) & 5.403(0.310) & 5.366(0.300)\\ & & 2000 & \textbf{5.719(0.262)} & 3.855(0.230) & 5.274(0.265) & 5.459(0.232) & 5.598(0.299) & 5.423(0.289)\\ \cline{3-9} &\multirow{3}{0pt}{4} & 500 & \textbf{6.117(0.328)} & 4.490(0.264) & 5.900(0.278) & 5.948(0.277) & 5.995(0.335) & 5.895(0.328)\\ & & 1000 & \textbf{6.374(0.319)} & 4.850(0.262) & 6.200(0.272) & 6.120(0.269) & 6.012(0.321) & 5.998(0.315)\\ & & 2000 & \textbf{6.732(0.310)} & 5.252(0.254) & 6.506(0.259) & 6.494(0.262) & 6.254(0.311) & 6.057(0.310)\\ \cline{3-9} \hline \multirow{12}{0pt}{PS-based}&\multirow{3}{0pt}{1} & 500 & \textbf{5.415(0.238)} & 4.048(0.198) & 5.061(0.215) & 4.897(0.233) & 5.218(0.273) & 5.013(0.305)\\ & & 1000 & \textbf{5.589(0.223)} & 4.087(0.198) & 5.099(0.213) & 4.959(0.231) & 5.223(0.272) & 5.018(0.298)\\ & & 2000 & \textbf{5.662(0.219)} & 4.224(0.183) & 5.178(0.201) & 4.898(0.230) & 5.238(0.279) & 5.017(0.299)\\ \cline{3-9} &\multirow{3}{0pt}{2} & 500 & \textbf{5.005(0.324)} & 3.980(0.236) & 4.815(0.254) & 4.629(0.279) & 4.635(0.336) & 4.886(0.352)\\ & & 1000 & \textbf{5.622(0.322)} & 4.042(0.235) & 4.906(0.249) & 4.700(0.276) & 4.913(0.334) & 5.021(0.341)\\ & & 2000 & \textbf{5.658(0.320)} & 4.104(0.229) & 5.005(0.245) & 4.672(0.274) & 4.998(0.326) & 5.054(0.331)\\ \cline{3-9} &\multirow{3}{0pt}{3} & 500 & \textbf{5.665(0.330)} & 3.384(0.258) & 3.540(0.269) & 5.401(0.302) & 5.505(0.352) & 5.476(0.338)\\ & & 1000 & \textbf{5.792(0.321)} & 4.560(0.248) & 5.009(0.268) & 5.519(0.303) & 5.784(0.348) & 5.676(0.308)\\ & & 2000 & \textbf{5.796(0.318)} & 5.273(0.246) & 5.307(0.259) & 5.582(0.300) & 5.788(0.338) & 5.774(0.299)\\ \cline{3-9} &\multirow{3}{0pt}{4} & 500 & \textbf{5.630(0.356)} & 4.462(0.285) & 3.816(0.305) & 5.090(0.321) & 5.028(0.405) & 5.108(0.387) \\ & & 1000 & \textbf{6.001(0.348)} & 5.432(0.355) & 5.822(0.302) & 5.134(0.319) & 5.384(0.400) & 5.338(0.379)\\ & & 2000 & \textbf{6.289(0.345)} & 5.808(0.344) & 6.141(0.299) & 5.294(0.318) & 5.684(0.389) & 5.589(0.378)\\ \hline \end{tabular} \end{table}
\begin{table} \caption{\label{tab: sim1_accuracy}Unconstrained simulation study: Comparisons of accuracies for the proposed method and existing methods including the $L_1$-penalized least square ($L_1$-pls, \citealt{qian2011performance}), the outcome weighted learning with multinomial deviance (OWL-MD, \citealt{huang2019multicategory}), the multicategory outcome weighted learning with linear decisions (MOWL-linear, \citealt{zhang2020multicategory}), the outcome weighted learning with deep learning (OWL-DL, \citealt{liang2018estimating}) and the treatment-agnostic representation network (TARNet, \citealt{shalit2017estimating}). Two treatment assignment schemes are presented: all treatments are uniformly assigned to subjects (Uniform), and treatments are assigned based on the propensity score model (\ref{sim: propensity_score_model}, PS-based).} \centering \scriptsize \begin{tabular}{p{15mm}p{12mm}p{10mm}p{16mm}p{16mm}p{16mm}p{16mm}p{16mm}p{16mm}p{16mm}} \hline \multicolumn{8}{c}{Accuracy} \\ \hline Treatment Assignment & Setting & Sample Size & \textbf{Proposed} & $L_1$-pls & OWL-MD & MOWL-linear & OWL-DL & TARNet\\ \hline \multirow{12}{0pt}{Uniform}&\multirow{3}{0pt}{1} & 500 & \textbf{0.622(0.052)} & 0.338(0.039) & 0.572(0.040) & 0.491(0.041) & 0.598(0.046) & 0.553(0.050)\\ & & 1000 & \textbf{0.707(0.050)} & 0.358(0.038) & 0.642(0.040) & 0.507(0.040) & 0.682(0.043) & 0.640(0.050) \\ & & 2000 & \textbf{0.738(0.050)} & 0.382(0.038) & 0.694(0.039) & 0.552(0.042) & 0.710(0.042) & 0.686(0.048) \\ \cline{3-9} &\multirow{3}{0pt}{2} & 500 & \textbf{0.544(0.058)} &0.350(0.037) & 0.515(0.041) & 0.474(0.035) & 0.488(0.045) & 0.432(0.052)\\ & & 1000 & \textbf{0.610(0.054)} & 0.367(0.034) & 0.573(0.038) & 0.505(0.033) & 0.532(0.040) & 0.462(0.051)\\ & & 2000 & \textbf{0.630(0.051)} & 0.389(0.033) & 0.605(0.036) & 0.516(0.034) & 0.568(0.041) & 0.506(0.049)\\ \cline{3-9} &\multirow{3}{0pt}{3} & 500 & \textbf{0.420(0.041)} & 0.102(0.025) & 0.148(0.031) & 0.251(0.034) & 0.271(0.047) & 0.205(0.057)\\ & & 1000 & \textbf{0.445(0.039)} & 0.099(0.024) & 0.187(0.027) & 0.285(0.033) & 0.300(0.046) & 0.268(0.053)\\ & & 2000 & \textbf{0.464(0.039)} & 0.116(0.021) & 0.254(0.027) & 0.331(0.031) & 0.353(0.048) & 0.311(0.045)\\ \cline{3-9} &\multirow{3}{0pt}{4} & 500 & \textbf{0.324(0.045)} & 0.146(0.031) & 0.205(0.032) & 0.215(0.032) & 0.154(0.047) & 0.162(0.041)\\ & & 1000 & \textbf{0.335(0.044)} & 0.191(0.031) & 0.279(0.030) & 0.229(0.032) & 0.189(0.045) & 0.193(0.043)\\ & & 2000 & \textbf{0.372(0.041)} & 0.222(0.029) & 0.323(0.029) & 0.241(0.029) & 0.228(0.044) & 0.225(0.042)\\ \cline{3-9} \hline \multirow{12}{0pt}{PS-based}&\multirow{3}{0pt}{1} & 500 & \textbf{0.571(0.048)} & 0.317(0.031) & 0.477(0.035) & 0.430(0.037) & 0.498(0.050) & 0.432(0.053)\\ & & 1000 & \textbf{0.648(0.044)} & 0.327(0.027) & 0.502(0.033) & 0.451(0.032) & 0.525(0.047) & 0.462(0.051)\\ & & 2000 & \textbf{0.699(0.044)} & 0.332(0.028) & 0.522(0.031) & 0.448(0.031) & 0.552(0.048) & 0.481(0.050)\\ \cline{3-9} &\multirow{3}{0pt}{2} & 500 & \textbf{0.492(0.052)} & 0.284(0.037) & 0.420(0.037) & 0.397(0.036) & 0.370(0.047) & 0.375(0.048)\\ & & 1000 & \textbf{0.566(0.051)} & 0.290(0.037) & 0.443(0.034) & 0.414(0.037) & 0.397(0.045) & 0.388(0.047)\\ & & 2000 & \textbf{0.618(0.048)} & 0.291(0.037) & 0.464(0.031) & 0.406(0.032) & 0.411(0.041) & 0.425(0.049)\\ \cline{3-9} &\multirow{3}{0pt}{3} & 500 & \textbf{0.378(0.041)} & 0.053(0.030) & 0.071(0.033) & 0.223(0.036) & 0.300(0.051) & 0.248(0.050)\\ & & 1000 & \textbf{0.439(0.041)} & 0.091(0.031) & 0.181(0.032) & 0.279(0.034) & 0.376(0.041) & 0.344(0.048)\\ & & 2000 & \textbf{0.444(0.040)} & 0.137(0.029) & 0.242(0.030) & 0.327(0.031) & 0.416(0.038) & 0.378(0.047)\\ \cline{3-9} &\multirow{3}{0pt}{4} & 500 & \textbf{0.223(0.056)} & 0.101(0.038) & 0.083(0.039) & 0.090(0.043) & 0.102(0.052) & 0.084(0.061)\\ & & 1000 & \textbf{0.267(0.053)} & 0.136(0.036) & 0.195(0.041) & 0.089(0.041) & 0.121(0.051) & 0.098(0.056)\\ & & 2000 & \textbf{0.279(0.052)} & 0.205(0.037) & 0.245(0.041) & 0.101(0.039) & 0.168(0.048) & 0.127(0.055)\\ \hline \end{tabular} \end{table}
\subsection{Budget-constrained ITR simulation} \label{sec: cstr_simulation}
\begin{table} \caption{\label{tab: sim2_value}Constrained simulation study: Comparisons of value functions for the proposed method and existing methods including the $L_1$-penalized least square ($L_1$-pls, \citealt{qian2011performance}), the outcome weighted learning with multinomial deviance (OWL-MD, \citealt{huang2019multicategory}), the multicategory outcome weighted learning with linear decisions (MOWL-linear, \citealt{zhang2020multicategory}), and the treatment-agnostic representation network (TARNet, \citealt{shalit2017estimating}). All treatments are uniformly assigned to subjects.} \centering \scriptsize \begin{tabular}{p{12mm}p{12mm}p{16mm}p{16mm}p{16mm}p{16mm}p{16mm}} \hline \multicolumn{7}{c}{Value} \\ \hline Sample Size & Constraints & \textbf{Proposed} & $L_1$-pls & OWL-MD & MOWL-linear & TARNet\\ \hline \multirow{4}{0pt}{500} & 20\% & \textbf{5.237(0.322)} & 3.445(0.268) & 4.918(0.283) & 4.909(0.279) & 4.874(0.335)\\ & 50\% & \textbf{5.527(0.320)} & 3.756(0.277) & 5.218(0.271) & 5.215(0.273) & 4.998(0.335)\\ & 80\% & \textbf{5.832(0.319)} & 4.108(0.267) & 5.510(0.280) & 5.499(0.276) & 5.425(0.329)\\ & 100\% & \textbf{6.117(0.328)} & 4.490(0.264) & 5.900(0.278) & 5.948(0.277) & 5.895(0.328)\\ \hline \multirow{4}{0pt}{1000} & 20\% & \textbf{5.498(0.321)} & 3.685(0.261) & 5.175(0.270) & 5.170(0.271) & 5.047(0.320)\\ & 50\% & \textbf{5.841(0.315)} & 4.014(0.265) & 5.487(0.269) & 5.318(0.275) & 5.274(0.318)\\ & 80\% & \textbf{6.102(0.320)} & 4.417(0.261) & 5.612(0.273) & 5.598(0.267) & 5.418(0.315)\\ & 100\% & \textbf{6.374(0.319)} & 4.850(0.262) & 6.200(0.272) & 6.120(0.269) & 5.998(0.315)\\ \hline \multirow{4}{0pt}{2000} & 20\% & \textbf{5.789(0.309)} & 4.108(0.249) & 5.356(0.209) & 5.317(0.266) & 5.015(0.308)\\ & 50\% & \textbf{6.015(0.315)} & 4.437(0.251) & 5.897(0.207) & 5.778(0.261) & 5.298(0.298)\\ & 80\% & \textbf{6.324(0.311)} & 4.847(0.255) & 6.215(0.208) & 6.117(0.264) & 5.598(0.315)\\ & 100\% & \textbf{6.732(0.310)} & 5.252(0.254) & 6.506(0.201) & 6.494(0.262) & 6.057(0.310)\\ \hline \end{tabular} \end{table}
In this subsection, we investigate the budget-constrained setting with the same data generation mechanism as in simulation setting 4 in Section \ref{sec: uncstr_simulation}. For a fair comparison, we compare the proposed method with competing methods which provide a score to measure the utility or effect of each treatment. We then apply the proposed MCKP method to all of these methods because the proposed framework (\ref{empirical_constrained_problem}) does not require specification for the treatment effects.
For the budget constraint, we let the second treatment be the most critical, or urgently needed, by the population. Thus, we constrain the amount of the second treatment so that only partial patients can be treated by the second treatment. The quantiles of the constrained populations are $20\%, 50\%, 80\%$, and $100\%$, where the constraints in the last case is trivial constraints as it is equivalent to an unconstrained setting.
The simulation results are provided in Tables \ref{tab: sim2_value}, which clearly indicate that the proposed method outperforms other methods in the constrained cases. Moreover, the proposed method achieves smaller reductions of value functions when budget constraints are imposed. Specifically, the value functions of the competing methods are reduced by about 0.9 when the budget is decreased from 100$\%$ to 20$\%$. More precisely, when the sample size is 2000, compared with the best performances of competing methods, the proposed method improves the value function by $8.08\%$ when the budget is $20\%$, and it achieves $3.47\%$ improvement in value function when the budget is $100\%$. The significant improvement of the value function under limited budget scenarios shows that the proposed method provides a more accurate estimation of treatment effects, and thus leads to a better individualized treatment rule estimation under restrictive constraints.
\section{Optimal Combination Treatments for Type-2 Diabetes Patients}
In this section, we apply our method to electronic health record data for type-2 diabetes patients from the Clinical Practice Research Datalink. In this study, subjects were Type-2 diabetes patients recruited based on ICD9 codes \footnote{\url{https://www.cdc.gov/nchs/icd/icd9cm.htm}} in the CPRD system \footnote{\url{https://cprd.com}} from 2015 to 2018. Each patient was followed for 6 months to measure their treatment effectiveness. If patients have less than 3 months follow-up, those patients were removed from the dataset; if patients do not have a 6 month measurement, we use the closest A1C measurement between 3 months to 1 year to impute 6 month measurement.
The dataset includes 4 treatments: dipeptidyl peptidase-4 (DPP4), sulfonylurea (SU), metformin (Met), and thiazolidinedione (TZD), and 16 combinations of these treatments are assigned to patients. Recent works \citep{salvo2016addition, ahren2008novel, mikhail2008combination, stafford2007treatment, staels2006metformin} show that interaction effects among these treatments exist. For example, \cite{salvo2016addition} suggest that SU combined with DDP4 induces a higher risk of hypoglycemia compared with using the SU treatment alone. Therefore, it is critical to incorporate interaction effects in estimating the optimal ITR for type-2 diabetes patients.
Due to the nature of the observational study, treatments are not randomly assigned, and the frequencies of the 16 combination treatments are shown in Table \ref{tab: real_data_overview}. For this unbalanced assignment problem, the positivity assumption \ref{causal_assumption}(c) is likely to be violated. Therefore, we only consider the treatments that are assigned to more than 60 subjects. To validate the positivity assumption, we compute the propensity scores for all subjects and all treatments. We find the common support of all treatments, which is (0, 0.6), and filter out subjects whose propensity scores for certain treatments are outside of the common support. After the above pre-processing, we obtain a dataset including 923 subjects assigned with 6 treatments.
\begin{table} \caption{\label{tab: real_data_overview}Number of subjects assigned various combinations of four medications: dipeptidyl peptidase-4 inhibitor(DPP4), sulfonylurea (SU), metformin (Met) and thiazolidinedione (TZD). Bold numbers are treatment combinations considered in our real data analysis for any combination treatment with more than 60 subjects.} \centering \begin{tabular}{p{10mm}p{10mm}p{10mm}p{10mm}p{20mm}p{10mm}p{10mm}p{10mm}p{10mm}p{20mm}} \hline
DPP4 & SU & Met & TZD & \# subject & DPP4 & SU & Met & TZD & \# subject \\ \hline
\textbf{0} & \textbf{0} & \textbf{0} & \textbf{0} & \textbf{202} & 1 & 0 & 0 & 0 & 4 \\
0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & 3 \\
\textbf{0} & \textbf{0} & \textbf{1} & \textbf{0} & \textbf{185} & 1 & 0 & 1 & 0 & 53 \\
0 & 0 & 1 & 1 & 6 & 1 & 0 & 1 & 1 & 20 \\
\textbf{0} & \textbf{1} & \textbf{0} & \textbf{0} & \textbf{72} & 1 & 1 & 0 & 0 & 31 \\
0 & 1 & 0 & 1 & 11 & 1 & 1 & 0 & 1 & 4 \\
\textbf{0} & \textbf{1} & \textbf{1} & \textbf{0} & \textbf{222} & \textbf{1} & \textbf{1} & \textbf{1} & \textbf{0} & \textbf{226} \\
\textbf{0} & \textbf{1} & \textbf{1} & \textbf{1} & \textbf{67} & 1 & 1 & 1 & 1 & 32 \\ \hline \end{tabular} \end{table}
In this study, 21 pre-treatment covariates were collected, including basic demographic information (e.g., age, BMI, gender, weight, height), diabetes-related health index (e.g., high-density lipoprotein, low-density lipoprotein, hematocrit), and medical history (e.g., congestive heart failure, stroke, stroke, hypertension). In our analysis, all of these covariates, except for lower extremity arteries (LEA), are used to control for confounding. In the supplementary, we also adopt the standardized mean difference (SMD) to evaluate the covariates balancing after the inverse probability weighting, and it verifies that these covariates are well balanced.
The primary interest of measurement for type-2 diabetes is the A1C, which measures average blood glucose levels \citep{kahn2008translating}. The normal A1C level is below 5.7$\%$, and type-2 diabetes patients are generally above 6.5$\%$ \citep{zhang2010a1c}. The A1C levels are likely to decrease after the treatments are applied. Thus, we use the negative change of A1C as our outcome measure, where a larger value indicates better performance.
To implement our proposed method and other competing methods, We randomly split the dataset into training (743), validation (60), and testing (120) sets. For each treatment, we randomly sample 10 subjects as the validation set and 20 subjects as the testing set and leave other subjects in the training sets. To validate the results, we repeat the random splitting procedure and run the experiment 100 times, then report the averaged value function evaluation on the test sets.
Furthermore, we evaluate the proposed constrained individualized treatment rule on the same dataset with synthetic budget constraints. Suppose the total budget for patients receiving treatments in the test sets (120 subjects) is limited, then some subjects may not have access to their optimal treatments. For example, the costs for DDP4, SU, Met and TZD are $\$533$, $\$13$, $\$6$, and $\$30$ per dose, respectively, and our hypothetical budgets are \$500, \$2,000, \$5,000, \$20,000, \$100,000, where the $\$100,000$ budget constraint is equivalent to an unconstrained scenario as the total cost is less than $\$100,000$ even under the scenario such that all patients are assigned to the combination of four treatments. Finally, we report the mean and standard deviations of the value functions under different budget constraints in Table \ref{tab: real_data}. For the unconstrained scenario ($\$100,000$ budget), the proposed method achieves great improvement in value maximization. As a reference, the value function under one-size-fits-all rules and assigned treatments are shown in Table \ref{tab: real_data_value_universal}, which shows that our proposed method improves the value function by 4.21\% to 260.58\% compared with one-size-fits-all rules. In comparison with the competing methods that also estimate the ITR, our proposed method improves the value function by 3.62\% to 37.6\%.
\begin{table}
\caption{\label{tab: real_data_value_universal}Value function under one-size-fits-all rules and assigned treatments}
\centering
\begin{tabular}{p{30mm}p{30mm}p{30mm}p{30mm}}
\hline
Treatment & Value function & Treatment & Value function \\
\hline
Null & 2.622(0.849) & Met & 2.926(0.795) \\
SU & 4.038(0.738) & SU + Met & 2.181(1.167) \\
Su + Met + TZD & 1.167(0.490) & DPP4 + SU + Met & 1.611(0.730) \\
\textit{Assigned} & 2.373(0.338) & & \\
\hline
\end{tabular} \end{table}
\begin{table} \caption{\label{tab: real_data}Mean and standard errors of value function under different budget constraints. The last row ($\$100,000$ budget) is for the unconstrained scenario.} \centering {\footnotesize
\begin{tabular}{p{15mm}p{20mm}p{15mm}p{15mm}p{15mm}p{15mm}p{15mm}p{15mm}}
\hline
Budgets & Proposed method & $L_1$-PLS & OWL-MD & MOWL-linear & MOWL-kernel & OWL-DL & TARNet \\
\hline
\$100,000 & \textbf{4.208(0.880)} & 3.912(0.707) & 4.061(0.702) & 3.200(0.563) & 3.947(0.934) & 3.058(0.908) & 3.868(0.870) \\
\$20,000 & \textbf{3.918(0.872)} & 3.568(0.628) & 3.729(0.705) & 3.001(0.540) & 3.609(0.924) & 2.786(0.905) & 3.376(0.845) \\
\$5,000 & \textbf{3.576(0.877)} & 3.187(0.612) & 3.215(0.685) & 2.890(0.548) & 3.113(0.923) & 2.408(0.916) & 3.036(0.846) \\
\$2,000 & \textbf{3.287(0.804)} & 2.960(0.597) & 2.906(0.599) & 2.678(0.540) & 2.805(0.887) & 2.206(0.879) & 2.667(0.898) \\
\$500 & \textbf{3.020(0.870)} & 2.709(0.589) & 2.512(0.580) & 2.387(0.558) & 2.578(0.687) & 2.089(0.657) & 2.338(0.865) \\
\hline
\end{tabular} } \end{table}
For the budget-constrained scenarios, the proposed method retains dominant advantages over the competing methods under varying budgets. Moreover, the reduction of the value function under the proposed method is relatively small. Specifically, the value function attained under the proposed method decreases by 1.188, while the competing methods decrease by 1.203, 1.549, 0.813, 1.369, 0.969, and 1.53, respectively. In summary, the proposed method is able to reduce glucose levels more than any other competing methods and one-size-fits-all rules, which might have great potential on improving therapy quality for type-2 diabetes patients.
\section{Discussion} In this paper, we broaden the scope of the individualized treatment rule (ITR) from binary and multi-arm treatments to combination treatments, which are combinations of multiple treatments where treatments within each combination can interact with each other. The proposed double encoder model is a nonparametric approach that effectively accounts for intricate treatment effects and interaction effects among treatments. Furthermore, it enhances estimation efficiency through the use of parameter-sharing treatment encoders. We also adapt the estimated ITR to account for budget constraints by employing a multi-choice knapsack framework, which strengthens our proposed method in situations with limited resources. Theoretically, we offer a value reduction bound with and without budget constraints and an improved convergence rate concerning the number of treatments under the DEM.
Several potential research directions could be worth further investigation. First of all, the proposed method employs the propensity score model to achieve the double robustness property. However, the inverse probability weighting method could be worrisome in observational studies considering the combination treatments, due to the potential violation of positivity assumptions. This phenomenon is also observed in the binary treatment scenario with high-dimensional covariates \citep{d2021overlap}. There are some existing works investigating this issue in the binary treatment setting which propose some alternatives to the propensity score, such as overlap weights\citep{li2019propensity, lifan2019addressing}, but the same issue in multi-arm or combination treatment problems has not been investigated yet. Therefore, exploring multi-arm or combination treatment problems in depth is worth consideration.
Second, compared with the binary treatments, combination treatments enable us to optimize multiple outcomes of interest simultaneously. The major challenge of multiple outcomes is that each combination treatment may only favor a few outcomes, so an optimal ITR is expected to achieve a trade-off among multiple outcomes. Some recent works have studied trade-offs between the outcome of interest and risk factors \citep{wang2018learning, huang2020estimating}. However, trade-offs among multiple outcomes could be a more complicated problem for further investigation.
\end{document} | arXiv |
\begin{document}
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\title[On the minimal number of small elements generating prime fields] {\bf On the minimal number of small elements generating prime fields}
\author{Marc Munsch} \address{5010 Institut f\"{u}r Analysis und Zahlentheorie 8010 Graz, Steyrergasse 30, Graz} \email{[email protected]}
\date{\today}
\subjclass[2010]{11A07, 11A15, 11L40, 11T99, 11N36} \keywords{Finite fields, generating set, primitive roots, Burgess inequality, sieve theory, character sums, algorithmic number theory}
\begin{abstract}
In this note, we give an upper bound for the number of elements from the interval $\[1,p^{1/4e^{1/2}+\epsilon}\right]$ necessary to generate the finite field $\mathbb{F}_{p}$ with $p$ an odd prime. The general result depends on the localization of the divisors of $p-1$ and can be for instance used to deduce easily results on a set of primes of density $1$.
\end{abstract}
\maketitle
\section{Introduction}
E. Artin conjectured in 1927 that any positive integer $n>1$, which is not a perfect square, is a primitive root modulo $p$ for infinitely many primes $p$. It remains open nowadays but was proved assuming the Generalized Riemann Hypothesis for some specific Dedekind zeta functions by Hooley in \cite{Hooley}. Using the development of large sieve theory leading to Bombieri-Vinogradov theorem, one can show that Artin's primitive root conjecture is true for almost all primes (see for instance \cite{HB} or \cite{Moreesurvey} for an extended survey about this conjecture). Another related classical problem is to bound the size $g(p)$ of the smallest primitive root modulo $p$. The best unconditional result is $g(p)=O(p^{1/4+\epsilon})$ and is due to Burgess in \cite{Burgess}, as a consequence of his famous character sum estimate. This is very far from what we expect, and assuming Generalized Riemann Hypothesis we can for instance show that $g(p)= O(\log^2 p)$ (see \cite{Montgomery} following \cite{Ankeny} or \cite{Tao} for results under the Elliott-Halberstam conjecture). Like before, as a consequence of the large sieve, the upper bound $g(p) = O ((\log p)^{2+\epsilon})$ is valid for almost all primes (see \cite{BurElliott}). The problem of improving unconditionnaly the bound on the least primitive root seems presently out of reach. For instance, we cannot perform directly the Vinogradov trick to show that there exists a primitive root less than $p^{1/4\sqrt{e}+\epsilon}$, however we can reach that range for this following variant question:
\begin{question*}How large should be $N$ (in terms of $p$) such that $<1,\dots,N>$ is a generating set of $\mathbb{F}_{p}^{*}$. \end{question*} Is is shown by Burthe in \cite{Burthegenerating} that $N=p^{1/4e^{1/2}+\epsilon}$ is sufficient\footnote{The result holds in fact for a composite $n$ as long as $8\nmid n$.} and seems to be the lower limit of what is possible unless Burgess character sum bound is improved. Nonetheless, in view of this result, several interesting related questions can be formulated. Harman and Shparlinski considered the problem of minimizing the value of $k$ such that for a sufficiently large prime $p$ and for any integer $a<p$, there is always a solution to the congruence $$ n_1 \dots n_k \equiv a \,\,(\bmod \,p),\,\,1 \leq n_1, \dots , n_k \leq N$$ and showed in \cite{HarShpar} that $k=14$ is an admissible value. \footnote{$k$=7 is admissible if we only ask that there is a solution for almost all values of $a$.} From an algorithmic point of view, another interesting question is to know precisely how many elements of $\{1,\dots,N\}$ are in fact necessary to generate the full multiplicative group. We consider in this note the problem of the size of a generating set consisting of small elements less than $N$. \begin{question*} How many elements of $\{1,\cdots,p^{1/4e^{1/2}+\epsilon}\}$ do we need in order to to generate $\mathbb{F}_{p}^{*}$?\end{question*}
$
$
Let $p$ be a prime such that $p-1=\# \mathbb{F}_{p}^{*}=q_1^{\alpha_1}\dots q_r^{\alpha_r}$ with $q_i,\, i=1,\dots, r$ the prime factors of $p-1$. We denote by $\omega(n)$ the number of prime factors of an integer $n$, therefore we have $\omega(p-1)=r$.
$
$
The first elementary result in this direction is the following:
\begin{lemma}\label{elementary} For every $\epsilon>0$, we need only $\omega(p-1)$ elements among $\{1,\cdots,p^{1/4e^{1/2}+\epsilon}\}$ to generate $\mathbb{F}_{p}^{*}$. \end{lemma}
\begin{proof} Classically, using Burgess' character sums inequality (see \cite{Burgess}) combined with ``Vinogradov trick" (see \cite{Vino2}, \cite{Vino1}), we can pick up $x_1,\dots,x_r<p^{1/4e^{1/2}+\epsilon}$ such that $x_i$ is not a $q_i^{th}$ residue modulo $p$. Fixing $g$ a primitive root, we have $x_i=g^{\beta_i}$ with $\gcd(\beta_i,q_i)=1$. Thus, $\gcd(\beta_1,\dots,\beta_r)$ is coprime to $p-1$. By Bezout's theorem, there exists integers $l_1,\dots l_r$ such that $\displaystyle{\sum_{i=1}^{r} l_i\beta_i}$ is coprime to $p-1$. Hence, $x_1^{l_1}\dots x_r^{l_r}$ is a primitive root of $\mathbb{F}_{p}^{*}$ and the statement is proved. \end{proof}
In this note, we wonder if we could improve on this bound which means require less small elements to generate the full group. We will not be able to do this in full generality, the result depending on the anatomy of $p-1$. To measure this, we introduce the following definition.
\begin{definition*} Let $l\geq 1$, we denote by $\omega_l(n)=\#\{q \text{ prime }, q\vert n, q\leq (\log (n+1))l^l\}$ the number of ``small" prime divisors of $n$. \end{definition*}
In the rest of the paper, $\log_k x$ will denote the $k$ times iterated logarithm when $k$ is an integer. Using a combinatorial argument and recent development in sieve theory in non regularly distributed sets, we prove in Section $3$:
\begin{theorem*} Let $l:=l(p)\geq 1$ a parameter tending to infinity with $p$ such that $l\leq \frac{\log p}{1000 \log_2 p}$. We need $O\left(\omega_l(p-1) + \frac{\omega(p-1)}{\log l}\right)$ elements smaller than $p^{1/4\sqrt{e}+\epsilon}$ to generate the multiplicative group $\mathbb{F}_{p}^{*}$ where the implied constant is effective. \end{theorem*} We will also give a more precise result of this type and deduce, in the last part of the paper, stronger results for almost all primes. In the next section, we recall some sieve results that we will use in our argument.
\section{Sieve fundamental result}
In this section, we will use the notations and recall the setting of \cite{Matomakisieve}. Let $\mathbb{P}$ be the set of all primes and let $\mathcal{P} \subseteq \mathbb{P}$ be a subset of the primes $\leq x$. The most basic sieving problem is to estimate $$ \Psi(x; \mathcal{P}) := \#\{ n \leq x \colon p \mid n \implies p \in \mathcal{P}\}. $$ In other words we sieve the integers in $[1, x]$ by the primes in $\mathcal{P}^c = (\mathbb{P} \cap [1,x]) \setminus \mathcal{P}$. A simple inclusion-exclusion argument suggests that $\Psi(x; \mathcal{P})$ should be approximated by $$ x \prod_{p \in \mathcal{P}^c} \left(1-\frac{1}{p}\right). $$ This is always an upper bound, up to a constant, and a lower bound, up to a constant, if $\mathcal{P}$ contains all the primes larger than $x^{1/2-o(1)}$.
On the other hand, there are examples where $\Psi(x; \mathcal{P})$ is much smaller than the expected lower bound. For instance if one fixes $u \geq 1$ and lets $\mathcal{P}$ consist of all the primes up to $x^{1/u}$, then the prediction is about $x/u$ whereas, by an estimate for the number of smooth numbers, we know that $\Psi(x; \mathcal{P}) = \rho(u) x$ with $\rho(u) = u^{-u(1+o(1))} $ as $u \rightarrow \infty$, which is much smaller for large $u$.
The first ones to study what happens if one also sieves out some primes from $[x^{1/2}, x]$ were Granville, Koukoulopoulos and Matom\"aki~\cite{sieveworks}. They conjectured that the critical issue is to understand what is the largest $y$ such that \begin{equation} \label{eq:sum1/pydef} \sum_{\substack{p \in \mathcal{P} \\ y \leq p \leq x^{1/u}}} \frac{1}{p} \geq \frac{1+\varepsilon}{u}. \end{equation} More precisely, they conjectured that when this inequality holds, the sieve works about as expected. On the other hand they gave examples with $$ \sum_{y \leq p \leq x^{1/u}} \frac{1}{p} = \frac{1-\varepsilon}{u} $$ such that $\Psi(x; \mathcal{P})$ is much smaller than expected.
We will use the following result proved by Matom\"aki and Shao confirming that conjecture:
\begin{theorem} \label{th:MT}\cite[Theorem 1.1]{Matomakisieve} Fix $\varepsilon>0$. If $x$ is large and $\mathcal{P}$ is a subset of the primes $\le x$ for which there are some $1 \leq u \leq v\le \frac{\log x}{1000 \log_2 x}$ with $$ \sum_{\substack{ p\in \mathcal{P} \\ x^{1/v}<p\leq x^{1/u}}} \frac 1p \geq \frac{1+\varepsilon}{u}, $$ then $$ \frac{\Psi(x;\mathcal{P})}{x} \geq A_v \prod_{p\in \mathcal{P}^c} \left( 1 -\frac 1p \right), $$ where $A_v$ is a constant with $A_v = v^{-v(1+o_\varepsilon(1))}$ as $v \to \infty$. If $u$ is fixed, one can take $A_v = v^{-e^{-1/u} v(1+o_\varepsilon(1))}$ as $v \to \infty$. \end{theorem}
\section{Idea of the method and main results}
\begin{definition*}We define $h(p)$ as the number of elements smaller than $p^{1/4\sqrt{e}+\epsilon}$ which are sufficient to generate the multiplicative group $\mathbb{F}_{p}^{*}$. \end{definition*}
The aim of this note is to give improvements on the size of $h(p)$. The main idea is the following: due to the sparsity of powers, for large divisors $q_1,q_2$ of $p-1$, a non $q_1^{th}$ residue will be more likely a non $q_2^{th}$ residue. Thus, we do not need to pick up a non-residue for every power as it is done in Lemma \ref{elementary} and we can further play this game with more divisors in order to decrease the number of necessary steps in the argument. In order to do that, we will use the result on the sieve recalled in previous section. The dependance on $v$ in the lower bound of Theorem \ref{th:MT} will prevent us to regroup as much divisors as we want, thus we will carefully split the set of prime divisors in blocks of size $k$ with an ``optimal" value of $k$ coming from the application of Theorem \ref{th:MT}.
Given a parameter $l \geq 1$, we obtain a bound for $h(p)$ depending on $\omega_l(p-1)$. If for some relatively large $l$, $\omega_l(p-1)$ is small, this will give a significant improvement on the trivial bound $\omega(p-1)$ coming from Lemma \ref{elementary}.
The next result is the main tool that we are going to use to deduce to derive these improvements. It shows that we can handle several large prime divisors of $p-1$ simultaneously.
\begin{prop}\label{blocsk}\textnormal{[\bf{Main proposition}]} Let $l:=l(p)\geq 1$ a parameter tending to infinity with $p$ and $k$ an integer verifying $k\leq \frac{\log l}{4}$. Moreover, assume that $l\leq \frac{\log p}{1000 \log_2 p}$. Suppose that $q_1,\dots,q_k$ are prime divisors of $p-1$ greater than $(\log p)l^l$. Then, for $p$ sufficiently large, there exists an integer $n\leq N=p^{1/4\sqrt{e}+\epsilon}$ which is a non $q_i^{th}$ residue for $i=1,\dots,k$. \end{prop}
\begin{proof}
Define $S=\{1\leq n\leq N \text{ s.t. } n\text{ is a non } q_i^{th} \text{-residue modulo }p\text{ for } i=1,\dots,k\}$ and suppose that $S=\emptyset$ which means that every integer in this interval is $q_i^{th}$ residue modulo $p$ for at least one $i$. Thus, we have in particular \begin{equation}\label{decompprimes}\mathcal{P}=\{q \text{ prime}, 1\leq q\leq N\}= \bigcup_{i=1}^{k} \mathcal{P}_{i}\end{equation} where $\mathcal{P}_{i}=\mathcal{P} \cap \{ q_i^{th} \text{-residue modulo }p \} .$ For $x$ sufficiently large and $u,v$ parameters to be specified later, we have by Mertens' Theorem that,
$$ \sum_{q\leq x } \frac{1}{q} = \log_2 x + O(1)$$ and thus $$ \sum_{\substack{ q\in \mathcal{P} \\ x^{1/v}<q\leq x^{1/u}}} \frac 1q \geq \frac{1}{2} \log(v/u) .$$ Consequently, using (\ref{decompprimes}) we get that there exists $i \in \{1,\dots,k\}$ such that
\begin{equation}\label{inverseprimes} \sum_{\substack{ q\in \mathcal{P}_{i} \\ x^{1/v}<q\leq x^{1/u}}} \frac 1q \geq \frac{1}{2k} \log(v/u) .\end{equation} We want to apply Theorem \ref{th:MT}, hence we need the right hand side of (\ref{inverseprimes}) to be larger than $\frac{1+\epsilon}{u}$ under the conditions $1 \leq u \leq v\le \frac{\log x}{1000 \log_2 x}$. Let us fix $u$ such that $\frac{1}{u}=\frac{1}{4\sqrt{e}}+\epsilon$ and $x=p$ so that $N=x^{1/u}$.
Thus the condition of Theorem \ref{th:MT} is verified as long $k\leq \frac{\log v}{4}$. Therefore, we get
$$\frac{\Psi(p;\mathcal{P}_{i})}{p} \geq A_v \prod_{q\in \mathcal{P}_{i}^c} \left( 1 -\frac 1q \right).$$ Using the third Mertens' Theorem, the product is trivially bounded from below by $$\displaystyle \prod_{q \leq p} \left( 1 -\frac 1q \right)\geq \frac{1}{\log p}$$ for $p$ large enough. Thus, we obtain the inequality $\Psi(p;\mathcal{P}_{i}) \gg A_v(\log p)^{-1}x \gg v^{-v}\frac{x}{\log p}$. On the other hand, we are counting integers less than $p$ which are $q_1^{th}$ residues and so there are at most $p/q_i$ of these. It leads to a contradiction when $v^{-v}(\log p)^{-1} \geq 1/q_i$ or equivalently $q_i\geq (\log p)v^v$. In this case, the set $S$ is non empty and this concludes the proof setting $v=l$.
\end{proof}
This proposition helps us to regroup the divisors in ``blocks" of size $k$. Using this idea in a simple way, we are able to deduce the result announced in the introduction:
\begin{theorem}\label{general} Let $l:=l(p)\geq 1$ a parameter tending to infinity with $p$ such that $l\leq \frac{\log p}{1000 \log_2 p}$. For $p$ a sufficiently large prime, we have the bound $$h(p) \ll \omega_l(p-1) + \frac{\omega(p-1)-\omega_l(p-1)}{\log l}$$ where the implied constant is effective.
\end{theorem}
\begin{proof} Consider the prime divisors of $p-1$ which are greater than $(\log p)l^l$. We can apply the Proposition \ref{blocsk} with $k=\frac{\log p}{4}$ and pick up an integer less than $p^{1/4\sqrt{e}+\epsilon}$ which is a non $q^{th}$ residue for $k$ different large $q$. Regrouping the large divisors of $p-1$ in blocks of size $k$, we have at most $\frac{\omega(p-1)-\omega_l(p-1)}{k}$ of such blocks. This concludes the proof including the contribution of small divisors treated individually using Burgess' character sums inequality combined with ``Vinogradov trick" as in Lemma \ref{elementary}. \end{proof}
\begin{remark} The value of the optimal parameter $l$ is not so clear for a general $p$, it will depends heavily on the repartition of the prime divisors of $p-1$. \end{remark} We can in fact iterate in some sense the argument used to prove Theorem \ref{general} and obtain the following stronger result:
\begin{theorem}\label{iteration} Let $l_n(p), n=0,\dots,N$ be a strictly decreasing sequence of parameters tending to infinity with $p$ such that $(\log p)l_0^{l_0}>p$ and that $l_1\leq \frac{\log p}{1000 \log_2 p}$. Then, for $p$ a sufficiently large prime, we have
$$h(p) \ll \omega_{l_{N}}(p-1)+\sum_{n=0}^{N-1} \frac{\omega_{l_n}(p-1)-\omega_{l_{n+1}}(p-1)}{\log(l_{n+1})}. $$
\end{theorem}
\begin{proof} We argue similarly as in Theorem \ref{general}, regrouping the divisors of $p-1$ lying in the interval $](\log p)l_{n+1}^{l_{n+1}},(\log p)l_n^{l_n}]$ in blocks of size $k_n \approx \log (l_{n+1})$. The contribution of the remaining small prime divisors is given by $\omega_{l_{N}}(p-1)$.
\end{proof}
Even though stronger results about primitive roots are known for almost all primes, a result on a set of primes of density $1$ follows as a consequence of Theorem \ref{iteration}.
\subsection{Results for almost all primes}
The next result gives a bound on the number of small prime divisors of $p-1$ for most of the primes $p$.
\begin{lemma}\label{omega} Let $A>1$ and $\epsilon>0$. Suppose $l$ is such that $l^l \ll x^{1/2-\epsilon}$. Then, the set of primes $p\leq x$ such that $p-1$ verifies $\omega_l(p-1) \ll \log l$ is asymptotically of density $1$. \end{lemma}
\begin{proof} We evaluate the average number of primes verifying the inequality of the lemma: \begin{equation*}\sum_{p\leq x \atop p \text{ prime }} \sum_{q\vert p-1 \atop q\leq (\log p)l^l, q \text{ prime }} 1= \sum_{q \leq (\log x)l^l} \sum_{p\equiv 1 \bmod q \atop p\leq x} 1 = \sum_{q\leq (\log x)l^l} \frac{x}{(q-1)\log x} + O\left(\frac{x}{\log^A x}\right) \end{equation*} where we used the Bombieri-Vinogradov theorem (see for instance \cite{Bombieri}). Thus, using Mertens' Theorem, it gives
$$\sum_{p\leq x \atop p \text{ prime }} \sum_{q\vert p-1 \atop q\leq (\log p)l^l, q \text{ prime }} 1=\frac{x}{\log x} (\log l + \log_2 l + M) + O\left( \frac{x}{\log^B x}\right)$$ where $M$ is the Meissel-Mertens constant and $B=\min\left\{A,2\right\}$. The conclusion follows easily.
\end{proof}
\begin{remark} We could obtain the normal order of $\omega_l(p-1)$ following the method of Turan (see \cite{Turan}) using the first two moments. It might be even possible to prove a more precise statement like an Erd\"{o}s-Kac version of this result using the method of Granville and Soundararajan (see \cite{Kac}) but it is not the purpose of this note. \end{remark}
\begin{corollary}\label{dyadicappli} For almost all primes $p$, we have $h(p)\ll (\log_3 p)^2$.\end{corollary}
\begin{proof}
In order to prove this result, we define dyadically some special parameters. Let $l_n=\exp\left(\frac{\log_2 p}{2^n\log_3 p}\right)$ for $1\leq n\leq N=\frac{\log_3 p-2\log_4 p}{\log 2}$. It is easy to see that this sequence fullfils the hypotheses of Theorem \ref{iteration}, thus we derive
$$ h(p) \ll \omega_{l_{N}}(p-1)+\sum_{n=1}^{N-1} \frac{\omega_{l_n}(p-1)-\omega_{l_{n+1}}(p-1)}{\log(l_{n+1})}+\frac{\omega(p-1)-\omega_{l_1}(p-1)}{\log (l_1)}.$$ We remark by using Lemma \ref{omega} that the bound $\omega_{l_n}(p)\leq \log (l_{n})(\log_3 p)$ holds for almost all primes $p\leq x$ with an exceptional set of ``bad" primes of size at most $\frac{x}{\log x \log_3 x}$. Applying $N$ times Lemma \ref{omega}, we end up with a set of primes of density $1$ verifying $\omega_{l_n}(p-1)\leq \log (l_{n})(\log_3 p)$ for all $1\leq n\leq N$ with a negligeable exceptional set of ``bad" primes. Using the trivial inequality $\frac{\log (l_{n})}{\log (l_{n+1})} \leq 2$ this leads to
$$ h(p) \leq \log (l_{N})\log_3 p + 2N \log_3 p+ \log_3 p \ll (\log_3 p)^2$$ on a set of primes of density $1$.
\end{proof}
\begin{remark} As an application of large sieve, Pappalardi obtained a similar flavour type of result. Precisely, in \cite{Pappagenerating}, he showed that the first $\frac{\log^2 p}{\log_2 p}$ primes generate a primitive root modulo $p$ for almost all primes $p$. \end{remark}
\end{document} | arXiv |
Last edited by Shagis
4 edition of Selected papers on algebra and topology found in the catalog.
Selected papers on algebra and topology
by Garrett Birkhoff
Published 1987 by Birkhäuser in Boston .
Algebra, Universal.,
Topology.,
Lattice theory.
Statement by Garrett Birkhoff ; edited by Gian-Carlo Rota, Joseph S. Oliveira.
Series Contemporary mathematicians
Contributions Rota, Gian-Carlo, 1932-, Oliveira, Joseph S.
LC Classifications QA251 .B4972 1987
Topology; Algebraic Topologists tend to study much less `rigid' geometric situations than other have also been significant interactions with many areas of Algebra, and indeed much of Algebraic Topology can be viewed as `applied algebra' . Topology has several di erent branches | general topology (also known as point-set topology), algebraic topology, di erential topology and topological algebra | the rst, general topology, being the door to the study of the others. I aim in this book to provide a thorough grounding in general topology. Anyone who conscientiously.
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Selected papers on algebra and topology by Garrett Birkhoff Download PDF EPUB FB2
Selected Papers on Algebra and Topology by Garrett Birkhoff. Editors: Oliveira, J.S., Rota, G.-C. (Eds.) Buy this book Hardcover ,19 *immediately available upon purchase as print book shipments may be delayed due to the COVID crisis.
ebook access is temporary and does not include ownership of the ebook. Only valid for books with an. It is not often that one gets to write a preface to a collection of one's own papers. The most urgent task is to thank the people who made this book possible.
That means first of all Hy Bass who, on behalf of Springer-Verlag, approached me about the idea. The late Walter Kaufmann-Biihler was very encouraging; Paulo Ribenboim helped in an important way; and Ina Lindemann saw the project through. Six papers on signatures, braids and Seifert surfaces. (edited by Etienne Ghys and Andrew Ranicki).
Ensaios matematicos, Volume 30 () Exotic Homology Manifolds, Oberwolfach edited by Frank Quinn and Andrew Ranicki. Geometry and Topology Monograph 9 () Noncommutative localization in algebra and topology. The meeting brought together distinguished researchers from a variety of areas related to algebraic topology and its applications.
Papers in the book present a wide range of subjects, reflecting the nature of the conference. Topics include moduli spaces, configuration spaces, surgery theory, homotopy theory, knot theory, group actions, and more. The selected papers consist of original research work and a survey paper.
They are intended for a large audience, including researchers and graduate students interested in algebraic geometry, combinatorics, topology, hyperplane arrangements and commutative algebra.
Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited Selected papers on algebra and topology book of carefully selected survey or expository papers are also included.
The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt. This proceedings volume centers on new developments in rational homotopy and on their influence on algebra and algebraic topology.
Most of the papers are original research papers dealing with rational. History of Functional Analysis presents functional analysis as a rather complex blend of algebra and topology, with its evolution influenced by the development of these two branches of mathematics.
The book adopts a narrower definition—one that is assumed to satisfy various algebraic and topological conditions. A moment of reflections shows that this already covers a large part of modern.
This is a list of important publications in mathematics, organized by field. Some reasons why a particular publication might be regarded as important: Topic creator – A publication that created a new topic; Breakthrough – A publication that changed scientific knowledge significantly; Influence – A publication which has significantly influenced the world or has had a massive impact on.
A downloadable textbook in algebraic topology. What's in the Book. To get an idea you can look at the Table of Contents and the Preface. Printed Version: The book was published by Cambridge University Press in in both paperback and hardback editions, but only the paperback version is currently available (ISBN ).
I have tried very hard to keep the price of the paperback. The book focuses on various aspects of the numerical solution of elliptic boundary value problems. The selection first offers information on building elliptic problem solvers with ELLPACK; presentation and evolution of the club module; and a fourth order accurate fast direct method for the Helmholtz equation.
Selected papers on algebra and. Check out details of GATE Mathematics Reference Suggested books are very important for GATE exam and Mathematics Preparation. Here below we have provided Best Recommended books for GATE Mathematics (गेट गणित संदर्भ पुस्तकें). you can also Check GATE Notification and the last date to apply for This book is a revised version of my PhD Thesis [5], supervised by Gabriel papers [55,76] were the starting point of our research.
May's notes contain investigate deep well-known problems in Topology, Algebra and Geometry. InAlexandroff [1] showed that finite spaces and finite partially. Book on operads.
A book of Markl, Shnider and Stasheff Operads in algebra, topology, and physics was the first book to provide a systematic treatment of operad theory, an area of mathematics that came to prominence in s and found many applications in algebraic topology, category theory, graph cohomology, representation theory, algebraic geometry, combinatorics, knot theory, moduli.
From the reviews: "Kaplansky's Selected Papers and Other Writings was originally published in The selection was presumably made by Kaplansky, who also adds 'afterthoughts' to each of the papers.
this volume is especially attractive. Algebra, Topology, and Category Theory: A Collection of Papers in Honor of Samuel Eilenberg is a collection of papers dealing with algebra, topology, and category theory in honor of Samuel Eilenberg.
Topics covered range from large modules over artin algebras to two-dimensional Poincaré duality groups, along with the homology of certain H. H.S.M. Coxeter is one of the worlds best-known mathematicians who wrote several papers and books on geometry, algebra and topology, and finite mathematics.
This book is being published in conjunction with the 50th anniversary of the Canadian Mathematical Society and it is a collection of 26 papers. Birkhoff, Garrett, Birkhoff, Garrett Birkhoff, G. (Garrett), Garrett Birkhoff American mathematician Garrett Birkhoff Birkhoff, Garrett, set topology, which is concerned with the more analytical and aspects of the theory.
Part II is an introduction to algebraic topology, which associates algebraic structures such as groups to topological spaces. We will follow Munkres for the whole course, with. This is a modern reproduction of the first published image of the Mandelbrot set, which appeared in in a technical paper on Kleinian groups by Robert W.
Brooks and Peter Matelski. The Mandelbrot set consists of the points c in the complex plane that generate a bounded sequence of values when the recursive relation z n+1 = z n 2 + c is repeatedly applied starting with z 0 = 0.
The goal of this part of the book is to teach the language of math-ematics. More specifically, one of its most important components: the language of set-theoretic topology, which treats the basic notions related to continuity. The term general topology means: this is the topology that is needed and used by most mathematicians.
A permanent.This original anthology collects 11 of Weyl's less-technical writings that address the broader scope and implications of mathematics.
Most have been long unavailable or not previously published in book form. Subjects include logic, topology, abstract algebra, relativity theory, and reflections on the work of Weyl's mentor, David Hilbert. edition.Topology I and II by Chris Wendl.
This note describes the following topics: Metric spaces, Topological spaces, Products, sequential continuity and nets, Compactness, Tychonoff's theorem and the separation axioms, Connectedness and local compactness, Paths, homotopy and the fundamental group, Retractions and homotopy equivalence, Van Kampen's theorem, Normal subgroups, generators and.
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Heritage Science
Virtual cleaning of works of art using a deep generative network: spectral reflectance estimation
Morteza Maali Amiri1 &
David W. Messinger1
Heritage Science volume 11, Article number: 16 (2023) Cite this article
65 Accesses
Generally applied to a painting for protection purposes, a varnish layer becomes yellow over time, making the painting undergo an appearance change. Upon this change, the conservators start a process that entails removing the old layer of varnish and applying a new one. As widely discussed in the literature, helping the conservators through supplying them with the probable outcome of the varnish removal can be of great value to them, aiding in the decision making process regarding varnish removal. This help can be realized through virtual cleaning, which in simple terms, refers to simulation of the cleaning process outcome. There have been different approaches devised to tackle the problem of virtual cleaning, each of which tries to develop a method that virtually cleans the artwork in a more accurate manner. Although successful in some senses, the majority of them do not possess a high level of accuracy. Prior approaches suffer from a range of shortcomings such as a reliance on identifying locations of specific colors on the painting, the need to access a large set of training data, or their lack of applicability to a wide range of paintings. In this work, we develop a Deep Generative Network to virtually clean the artwork. Using this method, only a small area of the painting needs to be physically cleaned prior to virtual cleaning. Using the cleaned and uncleaned versions of this small area, the entire unvarnished painting can be estimated. It should be noted that this estimation is performed in the spectral reflectance domain and herein it is applied to hyperspectral imagery of the work. The model is first applied to a Macbeth ColorChecker target (as a proof of concept) and then to real data of a small impressionist panel by Georges Seurat (known as 'Haymakers at Montfermeil' or just 'Haymakers'). The Macbeth ColorChecker is simulated in both varnished and unvarnished forms, but in the case of the 'Haymakers', we have real hyperspectral imagery belonging to both states. The results of applying the Deep Generative Network show that the proposed method has done a better job virtually cleaning the artwork compared to a physics-based method in the literature. The results are presented through visualization in the sRGB color space and also by computing Euclidean distance and spectral angle (calculated in the spectral reflectance domain) between the virtually cleaned artwork and the physically cleaned one. The ultimate goal of our virtual cleaning algorithm is to enable pigment mapping and identification after virtual cleaning of the artwork in a more accurate manner, even before the process of physical cleaning.
Even though there have been artists who did not intend for their works to be varnished, those works were sometimes varnished once out of the artists' hands. The varnish appearance on the surface of the artwork changes with time, causing the artwork visual properties to change as well [1,2,3]. This change of appearance is more substantial after a significant passage of time [4]. Many factors play into the appearance alteration of the painting subsequent to varnish application. Two of the most important factors are the varnish type and its age [4, 5]. Due to the irreversibility of artwork cleaning, it is regarded as one of the tasks of highest importance undertaken by the conservators. This cleaning process constitutes physically removing the unwanted deposits and the aged varnish from the surface of the artwork aiding in reestablishing the original look of the artwork [6,7,8]. This process of cleaning is referred to as physical cleaning [9]. Physical cleaning can sometimes have damaging effects on the artwork along with being very time-consuming [10,11,12]. The simulation of the result of the varnish removal from an artwork is termed virtual cleaning. Virtual cleaning potentially provides the conservators with a likely appearance change of the painting if the cleaning process is undertaken. In some cases, the painting is not likely to undergo a thorough cleaning process anytime soon. In most cases, in fact, a small part of a painting is first cleaned. Using that cleaned portion, by estimating a relationship between the cleaned and uncleaned data of the region, the virtually cleaned version of the whole painting is estimated [13,14,15,16]. Therefore, virtual cleaning provides a method to estimate the original appearance of the painting [13]. Performing virtual cleaning in the spectral reflectance domain could also enable conservators to perform pigment mapping and identification more accurately even before a thorough physical cleaning.
Barni et al. [14] might be the original authors to report their work in virtual cleaning, albeit for RGB data. In their work, they first physically cleaned a part of the painting. They then found a transformation matrix between the cleaned and corresponding uncleaned portion in the RGB domain. Afterwards, they applied the same transformation matrix to the rest of the painting leading to a virtually cleaned artwork. Papas and Pitas stated that virtual cleaning in the CIELAB color space leads to a better result as opposed to the RGB color space [15]. They associated this observation with the close correlation between CIELAB and human perception. Having access to varnish and pigments that Leonardo da Vinci utilized at the time, Elias and Cotte (2008) were able to virtually clean the Mona Lisa [17]. They first built a chart of colors made from classical paints employed in 16th century Italy in a varnished and unvarnished state. Making use of these charts enabled them to deduce a mean multiplicative factor for each wavelength, which was then applied to the Mona Lisa's image spectra resulting in a virtually cleaned version of the painting [17]. Palomero and Soriano were the first to apply a neural network to the field of virtual cleaning [18]. They first physically cleaned a part of the painting and trained a shallow neural network to go from the uncleaned painting to the cleaned one using the small physically cleaned part in the RGB domain. The network was then used to predict the RGB image of the cleaned painting. Using an estimation method, they were also able to estimate the spectral reflectance of the cleaned and uncleaned painting from RGB color data [18]. The estimation method was based on the Pseudo-Inverse (PI). This method is referred to as the PI because a multiplication of the pseudo-inverse matrix of RGB data (or any color data) and the reflectance data of the training samples is performed first in an attempt to recover the spectral reflectance of the testing samples from the RGB data. One point worth mentioning is that PI relies heavily on the training samples, similar to other supervised approaches. The PI works in a way that two sets of RGB and spectral training data are used to find the relationship between the RGB and spectral data. The same relationship is then applied to the testing data to estimate the spectral information[18, 19]. However, the important point is that the material types should be both the same in the training and testing data, and if the material type in the testing samples is not the same as the material in the training data, the spectral estimation of the testing data will not be accurate [19]. This is true even if the two spectra have the same RGB values, as visual colors might visually appear the same but the spectral reflectances could be different [20]. Let us assume, for example, the training data is a green plastic and its spectrum is also known. The PI is then used (or any supervised method for that matter) to extract the relationship between the spectral reflectance of the green plastic and its RGB color values. The same relationship is then applied to the testing data which is comprised of a green leaf, for which we only have the RGB values. The green leaf spectral reflectance estimated using this method will not be an accurate representation of its original spectral reflectance due to the spectral difference outside the visual portion of the specturm[19]. Going back to the paper reported by [18], they estimated the spectral reflectance of the artwork using Munsell color chips, neglecting that Munsell chips are not constructed from the same material (pigments) as the ones used in the artwork, hence making the estimation of the spectra of artwork less accurate. Trumpy et al. were the first to approach the problem of virtual cleaning from a physics standpoint [13]. Using Keubelka-Munk theory as the base, they were able to develop a model that estimates the spectral reflectance of the cleaned painting. In order to do that, they made some simplifying assumptions about how the light interacted between the varnish and the painting. Some of those simplifying assumptions are that pigment particles are immersed in the binding medium, that there is varnish wetting of the painting layer, the varnish surfaces are optically smooth while the exposed paint layer is rough, neglecting the reflection at the air/varnish interface as well as the paint/varnish interface, that a dark location completely absorbs the incident radiation, and finally, that the varnish body reflectance (which they assumed to be equal to the reflectance spectra of the uncleaned dark location of the painting) is wavelength independent [13]. This model is used below as a reference for comparison to our model. Kirchner et al. characterized the varnish layer through making a number of key measurements, particularly of the pure white on the painting using Kubelka-Munk two-constant theory [16]. Wan et al. using variational autoencoders and assuming that the image restoration can be treated as an image translation problem (in which the images are translated through three domains: the real image, a synthetic, artificially degraded image, and the ground truth with no degradation) were able to restore old images [21]. Linhares et al., using hyperspectral imaging technology, were able to characterize the varnish layer allowing them to virtually clean the artwork [22]. To do that, they measured the reflectance spectra of the painting before and after varnish removal and used that information for the subsequent characterization and virtual removal of the varnish layer [22]. The latest work in the area of virtual cleaning might belong to Maali Amiri and Messinger, where they trained a deep convolutional neural network in the RGB domain on an image database containing natural scenery and humans that were artificially yellowed [9]. They reported the ability to virtually clean artwork even though their network was trained on natural scenes [9]. Many of the previously reported works have shortcomings that are addressed in this work. Among them are the need to specify locations of pure black and white pigments, the need to have access to a large set of training data, low accuracy, and the inability to generalize the method and results to other works. The approach presented below overcomes these deficiencies while still providing a successful virtual cleaning of a painting.
In this paper, we develop a Deep Generative Network (DGN) to virtually clean artwork in the spectral reflectance domain. Similar to other approaches, this method requires a small part of the painting to be physically cleaned beforehand. Using that portion of the work, the DGN can virtually clean the entire painting. As a first test using simulated data, a Macbeth ColorChecker was synthetically yellowed using the method proposed in [9] in the spectral domain. It was then assumed that only a small part of it has been physically cleaned. Using the DGN, we successfully clean the rest of the color chart. The model is also applied to real hyperspectral imagery of a partially cleaned work referred to as 'Haymakers', the same painting used in the study by [13]Footnote 1. It is worth noting that in the case of the 'Haymakers', there is no need to simulate the varnished version of the painting as we have real data belonging to both varnished and unvarnished states of the painting. The results are shown in terms of Euclidean distance and spectral angle between the virtually cleaned and the physically cleaned artwork (computed in the spectral reflectance domain) along with visualizations of the results in the sRGB color space as well as the spectral domain. The results are compared with the physics-based model proposed by [13]. The comparison shows that the model proposed herein has outperformed the physics-based model.
The paper is laid out as follows: Sect. "Methodology" describes the data sets utilized in this work, the proposed deep generative network, and the experiments performed. Section "Results and discussion" demonstrates the results and discussions where our model results are compared with the physics-based model. We finish our paper with conclusions summarizing the paper's outcomes and contributions.
This section describes the data used and the network algorithm along with its architecture. The criteria used to evaluate the success of the method proposed here is also presented.
Hyperspectral imagery of the Macbeth ColorChecker and 'Haymakers at Montfermeil' are used in this work to test our model. The ColorChecker dataset consists of 24 color patches, and is a very suitable sample as it comprises a set of colors represented frequently works of art. Spectral reflectance data of this color chart from 400 to 700 nm at 5 nm intervals are available [23] and are used to simulate hyperspectral imagery, both "varnished" and "cleaned", of a ColorChecker target. The hyperspectral imagery of the 'Haymakers' is the same as used in [13], and was provided to us by the National Gallery of Art. There are two hyperspectral images of this work used in this study. One image is of the work after approximately 1/3 of the work was physically cleaned, while the other was collected after the full cleaning was completed. For our study, we use the small, pre-cleaned area to virtually clean the remaining 2/3 of the work, and then compare the result to the post-cleaned imagery. The hyperspectral image cubes of the 'Haymakers' contain reflectance spectra from 400 to 780 nm with a spectral sampling of 2.5 nm. A visualization (performed in the sRGB domain) of the data is shown in Fig. 1 for the varnished and unvarnished states (the pre-cleaned area of the painting is visible on the right-hand side).
Images of the 'Haymakers at Montfermeil' a before removal of varnish and b after removal of varnish. It should be noted that this data is from a real artwork and in no way, we have done any simulation as we have with Macbeth ColorChecker
Importantly for our results, we use the simulated Macbeth ColorChecker, which was artificially yellowed using the same method proposed in [9], to test the method on synthetic data, and then apply the approach to real hyperspectral imagery of a painting, both before and after physical cleaning was performed. The result of the virtual cleaning applied to the 'Haymakers' is compared with that of physics-based model. In the case of Macbeth ColorChecker, the results are not compared with the physics-based model, and it is only used as a way to test the feasibility of our model.
Deep generative network (DGN)
Neural Networks are generally able to learn non-linear transfer functions. Here, this approach learns the relationship between the spectra of those cleaned parts of the work, and the corresponding uncleaned parts. We then generalize the same relationship to other uncleaned areas. This precludes the need to use a physics-based model in which having access to samples of pure black and white on the painting is of great importance. Using the proposed method, we do not need to be concerned about the type of colors needed; literally any colors can be used for this purpose as long as they are representative of the entire work. It should also be noted that in this work we are assuming that the varnish effect is spatially uniform, an assumption that all virtual cleaning approaches make. Therefore, for only a small area of the painting we have both the cleaned and uncleaned spectral reflectance, and we use these spectra to learn the relationship between them. After discovering that relationship, we apply it to other parts of the painting that are still varnished and consequently we are able to virtually clean the artwork. For this we use the method called deep generative network [24].
The idea of a generative network is to learn the relationship \(x = f_\theta (z)\) mapping an image z to another image x. This approach will be applied to reconstruct the virtually cleaned hyperspectral image from the hyperspectral image of the uncleaned artwork. Our goal is to generate image \(X \in R^{B\times W\times H}\) (where, B is the number of spectral bands, W is the width and H is the height of the image, both in pixels) which is a virtually cleaned image of the varnished artwork. Through feeding the image cube \(Z \in R^{B\times W\times H}\) into the generator, an image with this characteristic will be attained. Here, Z is the hyperspectral image of the artwork before cleaning. As mentioned above, only a small area of the painting is pre-cleaned and we have the spectral images of that area for both cleaned and uncleaned conditions. Let us call the area of the painting for which we have both the cleaned and uncleaned spectra A. The spectral image of this area that is physically cleaned is called \(A_c\) and the corresponding spectral image of this area before cleaning is \(A_u\). When Z goes through the network, the portion corresponding to \(A_u\) is taken out and the pixel-wise error between \(A_u\) and \(A_c\) is calculated to compute the loss. This is then back-propagated to the generator, through which the parameter \(\theta\) of the mapping function is optimized. Fig. 2 shows the process described here.
Overview of the algorithm used by generative network
The generator consists of different layers that are described below:
Convolution layer: this layer is comprised of a block of neurons involving the multiplication of a set of weights and biases by the input. The convolution layer will extract a particular feature of the input image. Given a convolution layer \(C^{(i)}\) and biases \(B^{(i)}\) and the field of view (FoV) of the feature map of the previous convolution layer \(O^{(i-1)}\), \(O^{(i)}\) is written as
$$\begin{aligned} O^{(i)} = (O^{(i-1)}\cdot W^{(i)})_{f,l} + B^{(i)} = \sum _{m=1}^{k^{(i)}}\sum _{n=1}^{k^{(i)}}(o^{(i-1)}_{f-m,l-n}\cdot \omega ^{(i)}_{(m,n)}) + B^{(i)} \end{aligned}$$
where \(k^{(i)}\) is the size of the kernel, \(O_{(f,l)}^{(i-1)}\) is the feature (f, l) of the feature map \(O^{(i)}\) with \(f = 1,2,...,W\) and \(l = 1,2,...,H\) and \(\omega _{(m,n)}^{(i)}\) is the \((m, n)^{th}\) element of the weight matrix \(W^{(i)}\).
Batch normalization layer: this layer is a method for standardizing the inputs to the next layer, which has the impact of stabilizing the process of learning, and it is usually placed behind the convolution layer. The normalization is defined as
$$\begin{aligned} y = \frac{x - mean(x)}{\sqrt{Var(x) + \epsilon }}\cdot \gamma + \beta \end{aligned}$$
where \(\gamma\) and \(\beta\) are learnable parameters, and \(\epsilon\) is a parameter used for numerical stability. mean(x) and Var(x) are the mean and variance of x, respectively.
Activation layer: this layer is a nonlinear function that is attached to each neuron. It is a component of great importance as it specifies the computational efficiency of training a model and the convergence speed of the neural network. LeakyReLU is used here, defined as
$$\begin{aligned} f(x)= {\left\{ \begin{array}{ll} x,&{} \text {if } x > 0\\ \alpha x, &{} \text {if } x\le 0 \end{array}\right. } \end{aligned}$$
where, \(\alpha\) is a small nonzero parameter. We did not use ReLU as one of the problems of using ReLU is that its derivative is zero for negative part values which blocks the learning. However, in the case of the LeakyReLU, the derivative is a small fraction in the negative parts, which allows the gradients to flow on in the learning process.
The proposed generative network has an hourglass architecture, shown in Fig 3.
The proposed generative network architecture
To be more specific, an image cube \(Z \in R^{B\times W\times H}\), as input, goes through four main modules consisting of several blocks as follows:
The down-sampling block: \(d_{(i)}\) denotes the down-sampling blocks. Each \(d_{(i)}\) is comprised of an initial convolution layer \(C_d^{(1)}(i)\) that also performs the down-sampling step through setting the stride \(S = 2\). It is then followed by the batch normalization and the LeakyReLU activation layer. The output is fed into the second convolution layer \(C_d^{(2)}(i)\) with again \(S = 2\). The same as the first activation layer, the second activation layer is followed by a batch normalization layer and the LeakyReLU activation function as well. \(C_d^{(1)}(i)\) and \(C_d^{(2)}(i)\) can be set to different kernel sizes and different numbers of filters shown as \(k_d^{(1)}(i)\), \(k_d^{(2)}(i)\), \(n_d^{(1)}(i)\) and \(n_d^{(2)}(i)\).
The up-sampling block: \(u^{(i)}\) denotes the up-sampling blocks. Each \(u^{(i)}\) consists of a few stacked layers. Opposite to the down-sampling blocks, the first layer here is batch normalization. It is then followed by the first convolution layer \(C_u^{(1)}(i)\) with S = 1 and a batch normalization and LeakyReLU activation function. Its output is then fed into the second convolution layer \(C_u^{(2)}(i)\). The output, after batch normalization and non-linear activation, is fed into the bilinear up-sampling layer with a factor of 2. \(C_u^{(1)}(i)\) and \(C_u^{(2)}(i)\), similar to the down-sampling block, can be set to different kernel sizes and different numbers of filters shown as \(k_u^{(1)}(i)\), \(k_u^{(2)}(i)\), \(n_u^{(1)}(i)\) and \(n_u^{(2)}(i)\), respectively.
Skip connection block: \(s^{(i)}\) is utilized to denote the skip connection blocks. These blocks are used to connect the down-sampled data to the up-sampled data, so the residual information can be fully employed. It consists of one convolution layer, one batch normalization layer and one activation function. The number of filters and kernel size of convolution kernels in different layers can be set differently.
Output block: \(o^{(0)}\) denotes the output block. It is an up-sampling block that is modified such that the up-sampling layer is replaced with one convolution layer, which is followed by one Sigmoid activation layer.
As mentioned, the input to the network is the hyperspectral image of the uncleaned artwork \(Z \in R^{B \times W \times H}\) and the generated image is \(X \in R^{B\times W\times H}\). The cost function is defined as the pixel-wise difference between \(A_u\) and \(A_c\); defined above, \(A_c\) is the hyperspectral image of the area of the painting that is cleaned and \(A_u\) is the hyperspectral image of the same area but before cleaning. \(A_u\) belongs to X and therefore, it is changing in each iteration. Consequently, the cost function is given as
$$\begin{aligned} min\Vert A_u - A_c\Vert ^2. \end{aligned}$$
To iterate to the best solution, the input to the model should be replaced with the output of the model after each iteration. As mentioned, the network has an hourglass architecture as shown in Fig 3. Each down-sampling and up-sampling sections are comprised of 5 layers and 5 skip connections. The filter size is 3 \(\times\) 3 in the up-sampling and down-sampling blocks but it is 1 \(\times\) 1 in the last convolutional layer. There are 128 filters in each layer, both in the down-sampling and up-sampling blocks. There are 120 filters in the last convolutional layers equaling the spectral resolution of the hyperspectral images used. Overall there are 12 layers including input and output layers. The Adam optimization algorithm is used, chosen based on trial and error. The loss function, as mentioned before, is the Euclidean distance between the virtually cleaned area of the artwork and the physically cleaned one (\(A_u\) and \(A_c\)). The overall algorithm is shown in Algorithm 1
It should be noted that this is an unsupervised approach. There is no training in the traditional sense here because it is a generative network. The error computed between \(A_u\) and \(A_c\) is back propagated to the generator which cleans the entire image using the error coming from the loss function. This cleaning process takes place step by step at each epoch, until the network reaches the maximum number of epochs.
Evaluation metrics
The virtually cleaned result is transformed into the sRGB format for visual inspection and to evaluate the success of the process. For a quantitative evaluation the per-pixel spectral Euclidean Distance and Spectral Angle (SA) are also calculated between the hyperspectral image of the physically cleaned work and the virtually cleaned hyperspectral image [25]. The spectral angle is calculated between two vectors in the spectral reflectance space and it is reported in radians in the range [0, 3.142]. The spectral angle is defined as
$$\begin{aligned} SA_k = cos^{-1} \left( \frac{ \textbf{t}_k\cdot \textbf{r}_k}{|\textbf{t}_k||\mathbf {r_k}|} \right) \end{aligned}$$
where k represents the \(k\)th pixel, and \(\textbf{t}_k\) and \(\textbf{r}_k\) represent the two pixels belonging to the test and reference images. Also, the mean spectral reflectance of a few randomly chosen areas on the painting are compared between different approaches.
Experimental environment
Python 3.9.7 \(\vert\)Anaconda, Inc. is used as a base coding environment for the DGN algorithm. More specifically, the DGN codes were written and run in the TensorFlow environment, which was installed onto the Anaconda. In terms of hardware, the programs are run on a CPU that is 11th Gen Intel(R) Core(TM) i7-1165G7 @ 2.80GHz. The training of the DGN is performed using only one image and is consequently referred to as an unsupervised learning method [24]. As mentioned before, only a small area of the image is used to compute the loss function, and the same loss is then used for the whole image to virtually clean it. 10000 epochs are used to train the model. MATLAB R2022a, the package of mathematical software was also used for evaluation computations and processing the samples used in our model.
In this section, the results of applying the DGN to the problem of virtual cleaning are presented. This section is divided into two subsections in which the results for the Macbeth ColorChecker and the 'Haymakers' are presented separately.
Macbeth ColorChecker
A spectral representation of the Macbeth ColorChecker, as mentioned before, is used to test the DGN model that was developed in this work. As described above, a small part of the image of the Macbeth ColorChecker should be first physically cleaned. The rest of the painting will be virtually cleaned using hyperspectral imagery of that small part both before and after physical cleaning. A key question in this work is dependence of the performance on the choice of data that is pre-cleaned. In our simulation of the ColorChecker, we assumed that first, only white, red, green and blue patches are cleaned and the rest of it is uncleaned. In the second case, we assume that half of all the patches available on the Macbeth ColorChecker are cleaned. Through trial and error, we realized that at least four color patches are required to virtually clean the image successfully. Choosing half of all patches gives us a good idea about how the DGN works when it has data associated with all colors represented in the work. The results of these two cases are shown in Fig. 4.
a Unclean Macbeth ColorChecker, b clean Macbeth ColorChecker, c cleaned Macbeth ColorChecker using DGN (white, red, green and blue patches are used), d cleaned using DGN (half of all patches are used)
As observed from Fig. 4, the DGN has been able to virtually clean the Macbeth ColorChecker and is successful in replicating the original cleaned Macbeth Chart in Fig. 4b. It is difficult to distinguish between the results in Fig. 4 as they are visually similar.
Table 1 Euclidean distance and SA mean and standard deviation (SD) values between the original and virtually cleaned Macbeth color chart
However, Table 1 presents the mean and standard deviation of the Euclidean distance and spectral angle between the virtually cleaned Macbeth ColorChecker and the original one. Note that the data are in spectral reflectance in the range [0, 1]. As it is observed from the table, the DGN has been able to clean the Macbeth ColorChecker at a very acceptable manner signaling that this method could be potentially applied to real artworks as well. Moreover, the DGN has resulted in a better outcome when half of all patches present on the Macbeth ColorChecker are utilized. This is not surprising. When we use half of all the patches to compute the loss function, we are using all the possible samples present in our dataset resulting in a lower error in the virtual cleaning process as shown herein.
The Euclidean distance and spectral angle measured between the virtually cleaned Macbeth ColorChecker and the original one are also presented in Fig. 5. This representation helps us see where the method is falling short. It should be noted that all of these data are normalized between 0 and 1 across all results. This helps see visually which methods, and which patches, are cleaned better than others.
Visualization of Eulidean distance computed between a Virtually cleaned Macbeth ColorChecker using DGN (white, red, green and blue patches) and the original one and b virtually cleaned Macbeth ColorChecker using DGN (half of all the patches) and the original one. Visualization of Spectral angle computed between c virtually cleaned Macbeth ColorChecker using DGN (white, red, green and blue patches) and the original one and d virtually cleaned Macbeth ColorChecker using DGN (half of all the patches) and the original one. The data is normalized between 0 and 1
It is clear that using half of all the patches leads to a better result as opposed to using only white, red, green and blue patches. Interestingly, the white patch on the color chart output by the DGN when half of all patches are used looks even better than that of DGN when white, redm green and blue patches are used. The black patch, on the other hand, has been cleaned in a more acceptable way. It should be noted that these results are normalized with respect to the maximum value present in the Euclidean distance and spectral angle computations, separately.
'Haymakers at Montfermeil'
In this section, the results of applying the DGN to the 'Haymakers' are presented and described.
As observed above from applying the DGN to the Macbeth ColorChecker, the result varies depending on which patches are chosen on the chart as data to compute the loss function. We presented two different conditions: one with only 4 patches assumed to be physically cleaned, and the other assuming that half of all patches present are physically cleaned. For imagery of real artwork, these represent two use cases, one in which a small part of the painting is physically cleaned, and one in which an attempt is made to partially clean as many representative pigments as possible in the work. Using the data belonging to both the cleaned and uncleaned states of that small area, the DGN is able to estimate the virtually cleaned version of the whole work. To examine the same point (the impact of the area chosen to be physically cleaned), two different conditions are tested herein as well. In other words, two different experiments are performed differing only in cleaned small area the DGN uses as data for the computation of the loss function to estimate the cleaned version of the whole painting. Fig. 6 shows the two different areas chosen to perform these two experiments. The experiments are referred to as the "first" (Fig. 6a) and "second" (Fig. 6b) experiments. These two experiments differ in terms of \(A_c\), or the small area that is physically cleaned using which the network tries to virtually clean the whole painting.
Two different areas used in two experiments referred to as a first and b second experiments
In the first experiment, a contiguous small area of the painting is chosen to be physically cleaned. In the second experiment, a few small areas spread over the painting are chosen. We performed the second experiment in an attempt to include as many different pigments as possible. To do that, we chose different areas spread over the painting from different pigments. We actually want to test the impact of the \(A_c\) on the final outcome of the virtual cleaning. In total there are 357,594 reflectance spectral samples in the imagery of the painting. In the first experiment, 88,409 of those were used for training (almost 25 percent) and in the second experiment, 8860 were used for training (almost 2.5 percent).
a Virtually cleaned artwork using the physics-based model, b physically cleaned artwork, c virtually cleaned artwork in the second experiment and d virtually cleaned artwork in the first experiment
Figure 7 shows the results for these two experiments along with the results of applying the physics-based approach. Visually, the physics-based model has not been able to clean the artwork as well as the DGN. This is subtle, but can be seen by looking closely at the grass in the middle of the images and noticing that there is a tint of yellow in the output of the physics-based model, not seen in the output of the DGN. One reason for this result is that there is no true black color present on the 'Haymakers' making the prediction made by the physics-based method, which relies heavily on the presence of pure black and white paints, not as accurate [9]. It is also hard to tell the difference between the outputs of the two experiments performed using the DGN.
ED computed between the physically cleaned artwork and the virtually cleaned one using a physics-based model, b DGN (the first experiment) and c DGN (the second experiment). SA computed between the physically cleaned artwork and the virtually cleaned one using d the physics-based model, e DGN (the first experiment) and f DGN (the second experiment). The data has been normalized between 0 and 1
Figure 8 shows the Euclidean distance (ED) and spectral angle (SA) measures between the virtually cleaned artwork and the physically cleaned one for the experiments performed here as well as the physics-based method. It is clear that the physics-based approach has not led to a good result. The error, in terms of ED and SA, is much lower for both the first and second experiments in the case of the DGN, than the physics-based model. Looking more closely at Fig. 8, one sees that the second experiment has led to a slightly better result than the first experiment. The reason is that in the second experiment, the area chosen to compute the loss function by the DGN contains a better representation of the possible colors and paints in the work, helping the DGN learn the transfer function from the varnished version of the painting to the unvarnished one.
a Distribution of Euclidean distance calculated between physically cleaned artwork and the virtually cleaned one using (a) physics-based model, b DGN (the first experiment) and c DGN (the second experiment). Distribution of spectral angle calculated between the physically cleaned artwork and the virtually cleaned one using d the physics-based model, e DGN (the first experiment) and f DGN (the second experiment)
To further quantitatively examine the results shown in Fig. 8, the distributions of the spectral angle and Euclidean distance metrics are presented in Fig. 9. Overall, the distributions associated with DGN (both first and second experiment) have lower means and are narrower than those associated with the physics-based model.
Table 2 Euclidean distance and SA mean and standard deviation (SD) values between the physically and virtually cleaned 'Haymakers'
To see the difference between the first and second experiment more clearly, Table 2 shows that the second experiment has led to a better result than the first experiment, in terms of mean and standard deviation of the metrics, for the reasons explained before. These experiments show that the more representative the area chosen to be physically cleaned and used by the DGN to compute the loss, the better the overall virtual cleaning outcome. It is still worth noting that the DGN has cleaned the painting at an acceptable level even in the first experiment.
We end this section by showing some of the spectral reflectance curves from the physically cleaned painting, and virtually cleaned ones using the physics-based approach, and the DGN (the first and second experiments). The curves are obtained through computing the average spectral reflectance factor of randomly chosen small areas on the painting as shown in Fig. 10.
Averaged spectral reflectance curves of the physically cleaned painting and virtually cleaned ones using physics-based approach and DGN computed over different areas randomly selected on the painting
While for some pigments all of the methods perform well, in general, we can see from Fig. 10 that the DGN (especially the second experiment) has led to a better result compared to the physics-based model. As mentioned above, in the first experiment, a rigid area has been chosen which might not be a good representative of the whole painting. However, in the second experiment, the DGN is exposed to many more pigments in the work through choosing different areas of the painting to compute the loss function. Overall, in both cases, the DGN has done an acceptable job of cleaning the painting.
In this work, we applied a deep generative network (DGN) to the problem of virtual cleaning of artworks. We used a simulated Macbeth ColorChecker and real hyperspectral imagery of the 'Haymakers' painting for this purpose. The results were compared with a well-known physics-based model both visually and in terms of Euclidean distance and spectral angle, computed in the spectral reflectance domain, between the virtually cleaned and physically cleaned artworks. Different areas of the artwork were chosen for the DGN to compute the loss function, used to estimate the cleaned version of the whole painting. The results showed that the DGN was able to outperform the physics-based model. It was also observed that the choice of the small, pre-cleaned area used by the DGN is important, and the more representative the small area is of the entire painting, the better the virtual cleaning outcome. This could be seen as one of the limitations of this method. Nonetheless, the method was able to lead to an acceptable result even when the small area was not as representative of the entire painting. Another limitation of this method might be its reliance on a pre-cleaned small area of the painting, which might not always be possible to obtain. However, if this is available, the method described here is useful to perform virtual cleaning on the entire painting.
Availability of the data and materials
The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.
We wish to thank the National Gallery of Art for providing us with the data of the 'Haymakers'.
DGN:
Deep generative network
Euclidean distance
Spectral angle
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Maali Amiri M, Messinger DW. Virtual cleaning of works of art using deep convolutional neural networks. Herit Sci. 2021;9(1):1–19.
Al-Emam E, Soenen H, Caen J, Janssens K. Characterization of polyvinyl alcohol-borax/agarose (PVA-B/AG) double network hydrogel utilized for the cleaning of works of art. Herit Sci. 2020;8:106.
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This research was funded by the Xerox Chair in Imaging Science at Rochester Institute of Technology.
We would also like to express our gratitude to John Delaney and Kathryn Dooley from the National Gallery of Art for providing us with access to the hyperspectral images of the 'Haymakers' collected both before and after physical cleaning.
The work was done as a part of the author's Ph.D. research and was supported by the Xerox Chair in Imaging Science at the Rochester Institute of Technology.
Rochester Institute of Technology, Chester F Carlson Center for Imaging Science, 54 Lomb Memorial Drive, Rochester, NY, 14623, USA
Morteza Maali Amiri & David W. Messinger
Morteza Maali Amiri
David W. Messinger
MMA developed the principal aspects of the algorithm, implemented it, and trained/tested the DGN and applied it to the data. DWM oversaw and advised the research. Both authors wrote the article, read and approved of the final version of the manuscript.
Correspondence to Morteza Maali Amiri.
Maali Amiri, M., Messinger, D.W. Virtual cleaning of works of art using a deep generative network: spectral reflectance estimation. Herit Sci 11, 16 (2023). https://doi.org/10.1186/s40494-023-00859-x
Artwork virtual cleaning
Varnished and unvarnished artworks
Spectral reflectance | CommonCrawl |
International Congress of Mathematicians
The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU).
International Congress of Mathematicians
StatusActive
GenreMathematics conference
FrequencyQuadrennial
CountryVaries
Years active1897–present
Inaugurated9 August 1897 (1897-08-09)[1]
Founders
• Felix Klein
• Georg Cantor
Most recent6–14 July 2022
Previous event2022
Next event22–29 July 2026
ActivityActive
Websitewww.mathunion.org/activities/icm/
The Fields Medals, the IMU Abacus Medal (known before 2022 as the Nevanlinna Prize), the Gauss Prize, and the Chern Medal are awarded during the congress's opening ceremony. Each congress is memorialized by a printed set of Proceedings recording academic papers based on invited talks intended to be relevant to current topics of general interest. Being invited to talk at the ICM has been called "the equivalent ... of an induction to a hall of fame".[2]
History
Felix Klein
Georg Cantor
German mathematicians Felix Klein and Georg Cantor are credited with putting forward the idea of an international congress of mathematicians in the 1890s.[3][4]
The University of Chicago, which had opened in 1892, organized an International Mathematical Congress at the Chicago World's Fair in 1893, where Felix Klein participated as the official German representative.[5]
The first official International Congress of Mathematicians was held in Zurich in August 1897.[6] The organizers included such prominent mathematicians as Luigi Cremona, Felix Klein, Gösta Mittag-Leffler, Andrey Markov, and others.[7] The congress was attended by 208 mathematicians from 16 countries, including 12 from Russia and 7 from the US.[4] Only four were women: Iginia Massarini, Vera von Schiff, Charlotte Scott, and Charlotte Wedell.[8]
During the 1900 congress in Paris, France, David Hilbert announced his famous list of 23 unsolved mathematical problems, now termed Hilbert's problems. Moritz Cantor and Vito Volterra gave the two plenary lectures at the start of the congress.[9]
At the 1904 ICM Gyula Kőnig delivered a lecture where he claimed that Cantor's famous continuum hypothesis was false. An error in Kőnig's proof was discovered by Ernst Zermelo soon thereafter. Kőnig's announcement at the congress caused considerable uproar, and Klein had to personally explain to the Grand Duke of Baden (who was a financial sponsor of the congress) what could cause such an unrest among mathematicians.[10]
During the 1912 congress in Cambridge, England, Edmund Landau listed four basic problems about prime numbers, now called Landau's problems. The 1924 congress in Toronto was organized by John Charles Fields, initiator of the Fields Medal; it included a roundtrip railway excursion to Vancouver and ferry to Victoria. The first two Fields Medals were awarded at the 1936 ICM in Oslo.[10]
In the aftermath of World War I, at the insistence of the Allied Powers, the 1920 ICM in Strasbourg and the 1924 ICM in Toronto excluded mathematicians from the countries formerly part of the Central Powers. This resulted in a still unresolved controversy as to whether to count the Strasbourg and Toronto congresses as true ICMs. At the opening of the 1932 ICM in Zürich, Hermann Weyl said: "We attend here to an extraordinary improbable event. For the number of n, corresponding to the just opened International Congress of Mathematicians, we have the inequality 7 ≤ n ≤ 9; unfortunately our axiomatic foundations are not sufficient to give a more precise statement”.[10] As a consequence of this controversy, from the 1932 Zürich congress onward, the ICMs are not numbered.[10]
For the 1950 ICM in Cambridge, Massachusetts, Laurent Schwartz, one of the Fields Medalists for that year, and Jacques Hadamard, both of whom were viewed by the U.S. authorities as communist sympathizers, were only able to obtain U.S. visas after the personal intervention of President Harry Truman.[11][12]
The first woman to give an ICM plenary lecture, at the 1932 congress in Zürich, was Emmy Noether.[13] The second ICM plenary talk by a woman was delivered 58 years later, at the 1990 ICM in Kyoto, by Karen Uhlenbeck.[14]
The 1998 congress was attended by 3,346 participants. The American Mathematical Society reported that more than 4,500 participants attended the 2006 conference in Madrid, Spain. The King of Spain presided over the 2006 conference opening ceremony. The 2010 Congress took place in Hyderabad, India, on August 19–27, 2010. The ICM 2014 Archived 2014-12-29 at the Wayback Machine was held in Seoul, South Korea, on August 13–21, 2014. The 2018 Congress took place in Rio de Janeiro on August 1–9, 2018.
ICMs and the International Mathematical Union
The organizing committees of the early ICMs were formed in large part on an ad hoc basis and there was no single body continuously overseeing the ICMs. Following the end of World War I, the Allied Powers established in 1919 in Brussels the International Research Council (IRC). At the IRC's instructions, in 1920 the Union Mathematique Internationale (UMI) was created.[10] This was the immediate predecessor of the current International Mathematical Union. Under the IRC's pressure, UMI reassigned the 1920 congress from Stockholm to Strasbourg and insisted on the rule which excluded from the congress mathematicians representing the former Central Powers. The exclusion rule, which also applied to the 1924 ICM, turned out to be quite unpopular among mathematicians from the U.S. and Great Britain. The 1924 ICM was originally scheduled to be held in New York, but had to be moved to Toronto after the American Mathematical Society withdrew its invitation to host the congress, in protest against the exclusion rule.[4] As a result of the exclusion rule and the protests it generated, the 1920 and the 1924 ICMs were considerably smaller than the previous ones. In the run-up to the 1928 ICM in Bologna, IRC and UMI still insisted on applying the exclusion rule. In the face of the protests against the exclusion rule and the possibility of a boycott of the congress by the American Mathematical Society and the London Mathematical Society, the congress's organizers decided to hold the 1928 ICM under the auspices of the University of Bologna rather than of the UMI.[10] The 1928 congress and all the subsequent congresses have been open for participation by mathematicians of all countries. The statutes of the UMI expired in 1931 and at the 1932 ICM in Zurich a decision to dissolve the UMI was made, largely in opposition to IRC's pressure on the UMI.[10]
At the 1950 ICM the participants voted to reconstitute the International Mathematical Union (IMU), which was formally established in 1951. Starting with the 1954 congress in Amsterdam, the ICMs are held under the auspices of the IMU.
Soviet participation
The Soviet Union sent 27 participants to the 1928 ICM in Bologna and 10 participants to the 1932 ICM in Zurich.[13] No Soviet mathematicians participated in the 1936 ICM, although a number of invitations were extended to them. At the 1950 ICM there were again no participants from the Soviet Union, although quite a few were invited. Similarly, no representatives of other Eastern Bloc countries, except for Yugoslavia, participated in the 1950 congress. Andrey Kolmogorov had been appointed to the Fields Medal selection committee for the 1950 congress, but did not participate in the committee's work. However, in a famous episode, a few days before the end of the 1950 ICM, the congress' organizers received a telegram from Sergei Vavilov, President of the USSR Academy of Sciences. The telegram thanked the organizers for inviting Soviet mathematicians but said that they are unable to attend "being very much occupied with their regular work", and wished success to the congress's participants.[15] Vavilov's message was seen as a hopeful sign for the future ICMs and the situation improved further after Joseph Stalin's death in 1953. The Soviet Union was represented by five mathematicians at the 1954 ICM in Amsterdam, and several other Eastern Bloc countries sent their representatives as well. In 1957 the USSR joined the International Mathematical Union and the participation in subsequent ICMs by the Soviet and other Eastern Bloc scientists has been mostly at normal levels.[15] However, even after 1957, tensions between ICM organizers and the Soviet side persisted. Soviet mathematicians invited to attend the ICMs routinely experienced difficulties with obtaining exit visas from the Soviet Union and were often unable to come. Thus of the 41 invited speakers from the USSR for the 1974 ICM in Vancouver, only 20 actually arrived.[4] Grigory Margulis, who was awarded the Fields Medal at 1978 ICM in Helsinki, was not granted an exit visa and was unable to attend the 1978 congress.[4][16] Another, related, point of contention was the jurisdiction over Fields Medals for Soviet mathematicians. After 1978 the Soviet Union put forward a demand that the USSR Academy of Sciences approve all Soviet candidates for the Fields Medal, before it was awarded to them.[4][16] However, the IMU insisted that the decisions regarding invited speakers and Fields medalists be kept under exclusive jurisdiction of the ICM committees appointed for that purpose by the IMU.[4][16]
List of Congresses
Year City Country
2026Philadelphia United States
2022HelsinkiOnline event[lower-alpha 1]
2018Rio de Janeiro Brazil
2014Seoul South Korea
2010Hyderabad India
2006Madrid Spain
2002Beijing China
1998Berlin Germany
1994Zürich Switzerland
1990Kyoto Japan
1986Berkeley United States
1982 (met during 1983)Warsaw Poland
1978Helsinki Finland
1974Vancouver Canada
1970Nice France
1966Moscow Soviet Union
1962Stockholm Sweden
1958Edinburgh United Kingdom
1954Amsterdam Netherlands
1950Cambridge, Massachusetts United States
1936Oslo Norway
1932Zürich Switzerland
1928Bologna Italy
1924Toronto Canada
1920Strasbourg France
1912Cambridge United Kingdom
1908Rome Italy
1904Heidelberg German Empire
1900Paris France
1897Zürich Switzerland
1. Originally planned to be in Saint Petersburg, Russia, but was moved online following the 2022 Russian invasion of Ukraine. The IMU General Assembly took place in Helsinki, Finland, in early July, 2022.[17]
See also
• List of International Congresses of Mathematicians Plenary and Invited Speakers
References
1. "The International Congress of Mathematicians". Nature. Nature Publishing Group. 56 (1452): 395. 1897. doi:10.1038/056395a0. Retrieved August 8, 2023.
2. Castelvecchi, Davide (7 October 2015). "The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof". Nature. 526 (7572): 178–181. Bibcode:2015Natur.526..178C. doi:10.1038/526178a. PMID 26450038. S2CID 4403935.
3. The International Mathematical Union and The ICM Congresses. Archived 2021-02-23 at the Wayback Machine www.icm2006.org. Accessed December 23, 2009.
4. A. John Coleman. "Mathematics without borders": a book review. CMS Notes, vol 31, no. 3, April 1999, pp. 3–5
5. Robert de Boer (2009) Alexander Macfarlane in Chicago, 1893 from WebCite
6. C., Bruno, Leonard (2003) [1999]. Math and mathematicians : the history of math discoveries around the world. Baker, Lawrence W. Detroit, Mich.: U X L. pp. 56. ISBN 0787638137. OCLC 41497065.{{cite book}}: CS1 maint: multiple names: authors list (link)
7. In the section Vorgeschichte des Kongresses (prehistory of the congress) of the 1st ICM proceedings, 21 prominent organizers were cited: Hermann Bleuler, Heinrich Burkhardt, Luigi Cremona, Gustave Dumas, Jérôme Franel, Carl Friedrich Geiser, Alfred George Greenhill, Albin Herzog, George William Hill, Adolf Hurwitz, Felix Klein, Andrey Markov, Franz Mertens, Hermann Minkowski, Gösta Mittag-Leffler, Gabriel Oltramare, Henri Poincaré, Johann Jakob Rebstein, Ferdinand Rudio, Karl von der Mühll, and Heinrich Friedrich Weber. (See: Rudio, F., ed. (1898). Verhandlungen des ersten Internationalen Kongresses in Zürich vom 9. bis 11. August 1897. ICM proceedings. BG Teubner. p. 6.)
8. Curbera (2009), p. 16.
9. Scott, Charlotte Angas (1900). "The International Congress of Mathematicians in Paris" (PDF). Bull. Amer. Math. Soc. 7 (2): 57–79. doi:10.1090/s0002-9904-1900-00768-3.
10. G. Curbera. ICM through history. Newsletter of the European Mathematical Society, no. 63, March 2007, pp. 16–21. Accessed December 23, 2009.
11. Vladimir Maz'ya, Tatyana Shaposhnikova. Jacques Hadamard: a universal mathematician. American Mathematical Society, 1999. ISBN 0-8218-1923-2; p. 271
12. Michèle Audin, Correspondance entre Henri Cartan et André Weil (1928–1991), Documents Mathématiques 6, Société Mathématique de France, 2011, pp. 259–313
13. Guillermo Curbera. Mathematicians of the World, Unite!: The International Congress of Mathematicians: A Human Endeavor AK Peters, 2009. ISBN 1-56881-330-9; pp. 95–96
14. Sylvia Wiegand. Report on the Berlin ICM. AWM Newsletter, 28(6), November–December 1998, pp. 3–8
15. Guillermo Curbera. Mathematicians of the World, Unite!: The International Congress of Mathematicians: A Human Endeavor AK Peters, 2009. ISBN 1-56881-330-9; pp 149–150.
16. Olli Lehto. Mathematics without borders: a history of the International Mathematical Union. Springer-Verlag, 1998. ISBN 0-387-98358-9; pp. 205–206
17. "Decision of the Executive Committee of the IMU on the upcoming ICM 2022 and IMU General Assembly" (PDF).
Further reading
• Guillermo Curbera. Mathematicians of the World, Unite!: The International Congress of Mathematicians: A Human Endeavor AK Peters, 2009. ISBN 1-56881-330-9
• Olli Lehto. Mathematics without borders: a history of the International Mathematical Union Springer-Verlag, 1998. ISBN 0-387-98358-9
• Donald J. Albers, Gerald L. Alexanderson, Constance Reid. International Mathematical Congresses: An Illustrated History, 1893–1986, Springer-Verlag, 1986. ISBN 0-387-96409-6
• Yousef Alavi, Peter Hilton and Jean Pedersen. "Let's Meet at the Congress" American Mathematical Monthly, Vol. 93, No. 1 (Jan., 1986), pp. 3–8
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What do the local systems in Lusztig's perverse sheaves on quiver varieties look like?
In "Quivers, perverse sheaves and quantized enveloping algebras," Lusztig defines a category of perverse sheaves on the moduli stack of representations of a quiver. These perverse sheaves are defined as summands of the pushforwards of the constant sheaves on stacks of quiver representations along with a choice of invariant flag (and thus, by definition are supported on the nilpotent locus in the moduli stack). They're mostly of interest since they categorify the canonical basis.
My question is: Is there a stratum in this stack where the pull-back of one of these sheaves is not the trivial local system?
Now, in finite type, this is not a concern, since each stratum is the classifying space of a connected algebraic group, and thus simply connected. But I believe in affine or wild type this is no longer true; this was at least my takeaway from the latter sections of "Affine quivers and canonical bases." However, I got a little confused about the relationship between the results of the two papers mentioned above, since they use quite different formalisms, so I hold out some hope that the local systems associated to symmetric group representations aren't relevant to the perverse sheaves for the canonical basis. Am I just hoping in vain?
For affine quivers, except cyclic ones, there are always perverse sheaves attached to nontrivial local systems.
If you just need an example, I recommend you to read McGerty's paper math/0403279, before Lusztig's paper, where the Kronecker quiver case is studied in detail.
Crystal, canonical and PBW bases of quantum affine algebras, in Algebraic Groups and Homogeneous Spaces, Ed. V.B.Mehta, Narosa Publ House. 2007, 389–421.
Another thing perhaps worth mentioning is that, outside the finite type case, you have to do some real work to find a stratification with respect to which the perverse sheaves constituting the canonical basis are constructible: for affine types, such a stratification is pretty much implicit in Lusztig's Publ IHES paper which you cite, and it uses the classification of representations of tame quivers which he recovers via the McKay correspondence.
In general it is known that the characteristic cycles of the sheaves in the canonical basis lie in a certain Lagrangian variety (this is already established in the paper on quivers and canonical bases). This doesn't give you a stratification however, because they components are conormals to locally closed subvarieties of the moduli space whose union is not the whole space. The same phenomenon happens for character sheaves, though there Lusztig did produce a stratification of the group and show character sheaves have locally constant cohomology on the strata.
You see papers studying this sort of problem in terms of quiver representations at the level of functions on $\mathbb F_q$-points when people try and generalize the "existence of Hall polynomials" outside of finite type quivers and on the quantum group side when people look for "PBW" bases.
Not the answer you're looking for? Browse other questions tagged ag.algebraic-geometry rt.representation-theory perverse-sheaves quantum-groups quivers or ask your own question.
Are the strata of Nakajima quiver varieties simply-connected? Do they have odd cohomology?
What's known about the stalks of Lusztig's perverse sheaves on quiver varieties?
Auslander-Reiten theory of wild algebras known in examples?
Are Lusztig's perverse sheaves the only equivariant ones with nilpotent characteristic cycle?
Local counterpart of the NON-Hitchin Hecke eigen-sheaves ? | CommonCrawl |
\begin{document}
\title{Regular Languages are {Church-Rosser} Congruential}
\author{
Volker Diekert\,$^{*}$ \\
Manfred Kuf\-leitner\,$^{*}$
\\
\,\ Klaus Reinhardt\,$^{\dagger}$
\\
\,Tobias Walter\,$^{*}$ \\[4mm]
{\footnotesize \!\!$^*$\;Institut
f\"ur Formale Methoden der Informatik} \\[-3.5mm]
{\footnotesize University of Stuttgart, Germany} \\[0mm]
{\footnotesize \!\!$^{\dagger}$\;Wilhelm-Schickard-Institut f{\"u}r
Informatik} \\[-3.5mm]
{\footnotesize University of T{\"u}bingen, Germany} }
\date{}
\maketitle
\begin{abstract}
\noindent
\textbf{Abstract.}\,
This paper proves a long standing conjecture in formal language
theory. It shows that all regular languages are Church-Rosser
congruential. The class of Church-Rosser congruential languages was
introduced by \mbox{McNaughton}, Narendran, and Otto in 1988. A
language $L$ is Church-Rosser congruential, if there exists a finite
confluent, and length-reducing semi-Thue system $S$ such that $L$ is
a finite union of congruence classes modulo $S$. It was known that
there are deterministic linear context-free languages which are not
Church-Rosser congruential, but on the other hand it was
strongly believed that all regular language are of this form.
Actually, this paper proves a more general result.\,\footnote{ The
research on this paper was initiated during the program
\emph{Automata Theory and Applications} at the Institute for
Mathematical Sciences, National University of Singapore in
September~2011. The second author was supported by the German
Research Foundation (DFG) under grant \mbox{DI 435/5-1}.}
\noindent
\textbf{Keywords.}\,
String rewriting; Church-Rosser system; regular language;
finite monoid; finite semigroup; local divisor. \end{abstract}
\section{Introduction}\label{sec:intro}
It has been a long standing conjecture in formal language theory that all regular languages are Church-Rosser congruential. The class of Church-Rosser congruential languages was introduced by \mbox{McNaughton}, Narendran, and Otto in~1988~\cite{McNaughtonNO88}. A language $L$ is Church-Rosser congruential, if there exists a finite confluent, and length-reducing semi-Thue system $S$ such that $L$ is a finite union of congruence classes modulo $S$. One of the main motivations to consider this class of languages is that the membership problem for $L$ can be solved in linear time; this is done by computing normal forms using the system $S$, followed by a table look-up. For this it is not necessary that the quotient monoid $A^*/S$ is finite, it is enough that $L$ is a finite union of congruence classes modulo $S$. It is not hard to see that $\set{a^nb^n}{n\in \mathbb{N}}$ is Church-Rosser congruential, but $\set{a^mb^n}{m,n\in \mathbb{N} \text{ and } m \geq n }$ is not. This led the authors of~\cite{McNaughtonNO88} to the more technical notion of Church-Rosser languages; this class of languages captures all deterministic context-free languages. For more results about Church-Rosser languages see e.g.~\cite{BuntrockO98,Narendran84phd,Woinowski01Diss,Woinowski03IC}.
From the very beginning it was strongly believed that all regular languages are Church-Rosser congruential in the pure sense. However, after some significant initial progress~\cite{Narendran84phd,NiemannPhD02,NiemannO05,NiemannW02,reinhardtT03} there was some stagnation.
Before 2011 the most advanced result was the one announced in 2003 by Reinhardt and Th\'erien \cite{reinhardtT03}. According to this manuscript the conjecture is true for all regular languages where the syntactic monoid is a group. However, the manuscript has never been published as a refereed paper and there are some flaws in its presentation. The main problem with~\cite{reinhardtT03} has however been quite different for us. The statement is too weak to be useful in the induction for the general case. So, instead of being able to use~\cite{reinhardtT03} as a black box, we shall prove a more general result in the setting of weight-reducing systems. This part about group languages is a cornerstone in our approach.
The other ingredient to our paper has been established only very recently. Knowing that the result is true if the the syntactic monoid is a group, we started looking at aperiodic monoids. Aperiodic monoids correspond to star-free languages and the first two authors together with Weil proved that all star-free languages are Church-Rosser congruential~\cite{DiekertKW12tcs}. Our proof became possible by \emph{loading the induction hypothesis}. This means we proved a much stronger statement. We showed that for every star-free language $L \subseteq A^*$ there exists a finite confluent semi-Thue system $S\subseteq A^* \times A^*$ such that the quotient monoid $A^*/ S$ is finite (and aperiodic), $L$ is a union of congruence classes modulo $S$, and moreover all right-hand sides of rules appear as scattered subwords in the corresponding left-hand side. We called the last property \emph{subword-reducing}, and it is obvious that every subword-reducing system is length-reducing.
We have little hope that such a strong result could be true in general. Indeed here we step back from subword-reducing to weight-reducing systems.
We prove in \refthm{thm:main} the following result: Let $L \subseteq A^*$ be a regular language and $\Abs{a} \in \mathbb{N} \setminus \smallset{0}$ be a positive weight for every letter $a \in A$ (e.g.,{} $ \Abs{a} = \abs{a} =1$). Then we can construct for the given weight a finite, confluent and weight-reducing semi-Thue system $S\subseteq A^* \times A^*$ such that the quotient monoid $A^*/ S$ is finite and recognizes $L$. In particular, $L$ is a finite union of congruence classes modulo $S$.
Note that this gives us another characterization for the class of regular languages. By \refcor{cor:main} we see that a language $L \subseteq A^*$ is regular if and only if $L$ is recognized by a finite Church-Rosser system $S$ with finite index. As a consequence, a long standing conjecture about regular languages has been solved positively.
\section{Preliminaries}\label{sec:prelim}
\paragraph{Words and languages}
Throughout this paper, $A$ is a finite alphabet. An element of $A$ is called a \emph{letter}. The set $A^*$ is the free monoid generated by $A$. It consists of all finite sequences of letter from $A$. The elements of $A^*$ are called \emph{words}. The empty word is denoted by $1$. The \emph{length} of a word $u$ is denoted by $\abs{u}$. We have $\abs{u} = n$ for $u= a_1 \cdots a_n$ where $a_i \in A$. The empty word has length $0$, and it is the only word with this property. The set of word of length at most $n$ is denoted by $A^{\leq n}$, and the set of all nonempty words is $A^+$. We generalize the length of a word by introducing weights. A \emph{weighted alphabet} $(A,\Abs{\cdot})$ consists of an alphabet $A$ equipped with a weight function $\Abs{\cdot} : A \to \mathbb{N} \setminus \smallset{0}$. The \emph{weight} of a letter $a \in A$ is $\Abs{a}$ and the \emph{weight} $\Abs{u}$ of a word $u= a_1 \cdots a_n$ with $a_i \in A$ is $\Abs{a_1} + \cdots + \Abs{a_n}$. The weight of the empty word is $0$. The length is the special weight with $\Abs{a} = 1$ for all $a \in A$. A word $u$ is a \emph{factor} of a word $v$ if there exist $p,q \in A^*$ such that $puq = v$, and $u$ is a \emph{proper factor} of $v$ if $pq \neq 1$. The word $u$ is a \emph{prefix} of $v$ if $uq = v$ for some $q \in A^*$, and it is a \emph{suffix} of $v$ if $pu = v$ for some $p \in A^*$. We say that $u$ is a factor (resp.\ prefix, resp.\ suffix) of $v^+$ if there exists $n \in \mathbb{N}$ such that $u$ is a factor (resp.\ prefix, resp.\ suffix) of $v^n$. Two words $u,v \in A^*$ are \emph{conjugate} if there exist $p,q \in A^*$ such that $u = pq$ and $v = qp$. An integer $m > 0$ is a \emph{period} of a word $u = a_1 \cdots a_n$ with $a_i \in A$ if $a_i = a_{i+m}$ for all $1 \leq i \leq n-m$. A word $u \in A^+$ is \emph{primitive} if there exists no $v \in A^+$ such that $u = v^n$ for some integer $n > 1$. It is a standard fact that a word $u$ is not primitive if and only $u^2 = puq$ for some $p,q \in A^+$. This follows immediately from the result from combinatorics on words that $xy = yx$ if and only if\xspace $x$ and $y$ are powers of a common root; see e.g.~\cite[Section~1.3]{lot83}.
A monoid $M$ \emph{recognizes} a language $L \subseteq A^*$ if there exists a homomorphism $\varphi : A^* \to M$ such that $L = \varphi^{-1} \varphi(L)$. A language $L \subseteq A^*$ is \emph{regular} if it is recognized by a finite monoid. There are various other and well-known characterizations of regular languages; e.g., regular expressions, finite automata or monadic second order logic. Regular languages $L$ can be classified in terms of structural properties of the monoids recognizing $L$. In particular, we consider group languages; these are languages recognized by finite groups.
\paragraph{Semi-Thue systems}
A {\em semi-Thue system} over $A$ is a subset $S\subseteq A^*\times A^*$. In this paper, all semi-Thue systems are finite. The elements of $S$ are called {\em rules}. We frequently write $\ell \to r$ for rules $(\ell,r)$. A system $S$ is called {\em length-redu\-cing\xspace} if we have $\abs \ell > \abs r $ for all rules $\ell \to r$ in $S$. It is called \emph{weight-reducing} with respect to some weighted alphabet $(A,\Abs{\cdot})$, if $\Abs{\ell} > \Abs{r}$ for all rules $\ell \to r$ in $S$. Every system $S$ defines the rewriting relation $\RA{S} \subseteq A^* \times A^*$ by setting $u \RA{S} v$ if there exist $p,q,\ell,r \in A^*$ such that $u = p\ell q$, $v= prq$, and $\ell \to r$ is in $S$.
By $\RAS*{S}$ we mean the reflexive and transitive closure of $\RA{S}$. By $\DAS*{S}$ we mean the symmetric, reflexive, and transitive closure of $\RA{S}$. We also write $u \LAS*{S}v$ whenever $v\RAS*{S}u$. The system $S$ is {\em confluent} if for all $u\DAS*{S}v$ there is some $w$ such that $u\RAS*{S}w\LAS*{S}v$. It is \emph{locally confluent} if for all $v \LA{S} u \RA{S} v'$ there exists $w$ such that $v \RAS*{S} w \LAS*{S} v'$. If $S$ is locally confluent and weight-reducing for some weight, then $S$ is confluent; see e.g.~\cite{bo93springer,jan88eatcs}. Note that $u \RAS{}S v$ implies that $\Abs{u} > \Abs{v}$ for weight-reducing systems. The relation ${\DAS*{S}} \subseteq A^* \times A^*$ is a congruence, hence the congruence classes $[u]_S = \{v \in A^*\mid u \DAS*S v\}$ form a monoid which is denoted by $A^*/S $. The size of $A^*/S$ is called the \emph{index of
$S$}. A finite semi-Thue system $S$ can be viewed as a finite set of defining relations. Hence, $A^*/S $ becomes a finitely presented monoid. By $\mathrm{IRR}_S(A^*)$ we denote the set of irreducible words in $A^*$, i.e., the set of words where no left-hand side occurs as a factor.
Whenever the weighted alphabet $(A,\Abs{\cdot})$ is fixed, a finite semi-Thue system $S \subseteq A^* \times A^*$ is called a \emph{weighted Church-Rosser system} if it is finite, weight-reducing for $(A,\Abs{\cdot})$, and confluent. Hence, a finite semi-Thue system $S$ is a weighted Church-Rosser system if and only if\xspace (1) we have $\Abs \ell > \Abs r$ for all rules $\ell \to r$ in $S$ and (2) every congruence class has exactly one irreducible element. In particular, for weighted Church-Rosser systems $S$, there is a one-to-one correspondence between $A^* / S$ and $\mathrm{IRR}_S(A^*)$.
A \emph{Church-Rosser system} is a finite, length-reducing, and confluent semi-Thue system. In particular, every Church-Rosser system is a weighted Church-Rosser system. A language $L \subseteq A^*$ is called a {\em Church-Rosser congruential language} if there is a finite Church-Rosser system $S$ such that $L$ can be written as a finite union of congruence classes $[u]_S$.
\begin{definition}\label{def:cr}
Let $\varphi: A^* \to M$ be a homomorphism\xspace and let $S$ be a semi-Thue
system. We say that \emph{$\varphi$ factorizes through $S$} if for all
$u,v \in A^*$ we have:
\begin{equation*}
u \DAS*{S} v \quad \text{implies} \quad \varphi(u) = \varphi(v).
\end{equation*} \end{definition}
Note that if $S$ is a semi-Thue system and $\varphi : A^* \to M$ factorizes through $S$, then the following diagram commutes: \begin{center}
\begin{tikzpicture}[scale=0.75]
\draw (0,0) node (A) {$A^*$};
\draw (3,0) node (M) {$M$};
\draw (3,2) node (S) {$A^* / S$};
\draw[->] (A) -- node[pos=0.65,above] {$\varphi$} (M);
\draw[->] (A) -- node[above left,outer sep=-2pt] {$\pi$} (S);
\draw[->] (S) -- node[right] {$\psi$} (M);
\end{tikzpicture} \end{center} Here, $\pi(u) = [u]_S$ is the canonical homomorphism and $\psi([u]_S) = \varphi(u)$.
\section{Finite Groups}\label{sec:groups}
Our main result is that every homomorphism $\varphi : A^* \to M$ to finite monoid $M$ factorizes through a Church-Rosser system $S$. Our proof of this theorem distinguishes whether or not $M$ is a group. Thus, we first prove this result for groups. Before we turn to the general case, we show that for some particular groups, proving the claim is easy. The techniques developed here will also be used when proving the result for arbitrary finite groups.
\subsection{Groups without proper cyclic quotient groups}\label{sec:simple}
The aim of this section is to show that finding a Church-Rosser system is very easy for many cases. This list includes systems of all finite (non-cyclic) simple groups, but it goes far beyond this. Let $\varphi: A^* \to G$ be a homomorphism to a finite group, where $(A,\Abs{\cdot})$ is a weighted alphabet. This defines a regular language $L_G = \set{w \in A^*}{\varphi(w) = 1}$. Let us assume that the greatest common divisor $\gcd{\set{\Abs{w}}{w \in L_G}}$ is equal to one; e.g.{} $\oneset{6,10,15} \subseteq \set{\Abs{w}}{w \in
L_G}$. Then there are two words $u,v \in L_G$ such that $\Abs{u} -\Abs{v} = 1$. Now we can use these words to find a constant $d$ such that all $g\in G$ have a representing word $v_g$ with the exact weight $\Abs{v_g} = d$. To see this, start with some arbitrary set of representing words $v_g$. We multiply words $v_g$ with smaller weight with $u$ and words $v_g$ higher weights with $v$ until all weights are equal.
The final step is to define the following weight-reducing system $$S_G= \set{w \to v_{\varphi(w)}} {w \in A^{*}\; \text{Êand }Êd < \Abs w \leq d + \max\set{\Abs a}{a \in A}}.$$ Confluence of $S_G$ is trivial; and every language recognized by $\varphi$ is also recognized by the canonical homomorphism\xspace $A^* \to A^* / S_G$.
Now assume that we are not so lucky, i.e., $\gcd{\set{\Abs{w}}{w \in
L_G}} >1$. This means there is a prime number $p$ such that $p$ divides $\Abs{w}$ for all $w \in L_G$. Then, the homomorphism\xspace of $A^*$ to $\mathbb{Z}/p\mathbb{Z}$ defined by $a \mapsto \Abs{a} \bmod p$ factorizes through $\varphi$ and $\mathbb{Z}/p\mathbb{Z}$ becomes a quotient group of $G$. This can never happen if $G$ is simple and non-cyclic, because a simple group does not have any proper quotient group. But there are many other cases where a natural homomorphism $A^* \to G$ for some weighted alphabet $(A,\Abs{\cdot})$ satisfies the property $\gcd{\set{\Abs{w}}{w \in
L_G}= 1}$ although $G$ has a non-trivial cyclic quotient group. Just consider the length function and a presentation by standard generators for dihedral groups $D_{2n}$ or the permutation groups $\mathcal{S}_n$ where $n$ is odd.
For example, let $G = D_6 = \mathcal{S}_3$ be the permutation group of a triangle. Then~$G$ is generated by elements $\tau$ and $\rho$ with defining relations \begin{equation*}
\tau^2 = \rho^3 = 1 \text { and } \tau \rho \tau = \rho^2. \end{equation*} The following six words of length $3$ represent all six group elements: \begin{equation*}
1=\rho^3,\; \rho=\rho\tau^2,\; \rho^2 = \tau \rho \tau,\; \tau=\tau^3,\;
\tau\rho = \rho^2\tau,\; \tau\rho^2.
\end{equation*} The corresponding monoid $\smallset{\rho,\tau}^* / S_G$ has $15$ elements.
It is much harder to find a Church-Rosser system for the homomorphism\xspace $\varphi: \smallset{a,b,c}^* \to \mathbb{Z} / 3 \mathbb{Z}$ where $\varphi(a) = \varphi(b) = \varphi(c) = 1 \bmod 3$. In some sense this phenomenon suggests that finite cyclic groups or more general commutative groups are the obstacle to find a simple construction for Church-Rosser systems.
\subsection{The general case for group languages}
In this section, we consider arbitrary groups. We start with some simple properties of Church-Rosser systems. Then, in \refthm{thm:group}, we state and prove that group languages are Church-Rosser congruential.
\begin{lemma}\label{lem:irr}
Let $(A,\Abs{\cdot})$ be a weighted alphabet, let $d \in \mathbb{N}$, and
let $S \subseteq A^* \times A^*$ be a weighted Church-Rosser system
such that $\mathrm{IRR}_S(A^*)$ is finite. Then
\begin{equation*}
S_d = \set{u \ell v \to u r v}{u,v \in A^d \text{ and } \ell \to r \in S}
\end{equation*}
is a weighted Church-Rosser system satisfying:
\begin{enumerate}
\item\label{irri} $\mathrm{IRR}_{S_d}(A^*)$ is finite.
\item\label{irrii} All words of length at most $2d$ are irreducible
with respect to $S_d$.
\item\label{irriii} The mapping $[u]_{S_d} \mapsto [u]_S$ for $u \in
A^*$ is well-defined and yields a surjective homomorphism from
$A^* / S_d$ onto $A^* / S$.
\end{enumerate} \end{lemma}
\begin{proof}
First, one shows that local confluence of $S$ transfers to local
confluence of $S_d$. For ``\ref{irri}'' and ``\ref{irrii}'' note that
$\mathrm{IRR}_{S_d}(A^*) = A^{\leq 2d} \cup A^d \cdot \mathrm{IRR}_S(A^*) \cdot A^d$.
The remaining proof is straightforward and therefore left to the
reader.
\end{proof}
\begin{lemma}\label{lem:econfl}
Let $(A,\Abs{\cdot})$ be a weighted alphabet and let $\Delta
\subseteq A^+$ such that all words in $\Delta$ have length at most
$t$. Then, for every $n \geq 1$, the set of rules
\begin{equation*}
T = \set{\delta^{t+n} \to \delta^t}{
\delta \in \Delta,\; \delta \text{ is primitive}}
\end{equation*}
yields a weighted Church-Rosser system. \end{lemma}
\begin{proof}
Every rule in $T$ is weight-reducing. Thus it suffices to show that
$T$ is locally confluent. Let $\delta, \tilde{\delta} \in \Delta$ be
primitive with $\abs{\delta} \geq |\tilde{\delta}|$ and suppose $x
\delta^{t+n} = \tilde{\delta}^{t+n} y$. If $\delta^{t+n}$ is a
suffix of $\tilde{\delta}^{t} y$, then $\tilde{\delta}^{t+n}$ is a
prefix of $x \delta^t$; and the two $T$-rules $\delta^{t+n} \to
\delta^t$ and $\tilde{\delta}^{t+n} \to \tilde{\delta}^t$ can be
applied independently of one another. Thus we can assume
$\abs{\delta^{t+n}} > |\tilde{\delta}^{t} y|$. In particular,
$\tilde{\delta}^t$ is a factor of $\delta^+$. Note that
$|\tilde{\delta}^t| \geq \abs{\delta}$. Thus $|\tilde{\delta}|$ is
a period of $\delta$.
Let us first consider the case $\abs{\delta} > |\tilde{\delta}|$.
Since $\delta$ is primitive, $|\tilde{\delta}|$ cannot be a divisor
of $\abs{\delta}$. In particular, we have $|\tilde{\delta}| \geq 2$.
Suppose $|\tilde{\delta}| = 2$. Then $\delta = (ab)^ma$ for $a,b \in
A$ and some $m \geq 1$. We conclude that the suffix $a\delta$ or the
prefix $\delta a$ of $\delta^2$ is a factor of
$\tilde{\delta}^+$. Since both words $a\delta$ and $\delta a$ have a
factor $aa$ and $|\tilde{\delta}| = 2$, this contradicts
$\tilde{\delta}$ being primitive. Therefore, we can assume
$|\tilde{\delta}| \geq 3$ and hence, $|\tilde{\delta}^t| \geq
\abs{\delta^3}$. It follows that $\delta^2$ is a factor of
$\tilde{\delta}^+$ and $|\tilde{\delta}|$ is a period of $\delta^2$.
By shifting the prefix $\delta$ of $\delta^2$ by this period, we can
write $\delta^2 = p \delta q$ with $p,q \in A^+$ and $\abs{p} =
|\tilde{\delta}|$. We conclude that $\delta$ is not primitive,
which is a contradiction.
Let now $\abs{\delta} = |\tilde{\delta}|$. In this case, the words
$\delta$ and $\tilde{\delta}$ are conjugate. Therefore, applying
one of the rules $\delta^{t+n} \to \delta^t$ and
$\tilde{\delta}^{t+n} \to \tilde{\delta}^t$ yields the same word.
\end{proof} \begin{lemma}\label{lem:efin}
Let $\Delta \subseteq A^+$ be a set of words such that all words in
$\Delta$ have length at most $n$. If $u \in A^{> 2n}$ is not a
factor of some $\delta^+$ for $\delta \in \Delta$, then there is a
proper factor $v$ of $u$ which is also not a factor of some
$\delta^+$ for $\delta \in \Delta$. \end{lemma}
\begin{proof}
Assume that such a factor $v$ of $u$ does not exist. Let $u = a w b$
for $a,b \in A$. Then $aw$ is a factor of $\delta^+$ and $wb$ is a
factor of $\delta'^+$ for some $\delta,\delta' \in \Delta$. Let $p =
\abs{\delta}$ and $q = \abs{\delta'}$. Now, $p$ is a period of $aw$
and $q$ is a period of $wb$. Thus $p$ and $q$ are both periods of
$w$. Since $\abs{w} \geq 2n-1 \geq p + q - \gcd(p,q)$, we see that
$\gcd(p,q)$ is also a period of $w$ by the Periodicity Lemma of Fine
and Wilf~\cite[Section~1.3]{lot83}. The $(p+1)$-th letter in $aw$ is $a$.
Going in steps $\gcd(p,q)$ to the left or to the right in $w$, we see
that the $(q+1)$-th letter in $aw$ is $a$. Thus $awb$ is
a factor of $\delta'^+$, which is a contraction. \end{proof}
We are now ready to prove the main result of this section: Group languages are Church-Rosser congruential. An outline of the proof is as follows. By induction on the size of the alphabet, we show that every homomorphism $\varphi : A^* \to G$ factorizes through a weighted Church-Rosser system $S$ with finite index. Remove some letter $c$ from the alphabet $A$. This leads to a system $R$ for the remaining letters $B$. \reflem{lem:irr} allows to assume that certain words are irreducible. Then we consider $K = \mathrm{IRR}_R(B^*) c$ which is a prefix code in $A^*$. We consider $K$ as a new alphabet. Essentially, it is this situation where weighted alphabets come into play because we can choose the weight of $K$ such that it is compatible with the weight over the alphabet $A$. Over $K$, we introduce two sets of rules $T_\Delta$ and $T_\Omega$. The $T_\Delta$-rules reduce long repetitions of short words $\Delta$, and the $T_\Omega$-rules have the form $\omega \, u \, \omega \to \omega \, v_g \, \omega$. Here, $\Omega$ is some finite set of markers and $\omega \in \Omega$ is such a marker. The word $v_g$ is a normal form for the group element~$g$. The $T_\Omega$-rules reduce long words without long repetitions of short words. Then we show that $T_\Delta$ and $T_\Omega$ are confluent and that their union has finite index over~$K^*$. Here, the confluence of the $T_\Delta$-rules is \reflem{lem:econfl}. The confluence of the $T_\Omega$-rules relies on several combinatorial properties of the normal forms $v_g$ and the markers $\Omega$. Using \reflem{lem:efin}, we see that all sufficiently long words are reducible. Since by construction all rules in $T = T_\Delta \cup T_\Omega$ are weight-reducing, the system $T$ is a weighted Church-Rosser system over~$K^*$ with finite index such $\varphi : K^* \to G$ factorizes through $T$. Since $K \subseteq A^*$, we can translate the rules $\ell \to r$ in $T$ over $K^*$ to rules $c \ell \to c r$ over $A^*$. This leads to the set of $T'$-rules over $A^*$. The letter $c$ at the beginning of the $T'$-rules is require to shield from $R$-rules. Finally, we show that $S = R \cup T'$ is the desired system over~$A^*$.
\begin{theorem}\label{thm:group}
Let $(A,\Abs{\cdot})$ be a weighted alphabet and let $\varphi : A^*
\to G$ be a homomorphism to a finite group $G$. Then there exists a
weighted Church-Rosser system $S$ with finite index such that
$\varphi$ factorizes through $S$. \end{theorem}
\begin{proof}
In the following $n$ denotes the exponent of $G$; this is the least
positive integer $n$ such that $g^n = 1$ for all $g \in G$. The
proof is by induction on the size of the alphabet $A$. If $A =
\smallset{c}$, then we set $S = \oneset{c^n \to 1}$. Let now $A =
\oneset{a_0, \ldots, a_s, c}$ and let $a_0$ have minimal weight. We
set $B = A \setminus \smallset{c}$. Let
\begin{equation*}
\gamma_i = a_{i \bmod s}^{n + \floor{i/s}} \, c.
\end{equation*}
Since $A$ and $\oneset{a_0c,\ldots,a_sc, c}$ generate the same
subgroups of $G$ and since every element $a_j c\in G$ occurs infinitely
often as some $\gamma_i$, there exists $m > 0$ such that for every
$g \in G$ there exists a word
\begin{equation*}
v_g = \gamma_0^{n_0} \cdots \gamma_m^{n_m} \gamma_0
\end{equation*}
with $n_i > 0$ satisfying $\varphi(v_g) = g$ and $\Abs{v_g} -
\Abs{v_h} < n \Abs{a_0}$ for all $g,h \in G$. The latter property
relies on $\Abs{\gamma_0} + \Abs{a_0} = \Abs{\gamma_s}$ and pumping
with $\gamma_0^n$ and $\gamma_s^n$ which both map to the neutral
element of $G$: Assume $\Abs{v_g} - \Abs{v_h} \geq n \Abs{a_0}$
for some $g,h \in G$. Then we do the following. All $v_g$ with
maximal weight are multiplied by $\gamma_0^n$ on the left, and for
all other words $v_h$ the exponent $n_s$ of $\gamma_s$ is replaced
by $n_s + n$. After that, the maximal difference $\Abs{v_g} -
\Abs{v_h}$ has decreased at least by $1$ (and at most by $n
\Abs{a_0}$).
We can iterate this procedure until the weights of all
$v_g$ differ less than $n \Abs{a_0}$. Let
\begin{equation*}
\Gamma = \oneset{\gamma_0,\ldots,\gamma_m}
\end{equation*}
be the generators of the $v_g$.
By induction there exists a weighted Church-Rosser system $R$
for the restriction $\varphi : B^* \to G$ satisfying the statement
of the theorem. By \reflem{lem:irr}, we can assume $\Gamma
\subseteq \mathrm{IRR}_R(B^*)\,c$. Thus $v_g \in \mathrm{IRR}_R(A^*)$ for all $g \in
G$. Let
\begin{equation*}
K = \mathrm{IRR}_R(B^*)\,c.
\end{equation*}
The set $K$ is a prefix code in $A^*$. We consider $K$ as an
extended alphabet and its elements as extended letters. The weight
$\Abs{u}$ of $u \in K$ is its weight as a word over $A$. Each
$\gamma_i$ is a letter in $K$. The homomorphism $\varphi : A^* \to
G$ can be interpreted as a homomorphism $\varphi : K^* \to G$; it is
induced by $u \mapsto \varphi(u)$ for $u \in K$. The length
lexicographic order on $B^*$ induces a linear order $\leq$ on
$\mathrm{IRR}_R(B^*)$ and hence also on $K$. Here, we assume $a_0 < \cdots <
a_s$. The words $v_g$ can be read as words over the weighted
alphabet $(K,\Abs{\cdot})$ satisfying the following five properties:
First, $v_g$ starts with the extended letter
$\gamma_0$. Second, the last two extended letters of $v_g$ are
$\gamma_m \gamma_0$. Third, all extended letters in $v_g$ are in
non-decreasing order from left to right with respect to $\leq$, with
the sole exception of the last letter $\gamma_0$ which is smaller
than its predecessor $\gamma_m$. The fourth property is that all
extended letters in $v_g$ have a weight greater than $n \Abs{a_0}$.
And the last important property is that all differences $\Abs{v_g} -
\Abs{v_h}$ are smaller than $n \Abs{a_0}$. Let
\begin{equation*}
\Delta = \set{\delta \in K^+}{\delta \in K \text{ or }
\Abs{\delta} \leq n \Abs{a_0}}.
\end{equation*}
Note that $\Delta$ is closed under conjugation, i.e., if $uv \in
\Delta$ for $u,v \in K^*$, then $vu \in \Delta$. We can think of
$\Delta$ as the set of all ``short'' words. Choose $t \geq n$ such
that all normal forms $v_g$ have no factor $\delta^{t+n}$ for
$\delta \in \Delta$ and such that $\Abs{c^{t}} \geq \Abs{u}$ for all
$u \in K^{2n}$. Note that $c \in \Delta$ has the smallest
weight among all words in $\Delta$.
The first set of rules
over the extended alphabet $K$ deals with long repetitions of short
words: The $\Delta$-rules are
\begin{equation*}
T_\Delta = \set{\delta^{t+n} \to
\delta^{t}}{\delta \in \Delta \text{ and $\delta$ is primitive}}.
\end{equation*}
Let $F \subseteq K^*$ contain all words which are a factor of some
$\delta^+$ for $\delta \in \Delta$ and let $J \subseteq K^+$ be
minimal such that $K^* J K^* = K^* \setminus F$. By
\reflem{lem:efin}, we have $J \subseteq K^{2n}$. In particular, $J$
is finite. Since $J$ and $\Delta$ are disjoint, all words in $J$
have a weight greater than $n \Abs{a_0}$. Let $\Omega$ contain all
$\omega \in J$ such that $\omega \in \Gamma K^*$ implies $\omega
= \gamma \gamma'$ for some $\gamma > \gamma'$, i.e.,
\begin{equation*}
\Omega = J \cap \set{\omega \in K^*}{\omega \not\in \Gamma K^* \text{ or }
\omega =\gamma \gamma' \text{ for some } \gamma > \gamma'}.
\end{equation*}
As we will see below, every sufficiently long word without long
$\Delta$-repetitions contains a factor $\omega \in \Omega$.
\begin{claim}\label{clm:long:oo}
There exists a bound $t' \in \mathbb{N}$ such that every word $u \in K^*$
with $\Abs{u} \geq t'$ contains a factor $\omega \in \Omega$ or a
factor of the form $\delta^{t+n}$ for $\delta \in \Delta$.
\end{claim}
\noindent
\textit{Proof of \refclm{clm:long:oo}.}
Let $t'' = (t+n+2) \cdot \max\set{\Abs{v} \in \mathbb{N}}{v \in K}$. First,
suppose $u \in K^* \setminus K^* \Gamma K^*$ and $\Abs{u} \geq
t''$. If $u$ is a factor of $\delta^+$, then $\delta^{n+d}$ is a
factor of $u$ since $\Abs{\delta} \leq \max\set{\Abs{v} \in \mathbb{N}}{v
\in K}$. Thus we can assume $u \in K^* \setminus F$. By definition
of $J$, the word $u$ contains a factor $\omega \in J$. We have
$\omega \in \Omega$ because $u$ (and thus $\omega$) has no factor in
$\Gamma$.
If $u \in K^* b \gamma K^*$ for $b \in K \setminus \Gamma$ and
$\gamma \in \Gamma$, then $u$ contains a factor $\omega = b \gamma
\in \Omega$. Similarly, if $u \in K^* \gamma \gamma' K^*$ for
$\gamma,\gamma' \in \Gamma$ and $\gamma > \gamma'$, then $u$
contains a factor $\omega = \gamma \gamma' \in \Omega$. Thus, if $u
\in K^* \Gamma K^*$, then we can assume $u = \gamma_{i_1} \cdots
\gamma_{i_k} u'$ with
\begin{itemize}
\item $\gamma_{i_j} \in \Gamma$ and $\gamma_{i_1}
\leq \cdots \leq \gamma_{i_k}$, and
\item $u' \not\in K^* \Gamma K^*$ and $\Abs{u'} < t''$.
\end{itemize}
We set $t' = (t+n-1) \cdot \abs{\Gamma} \cdot \max\set{\Abs{v} \in
\mathbb{N}}{v \in \Gamma} + 1 + t''$. If $\Abs{u} \geq t'$, then $k \geq
(t+n-1) \cdot \abs{\Gamma} + 1$. By the pigeon hole principle, there
exists $\gamma \in \oneset{\gamma_{i_1}, \ldots, \gamma_{i_k}}
\subseteq \Delta$ such that $\gamma^{t+n}$ is a factor of $u$.
This completes the proof of \refclm{clm:long:oo}.~~~$\diamond$
Since $\Delta$ is closed under factors, $u$ contains no factor of
the form $\delta^{t+n}$ for $\delta \in \Delta$ if and only if $u
\in \mathrm{IRR}_{T_\Delta}(K^*)$. In particular, it is no restriction to
only allow primitive words from $\Delta$ in the rules $T_\Delta$.
Every sufficiently long word $u'$ can be written as $u' =
u_1 \cdots u_k$ with $\Abs{u_i} \geq t'$ and $k$ sufficiently large.
Thus, by repeatedly applying \refclm{clm:long:oo}, there exists a
non-negative integer $d_\Omega$ such that every word $u' \in
\mathrm{IRR}_{T_\Delta}(K^*)$ with $\Abs{u'} \geq t_\Omega$ contains two
occurrences of the same $\omega \in \Omega$ which are far
apart. More precisely, $u'$ has a factor $\omega\, u\, \omega$ with
$\Abs{u} > \Abs{v_g}$ for all $g \in G$.
This suggests rules of the form $\omega \, u \, \omega \to \omega\,
v_{\varphi(u)} \,\omega$; but in order to ensure confluence we have
to limit their use. For this purpose, we equip $\Omega$ with a
linear order $\preceq$ such that $\gamma_m \gamma_0$ is the smallest
element, and every element in $\Omega \cap K^+ \gamma_0$ is smaller
than all elements in $\Omega \setminus K^+ \gamma_0$. By making
$t_\Omega$ bigger, we can assume that every word $u'$ with $\Abs{u'}
\geq t_\Omega$ contains a factor $\omega\, u\, \omega$ such that
\begin{itemize}
\item $\Abs{u} > \Abs{v_g}$ for all $g \in G$, and
\item for every factor $\omega' \in \Omega$ of
$\omega\, u\, \omega$ we have $\omega' \preceq \omega$.
\end{itemize}
The following claim is one of the main reasons for using the above
definition of the normal forms $v_g$, and also for excluding all
words $\omega \in \Gamma K^*$ in the definition of~$\Omega$ except
for $\omega = \gamma\gamma' \in \Gamma^2$ with $\gamma > \gamma'$.
\begin{claim}\label{clm:oovoo}
Let $\omega,\omega' \in \Omega$ and $g \in G$. If $\omega \, v_g
\, \omega \in K^* \omega' K^*$, then $\omega' \preceq \omega$.
\end{claim}
\noindent
\textit{Proof of \refclm{clm:oovoo}.}
All normal forms $v_g$ have $\gamma_m \gamma_0$ as a suffix. In
addition, the word $\gamma_m \gamma_0$ is the only element in
$\Omega$ which is a factor of some $v_g$ for $g \in G$. The reason
is that all other letters in $v_g$ are in non-decreasing order
whereas all $\gamma \gamma' \in \Omega$ are in decreasing order. In
particular, if $\gamma_m \gamma_0 \, v_g \, \gamma_m \gamma_0 \in
K^* \omega' K^*$ for $\omega' \in \Omega$, then $\omega' = \gamma_m
\gamma_0$, i.e., $\gamma_m \gamma_0$ is the only factor of $\gamma_m
\gamma_0 \, v_g \, \gamma_m \gamma_0$ which is in $\Omega$.
Let now $\omega = b \gamma_0$ for $b \in K \setminus
\smallset{\gamma_0}$. Note that $\omega \in \Omega$ and that all
elements in $\Omega \cap K^+ \gamma_0$ have this form. Then the set of
factors of $\omega v_g \omega$ which are in $\Omega$ is
$\oneset{\gamma_m \gamma_0, \omega}$. Since $\gamma_m \gamma_0$ is
the smallest element with respect to $\preceq$, each of them
satisfies the claim.
Next, suppose $\omega \in K^+ b$ for $b \in K \setminus
\smallset{\gamma_0}$. Then the set of factors of $\omega v_g \omega$
which are in $\Omega$ is $\oneset{\gamma_m \gamma_0, b \gamma_0,
\omega}$. Since every element ending with $\gamma_0$ is smaller than
any other element in $\Omega$, the claim also holds in this case.
This completes the proof of \refclm{clm:oovoo}.~~~$\diamond$
We are now ready to define the second set of rules over the extended
alphabet $K$. They are reducing long words without long repetitions
of words in $\Delta$. We set
\begin{equation*}
T_\Omega' = \set{\omega \, u \, \omega \to \omega\,
v_{\varphi(u)} \,\omega}{
\parbox{7.1cm}{$\Abs{v_{\varphi(u)}}
< \Abs{u} \leq
t_\Omega \text{ and } $ \\
$\omega \,u \,\omega \text{ has no
factor } \omega' \in \Omega \text{ with } \omega \prec
\omega'$}}.
\end{equation*}
Whenever there is a shorter rule in $ T_{\Omega}' \cup T_\Delta$
then we want to give preference to this shorter rule. Thus the
$\Omega$-rules are
\begin{equation*}
T_\Omega = \set{\ell \to r \in T_\Omega'}{
\parbox{6.2cm}{$\text{there is no rule }
\ell' \to r' \in T_{\Omega}' \cup T_\Delta$ \\
$\text{such that }
\ell' \text{ is a proper factor of } \ell$}}.
\end{equation*}
Let now
\begin{equation*}
T = T_\Delta \cup T_\Omega \, .
\end{equation*}
\begin{claim}\label{clm:confl}
The system $T$ is locally confluent over $K^*$.
\end{claim}
\noindent
\textit{Proof of \refclm{clm:confl}.} The system $T_\Delta$ is
confluent by \reflem{lem:econfl}. Suppose we can apply two rules
$\ell \to r \in T_\Omega$ and $\ell' \to r' \in T_\Delta$. Then
$\ell'$ is not a factor of $\ell$. Let $\ell = \omega u
\omega$. Since $\omega$ is not a factor of $\ell'$, it is possible
to first apply $\ell \to r$ and then apply $\ell' \to r'$.
Moreover, by choice of $d$ we have $\Abs{\omega} \leq
\Abs{r'}$. Thus we also can first apply $\ell' \to r'$ and then
$\ell \to r$.
If $u \in \mathrm{IRR}_{T_\Delta}(K^*)$ and $u \RA{T_\Omega} v$, then $v \in
\mathrm{IRR}_{T_\Delta}(K^*)$ by definition of the normal forms $v_g$ and
the set $\Omega$.
Thus, it remains to show that $T_\Omega$ is locally confluent on
$\mathrm{IRR}_{T_\Delta}(K^*)$. By minimality of $J$, no $\omega \in \Omega$
is a proper factor of another word $\omega' \in \Omega$. Let $\omega
u \omega \to r$ and $\omega' u' \omega' \to r'$ be two
$\Omega$-rules with $\omega \neq \omega'$. By construction of
$T'_\Omega$, the left sides of both rules can overlap at most $\min
\oneset{\abs{\omega}, \abs{\omega'}} - 1$ positions. Thus the two
rules can always be applied independently of one another.
Let now $\omega u \omega \to \omega v_g \omega$ and $\omega u'
\omega \to \omega v_h \omega$ be two $\Omega$-rules. By construction
of $T_\Omega$, neither is $\omega u' \omega$ a proper factor of
$\omega u \omega$ nor vice versa. If $x \omega = \omega y$ for some
$x,y \in K^+$ with $\Abs{x} \leq n \Abs{a_0}$, then $x \in \Delta$
and $\omega$ is a prefix of $x^+$ which contradicts the definition
of $J \subseteq K^* \setminus F$. Therefore, whenever $x \omega =
\omega y$ for $x, y \in K^+$ then $\Abs{x} > n \Abs{a_0}$ and
$\Abs{y} > n \Abs{a_0}$.
Suppose now $x \omega u \omega = \omega u' \omega y = \omega u''
\omega$ for $x,y \in K^+$. If $\abs{x} \geq \abs{\omega u}$, then
the two rules can be applied independently of one another. Thus let
$\abs{x} < \abs{\omega u}$. As seen before, we have $\Abs{x} > n
\Abs{a_0}$ and $\Abs{y} > n \Abs{a_0}$. We will show
\begin{equation*}
x \,\omega\, v_g\, \omega \;\RAS*{T_\Omega}\;
\omega\, v_{\varphi(u'')}\, \omega
\;\LAS*{T_\Omega}\; \omega \, v_h \, \omega \, y.
\end{equation*}
If $x \,\omega\, v_g\, \omega \in K^* \omega' K^*$ or $\omega \, v_h
\, \omega \, y \in K^* \omega' K^*$, then by \refclm{clm:oovoo} we
have $\omega' \preceq \omega$.
We can write $x \omega = \omega x'$. Since $\Abs{x'} = \Abs{x} > n
\Abs{a_0}$, we have $\Abs{x' v_g} > n \Abs{a_0} + \Abs{v_g} >
\Abs{v_{g'}}$ for every $g' \in G$. This relies on the fact that the
weights all normal forms $v_{g'}$ differ less than $n \Abs{a_0}$.
This shows that the weight of $x' v_g$ is sufficiently high. If
$\Abs{x' v_g} > t_\Omega$, then by \refclm{clm:long:oo} we have $x'
v_g \RAS*{T_\Omega} x''$ such that $\Abs{v_{g'}} < \Abs{x''} \leq
t_\Omega$ for every $g' \in G$. Therefore, without loss of
generality we can assume that the weight of $x' v_g$ is not too
high, i.e., $\Abs{x' v_g} \leq t_\Omega$. Since $\varphi(x' v_g) =
\varphi(u'')$, we have $x \omega v_g \omega \RAS*{T_\Omega} \omega
v_{\varphi(u'')} \omega$. Similarly, $\omega v_h \omega y
\RAS*{T_\Omega} \omega v_{\varphi(u'')} \omega$.
This completes the proof of \refclm{clm:confl}.~~~$\diamond$
Since all rules in $T$ are weight-reducing, local confluence implies
confluence. Moreover, all rules $\ell \to r$
in $T$ satisfy $\varphi(\ell) = \varphi(r)$. We conclude that $T$ is
a weighted Church-Rosser system such that $K^* / T$ is finite
and $\varphi : K^* \to G$ factorizes through $T$. Remember that
every element in $K^*$ can be read as a sequence of elements in
$A^*$. Thus every $u \in K^*$ can be interpreted as a word $u \in
A^*$. We use this interpretation in order to apply the rules in $T$
to words in $A^*$; but in order to not destroy $K$-letters when
applying rules in $R$, we have to guard the first $K$-letter
of every $T$-rule by appending the letter $c$. This leads to the
system
\begin{equation*}
T' = \set{c\ell \to cr \in A^* \times A^*}{\ell \to r \in T}.
\end{equation*}
Combining the rules $R$ over the alphabet $B$ with the
$T'$-rules yields
\begin{equation*}
S = R \cup T'.
\end{equation*}
Since left sides of $R$-rules and of $T'$-rules can not overlap, the
system $S$ is confluent. By definition, each $S$-rule is
weight-reducing. This means that $S$ is a weighted
Church-Rosser system. We have
\begin{align*}
\mathrm{IRR}_S(A^*) \ = \
\mathrm{IRR}_R(B^*) \;\cup \;
\mathrm{IRR}_R(B^*) \cdot \mathrm{IRR}_{T'}\Big(
c\big(\mathrm{IRR}_R(B^*) c\big)^* \Big) \cdot \mathrm{IRR}_R(B^*).
\end{align*}
Therefore $\mathrm{IRR}_S(A^*)$ and $A^* / S$ are finite. Since all rules
$\ell \to r$ in $S$ satisfy $\varphi(\ell) = \varphi(r)$, the
homomorphism $\varphi$ factorizes through $S$.
\end{proof}
\section{Arbitrary Finite Monoids}\label{sec:arbitrary}
This section contains the main result of this paper. We show that every homomorphism $\varphi : A^* \to M$ to finite monoid factorizes through a weighted Church-Rosser system $S$ with finite index. The proof relies on \refthm{thm:group} and on a construction called local divisors.
\subsection{Local divisors}
The notion of {\em local divisor} has turned out to be a rather powerful tool when using inductive proofs for finite monoids, see e.g.~\cite{dg08SIWT:short,DiekertKS11,DiekertKW12tcs}. The same is true in this paper. The definition of a local divisor is as follows: Let $M$ be a monoid and let $c \in M$. We equip $cM \cap Mc$ with a monoid structure by introducing a new multiplication~$\circ$ as follows: \begin{equation*}
xc \circ cy = xcy. \end{equation*} It is straightforward to see that $\circ$ is well-defined and $(cM \cap Mc, \circ)$ is a monoid with neutral element $c$.
The following observation is crucial. If $1 \in {cM \cap Mc}$, then $c$ is a unit. Thus if the monoid $M$ is finite and $c$ is not a unit, then $\abs{cM \cap Mc} < \abs{M}$. The set $M' = \set{x}{cx \in Mc}$ is a submonoid of $M$, and $c{\cdot}: M' \to cM \cap Mc : x \mapsto cx$ is a surjective homomorphism. Since $(cM \cap Mc, \circ)$ is the homomorphic image of a submonoid, it is a divisor of $M$. We therefore call $(cM \cap Mc, \circ)$ the {\em local divisor} of $M$ at $c$.
\subsection{The main result}
We are now ready to prove our main result: Every homomorphism $\varphi : A^* \to M$ to a finite monoid factorizes through a weighted Church-Rosser system $S$ with finite index. The proof uses induction on the size of $M$ and the size of $A$. If $\varphi(A^*)$ is a group, then we apply \refthm{thm:group}; and if $\varphi(A^*)$ is not a group, then we find a letter $c \in A$ such that $c$ is not a unit. Thus in this case we can use local divisors.
\begin{theorem}\label{thm:main}
Let $(A,\Abs{\cdot})$ be a weighted alphabet and let $\varphi : A^*
\to M$ be a homomorphism to a finite monoid $M$. Then there exists a
weighted Church-Rosser system $S$ of finite index such that
$\varphi$ factorizes through $S$. \end{theorem}
\begin{proof}
The proof is by induction on $(\abs{M},\abs{A})$ with lexicographic
order. If $\varphi(A^*)$ is a group, then the claim follows by
\refthm{thm:group}. If $\varphi(A^*)$ is not a group, then there
exists $c \in A$ such that $\varphi(c)$ is not a unit. Let $B = A
\setminus \smallset{c}$. By induction on the size of the alphabet
there exists a weighted Church-Rosser system $R$ for the
restriction $\varphi : B^* \to M$ satisfying the statement of the
theorem. Let
\begin{equation*}
K = \mathrm{IRR}_R(B^*) c.
\end{equation*}
We consider the prefix code $K$ as a weighted alphabet. The weight
of a letter $uc \in K$ is the weight $\Abs{uc}$ when read as a word
over the weighted alphabet $(A,\Abs{\cdot})$. Let $M_c = \varphi(c)
M \cap M \varphi(c)$ be the local divisor of $M$ at $\varphi(c)$.
We let $\psi : K^* \to M_c$ be the homomorphism induced by $\psi(uc)
= \varphi(cuc)$ for $uc \in K$. By induction on the size of the
monoid there exists a weighted Church-Rosser system $T
\subseteq K^* \times K^*$ for $\psi$ satisfying the statement of the
theorem. Suppose $\psi(\ell) = \psi(r)$ for $\ell,r \in K^*$ and let
$\ell = u_1 c \cdots u_j c$ and $r = v_1 c \cdots v_k c$ with
$u_i,v_i \in \mathrm{IRR}_R(B^*)$. Then
\begin{align*}
\varphi(c \ell)
&= \varphi(cu_1c) \circ \cdots \circ \varphi(cu_jc) \\
&= \psi(u_1 c) \circ \cdots \circ \psi(u_j c) \\
&= \psi(\ell) = \psi(r) = \varphi(c r).
\end{align*}
This means that every $T$-rule $\ell \to r$ yields an
$\varphi$-invariant rule $c\ell \to cr$. Thus we can transform the
system $T \subseteq K^* \times K^*$ for $\psi$ into a system $T'
\subseteq A^* \times A^*$ for $\varphi$ by
\begin{equation*}
T' = \set{c\ell \to cr \in A^* \times A^*}{\ell \to r \in T}.
\end{equation*}
Since $T$ is confluent and weight-reducing over $K^*$, the system
$T'$ is confluent and weight-reducing over $A^*$. Combining $R$ and
$T'$ leads to
\begin{equation*}
S = R \cup T'.
\end{equation*}
The left sides of a rule in $R$ and a rule in $T'$ cannot overlap.
Therefore, $S$ is a weighted Church-Rosser system such that
$\varphi$ factorizes through $A^* / S$. Suppose that every word in
$\mathrm{IRR}_T(K^*)$ has length at most $k$. Here, the length is over the
extended alphabet $K$. Similarly, let every word in $\mathrm{IRR}_R(B^*)$
have length at most~$m$. Then
\begin{equation*}
\mathrm{IRR}_S(A^*) \subseteq \set{u_0 c u_1 \cdots c u_{k'+1}}{
u_i \in \mathrm{IRR}_R(B^*), \; k' \leq k}
\end{equation*}
and every word in $\mathrm{IRR}_S(A^*)$ has length at most $(k+2)m$. In
particular $\mathrm{IRR}_S(A^*)$ and $A^* / S$ are finite.
\end{proof}
The following corollary is a straightforward translation of the result in \refthm{thm:main} about homomorphisms to a statement about regular languages.
\begin{corollary}\label{cor:main}
A language $L \subseteq A^*$ is regular if and only if there exists
a Church-Rosser system $S$ of finite index such that $L =
\bigcup_{u \in L} [u]_S$. \end{corollary}
\begin{proof}
If $L$ is regular, then there exists a homomorphism $\varphi : A^*
\to M$ recognizing $L$. By \refthm{thm:main} there exists a finite
Church-Rosser system $S$ of finite index such that $\varphi$
factorizes through $S$. The latter property implies $\varphi^{-1}(x)
= \bigcup_{u \in \varphi^{-1}(x)} [u]_S$ for every $x \in M$. Thus
$L = \bigcup_{x \in \varphi(L)} \varphi^{-1}(x) = \bigcup_{u \in L}
[u]_S$.
The converse is trivial.
\end{proof}
In particular, we see that all regular languages are Church-Rosser congruential.
{\small \newcommand{\Ju}{Ju}\newcommand{\Ph}{Ph}\newcommand{\Th}{Th}\newcommand{\Ch}{Ch}\newcommand{\Yu}{Yu}\newcommand{\Zh}{Zh}
}
\end{document} | arXiv |
\begin{document}
\title{Loop Programming Practices that Simplify Quicksort Implementations}
\begin{abstract}
Quicksort algorithm with Hoare's partition scheme is traditionally implemented with
nested loops.
In this article, we present loop programming and refactoring techniques
that lead to simplified implementation for Hoare's quicksort algorithm consisting of a
single loop.
We believe that the techniques are beneficial for general programming and may be used for
the discovery of more novel algorithms.
\end{abstract}
\keywords{Quicksort \and Loop rotation \and Nested loop thinning \and Cascading conditional
\and Sentinel}
\section{Introduction}\label{sec:introduction}
Loops are one of the most widely used programming constructs featured
in almost all programming languages.
A loop is an productivity amplifier.
With nominal overheads (\textit{e.g.}, state-registering variables, \textit{etc.}),
the static body of a loop can be reused for unlimited number of times.
A key building block as loop, one would suppose its best practices should have been widely
known.
On the opposite, however, the best practices for loop has been so largely ignored that
haphazardly constructed loops with duplication issues is not uncommon even in production code.
Common problems in loop programming include, but are not limited to, duplicate code, nested
loops, leaky loop variables, and oversized initialization.
I will explain each of them next.
Duplicate code here refers to duplication between code in the loop body and code before (after)
the loop.
It is the biggest problem in loop programming and is the most common root causes for bugs.
Duplicate code anywhere is bad.
But duplicate code of this type is harder to realize or get rid of.
Uses of nested loops are sometimes controversial.
Many readers are ready to argue about this.
Anyhow, what is wrong with nested loop?
For almost all cases, nested loops bring complexity rather than convenience,
obstruct readability rather than facilitating it.
It turns what could have been coded as different components into monolithic mess and
discourages code reusing.
deepen the coupling rather than reducing it and discourage code reusability rather than encouraging it.
Loop variables refers to variables used to register loop state.
Many developers rely on exposed variable(s) to implicitly pass information from loop to
subsequent program.
However, uncontrolled exposure of loop variable to subsequent code is a violation of the
encapsulation principle.
While sometimes convenient, it usually does more harm than good.
Loop initialization is the prelude code needed for instantiating initial loop state.
It should be small and light-weight.
rather than light-weight, succinct ones.
However unseasoned programmers may code disproportionally heavy initializations.
With due skills and perseverance, loop initializations can be made succinct one-liner.
It is beyond the scope of this article to address all the topics.
Instead we will focus on duplicate code and nested loops.
We will present two pillar loop programming techniques---`loop rotation' and `Nested loop
thinning' which I found are effective in fighting against above programming foes.
These are the two pillars for loop programming.
Proper use of them help developers avoid commonly made mistakes.
We have tried these techniques on several case studies.
Particularly, we will apply them to simplify traditional quicksort algorithm to prove the
effectiveness of these techniques---the implementation of quicksort algorithms.
First invented in 1960, quicksort has been studied and analyzed well over the
years\cite{Hoare1961,Hoare1962,sedgewick:1978,Bentley:1986:PP}.
As one of the earliest `divide-n-conquer` algorithms, quicksort has become the
\textit{de facto} sorting algorithm in practice for its excellent expected performance.
Recently named one of the top $10$ algorithms of $20$th century, quicksort has a
profound influence on the history of programming\cite{top10algorithm:2000}.
We will walk you through the multi-stage refactor process that leads to the discovery of a
brand-new implementation of Hoare's quicksort algorithm.
We use Python as the working language in this article.
\section{Loop Rotation}\label{sec:loop-rotation}
\begin{figure}
\caption{ Using spool as a metaphor for loop construct in program,
duplicate code around a loop would be like tightening messy yarns into neat spool.
`Loop rotation' can be pictorially depicted as winding up loose yarns onto a spool.
\protect\subref{fig:loose-spool}: Duplicate, sloppy code is analogous to loose, unorganized
yarn around a spool.
\protect\subref{fig:tight-spool}: A neat loop is analogous to a tidy spool.}
\label{fig:spool-loop}
\end{figure}
Coming with the productivity amplification power is the coding complexity of loops. There are two key observations about loops. First, a loop seldom lives in vacuum. Loops are often used as an embedded subunits in a program just like an organelle in a living cell. So it has to get along with its neighbors. As a result, a large part of loop programming is to ``fit it in''. Secondly, there are many moving parts in a loop construct and they are tightly coupled, changes in one necessarily demand changes in others. As we discussed in ~\autoref{sec:introduction}, the biggest problem with loop programming is duplicate code in and around the loop. The foremost goal in implementing loop is to reduce duplicate code. `Loop rotation' is an important technique for that purpose.
It is beneficial to visualize a loop construct as a spool. Loops are to program as spools are to yarn---just as spools are used to organize yarn, loops are invented to organize program. Behind this analogy is an important symmetry with respect to circular shifts---the rotational symmetry shared by them. Aside from the point of entrance and one or more exits, a loop or a spool may be mathematically represented by a circular sequence, that is, a sequence with rotational symmetry. One can thus take this rotational degrees of freedom to one's advantage to decide where to enter and exit the loop. Coalescing duplicate code in a loop is analogous to winding up loose and messy yarn into a tidy spool (~\autoref{fig:spool-loop}).
Let's take a board game as an example to illustrate loop rotation. Imagine that the following subroutines are ready to use: \code{Init}: Initialize game; \code{BD}: Draw board; \code{PR}: Print legends and prompts; \code{UI}: Take user inputs; \code{EX}: Execute user inputs; \code{CM}: Compute moves; \code{PO}: Poll game status; \code{End}: End of game. ~\autoref{board_game_dupcode} shows pseudo-code for the driver program. The duplicate code between the loop and in the vicinity of the loop is conspicuous. With a ``loop rotation'' procedure (to be explained below), the code may be refactored into ~\autoref{board_game_rotated}.
\begin{minipage}[t]{.4\textwidth}
\lstinputlisting[language=python,
caption={\label{board_game_dupcode}Board game driver program}]{board_game_dupcode.py} \end{minipage}
\begin{minipage}[t]{.4\textwidth}
\lstinputlisting[language=python,
caption={\label{board_game_rotated}Refactored driver program}]{board_game_rotated.py} \end{minipage}
\begin{figure}
\caption{
Pictorial example of how loop rotation coalesces duplicate code.
Legend: blue are loop statements, green are non-loop statements.
The statements ``\code{BO}'', ``\code{PR}'', ``\code{UI}'', and ``\code{EX}''
are duplicated between pre-loop and loop.
They may be coalesced by a loop rotation.
\protect\subref{fig:ferris-wheel-dup-node}: Original program;
\protect\subref{fig:ferris-wheel-rotated}: Rotated program.}
\label{fig:loop-rotation}
\end{figure}
Looking at ~\autoref{fig:loop-rotation}, as we rotate the loop, the duplicate code can be aligned and coalesced line by line. As can been seen in ~\autoref{fig:ferris-wheel-dup-node}, the last statement in the loop and the last statement before the loop are verbatim duplicate and aligned. As such, we `roll up' the spool so that the two can be coalesced. This process can be repeated until all the duplicates are coalesced (~\autoref{fig:ferris-wheel-rotated}).
Getting rid of duplicate code is one of the main application scenario for ``loop rotation''. Another application scenario for the ``loop rotation'' procedure is for the effect of shifting code within a loop, for example, to move a portion of the code from the beginning to the end. This is often called ``reverse rotation'' of the loop. As a side effect, doing so will result in duplicate code. Given that this prepares ways for subsequent refactors, this adverse effect is often paid off with larger optimizations. In practice, one may start with the more malleable ``\code{while (true)}'' or ``\code{for (; ;)}'' loops which helps one focus on getting a correct program first. Once \emph{a} working and flexible code has been secured, the `loop rotation' technique can be used to fine-tune the program.
\section{Nested Loop thinning Technique}\label{sec:loop-thinning-technique}
Every developer writes nested loops now and then. Many times, nested loops appear compelling and inevitable. While they may solve our problem, nested loops inflict on a program unnecessary complexity, obstruct code readability, and bring in `soft duplication'. The presence of nested loops also thwart optimization at the compiler level. For these reasons, explicit use of loops at high level programming should be avoided altogether\cite{data-engineering-avoid-loop}. Moreover, getting rid of nested loops itself may not seem significant improvement. The optimizations that are made accessible after getting rid of nested loops dwarfs the improvement brought about by getting rid of nested loops itself. Like `Candy Crush Saga' game at certain critical point, a single move may unlock an avalanche of advantageous moves.
In this section, we are to present a process that coalesces nested loops into single-layered loops, which we will name as `Nested loop thinning technique'. For sake of argument, the body of the outer loop is divided into three sections---\emph{pre-inner-loop} section, \emph{inner-loop} section, and \emph{post-inner-loop} section---depending on their relative position in respect to the inner loop (as shown in ~\autoref{fig:segment-for-loop-thinning}). Cramming the functionalities of nested loops into a single loop is no easy task. As a price, the process almost always results in one or more of the following: \begin{itemize}
\item extra conditional constructs
\item extra tests added to existing conditional constructs
\item more dynamic loop pacing
\item more boundary-condition-handling logic
\item auxiliary data structure, such as queue or stack \end{itemize}
\begin{figure}
\caption{
Sectioning of the body of nested loops:
\emph{pre-inner-loop} statements, the inner loop itself, and the \emph{post-inner-loop}
statements.
Each section receives different treatment during the `nested loop thinning' process.}
\label{fig:segment-for-loop-thinning}
\end{figure}
The gist for nested loop thinning process is as follows: \begin{enumerate}
\item Preparation stage
\item Loop rotation to shift around components in the loop body
\item Reconstruction of loop body using conditionals \end{enumerate} But we will go through them one by one. One will see this pattern again and again for nested loop thinning in practice.
First, some preparatory measures may be taken beforehand to reduce the friction during the refactor process. Compound statement is a good example to be dealt with in this step. Compound statements may get in your way of refactoring for multiple reasons. The most obvious one is that a compound statement need to be broken up and sent to different places after refactoring. State-changing, non-idempotent conditional expressions, such as \code{if (--i < 0)}, are even more lethal because each evaluation of the condition ratchets the state of the loop. For example,
\begin{minipage}[c]{.3\textwidth}
\begin{lstlisting}[language=C]
while (++i < LIMIT);
\end{lstlisting} \end{minipage}
\begin{minipage}[c]{.3\textwidth}
shall be expanded to \end{minipage} \begin{minipage}[c]{.3\textwidth}
\lstinputlisting[language=C]{expanded-do-while.java} \end{minipage}
before the start of loop thinning refactor. Other types of compound statements, such as ``\code{if v := a[i]; v < LIMIT}'' in Golang, shall be preprocessed similarly. Because of the structural similarity between \code{while}-loops and conditionals. \code{while}-loop readily lends itself to ``nested loop thinning'' process. For this reason, \code{for}-loop are often converted to \code{while}-loop during preprocessing.
In the second stage, one is to move pre-loop statements, if any, out of the way. To that end, reverse loop rotation may be used to unwind the pre-loop statements (see ~\autoref{sec:loop-rotation}). The second step depends on the inner loop construct.
Finally, it the reconstruction of the loop body using conditional rather than nested loops. If the inner loop is an unconditional loop as `\code{while True}', \code{break} statements (or similar) are almost always present and most likely in a conditional statement somewhere in the loop body unless it is intended to be a non-typical loop construct. One should replace the \code{break} statement with the post-inner-loop statements. The inner loop can then be stripped away. Otherwise, if the inner loop comes with a non-trivial termination condition, then the inner loop can be converted to a conditional directly, \code{while} $\rightarrow$ \code{if}, while the post-inner-loop statements are wrapped away in an \code{else}-clause of it.
Regardless of the venue taken, new conditional statements are inevitably formed or extended and `cascading conditional construct' are the best way to organize them. `Cascading conditional construct' consists of ordered sequence of exclusive conditional statements such as ``\code{if .. elif .. else}''. For more information, please refer to relevant chapters in Reference ~\cite{Wan:book}. Throughout the process, one shall pay special attention to execution-path-shunting statements, such as \code{break} and \code{continue}, if any.
After `nested loop thinning', some cleanup may be performed to comply with convention, code style, or just for cosmetic reasons.
One disclaimer is that the `nested loop thinning' process does not always prevail. There are cases where the process is not applicable. Certain criteria must be met for `nested loop thinning' to be applicable. First, there must not be intermediate layers, such as conditional, between the inner and outer loops. Secondly, execution-path-shunting statements, such as \code{break}, cannot be present in the pre-inner-loop section. In what follows, we are going to demonstrate application of `nested loop thinning' technique on the quicksort algorithms.
\section{Case Study: quicksort}\label{sec:nested-loop-quick-sort}
Quicksort is one of the pivotal sorting algorithms widely used by modern software. The Hoare's scheme was the first partition scheme that came with the original invention of this algorithm\cite{Hoare1961,Hoare1962,sedgewick:1978}. Traditionally, Hoare's partition scheme has been implemented with nested loops. Later one, Lomuto's partitioning scheme was invented whose implementation is much simpler with only one loop\cite{Bentley:1986:PP}. However for certain edge cases, Lomuto's quicksort algorithm does not perform well. It is natural to ask if one can implement the Hoare's partition scheme with the simplicity of or close to the Lomuto's. Armed with the loop programming techniques presented in this article, let us give it a try. First let us lay the foundation of the implementations of the quicksort algorithm.
\subsection{Recursive implementation of quicksort}\label{subsec:recursive-impl-quicksort}
At the high level, a recursive quicksort implementation may be as follows \lstinputlisting[label={lst:QuickSortOutline},language=C, caption={An implementation of quicksort algorithm.}] {quicksort_outline.c} where arguments \code{s} and \code{e} are the starting and ending pointers to the input array. This function invokes `\code{int* part(int* s, int* e)}' which is a function stub for array partition which will be discussed in detail below. For single-element arrays, quicksort function is no-op which is a base case. For na\"{i}ve implementations, this the only base case and it is sufficient. But for more sophisticated implementations or to reduce partition overhead, more base cases are used to address under-sized arrays (\textit{e.g.}, arrays of size $\leq 5$ but $\geq 1$).
In plain English, this is how quicksort algorithm works: \begin{quote}
If the array contains fewer than $2$ elements (the base case), return as is.
Otherwise, invoke partition function to partition the array into two subarrays and an
element (the \emph{pivot}).
Each subarray is subject to the quicksort function again so on and so forth
until they are all reduced to the base case.
At the return of the function, the entire array is sorted. \end{quote} The implementation of the partition function is key to the quicksort algorithm. The partition function does three things: \begin{enumerate}
\item Pick a pivot from among the array and set it aside;
\item Use pivot as a benchmark, partition the rest of the array into two subarrays
with smaller (or equal) ones on the left and greater (or equal) ones on the right;
\item Put the pivot element back in between the subarrays. \end{enumerate}
The subarrays resulted from partition may not be equal sized which is called partition skewness. Partition skewness has an adverse impact on the performance of quicksort algorithm. In all practical implementations of the partition function for quicksort, some type of pivot selection strategy is needed to prevent partition skewness. Common and proven practices are random selection or ``median-of-three'' technique\cite{sedgewick:1978,sedgewick1977}. Of course, to apply the ``median-of-three'' technique, the array must exceed a minimum size.
With this said, we are ready to discuss implementation of quicksort and its partition function. Admittedly, implementing quicksort is quite tricky. Among the quicksort partition schemes, best known are Hoare's scheme and Lomuto's scheme\cite{Hoare1961,Bentley:1986:PP}. The main difference between them lies in how the array is traversed. In Hoare's scheme, two pointers, one from each end of the array, step toward each other; whereas in Lomuto's, two pointers, each on its own pace, start off the left end of the array and step rightward.
Among the quicksort algorithms, there are two main variants---those by Tony Hoare and by Nico Lomuto, respectively. Hoare's quicksort scheme has robust and optimal performance but its implementation has been quite involved. Lomuto's scheme, on the other hand, is straightforward to implement and easier to follow. However its performance may degrade catastrophically for certain edge cases. Comparing the two, one cannot help but wonder if there is an implementation that is as robust and performant as the traditional implementation of Hoare's quicksort algorithm and at the same time as succinct as that of Lomuto's. That is going to be the focus of rest of this article.
\subsection{Implementation of Lomuto's Partition Scheme}\label{subsec:impl-of-Lomuto-part-fun}
\lstinputlisting[label={lst:singleLoopLomuto},language=C, caption={Implementation of Lomuto's partition scheme for quicksort.}] {quicksort_partition_asym.c} Let's start with the relatively simpler Lomuto's partition scheme. In Lomuto's partition scheme, the two pointers have distinct tasks. The one running in front, variable \code{i}, is responsible to discover out-of-place elements. The one behind, \code{p}, guards the partition boundary. When \code{i} discovers an out-of-place element, \code{p} makes room and places it by a swap and the partition process continues. The code is shown in ~\autoref{lst:singleLoopLomuto}. Once one understands the code, implementation becomes highly consistent and intuitive. One seldom fails implementing even for a customized applications.
Notably, implementing this partition scheme only needs one loop. However the simplicity is no free lunch. In fact, for certain edge cases, Lomuto's partition scheme suffers severe performance penalty, \texttt{e.g.}, arrays with a large number of identical elements, in which the Lomuto's quicksort algorithm degrades close to quadratic runtime. The root cause leading to this degradation lies in the asymmetric traversal of the array which inevitably leads to partition skewness. With the Lomuto's quicksort partition function, let's come back and study the more sophisticated implementation---the symmetric Hoare's partition scheme.
\subsection{Implementation of Hoare's Partition Scheme}\label{subsec:impl-of-qsort-part-fun}
Invented along with the quicksort algorithm, the Hoare's partition scheme predates Lomuto's historically\cite{Hoare1961,Hoare1962,sedgewick:1978}. Because of its symmetric traversal, Hoare's partition scheme successfully avoids the drawback of Lomuto's.
Visually speaking, Hoare's partition scheme employs two pointers, \code{s} and \code{e}, starting off the opposite ends of the array, push through the array toward each other, given that a pivot element has been placed at the beginning of the array. As in Lomuto's partition function, these pointers also stop at `out-of-place' elements. When both stop, the `out-of-place' elements are swapped to where they belong. Then the pointers are on their way again so on and so forth until they meet or cross. At last the pivot element is swapped into its final position and a pointer to this final position is returned.
While it appears a minor change from Lomuto's scheme, the Hoare's partition comes with immense implementation complexity.
So much so that for a long time how Hoare's partition scheme work remained an enigma\cite{Bentley-Beautiful-code}. There are so many changing variables and so much coupling among them that once in a while, each attempt of implementing it may end up with a different solution. Even worse than that, when something goes wrong, one is often clueless as to what is wrong. Also, it is extremely hard, if not impossible, to devise a test case to hit an elusive bug.
But in stark contrast to the numerous slightly differing implementations, all known implementations have so far unanimously used $3$ nested loops: one outer loop and two sequential inner loops. This feature is so commonplace that it has become the stereotype of quicksort.
With the Hoare's partition scheme, a commonly found implementation for the partition function is as follows. \lstinputlisting[label={lst:nestedLoopHoareMethod},language=C, caption={Implementation of Hoare's partition scheme for quicksort.}] {quicksort_partition_sym.c} While we have given an outline of the working of Hoare's partition, many choices remain to be made in regard of ``how, when, and what''. As such, pitfalls lay in wait every now and then throughout the implementation process. We will leave the discussions of the implementation process of ~\autoref{lst:nestedLoopHoareMethod} encountered during this implementation in ~\autoref{sec:impl-varieties}.
\subsection{Thinning of Nested Loops}\label{subsec:unwind-hoare-part}
Now we are going to use the techniques presented earlier to transform the traditional implementation of Hoare's partition scheme and get rid of the nested loops.
The immediate difficulty is how to take apart the densely packed conditional construct: \begin{lstlisting}
while(s < e && *++s < *pivot);
while(pivot < e && *--e > *pivot); \end{lstlisting} The loop conditions here are awkwardly complicated and make the 'nested loop thinning' (outlined in ~\autoref{sec:loop-thinning-technique}) nontrivial. The main difficulty lies in the dilemma---when condition suits, we need to switch to alternative execute path; but when we do, the pre-incremental statements would have ratcheted the state variables `(\code{s}, \code{e})' one step too far.
Measure must be taken to break up the pre-incremental statements before we can proceed any further. We follow a two-step conversion procedure: first unfold to `\code{do-while}' and then, in turn, rotate to `\code{while}' as shown in ~\autoref{quicksort_inner_loop_do_while} and ~\autoref{quicksort_inner_loop_while}.
\begin{minipage}[t]{.4\textwidth}
\lstinputlisting[caption={Unfold to \code{do-while} loop},label={quicksort_inner_loop_do_while},
language=C]{quicksort_inner_loop_do_while.c} \end{minipage}
\begin{minipage}[t]{.4\textwidth}
\lstinputlisting[caption={Rotate to \code{while} loop},label={quicksort_inner_loop_while},
language=C]{quicksort_inner_loop_while.c} \end{minipage}
After these changes, our code becomes the listing on the left-hand side below. We have made slight adjustment so that the ++s and --e statements are gathered together into the pre-inner-loop section.
\begin{minipage}[t]{.4\textwidth}
\lstinputlisting[language=C,caption={After prep steps}]
{quicksort_partition_sym_expanded_inner_loop.c} \end{minipage}
\begin{minipage}[t]{.4\textwidth}
\lstinputlisting[language=C,caption={After \emph{pre-inner-loop} relocation}]
{quicksort_partition_sym_rotated_inner_loop.c} \end{minipage}
After these preparation steps, we are ready to follow the prescribed `loop rotation' procedure. Namely, relocate the \emph{pre-inner-loop} statements (shown as listing on the right-hand side above), convert inner loops into cascading conditionals, wrap up the \emph{post-inner-loop} statements into an \code{else} clause, and other cosmetic changes (refer to ~\fullref{sec:loop-thinning-technique}\footnote{Note that here we need to covert the second inner loop to an `\code{else if}' because the first inner loop is converted to an `\code{if}' clause}).
Shown in ~\autoref{lst:singleLoopHoarePart} is the Hoarse's partition function after completing the `loop thinning' procedure. During the refactor process, we relied heavily on the `loop rotation' technique and the `nested loop thinning' techniques. We also consciously employed skills for the construction of cascading conditionals. Comparing with where we started off ~\autoref{lst:nestedLoopHoareMethod}, the new quicksort implementation ~\autoref{lst:singleLoopHoarePart} retains the optimal and robust runtime as Hoare's algorithm but consists of just one loop as the Lomuto's partition function does. That's almost too good to believe. Our quest for a simple and performant partition scheme finally pays off.
\lstinputlisting[label={lst:singleLoopHoarePart},language=C,escapechar=\$, caption={Code after completing the `loop thinning' procedures.}] {quicksort_partition_single_loop.c}
Additionally, one may use sentinels to simplify the cluttered conditions in the cascading conditional constructs of ~\autoref{lst:singleLoopHoarePart}.
The end result is listed below. \lstinputlisting[label={lst:hoare-partition-sole-loop-sentinel}, caption={Hoare's partition function with use of sentinel}]{quicksort_partition_single_loop_sentinel.c} Note that this partition function requires at least $3$ elements in the array to work properly. Interested reader may refer to a more detailed discussion in ~\autoref{sec:quick-sort-sentinel}.
\section{Experiment}\label{sec:experiment}
Note that implementations ~\autoref{lst:singleLoopHoarePart} or ~\autoref{lst:hoare-partition-sole-loop-sentinel} for quicksort is just another way to implement the quicksort algorithm with Hoare's partitioning scheme. Their runtime complexity is expected to be the same as that of the traditional ones. As such, we have designed experiments to test this hypothesis. To prevent pivot skewness, we use the ``median-of-three'' technique in all the quicksort implementations use for this experiment\cite{sedgewick1977,sedgewick:1978}.
~\autoref{tab:quicksort-experiment} shows the runtime analysis and comparisons. The simplified implementation indeed comes with an overhead and is thus slower than its traditional counterpart of the Hoare's quicksort program. For sorted data sets (either ascending or descending), there is a $17\%$ slowdown. But for randomly shuffled data sets, the slow down is consistently around $7\%$. The experiment data seems to indicate that the simplified implementation of Hoare's quicksort algorithm shares same runtime complexity as its traditional counterpart. Their performance may differ by a multiplier.
\begin{table}[h]
\centering
\caption{\label{tab:quicksort-experiment}
Runtime measurement of Hoare's quicksort algorithms,
comparing the traditional implementation and the single-loop implementation.
Experiments are run against three data sets---the first is sorted ascending,
the second descending, and the third is randomly shuffled.
Courtesy: The experiment is conducted based on project ``Benchmarking Sorting
Algorithms''\cite{benchmarking-sorting-algo} with some modification.}
\vskip 10pt
\begin{tabular}{r|r|r|r}
\hline
\multicolumn{4}{l} {\mbox{Integer arrays in ascending order}}\\
array size & Traditional & Simplified & Percent difference\\
\hline
$10$ & $7$ & $8$ & $14\%$\\
$100$ & $18$ & $20$ & $11\%$\\
$1,000$ & $111$ & $126$ & $14\%$\\
$10,000$ & $1,071$ & $1,239$ & $16\%$\\
$50,000$ & $4,893$ & $5,867$ & $20\%$\\
$100,000$ & $9,479$ & $11,131$ & $17\%$\\
$1,000,000$ & $115,248$ & $137,658$ & $19\%$\\
\hline
\multicolumn{4}{l} {\mbox{Integer arrays in descending order}}\\
array size & Traditional & Simplified & Percent difference\\
\hline
$10$ & $7$ & $7$ & $0\%$\\
$100$ & $18$ & $19$ & $6\%$\\
$1,000$ & $106$ & $142$ & $34\%$\\
$10,000$ & $1,020$ & $1,167$ & $14\%$\\
$50,000$ & $4,757$ & $5,411$ & $14\%$\\
$100,000$ & $9,378$ & $11,132$ & $19\%$\\
$1,000,000$ & $118,696$ & $138,311$ & $17\%$\\
\hline
\multicolumn{4}{l} {\mbox{Integer arrays randomly shuffled}}\\
array size & Traditional & Simplified & Percent difference\\
\hline
$10$ & $6$ & $6$ & $0\%$\\
$100$ & $25$ & $25$ & $0\%$\\
$1,000$ & $264$ & $249$ & $-6\%$\\
$10,000$ & $2,949$ & $3,065$ & $4\%$\\
$50,000$ & $16,565$ & $17,943$ & $8\%$\\
$100,000$ & $34,005$ & $36,400$ & $7\%$\\
$1,000,000$ & $380,477$ & $406,659$ & $7\%$\\
\hline
\end{tabular} \end{table}
\section{Conclusion}\label{sec:conclusion}
In this article, we presented a couple of techniques for optimizing loop constructs in
high-level programming languages.
Taking advantage of the circular symmetry of loop constructs, loop rotation may be applied to a
loop to either reduce code duplication (forward rotation) or to shift certain part of the
code within the loop body (reverse rotation).
Another technique is loop thinning for simplifying nested loop complications.
These are two of the empirical techniques that helps programmers achieve software development
best practices.
As an example, we applied these techniques in simplifying a traditional implementation of
Hoare's quicksort algorithm.
We provided the simplified implementation in C++.
More generally, the programming techniques we developed in this article are applicable to all
programming languages that supports loop and conditional constructs.
\begin{appendices}
\appendix
\section{Hoare's Partitioning Functions}\label{sec:impl-varieties}
Below I compiled a partial list of frequently-made bugs.
\begin{enumerate}
\item In pivot election, failure to swap pivot element to the beginning of array
\item For the pointers \code{s} and \code{e}, there are two hesitating options:
`check-then-increment' or `increment-then-check'.
For the implementation shown in ~\autoref{lst:nestedLoopHoareMethod},
first option leads to infinite loop for certain cases.
\item Also a choice between `\code{<= pivot}' and `\code{< pivot}' for \code{s},
`\code{>= pivot}' and `\code{> pivot}' for \code{e}.
Incorrect choice may inadvertently cause pointer incrementation to be skipped under
obscure circumstances which will, in turn, cause infinite loop
(Remember that under no circumstances, should either pointer stops approaching each other in
any outer loop iteration before they meet or cross.)
\item\label{itm:quicksort-boundary-condition} In the second inner loop for pointer \code{e},
failing to check boundary condition causes `out-of-boundary' exception.
For certain solutions, the correct condition is \code{pivot < e}.
Incorrect boundary condition, such as \code{s < e}, again cause pointer \code{e} to stall
in the middle of the right partition.
This will cause the pivot is placed at the wrong place at the return of function.
\item Outer loop termination logic also needs deliberation.
The key decision to make is `where to break or return' rather than `when to break or return'.
For ~\autoref{lst:nestedLoopHoareMethod}, the choice of location of return
at the end of the loop body was made after quick a few failed attempts.
\item Fail to swap the pivot element to its final position.
This step often poses a stumbling block for beginners as well as other unsuspecting
developers.
\item Choice of subarray semantics with respect to inclusiveness/exclusiveness of end pointers
when using start and end pointers to denote a subarray. \end{enumerate} Any of the items in this list can easily take a good chunk of debugging time.
A large part of the complication of the problem comes from the fact that this implementation consists of many moving parts and the design decisions for them are intimately coupled---changing one would necessitate corresponding changes in others. For example, suppose we are going to change the subarray semantics of \code{e} from exclusive to inclusive, we know that the return pointer will have to be different. But what would it be changed to? \code{s}, \code{s+1}, \code{e}, or \code{e-1}? Also, with these changes, the recursive calls in ~\autoref{lst:nestedLoopHoareMethod} line $4$-$5$ need corresponding changes. What would that be? Would it be the LHS or RHS below?
\begin{minipage}[t]{.4\textwidth}
\begin{lstlisting}[language=C]
qsort(s, p - 1)
qsort(p + 1, e)
\end{lstlisting} \end{minipage}
\begin{minipage}[t]{.4\textwidth}
\begin{lstlisting}[language=C]
qsort(s, p + 1);
qsort(p + 2, e);
\end{lstlisting} \end{minipage}
Also one may have observed, most of these frequently-made bugs comes with a multiple choice question. The collection of them form a tree structure. Each of the leaf node of the tree structure represents either garbage code or a legitimate solution. From an existing solution, one may variational tunnel to nearby solutions. For more information on how more related implementations may be discovered, please refer to ~\autoref{sec:impl-varieties}.
~\autoref{sec:impl-varieties} where we listed a number of implementations variations for the partition function. While not all of them can be simplified into single-loop implementation, we have had success with a few. Exactly which one can and which one cannot or why are not completely clear and yet to be investigated with.
\lstinputlisting[label={lst:part_nested_loop_var1},language=C, caption={Partition scheme for quicksort variation 1. For elements equal to the pivot, we don't stop '<=' or '<' is not critical. But '<=' brings more saving on unnecessary swaps.}] {part_nested_loop_var1.c}
\lstinputlisting[label={lst:part_nested_loop_var2},language=C, caption={Partition scheme for quicksort variation 2. Found another variation of partition scheme based on part-nested-loop Diff wise, it lies between part-nested-loop and part-nested-loop-var - the order of the two while loops is identical to part-nested-loop - the boundary checking is identical to part-nested-loop-var - the return value is shifted toward the left by one unit }] {part_nested_loop_var2.c}
\lstinputlisting[label={lst:part_nested_loop_var3},language=C, caption={Partition scheme for quicksort variation 3. Variation of partition scheme. The two differs in three ways - order of the two while loops - increment condition for equal element pointer - return pointer choice between s and e }] {part_nested_loop_var3.c}
\lstinputlisting[label={lst:part_nested_loop_var4},language=C, caption={Partition scheme for quicksort variation 4. Find yet another variation of implementation where the second nested while-loop is made post-incremental. This will remove the difficulty encountered in loop-thinning.}] {part_nested_loop_var4.c}
All these listings are to be understood with some pivot-selection mechanism to avoid pivot skewness.
\section{Sentinels in quicksort}\label{sec:quick-sort-sentinel}
The cascading conditional in the loop body of ~\autoref{lst:singleLoopHoarePart} is still far from straightforward. A further simplification on that may achievable through the use of sentinels~\cite{NumericalRecipes:1992}.
For stark effect of the sentinels, we will first fix the traditional implementation of `Hoare's partition scheme': ~\autoref{lst:QuickSortOutline} gives the outline of the program and ~\autoref{lst:nestedLoopHoareMethod} gives a fully implemented partition function. As mentioned earlier, one can get involved in the thick of implementing `Hoare's quicksort algorithm'. Not only the problem itself is tricky but also the way we implement it. By picking Hoare's quicksort scheme over its alternative (such as Lomuto's), we insist on the robust $O(N log N)$ expected runtime.
We attempt to get rid of the boundary checking using sentinels. These boundary checking are there to ensure that the pointers `\code{s}' and `\code{e}' do not slip off the ends of the array. In large arrays, these boundary check operations may get in the way of performance of the algorithm. But a large part of the reason is to remove clutter from the code.
The main idea is to make boundary checking redundant by effecting some artifacts that catch the pointers before the `out of boundary' error happens. This is exactly what sentinel is good at! By deploying sought-after values, the `sentinels', at the ends of the array before the control hits the loop, the following will happen: \begin{enumerate}
\item the pointers, `\code{s}' and `\code{p}', would have to stop when they hit the ends of
array;
\item the subsequent logic will guarantee the termination of the outer loop and
thus guarantee the inner loop would not be executed again. \end{enumerate}
What are the sought-after values by `\code{s}' and `\code{e}'? The out-of-place values! Particularly, \code{s} is looking for values that are `$\geq$ the pivot'; \code{e} is looking for values `$\leq$ the pivot'. So then instead of randomly picking one value for pivot, we now randomly pick $3$ values, the median of which be elected as the pivot as usual, the minimum be deployed to the left end, and the maximum to the right end. We group these operations into a function called `\code{init_swap}': \lstinputlisting[]{qsort_init_swap.c}
Now back the function `\code{part}'. As we mentioned before, the deployment of sentinels in function `\code{init_swap}' makes the conditional expressions `\code{s < e}' and `\code{pivot < e}' semantically redundant. They can now be safely removed. Below is the function `\code{part}' after all the refactoring is done. \lstinputlisting[]{qsort_part_sentinel.c} Comparing with the implementation ~\autoref{lst:nestedLoopHoareMethod}, this code cleans out clutter in the loop conditions in the inner loops. We `float' the control all the way through the termination of the program on a `touchless' rail made by sentinels, without ever needing to check boundary condition. Of course, the onus is shifted to the initialization before the loop. Comparing with the inner loops, that section is non-critical. Not only that the code is made less cluttered, but also the number of such checks is reduced from $O(N)$ to $O(1)$. More importantly, by not checking boundary conditions at the most critical section, we avoid bugs that would otherwise cost us many hours of debugging time.
So by use of sentinels on the Hoare's quicksort algorithm, traditional nested loop implementation or the single-loop implementation ~\autoref{lst:singleLoopHoarePart} alike, we shifted the complexity among the nested loops to a non-critical part of the program, effectively reduced its coding complexity. Applied to the latter, we arrive at a new level of simplicity for the implementations of Hoare's quicksort (as shown in ~\autoref{lst:hoare-partition-sole-loop-sentinel}).
\end{appendices}
\end{document} | arXiv |
Sybilla Beckmann
Sybilla Beckmann is a Josiah Meigs Distinguished Teaching Professor of Mathematics, Emeritus, at the University of Georgia and a recipient of the Association for Women in Mathematics Louise Hay Award.
Sybilla Beckmann
NationalityAmerican
TitleJosiah Meigs Distinguished Teaching Professor of Mathematics
AwardsLouise Hay Award
Academic background
Alma materUniversity of Pennsylvania
Brown University
ThesisFields of Definition of Solvable Branched Coverings (1986)
Doctoral advisorDavid Harbater
Academic work
DisciplineMathematics
InstitutionsUniversity of Georgia
Yale University
Main interestsMathematical cognition
Mathematical education of teachers
Mathematics content for grades pre-K - 8
Biography
Sybilla Beckmann received her Sc.B. in Mathematics from Brown University in 1980[1] and her Ph.D. in Mathematics from the University of Pennsylvania under the supervision of David Harbater in 1986.[2] She taught at Yale University as a J.W. Gibbs Instructor of Mathematics, before becoming a Josiah Meigs Distinguished Teaching Professor of Mathematics at the University of Georgia.[3] She retired in 2020.[1]
Beckmann's main interests include mathematical cognition, mathematical education of teachers, and mathematics content for pre-Kindergarten through Grade 8.[4]
Publications
Beckmann's publications include the following.[5][6]
• Mathematics for Elementary Teachers: Making Sense by "Explaining Why", in Proceedings of the Second International Conference on the Teaching of Mathematics at the Undergraduate Level, J. Wiley & Sons, Inc., (2002).[7]
• What mathematicians should know about teaching math for elementary teachers. Mathematicians and Education Reform Newsletter, Spring 2004. Volume 16, number 2.
• Solving Algebra and Other Story Problems with Simple Diagrams: a Method Demonstrated in Grade 4 – 6 Texts Used in Singapore, The Mathematics Educator, 14, (1), pp. 42 – 46 (2004).[8]
• With Karen Fuson. Focal Points: Grades 5 and 6. Teaching Children Mathematics. May 2008. Volume 14, issue 9, pages 508 – 517.
• Focus in Grade 5, Teaching with Curriculum Focal Points. (2009). National Council of Teachers of Mathematics. This book elaborates on the Focal Points at grade 5, including discussions of the necessary foundations at grades 3 and 4.
• Thomas J. Cooney, Sybilla Beckmann, and Gwendolyn M. Lloyd. (2010). Developing Essential Understanding of Functions for Teaching Mathematics in Grades 9 – 12. National Council of Teachers of Mathematics.[9]
• Karen C. Fuson, Douglas Clements, and Sybilla Beckmann. (2010). Focus in Prekindergarten: Teaching with Curriculum Focal Points. National Council of Teachers of Mathematics.
• Karen C. Fuson, Douglas Clements, and Sybilla Beckmann. (2010). Focus in Kindergarten: Teaching with Curriculum Focal Points. National Council of Teachers of Mathematics.
• Karen C. Fuson, Douglas Clements, and Sybilla Beckmann. (2010). Focus in Grade 1: Teaching with Curriculum Focal Points. National Council of Teachers of Mathematics.[10]
• Karen C. Fuson, Douglas Clements, and Sybilla Beckmann. (2011). Focus in Grade 2: Teaching with Curriculum Focal Points. National Council of Teachers of Mathematics.
• Fuson, K. C. & Beckmann, S. (Fall/Winter, 2012–2013). Standard algorithms in the Common Core State Standards. National Council of Supervisors of Mathematics Journal of Mathematics Education Leadership, 14 (2), 14–30.[11]
• Mathematics for Elementary Teachers with Activities, 4th edition, published by Pearson Education, copyright 2014, publication date January 2013.[12]
• Beckmann, S., & Izsák, A. (2014). Variable parts: A new perspective on proportional relationships and linear functions. In Nicol, C., Liljedahl, P., Oesterle, S., & Allan, D. (Eds.) Proceedings of the Joint Meeting of Thirty-Eighth Conference of the International meeting of the Psychology of Mathematics Education and the Thirty-Sixth meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Vol. 2, pp. 113–120. Vancouver, Canada: PME.
• Beckmann, S. & Izsák, A. (2014). Why is slope hard to teach? American Mathematical Society Blog on Teaching and Learning Mathematics.[13]
• Beckmann, S., & Izsák, A. (2015). Two perspectives on proportional relationships: Extending complementary origins of multiplication in terms of quantities. Journal for Research in Mathematics Education 46(1), pp. 17–38.
• Beckmann, S., Izsák, A., & Ölmez, İ. B. (2015). From multiplication to proportional relationships. In X. Sun, B. Kaur, J. Novotna (Eds.), Conference proceedings of ICMI Study 23: Primary mathematics study on whole numbers, pp. 518 – 525. Macau, China: University of Macau.[14]
Awards
• Association for Women in Mathematics twenty-fourth annual Louise Hay Award (2014).[15]
• Mathematical Association of America fourth annual Mary P. Dolciani Award (2015).[16]
References
1. "About Me". Sybilla Beckmann, PhD. Retrieved October 10, 2022.
2. Sybilla Beckmann at the Mathematics Genealogy Project
3. "Sybilla Beckmann-Kazez". University of Georgia. Retrieved October 10, 2022.
4. "Biography | Sybilla Beckmann". faculty.franklin.uga.edu. Retrieved 2016-11-05.
5. "temrrg". temrrg. Retrieved 2016-11-05.
6. "Sybilla Beckmann".
7. Beckmann, Sybilla. "Mathematics for Elementary Teachers" (PDF).
8. "TME – Volume 14 Number 1". math.coe.uga.edu. Retrieved 2016-11-05.
9. "NCTM Store: Developing Essential Understanding of Functions for Teaching Mathematics in Grades 9-12". www.nctm.org. Retrieved 2017-04-05.
10. "NCTM Store: Focus in Grade 1: Teaching with Curriculum Focal Points". www.nctm.org. Retrieved 2017-04-05.
11. "Standard Algorithms in the Common Core State Standards" (PDF).
12. "Mathematics for Elementary Teachers with Activities, 4/e by Sybilla Beckmann | Pearson". www.pearsonhighered.com. Retrieved 2016-11-05.
13. "Why is Slope Hard to Teach? | On Teaching and Learning Mathematics". blogs.ams.org. Retrieved 2016-11-05.
14. "Primary Mathematics Study on Whole Numbers" (PDF).
15. "Sybilla Beckmann – AWM Association for Women in Mathematics". sites.google.com. Retrieved 2016-11-05.
16. "Dolciani Award | Mathematical Association of America". www.maa.org. Retrieved 2020-09-27.
Authority control
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• ISNI
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• United States
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• MathSciNet
• Mathematics Genealogy Project
• zbMATH
| Wikipedia |
\begin{document}
\sloppy
\title{ space{-1cm}
To Yu.L.Ershov on his seventy-fifth birthday
\begin{abstract}
A subgroup $H$ of a group $G$ is called {\it pronormal}, if the subgroups $H$ and $H^g$ are conjugate in $\langle H, H^g\rangle$ for every $g\in G$. It is proven that if a finite group $G$ possesses a $\pi$-Hall subgroup for a set of primes $\pi$, then its every normal subgroup (in particular, $G$ itself) has a $\pi$-Hall subgroup that is pronormal in~$G$. \end{abstract}
\section*{Introduction}
Throughout the paper the term ``group'' implies ``finite group''.
According to the definition by P.~Hall, a subgroup $H$ of a group $G$ is called {\it pronormal}, if the subgroups $H$ and $H^g$ are conjugate in~$\langle H, H^g\rangle$ for every~${g\in G}$.
The pronormality for subgroups is a more general property, than normality, and it plays an important role in the theory of groups. In particular, the Frattini argument holds for pronormal subgroups: {\sl If a pronormal subgroup $H$ of $G$ lies in a normal subgroup~$A$, then $G=AN_G(H).$} This statement is often important in inductive arguments. Notice that $G=AN_G(H)$ if and only if $ H^G=\{H^g\mid g\in G\}$
and $H^A=\{H^a\mid a\in A\}$ coincide.
In view of the Sylow theorem, the Sylow subgroups of a group, sa well as the Sylow subgroups of every normal subgroup are examples of pronormal subgroups. The goal of the paper is to study in what form this properties of Sylow subgroups can be reformulated for Hall subgroups. We recall the appropriate definitions.
Throughout the paper we suppose that $\pi$ is a fixed set of primes. We denote by $\pi'$ the set of all primes not in $\pi$; and by
$\pi(n)$, the set of all prime divisors of a natural number $n$, while for a group $G$ we denote the set $\pi(|G|)$ by $\pi(G)$. A natural $n$ with $\pi(n)\subseteq\pi$ is called a {\it $\pi$-number}, while a group $G$ with $\pi(G)\subseteq \pi$ is called a {\it $\pi$-group.} A subgroup $H$ of $G$ is called a {\it $\pi$-Hall} subgroup, if
$\pi(H)\subseteq\pi$ and $\pi(|G:H|)\subseteq \pi'$. Thus in case $\pi=\{p\}$ the definition of $\pi$-Hall subgroup coincides with the usual notion of Sylow $p$-subgroup. A subgroup is said to be a {\em Hall} subgroup, if it is a $\pi$-Hall subgroup for a set of primes~$\pi$, i.~e. if its index and order are coprime.
According to \cite{Hall} we say that $G$ {\it satisfies $E_\pi$} (or briefly $G\in E_\pi$), if $G$ possesses a $\pi$-Hall subgroup. If, at that, every two $\pi$-Hall subgroups are conjugate, then we say that $G$ {\it satisfies $C_\pi$} ($G\in C_\pi$). The group satisfying $E_\pi$ or $C_\pi$ we call also an $E_\pi$- or a $C_\pi$-{\it group} respectively.
The Hall theorem implies that Hall subgroups are pronormal in solvable groups. Also the $\pi$-Hall subgroups are known to be pronormal \begin{itemize}
\item in finite simple groups~\cite{VR3}; \item in $C_\pi$-groups~\cite{VR4}. \end{itemize} In~\cite{GR} some properties were specified of the groups such that $\pi$-Hall subgroups exist and all are pronormal.
At the same time it is known~\cite{VR4} that if a set $\pi$ of primes is such that $E_\pi\ne C_\pi$, then for every $X\in E_\pi\setminus C_\pi$ and $p\in\pi'$ the group $X\wr \mathbb{Z}_p$ possesses nonpronormal $\pi$-Hall subgroups, so we cannot transfer the property of pronormality of Sylow subgroups to Hall subgroups.
The following analog of the Frattini argument for $\pi$-Hall subgroups is proven in~\cite[Theorem~1]{VR5}: {\sl If $G\in E_\pi$, then each normal subgroup $A$ of $G$ possesses a $\pi$-Hall subgroup $H$ such that $G=AN_G(H)$.}
In the paper we prove the following statement, on using results from \cite{RevVdoArxive,ExCrit,VR1,VR2,VR3,VR5}.
\begin{Theo}\label{MainTheorem} Let $G\in E_\pi$ for a set of primes $\pi$ and $A\trianglelefteq G$. Then there is a $\pi$-Hall subgroup of $A$ pronormal in $G$. \end{Theo}
\begin{Cor}\label{cor} Let $\pi$ be a set of primes. Then every group possessing a $\pi$-Hall subgroup has a pronormal $\pi$-Hall subgroup. \end{Cor}
Notice that the result of~\cite{VR2} on the pronormality of $\pi$-Hall subgroups in $C_\pi$-groups (or, equivalently, on the inheriting of the $C_\pi$-property by overgroups of $\pi$-Hall subgroups) is a particular case of this statement.
Theorem~\ref{MainTheorem} generalizes the above mentioned result of~\cite{VR5}. Notice also the following statement that generalizes the useful Lemma~\ref{HallExist} (located below) and giving a criterion of existence of $\pi$-Hall subgroups in nonsimple groups.
\begin{Cor}\label{cor1} Let $A\trianglelefteq G$ and let $\pi$ be a set of primes. Then $G\in E_\pi$ if and only if $G/A\in E_\pi$ and $A$ has a $\pi$-Hall subgroup $H$ such that $H^A=H^G$. \end{Cor}
\section{Preliminary results}
The notation of the paper is standard. As we say in Introduction, $\pi$ always stands for a set of primes. Given $G$, the set of all $\pi$-Hall subgroups of $G$ is denoted by $\operatorname{Hall}_\pi(G)$. The notation $H\,\,{\rm prn}\,\, G$ means that $H$ is a pronormal subgroup of~$G$.
\begin{Lemma} \label{base} {\em \cite[Ch. IV, (5.11)]{Suz}} Let $A$ be a normal subgroup of $G$. If $H$ is a $\pi$-Hall subgroup of $G$, then $H \cap A$ is a $\pi$-Hall subgroup of $A$, while $HA/A$ is a $\pi$-Hall subgroup of~$G/A$. \end{Lemma}
Recall that a group is called {\it $\pi$-separable}, if it has a normal series with factors either $\pi$- or $\pi'$-groups.
\begin{Lemma} \label{base1} {\em \cite[Ch. V, Theorem 3.7]{Suz}} Every $\pi$-separable group satisfies~$C_\pi$. \end{Lemma}
\begin{Lemma}\label{HallExist} {\em \cite[Lemma~2.1(e)]{RevVdoArxive}} Let $A$ be a normal subgroup of $G$ such that $G/A$ is a $\pi$-group, $U$ a $\pi$-Hall subgroup of $A$. Then a $\pi$-Hall subgroup $H$ of $G$ with $H\cap A=U$ exists if and only if $U^G=U^A$. \end{Lemma}
\begin{Lemma} \label{Hall_Pron_Simple_Lemma} {\em \cite[Theorem~1]{VR3}} The Hall subgroups in simple groups are pronormal. \end{Lemma}
\begin{Lemma} \label{FrattiniArgument} {\em \cite[Theorem~1]{VR5}} Let $G\in E_\pi$ and $A\trianglelefteq G$. Then there exists $H\in\operatorname{Hall}_\pi(A)$ such that $G=AN_G(H)$. Moreover $N_G(H)\in E_\pi$ and $\operatorname{Hall}_\pi(N_G(H))\subseteq\operatorname{Hall}_\pi(G)$. \end{Lemma}
\begin{Lemma} \label{ReplWithConj} Let $H$ be a subgroup of $G$. Considering $g\in G$, $y\in \langle H, H^g\rangle$, assume that subgroups $H^y$ and $H^g$ are conjugate in $\langle H^y, H^g\rangle$. Then $H$ and $H^g$ are conjugate in~$\langle H, H^g\rangle$. \end{Lemma}
\begin{proof} Let $z\in \langle H^y, H^g\rangle$ and $H^{yz}=H^g$. Then $z\in \langle H, H^g\rangle$, since $\langle H^y, H^g\rangle\leqslant\langle H, H^g\rangle $. So $x=yz\in\langle H, H^g\rangle$ and $H^x=H^g$. \end{proof}
The statements of the following two lemmas are evident.
\begin{Lemma} \label{Quot} Let $\overline{\phantom{g}}:G\rightarrow G_1$ be a group homomorphism and $H\leq G$. Then $H\,\,{\rm prn}\,\, G$ implies $\overline{H}\,\,{\rm prn}\,\, \overline{G}$.\end{Lemma}
\begin{Lemma} \label{Over} Let $G$ be a group. Then $H\,\,{\rm prn}\,\, G$ implies $H\,\,{\rm prn}\,\, K$, for every subgroup $K$ of $G$ such that $H\leq K$. \end{Lemma}
\begin{Lemma} \label{CentralProd} {\em \cite[Lemma 7]{VR3}} Let $G$ be a finite group and let $G_1,\dots,G_n$ be normal subgroups of $G$ such that $[G_i,G_j]=1$ for $i\ne j$ and $G=G_1\cdot\ldots\cdot G_n$. Assume that for every $i=1,\dots, n$ a pronormal subgroup $H_i$ of $G_i$ is chosen, and let $H=\langle H_1,\dots, H_n\rangle$. Then $H\,\,{\rm prn}\,\, G$. \end{Lemma}
\begin{Lemma} \label{HallSubgrIn HomImage} {\em \cite[Corollary 9]{ExCrit}} Let $G\in E_\pi$ and $A\trianglelefteq G$. Then for every $K/A\in\operatorname{Hall}_\pi(G/A)$ there exists $H\in\operatorname{Hall}_\pi(G)$ such that $K=HA$. \end{Lemma}
\begin{Lemma} \label{ProCrit} Let $H\leq G$ and $A\trianglelefteq G$. The following are equivalent: \begin{itemize} \item[$(1)$] $H\,\,{\rm prn}\,\, G$. \item[$(2)$] $HA\,\,{\rm prn}\,\, G$ and $H\,\,{\rm prn}\,\, N_G(HA)$. \end{itemize} \end{Lemma}
\begin{proof} Assume (1). Then $HA\,\,{\rm prn}\,\, G$ by Lemma~\ref{Quot} and $H\,\,{\rm prn}\,\, N_G(HA)$ by Lemma~\ref{Over}. Conversely, assume (2). Take $g\in G$. We need to show that there exists $x\in\langle H,H^g\rangle$ such that $H^x=H^g$. Since $HA/A\,\,{\rm prn}\,\, G/A$, there exists $y\in \langle H,H^g\rangle$ with $H^yA=H^gA$. In accordance with Lemma~\ref{ReplWithConj} it is possible to replace $H$ by $H^y$, and to assume that $HA=H^yA=H^gA$, i.~e. $g\in N_G(HA)$. Since $H\,\,{\rm prn}\,\, N_G(HA)$, the existence of desired $x$ is evident. \end{proof}
\begin{Lemma} \label{PiSep} Let $A$ be a $\pi$-separable normal subgroup of $G$ and ${H\in\operatorname{Hall}_\pi(A)}$. Then $H\,\,{\rm prn}\,\, G$. \end{Lemma}
\begin{proof} The lemma follows since the subgroup $\langle H, H^g\rangle\leq A$ is $\pi$-separable for every $g\in G$, while if $H$ and $H^g$ are its $\pi$-Hall subgroups, then they are conjugate in $\langle H, H^g\rangle$ by Lemma~\ref{base1}. \end{proof}
\begin{Lemma} \label{SuffPro} Let $B$ be a normal subgroup of a finite group $G$. Then for every normal subgroup $A$ of $G$ including $B$, and for every $H\in\operatorname{Hall}_\pi(A)$ the conditions \begin{itemize} \item[$(1)$] $HB/B\,\,{\rm prn}\,\, G/B$; \item[$(2)$] $(H\cap B)\,\,{\rm prn}\,\, B$; \item[$(3)$] $(H\cap B)^G=(H\cap B)^B$ \end{itemize} imply $H\,\,{\rm prn}\,\, G$.. \end{Lemma}
\begin{proof} Since $HB/B\,\,{\rm prn}\,\, G/B$, we have $HB\,\,{\rm prn}\,\, G$. By Lemma~\ref{ProCrit} it is enough to show that $H\,\,{\rm prn}\,\, N_G(HB)$. So, without loss of generality, we may assue that $G=N_G(HB)$, i.~e. $HB\trianglelefteq G$, and $A=N_A(HB)$. Notice that in this case $A/B$ is $\pi$-separable, since the factors $A/HB$ and $HB/B$ of the normal series $A\trianglerighteq HB\trianglerighteq B$ are $\pi'$- and $\pi$-groups respectively.
Take $g\in G$ arbitrary . Since $(H\cap B)^G=(H\cap B)^B$, there exists $b\in B$ such that $H^g\cap B=H^b\cap B$. Since $(H\cap B)\,\,{\rm prn}\,\, B$, there exists $$ y\in \langle H\cap B, H^b\cap B\rangle =\langle H\cap B, H^g\cap B\rangle\leq \langle H,H^g\rangle $$ such that $H^y\cap B=H^b\cap B.$ By Lemma~ \ref{ReplWithConj} the conjugacy of $H$ and $H^b$ in $\langle H^,H^b\rangle$ follows from the conjugacy of $H^y$ and $H^b$ in $\langle H^y,H^b\rangle$. Thus we can replace $H$ by $H^y$ and suppose that $$ (H\cap B)^g=(H\cap B)^b=(H\cap B)^y=H\cap B, $$ i.~e. $g\in N_G(H\cap B)$. It is clear also that $H\leq N_A(H\cap B)$.
Since $(H\cap B)^G=(H\cap B)^B$, applying the Frattini argument we obtain $G=BN_G(H\cap B)$ and $A=BN_A(H\cap B)$. Therefore, $$ N_A(H\cap B)/N_B(H\cap B)\cong BN_A(H\cap B)/B=A/B $$ is $\pi$-separable. The group $N_B(H\cap B)$ is also $\pi$-separable, since it has the normal $\pi$-Hall subgroup $H\cap B$. Hence, $N_A(H\cap B)$ is $\pi$-separable as well. Then from $H\in\operatorname{Hall}_\pi(N_A(H\cap B))$ by Lemma~\ref{PiSep} applied to $N_G(H\cap B)$ and its normal $\pi$-separable subgroup $N_A(H\cap B)$ we have $H\,\,{\rm prn}\,\, N_G(H\cap B)$. Since $g\in N_G(H\cap B)$, for some $x\in \langle H, H^g\rangle$ the equality $H^x=H^g$ holds. Thus, $H\,\,{\rm prn}\,\, G$. \end{proof}
\section{Proof of the main results}
{\it Proof of Theorem} \ref{MainTheorem}. Let $G\in E_\pi$ and $A\trianglelefteq G$. We need to show that $A$ has a $\pi$-Hall subgroup $H$ such that
$H\,\,{\rm prn}\,\, G$. We proceed by induction on~$|G|$.
If $|G|=1$, we have nothing to prove.
Let $|G|>1$. Choose a minimal normal subgroup $B$ of $G$ lying in $A$ (note that the inequality $B\not=A$ is not assumed here). Since by Lemma~\ref{base} $G/B\in E_\pi$, the factor group $A/B$ has a $\pi$-Hall subgroup $K/B$ such that $K/B\,\,{\rm prn}\,\, G/B$ by induction. By Lemma~\ref{FrattiniArgument} it follows that $B$ possesses a $\pi$-Hall subgroup $V$ such that $G=BN_G(V)$ or, equivalently, $V^G=V^B$. This means, in particular, that $V^K=V^B$ and, by Lemma~6, there exists $H\in\operatorname{Hall}_\pi(K)$ such that $V=H\cap B$. Notice that
$|A:H|=|A:K||K:H|$ is a $\pi'$-number, and so $H\in\operatorname{Hall}_\pi(A)$. Let us show that $H\,\,{\rm prn}\,\, G$, so proving the theorem. We use Lemma~\ref{SuffPro}. By the choice of $K$ we have $HB/B=K/B \,\,{\rm prn}\,\, G/B$, which is equivalent to $HB=K\,\,{\rm prn}\,\, G$, and so (1) of Lemma~\ref{SuffPro} holds. Now $B$ is a direct product of simple groups $ B=S_1\times\dots\times S_n, $ since $B$ is a minimal normal subgroup of $G$, and $ V=\langle V\cap S_i\mid i=1,\dots, n\rangle. $
Since by Lemma~\ref{base}, $V\cap S_i\in\operatorname{Hall}_\pi(S_i)$ for every $i=1,\dots,n$, and by Lemma ~\ref{Hall_Pron_Simple_Lemma}, $(V\cap S_i)\,\,{\rm prn}\,\, S_i$, applying Lemma~\ref{CentralProd} we obtain $ H\cap B=V\,\,{\rm prn}\,\, B, $ and so (2) of Lemma~\ref{SuffPro} holds. Finally, $H\cap B=V$ and by of the choice of $V$ in $B$ we have $(H\cap B)^G=(H\cap B)^B$. So (3) of Lemma~\ref{SuffPro} holds. Thus $H\,\,{\rm prn}\,\, G$ by Lemma~\ref{SuffPro}.
{$\Box$}
\noindent {\it Proof of Corollary}~\ref{cor}. The corollary is immediate from Theorem~\ref{MainTheorem} for the case~${G=A}$.
{$\Box$}
\noindent{\it Proof of Corollary}~\ref{cor1}. Let $A\trianglelefteq G$. Assume that $G/A\in E_\pi$ and $A$ has a $\pi$-Hall subgroup $H$ such that $H^A=H^G$. Show that $G\in E_\pi$. Choose $X/A\in \operatorname{Hall}_\pi(G/A)$. Since $A\leq X\leq G$, we have $$ H^A\subseteq H^X\subseteq H^G; $$ whence $H^A\subseteq H^X$. Taking into account Lemma~\ref{HallExist} and the fact that
$X/A$ is a $\pi$-group, we obtain $X\in E_\pi$. Since $|G:X|=|G/A:X/A|$ is a $\pi'$-number, we have $$ \varnothing\ne\operatorname{Hall}_\pi(X) \subseteq\operatorname{Hall}_\pi(G)\ \text{ and }\ G\in E_\pi. $$
Conersely, let $G\in E_\pi$. Then by Lemma~\ref{base} $G/A\in E_\pi$. By Theorem~\ref{MainTheorem}, there exists a subgroup $H\in \operatorname{Hall}_\pi(A)$ such that $H\,\,{\rm prn}\,\, G$. In particular, for every $g\in G$ there exists $a\in\langle H, H^g\rangle\leq A$ such that $H^g= H^a$. So $H^A=H^G$.
{$\Box$}
\section{Remarks}
\newtheorem{Rem}{{\bfseries Remark}}
\begin{Rem} Theorem~\ref{MainTheorem} guarantees the existence of a pronormal $\pi$-Hall subgroup in every normal subgroup $A$ of $G\in E_\pi$. The condition $G\in E_\pi$ cannot be replaced by $A\in E_\pi$ which is weaker in view of Lemma~\ref{base}. Indeed, let $\pi=\{2,3\}$ and $A=\operatorname{GL}_3(2)=\operatorname{SL}_3(2)$. Then (ñì.~\cite[Theorem 1.2]{RevHallp}) $A$ has exactly the two classes of conjugate $\pi$-Hall subgroups with representatives $$ H_1=\left( \begin{array}{c@{}c} \fbox{ $\begin{array}{c} \!\!\! \GL_2(2) \!\!\! \\ \end{array}$ } & *\\ 0 &\fbox{1} \end{array} \right){\text{\,\,\, and\,\,\, }} H_2=\left( \begin{array}{c@{}c} \fbox{1}& *\\
0&\fbox{$\begin{array}{c} \! \GL_2(2) \! \\ \end{array}$} \end{array} \right) $$ respectively. The class $H_1^A$ consists of line stabilizers in the natural presentation of $G$, while $H_2^A$ consists of plane stabilizers. The map $\iota : x\in A\mapsto (x^t)^{-1}$ is an automorphism of order 2 of $A$ (here $x^t$ is the transpose of~$x$). This automorphism interchanges $H_1^A$ and $H_2^A$. Consider the natural split extension $G=A:\langle\iota\rangle$. Subgroups $H_1$ and $H_2$ are conjugate in $G$. At the same time $H_1$ and $H_2$ are not conjugate in $A$, cotaining both $H_1$ and $H_2$, and so they are not pronormal in $G$. We remain to notice that $G\notin E_\pi$ in this example by Lemma~\ref{HallExist}. \end{Rem}
\begin{Rem} In \cite{VR2,VR4} the definition of strongly pronormal subgroup is introduced. Recall that a subgroup $H$ of $G$ is called {\it strongly pronormal}, if for every $g\in G$ and $K\leq H$ there exists $x\in\langle H, K^g\rangle$ such that $K^{gx}\leq H$. In~\cite{VR4} the conjecture that every pronormal Hall subgroup should be strongly pronormal is formulated. In the light of Theorem~\ref{MainTheorem} and its corolary it is natural to formulate the weaker conjecture: Does every $E_\pi$-group have a strongly pronormal $\pi$-Hall subgroup? \end{Rem}
\begin{Rem} Lemma \ref{SuffPro} plays an important role in the proof of Theorem~\ref{MainTheorem}, and from the lemma, the following test for pronormality of Hall subgroups follows in particular: Let $A\trianglelefteq G$ for a group~$G$. A Hall subgroup $H$ of $G$ is pronormal if
\begin{itemize} \item[$(1)$] $HA/A\,\,{\rm prn}\,\, G/A$; \item[$(2)$] $(H\cap A)\,\,{\rm prn}\,\, A$; \item[$(3)$] $(H\cap A)^G=(H\cap A)^A$. \end{itemize}
The authors do not know, whether the converse is true; more precisely, whether the condition $H\,\,{\rm prn}\,\, G$, where $H$ is a Hall subgroup of $G$, implies (2) and (3)? Statement (1) follows from the pronormality of $H$ by Lemma~\ref{Quot}. \end{Rem}
\end{document} | arXiv |
\begin{document}
\title{Nuclear Magnetic Resonance model of an entangled sensor under noise}
\author{ Le Bin Ho} \affiliation{Department of Physics, Kindai University, Higashi-Osaka, 577-8502, Japan}
\author{Yuichiro Matsuzaki} \affiliation{Nanoelectronics Research Institute, National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan}
\author{Masayuki Matsuzaki} \affiliation{Department of Physics, Fukuoka University of Education, Munakata, Fukuoka 811-4192, Japan}
\author{ Yasushi Kondo} \affiliation{Department of Physics, Kindai University, Higashi-Osaka, 577-8502, Japan} \thanks{Electronic address: [email protected]}
\begin{abstract} Entangled sensors have been attracting much attention recently because they can achieve higher sensitivity than those of classical sensors. To exploit entanglement as a resource, it is important to understand the effect of noise, because the entangled state is highly sensitive to noise. In this paper, we present a Nuclear Magnetic Resonance (NMR) model of an entangled sensor in a controlled environment; one can implement the entangled sensor under various noisy environments. In particular, we experimentally investigate the performance of the entangled sensor under time-inhomogeneous noisy environments, where the entangled sensor has the potential to surpass classical sensors. Our ``entangled sensor'' consists of a multi-spin molecule solved in isotropic liquid, and we can perform, or simulate, quantum sensing by using NMR techniques. \end{abstract}
\maketitle
\section{Introduction}
Quantum sensing has been attracting much attention recently as an application of quantum mechanics like quantum information technology \cite{Nielsen2000}, because it may achieve better sensitivity than a classical sensor. Quantum sensing may be divided into three categories according to what aspect of quantum mechanics employed to improve the performance of measurements: (i) quantum objects such as electrons or nuclear spins, (ii) quantum coherence such as superposition states or matter-wave-like nature, and (iii) quantum entanglement, which cannot be described classically \cite{RevModPhys.89.035002}. The third may be the most quantum-like, and various efforts have been reported regarding this category. Among these, entanglement-enhanced magnetic field sensing with atomic vapors was reported, such as spin squeezing (entanglement) within the atomic internal structure \cite{PhysRevLett.101.073601}, and employing entanglement between two vapor cells \cite{PhysRevLett.104.133601} to reduce noise. A more direct approach to enhance the sensitivity of measurements using entanglement was presented \cite{Giovannetti:2011aa}, and experimental efforts using trapped ions \cite{Leibfried:2003aa,Leibfried1476,Leibfried:2005aa,PhysRevLett.86.5870}, ultra-cold atoms \cite{Orzel2386,Appel10960}, and NMR \cite{Jones1166,PhysRevA.82.022330} have been reported.
A potential problem involved in entangled sensors is their fragility to noise. In fact, it has been theoretically proven that an entangled sensor in a Markovian noisy environment, where relaxation is exponential, cannot overcome the standard quantum limit (SQL) \cite{huelga1997improvement}.
On the other hand, there are numerous theoretical predictions indicating that an entangled sensor can outperform a classical sensor under the effect of a time-inhomogeneous noisy environment, which induces a non-exponential decay \cite{PhysRevA.84.012103,chin2012quantum,tanaka2015proposed,RevModPhys.88.021002}. However, there have been no experimental demonstrations yet for the latter case. Therefore, it is very important to study the behavior of entangled sensors under the effect of time-inhomogeneous noisy environments.
In this work, we investigate the behavior of an entangled sensor in a controlled environment using NMR. Although the demonstration does not constitute a formal proof of the quantum-mechanical enhancement of an entangled sensor in a real environment, our achievements provide important information toward understanding the properties of an entangled sensor: (i) As the performance of the entangled sensor strongly depends on the properties of the environment, systematic experimental analysis of the entangled sensor operating under various types of noise is essential for the realization of quantum-enhanced sensing, and therefore our experimental investigation with the NMR model provides insights to assess the practicality of quantum sensors. (ii) Our NMR method can be implemented with commonly available experimental apparatus (which almost every university owns) at room temperature, and therefore experimentalists can use this model as a testbed or a simulator before attempting to construct a real entangled sensor. For example, experimentalists can first use our NMR model to evaluate the pulse sequences that are expected to apply to the entangled sensor in a real environment.
The remainder of the paper is organized as follows. In Section~\ref{sec2}, we closely follow References~\cite{Jones1166,PhysRevA.82.022330} and review how an entangled sensor is simulated using a star topology molecule. We then present a method demonstrating how to prepare a controlled environment following Ref.~\cite{Binho2019}. Note that an engineered noisy environment in Ref.~\cite{Binho2019} is equivalent to a controlled one in this paper. Finally, these two ideas are combined, and we simulate the entangled sensor in a controlled environment. We present experimental results in Section~\ref{sec3}, where the dynamics of the entangled sensor are evaluated in a time-inhomogeneous noisy environment, and a successful application of a dynamical decoupling technique~\cite{RevModPhys.88.041001} applied to the entangled sensor is demonstrated. Finally, Section~\ref{sec4} presents a summary of our findings.
\section{Theory}\label{sec2}
In this section, we describe our strategy to combine two ideas: (i) use of a star-topology molecule as an entangled magnetic sensor, and (ii) controlling the environment that surrounds the sensor.
\subsection{Molecules as a Simulator of an Entangled Magnetic Sensor} \label{TH_M_S} Assume an isolated nuclear spin with state \begin{eqnarray}
|+\rangle = \frac{|0\rangle + | 1\rangle}{\sqrt{2}}, \end{eqnarray}
where $|0\rangle$ and $|1\rangle$ are two eigenstates of the standard Pauli matrix $\sigma_z$. The system is exposed to a magnetic field $B\, \vec{z}$, where $\vec{z}$ is the unit vector along the $z$-axis, for a period $\tau$, and becomes \begin{eqnarray}
|+_\tau \rangle = \frac{|0\rangle + e^{i \gamma_{\rm G} B \,\tau}
| 1\rangle}{\sqrt{2}}, \end{eqnarray} where $\gamma_{\rm G}$ is the gyromagnetic ratio. Therefore, the acquired phase $\gamma_{\rm G} B \, \tau$ can be used to measure $B$. The sensitivity of a set of $N$ spins is proportional to $\sqrt{N}$, which is the SQL~\cite{PhysRevLett.79.3865,Giovannetti:2011aa}.
Now, if we assume that our sensor consists of $N$ spins and that the initial state is entangled, such that \begin{eqnarray}
|+_{\rm ent}\rangle = \frac{|00 \dots 0\rangle
+ | 11 \dots 1\rangle}{\sqrt{2}}, \end{eqnarray} then, this state will evolve to \begin{eqnarray}
|+_{\rm ent, \tau}\rangle = \frac{|00 \dots 0\rangle
+ e^{i N \gamma_{\rm G} B \tau}| 11 \dots 1\rangle}{\sqrt{2}}, \end{eqnarray} after time $\tau$ elapses, and thus the sensitivity is proportional to the number of spins $N$, which is the Heisenberg Limit~\cite{PhysRevLett.79.3865,Giovannetti:2011aa}.
Jones et al.\ demonstrated the above measurement scheme with star-topology molecules, as schematically shown in Fig.~\ref{fig:q_circuit}(a). They employed two molecules, trimethyl phosphite (TMP) \cite{Jones1166} and tetramethylsilane (TMS) \cite{PhysRevA.82.022330}. A TMP (or TMS) molecule consists of a center $^{31}$P ($^{29}$Si) and three (four) methyl groups, and thus the center $^{31}$P ($^{29}$Si) is surrounded by 9 (12) $^1$H spins. The highly symmetric structures of these molecules allow addressing of all surrounding spins (small open circles in Fig.~\ref{fig:q_circuit}(a)) globally, and thus the operations required to measure a magnetic field can be simplified.
In this work, we employed the simplest star topology molecule, which consists of a center spin ($\bigcirc$) and two side spins ($\circ$'s), as shown in Fig.~\ref{fig:q_circuit}(b).
\begin{figure}
\caption{ (a, b, and c) Three different interaction topologies among spins ($\bigcirc, \circ$, and $\bullet$) discussed in this study and (d) quantum circuit for an entangled magnetic field sensor simulation. (a) Sketch of a star-like interaction structure among spins, and (b) the simplest star topology structure. The small open circles ($\circ$'s) in (a) and (b) play the role of entangled sensors. (c) Two-step star topology structure used to conduct an entangled magnetic sensor simulation under a controlled environment. The large open circle ($\bigcirc$) is called the center spin, while the small open circles ($\circ$'s) are the side spins and play the role of an entangled sensor. The six surrounding solid circles ($\bullet$'s) are introduced in order to generate a time-inhomogeneous noisy environment acting on the entangled sensor. We refer to them as the environmental spins. (d) Basic quantum circuit used for measurements. Because of the symmetry of the interaction structure, all $\circ$ spins can be accessed globally. }
\label{fig:q_circuit}
\end{figure}
We take the initial density matrix
$\displaystyle |0\rangle \langle 0 | \otimes \frac{\sigma_0}{2} \otimes \frac{\sigma_0}{2} $ \cite{kondo2016} (see also the Appendix).
The state of the center spin is $|0\rangle \langle 0 |$, while the two side spins are in a fully mixed state. When a magnetic field is applied only to the center spin ($\bigcirc$), i.e., the star-topology structure is not effective, the density matrix becomes \begin{eqnarray} \rho_\bigcirc &=&
\frac{1}{8} {\small \left( \begin{array}{cc}
1 & e^{-i\theta}\\ e^{i\theta} & 1 \end{array} \right) } \otimes \sigma_0 \otimes \sigma_0, \end{eqnarray} after the magnetic field is applied. Here, $\theta = \gamma_{\rm G}B \tau$, and the subscript $\bigcirc$ stands for the case when the center spin is exposed to the field. This case corresponds to non-entangled sensor.
Next, consider the case when the magnetic field is applied to the side spins ($\circ$'s), as shown in Fig.~\ref{fig:q_circuit}(d). This case corresponds to an entangled sensor. In a frame that co-rotates with the center spin, it does not acquire a phase during the period $\tau$. The initial state
$\displaystyle \frac{1}{4} | 0 \rangle \langle 0 | \otimes \sigma_0 \otimes \sigma_0$ can be decomposed to \begin{eqnarray}
\frac{1}{4} \Bigl(
|000\rangle \langle 000|
+ |011\rangle \langle 011| +
|001\rangle \langle 001| + |010\rangle \langle 010| \Bigr), \end{eqnarray} and thus, the final density matrix after the measurement operation is \begin{eqnarray} \label{eq_meas_dm} \rho_\circ &=& {\small \frac{1}{8} \left( \begin{array}{cccccccc}
1 & 0 & 0 & 0 & e^{-2i\theta} & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & e^{2i\theta} \\
e^{2i\theta} & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & e^{-2i\theta} & 0 & 0 & 0 & 1 \\ \end{array} \right) }\nonumber \\ &=& \frac{1}{8} {\small \left( \begin{array}{cc}
1 & e^{-2i\theta}\\ e^{i2\theta} & 1 \end{array} \right) } \otimes
|00\rangle \langle 00| \nonumber \\ &+&\frac{1}{8} {\small \left( \begin{array}{cc}
1 & e^{2i\theta}\\ e^{-i2\theta} & 1 \end{array} \right) } \otimes
|11\rangle \langle 11| \nonumber \\ &+& \frac{1}{8} \left( {\small \begin{array}{cc}
1 & 1\\
1 & 1 \end{array}} \right) \otimes
\Bigl(|01\rangle \langle 01| + |10\rangle \langle 10| \Bigr), \end{eqnarray} where $\circ$ stands for the case when the magnetic field is applied to the side spins. Note that owing to the twice as large phase accumulation ($2\theta$, instead of $\theta$) of the
initial states of $|000\rangle \langle 000|$ and
$|011\rangle \langle 011|$, the sensitivity of the side spins (the entangled sensor) interacting with the magnetic fields becomes twice as large as that of the center spin (non-entangled sensor).
Next, the method of how to detect the acquired phases in the NMR is presented. The state $\rho_k$ ($k = \bigcirc, \circ$) is assumed to develop under the Hamiltonian \begin{eqnarray}
H&=& \omega_0 \frac{\sigma_z}{2} \otimes \sigma_0 \otimes \sigma_0\nonumber\\ &+& J \left(\frac{\sigma_z}{2} \otimes \frac{\sigma_z}{2} \otimes \sigma_0 + \frac{\sigma_z}{2} \otimes \sigma_0 \otimes \frac{\sigma_z}{2} \right), \end{eqnarray} where $\omega_0$ corresponds to a Larmor frequency of the center spin and $J$ is a coupling constant between the center spin and the two side spins. $\omega_0$ is introduced to identify the signal originating from the center spin in the frequency domain signal. The signal from the center spin is \cite{Cory1634,Gershenfeld350,liquidNMRQC} \begin{eqnarray}
S_k(t) &=& {\rm Tr}\left( \left((\sigma_x+i \sigma_y) \otimes \sigma_0 \otimes \sigma_0 \right) e^{-i H t}\rho_ke^{iHt} \right). \end{eqnarray} Therefore, the expected normalized signals are \begin{eqnarray}
S_\bigcirc(t)
&=& \frac{1}{4}e^{-t/T_2} \left( e^{-iJt} + 2 + e^{iJt} \right)\cos(\omega_0 t + \theta), \label{Sl}\\
S_\circ(t) &=& \frac{1}{4}e^{-t/T_2} \left( e^{-iJt-i2\theta} + 2 + e^{iJt+i2\theta} \right)\cos \omega_0 t \label{Ss}, \end{eqnarray} where $T_2$ is a phenomenological transverse relaxation time constant that is introduced for the signal to decrease exponentially. Equation~(\ref{Sl}) corresponds to the signal when the field is applied to the center spin (non-entangled sensor), while Eq.~(\ref{Ss}) corresponds to the case when the field is applied to the side spins (entangled sensor). Note the difference in the position of $\theta$ in $S_k(t)$. The three terms in parentheses in $S_k(t)$ correspond to three peaks that are observed when the $S_k(t)$'s are Fourier transformed (see Fig.~\ref{fig_fid_th}).
\begin{figure}
\caption{ Theoretical spectra calculated from $S_k(t)$ for the case when $J T_2 = 22$. (a) $\theta = 0^\circ$, (b) $\theta = 50^\circ$ for $S_{\bigcirc}(t)$, (c) $\theta = 50^\circ$ for $S_{\circ}(t)$. The frequency difference of these peaks is $J$, and the frequency of the center peaks is $\omega_0$. }
\label{fig_fid_th}
\end{figure}
\subsection{Controlled environment} \label{TH_E_E} Our idea to control the environment is demonstrated in Fig.~\ref{fig:model}. If System~I directly interacts with a Markovian environment, it decays exponentially. If it interacts with a Markovian environment through System~II, it exhibits various decay behaviors, because System~II acts as a memory of the controlled non-Markovian environment formed by System~II and the Markovian environment \cite{kondo2016,Iwakura2017,Kondo2018,kondo2007,PhysRevA.55.2290,PhysRevLett.120.030402}.
The chain of spins ($\circ-\bigcirc-\circ$ in Fig.~\ref{fig:q_circuit}(c)) is regarded as the sensor where the side spins (two $\circ$'s) accumulate the phase under the external field, and this phase is measured via the center spin ($\bigcirc$), as discussed in Sect.~\ref{TH_M_S}. The side spins ($\circ$'s) are surrounded by two sets of three spins (three $\bullet$'s), which we call environmental spins. We regard the side spins (the two $\circ$'s) as two independent System~I's, while we consider the two sets of $\bullet$'s as two System~II's. These environmental spins, with the Markovian environment outside them, act as a time-inhomogeneous noisy environment applied to the side spins (two $\circ$'s) as we previously discussed in Refs.~\cite{kondo2016,Iwakura2017,Kondo2018,kondo2007,PhysRevA.55.2290,PhysRevLett.120.030402}. Note that only the nearest neighbor interactions are assumed important in this discussion. After all, we can control the environment of the sensor ($\circ-\bigcirc-\circ$).
\begin{figure}
\caption{(Color online) Controlled environment. Herein, the Markovian environment indirectly interacts with System~I through System~II. }
\label{fig:model}
\end{figure}
\section{Experiments}\label{sec3} In this section, we describe our approach to simulate an entangled sensor in a controlled environment with a molecule solved in an isotropic liquid. First, we show how to prepare the Markovian controlled environment and then discuss how to simulate the entangled sensor in the controlled environment.
\subsection{ Markovian controlled environment}
Solute molecules in an isotropic liquid move rapidly and are influenced by a strong external magnetic field. Thus, the interactions among nuclear spins in the solute and solvent molecules are averaged out~\cite{Levitt2008}. In other words, the solute molecules are approximated to be isolated systems.
To control a Markovian environment, magnetic impurities, such as Fe(III), are added to the solution. Because the added magnetic impurities move rapidly and randomly, they produce a Markovian environment, which flip-flops the solute molecules' nuclear spins randomly. The flip-flopping rate is proportional to the concentration of the magnetic impurities~\cite{Iwakura2017}. Moreover, we emphasize that the nuclear spins of System~II are more strongly influenced by the magnetic impurities than the inner ones (System~I) because the dipole-dipole interactions are short-range~\cite{Binho2019}.
\subsection{Sample and controlled environment} The two-step star-topology molecule that we employ in this work is 2-propanol solved in acetone d6 with Fe(III) magnetic impurities added. The structure of this molecule is shown in Fig.~\ref{fig_st}(a). The three $^{13}$C spins correspond to $\circ-\bigcirc-\circ$ in Fig.~\ref{fig:q_circuit}(b, c), while the H spins are employed as System~II in Fig.~\ref{fig:model}. We can selectively nullify the H spins by dynamical decoupling techniques~\cite{Levitt2008,RevModPhys.88.041001}, as shown in Fig.~\ref{fig_st}(a, b, c): (a) without decoupling, (b) selective decoupling of the H spin attached to the center $^{13}$C (hereafter, referred to as the selective-decoupling case), and (c) full decoupling of all H spins (hereafter, referred to as the full-decoupling case). This implies that the behavior of an entangled magnetic sensor in three different controlled environments can be studied. Note that the interaction topology of Fig.~\ref{fig:q_circuit}(c) can be realized in the case of Fig.~\ref{fig_st}(b). The obtained spectra shown in Fig.~\ref{fig_sp} exhibit clear differences according to the interaction topology differences presented in Fig.~\ref{fig_st}(a, b, c).
\begin{figure}
\caption{(Color online) Sketch of a 2-propanol molecule in acetone-d6 with added magnetic impurities (Fe(III) ions). The chain of the $^{13}$C spins is surrounded by the H spins. These H spins can be selectively nullified by decoupling techniques, which provide three different controlled environments. The gray areas correspond to the System~II's and Markovian environments. We consider three cases: (a) without decoupling, (b) selective decoupling of the H spin attached to the center $^{13}$C, and (c) full decoupling of all H spins. }
\label{fig_st}
\end{figure}
\begin{figure}
\caption{ Spectra of the $^{13}$C spins in different decoupling conditions: (a) without decoupling, (b) selective decoupling by applying a small continuous RF excitation, whose frequency is the Larmor frequency of HC (the H spin attached to CC), and (c) full decoupling by applying the WALTZ sequence to all H spins. The sample is 0.41~M ${}^{13}$C-labeled 2-propanol in acetone-d6 with 12~mM of Fe(III)acac. The peaks at 62.6~ppm are originated from the center $^{13}$C of the chain, while those at 25.5~ppm are originated from the side $^{13}$C's. The spectra in (a) and (b) are magnified three times.
}
\label{fig_sp}
\end{figure}
The Hamiltonian governing the nuclear spin dynamics of 2-propanol is given as \begin{eqnarray} \label{eq_mol_ham}
{\mathcal H} &=& \sum_{j} \omega_0^{(j)} \frac{\sigma_z^{(j)}}{2}
+ \sum_{j<k} J^{(j,k)} \frac{\sigma_z^{(j)}\otimes\sigma_z^{(k)}}{4}, \end{eqnarray}
because $|\omega_{0}^{(j)} -\omega_{0}^{(k)}| \gg J^{(j,k)}$;
$\omega_{0}^{(j)}$ is the isotropic chemically shifted Larmor frequency of the $j$'th spin, and $J^{(j,k)}$ represents the interaction strength between the $j$'th and $k$'th spins~\cite{Levitt2008,Kondo2018}. $\omega_{0}^{(j)}$ and $J^{(j,k)}$ were measured from the spectra of a sample without magnetic impurities and are summarized in Table~\ref{H_parameters}.
\begin{table}[h] \caption{\label{H_parameters} $\omega_{0}^{(j)}$ and $J^{(j,k)}$ are summarized. We label the spins as CC (the center $^{13}$C in the C spin chain), CSs (the two side $^{13}$C's in the C spin chain), HC (the H spin attached to CC) and HSs (the H spins attached to a CS). Diagonal elements are the $\omega_{0}^{(j)}$'s in ppm, while the off-diagonal elements are those of $J^{(j,k)}$ in rad/s. NR implies that these values are too low to be measured reliably. } \begin{center} \small\addtolength{\tabcolsep}{0.pt}
\begin{tabular}[t]{|c|c|c|c|c|} \hline
\backslashbox{$j$}{$k$} &CC & CSs & HC & HSs \\ \hline CC &62.6 ppm & $ 2\pi \cdot 38.4$ rad/s & $2\pi\cdot 140$ rad/s & $2\pi\cdot 4.4$ rad/s \\ \hline CSs & & 25.5 ppm & NR &$2\pi \cdot124$ rad/s\\ \hline HC & & & 3.78 ppm & NR \\ \hline HSs &&&& 1.21 ppm \\ \hline \end{tabular} \end{center} \end{table}
\subsection{Simulation of Entangled Sensor} We implemented the quantum circuit shown in Fig.~\ref{fig:q_circuit}(d) under standard NMR pulse sequences~\cite{liquidNMRQC}. The rotation operations applied to CC are \begin{eqnarray}
R_{\rm C}(\phi,\theta) &=& R(\phi,\theta)\otimes \sigma_0 \otimes \sigma_0, \\
Z_{\rm C}(\theta) &=& Z(\theta)\otimes \sigma_0 \otimes \sigma_0, \end{eqnarray} where $ R(\phi,\theta)= e^{-i \theta(\sigma_x \cos \phi + \sigma_y\sin \phi)/2}$ and $Z(\theta) = e^{-i \theta \sigma_z/2}$. $\theta $ in $R(\theta,\phi)$ is the rotation angle and $\phi$ defines the rotation axis in the $xy$-plane from the $x$-axis, while $\theta$ in $Z(\theta)$ is the rotation angle about
the $z$-axis. The rotations that operate on the CSs can be implemented simultaneously because of the symmetry of the molecular structure and are defined as \begin{eqnarray}
R_{\rm S}(\phi,\theta) &=& \sigma_0 \otimes R(\phi,\theta)\otimes R(\phi,\theta), \\
Z_{\rm S}(\theta) &=& \sigma_0 \otimes Z(\theta)\otimes Z(\theta). \end{eqnarray} The Hadamard gate on CC was effectively implemented with an $R_{\rm C}(\pi/2,-\pi/2)$, while a pseudo-CNOT gate was realized as follows: \begin{eqnarray}
{\rm CNOT} &=& e^{ -i \frac{\pi}{4}} Z_{\rm C}(-\frac{\pi}{2}) Z_{\rm S}(-\frac{\pi}{2}) R_{\rm C}(0, \frac{\pi}{2}) U_{\rm E} R_{\rm S}(\frac{\pi}{2}, \frac{\pi}{2}) \nonumber \\ &=& {\small \left( \begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & i \\
0 & 0 & 0 & 0 & 0 & 0 & i & 0 \\
0 & 0 & 0 & 0 & 0 & i & 0 & 0 \\
0 & 0 & 0 & 0 & i & 0 & 0 & 0 \\ \end{array} \right) }, \end{eqnarray} where $U_{\rm E} = e^{-\pi(\sigma_z \otimes \sigma_z \sigma_0 + \sigma_z \otimes \sigma_0 \sigma_z)/4}$. $U_{\rm E}$ was implemented by waiting for a period of $\displaystyle \frac{n}{\Delta \omega_0}$, where $\Delta \omega_0$ is the Larmor frequency difference between CC and CSs. $n$ is an integer and is selected so that $ \displaystyle \frac{n}{\Delta \omega_0} \approx \frac{\pi}{J^{\rm (CC, CSs)}}$ (see Table~\ref{H_parameters}). All $Z_k(\theta)$'s ($k = $ C or S) were virtually implemented by controlling the $\phi$'s in the $R_k(\phi, \theta)$'s ($k = $ C or S)~\cite{Kondo2006}. We employed jump-and-return pulses~\cite{PhysRevA.74.052324} to realize $R_k(\phi, \theta)$ with concatenated composite pulses (reduced CinBB)~\cite{doi:10.7566/JPSJ.82.014004,SR_Comp_Pulse_2020} to reduce any errors caused by imperfect pulses.
Magnetic field sensing is equivalent to measuring the phase difference between the initial and final state of CC, as shown in Fig.~\ref{fig:q_circuit}. Therefore, we simulated the magnetic field by applying a $Z_k(\theta)$-rotation, \begin{align*} &\textbf{field on CC:}\\ &\left(R_{\rm C}(-\frac{\pi}{2}, \frac{\pi}{2})-Z_{\rm C}(\theta)\right)- {\rm CNOT}-(\frac{\tau}{2}-R_{\rm S}(0,\pi)-\frac{\tau}{2})-{\rm CNOT}, \\ &\textbf{field on CSs:}\\ &R_{\rm C}(-\frac{\pi}{2},\frac{\pi}{2})-\left({\rm CNOT}-Z_{\rm S}(\theta)\right)- (\frac{\tau}{2}-R_{\rm S}(0,\pi)-\frac{\tau}{2}) - {\rm CNOT}, \end{align*} where $(\tau/2-R_{\rm S}(0,\pi)-\tau/2)$ is the period when the entangled sensor acquires the phase $\theta = \gamma_{\rm G} B \tau$ in real measurements. In our simulations, the ``entangled magnetic sensor'' (CSs) is active in the controlled environment only during this period. $R_{\rm S}(0, \pi)$ in the middle of this period was added to nullify the time development caused by the interaction between CC and CSs. We started from the thermal state and observed only CC in our simulations.
We first demonstrate the ``entanglement-enhanced'' phase sensitivity~\cite{Jones1166,Liu:2015aa,Knott:2016aa} in the full-decoupling case (Fig.~\ref{fig_st}(c)). A sample consisting of 0.41~M ${}^{13}$C-labeled 2-propanol solved in acetone-d6 with 12~mM of Fe(acac) as a magnetic impurity was used. The $T_1$'s of $^{13}$C of this sample were measured to be 1.3~s, while the $T_1$'s of all H spins were approximately 100~ms. Figure~\ref{fig:p_meas} shows the spectra of CC as a function of $\theta$ (the ``field strength'') at $\tau = 3.4$~ms. When the ``field'' was applied to CC (Fig. \ref{fig:p_meas} (upper), the non-entangled sensor), the three peaks acquired the same amount of phase. These phases were the same as $\theta$ given by $Z_{\rm C}(\theta)$ within an acceptable experimental error range. Note the good agreement between the measured and calculated spectra. On the other hand, the acquired phases were depending on peaks when the ``field'' was applied to CSs (Fig. \ref{fig:p_meas} (lower), entangled sensor). The center peak does not acquire any phase, as one can see from the fact that it is symmetric, regardless of the $\theta$ values. The two side peaks acquire $\pm 2 \theta $; the $+$ sign is for the left peak and the $-$ sign is for the right. Again, we obtain reasonably good agreement between the measured and calculated spectra, although the measured spectra are not as sharp as the calculated spectra. The case of the non-entangled sensor appears to result in better agreement between the experiments and theory compared to that of the entangled sensor. We explain this result as follows. The phase (``field'') information is stored in the entangled spins ($\circ$'s) in the latter case and we therefore expect that this information is more fragile than that in the former case because of the fragility of the entangled state. All calculated spectra were obtained with the measured coupling constant of $J^{(\rm CC, CSs)}=2\pi \cdot 38.4$~rad/s and two fitting parameters: one being $T_2 = 0.1$~s and the other being the amplitude. The value of $T_2$ is quite reasonable based on the FID signal measurements. Therefore, the amplitude is the only fitting parameter required.
\begin{figure}
\caption{(Color online) Spectra as functions of $\theta$ (the strength of the ``magnetic field'') when all H spins were decoupled (full-decoupling case). The (red) solid lines are measured spectra, while the grey dots are the calculated theoretical spectra, such as in Fig.~\ref{fig_fid_th}. The left-most spectra is the reference without a ``magnetic field''. The upper spectra show cases when ``magnetic field''s were applied to CC (non-entangled sensor), while the lower ones correspond to CSs (the entangled sensor). The frequencies of the center peaks are 62.6~ppm and the frequency differences among the peaks are $2\pi\cdot 38.4$~rad/s, as listed in Table~\ref{H_parameters}. }
\label{fig:p_meas}
\end{figure}
Next, we show that the ``magnetic field sensing'' was affected by ``noise'' generated by the controlled environment and that the ``noise'' can be suppressed by a dynamical decoupling technique (XY-8~\cite{RevModPhys.88.041001}). Here, HS (corresponding to a $\bullet$ in Fig.~\ref{fig:q_circuit}(c)) and magnetic impurities combine to generate a time-inhomogeneous noisy environment acting on the entangled sensor (that corresponds to the $\circ$'s in Fig.~\ref{fig:q_circuit}(c)). We can nullify the time-inhomogeneous noisy environment by effectively removing the HSs with a decoupling technique. The experimental details are as follows. We consider two cases: the full-decoupling case (Fig.~\ref{fig_st}(c)) and the selective-decoupling case (Fig.~\ref{fig_st}(b)). In the full-decoupling case, the magnetic impurities produce weak effects on CC and CSs~\cite{Binho2019}, and thus we can state that ``magnetic field sensing'' was performed under an approximately noiseless enviroment in the short time scale of less than 50~ms in the experiment. This approximation was confirmed by the fact that the signal exhibited little decay in this time scale (see Fig.~\ref{fig:sen_rel}(a)). In the selective decoupling case, CSs should be affected by the time-inhomogeneous noisy environment formed by the HSs and the magnetic impurities. This noisy environment was also confirmed by the fact that the signal decays quickly, as shown in Fig.~\ref{fig:sen_rel}(b). The relaxation constant is approximately 30~ms. The spectra in Fig.~\ref{fig:sen_rel}(c) were measured when the HSs were not decoupled (the same as (b)), but the XY-8 sequence was applied to CC and CSs simultaneously. The signals decay much more slowly than those in (b), which indicates that the dynamical decoupling was effective in protecting the entangled sensor. When a dynamical decoupling technique is applied to a sensor, it cannot detect the DC component, but can measure the AC one, whose frequency is determined by the decoupling technique~\cite{RevModPhys.89.035002}. Therefore, it is possible to construct an entanglement-enhanced AC magnetic field sensor under time-inhomogeneous noise, as theoretically predicted in Refs.\ \cite{PhysRevA.84.012103,chin2012quantum,tanaka2015proposed,RevModPhys.88.021002}.
\begin{figure}
\caption{ Measurements in various noisy environments as a function of measurement time $\tau(=3.44~{\rm ms} \times n)$. (a) ``without noise'' by decoupling all H spins (full-decoupling case), (b) ``under noise'' without decoupling HSs (selective-decoupling), and (c) ``under noise,'' where the noise is suppressed by a dynamical decoupling technique (XY-8) during measurements. The frequency of the center peaks is 62.6~ppm and the frequency differences among the peaks are $2\pi\cdot 38.4$~rad/s, as listed in Table~\ref{H_parameters}.
}
\label{fig:sen_rel}
\end{figure}
\section{Summary}\label{sec4} We have successfully modeled an entangled magnetic field sensor in various noisy environments using NMR techniques. In our model, a sensor is a star-topology molecule, 2-propanol, solved in acetone-d6, and the magnetic field is simulated by rotational pulse sequences acting on the sensor. The environment surrounding a sensor can be controlled by adding Fe(III) as an impurity in the solvent and by selectively decoupling H spins in the 2-propanol molecule. We have demonstrated the entanglement-enhanced phase sensitivity and have discussed its enhancement mechanism. We further demonstrate that magnetic field sensing is affected by noise. Importantly, we have demonstrated that when the noise is time-inhomogeneous, its effect can be suppressed by a dynamical decoupling technique during the entanglement-enhanced magnetic field sensing period.
\section*{Acknowledgments} This work was supported by CREST(JPMJCR1774), JST. This work is also supported by Leading Initiative for Excellent Young Researchers MEXT Japan and MEXT KAKENHI (Grant No. 15H05870).
\appendix \section{} In previous work~\cite{Binho2019}, we systematically studied the three cases when the environments consisted of 1, 3, and 12 spins $+$ a Markovian environment generated by magnetic impurities. We add another case study here involving a
6 spins $+$ Markovian environment case by using 2-propanol solved in acetone d6.
The longitudinal relaxation times, $T_1$'s of the $^{13}$C spins of a 0.41~M {}$^{13}$C-labeled 2-propanol sample solved in acetone d-6 without magnetic impurities were measured to be 20~s (CC) and 8~s (CSs). Therefore, within a time scale much shorter than these $T_1$'s, the $^{13}$C chain in the 2-propanol molecules can be approximated as isolated systems. We added magnetic impurities (Fe(III) acac) and prepared four samples, as listed in Table~\ref{FID_tc}.
\begin{table}[h] \caption{\label{FID_tc} Measured $T_1$'s and $T_2$'s of CC, and $T_1$ of HSs are summarized. $C_{\rm m}$: concentration of the magnetic impurity (Fe(III)acac), $T_1$: longitudinal relaxation time constant of CC, $T_2^{(\rm f)}$: relaxation time constant of the signal in the full-decoupling case, $T_2^{(\rm s)}$: relaxation time constant of the signal in the selective-decoupling case, and $T_1$(HSs): longitudinal relaxation time constant of HSs' spins. } \begin{center} \small\addtolength{\tabcolsep}{-0.5pt}
\begin{tabular}[t]{|c|c|c|c|c|c|c|c|} \hline \multirow{2}{*}{Sample} &$C_{\rm m}$ &$T_1$ & $T_1 \cdot C_{\rm m}$ &$T_2^{(\rm f)}$ & $T_2^{(\rm f)}\cdot C_{\rm m}$& $T_2^{(\rm s)}$& $T_1$(HSs)\\ &(mM)&(s)&(mM$\cdot$s)&(ms)&(M$\cdot$s)&(ms)&(ms) \\ \hline 1 &12 & 1.3 & 15 & $3.0\times 10^2$ & $3.5 \times 10^{-3}$& $3.0 \times 10$ & 93\\ \hline 2 &26 & 0.64 & 17 & $1.0 \times 10^2$ & $2.6 \times 10^{-3}$& $3.9\times 10$ & 43\\ \hline 3 &47 & 0.36& 17& $9.9\times 10$ & $4.7 \times 10^{-3}$ & $3.8\times 10$ & 24\\ \hline 4 &94& 0.17& 16& $6.4\times 10$ & $6.0 \times 10^{-3}$& $3.9\times 10$ & 17\\ \hline \end{tabular} \end{center} \end{table}
In the full-decoupling case, a small but not negligible direct influence of the Markovian environment on CC should be observed. The $T_1$'s of CC in Table~\ref{FID_tc} are inversely proportional to the magnetic impurity concentration $C_{\rm m}$, which implies that in this case, $T_1$ is determined by the impurity concentration~\cite{Iwakura2017}. On the other hand, in the selective-decoupling case, the interaction between CC and the Markovian environment through the HSs (System~II) should be added, although it is expected to be small. Therefore, we obtain the controlled environment, which consists of 6 spins + Markovian environment, which causes a non-exponential decay of CC~\cite{Binho2019}. We note, however, that the large interactions between CC and CSs ($J^{\rm (CC, CSs)}=2\pi \cdot 34$~rad/s) compared with those between CC and the HSs ($J^{\rm (CC, HSs)}=2\pi \cdot 4.4$~rad/s) prevent direct observation of the subtle non-exponential dynamics.
\begin{figure}
\caption{(Color online) FID signals of Samples~1, 2, 3, and 4 in Table~\ref{FID_tc}. The initial states were
$\displaystyle |+\rangle \langle + | \otimes
\left( |01\rangle \langle 01 |
+ |10\rangle \langle 10 | \right). $ The real parts of the FID signals are shown in red, while the imaginary are shown in black. The full-decoupling cases are shown in the left panels and selective decoupling cases are presented in the right panels. The black dashed curves in the left panels are exponential fittings to the real parts of the FID signals. The green (blue) curves in the right panels are the calculated real (imaginary) parts of the FID signals \cite{Binho2019}. The blue curves overlap the experimental data and thus are hardly visible. }
\label{fig:fid}
\end{figure}
To successfully observe the above subtle non-exponential dynamics, let us re-examine the thermal state $\rho_{\rm th}$ of the three $^{13}$C's, which is~\cite{Cory1634,Gershenfeld350,liquidNMRQC} \begin{eqnarray}
\label{eq_thermal_state} \rho_{\rm th} &\approx &
\underbrace{\frac{ \sigma_0 + \epsilon |0\rangle \langle 0 |}{2}}_{\rm CC} \otimes \underbrace{\frac{\sigma_0}{2}\otimes\frac{\sigma_0}{2}}_{\rm CSs} \nonumber \\ &+& \underbrace{\frac{ \sigma_0}{2}}_{\rm CC} \otimes \underbrace{
\frac{ \sigma_0 + \epsilon |0\rangle \langle 0 |}{2} \otimes\frac{\sigma_0}{2}}_{\rm CSs} \nonumber\\ &+& \underbrace{\frac{ \sigma_0}{2}}_{\rm CC} \otimes \underbrace{ \frac{\sigma_0}{2} \otimes
\frac{ \sigma_0 + \epsilon |0\rangle \langle 0 |}{2}}_{\rm CSs} \end{eqnarray} Here, $\epsilon \sim 10^{-5}$ in NMR measurements. When we observe only CC, $\rho_{\rm th}$ is equivalent to \begin{eqnarray*}
\rho_{\rm th} \approx \frac{1}{8}
\left( \sigma_0 + \epsilon |0\rangle \langle 0 | \right)
\otimes \left( |0\rangle\langle 0|+ |1\rangle\langle 1|\right)
\otimes \left( |0\rangle\langle 0|+ |1\rangle\langle 1|\right). \end{eqnarray*} Moreover, $\sigma_0$ of CC is not observable in NMR and thus $\rho_{\rm th}$ can be re-normalized as \begin{eqnarray*}
\rho_{\rm th} \approx \frac{1}{8}
\left( |0\rangle \langle 0 | \right)
\otimes \left( |00\rangle\langle 00| + |01\rangle\langle 01|
+|10\rangle \langle 10| + |11\rangle \langle 11| \right). \end{eqnarray*}
The interaction effects on CC from the $ |01\rangle\langle 01|$ and
$|10\rangle \langle 10|$ states of CSs cancel each other out. Thus if we can prepare \begin{eqnarray} \label{eq_rho_i}
\rho_{\rm i} &=& |+\rangle \langle + | \otimes \left(
|01\rangle\langle 01|+|10\rangle \langle 10| \right), \end{eqnarray} we can observe the subtle non-exponential dynamics discussed previously.
This $\rho_{\rm i}$ can be prepared with a standard NMR technique called a soft pulse~\cite{Levitt2008}.
The results are summarized in Fig.~\ref{fig:fid}. We can successfully observe the exponential decays in the full decoupling cases (left panels), while the non-exponential decay dynamics are observed in the selective decoupling cases (right panels). We also calculated the decay dynamics from the data (summarized in Table~\ref{FID_tc}) as in our previous work~\cite{Binho2019}, which are plotted as green (blue) solid curves in the right panels in Fig.~\ref{fig:fid}. By taking into account that there are no fitting parameters except for the amplitude, we believe that the calculated dynamics reproduce the observations relatively well. However, the reproducibility may not be as good as in our preceding work~\cite{Binho2019}, which may be caused by the imperfect soft pulses employed for preparing the initial state or by the error in parameter determination summarized in Table~\ref{FID_tc}, especially regarding the interaction strength between CC and the HSs ($J^{\rm (CC, HSs)}$).
\end{document} | arXiv |
# Supervised learning: algorithms and techniques
Supervised learning is a type of machine learning where the model is trained on a labeled dataset. The goal is to learn a mapping from input data to output data, where the output data is based on the input data and a predefined function.
For example, if we have a dataset of house prices with features like the number of bedrooms, the square footage, and the location, we can train a supervised learning model to predict the house price based on these features.
There are several algorithms and techniques used in supervised learning. Some common ones include linear regression, decision trees, support vector machines, and neural networks.
## Exercise
Instructions:
- Write a Python function that trains a linear regression model using the `scikit-learn` library.
- Use the `boston` dataset from `scikit-learn` as the training data.
- Split the dataset into training and testing sets.
- Train the model and make predictions.
- Calculate the mean squared error of the model.
### Solution
```python
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error
from sklearn.datasets import load_boston
# Load the dataset
boston = load_boston()
X = boston.data
y = boston.target
# Split the dataset into training and testing sets
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
# Train the model
model = LinearRegression()
model.fit(X_train, y_train)
# Make predictions
y_pred = model.predict(X_test)
# Calculate the mean squared error
mse = mean_squared_error(y_test, y_pred)
print("Mean Squared Error:", mse)
```
# Linear regression: theory and implementation
Linear regression is a supervised learning algorithm that models the relationship between a dependent variable and one or more independent variables. It assumes a linear relationship between the variables, and it estimates the coefficients of the dependent variable based on the data.
For example, in a linear regression model, the equation is:
$$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \cdots + \beta_n x_n$$
where $y$ is the dependent variable, $\beta_i$ are the coefficients, and $x_i$ are the independent variables.
In Python, you can use the `scikit-learn` library to implement linear regression. The library provides a `LinearRegression` class that you can use to train and make predictions with a linear regression model.
## Exercise
Instructions:
- Write a Python function that trains a linear regression model using the `scikit-learn` library.
- Use the `boston` dataset from `scikit-learn` as the training data.
- Split the dataset into training and testing sets.
- Train the model and make predictions.
- Calculate the mean squared error of the model.
### Solution
```python
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error
from sklearn.datasets import load_boston
# Load the dataset
boston = load_boston()
X = boston.data
y = boston.target
# Split the dataset into training and testing sets
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
# Train the model
model = LinearRegression()
model.fit(X_train, y_train)
# Make predictions
y_pred = model.predict(X_test)
# Calculate the mean squared error
mse = mean_squared_error(y_test, y_pred)
print("Mean Squared Error:", mse)
```
# Decision trees: concepts and algorithms
Decision trees are a type of supervised learning algorithm that models the decision-making process. They are used for both classification and regression tasks. A decision tree consists of a set of nodes connected by edges, where each node represents a decision or a prediction.
For example, a decision tree can be used to classify emails as spam or not spam based on the presence of certain words or phrases in the email.
There are several algorithms for building decision trees, such as ID3, C4.5, and CART. These algorithms use different techniques to split the data at each node, such as information gain, Gini impurity, or chi-squared test.
## Exercise
Instructions:
- Write a Python function that trains a decision tree classifier using the `scikit-learn` library.
- Use the `iris` dataset from `scikit-learn` as the training data.
- Split the dataset into training and testing sets.
- Train the model and make predictions.
- Calculate the accuracy of the model.
### Solution
```python
from sklearn.model_selection import train_test_split
from sklearn.tree import DecisionTreeClassifier
from sklearn.metrics import accuracy_score
from sklearn.datasets import load_iris
# Load the dataset
iris = load_iris()
X = iris.data
y = iris.target
# Split the dataset into training and testing sets
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
# Train the model
model = DecisionTreeClassifier()
model.fit(X_train, y_train)
# Make predictions
y_pred = model.predict(X_test)
# Calculate the accuracy
accuracy = accuracy_score(y_test, y_pred)
print("Accuracy:", accuracy)
```
# Unsupervised learning: clustering algorithms
Unsupervised learning is a type of machine learning where the model is trained on an unlabeled dataset. The goal is to discover patterns or structures in the data without any prior knowledge.
For example, unsupervised learning can be used to group customers based on their purchasing behavior or to identify similar items in a dataset.
There are several clustering algorithms used in unsupervised learning, such as K-means, hierarchical clustering, and DBSCAN. These algorithms group the data into clusters based on their similarity or distance from each other.
## Exercise
Instructions:
- Write a Python function that performs K-means clustering on a dataset using the `scikit-learn` library.
- Use the `iris` dataset from `scikit-learn` as the training data.
- Set the number of clusters to 3.
- Calculate the silhouette score of the clustering.
### Solution
```python
from sklearn.cluster import KMeans
from sklearn.metrics import silhouette_score
from sklearn.datasets import load_iris
# Load the dataset
iris = load_iris()
X = iris.data
# Perform K-means clustering
model = KMeans(n_clusters=3)
model.fit(X)
# Calculate the silhouette score
score = silhouette_score(X, model.labels_)
print("Silhouette Score:", score)
```
# Clustering algorithms: K-means, hierarchical clustering
K-means is a popular clustering algorithm that aims to partition the data into K clusters. It iteratively assigns data points to the nearest cluster centroids and recalculates the centroids.
For example, K-means can be used to group customers based on their purchasing behavior.
Hierarchical clustering is another clustering algorithm that builds a tree-like structure of clusters. It starts with each data point as a separate cluster and iteratively merges the closest clusters until a stopping criterion is met.
## Exercise
Instructions:
- Write a Python function that performs hierarchical clustering on a dataset using the `scikit-learn` library.
- Use the `iris` dataset from `scikit-learn` as the training data.
- Set the linkage method to 'ward'.
- Calculate the silhouette score of the clustering.
### Solution
```python
from sklearn.cluster import AgglomerativeClustering
from sklearn.metrics import silhouette_score
from sklearn.datasets import load_iris
# Load the dataset
iris = load_iris()
X = iris.data
# Perform hierarchical clustering
model = AgglomerativeClustering(n_clusters=None, linkage='ward')
model.fit(X)
# Calculate the silhouette score
score = silhouette_score(X, model.labels_)
print("Silhouette Score:", score)
```
# Model evaluation and validation
Model evaluation and validation are essential steps in the machine learning process. They help to assess the performance of a model and avoid overfitting or underfitting.
For example, you can use cross-validation to estimate the performance of a model on unseen data.
There are several evaluation metrics and validation techniques used in machine learning, such as accuracy, precision, recall, F1 score, mean squared error, and hyperparameter tuning.
## Exercise
Instructions:
- Write a Python function that evaluates the performance of a linear regression model using the mean squared error.
- Use the `boston` dataset from `scikit-learn` as the training and testing data.
- Train the model and make predictions.
- Calculate the mean squared error of the model.
### Solution
```python
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error
from sklearn.datasets import load_boston
# Load the dataset
boston = load_boston()
X = boston.data
y = boston.target
# Split the dataset into training and testing sets
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
# Train the model
model = LinearRegression()
model.fit(X_train, y_train)
# Make predictions
y_pred = model.predict(X_test)
# Calculate the mean squared error
mse = mean_squared_error(y_test, y_pred)
print("Mean Squared Error:", mse)
```
# Feature engineering and selection
Feature engineering is the process of creating new features from the existing ones in the dataset. This can help improve the performance of a machine learning model.
For example, you can create a new feature by combining two existing features, such as the product or the ratio of two features.
Feature selection is the process of selecting the most relevant features from the dataset. This can help reduce the dimensionality of the data and improve the performance of a machine learning model.
## Exercise
Instructions:
- Write a Python function that performs feature selection using the `scikit-learn` library.
- Use the `boston` dataset from `scikit-learn` as the training and testing data.
- Train a linear regression model with and without feature selection.
- Calculate the mean squared error of the models.
### Solution
```python
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error
from sklearn.datasets import load_boston
from sklearn.feature_selection import SelectKBest
# Load the dataset
boston = load_boston()
X = boston.data
y = boston.target
# Split the dataset into training and testing sets
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
# Perform feature selection
selector = SelectKBest(k=5)
X_train_selected = selector.fit_transform(X_train, y_train)
X_test_selected = selector.transform(X_test)
# Train the model with feature selection
model_selected = LinearRegression()
model_selected.fit(X_train_selected, y_train)
# Make predictions
y_pred_selected = model_selected.predict(X_test_selected)
# Calculate the mean squared error of the model with feature selection
mse_selected = mean_squared_error(y_test, y_pred_selected)
print("Mean Squared Error with Feature Selection:", mse_selected)
# Train the model without feature selection
model = LinearRegression()
model.fit(X_train, y_train)
# Make predictions
y_pred = model.predict(X_test)
# Calculate the mean squared error of the model without feature selection
mse = mean_squared_error(y_test, y_pred)
print("Mean Squared Error without Feature Selection:", mse)
```
# Introduction to deep learning
Deep learning is a subfield of machine learning that focuses on neural networks with many layers. These networks can learn complex patterns and representations from large amounts of data.
For example, deep learning has been used to achieve state-of-the-art results in image recognition, speech recognition, and natural language processing.
There are several types of deep learning models, such as convolutional neural networks (CNNs) for image recognition and recurrent neural networks (RNNs) for sequence data.
## Exercise
Instructions:
- Write a Python function that trains a simple deep learning model using the `tensorflow` library.
- Use the `mnist` dataset from `tensorflow.keras.datasets` as the training data.
- Train the model and make predictions.
- Calculate the accuracy of the model.
### Solution
```python
import tensorflow as tf
from tensorflow.keras.datasets import mnist
from tensorflow.keras.models import Sequential
from tensorflow.keras.layers import Dense, Flatten
from tensorflow.keras.optimizers import Adam
# Load the dataset
(X_train, y_train), (X_test, y_test) = mnist.load_data()
# Normalize the data
X_train, X_test = X_train / 255.0, X_test / 255.0
# Create a simple deep learning model
model = Sequential([
Flatten(input_shape=(28, 28)),
Dense(128, activation='relu'),
Dense(10, activation='softmax')
])
# Compile the model
model.compile(optimizer=Adam(), loss='sparse_categorical_crossentropy', metrics=['accuracy'])
# Train the model
model.fit(X_train, y_train, epochs=5, validation_split=0.2)
# Make predictions
y_pred = model.predict(X_test)
# Calculate the accuracy
accuracy = model.evaluate(X_test, y_test)[1]
print("Accuracy:", accuracy)
```
# Neural networks: implementation and training
Neural networks are a type of deep learning model that is inspired by the structure and function of the human brain. They consist of interconnected nodes or neurons organized in layers.
For example, a neural network can be used to classify images of handwritten digits or to translate text from one language to another.
In Python, you can use the `tensorflow` library to implement and train neural networks. The library provides a `Sequential` class that you can use to create and train neural networks.
## Exercise
Instructions:
- Write a Python function that trains a simple neural network using the `tensorflow` library.
- Use the `mnist` dataset from `tensorflow.keras.datasets` as the training data.
- Train the model and make predictions.
- Calculate the accuracy of the model.
### Solution
```python
import tensorflow as tf
from tensorflow.keras.datasets import mnist
from tensorflow.keras.models import Sequential
from tensorflow.keras.layers import Dense, Flatten
from tensorflow.keras.optimizers import Adam
# Load the dataset
(X_train, y_train), (X_test, y_test) = mnist.load_data()
# Normalize the data
X_train, X_test = X_train / 255.0, X_test / 255.0
# Create a simple neural network
model = Sequential([
Flatten(input_shape=(28, 28)),
Dense(128, activation='relu'),
Dense(10, activation='softmax')
])
# Compile the model
model.compile(optimizer=Adam(), loss='sparse_categorical_crossentropy', metrics=['accuracy'])
# Train the model
model.fit(X_train, y_train, epochs=5, validation_split=0.2)
# Make predictions
y_pred = model.predict(X_test)
# Calculate the accuracy
accuracy = model.evaluate(X_test, y_test)[1]
print("Accuracy:", accuracy)
```
# Convolutional neural networks for image recognition
Convolutional neural networks (CNNs) are a type of neural network that is specifically designed for image recognition tasks. They use convolutional layers to learn local patterns and features from the input images.
For example, CNNs have been used to achieve state-of-the-art results in image classification tasks, such as recognizing objects in images.
In Python, you can use the `tensorflow` library to implement convolutional neural networks. The library provides a `Sequential` class that you can use to create and train CNNs.
## Exercise
Instructions:
- Write a Python function that trains a convolutional neural network using the `tensorflow` library.
- Use the `mnist` dataset from `tensorflow.keras.datasets` as the training data.
- Train the model and make predictions.
- Calculate the accuracy of the model.
### Solution
```python
import tensorflow as tf
from tensorflow.keras.datasets import mnist
from tensorflow.keras.models import Sequential
from tensorflow.keras.layers import Conv2D, MaxPooling2D, Flatten, Dense
from tensorflow.keras.optimizers import Adam
# Load the dataset
(X_train, y_train), (X_test, y_test) = mnist.load_data()
# Reshape the data
X_train = X_train.reshape(-1, 28, 28, 1)
X_test = X_test.reshape(-1, 28, 28, 1)
# Create a convolutional neural network
model = Sequential([
Conv2D(32, kernel_size=(3, 3), activation='relu', input_shape=(28, 28, 1)),
MaxPooling2D(pool_size=(2, 2)),
Flatten(),
Dense(128, activation='relu'),
Dense(10, activation='softmax')
])
# Compile the model
model.compile(optimizer=Adam(), loss='sparse_categorical_crossentropy', metrics=['accuracy'])
# Train the model
model.fit(X_train, y_train, epochs=5, validation_split=0.2)
# Make predictions
y_pred = model.predict(X_test)
# Calculate the accuracy
accuracy = model.evaluate(X_test, y_test)[1]
print("Accuracy:", accuracy)
```
# Recurrent neural networks for sequence data
Recurrent neural networks (RNNs) are a type of neural network that is specifically designed for sequence data, such as time series data or text data. They use recurrent layers to learn the dependencies between successive elements in the sequence.
For example, RNNs have been used to achieve state-of-the-art results in natural language processing tasks, such as language translation and sentiment analysis.
In Python, you can use the `tensorflow` library to implement recurrent neural networks. The library provides a `Sequential` class that you can use to create and train RNNs.
## Exercise
Instructions:
- Write a Python function that trains a recurrent neural network using the `tensorflow` library.
- Use the `imdb` dataset from `tensorflow.keras.datasets` as the training data.
- Train the model and make predictions.
- Calculate the accuracy of the model.
### Solution
```python
import tensorflow as tf
from tensorflow.keras.datasets import imdb
from tensorflow.keras.models import Sequential
from tensorflow.keras.layers import Embedding, LSTM, Dense
from tensorflow.keras.preprocessing.sequence import pad_sequences
from tensorflow.keras.optimizers import Adam
# Load the dataset
(X_train, y_train), (X_test, y_test) = imdb.load_data(num_words=10000)
# Pad the sequences
X_train = pad_sequences(X_train, maxlen=80)
X_test = pad_sequences(X_test, maxlen=80)
# Create a recurrent neural network
model = Sequential([
Embedding(10000, 128),
LSTM(128, return_sequences=True),
LSTM(128),
Dense(1, activation='sigmoid')
])
# Compile the model
model.compile(optimizer=Adam(), loss='binary_crossentropy', metrics=['accuracy'])
# Train the model
model.fit(X_train, y_train, epochs=5, validation_split=0.2)
# Make predictions
y_pred = model.predict(X_test)
# Calculate the accuracy
accuracy = model.evaluate(X_test, y_test)[1]
print("Accuracy:", accuracy)
``` | Textbooks |
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advanced_tools:fiber_bundles
Fiber Bundles
Phase factors $e^{i \theta(\vec x,t)}$, like they appear in quantum mechanics, are just complex numbers with amplitude $1$. Therefore, we can picture them as points on a circle with radius $1$:
The wave function that describes an electron has a specific phase $\Psi(\vec x,t)= |\Psi(\vec x,t)|e^{i \theta(\vec x,t)}$ at each point $\vec x$ at any given moment $t$. Each such phase $\theta$ can be represented by a dot on the unit circle.
Therefore, as an electron moves through space, we have above each point that it passes a specific point on the unit circle that denotes the specific phase that the electron wave function has at this location. A set of unit circles above each space point is like a notebook that keeps track of the phase of the electron at this location.
It is convenient to cut the circles such that they become lines. We only need to remember that the end points of these lines need to be identified.
The picture that then emerges, is that we have a line above each point in space. A dot in each such line represents the specific phase our electron has.
In other words, we now have a bundle of circles above the space in which our electron moves. This is an important tool since it allows us nicely to think about how phase factors evolve as a particle moves through space. It is especially useful, as soon as we are dealing with more than one electron.
In the lower part of the image we have the actual space where our electron moves. In the upper part of the image we have the "internal" space, which is our fiber bundle. As the particles follows a path in our actual space, it also traces out a path on the fiber bundle.
The actual tool that tells us which path in the fiber bundle our electron will follow is called the connection, and in physics corresponds to the gauge field. We can think of this connection like a family of ramps. Our electron starts at one specific location with one specific phase. Then, as it moves through space the ramps tell us how the phase changes, i.e. which path in the bundle electron traces out:
Now imagine two electrons that follow different path through space, for example, like they do in the Double Slit Experiment or the Aharonov-Bohm experiment. Depending on the physical situation, it can happen that a different path through space also leads to a different path through the fiber. As a result we get a phase difference between the two electrons that is measurable in terms of an interference pattern.
The best intuitive introduction to fiber bundles is "Fiber Bundles and Quantum Theory" by Herbert J. Bernstein and Anthony V. Phillips. (Can be found for free at Google.)
Source: Fiber Bundles and Quantum Theory by Bernstein and Phillips
A nice introduction to fiber bundles with many pictures is Gauge Theories and Fiber Bundles: Definitions, Pictures, and Results by Adam Marsh
A good textbook on the topic is Topology, Geometry, and Gauge Fields: Foundations by Naber
For a quick overview with many pictures, see http://gregnaber.com/wp-content/uploads/FIBER-BUNDLES-IN-MATHEMATICS-AND-PHYSICS.pdf
Bundles in Field Theory
A field configuration on a given spacetime Σ is meant to be some kind of quantity assigned to each point of spacetime (each event), such that this assignment varies smoothly with spacetime points. For instance an electromagnetic field configuration is at each point of spacetime a collection of vectors that encode the direction in which a charged particle passing through that point will feel a force (the Lorentz force). This is readily formalized: If
$$ F \in SmthMfd $$ is the smooth manifold of "values" that the the given kind of field may take at any spacetime point, then a field configuration $\Phi$ is modeled as a smooth function from spacetime to this space of values:
$$ \Phi \;\colon\; \Sigma \longrightarrow F \,. $$ It will be useful to unify spacetime and the space of field values into a single space, the Cartesian product
$$ E \; = : \; \Sigma \times F $$ and to think of this equipped with the projection map onto the first factor as a fiber bundle of spaces of field values over spacetime
This is then called the field bundle, which specifies the kind of values that the given field species may take at any point of spacetime. Since the space $F$ of field values is the fiber of this fiber bundle, it is sometimes also called the field fiber. https://ncatlab.org/nlab/show/field+%28physics%29#AFirstIdeaOfQuantumFields
Gravitation, Gauge Theories and Differential Geometry by Tohru Eguchi et. al.
Why is it interesting?
All of physics has two aspects: a local or even infinitesimal aspect, and a global aspect. Much of the standard lore deals just with the local and infinitesimal aspects – the perturbative aspects_ and fiber bundles play little role there. But they are the all-important structure that govern the global – the non-perturbative – aspect. Bundles are the global structure of physical fields and they are irrelevant only for the crude local and perturbative description of reality.
For instance the gauge fields in Yang-Mills theory, hence in EM, in QED and in QCD, hence in the standard model of the known universe, are not really just the local 1-forms $A^a_μ$ known from so many textbooks, but are globally really connections on principal bundles (or their associated bundles) and this is all-important once one passes to non-perturbative Yang-Mills theory, hence to the full story, instead of its infinitesimal or local approximation. https://physics.stackexchange.com/a/77412/37286
Fiber bundles are the appropriate mathematical tool to describe, for example, the physics around a magnetic monopole or also instanton effects. (This is described very nicely in chapter 1 of Topology, Geometry and Gauge fields - Part 1 Foundations by G. Naber). Moreover, non-local aspects like Gribov ambiguities can be understood much more clearly with fiber bundles.
They are also useful to show the similarity between general relativity and Yang-Mills theory, which we use in the standard model. The reformulation of the notions that we usually use in quantum field theory, like the notion of a quantum field, in the fibre bundle formalism, allows us to understand them geometrically. This makes them, contrary to what one may think at the beginning, much less abstract and lets us view at them from a completely new perspective.
See also: https://physics.stackexchange.com/questions/77368/intuitively-why-are-bundles-so-important-in-physics
Although gauge theory is introduced in the above inductive manner for historical and pedagogical reasons it is clear that the essential ingredients -the gauge potential, the gauge field, and the covariant derivative - have an intrinsic mathematical structure which is independent of the context.This structure has been well studied by mathematicians, in the context of differential geometry. In this context transformations g(x) are identified as sections of principal bundles, with Minkowski space J (as base and the Lie groups G as fibres, the scalar and fermion fields are identified as sections of vector bundles with base Jl, the gauge potential as a connection form for G{x), and i^"(x) as the components of the curvature. A review of these aspects is given by Daniel and Viallet (1980). The fibre-bundle formulation is not necessary for dealing with those aspects of gauge theory which are local in M, but it becomes important for understanding problems, such as the axial anomaly (next section) and the gauge-fixing ambiguity (Gribov, 1977; Singer, 1978) which are of global origin. It also shows that gauge theory, and thus the theory of strong, weak and electromagnetic interactions, is basically a geometrical theory. This is not only aesthetically pleasing but brings the unification of weak,electromagnetic and strong interactions with gravitation a step closer. Group Structure of Gauge Theories by Lochlainn O'Raifeartaigh
VECTOR BUNDLES AND CONNECTIONS IN PHYSICS AND MATHEMATICS: SOME HISTORICAL REMARKS by V. S. Varadarajan
advanced_tools/fiber_bundles.txt · Last modified: 2020/10/28 16:31 by 89.36.76.154 | CommonCrawl |
Results for 'Genia Sch��nbaumsfeld'
153 found
Antropo-genia o antropo-logia? Ernst Haeckel e Andrea Angiulli sulla pena di morte.M. Di Bartolo - 2004 - In Stefano Poggi (ed.), Natura Umana E Individualità Psichica: Scienza, Filosofia E Religione in Italia E Germania Tra Ottocento E Novecento. Unicopli.details
History of Biology in Philosophy of Biology
Big Data Dreams and Reality in Shenzhen: An Investigation of Smart City Implementation in China.Genia Kostka & Jelena Große-Bley - 2021 - Big Data and Society 8 (2).details
Chinese cities are increasingly using digital technologies to address urban problems and govern society. However, little is known about how this digital transition has been implemented. This study explores the introduction of digital governance in Shenzhen, one of China's most advanced smart cities. We show that, at the local level, the successful implementation of digital systems faces numerous hurdles in long-standing data management and bureaucratic practices that are at least as challenging as the technical problems. Furthermore, the study finds that (...) the digital systems in Shenzhen entail a creeping centralisation of data that potentially turns lower administrative government units into mere users of the city-level smart platforms rather than being in control of their own data resources. Smart city development and big data ambitions thereby imply shifting stakeholder relations at the local level and also pull non-governmental stakeholders, such as information technology companies and research institutions, closer to new data flows and smart governance systems. The findings add to the discussion of big data-driven smart systems and their implications for governance processes in an authoritarian context. (shrink)
Dreams in Philosophy of Mind
Koncepcia génia, nezištnosť a samostatné myslenie.Štefan Haško - forthcoming - Filosofia.details
Handbuch Nikolaus von Kues: Leben und Werk.Marco Brösch (ed.) - 2014 - Darmstadt: WBG, Wissenschaftliche Buchgesellschaft.details
Medienbildung im Zeitalter der Digitalisierung.Andreas Büsch - 2017 - In Ralph Bergold, Jochen Sautermeister & André Schröder (eds.), Dem Wandel eine menschliche Gestalt geben: sozialethische Perspektiven für die Gesellschaft von morgen: Festschrift zur Neueröffnung und zum 70-jährigen Bestehen des Katholisch-Sozialen Instituts. Herder.details
Literatur als philosophisches Erkenntnismodell: literarisch-philosophische Diskurse in Deutschland und Frankreich.Sebastian Hüsch & Sikander Singh (eds.) - 2016 - Tübingen: Narr Francke Attempto.details
Philosophy of Literature in Aesthetics
Langeweile bei Heidegger und Kierkegaard: zum Verhältnis philosophischer und literarischer Darstellung.Sebastian Hüsch - 2014 - Tübingen: Francke Verlag.details
$116.41 new View on Amazon.com
Power, knowledge, and dissent in Morgenthau's worldview.Felix Rösch - 2015 - New York, NY: Palgrave-Macmillan.details
This book provides a comprehensive investigation into Hans Morgenthau's life and work. Identifying power, knowledge, and dissent as the fundamental principles that have informed his worldview, this book argues that Morgenthau's lasting contribution to the discipline of International Relations is the human condition of politics.
$91.36 used $91.39 new $102.23 from Amazon View on Amazon.com
Cooperation between managers and the medical profession in the context of strategic decision making in non-profit hospitals : a manageable challenge?Stephanie Rüsch - 2016 - In Sabine Salloch & Verena Sandow (eds.), Ethics and Professionalism in Healthcare: Transition and Challenges. Routledge.details
Responding to the crisis of philosophy in modernity : from Nietzsche's perspectivism to Musil's essayism.Sebastian Hüsch - 2018 - In Brian Pines & Douglas Burnham (eds.), Understanding Nietzsche, Understanding Modernism. Bloomsbury Academic.details
Friedrich Nietzsche in 19th Century Philosophy
$59.00 used $92.92 new (collection) View on Amazon.com
The Illusion of Doubt.Genia Schönbaumsfeld - 2016 - Oxford, UK: Oxford University Press UK.details
The Illusion of Doubt confronts one of the most important questions in philosophy and beyond: what can we know? The radical sceptic's answer is 'not very much' if we cannot prove that we are not subject to deception. For centuries philosophers have been impressed by the radical sceptic's move, but this book shows that the radical sceptical problem turns out to be an illusion created by a mistaken picture of our evidential situation. This means that we don't need to answer (...) the radical sceptical problem 'head on', but rather to undermine the philosophical assumptions that it depends on. For without these assumptions, radical scepticism collapses all by itself. This result is highly significant, as it manages to dissolve one of the most intractable philosophical problems that has bedevilled some of the greatest minds until the present day. (shrink)
Replies to Skepticism, Misc in Epistemology
Scepticism About Scepticism or the Very Idea of a Global 'Vat-Language'.Genia Schönbaumsfeld - forthcoming - Topoi:1-15.details
This paper aims to motivate a scepticism about scepticism in contemporary epistemology. I present the sceptic with a dilemma: On one parsing of the BIV (brain-in-a-vat) scenario, the second premise in a closure-based sceptical argument will turn out false, because the scenario is refutable; on another parsing, the scenario collapses into incoherence, because the sceptic cannot even save the appearances. I discuss three different ways of cashing out the BIV scenario: 'Recent Envatment' (RE), 'Lifelong Envatment' (LE) and 'Nothing But Envatment' (...) (NBE). I show that RE scenarios are a kind of 'local' sceptical scenario that does not pose a significant threat to the possibility of perceptual knowledge as such. I then go on to consider the more radical (or global) LE and NBE scenarios, which do undermine the possibility of perceptual knowledge of an 'external' world by positing that it is conceivable that one has always been envatted and, hence, trapped in a 'global' illusion. I start by assuming that we could be in such a scenario (LE or NBE) and then spell out what we would need to presuppose for such scenarios to be capable of being actual. Drawing on some central insights from Wittgenstein's anti-private language considerations, I show that the truth of a global scepticism would presuppose the possibility of a private 'vat-language', a notion that cannot be rendered coherent. But, if so, then neither can the sceptical scenarios that presuppose such a conception. (shrink)
A Confusion of the Spheres: Kierkegaard and Wittgenstein on Philosophy and Religion.Genia Schönbaumsfeld - 2007 - Oxford University Press.details
As well as contributing to contemporary debate about how to read Kierkegaard's and Wittgenstein's work, A Confusion of the Spheres addresses issues which not ...
Ludwig Wittgenstein in 20th Century Philosophy
Søren Kierkegaard in 19th Century Philosophy
Hommage einer Autorin an GBL.Sch ne-Seifert M. Nster Bettina - 2008 - Ethik in der Medizin 20 (4):269-270.details
New questions about old heritability estimates.Peter H. Sch&öNemann - 1989 - Bulletin of the Psychonomic Society 27 (2):175-178.details
Heritability in Philosophy of Biology
Failures of SCH and Level by Level Equivalence.Arthur W. Apter - 2006 - Archive for Mathematical Logic 45 (7):831-838.details
We construct a model for the level by level equivalence between strong compactness and supercompactness in which below the least supercompact cardinal κ, there is a stationary set of cardinals on which SCH fails. In this model, the structure of the class of supercompact cardinals can be arbitrary.
Axioms of Set Theory in Philosophy of Mathematics
Cardinals and Ordinals in Philosophy of Mathematics
Logic and Philosophy of Logic, Miscellaneous in Logic and Philosophy of Logic
Technical rationality in Sch�n?s reflective practice: dichotomous or non-dualistic epistemological position.Elizabeth Anne Kinsella - 2007 - Nursing Philosophy 8 (2):102-113.details
Rationality in Epistemology
'Hinge Propositions' and the 'Logical' Exclusion of Doubt.Genia Schönbaumsfeld - 2016 - International Journal for the Study of Skepticism 6 (2-3):165-181.details
_ Source: _Volume 6, Issue 2-3, pp 165 - 181 Wittgenstein's notion of 'hinge propositions'—those propositions that stand fast for us and around which all empirical enquiry turns—remains controversial and elusive, and none of the recent attempts to make sense of it strike me as entirely satisfactory. The literature on this topic tends to divide into two camps: either a 'quasi-epistemic' reading is offered that seeks to downplay the radical nature of Wittgenstein's proposal by assimilating his thought to more mainstream (...) epistemological views, or a non-epistemic, 'quasi-pragmatic' conception is adopted that goes too far in the opposite direction by, for example, equating 'hinge propositions' with a type of 'animal' certainty. Neither interpretative strategy, I will argue, is promising for the reason that 'hinges' are best not conceived as certainties at all. Rather, what Wittgenstein says in respect to them is that doubt is "logically" excluded, and where there can be no doubt, I contend, there is no such thing as knowledge or certainty either. (shrink)
Skepticism, Misc in Epistemology
La Dynamique des phénomènes de la vie, Bibliothéque scientifique internationale.J. Loeb, H. Daudin, G. Schæffer & A. Giard - 1908 - Revue de Métaphysique et de Morale 16 (2):5-6.details
McDowellian Neo-Mooreanism?Genia Schönbaumsfeld - 2013 - International Journal for the Study of Skepticism 3 (3):202-217.details
In a series of recent articles, Duncan Pritchard argues for a 'neo-Moorean' interpretation of John McDowell's anti-sceptical strategy. Pritchard introduces a distinction between 'favouring' and 'discriminating' epistemic grounds in order to show that within the radical sceptical context an absence of 'discriminating' epistemic grounds allowing one to distinguish brain-in-a-vat from non-brain-in-a-vat scenarios does not preclude possessing knowledge of the denials of sceptical hypotheses. I argue that Pritchard's reading is mistaken for three reasons. First, the distinction between 'favouring' and 'discriminating' epistemic (...) grounds only works for 'mules-disguised-as zebras' examples, but breaks down in the radical sceptical case. Second, McDowellian disjunctivism neutralizes the radical sceptical threat, but does not refute it. Third, any attempt to refute scepticism is confronted by the following dilemma: either one accepts the 'highest common factor' conception of perceptual experience thus rendering radical scepticism in principle irrefutable, or one discards it in favour of the disjunctive conception, but then there is no radical sceptical scenario left to refute. So, whichever horn of this dilemma Pritchard grasps, refutation is either impossible or superfluous. (shrink)
Content Externalist Replies to Skepticism in Epistemology
A "resolute" later Wittgenstein?Genia Schönbaumsfeld - 2010 - Metaphilosophy 41 (5):649-668.details
Abstract: "Resolute readings" initially started life as a radical new approach to Wittgenstein's early philosophy, but are now starting to take root as a way of interpreting the later writings as well—a trend exemplified by Stephen Mulhall's Wittgenstein's Private Language (2007) as well as by Phil Hutchinson's "What's the Point of Elucidation?" (2007) and Rom Harré's "Grammatical Therapy and the Third Wittgenstein" (2008). The present article shows that there are neither good philosophical nor compelling exegetical grounds for accepting a resolute (...) reading of the later Wittgenstein's work. It is possible to make sense of Wittgenstein's philosophical method without either ascribing to him an incoherent conception of "substantial nonsense" or espousing the resolute readers' preferred option of nonsense austerity. If the interpretation here is correct, it allows us to recognize Wittgenstein's radical break with the philosophical tradition without having to characterize his achievements in purely therapeutic fashion. (shrink)
Social Media Use and Mental Health and Well-Being Among Adolescents – A Scoping Review.Viktor Schønning, Gunnhild Johnsen Hjetland, Leif Edvard Aarø & Jens Christoffer Skogen - 2020 - Frontiers in Psychology 11.details
Introduction: Social media has become an integrated part of daily life, with an estimated 3 billion social media users worldwide. Adolescents and young adults are the most active users of social media. Research on social media has grown rapidly, with the potential association of social media use and mental health and well-being becoming a polarized and much-studied subject. The current body of knowledge on this theme is complex and difficult-to-follow. The current paper presents a scoping review of the published literature (...) in the research field of social media use and its association with mental health and well-being among adolescents. Methods and analysis: First, relevant databases were searched for eligible studies with a vast range of relevant search terms for social media use and mental health and well-being over the past five years. Identified studies were screened thoroughly and included or excluded based on prior established criteria. Data from the included studies were extracted and summarized according to the previously published study protocol. Results: Among the 79 studies that met our inclusion criteria, the vast majority (94%) were quantitative, with a cross-sectional design (57%) being the most common study design. Several studies focused on different aspects of mental health, with depression (29%) being the most studied aspect. Almost half of the included studies focused on use of non-specified social network sites (43%). Of specified social media, Facebook (39%) was the most studied social network site. The most used approach to measuring social media use was frequency and duration (56%). Participants of both genders were included in most studies (92%) but seldom examined as an explanatory variable. 77% of the included studies had social media use as the independent variable. Conclusion: The findings from the current scoping review revealed that about ¾ of the included studies focused on social media and some aspect of pathology. Focus on the potential association between social media use and positive outcomes seems to be rarer in the current literature. Amongst the included studies, few separated between different forms of (inter)actions on social media, which are likely to be differentially associated with mental health and well-being outcomes. (shrink)
Wittgenstein and the 'Factorization Model' of Religious Belief.Genia Schönbaumsfeld - 2014 - European Journal for Philosophy of Religion 6 (1):93--110.details
In the contemporary literature Wittgenstein has variously been labelled a fideist, a non-cognitivist and a relativist of sorts. The underlying motivation for these attributions seems to be the thought that the content of a belief can clearly be separated from the attitude taken towards it. Such a "factorization model' which construes religious beliefs as consisting of two independent "factors' -- the belief's content and the belief-attitude -- appears to be behind the idea that one could, for example, have the religious (...) attitude alone or that religious content will remain broadly unaffected by a fundamental change in attitude. In this article I will argue that such a "factorization model' severely distorts Wittgenstein's conception of religious belief. (shrink)
Meaning and Conversational Impropriety in Sceptical Contexts.Genia Schönbaumsfeld - 2016 - Metaphilosophy 47 (3):431-448.details
According to "disjunctivist neo-Mooreanism"—a position Duncan Pritchard develops in a recent book—it is possible to know the denials of radical sceptical hypotheses, even though it is conversationally inappropriate to claim such knowledge. In a recent paper, on the other hand, Pritchard expounds an "überhinge" strategy, according to which one cannot know the denials of sceptical hypotheses, as "hinge propositions" are necessarily groundless. The present article argues that neither strategy is entirely successful. For if a proposition can be known, it can (...) also be claimed to be known. If the latter is not possible, this is not because certain propositions are either "intrinsically" conversationally inappropriate or else "rationally groundless", but rather that we are dealing with something that merely presents us with the appearance of being an epistemic claim. (shrink)
Kierkegaard and post-modernity.Robert John Sch Manning - 1993 - Philosophy Today 37 (2):133-152.details
'Meaning-dawning' in Wittgenstein's Notebooks: a Kierkegaardian reading and critique.Genia Schönbaumsfeld - 2018 - British Journal for the History of Philosophy 26 (3):540-556.details
ABSTRACTIn this paper, I am going to propose a new reading of Wittgenstein's cryptic talk of 'accession or loss of meaning' in the Notebooks that draws both on Wittgenstein's later work on aspect-perception, as well as on the thoughts of a thinker whom Wittgenstein greatly admired: Søren Kierkegaard. I will then go on to argue that, its merits apart, there is something existentially problematic about the conception that Wittgenstein is advocating. For the renunciation of the comforts of the world that (...) Wittgenstein proposes as a way of coping with the brute contingencies of life seems only to come as far as what Kierkegaard calls 'infinite resignation', and this falls far short of the joyful acceptance of existence that appears necessary for inhabiting what Wittgenstein calls a happy world. That is to say, I will show that what Wittgenstein's proposal lacks is a way of reconnecting with the finite after one has renounced it – the kind of transformation of existence achieved by the person Kierkegaard calls the 'knight of faith'. (shrink)
No New Kierkegaard.Genia Schönbaumsfeld - 2004 - International Philosophical Quarterly 44 (4):519-534.details
The aim of this paper is to contest an infl uential recent reading of one of Kierkegaard's most important books, the pseudonymously written Concluding Unscientific Postscript. According to the reading offered by James Conant, the Postscript is an "elaborate reductio" of the very philosophical project in which it itself appears to be engaged, namely, the project of attempting to clarify the nature of Christianity. I show that Conant's position depends upon four inter-related theses concerning Kierkegaard's text, and I argue that (...) noneof these theses is sustainable, either philosophically or exegetically. In the course of this critique, alternative and more convincing theses are developed, and I suggest that these theses are altogether better suited than Conant's to account for, and to provide a defense of, Kierkegaard's stature as a religious thinker. (shrink)
John Leslie Mackie beginnt sein berühmtes und breit diskutiertes Buch "Ethik. Die Erfindung des Richtigen und Falschen" mit dem Satz:"Es.Peter Sch Aber - 2003 - In P. Schaber & R. Huntelmann (eds.), Grundlagen der Ethik. pp. 9.details
$135.49 new $1496.00 used (collection) View on Amazon.com
'Objectively there is no truth' - Wittgenstein and Kierkegaard on religious belief.Genia Schönbaumsfeld - unknowndetails
Kierkegaard's influence on Wittgenstein's conception of religious belief was profound, but this hasn't so far been given the attention it deserves . Although Wittgenstein wrote comparatively little on the subject, while the whole of Kierkegaard's oeuvre has a religious theme, both philosophers have become notorious for refusing to construe religious belief in either of the two traditional ways: as a 'propositional attitude' on the one hand or as a mere 'emotional response' with no reference to the 'real world' on the (...) other. This refusal to play by the orthodox dichotomies, as it were, has led to gross misrepresentation of their thought by numerous commentators. Neither Wittgenstein nor Kierkegaard has been immune to allegations of both 'relativism' and 'fideism', although neither charge could be wider of the mark. It is not that Wittgenstein and Kierkegaard reject the role that reason has to play in both religion and philosophy, but that they try to undermine from within certain common assumptions about the nature of both religious faith and the point of philosophical activity that make us believe that the traditional dichotomies exhaust all the available options. What I hope to show in this paper is that more sense can be made of Wittgenstein's controversial remarks on religion, if we juxtapose them with Kierkegaard's religious thought, especially that of Kierkegaard's pseudonym , Johannes Climacus, in Concluding Unscientific Postscript. The focal point of this paper is going to be the attempt to read what little Wittgenstein has to say about this topic through the lens of Climacus' claim that 'objectively there is no truth; an objective knowledge about the truth or the truths of Christianity is precisely untruth.' I will begin by giving a brief exposition of Climacus' views, will then sketch out what Wittgenstein has to say on the matter and will then attempt to bring the two together. In the remainder of the paper I will assess the implications of Wittgenstein's and Kierkegaard's conception as well as address some of the problems that their account might be said to engender. (shrink)
Epistemology of Religion, Misc in Philosophy of Religion
International Research Ethics.Udo SchÜcklenk & Richard Ashcroft - 2000 - Bioethics 14 (2):158-172.details
Embryo research: destiny is what counts.Alex Polyakov & Genia Rozen - 2022 - Journal of Medical Ethics 48 (9):601-602.details
The paper by Savulescu et al is timely and the concepts illuminated deserve further reflection.1 Reproductive tissue which includes sperm, oocytes and embryos are commonly treated differently to other human tissue, even when the reproductive potential of these has no possibility of being realised. This unnecessary exceptionalism hampers research in human reproduction, disadvantaging patients and delaying life-changing treatments from being incorporated into clinical practice. In jurisdictions where embryo creation is permitted for clinical purposes, such as in vitro fertilisation, supernumerary embryos (...) are routinely discarded once they are no longer required to attempt pregnancy. We contend that research on such embryos, which will never realise their reproductive potential, is not substantially different to any other human tissue research and should not require any additional ethical or regulatory oversight. The only individuals who can possibly be harmed by such research are the tissue providers and their interests, including confidentiality and informed consent provision, should be protected. This protection once again is no different in its scope to any other type of research involving human cells and tissues. Most Western societies have moved away from the concept that 'every sperm is sacred' and it is time that the regulators and scientific community aligned with the prevailing communal norms and stop treating gametes and embryos as somehow holding higher moral value, compared with other human cell/tissue lines, provided that there is no intention to attempt a pregnancy …. (shrink)
From the Editors.Ruth Chadwick & Udo SchÜklenk - 2002 - Bioethics 16 (2).details
Wittgenstein's Private Language: Grammar, Nonsense, and Imagination in Philosophical Investigations, §§ 243–315, by Stephen Mulhall. Oxford: Oxford University Press, 2007. Pp. 148. H/b£ 19.99. [REVIEW]Genia Schoenbaumsfeld - 2008 - Mind 117 (468):1108-1112.details
Itp, isp, and sch.Sherwood Hachtman & Dima Sinapova - 2019 - Journal of Symbolic Logic 84 (2):713-725.details
Kierkegaard and the Tractatus.Genia Schoenbaumsfeld - unknowndetails
It is the object of this paper to investigate the parallels discernible between Wittgenstein's Tractatus and Kierkegaard's pseudonymous writings. While such attempts have, in the past, generally focussed on either trying to show that Kierkegaard's notion of paradox is similar to Wittgenstein's concept of the ineffable or that both thinkers seek to undermine the idea that there are things that cannot be put into words, I argue here that we must look for the affinities between the two philosophers in an (...) altogether different place, namely, in Kierkegaard's and Wittgenstein's concerted endorsement of an anti-consequentialist, quasi-religious conception of ethics. (shrink)
Précis of The Illusion of Doubt.Genia Schönbaumsfeld - forthcoming - International Journal for the Study of Skepticism:1-6.details
The Illusion of Doubt shows that radical scepticism is an illusion generated by a Cartesian picture of our evidential situation—the view that my epistemic grounds in both the 'good' and the 'bad' cases must be the same. It is this picture which issues both a standing invitation to radical scepticism and ensures that there is no way of getting out of it while agreeing to the sceptic's terms. The sceptical problem cannot, therefore, be answered 'directly'. Rather, the assumptions that give (...) rise to it, need to be undermined. These include the notion that radical scepticism can be motivated by the 'closure' principle for knowledge, that the 'Indistinguishability Argument' renders the Cartesian conception compulsory, that the 'New Evil Genius Thesis' is coherent, and the demand for a 'global validation' of our epistemic practices makes sense. Once these dogmas are undermined, the path is clear for a 'realism without empiricism' that allows us to re-establish unmediated contact with the objects and persons in our environment which an illusion of doubt had threatened to put forever beyond our cognitive grasp. (shrink)
Brains in Vats in Epistemology
Cartesian Skepticism in Epistemology
History: Skepticism in Epistemology
Reflection and not SCH with overlapping extenders.Moti Gitik - 2022 - Archive for Mathematical Logic 61 (5):591-597.details
We use the forcing with overlapping extenders to give a direct construction of a model of \SCH+Reflection.
Epistemic Angst, Intellectual Courage and Radical Scepticism.Genia Schönbaumsfeld - 2019 - International Journal for the Study of Skepticism 9 (3):206-222.details
The overarching aim of this paper is to persuade the reader that radical scepticism is driven less by independently plausible arguments and more by a fear of epistemic limitation which can be overcome. By developing the Kierkegaardian insight that knowledge requires courage, I show that we are not, as potential knowers, just passive recipients of a passing show of putatively veridical information, we also actively need to put ourselves in the way of it by learning to resist certain forms of (...) epistemic temptation: the Cartesian thought that we could be 'imprisoned' within our own representations, and, hence permanently 'out of touch' with an 'external' world, and the Reasons Identity Thesis, which has us believe that whether we are in the good case or in the bad case, our epistemic grounds are the same. (shrink)
Skepticism in Epistemology
Review of Hans-Johann Glock & John Hyman (eds.), Wittgenstein and Analytic Philosophy: Essays for P.M.S. Hacker. [REVIEW]Genia Schönbaumsfeld - forthcoming - Analysis.details
Epistemological Disjunctivism by Duncan Pritchard.Genia Schönbaumsfeld - 2015 - Analysis 75 (4):604-615.details
1. In this exciting and ambitious book, Duncan Pritchard defends a novel conception of perceptual epistemic grounds, which can both be factive and reflectively available to the agent. Pritchard calls this position the 'holy grail' of epistemology for its power to undercut two of contemporary epistemology's most central problems: the epistemic internalism/externalism controversy and radical scepticism. While Pritchard's book manages to make a convincing case for why one should accept epistemological disjunctivism (ED), the 'neo-Moorean' anti-sceptical strategy that he derives from (...) it is less compelling. This is not because ED does not provide us with the materials for a plausible anti-sceptical strategy, however, but rather because Pritchard misidentifies where ED's real power lies, namely, in its potential to prevent the radical sceptical problem from arising in the first place. If I am right, then there is no need to establish Moore-type anti-sceptical theses with the ground rules for doing so set by scepticism. For once (and as McDowell has maintained) the thought has been undermined that my perceptual epistemic grounds can only ever consist of the 'highest common factor' between the 'good case' and the 'bad case', the traditional route to radical scepticism is blocked. (shrink)
Disjunctivism in Philosophy of Mind
Perceptual Knowledge in Philosophy of Mind
Worlds or words apart? Wittgenstein on understanding religious language.Genia Schönbaumsfeld - 2007 - Ratio 20 (4):422–441.details
In this paper I develop an account of Wittgenstein's conception of what it is to understand religious language. I show that Wittgenstein's view undermines the idea that as regards religious faith only two options are possible – either adherence to a set of metaphysical beliefs (with certain ways of acting following from these beliefs) or passionate commitment to a 'doctrineless' form of life. I offer a defence of Wittgenstein's conception against Kai Nielsen's charges that Wittgenstein removes the 'content' from religious (...) belief and renders the religious form of life 'incommensurable' with other domains of discourse, thus immunizing it against rational criticism. (shrink)
Response to Critics.Genia Schönbaumsfeld - forthcoming - International Journal for the Study of Skepticism:1-17.details
In this paper I respond to the objections and comments made by Ranalli, Williams, and Moyal-Sharrock, participants in a symposium on my book on scepticism called The Illusion of Doubt.
Introspective Distinguishability.Genia Schönbaumsfeld - 2021 - Midwest Studies in Philosophy 45:241-256.details
It is generally thought that if introspective distinguishability were available, it would provide an answer to scepticism about perceptual knowledge by enabling us to tell the difference between a good case perceptual experience and a bad kind. This paper challenges this common assumption by showing that even if ID were available, it would not advance our case against scepticism. The conclusion to draw from this result is not to concede to scepticism, however, but rather to give up on the idea (...) that ID is required for knowledge. For if perception with ID turns out to get us no further than perception without ID, then the rational thing to do is to realize that the putative presence of ID is a red herring in the debate about scepticism and can make no difference to the question of whether or not perceptual knowledge is possible. (shrink)
Introspection and Introspectionism in Philosophy of Cognitive Science
The 'Default View' of Perceptual Reasons and 'Closure-Based' Sceptical Arguments.Genia Schönbaumsfeld - 2017 - International Journal for the Study of Skepticism 7 (2):114-135.details
_ Source: _Volume 7, Issue 2, pp 114 - 135 It is a commonly accepted assumption in contemporary epistemology that we need to find a solution to 'closure-based' sceptical arguments and, hence, to the 'scepticism or closure' dilemma. In the present paper I argue that this is mistaken, since the closure principle does not, in fact, do real sceptical work. Rather, the decisive, scepticism-friendly moves are made before the closure principle is even brought into play. If we cannot avoid the (...) sceptical conclusion, this is not due to closure's holding it in place, but because we've already been persuaded to accept a certain conception of perceptual reasons, which both issues a standing invitation to radical scepticism and is endemic in the contemporary literature. Once the real villain of the piece is exposed, it will become clear that the closure principle has been cast in the role of scapegoat in this debate. (shrink)
Closure of Knowledge in Epistemology
Simple proofs of $${\mathsf{SCH}}$$ SCH from reflection principles without using better scales.Hiroshi Sakai - 2015 - Archive for Mathematical Logic 54 (5-6):639-647.details
We give simple proofs of the Singular Cardinal Hypothesis from the Weak Reflection Principle and the Fodor-type Reflection Principle which do not use better scales.
Proof Theory in Logic and Philosophy of Logic
Rupert Read and Matthew A. Lavery , Beyond the Tractatus Wars: The New Wittgenstein Debate . xi + 200, price £24.99 pb.Genia Schönbaumsfeld - 2013 - Philosophical Investigations 36 (1):83-87.details
Introduction.Genia Schönbaumsfeld - 2019 - International Journal for the Study of Skepticism 9 (3):179-182.details
This introduction provides an overview of the content of the papers published in the special issue on epistemic vice and forms of scepticism.
The aesthetic as mirror of faith in Kierkegaard's Fear and Trembling.Genia Schönbaumsfeld - 2019 - European Journal of Philosophy 27 (3):661-674.details
European Journal of Philosophy, EarlyView.
Wittgenstein and Kierkegaard on religious belief.Genia Schönbaumsfeld - 2009 - In Ulrich Arnswald (ed.), In Search of Meaning: Ludwig Wittgenstein on Ethics, Mysticism and Religion. Universitätsverlag Karlsruhe.details
Wittgensteinian Approaches to Religion.Genia Schönbaumsfeld - 2015 - In Graham Oppy (ed.), Routledge Handbook for Contemporary Philosophy of Religion.details
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\begin{document}
\title[Vector lattices with a Hausdorff uo-Lebesgue topology]{Vector lattices with a Hausdorff uo-Lebesgue topology}
\author{Yang Deng} \address{Yang Deng; School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, Sichuan 611130, People's Republic of China} \email{[email protected]}
\author{Marcel de Jeu} \address{Marcel de Jeu; Mathematical Institute, Leiden University, P.O.\ Box 9512, 2300 RA Leiden, the Netherlands;
and Department of Mathematics and Applied Mathematics, University of Pretoria, Cor\-ner of Lynnwood Road and Roper Street, Hatfield 0083, Pretoria, South Africa} \email{[email protected]}
\keywords{Vector lattice, Banach lattice, unbounded order convergence, Lebesgue topology, uo-Lebesgue topology} \subjclass[2010]{Primary: 46A40. Secondary: 28A20, 46A16, 46B42}
\begin{abstract}
We investigate the construction of a Hausdorff uo-Lebesgue topology on a vector lattice from a Hausdorff (o)-Lebesgue topology on an order dense ideal, and what the properties of the topologies thus obtained are. When the vector lattice has an order dense ideal with a separating order continuous dual, it is always possible to supply it with such a topology in this fashion, and the restriction of this topology to a regular sublattice is then also a Hausdorff uo-Lebesgue topology. A regular vector sublattice of $\mathrm{L}_0(X,\Sigma,\mu)$ for a semi-finite measure $\mu$ falls into this category, and the convergence of nets in its Hausdorff uo-Lebesgue topology is then the convergence in measure on subsets of finite measure. When a vector lattice not only has an order dense ideal with a separating order continuous dual, but also has the countable sup property, we show that every net in a regular vector sublattice that converges in its Hausdorff uo-Lebesgue topology always contains a sequence that is uo-convergent to the same limit. This enables us to give satisfactory answers to various topological questions about uo-convergence in this context. \end{abstract}
\maketitle
\section{Introduction and overview}
\noindent In this paper, we investigate the construction of a Hausdorff uo-Lebesgue topology on a vector lattice from a Hausdorff (o)-Lebesgue topology\footnote{In the literature, what we call a o-Lebes\-gue topology\ is simply called a Lebesgue topology. Now that uo-Lebes\-gue topologies, with a completely analogous definition, have become objects of a more extensive study, it seems consistent to also add a prefix to the original term.}\ on an order dense ideal, and what the properties of the topologies thus obtained are.
After recalling the relevant notions and making the necessary preparations in \cref{sec:preliminaries}, the key construction is carried out in \cref{res:generated_topology} in \cref{sec:unbounded_topologie_generated_by_topologies_on_ideals}, below. The idea of starting with a topology on an order dense ideal originates from \cite{conradie:2005} but, whereas the construction in \cite{conradie:2005} to obtain a global topology is carried out using Riesz pseudo-norms, we follow an approach using neighbourhood bases of zero that is inspired by \cite{taylor_THESIS:2018, taylor:2019}. Using such neighbourhood bases, it is possible to perform the construction under minimal hypotheses on the initial data, and thus understand how these hypotheses are reflected in the properties of the resulting global topology. The remainder of \cref{sec:unbounded_topologie_generated_by_topologies_on_ideals} is mainly concerned with showing how the general theorem relates to existing results in the literature. Our working with neighbourhood bases of zero enables us to explain certain `pathologies' in the literature, where a topology of unbounded type is not Hausdorff, or not linear, from the general theorem.
In \cref{sec:uoLtops_going_up_and_going_down}, we move to the context where the initial ideal is actually order dense and admits a Hausdorff o-Lebes\-gue topology. In that case, every regular vector sublattice of the global vector lattice admits a Hausdorff uo-Lebes\-gue topology. The resulting overview \cref{res:overview}, below, mostly consists of a summary of results that are already in the literature, though not presented in this way. It is also recalled in that section that a regular vector sublattice admits a Hausdorff uo-Lebes\-gue topology\ when the global vector lattice admits one. Consequently, there is a going-up-going-down procedure: starting with a Hausdorff o-Lebes\-gue topology\ on an order dense ideal, one obtains a Hausdorff uo-Lebes\-gue topology\ on the global vector lattice, and then finally also one on every regular vector sublattice.
In view of the going-up-going-down construction, it is evidently desirable to have a class of vector lattices that admit Hausdorff o-Lebes\-gue topologies\ because such data can serve as `germs' for Hausdorff uo-Lebes\-gue topologies. The vector lattices with separating order continuous duals form such a class, and this is exploited in \cref{sec:separating_order_continuous_dual}.
\cref{sec:vector_lattices_of_equivalence_classes_of_measurable_functions} is concerned with regular vector sublattices of $\mathrm{L}_0(X,\Sigma,\mu)$ for a semi-finite measure $\mu$. Via the going-up-going-down principle, every regular vector sublattice of $\mathrm{L}_0(X,\Sigma,\mu)$ admits a Hausdorff uo-Lebes\-gue topology. We give a rigorous proof of the fact that the convergence of nets in such a topology is the convergence in measure on subsets of finite measure. For $\mathrm{L}_p(X,\Sigma,\mu)$, we also discuss how the (in fact) unique Hausdorff uo-Lebes\-gue topology\ on these spaces can be described in various seemingly different ways that are still equivalent. The relation between these topologies and minimal and smallest Hausdorff locally solid linear topologies on these spaces is explained.
\cref{sec:uo-convergent_subsequences_of_uoLt-convergent_nets} is concerned with convergent sequences that can always be found `within' nets that are convergent in a Hausdorff uo-Lebes\-gue topology\ on a vector lattice that has the countable sup property and that has an order dense ideal with a separating order continuous dual. The precise statement is in \cref{res:tau_m_to_sub_uo}, below; this is one of the main theorems in this paper. It is in the same spirit as the fact that a sequence that converges (globally) in measure always contains a subsequence that converges almost everywhere to the same limit.
Finally, in \cref{sec:topological aspects of uo-convergence}, we study topological aspects of uo-convergence. The relations between uo-convergence and various order topologies are not at all well understood, but when the global vector lattice has the countable sup property, and also has an order dense ideal with a separating order continuous dual, then a reasonably satisfactory picture emerges. In \cref{res:seven_sets_equal} and \cref{res:eleven_sets_equal}, below, various topological closures and (sequential) adherences are then seen to be equal. It is then also possible to give a necessary and sufficient criterion for sequential uo-convergence to be topological; see \cref{res:sequential_uo_convergence_topological_two}, below.
We have tried to be as complete in the development of this part of the theory of uo-convergence as we could, and also to relate to relevant existing results in the literature whenever possible. Any omissions at this point are unintentional.
\section{Preliminaries}\label{sec:preliminaries}
\noindent In this section, we collect a number of definitions, notations, conventions and preparatory results. We refer the reader to the textbooks \cite{abramovich_aliprantis_INVITATION_TO_OPERATOR_THEORY:2002}, \cite{aliprantis_border_INFINITE_DIMENSIONAL_ANALYSIS_THIRD_EDITION:2006}, \cite{aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003}, \cite{aliprantis_burkinshaw_POSITIVE_OPERATORS_SPRINGER_REPRINT:2006}, \cite{luxemburg_zaanen_RIESZ_SPACES_VOLUME_I:1971}, \cite{meyer-nieberg_BANACH_LATTICES:1991}, \cite{schaefer_BANACH_LATTICES_AND_POSITIVE_OPERATORS:1974}, \cite{zaanen_RIESZ_SPACES_VOLUME_II:1983}, and \cite{zaanen_INTRODUCTION_TO_OPERATOR_THEORY_IN_RIESZ_SPACES:1997} for general background information on vector lattices and Banach lattices.
\subsection{Vector lattices, operators, and (unbounded) order convergence}
\noindent All vector spaces are over the real numbers. Measures take their values in $[0,\infty]$ and are not supposed to satisfy any condition unless otherwise specified. All vector lattices are supposed to be Archimedean. The positive cone of a vector lattice $E$ is denoted by $\pos{E}$.
Let $E$ be a vector lattice, and let $F$ be a vector sublattice of $E$. Then $F$ is \emph{order dense in $E$} when, for every $x\in E$ with $x>0$, there exists a $y\in F$ such that $0<y\leq x$; $F$ is called \emph{super order dense in $E$} when, for every $x\in \pos{E}$, there exists a sequence $\seqxn\subseteq \pos{F}$ with $x_n\uparrow x$ in $E$. The vector sublattice $F$ of $E$ is order dense in $E$ if and only if, for every $x\in \pos{E}$, we have $x=\sup\{y\in F: 0\leq y\leq x\}$; see \cite[Theorem~1.34]{aliprantis_burkinshaw_POSITIVE_OPERATORS_SPRINGER_REPRINT:2006}, for example.
A vector sublattice $F$ of a vector lattice $E$ is called \emph{majorising in $E$} when, for every $x\in E$, there exists a $y\in F$ such that $x\leq y$. In some sources, such as \cite{conradie:2005}, $F$ is then said to be full in $E$.
A vector lattice $E$ \emph{has the countable sup property} when, for every non-empty subset $S$ of $E$ that has a supremum in $E$, there exists an at most countable subset of $S$ that has the same supremum in $E$ as $S$. In parts of the literature, such as in \cite{luxemburg_zaanen_RIESZ_SPACES_VOLUME_I:1971} and \cite{zaanen_INTRODUCTION_TO_OPERATOR_THEORY_IN_RIESZ_SPACES:1997}, $E$ is then said to be order separable. A vector lattice $E$ has the countable sup property if and only if, whenever a net $\net\subseteq\pos{E}$ and $x\in\pos{E}$ are such that $x_\alpha\uparrow x$ in $E$, there exists a sequence of indices $\seq{\alpha_n}{n}$ in ${\mathcal A}$ such that $x_{\alpha_n}\uparrow x$ in $E$; see \cite[Theorem~23.2.(iii)]{luxemburg_zaanen_RIESZ_SPACES_VOLUME_I:1971}.
Let $E$ be a vector lattice, and let $x\in E$. We say that a net $\net$ in $E$ is \emph{order convergent to $x\in E$} (denoted by $x_\alpha\convwithoverset{\mathrm{o}} x$) when there exists a net $(y_\beta)_{\beta\in \mathscr{B}}$ in $E$ such that $y_\beta\downarrow 0$ and with the property that, for every $\beta_0\in{\mathcal B}$, there exists an $\alpha_0\in {\mathcal A}$ such that $\abs{x-x_\alpha}\leq y_{\beta_0}$ whenever $\alpha$ in ${\mathcal A}$ is such that $\alpha\geq\alpha_0$. Note that the index sets ${\mathcal A}$ and ${\mathcal B}$ need not be equal; for a discussion of the difference between these two possible definitions we refer to \cite{abramovich_sirotkin:2005}, for example.
Let $E$ and $F$ be vector lattices. The order bounded operators from $E$ into $F$ will be denoted by $\mathcal{L}_{\ob}(E,F)$, and the regular operators from $E$ into $F$ by $\mathcal{L}_{\reg}(E,F)$. When $F$ is Dedekind complete, we have $\mathcal{L}_{\ob}(E,F)=\mathcal{L}_{\reg}(E,F)$, and this space is then a Dedekind complete vector lattice; see \cite[Theorem~1.18]{aliprantis_burkinshaw_POSITIVE_OPERATORS_SPRINGER_REPRINT:2006}, for example. We write $\odual{E}$ for $\mathcal{L}_{\ob}(E,\mathbb R)=\mathcal{L}_{\reg}(E,\mathbb R)$.
A linear operator $T: E\to F$ between two vector lattices $E$ and $F$ is \emph{order continuous} when, for every net $\net$ in $E$, the fact that $x_\alpha\convwithoverset{\mathrm{o}} 0$ in $E$ implies that $Tx_\alpha\convwithoverset{\mathrm{o}} 0$ in $F$. When $T$ is positive one can, equivalently, require that, for every net $\net$ in $E$, the fact that $x_\alpha\downarrow 0$ in $E$ imply that $Tx_\alpha\downarrow 0$ in $F$. An order continuous linear operator between two vector lattices is automatically order bounded; see \cite[Lemma~1.54]{aliprantis_burkinshaw_POSITIVE_OPERATORS_SPRINGER_REPRINT:2006}, for example. The order continuous linear operators from $E$ into $F$ will be denoted by $\mathcal{L}_{\oc}(E,F)$. In the literature, the notation $\mathcal L_{\mathrm n}(E,F)$ is often used. When $F$ is Dedekind complete, $\mathcal{L}_{\oc}(E,F)$ is a band in $\mathcal{L}_{\reg}(E,F)$; see \cite[Theorem~1.57]{aliprantis_burkinshaw_POSITIVE_OPERATORS_SPRINGER_REPRINT:2006}, for example. We write $\ocdual{E}$ for $\mathcal{L}_{\oc}(E,\mathbb R)$.
The following result is easily established using the Riesz-Kantorovich formulas and their `dual versions'; see \cite[Theorems~1.18 and~1.23]{aliprantis_burkinshaw_POSITIVE_OPERATORS_SPRINGER_REPRINT:2006}, for example. We shall be interested only in the case where the lattice $F$ in it is the real numbers and the band $B$ is the zero band, but the general case comes at no extra cost in the routine proof.
\begin{proposition}\label{res:polars}
Let $E$ and $F$ be vector lattices, where $F$ is Dedekind complete, and let $B$ be a band in $F$.
\begin{enumerate}
\item Let $I$ be an ideal of $E$. Then the subset
\[
\{T\in\mathcal{L}_{\reg}(E,F): Tx \in B \text{ for all }x\in I\}
\]
of $\mathcal{L}_{\reg}(E,F)$ is band in $\mathcal{L}_{\reg}(E,F)$. For every subset $S$ of $I$ that generates $I$, it is equal to
\[
\{T\in\mathcal{L}_{\reg}(E,F): \abs{T}\abs{x} \in B \text{ for all }x\in S\}.
\]
\item Let $\mathcal I$ be an ideal of $\mathcal{L}_{\reg}(E,F)$. Then the subset
\[
\{x\in E: Tx \in B \text{ for all }T\in \mathcal I\}
\]
of $E$ is an ideal of $E$. For every subset $\mathcal S$ of $\mathcal I$ that generates $\mathcal I$, it is equal to
\[
\{x\in E: \abs{T}\abs{x} \in B \text{ for all }T\in \mathcal S\}.
\]
It is a band in $E$ when $\mathcal I\subseteq\mathcal{L}_{\oc}(E,F)$.
\end{enumerate} \end{proposition}
Let $F$ be a vector sublattice of a vector lattice $E$. Then $F$ is a \emph{regular vector sublattice of $E$} when the inclusion map from $F$ into $E$ is order continuous. Equivalently, for every net $\net$ in $F$, the fact that $x_\alpha\downarrow 0$ in $F $ should imply that $x_\alpha\downarrow 0$ in $E$. It is immediate from the latter criterion that ideals are regular vector sublattices. It is also true that order dense vector sublattices are regular vector sublattices; see \cite[Theorem~1.23]{aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003}, for example.
Let $\net$ be a net in a vector lattice $E$, and let $x\in E$. We say that $(x_\alpha)$ is \emph{unbounded order convergent to $x$ in $E$} (denoted by $x_\alpha\convwithoverset{\uo} x$) when $\abs{x_\alpha-x}\wedge y\convwithoverset{\mathrm{o}} 0$ in $E$ for all $y\in \pos{E}$. Order convergence implies unbounded order convergence to the same limit. For order bounded nets, the two notions coincide. \footnote{Although we shall not need this, it would be less than satisfactory not to mention here that the uo-continuous dual of a vector lattice (defined in the obvious way) has a very concrete description, and is often trivial. According to \cite[Proposition~2.2]{gao_leung_xanthos:2018}, it is the linear span of the coordinate functionals corresponding to atoms.}
We shall repeatedly refer to the following collection of results; see \cite[Theorem~2.8, Corollary~2.12, and Theorem~3.2]{gao_troitsky_xanthos:2017}.
\begin{theorem}\label{res:local-global_for_o-convergence_and_uo-convergence}
Let $E$ be a vector lattice, and let $F$ be a vector sublattice of $E$. Take a net $\net$ in $F$.
\begin{enumerate}
\item Suppose that $F$ is order dense and majorising in $E$. Then $x_\alpha\convwithoverset{\mathrm{o}} 0$ in $F$ if and only if $x_\alpha\convwithoverset{\mathrm{o}} 0$ in $E$.
\item\label{part:order_convergence_and_regular_sublattices} Suppose that $F$ is a regular vector sublattice of $E$ and that $\net$ is order bounded in $F$. Then $x_\alpha\convwithoverset{\mathrm{o}} 0$ in $F$ if and only if $x_\alpha\convwithoverset{\mathrm{o}} 0$ in $E$.
\item\label{part:unbounded_order_convergence_and_regular_sublattices} The following are equivalent:
\begin{enumerate}
\item $F$ is a regular vector sublattice of $E$;
\item for every net $\net$ in $F$, the fact that $x_\alpha\convwithoverset{\uo} 0 $ in $F$ implies that $x_\alpha\convwithoverset{\uo} 0$ in $E$;
\item for every net $\net$ in $F$, $x_\alpha\convwithoverset{\uo} 0 $ in $F$ if and only if $x_\alpha\convwithoverset{\uo} 0$ in $E$.
\end{enumerate}
\end{enumerate} \end{theorem}
In the sequel of this paper, we shall encounter restrictions of order continuous linear functionals on vector lattices to vector sublattices. For this, we include the following result. It is based on a theorem of Veksler's. It contains quite a bit more than we shall actually need, but we use the opportunity to present the results in it, and its fourth and fifth parts in particular.
\begin{theorem}\label{res:veksler}
Let $E$ be a vector lattice, let $F$ be a vector sublattice of $E$, and let $G$ be a Dedekind complete vector lattice. Take $T\in \mathcal{L}_{\oc}(E,G)$.
\begin{enumerate}
\item Suppose that $F$ is a regular vector sublattice of $E$. Then the restriction $T|_F:F\to G$ of $T$ to $F$ is order continuous.\label{part:veksler_1}
\item Suppose that $F$ is a regular sublattice of $E$. When $\mathcal{L}_{\oc}(E,G)$ separates the points of $E$, then $\mathcal{L}_{\oc}(F,G)$ separates the points of $F$.\label{part:veksler_2}
\item Suppose that $F$ is an order dense vector sublattice of $E$. Then the restriction map $T\mapsto T|_F$ is a positive linear injection from $\mathcal{L}_{\oc}(E,G)$ into $\mathcal{L}_{\oc}(F,G)$.\label{part:veksler_3}
\end{enumerate}
Suppose that $F$ is an order dense and majorising vector sublattice of $E$. Then:
\begin{enumerate}[resume]
\item the restriction map $T\mapsto T|_F$ is a lattice isomorphism between $\mathcal{L}_{\oc}(E,G)$ and $\mathcal{L}_{\oc}(F,G)$;\label{part:veksler_4}
\item $\mathcal{L}_{\oc}(E,G)$ separates the points of $E$ if and only if $\mathcal{L}_{\oc}(F,G)$ separates the points of $F$.\label{part:veksler_5}
\end{enumerate} \end{theorem}
\begin{proof}
Part~\ref{part:veksler_1} is obvious, and then so is part~\ref{part:veksler_2}.
It is clear from part~\ref{part:veksler_1} that $\mathcal{L}_{\oc}(F,G)$ separates the points of $F$ whenever $\mathcal{L}_{\oc}(E,G)$ separates the points of $E$.
Suppose that $F$ is an order dense (hence regular) vector sublattice of $E$ and that $T\in\mathcal{L}_{\oc}(E,G)$ is such that $T|_F=0$. Take $x\in \pos{E}$. Then $\{y\in F: 0\leq y\leq x\}\uparrow x$ in $E$. Since $T|_F=0$, the order continuity of $T$ on $E$ then implies that $Tx=0$. Hence $T=0$, and we conclude that the restriction map $T\mapsto T|_F$ is a positive linear injection from $\mathcal{L}_{\oc}(E,G)$ into $\mathcal{L}_{\oc}(F,G)$.
Suppose that $F$ is order dense and majorising in $E$.
Take $S\in\mathcal{L}_{\oc}(F,G)$. In that case, according to a result of Veksler's (see \cite[Theorem~1.65]{aliprantis_burkinshaw_POSITIVE_OPERATORS_SPRINGER_REPRINT:2006}), each of $\pos{S}$ and $\nega{S}$ can be extended to a positive order continuous operator from $E$ into $G$. Hence $S$ itself can be extended to an order continuous operator $S^{\text{ext}}$ from $E$ into $G$. By what we have already observed in part~\ref{part:veksler_3}, such an order continuous extension is unique, and we conclude from this that the map $S\mapsto S^{\text{ext}}$ is a positive linear injection from $\mathcal{L}_{\oc}(F,G)$ into $\mathcal{L}_{\oc}(E,G)$. It is clear that the extension and restriction maps between $\mathcal{L}_{\oc}(E,G)$ and $\mathcal{L}_{\oc}(F,G)$ are each other's inverses. We conclude that the restriction map $T\mapsto T|_F$ is a bi-positive linear bijection between $\mathcal{L}_{\oc}(E,G)$ and $\mathcal{L}_{\oc}(F,G)$. Hence it is a lattice isomorphism, as required.
One direction of the equivalence in part~\ref{part:veksler_5} is clear from part~\ref{part:veksler_2}. For the converse direction, suppose that $\mathcal{L}_{\oc}(F,G)$ separates the points of $F$. Take $x\in E$ such that $Tx=0$ for all $T\in \mathcal{L}_{\oc}(E,G)$. Since $\mathcal{L}_{\oc}(E,G)$ is an ideal of $\mathcal{L}_{\reg}(E,F)$, \cref{res:polars} shows that $T\abs{x}=0$ for all $T\in \mathcal{L}_{\oc}(E,G)$. Suppose that $x\neq 0$. Then there exists a $y\in F$ such that $0<y\leq\abs{x}$, and we have $Ty=0$ for all positive $T\in \mathcal{L}_{\oc}(E,G)$, hence for all $T\in \mathcal{L}_{\oc}(E,G)$. In view of part~\ref{part:veksler_4}, this is the same as saying that $Sy=0$ for all $S\in\mathcal{L}_{\oc}(F,G)$. Our assumption yields that $y=0$; this contradiction shows that we must have $x=0$. \end{proof}
\subsection{Topologies on vector lattices}
\noindent When $E$ is a vector space, a \emph{linear topology on $E$} is a (not necessarily Hausdorff) topology that provides $E$ with the structure of a topological vector space. When $E$ is a vector lattice, a \emph{locally solid linear topology on $E$} is a linear topology on $E$ such that there exists a base of (not necessarily open) neighbourhoods of 0 that are solid subsets of $E$. For the general theory of locally solid linear topologies on vector lattices we refer to \cite{aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003}. A locally solid linear topology on $E$ that is also a locally convex linear topology is a \emph{locally convex-solid linear topology}. In that case, there exists a base of neighbourhoods of 0 that consists of absorbing, closed, convex, and solid subsets of $E$; see \cite[p.~59]{aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003}.
When $E$ is a vector lattice, a \emph{locally solid additive topology on $E$} is a topology that provides the additive group $E$ with the structure of a (not necessarily Hausdorff) topological group, such that there exists a base of (not necessarily open) neighbourhoods of 0 that are solid subsets of $E$.
Let $E$ be a vector lattice. We say that \emph{order convergence in $E$ is topological} when there exists a (evidently unique) topology on $E$ such that its convergent nets are precisely the order convergent nets, with preservation of limits. It follows from the properties of order convergence that such a topology is automatically a Hausdorff linear topology. Likewise, \emph{unbounded order convergence in $E$ is topological} when there exists a topology on $E$ such that its convergent nets are precisely the nets that are unbounded order convergent, with preservation of limits. Such a topology is again unique, and automatically a Hausdorff linear topology.
A topology $\tau$ on a vector lattice $E$ is an \emph{o-Lebes\-gue topology} when it is a (not necessarily Hausdorff) locally solid linear topology on $E$ such that, for a net $\net$ in $E$ and $x\in E$, the fact that $x_\alpha\convwithoverset{\mathrm{o}} x$ in $E$ implies that $x_\alpha\convwithoverset{\tau} x$. Equivalently, the fact that $x_\alpha\convwithoverset{\mathrm{o}} 0$ in $E$ should imply that $x_\alpha\convwithoverset{\tau} 0$. A vector lattice need not admit a Hausdorff o-Lebes\-gue topology. It can be shown, see \cite[Example~3.2]{aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003}, that $\mathrm{C}([0,1])$ does not even admit a Hausdorff locally solid linear topology such that \emph{sequential} order convergence implies topological convergence.
A topology $\tau$ on a vector lattice $E$ is a \emph{uo-Lebes\-gue topology} when it is a (not necessarily Hausdorff) locally solid linear topology on $E$ such that, for a net $\net$ in $E$ and $x\in E$, the fact that $x_\alpha\convwithoverset{\uo} x$ in $E$ implies that $x_\alpha\convwithoverset{\tau} x$. Equivalently, the fact that $x_\alpha\convwithoverset{\uo} 0$ in $E$ should imply that $x_\alpha\convwithoverset{\tau} 0$. Since order convergence implies unbounded order convergence, a uo-Lebes\-gue topology\ is an o-Lebes\-gue topology.
The following fundamental facts are from \cite[Proposition~3.2,~3.4, and~6.2, and Corollary~6.3]{conradie:2005} and \cite[Theorems~5.5,~5.9, and~6.4]{taylor:2019}.
\begin{theorem}[Conradie and Taylor]\label{res:conradie_taylor}
Let $E$ be a vector lattice. Then the following are equivalent:
\begin{enumerate}
\item $E$ admits a Hausdorff o-Lebes\-gue topology;
\item $E$ admits a Hausdorff uo-Lebes\-gue topology;\label{part:uoLtop}
\item the partially ordered set of all Hausdorff locally solid linear topologies on $E$ has a minimal element.\label{part:mintop}
\end{enumerate} When this is the case, the topologies in the parts~\ref{part:uoLtop} and~\ref{part:mintop} are both unique, they coincide, and they are the smallest Hausdorff o-Lebes\-gue topology\ on $E$. \end{theorem}
When $E$ admits a Hausdorff uo-Lebes\-gue topology, we shall denote the unique such topology by ${\widehat{\tau}}_E$. In \cite{conradie:2005}, it is denoted by $\tau_m$. For a given vector lattice, there may be several ways to obtain a Hausdorff uo-Lebes\-gue topology\ on it. This can then give criteria for the convergence of nets in the common resulting topology that are apparently equivalent, but not always immediately obviously so. See \cref{rem:various_descriptions} for this, for example.
\begin{remark}\label{rem:minimal_and_smallest}
Some caution is necessary when consulting the literature on minimal Hausdorff locally solid linear topologies because in \cite[Definition~7.64]{aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003} such a topology is defined as what would usually be called a \emph{smallest} Hausdorff locally solid linear topologies. When a vector lattice $E$ admits a complete metrisable o-Lebes\-gue topology, such as a Banach lattice with an order continuous norm, then it admits a smallest (in the usual sense of the word) Hausdorff locally solid linear topology; see \cite[Theorem~7.65]{aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003}. Combining this with \cref{res:conradie_taylor}, we see that $E$ then admits a (necessarily unique) Hausdorff uo-Lebes\-gue topology\ ${\widehat{\tau}}_E$, and that ${\widehat{\tau}}_E$ is then not just the smallest Hausdorff o-Lebes\-gue topology, but even the smallest Hausdorff locally solid linear topology on $E$. \end{remark}
\section{Unbounded topologies generated by topologies on ideals}\label{sec:unbounded_topologie_generated_by_topologies_on_ideals}
\noindent We shall now describe how topologies `of unbounded type' on vector lattices can be obtained from topologies on ideals. There are already several constructions in this vein and accompanying results in the literature; see \cite{conradie:2005,deng_o_brien_troitsky:2017,kandic_li_troitsky:2018,kandic_marabeh_troitsky:2017,taylor_THESIS:2018,taylor:2019}, for example. In the following result, we carry out such a construction in what appears to be the most general possible context. Starting from a locally solid (not necessarily linear or Hausdorff) additive topology on an ideal $F$ of a vector lattice $E$ and a non-empty subset of $F$, we define an `unbounded' locally solid additive topology on $E$. Various known results in more special cases can then be understood from the general theorem, as will be discussed in \crefrange{exam:taylor:2019}{exam:conradie:2005}, below.
It is important to note that, in our construction, the initial topology on the ideal $F$ need not be the restriction of a global topology on $E$. For such restricted topologies, several results are already available in \cite[Propositions~9.3 and~9.4]{taylor:2019}. It will, however, become clear in \cref{rem:essential_difference}, below, that such a global topology (of a suitable type) need not always exist. Hence there is an actual gain by starting from topologies on ideals, and a concrete illustration of this procedure can be found in \cref{sec:vector_lattices_of_equivalence_classes_of_measurable_functions}. The possibility of working in such a general setting was already observed in \cite[Remark~2.8, Example~2.9, and Remark~2.10]{taylor_THESIS:2018}, where it was also noted that the continuity of the scalar multiplication may fail to hold in the new topology. Part~\ref{part:last} of \cref{res:generated_topology}, below, gives necessary and sufficient conditions for this continuity.
The subset $S$ figuring in the construction can be replaced by the ideal that it generates without altering the result. Although it may conceptually be more natural to work with ideals than with subsets, working with arbitrary subsets has the advantage of keeping an eye on a small number of presumably relatively easily manageable `test elements'. It is for this reason that we carry this along to later results; see also \cref{rem:small_subset_criterion}, below. The convenience of this approach will become apparent in the proof of \cref{res:tau_m_is_convergence_in_measure}.
\begin{theorem}\label{res:generated_topology}
Let $E$ be a vector lattice, let $F$ be an ideal of $E$, and let $\tau_F$ be a \uppars{not necessarily Hausdorff} locally solid additive topology on $F$. Take a non-empty subset $S$ of $F$.
There exists a unique \uppars{possibly non-Hausdorff} additive topology $\mathrm{u}_S\!\tau_{F}$ on $E$ such that, for a net $\net$ in $E$, $x_\alpha\xrightarrow{\mathrm{u}_S\!\tau_{F}} 0$ in $E$ if and only if $\abs{x_\alpha}\wedge \abs{s}\xrightarrow {\tau_F}0$ in $F$ for all $s\in S$.
Let $I_S\subseteq F$ be the ideal generated by $S$ in $E$. For a net $\net$ in $E$, $x_\alpha\xrightarrow{\mathrm{u}_S\!\tau_{F}} 0$ in $E$ if and only if $\abs{x_\alpha}\wedge \abs{y}\xrightarrow {\tau_F}0$ in $F$ for all $y\in I_S$.
Furthermore:
\begin{enumerate}
\item\label{part:first} the inclusion map from $F$ into $E$ is $\tau_F$--$\mathrm{u}_S\!\tau_F$-continuous;
\item the topology $\mathrm{u}_S\!\tau_{F}$ on $E$ is a locally solid additive topology;
\item the following are equivalent:
\begin{enumerate}
\item $\mathrm{u}_S\!\tau_{F}$ is a Hausdorff topology on $E$;\label{test}
\item $\tau_F$ is a Hausdorff topology on $F$ and $I_S$ is order dense in $E$;
\end{enumerate}
\item\label{part:last} the following are equivalent:
\begin{enumerate}
\item\label{part:linear_condition_i} for all $x\in E$ and $s\in S$,
\begin{equation}\label{eq:absorbing_condition}
\abs{\varepsilon x}\wedge \abs{s}\xrightarrow{\tau_F} 0
\end{equation}
in $F$ as $\varepsilon\to 0$ in $\mathbb R$;
\item\label{part:linear_condition_ii} for all $x\in E$ and $y\in I_S$, $\abs{\varepsilon x}\wedge \abs{y}\xrightarrow{\tau_F} 0$ in $F$
as $\varepsilon\to 0$ in $\mathbb R$;
\item\label{part:linear_condition_iii} $\mathrm{u}_S\!\tau_F$ is a \uppars{possibly non-Hausdorff} linear topology on $E$.
\end{enumerate}
\end{enumerate} \end{theorem}
\begin{proof}
Suppose that $\tau_F$ is a (not necessarily Hausdorff) locally solid additive topology on $F$.
The uniqueness of $\mathrm{u}_S\!\tau_{F}$ is clear because the nets converging to 0 and then, by translation invariance of the topology, to arbitrary points of $E$ are prescribed.
We turn to the existence of such a topology $\mathrm{u}_S\!\tau_{F}$. Take a neighbourhood base $\{U_\lambda\}_{\lambda\in\Lambda}$ of zero in $F$ for $\tau_F$ consisting of solid subsets of $F$. For $y\in I_S$ and $\lambda\in\Lambda$, set
\begin{equation}\label{eq:local_base_1}
V_{\lambda,y}\coloneqq\{x\in E: \abs{x}\wedge\abs{y}\in U_\lambda\}.
\end{equation}
The $V_{\lambda,y}$ are solid subsets of $E$ since $F$ is an ideal of $E$ and the $U_\lambda$ are solid subsets of $F$. Set
\begin{equation}\label{eq:local_base_2}
\mathcal{N}_0\coloneqq \{V_{\lambda,y}: \lambda\in \Lambda, y\in I_S \}.
\end{equation}
We claim that $\mathcal{N}_0$ is a base of neighbourhoods of zero for a topology on $E$, which we shall already denote by $\mathrm{u}_S\!\tau_{F}$, that provides the additive group $E$ with the structure of a topological group. Necessary and sufficient conditions on $\mathcal N_0$ for this can be found in \cite[Theorem~3 on p.46]{husain_INTRODUCTION-TO_TOPOLOGICAL_GROUPS:1966}; we now verify these.
Take $V_{\lambda_1,y_1},V_{\lambda_2,y_2}\in \mathcal{N}_0$. There exists a $\lambda_3\in\Lambda$ such that $U_{\lambda_3}\subseteq U_{\lambda_1}\cap U_{\lambda_2}$. Take $x\in V_{\lambda_3,\abs{y_1}\vee\abs{y_2}}$. Then
\[
\abs{x}\wedge\abs{y_1}\leq\abs{x}\wedge\left({\abs{y_1}\vee\abs{y_2}}\right)\in U_{\lambda_3}\subseteq U_{\lambda_1}.
\]
Since $F$ is an ideal of $E$ and $U_{\lambda_1}$ is a solid subset of $F$, this implies that $\abs{x}\wedge\abs{y_1}\in U_{\lambda_1}$, so that $x\in V_{\lambda_1,y_1}$. Likewise, $x\in V_{\lambda_2,y_2}$, and we see that $V_{\lambda_3,\abs{y_1}\vee\abs{y_2}}\subseteq V_{\lambda_1,y_1}\cap V_{\lambda_2,y_2}$.
It is evident that $V_{\lambda,y}=-V_{\lambda,y}$ for all $V_{\lambda,y}\in\mathcal N_0$.
Take $V_{\lambda,y}\in\mathcal N_0$. There exists a $\mu\in\Lambda$ such that $ U_\mu + U_\mu \subseteq U_\lambda$. Then, for all $x_1,x_2\in V_{\mu,y}$, we have
\[
\abs{x_1+x_2}\wedge \abs{y}\leq \abs{x_1}\wedge\abs{y}+\abs{x_2}\wedge\abs{y}\in U_\mu+U_\mu\subseteq U_\lambda.
\]
Since $F$ is an ideal of $E$ and $U_\lambda$ is a solid subset of $F$, this implies that $\abs{x_1+x_2}\wedge \abs{y}\in U_{\lambda}$, so that $x_1+x_2\in V_{\lambda,y}$. Hence $V_{\mu,y}+V_{\mu,y}\subseteq V_{\lambda,y}$.
An appeal to \cite[p.~46, Theorem~3]{husain_INTRODUCTION-TO_TOPOLOGICAL_GROUPS:1966} now establishes our claim.
It is clear from the definition of $\mathrm{u}_S\!\tau_F$ that, for a net $\net$ in $E$, $x_\alpha\xrightarrow{\mathrm{u}_S\!\tau_{F}} 0$ in $E$ if and only if $\abs{x_\alpha}\wedge \abs{y}\xrightarrow {\tau_F}0$ in $F$ for all $y\in I_S$; this observation goes back to \cite[Remark~2.3]{taylor_THESIS:2018}.
Certainly, the fact that $\abs{x_\alpha}\wedge \abs{y}\xrightarrow {\tau_F}0$ in $F$ for all $y\in I_S$ implies that $\abs{x_\alpha}\wedge \abs{s}\xrightarrow {\tau_F}0$ in $F$ for all $s\in S$. Conversely, suppose that $\net$ is a net in $E$ such that $\abs{x_\alpha}\wedge \abs{s}\xrightarrow {\tau_F}0$ in $F$ for all $s\in S$. Take $y\in I_S$. There exist $s_1,\dotsc,s_n\in S$ and integers $k_1,\dotsc,k_n\geq 1$ such that $\abs{y}\leq \sum_{i=1}^{n} k_i\abs{s_i}$. Hence $\abs{x_\alpha}\wedge \abs{y}\leq \sum_{i=1}^{n} k_i \left(\abs{x_\alpha}\wedge \abs{s_i}\right)$. Since $\tau_F$ is a locally solid additive topology on $F$, this implies that $\abs{x_\alpha}\wedge \abs{y}\xrightarrow{\tau_F} 0$ in $F$.
We turn to the parts~\ref{part:first}--\ref{part:last}.
Since $F$ is an ideal of $E$ and the $U_\lambda$ are solid subsets of $F$, we have $U_\lambda\subseteq V_{\lambda,y}$ for all $\lambda\in\Lambda$ and $y\in I_S$. This implies that the inclusion map from $F$ into $E$ is $\tau_F$--$\mathrm{u}_S\!\tau_F$-continuous.
The topology $\mathrm{u}_S\!\tau_F$ is a locally solid additive topology on $E$ by construction.
Suppose that $\mathrm{u}_S\!\tau_{F}$ is a Hausdorff topology on $E$. Then so is the topology it induces on $F$, which is weaker than $\tau_F$. Hence $\tau_F$ is a Hausdorff topology on $F$. Take $x\in E$ with $x>0$. Then there exists a $V_{\lambda,y}\in \mathcal{N}_0$ with $x\notin V_{\lambda,y}$. In particular, $x\wedge\abs{y}\neq 0$. Hence $0<x\wedge\abs{y} \leq x$. Since $x\wedge\abs{y}\in I_S$, we see that $I_S$ is order dense in $E$.
Suppose, conversely, that $\tau_F$ is a Hausdorff topology on $F$ and that $I_S$ is order dense in $E$. Take $x\neq 0$ in $E$. There exists a $y\in I_S$ with $0<y\leq \abs{x}$. Pick $U_{\lambda_0}\in \{U_\lambda\}_{\lambda\in \Lambda}$ such that $y\notin U_{\lambda_0}$. Then $\abs{x}\wedge \abs{y}=y\notin U_{\lambda_0}$, so that $x\notin V_{\lambda_0,y}$. Hence $\bigcap_{V\in\mathcal N_0}V=\{0\}$. By \cite[p.~48, Theorem~4]{husain_INTRODUCTION-TO_TOPOLOGICAL_GROUPS:1966}, $\mathrm{u}_S\!\tau_F$ is a Hausdorff additive topology on the topological group $E$.
We shall now verify the equivalence of the parts~\ref{part:linear_condition_i}--\ref{part:linear_condition_iii} of part~\ref{part:last}.
We prove that part~\ref{part:linear_condition_i} implies part~\ref{part:linear_condition_ii}. Take $x\in E$ and $y\in I_S$. There exist $s_1,\dotsc,s_n\in S$ and integers $k_1,\dotsc,k_n\geq 1$ such that $\abs{y}\leq \sum_{i=1}^{n} k_i\abs{s_i}$, and it follows from this that $\abs{\varepsilon x}\wedge \abs{y}\leq \sum_{i=1}^{n} k_i\left(\abs{\varepsilon x}\wedge \abs{s_i}\right)$ for all $\varepsilon\in\mathbb R$. Since $\tau_F$ is a locally solid additive topology on $F$, it follows that $\abs{\varepsilon x}\wedge \abs{y}\xrightarrow{\tau_F} 0$ in $F$ as $\varepsilon\to 0$ in $\mathbb R$.
We prove that part~\ref{part:linear_condition_ii} implies part~\ref{part:linear_condition_iii}. Fix $\lambda\in\Lambda$ and $y\in I_S$, and take $x\in E$. Since $\abs{\varepsilon x}\wedge \abs{y}\xrightarrow{\tau_F} 0$ in $F$ as $\varepsilon\to 0$ in $\mathbb R$, there exists a $\delta>0$ such that $\abs{\varepsilon x}\wedge \abs{y}\in U_\lambda$ whenever $\abs{\varepsilon}\leq\delta$. That is, $\varepsilon x\in V_{\lambda,y}$ whenever $\abs{\varepsilon}\leq\delta$. This implies that $V_{\lambda,y}$ is absorbing. Furthermore, since $V_{\lambda,y}$ is a solid subset of $E$, it is clear that $\varepsilon x\in V_{\lambda,y}$ whenever $x\in V_{\lambda,y}$ and $\varepsilon\in[-1,1]$. Hence $V_{\lambda,y}$ is balanced.
Then \cite[Theorem 5.6]{aliprantis_border_INFINITE_DIMENSIONAL_ANALYSIS_THIRD_EDITION:2006} implies that $\mathrm{u}_S\!\tau_F$ is a linear topology on $E$.
We prove that part~\ref{part:linear_condition_iii} implies part~\ref{part:linear_condition_i}. Take $x\in E$. Then $\varepsilon x\xrightarrow{\mathrm{u}_S\!\tau_F} 0$ in $E$ as $\varepsilon\to 0$ in $\mathbb R$. By construction, this implies (and is, in fact, equivalent to) the fact that $\abs{\varepsilon x}\wedge \abs{s}\xrightarrow {\tau_F}0$ in $F$ for all $s\in S$.
This concludes the proof of the equivalence of the three parts of part~\ref{part:last}. The proof of the theorem is now complete.
\end{proof}
\begin{definition}
The topology $\mathrm{u}_S\!\tau_{F}$ in \cref{res:generated_topology,res:generated_topology} is called the \emph{unbounded topology on $E$ that is generated by $\tau_F$ via $S$}. \end{definition}
\begin{remark}\label{rem:two_sets_give_the_same_topology}
It is clear from the two equivalent criteria in \cref{res:generated_topology} for a net in $E$ to be $\mathrm{u}_S\!\tau_F$-convergent to zero that $\mathrm{u}_S\!\tau_F=\mathrm{u}_{I_S}\!\tau_F$ for every non-empty subset $S$ of $F$. Consequently, $\mathrm{u}_{S_1}\!\tau_F=\mathrm{u}_{S_2}\!\tau_F$ whenever $S_1,S_2$ are non-empty subsets of $F$ such that $I_{S_1}=I_{S_2}$. \end{remark}
\begin{remark}
In \cref{res:generated_topology}, suppose that the locally solid additive topology $F$ is the restriction $\tau_E|_F$ of a locally solid additive topology on $E$. It is then easy to see that $\mathrm{u}_S\!\left(\tau_E|_F\right)=\mathrm{u}_S\!\tau_E$ for every non-empty subset $S$ of $F$. \end{remark}
\begin{remark}\label{rem:explicit_neighbourhood_base}
In \cref{res:generated_topology}, and also in the remainder of this paper, the topologies of interest are characterised by their convergent nets. It should be noted, however, that in \cref{eq:local_base_1,eq:local_base_2} the proof of \cref{res:generated_topology} provides an explicit neighbourhood base of zero in $E$ for $\mathrm{u}_S\!\tau_F$, in terms of a neighbourhood base of zero in $F$ for $\tau_F$ and the ideal $I_S$. Suppose, for example that $\tau_F$ is a (possibly non-Hausdorff) locally convex-solid linear topology on $F$ that is generated by a family $\{\rho_\gamma: \gamma\in \Gamma\}$ of lattice semi-norms on $F$, as will be the case in \cref{sec:separating_order_continuous_dual}. Then the collection of subsets of $E$ of the form
\[
\{x\in E: \rho_i(\abs{x}\wedge\abs{y})<\varepsilon\text{ for } \rho_1,\dotsc\rho_n\in\Gamma\},
\]
where $y\in I_S$, $n\geq 1$, and $\varepsilon>0$ are arbitrary, is a neighbourhood base of zero in $E$ for $\mathrm{u}_S\!\tau_F$. \end{remark}
Our next result is concerned with iterating the construction in \cref{res:generated_topology}. It generalises what is in \cite[p.~997]{taylor:2019}.
\begin{proposition}\label{res:repeating_the_construction}
Let $E$ be a vector lattice, let $F_1$ be an ideal of $E$, and let $\tau_{F_1}$ be a \uppars{not necessarily Hausdorff} locally solid additive topology on $F_1$. Take a non-empty subset $S_1$ of $F_1$, and consider the unbounded topology $\mathrm{u}_{S_1}\!\!\tau_{F_1}$ on $E$ that is generated by $\tau_{F_1}$ via $S_1$. Let $F_2$ be an ideal of $E$, and let $\left(\mathrm{u}_{S_1}\!\!\tau_{F_1}\right)|_{F_2}$ denote the topology on $F_2$ that is induced on $F_2$ by $\mathrm{u}_{S_1}\!\!\tau_{F_1}$. Then $\left(\mathrm{u}_{S_1}\!\!\tau_{F_1}\right)|_{F_2}$ is a locally solid additive topology on $F_2$. Take a non-empty subset $S_2$ of $F_2$. Then $\mathrm{u}_{S_2}\!\left[\left(\mathrm{u}_{S_1}\!\!\tau_{F_1}\right)|_{F_2}\right]=\mathrm{u}_{I_{S_1}\cap I_{S_2}}\!\!\tau_{F_1}$.
In particular, when $S$ is a non-empty subset of $F_1\cap F_2$, then $\mathrm{u}_{S}\!\left[\left(\mathrm{u}_{S}\!\!\tau_{F_1}\right)|_{F_2}\right]=\mathrm{u}_{S}\!\!\tau_{F_1}$. \end{proposition}
\begin{proof}
It is clear from \cref{res:generated_topology} that $\left(\mathrm{u}_{S_1}\tau_{F_1}\right)|_{F_2}$ is a locally solid additive topology on $F_2$. Let $\net$ be a net in $E$. Then we have the following chain of equivalent statements: \begin{align*}
&x_\alpha\xrightarrow{\mathrm{u}_{S_2}\left[\left(\mathrm{u}_{S_1}\tau_{F_1}\right)|_{F_2}\right]}0 \text{ in }E\\\
&\iff \abs{x_\alpha}\wedge \abs{y_2}\xrightarrow{\left(\mathrm{u}_{S_1}\tau_{F_1}\right)|_{F_2}}0 \text{ in }F_2\text{ for all }y_2\in I_{S_2}\\ &\iff \abs{x_\alpha}\wedge \abs{y_2}\xrightarrow{\mathrm{u}_{S_1}\tau_{F_1} }0 \text{ in }E\text{ for all }y_2\in I_{S_2}\\ &\iff \abs{x_\alpha}\wedge \abs{y_2}\wedge\abs{y_1}\xrightarrow{\tau_{F_1}}0 \text{ in }F_1\text{ for all }y_1\in I_{S_1}\text{ and } y_2\in I_{S_2}\\ &\iff \abs{x_\alpha}\wedge \abs{y}\xrightarrow{\tau_{F_1}}0 \text{ in }F_1\text{ for all }y\in I_{S_1}\cap I_{S_2}\\ &\iff x_\alpha\xrightarrow{\mathrm{u}_{I_{S_1}\cap I_{S_2}}\tau_{F_1}}0 \text{ in }E. \end{align*}
Hence $\mathrm{u}_{S_2}\!\left[\left(\mathrm{u}_{S_1}\!\!\tau_{F_1}\right)|_{F_2}\right]=\mathrm{u}_{I_{S_1}\cap I_{S_2}}\!\!\tau_{F_1}$. \end{proof}
\begin{remark}
In \cref{res:repeating_the_construction}, suppose that $\tau_{F_1}$ is a \uppars{not necessarily Hausdorff} locally solid additive topology on $F_1$ such that, for all $x\in E$ and $s\in S_1$, $\abs{\varepsilon x}\wedge \abs{s}\xrightarrow{\tau_{F_1}} 0$ in $F_1$ as $\varepsilon\to 0$ in $\mathbb R$. It is then clear from \cref{res:generated_topology} that $\mathrm{u}_{S_1} \!\!\tau_{F_1}$, $\left(\mathrm{u}_{S_1}\!\!\tau_{F_1}\right)|_{F_2}$, and $\mathrm{u}_{I_{S_1}\cap I_{S_2}}\!\!\tau_{F_1}$ are \uppars{possibly non-Hausdorff} locally solid linear topologies on $E$, $F_2$, and $E$, respectively.
\end{remark}
We shall now explain how \cref{res:generated_topology} relates to various results already in the literature.
\begin{example}\label{exam:taylor:2019}
When $F=E$ and $\tau_E$ is a locally solid linear topology on $F=E$, the condition in \cref{eq:absorbing_condition} is automatically satisfied for any non-empty subset $S$ of $F=E$. According to \cref{res:generated_topology}, $\mathrm{u}_E\!\tau_E$ is a locally solid linear topology on $E$ that is Hausdorff if and only if $\tau_E$ is Hausdorff; this is \cite[Theorem~2.3]{taylor:2019}. Furthermore, when $A$ is an ideal of $E$, $\mathrm{u}_A\!\tau_E$ is a locally solid linear topology on $E$ that is Hausdorff if and only if $\tau_E$ is Hausdorff and $A$ is order dense in $E$; this is \cite[Propositions~9.3 and ~9.4]{taylor:2019}. \end{example}
\begin{example}\label{exam:deng_o'brien_troitsky:2018}
Let $E$ be a Banach lattice. In \cref{res:generated_topology}, we take $F=E$, for $\tau_F$ we take the norm topology $\tau_E$ on $F=E$, and for $S\subseteq F$ we take $S=F=E$. Then the condition in \cref{eq:absorbing_condition} is satisfied. According to \cref{res:generated_topology}, $\mathrm{u}_E\!\tau_E$ is a Hausdorff locally solid linear topology on $E$ and, for a net $\net$ in $E$, $x_\alpha\xrightarrow{\mathrm{u}_E\!\tau_E}0$ if and only if $\norm{\abs{x_\alpha}\wedge\abs{y}}\to 0$ for all $y\in E$. In \cite{deng_o_brien_troitsky:2017}, this type of convergence is called \emph{unbounded norm convergence}, or \emph{un-convergence} for short. It was already observed in \cite[Section~7]{deng_o_brien_troitsky:2017} that it is topological; in \cite[p.~746]{kandic_li_troitsky:2018}, $\mathrm{u}_F\!\tau_F$ is then called the \emph{un-topology}. \end{example}
\begin{example}\label{exam:kandic_li_troitsky:2018}
Let $E$ be a vector lattice, and let $F$ be an ideal of $E$ that is a normed vector lattice. In \cref{res:generated_topology}, we take for $\tau_F$ the norm topology on $F$, and for $S\subseteq F$ we take $S=F$. According to \cref{res:generated_topology}, $\mathrm{u}_F\!\tau_F$ is a (possibly non-Hausdorff) additive topology on $E$ and, for a net $\net$ in $E$, $x_\alpha\xrightarrow{\mathrm{u}_F\!\tau_F}0$ if and only if $\norm{\abs{x_\alpha}\wedge\abs{y}}\to 0$ for all $y\in F$. This type of convergence is called \emph{un-convergence with respect to $X$} in \cite{kandic_li_troitsky:2018}. It was already observed that it is topological in \cite[p.~747]{kandic_li_troitsky:2018}, where $\mathrm{u}_F\!\tau_F$ is called the \emph{un-topology on $E$ induced by $F$}.
In \cite[Example~1.3]{kandic_li_troitsky:2018}, it is shown that $\mathrm{u}_F\!\tau_F$ can fail to be a Hausdorff topology on $E$. Since $\tau_F$ is a Hausdorff topology on $F$, \cref{res:generated_topology} shows that the pertinent ideal $F$ in \cite[Example~1.3]{kandic_li_troitsky:2018} must fail to be order dense in $E$; this is indeed easily seen to be the case. \cref{res:generated_topology} implies that $\mathrm{u}_F\!\tau_F$ is Hausdorff if and only if $F$ is order dense in $F$; this is \cite[Proposition~1.4]{kandic_li_troitsky:2018}.
In \cite[Example~1.5]{kandic_li_troitsky:2018}, it is shown that $\mathrm{u}_F\!\tau_F$ can fail to be a linear topology on $E$. According to \cref{res:generated_topology}, the condition in \cref{eq:absorbing_condition} must fail to be satisfied in the context of \cite[Example~1.5]{kandic_li_troitsky:2018}; this is indeed easily seen to be the case. \cref{res:generated_topology} shows that $\mathrm{u}_F\!\tau_F$ always provides $E$ with an additive topology; this was also noted in \cite[p.~748]{kandic_li_troitsky:2018} in that particular context.
In \cite[p.~748]{kandic_li_troitsky:2018}, the authors observe that $\mathrm{u}_F\!\tau_F$ is a locally solid linear topology on the vector lattice $E$ whenever $E$ is a normed lattice and the norm on $E$ extends that on $F$, and also whenever the norm on $F$ is order continuous. Both facts follow from \cref{res:generated_topology} because \cref{eq:absorbing_condition} is then satisfied. This is clear when $E$ is a normed lattice and the norm on $E$ extends that on $F$. Suppose that the norm on $F$ is order continuous. Take $x\in E$ and $y\in F$. Then $\abs{\varepsilon x}\wedge \abs{y}\convwithoverset{\mathrm{o}} 0$ in $E$ as $\varepsilon\to 0$. Since the net $ \abs{\varepsilon x}\wedge \abs{y}$ is order bounded in the ideal $F$ of $E$, which is a regular vector sublattice of $E$, \cref{res:local-global_for_o-convergence_and_uo-convergence} implies that $\abs{\varepsilon x}\wedge \abs{y}\convwithoverset{\mathrm{o}} 0$ in $F$, and then $\abs{\varepsilon x}\wedge \abs{y}\xrightarrow{\tau_F}0$ as $\varepsilon\to 0$ by the order continuity of the norm on $F$. \end{example}
\begin{example}\label{exam:zabeti:2018} For a vector lattice $E$, we let $\abs{\sigma}(E,\odual{E})$ denote its absolute weak topology; the definition of this locally solid linear topology will be recalled in \cref{sec:separating_order_continuous_dual}. Taking $E=F=S$ in \cref{res:generated_topology} yields the so-called \emph{unbounded absolute weak topology $\mathrm{u}_E\abs{\sigma}(E,\odual{E})$} on $E$. It is a locally solid additive topology on $E$ that is Hausdorff if and only if $\odual{E}$ separates the points of $E$. When $E$ is a Banach lattice, $\mathrm{u}_E\abs{\sigma}(E,\odual{E})$ is a Hausdorff locally solid linear topology on $E$. It is studied in \cite{zabeti:2018}.
\end{example}
\begin{example}\label{exam:conradie:2005}
In \cite[p.~290]{conradie:2005}, a construction is given to obtain a locally solid linear topology on a vector lattice $E$ from a locally solid linear topology on an ideal $F$ of $E$. This is done using Riesz pseudo-norms, rather than by working with neighbourhood bases of zero as we have done. The key ingredient is to start with a Riesz pseudo-norm $p$ on $F$, take an element $u$ of $\pos{F}$, and introduce a map $p_u:E\to\mathbb R$ by setting $p_u(x)\coloneqq p(\abs{x}\wedge u)$ for $x\in E$. It is then remarked that $p_u$ is a Riesz pseudo-norm on $E$. This need not always be the case, however. By way of counter-example, take for $E$ the vector lattice of all real-valued functions on $\mathbb R$, and for $F$ the ideal of $E$ consisting of all bounded functions on $\mathbb R$. For $p$, we take the supremum norm on $F$. For $u\in\pos{F}$, we choose the constant function 1. We define $x\in E$ by setting $x(t)\coloneqq t$ for $t\in\mathbb R$. Then $p_u(\lambda x)=\norm{\abs{\lambda x}\wedge u}=1$ for all non-zero $\lambda\in\mathbb R$, whereas we should have that $\lim_{\lambda\to 0} p_u(\lambda x)=0$. This implies that the topologies on $E$ that are thus constructed, although locally solid additive topologies, need not be linear topologies. This `pathology' is similar to that in \cite[Example~1.5]{kandic_li_troitsky:2018} that was mentioned above; our example here is also quite similar to that in \cite[Example~1.5]{kandic_li_troitsky:2018}. Fortunately, in the continuation of the argument in \cite{conradie:2005}, $p$ is taken to be a Riesz pseudo-norm on $F$ that is continuous with respect to a Hausdorff o-Lebes\-gue topology\ $\tau_F$ on $F$. In this context, $p_u$ \emph{is} a Riesz pseudo-norm on $E$. Indeed, since $F$, being an ideal of $E$, is a regular vector sublattice of $E$, \cref{res:local-global_for_o-convergence_and_uo-convergence} easily yields that $\abs{\lambda x}\wedge u\convwithoverset{\mathrm{o}} 0$ in $F$ as $\lambda\to 0$. Since $\tau_F$ is an o-Lebes\-gue topology\ on $E$, we have $\abs{\lambda x}\wedge u\xrightarrow{\tau_F} 0$ in $F$ as $\lambda\to 0$, and then the continuity of $p$ on $F$ yields that $p_u(\lambda x)\to 0$ as $\lambda\to 0$. Thus the construction in \cite{conradie:2005} proceeds correctly after all. The results of our systematic investigation with minimal hypotheses in \cref{res:generated_topology}, however, are more comprehensive than those in \cite{conradie:2005}. \end{example}
\section{Hausdorff uo-Lebes\-gue topologies: going up and going down}\label{sec:uoLtops_going_up_and_going_down}
\noindent In this section, we investigate how, via a going-up-going-down construction, the existence of a Hausdorff o-Lebes\-gue topology\ on an order dense ideal of a vector lattice $E$ implies that every regular vector sublattice of $E$ admits a (necessarily unique) Hausdorff uo-Lebes\-gue topology.
We start by going up.
\begin{proposition}\label{res:preparation_uoLt}
Let $E$ be a vector lattice, and let $F$ be an ideal of $E$. Suppose that $F$ admits an o-Lebes\-gue topology\ $\tau_F$. Choose a non-empty subset $S$ of $F$. Then $\mathrm{u}_S\!\tau_F$ is a uo-Lebes\-gue topology\ on $E$. It is a \uppars{necessarily unique} Hausdorff uo-Lebes\-gue topology\ on $E$ if and only if $\tau_F$ is a Hausdorff topology on $F$ and the ideal $I_S$ that is generated by $S$ is order dense in $E$.
\end{proposition}
\begin{proof}
We know from \cref{res:generated_topology} that $\mathrm{u}_S\!\tau_F$ is a locally solid additive topology on $E$. In order to see that it is a linear topology on $E$, we verify the condition in \cref{eq:absorbing_condition}. Take $x$ in $E$ and $s$ in $S$. Then $\abs{\varepsilon x}\wedge \abs{s}\convwithoverset{\mathrm{o}} 0$ in $E$ as $\varepsilon\to 0$ in $\mathbb R$. Since $F$, being in ideal of $E$, is a regular vector sublattice of $E$, \cref{res:local-global_for_o-convergence_and_uo-convergence} shows that $\abs{\varepsilon x}\wedge\abs{s}\convwithoverset{\mathrm{o}} 0$ in $F$. Since $\tau_F$ is an o-Lebes\-gue topology\ on $F$, this implies that $\abs{\varepsilon x}\wedge\abs{s}\xrightarrow{\tau_F} 0$ in $F$ as $\varepsilon\to 0$ in $\mathbb R$, as required.
To conclude the proof, suppose that $\net$ is a net in $E$ such that $x_\alpha\convwithoverset{\uo} 0$ in $E$. Take $s\in S$. Then $\abs{x_\alpha}\wedge \abs{s}\convwithoverset{\mathrm{o}} 0$ in $E$. Again, since $F$ is a regular vector sublattice of $E$, \cref{res:local-global_for_o-convergence_and_uo-convergence} shows that $\abs{x_\alpha}\wedge \abs{s}\convwithoverset{\mathrm{o}} 0$ in $F$. Since $\tau_F$ is an o-Lebes\-gue topology\ on $F$, this implies that $\abs{x_\alpha}\wedge \abs{s}\xrightarrow{\tau_F} 0$ in $F$. It now follows from \cref{res:generated_topology} that $x_\alpha\xrightarrow{\mathrm{u}_S\!\tau_F}0$ in $E$, as required.
The uniqueness statement is clear from \cref{res:conradie_taylor}. \end{proof}
The combination of \cref{res:generated_topology} and \cref{res:preparation_uoLt} immediately yields the following.
\begin{theorem}\label{res:Hausdorff_oLt_on_order_dense_ideal_generates_global_Hausdorff_uoLt} Let $E$ be a vector lattice. Suppose that $E$ has an order dense ideal $F$ that admits a Hausdorff o-Lebes\-gue topology\ $\tau_F$. Then $E$ admits a \uppars{necessarily unique} Hausdorff uo-Lebes\-gue topology\ ${\widehat{\tau}}_E$. This topology ${\widehat{\tau}}_E$ is equal to $\mathrm{u}_S\!\tau_F$ for every subset $S$ of $F$ such that the ideal $I_S\subseteq F$ that is generated by $S$ is order dense in $E$.
For a net $\net$ in $E$, the following are equivalent: \begin{enumerate}
\item $x_\alpha\xrightarrow{{\widehat{\tau}}_E} 0$ in $E$;\label{part:convergence_in_uoLtop}
\item $\abs{x_\alpha}\wedge \abs{s}\xrightarrow{\tau_F}0$ in $F$ for all $s\in S$; \label{part:testing_against_subset}
\item $\abs{x_\alpha}\wedge \abs{y}\xrightarrow{\tau_F}0$ in $F$ for all $y\in F$. \label{part:convergence_criteria_order_dense_ideal} \end{enumerate} \end{theorem}
\begin{remark}\label{rem:essential_difference} For the case in \cref{res:Hausdorff_oLt_on_order_dense_ideal_generates_global_Hausdorff_uoLt} where $S=F$ and $\tau_F$ is the restriction of a Hausdorff o-Lebes\-gue topology\ on $E$, it was already established in \cite[Theorem~9.6]{taylor:2019} that $\mathrm{u}_F\!\tau_F$ is a Hausdorff uo-Lebes\-gue topology\ on $E$. It is, therefore, of some importance to point out that not every Hausdorff o-Lebes\-gue topology\ on an order dense ideal is the restriction of a Hausdorff o-Lebes\-gue topology\ on the enveloping vector lattice. By way of example, consider the order dense ideal $c_0$ of $\ell^\infty$. Since the supremum norm on $c_0$ is order continuous, the usual norm topology $\tau_{c_0}$ on $c_0$ is a Hausdorff o-Lebes\-gue topology. However, there does not even exist a possibly non-Hausdorff o-Lebes\-gue topology\ $\tau_{\ell^\infty}$ on $\ell^\infty$ that extends $\tau_{c_0}$. In order to see this, consider the sequence of standard unit vectors $\seq{e}{n}$ in $\ell^\infty$. We have $e_n\convwithoverset{\mathrm{o}} 0$ in $\ell^\infty$ , which would imply that $e_n\xrightarrow{\tau_{\ell^\infty}}0$ in $\ell^\infty$. Since $\tau_{\ell^\infty}$ extends $\tau_{c_0}$, we would have that $e_n\to 0$ in norm. This contradiction shows that such an extension does not exist.
Although the terminology is not used as such, the fact that $\mathrm{u}_F\!\tau_F$ is a Hausdorff uo-Lebes\-gue topology\ on $E$ is implicit in the construction in \cite[p.~290]{conradie:2005}. \end{remark}
\begin{remark}\label{rem:small_subset_criterion}
We are not aware of a reference where it is noted, as in part~\ref{part:testing_against_subset}, that convergence of a net in the Hausdorff uo-Lebes\-gue topology\ on $E$ can be established by using a (presumably small and manageable) subset $S$ of $F$ instead of the full ideal $F$. This non-trivial fact, which relies on the uniqueness of a Hausdorff uo-Lebes\-gue topology, appears to be of some practical value. \end{remark}
In view of the uniqueness of a Hausdorff uo-Lebesgue topology (see \cref{res:conradie_taylor}), the following is now clear from \cref{res:Hausdorff_oLt_on_order_dense_ideal_generates_global_Hausdorff_uoLt}.
\begin{corollary}\label{res:same_generated_topology}
Let $E$ be a vector lattice, and suppose that $E$ has order dense ideals $F_1$ and $F_2$, each of which admits a Hausdorff o-Lebes\-gue topology. For $i=1,2$, choose a Hausdorff o-Lebes\-gue topology\ $\tau_{F_i}$ on $F_i$, and choose a non-empty subset $S_i$ of $F_i$ such that the ideal $I_{S_i}\subseteq F_i$ that is generated by $S_i$ in $E$ is order dense in $E$. Then $\mathrm{u}_{S_1}\!\tau_{F_1}$ and $\mathrm{u}_{S_2}\!\tau_{F_2}$ are both equal to the \uppars{necessarily unique} uo-Lebes\-gue topology\ ${\widehat{\tau}}_E$ on $E$. \end{corollary}
\begin{remark}
The case in \cref{res:same_generated_topology} where $S_1=F_1$ and $S_2=F_2$ is \cite[Proposition~3.2]{conradie:2005}.
The case where, for $i=1,2$, $S_i=F_i$ and $\tau_{F_i}$ is the restriction to $F_i$ of a Hausdorff o-Lebes\-gue topology\ $\tau_i$ on $E$, is a part of \cite[Theorem~9.6]{taylor:2019}. Note, however, that our underlying proof in \cref{res:preparation_uoLt} that $\mathrm{u}_S\!\tau_F$ is a uo-Lebes\-gue topology\ is direct, whereas in the proof of \cite[Theorem~9.6]{taylor:2019} the identification of a Hausdorff uo-Lebes\-gue topology\ as a minimal Hausdorff locally solid topology as in \cref{res:conradie_taylor} is used. \end{remark}
Complementing the preceding going-up results, we cite the following going-down result; see \cite[Proposition 5.12]{taylor:2019}.
\begin{proposition}[Taylor]\label{res:going_down_one}
Suppose that the vector lattice $E$ admits a \uppars{necessarily unique} Hausdorff uo-Lebes\-gue topology\ ${\widehat{\tau}}_E$. Take a vector sublattice $F$ of $E$. Then $F$ is a regular vector sublattice of $E$ if and only if the restriction ${\widehat{\tau}}_E|_F$ of ${\widehat{\tau}}_E$ to $F$ is a \uppars{necessarily unique} Hausdorff uo-Lebes\-gue topology\ on $F$. \end{proposition}
A variation on this theme, with a wider range of topologies to use for testing the regularity of a vector sublattice, is the following.
\begin{proposition}\label{res:going_down_two}
Suppose that the vector lattice $E$ admits a Hausdorff o-Lebes\-gue topology\ $\tau_E$. Take a vector sublattice $F$ of $E$. Then $F$ is a regular vector sublattice of $E$ if and only if the restriction $\tau_E|_F$ of $\tau_E$ to $F$ is a Hausdorff o-Lebes\-gue topology\ on $F$. \end{proposition}
\begin{proof} Once one recalls that, by definition, order convergence of a net to 0 in the regular vector sublattice $F$ of $E$ implies order convergence of the net to 0 in $E$, the proof is a straightforward minor adaptation of that of \cite[Proposition 5.12]{taylor:2019}. \end{proof}
We now have the following overview theorem concerning Hausdorff o-Lebes\-gue topologies\ and Hausdorff uo-Lebes\-gue topologies\ on a vector lattice and on its order dense ideals. It is easily established by recalling that a uo-Lebes\-gue topology\ is an o-Lebes\-gue topology, that an ideal is a regular vector sublattice, and by using \cref{res:Hausdorff_oLt_on_order_dense_ideal_generates_global_Hausdorff_uoLt}, \cref{res:going_down_one}, and \cref{res:going_down_two}.
\begin{theorem}\label{res:overview}
Let $E$ be a vector lattice, and let $F$ be an order dense ideal of $E$.
\begin{enumerate}
\item Suppose that $E$ admits a Hausdorff o-Lebes\-gue topology\ $\tau_E$. Then the restricted topology $\tau_E|_F$ is a Hausdorff o-Lebes\-gue topology\ on $E$.
\item Suppose that $E$ admits a \uppars{necessarily unique} Hausdorff uo-Lebes\-gue topology\ ${\widehat{\tau}}_E$. Then the restricted topology ${\widehat{\tau}}_E|_F$ is a \uppars{necessarily unique} Hausdorff uo-Lebes\-gue topology\ on $F$.
\item The following are equivalent:
\begin{enumerate}
\item $F$ admits a Hausdorff o-Lebes\-gue topology;
\item $F$ admits a Hausdorff uo-Lebes\-gue topology;
\item $E$ admits a Hausdorff o-Lebes\-gue topology;
\item $E$ admits a Hausdorff uo-Lebes\-gue topology.
\end{enumerate}
In that case, the unique uo-Lebes\-gue topology\ ${\widehat{\tau}}_E$ on $E$ equals $\mathrm{u}_S\!\tau_F$ for every Hausdorff o-Lebes\-gue topology\ on $F$ and every subset $S$ of $F$ such that the ideal $I_S\subseteq F$ is order dense in $E$, and the following are equivalent:
\begin{enumerate_roman}
\item $x_\alpha\xrightarrow{{\widehat{\tau}}_E} 0$ in $E$;
\item $\abs{x_\alpha}\wedge \abs{s}\xrightarrow{\tau_F}0$ in $F$ for all $s\in S$;
\item $\abs{x_\alpha}\wedge \abs{y}\xrightarrow{\tau_F}0$ in $F$ for all $y\in F$.
\end{enumerate_roman}
\end{enumerate} \end{theorem}
We conclude this section with a short discussion of Banach lattices with order continuous norms. Evidently, the norm topologies on such Banach lattices are Hausdorff o-Lebes\-gue topologies. As already noted in \cite[p.~993]{taylor:2019}, \cref{res:Hausdorff_oLt_on_order_dense_ideal_generates_global_Hausdorff_uoLt} allows one to identify the so-called un-topologies (see \cite[Section~7]{deng_o_brien_troitsky:2017} and \cite[p.~746]{kandic_li_troitsky:2018}) on such lattices as the Hausdorff uo-Lebes\-gue topologies\ that these spaces apparently admit. Consequently, we have the following result. The case where $S=E$ can be found in \cite[p.~993]{taylor:2019}.
\begin{proposition}\label{res:banach_lattice_with_order_continuous_norm_one}
Let $E$ be a Banach lattice with an order continuous norm and norm topology $\tau_E$. Then $E$ admits a \uppars{necessarily unique} uo-Lebes\-gue topology.
Choose a subset $S$ of $E$ such that the ideal $I_S$ that is generated by $S$ in $E$ is order dense in $E$. Then:
\begin{enumerate}
\item $\mathrm{u}_S\!\tau_E$ is the uo-Lebesgue topology ${\widehat{\tau}}_E$ of $E$;
\item when $\net$ is a net in $E$, then $x_\alpha\xrightarrow{{\widehat{\tau}}_E} 0$ in $E$ if and only if $\norm{\abs{x_\alpha}\wedge \abs{s}}\xrightarrow{}0$ for all $s\in S$; equivalently, if and only if $\norm{\abs{x_\alpha}\wedge \abs{y}}\xrightarrow{}0$ for all $y\in E$.\label{part:convergence_criteria_order_dense_ideal_banac_lattice_with_oc_norm}
\end{enumerate} \end{proposition}
There is an alternative reason why Banach lattices with an order continuous norms admit Hausdorff uo-Lebes\-gue topologies, and this results in an alternative description of these topologies; see \cref{res:banach_lattices_with_order_continuous_norm_two}, below.
Finally, suppose that $E$ is a vector lattice that has order dense ideals $F_1$ and $F_2$ that are Banach lattices with order continuous norm topologies $\tau_{F_1}$ and $\tau_{F_2}$, respectively. Then it is immediate from \cref{res:same_generated_topology} that $E$ admits a Hausdorff uo-Lebes\-gue topology\ ${\widehat{\tau}}_E$, and that $\mathrm{u}_{F_1}\!\tau_{F_1}$ and $\mathrm{u}_{F_2}\!\tau_{F_2}$ are both equal to ${\widehat{\tau}}_E$. As discussed in \cref{exam:kandic_li_troitsky:2018}, this can, using the terminology in \cite{kandic_li_troitsky:2018}, be rephrased as stating that $F_1$ and $F_2$ induce the same un-topology on $E$. We have thus retrieved \cite[Theorem~2.6]{kandic_li_troitsky:2018}.
\section{uo-Lebes\-gue topologies\ generated by absolute weak topologies on order dense ideals}\label{sec:separating_order_continuous_dual}
\noindent In this section, we shall be concerned with vector lattices having order dense ideals with separating order continuous duals as a source for Hausdorff uo-Lebes\-gue topologies\ on the vector lattices themselves.
We start by recapitulating some facts from \cite[p.~63--64]{aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003}. Let $E$ be a vector lattice, and let $A$ be a non-empty subset of the order dual $\odual{E}$ of $E$. For $\varphi\in A$, define the lattice semi-norm $\rho_\varphi: E\to[0,\infty)$ by setting $\rho_\varphi(x)\coloneqq \abs{\varphi}(\abs{x})$ for $x\in E$. Then the locally convex-solid linear topology on $E$ that is generated by the family $\{\rho_\varphi: \varphi\in A\}$ is called the \emph{absolute weak topology generated by $A$ on $E$}; it is denoted by $\abs{\sigma}(E,A)$. With $I_A$ denoting the ideal generated by $A$ in $\odual{E}$, we have $\abs{\sigma}(E,A)=\abs{\sigma}(E,I_A)$. Using \cref{res:polars}, one easily concludes that $\abs{\sigma}(E,A)$ is Hausdorff if and only if $I_A$ separates the points of $E$. Although we shall not use it, let us still remark that it is not difficult to see that a net $\net$ in $E$ is $\abs{\sigma}(E,A)$-convergent to zero if and only if $\varphi(x_\alpha)\xrightarrow{} 0$ uniformly for $\varphi$ in each fixed order interval of $I_A$. Thus absolute weak topologies are more natural than is perhaps apparent from their definition.
The following is now clear.
\begin{lemma}\label{res:absolute_weak_topology_has_the_o-lebesgue_property}
Let $E$ be a vector lattice, and let $A$ be a non-empty subset of $\ocdual{E}$. Let $I_A$ denote the ideal that is generated by $A$ in $\ocdual{E}$. Then $\abs{\sigma}(E,A)=\abs{\sigma}(E,I_A)$ is an o-Lebes\-gue topology\ on $E$ that is even locally convex-solid. It is a Hausdorff topology if and only if $I_A$ separates the points of $E$. When $\net$ is a net in $E$, then $x_\alpha\xrightarrow{\abs{\sigma}(E,A)}0$ in $E$ if and only if $\abs{\varphi}\left(\abs{x_\alpha}\right)\to 0$ for all $\varphi\in A$; equivalently, if and only if $\abs{\varphi}\left(\abs{x_\alpha}\right)\to 0$ for all $\varphi\in I_A$. \end{lemma}
Now that \cref{res:absolute_weak_topology_has_the_o-lebesgue_property} provides a whole class of vector lattices admitting Hausdorff o-Lebes\-gue topologies, we can use these as input for \cref{res:Hausdorff_oLt_on_order_dense_ideal_generates_global_Hausdorff_uoLt}. Taking the convergence statements in \cref{res:absolute_weak_topology_has_the_o-lebesgue_property} into account, we arrive at the following.
\begin{theorem}\label{res:uo-lebesgue_topology_originating_from_separating_order_continuous_dual_of_order_dense_ideal}
Let $E$ be a vector lattice. Suppose that $E$ has an order dense ideal $F$ such that $\ocdual{F}$ separates the points of $F$. Then $E$ admits a \uppars{necessarily unique} Hausdorff uo-Lebes\-gue topology\ ${\widehat{\tau}}_E$.
Choose a subset $A$ of $\ocdual{F}$ such that the ideal $I_A$ that is generated by $A$ in $\ocdual{F}$ separates the points of $F$, and choose a subset $S$ of $F$ such that the ideal $I_S\subseteq F$ that is generated by $S$ is order dense in $E$. Then:
\begin{enumerate}
\item $\mathrm{u}_S\abs{\sigma}(F,A)$ and $\mathrm{u}_F\abs{\sigma}(F,I_A)$ are both equal to ${\widehat{\tau}}_E$;
\item for a net $\net$ in $E$, $x_\alpha\xrightarrow{{\widehat{\tau}}_E}0$ in $E$ if and only if $\abs{\varphi}\left(\abs{x_\alpha}\wedge\abs{s}\right)\to 0$ for all $\varphi\in A$ and $s\in S$; equivalently, if and only if $\abs{\varphi}\left(\abs{x_\alpha}\wedge\abs{y}\right)\to 0$ for all $\varphi\in \ocdual{F}$ and $y\in F$.
\end{enumerate} \end{theorem}
For the sake of completeness, we recall that a regular vector sublattice of a vector lattice $E$ as in the theorem also has a (necessarily unique) Hausdorff uo-Lebes\-gue topology, and that this topology is the restriction of ${\widehat{\tau}}_E$ to the vector sublattice.
\begin{remark} As noted in \cref{rem:explicit_neighbourhood_base}, one can give an explicit neighbourhood base at zero for the topology ${\widehat{\tau}}_E$ in \cref{res:uo-lebesgue_topology_originating_from_separating_order_continuous_dual_of_order_dense_ideal}. \end{remark}
For Banach lattices with order continuous norms, the order/norm dual consists of order continuous linear functionals only. Hence we have the following consequence of \cref{res:uo-lebesgue_topology_originating_from_separating_order_continuous_dual_of_order_dense_ideal}.
\begin{corollary}\label{res:banach_lattices_with_order_continuous_norm_two} A Banach lattice $E$ with an order continuous norm admits a \uppars{necessarily unique} Hausdorff uo-Lebes\-gue topology\ ${\widehat{\tau}}_E$, namely $\mathrm{u}_E\abs{\sigma}(E,E^\ast)$. \end{corollary}
\begin{remark} For a Banach lattice $E$ with an order continuous norm, \cref{res:banach_lattice_with_order_continuous_norm_one} also shows that the (necessarily unique) Hausdorff uo-Lebes\-gue topology\ ${\widehat{\tau}}_E$ on $E$ is the unbounded norm topology. The unbounded norm topology and the unbounded absolute weak topology therefore coincide for Banach lattices with order continuous norms. \end{remark}
The following gives a necessary condition for convergence in a Hausdorff uo-Lebes\-gue topology. It is essential in the proof of \cref{res:tau_m_to_sub_uo}, below.
\begin{proposition}\label{res:essential}
Let $E$ be a vector lattice that admits a \uppars{necessarily unique} Hausdorff uo-Lebes\-gue topology\ ${\widehat{\tau}}_E$, and let $\net$ be a net in $E$ such that $x_\alpha\xrightarrow{{\widehat{\tau}}_E}0$ in $E$. Take an ideal $F$ of $E$ such that $\ocdual{F}$ separates the points of $F$. Then $\abs{\varphi}(\abs{x_\alpha}\wedge \abs{y})\xrightarrow{}0$ for all $\varphi\in \ocdual{F}$ and $y\in F$. \end{proposition}
\begin{proof}
Take $\varphi\in\ocdual{F}$ and $y\in F$. Since ${\widehat{\tau}}_E$ is a locally solid topology, we have $\abs{x}_\alpha\wedge\abs{y}\xrightarrow{{\widehat{\tau}}_E}0$ in $E$. It follows from \cref{res:going_down_one} that $F$ has a (necessarily unique) Hausdorff uo-Lebes\-gue topology\ ${\widehat{\tau}}_F$ and that $\abs{x}_\alpha\wedge\abs{y}\xrightarrow{{\widehat{\tau}}_F}0$. Now we apply \cref{res:uo-lebesgue_topology_originating_from_separating_order_continuous_dual_of_order_dense_ideal} with $E=F$ to see that $\abs{\varphi}((\abs{x_\alpha}\wedge \abs{y})\wedge\abs{y})\xrightarrow{}0$. \end{proof}
We shall now consider the order dual $\odual{E}$ of a vector lattice $E$. For $x\in E$, we set \[ \varphi_x(\varphi)\coloneqq \varphi(x) \] for $\varphi\in \odual{E}$. Then $\varphi_x\in\ocdual{\left(\odual{E}\right)}$, and the map $\varphi:E\to \odual{E}$ is a lattice homomorphism; see \cite[p.~43]{aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003}. Since $\varphi(E)$ already separates the points of $\odual{E}$, we see that $\ocdual{\left(\odual{E}\right)}$ separates the points of $\odual{E}$.
We can now apply \cref{res:uo-lebesgue_topology_originating_from_separating_order_continuous_dual_of_order_dense_ideal} twice. In both cases, we replace $E$ with $\odual{E}$, and we choose $\odual{E}$ for both $F$ and $S$. In the first application, we choose $\ocdual{\left(\odual{E}\right)}$ for $A$; in the second, we choose $\varphi(E)$. The result is as follows.
\begin{corollary}\label{res:corollary_to_uniqueness_of_tau_m_for_dual} Let $E$ be a vector lattice. Then the order dual $\odual{E}$ of $E$ admits a \uppars{necessarily unique} Hausdorff uo-Lebes\-gue topology\ ${\widehat{\tau}}_{\odual{E}}$.
Moreover: \begin{enumerate}
\item $\mathrm{u}_{\odual{E}}\abs{\sigma}( \odual{E}, \ocdual{\left(\odual{E}\right)})$ and $\mathrm{u}_{\odual{E}}\abs{\sigma}( \odual{E}, E)$ are both equal to ${\widehat{\tau}}_{\odual{E}}$;
\item when $(\varphi_\alpha)_{\alpha\in{\mathcal A}}$ is a net in $\odual{E}$, then we have that $\varphi_\alpha\xrightarrow{{\widehat{\tau}}_{\odual{E}}}0$ in $E$ if and only if $\abs{\xi}\left(\abs{\varphi_\alpha}\wedge\abs{\varphi}\right)\to 0$ for all $\xi\in\ocdual{\left(\odual{E}\right)}$ and $\varphi\in \odual{E}$; equivalently, if and only if $(\abs{\varphi_\alpha}\wedge\abs{\varphi})(\abs{x})\to 0$ for all $x\in E$ and $\varphi\in \odual{E}$. \end{enumerate} \end{corollary}
\begin{remark}\quad
\begin{enumerate}
\item As in the case of \cref{res:uo-lebesgue_topology_originating_from_separating_order_continuous_dual_of_order_dense_ideal}, \cref{rem:explicit_neighbourhood_base} shows how to give an explicit neighbourhood base at zero for the topology ${\widehat{\tau}}_{\odual{E}}$ in \cref{res:corollary_to_uniqueness_of_tau_m_for_dual}.
\item By \cref{res:going_down_one}, every regular sublattice of the order dual of a vector lattice also admits a (necessarily unique) Hausdorff Lebesgue topology that can be described in two ways. For an ideal, one of these descriptions is already in \cite[Example~5.8]{taylor:2019}. \item \cref{res:corollary_to_uniqueness_of_tau_m_for_dual} shows that, in particular, the norm/order dual $E^\ast$ of a Banach lattice admits a (necessarily unique) Hausdorff uo-Lebes\-gue topology\ ${\widehat{\tau}}_{E^\ast}$, namely the so-called unbounded absolute weak $^\ast$-topology $\mathrm{u}_{E^\ast}\abs{\sigma}(E^\ast,E)$. This was already observed in \cite[Lemma~6.6]{taylor:2019}.
\end{enumerate} \end{remark}
\section{Regular vector sublattices of $\mathrm{L}_0(X,\Sigma,\mu)$ for semi-finite measures}\label{sec:vector_lattices_of_equivalence_classes_of_measurable_functions}
\noindent Let $(X,\Sigma,\mu)$ be a measure space, and write $\mathrm{L}_0(X,\Sigma,\mu)$ for the vector lattice of all real-valued $\Sigma$-measurable functions on $X$, with identification of two functions when they agree $\mu$-almost everywhere. In this section we show that, for semi-finite $\mu$, every regular sublattice of $\mathrm{L}_0(X,\Sigma,\mu)$ admits a (necessarily unique) Hausdorff uo-Lebes\-gue topology, and that a net converges in this topology if and only if it converges in measure on subsets of finite measure; see \cref{res:going_downlattice_of_L_0}, below.
For some regular sublattices of $\mathrm{L}_0(X,\Sigma,\mu)$, it is quite obvious that they admit a Hausdorff uo-Lebes\-gue topology. Recall that the spaces $\mathrm{L}_p(X,\Sigma,\mu)$ for $p$ such that $1\leq p<\infty$ have order continuous norms for all measures $\mu$; see \cite[Theorem~13.7]{aliprantis_border_INFINITE_DIMENSIONAL_ANALYSIS_THIRD_EDITION:2006}, for example. Hence their norm topologies are Hausdorff o-Lebes\-gue topologies, and then their un-topologies are the Hausdorff uo-Lebes\-gue topologies\ on these spaces. Alternatively, one can observe that their order continuous duals separate their points, and then also identify the Hausdorff uo-Lebes\-gue topologies\ on these spaces as the unbounded absolute weak topologies. In a similar vein, when $\mu$ is $\sigma$-finite, every ideal of $L_0(X,\Sigma,\mu)$ that can be supplied with a lattice norm has a separating order continuous dual. This result of Lozanovsky's (see \cite[Theorem~5.25]{abramovich_aliprantis_INVITATION_TO_OPERATOR_THEORY:2002}, for example) then implies that such a normed function space admits a Hausdorff uo-Lebes\-gue topology.
How about the spaces $\mathrm{L}_p(X,\Sigma,\mu)$ for $0\leq p<1$? There is no norm to work with, and it may well be the case that their order continuous duals are even trivial. Indeed, when $\mu$ is atomless, then, according to a results of Day's, the order continuous dual of $\mathrm{L}_p(X,\Sigma,\mu)$ is trivial for $0<p<1$; see \cite[Theorem~13.31]{aliprantis_border_INFINITE_DIMENSIONAL_ANALYSIS_THIRD_EDITION:2006}, for example. According to \cite[Exercise~25.2]{zaanen_INTRODUCTION_TO_OPERATOR_THEORY_IN_RIESZ_SPACES:1997}, the order continuous dual of $\mathrm{L}_0(X,\Sigma,\mu)$ is trivial for every $\sigma$-finite measure with the property that, for any measurable subset $A$ such that $0<\mu(A)<\infty$ and for any $\alpha$ such that $0<\alpha<\mu(A)$, there exists a measurable subset $A^\prime$ of $A$ such that $\mu(A^\prime)=\alpha$. Taking \cite[Exercise~10.12 on p.~67]{zaanen_INTEGRATION:1967} into account, we see that, in particular, the order continuous dual of $\mathrm{L}_0(X,\Sigma,\mu)$ is trivial for all atomless $\sigma$-finite measures.
In spite of the failure of the two obvious approaches, it is still possible to show that all spaces $\mathrm{L}_p(X,\Sigma,\mu)$ for $0\leq p<1$ admit Hausdorff uo-Lebes\-gue topologies, provided that the measure is semi-finite. For such $\mu$, this is even true for all regular vector sublattices of $\mathrm{L}_0(X,\Sigma,\mu)$. This can be seen via the going-up-going-down approach from \cref{sec:uoLtops_going_up_and_going_down}, and we shall now elaborate on this. We start with a few preliminary remarks.
Recall that a measure space $(X, \Sigma, \mu)$ is said to be \emph{semi-finite} if, for any $A\in \Sigma$ with $\mu(A)=\infty$, there exists a measurable subset $A^\prime$ of $A$ such that $0<\mu(A^\prime)<\infty$. Every $\sigma$-finite measure is semi-finite. For an arbitrary measure $\mu$ and an arbitrary $p$ such that $1\leq p<\infty$, it is easy to see that the ideal $\mathrm{L}_p(X,\Sigma,\mu)$ of $\mathrm{L}_0(X,\Sigma,\mu)$ is order dense in $\mathrm{L}_0(X,\Sigma,\mu)$ if and only if $\mu$ is semi-finite. In that case, the ideal that is generated in $\mathrm{L}_0(X,\Sigma,\mu)$ by the subset $S\coloneqq \{1_A: A\in\Sigma\text{ has finite measure}\}$ of $\mathrm{L}_p(X,\Sigma,\mu)$ is obviously also order dense in $\mathrm{L}_0(X,\Sigma,\mu)$.
Let $(X,\Sigma,\mu)$ be a measure space. Take $f\in\mathrm{L}_0(X,\Sigma,\mu)$. Then a net $(f_\alpha)_{\alpha\in{\mathcal A}}$ in $\mathrm{L}_0(X,\Sigma,\mu)$ \emph{converges to f in measure on subsets of finite measure} when, for all $A\in\Sigma$ such that $\mu(A)<\infty$ and for all $\varepsilon>0$, $\mu(\{x\in A: \abs{f_\alpha(x)-f(x)}\geq\varepsilon\})\xrightarrow{}0$. In that case, we write $f_\alpha\xrightarrow{{\mu^\ast}}f$, using as asterisk to distinguish this convergence from the perhaps more usual global convergence in measure.
The following is the core result of this section. We recall that, as already mentioned, the spaces $\mathrm{L}_p(X,\Sigma,\mu)$ have order continuous norms for all measures $\mu$ and for all $p$ such that $1\leq p<\infty$, so that their norm topologies are Hausdorff o-Lebes\-gue topologies.
\begin{theorem}\label{res:tau_m_is_convergence_in_measure}
Let $E=\mathrm{L}_0(X,\Sigma,\mu)$, where $\mu$ is a semi-finite measure. Then $G$ admits a \uppars{necessarily unique} Hausdorff uo-Lebes\-gue topology\ ${\widehat{\tau}}_E$.
Take a net $(f_\alpha)_{\alpha\in{\mathcal A}}$ in $E$. Then the following are equivalent for every $p$ such that $1\leq p<\infty$:
\begin{enumerate}
\item $f_\alpha\xrightarrow{{\widehat{\tau}}_E} 0$; \label{part:Ell_0_1}
\item
\[
\int_X\! \abs{f_\alpha}^p\wedge 1_A\,\text{d}\mu=\norm{\,\abs{f_\alpha}\wedge \abs{1_A}\,}_p^p\xrightarrow{}0
\]
for every measurable subset $A$ of $X$ with finite measure;\label{part:Ell_0_2}
\item
\[
\int_X\! \abs{f_\alpha}^p\wedge \abs{f}^p\,\text{d}\mu=\norm{\,\abs{f_\alpha}\wedge \abs{f}\,}_p^p\xrightarrow{}0
\]for every $f\in\mathrm{L}_p(X,\Sigma,\mu)$;\label{part:Ell_0_3}
\item $f_\alpha\xrightarrow{\mu^\ast} f$.\label{part:Ell_0_4}
\end{enumerate} \end{theorem}
\begin{proof}
We know from the semi-finiteness of $\mu$ that, for $p$ such that $1\leq p\leq \infty$, $L_p(X,\Sigma,\mu)$ is an order dense ideal of $\mathrm{L}_0(X,\Sigma,\mu)$. Since $L_p(X,\Sigma,\mu)$ admits a Hausdorff o-Lebes\-gue topology\ when $1\leq p<\infty$, \cref{res:Hausdorff_oLt_on_order_dense_ideal_generates_global_Hausdorff_uoLt} shows that $\mathrm{L}_0(X,\Sigma,\mu)$ admits a (necessarily unique) Hausdorff uo-Lebes\-gue topology, and also that the statements in the parts~\ref{part:Ell_0_1},~\ref{part:Ell_0_2}, and~\ref{part:Ell_0_3} of the present theorem are equivalent for all such $p$.
We show that part~\ref{part:Ell_0_3} implies part~\ref{part:Ell_0_4}.
Take a measurable subset $A$ of $X$ with finite measure, and let $\varepsilon>0$. Since $\varepsilon 1_A\in\mathrm{L}_p(X,\Sigma,\mu)$, we have, by assumption,
\[
\int_X\! \abs{f_\alpha}^p\wedge(\varepsilon^p 1_A)\,\text{d}\mu\xrightarrow{}0.
\]
Because
\[
\int_X \!\abs{f_\alpha}^p\wedge(\varepsilon^p 1_A)\,\text{d}\mu\geq \int_{\{x\in A: \abs{f_\alpha(x)}\geq\varepsilon\}}\!\!\! \varepsilon^p\,\text{d}\mu=\varepsilon^p\mu\left(\{x\in A: \abs{f_\alpha(x)}\geq\varepsilon\}\right)
\]
we conclude that $\mu(\{x\in A: \abs{f_\alpha(x)}\geq\varepsilon\})\xrightarrow{}0$. Hence $f_\alpha\xrightarrow{\mu^\ast} 0$.
We show that part~\ref{part:Ell_0_4} implies part~\ref{part:Ell_0_2}. Take a measurable subset $A$ of $X$ with finite measure, and take $\varepsilon>0$. Choose a $\delta>0$ such that $\delta^p\mu(A)<\varepsilon/2$. Then
\begin{align*}
\int_X\! \abs{f_\alpha}^p\wedge1_A\,\text{d}\mu&= \int_{\{x\in A: \abs{f_\alpha(x)}^p\geq\delta^p\}}\!\!\!\abs{f_\alpha}^p\wedge 1_A \,\text{d}\mu+\int_ {\{x\in A: \abs{f_\alpha(x)}^p<\delta^p\}}\!\!\!\abs{f_\alpha}^p\wedge 1_A\,\text{d}\mu\\
&\leq \int_{\{x\in A: \abs{f_\alpha(x)}^p\geq\delta^p\}}\!\!\!1 \,\text{d}\mu + \int_A\delta^p\,\text{d}\mu\\
&\leq \mu\left(\{x\in A: \abs{f_\alpha(x)}\geq\delta\}\right)+ \varepsilon/2.
\end{align*}
By our assumption, there exists an $\alpha_0\in{\mathcal A}$ such that $\mu\left(\{x\in A: \abs{f_\alpha(x)}\geq\delta\}\right)<\varepsilon/2$ for all $\alpha\geq\alpha_0$. Then $\int_X \abs{f_\alpha}^p\wedge1_A\,\text{d}\mu<\varepsilon$ for all $\alpha\geq\alpha_0$. Hence $\int_X \abs{f_\alpha}\wedge1_A\,\text{d}\mu\xrightarrow{}0$. \end{proof}
\begin{remark}\quad
\begin{enumerate}
\item We are not aware of a proof of \cref{res:tau_m_is_convergence_in_measure} in the literature. It is stated in \cite[p.~292]{conradie:2005} that the parts~\ref{part:Ell_0_1} and~\ref{part:Ell_0_4} are equivalent, but there only a reference is given to \cite[65K and~63L]{fremlin_TOPOLOGICAL_RIESZ_SPACES_AND_MEASURE_THEORY:1974}. Since \cite[63L]{fremlin_TOPOLOGICAL_RIESZ_SPACES_AND_MEASURE_THEORY:1974} relies on the solution of the non-trivial exercise \cite[Exercise~63M(j)]{fremlin_TOPOLOGICAL_RIESZ_SPACES_AND_MEASURE_THEORY:1974} for which a solution is not provided, we thought it appropriate to give an independent proof in the present paper.
\item The equivalence of the parts~\ref{part:Ell_0_3} and~\ref{part:Ell_0_4} for finite measures and sequences was also established by different methods in \cite[Example~23]{troitsky:2004}. Still earlier, this case was covered in \cite[Corollary~4.2]{deng_o_brien_troitsky:2017}, with a proof in the same spirit as our proof.
\end{enumerate} \end{remark}
As an immediate consequence of \cref{res:going_down_one} and \cref{res:tau_m_is_convergence_in_measure}, we obtain the following result via our going-up-going-down approach.
\begin{theorem}\label{res:going_downlattice_of_L_0}
Let $(X,\Sigma,\mu)$ be a measure space, where $\mu$ is a semi-finite measure. Take a regular vector sublattice $E$ of $\mathrm{L}_0(X,\Sigma,\mu)$. Then $E$ admits a \uppars{necessarily unique} Hausdorff uo-Lebes\-gue topology\ ${\widehat{\tau}}_E$. This
topology ${\widehat{\tau}}_E$ on $E$ is the restriction of the Hausdorff uo-Lebes\-gue topology\ on $\mathrm{L}_0(X,\Sigma,\mu)$. A net $(f_\alpha)_{\alpha\in{\mathcal A}}$ in $E$ converges to zero in ${\widehat{\tau}}_E$ if and only if it satisfies one of the three equivalent criteria in the parts~\ref{part:Ell_0_2},~\ref{part:Ell_0_3}, and~\ref{part:Ell_0_4} of \cref{res:tau_m_is_convergence_in_measure}. In particular, it is ${\widehat{\tau}}_E$-convergent to zero if and only if it converges to zero in measure on subsets of finite measure. \end{theorem}
\begin{remark}\label{rem:various_descriptions}
Let $p$ be such that $1\leq p<\infty$. For arbitrary measures, \cref{res:banach_lattice_with_order_continuous_norm_one,res:banach_lattices_with_order_continuous_norm_two} both give a description of the convergent nets in the Hausdorff uo-Lebes\-gue topology\ on $\mathrm{L}_p(X,\Sigma,\mu)$. The former as the convergent nets in the un-topology, and the latter as the convergent nets in the unbounded absolute weak topology, respectively. When $\mu$ is semi-finite, \cref{res:going_downlattice_of_L_0} gives a third description as the convergence in measure on subsets of finite measure.
Also for $p=\infty$, \cref{res:going_downlattice_of_L_0} shows that $\mathrm{L}_\infty(X,\Sigma,\mu)$ admits a (necessarily unique) Hausdorff uo-Lebes\-gue topology\ whenever $\mu$ is semi-finite, and gives a description of its convergent nets. When $\mu$ is a localisable measure, two more descriptions are possible. We refer to \cite[211G]{fremlin_MEASURE_THEORY_VOLUME_2:2003} for the definition of localisable measures, and note that $\sigma$-finite measures are localisable, and that localisable measures are semi-finite. Indeed, for localisable measures, $\mathrm{L}_\infty(X,\Sigma,\mu)$ is the order dual of $L_1(X,\Sigma,\mu)$; see \cite[243G(b)]{fremlin_MEASURE_THEORY_VOLUME_2:2003}. Hence \cref{res:corollary_to_uniqueness_of_tau_m_for_dual} shows once more that $\mathrm{L}_\infty(X,\Sigma,\mu)$ admits a Hausdorff uo-Lebes\-gue topology\ when $\mu$ is localisable, and gives a second and third description of its convergent nets. \end{remark}
\begin{remark}
Let $(X,\Sigma,\mu)$ be a measure space, where $\mu$ is a semi-finite measure.
Let $p$ be such that $0< p<\infty$. On combining \cref{res:going_downlattice_of_L_0} and \cref{rem:minimal_and_smallest}, we see that the topology of convergence in measure on subsets of finite measure is the \emph{smallest} Hausdorff locally solid linear topology on $\mathrm{L}_p(X,\Sigma,\mu)$.\footnote{For this conclusion, we should note here that the usual metric topology on $\mathrm{L}_p(X,\Sigma,\mu)$ is a complete o-Lebes\-gue topology\ for every measure $\mu$ and for every $p$ such that $0< p<\infty$. This is commonly known when $1\leq p<\infty$. When $0<p<1$, then the completeness is asserted in \cite[1.47]{rudin_FUNCTIONAL_ANALYSIS_SECOND_EDITION:1991}. The fact that the metric topology is an o-Lebes\-gue topology\ for such $p$ follows from what is stated on \cite[p.~211]{aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003} in the context of $\sigma$-finite measures. This implies the result for general measures. Indeed, suppose that $\netgen{f_\alpha}{\alpha\in {\mathcal A}}$ is a net in $\mathrm{L}_p(X,\Sigma,\mu)$ such that $f_\alpha\downarrow 0$. Passing to a tail, we may suppose that the net is bounded above by an $f_{\alpha_0}\in\mathrm{L}_p(X,\Sigma,\mu)$. The support of this $f_{\alpha_0}$ is $\sigma$-finite. Using the fact that the elements of $\mathrm{L}_p(X,\Sigma,\mu)$ that vanish off this support form an ideal of $\mathrm{L}_p(X,\Sigma,\mu)$, it is then easily seen from the $\sigma$-finite case that the chosen tail of the net converges to zero in the metric topology of $\mathrm{L}_p(X,\Sigma,\mu)$. } For $\sigma$-finite measures, this can already be found in \cite[Theorem~7.74]{aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003}, where it is also established that the usual metric topology is then the largest Hausdorff locally solid linear topology.
For $p=\infty$, the combination of \cref{res:going_downlattice_of_L_0} and \cref{res:conradie_taylor} shows that the topology of convergence in measure on subsets of finite measure is the unique \emph{minimal} Hausdorff locally solid linear topology on $\mathrm{L}_\infty(X,\Sigma,\mu)$. It seems worthwhile to note that, when $\mu$ is, in fact, $\sigma$-finite, and also non-atomic, \cite[Theorem~7.75]{aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003} shows that there is now no \emph{smallest} Hausdorff locally solid linear topology on $\mathrm{L}_\infty(X,\Sigma,\mu)$. \end{remark}
\begin{remark}
Let $\seq{x_n}{n}$ be a sequence in $\mathrm{L}_0(X,\Sigma,\mu)$, where $\mu$ is a semi-finite measure. Suppose that $f_n\xrightarrow{}0$ $\mu$-almost everywhere. Then $f_n\xrightarrow{\mu^\ast}0$. This is immediate from Egoroff's theorem (see \cite[Theorem~2.33]{folland_REAL_ANALYSIS_SECOND_EDITION:1999}, for example), but it can also be obtained (with a long detour) in the context of uo-convergence and uo-Lebes\-gue topologies. Indeed, by \cite[Proposition~3.1]{gao_troitsky_xanthos:2017}, almost everywhere convergence of a sequence in $\mathrm{L}_0(X,\Sigma,\mu)$ is, for arbitrary measures, equivalent to uo-convergence in $\mathrm{L}_0(X,\Sigma,\mu)$. Since, by definition, uo-convergence implies convergence in a uo-Lebes\-gue topology\ (when this exists), an appeal to \cref{res:tau_m_is_convergence_in_measure} also yields the desired result. \end{remark}
\section{uo-convergent sequences within ${\widehat{\tau}}_E$-convergent nets}\label{sec:uo-convergent_subsequences_of_uoLt-convergent_nets}
\noindent Let $E$ be a vector lattice that admits a (necessarily unique) Hausdorff uo-Lebes\-gue topology\ ${\widehat{\tau}}_E$. When $\net$ is a net in $E$ such that $x_\alpha\convwithoverset{\uo} 0$, then, by definition, $x_\alpha\xrightarrow{{\widehat{\tau}}_E} 0$. The present section is concerned with results that go in the opposite direction. The main result is \cref{res:tau_m_to_sub_uo}, below, which lies at the basis of topological considerations in \cref{sec:topological aspects of uo-convergence}, but we start with a few more elementary results.
For an atomic vector lattice $E$, the situation is as easy as can be. Recall that, by \cite[Theorem~1.78]{aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003}, the atomic vector lattices are precisely the order dense vector sublattices of $\mathbb R^X$ for some set $X$. Combining \cite[Lemma~3.1]{dabboorasad_emelyanov_marabeh:2020} and \cite[Lemma~7.4]{taylor:2019}, we have the following.
\begin{proposition}[Taylor]\label{res:atomic}
Let $E$ be an atomic vector lattice. Then $E$ admits a \uppars{necessarily unique} Hausdorff uo-Lebes\-gue topology\ ${\widehat{\tau}}_E$, and this topology is locally convex-solid. For a net in $E$, uo-convergence and ${\widehat{\tau}}_E$-convergence coincide, so that uo-convergence is topological. When $E$ is an order dense vector sublattice of $\mathbb R^X$ for some set $X$, then a net in $E$ is uo- and ${\widehat{\tau}}_E$-convergent if and only if it is pointwise convergent. \end{proposition}
For monotone nets, uo-convergence and ${\widehat{\tau}}_E$-convergence both coincide with order convergence, according to the following lemma that is a consequence of \cite[Theorem~2.21]{aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003}.
\begin{lemma}\label{res:monotone_tau_and_uo}
Let $E$ be a vector lattice, and suppose that $\tau$ is a Hausdorff locally solid linear topology on $E$. Let $\net$ be a monotone net in $E$ and let $x\in E$. When $x_\alpha\convwithoverset{\tau} x$ in $E$, then $x_\alpha\convwithoverset{\mathrm{o}} x$ in $E$, which is equivalent to $x_\alpha\convwithoverset{\uo} x$. When ${\widehat{\tau}}_E$ is a \uppars{necessarily unique} Hausdorff uo-Lebes\-gue topology\ on $E$, then $x_\alpha\xrightarrow{{\widehat{\tau}}_E} x$ in $E$ if and only if $x_\alpha\convwithoverset{\mathrm{o}} x$ in $E$ if and only if $x_\alpha\convwithoverset{\uo} x$. \end{lemma}
\begin{comment} \begin{lemma}\label{res:monotone_tau_and_uo}
Let $E$ be a vector lattice, and suppose that $\tau$ is a Hausdorff locally solid linear topology on $E$. Let $\net$ be a monotone net in $E$ and let $x\in E$. When $x_\alpha\convwithoverset{\tau} x$ in $E$, then $x_\alpha\convwithoverset{\uo} x$ in $E$. When ${\widehat{\tau}}_E$ is a \uppars{necessarily unique} Hausdorff uo-Lebes\-gue topology\ on $E$, then $x_\alpha\xrightarrow{{\widehat{\tau}}_E} x$ in $E$ if and only if $x_\alpha\convwithoverset{\uo} x$ in $E$. \end{lemma}
\begin{proof}
We may suppose that $x_\alpha \downarrow$. Take $y\in E$. Then $\abs{x_\alpha-x}\wedge\abs{y}\downarrow$ and
$\abs{x-x_\alpha}\wedge\abs{y}\convwithoverset{\tau} 0$. By \cite[Theorem~2.21]{aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003}, we have $\abs{x_\alpha-x}\wedge\abs{y}\downarrow 0$. Hence $x_\alpha\convwithoverset{\uo} x$. The final statement is clear. \end{proof} \end{comment}
For non-monotone nets in general vector lattices, it is not generally true that ${\widehat{\tau}}_E$-convergence implies uo-convergence. This can already fail for sequences in Banach lattices with order continuous norms. As an example, consider $E=\mathrm{L}_1([0,1])$. For $n=1,2,\dotsc$ and $k=1,2,\dotsc,n$, let $f_{nk}$ be the characteristic function of $[\frac{k-1}{n},\frac{k}{n}]$, and consider the sequence $f_{11}, f_{21}, f_{22}, f_{31}, f_{32}, f_{33}, f_{41},\dotsc$. It converges to zero in measure, so \cref{res:tau_m_is_convergence_in_measure} shows that it is ${\widehat{\tau}}_E$-convergent to zero. On the other hand, \cite[Proposition~3.1]{gao_troitsky_xanthos:2017} shows that uo-convergence of a sequence in $\mathrm{L}_1([0,1])$ is the same as almost everywhere convergence. Hence the sequence is not uo-convergent to zero.
Still, something can be salvaged in the general case. As a motivating example, suppose that $(X,\Sigma,\mu)$ is a measure space. It is well known that a sequence in $\mathrm{L}_0(X,\Sigma,\mu)$ that converges (globally) in measure has a subsequence that converges to the same limit almost everywhere; see \cite[Theorem 2.30]{folland_REAL_ANALYSIS_SECOND_EDITION:1999}, for example. When $\mu$ is finite, then, in view of \cref{res:tau_m_is_convergence_in_measure} and \cite[Proposition~3.1]{gao_troitsky_xanthos:2017}, this can be restated as saying that a ${\widehat{\tau}}_E$-convergent sequence in $\mathrm{L}_0(X,\Sigma,\mu)$ has a subsequence that is uo-convergent to the same limit. We shall now extend this formulation of the result to a more general context of nets and Hausdorff uo-Lebes\-gue topologies\ on vector lattices; see \cref{res:tau_m_to_sub_uo}, below. In \cref{res:convergence_in_measure_and_almost_everywhere}, below, we shall then obtain a stronger version of the motivating result for convergence in measure and convergence almost everywhere, as a specialisation of the general result.
We start with some preparations that appear to have some independent interest.
\begin{proposition}\label{res:prep_for_tau_m_to_sub_uo_1}
Let $E$ be a vector lattice with the countable sup property such that $\ocdual{E}$ separates the points of $E$. Take $e\in \pos{E}$, and let $I_e$ denote the ideal that is generated in $E$ by $e$. Then $\ocdual{(I_e)}$ separates the points of $I_e$. In fact, there even exists a $\varphi\in \ocdual{(I_e)}$ that is strictly positive on $I_e$. \end{proposition}
\begin{proof}
It is immediate from \cref{res:veksler} that $\ocdual{(I_e)}$ separates the points of $I_e$. It follows from \cref{res:polars} that the ideal of $\ocdual{(I_e)}$ that is generated by a strictly positive $\varphi$ in $\ocdual{(I_e)}$ would already separate the points of $E$. We turn to the existence of such a strictly positive $\varphi\in\ocdual{(I_e)}$.
Suppose first that $E$ is Dedekind complete.
For $\psi\in\pos{\left(\ocdual{E}\right)}$, we let
\[
N_\psi\coloneqq\{x\in E: \psi(\abs{x})=0\}
\]
denote its null ideal, and we let
\[
C_\psi\coloneqq \text{N}_\psi^\text{d}
\]
denote its carrier. Since $\psi$ is order continuous, $N_\psi$ is a band in $E$.
Let $B_0$ be the band that is generated by the subset $\{C_\psi: \psi\in \pos{\left(\ocdual{E}\right)} \}$ of $E$. Then
\[
B_0^\text{d}=\bigcap_{\psi\in\pos{\left(\ocdual{E}\right)}}C_\psi^{\text{d}}=\bigcap_{\psi\in\pos{\left(\ocdual{E}\right)}}N_\psi^{\text{dd}}=\bigcap_{\psi\in\pos{\left(\ocdual{E}\right)}}N_\psi=\{0\},
\]
where in the final step we have used \cref{res:polars} and the fact that $\ocdual{E}$ separates the points of $E$. We thus see that $B_0=E$.
For $\psi\in\pos{\left(\ocdual{E}\right)}$, let $P_{C_\psi}$ denote the band projection from $E$ onto $C_\psi$. When $\psi_1,\psi_2\in \pos{\left(\ocdual{E}\right)}$ and $\psi_1\leq\psi_2$, then $C_{\psi_1}\subseteq C_{\psi_2}$ which, by \cite[Theorem~1.46]{aliprantis_burkinshaw_POSITIVE_OPERATORS_SPRINGER_REPRINT:2006}, is equivalent to $P_{C_{\psi_1}}\leq P_{C_{\psi_2}}$. Therefore, the net $\{P_{C_\psi}:\psi \in \pos{\left(\ocdual{E}\right)}\}$ in $\mathcal{L}_{\reg}(E)$ is increasing. Set
\[
P\coloneqq \sup\, \{P_{C_\psi}:\psi \in \pos{\left(\ocdual{E}\right)}\},
\]
where the supremum is in $\mathcal{L}_{\reg}(E)$. From \cite[Theorem~30.5]{luxemburg_zaanen_RIESZ_SPACES_VOLUME_I:1971} we know that $P$ is a band projection with $B_0$ as its range space. Since $B_0=E$, it follows that $P=I$. This implies that $\{P_{C_\psi}e:\psi \in \pos{\left(\ocdual{E}\right)}\}\uparrow e$, and it follows from the fact that $E$ has the countable sup property that there exists a sequence $(\psi_n)_{n=1}^\infty$ in $\pos{\left(\ocdual{E}\right)}$ such that $P_{C_{\psi_n}}e\uparrow e$ in $E$.
Consider the ideal $I_e$ of $E$. Since $E$ is Dedekind complete it is uniformly complete, so that $I_e$ is a Banach lattice when supplied with its order unit norm $\norm{\,\cdot\,}_e$. Its order dual $\odual{I}_e$ coincides with its norm dual $E^\ast$ and is then a Banach lattice. Choose strictly positive real numbers $\alpha_1,\alpha_2,\ldots$ such that $\sum_{n=1}^{\infty} \alpha_n\norm{\psi_n|_{I_e}}<\infty$, and define $\varphi\in \odual{I}_e$ by setting
\[
\varphi\coloneqq \sum_{n=1}^{\infty} \alpha_n \psi_n|_{I_e}.
\]
Since $I_e$, being an ideal of $E$, is a regular vector sublattice of $E$, each $\psi_n|_{I_e}$ is order continuous. On observing that, being a band, $\odual{(I_e)}_{\mathrm{oc}}$ is an order closed and, therefore, norm closed subset of the Banach lattice $E^\ast$, we see that $\varphi$ is order continuous on $I_e$. Obviously, $\varphi$ is positive.
Suppose that $x\in I_e$ is positive and that $\varphi(x)=0$. Then $\psi_n(x)=0$ for all $n\geq 1$. That is, $x\in N_{\psi_n}$ for all $n\geq 1$, so that $P_{C_{\psi_n}}x=0$ for all $n\geq 1$.
Take $\lambda\geq 0$ such that $0\leq x\leq\lambda e$. Using \cite[Theorem~2.49, Theorem~2.44, and Definition~2.41]{aliprantis_burkinshaw_POSITIVE_OPERATORS_SPRINGER_REPRINT:2006}, we see that there exists an order continuous operator $T$ on $E$ that commutes with all band projections on $E$ and is such that $T(\lambda e)=x$. Since $P_{C_{\Psi_n}}(\lambda e)\uparrow \lambda e$ in $E$, we have $TP_{C_{\Psi_n}}(\lambda e)\uparrow T(\lambda e)=x$ in $E$. On the other hand, we know that $TP_{C_{\Psi_n}}(\lambda e)=P_{C_{\Psi_n}}T(\lambda e)=P_{C_{\Psi_n}}x=0$ for all $n$. We conclude that $x=0$. Hence $\varphi$ is strictly positive on $I_e$. This completes the proof when $E$ is Dedekind complete.
For general $E$, we note that its Dedekind completion $E^\delta$ also has the countable sup property; see \cite[Theorem~32.9 ]{luxemburg_zaanen_RIESZ_SPACES_VOLUME_I:1971}. Furthermore, \cref{res:veksler} shows that $\ocdual{\left(E^\delta\right)}$ separates the points of $E^\delta$. Let $I_{e,\delta}$ denote the ideal that is generated by $e$ in $E^\delta$. By what has been established above, there exists a $\varphi_\delta\in \ocdual{\left(I_{e,\delta}\right)}$ that is strictly positive on $I_{e, \delta}$. Hence its restriction $\varphi_\delta|_{I_e}$ to $I_e$ is strictly positive on $I_e$. This restriction is also order continuous on $I_e$. To see this, suppose that $\net$ is a net in $I_e$ and that $x_\alpha\convwithoverset{\mathrm{o}} 0$ in $I_e$. Since $I_e$, being an ideal of $E$, is a regular vector sublattice of $E$, and since $E$, being order dense in $E^\delta$, is a regular vector sublattice of $E^\delta$, $I_e$ is a regular vector sublattice of $E^\delta$. Thus $x_\alpha\convwithoverset{\mathrm{o}} 0$ in $E^\delta$. There exists an $\alpha_0\in{\mathcal A}$ such that the tail $(x_\alpha)_{\alpha\in{\mathcal A}, \alpha\geq\alpha_0}$ is order bounded in $I_e$. Since this tail is then evidently also order bounded in $I_{e,\delta}$, \cref{res:local-global_for_o-convergence_and_uo-convergence} shows that $x_\alpha\convwithoverset{\mathrm{o}} 0$ in $I_{e,\delta}$ for $\alpha\geq\alpha_0$. Then $\varphi_\delta|_{I_e}(x_\alpha)\xrightarrow{}0$ for $\alpha\geq\alpha_0$ by the order continuity of $\varphi$ on $I_{e,\delta}$. Consequently, $\varphi_\delta|_{I_e}(x_\alpha)\xrightarrow{}0$, as required. \end{proof}
Suppose that a vector lattice $E$ has an order unit $e$ and that $\net$ is a net in $E$. According to \cite[Corollary~3.5]{gao_troitsky_xanthos:2017}, the fact that $\abs{x_\alpha}\wedge e\convwithoverset{\mathrm{o}} 0$ is already enough to imply that $x_\alpha\convwithoverset{\uo} 0$. This is a special case of the following.
\begin{proposition}\label{res:prep_for_tau_m_to_sub_uo_3}
Let $E$ be a vector lattice, let $S$ be a non-empty subset of $E$, and let $B_S$ denote the band that is generated by $S$ in $E$. Suppose that $\net$ is a net in $B_S$ such that $\abs{x_\alpha}\wedge \abs{y}\convwithoverset{\mathrm{o}} 0$ in $E$ for all $y\in S$. Then $x_\alpha\convwithoverset{\uo} 0$ in $E$. \end{proposition}
\begin{proof} Since $B_S$ is a regular sublattice of $E$, part~\ref{part:order_convergence_and_regular_sublattices} of \cref{res:local-global_for_o-convergence_and_uo-convergence} shows that $\abs{x_\alpha}\wedge \abs{y}\convwithoverset{\mathrm{o}} 0$ in $B_S$ for all $y\in S$. From \cite[Lemma~2.2]{li_chen:2018} it then follows that $x_\alpha\convwithoverset{\uo} 0$ in $B_S$. Part~\ref{part:unbounded_order_convergence_and_regular_sublattices} of \cref{res:local-global_for_o-convergence_and_uo-convergence} then implies that $x_\alpha\convwithoverset{\uo} 0$ in $E$. \end{proof}
\begin{lemma}\label{res:countable_sup_property_and_generated_ideals} Let $E$ be a vector lattice, and let $F$ be an order dense vector sublattice of $E$. Then $F$ has the countable sup property if and only if the ideal $I_F$ of $E$ that is generated by $F$ has the countable sup property. \end{lemma}
\begin{proof} Since $F$ is an order dense and majorising vector sublattice of $I_F$, the Dedekind completions of $F$ and $I_F$ are isomorphic. The proof is then completed by using that the countable sup property of a vector lattice and that of its Dedekind completion are equivalent; see \cite[Theorem~32.9]{luxemburg_zaanen_RIESZ_SPACES_VOLUME_I:1971}. \end{proof}
\begin{proposition}\label{res:order_separability_equivalences}
Let $E$ be a vector lattice, and let $F$ be an order dense vector sublattice of $E$. The following are equivalent:
\begin{enumerate}
\item $E$ has the countable sup property;\label{part:order_separability_equivalences_1}
\item $F$ has the countable sup property and $F$ is super order dense in $E$;\label{part:order_separability_equivalences_2}
\end{enumerate} \end{proposition}
\begin{proof}
Suppose that $E$ has the countable sup property. Let the net $\net$ in $\pos{F}$ and $x$ in $\pos{F}$ be such that $x_\alpha\uparrow x$ in $F$. Since $F$, being order dense in $E$, is a regular vector sublattice of $E$, we also have that $x_\alpha\uparrow x$ in $E$. By hypothesis, there exists a sequence of indices $\seq{\alpha_n}{n}$ in ${\mathcal A}$ such that $x_{\alpha_n}\uparrow x$ in $E$. Then also $x_{\alpha_n}\uparrow x$ in $F$. Hence $F$ has the countable sup property. Take an $x\in F^+$. Since $F$ is order dense in $E$, \cite[Theorem~1.34]{aliprantis_burkinshaw_POSITIVE_OPERATORS_SPRINGER_REPRINT:2006} shows that $S\coloneqq\{y\in F:0\leq y\leq x\}\uparrow x$ in $E$. The fact that $E$ has the countable sup property then yields a sequence $\seq{x_n}{n}\subseteq S\subseteq F$ such that $x_n\uparrow x$ in $E$. Hence $F$ is super order dense in $E$.
Suppose that $F$ has the countable sup property and that $F$ is super order dense in $E$. According to \cref{res:countable_sup_property_and_generated_ideals}, the ideal $I_F$ in $E$ that is generated by $F$ also has the countable sup property. Since $I_F\supseteq F$ is evidently super order dense in $E$, it follows from \cite[Theorem~29.4]{luxemburg_zaanen_RIESZ_SPACES_VOLUME_I:1971} that $E$ has the countable sup property. \end{proof}
\begin{remark}\quad\begin{enumerate}
\item The assumption in part~\ref{part:order_separability_equivalences_2} that $F$ be super order dense in $E$ cannot be relaxed to requiring it to be merely order dense in $E$, not even when $F$ is an order dense ideal of $E$ rather than an order dense vector sublattice; see \cite[Example~4.3]{kandic_vavpetic:2018}.
\item When $E$ is a vector lattice and $F$ is a vector sublattice of $E$ that has the countable sup property, then the fact that $E$ has the countable sup property is equivalent to the super order density of the Dedekind completion of $F$ in the Dedekind completion of $E$. We refer to \cite[Theorem~4.5]{kandic_vavpetic:2018} for this result in the same spirit as \cref{res:order_separability_equivalences}.
\end{enumerate} \end{remark}
All preparations have now been made for the proof of the core result of this section.
\begin{theorem}\label{res:tau_m_to_sub_uo}
Let $E$ be a vector lattice with the countable sup property, and suppose that $E$ has an order dense ideal $F$ such that $\ocdual{F}$ separates the points of $F$. Let $G$ be a regular vector sublattice of $E$. Then $G$ admits a \uppars{necessarily unique} Hausdorff uo-Lebes\-gue topology\ ${\widehat{\tau}}_G$.
Let $\net$ be a net in $G$ and suppose that $x_\alpha\xrightarrow{{\widehat{\tau}}_G} x$ for some $x\in G$. Take a sequence $(\alpha^\prime_n)_{n=1}^\infty$ of indices in ${\mathcal A}$. Then there exists an increasing sequence $\alpha_1^\prime=\alpha_1\leq\alpha_2\leq\dotsb$ of indices in ${\mathcal A}$ such that $\alpha_n\geq\alpha^\prime_n$ for all $n\geq 1$ and $x_{\alpha_n}\convwithoverset{\uo} x$ in $G$. In particular, when a sequence $(x_n)_{n=1}^\infty $ in $G$ and $x\in G$ are such that $x_n\xrightarrow{{\widehat{\tau}}_G}x$ in $G$, then there exists a subsequence $(x_{n_k})_{k=1}^\infty$ of $(x_n)_{n=1}^\infty $ such that $x_{n_k}\convwithoverset{\uo} x$ in $G$. \end{theorem}
\begin{proof}
In view of \cref{res:going_down_one} and \cref{res:local-global_for_o-convergence_and_uo-convergence}, we may (and shall) suppose that $G=E$.
We know from \cref{res:uo-lebesgue_topology_originating_from_separating_order_continuous_dual_of_order_dense_ideal} that $E$ admits a (necessarily unique) Hausdorff uo-Lebes\-gue topology\ ${\widehat{\tau}}_E$.
The statement on subsequences is clear from the statement on nets, so we need only establish the existence of the $\alpha_n$ for $n\geq 1$. We may suppose that $x=0$.
Suppose first that $E$ is Dedekind complete.
For $y\in \pos{F}$, we let $I_y\subseteq F$ denote the ideal that is generated by $y$ in $E$. By \cref{res:order_separability_equivalences}, $F$ inherits the countable sup property from $E$. Hence \cref{res:prep_for_tau_m_to_sub_uo_1} applies to the vector lattice $F$. We then see that $(I_y)^\sim_\mathrm{oc}$ separates the points of $I_y$ and that there even exists a strictly positive order continuous linear functional on $I_y$. We choose and fix such a strictly positive $\varphi_y\in(I_y)^\sim_\mathrm{oc}$ for each $y\in \pos{F}$.
From \cref{res:essential} we know that
\begin{equation}\label{eq:converges_to_zero}
\varphi_y(\abs{x_\alpha}\wedge y)\to 0
\end{equation}
for all $y\in \pos{F}$.
Set $\alpha_1\coloneqq\alpha_1^\prime$. Since $F$ is super order dense in $E$ by \cref{res:order_separability_equivalences}, we can choose a sequence $\{y_m^1\}_{m=1}^\infty$ in $\pos{F}$ such that $0\leq y_m^1\uparrow_m \abs{x_{\alpha_1}}$.
For $n\geq 2$, we shall now inductively construct an index $\alpha_n\in{\mathcal A}$ and a sequence $\{y_m^n\}_{m=1}^\infty$ in $\pos{F}$ such that, for all $n\geq 2$:
\begin{enumerate_alpha}
\item $\alpha_n\geq\alpha_n^\prime$;
\item $\alpha_{n}\geq\alpha_{n-1}$;
\item $\varphi_{y_m^i}\bigl(\abs{x_{\alpha_n}}\wedge y_m^i\bigr)<2^{-n}$ for $i=1,2,\ldots,n-1$ and $m=1,2,\dotsc,n$;
\item $0\leq y_m^n\uparrow_m \abs{x_{\alpha_n}}$ in $E$.
\end{enumerate_alpha}
We start with $n=2$. The elements $y_m^1$ of $\pos{F}$ are already known for all $m\geq 1$, and $\varphi_{y_m^1}\bigl(\abs{x_\alpha}\wedge y_m^1\bigr)\xrightarrow{}0$ for all $m\geq 1$ by \cref{eq:converges_to_zero}. Therefore, we can choose an $\alpha_2\in{\mathcal A}$ such that $\varphi_{y_m^1}\bigl(\abs{x_{\alpha_2}}\wedge y_m^1\bigr)<2^{-2}$ for $m=1,2$. We can arrange that also $\alpha_2\geq\alpha_2^\prime$ and $\alpha_2\geq\alpha_1$. Finally, we choose a sequence $(y_m^2)_{m=1}^\infty$ in $F$ such that $0\leq y_m^2\uparrow_m \abs{x_{\alpha_2}}$. This completes the construction for $n=2$.
Suppose that $n\geq 2$ and that we have already constructed $\alpha_2,\ldots,\alpha_n\in{\mathcal A}$ and sequences $(y_m^1)_{m=1}^\infty,\dotsc, (y_m^n)_{m=1}^\infty$ in $\pos{F}$ satisfying the four requirements above. The elements $y_m^i$ of $\pos{F}$ are already known for all $i=1,2,\ldots,n$ and $m\geq 1$, and $\varphi_{y_m^i}\bigl(\abs{x_\alpha}\wedge y_m^i\bigr)\xrightarrow{}0$ for all such $i$ and $m$ by \cref{eq:converges_to_zero}. Therefore, we can choose $\alpha_{n+1}\in{\mathcal A}$ such that $\varphi_{y_m^i}\bigl(\abs{x_{\alpha_{n+1}}}\wedge y_m^i\bigr)<2^{-(n+1)}$ for all $i=1,2,\ldots,n$ and $m=1,2,\dotsc,n+1$. We can arrange that also $\alpha_{n+1}\geq\alpha_{n+1}^\prime$ and $\alpha_{n+1}\geq\alpha_n$. Finally, we choose a sequence $(y_m^{n+1})_{m=1}^\infty$ in $\pos{F}$ such that $0\leq y_m^{n+1}\uparrow_m \abs{x_{\alpha_{n+1}}}$ in $E$. This completes the construction for $n+1$.
Fix $i,m\geq 1$. Since $0\leq \abs{x_{\alpha_j}}\wedge y_m^i\leq y_m^i$ for all $j\geq 1$, we can define elements $z_n^{j,m}$ of $I_{y_m ^i}$ for $n\geq 1$ by setting $z_{n}^{i,m}\coloneqq\bigvee_{j=n}^{\infty}\bigl(\abs{x_{\alpha_j}}\wedge y_m^i\bigr)$. Here the supremum is in the ideal $I_{y_m ^i}$ in $E$ (which, although this is immaterial, happens to coincide with the supremum in $E$). It is clear that $z_n\geq 0$ for $n\geq 1$ and that $z_{n}^{i,m}\downarrow_n$; we shall show that $z_{n}^{i,m}\downarrow_n 0$ in $I_{y_m^i}$. For this, we start by noting that
the inequality in (c) shows that $\varphi_{y_m^i}\bigl(\abs{x_{\alpha_j}}\wedge y_m^i\bigr)<2^{-j}$ for all $j\geq \max(i+1,m)$. Therefore, for all $n\geq \max(i+1,m)$, we can use the order continuity of $\varphi_{y_m^i}$ on $I_{y_m^i}$ to see that
\begin{align*}
0&\leq \varphi_{y_m^i}(z_{n}^{i,m})
\\&=\varphi_{y_m^i}\left(\bigvee_{j=n}^{\infty}\bigl(\abs{x_{\alpha_j}}\wedge y_i^m\bigr)\right)
\\ &=\varphi_{y_m^i}\left(\sup_{k\geq n}\left(\bigvee_{j=n}^{k}\bigl(\abs{x_{\alpha_j}}\wedge y_i^m\bigr)\right)\right)
\\ &=\lim_{\overset{k\to\infty}{k\geq n}}\varphi_{y_m^i}\left(\bigvee_{j=n}^{k}\bigl(\abs{x_{\alpha_j}}\wedge y_i^m\bigr)\right)\\
\\ &\leq \limsup_{\overset{k\to\infty}{k\geq n}}\varphi_{y_m^i}\left(\sum_{j=n}^{k}\bigl(\abs{x_{\alpha_j}}\wedge y_i^m\bigr)\right)
\\ &\leq \limsup_{\overset{k\to\infty}{k\geq n}}\sum_{j=n}^k 2^{-j}
\\&\leq 2^{-n+1}.
\end{align*}
We see from this that for the infimum $\inf_{n\geq 1}z_n^{i,m}$ in $I_{y_m^i}$ (which, although again immaterial, happens to coincide with the infimum in $E$) we have $0\leq \varphi_{y_m^i}\left(\inf_{n\geq 1} z_n^{i,m}\right)\leq 2^{-n+1}$ for all $n\geq \max(i+1,m)$. Hence $\varphi_{y_m^i}\left(\inf_{n\geq 1} z_n^{i,m}\right)=0$. Since $\varphi_{y_m^i}$ is strictly positive on $I_{y_m^i}$, this implies that $\inf_{n\geq 1}z_n^{i,m}=0$ in $I_{y_m^i}$, as we wanted to show.
The inequalities $0\leq \abs{x_{\alpha_n}}\wedge y_m^i\leq z_n^{i,m}$ for all $n\geq 1$ now show that $\abs{x_{\alpha_n}}\wedge y_m^i\convwithoverset{\mathrm{o}} 0$ in $I_{y_m^i}$ as $n\to\infty$, and then also $\abs{x_{\alpha_n}}\wedge y_m^i\convwithoverset{\mathrm{o}} 0$ in $E$ as $n\to\infty$.
We have now shown that, for all $i,m\geq 1$, $\abs{x_{\alpha_n}}\wedge y_m^i\convwithoverset{\mathrm{o}} 0$ in $E$ as $n\to\infty$.
Let $B$ denote the band that is generated by $\{y_m^i:i,m\geq 1\}$ in $E$. In view of (d) above, it is clear that the sequence $(x_{\alpha_n})_{n=1}^\infty$ is a sequence in $B$. We can now conclude from \cref{res:prep_for_tau_m_to_sub_uo_3} that $x_{\alpha_n}\convwithoverset{\uo} 0$ in $E$. This concludes the proof when $E$ is Dedekind complete.
For a general vector lattice $E$, we pass to the Dedekind completion $E^\delta$ of $E$. By \cite[Theorem~32.9]{luxemburg_zaanen_RIESZ_SPACES_VOLUME_I:1971}, $E^\delta$ also has the countable sup property. We let $F^\delta$ denote the ideal that is generated in $E^\delta$ by $F$. Then $F$ is obviously majorising in $F^\delta$. Since $F$ is order dense in $E$ and $E$ is order dense in $E^\delta$, $F$ is order dense in $E^\delta$ and then also in $F^\delta$. We see from this that, as the notation already suggests, $F^\delta$ is the Dedekind completion of $F$, but what we actually need is that, by \cref{res:veksler}, $\ocdual{\left(F^\delta\right)}$ separates the points of $F^\delta$. The fact that $F$ is order dense in $E^\delta$ implies that $F^\delta\supseteq F$ is order dense in $E^\delta$. Hence $E^\delta$ also admits a (necessarily) unique Hausdorff o-Lebes\-gue topology\ ${\widehat{\tau}}_{E^\delta}$. Moreover, \cref{res:going_down_one} shows that $x_\alpha\xrightarrow{{\widehat{\tau}}_{E^\delta}}0$ in $E^\delta$.
By what has been established for the Dedekind complete case, there exist indices $\alpha_n$ as specified such that $x_{\alpha_n}\convwithoverset{\uo} 0$ in $E^\delta$. By \cref{res:local-global_for_o-convergence_and_uo-convergence}, $x_{\alpha_n}\convwithoverset{\uo} 0$ in $E$. \end{proof}
For comparison, we include the following; see \cite[Theorem~4.19]{aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003}. We recall that a topology on a vector lattice $E$ is a \emph{Fatou topology} when it is a (not necessarily Hausdorff) locally solid linear topology on $E$ that has a base of neighbourhoods of zero consisting of solid and order closed sets. A Lebesgue topology is a Fatou topology; see \cite[Lemma~4.1]{aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003}, for example.
\begin{theorem}\label{res:other_subsequence_result}
Let $E$ be a vector lattice with the countable sup property that is supplied with a Hausdorff locally solid linear topology $\tau$ with the Fatou property. Suppose that $\net$ is an order bounded net in $E$ and that $x_\alpha\xrightarrow{\tau} x$ for some $x\in E$. Then there exist indices $\alpha_1\leq\alpha_2\leq\dotsb$ in ${\mathcal A}$ such that $x_{\alpha_n}\convwithoverset{\mathrm{o}} x$. \end{theorem}
The hypotheses in \cref{res:other_subsequence_result} on the topology on the vector lattice are weaker than those in \cref{res:tau_m_to_sub_uo}, and its conclusion is stronger. The big difference is, however, that the net in \cref{res:other_subsequence_result} is supposed to be order bounded, whereas there is no such restriction in \cref{res:tau_m_to_sub_uo}.
\cref{res:other_subsequence_result} also holds when, instead of requiring $E$ to have the countable sup property, it is required that there exist an at most countably infinite subset of $E$ such that the band that it generates equals the carrier of $\tau$; see \cite[Theorem~6.7]{kandic_taylor:2018}. We refer to \cite[Definition~4.15]{aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003} for the definition of the carrier of a (not necessarily Hausdorff) locally solid topology on a vector lattice.
\begin{remark}
The hypothesis in \cref{res:tau_m_to_sub_uo} that $E$ have the countable sup property cannot be relaxed to merely requiring that $F$ have this property. As a counter-example, consider the situation where $F$ is a Banach lattice with an order continuous norm that is an order dense ideal of a vector lattice $E$. Then $\ocdual{F}=F^\ast$ separates the points of $F$, and it is easy to see that $F$ has the countable sup property; the latter also follows from a more general result in \cite[Theorem~4.26]{aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003}.
Since the norm topology on $F$ is a Hausdorff o-Lebes\-gue topology\ on $F$, $E$ has a (necessarily unique) Hausdorff uo-Lebes\-gue topology\ ${\widehat{\tau}}_E$. It is the topology of un-convergence with respect to $F$. It is possible to find such $F$ and $E$, and a sequence in $E$ that is ${\widehat{\tau}}_E$-convergent to zero in $E$, yet has no subsequence that is uo-convergent to zero in $E$; see \cite[Example 9.6]{kandic_li_troitsky:2018}. \end{remark}
We have the following consequence of \cref{res:tau_m_is_convergence_in_measure} and \cref{res:tau_m_to_sub_uo}.
\begin{theorem}\label{res:convergence_in_measure_and_almost_everywhere}
Let $(X,\Sigma,\mu)$ be a measure space where $\mu$ is $\sigma$-finite. Suppose that $(f_\alpha)_{\alpha\in{\mathcal A}}$ is a net in $\mathrm{L}_0(X,\Lambda,\mu)$ such that $f_\alpha\xrightarrow{\mu^\ast}0$. Take a sequence $(\alpha^\prime_n)_{n=1}^\infty$ of indices in ${\mathcal A}$. Then there exists an increasing sequence $\alpha_1^\prime=\alpha_1\leq\alpha_2\leq\dotsb$ of indices in ${\mathcal A}$ such that $\alpha_n\geq\alpha^\prime_n$ for all $n\geq 1$ and $f_{\alpha_n}\xrightarrow{} 0$ almost everywhere. In particular, when a sequence $(f_n)_{n=1}^\infty$ is a sequence in $\mathrm{L}_0(X,\Lambda,\mu)$ and $f_n\xrightarrow{\mu^\ast}0$, then there exists a subsequence $(f_{n_k})_{k=1}^\infty$ of $(f_n)_{n=1}^\infty $ such that $f_{n_k}\xrightarrow{}0$ almost everywhere. \end{theorem}
\begin{proof}
It is known that $\mathrm{L}_0(X,\Sigma,\mu)$ has the countable sup property for every $\sigma$-finite measure $\mu$; see \cite[Theorem~7.73]{aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003} or \cite[Lemma~2.6.1]{meyer-nieberg_BANACH_LATTICES:1991}, for example.
The combination of \cref{res:tau_m_is_convergence_in_measure} and \cref{res:tau_m_to_sub_uo} yields a sequence of indices $\alpha_n$ as specified such that $f_{\alpha_n}\convwithoverset{\uo} 0$. Since, for a general measure $\mu$, uo-convergence of a sequence in $\mathrm{L}_0(X,\Sigma,\mu)$ is equivalent to its convergence almost everywhere (see \cite[Proposition~3.1]{gao_troitsky_xanthos:2017}), the proof is complete. \end{proof}
\begin{remark}\quad
\begin{enumerate}
\item In view of its proof, the natural condition on $\mu$ in \cref{res:convergence_in_measure_and_almost_everywhere} is that $\mu$ be semi-finite and have the countable sup property. It is known, however, that this is equivalent to requiring that $\mu$ be $\sigma$-finite; see \cite[Proposition~6.5]{kandic_taylor:2018}.
\item For every measure $\mu$, a sequence in $\mathrm{L}_0(X,\Lambda,\mu)$ that converges (globally) in measure has a subsequence that converges almost everywhere to the same limit; see \cite[Theorem 2.30]{folland_REAL_ANALYSIS_SECOND_EDITION:1999}, for example. \cref{res:convergence_in_measure_and_almost_everywhere} does not imply this result for arbitrary measures, but once the measure is known to be $\sigma$-finite, it \emph{does} produce the desired subsequence, and it even does so under the weaker hypothesis of convergence in measure on subsets of finite measure.
\item Even for finite measures, we are not aware of an existing result that, as in \cref{res:convergence_in_measure_and_almost_everywhere}, is concerned with \emph{nets} that converge in measure. \end{enumerate}
\end{remark}
We conclude this section by extending another classical result from measure theory to the context of Hausdorff uo-Lebes\-gue topologies\ and uo-convergence. Suppose that $(X,\Sigma.\mu)$ is a measure space, where $\mu$ is $\sigma$-finite. Then a sequence in $\mathrm{L}_0(X,\Sigma,\mu)$ is convergent in measure on subsets of finite measure if and only if every subsequence has a further subsequence that converges to the same limit almost everywhere; see \cite[Exercise~18.14 on p.~132]{zaanen_INTEGRATION:1967}. This is a special case of the following.
\begin{theorem}\label{res:tau_m_to_sub_uo_iff}
Let $E$ be a vector lattice with the countable sup property, and suppose that $E$ has an order dense ideal $F$ such that $\ocdual{F}$ separates the points of $F$. Let $G$ be a regular sublattice of $E$. Then $G$ admits a \uppars{necessarily unique} Hausdorff uo-Lebes\-gue topology\ ${\widehat{\tau}}_G$.
For a sequence $\seq{x_n}{n}\subseteq G$, $x_n \xrightarrow{{\widehat{\tau}}_G} 0$ in $G$ if and only if every subsequence $\seq{x_{n_k}}{k}$ of $\seq{x_n}{n}$ has a further subsequence $\seq{x_{n_{k_i}}}{i}$ such that $x_{n_{k_i}}\convwithoverset{\uo} 0$ in $G$. \end{theorem}
\begin{proof}
In view of \cref{res:going_down_one} and \cref{res:local-global_for_o-convergence_and_uo-convergence}, we may (and shall) suppose that $G=E$.
The forward implication is clear from \cref{res:tau_m_to_sub_uo}. We now show the converse. When it fails that $x_n\convwithoverset{\uoLt} 0$ in $E$, then \cref{res:uo-lebesgue_topology_originating_from_separating_order_continuous_dual_of_order_dense_ideal} shows that there exists an $\varphi\in\ocdual{F}$, an $y\in F$, a subsequence $\seq{x_{n_k}}{k}$ of $\seq{x_n}{n}$ and an $\varepsilon>0$ such that $\abs{\varphi}(\abs{x_{n_k}}\wedge\abs{y})> \varepsilon$ for all $k$. It is then clear from the order continuity of $\varphi$ that it is impossible to find a further subsequence $\seq{x_{n_{k_i}}}{i}$ of $\seq{x_{n_k}}{k}$ such that $x_{n_{k_i}}\convwithoverset{\uo} 0$ in $E$. \end{proof}
As another special case of \cref{res:tau_m_to_sub_uo_iff}, we see that a sequence in a Banach lattice with an order continuous norm is un-convergent to zero if and only if every subsequence has a further subsequence that is uo-convergent to zero; we recall that a Banach lattice with an order continuous norm has the countable sup property by \cite[Theorem~4.26]{aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003}, for example. We have thus retrieved \cite[Theorem 4.4]{deng_o_brien_troitsky:2017}.
\section{Topological aspects of (unbounded) order convergence}\label{sec:topological aspects of uo-convergence}
\noindent In this section, we consider topological issues that are related to (sequential) order convergence and to (sequential) unbounded order convergence, with an emphasis on the latter. \cref{res:tau_m_to_sub_uo} will be seen to be an important tool.
Let $E$ be a vector lattice, and let $A\subseteq E$. We define the \emph{o-ad\-her\-ence\ of $A$} as the set of all order limits of nets in $A$, and denote it by $\oadh{A}$. The \emph{$\sigma$o-ad\-her\-ence\ of $A$} is the set of all order limits of sequences in $A$; it is denoted by $\soadh{A}$. \footnote{In \cite[p.~82]{luxemburg_zaanen_RIESZ_SPACES_VOLUME_I:1971}, our $\sigma$o-ad\-her\-ence\ is called the pseudo order closure. In \cite{gao_leung:2018}, our o-ad\-her\-ence\ of a subset $A$ is called the order closure of $A$, and it is denoted by $\overline{A}^\mathrm{o}$. These two terminologies, as well as the notation $\overline{A}^\mathrm{o}$, could suggest that taking the (pseudo) order closure is a (sequential) closure operation for a topology. Since this is hardly ever the case, we prefer a terminology and notation that avoid this possible confusion. It is inspired by \cite[Definition~1.3.1]{beattie_butzmann_CONVERGENCE_STRUCTURES_AND_APPLICATIONS_TO_FUNCTIONAL_ANALYSIS:2002}.} The \emph{uo-ad\-her\-ence} $\uoadh{A}$ and the \emph{$\sigma$uo-ad\-her\-ence} $\suoadh{A}$ of $A$ are similarly defined. The subset $A$ is \emph{o-closed} when $\oadh{A}=A$.\footnote{This definition is consistent with that in \cite{gao_leung:2018}.} The collection of all o-closed subsets of $E$ is easily seen to be the collection of closed sets of a topology that is called the \emph{o-topology on $E$}. The closure of a subset $A$ in the o-topology is denoted by $\oclos{A}$.\footnote{There is no notation for the closure operation in the o-topology in \cite{gao_leung:2018}.} We have $\oadh{A}\subseteq\oclos{A}$, with equality if and only if $\oadh{A}$ is o-closed. Likewise, there are $\sigma$o-closed subsets and a $\sigma$o-topology, uo-closed subsets and a uo-topology, and $\sigma$uo-closed subsets and a $\sigma$uo-topology, with similar notations and statements about inclusions and equalities of sets. Evidently, a uo-closed subset is o-closed, and a $\sigma$uo-closed subset is $\sigma$o-closed.
Order convergence in a vector lattice $E$ is hardly ever topological; according to \cite[Theorem~2.2]{dabboorasad_emelyanov_marabeh:2020} or \cite[Theorem~18.36]{taylor_THESIS:2018}, this is the case if and only if $E$ is finite-dimensional. It is not even true that the set map $A\mapsto\oadh{A}$ is always idempotent, i.e., that the o-ad\-her\-ence\ of a set is always o-closed. It is known, for example, that in every $\sigma$-order complete Banach lattice that does \emph{not} have an order continuous norm, there even exists a vector sublattice such that its o-ad\-her\-ence\ is not order closed; see \cite[Theorem~2.7]{gao_leung:2018}.
We know from \cref{res:atomic} that uo-convergence in atomic vector lattices is topological. According to \cite[Theorem~6.45]{taylor_THESIS:2018}, atomic vector lattices are, in fact, the only ones for which this is the case.
It appears to be open whether the uo-ad\-her\-ence\ of a subset of a vector lattice is always uo-closed. In \cite[Problem~2.5]{gao_leung:2018}, it is even asked whether the uo-ad\-her\-ence\ of a vector sublattice is always o-closed, which is asking for a weaker conclusion for a much more restrictive class of subsets.
Even though the topological aspects of uo-convergence are still not well understood in general, there is a class of vector lattices where we have a reasonably complete picture. In order to formulate this, we need some more notation. For a set $X$ with a topology $\tau$ and a subset $A\subseteq X$ of $X$, we let $\sadh{\tau}{A}$ denote the $\sigma\tau$-adherence of $A$, i.e., $\sadh{\tau}{A}$ is the set consisting of all $\tau$-limits of sequences in $A$. When $\sadh{\tau}{A}=A$, $A$ is said to be $\sigma\tau$-closed. The $\sigma\tau$-closed subsets of $X$ are the closed subsets of a topology on $X$ that is called the $\sigma\tau$-topology on $X$. We let $\overline{A}^\tau$ and $\overline{A}^{\sigma\tau}$ denote the $\tau$-closure and the $\sigma\tau$-closure of a subset $A$ of $X$, respectively. Then $\sadh{\tau}{A}\subseteq\overline{A}^{\sigma\tau}$, with equality if and only if $\sadh{\tau}{A}$ is $\sigma\tau$-closed.
\begin{theorem}\label{res:seven_sets_equal} Let $E$ be a vector lattice with the countable sup property, and suppose that $E$ has an order dense ideal $F$ such that $\ocdual{F}$ separates the points of $F$. Let $G$ be a regular vector sublattice of $E$. Then $G$ admits a \uppars{necessarily unique} Hausdorff uo-Lebes\-gue topology\ ${\widehat{\tau}}_G$. For a subset $A$ of $G$, the following seven subsets of $G$ are all equal:
\begin{enumerate}
\item $\sadh{{\widehat{\tau}}_G}{A}$ and $\overline{A}^{\sigma{\widehat{\tau}}_G}$;
\item $\suoadh{A}$ and $\suoclos{A}$;
\item $\uoadh{A}$ and $\uoclos{A}$;
\item $\overline{A}^{{\widehat{\tau}}_G}$. \end{enumerate}
In particular, the $\sigma{\widehat{\tau}}_G$-topology, the $\sigma$uo-topology, and the uo-topology on $G$ all coincide with ${\widehat{\tau}}_G$. \end{theorem}
In \cref{res:seven_sets_equal}, the topological closures and ($\sigma$-)adherences are to be taken with respect to the topologies and convergences in $G$.
\begin{proof} The existence and uniqueness of ${\widehat{\tau}}_G$ are clear from \cref{res:tau_m_to_sub_uo}. Using \cref{res:tau_m_to_sub_uo} for the first inclusions, we have, for an arbitrary subset $A$ of $G$, \[
\overline{A}^{{\widehat{\tau}}_G}\subseteq\suoadh{A}\subseteq\uoadh{A}\subseteq\overline{A}^{{\widehat{\tau}}_G} \] and \[ \sadh{{\widehat{\tau}}_G}{A}\subseteq \suoadh{A}\subseteq\sadh{{\widehat{\tau}}_G}{A}. \] This gives equality of $\sadh{{\widehat{\tau}}_G}{A}$, $\suoadh{A}$, $\uoadh{A}$, and $\overline{A}^{{\widehat{\tau}}_G}$. Since the set map $A\mapsto \overline{A}^{{\widehat{\tau}}_G}$ is idempotent, so is $A\mapsto\sadh{{\widehat{\tau}}_G}{A}$. Hence $\sadh{{\widehat{\tau}}_G}{A}$ is $\sigma{\widehat{\tau}}_G$-closed, so that it coincides with the $\sigma\tau$-closure $\overline{A}^{\sigma{\widehat{\tau}}_G}$ of $A$. A similar argument works for $\suoclos{A}$ and $\uoclos{A}$. \end{proof}
\begin{remark} Taking $G=E$ in \cref{res:seven_sets_equal}, the equality of $\overline{A}^{{\widehat{\tau}}_G}$ and $\suoadh{A}$ implies that, for a $\sigma$-finite measure $\mu$, a subset of $\mathrm{L}_0(X,\Sigma,\mu)$ is closed in the topology of convergence in measure on subsets of finite measure if and only if it contains the almost every limits of sequences in it. This is \cite[245L(b)]{fremlin_MEASURE_THEORY_VOLUME_2:2003}. \end{remark}
In the context of \cref{res:seven_sets_equal}, it is also possible to give a necessary and sufficient condition for sequential uo-convergence to be topological; see \cref{res:sequential_uo_convergence_topological_two}, below. The proof of the following preparatory lemma is an abstraction of the argument in \cite{ordman:1966}.
\begin{lemma}\label{res:ordman_argument}
Let $E$ be a vector lattice that is supplied with a topology $\tau$. Suppose that $\tau$ has the following properties:
\begin{enumerate}
\item for every sequence $\seq{x_n}{n}$ in $E$ and for every $x\in E$, the fact that $x_n\convwithoverset{\tau} x$ implies that there exists a subsequence $\seq{x_{n_k}}{k}$ of $\seq{x_n}{n}$ such that $x_{n_k}\convwithoverset{\uo} x$ as $k\to\infty$.
\item there exists a sequence $\seq{x_n}{n}$ in $E$ and an $x\in E$ such that $x_n\convwithoverset{\tau} x$ but $x_n\overset{\mathrm{uo}}{\nrightarrow} x$;
\end{enumerate} Then there does not exist a topology $\tau^\prime$ on $E$ such that, for every sequence $\seq{x_n}{n}$ in $E$ and for every $x\in E$, $x_n\convwithoverset{\uo} x$ if and only if $x_n\xrightarrow{\tau^\prime} x$. \end{lemma}
\begin{proof} Suppose that there were such a topology $\tau^\prime$. Take a sequence $\seq{x_n}{n}$ in $E$ and an $x\in E$ such that $x_n\convwithoverset{\tau} x$ but $x_n\overset{\mathrm{uo}}{\nrightarrow} x$. Then also $x_n\overset{\tau^\prime}{\nrightarrow} x$, so that there exists a $\tau^\prime$-neighbourhood $V$ of $x$ and a subsequence $\seq{x_{n_k}}{k}$ of $\seq{x_n}{n}$ such that $x_{n_k}\not\in V$ for all $k\geq 1$. Since also $x_{n_k}\convwithoverset{\tau} x$ as $k\to\infty$, there exists a subsequence $\seq{x_{n_{k_i}}}{i}$ of $\seq{x_{n_k}}{k}$ such that $x_{n_{k_i}}\convwithoverset{\uo} x$ as $i\to\infty$. Hence also $x_{n_{k_i}}\xrightarrow{\tau^\prime}x$ as $i\to\infty$. But this is impossible, since the entire sequence $\seq{x_{n_{k_i}}}{i}$ stays outside $V$. \end{proof}
The following is a direct consequence of \cref{res:ordman_argument}. The topology $\tau$ in it could be a uo-Lebes\-gue topology, but for the result to hold it need not even be a linear topology, nor need the topology $\tau^\prime$ be.
\begin{proposition}\label{res:sequential_uo_convergence_topological_one}
Let $E$ be a vector lattice that is supplied with a topology $\tau$. Suppose that $\tau$ has the following properties:
\begin{enumerate}
\item for every sequence $\seq{x_n}{n}$ in $E$ and for every $x\in E$, the fact that $x_n\convwithoverset{\uo} x$ implies that $x_n\convwithoverset{\tau} x$;
\item for every sequence $\seq{x_n}{n}$ in $E$ and for every $x\in E$, the fact that $x_n\convwithoverset{\tau} x$ implies that there exists a subsequence $\seq{x_{n_k}}{k}$ of $\seq{x_n}{n}$ such that $x_{n_k}\convwithoverset{\uo} x$ as $k\to\infty$.
\end{enumerate}
Then the following are equivalent;
\begin{enumerate}
\item there exists a topology $\tau^\prime$ on $E$ such that, for every sequence $\seq{x_n}{n}$ in $E$ and for every $x\in E$, $x_n\convwithoverset{\uo} x$ if and only if $x_n\xrightarrow{\tau^\prime} x$;
\item for every sequence $\seq{x_n}{n}$ in $E$ and for every $x\in E$, the fact that $x_n\convwithoverset{\tau} x$ implies that $x_n\convwithoverset{\uo} x$.
\end{enumerate} In that case, one can take $\tau$ for $\tau^\prime$. \end{proposition}
In the appropriate context, the combination of \cref{res:tau_m_to_sub_uo} and \cref{res:sequential_uo_convergence_topological_one} yields the following necessary and sufficient condition for sequential uo-convergence to be topological. Note that there are no assumptions at all on the topology $\tau$ in its first part.
\begin{corollary}\label{res:sequential_uo_convergence_topological_two} Let $E$ be a vector lattice with the countable sup property, and suppose that $E$ has an order dense ideal $F$ such that $\ocdual{F}$ separates the points of $F$. Let $G$ be a regular vector sublattice of $E$. Then $G$ admits a \uppars{necessarily unique} Hausdorff uo-Lebes\-gue topology\ ${\widehat{\tau}}_G$, and the following are equivalent: \begin{enumerate}
\item there exists a topology $\tau$ on $G$ such that, for every sequence $\seq{x_n}{n}$ in $G$ and for every $x\in G$, $x_n\convwithoverset{\uo} x$ in $G$ if and only if $x_n\xrightarrow{\tau} x$;\label{part:uo_convergence_implies_top_convergence}
\item for every sequence $\seq{x_n}{n}$ in $G$ and for every $x\in G$, the fact that $x_n\xrightarrow{{\widehat{\tau}}_G} x$ in $G$ implies that $x_n\convwithoverset{\uo} x$ in $G$.\label{part:top_convergence_implies_uo_convergence} \end{enumerate} In that case, one can take ${\widehat{\tau}}_G$ for $\tau$. \end{corollary}
The proof of the following result closely follows the one in \cite{ordman:1966}, where it is shown that sequential almost everywhere convergence in $L_\infty([0,1])$ is not topological.
\begin{corollary}\label{res:ordman_construction}
Let $(X,\Sigma,\mu)$ be a measure space, where $\mu$ is $\sigma$-finite. Suppose that there exists an $A\in\Sigma$ with the property that, for every $k\geq 1$, there exist finitely many mutually disjoint $A_{k,1},\dotsc, A_{k,N_k}\in\Sigma$ such that $0<\mu(A_{k,1}),\dotsc,\mu(A_{k,N_k})<1/k$ and $A=\bigcup_{l=1}^{N_k}A_{k,l}$.
Take a regular vector sublattice $G$ of $\mathrm{L}_0(X,\Sigma,\mu)$ that contains the characteristic functions $1_{A_{k,l}}$ of all sets $A_{k,l}$ for $k=1,2,\dotsc$ and $l=1,\dotsc,N_k$. Then there does not exist a topology $\tau$ on $G$ such that, for every sequence $\seq{x_n}{n}$ in $G$ and for every $x\in G$, $x_n\convwithoverset{\uo} x$ in $G$ if and only if $x_n\xrightarrow{\tau} x$. \end{corollary}
\begin{proof}
We are in the situation of \cref{res:sequential_uo_convergence_topological_two}, where ${\widehat{\tau}}_G$-convergence is convergence in measure on subsets of finite measure by \cref{res:tau_m_is_convergence_in_measure}, and sequential uo-convergence is almost everywhere convergence by \cite[Proposition~3.1]{gao_troitsky_xanthos:2017}. Consider the following sequence in $G$: \[ A_{1,1},\dotsc, A_{1,N_1},A_{2,1},\dotsc, A_{2,N_2},A_{3,1},\dotsc,A_{3,N_3},\dotsc. \] This sequence clearly converges to zero on subsets of finite measure, but it converges nowhere to zero on the subset $A$ of strictly positive measure. Hence the property in part~\ref{part:top_convergence_implies_uo_convergence} of \cref{res:sequential_uo_convergence_topological_two} does not hold, and then neither does the property in its part~ \ref{part:uo_convergence_implies_top_convergence}. \end{proof}
\begin{remark} \cref{res:ordman_construction} provides us with a large class of examples of vector lattices where sequential uo-convergence is not topological\textemdash so that uo-convergence is certainly not topological\textemdash but where, according to \cref{res:seven_sets_equal}, the set maps $A\mapsto\suoadh{A}$ and $A\mapsto\uoadh{A}$ are both still idempotent, so that $\suoadh{A}$ is $\sigma$uo-closed and $\uoadh{A}$ is uo-closed for every subset $A$ of $G$. For all $p$ such that $0\leq p\leq\infty$, the space $L_p([0,1])$ is such an example. \end{remark}
We conclude with a strengthened version of \cite[Theorem~2.2]{gao_leung:2018}. The improvement lies in the removal of the hypothesis that $E$ be Banach lattice, and by adding eight more equal, but not obviously equal, sets to the three equal sets in the original result.
\begin{theorem}\label{res:eleven_sets_equal}
Let $E$ be a vector lattice with the countable sup property, and suppose that $\ocdual{E}$ separates the points of $E$. Then $E$ admits a \uppars{necessarily unique} Hausdorff uo-Lebes\-gue topology\ ${\widehat{\tau}}_E$. Take an ideal $I$ of $\ocdual{E}$ that separates the points of $E$, and take a vector sublattice $F$ of $E$. Then the following eleven vector sublattices of $E$ are all equal:
\begin{enumerate}
\item $\sadh{{\widehat{\tau}}_E}{F}$ and $\overline{F}^{\sigma{\widehat{\tau}}_E}$;
\item $\suoadh{F}$ and $\suoclos{F}$;
\item $\uoadh{F}$ and $\uoclos{F}$;
\item $\overline{F}^{{\widehat{\tau}}_E}$, $\overline{F}^{\abs{\sigma}(E,I)}$, and $\overline{F}^{\sigma(E,I)}$;
\item\label{part:double_order_adherence_and_order_closure} $(\oadh{\oadh{F}})$ and $\oclos{F}$.
\end{enumerate}
\end{theorem}
The equality of $\uoadh{F}$, $\oadh{\oadh{F}}$, and $\overline{F}^{\sigma(E,I)}$ can already be found in \cite[Theorem~2.2]{gao_leung:2018}, where it also noted that these sets coincide with the smallest order closed vector sublattice of $E$ containing $F$.
\begin{proof}
The equality of the first seven subsets is clear from \cref{res:seven_sets_equal}. Since we know from \cref{res:uo-lebesgue_topology_originating_from_separating_order_continuous_dual_of_order_dense_ideal} that ${\widehat{\tau}}_E=\mathrm{u}_E\abs{\sigma}(E,I)$, it follows from \cite[Proposition~2.12]{taylor:2019} that $\overline{F}^{{\widehat{\tau}}_E}=\overline{F}^{\abs{\sigma}(E,I)}$. Furthermore, from Kaplan's theorem (see \cite[Theorem~2.33]{aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003}, for example) we know that $E$, when supplied with the Hausdorff locally convex $\abs{\sigma}(E,I)$-topology, has the same topological dual as when it is supplied with the Hausdorff locally convex $\sigma(E,I)$-topology. By the convexity of $F$, we have $\overline{F}^{\abs{\sigma}(E,I)}=\overline{F}^{\sigma(E,I)}$. This argument was already used in \cite[Proof of Lemma~2.1]{gao_leung:2018}.
We turn to the two sets in part~\ref{part:double_order_adherence_and_order_closure}. It was established in \cite[Lemma~2.1]{gao_leung:2018} that $
\uoadh{F}\subseteq\oadh{\oadh{F}}$; this is, in fact, valid for vector sublattices of general vector lattices. It was also observed there that, obviously, the fact that $I\subseteq\ocdual{E}$ implies that $\overline{F}^{\sigma(E,I)}$ is o-closed. Using also that we already know that $\uoadh{F}=\uoclos{F}$, we therefore have the following chain of inclusions:
\[
\uoclos{F}=\uoadh{F}\subseteq \oadh{\oadh{F}}\subseteq \oclos{F}\subseteq \overline{F}^{\sigma(E,I)}. \]
Since we also already know that $\uoclos{F}=\overline{F}^{\sigma(E,I)}$, the proof is complete.
\end{proof}
\urlstyle{same}
\end{document} | arXiv |
Padovan sequence
In number theory, the Padovan sequence is the sequence of integers P(n) defined[1] by the initial values
$P(0)=P(1)=P(2)=1,$
and the recurrence relation
$P(n)=P(n-2)+P(n-3).$
The first few values of P(n) are
1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, ... (sequence A000931 in the OEIS)
A Padovan prime is a Padovan number that is prime. The first Padovan primes are:
2, 3, 5, 7, 37, 151, 3329, 23833, 13091204281, 3093215881333057, 1363005552434666078217421284621279933627102780881053358473, 1558877695141608507751098941899265975115403618621811951868598809164180630185566719, ... (sequence A100891 in the OEIS).
The Padovan sequence is named after Richard Padovan who attributed its discovery to Dutch architect Hans van der Laan in his 1994 essay Dom. Hans van der Laan : Modern Primitive.[2] The sequence was described by Ian Stewart in his Scientific American column Mathematical Recreations in June 1996.[3] He also writes about it in one of his books, "Math Hysteria: Fun Games With Mathematics". [4]
The above definition is the one given by Ian Stewart and by MathWorld. Other sources may start the sequence at a different place, in which case some of the identities in this article must be adjusted with appropriate offsets.
Recurrence relations
In the spiral, each triangle shares a side with two others giving a visual proof that the Padovan sequence also satisfies the recurrence relation
$P(n)=P(n-1)+P(n-5)$
Starting from this, the defining recurrence and other recurrences as they are discovered, one can create an infinite number of further recurrences by repeatedly replacing $P(m)$ by $P(m-2)+P(m-3)$
The Perrin sequence satisfies the same recurrence relations as the Padovan sequence, although it has different initial values.
The Perrin sequence can be obtained from the Padovan sequence by the following formula:
$\mathrm {Perrin} (n)=P(n+1)+P(n-10).\,$
Extension to negative parameters
As with any sequence defined by a recurrence relation, Padovan numbers P(m) for m<0 can be defined by rewriting the recurrence relation as
$P(m)=P(m+3)-P(m+1),$
Starting with m = −1 and working backwards, we extend P(m) to negative indices:
P−20 P−19 P−18 P−17 P−16 P−15 P−14 P−13 P−12 P−11 P−10 P−9 P−8 P−7 P−6 P−5 P−4 P−3 P−2 P−1 P0 P1 P2
7 −7 4 0 −3 4 −3 1 1 −2 2 −1 0 1 −1 1 0 0 1 0 1 1 1
Sums of terms
The sum of the first n terms in the Padovan sequence is 2 less than P(n + 5), i.e.
$\sum _{m=0}^{n}P(m)=P(n+5)-2.$
Sums of alternate terms, sums of every third term and sums of every fifth term are also related to other terms in the sequence:
$\sum _{m=0}^{n}P(2m)=P(2n+3)-1$ OEIS: A077855
$\sum _{m=0}^{n}P(2m+1)=P(2n+4)-1$
$\sum _{m=0}^{n}P(3m)=P(3n+2)$ OEIS: A034943
$\sum _{m=0}^{n}P(3m+1)=P(3n+3)-1$
$\sum _{m=0}^{n}P(3m+2)=P(3n+4)-1$
$\sum _{m=0}^{n}P(5m)=P(5n+1).$ OEIS: A012772
Sums involving products of terms in the Padovan sequence satisfy the following identities:
$\sum _{m=0}^{n}P(m)^{2}=P(n+2)^{2}-P(n-1)^{2}-P(n-3)^{2}$
$\sum _{m=0}^{n}P(m)^{2}P(m+1)=P(n)P(n+1)P(n+2)$
$\sum _{m=0}^{n}P(m)P(m+2)=P(n+2)P(n+3)-1.$
Other identities
The Padovan sequence also satisfies the identity
$P(n)^{2}-P(n+1)P(n-1)=P(-n-7).\,$
The Padovan sequence is related to sums of binomial coefficients by the following identity:
$P(k-2)=\sum _{2m+n=k}{m \choose n}=\sum _{m=\lceil k/3\rceil }^{\lfloor k/2\rfloor }{m \choose k-2m}.$
For example, for k = 12, the values for the pair (m, n) with 2m + n = 12 which give non-zero binomial coefficients are (6, 0), (5, 2) and (4, 4), and:
${6 \choose 0}+{5 \choose 2}+{4 \choose 4}=1+10+1=12=P(10).\,$
Binet-like formula
The Padovan sequence numbers can be written in terms of powers of the roots of the equation[1]
$x^{3}-x-1=0.\,$
This equation has 3 roots; one real root p (known as the plastic number) and two complex conjugate roots q and r.[5] Given these three roots, the Padovan sequence can be expressed by a formula involving p, q and r :
$P(n)=ap^{n}+bq^{n}+cr^{n}$
where a, b and c are constants.[1]
Since the absolute values of the complex roots q and r are both less than 1 (and hence p is a Pisot–Vijayaraghavan number), the powers of these roots approach 0 for large n, and $P(n)-ap^{n}$ tends to zero.
For all $n\geq 0$, P(n) is the integer closest to ${\frac {p^{5}}{2p+3}}p^{n}$. Indeed, ${\frac {p^{5}}{2p+3}}$ is the value of constant a above, while b and c are obtained by replacing p with q and r, respectively.
The ratio of successive terms in the Padovan sequence approaches p, which has a value of approximately 1.324718. This constant bears the same relationship to the Padovan sequence and the Perrin sequence as the golden ratio does to the Fibonacci sequence.
Combinatorial interpretations
• P(n) is the number of ways of writing n + 2 as an ordered sum in which each term is either 2 or 3 (i.e. the number of compositions of n + 2 in which each term is either 2 or 3). For example, P(6) = 4, and there are 4 ways to write 8 as an ordered sum of 2s and 3s:
2 + 2 + 2 + 2 ; 2 + 3 + 3 ; 3 + 2 + 3 ; 3 + 3 + 2
• The number of ways of writing n as an ordered sum in which no term is 2 is P(2n − 2). For example, P(6) = 4, and there are 4 ways to write 4 as an ordered sum in which no term is 2:
4 ; 1 + 3 ; 3 + 1 ; 1 + 1 + 1 + 1
• The number of ways of writing n as a palindromic ordered sum in which no term is 2 is P(n). For example, P(6) = 4, and there are 4 ways to write 6 as a palindromic ordered sum in which no term is 2:
6 ; 3 + 3 ; 1 + 4 + 1 ; 1 + 1 + 1 + 1 + 1 + 1
• The number of ways of writing n as an ordered sum in which each term is odd and greater than 1 is equal to P(n − 5). For example, P(6) = 4, and there are 4 ways to write 11 as an ordered sum in which each term is odd and greater than 1:
11 ; 5 + 3 + 3 ; 3 + 5 + 3 ; 3 + 3 + 5
• The number of ways of writing n as an ordered sum in which each term is congruent to 2 mod 3 is equal to P(n − 4). For example, P(6) = 4, and there are 4 ways to write 10 as an ordered sum in which each term is congruent to 2 mod 3:
8 + 2 ; 2 + 8 ; 5 + 5 ; 2 + 2 + 2 + 2 + 2
Generating function
The generating function of the Padovan sequence is
$G(P(n);x)={\frac {x+x^{2}}{1-x^{2}-x^{3}}}.$
This can be used to prove identities involving products of the Padovan sequence with geometric terms, such as:
$\sum _{n=0}^{\infty }{\frac {P(n)}{2^{n}}}={\frac {12}{5}}.$
$\sum _{n=0}^{\infty }{\frac {P(n)}{\alpha ^{n}}}={\frac {\alpha ^{2}(\alpha +1)}{\alpha ^{3}-\alpha -1}}.$
Generalizations
In a similar way to the Fibonacci numbers that can be generalized to a set of polynomials called the Fibonacci polynomials, the Padovan sequence numbers can be generalized to yield the Padovan polynomials.
Padovan L-system
If we define the following simple grammar:
variables : A B C
constants : none
start : A
rules : (A → B), (B → C), (C → AB)
then this Lindenmayer system or L-system produces the following sequence of strings:
n = 0 : A
n = 1 : B
n = 2 : C
n = 3 : AB
n = 4 : BC
n = 5 : CAB
n = 6 : ABBC
n = 7 : BCCAB
n = 8 : CABABBC
and if we count the length of each string, we obtain the Padovan numbers:
1, 1, 1, 2, 2, 3, 4, 5, ...
Also, if you count the number of As, Bs and Cs in each string, then for the nth string, you have P(n − 5) As, P(n − 3) Bs and P(n − 4) Cs. The count of BB pairs and CC pairs are also Padovan numbers.
Cuboid spiral
Main article: Padovan cuboid spiral
A spiral can be formed based on connecting the corners of a set of 3-dimensional cuboids. This is the Padovan cuboid spiral. Successive sides of this spiral have lengths that are the Padovan numbers multiplied by the square root of 2.
Pascal's triangle
Erv Wilson in his paper The Scales of Mt. Meru[6] observed certain diagonals in Pascal's triangle (see diagram) and drew them on paper in 1993. The Padovan numbers were discovered in 1994. Paul Barry (2004) showed that these diagonals generate the Padovan sequence by summing the diagonal numbers.
References
1. Weisstein, Eric W. "Padovan Sequence". MathWorld..
2. Richard Padovan. Dom Hans van der Laan: modern primitive: Architectura & Natura Press, ISBN 9789071570407.
3. Ian Stewart, Tales of a Neglected Number, Scientific American, No. 6, June 1996, pp. 92-93.
4. Ian Stewart (2004), Math hysteria: fun and games with mathematics, Oxford University Press, p. 87, ISBN 978-0-19-861336-7.
5. Richard Padovan, "Dom Hans Van Der Laan and the Plastic Number", pp. 181-193 in Nexus IV: Architecture and Mathematics, eds. Kim Williams and Jose Francisco Rodrigues, Fucecchio (Florence): Kim Williams Books, 2002.
6. Erv Wilson (1993), Scales of Mt. Meru
• Ian Stewart, A Guide to Computer Dating (Feedback), Scientific American, Vol. 275, No. 5, November 1996, Pg. 118.
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• OEIS sequence A000931 (Padovan sequence)
• A Padovan sequence calculator
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| Wikipedia |
\begin{document}
\title[]{Constrained ergodic optimization for generic continuous functions}
\author[S.Motonaga]{Shoya Motonaga} \address{Research Organization of Science and Technology, Ritsumeikan University, 1-1-1 Noji-higashi, Kusatsu, Shiga 525-8577, Japan} \email{[email protected]}
\author[M. Shinoda]{Mao Shinoda} \address{Department of Mathematics, Ochanomizu University, 2-1-1 Otsuka, Bunkyo-ku, Tokyo, 112-8610, Japan} \email{[email protected]}
\subjclass[2010]{\textcolor{black}{Primary} 37E45, 37B10, 37A99}
\keywords{}
\begin{abstract} One of the fundamental results of ergodic optimization asserts that for any dynamical system on a compact metric space with the specification property and for a generic continuous function $f$ every invariant probability measure that maximizes the space average of $f$ must have zero entropy. We establish the analogical result in the context of constrained ergodic optimization, which is introduced by Garibaldi and Lopes (2007). \end{abstract}
\maketitle
\section{Introduction}
Let $T:X\rightarrow X$ be a continuous map on a compact metric space and $\mathcal{M}_T(X)$ be the space of Borel probability measures endowed with the weak*-topology.
Let $C(X, \mathbb{R})$ be the space of real-valued continuous functions on $X$ with the supremum norm $\|\cdot\|_\infty$. For each $f\in C(X,\mathbb{R})$ we consider the maximum ergodic average \begin{align}
\beta(f)=\sup_{\mu\in \mathcal{M}_T(X)}\int f\ d\mu
\label{beta} \end{align} and the set of all maximizing measures of $f$ \begin{align}
\mathcal{M}_{{\rm max}}(f)=\left\{\mu\in \mathcal{M}_T(X): \int f\ d\mu=\beta(f)\right\}.
\label{maxset} \end{align} The functional $\beta$ and the set $\mathcal{M}_{{\rm max}}(f)$ are main objects in ergodic optimization, which has been actively studied for a decade (see for more details \cite{Jen06survey, Jen2017}).
Constrained erogdic optimization, which is introduced by Garibaldi and Lopes in \cite{GarLop07}, investigates the analogical objects as \eqref{beta} and \eqref{maxset} under some constraint. Introducing a continuous $\mathbb{R}^d$-valued function $\varphi\in C(X,\mathbb{R}^d)$
playing the role of a constraint, we define the {\it rotation set} of $\varphi=(\varphi_1, \ldots, \varphi_d)$ by \begin{align*}
{\rm Rot}(\varphi)=\left\{{\rm rv}_{\varphi}(\mu)\in \mathbb{R}^d: \mu\in \mathcal{M}_T(X)\right\}, \end{align*} where ${\rm rv}_{\varphi}(\mu)=(\int \varphi_1 d\mu, \ldots, \int\varphi_d d\mu)$. For $h\in {\rm Rot}(\varphi)$, the fiber ${\rm rv}_{\varphi}^{-1}(h)$ is called the {\it rotation class} of $h$. A point $h\in {\rm Rot}(\varphi)$ is called a {\it rotation vector}. The terminology comes from Poincar\'e's rotation numbers for circle homeomorphisms. Many authors study the properties of rotation sets and characterize several dynamics in terms of rotation vectors \cite{GM99,Jen01rotation,K92, K95,KW14,KW16,KW19}.
We define the maximum ergodic average for a continuous function $f$ with constraint $h\in {\rm Rot}(\varphi)$ by \begin{align}
\beta^\varphi_h(f)=\sup_{\mu\in {\rm rv}_{\varphi}^{-1}(h)}\int f\ d\mu
\label{beta_constraint} \end{align} and the set of all relative maximizing measures of $f$ with a rotation vector $h\in {\rm Rot}(\varphi)$ by \begin{align*}
\mathcal{M}^\varphi_h(f)=\left\{\mu\in \mathcal{M}_{T}(X): \beta^\varphi_h(f)=\int f\ d\mu,\ {\rm rv}_{\varphi}(\mu)=h\right\}.
\end{align*} Note that this formulation is a generalization of (unconstrained) ergodic optimization, since for a constant constraint $\varphi\equiv h$ and its unique rotation vector $h$ the rotation set of $h$ is $\mathcal{M}_T(X)$.
We have much interest in the above constrained optimization because the constraints provide information about the asymptotic behavior of orbits. Such problem is originally studied by Mather \cite{Mather91} and Ma\~{n}\'{e} \cite{Mane96} for Euler-Lagrange flows, where the asymptotic homological position of the trajectory in the configuation space is given as a constraint. Recently, Bochi and Rams \cite{BochiRams16} proved that the Lyapunov optimizing measures for one-step cocycles of $2 \times 2$ matrices have zero entropy on the Mather sets under some conditions, which implies that a low complexity phenomena occurs in noncommutative setting such as a Lyapunov-optimization problem for one-step cocycles. A relative Lyapunov-optimization is mentioned for their future research but to the authors’ knowledge the classical commutative counterpart, typicality of zero entropy of relative maximizing measure, has not been established yet. We remark that Zhao \cite{Zhao19} studies constrained ergodic optimization for asymptotically additive potentials for the application in the study of multifractal analysis.
It is natural to extend the fundamental results of unconstrained ergodic optimization to constrained one. In \cite{GarLop07}, several general results on constrained ergodic optimization are provided, especially, uniqueness of maximizing measures with any constraint for generic continuous functions is asserted.
Moreover, prevalent uniqueness of maximizing measures for continuous functions in the constrained settings easily follows from \cite{Mor21} (see Appendix \ref{AppendixB}). However, in these studies, the differences between constrained ergodic optimization and unconstrained one are not mentioned explicitly. In some cases, constraints prevent existence of relatively maximizing periodic measures (see Remark \ref{Rmk:periodic}), which should give rise to the problem of existence of periodic measures in a given rotation class. Moreover, in contrast to Morris’ theorem \cite{Mor2010} which asserts that for any dynamical systems with the specification property every maximizing measure for a generic continuous function has zero entropy, we can easily verify that there exists a constraint such that for a generic potential its unique relatively maximizing measure has positive entropy (see Proposition~\ref{positive_entropy}). Thus it is important to investigate the condition that the statement of Morris’s theorem holds for constrained ergodic optimization.
In this paper, we study the structure of rotation classes and the generic property in constrained ergodic optimization for symbolic dynamics. Our first main result is the density of periodic measures with some rational constraints. This is an analogical result to Sigmund's work for a dynamical system with the specification \cite{Sigmund}. The difficulty in the constrained case comes from the existence of a measure which is a convex combination of ergodic measures with different rotation vectors. This prevents us to use the ergodic decomposition in a given rotation class. To overcome this difficulty, our approach requires a certain finiteness for both of a subshift and a constraint function. Moreover, a detailed analysis is needed to construct a periodic measure approximating a given invariant one with the same rotation vector.
Let $(\Omega, \sigma)$ be an irreducible sofic shift with finite alphabets (See \S\ref{symbolic} below). A function $\varphi\in C(\Omega, \mathbb{R}^d)$ is said to be {\it locally constant} if for each $i=1, \ldots, d$ there exists $k_i\geq0$ such that \begin{align*}
\varphi_i(x)=\varphi_i(y)
\quad \mbox{if}\quad
x_j=y_j\ \mbox{for}\ j=0, \ldots, k_i. \end{align*} Denote by $\mathcal{M}_\sigma^p(\Omega)$ the set of invariant measures supported on a single periodic orbit. \begin{maintheorem} \label{density_of_periodic_measures} Let $(\Omega, \sigma)$ be an irreducible sofic shift. Let $\varphi=(\varphi_1, \ldots, \varphi_d)\in C(\Omega, \mathbb{Q}^d)$ be a $\mathbb{Q}^d$-valued locally constant function and $h\in {\rm int}({\rm Rot}(\varphi))\cap \mathbb{Q}^d$. Then the set ${\rm rv}_{\varphi}^{-1}(h)\cap \mathcal{M}_\sigma^p(\Omega)$ is dense in ${\rm rv}_{\varphi}^{-1}(h)$.
\end{maintheorem}
\begin{remark} Theorem \ref{density_of_periodic_measures} is motivated by Theorem 10 in \cite{GarLop07}: It was shown that
for a Walters potential $A$ on a subshift of finite type $\Omega$, there exists a periodic measure $\mu$ whose action $\int A \ d\mu$ approximates $\int A \ d\nu$ for $\nu\in {\rm rv}_{\varphi}^{-1}(h)$ with ${\rm rv}_{\varphi}(\mu)=h$ if $\nu$ is ergodic and the locally constant constraint $\varphi\in C(\Omega, \mathbb{Q}^d)$ is joint recurrent in relation to $\nu$ (see \cite{GarLop07} for the definition of the joint recurrence and their precise statement).
Note that Theorem 10 in \cite{GarLop07} was suggested by the fact that a circle homeomorphism with rational rotation number has a periodic point and a result of this kind for symbolic dynamics was studied by Ziemian (see Theorem 4.2 of \cite{Zie95}).
\end{remark}
\begin{remark}{ Although the properties of rotation sets are well-studied in \cite{Zie95}, to the authors’ knowledge, that of rotation classes have attracted little attention. We emphasize that Theorem \ref{density_of_periodic_measures} provides a more detailed description of Theorem 4.2 in \cite{Zie95} under weaker assumptions. } \end{remark}
\begin{remark}\label{Rmk:periodic} If a constraint $\varphi\in C(\Omega,\mathbb{Q}^d)$ is locally constant, it is easy to see that the rotation vector of a periodic measure should be rational, i.e., \begin{align*}
\varphi(\mathcal{M}_\sigma^p(\Omega))\subset {\rm Rot}(\varphi)\cap \mathbb{Q}^d. \end{align*} Hence in Theorem \ref{density_of_periodic_measures} we need to choose a rotation vector $h$ in ${\rm Rot}(\varphi)\cap \mathbb{Q}^d$. \end{remark}
Applying Theorem \ref{density_of_periodic_measures}, we next investigate the property of relative maximizing measure for symbolic dynamics . Regarding \eqref{beta_constraint} as a functional on $C(\Omega,\mathbb{R})$, we can characterize a relative maximizing measure as a ``tangent" measure. This characterization allows us to adapt argument in \cite{Mor2010} for our constrained case (See \S\ref{generic} for more details). The following generic property is our second main result. \begin{maintheorem} \label{generic_zero_entropy} Let $(\Omega, \sigma)$ be an irreducible sofic shift. Let $\varphi=(\varphi_1, \ldots, \varphi_d)\in C(\Omega, \mathbb{Q}^d)$ be a $\mathbb{Q}^d$-valued locally constant function and $h\in {\rm int}({\rm Rot}(\varphi))\cap \mathbb{Q}^d$. Then for generic $f\in C(\Omega,\mathbb{R})$ every relative maximizing measure of $f$ with constraint $h\in {\rm Rot}(\varphi)$ has zero entropy. In particular, setting $K(f):=\bigcup_{\mu\in \mathcal{M}_h^\varphi(f)} {\rm supp}\mu $, we have $h_{{\rm top}}(K)=0$ for generic $f\in C(\Omega,\mathbb{R})$. \end{maintheorem}
The remainder of this paper is organized as follows. In \S\ref{density}, we study the structure of rotation classes. In particular, we clarify our definitions and notations for symbolic dynamics and prove Theorem \ref{density_of_periodic_measures} in \S\ref{symbolic}. In \S\ref{generic} we will illustrate that a relative maximizing measure is regarded as a tangent measure of \eqref{beta_constraint} and prove Theorem \ref{generic_zero_entropy}.
\section{Structure of rotation classes} \label{density} \subsection{Density of convex combinations of periodic measures}
We first investigate the structure of rotation classes
for a continuous map $T:X\rightarrow X$ on a compact metric space $X$
such that $\mathcal{M}_T^p(X)$ is dense in $\mathcal{M}_T(X)$.
As mentioned in Remark \ref{Rmk:periodic},
a rotation class does not contain a periodic measure in some cases.
Nevertheless, we have the density of convex combinations of periodic measures in a rotation class for every continuous constraint function and for every rotation vector in the interior of the rotation set.
For $\nu\in \mathcal{M}_T(X)$, a finite set $F\subset C(X, \mathbb{R})$ and $\varepsilon>0$ set \begin{align*}
U_{(F, \varepsilon)}(\nu):=\left\{\mu\in \mathcal{M}_T(X): \left|\int f d\mu-\int f d\nu\right|<\varepsilon, f\in F\right\}. \end{align*} Let \begin{align*}
\Delta^d&=\left\{(\lambda_1,\ldots,\lambda_{d})\in (0,1)^{d}: \sum_{i=1}^{d}\lambda_i=1\right\},\\
\mathcal{N}_T&=\Bigg\{\sum_{i=1}^{d+1} \lambda_i\mu_i:
\ \mu_i\in \mathcal{M}_T^p(X),\quad
(\lambda_1,\ldots,\lambda_{d+1})\in\Delta^{d+1},\\
&\qquad\quad
\dim {\rm span} \{{\rm rv}_{\varphi}(\mu_1)-{\rm rv}_{\varphi}(\mu_{d+1}), \ldots, {\rm rv}_{\varphi}(\mu_{d})-{\rm rv}_{\varphi}(\mu_{d+1})\}=d\Bigg\}, \end{align*}
where ${\rm span}\{h_1, \ldots, h_{d}\}$ is the vector space spanned by $\{h_1, \ldots, h_d\}$.
We begin with the following proposition.
\begin{proposition}\label{density_of_rational_convex_combinations}
Let $T:X\rightarrow X$ be a continuous map on a compact metric space $X$ such that $\mathcal{M}_T^p(X)$ is dense in $\mathcal{M}_T(X)$.
Let $\varphi=(\varphi_1, \ldots, \varphi_d)\in C(X, \mathbb{R}^d)$ and $h\in {\rm int}({\rm Rot}(\varphi))$. Then the set ${\rm rv}_{\varphi}^{-1}(h)\cap\mathcal{N}_T$ is dense in ${\rm rv}_{\varphi}^{-1}(h)$.
\end{proposition}
\begin{proof} Take $\nu\in {\rm rv}_{\varphi}^{-1}(h)$ and a open neighborhood $U_\nu$ of $\nu$. Then there exists a finite set $F\subset C(X, \mathbb{R})$ and $\varepsilon>0$ such that $U_{(F, \varepsilon)}(\nu)\subset U_\nu$.
Since $h\in{\rm int}({\rm Rot}(\varphi))$, there exists $\delta>0$ such that \begin{align*}
B_{\sqrt{d}\delta}(h)\subset {\rm Rot}(\varphi). \end{align*} where $B_{\sqrt{d}\delta}(h)$ is the open ball of radius $\sqrt{d}\delta$ centered at $h$. Let $h'_i=h+\delta e_i$ for $i=1,\ldots, d+1$ where $\{e_i\}_{i=1}^d$ is the standard basis of $\mathbb{R}^d$ and $e_{d+1}=-\sum_{i=1}^d e_i$. Then $\{h'_i\}_{i=1}^{d+1}$ forms a simplex with the barycenter $h$. Let $\xi_i\in{\rm rv}_\varphi^{-1}(h'_i)$.
Fix \begin{align*}
0<t< \frac{\varepsilon}{4\max_{f\in F}\|f\|_\infty} \end{align*} and define $\nu_i:=(1-t)\nu+t\xi_i$. Then for $f\in F$ we have \begin{align*}
\left|\int f d\nu_i-\int f d\nu\right|&\leq t\left|\int f d\xi_i-\int f d\nu\right|
\leq 2t\|f\|_\infty<\frac{\varepsilon}{2} \end{align*} and $\nu_i\in U_{(F, \varepsilon/2)}(\nu)$. By the definition of $\nu_i$ we have \begin{align*}
{\rm rv}_\varphi(\nu_i)=(1-t){\rm rv}_\varphi(\nu)+t{\rm rv}_\varphi(\xi_i)
=(1-t)h+th'_i=h+t\delta e_i. \end{align*} Thus $\frac{1}{d+1}\sum_{i=1}^{d+1}{\rm rv}_\varphi(\nu_i)=h$ holds.
Since $\mathcal{M}_T^p(X)$ is dense in $\mathcal{M}_T(X)$, for $r>0$, there exists \begin{align*}
\mu_i\in U_{(F, \varepsilon/2)}(\nu_i) \cap U_{(\{\varphi\}, r/\sqrt{d})}(\nu_i) \cap \mathcal{M}_T^p(X). \end{align*} Since ${\rm rv}_{\varphi}(\nu_1)-{\rm rv}_{\varphi}(\nu_{d+1}),{\rm rv}_{\varphi}(\nu_2)-{\rm rv}_{\varphi}(\nu_{d+1}),\ldots, {\rm rv}_{\varphi}(\nu_d)-{\rm rv}_{\varphi}(\nu_{d+1})$ are linearly independent, by the open property of linearly independence and the continuity of the map ${\rm rv}_\varphi$, we see that ${\rm rv}_{\varphi}(\mu_1)-{\rm rv}_{\varphi}(\mu_{d+1}),{\rm rv}_{\varphi}(\mu_2)-{\rm rv}_{\varphi}(\mu_{d+1}),\ldots, {\rm rv}_{\varphi}(\mu_d)-{\rm rv}_{\varphi}(\mu_{d+1})$ are also linearly independent for all sufficiently small $r>0$. Moreover, we obtain \begin{align*}
\left|\left|\frac{1}{d+1}\sum_{i=1}^{d+1}{\rm rv}(\mu_i)-h\right|\right|_{\mathbb{R}^d}=
\left|\left|\frac{1}{d+1}\sum_{i=1}^{d+1}{\rm rv}(\mu_i)-\frac{1}{d+1}\sum_{i=1}^{d+1}{\rm rv}(\nu_i)\right|\right|_{\mathbb{R}^d}\\
\le\frac{1}{d+1}\sum_{i=1}^{d+1}\left|\left|{\rm rv}_\varphi(\mu_i)-{\rm rv}_\varphi(\nu_i)\right|\right|_{\mathbb{R}^d}
<\frac{d+1}{d+1}r=r, \end{align*}
where $\|\cdot\|_{\mathbb{R}^d}$ is the standard norm of $\mathbb{R}^d$. Hence $h$ is contained in the open ball of radius $r$ centered at $\frac{1}{d+1}\sum_{i=1}^{d+1}{\rm rv}(\mu_i)$. Taking $r>0$ sufficiently small, we deduce that $h$ is an interior point of the simplex whose vertices are ${\rm rv}_\varphi(\mu_1),\ldots, {\rm rv}_\varphi(\mu_{d+1})$, which implies that there exists $(\lambda_1,\ldots,\lambda_{d+1})\in\Delta^{d+1}$ such that $h=\sum_{i=1}^{d+1} \lambda_i {\rm rv}_\varphi(\mu_i)$.
Let $\tilde{\nu}=\sum_{i=1}^{d+1} \lambda_i\mu_i$. Trivially, ${\rm rv}_\varphi(\tilde{\nu})=h$ holds. Then for every $f\in F$ we have \begin{align*}
\left|\int f d\tilde{\nu}-\int f d\nu\right|
&=\left|\sum_{i=1}^{d+1}\lambda_i\int f d\mu_i-\sum_{i=1}^{d+1}\lambda_i\int f d\nu\right|\\
&\leq \sum_{i=1}^{d+1}\lambda_i \left|\int f d\mu_i-\int f d\nu\right|\\
&\leq \sum_{i=1}^{d+1}\lambda_i \left(\left|\int f d\mu_i-\int f d\nu_i\right|+\left|\int f d\nu_i-\int f d\nu\right|\right)\\
&<\sum_{i=1}^{d+1}\lambda_i\left(\frac{\varepsilon}{2}+\frac{\varepsilon}{2}\right)
=\varepsilon, \end{align*} i.e., $\tilde{\nu}=\sum_{i=1}^{d+1} \lambda_i\mu_i\in U_{(F,\varepsilon)}$, and this is precisely the assertion of the proposition. \end{proof}
\begin{remark} \label{decomposition}
In the proof of Proposition \ref{density_of_rational_convex_combinations}, we have $\mu_i\in U_{(F, \varepsilon)}(\nu)$ for every $i\in \{1, \ldots, d+1\}$ since we have $\nu_i\in U_{(F, \varepsilon/2)}(\nu)$ and $\mu_i\in U_{(F,\varepsilon/2)}(\nu_i)$. We will use this fact in the proof of Theorem \ref{density_of_periodic_measures}. \end{remark}
\begin{corollary}\label{residual_set_of_zero_entropy}
Assume the hypotheses of Proposition \ref{density_of_rational_convex_combinations}. Let $H_\mu$ be the entropy map of $T$. Then the set
\begin{align*}
\mathcal{Z}=\{\mu\in {\rm rv}_{\varphi}^{-1}(h): H_\mu=0\}
\end{align*} is a residual subset of ${\rm rv}_{\varphi}^{-1}(h)$. \end{corollary} \begin{proof}
The argument is similar to \cite{DGS}[Proposition 22.16, p.223].
By Proposition \ref{density_of_rational_convex_combinations}, $\mathcal{N}_T\cap {\rm rv}_{\varphi}^{-1}(h)$ is dense in ${\rm rv}_{\varphi}^{-1}(h)$
and thus $\mathcal{Z}$ is also.
Moreover, upper semi-continuity of the entropy map $\mu\mapsto H_\mu$ implies for every $n\geq1$ \begin{align*}
\mathcal{Z}_n:=\left\{\mu\in {\rm rv}_{\varphi}^{-1} (h): 0\leq H_\mu<\frac{1}{n}\right\} \supset \mathcal{N}_T\cap {\rm rv}_{\varphi}^{-1} (h)\end{align*} is nonempty, open and dense in ${\rm rv}_{\varphi}^{-1}(h)$. Hence $\mathcal{Z}=\bigcap_{n\geq 1}\mathcal{Z}_n$ is a residual set in ${\rm rv}_{\varphi}^{-1}(h)$. \end{proof}
\begin{lemma}\label{lem:rational_coefficient}
Assume the hypotheses of Proposition \ref{density_of_rational_convex_combinations}.
Let $\mu_1,\ldots,\mu_{d+1}\in\mathcal{M}_T^p(X)$
such that ${\rm rv}_\varphi(\mu_1), \ldots, {\rm rv}_\varphi(\mu_{d+1})\in \mathbb{Q}^d$ and
\begin{align}\label{eqn:dimension}
\dim {\rm span} \{{\rm rv}_{\varphi}(\mu_1)-{\rm rv}_{\varphi}(\mu_{d+1}), \ldots, {\rm rv}_{\varphi}(\mu_{d})-{\rm rv}_{\varphi}(\mu_{d+1})\}=d.
\end{align}
Let $(\lambda_1,\ldots,\lambda_{d+1})\in\Delta^{d+1}$
and $\tilde{\nu}=\sum_{i=1}^{d+1}\lambda_i\mu_i\in\mathcal{M}_T(X)$.
Then $\lambda_i$ is rational for each $i=1,\ldots, d+1$ if ${\rm rv}_\varphi(\tilde{\nu})$ belongs to $\mathbb{Q}^d$. \end{lemma} \begin{proof}
Since $\tilde{\nu}=\sum_{i=1}^{d+1}\lambda_i\mu_i$ and $\sum_{i=1}^{d+1}\lambda_i=1$, we have
\begin{align}\label{eqn:rv}
{\rm rv}_\varphi(\tilde{\nu})={\rm rv}_\varphi(\mu_{d+1})+\sum_{i=1}^d \lambda_i\left({\rm rv}_\varphi(\mu_i)-{\rm rv}_\varphi(\mu_{d+1})\right).
\end{align}
Let $V$ be a $d\times d$-matrix given by \begin{align}\label{eqn:V}
V=\Big({\rm rv}(\mu_{d+1})-{\rm rv}(\mu_{1}), \ldots, {\rm rv}(\mu_{d+1})-{\rm rv}(\mu_{d})\Big) \end{align} and $\lambda$ be a column vector given by $(\lambda)_i=\lambda_i$ for $i=1,\ldots, d$. It follows from \eqref{eqn:dimension} that the matrix $V$ is invertible. Therefore, by \eqref{eqn:rv}, we obtain \begin{align}\label{eqn:lambda}
\lambda=V^{-1}\left({\rm rv}_\varphi(\mu_{d+1})-{\rm rv}_\varphi(\tilde{\nu})\right). \end{align} Since ${\rm rv}_\varphi(\mu_1), \ldots, {\rm rv}_\varphi(\mu_{d+1})$ and ${\rm rv}_\varphi(\tilde{\nu})$ are in $\mathbb{Q}^d$,
each component of $V$ and that of its inverse $V^{-1}$ are rational. Therefore, by \eqref{eqn:lambda}, we deduce that $\lambda_i$ is rational for each $i=1,\ldots, d+1$. \end{proof}
\subsection{Symbolic dynamics and density of periodic measures} \label{symbolic} We next consider symbolic dynamics. In this particular case, under some assumptions, we can prove the density of periodic measures in a given rotation class. Denote by $\mathbb{N}_0$ the set of all non-negative integers. For a finite set $\mathcal{A}$ we consider the one-sided infinite product $\mathcal{A}^{\mathbb{N}_0}$ equipped with the product topology of the discrete one.
Let $\sigma$ be the shift map on $\mathcal{A}^{\mathbb{N}_0}$
(i.e.,~$(\sigma (\ul{x}))_i= x_{i+1}$ for each $i\in {\mathbb{N}_0}$
and $\ul{x}= ( x_i)_{i\in {\mathbb{N}_0}} \in \mathcal{A}^{\mathbb{N}_0}$). When a subset $\Omega$ of $\mathcal{A}^{\mathbb{N}_0}$ is $\sigma$-invariant and closed, we call it a \textit{subshift}. Slightly abusing the notation we denote by $\sigma$ the shift map restricted on $\Omega$.
For a subshift $\Omega$, let $[u] = \left\{ \ul{x}\in \Omega : u=x_0\cdots x_{n-1}\right\}$ for each $u\in \mathcal{A}^n$, $n\geq 1$ and set $ \mathcal L(\Omega ) =\left\{ u\in \bigcup _{n\geq 1}\mathcal{A}^n : [u] \neq \emptyset \right\}
$. We also denote $\mathcal{L}_n(\Omega):=\{u\in\mathcal{L}(\Omega):|u|=n\}$ for $n\ge 1$, where $|u|$ denotes the length of $u$, i.e., $|u|=n$ if $u =u_0\cdots u_{n-1} \in \mathcal{A}^n$. For $u,v\in\mathcal{L}(\Omega)$, we use the juxtaposition $uv$ to denote the word obtained by the concatenation and $u^\infty$ means a one-sided infinite sequence $uuu\cdots \in \mathcal{A}^{\mathbb{N}_0}$. We say that $\Omega$ is \textit{irreducible} if for any $i,j\in A$, we can find $u\in\mathcal{L}(\Omega)$ such that $iuj\in\mathcal{L}(\Omega)$ holds. For each $u\in \mathcal{L}(\Omega)$ define its follower set $F_\Omega(u)$ by $F_\Omega(u)=\{v\in \mathcal{L}(\Omega): uv\in \mathcal{L}(\Omega)\}$. Note that the follower sets of different words may coincide. A subshift $\Omega$ is said to be {\it sofic} if the set of follower sets is finite: $\#\{F_\Omega(u): u\in \mathcal{L}(\Omega)\}<+\infty$.
We refer to \cite{LM} and references therein for additional details on sofic shifts.
It is easy to see that the finiteness of the set of the follower sets yields the following:
\begin{lemma} \label{finite_follower} Let $(\Omega,\sigma)$ be a sofic shift. Then there exists $\kappa\geq 1$ such that for every $u\in \mathcal{L}_\kappa(\Omega)$ and $v\in \mathcal{L}(\Omega)$, we have $F_\Omega(vu)=F_\Omega(u)$ provided $u\in F_\Omega(v)$. \end{lemma}
In order to prove Theorem \ref{density_of_periodic_measures} we shall show that on a sofic shift periodic orbits which share the same word can be concatenated without extra gap words.
\begin{lemma} \label{concatenation} Let $(\Omega,\sigma)$ be a sofic shift and $\kappa\geq1$ for which Lemma \ref{finite_follower} holds. Let $u\in \mathcal{L}_\kappa(\Omega)$ and $v,w\in \mathcal{L}(\Omega)$ such that $(vu)^\infty, (wu)^\infty\in \Omega$. Then for every $k\geq1$ and sequences $\{m_i\}_{i=1}^k, \{n_i\}_{i=1}^k\subset \mathbb{N}$, we have \begin{align*}
((vu)^{m_1}(wu)^{n_1}(vu)^{m_2}\cdots (vu)^{m_k}(wu)^{n_k})^\infty \in \Omega. \end{align*} \end{lemma} \begin{proof} Let $k\geq1$, $\{m_i\}_{i=1}^k$ and $\{n_i\}_{i=1}^k\subset \mathbb{N}$. Since $(vu)^\infty \in \Omega$, we see that $u\in F_\Omega((vu)^{m_1-1}v)$. By Lemma \ref{finite_follower}, $F_\Omega((vu)^{m_1})=F_\Omega(u)$ holds. Moreover, it follows from $(wu)^\infty \in \Omega$ that \begin{align*}
(wu)^{n_1}\in F(u)=F((vu)^{m_1}), \end{align*} which implies $(vu)^{m_1}(wu)^{n_1}\in \mathcal{L}(\Omega)$. We also have $F((vu)^{m_1}(wu)^{n_1})=F(u)$ by Lemma \ref{finite_follower}. Similarly, we obtain \begin{align*}
(vu)^{m_2}(wu)^{n_2}\in F(u)=F((vu)^{m_1}(wu)^{n_1}) \end{align*} and $(vu)^{m_1}(wu)^{n_1}(vu)^{m_2}(wu)^{n_2}\in \mathcal{L}(\Omega)$. Repeating this argument,
\begin{align*}
((vu)^{m_1}(wu)^{n_1}(vu)^{m_2}\cdots (vu)^{m_k}(wu)^{n_k})^l\in \mathcal{L}(\Omega) \end{align*} holds for every $\ell\geq1$, which implies \begin{align*} ((vu)^{m_1}(wu)^{n_1}(vu)^{m_2}\cdots (vu)^{m_k}(wu)^{n_k})^\infty\in \Omega. \end{align*} \end{proof} In addition, an irreducible sofic shift satisfies the specification property and thus $\mathcal{M}_\sigma^p(\Omega)$ is dense in $\mathcal{M}_\sigma (\Omega)$ (see for example \cite{Weiss73} and \cite{Sigmund}). Hence we have the following by Proposition \ref{density_of_rational_convex_combinations}. \begin{corollary}\label{cor:density_of_rational_convex_combinations} Let $(\Omega, \sigma)$ be an irreducible sofic shift. Let $\varphi=(\varphi_1, \ldots, \varphi_d)\in C(\Omega, \mathbb{R}^d)$ be a locally constant function and $h\in {\rm int}({\rm Rot}(\varphi))$. Then the set ${\rm rv}_\varphi^{-1}(h)\cap\mathcal{N}_\sigma$ is dense in ${\rm rv}_\varphi^{-1}(h)$. \end{corollary}
Using Lemma \ref{concatenation} and Corollary \ref{cor:density_of_rational_convex_combinations}, we can prove Theorem \ref{density_of_periodic_measures}. \begin{proof}[Proof of Theorem \ref{density_of_periodic_measures}]
Take $\nu\in {\rm rv}_{\varphi}^{-1}(h)$ and a open neighborhood $U_\nu$ of $\nu$. There exists a finite set $F\subset C(\Omega, \mathbb{R})$ and $\varepsilon>0$ such that $U_{(F, \varepsilon)}(\nu)\subset U_\nu$. Without loss of generality we may assume every $f\in F$ is locally constant. Take $\kappa$ for which Lemma \ref{finite_follower} holds and $u=u_0\cdots u_{\kappa-1}\in \mathcal{L}_\kappa(\Omega)$ such that $\nu([u])>0$. Replace $F$ and $\varepsilon$ with $F\cup\{\chi_{[u]}\}$ and $\min\{\varepsilon, \nu([u])/2\}$ respectively, where $\chi_{[u]}$ is the characteristic function of $[u]$.
By Corollary \ref{cor:density_of_rational_convex_combinations}, there is $\tilde{\nu}\in U_{(F,\varepsilon/2)}(\nu)$ of the form $\tilde{\nu}=\sum_{i=1}^{d+1} \lambda_i\mu_i$ where $\mu_i\in\mathcal{M}_\sigma^p(\Omega)$ and $(\lambda_1,\ldots,\lambda_{d+1})\in\Delta^{d+1}$ with \eqref{eqn:dimension}. Note that ${\rm rv}_{\varphi}(\mu_i)\in \mathbb{Q}^d\ (i=1,\ldots,d+1)$ as stated in Remark \ref{Rmk:periodic}. For each $i=1,\ldots,d+1$, we will denote by $a_i^\infty$ the corresponding periodic orbits to $\mu_i\in\mathcal{M}_\sigma^p(\Omega)$ where $a_i$ is the word with lengths of periods. Moreover, by Lemma \ref{lem:rational_coefficient}, each $\lambda_i$ can be written as $\lambda_i=q_i/Q$ where $q_1,\ldots, q_{d+1},Q \in \mathbb{N}$ with $q_1+\ldots+q_{d+1}=Q$ since $(\lambda_1,\ldots,\lambda_{d+1})\in\Delta^{d+1}\cap\mathbb{Q}^{d+1}$.
Let $\ell$ be the maximum length of the words on which elements of $F\cup\{\varphi_1, \ldots, \varphi_d\}$ depend. We can assume \begin{align*}
\ell<\min\{|a_1|,\ldots, |a_{d+1}|\} \end{align*}
by replacing $a_i\ (i=1,\ldots,d+1)$ with concatenations $a_i^k$ for some $k\in \mathbb{N}$ if necessary. As stated in Remark \ref{decomposition}, we have $\mu_i\in U_{(F,\varepsilon)}(\nu)$ for every $i=1, \ldots, d+1$, which implies $\mu_i([u])>\nu([u])-\varepsilon\geq \nu([u])/2>0$. Hence $u$ is a subword of each $a_i$
and without loss of generality we may assume $a_{i, 0}\cdots a_{i, |u| -1}=u$. For $g\in F\cup\{\varphi\}$ define $\delta_{g,k}\ (k=1,\ldots,d+1)$ by \begin{align*}
\delta_{g,k}=
\sum_{i=0}^{\ell-1}
\Big\{&g(\sigma^{|a_k|-\ell+i}(a_k a_{k+1}))
-g(\sigma^{|a_k|-\ell+i}(a_k a_k))
\Big\} \end{align*} where $a_{d+2}=a_1$. Set $\delta_{g}=\delta_{g,1}+\ldots+\delta_{g,d+1}$.
Let $V$ be a $d\times d$-matrix given by \eqref{eqn:V}. As stated in the proof of Lemma \ref{lem:rational_coefficient}, the matrix $V$ is invertible by \eqref{eqn:dimension}. Since each component of $V$ and $\delta_{\varphi}$ is rational, we denote $V^{-1}\delta_{\varphi}=v/R$ where $v\in \mathbb{Z}^d$ and $R\in \mathbb{N}$. Let $v_i=(v)_i\ (i=1,\ldots, d)$.
Now we construct a periodic measure near $\nu$ with the rotation vector $h$. Since $a_1, \ldots, a_{d+1}$ share the same word $u$, by Lemma \ref{concatenation}, we can concatenate them without extra gap words. Hence let \begin{align*}
A&=|a_1|\ldots|a_{d+1}|,\quad
C=\max_{f\in F,\ i=1,\ldots,d+1} \left|\int f \ d \mu_i\right|,\quad \delta^*=\max_{f\in F} |\delta_f|,\\
m_i'&=Av_i/|a_i|\ (i=1,\ldots,d),\quad
m_{d+1}'=-A(\sum_{i=1}^d v_i)/|a_{d+1}|,\\
M_j'&=Aq_j/|a_j|,\quad
M_j=tARM_j'+m_j'\ (j=1,\ldots, d+1),\\
y&=a_1^{tM_1'+m_1'}a_2^{tM_2'+m_2'}\cdots a_{d+1}^{tM_{d+1}'+m_{d+1}'},\\
z&=a_1^{tM_1'}a_2^{tM_2'}\cdots a_{d+1}^{tM_{d+1}'},\quad
x=yz^{(AR-1)}, \end{align*} where $t\in \mathbb N$ is large enough to satisfy \begin{align}\label{ineq_t}
\frac{AR}{|x|}\delta^*+C\sum_{i=1}^{d+1}\left|\frac{M_i|a_i|}{|x|}-\lambda_i\right|<\frac{\varepsilon}{2} \end{align}
and $tM_{d+1}'-A(\sum_{i=1}^d v_i)/|a_{d+1}|>0$. Note that such $t\in\mathbb N$ exists since \begin{align*}
|x|&=\sum_{i=1}^{d+1} M_i|a_i|=tAR\sum_{i=1}^{d+1} M_i'|a_i|,\\
\lambda_i&=\frac{q_i}{\sum_{j=1}^{d+1}q_j}=\frac{M_i'|a_i|}{\sum_{j=1}^{d+1} M_j'|a_j|} \end{align*} hold and the left hand side of \eqref{ineq_t} tends to $0$ as $t\to +\infty$. Let $\mu$ be the periodic measure supported on $x^\infty$.
First we check ${\rm rv}_{\varphi} (\mu)=h (={\rm rv}_{\varphi}(\tilde{\nu}))$. Since $\varphi$ is locally constant, \begin{align*}
{\rm rv}_{\varphi}(\mu)&=\frac{1}{|x|}S_{|x|}\varphi(x)\\
&=\frac{1}{|x|}\left(AR\delta_{\varphi}+\sum_{i=1}^{d+1} M_i|a_i|\int \varphi d\mu_i\right)\\
&=\frac{1}{|x|}AR\delta_{\varphi}+\frac{1}{|x|}\sum_{i=1}^{d+1} m_i'|a_i|{\rm rv}_{\varphi}(\mu_i)+\sum_{i=1}^{d+1}\lambda_i{\rm rv}_{\varphi}(\mu_i)\\
&=h+\frac{A}{|x|}Vv+\frac{1}{|x|}\sum_{i=1}^{d+1} m_i'|a_i|{\rm rv}_{\varphi}(\mu_i)\\
&=h+\frac{A}{|x|}\left\{Vv+\sum_{i=1}^{d}v_i{\rm rv}_{\varphi}(\mu_i)-(\sum_{j=1}^d v_j){\rm rv}_{\varphi}(\mu_{d+1})\right\}
=h. \end{align*}
Next we check $\mu\in U_{(F, \varepsilon)}(\nu)$. Let $f\in F$. We compute \begin{align*}
&\left|\frac{1}{|x|}S_{|x|}f-\int f\ d\nu\right|\\
&\leq \left|\frac{1}{|x|}S_{|x|}f-\int f\ d\tilde{\nu}\right|+\left|\int f\ d\tilde{\nu}-\int f\ d\nu\right|\\
&=\left|\frac{1}{|x|}\Big(AR\delta_{f}+\sum_{i=1}^{d+1} M_i|a_i|\int f d\mu_i\Big)-\sum_{j=1}^{d+1} \lambda_i\int f d\mu_i\right|+\frac{\varepsilon}{2}\\
&\leq \sum_{i=1}^{d+1}\left|\left(\frac{M_i|a_i|}{|x|}-\lambda_i\right)\int f d\mu_i\right|
+\frac{AR}{|x|}|\delta_f|+\frac{\varepsilon}{2}\\
&< \frac{\varepsilon}{2}+\frac{\varepsilon}{2}= \varepsilon, \end{align*} which completes the proof. \end{proof}
\begin{remark} In the proof of Theorem \ref{density_of_periodic_measures}, we think the word $y$ as a corrective one to attain the desired rotation vector. A similar approach is used in Theorem 5 of \cite{Jen01rotation} in a different setting but our construction is more explicit than it. \end{remark}
\begin{remark} Note that the error term $\delta_\varphi$ does not depend on $m'_i\ (i=1,\ldots, d+1)$ in our case. For a subshift with the specification condition, we can concatenate the words $a_1^{m'_1}, \ldots,a_{d+1}^{m'_{d+1}}$ with some gap words but the error term in such case depends on $m'_i\ (i=1,\ldots, d+1)$, which implies we cannot choose a suitable corrective word $y$ for the error term $\delta_\varphi$ in such case. So we have to overcome this difficulty to extend Theorem \ref{density_of_periodic_measures} to the case of a subshift with the specification condition. \end{remark}
\begin{remark} For a rotation vector in the boundary of a rotation set, there may exist no periodic measure in the rotation class.
Let $\Omega\subset\{1,2,3\}^{\mathbb{N}_0}$ be a Markov shift with an adjacency matrix \begin{align*}
A=\begin{pmatrix}1&1&0\\1& 1 &1\\0&1&1\end{pmatrix} \end{align*} (i.e., $\Omega=\{\ul{x}\in \{1,2,3\}^{\mathbb{N}_0}: A_{x_i x_{i+1}}=1\ \mbox{for all} \ i\in \mathbb{N}_0\}$) and define $\varphi=(\varphi_1,\varphi_2,\varphi_3):\Omega \rightarrow\mathbb{Q}^3$ by \begin{align*}
\varphi_i(\ul{x})=\left\{\begin{array}{cc}
1 & x_0=i \\
0 & \mbox{else.}
\end{array}
\right. \end{align*} Then its rotation set ${\rm Rot}(\varphi)$ is the polyhedron whose extremal points are $e_1, e_2$ and $e_3$
, where $\{e_1,e_2,e_3\}$ is the standard basis of $\mathbb{R}^3$.
Take a rotation vector $h$ from the open side whose vertices are $e_1$ and $e_3$, i.e., $h\in \{t e_1+(1-t)e_3: t\in (0,1)\}$. If there exists a periodic measure $\mu\in {\rm rv}^{-1}(h)$, the corresponding periodic orbit $u^\infty$ should contain both of $1$ and $3$. Since there is no sequence including $13$ and $31$ in $\Omega$, the word $u$ must contain the symbol $2$. Hence we have ${\rm rv}_{\varphi_2}(\mu)>0$ and $h={\rm rv}_\varphi(\mu)\notin \{t e_1+(1-t)e_3: t\in (0,1)\}$, which is a contradiction.
\end{remark}
\section{Generic property for constraint ergodic optimization} \label{generic}
\subsection{Characterization by tangency} In this subsection, we turn to a general dynamical system and state analogous results in \cite{Mor2010} for the constrained case. Let $T: X\rightarrow X$ be a continuous map on a compact metric space.
Denote by $\mathcal{M}_T^e(X)$ the set of all ergodic measures on $X$. By the Riesz representation theorem a Borel probability measure on $X$ can be regarded as a bounded linear functional on $C(X,\mathbb{R})$. Hence we use the operator norm $\|\mu\|=\sup\{|\mu(f)|: f\in C(X,\mathbb{R})\ \mbox{with}\ \|f\|_\infty=1\}$ for an invariant measure $\mu\in \mathcal{M}_T(X)$.
First we characterize a relative maximizing measures by tangency to \eqref{beta_constraint}. Let $\varphi\in C(X,\mathbb{R}^d)$ and $h\in {\rm Rot}(\varphi)$.
\begin{lemma} \label{tangent} $\mu\in \mathcal{M}^\varphi_h (f)$ iff $\mu$ is tangent to $\beta^\varphi_h$ at $f$. \end{lemma}
\begin{proof} Let $\mu\in \mathcal{M}^\varphi_h(f)$. Then for every $g\in C(X,\mathbb{R})$ we have \[ \beta^\varphi_h(f+g)-\beta^\varphi_h(f)\geq \int f+g\ d\mu-\int f\ d\mu \geq \int g\ d\mu. \]
Let $\mu$ be tangent to $\beta^\varphi_h$ at $f$. For every $g\in C(X,\mathbb{R})$ we have \begin{align*}
\int g\ d\mu &\leq \beta^\varphi_h(f+g)-\beta^\varphi_h(f) \nonumber\\
&=\beta^\varphi_h(g)+\beta^\varphi_h(f+g)-\beta^\varphi_h(f)-\beta^\varphi_h(g) \nonumber\\
&\leq \beta^\varphi_h(g). \label{bounded} \end{align*} Then we can show that $\mu$ is an invariant probability measure in the same way as unconstrained case (See for example \cite{Shi2018, Bre08}). We now see that $\mu$ takes the rotation vector $h$. For $i=1, \ldots, d$ we have
\begin{align*}
\int \varphi_i\ d\mu\leq \beta^\varphi_h(\varphi_i)=h_i,
\quad
-\int \varphi_i\ d\mu\leq \beta^\varphi_h(-\varphi_i)=-h_i,
\end{align*} which yields ${\rm rv}_{\varphi}(\mu)=h$.
Finally we check $\int f\ d\mu=\beta^\varphi_h(f)$. Indeed, \begin{align*}
\int -f\ d\mu \leq \beta^\varphi_h(f-f)-\beta^\varphi_h(f)=-\beta_h^\varphi(f). \end{align*} Multiplying $-1$, we have \begin{align*}
\int f d\mu\geq \beta^\varphi_h(f). \end{align*} \end{proof}
Checking that a relative maximizing measure is characterized by tangency of \eqref{beta_constraint}, we can adopt techniques in \cite{Mor2010}. \begin{lemma} \label{approx} Let $f\in C(X,\mathbb{R})$ and $\varepsilon>0$. If $\nu\in {\rm rv}_{\varphi}^{-1}(h)\cap \mathcal{M}_{T}^e(X)$ such that $\beta^\varphi_h(f)-\int f\ d\nu<\varepsilon$,
then there exists $g\in C(X,\mathbb{R})$ such that $\|f-g\|_\infty <\varepsilon$ and $\nu\in \mathcal{M}^\varphi_h(g)$. \end{lemma}
\begin{proof}
Applying Bishop-Phelps's theorem \cite[Theorem V.1.1.]{Israel} to $f$, $\nu$ and $\varepsilon'=1$ we have $g\in C(X,\mathbb{R})$, $\eta\in {\rm rv}_{\varphi}^{-1}(h)$ such that $\eta$ is tangent to $\beta^\varphi_h$ at $g$, $\|\nu-\eta\|<\varepsilon'=1$ and \[
\|f-g\|_\infty<\frac{1}{\varepsilon'}\left(\beta^\varphi_h(f)-\int f d\nu\right)<\varepsilon. \] By Lemma \ref{tangent} we have $\eta\in \mathcal{M}^\varphi_h(g)$. If $\eta=\nu$, the proof is complete.
Let $\eta \neq \nu$. We can conclude $\nu$ and $\eta$ are not mutually singular by $\|\nu-\eta\|<1$ in the same way as in \cite{Mor2010}[Lemma 2.2].
Hence by the Lebesgue decomposition theorem, there exist $\hat{\nu}, \hat{\eta}\in \mathcal{M}$ and $\lambda\in (0,1)$ such that \[
\eta=(1-\lambda)\hat{\eta}+\lambda\hat{\nu} \] where $\hat{\eta}\perp \nu$ and $\hat{\nu}\ll\nu$. By a standard argument using the Radon-Nikodym theorem, it is easy to see $\hat{\nu}=\nu$ (See for example \cite{Walters}).
Then for each $i=1, \ldots, d$ \begin{align*}
h_i=\int \varphi_i\ d\eta&=(1-\lambda)\int \varphi_i\ d\hat{\eta}+\lambda \int \varphi_i\ d\nu\\
&=(1-\lambda)\int \varphi_i\ d\hat{\eta}+\lambda h_i. \end{align*} Hence we have \begin{align*}
(1-\lambda)\left(\int \varphi_i\ d\hat{\eta}-h_i\right)=0
\quad (i=1, \ldots, m), \end{align*} which yields $\hat{\eta}\in {\rm rv}_{\varphi}^{-1}(h)$. Since $\hat{\eta}, \nu\in {\rm rv}_{\varphi}^{-1}(h)$, we have \[
\int g\ d\hat{\eta}\leq \beta^\varphi_h(g) \quad\mbox{and}\quad \int g\ d\nu\leq \beta^\varphi_h(g). \] They should be equality since $\int g\ d\eta =\beta^\varphi_h(g)$. Therefore, we obtain $\nu\in \mathcal{M}^\varphi_h(g)$. \end{proof}
Set $\mathcal{E}={\rm rv}_{\varphi}^{-1}(h)\cap \mathcal{M}^e_{T}(X)$. In the reminder of this subsection we assume \begin{align}
\overline{\mathcal{E}}={\rm rv}_{\varphi}^{-1}(h)
\label{e_dense}
\end{align}
Since proofs of Lemma \ref{open1} and Lemma \ref{open2} are slight modifications of Lemma 3.1 and Lemma 3.2 in \cite{Mor2010} respectively, we omit their proofs. \begin{lemma} \label{open1} Let $\mathcal{U}\subset {\rm rv}_{\varphi}^{-1}(h)$ be an open set. Then \[ U:=\{f\in C(X,\mathbb{R}): \mathcal{M}^\varphi_h(f)\subset \mathcal{U}\} \] is open in $C(X,\mathbb{R})$. \end{lemma}
\begin{lemma} \label{open2} Let $U\subset C(X,\mathbb{R})$ be an open set. Then \[ \mathcal{U}:=\mathcal{E}\cap \bigcup_{f\in U} \mathcal{M}^\varphi_h(f) \] is open in $\mathcal{E}$. \end{lemma}
While proofs of the next two lemmas are also similar to \cite{Mor2010}, attention should be paid to the constraint. Hence we give proofs. \begin{lemma} \label{dense1} Let $\mathcal{U}$ be a dense subset of $\mathcal{E}$. Then \[ U:=\{f\in C(X,\mathbb{R}): \mathcal{M}^\varphi_h(f)=\{\mu\}\ \mbox{for some}\ \mu\in \mathcal{U}\} \] is dense in $C(X,\mathbb{R})$. \end{lemma}
\begin{proof} Let $V\subset C(X,\mathbb{R})$ be an open set. Set \[ \mathcal{V}=\mathcal{E}\cap \bigcup_{f\in V}\mathcal{M}^\varphi_h(f). \] By Lemma \ref{open2}, $\mathcal{V}$ is open in $\mathcal{E}$. Since $\mathcal{U}$ is dense in $\mathcal{E}$, $\mathcal{U}\cap \mathcal{V}\neq \emptyset$. Take $\mu\in \mathcal{U}\cap \mathcal{V}$ and let $f\in V$ such that $\mu\in \mathcal{M}^\varphi_h(f)$. Note that $\mu\in{\rm rv}^{-1}(h)$. Since $\mu\in\mathcal{E}\subset \mathcal{M}_{T}^e(X)$, by Jenkinson's theorem \cite{Jen06}[Theorem 1], there exists $g\in C(X,\mathbb{R})$ such that $\mathcal{M}_{\rm max}(g)=\{\mu\}$. This implies $\mathcal{M}^\varphi_h(g)=\{\mu\}$ since $\mu\in{\rm rv}_{\varphi}^{-1}(h)$. Hence for $\delta>0$ we have $\mathcal{M}^\varphi_h(f+\delta g)=\{\mu\}$ and $f+\delta g\in U$. For sufficiently small $\delta$ we have $f+\delta g\in V$, which complete the proof. \end{proof}
\begin{lemma} \label{dense2} Let $U\subset C(X,\mathbb{R})$ be a dense subset. Then \[
\mathcal{U}:=\mathcal{E}\cap \bigcup_{f\in U} \mathcal{M}^\varphi_h(f) \] is dense in $\mathcal{E}$. \end{lemma} \begin{proof}
Take a nonempty open subset $\mathcal{V}\subset \overline{\mathcal{E}}$.
We show $\mathcal{V}\cap\mathcal{U}\neq\emptyset$.
By Lemma \ref{open1},
\[
V:=\{f\in C(X,\mathbb{R}): \overline{\mathcal{E}}\cap \mathcal{M}^\varphi_h(f)\subset \mathcal{V}\}
\]
is open in $C(X,\mathbb{R})$.
Since $\mathcal{V}$ is nonempty, there exists $\mu\in \mathcal{V}\cap\mathcal{E}\subset {\rm rv}^{-1}(h)\cap\mathcal{M}_{T}^e(X)$.
Since $\mu$ is ergodic, as stated in Lemma \ref{dense1}, there exists $g\in C(X,\mathbb{R})$ such that $\mathcal{M}_{\rm max}(g)=\{\mu\}$ and we have $\mathcal{M}^\varphi_h(g)=\{\mu\}\subset \mathcal{V}$ by Jenkinson's Theorem \cite{Jen06}[Theorem 1].
Hence $V$ is nonempty.
Since $U$ is dense in $C(X,\mathbb{R})$, $U\cap V\neq \emptyset$ and $\mathcal{U}\cap \mathcal{V}\neq \emptyset$. \end{proof}
\subsection{Generic zero entropy for a symbolic dynamics}
In this subsection, we apply the lemmas in the previous subsection to the symbolic case and give the proof of Theorem \ref{generic_zero_entropy}.
\begin{proof}[Proof of Theorem \ref{generic_zero_entropy}]
Suppose the hypotheses of Theorem \ref{generic_zero_entropy} hold. As in Corollary \ref{residual_set_of_zero_entropy}, for each $n\geq1$ \begin{align*}
\mathcal{Z}_n:=\left\{\mu\in {\rm rv}_{\varphi}^{-1} h: 0\leq h_\mu<\frac{1}{n}\right\} \supset \mathcal{N}_{\sigma}\cap {\rm rv}_{\varphi}^{-1} h\end{align*}
is nonempty, open and dense subset in ${\rm rv}_{\varphi}^{-1}(h)$. Moreover, by Theorem \ref{density_of_periodic_measures}, we have \begin{align*}
\overline{{\rm rv}_{\varphi}^{-1}(h)\cap \mathcal{M}_\sigma^p(\Omega)}={\rm rv}_{\varphi}^{-1}(h), \end{align*} which implies \eqref{e_dense}.
Fix $n\geq1$. Since \eqref{e_dense} holds, we can apply Lemmas \ref{open1} and \ref{dense1} to our symbolic case. Therefore, we see that \begin{align*}
U_n:=\{f\in C(\Omega,\mathbb{R}): \mathcal{M}^\varphi_h(f)\subset \mathcal{Z}_n\} \end{align*} is open in $C(\Omega,\mathbb{R})$ and \begin{align*}
\widehat{U}_n:=\{f\in C(\Omega,\mathbb{R}): \mathcal{M}_h^\varphi(f)=\{\mu\}\ \mbox{for some}\ \mu\in \mathcal{Z}_n\}
\subset U_n \end{align*} is dense in $C(\Omega,\mathbb{R})$. Hence $U_n$ is an open dense subset of $C(\Omega,\mathbb{R})$.
Then the set \begin{align*}
R:=\bigcap_{n\geq1} U_n
=\left\{f\in C(\Omega,\mathbb{R}): \mathcal{M}_h^\varphi(f)\subset \bigcap_{n\geq1}\mathcal{Z}_n\right\} \end{align*} is a residual subset of $C(\Omega,\mathbb{R})$, and the proof is complete. \end{proof}
\setcounter{equation}{0} \renewcommand{\Alph{section}.\arabic{equation}}{\Alph{section}.\arabic{equation}}
\appendix
\section{On positive entropy}\label{AppendixA}
In this appendix, we see that for generic continuous function every relative maximizing measure has positive entropy under some constraints, which is a trivial consequence of ergodic optimization.
\begin{proposition}
\label{positive_entropy}
Let $T:X\rightarrow X$ be a continuous map on a compact metric space $X$ with an ergodic invariant probability measure $\mu$ having positive entropy.
Then there exist $\varphi\in C(X,\mathbb{R})$ and $h\in {\rm Rot}(\varphi)$ such that
for generic $f\in C(X,\mathbb{R})$ every relative maximizing measure of $f$ with the constraint $h\in {\rm Rot}(\varphi)$ has positive entropy.
\end{proposition}
\begin{proof}
By Jenkinson's theorem \cite{Jen06}[Theorem 1], there exists $\varphi \in C(X,\mathbb{R})$ such that $\mathcal{M}_{\rm max}(\varphi)=\{\mu\}$.
Let $h:={\rm rv}_{\varphi}(\mu)$.
Then $\mathcal{M}^\varphi_h(f)\subset {\rm rv}_{\varphi}^{-1}(h)=\{\mu\}$ for all $f\in C(X,\mathbb{R})$.
Since $\mathcal{M}^\varphi_h(f)$ is not empty, we have $\mathcal{M}^\varphi_h(f)=\{\mu\}$, which implies the unique relative maximizing measure of $f$ with constraint $h\in {\rm Rot}(\varphi)$ has positive entropy.
\end{proof}
\section{Prevalent uniqueness}\label{AppendixB} In this paper we focus on generic property of continuous functions in constrained setting. On the other hand, measure-theoretic ``typicality" which is known as {\it prevalence} is also important. A subset $\mathcal{P}$ of $C(X,\mathbb{R})$ is {\it prevalent} if there exists a compactly supported Borel probability measure $m$ on $C(X,\mathbb{R})$ for every $f\in C(X,\mathbb{R})$ the set $f+\mathcal{P}$ has full measure.
Prevalent uniqueness of maximizing measures for continuous functions in the literature on (unconstrained) ergodic optimization is recently established by Morris \cite[Theorem 1]{Mor21}. Moreover the result is generalized to more abstract statement, which yields prevalent uniqueness in constrained setting.
\begin{corollary} Let $T:X\rightarrow X$ be a continuous map on a compact metric space $X$. For a continuous constraint $\varphi: X\rightarrow\mathbb{R}^d$ and a rotation vector $h\in {\rm Rot}(\varphi)$, the set \begin{align*}
\{f\in C(X,\mathbb{R}): \mathcal{M}^\varphi_h(f)\ \mbox{is a singleton}\} \end{align*} is prevalent. \end{corollary} \begin{proof}
Since ${\rm rv}_{\varphi}^{-1}(h)\subset \mathcal{M}_T(X)$ is a nonempty compact set in $C(X,\mathbb{R})^*$,
the statement follows from Theorem 2 in \cite{Mor21}. \end{proof}
\vspace*{33pt}
\noindent \textbf{Acknowledgement.}~ The first author was partially supported by JSPS KAKENHI Grant Number 22H01138 and the second author was partially supported by JSPS KAKENHI Grant Number 21K13816.
\noindent \textbf{Data Availability.}~ Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
\end{document} | arXiv |
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A photovoltaic (PV) module is an equipment that converts solar energy to electrical energy. A mathematical model should be presented to show the behavior of this device. The well-known single-diode and double-diode models are utilized to demonstrate the electrical behavior of the PV module. "Matlab/Simulink" is used to model and simulate the PV models because it is considered a major software for modeling, analyzing, and solving dynamic system real problems. In this work, a new modeling method based on the "Multiplexer and Functions blocks" in the "Matlab/Simulink Library" is presented. The mathematical analysis of single and double diodes is conducted on the basis of their equivalent circuits with simple modification. The corresponding equations are built in Matlab by using the proposed method. The unknown internal parameters of the PV panel circuit are extracted by using the PV array tool in Simulink, which is a simple method to obtain the PV parameters at certain weather conditions. Double-diode model results are compared with the single-diode model under various irradiances and temperatures to verify the performance and accuracy of the proposed method. The proposed method shows good agreement in terms of the I–V and P–V characteristics. A monocrystalline NST-120 W PV module is used to validate the proposed method. This module is connected to a variable load and tested for one summer day. The experimental voltage, current, and power are obtained under various irradiances and temperatures, and the I–V and P–V characteristics are obtained.
Renewable energy is the best source of electricity because it is free, clean, and highly abundant. Renewable energy gained by photovoltaic (PV) modules is the most common source1. A PV cell is a device that converts solar energy to electrical energy, and has a behavior similar to a P–N junction. The output voltage of these cells should be raised by connecting them in series to form a PV module. These cells are also connected in parallel to obtain a large output current. This connection is integrated to form a large PV system with a higher power, which is called an array2. The PV module's output power primarily depends on irradiation and cell temperature. The voltage drops, and the current rises slightly whenever the temperature increases. Thus, the PV system efficiency is decreased. Different types of PV cell technologies are currently made on the market, depending on the commercial maturity and manufacturing materials. These types can be summarized into three major technologies, namely, polycrystalline, monocrystalline, and thin-film technologies3,4. Polycrystalline technology is relatively inefficient due to the arrangement of crystals, which is random, and the cell's color is slightly bluer, reflecting more of the sunlight. Monocrystalline technology is extremely efficient. This technology absorbs extremely high sunlight radiation because it has a uniform black color. The efficiency of this technology is more than that of polycrystalline. The manufacturing cost of polycrystalline technology is lower than pure silicon wafers. Thin-film technology is more efficient than other technologies. This technology is made of a thin layer of amorphous film that elicits more energy from the available sunlight. These technologies of the PV cell are combined with the same point in the physical behavior of a diode P–N junction5,6,7,8.
The main drawback of the PV module is the lower efficiency. Thus, a maximum power point tracking (MPPT) was proposed9,10,11,12 to elicit the maximum power and enhance the efficiency of the PV module. The modeling representation of the PV panel is extremely important to show the PV characteristics under different weather conditions. Many researchers have developed single diode and double-diode PV models, and a large body of literature is found on this topic13,14. A simple PV model was explored by using a single-diode model with four parameters15,16. The electrical PV module circuit was built by using a photocurrent source, a diode, a series resistor, and an ideality diode constant. The accuracy of this model is lower compared with that proposed in17 that added a shunt resistor to achieve a new model with five parameters.
Several studies have developed PV panel models to solve this issue. In18,19,20,21,22, a double-diode PV model is presented to increase the accuracy of the panel model performance. In these studies, a photocurrent source, two diodes, a series resistor, a shunt resistor, and an ideality diode constant are used to form a PV model circuit. A more complex simulation model is utilized to represent the PV panel equivalent circuit and extract the PV characteristics of I–V and P–V curves under different irradiances and temperatures. The authors in23 presented a new method to determine the PV parameters of a double-diode circuit to enhance its efficiency and accuracy. The drawback of this method is the complex computational process to obtain circuit parameters, such as ideality constant, series, and shunt resistances. Therefore, the researchers in24,25,26,27 reduced the complexity of the PV circuit by using a five-parameter model. In24,25,26,27, this model is insufficient to depict a real PV circuit although the accuracy of the I–V and P–V characteristics are not mentioned. A more accurate PV circuit model based on two-diode model is proposed in28. The seven parameters of this model are computed by using a hybrid method consists of numerical and analytical methods. Single- and double-diode models are proposed in29. They are simulated to determine their difference under various irradiances and temperatures. In this work, an optimization method is used to compute the circuit parameters. Although these studies have achieved good results in terms of PV performance, they use a complicated and difficult modeling method.
Reference30 conducted implicit modeling of two-diode model for PV array configuration. Each subpanel is considered by the implicit expression derived to represent a series–parallel array on the basis of double-diode cell circuit. The resultant system equations are solved by using trust-region dogleg method to extract the PV circuit parameters used in every array. This method achieves satisfactory results are achieved in terms of I–V and P–V characteristics for different weather conditions. The researchers in31,32 proposed a new method to model and extract the parameters of PV cell or module by using flower pollination method. A double-diode PV circuit is used to obtain a high accuracy PV model. The results obtained are compared with other results that used optimization techniques, such as artificial bee swarm optimization, pattern search, and harmony search, to prove the effectiveness of the proposed method. However, most of these studies require a high computational cost, making it extremely difficult for users. The authors in33 presented a simple representation for PV circuit model. The proposed method is based on the stepwise simplification of the total current equation of the PV equivalent circuit for one and two diodes. The proposed method is investigated by using LTspice simulator tool. The authors in34,35,36 developed a double-diode model by adding another parallel diode in PV equivalent circuit. This model increases the power losses due to the leakage current in the third diode and reduces the total output power. Thus, the accuracy of the I–V and P–V characteristics of the PV panel is affected, especially when the PV panel works under lower irradiance levels.
In this study, a simple and new method for modeling the double-diode PV model in Matlab is presented. This method is built on the basis of the simple mathematical equations of reverse saturation diode currents. The double-diode PV parameters are extracted with a simple PV array tool presented in a new version of Simulink (2016), which is released by Mathworks. The simulation results obtained by this method are compared with that obtained by the single-diode model at standard test conditions (STCs). The I–V and P–V curves of the two models are achieved and validated with a 120 W-PV module to test the performance under different weather conditions.
The rest of this paper is organized as follows: Sect. 2 presents the PV cell models. Section 3 introduces the modeling of PV panel. Section 4 highlights the hardware implementation adopted in this research. Section 5 discusses the results. Section 6 presents the conclusion.
PV cell models
Single-diode model
As shown in Fig. 1, the PV cell model is a single-diode model because it is built on the assumption that the recombination failure in the depletion area is negligible. The loss of the P–N junction's depletion area is important, which is invisible in the single-diode configuration. The basic equation of semiconductor diode theory represents the characteristics of an ideal PV cell as follows9,13:
Single-diode model of the PV cell.
$${\mathrm{I}}_{\mathrm{pv}}={\mathrm{I}}_{\mathrm{ph}}-{\mathrm{I}}_{0}\left[\mathrm{exp}\left(\frac{\mathrm{q }{\mathrm{V}}_{\mathrm{pv}}}{\mathrm{\alpha K T}}\right)-1\right]$$
Equation (1) does not symbolize the real behavior of the PV cell. For this reason, a small milliohm of a series resistor with a high value of a parallel resistor is inserted into the equivalent circuit of the PV cell circuit. The PV cell current can be expressed as17,18
$${\mathrm{I}}_{\mathrm{pv}}={\mathrm{I}}_{\mathrm{ph}}-{\mathrm{I}}_{0}\left[\mathrm{exp}\left(\frac{\mathrm{q }{\mathrm{V}}_{\mathrm{pv}}}{\mathrm{\alpha }{\mathrm{V}}_{\mathrm{T}}}\right)-1\right]-\frac{{\mathrm{V}}_{\mathrm{pv}}+{\mathrm{R}}_{\mathrm{s }}{\mathrm{I}}_{\mathrm{pv}}}{{\mathrm{R}}_{\mathrm{P}}}$$
The source of photocurrent is linearly proportional to irradiance and is influenced by temperature, as shown in Eq. (3). 20.
$${\mathrm{I}}_{\mathrm{ph}}=\left({\mathrm{I}}_{\mathrm{phn}}+{\mathrm{K}}_{\mathrm{i}}\Delta \mathrm{T}\right)\frac{\mathrm{G}}{{\mathrm{G}}_{\mathrm{n}}}$$
where \(\Delta \mathrm{T}=\mathrm{T}-{\mathrm{T}}_{\mathrm{n}}\) (Tn = 25 °C), \(\mathrm{G}\) is the incident of irradiation on the module, and \({\mathrm{G}}_{\mathrm{n}}\)(1000 W/m2) at \(\mathrm{STC}\). The diode saturation current can be written as16
$${\mathrm{I}}_{0}={\mathrm{I}}_{0\mathrm{n}}{\left(\frac{{\mathrm{T}}_{\mathrm{n}}}{\mathrm{T}}\right)}^{3}\mathrm{exp}\left[\frac{{\mathrm{qE}}_{\mathrm{g}}}{\mathrm{\alpha K}}\left(\frac{1}{{\mathrm{T}}_{\mathrm{n}}}-\frac{1}{\mathrm{T}}\right)\right]$$
A modified equation that describes the current in the saturation case is shown below6.
$${\mathrm{I}}_{0}=\frac{\left({\mathrm{I}}_{\mathrm{scn}}+{\mathrm{K}}_{\mathrm{i}}\Delta \mathrm{T}\right)}{\mathrm{exp}\left[\frac{\left({\mathrm{V}}_{\mathrm{ocn}}+{\mathrm{K}}_{\mathrm{v}} \Delta \mathrm{T}\right)}{\mathrm{\alpha }{\mathrm{V}}_{\mathrm{T}}}\right]-1}$$
This modification aims to make the open-circuit voltage of the model match that of the experiment. The saturation current is influenced by the variation of the temperate. This modification facilitates the model and eliminates the model's error on open-circuit voltages for the regions of the I–V characteristic. The terms of the previous equations are presented as:
\({\mathrm{I}}_{\mathrm{pv}}\) is the PV output current.
\({\mathrm{V}}_{\mathrm{pv}}\) is the output voltage of the PV module.
\({\mathrm{I}}_{\mathrm{D}}\) is the diode current.
\({\mathrm{I}}_{\mathrm{ph}}\) is the photocurrent source.
\({\mathrm{I}}_{\mathrm{o}}\) is the saturation diode current.
\({\mathrm{I}}_{\mathrm{on}}\) represents the saturation current at STC condition.
\({\mathrm{I}}_{\mathrm{phn}}\) is the photocurrent at STC condition.
\(\mathrm{T}\) is the ambient temperature
\({\mathrm{V}}_{\mathrm{T}}\left(={\mathrm{N}}_{\mathrm{S}}\mathrm{K T}/\mathrm{q}\right)\) is the thermal voltage.
\({\mathrm{N}}_{\mathrm{S}}\) is the number of cells per module.
\({\mathrm{R}}_{\mathrm{P}}\) and \({\mathrm{R}}_{\mathrm{S}}\) are the parallel and series resistances, respectively.
\({\mathrm{K}}_{\mathrm{i}}\) is the thermal coefficient at \({\mathrm{I}}_{\mathrm{sc}}\).
\(\mathrm{\alpha }\) is the diode ideality factor.
\({\mathrm{K}}_{\mathrm{V}}\) is the thermal coefficient at \({\mathrm{V}}_{\mathrm{oc}}\).
\({\mathrm{E}}_{\mathrm{g}}\) is the band gap energy.
\(\mathrm{q}\) is the electron charge (1.602 × 10−19 °C).
\(\mathrm{K}\) is the Boltzmann constant (\(1.3806\times {10}^{-23}\mathrm{J}/\mathrm{K}).\)
Double-diode model
The lack of recombination that is ignored in the single–diode model causes the inaccuracy of the PV model parameters. Therefore, the double-diode model is chosen to represent the physical form of the PV cell, as shown in Fig. 2. A precise model is achieved by considering recombination loss. The diffusion current is focused in the p–n junction material by using one of the double diodes, and the other is added to account for recombination loss20,21. Hence, the PV module output current can be defined as:
Double-diode PV model circuit.
$${\mathrm{I}}_{\mathrm{pv}}={\mathrm{I}}_{\mathrm{ph}}-{\mathrm{I}}_{\mathrm{D}1}-{\mathrm{I}}_{\mathrm{D}2}-\left[\frac{{\mathrm{V}}_{\mathrm{pv}}+{\mathrm{I}}_{\mathrm{pv}}{\mathrm{R}}_{\mathrm{s}}}{{\mathrm{R}}_{\mathrm{P}}}\right]$$
$${\mathrm{I}}_{\mathrm{D}1}={\mathrm{I}}_{01}\left[\mathrm{exp}\left(\frac{{\mathrm{V}}_{\mathrm{pv}}+{\mathrm{I}}_{\mathrm{pv}}{\mathrm{R}}_{\mathrm{s}}}{{\mathrm{\alpha }}_{1}{\mathrm{V}}_{\mathrm{T}1}}\right)-1\right]$$
$${\mathrm{I}}_{\mathrm{D}2}={\mathrm{I}}_{02}\left[\mathrm{exp}\left(\frac{{\mathrm{V}}_{\mathrm{pv}}+{{\mathrm{I}}_{\mathrm{pv}}\mathrm{R}}_{\mathrm{s}}}{{\mathrm{\alpha }}_{2}{\mathrm{V}}_{\mathrm{T}2}}\right)-1\right]$$
As mentioned previously, a more accurate model can be realized by using a two-diode model. Seven parameters, namely, \({I}_{\mathrm{ph}}\), \({I}_{01}\), \({I}_{02}\),\({\alpha }_{1}\), \({\alpha }_{2}\),\({R}_{s}\), and \({R}_{\mathrm{P}}\) must be calculated. Some studies have used iteration methods to calculate the values of saturation currents for double-diode models (\({I}_{01}\) and \({I}_{02}\))7,8. \({I}_{01}\) is approximately 3–7 orders larger than that of \({I}_{02}\). \({\alpha }_{1}\) and \({\alpha }_{2}\) are taken as 1 and 2, respectively, to simplify the calculation. Thus, the saturation currents can be expressed as14,19
$${\mathrm{I}}_{01}=\frac{\left({{\mathrm{I}}_{\mathrm{scn}}}_{ }+{\mathrm{K}}_{\mathrm{i}} \Delta \mathrm{T}\right)}{\mathrm{exp}\left[\frac{\left({\mathrm{V}}_{\mathrm{ocn}}+{\mathrm{K}}_{\mathrm{v}} \Delta \mathrm{T}\right)}{{\mathrm{\alpha }}_{1}{\mathrm{V}}_{\mathrm{T}1}}\right]-1}$$
When the terminals of the PV panel are tested under open-circuit operation, the ambient temperature should be considered, which is affected on the I–V and P–V characteristics during different temperature conditions. Thus, the output voltage can be expressed as3
$${\mathrm{V}}_{\mathrm{oc}}={\mathrm{V}}_{\mathrm{ocn}}+{\mathrm{K}}_{\mathrm{V}}{ \Delta }_{\mathrm{T}}$$
Extracted PV parameters
The PV panel datasheet does not contain some important parameters, such as \({R}_{\mathrm{s}}\), \({R}_{\mathrm{P}}\), \({I}_{\mathrm{o}}\), and \(\alpha \), that are used in the modeling process. Therefore, these parameters are extremely important in modeling PV panels. They are used to represent the real circuit of the PV Panel. However, several methods are presented and reviewed to determine these parameters. This study utilized a simple and sufficient method of PV array Matlab/tool to compute these parameters. We use this tool provided in Matlab 2016 to set the datasheet parameters by simply clicking on the tool window, and the extracted parameters are obtained, as shown in Fig. 3. Block (1) refers to the datasheet parameters, and block (2) presents the extracted PV parameters. Table 1 demonstrates the datasheet parameters of the NST-120 PV panel that is utilized in this study. The extracted parameters are shown in Table 2.
Extracted parameters of the NST-120 panel.
Table 1 Parameters of the NST-120 PV panel at STC from the datasheet.
Table 2 Extracted parameters of the NST-120 PV panel by using PV array tool.
Modeling of PV panel
The datasheet and extracted parameters of the NST panel are used to simulate the single- and double-diode models for representing the P–V and I–V panel characteristics, as shown in Fig. 4. A simple simulation method of Matlab is used to obtain the PV graphs. This method is based on two utilized tools to represent the equations given in Sect. (2.1) of the single-diode model and the equations given in Sect. (2.2). These models are shown in Figs. 5 and 6. The two main tools utilized in this work are as follows:
Multiplexer block (Mux): this block is utilized from "library Simulink/Signal Routing".
Function block (Fcn): this block is utilized from "library Simulink/User-Defined Functions".
Simulink/Matlab block diagram of the NST-120 PV panel.
Single-diode model in Matlab.
Double-diode model in Matlab.
The NST panel is integrated with practical measurement devices to validate its performance under different values of irradiance and temperature, as shown in Fig. 7. This system consists of a lux meter to measure solar irradiance, a thermometer to sense temperature, ammeter, and voltmeter. A lux meter is used to measure irradiance practically and show the influence of solar irradiance values on the PV panel performance. This device shows the solar irradiance in Lux unit (1–50,000 Lux), where 1 Lux equals 0.79 W/m2. The ambient temperature in the experiment is measured by using a thermometer. This device offers an additional feature of humidity measurement, which is a low-cost simplicity for the user. The PV panel is connected to the variable resistive load and the corresponding voltage, and the current is extracted during a sunny day, as shown in Sect. 5.2.
Proposed PV system components.
Simulation results
The simple Matlab method combined from Fcn and Mux is used to verify the proposed method. The single-diode model is represented, and its simulation results are achieved to show the real PV module characteristics under different solar insulations and ambient temperatures. Figures 8 and 9 depict the I–V and P–V graphs for the single-diode model at STC of \(T\) = 25 °C and \(G\) = 1000 W/m2. The double-diode model results are shown in Figs. 10 and 11, showing the I–V and P–V graphs at STC. The double-diode model presents higher accuracy in point short-circuit current region, MPP region, and open-circuit voltage region. The double-diode model is simulated under various irradiances and ambient temperatures, as shown in Figs. 12 and 13. When the irradiation level is high, the open-circuit voltage is increased logarithmically, and the current is increased linearly in accordance with Eq. (3) in Sect. 2.1.If the ambient temperature is high, the open-circuit voltage becomes low due to the sign of \({K}_{\mathrm{v}}\), which is negative, as presented in Eq. (11) in Sect. 2.2. The PV current increases slightly in accordance with \({K}_{\mathrm{i}}\) constant, which is positive and extremely small. In this simulation, the value of the optimal resistance load is used on the basis of the MPP point at STC, \({R}_{\mathrm{opt}}=\frac{{V}_{\mathrm{mp}}}{{I}_{\mathrm{mp}}}\).
I–V graph of single-diode model at STC.
P–V graph of single-diode model at STC.
I–V graph of double-diode model at STC.
P–V graph of double-diode PV model at STC.
I–V and P–V graphs of double-diode model at different values of temperature and fixed irradiation, \(G=1000\) W/m2 (a) I–V graph and (b) P–V graph.
I–V and P–V graphs of double-diode model under different values of irradiation and constant temperature, T = 25 °C (a) I–V graph and (b) P–V graph.
The experiment is conducted to validate the effectiveness of proposed method. The PV panel measurement data (voltage, current, temperature and irradiation) in one summer day for NST-120 W PV panel are obtained for different weather conditions. The experimental results are extracted through a variable load to obtain the I–V and P–V graphs. The experimental system components of the PV system are presented in Fig. 14. The photography of measurements in this experiment are shown in Figs. 15 and 16. The collected data of the real implementation are shown in Table 3. The experimental results of the voltage, current, and power of the NST-PV module for one day are reported in Figs. 17, 18, 19, respectively. The output voltage of the PV module is approximately constant due to the lower change in the ambient temperature during the experiment, except for the temperature of 20 °C in the morning. The total PV module current is mainly proportional to the irradiance and reaches to peak current at irradiance of 600 W/m2. Figures 20 and 21 present the I–V and P–V characteristics for one summer day, respectively, to show the NST-120 W PV module characteristics in this experiment.
Experimental components of the proposed PV system.
Experimental current and voltage of the proposed PV system.
Measurements of irradiance and temperature for the proposed PV system.
Table 3 Experimental data for various weather conditions every 1 h for one summer day in Baghdad City, Iraq.
Experimental voltage of the NST-PV panel.
Experimental current of the NST-PV module.
Experimental power of the NST PV module.
Experimental I–V graph of the NST PV module.
Experimental P–V graph of the NST PV module.
In this study, a simple and new method for modeling a double-diode PV model is presented. The theoretical analysis for single- and double-diode circuits is conducted. The unknown internal PV parameters are computed by using PV array tool in Simulink, and the models are modeled on the basis of their mathematical equations. The new method used in this work for modeling PV module is based on two main functions of "Multiplexer and Functions blocks" that are presented in the Simulink library. The proposed model is validated experimentally by using a monocrystalline NST-120 W PV module. The experimental results for one summer day are obtained, and the corresponding I–V and P–V characteristics are achieved accurately. The simulation and experimental results show that the double-diode model is more efficient than the single-diode model in terms of accuracy.
A Correction to this paper has been published: https://doi.org/10.1038/s41598-021-01357-5
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This work was supported in part by the Robotics and Internet of Things Laboratory, Prince Sultan University, Riyadh, Saudi Arabia.
Authority of the Popular Crowd, Baghdad, Iraq
Salam J. Yaqoob
College of Engineering, Misan University, Amarah, Iraq
Ameer L. Saleh
Ecole National des sciences appliquées, Université Sidi Med Ben Abdellah, Fes, Morocco
Saad Motahhir
Department of Nuclear and Renewable Energy, Ural Federal University, 19 Mira Street, Ekaterinburg, Russia
Ephraim B. Agyekum
Graduate School, Duy Tan University, Da Nang, Vietnam
Anand Nayyar
College of Computer & Info Sc., Prince Sultan University, Riyadh, Saudi Arabia
Basit Qureshi
S.J.Y.: Conceived and designed the analysis, Collected the data, Wrote the paper. A.L.S.: Performed the analysis. S.M.: Improve the writing of the paper, Performed the analysis. E.B.A.: Improve the writing of the paper, Prepared the figures. A.N.: Improve the writing of the paper, Prepared the figures. B.Q.: Improve the writing of the paper, Funding acquisition.
Correspondence to Salam J. Yaqoob.
The original online version of this Article was revised: The original version of this Article contained an error in the Acknowledgements section. "This research was part of solar PV system project that received funding from the Middle Technical University for Research and Innovation Program 2020, Iraq." now reads: "This work was supported in part by the Robotics and Internet of Things Laboratory, Prince Sultan University, Riyadh, Saudi Arabia."
Yaqoob, S.J., Saleh, A.L., Motahhir, S. et al. Comparative study with practical validation of photovoltaic monocrystalline module for single and double diode models. Sci Rep 11, 19153 (2021). https://doi.org/10.1038/s41598-021-98593-6
Improved Arithmetic Optimization Algorithm for Parameters Extraction of Photovoltaic Solar Cell Single-Diode Model
Abdelkader Abbassi
Rached Ben Mehrez
Maryam Altalhi
Arabian Journal for Science and Engineering (2022)
Parameters Estimation of PV Models Using Artificial Neural Network
Hussein Abdellatif
Md Ismail Hossain
Mohammad A. Abido | CommonCrawl |
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Based on the idea and the provided source code of Andrej Karpathy (arxiv-sanity)
The Contribution of Stellar Winds to Cosmic Ray Production (1804.07486)
Jeongbhin Seo, Hyesung Kang, Dongsu Ryu
April 20, 2018 astro-ph.SR, astro-ph.HE
Massive stars blow powerful stellar winds throughout their evolutionary stages from the main sequence to Wolf-Rayet phases. The amount of mechanical energy deposited in the interstellar medium by the wind from a massive star can be comparable to the explosion energy of a core-collapse supernova that detonates at the end of its life. In this study, we estimate the kinetic energy deposition by massive stars in our Galaxy by considering the integrated Galactic initial mass function and modeling the stellar wind luminosity. The mass loss rate and terminal velocity of stellar winds during the main sequence, red supergiant, and Wolf-Rayet stages are estimated by adopting theoretical calculations and observational data published in the literature. We find that the total stellar wind luminosity due to all massive stars in the Galaxy is about $L_w\approx 1.1\times 10^{41}$ erg/s, which is about 1/4 of the power of supernova explosions, $L_{SN} \approx 4.8\times 10^{41}$ erg/s. If we assume that $\sim 1-10$ % of the wind luminosity could be converted to Galactic cosmic rays (GCRs) through collisonless shocks such as termination shocks in stellar bubbles and superbubbles, colliding-wind shocks in binaries, and bow-shocks of massive runaway stars, stellar winds might be expected to make a significant contribution to GCR production, though lower than that of supernova remnants.
Acceleration of Cosmic Ray Electrons at Weak Shocks in Galaxy Clusters (1707.07085)
Hyesung Kang, Dongsu Ryu, T. W. Jones
July 25, 2017 astro-ph.HE
According to structure formation simulations, weak shocks with typical Mach number, $M_{\rm s}\lesssim 3$, are expected to form in merging galaxy clusters. The presence of such shocks has been indicated by X-ray and radio observations of many merging clusters. In particular, diffuse radio sources known as radio relics could be explained by synchrotron-emitting electrons accelerated via diffusive shock acceleration (Fermi I) at quasi-perpendicular shocks. Here we also consider possible roles of stochastic acceleration (Fermi II) by compressive MHD turbulence downstream of the shock. Then we explore a puzzling discrepancy that for some radio relics, the shock Mach number inferred from the radio spectral index is substantially larger than that estimated from X-ray observations. This problem could be understood, if shock surfaces associated with radio relics consist of multiple shocks with different strengths.In that case, X-ray observations tend to pick up the part of shocks with lower Mach numbers and higher kinetic energy flux, while radio emissions come preferentially from the part of shocks with higher Mach numbers and higher cosmic ray (CR) production. We also show that the Fermi I reacceleration model with preexisting fossil electrons supplemented by Fermi II acceleration due to postshock turbulence could reproduce observed profiles of radio flux densities and integrated radio spectra of two giant radio relics. This study demonstrates the CR electrons can be accelerated at collisionless shocks in galaxy clusters just like supernova remnant shock in the interstellar medium and interplanetary shocks in the solar wind.
Particle Acceleration at Structure Formation Shocks (1706.03548)
Hyesung Kang
June 12, 2017 astro-ph.HE
Cosmological hydrodynamic simulations have demonstrated that shock waves could be produced in the intergalactic medium by supersonic flow motions during the course of hierarchical clustering of the large-scale-structure in the Universe. Similar to interplanetary shocks and supernova remnants (SNRs), these structure formation shocks can accelerate cosmic ray (CR) protons and electrons via diffusive shock acceleration. External accretion shocks, which form in the outermost surfaces of nonlinear structures, are as strong as SNR shocks and could be potential accelerations sites for high energy CR protons up to $10^{18}$ eV. But it could be difficult to detect their signatures due to extremely low kinetic energy flux associated with those accretion shocks. On the other hand, radiative features of internal shocks in the hot intracluster medium have been identified as temperature and density discontinuities in X-ray observations and diffuse radio emission from accelerated CR electrons. However, the non-detection of gamma-ray emission from galaxy clusters due to $\pi^0$ decay still remains to be an outstanding problem.
Shock Acceleration Model with Postshock Turbulence for Giant Radio Relics (1706.03201)
We explore the shock acceleration model for giant radio relics, in which relativistic electrons are accelerated via diffusive shock acceleration (DSA) by merger-driven shocks in the outskirts of galaxy clusters. In addition to DSA, turbulent acceleration by compressive MHD mode downstream of the shock is included as well as energy losses of postshock electrons by Coulomb scattering, synchrotron emission, and inverse Compton scattering off the cosmic background radiation. Considering that only a small fraction of merging clusters host radio relics, we favor the reacceleration scenario in which radio relics are generated preferentially when shocks encounter the regions containing low-energy ($\gamma_{\rm e} \lesssim 300$) cosmic ray electrons (CRe). We perform time-dependent DSA simulations of spherically expanding shocks with physical parameters relevant for the Sausage radio relic, and calculate the radio synchrotron emission from the accelerated CRe. We find that significant level of postshock turbulent acceleration is required in order to reproduce broad profiles of the observed radio flux densities of the Sausage relic. Moreover, the spectral curvature in the observed integrated radio spectrum can be explained, if the putative shock should have swept up and exited out of the preshock region of fossil CRe about 10~Myr ago.
Shock Acceleration Model for the Toothbrush Radio Relic (1703.00171)
June 5, 2017 astro-ph.HE
Although many of the observed properties of giant radio relics detected in the outskirts of galaxy clusters can be explained by relativistic electrons accelerated at merger-driven shocks, significant puzzles remain. In the case of the so-called Toothbrush relic, the shock Mach number estimated from X-ray observations ($M_{\rm X}\approx1.2-1.5$) is substantially weaker than that inferred from the radio spectral index ($M_{\rm rad}\approx2.8$).Toward understanding such a discrepancy, we here consider the following diffusive shock acceleration (DSA) models:(1) weak-shock models with $M_{\rm s}\lesssim 2$ and a preexisting population of cosmic-ray electrons (CRe) with a flat energy spectrum,and (2) strong-shock models with $M_{\rm s}\approx3$ and either shock-generated suprathermal electrons or preexisting fossil CRe. We calculate the synchrotron emission from the accelerated CRe, following the time evolution of the electron DSA, and subsequent radiative cooling and postshock turbulent acceleration (TA). We find that both models could reproduce reasonably well the observed integrated radio spectrum of the Toothbrush relic, but the observed broad transverse profile requires the stochastic acceleration by downstream turbulence, which we label "turbulent acceleration" or TA to distinguish it from DSA. Moreover, to account for the almost uniform radio spectral index profile along the length of the relic, the weak-shock models require a preshock region over 400~kpc with a uniform population of preexisting CRe with a high cutoff energy ($\gtrsim 40$ GeV). Due to the short cooling time, it is challenging to explain the origin of such energetic electrons. Therefore, we suggest the strong-shock models with low-energy seed CRe ($\lesssim 150$~MeV) are preferred for the radio observations of this relic.
The Case for Electron Re-Acceleration at Galaxy Cluster Shocks (1701.01439)
Reinout J. van Weeren, Felipe Andrade-Santos, William A. Dawson, Nathan Golovich, Dharam V. Lal, Hyesung Kang, Dongsu Ryu, Marcus Brüggen, Georgiana A. Ogrean, William R. Forman, Christine Jones, Vinicius M. Placco, Rafael M. Santucci, David Wittman, M. James Jee, Ralph P. Kraft, David Sobral, Andra Stroe, Kevin Fogarty
Jan. 10, 2017 astro-ph.CO, astro-ph.HE
On the largest scales, the Universe consists of voids and filaments making up the cosmic web. Galaxy clusters are located at the knots in this web, at the intersection of filaments. Clusters grow through accretion from these large-scale filaments and by mergers with other clusters and groups. In a growing number of galaxy clusters, elongated Mpc-size radio sources have been found, so-called radio relics. These relics are thought to trace relativistic electrons in the intracluster plasma accelerated by low-Mach number collisionless shocks generated by cluster-cluster merger events. A long-standing problem is how low-Mach number shocks can accelerate electrons so efficiently to explain the observed radio relics. Here we report on the discovery of a direct connection between a radio relic and a radio galaxy in the merging galaxy cluster Abell 3411-3412. This discovery indicates that fossil relativistic electrons from active galactic nuclei are re-accelerated at cluster shocks. It also implies that radio galaxies play an important role in governing the non-thermal component of the intracluster medium in merging clusters.
Re-acceleration model for the `Sausage' Radio Relic (1607.00459)
July 2, 2016 astro-ph.HE
The Sausage radio relic is the arc-like radio structure in the cluster CIZA J2242.8+5301, whose observed properties can be best understood by synchrotron emission from relativistic electrons accelerated at a merger-driven shock. However, there remain a few puzzles that cannot be explained by the shock acceleration model with only in-situ injection. In particular, the Mach number inferred from the observed radio spectral index, $M_{\rm radio}\approx 4.6$, while the Mach number estimated from X-ray observations, $M_{\rm X-ray}\approx 2.7$. In an attempt to resolve such a discrepancy, here we consider the re-acceleration model in which a shock of $M_s\approx 3$ sweeps through the intracluster gas with a pre-existing population of relativistic electrons. We find that observed brightness profiles at multi frequencies provide strong constraints on the spectral shape of pre-existing electrons. The models with a power-law momentum spectrum with the slope, $s\approx 4.1$, and the cutoff Lorentz factor, $\gamma_{e,c}\approx 3-5\times 10^4$ can reproduce reasonably well the observed spatial profiles of radio fluxes and integrated radio spectrum of the Sausage relic. The possible origins of such relativistic electrons in the intracluster medium remain to be investigated further.
Re-acceleration model for the "Toothbrush" Radio Relic (1603.07444)
March 24, 2016 astro-ph.HE
The Toothbrush radio relic associated the merging cluster 1RXS J060303.3 is presumed to be produced by relativistic electrons accelerated at merger-driven shocks. Since the shock Mach number inferred from the observed radio spectral index, $M_{radio}\approx 2.8$, is larger than that estimated from X-ray observations, $M_{X-ray}\lesssim 1.5$, we consider the re-acceleration model in which a weak shock of $M_s\approx 1.2-1.5$ sweeps through the intracluster plasma with a preshock population of relativistic electrons. We find the models with a power-law momentum spectrum with the slope, $s\approx 4.6$, and the cutoff Lorentz factor, $\gamma_{e,c}\approx 7-8\times 10^4$ can reproduce reasonably well the observed profiles of radio fluxes and integrated radio spectrum of the head portion of the Toothbrush relic. This study confirms the strong connection between the ubiquitous presence of fossil relativistic plasma originated from AGNs and the shock-acceleration model of radio relics in the intracluster medium.
Radio emission from weak spherical shocks in the outskirts of galaxy clusters (1502.02780)
Feb. 10, 2015 astro-ph.HE
In Kang (2015) we calculated the acceleration of cosmic-ray electrons and the ensuing radio synchrotron emission at weak spherical shocks that are expected to form in the outskirts of galaxy clusters.There we demonstrated that, at decelerating spherical shocks, the volume integrated spectra of both electrons and radiation deviate significantly from the test-particle power-laws predicted for constant planar shocks, because the shock compression ratio and the flux of injected electrons decrease in time. In this study, we consider spherical blast waves propagating into a constant density core surrounded by an isothermal halo with a decreasing density profile in order to explore how the deceleration rate of the shock speed affects the radio emission from accelerated electrons. The surface brightness profile and the volume-integrated radio spectrum of the model shocks are calculated by assuming a ribbon-like shock surface on a spherical shell and the associated downstream region of relativistic electrons. If the postshock magnetic field strength is about 7 microgauss, at the shock age of ~50 Myr, the volume-integrated radio spectrum steepens gradually with the spectral index from alpha_{inj} to alpha_{inj}+0.5 over 0.1-10 GHz, where alpha_{inj} is the injection index at the shock positionexpected from the diffusive shock acceleration theory. Such gradual steepening could explain the curved radio spectrum of the radio relic in cluster A2266, which was interpreted as a broken power-law by Trasatti et al. (2014), if the relic shock is young enough so that the break frequency falls in around 1 GHz.
Nonthermal radiation from relativistic electrons accelerated at spherically expanding shocks (1411.7513)
Feb. 7, 2015 astro-ph.HE
We study the evolution of the energy spectrum of cosmic-ray electrons accelerated at spherically expanding shocks with low Mach numbers and the ensuing spectral signatures imprinted in radio synchrotron emission. Time-dependent simulations of diffusive shock acceleration (DSA) of electrons in the test-particle limit have been performed for spherical shocks with parameters relevant for typical shocks in the intracluster medium. The electron and radiation spectra at the shock location can be described properly by the test-particle DSA predictions with instantaneous shock parameters. However, the volume integrated spectra of both electrons and radiation deviate significantly from the test-particle power-laws, because the shock compression ratio and the flux of injected electrons at the shock gradually decrease as the shock slows down in time.So one needs to be cautious about interpreting observed radio spectra of evolving shocks based on simple DSA models in the test-particle regime.
Injection of $\kappa$-like Suprathermal Particles into Diffusive Shock Acceleration (1405.0557)
Hyesung Kang, Vahe Petrosian, Dongsu Ryu, T. W. Jones
May 3, 2014 astro-ph.HE
We consider a phenomenological model for the thermal leakage injection in the diffusive shock acceleration (DSA) process, in which suprathermal protons and electrons near the shock transition zone are assumed to have the so-called $\kappa$-distributions produced by interactions of background thermal particles with pre-existing and/or self-excited plasma/MHD waves or turbulence. The $\kappa$-distribution has a power-law tail, instead of an exponential cutoff, well above the thermal peak momentum. So there are a larger number of potential seed particles with momentum, above that required for participation in the DSA process. As a result, the injection fraction for the $\kappa$-distribution depends on the shock Mach number much less severely compared to that for the Maxwellian distribution. Thus, the existence of $\kappa$-like suprathermal tails at shocks would ease the problem of extremely low injection fractions, especially for electrons and especially at weak shocks such as those found in the intracluster medium. We suggest that the injection fraction for protons ranges $10^{-4}-10^{-3}$ for a $\kappa$-distribution with $10 < \kappa_p < 30$ at quasi-parallel shocks, while the injection fraction for electrons becomes $10^{-6}-10^{-5}$ for a $\kappa$-distribution with $\kappa_e < 2$ at quasi-perpendicular shocks. For such $\kappa$ values the ratio of cosmic ray electrons to protons naturally becomes $K_{e/p}\sim 10^{-3}-10^{-2}$, which is required to explain the observed ratio for Galactic cosmic rays.
Nonthermal Radiation from Supernova Remnants: Effects of Magnetic Field Amplification and Particle Escape (1308.6652)
Hyesung Kang, T. W. Jones, Paul P. Edmon
Aug. 30, 2013 astro-ph.HE
We explore nonlinear effects of wave-particle interactions on the diffusive shock acceleration (DSA) process in Type Ia-like, SNR blast waves, by implementing phenomenological models for magnetic field amplification, Alfv'enic drift, and particle escape in time-dependent numerical simulations of nonlinear DSA. For typical SNR parameters the CR protons can be accelerated to PeV energies only if the region of amplified field ahead of the shock is extensive enough to contain the diffusion lengths of the particles of interest. Even with the help of Alfv'enic drift, it remains somewhat challenging to construct a nonlinear DSA model for SNRs in which order of 10 % of the supernova explosion energy is converted to the CR energy and the magnetic field is amplified by a factor of 10 or so in the shock precursor, while, at the same time, the energy spectrum of PeV protons is steeper than E^{-2}. To explore the influence of these physical effects on observed SNR emissions, we also compute resulting radio-to-gamma-ray spectra. Nonthermal emission spectra, especially in X-ray and gamma-ray bands,depend on the time dependent evolution of CR injection process, magnetic field amplification, and particle escape, as well as the shock dynamic evolution. This result comes from the fact that the high energy end of the CR spectrum is composed of the particles that are injected in the very early stages of blast wave evolution. Thus it is crucial to understand better the plasma wave-particle interactions associated with collisionless shocks in detail modeling of nonthermal radiation from SNRs.
Diffusive Shock Acceleration at Cosmological Shock Waves (1212.3246)
Hyesung Kang, Dongsu Ryu
Dec. 13, 2012 astro-ph.CO, astro-ph.HE
We reexamine nonlinear diffusive shock acceleration (DSA) at cosmological shocks in the large scale structure of the Universe, incorporating wave-particle interactions that are expected to operate in collisionless shocks. Adopting simple phenomenological models for magnetic field amplification (MFA) by cosmic-ray (CR) streaming instabilities and Alfv'enic drift, we perform kinetic DSA simulations for a wide range of sonic and Alfv'enic Mach numbers and evaluate the CR injection fraction and acceleration efficiency. In our DSA model the CR acceleration efficiency is determined mainly by the sonic Mach number Ms, while the MFA factor depends on the Alfv'enic Mach number and the degree of shock modification by CRs. We show that at strong CR modified shocks, if scattering centers drift with an effective Alfv'en speed in the amplified magnetic field, the CR energy spectrum is steepened and the acceleration efficiency is reduced significantly, compared to the cases without such effects. As a result, the postshock CR pressure saturates roughly at ~ 20 % of the shock ram pressure for strong shocks with Ms>~ 10. In the test-particle regime (Ms<~ 3), it is expected that the magnetic field is not amplified and the Alfv'enic drift effects are insignificant, although relevant plasma physical processes at low Mach number shocks remain largely uncertain.
Diffusive shock acceleration with magnetic field amplification and Alfvenic drift (1209.5203)
Sept. 24, 2012 astro-ph.HE
We explore how wave-particle interactions affect diffusive shock acceleration (DSA) at astrophysical shocks by performing time-dependent kinetic simulations, in which phenomenological models for magnetic field amplification (MFA), Alfvenic drift, thermal leakage injection, Bohm-like diffusion, and a free escape boundary are implemented. If the injection fraction of cosmic-ray (CR) particles is greater than 2x10^{-4}, for the shock parameters relevant for young supernova remnants, DSA is efficient enough to develop a significant shock precursor due to CR feedback, and magnetic field can be amplified up to a factor of 20 via CR streaming instability in the upstream region. If scattering centers drift with Alfven speed in the amplified magnetic field, the CR energy spectrum can be steepened significantly and the acceleration efficiency is reduced. Nonlinear DSA with self-consistent MFA and Alfvenic drift predicts that the postshock CR pressure saturates roughly at 10 % of the shock ram pressure for strong shocks with a sonic Mach number ranging 20< M_s< 100. Since the amplified magnetic field follows the flow modification in the precursor, the low energy end of the particle spectrum is softened much more than the high energy end. As a result, the concave curvature in the energy spectra does not disappear entirely even with the help of Alfvenic drift. For shocks with a moderate Alfven Mach number (M_A<10), the accelerated CR spectrum can become as steep as E^{-2.1}-E^{-2.3}, which is more consistent with the observed CR spectrum and gamma-ray photon spectrum of several young supernova remnants.
Diffusive Shock Acceleration Simulations of Radio Relics (1205.1895)
Hyesung Kang, Dongsu Ryu (Chungnam National University, Korea), T. W. Jones
May 9, 2012 astro-ph.CO, astro-ph.HE
Recent radio observations have identified a class of structures, so-called radio relics, in clusters of galaxies. The radio emission from these sources is interpreted as synchrotron radiation from GeV electrons gyrating in microG-level magnetic fields. Radio relics, located mostly in the outskirts of clusters, seem to associate with shock waves, especially those developed during mergers. In fact, they seem to be good structures to identify and probe such shocks in intracluster media (ICMs), provided we understand the electron acceleration and re-acceleration at those shocks. In this paper, we describe time-dependent simulations for diffusive shock acceleration at weak shocks that are expected to be found in ICMs. Freshly injected as well as pre-existing populations of cosmic-ray (CR) electrons are considered, and energy losses via synchrotron and inverse Compton are included. We then compare the synchrotron flux and spectral distributions estimated from the simulations with those in two well-observed radio relics in CIZA J2242.8+5301 and ZwCl0008.8+5215. Considering that the CR electron injection is rather inefficient at weak shocks with Mach number M <~ a few, the existence of radio relics could indicate the pre-existing population of low-energy CR electrons in ICMs. The implication of our results on the merger shock scenario of radio relics is discussed.
Cosmic Ray Spectrum in Supernova Remnant Shocks (1102.3123)
May 19, 2011 astro-ph.HE
We performed kinetic simulations of diffusive shock acceleration in Type Ia supernova remnants (SNRs) expanding into a uniform interstellar medium (ISM). The preshock gas temperature is the primary parameter that governs the cosmic ray (CR) acceleration, while magnetic field strength and CR injection rate are secondary parameters. SNRs in the hot ISM, with an injection fraction smaller than 10^{-4}, are inefficient accelerators with less than 10 % energy getting converted to CRs. The shock structure is almost test-particle like and the ensuing CR spectrum can be steeper than E^{-2}. Although the particles can be accelerated to the knee energy of 10^{15.5}Z eV with amplified magnetic fields in the precursor, Alfv'enic drift of scattering centers softens the source spectrum as steep as E^{-2.1} and reduces the CR acceleration efficiency.
Nonthermal Radiation from Type Ia Supernova Remnants (1103.0963)
Paul P. Edmon, Hyesung Kang, T. W. Jones, Renyi Ma
May 12, 2011 astro-ph.GA, astro-ph.SR, astro-ph.HE
We present calculations of expected continuum emissions from Sedov-Taylor phase Type Ia supernova remnants (SNRs), using the energy spectra of cosmic ray (CR) electrons and protons from nonlinear diffusive shock acceleration (DSA) simulations. A new, general-purpose radiative process code, Cosmicp, was employed to calculate the radiation expected from CR electrons and protons and their secondary products. These radio, X-ray and gamma-ray emissions are generally consistent with current observations of Type Ia SNRs. The emissions from electrons in these models dominate the radio through X-ray bands. Decays of \pi^0 s from p-p collisions mostly dominate the gamma-ray range, although for a hot, low density ISM case (n_{ISM}=0.003 cm^{-3}), the pion decay contribution is reduced sufficiently to reveal the inverse Compton contribution to TeV gamma-rays. In addition, we present simple scalings for the contributing emission processes to allow a crude exploration of model parameter space, enabling these results to be used more broadly. We also discuss the radial surface brightness profiles expected for these model SNRs in the X-ray and gamma-ray bands.
Comparison of CORSIKA and COSMOS simulations (1104.1005)
Soonyoung Roh, Jihee Kim, Katsuaki Kasahara, Eiji Kido, Akimichi Taketa, Dongsu Ryu, Hyesung Kang
April 6, 2011 astro-ph.HE
Ultra-high-energy cosmic rays (UHECRs) refer to cosmic rays with energy above 10^{18} eV. UHECR experiments utilize simulations of extensive air shower to estimate the properties of UHECRs. The Telescope Array (TA) experiment employs the Monte Carlo codes of CORSIKA and COSMOS to obtain EAS simulations. In this paper, we compare the results of the simulations obtained from CORSIKA and COSMOS and report differences between them in terms of the longitudinal distribution, Xmax-value, calorimetric energy, and energy spectrum at ground.
Energy Spectrum Of Nonthermal Electrons Accelerated At A Plane Shock (1102.3109)
We calculate the energy spectra of cosmic ray (CR) protons and electrons at a plane shock with quasi-parallel magnetic fields, using time-dependent, diffusive shock acceleration (DSA) simulations, including energy losses via synchrotron emission and Inverse Compton (IC) scattering. A thermal leakage injection model and a Bohm type diffusion coefficient are adopted. The electron spectrum at the shock becomes steady after the DSA energy gains balance the synchrotron/IC losses, and it cuts off at the equilibrium momentum p_{eq}. In the postshock region the cutoff momentum of the electron spectrum decreases with the distance from the shock due to the energy losses and the thickness of the spatial distribution of electrons scales as p^{-1}. Thus the slope of the downstream integrated spectrum steepens by one power of p for p_{br}<p<p_{eq}, where the break momentum decrease with the shock age as p_{br}\propto t^{-1}. In a CR modified shock, both the proton and electron spectrum exhibit a concave curvature and deviate from the canonical test-particle power-law, and the upstream integrated electron spectrum could dominate over the downstream integrated spectrum near the cutoff momentum. Thus the spectral shape near the cutoff of X-ray synchrotron emission could reveal a signature of nonlinear DSA.
Re-acceleration of Nonthermal Particles at Weak Cosmological Shock Waves (1102.2561)
Feb. 13, 2011 astro-ph.CO
We examine diffusive shock acceleration (DSA) of the pre-exisiting as well as freshly injected populations of nonthermal, cosmic-ray (CR) particles at weak cosmological shocks. Assuming simple models for thermal leakage injection and Alfv\'enic drift, we derive analytic, time-dependent solutions for the two populations of CRs accelerated in the test-particle regime. We then compare them with the results from kinetic DSA simulations for shock waves that are expected to form in intracluster media and cluster outskirts in the course of large-scale structure formation. We show that the test-particle solutions provide a good approximation for the pressure and spectrum of CRs accelerated at these weak shocks. Since the injection is extremely inefficient at weak shocks, the pre-existing CR population dominates over the injected population. If the pressure due to pre-existing CR protons is about 5 % of the gas thermal pressure in the upstream flow, the downstream CR pressure can absorb typically a few to 10 % of the shock ram pressure at shocks with the Mach number $M \la 3$. Yet, the re-acceleration of CR electrons can result in a substantial synchrotron emission behind the shock. The enhancement in synchrotron radiation across the shock is estimated to be about a few to several for $M \sim 1.5$ and $10^2-10^3$ for $M \sim 3$, depending on the detail model parameters. The implication of our findings for observed bright radio relics is discussed.
Diffusive Shock Acceleration in Test-Particle Regime (1008.0429)
Aug. 3, 2010 astro-ph.HE
We examine the test-particle solution for diffusive shock acceleration, based on simple models for thermal leakage injection and Alfv'enic drift. The critical injection rate, \xi_c, above which the cosmic ray (CR) pressure becomes dynamically significant, depends mainly on the sonic shock Mach number, M, and preshock gas temperature, T_1. In the hot-phase interstellar medium (ISM) and intracluster medium, \xi_c < 10^{-3} for shocks with M < 5, while \xi_c ~ 10^{-4}(T_1/10^6 K)^{1/2} for shocks with M > 10. For T_1=10^6 K, for example, the test-particle solution would be valid if the injection momentum, p_{inj} > 3.8 p_{th}. This leads to the postshock CR pressure less than 10% of the shock ram pressure. If the Alfv'en speed is comparable to the sound speed in the preshock flow, as in the hot-phase ISM, the power-law slope of CR spectrum can be significantly softer than the canonical test-particle slope. Then the CR spectrum at the shock can be approximated by the revised test-particle power-law with an exponential cutoff at the highest accelerated momentum, p_{max}(t). An analytic form of the exponential cutoff is also suggested.
Comparison of Different Methods for Nonlinear Diffusive Shock Acceleration (1005.2127)
D. Caprioli, Hyesung Kang, A. Vladimirov, T.W. Jones
We provide a both qualitative and quantitative comparison among different approaches aimed to solve the problem of non-linear diffusive acceleration of particles at shocks. In particular, we show that state-of-the-art models (numerical, Monte Carlo and semi-analytical), even if based on different physical assumptions and implementations, for typical environmental parameters lead to very consistent results in terms of shock hydrodynamics, cosmic ray spectrum and also escaping flux spectrum and anisotropy. Strong points and limits of each approach are also discussed, as a function of the problem one wants to study.
We perform kinetic simulations of diffusive shock acceleration (DSA) in Type Ia supernova remnants (SNRs) expanding into a uniform interstellar medium (ISM). Bohm-like diffusion assumed, and simple models for Alfvenic drift and dissipation are adopted. Phenomenological models for thermal leakage injection are considered as well. We find that the preshock gas temperature is the primary parameter that governs the cosmic ray (CR) acceleration efficiency and energy spectrum, while the CR injection rate is a secondary parameter. For SNRs in the warm ISM, if the injection fraction is larger than 10^{-4}, the DSA is efficient enough to convert more than 20 % of the SN explosion energy into CRs and the accelerated CR spectrum exhibits a concave curvature flattening to E^{-1.6}. Such a flat source spectrum near the knee energy, however, may not be reconciled with the CR spectrum observed at Earth. On the other hand, SNRs in the hot ISM, with an injection fraction smaller than 10^{-4}, are inefficient accelerators with less than 10 % of the explosion energy getting converted to CRs. Also the shock structure is almost test-particle like and the ensuing CR spectrum can be steeper than E^{-2}. With amplified magnetic field strength of order of 30 microG, Alfven waves generated by the streaming instability may drift upstream fast enough to make the modified test-particle power-law as steep as E^{-2.3}, which is more consistent with the observed CR spectrum.
Self-Similar Evolution of Cosmic-Ray Modified Shocks: The Cosmic-Ray Spectrum (0901.1702)
Jan. 13, 2009 astro-ph.HE
We use kinetic simulations of diffusive shock acceleration (DSA) to study the time-dependent evolution of plane, quasi-parallel, cosmic-ray (CR) modified shocks. Thermal leakage injection of low energy CRs and finite Alfv\'en wave propagation and dissipation are included. Bohm diffusion as well as the diffusion with the power-law momentum dependence are modeled. As long as the acceleration time scale to relativistic energies is much shorter than the dynamical evolution time scale of the shocks, the precursor and subshock transition approach the time-asymptotic state, which depends on the shock sonic and Alfv\'enic Mach numbers and the CR injection efficiency. For the diffusion models we employ, the shock precursor structure evolves in an approximately self-similar fashion, depending only on the similarity variable, x/(u_s t). During this self-similar stage, the CR distribution at the subshock maintains a characteristic form as it evolves: the sum of two power-laws with the slopes determined by the subshock and total compression ratios with an exponential cutoff at the highest accelerated momentum, p_{max}(t). Based on the results of the DSA simulations spanning a range of Mach numbers, we suggest functional forms for the shock structure parameters, from which the aforementioned form of CR spectrum can be constructed. These analytic forms may represent approximate solutions to the DSA problem for astrophysical shocks during the self-similar evolutionary stage as well as during the steady-state stage if p_{max} is fixed.
Propagation of UHE Protons through Magnetized Cosmic Web (0801.0371)
Santabrata Das, Hyesung Kang, Dongsu Ryu, Jungyeon Cho
May 16, 2008 astro-ph
If ultra-high-energy cosmic rays (UHECRs) originate from extragalactic sources, understanding the propagation of charged particles through the magnetized large scale structure (LSS) of the universe is crucial in the search for the astrophysical accelerators. Based on a novel model of the turbulence dynamo, we estimate the intergalactic magnetic fields (IGMFs) in cosmological simulations of the formation of the LSS. Under the premise that the sources of UHECRs are strongly associated with the LSS, we consider a model in which protons with E >10^{19} eV are injected by sources that represent active galactic nuclei located inside clusters of galaxies. With the model IGMFs, we then follow the trajectories of the protons, while taking into account the energy losses due to interactions with the cosmic background radiation. For observers located inside groups of galaxies like ours, about 70% and 35% of UHECR events above 60 EeV arrive within ~15 degree and ~5 degree, respectively, of the source position with time delays of less than ~10^7 yr. This implies that the arrival direction of super-GZK protons might exhibit a correlation with the distribution of cosmological sources on the sky. In this model, nearby sources (within 10 - 20 Mpc) should contribute significantly to the particle flux above ~10^{20} eV. | CommonCrawl |
\begin{document}
\maketitle
\begin{abstract} We prove the full range of estimates for a five-linear singular integral of Brascamp-Lieb type. The study is methodology-oriented with the goal to develop a sufficiently general technique to estimate singular integral variants of Brascamp-Lieb inequalities that do not obey H\"older scaling. The invented methodology constructs localized analysis on the entire space from local information on its subspaces of lower dimensions and combines such tensor-type arguments with the generic localized analysis. A direct consequence of the boundedness of the five-linear singular integral is a Leibniz rule which captures nonlinear interactions of waves from transversal directions.
\end{abstract}
\section{Introduction} \label{section_intro}
\subsection{Background and Motivation} \label{section_intro_background} Brascamp-Lieb inequalities refer to inequalities of the form \begin{align} \label{classical_bl}
\displaystyle \int_{\mathbb{R}^n} \big|\prod_{j=1}^{m}F_j(L_j(x))\big| dx \leq \text{BL}(\textbf{L,p})\prod_{j=1}^{m}\left(\int_{\mathbb{R}^{k_j}}|F_j|^{p_j}\right)^{\frac{1}{p_j}}, \end{align}
where $\text{BL}(\textbf{L,p})$ represents the Brascamp-Lieb constant depending on $\textbf{L} := (L_j)_{j=1}^m$ and $\textbf{p} := (p_j)_{j=1}^m$. For each $1 \leq j \leq m$, $L_j: R^{n} \rightarrow R^{k_j}$ is a linear surjection and $p_j \geq 0$. One equivalent formulation of (\ref{classical_bl}) is \begin{align} \label{classical_bl_exp}
\displaystyle \bigg(\int_{\mathbb{R}^n} \big|\prod_{j=1}^{m}F_j(L_j(x))\big|^r dx\bigg)^{\frac{1}{r}} \leq \text{BL}(\textbf{L},r\textbf{p})\prod_{j=1}^{m}\left(\int_{\mathbb{R}^{k_j}}|F_j|^{rp_j}\right)^{\frac{1}{rp_j}}, \end{align}
for any $r > 0$. Brascamp-Lieb inequalities have been well-developed in \cite{bl}, \cite{bcct}, \cite{bbcf}, \cite{bbbf}, \cite{chv}. Examples of Brascamp-Lieb inequalities consist of H\"older's inequality and the Loomis-Whitney inequality.
We adopt the informal definition that an $n$-linear singular integral operator $T$ is \textit{of Brascamp-Lieb type} if $T(F_1,\cdots, F_n)$ is reduced to the integrand on the left hand side of a classical Brascamp-Lieb inequality when the kernels are replaced by Dirac distributions $\delta_0$. As a consequence of the definition, a singular integral operator of Brascamp-Lieb type obeys the same scaling property as the corresponding classical Brascamp-Lieb inequality, which will be called the basic inequality. We will also refer to the estimate for the singular integral operator of Brascamp-Lieb type as the \textit{singular integral estimate of Brascamp-Lieb type}. For the readers familiar with the recent expository work of Durcik and Thiele in \cite{dt2}, this is similar to the generic estimate for the singular Brascamp-Lieb form described in (2.3) from \cite{dt2}.
Singular integral estimates with H\"older scaling have been studied extensively, including boundedness of single-parameter paraproducts \cite{cm} and multi-parameter paraproducts \cite{cptt}, \cite{cptt_2}, single-parameter flag paraproducts \cite{c_flag}, bilinear Hilbert transform \cite{lt}, multilinear operators of arbitrary rank \cite{mtt2002}, etc.
But it is of course natural to ask if there are similar singular integral estimates corresponding to Brascamp-Lieb inequalities that do not satisfy H\"older scaling.
So far, to the best of our knowledge, the only research article in the literature where the term "singular Brascamp-Lieb" has been used is the recent work by Durcik and Thiele \cite{dt}. However, we would like to emphasize that the basic inequalities corresponding to the "cubic singular expressions" considered in \cite{dt} are still H\"older's inequality, and the term "singular Brascamp-Lieb" was used to underline that the necessary and sufficient boundedness condition (1.6) of \cite{dt} is of the same flavor as the one for classical Brascamp-Lieb inequalities stated as (8) in \cite{bcct}.
Techniques to tackle multilinear singular integral operators with H\"older scaling \cite{cm}, \cite{cptt}, \cite{cptt_2}, \cite{c_flag}, \cite{lt}, \cite{mtt2002} usually involve localizations on phase space subsets of the full-dimension. In contrast, the understanding of singular integral operators of Brascamp-Lieb type with input functions defined on subspaces of the domain of the output function
(and thus with non-H\"older scaling) is in its primitive stage. The ultimate goal would be to develop a general methodology to treat a large class of singular Brascamp-Lieb estimates with non-H\"older scaling. It is natural to believe that such an approach would need to extract and integrate local information on subspaces of lower dimensions. Also due to its multilinear structure, localizations on the entire space could be necessary as well and a hybrid of both localized analyses would be demanded.
The subject of our study in this present paper is one of the simplest multilinear operators, whose complete understanding cannot be reduced to earlier results\footnote{Many cases of arbitrary complexity follow from the mixed-norm estimates for vector-valued inequalities in the paper by Benea and the first author \cite{bm2}.} and which requires such a new type of analysis. More precisely, it is the five-linear operator defined by
\begin{align} \label{bi_flag_int} & T_{K_1K_2}(f_1, f_2, g_1, g_2, h)(x,y) \nonumber \\ := & p.v. \displaystyle \int_{\mathbb{R}^{10}} K_1\big((t_1^1, t_1^2),(t_2^1,t_2^2))K_2((s_1^1,s_1^2), (s_2^1,s_2^2), (s_3^1,s_3^2)\big) \cdot \nonumber \\ &\quad \quad \quad f_1(x-t_1^1-s_1^1)f_2(x-t_2^1-s_2^1)g_1(y-t_1^2-s_1^2)g_2(y-t_2^2-s_2^2)h(x-s_3^1,y-s_3^2) d\vec{t_1} d\vec{t_2} d \vec{s_1} d\vec{s_2} d\vec{s_3}, \end{align} where $\vec{t_i} = (t_i^1, t_i^2)$, $\vec{s_j} = (s_j^1,s_j^2)$ for $i = 1, 2$ and $j = 1,2,3$. In (\ref{bi_flag_int}), $K_1$ and $K_2$ are Calder\'on-Zygmund kernels that satisfy \begin{align*}
& |\nabla K_1(\vec{t_1}, \vec{t_2})| \lesssim \frac{1}{|(t_1^1,t_2^1)|^{3}}\frac{1}{|(t_1^2,t_2^2)|^{3}}, \nonumber \\
& |\nabla K_2(\vec{s_1}, \vec{s_2}, \vec{s_3})| \lesssim \frac{1}{|(s_1^1,s_2^1,s_3^1)|^{4}}\frac{1}{|(s_1^2,s_2^2, s_3^2)|^{4}} . \end{align*}
As one can see, the operator $T_{K_1K_2}$ takes two functions depending on the $x$ variable ($f_1$ and $f_2$), two functions depending on the $y$ variable ($g_1$ and $g_2$) and one depending on both $x$ and $y$ (namely $h$) into another function of $x$ and $y$. Our goal is to prove that $T_{K_1K_2}$ satisfies the mapping property $$ L^{p_1}(\mathbb{R}) \times L^{q_1}(\mathbb{R}) \times L^{p_2}(\mathbb{R}) \times L^{q_2}(\mathbb{R}) \times L^{s}(\mathbb{R}^2) \rightarrow L^{r}(\mathbb{R}^2)
$$ for $1 < p_1, p_2, q_1, q_2, s \leq \infty$, $r >0$, $(p_1,q_1), (p_2,q_2) \neq (\infty, \infty)$ with \begin{equation} \label{bl_exp} \frac{1}{p_1} + \frac{1}{q_1} + \frac{1}{s} = \frac{1}{p_2} + \frac{1}{q_2} + \frac{1}{s} = \frac{1}{r}. \end{equation} To verify that the boundedness of $T_{K_1K_2}$ qualifies to be a singular integral estimate of Brascamp-Lieb type, one can remove the singularities by setting \begin{align*} & K_1(\vec{t_1}, \vec{t_2}) = \delta_{\textbf{0}}(\vec{t_1}, \vec{t_2}), \nonumber \\ & K_2(\vec{s_1}, \vec{s_2}, \vec{s_3}) = \delta_{\textbf{0}}(\vec{s_1}, \vec{s_2}, \vec{s_3}), \end{align*} and express its boundedness explicitly as
\begin{align} \label{flag_bl}
\|f_1(x) f_2(x) g_1(y) g_2(y) h(x,y)\|_{r} \lesssim \|f_1\|_{L^{p_1}(\mathbb{R})}\|f_2\|_{L^{q_1}(\mathbb{R})}\|g_1\|_{L^{p_2}(\mathbb{R})} \|g_4\|_{L^{q_2}(\mathbb{R})}\|h\|_{L^{s}(\mathbb{R}^2)}. \end{align}
The above inequality follows from H\"older's inequality and the Loomis-Whitney inequality, which, in this simple two dimensional case, is the same as Fubini's theorem. Clearly, it is an inequality of the same type as (\ref{classical_bl_exp}), with a different homogeneity than H\"older. Moreover, this reduction shows that (\ref{bl_exp}) is indeed a necessary condition for the boundedness exponents of (\ref{flag_bl}) and thus of (\ref{bi_flag_int}).
\subsection{Connection with Other Multilinear Objects} \label{section_intro_connection} The connection with other well-established multilinear operators that we will describe next justifies that $T_{K_1K_2}$ defined in (\ref{bi_flag_int}) is a reasonably simple and interesting operator to study, with the hope of inventing a general method that can handle a large class of singular integral estimates of Brascamp-Lieb type with non-H\"older scaling.
Let $\mathcal{M}(\mathbb{R}^d)$ denote the set of all bounded symbols $m \in L^{\infty}(\mathbb{R}^d)$ smooth away from the origin and satisfying the Marcinkiewicz-H\"ormander-Mihlin condition
\begin{equation*}
\left|\partial^{\alpha} m(\xi) \right| \lesssim \frac{1}{|\xi|^{|\alpha|}} \end{equation*} for any $\xi \in \mathbb{R}^d \setminus \{0\}$ and sufficiently many multi-indices $\alpha$. The simplest singular integral operator which corresponds to the two-dimensional Loomis-Whitney inequality would be \begin{equation} \label{tensor_ht} T_{m_1m_2}(f, g)(x,y) := \int_{\mathbb{R}^2} m_1(\xi)m_2(\eta) \widehat{f}(\xi) \widehat{g}(\eta) e^{2 \pi i x \xi} e^{2\pi i y\eta}d\xi d\eta, \end{equation}
where $m_1, m_2 \in \mathcal{M}(\mathbb{R})$. (\ref{tensor_ht}) is a tensor product of Hilbert transforms whose boundedness are well-known. The bilinear variant of (\ref{tensor_ht}) can be expressed as
\begin{align} \label{tensor_para} &T_{m_1m_2}(f_1,f_2, g_1, g_2)(x,y) \nonumber \\ := & \int_{\mathbb{R}^4} m_1(\xi_1,\xi_2) m_2(\eta_1,\eta_2) \widehat{f_1}(\xi_1) \widehat{f_2}(\xi_2)\widehat{g_1}(\eta_1) \widehat{g_2}(\eta_2)e^{2 \pi i x(\xi_1+\xi_2)}e^{2 \pi i y(\eta_1+\eta_2)} d\xi_1 d\xi_2 d\eta_1 d\eta_2, \end{align}
where $m_1, m_2 \in \mathcal{M}(\mathbb{R}^2)$. It can be separated as a tensor product of single-parameter paraproducts whose boundedness are proved by Coifman-Meyer's theorem \cite{cm}. To avoid trivial tensor products of single-parameter operators, one then completes (\ref{tensor_para}) by adding a generic function of two variables thus obtaining \begin{align} \label{bi_pp} &T_{b}(f_1, f_2, g_1, g_2, h)(x,y) \nonumber \\ :=& \int_{\mathbb{R}^6} b((\xi_1,\eta_1),(\xi_2,\eta_2),(\xi_3,\eta_3)) \widehat{f_1 \otimes g_1}(\xi_1, \eta_1) \widehat{f_2 \otimes g_2}(\xi_2, \eta_2) \widehat{h}(\xi_3,\eta_3) \nonumber \\ & \quad \quad \cdot e^{2 \pi i x(\xi_1+\xi_2+ \xi_3)}e^{2 \pi i y(\eta_1+\eta_2+ \eta_3)} d\xi_1 d\xi_2 d\xi_3 d\eta_1 d\eta_2 d\eta_3, \end{align} where \begin{align*}
& \left|\partial^{\alpha}_{(\xi_1,\xi_2,\xi_3)} \partial^{\beta}_{(\eta_1,\eta_2, \eta_3)} b \right| \lesssim \frac{1}{|(\xi_1,\xi_2,\xi_3)|^{|\alpha|}|(\eta_1,\eta_2,\eta_3)|^{|\beta|}} \end{align*} for sufficiently many multi-indices $\alpha$ and $\beta$. Such a multilinear operator is indeed a bi-parameter paraproduct whose theory has been developed by Muscalu, Pipher, Tao and Thiele \cite{cptt}. It also appeared naturally in nonlinear PDEs, such as Kadomtsev-Petviashvili equations studied by Kenig \cite{k}. To reach beyond bi-parameter paraproducts, one then replaces the singularity in each subspace by a flag singularity. In one dimension, the corresponding trilinear operator takes the form \begin{equation} \label{flag} T_{m_1m_2}(f_1,f_2,f_3)(x) := \int_{\mathbb{R}^3} m_1(\xi_1,\xi_2)m_2(\xi_1,\xi_2,\xi_3) \widehat{f_1}(\xi_1) \widehat{f_2}(\xi_2) \widehat{f_3}(\xi_3) e^{2 \pi i x(\xi_1+\xi_2+\xi_3)} d\xi_1 d\xi_2 d\xi_3, \end{equation} where $m_1 \in \mathcal{M}(\mathbb{R}^2)$ and $m_2 \in \mathcal{M}(\mathbb{R}^3)$. The operator (\ref{flag}) was studied by Muscalu \cite{c_flag} using time-frequency analysis which applies not only to the operator itself, but also to all of its adjoints. Miyachi and Tomita \cite{mt} extended the $L^p$-boundedness for $p>1$ established in \cite{c_flag} to all Hardy spaces $H^p$ with $p > 0$. The single-parameter flag paraproduct and its adjoints are closely related to various nonlinear partial differential equations, including nonlinear Schr\"odinger equations and water wave equations as discovered by Germain, Masmoudi and Shatah \cite{gms}. Its bi-parameter variant is indeed related to the subject of our study and is equivalent to (\ref{bi_flag_int}): \begin{align} \label{bi_flag_mult} T_{ab}(f_1, f_2, g_1, g_2, h)(x,y) := \int_{\mathbb{R}^6} & a((\xi_1,\eta_1),(\xi_2,\eta_2)) b((\xi_1,\eta_1),(\xi_2,\eta_2),(\xi_3,\eta_3)) \widehat{f_1 \otimes g_1}(\xi_1, \eta_1) \widehat{f_2 \otimes g_2}(\xi_2, \eta_2) \nonumber \\ & \cdot \widehat{h}(\xi_3,\eta_3) e^{2\pi i x(\xi_1+\xi_2+\xi_3)}e^{2\pi i y(\eta_1+\eta_2+\eta_3)} d \xi_1 d\xi_2 d\xi_3 d\eta_1 d\eta_2 d\eta_3, \end{align} where \begin{align*}
& \left|\partial^{\alpha_1}_{(\xi_1,\xi_2)} \partial^{\beta_1}_{(\eta_1,\eta_2)} a\right| \lesssim \frac{1}{|(\xi_1,\xi_2)|^{|\alpha_1|}|(\eta_1,\eta_2)|^{|\beta_1|}}, \nonumber \\
& \left|\partial^{\alpha_2}_{(\xi_1,\xi_2,\xi_3)} \partial^{\beta_2}_{(\eta_1,\eta_2, \eta_3)} b \right| \lesssim \frac{1}{|(\xi_1,\xi_2,\xi_3)|^{|\alpha_2|}|(\eta_1,\eta_2,\eta_3)|^{|\beta_2|}}, \end{align*} for sufficiently many multi-indices $\alpha_1, \beta_1, \alpha_2$ and $\beta_2$. The equivalence can be derived with \begin{align*} & a = \widehat{K_1}, \nonumber \\ & b = \widehat{K_2}. \end{align*}
\begin{comment} An equivalent expression of (\ref{bi_flag_int}) as a multiplier operator is \begin{equation*} T_{ab}(f_1, f_2, f_3^x, g_1, g_2, g_3^y)(x,y) := \int_{\mathbb{R}^4} b_1(\xi_1,\xi_2) b_2(\eta_1,\eta_2) \widehat{f}(\xi_1) \widehat{g}(\eta_1) \widehat{h}(\xi_2,\eta_2)e^{2 \pi i x(\xi_1+\xi_2)}e^{2 \pi i y(\eta_1+\eta_2)} d\xi_1 d\xi_2 d\eta_1 d\eta_2. \end{equation*} \end{comment}
\begin{comment} The single-parameter variant of the multiplier operator defined in (\ref{bi_flag_mult}) takes the form \begin{equation} \label{flag} T_{a_1b_1}(f_1,f_2,f_3)(x) := \int a_1(\xi_1,\xi_2)b_1(\xi_1,\xi_2,\xi_3) \widehat{f_1}(\xi_1) \widehat{f_2}(\xi_2) \widehat{f_3}(\xi_3) e^{2 \pi i x(\xi_1+\xi_2+\xi_3)} d\xi_1 d\xi_2 d\xi_3, \end{equation} where $a_1 \in \mathcal{M}(\mathbb{R}^2)$, $b_1 \in \mathcal{M}(\mathbb{R}^3)$ are Coifman-Meyer symbols. The operator (\ref{flag}) was studied by Muscalu \cite{c_flag} using time-frequency analysis which applies to not only the operator itself, but also all of its adjoints. Miyachi and Tomita \cite{mt} extended the $L^p$-boundedness for $p>1$ established in \cite{c_flag} to all Hardy spaces $H^p$ with $p > 0$. The single-parameter flag paraproduct and its adjoints are closely related to various nonlinear partial differential equations, including nonlinear Schrodinger equations and water wave equations as discovered by Germain, Masmoudi and Shatah \cite{gms}. As can be seen, (\ref{flag}) is a special case of (\ref{bi_flag_mult}) when
$$
a((\xi_1,\eta_1),(\xi_2,\eta_2)) = a_1(\xi_1,\xi_2) $$ $$ b((\xi_1,\eta_1),(\xi_2,\eta_2),(\xi_3,\eta_3)) = b_1(\xi_1,\xi_2,\xi_3) $$ \end{comment}
\begin{comment} Another multilinear operator related to (\ref{bi_flag_mult}) with wide applications in PDEs is the single-parameter paraproduct: \begin{equation} \label{pp} T_{b_1}(f_1,f_2)(x) := \int b_1(\xi_1,\xi_2,\xi_3) \widehat{f_1}(\xi_1) \widehat{f_2}(\xi_2) \widehat{f_3}(\xi_3) e^{2 \pi i x(\xi_1+\xi_2+\xi_3)} d\xi_1 d\xi_2 d\xi_3, \end{equation} where $b_1(\xi_1,\xi_2,\xi_3) \in \mathcal{M}(\mathbb{R}^3)$ is a Coifman-Meyer symbol. Its boundedness is proved by Coifman-Meyer's theorem on paraproducts \cite{cm}. One may notice that (\ref{pp}) can be deduced from (\ref{bi_flag_mult}) by choosing $$
a((\xi_1,\eta_1),(\xi_2,\eta_2)) = 1 $$ $$ b((\xi_1,\eta_1),(\xi_2,\eta_2),(\xi_3,\eta_3)) = b_1(\xi_1,\xi_2,\xi_3) $$
The bi-parameter variant of (\ref{pp}) is called bi-parameter paraproducts whose theory has been developed by Muscalu, Pipher, Tao and Thiele \cite{cptt}. It also appeared naturally in nonlinear PDEs, such as Kadomtsev-Petviashvili equations studied by Kenig \cite{k}. It is defined as \begin{align} \label{bi_pp} T_b(F_1,F_2,F_3)(x,y) := \int & b((\xi_1,\eta_1),(\xi_2,\eta_2), (\xi_3,\eta_3)) \widehat{F_1}(\xi_1,\eta_1) \widehat{F_2}(\xi_2,\eta_2)\widehat{F_3}(\xi_3,\eta_3) \nonumber \\ & e^{2\pi i x(\xi_1+\xi_2+\xi_3)}e^{2\pi i y(\eta_1+\eta_2+\eta_3)} d \xi_1 d\xi_2 d\xi_3 d\eta_1 d\eta_2 d\eta_3, \end{align} where $b((\xi_1,\eta_1),(\xi_2,\eta_2), (\xi_3,\eta_3))$. Suppose that one again restricts the function space to tensor product space, it is not difficult to observe that (\ref{bi_pp}) is a sub-case of (\ref{bi_flag_mult}) when $$
a((\xi_1,\eta_1),(\xi_2,\eta_2)) = 1 $$ \end{comment}
The general bi-parameter trilinear flag paraproduct is defined on larger function spaces where the tensor products are replaced by general functions in the plane.\footnote{Its boundedness is at present an open question, raised by the first author of the article on several occasions.}
From this perspective, $T_{ab}$ or equivalently $T_{K_1K_2}$ defined in (\ref{bi_flag_mult}) and (\ref{bi_flag_int}) respectively can be viewed as a trilinear operator with the desired mapping property $$ T_{ab}: L^{p_1}_{x}(L^{p_2}_y) \times L^{q_1}_{x}(L^{q_2}_y) \times L^{s}(\mathbb{R}^2) \rightarrow L^{r}(\mathbb{R}^2) $$ for $ 1 < p_1, p_2, q_1, q_2, s \leq \infty$, $r > 0$, $(p_1, q_1), (p_2,q_2) \neq (\infty, \infty)$ and $\frac{1}{p_1} + \frac{1}{q_1} + \frac{1}{s} = \frac{1}{p_2} + \frac{1}{q_2} + \frac{1}{s} = \frac{1}{r}$, where the first two function spaces are restricted to be tensor product spaces. The condition that $(p_1, q_1), (p_1,q_2) \neq (\infty, \infty)$ is inherited from single-parameter flag paraproducts and can be verified by the unboundedness of the operator when $f_1, f_2 \in L^{\infty}(\mathbb{R})$ are constant functions. Lu, Pipher and Zhang \cite{lpz} showed that the general bi-parameter flag paraproduct can be reduced to an operator given by a symbol with better singularity using an argument inspired by Miyachi and Tomita \cite{mt}. The boundedness of the reduced multiplier operator still remains open. The reduction allows an alternative proof of $L^p$-boundedness for (\ref{bi_flag_mult}) as long as $p \neq \infty$. However, we emphasize again that we will not take this point of view now, and instead, we treat our operator $T_{ab}$ as a five-linear operator.
\begin{comment} Alternatively, one can also perceive (\ref{bi_flag_int}) as a bi-parameter flag paraproduct. More precisely, the bi-parameter flag paraproduct is a multilinear operator defined as $$ T_{m} (F_1,F_2,F_3)(x,y) := \int m(\vec{\zeta_1},\ldots,\vec{\zeta_n})\widehat{F_1}(\vec{\zeta_1})\ldots \widehat{F_n}(\vec{\zeta_n}) e^{2 \pi i (x,y)\cdot(\vec{\zeta_1} + \ldots + \vec{\zeta_n})}d\vec{\zeta_1}\ldots d\vec{\zeta_n}$$ where $\zeta_i = (\xi_i,\eta_i) \in \mathbb{R}^2$ and \begin{equation} \label{bi_flag_sym} m(\vec{\zeta_1},\ldots,\vec{\zeta_n}):= \prod_{S\subseteq \{1,\ldots, n \}} m_S(\vec{\zeta}_S) \end{equation} with $m_S$ being a symbol in $\mathbb{R}^{\text{card}(S)} \times \mathbb{R}^{\text{card}(S)}$ that is bounded, smooth away from the subspaces $\{(\xi_i)_{i \in S } = 0 \} \cup \{(\eta_i)_{i \in S} = 0 \} $ and satisfying the Marcinkiewicz condition $$
\left|\partial^{\vec{\alpha}}_{(\xi_i)_{i \in S}} \partial^{\vec{\beta}}_{(\eta_i)_{i \in \S}} m_S\right| \lesssim \frac{1}{|(\xi_i)_{i \in S}|^{|\vec{\alpha}|}|(\eta_i)_{i \in S}|^{|\vec{\beta}|}} $$ where $\vec{\alpha}$ and $\vec{\beta}$ denote multi-indices with nonnegative entries. With this formulation, (\ref{bi_flag_int}) is a special case when $$ m = m_{S_1}\cdot m_{S_2} $$
where \begin{align*} &S_1 = \{1,2\} \nonumber \\ &S_2 = \{1,2,3\}. \end{align*} And the function spaces are restricted in the sense that \begin{align*} &F_1 = f_1 \otimes g_1 \nonumber \\ &F_2 = f_2 \otimes g_2. \end{align*} \end{comment}
\subsection{Methodology} \label{section_intro_method}
As one may notice from the last section, the five-linear operator $T_{ab}$ ( or $T_{K_1K_2}$) contains the features of the bi-parameter paraproduct defined in (\ref{bi_pp}) and the single-parameter flag paraproduct defined in (\ref{flag}), which hints that the methodology would embrace localized analyses of both operators. Nonetheless, it is by no means a simple concatenation of two existing arguments. The methodology includes \begin{enumerate} \item \textbf{tensor-type stopping-time decomposition} which refers to an algorithm that first implements a one-dimensional stopping-time decomposition for each variable and then combines information for different variables to obtain estimates for operators involving several variables;
\item \textbf{general two-dimensional level sets stopping-time decomposition} which refers to an algorithm that partitions the collection of dyadic rectangles such that the dyadic rectangles in each sub-collection intersect with a certain level set non-trivially; \end{enumerate} and the main novelty lies in \begin{enumerate}[(i)] \item the construction of two-dimensional stopping-time decompositions from stopping-time decompositions on one-dimensional subspaces; \item the hybrid of tensor-type and general two-dimensional level sets stopping-time decompositions in a meaningful fashion. \end{enumerate}
The methodology outlined above is considered to be robust in the sense that it captures all local behaviors of the operator. The robustness may also be verified by the entire range of estimates obtained. After closer inspection of the technique, it would not be surprising that the technique gives estimates involving $L^{\infty}$ norms. In particular, the tensor-type stopping-time decompositions process information on each subspaces independently. As a consequence, when some function defined on some subspace lies in $L^{\infty}$, one simply ``forgets" about that function and glues the information from subspaces in an intelligent way specified later.
\subsection{Structure} \label{section_intro_structure} The paper is organized as follows: main theorems are stated in Section \ref{section_result} followed by preliminary definitions and theorems introduced in Section \ref{section_prelim}. Section \ref{section_discrete_model} describes the reduced discrete model operators and estimates one needs to obtain for the model operators while the reduction procedure is postponed to Appendix II. Section \ref{section_size_energy} gives the definition and estimates for the building blocks in the argument - sizes and energies. Sections \ref{section_thm_haar_fixed}, \ref{section_thm_haar}, \ref{section_thm_inf_fixed_haar} and \ref{section_thm_inf_haar} focus on estimates for the model operators in the Haar case. All four sections start with a specification of the stopping-time decompositions used. Section \ref{section_fourier} extends all the estimates in the Haar setting to the general Fourier case.
It is also important to notice that Section \ref{section_thm_haar_fixed} develops an argument for one of the simpler model operators with emphasis on the key geometric feature implied by a stopping-time decomposition, that is the sparsity condition. Section \ref{section_thm_haar} focuses on a more complicated model which requires not only the sparsity condition, but also a Fubini-type argument which is discussed in details. Sections \ref{section_thm_inf_fixed_haar} and \ref{section_thm_inf_haar} are devoted to estimates involving $L^{\infty}$ norms and the arguments for those cases are similar to the ones in Section \ref{section_thm_haar_fixed}, in the sense that the sparsity condition is sufficient to obtain the results.
\subsection{Acknowledgements.} We thank Jonathan Bennett for the inspiring conversation we had in Matsumoto, Japan, in February 2016, that triggered our interest in considering and understanding singular integral generalizations of Brascamp-Lieb inequalities, and, in particular, the study of the present paper. We also thank Guozhen Lu, Jill Pipher and Lu Zhang for discussions about their recent work in \cite{lpz}. Finally, we thank Polona Durcik and Christoph Thiele for the recent conversation which clarified the similarities and differences between the results in \cite{dt} and those in our paper and \cite{bm2}.
The first author was partially supported by a Grant from the Simons Foundation. The second author was partially supported by the ERC Project FAnFArE no. 637510.
\begin{comment} The following Leibniz rule, originally studied by Kato and Ponce \cite{kp}, plays an important role in the study of nonlinear PDE. Suppose $f_1,f_2 \in \mathcal{S}(\mathbb{R}^2).$ Then for $\alpha \geq 0$, \begin{equation}\label{lb1}
\| D^{\alpha} (f_1f_2)\|_r \lesssim \|D^{\alpha} f_1 \|_{p_1} \|f_2 \|_{q_1} + \| f_1 \|_{p_2} \|D^{\alpha}f_2 \|_{q_2} \end{equation}
with $1 < p_i, q_i < \infty, \frac{1}{p_i}+ \frac{1}{q_i} = \frac{1}{r}, i= 1,2.$ Here $D^{\alpha} F := \mathcal{F}^{-1}(|(\xi,\eta)|^{\alpha}\hat{F}(\xi,\eta))$, where $\xi, \eta \in \mathbb{R}$.
The following Leibniz rule with one more level of intricacy arises naturally in the study of nonlinear Schr\"{o}dinger equation and 3-dimensional water wave equation by Germain, Masmoudi and Shatah \cite{gms}. Details of the derivation can be found in \cite{cm}. Suppose $f_1,f_2,f_3 \in \mathcal{S}(\mathbb{R}^2).$ Then for $\alpha, \beta \geq 0$, \begin{align}\label{lb2}
\|D^{\beta} (D^{\alpha}(f_1 f_2) f_3) \| _r \lesssim & \|D^{\alpha + \beta} f_1\|_{p_1}\|f_2\|_{q_1} \|f_3\|_{s_1} + \|f_1\|_{p_2}\|D^{\alpha + \beta} f_2\|_{q_2} \|f_3\|_{s_2} +
\nonumber \\ & \|D^{\alpha} f_1\|_{p_3}\|f_2\|_{q_3} \|D^{\beta}f_3\|_{s_3} + \|f_1\|_{p_4}\|D^{\alpha} f_2\|_{q_4} \|D^{\beta}f_3\|_{s_4}. \end{align} with $1 < p_i, q_i,s_i < \infty, \frac{1}{p_i}+ \frac{1}{q_i}+\frac{1}{s_i} = \frac{1}{r}, i= 1,2,3,4.$ One remark that we will explain in more details later is while some range of estimates can be obtained for (\ref{lb2}) by repeatedly applying (\ref{lb1}), the nontrivial range of estimates $1<p_i, q_i <2$ requires more delicate treatment \cite{cmf}.
The above Leibniz rule can be extended to involve partial fractional derivatives by Muscalu, Piper, Tao and Thiele \cite{cptt}. Such bi-parameter variant of appears in Kadomtsev-Petviashvili estimates studied by Kenig \cite{k}:
Suppose $f_1,f_2 \in \mathcal{S}(\mathbb{R}^2).$ Then for $\alpha_1,\alpha_2 > 0$ sufficiently large and $1 < p_i, q_i < \infty, \frac{1}{p_i}+ \frac{1}{q_i} = \frac{1}{r}, i= 1,2,3,4,$ \begin{align*}\label{lb3}
\| D_1^{\alpha_1}D_2^{\alpha_2} (f_1f_2)\|_r \lesssim & \|D_1^{\alpha_1}D_2^{\alpha_2} f_1 \|_{p_1} \|f_2 \|_{q_1} + \| f_1 \|_{p_2} \|D_1^{\alpha_1}D_2^{\alpha_2}f_2 \|_{q_2} +
\nonumber \\ & \|D_1^{\alpha_1} f_1 \|_{p_3} \|D_2^{\alpha_2}f_2 \|_{q_3} + \|D_2^{\alpha_2} f_1 \|_{p_4} \|D_1^{\alpha_1}f_2 \|_{q_4}. \end{align*}
Here $D_1^{\alpha_1} F(x,y) := \mathcal{F}^{-1}(|\xi|^{\alpha_1}\hat{F}(\xi,\eta))$, $D_2^{\alpha_2} F(x,y) := \mathcal{F}^{-1}(|\eta|^{\alpha_2}\hat{F}(\xi,\eta))$, where $\xi, \eta \in \mathbb{R}$.
One natural question is to extend Leibniz rule \ref{lb2} to allow for partial fractional derivatives on functions. \begin{problem}\label{lb4} Suppose $F_1, F_2, F_3 \in \mathcal{S}(\mathbb{R}^2)$. Is it true that for $\beta_1, \beta_2, \alpha_1, \alpha_2 > 0$ sufficiently large and $1< p_i, q_i, s_i <\infty, \frac{1}{p_i}+\frac{1}{q_i} + \frac{1}{s_i} = \frac{1}{r},i = 1, \ldots, 16,$ \begin{align*}
\|D_1^{\beta_1} D_2^{\beta_2}(D_1^{\alpha_1}D_2^{\alpha_2}(F_1F_2) F_3)\|_{r} \lesssim & \ \ 16 \text{\ \ terms of the types: \ \ }
\nonumber \\ & \|D_1^{\alpha_1+\beta_1}D_2^{\alpha_2 + \beta_2 }F_1\|_{p_1} \|F_2\|_{q_1} \| \|h\|_{s_1},
\nonumber \\ & \|D_2^{\alpha_2 + \beta_2}F_1\|_{p_2} \|D_1^{\alpha_1+\beta_1}F_2\|_{q_2} \|h\|_{s_2},
\nonumber \\ & \|D_1^{\alpha_1+\beta_1}D_2^{\alpha_2}F_1\|_{p_3} \|F_2\|_{q_3}\|D_2^{\beta_2}h\|_{s_3}\ldots \end{align*}
\end{problem} The above Leibniz rule generates question of boundedness of a bi-parameter multilinear operator which requires localization of two-dimensional objects. One intermediate step to understand such Leibniz rule is to study Leibniz rule \ref{lb4} on functions of tensor product type.
\begin{theorem} \label{lb_main} Suppose $f_1 \otimes g_1,f_2\otimes g_2, h \in \mathcal{S}(\mathbb{R}^2).$ Is it true that for $\beta_1, \beta_2, \alpha_1, \alpha_2 > 0$ sufficiently large and H\H older-type exponents $1 < p^j_i, q^j_i, s^j_i < \infty, i = 1,2, j = 1, \ldots, 16 $, \begin{align*} D_1^{\beta_1} D_2^{\beta_2}(D_1^{\alpha_1}D_2^{\alpha_1}(f_1^x \otimes g_1 f_2 \otimes g_2) h) \lesssim & \ \ 16 \text{\ \ terms of the types: \ \ }
\nonumber \\ & \|D_1^{\alpha_1+\beta_1}f_1\|_{p^1_1} \|f_2\|_{p^1_2} \|D_2^{\alpha_2 + \beta_2}g_1\|_{q^1_1} \|g_2\|_{q^1_2} \|h\|_{L^{s^1_1}(L^{s^1_2})} ,
\nonumber \\ & \|f_1\|_{p^2_1} \|D_1^{\alpha+\beta_1}f_2\|_{p^2_2} \|D_2^{\alpha_2 + \beta_2}g_1\|_{q^2_1} \|g_2\|_{q^2_2} \|h\|_{L^{s^2_1}(L^{s^2_2})},
\nonumber \\ & \|D_1^{\alpha+\beta_1}f_1\|_{p^3_1} \|f_2\|_{p^3_2} \|D_2^{\alpha_2}g_1\|_{q^3_1} \|g_2\|_{q^3_2} \|D_2^{\beta_2}h\|_{L^{s^3_1}(L^{s^3_2})} \ldots \end{align*} \end{theorem}
\subsection{Singular Brascamp-Lieb Inequalities} Whereas Leibniz rule \ref{lb6} can be viewed as a bi-parameter variant of Leibniz rule \ref{lb3} for functions of two variables, it can also be perceived as a Leibniz rule about mixed norms with $L^{\infty}$ norm involved. This perspective is closely related to singular Brascamp-Lieb inequalities through the singular integral formulation of Leibniz rule \ref{lb4}: \begin{theorem} Suppose $K^1_1, K^1_2, K^2$ are Calder$\acute{o}$n-Zygmund kernels on $\mathbb{R}^2$. Let \begin{align*} \displaystyle T_{K^1_1,K^1_2, K^2} (f^x_1,f^x_2,g^y_1,g^y_2,h)(x,y) & := p.v. \int_{\mathbb{R}^7} K^1_1(t_1,t_2) K^1_2(t_3,t_4) f_1(x - t_1) f_2(x-t_2-t_3) \nonumber \\ & \ \ \ \ \ \ \ \cdot K^2(s_1,s_2,s_3) g_1(y - s_1) g_2(y - s_2) h(x - t_4, y - s_3) d\mathbf{t} d\mathbf{s}, \end{align*} Then for $ 1 < p_i, q_i ,s < \infty$, $\frac{1}{p_i}+\frac{1}{q_i} + \frac{1}{s} = \frac{1}{r}, i = 1,2$,
$$ \|T_{K^1_1,K^1_2, K^2} (f^x_1,f^x_2,g^y_1,g^y_2,h)\|_{L^r_x(L^r_y)} \lesssim \|f^x_1\|_{p_1} \|f^x_2\|_{p_2} \|g^y_1\|_{q_1} \|g^y_2\|_{q_2} \|h\|_{L^s_x(L^s_y)}.$$ \end{theorem} The machinery invented for studying Leibniz rule \ref{lb6} incorporates information from different subspaces to obtain information on the entire space, which sheds light on the study of singular variants of Brascamp-Lieb inequalities. One good candidate is a singular integral variant of the Loomis-Whitney inequality:
\begin{problem} Suppose $F_1, F_2, F_3 \in \mathcal{S}(\mathbb{R}^2)$, For $\beta_1, \beta_2, \alpha_1, \alpha_2 > 0$ sufficiently large, what is the full range of estimates for the Leibniz rule concerning $D_1^{\beta_1}D_2^{\beta_2} D_3^{\beta_3}((D_1^{\alpha_1}F_1^{x,y} F_2^{x,y} ) G_1^{y,z} G_2^{y,z} H_1^{z,x} H_2^{z,x})$? \end{problem} \end{comment}
\section{Main Results} \label{section_result}
We state the main results in Theorem \ref{main_theorem} and \ref{main_thm_inf}. Theorem \ref{main_theorem} proves the boundedness when $p_i, q_i$ are strictly between $1$ and infinity whereas Theorem \ref{main_thm_inf} deals with the case when $p_i = \infty$ or $q_j = \infty$ for some $i\neq j$.
\begin{theorem} \label{main_theorem} Suppose $a \in L^{\infty}(\mathbb{R}^4)$, $b\in L^{\infty}(\mathbb{R}^6)$, where $a$ and $b$ are smooth away from $\{(\xi_1,\xi_2) = 0 \} \cup \{(\eta_1,\eta_2) = 0 \}$ and $\{(\xi_1, \xi_2,\xi_3) = 0 \} \cup \{(\eta_1,\eta_2,\eta_3) = 0\}$ respectively and satisfy the following Marcinkiewicz conditions: \begin{align*}
& |\partial^{\alpha_1}_{\xi_1} \partial^{\alpha_2}_{\eta_1} \partial^{\beta_1}_{\xi_2} \partial^{\beta_2}_{\eta_2} a(\xi_1,\eta_1, \xi_2,\eta_2)| \lesssim \frac{1}{|(\xi_1,\xi_2)|^{\alpha_1 + \beta_1}} \frac{1}{|(\eta_1,\eta_2)|^{\alpha_2+\beta_2}}, \nonumber \\
& |\partial^{\bar{\alpha_1}}_{\xi_1} \partial^{\bar{\alpha_2}}_{\eta_1} \partial^{\bar{\beta_1}}_{\xi_2} \partial^{\bar{\beta_2}}_{\eta_2}\partial^{\bar{\gamma_1}}_{\xi_3} \partial^{\bar{\gamma_2}}_{\eta_3}b(\xi_1,\eta_1, \xi_2,\eta_2, \xi_3, \eta_3)| \lesssim \frac{1}{|(\xi_1,\xi_2, \xi_3)|^{\bar{\alpha_1} + \bar{\beta_1}+\bar{\gamma_1}}} \frac{1}{|(\eta_1,\eta_2, \eta_3)|^{\bar{\alpha_2}+\bar{\beta_2}+ \bar{\gamma_2}}} \end{align*} for sufficiently many multi-indices $\alpha_1,\alpha_2,\beta_1,\beta_2, \bar{\alpha_1}, \bar{\alpha_2},\bar{\beta_1},\bar{\beta_2}, \bar{\gamma_1}, \bar{\gamma_2} \geq 0$. For $f_1, f_2, g_1,g_2 \in \mathcal{S}(\mathbb{R})$ and $h \in \mathcal{S}(\mathbb{R}^2)$ where $\mathcal{S}(\mathbb{R})$ and $\mathcal{S}(\mathbb{R}^2)$ denote the Schwartz spaces, define \begin{align} \label{bi_flag} \displaystyle T_{ab}(f_1, f_2, g_1 ,g_2,h)(x,y) := \int_{\mathbb{R}^6} & a(\xi_1,\eta_1,\xi_2,\eta_2) b(\xi_1,\eta_1,\xi_2,\eta_2,\xi_3,\eta_3) \nonumber \\ & \hat{f_1}(\xi_1)\hat{f_2}(\xi_2)\hat{g_1}(\eta_1)\hat{g_2}(\eta_2)\hat{h}(\xi_3,\eta_3) \nonumber \\ & e^{2\pi i x(\xi_1+\xi_2+\xi_3)}e^{2\pi i y(\eta_1+\eta_2+\eta_3)} d \xi_1 d\xi_2 d\xi_3 d\eta_1 d\eta_2 d\eta_3. \end{align} Then for $1< p_1, p_2, q_1, q_2 < \infty, 1 < s \leq \infty$, $r > 0$, $\frac{1}{p_1} + \frac{1}{q_1} + \frac{1}{s} =\frac{1}{p_2} + \frac{1}{q_2} + \frac{1}{s} = \frac{1}{r} $, $T_{ab}$ satisfies the following mapping property $$ T_{ab}: L^{p_1}(\mathbb{R}) \times L^{q_1}(\mathbb{R}) \times L^{p_2}(\mathbb{R}) \times L^{q_2}(\mathbb{R}) \times L^{s}(\mathbb{R}^2) \rightarrow L^{r}(\mathbb{R}^2). $$ \end{theorem}
\begin{theorem} \label{main_thm_inf} Let $T_{ab}$ be defined as (\ref{bi_flag}). Then for $1< p < \infty$, $1 < s \leq \infty$, $r >0$, $\frac{1}{p} + \frac{1}{s} = \frac{1}{r}$, $T_{ab}$ satisfies the following mapping property \begin{align*} T_{ab}: & L^{p}(\mathbb{R}) \times L^{\infty}(\mathbb{R}) \times L^{p}(\mathbb{R}) \times L^{\infty}(\mathbb{R}) \times L^{s}(\mathbb{R}^2) \rightarrow L^{r}(\mathbb{R}^2) \nonumber
\end{align*} where $p_1 = p_2 = p$ as imposed by (\ref{bl_exp}). \end{theorem} \begin{remark}
The cases $(i) q_1 = q_2 < \infty$ and $p_1= p_2= \infty$ $(ii) p_1 = q_2 < \infty$ and $p_2 = q_1 = \infty$ $(iii) q_1 = p_2 < \infty$ and $p_1 = q_2 = \infty$ follows from the same argument by symmetry.
\end{remark}
\subsection{Restricted Weak-Type Estimates} \label{section_result_rw} For the Banach estimates when $r > 1$, H\"older's inequality involving maximal function operator, square function operator and hybrid operators (Definition \ref{def_hybrid}) is sufficient. The argument resembles the Banach estimates for the single-parameter flag paraproduct. The quasi-Banach estimates when $r < 1$ is trickier and requires a careful treatment. In this case, we use multilinear interpolations and reduce the desired estimates specified in Theorem \ref{main_theorem} and Theorem \ref{main_thm_inf} to the following restricted weak-type estimates for the associated multilinear form\footnote{Multilinear form, denoted by $\Lambda$, associated to an n-linear operator $T(f_1, \ldots, f_n)$ is defined as $\Lambda(f_1, \ldots, f_n, f_{n+1}) := \langle T(f_1,\ldots, f_n), f_{n+1}\rangle $.}. \begin{theorem}\label{thm_weak}
Let $T_{ab}$ denote the operator defined in (\ref{bi_flag}). Suppose that $1< p_1, p_2, q_1, q_2 < \infty, 1 < s <2$, $0 < r <1$, $\frac{1}{p_1} + \frac{1}{q_1} + \frac{1}{s} =\frac{1}{p_2} + \frac{1}{q_2} + \frac{1}{s} = \frac{1}{r}$. Then for every measurable set $F_1, F_2, G_1, G_2 \subseteq \mathbb{R}, E \subset \mathbb{R}^2$ of positive and finite measure and every measurable function $|f_1(x)| \leq \chi_{F_1}(x)$, $|f_2(x)| \leq \chi_{F_2}(x)$, $|g_1(y)| \leq \chi_{G_1}(y)$, $|g_2(y)| \leq \chi_{G_2}(y)$, $h \in L^{s}(\mathbb{R}^2)$, there exists $E' \subseteq E$ with $|E'| > |E|/2$ such that the multilinear form associated to $T_{ab}$ satisfies \begin{equation} \label{thm_weak_explicit}
|\Lambda(f_1, f_2, g_1, g_2, h,\chi_{E'}) | \lesssim |F_1|^{\frac{1}{p_1}} |G_1|^{\frac{1}{p_2}} |F_2|^{\frac{1}{q_1}} |G_2|^{\frac{1}{q_2}} \|h\|_{L^{s}(\mathbb{R}^2)}|E|^{\frac{1}{r'}}, \end{equation} where $r'$ represents the conjugate exponent of $r$. \end{theorem} \vskip.15in \begin{theorem} \label{thm_weak_inf} Let $T_{ab}$ denote the operator defined in (\ref{bi_flag}). Suppose that $1< p < \infty$, $1 < s < 2$, $0 < r < 1$, $\frac{1}{p} + \frac{1}{s} = \frac{1}{r}$. Then for every measurable set
$E \subset \mathbb{R}^2$ of positive and finite measure and every measurable function
$f_1,g_1 \in L^p(\mathbb{R})$, $f_2, g_2 \in L^{\infty}(\mathbb{R})$, $h \in L^{s}(\mathbb{R}^2)$, there exists $E' \subseteq E$ with $|E'| > |E|/2$ such that the multilinear form associated to $T_{ab}$ satisfies \begin{equation} \label{thm_weak_inf_explicit}
|\Lambda(f_1, f_2, g_1, g_2, h,\chi_{E'}) | \lesssim \|f_1\|_{L^p(\mathbb{R})} \|f_2\|_{L^{\infty}(\mathbb{R})} \|g_1\|_{L^p(\mathbb{R})} \|g_2\|_{L^{\infty}(\mathbb{R})} \|h\|_{L^s(\mathbb{R}^2)}|E|^{\frac{1}{r'}}, \end{equation} where $r'$ represents the conjugate exponent of $r$. \end{theorem}
\begin{remark}
Theorem \ref{thm_weak} and \ref{thm_weak_inf} hint the necessity of localization and the major subset $E'$ of $E$ is constructed based on the philosophy to localize the operator where it is well-behaved.
\end{remark} The reduction of Theorem \ref{main_theorem} and \ref{main_thm_inf} to Theorem \ref{thm_weak} and \ref{thm_weak_inf} respectively will be postponed to Appendix I. In brief, it depends on the interpolation of multilinear forms described in Lemma 9.6 of \cite{cw} and a tensor product version of Marcinkiewicz interpolation theorem.
\vskip .15in
\begin{comment} \begin{proof}[Steps of Interpolation] \begin{enumerate}[Step 1.]
\item One first freezes $f_1, f_2, g_1, g_2$ where $|f_i| \leq \chi_{F_i}$ and $|g_j| \leq \chi_{G_j}$ for $i, j = 1, 2$. By \end{enumerate} \end{proof} \end{comment}
\begin{comment} \subsection{Tensor Product of Flag Paraproduct and Paraproduct on Restricted Function Spaces} \begin{theorem} Suppose $a \in \mathcal{M}(\mathbb{R}^2)$ and $b,c\in \mathcal{M}(\mathbb{R}^3)$. For $f_1, f_2, g_1,g_2 \in \mathcal{S}(\mathbb{R})$ and $h \in\mathcal{S}(\mathbb{R}^2)$, define \begin{align*} \displaystyle T_{abc}(f^x_1 \otimes g^y_1, f^x_2 \otimes g^y_2,h) := \int_{\mathbb{R}^6} & a(\xi_1,\xi_2)b(\xi_1,\xi_2,\xi_3)c(\eta_1,\eta_2,\eta_3)\hat{f_1}(\xi_1)\hat{f_2}(\xi_2)\hat{g_1}(\eta_1)\hat{g_2}(\eta_2)\hat{h}(\xi_3,\eta_3) \nonumber \\ & e^{2\pi i x(\xi_1+\xi_2+\xi_3)}e^{2\pi i y(\eta_1+\eta_2+\eta_3)} d \xi_1 d\xi_2 d\xi_3 d\eta_1 d\eta_2 d\eta_3. \end{align*} Then \newline (1) for $1< p, q, s < \infty$, $\frac{1}{p} + \frac{1}{q} + \frac{1}{s} = \frac{1}{r}$, $$ T_{abc}: L^{p} \times L^{q} \times L^{s} \rightarrow L^{r} $$ (2) for $1< p,s < \infty$, $\frac{1}{p} + \frac{1}{q} + \frac{1}{s} = \frac{1}{r}$, \begin{align*} T_{abc}: & L^{p} \times L^{\infty} \times L^{s} \rightarrow L^{r} \nonumber \\ & L^{\infty} \times L^{p} \times L^{s} \rightarrow L^{r}. \end{align*} (3) for $1< p,s < \infty$, $\frac{1}{p} + \frac{1}{q} + \frac{1}{s} = \frac{1}{r}$, \begin{align*} T_{abc}: & L_x^{p}(L_y^{\infty}) \times L_x^{\infty}(L^p_y) \times L^{s} \rightarrow L^{r} \nonumber \\ & L_x^{\infty}(L^p_y) \times L_x^{p}(L_y^{\infty}) \times L^{s} \rightarrow L^{r} \nonumber \\ & L_x^{p}(L_y^{\infty}) \times L_x^{p}(L^{\infty}_y) \times L^{s} \rightarrow L^{r} \nonumber \\ & L_x^{\infty}(L^p_y) \times L_x^{\infty}(L_y^{p}) \times L^{s} \rightarrow L^{r} \nonumber \\ \end{align*}. \end{theorem} \end{comment}
\subsection{Application - Leibniz Rule} \label{section_result_lr} A direct corollary of Theorem \ref{main_theorem} is a Leibniz rule which captures the nonlinear interaction of waves coming from transversal directions. In general, Leibniz rules refer to inequalities involving norms of derivatives. The derivatives are defined in terms of Fourier transforms. More precisely, for $\alpha \geq 0$ and $f \in \mathcal{S}(\mathbb{R}^d)$ a Schwartz function in $\mathbb{R}^d$, define the homogeneous derivative of $f$ as \begin{equation*}
D^{\alpha}f := \mathcal{F}^{-1}\left(|\xi|^{\alpha}\widehat{f}(\xi)\right). \end{equation*} Leibniz rules are closely related to boundedness of multilinear operators discussed in Section \ref{section_intro_connection}. For example, the boundedness of one-parameter paraproducts give rise to a Leibniz rule by Kato and Ponce \cite{kp}. For $f, g \in \mathcal{S}(\mathbb{R}^d)$ and $\alpha > 0$ sufficiently large, \begin{equation} \label{lb_para}
\| D^{\alpha} (fg)\|_r \lesssim \|D^{\alpha} f \|_{p_1} \|g \|_{q_1} + \| f \|_{p_2} \|D^{\alpha}g \|_{q_2} \end{equation} with $1 < p_i, q_i < \infty, \frac{1}{p_i}+ \frac{1}{q_i} = \frac{1}{r}, i= 1,2.$ The inequality in (\ref{lb_para}) generalizes the trivial and well-known Leibniz rule when $\alpha = 1$ and states that the derivative for a product of two functions can be dominated by the terms which involve the highest order derivative hitting on one of the functions. The reduction of (\ref{lb_para}) to the boundedness of one-parameter paraproducts is routine (see Section 2 in \cite{cw} for details) and can be applied to other Leibniz rules with their corresponding multilinear operators, including the boundedness of our operator $T_{ab}$ and its Leibniz rule stated in Theorem \ref{lb_main} below. The Leibniz rule stated in Theorem \ref{lb_main} deals with partial derivatives, where the partial derivative of $f \in \mathcal{S}(\mathbb{R}^d)$ is defined, for $(\alpha_1,\ldots, \alpha_d)$ with $\alpha_1, \ldots, \alpha_d \geq 0$, as \begin{equation*}
D_1^{\alpha_1}\cdots D_d^{\alpha_d}f := \mathcal{F}^{-1}\left(|\xi_1|^{\alpha_1} \cdots |\xi_d|^{\alpha_d}\widehat{f}(\xi_1,\ldots, \xi_d)\right). \end{equation*} For the statement of the Leibniz rule, we will adopt the notation $f^x$ for the function $f$ depending on the variable $x$. \begin{theorem} \label{lb_main} Suppose $f_1, f_2 \in \mathcal{S}(\mathbb{R})$, $ g_1, g_2 \in \mathcal{S}(\mathbb{R})$ and $h \in \mathcal{S}(\mathbb{R}^2).$ Then for $\beta_1, \beta_2, \alpha_1, \alpha_2 > 0$ sufficiently large and $1 < p^j_1, p^j_2, q^j_1, q^j_2, s^j \leq \infty$, $r >0$, $(p^j_1, q^j_1), (p^j_2, q^j_2) \neq (\infty, \infty)$, $\frac{1}{p^j_1} + \frac{1}{q^j_1} + \frac{1}{s^j}= \frac{1}{p^j_2} + \frac{1}{q^j_2} + \frac{1}{s^j}= \frac{1}{r} $ for each $j = 1, \ldots, 16 $, \begin{align*}
& \|D_1^{\beta_1} D_2^{\beta_2}(D_1^{\alpha_1}D_2^{\alpha_2}(f_1^x f_2^x g_1^y g_2^y) h^{x,y})\|_{L^r(\mathbb{R}^2)} \nonumber \\
\lesssim & \ \ \text{sum of \ \ }16 \text{\ \ terms of the forms: \ \ }
\nonumber \\ & \|D_1^{\alpha_1+\beta_1}f_1\|_{L^{p^1_1}(\mathbb{R})} \|f_2\|_{L^{q^1_1}(\mathbb{R})} \|D_2^{\alpha_2 + \beta_2}g_1\|_{L^{p^1_2}(\mathbb{R})} \|g_2\|_{L^{q^1_2}(\mathbb{R})} \|h\|_{L^{s^1}(\mathbb{R}^2)} +
\nonumber \\ & \|f_1\|_{L^{p^2_1}(\mathbb{R})} \|D_1^{\alpha+\beta_1}f_2\|_{L^{q^2_1}(\mathbb{R})} \|D_2^{\alpha_2 + \beta_2}g_1\|_{L^{p^2_2}(\mathbb{R})} \|g_2\|_{L^{q^2_2}(\mathbb{R})} \|h\|_{L^{s^2}(\mathbb{R}^2)} +
\nonumber \\ & \|D_1^{\alpha+\beta_1}f_1\|_{L^{p^3_1}(\mathbb{R})} \|f_2\|_{L^{q^3_1}(\mathbb{R})} \|D_2^{\alpha_2}g_1\|_{L^{p^3_2}(\mathbb{R})} \|g_2\|_{L^{q^3_2}(\mathbb{R})} \|D_2^{\beta_2}h\|_{L^{s^3}(\mathbb{R}^2)} + \ldots \end{align*} \end{theorem} \begin{remark} The reasoning for the number ``16'' is that \begin{enumerate} [(i)] \item for $\alpha_1$, there are $2$ possible distributions of highest order derivatives thus yielding 2 terms; \item for $\alpha_2$, there are $2$ terms for the same reason in (i);
\item for $\beta_1$, it can hit $h$ or some function which comes from the dominant terms of $D^{\alpha_1}(f_1 f_2)$ and which have two choices as illustrated in (i), thus generating $2 \times 2 = 4$ terms; \item for $\beta_2$, there would be $4$ terms for the same reason in (iii). \end{enumerate} By summarizing (i)-(iv), one has the count $4 \times 4 = 16$. \end{remark} \begin{remark} As commented in the beginning of this section, $f_1$ and $f_2$ in Theorem \ref{lb_main} can be viewed as waves coming from one direction while $g_1$ and $g_2$ are waves from the orthogonal direction. The presence of $h$, as a generic wave in the plane, makes the interaction nontrivial. \end{remark}
\section{Preliminaries} \label{section_prelim} \subsection{Terminology} \label{section_prelim_term} We will first introduce the notation which will be useful throughout the paper. \begin{definition} \label{bump} Suppose $I \subseteq \mathbb{R}$ is an interval. Then we say a smooth function $\phi$ is a \textit{bump function adapted to $I$} if $$
|\phi^{(l)}(x)| \leq C_l C_M \frac{1}{|I|^l} \frac{1}{\big(1+\frac{|x-x_I|}{|I|}\big)^M} $$ for sufficiently many derivatives $l$, where $x_I$ denotes the center of the interval $I$ and $M$ is a large positive number. Moreover, suppose $\mathcal{I}$ is a collection of dyadic intervals and if for any $I \in \mathcal{I}$, $\phi_I$ is an $L^2$-normalized bump function adapted to $I$, then $(\phi_I)_{I \in \mathcal{I}}$ is denoted by \textit{a family of $L^2$-normalized adapted bump functions}. \end{definition}
\begin{definition} \label{def_lacunary} The family of $L^2$-normalized adapted bump functions $(\phi_I)_{I \in \mathcal{I}}$ is \textit{lacunary} if and only if for every $ I \in \mathcal{I}$, $$
\text{supp}\ \ \widehat{\phi_I} \subseteq [-4|I|^{-1}, -\frac{1}{4}|I|^{-1}] \cup [\frac{1}{4}|I|^{-1}, 4|I|^{-1}]. $$ A family of $L^2$-normalized bump functions $(\phi_I)_{I \in \mathcal{I}}$ is \textit{non-lacunary} if and only if for every $ I \in \mathcal{I}$, $$
\text{supp}\ \ \widehat{\phi_I} \subseteq [-\frac{1}{4}|I|^{-1}, \frac{1}{4}|I|^{-1}]. $$ We usually denote bump functions in lacunary family by $(\psi_I)_I$ and those in non-lacunary family by $(\varphi_I)_I$.
\end{definition}
We will now define cutoff functions correspond to adapted bump functions given in Definition \ref{bump}, which can be viewed as simplified variants of bump functions.
\begin{definition} \label{bump_walsh} Define $$ \psi^H(x) := \begin{cases} 1 \ \ \text{for}\ \ x \in [0,\frac{1}{2})\\ -1 \ \ \text{for}\ \ x \in [\frac{1}{2},1).\\ \end{cases} $$ Let $I := [n2^{k},(n+1)\cdot2^k)$ denote a dyadic interval. Then the Haar wavelet on $I$ is defined as $$ \psi^H_I(x) := 2^{-\frac{k}{2}}\psi^H(2^{-k}x-n). $$ The $L^2$-normalized indicator function on $I$ is expressed as $$
\varphi^H_I(x) := |I|^{-\frac{1}{2}}\chi_{I}(x). $$ We adopt the notation $\phi^{H}_I$ for both $\psi^H_I$ and $\varphi_I^H$ and differentiate them by referring $\psi_I^H$ as the \textit{lacunary} case of the cutoff function $\phi_I^H$ and $\varphi_I^H$ as the \textit{non-lacunary} case. We thus have an analogous and discontinuous variant of adapted bump functions. In particular, Haar wavelets correspond to lacunary (or equivalently having Fourier support away from the origin) family of bump functions and $L^2$-normalized indicator functions correspond to non-lacunary family of bump functions. \end{definition}
We shall remark that the boundedness of the multilinear form described in Theorem \ref{thm_weak} and \ref{thm_weak_inf} can be reduced to the estimates of discrete model operators which are defined in terms of bump functions of the form specified in Definition \ref{bump}. The precise statements are included in Theorem \ref{thm_weak_mod} and \ref{thm_weak_inf_mod} and the proof is discussed in Appendix II. However, we will first study the simplified model operators with the general bump functions replaced by Haar wavelets and indicator functions defined in Definition \ref{bump_walsh}. The arguments for the simplified models capture the main challenges while avoiding some technical aspects. We will leave the generalization and the treatment of the technical details to Section \ref{section_fourier}. The simplified models would be denoted as Haar models and we will highlight the occasions when the Haar models are considered.
\ \begin{comment} Suppose $\mathcal{I}$ is a collection of dyadic intervals. Then a family of $L^2$-normalized bump functions $(\phi_I)_{I \in \mathcal{I}}$ is \textit{non-lacunary} if and only if for every $ I \in \mathcal{I}$, $$
\text{supp}\ \ \widehat{\phi_I} \subseteq [-4|I|^{-1}, 4|I|^{-1}] $$ A family of $L^2$-normalized bump functions $(\phi_I)_{I \in \mathcal{I}}$ is \textit{lacunary} if and only if for every $ I \in \mathcal{I}$, $$
\text{supp}\ \ \widehat{\phi_I} \subseteq [-4|I|^{-1}, \frac{1}{4}|I|^{-1}] \cup [\frac{1}{4}|I|^{-1}, 4|I|^{-1}] $$ We usually denote bump functions in non-lacunary family by $(\varphi_I)_I$ and those in lacunary family by $(\psi_I)_I$. \end{comment}
\subsection{Useful Operators - Definitions and Theorems} \label{section_prelim_useful_op} We also give explicit definitions for the Hardy-Littlewood maximal function, the discretized Littlewood-Paley square function and the hybrid square-and-maximal functions that will appear naturally in the argument. \begin{definition} The \textit{Hardy-Littlewood maximal operator} $M$ is defined as $$
Mf(\vec{x}) = \sup_{\vec{x} \in B} \frac{1}{|B|} \int_{B}|f(\vec{u})|d\vec{u} $$ where the supremum is taken over all open balls $B \subseteq \mathbb{R}^d$ containing $\vec{x}$. \end{definition}
\begin{definition} Suppose $\mathcal{I}$ is a finite family of dyadic intervals and $(\psi_I)_I$ a lacunary family of $L^2$-normalized bump functions. The \textit{discretized Littlewood-Paley square function operator} $S$ is defined as
$$
Sf(x) = \bigg(\sum_{I \in \mathcal{I}}\frac{|\langle f, \psi_I\rangle|^2 }{|I|}\chi_{I}(x)\bigg)^{\frac{1}{2}}.
$$
\end{definition}
\begin{definition} \label{def_hybrid} Suppose $\mathcal{R}$ is a finite collection of dyadic rectangles. Let $(\phi_R)_{R \in \mathcal{R}}$ denote the family of $L^2$-normalized bump functions with $\phi_R = \phi_I \otimes \phi_J$ where $R= I \times J$. Let $(\phi^H_R)_{R \in \mathcal{R}}$ denote the family of cutoff functions with $\phi_R^H = \phi^H_I \otimes \phi^H_J$ where $R= I \times J$. \begin{enumerate} \item the \textit{double square function operator} $SS$ is defined as $$
\displaystyle SSh(x,y) = \bigg(\sum_{I \times J } \frac{|\langle h, \psi_{I} \otimes \psi_J \rangle|^2 }{|I||J|} \chi_{I \times J} (x,y)\bigg)^{\frac{1}{2}} $$ and the \textit{Haar double square function operator} $(SS)^H$ is defined as $$
\displaystyle (SS)^Hh(x,y) = \bigg(\sum_{I \times J } \frac{|\langle h, \psi^H_{I} \otimes \psi^H_J \rangle|^2 }{|I||J|} \chi_{I \times J} (x,y)\bigg)^{\frac{1}{2}}; $$
\item the \textit{hybrid maximal-square operator} $MS$ is defined as $$
MSh(x,y) = \sup_{I}\frac{1}{|I|^{\frac{1}{2}}} \bigg(\sum_{J} \frac{|\langle h, \varphi_I \otimes \psi_J \rangle|^2}{|J|} \chi_{J}(y)\bigg)^{\frac{1}{2}}\chi_I(x) $$ and the \textit{Haar hybrid maximal-square operator} $(MS)^H$ is defined as $$
(MS)^Hh(x,y) = \sup_{I}\frac{1}{|I|^{\frac{1}{2}}} \bigg(\sum_{J} \frac{|\langle h, \varphi^H_I \otimes \psi^H_J \rangle|^2}{|J|} \chi_{J}(y)\bigg)^{\frac{1}{2}}\chi_I(x); $$
\item the \textit{hybrid square-maximal operator} $SM$ is defined as
$$ \displaystyle SMh(x,y) = \bigg(\sum_{I} \frac{\big(\sup_{J}\frac{|\langle h,\psi_I \otimes \varphi_J \rangle|}{|J|}\chi_J(y) \big)}{|I|}\chi_{I}(x)\bigg)^{\frac{1}{2}} $$ and the \textit{Haar hybrid square-maximal operator} $(SM)^H$ is defined as $$ \displaystyle
(SM)^Hh(x,y) = \bigg(\sum_{I} \frac{\big(\sup_{J}\frac{|\langle h,\psi^H_I \otimes \varphi^H_J \rangle|}{|J|}\chi_J(y) \big)}{|I|}\chi_{I}(x)\bigg)^{\frac{1}{2}}; $$
\item the \textit{double maximal function} $MM$ is defined as $$
MM h(x,y) = \sup_{(x,y) \in R} \frac{1}{|R|}\int_{R}|h(s,t)| ds dt, $$ where the supremum is taken over all dyadic rectangles in $\mathcal{R}$ containing $(x,y)$. \end{enumerate} \end{definition}
The following theorem about the operators defined above is used frequently in the arguments. The proof of the theorem and other contexts where the hybrid operators appear can be found in \cite{cw}, \cite{cf} and \cite{fs}. \begin{theorem} \label{maximal-square} \noindent \begin{enumerate} \item $M$ is bounded in $L^{p}(\mathbb{R}^{d})$ for $1< p \leq \infty$ and $M: L^{1} \longrightarrow L^{1,\infty}$. \item $S$ is bounded in $L^{p}(\mathbb{R})$ for $1< p < \infty$. \item The operators $SS, (SS)^H, MS, (MS)^H, SM, (SM)^H, MM$ are bounded in $L^{p}(\mathbb{R}^2)$ for $1 < p < \infty$. \end{enumerate} \end{theorem} \vskip.25in
\section{Discrete Model Operators} \label{section_discrete_model}
In this section, we will introduce the discrete model operators whose boundedness implies the estimates specified in Theorem \ref{thm_weak} and Theorem \ref{thm_weak_inf}. The reduction procedure follows from a routine treatment which has been discussed in \cite{cw}. The details will be enclosed in Appendix II for the sake of completeness. The model operators are usually more desirable because they are more ``localizable''. We will first introduce the definition of some ``pararproduct''-type operators - all of which will be useful in the proof of the main theorems.
\begin{definition} \label{B_definition} Let $\mathcal{Q}$ denote a finite collection of dyadic intervals. Suppose that $(\phi^i_Q)_{Q \in \mathcal{Q}}$ for $i = 1, 2, 3$ are families of $L^2$-normalized adapted bump functions (Definition \ref{bump}) such that at least two families are lacunary (Definition \ref{def_lacunary}). We define the bilinear operator $B_{\mathcal{Q}}$ by \begin{align} \label{B_global_definition}
& B_{\mathcal{Q}}(v_1,v_2) := \sum_{Q \in \mathcal{Q}}\frac{1}{|Q|^{\frac{1}{2}}}\langle v_1, \phi_Q^1 \rangle \langle v_2, \phi_Q^2 \rangle \phi_Q^3. \end{align} Fix a dyadic interval $P$ and a non-negative number $\#$. We define localized bilinear operators $B_{\mathcal{Q}, P}$ and $B^{\#}_{\mathcal{Q}, P}$ by \begin{align}
B_{\mathcal{Q}, P}(v_1,v_2) :=& \sum_{\substack{Q \in \mathcal{Q} \\ |Q| \geq |P|}}\frac{1}{|Q|^{\frac{1}{2}}}\langle v_1, \phi_Q^1 \rangle \langle v_2, \phi_Q^2 \rangle \phi_Q^3, \label{B_local0_haar} \\
B_{\mathcal{Q}, P}^{\#}(v_1,v_2) :=& \sum_{\substack{Q \in \mathcal{Q} \\ |Q| \sim 2^{\#}|P|}}\frac{1}{|Q|^{\frac{1}{2}}}\langle v_1, \phi_Q^1 \rangle \langle v_2, \phi_Q^2 \rangle \phi_Q^3. \label{B_fixed_fourier} \end{align} \end{definition} We define the analogous localized bilinear operators in the Haar model as follows. \begin{definition} \label{B_definition_haar} Let $\mathcal{Q}$ denote a finite collection of dyadic intervals. Suppose that $(\phi^i_Q)_{Q \in \mathcal{Q}}$ for $i = 1, 2$ are families of $L^2$-normalized adapted bump functions and $(\phi^{3,H}_Q)_{Q \in \mathcal{Q}}$ is a family of $L^2$-normalized cutoff functions (Definition \ref{bump_walsh}) such that at least two of the three families are lacunary. Fix a dyadic interval $P$ and a non-negative number $\#$. We define the bilinear operators \begin{align}
B_{\mathcal{Q}}^H(v_1,v_2) :=& \sum_{\substack{Q \in \mathcal{Q} \\}}\frac{1}{|Q|^{\frac{1}{2}}}\langle v_1, \phi_Q^1 \rangle \langle v_2, \phi_Q^2 \rangle \phi_Q^{3,H}, \label{B_global_haar}\\
B_{\mathcal{Q}, P}^H(v_1,v_2) :=& \sum_{\substack{Q \in \mathcal{Q} \\ |Q| \geq |P|}}\frac{1}{|Q|^{\frac{1}{2}}}\langle v_1, \phi_Q^1 \rangle \langle v_2, \phi_Q^2 \rangle \phi_Q^{3,H}, \label{B_local_definition_haar} \\
B_{\mathcal{Q}, P}^{\#,H}(v_1,v_2) :=& \sum_{\substack{Q \in \mathcal{Q} \\ |Q| \sim 2^{\#}|P|}}\frac{1}{|Q|^{\frac{1}{2}}}\langle v_1, \phi_Q^1 \rangle \langle v_2, \phi_Q^2 \rangle \phi_Q^{3,H}. \label{B_local_definition_haar_fix_scale} \end{align}
\end{definition}
The discrete model operators are defined as follows.
\begin{definition} \label{discrete_model_op} Suppose $\mathcal{I}, \mathcal{J}, \mathcal{K}$, $\mathcal{L}$ are finite collections of dyadic intervals. Suppose $\displaystyle(\phi^i_I)_{I\in \mathcal{I}}$, $ (\phi^j_J)_{J \in \mathcal{J}}$, $i, j,
= 1, 2, 3$ are families of $L^2$-normalized adapted bump functions.
We further assume that at least two families of $(\phi^i_I)_{I\in \mathcal{I}}$ for $i = 1, 2, 3$ are lacunary. Same conditions are assumed for families $ (\phi^j_J)_{J \in \mathcal{J}}$ for $j = 1, 2, 3$.
In some models, we specify the lacunary and non-lacunary families by explicitly denoting the functions in the lacunary family as $\psi$ and those in the non-lacunary family as $\varphi$. Let $\#_1, \#_2$ denote some positive integers. Define \newline \begin{enumerate} \item
$$ \Pi_{\text{flag}^0 \otimes \text{paraproduct}}(f_1, f_2, g_1, g_2, h)(x,y) := \displaystyle \sum_{I \times J \in \mathcal{I} \times \mathcal{J}} \frac{1}{|I|^{\frac{1}{2}} |J|} \langle B_{\mathcal{K},I}(f_1,f_2),\varphi_I^1 \rangle \langle g_1,\phi^1_J \rangle \langle g_2, \phi^2_J \rangle \langle h, \psi_I^{2} \otimes \phi_{J}^2 \rangle \psi_I^{3} \otimes \phi_{J}^3(x,y);$$
\item
$$ \Pi_{\text{flag}^{\#_1} \otimes \text{paraproduct}}(f_1, f_2, g_1, g_2, h)(x,y) := \displaystyle \sum_{I \times J \in \mathcal{I} \times \mathcal{J}} \frac{1}{|I|^{\frac{1}{2}} |J|} \langle B^{\#_1}_{\mathcal{K},I}(f_1,f_2),\varphi_I^1 \rangle \langle g_1,\phi^1_J \rangle \langle g_2, \phi^2_J \rangle \langle h, \psi_I^{2} \otimes \phi_{J}^2 \rangle \psi_I^{3}\otimes \phi_{J}^3(x,y) ;$$
\item
$$ \Pi_{\text{flag}^0 \otimes \text{flag}^0}(f_1, f_2, g_1, g_2, h)(x,y) := \displaystyle \sum_{I \times J \in \mathcal{I} \times \mathcal{J}} \frac{1}{|I|^{\frac{1}{2}} |J|^{\frac{1}{2}}} \langle B_{\mathcal{K},I}(f_1,f_2),\varphi_I^1 \rangle \langle B_{\mathcal{L},J}(g_1, g_2), \varphi_J^1 \rangle \langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle \psi_I^{3} \otimes \psi_J^{3}(x,y);$$
\item
$$ \Pi_{\text{flag}^0 \otimes \text{flag}^{\#_2}}(f_1, f_2, g_1, g_2, h)(x,y) := \displaystyle \sum_{I \times J \in \mathcal{I} \times \mathcal{J}} \frac{1}{|I|^{\frac{1}{2}} |J|^{\frac{1}{2}}} \langle B_{\mathcal{K},I}(f_1,f_2),\varphi_I^1 \rangle \langle B_{\mathcal{L},J}^{\#_2}(g_1, g_2), \varphi_J^1 \rangle \langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle \psi_I^{3} \otimes \psi_J^{3}(x,y);$$
\item
$$\Pi_{\text{flag}^{\#_1}\otimes \text{flag}^{\#_2}}(f_1, f_2, g_1, g_2, h)(x,y) := \displaystyle \sum_{I \times J \in \mathcal{I} \times \mathcal{J}} \frac{1}{|I|^{\frac{1}{2}} |J|^{\frac{1}{2}}} \langle B^{\#_1}_{\mathcal{K},I}(f_1,f_2),\varphi_I^1 \rangle \langle B^{\#_2}_{\mathcal{L},J}(g_1, g_2), \varphi_J^1 \rangle \langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle \psi_I^{3} \otimes \psi_J^{3} (x,y).$$
\end{enumerate}
\end{definition}
The mapping properties of the discrete model operators are stated as follows.
\begin{theorem} \label{thm_weak_mod} Let $\Pi_{\text{flag}^0 \otimes \text{paraproduct}}$, $ \Pi_{\text{flag}^{\#_1} \otimes \text{paraproduct}}$, $\Pi_{\text{flag}^0 \otimes \text{flag}^0}$, $\Pi_{\text{flag}^0 \otimes \text{flag}^{\#_2}}$ and $\Pi_{\text{flag}^{\#_1}\otimes \text{flag}^{\#_2}}$ be multilinear operators specified in Definition \ref{discrete_model_op}. Then all of them satisfy the mapping property stated in Theorem \ref{thm_weak}, where the constants are independent of $\#_1,\#_2$ and the cardinalities of the collections $\mathcal{I}, \mathcal{J}, \mathcal{K}$ and $\mathcal{L}$.
\end{theorem}
\begin{theorem} \label{thm_weak_inf_mod} Let $\Pi_{\text{flag}^0 \otimes \text{paraproduct}}$, $ \Pi_{\text{flag}^{\#_1} \otimes \text{paraproduct}}$, $\Pi_{\text{flag}^0 \otimes \text{flag}^0}$, $\Pi_{\text{flag}^0 \otimes \text{flag}^{\#_2}}$ and $\Pi_{\text{flag}^{\#_1}\otimes \text{flag}^{\#_2}}$ be multilinear operators specified in Definition \ref{discrete_model_op}. Then all of them satisfy the mapping property stated in Theorem \ref{thm_weak_inf}, where the constants are independent of $\#_1,\#_2$ and the cardinalities of the collections $\mathcal{I}, \mathcal{J}, \mathcal{K}$ and $\mathcal{L}$. \end{theorem}
\begin{comment} We will further specify some models for $\Pi_{\text{flag}^0 \otimes \text{flag}^0}$ generated from different lacunary and non-lacunary positions. The main difference of the following two models are that $\Pi^1_{\text{flag}^0 \otimes \text{flag}^0}$ has $(\phi_K^3)_K$ as a lacunary family whereas $\Pi^0_{\text{flag}^0 \otimes \text{flag}^0}$ has $(\phi_K^3)_K$ as a non-lacunary family. Our plan would be to prove estimates for $\Pi^1_{\text{flag}^0 \otimes \text{flag}^0}$ and modify the argument to study $\Pi^1_{\text{flag}^0 \otimes \text{flag}^0}$. It turns out this strategy can be applied to all other models with $(\phi_K^3)_K$ being specified as lacunary or non-lacunary family.
\begin{enumerate}[Model 1:] \item $$
\Pi^1_{\text{flag}^0 \otimes \text{flag}^0}(f_1, f_2, g_1, g_2, h) := \displaystyle \sum_{I \times J \in \mathcal{I} \times \mathcal{J}} \frac{1}{|I|^{\frac{1}{2}} |J|^{\frac{1}{2}}} \langle B_I(f_1,f_2),\varphi_I^1 \rangle \langle \tilde{B_J}(g_1, g_2), \varphi_J^1 \rangle \langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle \psi_I^{3} \otimes \psi_J^{3} $$ where
$$B_I(f_1,f_2)(x) := \displaystyle \sum_{K \in \mathcal{K}:|K| \geq |I|} \frac{1}{|K|^{\frac{1}{2}}}\langle f_1, \varphi_K^1 \rangle \langle f_2, \psi_K^2 \rangle \psi_K^3(x),$$
$$\tilde{B}_J(g_1,g_2)(y) := \displaystyle \sum_{L \in \mathcal{L}:|L| \geq |J|} \frac{1}{|L|^{\frac{1}{2}}}\langle g_1, \varphi_L^1 \rangle \langle g_2, \psi_L^2 \rangle \psi_L^3(y). $$
\item $$
\Pi^0_{\text{flag}^0 \otimes \text{flag}^0}(f_1, f_2, g_1, g_2, h) := \displaystyle \sum_{I \times J \in \mathcal{I} \times \mathcal{J}} \frac{1}{|I|^{\frac{1}{2}} |J|^{\frac{1}{2}}} \langle B_I(f_1,f_2),\varphi_I^1 \rangle \langle \tilde{B_J}(g_1, g_2), \varphi_J^1 \rangle \langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle \psi_I^{3} \otimes \psi_J^{3} $$ where
$$B_I(f_1,f_2)(x) := \displaystyle \sum_{K \in \mathcal{K}:|K| \geq |I|} \frac{1}{|K|^{\frac{1}{2}}}\langle f_1, \psi_K^1 \rangle \langle f_2, \psi_K^2 \rangle \varphi_K^3(x),$$
$$\tilde{B}_J(g_1,g_2)(y) := \displaystyle \sum_{L \in \mathcal{L}:|L| \geq |J|} \frac{1}{|L|^{\frac{1}{2}}}\langle g_1, \psi_L^1 \rangle \langle g_2, \psi_L^2 \rangle \varphi_L^3(y). $$ \end{enumerate} \end{comment}
The following sections are devoted to the proofs of Theorem \ref{thm_weak_mod} and \ref{thm_weak_inf_mod} which would imply Theorem \ref{thm_weak} and \ref{thm_weak_inf}. We will mainly focus on discrete model operators defined in $(3)$ (Section \ref{section_thm_haar}) and $(5)$ (Section \ref{section_thm_haar_fixed}), whose arguments consist of all the essential tools that are needed for other discrete models.
\begin{comment} \begin{align*} f_1\otimes g_1(x,y) f_2 \otimes g_2(x,y)h(x,y) & = \sum_{\substack{k_1,k_2,k_3 \\ l_1,l_2,l_3}} f_1*\psi_{k_1}(x) g_1 * \psi_{l_1}(y) f_2 * \psi_{k_2}(x) g_2 * \psi_{l_2}(y) h * \psi_{k_3} \otimes \psi_{l_3}(x,y) \nonumber \\& = \sum_{\substack{k_1 \ll k_2,k_2 \approx k_3 \\ l_1,l_2,l_3}} + \sum_{\substack{k_1 \ll k_2,k_3 \ll k_2 \\ l_1,l_2,l_3}} + \sum_{\substack{k_1 \ll k_2 \ll k_3 \\ l_1,l_2,l_3}} + \nonumber \\& \ \ \ \sum_{\substack{k_2 \ll k_1,k_1 \approx k_3 \\ l_1,l_2,l_3}} + \sum_{\substack{k_2 \ll k_1,k_3 \ll k_1 \\ l_1,l_2,l_3}} + \sum_{\substack{k_2 \ll k_1 \ll k_3 \\ l_1,l_2,l_3}} + \nonumber \\& \ \ \ \sum_{\substack{k_1 \approx k_2,k_2 \approx k_3 \\ l_1,l_2,l_3}} + \sum_{\substack{k_1 \approx k_2,k_3 \ll k_2 \\ l_1,l_2,l_3}} + \sum_{\substack{k_1 \approx k_2 \ll k_3 \\ l_1,l_2,l_3}} \end{align*} The same separation of cases for indices can be applied to $l_1, l_2, l_3$ in $y$-variables. One simple case that can be reduced to a trilinear bi-parameter paraproduct is \begin{align} & \sum_{\substack{k_1 \ll k_2,k_3 \ll k_2 \\ l_1 \ll l_2, l_3 \ll l_2}} f_1*\psi_{k_1}(x) g_1 * \psi_{l_1}(y) f_2 * \psi_{k_2}(x) g_2 * \psi_{l_2}(y) h * \psi_{k_3} \otimes \psi_{l_3}(x,y) \nonumber \\& = \sum_{k_2,l_2}(f_1*\varphi_{k_2}) (f_2 * \psi_{k_2}) (g_1 * \varphi_{l_2}) (g_2 * \psi_{l_2}) (h * \varphi_{k_2} \otimes \varphi_{l_2}) * \tilde{\psi}_{k_2} \otimes \tilde{\psi}_{l_2} (x,y) \end{align} By similar computation, one can see that all cases other than the ones involving $k_1 \ll k_2 \ll k_3$, $k_2 \ll k_1 \ll k_3$, $k_1 \approx k_2 \ll k_3$ and/or $l_1 \ll l_2 \ll l_3$, $l_2 \ll l_l \ll l_3$, $l_1 \approx l_2 \ll l_3$ can be reduced to trilinear bi-parameter paraproducts. For those tricky cases, \begin{align} & \sum_{\substack{k_1 \ll k_2 \ll k_3 \\ l_1 \ll l_2 \ll l_3}} f_1*\psi_{k_1}(x) g_1 * \psi_{l_1}(y) f_2 * \psi_{k_2}(x) g_2 * \psi_{l_2}(y) h * \psi_{k_3} \otimes \psi_{l_3}(x,y) \nonumber \\& = \sum_{\substack{k_2 \ll k_3\\l_2 \ll l_3}} f_1 * \varphi_{k_2}(x) f_2 * \psi_{k_2}(x) g_1 * \varphi_{l_2}(y) g_2 * \psi_{l_2}(y) h * \psi_{k_3} \otimes \psi_{l_3}(x,y) \nonumber \\& = \sum_{\substack{k_2 \ll k_3\\l_2 \ll l_3}} (f_1 * \varphi_{k_2})(f_2 * \psi_{k_2})*\tilde{\psi}_{k_2} (x) \cdot (g_1 * \varphi_{l_2}) (g_2 * \psi_{l_2})* \tilde{\psi}_{l_2}(y) \cdot h * \psi_{k_3} \otimes \psi_{l_3}(x,y) \nonumber \\& = \sum_{\substack{k_2 \ll k_3\\l_2 \ll l_3}} \Big((f_1 * \varphi_{k_2})(f_2 * \psi_{k_2})*\tilde{\psi}_{k_2}\Big) *\varphi_{k_3} (x) \cdot \Big((g_1 * \varphi_{l_2}) (g_2 * \psi_{l_2})* \tilde{\psi}_{l_2}\Big) * \varphi_{l_3} (y) \cdot h * \psi_{k_3} \otimes \psi_{l_3}(x,y) \nonumber \\& = \sum_{\substack{k_2 \ll k_3\\l_2 \ll l_3}}\bigg( \Big((f_1 * \varphi_{k_2})(f_2 * \psi_{k_2})*\tilde{\psi}_{k_2}\Big) *\varphi_{k_3} \cdot \Big((g_1 * \varphi_{l_2}) (g_2 * \psi_{l_2})* \tilde{\psi}_{l_2}\Big) * \varphi_{l_3} \cdot \Big(h * \psi_{k_3} \otimes \psi_{l_3}\Big) \bigg) * \tilde{\tilde{\psi}}_{k_3} \otimes \tilde{\tilde{\psi}}_{l_3} (x,y) \end{align}
\subsubsection{Case I: Bi-parameter paraproducts} Now we will first consider the simple case and explain how to mollify ... \begin{align*} &D_1^{\alpha_1} D_2^{\alpha_2}\Big(\sum_{\substack{k_1,k_2:k_1\ll k_2,k_3 \ll k_2 \\ l_1,l_2: l_1 \ll l_2,l_3 \ll l_2}} f_1*\psi_{k_1}(x) g_1 * \psi_{l_1}(y) f_2 * \psi_{k_2}(x) g_2 * \psi_{l_2}(y) \Big) \nonumber \\&= D_1^{\alpha_1}D_2^{\alpha_2}\Big(\sum_{\substack{k_2:k_2 \ll k_3 \\ l_2: l_2 \ll l_3}}f_1*\varphi_{k_2}(x) g_1 * \varphi_{l_2}(y) f_2 * \psi_{k_2}(x) g_2 * \psi_{l_2}(y)\Big) \nonumber \\& = D_1^{\alpha_1}D_2^{\alpha_2}\Big(\sum_{\substack{k_2:k_2 \ll k_3 \\ l_2: l_2 \ll l_3}}\big(f_1*\varphi_{k_2} \cdot f_2 * \psi_{k_2}\big) *\tilde{\tilde{ \psi}}_{k_2} (x) \big(g_1 * \varphi_{l_2} \cdot g_2 * \psi_{l_2} \big) * \tilde{\tilde{\psi}}_{l_2}(y) \Big) \nonumber \\& = \sum_{\substack{k_2:k_2 \ll k_3 \\ l_2: l_2 \ll l_3}}\big(f_1*\varphi_{k_2} \cdot f_2 * \psi_{k_2}\big) * D_1^{\alpha_1}\tilde{\tilde{\psi}}_{k_2} (x) \big(g_1 * \varphi_{l_2} \cdot g_2 * \psi_{l_2} \big) * D_2^{\alpha_2}\tilde{\tilde{\psi}}_{l_2}(y) \nonumber \\& = \sum_{\substack{k_2:k_2 \ll k_3 \\ l_2: l_2 \ll l_3}}\big(f_1*\varphi_{k_2} \cdot f_2 * D_1^{\alpha_1}\tilde{\tilde{\tilde{\tilde{\psi}}}}_{k_2}\big) * \tilde{\tilde{\tilde{\psi}}}_{k_2} (x) \big(g_1 * \varphi_{l_2} \cdot g_2 * D_2^{\alpha_2}\tilde{\tilde{\tilde{\tilde{\psi}}}}_{l_2} \big) * \tilde{\tilde{\tilde{\psi}}}_{l_2}(y) \nonumber \\& = \sum_{\substack{k_2:k_2 \ll k_3 \\ l_2: l_2 \ll l_3}}\big(f_1*\varphi_{k_2} \cdot D_1^{\alpha_1}f_2 * \tilde{\tilde{\tilde{\tilde{\psi}}}}_{k_2}\big) * \tilde{\tilde{\tilde{\psi}}}_{k_2} (x) \big(g_1 * \varphi_{l_2} \cdot D_2^{\alpha_2}g_2 * \tilde{\tilde{\tilde{\tilde{\psi}}}}_{l_2} \big) * \tilde{\tilde{\tilde{\psi}}}_{l_2}(y) \nonumber \\& = \sum_{\substack{k_2:k_2 \ll k_3 \\ l_2: l_2 \ll l_3}} f_1*\varphi_{k_2}(x) D_1^{\alpha_1}f_2 * \tilde{\tilde{\tilde{\tilde{\psi}}}}_{k_2}(x) g_1 * \varphi_{l_2}(y) D_2^{\alpha_2}g_2 * \tilde{\tilde{\tilde{\tilde{\psi}}}}_{l_2} (y) \end{align*} Now \begin{align*} & D_1^{\beta_1}D_2^{\beta_2}\Big(\sum_{\substack{k_3\\ l_3}} D_1^{\alpha_1}D_2^{\alpha_2}\big(\sum_{\substack{k_2:k_2 \ll k_3 \\ l_2: l_2 \ll l_3}}f_1*\varphi_{k_2}(x) g_1 * \varphi_{l_2}(y) f_2 * \psi_{k_2}(x) g_2 * \psi_{l_2}(y)\big) h* \psi_{k_3} \otimes \psi_{l_3}(x,y) \Big) \nonumber \\& = D_1^{\beta_1}D_2^{\beta_2} \Big( \sum_{\substack{k_3\\ l_3}}\sum_{\substack{k_2:k_2 \ll k_3 \\ l_2: l_2 \ll l_3}} f_1*\varphi_{k_2}(x) D_1^{\alpha_1}f_2 * \tilde{\tilde{\tilde{\tilde{\psi}}}}_{k_2}(x) g_1 * \varphi_{l_2}(y) D_2^{\alpha_2}g_2 * \tilde{\tilde{\tilde{\tilde{\psi}}}}_{l_2} (y) h* \psi_{k_3} \otimes \psi_{l_3}(x,y)\Big) \nonumber \\& = D_1^{\beta_1}D_2^{\beta_2} \Big(\sum_{k_2,l_2} f_1*\varphi_{k_2}(x) D_1^{\alpha_1}f_2 * \tilde{\tilde{\tilde{\tilde{\psi}}}}_{k_2}(x) g_1 * \varphi_{l_2}(y) D_2^{\alpha_2}g_2 * \tilde{\tilde{\tilde{\tilde{\psi}}}}_{l_2} (y) h* \varphi_{k_2} \otimes \varphi_{l_2}(x,y)\Big) \nonumber \\& = D_1^{\beta_1}D_2^{\beta_2} \Big( \sum_{k_2,l_2} (f_1*\varphi_{k_2}) (D_1^{\alpha_1}f_2 * \tilde{\tilde{\tilde{\tilde{\psi}}}}_{k_2}) (g_1 * \varphi_{l_2}) (D_2^{\alpha_2}g_2 * \tilde{\tilde{\tilde{\tilde{\psi}}}}_{l_2}) (h* \varphi_{k_2} \otimes \varphi_{l_2}) * \tilde{\tilde{\tilde{\tilde{\tilde {\psi}}}}}_{k_2} \otimes \tilde{\tilde{\tilde{\tilde{\tilde {\psi_{l_2}}}}}} (x,y) \Big) \nonumber \\& = \sum_{k_2,l_2} (f_1*\varphi_{k_2}) (D_1^{\alpha_1}f_2 * \tilde{\tilde{\tilde{\tilde{\psi}}}}_{k_2}) (g_1 * \varphi_{l_2}) (D_2^{\alpha_2}g_2 * \tilde{\tilde{\tilde{\tilde{\psi}}}}_{l_2}) (h* \varphi_{k_2} \otimes \varphi_{l_2}) * D_1^{\beta_1}\tilde{\tilde{\tilde{\tilde{\tilde {\psi}}}}}_{k_2} \otimes D_2^{\beta_2} \tilde{\tilde{\tilde{\tilde{\tilde {\psi_{l_2}}}}}} (x,y) \nonumber \\& = \sum_{k_2,l_2} (f_1*\varphi_{k_2}) (D_1^{\alpha_1}f_2 *D_1^{\beta_1} \tilde{\tilde{\tilde{\tilde{\tilde{\tilde{\tilde{\psi}}}}}}}_{k_2}) (g_1 * \varphi_{l_2}) (D_2^{\alpha_2}g_2 * D_2^{\beta_2} \tilde{\tilde{\tilde{\tilde{\tilde{\tilde{\tilde{\psi}}}}}}}_{l_2}) (h* \varphi_{k_2} \otimes \varphi_{l_2}) * \tilde{\tilde{\tilde{\tilde{\tilde{\tilde {\psi}}}}}}_{k_2} \otimes \tilde{\tilde{\tilde{\tilde{\tilde{\tilde {\psi_{l_2}}}}}}}(x,y) \nonumber \\& = \sum_{k_2,l_2} (f_1*\varphi_{k_2}) (D_1^{\alpha_1+\beta_1}f_2 * \tilde{\tilde{\tilde{\tilde{\tilde{\tilde{\tilde{\psi}}}}}}}_{k_2}) (g_1 * \varphi_{l_2}) (D_2^{\alpha_2+\beta_2}g_2 * \tilde{\tilde{\tilde{\tilde{\tilde{\tilde{\tilde{\psi}}}}}}}_{l_2}) (h* \varphi_{k_2} \otimes \varphi_{l_2}) * \tilde{\tilde{\tilde{\tilde{\tilde{\tilde {\psi}}}}}}_{k_2} \otimes \tilde{\tilde{\tilde{\tilde{\tilde{\tilde{\psi_{l_2}}}}}}} (x,y) \end{align*} \subsubsection{Case II: Bi-parameter flag paraproduct} \begin{align*} & D_1^{\beta_1} D_2^{\beta_2}\Big(\sum_{\substack{k_3 \\ l_3}}D_1^{\alpha_1}D_2^{\alpha_2}\big(\sum_{\substack{ k_1 \ll k_2 \ll k_3 \\ l_1,l_2 l_1 \ll l_2 \ll l_3}} f_1*\psi_{k_1}(x) g_1 * \psi_{l_1}(y) f_2 * \psi_{k_2}(x) g_2 * \psi_{l_2}(y) \big) h * \psi_{k_3} \otimes \psi_{l_3}(x,y)\Big) \nonumber \\& = D_1^{\beta_1} D_2^{\beta_2}\Big(\sum_{\substack{k_3 \\ l_3}}D_1^{\alpha_1} D_2^{\alpha_2}\big(\sum_{\substack{k_2 \ll k_3\\l_2 \ll l_3}} f_1 * \varphi_{k_2}(x) f_2 * \psi_{k_2}(x) g_1 * \varphi_{l_2}(y) g_2 * \psi_{l_2}(y)\big) h * \psi_{k_3} \otimes \psi_{l_3}(x,y)\Big) \nonumber \\& = D_1^{\beta_1} D_2^{\beta_2}\Big(\sum_{\substack{k_3 \\ l_3}} \big(\sum_{\substack{k_2 \ll k_3\\l_2 \ll l_3}} f_1 * \varphi_{k_2}(x) D_1^{\alpha_1}f_2 * \tilde{\psi}_{k_2}(x) g_1 * \varphi_{l_2}(y) D_2^{\alpha_2}g_2 * \tilde{\psi}_{l_2}(y)\big) h * \psi_{k_3} \otimes \psi_{l_3}(x,y)\Big) \nonumber \\& = D_1^{\beta_1}D_2^{\beta_2}\Big(\sum_{\substack{k_2 \ll k_3\\l_2 \ll l_3}} (f_1 * \varphi_{k_2})(D_1^{\alpha_1}f_2 * \tilde{\psi}_{k_2})*\tilde{\tilde{\psi}}_{k_2} (x) \cdot (g_1 * \varphi_{l_2}) (D_2^{\alpha_2}g_2 * \tilde{\psi}_{l_2})* \tilde{\tilde{\psi}}_{l_2}(y) \cdot h * \psi_{k_3} \otimes \psi_{l_3}(x,y)\Big) \nonumber \\& = D_1^{\beta_1} D_2^{\beta_2} \bigg[\sum_{\substack{k_2 \ll k_3\\l_2 \ll l_3}} \bigg(\Big((f_1 * \varphi_{k_2})(D_1^{\alpha_1}f_2 * \tilde{\psi}_{k_2})*\tilde{\tilde{\psi}}_{k_2}\Big) *\varphi_{k_3} \cdot \Big((g_1 * \varphi_{l_2}) (D_2^{\alpha_2}g_2 * \tilde{\psi}_{l_2})* \tilde{\tilde{\psi}}_{l_2}\Big) * \varphi_{l_3} \nonumber \\& \ \ \ \ \ \ \ \ \cdot \Big(h * \psi_{k_3} \otimes \psi_{l_3}\Big) \bigg) * \tilde{\psi}_{k_3} \otimes \tilde{\psi}_{l_3} (x,y)\bigg] \nonumber \\& = \sum_{\substack{k_2 \ll k_3\\l_2 \ll l_3}} \bigg(\Big((f_1 * \varphi_{k_2})(D_1^{\alpha_1}f_2 * \tilde{\psi}_{k_2})*\tilde{\tilde{\psi}}_{k_2}\Big) *\varphi_{k_3} \cdot \Big((g_1 * \varphi_{l_2}) (D_2^{\alpha_2}g_2 * \tilde{\psi}_{l_2})* \tilde{\tilde{\psi}}_{l_2}\Big) * \varphi_{l_3} \nonumber \\& \ \ \ \ \ \ \ \ \cdot \Big(D_1^{\beta_1}D_2^{\beta_2} h * \tilde{\tilde{\tilde{\psi}}}_{k_3} \otimes \tilde{\tilde{\tilde{\psi}}}_{l_3}\Big) \bigg) * \tilde{\tilde{\psi}}_{k_3} \otimes \tilde{\tilde{\psi}}_{l_3} (x,y) \end{align*} \end{comment}
\begin{comment} \section{Key Tool} The stopping-time decomposition is an algorithm that partitions intervals (or rectangles) such that on intervals (or rectangles) in the same family, the average of the given function is of a certain level. In addition to information about the average, such algorithm also allows estimates for the measure of the corresponding intervals (or rectangles) given that the function is in some $L^p$ space. Our key tool refers to a tensor-like stopping-time decomposition.
To introduce the machinery, we focus on the simple case. Let us recall the model operator
\begin{theorem} Let
$$\Pi (f_1 \otimes g_1, f_2 \otimes g_2, h) := \displaystyle \sum_{I \times J \in \mathcal{I} \times \mathcal{J}} \frac{1}{|I| |J|} \langle f_1,\varphi_I^{1} \rangle \langle f_2, \psi^{2}_J \rangle \langle g_1, \varphi_J^{1} \rangle \langle g_2, \psi_J^{2} \rangle \langle h, \varphi_{I}^3 \otimes \varphi_{J}^3 \rangle \psi_{I}^4 \otimes \psi_{J}^4$$ Then $$\Pi: L_x^{p_1}(L_y^{q_1}) \times L_x^{p_2}(L_y^{q_2}) \times L^s \rightarrow L^r.$$ \end{theorem} \end{comment}
\begin{comment} The main challenge can then be clarified as combining two one-dimensional stopping-time decompositions to gain estimates for a bi-parameter object. Moreover, the difficulty arises when such tensor-like stopping-time decomposition needs to be combined with the two-dimensional stopping-time decomposition.
Our construction of the tensor-like stopping-time decomposition follows from the ideology that our ultimate goal is to control the product of the average of a function in $x$-direction and the average of a function in $y$-direction. This suggests that when performing such stopping-time decomposition, we group rectangles which are formed by $x$-intervals and $y$-intervals from one-dimensional stopping-time decompositions such that the product of two averages are of a certain level. Meanwhile, there are infinitely many levels for $x$-intervals and $y$-intervals that satisfy this condition. For example, for a fixed $n$, there is a nested collection of $I \times J$, such that $\text{ave}_I(f_1) \sim 2^{-n_1}$ and $\text{ave}_J(g_1) \sim 2^{-n+n_1},\ \ \forall n_1 \in \mathbb{Z}$. The issue of how to sum measures of nested rectangles then arises.
Such issue intensifies when the two-dimensional stopping-time decomposition mingles in. More precisely, nested sum of measures of arbitrary unions of rectangles will naturally appear. We resolve the issue by observing that the nested sum of measures of unions of rectangles (generated from stopping-time decompositions) is comparable to the measure of the total union of those rectangles. This observation is drawn based on the sparsity generated from the one-dimensional stopping-time decompositions. \end{comment}
\section{Sizes and Energies} \label{section_size_energy} The notion of sizes and energies appear first in \cite{mtt} and \cite{mtt2}. Since they will play important roles in the main arguments, the explicit definitions of sizes and energies are introduced and some useful properties are highlighted in this section. \begin{definition} \label{def_size_energy} Let $\mathcal{I}$ be a finite collection of dyadic intervals. Let $(\psi_I)_{I \in \mathcal{I}}$ denote a lacunary family of $L^2$-normalized bump functions and $(\varphi_I)_{I \in \mathcal{I}}$ a non-lacunary family of $L^2$-normalized bump functions. Define \begin{enumerate}[(1)] \item
$$\text{size}_{\mathcal{I}}((\langle f, \varphi_I \rangle)_{I \in \mathcal{I}}) := \sup_{I \in \mathcal{I}} \frac{|\langle f, \varphi_I\rangle|}{|I|^{\frac{1}{2}}};$$
\item
$$\text{size}_{\mathcal{I}}((\langle f, \psi_I \rangle)_{I \in \mathcal{I}}) := \sup_{I_0 \in \mathcal{I}} \frac{1}{|I_0|}\left\Vert \bigg(\sum_{\substack{I \subseteq I_0 \\ I \in \mathcal{I}}} \frac{|\langle f, \psi_I \rangle|^2}{|I|} \chi_{I}\bigg)^{\frac{1}{2}}\right\Vert_{1,\infty};$$ \item
$$\text{energy}^{1,\infty} _{\mathcal{I}}((\langle f, \varphi_I \rangle)_{I \in \mathcal{I}}) := \sup_{n \in \mathbb{Z}} 2^{n} \sup_{\mathbb{D}_n} \sum_{I \in \mathbb{D}_n} |I|$$ where $\mathbb{D}_n$ ranges over all collections of disjoint dyadic intervals in $\mathcal{I}$ satisfying $$
\frac{|\langle f,\varphi_I \rangle|}{|I|^{\frac{1}{2}}} > 2^n; $$ \item
$$\text{energy}^{1,\infty} _{\mathcal{I}}((\langle f, \psi_I \rangle)_{I \in \mathcal{I}}) := \sup_{n \in \mathbb{Z}} 2^{n} \sup_{\mathbb{D}_n} \sum_{I \in \mathbb{D}_n} |I|$$ where $\mathbb{D}_n$ ranges over all collections of disjoint dyadic intervals in $\mathcal{I}$ satisfying $$
\frac{1}{|I|}\left\Vert \bigg(\sum_{\substack{\tilde{I} \subseteq I \\ \tilde{I} \in \mathcal{I}}} \frac{|\langle f, \psi_{\tilde{I}} \rangle|^2}{|\tilde{I}|} \chi_{\tilde{I}}\bigg)^{\frac{1}{2}}\right\Vert_{1,\infty} > 2^{n}; $$ \item For $t>1$, define
$$\text{energy}^{t}_{\mathcal{I}}((\langle f, \varphi_I \rangle)_{I \in \mathcal{I}}) := \left(\sum_{n \in \mathbb{Z}}2^{tn}\sup_{\mathbb{D}_n}\sum_{I \in \mathbb{D}_n}|I| \right)^{\frac{1}{t}}$$ where $\mathbb{D}_n$ ranges over all collections of disjoint dyadic intervals in $\mathcal{I}$ satisfying $$
\frac{|\langle f,\varphi_I \rangle|}{|I|^{\frac{1}{2}}} > 2^n. $$ \end{enumerate} \end{definition} The sizes and energies involving cutoff functions in the Haar model can be defined analogously. \begin{definition} \label{def_size_energy_haar} Let $\mathcal{I}$ be a finite collection of dyadic intervals. Let $(\psi^H_I)_{I \in \mathcal{I}}$ denote a family of Haar wavelets and $(\varphi^H_I)_{I \in \mathcal{I}}$ a family of $L^2$-normalized indicator functions. Define $$ \text{size}_{\mathcal{I}}((\langle f, \varphi^H_I \rangle)_{I \in \mathcal{I}}), \ \ \text{size}_{\mathcal{I}}((\langle f, \psi^H_I \rangle)_{I \in \mathcal{I}}) $$ and $$ \text{energy}^{1,\infty} _{\mathcal{I}}((\langle f, \psi^H_I \rangle)_{I \in \mathcal{I}}), \ \ \text{energy}^{1,\infty} _{\mathcal{I}}((\langle f, \varphi^H_I \rangle)_{I \in \mathcal{I}}), \ \ \text{energy}^{t}_{\mathcal{I}}((\langle f, \varphi^H_I \rangle)_{I \in \mathcal{I}})\ \ \text{for} \ \ t > 1 $$ by substituting $\varphi_I$ with $\varphi^H_I$ and $\psi_I$ with $\psi^H_I$ in Definition \ref{def_size_energy} respectively. \begin{comment} \begin{enumerate}[(1)] \item
$$\text{size}_{\mathcal{I}}((\langle f, \varphi_I \rangle)_{I \in \mathcal{I}}) := \sup_{I \in \mathcal{I}} \frac{|\langle f, \varphi_I\rangle|}{|I|^{\frac{1}{2}}};$$
\item
$$\text{size}_{\mathcal{I}}((\langle f, \psi_I \rangle)_{I \in \mathcal{I}}) := \sup_{I_0 \in \mathcal{I}} \frac{1}{|I_0|}\left\Vert \bigg(\sum_{\substack{I \subseteq I_0 \\ I \in \mathcal{I}}} \frac{|\langle f, \psi_I \rangle|^2}{|I|} \chi_{I}\bigg)^{\frac{1}{2}}\right\Vert_{1,\infty};$$ \item
$$\text{energy}^{1,\infty} _{\mathcal{I}}((\langle f, \varphi_I \rangle)_{I \in \mathcal{I}}) := \sup_{n \in \mathbb{Z}} 2^{n} \sup_{\mathbb{D}_n} \sum_{I \in \mathbb{D}_n} |I|$$ where $\mathbb{D}_n$ ranges over all collections of disjoint dyadic intervals in $\mathcal{I}$ satisfying $$
\frac{|\langle f,\varphi_I \rangle|}{|I|^{\frac{1}{2}}} > 2^n; $$ \item
$$\text{energy}^{1,\infty} _{\mathcal{I}}((\langle f, \psi_I \rangle)_{I \in \mathcal{I}}) := \sup_{n \in \mathbb{Z}} 2^{n} \sup_{\mathbb{D}_n} \sum_{I \in \mathbb{D}_n} |I|$$ where $\mathbb{D}_n$ ranges over all collections of disjoint dyadic intervals in $\mathcal{I}$ satisfying $$
\frac{1}{|I|}\left\Vert \bigg(\sum_{\substack{\tilde{I} \subseteq I \\ \tilde{I} \in \mathcal{I}}} \frac{|\langle f, \psi_{\tilde{I}} \rangle|^2}{|\tilde{I}|} \chi_{\tilde{I}}\bigg)^{\frac{1}{2}}\right\Vert_{1,\infty} > 2^{n}; $$ \item For $t>1$, define
$$\text{energy}^{t}_{\mathcal{I}}((\langle f, \varphi_I \rangle)_{I \in \mathcal{I}}) := \left(\sum_{n \in \mathbb{Z}}2^{tn}\sup_{\mathbb{D}_n}\sum_{I \in \mathbb{D}_n}|I| \right)^{\frac{1}{t}}$$ where $\mathbb{D}_n$ ranges over all collections of disjoint dyadic intervals in $\mathcal{I}$ satisfying $$
\frac{|\langle f,\varphi_I \rangle|}{|I|^{\frac{1}{2}}} > 2^n. $$ \end{enumerate} \end{comment} \end{definition}
\subsection{Useful Facts about Sizes and Energies} \label{section_size_energy_fact} The following propositions describe facts about sizes and energies which will be heavily employed later on. Propositions \ref{JN} and \ref{size} are routine and the proofs can be found in Section 2 of \cite{cw}. Proposition \ref{energy_classical} consists of two parts - the first part is discussed in \cite{cw} while the second part is less standard and the proof will be included in Section \ref{Proof_prop_energy_classical}.
Proposition \ref{B_en_global} describes a ``global'' energy estimate for the bilinear operators $B_{\mathcal{Q}}$ defined in (\ref{B_global_definition}) of Definition \ref{B_definition} and the proof follows from the boundedness of paraproducts (\cite{cm}, \cite{cw}). Propositions \ref{size_cor} and \ref{B_en} highlight the ``local'' size and energy estimates for the analogous operators defined under the Haar model assumption, namely $B_{\mathcal{Q},P}^H$ and $B_{\mathcal{Q},P}^{\#, H}$
defined in (\ref{B_local_definition_haar}) and (\ref{B_local_definition_haar_fix_scale}) respectively. More precisely, Propositions \ref{size_cor} and \ref{B_en} take into the consideration that the operators $B_{\mathcal{Q},P}^H$ and $B_{\mathcal{Q},P}^{\#, H}$ are localized to intersect certain level sets which carry crucial information for the size and energy estimates. The emphasis on the Haar model assumption keeps track of the arguments we need to modify for the general Fourier case.
\begin{proposition}[John-Nirenberg] \label{JN} Let $\mathcal{I}$ be a finite collection of dyadic intervals. For any sequence $(a_I)_{I \in \mathcal{I}}$ and $r > 0$, define the BMO-norm for the sequence as $$
\|(a_I)_I\|_{\text{BMO}(r)} := \sup_{I_0 \in \mathcal{I}}\frac{1}{|I_0|^{\frac{1}{r}}} \left\Vert \left(\sum_{I \subseteq I_0} \frac{|a_I|^2}{|I|}\chi_{I}(x)\right)^{\frac{1}{2}}\right\Vert_r. $$ Then for any $0 < p < q < \infty$, $$
\|(a_I)_I\|_{\text{BMO}(p)} \simeq \|(a_I)_I \|_{\text{BMO}(q)}. $$ \end{proposition}
\begin{proposition} \label{size} Suppose $f \in L^1(\mathbb{R})$. Then
$$\text{size}_{\mathcal{I}}\big((\langle f,\phi_I \rangle)_{I \in \mathcal{I}}\big), \text{size}_{\mathcal{I}}\big((\langle f,\phi^H_I \rangle)_{I \in \mathcal{I}} \big) \lesssim \sup_{I \in \mathcal{I}}\int_{\mathbb{R}}|f|\tilde{\chi}_I^M dx$$ for $M > 0$ and the implicit constant depends on $M$. $\tilde{\chi}_I$ is an $L^{\infty}$-normalized bump function adapted to $I$. \end{proposition}
\begin{proposition} \label{energy_classical} \noindent \begin{enumerate} \item Suppose $f \in L^1(\mathbb{R})$. Then $$
\text{energy}^{1,\infty}_{\mathcal{I}}((\langle f, \phi_I \rangle))_{I \in \mathcal{I}}, \text{energy}^{1,\infty}_{\mathcal{I}}((\langle f, \phi^H_I \rangle))_{I \in \mathcal{I}} \lesssim \|f\|_1. $$ \item Suppose $f \in L^t(\mathbb{R})$ for $t >1$. Then $$
\text{energy}^t_{\mathcal{I}}((\langle f, \varphi_I \rangle))_{I\in \mathcal{I}}, \text{energy}^t_{\mathcal{I}}((\langle f, \varphi^H_I \rangle))_{I \in \mathcal{I}}\lesssim \|f\|_t. $$ \end{enumerate} \end{proposition}
\begin{proposition}[Global Energy] \label{B_en_global}
Suppose that $V_1, V_2 \subseteq \mathbb{R}$ are sets of finite measure and $|v_i| \leq \chi_{V_i}$ for $i = 1,2$. Let $\mathcal{P}$
denote a finite collection of dyadic intervals. Let $B_{\mathcal{Q}}$ and $B^H_{\mathcal{Q}}$ denote the bilinear operators defined in (\ref{B_global_definition}) of Definition \ref{B_definition} and (\ref{B_global_haar}) of Definition \ref{B_definition_haar} respectively.
\begin{enumerate} \item Then for any $0 < \rho <1$, one has \begin{align*}
& \text{energy}^{1,\infty}_{\mathcal{P}}((\langle B_{\mathcal{Q}}(v_1,v_2), \varphi_P \rangle)_{P \in \mathcal{P}}), \text{energy}^{1,\infty}_{\mathcal{P}}((\langle B^H_{\mathcal{Q}}(v_1,v_2), \varphi_P \rangle)_{P \in \mathcal{P}}) \lesssim |V_1|^{\rho}|V_2|^{1-\rho}.
\end{align*} \item Suppose that $t >1$. Then for any $0 \leq \theta_1, \theta_2 <1$, with $\theta_1 + \theta_2 = \frac{1}{t}$, one has \begin{align*}
& \text{energy}^t_{\mathcal{P}}((\langle B_{\mathcal{Q}}(v_1,v_2), \varphi_P \rangle)_{P \in \mathcal{P}}), \text{energy}^t_{\mathcal{P}}((\langle B^H_{\mathcal{Q}}(v_1,v_2), \varphi_P \rangle)_{P \in \mathcal{P}}) \lesssim |V_1|^{\theta_1}|V_2|^{\theta_2}.
\end{align*} \end{enumerate} \end{proposition}
It is not difficult to observe that Proposition \ref{B_en_global} follows immediately from Proposition \ref{energy_classical} and the following lemma.
\begin{lemma}\label{B_global_norm}
Let $B_{\mathcal{Q}}$ and $B^H_{\mathcal{Q}}$ denote the bilinear operators defined in (\ref{B_global_definition}) and (\ref{B_global_haar}) respectively.
Then for any $v_1 \in L^{p}$, $v_2 \in L^{q}$ with $1 < p,q \leq \infty$, $0 < t < \infty$ and $\frac{1}{t} = \frac{1}{p}+ \frac{1}{q} $, \begin{align*}
& \|B_{\mathcal{Q}}(v_1,v_2)\|_{t}, \|B^H_{\mathcal{Q}}(v_1,v_2)\|_{t} \lesssim \|v_1\|_{L^{p}} \|v_2\|_{L^{q}}. \nonumber
\end{align*}
\end{lemma} By identifying that $B_{\mathcal{Q}}$ (\ref{B_global_definition}) and $B^H_{\mathcal{Q}}$ (\ref{B_global_haar}) are one-parameter paraproducts, Lemma \ref{B_global_norm} is a restatement of Coifman-Meyer's theorem on the boundedness of paraproducts \cite{cm}.
We will now turn our attention to the local size estimate for $(\langle B_{\mathcal{Q}, P }^{\#,H}(v_1, v_2), \varphi_{P} \rangle)_{P \in \mathcal{P}}$
and the local energy estimate for $(\langle B_{\mathcal{Q}, P}^H(v_1, v_2), \varphi_P \rangle )_{P \in \mathcal{P}}$.
\begin{comment} The precise definitions for the operators $B_I^{\#_1,H}, \tilde{B}_J^{\#_2,H}, B_I^H$ and $B_J^H$ are stated as follows. \begin{definition} \label{B_def} Suppose that $I$ and $J$ are fixed dyadic intervals and $\mathcal{K}$ and $\mathcal{L}$ are finite collections of dyadic intervals. Suppose that $(\phi_{K}^{i})_{K \in \mathcal{K}}, (\phi_{L}^{j})_{L \in \mathcal{L}}$ for $i, j = 1,2$ are families of $L^2$-normalized bump functions. Further assume that $(\phi_K^{3,H})_{K \in \mathcal{K}}$ and $(\phi_L^{3,H})_{L \in \mathcal{L}} $ are families of Haar wavelets or $L^2$-normalized indicator functions. A family of Haar wavelets are considered to be a lacunary family and a family of $L^2$-normalized indicator functions to be a non-lacunary family. Suppose that at least two families of $(\phi_{K}^{1})_K, (\phi_{K}^2)_K$ and $(\phi_{K}^{3,H})_K$ are lacunary and that at least two families of $(\phi_{L}^{1})_L, (\phi_{L}^2)_L$ and $(\phi_{L}^{3,H})_L$ are lacunary. Let \begin{enumerate}[(i)] \item \begin{align} \label{B_size_haar}
& B_I^{\#_1,H}(f_1, f_2)(x) := \displaystyle \sum_{K \in \mathcal{K}:|K| \sim 2^{\#_1} |I|} \frac{1}{|K|^{\frac{1}{2}}}\langle f_1, \phi_K^1 \rangle \langle f_2, \phi_K^2 \rangle \phi_K^{3,H}(x), \nonumber \\
& \tilde{B}_J^{\#_2,H}(g_1,g_2) (y) := \displaystyle \sum_{L \in \mathcal{L}:|L| \sim 2^{\#_2} |J|} \frac{1}{|L|^{\frac{1}{2}}}\langle g_1, \phi_L^1 \rangle \langle g_2, \phi_L^2 \rangle \phi_L^{3,H}(y); \end{align} \item \begin{align}\label{B_haar_def}
& B_{I}^H(f_1, f_2)(x) := \sum_{K: |K| \geq |I|} \frac{1}{|K|^{\frac{1}{2}}}\langle f_1, \phi_K^1\rangle \langle f_2, \phi_K^2 \rangle \phi_K^{3,H} (x), \nonumber \\
& \tilde{B}_{J}^H(g_1, g_2)(y) := \sum_{L: |L| \geq |J|} \frac{1}{|L|^{\frac{1}{2}}}\langle g_1, \phi_L^1\rangle \langle g_2, \phi_L^2 \rangle \phi_L^{3,H} (y). \end{align} \end{enumerate} \end{definition} \end{comment}
In the Haar model, for any fixed dyadic intervals $Q$ and $P$ with $|Q|\geq |P|$, the only non-degenerate case $\langle \phi_Q^{3,H}, \varphi_{P}^H \rangle \neq 0$ is that $Q \supseteq P$. Such observation provides natural localizations for the sequence $(\langle B_{\mathcal{Q}, P}^{\#,H}(v_1, v_2), \varphi^H_P \rangle)_{P \in \mathcal{P}}$ and thus for the sequences $(\langle v_1, \phi_Q^1 \rangle)_{Q \in \mathcal{Q}}$ and $(\langle v_2, \phi_Q^2 \rangle)_{Q \in \mathcal{Q}}$ as explicitly stated in the following lemma.
\begin{lemma}[Localization of Sizes in the Haar Model] \label{B_size}
Let $S$ denote a measurable subset of $\mathbb{R}$ and $\mathcal{P}'$ a finite collection of dyadic intervals
such that for any $P \in \mathcal{P}'$, $P \cap S \neq \emptyset$.
For each $P \in \mathcal{P}'$, let $B_{\mathcal{Q},P}^{\#, H}$ denote the bilinear operator defined in (\ref{B_local_definition_haar_fix_scale}) of Definition \ref{B_definition}. Then for any measurable functions $v_1$ and $v_2$, $$
\text{size}_{\mathcal{P'}}((\langle B_{\mathcal{Q},P}^{\#,H}(v_1, v_2), \varphi^H_P \rangle)_{P \in \mathcal{P}'}) \lesssim \sup_{\substack{Q \in \mathcal{Q} \\ Q \cap S \neq \emptyset}}\frac{|\langle v_1, \phi_Q^1 \rangle|}{|Q|^{\frac{1}{2}}} \sup_{\substack {Q \in \mathcal{Q} \\ Q \cap S \neq \emptyset}}\frac{|\langle v_2, \phi_Q^2 \rangle|}{|Q|^{\frac{1}{2}}}. $$
\end{lemma}
The localization generates more quantitative and useful estimates for the sizes involving $B_{\mathcal{Q}, P}^{\#,H}$ when $S$ is a level set of the Hardy-Littlewood maximal functions $Mv_1$ and $Mv_2$ as elaborated in the following proposition.
\begin{remark} One notational comment is that $C$, $C_1, C_2$ and $C_3$ used throughout the paper denote some sufficiently large constants greater than 1. \end{remark}
\begin{proposition} [Local Size Estimates in the Haar Model]\label{size_cor}
Suppose that $V_1, V_2 \subseteq \mathbb{R}$ are sets of finite measure and $|v_i| \leq \chi_{V_i}$ for $i = 1,2$. Let $\tilde{n}, \tilde{m}$ denote some integers and $\ \mathcal{U}_{\tilde{n},\tilde{m}}:=\{z: Mv_1(z) \leq C2^{\tilde{n}} |V_1|\} \cap \{ z: Mv_2(z) \leq C 2^{\tilde{m}} |V_2|\}$. Further assume that $ \mathcal{P}' $ is a finite collection of dyadic intervals such that $P \cap \mathcal{U}_{\tilde{n},\tilde{m}} \neq \emptyset$ for any $P \in \mathcal{P}'$. For each $P \in \mathcal{P}'$, $B_{\mathcal{Q},P}^{\#, H}$ denote the bilinear operator defined in (\ref{B_local_definition_haar_fix_scale}).
Then $$
\text{size}_{\mathcal{P}'}((\langle B_{\mathcal{Q}, P}^{\#,H}(v_1, v_2), \varphi^H_P \rangle)_{P \in \mathcal{P}'}) \lesssim (C_1 2^{\tilde{n}}|V_1|)^{\alpha} (C_1 2^{\tilde{m}}|V_2|)^{\beta},$$
for any $ 0 \leq \alpha, \beta \leq 1$. \end{proposition} The proof of the proposition follows directly from Lemma \ref{B_size} and the trivial estimates $$
\sup_{Q \cap \mathcal{U}_{\tilde{n},\tilde{m}} \neq \emptyset}\frac{|\langle v_1, \phi_Q^1 \rangle|}{|Q|^{\frac{1}{2}}} \lesssim \min(C 2^{\tilde{n}}|V_1|,1) \leq (C 2^{\tilde{n}}|V_1|)^{\alpha} $$ and $$
\sup_{Q \cap \mathcal{U}_{\tilde{n},\tilde{m}} \neq \emptyset}\frac{|\langle v_2, \phi_Q^2 \rangle|}{|Q|^{\frac{1}{2}}} \lesssim \min(C 2^{\tilde{m}}|V_2|,1) \leq (C 2^{\tilde{m}}|V_2|)^{\beta} $$ for any $0 \leq \alpha,\beta \leq 1$.
We will also explore the local energy estimates which are ``stronger'' than the global energy estimates. Heuristically, in the case when $v_1 \in L^{p}$ and $ v_2 \in L^{q}$ with $|v_1| \leq \chi_{V_1}$ and $|v_2| \leq \chi_{V_2}$ for $p,q>1$ and close to $1$, the global energy estimates would not yield the desired boundedness exponents for $|V_1|$ and $|V_2| $ whereas one could take advantages of the local energy estimates to obtain the result. In the Haar model, a perfect localization can be achieved for energy estimates involving bilinear operators $B^H_{\mathcal{Q},P}$ specified in (\ref{B_local_definition_haar}) of Definition \ref{B_definition}. In particular, the corresponding energy estimates can be compared to the energy estimates for
localized operators defined as follows.
\begin{comment} \begin{definition} Suppose that $\phi_{K}^{i}, \phi_{L}^{j}$ for $i, j = 1,2$ are $L^2$-normalized bump functions adapted to $I$ and $J$ respectively. Further assume that $\phi_K^{3,H}$ and $\phi_L^{3,H} $ are Haar wavelets or $L^2$-normalized indicator functions on $K$ and $L$. One will consider Haar wavelets to be a lacunary family and $L^2$-normalized indicator functions to be a non-lacunary family. Suppose that at least two families of $(\phi_{K}^{1})_K, (\phi_{K}^2)_K$ and $(\phi_{K}^{3,H})_K$ are lacunary and that at least two families of $(\phi_{L}^{1})_L, (\phi_{L}^2)_L$ and $(\phi_{L}^{3,H})_L$ are lacunary. Let \begin{align}
& B_{I}^H(f_1, f_2)(x) := \sum_{K: |K| \geq |I|} \frac{1}{|K|^{\frac{1}{2}}}\langle f_1, \phi_K^1\rangle \langle f_2, \phi_K^2 \rangle \phi_K^{3,H} (x) \nonumber \\
& \tilde{B}_{J}^H(g_1, g_2)(y) := \sum_{L: |L| \geq |J|} \frac{1}{|L|^{\frac{1}{2}}}\langle g_1, \phi_L^1\rangle \langle g_2, \phi_L^2 \rangle \phi_L^{3,H} (y). \end{align}
\end{definition} \end{comment}
\begin{definition}\label{B^0_def} Let $\mathcal{U}_{\tilde{n},\tilde{m}}$ be defined as the level set described in Proposition \ref{size_cor}. Let $\mathcal{Q}$ denote a finite collection of dyadic intervals. \begin{enumerate} \item Suppose that $(\phi^i_Q)_{Q \in \mathcal{Q}}$ for $i = 1, 2$ are families of $L^2$-normalized adapted bump functions and $(\phi^{3,H}_Q)_{Q \in \mathcal{Q}}$ is a family of $L^2$-normalized cutoff functions
such that the family $(\phi_Q^{3,H})_{Q \in \mathcal{Q}}$ is lacunary while at least one of the families $(\phi^1_Q)_{Q \in \mathcal{Q}}$ and $(\phi^2_Q)_{Q \in \mathcal{Q}}$ is lacunary.
Define the bilinear operator
\begin{align}
B^{\tilde{n},\tilde{m},0}_{\mathcal{Q},\text{lac}}(v_1,v_2):= &\displaystyle \sum_{\substack{Q \in \mathcal{Q} \\ Q \cap \mathcal{U}_{\tilde{n},\tilde{m}} \neq \emptyset}} \frac{1}{|Q|^{\frac{1}{2}}}\langle v_1, \phi_Q^1\rangle \langle v_2, \phi_Q^2 \rangle \psi_Q^{3,H}. \label{B^0_lac}
\end{align}
\item Suppose that $(\phi^i_Q)_{Q \in \mathcal{Q}}$ for $i = 1, 2$ are families of $L^2$-normalized adapted bump functions and $(\phi^{3,H}_Q)_{Q \in \mathcal{Q}}$ is a family of $L^2$-normalized cutoff functions such that the family $(\phi^{3,H}_Q)_{Q \in \mathcal{Q}}$ is nonlacunary while the families $(\phi^i_Q)_{Q \in \mathcal{Q}}$ for $i \neq 3$ are both lacunary. Define the bilinear operator
\begin{align} B^{\tilde{n},\tilde{m},0}_{\mathcal{Q},\text{nonlac}}(v_1,v_2) := &
\sum_{\substack{ Q \in \mathcal{Q} \\Q: Q \cap \mathcal{U}_{\tilde{n},\tilde{m}} \neq \emptyset}} \frac{1}{|Q|^{\frac{1}{2}}}|\langle v_1, \psi_Q^1\rangle| |\langle v_2, \psi_Q^2 \rangle| |\varphi_Q^{3,H} |. \label{B^0_nonlac} \end{align} \end{enumerate}
\end{definition}
\begin{remark} We would like to emphasize that $B^{\tilde{n},\tilde{m},0}_{\mathcal{Q},\text{lac}}$ and $B^{\tilde{n},\tilde{m},0}_{\mathcal{Q},\text{nonac}}$ are localized to intersect level sets $\mathcal{U}_{\tilde{n},\tilde{m}}$ nontrivially. It is not difficult to imagine that the energy estimates for $(\langle B^{\tilde{n},\tilde{m},0}_{\mathcal{Q},\text{lac}}(v_1,v_2), \varphi_P \rangle )_{P \in \mathcal{P}'}$ and $(\langle B^{\tilde{n},\tilde{m},0}_{\mathcal{Q},\text{nonlac}}(v_1,v_2), \varphi_P \rangle )_{P \in \mathcal{P}'}$ would be better than the ``global'' energy estimates
stated in Proposition \ref{B_en_global} since one can now employ the information about intersections with level sets to control $
\frac{|\langle v_1, \phi_Q^1 \rangle|}{|Q|^{\frac{1}{2}}} \text{ and } \frac{|\langle v_2, \phi_Q^2 \rangle|}{|Q|^{\frac{1}{2}}}. $ The energy estimates for $(\langle B^{H}_{\mathcal{Q},P}(v_1,v_2), \varphi_P \rangle )_{P \in \mathcal{P}'}$ can indeed be reduced to the energy estimates for $(\langle B^{\tilde{n},\tilde{m},0}_{\mathcal{Q}, \text{lac}}, \varphi_P \rangle)_{P \in \mathcal{P}'}$ or $(\langle B^{\tilde{n},\tilde{m},0}_{\mathcal{Q}, \text{nonlac}}, \varphi_P \rangle)_{P \in \mathcal{P}'}$ as stated in Lemma \ref{localization_haar}. \end{remark}
\begin{lemma}[Localization of Energies in the Haar Model] \label{localization_haar} Suppose that $\mathcal{P}'$ is a finite collection of dyadic intervals such that $P \cap \mathcal{U}_{\tilde{n},\tilde{m}} \neq \emptyset$ for any $P \in \mathcal{P}'$. For each $P \in \mathcal{P}'$, let $B^{H}_{\mathcal{Q},P}$ denote the bilinear operators defined in (\ref{B_local_definition_haar}).
Let $B^{\tilde{n},\tilde{m},0}_{\mathcal{Q}, \text{lac}}$ and $B^{\tilde{n},\tilde{m},0}_{\mathcal{Q}, \text{nonlac}}$ denote the bilinear operators defined in Definition \ref{B^0_def} where the bump functions in $ B^{\tilde{n},\tilde{m},0}_{\mathcal{Q}, \text{lac}}$ or $B^{\tilde{n},\tilde{m},0}_{\mathcal{Q}, \text{nonlac}}$ are the same as the ones in $B^H_{\mathcal{Q},P}$.
Then for $t > 1$ and any measurable functions $v_1$ and $v_2$, \begin{align} & \text{energy}^t_{\mathcal{P}'}((\langle B^H_{\mathcal{Q},P}(v_1,v_2), \varphi^H_P \rangle)_{P \in \mathcal{P}'})\nonumber \\ \leq & \begin{cases} \text{energy}^t_{\mathcal{P}'}((\langle B^{\tilde{n},\tilde{m},0}_{\mathcal{Q}, \text{lac}}(v_1,v_2), \varphi^H_P \rangle)_{P \in \mathcal{P}'}) \ \ \ \ \ \text{if} \ \ (\phi^{3,H}_Q)_{Q \in \mathcal{Q}} \text{ is a lacunary family} \\ \\ \text{energy}^t_{\mathcal{P}'}((\langle B^{\tilde{n},\tilde{m},0}_{\mathcal{Q}, \text{nonlac}}(v_1,v_2), \varphi^H_P \rangle)_{P \in \mathcal{P}'}) \ \ \text{if} \ \ (\phi^{3,H}_Q)_{Q \in \mathcal{Q}} \text{ is a non-lacunary family},
\end{cases} \end{align} and \begin{align} & \text{energy}^{1,\infty}_{\mathcal{P}'}((\langle B^H_{\mathcal{Q},P}(v_1,v_2), \varphi^H_P \rangle)_{P \in \mathcal{P}'}) \nonumber \\ \leq & \begin{cases} \text{energy}^{1,\infty}_{\mathcal{P}'}((\langle B^{\tilde{n},\tilde{m},0}_{\mathcal{Q}, \text{lac}}(v_1,v_2), \varphi^H_P \rangle)_{P \in \mathcal{P}'}) \ \ \ \ \ \text{if} \ \ (\phi^{3,H}_Q)_{Q \in \mathcal{Q}} \text{ is a lacunary family} \\ \\ \text{energy}^{1,\infty}_{\mathcal{P}'}((\langle B^{\tilde{n},\tilde{m},0}_{\mathcal{Q}, \text{nonlac}}(v_1,v_2), \varphi^H_P \rangle)_{P \in \mathcal{P}'}) \ \text{if} \ \ (\phi^{3,H}_Q)_{Q \in \mathcal{Q}} \text{ is a non-lacunary family}.
\end{cases} \end{align}
\end{lemma}
The following local energy estimates will play a crucial role in the proof of our main theorem. \begin{proposition}[Local Energy Estimates in the Haar Model] \label{B_en}
Suppose that $V_1, V_2 \subseteq \mathbb{R}$ are sets of finite measure and $|v_i| \leq \chi_{V_i}$ for $i= 1,2$.
Let $\mathcal{P}'$ denote the collection of dyadic intervals satisfying the condition described in Lemma \ref{localization_haar} and $B_{\mathcal{Q},P}^{ H}$ the bilinear operator defined in (\ref{B_local_definition_haar}) for each $P \in \mathcal{P}'$.
Further assume that $\frac{1}{p} + \frac{1}{q} > 1$. \begin{enumerate}[(i)] \item Suppose that $t >1$. Then for any $0 \leq \theta_1,\theta_2 <1$ with $\theta_1 + \theta_2 = \frac{1}{t}$, one has
\begin{align} \label{B_en_t}
& \text{energy}^{t} _{\mathcal{P}'}((\langle B^H_{\mathcal{Q},P}(v_1,v_2), \varphi_P^H \rangle)_{P \in \mathcal{P}'}) \lesssim C_1^{\frac{1}{p}+ \frac{1}{q} - \theta_1 - \theta_2}2^{\tilde{n}(\frac{1}{p} - \theta_1)}2^{\tilde{m}(\frac{1}{q} - \theta_2)}|V_1|^{\frac{1}{p}}|V_2|^{\frac{1}{q}}.
\end{align}
\item For any $0 < \theta_1,\theta_2 <1$ with $\theta_1 + \theta_2 = 1$,
\begin{align} \label{B_en_1}
&\text{energy}^{1,\infty}_{\mathcal{P}'}((\langle B^H_P(v_1,v_2), \varphi_P^H \rangle)_{P \in \mathcal{P}'}) \lesssim C_1^{\frac{1}{p}+ \frac{1}{q} - \theta_1 - \theta_2} 2^{\tilde{n}(\frac{1}{p} - \theta_1)} 2^{\tilde{m}(\frac{1}{q} - \theta_2)} |V_1|^{\frac{1}{p}} |V_2|^{\frac{1}{q}}.
\end{align}
\end{enumerate} \end{proposition}
\begin{remark} The condition that \begin{equation} \label{diff_exp} \frac{1}{p} + \frac{1}{q} > 1
\end{equation} is required in the proof the proposition. Moreover, the energy estimates in Proposition \ref{B_en} are useful for the proof of the main theorems in the range of exponents specified as (\ref{diff_exp}). A simpler argument without the use of Proposition \ref{B_en} can be applied for the other case $$ \frac{1}{p} + \frac{1}{q} \leq 1. $$ \end{remark}
\begin{remark}\label{loc_easy_haar}
Thanks to the localization specified in Lemma \ref{localization_haar}, it suffices to prove that $$\text{energy}^t_{\mathcal{P}'}((\langle B^{\tilde{n},\tilde{m},0}_{\mathcal{Q},\text{lac}}(v_1,v_2), \varphi_P^H \rangle)_{P \in \mathcal{P}'}), \ \ \ \text{energy}^t_{\mathcal{P}'}((\langle B^{\tilde{n},\tilde{m},0}_{\mathcal{Q},\text{nonlac}}(v_1,v_2), \varphi_P^H \rangle)_{P \in \mathcal{P}'}),$$ and $$\text{energy}^{1,\infty}_{\mathcal{P}'}((\langle B^{\tilde{n},\tilde{m},0}_{\mathcal{Q},\text{lac}}(v_1,v_2), \varphi_P^H \rangle)_{P \in \mathcal{P}'}), \ \ \ \text{energy}^{1,\infty}_{\mathcal{P}'}((\langle B^{\tilde{n},\tilde{m},0}_{\mathcal{Q},\text{nonlac}}(v_1,v_2), \varphi_P^H \rangle)_{P \in \mathcal{P}'}),$$
satisfy the same estimates on the right hand side of the inequalities in Proposition \ref{B_en}. Equivalently,
\begin{enumerate}[(i')] \item for any $0 \leq \theta_1,\theta_2 <1$ with $\theta_1 + \theta_2 = \frac{1}{t}$, \begin{align} \label{prop_en_equiv}
& \text{energy}^{t} _{\mathcal{P}'}((\langle B^{\tilde{n},\tilde{m},0}_{\mathcal{Q},\text{lac}}(v_1,v_2), \varphi_P^H \rangle)_{P \in \mathcal{P}'}), \text{energy}^{t} _{\mathcal{P}'}((\langle B^{\tilde{n},\tilde{m},0}_{\mathcal{Q},\text{nonlac}}(v_1,v_2), \varphi_P^H \rangle)_{P \in \mathcal{P}'}) \nonumber\\
\lesssim & C^{\frac{1}{p}+ \frac{1}{q} - \theta_1 - \theta_2}2^{\tilde{n}(\frac{1}{p} - \theta_1)}2^{\tilde{m}(\frac{1}{q} - \theta_2)}|V_1|^{\frac{1}{p}}|V_2|^{\frac{1}{q}};
\end{align} \item for any $0 < \theta_1,\theta_2 <1$ with $\theta_1 + \theta_2 = 1$, \begin{align}\label{prop_en_1_equiv} & \text{energy}^{1,\infty} _{\mathcal{P}'}((\langle B^{\tilde{n},\tilde{m},0}_{\mathcal{Q},\text{lac}}(v_1,v_2), \varphi_P^H \rangle)_{P \in \mathcal{P}'}), \text{energy}^{1,\infty} _{\mathcal{P}'}((\langle B^{\tilde{n},\tilde{m},0}_{\mathcal{Q},\text{nonlac}}(v_1,v_2), \varphi_P^H \rangle)_{P \in \mathcal{P}'}) \nonumber \\
\lesssim & C^{\frac{1}{p}+ \frac{1}{q} - \theta_1 - \theta_2}2^{\tilde{n}(\frac{1}{p} - \theta_1)}2^{\tilde{m}(\frac{1}{q} - \theta_2)}|V_1|^{\frac{1}{p}}|V_2|^{\frac{1}{q}}. \end{align} \end{enumerate} \end{remark}
Due to Proposition \ref{energy_classical}, the proof of (\ref{prop_en_equiv}) and (\ref{prop_en_1_equiv}) and thus of Proposition \ref{B_en} can be reduced to verifying Lemma \ref{B_loc_norm}.
\begin{lemma} \label{B_loc_norm}
Suppose that $V_1, V_2 \subseteq \mathbb{R}$ are sets of finite measure and $|v_i| \leq \chi_{V_i}$ for $i= 1,2$. Let $B^{\tilde{n},\tilde{m},0}_{\mathcal{Q},\text{lac}}$ and $B^{\tilde{n},\tilde{m},0}_{\mathcal{Q},\text{nonlac}}$ denote the bilinear operators defined in (\ref{B^0_lac}) and (\ref{B^0_nonlac}) of Definition \ref{B^0_def}. Then for $t \geq 1$, \begin{align} \label{B^0_norm}
& \|B^{\tilde{n},\tilde{m},0}_{\mathcal{Q},\text{lac}}(v_1,v_2)\|_t, \|B^{\tilde{n},\tilde{m},0}_{\mathcal{Q},\text{nonlac}}(v_1,v_2)\|_t \lesssim C_1^{\frac{1}{p}+ \frac{1}{q} - \theta_1 - \theta_2} 2^{\tilde{n}(\frac{1}{p} - \theta_1)} 2^{\tilde{m}(\frac{1}{q} - \theta_2)} |V_1|^{\frac{1}{p}} |V_2|^{\frac{1}{q}}.
\end{align} where $0 \leq \theta_1,\theta_2 <1$ and $\theta_1 + \theta_2 = \frac{1}{t}$. \end{lemma} \vskip .15in \subsection{Proof of Proposition \ref{energy_classical} $(2)$} \label{Proof_prop_energy_classical}
One observes that for each $n$, there exists a disjoint collection of intervals, denoted by $\mathbb{D}^{0}_n$ such that \begin{equation}
\text{energy}^{t} _{\mathcal{I}'}((\langle f, \varphi_I \rangle)_{I \in \mathcal{I}'}) = \bigg(\sum_{n}2^{tn} \sum_{\substack{I \in \mathbb{D}_n^{0}\\ I \in \mathcal{I'}}}|I|\bigg)^{\frac{1}{t}}\label{energy_p} \end{equation}\label{B_energy}
where for any $I \in \mathbb{D}^0_n$, \begin{equation} \label{energy_interval}
\frac{|\langle f, \varphi_I \rangle|}{|I|^{\frac{1}{2}}} > 2^{n}. \end{equation}
Meanwhile for any $x \in I$, \begin{equation*}
Mf(x) \geq \frac{|\langle f, \varphi_I \rangle|}{|I|^{\frac{1}{2}}}, \end{equation*} which implies that $$I \subseteq \{Mf(x) > 2^{n}\}.$$ for any $I \in \mathcal{I}'$ satisfying (\ref{energy_interval}). Then by the disjointness of $\mathbb{D}^0_n$, one can estimate the energy as follows: $$
\text{energy}^{t} _{\mathcal{I}'}((\langle f, \varphi_I \rangle)_{I \in \mathcal{I}'} \leq \big(\sum_{n}2^{tn } |\{Mf(x) > 2^{n}\}|\big)^{\frac{1}{t}} \lesssim \|Mf\|_{t}. $$ One can then apply the fact that the mapping property of maximal operator $M: L^{t} \rightarrow L^{t}$ for $t >1$ and derive $$
\|Mf\|_{t} \lesssim \|f\|_{t}. $$ \begin{remark} To prove the same estimate for $$ \text{energy}^{t} _{\mathcal{I}'}((\langle f, \varphi^H_I \rangle)_{I \in \mathcal{I}'}), $$ one can replace $\varphi_I$ by $\varphi^H_I$ and the proof above still goes through. \end{remark}
\subsection{Proof of Lemma \ref{B_size}} \label{section_proof_B_size}
One recalls the definition of $$
\text{size}_{\mathcal{P'}}((\langle B_{\mathcal{Q}, P}^{\#,H}(v_1,v_2), \varphi^H_P \rangle)_{P \in \mathcal{P}'} = \frac{|\langle B^{\#,H}_{\mathcal{Q}, P_0}(v_1,v_2),\varphi_{P_0}^H \rangle|}{|P_0|^{\frac{1}{2}}} $$ for some $P_0 \in \mathcal{P}'$ with the property that $P_0 \cap S \neq \emptyset$ by the assumption. Then \begin{align} \label{B^no_local_size}
\frac{|\langle B^{\#,H}_{\mathcal{Q},P_0}(v_1,v_2),\varphi_{P_0}^H \rangle|}{|P_0|^{\frac{1}{2}}} \leq & \frac{1}{|P_0|}\sum_{Q:|Q|\sim 2^{\#}|P_0|}\frac{1}{|Q|^{\frac{1}{2}}}|\langle v_1, \phi_Q^1 \rangle| |\langle v_2, \phi_Q^2 \rangle| |\langle |P_0|^{\frac{1}{2}}\varphi^H_{P_0},\phi_Q^{3,H} \rangle| \nonumber \\
= & \frac{1}{|P_0|}\sum_{Q:|Q|\sim 2^{\#}|P_0|}\frac{|\langle v_1, \phi_Q^1 \rangle|}{|Q|^{\frac{1}{2}}} \frac{|\langle v_2, \phi_Q^2 \rangle|}{|Q|^{\frac{1}{2}}} |\langle |P_0|^{\frac{1}{2}}\varphi_{P_0}^H, |Q|^{\frac{1}{2}}\phi_Q^{3,H} \rangle|. \end{align}
Since $ \varphi_{P_0}^H$ and $\phi_Q^{3,H}$ are compactly supported on $P_0$ and $Q$ respectively with $|P_0| \leq |Q|$, one has $$
\langle |P_0|^{\frac{1}{2}}\varphi_{P_0}^H, |Q|^{\frac{1}{2}}\phi_Q^{3,H} \rangle \neq 0 $$ if and only if $$ P_0 \subseteq Q. $$ By the hypothesis that $P_0 \cap S \neq \emptyset$, one derives that $Q \cap S\neq \emptyset$ and \begin{align*}
(\ref{B^no_local_size}) \leq &\frac{1}{|P_0|} \sup_{Q \cap S \neq \emptyset}\frac{|\langle v_1, \phi_Q^1 \rangle|}{|Q|^{\frac{1}{2}}} \sup_{Q \cap S \neq \emptyset}\frac{|\langle v_2, \phi_Q^2 \rangle|}{|Q|^{\frac{1}{2}}}\sum_{Q:|Q|\sim 2^{\#_1}|P_0|}|\langle |P_0|^{\frac{1}{2}}\varphi^H_{P_0}, |Q|^{\frac{1}{2}}\phi_Q^{3,H} \rangle| \nonumber \\
\lesssim & \frac{1}{|P_0|} \sup_{Q \cap S \neq \emptyset}\frac{|\langle v_1, \phi_Q^1 \rangle|}{|Q|^{\frac{1}{2}}} \sup_{Q \cap S\neq \emptyset}\frac{|\langle v_2, \phi_Q^2 \rangle|}{|Q|^{\frac{1}{2}}} \cdot |P_0|, \end{align*}
where the last inequality holds trivially given that $|P_0|^{\frac{1}{2}}\varphi^H_{P_0}$ is indeed an indicator function of $P_0$ and $|Q|^{\frac{1}{2}}\phi_Q^{3,H}$ is majorized by the indicator function of $Q$. This completes the proof of the proposition.
\subsection{Proof of Lemma \ref{localization_haar}} \label{section_proof_localization_haar}
According to the definition of energy (Definition \ref{def_size_energy_haar}), it suffices to prove that for any $P \in \mathcal{P}'$,
\begin{equation} \label{goal_localization_en}
|\langle B^H_{\mathcal{Q},P}(v_1,v_2), \varphi^H_P \rangle| \leq \begin{cases}
|\langle B^{\tilde{n},\tilde{m},0}_{\mathcal{Q},\text{lac}}(v_1,v_2), \varphi^H_{P} \rangle| \ \ \ \ \ \ \ \text{ if } (\phi_Q^{3,H})_{Q \in \mathcal{Q}} \text{ is a lacunary family}\\
\\
|\langle B^{\tilde{n},\tilde{m},0}_{\mathcal{Q},\text{nonac}}(v_1,v_2), \varphi^H_{P} \rangle| \ \ \ \ \text{ if } (\phi_Q^{3,H})_{Q \in \mathcal{Q}} \text{ is a nonlacunary family},
\end{cases} \end{equation} where the bump functions in $ B^{\tilde{n},\tilde{m},0}_{\mathcal{Q}, \text{lac}}$ or $B^{\tilde{n},\tilde{m},0}_{\mathcal{Q}, \text{nonlac}}$ are the same as the ones in $B^H_{\mathcal{Q},P}$. \vskip .15in \noindent \textbf{Case I. $(\phi^3_K)_K$ is lacunary. } One recalls that in the Haar model, $$
\langle B^H_{\mathcal{Q},P}(v_1,v_2), \varphi_P^H \rangle := \frac{1}{|P|^{\frac{1}{2}}} \sum_{\substack{Q \in \mathcal{Q} \\ |Q| \geq |P|}} \frac{1}{|Q|^{\frac{1}{2}}} \langle v_1, \phi_Q^1\rangle \langle v_2, \phi_Q^2 \rangle \langle \varphi^H_P,\psi_Q^{3,H} \rangle $$ where $\varphi^H_P$ is an $L^2$-normalized indicator function of $P$ and $\psi_Q^{3,H}$ is a Haar wavelet on $Q$ with $P$ and $Q$ being dyadic intervals. It is not difficult to observe that \begin{equation} \label{haar_biest_cond} \langle \varphi^H_P,\psi_Q^{3,H} \rangle \neq 0 \iff Q \supseteq P. \end{equation} Given $P \cap \mathcal{U}_{\tilde{n},\tilde{m}} \neq \emptyset$, one can deduce that $Q \cap \mathcal{U}_{\tilde{n},\tilde{m}} \neq \emptyset$. As a consequence, \begin{align} \label{haar_biest}
\langle B^H_{\mathcal{Q}, P}(v_1,v_2), \varphi^H_P \rangle =& \sum_{\substack{Q \in \mathcal{Q} \\ Q \cap \mathcal{U}_{\tilde{n},\tilde{m}} \neq \emptyset \\ |Q| \geq |P|}} \frac{1}{|Q|^{\frac{1}{2}}} \langle v_1, \phi_Q^1\rangle \langle v_2, \phi_Q^2 \rangle \langle \varphi^H_P,\psi_Q^{3,H} \rangle \nonumber \\
= &\sum_{\substack{Q \in \mathcal{Q} \\ Q \cap \mathcal{U}_{\tilde{n},\tilde{m}} \neq \emptyset}} \frac{1}{|Q|^{\frac{1}{2}}} \langle v_1, \phi_Q^1\rangle \langle v_2, \phi_Q^2 \rangle \langle \varphi^H_P,\psi_Q^{3,H} \rangle. \end{align}
By the definition of $B^{\tilde{n},\tilde{m},0}_{\mathcal{Q},\text{lac}}$ (\ref{B^0_lac}) with the choice the bump functions to be the same as the ones in $B^H_{\mathcal{Q},P}$, one can conclude that \begin{equation*} \langle B^H_P(v_1,v_2), \varphi^H_P \rangle = \langle B^{\tilde{n},\tilde{m},0}_{\mathcal{Q},\text{lac}}(v_1,v_2), \varphi^H_P \rangle, \end{equation*} which is the desired estimate in Case I highlighted in (\ref{goal_localization_en}). \begin{remark}[Biest trick]\label{biest_trick_rmk} In the Haar model, equation (\ref{haar_biest}) trivially holds due to (\ref{haar_biest_cond}). Such technique of replacing the operator defined in terms of $P$ (namely $B_P^H$) by another operator independent of $P$ (namely $B^{\tilde{n},\tilde{m}}_0$) is called \textbf{biest trick} which allows neat energy estimates for \begin{align*} & \text{energy}_{\mathcal{P}'}^t((\langle B^{\tilde{n},\tilde{m},0}_{\mathcal{Q},\text{lac}}(v_1,v_2), \varphi_P^H \rangle)_{P \in \mathcal{P}'}) \text{ and } \text{energy}^{1,\infty}_{\mathcal{P}'}((\langle B^{\tilde{n},\tilde{m},0}_{\mathcal{Q},\text{lac}}(v_1,v_2), \varphi_P^H \rangle)_{P \in \mathcal{P}'}) \end{align*} and yields local energy estimates (\ref{B_en_t}) and (\ref{B_en_1}) described in Proposition \ref{B_en}. \end{remark} \noindent \textbf{Case II: $(\phi^3_Q)_Q$ is non-lacunary. }
Since $\varphi^{3,H}_Q$ and $\varphi_P^H$ are $L^2$-normalized indicator functions of $Q$ and $P$ respectively, $|Q| \geq |P|$ implies that $Q \supseteq P$. As a result, $Q \cap \mathcal{U}_{\tilde{n},\tilde{m}} \neq \emptyset$ given $P \cap \mathcal{U}_{\tilde{n},\tilde{m}} \neq \emptyset$. Then
\begin{align*}
\frac{|\langle B_{\mathcal{Q},P}^H(v_1,v_2), \varphi_P^H \rangle|}{|P|^{\frac{1}{2}}} = & \frac{1}{|P|^{\frac{1}{2}}} \bigg|\sum_{\substack{Q \in \mathcal{Q} \\ Q \supseteq P \\ Q \cap \mathcal{U}_{\tilde{n},\tilde{m}} \neq \emptyset}} \frac{1}{|Q|^{\frac{1}{2}}} \langle v_1, \psi_Q^1\rangle \langle v_2, \psi_Q^2 \rangle \langle \varphi^{H}_P,\varphi_Q^{3,H} \rangle \bigg| \nonumber \\
\leq & \frac{1}{|P|^{\frac{1}{2}}} \sum_{\substack{Q \in \mathcal{Q} \\ Q \supseteq P \\ Q \cap \mathcal{U}_{\tilde{n},\tilde{m}} \neq \emptyset}} \frac{1}{|Q|^{\frac{1}{2}}} |\langle v_1, \psi_Q^1\rangle| |\langle v_2, \psi_Q^2 \rangle| \langle \varphi^{H}_P,|\varphi_Q^{3,H}| \rangle, \end{align*} where the last inequality follows from the fact that $\varphi_P^H$ is an indicator function and thus non-negative. One can drop the condition $Q \supseteq P$ in the sum and bound the above expression by $$
\frac{|\langle B_{\mathcal{Q}, P}^H(v_1,v_2), \varphi_P^H \rangle|}{|P|^{\frac{1}{2}}} \leq \frac{1}{|P|} \sum_{\substack{Q \in \mathcal{Q}\\ Q \cap \mathcal{U}_{\tilde{n},\tilde{m}} \neq \emptyset}} \frac{1}{|Q|^{\frac{1}{2}}} |\langle v_1, \psi_Q^1\rangle| |\langle v_2, \psi_Q^2 \rangle| \langle \varphi_P^H,|\varphi_Q^{3,H}| \rangle. $$
One recalls the definition of $B_{\mathcal{Q},\text{nonlac}}^{\tilde{n},\tilde{m},0}$ (\ref{B^0_nonlac}) with the choice the bump functions to be the same as the ones in $B^H_{\mathcal{Q},P}$ and deduces
$$
\frac{|\langle B_{\mathcal{Q}, P}^H(v_1,v_2), \varphi_P^H \rangle|}{|P|^{\frac{1}{2}}} \leq \frac{|\langle B_{\mathcal{Q},\text{nonlac}}^{\tilde{n},\tilde{m},0}(v_1,v_2), \varphi_P^H \rangle|}{|P|^{\frac{1}{2}}} $$ which agrees with the estimate described in (\ref{goal_localization_en}).
This completes the proof of the lemma. \begin{remark} $B^{\tilde{n},\tilde{m},0}_{\mathcal{Q},\text{lac}}$ and $B^{\tilde{n},\tilde{m},0}_{\mathcal{Q},\text{nonlac}}$ are perfectly localized in the sense that the dyadic intervals (that matter) intersect with $\mathcal{U}_{\tilde{n},\tilde{m}}$ nontrivially given that $P \cap \mathcal{U}_{\tilde{n},\tilde{m}} \neq \emptyset$. As will be seen from the proof of Lemma \ref{B_loc_norm}, such localization is essential in deriving desired estimates. In the general Fourier case, more efforts are needed to create similar localizations as will be discussed in Section \ref{section_fourier}. \end{remark}
\subsection{Proof of Lemma \ref{B_loc_norm}} \label{section_proof_B_loc_norm} The estimates described in Lemma \ref{B_loc_norm} can be obtained by a very similar argument for proving the boundedness of one-parameter paraproducts discussed in Section 2 of \cite{cw}. We would include the customized proof here since the argument depends on a one-dimensional stopping-time decomposition which is also an important ingredient for our tensor-type stopping-time decompositions that will be introduced in later sections. \subsubsection{One-dimensional stopping-time decomposition - maximal intervals} \label{section_size_energy_one_dim_st_maximal} For a sequence $$(a_Q)_{Q \in \mathcal{Q}} := (v, \varphi_Q)_{Q \in \mathcal{Q}} \ \ \text{or} \ \ (v,\varphi^H_Q)_{Q \in \mathcal{Q}},$$ we perform the the following stopping-time decomposition. Given finiteness of the collection of dyadic intervals $\mathcal{Q}$, there exists some $K_1 \in \mathbb{Z}$ such that
$$\frac{|a_Q|}{|Q|^{\frac{1}{2}}} \leq C_1 2^{K_1} \text{energy}_{\mathcal{Q}}((a_Q)_{Q \in \mathcal{Q}}).$$ We can pick the largest interval $Q_{\text{max}}$ such that
$$\frac{|a_{Q_{\text{max}}} |}{|Q_{\text{max}}|^{\frac{1}{2}}} > C_1 2^{K_1-1}\text{energy}_{\mathcal{Q}}((a_Q)_{Q \in \mathcal{Q}}).$$ Then we define a tree $$U:= \{Q \in \mathcal{Q}: Q \subseteq Q_{\text{max}}\},$$ and let $$Q_U := Q_{\text{max}},$$ usually called \textit{tree-top}. Now we look at $\mathcal{Q} \setminus U$ and repeat the above step to choose maximal intervals and collect their subintervals in their corresponding sets. Since $\mathcal{Q}$ is finite, the process will eventually end. We then collect all $U$'s in a set $\mathbb{U}_{K_1-1}$. Next we repeat the above algorithm to $\displaystyle \mathcal{Q} \setminus \bigcup_{U \in \mathbb{U}_{K_1-1}} U$. We thus obtain a decomposition $$\displaystyle \mathcal{Q} = \bigcup_{k}\bigcup_{U \in \mathbb{U}_{k}}U.$$ If, otherwise, the sequence is formed in terms of bump or cutoff functions in a lacunary family, namely $$ (a_Q)_{Q \in \mathcal{Q}} := (v, \psi_Q)_{Q \in \mathcal{Q}} \ \ \text{or} \ \ (v,\psi^H_Q)_{Q \in \mathcal{Q}}, $$ then the same procedure can be performed to $$
\frac{1}{|Q|} \left\Vert \bigg(\sum_{\substack{Q' \in \mathcal{Q} \\ Q' \subseteq Q}}\frac{|a_{Q'}|^2 }{|Q'|}\chi_{Q'}\bigg)^{\frac{1}{2}}\right\Vert_{1,\infty}. $$
\vskip .15in
The next proposition summarizes the information from the stopping-time decomposition and the details of the proof are included in Section 2 of \cite{cw}. \begin{proposition}\label{st_prop} Suppose $\displaystyle \mathcal{Q} = \bigcup_{k}\bigcup_{U \in \mathbb{U}_{k}}U$ is a decomposition obtained from the stopping-time algorithm specified above, then for any $k \in \mathbb{Z}$, one has \begin{equation*} \displaystyle 2^{k-1}\text{energy}_{\mathcal{Q}}((a_Q)_{Q \in \mathcal{Q}}) \leq \text{size}_{\bigcup_{U \in \mathbb{U}_k}U}((a_Q)_{Q \in \mathcal{Q}}) \leq \min(2^{k}\text{energy}_{\mathcal{Q}}((a_Q)_{Q \in \mathcal{Q}}),\text{size}_{\mathcal{Q}}((a_Q)_{Q \in \mathcal{Q}})). \end{equation*} In addition, $$
\sum_{U \in \mathbb{U}_k} |Q_{U}| \lesssim 2^{-k}. $$ \end{proposition}
The next lemma follows from the stopping-time decomposition, Proposition \ref{st_prop} and Proposition \ref{JN}, whose proof is discussed carefully in Section 2.9 of \cite{cw}. It plays an important role in proving Lemma \ref{B_loc_norm} as can be seen in Section \ref{section_pf_B_loc_norm}. Suppose that $\mathcal{Q}$ is a finite collection of dyadic intervals and we would like to estimate \begin{equation} \label{1-parameter-paraproduct}
\sum_{Q \in \mathcal{Q}}\frac{1}{|Q|^{\frac{1}{2}}} a_Q^1 a_Q^2 a_Q^3 \end{equation} where for $1 \leq i \leq 3$, \begin{align*} a_Q^i := \langle v_i,\phi_Q^i \rangle \ \ \text{or} \ \ \langle v_i,\phi_Q^{i,H} \rangle \end{align*} and at least two of the three families of $L^2$-normalized bump or cutoff functions are lacunary. \begin{lemma} \label{s-e}
The trilinear form (\ref{1-parameter-paraproduct}) can be estimated by $$
\bigg|\sum_{Q \in \mathcal{Q}}\frac{1}{|Q|^{\frac{1}{2}}}a_Q^1 a_Q^2 a_Q^3 \bigg| \lesssim \prod_{i=1}^3 \text{size}_{\mathcal{Q}} \big((a_Q^i)_{Q \in \mathcal{Q}} \big)^{1-\theta_i}\text{energy}^{1,\infty}_{\mathcal{Q}}\big((a_Q^i)_{Q \in \mathcal{Q}} \big)^{\theta_i}, $$ for any $0 \leq \theta_1, \theta_2, \theta_3 < 1$ and $\theta_1 + \theta_2 + \theta_3 = 1$. The implicit constant depends on $\theta_1,\theta_2$ and $\theta_3$. \end{lemma}
\begin{comment} \begin{proof}[Proof of Lemma \ref{s-e}] Let $$\displaystyle \mathcal{K} = \bigcup_{k_1,k_2,k_3}\bigcup_{\mathbb{U}_{k_1,k_2,k_3}}U$$ where $\mathbb{U}_{k_1,k_2,k_3}$ contains trees $U$ of the type $\{U_{k_1} \cap U_{k_2}\cap U_{k_3} \}$ such that $U_{k_i}$ is a tree obtained from the stopping-time decomposition with respect to $(\langle f_i, \phi_K \rangle)_{K \in \mathcal{K}} $. For any fixed tree $U$ \end{proof} \end{comment} \begin{comment} \subsubsection{Proof of Lemma \ref{B_global_norm}} \begin{enumerate}[(1)] \item
\textbf{Estimate of $\|B\|_1$.} It suffices to prove that for any $\eta \in L^{\infty}$, the corresponding multilinear form \begin{equation}
|\langle B, \eta\rangle| \lesssim |F_1|^{\theta_1}|F_2|^{\theta_2} \| \eta\|_{\infty} \end{equation} for any $0 \leq \theta_1, \theta_2 <1$ with $\theta_1 + \theta_2 = 1$. We recall the definition of $B$ to expand the multilinear form as follows: \begin{align} \label{B_inf}
|\langle B, \eta\rangle| & \leq \sum_{K}\frac{1}{|K|^{\frac{1}{2}}} |\langle v_1, \phi^1_K\rangle| |\langle v_2, \phi^2_K \rangle| |\langle \eta,\phi^3_K \rangle| \nonumber \\ &\lesssim \text{size}((\langle v_1,\phi^1_K \rangle)_{K \in \mathcal{K}})^{1-\theta_1} \text{size}((\langle v_2,\phi^2_K \rangle)_{K \in \mathcal{K}}) ^{1-\theta_2} \text{size}((\langle \eta,\phi^3_K \rangle)_{K \in \mathcal{K}})^{1-\theta_3} \nonumber \\ & \quad \text{energy}((\langle v_1,\phi^1_K \rangle)_{K \in \mathcal{K}})^{\theta_1}\text{energy}((\langle v_2,\phi^2_K \rangle)_{K \in \mathcal{K}})^{\theta_2} \text{energy}((\langle \eta,\phi^3_K \rangle)_{K \in \mathcal{K}})^{\theta_3}, \end{align} for any $0 \leq \theta_1,\theta_2,\theta_3 < 1$ with $\theta_1 + \theta_2 + \theta_3 = 1$. One now uses the trivial estimates $$ \text{size}((\langle v_1,\phi^1_K \rangle)_{K \in \mathcal{K}}), \text{size}((\langle v_2,\phi^2_K \rangle)_{K \in \mathcal{K}}) \leq 1 $$
since $|v_1| \leq \chi_{F_1}$ and $v_1 \leq |F_2|$ and $$
\text{size}((\langle \chi_S,\phi^3_K \rangle)_{K \in \mathcal{K}}) \lesssim \|\eta\|_{\infty} $$ Moreover, Proposition \ref{energy} implies that \begin{align*}
& \text{energy}((\langle v_1,\phi^1_K \rangle)_{K \in \mathcal{K}}) \lesssim \|v_1\|_1 = |F_1| \nonumber \\
& \text{energy}((\langle v_2,\phi^2_K \rangle)_{K \in \mathcal{K}}) \lesssim \|v_2\|_1 = |F_2|. \nonumber \\ \end{align*} By taking $\theta_3 = 0$, $0 \leq \theta_1, \theta_2 < 1$ with $\theta_1 + \theta_2 = 1$, one has $$
|\langle B, \eta \rangle| \lesssim |F_1|^{\theta_1}|F_2|^{\theta_2} \| \eta\|_{\infty} $$ as desired. \item
\textbf{Estimate of $\|B\|_t$ for $t >1$.}
A similar argument can be applied to estimate the weak norm $\|B\|_{t,\infty}$ and the bound for the strong norm will follow from standard interpolations.
\begin{claim} \label{B_weak} \begin{enumerate}
\item Suppose that $t >1$. Then for every measurable set $F_1, F_2 \subseteq \mathbb{R}_x$ of positive and finite measure and every measurable function $|v_1| \leq \chi_{F_1}$ and $|v_2| \leq \chi_{F_2}$,
$$\|B(v_1,v_2)\|_{t,\infty} \lesssim |F_1|^{\theta_1} |F_2|^{\theta_2}.$$ for any $0 \leq \theta_1, \theta_2 < 1$ and $\theta_1 + \theta_2= \frac{1}{t}$. \item
Suppose that $1 < t < \infty$. Then for every measurable set $F_1\subseteq \mathbb{R}_x$ of positive and finite measure and every measurable function $|v_1| \leq \chi_{F_1}$ and $v_2 \in L^{\infty}$,
$$\|B(v_1,v_2)\|_{t,\infty} \lesssim |F_1|^{\frac{1}{t}} \|v_2\|_{\infty}.$$ \end{enumerate} \end{claim}
\begin{claim} \label{B_strong} Suppose that $1 < p_1 < \infty $ and $ 1 < q_1 \leq \infty$ and $\frac{1}{t} := \frac{1}{p_1}+ \frac{1}{q_1} <1$. Then for any $v_1 \in L^{p_1}$ and $v_2 \in L^{q_1}$,
$$\|B(v_1,v_2)\|_{t} \lesssim \|v_1\|_{L^{p_1}} \|v_2\|_{L^{q_1}}.$$ \end{claim}
\begin{proof}[Proof of Claim (\ref{B_weak})]
\begin{enumerate} \item For any $\chi_S \in L^{t'}$, one consider the corresponding multilinear form and apply the size-energy estimates as follows: \begin{align} \label{B_inf}
|\langle B, \chi_S \rangle| & \leq \sum_{K}\frac{1}{|K|^{\frac{1}{2}}} |\langle f_1, \phi^1_K\rangle| |\langle v_2, \phi^2_K \rangle| |\langle \chi_S,\phi^3_K \rangle| \nonumber \\ &\lesssim \text{size}((\langle f_1,\phi^1_K \rangle)_{K \in \mathcal{K}})^{1-\theta_1} \text{size}((\langle v_2,\phi^2_K \rangle)_{K \in \mathcal{K}})^{1-\theta_2} \text{size}((\langle \chi_S,\phi^3_K \rangle)_{K \in \mathcal{K}})^{1-\theta_3} \nonumber \\ & \quad \text{energy}((\langle f_1,\phi^1_K \rangle)_{K \in \mathcal{K}})^{\theta_1}\text{energy}((\langle v_2,\phi^2_K \rangle)_{K \in \mathcal{K}})^{\theta_2} \text{energy}((\langle \chi_S,\phi^3_K \rangle)_{K \in \mathcal{K}})^{\theta_3}, \end{align} for any $0 \leq \theta_1,\theta_2,\theta_3 < 1$ with $\theta_1 + \theta_2 + \theta_3 = 1$. One now uses the trivial estimates \begin{align*} &\text{size}((\langle f_1,\phi^1_K \rangle)_{K \in \mathcal{K}}), \text{size}((\langle v_2,\phi^2_K \rangle)_{K \in \mathcal{K}}), \text{size}((\langle \chi_S,\phi^3_K \rangle)_{K \in \mathcal{K}}) \nonumber \leq 1 \\
& \text{energy}((\langle f_1,\phi^1_K \rangle)_{K \in \mathcal{K}}) \lesssim \|f_1\|_1 = |F_1| \nonumber \\
& \text{energy}((\langle v_2,\phi^2_K \rangle)_{K \in \mathcal{K}}) \lesssim \|v_2\|_1 = |F_2| \nonumber \\
& \text{energy}((\langle \chi_S,\phi^3_K \rangle)_{K \in \mathcal{K}}) \lesssim \|\chi_S\|_1 = |S|, \end{align*}
since $|f_1| \leq \chi_{F_1}$ and $f_1 \leq |F_2|$. By taking $0 \leq \theta_1, \theta_2 <1$, $\theta_3 = \frac{1}{t'}$ and $\theta_1 + \theta_2 + \theta_3 = 1$, which is possible given that $0 < \frac{1}{t} \leq 1$ and equivalently $0 \leq \frac{1}{t'} < 1$, one has $$
|\langle B, \chi_S \rangle| \lesssim |F_1|^{\theta_1}|F_2|^{\theta_2}|S|^{\frac{1}{t'}} $$ which implies that $$
\|B\|_{t,\infty} \lesssim |F_1|^{\theta_1} |F_2|^{\theta_2}. $$ where $\theta_1 + \theta_2 = \frac{1}{t}$.
\item The second statement can be proved by the same argument with slight modification on $\text{size}_{\mathcal{K}}(\langle v_2, \phi_K^2 \rangle)_{K \in \mathcal{K}}$. Since $v_2$ is a generic function in $L^{\infty}$, one has the trivial estimate $$
\text{size}_{\mathcal{K}}(\langle v_2, \phi_K^2 \rangle)_{K \in \mathcal{K}} \leq \|v_2\|_{\infty} $$ By choosing $\theta_1 = \frac{1}{p}$, $\theta_2 = 0$ and $\theta_3 = \frac{1}{p'}$, one can reduce (\ref{B_inf}) to
$$|\langle B, \chi_S \rangle| \lesssim |F_1|^{\frac{1}{p}}\|v_2\|_{\infty}|S|^{\frac{1}{p'}}$$ which gives the desired result. \end{enumerate} \end{proof} \end{enumerate} \end{comment}
\subsubsection{Proof of Lemma \ref{B_loc_norm}} \label{section_pf_B_loc_norm} \begin{comment} \begin{enumerate} \item
\textbf{Estimate of $ \|B_0^{\tilde{m}\tilde{n},\tilde{m}}\|_1$.}
For any $\eta \in L^{\infty}$ one has \begin{align*}
|\langle B_0^{\tilde{n},\tilde{m}},\eta \rangle |\leq & \sum_{\substack{K \in \mathcal{K} \\ K \cap \mathcal{U}_{\tilde{n},\tilde{m}} \neq \emptyset}} \frac{1}{|K|^{\frac{1}{2}}} |\langle f_1, \phi_K^1\rangle| |\langle v_2, \phi_K^2 \rangle| |\langle \eta, \phi_K^{3} \rangle |, \nonumber \\ \end{align*} where \begin{equation*} \phi^{3}_{K}:=\begin{cases} \psi^{3,H}_K \quad \quad \ \ \text{in Case}\ \ I\\
|\varphi^{3,H}_{K}| \quad \quad \text{in Case} \ \ II. \end{cases} \end{equation*} Let $\mathcal{K}'$ denote the sub-collection $$\mathcal{K'}:= \{K \in \mathcal{K}: K \cap \mathcal{U}_{\tilde{n},\tilde{m}} \neq \emptyset \}.$$ Then, one can now apply Lemma \ref{s-e} to obtain \begin{align*}
& |\langle B_0^{\tilde{n},\tilde{m}}, \eta \rangle | \nonumber \\ \lesssim & \text{\ \ size}_{\mathcal{K}'} ((\langle f_1, \phi^1_K \rangle)_{K})^{1-\theta_1}\text{size}_{\mathcal{K}'}((\langle v_2, \phi^2_K \rangle)_{K})^{1-\theta_2} \text{size}_{\mathcal{K}'}((\langle \eta, \phi^3_K \rangle)_{K})^{1-\theta_3} \nonumber \\ & \text{\ \ energy} _{\mathcal{K}'}((\langle f_1, \phi^1_K\rangle)_{K})^{\theta_1}\text{energy} _{\mathcal{K}'}((\langle v_2, \phi^2_K\rangle)_{K})^{\theta_2} \text{energy} _{\mathcal{K}'}((\langle \eta, \phi^3_K\rangle)_{K})^{\theta_3}, \end{align*} for any $0 \leq \theta_1,\theta_2, \theta_3 <1$ with $\theta_1 + \theta_2 + \theta_3 = 1$. By applying Proposition \ref{size} and the fact that $K \cap \mathcal{U}_{\tilde{n},\tilde{m}} \neq \emptyset$ for any $K \in \mathcal{K}'$, one deduces that \begin{align} \label{f_size}
& \text{size}_{\mathcal{K}'}((\langle f_1, \phi^1_K \rangle)_{K}) \lesssim 2^{\tilde{n}}|F_1|, \nonumber \\
& \text{size}_{\mathcal{K}'}((\langle f_1, \phi^1_K \rangle)_{K}) \lesssim 2^{\tilde{n}}|F_1|. \end{align} One also recalls that $\eta \in L^{\infty}$, which gives \begin{equation} \label{inf_size} \text{size}_{K \in \mathcal{K}}((\langle \eta, \phi^3_K \rangle)_{K}) \lesssim 1. \end{equation} By choosing $\theta_3 = 0$ and combining the estimates (\ref{f_size}), (\ref{inf_size}) with the energy estimates described in Proposition \ref{energy_classical}, one obtains \begin{align}
|\langle B_0^{\tilde{n},\tilde{m}}, \eta \rangle |\lesssim & (C_12^{\tilde{n}}|F_1|)^{\alpha_1(1-\theta_1)} (C_1 2^{\tilde{m}}|F_2|)^{\beta_1(1-\theta_2)} \|\eta\|_{L^{\infty}}|F_1|^{\theta_1}|F_2|^{\theta_2} \nonumber \\
= & C_1^{\alpha_1(1-\theta_1)+ \beta_1(1-\theta_2)}2^{\tilde{n}\alpha_1(1-\theta_1)}2^{\tilde{m}\beta_1(1-\theta_2)}|F_1|^{\alpha_1(1-\theta_1)+\theta_1}|F_2|^{\beta_1(1-\theta_2)+\theta_2} \|\eta\|_{L^{\infty}}, \end{align} where $\theta_1 + \theta_2 = 1$, $ 0 \leq \alpha_1, \beta_1 \leq 1$. Therefore, one can conclude that \begin{align*}
& \|B_0^{\tilde{n},\tilde{m}}\|_1 \lesssim C_1^{\alpha_1(1-\theta_1)+ \beta_1(1-\theta_2)}2^{\tilde{n}\alpha_1(1-\theta_1)}2^{\tilde{m}\beta_1(1-\theta_2)}|F_1|^{\alpha_1(1-\theta_1)+\theta_1}|F_2|^{\beta_1(1-\theta_2)+\theta_2}.
\end{align*} By choosing $\alpha_1(1-\theta_1)+\theta_1 = \frac{1}{p_1}$ and $\beta_1(1-\theta_2)+\theta_2 = \frac{1}{q_1}$ which is possible given $\frac{1}{p_1} + \frac{1}{q_1} > 1$, one obtains the desired result.
\vskip 0.25 in \end{comment}
We will focus on proving (\ref{B^0_norm}) for $\|B_{\mathcal{Q}, \text{lac}}^{\tilde{n},\tilde{m},0}(v_1,v_2)\|_t$ for $t \geq 1$ and the estimate for $\|B_{\mathcal{Q}, \text{nonlac}}^{\tilde{n},\tilde{m},0}(v_1,v_2)\|_t$ follows from the exactly same argument. \begin{enumerate} \item
\textbf{Estimate of $ \|B_{\mathcal{Q}, \text{lac}}^{\tilde{n},\tilde{m},0}(v_1,v_2)\|_1$.}
For any $\eta \in L^{\infty}$ one has \begin{align*}
|\langle B_{\mathcal{Q}.\text{lac}}^{\tilde{n},\tilde{m},0}(v_1,v_2),\eta \rangle |\leq & \sum_{\substack{Q \in \mathcal{Q} \\ Q \cap \mathcal{U}_{\tilde{n},\tilde{m}} \neq \emptyset}} \frac{1}{|Q|^{\frac{1}{2}}} |\langle v_1, \phi_Q^1\rangle| |\langle v_2, \phi_Q^2 \rangle| |\langle \eta, \psi_Q^{3,H} \rangle |. \end{align*}
Let $\mathcal{Q}'$ denote the sub-collection $$\mathcal{Q'}:= \{Q \in \mathcal{Q}: Q \cap \mathcal{U}_{\tilde{n},\tilde{m}} \neq \emptyset \}.$$ Then, one can apply Lemma \ref{s-e} to obtain \begin{align*}
& |\langle B_{\mathcal{Q},\text{lac}}^{\tilde{n},\tilde{m},0}(v_1.v_2), \eta \rangle | \nonumber \\ \lesssim & \text{\ \ size}_{\mathcal{Q}'} ((\langle v_1, \phi^1_Q \rangle)_{Q \in \mathcal{Q}'})^{1-\theta_1}\text{size}_{\mathcal{Q}'}((\langle v_2, \phi^2_Q \rangle)_{Q \in \mathcal{Q}'})^{1-\theta_2} \text{size}_{\mathcal{Q}'}((\langle \eta, \psi^{3,H}_Q \rangle)_{Q \in \mathcal{Q}'})^{1-\theta_3} \nonumber \\ & \text{\ \ energy} _{\mathcal{Q}'}((\langle v_1, \phi^1_Q\rangle)_{Q \in \mathcal{Q}'})^{\theta_1}\text{energy} _{\mathcal{Q}'}((\langle v_2, \phi^2_Q\rangle)_{Q \in \mathcal{Q}'})^{\theta_2} \text{energy} _{\mathcal{Q}'}((\langle \eta, \psi^{3,H}_Q\rangle)_{Q \in \mathcal{Q}'})^{\theta_3}, \end{align*} for any $0 \leq \theta_1,\theta_2, \theta_3 <1$ with $\theta_1 + \theta_2 + \theta_3 = 1$. By Proposition \ref{size} and the fact that $Q \cap \mathcal{U}_{\tilde{n},\tilde{m}} \neq \emptyset$ for any $Q \in \mathcal{Q}'$, one deduces that \begin{align}
& \text{size}_{\mathcal{Q}'}((\langle v_1, \phi^1_Q \rangle)_{Q \in \mathcal{Q}'}) \lesssim \min(2^{\tilde{n}}|V_1|,1) \leq (2^{\tilde{n}}|V_1|)^{\alpha}, \label{f_size1} \\
& \text{size}_{\mathcal{Q}'}((\langle v_2, \phi^2_Q \rangle)_{Q \in \mathcal{Q}'}) \lesssim \min( 2^{\tilde{m}}|V_2|,1) \leq ( 2^{\tilde{m}}|V_2|)^{\beta}. \label{f_size} \end{align} for any $0 \leq \alpha, \beta \leq 1$. One also recalls that $\eta \in L^{\infty}$, which gives \begin{equation} \label{inf_size} \text{size}_{Q \in \mathcal{Q}}((\langle \eta, \psi^{3,H}_Q \rangle)_{Q \in \mathcal{Q}'}) \lesssim 1. \end{equation} \begin{comment} \begin{align*}
& |\langle B_0^{\tilde{n},\tilde{m}}, \eta \rangle | \nonumber \\ \lesssim & \text{\ \ size}_{Q \in \mathcal{Q}: Q \cap \mathcal{U}_{\tilde{n},\tilde{m}} \neq \emptyset}((\langle v_1, \phi^1_Q \rangle)_{Q})^{1-\theta_1}\text{size}_{Q \in \mathcal{Q}:Q \cap \mathcal{U}_{\tilde{n},\tilde{m}} \neq \emptyset}((\langle v_2, \phi^2_Q \rangle)_{Q})^{1-\theta_2} \text{size}_{Q \in \mathcal{Q}}((\langle \eta, \phi^3_Q \rangle)_{Q})^{1-\theta_3} \nonumber \\ & \text{\ \ energy} _{\mathcal{Q}}((\langle v_1, \phi^1_Q\rangle)_{Q})^{\theta_1}\text{energy} _{\mathcal{Q}}((\langle v_2, \phi^2_Q\rangle)_{Q})^{\theta_2} \text{energy} _{\mathcal{Q}}((\langle \eta, \phi^3_Q\rangle)_{Q})^{\theta_3}, \end{align*} for any $0 \leq \theta_1,\theta_2, \theta_3 <1$ with $\theta_1 + \theta_2 + \theta_3 = 1$. By applying Proposition \ref{size}, one deduces that $$ \text{size}_{Q \in \mathcal{Q}}((\langle \eta, \phi^3_Q \rangle)_{Q}) \lesssim 1. $$
and using the fact that $\eta \in L^{\infty}$, one has $$ \text{size}_{Q \in \mathcal{Q}}((\langle \eta, \phi^3_Q \rangle)_{Q}) \lesssim 1. $$ \end{comment}
By choosing $\theta_3 = 0$ and combining the estimates (\ref{f_size1}), (\ref{f_size}) and (\ref{inf_size}) with the energy estimates described in Proposition \ref{energy_classical}, one obtains \begin{align}
|\langle B_{\mathcal{Q},\text{lac}}^{\tilde{n},\tilde{m},0}(v_1,v_2), \eta \rangle |\lesssim & (C_12^{\tilde{n}}|V_1|)^{\alpha(1-\theta_1)} (C_1 2^{\tilde{m}}|V_2|)^{\beta(1-\theta_2)}|V_1|^{\theta_1}|V_2|^{\theta_2} \|\eta\|_{L^{\infty}} \nonumber \\
= & C_1^{\alpha(1-\theta_1)+ \beta(1-\theta_2)}2^{\tilde{n}\alpha(1-\theta_1)}2^{\tilde{m}\beta(1-\theta_2)}|V_1|^{\alpha(1-\theta_1)+\theta_1}|V_2|^{\beta(1-\theta_2)+\theta_2} \|\eta\|_{L^{\infty}}, \end{align} where $0 < \theta_1,\theta_2 < 1$ with $\theta_1 + \theta_2 = 1$ and $ 0 \leq \alpha, \beta \leq 1$. Therefore, one can conclude that \begin{align*}
& \|B_{\mathcal{Q},\text{lac}}^{\tilde{n},\tilde{m},0}(v_1,v_2)\|_1 \lesssim C_1^{\alpha(1-\theta_1)+ \beta(1-\theta_2)}2^{\tilde{n}\alpha(1-\theta_1)}2^{\tilde{m}\beta(1-\theta_2)}|V_1|^{\alpha(1-\theta_1)+\theta_1}|V_2|^{\beta(1-\theta_2)+\theta_2}.
\end{align*} By choosing $\alpha(1-\theta_1)+\theta_1 = \frac{1}{p}$ and $\beta(1-\theta_2)+\theta_2 = \frac{1}{q}$ which is possible given $\frac{1}{p} + \frac{1}{q} > 1$, one obtains the desired result.
\vskip 0.25 in \item
\noindent
\textbf{Estimate of $\| B_{\mathcal{Q}, \text{lac}}^{\tilde{n},\tilde{m},0}(v_1,v_2)\|_{t}$ for $t >1$.} We will first prove restricted weak-type estimates for $B_{\mathcal{Q},\text{lac}}^{\tilde{n},\tilde{m},0}$ specified in Claim \ref{en_weak_p} and then the strong-type estimates in Claim \ref{en_strong_p} follow from the standard interpolation technique. \begin{claim} \label{en_weak_p}
$\| B_{\mathcal{Q}, \text{lac}}^{\tilde{n},\tilde{m},0}(v_1,v_2)\|_{\tilde{t},\infty} \lesssim C_1^{\frac{1}{p} + \frac{1}{q}-\theta_1 -\theta_2}2^{\tilde{n}(\frac{1}{p}-\theta_1)}2^{\tilde{m}(\frac{1}{q}-\theta_2)}|V_1|^{\frac{1}{p}}|V_2|^{\frac{1}{q}},$ \newline where $0 \leq \theta_1,\theta_2 < 1$ with $\theta_1 + \theta_2 = \frac{1}{\tilde{t}}$ and $\tilde{t} \in (t-\delta, t+ \delta)$ for some $\delta > 0 $ sufficiently small. \end{claim}
\begin{claim} \label{en_strong_p}
$\| B_{\mathcal{Q},\text{lac}}^{\tilde{n},\tilde{m},0}(v_1,v_2)\|_{\tilde{t}} \lesssim C_1^{\frac{1}{p} + \frac{1}{q}-\theta_1-\theta_2} 2^{\tilde{n}(\frac{1}{p}-\theta_1)}2^{\tilde{m}(\frac{1}{q}-\theta_2)}|V_1|^{\frac{1}{p}}|V_2|^{\frac{1}{q}},$ \newline where $0 \leq \theta_1,\theta_2 < 1$ with $\theta_1 + \theta_2 = \frac{1}{t}$. \end{claim}
\begin{proof}[Proof of Claim \ref{en_weak_p}] It suffices to apply the dualization and prove that for any $\chi_S \in L^{\tilde{t}'}$, $$
|\langle B_{\mathcal{Q},\text{lac}}^{\tilde{n},\tilde{m},0}(v_1,v_2), \chi_S \rangle| \lesssim 2^{\tilde{n}(\frac{1}{p}-\theta_1)}2^{\tilde{m}(\frac{1}{q}-\theta_2)}|V_1|^{\frac{1}{p}}|V_2|^{\frac{1}{q}}|S|^{\frac{1}{\tilde{t}'}} $$ where $0 \leq \theta_1,\theta_2 < 1$ with $\theta_1 + \theta_2 = \frac{1}{\tilde{t}}$.
The multilinear form can be estimated using a similar argument developed for $\| B_{\mathcal{Q}, \text{lac}}^{\tilde{n},\tilde{m},0}(v_1,v_2)\|_1$. In particular,
\begin{align} \label{linear_form_p}
& |\langle B_{\mathcal{Q},\text{lac}}^{\tilde{n},\tilde{m},0}(v_1,v_2), \chi_S \rangle| \nonumber \\
\lesssim & \text{\ \ size}_{\mathcal{Q}'}((\langle v_1, \phi^1_Q \rangle)_{Q\in \mathcal{Q}'})^{1-\theta_1}\text{size}_{\mathcal{Q}'}((\langle v_2, \phi^2_Q \rangle)_{Q \in \mathcal{Q}'})^{1-\theta_2} \text{size}_{\mathcal{Q}'}((\langle \chi_S, \psi^{3,H}_Q \rangle)_{Q \in \mathcal{Q}'})^{1-\theta_3} \nonumber \\ & \text{\ \ energy} _{\mathcal{Q}'}((\langle v_1, \phi^1_Q\rangle)_{Q \in \mathcal{Q}'})^{\theta_1}\text{energy} _{\mathcal{Q}'}((\langle v_2, \phi^2_Q\rangle)_{Q \in \mathcal{Q}'})^{\theta_2} \text{energy} _{\mathcal{Q}'}((\langle \chi_S , \psi^{3,H}_Q\rangle)_{Q \in \mathcal{Q}'})^{\theta_3}, \end{align} for any $0 \leq \theta_1,\theta_2, \theta_3 <1$ with $\theta_1 + \theta_2 + \theta_3 = 1$. The size and energy estimates involving $v_1, v_2$, namely (\ref{f_size}) and Proposition \ref{energy_classical}, are still valid. One also applies the same straightforward estimates that \begin{align} & \text{size}_{\mathcal{Q}'}((\langle \chi_S , \psi^{3,H}_Q \rangle)_{Q \in \mathcal{Q}'}) \lesssim 1, \label{set_size_en1}\\
& \text{energy} _{\mathcal{Q}'}((\langle \chi_S , \psi^{3,H}_Q\rangle)_{Q \in \mathcal{Q}'}) \lesssim |S|. \label{set_size_en} \end{align} By plugging in the above estimates (\ref{set_size_en1}), (\ref{set_size_en}), (\ref{f_size1}) and (\ref{f_size}) into (\ref{linear_form_p}), one has \begin{align*}
|\langle B_{\mathcal{Q},\text{lac}}^{\tilde{n},\tilde{m},0}(v_1,v_2), \chi_S \rangle| \lesssim & (C_1 2^{\tilde{n}}|V_1|)^{\alpha(1-\theta_1)} (C_1 2^{\tilde{m}}|V_2|)^{\beta(1-\theta_2)}|V_1|^{\theta_1}|V_2|^{\theta_2}|S|^{\theta_3} \nonumber \\
= & C_1^{\alpha(1-\theta_1) + \beta(1-\theta_2)}2^{\tilde{n}\alpha(1-\theta_1)}2^{\tilde{m}\beta(1-\theta_2)}|V_1|^{\alpha(1-\theta_1) + \theta_1} |V_2|^{\beta(1-\theta_2)+ \theta_2} |S|^{\theta_3},
\end{align*} for any $0 \leq \alpha, \beta \leq 1$. Let $\theta_3 = \frac{1}{\tilde{t}'}$, then $\theta_1 + \theta_2 = \frac{1}{\tilde{t}}$. One can then conclude \begin{align*}
\|B_{\mathcal{Q},\text{lac}}^{\tilde{n},\tilde{m},0}(v_1,v_2)\|_{\tilde{t},\infty} \lesssim & C_1^{\alpha(1-\theta_1) + \beta(1-\theta_2)}2^{\tilde{n}\alpha(1-\theta_1)}2^{\tilde{m}\beta(1-\theta_2)}|V_1|^{\alpha(1-\theta_1) + \theta_1} |V_2|^{\beta(1-\theta_2)+ \theta_2}.
\end{align*} Since $\frac{1}{p} + \frac{1}{q} >1$, one can choose $0 \leq \alpha, \beta \leq 1$ and $0 \leq \theta_1,\theta_2 <1$ with $\theta_1 + \theta_2 = \frac{1}{\tilde{t}} \sim \frac{1}{t}$ such that $$ \alpha(1-\theta_1) + \theta_1 = \frac{1}{p}, $$ $$
\beta(1-\theta_2)+ \theta_2 = \frac{1}{q}, $$ the claim then follows. \end{proof} \end{enumerate}
\begin{comment} \begin{claim} \label{en_weak_p}
$\| B_0^{\tilde{n},\tilde{m}}(v_1,v_2)\|_{\tilde{t},\infty} \lesssim C_1^{\frac{1}{p} + \frac{1}{q}-\theta_1 -\theta_2}2^{\tilde{n}(\frac{1}{p}-\theta_1)}2^{\tilde{m}(\frac{1}{q}-\theta_2)}|V_1|^{\frac{1}{p_1}}|V_2|^{\frac{1}{q}},$ \newline where $\theta_1 + \theta_2 = \frac{1}{\tilde{t}}$ and $\tilde{t} \in (t-\delta, t+ \delta)$ for some $\delta > 0 $ sufficiently small. \end{claim}
\begin{claim} \label{en_strong_p}
$\| B_0^{\tilde{n},\tilde{m}}(v_1,v_2)\|_{t} \lesssim C_1^{\frac{1}{p_1} + \frac{1}{q_1}-\theta_1-\theta_2} 2^{\tilde{n}(\frac{1}{p_1}-\theta_1)}2^{\tilde{m}(\frac{1}{q_1}-\theta_2)}|V_1|^{\frac{1}{p_1}}|V_2|^{\frac{1}{q_1}},$ \newline where $\theta_1 + \theta_2 = \frac{1}{t}$. \end{claim}
\begin{proof}[Proof of Claim \ref{en_weak_p}] It suffices to apply the dualization and prove that for any $\chi_S \in L^{\tilde{t}'}$, $$
|\langle B_0^{\tilde{n},\tilde{m}}, \chi_S \rangle| \lesssim 2^{\tilde{n}(\frac{1}{p_1}-\theta_1)}2^{\tilde{m}(\frac{1}{q_1}-\theta_2)}|V_1|^{\frac{1}{p_1}}|V_2|^{\frac{1}{q_1}}|S|^{\frac{1}{\tilde{t}'}} $$ where $\theta_1 + \theta_2 = \frac{1}{\tilde{t}}$.
Consider \begin{align*}
|\langle B_0^{\tilde{n},\tilde{m}},\chi_S \rangle |\leq & \sum_{\substack{Q \in \mathcal{Q} \\ Q \cap \mathcal{U}_{\tilde{n},\tilde{m}} \neq \emptyset}} \frac{1}{|Q|^{\frac{1}{2}}} |\langle v_1, \phi_Q^1\rangle| |\langle v_2, \phi_Q^2 \rangle| |\langle \chi_S, \phi_Q^{3} \rangle |, \nonumber \\ \end{align*} where \begin{equation*} \phi^{3}_{Q}:=\begin{cases} \psi^{3,H}_Q \quad \quad \ \ \text{in Case}\ \ I\\
|\varphi^{3,H}_{Q}| \quad \quad \text{in Case} \ \ II. \end{cases} \end{equation*} Let $\mathcal{Q}'$ denote the sub-collection $$ \mathcal{Q}' := \{Q \in \mathcal{Q}: Q \cap \mathcal{U}_{\tilde{n},\tilde{m}} \neq \emptyset\}. $$ Then one can now apply Lemma \ref{s-e} to obtain \begin{align} \label{linear_form_p}
& |\langle B_0^{\tilde{n},\tilde{m}}, \chi_S \rangle| \nonumber \\
\lesssim & \text{\ \ size}_{\mathcal{Q}'}((\langle v_1, \phi^1_Q \rangle)_{Q})^{1-\theta_1}\text{size}_{\mathcal{Q}'}((\langle v_2, \phi^2_Q \rangle)_{Q})^{1-\theta_2} \text{size}_{\mathcal{Q}'}((\langle \chi_S, \phi^3_Q \rangle)_{Q})^{1-\theta_3} \nonumber \\ & \text{\ \ energy}^{1,\infty} _{\mathcal{Q}'}((\langle v_1, \phi^1_Q\rangle)_{Q})^{\theta_1}\text{energy}^{1,\infty} _{\mathcal{Q}'}((\langle v_2, \phi^2_Q\rangle)_{Q})^{\theta_2} \text{energy}^{1,\infty} _{\mathcal{Q}'}((\langle \chi_S , \phi^3_Q\rangle)_{Q})^{\theta_3}, \end{align} for any $0 \leq \theta_1,\theta_2, \theta_3 <1$ with $\theta_1 + \theta_2 + \theta_3 = 1$. The size and energy estimates involving $v_1, v_2$ in part (1) are still valid. Here $\phi^3_Q$ are defined differently in Case $I$ and $II$. By applying Proposition \ref{size} and the fact that $Q \cap \mathcal{U}_{\tilde{n},\tilde{m}} \neq \emptyset$ for any $Q \in \mathcal{Q}'$, one deduces that \begin{align} \label{f_size}
& \text{size}_{\mathcal{Q}'}((\langle v_1, \phi^1_Q \rangle)_{Q}) \lesssim \min(1,2^{\tilde{n}}|V_1|), \nonumber \\
& \text{size}_{\mathcal{Q}'}((\langle v_2, \phi^1_Q \rangle)_{Q}) \lesssim \min(1,2^{\tilde{m}}|V_2|). \end{align} One also recalls that $\chi_S \in L^{\infty}$, which gives \begin{align} \label{set_size_en} & \text{size}_{\mathcal{Q}'}((\langle \chi_S , \phi^3_Q \rangle)_{Q}) \lesssim 1. \end{align} By plugging in (\ref{set_size_en}), (\ref{f_size}) and the energy estimates described in Proposition \ref{energy_classical} into (\ref{linear_form_p}), one has \begin{align*}
|\langle B_0^{\tilde{n},\tilde{m}}, \chi_S \rangle| \lesssim & (C_1 2^{\tilde{n}}|V_1|)^{\alpha(1-\theta_1)} (C_1 2^{\tilde{m}}|V_2|)^{\beta(1-\theta_2)}|V_1|^{\theta_1}|V_2|^{\theta_2}|S|^{\theta_3} \nonumber \\
= & C_1^{\alpha(1-\theta_1) + \beta(1-\theta_2)}2^{\tilde{n}\alpha(1-\theta_1)}2^{\tilde{m}\beta(1-\theta_2)}|V_1|^{\alpha(1-\theta_1) + \theta_1} |V_2|^{\beta(1-\theta_2)+ \theta_2} |S|^{\theta_3},
\end{align*} for any $0 \leq \alpha, \beta \leq 1$. Let $\theta_3 = \frac{1}{\tilde{t}'}$, then $\theta_1 + \theta_2 = \frac{1}{\tilde{t}}$. One can then conclude \begin{align*}
\|B_0^{\tilde{n},\tilde{m}}\|_{\tilde{t},\infty} \lesssim & C_1^{\alpha(1-\theta_1) + \beta(1-\theta_2)}2^{\tilde{n}\alpha(1-\theta_1)}2^{\tilde{m}\beta(1-\theta_2)}|V_1|^{\alpha(1-\theta_1) + \theta_1} |V_2|^{\beta(1-\theta_2)+ \theta_2}.
\end{align*} Since $\frac{1}{p_1} + \frac{1}{q_1} >1$, one can choose $0 \leq \alpha, \beta \leq 1$ and $\theta_1,\theta_2$ with $\theta_1 + \theta_2 = \frac{1}{\tilde{t}} \sim \frac{1}{t}$ such that $$ \alpha(1-\theta_1) + \theta_1 = \frac{1}{p_1}, $$ $$
\beta(1-\theta_2)+ \theta_2 = \frac{1}{q_1}, $$ the claim then follows. \end{proof} \end{comment}
\begin{comment} \subsection{Proof of Proposition \ref{B_en} (i)} By the definition of $L^{1,\infty}$-energy, given that $\mathcal{I}$ is a finite collection of intervals, there exists $n \in \mathbb{Z}$ and a disjoint collection of dyadic intervals $\mathbb{D}^0_{n}$ such that \begin{align} \label{proof_B_en_1}
\text{energy}^{1,\infty}((\langle B_0^{\tilde{n},\tilde{m}}, \varphi_I \rangle)_{I \in \mathcal{I}''}) = & 2^n \sum_{I \in \mathbb{D}^0} |I| = 2^n \|\sum_{I \in \mathbb{D}_n^0} \chi_I \|_1 \leq \left\| \sum_{I \in \mathbb{D}^0_{n}} \frac{|\langle B^{\tilde{n},\tilde{m}}_0, \varphi_I^H \rangle|}{|I|^{\frac{1}{2}}} \chi_I \right\|_1 \nonumber \\
= & \left\| \sum_{I \in \mathbb{D}^0_{n}}\frac{1}{|I|^{\frac{1}{2}}}\sum_{\substack{K \in \mathcal{K}' \\ }} \frac{1}{|K|^{\frac{1}{2}}} |\langle v_1, \phi_K^1\rangle| |\langle f_2, \phi_K^2\rangle| |\langle \varphi_I^H, \phi_K^{3,H}\rangle| \chi_I\right\|_1 \nonumber \\
\leq &\sum_{I \in \mathbb{D}^0_{n}}\frac{1}{|I|}\sum_{\substack{K \in \mathcal{K}' \\ }} \frac{1}{|K|^{\frac{1}{2}}} |\langle v_1, \phi_K^1\rangle| |\langle f_2, \phi_K^2\rangle| |\langle \varphi_I^H |I|^{\frac{1}{2}}, \phi_K^{3,H}\rangle| \left\| \chi_I\right\|_1 \nonumber \\
\leq & \sup_{K \in \mathcal{K}'} \frac{\langle v_1, \phi_K^1\rangle|}{|K|^{\frac{1}{2}}}\sup_{K \in \mathcal{K}'} \frac{\langle f_2, \phi_K^2\rangle|}{|K|^{\frac{1}{2}}} \sum_{I \in \mathbb{D}^0_{n}}\sum_{K\in \mathcal{K}'}|\langle \varphi_I^H |I|^{\frac{1}{2}}, \phi_K^{3,H}|K|^{\frac{1}{2}}\rangle| \nonumber\\
\lesssim & \sup_{K \in \mathcal{K}'} \frac{\langle v_1, \phi_K^1\rangle|}{|K|^{\frac{1}{2}}}\sup_{K \in \mathcal{K}'} \frac{\langle f_2, \phi_K^2\rangle|}{|K|^{\frac{1}{2}}} \sum_{I \in \mathbb{D}^0_{n}}|I| \end{align}
where the second to last inequality follows from the triangle inequality and the last inequality follows from the fact that $\varphi_I^H |I|^{\frac{1}{2}}$ and $\phi_K^{3,H}|K|^{\frac{1}{2}}$ are $L^{\infty}$-normalized functions supported on $I$ and $K$ respectively. We recall that $$ \mathcal{K}' := \{K \in \mathcal{K}: K \cap \mathcal{U}_{\tilde{n},\tilde{m}} \neq \emptyset\}, $$ which yields the estimate (\ref{f_size}). Furthermore, one uses the disjointness of $\mathbb{D}^0_{n}$ and derive \begin{equation*}
\sum_{I \in \mathbb{D}^0_{n}}|I| = \big|\bigcup_{I \in \mathbb{D}^0_{n}}I \big|. \end{equation*} Since $I \in \mathcal{I}''$ and satisfies (\ref{I_intersect_level}), \begin{equation} \label{I_measure}
\big|\bigcup_{I \in \mathbb{D}^0_{n}}I \big| \lesssim \min(|\{Mv_1 >C_12^{\tilde{n}-1}|V_1| \}|,|\{Mf_2 >C_12^{\tilde{m}-1}|V_2| \}|) \lesssim \min(2^{-\tilde{n}}, 2^{-\tilde{m}}). \end{equation} Applying (\ref{I_measure}) to (\ref{proof_B_en_1}), one can obtain \begin{align*}
(2^{\tilde{n}}|V_1|)^{\alpha}(2^{\tilde{m}}|V_2|)^{\beta}\sum_{I \in \mathbb{D}^0_{n}} |I| \lesssim & (2^{\tilde{n}}|V_1|)^{\alpha}(2^{\tilde{m}}|V_2|)^{\beta} (2^{-\tilde{n}})^{\theta_1} (2^{-\tilde{m}})^{\theta_2} \end{align*} for any $0 \leq \alpha, \beta \leq 1$ and $0 \leq \theta_1, \theta_2 \leq 1$ with $\theta_1 + \theta_2 = 1$. By letting \begin{align*} \alpha_1 &:= \frac{1}{p_1} \\ \beta_1 &:= \frac{1}{q_1}, \end{align*} we complete the proof of (\ref{B_en_1}). \end{comment}
\section{Proof of Theorem \ref{thm_weak_mod} for $\Pi_{\text{flag}^{\#1} \otimes \text{flag}^{\#2}}$ - Haar Model} \label{section_thm_haar_fixed} \begin{comment}
$$\Pi_1^{\#_1,\#_2}(f_1 \otimes g_1, f_2 \otimes g_2, h) := \displaystyle \sum_{I \times J \in \mathcal{I} \times \mathcal{J}} \frac{1}{|I|^{\frac{1}{2}} |J|^{\frac{1}{2}}} \langle B_I^{\#_1,H}(f_1,f_2),\varphi_I^{1,H} \rangle \langle \tilde{B}_J^{\#_2,H}(g_1, g_2), \varphi_J^1 \rangle \langle h, \psi_I^{2,H} \otimes \psi_J^{2,H} \rangle \psi_I^{3,H} \otimes \psi_J^{3,H}$$ where
$$B_I^{\#_1,H}(f_1,f_2)(x) := \displaystyle \sum_{K \in \mathcal{K}:|K| \sim 2^{\#_1} |I|} \frac{1}{|K|^{\frac{1}{2}}}\langle f_1, \phi_K^1 \rangle \langle f_2, \phi_K^2 \rangle \phi_K^3(x),$$
$$\tilde{B}_J^{\#_2,H}(g_1,g_2)(y) := \displaystyle \sum_{L \in \mathcal{L}:|L| \sim 2^{\#_2} |J|} \frac{1}{|L|^{\frac{1}{2}}}\langle g_1, \phi_L^1 \rangle \langle g_2, \phi_L^2 \rangle \phi_L^3(y).$$ \end{comment}
\begin{comment} \begin{theorem} \label{fixed_scale_p} For $ 1 < p_i, q_i < \infty$, $\displaystyle \sum_{i=1}^{3}\frac{1}{p_i} = \sum_{i=1}^{3}\frac{1}{q_i} = \frac{1}{r}$, $$\Pi^l: L_x^{p_1}(L_y^{q_1}) \times L_x^{p_2}(L_y^{q_2}) \times L_x^{p_3}(L_y^{q_3}) \rightarrow L^r.$$ \end{theorem}
\begin{theorem} \label{fixed_scale_inf} For $1<p,q \leq \infty$, $1<s<\infty$, $\displaystyle \frac{1}{p} + \frac{1}{s} = \frac{1}{r}$, $$\Pi^l: L^{p}(\mathbb{R}^2) \times L^{q}(\mathbb{R}^2) \times L^{s}(\mathbb{R}^2) \rightarrow L^r.$$ $$\Pi^l: L_x^{p}(L_y^{\infty}) \times L_x^{\infty}(L_y^{p}) \times L^{s}(\mathbb{R}^2) \rightarrow L^r.$$ \end{theorem} \end{comment}
In this section, we will study the model operator $\Pi_{\text{flag}^{\#1} \otimes \text{flag}^{\#2}}$ in the Haar case. In particular, we will focus on \begin{align} \label{Pi_fixed_haar}
\Pi^H_{\text{flag}^{\#1} \otimes \text{flag}^{\#2}}:= & \displaystyle \sum_{I \times J \in \mathcal{R}} \frac{1}{|I|^{\frac{1}{2}} |J|^{\frac{1}{2}}} \langle B^{\#_1,H}_{\mathcal{K},I}(f_1,f_2),\varphi_I^{1,H} \rangle \langle B^{\#_2,H}_{\mathcal{L},J}(g_1, g_2), \varphi_J^{1,H} \rangle \langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle \psi_I^{3,H} \otimes \psi_J^{3,H}. \end{align} We will first specify the localization for the operator $\Pi^H_{\text{flag}^{\#1} \otimes \text{flag}^{\#2}}$ (\ref{Pi_fixed_haar}),
which can be viewed as a starting point for the stopping-time decompositions. Then we will introduce different stopping-time decompositions used in the estimates.
Finally, we will discuss how to apply information from the multiple stopping-time decompositions to obtain estimates. The organizations of Sections \ref{section_thm_haar}, \ref{section_thm_inf_fixed_haar} and \ref{section_thm_inf_haar} will follow the similar scheme.
\begin{notation} Recall that $B_{\mathcal{K},I}$, $B_{\mathcal{L},J}$, $B_{\mathcal{K},I}^{\#_1}$ and $B^{\#_2}_{\mathcal{L},J}$ are bilinear operators (Definition \ref{B_definition}) that appear in the discrete model stated in Definition \ref{discrete_model_op}. To avoid being overloaded with heavy notation in further discussions, we introduce simplified notation so that the dependence on the collections of dyadic intervals $\mathcal{K}$ and $\mathcal{L}$ are implicit. In particular, let \begin{align} &B_I(f_1,f_2):= B_{\mathcal{K},I}(f_1,f_2), \ \ \tilde{B}_J(g_1,g_2):= B_{\mathcal{L},J}(g_1,g_2), \label{B_local_fourier_simple} \\ &B^{\#_1}_I(f_1,f_2):= B_{\mathcal{K},I}^{\#_1}(f_1,f_2), \ \ \tilde{B}^{\#_2}_J(g_1,g_2):= B^{\#_2}_{\mathcal{L},J}(g_1,g_2).\label{B_fixed_local_fourier_simple} \end{align} Similarly, for the other bilinear operators in Definition \ref{B_definition} and Definition \ref{B_definition_haar}, we adopt the following notation: \begin{align} & B(f_1,f_2):= B_{\mathcal{K}}(f_1,f_2), \ \ \tilde{B}(g_1,g_2):= B_{\mathcal{L}}(g_1,g_2), \label{B_global_proof}\\ & B^H(f_1,f_2):= B^H_{\mathcal{K}}(f_1,f_2), \ \ \tilde{B}^H(g_1,g_2):= B^H_{\mathcal{L}}(g_1,g_2), \label{B_global_haar_simplified} \\ & B_I^H(f_1,f_2):= B^H_{\mathcal{K},I}(f_1,f_2), \ \ \tilde{B}^H_J(g_1,g_2):= B^H_{\mathcal{L},J}(g_1,g_2), \label{B_0_local_haar_simplified} \\ & B^{\#_1,H}_I(f_1,f_2):= B_{\mathcal{K},I}^{\#_1,H}(f_1,f_2), \ \ \tilde{B}^{\#_2,H}_J(g_1,g_2):= B^{\#_2,H}_{\mathcal{L},J}(g_1,g_2). \label{B_fixed_local_haar_simplified} \end{align} \end{notation}
\subsection{Localization} \label{section_thm_haar_fixed_localization} The definition of the exceptional set (the set that would be taken away) settles the starting point for the stopping-time decompositions and thus is expected to be compatible with the stopping-time algorithms involved. There would be two types of stopping-time decompositions undertaken for the estimates of $\Pi^H_{\text{flag}^{\#1} \otimes \text{flag}^{\#2}}$ - one is the \textit{tensor-type stopping-time decomposition} and the other one the \textit{general two-dimensional level sets stopping-time decomposition}. While the second algorithm is related to a generic exceptional set (denoted by $\Omega^2$), the first algorithm aims to integrate information from two one-dimensional decompositions, which corresponds to the creation of a two-dimensional exceptional set (denoted by $\Omega^1$) as a Cartesian product of two one-dimensional exceptional sets.
One defines the exceptional set as follows. Let $$\Omega := \Omega^1 \cup \Omega^2,$$ where \begin{align*}
\displaystyle \Omega^1 := &\bigcup_{\mathfrak{n}_1 \in \mathbb{Z}}\{x: Mf_1(x) > C_1 2^{\mathfrak{n}_1}|F_1|\} \times \{y: Mg_1(y) > C_2 2^{-\mathfrak{n}_1}|G_1|\}\cup \nonumber \\
& \bigcup_{\mathfrak{n}_2 \in \mathbb{Z}}\{x:Mf_2(x) > C_1 2^{\mathfrak{n}_2}|F_2|\} \times \{y:Mg_2(y) > C_2 2^{-\mathfrak{n}_2}|G_2|\}\cup \nonumber \\
&\bigcup_{\mathfrak{n}_3 \in \mathbb{Z}}\{x:Mf_1(x) > C_1 2^{\mathfrak{n}_3}|F_1|\} \times \{y:Mg_2(y) > C_2 2^{-\mathfrak{n}_3}|G_2|\}\cup \nonumber \\
& \bigcup_{\mathfrak{n}_4 \in \mathbb{Z}}\{x:Mf_2 (x)> C_1 2^{\mathfrak{n}_4 }|F_2|\} \times \{y:Mg_1(y) > C_2 2^{-\mathfrak{n}_4 }|G_1|\}, \nonumber \\
\Omega^2 := & \{(x,y) \in \mathbb{R}^2: SSh(x,y) > C_3 \|h\|_{L^s(\mathbb{R}^2)}\}. \end{align*} For technical reason that would become clear later on, we would need to get rid of the enlargement of $\Omega$ defined by $$ Enl(\Omega) := \{(x,y)\in \mathbb{R}^2: MM\chi_{\Omega}(x,y) > \frac{1}{100}\}. $$
\begin{remark} \label{subset} Given by the boundedness of the Hardy-Littlewood maximal operator and the double square function operator, it is not difficult to check that if $C_1, C_2, C_3 \gg 1$, then $
|Enl(\Omega)| \ll 1. $ For different model operators, we will define different exceptional sets based on different stopping-time decompositions to employ. Nevertheless, their measures can be controlled similarly using the boundedness of the operators enclosed in Theorem \ref{maximal-square}.
\end{remark}
By scaling invariance, we will assume without loss of generality that $|E| = 1$ throughout the paper. Let \begin{equation} \label{set_E'} E' := E \setminus Enl(\Omega), \end{equation} then $
|E'| \sim |E|
$ and thus $|E'| \sim 1$.
Our goal is to show that (\ref{thm_weak_explicit}) holds with the corresponding subset (which will be different for each discrete model operator) $E' \subseteq E$.
In the current setting, this is equivalent to proving that the multilinear form \begin{equation} \label{form_haar_larger} \Lambda^H_{\text{flag}^{\#1} \otimes \text{flag}^{\#2}}(f_1, f_2, g_1, g_2, h, \chi_{E'}) := \langle \Pi^H_{\text{flag}^{\#1} \otimes \text{flag}^{\#2}}(f_1, f_2, g_1, g_2, h), \chi_{E'} \rangle \end{equation} satisfies the following restricted weak-type estimate \begin{equation} \label{form_haar_larger_goal}
|\Lambda^H_{\text{flag}^{\#1} \otimes \text{flag}^{\#2}}(f_1, f_2, g_1, g_2, h, \chi_{E'})| \lesssim |F_1|^{\frac{1}{p_1}} |G_1|^{\frac{1}{p_2}} |F_2|^{\frac{1}{q_1}} |G_2|^{\frac{1}{q_2}} \|h\|_{L^{s}(\mathbb{R}^2)}. \end{equation}
\begin{remark}\label{localization_haar_fixed} It is noteworthy that the discrete model operators are perfectly localized to $E'$ in the Haar model. More precisely, \begin{align}\label{haar_local} & \Lambda^H_{\text{flag}^{\#1} \otimes \text{flag}^{\#2}}(f_1, f_2, g_1, g_2, h, \chi_{E'}) \nonumber\\
= & \displaystyle \sum_{I \times J \in \mathcal{R}} \frac{1}{|I|^{\frac{1}{2}} |J|^{\frac{1}{2}}} \langle B^{\#_1,H}_I(f_1,f_2),\varphi_I^{1,H} \rangle \langle \tilde{B}^{\#_2,H}_J(g_1, g_2), \varphi_J^{1,H} \rangle \langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle \langle \chi_{E'}, \psi_I^{3,H} \otimes \psi_J^{3,H} \rangle \nonumber \\
= & \displaystyle \sum_{\substack{I \times J \in \mathcal{R} \\ I \times J \cap Enl(\Omega)^c \neq \emptyset}} \frac{1}{|I|^{\frac{1}{2}} |J|^{\frac{1}{2}}} \langle B^{\#_1,H}_I(f_1,f_2),\varphi_I^{1,H} \rangle \langle \tilde{B}^{\#_2,H}_J(g_1, g_2), \varphi_J^{1,H} \rangle \langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle \langle \chi_{E'}, \psi_I^{3,H} \otimes \psi_J^{3,H} \rangle, \end{align} because for $I \times J \cap Enl(\Omega)^c = \emptyset$, $I \times J \cap E' = \emptyset$ and thus $ \langle \chi_{E'}, \psi_I^{3,H} \otimes \psi_J^{3,H} \rangle = 0, $ which means that dyadic rectangles satisfying $I \times J \cap Enl(\Omega)^c = \emptyset$ do not contribute to the multilinear form. In the Haar model, we would heavily rely on the localization (\ref{haar_local}) and consider only the dyadic rectangles $I \times J \in \mathcal{R}$ such that $I \times J \cap Enl(\Omega)^c \neq \emptyset$. \end{remark}
\begin{comment}
Same procedure can be applied to sequences indexed by $J \in \mathcal{J}$, namely $\Big(\frac{|\langle g_1, \varphi^1_J \rangle|}{|J|^{\frac{1}{2}}}\Big)_{J \in \mathcal{J}}$ and $\Big(\frac{|\langle g_2, \varphi^2_J \rangle|}{|J|^{\frac{1}{2}}}\Big)_{J \in \mathcal{J}}$. We denote the decomposition with respect to the former as $\displaystyle \mathcal{J} = \bigcup_{-n_1}\bigcup_{S \in \mathbb{S}_{-n_1}}S$ and the latter as $\displaystyle \mathcal{J} = \bigcup_{-n_2}\bigcup_{S \in \mathbb{S}_{-n_2}}S$. To combine the two decompositions for $\mathcal{J}$, one defines $$\mathbb{S}_{-n_1,-n_2} := \{S _1 \cap S_2: S\in \mathbb{S}_{-n_1}, S_2 \in \mathbb{S}_{-n_2}\}$$
and write $$\displaystyle \mathcal{J} = \bigcup_{\substack{-n_1\\-n_2}}\bigcup_{S\in \mathbb{S}_{-n_1,-n_2}}S.$$ If one assumes that $I \times J \cap \Omega^{c} \neq \emptyset$, then $I \times J \in T \times S$ with $T \in \mathbb{T}_{l}$, $S \in \mathbb{S}_{-n_1,-n_2}$ such that $l-n_1\leq 0$, $l-n_2 \leq 0$. \end{comment}
\begin{comment} One can perform the following \textit{tensor-type stopping-time decomposition} for dyadic rectangles. Since the collection of the dyadic rectangles is finite, there exists some $N \in \mathbb{Z}$ such that
$$\Omega^{y}_{N} := \{ Mg_1 > C_2 2^{N}|G_1|\}$$ and define
$$\mathcal{J}_{N} := \{J \in \mathcal{J}: |J \cap \Omega^{y}_{N}| > \frac{1}{100}|J| \}.$$ Iteratively define
$$\Omega^{y}_{N-1} := \{ Mg_1 > C_2 2^{N-1}|G_1|\}$$ and
$$\mathcal{J}_{N-1} := \{J \in \mathcal{J} \setminus \mathcal{J}^{N}: |J \cap \Omega^{y}_{N-1}| > \frac{1}{100}|J| \}.$$ \end{comment}
\subsection{Tensor-type stopping-time decomposition I - level sets} \label{section_thm_haar_fixed_tensor}
The first tensor-type stopping time decomposition, denoted by the \textit{tensor-type stopping-time decomposition I}, will be performed to obtain estimates for $\Pi_{\text{flag}^{\#1} \otimes \text{flag}^{\#2}}^H$. It aims to recover intersections with two-dimensional level sets from intersections with one-dimensional level sets for each variable. Another tensor-type stopping-time decomposition, denoted by the \textit{tensor-type stopping-time decomposition II}, involves maximal intervals and plays an important role in the discussion for $\Pi_{\text{flag}^0 \otimes \text{flag}^0}$. We will focus on the \textit{tensor-type stopping-time decomposition I} in this section.
\subsubsection{One-dimensional stopping-time decompositions - level sets} One can perform a one-dimensional stopping-time decomposition on $\mathcal{I} := \{I: I \times J \in \mathcal{R}\}$, or equivalently the collection of $x$-intervals of all dyadic rectangles in $\mathcal{R}$. Since $\mathcal{I}$ is a finite collection of dyadic intervals, there exists $N_1 \in \mathbb{Z}$ such that for any $I \in \mathcal{I}$, \begin{equation*}
|I \cap \{x: Mf_1(x) \leq C_1 2^{N_1+1}|F_1| \}. \end{equation*} Now let
$$\Omega^{}_{N_1} := \{x: Mf_1(x) > C_1 2^{N_1}|F_1|\},$$ and
$$\mathcal{I}_{N_1} := \{I \in \mathcal{I}: |I \cap \Omega^{}_{N_1}| > \frac{1}{10}|I| \}.$$ Define
$$\Omega^{}_{N_1-1} := \{x: Mf_1(x) > C_1 2^{N_1-1}|F_1|\},$$ and
$$\mathcal{I}_{N_1-1} := \{I \in \mathcal{I} \setminus \mathcal{I}_{N_1}: |I \cap \Omega^{}_{N_1-1}| > \frac{1}{10}|I| \}.$$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \vdots \par The procedure generates the sets $(\Omega^{}_{n_1})_{n_1 \in \mathbb{Z}}$ and $(\mathcal{I}_{n_1})_{n_1 \in \mathbb{Z}}$. Independently define
$$\Omega'^{}_{M_1} := \{x: Mf_2(x) > C_1 2^{M_1}|F_2|\},$$ for some $M_1 \in \mathbb{Z}$ and
$$\mathcal{I}'_{M_1} := \{I \in \mathcal{I}: |I \cap \Omega'^{}_{M_1}| > \frac{1}{10}|I| \}.$$ Define
$$\Omega'^{}_{M_1-1} := \{x: Mf_2(x) > C_1 2^{M_1-1}|F_2|\},$$ and
$$\mathcal{I}'_{M_1-1} := \{I \in \mathcal{I} \setminus \mathcal{I}'_{M_1}: |I \cap \Omega'^{}_{M_1-1}| > \frac{1}{10}|I| \}.$$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \vdots \par The procedure generates the sets $(\Omega'^{}_{m_1})_{m_1 \in \mathbb{Z}}$ and $(\mathcal{I}'_{m_1})_{m_1 \in \mathbb{Z}}$. Now define $$\mathcal{I}_{n_1,m_1} := \mathcal{I}_{n_1} \cap \mathcal{I}'_{m_1}$$
and one achieves the decomposition on $\displaystyle \mathcal{I} = \bigcup_{n_1,m_1 \in \mathbb{Z}}\mathcal{I}_{n_1,m_1}$. \par Same algorithm can be applied to $\mathcal{J}:= \{J: I \times J \in \mathcal{R}\}$ with respect to the level sets in terms of $Mg_1$ and $Mg_2$,
which produces the sets \begin{enumerate} [(i)] \item $(\tilde{\Omega}^{}_{n_2})_{n_2 \in \mathbb{Z}}$ and $(\mathcal{J}_{n_2})_{n_2 \in \mathbb{Z}}$, where
$$\tilde{\Omega}^{}_{n_2 } := \{y: Mg_1(y) > C_2 2^{n_2}|G_1|\},$$ and
$$\mathcal{J}_{n_2} := \{J \in \mathcal{J} \setminus \mathcal{J}_{n_2+1}: |J \cap \tilde{\Omega}^{}_{n_2}| > \frac{1}{10}|J| \}.$$ \item $(\tilde{\Omega}'^{}_{m_2})_{m_2 \in \mathbb{Z}}$ and $(\mathcal{J}'_{m_2})_{m_2 \in \mathbb{Z}}$, where
$$\tilde{\Omega}'^{}_{m_2} := \{ y: Mg_2(y) > C_2 2^{m_2}|G_2|\},$$ and
$$\mathcal{J}'_{m_2} := \{J \in \mathcal{J} \setminus \mathcal{J}'_{m_2+1}: |J \cap \tilde{\Omega}'^{}_{m_2}| > \frac{1}{10}|J| \}.$$ \end{enumerate} One thus obtains the decomposition $\displaystyle \mathcal{J} = \bigcup_{n_2, m_2 \in \mathbb{Z}} \mathcal{J}_{n_2,m_2}$, where $\mathcal{J}_{n_2,m_2} := \mathcal{J}_{n_2} \cap \mathcal{J}'_{m_2}$. \begin{comment} Let
$$\Omega^{y}_{N_2} := \{ Mg_1 > C_2 2^{N_2}|G_1|\}$$ for some $N_2 \in \mathbb{Z}$ and define
$$\mathcal{J}_{N_2} := \{J \in \mathcal{J}: |J \cap \Omega^{y}_{N_2}| > \frac{1}{10}|J| \}.$$ Iteratively define
$$\Omega^{y}_{N_2-1} := \{ Mg_1 > C_2 2^{N_2-1}|G_1|\}$$ and
$$\mathcal{J}_{N_2-1} := \{J \in \mathcal{J} \setminus \mathcal{J}^{N_2}: |J \cap \Omega^{y}_{N_2-1}| > \frac{1}{10}|J| \}.$$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \vdots \par The procedure produces the sets $(\Omega^{y}_{n_2})_{n_2}$ and $(\mathcal{J}_{n_2})_{n_2}$. \par Independently define
$$\Omega^{y}_{M_2} := \{ Mg_2 > C_2 2^{M_2}|G_2|\}$$ for some $M_2 \in \mathbb{Z}$ and define
$$\mathcal{J}_{M_2} := \{J \in \mathcal{J}: |J \cap \Omega^{y}_{M_2}| > \frac{1}{10}|J| \}.$$ Iteratively define
$$\Omega^{y}_{M_2-1} := \{ Mg_2 > C_2 2^{M_2-1}|G_2|\}$$ and
$$\mathcal{J}_{M_2-1} := \{J \in \mathcal{J} \setminus \mathcal{J}^{M_2}: |J \cap \Omega^{y}_{M_2-1}| > \frac{1}{10}|J| \}.$$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \vdots \par The procedure produces the sets $(\Omega^{y}_{m_2})_{m_2}$ and $(\mathcal{J}_{m_2})_{m_2}$. Therefore $\displaystyle \mathcal{J} = \bigcup_{n_2, m_2} \mathcal{J}_{n_2,m_2}$, where $\mathcal{J}_{n_2,m_2} := \mathcal{J}_{n_2} \cap \mathcal{J}_{m_2}$. \end{comment} \newline \subsubsection{Tensor product of two one-dimensional stopping-time decompositions - level sets} \label{section_thm_haar_fixed_tensor_1d_level} If we assume that all dyadic rectangles satisfy $I \times J \cap Enl(\Omega)^{c} \neq \emptyset$ as in the Haar model, then we have the following observation. \begin{obs} \label{obs_indice} If $I \times J \in \mathcal{I}_{n_1,m_1} \times \mathcal{J}_{n_2,m_2}$, then $n_1 , m_1 ,n_2, m_2 \in \mathbb{Z}$ satisfies $n_1+n_2 < 0$ and $m_1 + m_2 < 0$. (Equivalently, $\forall I \times J \cap Enl(\Omega)^{c} \neq \emptyset$, $I \times J \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2}$, for some $n_2, m_2 \in \mathbb{Z}$ and $n, m > 0$.) \begin{remark} The observation shows that how a rectangle $I \times J$ intersects with a two-dimensional level sets is closely related to how the corresponding intervals intersect with one-dimensional level sets (namely $I \in \mathcal{I}_{n_1.m_1}$ and $J \in \mathcal{J}_{n_2,m_2}$ with $n_1 + n_2 < 0$ and $m_1 + m_2 < 0$), as commented in the beginning of the section. \end{remark} \begin{proof} Given $I \in \mathcal{I}_{n_1}$, one has
$|I \cap \{x: Mf_1(x) > C_1 2^{n_1}|F_1|\}| > \frac{1}{10} |I|$; similarly, $J \in \mathcal{J}_{n_2}$ implies that $|J \cap \{y: Mg_1(y) > C_2 2^{n_2}|G_1|\}| > \frac{1}{10}|J|$. If $n_1 + n_2 \geq 0$ , then $\{x: Mf_1(x) > C_1 2^{n_1}|F_1|\} \times\{y: Mg_1(y) > C_2 2^{n_2}|G_1|\} \subseteq \Omega^1 \subseteq \Omega$. Then $|I \times J \cap \Omega| > \frac{1}{100}|I \times J|$, which implies that $I \times J \subseteq Enl(\Omega)$ and contradicts with the assumption. Same reasoning applies to the pairs $(m_1,m_2)$, $(n_1,m_2)$ and $(m_1,n_2)$. \end{proof} \end{obs} \begin{remark} Thanks to Obervation \ref{obs_indice}, we conclude that in the Haar model when $I \times J \cap Enl(\Omega)^{c} \neq \emptyset$, \begin{equation*} \mathcal{R} = \bigcup_{\substack{n_1,m_1,n_2,m_2 \in \mathbb{Z} \\ n_1 + n_2 < 0 \\ m_1 + m_2 < 0 \\ n_1 + m_2 < 0 \\ m_1 + n_2 < 0}}\mathcal{I}_{n_1,m_1} \times \mathcal{J}_{n_2,m_2}. \end{equation*} \end{remark}
\begin{comment} Same procedure can be applied with respect to $Mf_2$ and $Mg_2$. The definition of the exceptional set then implies that $I \times J \in \mathcal{I}_{-m+m_1} \times \mathcal{J}_{-m_1}$ for some $m_1 \in \mathbb{Z}$ and $m \geq 0$. \end{comment}
\subsection{General two-dimensional level sets stopping-time decomposition} \label{section_thm_haar_fixed_level} With the assumption that $R \cap Enl(\Omega)^c \neq \emptyset$, one has that $$
|R\cap \Omega^2| \leq \frac{1}{100}|R|, $$ where $$
\Omega^2 = \{(x,y)\in \mathbb{R}^2: SSh(x,y) >C_3 \|h\|_{L^s(\mathbb{R}^2)}\}. $$ Then define
$$\Omega^2_{-1}:= \{(x,y)\in \mathbb{R}^2: SSh(x,y) > C_3 2^{-1}\|h\|_{L^s(\mathbb{R}^2)}\}$$ and
$$\mathcal{R}_{-1} := \{R \in \mathcal{R}: |R \cap \Omega^2_{-1}| > \frac{1}{100}|R|\}.$$ Successively define
$$\Omega^2_{-2}:= \{(x,y)\in \mathbb{R}^2: SSh(x,y) > C_3 2^{-2}\|h\|_{L^s(\mathbb{R}^2)}\} $$ and
$$\mathcal{R}_{-2} := \{R \in \mathcal{R} \setminus \mathcal{R}_{-1}: |R \cap \Omega^2_{-2}| > \frac{1}{100}|R|\}.$$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \vdots \newline This two-dimensional stopping-time decomposition generates the sets $(\Omega^{2}_{k_1})_{k_1 \leq 0}$ and $(\mathcal{R}_{k_1})_{k_1 \leq 0}$. \par
Independently one can apply the same algorithm involving $(SS)^H\chi_{E'}$ which generates $(\Omega'^{2}_{k_2})_{k_2 \leq K}$ and $(\mathcal{R}'_{k_2})_{k_2 \leq K}$ where $K$ can be arbitrarily large. The existence of $K$ is guaranteed by the finite cardinality of the collection of dyadic rectangles. Then define $$ \mathcal{R}_{k_1,k_2} := \mathcal{R}_{k_1} \cap \mathcal{R}'_{k_2}. $$ and thus \begin{equation*} \displaystyle \mathcal{R} = \bigcup_{k_1\leq 0, k_2 \leq K} \mathcal{R}_{k_1,k_2}. \end{equation*}
\subsection{Sparsity condition} \label{section_thm_haar_fixed_sparsity} One important property followed from the \textit{tensor-type stopping-time decomposition I - level sets} is the sparsity of dyadic intervals at different levels. Such geometric property plays an important role in the arguments for the main theorems. \begin{proposition} \label{sparsity} Suppose that $\displaystyle \mathcal{J} = \bigcup_{n_2 \in \mathbb{Z}} \mathcal{J}_{n_2}$ is a decomposition of dyadic intervals with respect to $Mg_1$ as specified in Section \ref{section_thm_haar_fixed_tensor}. For any fixed $n_2 \in \mathbb{Z}$, suppose that $J_0 \in \mathcal{J}_{n_2 - 10}$. Then $$\displaystyle
\sum_{\substack{J \in \mathcal{J}_{n_2}\\J \cap J_0 \neq \emptyset}} |J| \leq \frac{1}{2}|J_0|. $$ \end{proposition} To prove the proposition, one would need the following claim about pointwise estimates for $Mg_1$ on $J \in \mathcal{J}_{n_2}$: \begin{claim} \label{ptwise} Suppose that $\bigcup_{n_2}\mathcal{J}_{n_2}$ is a partition of dyadic intervals generated from the stopping-time decomposition described above. If $J \in \mathcal{J}_{n_2}$, then for any $y \in J,$ $$
Mg_1(y)> 2^{-7} \cdot C_2 2^{n_2}|G_1|. $$ \begin{comment}
Similarly, $J \in \mathcal{J}_{m_2} \Longrightarrow Mg_2 \gtrsim 2^{m_2}|G_2|$ on $J$. \end{comment} \end{claim} \begin{proof}[Proof on Claim $\Longrightarrow$ Proposition \ref{sparsity}] We will first explain why the proposition follows from the claim and then prove the claim. One recalls that all the intervals are dyadic, which means if $J \cap J_0 \neq \emptyset$, then either $$J \subseteq J_0$$ or $$J_0 \subseteq J.$$ If $J_0 \subseteq J$, then the claim implies that
$$J_0 \subseteq J \subseteq \{ Mg_1 > C_2 2^{n_2-7}|G_1|\}.$$ But $J_0 \in \mathcal{J}_{n_2-10}$ infers that
$$ \big|J_0 \cap \{ Mg_1 > C_2 2^{n_2 - 7}|G_1|\}\big| < \frac{1}{10}|J_0|,$$ which is a contradiction. If $J \subseteq J_0$ and suppose that $$ \displaystyle
\sum_{\substack{J \in \mathcal{J}_{n_2}\\J \subseteq J_0}} |J| > \frac{1}{2}|J_0|. $$
Then one can derive from $J \in \mathcal{J}_{n_2}$ that $$\big|J\cap \{Mg_1 > C_2 2^{n_2}|G_1| \} \big| > \frac{1}{10}|J|.$$ Therefore
$$\sum_{\substack{J \in \mathcal{J}_{n_2}\\J \subseteq J_0}} \big|J\cap \{Mg_1 > C_2 2^{n_2}|G_1| \} \big| > \frac{1}{10}\sum_{\substack{J \in \mathcal{J}_{n_2}\\J \subseteq J_0}}|J| > \frac{1}{20}|J_0|.$$ But by the disjointness of $(J)_{J \in \mathcal{J}_{n_2}}$,
$$\sum_{\substack{J \in \mathcal{J}_{n_2}\\J \subseteq J_0}} \big|J\cap \{Mg_1 > C_2 2^{n_2}|G_1| \} \big| \leq \big|J_0\cap \{Mg_1 > C_2 2^{n_2}|G_1| \} \big|.$$ Thus $$
\big|J_0\cap \{Mg_1 > C_2 2^{n_2}|G_1| \} \big| > \frac{1}{20}|J_0|, $$
Now the claim, with slight modifications, implies that $J_0 \subseteq \{Mg_1 > C_2 2^{n_2-8}|G_1| \}$. But $J_0 \in \mathcal{J}_{n_2-10}$, which gives the necessary condition that $$
\big|J_0\cap \{Mg_1 > C_2 2^{n_2}|G_1| \} \big| \leq \frac{1}{10}|J_0| $$ and reaches a contradiction. \end{proof}
We will now prove the claim. \begin{proof}[Proof of Claim]
Without loss of generality, we assume that $g_1$ is non-negative since if it is not, we can always replace it by $|g_1|$ where $Mg_1 = M(|g_1|)$. We prove the claim case by case: \newline Case (i): $\forall y \in \{Mg_1 > C_2 2^{n_2}|G_1|\}$, there exists $J_{y} \subseteq J$ such that $\text{ave}_{J_y}(g_1)\footnote{$\text{ave}_{J} (g_1) := \frac{1}{|J|}\int_J g(s) ds$.} > C_2 2^{n_2}|G_1|;$ \newline Case (ii): There exists $y_0 \in \{Mg_1 > C_2 2^{n_2}|G_1|\}$ and $J_{y_0} \nsubseteq J$ such that $\text{ave}_{J_{y_0}}(g_1) > C_2 2^{n_2}|G_1|$ and \newline
\indent Case (iia): $\frac{1}{40}|J| \leq |J_{y_0} \cap J|$ and $|J_{y_0}| \leq |J|$; \newline
\indent Case (iib): $\frac{1}{40}|J| \leq |J_{y_0} \cap J|$ and $|J_{y_0}| > |J|$; \newline
\indent Case (iic): $|J_{y_0} \cap J| < \frac{1}{40}|J|$. \newline
\textit{Proof of (i):} In Case (i), one observes that $\{Mg_1 > C_2 2^{n_2}|G_1|\} \cap J$ can be rewritten as $\{M(g_1\cdot \chi_J) > C_2 2^{n_2}|G_1|\} \cap J$. Thus
$$C_2 2^{n_2}|G_1||\{Mg_1 > C_2 2^{n_2}|G_1|\} \cap J| = C_2 2^{n_2}|G_1||\{M(g_1\chi_J) > C_2 2^{n_2}|G_1|\} \cap J| \leq \|g_1\chi_J\|_1.$$
One recalls that $|\{Mg_1 > C_2 2^{n_2}|G_1|\} \cap J| > \frac{1}{10}|J|$, which implies that
$$C_2 2^{n_2}|G_1|\cdot \frac{1}{10}|J| \leq \|g_1\chi_J\|_1,$$ or equivalently,
$$\frac{\|g_1\chi_J\|_1}{|J|} \geq \frac{1}{10}C_2 2^{n_2}|G_1|.$$
Therefore $Mg_1 > 2^{-4} C_2 2^{n_2}|G_1|$. \newline
\textit{Proof of (ii)}: We will prove that if either (iia) or (iib) holds, then $Mg_1 > 2^{-7} C_2 2^{n_2}|G_1|$. If neither (iia) nor (iib) happens, then (iic) has to hold and in this case, $Mg_1 > 2^{-7} C_2 2^{n_2}|G_1|$. \par If there exists $y_0 \in \{Mg_1 > C_2 2^{n_2}|G_1|\}$ such that (iia) holds, then
$$\frac{\|g_1 \chi_{J_{y_0}} \|_1}{|J_{y_0}|} \leq \frac{\|g_1 \chi_{J_{y_0} \cup J} \|_1}{|J_{y_0}|} \leq \frac{\|g_1 \chi_{J_{y_0} \cup J} \|_1}{|J_{y_0}\cap J|} \leq \frac{\|g_1 \chi_{J_{y_0} \cup J} \|_1}{\frac{1}{40}|J|},$$
where the last inequality follows from $\frac{1}{40}|J| \leq |J_{y_0} \cap J|$. Moreover, $|J_{y_0}| \leq |J|$ and $y \in J_{y_0} \cap J \neq \emptyset$ infer that $|J_{y_0} \cup J| \leq 2|J|$. Thus
$$\frac{\|g_1 \chi_{J_{y_0} \cup J} \|_1}{\frac{1}{20}|J|} \leq \frac{\|g_1 \chi_{J_{y_0} \cup J} \|_1}{\frac{1}{40}\frac{1}{2}|J_{y_0} \cup J|},$$ which implies
$$\frac{\|g_1 \chi_{J_{y_0} \cup J} \|_1}{|J_{y_0} \cup J|} > \frac{1}{80}C_2 2^{n_2}|G_1|,$$
and as a result $Mg_1 > 2^{-7} C_2 2^{n_2}|G_1|$ on $J$. \par If there exists $y \in \{Mg_1 > C_2 2^{n_2}|G_1|\}$ such that (iib) holds, then
$$\frac{\|g_1 \chi_{J_{y_0}} \|_1}{|J_{y_0}|} \leq \frac{\|g_1 \chi_{J_{y_0} \cup J} \|_1}{|J_{y_0}|} = \frac{2\|g_1 \chi_{J_{y_0} \cup J} \|_1}{2|J_{y_0}|} \leq \frac{2\|g_1 \chi_{J_{y_0} \cup J} \|_1}{|J_{y_0} \cup J|},$$
where the last inequality follows from $|J_{y_0}| > |J|$. As a consequence,
$$\frac{2\|g_1 \chi_{J_{y_0} \cup J} \|_1}{|J_{y_0} \cup J|} > C_2 2^{n_2}|G_1|,$$
and $Mg_1 > 2^{-1} C_22^{n_2}|G_1|$ on $J$. \par
If neither (i), (iia) nor (iib) happens, then for $\mathcal{S}_{(iic)} := \{y: Mg_1(y) > C_2 2^{n_2}|G_1| \text{\ \ and\ \ } (i) \text{\ \ does not hold}\}$,
one direct geometric observation is that $|\mathcal{S}_{(iic)} \cap J| \leq \frac{1}{20}|J|$. In particular, suppose $y \in \mathcal{S}_{(iic)}$, then any $J_{y_0}$ with $\text{ave}_{J_{y_0}}(g_1) > C_2 2^{n_2}|G_1|$ has to contain the left endpoint or right endpoint of $J$, which we denote by $J_{\text{left}}$ and $J_{\text{right}}$. If $J_{\text{left}} \in J_{y_0}$, then the assumption that neither (iia) nor (iib) holds implies that
$$|J_{y_0} \cap J| < \frac{1}{40} |J|,$$ and thus
$$|[J_{\text{left}}, y]| < \frac{1}{40}|J|.$$
Same implication holds true for $y \in \mathcal{S}_{(iic)}$ with $J_{\text{right}} \in J_{y_0}$. Therefore, for any $y \in \mathcal{S}_{(iic)}$, $|[J_{\text{left}}, y]| < \frac{1}{40}|J|$ or $|[y, J_{\text{right}}]| < \frac{1}{40}|J|$, which can be concluded as
$$\big|\mathcal{S}_{(iic)} \cap J\big|< \frac{1}{20}|J|.$$
Since $\big|\{Mg_1> C_2 2^{n_2}|G_1|\} \cap J\big| > \frac{1}{10}|J|$,
$$\bigg|\big(\{Mg_1> C_2 2^{n_2}|G_1|\} \setminus \mathcal{S}_{(iic)}\big) \cap J \bigg| > \frac{1}{20}|J|,$$
in which case one can apply the argument for (i) with $\{Mg_1> C_2 2^{n_2}|G_1|\}$ replaced by $\{Mg_1> C_2 2^{n_2}|G_1|\} \setminus \mathcal{S}_{(iic)}$ to conclude that $$Mg_1 > 2^{-5}C_2 2^{n_2} |G_1|.$$ This ends the proof for the claim. \end{proof} \begin{comment} \begin{remark} The non-negativity of the function is not only a sufficient condition as can be seen in the proof, but also a necessary condition. Let $$ g_1(y) = \begin{cases}
C_2 |G_2| \quad \quad \ \ \ \text{for}\ \ y \in [0,\frac{10}{11}] \\
-10^5 C_2|G_2| \ \ \ \text{for}\ \ y \in (\frac{10}{11}, 1] \\ 0 \quad \quad \quad \quad \quad \ \ \text{elsewhere} \end{cases} $$ Then clearly the dyadic interval $$ [0,1] \in \mathcal{J}_{0}$$ However, $$
Mg_1(y) \leq C_22^{n_2}|G_1| $$ for $y \in (\frac{10}{11}, 1]$. Such observation is important in the sense that the non-negativity is required for the application of the sparsity condition in Proposition \ref{sparsity} and its corollary - Proposition \ref{sp_2d} as specified below. \end{remark} \end{comment}
\begin{proposition}\label{sp_2d} Given an arbitrary finite collection of dyadic rectangles $\mathcal{R}_0$. Define $\mathcal{J}:= \{J: I \times J \in \mathcal{R}_0 \}$. Suppose that $\displaystyle \mathcal{J} = \bigcup_{n_2 \in \mathbb{Z}} \mathcal{J}_{n_2}$ is a decomposition of dyadic intervals with respect to $Mg_1$ as specified in Section \ref{section_thm_haar_fixed_tensor} so that $\displaystyle \mathcal{R}_0 = \bigcup_{n_2 \in \mathbb{Z}} \bigcup_{\substack{R= I \times J \in \mathcal{R}_0 \\ J \in \mathcal{J}_{n_2} \\ }} R $ is a decomposition of dyadic rectangles in $\mathcal{R}_0$. Then
$$
\sum_{n_2 \in \mathbb{Z}} \bigg|\bigcup_{\substack{R = I \times J \in \mathcal{R}_0 \\ J \in \mathcal{J}_{n_2}}}R\bigg| \lesssim \bigg|\bigcup_{R \in \mathcal{R}_0} R \bigg|.
$$ \end{proposition} \begin{proof}[Proof of Proposition \ref{sp_2d}] Proposition \ref{sparsity} gives a sparsity condition for intervals in the $y$-direction, which is sufficient to generate sparsity for dyadic rectangles in $\mathbb{R}^2$. In particular, \begin{align*}
\sum_{n_2 \in \mathbb{Z}} \bigg|\bigcup_{\substack{R= I \times J \in \mathcal{R}_0 \\ J \in \mathcal{J}_{n_2}}}R\bigg| =& \sum_{i = 0}^9 \sum_{n_2 \equiv i \ \ \text{mod} \ \ 10} \bigg|\bigcup_{\substack{R= I \times J \in \mathcal{R}_0 \\ J \in \mathcal{J}_{n_2}}}R\bigg| \nonumber \\
\lesssim & \sum_{i = 0}^9 \bigg|\bigcup_{n_2 \equiv i \ \ \text{mod} \ \ 10}\bigcup_{\substack{R= I \times J \in \mathcal{R}_0 \\ J \in \mathcal{J}_{n_2}}} R \bigg| \nonumber \\
\leq &10 \bigg|\bigcup_{n_2 \in \mathbb{Z}}\bigcup_{\substack{R= I \times J \in \mathcal{R}_0 \\ J \in \mathcal{J}_{n_2}}} R \bigg| \nonumber\\
= & 10 \big|\bigcup_{R \in \mathcal{R}_0} R \big|, \end{align*} where the second inequality follows from the sparsity condition in Proposition \ref{sparsity}. \end{proof} \begin{remark} The picture below illustrates from a geometric point of view why the two-dimensional sparsity condition (Proposition \ref{sp_2d}) follows naturally from the one-dimensional sparsity (Proposition \ref{sparsity}). In the figure, $A_1, A_2 \in \mathcal{I} \times \mathcal{J}_{n_2+20}$, $B \in \mathcal{I} \times \mathcal{J}_{n_2+10}$ and $C \in \mathcal{I} \times \mathcal{J}_{n_2}$ for some $n_2 \in \mathbb{Z}$. \end{remark} \begin{comment} \begin{center} \begin{tikzpicture} \draw[red] (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0); \draw[line width=0.4mm, dashed, blue] (1/2,-2) -- (1/2,4) -- (1,4) -- (1,-2) -- (1/2,-2);
\draw[line width=0.6mm] (1/2,-2) -- (1,-2) -- (1,4) -- (1/2,4) -- (1/2,-2); \draw[line width=0.6mm] (5/4,-4) -- (3/2,-4) -- (3/2,4) -- (5/4,4) -- (5/4,-4); \draw(0.75,3)node{$A_1$}; \draw(1.5,3)node{$A_2$}; \draw(1.75,1.5)node[red]{B}; \draw(3,0.7)node[blue]{C}; \end{tikzpicture} \end{center} \end{comment}
\begin{center} \begin{tikzpicture} \draw[red] (0,0) -- (2,0) -- (2,2) -- (0,2) -- (0,0); \draw[line width=0.6mm] (-2,1/2) -- (4,1/2) -- (4,1) -- (-2,1) -- (-2,1/2); \draw[line width=0.6mm] (-4,5/4) -- (8,5/4) -- (8,3/2) -- (-4,3/2) -- (-4,5/4); \draw[line width=0.4mm, dashed, blue] (1/2,-2) -- (1,-2) -- (1,4) -- (1/2,4) -- (1/2,-2);
\draw(0.75,3)node[blue]{$C$}; \draw(3,1.45)node{$A_2$}; \draw(1.75,1.75)node[red]{$B$}; \draw(3,0.7)node{$A_1$}; \end{tikzpicture} \end{center} \vskip .15in \subsection{Summary of stopping-time decompositions} \label{section_thm_haar_fixed_summary}
\ \ {\fontsize{10}{10} \begin{center} \begin{tabular}{ l l l } I. Tensor-type stopping-time decomposition I on $\mathcal{I} \times \mathcal{J}$ & $\longrightarrow$ & $I \times J \in \mathcal{I}_{n_1,m_1} \times \mathcal{J}_{n_2,m_2}$ \\
& & $(n_1 + n_2 < 0, m_1 + m_2 < 0, $ \\
& & $n_1 + m_2 < 0 , m_1 + n_2 < 0)$ \\ II. General two-dimensional level sets stopping-time decomposition & $\longrightarrow$ & $I \times J \in \mathcal{R}_{k_1,k_2} $ \\ \ \ \ \ on $\mathcal{I} \times \mathcal{J}$ & & $(k_1 <0, k_2 \leq K) $ \end{tabular} \end{center}}
\begin{comment} \subsection{Hybrid of stopping-time decompositions} \begin{center} \begin{tabular}{ c c c } Tensor-type stopping-time decomposition I & $\longrightarrow$ & $I \times J \in \mathcal{I}_{n_1,m_1} \times \mathcal{J}_{n_2,m_2}$ \\ on $\mathcal{I} \times \mathcal{J}$ & & $(n_1 + n_2 < 0, m_1 + m_2 < 0, $ \\
$\Downarrow$ & & $n_1 + m_2 < 0 , m_1 + n_2 < 0)$ \\
General two-dimensional level sets stopping-time decomposition& $\longrightarrow$ & $I \times J \in \mathcal{I}_{n_1,m_1} \times \mathcal{J}_{n_2,m_2} \cap \mathcal{R}_{k_1,k_2} $ \\ on $ \mathcal{I}_{n_1,m_1} \times \mathcal{J}_{n_2,m_2}$ & & $(n_1 + n_2 < 0, m_1 + m_2 < 0,$ \\ & & $n_1 + m_2 < 0 , m_1 + n_2 < 0, $ \\ & & $k_1 <0, k_2 \leq K) $ \end{tabular} \end{center} \end{comment}
\vskip .25in \subsection{Application of stopping-time decompositions} \label{section_thm_haar_fixed_application_st} With the stopping-time decompositions specified above, one can rewrite (\ref{haar_local}) as {\fontsize{9.5}{9.5}\begin{align} \label{form_st_fixed}
&\bigg|\displaystyle \sum_{\substack{n_1 + n_2 < 0 \\ m_1 + m_2 < 0 \\ n_1 + m_2 < 0 \\ m_1 + n_2 < 0 \\ k_1 < 0 \\ k_2 \leq K}} \sum_{\substack{I \times J \in \mathcal{I}_{n_1, m_1} \times \mathcal{J}_{n_2, m_2} \\ \cap \mathcal{R}_{k_1,k_2} \\}} \frac{1}{|I|^{\frac{1}{2}} |J|^{\frac{1}{2}}} \langle B_I^{\#_1,H}(f_1,f_2),\varphi_I^{1,H} \rangle \langle \tilde{B}_J^{\#_2,H}(g_1,g_2),\varphi_J^{1,H} \rangle \langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle \langle \chi_{E'},\psi_I^{3,H} \otimes \psi_J^{3,H} \rangle \bigg| \nonumber \\
\leq & \sum_{\substack{n_1 + n_2 < 0 \\ m_1 + m_2 < 0 \\ n_1 + m_2 < 0 \\ m_1 + n_2 < 0 \\ k_1 < 0 \\ k_2 \leq K}} \sum_{\substack{I \times J \in \mathcal{I}_{n_1,m_1} \times \mathcal{J}_{n_2,m_2} \\ \cap \mathcal{R}_{k_1,k_2}\\ }} \frac{|\langle B_I^{\#_1,H}(f_1,f_2),\varphi_I^{1,H} \rangle|}{|I|^{\frac{1}{2}}} \frac{|\langle \tilde{B}_J^{\#_2,H}(g_1,g_2), \varphi_J^{1,H} \rangle|}{|J|^{\frac{1}{2}}} \cdot \frac{|\langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle|}{|I|^{\frac{1}{2}}|J|^{\frac{1}{2}}} \frac{|\langle \chi_{E'},\psi_I^{3,H} \otimes \psi_J^{3,H} \rangle|}{|I|^{\frac{1}{2}}|J|^{\frac{1}{2}}} |I| |J|.\nonumber\\ \end{align}}
One recalls the \textit{general two-dimensional level sets stopping-time decomposition} that $I \times J \in \mathcal{R}_{k_1,k_2} $ only if
$$ |I\times J \cap (\Omega^2_{k_1})^c | \geq \frac{99}{100}|I\times J|$$
$$ |I\times J \cap (\Omega'^2_{k_2})^{c} | \geq \frac{99}{100}|I \times J|$$
with $\Omega^2_{k_1} := \{SSh(x,y) > C_3 2^{k_1}\|h\|_{L^s(\mathbb{R}^2)}\}$, and $\Omega'^2_{k_2}:= \{(SS)^H\chi_{E'}(x,y) > C_3 2^{k_2}\}$. As a result,
$$|I \times J| \sim |I \times J \cap (\Omega^2_{k_1})^c \cap (\Omega'^2_{k_2})^{c}|.$$ One can therefore majorize (\ref{form_st_fixed}) by {\fontsize{10}{10}\begin{align}\label{form12}
& \sum_{\substack{n_1 + n_2 < 0 \\ m_1 + m_2 < 0 \\ n_1 + m_2 < 0 \\ m_1 + n_2 < 0 \\ k_1 < 0 \\ k_2 \leq K}} \sum_{\substack{I \times J \in \mathcal{I}_{n_1,m_1} \times \mathcal{J}_{n_2,m_2} \cap\mathcal{R}_{k_1,k_2}}} \frac{|\langle B_I^{\#_1,H}(f_1,f_2),\varphi_I^{1,H} \rangle|}{|I|^{\frac{1}{2}}} \frac{|\langle \tilde{B}_J^{\#_2,H}(g_1,g_2), \varphi_J^{1,H} \rangle|}{|J|^{\frac{1}{2}}} \cdot \nonumber \\
&\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad\frac{|\langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle|}{|I|^{\frac{1}{2}}|J|^{\frac{1}{2}}} \frac{|\langle \chi_{E'},\psi_I^{3,H} \otimes \psi_J^{3,H} \rangle|}{|I|^{\frac{1}{2}}|J|^{\frac{1}{2}}} |I\times J \cap (\Omega^2_{k_1})^c \cap (\Omega'^2_{k_2})^c| \nonumber \\
\leq & \sum_{\substack{n_1 + n_2 < 0 \\ m_1 + m_2 < 0 \\ n_1 + m_2 < 0 \\ m_1 + n_2 < 0 \\ k_1 < 0 \\ k_2 \leq K}} \displaystyle \sup_{I \in \mathcal{I}_{n_1,m_1}} \frac{|\langle B_I^{\#_1,H}(f_1,f_2),\varphi_I^{1,H} \rangle|}{|I|^{\frac{1}{2}}} \sup_{J \in \mathcal{J}_{n_2,m_2}} \frac{|\langle \tilde{B}_J^{\#_2,H}(g_1,g_2),\varphi_J^{1,H} \rangle|}{|J|^{\frac{1}{2}}}\cdot \nonumber \\
&\quad \quad \quad \quad \quad \int_{(\Omega^2_{k_1})^c \cap (\Omega'^2_{k_2})^{c}} \sum_{\substack{I \times J \in \mathcal{I}_{n_1,m_1} \times \mathcal{J}_{n_2,m_2} \cap \mathcal{R}_{k_1,k_2}}} \frac{|\langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle|}{|I|^{\frac{1}{2}}|J|^{\frac{1}{2}}} \frac{|\langle \chi_{E'},\psi_I^{3,H} \otimes \psi_J^{3,H} \rangle|}{|I|^{\frac{1}{2}}|J|^{\frac{1}{2}}}\chi_{I}(x) \chi_{J}(y) dx dy. \nonumber\\ \end{align}} We will now estimate each components in (\ref{form12}) separately for clarity. \subsubsection{Estimate for the $integral$} \label{section_thm_haar_fixed_est_integral} One can apply the Cauchy-Schwarz inequality to the integrand and obtain \begin{align} \label{integral12}
& \int_{(\Omega^2_{k_1})^c \cap (\Omega'^2_{k_2})^{c}} \sum_{\substack{I \times J \in \mathcal{I}_{n_1,m_1} \times \mathcal{J}_{n_2,m_2} \cap \mathcal{R}_{k_1,k_2}}} \frac{|\langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle|}{|I|^{\frac{1}{2}}|J|^{\frac{1}{2}}} \frac{|\langle \chi_{E'},\psi_I^{3,H} \otimes \psi_J^{3,H} \rangle|}{|I|^{\frac{1}{2}}|J|^{\frac{1}{2}}}\chi_{I}(x) \chi_{J}(y) dx dy \nonumber \\
\leq & \int_{(\Omega^2_{k_1})^c \cap (\Omega'^2_{k_2})^{c}} \bigg(\sum_{\substack{I \times J \in \mathcal{I}_{n_1,m_1} \times \mathcal{J}_{n_2,m_2}\cap \mathcal{R}_{k_1,k_2}}}\frac{|\langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle|^2}{|I||J|} \chi_I(x)\chi_J(y)\bigg)^{\frac{1}{2}} \nonumber \\
&\quad \quad \quad \quad \quad \quad \bigg(\sum_{\substack{I \times J \in \mathcal{I}_{n_1,m_1} \times \mathcal{J}_{n_2,m_2}\cap \mathcal{R}_{k_1,k_2}}}\frac{|\langle \chi_{E'},\psi_I^{3,H} \otimes \psi_J^{3,H} \rangle|^2}{|I||J|}\chi_{I}(x) \chi_{J}(y)\bigg)^{\frac{1}{2}} dxdy \nonumber \\ \leq &\displaystyle \int_{(\Omega^2_{k_1})^c \cap (\Omega'^2_{k_2})^{c}} SSh(x,y) (SS)^H\chi_{E'}(x,y) \cdot \chi(\displaystyle\bigcup_{\substack{I \times J \in \mathcal{I}_{n_1,m_1} \times \mathcal{J}_{n_2,m_2} \cap \mathcal{R}_{k_1,k_2}}}I \times J)(x,y) dxdy. \end{align} Based on the \textit{general two-dimensional level sets stopping-time decomposition}, the hybrid functions have pointwise control on the domain for integration. In particular, for any $(x,y) \in (\Omega^2_{k_1})^c \cap (\Omega'^2_{k_2})^{c}$, \begin{align*}
& SSh(x,y) \lesssim C_3 2^{k_1} \|h\|_{L^s(\mathbb{R}^2)}, \nonumber \\ & (SS)^H\chi_{E'}(x,y) \lesssim C_3 2^{k_2}. \end{align*} As a result, the integral can be estimated by \begin{align}\label{h_integral}
C_3^2 2^{k_1}\|h\|_{L^s(\mathbb{R}^2)} 2^{k_2} \bigg| \bigcup_{\substack{I \times J \in \mathcal{I}_{n_1,m_1} \times \mathcal{J}_{n_2,m_2} \cap \mathcal{R}_{k_1,k_2}}}I \times J \bigg|. \end{align}
\begin{comment} For each fixed interval $I \in \mathcal{I}_{-n-n_2,-m-m_2}$, the term $
\frac{|\langle B^l_I(f_1,f_2),\varphi_I^1 \rangle|}{|I|^{\frac{1}{2}}} $ is well-behaved, which generates a better estimate for $\text{size}((\langle B^l_I(f_1,f_2),\varphi_I^1 \rangle)_{I})$ than for $\text{size}((\langle B_I(f_1,f_2),\varphi_I^1 \rangle)_{I \in \mathcal{I}_{-n-n_2,-m-m_2}})$. In particular, \begin{align*}
\frac{|\langle B^l_I(f_1,f_2),\varphi_I^1 \rangle|}{|I|^{\frac{1}{2}}} \leq & \frac{1}{|I|}\sum_{K:|K|\sim 2^l|I|}\frac{1}{|K|^{\frac{1}{2}}}|\langle f_1, \varphi_K^1 \rangle| |\langle f_2, \psi_K^2 \rangle| |\langle \tilde{\chi}_I,\psi_K^3 \rangle| \nonumber \\
= & \frac{1}{|I|}\sum_{K:|K|\sim 2^l|I|}\frac{|\langle f_1, \varphi_K^1 \rangle|}{|K|^{\frac{1}{2}}} \frac{|\langle f_2, \psi_K^2 \rangle|}{|K|^{\frac{1}{2}}} |\langle \tilde{\chi}_I, |K|^{\frac{1}{2}}\psi_K^3 \rangle|. \nonumber \\ \end{align*}
In the Haar model, $|K|>|I|$ implies that $I \subseteq K$. Given $I \in \mathcal{I}_{-n-n_2,-m-m_2}$, one has
$$I \cap \{ Mf_1 < C_1 2^{-n-n_2}|F_1|\} \neq \emptyset$$
$$I \cap \{Mf_2 < C_1 2^{-m-m_2}|F_2|\} \neq \emptyset$$
Thus $I \subseteq K$ gives that
$$K \cap \{ Mf_1 < C_1 2^{-n-n_2}|F_1|\} \neq \emptyset$$
$$K \cap \{Mf_2 < C_1 2^{-m-m_2}|F_2|\} \neq \emptyset$$ The expression can be majorized by
\begin{align*}
& \frac{1}{|I|}2^{-n-n_2}|F_1|2^{-m-m_2}|F_2| \sum_{K:|K|\sim 2^l|I|} |\langle \tilde{\chi}_I, |K|^{\frac{1}{2}}\psi_K^3 \rangle| \nonumber \\
\lesssim & \frac{1}{|I|}2^{-n-n_2}|F_1|2^{-m-m_2}|F_2| |I|\nonumber \\
= & 2^{-n-n_2}|F_1|2^{-m-m_2}|F_2|,
\end{align*}
where the second inequality follows from the fact that fix an interval $I$, $K$'s in our sum are of the same length and thus are disjoint. \end{comment}
\subsubsection{Estimate for $ \displaystyle \sup_{I \in \mathcal{I}_{n_1,m_1}} \frac{|\langle B_I^{\#_1,H}(f_1,f_2),\varphi_I^{1,H} \rangle|}{|I|^{\frac{1}{2}}} $ and $
\displaystyle \sup_{J \in \mathcal{J}_{n_2,m_2}} \frac{|\langle \tilde{B}_J^{\#_2,H}(g_1,g_2),\varphi_J^{1,H} \rangle|}{|J|^{\frac{1}{2}}}$} One recalls the algorithm in the \textit{tensor type stopping-time decomposition I - level sets}, which incorporates the following information. $$ I \in \mathcal{I}_{n_1, m_1}$$ implies that
$$|I \cap \{x: Mf_1(x) < C_1 2^{n_1}|F_1|\}| \geq \frac{9}{10}|I|,$$
$$|I \cap \{x: Mf_2(x) < C_1 2^{m_1}|F_2|\}| \geq \frac{9}{10}|I|,$$ which translates into $$
I \cap \{x: Mf_1(x) < C_1 2^{n_1}|F_1|\} \cap \{x:Mf_2(x) < C_1 2^{m_1}|F_2|\} \neq \emptyset. $$ Then one can recall Proposition \ref{size_cor} with \begin{equation*}
\mathcal{U}_{n_1,m_1}:= \{x: Mf_1 (x)< C_1 2^{n_1}|F_1|\} \cap \{x:Mf_2 (x)< C_1 2^{m_1}|F_2|\} \end{equation*} to estimate \begin{comment}
apply Lemma \ref{B_size} with $S: = \{ Mf_1 < C_1 2^{-n-n_2}|F_1|\} \cap \{Mf_2 < C_1 2^{-m-m_2}|F_2|\}$ together with the estimates: $$
\sup_{K \cap S \neq \emptyset} \frac{\langle f_1, \phi_K^1\rangle }{|K|^{\frac{1}{2}}} \leq C_1 2^{n_1}|F_1| $$ $$
\sup_{K \cap S \neq \emptyset} \frac{\langle f_2, \phi_K^1\rangle }{|K|^{\frac{1}{2}}} \leq C_1 2^{m_1}|F_2| $$
and the trivial estimates followed from the fact that $|f_i| \leq \chi_{F_i}$ for $ i = 1,2$: $$
\sup_{K \cap S \neq \emptyset} \frac{\langle f_1, \phi_K^1\rangle }{|K|^{\frac{1}{2}}} \leq 1 $$ $$
\sup_{K \cap S \neq \emptyset} \frac{\langle f_2, \phi_K^1\rangle }{|K|^{\frac{1}{2}}} \leq 1 $$ One can derive that \end{comment} $$
\sup_{I \in \mathcal{I}_{n_1,m_1}} \frac{|\langle B_I^{\#_1,H}(f_1,f_2),\varphi_I^{1,H} \rangle|}{|I|^{\frac{1}{2}}} \lesssim C_1^2 (2^{n_1}|F_1|)^{\alpha_1} (2^{m_1}|F_2|)^{\beta_1}, $$ for any $0 \leq \alpha_1,\beta_1 \leq 1$. Similarly, one can apply Proposition \ref{size_cor} with \begin{equation*}
\tilde{\mathcal{U}}_{n_2,m_2}:= \{y: Mg_1 (y)< C_2 2^{n_2}|G_1|\} \cap \{y:Mg_2(y) < C_2 2^{m_2}|G_2|\} \end{equation*} to conclude that $$
\sup_{J \in \mathcal{J}_{n_2,m_2}} \frac{|\langle \tilde{B}_J^{\#_2,H}(g_1,g_2),\varphi_J^{1,H} \rangle|}{|J|^{\frac{1}{2}}} \lesssim C_2^2 (2^{n_2}|G_1| )^{\alpha_2}(2^{m_2}|G_2|)^{\beta_2}, $$ for any $0 \leq \alpha_2,\beta_2 \leq 1$. By choosing $$\alpha_1 = \frac{1}{p_1}, \beta_1 = \frac{1}{q_1}, \alpha_2 = \frac{1}{p_2}, \beta_2 = \frac{1}{q_2}, $$ and applying (\ref{h_integral}), (\ref{form12}) can therefore be estimated by \begin{align} \label{linear_form_fixed_scale}
& C_1^2 C_2^2 C_3^2\sum_{\substack{n_1 + n_2 < 0 \\ m_1 + m_2 < 0 \\ n_1 + m_2 < 0 \\ m_1 + n_2 < 0 \\ k_1 < 0 \\ k_2 \leq K}} 2^{n_1 \frac{1}{p_1}}2^{m_1\frac{1}{q_1}}2^{n_2 \frac{1}{p_2}} 2^{m_2 \frac{1}{q_2}}|F_1|^{\frac{1}{p_1}}|F_2|^{\frac{1}{q_1}}|G_1|^{\frac{1}{p_2}}|G_2|^{\frac{1}{q_2}}\cdot 2^{k_1} \| h \|_{L^s} 2^{k_2} \cdot \bigg|\bigcup_{\substack{ R \in \mathcal{I}_{n_1,m_1} \times \mathcal{J}_{n_2,m_2} \cap \mathcal{R}_{k_1,k_2} }} R\bigg| . \end{align} One recalls that $$ \frac{1}{p_1} + \frac{1}{q_1} = \frac{1}{p_2} + \frac{1}{q_2}, $$ then \begin{align} \label{exp_size} 2^{n_1 \frac{1}{p_1}}2^{m_1\frac{1}{q_1}}2^{n_2 \frac{1}{p_2}} 2^{m_2 \frac{1}{q_2}} = & 2^{n_1\frac{1}{p_2}} 2^{n_1(\frac{1}{q_2} - \frac{1}{q_1})}2^{m_1 \frac{1}{q_1}} 2^{n_2\frac{1}{p_2}}2^{m_2(\frac{1}{q_2} - \frac{1}{q_1})} 2^{m_2\frac{1}{q_1}} \nonumber \\ = & (2^{n_1 + n_2})^{\frac{1}{p_2}} (2^{n_1+m_2})^{\frac{1}{q_2} - \frac{1}{q_1}}(2^{m_1+m_2})^{\frac{1}{q_1}}. \end{align} By the definition of exceptional sets, $ 2^{n_1 + n_2} \lesssim 1, 2^{n_1 + m_2} \lesssim 1, 2^{m_1 + n_2} \lesssim 1, 2^{m_1 + m_2} \lesssim 1 $. Then $$ n := -(n_1 + n_2) \geq 0, $$ $$ m := -(m_1 + m_2) \geq 0. $$ Without loss of generality, one further assumes that $\frac{1}{q_2} \geq \frac{1}{q_1}$ (with $q_1$ and $q_2$ swapped in the opposite case), which implies that $$ (2^{n_1+m_2})^{\frac{1}{q_2}- \frac{1}{q_1}} \lesssim 1. $$ Now (\ref{linear_form_fixed_scale}) can be bounded by \begin{align} \label{linear_almost}
& C_1^2 C_2^2 C_3^2\sum_{\substack{n > 0 \\ m > 0 \\ k_1 < 0 \\ k_2 \leq K}} \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}} 2^{-n \frac{1}{p_2}}2^{-m \frac{1}{q_1}}|F_1|^{\frac{1}{p_1}}|F_2|^{\frac{1}{q_1}}|G_1|^{\frac{1}{p_2}}|G_2|^{\frac{1}{q_2}}\cdot 2^{k_1} \| h \|_{L^s} 2^{k_2} \cdot \bigg|\bigcup_{\substack{R \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2} \cap \mathcal{R}_{k_1,k_2} \\ }} R\bigg|. \end{align} With $k_1, k_2, n, m$ fixed, one can apply the sparsity condition (Proposition \ref{sp_2d}) repeatedly and obtain the following bound for the expression {\fontsize{8.5}{8.5} \begin{align} \label{nested_area}
\sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}} \bigg|\bigcup_{\substack{R\in \mathcal{R}_{k_1,k_2}\cap \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2}}} R \bigg|
\lesssim \sum_{m_2 \in \mathbb{Z}} \bigg| \bigcup_{\substack{R \in \mathcal{R}_{k_1,k_2} \cap \mathcal{I}_{-m-m_2} \times \mathcal{J}_{m_2}}} R \bigg| \lesssim& \bigg|\bigcup_{R\in \mathcal{R}_{k_1,k_2}} R\bigg| \leq \min\left( \big|\bigcup_{R\in \mathcal{R}_{k_1}} R\big|, \big|\bigcup_{R\in \mathcal{R}_{k_2}} R\big|\right).
\end{align}} \begin{remark} The arbitrariness of the collection of rectangles in Proposition \ref{sp_2d} provides the compatibility of different stopping-time decompositions. In the current setting, the notation $\mathcal{R}_0$ in Proposition \ref{sp_2d} is chosen to be $\mathcal{R}_{k_1, k_2}$. The sparsity condition allows one to combine the \textit{tensor-type stopping-time decomposition I} and \textit{general two-dimensional level sets stopping-time decomposition} and to obtain information from both stopping-time decompositions. \end{remark} \begin{remark} The readers who are familiar with the proof of single-parameter paraproducts \cite{cw} or bi-parameter paraproducts \cite{cptt}, \cite{cw} might recall that (\ref{nested_area}) employs a different argument from the previous ones \cite{cptt}, \cite{cw}. In particular, by previous reasonings, one would fix $n_2, m_2 \in \mathbb{Z}$ and obtain \begin{equation} \label{old}
\bigg|\bigcup_{\substack{R\in \mathcal{R}_{k_1,k_2} \cap \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2}}} R \bigg| \lesssim \min \left( \big|\bigcup_{R\in \mathcal{R}_{k_1}} R\big|, \big|\bigcup_{R\in \mathcal{R}_{k_2}} R\big| \right). \end{equation} However, the expression on the right hand side of (\ref{old}) is independent of $n_2$ or $m_2$, which gives a divergent series when the sum is taken over all $n_2, m_2 \in \mathbb{Z}$. This explains the novelty and necessity of the sparsity condition (Proposition \ref{sp_2d}) for our argument. \end{remark} To estimate the right hand side of (\ref{nested_area}), one recalls from the \textit{general two-dimensional level sets stopping-time decomposition} that $R \in \mathcal{R}_{k_1}$ implies
$$\big|R \cap \Omega^2_{k_1-1} \big| > \frac{1}{100}|R|,$$ or equivalently
$$\displaystyle \bigcup_{R\in \mathcal{R}_{k_1}} R \subseteq \{(x,y)\in \mathbb{R}^2: MM (\chi_{\Omega^2_{k_1-1}})(x,y) > \frac{1}{100}\}.$$ As a result, \begin{align} \label{rec_area_1}
\bigg|\bigcup_{R\in \mathcal{R}_{k_1}} R\bigg| \leq & \big|\{(x,y): MM (\chi_{\Omega^2_{k_1-1}})(x,y) > \frac{1}{100}\}\big| \lesssim |\Omega^2_{k_1-1}|=|\{(x,y): SSh(x,y) > C_3 2^{k_1} \|h\|_{L^s(\mathbb{R}^2)}\}| \lesssim C_3^{-s}2^{-k_1s}, \end{align}
where the second and last inequalities follow from the boundedness of the double maximal function and the double square function described in Theorem \ref{maximal-square}. By a similar reasoning and the fact that $|E'| \sim 1$, \begin{align} \label{rec_area_2}
\bigg|\bigcup_{R\in \mathcal{R}_{k_2}} R\bigg|
\lesssim
|\{(x,y): (SS)^H(\chi_{E'})(x,y) > C_3 2^{k_2}\}| \lesssim C_3^{-\gamma}2^{-k_2\gamma}, \end{align} for any $\gamma >1$. Interpolation between (\ref{rec_area_1}) and (\ref{rec_area_2}) yields \begin{equation} \label{int_area}
\bigg|\bigcup_{R\in \mathcal{R}_{k_1,k_2}} R\bigg| \lesssim 2^{-\frac{k_1s}{2}}2^{-\frac{k_2\gamma}{2}}, \end{equation} and by plugging (\ref{int_area}) into (\ref{nested_area}), one has \begin{equation} \label{rec_area_hybrid}
\sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}} \bigg|\bigcup_{\substack{R\in \mathcal{R}_{k_1,k_2} \cap\mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2}}} R \bigg| \lesssim 2^{-\frac{k_1s}{2}}2^{-\frac{k_2\gamma}{2}}, \end{equation} for any $\gamma >1$. One combines the estimates (\ref{rec_area_hybrid}) and (\ref{linear_almost}) to obtain \begin{align*}
|\Lambda^H_{\text{flag}^{\#1} \otimes \text{flag}^{\#2}}(f_1, f_2, g_1, g_2, h, \chi_{E'})| \lesssim &C_1^2 C_2^2 C_3^2 \sum_{\substack{n > 0 \\ m > 0 \\ k_1 < 0 \\ k_2 \leq K}} 2^{-n \frac{1}{p_2}}2^{-m \frac{1}{q_1}}|F_1|^{\frac{1}{p_1}}|F_2|^{\frac{1}{p_2}}|G_1|^{\frac{1}{q_1}}|G_2|^{\frac{1}{q_2}}\cdot 2^{k_1(1-\frac{s}{2})} \| h \|_{L^s} 2^{k_2(1-\frac{\gamma}{2})}. \nonumber \end{align*} The geometric series $\displaystyle \sum_{k_1<0}2^{k_1(1-\frac{s}{2})}$ is convergent given that $s <2$. For $\displaystyle\sum_{k_2 \leq K}2^{k_2(1-\frac{\gamma}{2})}$, one can choose $\gamma >1$ to be sufficiently large for the range $0 \leq k_2 \leq K$ and $\gamma >1$ and close to $1$ for $k_2 <0$. One thus concludes that \begin{align*}
|\Lambda^H_{\text{flag}^{\#1} \otimes \text{flag}^{\#2}}(f_1, f_2, g_1, g_2, h, \chi_{E'})| \lesssim &C_1^2 C_2^2 C_3^2 |F_1|^{\frac{1}{p_1}}|F_2|^{\frac{1}{p_2}}|G_1|^{\frac{1}{q_1}}|G_2|^{\frac{1}{q_2}}\| h \|_{L^s}. \end{align*}
\begin{remark} One important observation is that thanks to Lemma \ref{B_size}, the sizes $$
\sup_{I \in \mathcal{I}_{n_1,m_1}} \frac{|\langle B_I^{\#_1,H}(f_1,f_2),\varphi_I^{1,H} \rangle|}{|I|^{\frac{1}{2}}} $$ and $$
\sup_{J \in \mathcal{J}_{n_2,m_2}} \frac{|\langle \tilde{B}_J^{\#_2,H}(g_1,g_2),\varphi_J^{1,H} \rangle|}{|J|^{\frac{1}{2}}} $$ can be estimated in the exactly same way as $$ \text{size}_{\mathcal{I}_{n_1}}\big( (f_1,\phi_I)_I \big) \cdot \text{size}_{\mathcal{I}_{m_1}}\big( (f_2,\phi_I)_I \big) $$ and $$ \text{size}_{\mathcal{J}_{n_2}}\big( (g_1,\phi_J)_J \big) \cdot \text{size}_{\mathcal{J}_{m_2}}\big( (g_2,\phi_J)_J \big) $$ respectively. Based on this observation, it is not difficult to verify that under the Haar assumption, the discrete models $\Pi_{\text{flag}^{\#1} \otimes \text{paraproduct}}$ and $\Pi_{\text{paraproduct}\otimes \text{paraproduct}}$
can be estimated by an essentially same argument as $\Pi_{\text{flag}^{\#_1}\otimes \text{flag}^{\#_2}}$ while $\Pi_{\text{flag}^0 \otimes \text{flag}^{\#_2}}$ can be studied similarly as $\Pi_{\text{flag}^0 \otimes \text{paraproduct}}$. \end{remark}
\vskip .15in \section{Proof of Theorem \ref{thm_weak_mod} for $\Pi_{\text{flag}^0 \otimes \text{flag}^0}$ - Haar Model} \label{section_thm_haar} \begin{comment}
$$\Pi (f_1 \otimes g_1, f_2 \otimes g_2, h) := \displaystyle \sum_{I \times J \in \mathcal{I} \times \mathcal{J}} \frac{1}{|I|^{\frac{1}{2}} |J|^{\frac{1}{2}}} \langle B_I^H(f_1,f_2),\varphi_I^1 \rangle \langle \tilde{B_J}(g_1, g_2), \varphi_J^1 \rangle \langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle \psi_I^{3,H} \otimes \psi_J^{3,H}$$ where
$$B_I^H(f_1,f_2)(x) := \displaystyle \sum_{K \in \mathcal{K}:|K| \geq |I|} \frac{1}{|K|^{\frac{1}{2}}}\langle f_1, \varphi_K^1 \rangle \langle f_2, \psi_K^2 \rangle \psi_K^3(x),$$
$$\tilde{B}_J(g_1,g_2)(y) := \displaystyle \sum_{L \in \mathcal{L}:|L| \geq |J|} \frac{1}{|L|^{\frac{1}{2}}}\langle g_1, \varphi_L^1 \rangle \langle g_2, \psi_L^2 \rangle \psi_L^3(y).$$ \end{comment}
\begin{comment} Then for $ 1 < p_i, q_i < \infty$, $\displaystyle \sum_{i=1}^{3}\frac{1}{p_i} = \sum_{i=1}^{3}\frac{1}{q_i} = \frac{1}{r}$, $$\Pi: L_x^{p_1}(L_y^{q_1}) \times L_x^{p_2}(L_y^{q_2}) \times L_x^{p_3}(L_y^{q_3}) \rightarrow L^r.$$ \end{comment} We would now derive estimates for the operator $\Pi_{\text{flag}^0 \otimes \text{flag}^0}$ in the Haar model, namely \begin{equation} \label{Pi_larger_haar}
\Pi^H_{\text{flag}^0 \otimes \text{flag}^0}(f_1,f_2,g_1,g_2,h):= \sum_{I \times J \in \mathcal{R}} \frac{1}{|I|^{\frac{1}{2}} |J|^{\frac{1}{2}}} \langle B_I^H(f_1,f_2),\varphi_I^{1,H} \rangle \langle \tilde{B}_J^H(g_1,g_2),\varphi_J^{1,H} \rangle \langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle \psi_I^{3,H} \otimes \psi_J^{3,H}. \end{equation} where $B_I^H$ and $B^H_J$ are defined in (\ref{B_0_local_haar_simplified}) and (\ref{B_local_definition_haar}). The argument in Section \ref{section_thm_haar_fixed} is not sufficient for $\Pi^H_{\text{flag}^0 \otimes \text{flag}^0}$ (\ref{Pi_larger_haar}) because the localized size \begin{align*}
& \sup_{I \cap S \neq \emptyset} \frac{|\langle B_I^H(f_1,f_2), \varphi^{1,H}_I \rangle |}{|I|^{\frac{1}{2}}}, \nonumber \\
& \sup_{J \cap S' \neq \emptyset} \frac{|\langle \tilde{B}_J^H(g_1,g_2), \varphi^{1,H}_J \rangle }{|J|^{\frac{1}{2}}} \end{align*} cannot be controlled without information about corresponding level sets. In particular, one needs to impose the additional assumption that \begin{align*}
&I \cap \{x: MB^H(f_1,f_2)(x) \leq C_1 2^{l_1}\|B^H(f_1,f_2)\|_1\} \neq \emptyset, \nonumber \\
&J \cap \{y: M\tilde{B}^H(g_1,g_2)(y) \leq C_2 2^{l_2}\|\tilde{B}^H(g_1,g_2)\|_1\} \neq \emptyset, \end{align*} where $B^H$ and $\tilde{B}^H$ are defined in (\ref{B_global_haar_simplified}) and (\ref{B_global_haar}).
However, while the sizes of $B^H(f_1,f_2)$ and $\tilde{B}^H(g_1,g_2)$ can be controlled in this way, they lose the information from the localization (e.g. $K \cap \{x: Mf_1(x) \leq C_1 2^{n_1}|F_1|\} \neq \emptyset$ for some $n_1 \in \mathbb{Z}$) and are thus far away from satisfaction. It is indeed the energies which capture such local information and compensate for the loss from size estimates in this scenario. \subsection{Localization} \label{section_thm_haar_localization} As one would expect from the definition of the exceptional set, the \textit{tensor-type stopping-time decompositions} and the \textit{general two-dimensional level sets stopping-time decomposition} are involved in the argument. We define the set $$\Omega := \Omega^1 \cup \Omega^2,$$ where \begin{align*}
\displaystyle \Omega^1 := &\bigcup_{n_1 \in \mathbb{Z}}\{x:Mf_1(x) > C_1 2^{n_1}|F_1|\} \times \{y:Mg_1(y) > C_2 2^{-n_1}|G_1|\}\cup \nonumber \\
& \bigcup_{m_1 \in \mathbb{Z}}\{x: Mf_2(x) > C_1 2^{m_1}|F_2|\} \times \{y:Mg_2(y) > C_2 2^{-m_1}|G_2|\}\cup \nonumber \\
&\bigcup_{l_1 \in \mathbb{Z}} \{x: MB^H(f_1,f_2)(x) > C_1 2^{l_1}\| B^H(f_1,f_2)\|_1\} \times \{y: M\tilde{B}^H(g_1,g_2)(y) > C_2 2^{-l_1}\| \tilde{B}^H(g_1,g_2) \|_1\},\nonumber \\
\Omega^2 := & \{(x,y) \in \mathbb{R}^2: SSh(x,y) > C_3 \|h\|_{L^s(\mathbb{R}^2)}\}, \nonumber \\ \end{align*} and $$Enl(\Omega) := \{(x,y)\in \mathbb{R}^2: MM\chi_{\Omega}(x,y) > \frac{1}{100}\}.$$ Let
$$E' := E \setminus Enl(\Omega).$$ Then the argument in Remark \ref{subset} yields that $|E'| \sim |E|$ where $|E|$ can be assumed to be 1 by scaling invariance. We aim to prove that the multilinear form \begin{equation} \Lambda^H_{\text{flag}^{0} \otimes \text{flag}^{0}}(f_1, f_2, g_1, g_2, h, \chi_{E'}) := \langle \Pi^H_{\text{flag}^{0} \otimes \text{flag}^{0}}(f_1, f_2, g_1, g_2, h), \chi_{E'} \rangle \end{equation} satisfies the following restricted weak-type estimate \begin{equation}
|\Lambda^H_{\text{flag}^{0} \otimes \text{flag}^{0}}(f_1, f_2, g_1, g_2, h, \chi_{E'})| \lesssim |F_1|^{\frac{1}{p_1}} |G_1|^{\frac{1}{p_2}} |F_2|^{\frac{1}{q_1}} |G_2|^{\frac{1}{q_2}} \|h\|_{L^{s}(\mathbb{R}^2)}. \end{equation} The localization argument in Remark \ref{localization_haar_fixed} can be applied so that \begin{align} \label{form_localized}
& |\Lambda^H_{\text{flag}^{0} \otimes \text{flag}^{0}}(f_1, f_2, g_1, g_2, h, \chi_{E'})| \nonumber\\
= &\displaystyle \sum_{\substack{I \times J \in \mathcal{R} \\ I \times J \cap Enl(\Omega)^c \neq \emptyset}} \frac{1}{|I|^{\frac{1}{2}} |J|^{\frac{1}{2}}} \langle B^{H}_I(f_1,f_2),\varphi_I^{1,H} \rangle \langle \tilde{B}^{H}_J(g_1, g_2), \varphi_J^{1,H} \rangle \langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle \langle \chi_{E'}, \psi_I^{3,H} \otimes \psi_J^{3,H} \rangle. \end{align} \vskip .15in \subsection{Tensor-type stopping-time decomposition II - maximal intervals} \label{section_thm_haar_tensor} \subsubsection{One-dimensional stopping-time decomposition - maximal intervals} \begin{comment} A one-dimensional stopping-time decomposition described in Section 5.4.1 can be performed to the sequence $(\langle B_I^H, \varphi_I\rangle)_{I}$. We denote the decomposition as $$\displaystyle \mathcal{I} = \bigcup_{l_1}\bigcup_{T \in \mathbb{T}_{l_1}}T$$ where $$T := \{I \in \mathcal{I}: I \subseteq I_{T}\}$$ with $I_T$ being the corresponding tree-top. Similarly, \end{comment} One applies the stopping-time decomposition described in Section \ref{section_size_energy_one_dim_st_maximal} to the sequences $$
\big(\frac{|\langle B^H_{I}(f_1,f_2), \varphi^{1,H}_I \rangle|}{|I|^{\frac{1}{2}}}\big)_{I \in \mathcal{I}} $$ and $$
\big(\frac{|\langle \tilde {B}^H_{J}(g_1,g_2), \varphi^{1,H}_J \rangle|}{|J|^{\frac{1}{2}}}\big)_{J \in \mathcal{J}} $$
We will briefly recall the algorithm and introduce some necessary notations for the sake of clarity. Since $\mathcal{I}$ is finite, there exists some $L_1 \in \mathbb{Z}$ such that for any $I \in \mathcal{I}$, $\frac{|\langle B^H_{I}(f_1,f_2), \varphi^{1,H}_I \rangle|}{|I|^{\frac{1}{2}}} \leq C_1 2^{L_1} \|B^H(f_1,f_2)\|_1$. There exists a largest interval $I_{\text{max}}$ such that
$$\frac{|\langle B^H_{I_{\text{max}}}(f_1,f_2), \varphi^{1,H}_{I_{\text{max}}} \rangle|}{|I_{\text{max}}|^{\frac{1}{2}}} \geq C_1 2^{L_1-1}\|B^H(f_1,f_2)\|_1.$$ Then we define a \textit{tree} $$T := \{I \in \mathcal{I}: I \subseteq I_{\text{max}}\},$$ and the corresponding \textit{tree-top} $$I_T := I_{\text{max}}.$$ Now we repeat the above step on $\mathcal{I} \setminus T$ to choose maximal intervals and collect their subintervals in their corresponding sets, which will end thanks to the finiteness of $\mathcal{I}$. Then collect all $T$'s in a set $\mathbb{T}_{L_1-1}$ and repeat the above algorithm to $\displaystyle \mathcal{I} \setminus \bigcup_{T \in \mathbb{T}_{L_1-1}} T$. Eventually the algorithm generates a decomposition $$\displaystyle \mathcal{I} = \bigcup_{l_1}\bigcup_{T \in \mathbb{T}_{l_1}}T.$$ One simple observation is that the above procedure can be applied to general sequences indexed by dyadic intervals. One can thus apply the same algorithm to $\mathcal{J} := \{J: I \times J \in \mathcal{R}\}$. We denote the decomposition as $$\displaystyle \mathcal{J} = \bigcup_{l_2}\bigcup_{S \in \mathbb{S}_{l_2}}S$$ with respect to the sequence
$$\big(\frac{|\langle \tilde {B}^H_{J}(g_1,g_2), \varphi^{1,H}_J \rangle|}{|J|^{\frac{1}{2}}}\big)_{J \in \mathcal{J}},$$ where $S$ is a collection of dyadic intervals analogous to $T$ and is denoted by \textit{tree}. And $J_S$ represents the corresponding \textit{tree-top} analogous to $I_T$.
\vskip.15in \subsubsection{Tensor product of two one-dimensional stopping-time decompositions - maximal intervals}\label{section_thm_haar_tensor_1d_maximal} \begin{obs} \label{obs_st_B} If $I \times J \cap Enl(\Omega)^{c} \neq \emptyset$ and $I \times J \in T \times S$ with $T \in \mathbb{T}_{l_1}$ and $S \in \mathbb{S}_{l_2}$, then $l_1, l_2 \in \mathbb{Z}$ satisfies $l_1 + l_2 < 0$. Equivalently, $I \times J \in T \times S$ with $T \in \mathbb{T}_{-l - l_2}$ and $S \in \mathbb{S}_{l_2}$ for some $l_2 \in \mathbb{Z}$, $l> 0$.
\begin{proof}
$I \in T$ with $T \in \mathbb{T}_{l_1}$ means that $I \subseteq I_T$ where $\frac{|\langle B^H_{I_T}(f_1,f_2), \varphi^{1,H}_{I_T} \rangle|}{|I_T|^{\frac{1}{2}}} > C_12^{l_1} \|B^H(f_1,f_2)\|_1$. By the biest trick (Remark \ref{biest_trick_rmk}),
$$\frac{|\langle B^H_{I_T}(f_1,f_2), \varphi^{1,H}_{I_T} \rangle|}{|I_T|^{\frac{1}{2}}} = \frac{|\langle B^H(f_1,f_2), \varphi^{1,H}_{I_T} \rangle|}{|I_T|^{\frac{1}{2}}} \leq MB^H(f_1,f_2)(x)$$ for any $x \in I_T$. Thus
$$I_T \subseteq \{x: MB^H(f_1,f_2)(x) > C_12^{l_1} \|B^H(f_1,f_2)\|_1\}.$$ By a similar reasoning, $J \in S$ with $S \in \mathbb{S}_{l_2}$ implies that
$$J \subseteq J_S \subseteq \{y: M\tilde{B}^H(g_1,g_2) (y)> C_22^{l_2} \|\tilde{B}^H(g_1,g_2)\|_1\}.$$ If $l_1 + l_2 \geq 0$, then
$$\{x: MB^H(f_1,f_2)(x) > C_12^{l_1} \|B^H(f_1,f_2)\|_1\} \times \{y: M\tilde{B}^H(g_1,g_2)(y) > C_2 2^{l_2}\| \tilde{B}^H(g_1,g_2)\|_1\} \subseteq \Omega^1 \subseteq \Omega.$$ As a consequence, $I \times J \subseteq \Omega \subseteq Enl(\Omega)$, which is a contradiction. \end{proof} \end{obs} \vskip .15in
\subsection{Summary of stopping-time decompositions} \label{section_thm_haar_summary} The notions of \textit{tensor-type stopping-time decomposition I} and \textit{general two-dimensional level sets stopping-time decomposition} introduced in Section \ref{section_thm_haar_fixed} will be applied without further specifications.
{\fontsize{9.5}{9.5} \begin{center} \begin{tabular}{ l l l } I. Tensor-type stopping-time decomposition I on $\mathcal{I} \times \mathcal{J}$& $\longrightarrow$ & $I \times J \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2}$ \\ & & $(n_2, m_2 \in \mathbb{Z}, n > 0, m>0)$\\ II. Tensor-type stopping-time decomposition II on $\mathcal{I} \times \mathcal{J}$ & $\longrightarrow$ & $I \times J \in T \times S$,with $T \in \mathbb{T}_{-l-l_2}$, $S \in \mathbb{S}_{l_2}$\\ & & $(l_2 \in \mathbb{Z}, l> 0)$\\ III. General two-dimensional level sets stopping-time decomposition& $\longrightarrow$ & $I \times J \in \mathcal{R}_{k_1,k_2} $ \\ \ \ \ \ \ on $\mathcal{I} \times \mathcal{J}$& & $(k_1 <0, k_2 \leq K)$\\ \end{tabular} \end{center}}
\begin{comment} \subsection{Hybrid of stopping-time decompositions} \begin{center} \begin{tabular}{ c c c } One-dimensional stopping-time decomposition& $\longrightarrow$ & $K \in \mathcal{K}_{n_0}$ \\ on $\mathcal{K}$ & & $(n_0 \in \mathbb{Z})$ \\ \end{tabular} \end{center}
\begin{center} \begin{tabular}{ c c c } One-dimensional stopping-time decomposition& $\longrightarrow$ & $L \in \mathcal{L}_{n'_0}$ \\ on $\mathcal{L}$ & & $(n'_0 \in \mathbb{Z})$ \\ \end{tabular} \end{center}
\begin{table}[h!] \begin{tabular}{ c c c } Tensor-type stopping-time decomposition I& $\longrightarrow$ & $I \times J \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2}$ \\ on $\mathcal{I} \times \mathcal{J}$ && \\ & & $(n_2, m_2 \in \mathbb{Z}, n > 0)$\\ $\Downarrow$ & & \\ General two-dimensional level sets stopping-time decomposition& $\longrightarrow$ & $I \times J \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2} \cap \mathcal{R}_{k_1,k_2} $ \\ on $\mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2}$ & & \\ & & $(n_2, m_2 \in \mathbb{Z}, n > 0,k_1 <0, k_2 \leq K)$\\ $\Downarrow$& & \\ Tensor-type stopping-time decomposition II & $\longrightarrow$ & $I \times J \in \big(\mathcal{I}_{-n-n_2,-m-m_2} \cap T\big) \times \big(\mathcal{J}_{n_2,m_2} \cap S\big) \cap \mathcal{R}_{k_1,k_2} $\\ on $I \times J \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2} \cap \mathcal{R}_{k_1,k_2} $ & & with $T \in \mathbb{T}_{-l-l_2}$, $S \in \mathbb{S}_{l_2}$ \\ & & $(n_2, m_2, l_2 \in \mathbb{Z}, n, l > 0,k_1 <0, k_2 \leq K, )$\\ \end{tabular} \end{table} \end{comment}
\subsection{Application of stopping-time decompositions} \label{section_thm_haar_application_st} One first rewrites (\ref{form_localized}) with the partition of dyadic rectangles specified in the stopping-time algorithm: \begin{align} \label{form_decomposed}
&\displaystyle \sum_{\substack{n > 0 \\ m > 0 \\ l > 0 \\ k_1 < 0 \\ k_2 \leq K}} \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z} \\ l_2 \in \mathbb{Z}}}\sum_{\substack{T \in \mathbb{T}_{-l-l_2} \\ S \in \mathbb{S}_{l_2}}}\sum_{\substack{I \times J \in T \times S \\ I \times J \in \mathcal{I}_{-n - n_2, -m - m_2} \times \mathcal{J}_{n_2, m_2} \\I \times J \in \mathcal{R}_{k_1,k_2}}} \frac{1}{|I|^{\frac{1}{2}} |J|^{\frac{1}{2}}}| \langle B_I^H(f_1,f_2),\varphi_I^{1,H} \rangle| |\langle \tilde{B}_J^H(g_1,g_2),\varphi_J^{1,H} \rangle| \cdot \nonumber \\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ |\langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle| |\langle \chi_{E'},\psi_I^{3,H} \otimes \psi_J^{3,H} \rangle|.
\end{align}
One can now apply the exactly same argument in Section \ref{section_thm_haar_fixed_est_integral} to estimate (\ref{form_decomposed}) by \begin{align} \label{form_00}
&\sum_{\substack{n > 0 \\ m > 0 \\ l > 0 \\ k_1 < 0 \\ k_2 \leq K}} \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z} \\ l_2 \in \mathbb{Z}}} \displaystyle & \sup_{I \in T} \frac{|\langle B_I^H(f_1,f_2),\varphi_I^{1,H} \rangle|}{|I|^{\frac{1}{2}}} \sup_{J \in S} \frac{|\langle \tilde{B}_J^H(g_1,g_2),\varphi_J^{1,H} \rangle|}{|J|^{\frac{1}{2}}} \cdot 2^{k_1} \| h \|_{L^s} 2^{k_2} \sum_{\substack{T \in \mathbb{T}_{-l-l_2}\\S \in \mathbb{S}_{l_2}}} \bigg|\bigcup_{\substack{I \times J \in T \times S \\ I \times J \in \mathcal{I}_{-n - n_2, -m - m_2} \times \mathcal{J}_{n_2,m_2} \\I \times J \in \mathcal{R}_{k_1,k_2}}} I \times J \bigg|.
\end{align}
\begin{comment} \begin{align}\label{form00}
|\Lambda_{\text{flag}^{0} \otimes \text{flag}^{0}}| \lesssim & \displaystyle \sum_{\substack{n > 0 \\ m > 0 \\ l > 0 \\ k_1 < 0 \\ k_2 \leq K}} \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z} \\ l_2 \in \mathbb{Z}}}\sum_{\substack{T \in \mathbb{T}_{-l-l_2} \\ S \in \mathbb{S}_{l_2}}}\sum_{\substack{I \times J \in \mathcal{I}_{-n - n_2, -m - m_2} \cap{T} \times \mathcal{J}_{n_2, m_2} \cap S \\I \times J \in \mathcal{R}_{k_1,k_2}}} \frac{|\langle B_I^H(f_1,f_2),\varphi_I^1 \rangle|}{|I|^{\frac{1}{2}} |J|^{\frac{1}{2}}} \frac{|\langle \tilde{B}_J(g_1,g_2),\varphi_J^1 \rangle| }{|I|^{\frac{1}{2}} |J|^{\frac{1}{2}}}\nonumber \\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot \frac{|\langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle|}{|I|^{\frac{1}{2}} |J|^{\frac{1}{2}}} \frac{|\langle \chi_{E'},\psi_I^{3,H} \otimes \psi_J^{3,H} \rangle|}{|I|^{\frac{1}{2}} |J|^{\frac{1}{2}}} |I \times J \cap (\Omega^2_{k_1})^c \cap (\Omega^2_{k_2})^c| \nonumber \\
= & \sum_{\substack{n > 0 \\ m > 0 \\ l > 0 \\ k_1 < 0 \\ k_2 \leq K}} \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z} \\ l_2 \in \mathbb{Z}}}\sum_{\substack{T \in \mathbb{T}_{-l-l_2}\\S \in \mathbb{S}_{l_2}}} \displaystyle \sup_{I \in T} \frac{|\langle B_I^H(f_1,f_2),\varphi_I^1 \rangle|}{|I|^{\frac{1}{2}}} \sup_{J \in S} \frac{|\langle \tilde{B}_J(g_1,g_2),\varphi_J^1 \rangle|}{|J|^{\frac{1}{2}}} \cdot \nonumber \\
&\int_{(\Omega^2_{k_1})^c \cap (\Omega^2_{k_2})^c} \sum_{\substack{I \times J \in \mathcal{I}_{-n - n_2, -m - m_2} \cap{T} \times \mathcal{J}_{n_2,m_2} \cap S\\I \times J \in \mathcal{R}_{k_1,k_2}}} \frac{|\langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle|}{|I|^{\frac{1}{2}}|J|^{\frac{1}{2}}} \frac{|\langle \chi_{E'},\psi_I^{3,H} \otimes \psi_J^{3,H} \rangle|}{|I|^{\frac{1}{2}}|J|^{\frac{1}{2}}}\chi_{I}(x) \chi_{J}(y) dx dy. \nonumber \\ \end{align}
\subsubsection{Estimate for the integral} Same argument used in the previous section can be applied here thanks to the same two-dimensional general level-sets stopping-time decomposition for both models: \begin{align} \label{form00}
& \int_{(\Omega^2_{k_1})^c \cap (\Omega^2_{k_2})^c} \sum_{\substack{I \times J \in \mathcal{I}_{-n - n_2, -m - m_2} \cap{T} \times \mathcal{J}_{n_2,m_2} \cap S\\I \times J \in \mathcal{R}_{k_1,k_2}}} \frac{|\langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle|}{|I|^{\frac{1}{2}}|J|^{\frac{1}{2}}} \frac{|\langle \chi_{E'},\psi_I^{3,H} \otimes \psi_J^{3,H} \rangle|}{|I|^{\frac{1}{2}}|J|^{\frac{1}{2}}}\chi_{I}(x) \chi_{J}(y) dx dy \nonumber \\ \leq & \int_{(\Omega^2_{k_1})^c \cap (\Omega^2_{k_2})^c} SSh(x,y) (SS)^H\chi_{E'}(x,y) \cdot \chi_{\bigcup_{\substack{I \times J \in \mathcal{I}_{-n - n_2, -m - m_2} \cap{T} \times \mathcal{J}_{n_2,m_2} \cap S\\I \times J \in \mathcal{R}_{k_1,k_2}}}I \times J} dxdy \nonumber \\
\lesssim & C_3^2 2^{k_1}\|h\|_{L^s(\mathbb{R}^2)} 2^{k_2} \bigg| \bigcup_{\substack{I \times J \in \mathcal{I}_{-n - n_2, -m - m_2} \cap{T} \times \mathcal{J}_{n_2,m_2} \cap S\\I \times J \in \mathcal{R}_{k_1,k_2}}}I \times J \bigg|.\nonumber \\ \end{align} The last inequality follows from the pointwise bounds for $SSh$ and $(SS)^H\chi_{E'}$ on $(\Omega^2_{k_1})^c \cap (\Omega^2_{k_2})^c$. \end{comment}
Fix $-l-l_2$ and $T \in \mathbb{T}_{-l-l_2}$, one recalls the \textit{tensor-type stopping-time decomposition II} to conclude that \begin{equation} \label{ave_1}
\sup_{I \in T } \frac{|\langle B_I^H(f_1,f_2),\varphi_I^{1H} \rangle|}{|I|^{\frac{1}{2}}} \lesssim C_1 2^{-l-l_2} \|B^H(f_1,f_2)\|_1. \end{equation} By the similar reasoning, \begin{equation} \label{ave_2}
\sup_{J \in S } \frac{|\langle \tilde{B}^H_J(g_1,g_2),\varphi_J^{1,H} \rangle|}{|J|^{\frac{1}{2}}} \lesssim C_2 2^{l_2} \|\tilde{B}^H(g_1,g_2)\|_1. \end{equation} By applying the estimates (\ref{ave_1}) and (\ref{ave_2}) to (\ref{form_00}), one derives \begin{align} \label{form00_set}
&C_1 C_2 C_3^2 \sum_{\substack{n > 0 \\ m > 0 \\ l > 0 \\ k_1 < 0 \\ k_2 \leq K}} 2^{-l}\|B^H(f_1,f_2)\|_1 \|\tilde{B}^H(g_1,g_2) \|_1\cdot 2^{k_1} \| h \|_{L^s} 2^{k_2} \cdot \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z} \\ l_2 \in \mathbb{Z}}} \sum_{\substack{T \in \mathbb{T}_{-l-l_2}\\S \in \mathbb{S}_{l_2}}} \bigg|\bigcup_{\substack{I \times J \in T \times S \\ I \times J \in \mathcal{I}_{-n - n_2, -m - m_2} \times \mathcal{J}_{n_2,m_2} \\I \times J \in \mathcal{R}_{k_1,k_2}}} I \times J \bigg|.
\end{align}
\vskip 0.15in \subsection{Estimate for nested sum of dyadic rectangles} \label{section_thm_haar_nestsum} One can estimate the nested sum (\ref{ns}) in two approaches - one with the application of the sparsity condition and the other with a Fubini-type argument which will be introduced in Section \ref{section_thm_haar_ns_fubini}. \begin{equation}\label{ns}
\sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z} \\ l_2 \in \mathbb{Z}}}\sum_{\substack{T \in \mathbb{T}_{-l-l_2}\\S \in \mathbb{S}_{l_2}}} \bigg|\bigcup_{\substack{I \times J \in T\times S \\ I \times J \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2} \\ I \times J \in \mathcal{R}_{k_1,k_2}}} I \times J \bigg|. \end{equation}
Both arguments aim to combine different stopping-time decompositions and to extract useful information from them. Generically, the sparsity condition argument employs the geometric property, namely Proposition \ref{sp_2d}, of the \textit{tensor-type stopping-time decomposition I} and applies the analytical implication from the \textit{general two-dimensional level sets stopping-time decomposition}. Meanwhile, the Fubini-type argument focuses on the hybrid of the \textit{tensor-type stopping time decomposition I - level sets} and the \textit{tensor-type stopping-time decomposition II - maximal intervals}. As implied by the name, the Fubini-type argument attempts to estimate the measure of a two dimensional set by the measures of its projected one-dimensional sets. The approaches to estimate projected one-dimensional sets are different depending on which tensor-type stopping decomposition is in consideration.
\subsubsection{Sparsity condition.} The first approach relies on the sparsity condition which mimics the argument in the Section \ref{section_thm_haar_fixed}. In particular, fix $n, m, l , k_1$ and $k_2$, one estimates (\ref{ns}) as follows. \begin{align*}
& \sum_{l_2}\sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\sum_{\substack{T \in \mathbb{T}_{-l-l_2}\\S \in \mathbb{S}_{l_2}}} \bigg|\bigcup_{\substack{I \times J \in T \times S \\ I \times J \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2} \\ I \times J \in \mathcal{R}_{k_1,k_2}}} I \times J \bigg| \nonumber \\
\leq & \underbrace{ \sup_{l_2}\Bigg(\sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\sum_{\substack{T \in \mathbb{T}_{-l-l_2}\\S \in \mathbb{S}_{l_2}}} \bigg|\bigcup_{\substack{I \times J \in T \times S \\ I \times J \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2} \\ I \times J\in \mathcal{R}_{k_1,k_2}}} I \times J \bigg|\Bigg)^{\frac{1}{2}}}_{SC-I} \nonumber \\
& \cdot \underbrace{\sum_{l_2}\Bigg(\sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\sum_{\substack{T \in \mathbb{T}_{-l-l_2}\\S \in \mathbb{S}_{l_2}}}\bigg|\bigcup_{\substack{I \times J \in T \times S \\ I \times J \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2}\\ I \times J \in \mathcal{R}_{k_1,k_2}}} I \times J\bigg| \Bigg)^{\frac{1}{2}}}_{SC-II}. \end{align*}
\begin{comment} \begin{remark}\label{dis_sp} \begin{enumerate} \item One important observation is that based on the order that each stopping-time decomposition is performed, for any fixed $n_1$ and $m_1$, and for any fixed $l_1$, the collection of maximal dyadic intervals $$ \mathcal{T}^{n_1,m_1,l_1}:= \{I_T: I_T \in \mathcal{I}_{n_1,m_1} \text{and }T \in \mathbb{T}^{l_1}\} $$ is disjoint. This fact is implied from the stopping-time algorithm that the tensor-type stopping-time decomposition-II is applied to each sub-collection of dyadic intervals $\mathcal{I}_{n_1,m_1}$. And the disjointness is obtained thanks to the tensor-type stopping-time decomposition-II. \item In contrast, for any fixed $l_1$, if one does not specify $n_1$ and $m_1$, then the collection of maximal dyadic intervals $$ \mathcal{T}^{l_1}:= \{I_T: T \in \mathbb{T}^{l_1}\} $$ is not disjoint. It is possible that $I_{T} \in \mathcal{I}_{n_1}$ and $I_{T'} \in \mathcal{I}_{n_1'}$ are both maximal intervals that satisfy $$
\frac{|\langle B^H_{I_T},\varphi^H_{I_{T}} \rangle|}{|I_T|^{\frac{1}{2}}} \sim 2^{l_1} \|B^H(f_1,f_2)\|_1 $$ and $$
\frac{|\langle B^H_{I_{T'}},\varphi^H_{I_{T'}} \rangle|}{|I_{T'}|^{\frac{1}{2}}} \sim 2^{l_1} \|B^H(f_1,f_2)\|_1 $$ If $n_1 = n_1'$, then the discussion in the first case allows us to conclude that $I_T \cap I_{T'} = \emptyset$. However, if $n_1 \neq n_1'$, then $I_T$ and $I_{T'}$ are collected in two independent tensor-type stopping decompositions-II. As a consequence, it is possible that $I_T \cap I_{T'} \neq \emptyset$. Conversely, if $I_T \cap I_{T'} \neq \emptyset$, then we know that they must be collected in $\mathcal{I}_{n_1}$ and $\mathcal{I}_{n_1'}$ respectively with $n_1 \neq n_1'$. In this scenario, one needs to invoke the sparsity condition to continue the estimates that we will delve into now. \end{enumerate} \end{remark} We will estimate the two parts, namely $SC-I$ and $SC-II$, separately. We recall that $l>0, k_1 < 0$ and $k_2 \leq K$ are fixed for our discussion below. \vskip .15in \end{comment}
\noindent \textbf{Estimate of $SC-I$.} Fix $l, n, m, k_1, k_2$ and $l_2$. Then by the \textit{one-dimensional stopping-time decomposition - maximal intervals}, for any $I \in T$ and $I' \in T'$ such that $T, T' \in \mathbb{T}^{-l-l_2}$ and $T \neq T'$, one has $I \cap I' = \emptyset$.
Hence for any fixed $n_2$ and $m_2$, one can rewrite \begin{align} \label{SC-I}
\sum_{\substack{T \in \mathbb{T}_{-l-l_2}\\S \in \mathbb{S}_{l_2}}} \bigg|\bigcup_{\substack{I \times J \in T \times S \\ I \times J \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2} \\ I \times J \in \mathcal{R}_{k_1,k_2}}} I \times J \bigg|= & \bigg|\bigcup_{\substack{T \in \mathbb{T}_{-l-l_2}\\S \in \mathbb{S}_{l_2}}} \bigcup_{\substack{I \times J \in T \times S \\ I \times J \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2} \\ I \times J \in \mathcal{R} _{k_1,k_2} }} I \times J \bigg|, \end{align} where the right hand side of (\ref{SC-I}) can be trivially bounded by $$
\bigg|\bigcup_{\substack{I \times J \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2} \\ I \times J \in \mathcal{R}_{k_1,k_2}}} I \times J \bigg|. $$
One can then recall the sparsity condition highlighted as Proposition \ref{sp_2d} and reduce the nested sum of measures of unions of rectangles to the measure of the corresponding union of rectangles. More precisely, \begin{equation}\label{compare_nested_union}
\sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\bigg| \bigcup_{\substack{I \times J \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2} \\ I \times J \in \mathcal{R}_{k_1,k_2} }} I \times J \bigg| \sim \bigg|\bigcup_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\bigcup_{\substack{I \times J \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2} \\ I \times J \in \mathcal{R}_{k_1,k_2}}} I \times J\bigg|, \end{equation} where the right hand side of (\ref{compare_nested_union}) can be estimated by $$
\bigg|\bigcup_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\bigcup_{\substack{ I \times J \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2} \\ I \times J \in \mathcal{R}_{k_1,k_2}}} I \times J\bigg| \leq \bigg| \bigcup_{I \times J \in \mathcal{R}_{k_1,k_2}}I \times J\bigg| \lesssim \min(2^{-k_1s},2^{-k_2\gamma}), $$ for any $\gamma >1$. The last inequality follows directly from (\ref{rec_area_hybrid}). Since the above estimates hold for any $l_2 \in \mathbb{Z}$, one can conclude that \begin{equation} \label{SC-I-final} SC-I \lesssim \min(2^{-\frac{k_1s}{2}},2^{-\frac{k_2\gamma}{2}}). \end{equation} \vskip .15in \noindent \textbf{Estimate of $SC-II$.} One invokes (\ref{SC-I}) and Proposition \ref{sp_2d} to obtain \begin{equation} \label{SC-II}
\sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\sum_{\substack{T \in \mathbb{T}_{-l-l_2}\\S \in \mathbb{S}_{l_2}}}\bigg|\bigcup_{\substack{I \times J \in T \times S \\ I \times J \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2} \\ I \times J \in \mathcal{R}_{k_1,k_2}}} I \times J\bigg| \sim \bigg|\bigcup_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\bigcup_{\substack{T \in \mathbb{T}_{-l-l_2}\\S \in \mathbb{S}_{l_2}}} \bigcup_{\substack{I \times J \in T \times S \\ I \times J \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2} \\ I \times J \in \mathcal{R}_{k_1,k_2}}} I \times J\bigg|. \end{equation} One can enlarge the collection of the rectangles by forgetting about the restriction that the rectangles lie in $\mathcal{R}_{k_1,k_2}$ and estimate the right hand side of (\ref{SC-II}) by \begin{equation} \label{SC-II2}
\bigg|\bigcup_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\bigcup_{\substack{T \in \mathbb{T}_{-l-l_2}\\S \in \mathbb{S}_{l_2}}} \bigcup_{\substack{I \times J \in T \times S \\ I \times J \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2} }} I \times J\bigg|, \end{equation} which is indeed the measure of the union of the rectangles collected in the \textit{tensor-type stopping-time decomposition II - maximal intervals} at a certain level. In other words, $$
(\ref{SC-II2}) = \bigg| \bigcup_{T \times S \in \mathbb{T}_{-l-l_2} \times \mathbb{S}_{l_2}} I_T \times J_S\bigg|. $$ \begin{comment} One could then re-define the collection trees and tree-tops for this more universal stopping-time decomposition. More rigorously, for any two trees $T$ and $T'$, if $I_T \cap I_{T'} \neq \emptyset$, then define the new tree by $$T:= T \cup T'.$$ Without loss of generality, suppose that $I_{T'} \subset I_{T}$, then define the new tree-top by $$ I_{T} := I_{T} $$ The collection of trees at each level can be defined correspondingly. And the same mechanism can be applied to the dyadic intervals in $\mathcal{J}$. It follows from the new definition of the trees and tree-tops that for any fixed $-l-l_2$ and $l_2$, $$ \bigcup_{T\times S \in \mathbb{T}_{-l-l_2} \times \mathbb{S}_{l_2}} I_T \times J_S = \bigcup_{T \times S \in \mathbb{T}_{-l-l_2} \times \mathbb{S}_{l_2}} I_{T} \times J_{S} $$ where \begin{align*} &\{ I_{T}: T \in \mathbb{T}_{-l-l_2} \} \nonumber \\ & \{ J_{S}: S \in \mathbb{S}_{l_2} \} \end{align*} are disjoint collections of dyadic intervals. \begin{remark} The re-definition of trees and tree-tops creates disjointness highlighted above, which is essential for the future application of energy estimates as can be seen in the proof of Proposition \ref{energy_classical}. \end{remark} \end{comment} Then \begin{equation} \label{fb_simple}
SC-II \leq \sum_{l_2 \in \mathbb{Z}}\bigg| \bigcup_{T \times S \in \mathbb{T}_{-l-l_2} \times \mathbb{S}_{l_2}} I_{T} \times J_{S}\bigg|^{\frac{1}{2}}, \end{equation} whose estimate follows a Fubini-type argument that plays an important role in the proof. We will focus on the development of this Fubini-type argument in a separate section and discuss its applications in other useful estimates for the proof.
\subsubsection{Fubini argument.} \label{section_thm_haar_ns_fubini} Alternatively, one can apply a Fubini-type argument to estimate (\ref{ns}) in the sense that the measure of some two-dimensional set is estimated by the product of the measures of its projected one-dimensional sets. To introduce this argument, we will first look into (\ref{fb_simple}) which requires a simpler version of the argument. \vskip .05in \noindent \textbf{Estimate of (\ref{fb_simple}) - Introduction of Fubini argument.} As illustrated before, one first rewrites the measure of two dimensional-sets in terms of the measures of two one-dimensional sets as follows. \begin{align} \label{fb_simple_2d}
\text{Right hand side of } (\ref{fb_simple}) \leq & \bigg( \sum_{l_2 \in \mathbb{Z}}\big|\bigcup_{T\in \mathbb{T}_{-l-l_2}} I_{T} \big|\bigg)^{\frac{1}{2}}\bigg( \sum_{l_2 \in \mathbb{Z}}\big|\bigcup_{S\in \mathbb{S}_{l_2}} J_{S} \big|\bigg)^{\frac{1}{2}}, \end{align} where the last step follows from the Cauchy-Schwarz inequality. To estimate the measures of the one-dimensional sets appearing above, one can convert them to the form of ``global'' energies and apply the energy estimates specified in Proposition \ref{B_en_global}. In particular, (\ref{fb_simple_2d}) can be rewritten up to a constant as \begin{align} \label{SC-II-en}
& \bigg( \sum_{l_2 \in \mathbb{Z}}(C_12^{-l-l_2} \|B^H(f_1,f_2)\|_1)^{1+\delta}\big|\bigcup_{T\in \mathbb{T}_{-l-l_2}} I_{T} \big|\bigg)^{\frac{1}{2}}\bigg( \sum_{l_2 \in \mathbb{Z}}(C_22^{l_2} \|\tilde{B}^H(g_1,g_2)\|_1)^{1+\delta}\big|\bigcup_{S\in \mathbb{S}_{l_2}} J_{S} \big|\bigg)^{\frac{1}{2}} \cdot \nonumber\\
& \ \ 2^{l\frac{(1+\delta)}{2}}\|B^H(f_1,f_2)\|_1^{-\frac{1+\delta}{2}}\|\tilde{B}^H(g_1,g_2)\|_{1}^{-\frac{1+\delta}{2}},
\end{align} for any $\delta >0$. One notices that for fixed $l$ and $l_2$, $$ \{I_T: T \in \mathbb{T}_{-l-l_2} \} $$ is a disjoint collection of dyadic intervals according to the \textit{one-dimensional stopping-time decomposition - maximal interval}. Thus the first sum in (\ref{SC-II-en}) can be rewritten as \begin{equation} \label{en_global}
\sum_{l_2 \in \mathbb{Z}}(C_12^{-l-l_2} \|B^H(f_1,f_2)\|_1)^{1+\delta}\big|\bigcup_{T\in \mathbb{T}_{-l-l_2}} I_{T} \big| = \sum_{l_2 \in \mathbb{Z}}(C_12^{-l-l_2} \|B^H(f_1,f_2)\|_1)^{1+\delta}\sum_{T\in \mathbb{T}_{-l-l_2}}|I_{T}|, \end{equation} which is indeed a ``global'' $L^{1+\delta}$ energy, namely $$ \left(\text{energy}^{1+\delta}_{\mathcal{I}}((\langle B^H(f_1,f_2), \varphi_I^{1,H} \rangle)_{I \in \mathcal{I}})\right)^{1+\delta} $$
so that one can apply the energy estimates described in Proposition \ref{B_en_global} to obtain the bound $$
\left(\text{energy}^{1+\delta}_{\mathcal{I}}((\langle B^H(f_1,f_2), \varphi_I^{1,H} \rangle)_{I \in \mathcal{I}})\right)^{1+\delta} \lesssim |F_1|^{\mu_1(1+\delta)}|F_2|^{\mu_2(1+\delta)}, $$ where $\delta, \mu_1, \mu_2 >0$ with $\mu_1 + \mu_2 = \frac{1}{1+\delta}$. Similarly, one can apply the same reasoning to the second sum in (\ref{SC-II-en}) to derive \begin{equation} \label{SC-II-y}
\sum_{l_2 \in \mathbb{Z}}(C_22^{l_2} \|\tilde{B}^H(g_1,g_2)\|_1)^{1+\delta}\big|\bigcup_{S\in \mathbb{S}_{l_2}} J_{S} \big| \lesssim |G_1|^{\nu_1(1+\delta)}|G_2|^{\nu_2(1+\delta)}, \end{equation} for any $\nu_1, \nu_2 > 0$ with $\nu_1 + \nu_2 = \frac{1}{1+\delta}$. By applying (\ref{en_global}) and (\ref{SC-II-y}) to (\ref{SC-II-en}), one has that \begin{equation}\label{SCiII-final}
(\ref{SC-II-en}) \lesssim 2^{l \frac{(1+\delta)}{2}} |F_1|^{\frac{\mu_1(1+\delta)}{2}}|F_2|^{\frac{\mu_2(1+\delta)}{2}}|G_1|^{\frac{\nu_1(1+\delta)}{2}}|G_2|^{\frac{\nu_2(1+\delta)}{2}}\|B^H(f_1,f_2)\|_1^{-\frac{1+\delta}{2}}\|\tilde{B}^H(g_1,g_2)\|_1^{-\frac{1+\delta}{2}}, \end{equation} for any $\delta,\mu_1,\mu_2,\nu_1,\nu_2 >0$ with $\mu_1+ \mu_2 = \nu_1+ \nu_2 = \frac{1}{1+\delta}$. \begin{remark}
The reason for leaving the expressions $\|B^H(f_1,f_2)\|_1^{-\frac{1+\delta}{2}}$ or $\|\tilde{B}^H(g_1,g_2)\|_1^{-\frac{1+\delta}{2}}$ will become clear later. In short, $\|B^H(f_1,f_2)\|_1$ and $\|\tilde{B}^H(g_1,g_2)\|_1$ will appear in estimates for other parts. We will keep them as they are for the exponent-counting and then use the estimates for $\|B^H(f_1,f_2)\|_1$ and $\|\tilde{B}^H(g_1,g_2)\|_1$ at last. \end{remark} By combining the estimates for $SC-I$ (\ref{SC-I-final}) and $SC-II$ (\ref{SCiII-final}), one can conclude that (\ref{ns}) is majorized by \begin{equation} \label{ns_sp}
2^{-\frac{k_2\gamma}{2}}2^{l \frac{(1+\delta)}{2}} |F_1|^{\frac{\mu_1(1+\delta)}{2}}|F_2|^{\frac{\mu_2(1+\delta)}{2}}|G_1|^{\frac{\nu_1(1+\delta)}{2}}|G_2|^{\frac{\nu_2(1+\delta)}{2}}\|B^H(f_1,f_2)\|_1^{-\frac{1+\delta}{2}}\|\tilde{B}^H(g_1,g_2)\|_1^{-\frac{1+\delta}{2}}, \end{equation} for $\gamma >1 $, $\delta,\mu_1,\mu_2,\nu_1,\nu_2 >0$ with $\mu_1+ \mu_2 = \nu_1+ \nu_2 = \frac{1}{1+\delta}$.
\begin{remark} The framework for estimating the measure of a two-dimensional set by its corresponding one-dimensional sets, as illustrated by (\ref{fb_simple_2d}), is the so-called ``Fubini-type'' argument which we will heavily employ from now on. \end{remark}
\vskip .15in \noindent \textbf{Estimate of (\ref{ns}) - Application of Fubini argument.} It is not difficult to observe that (\ref{ns}) can also be estimated by \begin{equation} \label{set_00}
\sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z} \\ l_2 \in \mathbb{Z}}}\left(\sum_{\substack{T \in \mathbb{T}_{-l-l_2}}}\bigg|\bigcup_{\substack{I \in T \\ I \in \mathcal{I}_{-n-n_2,-m-m_2}}} I \bigg|\right)\left(\sum_{\substack{S \in \mathbb{S}_{l_2}}}\bigg|\bigcup_{\substack{J \in S \\ J \in \mathcal{J}_{n_2,m_2} }} J \bigg|\right). \end{equation} One now rewrites the above expression and separates it into two parts. Both parts can be estimated by the Fubini-type argument whereas the methodologies to estimate projected one-dimensional sets are different. More precisely, \begin{align} \label{ns_AB}
(\ref{set_00}) = &\underbrace{\sup_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}} \sum_{l_2 \in \mathbb{Z}}\bigg(\sum_{\substack{T \in \mathbb{T}_{-l-l_2}}}\bigg|\bigcup_{\substack{I \in T \\ I \in \mathcal{I}_{-n-n_2,-m-m_2}}} I \bigg|\bigg)^{\frac{1}{2}}\bigg(\sum_{\substack{S \in \mathbb{S}_{l_2}}} \bigg|\bigcup_{\substack{J \in S \\ J \in \mathcal{J}_{n_2,m_2} }} J \bigg|\bigg)^{\frac{1}{2}}}_{\mathcal{A}} \times \nonumber \\
& \underbrace{\sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\sup_{l_2 \in \mathbb{Z}}\bigg(\sum_{\substack{T \in \mathbb{T}_{-l-l_2}}}\bigg|\bigcup_{\substack{I \in T \\ I \in \mathcal{I}_{-n-n_2,-m-m_2}}} I \bigg|\bigg)^{\frac{1}{2}}\bigg(\sum_{\substack{S \in \mathbb{S}_{l_2}}}\bigg|\bigcup_{\substack{J \in S \\ J \in \mathcal{J}_{n_2,m_2} }} J \bigg|\bigg)^{\frac{1}{2}}}_{\mathcal{B}}. \end{align} To estimate $\mathcal{A}$, one first notices that for for any fixed $n, m, n_2, m_2, l, l_2$ and a fixed tree $T \in \mathbb{T}^{-l-l_2}$, a dyadic interval $I \in T \cap \mathcal{I}_{-n-n_2.-m-m_2}$ means that \begin{enumerate}[(i)] \item $I \subseteq I_T$ where $I_T$ is the tree-top interval as implied by the \textit{one-dimensional stopping-time decomposition - maximal interval}; \item $I \cap \mathcal{U}_{-n-n_2+1,-m-m_2+1} \neq \emptyset$, where $$
\mathcal{U}_{-n-n_2+1,-m-m_2+1} := \{x:Mf_1(x) \leq C_1 2^{-n-n_2+1}|F_1| \} \cap \{x:Mf_2(x) \leq C_1 2^{-m-m_2+1}|F_2| \}. $$ \end{enumerate} By (i) and (ii), one can deduce that $$ I_T \cap \mathcal{U}_{-n-n_2+1,-m-m_2+1} \neq \emptyset. $$ As a consequence, \begin{equation} \label{a_x}
\sum_{\substack{T \in \mathbb{T}_{-l-l_2}}}\bigg|\bigcup_{\substack{I \in T \\ I \in \mathcal{I}_{-n-n_2,-m-m_2}}} I \bigg| \leq \sum_{\substack{T \in \mathbb{T}_{-l-l_2} \\ I_T \cap \mathcal{U}_{-n-n_2+1,-m-m_2+1} \neq \emptyset}}|I_T|. \end{equation} A similar reasoning applies to the term involving intervals in the $y$-direction and generates \begin{equation} \label{a_y}
\sum_{\substack{S \in \mathbb{S}_{l_2}}}\bigg|\bigcup_{\substack{J \in S \\ I \in \mathcal{J}_{n_2,m_2}}} J \bigg| \leq \sum_{\substack{S \in \mathbb{S}_{l_2} \\J_S \cap \tilde{\mathcal{U}}_{n_2+1,m_2+1} \neq \emptyset}}|J_S|, \end{equation} where $$
\tilde{\mathcal{U}}_{n_2+1,m_2+1} := \{y:Mg_1(y) \leq C_2 2^{n_2+1}|G_1| \} \cap \{y:Mg_2(y) \leq C_2 2^{m_2+1}|G_2| \}. $$ By applying the Cauchy-Schwarz inequality together with (\ref{a_x}) and (\ref{a_y}), one obtains \begin{align} \label{a_pre_en}
\mathcal{A} \leq & \sup_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}} \bigg(\sum_{l_2 \in \mathbb{Z}}\sum_{\substack{T \in \mathbb{T}_{-l-l_2} \\ I_T \cap \mathcal{U}_{-n-n_2+1,-m-m_2+1} \neq \emptyset}}|I_T|\bigg)^{\frac{1}{2}}\cdot \bigg(\sum_{l_2 \in \mathbb{Z}}\sum_{\substack{S \in \mathbb{S}_{l_2} \\J_S \cap \tilde{\mathcal{U}}_{n_2+1,m_2+1}\neq \emptyset}}|J_S|\bigg) ^{\frac{1}{2}}. \end{align} One then ``completes'' the expression (\ref{a_pre_en}) to produce localized energy-like terms as follows. \begin{align} \label{A_energy}
& \sup_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}} \underbrace{\bigg[\sum_{l_2 \in \mathbb{Z}}(C_1 2^{-l-l_2}\|B^H(f_1,f_2)\|_1)^2\sum_{\substack{T \in \mathbb{T}_{-l-l_2}\\ I_T \cap \mathcal{U}_{-n-n_2+1,-m-m_2+1} \neq \emptyset}}|I_{T}|\bigg]^{\frac{1}{2}}}_{\mathcal{A}^1}\cdot \underbrace{\bigg[\sum_{l_2 \in \mathbb{Z}}(C_2 2^{l_2}\|\tilde{B}^H(g_1,g_2)\|_1)^{2} \sum_{\substack{S \in \mathbb{S}_{l_2}\\ J_S \cap \tilde{\mathcal{U}}_{n_2+1,m_2+1}\neq \emptyset}} |J_S |\bigg]^{\frac{1}{2}}}_{\mathcal{A}^2} \nonumber \\
&\cdot 2^{l}\|B^H(f_1,f_2)\|_1^{-1}\|\tilde{B}^H(g_1,g_2)\|_1^{-1}.\nonumber \\
\end{align}
\begin{comment} \begin{remark} \label{loc_for_energy} \begin{enumerate} \item It is noteworthy that the estimates for $a^1$ and $a^2$ proceed under the condition that $-n-n_2, -m-m_2, n_2, m_2$ are fixed. This condition is essential because those indices specify the sub-collection of dyadic intervals, namely $\mathcal{I}_{-n-n_2,-m-m_2}$ and $\mathcal{J}_{n_2,m_2}$, that the tensor-type stopping-time decomposition is performed on. One can then apply the ``localization of energy'' described in Section 5.5 so that \begin{equation}\label{alt_loc} \text{energy}(\langle B_I, \varphi_I^1\rangle )_{I \in \mathcal{I}_{-n-n_2,-m-m_2}} \lesssim \text{energy}(\langle B_0^{-n-n_2,-m-m_2}, \varphi_I^1\rangle )_{I \in \mathcal{I}}. \end{equation}
\item One needs to keep in mind the hypothesis for the ``localization of energy'' used to obtain (\ref{alt_loc}). For non-lacunary family $(\phi_K^3)_K$, one assumes the compactness of supports of the bump functions; for lacunary family $(\phi_K^3)_K$, one has imposed the assumption (\ref{Haar_cond}). \end{enumerate} \end{remark} \end{comment}
It is not difficult to recognize that $\mathcal{A}^1$ and $\mathcal{A}^2$ are $L^2$ energies. Moreover, they follow stronger local energy estimates described in Proposition \ref{B_en}.
$\mathcal{A}^1$ is indeed an $L^2$ energy localized to $\mathcal{U}_{-n-n_2+1,-m-m_2+1}$. Then Proposition \ref{B_en} gives the estimate \begin{align} \label{a_1}
\mathcal{A}^1 \leq & \text{energy}^2\left((\langle B_I^H(f_1,f_2), \varphi_I^{1,H} \rangle)_{I \cap \mathcal{U}_{-n-n_2+1,-m-m_2+1} \neq \emptyset }\right) \lesssim & (C_1 2^{-n-n_2})^{\frac{1}{p_1}-\theta_1} (C_1 2^{-m-m_2})^{\frac{1}{q_1} - \theta_2}|F_1|^{\frac{1}{p_1}}|F_2|^{\frac{1}{q_1}}, \end{align} for any $0 \leq \theta_1, \theta_2 < 1$ satisfying $\theta_1 + \theta_2 = \frac{1}{2}$.
By the same reasoning, \begin{equation} \label{a_2}
\mathcal{A}^2 \lesssim C_2^{2}2^{n_2(\frac{1}{p_2} - \zeta_1)}2^{m_2(\frac{1}{q_2}- \zeta_2)}|G_1|^{\frac{1}{p_2}}|G_2|^{\frac{1}{q_2}}, \end{equation} where $0 \leq \zeta_1, \zeta_2 < 1$ and $\zeta_1 + \zeta_2 = \frac{1}{2}$. One can now apply the estimates for $\mathcal{A}^1$ (\ref{a_1}) and $\mathcal{A}^2$ (\ref{a_2}) to (\ref{A_energy}) and derive \begin{align} \label{A_energy_almostfinal} (\ref{A_energy}) \lesssim & C_1^{2}C_2^{2}\sup_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}2^{(-n-n_2)(\frac{1}{p_1}- \theta_1)}2^{(-m-m_2)(\frac{1}{q_1} - \theta_2)}2^{n_2(\frac{1}{p_2} - \zeta_1)}2^{m_2(\frac{1}{q_2}- \zeta_2)} \cdot \nonumber \\
& |F_1|^{\frac{1}{p_1}}|F_2|^{\frac{1}{q_1}}|G_1|^{\frac{1}{p_2}}|G_2|^{\frac{1}{q_2}}\cdot 2^{l}\|B^H(f_1,f_2)\|_1^{-1}\|\tilde{B}^H(g_1,g_2)\|_1^{-1}. \end{align} One observes that the following two conditions are equivalent: \begin{equation} \label{exp_1} \frac{1}{p_1} - \theta_1 = \frac{1}{p_2} - \zeta_1 \iff
\frac{1}{q_1} - \theta_2 = \frac{1}{q_2} - \zeta_2. \end{equation} The equivalence is imposed by the fact that \begin{align} \label{exp_2} &\frac{1}{p_1} + \frac{1}{q_1} = \frac{1}{p_2} + \frac{1}{q_2}, \nonumber \\ &\theta_1 + \theta_2 = \zeta_1 + \zeta_2 = \frac{1}{2} . \end{align}
With the choice $ 0 \leq \theta_1, \zeta_1 < 1$ with $\theta_1- \zeta_1 = \frac{1}{p_1} - \frac{1}{p_2}$, one can simplify (\ref{A_energy_almostfinal}) and conclude \begin{equation} \label{a_estimate}
\mathcal{A} \lesssim C_1^2 C_2^{2}2^{-n(\frac{1}{p_1} - \theta_1)}2^{-m(\frac{1}{q_1} - \theta_2)} |F_1|^{\frac{1}{p_1}}|F_2|^{\frac{1}{q_1}}|G_1|^{\frac{1}{p_2}}|G_2|^{\frac{1}{q_2}}\cdot 2^{l}\|B^H(f_1,f_2)\|_1^{-1}\|\tilde{B}^H(g_1,g_2)\|_1^{-1}. \end{equation}
\begin{remark} (\ref{exp_1}) and (\ref{exp_2}) together imposes a condition that \begin{equation} \label{pair_exp}
\left|\frac{1}{p_1} - \frac{1}{p_2}\right| = \left|\frac{1}{q_1} - \frac{1}{q_2}\right| < \frac{1}{2}. \end{equation} Without loss of generality, one can assume that $\frac{1}{p_1} \geq \frac{1}{p_2}$ and $\frac{1}{q_1} \leq \frac{1}{q_2}$. Then either (\ref{pair_exp}) holds or $$ \frac{1}{p_1} - \frac{1}{p_2} = \frac{1}{q_2} - \frac{1}{q_1} > \frac{1}{2}, $$ which implies $$
\left|\frac{1}{p_1} - \frac{1}{q_2}\right| = \left|\frac{1}{p_2} - \frac{1}{q_1}\right| < \frac{1}{2}. $$ Then one can switch the role of $g_1$ and $g_2$ to ``pair`` the functions as $f_1$ with $g_2$ and $f_2$ with $g_1$. A parallel argument can be applied to obtain the desired estimates. \end{remark}
One can apply another Fubini-type argument to estimate $\mathcal{B}$ with $l, n$ and $m$ fixed. Such argument again relies heavily on the localization. First of all, for any fixed $l_2 \in \mathbb{Z}$, $$ \{I: I \in T, T \in \mathbb{T}_{-l-l_2} \} $$ is a disjoint collection of dyadic intervals. Thus $$
\sum_{\substack{T \in \mathbb{T}_{-l-l_2}}}\bigg|\bigcup_{\substack{I \in T \\ I \in \mathcal{I}_{-n-n_2,-m-m_2}}} I \bigg| \leq \bigg|\bigcup_{\substack{ I \in \mathcal{I}_{-n-n_2,-m-m_2}}} I \bigg|. $$ One then recalls the pointwise estimate stated in Claim \ref{ptwise} to deduce $$
\bigcup_{\substack{ I \in \mathcal{I}_{-n-n_2,-m-m_2}}} I \subseteq \{x: Mf_1(x) > C_1 2^{-n-n_2-10}|F_1|\} \cap \{x: Mf_2(x) > C_1 2^{-m-m_2-10}|F_2|\}, $$ and for arbitrary but fixed $l_2 \in \mathbb{Z}$, \begin{equation} \label{b_x}
\sum_{\substack{T \in \mathbb{T}_{-l-l_2}}}\bigg|\bigcup_{\substack{I \in T \\ I \in \mathcal{I}_{-n-n_2,-m-m_2}}} I \bigg|\leq \big|\{x: Mf_1(x) > C_1 2^{-n-n_2-10}|F_1|\} \cap \{x: Mf_2(x) > C_1 2^{-m-m_2-10}|F_2|\} \big|. \end{equation}
A similar reasoning applies to the intervals in the $y$-direction and yields that for any fixed $l_2 \in \mathbb{Z}$, \begin{equation} \label{b_y}
\sum_{\substack{S \in \mathbb{S}_{l_2}}}\bigg|\bigcup_{\substack{J \in S \\ J \in \mathcal{J}_{n_2,m_2} }} J \bigg| \leq
\big|\{y: Mg_1(y) > C_2 2^{n_2-10}|G_1|\} \cap \{y: Mg_2(y) > C_2 2^{m_2-10}|G_2|\} \big|. \end{equation} To apply the above estimates, one notices that the finite collection of dyadic rectangles guarantees the existence of some $\tilde{l}_2 \in \mathbb{Z}$ possibly depending $n, m, l, n_2, m_2$ such that \begin{align*}
\mathcal{B} =& \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\bigg(\sum_{T \in \mathbb{T}_{-l-\tilde{l}_2}}\bigg|\bigcup_{\substack{ I \in T \\ I \in \mathcal{I}_{-n-n_2,-m-m_2}}} I \bigg| \bigg)^{\frac{1}{2}}\bigg(\sum_{S \in \mathbb{S}_{\tilde{l}_2}}\bigg|\bigcup_{\substack{ J \in S \\ J \in \mathcal{I}_{n_2,m_2}}} J \bigg| \bigg)^{\frac{1}{2}}. \nonumber \end{align*} One can further``complete'' $\mathcal{B}$ in the following manner for an appropriate use of the Cauchy-Schwarz inequality. \begin{align}
\mathcal{B} = & \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\bigg((C_12^{-n-n_2}|F_1|)^{\mu(1+\epsilon)}(C_12^{-m-m_2}|F_2|)^{(1-\mu)(1+\epsilon)}\sum_{T \in \mathbb{T}_{-l-\tilde{l}_2}}\bigg|\bigcup_{\substack{ I \in T \\ I \in \mathcal{I}_{-n-n_2,-m-m_2}}} I \bigg| \bigg)^{\frac{1}{2}} \cdot \nonumber \\
& \quad \quad \ \ \bigg((C_22^{n_2}|G_1|)^{\mu(1+\epsilon)}(C_2 2^{m_2}|G_2|)^{(1-\mu)(1+\epsilon)}\sum_{S \in \mathbb{S}_{\tilde{l}_2}}\bigg|\bigcup_{\substack{ J \in S \\ J \in \mathcal{I}_{n_2,m_2}}} J \bigg|\bigg)^{\frac{1}{2}}\nonumber \\
&\quad \quad \ \ \cdot 2^{n\cdot\frac{1}{2}\mu(1+\epsilon)}2^{m\cdot \frac{1}{2}(1-\mu)(1+\epsilon)}|F_1|^{-\frac{1}{2}\mu(1+\epsilon)}|F_2|^{-\frac{1}{2}(1-\mu)(1+\epsilon)}|G_1|^{-\frac{1}{2}\mu(1+\epsilon)}|G_2|^{-\frac{1}{2}(1-\mu)(1+\epsilon)} \nonumber \\
\leq & \underbrace{\bigg[\sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}(C_12^{-n-n_2}|F_1|)^{\mu(1+\epsilon)}(C_12^{-m-m_2}|F_2|)^{(1-\mu)(1+\epsilon)}\sum_{T \in \mathbb{T}_{-l-\tilde{l}_2}}\bigg|\bigcup_{\substack{ I \in T \\ I \in \mathcal{I}_{-n-n_2,-m-m_2}}} I \bigg|\bigg]^{\frac{1}{2}}}_{\mathcal{B}^1} \cdot \nonumber \\
&\underbrace{\bigg[\sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}(C_22^{n_2}|G_1|)^{\mu(1+\epsilon)}(C_2 2^{m_2}|G_2|)^{(1-\mu)(1+\epsilon)}\sum_{S \in \mathbb{S}_{\tilde{l}_2}}\bigg|\bigcup_{\substack{ J \in S \\ J \in \mathcal{I}_{n_2,m_2}}} J \bigg|\bigg]^{\frac{1}{2}}}_{\mathcal{B}^2}\nonumber \\
&\ \ \cdot 2^{n\cdot\frac{1}{2}\mu(1+\epsilon)}2^{m\cdot \frac{1}{2}(1-\mu)(1+\epsilon)}|F_1|^{-\frac{1}{2}\mu(1+\epsilon)}|F_2|^{-\frac{1}{2}(1-\mu)(1+\epsilon)}|G_1|^{-\frac{1}{2}\mu(1+\epsilon)}|G_2|^{-\frac{1}{2}(1-\mu)(1+\epsilon)}, \label{B_completed}
\end{align} for any $\epsilon > 0$, $0 < \mu <1$, where the second inequality follows from the Cauchy-Schwarz inequality.
To estimate $\mathcal{B}^1$, one recalls (\ref{b_x}) - which holds for any fixed $l_2 \in \mathbb{Z}$ - to obtain \begin{align} \label{B_1}
\mathcal{B}^1 \lesssim & \bigg[\sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}(C_1 2^{-n-n_2}|F_1|)^{\mu(1+\epsilon)}(C_1 2^{-m-m_2}|F_2|)^{(1-\mu)(1+\epsilon)}\big| \{ Mf_1(x) > C_1 2^{-n-n_2}|F_1|\} \cap \{ Mf_2(x) > C_1 2^{-m-m_2}|F_2|\} \big|\bigg]^{\frac{1}{2}} \nonumber \\ \leq & \bigg[\int (Mf_1(x))^{\mu(1+\epsilon)}(Mf_2(x))^{(1-\mu)(1+\epsilon)} dx\bigg]^{\frac{1}{2}} \nonumber \\ \leq & \Bigg[\bigg(\int (Mf_1(x))^{\mu(1+\epsilon) \frac{1}{\mu}} dx\bigg)^{\mu}\bigg(\int (Mf_2(x))^{(1-\mu)(1+\epsilon) \frac{1}{1-\mu}} dx\bigg)^{1-\mu}\Bigg]^{\frac{1}{2}}, \end{align} where the second inequality holds by the definition of Lebesgue integration and the last step follows from H\"older's inequality. One can now use the mapping property for the Hardy-Littlewood maximal operator $M: L^{p} \rightarrow L^{p}$ for any $p >1$ and deduces that \begin{align}
&\bigg(\int (Mf_1(x))^{1+\epsilon} dx\bigg)^{\mu} \lesssim \|f_1\|_{1+\epsilon}^{(1+\epsilon)\mu} \leq |F_1|^{\mu}, \label{piece1} \\
&\bigg(\int (Mf_2(x))^{1+\epsilon} dx\bigg)^{1-\mu} \lesssim \|f_2\|_{1+\epsilon}^{(1+\epsilon)(1-\mu)} \leq |F_2|^{1-\mu}. \label{piece2} \end{align}
By plugging the estimate (\ref{piece1}) and (\ref{piece2}) into (\ref{B_1}), \begin{equation} \label{B_1_final}
\mathcal{B}^1 \lesssim |F_1|^{\frac{\mu}{2}}|F_2|^{\frac{1-\mu}{2}}. \end{equation} By the same argument with $-n-n_2$ and $-m-m_2$ replaced by $n_2$ and $m_2$ correspondingly, one obtains \begin{equation} \label{B_2_final}
\mathcal{B}^2 \lesssim |G_1|^{\frac{\mu}{2}}|G_2|^{\frac{1-\mu}{2}}. \end{equation} Application of the estimates for $\mathcal{B}^1$ (\ref{B_1_final}) and $\mathcal{B}^2$ (\ref{B_2_final}) to (\ref{B_completed}) yields \begin{equation}\label{b_estimate}
\mathcal{B} \lesssim |F_1|^{-\frac{\mu}{2}\epsilon}|F_2|^{-\frac{1-\mu}{2}\epsilon}|G_1|^{-\frac{\mu}{2}\epsilon}|G_2|^{-\frac{1-\mu}{2}\epsilon}2^{n\cdot\frac{1}{2}\mu(1+\epsilon)}2^{m\cdot \frac{1}{2}(1-\mu)(1+\epsilon)}. \end{equation} By combining the results for both $\mathcal{A}$ (\ref{a_estimate}) and $\mathcal{B}$ (\ref{b_estimate}), one concludes with the following estimate for (\ref{ns_AB}).
\begin{comment}}
Case III follows the same argument as in Case I except that one chooses $\theta_3 > \frac{\epsilon}{1+\epsilon}$ in the expression and brings in the contribution of $|S_2|$, which is supposed to be small given $n_2, m_2 < 0 $. In particular,
$$
a \lesssim \big(2^{(-n-n_2)(1-\theta_1)}2^{(-m-m_2)(1-\theta_2)}|F_1||F_2| |S_2|^{ \theta_3 - \frac{\epsilon}{1+\epsilon}}\big)^{1+\epsilon} $$ where one recalls the definition of $$
|S_2| = \big|\{ Mf_1 > C_1 2^{-n-n_2-10}|F_1|\} \cap \{Mf_2 > C_1 2^{-m-m_2-10}|F_2|\}\big| \lesssim 2^{\frac{n+n_2}{2}}2^{\frac{m+m_2}{2}} $$ One then has the following estimate for $a$: $$
\big(2^{(-n-n_2)(1-\theta_1)}2^{(-m-m_2)(1-\theta_2)}|F_1||F_2| 2^{(n+n_2)\frac{1}{2}(\theta_3-\frac{\epsilon}{1+\epsilon})}2^{(m+m _2)\frac{1}{2}(\theta_3-\frac{\epsilon}{1+\epsilon})}\big)^{1+\epsilon} $$ which can be simplified to $$
a \lesssim \big(2^{-n\big((1-\theta_1)-\frac{1}{2}(\theta_3- \frac{\epsilon}{1+\epsilon})\big)}2^{-m\big((1-\theta_2)-\frac{1}{2}(\theta_3- \frac{\epsilon}{1+\epsilon})\big)}2^{n_2\big(\frac{1}{2}(\theta_3- \frac{\epsilon}{1+\epsilon}) - (1-\theta_1)\big)}2^{m_2\big(\frac{1}{2}(\theta_3- \frac{\epsilon}{1+\epsilon}) - (1-\theta_2)\big)}|F_1||F_2| \big)^{1+\epsilon} $$
By letting $\theta\theta_3 = \frac{\epsilon}{1+\epsilon}$ (equivalently $\theta_1 + \theta_2 = 1- \frac{1}{\theta}\frac{\epsilon}{1+\epsilon}$) and $\kappa = \frac{1}{2}$, one can simplify the above expression and derive $$
\|B_{\text{loc}}\|_{1+\epsilon} \lesssim 2^{-n(1-\theta_1-\frac{1}{2}(1-\theta)\theta_3)}2^{-m(1-\frac{1}{\theta}\frac{\epsilon}{1+\epsilon}+\theta_1-\frac{1}{2}(1-\theta)\theta_3)}2^{n_2(\frac{1}{2}(1-\theta)\theta_3-(1-\theta_1))}2^{m_2({\frac{1}{2}}(1-\theta)\theta_3-(1-\frac{1}{\theta}\frac{\epsilon}{1+\epsilon}+\theta_1))}|F_1||F_2| $$ Thus $$
a \lesssim \big(2^{-n(1-\theta_1-\frac{1}{2}(1-\theta)\theta_3)}2^{-m(1-\frac{1}{\theta}\frac{\epsilon}{1+\epsilon}+\theta_1-\frac{1}{2}(1-\theta)\theta_3)}2^{n_2(\frac{1}{2}(1-\theta)\theta_3-(1-\theta_1))}2^{m_2({\frac{1}{2}}(1-\theta)\theta_3-(1-\frac{1}{\theta}\frac{\epsilon}{1+\epsilon}+\theta_1))}|F_1||F_2|\big)^{1+\epsilon} $$
One applies the estimate for $b$ with $\theta_3' = \frac{\epsilon}{1+\epsilon}$ as in Case $I$ and combines it with the above estimates for $a$: \begin{align*}
a\cdot b \lesssim & \bigg(2^{-n\big((1-\theta_1)-\frac{1}{2}(\theta_3- \frac{\epsilon}{1+\epsilon})\big)}2^{-m\big((1-\theta_2)-\frac{1}{2}(\theta_3- \frac{\epsilon}{1+\epsilon})\big)}2^{n_2\big(\frac{1}{2}(\theta_3- \frac{\epsilon}{1+\epsilon}) - (1-\theta_1)\big)}2^{m_2\big(\frac{1}{2}(\theta_3- \frac{\epsilon}{1+\epsilon}) - (1-\theta_2)\big)}|F_1||F_2| \nonumber \\
& \cdot 2^{n_2(1-\zeta_1)} 2^{m_2(1-\frac{1}{1+\epsilon}+\theta_1')}|G_1||G_2|\bigg)^{1+\epsilon}. \end{align*}
By letting $\theta_1 = \theta_1'$, one can simplify the above expression: $$p
\big(2^{-n(1-\theta_1-\frac{1}{2}(1-\theta)\theta_3)}2^{-m(1-\frac{1}{\theta}\frac{\epsilon}{1+\epsilon}+\theta_1-\frac{1}{2}(1-\theta)\theta_3)}2^{n_2(\frac{1}{2}(1-\theta)\theta_3)}2^{m_2({\frac{1}{2}}(1-\theta)\theta_3+(\frac{1}{\theta}-1)\frac{\epsilon}{1+\epsilon}))}|F_1||F_2||G_1||G_2|\big)^{1+\epsilon} $$ For any $0 <\theta, \theta_3 < 1$ and $\epsilon >0$ one has $$
III \lesssim \big(2^{-n(1-\theta_1-\frac{1}{2}(1-\theta)\theta_3)}2^{-m(1-\frac{1}{\theta}\frac{\epsilon}{1+\epsilon}+\theta_1-\frac{1}{2}(1-\theta)\theta_3)}\big)^{1+\epsilon}|F_1|^{1+\epsilon}|F_2|^{1+\epsilon}|G_1|^{1+\epsilon}|G_2|^{1+\epsilon}2^{l(1+\epsilon)}\|B^H(f_1,f_2)\|_1^{-1-\epsilon}\|\tilde{B}^H(g_1,g_2)\|_1^{-1-\epsilon} $$ One can choose $\theta_3$ close to $0$, $\theta$ close to 1 and $\theta_1$ close to 0 to ensure the positivity of the exponents for $2^{-n}$ and $2^{-m}$. One particular choice would be $\theta_3 = \epsilon$, $\theta = \frac{1}{1+\epsilon}$ and $\theta_1 = \frac{1}{2}$: $$
III \lesssim 2^{-n(\frac{1}{2}-\frac{1}{2}\frac{\epsilon^2}{1+\epsilon})}2^{-m(\frac{3}{2}-\epsilon-\frac{1}{2}\frac{\epsilon^2}{1+\epsilon})}|F_1|^{1+\epsilon}|F_2|^{1+\epsilon}|G_1|^{1+\epsilon}|G_2|^{1+\epsilon}2^{l(1+\epsilon)}\|B^H(f_1,f_2)\|_1^{-1-\epsilon}\|\tilde{B}^H(g_1,g_2)\|_1^{-1-\epsilon} $$ Then by choosing $\epsilon>0$ small enough, one obtains the desired estimate. \end{comment}
\begin{align} \label{ns_fb}
& C_1^{2} C_2^{2} 2^{-n(\frac{1}{p_1} - \theta_1-\frac{1}{2}\mu(1+\epsilon))}2^{-m(\frac{1}{q_1}- \theta_2-\frac{1}{2}(1-\mu)(1+\epsilon))} \cdot \nonumber\\
& |F_1|^{\frac{1}{p_1}-\frac{\mu}{2}\epsilon}|F_2|^{\frac{1}{q_1}-\frac{1-\mu}{2}\epsilon}|G_1|^{\frac{1}{p_2}-\frac{\mu}{2}\epsilon}|G_2|^{\frac{1}{q_2}-\frac{1-\mu}{2}\epsilon}\cdot 2^{l}\|B^H(f_1,f_2)\|_1^{-1}\|\tilde{B}^H(g_1,g_2)\|_1^{-1}, \end{align} for any $0 \leq \theta_1, \theta_2 < 1$ with $\theta_1 + \theta_2 = \frac{1}{2}$, $0 <\mu<1$ and $\epsilon > 0$.
One can now interpolate between the estimates obtained with two different approaches, namely (\ref{ns_sp}) and (\ref{ns_fb}), to derive the following bound for (\ref{ns}). \begin{align} \label{ns_sum_result} &C_1^{2}C_2^{2} 2^{-\frac{k_2\gamma\lambda}{2}}2^{-n(\frac{1}{p_1}-\theta_1-\frac{1}{2}\mu(1+\epsilon))(1-\lambda)}2^{-m(\frac{1}{q_1} - \theta_2-\frac{1}{2}(1-\mu)(1+\epsilon))(1-\lambda)} \cdot \nonumber \\
& (2^{l})^{\lambda\frac{(1+\delta)}{2}+(1-\lambda)}\|B^H(f_1,f_2)\|_1^{-\lambda\frac{(1+\delta)}{2}-(1-\lambda)}\|\tilde{B}^H(g_1,g_2)\|_1^{-\lambda\frac{(1+\delta)}{2}-(1-\lambda)} \cdot \nonumber \\
& |F_1|^{\lambda \frac{\mu_1(1+\delta)}{2} + (1-\lambda)(\frac{1}{p_1}-\frac{\mu}{2}\epsilon)}|F_2|^{\lambda \frac{\mu_2(1+\delta)}{2} + (1-\lambda)(\frac{1}{q_1}-\frac{1-\mu}{2}\epsilon)}|G_1|^{\lambda \frac{\nu_1(1+\delta)}{2}+(1-\lambda)(\frac{1}{p_2}-\frac{\mu}{2}\epsilon)}|G_2|^{\lambda\frac{\nu_2(1+\delta)}{2}+(1-\lambda)(\frac{1}{q_2}-\frac{1-\mu}{2}\epsilon)}, \nonumber\\ \end{align} for some $0 \leq \lambda \leq 1$. By applying (\ref{ns_sum_result}) to (\ref{form00_set}), one has \begin{align}
& C_1^{3}C_2^{3} C_3^{3} \| h \|_{L^s}\sum_{\substack{n > 0 \\ m > 0 \\ l > 0 \\ k_1 < 0 \\ k_2 \leq K}}2^{-l\lambda(1-\frac{1+\delta}{2})}2^{k_1}2^{k_2(1-\frac{\lambda\gamma}{2})}2^{-n(\frac{1}{p_1}-\theta_1-\frac{1}{2}\mu(1+\epsilon))(1-\lambda)}2^{-m(\frac{1}{q_1}-\theta_2-\frac{1}{2}(1-\mu)(1+\epsilon))(1-\lambda)} \nonumber \\
&\cdot |F_1|^{\lambda \frac{\mu_1(1+\delta)}{2} + (1-\lambda)(\frac{1}{p_1}-\frac{\mu}{2}\epsilon)}|F_2|^{\lambda \frac{\mu_2(1+\delta)}{2} + (1-\lambda)(\frac{1}{q_1}-\frac{1-\mu}{2}\epsilon)}|G_1|^{\lambda \frac{\nu_1(1+\delta)}{2}+(1-\lambda)(\frac{1}{p_2}-\frac{\mu}{2}\epsilon)}|G_2|^{\lambda\frac{\nu_2(1+\delta)}{2}+(1-\lambda)(\frac{1}{q_2}-\frac{1-\mu}{2}\epsilon)} \nonumber \\
& \cdot \|B^H(f_1,f_2)\|_1^{\lambda(1-\frac{1+\delta}{2})}\|\tilde{B}^H(g_1,g_2)\|_1^{\lambda(1-\frac{1+\delta}{2})}. \label{form_nofixed_almost} \end{align} One notices that there exists $\epsilon > 0$, $0 < \mu < 1$ and $0 <\theta_1<\frac{1}{2}$ such that \begin{align} \label{nec_condition} &\frac{1}{p_1}-\theta_1-\frac{1}{2}\mu(1+\epsilon) > 0, \nonumber \\ &\frac{1}{q_1} - \theta_2-\frac{1}{2}(1-\mu)(1+\epsilon) > 0. \end{align}
\begin{remark} \label{rmk_easyhard_exponent} One realizes that (\ref{nec_condition}) imposes a necessary condition on the range of exponents. In particular, \begin{equation} \label{>1/2} \frac{1}{p_1} + \frac{1}{q_1} - (\theta_1 + \theta_2) > \frac{1}{2}\mu(1+ \epsilon) + \frac{1}{2}(1-\mu)(1+\epsilon). \end{equation} Using the fact that $\theta_1 + \theta_2 = \frac{1}{2}$, one can rewrite (\ref{>1/2}) as $$ \frac{1}{p_1} + \frac{1}{q_1} > 1+ \frac{\epsilon}{2}. $$ As a consequence, the case \begin{equation}\label{hard_exponent} 1< \frac{1}{p_1} + \frac{1}{q_1} = \frac{1}{p_2} + \frac{1}{q_2} < 2 \end{equation}
can be treated by the current argument. Meanwhile, the case $0 < \frac{1}{p_1} + \frac{1}{q_1} = \frac{1}{p_2} + \frac{1}{q_2} \leq 1$ follows a simpler argument which resembles the one for the estimates involving $L^{\infty}$ norms and will be postponed to Section \ref{section_thm_inf_haar}. \end{remark}
Imposed by (\ref{nec_condition}), the geometric series involving $2^{-n}$ and $2^{-m}$ are convergent. The convergence of series involving $2^{k_1}$ is trivial. One also observes that for any $0 < \lambda < 1$ and $0 < \delta < 1$, $$ \lambda(1-\frac{1+\delta}{2}) > 0, $$ which implies that the series involving $2^{-l}$ is convergent. One can separate the cases when $k_2 > 0 $ and $k_2 \leq 0$ and select $\gamma >1$ in each case to make the series about $2^{k_2}$ convergent. Therefore, one can estimate (\ref{form_nofixed_almost}) by \begin{align}
& C_1^3 C_2^3 C_3^{3}\| h \|_{L^s} \|B^H(f_1,f_2)\|_1^{\lambda(1-\frac{1+\delta}{2})}\|\tilde{B}^H(g_1,g_2)\|_1^{\lambda(1-\frac{1+\delta}{2})} \nonumber \\
&\cdot |F_1|^{\lambda \frac{\mu_1(1+\delta)}{2} + (1-\lambda)(\frac{1}{p_1}-\frac{\mu}{2}\epsilon)}|F_2|^{\lambda \frac{\mu_2(1+\delta)}{2} + (1-\lambda)(\frac{1}{q_1}-\frac{1-\mu}{2}\epsilon)}|G_1|^{\lambda \frac{\nu_1(1+\delta)}{2}+(1-\lambda)(\frac{1}{p_2}-\frac{\mu}{2}\epsilon)}|G_2|^{\lambda\frac{\nu_2(1+\delta)}{2}+(1-\lambda)(\frac{1}{q_2}-\frac{1-\mu}{2}\epsilon)}, \label{form_nofixed_exponent} \end{align} where one can apply Lemma \ref{B_global_norm} to derive \begin{align}
& \|B^H(f_1,f_2)\|_1 \lesssim |F_1|^{\rho}|F_2|^{1-\rho}, \label{B_norm1} \\
& \|\tilde{B}^H(g_1,g_2)\|_1 \lesssim |G_1|^{\rho'}|G_2|^{1-\rho'}, \label{B_norm2} \end{align} with the corresponding exponent to be positive as guaranteed by the fact that $0 < \lambda,\delta < 1$. From , (\ref{form_nofixed_exponent}), (\ref{B_norm1}) and (\ref{B_norm2}), one thus obtains \begin{align} \label{exp00}
|\Lambda^H_{\text{flag}^{0} \otimes \text{flag}^{0}}|\lesssim& C_1^3 C_2^3 C_3^{3}\|h\|_{L^s(\mathbb{R}^2)} |F_1|^{\lambda \frac{\mu_1(1+\delta)}{2} + (1-\lambda)(\frac{1}{p_1}-\frac{\mu}{2}\epsilon) + \rho\lambda(1-\frac{1+\delta}{2})}|F_2|^{\lambda \frac{\mu_2(1+\delta)}{2} + (1-\lambda)(\frac{1}{q_1}-\frac{1-\mu}{2}\epsilon) + (1-\rho)\lambda(1-\frac{1+\delta}{2})} \nonumber \\
& \cdot |G_1|^{\lambda \frac{\nu_1(1+\delta)}{2}+(1-\lambda)(\frac{1}{p_2}-\frac{\mu}{2}\epsilon)+ \rho'\lambda(1-\frac{1+\delta}{2})}|G_2|^{\lambda\frac{\nu_2(1+\delta)}{2}+(1-\lambda)(\frac{1}{q_2}-\frac{1-\mu}{2}\epsilon)+(1-\rho')\lambda(1-\frac{1+\delta}{2})}.
\end{align} With a little abuse of notation, we use $\tilde{p_i}$ and $\tilde{q_i}$, $i = 1,2$ to represent $p_i$ and $q_i$ in the above argument. And from now on, $p_i$ and $q_i$ stand for the boundedness exponents specified in the main theorem. One has the freedom to choose $1 < \tilde{p_i}, \tilde{q_i}< \infty$, $0 < \mu,\lambda < 1$ and $\epsilon > 0 $ such that \begin{align} \label{exp_tilde} & \lambda \frac{\mu_1(1+\delta)}{2} + (1-\lambda)(\frac{1}{\tilde{p_1}}-\frac{\mu}{2}\epsilon) + \rho\lambda(1-\frac{1+\delta}{2}) = \frac{1}{p_1} \nonumber \\ & \lambda \frac{\mu_2(1+\delta)}{2} + (1-\lambda)(\frac{1}{\tilde{q_1}}-\frac{1-\mu}{2}\epsilon) + (1-\rho)\lambda(1-\frac{1+\delta}{2}) = \frac{1}{q_1} \nonumber \\ & \lambda \frac{\nu_1(1+\delta)}{2}+(1-\lambda)(\frac{1}{\tilde{p_2}}-\frac{\mu}{2}\epsilon)+ \rho'\lambda(1-\frac{1+\delta}{2}) = \frac{1}{p_2} \nonumber \\ & \lambda\frac{\nu_2(1+\delta)}{2}+(1-\lambda)(\frac{1}{\tilde{q_2}}-\frac{1-\mu}{2}\epsilon)+(1-\rho')\lambda(1-\frac{1+\delta}{2}) = \frac{1}{q_2}. \end{align} \begin{remark} To see that above equations can hold, one can view the parts without $\tilde{p_i}$ and $\tilde{q_i}$ as perturbations which can be controlled small. More precisely, when $0 < \delta < 1$ is close to $1$, $$ \lambda(1-\frac{1+\delta}{2}) \ll 1. $$ When $0 < \lambda < 1$ is close to $0$, one has $$ \lambda \frac{\mu_1(1+\delta)}{2}, \lambda \frac{\mu_2(1+\delta)}{2}, \lambda \frac{\nu_1(1+\delta)}{2}, \lambda\frac{\nu_2(1+\delta)}{2} \ll 1 $$ and \begin{align*} &\frac{1}{p_i} - (1-\lambda)(\frac{1}{\tilde{p_i}}-\frac{\mu}{2}\epsilon) \ll 1,\nonumber \\ & \frac{1}{q_i} - (1-\lambda)(\frac{1}{\tilde{q_i}}-\frac{1-\mu}{2}\epsilon)\ll 1, \end{align*} for $ i = 1,2$. \end{remark} It is also necessary to check is that $\tilde{p_i}$ and $\tilde{q_i}$ satisfy the conditions which have been used to obtain (\ref{exp00}), namely \begin{align} \frac{1}{\tilde{p_1}} + \frac{1}{\tilde{q_1}} = & \frac{1}{\tilde{p_2}} + \frac{1}{\tilde{q_2}} > 1. \end{align} One can easily verify the first equation and the second inequality by manipulating (\ref{exp_tilde}). As a result, we have derived that
$$|\Lambda^H_{\text{flag}^{0} \otimes \text{flag}^{0}}(f_1,f_2,g_1,g_2,h, \chi_{E'})| \lesssim C_1^3 C_2^3 C_3^{3}|F_1|^{\frac{1}{p_1}}|F_2|^{\frac{1}{q_1}}|G_1|^{\frac{1}{p_2}}|G_2|^{\frac{1}{q_2}} \|h\|_{L^s(\mathbb{R}^2)}. $$
\begin{comment} We will separate the cases when \begin{enumerate}[I.] \item $(\phi^3_K)_K$ and $(\phi^3_L)_L$ are both lacunary; \item $(\phi^3_K)_K$ and $(\phi^3_L)_L$ are both non-lacunary; \item one of $(\phi^3_K)_K$ and $(\phi^3_L)_L$ is lacunary while the other is non-lacunary. \end{enumerate}
The main difference lies in the fact that the biest trick specified in Section 5 only works for the case when $(\phi^3_K)_K$ or $(\phi^3_L)_L$ is lacunary and for the other cases modifications on $B$ or $\tilde{B}$ are necessary as hinted in the proof of Proposition \ref{B_en}. The section will mainly focus on Case I while the latter cases will follow a similar argument with slight differences that will be highlighted at the end of the section.
\subsection{$(\phi^3_K)_K$ and $(\phi^3_L)_L$ are non-lacunary} As commented at the beginning of the section, the failure of biest trick requires some modifications on $B$ and $\tilde{B}$ as revealed in Section 5.5. In particular, one would dream of \begin{equation} \label{biest}
\frac{|\langle B_I,\varphi^1_I \rangle|}{|I|^{\frac{1}{2}}} \leq M(B)(x) \end{equation} for any $x \in I$. It is not difficult to check that (\ref{biest}) is no longer true in the case when $(\phi_K^3)_K$ is non-lacunary for
$$ \displaystyle B(x):= \sum_{K} \frac{1}{|K|^{\frac{1}{2}}} \langle f_1, \psi^1_K \rangle \langle f_2, \psi^2_K \rangle \varphi^3_K(x) $$ Instead, one introduces a variant of $B$: $$
\displaystyle B^{+}(x) := \sum_{K}\frac{1}{|K|^{\frac{1}{2}}}|\langle f_1, \psi^1_K \rangle| |\langle f_2, \psi^2_K \rangle| |\varphi^3_K|(x) $$ The same computation carried out in Section 5.5 shows that for any $x \in I$, \begin{equation} \label{mod_biest}
\frac{|\langle B_I,\varphi^1_I \rangle|}{|I|^{\frac{1}{2}}} \leq M(B^{+})(x). \end{equation}
Therefore, it is natural to define the exceptional set as follows: \begin{align*}
\displaystyle \Omega^1 := &\bigcup_{n_1 \in \mathbb{Z}}\{Mf_1 > C_1 2^{n_1}|F_1|\} \times \{Mg_1 > C_2 2^{-n_1}|G_1|\}\cup \nonumber \\
& \bigcup_{m_1 \in \mathbb{Z}}\{Mf_2 > C_1 2^{m_1}|F_2|\} \times \{Mg_2 > C_2 2^{-m_1}|G_2|\}\cup \nonumber \\
&\bigcup_{l_1 \in \mathbb{Z}} \{MB^{+} > C_1 2^{l_1}\| B^{+}\|_1\} \times \{M\tilde{B^{+}} > C_2 2^{-l_1}\| \tilde{B^{+}}\|_1\}\nonumber \\
\Omega^2 := & \{SSh > C_3 \|h\|_{L^s(\mathbb{R}^2)}\} \nonumber \\ \end{align*} where $$
\displaystyle B^{+} := \sum_{K}\frac{1}{|K|^{\frac{1}{2}}}|\langle f_1, \psi^1_K \rangle| |\langle f_2, \psi^2_K \rangle| |\varphi^3_K| $$ $$
\displaystyle \tilde{B^{+}} := \sum_{L}\frac{1}{|L|^{\frac{1}{2}}}|\langle g_1, \psi^1_L \rangle| |\langle g_2, \psi^2_L \rangle| |\varphi^3_L| $$ Then define $$ \Omega := \Omega^1 \cup \Omega^2$$ $$\tilde{\Omega} := \{ MM\chi_{\Omega} > \frac{1}{100}\}.$$ Let $$ E' := E \setminus \tilde{\Omega}. $$ \subsubsection{Tensor-type stopping-time decompositions II - maximal intervals}
One can perform a similar stopping-time decomposition as in the previous section with $\|B\|_1$ replaced by $\|B^{+}\|_1$. In particular, $$ \mathcal{I} = \displaystyle \bigcup_{l_1}\bigcup_{T^{+} \in \mathbb{T}^{+}_{l_1}}T^{+}, $$
where $T^{+}$ represents for the tree with the tree top $I_{T^{+}}$ satisfying
$$
\frac{|\langle B_{I_{T^{+}}}, \varphi^1_{I_{T+}}\rangle|}{|I_{T^{+}}|^{\frac{1}{2}}} > C_1 2^{l_1}\|\tilde{B^{+}}\|_1
$$
In addition,
$$
\mathcal{J} = \displaystyle \bigcup_{l_2}\bigcup_{S^{+} \in \mathbb{S}^{+}_{l_2}}S^{+},
$$
where $S^{+}$ represents for the tree with the tree top $J_S$ satisfying
$$
\frac{|\langle \tilde{B}_{J_{S^{+}}}, \varphi^1_{J_{S^{+}}}\rangle|}{|J_{S^{+}}|^{\frac{1}{2}}} > C_2 2^{l_2}\|\tilde{B^{+}}\|_1
$$
\begin{obs} \label{obs_exp_non}
If $I \times J \cap \tilde{\Omega}^{c} \neq \emptyset$ and $I \times J \in T^{+} \times S^{+}$ with $T^{+} \in \mathbb{T}^{+}_{l_1}$ and $S^{+} \in \mathbb{S}^{+}_{l_2}$, then $l_1, l_2 \in \mathbb{Z}$ satisfies $l_1 + l_2 < 0$. Equivalently, $I \times J \in T^{+} \times S^{+}$ with $T^{+} \in \mathbb{T}^{+}_{-l - l_2}$ and $S^{+} \in \mathbb{S}^{+}_{l_2}$ for some $l_2 \in \mathbb{Z}$, $l> 0$.
\end{obs}
The proof of Observation \ref{obs_exp_non} follows from the same argument as the one for Observation \ref{} and the fact that
$$
\frac{|\langle B_{I_{T^{+}}}(f_1,f_2), \varphi^1_{I_{T^{+}}} \rangle|}{|I_{T^{+}}|^{\frac{1}{2}}} \leq MB^{+}(x)
$$
$$
\frac{|\langle B_{J_{S^{+}}}(g_1,g_2), \varphi^1_{J_{S^{+}}} \rangle|}{|J_{S^{+}}|^{\frac{1}{2}}} \leq M\tilde{B}^{+}(y)
$$
for any $x \in I_{T^+}$ and $y \in J_{S^+}$.
\vskip 0.25in One can apply the same hybrid of stopping-time decompositions in Section 6 with the tensor-type stopping-time decompositions II specified above. The only differences from the argument in Section 6 are \begin{enumerate} \item $\text{energy}((\langle B_I,\varphi^1_I \rangle)_I)$ and $\text{energy}((\langle\tilde{B}_J,\varphi^1_J\rangle)_J)$ with non-lacunary families $(\phi^3_K)_K$ and $(\phi^3_L)_L$ respectively \item
$\|B^{+}\|_1$ and $\|\tilde{B}^+\|_1$ \end{enumerate}
For (1), one notices that the localization of energy in Section 5.5 includes the case when $(\phi^3_K)_K$ and $(\phi^3_L)_L$ are non-lacunary and thus \ref{alt_loc} still holds. Moreover, the local energy estimates in Proposition \ref{B_en} cover this case and generate the same result as the case when when $(\phi^3_K)_K$ and $(\phi^3_L)_L$ are lacunary.
For (2), one recalls that the estimates in Proposition \ref{B_global_norm} are valid for all possible lacunary and non-lacunary positions of bump functions. This ends the discussion of Case II.
\begin{remark} We will spare the full discussion of Case III as one can easily mix and match the arguments for lacunary and non-lacunary families appearing in the operator $B$ and $\tilde{B}$. Since in either case one obtains the same result, desired estimates follow immediately. This completes the proof of Theorem \ref{thm_weak} for the model $\Pi_{\text{flag}^0 \otimes \text{flag}^0}$ in the Haar setting. \end{remark} \end{comment}
\begin{comment} \subsubsection{Hybrid of stopping-time decompositions}
\begin{center} \begin{tabular}{ c c c } One-dimensional stopping-time decomposition& $\longrightarrow$ & $K \in \mathcal{K}_{n_0}$ \\ on $\mathcal{K}$ & & $(n_0 \in \mathbb{Z})$ \\ \end{tabular} \end{center}
\begin{center} \begin{tabular}{ c c c } One-dimensional stopping-time decomposition& $\longrightarrow$ & $L \in \mathcal{L}_{n'_0}$ \\ on $\mathcal{L}$ & & $(n'_0 \in \mathbb{Z})$ \\ \end{tabular} \end{center}
\begin{table}[h!] \begin{tabular}{ c c c } Tensor-type stopping-time decomposition I& $\longrightarrow$ & $I \times J \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2}$ \\ on $\mathcal{I} \times \mathcal{J}$ && \\ & & $(n_2, m_2 \in \mathbb{Z}, n > 0)$\\ $\Downarrow$ & & \\ General two-dimensional level sets stopping-time decomposition& $\longrightarrow$ & $I \times J \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2} \cap \mathcal{R}_{k_1,k_2} $ \\ on $\mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2}$ & & \\ & & $(n_2, m_2 \in \mathbb{Z}, n > 0,k_1 <0, k_2 \leq K)$\\ $\Downarrow$& & \\ Tensor-type stopping-time decomposition II & $\longrightarrow$ & $I \times J \in \big(\mathcal{I}_{-n-n_2,-m-m_2} \cap T^{+}\big) \times \big(\mathcal{J}_{n_2,m_2} \cap S^{+} \big)$\\ on $I \times J \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2} \cap \mathcal{R}_{k_1,k_2} $ & & $ \cap \ \ \ \ \mathcal{R}_{k_1,k_2} $ \\ & & with $T^{+} \in \mathbb{T}^{+}_{-l-l_2}$, $S^{+} \in \mathbb{S}^{+}_{l_2}$ \\ & & $(n_2, m_2, l_2 \in \mathbb{Z}, n, l > 0,k_1 <0, k_2 \leq K, )$\\ \end{tabular} \end{table} \end{comment}
\begin{comment} \section{Proof of Theorem \ref{thm_weak} for $\Pi_{\text{flag}^0 \otimes \text{paraproduct}}$ - Haar Model}
We assume that $f_i \leq \chi_{E_i}$, $g_j \leq \chi_{F_j}$, for $1 \leq i,j \leq 2$. We define the exceptional set $$\Omega := \Omega^1 \cup \Omega^2,$$ where \begin{align*}
\displaystyle \Omega^1 := &\bigcup_{n_1 \in \mathbb{Z}}\{Mf_1 > C_1 2^{n_1}|F_1|\} \times \{Mg_1 > C_2 2^{-n_1}|G_1|\}\cup \nonumber \\
& \bigcup_{m_1 \in \mathbb{Z}}\{Mf_2 > C_1 2^{m_1}|F_2|\} \times \{Mg_2 > C_2 2^{-m_1}|G_2|\}\cup \nonumber \\
&\bigcup_{l_1 \in \mathbb{Z}} \{MB > C_1 2^{l_1}\| B\|_1\} \times \{Mg_1 > C_2 2^{-l_1}|G_1|\}\cup \nonumber \\
&\bigcup_{l_2 \in \mathbb{Z}} \{MB > C_1 2^{l_2}\| B\|_1\} \times \{Mg_2 > C_2 2^{-l_2}|G_2|\};\nonumber \\
\Omega^2 := & \{SSh > C_3 \|h\|_{L_{x}^{p_3}(L_{y}^{q_3})}\} \nonumber \\ \end{align*} and $$\tilde{\Omega} := \{ MM\chi_{\Omega} > \frac{1}{100}\}.$$
\subsection{Stopping-time decompositions} \begin{center} \begin{tabular}{ c c c } One-dimensional stopping-time decomposition& $\longrightarrow$ & $K \in \mathcal{K}_{n_0}$ \\ on $\mathcal{K}$ & & $(n_0 \in \mathbb{Z})$ \\ \end{tabular} \end{center}
\begin{center} \begin{tabular}{ c c c } Two-dimensional tensor-type & $\longrightarrow$ & $I \times J \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2}$ \\ stopping-time decomposition I & & $(n_2, m_2 \in \mathbb{Z}, n > 0)$\\ on $\mathcal{I} \times \mathcal{J}$ && \\ $\Downarrow$ & & \\ Two-dimensional level general sets decomposition& $\longrightarrow$ & $I \times J \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2} \cap \mathcal{R}_{k_1,k_2} $ \\ on $\mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2}$ & & $(n_2, m_2 \in \mathbb{Z}, n > 0,k_1 <0, k_2 \leq K)$\\ $\Downarrow$& & \\ One-dimensional stopping-time decomposition & $\longrightarrow$ & $I \times J \in \big(\mathcal{I}_{-n-n_2,-m-m_2} \cap T\big) \times \mathcal{J}_{n_2,m_2} \cap \mathcal{R}_{k_1,k_2} $\\
- maximal intervals & & with $T \in \mathbb{T}_{-l-n_2}$ \\ on $I \in \mathcal{I}_{-n-n_2,-m-m_2} \cap \{I: I \times J \in \mathcal{R}_{k_1,k_2}\}$& & $(n_2, m_2 \in \mathbb{Z}, n, l > 0,k_1 <0, k_2 \leq K )$\\ \end{tabular} \end{center}
\begin{obs} If $I \times J \in T \times \mathcal{J}_{n_2,m_2}$ with $T \in \mathbb{T}_{l_1}$, then $l_1, m_1, m_2 \in \mathbb{Z}$ satisfies $l_1 + m_1 < 0$ and $l_1 + m_2 < 0$. Equivalently, $\forall I \times J \cap \tilde{\Omega}^{c} \neq \emptyset$, $I \times J \in T \times \mathcal{J}_{n_2,m_2}$, with $T \in \mathbb{T}_{-l - n_2}$ and $T \in \mathbb{T}_{-l' -m_2}$ for some $n_2, m_2 \in \mathbb{Z}$, $l, l' > 0$ and $-l-n_2 = -l' - m_2$. \begin{proof}
$I \in T$ with $T \in \mathbb{T}_{l_1}$ means that $\frac{|\langle B_{I}(f_1,f_2), \varphi^1_I \rangle|}{|I|^{\frac{1}{2}}} > C_12^{l_1} \|B\|_1$. By the biest trick, $\frac{|\langle B_{I}(f_1,f_2), \varphi^1_I \rangle|}{|I|^{\frac{1}{2}}} = \frac{|\langle B(f_1,f_2), \varphi^1_I \rangle|}{|I|^{\frac{1}{2}}} \leq MB(x)$ for any $x \in I$.
Thus $I \subseteq \{ MB > C_12^{l_1} \|B\|_1\}$
Meanwhile $J \in \mathcal{J}_{n_2}$ implies that $|J \cap \{ Mg_1 > C_2 2^{n_2}|G_1|\}| > \frac{1}{10}|J|$. If $l_1 + n_2 \geq 0$, then $\{ MB > C_12^{l_1} \|B\|_1\} \times \{ Mg_1 > C_2 2^{n_2}|G_1|\} \subseteq \Omega^1 \subseteq \Omega$. As a consequence, $|I \times J \cap \Omega| > \frac{1}{10} |I\times J|$ and $I \times J \subseteq \tilde{\Omega}$, which is a contradiction. Same reasoning applies to $l_1$ and $m_2$. \end{proof} \end{obs}
\subsubsection{Combination of two stopping-time algorithms}
For any $E \subseteq \mathbb{R}^2$ with $|E| < \infty$, one can define $E':= E \setminus \tilde{\Omega}$. By the dualization theorem, it suffices to prove that the multilinear form corresponding to our model operator
$$\Lambda_{\text{flag}^{0} \otimes \text{flag}^{0}}(f_1\otimes g_1, f_2 \otimes g_2, h, \chi_{E'}) := \displaystyle \sum_{I \times J \in \mathcal{I} \times \mathcal{J}} \frac{1}{|I|^{\frac{1}{2}} |J|} \langle B_I(f_1,f_2),\varphi_I^1 \rangle \langle g_1, \varphi_J^1 \rangle \langle g_2, \varphi_J^2 \rangle \langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle \langle \chi_{E'},\psi_I^{3,H} \otimes \psi_J^{3,H} \rangle$$ satisfies
$$|\Lambda(f_1\otimes g_1, f_2 \otimes g_2, h, \chi_{E'})| \lesssim \|f_1\|_{L_x^{p_1}} \|g_1\|_{L_y^{q_1}} \|f_2\|_{L_x^{p_2}}\|g_2\|_{L_y^{q_2}}\|h\|_{L^s(\mathbb{R}^2)}|E|^{\frac{1}{r'}}.$$ First assume that $I \times J \cap \tilde{\Omega} \neq \emptyset$.
\subsection{Application of stopping-time decompositions} We combine the stopping-time decompositions of dyadic rectangles and rewrite the multilinear form as \begin{align*}
&\displaystyle \big|\sum_{\substack{n > 0 \\ m > 0 \\ l > 0 \\ k_1 < 0 \\ k_2 \leq K}} \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\sum_{T \in \mathbb{T}_{-l-n_2}}\sum_{\substack{I \times J \in \mathcal{I}_{-n - n_2, -m - m_2} \cap{T} \times \mathcal{J}_{n_2, m_2}\\I \times J \in \mathcal{R}_{k_1,k_2}}} \frac{1}{|I|^{\frac{1}{2}} |J|} \langle B_I(f_1,f_2),\varphi_I^1 \rangle \langle g_1, \varphi_J^1 \rangle \langle g_2, \psi_J^2 \rangle \nonumber \\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \langle h, \psi_I^{2,H} \otimes \varphi_{J}^2 \rangle \langle \chi_{E'},\psi_I^{3,H} \otimes \psi_J^{3,H} \rangle \big| \nonumber \\
& = \sum_{\substack{n > 0 \\ m > 0 \\ l > 0 \\ k_1 < 0 \\ k_2 \leq K}} \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\sum_{T \in \mathbb{T}_{-l-n_2}}\sum_{\substack{I \times J \in \mathcal{I}_{-n - n_2, -m - m_2} \cap{T} \times \mathcal{J}_{n_2, m_2}\\I \times J \in \mathcal{R}_{k_1,k_2}}} \displaystyle \frac{|\langle B_I(f_1,f_2),\varphi_I^1 \rangle|}{|I|^{\frac{1}{2}}}\frac{|\langle g_1, \varphi_J^1 \rangle|}{|J|^{\frac{1}{2}}} \frac{|\langle g_2, \psi_J^2 \rangle|}{|J|^{\frac{1}{2}}} \cdot \nonumber \\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot \frac{|\langle h, \psi_I^{2,H} \otimes \varphi_{J}^2 \rangle|}{|I|^{\frac{1}{2}}|J|^{\frac{1}{2}}} \frac{|\langle \chi_{E'},\psi_I^{3,H} \otimes \psi_J^{3,H} \rangle|}{|I|^{\frac{1}{2}}|J|^{\frac{1}{2}}}|I||J|. \nonumber \\ \end{align*}
One notices that for any $I \times J \in \mathcal{R}_{k_1,k_2}$, $$
|I \times J \cap (\Omega^2_{k_1})^c| \geq \frac{99}{100}|I\times J| $$ $$
|I \times J \cap (\Omega^2_{k_2})^c| \geq \frac{99}{100}|I\times J| $$
where $\Omega^2_{k_1}:=\{SMh(x,y) > C_3 2^{k_1+1}\|h\|_{L^s(\mathbb{R}^2)}\} $ and $\Omega^2_{k_2} := \{(SS)^H\chi_{E'}(x,y) > C_3 2^{k_2+1}\}$.
As a result, one can restrict $I \times J$ in the sum to its smaller subset: \begin{align*}
|\Lambda | & \lesssim \sum_{\substack{n > 0 \\ m > 0 \\ l > 0 \\ k_1 < 0 \\ k_2 \leq K}} \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\sum_{T \in \mathbb{T}_{-l-n_2}}\sum_{\substack{I \times J \in \mathcal{I}_{-n - n_2, -m - m_2} \cap{T} \times \mathcal{J}_{n_2, m_2}\\I \times J \in \mathcal{R}_{k_1,k_2}}} \displaystyle \frac{|\langle B_I(f_1,f_2),\varphi_I^1 \rangle|}{|I|^{\frac{1}{2}}}\frac{|\langle g_1, \varphi_J^1 \rangle|}{|J|^{\frac{1}{2}}} \frac{|\langle g_2, \psi_J^2 \rangle|}{|J|^{\frac{1}{2}}} \cdot \nonumber \\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot \frac{|\langle h, \psi_I^{2,H} \otimes \varphi_{J}^2 \rangle|}{|I|^{\frac{1}{2}}|J|^{\frac{1}{2}}} \frac{|\langle \chi_{E'},\psi_I^{3,H} \otimes \psi_J^{3,H} \rangle|}{|I|^{\frac{1}{2}}|J|^{\frac{1}{2}}}|I \times J \cap (\Omega^2_{k_1})^c \cap (\Omega^2_{k_2})^c | \nonumber \\
& = \sum_{\substack{n > 0 \\ m > 0 \\ l > 0 \\ k_1 < 0 \\ k_2 \leq K}} \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\sum_{T \in \mathbb{T}_{-l-n_2}}\sum_{\substack{I \times J \in \mathcal{I}_{-n - n_2, -m - m_2} \cap{T} \times \mathcal{J}_{n_2, m_2}\\I \times J \in \mathcal{R}_{k_1,k_2}}} \displaystyle \int_{ (\Omega^2_{k_1})^c \cap (\Omega^2_{k_2})^c }\frac{|\langle B_I(f_1,f_2),\varphi_I^1 \rangle|}{|I|^{\frac{1}{2}}}\frac{|\langle g_1, \varphi_J^1 \rangle|}{|J|^{\frac{1}{2}}} \frac{|\langle g_2, \psi_J^2 \rangle|}{|J|^{\frac{1}{2}}} \cdot \nonumber \\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot \frac{|\langle h, \psi_I^{2,H} \otimes \varphi_{J}^2 \rangle|}{|I|^{\frac{1}{2}}|J|^{\frac{1}{2}}} \frac{|\langle \chi_{E'},\psi_I^{3,H} \otimes \psi_J^{3,H} \rangle|}{|I|^{\frac{1}{2}}|J|^{\frac{1}{2}}}\chi_{I}(x) \chi_{J}(y) dx dy. \nonumber \\ \end{align*}
By applying the Cauchy-Schwarz inequality twice with respect to the sums over $I$ and $J$, one obtains \begin{align*}
& \sum_{\substack{n > 0 \\ m > 0 \\ l > 0 \\ k_1 < 0 \\ k_2 \leq K}} \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\sum_{T \in \mathbb{T}_{-l-n_2}} \displaystyle \int_{ (\Omega^2_{k_1})^c \cap (\Omega^2_{k_2})^c } \sup_{\substack{I \in T \\ I \in \mathcal{I}_{-n - n_2, -m - m_2}}} \frac{|\langle B_I(f_1,f_2),\varphi_I^1 \rangle|}{|I|^{\frac{1}{2}}}\chi_{I}(x) \sum_{\substack{J \in \mathcal{J}_{n_2, m_2}\\ J \in \{J: I \times J \in \mathcal{R}_{k_1,k_2} \}}}\frac{|\langle g_1, \varphi_J^1 \rangle|}{|J|^{\frac{1}{2}}} \frac{|\langle g_2, \psi_J^2 \rangle|}{|J|^{\frac{1}{2}}}\chi_{J}(y) \cdot \nonumber \\
& \bigg(\sum_{\substack{I \in \mathcal{I}_{-n - n_2, -m - m_2} \cap{T} \\I \in \{I: I \times J \in \mathcal{R}_{k_1,k_2}\}}} \frac{|\langle h, \psi_I^{2,H} \otimes \varphi_{J}^2 \rangle|^2}{|I||J|}\chi_{I}(x) \chi_{J}(y)\bigg)^{\frac{1}{2}} \bigg(\sum_{\substack{I \in \mathcal{I}_{-n - n_2, -m - m_2} \cap{T} \\I \in \{I: I \times J \in \mathcal{R}_{k_1,k_2}\}}}\frac{|\langle \chi_{E'},\psi_I^{3,H} \otimes \psi_J^{3,H} \rangle|^2}{|I||J|}\chi_{I}(x) \chi_{J}(y)\bigg)^{\frac{1}{2}} dx dy \nonumber \\
& \leq \sum_{\substack{n > 0 \\ m > 0 \\ l > 0 \\ k_1 < 0 \\ k_2 \leq K}} \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\sum_{T \in \mathbb{T}_{-l-n_2}} \displaystyle \int_{ (\Omega^2_{k_1})^c \cap (\Omega^2_{k_2})^c }\sup_{\substack{I \in T \\ I \in \mathcal{I}_{-n - n_2, -m - m_2}}} \frac{|\langle B_I(f_1,f_2),\varphi_I^1 \rangle|}{|I|^{\frac{1}{2}}} \chi_{I}(x) \sup_{J \in \mathcal{J}_{n_2}}\frac{|\langle g_1, \varphi_J^1 \rangle|}{|J|^{\frac{1}{2}}} \cdot \nonumber \\
& \bigg(\sum_{\substack{J \in \mathcal{J}_{m_2}}} \frac{|\langle g_2, \psi_J^2 \rangle|^2}{|J|}\chi_{J}(y)\bigg)^{\frac{1}{2}} \cdot\sup_{\substack{J \in \mathcal{J}_{n_2, m_2}\\ J \in \{J: I \times J \in \mathcal{R}_{k_1,k_2} \}}}\frac{1}{|J|^{\frac{1}{2}}} \bigg(\sum_{\substack{I \in \mathcal{I}_{-n - n_2, -m - m_2} \cap{T} \\I \in \{I: I \times J \in \mathcal{R}_{k_1,k_2}\}}} \frac{|\langle h, \psi_I^{2,H} \otimes \varphi_{J}^2 \rangle|^2}{|I||J|}\chi_{I}(x)\bigg)^{\frac{1}{2}} \chi_{J}(y) \nonumber \\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \bigg(\sum_{\substack{I \times J \in \mathcal{I}_{-n - n_2, -m - m_2} \cap{T} \times \mathcal{J}_{n_2,m_2} \\I \times J \in \mathcal{R}_{k_1,k_2}}}\frac{|\langle \chi_{E'},\psi_I^{3,H} \otimes \psi_J^{3,H} \rangle|^2}{|I||J|}\chi_{I}(x) \chi_{J}(y)\bigg)^{\frac{1}{2}} dx dy. \nonumber \\ \end{align*} One can then use H\"older's inequality to obtain: \begin{align*}
& \sum_{\substack{n > 0 \\ m > 0 \\ l > 0 \\ k_1 < 0 \\ k_2 \leq K}} \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\sum_{T \in \mathbb{T}_{-l-n_2}} \displaystyle \sup_{I \in T} \frac{|\langle B_I(f_1,f_2),\varphi_I^1 \rangle|}{|I|^{\frac{1}{2}}} \sup_{J \in \mathcal{J}_{n_2}}\frac{|\langle g_1, \varphi_J^1 \rangle|}{|J|^{\frac{1}{2}}}\cdot \nonumber \\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \Bigg[\int \displaystyle \chi_{\bigcup_{I \in \mathcal{I}_{-n-n_2,-m-m_2} \cap T}I}(x) \sum_{\substack{J \in \mathcal{J}_{n_2,m_2}}} \frac{|\langle g_2, \psi_J^2 \rangle|^2}{|J|}\chi_{J}(y) dxdy\Bigg]^{\frac{1}{2}} \nonumber \\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \Bigg[\int_{ (\Omega^2_{k_1})^c \cap (\Omega^2_{k_2})^c } \bigg(\sup_{\substack{J \in \mathcal{J}_{n_2, m_2}\\ J \in \{J: I \times J \in \mathcal{R}_{k_1,k_2} \}}}\frac{1}{|J|^{\frac{1}{2}}} \sum_{\substack{I \in \mathcal{I}_{-n - n_2, -m - m_2} \cap{T} \\I \in \{I: I \times J \in \mathcal{R}_{k_1,k_2}\}}} \frac{|\langle h, \psi_I^{2,H} \otimes \varphi_{J}^2 \rangle|^2}{|I||J|}\chi_{I}(x)\chi_{J}(y)\bigg)^2 \nonumber \\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \bigg(\sum_{\substack{I \times J \in \mathcal{I}_{-n - n_2, -m - m_2} \cap{T} \times \mathcal{J}_{n_2,m_2} \\I \times J \in \mathcal{R}_{k_1,k_2}}}\frac{|\langle \chi_{E'},\psi_I^{3,H} \otimes \psi_J^{3,H} \rangle|^2}{|I||J|}\chi_{I}(x) \chi_{J}(y)\bigg)^{\frac{1}{2} \cdot 2} dx dy \Bigg]^{\frac{1}{2}} \nonumber \\
\leq & \sum_{\substack{n > 0 \\ m > 0 \\ l > 0 \\ k_1 < 0 \\ k_2 \leq K}} \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\sum_{T \in \mathbb{T}_{-l-n_2}} \displaystyle \sup_{I \in T} \frac{|\langle B_I(f_1,f_2),\varphi_I^1 \rangle|}{|I|^{\frac{1}{2}}} \sup_{J \in \mathcal{J}_{n_2}}\frac{|\langle g_1, \varphi_J^1 \rangle|}{|J|^{\frac{1}{2}}}\cdot \nonumber \\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \Bigg[\int \displaystyle \chi_{\bigcup_{I \in \mathcal{I}_{-n-n_2,-m-m_2} \cap T}I}(x) \sum_{\substack{J \in \mathcal{J}_{n_2,m_2}}} \frac{|\langle g_2, \psi_J^2 \rangle|^2}{|J|}\chi_{J}(y) dxdy\Bigg]^{\frac{1}{2}} \nonumber \\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \Bigg[\int_{\substack{(\Omega^2_{k_1})^c \cap (\Omega^2_{k_2})^c \cap \\ \bigcup_{I \times J \in \mathcal{I}_{-n-n_2,-m-m_2}\cap T \times \mathcal{J}_{n_2,m_2}} I \times J \cap \\ \bigcup_{I \times J \in \mathcal{R}_{k_1,k_2}}I \times J}}(SMh(x,y))^2 ((SS)^H\chi_{E'}(x,y))^2 dx dy \Bigg]^{\frac{1}{2}}, \nonumber \\ \end{align*} where the last step follows from the definition of the hybrid maximal and square functions. One can also simplify the other integral term $$
\Bigg[\int \displaystyle \chi_{\bigcup_{I \in \mathcal{I}_{-n-n_2,-m-m_2} \cap T}I}(x) \sum_{\substack{J \in \mathcal{J}_{n_2,m_2}}} \frac{|\langle g_2, \psi_J^2 \rangle|^2}{|J|}\chi_{J}(y) dxdy\Bigg]^{\frac{1}{2}} = \bigg|\bigcup_{I \in \mathcal{I}_{-n-n_2,-m-m_2} \cap T}I \bigg|^{\frac{1}{2}}\bigg(\sum_{\substack{J \in \mathcal{J}_{n_2,m_2}}}|\langle g_2, \psi_J^2 \rangle|^2\bigg)^{\frac{1}{2}} $$ By the John-Nirenberg inequality described in Proposition \ref{JN}, one has $$
\bigg(\sum_{\substack{J \in \mathcal{J}_{n_2,m_2}}}|\langle g_2, \psi_J^2 \rangle|^2\bigg)^{\frac{1}{2}} \lesssim \sup_{J_0 \in \mathcal{J}_{n_2,m_2}}\frac{1}{|J_0|}\bigg\|\sum_{J\subseteq J_0} \frac{\langle g_2,\psi^2_J \rangle }{|J|}\chi_J \bigg\|_{1,\infty} \cdot \sup_{J_0 \in \mathcal{J}_{n_2,m_2}} |J_0|^{\frac{1}{2}} $$ Proposition \ref{size} provides estimates for the $L^{1,\infty}$-size: \begin{align*}
\sup_{J_0 \in \mathcal{J}_{n_2,m_2}}\frac{1}{|J_0|}\bigg\|\sum_{J\subseteq J_0} \frac{\langle g_2,\psi^2_J \rangle }{|J|}\chi_J \bigg\|_{1,\infty} \sup_{J_0 \in \mathcal{J}_{n_2,m_2}} |J_0| \lesssim & \sup_{J_0 \in \mathcal{J}_{n_2,m_2}}\frac{1}{|J_0|}\int |g_2(y) \tilde{\chi}_{J_0}(y)| dy \sup_{J_0 \in \mathcal{J}_{n_2,m_2}} |J_0|^{\frac{1}{2}} \nonumber \\
\leq & \sup_{J_0 \in \mathcal{J}_{n_2,m_2}}\frac{1}{|J_0|}\int |g_2(y) \tilde{\chi}_{J_0}(y)| dy \bigg|\bigcup_{J_0 \in \mathcal{J}_{n_2,m_2}}J_0\bigg|^{\frac{1}{2}}, \end{align*} where $\tilde{\chi}_{J_0}$ denotes an $L^{\infty}$-normalized bump function adapted to $J_0$. One therefore has the following estimate: \begin{align*}
& \Bigg[\int \displaystyle \chi_{\bigcup_{I \in \mathcal{I}_{-n-n_2,-m-m_2} \cap T}I}(x) \sum_{\substack{J \in \mathcal{J}_{n_2,m_2}}} \frac{|\langle g_2, \psi_J^2 \rangle|^2}{|J|}\chi_{J}(y) dxdy\Bigg]^{\frac{1}{2}} \nonumber \\
& \lesssim \sup_{J_0 \in \mathcal{J}_{n_2,m_2}}\frac{1}{|J_0|}\int |g_2(y) \tilde{\chi}_{J_0}(y)| dy\bigg|\bigcup_{I \in \mathcal{I}_{-n-n_2,-m-m_2} \cap T}I \bigg|^{\frac{1}{2}} \bigg|\bigcup_{J_0 \in \mathcal{J}_{n_2,m_2}}J_0\bigg|^{\frac{1}{2}}. \end{align*}
By applying the above result to the estimate for the multilinear form, one has \begin{align} \label{form0p}
|\Lambda| \lesssim & \sum_{\substack{n > 0 \\ m > 0 \\ l > 0 \\ k_1 < 0 \\ k_2 \leq K}} \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\sum_{T \in \mathbb{T}_{-l-n_2}} \displaystyle \sup_{I \in T} \frac{|\langle B_I(f_1,f_2),\varphi_I^1 \rangle|}{|I|^{\frac{1}{2}}} \sup_{J \in \mathcal{J}_{n_2}}\frac{|\langle g_1, \varphi_J^1 \rangle|}{|J|^{\frac{1}{2}}} \sup_{J_0 \in \mathcal{J}_{n_2,m_2}}\frac{1}{|J_0|}\int |g_2(y) \tilde{\chi}_{J_0}(y)| dy\cdot \nonumber \\
& \bigg|\bigcup_{I \in \mathcal{I}_{-n-n_2,-m-m_2} \cap T}I \bigg|^{\frac{1}{2}} \bigg|\bigcup_{J_0 \in \mathcal{J}_{n_2,m_2}}J_0\bigg|^{\frac{1}{2}} \Bigg[\int_{\substack{(\Omega^2_{k_1})^c \cap (\Omega^2_{k_2})^c \cap \\ \bigcup_{I \times J \in \mathcal{I}_{-n-n_2,-m-m_2}\cap T \times \mathcal{J}_{n_2,m_2}} I \times J \cap \\ \bigcup_{I \times J \in \mathcal{R}_{k_1,k_2}}I \times J}}(SMh(x,y))^2 ((SS)^H\chi_{E'}(x,y))^2 dx dy \Bigg]^{\frac{1}{2}}, \end{align} where, by recalling the definition of $T \in \mathbb{T}_{-l-n_2}$, $\mathcal{I}_{-n-n_2,-m-m_2}$, and $\mathcal{J}_{n_2,m_2}$ in the tensor-product type stopping-time decomposition, $$
\sup_{I \in T} \frac{|\langle B_I(f_1,f_2),\varphi_I^1 \rangle|}{|I|^{\frac{1}{2}}} \lesssim C_1 2^{-l-n_2}\|B\|_1 $$ $$
\sup_{J \in \mathcal{J}_{n_2}}\frac{|\langle g_1, \varphi_J^1 \rangle|}{|J|^{\frac{1}{2}}} \lesssim C_22^{n_2}|G_1| $$ $$
\sup_{J_0 \in \mathcal{J}_{n_2,m_2}}\frac{1}{|J_0|}\int |g_2(y) \tilde{\chi}_{J_0}(y)| dy \leq \sup_{J_0 \in \mathcal{J}_{m_2}}\frac{1}{|J_0|}\int |g_2(y) \tilde{\chi}_{J_0}(y)| dy \lesssim C_22^{m_2}|G_2| $$ Meanwhile, since $g_j \leq \chi_{G_j}$ for $j = 1,2$,
$$\sup_{J \in \mathcal{J}}\frac{|\langle g_1, \varphi_J^1 \rangle|}{|J|^{\frac{1}{2}}} \leq 1,$$
$$ \sup_{J \in \mathcal{J}} \frac{|\langle g_2, \varphi_J^1 \rangle|}{|J|^{\frac{1}{2}}} \leq1.$$ One way to integrate the above estimates is
$$\sup_{J \in \mathcal{J}_{n_2}} \frac{|\langle g_1, \varphi_J^1 \rangle|}{|J|^{\frac{1}{2}}} \leq \min(1, C_2 2^{n_2}|G_1|),$$
$$ \sup_{J \in \mathcal{J}_{m_2}} \frac{|\langle g_2, \varphi_J^1 \rangle|}{|J|^{\frac{1}{2}}} \leq \min(1, C_2 2^{m_2}|G_2|).$$ Moreover, by the \textit{general two-dimensional level sets stopping-time decomposition}, for any $(x,y) \in (\Omega^2_{k_1})^c \cap (\Omega^2_{k_2})^c$, $$
SMh(x,y) \lesssim C_3 2^{k_1}\|h\|_{L^s(\mathbb{R}^2)} $$ $$ (SS)^H\chi_{E'}(x,y) \lesssim C_3 2^{k_2} $$ By plugging in the above estimates into expression (\ref{form0p}), one obtains \begin{align*}
|\Lambda | \lesssim & C_1 C_3^2 \sum_{\substack{n > 0 \\ m > 0 \\ l > 0 \\ k_1 > 0 \\ k_2 \geq -K}} \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\sum_{T \in \mathbb{T}_{-l-n_2}} \displaystyle 2^{-l - n_2} \|B\|_1\cdot \min(1, C_2 2^{n_2}|G_1|) \cdot \min(1, C_2 2^{m_2}|G_2|) 2^{k_1}\|h\|_{L^s(\mathbb{R}^2)}2^{k_2}\cdot \nonumber \\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \bigg|\bigcup_{I \in \mathcal{I}_{-n-n_2,-m-m_2} \cap T}I \bigg|^{\frac{1}{2}} \bigg|\bigcup_{J_0 \in \mathcal{J}_{n_2,m_2}}J_0\bigg|^{\frac{1}{2}} \bigg|\bigcup_{\substack{I \times J \in \mathcal{I}_{-n-n_2,-m-m_2}\cap T \times \mathcal{J}_{n_2,m_2} \\ I \times J \in \mathcal{R}_{k_1,k_2}}} I \times J \bigg|^{\frac{1}{2}}.\nonumber \\ \end{align*}
Now one can apply the Cauchy-Schwarz inequality to obtain \begin{align}
&C_1 C_3^2 \sum_{\substack{n > 0 \\ m > 0 \\ l > 0 \\ k_1 > 0 \\ k_2 \geq -K}} \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}} \displaystyle 2^{-l - n_2} \|B\|_1\cdot \min(1, C_2 2^{n_2}|G_1|) \cdot \min(1, C_2 2^{m_2}|G_2|) 2^{k_1}\|h\|_{L^s(\mathbb{R}^2)}2^{k_2}\cdot \nonumber \\
&\ \ \ \ \ \ \ \ \ \ \ \ \Bigg(\sum_{T \in \mathbb{T}_{-l-n_2}}\bigg|\bigcup_{I \in \mathcal{I}_{-n-n_2,-m-m_2} \cap T}I \bigg| \bigg|\bigcup_{J_0 \in \mathcal{J}_{n_2,m_2}}J_0\bigg|\Bigg)^{\frac{1}{2}} \Bigg(\sum_{T \in \mathbb{T}_{-l-n_2}}\bigg|\bigcup_{\substack{I \times J \in \mathcal{I}_{-n-n_2,-m-m_2}\cap T \times \mathcal{J}_{n_2,m_2} \\ I \times J \in \mathcal{R}_{k_1,k_2}}} I \times J \bigg|\Bigg)^{\frac{1}{2}}\nonumber \\
& = C_1 C_3^2 \sum_{\substack{n > 0 \\ m > 0 \\ l > 0 \\ k_1 < 0 \\ k_2 \leq K}}\|B \|_1 \| h \|_{L^s} 2^{k_1}2^{k_2} \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\# \cdot \nonumber \\
&\ \ \ \ \ \ \ \ \ \ \ \ \Bigg(\sum_{T \in \mathbb{T}_{-l-n_2}}\bigg|\bigcup_{I \in \mathcal{I}_{-n-n_2,-m-m_2} \cap T}I \bigg| \bigg|\bigcup_{J_0 \in \mathcal{J}_{n_2,m_2}}J_0\bigg|\Bigg)^{\frac{1}{2}} \Bigg(\sum_{T \in \mathbb{T}_{-l-n_2}}\bigg|\bigcup_{\substack{I \times J \in \mathcal{I}_{-n-n_2,-m-m_2}\cap T \times \mathcal{J}_{n_2,m_2} \\ I \times J \in \mathcal{R}_{k_1,k_2}}} I \times J \bigg|\Bigg)^{\frac{1}{2}},\nonumber \\
\label{form0p_1} \end{align} where \begin{align*}
\# &:= 2^{-l - n_2} \min(1, C_2 2^{n_2}|G_1|) \min(1, C_2 2^{m_2}|G_2|) \nonumber \\
& \leq \min\left(2^{-l-n_2} (C_2 2^{n_2}G_1)^{1-\theta} (C_2 2^{m_2}|G_2|)^{\theta}, C_2^2 2^{-l-n_2}2^{n_2}|G_1|2^{m_2}|G_2| \right) \nonumber \\
& \leq \min(C_2|G_1|^{1-\theta}|G_2|^{\theta}, C_2^2 2^{-l-n_2}2^{n_2}2^{m_2}|G_1||G_2|).\nonumber \\ \end{align*} where the last step uses the fact that $l > 0 $ and $l + n_2 - m_2 > 0$ by the definition of our exceptional set. Now \ref{form0p_1} can be estimated by \begin{align}
& C_1 {C_3}^2 \sum_{\substack{n > 0 \\ m > 0 \\ l > 0 \\ k_1 < 0 \\ k_2 \leq K}}\|B \|_1 \| h \|_{L^s} 2^{k_1}2^{k_2} \cdot \nonumber \\
& \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\Bigg[ \#\sum_{T \in \mathbb{T}_{-l-n_2}}\bigg|\bigcup_{I \in \mathcal{I}_{-n-n_2,-m-m_2} \cap T}I \bigg| \bigg|\bigcup_{J_0 \in \mathcal{J}_{n_2,m_2}}J_0\bigg| \Bigg]^{\frac{1}{2}}\cdot \Bigg[ \#\sum_{T \in \mathbb{T}_{-l-n_2}}\bigg|\bigcup_{\substack{I \times J \in \mathcal{I}_{-n-n_2,-m-m_2}\cap T \times \mathcal{J}_{n_2,m_2} \\ I \times J \in \mathcal{R}_{k_1,k_2}}} I \times J \bigg|\Bigg]^{\frac{1}{2}} \nonumber \\
& \leq C_1 {C_3}^2 \sum_{\substack{n > 0 \\ m > 0 \\ l > 0 \\ k_1 < 0 \\ k_2 \leq K}} \|B \|_1 \| h \|_{L^s} 2^{k_1}2^{k_2} \cdot \nonumber \\
& \Bigg[\sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}} \#\sum_{T \in \mathbb{T}_{-l-n_2}}\bigg|\bigcup_{I \in \mathcal{I}_{-n-n_2,-m-m_2} \cap T}I \bigg| \bigg|\bigcup_{J_0 \in \mathcal{J}_{n_2,m_2}}J_0\bigg|\Bigg]^{\frac{1}{2}}\cdot \Bigg[ \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}} \#\sum_{T \in \mathbb{T}_{-l-n_2}}\bigg|\bigcup_{\substack{I \times J \in \mathcal{I}_{-n-n_2,-m-m_2}\cap T \times \mathcal{J}_{n_2,m_2} \\ I \times J \in \mathcal{R}_{k_1,k_2}}} I \times J \bigg|\Bigg]^{\frac{1}{2}}, \nonumber \\ \end{align} \label{form0p_set} where the last inequality follows from the Cauchy-Schwarz inequality.
\subsection{Sparsity Condition} One can apply the sparsity condition to capture the contribution from the stopping-time decompositions $\displaystyle \mathcal{R} = \bigcup_{k_1,k_2}\bigcup_{R \in \mathcal{R}_{k_1,k_2}}R$. In particular, one first uses the fact that $\{I : I \in T, T \in \mathbb{T}_{-l-n_2}\}$ is a disjoint collection of intervals for any fixed $-l-n_2$ to simplify \begin{align*}
\sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\sum_{T \in \mathbb{T}_{-l-n_2}}\bigg|\bigcup_{\substack{I \times J \in \mathcal{I}_{-n-n_2,-m-m_2}\cap T \times \mathcal{J}_{n_2,m_2} \\ I \times J \in \mathcal{R}_{k_1,k_2}}} I \times J \bigg|
=\sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\bigg|\bigcup_{\substack{I \times J \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2} \\ I \times J \in \mathcal{R}_{k_1,k_2}}} I \times J \bigg|. \end{align*} One can now apply Proposition \ref{sp_2d} to deduce that \begin{align*}
\sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\bigg|\bigcup_{\substack{I \times J \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2} \\ I \times J \in \mathcal{R}_{k_1,k_2}}} I \times J \bigg| \lesssim & \sum_{m_2 \in \mathbb{Z}}\bigg|\bigcup_{n_2 \in \mathbb{Z}}\bigcup_{\substack{I \times J \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2} \\ I \times J \in \mathcal{R}_{k_1,k_2}}} I \times J \bigg| \nonumber \\
\leq & \sum_{m_2 \in \mathbb{Z}}\bigg|\bigcup_{\substack{I \times J \in \mathcal{I}_{-m-m_2} \times \mathcal{J}_{m_2} \\ I \times J \in \mathcal{R}_{k_1,k_2}}} I \times J \bigg| \nonumber \\
\lesssim & \bigg|\bigcup_{m_2 \in \mathbb{Z}}\bigcup_{\substack{I \times J \in \mathcal{I}_{-m-m_2} \times \mathcal{J}_{m_2} \\ I \times J \in \mathcal{R}_{k_1,k_2}}} I \times J \bigg| \nonumber \\
\leq & \bigg|\bigcup_{\substack{I \times J \in \mathcal{R}_{k_1,k_2}}} I \times J \bigg| \nonumber \\ \lesssim & \min(2^{-k_1}, 2^{-k_2\gamma}), \end{align*} for any $\gamma >1$.
\noindent \textbf{Fubini Argument.} One also observes that a trivial bound can be obtained for $$
\sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}} \#\sum_{T \in \mathbb{T}_{-l-n_2}}\bigg|\bigcup_{\substack{I \times J \in \mathcal{I}_{-n-n_2,-m-m_2}\cap T \times \mathcal{J}_{n_2,m_2} \\ I \times J \in \mathcal{R}_{k_1,k_2}}} I \times J \bigg| $$ by forgetting the information from the stopping-time decompositions with respect to $SM(h)$ and $(SS)^H(\chi_{E'})$. In particular, the above expression can be estimated by $$
\sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}} \#\sum_{T \in \mathbb{T}_{-l-n_2}}\bigg|\bigcup_{I \in \mathcal{I}_{-n-n_2,-m-m_2} \cap T}I \bigg| \bigg|\bigcup_{J \in \mathcal{J}_{n_2,m_2}}J\bigg|. $$ which already appear in the expression (\ref{form0p_set}).
One can apply the bound for $\#$ to estimate \begin{align}\label{fb0p}
& \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}} \#\sum_{T \in \mathbb{T}_{-l-n_2}}\bigg|\bigcup_{I \in \mathcal{I}_{-n-n_2,-m-m_2} \cap T}I \bigg| \bigg|\bigcup_{J \in \mathcal{J}_{n_2,m_2}}J\bigg| \nonumber \\
\leq & C_2^2 \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}} 2^{-l-n_2}2^{n_2}|G_1|2^{m_2}|G_2| \sum_{T \in \mathbb{T}_{-l-n_2}}\bigg|\bigcup_{I \in \mathcal{I}_{-n-n_2,-m-m_2} \cap T}I \bigg| \bigg|\bigcup_{J \in \mathcal{J}_{n_2,m_2}}J\bigg| \nonumber \\
\leq & \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}} C_1 2^{-l-n_2}\|B\|_1 \sum_{T \in \mathbb{T}_{-l-n_2}}\bigg|\bigcup_{I \in \mathcal{I}_{-n-n_2,-m-m_2} \cap T}I \bigg| \cdot C_2 2^{n_2}|G_1|C_2 2^{m_2}|G_2| \bigg|\bigcup_{J \in \mathcal{J}_{n_2,m_2}}J\bigg| \cdot \|B\|_1^{-1} C_1^{-1}\nonumber \\
\leq & \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}} C_12^{-l-n_2}\|B\|_1 \sum_{\substack{T \in \mathbb{T}_{-l-n_2} \\ I_T \in \mathcal{I}_{-n-n_2,-m-m_2}}}|I_T| \cdot C_2 2^{n_2}|G_1|C_2 2^{m_2}|G_2| \bigg|\bigcup_{J \in \mathcal{J}_{n_2,m_2}}J\bigg| \cdot \|B\|_1^{-1}C_1^{-1}. \end{align}
For any fixed $n_2, m_2$, one can apply Proposition \ref{B_en} with $S:= \{Mf_1 \leq C_1 2^{-n-n_2}|F_1| \} \cap \{ Mf_2 \leq C_1 2^{-m-m_2} |F_2| \}$ and $\theta_3 = 0$: \begin{align*}
C_1 2^{-l-n_2}\|B\|_1 \sum_{\substack{T \in \mathbb{T}_{-l-n_2} \\ I_T \in \mathcal{I}_{-n-n_2,-m-m_2}}}|I_T| \lesssim &\min(1,C_1 2^{-n-n_2}|F_1|)^{1-\theta_1}\min(1,C_12^{-m-m_2}|F_2|)^{\theta_1}|F_1|^{\theta_1}|F_2|^{1-\theta_1} \nonumber \\
\leq & C_1^{\alpha_1(1-\theta_1)+ \alpha_2\theta_1}2^{(-n-n_2)\alpha_1(1-\theta_1)}2^{(-m-m_2)\alpha_2\theta_1}|F_1|^{\alpha_1(1-\theta_1) + \theta_1}|F_2|^{\alpha_2\theta_1 + (1-\theta_1)}. \end{align*}
By applying the above energy estimate into (\ref{fb0p}): \begin{align*}
& C_1^{\alpha_1(1-\theta_1)+ \alpha_2\theta_1-1}\sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}} 2^{(-n-n_2)\alpha_1(1-\theta_1)}2^{(-m-m_2)\alpha_2\theta_1}|F_1|^{\alpha_1(1-\theta_1) + \theta_1}|F_2|^{\alpha_2\theta_1 + (1-\theta_1)} \nonumber \\
& \quad \quad \quad \quad \quad \quad \quad \quad \ \ \ \cdot C_2 2^{n_2}|G_1|C_2 2^{m_2}|G_2| \bigg|\bigcup_{J \in \mathcal{J}_{n_2,m_2}}J\bigg| \cdot \|B\|_1^{-1} \nonumber \\
= & C_1^{\alpha_1(1-\theta_1)+ \alpha_2\theta_1-1}\sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}} 2^{(-n-n_2)\alpha_1(1-\theta_1)}2^{(-m-m_2)\alpha_2\theta_1}|F_1|^{\alpha_1(1-\theta_1) + \theta_1}|F_2|^{\alpha_2\theta_1 + (1-\theta_1)}\nonumber \\
& \quad \quad \quad \quad \quad \quad \quad \quad \ \ \ \cdot (C_2 2^{n_2}|G_1|)^{(1-\theta_1')(1+\epsilon)}(C_2 2^{m_2}|G_2|)^{\theta_1'(1+\epsilon)} \bigg|\bigcup_{J \in \mathcal{J}_{n_2,m_2}}J\bigg| \cdot \nonumber \\
&\quad \quad \quad \quad \quad \quad \quad \quad \ \ \ \|B\|_1^{-1} (C_2 2^{n_2}|G_1|)^{1-(1-\theta_1')(1+\epsilon)}(C_22^{m_2}|G_2|)^{1-\theta_1'(1+\epsilon)} \nonumber \\
\leq& C_1 C_2^2 \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}} 2^{-n\alpha_1(1-\theta_1)}2^{-m\alpha_2\theta_1}|F_1|^{\alpha_1(1-\theta_1) + \theta_1}|F_2|^{\alpha_2\theta_1 + (1-\theta_1)} \nonumber \\
& \quad \quad \cdot (C_2 2^{n_2}|G_1|)^{(1-\theta_1')(1+\epsilon)}(C_2 2^{m_2}|G_2|)^{\theta_1'(1+\epsilon)} \bigg|\bigcup_{J \in \mathcal{J}_{n_2,m_2}}J\bigg| \cdot \nonumber \\
&\quad \quad\quad \|B\|_1^{-1} |G_1|^{1-(1-\theta_1')(1+\epsilon)}|G_2|^{1-\theta_1'(1+\epsilon)} 2^{n_2(1-(1-\theta_1')(1+\epsilon)-\alpha_1(1-\theta_1) )} 2^{m_2(1-\theta_1'(1+\epsilon) - \alpha_2\theta_1)}.\nonumber \\ \end{align*} If \begin{align} \label{exp} 1-(1-\theta_1')(1+\epsilon) & = \alpha_1(1-\theta_1) \nonumber \\ 1-\theta_1'(1+\epsilon) & = \alpha_2\theta_1, \nonumber \\ \end{align} then the above expression can be estimated by \begin{align} \label{fb_0p_2}
& C_1 C_2^2 \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}} (2^{n_2}|G_1|)^{(1-\theta_1')(1+\epsilon)}(2^{m_2}|G_2|)^{\theta_1'(1+\epsilon)} \bigg|\bigcup_{J \in \mathcal{J}_{n_2,m_2}}J\bigg| \cdot \nonumber \\
&\quad \quad\quad \|B\|_1^{-1} 2^{-n\alpha_1(1-\theta_1)}2^{-m\alpha_2\theta_1}|F_1|^{\alpha_1(1-\theta_1) + \theta_1}|F_2|^{\alpha_2\theta_1 + (1-\theta_1)}|G_1|^{1-(1-\theta_1')(1+\epsilon)}|G_2|^{1-\theta_1'(1+\epsilon)}, \end{align} where by applying the pointwise estimate specified in Claim \ref{ptwise}: \begin{align*}
& \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}} (C_22^{n_2}|G_1|)^{(1-\theta_1')(1+\epsilon)}(C_22^{m_2}|G_2|)^{\theta_1'(1+\epsilon)} \bigg|\bigcup_{J \in \mathcal{J}_{n_2,m_2}}J\bigg| \nonumber \\
& \leq \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}} (C_2 2^{n_2}|G_1|)^{(1-\theta_1')(1+\epsilon)}(C_2 2^{m_2}|G_2|)^{\theta_1'(1+\epsilon)} \left|\{ Mg_1 > C_2 2^{n_2}|G_1|\} \cap \{Mg_2 > C_2 2^{m_2}|G_2| \}\right| \nonumber\\ & \lesssim \int (Mg_1(y))^{(1-\theta_1')(1+\epsilon)} (Mg_2(y))^{\theta_1'(1+\epsilon)} dy \nonumber \\ & \leq \left[\int (Mg_1(y))^{(1-\theta_1')(1+\epsilon)\frac{1}{1-\theta_1'}}dy\right]^{1-\theta_1'}\left[\int (Mg_2(y))^{(\theta_1')(1+\epsilon)\frac{1}{\theta_1'}}dy\right]^{\theta_1'}. \end{align*} The last inequality is the application of H\"older's inequality. Now one can apply the mapping property for Hardy-Littlewood maximal operator to derive the following bound $$
\|g_1\|_{1+\epsilon}^{(1+\epsilon)(1-\theta_1')}\|g_2\|_{1+\epsilon}^{(1+\epsilon)\theta_1'} = |G_1|^{1-\theta_1'}|G_2|^{\theta_1'} $$ As a result, (\ref{fb_0p_2}) can be estimated by $$
C_1 C_2^2 \|B\|_1^{-1} 2^{-n\alpha_1(1-\theta_1)}2^{-m\alpha_2\theta_1}|F_1|^{\alpha_1(1-\theta_1) + \theta_1}|F_2|^{\alpha_2\theta_1 + (1-\theta_1)} |G_1|^{1-(1-\theta_1')\epsilon}|G_2|^{1-\theta_1'\epsilon} $$
Assuming the following equations hold: $$1-(1-\theta_1')\epsilon = \alpha_1(1-\theta_1) + \theta_1$$ $$1-\theta_1'\epsilon = \alpha_2\theta_1 + (1-\theta_1) $$ one would derive equations (\ref{exp}) if $\theta_1 = 1-\theta_1'$.
As a result, one has $$
C_1 C_2^2 \|B\|_1^{-1} 2^{-n(1-\theta_1(1+\epsilon))}2^{-m(1-(1-\theta_1)(1+\epsilon))}|F_1|^{1-\theta_1 \epsilon}|F_2|^{1-(1-\theta_1)\epsilon} |G_1|^{1-\theta_1 \epsilon}|G_2|^{1-(1-\theta_1)\epsilon} $$ \newline \newline \noindent \textbf{Combination of Sparsity Condition and Fubini Argument.} Now, one can combine both estimates by interpolating between them: \begin{align*}
& \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}} \#\sum_{T \in \mathbb{T}_{-l-n_2}}\bigg|\bigcup_{\substack{I \times J \in \mathcal{I}_{-n-n_2,-m-m_2}\cap T \times \mathcal{J}_{n_2,m_2} \\ I \times J \in \mathcal{R}_{k_1,k_2}}} I \times J \bigg| \nonumber \\
\lesssim & \left(C_2|G_1|^{1-\theta} |G_2|^{\theta} \min(C_3^{-1}2^{-k_1},C_3^{-\gamma}2^{-k_2\gamma})\right)^{\lambda} \nonumber \\
&\cdot \left(C_1 C_2^2\|B\|_1^{-1} 2^{-n(1-\theta_1(1+\epsilon))}2^{-m(1-(1-\theta_1)(1+\epsilon))}|F_1|^{1-\theta_1 \epsilon}|F_2|^{1-(1-\theta_1)\epsilon} |G_1|^{1-\theta_1 \epsilon}|G_2|^{1-(1-\theta_1)\epsilon} \right)^{1-\lambda}. \end{align*}
One can now plug in the above estimates into the expression (\ref{form0p_set}) and obtain: \begin{align*}
|\Lambda| \lesssim & C_1^2 C_2^2 C_3^2 \sum_{\substack{n > 0 \\ m > 0 \\ l > 0 \\ k_1 < 0 \\ k_2 \leq K}} \|B\|_1\| h \|_{L^s} 2^{k_1}2^{k_2} \left(|G_1|^{1-\theta} |G_2|^{\theta}2^{-\frac{1}{2}k_1}2^{-\frac{1}{2}k_2\gamma}\right)^{\frac{\lambda}{2}}\nonumber \\
&\ \ \quad \quad \quad\quad \cdot \left(\|B\|_1^{-1} 2^{-n(1-\theta_1(1+\epsilon))}2^{-m(1-(1-\theta_1)(1+\epsilon))}|F_1|^{1-\theta_1 \epsilon}|F_2|^{1-(1-\theta_1)\epsilon} |G_1|^{1-\theta_1 \epsilon}|G_2|^{1-(1-\theta_1)\epsilon} \right)^{\frac{1-\lambda}{2}+\frac{1}{2}} \nonumber \\
= & C_1^2 C_2^2 C_3^2 \sum_{\substack{n > 0 \\ m > 0 \\ l > 0 \\ k_1 < 0 \\ k_2 \leq K}} \| h \|_{L^s} 2^{k_1(1-\frac{\lambda}{4})}2^{k_2(1-\frac{\lambda}{4}\gamma)}2^{-n(1-\theta_1(1+\epsilon))(1-\frac{\lambda}{2})}2^{-m(1-(1-\theta_1)(1+\epsilon))(1-\frac{\lambda}{2})}\nonumber \\
& \quad \quad \quad \cdot |F_1|^{(1-\theta_1 \epsilon)(\frac{1}{2}+ \frac{1-\lambda}{2})}|F_2|^{(1-(1-\theta_1)\epsilon)(\frac{1}{2}+ \frac{1-\lambda}{2})} |G_1|^{(1-\theta_1 \epsilon)(\frac{1}{2}+ \frac{1-\lambda}{2})}|G_2|^{(1-(1-\theta_1)\epsilon)(\frac{1}{2}+ \frac{1-\lambda}{2})} \|B\|_1^{\frac{\lambda}{2}}, \end{align*}
where one can use the trivial estimate for $\|B\|_1$:
$$\|B\|_1 \lesssim |F_1|^{\rho} |F_2|^{1-\rho},$$ for any $0 < \rho < 1$. One can then simplify the expression further as \begin{align*}
& C_1^2 C_2^2 C_3^2 \sum_{\substack{n > 0 \\ m > 0 \\ l > 0 \\ k_1 < 0 \\ k_2 \leq K}} \| h \|_{L^s} 2^{k_1(1-\frac{\lambda}{4})}2^{k_2(1-\frac{\lambda}{4}\gamma)} 2^{-n(1-\theta_1(1+\epsilon))(1-\frac{\lambda}{2})}2^{-m(1-(1-\theta_1)(1+\epsilon))(1-\frac{\lambda}{2})} \nonumber \\
& \quad \quad \quad \ \ \cdot |F_1|^{(1-\rho)\frac{\lambda}{2} +(1-\theta_1 \epsilon)(\frac{1}{2}+ \frac{1-\lambda}{2})} |F_2|^{\rho
\frac{\lambda}{2} + (1-(1-\theta_1)\epsilon)(\frac{1}{2}+ \frac{1-\lambda}{2})} |G_1|^{(1-\theta)\frac{\lambda}{2} + (1-\theta_1\epsilon)(\frac{1}{2} + \frac{1-\lambda}{2})} |G_2|^{\theta \frac{\lambda}{2} + (1-(1-\theta_1)\epsilon)(\frac{1}{2}+ \frac{1-\lambda}{2})}. \end{align*}
One first observes that for $0 < \lambda < 1$,$ \epsilon > 0 $, $0 < \theta_1 < 1$ such that \begin{align} \label{exp2} & 1-\theta_1(1+\epsilon) > 0 \nonumber \\ & 1-(1-\theta_1)(1+\epsilon) > 0, \end{align} the series involving $2^{k_1}$, $2^{-n}$ and $2^{-m}$ are convergent. Also, for $k_2 > 0$, as long as $0 < \lambda < 1$ and $\gamma > 1$ sufficiently large, the series converges. For $k_2 < 0$, if $0 < \lambda < 1$ and $\gamma >1$ close to $1$, one has a convergent series as well. Thus \begin{align*}
& |\Lambda | \nonumber \\
\lesssim & C_1^2 C_2^2 C_3^2 |F_1|^{(1-\rho)\frac{\lambda}{2} +(1-\theta_1 \epsilon)(\frac{1}{2}+ \frac{1-\lambda}{2})} |F_2|^{\rho
\frac{\lambda}{2} + (1-(1-\theta_1)\epsilon)(\frac{1}{2}+ \frac{1-\lambda}{2})} |G_1|^{(1-\theta)\frac{\lambda}{2} + (1-\theta_1\epsilon)(\frac{1}{2} + \frac{1-\lambda}{2})} |G_2|^{\theta \frac{\lambda}{2} + (1-(1-\theta_1)\epsilon)(\frac{1}{2}+ \frac{1-\lambda}{2})}. \end{align*} One can choose $0 < \rho = \theta < 1$, $0 < \lambda < 1$ close to $0$, and $\epsilon > 0$ close to 0 such that (\ref{exp2}) hold and $$ (1-\rho)\frac{\lambda}{2} + (1-\theta_1\epsilon)(\frac{1}{2}+ \frac{1-\lambda}{2}) = (1-\theta)\frac{\lambda}{2} + (1-\theta_1\epsilon)(\frac{1}{2} + \frac{1-\lambda}{2}) = \frac{1}{p} $$ $$ \rho\frac{\lambda}{2} + (1-(1-\theta_1)\epsilon)(\frac{1}{2}+ \frac{1-\lambda}{2})= \theta \frac{\lambda}{2} + (1-(1-\theta_1)\epsilon)(\frac{1}{2} + \frac{1-\lambda}{2}) = \frac{1}{q} $$
As a consequence, $$
\Lambda \lesssim C_1^2 C_2^2 C_3^2|F_1|^{\frac{1}{p}}|F_2|^{\frac{1}{q}}|G_1|^{\frac{1}{p}}|G_2|^{\frac{1}{q}}\|h\|_{L^{s}(\mathbb{R}^2)}. $$
\end{comment} \vskip .25in
\begin{comment} \section{Proof of Theorem \ref{thm_weak} for $\Pi^0_{\text{flag}^0 \otimes \text{flag}^0}$ and $\Pi^0_{\text{flag}^0 \otimes \text{flag}^{\#_2}}$ - Haar Model} One defines the exceptional set in a similar fashion as in the previous section with a modification on $B$: \begin{align*}
\displaystyle \Omega^1 := &\bigcup_{n_1 \in \mathbb{Z}}\{Mf_1 > C_1 2^{n_1}|F_1|\} \times \{Mg_1 > C_2 2^{-n_1}|G_1|\}\cup \nonumber \\
& \bigcup_{m_1 \in \mathbb{Z}}\{Mf_2 > C_1 2^{m_1}|F_2|\} \times \{Mg_2 > C_2 2^{-m_1}|G_2|\}\cup \nonumber \\
&\bigcup_{l_1 \in \mathbb{Z}} \{MB^{+} > C_1 2^{l_1}\| B^{+}\|_1\} \times \{M\tilde{B^{+}} > C_2 2^{-l_1}\| \tilde{B^{+}}\|_1\}\nonumber \\
\Omega^2 := & \{SSh > C_3 \|h\|_{L^s(\mathbb{R}^2)}\}, \nonumber \\ \end{align*} where $$
\displaystyle B^{+} := \sum_{K}\frac{1}{|K|^{\frac{1}{2}}}|\langle f_1, \psi^1_K \rangle| |\langle f_2, \psi^2_K \rangle| |\varphi^3_K| $$ $$
\displaystyle \tilde{B^{+}} := \sum_{L}\frac{1}{|L|^{\frac{1}{2}}}|\langle g_1, \psi^1_L \rangle| |\langle g_2, \psi^2_L \rangle| |\varphi^3_L| $$ Then define $$ \Omega := \Omega^1 \cup \Omega^2$$ $$\tilde{\Omega} := \{ MM\chi_{\Omega} > \frac{1}{100}\}.$$ \subsection{Two-dimensional stopping-time decompositions II - maximal intervals}
One can perform a same stopping-time decomposition as in the previous section with $\|B\|_1$ replaced by $\|B^{+}\|_1$. In particular, $$ \mathcal{I} = \displaystyle \bigcup_{l_1}\bigcup_{T^{+} \in \mathbb{T}^{+}_{l_1}}T^{+}, $$
where $T^{+}$ represents for the tree with the tree top $I_{T^{+}}$ satisfying
$$
\frac{|\langle B_{I_{T^{+}}}, \varphi^1_{I_{T+}}\rangle|}{|I_{T^{+}}|^{\frac{1}{2}}} > C_1 2^{l_1}\|\tilde{B^{+}}\|_1
$$
In addition,
$$
\mathcal{J} = \displaystyle \bigcup_{l_2}\bigcup_{S^{+} \in \mathbb{S}^{+}_{l_2}}S^{+},
$$
where $S^{+}$ represents for the tree with the tree top $J_S$ satisfying
$$
\frac{|\langle \tilde{B}_{J_{S^{+}}}, \varphi^1_{J_{S^{+}}}\rangle|}{|J_{S^{+}}|^{\frac{1}{2}}} > C_2 2^{l_2}\|\tilde{B^{+}}\|_1
$$
\begin{obs}
If $I \times J \cap \tilde{\Omega}^{c} \neq \emptyset$ and $I \times J \in T^{+} \times S^{+}$ with $T^{+} \in \mathbb{T}^{+}_{l_1}$ and $S^{+} \in \mathbb{S}^{+}_{l_2}$, then $l_1, l_2 \in \mathbb{Z}$ satisfies $l_1 + l_2 < 0$. Equivalently, $I \times J \in T^{+} \times S^{+}$ with $T^{+} \in \mathbb{T}^{+}_{-l - l_2}$ and $S^{+} \in \mathbb{S}^{+}_{l_2}$ for some $l_2 \in \mathbb{Z}$, $l> 0$.
\end{obs}
\begin{proof}[Proof of Proposition] One has \begin{align*}
\frac{|\langle B_{I_{T^{+}}}(f_1,f_2), \varphi^1_{I_{T^{+}}} \rangle|}{|I_{T^{+}}|^{\frac{1}{2}}} = & \frac{1}{|I_{T^{+}}|^{\frac{1}{2}}} \bigg|\sum_{|K| \geq |I|}\frac{1}{|K|^{\frac{1}{2}}}\langle f_1, \psi^1_K \rangle \langle f_2, \psi^2_K \rangle \langle \varphi^3_K, \varphi^1_{I_{T^{+}}} \rangle \bigg| \nonumber \\
\leq & \frac{1}{|I_{T^{+}}|^{\frac{1}{2}}} \sum_{K}\frac{1}{|K|^{\frac{1}{2}}}|\langle f_1, \psi^1_K \rangle| |\langle f_2, \psi^2_K \rangle| \langle |\varphi^3_K|, |\varphi^1_{I_{T^{+}}}| \rangle \nonumber \\
= & \frac{|\langle B^{+}(f_1,f_2), |\varphi^1_{I_{T^{+}}}| \rangle|}{|I_{T^{+}}|^{\frac{1}{2}}} \leq MB^{+}(x), \end{align*}
for any $x \in I_T$, where $\displaystyle B^{+} := \sum_{K}\frac{1}{|K|^{\frac{1}{2}}}|\langle f_1, \psi^1_K \rangle| |\langle f_2, \psi^2_K \rangle| |\varphi^3_K|$. As a consequence,
$$I_T \subseteq \{ MB^{+} > C_12^{l_1} \|B^{+}\|_1\}.$$ Similarly, from the stopping-time decomposition on $\mathcal{J}$, one can deduce that $$
J_S \subseteq \{ M\tilde{B^{+}} > C_2 2^{l_2} \|\tilde{B^{+}}\|_1\}. $$
If $l_1 + l_2 \geq 0$, then $\{ MB^{+} > C_12^{l_1} \|B^{+}\|_1\} \times \{ M\tilde{B^{+}} > C_2 2^{l_2}\| \tilde{B^{+}}\|_1\} \subseteq \Omega^1 \subseteq \Omega$, which implies that $I \times J \subseteq \Omega \subseteq \tilde{\Omega}$ and contradicts the assumption. \end{proof}
\vskip 0.25in \subsection{Hybrid of stopping-time decompositions}
\begin{center} \begin{tabular}{ c c c } One-dimensional stopping-time decomposition& $\longrightarrow$ & $K \in \mathcal{K}_{n_0}$ \\ on $\mathcal{K}$ & & $(n_0 \in \mathbb{Z})$ \\ \end{tabular} \end{center}
\begin{center} \begin{tabular}{ c c c } One-dimensional stopping-time decomposition& $\longrightarrow$ & $L \in \mathcal{L}_{n'_0}$ \\ on $\mathcal{L}$ & & $(n'_0 \in \mathbb{Z})$ \\ \end{tabular} \end{center}
\begin{table}[h!] \begin{tabular}{ c c c } Tensor-type stopping-time decomposition I& $\longrightarrow$ & $I \times J \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2}$ \\ on $\mathcal{I} \times \mathcal{J}$ && \\ & & $(n_2, m_2 \in \mathbb{Z}, n > 0)$\\ $\Downarrow$ & & \\ General two-dimensional level sets stopping-time decomposition& $\longrightarrow$ & $I \times J \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2} \cap \mathcal{R}_{k_1,k_2} $ \\ on $\mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2}$ & & \\ & & $(n_2, m_2 \in \mathbb{Z}, n > 0,k_1 <0, k_2 \leq K)$\\ $\Downarrow$& & \\ Tensor-type stopping-time decomposition II & $\longrightarrow$ & $I \times J \in \big(\mathcal{I}_{-n-n_2,-m-m_2} \cap T^{+}\big) \times \big(\mathcal{J}_{n_2,m_2} \cap S^{+} \big)$\\ on $I \times J \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2} \cap \mathcal{R}_{k_1,k_2} $ & & $ \cap \ \ \ \ \mathcal{R}_{k_1,k_2} $ \\ & & with $T^{+} \in \mathbb{T}^{+}_{-l-l_2}$, $S^{+} \in \mathbb{S}^{+}_{l_2}$ \\ & & $(n_2, m_2, l_2 \in \mathbb{Z}, n, l > 0,k_1 <0, k_2 \leq K, )$\\ \end{tabular} \end{table}
One can apply the essentially same argument for $\Pi$ with $\|B\|_1$ replaced by $\|B^{+}\|_1$. For the sake completeness, we provides outlines of the proof: \begin{align*}
|\Lambda| \lesssim &\displaystyle \sum_{\substack{n > 0 \\ m > 0 \\ l > 0 \\ k_1 < 0 \\ k_2 \leq K}} \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z} \\ l_2 \in \mathbb{Z}}}\sum_{\substack{T^{+} \in \mathbb{T}^{+}_{-l-l_2} \\ S^{+} \in \mathbb{S}^{+}_{l_2}}}\sum_{\substack{I \times J \in \mathcal{I}_{-n - n_2, -m - m_2} \cap{T^{+}} \times \mathcal{J}_{n_2, m_2} \cap S^{+} \\I \times J \in \mathcal{R}_{k_1,k_2}}} \frac{1}{|I|^{\frac{1}{2}} |J|^{\frac{1}{2}}}| \langle B_I(f_1,f_2),\varphi_I^1 \rangle| |\langle \tilde{B}_J(g_1,g_2),\varphi_J^1 \rangle| \nonumber \\
&\quad \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot |\langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle| |\langle \chi_{E'},\psi_I^{3,H} \otimes \psi_J^{3,H} \rangle| \nonumber \\
\lesssim & \sum_{\substack{n > 0 \\ m > 0 \\ l > 0 \\ k_1 < 0 \\ k_2 \leq K}} \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z} \\ l_2 \in \mathbb{Z}}}\sum_{\substack{T^{+} \in \mathbb{T}^{+}_{-l-l_2}\\S^{+} \in \mathbb{S}^{+}_{l_2}}} \displaystyle \sup_{I \in T^{+}} \frac{|\langle B_I(f_1,f_2),\varphi_I^1 \rangle|}{|I|^{\frac{1}{2}}} \sup_{J \in S^{+}} \frac{|\langle \tilde{B}_J(g_1,g_2),\varphi_J^1 \rangle|}{|J|^{\frac{1}{2}}} \cdot \nonumber \\
&\int_{(\Omega^2_{k_1})^c \cap (\Omega^2_{k_2})^c} \sum_{\substack{I \times J \in \mathcal{I}_{-n - n_2, -m - m_2} \cap{T^{+}} \times \mathcal{J}_{n_2,m_2} \cap S^{+} \\I \times J \in \mathcal{R}_{k_1,k_2}}} \frac{|\langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle|}{|I|^{\frac{1}{2}}|J|^{\frac{1}{2}}} \frac{|\langle \chi_{E'},\psi_I^{3,H} \otimes \psi_J^{3,H} \rangle|}{|I|^{\frac{1}{2}}|J|^{\frac{1}{2}}}\chi_{I}(x) \chi_{J}(y) dx dy, \nonumber \\ \end{align*}
where $\Omega^2_{k_1}:=\{SSh(x,y) > C_3 2^{k_1+1}\|h\|_{L^s(\mathbb{R}^2)}\} $ and $\Omega^2_{k_2} := \{(SS)^H\chi_{E'}(x,y) > C_3 2^{k_2+1}\}$.
Based on the tensor-type stopping-time decomposition, $$
\sup_{I \in T^{+}} \frac{|\langle B_I(f_1,f_2),\varphi_I^1 \rangle|}{|I|^{\frac{1}{2}}} \lesssim C_1 2^{-l-l_2}\|B^{+}\|_1 $$ $$
\sup_{J \in S^{+}} \frac{|\langle \tilde{B}_J(g_1,g_2),\varphi_J^1 \rangle|}{|J|^{\frac{1}{2}}} \lesssim C_2 2^{l_2} \|\tilde{B^{+}}\|_1 $$
Meanwhile, the integral can be estimated in the same manner as in the previous section: \begin{align*}
& \int_{(\Omega^2_{k_1})^c \cap (\Omega^2_{k_2})^c} \sum_{\substack{I \times J \in \mathcal{I}_{-n - n_2, -m - m_2} \cap{T^{+}} \times \mathcal{J}_{n_2,m_2} \cap S^{+} \\I \times J \in \mathcal{R}_{k_1,k_2}}} \frac{|\langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle|}{|I|^{\frac{1}{2}}|J|^{\frac{1}{2}}} \frac{|\langle \chi_{E'},\psi_I^{3,H} \otimes \psi_J^{3,H} \rangle|}{|I|^{\frac{1}{2}}|J|^{\frac{1}{2}}}\chi_{I}(x) \chi_{J}(y) dx dy \nonumber \\
\leq & \int_{(\Omega^2_{k_1})^c \cap (\Omega^2_{k_2})^c} \bigg(\sum_{\substack{I \times J \in \mathcal{I}_{-n - n_2, -m - m_2} \cap{T^{+}} \times \mathcal{J}_{n_2,m_2} \cap S^{+} \\I \times J \in \mathcal{R}_{k_1,k_2}}}\frac{|\langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle|^2}{|I||J|} \chi_I(x)\chi_J(y)\bigg)^{\frac{1}{2}} \nonumber \\
&\quad \quad \quad \quad \quad \quad \bigg(\sum_{\substack{I \times J \in \mathcal{I}_{-n - n_2, -m - m_2} \cap{T^{+}} \times \mathcal{J}_{n_2,m_2} \cap S^{+}\\I \times J \in \mathcal{R}_{k_1,k_2}}}\frac{|\langle \chi_{E'},\psi_I^{3,H} \otimes \psi_J^{3,H} \rangle|^2}{|I||J|}\chi_{I}(x) \chi_{J}(y)\bigg)^{\frac{1}{2}} dxdy \nonumber \\
\lesssim & C_3^2 2^{k_1}\|h\|_{L^s(\mathbb{R}^2)} 2^{k_2} \bigg| \bigcup_{\substack{I \times J \in \mathcal{I}_{-n - n_2, -m - m_2} \cap{T^{+}} \times \mathcal{J}_{n_2,m_2} \cap S^{+}\\I \times J \in \mathcal{R}_{k_1,k_2}}}I \times J \bigg|,\nonumber \\ \end{align*} where the last inequality follows from the pointwise estimates on the set $(\Omega^2_{k_1})^c \cap (\Omega^2_{k_2})^c$. \vskip 0.25 in \subsection{Sparsity Condition.} One can apply the sparsity condition as in the previous section to derive that $$
\sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z} \\ l_2 \in \mathbb{Z}}}\sum_{\substack{T^{+} \in \mathbb{T}^{+}_{-l-l_2}\\S^{+} \in \mathbb{S}^{+}_{l_2}}} \bigg|\big(\bigcup_{\substack{R\in \mathcal{R}_{k_1,k_2} \\ R \in \mathcal{I}_{-n-n_2,-m-m_2} \times \mathcal{J}_{n_2,m_2}}} R \big) \cap \big(I_{T^{+}} \times J_{S^{+}} \big)\bigg| \lesssim \min(C_3^{-1}2^{-k_1},C_3^{-\gamma}2^{-k_2 \gamma}) $$ \vskip 0.25in \subsection{Fubini Argument.} \begin{align*}
& \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z} \\ l_2 \in \mathbb{Z}}}\sum_{\substack{T^{+} \in \mathbb{T}^{+}_{-l-l_2}\\S^{+} \in \mathbb{S}^{+}_{l_2}}} \big|I_{T^{+}} \times J_{S^{+}} \big| \nonumber \\
\leq &\underbrace{\sup_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}} \sum_{l_2 \in \mathbb{Z}}\bigg(\sum_{\substack{T^{+} \in \mathbb{T}^{+}_{-l-l_2}\\ I_{T^{+}} \in \mathcal{I}_{-n-n_2,-m-m_2}}}|I_{T^{+}}|\bigg)^{\frac{1}{2}}\bigg(\sum_{\substack{S^{+} \in \mathbb{S}^{+}_{l_2}\\ J_{S^{+}} \in \mathcal{J}_{n_2,m_2}}} |J_{S^{+}} |\bigg)^{\frac{1}{2}}}_{a} \cdot \nonumber \\
& \underbrace{\sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\sup_{l_2 \in \mathbb{Z}}\bigg(\sum_{\substack{T^{+} \in \mathbb{T^+}_{-l-l_2}\\ I_{T^+} \in \mathcal{I}_{-n-n_2,-m-m_2}}}|I_{T^+}|\bigg)^{\frac{1}{2}}\bigg(\sum_{\substack{S^+ \in \mathbb{S}^+_{l_2}\\ J_{S^+} \in \mathcal{J}_{n_2,m_2}}} |J_{S^+} |\bigg)^{\frac{1}{2}}}_{b}. \nonumber \\ \end{align*} One first notices that $b$ can be estimated by the exactly same argument as in the previous section as the only fact about $T^{+}$ and $S^+$ is that $\{I_{T^+}: T^+ \in \mathbb{T}^+_{-l-l_2} \}$, $\{J_{S^+}: S^+ \in \mathbb{S}^+_{l_2} \}$ form disjoint collections of intervals. As a result, the following estimate holds for $b$:
$$b \lesssim |F_1|^{-\frac{\mu}{2}\epsilon}|F_2|^{-\frac{1-\mu}{2}\epsilon}|G_1|^{-\frac{\mu}{2}\epsilon}|G_2|^{-\frac{1-\mu}{2}\epsilon}2^{n\cdot\frac{1}{2}\mu(1+\epsilon)}2^{m\cdot \frac{1}{2}(1-\mu)(1+\epsilon)}$$ with $0 < \mu < 1$, $\epsilon > 0$.
The estimate for $a$ requires slight modification: \begin{align*}
a \leq & \sup_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}} \sum_{l_2 \in \mathbb{Z}}(C_12^{-l-l_2}\|B^+\|_1)\bigg(\sum_{\substack{T^+ \in \mathbb{T}^+_{-l-l_2}\\ I_{T^+} \in \mathcal{I}_{-n-n_2,-m-m_2}}}|I_{T^+}|\bigg)^{\frac{1}{2}}\cdot (C_22^{l_2}\|\tilde{B^+}\|_1) \bigg(\sum_{\substack{S^+ \in \mathbb{S}^+_{l_2}\\ J_{S^+} \in \mathcal{J}_{n_2,m_2}}} |J_{S^+} |\bigg) ^{\frac{1}{2}} \nonumber \\
& \cdot 2^{l}\|B^+\|_1^{-1}\|\tilde{B^+}\|_1^{-1}C_1^{-1} C_2^{-1}\nonumber \\
\leq & \sup_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}} \underbrace{\bigg[\sum_{l_2 \in \mathbb{Z}}(C_12^{-l-l_2}\|B^+\|_1)^2\sum_{\substack{T^+ \in \mathbb{T}^+_{-l-l_2}\\ I_{T^+} \in \mathcal{I}_{-n-n_2,-m-m_2}}}|I_{T^+}|\bigg]^{\frac{1}{2}}}_{a^1}\cdot \underbrace{\bigg[\sum_{l_2 \in \mathbb{Z}}(C_22^{l_2}\|\tilde{B^+}\|_1)^{2} \sum_{\substack{S^+ \in \mathbb{S}^+_{l_2}\\ J_{S^+} \in \mathcal{J}_{n_2,m_2}}} |J_{S^+} |\bigg]^{\frac{1}{2}}}_{a^2} \nonumber \\
& \cdot 2^{l}\|B^+\|_1^{-1}\|\tilde{B^+}\|_1^{-1}C_1^{-1} C_2^{-1}.\nonumber \\
\end{align*}
With the same argument in Section 7 and Proposition \ref{B_en} which gives energy estimates for the case when $(\phi_K^3)_K$ is non-lacunary, one has \begin{align*}
a^1 \lesssim & 2^{(-n-n_2)(\frac{1}{p_1} - \theta_1)} 2^{(-m-m_2)(\frac{1}{q_1} - \theta_2)}|F_1|^{\frac{1}{p_1}}|F_2|^{\frac{1}{q_1}} \nonumber \\
a^2 \lesssim & 2^{n_2 (\frac{1}{p_2} - \theta_1')}2^{m_2(\frac{1}{q_2} - \theta_2')} |G_1|^{\frac{1}{p_2}}|G_2|^{\frac{1}{q_2}}, \end{align*} for any $0 < \theta_1,\theta_2, \theta_1', \theta_2' < 1$ with $\theta_1 + \theta_2 = \theta_1' + \theta_2' = \frac{1}{2}$.
Then by choosing $0 < \theta_1, \theta_1' < 1$ such that $$ \frac{1}{p_1} - \theta_1 = \frac{1}{p_2} - \theta_1' $$ $$ \frac{1}{q_1} - \theta_2 = \frac{1}{q_2} - \theta_2' $$ one obtains $$
a \lesssim 2^{-n(\frac{1}{p_1}-\theta_1)}2^{-m(\frac{1}{q_1} - \theta_2)}|F_1|^{\frac{1}{p_1}}|F_2|^{\frac{1}{p_2}}|G_1|^{\frac{1}{q_1}}|G_2|^{\frac{1}{q_2}}\cdot 2^{l}\|B^+\|_1^{-1}\|\tilde{B^+}\|_1^{-1} $$
One observes that the estimates for the current model match with the ones for $\Pi_{\text{flag}^0\otimes \text{flag}^0}$ except that $\|B\|_1$ and $\|\tilde{B}\|_1$ are replaced by $\|B^+\|_1$ and $\|\tilde{B^+}\|_1$. One can therefore apply the estimates for the multilinear form of $\Pi_{\text{flag}^0\otimes \text{flag}^0}$ with appropriate modifications: $$
|\Lambda| \lesssim C_1 C_2 C_3^2 \| h \|_{L^s}
\cdot \big(|F_1|^{\frac{1}{p_1}-\frac{\mu}{2}\epsilon}|F_2|^{\frac{1}{q_1}-\frac{1-\mu}{2}\epsilon}|G_1|^{\frac{1}{p_2}-\frac{\mu}{2}\epsilon}|G_2|^{\frac{1}{q_2}-\frac{1-\mu}{2}\epsilon}\big)^{1-\lambda}\|B^+\|_1^{\lambda}\|\tilde{B^+}\|_1^{\lambda} $$ where $$
\|B^+\|_1 \lesssim |F_1|^{\rho}|F_2|^{1-\rho} $$ $$
\|\tilde{B^+}\|_1 \lesssim |G_1|^{\rho'}|G_2|^{1-\rho'} $$
The above estimates for $\|B^+\|_1$, $\|\tilde{B^+}\|_1$ agree with the estimates for $\|B\|_1$, $\|\tilde{B}\|_1$, which allow one to conclude that $$
|\Lambda | \lesssim |F_1| ^{\frac{1}{p_1}} |F_2|^{\frac{1}{q_1}} |G_1|^{\frac{1}{p_2}} |G_2|^{\frac{1}{q_2}} $$ with proper choice of $\mu, \epsilon, \lambda, \rho, \rho' $ which agrees with the choice in expression (\ref{exp00}). \end{comment}
\section{Proof of Theorem \ref{thm_weak_inf_mod} for $\Pi_{\text{flag}^{\#_1} \otimes \text{flag}^{\#_2}}$ - Haar Model} \label{section_thm_inf_fixed_haar}
One can mimic the proof in Section \ref{section_thm_haar_fixed} with a change of perspectives on size estimates. More precisely, one applies some trivial size estimates for functions $f_2$ and $g_2$ lying in $L^{\infty}$ spaces while one needs to pay respect to the fact that $f_1$ and $g_1$ could lie in $L^p$ space for any $p >1$. Such perspective is demonstrated in the stopping-time decomposition and in the definition of the exceptional set. \subsection{Localization} \label{section_thm_inf_fixed_haar_localization} One defines $$\Omega := \Omega^1 \cup \Omega^2,$$ where \begin{align*}
\Omega^1 := &\bigcup_{n_1 \in \mathbb{Z}}\{x:Mf_1(x) > C_1 2^{n_1}\|f_1\|_{p}\} \times \{y:Mg_1(y) > C_2 2^{-n_1}\|g_1\|_{p}\}, \nonumber \\
\Omega^2 := & \{(x,y) \in \mathbb{R}^2: SSh(x,y) > C_3 \|h\|_{L^s(\mathbb{R}^2)}\}, \nonumber \\ \end{align*} and $$Enl(\Omega) := \{(x,y) \in \mathbb{R}^2: MM\chi_{\Omega}(x,y) > \frac{1}{100}\}.$$ Let $$E' := E \setminus Enl(\Omega).$$
It is not difficult to check that given $C_1, C_2$ and $C_3$ are sufficiently large, $|E'| \sim |E|$ where $|E|$ can be assumed to be 1. It suffices to prove that the multilinear form defined in (\ref{form_haar_larger})
satisfies \begin{equation}
|\Lambda^H_{\text{flag}^{\#1} \otimes \text{flag}^{\#2}}(f_1,f_2,g_1,g_2,h, \chi_{E'})| \lesssim \|f_1\|_{L^p(\mathbb{R})}\|f_2\|_{L^{\infty}(\mathbb{R})} \|g_1\|_{L^p(\mathbb{R})}\|g_2\|_{L^{\infty}(\mathbb{R})} \|h\|_{L^{s}(\mathbb{R}^2)}. \end{equation} \subsection{Summary of stopping-time decompositions.} \label{section_thm_inf_fixed_haar_summary} {\fontsize{10}{10} \begin{center} \begin{tabular}{ l l l } I. Tensor-type stopping-time decomposition I on $\mathcal{I} \times \mathcal{J}$& $\longrightarrow$ & $I \times J \in \mathcal{I}_{-n-n_2}^{generic} \times \mathcal{J}_{n_2}^{generic}$ \\
& & $(n_2 \in \mathbb{Z}, n > 0)$ \\ II. General two-dimensional level sets stopping-time decomposition & $\longrightarrow$ & $I \times J \in \mathcal{R}_{k_1,k_2} $ \\ \ \ \ \ on $\mathcal{I} \times \mathcal{J}$& & $(k_1 <0, k_2 \leq K)$ \end{tabular} \end{center}} where \begin{align*}
\mathcal{I}_{-n-n_2}^{generic} := & \{ I \in \mathcal{I} \setminus \mathcal{I}_{-n-n_2+1}: \left| I \cap \Omega^{generic}_{-n-n_2}\right| > \frac{1}{10}|I| \}, \nonumber \\
\mathcal{J}_{n_2}^{generic} := & \{ J \in \mathcal{J} \setminus \mathcal{J}_{n_2+1}: \left| I \cap \tilde{\Omega}^{generic}_{n_2}\right| > \frac{1}{10}|J| \}, \end{align*} with \begin{align*}
\Omega_{-n-n_2}^{generic} := &\{x: Mf_1(x)> C_1 2^{-n-n_2}\|f_1\|_p \}, \nonumber \\
\tilde{\Omega}_{n_2}^{generic} := & \{y: Mg_1(y)> C_2 2^{n_2} \|g_1\|_p \}. \end{align*} \subsection{Application of stopping-time decompositions} \label{section_thm_inf_fixed_haar_application_st} One can now apply the stopping-time decompositions and follow the same argument in Section \ref{section_thm_haar_fixed} to deduce that {\fontsize{10}{10}\begin{align} \label{form11_inf}
& |\Lambda^H_{\text{flag}^{\#1} \otimes \text{flag}^{\#2}}(f_1,f_2,g_1,g_2,h, \chi_{E'})| \nonumber\\
\lesssim &\bigg|\displaystyle \sum_{\substack{n> 0 \\ k_1 < 0 \\ k_2 \leq K}} \sum_{n_2 \in \mathbb{Z}}\sum_{\substack{I \times J \in \mathcal{I}^{generic}_{-n-n_2} \times \mathcal{J}^{generic}_{n_2}\\I \times J \in \mathcal{R}_{k_1,k_2}}} \frac{1}{|I|^{\frac{1}{2}} |J|^{\frac{1}{2}}} \langle B_I^{\#_1,H}(f_1,f_2),\varphi_I^{1,H} \rangle \langle \tilde{B}_J^{\#_2,H} (g_1,g_2),\varphi_J^{1,H} \rangle \langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle \langle \chi_{E'},\psi_I^{3,H} \otimes \psi_J^{3,H} \rangle \bigg| \nonumber \\\nonumber \\
\lesssim & \sum_{\substack{n> 0 \\ k_1 < 0 \\ k_2 \leq K}} \sum_{n_2 \in \mathbb{Z}}\sup_{I \in \mathcal{I}^{generic}_{-n-n_2}} \frac{|\langle B_I^{\#_1,H}(f_1,f_2),\varphi_I^{1,H} \rangle|}{|I|^{\frac{1}{2}}} \cdot \sup_{J \in \mathcal{J}^{generic}_{n_2}}\frac{| \langle \tilde{B}_J^{\#_2,H} (g_1,g_2),\varphi_J^{1,H} \rangle|}{|J|^{\frac{1}{2}}}\cdot C_3 2^{k_1} \| h \|_{L^s} 2^{k_2} \cdot \nonumber \\
&\quad \quad \quad \quad \quad \quad \bigg|\big(\bigcup_{R\in \mathcal{R}_{k_1,k_2}} R\big) \cap \big(\bigcup_{I \in \mathcal{I}'_{-n-n_2}} I \times \bigcup_{J \in \mathcal{J}'_{n_2}}J\big)\bigg|. \end{align}}
To estimate $\displaystyle \sup_{I \in \mathcal{I}^{generic}_{-n-n_2}} \frac{|\langle B_I^{\#_1,H}(f_1,f_2),\varphi_I^{1,H} \rangle|}{|I|^{\frac{1}{2}}} $, one can now apply Lemma \ref{B_size} with $S:= \{Mf_1 \leq C_1 2^{-n-n_2+1}|F_1|^{\frac{1}{p}} \}$ and obtain $$
\sup_{I \in \mathcal{I}^{generic}_{-n-n_2}}\frac{|\langle B_I^{\#_1,H}(f_1,f_2),\varphi_I^{1,H} \rangle|}{|I|^{\frac{1}{2}}} \lesssim \sup_{K \cap S \neq \emptyset}\frac{|\langle f_1, \varphi^1_K \rangle|}{|K|^{\frac{1}{2}}} \sup_{K \cap S \neq \emptyset} \frac{|\langle f_2, \phi_K^2 \rangle|}{|K|^{\frac{1}{2}}}, $$ where by the definition of $S$, $$
\sup_{K \cap S \neq \emptyset}\frac{|\langle f_1, \varphi^1_K \rangle|}{|K|^{\frac{1}{2}}} \lesssim C_12^{-n-n_2}\|f_1\|_p, $$ and by the fact that $f_2 \in L^{\infty}$, $$
\sup_{K \cap S \neq \emptyset} \frac{|\langle f_2, \phi_K^2 \rangle|}{|K|^{\frac{1}{2}}} \lesssim \|f_2\|_{\infty}. $$ As a result, \begin{equation} \label{est_x}
\sup_{I \in \mathcal{I}^{generic}_{-n-n_2}}\frac{|\langle B_I^{\#_1,H}(f_1,f_2),\varphi_I^{1,H} \rangle|}{|I|^{\frac{1}{2}}} \lesssim C_12^{-n-n_2}\|f_1\|_p\|f_2\|_{\infty}. \end{equation} By a similar reasoning, \begin{equation} \label{est_y}
\sup_{J \in \mathcal{J}^{generic}_{n_2}}\frac{|\langle \tilde{B}_J^{\#_2,H}(g_1,g_2),\varphi_J^{1,H} \rangle|}{|J|^{\frac{1}{2}}} \lesssim C_2 2^{n_2}\|g_1\|_p\|g_2\|_{\infty}. \end{equation} When applying the estimates (\ref{est_x}) and (\ref{est_y}) to (\ref{form11_inf}), one concludes that \begin{align*}
|\Lambda^{H}_{\text{flag}^{\#1} \otimes \text{flag}^{\#2}}| \lesssim
&C_1 C_2 C_3^2 \|f_1\|_{L^p(\mathbb{R})}\|f_2\|_{L^{\infty}(\mathbb{R})} \|g_1\|_{L^p(\mathbb{R})}\|g_2\|_{L^{\infty}(\mathbb{R})} \|h\|_{L^{s}(\mathbb{R}^2)} \cdot \\
& \sum_{\substack{n > 0 \\ k_1 < 0 \\ k_2 \leq K}} 2^{-n}2^{k_1} 2^{k_2} \sum_{n_2 \in \mathbb{Z}}\bigg|\big(\bigcup_{R\in \mathcal{R}_{k_1,k_2}} R\big) \cap \big(\bigcup_{I \in \mathcal{I}^{generic}_{-n-n_2,-m-m_2}} I \times \bigcup_{J \in \mathcal{J}^{generic}_{n_2,m_2}}J\big)\bigg|\nonumber \\
\lesssim &C_1 C_2 C_3^2 \|f_1\|_{L^p(\mathbb{R})}\|f_2\|_{L^{\infty}(\mathbb{R})} \|g_1\|_{L^p(\mathbb{R})}\|g_2\|_{L^{\infty}(\mathbb{R})} \|h\|_{L^{s}(\mathbb{R}^2)} \sum_{\substack{n > 0 \\ k_1 < 0 \\ k_2 \leq K}} 2^{-n}2^{k_1(1-\frac{s}{2})} 2^{k_2(1-\frac{\gamma}{2})}, \nonumber \end{align*} where the last inequality follows from the sparsity condition. With proper choice of $\gamma >1$, one obtains the desired estimate.
\section{Proof of Theorem \ref{thm_weak_inf_mod} for $\Pi_{\text{flag}^0 \otimes \text{flag}^0}$ - Haar Model}\label{section_thm_inf_haar} One interesting fact is that when \begin{equation} \label{easy_case} \frac{1}{p_1} + \frac{1}{q_1} = \frac{1}{p_2} + \frac{1}{q_2} \leq 1, \end{equation} Theorem \ref{thm_weak_mod} can be proved by a simpler argument as commented in Remark \ref{rmk_easyhard_exponent} of Section \ref{section_thm_haar}. And Theorem \ref{thm_weak_inf_mod} for the model $\Pi_{\text{flag}^0 \otimes \text{flag}^0}$ can be viewed as a sub-case and proved by the same argument. The key idea is that in the case specified in (\ref{easy_case}), one no longer needs the localized operators $B^H_I$ and $\tilde{B}^H_J$ (defined in (\ref{B_0_local_haar_simplified}) and (\ref{B_local_definition_haar})) in the proof and the operators $B$ and $\tilde{B}$ (defined in (\ref{B_global_proof}) and (\ref{B_global_definition})) will be involved. In particular, we would consider the operator \begin{align} \label{Pi_nofixed_easy}
\Pi^{H,3}_{\text{flag}^0 \otimes \text{flag}^0}(f_1,f_2,g_1,g_2,h):= \sum_{I \times J \in \mathcal{R}} \frac{1}{|I|^{\frac{1}{2}} |J|^{\frac{1}{2}}} \langle B_I(f_1,f_2),\varphi_I^{1} \rangle \langle \tilde{B}_J(g_1,g_2),\varphi_J^{1} \rangle \langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle \psi_I^{3,H} \otimes \psi_J^{3,H}, \end{align} where the notation for the operator represents the Haar assumption on the third families of the functions $(\psi_I^{3,H})_{I \in \mathcal{I}}$ and $(\psi_J^{3,H})_{J \in \mathcal{J}}$.
Let $$ \frac{1}{t} := \frac{1}{p_1} + \frac{1}{q_1} = \frac{1}{p_2} + \frac{1}{q_2}, $$ and the condition of the exponents (\ref{easy_case}) translates to $$ t \geq 1. $$
\subsection{Localization.} \label{section_thm_inf_haar_localization} One first defines $$\Omega := \Omega^1 \cup \Omega^2,$$ where \begin{align*}
\displaystyle \Omega^1 := &\bigcup_{l_2 \in \mathbb{Z}} \{x:MB(f_1,f_2)(x) > C_1 2^{-l_2}\| B\|_t\} \times \{y: M\tilde{B}(g_1,g_2)(y) > C_2 2^{l_2}\|\tilde{B}\|_t\}, \nonumber \\
\Omega^2 := & \{(x,y) \in \mathbb{R}^2: SSh (x,y)> C_3 \|h\|_{L^s(\mathbb{R}^2)}\}, \nonumber \\ \end{align*} and $$Enl(\Omega) := \{(x,y) \in \mathbb{R}^2: MM\chi_{\Omega}(x,y) > \frac{1}{100}\}.$$ Let $$ E' := E \setminus Enl(\Omega). $$ \begin{remark}
We shall notice that $t \geq 1$ allows one to use the mapping property of the Hardy-Littlewood maximal operator, which plays an essential role in the estimate of $|\Omega|$. \end{remark}
A straightforward computation shows $|E'| \sim |E|$ given that $C_1, C_2$ and $C_3$ are sufficiently large. It suffices to assume that $|E'| \sim |E| = 1$ and to prove that the multilinear form \begin{equation} \Lambda^{H,3}_{\text{flag}^{0} \otimes \text{flag}^{0}}(f_1, f_2, g_1, g_2, h, \chi_{E'}) := \langle \Pi^{H,3}_{\text{flag}^{0} \otimes \text{flag}^{0}}(f_1, f_2, g_1, g_2, h), \chi_{E'} \rangle \end{equation} satisfies \begin{equation}
|\Lambda^{H,3}_{\text{flag}^{0} \otimes \text{flag}^{0}}(f_1,f_2,g_1,g_2,h, \chi_{E'})| \lesssim \|f_1\|_{L^p(\mathbb{R})}\|f_2\|_{L^{\infty}(\mathbb{R})} \|g_1\|_{L^p(\mathbb{R})}\|g_2\|_{L^{\infty}(\mathbb{R})} \|h\|_{L^{s}(\mathbb{R}^2)}.
\end{equation} \subsection{Summary of stopping-time decompositions.} \label{section_thm_inf_haar_summary}
\begin{center} \begin{tabular}{ c c c } General two-dimensional level sets stopping-time decomposition& $\longrightarrow$ & $I \times J \in \mathcal{R}_{k_1,k_2} $ \\ on $ \mathcal{I} \times \mathcal{J}$ & & $(k_1 <0, k_2 \leq K)$ \end{tabular} \end{center}
\begin{comment} \begin{center} \begin{tabular}{ c c c } Tensor-type stopping-time decomposition I & $\longrightarrow$ & $I \times J \in \mathcal{I}''_{-l-l_2} \times \mathcal{J}''_{l_2}$ \\ on $\mathcal{I} \times \mathcal{J}$ & & $(l_2 \in \mathbb{Z}, l > 0)$ \\
$\Downarrow$ & & \\
General two-dimensional level sets stopping-time decomposition& $\longrightarrow$ & $I \times J \in \mathcal{I}''_{-l-l_2} \times \mathcal{J}''_{l_2} \cap \mathcal{R}_{k_1,k_2} $ \\ on $ \mathcal{I}''_{-l-l_2} \times \mathcal{J}''_{l_2}$ & & $(l_2 \in \mathbb{Z}, l > 0,k_1 <0, k_2 \leq K)$ \end{tabular} \end{center} where \begin{align*}
\mathcal{I}''_{-l-l_2} := & \{ I \in \mathcal{I} \setminus \mathcal{I}''_{-l-l_2+1}: \left| I \cap \Omega''^x_{-l-l_2}\right| > \frac{1}{10}|I| \} \nonumber \\
\mathcal{J}''_{l_2} := & \{ J \in \mathcal{J} \setminus \mathcal{J}''_{l_2+1}: \left| I \cap \Omega''^y_{l_2}\right| > \frac{1}{10}|J| \} \end{align*} with \begin{align*}
\Omega''^x_{-l-l_2} := &\{MB> C_1 2^{-l-l_2}\|B\|_{t} \} \nonumber \\
\Omega''^y_{l_2} := & \{M\tilde{B}> C_2 2^{l_2}\|\tilde{B}\|_{t} \} \end{align*} \end{comment}
\begin{comment} One can then perform a tensor-type stopping-time decompositions for $I \times J$ with respect to $Mf_1$ and $Mg_1$ as before and a general 2-dimensional level sets decomposition with respect to $SSh$ as before. The only difference here is that one "forgets about" $f_2$ and $g_2$ and uses the trivial estimates $$
\frac{|\langle f_2, \varphi^1_I \rangle |}{|I|^{\frac{1}{2}}} \lesssim 1 $$ $$
\frac{|\langle g_2, \varphi^1_J \rangle |}{|I|^{\frac{1}{2}}} \lesssim 1 $$ for any $I$ and $J$. \end{comment} \vskip .25in
\begin{comment}}
As before, one defines $E' := E \setminus \tilde{\Omega}$ for any $E \subseteq \mathbb{R}^2$ with $|E| < \infty$. It suffices to prove that
$$\Lambda(f_1\otimes g_1, f_2 \otimes g_2, h, \chi_{E'}) := \displaystyle \sum_{I \times J \in \mathcal{I} \times \mathcal{J}} \frac{1}{|I|^{\frac{1}{2}} |J|} \langle B_I(f_1,f_2),\varphi_I^1 \rangle \langle g_1, \varphi_J^1 \rangle \langle g_2, \varphi_J^2 \rangle \langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle \langle \chi_{E'},\psi_I^{3,H} \otimes \psi_J^{3,H} \rangle$$ satisfies
$$|\Lambda(f_1\otimes g_1, f_2 \otimes g_2, h, \chi_{E'})| \lesssim \|f_1\|_{L_x^{p}} \|g_1\|_{L_y^{p}}\|h\|_{L^s(\mathbb{R}^2)}.$$ \end{comment} \subsection{Application of the stopping-time decomposition} \label{section_thm_inf_haar_application_st} One performs the \textit{general two-dimensional level sets stopping-time decomposition} with respect to the hybrid operators as specified in the definition of the exceptional set. It would be evident from the argument below that there is no stopping-time decomposition necessary for the maximal functions of $B$ and $\tilde{B}$ (defined in (\ref{B_global_definition}) and (\ref{B_global_proof})). One brief explanantion is that only ``averages'' for $B$ and $\tilde{B}$ are required while the measurement of the set where the averages are attained is not. As a consequence, the macro-control of the averages would be sufficient and the stopping-time decomposition, which can be seen as a more delicate ``slice-by-slice'' or ``level-by-level'' partition, is not compulsory. More precisely, \begin{align} \label{form00_inf}
&|\Lambda^{H,3}_{\text{flag}^{0} \otimes \text{flag}^{0}}(f_1,f_2,g_1,g_2,h, \chi_{E'})|\nonumber \\
= &\displaystyle \bigg|\sum_{\substack{ k_1 < 0 \\ k_2 \leq K}} \sum_{\substack{I \times J \in \mathcal{R}_{k_1,k_2}}} \frac{1}{|I|^{\frac{1}{2}} |J|^{\frac{1}{2}}} \langle B_I(f_1,f_2),\varphi_I^{1} \rangle \langle \tilde{B}_J(g_1, g_2), \varphi_J^{1} \rangle \langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle \langle \chi_{E'},\psi_I^{3,H} \otimes \psi_J^{3,H} \rangle \bigg|\nonumber \\
\lesssim & \sum_{\substack{k_1 < 0 \\ k_2 \leq K}}\displaystyle \sup_{I \times J \in \mathcal{I}\times \mathcal{J}} \bigg(\frac{|\langle B_I(f_1,f_2),\varphi_I^{1} \rangle|}{|I|^{\frac{1}{2}}} \frac{|\langle \tilde{B}_J(g_1, g_2), \varphi_J^{1} \rangle|}{|J|^{\frac{1}{2}}} \bigg) \cdot C_3^2 2^{k_1}\|h\|_{L^s(\mathbb{R}^2)} 2^{k_2} \bigg|\bigcup_{\substack{I \times J \in \mathcal{R}_{k_1,k_2}}}I \times J\bigg|. \end{align}
By the same esimate applied in previous sections, namely (\ref{int_area}), one has \begin{equation} \label{measure_ch9}
\bigg|\bigcup_{\substack{I \times J \in \mathcal{R}_{k_1,k_2}}}I \times J\bigg| \lesssim \min(C_3^{-1}2^{-k_1s}, C_3^{-\gamma}2^{-k_2 \gamma}), \end{equation} for any $\gamma >1$. Meanwhile, an argument similar to the proof of Observation \ref{obs_st_B} in Section \ref{section_thm_haar_tensor_1d_maximal} implies that \begin{equation} \label{B_global_ch9}
\frac{|\langle B_I(f_1,f_2),\varphi_I^{1,H} \rangle|}{|I|^{\frac{1}{2}}} \frac{|\langle \tilde{B}_J(g_1, g_2), \varphi_J^{1,H} \rangle|}{|J|^{\frac{1}{2}}} \lesssim C_1 C_2 \|B(f_1,f_2)\|_t \|\tilde{B}(g_1,g_2)\|_t, \end{equation} for any $I \times J \cap Enl(\Omega)^c \neq \emptyset$ as assumed in the Haar model. By applying (\ref{measure_ch9}) and (\ref{B_global_ch9}) to (\ref{form00_inf}), one has \begin{equation} \label{form00_inf_almost}
C_1 C_2 C_3^2 \| h \|_{L^s(\mathbb{R}^2)} \|B(f_1,f_2)\|_t \|\tilde{B}(g_1,g_2)\|_t \sum_{\substack{k_1 < 0 \\ k_2 \leq K}} C_3 2^{k_1(1-\frac{s}{2})}2^{k_2(1-\frac{\gamma}{2})} \lesssim \| h \|_{L^s(\mathbb{R}^2)} \|B(f_1,f_2)\|_t \|\tilde{B}(g_1,g_2)\|_t, \end{equation} with appropriate choice of $\gamma>1$. One can now invoke Lemma \ref{B_global_norm} to conclude the estimate of (\ref{form00_inf_almost}). In particular, for $1 < p_i, q_j \leq \infty$ for $i, j = 1, 2$ satisfying (\ref{easy_case}), \begin{align} \label{B_easy}
\|B(f_1,f_2)\|_t \lesssim & \|f_1\|_{p_1} \|f_2\|_{q_1} \ \ \text{and} \ \ \|\tilde{B}(g_1,g_2)\|_t \lesssim \|g_1\|_{p_2} \|g_2\|_{q_2}, \end{align}
Therefore, \begin{equation*}
|\Lambda^{H,3}_{\text{flag}^{0} \otimes \text{flag}^{0}}(f_1,f_2,g_1,g_2,h, \chi_{E'})| \lesssim \|f_1\|_{p_1} \|f_2\|_{q_1}\|g_1\|_{p_2} \|g_2\|_{q_2} \| h \|_{L^s(\mathbb{R}^2)}. \end{equation*} \begin{comment}
\textbf{Estimate of $\|B\|_t$.}
Without loss of generality, we focuses on the estimate for $\|B\|_p$, which would apply to $\|\tilde{B}\|_p$ as well. \begin{claim} \label{B_weak}
Suppose that $1 < p < \infty$. Then for any $|f_1| \leq \chi_{F_1}$ with $|F_1| < \infty$ and $f_2 \in L^{\infty}$,
$$\|B(f_1,f_2)\|_{p,\infty} \lesssim |F_1|^{\frac{1}{p}} \|f_2\|_{L^{\infty}}.$$ \end{claim}
\begin{claim} \label{B_strong} Suppose that $1 < p < \infty$. Then for any $f_1 \in L^{p}$ and $f_2 \in L^{\infty}$,
$$\|B(f_1,f_2)\|_{p} \lesssim \|f_1\|_{L^p} \|f_2\|_{L^{\infty}}.$$ \end{claim}
One observes that Claim (\ref{B_weak}) implies Claim (\ref{B_strong}) by multilinear interpolation.
The remaining of the section will be devoted to the proof of Claim (\ref{B_weak}), which is similar to the estimate of $\|B\|_1$ described in Section 7. \begin{proof}[Proof of Claim (\ref{B_weak})]
For any $\chi_S \in L^{p'}$, one consider the corresponding multilinear form and apply the size-energy estimates as follows: \begin{align} \label{B_inf}
|\langle B, \chi_S \rangle| & \leq \sum_{K}\frac{1}{|K|^{\frac{1}{2}}} |\langle f_1, \phi^1_K\rangle| |\langle f_2, \phi^2_K \rangle| |\langle \chi_S,\phi^3_K \rangle| \nonumber \\ &\lesssim \text{size}((\langle f_1,\varphi^1_K \rangle)_{K \in \mathcal{K}})^{1-\theta_1} \text{size}((\langle f_2,\psi^2_K \rangle)_{K \in \mathcal{K}})^{1-\theta_2} \text{size}((\langle \chi_S,\psi^3_K \rangle)_{K \in \mathcal{K}})^{1-\theta_3} \nonumber \\ & \quad \text{energy}((\langle f_1,\phi^1_K \rangle)_{K \in \mathcal{K}})^{\theta_1}\text{energy}((\langle f_2,\phi^2_K \rangle)_{K \in \mathcal{K}})^{\theta_2} \text{energy}((\langle \chi_S,\phi^3_K \rangle)_{K \in \mathcal{K}})^{\theta_3}, \end{align} where one uses the trivial estimates $$ \text{size}((\langle f_1,\varphi^1_K \rangle)_{K \in \mathcal{K}}), \text{size}((\langle \chi_S,\psi^3_K \rangle)_{K \in \mathcal{K}}) \leq 1 $$
since $|f_1| \leq \chi_{F_1}$ and $f_2 \in L^{\infty}$. Also, since $f_2 \in L^{\infty}$, $$
\text{size}((\langle f_2,\psi^2_K \rangle)_{K \in \mathcal{K}}) \leq \|f_2\|_{L^{\infty}} $$ Moreover, $$
\text{energy}((\langle f_1,\varphi^1_K \rangle)_{K \in \mathcal{K}}) \lesssim \|f_1\|_1 = |F_1| $$ and $$
\text{energy}((\langle \chi_S,\psi^3_K \rangle)_{K \in \mathcal{K}}) \lesssim \|\chi_S\|_1 = |S| $$ By taking $\theta_2 = 0$, $\theta_3 = \frac{1}{p'}$ and applying the above estimates to (\ref{B_inf}), one has $$
|\langle B, \chi_S \rangle| \lesssim |F_1|^{\frac{1}{p}}\|f_2\|_{L^{\infty}}|S|^{\frac{1}{p'}} $$ which implies that $$
\|B\|_{p,\infty} \lesssim |F_1|^{\frac{1}{p}} \|f_2\|_{L^{\infty}} $$ as desired. \end{proof} \end{comment} \begin{remark} \begin{enumerate} \noindent \item One notices that when \begin{align*} & p_1 = p_2 = p \\ & q_1 = q_2 = \infty, \end{align*} Theorem \ref{thm_weak_inf_mod} in the Haar model is proved. Indeed Theorem \ref{thm_weak_inf_mod} is verified for generic functions in $L^p$ and $L^s$ spaces for $1< p < \infty, 1 < s < 2$. \item With the range of exponents described as (\ref{easy_case}), the above argument proves Theorem \ref{thm_weak_mod} for the operator $\Pi^{H,3}_{\text{flag}^0 \otimes \text{flag}^0}$ (\ref{Pi_nofixed_easy}) whereas for the exponents satisfying (\ref{hard_exponent}), we would need a more delicate and localized model operator $\Pi^{H}_{\text{flag}^0 \otimes \text{flag}^0}$ (\ref{Pi_larger_haar}) whose boundedness is proved in Section \ref{section_thm_haar}.
\end{enumerate} \end{remark}
\begin{comment} \section{$\Pi_{\text{flag}^0 \otimes \text{paraproduct}}$ involving $L^{\infty}$-norms} One first defines $$\Omega := \Omega^1 \cup \Omega^2,$$ where \begin{align*}
\displaystyle \Omega^1 := &\bigcup_{n_2 \in \mathbb{Z}} \{MB > C_1 2^{-n_2}\| B\|_p\} \times \{Mg_1 > C_2 2^{n_2}|G_1|^{\frac{1}{p}}\}\nonumber \\
\Omega^2 := & \{SSh > C_3 \|h\|_{L_{x}^{p_3}(L_{y}^{q_3})}\} \nonumber \\ \end{align*} and $$\tilde{\Omega} := \{ MM\chi_{\Omega} > \frac{1}{100}\}.$$
\subsection{Stopping-time decompositions.}
\begin{center} \begin{tabular}{ c c c } Tensor-type stopping-time decomposition I & $\longrightarrow$ & $I \times J \in \mathcal{I}''_{-l-n_2} \times \mathcal{J}'_{n_2}$ \\ on $\mathcal{I} \times \mathcal{J}$ & & $(n_2 \in \mathbb{Z}, l > 0)$ \\
$\Downarrow$ & & \\
General two-dimensional level sets stopping-time decomposition& $\longrightarrow$ & $I \times J \in \mathcal{I}''_{-l-n_2} \times \mathcal{J}'_{n_2} \cap \mathcal{R}_{k_1,k_2} $ \\ on $ \mathcal{I}''_{-l-n_2} \times \mathcal{J}'_{n_2}$ & & $(n_2 \in \mathbb{Z}, l > 0,k_1 <0, k_2 \leq K)$ \end{tabular} \end{center} where \begin{align*}
\mathcal{I}''_{-l-n_2} := & \{ I \in \mathcal{I} \setminus \mathcal{I}''_{-l-n_2+1}: \left| I \cap \Omega''^x_{-l-n_2}\right| > \frac{1}{10}|I| \} \nonumber \\
\mathcal{J}'_{n_2} := & \{ J \in \mathcal{J} \setminus \mathcal{J}'_{n_2+1}: \left| I \cap \Omega'^y_{n_2}\right| > \frac{1}{10}|J| \} \end{align*} with \begin{align*}
\Omega''^x_{-l-n_2} := &\{MB> C_1 2^{-l-n_2}\|B\|_{p} \} \nonumber \\
\Omega'^y_{n_2} := & \{Mg_1> C_2 2^{n_2}|G_1|^{\frac{1}{p}} \} \end{align*}
With the tensor-type stopping-time decomposition and the general 2-dimensional level sets stopping-time decomposition, the multilinear form can be rewritten as \begin{align} \label{form0p_inf}
&\bigg| \displaystyle \sum_{\substack{l > 0 \\ k_1 < 0 \\ k_2 \leq K}} \sum_{\substack{n_2 \in \mathbb{Z} \\}}\sum_{T \in \mathbb{T}_{-l-n_2}}\sum_{\substack{I \times J \in T \times \mathcal{J}_{n_2}\\I \times J \in \mathcal{R}_{k_1,k_2}}} \frac{1}{|I|^{\frac{1}{2}} |J|} \langle B_I(f_1,f_2),\varphi_I^1 \rangle \langle g_1, \varphi_J^1 \rangle \langle g_2, \varphi_J^2 \rangle \langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle \langle \chi_{E'},\psi_I^{3,H} \otimes \psi_J^{3,H} \rangle\bigg| \nonumber \\
& \lesssim \sum_{\substack{l > 0 \\ k_1 < 0 \\ k_2 \leq K}} \sum_{\substack{n_2 \in \mathbb{Z}}}\sum_{T \in \mathbb{T}_{-l-n_2}} \displaystyle \sup_{I \in T} \frac{|\langle B_I(f_1,f_2),\varphi_I^1 \rangle|}{|I|^{\frac{1}{2}}} \sup_{J \in \mathcal{J}_{n_2}} \frac{|\langle g_1, \varphi_J^1 \rangle|}{|J|^{\frac{1}{2}}} \frac{|\langle g_2, \varphi_J^2 \rangle|}{|J|^{\frac{1}{2}}} \cdot \nonumber \\
&\int \sum_{\substack{I \times J \in T \times \mathcal{J}_{n_2}\\I \times J \in \mathcal{R}_{k_1,k_2}}} \frac{|\langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle|}{|I|^{\frac{1}{2}}|J|^{\frac{1}{2}}} \frac{|\langle \chi_{E'},\psi_I^{3,H} \otimes \psi_J^{3,H} \rangle|}{|I|^{\frac{1}{2}}|J|^{\frac{1}{2}}}\chi_{I}(x) \chi_{J}(y) dx dy \nonumber \\
\lesssim & C_1 C_2\sum_{\substack{l > 0 \\ k_1 < 0 \\ k_2 \leq K}} \sum_{\substack{n_2 \in \mathbb{Z}}}\sum_{T \in \mathbb{T}_{-l-n_2}} 2^{-l-n_2} \|B\|_p 2^{n_2} |G_1|^{\frac{1}{p}} \cdot 1 \cdot 1\nonumber \\
& \cdot \int \sum_{\substack{I \times J \in T \times \mathcal{J}_{n_2}\\I \times J \in \mathcal{R}_{k_1,k_2}}} \frac{|\langle h, \psi_I^{2} \otimes \psi_J^{2} \rangle|}{|I|^{\frac{1}{2}}|J|^{\frac{1}{2}}} \frac{|\langle \chi_{E'},\psi_I^{3,H} \otimes \psi_J^{3,H} \rangle|}{|I|^{\frac{1}{2}}|J|^{\frac{1}{2}}}\chi_{I}(x) \chi_{J}(y) dx dy, \end{align} where the integral can be estimated as \begin{align*}
& C_3^2 2^{k_1} \| h \|_{L^s(\mathbb{R}^2)} 2^{k_2} \cdot \bigg|\big(\bigcup_{R\in \mathcal{R}_{k_1,k_2}} R\big) \cap \big(I_{T} \cap \bigcup_{I \in \mathcal{I}_{-n-n_2}} I \times \bigcup_{J \in \mathcal{J}_{n_2}}J\big)\bigg|.\nonumber \\ \end{align*} Now one combines the above estimate into expression (\ref{form0p_inf}): \begin{align*}
C_1 C_2 C_3^2 \sum_{\substack{ l > 0 \\ k_1 < 0 \\ k_2 \leq K}} & \|B\|_p |G_1|^{\frac{1}{p}} C_3 2^{k_1} \| h \|_{L^s(\mathbb{R}^2)} 2^{k_2} \cdot \sum_{\substack{n_2 \in \mathbb{Z}}}\sum_{T \in \mathbb{T}_{-l-n_2}} 2^{-l-n_2} 2^{n_2} \cdot \bigg|\big(\bigcup_{R\in \mathcal{R}_{k_1,k_2}} R\big) \cap \big(I_{T} \cap \bigcup_{I \in \mathcal{I}_{-n-n_2}} I \times \bigcup_{J \in \mathcal{J}_{n_2}}J\big)\bigg|.\nonumber \\ \end{align*} One can again use the sparsity condition as before and estimate \begin{align*}
& \sum_{n_2\in \mathbb{Z}}\sum_{T \in \mathbb{T}_{-l-n_2}} 2^{-l-n_2} 2^{n_2} \bigg|\big(\bigcup_{R\in \mathcal{R}_{k_1,k_2}} R\big) \cap \big(I_{T} \times \bigcup_{J \in \mathcal{J}_{n_2}}J\big)\bigg| \nonumber \\
\lesssim & 2^{-l}\big|\bigcup_{R \in \mathcal{R}_{k_2}}R \big| \nonumber \\
\lesssim & 2^{-l}C_3^{\gamma}2^{-k_2\gamma}, \nonumber \\ \end{align*} for any $\gamma >1$. \par
One can apply the above estimates to derive the following bound for the multilinear form \begin{align*}
|\Lambda_{\text{flag}^{0} \otimes \text{flag}^{0}}| \lesssim C_1 C_2 C_3^2 \sum_{\substack{l>0 \\ k_1 < 0 \\ k_2 \leq K}} 2^{-l}2^{k_1}2^{k_2(1-\gamma}\|h\|_{L^s(\mathbb{R}^2)}\|B\|_{p}|G_1|^{\frac{1}{p}}, \end{align*}
where one can again apply the estimate $\|B\|_p \lesssim |F_1|^{\frac{1}{p}}$. Thus $$
|\Lambda_{\text{flag}^{0} \otimes \text{flag}^{0}}| \lesssim C_1 C_2 C_3^2 \sum_{\substack{l>0 \\ k_1 < 0 \\ k_2 \leq K}} 2^{-l} 2^{k_1(1-\frac{\alpha}{2})}2^{k_2(1-\frac{\alpha\gamma}{2})}\|h\|_{L^s(\mathbb{R}^2)}|F_1|^{\frac{1}{p}}|G_1|^{\frac{1}{p}}. $$ As before, one can separate the case when $0 \leq k_2 \leq K $ and $k_2 < 0$, where in the former one lets $\gamma >1$ to be sufficiently large and the latter $\gamma > 1$ close to 1. As a consequence, $$
|\Lambda_{\text{flag}^{0} \otimes \text{flag}^{0}}| \lesssim C_1 C_2 C_3^2 |F_1|^{\frac{1}{p}}|G_1|^{\frac{1}{p}}\|h\|_{L^s(\mathbb{R}^2)} $$ as desired. \end{comment}
\section{Generalization to Fourier Case} \label{section_fourier} We will first highlight where we have used the assumption about the Haar model in the proof and then modify those partial arguments to prove the general case.
We have used the following implications specific to the Haar model. \begin{enumerate}[H(I).] \item Let $\chi_{E'} := \chi_{E \setminus Enl(\Omega)}$. Then \begin{equation} \label{Haar_loc_bipara} \langle \chi_{E'}, \phi^H_{I} \otimes \phi^H_J \rangle \neq 0 \iff I \times J \cap Enl(\Omega)^c \neq \emptyset. \end{equation} As a result, what contributes to the multilinear forms in the Haar model are the dyadic rectangles $I \times J \in \mathcal{R}$ satisfying $I \times J \cap Enl(\Omega)^c \neq \emptyset$, which is a condition we heavily used in the proofs of the theorems in the Haar model.
\item
For any dyadic intervals $K$ and $I$ with $|K| \geq |I|$,
$$ \langle \phi^{3,H}_K, \phi^H_I \rangle \neq 0 $$ if and only if $$ K \supseteq I. $$ Therefore, the non-degenerate case imposes the condition on the geometric containment we have employed for localizations of the operators $B^H_I$, $B^H_J$, $B^{\#_1,H}_I$ and $B^{\#_2,H}_J$ defined in (\ref{B_local_definition_haar}), (\ref{B_local_definition_haar_fix_scale}), (\ref{B_0_local_haar_simplified}) and (\ref{B_fixed_local_haar_simplified}).
\item In the case $(\phi^{3,H}_K)_K$ is a family of Haar wavelets, the observation highlighted as (\ref{haar_biest_cond}) generates the biest trick (\ref{haar_biest}) which is essential in the energy estimates.
\end{enumerate}
We will focus on how to generalize proofs of Theorem \ref{thm_weak_mod} for $\Pi^H_{\text{flag}^{\#_1} \otimes \text{flag}^{\#_2}} $ (\ref{Pi_fixed_haar}) and $\Pi^H_{\text{flag}^{0} \otimes \text{flag}^{0}}$ (\ref{Pi_larger_haar}) and discuss how to tackle restrictions listed as $H(I), H(II)$ and $H(III)$. The generalizations of arguments for other model operators and for Theorem \ref{thm_weak_inf_mod} follow from the same ideas.
\subsection{Generalized Proof of Theorem \ref{thm_weak_mod} for $\Pi_{\text{flag}^{\#_1} \otimes \text{flag}^{\#_2}} $} \label{section_fourier_fixed}
\subsubsection{Localization and generalization of $H(I)$} The argument for $\Pi_{\text{flag}^{\#_1} \otimes \text{flag}^{\#_2}} $ in Section \ref{section_thm_haar_fixed} takes advantages of the localization of spatial variables, as stated in $H(I)$. The following lemma allows one to decompose the original bump function into bump functions of compact supports so that a perfect localization in spatial variables can be achieved, which can be viewed as generalized $H(I)$ and whose proof is included in \cite{cptt_2} and Section 3 of \cite{cw}. \begin{lemma} \label{decomp_compact}
Let $I \subseteq \mathbb{R}$ be an interval. Then any smooth bump function $\phi_I$ adapted to $I$ can be decomposed as $$ \phi_I = \sum_{\tau \in \mathbb{N}} 2^{-100 \tau} \phi_I^{\tau} $$ where for each $\tau \in \mathbb{N}$, $\phi_I^{\tau}$ is a smooth bump function adapted to $I$ and $\text{supp}(\phi_I^{\tau}) \subseteq 2^{\tau} I$. If $\int \phi_I = 0$, then the functions $\phi_I^{\tau}$ can be chosen such that $\int \phi_I^{\tau} = 0$. \end{lemma} The multilinear form associated to $\Pi_{\text{flag}^{\#_1} \otimes \text{flag}^{\#_2}} $ in the general case can now be rewritten as \begin{align} \label{compact} & \Lambda_{\text{flag}^{\#1} \otimes \text{flag}^{\#_2}}(f_1, f_2, g_1, g_2, h, \chi_{E'}) = \displaystyle \sum_{\tau_1,\tau_2 \in \mathbb{N}}2^{-100(\tau_1+\tau_2)} \Lambda_{\text{flag}^{\#1} \otimes \text{flag}^{\#_2}}^{\tau_1, \tau_2} (f_1, f_2, g_1, g_2, h, \chi_{E'}),
\end{align} where for any fixed $\tau_1, \tau_2 \in \mathbb{N}$, {\fontsize{10}{10}\begin{align} & \Lambda_{\text{flag}^{\#1} \otimes \text{flag}^{\#_2}}^{\tau_1, \tau_2} (f_1, f_2, g_1, g_2, h, \chi_{E'}) \nonumber\\
:= & \sum_{I \times J \in \mathcal{R}} \frac{1}{|I|^{\frac{1}{2}} |J|^{\frac{1}{2}}} \langle B^{\#_1}_I(f_1,f_2),\phi_I^1 \rangle \langle \tilde{B}^{\#_2}_J(g_1,g_2), \phi_J^1 \rangle \cdot \langle h, \phi_{I}^2 \otimes \phi_{J}^2 \rangle \langle \chi_{E'},\phi_{I}^{3,\tau_1} \otimes \phi^{3, \tau_2}_{J} \rangle. \end{align}} with $B^{\#_1}_I$ and $\tilde{B}^{\#_2}_J$ defined in (\ref{B_fixed_local_fourier_simple}) and (\ref{B_fixed_fourier}).
To show that the multilinear form $\Lambda_{\text{flag}^{\#1} \otimes \text{flag}^{\#_2}} (f_1, f_2, g_1, g_2, h, \chi_{E'}) $ satisfies the same estimate as \newline $\Lambda^H_{\text{flag}^{\#1} \otimes \text{flag}^{\#_2}}(f_1, f_2, g_1, g_2, h, \chi_{E'})$ (\ref{form_haar_larger_goal}), it suffices to prove that for any fixed $\tau_1,\tau_2 \in \mathbb{N}$, \begin{equation} \label{linear_fix_fourier}
|\Lambda_{\text{flag}^{\#1} \otimes \text{flag}^{\#_2}}^{\tau_1, \tau_2}(f_1,f_2,g_1,g_2,h, \chi_{E'})| \lesssim (2^{\tau_1+ \tau_2})^{\Theta}|F_1|^{\frac{1}{p_1}}|F_2|^{\frac{1}{q_1}}|G_1|^{\frac{1}{p_2}}|G_2|^{\frac{1}{q_2}}\|h\|_{L^s(\mathbb{R}^2)} \end{equation} for some $0 < \Theta < 100$, thanks to the fast decay $2^{-100(\tau_1+\tau_2)}$ in the decomposition of the original multilinear form (\ref{compact}).
One first redefines the exceptional set with the replacement of $C_1$, $C_2$ and $C_3$ by $C_12^{10\tau_1}$, $C_2 2^{10\tau_2}$ and $C_3 2^{10\tau_1+10\tau_2}$ respectively. In particular, let \begin{align*} & C_1^{\tau_1} := C_12^{10\tau_1}, \nonumber\\ & C_2^{\tau_2} := C_22^{10\tau_2}, \nonumber\\ & C_3^{\tau_1,\tau_2} := C_32^{10\tau_1+10\tau_2}. \nonumber \end{align*} Then define \begin{align*}
\displaystyle \Omega_1^{\tau_1, \tau_2} := &\bigcup_{\mathfrak{n}_1 \in \mathbb{Z}}\{x: Mf_1(x) > C_1^{\tau_1} 2^{\mathfrak{n}_1}|F_1|\} \times \{y: Mg_1(y) > C_2^{\tau_2} 2^{-\mathfrak{n}_1}|G_1|\}\cup \nonumber \\
& \bigcup_{\mathfrak{n}_2 \in \mathbb{Z}}\{x:Mf_2(x) > C_1^{\tau_1} 2^{\mathfrak{n}_2}|F_2|\} \times \{y: Mg_2(y) > C_2^{\tau_2} 2^{-\mathfrak{n}_2}|G_2|\}\cup \nonumber \\
&\bigcup_{\mathfrak{n}_3 \in \mathbb{Z}}\{x: Mf_1(x) > C_1^{\tau_1} 2^{\mathfrak{n}_3}|F_1|\} \times \{y: Mg_2(y) > C_2^{\tau_2} 2^{-\mathfrak{n}_3}|G_2|\}\cup \nonumber \\
& \bigcup_{\mathfrak{n}_4 \in \mathbb{Z}}\{x: Mf_2(x) > C_1^{\tau_1} 2^{\mathfrak{n}_4 }|F_2|\} \times \{y: Mg_1(y) > C_2^{\tau_2} 2^{-\mathfrak{n}_4 }|G_1|\},\nonumber \\
\Omega_2^{\tau_1,\tau_2} := & \{(x,y) \in \mathbb{R}^2: SSh(x,y) > C_3^{\tau_1, \tau_2} \|h\|_{L^s(\mathbb{R}^2)}\}. \nonumber \\ \end{align*}
One also defines \begin{align*} & \Omega^{\tau_1,\tau_2} := \Omega_1^{\tau_1,\tau_2} \cup \Omega_2^{\tau_1,\tau_2}, \nonumber \\ & Enl(\Omega^{\tau_1,\tau_2}) := \{(x,y) \in \mathbb{R}^2: MM(\chi_{\Omega^{\tau_1,\tau_2}})(x,y)> \frac{1}{100} \}, \nonumber\\ & Enl_{\tau_1,\tau_2}(Enl(\Omega^{\tau_1,\tau_2})) := \{(x,y) \in \mathbb{R}^2: MM(\chi_{Enl(\Omega^{\tau_1,\tau_2})}(x,y)> \frac{1}{2^{2\tau_1+ 2\tau_2}} \}, \end{align*} and finally $$ \mathbf{\Omega}:= \bigcup_{\tau_1,\tau_2\in \mathbb{N}} Enl_{\tau_1,\tau_2}(Enl(\Omega^{\tau_1,\tau_2})) . $$ \begin{remark}
It is not difficult to verify that $|\mathbf{\Omega}| \ll 1$ given that $C_1, C_2$ and $C_3$ are sufficiently large. One can then define $E' := E \setminus \mathbf{\Omega}$, where $|E'| \sim |E|$ as desired. For such $E'$, one has the following simple but crucial observation. \end{remark} \begin{obs} \label{start_point} For any fixed $\tau_1, \tau_2 \in \mathbb{N}$ and any dyadic rectangle $I \times J$, $$ \langle \chi_{E'},\phi_{I}^{3,\tau_1} \otimes \phi^{3, \tau_2}_{J} \rangle \neq 0 $$ implies that $$ I \times J \cap \left(Enl(\Omega^{\tau_1,\tau_2})\right)^c \neq \emptyset. $$
\end{obs} \begin{proof} We will prove the equivalent contrapositive statement. Suppose that $I \times J \cap \left(Enl(\Omega^{\tau_1,\tau_2})\right)^c = \emptyset$, or equivalently $I \times J \subseteq Enl(\Omega^{\tau_1,\tau_2})$, then
$$|(2^{\tau_1}I \times 2^{\tau_2}J) \cap Enl(\Omega^{\tau_1,\tau_2})| > \frac{1}{2^{2\tau_1+2\tau_2}}|2^{\tau_1}I \times 2^{\tau_2}J|, $$ which infers that $$ (2^{\tau_1}I \times 2^{\tau_2}J) \subseteq Enl_{\tau_1,\tau_2}(Enl(\Omega^{\tau_1,\tau_2})) \subseteq \mathbf{\Omega}. $$ Since $E' \cap \mathbf{\Omega} = \emptyset$, one can conclude that $$ \langle \chi_{E'},\phi_{I}^{3,\tau_1} \otimes \phi^{3, \tau_2}_{J} \rangle = 0, $$ which completes the proof of the observation. \end{proof}
\begin{remark}\label{st_general} Observation \ref{start_point} settles a starting point for the stopping-time decompositions with fixed parameters $\tau_1$ and $\tau_2$. More precisely, suppose that $\mathcal{R}$ is an arbitrary finite collection of dyadic rectangles. Then with fixed $\tau_1, \tau_2 \in \mathbb{N}$, let $\displaystyle \mathcal{R} := \bigcup_{n_1, n_2 \in \mathbb{Z}}\mathcal{I}^{\tau_1}_{n_1} \times \mathcal{J}^{\tau_2}_{n_2}$ denote the \textit{tensor-type stopping-time decomposition I - level sets} introduced in Section \ref{section_thm_haar_fixed_tensor}. Now $\mathcal{I}^{\tau_1}_{n_1}$ and $\mathcal{J}^{\tau_2}_{n_2}$ are defined in the same way as $\mathcal{I}_{n_1}$ and $\mathcal{J}_{n_2}$ with $C_1$ and $C_2$ replaced by $C_1^{\tau_1}$ and $C_2^{\tau_2}$. By the argument for Observation \ref{obs_indice} in Section \ref{section_thm_haar_fixed_tensor_1d_level}, one can deduce the same conclusion that if for any $I \times J \in \mathcal{R}$, $I \times J \cap \mathbf{\Omega}^c \neq \emptyset$, then $n_1 + n_2 < 0$. \end{remark} Due to Remark \ref{st_general}, one can perform the stopping-time decompositions specified in Section \ref{section_thm_haar_fixed} with $C_1, C_2$ and $C_3$ replaced by $C_1^{\tau_1}$, $C_2^{\tau_2}$ and $C_3^{\tau_1,\tau_2}$ respectively and adopt the argument without issues. The difference that lies in the resulting estimate is the appearance of $O(2^{50\tau_1})$, $O(2^{50\tau_2})$ and $O(2^{50\tau_1+50\tau_2})$, which is not of concerns as illustrated in (\ref{linear_fix_fourier}). The only ``black-box'' used in Section \ref{section_thm_haar_fixed} is the local size estimates (Proposition \ref{size_cor}), which needs a more careful treatment and will be explored in the next subsection.
\subsubsection{Local size estimates and generalization of H(II)} \begin{comment} The partial arguments which involve the localization of $B$ need to be studied more carefully, namely \begin{enumerate} \item Proof of Theorem \ref{thm_weak_mod} for $\Pi_{\text{flag}^{\#_1} \otimes \text{flag}^{\#_2}} $ \item Proof of Theorem \ref{thm_weak_mod} for $\Pi_{\text{flag}^{0} \otimes \text{flag}^{0}} $ in the case $\frac{1}{p_1} + \frac{1}{q_1} = \frac{1}{p_2} + \frac{1}{q_2} > 1$ \item Proof of Theorem \ref{thm_weak_inf_mod} for $\Pi_{\text{flag}^{\#_1} \otimes \text{flag}^{\#_2}} $ \end{enumerate}
One can apply Lemma \ref{decomp_compact} for localizations of \begin{enumerate}[(i)] \item $\text{size}(\langle B^{\#_1}_I, \varphi_I\rangle)_I$ and $\text{size}(\langle \tilde{B}^{\#_2}_J, \varphi_J\rangle)_J$ \item $\text{energy}(\langle B_I, \varphi_I\rangle)_I$ and $\text{energy}(\langle \tilde{B}_J, \varphi_I\rangle)_J$ with non-lacunary families $(\varphi^3_K)_K$ and $(\varphi^3_L)_L$ respectively \end{enumerate} Thanks to the localization, $\text{size}(\langle B^{\#_1}_I, \varphi_I\rangle)_I$ and $\text{size}(\langle \tilde{B}^{\#_2}_J, \varphi_J\rangle)_J$ can be majorized by the same bounds in Lemma \ref{B_size} which will be clarified in Section 10.1. As a consequence, proofs of Theorem \ref{thm_weak_mod} and \ref{thm_weak_inf_mod} for $\Pi_{\text{flag}^{\#_1} \otimes \text{flag}^{\#_2}}$ are generalized to the Fourier case. A similar modification can be performed to the proof of Theorem \ref{thm_weak_mod} for $\Pi_{\text{flag}^{0} \otimes \text{flag}^{0}} $ when $(\varphi^3_K)_K$ and $(\varphi^3_L)_L$ are non-lacunary families in the case $\frac{1}{p_1} + \frac{1}{q_1} = \frac{1}{p_2} + \frac{1}{q_2} > 1$. It remains to fix the proof of Theorem \ref{thm_weak_mod} for $\Pi_{\text{flag}^{0} \otimes \text{flag}^{0}} $ when either $(\phi_K^3)_K$ or $(\phi^3_L)_L$ are lacunary families in the case $\frac{1}{p_1} + \frac{1}{q_1} = \frac{1}{p_2} + \frac{1}{q_2} > 1$, which requires a more delicate treatment. \end{comment}
We will focus on the estimates of $\text{size}(\langle B^{\#_1}_I, \varphi_I\rangle)_I$, whose argument applies to $\text{size}(\langle \tilde{B}^{\#_2}_J, \varphi_J\rangle)_J$ as well. It suffices to prove Lemma \ref{B_size} in the Fourier case and the local size estimates described in Proposition \ref{size_cor} follow immediately. We will verify Lemma \ref{B_size} for the bilinear operator $B^{\#_1}_I = B^{\#_1}_{\mathcal{K},I} $ directly without going through a general formalization in terms of $\mathcal{Q}$, $P$ and $v_i$, $i = 1,2$ and then specifying $\mathcal{Q} := \mathcal{K}$, $P:= I$ and $v_i := f_i$ for $i= 1,2$. One first attempts to apply Lemma \ref{decomp_compact} to create a setting of compactly supported bump functions so that the same localization described in Section \ref{section_size_energy} can be achieved. Suppose that for any $I \in \mathcal{I}'$, $I \cap S \neq \emptyset$. Then for some $I_0 \in \mathcal{I}'$ such that $I_0 \cap S \neq \emptyset$, \begin{comment} \begin{align} \label{loc_size}
\frac{|\langle B_{I}, \varphi^1_{I}\rangle|}{|I|^{\frac{1}{2}}} = & \frac{1}{|I|^{\frac{1}{2}}}\bigg|\sum_{\substack{K: |K| \sim 2^{\#_1}|I|}}\frac{1}{|K|^{\frac{1}{2}}} \langle f_1, \phi^1_K \rangle \langle f_2, \phi^2_K \rangle \langle \varphi^1_{I} ,\phi^3_K \rangle \bigg| \nonumber \\
= & \frac{1}{|I|^{\frac{1}{2}}}\bigg|\sum_{\tau_3,\tau_4 \in \mathbb{N}}2^{-100\tau_3}2^{-100{\tau_4}}\sum_{K: |K| \sim 2^{\#_1} |I|} \frac{1}{|K|^{\frac{1}{2}}} \langle f_1, \phi^1_K \rangle \langle f_2, \phi^2_K \rangle \langle \varphi^{1,\tau_3}_{I}, \phi^{3,\tau_4}_K\rangle\bigg|. \end{align}
One recalls the definition of \end{comment}
\begin{align*}
\text{size}_{\mathcal{I}'}((\langle B_I^{\#_1}(f_1,f_2), \varphi_I \rangle)_{I \in \mathcal{I}'} =& \frac{|\langle B^{\#_1}_{I_0}(f_1,f_2),\varphi_{I_0}^1 \rangle|}{|I_0|^{\frac{1}{2}}} =\sum_{K: |K| \sim 2^{\#_1}|I_0|} \frac{1}{|K|^{\frac{1}{2}}} \langle f_1, \phi^1_K \rangle \langle f_2, \phi^2_K \rangle \langle \varphi^{1}_{I_0}, \phi^{3}_K\rangle . \nonumber \end{align*} One can now invoke Lemma \ref{decomp_compact} to decompose the bump functions $\varphi^1_{I_0}$ and $\phi^3_K$ to obtain \begin{align} \label{form_f}
\frac{1}{|I_0|^{\frac{1}{2}}}\bigg|\sum_{\tau_3,\tau_4 \in \mathbb{N}}2^{-100 (\tau_3+\tau_4)}\sum_{K: |K| \sim 2^{\#_1}|I_0|} \frac{1}{|K|^{\frac{1}{2}}} \langle f_1, \phi^1_K \rangle \langle f_2, \phi^2_K \rangle \langle \varphi^{1,\tau_3}_{I_0}, \phi^{3,\tau_4}_K\rangle\bigg|, \end{align} where $\varphi^{1,\tau_3}_{I_0}$ is an $L^{2}$-normalized bump function adapted to $I_0$ with $\text{supp}(\varphi^{1,\tau_3}_{I_0}) \subseteq 2^{\tau_3}I_0$, and $\phi^{3,\tau_4}_K$ is an $L^2$-normalized bump function with $\text{supp}(\phi^{3,\tau_4}_K)\subseteq 2^{\tau_4}K$. With the property of being compactly supported, one has that if $$ \langle \varphi^{1,\tau_3}_{I_0}, \phi^{3, \tau_4}_K\rangle \neq 0, $$ then $$2^{\tau_3} I_0 \cap 2^{\tau_4}K \neq \emptyset.$$
One also recalls that $I_0 \cap S \neq \emptyset$ and $|I_0| \leq |K|$, it follows that $\text{dist}(K,I)$\footnote{For any measurable sets $A,B \subseteq \mathbb{R}$, $\text{dist}(A,B) := \inf\{|a-b|: a \in A, b \in B \}$.} satisfies \begin{equation} \label{geometry_fourier}
\frac{\text{dist}(K,S)}{|K|} \lesssim 2^{\tau_3 + \tau_4}. \end{equation} Therefore, one can apply (\ref{geometry_fourier}) and rewrite (\ref{form_f}) as
\begin{align} \label{size_B_f}
&\sum_{\tau_3,\tau_4 \in \mathbb{N}} 2^{-100(\tau_3+\tau_4)} \frac{1}{|I_0|} \sum_{K:|K|\sim 2^{\#_1}|I_0|}\frac{|\langle f_1, \phi_K^1 \rangle|}{|K|^{\frac{1}{2}}} \frac{|\langle f_2, \phi_K^2 \rangle|}{|K|^{\frac{1}{2}}} |\langle |I_0|^{\frac{1}{2}}\varphi^{1,\tau_3}_{I_0}, |K|^{\frac{1}{2}}\phi_K^3 \rangle| \nonumber \\
\leq & \sum_{\tau_3,\tau_4 \in \mathbb{N}} 2^{-100(\tau_3+\tau_4)}\frac{1}{|I_0|} \sup_{K:\frac{\text{dist}(K,S)}{|K|} \lesssim 2^{\tau_3 + \tau_4}}\frac{|\langle f_1, \phi_K^1 \rangle|}{|K|^{\frac{1}{2}}} \sup_{K:\frac{\text{dist}(K,S)}{|K|} \lesssim 2^{\tau_3 + \tau_4}}\frac{|\langle f_2, \phi_K^2 \rangle|}{|K|^{\frac{1}{2}}} \sum_{K:|K|\sim 2^{\#_1}|I_0|}|\langle |I_0|^{\frac{1}{2}}\varphi^{1,\tau_3}_{I_0}, |K|^{\frac{1}{2}}\phi_K^3 \rangle|.\nonumber \\ \end{align} One notices that \begin{align} \label{size_f_1}
& \sup_{K:\frac{\text{dist}(K,S)}{|K|} \lesssim 2^{\tau_3 + \tau_4}}\frac{|\langle f_1, \phi_K^1 \rangle|}{|K|^{\frac{1}{2}}} \lesssim 2^{\tau_3+\tau_4}\sup_{K:\frac{\text{dist}(K,S)}{|K|} \lesssim 2^{\tau_3 + \tau_4}}\frac{|\langle f_1, \phi_{2^{\tau_3+\tau_4}K}^1 \rangle|}{|2^{\tau_3+\tau_4}K|^{\frac{1}{2}}} \leq 2^{\tau_3+ \tau_4} \sup_{K' \cap S \neq \emptyset} \frac{|\langle f_1, \phi_{K'}^1 \rangle|}{|K'|^{\frac{1}{2}}}, \end{align}
where $K' := 2^{\tau_3+\tau_4}K$ represents the interval with the same center as $K$ and the radius $2^{\tau_3 + \tau_4}|K|.$ Similarly, \begin{equation} \label{size_f_2}
\sup_{K:\frac{\text{dist}(K,S)}{|K|} \lesssim 2^{\tau_3 + \tau_4}}\frac{|\langle f_2, \phi_K^2 \rangle|}{|K|^{\frac{1}{2}}} \lesssim 2^{\tau_3+ \tau_4} \sup_{K' \cap S \neq \emptyset} \frac{|\langle f_2, \phi_{K'}^2 \rangle|}{|K'|^{\frac{1}{2}}}. \end{equation} Moreoever, \begin{align} \label{disjoint}
& \sum_{K:|K|\sim 2^{\#_1}|I_0|}|\langle |I_0|^{\frac{1}{2}}\varphi^{1,\tau_3}_{I_0}, |K|^{\frac{1}{2}}\phi_K^3 \rangle| \leq \sum_{K:|K|\sim 2^{\#_1}|I_0|}\frac{1}{\left(1+\frac{\text{dist}(K,I_0)}{|K|}\right)^{100}} |I_0| \leq |I_0| \sum_{k \in \mathbb{N}}k^{-100} \leq |I_0|. \end{align} By combining (\ref{size_f_1}), (\ref{size_f_2}) and (\ref{disjoint}), one can estimate (\ref{size_B_f}) by \begin{align*}
& \frac{1}{|I_0|}\sum_{\tau_3,\tau_4 \in \mathbb{N}}2^{-100 \tau_3}2^{-100{\tau_4}} 2^{2(\tau_3+ \tau_4)} \sup_{K' \cap S \neq \emptyset} \frac{|\langle f_1, \phi_{K'}^1 \rangle|}{|K'|^{\frac{1}{2}}} \sup_{K' \cap S \neq \emptyset}\frac{|\langle f_2, \phi_{K'}^2 \rangle|}{|K'|^{\frac{1}{2}}} |I_0|\nonumber \\
\lesssim & \sup_{K' \cap S \neq \emptyset} \frac{|\langle f_1, \phi_{K'}^1 \rangle|}{|K'|^{\frac{1}{2}}} \sup_{K' \cap S \neq \emptyset}\frac{|\langle f_2, \phi_{K'}^2 \rangle|}{|K'|^{\frac{1}{2}}}, \end{align*} which is exactly the same estimate for the corresponding term in Lemma \ref{B_size}. This completes the proof of Theorem \ref{thm_weak_mod}
for $\Pi_{\text{flag}^{\#_1} \otimes \text{flag}^{\#_2}}$.
\subsection{Generalized Proof of Theorem \ref{thm_weak_mod} for $\Pi_{\text{flag}^0 \otimes \text{flag}^0}$} \label{section_fourier_0} \subsubsection{Local energy estimates and generalization of H(III)}
The delicacy of the argument for $\Pi_{\text{flag}^0 \otimes \text{flag}^0}$ with the lacunary family $(\phi_K^3)_K$ lies in the localization and the application of the biest trick for the energy estimates. It is worthy to note that Lemma \ref{decomp_compact} fails to generate the local energy estimates. In particular, one can decompose
$$ \langle B_I(f_1,f_2), \varphi_I^1\rangle =
\sum_{\tau_3,\tau_4 \in \mathbb{N}}2^{-100 \tau_3}2^{-100{\tau_4}}\sum_{K: |K| \geq |I|} \frac{1}{|K|^{\frac{1}{2}}} \langle f_1, \phi^1_K \rangle \langle f_2, \phi^2_K \rangle \langle \varphi^{1,\tau_3}_{I}, \psi^{3,\tau_4}_K\rangle, $$ where $B_I$ is a bilinear operator defined in (\ref{B_local_fourier_simple}) and (\ref{B_local0_haar}). Without loss of generality, we further assume that $(\phi^1_K)_{K \in \mathcal{K}}$ is non-lacunary and $(\phi^2_K)_{K \in \mathcal{K}}$ is lacunary. Then by the geometric observation (\ref{geometry_fourier}) implied by the non-degenerate condition $ \langle \varphi^{1,\tau_3}_{I}, \psi^{3,\tau_4}_K\rangle \neq 0$, \begin{equation} \label{loc_attempt_fourier} \langle B_I(f_1,f_2), \varphi_I^1\rangle =
\sum_{\tau_3,\tau_4 \in \mathbb{N}}2^{-100 \tau_3}2^{-100{\tau_4}}\sum_{\substack{K:|K| \geq |I| \\ K: \frac{\text{dist}(K,I)}{|K|} \lesssim 2^{\tau_3 + \tau_4}}} \frac{1}{|K|^{\frac{1}{2}}} \langle f_1, \varphi^1_K \rangle \langle f_2, \psi^2_K \rangle \langle \varphi^{1,\tau_3}_{I}, \psi^{3,\tau_4}_K\rangle. \end{equation} The localization has been obtained in (\ref{loc_attempt_fourier}). Nonetheless, for each fixed $\tau_3$ and $\tau_4$, one cannot equate the terms \begin{equation*}
\sum_{\substack{K:|K| \geq |I| \\ K: \frac{\text{dist}(K,I)}{|K|} \lesssim 2^{\tau_3 + \tau_4}}} \frac{1}{|K|^{\frac{1}{2}}} \langle f_1, \varphi^1_K \rangle \langle f_2, \psi^2_K \rangle \langle \varphi^{1,\tau_3}_{I}, \psi^{3,\tau_4}_K\rangle \neq \sum_{\substack{ \\ K: \frac{\text{dist}(K,I)}{|K|} \lesssim 2^{\tau_3 + \tau_4}}} \frac{1}{|K|^{\frac{1}{2}}} \langle f_1, \varphi^1_K \rangle \langle f_2, \psi^2_K \rangle \langle \varphi^{1,\tau_3}_{I}, \psi^{3,\tau_4}_K\rangle. \end{equation*} The reason is that $\varphi_I^{1,\tau_3}$ and $\psi_K^{3,\tau_4}$ are general $L^2$-normalized bump functions instead of Haar wavelets and $L^2$-normalized indicator functions. The ``if and only if'' condition \begin{equation} \label{haar_biest_tri}
\langle \varphi^{1,\tau_3}_{I}, \psi^{3,\tau_4}_K\rangle \neq 0 \iff |K| \geq |I| \end{equation} is no longer valid and is insufficient to derive the biest trick. As a consequence, one cannot simply apply Lemma \ref{decomp_compact} to localize the energy term and compare $$ \text{energy}((\langle B^{\tau_4}_I(f_1,f_2), \varphi_I^{1,\tau_3} \rangle)_{I \cap S \neq \emptyset}) $$ with $$ \text{energy}((\langle B^{S,\tau_3,\tau_4}_{\mathcal{K},\text{lac}}(f_2,f_2), \varphi_{I,\tau_3}^1 \rangle)_{I \cap S \neq \emptyset}), $$ where $$
B^{\tau_4}_I(f_1,f_2):= \sum_{\substack{K: K \in \mathcal{K} \\ |K| \geq |I| \\ }} \frac{1}{|K|^{\frac{1}{2}}} \langle f_1, \varphi^1_K \rangle \langle f_2, \psi^2_K \rangle \psi^{3,\tau_4}_K $$ and $$
B^{S,\tau_3,\tau_4}_{\mathcal{K},\text{lac}}(f_2,f_2):= \sum_{\substack{K \in \mathcal{K} \\ K: \frac{\text{dist}(K,S)}{|K|} \lesssim 2^{\tau_3 + \tau_4}}} \frac{1}{|K|^{\frac{1}{2}}} \langle f_1, \varphi^1_K \rangle \langle f_2, \psi^2_K \rangle \psi^{3,\tau_4}_K. $$ The biest trick is crucial in the local energy estimates which shall be evident from the previous analysis. In order to use the biest trick in the Fourier case, one needs to exploit the compact Fourier supports instead of the compact supports for spatial variables in the Haar model.
One first recalls that the Littlewood-Paley decomposition imposes that $\text{supp}(\widehat{\varphi}_I^1) \subseteq [-\frac{1}{4}|I|^{-1}, \frac{1}{4}|I|^{-1}]$
and $\text{supp}(\widehat{\psi}_K^3) \subseteq [-4|K|^{-1}, -\frac{1}{4}|K|^{-1}] \cup [\frac{1}{4}|K|^{-1},4|K|^{-1}]$. The interaction between the two Fourier supports is shown in the following figure:
\vskip.25 in
\begin{tikzpicture}[scale=0.3, decoration=brace]
\draw (-20,0)-- (20,0);
\foreach \x/\xtext in {-20/$-\frac{1}{4}|I|^{-1}$, -10/$-4|K|^{-1}$, -5/$-\frac{1}{4}|K|^{-1}$, 0/0,5/$\frac{1}{4}|K|^{-1}$,10/$4|K|^{-1}$,20/$\frac{1}{4}|I|^{-1}$} {
\draw (\x,0.5) -- (\x,-0.5) node[below] {\xtext};
}
\draw[decorate, yshift=2ex] (-10,0) -- node[above=0.4ex] {} (-5,0);
\draw[decorate, yshift=2ex] (5,0) -- node[above=0.4ex] {} (10,0);
\draw[decorate, yshift=12ex] (-20,0) -- node[above=0.8ex] {} (20,0);
\draw (-20,-10)-- (20,-10);
\foreach \x/\xtext in {-5/$-\frac{1}{4}|I|^{-1}$, -20/$-4|K|^{-1}$, -10/$-\frac{1}{4}|K|^{-1}$, 0/0,10/$\frac{1}{4}|K|^{-1}$,20/$4|K|^{-1}$,5/$\frac{1}{4}|I|^{-1}$} {
\draw (\x,-9.5) -- (\x,-10.5) node[below] {\xtext};
}
\draw[decorate, yshift=2ex] (-20,-10) -- node[above=0.4ex] {} (-10,-10);
\draw[decorate, yshift=2ex] (10,-10) -- node[above=0.4ex] {} (20,-10);
\draw[decorate, yshift=2ex] (-5,-10) -- node[above=0.4ex] {} (5,-10);
\end{tikzpicture}
\newline
\noindent As one may notice, \begin{equation} \label{biest_fourier}
\langle \varphi^{1}_{I}, \psi^{3}_K \rangle \neq 0 \iff |K| \geq |I|, \end{equation} which yields the biest trick as desired. Meanwhile, we would like to attain some generalized localization for the energy. In particular, fix any $n_1, m_1$, define the set \begin{equation} \label{level_set_fourier}
\mathcal{U}_{n_1,m_1} := \{x: Mf_1(x) \lesssim C_12^{n_1}|F_1|\} \cap \{x: Mf_2(x) \lesssim C_12^{m_1}|F_2|\}. \end{equation} Suppose that $\mathcal{I}'$ is a finite collection of dyadic intervals satisfying \begin{equation} \label{condition_collection} I \cap \mathcal{U}_{n_1,m_1} \neq \emptyset, \ \ \text{for any} \ \ I \in \mathcal{I}'. \end{equation} Then one would like to reduce $$\text{energy}_{\mathcal{I}'}(\langle B_I(f_1,f_2), \varphi_I\rangle)_{I \in \mathcal{I}'}$$ to $$\text{energy}_{\mathcal{I}'}(\langle B^{n_1,m_1,0}_{\mathcal{K}, \text{generic lac}}(f_1,f_2), \varphi_I\rangle)_{I \in \mathcal{I}'}$$ (or $\text{energy}(\langle B^{n_1,m_1,0}_{\mathcal{K}, \text{generic nonlac}}(f_1,f_2), \varphi_I\rangle)_{I}$ depending on whether the third family of bump functions defining $B_I$ is lacunary or not), where \begin{align}
B^{n_1,m_1,0}_{\mathcal{K},\text{generic lac}}(f_1,f_2) := & \sum_{\substack{K \in \mathcal{K}\\ K \cap \mathcal{U}_{n_1,m_1} \neq \emptyset}} \frac{1}{|K|^{\frac{1}{2}}} \langle f_1, \varphi_K^1\rangle \langle f_2, \psi_K^2 \rangle \psi_K^3, \\
B^{n_1,m_1,0}_{\mathcal{K},\text{generic nonlac}}(f_1,f_2) := & \sum_{\substack{K \in \mathcal{K}\\ K \cap \mathcal{U}_{n_1,m_1} \neq \emptyset}} \frac{1}{|K|^{\frac{1}{2}}} \langle f_1, \psi_K^1\rangle \langle f_2, \psi_K^2 \rangle \varphi_K^3. \end{align}
One observes that since $\psi_K^3$ and $\varphi_I^1$ are not compactly supported in $K$ and $I$ respectively, one cannot deduce that $K \cap \mathcal{U}_{n_1,m_1} \neq \emptyset$ given that $I \cap \mathcal{U}_{n_1,m_1} \neq \emptyset$ and $|K| \geq |I|$. The localization in the Fourier case is attained in a more analytic fashion. One decomposes the sum \begin{align} \label{d_0}
&\frac{|\langle B_I(f_1,f_2), \varphi^1_I \rangle|}{|I|^{\frac{1}{2}}} \nonumber\\
= & \frac{1}{|I|^{\frac{1}{2}}}\bigg|\sum_{K: |K| \geq |I|} \frac{1}{|K|^{\frac{1}{2}}}\langle f_1, \varphi_K^1 \rangle \langle f_2, \psi_K^2 \rangle \langle \varphi^1_I, \psi_K^3 \rangle\bigg| \nonumber \\
= & \frac{1}{|I|^{\frac{1}{2}}}\bigg|\sum_{d >0} \sum_{\substack{|K| \geq |I| \\ K \in \mathcal{K}^{n_1,m_1}_{d}}} \frac{1}{|K|^{\frac{1}{2}}}\langle f_1, \varphi_K^1 \rangle \langle f_2, \psi_K^2 \rangle \langle \varphi^1_I, \psi_K^3 \rangle + \sum_{\substack{|K| \geq |I| \\ K \in \mathcal{K}^{n_1,m_1}_0}}\frac{1}{|K|^{\frac{1}{2}}}\langle f_1, \varphi_K^1 \rangle \langle f_2, \psi_K^2 \rangle \langle \varphi_I^1, \psi_K^3 \rangle \bigg|, \end{align} where for $d > 0$, $$
\mathcal{K}_d^{n_1,m_1} := \{K: 1 + \frac{\text{dist}(K,\mathcal{U}_{n_1,m_1})}{|K|} \sim 2^{d} \}, $$ and $$ \mathcal{K}_0^{n_1,m_1} := \{K : K \cap \mathcal{U}_{n_1,m_1} \neq \emptyset\}. $$
\begin{comment} We will now separate the case when the family $(\phi^3_K)_K$ is lacunary or non-lacunary. \newline \noindent \textbf{Case I: $(\phi^3_K)_K$ is lacunary.} One notices that the only degenerate case $\langle \psi^3_K, \varphi^1_I \rangle \neq 0 $ is when the Fourier supports of $\psi_K^3$ and $\tilde{\chi}_I^1$, denoted by $\omega_K$ and $\omega_I$ respectively, satisfy $$ \omega_K \subseteq \omega_I $$
or equivalently $|K| \geq |I|$. As a consequence, one can drop the condition that $|K| \geq |I|$ and rewrite $$
\frac{|\langle B_I, \varphi_I \rangle|}{|I|^{\frac{1}{2}}} = \frac{1}{|I|} \bigg|\sum_{\substack{K \in \mathcal{K}\\ }} \frac{1}{|K|^{\frac{1}{2}}} \langle f_1, \phi_K^1\rangle \langle f_2, \phi_K^2 \rangle \langle \tilde{\chi}_I,\psi_K^3 \rangle \bigg| $$ This is usually called the biest trick.
We would like to preserve the compactness of Fourier supports of the bump functions for the application of the biest trick. To create a ``pseudo''- Haar setting, we first decompose \begin{align} \label{B_decomp}
\frac{|\langle B_I, \varphi^1_I \rangle|}{|I|^{\frac{1}{2}}} = & \frac{1}{|I|}\bigg|\sum_{K: |K| \geq |I|} \frac{1}{|K|^{\frac{1}{2}}}\langle f_1, \varphi_K^1 \rangle \langle f_2, \psi_K^2 \rangle \langle \tilde{\chi}^1_I, \psi_K^3 \rangle\bigg| \nonumber \\
= & \frac{1}{|I|}\bigg|\sum_{d >0} \sum_{\substack{|K| \geq |I| \\ K \in \mathcal{K}_{d}}} \frac{1}{|K|^{\frac{1}{2}}}\langle f_1, \varphi_K^1 \rangle \langle f_2, \psi_K^2 \rangle \langle \tilde{\chi}_I, \psi_K^3 \rangle + \sum_{\substack{|K| \geq |I| \\ K \in \mathcal{K}_0}}\frac{1}{|K|^{\frac{1}{2}}}\langle f_1, \varphi_K^1 \rangle \langle f_2, \psi_K^2 \rangle \langle \tilde{\chi}_I, \psi_K^3 \rangle \bigg|. \end{align} We have defined for any $d \geq 0$, $$
\mathcal{K}_d^{n_1,m_1}:= \{K: \frac{\text{dist}(K,S)}{|K|} +1 \sim2^{d} \} $$ where
$$S:=\{Mf_1 \leq C_12^{n_1} |F_1|\} \cap \{ Mf_2 \leq C_1 2^{m_1} |F_2|\}.$$ We further impose the following extra assumption: \begin{equation} \label{Haar_cond}
\frac{1}{|I|} \bigg|\sum_{d >0 }\sum_{\substack{K \in \mathcal{K}_d^{n_1,m_1}\\ }} \frac{1}{|K|^{\frac{1}{2}}} \langle f_1, \phi_K^1\rangle \langle f_2, \phi_K^2 \rangle \langle \tilde{\chi}_I,\psi_K^3 \rangle \bigg| \ll \frac{1}{|I|} \bigg|\sum_{\substack{K \in \mathcal{K}_0^{n_1,m_1}\\ }} \frac{1}{|K|^{\frac{1}{2}}} \langle f_1, \phi_K^1\rangle \langle f_2, \phi_K^2 \rangle \langle \tilde{\chi}_I,\psi_K^3 \rangle \bigg| \end{equation} for any $I \in \mathcal{I}$ and any $n_1,m_1 \in \mathbb{Z}$. The condition (\ref{Haar_cond}) implies that $$
\frac{|\langle B_I, \varphi_I \rangle|}{|I|^{\frac{1}{2}}} \sim \frac{1}{|I|} \bigg|\sum_{\substack{K \in \mathcal{K}_0^{n_1,m_1}\\ }} \frac{1}{|K|^{\frac{1}{2}}} \langle f_1, \phi_K^1\rangle \langle f_2, \phi_K^2 \rangle \langle \tilde{\chi}_I,\psi_K^3 \rangle \bigg| $$ which artificially creates the Haar setting.
\begin{remark} With some abuse of notations, we will refer to the estimates obtained with condition (\ref{Haar_cond}) as Haar model. The general Fourier case without the condition will be discussed in the last section. \end{remark}
One can then define the localized operator
$$\displaystyle B^{n_1,m_1}_0(x) := \sum_{\substack{K \in \mathcal{K}\\ K \cap S \neq \emptyset}} \frac{1}{|K|^{\frac{1}{2}}} \langle f_1, \phi_K^1\rangle \langle f_2, \phi_K^2 \rangle \psi_K^3 (x).$$
\textbf{Case I: $\phi^3_K$ is non-lacunary. } $$
\frac{|\langle B_I, \varphi_I \rangle|}{|I|^{\frac{1}{2}}} := \frac{1}{|I|} \bigg| \sum_{\substack{K \in \mathcal{K} \\ K \supseteq I}} \frac{1}{|K|^{\frac{1}{2}}} \langle f_1, \phi_K^1\rangle \langle f_2, \phi_K^2 \rangle \langle \tilde{\chi}_I,\varphi_K^3 \rangle \bigg| $$
where $ \tilde{\chi}_I := \frac{\varphi_I}{|I|^{\frac{1}{2}}}$ is a $L^{\infty}$-normalized bump function. By the assumption that $I \cap S \neq \emptyset$ for any $I \in \mathcal{I}'$, one can derive that $K \cap S \neq \emptyset$ given $K \supseteq I$. Therefore, one can rewrite \begin{align*}
\frac{|\langle B_I, \varphi_I \rangle|}{|I|^{\frac{1}{2}}} = & \frac{1}{|I|} \bigg|\sum_{\substack{K \in \mathcal{K} \\ K \supseteq I \\ K \cap S \neq \emptyset}} \frac{1}{|K|^{\frac{1}{2}}} \langle f_1, \phi_K^1\rangle \langle f_2, \phi_K^2 \rangle \langle \varphi^{\infty}_I,\varphi_K^3 \rangle \bigg| \nonumber \\
\leq & \frac{1}{|I|} \sum_{\substack{K \in \mathcal{K} \\ K \supseteq I \\ K \cap S \neq \emptyset}} \frac{1}{|K|^{\frac{1}{2}}} |\langle f_1, \phi_K^1\rangle| |\langle f_2, \phi_K^2 \rangle| |\langle |\varphi^{\infty}_I|,|\varphi_K^3| \rangle. \end{align*}
One can drop the condition $K \supseteq I$ in the sum and bound the above expression by $$
\frac{|\langle B_I, \varphi_I \rangle|}{|I|^{\frac{1}{2}}} \leq \frac{1}{|I|} \sum_{\substack{K \in \mathcal{K}\\ K \cap S \neq \emptyset}} \frac{1}{|K|^{\frac{1}{2}}} |\langle f_1, \phi_K^1\rangle| |\langle f_2, \phi_K^2 \rangle| \langle |\tilde{\chi}_I|,|\varphi_K^3| \rangle $$
One can define another localized operator in this case $$
B^{n_1,m_1}_0(x) := \displaystyle \sum_{\substack{K \in \mathcal{K}\\ K \cap S \neq \emptyset}} \frac{1}{|K|^{\frac{1}{2}}} |\langle f_1, \phi_K^1\rangle| |\langle f_2, \phi_K^2 \rangle| |\varphi_K^3|(x).$$
\begin{comment} Now one can summarize both cases as follows: \begin{equation*}
2^{n} < \frac{|\langle B_I, \varphi_I \rangle|}{|I|^{\frac{1}{2}}} \begin{cases}
\stackrel{\text{Case} I}{\lesssim} \frac{|\langle B_0^{n_1,m_1}, \varphi_I \rangle|}{|I|^{\frac{1}{2}}} \quad \ \ \text{where} \ \ \displaystyle B^{n_1,m_1}_0 := \sum_{\substack{K \in \mathcal{K}\\ K \cap S \neq \emptyset}} \frac{1}{|K|^{\frac{1}{2}}} \langle f_1, \phi_K^1\rangle \langle f_2, \phi_K^2 \rangle \psi_K^3 \\
\stackrel{\text{Case} II}{\leq} \frac{|\langle B_0^{n_1,m_1}, |\varphi_I| \rangle|}{|I|^{\frac{1}{2}}} \quad \text{where}\ \ B_0^{n_1,m_1}:= \displaystyle \sum_{\substack{K \in \mathcal{K}\\ K \cap S \neq \emptyset}} \frac{1}{|K|^{\frac{1}{2}}} |\langle f_1, \phi_K^1\rangle| |\langle f_2, \phi_K^2 \rangle| |\varphi_K^3|
\end{cases} \end{equation*}
for any $I \in \mathbb{D}^{0}_{n}$. \end{comment}
\noindent Ideally, one would like to ''omit'' the former term, which is reasonable once \begin{equation} \label{energy_needed}
\bigg|\sum_{d >0} \sum_{\substack{|K| \geq |I| \\ K \in \mathcal{K}^{n_1,m_1}_{d}}} \frac{1}{|K|^{\frac{1}{2}}}\langle f_1, \varphi_K^1 \rangle \langle f_2, \psi_K^2 \rangle \langle \varphi^1_I, \psi_K^3 \rangle\bigg| \ll \bigg|\sum_{\substack{|K| \geq |I| \\ K \in \mathcal{K}^{n_1,m_1}_0}}\frac{1}{|K|^{\frac{1}{2}}}\langle f_1, \varphi_K^1 \rangle \langle f_2, \psi_K^2 \rangle \langle \varphi^1_I, \psi_K^3 \rangle \bigg| \end{equation} so that one can apply the previous argument discussed in Section \ref{section_thm_haar_fixed}. In the other case when \begin{equation*}
\bigg|\sum_{d >0} \sum_{\substack{|K| \geq |I| \\ K \in \mathcal{K}^{n_1,m_1}_{d}}} \frac{1}{|K|^{\frac{1}{2}}}\langle f_1, \varphi_K^1 \rangle \langle f_2, \psi_K^2 \rangle \langle \varphi^1_I, \psi_K^3 \rangle \bigg| \gtrsim \bigg|\sum_{\substack{|K| \geq |I| \\ K \in \mathcal{K}^{n_1,m_1}_0}}\frac{1}{|K|^{\frac{1}{2}}}\langle f_1, \varphi_K^1 \rangle \langle f_2, \psi_K^2 \rangle \langle \varphi^1_I, \psi_K^3 \rangle \bigg|, \end{equation*} local energy estimates are not necessary to achieve the result. The following lemma generates estimates for the former term and provides a guideline about the separation of cases. The notation in the lemma is consistent with the previous discussion. \begin{lemma} \label{en_loc} Suppose that $d >0$ and $I$ is a fixed dyadic interval such that $I \cap \mathcal{U}_{n_1,m_1} \neq \emptyset$ for $\mathcal{U}_{n_1,m_1}$ defined in (\ref{level_set_fourier}). Then $$
\frac{1}{|I|^{\frac{1}{2}}}\bigg| \sum_{\substack{|K| \geq |I| \\ K \in \mathcal{K}_{d}^{n_1,m_1}}} \frac{1}{|K|^{\frac{1}{2}}}\langle f_1, \varphi_K^1 \rangle \langle f_2, \psi_K^2 \rangle \langle \varphi^1_I, \psi_K^3 \rangle \bigg| \lesssim 2^{-Nd}(C_12^{n_1}|F_1|)^{\alpha_1} (C_12^{m_1}|F_2|)^{\beta_1}, $$ for any $0 \leq \alpha_1,\beta_1 \leq 1$ and some $N \gg 1$. \end{lemma} \begin{remark} \begin{enumerate} \noindent \item One simple but important fact is that for any fixed $d>0$, $n_1$ and $m_1$, $\mathcal{K}_d^{n_1,m_1}$ is a disjoint collection of dyadic intervals. \item Aware of the first comment, one can apply the exactly same argument in Section \ref{section_fourier_fixed} to prove the lemma. \end{enumerate}
\end{remark}
Based on the estimates described in Lemma \ref{en_loc}, one has that \begin{align} \label{threshold}
\frac{1}{|I|^{\frac{1}{2}}}\bigg|\sum_{d >0} \sum_{\substack{|K| \geq |I| \\ K \in \mathcal{K}_{d}^{n_1,m_1}}} \frac{1}{|K|^{\frac{1}{2}}}\langle f_1, \varphi_K^1 \rangle \langle f_2, \psi_K^2 \rangle \langle \varphi^1_I, \psi_K^3 \rangle \bigg| & \lesssim \sum_{d>0} 2^{-Nd} (C_12^{n_1}|F_1|)^{\alpha_1} (C_12^{m_1}|F_2|)^{\beta_1} \nonumber \\
& \lesssim (C_12^{n_1}|F_1|)^{\alpha_1} (C_12^{m_1}|F_2|)^{\beta_1}, \end{align} for any $0 \leq \alpha_1, \beta_1 \leq 1$. One can then use the upper bound in (\ref{threshold}) to proceed the discussion case by case. \vskip .15in \noindent
\textbf{Case I: There exists $0 \leq \alpha_1, \beta_1 \leq 1$ such that $\frac{|\langle B_I, \varphi^1_I \rangle|}{|I|^{\frac{1}{2}}} \gg (C_12^{n_1}|F_1|)^{\alpha_1} (C_12^{m_1}|F_2|)^{\beta_1}.$} \vskip .1in
In Case I, (\ref{energy_needed}) holds and the dominant term in expression (\ref{d_0}) has to be $$
\frac{1}{|I|^{\frac{1}{2}}}\sum_{\substack{|K| \geq |I| \\ K \in \mathcal{K}_0^{n_1,m_1}}}\frac{1}{|K|^{\frac{1}{2}}}\langle f_1, \varphi_K^1 \rangle \langle f_2, \psi_K^2 \rangle \langle \varphi^1_I, \psi_K^3 \rangle, $$ which provides a localization for energy estimates. More precisely, in the current case, \begin{align*}
\text{energy}_{\mathcal{I}'}^{1,\infty}((\langle B_I(f_1,f_2), \varphi_I^1\rangle)_{I \in \mathcal{I}'} \lesssim &\text{energy}_{\mathcal{I}'}^{1,\infty}(\langle B^{n_1,m_1,0}_{\mathcal{K}, \text{generic lac}}(f_1,f_2), \varphi_I\rangle)_{I \in \mathcal{I}'}, \nonumber \\
\text{energy}_{\mathcal{I}'}^t((\langle B_I(f_1,f_2), \varphi_I^1\rangle)_{I \in \mathcal{I}'}) \lesssim & \text{energy}_{\mathcal{I}'}^t(\langle B^{n_1,m_1,0}_{\mathcal{K},\text{generic lac}}(f_1,f_2), \varphi_I\rangle)_{I \in \mathcal{I}'}, \nonumber \end{align*} for any $t >1$, where one recalls that $\mathcal{I}'$ is a finite collection of dyadic intervals satisfying (\ref{condition_collection}). Furthermore, \begin{align*}
\text{energy}_{\mathcal{I}'}^{1,\infty}(\langle B^{n_1,m_1,0}_{\mathcal{K},\text{generic lac}}(f_1,f_2), \varphi_I\rangle)_{I \in \mathcal{I}'} \lesssim& \|B^{n_1,m_1,0}_{\mathcal{K}, \text{generic lac}}(f_1,f_2)\|_1, \nonumber \\
\text{energy}_{\mathcal{I}'}^t(\langle B^{n_1,m_1,0}_{\mathcal{K},\text{generic lac}}(f_1,f_2), \varphi_I\rangle)_{I \in \mathcal{I}'} \lesssim & \|B^{n_1,m_1, 0}_{\mathcal{K}, \text{generic lac}}(f_1,f_2)\|_t, \end{align*}
for any $t > 1$. It is not difficult to verify that $\|B^{n_1,m_1,0}_{\mathcal{K}, \text{generic lac}}(f_1,f_2)\|_t$ for $t \geq 1$ follows from the same estimates for its Haar variant described in Lemma \ref{B_loc_norm}. We will explicitly state the local energy estimates in this case.
\begin{proposition}[Local Energy Estimates in Fourier Case in $x$-Direction] \label{localized_energy_fourier_x} Suppose that $n_1, m_1 \in \mathbb{Z}$ are fixed and suppose that $\mathcal{I}'$ is a finite collection of dyadic intervals such that for any $I \in \mathcal{I}'$, $I $ satisfies \begin{enumerate} \item $I \cap \mathcal{U}_{n_1,m_1} \neq \emptyset$; \item $I \in T $ with $T \in \mathbb{T}_{l_1}$ for some $l_1$ satisfying the condition that there exists some $ 0 \leq \alpha_1, \beta_1 \leq 1$ such that \begin{equation} \label{loc_condition_x}
2^{l_1}\|B\|_1 \gg (C_12^{n_1}|F_1|)^{\alpha_1} (C_12^{m_1}|F_2|)^{\beta_1}. \end{equation} \end{enumerate}
\begin{enumerate}[(i)] \item Then for any $0 < \theta_1,\theta_2 <1$ with $\theta_1 + \theta_2 = 1$, one has \begin{align*}
&\text{energy}^{1,\infty}_{\mathcal{I}'}((\langle B_I, \varphi_I\rangle)_{I \in \mathcal{I}'}) \lesssim C_1^{\frac{1}{p_1}+ \frac{1}{q_1} - \theta_1 - \theta_2} 2^{n_1(\frac{1}{p_1} - \theta_1)} 2^{m_1(\frac{1}{q_1} - \theta_2)} |F_1|^{\frac{1}{p_1}} |F_2|^{\frac{1}{q_1}}.\nonumber
\end{align*}
\item Suppose that $t >1$. Then for any $0 \leq \theta_1, \theta_2 <1$ with $\theta_1 + \theta_2 = \frac{1}{t}$, one has
\begin{align*}
& \text{energy}^{t} _{\mathcal{I}'}((\langle B_I, \varphi_I\rangle)_{I \in \mathcal{I}'}) \lesssim C_1^{\frac{1}{p_1}+ \frac{1}{q_1} - \theta_1 - \theta_2}2^{n_1(\frac{1}{p_1} - \theta_1)}2^{m_1(\frac{1}{q_1} - \theta_2)}|F_1|^{\frac{1}{p_1}}|F_2|^{\frac{1}{q_1}}. \nonumber
\end{align*} \end{enumerate} \end{proposition} A parallel statement holds for dyadic intervals in the $y$-direction, which will be stated for the convenience of reference later on.
\begin{proposition}[Local Energy Estimates in Fourier Case in $y$-Direction] \label{localized_energy_y} Suppose that $ n_2, m_2 \in \mathbb{Z}$ are fixed and suppose that $\mathcal{J}'$ is a finite collection of dyadic intervals such that for any $J \in \mathcal{J} '$, $J $ satisfies \begin{enumerate} \item
$J \cap \tilde{\mathcal{U}}_{n_2,m_2}$ where $\tilde{\mathcal{U}}_{n_2,m_2}:= \{ y: Mg_1(y) \lesssim C_2 2^{n_2}|G_1| \} \cap \{ y: Mg_2(y) \lesssim C_2 2^{m_2}|G_2|\} $. \item $J \in S $ with $S \in \mathbb{S}_{l_2}$ for some $l_2$ satisfying the condition that there exists some $0 \leq \alpha_2, \beta_2 \leq 1$ such that \begin{equation} \label{loc_condition_y}
2^{l_2}\|\tilde{B}\|_1 \gg (C_22^{n_2}|G_1|)^{\alpha_2} (C_22^{m_2}|G_2|)^{\beta_2}. \end{equation} \end{enumerate} \begin{enumerate}[(i)] \item Then for any $0 < \zeta_1,\zeta_2 <1$ with $\zeta_1 + \zeta_2= 1$, one has \begin{align*}
& \text{energy}^{1,\infty}_{\mathcal{J}'}((\langle \tilde{B}_J, \varphi_J \rangle)_{J \in \mathcal{J}'}) \lesssim C_2^{\frac{1}{p_2}+ \frac{1}{q_2} - \zeta_1 - \zeta_2} 2^{n_2(\frac{1}{p_2} - \zeta_1)} 2^{m_2(\frac{1}{q_2} - \zeta_2)} |G_1|^{\frac{1}{p_2}} |G_2|^{\frac{1}{q_2}}.
\end{align*}
\item Suppose that $s >1$. Then for any $0 \leq \zeta_1, \zeta_2 <1$ with $\zeta_1 + \zeta_2= \frac{1}{s}$, one has
\begin{align*}
& \text{energy}^{t} _{\mathcal{J}'}((\langle \tilde{B}_J, \varphi_J \rangle)_{J \in \mathcal{J}'}) \lesssim C_2^{\frac{1}{p_2}+ \frac{1}{q_2} - \zeta_1 - \zeta_2}2^{n_2(\frac{1}{p_2} - \zeta_1)}2^{m_2(\frac{1}{q_2} - \zeta_2)}|G_1|^{\frac{1}{p_2}}|G_2|^{\frac{1}{q_2}}.
\end{align*} \end{enumerate} \end{proposition} \begin{remark} We would like to highlight that the localization of energies is attained under the additional conditions (\ref{loc_condition_x}) and (\ref{loc_condition_y}), in which case one obtains the local energy estimates stated in Proposition \ref{localized_energy_fourier_x} and \ref{localized_energy_y} that can be viewed as analogies of Proposition \ref{B_en}. \end{remark} \vskip .15in \noindent
\textbf{Case II: For any $0 \leq \alpha_1, \beta_1 \leq 1$, $ \frac{|\langle B_I, \varphi^1_I \rangle|}{|I|^{\frac{1}{2}}} \lesssim (C_12^{n_1}|F_1|)^{\alpha_1} (C_12^{m_1}|F_2|)^{\beta_1}.$} \vskip .1in
In this alternative case, the size estimates are favorable and a simpler argument can be applied without invoking the local energy estimates.
\begin{comment} More precisely, let $$
B_0^{-n-n_2,-m-m_2} := \sum_{\substack{ K \in \mathcal{K}_0}}\frac{1}{|K|^{\frac{1}{2}}}\langle f_1, \varphi_K^1 \rangle \langle f_2, \psi_K^2 \rangle \psi_K^3, $$ then (\ref{dom}) coupled with the lemma gives that \begin{equation}\label{compare}
\frac{|\langle B_0^{-n-n_2,-m-m_2}, \varphi^1_I \rangle|}{|I|^{\frac{1}{2}}} \sim \frac{|\langle B_I, \varphi^1_I \rangle|}{|I|^{\frac{1}{2}}}. \end{equation} Suppose that $I \in \mathcal{I}_{-n-n_2,-m-m_2}$. Define $\mathcal{EXP}^{-n-n_2,-m-m_2,\alpha_1, \beta_1}$ to be the collection of $l_1$ satisfying \begin{enumerate}[(1)] \item $
\displaystyle \frac{|\langle B_I, \varphi^1_I \rangle|}{|I|^{\frac{1}{2}}} \sim 2^{l_1}\|B\|_1; $ \item $
2^{l_1}\|B\|_1 \gg (C_12^{-n-n_2}|F_1|)^{\alpha_1} (C_12^{-m-m_2}|F_2|)^{\beta_1} $ \end{enumerate} The following observation articulates how the localization can be achieved. The proof of it is solely based on the comparison indicated in (\ref{compare}). \begin{obs} For any $l_1 \in \mathcal{EXP}^{-n-n_2,-m-m_2,\alpha_1, \beta_1}$, if $I_T \in T$ is a maximal interval with $T \in \mathbb{T}^{l_1}$, then $$
I \subseteq \{ MB_0^{-n-n_2,-m-m_2} > 2^{l_1}\|B\|_1\} $$ \end{obs} The localization property generates the following energy estimates which mimic the corresponding ones discussed in Section 5. \begin{enumerate} \item Localized $L^{1,\infty}$-energy: \begin{align} \label{B_en_1}
& \displaystyle \sup_{l_1 \in \mathcal{EXP}^{-n-n_2,-m-m_2,\alpha_1, \beta_1}}2^{l_1}\|B\|_1\sum_{T \in \mathbb{T}^{l_1}}|I_T| \nonumber \\
\lesssim & \sup_{l_1 \in \mathcal{EXP}^{-n-n_2,-m-m_2,\alpha_1, \beta_1}}2^{l_1}\|B\|_1 \big| \{ MB_0^{-n-n_2,-m-m_2} > 2^{l_1} \| B\|_1\}\big| \nonumber \\
\leq & \|MB_0^{-n-n_2,-m-m_2}\|_{1,\infty} \nonumber\\
\lesssim & \|B_0^{-n-n_2,-m-m_2}\|_1 \nonumber \\
\lesssim & (C_12^{-n-n_2})^{\alpha_1(1-\theta_1)}(C_12^{-m-m_2})^{\beta_1\theta_1}|F_1|^{\alpha_1(1-\theta_1)+\theta_1}|F_2|^{\beta_1\theta_1+(1-\theta_1)}, \end{align} for any $0 < \theta_1 < 1$ and $0 \leq \alpha_1, \beta_1 \leq 1$. \item Localized $L^2$-energy: \begin{align} \label{B_en_2}
& \bigg(\sum_{l_1 \in \mathcal{EXP}^{-n-n_2,-m-m_2,\alpha_1,\beta_1}}(2^{l_1}\|B\|_1)^2\sum_{T \in \mathbb{T}^{l_1}}|I_T|\bigg)^{\frac{1}{2}} \nonumber \\
\leq & \bigg(\sum_{l_1 \in \mathcal{EXP}^{-n-n_2,-m-m_2,\alpha_1,\beta_1}}(2^{l_1}\|B\|_1)^2 \big|\{MB_0^{-n-n_2,-m-m_2} > 2^{l_1}\|B\|_1 \}\big|\bigg)^{\frac{1}{2}} \nonumber \\
\lesssim & \|MB_0^{-n-n_2,-m-m_2}\|_{2} \nonumber \\
\lesssim & \|B_0^{-n-n_2,-m-m_2}\|_2 \nonumber \\
\lesssim & (C_12^{-n-n_2})^{\alpha_1(1-\theta_1)}(C_12^{-m-m_2})^{\beta_1(1-\theta_2)}|F_1|^{\alpha_1(1-\theta_1)+\theta_1}|F_2|^{\beta_1(1-\theta_2)+\theta_2}, \end{align} for any $0 \leq \theta_1, \theta_2 < 1$ with $\theta_1 + \theta_2 = \frac{1}{2}$ and $0 \leq \alpha_1, \beta_1 \leq 1$. \end{enumerate}
\begin{remark} In the other case when $$
\frac{|\langle B_I, \varphi^1_I \rangle|}{|I|^{\frac{1}{2}}} \lesssim (C_12^{-n-n_2}|F_1|)^{\alpha_1} (C_12^{-m-m_2}|F_2|)^{\beta_1}, $$ the localization cannot be achieved. However, it turns out that in this case a simpler argument can be applied without invoking the energy estimates. \end{remark} \end{comment} \vskip .25in \subsubsection{Proof Part 1 - Localization} In this last section, we will explore how to implement the case-by-case analysis and develop a generalized proof of Theorem \ref{thm_weak_mod} for $\Pi_{\text{flag}^0 \otimes \text{flag}^0}$ when $(\phi^3_K)_K$ and $(\phi^3_L)_L$ are \textbf{lacunary} families and $\frac{1}{p_1} + \frac{1}{q_1} = \frac{1}{p_2} + \frac{1}{q_2} >1 $, which is the most tricky part to generalize from the argument in Section \ref{section_thm_haar}.
The generalized argument can be viewed as a combination of the discussions in Sections \ref{section_thm_haar_fixed} and \ref{section_thm_haar}. One first defines the exceptional set $\Omega$ as follows. For any $\tau_1,\tau_2 \in \mathbb{N}$, define $$\Omega^{\tau_1,\tau_2} := \Omega_1^{\tau_1,\tau_2}\cup \Omega_2^{\tau_1,\tau_2} $$ with \begin{align*}
\displaystyle \Omega_1^{\tau_1,\tau_2} := &\bigcup_{\mathfrak{n}_1 \in \mathbb{Z}}\{x: Mf_1(x) > C_1^{\tau_1} 2^{\mathfrak{n}_1}|F_1|\} \times \{y: Mg_1(y) > C_2^{\tau_2} 2^{-\mathfrak{n}_1}|G_1|\}\cup \nonumber \\
& \bigcup_{\mathfrak{n}_2 \in \mathbb{Z}}\{x: Mf_2(x) > C_1^{\tau_1} 2^{\mathfrak{n}_2}|F_2|\} \times \{y: Mg_2(y) > C_2^{\tau_2} 2^{-\mathfrak{n}_2}|G_2|\}\cup \nonumber \\
&\bigcup_{\mathfrak{n}_3 \in \mathbb{Z}}\{x: Mf_1(x) > C_1^{\tau_1} 2^{\mathfrak{n}_3}|F_1|\} \times \{y: Mg_2(y) > C_2^{\tau_2} 2^{-\mathfrak{n}_3}|G_2|\}\cup \nonumber \\
& \bigcup_{\mathfrak{n}_4 \in \mathbb{Z}}\{x: Mf_2(x) > C_1^{\tau_1} 2^{\mathfrak{n}_4 }|F_2|\} \times \{y: Mg_1(y) > C_2^{\tau_2} 2^{-\mathfrak{n}_4 }|G_1|\}\cup \nonumber \\
& \bigcup_{l_2 \in \mathbb{Z}}\{x: MB(f_1,f_2)(x) > C_1^{\tau_1}2^{-l_2}\|B(f_1,f_2)\|_1\} \times \{y: M\tilde{B}(g_1,g_2)(y) > C_2^{\tau_2} 2^{l_2} \|\tilde{B}(g_1,g_2)\|_1\}, \nonumber \\
\Omega_2^{\tau_1,\tau_2} := & \{(x,y) \in \mathbb{R}^2: SSh(x,y) > C_3^{\tau_1,\tau_2} \|h\|_{L^s(\mathbb{R}^2)}\}, \nonumber \\ \end{align*} and \begin{align*} & Enl(\Omega^{\tau_1,\tau_2}) := \{(x,y) \in \mathbb{R}^2: MM(\chi_{\Omega^{\tau_1,\tau_2}})(x,y) > \frac{1}{100}\}, \nonumber \\ & Enl_{\tau_1,\tau_2}(Enl(\Omega^{\tau_1,\tau_2})) := \{(x,y) \in \mathbb{R}^2: MM(\chi_{Enl(\Omega^{\tau_1,\tau_2})}(x,y)> \frac{1}{2^{2\tau_1+ 2\tau_2}} \}, \end{align*} and lastly $$\mathbf{\Omega} := \bigcup_{\tau_1,\tau_2 \in \mathbb{N}}Enl_{\tau_1,\tau_2}(Enl(\Omega^{\tau_1,\tau_2})).$$ Let $$E' := E \setminus \mathbf{\Omega},$$
where $|E'| \sim |E| =1$ given that $C_1, C_2$ and $C_3$ are sufficiently large constants. For any $\tau_1, \tau_2 \in \mathbb{N}$ fixed, let \begin{equation}
\Lambda_{\text{flag}^{0} \otimes \text{flag}^{0}}^{\tau_1, \tau_2} (f_1, f_2, g_1, g_2, h, \chi_{E'}):= \sum_{I \times J \in \mathcal{R}} \frac{1}{|I|^{\frac{1}{2}} |J|^{\frac{1}{2}}} \langle B_I(f_1,f_2),\phi_I^1 \rangle \langle \tilde{B}_J(g_1,g_2), \phi_J^1 \rangle \cdot \langle h, \phi_{I}^2 \otimes \phi_{J}^2 \rangle \langle \chi_{E'},\phi_{I}^{3,\tau_1} \otimes \phi^{3, \tau_2}_{J} \rangle, \end{equation} where $B_I$ and $\tilde{B}_J$ are bilinear operators defined in (\ref{B_local_fourier_simple}) and (\ref{B_local0_haar}). Our goal is to prove that \begin{align} \label{compact} \Lambda_{\text{flag}^{0} \otimes \text{flag}^{0}}(f_1, f_2, g_1, g_2, h, \chi_{E'}) = \displaystyle \sum_{\tau_1,\tau_2 \in \mathbb{N}}2^{-100(\tau_1+\tau_2)}\Lambda_{\text{flag}^{0} \otimes \text{flag}^{0}}^{\tau_1, \tau_2} (f_1, f_2, g_1, g_2, h, \chi_{E'}) \nonumber \end{align} satisfies the restricted weak-type estimates \begin{equation} \label{final_linear}
|\Lambda_{\text{flag}^{0} \otimes \text{flag}^{0}}(f_1, f_2, g_1, g_2, h, \chi_{E'})| \lesssim |F_1|^{\frac{1}{p_1}}|F_2|^{\frac{1}{q_1}}|G_1|^{\frac{1}{p_2}}|G_2|^{\frac{1}{q_2}}\|h\|_{L^s(\mathbb{R}^2)}, \end{equation} which can be reduced to proving that for any fixed $\tau_1, \tau_2 \in \mathbb{N}$, \begin{equation} \label{linear_fix_0_fourier}
|\Lambda_{\text{flag}^0 \otimes \text{flag}^{0}}^{\tau_1, \tau_2}(f_1, f_2, g_1, g_2, h, \chi_{E'})| \lesssim (2^{\tau_1+ \tau_2})^{\Theta}|F_1|^{\frac{1}{p_1}}|F_2|^{\frac{1}{q_1}}|G_1|^{\frac{1}{p_2}}|G_2|^{\frac{1}{q_2}}\|h\|_{L^s(\mathbb{R}^2)} \end{equation} for some $0 < \Theta < 100$.
\subsubsection{Proof Part 2 - Summary of stopping-time decompositions.} \label{fourier_summary_st} For any fixed $\tau_1,\tau_2 \in \mathbb{N}$, one can carry out the exactly same stopping-time algorithms in Section \ref{section_thm_haar} with the replacement of $C_1, C_2$ and $C_3$ by $C_1^{\tau_1,\tau_2}$, $C_2^{\tau_1,\tau_2}$ and $C_3^{\tau_1,\tau_2}$ respectively. The resulting level sets, trees and collections of dyadic rectangles will follow the similar notation as before with extra indications of $\tau_1$ and $\tau_2$. \vskip .15in
{\fontsize{9.5}{9.5} \begin{table}[h!] \begin{tabular}{ l l l }
I. Tensor-type stopping-time decomposition I on $\mathcal{I} \times \mathcal{J}$& $\longrightarrow$ & $I \times J \in \mathcal{I}^{\tau_1}_{-n-n_2,-m-m_2} \times \mathcal{J}^{\tau_2}_{n_2,m_2}$ \\ & & $(n_2, m_2 \in \mathbb{Z}, n > 0, m>0)$\\ II. Tensor-type stopping-time decomposition II on $\mathcal{I} \times \mathcal{J}$ & $\longrightarrow$ & $I \times J \in T \times S $ with $T \in \mathbb{T}_{-l-l_2}^{\tau_1}$, $S \in \mathbb{S}_{l_2}^{\tau_2}$\\ & & $(l_2 \in \mathbb{Z}, l > 0)$\\ III. General two-dimensional level sets stopping-time & $\longrightarrow$ & $I \times J \in \mathcal{R}_{k_1,k_2}^{\tau_1,\tau_2} $ \\ \ \ \ \ \ decomposition on $\mathcal{I} \times \mathcal{J}$& & $(k_1 <0, k_2 \leq K)$\\
\end{tabular} \end{table}} \noindent
\begin{comment} The ranges of exponents $l_2$ are defined as follows: \begin{align*}
\mathcal{EXP}_1^{l,-n-n_2,n_2,-m-m_2,m_2\alpha_1,\beta_1,\alpha_2,\beta_2} := & \{l_2: 2^{-l-l_2} \| B\|_1\lesssim (C_1^{\tau_1}2^{-n-n_2}|F_1|)^{\alpha_1} (C_1^{\tau_1}2^{-m-m_2}|F_2|)^{\beta_1} \ \ \text{and} \nonumber \\
& \quad \quad 2^{l_2} \|\tilde{B}\|_1 \lesssim (C_2^{\tau_2}2^{n_2}|G_1|)^{\alpha_2} (C_2^{\tau_2}2^{m_2}|G_2|)^{\beta_2}\} \nonumber \\
\mathcal{EXP}_2^{l,-n-n_2,n_2, -m-m_2,m_2, \alpha_1,\beta_1,\alpha_2,\beta_2} := & \{l_2: 2^{-l-l_2} \| B\|_1 \gg (C_1^{\tau_1}2^{-n-n_2}|F_1|)^{\alpha_1} (C_1^{\tau_1}2^{-m-m_2}|F_2|)^{\beta_1} \ \ \text{and} \nonumber \\
& \quad \quad 2^{l_2} \|\tilde{B}\|_1 \lesssim (C_2^{\tau_2}2^{n_2}|G_1|)^{\alpha_2} (C_2^{\tau_2}2^{m_2}|G_2|)^{\beta_2}\} \nonumber \\
\mathcal{EXP}_3^{l,-n-n_2,n_2,-m-m_2,m_2\alpha_1,\beta_1,\alpha_2,\beta_2} := & \{l_2: 2^{-l-l_2} \| B\|_1 \lesssim (C_1^{\tau_1}2^{-n-n_2}|F_1|)^{\alpha_1} (C_1^{\tau_1}2^{-m-m_2}|F_2|)^{\beta_1} \ \ \text{and} \nonumber \\
& \quad \quad 2^{l_2} \|\tilde{B}\|_1 \gg (C_2^{\tau_2}2^{n_2}|G_1|)^{\alpha_2} (C_2^{\tau_2}2^{m_2}|G_2|)^{\beta_2}\} \nonumber \\
\mathcal{EXP}_4^{l,-n-n_2,n_2,-m-m_2,m_2, \alpha_1,\beta_1,\alpha_2,\beta_2} := & \{l_2: 2^{-l-l_2} \| B\|_1 \gg (C_1^{\tau_1}2^{-n-n_2}|F_1|)^{\alpha_1} (C_1^{\tau_1}2^{-m-m_2}|F_2|)^{\beta_1} \ \ \text{and} \nonumber \\
& \quad \quad 2^{l_2} \|\tilde{B}\|_1 \gg (C_2^{\tau_2}2^{n_2}|G_1|)^{\alpha_2} (C_2^{\tau_2}2^{m_2}|G_2|)^{\beta_2}\}. \nonumber \\ \end{align*} \end{comment}
\subsubsection{Proof Part 3 - Application of stopping-time decompositions.} As one may recall, in Section \ref{section_thm_haar} the multilinear form is estimated based on the stopping-time decompositions, the sparsity condition and the Fubini-type argument. Analogously, we first apply the stopping-time decompositions specified in Section \ref{fourier_summary_st} and denote the range of exponents \begin{equation} \mathcal{W} := \{(l,n,m,k_1,k_2) \in \mathbb{Z}^5: l >0, n> 0, m> 0, k_1 < 0, k_2 \leq K \} \end{equation} so that the multilinear form can be estimated as \begin{align}
&|\Lambda_{\text{flag}^0 \otimes \text{flag}^{0}}^{\tau_1, \tau_2}(f_1,f_2,g_1,g_2,h, \chi_{E'})| \nonumber \\
= & \bigg| \sum_{(l,n,m,k_1,k_2) \in \mathcal{W}}\sum_{\substack{n_2 \in \mathbb{Z}\\ m_2 \in \mathbb{Z}\\ l_2\in \mathbb{Z}}} \sum_{\substack{T \in \mathbb{T}_{-l-l_2}^{\tau_1} \\ S \in \mathbb{S}_{l_2}^{\tau_2}}}\sum_{\substack{I \times J \in \mathcal{I}_{-n-n_2,-m-m_2}^{\tau_1} \times \mathcal{J}_{n_2,m_2}^{\tau_2} \\ I \times J \in T \times S \\ I \times J \in \mathcal{R}_{k_1,k_2}^{\tau_1,\tau_2}}}\frac{1}{|I|^{\frac{1}{2}} |J|^{\frac{1}{2}}} \langle B_I(f_1,f_2),\phi_I^1 \rangle \langle \tilde{B}_J(g_1,g_2), \phi_J^1 \rangle \cdot \nonumber\\
& \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \langle h, \phi_I^2 \otimes \phi_J^2 \rangle \langle \chi_{E'}, \phi_I^{3,\tau_1} \otimes \phi_J^{3,\tau_2} \rangle\bigg| \nonumber \\
\lesssim &C_1^{\tau_1}C_2^{\tau_2}(C_3^{\tau_1,\tau_2})^2\sum_{(l,n,m,k_1,k_2) \in \mathcal{W}}2^{k_1} \|h\|_{L^s(\mathbb{R}^2)} 2^{k_2} \cdot \sum_{\substack{n_2 \in \mathbb{Z}\\ m_2 \in \mathbb{Z}\\ l_2\in \mathbb{Z}}} 2^{-l-l_2} \|B(f_1,f_2)\|_1 2^{l_2} \|\tilde{B}(g_1,g_2)\|_1 \cdot \nonumber\\
& \quad \quad \quad\quad \quad \quad\quad \quad \quad \quad \quad \quad\quad \quad \quad\quad \quad \quad \quad \quad \quad\quad \sum_{\substack{T \in \mathbb{T}_{-l-l_2}^{\tau_1} \\ S \in \mathbb{S}_{l_2}^{\tau_2}}} \bigg|\bigcup_{\substack{I \times J \in \mathcal{I}_{-n-n_2,-m-m_2}^{\tau_1} \cap T \times \mathcal{J}_{n_2,m_2}^{\tau_2} \cap S \\ I \times J \in \mathcal{R}_{k_1,k_2}^{\tau_1,\tau_2}}}I \times J\bigg|. \end{align}
The nested sum \begin{align} \label{ns_fourier}
& \sum_{\substack{n_2 \in \mathbb{Z}\\ m_2 \in \mathbb{Z}\\ l_2\in \mathbb{Z}}} 2^{-l-l_2} \|B(f_1,f_2)\|_1 2^{l_2}\|\tilde{B}(g_1,g_2)\|_1 \sum_{\substack{T \in \mathbb{T}_{-l-l_2}^{\tau_1} \\ S \in \mathbb{S}_{l_2}^{\tau_2}}} \bigg|\bigcup_{\substack{I \times J \in \mathcal{I}_{-n-n_2,-m-m_2}^{\tau_1} \cap T \times \mathcal{J}_{n_2,m_2}^{\tau_2} \cap S \\ I \times J \in \mathcal{R}_{k_1,k_2}^{\tau_1,\tau_2}}}I \times J \bigg| \end{align} can be estimated using the sparsity condition (Proposition \ref{sp_2d}) and a modified Fubini argument as discussed in the following two subsections. \subsubsection{Proof Part 4 - Sparsity condition} One invokes the sparsity condition and the argument in Section \ref{section_thm_haar_fixed} to obtain the following estimate for (\ref{ns_fourier}) analogous to (\ref{ns_sp}) . \begin{align} \label{ns_fourier_sp}
(\ref{ns_fourier}) \lesssim & 2^{-\frac{k_2\gamma}{2}}2^{-l(1- \frac{(1+\delta)}{2})} |F_1|^{\frac{\mu_1(1+\delta)}{2}}|F_2|^{\frac{\mu_2(1+\delta)}{2}}|G_1|^{\frac{\nu_1(1+\delta)}{2}}|G_2|^{\frac{\nu_2(1+\delta)}{2}}\|B(f_1,f_2)\|_1^{1-\frac{1+\delta}{2}}\|\tilde{B}(g_1,g_2)\|_1^{1-\frac{1+\delta}{2}}. \end{align} where $\gamma >1 $, $\delta,\mu_1,\mu_2,\nu_1,\nu_2 >0$ with $\mu_1+ \mu_2 = \nu_1+ \nu_2 = \frac{1}{1+\delta}$. For any $0 < \delta \ll1$, Lemma \ref{B_global_norm} implies that \begin{align}
& \|B(f_1,f_2)\|_1^{1-\frac{1+\delta}{2}} \lesssim |F_1|^{\rho(1-\frac{1+\delta}{2})}|F_2|^{(1-\rho)(1-\frac{1+\delta}{2})}, \label{B_norm_fourier1} \\
& \|\tilde{B}(g_1,g_2)\|_1^{1-\frac{1+\delta}{2}} \lesssim |G_1|^{\rho'(1-\frac{1+\delta}{2})}|G_2|^{(1-\rho')(1-\frac{1+\delta}{2})}. \label{B_norm_fourier2} \end{align}
By applying (\ref{B_norm_fourier1}) and (\ref{B_norm_fourier2}) to (\ref{ns_fourier_sp}), one derives the following bound: \begin{equation} \label{ns_fourier_sp_final}
2^{-\frac{k_2\gamma}{2}}2^{-l(1- \frac{(1+\delta)}{2})} |F_1|^{\frac{\mu_1(1+\delta)}{2}+\rho(1-\frac{1+\delta}{2})}|F_2|^{\frac{\mu_2(1+\delta)}{2}+(1-\rho)(1-\frac{1+\delta}{2})}|G_1|^{\frac{\nu_1(1+\delta)}{2}+\rho'(1-\frac{1+\delta}{2})}|G_2|^{\frac{\nu_2(1+\delta)}{2}+(1-\rho')(1-\frac{1+\delta}{2})}. \end{equation}
\vskip .15 in \subsubsection{Proof Part 5 - Fubini argument} The separation of cases based on the levels of the stopping-time decompositions for $
\big(\frac{|\langle B_{I}(f_1,f_2), \varphi^{1}_I \rangle|}{|I|^{\frac{1}{2}}}\big)_{I \in \mathcal{I}} $ and $
\big(\frac{|\langle \tilde {B}_{J}(g_1,g_2), \varphi^{1}_J \rangle|}{|J|^{\frac{1}{2}}}\big)_{J \in \mathcal{J}} $. In particular, the ranges of $l_2$ in the \textit{tensor-type stopping-time decomposition I}, plays an important role in the modified Fubini-type argument. With $l \in \mathbb{N}$ fixed, the ranges of exponents $l_2$ are defined as follows: \begin{align*} \mathcal{EXP}_1^{l,n,m,n_2,m_2} := & \{l_2 \in \mathbb{Z}: \text{for any}\ \ 0 \leq \alpha_1, \beta_1, \alpha_2, \beta_2 \leq 1, \nonumber \\
& \quad \quad \quad \quad 2^{-l-l_2} \| B(f_1,f_2)\|_1\lesssim (C_1^{\tau_1}2^{-n-n_2}|F_1|)^{\alpha_1} (C_1^{\tau_1}2^{-m-m_2}|F_2|)^{\beta_1} \ \ \text{and} \nonumber \\
& \quad \quad \quad \quad 2^{l_2} \|\tilde{B}(g_1,g_2)\|_1 \lesssim (C_2^{\tau_2}2^{n_2}|G_1|)^{\alpha_2} (C_2^{\tau_2}2^{m_2}|G_2|)^{\beta_2} \}, \nonumber \\ \mathcal{EXP}_2^{l,n,m,n_2,m_2} := & \{l_2 \in \mathbb{Z}: \text{there exists} \ \ 0 \leq \alpha_1, \beta_1 \leq 1 \ \ \text{such that} \nonumber \\
& \quad \quad \quad \quad 2^{-l-l_2} \| B(f_1,f_2)\|_1 \gg (C_1^{\tau_1}2^{-n-n_2}|F_1|)^{\alpha_1} (C_1^{\tau_1}2^{-m-m_2}|F_2|)^{\beta_1} \ \ \text{and} \nonumber \\ & \quad \quad \quad \quad \text{for any} \ \ 0 \leq \alpha_2, \beta_2 \leq 1, \nonumber \\
& \quad \quad \quad \quad 2^{l_2} \|\tilde{B}(g_1,g_2)\|_1 \lesssim (C_2^{\tau_2}2^{n_2}|G_1|)^{\alpha_2} (C_2^{\tau_2}2^{m_2}|G_2|)^{\beta_2}\}, \nonumber \\ \mathcal{EXP}_3^{l,n,m,n_2,m_2} := & \{l_2 \in \mathbb{Z}: \text{for any} \ \ 0 \leq \alpha_1, \beta_1 \leq 1, \nonumber \\
& \quad \quad \quad \quad 2^{-l-l_2} \| B(f_1,f_2)\|_1 \lesssim (C_1^{\tau_1}2^{-n-n_2}|F_1|)^{\alpha_1} (C_1^{\tau_1}2^{-m-m_2}|F_2|)^{\beta_1} \ \ \text{and} \nonumber \\
& \quad \quad \quad \quad \text{there exists} \ \ 0 \leq \alpha_2, \beta_2 \leq 1 \ \ \text{such that} \nonumber \\
& \quad \quad \quad \quad 2^{l_2} \|\tilde{B}(g_1,g_2)\|_1 \gg (C_2^{\tau_2}2^{n_2}|G_1|)^{\alpha_2} (C_2^{\tau_2}2^{m_2}|G_2|)^{\beta_2}\}, \nonumber \\ \mathcal{EXP}_4^{l,n,m,n_2,m_2} := & \{l_2 \in \mathbb{Z}: \text{there exists} \ \ 0 \leq \alpha_1, \beta_1, \alpha_2, \beta_2 \leq 1 \ \ \text{such that} \nonumber \\
& \quad \quad \quad \quad 2^{-l-l_2} \| B(f_1,f_2)\|_1 \gg (C_1^{\tau_1}2^{-n-n_2}|F_1|)^{\alpha_1} (C_1^{\tau_1}2^{-m-m_2}|F_2|)^{\beta_1} \ \ \text{and} \nonumber \\
& \quad \quad \quad \quad 2^{l_2} \|\tilde{B}(g_1,g_2)\|_1 \gg (C_2^{\tau_2}2^{n_2}|G_1|)^{\alpha_2} (C_2^{\tau_2}2^{m_2}|G_2|)^{\beta_2}\}. \nonumber \end{align*}
With little abuse of notation, we will simplify $\mathcal{EXP}_i^{l,n,m,n_2,m_2}$ by $\mathcal{EXP}_i$, for $i = 1,2,3,4$. We will then decompose the sum into four parts based on the ranges specified above: \begin{align*}
& \sum_{\substack{n_2 \in \mathbb{Z}\\ m_2 \in \mathbb{Z}\\ l_2\in \mathbb{Z}}} 2^{-l-l_2} \|B\|_1 2^{l_2} \|\tilde{B}\|_1\sum_{\substack{S \in \mathbb{S}^{-l-l_2} \\ T \in \mathbb{T}^{l_2}}} \bigg|\bigcup_{\substack{I \times J \in \mathcal{I}_{-n-n_2,-m-m_2} \cap S \times \mathcal{J}_{n_2,m_2} \cap T \\ I \times J \in \mathcal{R}_{k_1,k_2}}}I \times J\bigg| \nonumber \\ = &\underbrace{ \sum_{\substack{n_2 \in \mathbb{Z}\\ m_2 \in \mathbb{Z}\\ }} \sum_{l_2\in \mathcal{EXP}_1^{}}}_I+ \underbrace{\sum_{\substack{n_2 \in \mathbb{Z}\\ m_2 \in \mathbb{Z}\\ }} \sum_{l_2\in \mathcal{EXP}_2^{}}}_{II} + \underbrace{\sum_{\substack{n_2 \in \mathbb{Z}\\ m_2 \in \mathbb{Z}\\ }} \sum_{_2\in \mathcal{EXP}_3^{}} }_{III}+ \underbrace{\sum_{\substack{n_2 \in \mathbb{Z}\\ m_2 \in \mathbb{Z}\\ }} \sum_{l_2\in \mathcal{EXP}_4^{}}}_{IV}. \end{align*} One denotes the four parts by $I$, $II$, $III$ and $IV$ and will derive estimates for each part separately. The multilinear form can thus be decomposed correspondingly as follows: \begin{align*}
&|\Lambda_{\text{flag}^0 \otimes \text{flag}^{0}}^{\tau_1, \tau_2}| \nonumber\\
\lesssim & C_1^{\tau_1}C_2^{\tau_2} (C_3^{\tau_1,\tau_2})^2 \bigg(\underbrace{\sum_{(l,n,m,k_1,k_2) \in \mathcal{W}} 2^{k_1}\|h\|_{L^s(\mathbb{R}^2)} 2^{k_2} \cdot I}_{\Lambda_{I}^{\tau_1, \tau_2}} + \underbrace{ \sum_{(l,n,m,k_1,k_2) \in \mathcal{W}} 2^{k_1}\|h\|_{L^s(\mathbb{R}^2)} 2^{k_2} \cdot II}_{\Lambda_{II}^{\tau_1, \tau_2}} + \nonumber \\
& \quad \quad \quad \quad \quad \quad \quad \quad \underbrace{ \sum_{(l,n,m,k_1,k_2) \in \mathcal{W}} 2^{k_1}\|h\|_{L^s(\mathbb{R}^2)} 2^{k_2} \cdot III}_{\Lambda_{III}^{\tau_1, \tau_2}} + \underbrace{\sum_{(l,n,m,k_1,k_2) \in \mathcal{W}} 2^{k_1}\|h\|_{L^s(\mathbb{R}^2)} 2^{k_2} \cdot IV}_{\Lambda_{IV}^{\tau_1, \tau_2}}\bigg). \end{align*} It would be sufficient to prove that each part satisfies the bound on the right hand side of (\ref{linear_fix_0_fourier}).
\vskip .15in \noindent \textbf{Estimate of $\Lambda_I^{\tau_1,\tau_2}$.} Though for $I$, the localization of energies cannot be applied at all, one observes that energy estimates are indeed not necessary. In particular, \begin{align} \label{I} &I \nonumber \\
\lesssim & \sum_{\substack{n_2 \in \mathbb{Z}\\ m_2 \in \mathbb{Z}}} \sum_{l_2\in \mathcal{EXP}_1}2^{-l-l_2}\|B(f_1,f_2)\|_1 2^{l_2}\|\tilde{B}(g_1,g_2)\|_1 \sum_{\substack{T \in \mathbb{T}_{-l-l_2}^{\tau_1} \\ S \in \mathbb{S}_{l_2}^{\tau_2}}} \bigg|\bigcup_{\substack{I \times J \in \mathcal{I}_{-n-n_2,-m-m_2}^{\tau_1} \cap T \times \mathcal{J}_{n_2,m_2}^{\tau_2} \cap S \\ I \times J \in \mathcal{R}_{k_1,k_2}^{\tau_1,\tau_2}}}I \times J\bigg| \nonumber \\
\leq & \sum_{\substack{n_2 \in \mathbb{Z}\\ m_2 \in \mathbb{Z}}} \bigg(\sup_{l_2\in \mathcal{EXP}_1} 2^{-l-l_2}\|B(f_1,f_2)\|_1\bigg)\bigg(\sup_{l_2\in \mathcal{EXP}_1} \sum_{\substack{T \in \mathbb{T}_{-l-l_2}^{\tau_1} \\ S \in \mathbb{S}_{l_2}^{\tau_2}}} \bigg|\bigcup_{\substack{I \times J \in \mathcal{I}_{-n-n_2,-m-m_2}^{\tau_1} \cap T \times \mathcal{J}_{n_2,m_2}^{\tau_2} \cap S \\ I \times J \in \mathcal{R}_{k_1,k_2}^{\tau_1,\tau_2}}}I \times J\bigg|\bigg)\cdot \nonumber\\
& \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad \bigg(\sum_{l_2\in \mathcal{EXP}_1}2^{l_2}\|\tilde{B}(g_1,g_2)\|_1\bigg).
\end{align} We will estimate the expressions in the parentheses separately. \begin{enumerate}[(i)] \item It is trivial from the definition of $\mathcal{EXP}_1$ that for any $0 \leq \alpha_1, \beta_1 \leq 1$, \begin{equation} \label{I_i}
\sup_{l_2\in \mathcal{EXP}_1} 2^{-l-l_2}\|B(f_1,f_2)\|_1 \lesssim (C_1^{\tau_1}2^{-n-n_2}|F_1|)^{\alpha_1} (C_1^{\tau_1}2^{-m-m_2}|F_2|)^{\beta_1}. \end{equation} \item The last expression is a geometric series with the largest term bounded by \begin{equation}\label{I_ii}
(C_2^{\tau_2}2^{n_2}|G_1|)^{\alpha_2} (C_2^{\tau_2}2^{m_2}|G_2|)^{\beta_2}, \end{equation} for any $0 \leq \alpha_2, \beta_2 \leq 1$ according to the definition of $\mathcal{EXP}_1$. As a result, $$
\sum_{l_2 \in \mathcal{EXP}_1} 2^{l_2} \|\tilde{B}(g_1,g_2)\|_1 \lesssim (C_2^{\tau_2}2^{n_2}|G_1|)^{\alpha_2} (C_2^{\tau_2},2^{m_2}|G_2|)^{\beta_2}, $$ for any $0 \leq \alpha_2, \beta_2 \leq 1$. \item For any fixed $-n-n_2, -m-m_2, n_2,m_2, l_2, \tau_1,\tau_2$, \begin{align*} &\{I_T: I_T \in \mathcal{I}_{-n-n_2,-m-m_2}^{\tau_1} \ \ \text{and} \ \ T \in \mathbb{T}^{\tau_1}_{-l-l_2} \}, \nonumber \\ & \{J_S: J_S \in \mathcal{I}_{n_2,-m_2}^{\tau_2} \ \ \text{and} \ \ S \in \mathbb{S}^{\tau_2}_{l_2} \} \end{align*} are disjoint collections of dyadic intervals. Therefore, for some fixed $\tilde{l}_2 \in \mathbb{Z}$, \begin{align} \label{I_iii}
\sup_{l_2\in \mathcal{EXP}_1} \sum_{\substack{T \in \mathbb{T}_{-l-l_2}^{\tau_1} \\ S \in \mathbb{S}_{l_2}^{\tau_2}}} \bigg|\bigcup_{\substack{I \times J \in \mathcal{I}_{-n-n_2,-m-m_2}^{\tau_1} \cap T \times \mathcal{J}_{n_2,m_2}^{\tau_2} \cap S \\ I \times J \in \mathcal{R}_{k_1,k_2}^{\tau_1,\tau_2}}}I \times J\bigg| = & \bigg|\bigcup_{\substack{T \in \mathbb{T}_{-l-\tilde{l}_2}^{\tau_1} \\ S \in \mathbb{S}_{\tilde{l}2}^{\tau_2}}}\bigcup_{\substack{I \times J \in \mathcal{I}_{-n-n_2,-m-m_2}^{\tau_1} \cap T \times \mathcal{J}_{n_2,m_2}^{\tau_2} \cap S \\ I \times J \in \mathcal{R}_{k_1,k_2}^{\tau_1,\tau_2}}}I \times J\bigg| \nonumber\\
\leq & \bigg|\bigcup_{\substack{I \times J \in \mathcal{I}_{-n-n_2,-m-m_2}^{\tau_1} \times \mathcal{J}_{n_2,m_2}^{\tau_2} \\ I \times J \in \mathcal{R}_{k_1,k_2}^{\tau_1,\tau_2}}}I \times J \bigg|. \end{align} \end{enumerate} One can now plug in the estimates (\ref{I_i}), (\ref{I_ii}) and (\ref{I_iii}) into (\ref{I}) and derive that for any $0 \leq \alpha_1, \beta_1, \alpha_2, \beta_2 \leq 1$, \begin{align} & I \nonumber \\
\lesssim & (C_1^{\tau_1})^2(C_2^{\tau_2})^2\sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}(2^{-n-n_2}|F_1|)^{\alpha_1} (2^{-m-m_2}|F_2|)^{\beta_1}(2^{n_2}|G_1|)^{\alpha_2} (2^{m_2}|G_2|)^{\beta_2}\bigg|\bigcup_{\substack{I \times J \in \mathcal{I}_{-n-n_2,-m-m_2}^{\tau_1} \times \mathcal{J}_{n_2,m_2}^{\tau_2} \\ I \times J \in \mathcal{R}_{k_1,k_2}^{\tau_1,\tau_2}}}I \times J \bigg| \nonumber \\
\leq & (C_1^{\tau_1})^2(C_2^{\tau_2})^2 \sup_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}(2^{-n-n_2}|F_1|)^{\alpha_1} (2^{-m-m_2}|F_2|)^{\beta_1}(2^{n_2}|G_1|)^{\alpha_2} (2^{m_2}|G_2|)^{\beta_2}\sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\bigg|\bigcup_{\substack{I \times J \in \mathcal{I}_{-n-n_2,-m-m_2}^{\tau_1} \times \mathcal{J}_{n_2,m_2}^{\tau_2} \\ I \times J \in \mathcal{R}_{k_1,k_2}^{\tau_1,\tau_2}}}I \times J \bigg|. \end{align} By letting $\alpha_1 = \frac{1}{p_1}$, $\beta_2 = \frac{1}{q_1}$, $\alpha_2 = \frac{1}{p_2}$ and $\beta_2 = \frac{1}{q_2}$ and the argument for the choice of indices in Section \ref{section_thm_haar_fixed}, one has $$
I \lesssim (C_1^{\tau_1})^2(C_2^{\tau_2})^2 2^{-n\frac{1}{p_2}}2^{-m\frac{1}{q_1}} |F_1|^{\frac{1}{p_1}} |F_2|^{\frac{1}{q_1}} |G_1|^{\frac{1}{p_2}} |G_2|^{\frac{1}{q_2}}\sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\bigg|\bigcup_{\substack{I \times J \in \mathcal{I}_{-n-n_2,-m-m_2}^{\tau_1} \times \mathcal{J}_{n_2,m_2}^{\tau_2} \\ I \times J \in \mathcal{R}_{k_1,k_2}^{\tau_1,\tau_2}}}I \times J \bigg| . $$ where \begin{align} \label{I_measure}
& \sum_{\substack{n_2 \in \mathbb{Z}\\ m_2 \in \mathbb{Z}}}\bigg|\bigcup_{\substack{I \times J \in \mathcal{I}_{-n-n_2,-m-m_2}^{\tau_1} \times \mathcal{J}_{n_2,m_2}^{\tau_2} \\ I \times J \in \mathcal{R}_{k_1,k_2}^{\tau_1,\tau_2}}}I \times J \bigg| \lesssim \min(2^{-k_1s},2^{-k_2\gamma}), \end{align} for any $\gamma >1$. The estimate is a direct application of the sparsity condition described in Proposition \ref{sp_2d} that has been extensively used before (see (\ref{rec_area_hybrid})). One can now apply (\ref{I_measure}) to conclude that \begin{align*}
|\Lambda_I^{\tau_1,\tau_2}| = & \sum_{(l,n,m,k_1,k_2) \in \mathcal{W}} 2^{k_1}\|h\|_{L^s(\mathbb{R}^2)} 2^{k_2} \cdot I \nonumber \\
\lesssim & (C_1^{\tau_1}C_2^{\tau_2}C_3^{\tau_1,\tau_2})^{4} |F_1|^{\frac{1}{p_1}} |F_2|^{\frac{1}{q_1}} |G_1|^{\frac{1}{p_2}} |G_2|^{\frac{1}{q_2}}\|h\|_{L^s(\mathbb{R}^2)} \sum_{(l,n,m,k_1,k_2) \in \mathcal{W}} 2^{k_1(1-\frac{s}{2})}2^{k_2(1-\frac{\gamma}{2})} 2^{-n\frac{1}{p_2}}2^{-m\frac{1}{q_1}}, \end{align*} and achieve the desired bound with the appropriate choice of $\gamma>1$.
\vskip .15in \noindent \textbf{Estimate of $\Lambda_{II}^{\tau_1,\tau_2}$.} One first observes that the estimates for $\Lambda_{II}^{\tau_1,\tau_2}$ apply to $\Lambda_{III}^{\tau_1,\tau_2}$ due to symmetry. One shall notice that \begin{align} \label{II}
II \leq & \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}\bigg( \sum_{l_2 \in \mathcal{EXP}_2} 2^{l_2} \|\tilde{B}(g_1,g_2)\|_1\bigg) \bigg(\sup_{l_2 \in \mathcal{EXP}_2}2^{-l-l_2}\|B(f_1,f_2)\|_1\sum_{\substack{T \in \mathbb{T}_{-l-l_2}^{\tau_1} \\ I_T \in \mathcal{I}^{\tau_1}_{-n-n_2,-m-m_2}}}|I_T|\bigg)\bigg(\sup_{l_2}\sum_{\substack{S \in \mathbb{S}_{l_2}^{\tau_2} \\ J_S \in \mathcal{J}_{n_2,m_2}^{\tau_2}}}|J_S|\bigg). \end{align} \begin{enumerate}[(i)] \item The first expression is a geometric series which can be bounded by \begin{equation} \label{II_i}
(C_2^{\tau_2}2^{n_2}|G_1|)^{\alpha_2} (C_2^{\tau_2}2^{m_2}|G_2|)^{\beta_2}, \end{equation} for any $0 \leq \alpha_2, \beta_2 \leq 1$ (up to some constant as discussed in the estimate of $I$). \item The second term in (\ref{II}) can be considered as a localized $L^{1,\infty}$ energy. In addition, given by the restriction that $l_2 \in \mathcal{EXP}_2$, one can apply the localization and the corresponding energy estimates described in Proposition \ref{localized_energy_fourier_x}. In particular, for any $0 < \theta_1, \theta_2 < 1$ with $\theta_1 + \theta_2 = 1$, \begin{align} \label{II_ii}
& \sup_{l_2 \in \mathcal{EXP}_2}2^{-l-l_2}\|B(f_1,f_2)\|_1\sum_{\substack{T \in \mathbb{T}_{-l-l_2}^{\tau_1} \\ I_T \in \mathcal{I}_{-n-n_2,-m-m_2}}}|I_T|\lesssim & (C_1^{\tau_1}2^{-n-n_2})^{\frac{1}{p_1}-\theta_1}(C_1^{\tau_1} 2^{-m-m_2})^{\frac{1}{q_1}- \theta_2} |F_1|^{\frac{1}{p_1}}|F_2|^{\frac{1}{q_1}}. \end{align} \item For any fixed $n_2,m_2,l_2$ and $\tau_2$, $\{J_S: J_S \in \mathcal{J}_{n_2,m_2}^{\tau_2} \ \ \text{and } \ \ S \in \mathbb{S}_{l_2}^{\tau_2}\}$ is a disjoint collection of dyadic intervals, which implies that \begin{align} \label{II_iii}
\sup_{l_2}\sum_{\substack{S \in \mathbb{S}^{l_2} \\ J_S \in \mathcal{J}_{n_2,m_2}}}|J_S| & \leq \big| \bigcup_{\substack{J_S \in \mathcal{J}_{n_2,m_2}}}J_S\big| \lesssim \big|\{ Mg_1 > C_2^{\tau_2} 2^{n_2-10}|G_1| \} \cap \{Mg_2 > C_2^{\tau_2} 2^{m_2-10}|G_2| \}\big|, \end{align} where the last inequality follows from the pointwise estimates indicated in Claim \ref{ptwise}. \end{enumerate} By combining (\ref{II_i}), (\ref{II_ii}) and (\ref{II_iii}), one can majorize (\ref{II}) by \begin{align}\label{II_final}
II \lesssim & \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}(C_2^{\tau_2} 2^{n_2}|G_1|)^{\alpha_2} (C_2^{\tau_2} 2^{m_2}|G_2|)^{\beta_2}(C_1^{\tau_1}2^{-n-n_2})^{\frac{1}{p_1}- \theta_1}(C_1^{\tau_1} 2^{-m-m_2})^{\frac{1}{q_1} - \theta_2} |F_1|^{\frac{1}{p_1}}|F_2|^{\frac{1}{q_1}} \nonumber \\
& \quad \quad \cdot \big|\{ Mg_1 > C_2^{\tau_2} 2^{n_2-10}|G_1| \} \cap \{Mg_2 > C_2^{\tau_2} 2^{m_2-10}|G_2| \}\big| \nonumber \\
& \leq \sup_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}} (C_1^{\tau_1}2^{-n-n_2})^{\frac{1}{p_1} - \theta_1}(C_1^{\tau_1} 2^{-m-m_2})^{\frac{1}{q_1} - \theta_2} |F_1|^{\frac{1}{p_1}}|F_2|^{\frac{1}{q_1}}\nonumber \\
& \quad \quad \quad \cdot (C_2^{\tau_2} 2^{n_2})^{\alpha_2 - (1+\epsilon)(1-\mu)}(C_2^{\tau_2 }2^{m_2})^{\beta_2-(1+\epsilon)\mu}|G_1|^{\alpha_2- (1+\epsilon)(1-\mu)} |G_2|^{\beta_2-(1+\epsilon)\mu} \cdot \nonumber \\
&\quad \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}(C_2^{\tau_2}2^{n_2}|G_1|)^{(1+\epsilon)(1-\mu)}(C_2^{\tau_2}2^{m_2}|G_2|)^{(1+\epsilon)\mu}\big|\{ Mg_1 > C_2^{\tau_2} 2^{n_2-10}|G_1| \} \cap \{Mg_2 > C_2^{\tau_2} 2^{m_2-10}|G_2| \}\big|. \end{align} By the H\"older-type argument introduced in Section \ref{section_thm_haar}, namely (\ref{B_2_final}), one can estimate the expression \begin{align} \label{II_fub}
& \sum_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}}(C_2^{\tau_2}2^{n_2}|G_1|)^{(1+\epsilon)(1-\mu)}(C_2^{\tau_2}2^{m_2}|G_2|)^{(1+\epsilon)\mu}\big|\{ Mg_1 > C_2^{\tau_2} 2^{n_2-10}|G_1| \} \cap \{Mg_2 > C_2^{\tau_2} 2^{m_2-10}|G_2| \}\big| \nonumber \\
\lesssim & |G_1|^{1-\mu} |G_2|^{\mu}. \end{align} Therefore, by plugging in (\ref{II_fub}) and some simplifications, (\ref{II_final}) can be bounded by \begin{align*}
& (C_1^{\tau_1} C_2^{\tau_2})^2\sup_{\substack{n_2 \in \mathbb{Z} \\ m_2 \in \mathbb{Z}}} (2^{-n-n_2})^{\frac{1}{p_1} - \theta_1}( 2^{-m-m_2})^{\frac{1}{q_1} - \theta_2} |F_1|^{\frac{1}{p_1}}|F_2|^{\frac{1}{q_1}} (2^{n_2})^{\alpha_2 - (1+\epsilon)(1-\mu)}(2^{m_2})^{\beta_2-(1+\epsilon)\mu}|G_1|^{\alpha_2-\epsilon(1-\mu)} |G_2|^{\beta_2-\epsilon\mu}.
\end{align*} One would like to choose $0 \leq \alpha_2, \beta_2 \leq 1, 0 < \mu < 1$ and $\epsilon>0$ such that \begin{align} \label{exp_cond_fourier} & \alpha_2-\epsilon(1-\mu) = \frac{1}{p_2}, \nonumber \\ & \beta_2 - \epsilon\mu = \frac{1}{q_2}. \end{align} Meanwhile, one can also achieve the equalities \begin{align*} & \frac{1}{p_1} - \theta_1 = \alpha_2-(1+\epsilon)(1-\mu), \nonumber \\ & \frac{1}{q_1} - (1-\theta_1) = \beta_2-(1+\epsilon)\mu, \end{align*} which combined with (\ref{exp_cond_fourier}), yield \begin{align*} & \frac{1}{p_1} - \theta_1 = \frac{1}{p_2} - (1-\mu), \nonumber \\ & \frac{1}{q_1} - (1-\theta_1) = \frac{1}{q_2} -\mu. \end{align*} Thanks to the condition that $$ \frac{1}{p_1} + \frac{1}{q_1} = \frac{1}{p_2} + \frac{1}{q_2}, $$ one only needs to choose $0 < \theta_1, \mu < 1$ such that $$ \frac{1}{p_1} - \frac{1}{p_2} = \theta_1- (1-\mu). $$
To sum up, one has the following estimate for II: \begin{equation} \label{II_ns}
II \lesssim (C_1^{\tau_1} C_2^{\tau_2})^2 2^{-n(\frac{1}{p_1}- \theta_1)}2^{-m(\frac{1}{q_1}- (1-\theta_1))}|F_1|^{\frac{1}{p_1}} |F_2|^{\frac{1}{q_1}}|G_1|^{\frac{1}{p_2}} |G_2|^{\frac{1}{q_2}}. \end{equation} Last but not least, one can interpolate between the estimates (\ref{II_ns}) and (\ref{ns_fourier_sp_final}) obtained from the sparsity condition to conclude that \begin{align} \label{ns_fourier_fb_final}
|\Lambda_{II}^{\tau_1,\tau_2}| = & \sum_{(l,n,m,k_1,k_2) \in \mathcal{W}} 2^{k_1}\|h\|_{L^s(\mathbb{R}^2)} 2^{k_2} \cdot II \nonumber \\
\lesssim & (C_1^{\tau_1} C_2^{\tau_2}C_3^{\tau_1,\tau_2})^4 \sum_{(l,n,m,k_1,k_2) \in \mathcal{W}} 2^{k_1}\|h\|_{L^s(\mathbb{R}^2)} 2^{k_2(1-\frac{\lambda\gamma}{2})} 2^{-l\lambda(1- \frac{(1+\delta)}{2})}2^{-n(1-\lambda)(\frac{1}{p_1}- \theta_1)}2^{-m(1-\lambda)(\frac{1}{q_1}- (1-\theta_1))} \nonumber \\
& \cdot |F_1|^{(1-\lambda)\frac{1}{p_1}+\lambda\frac{\mu_1(1+\delta)}{2}+\lambda\rho(1-\frac{1+\delta}{2})} |F_2|^{(1-\lambda)\frac{1}{q_1}+\lambda\frac{\mu_2(1+\delta)}{2}+\lambda(1-\rho)(1-\frac{1+\delta}{2})} \nonumber \\
&\cdot |G_1|^{(1-\lambda)\frac{1}{p_2}+\lambda\frac{\nu_1(1+\delta)}{2}+\lambda\rho'(1-\frac{1+\delta}{2})} |G_2|^{(1-\lambda)\frac{1}{q_2}+\lambda\frac{\nu_2(1+\delta)}{2}+\lambda(1-\rho')(1-\frac{1+\delta}{2})}. \end{align}
One has enough degree of freedom to choose the indices and obtain the desired estimate: \begin{enumerate}[(i)] \item for any $0 < \lambda,\delta < 1$, the series $\displaystyle \sum_{l>0}2^{-l\lambda(1- \frac{(1+\delta)}{2})}$ is convergent; \item one notices that for $0 < \theta_1 < 1$, $\displaystyle \sum_{n>0}2^{-n(1-\lambda)(\frac{1}{p_1}- \theta_1)}$ and $\displaystyle \sum_{m>0}2^{-m(1-\lambda)(\frac{1}{q_1}- (1-\theta_1))}$ converge if \begin{align*} &\frac{1}{p_1} - \theta_1>0, \nonumber \\ &\frac{1}{q_1} - (1-\theta_1)>0, \end{align*} which implies that $$ \frac{1}{p_1} + \frac{1}{q_1} > 1. $$ This would be the condition we impose on the exponents $p_1$ and $q_1$. The proof for the range $\frac{1}{p_1} + \frac{1}{q_1} \leq 1$ follows a simpler argument.
\item One can identify (\ref{ns_fourier_fb_final}) with (\ref{exp00}) and choose the indices to match the desired exponents for $|F_1|,|F_2|, |G_1|$ and $|G_2|$ in the exactly same fashion. \end{enumerate} \vskip .15 in \noindent \textbf{Estimate of $\Lambda_{IV}^{\tau_1,\tau_2}$.} When $l_2 \in \mathcal{EXP}_4$, one has the localization that the main contribution of $$
\sum_{|K| \geq |I|}\frac{1}{|K|^{\frac{1}{2}}}\langle f_1, \varphi_K^1 \rangle \langle f_2, \psi_K^2 \rangle \langle \chi_{E'}, \psi_K^3\rangle $$ comes from $$
\sum_{K \supseteq I}\frac{1}{|K|^{\frac{1}{2}}}\langle f_1, \varphi_K^1 \rangle \langle f_2, \psi_K^2 \rangle \langle \chi_{E'}, \psi_K^3\rangle $$ as in the Haar model. As a consequence, it is not difficult to check that the argument in Section \ref{section_thm_haar} applies to the estimate of $IV$, where one employs the local energy estimates stated in Proposition \ref{localized_energy_fourier_x} and \ref{localized_energy_y} instead of Proposition \ref{B_en} and derive that \begin{align} \label{ns_fourier_fb_iv}
IV \lesssim (C_1^{\tau_1} C_2^{\tau_2})^2 2^{-n(\frac{1}{p_1} - \theta_1-\frac{1}{2}\mu(1+\epsilon))}2^{-m(\frac{1}{q_1}- \theta_2-\frac{1}{2}(1-\mu)(1+\epsilon))} |F_1|^{\frac{1}{p_1}-\frac{\mu}{2}\epsilon}|F_2|^{\frac{1}{q_1}-\frac{1-\mu}{2}\epsilon}|G_1|^{\frac{1}{p_2}-\frac{\mu}{2}\epsilon}|G_2|^{\frac{1}{q_2}-\frac{1-\mu}{2}\epsilon}.
\end{align}
By interpolating between (\ref{ns_fourier_fb_iv}) and (\ref{ns_fourier_sp}) which correspond to the estimates for the nested sum using the Fubini argument and the sparsity condition developed in Section \ref{section_thm_haar}, namely (\ref{ns_fb}) and (\ref{ns_sp}), one achieves the desired bound. \begin{remark} When only one of the families $(\phi^3_K)_{K \in \mathcal{K}}$ and $(\phi^3_L)_{L \in \mathcal{L}}$ is lacunary, a simplified argument is sufficient. Without loss of generality, we assume that $(\varphi^3_K)_{K \in \mathcal{K}}$ is a non-lacunary family while $(\psi^3_L)_{L \in \mathcal{L}}$ is a lacunary family. One can then split the argument into two parts depending on the range of the exponents $l_2$: \begin{enumerate}[(i)] \item
$l_2 \in \{l_2 \in \mathbb{Z}: 2^{l_2}\|\tilde{B}(g_1,g_2)\|_1 \lesssim (C_2^{\tau_2}2^{n_2}|G_1|)^{\alpha_2}(C_2^{\tau_2}2^{m_2}|G_2|)^{\beta_2} \}$; \item
$l_2 \in \{l_2 \in \mathbb{Z}: 2^{l_2}\|\tilde{B}(g_1,g_2)\|_1 \gg (C_2^{\tau_2}2^{n_2}|G_1|)^{\alpha_2}(C_2^{\tau_2}2^{m_2}|G_2|)^{\beta_2} \}$; \end{enumerate} where Case (i) can be treated by the same argument for $II$ and Case (ii) by the reasoning for $IV$. This completes the proof of Theorem \ref{thm_weak_mod} for $\Pi_{\text{flag}^0 \otimes \text{flag}^0}$ in the general case. \end{remark}
As commented in the beginning of this section, the argument for Theorem \ref{thm_weak_mod} and \ref{thm_weak_inf_mod} developed in the Haar model can be generalized to the Fourier setting, which ends the proof of the main theorems.
\section{Appendix I - Multilinear Interpolations} This section is devoted to various multilinear interpolations that allow one to reduce Theorem \ref{main_theorem} to \ref{thm_weak} (and Theorem \ref{main_thm_inf} to \ref{thm_weak_inf} correspondingly). We will start from the statement in Theorem \ref{thm_weak} and implement interpolations step by step to reach Theorem \ref{main_theorem}. Throughout this section, we will consider $T_{ab}$ as a trilinear operator with first two function spaces restricted to tensor product spaces. \subsection{Interpolation of Multilinear Forms} One may recall that Theorem \ref{thm_weak} covers all the restricted weak-type estimates except for the case $2 \leq s \leq \infty$. We will apply the interpolation of multilinear forms to fill in the gap. In particular, let $T^*_{ab}$ denote the adjoint operator of $T_{ab}$ such that $$
\langle T_{ab}(f_1 \otimes g_1, f_2 \otimes g_2, h),l\rangle = \langle T^*_{ab}(f_1 \otimes g_1, f_2 \otimes g_2, l), h \rangle $$ Due to the symmetry between $T_{ab}$ and $T^*_{ab}$, one concludes that the multilinear form associated to $T^{*}_{ab}$ satisfies $$
|\Lambda(f_1 \otimes g_1, f_2 \otimes g_2, h, l)| \lesssim |F_1|^{\frac{1}{p_1}} |G_1|^{\frac{1}{p_2}} |F_2|^{\frac{1}{q_1}} |G_2|^{\frac{1}{q_2}} |H|^{\frac{1}{r'}} |L|^{\frac{1}{s}} $$
for every measurable set $F_1, F_2 \subseteq \mathbb{R}_x$, $G_1, G_2 \subseteq \mathbb{R}_y$, $H, L \subseteq \mathbb{R}^2$ of positive and finite measure and every measurable function $|f_i| \leq \chi_{F_i}$, $|g_j| \leq \chi_{G_j}$, $|h| \leq \chi_{H}$ and $|l| \leq \chi_{L}$ for $i, j = 1, 2$. The notation and the range of exponents agree with the ones in Theorem \ref{thm_weak}. One can now apply the interpolation of multilinear forms described in Lemma 9.6 of \cite{cw} to attain the restricted weak-type estimate with $1 < s \leq \infty $: \begin{equation} \label{s=inf}
|\Lambda(f_1 \otimes g_1, f_2 \otimes g_2, h, l)| \lesssim |F_1|^{\frac{1}{p_1}} |G_1|^{\frac{1}{p_2}} |F_2|^{\frac{1}{q_1}} |G_2|^{\frac{1}{q_2}}|H|^{\frac{1}{s}} |L|^{\frac{1}{r'}} \end{equation} where $\frac{1}{s} = 0$ if $s= \infty$. For $1 \leq s < \infty$, one can fix $f_1, g_1, f_2, g_2 $ and apply linear Marcinkiewiecz interpolation theorem to prove the strong-type estimates for $h \in L^s(\mathbb{R}^2)$ with $1 < s < \infty$. The next step would be to validate the same result for $h \in L^{\infty}$. One first rewrites the multilinear form associated to $T_{ab}(f_1 \otimes g_1, f_2 \otimes g_2, h)$ as \begin{align}\label{linear_form_interp} \Lambda(f_1 \otimes g_1, f_2 \otimes g_2, h, \chi_{E'}) := & \langle T_{ab}(f_1 \otimes g_1, f_2 \otimes g_2, h), \chi_{E'}\rangle \nonumber \\ = & \langle T^*_{ab}(f_1 \otimes g_1, f_2 \otimes g_2, \chi_{E'}), h\rangle. \end{align} Let $Q_N := [ -N,N]^2$ denote the cube of length $2N$ centered at the origin in $\mathbb{R}^2$, then (\ref{linear_form_interp}) can be expressed as \begin{align*} & \displaystyle \lim_{N \rightarrow \infty} \int_{Q_N} T^*_{ab}(f_1 \otimes g_1, f_2 \otimes g_2, \chi_{E'})(x) h(x) dx \nonumber \\ = & \lim_{N \rightarrow \infty}\int T^*_{ab}(f_1 \otimes g_1, f_2 \otimes g_2, \chi_{E'})(x) (h\cdot\chi_{Q_N})(x) dx \nonumber \\ = & \lim_{N \rightarrow \infty}\int T_{ab}(f_1 \otimes g_1, f_2 \otimes g_2, h\cdot\chi_{Q_N})(x) \chi_{E'}(x) dx \nonumber \\ = & \lim_{N \rightarrow \infty}\Lambda(f_1 \otimes g_1, f_2 \otimes g_2, h\cdot\chi_{Q_N}, \chi_{E'}). \end{align*}
Let $\tilde{h}:= \frac{h \chi_{Q_N}}{\|h\|_{\infty}}$, where $|\tilde{h}| \leq \chi_{Q_N}$ with $|Q_N| \leq N^2$. One can thus invoke (\ref{s=inf}) to conclude that \begin{align*}
|\Lambda(f_1 \otimes g_1, f_2 \otimes g_2, h\chi_{Q_N}, \chi_{E'})| =& \|h\|_{\infty} \cdot |\Lambda(f_1 \otimes g_1, f_2 \otimes g_2, \tilde{h}, \chi_{E'})| \nonumber \\
\lesssim & |F_1|^{\frac{1}{p_1}}|G_1|^{\frac{1}{p_2}}|F_2|^{\frac{1}{q_1}}|G_2|^{\frac{1}{q_2}}\|h\|_{\infty}|E|^{\frac{1}{r'}}. \end{align*} As the bound for the multilinear form is independent of $N$, passing to the limit when $N \rightarrow \infty$ yields that $$
|\Lambda(f_1 \otimes g_1, f_2 \otimes g_2, h, \chi_{E'})| \lesssim |F_1|^{\frac{1}{p_1}}|G_1|^{\frac{1}{p_2}}|F_2|^{\frac{1}{q_1}}|G_2|^{\frac{1}{q_2}}\|h\|_{\infty}|E|^{\frac{1}{r'}}. $$ Combined with the statement in Theorem \ref{thm_weak}, one has that for any $1 < p_1,p_2, q_1,q_2 < \infty$, $1<s \leq \infty$, $0 < r < \infty$, $\frac{1}{p_1} + \frac{1}{q_1} = \frac{1}{p_2} + \frac{1}{q_2} = \frac{1}{r} - \frac{1}{s}$, \begin{equation} \label{restricted_weak}
\|T_{ab}(f_1 \otimes g_1, f_2 \otimes g_2, h)\|_{r,\infty} \lesssim |F_1|^{\frac{1}{p_1}}|G_1|^{\frac{1}{p_2}}|F_2|^{\frac{1}{q_1}}|G_2|^{\frac{1}{q_2}}\|h\|_{L^s(\mathbb{R}^2)} \end{equation}
for every measurable set $F_1, F_2 \subseteq \mathbb{R}_x$, $G_1, G_2 \subseteq \mathbb{R}_y$ of positive and finite measure and every measurable function $|f_i| \leq \chi_{F_i}$, $|g_j| \leq \chi_{G_j}$ for $i, j = 1, 2$. \subsection{Tensor-type Marcinkiewiecz Interpolation} The next and final step would be to attain strong-type estimates for $T_{ab}$ from (\ref{restricted_weak}). We first fix $h \in L^{s}$ and define $$T^{h}(f_1 \otimes g_1, f_2 \otimes g_2) := T_{ab}(f_1 \otimes g_1, f_2 \otimes g_2,h)$$ One can then apply the following tensor-type Marcinkiewiecz interpolation theorem to each $T^h$ so that Theorem \ref{main_theorem} follows. \begin{theorem}\label{tensor_interpolation} Let $1 < p_1,p_2, q_1, q_2< \infty$ and $ 0 < t < \infty$ such that $\frac{1}{p_1} + \frac{1}{q_1} = \frac{1}{p_2} + \frac{1}{q_2} = \frac{1}{t}$. Suppose a multilinear tensor-type operator $T(f_1 \otimes g_1, f_2 \otimes g_2)$ satisfies the restricted weak-type estimates for any $\tilde{p_1}, \tilde{p_2}, \tilde{q_1}, \tilde{q_2}$ in a neighborhood of $p_1, p_2, q_1, q_2$ respectively with $ \frac{1}{\tilde{p_1}} + \frac{1}{\tilde{q_1}} = \frac{1}{\tilde{p_2}} + \frac{1}{\tilde{q_2}} = \frac{1}{\tilde{t}}$, equivalently $$
\|T(f_1 \otimes g_1, f_2 \otimes g_2) \|_{\tilde{t},\infty} \lesssim |F_1|^{\frac{1}{\tilde{p_1}}} |G_1|^{\frac{1}{\tilde{p_2}}} |F_2|^{\frac{1}{\tilde{q_1}}} |G_2|^{\frac{1}{\tilde{q_2}}} $$
for every measurable set $F_1 \subseteq \mathbb{R}_{x} , F_2 \subseteq \mathbb{R}_{x}, G_1\subseteq \mathbb{R}_y, G_2\subseteq \mathbb{R}_y$ of positive and finite measure and every measurable function $|f_1(x)| \leq \chi_{F_1}(x)$, $|f_2(x)| \leq \chi_{F_2}(x)$, $|g_1(y)| \leq \chi_{G_1}(y)$, $|g_2(y)| \leq \chi_{G_2}(y)$. Then $T$ satisfies the strong-type estimate $$
\|T(f_1 \otimes g_1, f_2 \otimes g_2) \|_{t} \lesssim \|f_1\|_{p_1} \|g_1\|_{p_2} \|f_2\|_{q_1} \|g_2\|_{q_2} $$ for any $f_1 \in L^{p_1}(\mathbb{R}_x)$, $f_2 \in L^{q_1}(\mathbb{R}_x)$, $g_1 \in L^{p_2}(\mathbb{R}_y)$ and $g_2 \in L^{q_2}(\mathbb{R}_y)$. \end{theorem} \begin{remark} The proof of the theorem resembles the argument for the multilinear Marcinkiewiecz interpolation(see \cite{bm}) with small modifications. \end{remark}
\section{Appendix II - Reduction to Model Operators} \subsection{Littlewood-Paley Decomposition} \subsubsection{Set up} Let $\varphi \in \mathcal{S}(\mathbb{R})$ be a Schwartz function with $\text{supp} (\widehat{\varphi}) \subseteq [-2,2]$ and $\widehat{\varphi}(\xi) = 1$ on $[-1,1]$. Let $$ \widehat{\psi}(\xi) = \widehat{\varphi}(\xi) - \widehat{\varphi}(2\xi) $$ so that $\text{supp} \widehat{\psi} \subseteq [-2,-\frac{1}{2}] \cup [-\frac{1}{2}, 2]$. Now for every $k \in \mathbb{Z}$, define $$ \widehat{\psi}_{k}: = \widehat{\psi}(2^{-k}\xi) $$ One important observation is that $$ \sum_{k \in \mathbb{Z}} \widehat{\psi}_k(\xi) = 1 $$ Let $$ \widehat{\varphi}_{k}(\xi) := \sum_{k' \leq k - 10} \widehat{\psi}_{k'}(\xi) $$ \begin{notation} We say that the family $({\psi}_k)_k$ is \textit{$\psi$ type} and the family $({\varphi}_k)_k$ is \textit{$\varphi$ type}. \end{notation}
\subsubsection{Special Symbols} We will first focus on a special case of the symbols and the general case will be studied as an extension afterwards. Suppose that $$ a(\xi_1,\eta_1,\xi_2,\eta_2) = a_1(\xi_1,\xi_2)a_2(\eta_1,\eta_2) $$ $$ b(\xi_1,\eta_1,\xi_2,\eta_2,\xi_3,\eta_3) = b_1(\xi_1,\xi_2,\xi_3) b_2(\eta_1,\eta_2,\eta_3) $$ where \begin{align} a_1(\xi_1,\xi_2) = & \sum_{k_1} \widehat{\phi}^{1}_{k_1}(\xi_1) \widehat{\phi}^{2}_{k_1}(\xi_2) \label{special_symb_a_1} \\ b_1(\xi_1,\xi_2,\xi_3) = & \sum_{k_2} \widehat{\phi}^{1}_{k_2}(\xi_1) \widehat{\phi}^{2}_{k_2}(\xi_2) \widehat{\phi}^{3}_{k_2}(\xi_3) \label{special_symb_b_1} \end{align} At least one of the families $({\phi}^1_{k_1})_{k_1}$ and $({\phi}^2_{k_1})_{k_1}$ is $\psi$ type and at least one of the families $({\phi}^1_{k_2})_{k_2}$, $({\phi}^2_{k_2})_{k_2}$ and $({\phi}^3_{k_2})_{k_2}$ is $\psi$ type. Similarly, $$ a_2(\eta_1,\eta_2) = \sum_{j_1} \widehat{{\phi}}^1_{j_1}(\eta_1) \widehat{{\phi}}^2_{j_1}(\eta_2) $$ $$ b_2(\eta_1,\eta_2,\eta_3) = \sum_{j_2} \widehat{{\phi}}^1_{j_2}(\eta_1) \widehat{{\phi}}^2_{j_2}(\eta_2) \widehat{{\phi}}^3_{j_2}(\eta_3) $$ where at least one of the families $({\phi}^1_{j_1})_{j_1}$ and $({\phi}^2_{j_1})_{j_1}$ is $\psi$ type and at least one of the families $({\phi}^1_{j_2})_{j_2}$, $({\phi}^2_{j_2})_{j_2}$ and $({\phi}^3_{j_2})_{j_2}$ is $\psi$ type.
Then \begin{align} a_1(\xi_1,\xi_2) b_1(\xi_1,\xi_2,\xi_3) = & \sum_{k_1,k_2} \widehat{\phi}^1_{k_1}(\xi_1) \widehat{\phi}^2_{k_1}(\xi_2) \widehat{\phi}^1_{k_2}(\xi_1) \widehat{\phi}^2_{k_2}(\xi_2) \widehat{\phi}^3_{k_2}(\xi_3) \nonumber \\ = & \underbrace{\sum_{k_1 \approx k_2}}_{I^1} + \underbrace{\sum_{k_1 \ll k_2}}_{II^1} + \underbrace{\sum_{k_1 \gg k_2}}_{III^1}, \end{align} where $k_1 \approx k_2$ means $k_2-100 \leq k_1 \leq k_2 + 100$ and $k_1 \ll k_2$ means $k_1 < k_2 -100$.
Case $I^1$ gives rise to the symbol of a paraproduct. More precisely, \begin{equation}\label{I^1_step1} I^1 = \sum_{k} \widehat{\tilde{\phi}^1_k}(\xi_1) \widehat{\tilde{\phi}^2_k}(\xi_2) \widehat{\phi}^3_k(\xi_3), \end{equation} where $\widehat{\tilde{\phi}^1_k}(\xi_1) := \widehat{\phi}^1_{k_1}(\xi_1) \widehat{\phi}^1_{k_2}(\xi_1)$ and $\widehat{\tilde{\phi}^2_k}(\xi_2) := \widehat{\phi}^2_{k_1}(\xi_2) \widehat{\phi}^2_{k_2}(\xi_2)$ when $k := k_1 \approx k_2$. (\ref{I^1_step1}) can be completed as \begin{equation} \label{I^1_completed} I^1 = \sum_{k} \widehat{\tilde{\phi}^1_k}(\xi_1) \widehat{\tilde{\phi}^2_k}(\xi_2) \widehat{\phi}^3_{k}(\xi_3) \widehat{\tilde{\phi}^4_{k}}(\xi_1 + \xi_2 + \xi_3) \end{equation} and at least two of the families $ ({\tilde{\phi}}^1_{k})_{k}$, $({\tilde{\phi}}^2_{k})_{k}$, $ ({\phi}^3_{k})_{k}$, $(\tilde{{\phi}}^4_{k})_{k}$ are $\psi$ type.
Case $II^1$ and $III^1$ can be treated similarly. In Case $II^1$, the sum is non-degenerate when $(\phi_{k_2}^1)_{k_2}$ and $(\phi_{k_2}^2)_{k_2}$ are $\varphi$ type. In particular, one has \begin{equation} \label{II^1_step1} II ^1 = \sum_{k_1 \ll k_2} \widehat{\phi}^1_{k_1}(\xi_1) \widehat{\phi}^2_{k_1}(\xi_2) \widehat{\varphi}^1_{k_2}(\xi_1) \widehat{\varphi}^2_{k_2}(\xi_2) \widehat{\psi}^3_{k_2}(\xi_3), \end{equation} where at least one of the families $(\phi^1_{k_1})_{k_1}$ and $(\phi^2_{k_1})_{k_1}$ is $\psi$ type. In the case when the symbols are assumed to take the special form (namely (\ref{special_symb_a_1}) and (\ref{special_symb_b_1})), (\ref{II^1_step1}) can be rewritten as \begin{equation} \label{simp_special_symbol} \sum_{k_1 \ll k_2} \widehat{\phi}^1_{k_1}(\xi_1) \widehat{\phi}^2_{k_1}(\xi_2) \widehat{\psi}^3_{k_2}(\xi_3), \end{equation} which can be ``completed" as \begin{equation} \label{completion} II^1 = \sum_{k_1 \ll k_2} \widehat{\phi}^1_{k_1}(\xi_1) \widehat{\phi}^2_{k_1}(\xi_2)\widehat{\tilde{{\phi}}^3_{k_1}}(\xi_1+\xi_2) \widehat{\tilde{\varphi}^1_{k_2}}(\xi_1+\xi_2) \widehat{\tilde{\psi}^2_{k_2}}(\xi_3)\widehat{\tilde{\psi}^3_{k_2}}(\xi_1+\xi_2+\xi_3), \end{equation} where ${\tilde{\psi}^2_{k_2}}(\xi_3):= {\psi}^3_{k_2}(\xi_3)$ and at least two of the families $(\phi^1_{k_1})_{k_1}$, $(\phi^2_{k_1})_{k_1}$ and $(\tilde{\phi}^3_{k_1})_{k_1}$ are $\psi$ type.
The exact same argument can be applied to $a_2(\eta_1,\eta_2)b_2(\eta_1,\eta_2,\eta_3)$ so that the symbol can be decomposed as $$ \underbrace{\sum_{j_1 \approx j_2}}_{I^2} + \underbrace{\sum_{j_1 \ll j_2}}_{II^2} + \underbrace{\sum_{j_1 \gg j_2}}_{III^2} $$ where $$ I^2 = \sum_{j} \widehat{\tilde{\phi}^1_j}(\eta_1) \widehat{\tilde{\phi}^2_j}(\xi_2) \widehat{\phi}^3_{j}(\eta_3) \widehat{\tilde{\phi}^4_{j}}(\eta_1+\eta_2+\eta_3) $$ and at least two of the families $({\tilde{\phi}}^1_{j})_{j}$, $( {\tilde{\phi}}^2_{j})_{j}$, $({\phi}^3_{j})_{j}$ and $(\tilde{\phi}^4_{j})_j$ are $\psi$ type. Case $II^2$ and $III^2$ have similar expressions, where $$ II ^2 = \sum_{j_1 \ll j_2} \widehat{\phi}^1_{j_1}(\eta_1) \widehat{\phi}^2_{j_1}(\eta_2)\widehat{\tilde{\phi}^3_{j_1}}(\eta_1+\eta_2) \widehat{\tilde{\varphi}^1_{j_2}}(\eta_1+\eta_2) \widehat{\tilde{\psi}^2_{j_2}}(\eta_3)\widehat{\tilde{\psi}^3_{j_2}}(\eta_1+\eta_2+\eta_3) $$ and at least two of the families $(\phi^1_{j_1})_{j_1}$, $(\phi^2_{j_1})_{j_1}$ and $(\tilde{\phi}^3_{j_1})_{j_1}$ are $ \psi$ type.
One can now combine the decompositions and analysis for $a_1,a_2,b_1$ and $b_2$ to study the original operator: \begin{align*} & T_{ab}(f_1 \otimes g_1,f_2 \otimes g_2, h) \nonumber\\ = & T_{ab}^{I^1I^2} + T_{ab}^{I^1 II^2} + T_{ab}^{I^1 III^2} + T_{ab}^{II^1 I^2} + T_{ab}^{II^1 II^2} + T_{ab}^ {II^1 III^2} + T_{ab}^{III^1 I^1} + T_{ab}^{III^2 II^2} + T_{ab}^{III^1 III^2}, \end{align*} where each operator in the sum is defined to be the operator taking the form of $T_{ab}$ (\ref{bi_flag}) with the replacement of the symbol $a\cdot b$ by the symbol specified in the superscript. Because of the symmetry between frequency variables $(\xi_1,\xi_2,\xi_3)$ and $(\eta_1,\eta_2,\eta_3)$ and the symmetry between cases for frequency scales $k_1 \ll k_2$ and $k_1 \gg k_2$, $j_1 \ll j_2$ and $j_1 \gg j_2$, it suffices to consider the following operators and others can be proved using the same argument.
\begin{enumerate} \item $T_{ab}^{I^1 I^2}$ is a bi-parameter paraproduct; \vskip 0.15in \item $ T_{ab}^{II^1 I^2}$ defined by \begin{align*} & T_{ab}^{II^1 I^2} \nonumber\\ := & \displaystyle \sum_{\substack{k_1 \ll k_2 \\ j \in \mathbb{Z}}} \int \widehat{\phi}^1_{k_1}(\xi_1) \widehat{\phi}^2_{k_1}(\xi_2)\widehat{\tilde{\phi}^3_{k_1}}(\xi_1+\xi_2) \widehat{\tilde{\varphi}^1_{k_2}}(\xi_1+\xi_2) \widehat{\tilde{\psi}^2_{k_2}}(\xi_3)\widehat{\tilde{\psi}^3_{k_2}}(\xi_1+\xi_2+\xi_3) \widehat{\tilde{\phi}^1_j}(\eta_1) \widehat{\tilde{\phi}^2_{j}}(\eta_2) \widehat{\phi}^3_{j}(\eta_3) \widehat{\tilde{\phi}^4_{j}}(\eta_1+\eta_2+\eta_3) \nonumber \\ & \quad \quad \quad \ \ \widehat{f_1}(\xi_1) \widehat{f_2}(\xi_2) \widehat{g_1}(\eta_1) \widehat{g_2}(\eta_2) \widehat{h}(\xi_3,\eta_3) e^{2\pi i x(\xi_1+\xi_2+\xi_3)} e^{2\pi i y(\eta_1+\eta_2+\eta_3)}d\xi_1 d\xi_2 d\xi_3 d\eta_1 d\eta_2 d\eta_3 \nonumber \\ = & \sum_{\substack{k_1 \ll k_2 \\ j \in \mathbb{Z}}}\left(\bigg(\big(( f_1 * \phi^1_{k_1}) (f_2 * \phi^2_{k_1}) * \tilde{\phi}^3_{k_1}\big) * \tilde{\varphi}^1_{k_2} \bigg) ( g_1 * \tilde{\phi}^1_{j}) (g_2 * \tilde{\phi}^2_{j}) (h * \tilde{\psi}_{k_2}^2\otimes \phi^3_{j})\right) * \tilde{\psi}^3_{k_2}\otimes \tilde{\phi}^4_{j}, \nonumber \end{align*} where at least two of the families $(\phi^1_{k_1})_{k_1}$, $(\phi^2_{k_1})_{k_1}$ and $(\tilde{\phi}^3_{k_1})_{k_1}$ are $\psi$ type and at least two of the families $(\tilde{\phi}^1_{j})_{j}$, $(\tilde{\phi}^2_{j})_{j}$, $({\phi}^3_{j})_{j}$ and $(\tilde{\phi}^4_{j})_{j}$ are $\psi$ type. \vskip .15 in \item $T_{ab}^{II^1 II^2}$ defined by \begin{align*} & T_{ab}^{II^1 II^2} \nonumber\\ := & \displaystyle \sum_{\substack{k_1 \ll k_2 \\ j_1 \ll j_2}} \int \widehat{\phi}^1_{k_1}(\xi_1) \widehat{\phi}_{k_1}^2(\xi_2)\widehat{\tilde{\phi}^3_{k_1}}(\xi+\xi_2) \widehat{\tilde{\varphi}^1_{k_2}}(\xi_1+\xi_2) \widehat{\tilde{\psi}^2_{k_2}}(\xi_3)\widehat{\tilde{\psi}^3_{k_2}}(\xi_1+\xi_2+\xi_3) \cdot \nonumber \\ & \quad \quad \quad \ \ \widehat{\phi}^1_{j_1}(\eta_1) \widehat{\phi}^2_{j_1}(\eta_2)\widehat{\tilde{\phi}^3_{j_1}}(\eta_1+\eta_2) \widehat{\tilde{\varphi}^1_{j_2}}(\eta_1+\eta_2) \widehat{\tilde{\psi}^2_{j_2}}(\eta_3)\widehat{\tilde{\psi}^3_{j_2}}(\eta_1+\eta_2+\eta_3) \cdot \nonumber \\ & \quad \quad \quad\ \ \widehat{f_1}(\xi_1) \widehat{f_2}(\xi_2) \widehat{g_1}(\eta_1) \widehat{g_2}(\eta_2) \widehat{h}(\xi_3,\eta_3) \cdot e^{2\pi i x(\xi_1+\xi_2+\xi_3)} e^{2\pi i y(\eta_1+\eta_2+\eta_3)}d\xi_1 d\xi_2 d\xi_3 d\eta_1 d\eta_2 d\eta_3 \nonumber \\ = & \sum_{\substack{k_1 \ll k_2 \\ j_1 \ll j_2}} \left(\bigg(\big(( f_1 * \phi^1_{k_1}) (f_2 * \phi^2_{k_1}) * \tilde{\phi}^3_{k_1}\big) * \tilde{\varphi}^1_{k_2} \bigg) \bigg(\big(( g_1 * \phi^1_{j_1}) (g_2 * \phi^2_{j_1}) * \tilde{\phi}^3_{j_1}\big) * \tilde{\varphi}^1_{j_2} \bigg) \cdot (h * \tilde{\psi}^2_{k_2}\otimes \tilde{\psi}^2_{j_2}) \right)* \tilde{\psi}^3_{k_2}\otimes \tilde{\psi}^3_{j_2},\nonumber
\end{align*} where at least two of the families $(\phi^1_{k_1})_{k_1}$, $(\phi^2_{k_1})_{k_1}$ and $(\tilde{\phi}^3_{k_1})_{k_1}$ are $\psi$ type and at least two of the families $(\phi^1_{j_1})_{j_1}$, $(\phi^2_{j_1})_{j_1}$ and $(\tilde{\phi}^3_{j_1})_{j_1}$ are $\psi$ type. \end{enumerate}
\vskip .15in
\subsubsection{General Symbols} The extension from special symbols to general symbols can be treated as specified in Section 2.13 of \cite{cw}. Due to the fact that generalized bump functions involved for general symbols do not necessarily equal to $1$ on their supports, simple manipulations in the previous section cannot be performed. In particular, for generic symbols \begin{align*} & a_1(\xi_1,\xi_2) = \sum_{k \in \mathbb{Z}}\sum_{n_1,n_2 \in \mathbb{Z}} C^k_{n_1,n_2} \widehat{\phi}_{k,n_1}^1(\xi_1)\widehat{\phi}_{k,n_2}^2(\xi_2) \\ & b(\xi_1, \xi_2, \xi_3) = \sum_{j \in \mathbb{Z}}\sum_{m_1,m_2,m_3 \in \mathbb{Z}} C^j_{m_1,m_2,m_3} \widehat{\phi}_{j,m_1}^1(\xi_1)\widehat{\phi}_{j,m_2}^2(\xi_2) \widehat{\phi}_{j,m_3}^3(\xi_3) \end{align*} where \begin{align} C^k_{n_1,n_2} := & \frac{1}{2^{2k}}\int_{\mathbb{R}^2} a(\xi_1, \xi_2) \widehat{\phi}_k^1(\xi_1) \widehat{\phi}_k^2(\xi_2) e^{-2\pi i n_1 \frac{\xi_1}{2^k}} e^{-2\pi i n_2 \frac{\xi_2}{2^k}} d\xi_1 d\xi_2\nonumber \\
C^j_{m_1,m_2,m_3} := & \frac{1}{2^{3j}}\int_{\mathbb{R}^3} b(\xi_1, \xi_2,\xi_3) \widehat{\phi}_j^1(\xi_1) \widehat{\phi}_j^2(\xi_2) \widehat{\phi}_j^3(\xi_3) e^{-2\pi i m_1 \frac{\xi_1}{2^j}} e^{-2\pi i m_2 \frac{\xi_2}{2^j}} e^{-2\pi i m_3 \frac{\xi_3}{2^j}} d\xi_1 d\xi_2 d\xi_3 \nonumber \\
\widehat{\phi}^{\mathfrak{i}}_{k,n_{\mathfrak{i}}}(\xi_{\mathfrak{i}}) := &e^{2\pi i n_{\mathfrak{i}} \frac{\xi_{\mathfrak{i}}}{2^{k}}} \widehat{\phi}_k^{\mathfrak{i}}(\xi_{\mathfrak{i}}), \mathfrak{i} = 1,2, \\ \widehat{\phi}^{\mathfrak{l}}_{j,m_{\mathfrak{l}}}(\xi_{\mathfrak{l}}) := & e^{2\pi i m_{\mathfrak{l}} \frac{\xi_{\mathfrak{l}}}{2^{j}}} \widehat{\phi}_j^{\mathfrak{l}}(\xi_{\mathfrak{l}}), \mathfrak{l} = 1,2,3.
\end{align} We denote $ \widehat{\phi}^{\mathfrak{i}}_{k,n_{\mathfrak{i}}}$ and $\widehat{\phi}^{\mathfrak{l}}_{j,m_{\mathfrak{l}}}$ by generalized bump functions. As a consequence, one cannot equate the following two terms: \begin{equation*}
\widehat{\phi}_{k,n_1}^1(\xi_1)\widehat{\phi}_{k,n_2}^2(\xi_2)\widehat{\varphi}_{j,m_1}^1(\xi_1)\widehat{\varphi}_{j,m_2}^2(\xi_2) \widehat{\psi}_{j,m_3}^3(\xi_3) \neq \widehat{\phi}_{k,n_1}^1(\xi_1)\widehat{\phi}_{k,n_2}^2(\xi_2)\widehat{\psi}_{j,m_3}^3(\xi_3). \end{equation*} With abuse of notations, we will proceed the discussion as in the previous section with recognition of the presence of generalized bump functions. One notices that $I^1$ generates bi-parameter paraproduct as previously. In Case $II^1$, since $k_1 \ll k_2$, $\widehat{\varphi}_{k_2}(\xi_1)$ and $\widehat{\varphi}_{k_2}(\xi_2)$ behave like $\widehat{\varphi}_{k_2}(\xi_1 + \xi_2)$. One could obtain (\ref{completion}) as a result. To make the argument rigorous, one considers the Taylor expansions $$ \widehat{\varphi}_{k_2}^1(\xi_1) = \widehat{\varphi}^1_{k_2}(\xi_1 + \xi_2) + \sum_{0 < l_1 < M_1} \frac{(\widehat{\varphi}^1_{k_2})^{(l_1)}(\xi_1+ \xi_2)}{{l_1}!}(-\xi_2)^{l_1} + R^1_{M_1}(\xi_1,\xi_2), $$ $$ \widehat{\varphi}^2_{k_2}(\xi_2) = \widehat{\varphi}^2_{k_2}(\xi_1 + \xi_2) + \sum_{0 < l _2 <M_2} \frac{(\widehat{\varphi}^2_{k_2})^{(l_2)}(\xi_1+ \xi_2)}{{l_2}!}(-\xi_1)^{l_2} + R^2_{M_2}(\xi_1,\xi_2), $$
where $({\varphi}_{k_2}^1)_{k_2}$ and $({\varphi}_{k_2}^2)_{k_2}$ are families of $\varphi$ type bump functions. Let \begin{equation*} \widehat{\tilde{\varphi}^1_{k_2}}(\xi_1+\xi_2) := \widehat{\varphi}^1_{k_2}(\xi_1 + \xi_2) \cdot \widehat{\varphi}^2_{k_2}(\xi_1 + \xi_2) \end{equation*} and one can rewrite $II^1$ as \begin{align*} &\underbrace{\sum_{k_1 \ll k_2} \widehat{\phi}^1_{k_1}(\xi_1) \widehat{\phi}_{k_1}^2(\xi_2) \widehat{\tilde{\varphi}^1_{k_2}}(\xi_1 + \xi_2)\widehat{\tilde{\psi}^2_{k_2}}(\xi_3)}_{II^1_0} + \nonumber \\ & \underbrace{\sum_{\substack{0 < l_1+l_2 \leq M}}\sum_{k_1 \ll k_2}\widehat{\phi}^1_{k_1}(\xi_1) \widehat{\phi}^2_{k_1}(\xi_2) \frac{(\widehat{\varphi}_{k_2}^1)^{(l_1)}(\xi_1 + \xi_2)}{{l_1}!} \frac{(\widehat{\varphi}^2_{k_2})^{(l_2)}(\xi_1 + \xi_2)}{{l_2}!} (-\xi_1)^{l_2}(-\xi_2)^{l_1} \widehat{\tilde{\psi}^2_{k_2}}(\xi_3)}_{II^1_1} + \underbrace{R_M(\xi_1,\xi_2)}_{II^1_{\text{rest}}}, \nonumber
\end{align*}
where $M \gg |\alpha_1|$.
One observes that $II^1_0$ can be ``completed" to obtain (\ref{completion}) as desired.
One can simplify $II^1_1$ as \begin{align} \label{II^1_generic} II^1_1 & = \sum_{\substack{0 < l_1 + l_2\leq M}} \sum_{\mu=100}^{\infty} \sum_{k_2 = k_1 + \mu} \widehat{\phi}^1_{k_1}(\xi_1) \widehat{\phi}^2_{k_1}(\xi_2) \frac{(\widehat{\varphi}^1_{k_2})^{(l_1)}(\xi_1 + \xi_2)}{{l_1}!} \frac{(\widehat{\varphi}^2_{k_2})^{(l_2)}(\xi_1 + \xi_2)}{{l_2}!} (-\xi_1)^{l_2}(-\xi_2)^{l_1} \widehat{\tilde{\psi}^2_{k_2}}(\xi_3) . \end{align} Let \begin{align*} \widehat{\tilde{\phi}^1_{k_1}}(\xi_1) :=& \frac{(-\xi_1)^{l_2}}{2^{k_1l_2}}\widehat{\phi}^1_{k_1}(\xi_1), \\ \widehat{\tilde{\phi}^2_{k_1}}(\xi_2) :=& \frac{(-\xi_2)^{l_1}}{2^{k_1l_1}}\widehat{\phi}^2_{k_1}(\xi_2), \\ \widehat{\tilde{\varphi}^1_{k_2,l_1,l_2}}(\xi_1 + \xi_2) := & 2^{k_2(l_1+l_2)} \frac{(\widehat{\varphi}^1_{k_2})^{(l_1)}(\xi_1 + \xi_2)}{{l_1}!} \frac{(\widehat{\varphi}^2_{k_2})^{(l_2)}(\xi_1 + \xi_2)}{{l_2}!} \end{align*} where ${\tilde{\varphi}}^1_{k_2,l_1,l_2} $ represents an $L^{\infty}$-normalized $\varphi$ type bump function with Fourier support at scale $2^{k_2}$. Then (\ref{II^1_generic}) can be rewritten as \begin{align*}
& \sum_{\substack{0 < l_1+l_2 \leq M }} 2^{(k_1-k_2)(l_1+l_2)} \widehat{\tilde{\phi}^1_{k_1}}(\xi_1) \widehat{\tilde{\phi}^2_{k_1}}(\xi_2) \widehat{\tilde{\varphi}^1_{k_2,l_1,l_2}}(\xi_1 + \xi_2) \widehat{\tilde{\psi}^2_{k_2}}(\xi_3) \nonumber \\ & = \sum_{\substack{0 < l_1+l_2 \leq M }} \sum_{\mu=100}^{\infty} 2^{-\mu(l_1+\l_2)}\underbrace{\sum_{k_2 = k_1 + \mu} \widehat{\tilde{\phi}^1_{k_1}}(\xi_1) \widehat{\tilde{\phi}^2_{k_1}}(\xi_2)\widehat{\tilde{\varphi}^1_{k_2,l_1,l_2}}(\xi_1 + \xi_2) \widehat{\tilde{\psi}^2_{k_2}}(\xi_3)}_{II_{1,\mu}^{1}}, \nonumber
\end{align*}
One notices that $II_{1,\mu}^{1}$ has a form similar to (\ref{completion}) and can be rewritten as $$ \sum_{k_2 = k_1 + \mu} \widehat{\tilde{\phi}^1_{k_1}}(\xi_1) \widehat{\tilde{\phi}^2_{k_1}}(\xi_2)\widehat{\tilde{\phi}^3_{k_1}}(\xi_1+\xi_2) \widehat{\tilde{\varphi}^1_{k_2,l_1,l_2}}(\xi_1+\xi_2) \widehat{\tilde{\psi}^2_{k_2}}(\xi_3)\widehat{\tilde{\psi}^3_{k_2}}(\xi_1+\xi_2+\xi_3). $$ Meanwhile, one can decompose $II^1_{\text{rest}}$ such that \begin{align*} R_M(\xi_1,\xi_2)= &\sum_{\mu = 100}^{\infty}2^{-\mu M} \cdot II^1_{\text{rest},\mu}
\end{align*} where $II^1_{\text{rest},\mu}$ is a Coifman-Meyer symbol satisfying $$
\left|\partial^{\alpha_1} II^1_{\text{rest},\mu}(\xi_1,\xi_2,\xi_3)\right| \lesssim 2^{\mu |\alpha_1|}\frac{1}{|(\xi_1,\xi_2,\xi_3)|^{|\alpha_1|}} $$ for sufficiently many multi-indices $\alpha_1$.
Same procedure can be applied to study $a_2(\eta_1,\eta_2)b_2(\eta_1 \eta_2,\eta_3)$. One can now combine all the arguments above to decompose and study $$ T_{ab} = T_{ab}^{I^1I^2} + T_{ab}^{I^1 II^2} + T_{ab}^{I^1 III^2} + T_{ab}^{II^1 I^2} + T_{ab}^{II^1 II^2} + T_{ab}^ {II^1 III^2} + T_{ab}^{III^1 I^1} + T_{ab}^{III^2 II^2} + T_{ab}^{III^1 III^2} $$ where each operator takes the form $$ \displaystyle \int_{\mathbb{R}^6} \text{symbol} \cdot \widehat{f_1}(\xi_1) \widehat{f_2}(\xi_2) \widehat{g_1}(\eta_1) \widehat{g_2}(\eta_2) \widehat{h}(\xi_3,\eta_3)e^{2\pi i x(\xi_1+\xi_2+\xi_3)} e^{2\pi i y(\eta_1+\eta_2+\eta_3)}d\xi_1 d\xi_2 d\xi_3 d\eta_1 d\eta_2 d\eta_3 $$ with the symbol for each operator specified as follows. \begin{enumerate} \item $T_{ab}^{I^1I^2}$ is a bi-parameter paraproduct as in the special case. \vskip .15in \item $T_{ab}^{II^1 I^2}$: $(II^{1}_0 + II^{1}_{1} + II^{1}_{\text{rest}}) \otimes I^2$ \newline where the operator associated with each symbol can be written as \begin{enumerate}[(i)] \item $$ T^{II_0^1 I^2} := \sum_{\substack{k_1 \ll k_2 \\ j \in \mathbb{Z}}}\left(\bigg(\big(( f_1 * \phi^1_{k_1}) (f_2 * \phi^2_{k_1}) * \tilde{\phi}^3_{k_1}\big) * \tilde{\varphi}^1_{k_2} \bigg) ( g_1 * \tilde{\phi}^1_{j}) (g_2 * \tilde{\phi}^2_{j}) (h * \tilde{\psi}^2_{k_2}\otimes \phi^3_{j})\right) * \tilde{\psi}^3_{k_2}\otimes \tilde{\phi}^4_{j} $$ \item $$ T^{II^1_1 I^2} := \sum_{\substack{0 < l_1+l_2 \leq M }}\sum_{\mu= 100}^{\infty} 2^{-\mu(l_1+\l_2)} T^{II^1_{1,\mu}I^2} $$ with \begin{align*} & T^{II^1_{1,\mu}I^2}\\ :=& \sum_{\substack{k_2 = k_1 + \mu \\ j \in \mathbb{Z}}}\left(\bigg(\big(( f_1 * \tilde{\phi}^1_{k_1}) (f_2 * \tilde{\phi}^2_{k_1}) * \tilde{\phi}^3_{k_1}\big) * \tilde{\varphi}^1_{k_2,l_1,l_2} \bigg)( g_1 * \tilde{\phi}^1_{j}) (g_2 * \tilde{\phi}^2_{j}) (h * \tilde{\psi}^2_{k_2}\otimes \phi^3_{j}) \right)* \tilde{\psi}^3_{k_2}\otimes \tilde{\phi}^4_{j} \end{align*} \item $$ T^{II^1_{\text{rest}}I^2} := \sum_{\mu= 100}^{\infty}2^{\mu M} T^{II^1_{\text{rest},\mu}I^2} $$
One notices that $II^1_{\text{rest},\mu}$ and $I^2$ are Coifman-Meyer symbols. $T^{II^1_{\text{rest},\mu}I^2}$ is therefore a bi-parameter paraproduct and one can apply the argument in \cite{cptt} to derive the bound of type $O(2^{|\alpha_1|\mu})$, which would suffice due to the decay factor $2^{-\mu M}$. \end{enumerate}
\begin{comment} \begin{align*} T_{ab}^{II^1 I^2} = & \displaystyle \sum_{\substack{k_1 \ll k_2 \\ j \in \mathbb{Z}}} \int \widehat{\phi}_{k_1}(\xi_1) \widehat{\phi}_{k_1}(\xi_2)\widehat{\phi}_{k_1}(\xi+\xi_2) \widehat{\varphi}_{k_2}(\xi_1+\xi_2) \widehat{\psi}_{k_2}(\xi_3)\widehat{\psi}_{k_2}(\xi_1+\xi_2+\xi_3) \nonumber \\ & \quad \quad \quad \widehat{\tilde{\phi}}_{j}(\eta_1) \widehat{\tilde{\phi}}_{j}(\eta_2) \widehat{\phi}_{j}(\eta_3) \widehat{\phi}_{j}(\eta_1+\eta_2+\eta_3) \widehat{f_1}(\xi_1) \widehat{f_2}(\xi_2) \widehat{g_1}(\eta_1) \widehat{g_2}(\eta_2) \widehat{h}(\xi_3,\eta_3)\nonumber \\ & \quad \quad \quad \cdot e^{2\pi i x(\xi_1+\xi_2+\xi_3)} e^{2\pi i y(\eta_1+\eta_2+\eta_3)}d\xi_1 d\xi_2 d\xi_3 d\eta_1 d\eta_2 d\eta_3 \nonumber \\ &+ \sum_{\substack{0 < l_1 \leq M \\ 0 < l_2 \leq N}}\sum_{\mu=100}^{\infty} 2^{-\mu(l_1+\l_2)} \sum_{\substack{k_2 = k_1 + \mu \\ j \in \mathbb{Z}}} \int \widehat{\phi}_{k_1}(\xi_1) \widehat{\phi}_{k_1}(\xi_2)\widehat{\phi}_{k_1}(\xi+\xi_2) \widehat{\tilde{\varphi}}_{k_2,l_1,l_2}(\xi_1+\xi_2) \widehat{\psi}_{k_2}(\xi_3)\widehat{\psi}_{k_2}(\xi_1+\xi_2+\xi_3) \nonumber \\ & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \widehat{\tilde{\phi}}_{j}(\eta_1) \widehat{\tilde{\phi}}_{j}(\eta_2) \widehat{\phi}_{j}(\eta_3) \widehat{\phi}_{j}(\eta_1+\eta_2+\eta_3) \widehat{f_1}(\xi_1) \widehat{f_2}(\xi_2) \widehat{g_1}(\eta_1) \widehat{g_2}(\eta_2) \widehat{h}(\xi_3,\eta_3)\nonumber \\ &\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \cdot e^{2\pi i x(\xi_1+\xi_2+\xi_3)} e^{2\pi i y(\eta_1+\eta_2+\eta_3)}d\xi_1 d\xi_2 d\xi_3 d\eta_1 d\eta_2 d\eta_3 + \nonumber \\ & + \sum_{\mu=100}^{\infty} 2^{-\mu(M+N)} \int m^{1}_{\mu} (\xi_1,\xi_2,\xi_3) \sum_{j \in \mathbb{Z}}\widehat{\tilde{\phi}}_{j}(\eta_1) \widehat{\tilde{\phi}}_{j}(\eta_2) \widehat{\phi}_{j}(\eta_3) \widehat{\phi}_{j}(\eta_1+\eta_2+\eta_3) \nonumber \\ & \quad \quad \quad \quad \quad \quad \quad \quad \quad \widehat{f_1}(\xi_1) \widehat{f_2}(\xi_2) \widehat{g_1}(\eta_1) \widehat{g_2}(\eta_2) \widehat{h}(\xi_3,\eta_3) e^{2\pi i x(\xi_1+\xi_2+\xi_3)} e^{2\pi i y(\eta_1+\eta_2+\eta_3)}d\xi_1 d\xi_2 d\xi_3 d\eta_1 d\eta_2 d\eta_3 \nonumber \\ = & \underbrace{\sum_{\substack{k_1 \ll k_2 \\ j \in \mathbb{Z}}}\bigg(\big(( f_1 * \phi_{k_1}) (f_2 * \phi_{k_1}) * \phi_{k_1}\big) * \varphi_{k_2} \bigg) ( g_1 * \tilde{\phi}_{j}) (g_2 * \tilde{\phi}_{j}) (h * \psi_{k_2}\otimes \phi_{j}) * \psi_{k_2}\otimes \phi_{j} }_{T_{ab}^{II_0^1 I^2}} \nonumber \\ & + \sum_{\substack{0 < l_1 \leq M \\ 0 < l_2 \leq N}}\sum_{\mu= 100}^{\infty} 2^{-\mu(l_1+\l_2)} \underbrace{\sum_{\substack{k_2 = k_1 + \mu \\ j \in \mathbb{Z}}}\bigg(\big(( f_1 * \phi_{k_1}) (f_2 * \phi_{k_1}) * \phi_{k_1}\big) * \tilde{\varphi}_{k_2,l_1,l_2} \bigg)}_{} \nonumber \\ &\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \ \ \underbrace{( g_1 * \tilde{\phi}_{j}) (g_2 * \tilde{\phi}_{j}) (h * \psi_{k_2}\otimes \phi_{j}) * \psi_{k_2}\otimes \phi_{j}}_{T_{ab}^{II^1_1 I^2}} \nonumber \\ & + \sum_{\mu=100}^{\infty} 2^{-\mu(M+N)} \int m^{1}_{\mu} (\xi_1,\xi_2,\xi_3) \underbrace{\sum_{j \in \mathbb{Z}}\widehat{\tilde{\phi}}_{j}(\eta_1) \widehat{\tilde{\phi}}_{j}(\eta_2) \widehat{\phi}_{j}(\eta_3) \widehat{\phi}_{j}(\eta_1+\eta_2+\eta_3)}_{m^2} \nonumber \\ & \quad \quad \quad \quad \quad \quad \quad \quad \quad \widehat{f_1}(\xi_1) \widehat{f_2}(\xi_2) \widehat{g_1}(\eta_1) \widehat{g_2}(\eta_2) \widehat{h}(\xi_3,\eta_3) e^{2\pi i x(\xi_1+\xi_2+\xi_3)} e^{2\pi i y(\eta_1+\eta_2+\eta_3)}d\xi_1 d\xi_2 d\xi_3 d\eta_1 d\eta_2 d\eta_3. \nonumber \\ \end{align*} \end{comment}
\item $T^{II^1 II^2}$: $(II^1_0 + II^1_1 + II^1_{\text{rest}}) \otimes (II^2_0 + II^2_1 + II^2_{\text{rest}})$ \newline where the operator associated with each symbol can be written as \begin{enumerate}[(i)] \item \begin{align*} & T^{II_0^1 II_0^2} \\ :=& \sum_{\substack{k_1 \ll k_2 \\ j_1 \ll j_2}}\left(\bigg(\big(( f_1 * \phi^1_{k_1}) (f_2 * \phi^2_{k_1}) * \tilde{\phi}^3_{k_1}\big) * \tilde{\varphi}^1_{k_2} \bigg) \bigg(\big(( g_1 * \phi^1_{j_1}) (g_2 * \phi^2_{j_1}) * \tilde{\phi}^3_{j_1}\big) * \tilde{\varphi}^1_{j_2} \bigg) (h * \tilde{\psi}^2_{k_2}\otimes \tilde{\psi}^2_{j_2}) \right)* \tilde{\psi}^3_{k_2}\otimes \tilde{\psi}^3_{j_2} \end{align*}
\item $$ T^{II^1_1 II_0^2} := \sum_{\substack{0 < l_1+l_2 \leq M }}\sum_{\mu= 100}^{\infty} 2^{-\mu(l_1+\l_2)} T^{II^1_{1,\mu}II^2_0} $$ with \begin{align*} & T^{II^1_{1,\mu}II^2_0} \\ :=& \sum_{\substack{k_2 = k_1 + \mu \\ j_1 \ll j_2}}\left(\bigg(\big(( f_1 * \tilde{\phi}^1_{k_1}) (f_2 * \tilde{\phi}^2_{k_1}) * \tilde{\phi}^3_{k_1}\big) * \tilde{\varphi}^1_{k_2,l_1,l_2} \bigg)\bigg(\big(( g_1 * \phi^1_{j_1}) (g_2 * \phi^2_{j_1}) *\tilde{\phi}^3_{j_1}\big) * \tilde{\varphi}^1_{j_2} \bigg) (h * \tilde{\psi}^2_{k_2}\otimes \tilde{\psi}^2_{j_2}) \right)* \tilde{\psi}^3_{k_2}\otimes \tilde{\psi}^3_{j_2} \end{align*}
\item $$ T^{II^1_{\text{rest}}II^2_0}:= \sum_{\mu= 100}^{\infty}2^{\mu M} T^{II^1_{\text{rest},\mu}II^2_0} $$ where $T^{II^1_{\text{rest},\mu}II^2_0}$ is a multiplier operator with the symbol $$ II^1_{\text{rest},\mu}\otimes II^2_0 $$ which generates a model similar as $T^{I^1 II^2_0}$ or, by symmetry, $T^{II^1_0 I^2}$. \item $$ T^{II^1_1 II^2_1}:= \sum_{\substack{0 < l_1+l_2 \leq M \\ l_1' + l_2' \leq M'}}\sum_{\mu,\mu'= 100}^{\infty} 2^{-\mu(l_1+\l_2)} 2^{\mu'(l_1'+l_2')}T^{II^1_{1,\mu}II^2_{1,\mu'}} $$ with \begin{align*} & T^{II^1_{1,\mu}II^2_{1,\mu'}} \\ :=& \sum_{\substack{k_2 = k_1 + \mu \\ j_2 = j_1 + \mu'}}\left(\bigg(\big(( f_1 * \tilde{\phi}^1_{k_1}) (f_2 * \tilde{\phi}^2_{k_1}) * \tilde{\phi}^3_{k_1}\big) * \tilde{\varphi}^1_{k_2,l_1,l_2} \bigg)\bigg(\big(( g_1 * \tilde{\phi}^1_{j_1}) (g_2 * \tilde{\phi}^2_{j_1}) * \tilde{\phi}^3_{j_1}\big) * \tilde{\varphi}^1_{j_2,l_1',l_2'} \bigg) (h *\tilde{\psi}^2_{k_2}\otimes \tilde{\psi}^2_{j_2}) \right)* \tilde{\psi}^3_{k_2}\otimes \tilde{\psi}^3_{j_2} \end{align*}
\item $$ T^{II^1_{\text{rest}} II^2_1} := \sum_{\mu= 100}^{\infty}2^{\mu M} T^{II^1_{\text{rest},\mu}II^2_1} $$ where $T^{II^1_{\text{rest},\mu}II^2_1}$ has the symbol $$ II^1_{\text{rest},\mu}\otimes II^2_1 $$ which generates a model similar as $T^{I^1 II^2_1}$ or $T^{II^1_1 I^2}$.
\item $$ T^{II^1_{\text{rest}}II^2_{\text{rest}}} := \sum_{\mu,\mu'= 100}^{\infty}2^{\mu M}2^{\mu'M'} T^{II^1_{\text{rest},\mu}II^2_{\text{rest},\mu'}} $$ where $ T^{II^1_{\text{rest},\mu}II^2_{\text{rest},\mu'}}$ is associated with the symbol $$ II^1_{\text{rest},\mu}\otimes II^2_{\text{rest},\mu'} $$ which generates a model similar as $T^{II^1_{\text{rest},\mu}I^2}$, $T^{I^1 II^2_{\text{rest},\mu'}}$ or $T^{I^1I^2}$.
\end{enumerate}
\item $T^{III^1 II^2}$, $T^{III^1 I^2}$ and $T^{III^1 III^2}$ can be studied by the exact same reasoning for $T^{II^1II^2}$, $T^{II^1 I^2}$ and $T^{II^1 II^2}$ by the symmetry between symbols $II$ and $III$. \end{enumerate}
\vskip .15in \subsection{Discretization} With discretization procedure specified in Section 2.2 of \cite{cw}, one can reduce the above operators into the following discrete model operators listed in Theorem (\ref{thm_weak}): \begin{center} \begin{tabular}{ c c c } $T^{II^1_0 I^2}$ & $ \longrightarrow$ & $\Pi_{\text{flag}^0 \otimes \text{paraproduct}}$ \\ $T^{II^1_{1,\mu} I^2}$ & $ \longrightarrow$ & $ \Pi_{\text{flag}^{\mu} \otimes \text{paraproduct}}$ \\ $T^{II^1_0 II^2_0}$ & $ \longrightarrow$ & $ \Pi_{\text{flag}^0 \otimes \text{flag}^0} $ \\ $T^{II^1_0 II^2_{1,\mu'}}$ & $ \longrightarrow$ & $\Pi_{\text{flag}^0 \otimes \text{flag}^{\mu'}} $ \\ $T^{II^1_{1,\mu} II^2_{1,\mu'}}$ & $ \longrightarrow$ & $\Pi_{\text{flag}^{\mu} \otimes \text{flag}^{\mu'}} $ \\ \end{tabular} \end{center} \begin{comment} ``localize" not only frequency variables but also spatial variables To be more precise, we cannot exactly ``localize'' spatial variables because of the uncertainty principle, and instead we obtain decay away from corresponding dyadic intervals. Littlewood-Paley + discretization \end{comment}
\begin{comment} \noindent
\textbf{Estimate for $\|B^{\tau_3,\tau_4}_{-n-n_2,-m-m_2}\|_1$.} Similarly, one can find $\eta \in L^{\infty}$ with $\|\eta\|_{\infty} = 1$ such that \begin{align*}
|\langle B^{\tau_3,\tau_4}_{-n-n_2,-m-m_2}, \eta \rangle| = & \bigg|\int \sum_{\substack{K: \frac{\text{dist}(K,\Omega_{-n-n_2+1,-m-m_2+1}^{c})}{|K|} \lesssim 2^{\tau_3 + \tau_4}}} \frac{1}{|K|^{\frac{1}{2}}} \langle f_1, \varphi^1_K \rangle \langle f_2, \psi^2_K \rangle \psi^{3,\tau_4}_K(x)\eta(x) dx \bigg|
\leq & \sum_{\substack{K: \frac{\text{dist}(K,\Omega_{-n-n_2+1,-m-m_2+1}^{c})}{|K|} \lesssim 2^{\tau_3 + \tau_4}}} \frac{1}{|K|^{\frac{1}{2}}} |\langle f_1, \varphi^1_K \rangle| |\langle f_2, \psi^2_K \rangle| |\langle \eta, \psi^{3,\tau_4}_K \rangle| \nonumber \\ \end{align*}
Now the following size-energy estimates can be obtained by a similar argument as in Case II:
$$\text{size}\big((\langle f_1, \varphi^1_K \rangle)_{K: \frac{\text{dist}(K,\Omega_{-n-n_2+1,-m-m_2+1}^{c})}{|K|}} \lesssim 2^{\tau_3 + \tau_4}\big) \lesssim \min(1,C_1 2^{\tau_3 + \tau_4}2^{-n-n_2}|F_1|) \leq (C_1 2^{\tau_3 + \tau_4}2^{-n-n_2}|F_1|)^{\mu_1} $$
$$\text{size}\big((\langle f_2, \psi^2_K \rangle)_{K: \frac{\text{dist}(K,\Omega_{-n-n_2+1,-m-m_2+1}^{c})}{|K|}} \lesssim 2^{\tau_3 + \tau_4}\big) \lesssim \min(1,C_1 2^{\tau_3 + \tau_4}2^{-m-m_2}|F_2|) \leq (C_1 2^{\tau_3 + \tau_4}2^{-m-m_2}|F_2|)^{\mu_2}$$ Also, using the fact that $X_S$ is a characteristic function, one has $$\text{size}\big((\langle \chi_S, \psi^3_K\rangle)_{K}) \leq 1$$ Moreover,
$$\text{energy}\big((\langle f_1, \varphi^1_K \rangle)_{K}\big) \lesssim |F_1|$$
$$\text{energy}\big((\langle f_2, \psi^2_K \rangle)_{K}\big) \lesssim |F_2|$$
$$\text{energy}\big((\langle \chi_S, \psi^3_K\rangle)_{K}) \lesssim |S|$$ Plugging in all the size-energy estimates ({\color{blue} which are essentially the same for the other model}), \begin{align*}
& \sum_{\substack{K: \frac{\text{dist}(K,\Omega_{-n-n_2+1,-m-m_2+1}^{c})}{|K|}} \lesssim 2^{\tau_3 + \tau_4}} \frac{1}{|K|^{\frac{1}{2}}} |\langle f_1, \varphi^1_K \rangle| |\langle f_2, \psi^2_K \rangle| |\langle \eta, \psi^3_K\rangle| \nonumber \\
\lesssim & (C_1 2^{\tau_3 + \tau_4}2^{-n-n_2}|F_1|)^{\theta_2} (C_1 2^{\tau_3 + \tau_4}2^{-m-m_2}|F_2|)^{1-\theta_2} \cdot 1 \cdot |F_1|^{1-\theta_2}|F_2|^{\theta_2} \nonumber \\
= & C_1 2^{\tau_3+\tau_4}(2^{-n-n_2})^{\theta_2}(2^{-m-m_2})^{1-\theta_2} |F_1||F_2| \nonumber \\ \end{align*}
Then for $0 < \alpha < 1$ \begin{align*}
\mathcal{E}_{\alpha} \lesssim & \sum_{\tau_3,\tau_4 \in \mathbb{N}} 2^{-80\tau_3} 2^{80 \tau_4}\|B^{\tau_3,\tau_4}_{-n-n_2,-m-m_2}\|_{\frac{1}{\alpha}} \nonumber \\
\lesssim & \sum_{\tau_3,\tau_4 \in \mathbb{N}} 2^{-80\tau_3} 2^{80 \tau_4} C_1^{2-\alpha} 2^{(\tau_3+\tau_4)(2-\alpha)}(2^{-n-n_2})^{1-\alpha+\theta_2}(2^{-m-m_2})^{1-\theta_2} |F_1||F_2|\nonumber \\
\lesssim & C_1^{2-\alpha} (2^{-n-n_2})^{1-\alpha+\theta_2}(2^{-m-m_2})^{1-\theta_2} |F_1||F_2|, \end{align*} which is exactly the same estimate for the corresponding energy term in Case I. \newline For $\alpha = 1$, \begin{align*}
\mathcal{E}_1 \lesssim & \sum_{\tau_3,\tau_4 \in \mathbb{N}} 2^{-80\tau_3} 2^{80 \tau_4}\|B^{\tau_3,\tau_4}_{-n-n_2,-m-m_2}\|_1 \nonumber \\
\lesssim & \sum_{\tau_3,\tau_4 \in \mathbb{N}} 2^{-80\tau_3} 2^{80 \tau_4} C_1 2^{\tau_3+\tau_4}(2^{-n-n_2})^{\theta_2}(2^{-m-m_2})^{1-\theta_2} |F_1||F_2|\nonumber \\
\lesssim & C_1(2^{-n-n_2})^{\theta_2}(2^{-m-m_2})^{1-\theta_2} |F_1||F_2|, \end{align*} which is exactly the same estimate for the corresponding energy term in Case II, III and IV. \newline \end{comment}
\begin{comment} \section{Appendix- Reduction to discrete models} $$ \sum_{k \ll l}\hat{\varphi}_{k}(\xi_1)\hat{\psi}_{k}(\xi_2)\hat{\phi}_{l}(\xi_2) \hat{\psi}_{l}(\xi_3) = \sum_{k \ll l}\hat{\varphi}_{k}(\xi_1)\hat{\psi}_{k}(\xi_2) \hat{\psi}_{l}(\xi_3) $$
The equation holds as for $k \ll l$, $\hat{\phi}_{I}(\xi_2) = 1$ on $\text{supp}(\hat{\psi}_k)$ and $\hat{\psi}_{k} = 0$ on $|\xi_2| > 2^{-k+3}$. Then one can perform a``completion" and use the fact that $k \ll l$ to obtain $$ \sum_{k \ll l}\hat{\varphi}_{k}(\xi_1)\hat{\psi}_{k}(\xi_2) \hat{\psi}_{l}(\xi_3) = \sum_{k \ll l}\hat{\varphi}_{k}(\xi_1)\hat{\psi}_{k}(\xi_2)\hat{\psi}_{k}(\xi_1+\xi_2) \hat{\varphi}_{l}(\xi_2+\xi_3)\hat{\psi}_{l}(\xi_3) \hat{\psi}_{l}(\xi_1+\xi_2+\xi_3) $$ \end{comment}
\end{document} | arXiv |
Robin Gandy
Robin Oliver Gandy (22 September 1919 – 20 November 1995) was a British mathematician and logician.[4] He was a friend, student, and associate of Alan Turing, having been supervised by Turing during his PhD at the University of Cambridge,[1] where they worked together.[5][6][7]
Robin Gandy
Born
Robin Oliver Gandy
(1919-09-22)22 September 1919
Rotherfield Peppard, Oxfordshire, England
Died20 November 1995(1995-11-20) (aged 76)
Oxford, England
NationalityBritish
EducationAbbotsholme School
Alma materUniversity of Cambridge (PhD)
Known forRecursion theory
Scientific career
FieldsMathematical logic
Institutions
• University of Manchester
• University of Leicester
• University of Leeds
• Stanford University
• University of California, Los Angeles
• University of Oxford
ThesisOn Axiomatic Systems in Mathematics and Theories in Physics (1953)
Doctoral advisorAlan Turing[1][2]
Doctoral students
• Martin Hyland[3]
• Jane Kister[2]
• Jeff Paris[2]
Education and early life
Robin Gandy was born in the village of Rotherfield Peppard, Oxfordshire, England.[4] He was the son of Thomas Hall Gandy (1876–1948), a general practitioner, and Ida Caroline née Hony (1885–1977), a social worker and later an author.[8] He was a great-great-grandson of the architect and artist Joseph Gandy (1771–1843).
Educated at Abbotsholme School in Derbyshire, Gandy took two years of the Mathematical Tripos, at King's College, Cambridge, before enlisting for military service in 1940. During World War II he worked on radio intercept equipment at Hanslope Park, where Alan Turing was working on a speech encipherment project, and he became one of Turing's lifelong friends and associates. In 1946, he completed Part III of the Mathematical Tripos, then began studying for a PhD under Turing's supervision. He completed his thesis, On axiomatic systems in mathematics and theories in Physics, in 1952.[1] He was a member of the Cambridge Apostles.[9]
Career and research
Gandy held positions at the University of Leicester, the University of Leeds, and the University of Manchester. He was a visiting associate professor at Stanford University from 1966 to 1967, and held a similar position at University of California, Los Angeles in 1968. In 1969, he moved to Wolfson College, Oxford, where he became Reader in Mathematical Logic.
Gandy is known for his work in recursion theory. His contributions include the Spector–Gandy theorem, the Gandy Stage Comparison theorem, and the Gandy Selection theorem. He also made a significant contribution to the understanding of the Church–Turing thesis, and his generalisation of the Turing machine is called a Gandy machine.[10]
Gandy died in Oxford, England on 20 November 1995.[4][11]
Legacy
The Robin Gandy Buildings, a pair of accommodation blocks at Wolfson College, Oxford, are named after Gandy.[12][13] A one-day centenary Gandy Colloquium was held on 22 February 2020 at the College in Gandy's honour, including contributions by some of his students;[14][15] the speakers were Marianna Antonutti Marfori (Munich), Andrew Hodges (Oxford), Martin Hyland (Cambridge), Jeff Paris (Manchester), Göran Sundholm (Leiden), Christine Tasson (Paris), and Philip Welch (Bristol).
References
1. Gandy, Robin Oliver (1953). On axiomatic systems in mathematics and theories in physics. repository.cam.ac.uk (PhD thesis). University of Cambridge. doi:10.17863/CAM.16125. EThOS uk.bl.ethos.590164.
2. Robin Gandy at the Mathematics Genealogy Project
3. Hyland, John Martin Elliott (1975). Recursion Theory on the Countable Functionals. bodleian.ox.ac.uk (DPhil thesis). University of Oxford. EThOS uk.bl.ethos.460247.
4. Yates, Mike (24 November 1995). "Obituary: Robin Gandy". The Independent. Retrieved 1 January 2012.
5. Hodges, Andrew (1983). Alan Turing: The Enigma. Simon & Schuster. ISBN 0-671-49207-1.
6. "Notices". The Bulletin of Symbolic Logic. 2 (1): 121–125. March 1996. doi:10.1017/s1079898600007988. JSTOR 421052. S2CID 246638427.
7. Moschovakis, Yannis & Yates, Mike (September 1996). "In Memoriam: Robin Oliver Gandy, 1919–1995". The Bulletin of Symbolic Logic. 2 (3): 367–370. doi:10.1017/s1079898600007873. JSTOR 420996. S2CID 120785678.
8. "Ida Gandy - Writer". Aldbourne Heritage Centre. Retrieved 7 April 2021.
9. "Wolfson College salutes Robin Gandy on his centenary | Wolfson College, Oxford".
10. Wilfried Sieg, 2005, Church without dogma: axioms for computability, Carnegie Mellon University
11. Robin Gandy — The Alan Turing Scrapbook, archived at Archive.Today
12. "Accommodation types – Robin Gandy Buildings". UK: Wolfson College, Oxford. Retrieved 14 April 2020.
13. "Robin Gandy Buildings, Wolfson". Flickr. 6 April 2008. Retrieved 1 January 2012.
14. "The Gandy Colloquium". UK: Wolfson College, Oxford. 22 February 2020. Retrieved 14 April 2020.
15. Isaacson, Daniel (2020). "Wolfson College salutes Robin Gandy on his centenary". UK: Wolfson College, Oxford. Retrieved 14 April 2020.
Authority control
International
• ISNI
• VIAF
National
• Norway
• Catalonia
• Germany
• Israel
• United States
• Czech Republic
• Netherlands
Academics
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• MathSciNet
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• zbMATH
Other
• IdRef
| Wikipedia |
\begin{document}
\title{On a classification of fat bundles over compact homogeneous spaces}{}
\abstract{This article deals with fat bundles. B\'erard-Bergery classified all homogeneous bundles of that type. We ask a question of a possibility to generalize his description in the case of arbitrary $G$-structures over homogeneous spaces. We obtain necessary conditions for the existence of such bundles. These conditions yield a kind of classification of fat bundles associated with $G$-structures over compact homogeneous spaces provided that the connection in a $G$-structure is canonical.}
\section{Introduction} Let $P(M,G)$ be a principal bundle endowed with a connection form $\theta$ and its curvature form $\Omega$. Let $\operatorname{Ker}\theta=\mathcal{H}\subset TP$ be the corresponding horizontal distribution. Assume that the Lie algebra $\mathfrak{g}$ of $G$ is endowed with an invariant non-degenerate bilinear form $B_{\mathfrak{g}}$. We say that a vector $u\in\mathfrak{g}$ is {\it fat} or that the connection form $\theta$ is $u$-fat, if the bilinear two-form $B_{\mathfrak{g}}(\Omega(\cdot ,\cdot ),u)$ is non-degenerate on $\mathcal{H}$. If the fatness condition is fulfilled for every non-zero $u\in \mathfrak{g}$ then we say that the connection form $\theta $ is fat or that the principal bundle is fat. The role of the fatness condition in Riemannian geometry follows from its relation with the O'Neil tensor \cite{B}, \cite{Z} of a specific fiberwise metric on the associated bundle. In greater detail, consider an associated bundle
$$F\rightarrow P\times_GF\rightarrow M.$$
Endow $F$ with a $G$-invariant Riemannian metric $g_F$, and $M$ with a Riemannian metric $g_M$. Equip $P\times _G F$ with the {\it connection metric} defining it to be the Riemannian metric which equals $g_F$ on $F$, $g(X^*,Y^*)=g_M(X,Y)$ for horizontal lifts $X^*,Y^*$ of $X,Y\in TM$ respectively, and declaring $TF$ and $\mathcal{H}$ to be orthogonal with respect to $g$. For this metric the following holds. Let $A_X$ denote the O'Neil tensor.
\begin{theorem}{\rm \cite{Z}}\label{thm:o'neil} The connection metric $g$ on $P\times_GF$ is complete and defines a Riemannian submersion $\pi: P\times_GF\rightarrow M$ with totally geodesic fiber $F$ and with holonomy group a subgroup of $G$. Conversely, every Riemannian submersion over $M$ with totally geodesic fibers arises in this fashion. Moreover, the O'Neil tensor for such submersion satisfies the equality
$$\theta(A_XY)=-\Omega(X,Y).$$
\end{theorem}
The condition of fatness is an important tool of constructing manifolds of positive and non-negative curvature \cite{GZ},\cite{W}.
In the Riemannian context, the following definition of fatness is used: a Riemannian submersion $\pi: E\rightarrow M$ with totally geodesic fibers is fat, if $A_XU\not=0$ for all horizontal vector fields $X$ and vertical vector fields $U$ (``all vertizontal curvatures are positive'').
For the associated bundles, the characterization of fatness of any connection is known \cite{Z} and can be formulated as follows.
\begin{theorem}{\rm \cite{Z}}\label{thm:asso} The Riemannian submersion $P\times_GF\rightarrow M$ with totally geodesic fibers $G/L$ is fat if and only if the 2-form
$B_{\mathfrak{g}}(\Omega(X,Y),u)$
is non-degenerate on the horizontal distribution for all $u\in \mathfrak{l}^{\perp}$.
\end{theorem}
Keeping the above theorem in mind, from now on whenever we say that an associated bundle $G/L \rightarrow P/L \rightarrow M$ is fat, we mean that the set $\frak{l}^\perp $ consists of fat vectors.
In this paper we consider associated bundles with homogeneous fibers since it was proved (see Proposition 2.6 in \cite{Z}) that every fat submersion necessarily has a homogeneous fiber.
Note that the connection metric $g$ depends on a (chosen) principal connection. The following is known.
\begin{enumerate}
\item There is a an algebraic condition on the curvature tensor of the associated metric connection of the sphere bundles of the form
$$SO(n+1)/SO(n)\rightarrow P/SO(n)\rightarrow M$$
ensuring fatness (see Proposition 2.21 in \cite{Z}).
\item A theorem in \cite{DR} shows that the only fat $SO(4)/SO(3)$-bundle over $S^4$ is the Hopf bundle $S^7\rightarrow S^4$.
\item A theorem of B\'erard-Bergery \cite{BB} which classifies all homogeneous fat bundles, that is, associated bundles of the form $H/L\rightarrow K/L\rightarrow K/H$, where $K,H,L$ are compact Lie groups. Note that in this case the classification is obtained for {\it any} invariant connection.
\item There are necessary conditions for fatness, see for example \cite{FZ}.
\end{enumerate}
To stress the fact that in general the fatness condition is dependent on the choice of the connection, we will always refer to ``fatness with respect to a connection''. Here it seems to be instructive to compare \cite{BB} with the general problem. In the case of principal bundles $K\rightarrow K/H$, there is a one-to-one correspondence between the invariant connections and the linear maps $\Lambda:\mathfrak{k}\rightarrow\mathfrak{h}$ satisfying the conditions
$\Lambda (X)=X$ for any $X\in \mathfrak{h}$, and $\Lambda ([X,Y])=[X,\Lambda (Y)]$ for all $X\in\mathfrak{h}, Y\in \mathfrak{k}$. If, for example, $\mathfrak{k}$ is semisimple the fatness condition can be expressed as follows: for any $X\in\mathfrak{h}^{\perp}$ and any $Y\in \operatorname{Ker}\Lambda$ there exists $Z\in\operatorname{Ker}\Lambda$ such that $\langle X,\Lambda ([Y,Z])\rangle\not=0$. Here $\langle \cdot , \cdot \rangle$ denotes the Killing form. The latter condition can be expressed entirely in terms of the Lie brackets of $\mathfrak{k}$. In general, the problem becomes much more complicated even for the invariant connections in $G$-structures. In this work we will adopt the following terminology. A homogeneous bundle
$$H/L\rightarrow K/L\rightarrow K/H$$
endowed with an invariant fat connection will be called the {\it B\'erard-Bergery} bundle. The triples $(K,H,L)$ (classified in \cite{BB}) will be called the {\it B\'erard-Bergery triples}, and the pairs $(H,H\cap L)$ will be called the {\it B\'erard-Begery pairs}. Note that all $(K,H,L)$ and $(H,H\cap L)$ are known \cite{BB} (see Theorem 2 in this paper). In this work we use this classification, therefore, we reproduce part of it in the last section.
In this article we want to follow the line of reasoning of \cite{BB} and ask a more general question: {\it can one classify bundles associated with a $G$-structure over a compact homogeneous space $K/H$ which are fat with respect to invariant connections?} The purpose of this paper is to solve this problem for the class of fat bundles over homogeneous spaces determined by the canonical invariant connections (Theorem \ref{thm:main}). Our result yields necessary conditions, which can be considered as a kind of classification, because we show that the bases of such bundles are determined by the B\'erard-Bergery triples, and they are determined by a fiber bundle, which is naturally related to the B\'erard-Bergery bundle. For example, if $G$ is simple, there are no other fat bundles than the B\'erard-Bergery homogeneous bundles. Consider the B\'erard-Bergery pair $(H,H\cap L)$. Let $\mathfrak{n}'$ be a maximal common ideal in $\mathfrak{h}$ and $\mathfrak{h}\cap\mathfrak{l}$. Then, clearly, $$\mathfrak{h}=\mathfrak{h}'\oplus\mathfrak{n}',\mathfrak{h}\cap\mathfrak{l}=(\mathfrak{h}\cap\mathfrak{l})'\oplus\mathfrak{n}'.$$ Analogously, let $\mathfrak{n}$ be a common maximal ideal in $\mathfrak{g}$ and $\mathfrak{l}$. \begin{theorem}\label{thm:main} Let $G\rightarrow P\rightarrow K/H$ be a $G$-structure over a homogeneous space $K/H$ of a compact semisimple Lie group $K$. Assume that $G$ is compact and semisimple. Let $$G/L\rightarrow P/L\rightarrow K/H$$ be an associated bundle. If it is fat with respect to the canonical connection, then: \begin{enumerate} \item $\mathfrak{g}=\mathfrak{h}+\mathfrak{l}$ and $(\mathfrak{h},\mathfrak{h}\cap\mathfrak{l})$ is a B\'erard-Bergery pair; \item $K/H$ is a symmetric space of maximal rank; \item $\mathfrak{h}=\mathfrak{h}'\oplus\mathfrak{n}'$ and $\mathfrak{g}=\mathfrak{h}'\oplus\mathfrak{n}$ \end{enumerate} In particular, if $G$ is simple, then the associated bundle is fat with respect to the canonical connection only if $G=H, P=K$. Therefore, the fiber bundle is a B\'erard-Bergery bundle. \end{theorem}
Let us mention a recent publication \cite{FP1}, where the authors consider a particular class of fat bundles over 4-manifolds $M$ endowed with an $S^1$-action which lifts to an $SO(3)$-bundle and leaves the connection invariant. Thus, the case considered in \cite{FP1} is an opposite to the case we deal with. It would be interesting to get deeper understanding of the symmetry of fat connections. Finally, let us mention that there is a different but related notion of symplectic fatness \cite{W}, and there is an interplay between these two \cite{FP},\cite{bstw}, \cite{KTW}.
\section{Preliminaries} \subsection{Notation, terminology and assumptions} The basic tools in this work are invariant connections and the Lie group theory. Therefore, we closely follow the terminology and notation in \cite{KN1},\cite{KN2}, \cite{H} and \cite{OV} without further explanations. We denote the Lie algebras of Lie groups by the corresponding Gothic letters, so the Lie algebra of the Lie group $G$ is denoted by $\mathfrak{g}$, etc. We consider principal bundles $G\rightarrow P\rightarrow M$ and in the most cases use the notation $P(M,G)$. Throughout the whole article we consider homogeneous spaces $K/H$ and assume that both $K$ and $H$ are compact Lie groups. In this note we will use the following result. \begin{proposition} \label{prop:bb}{\rm \cite{BB},\cite{Z}}. If $H/L\rightarrow K/L\rightarrow K/H$ is a fat bundle with $\dim\,(H/L)>1$, then $K/H$ is a Riemannian symmetric space and $\operatorname{rank}\,K=\operatorname{rank}\,H$. \end{proposition}
\subsection{ Invariant connections on reductive homogeneous spaces} In this subsection we will recall the theory of invariant connections on homogeneous spaces in the form presented in Sections 1 and 2 of Chapter X of \cite{KN2}. Let $M$ be a smooth manifold of dimension $n$, and let $$G\rightarrow P\rightarrow M $$ be a $G$-structure, that is, a reduction of the frame bundle $L(M)\rightarrow M$ to a Lie group $G$. Any diffeomorphism $f\in \operatorname{Diff}(M)$ acts on $L(M)$ by the formula $$f(u):=(df_xX_1,...,df_xX_n)$$ for any frame $u=(X_1,...,X_n), X_i\in T_xM$ over a point $x\in M$. By definition, $f$ is called an automorphism of the given $G$-structure, if this action commutes with the action of $G$.
Let $M=K/H$ be a homogeneous space of a connected Lie group $K$. Assume that $M$ is equipped with a $K$-invariant $G$-structure. The latter means that any left translation $\tau(k): K/H\rightarrow K/H$, $\tau(k)(aH)=kaH$ lifts to an automorphism. Let $o=H\in K/H$. Consider the linear isotropy representation of $H,$ that is, a homomorphism $H\rightarrow GL(T_oM)$ given by the formula $$h \mapsto d\tau(h)_o,\,\text{for}\,h\in H,o=H\in K/H.$$ It is important to observe that we can fix a frame $u_o:\mathbb{R}^n\rightarrow T_oM,$ $u_{o}\in P$ and identify the linear isotropy representation of $H$ with a homomorphism $\lambda :H \rightarrow G$ $$\lambda(h)=u_o^{-1}d\tau(h)_ou_o,\,h\in H.$$ One can see this as follows. Denote by $P_{o}\subset P$ the fiber over the point $o,$ then $u_{o} \in P_{o}$. The action of $H$ lifted to $P$ preserves $P_{o}$, hence $h(u_{o}) \in P_{o}.$ Since the structure group $G$ acts transitively on $P_{o}$, there exists exactly one $g\in G$ such that $$h(u_{o})=u_{o}g.$$ It is easy to see that $\lambda (h)=g.$
In the sequel we assume that $K/H$ is reductive. In this case $\mathfrak{k}$ can be decomposed into a direct sum $$\mathfrak{k}=\mathfrak{h}\oplus\mathfrak{m}$$ such that $Ad_{H}(\mathfrak{m})\subset\mathfrak{m}$. Note that the latter implies $[\mathfrak{h},\mathfrak{m}]\subset\mathfrak{m}$. Note that since we consider compact Lie groups, the reductivity can be assumed (see, for example, \cite{KN2}, Chapter X). Also, we assume that the isotropy representation is faithful. Let us make one more straightforward but important observation. One can identify $\lambda$ with the restriction of the adjoint representation of $H$ on $\mathfrak{m}$ (which we also denote by $\lambda$).
We say that a connection $\theta$ in $P\rightarrow M$ is $K$-invariant, if for any $k\in K$ the lift of $\tau(k)$ preserves it. We need the following description of the set of invariant connections in the principal bundle $P\rightarrow M$ from \cite{KN2}, Chapter X.
\begin{theorem}\label{thm:inv-connections} Let $M=K/H$ be a reductive homogeneous space equipped with a $K$-invariant $G$-structure. Then there is a one-to-one correspondence between the $K$-invariant connections in it, and $Ad_H$-invariant linear maps
$$\Lambda: \mathfrak{k}\rightarrow \mathfrak{g}, \Lambda|_{\mathfrak{h}}=\lambda.$$ \end{theorem}
\noindent Let $\Lambda_{\mathfrak{m}}=\Lambda |_{\mathfrak{m}}$. Note that here $Ad_H$-invariance means that $$\Lambda_\mathfrak{m}(Ad\,h(Z))=\lambda(h)(\Lambda_\mathfrak{m}(Z)),\,Z\in\mathfrak{m},h\in H.$$ \begin{definition} {\rm Recall that a connection in the given $K$-invariant $G$-structure is called {\it canonical} if it corresponds to the map $\Lambda$ with $\Lambda_{\mathfrak{m}}=0$. } \label{r2} \end{definition}
In this article we will need the description of the holonomy Lie algebra of the canonical connection. \begin{theorem}{\rm \cite{KN2}}\label{thm:holonomy-alg} The holonomy Lie algebra of the canonical connection of a reductive homogeneous space $K/H$ is given by the formula $$\mathfrak{h}^*=\lambda(ad([X,Y]_{\mathfrak{h}})),\,\forall\, X,Y\in\mathfrak{m}.$$ \end{theorem} \subsection{Factorizations of Lie groups and Lie algebras} Following Onishchik \cite{O},\cite{O1} we say that a triple of Lie algebras $(\mathfrak{g},\mathfrak{h},\mathfrak{l})$ is a {\it factorization}, if $\mathfrak{h}$ and $\mathfrak{l}$ are subalgebras of $\mathfrak{g}$, $\mathfrak{g}=\mathfrak{h}+\mathfrak{l}$, and $\mathfrak{h}\not=\mathfrak{g},\mathfrak{l}\not=\mathfrak{g}$. In the same way, we say that $(G,H,L)$ is a factorization, if $H$ and $L$ are Lie subgroups of $G$, $G=H\cdot L$ and $G\not=H$, $G\not=L$. It is proved in \cite{O},\cite{O1} that, for compact Lie groups, $(G,H,L)$ is a factorization if and only if $(\mathfrak{g},\mathfrak{h},\mathfrak{l})$ is a factorization. All factorizations of simple compact Lie algebras are classified in \cite{O1}. The proof of our result is essentially based on this classification, and we reproduce it here.
As usual, we consider the complexification $\mathfrak{g}^c$ and a Cartan subalgebra $\mathfrak{j}\subset\mathfrak{g}^c$. Consider the corresponding root system $\Delta$, and choose the set of simple roots $\alpha_1,...,\alpha_m$. It is known that any irreducible linear representation of $\mathfrak{g}$ is determined (up to an equivalence) by the highest weight of the representation. This is a vector $\Phi\in\mathfrak{j}$, which can be described by the integers
$$\Phi_i={2\langle\Phi,\alpha_i\rangle\over\langle\alpha_i,\alpha_i\rangle},\,i=1,...,m.$$
We write $\Phi=(\Phi_1,...,\Phi_m)$ and denote by $\varphi_i$ the irreducible representation of a simple Lie algebra $\mathfrak{g}$ with the highest weight $\Phi$ which has $\Phi_i=1$ and $\Phi_j=0,i\not=j$. The following classification result is proved by Onishchik.
\begin{theorem}\label{thm:onishchik}{\rm \cite{O}} Let $\mathfrak{g}$ be a compact simple Lie algebra. All factorizations
$(\mathfrak{g},\mathfrak{l},\mathfrak{h})$ and possible embeddings
$$i': \mathfrak{h}\rightarrow\mathfrak{g}, i'':\mathfrak{l}\rightarrow\mathfrak{g}$$ are given in Table 1.
\end{theorem} \begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|} \hline $\mathfrak{g}$ & $\mathfrak{h}$ & $i'$ & $\mathfrak{l}$ & $i''$ & $\mathfrak{h} \cap \mathfrak{l}$ & restrictions \\ \hline \hline $A_{2n-1}$ & $C_n$ & $\varphi _1$ & $A_{2n-2}$ & $\varphi _1+N$ & $C_{n-1}$ & $n>1$\\ \hline $A_{2n-1}$ & $C_n$ & $\varphi _1$ & $A_{2n-2}\oplus T$ & $\varphi _1+N$ & $C_{n-1}\oplus T$ & $n>1$\\ \hline $B_3$ & $G_2$ & $\varphi _2$ & $B_2$ & $\varphi _1+2N$ & $A_{1}$ & \null \\ \hline $B_3$ & $G_2$ & $\varphi _2$ & $B_2\oplus T$ & $\varphi _1+2N$ & $A_{1}\oplus T$ & \null \\ \hline $B_3$ & $G_2$ & $\varphi _2$ & $D_3$ & $\varphi _1+N$ & $A_{2}$ & \null \\ \hline $D_{n+1}$ & $B_n$ & $\varphi _1+N$ & $A_n$ & $\varphi _1+\varphi _n$ & $A_{n-1}$ & $n>2$\\ \hline $D_{n+1}$ & $B_n$ & $\varphi _1+N$ & $A_n\oplus T$ & $\varphi _1+\varphi _n$ & $A_{n-1}\oplus T$ & $n>2$ \\ \hline $D_{2n}$ & $B_{2n-1}$ & $\varphi _1+N$ & $C_n$ & $\varphi _1+\varphi _1$ & $C_{n-1}$ & $n>1$\\ \hline $D_{2n}$ & $B_{2n-1}$ & $\varphi _1+N$ & $C_n\oplus T$ & $\varphi _1+\varphi _1$ & $C_{n-1}\oplus T$ & $n>1$ \\ \hline $D_{2n}$ & $B_{2n-1}$ & $\varphi _1+N$ & $C_n\oplus A_1$ & $\varphi _1+\varphi _1$ & $C_{n-1}\oplus A_{1}$ & $n>1$ \\ \hline $D_8$ & $B_7$ & $\varphi _1+N$ & $B_4$ & $\varphi _4$ & $B_{3}$ & \null \\ \hline $D_{4}$ & $B_3$ & $\varphi _3$ & $B_2$ & $\varphi _1+3N$ & $A_{1}$ & \null \\ \hline $D_{4}$ & $B_3$ & $\varphi _3$ & $B_2\oplus T$ & $\varphi _1+3N$ & $A_{1}\oplus T$ & \null\\ \hline $D_{4}$ & $B_3$ & $\varphi _3$ & $B_2\oplus A_1$ & $\varphi _1+3N$ & $A_{1}\oplus A_{1}$ & \null \\ \hline $D_{4}$ & $B_3$ & $\varphi _3$ & $D_3$ & $\varphi _1+2N$ & $A_{2}$ & \null \\ \hline $D_{4}$ & $B_3$ & $\varphi _3$ & $D_3\oplus T$ & $\varphi _1+2N$ & $A_{2}\oplus T$ & \null \\ \hline $D_{4}$ & $B_3$ & $\varphi _3$ & $B_3$ & $\varphi _1+N$ & $G_{2}$ & \null \\ \hline \end{tabular} \end{center}
\vskip6pt \centerline{TABLE 1} \begin{remark} {\rm In Table 1 above $N$ stands for the trivial representation. The types of simple Lie algebras are denoted as usual, following \cite{OV}}. \end{remark} \section{Proof of Theorem \ref{thm:main}} \subsection{Preliminary statements} \begin{proposition}\label{prop:onishchik} Let $P(M,G)$ be a principal fiber bundle with a connection form $\theta$ and the curvature form $\Omega$. Assume that $M=K/H$ is a homogeneous space of a compact Lie group $K$. Fix a point $p\in P$. Let $\mathfrak{h}'$ be the Lie subalgebra of $\mathfrak{g}$ generated by all $\Omega_p(X,Y), X,Y\in\mathcal{H}_p$. Then, if there exists a fat associated bundle with fiber $G/L$, then $\mathfrak{g}=\mathfrak{h}'+\mathfrak{l}$. Moreover, if the bundle is fat with respect to the canonical connection, then $\mathfrak{g}=\mathfrak{h}+\mathfrak{l}$. \end{proposition} \begin{proof} Notice that the fatness condition is equivalent to the condition that all 2-forms $B_{\mathfrak{g}}(\Omega_p(X,Y),u)$ are non-degenerate for all $u\in\mathfrak{l}^{\perp}$, all $p\in P$ and $X,Y\in \mathcal{H}_p$. The latter implies $(\mathfrak{h}')^{\perp} \cap \mathfrak{l}^{\perp}=\{ 0 \} .$
But $(\mathfrak{h}')^{\perp} \cap \mathfrak{l}^{\perp}=\{ 0 \} \ \textrm{ iff} \ \mathfrak{g}=\mathfrak{h}'+ \mathfrak{l}. $ We complete the proof by noticing that $\mathfrak{h}'\subset\mathfrak{h}^*$, where $\mathfrak{h}^*$ denotes the holonomy Lie algebra, and that for the canonical connection $\mathfrak{h}^*\subset \mathfrak{h}\cong\lambda (\mathfrak{h})$, by Theorem \ref{thm:holonomy-alg}. \end{proof} Let there be given a principal $G$-bundle $G\rightarrow P \rightarrow K/H$ endowed with a principal $K$-invariant connection $\theta .$ Use Theorem \ref{thm:inv-connections}. On the Lie algebra level we get that a connection $\theta$ is given by a linear mapping $\Lambda:\mathfrak{k} \rightarrow \mathfrak{g}$ such that
\begin{itemize}
\item $\Lambda (X)=X$ for every $X\in \mathfrak{h},$
\item $\Lambda [X,Y]=[X,\Lambda (Y)]$ for every $X\in \mathfrak{h}$ and $Y\in \mathfrak{k}$. \end{itemize} Here we identify $X$ with $\lambda(X)$. Let $L\subset G$ be a closed subgroup and $A:=\mathfrak{l}^{\perp}\subset \mathfrak{g}$ be the orthogonal complement of $\mathfrak{l}$ in $\mathfrak{g}$ with respect to $B_{\mathfrak{g}}$ (where $B_{\mathfrak{g}}$ denotes an invariant, negative-definite 2-form on $\mathfrak{g}$). Let also $B:=\mathfrak{h}\cap \mathfrak{l}$ and $C:=(\mathfrak{h}\cap \mathfrak{l})^{\perp}_{\mathfrak{h}} \subset \mathfrak{h}$ be the orthogonal complement of $\mathfrak{h}\cap \mathfrak{l}$ in $\mathfrak{h}$ with respect to $B_{\mathfrak{g}}.$ Throughout this note we use the following notation: if $W=U\oplus V$, is a direct sum of vector spaces, the projection onto $V$ is denoted by $proj_V$.
\begin{lemma} If $\mathfrak{g}=\mathfrak{h}+\mathfrak{l}$ then for any subspace $V\subset \mathfrak{k}$ the conditions $(1)$ and $(2)$ below are equivalent \begin{equation} \forall_{0\neq X\in A} \ \forall_{0\neq Y\in V} \ \exists_{0\neq Z\in V} \ B_{\mathfrak{g}} (X, [Y,Z]_{\mathfrak{h}})\neq 0. \label{lr2} \end{equation} \begin{equation} \forall_{0\neq X\in C} \ \forall_{0\neq Y\in V} \ \exists_{0\neq Z\in V} \ B_{\mathfrak{g}} (X, [Y,Z]_{\mathfrak{h}})\neq 0. \label{lr3} \end{equation} \label{lemma1} \end{lemma} \begin{proof}
We have the (orthogonal) direct sum decompositions $\mathfrak{h}=B\oplus C,$ $\mathfrak{g}=\mathfrak{h}\oplus \mathfrak{h}^{\perp}$ and thus $\mathfrak{g}=B\oplus C\oplus \mathfrak{h}^{\perp}$. For any vector $X\in \mathfrak{g}$, denote its $A$-component ($B,C$-component, etc) by $X_{A}$ ($X_{B},X_{C},$ etc). Note that for $X\in A\subset \mathfrak{g}$ and $H\in \mathfrak{h}$ we have $H=H_{B}+H_{C}$ and $$B_{\mathfrak{g}} (X,H)=B_{\mathfrak{g}} (X,H_{B}+H_{C})=B_{\mathfrak{g}} (X,H_{C})=B_{\mathfrak{g}} (X_{\mathfrak{h}^{\perp}}+X_{\mathfrak{h}},H_{C})=$$ $$B_{\mathfrak{g}} (X_{\mathfrak{h}},H_{C})=B_{\mathfrak{g}} (X^{B}_{\mathfrak{h}}+X^{C}_{\mathfrak{h}},H_{C})=B_{\mathfrak{g}} (X^{C}_{\mathfrak{h}},H_{C})=B_{\mathfrak{g}} (X_{C},H_{C})=$$ $$B_{\mathfrak{g}} (X_{C},H_{C}+H_{B})=B_{\mathfrak{g}} (X_{C},H)$$ where $X^{B}_{\mathfrak{h}}$ ($X^{C}_{\mathfrak{h}}$) is the $B$-component ($C$-component) of $X_{\mathfrak{h}}.$
Thus the condition (\ref{lr2}) implies that \begin{equation}
\operatorname{Ker}(proj_C)|_{A}=\{ 0 \} \label{lr4} \end{equation} But $$\mathfrak{g}=\mathfrak{l}\oplus A=\mathfrak{l}\oplus C$$
are the direct sum decompositions and therefore $\dim A=\dim C$, so $proj_C|_{A}:A\rightarrow C$ is an isomorphism. Thus, $$B_{\mathfrak{g}} (X, [Y,Z]_{\mathfrak{h}})=B_{\mathfrak{g}} (X_{C}, [Y,Z]_{\mathfrak{h}})=B_{\mathfrak{g}} (proj_{C}(X), [Y,Z]_{\mathfrak{h}})$$ therefore condition (\ref{lr2}) implies condition (\ref{lr3})..
Now we will show that condition (\ref{lr3}) implies condition (\ref{lr2}). Assume that (\ref{lr2}) is not satisfied. Then there exists $0\neq X\in A$ and $0\neq Y\in V$ such that $$\forall_{0\neq Z\in V} \ B_{\mathfrak{g}} (X, [Y,Z]_{\mathfrak{h}})=0$$ therefore $$\forall_{0\neq Z\in V} \ B_{\mathfrak{g}} (X_{C}, [Y,Z]_{\mathfrak{h}})=0.$$ If $X_{C}\neq 0$ then the lemma is proved. Assume that $X_{C}=0$. As $X\in A$ thus $$X=X_{C}+X_{\mathfrak{h}^{\perp}}=X_{\mathfrak{h}^{\perp}},$$ that is $0\neq X\in X_{\mathfrak{h}^{\perp}},$ so $X$ is a non-zero vector which is orthogonal to $\mathfrak{h}$ and to $\mathfrak{l}.$ This is impossible since $\mathfrak{g}=\mathfrak{h}+\mathfrak{l}.$ \end{proof} \begin{lemma} Let $\Lambda_{\mathfrak{h}}:\mathfrak{k}\rightarrow \mathfrak{h}$ be the linear operator defined by $$\Lambda_{\mathfrak{h}} (X):=proj_{\mathfrak{h}} \circ \Lambda(X).$$ The map $\Lambda_{\mathfrak{h}}$ induces an invariant connection in the principal bundle $K\rightarrow K/H$. The associated bundle $$G/L\rightarrow P/L\rightarrow K/H$$ is fat (with respect to an invariant connection) if and only if the bundle $$H/(H\cap L)\rightarrow K/L\rightarrow K/H$$ is fat. \label{lemmabb} \end{lemma} \begin{proof} Note that in the proof of this Lemma, for brevity, we use a terminology which is somewhat not precise but clear: we say that $\Lambda$ or $\Lambda_{\mathfrak{h}}$ is fat, if it yields a fat associated bundle in our usual sense. The condition $\Lambda_{\mathfrak{h}}(X)=X$ for $X\in \mathfrak{h}$ is obviously satisfied. Also $$\Lambda([X,Y])=\Lambda_{\mathfrak{h}}([X,Y])+\Lambda_{\mathfrak{h}^{\perp}}([X,Y])$$ and $$[X,\Lambda (Y)]=[X,\Lambda_{\mathfrak{h}}(Y)+\Lambda_{\mathfrak{h}^{\perp}}(Y)]=[X,\Lambda_{\mathfrak{h}}(Y)]+[X,\Lambda_{\mathfrak{h}^{\perp}}(Y)],$$ for $X\in \mathfrak{h}$ and $Y\in \mathfrak{k}.$ But $[X,\Lambda_{\mathfrak{h}}(Y)]\in \mathfrak{h}$ and $[X,\Lambda_{\mathfrak{h}^{\perp}}(Y)]\in \mathfrak{h}^{\perp}$ since $K/H$ is reductive and, therefore, we obtain $$\Lambda_{\mathfrak{h}}([X,Y])=[X,\Lambda_{\mathfrak{h}}(Y)],$$ as desired.
It follows from Lemma \ref{lemma1} that if $\Lambda$ is fat then $$\forall_{0\neq X\in (\mathfrak{h} \cap \mathfrak{l})^{\perp}_{\mathfrak{h}}} \ \forall_{0\neq Y\in \operatorname{Ker}\Lambda} \ \exists_{Z\in \operatorname{Ker}\Lambda} \ B_{\mathfrak{g}}(X,\Lambda([Y,Z]))\neq 0.$$ But $$B_{\mathfrak{g}}(X,\Lambda([Y,Z]))=B_{\mathfrak{g}}(X,\Lambda_{\mathfrak{h}}([Y,Z])+\Lambda_{\mathfrak{h}^{\perp}}([Y,Z]))=$$ $$B_{\mathfrak{g}}(X,\Lambda_{\mathfrak{h}}([Y,Z]))+B_{\mathfrak{g}}(X,\Lambda_{\mathfrak{h}^{\perp}}([Y,Z]))=B_{\mathfrak{g}}(X,\Lambda_{\mathfrak{h}}([Y,Z]))$$ and so if $\Lambda$ is fat then $$\forall_{0\neq X\in (\mathfrak{h} \cap \mathfrak{l})^{\perp}_{\mathfrak{h}}} \ \forall_{0\neq Y\in \operatorname{Ker}\Lambda} \ \exists_{Z\in \operatorname{Ker}\Lambda} \ B_{\mathfrak{g}}(X,\Lambda_{\mathfrak{h}}([Y,Z]))\neq 0.$$ On the other hand if $\Lambda_{\mathfrak{h}}$ is fat then again by Lemma \ref{lemma1} it is fat as a map $\mathfrak{k} \rightarrow \mathfrak{g}$. \end{proof} Now we are ready to get the main part of the proof of Theorem \ref{thm:main}. \begin{lemma}\label{thm:sub-main} Let $G\rightarrow P\rightarrow K/H$ be a $G$-structure over a homogeneous space $K/H$ of a compact Lie group $K$. Assume that $G$ is compact. An associated bundle $$G/L\rightarrow P/L\rightarrow K/H$$ is fat with respect to a connection metric determined by the $K$-invariant canonical connection if and only if the following three conditions are satisfied: \begin{enumerate} \item $G=HL$, \item $K/H$ is a Riemannian symmetric space; \item the pair ($\mathfrak{h},\mathfrak{h}\cap\mathfrak{l})$ is a B\'erard-Bergery pair. \end{enumerate} \end{lemma} \begin{proof} By proposition \ref{prop:onishchik}, we have $\mathfrak{g}=\mathfrak{h}+\mathfrak{l}$. It follows from Lemma \ref{lemmabb} that $$G/L \rightarrow P/L \rightarrow K/H$$ is fat if and only if the homogeneous bundle $H/H\cap L\rightarrow K/L\rightarrow K/H$ is fat, hence, $K/H$ must be symmetric by Proposition \ref{prop:bb}, the triple $(K,H,L)$ must be the B\'erard-Bergery triple, and the pair $(H,H\cap L)$ must be the B\'erard-Bergery pair. \end{proof} \subsection{Proof of Theorem \ref{thm:main}} We complete the proof in two steps. Assume, first, that $G$ is simple. If $G$ is simple then $G/L \rightarrow P/L \rightarrow K/H$ is fat if and only if it is B\'erard-Bergery bundle. Indeed, by Lemma \ref{thm:sub-main} $\mathfrak{g}=\mathfrak{h}+\mathfrak{l}$ and the pair $(\mathfrak{h},\mathfrak{h}\cap\mathfrak{l})$ is a B\'erard-Bergery pair. Thus, either $(\mathfrak{g},\mathfrak{h},\mathfrak{l})$ is a factorization or $\mathfrak{g}=\mathfrak{h}$. In the first case, the triple $(\mathfrak{g},\mathfrak{h},\mathfrak{l})$ and $\mathfrak{h}\cap\mathfrak{l}$ are contained in Table 1.
However, the comparison of Table 1 and Table 2 shows that none of the pairs $(\mathfrak{h},\mathfrak{h}\cap\mathfrak{l})$ is contained in both tables. Hence, $G=H$. We get the commutative diagram $$ \CD H @>{=}>> H\\ @VVV @VVV\\ K @>{i}>> P\\ @VVV @VVV\\ K/H @>{=}>>K/H \endCD $$ where $i$ denotes the evaluation map. Clearly, under the conditions of Theorem \ref{thm:main}, we get an isomorphism of the principal bundles. This follows since the isotropy representation is faithful, and, therefore, $i$ must be an embedding, hence a diffeomorphism. This completes the proof when $G$ is simple. Note that the argument also goes through for any $\mathfrak{g}=\mathfrak{h}$.
In the general case, the proof follows from Lemma \ref{lemma:extension} below. We need to consider the case when $\mathfrak{g}$ is not simple, but $(\mathfrak{g},\mathfrak{h},\mathfrak{l})$ is a factorization.
\begin{lemma}\label{lemma:extension} Under the assumptions of Theorem \ref{thm:main}, the following holds $$\mathfrak{g}=\mathfrak{h}'+\mathfrak{l}, \ \mathfrak{g}=\mathfrak{h}'+\mathfrak{n} \ \textrm{and} \ \mathfrak{l}=\mathfrak{h}'+\mathfrak{n}.$$ \end{lemma} \begin{proof} We know that $\mathfrak{g}=\mathfrak{h}+\mathfrak{l}$ and $\mathfrak{h}=\mathfrak{h}'+\mathfrak{n}'.$ But $\mathfrak{n}'\subset \mathfrak{h}\cap \mathfrak{l}$ and therefore $$\mathfrak{g}=\mathfrak{h}'+\mathfrak{l}.$$ It follows from Table 2 that $\mathfrak{h}'$ is a simple Lie algebra or $\mathfrak{h}'=\mathbb{R}+\mathfrak{su}(2).$ Take a connected Lie subgroup $H'$ corresponding to $\mathfrak{h}'$ (it exists if $\mathfrak{h}'$ is simple; if $\mathfrak{h}'$ is not simple then $\mathbb{R}$ corresponds to a circle and $H'/(H\cap)L$ is not simply connected). So assume that $H'$ is simple. In this case it follows from Table 2 and (see comments to Corollary to Theorem 5.1 and Corollary to Theorem 6.1 in \cite{O}) that only $H'$ acts effectively on $H'/(H\cap L)'$ and therefore $\mathfrak{g}=\mathfrak{h}'+\mathfrak{n}$ and $\mathfrak{l}=\mathfrak{h}'+\mathfrak{n}$ for some ideal $\mathfrak{n} \subset \mathfrak{g}.$ \end{proof} Now, we complete the proof of Theorem \ref{thm:main} by applying Lemma \ref{thm:sub-main} and Lemma \ref{lemma:extension}.
\section{Addendum: the B\'erard-Bergery pairs} \subsection{Table 2} For the convenience of the reader we reproduce a part of the B\'erad-Bergery classification \cite{BB} (see explanations below Table 2). \vskip6pt \begin{center} \renewcommand{1.0}{1.2}
\begin{tabular}{|c|c|c|} \hline $\mathfrak{h}$ & $\mathfrak{h} \cap \mathfrak{l}$ & restrictions \\ \hline \hline $A_{n-2} \oplus \mathbb{R} \oplus A_{1}$ & $A_{n-2} \oplus \mathbb{R} \oplus \mathbb{R}_{1}$ & $n \geq 2$ \\ \hline $B_{n-2} \oplus A_{1} \oplus A_{1}$ & $B_{n-2} \oplus A_{1} \oplus \mathbb{R}_{1}$ & $n \geq 2$ \\ \hline $C_{n-1} \oplus A_{1}$ & $C_{n-1} \oplus \mathbb{R}_{1}$ & $n \geq 2$ \\ \hline $D_{n-2} \oplus A_{1} \oplus A_{1}$ & $D_{n-2} \oplus A_{1} \oplus \mathbb{R}_{1}$ & $n \geq 3$ \\ \hline $A_{5} \oplus A_{1}$ & $A_{5} \oplus \mathbb{R}_{1}$ & \\ \hline $D_{6} \oplus A_{1}$ & $D_{6} \oplus \mathbb{R}_{1}$ & \\ \hline $E_{7} \oplus A_{1}$ & $E_{7} \oplus \mathbb{R}_{1}$ & \\ \hline $C_{3} \oplus A_{1}$ & $C_{3} \oplus \mathbb{R}_{1}$ & \\ \hline $\overline{A}_{1} \oplus A_{1}$ & $\overline{A}_{1} \oplus \mathbb{R}_{1}$ & \\ \hline $A_{1} \oplus \overline{A}_{1}$ & $A_{1} \oplus \mathbb{R}_{1}$ & \\ \hline $A_{n-4} \oplus \mathbb{R} \oplus A_{3}$ & $A_{n-4} \oplus \mathbb{R} \oplus \overline{C}_{2}$ & $n \geq 4$ \\ \hline $B_{n-4} \oplus D_{4}$ & $B_{n-4} \oplus \overline{B}_{3}$ & $n \geq 4$ \\ \hline $C_{n-2} \oplus C_{2}$ & $C_{n-2} \oplus A_{1} \oplus A_{1}$ & $n \geq 3$ \\ \hline $D_{n-4} \oplus D_{4}$ & $D_{n-4} \oplus \overline{B}_{3}$ & $n\geq 5$ \\ \hline $\mathbb{R} \oplus D_{5}$ & $\mathbb{R} \oplus B_{4}$ & \\ \hline $A_{1} \oplus D_{6}$ & $A_{1} \oplus B_{5}$ & \\ \hline $D_{8}$ & $B_{7}$ & \\ \hline $B_{4}$ & $D_{4}$ & \\ \hline $A_{n-2} \oplus \mathbb{R} \oplus A_{1}$ & $A_{n-2} \oplus \mathbb{R}_{a}$ & $n \geq 2$ \\ \hline $\mathbb{R} \oplus A_{1} \oplus A_{1}$ & $A_{1} \oplus \mathbb{R}_{a}$ & \\ \hline \end{tabular} \renewcommand{1.0}{1.0} \end{center} \vskip6pt \centerline{TABLE 2}
\subsection{Explanations} We write down only the B\'erard-Bergery pairs $(\mathfrak{h},\mathfrak{h}\cap\mathfrak{l})$ which come from the {\it semisimple} Lie algebras $\mathfrak{k}$. The symbols $A_1$ and $\bar A_1$ denote subalgebras associated to the highest and the lowest root. Note that our notation differs from that of \cite{BB}: the triples $(\mathfrak{k},\mathfrak{h},\mathfrak{h}\cap \mathfrak{l})$ are denoted by $(\mathfrak{g},\mathfrak{h},\mathfrak{k})$ in \cite{BB}. Also, we have reproduced only cases (A), (C) and (D) from the classification. Cases (B), (E) and (F) are not essential in our considerations. This follows from the description of the classification in \cite{BB}: \begin{enumerate} \item case (B) is obtained from case (A) by removing $\mathbb{R}_1$ from $\mathfrak{h}\cap\mathfrak{l}$ \item case (E) consists of the pairs $$(A_1'\oplus\mathfrak{h}'\oplus A_1''\oplus\mathfrak{h}'',\bar A_1\oplus\mathfrak{h}'\oplus\mathfrak{h}''),$$ where $(A_1'\oplus\mathfrak{h}',\mathfrak{h}')$ is of type (B), and $(A_1''\oplus\mathfrak{h}'',\mathfrak{h}'')$ is either of type (B), or $(\mathfrak{h},\mathfrak{h}\cap\mathfrak{l})$ has the form $(A_1\oplus A_1,\bar A_1)$, where $\bar A_1$ is a diagonal subalgebra in $A_1\oplus A_1$. \item case (F) is eliminated by the assumption that $\mathfrak{k}$ is semisimple, \item the dual triples in case (C) yield the same pair $(\mathfrak{h},\mathfrak{h}\cap\mathfrak{l})$ (see Lemma 12 in \cite{BB}). \end{enumerate} \vskip6pt
\begin{remark} {\rm We consider only the case of the canonical connection in a $G$-structure. We conjecture that the same holds for any invariant connection, but the problem seems to be much more difficult. It would be interesting to try some other better known invariant connections, for example, metric connections described in spirit of \cite{AFF}.} \end{remark} \vskip6pt \noindent {\bf Acknowledgement.} We thank Ilka Agricola for answering our questions and Jarek K\c edra for valuable discussions.
MB, AS, AT, AW: Department of Mathematics and Computer Science
\vskip6pt
University of Warmia and Mazury
\vskip6pt
S\l\/oneczna 54, 10-710 Olsztyn, Poland
\vskip6pt
e-mail addresses:
\vskip6pt
MB: [email protected]
\vskip6pt
AS: [email protected]
\vskip6pt
AT: [email protected]
\vskip6pt
AW: [email protected]
\vskip6pt
\end{document} | arXiv |
\begin{document}
\begin{abstract} Let $M$ be a manifold with pinched negative sectional curvature. We show that when $M$ is geometrically finite and the geodesic flow on $T^1 M$ is topologically mixing then the set of mixing invariant measures is dense in the set ${\mathscr M}^1(T^1M)$ of invariant probability measures. This implies that the set of weak-mixing measures which are invariant by the geodesic flow is a dense $G_{\delta}$ subset of ${\mathscr M}^1(T^1 M)$. We also show how to extend these results to manifolds with cusps or with constant negative curvature. \end{abstract} \maketitle
\section{Introduction}
Let $\widetilde{M}$ be a complete, simply connected manifold with pinched sectional curvature (\textit{i.e} there exists $b > a > 0$ such that $-b^2 \leq \kappa \leq -a^2$) and $\Gamma$ a non elementary group of isometries of $\widetilde{M}$. We will denote by $M$ the quotient manifold $\widetilde{M} / \Gamma$ and $\phi_t$ the geodesic flow on the unit tangent bundle $T^1 M$. The interesting behavior of this flow occurs on its non-wandering set $\Omega$.\\ Let us recall a few definitions related to the mixing property of the geodesic flow.
\begin{enumerate}
\item $\phi_t$ is topologically mixing if for all open sets ${\mathscr U}, {\mathscr V} \subset \Omega$ there exists $T>0$ such that \begin{center}
$\phi_t({\mathscr U}) \cap {\mathscr V} \neq \emptyset$ for all $|t| >T$, \end{center} \item Given a finite measure $\mu$ the geodesic flow is mixing with respect to $\mu$ if for all $f \in L^2(T^1 M,\mu)$, $$ \lim\limits_{t \to \infty} \int_{T^1M} f \circ \phi_t \cdot f d\mu = \left( \int_{T^1 M} f d\mu \right)^2, $$ \item it is weakly-mixing if for all continuous function $f$ with compact support we have
$$ \lim\limits_{T \to \infty} \frac{1}{T} \int_0^T \left| { \int_{T^1 M} f \circ \phi_t (x)f(x) d\mu(x) - \left( \int_{T^1 M} f d\mu \right) ^2 } \right| dt = 0.$$ \end{enumerate}
Finally, let us recall what is the weak topology: a sequence $\mu_n$ of probability measures converges to a probability measure $\mu$ if for all bounded continuous functions $f$, $$\int_{T^1 M} f d\mu_n \xrightarrow{n \to \infty} \int_{T^1 M} f d\mu.$$\\
In this article, we will show that the weak-mixing property is generic in the set ${\mathscr M}^1(T^1 M)$ of probability measures invariant by the geodesic flow and supported on $\Omega$. We endow ${\mathscr M}^1(T^1 M)$ with the weak topology.\\ In \cite{Si72}, K. Sigmund studies this question for Anosov flows defined on compact manifolds and shows that the set of ergodic probability measures is a dense $G_{\delta}$ set (\textit{i.e} a countable intersection of dense sets) in ${\mathscr M}^1 (T^1 M)$. On non-compact manifolds, the question has been studied by Y. Coudène and B. Schapira in \cite{MR2735038} and \cite{MR3322793}. It appears that ergodicity and zero entropy are typical properties for the geodesic flow on negatively curved manifolds.\\ Since the set of mixing measures with respect to the geodesic flow is contained in a meager set (see \cite{MR3322793}) it is natural to consider the set of weak-mixing measures from the generic point of view.\\
Let us recall that a manifold $M$ is geometrically finite if it is negatively curved, complete and has finitely many ends, all of which are cusps of finite volume and funnels.\\
Here is our main result. \begin{thm}\label{thm:princ}
Let $M$ be a geometrically finite manifold with pinched curvature and $\phi_t$ the geodesic flow defined on its unit tangent bundle $T^1 M$.\\
If $\phi_t$ is topologically mixing on $\Omega$, then the set of mixing probability measures invariant by the geodesic flow is dense in ${\mathscr M}^1 (T^1 M)$
for the weak topology. \end{thm} \begin{cor}\label{cor:gef}
Let $M$ be a geometrically finite manifold with pinched negative curvature and $\phi_t$ the geodesic flow defined on its unit tangent bundle $T^1M$.\\
If $\phi_t$ is topologically mixing on the non-wandering set of $T^1M$, then the set of invariant weak-mixing probability measures with
full support on $\Omega$ is a dense $G_{\delta}$ subset of ${\mathscr M}^1 (T^1 M)$. \end{cor} To prove theorem \ref{thm:princ}, we will use the fact that Dirac measures supported on periodic orbits are dense in ${\mathscr M}^1(T^1 M)$. This comes from \cite{MR2735038} where the result is shown for any metric space $X$ admitting a local product structure and satisfying the closing lemma.\\ The rest of the proof relies on the approximation of a single Dirac measure supported on a periodic orbit ${\mathscr O}(p)$ using a sequence of Gibbs measures associated to $(\Gamma,F_n)$ where $F_n:T^1 M \to {\Bbb R}$ is a Hölder-continuous potential.\\ The notion of Gibbs measures which is related to the construction of \\ Patterson-Sullivan densities on the boundary at infinity of $\widetilde{M}$ will be recalled in $\S. 2$ .\\ In $\S. 4$ we will prove a criterion connecting the divergence of some subgroups of $\Gamma$ with the finiteness of the Gibbs measures which comes from \cite{MR1776078} for the potential $F=0$.\\ After this, we will construct in $\S. 5$ a sequence of bounded potentials satisfying the desired property. The main step of this paragraph builds on a result of \cite{C04} which claims that there exists a bounded potential such that the Gibbs measure is finite.\\ Finally, we will prove in $\S. 6$ the convergence of the Gibbs measures using the variational principle which is recalled in $\S. 2$.\\ Now, assume that Theorem \ref{thm:princ} is true. The proof of corollary \ref{cor:gef} is a consequence of \cite{MR3322793} where it is shown that the set of weak-mixing measures with full support is a $G_{\delta}$ subset of the set of invariant Borel probability measures supported on $\Omega$.\\
In the previous theorem, we restricted ourselves to the case of geometrically finite manifolds but we make the following conjecture: the result is still true for non geometrically finite manifolds as soon as the geodesic flow is topologically mixing on its non wandering set.\\ The conjecture is supported by the following two results.
\begin{cor}\label{cor:geinf}
Let $M$ be a connected, complete pinched negatively curved manifold with a cusp then the set of probability measures
fully supported on $\Omega$ that are weakly mixing with respect to the geodesic flow is a dense $G_{\delta}$ set of ${\mathscr M}^1 (T^1 M)$. \end{cor}
\begin{cor}\label{cor:surf}
Let $S$ be a pinched negatively curved surface or a manifold with constant negative curvature then the set of probability measures
fully supported on $\Omega$ that are weakly mixing with respect to the geodesic flow is a dense $G_{\delta}$ set of ${\mathscr M}^1 (T^1 M)$. \end{cor} Let $M$ be a manifold such that $dim(M)= 2$, $\kappa_M = -1$ or $M$ possesses a cusp and denote by $\phi_t$ the geodesic flow on $T^1M$. We will show in $\S.7$ that we can find a geometrically finite manifold $\hat{M}$ on which theorem \ref{thm:princ} applies and for which $\phi_t$ is a factor of the geodesic flow $\hat{\phi}_t$ on $T^1 \hat{M}$.\\
One way to confirm the conjecture is to find a positive answer to the following question:\\ Let $M$ be a connected, complete manifold with pinched negative curvature. We will suppose that $M$ is not geometrically finite and has no cusp. Does there exist a potential $F: T^1 M \to {\Bbb R}$ such that the Gibbs measure associated with $(\Gamma, F)$ is finite?
\section{Preliminaries} \subsection{Geometry on $T^1 \widetilde{M}$} We first recall a few notations and results related to the geometry of negatively curved manifolds.\\ Let $\partial_{\infty} \widetilde{M}$ be the boundary at infinity of $\widetilde{M}$, we define the limit set of $\Gamma$ by $$\Lambda \Gamma = \overline{ \Gamma x} \cap \partial_{\infty} \widetilde{M},$$ where $x$ is any point of $\widetilde{M}$.\\ An element $\xi_p \in \Lambda \Gamma$ is \textbf{parabolic} if there exists a parabolic isometry $\gamma \in \Gamma$ satisfying $\gamma \xi = \xi$.\\ A parabolic point $\xi \in \Lambda \Gamma$ is \textbf{bounded} if $\Lambda \Gamma / \Gamma_{\xi_p}$ is compact where $\Gamma_{\xi_p}$ is the maximal subgroup of $\Gamma$ fixing $\xi_p$. In this case, let ${\mathscr H}_{\xi}$ be a horoball centered at $\xi$ then, $$ {\mathscr C}_{\xi} = {\mathscr H}_{\xi} / \Gamma_{\xi_p} $$ is called the cusp associated with $\xi$.\\ We say that $\Lambda_c \Gamma \subset \Lambda \Gamma$ is the conical limit set if for all $\xi \in \Lambda_c \Gamma$ for some $\widetilde{x} \in \widetilde{M}$ there exists $\epsilon >0$ and a sequence $(\gamma_n)_{n \in {\Bbb N}}$ such that $(\gamma_n (\widetilde{x}))_{n \in {\Bbb N}}$ converges to $\xi$ and stays at distance at most $\epsilon$ from the geodesic ray $(\widetilde{x} \xi)$.\\ We define the parabolic limit set as follows.
$$\Lambda_p \Gamma = \{ \eta \in \Lambda \Gamma : \exists \gamma \in \Gamma \text{ parabolic }, \gamma \cdot \eta = \eta \}$$
Let us choose an origin $\widetilde{x}_0$ in $\widetilde{M}$ once and for all. We define the Dirichlet domain of $\Gamma$, centered on $\widetilde{x}_0$ as follows. $$ {\mathscr D} = \bigcap\limits_{\gamma \in \Gamma, \gamma \neq Id} \{ \widetilde{x} \in \widetilde{M} : d(\widetilde{x},\widetilde{x}_0) \leq d(\widetilde{x},\gamma \widetilde{x}_0) \}.$$
It is a convex domain having the following properties. \begin{itemize}
\item $\bigcup\limits_{\gamma \in \Gamma} \gamma {\mathscr D} = \widetilde{M},$
\item for all $\gamma \in \Gamma \backslash \{Id\}, \mathring{{\mathscr D}} \cap \gamma \mathring{{\mathscr D}} = \emptyset.$ \end{itemize}
We define the diagonal of $\Lambda \Gamma \times \Lambda \Gamma$ the set of points $(x,y) \in \Lambda \Gamma \times \Lambda \Gamma$ such that $x=y$. We denote by $\Delta$ this set.\\ For all $\xi, \eta \in \partial_{\infty} \widetilde{M}$, we denote by $(\xi \eta)$ the geodesic joining $\xi$ to $\eta$. We define the lift of the non wandering set on $T^1 \widetilde{M}$ by $$\widetilde{\Omega} = \{\widetilde{x} \in T^1 \widetilde{M}: \exists (\xi,\eta)\in \Lambda \Gamma \times \Lambda \Gamma \backslash \Delta, \text{ } \widetilde{x} \in
(\xi\eta) \}.$$ Let $\pi:T^1\widetilde{M} \to \widetilde{M}$ be the natural projection of the unit tangent bundle to the associated manifold. We denote by ${\mathscr C}\Lambda\Gamma$ the smallest convex set in $\widetilde{M}$ containing $\pi(\widetilde{\Omega})$.
$M$ is geometrically finite if one of the following equivalent conditions is satisfied \begin{enumerate}
\item $ \Lambda \Gamma = \Lambda_c \Gamma \cup \Lambda_p \Gamma $ = $\Lambda_c \Gamma \cup \{ \text{bounded parabolic fixed points} \}$,
\item For some $\epsilon >0$, the $\epsilon-$neighborhood of ${\mathscr C} \Lambda\Gamma / \Gamma$ has finite volume,
\item $M$ has finitely many ends, all of which are funnels and cusps of finite volume. \end{enumerate}
We define a map $$ C_F: \partial_{\infty} \widetilde{M} \times \widetilde{M}^2 \to {\Bbb R} $$ called the Gibbs cocycle of $(\Gamma,F)$ by $$ C_{F,\xi}(x,y)= C_F (\xi,x,y) = \lim\limits_{t \to \infty} \int_y^{\xi(t)}\widetilde{F} - \int_x^{\xi(t)} \widetilde{F}$$ where $t \mapsto \xi(t)$ is any geodesic ending at $\xi$.\\
Here is a technical lemma of \cite{PPS} giving estimates for the Gibbs cocycle. \begin{prop}\label{prop:major}(\cite{PPS})
For every $r_0>0$, there exists $c_1,c_2,c_3,c_4>0$ with $c_2,c_4\leq 1$ such that the following assertions hold.
(1) For all $x,y\in \widetilde{M}$ and $\xi\in\partial_\infty \widetilde{M}$, $$
|\;C_{F,\,\xi}(x,y)\;|\leq
\;c_1\,e^{d(x,\,y)} \;+\; d(x,y)\max_{\pi^{-1}(B(x,\,d(x,\,y)))}|\widetilde{F}|\;, $$ and if furthermore $d(x,y)\leq r_0$, then $$
|\;C_{F,\,\xi}(x,y)\;|\leq c_1\,d(x,y)^{c_2}+ d(x,y)\max_{\pi^{-1}(B(x,\,d(x,\,y)))}|\widetilde{F}|\;. $$
(2) For every $r\in\mathopen{[}0,r_0\mathclose{]}$, for all $x,y'$ in $\widetilde{M}$, for every $\xi$ in the shadow ${\mathscr O}_xB(y',r)$ of the ball $B(y',r)$ seen from $x$, we have $$
\Big|\;C_{F,\,\xi}(x,y')+\int_x^{y'} \widetilde{F}\;\Big|\leq
c_3 \;r^{c_4}+2r\max_{\pi^{-1}(B(y',\,r))}|\widetilde{F}|\;. $$ \end{prop} \subsection{Thermodynamic formalism for negatively curved manifolds} We start by recalling a few facts on Gibbs measures on negatively curved manifolds. The results of this paragraph come from \cite{PPS}.
Let $\widetilde{F}: T^1 \widetilde{M} \to {\Bbb R}$ be a $\Gamma-$invariant Hölder function. We will say that the induced function $F$ on $T^1 M = T^1 \widetilde{M} / \Gamma$ is a potential.\\
The Poincaré series associated with $(\Gamma,F)$ is defined by $$ P_{x,\Gamma,F} (s) = \sum\limits_{\gamma \in \Gamma} e^{\int_x^{\gamma x}\widetilde{F}-s}.$$ Its critical exponent is given by $$ \delta_{\Gamma,F} = \limsup\limits_{n \to \infty} \frac{1}{n} \log (\sum\limits_{\gamma \in \Gamma, n-1 \leq d(x,\gamma x)\leq n} e^{\int_x^{\gamma x} \widetilde{F}}). $$ We say that $(\Gamma,F)$ is of divergence type if\\ $P_{x,\Gamma,F}(\delta_{\Gamma,F})$ diverges.\\ When $F=0$ on $T^1M$, we will denote by $\delta_{\Gamma}$ the critical exponent associated to $(\Gamma,F)$.
\begin{prop} \label{prop:exp}
Let $F$ be the potential on $T^1 M = (T^1 \widetilde{M}) / \Gamma$ induced by the $\Gamma-$invariant potential
$\widetilde{F}:T^1 \widetilde{M} \to {\Bbb R}$.
\begin{enumerate}
\item The Poincaré series associated with $(\Gamma,F)$ converges if $s >\delta_{\Gamma,F}$ and diverges if $s<\delta_{\Gamma,F}$,
\item We have the upper bound
$$ \delta_{\Gamma,F} \leq \delta_{\Gamma} + \sup\limits_{\pi^{-1}({\mathscr C} \Lambda \Gamma)} \widetilde{F},$$
\item For every $c>0$, we have
$$ \delta_{\Gamma,F} = \limsup\limits_{n \to \infty} \frac{1}{n} \log (\sum\limits_{\gamma \in \Gamma, n-c \leq d(x,\gamma x)\leq n} e^{\int_x^{\gamma x} \widetilde{F}}).$$
\end{enumerate} \end{prop}
We define a set of measures on $\partial_{\infty} \widetilde{M}$ as the limit points when $s \to \delta_{\Gamma,F}^+$ of $$ \frac{1}{P_{x,\Gamma,F}(s)} \sum\limits_{\gamma \in \Gamma} e^{\int_x^{\gamma x} \widetilde{F}-s} h(d(x,\gamma x)) D_{\gamma x} = \mu^F_{x,s}.$$ where $h: {\Bbb R}_+ \to {\Bbb R}_+^*$ is a well chosen non-decreasing map and $D_{\gamma x}$ is the Dirac measure supported on $\gamma x$.
\begin{prop}
If $\delta_{\Gamma,F}< \infty$ then
\begin{enumerate}
\item $\{ \mu^F_{x,s} \}$ has at least one limit point when $s \to \delta_{\Gamma,F}^+$ with support $\Lambda \Gamma$,
\item If $\mu^F_x$ is a limit point then it is a Patterson density \textit{i.e}\\
$\forall \gamma \in \Gamma, x,y \in \widetilde{M}, \xi \in \partial_{\infty} \widetilde{M}$\\
$$ \gamma_* \mu^F_x = \mu^F_{\gamma x}, $$
$$ d\mu^F_x (\xi) = e^{-C_{F-\delta_{\Gamma,F}, \xi}(x,y)} d\mu^F_y (\xi). $$
\end{enumerate} \end{prop}
Using the Hopf parametrization on $T^1 \widetilde{M}$, each unit tangent vector $v$ can be written as $v= (v_+,v_-,t) \in \partial_{\infty} \widetilde{M} \times \partial_{\infty} \widetilde{M} \times {\Bbb R}$. We define a measure on $T^1 \widetilde{M}$ by $$ d \widetilde{m}_{\widetilde{F}} (v) = \frac{d \widetilde{\mu}_x^{\widetilde{F} \circ \iota} (v_-) d \widetilde{\mu}_x^{\widetilde{F}} (v) dt}{D_{F,x} (v_+,v_-) } $$
where $$D_{F,x} (v_+,v_-) = e^{- \frac{1}{2} (C_{F,v_-} (x, \pi (v)) + (C_{F\circ \iota,v_+} (x,\pi(v)))} $$ is the potential gap and $$ \iota \left\{
\begin{aligned}
T^1 \widetilde{M} &\to &T^1 \widetilde{M}\\
v &\mapsto& -v
\end{aligned} \right.$$ is the antipodal map.\\ This measure is called the Gibbs measure associated to $(\Gamma,F)$.\\ It is a measure independent of $x$, invariant under the action of $\Gamma$ and invariant by the geodesic flow. Hence it defines a measure $m^F$ on $T^1M$ invariant by the geodesic flow.\\
Let $m \in {\mathscr M}^1 (T^1 M)$ be a measure with finite entropy $h_{m}(\phi_t)$. We define the \textbf{metric pressure} of a potential $F$ with respect to the measure $m$ as the quantity $$ P_{\Gamma,F}(m) = h_{m}(\phi_t) +\int_{T^1 M} F dm .$$ We say that the supremum $$ P(\Gamma,F) = \sup\limits_{m \in {\mathscr M}(T^1M)} P_{\Gamma,F}(m) $$ is the \textbf{topological pressure} of the potential $F$. An element realizing this upper bound is called an \textbf{equilibrium state} for $(\Gamma,F)$.
\begin{thm}\cite{otal2004, PPS}\label{thm:varp}
Let $\widetilde{M}$ be a complete, simply connected Riemannian manifold with pinched negative curvature, $\Gamma$ a non-elementary discrete group of isometries of $\widetilde{M}$
and $\widetilde{F}: T^1 \widetilde{M} \to {\Bbb R}$ a Hölder-continuous $\Gamma$-invariant map with $\delta_{\Gamma,F} < \infty$.
\begin{enumerate}
\item We have
$$ P(\Gamma,F) = \delta_{\Gamma,F}.$$
\item If there exists a finite Gibbs measure $m_F$ for $(\Gamma,F)$ such that the negative part of $F$ is $m_F -$integrable, then
$m^F = \frac{m_F}{\norm{m_F}}$ is the unique equilibrium state for $(\Gamma,F)$. Otherwise, there exists no equilibrium state for
$(\Gamma,F)$.
\end{enumerate} \end{thm}
\section{Mixing property for the geodesic flow} The question of the topological mixing of the geodesic flow on a negatively curved manifold is still open in full generality. This question is closely related to mixing with respect to a Gibbs measure (see \cite{PPS}).\\ We define the length of an element $\gamma \in \Gamma$ by $\ell (\gamma) = \inf\limits_{z \in M} d(z, \gamma z)$. \begin{thm}
If $\delta_{\Gamma,F} < \infty$ and $m^F$ is finite then the following propositions are equivalent.
\begin{enumerate}
\item The geodesic flow is topologically mixing on $\Omega $,
\item The geodesic flow is mixing with respect to $m^F$,
\item $L(\Gamma) = \{ \ell(\gamma); \gamma \in \Gamma \}$ is not contained in a discrete subgroup of ${\Bbb R}$.
\end{enumerate} \end{thm}
Here are some cases where the geodesic flow is known to be topologically mixing \cite{MR1617430},\cite{MR1703039},\cite{MR1779902}.
\begin{lem}\label{lem:mix}
Let $\Gamma$ be a non elementary group of isometries of a Hadamard manifold $\widetilde{M}$ with pinched negative curvature.
If $M = \widetilde{M} / \Gamma$ satisfies one of the following properties then the restriction of the geodesic flow to its non-wandering set is topologically mixing.
\begin{enumerate}
\item The curvature of $M$ is constant,
\item $\dim M = 2$,
\item There exists a parabolic isometry in $\Gamma$,
\item $\Omega = T^1 M$.
\end{enumerate} \end{lem}
To conclude this section, let us recall the Hopf-Tsuji-Sullivan criterion for the ergodicity of the geodesic flow with respect to the Gibbs measure (see \cite{PPS} for a proof)
\begin{thm}\label{thm:HTS}
The following assertions are equivalent:
\begin{enumerate}
\item $(\Gamma,F)$ is of divergence type,
\item $\forall x\in \widetilde{M}, \mu_x^{F} (\partial_{\infty} \widetilde{M} \backslash \Lambda_c \Gamma) = 0$,
\item The dynamical system $(T^1M, (\phi_t)_{t \in {\Bbb R}},m_F)$ is ergodic.
\end{enumerate} \end{thm}
As a consequence of this theorem, one can show that if $\delta_{\Gamma,F} < \infty$, the Patterson density $(\mu_x^{F})_{x \in \widetilde{M}}$ associated with $(\Gamma,F)$ is non-atomic (see \cite{PPS} Proposition 5.13).
\section{A finiteness criterion}
First, let us give a criterion for the finiteness of the Gibbs measure. This result comes from \cite{MR1776078} for a potential $F=0$. For
the general case where $F$ is an Hölder potential, the proof is given in \cite{PPS}.
\begin{thm}
Suppose that $\Gamma$ is a geometrically finite group with $(\Gamma,F)$ of divergence type and $\delta_{\Gamma,F}< \infty$.
The Gibbs measure $m_{F}$ is finite if and only if for every parabolic fixed point $\xi_p$
\begin{center}
$\sum\limits_{\gamma \in \Gamma_{\xi_p}} d(x,\gamma x) e^{\int_x^{\gamma x} (\widetilde{F}-\delta_{\Gamma,F})} $
\end{center}
converges. \end{thm}
\begin{Def}
$(\Gamma ,F)$ satisfies the spectral gap property if for all parabolic points $ \xi_p \in \partial_{\infty} \widetilde{M}$,
$$ \delta_{\Gamma_{\xi_p},F} < \delta_{\Gamma,F}. $$ \end{Def}
Proposition 2 of \cite{MR1776078} gives a criterion for this property for the zero potential. The following proposition is more general and applies to all Hölderian potentials.
\begin{prop}
Let $\widetilde{M}$ be a Hadamard manifold with pinched negative curvature and $\Gamma$ a geometrically finite discrete group acting on
it. Suppose there exists a bounded Hölderian potential $\widetilde{F}: T^1 \widetilde{M} \to {\Bbb R}$.\\
If for all parabolic fixed point $\xi_p$ the couple $(\Gamma_{\xi_p},F)$ is of divergence type, then $(\Gamma,F)$ satisfies the spectral gap property.
Moreover, the Gibbs measure $m^F$ associated to $(\Gamma,F)$ is finite. \end{prop}
\begin{proof} To prove the first claim, we follow the ideas of \cite{MR1776078} when $F=0$.\\ Since the action of $\Gamma_{\xi_p}$ on $\partial_{\infty} \widetilde{M}$ has a fundamental domain ${\mathscr G}$ in $\partial_{\infty} \widetilde{M}$, we have $$ \mu_x^F (\partial_{\infty} \widetilde{M}) = \sum\limits_{g \in \Gamma_{\xi_p}} \mu_x^F (g {\mathscr G}) + \mu_x^F (\Lambda \Gamma_{\xi_p}). $$ Moreover, since there exists $K\in {\Bbb R}$ such that $$ \left\{
\begin{aligned}
\mu_x^F (g {\mathscr G}) = \int_{{\mathscr G}} e^{-C_{F-\delta_{\Gamma,F}, \xi}(x,gx)} d\mu_{gx} (\xi)\\
|C_{F,\xi} (x,gx) + \int_x^{gx} \widetilde{F}| \leq K,
\end{aligned} \right. $$ we have $$ \mu_x^F (g {\mathscr G}) \geq (e^{\int_x^{gx} \widetilde{F} - \delta_{\Gamma,F}}) ( e^{-K} \mu_x^F({\mathscr G}) ) $$ and $$ \infty > \mu_x^F (\partial_{\infty} \widetilde{X}) \geq \sum\limits_{g \in \Gamma_{\xi_p}} \mu_x^F (g {\mathscr G})
\geq C_0 \cdot P_{x,\Gamma_{\xi_p},F}(\delta_{\Gamma,F}). $$ So, $\delta_{\Gamma,F} > \delta_{\Gamma_{\xi_p},F}.$\\
For the second claim, since $(\Gamma,F)$ satisfies the spectral gap property for $M = \widetilde{M} / \Gamma$ and $$ \forall \epsilon >0, \exists C_{\epsilon}>0, d(x,\gamma x) \geq C_{\epsilon} \Rightarrow
e^{\epsilon d(x,\gamma x)} > d(x,\gamma x), $$ we have
$$ \sum\limits_{\gamma \in \Gamma_{\xi_p}} d(x,\gamma x) e^{\int_x^{\gamma x} \widetilde{F} - \delta_{\Gamma, F}} \leq \sum\limits_{\gamma \in \Gamma_{\xi_p}} e^{\int_x^{\gamma x} \widetilde{F} - (\delta_{\Gamma, F} - \epsilon)} $$
Choosing $\epsilon$ small enough such that $\frac{\delta_{\Gamma,F} - \delta_{\Gamma_{\xi_p},F}}{2} > \epsilon$, the series converges and the Gibbs measure is finite. \end{proof}
\section{Construction of the potentials} We now construct a $\Gamma-$invariant potential $\widetilde{H}: T^1 \widetilde{M} \to {\Bbb R}$ such that the associated Gibbs measure is finite and which critical exponent associated to $(\Gamma_{\xi_p},H)$ is of divergence type.\\
Since $M$ is geometrically finite, the set $Par_{\Gamma}$ of parabolic points $\xi_p \in \partial_{\infty} \widetilde{M}$ intersecting the boundary of the Dirichlet domain is finite.\\ We define for those parabolic points a family of disjoint horoballs $\{{\mathscr H}_{\xi_p} (u_{0,\xi_p}) \}_{\xi_p}$ on $\widetilde{M}$ passing through a well chosen point $\widetilde{u}_{0,\xi_p}$ of the cusp.
For any $\widetilde{u}$ in ${\mathscr H}_{\xi_p} (\widetilde{u}_{0,\xi_p})$, we define a height function $\rho: \widetilde{M} \to {\Bbb R}$ by the Buseman cocycle at $\widetilde{u}_{0,\xi_p}$: $$ \rho(\widetilde{u}) = \beta_{\xi_p}(\widetilde{u},\widetilde{u}_{0,\xi_p}).$$ This cocycle coincides with the Gibbs cocycle for the potential $F= -1$.\\ The curve levels of this function are the horocycles based on $\xi_p$ passing through $\widetilde{u}$.\\
Let $t_n$ be a decreasing sequence of positive numbers converging to $0$.\\ One can construct a sequence $Y_n$ of positive numbers such that
$$ \left\{
\begin{aligned}
Y_{n+1} \geq Y_n + t_n - t_{n+1}, \\
\sum \limits_{p \in \rho^{-1}(]Y_n , Y_{n+1}])} e^{d(x_0,px_0)(t_n - \delta_{\Gamma_{\xi_p}})} \geq 1.
\end{aligned} \right. $$
for all $\xi_p \in Par_{\Gamma}$, we define a $\Gamma$-invariant map on ${\mathscr D} \cap {\mathscr H}_{\xi_p}(u_{0,\xi_p})$ as follows:\\ for every $\widetilde{u}$ in ${\mathscr H}_{\xi_p} (u_{0,\xi_p})$,
$$ \widetilde{H}_1(\widetilde{u})= \left\{
\begin{aligned}
t_n + Y_n -\rho (\widetilde{u}) \text{on } \rho^{-1}(]Y_n,Y_n+t_n-t_{n+1}]) , \\
t_{n+1} \text{on } \rho^{-1}(]Y_n+t_n-t_{n+1},Y_{n+1}]).
\end{aligned} \right. $$ We extend $\widetilde{H}_1$ to $\widetilde{M}$ as follows.\\ First, we extend it to ${\mathscr D}$ by a constant function such that $$ \widetilde{H}_1: {\mathscr D} \to {\Bbb R}$$ is Hölder-continuous. Next, we extend it to $\widetilde{M}$ as follows. For all $\gamma \in \Gamma$, if $\widetilde{x} \in \gamma {\mathscr D}$ then $$ \widetilde{H}_1 (\widetilde{x}) = \widetilde{H}_1 (\gamma^{-1} \widetilde{x}). $$
Let $\widetilde{H}$ be the $\Gamma-$invariant potential obtained by pulling back ${H}_1$ on $T^1 \widetilde{M}$. We denote by $H: T^1 M \to {\Bbb R}$ the map induced by $\widetilde{H}$ on $T^1 M$.
\begin{lem}(Coudène \cite{C04})
$\widetilde{H}$ is a $\Gamma-$invariant Hölderian potential such that for all parabolic fixed point $\xi_p$, the critical exponent associated with
$(\Gamma_{\xi_p},H)$ is of divergence type. \end{lem}
We have therefore constructed a $\Gamma$-invariant Hölderian potential on $T^1 M$ such that the associated Gibbs measure is finite and is supported on $\Omega$.\\
Remark that the divergence of the parabolic subgroups only depends on the value of the potential in the cusp. In the previous construction, we made the assumption that the potential was constant on $$T^1 M_0 = T^1 ({\mathscr C} \Lambda \Gamma \bigcup\limits_{\xi_p \in Par_{\Gamma}} {\mathscr H}_{\xi_p}) / \Gamma .$$ However, taking any bounded potential such that the resulting function on $T^1 M_0$ is Hölder-continuous, the associated Gibbs measure will still be finite.\\ We now choose $p\in T^1 M = T^1\widetilde{M} / \Gamma$ such that $\phi_{t} (p)$ is periodic. We will denote by ${\mathscr O}(p)$ the closed curve $\phi_{{\Bbb R}}(p)$ and assume that ${\mathscr O}(p) \subset T^1 M_0$.\\ For all $n \in {\Bbb N}$, we define a Lipschitz-continuous potential by
$$ F_n (x) = \max \{ c_n - c_n d ({\mathscr O}(p),x) ; H(x) \}.$$
\begin{prop}
For all $n \in {\Bbb N}$, the critical exponent $\delta_{\Gamma,F_n}$ is finite. Moreover, we have
$$c_n \leq \delta_{\Gamma,F_n} \leq \delta_{\Gamma} + c_n.$$ \end{prop} \begin{proof} Since $c_k = \sup\limits_{x \in T^1 M} F_k(x)$, the upper bound $\delta_{\Gamma,F_k} \leq \delta_{\Gamma} + c_k$ is evident by the second claim of Proposition \ref{prop:exp}. \\ Let $p$ be the periodic point of $T^1M$ defined above and $h\in \Gamma$ the generator of the isometry group fixing the periodic orbit $\phi_{{\Bbb R}}(p)$. Let $H= <h>$ and denote by $\ell$ the length of $h$. By the very definition of critical exponents, we have
\begin{center}\begin{align}
\delta_{\Gamma,F_k} &= & \limsup\limits_{n \to \infty} \frac{1}{n} \log \sum\limits_{\gamma \in \Gamma,\;n- \ell< d(x,\,\gamma x)\leq n} e^{\int_x^{\gamma x}\widetilde{F}_k} \\
&\geq & \limsup\limits_{n \to \infty} \frac{1}{n} \log \sum\limits_{\gamma \in H ,\;n-\ell< d(x,\,\gamma x)\leq n} e^{\int_x^{\gamma x}\widetilde{F}_k}. \end{align}\end{center} Since the critical exponent does not depend on the choice of a base point, one can choose $x = \widetilde{p}$ where $\widetilde{p}$ is a lift of $p$ on $T^1 \widetilde{M}$. Therefore, the fact that the value of the potential on ${\mathscr O}(p)$ is constant, equal to $c_k$ implies that $$ \int_{\widetilde{p}}^{\gamma {\widetilde{p}}}\widetilde{F}_k = d(x, \gamma x ) c_k \geq (n- \ell) c_k$$ which gives \begin{center}\begin{align}
\delta_{\Gamma,F_k} &\geq \limsup\limits_{n \to \infty} \frac{1}{n} \log \sum\limits_{\gamma \in H ,\;n-\ell< d(x,\,\gamma x)\leq n} e^{(n-\ell) c_k}
\geq \limsup\limits_{n \to \infty} \frac{1}{n} \log \left( A_x e^{(n- \ell)c_k} \right), \end{align}\end{center} where $A_x = \sharp \{\gamma \in H : n- \ell < d(x,\gamma x) \leq n \}$.\\ Since the group $H$ is generated by a hyperbolic isometry $h$, for all $\gamma \in H$, there exists $i \in {\Bbb N}$ such that $\gamma = h^i$ and $\ell (\gamma)= \ell (h^i) = i \ell(h).$ Therefore the quantity $A_x$ does not depend on $n$ and $$ \delta_{\Gamma,F_k} \geq c_k.$$ which concludes the proof. \end{proof} Therefore, the Gibbs measure $m_{F_n}$ associated with $(\Gamma,F_n)$ of dimension $\delta_{\Gamma,F_n}$ exists for all $n\in {\Bbb N}$.\\
\section{Proof of the main theorem} Let $D_{{\mathscr O}(p)}$ be the Dirac measure supported on ${\mathscr O}(p)$. We prove Theorem \ref{thm:princ} which states the following.\\
\textit{
Let $M$ be a geometrically finite, negatively curved manifold and $\phi_t$ its geodesic flow.
If $\phi_t$ is topologically mixing on $\Omega$, then the set of probability measures that are mixing with respect
to the geodesic flow is dense in ${\mathscr M}^1 (T^1 M)$ for the weak topology.}\\
Here is our strategy: since the geodesic flow on a manifold with pinched negative curvature satisfies the
closing lemma (see \cite{MR1441541}) and admits a local product structure, we use the following result.
\begin{prop}(Coudène-Schapira \cite{MR2735038})
Let $M$ be a complete, connected Riemannian manifold with pinched negative curvature and $\phi_t$ its geodesic flow.
Then the set of normalized Dirac measures on periodic orbits is dense in the set of all invariant measures ${\mathscr M}^1 (T^1 M)$.
\end{prop}
It is therefore clear that the following proposition implies Theorem \ref{thm:princ}.
\begin{prop} \label{prop:conv}
For all $p \in T^1 M$ such that ${\mathscr O}(p)= \phi_{{\Bbb R}} (p)$ is periodic, there exists a sequence $\{m_k \}_{k \in {\Bbb N}}$ of measures satisfying
the following properties
\begin{enumerate}
\item $m_k$ is a probability measure which is mixing with respect to the geodesic flow,
\item $m_k \rightharpoonup D_{{\mathscr O}(p)}$.
\end{enumerate}
\end{prop}
Let ${\rm h_{\rm top}\,}(\phi_t)$ be the topological entropy of the geodesic flow on $T^1 M$ and $h_{\mu}(\phi_t)$ the measure theoretic entropy of the geodesic flow with respect to $\mu$. D. Sullivan proved in \cite{Su} that if the Bowen-Margulis measure $m_{BM}$ (\textit{i.e} the Gibbs measure associated with the potential $F=0$) is finite then $$ \delta_{\Gamma} = h_{m_{BM}}(\phi_t) .$$
Using a result of C.J.Bishop and P.W.Jones \cite{BJ} connecting the critical exponent $\delta_{\Gamma}$ with the Hausdorff dimension of the conical limit set of $\Gamma$, J.P.Otal and M.Peigné proved that for all $\phi_t-$invariant probability measures $m \in {\mathscr M}^1(T^1\widetilde{M})$ which are not the Bowen-Margulis measure, we have the strict inequality, $$ h_{\mu}(\phi_t) < \delta_{\Gamma} .$$ We refer to F. Ledrappier \cite{ENSML} for a survey of these results.
\begin{thm} \cite{otal2004} Let $\widetilde{M}$ be a simply connected, complete Riemannian manifold with pinched negative curvature and $\Gamma$ be a non-elementary discrete group of isometries of $\widetilde{M}$, then $$ {\rm h_{\rm top}\,} (\phi_t)= \delta_{\Gamma}. $$ Moreover, there exists a probability measure $\mu$ maximizing the entropy if and only if the Bowen-Margulis measure is finite and $\mu= m_{BM}$. \end{thm} We begin the proof of the main theorem. First, we state a general result which holds true for any metric space $X$ satisfying a variational principle. In the next claim, we suppose that $F: X \to {\Bbb R}$ is a measurable function such that there exist an invariant compact set $K \subset X$ and a neighborhood $V$ of $K$ satisfying the following assumptions: \begin{center}
$ \left\{
\begin{aligned}
\forall x \in K, F(x) = c = \sup\limits_{x \in X} F(x)\\
\sup\limits_{x\in X \backslash V} F (x)= c' < c.
\end{aligned}
\right.$
\end{center} We say that a probability measure $\mu$ with finite entropy is an equilibrium state for a potential $F:X \to {\Bbb R}$ if it achieves the supremum of
$$ m \mapsto h_m (\phi_t) + \int_X F dm $$ over all invariant probability measures with finite entropy.
\begin{lem}\label{lem:majvar}
Let $X$ be a metric space, $\phi_t$ a flow defined on $X$ and $F:X \to {\Bbb R}$, $K$, $V$ defined as above.\\
Suppose there exists an equilibrium state $\mu$ for $F$, then
$$ \mu(X \backslash V) \leq \frac{h_{\mu}(\phi_t)}{c-c'}. $$ \end{lem}
\begin{proof}
Let $m_K$ be a probability measure supported on $K$. Since $\mu$ realises the supremum of
$$ m \in {\mathscr M}^1(X) \mapsto h_m (\phi_t) + \int_X F dm $$
we have
$$ h_{\mu} (\phi_t) + \int_X F d\mu \geq h_{m_K} (\phi_t) + \int_X F dm_K $$
which implies
\begin{equation}\label{eqn:1}
h_{\mu} (\phi_t) + \int_X F d\mu \geq \int_X F dm_K
\end{equation}
since $h_{m_K} (\phi_t) \geq 0$. Moreover, since the potential $F$ is constant on $K$ we have
$$ \int_X F dm_K = c $$
which can be written as
\begin{equation}\label{eqn:2}
\int_X F dm_K = c (\mu(V) + \mu(X \backslash V)).
\end{equation}
Combining equations \ref{eqn:1} and \ref{eqn:2}, we obtain
$$
\begin{array}{rcl}
h_{\mu} (\phi_t) &\geq& \int_X F dm_K - \int_X F d\mu \\
&\geq & c (\mu(V) + \mu(X \backslash V)) - c\mu(V) - c' \mu(X \backslash V) )
\end{array}
$$
which finally gives
$$ \frac{h_{\mu}(\phi_t)}{c-c'} \geq \mu(X \backslash V). $$ \end{proof}
Recall that a sequence $\{\mu_n\}_{n \in {\Bbb N}}$ of probability measures on a Polish space $X$ is tight if for all $\epsilon >0$ there exists a compact set $K \subset X$ such that $$ \forall n \in {\Bbb N}, \text{ } \mu_n(X \backslash K)< \epsilon .$$ We give a criterion for the convergence of a sequence of probability measures to the Dirac measure supported on the periodic orbit ${\mathscr O}(p)$. We denote by $V_{\epsilon}$ the subset of $T^1 M$ defined by
$$ V_{\epsilon}= \{ x \in T^1M : d(x, {\mathscr O}(p)) \leq \epsilon \}.$$
\begin{lem} \label{lem:cv}
The following assertions are equivalent:
\begin{enumerate}
\item The sequence of probability measures $\{m^{F_n} \}_{n \in {\Bbb N}}$ converges to the Dirac measure supported on ${\mathscr O}(p)$,
\item for all $\epsilon >0$,
$$ \lim\limits_{n\to \infty} m^{F_n}(T^1 M \backslash V_{\epsilon}) =0 . $$
\end{enumerate} \end{lem} \begin{proof}It is clear that $(1) \Rightarrow (2)$. Let us show that $(2) \Rightarrow (1)$.\\
We first notice that $\{m^{F_n} \}_{n \in {\Bbb N}}$ is tight. Indeed,
let $V$ be a compact subset of $T^1 M$ containing ${\mathscr O}(p)$. Since condition $(2)$ is satisfied, for all $\epsilon >0$, there exists $N_0 >0$ such that
for all $n \geq N_0,$
$$ m^{F_n} (T^1 M \backslash V ) < \epsilon. $$
For all $n \in \{1,..,N_0-1 \}$, we can also find a compact set $K_n$ such that
$$ m^{F_n} (T^1 M \backslash K_n) < \epsilon.$$
Define the compact set $K = (\bigcup\limits_{n = 1}^{N_0 -1} K_n) \cup V$.\\
For all $n \in {\Bbb N}$, $K$ satisfies
$$ m^{F_n}(T^1 M \backslash K) \leq \epsilon. $$
Therefore, the sequence $\{m^{F_n} \}_{n \in {\Bbb N}}$ is tight.\\
Since condition $(2)$ is satisfied and using the fact that the unique invariant probability measure supported on ${\mathscr O}(p)$ is $D_{{\mathscr O}(p)}$, each converging subsequence
of $\{m^{F_n}\}$ converges to $D_{{\mathscr O}(p)}$.\\
Therefore, by Prokhorov's Theorem \cite{Pro}, each sub sequence of $\{m^{F_n}\}$ possesses a further subsequence converging weakly to $D_{{\mathscr O}(p)}$ so the sequence
$m^{F_n}$ converges weakly to $D_{{\mathscr O}(p)}. $ \end{proof}
We are now able to prove our main result of convergence. \begin{thm}
The sequence $\{m^{F_n}\}_{n \in {\Bbb N}}$ of probability measures converges to the Dirac measure supported on ${\mathscr O}(p)$. \end{thm} \begin{proof}
Let $D_{{\mathscr O}(p)}$ be the Dirac measure supported on the periodic orbit ${\mathscr O}(p)$. Recall that $c_n = \sup\limits_{x \in T^1 M} F_n(x)$.\\
By lemma \ref{lem:cv}, we
have to show that
$$\lim\limits_{n\to \infty} m^{F_n}(T^1 M \backslash V_{\epsilon}) =0 .$$
Using the variational principle described in Theorem \ref{thm:varp}, we have
$$ \delta_{\Gamma,F_n} = \sup\limits_{\mu \in {\mathscr M}^1(T^1M)} P_{\Gamma,F_n}(\mu) = P_{\Gamma,F_n}(m^{F_n})$$
and
$$ \delta_{\Gamma,F_n} \geq P_{\Gamma,F_n}(D_{{\mathscr O}(p)}) = h_{D_{{\mathscr O}(p)}}(\phi_t) +\int_{T^1 M} F_n dD_{{\mathscr O}(p)} .$$
Since ${\mathscr O}(p)$ is invariant by the action of the geodesic flow and
$$h_{m_{F_n}} (\phi_t) \leq \delta_{\Gamma} < \infty,$$
one can use the claim of lemma \ref{lem:majvar} to obtain the following inequality
$$ m^{F_n}(T^1 M \backslash V_{\epsilon}) \leq \frac{\delta_{\Gamma}}{c_n -c'_n}$$
where
$$c'_n = \sup\limits_{x \in T^1M \backslash V_{\epsilon}} F_n(x) .$$
By the definition of the potential $F_n$, we know that
$$c'_n \leq \max\{c_0,c_n(1-\epsilon) \}.$$
Therefore, for all $\epsilon >0$ and $n$ large enough,
$$m^{F_n}(T^1 M \backslash V_{\epsilon}) \leq \frac{\delta_{\Gamma}}{c_n \epsilon}$$
implies that
$$ \lim\limits_{n\to \infty} m^{F_n}(T^1 M \backslash V_{\epsilon}) =0 .$$
Which concludes the proof. \end{proof}
Finally, we are able to prove corollary \ref{cor:gef} which states that the set of weak-mixing measures is a dense $G_{\delta}$ subset
of ${\mathscr M}^1(T^1 M)$.\\
Our proof relies on the following theorem.
\begin{thm}(Coudène-Schapira \cite{MR3322793})
Let $(\varphi^t)_{t\in {\Bbb R}}$ be a continuous flow on a Polish space. The set of weak-mixing measures on $X$ is a $G_{\delta}$ subset
of the set of Borel invariant probability measures on $X$.
\end{thm}
\begin{proof}(of corollary \ref{cor:gef})\\
Since mixing measures are weak-mixing, Theorem \ref{thm:princ} implies that weak-mixing measures are dense in the set of probability
measures on $\Omega$. The previous theorem insures us that it is a $G_{\delta}$ set.
Since negatively curved manifolds with pinched curvature satisfy the closing lemma, it is shown in \cite{MR2735038} that the set of
invariant measures with full support on $\Omega$ is a dense $G_{\delta}$ subset of the set of invariant probability measures on $\Omega$.
The intersection of those two dense $G_{\delta} $ sets is a dense $G_{\delta} $ set by the Baire Category Theorem.
\end{proof}
\section{Non-geometrically finite manifolds with cusps}
We now prove corollaries \ref{cor:geinf} and \ref{cor:surf}. First of all, remark that since the manifold possesses a cusp then from lemma \ref{lem:mix}, the geodesic flow is topologically mixing on $T^1M= T^1\widetilde{M} / \Gamma$. \\ Let $\gamma \in \Gamma$ and $D(\gamma)$ the subset of $\widetilde{M}$ bounded by $$\{ \widetilde{x} \in \widetilde{M} : d(\widetilde{x},\widetilde{x}_0) = d(\widetilde{x},\gamma \widetilde{x}_0) \}$$ and containing $\gamma \widetilde{x_0}$. We define the subset $C_{\gamma}$ on $\partial_{\infty} \widetilde{M}$ as follows. $$ C_{\gamma} = \partial_{\infty} \widetilde{M} \cap \overline{D(\gamma)}.$$
The proof of corollary \ref{cor:geinf} is deduced from the following result. \begin{lem}\label{lem:V_0}
Let $M$ be a connected, complete pinched negatively curved manifold with a cusp and $\phi_t$ the geodesic flow defined on $T^1 M$.
There exists a geometrically finite manifold $\hat{M}$ such that its geodesic flow $\hat{\phi}_t$ is topologically mixing and a
covering map $ \rho: T^1 \hat{M} \to T^1M $ such that the diagram
$$
\begin{CD}
T^1 \hat{M} @>{\hat{\phi}_t}>> T^1 \hat{M} \\
@V{\rho}VV @VV{\rho}V \\
T^1 M @>{\phi_t}>> T^1 M
\end{CD}
$$
commutes. \end{lem}
\begin{proof} Let $\xi_p \in \partial_{\infty}\widetilde{M}$ be a bounded parabolic point fixed by a parabolic isometry $\gamma_p \in \Gamma$ and $h \in \Gamma$ be a hyperbolic transformation.\\ Let $N>0$ be defined such that the sets $C_{\gamma_p^N} , C_{h^N}, C_{\gamma_p^{-N}} , C_{h^{-N}}$ have disjoint interiors. We define $\Gamma_0 =$ $<\gamma_p^N , h^N>$, a subgroup of $\Gamma$. The ping-pong lemma shows that $\Gamma_0$ is a discrete group which acts freely discontinuously on $\widetilde{M}$. So, the quotient $\widetilde{M} / \Gamma_0$ is a geometrically finite manifold. \end{proof}
\begin{proof}(of corollary \ref{cor:geinf}) Let $\hat{M}$ be a geometrically finite manifold constructed as in lemma \ref{lem:V_0} and $\rho$ its associated covering map. We can use the proof of Theorem \ref{thm:princ} on $\hat{M}$ and construct a sequence $\hat{m}_k$ of invariant mixing measures for $\hat{\phi}_t$ such that $\hat{m}_k \rightharpoonup D_{{\mathscr O}(p)}$.\\ Since the geodesic flow $\phi_t$ is a factor of $\hat{\phi}_t$, we can define $\nu_k$ to be the push-forward by $\rho$ of $\hat{m}_k$, then it is an invariant mixing measure on $T^1M$ and for all bounded continuous function $g$ on $T^1 M$, $$ \begin{array}{rcl}
\lim\limits_{k \to \infty} \int_{T^1M} g d\nu_k &=& \lim\limits_{k \to \infty} \int_{T^1\hat{M}} g\circ \rho dm_k \\
&=& \int_{T^1 \hat{M}} g\circ \rho dD_{{\mathscr O}(p)} \\
&=& \int_{T^1 M} g d(\rho_* D_{{\mathscr O}(p)}). \end{array} $$ \end{proof}
We end up by the proof of corollary \ref{cor:surf}. In the case of a surface $S$ (or a constant negatively curved manifold), we don't need to ask for the existence of a bounded parabolic point. Since the geodesic flow is always topologically mixing in restriction to its non wandering set, we can choose two hyperbolic isometries $h_1, h_2$ in $\Gamma$ such that the subgroup $\Gamma_0 = <h_1, h_2>$ is convex-cocompact.\\ The same proof as lemma \ref{lem:V_0} shows us that the geodesic flow $\phi_t$ on $S$ is a factor of the geodesic flow $\hat {\phi_t}$ on the convex-cocompact manifold $T^1 S_0 = T^1\widetilde{S} / \Gamma_0 $. Therefore, the previous proof gives us the density result.
\end{document} | arXiv |
Square $ABCD$ and equilateral triangle $AED$ are coplanar and share $\overline{AD}$, as shown. What is the measure, in degrees, of angle $BAE$? [asy] size(3cm);
pair A,B,C,D,E;
A = (0,0);
B = (0,1);
C = (1,1);
D = (1,0);
E = dir(60);
draw(A--B--C--D--E--A--D);
label("$A$",A,dir(180));
label("$B$",B,dir(180));
label("$C$",C,dir(0));
label("$D$",D,dir(0));
label("$E$",E,dir(0));
[/asy]
The angles in a triangle sum to 180 degrees, so the measure of each angle of an equilateral triangle is 60 degrees. Therefore, the measure of angle $EAD$ is 60 degrees. Also, angle $BAD$ measures 90 degrees. Therefore, the measure of angle $BAE$ is $90^\circ-60^\circ=\boxed{30}$ degrees. | Math Dataset |
\begin{document}
\title{An Upper Bound for the Number of Solutions of Ternary Purely Exponential Diophantine Equations \uppercase\expandafter{\romannumeral2} }
\author{ Yongzhong Hu and Maohua Le } \maketitle
\maketitle \edef \tmp {\the \catcode`@}
\catcode`@=11
\def \@thefnmark {}
\@footnotetext {Supported by the National Natural Science Foundation of China(No.10971184)}
\catcode`@=\tmp
\let\tmp = \undefined
\begin{abstract} Let $a,b,c$ be fixed coprime positive integers with $\min\{a,b,c\}>1$. In this paper, by analyzing the gap rule for solutions of the ternary purely exponential diophantine equation $a^x+b^y=c^z$, we prove that if $\max\{a,b,c\}\geq 10^{62}$, then the equation has at most two positive integer solutions $(x,y,z)$. \end{abstract}
{\bf Keywords}: ternary purely exponential diophantine equation; upper bound for solution number; gap rule for solutions
{\bf 2010 Mathematics Subject Classification:} 11D61
\section {Introduction} \quad Let $\mathbb{Z,N}$ be the sets of all integers and positive integers respectively. Let $a,b,c$ be fixed coprime positive integers with $\min\{a,b,c\}>1$. In this paper we discuss the number of solutions $(x,y,z)$ of the ternary purely exponential diophantine equation
\begin{equation}\label{1.1} a^x+b^y=c^z, x,y,z\in\mathbb{N}. \end{equation}
In 1933, K. Mahler \cite{mah8} used his $p-$adic analogue of the Thue-Siegel method to prove that $(\ref{1.1})$ has only finitely many solutions $(x,y,z)$. His method is ineffective. Later, an effective result for solutions of $(\ref{1.1})$ was given by A.O. Gel'fond \cite{gel4}. Let $N(a,b,c)$ denote the number of solutions $(x,y,z)$ of $(\ref{1.1})$. As a straightforward consequence of an upper bound for the number of solutions of binary $S-$unit equations due to F. Beukers and H. P. Schlickewei \cite{beu2}, we have $N(a,b,c)\leq2^{36}$. In nearly two decades, many papers investigated the exact values of $N(a,b,c)$. The known results showed that $(\ref{1.1})$ has only a few solutions for some special cases(see the references of \cite{hu5} and \cite{hu6} ).
Recently, Y.-Z. Hu and M.-H. Le \cite{hu5,hu6} successively proved that (\rmnum 1) if $a,b,c$ satisfy certain divisibility conditions and $\max\{a,b,c\}$ is large enough, then $(\ref{1.1})$ has at most one solution $(x,y,z)$ with $\min\{x,y,z\}>1$. (\rmnum 2) If $\max\{a,b,c\}>5\times 10^{27}$, then $N(a,b,c)\leq3$. R. Scott and R. Styer \cite{sco9} proved that if $2\nmid c$, then $N(a,b,c)\leq2$. The proofs of the first two results are using the Gel'fond-Baker method with an elementary approach, and the proof of the last result is using some elementary algebraic number theory methods. In this paper, by analyzing the gap rule for solutions of $(\ref{1.1})$ along the approach given in \cite{hu6}, we prove a general result as follows:
\begin{theo} If $\max\{a,b,c\}\geq 10^{62}$, then $N(a,b,c)\leq2$. \end{theo}
Notice that, for any positive integer $k$ with $k>1$, if $(a,b,c)=(2,2^k-1,2^k+1)$, then $(\ref{1.1})$ has solutions $(x,y,z)=(1,1,1)$ and $(k+2,2,2)$. It implies that there exist infinitely many triples $(a,b,c)$ which make $N(a,b,c)=2$. Therefore, in general, $N(a,b,c)\leq 2$ should be the best upper bound for $N(a,b,c)$.
\section {Preliminaries}
\begin{lemma}\label{2l1}
Let $t$ be a real number. If $t\geq10^{62}$, then $t>6500^6(\log t)^{18}$. \end{lemma}
{\bf {\it Proof.}}\ Let $F(t)=t-6500^6(\log t)^{18}$. Then we have $F'(t)=1-18\times 6500^6(\log t)^{17}/t$ and $F''(t)=18\times 6500^6(\log t)^{16}(\log t-17)/t^2$, where $F'(t)$ and $F''(t)$ are the derivative and the second derivative of $F(t)$. Since $F'(10^{62})>0$ and $F''(t)>0$ for $t\geq10^{62}$, we get $F'(t)>0$ for $t\geq10^{62}$. Further, since $F(10^{62})>0$, we obtain $F(t)>0$ for $t\geq10^{62}$. The lemma is proved.$\Box$
Let $\alpha$ be a fixed positive irrational number, and let $\alpha=[a_0,a_1,\dots]$ denote the simple continuous fraction of $\alpha$. For any nonnegative integer $i$, let $p_i/q_i$ be the $i-$th convergent of $\alpha$. By Chapter 10 of \cite{hua7}, we obtain the following two lemmas immediately.
\begin{lemma}\label{2l2} \begin{enumerate} \rm \item {\it The convergents $p_i/q_i(i=0,1,\dots)$ satisfy} $$\begin{array}{cc} p_{-1}=1, p_0=a_0,p_{i+1}=a_{i+1}p_i+p_{i-1}, \\ q_{-1}=0, q_0=1,q_{i+1}=a_{i+1}q_i+q_{i-1}, \end{array}i\geq 0.$$ \rm \item { $p_0/q_0<p_2/q_2<\dots<p_{2i}/q_{2i}<p_{2i+2}/q_{2i+2}<\dots<\alpha$ $$<\dots<p_{2i+3}/q_{2i+3}<p_{2i+1}/q_{2i+1}<\dots<p_3/q_3<p_1/q_1, i\geq 0.$$ }
\rm \item {$1/q_i(q_{i+1}+q_i)<|\alpha-p_i/q_i|<1/q_iq_{i+1}, i\geq 0.$} \end{enumerate} \end{lemma}
\begin{lemma}\label{2l3}
Let $p$ and $q$ be positive integers. If $|\alpha-p/q|<1/2q^2$, then $(p/d)/(q/d)$ is a convergent of $\alpha$, where $d=\gcd(p,q)$. \end{lemma}
Let $u,v,k$ be fixed positive integers such that $\min\{u,v,k\}>1$ and $\gcd(u,v)=1$.
\begin{lemma}\label{2l4}{\rm(\cite{hu6},Lemma 4.3)}
The equation \begin{equation}\label{2.1} u^l+v^m=k, l,m\in\mathbb{N}. \end{equation} has at most two solutions $(l,m)$. \end{lemma}
\begin{lemma}\label{2l5}
Let $(l_1,m_1)$ and $(l_2,m_2)$ be two solutions of $(\ref{2.1})$. If $l_1<l_2$, then $m_1>m_2$,
\begin{equation}\label{2.2} \max\{u^{l_2-l_1},v^{m_1-m_2}\}>\sqrt{k}. \end{equation}
and
\begin{equation}\label{2.3} u^{l_2-l_1}=v^{m_2}t+1,v^{m_1-m_2}=u^{l_1}t+1, t\in \mathbb N. \end{equation} \end{lemma}
{\bf {\it Proof.}}\, Since
\begin{equation}\label{2.4} u^{l_1}+v^{m_1}=k, u^{l_2}+v^{m_2}=k, \end{equation}
we have
\begin{equation}\label{2.5} u^{l_1}\equiv-v^{m_1}\pmod k, u^{l_2}\equiv-v^{m_2}\pmod k. \end{equation}
If $l_1<l_2$ and $m_1\leq m_2$, then from $(\ref{2.5})$ we get
\begin{equation}\label{2.6} u^{l_2-l_1}\equiv v^{m_2-m_1}\pmod k. \end{equation}
Since $\gcd(u,v)=1$ and $\min\{u,v\}>1$, we have $u^{l_2-l_1}\not= v^{m_2-m_1}$. Hence, by $(\ref{2.4})$ and $(\ref{2.6})$, we get
\begin{equation}\label{2.7} k>\max\{u^{l_2},v^{m_2}\}>\max\{u^{l_2-l_1}, v^{m_2-m_1}\}>k, \end{equation}
a contradiction. Therefore, if $l_1<l_2$, then $m_1> m_2$. Moreover, by $(\ref{2.5})$, we get $u^{l_2-l_1} v^{m_1-m_2}\equiv 1\pmod k$ and $(\ref{2.2})$.
On the other hand, by $(\ref{2.4})$, we have
\begin{equation}\label{2.8} u^{l_1}(u^{l_2-l_1}-1)=v^{m_2}(v^{m_1-m_2}-1). \end{equation} Therefore, since $\gcd(u,v)=1$, by $(\ref{2.8})$, we get $(\ref{2.3})$. The lemma is proved.$\Box$
\begin{lemma}\label{2l6}{\rm(\cite{ben1})}
The equation \begin{equation}\label{2.9} u^l-v^m=k, l,m\in\mathbb{N} \end{equation} has at most two solutions $(l,m)$. \end{lemma}
\begin{lemma}\label{2l7}
Let $(l_1,m_1)$ and $(l_2,m_2)$ be two solutions of $(\ref{2.9})$. If $l_1<l_2$, then $m_1<m_2$,
\begin{equation}\label{2.10} u^{l_2-l_1}=v^{m_1}t+1,v^{m_2-m_1}=u^{l_1}t+1, t\in \mathbb N, \end{equation}
\begin{equation}\label{2.11} v^{m_2-m_1}>u^{l_2-l_1}>v^{m_1} \end{equation} and \begin{equation}\label{2.12} v^{m_2-m_1}>k. \end{equation} \end{lemma}
{\bf {\it Proof.}}\, Since
\begin{equation}\label{2.13} u^{l_1}-v^{m_1}=k, u^{l_2}-v^{m_2}=k, \end{equation} if $l_1<l_2$, then from $(\ref{2.13})$ we get $v^{m_2}+k=u^{l_2}>u^{l_1}=v^{m_1}+k$ and $m_1<m_2$. Hence, by $(\ref{2.13})$, we have
\begin{equation}\label{2.14} u^{l_1}(u^{l_2-l_1}-1)=v^{m_1}(v^{m_2-m_1}-1), \end{equation} whence we obtain $(\ref{2.10})$, since $\gcd(u,v)=1$. Further, by $(\ref{2.10})$ and $(\ref{2.13})$, we have
\begin{equation}\label{2.15} v^{m_2-m_1}-u^{l_2-l_1}=(u^{l_1}-v^{m_1})t=kt. \end{equation} Therefore, by $(\ref{2.10})$ and $(\ref{2.15})$, we obtain $(\ref{2.11})$ and $(\ref{2.12})$. The lemma is proved.$\Box$
Let $r,s$ be fixed coprime positive integers with $\min\{r,s\}>1$.
\begin{lemma}\label{2l8}{\rm(\cite{car3})}
There exist positive integers $n$ such that \begin{equation}\label{2.16} r^n\equiv \delta\pmod s,\delta\in\{1,-1\}. \end{equation} \end{lemma} Let $n_1$ be the least value of $n$ with $(\ref{2.16})$. Then we have $r^{n_1}\equiv \delta_1\pmod s$ and
\begin{equation}\label{2.17} r^{n_1}=sf+\delta_1,\delta_1\in\{1,-1\}, f\in\mathbb N. \end{equation}
A positive integer $n$ satisyies $(\ref{2.16})$ if and only if $n_1|n$. Moreover, if $n_1|n$, then $r^{n_1}-\delta_1|r^n-\delta$.
Obviously, for any fixed $r$ and $s$, the corresponding $n_1,\delta_1$ and $f$ are unique.
\begin{lemma}\label{2l9}
Let $t$ be a positive integer such that $t>1$ and $s$ is divisble by every prime divisor of $t$. Let $n'$ be a positive integer satisfies \begin{equation}\label{2.18} r^{n'}\equiv\delta'\pmod {st},\delta'\in\{1,-1\}. \end{equation}
If $s$ satisfies either $2\nmid s$ or $4|s$, then $n_1|n'$ and \begin{equation}\label{2.19} \frac{n'}{n_1}\equiv 0\pmod{\frac{t}{\gcd(t,f)}}. \end{equation} \end{lemma}
{\bf {\it Proof.}}\, Notice that $\gcd(r,s)=1$ and $s$ is divisble by every prime divisor of $t$. We have $\gcd(r,st)=1$. Hence, by Lemma $\ref{2l8}$, there exist positive integers $n'$ satisfy $(\ref{2.18})$. Further, since $r^{n'}\equiv\delta'\pmod s$ by $(\ref{2.18})$, we get $n_1|n'$ and \begin{equation}\label{2.20} n'=n_1n_2, n_2\in\mathbb N. \end{equation}
Since either $2\nmid s$ or $4|s$, we have
\begin{equation}\label{2.21} s>2. \end{equation} By $(\ref{2.17})$, $(\ref{2.18})$ and $(\ref{2.20})$, we get
$$r^{n'}\equiv(r^{n_1})^{n_2}\equiv(sf+\delta_1)^{n_2}\equiv \delta_1^{n_2}+n_2\delta_1^{n_2-1}sf+$$
\begin{equation}\label{2.22} \sum\limits_{i=2}^{n_2}\left(\begin{array}{cc} n_2 \\ i \end{array} \right)\delta_1^{n_2-i}(sf)^i\equiv\delta'\pmod{st}. \end{equation} We see from $(\ref{2.22})$ that $\delta_1^{n_2}\equiv\delta'\pmod s$. Hence, by $(\ref{2.21})$, we get $\delta_1^{n_2}=\delta'$, and by $(\ref{2.22})$,
\begin{equation}\label{2.23} f\left(n_2+\sum\limits_{i=2}^{n_2}\left(\begin{array}{cc} n_2 \\ i \end{array} \right)(\delta_1sf)^{i-1}\right)\equiv 0\pmod t. \end{equation} Further, by $(\ref{2.23})$, we obtain
\begin{equation}\label{2.24} n_2+\sum\limits_{i=2}^{n_2}\left(\begin{array}{cc} n_2 \\ i \end{array} \right)(\delta_1sf)^{i-1}\equiv 0\pmod {\frac{t}{\gcd(t,f)}}. \end{equation}
Obviously, if $t/\gcd(t,f)=1$, then $(\ref{2.19})$ holds. We just have to consider the case that $t/\gcd(t,f)>1$. Let $p$ be a prime divisor of $t/\gcd(t,f)$. Since $p|t$ and $p|s$, we see from $(\ref{2.24})$ that $p|n_2$. Let
\begin{equation}\label{2.25}
p^\alpha\big|\big|n_2, p^\beta\big|\big|sf, p^\gamma\big|\big|\frac{t}{\gcd(t,f)}, p^{\pi_i}\big|\big|i, i\geq 2. \end{equation} Then, $\alpha,\beta$ and $\gamma$ are positive integers with $\beta\geq 2$ if $p=2$, $\pi_i(i\geq 2)$ are nonnegative integers satisfy
\begin{equation}\label{2.26} \pi_i\leq\frac{\log i}{\log p}\left\{\begin{array}{cc} \leq i-1<2(i-1)\leq\beta(i-1), &{{\rm if}\ \ p=2,} \\ <i-1, &{\ \ {\rm otherwise.}} \end{array} \right. \end{equation} Hence, by $(\ref{2.25})$ and $(\ref{2.26})$, we have
\begin{equation}\label{2.27} \left(\begin{array}{cc} n_2 \\ i \end{array} \right)(\delta_1sf)^{i-1}\equiv n_2\left(\begin{array}{cc} n_2-1 \\ i-1 \end{array} \right)\frac{(\delta_1sf)^{i-1}}{i}\equiv0 \pmod {p^{\alpha+1}} \end{equation} for $i\geq2$.
By $(\ref{2.25})$ and $(\ref{2.27})$, we get
\begin{equation}\label{2.28}
p^\alpha\big|\big|n_2+\sum\limits_{i=2}^{n_2}\left(\begin{array}{cc} n_2 \\ i \end{array} \right)(\delta_1sf)^{i-1}. \end{equation} Further, we see from $(\ref{2.24})$, $(\ref{2.25})$ and $(\ref{2.28})$ that \begin{equation}\label{2.29} \alpha\geq\gamma. \end{equation} Therefore, take $p$ through all prime divisors of $t/\gcd(t,f)$, by $(\ref{2.20})$, $(\ref{2.25})$ and $(\ref{2.29})$, we obtain $(\ref{2.19})$. The lemma is proved.$\Box$
\section {Further lemmas on solutions of $(\ref{1.1})$}
\begin{lemma}\label{3l1}{\rm(\cite{hu6},Theorem 2.1)}
All solutions $(x,y,z)$ of $(\ref{1.1})$ satisfy $\max\{x,y,z\}<6500(\log \max \{a,b,c\})^3$. \end{lemma} \begin{lemma}\label{3l2}
Let $(x,y,z)$ be a solution of $(\ref{1.1})$ with $a^{2x}<c^z$. If $b\geq3$ and $c\geq 16$, then $y/z$ is a convergent of $\log c/\log b$ with \begin{equation}\label{3.1} 0<\frac{\log c}{\log b}-\frac{y}{z}<\frac{2}{zc^{z/2}\log b}. \end{equation} \end{lemma} {\bf {\it Proof.}}\, Since $\min\{b,c\}>1$ and $\gcd(b,c)=1$,\,$\log c/\log b$ is a positive irratrional number. Let $d=\gcd(y,z)$. Since $a^{2x}<c^z$, if $d\geq 2$, then from $(\ref{1.1})$ we get
\begin{equation}\label{3.2} c^{z/2}>a^x=c^z-b^y=(c^{z/d}-b^{y/d})\sum\limits_{i=0}^{d-1}c^{(d-1-i)z/d}b^{iy/d}>c^{(d-1)z/d}\geq c^{z/2}, \end{equation} a contradiction. So we have $d=1$ and $\gcd(y,z)=1$.
Since $a^x<c^{z/2}$, we have $a^x<b^y$. Hence, by $(\ref{1.1})$, we get
\begin{equation}\label{3.3} z\log c=\log(b^y(1+\frac{a^x}{b^y}))<y\log b+\frac{a^x}{b^y}. \end{equation} Since $a^x<b^y$, by $(\ref{1.1})$, we have $c^z<2b^y$ and \begin{equation}\label{3.4} \frac{a^x}{b^y}<\frac{2a^x}{c^z}<\frac{2c^{z/2}}{c^z}=\frac{2}{c^{z/2}}. \end{equation} Hence, by $(\ref{3.3})$ and $(\ref{3.4})$, we get
\begin{equation}\label{3.5} 0<z\log c-y\log b<\frac{2}{c^{z/2}}, \end{equation} whence we obtain $(\ref{3.1})$. On the other hand, since $b\geq3$ and $c\geq16$, we have $2/zc^{z/2}\log b<1/2z^2$. It implies that $0<\log c/\log b-y/z<1/2z^2$ by $(\ref{3.5})$. Therefore, applying Lemma $\ref{2l3}$, $y/z$ is a convergent of $\log c/\log b$ with $(\ref{3.1})$. Thus, the lemma is proved.$\Box$
Using the same method as in the proof of Lemma $\ref{3l2}$, we can obtain the following lemma immediately.
\begin{lemma}\label{3l3}
Let $(x,y,z)$ be a solution of $(\ref{1.1})$ with $b^{2y}<c^z$. If $a\geq10^{62}$, then $x/z$ is a convergent of $\log c/\log a$ with
\begin{equation}\label{3.6} 0<\frac{\log c}{\log a}-\frac{x}{z}<\frac{2}{zc^{z/2}\log a}. \end{equation} \end{lemma}
\begin{lemma}\label{3l4}
Let $(x,y,z)$ and$(x',y',z')$ be two solutios of $(\ref{1.1})$ such that $x>x'$ and $z<z'$. If $c=\max\{a,b,c\}\geq10^{62}$, then $(y'/d)/(z'/d)$ is a convergent of $\log c/\log b$ with \begin{equation}\label{3.7} 0<\frac{\log c}{\log b}-\frac{y'/d}{z'/d}<\frac{2}{z'ac\log b}, \end{equation} where $d=\gcd(y',z')$. \end{lemma}
{\bf {\it Proof.}}\, Since $x>x'$ and $z<z'$, if $a^{x'}>b^{y'}$, then we get $2a^{x'}>c^{z'}>c^z>a^x\geq a^{x'+1}\geq 2a^{x'}$, a contradiction. So we have $a^{x'}<b^{y'}$ and
\begin{equation}\label{3.8} z'\log c=\log(b^{y'}(1+\frac{a^{x'}}{b^{y'}}))<y'\log b+\frac{a^{x'}}{b^{y'}}. \end{equation} Since $2b^{y'}>c^{z'}$, we get \begin{equation}\label{3.9} \frac{a^{x'}}{b^{y'}}<\frac{2a^{x'}}{c^{z'}}=\frac{2}{a^{x-x'}c^{z'-z}}\cdot\frac{a^x}{c^z}<\frac{2}{ac}. \end{equation} Hence, by $(\ref{3.8})$ and $(\ref{3.9})$, we obtain
\begin{equation}\label{3.10} 0<\frac{\log c}{\log b}-\frac{y'}{z'}<\frac{2}{z'ac\log b}. \end{equation}
If $|\log c/\log b-y'/z'|\geq 1/2z'^2$, then from $(\ref{3.10})$ we get
\begin{equation}\label{3.11} z'>\frac{1}{4}ac\log b. \end{equation} Since $c=\max\{a,b,c\}$, by Lemma $\ref{3l1}$, we have $z'<6500(\log c)^3$. Since $a\log b\geq\min\{2\log3,3\log2\}>2$, by $(\ref{3.11})$, we get
\begin{equation}\label{3.12} 13000(\log c)^3>c. \end{equation} But, since $c\geq10^{62}$, by Lemma $\ref{2l1}$, $(\ref{3.12})$ is false. Therefore, we have
\begin{equation}\label{3.13}
\left|\frac{\log c}{\log b}-\frac{y'}{z'}\right|<\frac{1}{2z'^2}. \end{equation} Applying Lemma $\ref{2l3}$ to $(\ref{3.13})$, we find from $(\ref{3.10})$ that $(y'/d)/(z'/d)$ is a convergent of $\log c/\log b$ with $(\ref{3.7})$. Thus, the lemma is proved. $\Box$
\begin{lemma}\label{3l5}
Let $(x,y,z)$ and$(x',y',z')$ be two solutios of $(\ref{1.1})$ such that $y>y'$ and $z\leq z'$. If $a=\max\{a,b,c\}\geq10^{62}$, then $(x'/d)/(z'/d)$ is a convergent of $\log c/\log a$ with \begin{equation}\label{3.14} 0<\frac{\log c}{\log a}-\frac{x'/d}{z'/d}<\frac{2}{z'a\log a}, \end{equation} where $d=\gcd(x',z')$. \end{lemma}
{\bf {\it Proof.}}\, The proof of this lemma is similar to Lemma $\ref{3l4}$ . Since $y>y'$ and $z\leq z'$, we see from \begin{equation}\label{3.15} a^x+b^y=c^z,a^{x'}+b^{y'}=c^{z'} \end{equation} that
\begin{equation}\label{3.16} x<x', \end{equation} $a^{x'}>b^{y'}$ and $2a^{x'}>c^{z'}$ . Hence, by the second equality of $(\ref{3.15})$, we have
\begin{equation}\label{3.17} z'\log c=\log(a^{x'}(1+\frac{b^{y'}}{a^{x'}}))<x'\log a+\frac{b^{y'}}{a^{x'}} \end{equation} and \begin{equation}\label{3.18} \frac{b^{y'}}{a^{x'}}<\frac{2b^{y'}}{c^{z'}}=\frac{2}{b^{y-y'}c^{z'-z}}\cdot\frac{b^y}{c^z}<\frac{2}{b^{y-y'}c^{z'-z}}. \end{equation}
By $(\ref{3.15})$ and $(\ref{3.16})$, we have $b^y\equiv c^z\pmod {a^x}$ and $b^{y'}\equiv c^{z'}\pmod {a^{x'}}$, whence we get \begin{equation}\label{3.19} b^{y-y'}c^{z'-z}\equiv 1\pmod{a^x}. \end{equation} Further, since $y>y'$, we have $b^{y-y'}c^{z'-z}>1$. Hence, by $(\ref{3.19})$, we get \begin{equation}\label{3.20} b^{y-y'}c^{z'-z}>a^x. \end{equation} Therefore, by $(\ref{3.17})$, $(\ref{3.18})$ and $(\ref{3.20})$, we obtain $b^{y'}/a^{x'}<2/a^x$ and
\begin{equation}\label{3.21} 0<\frac{\log c}{\log a}-\frac{x'}{z'}<\frac{2}{z'a^x\log a}. \end{equation} Since $a^x\geq a=\max\{a,b,c\}\geq 10^{62}$, by Lemma $\ref{3l1}$, we can deduce that
\begin{equation}\label{3.22} \frac{2}{z'a^x\log a}<\frac{1}{2z'^2}. \end{equation} Thus, by Lemma $\ref{2l3}$, we fond from $(\ref{3.21})$ and $(\ref{3.22})$ that $(x'/d)/(z'/d)$ is a convergent of $\log c/\log a$ with $(\ref{3.14})$. The lemma is proved.$\Box$
\section {The equation $A^X+\lambda B^Y=C^Z$}
\quad For any fixed triple $(a,b,c)$, put \begin{equation}\label{4.1} P(a,b,c)=\{(a,b,c,1),(c,a,b,-1),(c,b,a,-1)\}. \end{equation}
Obviously, for any element in $P(a,b,c)$, say $(A,B,C,\lambda)$, $(\ref{1.1})$ has a solution $(x,y,z)$ is equivalent to the equation
\begin{equation}\label{4.2} A^X+\lambda B^Y=C^Z, X,Y,Z\in\mathbb{N} \end{equation} has the solution
$$(X,Y,Z)=\left\{\begin{array}{cc} (x,y,z), &{{\rm if}\ \ (A,B,C,\lambda)=(a,b,c,1) ,} \\ (z,x,y), &{{\rm if}\ \ (A,B,C,\lambda)=(c,a,b,-1),}\\ (z,y,x),&{{\rm if}\ \ (A,B,C,\lambda)=(c,b,a,-1).} \end{array} \right.$$ It implies that, for any $(A,B,C,\lambda)\in P(a,b,c)$, the numbers of solutions of $(\ref{1.1})$ and $(\ref{4.2})$ are equal. Moreover, by Lemma $\ref{3l1}$, we have
\begin{lemma}\label{4l1} All solutions $(X,Y,Z)$ of $(\ref{4.2})$ satisfy $\max\{X,Y,Z\}<6500(\log\max\{a,b,c\})^3$. \end{lemma}
Here and below, we always assume that $(\ref{1.1})$ has solutions $(x,y,z)$. Then, for any $(A,B,C,\lambda)\in P(a,b,c)$, $(\ref{4.2})$ has solutions $(X,Y,Z)$.
For a fixed element $(A,B,C,\lambda)\in P(a,b,c)$, $(\ref{4.2})$ is sure to have a solution $(X_1,Y_1,Z_1)$ such that $Z_1\leq Z$, where $Z$ through all solutions $(X,Y,Z)$ of $(\ref{4.2})$ for this $(A,B,C,\lambda)$. Since $\gcd(A,C)=1$ and $\min\{A,C\}>1$, by Lemma $\ref{2l8}$, there exist positive integers $n$ such that
\begin{equation}\label{4.3} A^n\equiv \delta\pmod{C^{Z_1}},\delta\in\{1,-1\}. \end{equation} Let $n_1$ be the least value of $n$ with $(\ref{4.3})$, and let \begin{equation}\label{4.4} A^{n_1}\equiv \delta_1\pmod{C^{Z_1}},\delta_1\in\{1,-1\}. \end{equation} Then we have \begin{equation}\label{4.5} A^{n_1}=C^{Z_1}f+\delta_1,f\in\mathbb{N}. \end{equation} Obviously, for any fixed triple $(A,B,C,\lambda)\in P(a,b,c)$, the parameters $Z_1,n_1,\delta_1$ and $f$ are unique.
\begin{lemma}\label{4l2}
$(\ref{4.2})$ has at most two solutions $(X,Y,Z)$ with the same value $Z$. \end{lemma}
{\bf {\it Proof .}}\,\, By Lemmas $\ref{2l4}$ and $\ref{2l6}$, we obtain the lemma immediately.$\Box$
\begin{lemma}\label{4l3}{\rm (\cite{hu5}, Lemma 3.3)} Let $(X,Y,Z)$ and $(X^\prime,Y^\prime,Z^\prime)$ be two solutions of $(\ref{4.2})$ with $Z\leq Z^\prime$. Then we have $XY^\prime-X^\prime Y\not=0$ and
$A^{|XY^\prime-X^\prime Y|}\equiv(-\lambda)^{Y+Y^\prime}\pmod {C^Z}.$ \end{lemma}
\begin{lemma}\label{4l4}. Let $(X_1,Y_1,Z_1)$ and $(X_2,Y_2,Z_2)$ be two solutions of $(\ref{4.2})$ such that $Z_1<Z_2$ and $Z_1\leq Z$, where $Z$ through all solutions $(X,Y,Z)$ of $(\ref{4.2})$ for this $(A,B,C,\lambda)$. If $C$ satisfies
\begin{equation}\label{4.6}
2\nmid C \,\,\,or\,\,\, 4|C^{Z_1}, \end{equation} then
\begin{equation}\label{4.7}
\gcd (C^{Z_2-Z_1},f)|Y_2, \end{equation} where $f$ is defined as in $(\ref{4.5})$. \end{lemma} {\bf {\it Proof .}}\,\, The proof of this lemma is similar to Lemma $\ref{2l9}$. Since $A^{X_1}+\lambda B^{Y_1}=C^{Z_1}$, $A^{X_2}+\lambda B^{Y_2}=C^{Z_2}$ and $Z_1<Z_2$, we have
$$A^{X_1Y_2}=(-\lambda)^{Y_2} B^{Y_1Y_2}+C^{Z_1}\sum\limits_{i=1}^{Y_2}\left(\begin{array}{cc} Y_2 \\ i \end{array} \right)(-\lambda B^{Y_1})^{Y_2-i}C^{Z_1(i-1)},$$
\begin{equation}\label{4.8} A^{X_2Y_1}\equiv(-\lambda)^{Y_1}B^{Y_1Y_2}\pmod {C^{Z_2}}. \end{equation} Eliminating $B^{Y_1Y_2}$ from $(\ref{4.8})$£¬ we get
$$\lambda^\prime A^{\min\{X_1Y_2,X_2Y_1\}}\left (A^{\left |X_1Y_2-X_2Y_1\right |}-(-\lambda)^{Y_1+Y_2}\right )$$
\begin{equation}\label{4.9} \equiv Y_2B^{Y_1(Y_2-1)}C^{Z_1}+\sum\limits_{i=2}^{Y_2}(-\lambda)^{i+1}\left(\begin{array}{cc} Y_2 \\ i \end{array} \right)B^{Y_1(Y_2-i)}C^{Z_1i}\pmod{C^{Z_2}}, \end{equation} where $\lambda^\prime\in\{1,-1\}$.
By Lemma $\ref{4l3}$ , we have $X_1Y_2-X_2Y_1\not=0$. It implies that $\left |X_1Y_2-X_2Y_1\right|$ is a positive integer. Since $Z_1<Z_2$, using Lemma $\ref{4l3}$ again, wer have \begin{equation}\label{4.10}
A^{\left |X_1Y_2-X_2Y_1\right |}\equiv(-\lambda)^{Y_1+Y_2}\pmod{C^{Z_1}}. \end{equation}
Therefore, by Lemma $\ref{2l8}$, we get from $(\ref{4.4})$, $(\ref{4.5})$ and $(\ref{4.10})$ that $A^{n_1}-\delta_1\big |A^{\left |X_1Y_2-X_2Y_1\right |}-(-\lambda)^{Y_1+Y_2}$ and
\begin{equation}\label{4.11}
A^{\left |X_1Y_2-X_2Y_1\right |}-(-\lambda)^{Y_1+Y_2}=C^{Z_1}fg, g\in\mathbb N. \end{equation} Substitute $(\ref{4.11})$ into $(\ref{4.9})$, we have $$\lambda^\prime A^{\min\{X_1Y_2,X_2Y_1\}}fg\equiv Y_2B^{Y_1(Y_2-1)}$$ \begin{equation}\label{4.12} +\sum\limits_{i=2}^{Y_2}(-\lambda)^{i+1}\left(\begin{array}{cc} Y_2 \\ i \end{array} \right)B^{Y_1(Y_2-i)}C^{Z_1(i-1)}\pmod{C^{Z_2-Z_1}}. \end{equation}
Obviously, if $\gcd(C^{Z_2-Z_1},f)=1$, then $(\ref{4.7})$ holds. We just have to consider the case that $\gcd(C^{Z_2-Z_1},f)>1$. Let $p$ be a prime divisor of $\gcd(C^{Z_2-Z_1},f)$. Since $p\big |C$ and $\gcd(B,C)=1$, we see from $(\ref{4.12})$ that $p\big | Y_2$. Let
\begin{equation}\label{4.13}
p^\alpha\big|\big|Y_2, p^\beta\big|\big|C^{Z_1}, p^\gamma\big|\big|\gcd(C^{Z_2-Z_1},f), p^{\pi_i}\big|\big|i, i\geq 2. \end{equation} Then, by $(\ref{4.6})$, $\alpha,\beta$ and $\gamma$ are positive integers with $\beta\geq2$ if $p=2, \pi_i(i\geq2)$ are nonnegative integers satisfy $(\ref{2.26})$. By $(\ref{2.26})$ and $(\ref{4.13})$, we have $$\left(\begin{array}{cc} Y_2 \\ i \end{array} \right)B^{Y_1(Y_2-i)}C^{Z_1(i-1)}\equiv Y_2\left(\begin{array}{cc} Y_2-1 \\ i-1 \end{array} \right)\frac{B^{Y_1(Y_2-i)}C^{Z_1(i-1)}}{i}$$ \begin{equation}\label{4.14} \equiv0\pmod{p^{\alpha+1}}, i\geq 2. \end{equation} Hence, by $(\ref{4.13})$ and $(\ref{4.14})$, we get
\begin{equation}\label{4.15}
p^\alpha\big|\big|Y_2B^{Y_1(Y_2-1)}+\sum\limits_{i=2}^{Y_2}(-\lambda)^{i+1}\left(\begin{array}{cc} Y_2 \\ i \end{array} \right)B^{Y_1(Y_2-i)}C^{Z_1(i-1)}. \end{equation}
Therefore, since $\gcd(C^{Z_2-Z_1},f)\big|f$ and $\gcd(C^{Z_2-Z_1},f)\big|C^{Z_2-Z_1}$, we find from $(\ref{4.12})$, $(\ref{4.13})$ and $(\ref{4.15})$ that $\alpha$ and $\gamma$ satisfies $(\ref{2.29})$. Thus, take $p$ through all prime divisors of $\gcd(C^{Z_2-Z_1},f)$, by $(\ref{2.29})$ and $(\ref{4.13})$, we obtain $(\ref{4.7})$. The lemma is proved.$\Box$
\begin{lemma}\label{4l5}{\rm (\cite{hu6}, Lemma 4.7)} Let $(X_j,Y_j,Z_j)(j=1,2,3)$ be three solutions of $(\ref{4.2})$ with $Z_1<Z_2\leq Z_3$. If $C=\max\{a,b,c\}$, then $\max\{a,b,c\}<5\times10^{27}.$ \end{lemma}
\begin{lemma}\label{4l6} Let $(X_j,Y_j,Z_j)(j=1,2,3)$ be three solutions of $(\ref{4.2})$ with $Z_1<Z_2\leq Z_3$. If $C^{Z_2-Z_1}>(\max\{a,b,c\})^{1/2}$ and $C$ satisfies $(\ref{4.6})$, then $\max\{a,b,c\}<10^{62}$. \end{lemma} {\bf {\it Proof .}}\,\, Since $Z_2\leq Z_3$, by Lemma $\ref{4l3}$, we have $X_2Y_3-X_3Y_2\not=0$ and
\begin{equation}\label{4.16}
A^{\left |X_2Y_3-X_3Y_2\right |}\equiv(-\lambda)^{Y_2+Y_3}\pmod{C^{Z_2}}. \end{equation} Further, since $Z_1<Z_2$ and $C$ satisfies $(\ref{4.6})$, by Lemma $\ref{2l9}$, we get from $(\ref{4.4})$, $(\ref{4.5})$ and $(\ref{4.16})$ that \begin{equation}\label{4.17}
\left |X_2Y_3-X_3Y_2\right |\equiv0\pmod{\frac{C^{Z_2-Z_1}}{\gcd(C^{Z_2-Z_1},f)}}, \end{equation} where $f$ is defined as in $(\ref{4.5})$. Recall that $X_2Y_3-X_3Y_2\not=0$. By $(\ref{4.17})$, we have \begin{equation}\label{4.18}
\left |X_2Y_3-X_3Y_2\right |\gcd(C^{Z_2-Z_1},f)\geq C^{Z_2-Z_1}. \end{equation} Furthermore, by Lemma $\ref{4l4}$, we have $\gcd(C^{Z_2-Z_1},f)\leq Y_2$. Hence, we get from $(\ref{4.18})$ that \begin{equation}\label{4.19}
Y_2\left |X_2Y_3-X_3Y_2\right |\geq C^{Z_2-Z_1}. \end{equation}
By Lemma $\ref{4l1}$, we have
$$Y_2\left |X_2Y_3-X_3Y_2\right |< Y_2\max\{X_2Y_3,X_3Y_2\}\leq\left(\max\{X_2,Y_2,X_3,Y_3\}\right)^3$$ \begin{equation}\label{4.20} <6500^3\left(\log\max\{a,b,c\}\right)^9. \end{equation} Therefore, if $C^{Z_2-Z_1}>\left(\max\{a,b,c\}\right)^{1/2}$, then from $(\ref{4.19})$ and $(\ref{4.20})$ we get \begin{equation}\label{4.21} 6500^6\left(\log\max\{a,b,c\}\right)^{18}>\max\{a,b,c\}. \end{equation} Thus, applying Lemma $\ref{2l1}$ to $(\ref{4.21})$, we obtain $\max\{a,b,c\}<10^{62}$. The lemma is proved.$\Box$
\section {Proof of Theorem 1.1 for $c=\max\{a,b,c\}$ }
\quad By \cite{sco9}, Theorem 1.1 holds for $2\nmid c$. Therefore, we just have to consider the case that \begin{equation}\label{5.1}
2\big|c. \end{equation} Since $\gcd(ab,c)=1$, by $(\ref{5.1})$, we have
\begin{equation}\label{5.2} 2\nmid a,\,\,\,2\nmid b. \end{equation}
In this section we will prove the theorem for the case that
\begin{equation}\label{5.3} c=\max\{a,b,c\}\geq10^{62}. \end{equation} We now assume that $(\ref{1.1})$ has three solutions $(x_j,y_j,z_j)(j=1,2,3)$ with $z_1\leq z_2\leq z_3$. Then, $(\ref{4.2})$ has three solutions $(X_j,Y_j,Z_j)=(x_j,y_j,z_j)(j=1,2,3)$ for $(A,B,C,\lambda)=(a,b,c,1)$ with $Z_1\leq Z_2\leq Z_3$. By Lemma $\ref{4l2}$, we can remove the case $z_1= z_2= z_3$. Since $C=c=\max\{a,b,c\}\geq10^{62}$, by Lemma $\ref{4l5}$, we can remove the case $z_1<z_2\leq z_3$. So we have \begin{equation}\label{5.4} z_1=z_2< z_3. \end{equation}
Since $z_1=z_2$ and
\begin{equation}\label{5.5} a^{x_1}+b^{y_1}=a^{x_2}+b^{y_2}=c^{z_1}, \end{equation} $(\ref{2.1})$ has two solutions $(l,m)=(x_j,y_j)(j=1,2)$ for $(u,v,k)=(a,b,c^{z_1})$. Since $(x_1,y_1)\not=(x_2,y_2)$ by $(\ref{5.5})$, we may therefore assume that \begin{equation}\label{5.6} x_1<x_2. \end{equation} Then, by Lemma $\ref{2l5}$, we have \begin{equation}\label{5.7} y_1>y_2, \end{equation} \begin{equation}\label{5.8} a^{x_2-x_1}=b^{y_2}t_1+1, b^{y_1-y_2}=a^{x_1}t_1+1,t_1\in\mathbb N \end{equation} and \begin{equation}\label{5.9} \max\{a^{x_2-x_1}, b^{y_1-y_2}\}>c^{z_1/2}. \end{equation} Accordind to the symmetry of $a$ and $b$ in $(\ref{5.5})$, we may assume that \begin{equation}\label{5.10} a^{x_2-x_1}> b^{y_1-y_2}. \end{equation} Hence, by $(\ref{5.3})$, $(\ref{5.9})$ and $(\ref{5.10})$, we have \begin{equation}\label{5.11} a^{x_2-x_1}> c^{z_1/2}\geq\sqrt{c}=\left(\max\{a,b,c\}\right)^{1/2}. \end{equation}
By $(\ref{5.6})$, if $x_3\geq x_2$, then $(\ref{4.2})$ has three solutions $(X_j,Y_j,Z_j)=(z_j,y_j,x_j)(j=1,2,3)$ for $(A,B,C,\lambda)=(c,b,a,-1)$ with $Z_1< Z_2\leq Z_3$. Since $C^{Z_2-Z_1}=a^{x_2-x_1}> \left(\max\{a,b,c\}\right)^{1/2}$ by $(\ref{5.11})$, using Lemma $\ref{4l6}$, we get from $(\ref{5.2})$ that $\max\{a,b,c\}<10^{62}$, which contradicts $(\ref{5.3})$. Therefore, we have
\begin{equation}\label{5.12} x_3<x_2.
\end{equation}
By $(\ref{5.8})$ and $(\ref{5.10})$ we get $a^{x_2-x_1}>b^{y_1-y_2}=a^{x_1}t_1+1>a^{x_1}$, and by $(\ref{5.5})$, $c^{z_1}>a^{x_2}>a^{2x_1}$. It implies that $(x,y,z)=(x_1,y_1,z_1)$ is a solution of $(\ref{1.1})$ with $a^{2x}<c^{z}$. Notice that $b\geq3$ and $c\geq16$ by $(\ref{5.2})$ and $(\ref{5.3})$. Using Lemma $\ref{3l2}$, $y_1/z_1$ is a convergent of $\log c/\log b$ with
\begin{equation}\label{5.13} 0<\frac{\log c}{\log b}-\frac{y_1}{z_1}<\frac{2}{z_1c^{z_1/2}\log b}. \end{equation}
On the other hand, by $(\ref{5.4})$ and $(\ref{5.12})$, $(x_2,y_2,z_2)$ and $(x_3,y_3,z_3)$ are two solutions of $(\ref{1.1})$ such that $x_2>x_3$ and $z_2<z_3$. Since $c=\max\{a,b,c\}\geq10^{62}$, by Lemma $\ref{3l4}$, $(y_3/d)/(z_3/d)$ is also a convergent of $\log c/\log b$ with
\begin{equation}\label{5.14} 0<\frac{\log c}{\log b}-\frac{y_3/d}{z_3/d}<\frac{2}{z_3ac\log b}, \end{equation}
where $d=\gcd(y_3,z_3)$.
By $(\ref{5.4})$,$(X_1,Y_1,Z_1)=(z_1,y_1,x_1)$ and $(X_3,Y_3,Z_3)=(z_3,y_3,x_3)$ are two distinct solutions of $(\ref{4.2})$ for $(A,B,C,\lambda)=(c,b,a,-1)$. Hence, by Lemma $\ref{4l3}$, we have $z_1y_3-z_3y_1\not=0$. It implies that $y_1/z_1$ and $(y_3/d)/(z_3/d)$ are two distinct convergents of $\log c/\log b$. Therefore, by (\romannumeral2) of Lemma $\ref{2l2}$, we see from $(\ref{5.13})$ and $(\ref{5.14})$ that
\begin{equation}\label{5.15} \frac{y_1}{z_1}=\frac{p_{2s}}{q_{2s}}, \frac{y_3/d}{z_3/d}=\frac{p_{2t}}{q_{2t}}, s,t\in\mathbb Z, s\not=t,\min\{s,t\}\geq0. \end{equation}
If $s<t$, by (\romannumeral1) and (\romannumeral3) of Lemma $\ref{2l2}$, then from $(\ref{5.13})$ and $(\ref{5.15})$ we get
$$z_3\geq\frac{z_3}{d}=q_{2t}\geq q_{2s+2}=a_{2s+2}q_{2s+1}+q_{2s}\geq q_{2s+1}+q_{2s}$$
$$>\left(q_{2s}\left |\frac{\log c}{\log b}-\frac{p_{2s}}{q_{2s}}\right|\right)^{-1}=\left(z_1\left (\frac{\log c}{\log b}-\frac{y_1}{z_1}\right)\right)^{-1}$$
\begin{equation}\label{5.16} >\frac{1}{2}c^{z_1/2}\log b>\frac{\sqrt c}{2}. \end{equation}
Since $c=\max\{a,b,c\}$, by Lemma $\ref{3l1}$, we have $z_3<6500(\log c)^3$. Therefore, by $(\ref{5.16})$, we get
\begin{equation}\label{5.17} 13000^2\left(\log c\right)^6>c. \end{equation}
However, since $c\geq 10^{62}$, by Lemma $\ref{2l1}$, $(\ref{5.17})$ is false.
Similarly, if $s>t$, then from $(\ref{5.14})$ and $(\ref{5.15})$ we get
$$z_1=q_{2s}\geq q_{2t+2}\geq q_{2t+1}+q_{2t}>\left(q_{2t}\left |\frac{\log c}{\log b}-\frac{p_{2t}}{q_{2t}}\right|\right)^{-1}$$
\begin{equation}\label{5.18} =\left(\frac{z_3}{d}\left (\frac{\log c}{\log b}-\frac{y_3/d}{z_3/d}\right)\right)^{-1}>\frac{1}{2}ac\log b>c. \end{equation}
Further, by Lemma $\ref{3l1}$, we have $z_1<6500(\log c)^3$. Therefore, by $(\ref{5.18})$, we get
\begin{equation}\label{5.19} 6500\left(\log c\right)^3>c. \end{equation}
However, since $c\geq 10^{62}$, by Lemma $\ref{2l1}$, $(\ref{5.19})$ is false.
Thus, we have $N(a,b,c)\leq2$ for the case $(\ref{5.3})$.
\section {Proof of Theorem 1.1 for $c\not=\max\{a,b,c\}$ }
\quad In this section we will prove Theorem 1.1 for the case that $c\not=\max\{a,b,c\}$. Then, by the symmetry of $a$ and $b$ in $(\ref{1.1})$, we may assume that
\begin{equation}\label{6.1} a=\max\{a,b,c\}\geq10^{62}. \end{equation}
For any solution $(x,y,z)$ of $(\ref{1.1})$, since $a^z>c^z=a^x+b^y>a^x\geq a$ by $(\ref{6.1})$, we have
\begin{equation}\label{6.2} z\geq2. \end{equation}
Hence, by $(\ref{5.1})$ and $(\ref{6.2})$, we get
\begin{equation}\label{6.3}
4\big |c^z. \end{equation}
We now assume that $(\ref{1.1})$ has three solutions $(x_j,y_j,z_j)(j=1,2,3)$ with $x_1\leq x_2\leq x_3$. Then, $(\ref{4.2})$ has three solutions $(X_j,Y_j,Z_j)=(z_j,y_j,x_j)(j=1,2,3)$ for $(A,B,C,\lambda)=(c,b,a,-1)$ with $Z_1\leq Z_2\leq Z_3$. By Lemma $\ref{4l2}$, we can remove the case
$x_1= x_2= x_3$. Since $C=a=\max\{a,b,c\}\geq10^{62}$, by Lemma $\ref{4l5}$, we an remove the case $x_1<x_2\leq x_3$. So we have
\begin{equation}\label{6.4} x_1=x_2<x_3.
\end{equation}
Since $x_1=x_2$, we have
\begin{equation}\label{6.5} c^{z_1}-b^{y_1}=c^{z_2}-b^{y_2}=a^{x_1}.
\end{equation}
It implies that $(\ref{2.9})$ has two solutions $(l,m)=(z_j,y_j)(j=1,2)$ for $(u,v,k)=(c,b,a^{x_1})$. Since $(z_1,y_1)\not=(z_2,y_2)$, we may assume that
\begin{equation}\label{6.6} z_1<z_2.
\end{equation}
Then, by Lemma $\ref{2l7}$, we get from $(\ref{6.6})$ that
\begin{equation}\label{6.7} y_1<y_2 \end{equation} and
\begin{equation}\label{6.8} b^{y_2-y_1}=c^{z_1}t_2+1, c^{z_2-z_1}=b^{y_1}t_2+1, t_2\in\mathbb N. \end{equation} By the first equality of $(\ref{6.8})$, we have
\begin{equation}\label{6.10} b^{y_2-y_1}>c^{z_1}>a^{x_1}. \end{equation}
If $y_3\geq y_2$, by $(\ref{6.7})$, then $(\ref{4.2})$ has three solutions $(X_j,Y_j,Z_j)=(z_j,x_j,y_j)(j=1,2,3)$ for $(A,B,C,\lambda)=(c,a,b,-1)$ with
$Z_1< Z_2\leq Z_3$. However, since $2\nmid b=C$ and $C^{Z_2-Z_1}=b^{y_2-y_1}>a^{x_1}\geq a=\max\{a,b,c\}\geq10^{62}$ by $(\ref{5.2})$, $(\ref{6.1})$ and $(\ref{6.10})$, using Lemma $\ref{4l6}$, it is impossible. Therefore, we obtain
\begin{equation}\label{6.11} y_3<y_2. \end{equation}
If $z_3\geq z_2$, by $(\ref{6.6})$, then $(\ref{4.2})$ has three solutions $(X_j,Y_j,Z_j)=(x_j,y_j,z_j)(j=1,2,3)$ for $(A,B,C,\lambda)=(a,b,c,1)$ with
$Z_1< Z_2\leq Z_3$. Since $a=\max\{a,b,c\}\geq10^{62}$, by Lemma $\ref{4l6}$ with $(\ref{4.3})$, we obtain
\begin{equation}\label{6.12} C^{Z_2-Z_1}=c^{z_2-z_1}<\left(\max\{a,b,c\}\right)^{1/2}=\sqrt a. \end{equation} Hence, by the second equality of $(\ref{6.8})$ and $(\ref{6.12})$, we have \begin{equation}\label{6.13} b^{y_1}<b^{y_1}t_2+1= c^{z_2-z_1}<\sqrt a\leq a^{x_1/2}<c^{z_1/2}. \end{equation}
It implies that $(x,y,z)=(x_1,y_1,z_1)$ is a solution of $(\ref{1.1})$ with $b^{2y}<c^z$. Therefore, by Lemma $\ref{3l3}$, $x_1/z_1$ is a convergent of
$\log c/\log a$ with
\begin{equation}\label{6.14} 0<\frac{\log c}{\log a}-\frac{x_1}{z_1}<\frac{2}{z_1c^{z_1/2}\log a}. \end{equation}
On the other hand, by $(\ref{6.11})$, $(\ref{1.1})$ has two solutions $(x_2,y_2,z_2)$ and $(x_3,y_3,z_3)$ such that $y_2> y_3$ and $z_2\leq z_3$. Since $a=\max\{a,b,c\}\geq10^{62}$, by Lemma $\ref{3l5}$, $(x_3/d)/(z_3/d)$ is also a convergent of
$\log c/\log a$ with
\begin{equation}\label{6.15} 0<\frac{\log c}{\log a}-\frac{x_3/d}{z_3/d}<\frac{2}{z_3a\log a}, \end{equation}
where $d=\gcd(x_3,z_3)$. Further, by Lemma $\ref{4l3}$, we have $x_1z_3-x_3z_1\not=0$. It implies that $x_1/z_1$ and $(x_3/d)/(z_3/d)$ are two distinct convergents of
$\log c/\log a$. Hence, by (\romannumeral2) of Lemma $\ref{2l2}$, we see from $(\ref{6.14})$ and $(\ref{6.15})$ that
\begin{equation}\label{6.16} \frac{x_1}{z_1}=\frac{p_{2s}}{q_{2s}}, \frac{x_3/d}{z_3/d}=\frac{p_{2t}}{q_{2t}}, s,t\in\mathbb Z, s\not=t, \min\{s,t\}\geq 0. \end{equation}
Since $a=\max\{a,b,c\}$, by Lemma $\ref{2l2}$ and $\ref{3l1}$, we get from $(\ref{6.14})$, $(\ref{6.15})$ and $(\ref{6.16})$ that
$$6500\left(\log a\right)^3>\left\{\begin{array}{cc} z_3\geq\frac{z_3}{d}=q_{2t}\geq q_{2s+2} \geq q_{2s+1}+q_{2s}\\ z_1=q_{2s}\geq q_{2t+2} \geq q_{2t+1}+q_{2t} \end{array} \right.$$
$$\begin{array}{cc}
>\left(q_{2s}\left |\frac{\log c}{\log a}-\frac{p_{2s}}{q_{2s}}\right|\right)^{-1}=\left(z_1\left (\frac{\log c}{\log a}-\frac{x_1}{z_1}\right)\right)^{-1}\\
>\left(q_{2t}\left |\frac{\log c}{\log a}-\frac{p_{2t}}{q_{2t}}\right|\right)^{-1}=\left(\frac{z_3}{d}\left (\frac{\log c}{\log a}-\frac{x_3/d}{z_3/d}\right)\right)^{-1} \end{array} $$
\begin{equation}\label{6.17}
\left.\begin{array}{cc} >\frac{1}{2}c^{z_1/2}\log a, &{{\rm if}\ \ s<t} \\ >\frac{1}{2}a\log a, &{{\rm if}\ \ s>t} \end{array} \right\}>\frac{1}{2}\sqrt a\log a.
\end{equation}
But, since $a\geq10^{62}$, by Lemma $\ref{2l1}$, $(\ref{6.17})$ is false. Therefore, we obtain
\begin{equation}\label{6.18}
z_3<z_2.
\end{equation}
Finally, by the known results $(\ref{6.4})$, $(\ref{6.6})$, $(\ref{6.7})$, $(\ref{6.11})$ and $(\ref{6.18})$, we can complete the proof in the following four cases.
Case \uppercase\expandafter{\romannumeral1:} $y_3\leq y_1<y_2$ and $z_3\leq z_1<z_2$
In this case, since $x_3>x_1=x_2$ and
\begin{equation}\label{6.19}
a^{x_3}=c^{z_3}-b^{y_3}=a^{x_3-x_1}c^{z_1}-a^{x_3-x_1}b^{y_1},
\end{equation} we get
\begin{equation}\label{6.20}
c^{z_3}(a^{x_3-x_1}c^{z_1-z_3}-1)=b^{y_3}(a^{x_3-x_1}b^{y_1-y_3}-1).
\end{equation}
Since $\gcd(b,c)=1$, by $(\ref{6.20})$, we have
\begin{equation}\label{6.21}
a^{x_3-x_1}c^{z_1-z_3}=b^{y_3}t_3+1, a^{x_3-x_1}b^{y_1-y_3}=c^{z_3}t_3+1, t_3\in\mathbb N.
\end{equation}
Hence, we see from the second equality of $(\ref{6.21})$ that
$a^{x_3-x_1}b^{y_1-y_3}>c^{z_3}= a^{x_3}+b^{y_3}>a^{x_3}$ and
\begin{equation}\label{6.22}
b^{y_1-y_3}>a^{x_1}.
\end{equation}
It implies that $y_3<y_1$. Therefore, $(\ref{4.2})$ has three solutions $(X_j,Y_j,Z_j)(j=1,2,3)$ for $(A,B,C,\lambda)=(c,a,b,-1)$ such that $(X_1,Y_1,Z_1)=(z_3,x_3,y_3)$, $(X_2,Y_2,Z_2)=(z_1,x_1,y_1)$, $(X_3,Y_3,Z_3)=(z_2,x_1,y_2)$ and $Z_1< Z_2<Z_3$. However, since $2\nmid b=C$ and
$C^{Z_2-Z_1}=b^{y_1-y_3}>a^{x_1}\geq a=\max\{a,b,c\}\geq10^{62}$, by Lemma $\ref{4l6}$, it is impossible.
Case \uppercase\expandafter{\romannumeral2:} $y_3\leq y_1<y_2$ and $z_1< z_3<z_2$
Since
\begin{equation}\label{6.23}
c^{z_3}=a^{x_3}+b^{y_3}=c^{z_3-z_1}a^{x_1}+c^{z_3-z_1}b^{y_1},
\end{equation} we have
\begin{equation}\label{6.24}
b^{y_3}(c^{z_3-z_1}b^{y_1-y_3}-1)=a^{x_1}(a^{x_3-x_1}-c^{z_3-z_1}),
\end{equation}
whence we get
\begin{equation}\label{6.25}
c^{z_3-z_1}b^{y_1-y_3}=a^{x_1}t_4+1, a^{x_3-x_1}=b^{y_3}t_4+c^{z_3-z_1}, t_4\in\mathbb N.
\end{equation}
By the first equality of $(\ref{6.25})$, we have
$c^{z_3-z_1}b^{y_1-y_3}>a^{x_1}$. It implies that
\begin{equation}\label{6.26} \max\{c^{z_3-z_1},b^{y_1-y_3}\}>a^{x_1/2} \geq \sqrt a.
\end{equation}
If $c^{z_3-z_1}>b^{y_1-y_3}$, by $(\ref{6.26})$, then $c^{z_3-z_1}>\sqrt a$. Notice that $(\ref{4.2})$ has three solutions $(X_j,Y_j,Z_j)(j=1,2,3)$ for $(A,B,C,\lambda)=(a,b,c,1)$ such that
$(X_1,Y_1,Z_1)=(x_1,y_1,z_1)$, $(X_2,Y_2,Z_2)=(x_3,y_3,z_3)$, $(X_3,Y_3,Z_3)=(x_1,y_2,z_2)$ and $Z_1< Z_2<Z_3$. However, because of $4\big |c^z=C^Z$ for any solutions $(x,y,z)$ and $(X,Y,Z)$ of $(\ref{1.1})$ and $(\ref{4.2})$ respectively, $C^{Z_2-Z_1}=c^{z_3-z_1}>\sqrt a=(\max\{a,b,c\})^{1/2}$ and $\max\{a,b,c\}\geq10^{62}$, by Lemma $\ref{4l6}$, it is impossible.
Similarly, if $c^{z_3-z_1}<b^{y_1-y_3}$, then $y_3<y_1$ and $b^{y_1-y_3}>\sqrt a$.
In this case, $(\ref{4.2})$ has three solutions $(X_j,Y_j,Z_j)(j=1,2,3)$ for $(A,B,C,\lambda)=(c,a,b,-1)$ such that $(X_1,Y_1,Z_1)=(z_3,x_3,y_3)$, $(X_2,Y_2,Z_2)=(z_1,x_1,y_1)$, $(X_3,Y_3,Z_3)=(z_2,x_1,y_2)$ and $Z_1< Z_2<Z_3$. However, since $2\nmid b=C,C^{Z_2-Z_1}=b^{y_1-y_3}>\sqrt a=(\max\{a,b,c\})^{1/2}$ and $\max\{a,b,c\}\geq10^{62}$, by Lemma $\ref{4l6}$, it is impossible.
Case \uppercase\expandafter{\romannumeral3:} $y_1<y_3<y_2$ and $z_3\leq z_1<z_2$
Since $x_3>x_1$, we have
\begin{equation}\label{6.27}
a^{x_1}+b^{y_1}=c^{z_1}\geq c^{z_3}=a^{x_3}+b^{y_3}>a^{x_1}+b^{y_1},
\end{equation} a contradiction.
Case \uppercase\expandafter{\romannumeral4:} $y_1<y_3<y_2$ and $z_1< z_3<z_2$
By $(\ref{6.23})$, we have
\begin{equation}\label{6.28}
a^{x_1}(c^{z_3-z_1}-a^{x_3-x_1})=b^{y_1}(b^{y_3-y_1}-c^{z_3-z_1}).
\end{equation}
Since $\gcd(a,b)=1$, we get from $(\ref{6.28})$ that
\begin{equation}\label{6.29}
c^{z_3-z_1}-a^{x_3-x_1}=b^{y_1}t_5, b^{y_3-y_1}-c^{z_3-z_1}=a^{x_1}t_5, t_5\in\mathbb N, t_5\not=0.
\end{equation}
If $t_5>0$, then from the second equality of $(\ref{6.29})$ we get $b^{y_3-y_1}>a^{x_1}\geq a=\max\{a,b,c\}\geq10^{62}$, In this case, $(\ref{4.2})$ has three solutions $(X_j,Y_j,Z_j)(j=1,2,3)$ for $(A,B,C,\lambda)=(c,a,b,-1)$ such that $(X_1,Y_1,Z_1)=(z_1,x_1,y_1)$, $(X_2,Y_2,Z_2)=(z_3,x_3,y_3)$, $(X_3,Y_3,Z_3)=(z_2,x_1,y_2)$ and $Z_1< Z_2<Z_3$. However, since $2\nmid b=C,C^{Z_2-Z_1}=b^{y_3-y_1}>a=\max\{a,b,c\}\geq10^{62}$, by Lemma $\ref{4l6}$, it is impossible.
Similarly,if $t_5<0$, then from the second equality of $(\ref{6.29})$ we get $c^{z_3-z_1}>a^{x_1}\geq a=\max\{a,b,c\}\geq10^{62}$, In this case, $(\ref{4.2})$ has three solutions $(X_j,Y_j,Z_j)(j=1,2,3)$ for $(A,B,C,\lambda)=(a,b,c,1)$ such that
$(X_1,Y_1,Z_1)=(x_1,y_1,z_1)$, $(X_2,Y_2,Z_2)=(x_3,y_3,z_3)$, $(X_3,Y_3,Z_3)=(x_1,y_2,z_2)$ and $Z_1< Z_2<Z_3$. However, since $4\big |c^z=C^Z$ for any solutions $(x,y,z)$ and $(X,Y,Z)$ of $(\ref{1.1})$ and $(\ref{4.2})$ respectively, and $C^{Z_2-Z_1}=c^{z_3-z_1}>\max\{a,b,c\}\geq10^{62}$, by Lemma $\ref{4l6}$, it is impossible.
Thus, Theorem 1.1 holds for the case $c\not=\max\{a,b,c\}$. To sum up, the theorem is proved. $\Box$
\begin{flushleft} Yongzhong Hu\\ Department of Mathematics \\ Foshan University\\ Foshan,Guangdong 528000,China\\ E-mail:[email protected] \end{flushleft}
\begin{flushleft} Maohua Le\\ Institute of Mathematics\\ Lingnan Normal University\\ Zhanjiang,Guangdong 524048,China\\ E-mail:[email protected] \end{flushleft}
\end{document} | arXiv |
Munishvara
Munishvara or Munīśvara Viśvarūpa (born 1603) was an Indian mathematician who wrote several commentaries including one on astronomy Siddhanta Sarvabhauma (1646) which included descriptions of astronomical instruments such as the pratoda yantra.[1] Another commentary was Lilavativivruti.[2] Very little is known about his other than that he came from a family of astronomers including his father Ranganatha who wrote a commentary called Gụ̄hārthaprakaśa/Gūḍhārthaprakāśikā,[3] a commentary on the Suryasiddhanta. His grandfather Ballala had his origins in Dadhigrama in Vidharba and had moved to Benares and several sons wrote commentaries on astronomy and mathematics. Munisvara's Siddhantasarvabhauma had the patronage of Shah Jahan like his paternal uncle Krishna Daivagna. He was opposed to fellow mathematician Kamalakara whose brother also wrote a critique of Munisvara's bhangi-vibhangi method for planetary motions. He was also opposed to the adoption of some mathematical ideas in spherical trigonometry from Arab scholars.[4] An edition of his Siddhanta Sarvabhauma was published from the Princess of Wales Sarasvati Bhavana Granthamala edited by Gopinath Kaviraj.[5] Munisvara's book had twelve chapters in two parts. The second part had notes on astronomical instruments. He was a follower of Bhaskara II.[6]
Munishvara
Born1608
Other namesMunīśvara Viśvarūpa
Occupationmathematician
Known for
• Siddhanta Sarvabhauma
• Lilavativivruti
Parent
• Ranganatha (father)
RelativesBallala (grandfather)
References
1. Ohashi, Yukio (1987). "A note on some Sanskrit manuscripts on astronomical instruments". International Astronomical Union Colloquium. 91: 191–196. doi:10.1017/S0252921100106037. S2CID 128504497.
2. Sinha, Kripa Nath (1985). "Śrîpati: An eleventh-century Indian mathematician". Historia Mathematica. 12 (1): 25–44. doi:10.1016/0315-0860(85)90066-7.
3. Hall, F.E., ed. (1859). The Gudhartha-Prakasaka.
4. Plofker, Kim (2002), Ansari, S. M. Razaullah (ed.), "Spherical Trigonometry and the Astronomy of the Medieval Kerala School", History of Oriental Astronomy, Astrophysics and Space Science Library, Dordrecht: Springer Netherlands, vol. 275, pp. 83–93, doi:10.1007/978-94-015-9862-0_8, ISBN 978-90-481-6033-4, retrieved 18 July 2022
5. Gopinath Kaviraj, Munishvara (1932). Siddhanta Sarvabhauma. Benaras: Sarasvati Bhavana Granthamala, No, 41.
6. Sarma, K.V. (2008). "Munisvara". In Selin, Helaine (ed.). Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. Springer. p. 752.
External links
• Manuscript version of Siddhant Sarvabhauma (1627) from the Asiatic Society of Bombay
• Manuscript from Raghunath Temple, Jammu
• Siddhanta Sarvabhauma - Saraswati Bhavan - (Part 1) (Part 2)
Indian mathematics
Mathematicians
Ancient
• Apastamba
• Baudhayana
• Katyayana
• Manava
• Pāṇini
• Pingala
• Yajnavalkya
Classical
• Āryabhaṭa I
• Āryabhaṭa II
• Bhāskara I
• Bhāskara II
• Melpathur Narayana Bhattathiri
• Brahmadeva
• Brahmagupta
• Govindasvāmi
• Halayudha
• Jyeṣṭhadeva
• Kamalakara
• Mādhava of Saṅgamagrāma
• Mahāvīra
• Mahendra Sūri
• Munishvara
• Narayana
• Parameshvara
• Achyuta Pisharati
• Jagannatha Samrat
• Nilakantha Somayaji
• Śrīpati
• Sridhara
• Gangesha Upadhyaya
• Varāhamihira
• Sankara Variar
• Virasena
Modern
• Shanti Swarup Bhatnagar Prize recipients in Mathematical Science
Treatises
• Āryabhaṭīya
• Bakhshali manuscript
• Bijaganita
• Brāhmasphuṭasiddhānta
• Ganita Kaumudi
• Karanapaddhati
• Līlāvatī
• Lokavibhaga
• Paulisa Siddhanta
• Paitamaha Siddhanta
• Romaka Siddhanta
• Sadratnamala
• Siddhānta Shiromani
• Śulba Sūtras
• Surya Siddhanta
• Tantrasamgraha
• Vasishtha Siddhanta
• Veṇvāroha
• Yuktibhāṣā
• Yavanajataka
Pioneering
innovations
• Brahmi numerals
• Hindu–Arabic numeral system
• Symbol for zero (0)
• Infinite series expansions for the trigonometric functions
Centres
• Kerala school of astronomy and mathematics
• Jantar Mantar (Jaipur, New Delhi, Ujjain, Varanasi)
Historians of
mathematics
• Bapudeva Sastri (1821–1900)
• Shankar Balakrishna Dikshit (1853–1898)
• Sudhakara Dvivedi (1855–1910)
• M. Rangacarya (1861–1916)
• P. C. Sengupta (1876–1962)
• B. B. Datta (1888–1958)
• T. Hayashi
• A. A. Krishnaswamy Ayyangar (1892– 1953)
• A. N. Singh (1901–1954)
• C. T. Rajagopal (1903–1978)
• T. A. Saraswati Amma (1918–2000)
• S. N. Sen (1918–1992)
• K. S. Shukla (1918–2007)
• K. V. Sarma (1919–2005)
Translators
• Walter Eugene Clark
• David Pingree
Other regions
• Babylon
• China
• Greece
• Islamic mathematics
• Europe
Modern
institutions
• Indian Statistical Institute
• Bhaskaracharya Pratishthana
• Chennai Mathematical Institute
• Institute of Mathematical Sciences
• Indian Institute of Science
• Harish-Chandra Research Institute
• Homi Bhabha Centre for Science Education
• Ramanujan Institute for Advanced Study in Mathematics
• TIFR
| Wikipedia |
# Chapter 1
## Fourier Series
We will consider complex valued periodic functions with period $2 \pi$. We can view them as functions defined on the circumference $S$ of the unit circle in the complex plane or equivalently as function $f$ defined on $[-\pi, \pi]$ with $f(-\pi)=f(\pi)$. The Fourier Coefficients of the function $f$ are defined by
$$
a_{n}=\frac{1}{2 \pi} \int_{-\pi}^{\pi} f(x) e^{-i n x} d x
$$
and formally
$$
f(x) \simeq \sum a_{n} e^{i n x}
$$
If we assume that $f \in L_{1}[0,2 \pi]$ then clearly $a_{n}$ is well defined and
$$
\left|a_{n}\right| \leq \frac{1}{2 \pi} \int_{-\pi}^{\pi}|f(x)| d x
$$
It is not clear that the sum on right hand side of equation 1.2 converges and even if it does it is not clear that it is actually equal to the the function $f(x)$. It is relatively easy to find conditions on $f(\cdot)$ so that the sum in 1.2 is convergent. If $f(x)$ is assumed to be $k$ times continuously differentiable on $S$, integrating by parts $k$ times one gets, for $n \neq 0$,
$$
\left|a_{n}\right| \leq \frac{1}{n^{k}} \sup _{x}\left|f^{\{k\}}(x)\right|
$$
From the estimate 1.3 it is easily seen that the sum is convergent if $f$ is twice continuosly differentiable.
Another important but elementary fact is Theorem 1 (Riemann-Lebesgue). For every $f \in L_{1}[-\pi, \pi]$,
$$
\lim _{n \rightarrow \pm \infty} a_{n}=0
$$
Let us define the partial sums
$$
s_{N}(f, x)=\sum_{|n| \leq N} a_{n} e^{i n x}
$$
and the Fejer sum
$$
S_{N}(f, x)=\frac{1}{N+1} \sum_{0 \leq n \leq N} s_{n}(f, x)
$$
We can calculate
$$
\begin{aligned}
s_{n}(f, x)= & \frac{1}{2 \pi} \sum_{|j| \leq n} e^{i j x} \int_{-\pi}^{\pi} e^{-i j y} f(y) d y \\
& =\frac{1}{2 \pi} \int_{-\pi}^{\pi} f(y)\left[\sum_{|j| \leq n} e^{i j(x-y)}\right] d y \\
& =\frac{1}{2 \pi} \int_{-\pi}^{\pi} f(y) \frac{e^{-i n(x-y)}\left(e^{i(2 n+1)(x-y)}-1\right)}{e^{i(x-y)}-1} d y \\
& =\int_{0}^{2 \pi} f(y) k_{n}(x-y) d y
\end{aligned}
$$
where
$$
k_{n}(z)=\frac{1}{2 \pi} \frac{e^{-i n z}\left(e^{i(2 n+1) z}-1\right)}{e^{i z}-1}=\frac{1}{2 \pi} \frac{\sin \left(n+\frac{1}{2}\right) z}{\sin \frac{z}{2}}
$$
A similar calculation reveals
$$
S_{N}(f, x)=\int_{-\pi}^{\pi} f(y) K_{N}(x-y) d y
$$
where
$$
\begin{aligned}
K_{N}(z)= & \frac{1}{2 \pi} \frac{1}{(N+1)} \frac{1}{\sin \frac{z}{2}} \sum_{0 \leq n \leq N}\left[e^{i(n+1) z}-e^{-i n z}\right] \\
& =\frac{1}{2 \pi} \frac{1}{(N+1)} \frac{1}{\sin \frac{z}{2}} \frac{1}{e^{i z}-1}\left[\left(e^{i z}-e^{-i N z}\right)\left(e^{i(N+1) z}-1\right)\right] \\
& =\frac{1}{\pi} \frac{1}{(N+1)}\left[\frac{\sin \left(N+\frac{1}{2}\right) z}{\sin \frac{z}{2}}\right]^{2}
\end{aligned}
$$
Notice that for every $N$,
$$
\int_{-\pi}^{\pi} k_{N}(z) d z=\int_{-\pi}^{\pi} K_{N}(z) d z=1
$$
The following observations are now easy to make.
1. Nonnegativity.
$$
K_{N}(z) \geq 0
$$
2. For any $\delta>0$,
$$
\lim _{N \rightarrow \infty} \sup _{|z| \geq \delta} K_{N}(z)=0
$$
3. Therefore
$$
\lim _{N \rightarrow \infty} \int_{|z| \geq \delta} K_{N}(z) d z=0
$$
It is now an easy exercise to prove
Theorem 2. For any $f$ that is bounded and continuous on $S$
$$
\lim _{N \rightarrow \infty} \sup _{x \in S}\left|S_{N}(f, x)-f(x)\right|=0
$$
Theorem 3. For any $f \in L_{p}[-\pi, \pi]$
$$
\left\|S_{N}(f, \cdot)\right\|_{p} \leq\|f\|_{p}
$$
and therefore for $1 \leq p<\infty$,
$$
\lim _{N \rightarrow \infty}\left\|S_{N}(f, \cdot)-f(\cdot)\right\|_{p}=0
$$
The behavior of $s_{N}(f, x)$ is more complicated. However it is easy enough to see that
Theorem 4. For $f \in C^{2}(S)$,
$$
\lim _{N \rightarrow \infty} \sup _{x}\left|s_{N}(f, x)-f(x)\right|=0
$$
The series converges and so $s_{N}(f, \cdot)$ has a uniform limit $g . S_{N}(f, \cdot)$ has the same limit, but has just been shown to converge to $f$. Therefore $f=g$. The following Theorem is fairly easy.
Theorem 5. If $f$ is a function in $C^{\alpha}(S)$ i.e Hölder continuous with some exponent $\alpha>0$ then
$$
\lim _{N \rightarrow \infty} s_{N}(f, x)=f(x)
$$
Proof. We can assume thatwith out loss of generality that $x=0$ and let $f(0)=a$. We need to show that
$$
\lim _{N \rightarrow \infty} \frac{1}{2 \pi} \int_{-\pi}^{\pi} f(y) \frac{\sin \left(N+\frac{1}{2}\right) y}{\sin \frac{y}{2}} d y=a
$$
Because $\frac{f(y)-a}{\sin \frac{y}{2}}$ is integrable, 1.12 is consequence of the Riemann-Lebesgue Theorem, i.e. Theorem 1.
If $f$ is a bounded function, then one can replace $\sin \frac{y}{2}$ by $\frac{y}{2}$ and the problem reduces to calculating
$$
\lim _{\lambda \rightarrow \infty} \frac{1}{\pi} \int_{-\pi}^{\pi} f(y) \frac{\sin \lambda y}{y} d y
$$
Let us now assume that $f$ is a function of bounded variation on $S$ which has left and right limits $a_{l}$ and $a_{r}$ at 0 . By a change of variables one can reduce the above to calculating
$$
\lim _{\lambda \rightarrow \infty} \frac{1}{\pi} \int_{-\lambda \pi}^{\lambda \pi} f\left(\frac{y}{\lambda}\right) \frac{\sin y}{y} d y
$$
If we denote by
$$
G(y)=\int_{y}^{\infty} \frac{\sin x}{x} d x
$$
then
$$
\begin{aligned}
a_{r}(\lambda) & =\frac{1}{\pi} \int_{0}^{\lambda \pi} f\left(\frac{y}{\lambda}\right) \frac{\sin y}{y} d y=-\frac{1}{\pi} \int_{0}^{\lambda \pi} f\left(\frac{y}{\lambda}\right) d G(y) \\
& =\frac{1}{2} a_{r}+\frac{1}{\pi} \int_{0}^{\lambda \pi} G(y) d f\left(\frac{y}{\lambda}\right)=\frac{1}{2} a_{r}+\frac{1}{\pi} \int_{0}^{\pi} G(\lambda y) d f(y) \\
& \rightarrow \frac{1}{2} a_{r}
\end{aligned}
$$
by the bounded convergence theorem. This establishes the following Theorem 6. If $f$ is of bounded variation on $S$
$$
\lim _{N \rightarrow \infty} s_{N}(f, x)=\frac{f(x+0)+f(x-0)}{2}
$$
The behavior of $s_{N}(f, x)$ for $f$ in $L_{p}[-\pi, \pi]$ for $1 \leq p<\infty$ is more complex. Let us define the linear operator
$$
\left(T_{\lambda} f\right)(x)=\int_{-\pi}^{\pi} f(x+y) \frac{\sin \lambda y}{\sin \frac{y}{2}} d y
$$
on smooth functions $f$. If $s_{N}(f, x)$ were to converge uniformly to $f$ for every bounded continuous function it would follow by the uniform boundedness principle that
$$
\sup _{x}\left|\left(T_{\lambda} f\right)(x)\right| \leq C \sup _{x}|f(x)|
$$
with a constant independent of $\lambda$, atleast for $\lambda=N+\frac{1}{2}$ for positive integers $N$. Let us show that this is false. The best possible bound $C=C_{\lambda}$ is seen to be
$$
C_{\lambda}=\frac{1}{2 \pi} \int_{-\pi}^{\pi} \frac{|\sin \lambda y|}{\left|\sin \frac{y}{2}\right|} d y
$$
and because
$$
\left|\frac{1}{\sin \frac{y}{2}}-\frac{2}{y}\right|
$$
is integrable, $C_{\lambda}$ differs from
$$
\frac{1}{2 \pi} \int_{-\pi}^{\pi} \frac{|\sin \lambda y|}{|y|} d y=\frac{1}{2 \pi} \int_{-\lambda \pi}^{\lambda \pi} \frac{|\sin y|}{|y|} d y
$$
by a uniformly bounded amount. The divergence of
$$
\frac{1}{2 \pi} \int_{-\infty}^{\infty} \frac{|\sin y|}{|y|} d y
$$
implies that $C_{\lambda} \rightarrow \infty$ as $\lambda \rightarrow \infty$. By duality theis means that $T_{\lambda} f$ is not uniformly bounded as an operator from $L_{1}[-\pi, \pi]$ into itself either. Again from uniform boudedness principle one cannot expect that $s_{N}(f, \cdot)$ tends to $f(\cdot)$ in $L_{1}[-\pi, \pi]$ for evrery $f \in L_{1}[-\pi, \pi]$. However we will prove that for $1<p<\infty$, for $f \in L_{p}[-\pi, \pi]$
$$
\lim _{\lambda \rightarrow \infty}\left\|T_{\lambda} f-f\right\|_{p}=0
$$
By standard arguments involving the approximation of an $L_{p}$ function by a continuous function it is sufficient to prove a uniform bound of the form
$$
\left\|T_{\lambda} f\right\|_{p} \leq C_{p}\|f\|_{p}
$$
with a constant $C_{p}$ depending only on $p$ for smooth functions $f$ and $\lambda \geq 1$ First we establtsh what is known as weak type inequality.
Theorem 7. There is a constant $C$ such that for all $\lambda \geq 1$ and smooth $f$
$$
\operatorname{mes}\left\{x:\left|\left(T_{\lambda} f\right)\right| \geq \ell\right\} \leq \frac{C}{\ell}\|f\|_{1}
$$
## Singular Integrals
We start with a very useful covering lemma.
Lemma 1. Suppose $K \subset S$ is a compact subset and $I_{\alpha}$ is a covering of $K$. There is a finite subcollection $\left\{I_{j}\right\}$ such that
1. $\left\{I_{j}\right\}$ are disjoint.
2. The intervals $\left\{3 I_{j}\right\}$ that have the same midpoints as $\left\{I_{j}\right\}$ but three times the lenghth cover $K$.
Proof. We first choose a finite subcover. From the finite subcover we pick the largest interval. In case of a tie pick any of the competing ones. Then, at any stage, of the remaining intervals from our finite subcollection we pick the largest one that is disjoint from the ones already picked. We stop when we cannot pick any more. The collection that we end up with is clearly disjoint and finite. Let $x \in K$. This is covered by one of the intervals $I$ from our finite subcollection covering $K$. If $I$ was picked there is nothing to prove. If $I$ is not picked it must intersect some $I_{j}$ already picked. Let us look at the first such interval and call it $I_{j} . \quad I$ is disjoint from all the previously picked ones and $I$ was passed over when we picked $I_{j}$. Therefore inaddition to intersecting $I_{j}, I$ is not larger than $I_{j}$. Therefore $3 I_{j} \supset I \ni x$.
This lemma is used in proving maximal inequalities. For instance, for the Hardy-Littlewood maximal function we have
Theorem 1. Let $f \in L_{1}(S)$. Define
$$
\begin{gathered}
M_{f}(x)=\sup _{0<r<\frac{\pi}{2}} \frac{1}{2 r} \int_{|y-x|<r}|f(y)| d y \\
\mu\left[x: M_{f}(x)>\ell\right] \leq \frac{3 \int|f(y)| d y}{\ell}
\end{gathered}
$$
Proof. Let us denote by $E_{\ell}$ the set
$$
E_{\ell}=\left\{x: M_{f}(x)>\ell\right\}
$$
and let $K \subset E_{\ell}$ be an arbitrary compact set. For each $x \in K$ there is an interval $I_{x}$ such that
$$
\int_{I_{x}}|f(y)| d y \geq \ell \mu\left(I_{x}\right)
$$
Clearly $\left\{I_{x}\right\}$ is a covering of $K$ and by lemma we get a finite disjoint sub collection $\left\{I_{j}\right\}$ such that $\left\{3 I_{j}\right\}$ covers $K$. Adding them up
$$
\int|f(y)| d y \geq \sum_{j} \mu\left(I_{j}\right) \geq \frac{1}{3} \sum_{j} \mu\left(3 I_{j}\right) \geq \mu(K)
$$
Sine $K \subset E_{\ell}$ is arbitrary we are done.
There is no problem in replacing $\left\{x:\left|M_{f}(x)\right|>\ell\right\}$ by $\left\{x:\left|M_{f}(x)\right| \geq \ell\right\}$. Replace $\ell$ by $\ell-\epsilon$ and let $\epsilon \rightarrow 0$.
This theorem can be used to prove the Labesgue diffrentiability theorem.
Theorem 2. For any $f \in L_{1}(S)$,
$$
\lim _{h \rightarrow 0} \frac{1}{2 h} \int_{|x-y| \leq h}|f(y)-f(x)| d y=0 \quad \text { a.e. } \quad x
$$
Proof. It is sufficient to prove that for any $\delta>0$
$$
\mu\left[x: \limsup _{h \rightarrow 0} \frac{1}{2 h} \int_{|x-y| \leq h}|f(y)-f(x)| d y \geq \delta\right]=0
$$
Given $\epsilon>0$ we can write $f=f_{1}+g$ with $f_{1}$ continuous and $\|g\|_{1} \leq \epsilon$ and
$$
\begin{aligned}
\mu\left[x: \limsup _{h \rightarrow 0} \frac{1}{2 h}\right. & \left.\int_{|x-y| \leq h}|f(y)-f(x)| d y \geq \delta\right] \\
& =\mu\left[x: \limsup _{h \rightarrow 0} \frac{1}{2 h} \int_{|x-y| \leq h}|g(y)-g(x)| d y \geq \delta\right] \\
& \leq \mu\left[x: \sup _{h>0} \frac{1}{2 h} \int_{|x-y| \leq h}|g(y)-g(x)| d y \geq \delta\right] \\
& \leq \frac{3\|h\|_{1}}{\delta} \leq \frac{3 \epsilon}{\delta}
\end{aligned}
$$
Since $\epsilon>0$ is arbitrary we are done. In other words the maximal inequality is useful to prove almost sure convergence. Typically almost sure convergence will be obvious for a dense set and the maximal inequality will be used to interchange limits in the approximation.
Another summability method, like the Fejer sum that is often considred is the Poisson sum
$$
S(\rho, x)=\sum_{n} a_{n} \rho^{|n|} e^{i n x}
$$
and the kernel corresponding to it is the Poisson kernel
$$
p(\rho, z)=\frac{1}{2 \pi} \sum_{n} \rho^{|n|} e^{i n z}=\frac{1}{2 \pi} \frac{1-\rho^{2}}{\left(1-2 \rho \cos z+\rho^{2}\right)}
$$
so that
$$
P(\rho, x)=\int f(y) p(\rho, x-y) d y
$$
It is left as an exercise to prove that for for $1 \leq p<\infty$, every $f \in L_{p}$ $P(\rho, \cdot) \rightarrow f(\cdot)$ in $L_{p}$ as $\rho \rightarrow 1$. We will prove a maximal inequality for the Poisson sum, so that as a consequence we will get the almost sure convergence of $P(\rho, x)$ to $f$ for every $f$ in $L_{1}$.
Theorem 3. For every $f$ in $L_{1}$
$$
\mu\left[x: \sup _{0 \leq \rho<1} P(\rho, x) \geq \ell\right] \leq \frac{C\|f\|_{1}}{\ell}
$$
Proof. The proof consists of estimating the Poisson maximal function interms of the Hardy-Littlewood maximal function $M_{f}(x)$. We begin with some simple estimates for the Poisson kernel $p(\rho, z)$.
$$
\begin{aligned}
p(\rho, z)= & \frac{1}{2 \pi} \frac{1-\rho^{2}}{(1-\rho)^{2}+2 \rho(1-\cos z)} \leq \frac{1}{2 \pi} \frac{1-\rho^{2}}{(1-\rho)^{2}} \\
& =\frac{1}{2 \pi} \frac{1+\rho}{1-\rho} \leq \frac{1}{\pi} \frac{1}{1-\rho}
\end{aligned}
$$
The problem therefore is only as $\rho \rightarrow 1$. Lets us assume that $\rho \geq \frac{1}{2}$. For any symmetric function $\phi(z)$ the intgral
$$
\begin{aligned}
\mid \int_{-\pi}^{\pi} & f(z) \phi(z) d z \mid \\
& =\left|\int_{0}^{\pi}[f(z)+f(-z)] \phi(z) d z\right| \\
& =\left|\int_{0}^{\pi} \phi(z)\left[\frac{d}{d z} \int_{-z}^{z} f(y) d y\right] d z\right| \\
& \leq\left|\int_{0}^{\pi} \phi^{\prime}(z)\left[\int_{-z}^{z} f(y) d y\right] d z\right|+|\phi(\pi)| \int_{-\pi}^{\pi} f(z) d z \mid \\
& \leq \int_{0}^{\pi} 2\left|z \phi^{\prime}(z)\right|\left[\int|f(y)| \lambda_{z}(d y)\right] d z+\phi(\pi)\left|\int_{-\pi}^{\pi}\right| f(z) \mid d z \\
& \leq 2 M_{f}(0) \int_{0}^{\pi}\left|z \phi^{\prime}(z)\right| d z+\phi(\pi)|| M_{f}(0) \mid
\end{aligned}
$$
For the Poisson kernel
$$
\begin{aligned}
\left|z \frac{d}{d z} p(\rho, z)\right| & =\frac{1}{2 \pi} \frac{1-\rho^{2}}{\left(1-2 \rho \cos z+\rho^{2}\right)^{2}} 2 \rho|z \sin z| \\
& \leq \frac{1}{\pi} \frac{(1-\rho) z^{2}}{(1-\rho)^{4}+(1-\cos z)^{2}} \\
& \leq C \frac{(1-\rho) z^{2}}{(1-\rho)^{4}+z^{4}}
\end{aligned}
$$
and
$$
\begin{aligned}
\int_{-\pi}^{\pi}\left|z \frac{d}{d z} p(\rho, z)\right| d z & \leq C \int_{-\pi}^{\pi} \frac{(1-\rho) z^{2}}{(1-\rho)^{4}+z^{4}} d z \\
& =\int_{-\frac{\pi}{1-\rho}}^{\frac{\pi}{1-\rho}} \frac{z^{2}}{1+z^{4}} d z \\
& \leq \int_{-\infty}^{\infty} \frac{z^{2}}{1+z^{4}} d z \leq C_{1}
\end{aligned}
$$
uniformly in $\rho$.
Interpolation theorems play a very important role in Harmonic Analysis. An example is the following Theorem 4 (Marcinkiewicz). Let $T$ be a sublinear map defiened on $L_{p} \cap$ $L_{q}$ that satisfies weak type inequlities
$$
\mu[|x|:|(T f)(x)| \geq \ell] \leq \frac{C_{i}\|f\|_{p_{i}}^{p_{i}}}{\ell^{p_{i}}}
$$
for $i=1,2$ where $1 \leq p_{1}<p_{2}<\infty$. Then for $p_{1}<p<p_{2}$, there are constants $C_{p}$ such that
$$
\|T f\|_{p} \leq C_{p}\|f\|_{p}
$$
Note that $T$ need not be linear. It need only satisfy
$$
|T(f+g)|(x) \leq|T f|(x)+|T g|(x)
$$
Proof. Let $p \in\left(p_{1}, p_{2}\right)$ be fixed. For any function $f \in L_{p}$ and for any positive number $a$ we deine $f_{a}=f \chi_{\{|f| \leq a\}}$ and $f^{a}=\chi_{\{|f|>a\}}$. Clearly $f_{a} \in L_{p_{2}}$ and $f^{a} \in L_{p_{1}}$
$$
\begin{aligned}
\mu[x:|T f(x)| \mid \geq 2 \ell] & \leq \mu\left[x:\left|T f_{a}(x)\right| \mid \geq \ell\right]+\mu\left[x: \mid T f^{a}(x) \| \geq \ell\right] \\
& \leq \frac{C_{2}}{\ell^{p_{2}}} \int_{|f(x)| \leq a}|f(x)|^{p_{2}} d \mu+\frac{C_{1}}{\ell^{p_{1}}} \int_{|f(x)|>a}|f(x)|^{p_{1}} d \mu
\end{aligned}
$$
Take $a=\ell$, multiply by $\ell^{p-1}$ and integrate with respect to $\ell$ from 0 to $\infty$. Use Fubini's theorem. We get
$$
\int_{0}^{\infty} \ell^{p-1} \mu[x:|T f(x)| \mid \geq 2 \ell] d \ell \leq\left[\frac{C_{2}}{p_{2}-p}+\frac{C_{1}}{p-p_{1}}\right] \int|f(x)|^{p} d \mu
$$
Since the left hand side is $\frac{\|T f\|_{p}^{p}}{p}$ we are done.
There is a slight variation of the argument that allows $p_{2}$ to be infinite provided $T$ is bounded on $L_{\infty}$. If we denote the norm by $C_{2}$ we use
$$
\mu[x:|T f(x)| \mid \geq(C+1) \ell] \leq \mu\left[x:\left|T f^{a}(x)\right| \mid \geq \ell\right]
$$
and proceed as before.
A different interpolation theorem for linear maps $T$ is the following Theorem 5 (Riesz-Thorin). If a linear map $T$ is bounded from $L_{p_{i}}$ into $L_{p_{i}}$ with a bound $C_{i}$ for $i=1,2$ then for $p_{1} \leq p \leq p_{2}$ it is bounded from $L_{p}$ into $L_{p}$ with a bound $C_{p}$ that can be taken to be
$$
C_{p}=C_{1}^{t} C_{2}^{1-t}
$$
where $t$ is determined by
$$
\frac{1}{p}=t \frac{1}{p_{1}}+(1-t) \frac{1}{p_{2}}
$$
Proof. The proof uses methods from the theory of functions of a complex variable. The starting point is the maximum modulus principle. Let us assume that $u(z)$ is analytic in the open strip $a<R e z<b$ and bounded and continuous in the closed strip $a \leq R e z \leq b$. Let $M(x)$ be the maximum modulus of the function on the line $\operatorname{Re} z=x$. Then $\log M(x)$ is a convex function of $x$. This is not hard to see. Clearly the maximum principle dictates that
$$
M(x) \leq \max [M(a), M(b)]
$$
If one is worried about the maximum being attained, one can always mutiply by $e^{\epsilon z^{2}}$ and let $\epsilon$ go to 0 . Replacing $u(z)$ by $u(z) e^{t z}$ yields the inequality
$$
M(x) \leq \max \left[M(a) e^{t(a-x)}, M(b) e^{t(b-x)}\right]
$$
optimizing with respect to $t$ we get,
$$
M(x) \leq \max \left[[M(a)]^{\frac{b-x}{b-a}},[M(b)]^{\frac{x-a}{b-a}}\right]
$$
which is the required convexity.
We note that the maximum of any collection of convex functions is again convex. The proof is completed by representing $\log F(p)$, where $F(p)$ is the norm of $T$ from $L_{p}$ to $L_{p}$, as the supremum of a bunch of functions that are convex in $x=\frac{1}{p}$.
$$
\begin{aligned}
\|T\|_{p, p} & =\sup _{\substack{\|f\|_{p} \leq 1 \\
\|g\|_{q} \leq 1}}\left|\int g(T f) d \mu\right| \\
& =\sup _{\substack{\|f\|_{p} \leq 1, f \geq 0,|\phi|=1 \\
\|g\|_{q} \leq 1, g \geq 0,|\psi|=1}}\left|\int(g \psi)(T(f \phi)) d \mu\right| \\
& =\sup _{\substack{\|f\|_{1} \leq 1, f>0,|\phi|=1 \\
\|g\|_{1} \leq 1, g>0,|\psi|=1}}\left|\int\left(g^{x} \psi\right)\left(T\left(f^{1-x} \phi\right)\right) d \mu\right| \\
& =\sup _{\substack{\|f\|_{1} \leq 1, f>0,|\phi|=1 \\
\|g\|_{1} \leq 1, g>0,|\psi|=1 \\
\operatorname{Rez}=x}}\left|\int\left(g^{z} \psi\right)\left(T\left(f^{1-z} \phi\right)\right) d \mu\right| \\
& =\sup _{\substack{\|f\|_{1} \leq 1, f>0,|\phi|=1 \\
\|g\|_{1} \leq 1, g>0,|\psi|=1}} \sup _{\operatorname{Rez}=x}|u(f, g, \phi, \psi, z)|
\end{aligned}
$$
In particular for the Hardy-Littlewood or Poisson maximal function the $L_{\infty}$ bound is trivial and we now have a bound for the $L_{p}$ norm of the maximal function in terms of the $L_{p}$ norm of the original function provided $p>1$.
For a convolution operator of the form
$$
(T f)(x)=\int_{-\pi}^{\pi} f(y) k(x-y) d y
$$
we saw that for it to be bounded as an operator from $L_{1}$ into itself we need $k$ to be in $L_{1}$. However for $1<p<\infty$ the operator can some times be bounded even if $k$ is not in $L_{1}$. This is proved by establishing a bound from $L_{2}$ to $L_{2}$ and a weak type inequality in $L_{1}$. We can then use Marcinkiewicz interpolation, followed by Riesz-Thorin interpolation.
Theorem 6. If
$$
\hat{k}(n)=\int e^{i n z} k(z) d z
$$
is bounded in absolute value by $C$, then the convolution operator given by equation (12) is bounded by $C$ as an operator from $L_{2}$ to $L_{2}$. Proof. Use the the orthonormal basis $e^{i n x}$ to diagonalize $T$
$$
T e^{i n x}=\hat{k}(n) e^{i n x}
$$
We now proceed to establish weak type $(1,1)$ estimate. We shall assune that we have a kernel $k$ in $L_{1}$ that satisfies
1.
$$
\sup _{n}\left|\int k(y) e^{i n y} d y\right|=C_{1}<\infty
$$
2.
$$
\sup _{y} \int_{x:|x-y|>2|y|}|k(x-y)-k(x)| d x=C_{2}<\infty
$$
Although we have assumed that $k$ is in $L_{1}$ we will prove a weak type $(1,1)$ bound.
Theorem 7. The operator of convolution by $k$
$$
\left(T_{k} f\right)(x)=\int_{-\pi}^{\pi} k(x-y) f(y) d y
$$
satisfies the weak type inequality $(1,1)$
$$
\mu[|x|:|(T f)(x)| \geq \ell] \leq \frac{C}{\ell}\|f\|_{1}
$$
with a constant $C$ that depends only on $C_{1}$ and $C_{2}$.
Proof. Proof involves several steps.
- First we observe that the Hardy-Littlewood maximal function given by (1) satisfies equation 2). The set $G=\left[x: M_{f}(x) \geq \ell\right]$ is an open set in $[-\pi, \pi]$ and has Lebsgue measure atmost $\frac{3\|f\|_{1}}{\ell}$. We assume that $\ell>\frac{3\|f\|_{1}}{2 \pi}$ so that $B=G^{c}$ is nonempty. We write the open set $G$ as a possible countable union of disjoint open intervals $I_{j}$ of length $r_{j}$ centered at $x_{j}$. Note that the end points $x_{j} \pm \frac{1}{2} r_{j}$ necessarily belong to $B$. The maximal inequality assures us that
$$
\sum_{j} r_{j} \leq \frac{3\|f\|_{1}}{\ell}
$$
- Let us define the averages
$$
m_{j}=\frac{1}{r_{j}} \int_{I_{j}} f(y) d y
$$
and write $f$ in the form
$$
\begin{aligned}
f(x) & =\left[f(x) 1_{B}(x)+\sum_{j} m_{j} 1_{I_{j}}(x)\right]+\sum_{j}\left[f(x)-m_{j}\right] 1_{I_{j}}(x) \\
& =g(x)+\sum_{j} h_{j}(x)
\end{aligned}
$$
- We have the bounds
$$
\begin{aligned}
\left|m_{j}\right| & \leq \frac{1}{r_{j}} \int_{I_{j}}|f(y)| d y \leq \frac{1}{r_{j}} \int_{\tilde{I}_{j}}|f(y)| d y \\
& \leq 2 \frac{1}{2 r_{j}} \int_{\tilde{I}_{j}}|f(y)| d y \leq 2 M_{f}\left(x_{j} \pm r_{j}\right) \leq 2 \ell
\end{aligned}
$$
Here $\tilde{I}_{j}$ is the interval centered around $x_{j} \pm \frac{r_{j}}{2}$ of length $2 r_{j}$. In particular $\|g\|_{\infty} \leq 2 \ell$. On the other hand
$$
\sum_{j}\left\|h_{j}\right\|_{1}=\sum_{j} \int_{I_{j}}\left|f(y)-m_{j}\right| d y \leq 2 \sum_{j} \int_{I_{j}}|f(y)| d y \leq 2\|f\|_{1}
$$
We therefore have
$$
\|g\|_{1} \leq 3\|f\|_{1}
$$
Note that the decomposition depends on $\ell$. Let us write the corresponding sum
$$
u=T_{k} f=T_{k} g+\sum_{j} T_{k} h_{j}=v+\sum_{j} w_{j}=v+w
$$
- We estimate the $L_{2}$ norm of $v$ and the $L_{1}$ norm of $w$ on large enough set. Then use Tchebychev's inequality.
$$
\mu\left[x:|v(x)| \geq \frac{\ell}{2}\right] \leq \frac{\|v\|_{2}^{2}}{\ell^{2}} \leq \frac{C_{1}\|g\|_{2}^{2}}{\ell^{2}} \leq \frac{2 \ell C_{1}\|g\|_{1}}{\ell^{2}}=\frac{6 C_{1}\|f\|_{1}}{\ell}
$$
Let us denote by $\hat{I}_{j}$ the interval centered around $x_{j}$ of length $3 r_{j}$ and by $U=\cup_{j} \hat{I}_{j}$. We begin by estimating $\left\|w \cdot 1_{U^{c}}\right\|_{1}$.
$$
\begin{aligned}
\left\|w \cdot 1_{U^{c}}\right\|_{1} & \leq \int_{U^{c}} \sum_{j}\left|\int_{I_{j}} k(x-y)\left[f(y)-m_{j}\right] d y\right| d x \\
& =\int_{U^{c}} \sum_{j}\left|\int_{I_{j}}\left[k(x-y)-k\left(x-x_{j}\right)\right]\left[f(y)-m_{j}\right] d y\right| d x \\
& \leq \int_{U^{c}} \sum_{j} \int_{I_{j}}\left|k(x-y)-k\left(x-x_{j}\right)\right|\left|f(y)-m_{j}\right| d y d x \\
& =\sum_{j} \int_{I_{j}}\left|f(y)-m_{j}\right| d y \int_{U^{c}}\left|k(x-y)-k\left(x-x_{j}\right)\right| d x \\
& \leq \sum_{j} \int_{I_{j}}\left|f(y)-m_{j}\right| d y \int_{\hat{I}_{j}^{c}}\left|k(x-y)-k\left(x-x_{j}\right)\right| d x \\
& \leq \sum_{j} \int_{I_{j}}\left|f(y)-m_{j}\right| d y \int_{x:|x-y| \geq 2\left|y-x_{j}\right|}\left|k(x-y)-k\left(x-x_{j}\right)\right| d x \\
& \leq C_{2} \sum_{j} \int_{I_{j}}\left|f(y)-m_{j}\right| d y \\
& \leq 2 C_{2}\|f\|_{1}
\end{aligned}
$$
We have used here two facts. $f(y)-m_{j}$ has mean zero on $I_{j}$. If $y \in I_{j}$ and $x \in \tilde{I}_{j}^{c}$, then $|y-x| \geq r_{j} \geq 2\left|y-x_{j}\right|$. On the other hand
$$
\mu(U) \leq \sum \mu\left(\tilde{I}_{j}\right) \leq 3 \sum \mu\left(I_{j}\right)=3 \sum_{j} r_{j} \leq \frac{9\|f\|_{1}}{\ell}
$$
- Finally we can put the pieces together.
$$
\begin{aligned}
\mu(x:|u(x)| \geq 2 \ell) & \leq \mu(x:|v(x)| \geq \ell)+\mu(x:|w(x)| \geq \ell) \\
& \leq \frac{6 C_{1}\|f\|_{1}}{\ell}+\frac{9\|f\|_{1}}{\ell}+\frac{2 C_{2}\|f\|_{1}}{\ell}
\end{aligned}
$$
or
$$
\mu(x:|u(x)| \geq \ell) \leq \frac{\left(12 C_{1}+18+4 C_{2}\right)\|f\|_{1}}{\ell}=\frac{C\|f\|_{1}}{\ell}
$$
There is one point that we should note. For the interval doubling construction on the circle we should be sure that we do not see for instance any interval of lenghth larger than $\frac{\pi}{2}$ in $G$. This can be ensured if we take $\ell>\frac{6\|f\|_{1}}{\pi}$. The inequality is however satisfied for all $\ell$ because $C \geq 12$.
We want to look at the special kernel $k(y)=\frac{1}{y}$ which is not in $L_{1}$. We consider its truncation
$$
k_{\delta}(y)=\frac{1}{y} \mathbf{1}_{\{|y| \geq \delta\}}(y)
$$
First we estimate the Fourier transform
$$
\begin{aligned}
\left|\int_{|y| \geq \delta} \frac{e^{i n y}}{y} d y\right| & =2\left|\int_{\delta}^{\pi} \frac{\sin n y}{y} d y\right| \\
& =2\left|\int_{n \delta} n \pi \frac{\sin y}{y} d y\right| \leq 4 \sup _{0<a<\infty}\left|\int_{0}^{a} \frac{\sin y}{y} d y\right| \leq C_{1}
\end{aligned}
$$
Next in order to verify the condition (15) we need to estimate the following quantity uniformly in $y$ and $\delta$.
$$
\int_{x:|x-y|>2|y|}\left|k_{\delta}(x-y)-k_{\delta}(x)\right| d x
$$
There are three sets over which the integral does not vanish.
$$
\begin{aligned}
& F_{1}=\{x:|x-y|>2|y|,|x-y| \geq \delta,|x| \geq \delta\} \\
& F_{2}=\{x:|x-y|>2|y|,|x-y| \leq \delta,|x| \geq \delta\} \\
& F_{3}=\{x:|x-y|>2|y|,|x-y| \geq \delta,|x| \leq \delta\}
\end{aligned}
$$
We consider
$$
\begin{aligned}
\int_{F_{1}}\left|\frac{1}{x-y}-\frac{1}{x}\right| d x & \leq \int_{x:|x-y| \geq 2|y|}\left|\frac{1}{x-y}-\frac{1}{x}\right| d x \\
& \leq \int_{|z-1| \geq 2}\left|\frac{1}{z-1}-\frac{1}{z}\right| d z \\
& =C_{3}
\end{aligned}
$$
It is clear that $F_{2} \subset[-2 \delta, 2 \delta]$. Therefore
$$
\int_{F_{2}} \frac{1}{|x|} d x \leq 2 \int_{\delta}^{2 \delta} \frac{d x}{x}=C_{4}
$$
Finally $F_{3} \subset[x:|x-y| \leq 2 \delta]$ and works similarly. With $C_{2}=C_{3}+2 C_{4}$ we are done.
We are now ready to prove
Theorem 8. For any $f \in L_{p}$ the partial sums $s_{N}(f, x)$ converge to $f$ in $L_{p}$ provided $1<p<\infty$.
Proof. We need only prove, for $1<p<\infty$, a bound from $L_{p}$ to $L_{p}$, for the partial sum operators
$$
\left(T_{N} f\right)(x)=\int f(x-y) k_{N}(y) d y
$$
with
$$
k_{N}(z)=\frac{1}{2 \pi} \frac{\sin \left(N+\frac{1}{2}\right) z}{\sin \frac{z}{2}}
$$
that is uniform in $N$. In terms of multipliers we are looking at a uniform $L_{p}$ bound for the operators defined by
$$
\hat{k}_{N}(n)=\mathbf{1}_{\{|n| \leq N\}}(n)
$$
Let us define the operators $M_{k}$ as multiplication by $e^{i k x}$ which are isometries in every $L_{p} . P_{0}$ is the operator of projection to constants, i.e. the operator with multiplier $\mathbf{1}_{\{0\}}(n)$ which is clearly bounded in every $L_{p}$. Finally the Hilbert transform $S$ is the one with multiplier signum $n$. It is easy to verify that
$$
\left.\left.T_{N}=M_{-N} \frac{1}{2}\left[(S+I)+P_{0}\right] M_{N}-M_{(N+1)}\right] \frac{1}{2}\left[(S+I)+P_{0}\right] M_{-(N+1)}\right]
$$
This reduces the problem to proving that a single operator $S$ is bounded on $L_{p}$. The kernel is calculated to be
$$
s(z)=\frac{1}{2 \pi} \cot \frac{z}{2}
$$
This can be replaced by the modified kernel
$$
k(z)=\frac{1}{\pi z}
$$
and we are done.
## Multidimensional Versions
The problem of convergence of Fourier Series in several dimensions is more complicated because there is no natural truncation. If $n=\left\{n_{1}, \ldots, n_{d}\right\}$ is a multi-index, then the sum
$$
\sum_{n} a_{n} e^{i n \cdot x}
$$
is natuarally computed by summing over finite stes $D_{N}$ which are allowed to increase to $Z^{d}$. One tries to recover the function $f$ by
$$
f=\lim _{N \rightarrow \infty} \sum_{n \in D_{N}} a_{n} e^{i n \cdot x}
$$
For smooth functions there is no problem because $a_{n}$ decays fast. The degree of smoothness needed gets worse as dimension goes up. In $d$ dimnsions we need $\left|a_{n}\right|$ to decay like $|n|^{-d+\delta}$ for some $\delta>0$ to be sure of uniform convergence of the Fourier Series. On the other hand the orthogonality relations imply that in $f \in L_{2}$, the series converges in $L_{2}$ and again $D_{N}$ can be arbitrary. However for $1<p<\infty$ but different from 2 the situation is far from clear.
If we take $D_{N}=\left\{n:\left|n_{j}\right| \leq N, j=1, \ldots, d\right\}$ the partial sum operator we need to look at is convolution by
$$
\begin{aligned}
& {\left[\frac{1}{2 \pi}\right]^{d} \sum_{\substack{\left|n_{j}\right| \leq N \\
j=1, \ldots, d}} e^{i<n, x>}=\prod_{j=1}^{d} \frac{\sin \left(N+\frac{1}{2}\right) x_{j}}{2 \pi \sin \frac{x_{j}}{2}}} \\
& =\prod_{j=1}^{d} t_{N}\left(x_{j}\right)
\end{aligned}
$$
The partial sum operator $S_{N}$ is therefore the product
$$
T^{N}=\prod_{j=1}^{d} T_{j}^{N}
$$
where $T_{j}^{N}$ is the convolution in the variable $x_{j}$ by the kernel $t_{N}\left(x_{j}\right)$. It is easy to see that as operators $T_{j}^{N}$ have a bound that is uniform in $N$. The bound in the context of a single variable extends to $d$ variables because $t_{j}^{N}$ acts only on the single variable $x_{j}$. Therefore $T_{N}$ have a uniform bound as well. Therefore we have with the choice of the cube $D_{N}=\left\{n:\left|n_{j}\right| \leq N, j+1, \ldots d\right\}$, we have convergence in $L_{p}$ of the partial sums to $f$, for every $f \in L_{p}$ provided $1<p<\infty$.
It is known that the result is false for any $p \neq 2$ if we choose $D_{N}=\{n$ : $n_{1}^{2}+\cdots+n_{d}^{2} \leq N^{2}$.
We now look at Fourier Transforms on $R^{d}$. If $f(x)$ is a function in $L_{1}\left(R^{d}\right)$ its Fourier transform $\hat{f}(y)$ is defined by
$$
\hat{f}(y)=\left(\frac{1}{\sqrt{2 \pi}}\right)^{d} \int_{R^{d}} e^{i<x, y>} f(x) d x
$$
We denote by $\mathcal{S}$ the class of all functions $f$ on $R^{d}$ that are infinitely differentiable such that the function and its derivitives of all orders decay faster than any power, i.e. for every $n_{1}, n_{2}, \ldots, n_{d} \geq 0$ and $k \geq 0$ there are constants $C_{n_{1}, n_{2}, \ldots, n_{d}, k}$ such that
$$
\left|\left[\left(\frac{d}{d x_{1}}\right)^{n_{1}}\left(\frac{d}{d x_{1}}\right)^{n_{2}} \ldots\left(\frac{d}{d x_{d}}\right)^{n_{d}} f\right](x)\right| \leq C_{n_{1}, n_{2}, \cdots, n_{d}, k}(1+\|x\|)^{-k}
$$
It is easy to show by repeated integration by parts that if $f \in \mathcal{S}$ so does $\hat{f}$.
Theorem 1. The Fourier transform has the inverse
$$
f(x)=\left(\frac{1}{\sqrt{2 \pi}}\right)^{d} \int_{R^{d}} e^{-i<x, y>} \hat{f}(y) d y
$$
proving that the Fourier transform is a one to one mapping of $\mathcal{S}$ onto itself.
In addition the Fourier transform extends as a unitary map from $L_{2}\left(R^{d}\right)$ onto $L_{2}\left(R^{d}\right)$.
Proof. Clearly
$$
g(x)=\left(\frac{1}{\sqrt{2 \pi}}\right)^{d} \int_{R^{d}} e^{-i<x, y>} \hat{f}(y) d y
$$
is well defined as a function in $\mathcal{S}$. We only have to identify it. We compute $g$ as
$$
\begin{aligned}
g(x) & =\left(\frac{1}{\sqrt{2 \pi}}\right)^{d} \int_{R^{d}} e^{-i<x, y>} \hat{f}(y) d y \\
& =\lim _{\epsilon \rightarrow 0}\left(\frac{1}{\sqrt{2 \pi}}\right)^{d} \int_{R^{d}} e^{-i<x, y>} \hat{f}(y) e^{-\epsilon \frac{\|y\|^{2}}{2}} d y \\
& =\lim _{\epsilon \rightarrow 0}\left(\frac{1}{\sqrt{2 \pi}}\right)^{d} \int_{R^{d}}\left[\left(\frac{1}{\sqrt{2 \pi}}\right)^{d} \int_{R^{d}} e^{i<z, y>} f(z) d z\right] e^{-i<x, y>} e^{-\epsilon \frac{\|y\|^{2}}{2}} d y \\
& =\lim _{\epsilon \rightarrow 0}\left(\frac{1}{2 \pi}\right)^{d} \int_{R^{d}} \int_{R^{d}} e^{i<z-x, y>} f(z) e^{-\epsilon \frac{\|y\|^{2}}{2}} d y d z \\
& =\lim _{\epsilon \rightarrow 0}\left(\frac{1}{2 \pi}\right)^{d} \int_{R^{d}} f(z)\left[\int_{R^{d}} e^{i<z-x, y>} e^{-\epsilon \frac{\|y\|^{2}}{2}} d y\right] d z \\
& =\lim _{\epsilon \rightarrow 0}\left(\frac{1}{\sqrt{2 \pi \epsilon}}\right)^{d} \int_{R^{d}} f(z) e^{-\frac{\|z-x\|^{2}}{2 \epsilon}} d z \\
& =f(x)
\end{aligned}
$$
Here we have used the identity
$$
\frac{1}{\sqrt{2 \pi}} \int_{R} e^{i x y} e^{-\frac{x^{2}}{2}} d x=e^{-\frac{y^{2}}{2}}
$$
We now turn to the computation of $L_{2}$ norm of $\hat{f}$. We calculate it as
$$
\begin{aligned}
\|\hat{f}\|_{2}^{2} & =\lim _{\epsilon \rightarrow 0} \int_{R_{d}}|\hat{f}(y)|^{2} e^{-\frac{\epsilon\|y\|^{2}}{2}} d y \\
& =\lim _{\epsilon \rightarrow 0} \int_{R_{d}} \int_{R_{d}} \int_{R_{d}} f(x) \bar{f}(z) e^{i<x-z, y>} e^{-\frac{\epsilon\|y\|^{2}}{2}} d y d x d z \\
& =\lim _{\epsilon \rightarrow 0}\left(\frac{1}{\sqrt{2 \pi \epsilon}}\right)^{d} \int_{R_{d}} \int_{R_{d}} f(x) \bar{f}(z) e^{-\frac{\|x-z\|^{2}}{2 \epsilon}} d x d z \\
& =\lim _{\epsilon \rightarrow 0} \int_{R^{d}} f(x)\left[K_{\epsilon} \bar{f}\right](x) d x \\
& =\int_{R^{d}}|f(x)|^{2} d x
\end{aligned}
$$
We see that the Fourier transform is a bounded linear map from $L_{1}$ to $L_{\infty}$ as well as $L_{2}$ to $L_{2}$ with corresponding bounds $C=\left(\frac{1}{\sqrt{2 \pi}}\right)^{d}$ and 1 . By the Riesz-Thorin interpolation theorem the Fourier transform is bounded from $L_{p}$ into $L_{\frac{p}{p-1}}$ for $1 \leq p \leq 2$. If $\frac{1}{p}=1 . t+\frac{1}{2}(1-t)$ then $\frac{1}{2}(1-t)=1-\frac{1}{p}=\frac{p-1}{p}$. See exercise to show that for $f \in L_{p}$ with $p>2$ the Fourier Transform need not exist.
For convolution operators of the form
$$
(T f)(x)=(k * f)(x)=\int_{R^{d}} k(x-y) f(y) d y
$$
we want to estimate $\|T\|_{p}$, the operator norm from $L_{p}$ to $L_{p}$ for $1 \leq p \leq \infty$. As before for $p=1, \infty$,
$$
\|T\|_{p}=\int_{R^{d}}|k(y)| d y
$$
Let us suppose that for some constant $C$,
1. The Fourier transform $\hat{k}(y)$ of $k(\cdot)$ satisfies
$$
\sup _{y \in R^{d}}|\hat{k}(y)| \leq C<\infty
$$
2. In addition,
$$
\sup _{x \in R^{d}} \int_{\{y:\|x-y\| \geq C\|x\|\}}|k(y-x)-k(y)| d y \leq C<\infty
$$
We will estimate $\|T\|_{p}$ in terms of $C$. The main step is to establish a weak type $(1,1)$ inequality. Then we will use the interpolation theorems to get boundedness in the range $1<p \leq 2$ and duality to reach the interval $2 \leq p<\infty$.
Theorem 2. The function $g(x)=(T f)(x)=(k * f)(x)$ satisfies a weak type $(1,1)$ inequality
$$
\mu\{x:|g(x)| \geq \ell\} \leq C_{0} \frac{\|f\|_{1}}{\ell}
$$
with a constant $C_{0}$ that depends only on $C$. We first prove a decomposition lemma that we will need for the proof of the theorem.
Lemma 1. Given any open set $G \in R^{d}$ of finite Lebesgue measure we can find a countable set of balls $\left\{S\left(x_{j}, r_{j}\right)\right\}$ with the following properties. The balls are all disjoint. $G=\cup_{j} S\left(x_{j}, 2 r_{j}\right)$ is the countable union of balls with the same centers but twice the radius. More over each point of $G$ is covered at most $9^{d}$ times by the covering $G=\cup_{j} S\left(x_{j}, 2 r_{j}\right)$. Finally each of the balls $S\left(x_{j}, 8 r_{j}\right)$ has a nonempty intersection with $G^{c}$.
Basically, the lemma says that it is possible to write $G$ as a nearly disjoint countable union of balls each having a radius that is comparable to the distance of the center from the boundary.
Proof. Suppose $G$ is an open set in the plane of finite volume. Let $d(x)=$ $d\left(x, G^{c}\right)$ be the distance from $x$ to $G^{c}$ or the boundary of $G$. Let $d_{0}=$ $\sup _{x \in G} d(x)$. Since the volume of $G$ is finite, $G$ cannot contain any large balls and consequently $d_{0}$ cannot be infinite. We consider balls $S(x, r(x))$ around $x$ of radius $r(x)=\frac{d(x)}{4}$. They are contained in $G$ and provide a covering of $G$ as $x$ varies over $G$. All these balls have the property that $S(x, 5 r(x))$ intersects $G^{c}$. We select a countable subcover from this covering $\cup_{x \in G} S(x, r(x))$. We choose $x_{1}$ such that $d\left(x_{1}\right)>\frac{d_{0}}{2}$. Having chosen $x_{1}, \ldots, x_{k}$ the choice of $x_{k+1}$ is made as follows. We consider the balls $S\left(x_{i}, r\left(x_{i}\right)\right)$ for $i=1,2, \ldots, k$. Look at the set $G_{k}=\left\{x: S(x, r(x)) \cap S\left(x_{i}, r\left(x_{i}\right)\right)=\emptyset\right.$ for $\left.1 \leq i \leq k\right\}$ and define $d_{k}=\sup _{x \in G_{k}} d(x)$. We pick $x_{k+1} \in G_{k}$ such that $d\left(x_{k+1}\right)>\frac{d_{k}}{2}$. We proceed in this fashion to get a countable collection of balls $\left\{S\left(x_{j}, r\left(x_{j}\right)\right)\right\}$. By construction, they are disjoint balls contained in the set $G$ of finite volume and therefore $r\left(x_{j}\right) \rightarrow 0$ as $j \rightarrow \infty$. Since, $d_{j} \leq 2 d\left(x_{j+1}\right) \leq 8 r\left(x_{j+1}\right)$ it must also necessarily go to 0 as $j \rightarrow \infty$. Every $S\left(x_{j}, 5 r\left(x_{j}\right)\right)$ intersects $G^{c}$. We now worry about how much of $G$ they cover. First we note that $G_{0} \supset G_{1} \supset$ $\cdots \supset G_{k} \supset G_{k+1} \supset \cdots$. We claim that $\cap_{k} G_{k}=\emptyset$. If not let $x \in G_{k}$ for every $k$. Then $d_{k} \geq d(x)>0$ for every $k$ contradicting the convergence of $d_{k}$ to 0 . Since $x \in G_{0}=G$, we can find $k \geq 1$ be such that $x \notin G_{k}$ but $x \in G_{k-1}$. Then $S(x, r(x))$ must intersect $S\left(x_{k}, r\left(x_{k}\right)\right)$ giving us the inequality $\left|x-x_{k}\right| \leq$ $r(x)+r\left(x_{k}\right) \leq \frac{d(x)}{4}+r\left(x_{k}\right) \leq \frac{d_{k-1}}{4}+r\left(x_{k}\right) \leq \frac{d\left(x_{k}\right)}{2}+r\left(x_{k}\right)=\frac{3}{2} r\left(x_{k}\right)$. Clearly $S\left(x_{k}, 2 r\left(x_{k}\right)\right.$ will contain $x$. Since $\frac{3}{2} r(x)<d(x)$ the enlarged ball is still within $G$. This means $G=\cup_{k} S\left(x_{k}, 2 r\left(x_{k}\right)\right)$. Now we worry about how often a point $x$ can be covered by $\left\{S\left(x_{k}, 2 r\left(x_{k}\right)\right\}\right.$. Let for some $k,\left|x-x_{k}\right| \leq 2 r\left(x_{k}\right)$. Then by the triangle inequality $\left|d(x)-d\left(x_{k}\right)\right| \leq 2 r\left(x_{k}\right)=\frac{1}{2} d\left(x_{k}\right)$. This implies that for the ratio $\frac{r(x)}{r\left(x_{k}\right)}=\frac{d(x)}{d\left(x_{k}\right)}$ we have $\frac{1}{2} \leq \frac{r(x)}{r\left(x_{k}\right)} \leq \frac{3}{2}$ In particular any ball $S\left(x_{j}, 2 r\left(x_{j}\right)\right.$ that covers $x$, must have its center with in a distance of $4 r(x)$ and the corresponding $r\left(x_{j}\right)$ must be in the range $\frac{2}{3} r(x) \leq r\left(x_{j}\right) \leq 2 r(x)$. The balls $S\left(x_{j}, r\left(x_{j}\right)\right.$ are then contained in $S(x, 6 r(x))$ are disjoint and have a radius of atleast $\frac{2}{3} r(x)$. There can be atmost $9^{d}$ of them by considering the total volume. We can choose our norm in $R^{d}$ to be $\max _{i}\left|x_{i}\right|$ and force the spheres to be cubes.
Proof of theorem. The proof is similar to the one-dimensional case with some modifications.
1. We let $G_{\ell}$ be the open set where the maximal function $M_{f}(x)$ satisfies $\left|M_{f}(x)\right|>\ell$. From the maximal inequality
$$
\mu\left[G_{\ell}\right] \leq C \frac{\|f\|_{1}}{\ell}
$$
2. We write $G_{\ell}=\cup_{j} B_{j}=\cup_{j} S\left(x_{j}, 2 r_{j}\right)$, a countable union of cubes according to the lemma.
3. If we let
$$
\phi(x)=\sum_{j} \mathbf{1}_{B_{j}}(x)
$$
then $1 \leq \phi(x) \leq 9^{d}$ on $G_{\ell}$.
4. Let us define a weighted average $m_{j}$ of $f(y)$ on $B_{j}$ by
$$
\int_{B_{j}}\left[f(y)-m_{j}\right] \frac{d y}{\phi(y)}=0
$$
and write
$$
\begin{aligned}
f(x) & =f(x) \mathbf{1}_{G_{\ell}^{c}}(x)+\frac{1}{\phi(x)} \sum_{j} f(x) \mathbf{1}_{B_{j}}(x) \\
& =f(x) \mathbf{1}_{G_{\ell}^{c}}(x)+\frac{1}{\phi(x)} \sum_{j} m_{j} \mathbf{1}_{B_{j}}(x)+\frac{1}{\phi(x)} \sum_{j}\left[f(x)-m_{j}\right] \mathbf{1}_{B_{j}}(x) \\
& =h_{0}(x)+\sum_{j} h_{j}(x)
\end{aligned}
$$
5. For any cube $B_{j}$ with center $x_{j}$ there is a cube with 4 times its size and with the same center that contains a point $x_{j}^{\prime} \in G_{\ell}^{c}$ with $\left|M_{f}\left(x_{j}^{\prime}\right)\right| \leq$ $\ell$. The cube $S\left(x_{j}^{\prime}, 10 r_{j}\right)$ contains $B_{j}$. Therefore with some constant depending only on the dimension
$$
\left|m_{j}\right| \leq C_{d} \ell
$$
Moreover on $G_{\ell}^{c},|f(x)| \leq M_{f}(x) \leq \ell$. Hence
$$
\left\|h_{0}\right\|_{\infty} \leq \ell+C_{d} \ell=\left(C_{d}+1\right) \ell
$$
On the other hand
$$
\begin{aligned}
\left\|h_{0}\right\|_{1} & \leq\|f\|_{1}+C_{d} \ell \sum_{j} \mu\left[B_{j}\right] \\
& \leq\|f\|_{1}+C_{d}^{2} \ell \mu\left[G_{\ell}\right] \\
& \leq\left(1+C C_{d}^{2}\right)\|f\|_{1}
\end{aligned}
$$
and therefore
$$
\left\|h_{0}\right\|_{2}^{2} \leq\left(C_{d}+1\right) \ell\left\|h_{0}\right\|_{1} \leq C_{1} \ell\|f\|_{1}
$$
From the boundedness of $T$ from $L_{2}$ to $L_{2}$ this gives
$$
\mu\left\{x:\left|\left(T h_{0}\right)(x)\right| \geq \ell\right\} \leq C_{2} \frac{\|f\|_{1}}{\ell}
$$
6. We now turn our attention to the functions $\left\{h_{j}\right\}$
$$
\begin{aligned}
w & =T\left[\sum_{j} h_{j}\right]=\sum_{j} \int_{B_{j}}\left[f(y)-m_{j}\right] k(x-y) \frac{d y}{\phi(y)} \\
& =\sum_{j} \int_{B_{j}}\left[f(y)-m_{j}\right]\left[k(x-y)-k\left(x-x_{j}\right)\right] \frac{d y}{\phi(y)} \\
& \leq \sum_{j} \int_{B_{j}}\left|f(y)-m_{j}\right|\left|k(x-y)-k\left(x-x_{j}\right)\right| d y
\end{aligned}
$$
We estimate $|w(x)|$ for $x \notin \cup_{j} U_{j}$ where $U_{j}$ is the cube with the same center $x_{j}$ as $B_{j}$ but enlarged by a factor $C+1$. In particular if $y \in B_{j}$ and $x \in U_{j}^{c}$, then $|y-x| \geq\left|x-x_{j}\right|-\left|y-x_{j}\right| \geq C\left|y-x_{j}\right|$.
$$
\begin{aligned}
\int_{\cap_{j} U_{j}^{c}}|w(x)| d x & \leq \sum_{j} \int_{\cap_{j} U_{j}^{c}}\left[\int_{B_{j}}\left|f(y)-m_{j}\right|\left|k(x-y)-k\left(x-x_{j}\right)\right| d y\right] d x \\
& \leq \sum_{j} \int_{B_{j}}\left|f(y)-m_{j}\right|\left[\int_{E_{j}}\left|k(x-y)-k\left(x-x_{j}\right)\right| d x\right] d y
\end{aligned}
$$
where $E_{j} \subset\left\{x:|x-y| \geq C\left|y-x_{j}\right|\right\}$. Therefore,
$$
\begin{aligned}
& \int_{E_{j}}\left|k(x-y)-k\left(x-x_{j}\right)\right| d x \\
& \quad \leq \sup _{y, j} \int_{\left\{x:|x-y| \geq C\left|y-x_{j}\right|\right\}}\left|k(x-y)-k\left(x-x_{j}\right)\right| d x \\
& \quad \leq \sup _{y} \int_{\{x:|x-y| \geq C|y|\}}|k(x-y)-k(x)| d x \\
& \quad \leq C
\end{aligned}
$$
giving us the estimate
$$
\begin{aligned}
\int_{\cap_{j} U_{j}^{c}}|w(x)| d x & \leq C \sum_{j} \int_{B_{j}}\left|f(y)-m_{j}\right| d y \\
& \leq C\left(\|f\|_{1}+\left[\sup _{j} m_{j}\right] \sum_{j} \mu\left[B_{j}\right]\right) \\
& \leq C_{1}\|f\|_{1}
\end{aligned}
$$
7. We put the pieces together and we are done.
## Riesz Kernels.
A natural generalization of the Hilbert transform to higher dimension is mutiplication of the Fourier Transform by homogeneous functions of degree 0 , the simplest ones being
$$
\widehat{R_{i} f}(\xi)=\frac{\xi_{i}}{|\xi|} \hat{f}(\xi)
$$
Since the functions $k_{i}(\xi)=\frac{\xi_{i}}{|\xi|}$ are bounded functions it is clear that $R_{i}$ are bounded operators from $L_{2}\left(R^{d}\right)$ into $L_{2}\left(R^{d}\right)$. On the other hand $k_{i}$ are not continuous at $\xi=0$, and therefore the formal kernel $K_{i}$ with the representation
$$
R_{i} f(x)=\int_{R^{d}} K_{i}(x-y) f(y) d y
$$
can not be in $L_{1}\left(R^{d}\right)$.
Lemma 1. The kernels $K_{i}(\cdot)$ are given by
$$
K_{i}(x)=c_{d} \frac{x_{i}}{|x|^{d+1}}
$$
where $c_{d}$ is a constant depending on the dimension.
Proof. We will begin with the following calculation. For any $\epsilon>0, d>\delta>0$
$$
\begin{aligned}
\int_{R^{d}} e^{i<x, \xi>} \frac{1}{|x|^{d-\delta}} e^{-\epsilon|x|^{2}} d x & =\frac{1}{\Gamma\left(\frac{d-\delta}{2}\right)} \int_{R^{d}} \int_{0}^{\infty} e^{i<x, \xi>} t^{\frac{d-\delta}{2}-1} e^{-(t+\epsilon)|x|^{2}} d x d t \\
& =\frac{c_{d}}{\Gamma\left(\frac{d-\delta}{2}\right)} \int_{0}^{\infty} e^{-\frac{|\xi|^{2}}{4(t+\epsilon)} t^{\frac{d-\delta}{2}}-1}(t+\epsilon)^{-\frac{d}{2}} d t
\end{aligned}
$$
If we let $\epsilon \rightarrow 0$ in equation (??)
$$
\lim _{\epsilon \rightarrow 0} \int_{R^{d}} e^{i<x, \xi>} \frac{1}{|x|^{d-\delta}} e^{-\epsilon|x|^{2}} d x=\frac{c_{d}}{\Gamma\left(\frac{d-\delta}{2}\right)} \int_{0}^{\infty} e^{-\frac{|\xi|^{2}}{4 t}} t^{-\frac{\delta}{2}-1} d t=\frac{c_{d} \Gamma\left(\frac{\delta}{2}\right)}{\Gamma\left(\frac{d-\delta}{2}\right)}|\xi|^{-\delta}
$$
If we let $f_{\epsilon, \delta}(x)=\frac{(d-\delta) x_{j}}{|x|^{d+2-\delta}} e^{-\epsilon|x|^{2}}, \lim _{\epsilon \rightarrow 0} \widehat{f}_{\epsilon, \delta}(x)=i c_{d} \frac{\Gamma\left(\frac{\delta}{2}\right)}{\Gamma\left(\frac{d-\delta}{2}\right)} \xi_{j}|\xi|^{-\delta}$. Finally we let $\delta>1 \rightarrow 1$.
It is not difficult to see that for any smooth function $f(x)$ with compact support
$$
\left(R_{j} f\right)(x)=\int_{|y| \leq \ell} \frac{y_{j}}{|y|^{d+1}}[f(x+y)-f(x)] d y+\int_{|y| \geq \ell} \frac{y_{j}}{|y|^{d+1}} f(x+y) d y
$$
is independent of $\ell$ because $\int_{S} \frac{y_{j}}{|y|^{d+1}} d y=0$ for any shell $S=\left\{\ell_{1} \leq|y| \leq \ell_{2}\right\}$. It is a smooth function of $x$. For large $x$, the first term is 0 , and the second integral can be estimated by,
$$
\int_{R^{d}}\left|\left[\frac{x_{j}-y_{j}}{|y-x|^{d+1}}-\frac{x_{j}}{|x|^{d+1}}\right] f(y)\right| d y \leq \frac{C}{|x|^{d+1}}
$$
if we use that $f$ has compact support and satisfies $\int_{R^{d}} f(y) d y=0$. It is now easy to compute
$$
\widehat{R_{j} f}(\xi)=\frac{1}{c_{d}} \frac{\xi_{j}}{|\xi|} \hat{f}(\xi)
$$
The next step is to show that the kernels $K_{i}$ satisfy condition of equation (??). Let us take $x, y \in R^{d}$ and write $y=r \omega$ where $r=|y|$ and $\omega=\frac{y}{|y|} \in S^{d-1}$.
$$
\begin{aligned}
\int_{|x-y| \geq C|y|}\left|\frac{x_{i}-y_{i}}{|x-y|^{d+1}}-\frac{x_{i}}{|x|^{d+1}}\right| d x= & \int_{|x-y| \geq C|y|}\left|\frac{\sigma(x-y)}{|x-y|^{d}}-\frac{\sigma(x)}{|x|^{d}}\right| d x \\
\leq & \int_{|x-r \omega| \geq C r \mid}\left|\frac{\sigma(x-r \omega)}{|x-r \omega|^{d}}-\frac{\sigma(x-r \omega)}{|x|^{d}}\right| d x \\
& +\int_{|x-r \omega| \geq C r} \frac{|\sigma(x-r \omega)-\sigma(x)|}{|x|^{d}} d x \\
\leq & C_{1} \int_{|x-r \omega| \geq C r} \frac{\mid}{|x-r \omega|^{d}}-\frac{1}{|x|^{d}} \mid d x \\
& +\int_{|x-r \omega| \geq C r} \frac{|\sigma(x-r \omega)-\sigma(x)|}{|x|^{d}} d x
\end{aligned}
$$
where $\sigma(x)=\frac{x_{i}}{|x|}$. If we make the substitution $x=r x^{\prime}$ we get
$$
\begin{aligned}
\int_{|x-y| \geq C|y|}\left|\frac{x_{i}-y_{i}}{|x-y|^{d+1}}-\frac{x_{i}}{|x|^{d+1}}\right| d x \leq & C_{1} \int_{\left|x^{\prime}-\omega\right| \geq C}\left|\frac{1}{\left|x^{\prime}-\omega\right|^{d}}-\frac{1}{\left|x^{\prime}\right| d}\right| d x^{\prime} \\
& +\int_{\left|x^{\prime}-\omega\right| \geq C} \frac{\left|\sigma\left(x^{\prime}-\omega\right)-\sigma\left(x^{\prime}\right)\right|}{\left|x^{\prime}\right|^{d}} d x^{\prime}
\end{aligned}
$$
The estimate is clearly uniform in $r$. If $C$ is large enough 0 and $\omega$ are excluded from the domain of integrartion. For large $x$ we get an extra cancellation in both the integrals to make them converge with a bound that is uniform in $\omega$. For the second integral we need only that $\sigma$ satisfies a Hölder condition on $S^{d-1}$.
We have therefore proved the following theorem.
Theorem 4.1. If the kernel $K(x)$ is given by
$$
K(x)=\frac{\sigma\left(\frac{x}{|x|}\right)}{|x|^{d}}
$$
and $\sigma(\cdot)$ satisfies a Hölder condition on $S^{d-1}$ and has mean 0 on $S^{d-1}$, then convolution by $K$ defines a bounded operator from $L_{p}\left(R^{d}\right)$ into $L_{p}\left(R^{d}\right)$ for all $p$ in the range $1<p<\infty$. In particular the Riesz transforms 4.1 given by 4.2 with kernels 4.3 are bounded operators in every $L_{p}$ in the same range.
## Sobolev Spaces.
In dealing with differential equations we often come across solutions that do not have the smoothness necessary to be a solution in the ordinary sense. To illustrate it by an example, suppose we want to solve the equation
$$
\Delta u=\sum_{i} u_{x_{i} x_{i}}=f
$$
on $R^{d}$. If $d=1$ the equation reduces to $u_{x x}=f$ which is easy to solve. We need only to integrate $f$ twice, and if $f$ has $d$ continuous derivatives $u$ will have $d+2$ continuous derivatives. On $R^{d}$ it is conceivable that each $u_{x_{i} x_{i}}$ may be singular, but somehow the singularities cancel miraculously to produce a much nicer $f$. Working formally with Fourier trnasforms
$$
-|\xi|^{2} \widehat{u}(\xi)=\widehat{f}(\xi)
$$
and
$$
\widehat{u}_{x_{i} x_{j}}(\xi)=-\frac{\xi_{i} \xi_{j}}{|\xi|^{2}} \widehat{f}(\xi)
$$
In other words
$$
u_{x_{i} x_{j}}=-R_{i} R_{j} f
$$
It says that for $1<p<\infty$, if $f \in L_{p}$ we can expect $u$ to have two derivatives in $L_{p}$, but if $f$ is bounded and continuous one should not expect $u$ to have two continuous derivatives. In fact on $d=2$, one can construct a counter example, i.e. a function $f$ which is continuous such that the soulution $u$ of Poisson's equation exhibits a singularity of the individual second derivatives at 0 , that of course cancel to produce a continuous $f$.
The Sobolev spaces $W_{k}^{p}\left(R^{d}\right)$ are defined as the space of functions $u$ on $R^{d}$ such that $u$ and all its partial derivatives $D_{x_{1}}^{n_{1}} \cdots D_{x_{d}}^{n_{d}} u$ of order $n=$ $n_{1}+\cdots+n_{d}$ are in $L_{p}$. We could start with $C^{\infty}$ functions with compact support on $R^{d}$ and complete it in the norm
$$
\|u\|_{k, p}=\sum_{\substack{n_{1}, \ldots n_{d} \\ n=n_{1}+\cdots+n_{d} \leq k}}\left\|D_{x_{1}}^{n_{1}} \cdots D_{x_{d}}^{n_{d}} u\right\|_{p}
$$
If $u \in L_{p}$ and $D_{i} u=D_{x_{i}} u \in L_{p} u$ should be more regular than an $L_{p}$ function.
Let us consider the operator
$$
\widehat{A u}(\xi)=\frac{1}{\left(1+|\xi|^{2}\right)^{\frac{1}{2}}} \widehat{u}(\xi)
$$
and consider its represenation by the kernel
$$
(A u)(x)=\int_{R^{d}} u(x+y) a(y) d y
$$
where
$$
\begin{aligned}
a(x) & =c_{d} \int_{R_{d}} \frac{e^{-i<x, \xi>}}{\left(1+|\xi|^{2}\right)^{\frac{1}{2}}} d x=\frac{c_{d}}{\sqrt{\pi}} \int_{R^{d}} \int_{0}^{\infty} e^{-i<x, \xi>} e^{-t\left(1+|\xi|^{2}\right)} \frac{1}{\sqrt{t}} d t \\
& =k_{d} \int_{0}^{\infty} \frac{e^{-t}}{t^{\frac{d+1}{2}}} e^{-\frac{|x|^{2}}{4 t}} d t=\frac{k_{d}}{|y|^{d-1}} \int_{0}^{\infty} e^{-t|x|^{2}} e^{-\frac{1}{4 t}} \frac{d t}{t^{\frac{d+1}{2}}}
\end{aligned}
$$
decays very rapidly at $\infty$, is smooth for $x \neq 0$ and has a singularity of $|x|^{1-d}$ near the origin for $d \geq 2$ and a logarithmic singularity at 0 when $d=1$. In particular $a(\cdot) \in L_{q}$ for $q<\frac{d}{d-1}$. By Hölder's inequality, $A$ will map $L_{p}$ into $L_{\infty}$ for $p>d$. If $d=p>1$ the result is false. Let us take $d=2$ and a nonnegative function $f$ with compact support such that $f \in L_{2}$ but $\int_{R^{d}} \frac{f(x)}{|x|} d x=\infty$. We saw that $A f$ has a singularity at 0 . Let us consider $u=D_{1}(A f)$. Clearly
$$
\|u\|_{2}^{2}=\|\hat{u}\|_{2}^{2}=\int_{R^{2}} \frac{\xi_{1}^{2}}{1+|\xi|^{2}}|\hat{f}(\xi)|^{2} d \xi \leq\|\hat{f}\|_{2}^{2}=\|f\|_{2}^{2}
$$
By Young's inequality any $K \in L_{q}$ maps $L_{p} \rightarrow L_{p^{\prime}}$ provided $\frac{1}{p}-\frac{1}{p^{\prime}}=1-\frac{1}{q}$. Therefore $f \in W_{1, p}$ implies $f \in L_{p^{\prime}}$ so long as $\frac{1}{p}-\frac{1}{p^{\prime}}<\frac{1}{d}$. By induction $f \in W_{k, p}$ implies that $f \in W_{1, p}$ implies $f \in L_{p^{\prime}}$ so long as $\left.\frac{1}{p}-\frac{1}{p^{\prime}}<\frac{k}{d}\right)$. Therefore on $R^{d}, f \in W_{k, p}$ implies the continuity of $f$ if $k>\frac{d}{p}$.
Actually one can prove a stronger result to the effect that if $\frac{1}{p}-\frac{1}{p^{\prime}}=\frac{1}{d}$. then $W_{1, p} \subset L_{p^{\prime}}$ as long as $1<p^{\prime}<\infty$. This requires the following theorem.
Theorem 5.1. Let $T_{a}$ be the operator of convolution by the kernel $|x|^{a-d}$ on $R^{d}$.
$$
\left(T_{a} f\right)(x)=\int_{R^{d}}|y|^{a-d} f(x+y) d y
$$
Then $T_{a}$ is bounded from $L_{p}$ to $L_{p^{\prime}}$ provided $1<p<\frac{d}{a}$ and $\frac{1}{p^{\prime}}=\frac{1}{p}-\frac{a}{d}$.
Proof. First, we note that for $a>0, T_{a}$ is well defined on bounded functions with compact support. We start by proving a weak type inequlity of the form
$$
\mu\left[x:\left|\left(T_{a} f\right)(x)\right| \geq \ell\right] \leq C \frac{\|f\|_{p}^{q}}{\ell^{q}}
$$
For any choice of $1<p<\frac{d}{a}$ let $f \in L_{p}$. We can assume without loss of generality that $f \geq 0$. We write
$$
\begin{aligned}
\left(T_{a} f\right)(x) & =\int_{|y| \leq \rho}|y|^{a-d} f(x+y) d y+\int_{|y| \geq \rho}|y|^{a-d} f(x+y) d y \\
& \leq u_{1}+u_{2}
\end{aligned}
$$
and estimate $u_{1}, u_{2}$ by
$$
\begin{aligned}
\left\|u_{1}\right\|_{p} & \leq C_{1} \rho^{a}\|f\|_{p} \\
\left\|u_{2}\right\|_{\infty} & \leq\left(\int_{|y| \geq \rho}|y|^{p^{*}(a-d)} d y\right)^{\frac{1}{p^{*}}}\|f\|_{p}=C_{2} \rho^{a-d+\frac{d}{p^{*}}}\|f\|_{p}
\end{aligned}
$$
We can now pick $\rho=\left(\frac{2 C_{2}\|f\|_{p}}{\ell}\right)^{\frac{p}{d-a p}}$ and estimate
$$
\begin{aligned}
\sup _{x} u_{2}(x) & \leq \frac{\ell}{2} \\
\mu\left[x: u_{1}(x) \geq \frac{\ell}{2}\right] & \leq 2^{p} C_{1}^{p} \rho^{a p} \frac{\|f\|_{p}^{p}}{\ell^{p}} \\
& =C_{3}\left(\frac{\|f\|_{p}}{\ell}\right)^{\frac{a p^{2}}{d-a p}+p} \\
& =C_{3}\left(\frac{\|f\|_{p}}{\ell}\right)^{q}
\end{aligned}
$$
where $q=\frac{p d}{d-a p}$ or $\frac{1}{q}=\frac{1}{p}-\frac{a}{d}$.
Now, an application of Marcinkiewicz interpolation gives boundedness from $L_{p}$ to $L_{q}$ in the same range and with the same relation between $p$ and $q$.
We can also define the fractional derivative operarors
$$
\left(|D|^{a} f\right)(x)=\int_{R^{d}} \frac{f(x+y)-f(x)}{|y|^{d+a}} d y
$$
for $0<a<2$. A calculation shows that in terms of Foirier transforms it is multiplication by
$$
\int_{R^{d}} \frac{e^{i<\xi, y>}-1}{|y|^{d+a}} d y=c_{d, a}|\xi|^{a}
$$
Therefore $|D|^{a}$ and $T_{a}$ are essentially (upto a constant) inverses of each other. If $r>0$ is written as $k+a$, where $k$ is a nonnegative integer and $0 \leq a<1$, then one defines the norm corresponding to $r^{\text {th }}$ derivative by
$$
\|u\|_{r, p}=\sum_{\sum_{i} n_{i} \leq k}\left\|D_{1}^{n_{1}} \cdots D_{d}^{n_{d}} u\right\|_{p}+\sum_{\sum_{i} n_{i}=k}\left\|D_{1}^{n_{1}} \cdots D_{d}^{n_{d}} u\right\|_{a, p}
$$
well.
This way the Sobolev spaces $W_{r, p}$ are defined for fractional derivatives as Theorem 5.2. The inclusion map is well defined and bounded from $W_{r, p}$ into $W_{s, q}$ provided $s<r, 1<p<q<\infty$, and $\frac{1}{q} \geq \frac{1}{p}-\frac{r-s}{d}$. The extreme value of $q=\infty$ is allowed if $\frac{1}{q}>\frac{1}{p}-\frac{r-s}{d}$.
Proof. We can assume without loss of generality that $0<r-s<1$. We can go from $W_{r, p}$ to $W_{s, q}$ in a finite number of steps, with $0<r-s<1$ at each step. We write $\mathcal{I}=c_{d, a} T_{a}|D|^{a}$ where $a=r-s$. By definition $|D|^{a}$ maps $W_{r, p}$ boundedly into $W_{s, p}$. By the earlier theorem $T_{a}$ maps $W_{s, p}$ boundedly into $W_{s, q}$. Although we proved it for $s=0$, it is true for every $s$ because $T_{a}$ commutes with $|D|^{a}$. The cae $q=\infty$ is covered as well by this argument.
## Generalized Functions.
Let us begin with the space $W_{1,2}$. This is a Hilbert Space with the inner product
$$
<f, g>_{1}=\int_{R^{d}}\left[f \bar{g}+\sum_{1}^{d} f_{x_{i}} \bar{g}_{x_{i}}\right] d x=\int_{R^{d}} f \bar{h} d x
$$
where $h=g-\sum_{1}^{d} g_{x_{i} x_{i}}$. Since $g \in W_{1,2}, g_{x_{i}} \in L_{2}$ and $g_{x_{i} x_{i}}$ is the derivative of an $L_{2}$ function. In fact since we can write $\int f g_{x_{i}} d x$ as $-\int f_{x_{i}} g d x$, Any derivative of an $L_{2}$ function can be thought of as a bounded linear functional on the space $W_{1,2}$. A simlar reasoning applies to all the spaces $W_{r, p}$. The dual space of $W_{r, p}$ is $W_{-r, q}$ where $\frac{1}{p}+\frac{1}{q}=1$.
For a function to be in $L_{p}$ its singularities as well as decay at $\infty$ must be controlled. We can get rid of the condition at $\infty$ by demaniding that $f$ be in $L_{p}(K)$ for every bounded set $K$ or equivalently by insisting that $\phi f \in L_{p}$ for every $C^{\infty}$ function $\phi$ with compact support. This definition makes sense for $W_{r, p}$ as well. We say that $f \in W_{r, p}^{\text {loc }}$ if $\phi f \in W_{r, p}$ for every $C^{\infty}$ function $\phi$ with compact support. One needs to check that on $W_{r, p}$ mutiplication by a smooth function is a bounded linear map. One can use Leibnitz's rule if $r$ is an integer. For $0<r<1$ we need the following lemma.
Lemma 2. If $f \in W_{r, p}$ and $\phi \in C^{r^{\prime}}$ with $r<r^{\prime} \leq 1$ i.e. $\phi$ is a bounded function satisfying $|\phi(x)-\phi(y)| \leq C|x-y|^{r^{\prime}}$, for all $x, y$, then $\phi f \in W_{r, p}$.
Proof. We need to prove
$$
g(x)=\int_{R^{d}} \frac{\phi(y) f(y)-\phi(x) f(x)}{|y-x|^{d+r}} d y
$$
is in $L_{p}$. We can write
$$
\phi(y) f(y)-\phi(x) f(x)=\phi(x)[f(y)-f(x)]+[\phi(y)-\phi(x)] f(y) .
$$
The contribution of first term is easy to control. To control the second term it is sufficient to show that
$$
\sup _{x} \int_{R^{d}} \frac{|\phi(y)-\phi(x)|}{|y-x|^{d+r}} d y<\infty
$$
which is not hard. We split the integral into two regions $|x-y| \leq 1$ and $|x-y|>1$, use the Hölder property of $\phi$ to obtain an estimate on the integral over $|x-y| \leq 1$ and the boundedness of $\phi$ to get an estimate over $|x-y|>1$, both of which are uniform in $x$.
## Hardy Spaces.
For $0<p<\infty$, the Hardy Space $\mathcal{H}_{p}$ in the unit disc $D$ with boundary $S=\partial D$ consists of functions $u(z)$ that are analytic in the $\operatorname{disc}\{z:|z|<1\}$, that satisfy
$$
\sup _{0 \leq r<1} \frac{1}{2 \pi} \int_{0}^{2 \pi}\left|u\left(r e^{i \theta}\right)\right|^{p} d \theta<\infty
$$
From the Poisson representation formula, valid for $1>r^{\prime}>r \geq 0$
$$
u\left(r e^{i \theta}\right)=\frac{{r^{\prime}}^{2}-r^{2}}{2 \pi} \int_{0}^{2 \pi} \frac{u\left(r^{\prime} e^{i(\theta-\varphi)}\right)}{{r^{\prime}}^{2}-2 r r^{\prime} \cos \varphi+r^{2}} d \varphi
$$
we get the monotonicity of the quantity $M(r)=\int_{0}^{2 \pi}\left|u\left(r e^{i \theta}\right)\right|^{p} d \theta$, which is obvious for $p=1$ and requires an application of Hölder's inequality for $p>1$. Actually $M(r)$ is monotonic in $r$ for $p>0$. To see this we note that $g\left(r e^{i \theta}\right)=\log \left|u\left(r e^{i \theta}\right)\right|$ is subharmonic and therefore, using Jensen's inequality,
$$
\begin{aligned}
\frac{r^{\prime 2}-r^{2}}{2 \pi} & \int_{0}^{2 \pi} \frac{\exp \left[p g\left(r^{\prime} e^{i(\theta-\varphi)}\right)\right]}{r^{\prime 2}-2 r r^{\prime} \cos \varphi+r^{2}} d \varphi \\
& \geq \exp \left[p \frac{{r^{\prime}}^{2}-r^{2}}{2 \pi} \int_{0}^{2 \pi} \frac{g\left(r^{\prime} e^{i(\theta-\varphi)}\right)}{r^{\prime 2}-2 r r^{\prime} \cos \varphi+r^{2}} d \varphi\right] \\
& \geq \exp \left[p g\left(r e^{i \theta}\right]\right.
\end{aligned}
$$
If $1<p<\infty$ and $u(x, y)$ is a Harmonic function in $D$, from the bound (7.1), we can get a weak radial limit $f$ (along a subsequence if necessary) of $u\left(r^{\prime} e^{i \theta}\right)$ as $r^{\prime} \rightarrow 1$. In (7.2) we can let $r^{\prime} \rightarrow 1$ keeping $r$ and $\theta$ fixed. The Poisson kernel converges strongly in $L_{q}$ to
$$
\frac{1}{2 \pi} \frac{1-r^{2}}{1-2 r \cos \varphi+r^{2}}
$$
and we get the representation (7.2) for $u\left(r e^{i \theta}\right)$ (with $r^{\prime}=1$ ) in terms of the boundary function $f$ on $S$.
$$
u\left(r e^{i \theta}\right)=\frac{1-r^{2}}{2 \pi} \int_{0}^{2 \pi} \frac{f\left(e^{i(\theta-\varphi)}\right)}{1^{2}-2 r \cos \varphi+r^{2}} d \varphi
$$
Now it is clear that actually
$$
\lim _{r \rightarrow 1} u\left(r e^{i \theta}\right)=f(\theta)
$$
in $L_{p}$. Since we can consider the real and imaginary parts seperately, these considerations apply to Hardy functions in $\mathcal{H}_{p}$ as well. The Poisson kernel is harmonic as a function of $r, \theta$ and has as its harmonic conjugate the function
$$
\frac{1}{2 \pi} \frac{2 R \sin \theta}{1-R \cos \theta+R^{2}}
$$
with $R=\frac{r}{r^{\prime}}$. Letting $R \rightarrow 1$, the imaginary part is see to be given by convolution of the real part by
$$
\frac{1}{2 \pi} \frac{2 \sin \theta}{2(1-\cos \theta)}=\frac{1}{2 \pi} \cot \frac{\theta}{2}
$$
which tells us that the real and imaginary parts at any level $|z|=r$ are related through the Hilbert transform in $\theta$. We need to normalize so that $\operatorname{Im} u(0)=0$. It is clear that any function in the Hardy Spaces is essentially determined by the boundary value of its real (or imaginary part) on $S$. The conjugate part is then determined through the Hilbert transform and to be in the Hardy class $\mathcal{H}_{p}$, both the real and imaginary parts should be in $L_{p}(R)$. For $p>1$, since the Hilbert transform is bounded on $L_{p}$, this is essentially just the condition that the real part be in $L_{p}$. However, for $p \leq 1$, to be in $\mathcal{H}_{p}$ both the real and imaginary parts should be in $L_{p}$, which is stronger than just requiring that the real part be in $L_{p}$.
We prove a factorization theorem for functions $u(z) \in \mathcal{H}_{p}$ for $p$ in the range $0<p<\infty$.
Theorem 7.1. Let $u(z) \in \mathcal{H}_{p}$ for some $p \in(0, \infty)$. Then there exists a factorization $u(z)=v(z) F(z)$ of $u$ into two analytic functions $v$ and $F$ on $D$ with the following properties. $|F(z)| \leq 1$ in $D$ and the boundary value $F^{*}\left(e^{i \theta}\right)=\lim _{r \rightarrow 1} F\left(r e^{i \theta}\right)$ that exists in every $L_{p}(S)$ satisfies $\left|F^{*}\right|=1$ a.e. on $S$. Moreover $F$ contains all the zeros of $u$ so that $v$ is zero free in $D$. Proof. Suppose $u$ has just a zero at the origin of order $k$ and no other zeros. Then we take $F(z)=z^{k}$ and we are done. In any case, we can remove the zero if any at 0 and are therefore free to assume that $u(z) \neq 0$. Suppose $u$ has a finite number of zeros, $z_{1}, \ldots, z_{n}$. For each zero $z_{j}$ consider $f_{z_{j}}(z)=\frac{z-z_{j}}{1-z \bar{z}_{j}}$. A simple calculation yields $\left|z-z_{j}\right|=\left|1-z \bar{z}_{j}\right|$ for $|z|=1$. Therefore $\left|f_{z_{j}}(z)\right|=1$ on $S$ and $\left|f_{z_{j}}(z)\right|<1$ in $D$. We can write $u(z)=v(z) \prod_{i=1}^{n} f_{z_{j}}(z)$. Clearly the factorization $u=F v$ works with $F(z)=\Pi f_{z_{i}}(z)$. If $u(z)$ is analytic in $D$, we can have a countable number of zeros accumulating near $S$. We want to use the fact that $u \in \mathcal{H}_{p}$ for some $p>0$ to control the infinite product $\Pi_{i=1}^{\infty} f_{z_{i}}(z)$, that we may now have to deal with. Since $\log |u(z)|$ is subharmonic and we can assume that $u(0) \neq 0$
$$
-\infty<c=\log |u(0)| \leq \frac{1}{2 \pi} \int_{0}^{2 \pi} \log \left|u\left(r e^{i \theta}\right)\right| d \theta
$$
for $r<1$. If we take a finite number of zeros $z_{1}, \ldots, z_{k}$ and factor $u(z)=$ $F_{k}(z) v_{k}(z)$ where $F_{k}(z)=\Pi_{1}^{k} f_{z_{i}}(z)$ is continuous on $D \cup S$ and $\left|F_{k}(z)\right|=1$ on $S$, we get
$$
\begin{aligned}
\log \left|v_{k}(0)\right| & \leq \limsup _{r \rightarrow 1} \frac{1}{2 \pi} \int_{0}^{2 \pi} \log \left|v_{k}\left(r e^{i \theta}\right)\right| d \theta \\
& =\limsup _{r \rightarrow 1} \frac{1}{2 \pi} \int_{0}^{2 \pi} \log \left|u\left(r e^{i \theta}\right)\right| d \theta \\
& \leq \sup _{0<r<1} \frac{1}{2 \pi} \int_{0}^{2 \pi}\left|u\left(r e^{i \theta}\right)\right| d \theta \\
& \leq C
\end{aligned}
$$
uniformly in $k$. In other words
$$
-\sum \log \left|f_{z_{i}}(0)\right| \leq-\log |u(0)|+C
$$
Denoting $C-c$ by $C_{1}$,
$$
\sum\left(1-\left|z_{j}\right|\right) \leq \sum-\log \left|z_{j}\right| \leq C_{1}
$$
One sees from this that actually the infinite product $F(z)=\Pi_{j} f_{z_{j}}(z) e^{-i a_{j}}$ converges. with proper phase factors $a_{j}$. We write $-z_{j}=\left|z_{j}\right| e^{-i a_{j}}$. Then
$$
\begin{aligned}
1-f_{z_{i}}(z) e^{-i a_{j}} & =1+\frac{z-z_{j}}{1-z_{\bar{z}_{j}}} \frac{\left|z_{j}\right|}{z_{j}} \\
& =\frac{z_{j}-z\left|z_{j}\right|^{2}+z\left|z_{j}\right|-z_{j}\left|z_{j}\right|}{z_{j}\left(1-z \bar{z}_{j}\right)} \\
& =\frac{\left(1-\left|z_{j}\right|\right)\left(z_{j}+z\left|z_{j}\right|\right)}{z_{j}\left(1-z \bar{z}_{j}\right)}
\end{aligned}
$$
Therefore $\left|1-f_{z_{j}}(z) e^{-i a_{j}}\right| \leq C\left(1-\left|z_{j}\right|\right)(1-|z|)^{-1}$ and if we redefine $F_{n}(z)$ by
$$
F_{n}(z)=\prod_{j=1}^{n} f_{z_{j}}(z) e^{-i a_{j}}
$$
we have the convergence
$$
\lim _{n \rightarrow \infty} F_{n}(z)=F(z)=\prod_{j=1}^{\infty} f_{z_{j}}(z) e^{-i a_{j}}
$$
uniformly on compact subsets of $D$ as $n \rightarrow \infty$. It follows from $\left|F_{n}(z)\right| \leq 1$ on $D$ that $|F(z)| \leq 1$ on $D$. The functions $v_{n}(z)=\frac{u(z)}{F_{n}(z)}$ are analytic in $D$ (as the only zeros of $F_{n}$ are zeros of $u$ ) and are seen easily to converge to the limit $v=\frac{u}{F}$ so that $u=F v$. Moreover $F_{n}(z)$ are continuous near $S$ and $\left|F_{n}(z)\right| \equiv 1$ on $S$. Therefore,
$$
\begin{aligned}
\sup _{0<r<1} \frac{1}{2 \pi} \int_{0}^{2 \pi}\left|v_{n}\left(r e^{i \theta}\right)\right|^{p} d \theta & =\limsup _{r \rightarrow 1} \frac{1}{2 \pi} \int_{0}^{2 \pi}\left|v_{n}\left(r e^{i \theta}\right)\right|^{p} d \theta \\
& =\limsup _{r \rightarrow 1} \frac{1}{2 \pi} \int_{0}^{2 \pi} \frac{\left|u\left(r e^{i \theta}\right)\right|^{p}}{\left|F_{n}\left(r e^{i \theta}\right)\right|^{p}} d \theta \\
& =\limsup _{r \rightarrow 1} \frac{1}{2 \pi} \int_{0}^{2 \pi}\left|u\left(r e^{i \theta}\right)\right|^{p} d \theta \\
& =\sup _{0<r<1} \frac{1}{2 \pi} \int_{0}^{2 \pi}\left|u\left(r e^{i \theta}\right)\right|^{p} d \theta
\end{aligned}
$$
Since $v_{n}(z) \rightarrow v(z)$ uniformly on compact subsets of $D$, by Fatou's lemma,
$$
\sup _{0<r<1} \frac{1}{2 \pi} \int_{0}^{2 \pi}\left|v\left(r e^{i \theta}\right)\right|^{p} d \theta \leq \sup _{0<r<1} \frac{1}{2 \pi} \int_{0}^{2 \pi}\left|u\left(r e^{i \theta}\right)\right|^{p} d \theta
$$
In other words we have succeeded in writing $u=F v$ with $|F(z)| \leq 1$, removing all the zeros of $u$, but $v$ still satsfying (7.4). In order to complete the proof of the theorem it only remains to prove that $|F(z)|=1$ a.e. on $S$. From (7.4) and the relation $u=v F$, it is not hard to see that
$$
\lim _{r \rightarrow 1} \frac{1}{2 \pi} \int_{0}^{2 \pi}\left|v\left(r e^{i \theta}\right)\right|^{p}\left(1-\left|F\left(r e^{i \theta}\right)\right|^{p}\right) d \theta=0
$$
Since $F\left(r e^{i \theta}\right)$ is known to have a boundary limit $F^{*}$ to show that $\left|F^{*}\right|=1$ a.e. all we need is to get uniform control on the Lebesgue measure of the set $\left\{\theta:\left|v\left(r e^{i \theta}\right)\right| \leq \delta\right\}$. It is clearly sufficient to get a bound on
$$
\sup _{0<r<1} \frac{1}{2 \pi} \int_{0}^{2 \pi}|\log | v\left(r e^{i \theta}\right)|| d \theta
$$
Since $\log ^{+} v$ can be dominated by $|v|^{p}$ with any $p>0$, it is enough to get a lower bound on $\frac{1}{2 \pi} \int_{0}^{2 \pi} \log \left|v\left(r e^{i \theta}\right)\right| d \theta$ that is uniform as $r \rightarrow 1$. Clearly
$$
\frac{1}{2 \pi} \int_{0}^{2 \pi} \log \left|v\left(r e^{i \theta}\right)\right| d \theta \geq \log |u(0)|
$$
is sufficient.
Theorem 7.2. Suppose $u \in \mathcal{H}_{p}$. Then $\lim _{r \rightarrow 1} u\left(r e^{i \theta}\right)=u^{*}\left(e^{i \theta}\right)$ exists in the following sense
$$
\lim _{r \rightarrow 1} \int_{0}^{2 \pi}\left|u\left(r e^{i, \theta}\right)-u^{*}\left(e^{i \theta}\right)\right|^{p} d \theta=0
$$
Moreover, if $p \geq 1, u$ has the Poisson kernel representation in terms of $u^{*}$.
Proof. If $u \in \mathcal{H}_{p}$, according to Theorem 7.1 , we can write $u=v F$ with $v \in \mathcal{H}_{p}$ which is zero free and $|F| \leq 1$. Choose an integer $k$ such that $k p>1$. Since $v$ is zero free $v=w^{k}$ for some $w \in \mathcal{H}_{k p}$. Now $w\left(r e^{i \theta}\right)$ has a limit $w^{*}$ in $L_{k p}(S)$. Since $|F| \leq 1$ and has a radial limit $F^{*}$ it is clear the $u$ has a limit $u^{*} \in L_{p}(S)$ given by $u^{*}=\left(w^{*}\right)^{k} F^{*}$. If $0<p \leq 1$ to show convergence in the sense claimed above, we only have to prove the uniform integrability of $\left|u\left(r e^{i \theta}\right)\right|^{p}=\left|w\left(r e^{i \theta}\right)\right|^{k p}$ which follows from the convergence of $w$ in $L_{k p}(S)$. If $p \geq 1$ it is easy to obtain the Poisson representation on $S$ by taking the limit as $r \rightarrow 1$ from the representation on $|z|=r$ which is always valid. We can actually prove a better version of Theorem 7.1. Let $u \in \mathcal{H}_{p}$ for some $p>0$, be arbitrary but not identically zero. We can start with the inequality
$$
-\infty<\log \left|u\left(r_{0} e^{i \theta_{0}}\right)\right| \leq \frac{r^{2}-r_{0}^{2}}{2 \pi} \int_{0}^{2 \pi} \frac{\log \left|u\left(r e^{i\left(\theta_{0}-\varphi\right)}\right)\right|}{r^{2}-2 r r_{0} \cos \varphi+r_{0}^{2}} d \varphi
$$
where $z_{0}=r_{0} e^{i \theta_{0}}$ is such that $r_{0}=\left|z_{0}\right|<1$ and $\left|u\left(z_{0}\right)\right|>0$. We can use the uniform integrability of $\log ^{+}\left|u\left(r e^{i \theta}\right)\right|$ as $r \rightarrow 1$, and conclude from Fatou's lemma that
$$
\int_{0}^{2 \pi} \frac{|\log | u\left(e^{i\left(\theta_{0}-\varphi\right)}\right)||}{1-2 r_{0} \cos \varphi+r_{0}^{2}} d \varphi<\infty
$$
Since the Poisson kernel is bounded above as well as below (away from zero) we conclude that the boundary function $u\left(e^{i \theta}\right)$ satisfies
$$
\int_{0}^{2 \pi}|\log | u\left(e^{i \theta}\right)|| d \theta<\infty
$$
We define $f\left(r e^{i \theta}\right)$ by the Poisson integral
$$
f\left(r e^{i \theta}\right)=\frac{1-r^{2}}{4 \pi} \int_{0}^{2 \pi} \frac{\log \left|u\left(e^{i(\theta-\varphi)}\right)\right|}{1-2 r \cos \varphi+r^{2}} d \varphi
$$
to be Harmonic with boundary value $\log \left|u\left(e^{i \theta}\right)\right|$. From the inequality (7.5) it follows that $f\left(r e^{i \theta}\right) \geq \log \left|u\left(r e^{i \theta}\right)\right|$ We then take the conjugate harmonic function $g$ so that $w(\cdot)$ given by $w\left(r e^{i \theta}\right)=f\left(r e^{i \theta}\right)+i g\left(r e^{i \theta}\right)$ is analytic. We define $v(z)=e^{w(z)}$ so that $\log |v|=f$. We can write $u=F v$ that produces a factorization of $u$ with a zero free $v$ and $F$ with $|F(z)| \leq 1$ on $D$. Since the boundaru values of $\log |u|$ and $\log |v|$ match on $S$, the boundary values of $F$ which exist must satisfy $|F|=1$ a.e. on $S$. We have therefore proved
Theorem 7.3. Any $u$ in $\mathcal{H}_{p}$, with $p>0$, can be factored as $u=F v$ with the following properties: $|F| \leq 1$ on $D,|F|=1$ on $S, v$ is zero free in $D$ and $\log |v|$, which is harmonic in $D$, is given by the Poisson formula in terms of its boundary value $\log \left|v\left(e^{i \theta}\right)\right|=\log \left|u\left(e^{i \theta}\right)\right|$ which is in $L_{1}(S)$. Such a factorization is essentially unique, the only ambiguity being a mutiplicatve constant of absolute value 1. Remark. The improvement over Theorem 7.1 is that we have made sure that $\log |v|$ is not only Harmonic in $D$ but actually takes on its boundary value in the sense $L_{1}(S)$. This provides the uniqueness that was missing before. As an example consider the Poisson kernel itself.
$$
u(z)=e^{\frac{z+1}{z-1}}
$$
$|u(z)|<1$ on $D, u\left(r e^{i \theta}\right) \rightarrow e^{i \cot \frac{\theta}{2}}$ as $r \rightarrow 1$. Such a factor is without zeros and would be left alone in Theorem 7.1, but removed now.
There are characterizations of the factor $F$ that occurs in $u=v F$. Let us suppose that $u \in \mathcal{H}_{2}$ is not identically zero.. If we denote by $\mathcal{H}_{\infty}$, the space of all bounded analytic functions in $D$, clearly if $H \in \mathcal{H}_{\infty}$ and $u \in \mathcal{H}_{2}$, then $H u \in \mathcal{H}_{2}$. We denote by $\mathcal{K}$ the closure in $\mathcal{H}_{2}$ of $H u$ as $H$ varies over $\mathcal{H}_{\infty}$. It is clear that $\mathcal{K}=\mathcal{H}_{2}$ if and only if $\mathcal{K}$ contains any and therefore all of the units i.e. invertible elements in $\mathcal{H}_{2}$. In any case since $u \equiv 0$ is ruled out, let us pick $a \in D, a \neq 0$ such that $|u(a)|>0$ and take $k_{a} \in K$ to be the orthogonal projection of $f_{a}(z)=\frac{1}{1-\bar{a} z}$ in $\mathcal{K}$. Note that by Cauchy's formula for any $v \in \mathcal{H}_{2}$,
$$
\frac{1}{2 \pi} \int_{0}^{2 \pi} \overline{f_{a}\left(e^{i \theta}\right)} v\left(e^{i \theta}\right) d \theta=\frac{1}{2 \pi i} \int_{0}^{2 \pi} \frac{1}{e^{i \theta}-a} v\left(e^{i \theta}\right) d e^{i \theta}=v(a)
$$
Then $\left(f_{a}-k_{a}\right) \perp \mathcal{K}$. Writing the orthogonality relations in terms of the boundary values, and noting that $z^{n} k_{a} \in \mathcal{K}$ for $n \geq 0$,
$$
\int_{0}^{2 \pi}\left[\overline{f_{a}\left(e^{i \theta}\right)-k_{a}\left(e^{i \theta}\right)}\right] e^{i n \theta} k_{a}\left(e^{i \theta}\right) d \theta=<f_{a}-k_{a}, z^{n} k_{a}>=0
$$
On the other hand for $n \geq 0$, since $z^{n} k_{a} \in \mathcal{H}_{2}$, by (7.6)
$$
\int_{0}^{2 \pi} \overline{f_{a}\left(e^{i \theta}\right)} e^{i n \theta} k_{a}\left(e^{i \theta}\right) d \theta=2 \pi a^{n} k_{a}(a)
$$
Combining with equation (7.7) we get for $n \geq 0$,
$$
\int_{0}^{2 \pi} e^{i n \theta}\left|k\left(e^{i \theta}\right)\right|^{2} d \theta=2 \pi k_{a}(a) a^{n}
$$
But $|k|^{2}$ is real and therefore $k_{a}(a)$ must be real and
$$
\int_{0}^{2 \pi} e^{i n \theta}\left|k\left(e^{i \theta}\right)\right|^{2} d \theta= \begin{cases}2 \pi k_{a}(a) a^{n} & \text { if } n>0 \\ 2 \pi k_{a}(a) & \text { if } n=0 \\ 2 \pi k_{a}(a) \bar{a}^{n} & \text { if } n<0\end{cases}
$$
This implies that $\left|k_{a}\left(e^{i \theta}\right)\right|^{2} \equiv c P_{a}\left(e^{i \theta}\right)$ on $S$ where $P_{a}$ is the Poisson kernel. If $c=0$, it follows that $f_{a} \perp \mathcal{K}$, which in turn implies by (7.6) that
$$
<f_{a}, u>=2 \pi u(a)=0
$$
which is not possible because of the choice of $a$. We claim that $\left\{k_{a} H\right\}$ as $H$ varies over $\mathcal{H}_{2}$ is all of $\mathcal{K}$. If not, let $v \in \mathcal{K}$ be such that $v \perp k_{a} H$ for all $H \in \mathcal{H}_{2}$. We have then, for $n \geq 0$, taking $H=z^{n}$,
$$
\int_{0}^{2 \pi} \overline{k_{a}\left(e^{i \theta}\right)} e^{-i n \theta} v\left(e^{i \theta}\right) d \theta=<v, k_{a} z^{n}>=0
$$
For $n=-m<0, z^{m} v \in \mathcal{K}$ and
$$
\int_{0}^{2 \pi} \overline{k_{a}\left(e^{i \theta}\right)} e^{-i n \theta} v\left(e^{i \theta}\right) d \theta=<z^{m} v, k_{a}>=<z^{m} v, f_{a}>=2 \pi a^{m} v(a)
$$
Now Fourier inversion gives
$$
\begin{aligned}
\overline{k_{a}\left(e^{i \theta}\right)} v\left(e^{i \theta}\right) & =v(a) \sum_{m=1}^{\infty} a^{m} e^{-i m \theta}=v(a) \frac{a e^{-i \theta}}{1-a e^{-i \theta}} \\
& =c_{1}(a) \frac{1}{e^{i \theta}-a}=c_{2}(a) P_{a}\left(e^{i \theta}\right)\left(e^{-i \theta}-\bar{a}\right)
\end{aligned}
$$
Multiplying by $k_{a}$ and remembering that $\left|k_{a}\right|^{2}=c P_{a}$, we obtain $\left(k_{a} v\right)\left(e^{i \theta}\right)=$ $c_{3}(a)\left(e^{-i \theta}-\bar{a}\right)$ This leads to
$$
v\left(e^{i \theta}\right)=\frac{k_{a}\left(e^{i \theta}\right)}{e^{-i \theta}-\bar{a}}=\frac{k_{a}\left(e^{i \theta}\right) e^{i \theta}}{1-\bar{a} e^{i \theta}}
$$
Therefore $v=k_{a} H$ with $H(z)=\frac{z}{1-\bar{a} z} \in \mathcal{H}_{2}$ contradicting $v \perp H k_{a}$ for all $H \in \mathcal{H}_{2}$ and forcing $v$ to be 0 . We are nowready to prove the following theorem.
Theorem 7.4. Let $u \in \mathcal{H}_{2}$ be arbitrary and nontrivial. Then 1 belongs to the span of $\left\{z^{n} u: n \geq 0\right\}$ if and only if
$$
\log |u(0)|=\frac{1}{2 \pi} \int_{0}^{2 \pi} \log \left|u\left(e^{i \theta}\right)\right| d \theta
$$
Proof. Let $\left\|p_{n}(z) u(z)-1\right\|_{\mathcal{H}_{2}} \rightarrow 0$ for some polynomials $p_{n}(\cdot)$. Then
$$
\log \left|p_{n}(0)\right| \leq \frac{1}{2 \pi} \int_{0}^{2 \pi} \log \left|p_{n}\left(e^{i \theta}\right)\right| d \theta
$$
Since $\log \left|p_{n}\left(e^{i \theta}\right) u\left(e^{i \theta}\right)\right| \rightarrow 0$ as $n \rightarrow \infty$ in measure on $S$ and $\log ^{+}\left|p_{n}\left(e^{i \theta}\right) u\left(e^{i \theta}\right)\right|$ is uniformly integrable,
$$
\limsup _{n \rightarrow \infty} \frac{1}{2 \pi} \int_{0}^{2 \pi} \log \left|p_{n}\left(e^{i \theta}\right) u\left(e^{i \theta}\right)\right| d \theta \leq 0
$$
This implies
$$
\log |u(0)| \geq \frac{1}{2 \pi} \int_{0}^{2 \pi} \log \left|u\left(e^{i \theta}\right)\right| d \theta
$$
The reverse inequality is always valid and we are done with one half. As for the converse, If the span of $\left\{z^{n} u: n \geq 0\right.$ is $\mathcal{K} \subset \mathcal{H}_{2}$ is a proper subspace, there is $k$ such that $u=k v$ for some $v \in \mathcal{H}_{2}$ with $|k|^{2}\left(e^{i \theta}\right)=c P_{a}\left(e^{i \theta}\right)$, the Poisson kernel for some $a \in D$. For the Poisson kernel it is easy to verify that
$$
\log \left|P_{a}(0)\right|<\frac{1}{2 \pi} \int_{0}^{2 \pi} \log \left|P_{a}\left(e^{i \theta}\right)\right| d \theta
$$
for any $a \in D$. Therefore we cannot have (7.8) satisfied.
Suppose $f\left(e^{i \theta}\right) \geq 0$ is a weight that is in $L_{1}(S)$. We consider the Hilbert Space $H=L_{2}(S, f)$ of functions $u$ that are square integrable with respect to the weight $f$, i.e. $g$ such that $\int_{0}^{2 \pi}\left|g\left(e^{i \theta}\right)\right|^{2} f\left(e^{i \theta}\right) d \theta<\infty$. The trigonometric functions $\left\{e^{i n \theta}:-\infty<n<\infty\right\}$ are still a basis for $H$, though they may no longer orthogonal. We define $H_{k}=\operatorname{span}\left\{e^{i n \theta}: n \geq k\right\}$. It is clear the $H_{k} \supset H_{k+1}$ and mutiplication by $e^{ \pm i \theta}$ is a unitary map $U^{ \pm 1}$ of $H$ onto itself that sends $H_{k}$ onto $H_{k \pm 1}$. We are interested in calculating the orthogonal projection $e_{0}\left(e^{i \theta}\right)$ of 1 into $H_{1}$ along with the residual error $\left\|e_{1}\left(e^{i \theta}\right)-1\right\|_{2}^{2}$. There are two possibilities. Either $1 \in H_{1}$ in which case $H_{0}=H_{1}$ and hence $H_{k}=H$ for all $k$, or $H_{0}$ is spanned by $H_{1}$ and a unit vector $u_{0} \in H_{0}$ that is orthogonal to $H_{1}$. If we define $u_{k}=U^{k} u_{0}$, then $H=\oplus_{j=-\infty}^{\infty} u_{j} \oplus H_{\infty}$ where $H_{\infty}=\cap_{k} H_{k}$. In a nice situation we expect that $H_{\infty}=\{0\}$. However if $1 \in H_{1}$ as we saw $H_{\infty}=H$. If $f\left(e^{i \theta}\right) \equiv c$ then of course $u_{k}=e^{i k \theta}$. Theorem 7.5. Let us suppose that
$$
\int_{0}^{2 \pi} \log f\left(e^{i \theta}\right) d \theta>-\infty
$$
Then $H_{\infty}=\{0\}$ and the residual error is given by
$$
\left\|e_{0}\left(e^{i \theta}\right)-e^{i \theta}\right\|_{2}^{2}=2 \pi \exp \left[\frac{1}{2 \pi} \int_{0}^{2 \pi} \log f\left(e^{i \theta}\right) d \theta\right]>0
$$
Proof. We will split the proof into several steps.
Step 1. We write $\left.f\left(e^{i \theta}\right)=\mid u e^{i \theta}\right)\left.\right|^{2}$, where $u$ is the boundary value of a function $\left.u\left(r e^{i \theta}\right)\right)$ in $\mathcal{H}_{2}$. Note that, if this were possible. according to Theorem 7.1 one can assume with out loss of generality that $u(0) \neq 0$ and for $0<r<1$
$$
-\infty<\log |u(0)| \leq \frac{1}{2 \pi} \int_{0}^{2 \pi} \log \left|u\left(r e^{i \theta}\right)\right| d \theta
$$
We can let $r \rightarrow 1$, use the domination of $\log ^{+}|u|$ by $|u|$ and Fatou's lemma on $\log ^{-}|u|$. We get
$$
-\infty<\log |u(0)| \leq \frac{1}{2 \pi} \int_{0}^{2 \pi} \log \left|u\left(r e^{i \theta}\right)\right| d \theta=\frac{1}{4 \pi} \int_{0}^{2 \pi} \log \left|f\left(r e^{i \theta}\right)\right| d \theta
$$
We see that the condition (7.9) is necessary for the representation that we seek. We begin with the function $\frac{1}{2} \log f \in L_{1}(S)$ and construct $u\left(r e^{i \theta}\right)$ given by the Poisson formula
$$
F\left(r e^{i \theta}\right)=\frac{1-r^{2}}{4 \pi} \int_{0}^{2 \pi} \frac{\log f\left(e^{i(\theta-\varphi)}\right)}{1-2 r \cos \varphi+r^{2}} d \varphi
$$
to be Harmonic with boundary value $\frac{1}{2} \log f$. We then take the conjugate harmonic function $G$ so that $w(\cdot)$ given by $w\left(r e^{i \theta}\right)=F\left(r e^{i \theta}\right)+i G\left(r e^{i \theta}\right)$ is analytic. We define $u(z)=e^{w(z)}$.
$$
\begin{aligned}
\int_{0}^{2 \pi}\left|u\left(r e^{i \theta}\right)\right|^{2} d \theta & =\int_{0}^{2 \pi} \exp \left[2 F\left(r e^{i \theta}\right)\right] d \theta \\
& \leq \int_{0}^{2 \pi} \frac{1-r^{2}}{2 \pi} \int_{0}^{2 \pi} \frac{f\left(e^{i(\theta-\varphi)}\right)}{1-2 r \cos \varphi+r^{2}} d \varphi d \theta \\
& =\int_{0}^{2 \pi} f\left(e^{i \theta}\right) d \theta
\end{aligned}
$$
Therefore $u \in \mathcal{H}_{2}$ and $\lim _{r \rightarrow 1} u\left(r e^{i \theta}\right)=u\left(e^{i \theta}\right)$ exists in $L_{2}(S)$. Clearly
$$
\left|u\left(e^{i \theta}\right)\right|=\exp \left[\lim _{r \rightarrow 1} F\left(r e^{i \theta}\right)\right]=\sqrt{f\left(e^{i \theta}\right)}
$$
and $f=|u|^{2}$ on $S$. It is easily seen that $u(z)=\sum_{n \geq 0} a_{n} z^{n}$ with
$$
\sum_{n \geq 0}\left|a_{n}\right|^{2}=\frac{1}{2 \pi} \int_{0}^{2 \pi} f\left(e^{i \theta}\right) d \theta
$$
Step 2. Our representation has the additional property that $u(z)$ is zero free in $D$ and satifies (7.8). Suppose $h\left(r e^{i \theta}\right)$ is any function in $\mathcal{H}_{2}$ with boundary value $h\left(e^{i \theta}\right)$ with $|h|=\sqrt{f}$ that also satsifies
$$
\log |h(0)|=\frac{1}{2 \pi} \int_{0}^{2 \pi} \frac{1}{2} \log f d \theta
$$
then
$$
\log |h(0)| \leq \frac{1}{2 \pi} \int_{0}^{2 \pi} \log \left|h\left(r e^{i \theta}\right)\right| d \theta
$$
By Fatou's lemma applied to $\log ^{-}|h|$ as $r \rightarrow 1$ we get
$$
\limsup _{r \rightarrow 1} \frac{1}{2 \pi} \int_{0}^{2 \pi} \log \left|h\left(r e^{i \theta}\right)\right| d \theta \leq \frac{1}{2 \pi} \int_{0}^{2 \pi} \frac{1}{2} \log f\left(e^{i \theta}\right) d \theta
$$
Therefore equality holds in Fatou's lemma implying the uniform integrabilty as well as the convergence in $L_{1}(S)$ of $\log \left|h\left(r e^{i \theta}\right)\right|$ to $\frac{1}{2} \log f\left(e^{i \theta}\right)$ as $r \rightarrow 1$. In particular for $0<r<1$,
$$
\log |h(0)|=\frac{1}{2 \pi} \int_{0}^{2 \pi} \log \left|h\left(r e^{i \theta}\right)\right| d \theta
$$
and hence $h$ is zero free in $D$. Consequently, for $0 \leq r<r^{\prime}<1$
$$
\log \left|h\left(r e^{i \theta}\right)\right|=\frac{r^{\prime 2}-r^{2}}{2 \pi} \int_{0}^{2 \pi} \frac{\log \left|h\left(r e^{i(\theta-\varphi)}\right)\right|}{r^{\prime 2}-2 r^{\prime} r \cos d \varphi+r^{2}} d \varphi
$$
We can let $r^{\prime} \rightarrow 1$ use the convergence of $\log \left|h\left(r e^{i \theta}\right)\right|$ to $\frac{1}{2} \log f$ in $L_{1}(S)$ to conclude
$$
\log \left|h\left(r e^{i \theta}\right)\right|=\frac{1-r^{2}}{4 \pi} \int_{0}^{2 \pi} \frac{\log f(\theta-\varphi)}{1-2 r \cos d \varphi+r^{2}} d \varphi
$$
Therfore the representation of $f\left(e^{i \theta}\right)=\left|u\left(e^{i \theta}\right)\right|^{2}$, with $u\left(e^{i \theta}\right)$ the boundary value of $u \in \mathcal{H}_{2}$ that satisfies condition (7.8) is unique to within a multiplicative constant of absolute value 1 . The significance of making the choice of $u$ so that the condition (7.8) is valid, is that we can conclude that $\left\{z^{j} u(z)\right\}$ spans all of $\mathcal{H}_{2}$.
Step 3. Consider the mapping from $L_{2}(S, d \theta)$ into $L_{2}(S, f)$ that sends $g\left(e^{i \theta}\right) \rightarrow \frac{g\left(e^{i \theta}\right)}{u\left(e^{i \theta}\right)}$. Since the integrability of $\log f$ implies that $f$ and therefore $u$ is almost surely nonzero on $S$, this map is a unitary isomorphism. Whereas any $u$ with $|u|^{2}=f$ would be enough, our $u$ has a special property. It is the boundary value of a function $u\left(r e^{i \theta}\right) \in \mathcal{H}_{2}$, that satisfies (7.8). Consider $g\left(e^{i \theta}\right)=a_{0}$. In the isomorphism it goes over to $\frac{a_{0}}{u}$. Its inner product with $e^{i k \theta}$ with $k \geq 1$ is given by
$$
\int_{0}^{2 \pi}\left[\frac{a_{0}}{u\left(e^{i \theta}\right)}\right] e^{-i k \theta} u\left(e^{i \theta}\right) \bar{u}\left(e^{i \theta}\right) d \theta=a_{0} \int_{0}^{2 \pi} \bar{u}\left(e^{i \theta}\right) e^{-i k \theta} d \theta=0
$$
This shows that $\frac{a_{0}}{u} \perp H_{1}$. We claim that the decomposition $1=\frac{a_{0}}{u}+\frac{u-a_{0}}{u}$ is the decomposition of 1 into its components in $\left(H_{0} \cap H_{1}^{\perp}\right) \oplus H_{1}$. The residual error is given by $2 \pi\left|a_{0}\right|^{2}$ which is equal to $2 \pi \exp [2 u(0)]$ and agrees with (7.10). We now establish our claim to complete the proof. We need to check that $\frac{a_{0}}{u} \in H_{0}$ and $\frac{u-a_{0}}{u} \in H_{1}$. In our isomorphism $1 \rightarrow \frac{1}{u}$ and for $n \geq 0$, $z^{n} u \rightarrow e^{i n \theta}$. We know that the span of $\left\{z^{n} u(z): n \geq 0\right\}$ in $\mathcal{H}_{2}$ is all of $\mathcal{H}_{2}$. Therefore $\frac{1}{u} \in H_{0}$. To complete the proof of our claim we need to verify that $u-a_{0}$ is in the span of $\left\{z^{n} u: n \geq 1\right\}$. This is easy because $u-a_{0}=z v(z)$ for some $v \in \mathcal{H}_{2}$. Finally the same agument shows that the norm of the projection of 1 onto $H_{k}$ equals $2 \pi \sum_{i=k}^{\infty}\left|a_{i}\right|^{2}$ which tends to 0 as $k \rightarrow \infty$. This proves $1 \perp H_{\infty}$. In fact since $U^{ \pm n} H_{\infty}=H_{\infty}$, it follows that $e^{i n \theta} \perp H_{\infty}$ for every $n$. Therefore $H_{\infty}=\{0\}$.
Consider the problem of approximating a function $f_{0} \in L_{2}(\mu)$ by linear cominations $\sum_{j \leq-1} a_{j} f_{j}$. We assume a stationarity in the form $\rho_{n}=\int f_{j} f_{n+j} d \mu$ which is independent of $j$. Of course $\rho_{n}=\rho_{-n}$ is a positive definite function and by Bochner's theorem $\rho_{n}=\int_{0}^{2 \pi} e^{i n \theta} d F(\theta)$ for some nonnegative measure $F$ on $S$. The object to be minimized is $\int\left|f_{0}-\sum_{j \leq-1} a_{j} f_{j}\right|^{2} d \mu$ over all possible $a_{-1}, \ldots, a_{-k}, \ldots$ By Bochner's theorem this is equal to
$$
\inf _{\left\{a_{j}: j \leq-1\right\}} \int_{0}^{2 \pi}\left|1-\sum_{j \leq-1} a_{j} e^{i j \theta}\right|^{2} d F(\theta)
$$
If $d F(\theta)=f(\theta) d \theta$, by the reality of $\rho_{n}, f(\theta)$ is symmetric, and we can replace $j \leq-1$ by $j \geq 1$. Then this is exactly the problem we considered. The minimum is equal to $2 \pi \exp \left[\frac{1}{2 \pi} \int_{0}^{2 \pi} \log f(\theta) d \theta\right]$ and we know how to find the minimizer.
Suppose now $f(t) \geq 0$ is a weight on $R$ with $\int_{-\infty}^{\infty} f(t) d t<\infty$. Let $H_{a}$ be the span of $\left\{e^{i b t}: b \leq a\right\}$ in $L_{2}[R, f]$. For $a<0$, what is the projection $e_{a}(\cdot)$ of 1 in $H_{a}$ and what is the value of $\int_{-\infty}^{\infty}\left|1-e_{a}(t)\right|^{2} f(t) d t ?$
The natural condition on $f$ is $\int_{-\infty}^{\infty} \frac{\log f(t)}{1+t^{2}} d t>-\infty$. Then the Poisson integral
$$
F(x, y)=\frac{y}{2 \pi} \int_{-\infty}^{\infty} \frac{f(t)}{y^{2}+(x-t)^{2}} d t
$$
defines a harmonic function $F$ in the upper half plane $C^{+}=\{(x, y): y>0\}$ with boundary value $\frac{1}{2} \log f$ on $R=\{(x, y): y=0\}$ and $F$ can be the real part of an analytic function $w=F+i G$ on $C^{+}$. The function $u=e^{w}$ defines an analytic function on $C^{+}$with boundary value $u^{*}$ and $f(t)=\left|u^{*}\right|^{2}$. Moreover $u^{*}$ is the Fourier transform of $v$ in $L_{2}(R)$ that is supported on $(-\infty, 0)$. One can again set up an isomorphism between $L_{2}[R, 1]$ and $L_{2}[R, f]$ by sending $g \rightarrow \frac{\widehat{g}}{u^{*}}(\widehat{g}$ is the Fourier transform of $g$ ). This maps $v \rightarrow 1$ and $v(\cdot-a) \rightarrow e^{i a t}$. The projecton is seen to be the image of $v \mathbf{1}_{(-\infty, a)}(\cdot)$ with the error being $\int_{a}^{0}|v|^{2}(t) d t$.
Example: Suppose $f(t)=\frac{1}{1+t^{2}}$. The factorization $f=\left|u^{*}\right|^{2}$ is produced by $u^{*}(t)=(i+t)^{-1}$. This produces $v(t)=e^{t} \mathbf{1}_{(-\infty, 0)}(t)$. The projection is the Fourier transform of $v_{a}(t)=e^{t} \mathbf{1}_{(-\infty, a)}(t)$ which is seen to be $e^{a} e^{i a t}$.
## BMO
The space of functions of Bounded Mean Oscillation (BMO) plays an important role in Harmonic Analysis.
A function $f$, in $L_{1}(l o c)$ in $R^{d}$ is said to be a $\mathbf{B M O}$ function if
$$
\sup _{x, r} \inf _{a} \frac{1}{\left|B_{x, r}\right|} \int_{y \in B_{x, r}}|f(y)-a| d y=\|u\|_{B M O}<\infty
$$
where $B_{x, r}$ is the ball of radius $r$ centered at $x$, and $\left|B_{x, r}\right|$ is its volume. Remark. The infimum over $a$ can be replaced by the choice of
$$
a=\bar{a}=\frac{1}{\left|B_{x, r}\right|} \int_{y \in B_{x, r}} f(y) d y
$$
giving us an equivalent definition. We note that for any $a$,
$$
|a-\bar{a}| \leq \frac{1}{\left|B_{x, r}\right|} \int_{y \in B_{x, r}}|f(y)-a| d y
$$
and therefore if $a^{*}$ is the optimal $a$,
$$
\left|\bar{a}-a^{*}\right| \leq\|f\|_{B M O}
$$
Remark. Any bounded function is in the class BMO and $\|f\|_{B M O} \leq\|f\|_{\infty}$.
Theorem 8.1 (John-Nirenberg). Let $f$ be a BMO function on a cube $Q$ of volume $|Q|=1$ satisfying $\int_{Q} f(x) d x=0$ and $\|f\|_{B M O} \leq 1$. Then there are finite positive constants $c_{1}, c_{2}$, independent of $f$, such that, for any $\ell>0$
$$
|\{x:|f(x)| \geq \ell\}| \leq c_{1} \exp \left[-\frac{\ell}{c_{2}}\right]
$$
Proof. Let us define
$$
F(\ell)=\sup _{f} \mid\{x:|f(x)| \geq \ell\}
$$
where the supremum is taken over all functions with $\|f\|_{B M O} \leq 1$ and $\int_{Q} f(x) d x=0$. Since $\int_{Q} f(x) d x=0$ implies that $\|f\|_{1} \leq\|f\|_{B M O} \leq 1$, $F(\ell) \leq \frac{1}{\ell}$. Let us subdivide the cube into $2^{d}$ subcubes with sides one half the original cube. We pick a number $a>1$ and keep the cubes $Q_{i}$ with $\frac{1}{\left|Q_{i}\right|} \int_{Q_{i}}|f(x)| d x \geq a$. We subdivide again those with $\frac{1}{\left|Q_{i}\right|} \int_{Q_{i}}|f(x)| d x<a$ and keep going. In this manner we get an atmost countable collection of disjoint cubes that we enumerate as $\left\{Q_{j}\right\}$, that have the following properties: 1. $\frac{1}{\left|Q_{j}\right|} \int_{Q_{j}}|f(x)| d x \geq a$.
2. Each $Q_{j}$ is contained in a bigger cube $Q_{j}^{\prime}$ with sides double the size of the sides of $Q_{j}$ and $\frac{1}{\left|Q_{j}^{\prime}\right|} \int_{Q_{j}^{\prime}}|f(x)| d x<a$.
3. By the Lebesgue theorem $|f(x)| \leq a$ on $Q \cap\left(\cup_{j} Q_{j}\right)^{c}$.
If we denote by $a_{j}=\frac{1}{\left|Q_{j}\right|} \int_{Q_{j}} f(x) d x$, we have
$$
\left|a_{j}\right| \leq \frac{1}{\left|Q_{j}\right|} \int_{Q_{j}}|f(x)| d x \leq \frac{2^{d}}{\left|Q_{j}^{\prime}\right|} \int_{Q_{j}^{\prime}}|f(x)| d x \leq 2^{d} a
$$
by property 2 ). On the other hand $f-a_{j}$ has mean 0 on $Q_{j}$ and BMO norm at most 1 . Therefore (scaling up the cube to standard size)
$$
\begin{aligned}
\left|Q_{j} \cap\left\{x:|f(x)| \geq 2^{d} a+\ell\right\}\right| & \leq\left|Q_{j} \cap\left\{x:\left|f(x)-a_{j}\right| \geq \ell\right\}\right| \\
& \leq\left|Q_{j}\right| F(\ell)
\end{aligned}
$$
Summing over $j$, because of property 3 ),
$$
\left|\left\{x:|f(x)| \geq 2^{d} a+\ell\right\}\right| \leq F(\ell) \sum_{j}\left|Q_{j}\right|
$$
On the other hand property 1) implies that $\sum_{j}\left|Q_{j}\right| \leq \frac{1}{a}$ giving us
$$
F\left(2^{d} a+\ell\right) \leq \frac{1}{a} F(\ell)
$$
which is enough to prove the theorem.
Corollary 8.1. For any $p>1$ there is a constant $C_{d, p}$ depending only on the dimension $d$ and $p$ such that
$$
\sup _{Q} \frac{1}{|Q|} \int_{Q}\left|f(x)-\frac{1}{|Q|} \int_{Q} f(x) d x\right|^{p} d x \leq C_{d, p}\|f\|_{B M O}^{p}
$$
The importance of BMO, lies partly in the fact that it is dual to $\mathcal{H}_{1}$.
Theorem 8.2. There are constants $0<c \leq C<\infty$ such that
$$
c\|f\|_{B M O} \leq \sup _{g:\|g\|_{\mathcal{H}_{1}} \leq 1}\left|\int f(x) g(x) d x\right| \leq C\|f\|_{B M O}
$$
and every bounded linear functional on $\mathcal{H}_{1}$ is of the above type. The proof of the theorem depends on some lemmas.
Lemma 8.1. The Riesz transforms $R_{i}$ map $L_{\infty} \rightarrow B M O$ boundedly. In fact convolution by any kernel of the form $K(x)=\frac{\Omega(x)}{|x|^{d}}$ where $\Omega(x)$ is homogeneous of degree zero, has mean 0 on $S^{d-1}$ and satisfies a Hölder condition on $S^{d-1}$ maps $L_{\infty} \rightarrow B M O$ boundedly.
Proof. Let us suppose that $Q$ is the unit cube centered around the origin and denote by $2 Q$ the doubled cube. We write $f=f_{1}+f_{2}$ where $f_{1}=f \mathbf{1}_{2 Q}$ and $f_{2}=f-f_{1}=f \mathbf{1}_{(2 Q)^{c}}$.
$$
g(x)=g_{1}(x)+g_{2}(x)
$$
where
$$
\begin{gathered}
g_{i}(x)=\int_{R^{d}} K(x-y) f_{i}(y) d y \\
\int_{Q}\left|g_{1}(x)\right| d x \leq\left\|g_{1}\right\|_{2} \leq \sup _{\xi}|\widehat{K}(\xi)|\left\|f_{1}\right\|_{2} \leq 2^{\frac{d}{2}} \sup _{\xi}|\widehat{K}(\xi)|\|f\|_{\infty}
\end{gathered}
$$
On the other hand with $a_{Q}=\int_{Q} K(-y) f_{2}(y) d y$
$$
\begin{aligned}
\int_{Q} \mid g_{2}(x) & -a_{Q} \mid d x \\
& \leq \int_{Q} d x \int_{R^{d}}|K(x-y)-K(-y)| f_{2}(y) d y \\
& \leq\|f\|_{\infty} \iint_{\substack{z \in Q \\
y \notin 2 Q}}|K(x-y)-K(-y)| d x d y \\
& \leq\|f\|_{\infty} \sup _{x} \int_{|y| \geq 2|x|}|K(x-y)-K(-y)| d y \\
& \leq B\|f\|_{\infty}
\end{aligned}
$$
The proof for arbitrary cube is just a matter of translation and scaling. The Hölder continuity is used to prove the boundedness of $\widehat{K}(\xi)$.
Lemma 8.2. Any bounded linear function $\Lambda$ on $\mathcal{H}_{1}$ is given by
$$
\Lambda(f)=\sum_{i=0}^{d} \int\left(R_{i} f\right)(x) g_{i}(x) d x=-\int f(x) \sum_{i=0}^{d}\left(R_{i} g_{i}\right)(x) d x
$$
where $R_{0}=\mathcal{I}$ and $R_{i}$ for $1 \leq i \leq d$ are the Riesz transforms. Proof. The space $\mathcal{H}_{1}$ is a closed subspace of the direct sum $\oplus L_{1}\left(R^{d}\right)$ of $d+1$ copies of $L_{1}\left(R^{d}\right)$. Hahn-Banach theorem allows us to extend $\Lambda$ boundedly to $\oplus L_{1}\left(R^{d}\right)$ and the Riesz representation theorem gives us $\left\{g_{i}\right\}$. Finally $g_{0}+\sum_{i=1}^{d} R_{i} g_{i}$ is in $\mathrm{BMO}$.
Lemma 8.3. If $g \in B M O$ then
$$
\int_{R^{d}} \frac{|g(y)|}{1+|y|^{d+1}} d y<\infty
$$
and
$$
G(t, x)=\int g(y) p(t, x-y) d y
$$
exists where $p(\cdot, \cdot)$ is the Poisson kernel for the half space $t>0$. Moreover $g(t, x)$ satisfies
$$
\sup _{x} \int_{\substack{|y-x|<h \\ 0<t<h}} t|\nabla G(t, y)|^{2} d t d y \leq A\|g\|_{B M O}^{2} h^{d}
$$
for some constant independent of $g$. Here $\nabla G$ is the full gradient in $t$ and $x$.
Proof. First let us estimate $\int_{R^{d}} \frac{|g(x)|}{1+|x|^{d+1}} d x$. If we denote by $Q_{n}$ the cube of side $2^{n}$ around the origin
$$
\begin{aligned}
\int_{R^{d}} \frac{|g(x)|}{1+|x|^{d+1}} d x & \leq \int_{Q_{0}} \frac{|g(x)|}{1+|x|^{d+1}} d x+\sum_{n} \int_{Q_{n+1} \cap Q_{n}^{c}} \frac{|g(x)|}{1+|x|^{d+1}} d x \\
& \leq \int_{Q_{0}}|g(x)| d x+\sum_{n} \frac{1}{2^{n(d+1)}} \int_{Q_{n+1}}|g(x)| d x \\
& \leq \int_{Q_{0}}|g(x)| d x+\sum_{n} \frac{1}{2^{n(d+1)}} \int_{Q_{n+1}}\left|g(x)-a_{n+1}\right| d x \\
& \leq \sum_{n} \frac{\left|a_{n+1}\right|}{2^{n}}|g(x)| d x+\|g\|_{B M O} \sum_{n} \frac{2^{(n+1) d}}{2^{n(d+1)}}+\sum_{n} \frac{\left|a_{n+1}\right|}{2^{n}}
\end{aligned}
$$
where
$$
a_{n+1} \leq \frac{1}{2^{(n+1) d}} \int_{Q_{n+1}} g(x) d x
$$
Moreover
$$
\left|a_{2 Q}-a_{Q}\right|=\frac{1}{|Q|} \int_{Q}\left|g(x)-a_{2 Q}\right| d x \leq 2^{d}\|g\|_{B M O}
$$
and this provides a bound of the form
$$
\left|a_{Q_{n}}\right| \leq C n\|g\|_{B M O}+\int_{Q_{0}}|g(x)| d x
$$
establishing (8.4). We now turn to proving (8.5). Again because of the homogeneity under translations and rescaling, we can assume that $x=0$ and $h=1$. So we only need to control
$$
\int_{\substack{|y|<1 \\ 0<t<1}} t|\nabla G(t, y)|^{2} d t d y \leq A\|g\|_{B M O}^{2}
$$
We denote by $Q_{4}$ the cube $|x| \leq 2$ and write $g$ as
$$
g=a_{Q_{4}}+\left(g_{1}-a_{Q_{4}}\right)+g_{2}
$$
where $g_{1}=g \mathbf{1}_{Q_{4}}, g_{2}=g-g_{1}=g \mathbf{1}_{Q_{4}^{c}}$. Since constants do not contribute to (8.5), we can assume that $a_{Q_{4}}=0$, and therefore the integral $\int_{Q_{4}}|g(x)| d x$ can be estimated in terms of $\|g\|_{B M O}$. An easy calculation, writing $G=G_{1}+G_{2}$ yields
$$
\left|\nabla G_{2}(t, y)\right| \leq \int_{Q_{4}^{c}} \frac{|g(x)|}{1+|x|^{d+1}} d x \leq A\|g\|_{B M O}
$$
As for the $G_{1}$ contribution in terms of the Fourier transform we can control it by
$$
\int_{0}^{\infty} \int_{R^{d}} t|\nabla G|^{2} d t d y=\int_{0}^{\infty} \int_{R^{d}} t|\xi|^{2} e^{-2 t|\xi|}\left|\widehat{g}_{1}(\xi)\right|^{2} d \xi d t=\int_{R^{d}}\left|\widehat{g}_{1}(\xi)\right|^{2} d \xi
$$
which is controlled by $\|g\|_{B M O}$ because of the John-Nirenberg theorem. Lemma 8.4. Any functiong whose Poisson intgeral $G$ satisfies (8.5) defines a bounded linear functional on $\mathcal{H}_{1}$.
Proof. The idea of the proof is to write
$$
\begin{aligned}
2 \int_{0}^{\infty} \int_{R^{d}} t \nabla G(t, x) \nabla F(t, x) d t d x & =4 \int_{0}^{\infty} \int_{R^{d}} t e^{-2 t|\xi|}|\xi|^{2} \widehat{f}(\xi) \overline{\widehat{g}}(\xi) d \xi d t \\
& =\int_{R^{d}} \widehat{f}(\xi) \overline{\widehat{g}}(\xi) d \xi \\
& =\int_{R^{d}} f(x) g(x) d x
\end{aligned}
$$
and concentrate on
$$
\int_{0}^{\infty} \int_{R^{d}} t\left|\nabla_{x} G(t, x)\right|\left|\nabla_{x} F(t, x)\right| d t d x
$$
We need the auxiliary function
$$
\left(S_{h} u\right)(x)=\left[\iint_{|x-y|<t<h} t^{1-d}|\nabla u|^{2} d y d t\right]^{\frac{1}{2}}
$$
Clearly $\left(S_{h} u\right)(x)$ is increasing in $h$ and we show in the next lemma that
$$
\left\|S_{\infty} F\right\|_{1} \leq C\|f\|_{\mathcal{H}_{1}}
$$
Let us assume it and complete the proof. Define
$$
h(x)=\sup \left\{h:\left(S_{h} F\right)(x) \leq M C\right\}
$$
then
$$
\left(S_{h(x)} F\right)(x) \leq M C
$$
In addition it follows from (8.5) that
$$
\sup _{y, h} \int_{|y-x| \leq h}\left|\left(S_{h} F\right)(x)\right|^{2} d x \leq C h^{d}
$$
Now $h(x)<h$ means $\left(S_{h} F\right)(x)>M C$ and therefore
$$
|\{x:|x-y|<h, h(x)<h\}| \leq \frac{C h^{d}}{M^{2}}
$$
By the proper choice of $M$, we can be sure that
$$
|\{x:|x-y|<h, h(x) \geq h\}| \geq c h^{d}
$$
Now we complete the proof.
$$
\begin{aligned}
\int_{0}^{\infty} \int_{R^{d}} t & \nabla_{x} G(t, x)|| \nabla_{x} F(t, x) \mid d t d x \\
& \leq C \int_{0}^{\infty} \int_{R^{d}} \int_{|y-x|<t \leq h(y)} t^{1-d}\left|\nabla_{x} G(t, x)\right|\left|\nabla_{x} F(t, x)\right| d t d x d y \\
& \leq \int_{R^{d}} d y\left(\int_{0}^{\infty} \int_{|y-x|<t \leq h(y)} t^{1-d}\left|\nabla_{x} G(t, x)\right|^{2} d x d t\right)^{\frac{1}{2}} \\
& \times\left(\int_{0}^{\infty} \int_{|y-x|<t \leq h(y)} t^{1-d}\left|\nabla_{x} F(t, x)\right|^{2} d x d t\right)^{\frac{1}{2}} \\
\leq & =M \int_{R^{d}}\left(S_{\infty} F\right)(y) d y \leq M\|f\|_{\mathcal{H}_{1}}
\end{aligned}
$$
Lemma 8.5. If $f \in \mathcal{H}_{1}$ then $\left|\left(S_{\infty} F\right)(x)\right|_{1} \leq C\|f\|_{\mathcal{H}_{1}}$.
Proof. This is done in two steps.
Step 1. We control the nontangential maximal function
$$
U^{*}(x)=\sup _{y, t:|x-y| \leq k t}|U(t, y)|
$$
by
$$
\left\|U^{*}\right\|_{1} \leq C_{k}\|u\|_{\mathcal{H}_{1}}
$$
If $U_{0}(x) \in \mathcal{H}_{1}$ then $U_{0}$ and its $n$ Riesz transforms $U_{1}, \ldots, U_{n}$ can be recognized as the full gradient of a Harmonic function $W$ on $R_{+}^{n+1}$. Then $V=\left(U_{0}^{2}+\cdots+U_{n}^{2}\right)^{\frac{p}{2}}$ can be verified to be subharmonic provided $p>\frac{n-1}{n-2}$. This depends on the calculation
$$
\begin{aligned}
\Delta V & =\frac{p}{2} \frac{p-2}{2} V^{\frac{p}{2}-2}\|\nabla V\|^{2}+\frac{p}{2} V^{\frac{p}{2}-1} \Delta V \\
& =p V^{\frac{p}{2}-2}\left[(p-2)\left\|\sum U_{j} \nabla U_{j}\right\|^{2}+V \sum_{j}\left\|\nabla U_{j}\right\|^{2}\right] \\
& \geq 0
\end{aligned}
$$
provided either $p \geq 2$, or if $0<p<2$,
$$
\|H \xi\|^{2} \leq \frac{1}{2-p} \operatorname{Tr}\left(H^{*} H\right)\|\xi\|^{2}
$$
where $H$ is the Hessian of $W$ with trace 0 and $\xi=\left(U_{0}, \ldots, U_{n}\right)$. Then if $\left\{\lambda_{j}\right\}$ are the $n+1$ eigenvalues of $H$, and $\lambda_{0}$ is the one with largest modulus, the remaining ones have an average of $-\frac{\lambda_{0}}{n}$ and therefore
$$
\operatorname{Tr}\left(H^{*} H\right)=\sum \lambda_{j}^{2} \geq\left(1+\frac{1}{n}\right) \lambda_{0}^{2}
$$
This means that for equation (8.6) to hold we only need $\frac{n}{n+1} \leq \frac{1}{2-p}$ or $p \geq$ $\frac{n-1}{n+1}$. In any case there is a choice of $p=p_{n}<1$ that is allowed.
Now consider the subharmonic function $V$. If we denote by $h(t, x)$ the Poisson integral of the boundary values of $h(0, x)=V(0, x)$,
$$
V(t, x) \leq h(t, x)
$$
and we have
$$
U^{*}(x)=\sup _{(y, t):\|x-y\| \leq k t} U(t, y) \leq \sup _{(y, t):\|x-y\| \leq k t} V[(t, y)]^{\frac{1}{p}} \leq \sup _{(y, t):\|x-y\| \leq k t} h[(t, y)]^{\frac{1}{p}}
$$
By maximal inequality, valid because $\frac{1}{p}>1$,
$$
\left\|U^{*}\right\|_{1} \leq\left\|h^{*}\right\|_{\frac{1}{p}}^{p} \leq C_{k, p}\|h(0, x)\|_{\frac{1}{p}}^{p}=C_{k, p}\|V(0, x)\|_{\frac{1}{p}}^{p} \leq C_{k}\|U\|_{\mathcal{H}_{1}}
$$
Step 2. It is now left to control $\left\|\left(S_{\infty} U\right)(x)\right\|_{1} \leq C\left\|U^{*}\right\|_{1}$. We use the room between the regions $|x-y| \leq t$ in the defintion of $S$ and the larger regions $|x-y| \leq k t$ used in the definition of $U^{*}$ to control $S$ through $U$. Let us pick $k=4$. Let $\alpha>0$ be a number. Consider the set $E=\left\{x:\left|U^{*}(x)\right| \leq \alpha\right.$ and $B=E^{c}=\left\{x:\left|U^{*}(x)\right|>\alpha\right\}$. We denote by $G$ the union $G=\cup_{x \in E}\{(t, y)$ : $|x-y| \leq t\}$. We want to estimate
$$
\begin{aligned}
\int_{E}\left|S_{\infty} U\right|^{2}(x) d x & =\iiint_{\substack{x \in E \\
|x-y| \leq t}} t^{1-d}|\nabla U|^{2}(t, y) d x d t d y \\
& \leq C \int_{G} t|\nabla U|^{2}(t, y) d t d y \\
& \leq C \int_{G} t\left(\Delta U^{2}\right)(t, y) d t d y \\
& \leq C \int_{\partial G}\left[\left|t \frac{\partial U^{2}}{\partial n}(t, y)\right|+\left|U^{2}(t, y) \frac{\partial t}{\partial n}(t, y)\right|\right] d \sigma
\end{aligned}
$$
by Greens's theorem. We have cheated a bit. We have assumed some smoothness on $\partial G$. We have assumed decay at $\infty$ so there are no contributions from $\infty$. We can assume that we have initially $U(0, x) \in L_{2}$ so the decay is valid. We can approximate $G$ from inside by regions $G_{\epsilon}$ with smooth boundary. The boundary consists of two parts. $B_{1}=\{t=0, x \in E\}$ and $B_{2}=\left\{x \in E^{c}, t=\phi(x)\right\}$. Moreover $|\nabla \phi| \leq 1$. We will show below that $t|\nabla U(t, y)| \leq C \alpha$ in $G$. On $B_{1}$ one can show that $t|U||\nabla U| \rightarrow 0$ and $U^{2} \frac{\partial t}{\partial n} \rightarrow U^{2}$. Moreover $d \sigma \simeq d x$. The contribution from $B_{1}$ is therefore bounded by $\int_{E}|U(0, x)|^{2} d x \leq \int_{E}\left|U^{*}(0, x)\right|^{2} d x$. On the other hand on $B_{2}$ since it is still true that $d \sigma=d x$, using the bound $t|\nabla U| \leq C \alpha,\left|\frac{\partial t}{\partial n}\right| \leq 1$, we see that the contribution is bounded by $C \alpha^{2}\left|E^{c}\right|$. Putting the pieces together we get
$$
\begin{aligned}
\int_{E}\left|S_{\infty} U\right|^{2}(x) d x & \leq C \alpha^{2} T_{U^{*}}(\alpha)+C \int_{E}\left|U^{*}\right|^{2}(x) d x \\
& \leq C \alpha^{2} T_{U^{*}}(\alpha)+C \int_{0}^{\alpha} z T_{U^{*}}(z) d z
\end{aligned}
$$
where $T_{U^{*}}(z)=\operatorname{mes}\left\{x:\left|U^{*}(x)\right|>z\right\}$. Finally since $\operatorname{mes}\left(E^{c}\right)=T_{U^{*}}(\alpha)$
$$
\operatorname{mes}\left\{x:\left|S_{\infty} U(x)\right|>\alpha\right\} \leq C T_{U^{*}}(\alpha)+\frac{C}{\alpha^{2}} \int_{0}^{\alpha} z T_{U^{*}}(z) d z
$$
Integrating with respect to $\alpha$ we obtain
$$
\left\|S_{\infty} U\right\|_{1} \leq C\left\|U^{*}\right\|_{1}
$$
Step 3. To get the bound $t|\nabla U| \leq C \alpha$ in $G$, we note that any $(t, x) \in G$ has a ball around it of radius $t$ contained in the set $\cup_{x \in E}\{y:|x-y| \leq 4 t\}$ where $|U| \leq \alpha$ and by standard estimates, if a Harmonic function is bounded by $\alpha$ in a ball of radius $t$ then its gradient at the center is bounded by $\frac{C \alpha}{t}$.
## Elliptic PDE's
We will apply the results of singular integrals particularly the estimate that the Riesz transforms are bounded on evry $L_{p}\left(R^{d}\right)$ for $1<p<\infty$ to prove existence of solutions $u \in W_{2, p}\left(R^{d}\right)$ for the equation
$$
u(x)-\sum_{i, j} a_{i, j}(x) \frac{\partial^{2} u}{\partial x_{i} \partial x_{j}}-\sum_{j} b_{j}(x) \frac{\partial u}{\partial x_{j}}=f(x)
$$
provided $f \in L_{p}$ and the coefficients of
$$
L=\sum_{i, j} a_{i, j}(x) \frac{\partial^{2}}{\partial x_{i} \partial x_{j}}+\sum_{j} b_{j}(x) \frac{\partial}{\partial x_{j}}
$$
satisfy
1. The coefficients $\left\{a_{i, j}(x)\right\}$, (assumed to satisfy with out loss of generality the symmetry condition $\left.a_{i, j}(x) \equiv a_{j, i}(x)\right)$, are unfiformly continuous on $R^{d}$ and satisfy
$$
c \sum_{j} \xi_{j}^{2} \leq \sum_{i, j} a_{i, j}(x) \xi_{i} \xi_{j} \leq C \sum_{x} i_{j}^{2}
$$
for some $0<c \leq C<\infty$.
2. The coefiicients $\left\{b_{j}(x)\right\}$ are measurable and satisfy
$$
\sum_{j}\left|b_{j}(x)\right|^{2} \leq C<\infty
$$
We first derive apriori bounds. We asume that $p$ is arbitrary in the range $1<p<\infty$ but fixed. Let $A_{p}$ be a bound for the Riesz transforms in $L_{p}\left(R^{d}\right)$. If we look at all constant coefficent operators
$$
L_{Q}=\sum q_{i, j} \frac{\partial^{2}}{\partial x_{i} \partial x_{j}}
$$
with symmetric matrces $Q$ satsfying the bounds (9.1) by a linear transformation they can be reduced to the operator $\Delta$ and if $\Delta u=f$ and $f \in L_{p}\left(R^{d}\right)$ we have the bounds
$$
\left\|u_{x_{i}, x_{j}}\right\|_{p} \leq A_{p}^{2}\|f\|_{p}
$$
and factoring in the constants coming from the linear transformation we can still conclude that there is a constant $A=A(p, c, C, d)$ such that if $L_{Q} u=f$, then
$$
\left\|u_{x_{i}, x_{j}}\right\|_{p} \leq A\|f\|_{p}
$$
Lemma 9.1. If $\epsilon \leq \epsilon_{0}$ is small enough and $\sup _{x \in R^{d}}\left|a_{i, j}(x)-q_{i, j}\right| \leq \epsilon$ for some $Q$ satisfying (9.1) we can still conclude that for any $u \in W_{2, p}$ that satisfies
$$
\sum a_{i, j}(x) \frac{\partial^{2} u}{\partial x_{i} \partial x_{j}}(x)=f(x)
$$
we must necessarily have a bound
$$
\left\|u_{x_{i}, x_{j}}\right\|_{p} \leq C\|f\|_{p}
$$
for some $C=C\left(A, d, \epsilon_{0}\right)$ independently of $u$. Consequently if $u$ is supported in a ball where $\left|a_{i, j}(x)-q_{i, j}\right| \leq \epsilon_{0}$ and
$$
\sum a_{i, j}(x) \frac{\partial^{2} u}{\partial x_{i} \partial x_{j}}(x)=f(x)
$$
then again
$$
\left\|u_{x_{i}, x_{j}}\right\|_{p} \leq C\|f\|_{p}
$$
Proof. Let us compute
$$
\begin{gathered}
L_{Q} u=\sum_{i, j} q_{i, j} u_{i, j}=\sum_{i . j} a_{i, j}(x) u_{i, j}(x)-\sum_{i, j}\left[a_{i, j}(x)-q_{i . j}\right] u_{i, j}(x) \\
=f-\sum_{i, j} \epsilon_{i, j}(x) u_{i, j}(x) \\
\left\|L_{Q} u\right\|_{p} \leq\|f\|_{p}+\epsilon_{0} d^{2} \sup _{i, j}\left\|u_{i, j}\right\|_{p}
\end{gathered}
$$
On the other hand
$$
\sup _{i, j}\left\|u_{i, j}\right\|_{p} \leq A\left\|L_{Q} u\right\|_{p} \leq A\|f\|_{p}+A \epsilon_{0} d^{2} \sup _{i, j}\left\|u_{i, j}\right\|_{p}
$$
If $\epsilon_{0}$ is chosen so that $A \epsilon_{0} d^{2} \leq \frac{1}{2}$, then
$$
\sup _{i, j}\left\|u_{i, j}\right\|_{p} \leq 2 A\|f\|_{p}
$$
For the second part we alter the coeffecients outside the support of $u$ so that we are back in a situation where we can apply the first part. We now consider a ball of radius $\delta<1$ small enough that if $x_{0}$ is the center of the ball and $x$ is any point in the ball, then $\left|a_{i, j}(x)-a_{i, j}\left(x_{0}\right)\right| \leq \epsilon_{0}$. This is possible because of uniform continuity of the coeffecients $\left\{a_{i, j}(x)\right\}$. Let $B_{\delta}$ be such a ball, and let
$$
L u=f \text { in } B_{\delta}
$$
Theorem 9.1. There is a constant $C$ such that for any $\rho<1$
$$
\sup _{i, j}\left\|u_{i, j}\right\|_{p, B_{\rho \delta}} \leq C\left[\|f\|_{p, B_{\delta}}+\delta^{-1}(1-\rho)^{-1}\|\nabla u\|_{p, B_{\delta}}+\delta^{-2}(1-\rho)^{-2}\|u\|_{p, B_{\delta}}\right]
$$
Proof. Let us for the moment take $\delta=1$ and construct a smooth function $\phi=\phi_{\rho}$ such that $\phi=1$ on $B_{\rho}$ and 0 outside $B_{1}$. We can assume that $|\nabla \phi| \leq C(1-\rho)^{-1}$ and $|\nabla \nabla \phi| \leq C(1-\rho)^{-2}$. We take $v=u \phi$ and compute
$$
\begin{aligned}
& g=\sum_{i, j} a_{i, j}(x) v_{i, j}(x)=\sum_{i, j} a_{i, j}(x)(\phi u)_{i, j}(x) \\
&= \phi \sum_{i, j} a_{i, j}(x) u_{i, j}(x)+2 \sum a_{i, j}(x) \phi_{i}(x) u_{j}(x)+u(x) \sum_{i, j} a_{i, j}(x) \phi_{i, j}(x) \\
&=\phi(x) f(x)-\phi(x) \sum b_{j}(x) u_{j}(x)+2 \sum a_{i, j}(x) \phi_{i}(x) u_{j}(x) \\
& \quad+u(x) \sum_{i, j} a_{i, j}(x) \phi_{i, j}(x)
\end{aligned}
$$
We can bound
$$
|g| \leq|f(x)|+C(1-\rho)^{-1}\|\nabla u\|(x)+C(1-\rho)^{-2}|u|(x)
$$
From the previous lemma we can get
$$
\sup _{i, j}\left\|v_{i, j}\right\|_{p, B_{1}} \leq A\|g\|_{p, B_{1}} \leq C\left[\|f\|_{p, B_{1}}+(1-\rho)^{-1}\|\nabla u\|_{p, B_{1}}+(1-\rho)^{-2}\|u\|_{p, B_{1}}\right]
$$
Since $v=u$ on $B_{\rho}$ we get
$$
\sup _{i, j}\left\|u_{i, j}\right\|_{p, B_{\rho}} \leq C\left[\|f\|_{p, B_{1}}+(1-\rho)^{-1}\|\nabla u\|_{p, B_{1}}+(1-\rho)^{-2}\|u\|_{p, B_{1}}\right]
$$
If $\delta<1$ we can redefine all functions involved as $u(\delta x), f(\delta x), a_{i, j}(\delta x)$ and $\delta b_{j}(\delta x)$. With the new operator
$$
L_{\delta}=\sum a_{i, j}(\delta x) \frac{\partial^{2}}{\partial x_{i} \partial x_{j}}+\sum \delta b_{j}(\delta x) \frac{\partial}{\partial x_{j}}
$$
we see that
$$
L_{\delta} u(\delta x)=\delta^{2} f(\delta x)
$$
We can now apply our estimate with $\delta=1$ and obtain
$$
\sup _{i, j}\left\|u_{i, j}\right\|_{p, B_{\delta \rho}} \leq C\left[\|f\|_{p, B_{\delta}}+\delta^{-1}(1-\rho)^{-1}\|\nabla u\|_{p, B_{\delta}}+\delta^{-2}(1-\rho)^{-2}\|u\|_{p, B_{\delta}}\right]
$$
At this point we can do one of two things. If we are interested only in dealing with all of $R^{d}$ we can raise the estimate (9.3) to the power $p$ and sum over a fine enough grid so that
$$
0<a<\sum_{\alpha} \mathbf{1}_{B\left(x_{\alpha}, \delta \rho\right)} \leq \sum_{\alpha} \mathbf{1}_{B\left(x_{\alpha}, \delta\right)} \leq A<\infty
$$
and we will get
$$
\sup _{i, j}\left\|u_{i, j}\right\|_{p} \leq C\left[\|f\|_{p}+\delta^{-1}(1-\rho)^{-1}\|\nabla u\|_{p}+\delta^{-2}(1-\rho)^{-2}\|u\|_{p}\right]
$$
Since $\delta>0$ is fixed (depending on the modulus of continuity of $\left\{a_{i, j}(x)\right\}$ ) and we could have fixed $\rho=\frac{1}{2}$, we have the following global estimate for any $u \in W_{2, p}$ satisfyng $L u=f$. The constant $C$ depends only on the ellipticity bounds in (9.1), the bounds in (9.2) and the modulus of continuty estimates of $\left\{a_{i, j}(x)\right\}$.
$$
\sup _{i, j}\left\|u_{i, j}\right\|_{p} \leq C\left[\|f\|_{p}+\|\nabla u\|_{p}+\|u\|_{p}\right]
$$
Lemma 9.2. For any constant $\epsilon>0$, there is a constant $C_{\epsilon}$ such that for any $u \in W_{2, p}$
$$
\|\nabla u\|_{p} \leq \epsilon \sup _{i, j}\left\|u_{i, j}\right\|_{p}+C \epsilon^{-1}\|u\|_{p}
$$
Proof. First we note that it is sufficient to prove an estimate of the type
$$
\|\nabla u\|_{p} \leq C\left[\sup _{i, j}\left\|u_{i, j}\right\|_{p}+\|u\|_{p}\right]
$$
We can then replace $u(x)$ by $u(\lambda x)$ and the estimate takes the form
$$
\lambda\|\nabla u\|_{p} \leq C\left[\lambda^{2} \sup _{i, j}\left\|u_{i, j}\right\|_{p}+\|u\|_{p}\right]
$$
If choosing $\lambda=C \epsilon$ the lemma is seen to be true. To prove (9.6) we basically need a one dimensional estimate. If we have
$$
\int_{-\infty}^{\infty}\left|g^{\prime}(x)\right|^{p} d x \leq C \int_{-\infty}^{\infty}\left|g^{\prime \prime}(x)\right|^{p} d x+C \int_{-\infty}^{\infty}|g(x)|^{p} d x
$$
on $R$, we could get the estimate on each line and then integrate it. The inequality itself needs to be proved only for the unit interval $[0,1]$. We can then translate and sum. It is quite easy to prove
$$
\sup _{0 \leq x \leq 1}\left|g^{\prime}(x)\right| \leq C\left[\int_{0}^{1}\left|g^{\prime \prime}(x)\right| d x+\int_{0}^{1}|g(x)| d x\right]
$$
Our basic apriori estmate becomes
Theorem 9.2. Any function $u \in W_{2, p}$ with $L u=f$ satisfies
$$
\sup _{i, j}\left\|u_{i, j}\right\|_{p} \leq C\left[\|f\|_{p}+\|u\|_{p}\right]
$$
Proof. Just choose $\epsilon$ in (9.5) so that $C \epsilon<\frac{1}{2}$ where $C$ is the constant in $(9.4)$.
We have to work a little harder If we want to prove a local regularity estimate of the form
Theorem 9.3. Let $\Omega \subset \bar{\Omega} \subset \Omega^{\prime}$ be bounded sets. For any $u \in W_{2, p}\left(\Omega^{\prime}\right)$ with $L u=f$, we have the bounds
$$
\left\|u_{i, j}\right\|_{p, \Omega} \leq C\left(\Omega, \Omega^{\prime}\right)\left[\|f\|_{p, \Omega^{\prime}}+\|u\|_{p, \Omega^{\prime}}\right]
$$
Proof. The trick is to go back and change the definition of $\phi_{\rho}$ so that it vanishes outside the ball of radius $\frac{1+\rho}{2}$ rather than outside the ball of radius 1. It does not change much since $\left(1-\frac{1+\rho}{2}\right)=\frac{1}{2}(1-\rho)$. We start with the modified version of (9.3)
$$
\sup _{i, j}\left\|u_{i, j}\right\|_{p, B_{\rho}} \leq C\left[\|f\|_{p, B_{1}}+(1-\rho)^{-1}\|\nabla u\|_{p, B_{1+\frac{\rho}{2}}}+(1-\rho)^{-2}\|u\|_{p, B_{1}}\right]
$$
and define
$$
\begin{aligned}
& T_{2}=\sup _{\frac{1}{2}<\rho<1}(1-\rho)^{2} \sup _{i, j}\left\|u_{i, j}\right\|_{p, B_{\rho}} \\
& T_{1}=\sup _{\frac{1}{2}<\rho<1}(1-\rho)\|\nabla u\|_{p, B_{\rho}} \\
& T_{0}=\sup _{\frac{1}{2}<\rho<1}\|u\|_{p, B_{\rho}}=\|u\|_{p, B_{1}}
\end{aligned}
$$
We see that
$$
T_{2} \leq C\left[\|f\|_{p, B_{1}}+T_{1}+T_{0}\right]
$$
Assume a uniform interpolation inequality for all balls of radius $\frac{1}{2} \leq \rho \leq 1$ of the type,
$$
\|\nabla u\|_{p, B_{\rho}} \leq \epsilon \sup _{i, j}\left\|u_{i, j}\right\|_{p, B_{\rho}}+C \epsilon^{-1}\|u\|_{p, B_{\rho}}
$$
for any choice of $\epsilon>0$, that translates to
$$
T_{1} \leq \epsilon T_{2}+C \epsilon^{-1} T_{0}
$$
and with the right chice of $\epsilon$ we get
$$
T_{2} \leq C\left[\|f\|_{p, B_{1}}+\|u\|_{p, B_{1}}\right]
$$
In particular
$$
\left\|u_{i, j}\right\|_{p, B_{\rho}} \leq C(1-\rho)^{-2}\left[\|f\|_{p, B_{1}}+\|u\|_{p, B_{1}}\right]
$$
With rescaling for $\delta_{1}<\delta_{2}<\delta_{0}$,
$$
\left\|u_{i, j}\right\|_{p, B_{\delta_{1}}} \leq C\left(\delta_{1}, \delta_{2}\right)\left[\|f\|_{p, B_{\delta_{2}}}+\|u\|_{p, B_{\delta_{2}}}\right]
$$
Covering $\bar{\Omega}$ by a finite number of balls of radius $\delta_{1}$, such that the concentric balls of radius $\delta_{2}$ are still contained in $\Omega^{\prime}$ we get our result.
Finally we prove the interpolation lemma for balls.
Lemma 9.3. Given $u \in W_{2, p, B_{1}}$ it can be extended as a function $v$ on $R^{d}$ supported on $B_{2}$ such that
$$
\begin{aligned}
\|\nabla \nabla v\|_{p, R^{d}} & \leq C\left[\|\nabla \nabla u\|_{p, B_{1}}+\|\nabla u\|_{p, B_{1}}\right] \\
\|\nabla v\|_{p, R^{d}} & \leq C\|\nabla u\|_{p, B_{1}} \\
\|v\|_{p, R^{d}} & \leq C\|u\|_{p, B_{1}}
\end{aligned}
$$
Proof. Basically if we want a function which is smooth inside $B_{1}$ and outside $B_{1}$ to be globally in $W_{2, p}$ the function and its derivatives have to match on the boundary. The usual reflection with $v(1+r, s)=u(1-r, s)$ for small $r$ matches the function and tangential derivatives but not the normal derivative. $v(1+r, s)=c_{1} u(1-r, s)+c_{2} u(1-2 r, s)$ works for a proper choice of $c_{1}$ and $c_{2}$. We use it to extend to $B_{\frac{3}{2}}$ and then a radial cutoff to kill it outside $B_{2}$. For the extended function $v$ we have the interpolation inequality
$$
\|\nabla v\|_{p, R^{d}} \leq \epsilon\|\nabla \nabla u\|_{p, R^{d}}+C \epsilon^{-1}\|u\|_{p, R^{d}}
$$
and this implie for the original $u$
$$
\|\nabla u\|_{p, B_{1}} \leq C \epsilon\|\nabla \nabla u\|_{p, B_{1}}+C \epsilon\|\nabla u\|_{p, B_{1}}+C \epsilon^{-1}\|u\|_{p, B_{1}}
$$
which is easily turned into
$$
\|\nabla u\|_{p, B_{1}} \leq \epsilon\|\nabla \nabla u\|_{p, B_{1}}+C \epsilon^{-1}\|u\|_{p, B_{1}}
$$
Finally we prove an existence theorem for solutions of $u-L u=f$.
Theorem 9.4. The equation
$$
u-L u=f
$$
has a solution in $W_{2, p}$ for each $f \in L_{p}$.
Proof. We wish to invert $(I-L)$. Suppose we can invert $\left(I-L_{1}\right)$. Then
$\left(I-L_{2}\right)^{-1}=\left[\left(I-L_{1}\right)-\left(L_{2}-L_{1}\right)\right]^{-1}=\left[I-L_{1}\right]^{-1}\left[I-\left(L_{2}-L_{1}\right)\left(I-L_{1}\right)^{-1}\right]^{-1}$
As long as $\left\|\left(L_{2}-L_{1}\right)\left(I-L_{1}\right)^{-1}\right\|<1$ as an operator mapping $L_{p} \rightarrow L_{p}$, $\left(I-L_{2}\right)^{-1}$ will map $L_{p}$ into $W_{2, p}$. We can perturb the operators from $\Delta$ to any $L$ nicely in small steps so that $\left\|L_{1}-L_{2}\right\|<\delta$ as operators from $W_{2, p} \rightarrow L_{p}$. All we need are uniform apriori bounds on $\left\|(I-L)^{-1} f\right\|_{p}$.
Theorem 9.5. Any solution $u$ of $u-L u=f$ with $L$ satisfying (9.1) and (9.2) also satisfies a bound of the form
$$
\|u\|_{p} \leq C\|f\|_{p}
$$
with a constant that does not depend on $L$ or $f$. The proof depend on lemmas.
Lemma 9.4 (Maximum Principle). Suppose $u \in W_{2, p}$ satisfies $u-L u \geq$ 0 in a possibly unbounded region $G$ and $p$ is large enough that Sobolev imbedding applies and $u$ is bounded and continuous on $\bar{G}$. If in addition $u \geq 0$ on $\partial G$ then $u \geq 0$ on $\bar{G}$. In particular if $u$ and $v$ are two functions wth $u-L u \geq v-L v$ in $G$ and $u \geq v$ on $\partial G$, then $u \geq v$ on $\bar{G}$.
Lemma 9.5. If $u-L u=0$ in a ball $B(x, \delta)$ of radius $\delta$, then
$$
|u(x)| \leq \rho(\delta) \sup _{y:|y-x|=\delta}|u(y)|
$$
Proof. We can cssume with out loss of generality that $x=0$. Consider the function
$$
\phi(x)=\exp \left[-c\left(1-\frac{|x|^{2}}{\delta^{2}}\right)\right]
$$
For some $c=c(\delta)>0$ small enough, $\phi-L \phi \geq 0$ and $\phi=1$ on the boundary. Therefore
$$
|u(0)| \leq \phi(0) \sup _{y:|y|=\delta}|u(y)|
$$
and we can take $\rho(\delta)=\exp [-c(\delta)]$.
Lemma 9.6. If $u$ is a bounded solution of $u-L u=0$ outside some ball $|x| \geq r$ then for some $c>0$ and $C<\infty$,
$$
\sup _{|x|=R}|u(x)| \leq C e^{-c(R-r)} \sup _{|x|=r}|u(x)|
$$
Proof. By the previous two lemmas
$$
\sup _{|x|=r+\delta}|u(x)| \leq \rho(\delta) \sup _{|x|=r}|u(x)|
$$
The lemma is now easily proved by induction.
Suppose $L$ is given and we modify $L$ outside a ball $B\left(x_{0}, 4 \delta\right)$ to get $L^{\prime}$ which has coeffecients $\left\{a_{i, j}^{\prime}(x)\right\}$ that are uniformly close to some constant $c_{i, j}$ and $\left\{b_{j}^{\prime}(x)\right\}$ are 0 in the complement of the ball. For $\delta$ small it is easy to see that our basic perturbation argument works for $L^{\prime}$ and
$$
u-L^{\prime} u=f
$$
has a solution in $W_{2, p}$ for $f \in L_{p}$. In particular if $f \geq 0$ is supported inside $B\left(x_{0}, \delta\right), L^{\prime} u=u$ outside the ball and if $u \in W_{2, p}$, it is in some better $L_{p_{1}}$ by Sobolev's lemma. We can iterate this process and obtain eventually an $L_{\infty}$ bound of the form
$$
\sup _{\left|x-x_{0}\right|=2 \delta}|u(x)| \leq C\|f\|_{p}
$$
If we now compare the solutions $u-L^{\prime} u=f$ and $v-L v=f$, both of which are nonnegative, since $L=L^{\prime}$ inside $B\left(x_{0}, 4 \delta\right)$, we have
$$
(u-v)-L(u-v)=0
$$
Therefore
$$
\sup _{\left|x-x_{0}\right|=2 \delta}|u(x)-v(x)| \leq \rho(\delta) \sup _{\left|x-x_{0}\right|=4 \delta}|u(x)-v(x)|
$$
From this we conclude that
$$
\sup _{\left|x-x_{0}\right|=2 \delta} v(x) \leq \sup _{\left|x-x_{0}\right|=2 \delta} u(x)+\rho(\delta)\left[\sup _{\left|x-x_{0}\right|=4 \delta} u(x)+\sup _{\left|x-x_{0}\right|=4 \delta} v(x)\right]
$$
But
$$
\sup _{\left|x-x_{0}\right|=4 \delta} v(x) \leq \rho(\delta) \sup _{\left|x-x_{0}\right|=2 \delta} v(x)
$$
and
$$
\sup _{\left|x-x_{0}\right|=4 \delta} u(x) \leq \rho(\delta) \sup _{\left|x-x_{0}\right|=2 \delta} u(x)
$$
We see now that
$$
\sup _{\left|x-x_{0}\right|=2 \delta} v(x) \leq C(\delta) \sup _{\left|x-x_{0}\right|=2 \delta} u(x) \leq C\|f\|_{p}
$$
Now one can estimate $\|v\|_{p} \leq C\|f\|_{p}$.
## Banach Alegebras, Wiener's Theorem
There is a theorem due to Wiener that asserts the following.
Theorem 10.1. Suppose $f(x)$ on the d-torus has an absoutely convergent Fourier Series and $f(x)$ is nonzero on the d-torus. Then the function $g(x)=$ $\frac{1}{f(x)}$ also has an absolutely convergent Fourier Series.
Wiener's original proof involves direct estimation. We will give a "soft" proof using functional analysis techniques developed by Naimark. The proof will be broken up in to several steps as we develop the theory.
A (commutative) Banach algebra $X$ is a Banach space with (associative) mutiplication of two elements $u, v$ defined as $u v$ satisfying $\|u v\| \leq\|u\|\|v\|$. It is commutatitve if $u v=v u$. A Banach algebra with a unit is one which has a special element called 1 such that $1 u=u$ for all $u$. Such a unit is unique because if $1,1^{\prime}$ are two units then $11^{\prime}=1=1^{\prime}$. An element $u$ is invertible if there is a $v$ such that $u v=1$. The element $v$, which is unique if it exists, is called the inverse of $u$. The unit 1 is its own inverse. We can assume with out loss of generality that $\|1\|=1$ by replacing $\|u\|$ by the equivalent norm of the operator $T_{u}: T_{u} v=u v$. An ideal $I$ is a subspace with the property that whenever $x \in X, y \in I$ it follows that $x y \in I$. An ideal is proper if it is not $X$ and not just the 0 element. A proper ideal can not contain 1 or any invertible element. A proper ideal is maximal if it is not contained in any other proper ideal. The closure of a proper ideal is proper. This needs proof. By a power series expansion if $\|1-u\|<1$, then $u$ has an inverse $v=\sum_{0}^{\infty}(1-u)^{n}$. Therefore any proper ideal is disjoint from the open unit ball around 1. So does its closure. We can therefore assume that all our ideals are closed. Any ideal can be enlarged to a maximal ideal. Just apply Zorn's lemma and take the maximal element among those that do not intersect the unit ball around 1. If $I$ is any (closed) ideal $X \backslash I$ is again a Banach Algebra, called the quotient algebra. If the ideal $I$ is maximal then the quotient $Y=X \backslash I$ has no proper ideals. In such an algebra every nonzero element is invertible. Just look at the range of $y Y$. It is an ideal. If $y \neq 0$, since it can not be proper, it must be all of $Y$, and therefore contains 1 .
Theorem 10.2. A Banach algebra with a unit and with out proper ideals over the complex numbers is the complex numbers.
Proof. Since every non zero element is invertible, if there is an element $u$ which is not a complex mutiple of $1,(z 1-u)^{-1}=f(z)$ exists for all $z \in \mathbf{C}$ and is an entire function with values in $Y$. For $z>\|u\|$ we can represent $f(z)=\sum_{n \geq 0} \frac{u^{n}}{z^{n+1}}$. Therefore $\|f(z)\| \rightarrow 0$ as $z \rightarrow \infty$ and by Liouville's theorem must be identically zero. Contradiction.
We now know that for any maximal ideal $I$, the map $X \rightarrow Y$ that sends $x \rightarrow x+I$ is a homomorphism onto $\mathbf{C}$.
Theorem 10.3. If $u \in X$ is not invertible, then there is a homomorphism, i.e a mutiplicatve bounded linear functional, $h$ such that $h(u)=0$.
Proof. Consider the ideal $u X$ and enlarge it to a maximal ideal $I$, and then take the natuaral homomorphism into $\mathbf{C}=X \backslash I$.
Corollary 10.1. $u \in X$ is invertible if and only if $h(u) \neq 0$ for every homomorphism $h$.
Consider the Banach algebra $X$ of absolutely convergent Fourier Series $\sum_{n \in Z^{d}} a_{n} e^{i<n, x>}$ with norm $\sum_{n \in Z^{d}}\left|a_{n}\right|$. Wiener's theorem will be proved if we show
Theorem 10.4. Every homomorphism $h$ on $X$ is given by
$$
h(u)=\sum_{n \in Z^{d}} a_{n} e^{i<n, \theta>}
$$
for some $\theta$ on the d-torus.
Proof. Let $h\left(u_{j}\right)=z_{j} \in \mathbf{C}$ where $u_{j}=e^{i x_{j}}$. Since $\left\|u_{j}^{k}\right\|=1$ for all positive and negative integrs $k$ and $h$ is a homomorphism $z_{j}^{k}$ must be bounded and therefore $\left|z_{j}\right|=1$. If we write $z_{j}=e^{i \theta_{j}}$, then $h(u)=\sum_{n \in Z^{d}} a_{n} e^{i<n, \theta>}$ for finite sums and since they are dense we are done.
## Compact Groups. Haar Measure.
A group is a set $G$ with a binary operation $G \times G \rightarrow G$ called multiplication written as $g h \in G$ for $g, h \in G$. It is associative in the sense that $(g h) k=$ $g(h k)$ for all $g, h, k \in G$. A group also has a special element $e$ called the identity that satisfies $e g=g e=g$ for all $g \in G$. It is easy to verify that $e$ is unique. A group also has the property that for each $g \in G$ there is an element $h=g^{-1}$ such that $g h=h g=e$. In general it need not be commutative i.e. $g h \neq h g$. If $g h=h g$ for all $g, h \in G$ the group is called abelian or commutative.
A topological group is a group with a topology such that the binary operation $G \times G \rightarrow G$ that sends $g, h \rightarrow g h^{-1}$ is continuous. This is seen to be equivalent to the assumption that the operations $G \times G \rightarrow G$ defined by $g, h \rightarrow g h$ and $G \rightarrow G$ defined by $g \rightarrow g^{-1}$ are continuous.
We will assume that our group $G$ as a topological space is a compact metric space. Let $\mathcal{B}$ be the class of Borel sets. A measure on $G$ is a nonnegative measure of total mass 1 on $(G, \mathcal{B})$. A left invariant (right invariant) Haar measure on $G$ is a measure $\lambda$ such that $\lambda\left(g^{-1} A\right)=\lambda(A)\left(\lambda\left(A g^{-1}\right)=\lambda(A)\right)$ for all $A \in \mathcal{B}$. Here $g^{-1} A$ is the set of elemnts of the form $g^{-1} h$ with $h \in A$. The set $A g^{-1}$ is defined similarly.
For any two measures $\alpha, \beta$ on $(G, \mathcal{B})$ the measure $\alpha * \beta$ is defined by
$$
\alpha * \beta(A)=\int_{G} \alpha\left(A g^{-1}\right) d \beta(g)=\int_{G} \beta\left(g^{-1} A\right) d \alpha(g)
$$
In terms of integrals
$$
\int_{G} f(k) d(\alpha * \beta)(k)=\int_{G} \int_{G} f(g h) d \alpha(g) d \beta(h)
$$
Theorem 11.1. The following are equivalent.
1. $\lambda$ is a left invariant Haar measure on $G$.
2. $\lambda$ is a right invariant Haar measure on $G$.
3. $\lambda$ is an idempotent i.e $\lambda * \lambda=\lambda$ and $\lambda(U)>0$ for every open set $U$.
The proof will depend on the following Lemma 11.1. $\alpha * \lambda=\lambda$ for an $\alpha$ with $\alpha(U)>0$ for all open sets $U \subset G$ if and only if $\lambda\left(g^{-1} A\right)=\lambda(A)$ for all $g \in G$ i.e. $\lambda$ is a left invariant Haar measure.
Proof. For any bounded continuous function $u$ on $G$ let us define
$$
v(g)=(u * \lambda)(g)=\int_{G} u(g h) d \lambda(h)
$$
Then for any $a \in G$
$$
\begin{aligned}
\int v(a g) d \alpha(g) & =\int_{G} \int_{G} u(a g h) d \alpha(g) d \lambda(h) \\
& =\int_{G} u(a k) d(\alpha * \lambda)(k) \\
& =\int_{G} u(a k) d \lambda(k) \\
& =v(a)
\end{aligned}
$$
We can take $a$ to be the element where the maximum of $v$ is attained. Then $v(a g)=v(a)$ for all $g$ in the support of $\alpha$. In particular $v(a g)$ and therefore $v$ is a constant. Hence $\delta_{g} * \lambda=\lambda$ or $\lambda$ is left invariant.
Proof. (of Theorem). If $\lambda$ is right invariant then $\lambda * \delta_{g}=\lambda$ for all $g \in G$ and by integrating $\lambda * \lambda=\lambda$ and that implies that $\lambda$ is left invariant as well. We note that any left or right invriant meausre cannot give zero mass to any open set because by compactness $G$ can be covered by a finite number of translates of $U$.
Theorem 11.2. A left (or right) invariant Haar meausre exists, is unique and is invariant from the right (left) as well.
Proof. Let us start with any $\alpha$ that gives positive mass to every open set and consider
$$
\lambda_{n}=\frac{\alpha+\alpha^{2}+\alpha^{n}}{n}
$$
Any weak limit $\lambda$ of $\lambda_{n}$ satisfies $\alpha * \lambda=\lambda$ and by lemma such a $\lambda$ is left and therefore right invariant. If $\lambda$ is left invariant then $\alpha * \lambda=\lambda$ for any $\alpha$ and if $\alpha$ is also right invariant then $\alpha * \lambda=\alpha$ proving uniqueness.
## Representations of a Group
Given a group $G$, a representation $\pi(g)$ of the group is a continuous mapping $\pi(\cdot)$ of $G$ into nonsingular linear transformations of a finite dimensional complex vector space $V$ such that $\pi(e)=I$ and $\pi(g h)=\pi(g) \pi(h)$ for all $g, h \in G$.
Theorem 12.1. Given a representation $\pi$ of $G$ on a finite dimensional vector space $V$, there is an inner product $\left\langle v_{1}, v_{2}\right\rangle$ on $V$, such that each $\pi(g)$ is a unitary transformation.
Proof. Let $<v_{1}, v_{2}>_{0}$ be any inner product. We define a new inner product
$$
<v_{1}, v_{2}>=\int_{G}<\pi(h) v_{1}, \pi(h) v_{2}>_{0} d h
$$
where $d h$ is the unique Haar measure on $G$. It is seen that
$$
\begin{aligned}
<\pi(g) v_{1}, \pi(g) v_{2}> & =\int_{G}<\pi(h) \pi(g) v_{1}, \pi(h) \pi(g) v_{2}>_{0} d h \\
& =\int_{G}<\pi(h g) v_{1}, \pi(h g) v_{2}>_{0} d h \\
& =\int_{G}<\pi(h) v_{1}, \pi(h) v_{2}>_{0} d h \\
& =<v_{1}, v_{2}>
\end{aligned}
$$
which proves that $\pi(\cdot)$ are unitary with respect to $\langle\cdot, \cdot\rangle$.
A representation $\pi(\cdot)$ of $G$ on $V$ is irreducible if there is no proper subspace $W$ of $U$ other than $U$ itself and the subspace $\{0\}$ that is left invariant by $\{\pi(g): g \in G\}$. Since any finite dimensional representation of $V$ can be made unitary, if $W \subset U$ is invariant so is $W^{\perp}$ and $\pi(\cdot)$ on $V$ is the direct sum of $\pi(\cdot)$ on $W$ and $W^{\perp}$. It is clear that any finite dmensional representation is a direct sum of irreducible representations.
Two unitary representations $\pi_{1}$ and $\pi_{2}$ of $G$ on two vector spaces $V_{1}$ and $V_{2}$ are said to be equivalent if there is a linear isomorphsim $T: V_{1} \rightarrow V_{2}$ such that $\pi_{2}(g) T=T \pi_{1}(g)$ for all $g \in G$. It is clear that $\pi_{1}$ and $\pi_{2}$ are equivalent then either they are both irreducble or neither is. The set of irreducible representations is naturally divided into equivalence classes. We denote them by $\omega \in \Omega$. Each $\omega$ is an equivalence class and $\Omega$ is the set of all equivalence classes. Lemma 12.1. If $\pi$ is an irreducible representation of $G$ on a finite dimensional vector space $V$, then the inner product that makes the representation unitary is unique upto a scalar multiple.
Proof. If $<\cdot, \cdot>_{i}: i=1,2$ are two inner products on $V$ that make $\pi(g)$ unitary for all $g \in G$, then
$$
<\pi(g) u, \pi(g) v>_{i}=<u, v>_{i}
$$
and if we represent by $T$ any unitary isomorphism between the two inner product spaces $\left\{V,<\cdot, \cdot>_{1}\right\}$ and $\left\{V,<\cdot, \cdot>_{2}\right\}$ so that $<u, v>_{1}=<$ $T u, T v>_{2}$ then
$$
<T u, T v>_{2}=<u, v>_{1}=<\pi(g) u, \pi(g) v>_{1}=<T \pi(g) u, T \pi(g) v>_{2}
$$
In other words if we denote by $T^{*}$ the adjoint of $T$, on the inner product space $<\cdot, \cdot>_{2}$
$$
\left(T^{*} T\right) \pi(g) \equiv \pi(g)\left(T^{*} T\right)
$$
for all $g \in G . T^{*} T$ is Hermitian and its eigenspaces are left invariant by the $\mathrm{i} \pi(g)$ that commute wth it. These eigenspaces have to be trivial because of rreducibility. That forces $T^{*} T$ to be a positive mutiple of identity. The two inner products are then essentially the same.
Given two representations $\pi_{1}$ and $\pi_{2}$ on $V_{1}$ and $V_{2}$ an intertwining operator from $V_{1} \rightarrow V_{2}$ is a linear map $T$ such that $T \pi_{1}(g)=\pi_{2}(g) T$ for all $g \in G$.
Theorem 12.2. Schur's Lemma. Given two irreducible representaions $\pi_{i}$ on $V_{i}$ any intertwining operator $T$, it is eiher an isomorphism which makes the two representations equivalent or $T=0$.
Proof. Suppose $T$ has a null space $W \subset V_{1}$. Then if $u \in W, T \pi_{1}(g) u=$ $\pi_{2}(g) T u=0$ so that $\pi(g) W \subset W$. Either $W=\{0\}$ or $W=U_{1}$ making $W$ either 0 or one-to-one. By a similar argument the range of $T$ is left invariant by $\pi_{2}$ making $T$ either 0 or onto. Therefore if $T$ is not an isomorpihsm it is 0 . Any isomorphism is essentially a unitary isomorphism.
## Representations of a Compact group.
A natural infinite dimensional representation of a compact group is the (left) regular representation $L_{g}$ on $L_{2}(G, d g)$ defined by $\left(L_{g} u\right)(h)=u\left(g^{-1} h\right)$ satisfying $L_{g_{1}} L_{g_{2}}=L_{g_{1} g_{2}}$. From the invariance of the Haar measure $L_{g}$ is unitary. First we prove some facts regarding finite dimensional representations of compact groups.
Theorem 13.1. Let $\pi$ be a finite dimnsional irreducible representation of $G$ in a space of dimension $d$. Then there is subspace of dimension $d^{2}$ in $L_{2}(G, d g)$ that is invariant under both left and right regular representations and either one on this subspace decomposes into d copies of $\pi$. The representation of this type, i.e. any equivalent representation does not occur in the orthogonal complement of this $d^{2}$ dimensional subspace.
Proof. Let $\pi$ be a representation of $G$ in a finite dimensional space $V$. Pick a basis for $V$ and represent $\pi(g)$ as a unitary matrix $\left\{t_{i, j}(g)\right\}$. The functions $t_{i, j}(\cdot)$ are continuous and are in $L_{2}(G, d g)$. Let us see what the left regular representation $L_{h}$ does to them.
$$
t_{i, j}(h g)=[\pi(h) \pi(g)]_{i, j}=\sum_{r} t_{i, r}(h) t_{r, j}(g)
$$
which is the same as
$$
L_{h} t_{i, j}(\cdot)=\sum_{r} t_{i, r}(h) t_{r, j}(\cdot)
$$
or for each $j$ the space spanned by $\left\{t_{r, j}(\cdot): 1 \leq r \leq d\right\}$ is invariant under $L_{h}$ and transforms like $\pi$. Similarly under $R_{h^{-1}}$, the rows $\left\{t_{j, r}(\cdot): 1 \leq r \leq d\right\}$ will again transform like $\pi$. If we can show that $\left\{t_{i, j}(\cdot)\right\}$ are linearly independent then $d^{2}$ dimensional space will transform like $d$ copies of $\pi$ under $L_{h}$ and $R_{h}$. Consider two representations $\pi_{1}, \pi_{2}$ that are irreducible on $V_{1}, V_{2}$ and vectors $u_{1}, v_{1}$ and $u_{2}, v_{2}$ in their respective spaces. Then for fixed $v_{1}, v_{2}$
$$
\int_{G}<\pi_{1}(g) u_{1}, v_{1}>_{1} \overline{<\pi_{2}(g) u_{2}, v_{2}>_{2}} d g=<B u_{1}, u_{2}>
$$
defines an operator from $V_{1} \rightarrow V_{2}$ and a calculation
$$
\begin{aligned}
<B \pi_{1}(h) u_{1}, u_{2}> & =\int_{G}<\pi_{1}(g) \pi_{1}(h) u_{1}, v_{1}>_{1} \overline{<\pi_{2}(g) u_{2}, v_{2}>_{2}} d g \\
& =\int_{G}<\pi_{1}(g h) u_{1}, v_{1}>_{1} \overline{<\pi_{2}(g) u_{2}, v_{2}>_{2}} d g \\
& =\int_{G}<\pi_{1}(g) u_{1}, v_{1}>_{1} \overline{<\pi_{2}\left(g h^{-1}\right) u_{2}, v_{2}>_{2}} d g \\
& =\int_{G}<\pi_{1}(g) u_{1}, v_{1}>_{1} \overline{<\pi_{2}(g) \pi_{2}\left(h^{-1}\right) u_{2}, v_{2}>_{2}} d g \\
& =\int_{G}<\pi_{1}(g) u_{1}, v_{1}>_{1} \overline{<\pi_{2}(g) \pi_{2}^{*}(h) u_{2}, v_{2}>_{2}} d g \\
& =<B u_{1}, \pi_{2}^{*}(h) u_{2}> \\
& =<\pi_{2}(h) B u_{1}, u_{2}>
\end{aligned}
$$
Showing that $B$ intertwines $\pi_{1}$ and $\pi_{2}$. If the representations are inequivalent then $B=0$. Otherwise, $B=c\left(v_{1}, v_{2}\right) T$ where $T$ is the isomorphism between $V_{1}$ and $V_{2}$ that intertwines $\pi_{1}$ and $\pi_{2}$ and a further calculation of the same nature shows that $c\left(v_{1}, v_{2}\right)=c<T v_{1}, v_{2}>_{2}$. Therefore
$$
\int_{G}<\pi_{1}(g) u_{1}, v_{1}>_{1} \overline{<\pi_{2}(g) u_{2}, v_{2}>_{2}} d g=c<T u_{1}, u_{2}>_{2}<T v_{1}, v_{2}>_{2}
$$
where $c=0$ for inequivalent representations. In the equivalent case taking $\pi \equiv \pi_{1} \equiv \pi_{2}, V=V_{1}=V_{2}$ and $T=I$,
$$
\int_{G}<\pi(g) u_{1}, v_{1}>\overline{<\pi(g) u_{2}, v_{2}>} d g=c<u_{1}, u_{2}>_{2}<v_{1}, v_{2}>
$$
To calculate $c$ we take $u=u_{1}=u_{2}$ and $v=v_{1}=v_{2}$
$$
c\|u\|^{2}\|v\|^{2}=\int_{G} \mid\left\langle\pi(g) u, v>\left.\right|^{2} d g=\int_{G}\left|\sum_{i, j} t_{i, j}(g) u_{i} v_{j}\right|^{2} d g\right.
$$
Thus
$$
\int_{G} t_{i, j}(g) \overline{t_{r s}(g)} d g=c \delta_{i r} \delta_{j s}
$$
On the other hand
$$
\sum_{i, j}\left|t_{i, j}\right|^{2} \equiv d
$$
so that $c=\frac{1}{d}$. The character of the represenation defined as $\chi_{\pi}(g)=$ trace $\pi(g)$ is independent of concrete vector space used for the representation and $\left\|\chi_{\pi}\right\|_{L_{2}(G, d g)}=1$. If $\pi$ occured again in the orthogonal complement of the $d$ dimensional space, then that space would have to contain $\chi_{\pi}$ again and that is not possible.
Now we show that there are lots of finite dimensional representations.
Theorem 13.2. Any right trnaslation $R_{k}$ defined by $\left(R_{k} u\right)(h)=u(h k)$ commutes with $L_{g}$. There are compact self adjoint operators that commute with the family $\left\{L_{g}\right\}$. Hence the representation $L_{g}$ on $L_{2}(G, d g)$ decomposes into a sum of irreducible finite dimensional represenations. In particular a compact group has sufficiently many irreducible finite dimensional representations. More precisely given $g \neq e$ there is one for which $\pi(g) \neq I$.
Proof. Clearly
$$
\left(R_{k} L_{g} u\right)(h)=\left(R_{k} L_{g} u\right)(h)=u\left(g^{-1} h k\right)
$$
The integral operators
$$
(T u)(h)=\int_{G} u(h g) \tau(g) d g=\int_{G} u(g) \tau\left(h^{-1} g\right) d g
$$
commute with $L_{g}$ and are compact (in fact Hilbert-Schmidt) and self adjoint provided
$$
\int_{G} \int_{G}\left|\tau\left(h^{-1} g\right)\right|^{2} d g d h<\infty
$$
and for all $k$,
$$
\tau\left(k^{-1}\right)=\overline{\tau(k)}
$$
The eigenspaces of $T$ of finite dimension provide lots of finite dimensional representations that can then be split up into irreducible pieces. We can check that the only possible infinite dimensional piece is the null space of $T$. Let us write $L_{2}(G, d g)=\oplus_{j} V_{j} \oplus V_{\infty}$ as the direct sum of finite dimensional pieces that are invariant under both $L_{g}$ and $R_{g}$ if possible an infinite dimensional piece $V_{\infty}$ that is also invariant under $L_{g}$ and $R_{g}$ and has no such nontrivial finite dimensional invariant subspace. The earlier argument of convolution by $\tau$ can be repeated on $V_{\infty}$ and produces a finite dimensional $L_{g}$ invariant subspace, which is a contradiction unless such a convolution is identically 0 on $V_{\infty}$ for all $\tau$. This is seen to be impossible. The character $\chi_{\pi}(g)$ determine $\pi$ completely and for ineuqivalent representations they are orthogonal. The cahracters have the additonal property that $\chi_{\pi}\left(h g h^{-1}\right)=\chi_{\pi}(g)$.
Theorem 13.3. Any function $u(g) \in L_{2}(G, d g)$ such that $u\left(h g h^{-1}\right)=u(g)$ i.e. $L_{h} u=R_{h} u$ for all $h \in G$ is spanned by the characters.
Proof. We need to show that if any such $u$ is orthogonal to $\chi_{\pi}$ it is also orthogonal to all the matrix elements.
$$
\begin{aligned}
\int_{G} u(g) \overline{t_{r, s}(g)} d g & =\int u\left(h g h^{-1}\right) \overline{t_{r, s}(g)} d g \\
& =\int_{G} u(g) \overline{t_{r, s}\left(h^{-1} g h\right)} d g \\
& =\int_{G} u(g) \overline{\left[\pi^{*}(h) \pi(g) \pi(h)\right]_{r, s}} d g \\
& =\int_{G} u(g) \sum_{i, j} t_{i, r}(h) \overline{t_{i, j}(g) t_{j, s}(h)} d g \\
& =\int_{G} \int_{G} u(g) \overline{t_{i, j}(g)} t_{i, r}(h) \overline{t_{j, s}(h)} d g d h \\
& =\sum_{i, j} \frac{1}{d} \delta_{i, j} \delta_{r, s} \int_{G} u(g) \overline{t_{i, j}(g)} d g \\
& =\frac{1}{d} \delta_{r, s} \int_{G} u(g) \overline{\chi_{\pi}(g)} d g=0
\end{aligned}
$$
## Representations of the permutation group.
Permutations on $n$ symbols is the set of one to one mappings $\sigma$ of a set $X=\left\{x_{1}, x_{2}, \ldots, x_{n}\right\}$ of $n$ elemnts on to itself. It is a finite group $G$ with $n$ ! elements. Given $\sigma \in G$ we can look at the orbits of $\sigma^{n} x$ and it will partition the speace $X$ into orbits $A_{1}, A_{2}, \ldots, A_{k}$ consisting of $n_{1} \geq \ldots \geq n_{k}$ points so that $n=n_{1}+\cdots+n_{k}$. If $\hat{\sigma}=s \sigma s^{-1}$ is conjugate to $\sigma$ then the orbits of $\hat{\sigma}$ will be $s A_{1}, s A_{2}, \ldots, s A_{k}$ so that the partition $n_{1} \geq \ldots \geq n_{k}$ of $n$ into $k$ numbers will be the same for $\sigma$ and $\hat{\sigma}$. Conversely if $\sigma$ and $\hat{\sigma}$ has the same partition of $n$, then the orbits $A_{1}, \ldots, A_{k}$ and $B_{1}, \ldots, B_{k}$ have the same cardinalities. We can find $s \in G$ that maps $A_{i} \rightarrow B_{i}$ in a one-to-one and onto manner and we can reduce the problem to the case where $A_{i}=B_{i}$ for every $i$. We can now relable the points with in each $A_{i}$ i.e. find a permutation of $n_{i}$ elements, such that bothe $\sigma$ and $\hat{\sigma}$ look the same on each $A_{i}$. We have therefore proved
Theorem 14.1. The number of distinct inequivalent irreducible representations of $G$ is the same as the number of distinct partitions $\mathcal{P}(n)$ of the integer $n$.
Just note that they are both equal to the dimension of the subspace $\mathcal{X}$ of functions $u$ satisfying $u\left(h^{-1} g h\right)=u(g)$ for all $h, g \in G$. We know that the characters $\chi_{\pi}(g)=$ trace $\pi(g)$ of all the equivalence classes of representations span $\mathcal{X}$ so that the number of such equivalence classes is also $\mathcal{P}(n)$. We will now construct a distinct representation for each distinct partition of $n$. Given a partition $\lambda$ of $n$ into $n_{1} \geq n_{2} \geq \cdots \geq n_{k}$, we associate a diagram called the Young diagram corresponding to $\lambda$. It looks like
when $n=12$ and the partition is $4,3,3,2$. A Young tableau $t$ is a diagram $\lambda$ with the boxes filled in arbitrarily by the numbers $1,2, \cdots, 8$, like
| 7 | 4 | 1 |
| :--- | :--- | :--- |
| 6 | 5 | |
| 8 | 3 | |
| 2 | | |
The rows of the array are of length $n_{1} \geq \cdots \geq n_{k}$. For any diagram there are $n$ ! tableaux. A tabloid is when the order of entries are not relevent. The tabloid $\{t\}$ consists of $\{7,4,1\},\{6,5\},\{8,3\},\{2\}$. There are $\frac{n !}{n_{1} ! \cdots n_{k} !}$ tabloids corresponding to any tableau $t$ of the diagram $\lambda$. The subgroup $C_{t}$ of the permutaion group consists of permutaions within each column. In our case it consists of 72 elements an arbitray permutation of 7,6,8,2 and an arbitrary permutation of $4,5,3$. For any permutaion $s, \sigma(s)= \pm 1$ is the sign of the permutation. We define an abstract inner product space $V$ of dimension $\frac{n !}{n_{1} ! \cdots n_{k} !}$ with the orthonormal basis $e_{\{t\}}$ as $\{t\}$ varies over the tabloids of the tableau $t$ corresponding to $\lambda$. We define $n$ ! vectors $e_{t}$ in $V$ by
$$
e_{t}=\sum_{s \in C_{t}} \sigma(s) s e_{\{t\}}
$$
The $e_{t}$ may not be linearly independent and the span of $\left\{e_{t}\right\}$ is denoted by $W$. One defines a representation of $\pi_{\lambda}(g)$ of the permutation group on $W$ corresponding to the diagram $\lambda$ by defining
$$
\pi(g) e_{t}=g e_{t}
$$
Theorem 14.2. Each $\pi_{\lambda}$ is irreducible. For two distinct diagrams they are inequivalent. We therefore have all the representations.
Proof is broken up into lemmas.
## Lemma 14.1.
$$
\begin{aligned}
\sum_{s \in C_{t}} \sigma(s) g s e_{\{t\}} & =\sum_{s \in C_{t}} \sigma(s) g s g^{-1} g e_{\{t\}}=\sum_{g s g^{-1} \in C_{g t}} \sigma(s) g s g^{-1} g e_{\{t\}} \\
& =\sum_{s \in C_{g t}} \sigma(s) s e_{\{g t\}}=e_{g t}
\end{aligned}
$$
Lemma 14.2. Suppose $\lambda, \mu$ are two different diagrams $t$ a $\lambda$-tableau and $\tau$ a $\mu$-tableau. Suppose
$$
\sum_{s \in C_{t}} \sigma(s) e_{\{s \tau\}} \neq 0
$$
Then $n_{1} \geq m_{1}, n_{1}+n_{2} \geq m_{1}+m_{2}, \cdots$ where $n_{1}, \ldots, n_{k}$ and $m_{1}, \ldots, m_{\ell}$ are the two partitions corresponding to $\lambda$ and $\mu$. We say then that $\lambda \geq_{1} \mu$. If $\lambda=\mu$ then the sum is $\pm e_{t}$.
Proof. Suppose that two elements $a, b$ are in the same row of $\tau$ and in the same column of $t$. Then the permutation $p=\{a \leftrightarrow b\}$ is in $C_{t}, s p e_{\{t\}}=s e_{\{t\}}$, with $\sigma(s p)+\sigma(s)=0$. So the sum adds up to 0 which is ruled out. Hence, no two elements in the same row of $\tau$ can be in the same column of $t$. In particular $t$ must have atleast as many columns as the number of elements in the first row of $\tau$ proving $n_{1} \geq m_{1}$. A variant of this argument proves $\lambda \geq_{1} \mu$. Suppose now that $\lambda=\mu$. Again all the elments in any row of $\tau$ appear in different columns of $t$. So there is a permutation $s^{*} \in C_{t}$ such that $s^{*} t=\tau$. The sum is unaltered if we replace $\tau$ by $s^{*} t$ except for $\sigma\left(s^{*}\right)= \pm 1$. Lemma 14.3. Let $u \in W$ corresponding to a diagram $\mu$. Let $t$ be any $\mu$ tableau. Then
$$
\sum_{s \in C_{t}} \sigma(s) \pi(s) u=c e_{t}
$$
Proof. $u$ is a linear combination of $e_{\tau}$ for different $\lambda$-tableau $\tau$. Each one from the previous lemma yields $c e_{\tau}$ with $c=0, \pm 1$. Add them up!
Let us define
$$
A_{t}=\sum_{s \in C_{t}} \sigma(s) \pi(s)
$$
We alraedy have an inner product that makes $\pi(s)$ orthogonal.
$$
\begin{aligned}
<A_{t} u, v> & =\sum_{s \in C_{t}} \sigma(s)<\pi(s) u, v> \\
& =\sum_{s \in C_{t}} \sigma(s)<u, \pi\left(s^{-1}\right) v> \\
& =\sum_{s \in C_{t}} \sigma(s)<u, \pi(s) v> \\
& =<u, A_{t} v>
\end{aligned}
$$
because $\operatorname{sigma}\left(s^{-1}\right)=\sigma(s)$.
Lemma 14.4. If $U$ is any invariant subspace of $V$ then either $U \supset W$ or $U \perp W$. This proves the irreducibility of $W$.
Proof. Suppose $u \in W$ and $t$ is a $\lambda$-tableau. We saw that $A_{t} u=c_{t} e_{t}$ for some constant $c_{t}$. Suppose for some $t, c_{t} \neq 0$. Then $e_{t} \in U$ and hence $W \subset U$. If $c_{t}=0$ for all $t, 0=<A_{t} u, e_{\{t\}}>=<u, A_{t} e_{\{t\}}>=<u, e_{t}>$ and $u \in W^{\perp}$.
Lemma 14.5. Let $T$ intertwine the representations on $V^{\lambda}$ and $V^{\mu}$. Suppose $W^{\lambda}$ is not contained in $\operatorname{Ker} T$. Then $\lambda \geq_{1} \mu$.
Proof. KerT is invariant under $\pi(g)$ and if it does not contain $W^{\lambda}$ it is orthogonal to it.
$$
0 \neq T e_{t}=T A_{t} e_{\{t\}}=A_{t} T e_{\{t\}}
$$
$T e_{\{t\}}$ is a combination of $e_{\{\tau\}}$ of $\mu$-tableaux $\tau$. So atleast one of them $A_{t} e_{\{\tau\}}$ is nonzero forcing $\lambda \geq_{1} \mu$.
Lemma 14.6. If $T \neq 0$ intertwines $W^{\lambda}$ and $W^{\mu}$, then $\lambda=\mu$.
Proof. Extend $T$ by making it 0 on $\left(W^{\lambda}\right)^{\perp}$ and we see that $\lambda \geq_{1} \mu$. The argument is symmetric.
## Representations $\mathrm{SO}(3)$
We will consider the irreducible representations of the group $G$ of rotations in $R^{3}$. These are orthogonal transformations of determinant 1 , i.e. that preserve orientation. An element $g \in G$ is represented as the matrix
$$
\left[\begin{array}{ccc}
t_{1,1}(g) & t_{1,2}(g) & t_{1,3}(g) \\
t_{2,1(g)} & t_{2,2}(g) & t_{2,3}(g) \\
t_{3,1(g)} & t_{3,2}(g) & t_{3,3}(g)
\end{array}\right]
$$
There is the trivial representation $\pi_{0}(g) \equiv I$. Then there is a natural three dimensional representation where $\pi_{1}(g)=t(g)=\left\{t_{i, j}(g)\right\}$ and it can be viewed as a unitary representation in $\mathcal{C}^{3}$. This representation is irreducible and faithful, i.e. it seperates points of $G$.
As we saw in the general theory, the characters can be used to identify the irreducible representations. It helps to know what the conjugacy classes are. Given two orthogonal matrices $g_{1}$ and $g_{2}$, when can we find a $g$ such that $g g_{1} g^{-1}=g_{2}$ ? The eigen values of $g_{1}$ are $1, e^{ \pm i \theta_{1}}$ and therefore in order for $g_{1}$ and $g_{2}$ to be mutually conjugate we neeed $\theta_{1}= \pm \theta_{2}$ or $\cos \theta_{1}=\cos \theta_{2}$. Conversely one can show that that if $g_{1}$ and $g_{2}$ have the same eigenvalues then they are indeed conjugate. If we use a $g$ to align the eigenspace correponding to 1 , then we need to show essentially that rotation by $\theta$ and $-\theta$ are conjugate. We can use the matrix
$$
\left[\begin{array}{ccc}
-1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & -1
\end{array}\right]
$$
to achieve this.
We will use the infinitesimal method to study irreducible representations. If $A=\left\{a_{i, j}\right\}$ is a real skewsymmetric matrix then $g_{t}=e^{t A}$ defines a one parameter curve in $G$, and if $\pi$ is a unitary representation on a complex vector space $V$, then $U_{t}=\pi\left(g_{t}\right)=e^{i t \sigma(A)}$ for some skew symmetric $\sigma(A)$. This way we get a map $A \rightarrow \sigma(A)$ from the space of real skewsymmetric $3 \times 3$ matrices into complex skewhermitian matrices on $V$.
The way to understand this map is to think of $G$ as three dimensional manifold and the vector space of real skewsymmetric $3 \times 3$ matrices as the tangent space at $e$. In fact there are global vector fields acting on functions defined on $G$ corresponding to any skew symmetric $A$,
$$
\left(X_{A}\right) f(g)=\left.\frac{d}{d t} f\left(g e^{t A}\right)\right|_{t=0}
$$
Then
$$
\sigma(A)=\left(X_{A}\right) \pi(e)
$$
and from the representation property
$$
\begin{gathered}
\left(X_{A}\right) \pi(g)=\pi(g) \sigma(A) \\
X_{A} X_{B}=\sigma(A) \sigma(B)
\end{gathered}
$$
The Poisson bracket $\left[X_{A}, X_{B}\right]=X_{A} X_{B}-X_{B} X_{A}$ is to equal $X_{[A B-B A]}$ and we get this a way a representation $\sigma$ of the "Lie Algebra" of $3 \times 3$ skewsymmetric matices in the space of skewhermitian trnasfromations on $V$. Moreover $\sigma([A, B])=[\sigma(A), \sigma(B)]$. $G$ acts irreducibly on $V$ if and only if $\sigma(A)$ acts irreducibly. We pick a basis $A_{1}, A_{2}, A_{3}$ where
$$
A_{1}=\left[\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & 1 \\
0 & -1 & 0
\end{array}\right] \quad A_{2}=\left[\begin{array}{ccc}
0 & 0 & -1 \\
0 & 0 & 0 \\
1 & 0 & 0
\end{array}\right] \quad A_{3}=\left[\begin{array}{ccc}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0
\end{array}\right]
$$
Let us note that
$$
\left[A_{1}, A_{2}\right]=-A_{3},\left[A_{2}, A_{3}\right]=-A_{1},\left[A_{3}, A_{1}\right]=-A_{2}
$$
If we define $\sigma\left(A_{1}\right)=H$ and $Z_{1}=\sigma\left(A_{2}\right)+i \sigma\left(A_{3}\right), Z_{2}=\sigma\left(A_{2}\right)-i \sigma\left(A_{3}\right)$, we can calculate
$$
\begin{aligned}
& {\left[H, Z_{1}\right]=\sigma\left(\left[A_{1}, A_{2}\right]\right)+i \sigma\left(\left[A_{1}, A_{3}\right]\right)=-\sigma\left(A_{3}\right)+i \sigma\left(A_{2}\right)=i Z_{1}} \\
& {\left[H, Z_{2}\right]=\sigma\left(\left[A_{1}, A_{2}\right]\right)-i \sigma\left(\left[A_{1}, A_{3}\right]\right)=-\sigma\left(A_{3}\right)-i \sigma\left(A_{2}\right)=-i Z_{2}}
\end{aligned}
$$
$H$ being skewhermitian on $V$, it has purely imaginary eigenvalues and a complete set of eigenvectors. Let $V=\oplus_{\lambda} V_{i \lambda}$ be the decomposition of $V$ into eigenspaces of $H$. Moreover $e^{2 \pi H}=\pi\left(e^{2 \pi A_{1}}\right)=\pi(e)=I$ The values $\lambda$ are therefore all integers. If $H v=i \lambda v$, then $H Z_{1} v=Z_{1} H v+\left[H, Z_{1}\right] v=$ $i \lambda Z_{1} v+i Z_{1} v=i(\lambda+1) Z_{1} v$. Therefore $Z_{1}$ maps $V_{i \lambda} \rightarrow V_{i(\lambda+1)}$ and similarly $Z_{2}$ maps $V_{i \lambda} \rightarrow V_{i(\lambda-1)}$. It is clear that if we start with some $v_{0} \in V_{i \lambda}$ then $v_{0},\left\{Z_{1}^{k} v_{0}: k \geq 1\right\},\left\{Z_{2}^{k} v_{0}: k \geq 1\right\}$ are all mutually orthogonal. Since the space is finite dimensional, $Z_{1}^{r} v_{0}=Z_{2}^{s} v_{0}=0$ for some $r, s$. If we take $r, s$ to be the smallest such values, then the subspace generated by them has dimension $r+s-1$ and is invariant under $H, Z_{1}, Z_{2}$. Since the representation is irreducible, this must be all of $V$. Another piece of information is that $H$ and $-H$ are conjugate. The set of $\lambda$ 's is therefore symmetric around the origin. Hence $V$ is odd dimensional and is $\{\lambda\}=\{-k, \ldots, 0, \ldots, k\}$ for some integer $k \geq 0$. This exhausts all possible irreducible representations in the infinitesimal sense and therefore the set of irreducible representations of $G$ cannot be larger. The character of such a representation if it exists is seen to be
$$
\chi_{k}(g)=\hat{\chi}_{k}(\theta)=\sum_{j=-k}^{k} \exp [i j \theta]
$$
where $1, e^{ \pm i \theta}$ are the eigenvalues of $g$. We will try construct them as the natural action of $G$ on the space of homogeneous harmonic polynomials of degree $k$. This dimension is calculated as $\frac{(k+1)(k+2)}{2}-\frac{k(k-1)}{2}=2 k+1 . \quad H$ which is the infinitesimal rotation around $x$-axis is calculated as
$$
H=z \frac{\partial}{\partial y}-y \frac{\partial}{\partial z}
$$
The polynomials $p_{k}^{ \pm}=(y \pm i z)^{k}$ are harmonic in two and therefore three variables and $H p_{k}^{ \pm}= \pm i k p_{k}^{ \pm}$. Therefore this representation has the eigenvlaues $\pm i k$ for $H$ and cannot be decomposed totally in terms of representations of dimension $(2 k-1)$ or less. On the other hand its dimension is only $(2 k+1)$. This is it.
Since we know that $\chi_{k}(g) \chi_{\ell}(g) d g=\delta_{k, \ell}$ it is convenient to determine the weight $w(\theta)$ on $[0, \pi]$ such that it is the probability density of $\theta(g)$ of a random $g$. Then
$$
\int_{0}^{\pi} \hat{\chi}_{k}(\theta) \hat{\chi}_{\ell}(\theta) w(\theta) d \theta=\delta_{k, \ell}
$$
In particular for $k \geq 2$
$$
\int_{0}^{\pi}\left[\hat{\chi}_{k}(\theta)-\hat{\chi}_{k-1}(\theta)\right] w(\theta) d \theta=\delta_{k, \ell}
$$
or
$$
w(\theta)=a+b \cos \theta
$$
Normalization of $\int_{0}^{\pi} w(\theta) d \theta=1$ gives $a=\frac{1}{\pi}$. The orthogonality relation $\int_{0}^{\pi} 1 .(1+2 \cos \theta) w(\theta) d \theta=0$ provides $a+b=0$ or
$$
w(\theta)=\frac{1-\cos \theta}{\pi}
$$
| Textbooks |
\begin{document}
\title{Spectrum Generating on Twistor Bundle}
\abstract{We give explicit formulas for the intertwinors of all orders on the twistor bundle over $S^1\times S^{n-1}$ using spectrum generating technique introduced in \cite{BOO:96}.}
\section{Introduction}
It was shown in \cite{BOO:96} that one can construct intertwining operators of some representations without too much effort when eigenspaces occur with multiplicity one. On the differential form bundle over $S^1\times S^{n-1}$, the double cover of the compactified Minkowski space, some $K$-type eigenspaces occur with multiplicity two. After some additional computation, Branson also showed spectral function for these operators.
Intertwinors on spinors like the Dirac operator have eigenspaces with multiplicity one over $S^1\times S^{n-1}$and explicit spectral function was given in \cite{Hong:04}. But on twistors, the eigenspaces of the intertwinors including Rarita Schwinger operator have multiplicity two on some $K$-type. In this paper, we present the spectral function for these operators.
We briefly review conformal covariance and intertwining relation (for more details, see \cite{Branson:96}, \cite{BOO:96}). \\ Let $M$ be an n-dimensional spin manifold. We enlarge the structure group $\operatorname{Spin}(n)$ to $\operatorname{Spin}(n)\times {\mathbb R}_+$ in conformal geometry. $(V(\l),\l^r)$ are finite dimensional $\operatorname{Spin}(n)\times {\mathbb R}_+$ representations, where $(V(\l),\l)$ are finite dimensional representations of $\operatorname{Spin}(n)$ and $\l^r(h,\a)=\a^r\l(h)$ for $h\in \operatorname{Spin}(n)$ and $\a\in {\mathbb R}_+$. The corresponding associated vector bundles are ${\mathbb V}(\l)=P_{\operatorname{Spin}(n)}\times_{\l}V(\l)$ and ${\mathbb V}^r(\l)=P_{\operatorname{Spin}(n)\times {\mathbb R}_+}\times_{\l^r}V(\l)$ with structure groups $\operatorname{Spin}(n)$ and $\operatorname{Spin}(n)\times {\mathbb R}_+$. $r$ is called the conformal weight of ${\mathbb V}^r$. Tangent bundle $TM$ carries conformal weight $-1$ and cotangent bundle $T^*M$ carries conformal weight $+1$. In general, if $V$ is a subbundle of $(TM)^{\otimes p}\otimes (T^*M)^{\otimes q} \otimes(\Sigma M)^{\otimes r}\otimes (\Sigma^* M)^{\otimes s}$, then $V$ carries conformal weight $q-p$, where $\Sigma M$ is the contravariant spinor bundle. \\ A conformal covariant of bidegree $(a,b)$ is a $\operatorname{Spin}(n)\times {\mathbb R}_+$-equivariant differential operator $D: {\mathbb V}^r(\l) \rightarrow {\mathbb V}^s(\sigma)$ which is a polynomial in the metric $g$, its inverse $g^{-1}$, the volume element $E$, and the fundamental tensor-spinor $\g$ with a conformal covariance law $$ \omega \in C^{\infty}, \quad \overline{g}=e^{2\omega}g, \quad \overline{E}=e^{n\omega}E, \quad \overline{\g}=e^{-\omega}\g \Rightarrow \overline{D}=e^{-b\omega}D\m(e^{a\omega}) , $$ where $\m(e^{a\omega})$ is multiplication of $e^{a\omega}$.
\noindent Given a conformal covariant of bidegree $(a,b)$, $D:{\mathbb V}^r(\l) \rightarrow {\mathbb V}^s(\sigma)$, we can assign new conformal weights to get $D:{\mathbb V}^{r'}(\l) \rightarrow {\mathbb V}^{s'}(\sigma)$ whose bidegree is then $(a-r'+r,b-s'+s)$. Calling this $D$ again is an abuse of notation. If $r'=r+a$ and $s'=s+b$, then $D:{\mathbb V}^{r+a}(\l) \rightarrow {\mathbb V}^{s+b}(\sigma)$ becomes conformally invariant and we call $(a+r,b+r)$ the reduced conformal bidegree of $D$. To see how conformal covariants behave under a conformal transformation and a conformal vector field, we recall followings.\\ A diffeomorphism $h:M\rightarrow M$ is called a conformal transformation if $h\cdot g=e^{2\omega_h}g$, where $\cdot$ is the natural action of $h$ on tensor fields. A conformal vector field is a vector field $X$ with ${\cal L}_Xg=2\omega_Xg$ for some $\omega_X \in C^{\infty}(M)$. A conformal covariant $D:{\mathbb V}^0(\l) \rightarrow {\mathbb V}^0(\sigma)$ of reduced bidegree $(a,b)$ satisfies $$ D(e^{a\omega_h}h\cdot\f)=e^{b\omega_h}h\cdot (D(\f)) \quad {\rm and} \quad D({\cal L}_X+a\omega_X)\f=({\cal L}_X+b\omega_X)D\f . $$ for all $\f \in \Gamma({\mathbb V}^0(\l))$. Thus if $D:{\mathbb V}^r(\l) \rightarrow {\mathbb V}^s(\sigma)$ of reduced bidegree $(a,b)$, then \begin{equation}\label{inter} D({\cal L}_X+(a-r)\omega_X)\f=({\cal L}_X+(b-s)\omega_X)D\f \end{equation} for $\f\in \Gamma({\mathbb V}^r(\l))$ and $D\f\in \Gamma({\mathbb V}^s(\sigma))$.\\ Note that conformal vector fields form a Lie algebra $\mathfrak{c}(M,g)$ and give rise to the principal series representation $$ U_a^{\l}:\mathfrak{c}(M,g)\rightarrow {\rm End}\Gamma({\mathbb V}^0(\l)) \quad {\rm by} \quad X \mapsto {\cal L}_X+a\omega_X . $$ So a conformal covariant $D:{\mathbb V}^r(\l) \rightarrow {\mathbb V}^s(\sigma)$ of reduced bidegree $(a,b)$ intertwines the principal series representation $$ DU_{a-r}^{\l}\f=U_{b-s}^{\sigma}D\f $$ for $\f\in \Gamma({\mathbb V}^r(\l))$ and $D\f\in \Gamma({\mathbb V}^s(\sigma))$.
\section{Spinors and Twistors}
Let $M=S^1\times S^{n-1}$, $n$ even, be a manifold endowed with the Lorentz metric $-dt^2+g_{S^{n-1}}$.
To get a fundamental tensor-spinor $\a$ for $M$ from the corresponding object $\g$ on $S^{n-1}$, let $$ \a^j=\left(\begin{array}{cc} \g^j & 0 \\ 0 & -\g^j \end{array}\right)\,,\qquad j=1,\ldots,n-1, $$ and $$ \a^0=\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right)\,. $$
Since $M$ is even-dimensional, there is a {\em chirality} operator $\c_M\,$, equal to some complex unit times $\a^0\tilde\c_S\,$, where $$ \tilde\c_S=\left(\begin{array}{cc} \c_S & 0 \\ 0 & -\c_S \end{array}\right)\,, $$ $\c_S$ being the chirality operator on $S$. The chirality operator is always normalized to have square $1$; thus $(\c_S)^2$ and $(\tilde\c_S)^2$ are identity operators, and since $\a^0\a^0=1$, we have $(\a^0\tilde\c_S)^2=-1$. As a result, we may take $$ \c_M=\pm\sqrt{-1}\a^0\tilde\c_S. $$ A {\em spinor} on $M$ can be viewed as a pair of time-dependent spinors on $S^{n-1},$ i.e., $ \left(\begin{array}{c}
\varphi \\
\psi
\end{array} \right), $ where $\varphi$ and $\psi$ are $t$-dependent spinors on $S^{n-1}.$ But by chirality consideration (\cite{Hong:00}), we get $\Xi=\pm 1$ spinors: $$ \left(\begin{array}{c}
\varphi \\
\psi
\end{array} \right) = \left(\begin{array}{c}
\Xi\psi/\sqrt{-1} \\
\psi
\end{array} \right)\, . $$ Recall that {\em twistors} are spinor-one-forms $\F_\l$ with $\a^\l\F_\l=0$. Given a chirality $\Xi$, a twistor $\Y$ is determined by a $t$-dependent spinor-one-form $\y_j$ on $S^{n-1}$ via $$ \Y=dt\wedge\twostac{\f_0}{\y_0}+\twostac{\f_j}{\y_j}\,, $$ where $$ \begin{array}{rl} \f_j&=-\Xi\sqrt{-1}\y_j, \\ \y_0&=\Xi\sqrt{-1}\g^k\y_k, \\ \f_0&=\g^k\y_k\,. \end{array} $$ Furthermore, by Hodge theoretic consideration (\cite{Hong:00}), twistors on $M$ can be decomposed into three pieces
\begin{equation}\label{decom} \left( \begin{array}{cc}
-(n-1) \theta & -\X\sqrt{-1}\g_{i}\theta \\
-(n-1)\X\sqrt{-1}\theta & \g_{i}\theta
\end{array} \right) + \left( \begin{array}{cc}
0 & -\X\sqrt{-1} T_{i}\tau \\
0 & T_{i}\tau
\end{array} \right) + \left( \begin{array}{cc}
0 & -\X\sqrt{-1}\nabla^{j}\eta_{ji} \\
0 & \nabla^{j}\eta_{ji}
\end{array} \right) \end{equation}
$$ =:\langle\theta\rangle+\{\tau\}+[\eta]\, . $$
\section{Intertwining relation on twistors}
Consider the standard conformal vector field (\cite{Branson:87, Orsted:81}) $$ T:=\cos\r\sin t\partial_t+\cos t\sin\r\partial_\r\, . $$ Here $\r$ is the azimuthal angle on $S^{n-1}$. The conformal factor of $T$ is $$ \varpi:=\cos t\cos\r. $$ Let $A=A_{2r}$ be an intertwinor of order $2r$. The intertwining relation says ((\ref{inter}), \cite{Branson:96, Branson:97, BOO:96})
\begin{equation}\label{rel-1} A\left(\tilde{\cal L}_T+\left(\frac{n}2-r\right)\varpi\right)= \left(\tilde{\cal L}_T+\left(\frac{n}2+r\right)\varpi\right)A, \end{equation}
where $\tilde{\cal L}_T$ is the {\em reduced Lie derivative}. On a tensor-spinor with $\twostac{p}{q}$ tensor content, this is $$ \tilde{\cal L}_T={\cal L}_T+(p-q)\varpi. $$ So here (with only 1-form content), it is ${\cal L}_T-\varpi$. Note that we are using the convention where spinors do not have an internal weight; otherwise the spinor content would influence the reduction.\\ Since intertwinors change chirality, we want to consider an exchange operator $$ \begin{array}{rl} E:&=\a^0(\i(\partial_t)\e(dt)-\e(dt)\i(\partial_t)) \\ &=\a^0(1-2\e(dt)\i(\partial_t)). \end{array} $$ It is immediate that $E^2={\rm Id}$. Because of the $\a^0$ factor, $E$ reverses chirality. To see that $E$ takes twistors to twistors, note that $$ \i(\partial_t)\e(dt) -\e(dt)\i(\partial_t):\Phi_\l\mapsto\Phi_\l -2\delta_\l{}^0\Phi_0\,. $$ Thus $$ \begin{array}{rl} \a^\l(E\F)_\l&=\a^\l\a^0(\F_\l-2\d_\l{}^0\F_0) \\ &=-2g^{\l 0}(\F_\l-2\d_\l{}^0\F_0)+2\a^0\a^\l\d_\l{}^0\F_0 \\ &=\underbrace{-2\F^0}_{2\F_0} +4\underbrace{g^{00}}_{-1}\F_0+2\underbrace{\a^0\a^0}_1\F_0 \\ &=0, \end{array} $$ as desired.\\ We want to convert the relation (\ref{rel-1}) for $EA$. So we will eventually need ${\cal L}_TE$. We have: $$ \begin{array}{rl} {\cal L}_TE&={\cal L}_T\left\{\a(dt)(1-2\e(dt)\i(\partial_t))\right\} \\ &=\{-\varpi\a(dt)+\a(d(Tt))\}(1-2\e^0\i_0) \\ &\qquad-2\a^0\{\e(dt)\i([T,\partial_t])+\e(d(Tt)\i(\partial_t)\}. \end{array} $$ But $$ \begin{array}{rl} Tt&=\cos\r\sin t, \\ d(Tt)&=-\sin\r\sin t\,d\r+\cos\r\cos t\,dt, \\ {}[T,\partial_t]&=-\cos\r\cos t\,\partial_t+\sin t\sin\r\,\partial_\r\,. \end{array} $$ This reduces the above to
\begin{equation}\label{lte} \begin{array}{rl} {\cal L}_TE&=\sin t\a(d\w)(1-2\e^0\i_0)-2\sin t\a^0(\e^0\i(Y)+\e(d\w)\i_0) \\ &=\sin t\sin\r\{-\a^1(1-2\e^0\i_0)-2\a^0(\e^0\i_1-\e^1\i_0)\}. \end{array} \end{equation}
By Kosmann (\cite{Kosmann:72}, eq(16)), the Lie and covariant derivatives on spinors are related by $$ {\cal L}_X-\nd_X=-\tfrac14\nd_{[a}X_{b]}\g^a\g^b =-\tfrac18(dX_\flat)_{ab}\g^a\g^b. $$ Note that $$ \begin{array}{rl} T_\flat&=-\cos\r\sin t\,dt+\cos t\sin\r\,d\r, \\ dT_\flat&=2\sin\r\sin t\,d\r\wedge dt. \end{array} $$ and $$ d\varpi=-T_{\flat,{\rm R}}\,, $$ where $\flat,$R is the musical isomorphism in the ``Riemannian" metric. According to the above,
\begin{equation}\label{ltandnt} {\cal L}_T-\nd_T=-\frac12\sin\r\sin t\a^1\a^0 \end{equation}
on spinors.\\ On a 1-form $\eta$, $$ \langle({\cal L}_T-\nd_T)\eta,X\rangle=-\langle\eta,({\cal L}_T-\nd_T)X\rangle, $$ since ${\cal L}_T-\nd_T$ kills scalar functions. But by the symmetry of the pseudo-Riemannian connection, $$ [T,X]-\nd_TX=-\nd_XT. $$ We conclude that $$ ({\cal L}_T-\nd_T)\eta=\langle\eta,\nd T\rangle, $$ where in the last expression, $\langle\cdot,\cdot\rangle$ is the pairing of a 1-form with the contravariant part of a $\twostac{1}{1}$-tensor: $$ (({\cal L}_T-\nd_T)\eta)_\l=\eta_\m\nd_\l T^\m. $$ Combining this with what we derived above for spinors (\ref{ltandnt}), for a spinor-1-form $\F_\l$, we have $$ (({\cal L}_T-\nd_T)\F)_\l=\F_\m\nd_\l T^\m -\frac12\sin\r\sin t\a^1\a^0\F_\l\,. $$ But $\nd T$ {\em a priori} has projections in 3 irreducible bundles, TFS${}^2$, $\L^0$, and $\L^2$ (after using the musical isomorphisms). By conformality, the TFS${}^2$ part is gone. We expect a $\L^0$ part, essentially $\varpi$. We also found the $\L^2$ part above, $$ dT_\flat=2\sin\r\sin t\,d\r\wedge dt. $$ More precisely, tracking the normalizations, $$ (\nd T_\flat)_{\l\m}=(\nd T_\flat)_{(\l\m)} +(\nd T_\flat)_{[\l\m]} =(\varpi g+\frac12dT_\flat)_{\l\m}\,. $$ Now note that $$ \begin{array}{rl} \F_\m\nd_\l T^\m&=(\nd_\bullet T^\bullet\sharp\F)_\l \\ &=\varpi(g\sharp \F)_\l +\frac12((dT_\flat)_{\nu\m}\e^\nu\i^\m\F)_\l \\ &=\varpi\F_\l+\frac12(((dT_\flat)_{01}\e^0\i^1 +(dT_\flat)_{10}\e^1\i^0)\F)_\l \\ &=\varpi\F_\l+\frac12((-2\sin\r\sin t\e^0\i^1 +2\sin\r\sin t\e^1\i^0)\F)_\l \\ &=\varpi\F_\l-\sin\r\sin t((\e^0\i^1-\e^1\i^0)\F)_\l \\ &=\varpi\F_\l-\sin\r\sin t((\e^0\i_1+\e^1\i_0)\F)_\l\,. \end{array} $$ As a result, $$ \begin{array}{rl} {\cal L}_T-\nd_T&=\varpi-\sin\r\sin t\left(\tfrac12\a^1\a^0+\e^0\i_1 +\e^1\i_0\right) \\ &=:\varpi-\sin\r\sin t P \\ &=:\varpi-{\cal P}, \end{array} $$ and $$ \tilde{\cal L}_T-\nd_T=-{\cal P}. $$ An explicit calculation using \nnn{lte} gives $$ ({\cal L}_TE)E=-2{\cal P}. $$ Since $E^2={\rm Id}$, we conclude that $$ {\cal L}_TE=-2{\cal P} E. $$ With the above, the intertwining relation for $EA$ becomes $$ \begin{array}{rl} \left(\tilde{\cal L}_T+\left(\frac{n}2+r\right)\varpi\right)EA &=E\left(\tilde{\cal L}_T+\left(\frac{n}2+r\right)\varpi\right)A+({\cal L}_TE)A \\ &=EA\left(\tilde{\cal L}_T+\left(\frac{n}2-r\right)\varpi\right)-2{\cal P} EA, \end{array} $$ so that, with $B=EA$, $$ B\left(\nd_T+\left(\frac{n}2-r\right)\varpi-{\cal P}\right)= \left(\nd_T+\left(\frac{n}2+r\right)\varpi+{\cal P}\right)B. $$ To see what $P$ does let us define two convenient operations. $$ \y_j\stackrel{{\bf expa}}{\longmapsto} \HHmx{u}{\X\y_j/\sqrt{-1}}{-\X u/\sqrt{-1}}{\y_j} \stackrel{{\bf slot}}{\longmapsto}{\y_j}\, , $$ where $u=\g^k\y_k$.\\ Note that $$ \begin{array}{l} \y_j\stackrel{{\bf expa}}{\longmapsto} \HHmx{u}{\X\y_j/\sqrt{-1}}{-\X u/\sqrt{-1}}{\y_j} \stackrel{\i_0}{\longmapsto}\HTmx{u}{-\X u/\sqrt{-1}} \\ \stackrel{\e^1}{\longmapsto} \HHmx{0}{\e^1 u}{0}{-\X\e^1 u/\sqrt{-1}} \stackrel{{\bf slot}}\longmapsto-\X\e^1 u/\sqrt{-1}. \end{array} $$ As for the $\e^0\i_1$ term, anything in the range of $\e^0$ has a {\bf slot} of $0$.
Finally, $$ \begin{array}{l} \y_j\stackrel{{\bf expa}}{\longmapsto} \HHmx{u}{\X\y_j/\sqrt{-1}}{-\X u/\sqrt{-1}}{\y_j} \stackrel{\a^0}{\longmapsto}\HHmx{0}{1}{1}{0} \HHmx{u}{\X\y_j/\sqrt{-1}}{-\X u/\sqrt{-1}}{\y_j} \\ =\HHmx{-\X u/\sqrt{-1}}{\y_j}{u}{\X\y_j/\sqrt{-1}} \stackrel{\a^1}{\longmapsto} \HHmx{-\X\g^1 u/\sqrt{-1}}{\g^1\y_j}{-\g^1 u}{-\X\g^1\y_j/\sqrt{-1}}. \end{array} $$ So $$ \begin{array}{l} {\bf slot}\,P\,{\bf expa}:\y_j\mapsto -\frac12\X\g^1\y_j/\sqrt{-1}-\X(\e^1 u)_j/\sqrt{-1} \\ =-\frac{\X}{\sqrt{-1}}(\frac12\g^1\y_j+(\e^1 u)_j) =-\frac{\X}{\sqrt{-1}}(\frac12\g^1\y_j+\d_j{}^1 u). \end{array} $$
Up to a factor of a complex unit, ${\bf slot}\,P\,{\bf expa}$ is $$ \frac12\g^1\y_j+\d_j{}^1\g^k\y_k\,. $$ We can also get this expression by successively taking the commutator of $\varpi$ with $\partial_t$ and $$ {\bf slot}\,{\cal D}\,{\bf expa}:\y_j\mapsto\frac12\g^k\N_k\y_j+\g^k\N_j\y_k\,. $$ That is, $$ {\cal P}=\Xi\sqrt{-1}[\partial_t,[{\cal D},\varpi]]\,. $$ Recall that ${\cal P}=\sin \r \sin t P$.\\ After some straightforward computation, we get the block matrix for ${\cal D}$ relative to the decomposition $\{\langle \theta \rangle, \{\tau \}, [\eta] \}$ (\ref{decom}) as follows. $$ \left(\begin{array}{ccc}
\displaystyle{\frac{n+1}{2(n-1)}J_\theta} &
\displaystyle{\frac{n-2}{4}-\frac{n-2}{(n-1)^2}J_\t^2} & 0\\
-n & \displaystyle{\frac{n-3}{2(n-1)}J_\t} & 0 \\
0 & 0 & \displaystyle{\frac{1}{2}L} \end{array}\right) \, , $$ where $J_\theta$ and $J_\t$ are the Dirac eigenvalues of $\theta$ and $\tau$ on $S^{n-1}$, respectively and $L$ is the Rarita-Schwinger eigenvalue of [$\eta$] on $S^{n-1}$.\\ The spectrum generating relation takes the following form:\\ $$ [N,\varpi]=2\left(\nd_T+\frac{n}2\varpi\right)\,, $$ where $\nd^{*,{\rm R}}\nd:=N$ is the Riemannian Bochner Laplacian. Therefore the relation (\ref{rel-1}) becomes
\begin{equation}\label{rel} B\left(\frac{1}{2}[N,\vp]-r\vp-\X\sqrt{-1}[\partial_t,[{\cal D},\vp]]\right) =\left(\frac{1}{2}[N,\vp]+r\vp+\X\sqrt{-1}[\partial_t,[{\cal D},\vp]]\right)B\, . \end{equation} As explained in detail in (\cite{Branson:97}), the recursive numerical spectral data come from the compressed relation of the above.
\section{Projections into isotypic summands}
Let us denote the $K=\mbox{Spin}(2)\times\mbox{Spin}(n)$-type with highest weight $$ (f)\otimes (j,\tfrac12+q\,\tfrac12,\ldots,\tfrac12\,,\frac{\e}{2})\, , $$ where $j\in\tfrac12+q+{\mathbb N}$, $\e=\pm 1$, and $q=0$ or $1\,$, by $$ {\cal V}_\Xi(f,j,\tfrac12+q\,\tfrac12,\ldots,\tfrac12\,,\frac{\e}{2})\, . $$ An $\frak{s}$-map from such a $K$-type lands in the direct sum of neighboring $K$-types (\cite{Branson:87}).\\ Consider a $\X$ spinor $\HTmx{\f}{\y}$. Since $\f=\Xi\y/\sqrt{-1}$, we have $$ \a^0\twostac{\bullet}{\y}=\twostac{\bullet}{\Xi\y/\sqrt{-1}}\, . $$ Here $\bullet$ denotes a top entry that is computable from the bottom entry, but whose value is not needed at the moment.\\ In addition, $$ \begin{array}{rl} \sin t\twostac{\bullet}{\y}&=\twostac{\bullet}{\sin t\y} =\twostac{\bullet}{-[\ptl_t,\cos t]\y}\,, \\ {\rm Proj}_{f'}\sin t\twostac{\bullet}{\y}
&=\twostac{\bullet}{\frac{f'-f}{\sqrt{-1}}\cos t|^{f'}_f\y}\,, \\ \sin\r\a^1\twostac{\bullet}{\y}&=\twostac{\bullet}{-\sin\r\g^1\y} =\twostac{\bullet}{[D,\cos\r]\y}\,, \\ {\rm Proj}_b \sin\r\a^1\twostac{\bullet}{\y}&=\twostac{\bullet}{-{\rm Proj}_b\sin\r\g^1\y}
=\twostac{\bullet}{D|^b_a\cos\r\,\y}\,, \end{array} $$ where $D=\g^i\nd_i$ is the Dirac operator on $S^{n-1}$. Here $a$ and $b$ (resp., $f$ and $f'$) are abbreviated labels for the Spin$(n)$-types (resp., Spin$(2)$-types) in question.\\ Note also that the compressed relations of $\vp$ between Clifford range part, twistor range part, and divergence part look (\ref{decom}):
\begin{equation}\label{w} \begin{array}{l}
\vp\HTTmx{\langle \theta \rangle}{0}{0}=\HTTmx{\langle|\vp|\theta\rangle} {0}{0} \stackrel{\mbox{Proj}}{\longmapsto} :\HTTmx{\langle\tilde\theta\rangle}{0}{0}\, ,\\
\vp\HTTmx{0}{\{\tau\}}{0}=\HTTmx{0}{|\vp|\{\tau\}}{|\vp|\{\tau\}}
=\HTTmx{0}{C\{|\vp|\tau\}}{|\vp|\{\tau\}} \stackrel{\mbox{Proj}}{\longmapsto}:\HTTmx{0}{C\{\tilde\tau\}}{[\eta]}\, ,\\
\vp\HTTmx{0}{0}{[\eta]}=\HTTmx{0}{|\vp|[\eta]}{|\vp|[\eta]} \stackrel{\mbox{Proj}}{\longmapsto}:\HTTmx{0}{\{\bar{\tau}\}} {[\tilde{\eta}]}\, , \end{array} \end{equation}
where $C$ is a quantity we will compute in the following lemma.\\ Note that $\vp\langle \theta \rangle$ has only Clifford range pieces, since it is made of a spinor and fundamental tensor-spinor on $S^{n-1}$. On the other hand, $\vp\{\tau\}$ and $\vp[\eta]$ have no Clifford range pieces, since they are made of twistors on $S^{n-1}$ (See \cite{Branson:96, Branson:97}).
\begin{lemma} Let $\a={\cal V}_\X(f;j,\frac{1}{2},\cdots,\frac{1}{2}, \frac{\e}{2})$ and $\b={\cal V}_\X(f';j',\frac{1}{2},\cdots,\frac{1}{2},\frac{\e'}{2})$, $\e=\pm 1$. Then we have $$
|_\b\vp|_\a\{\tau\}=C_{ba}\{|_\b\vp|_\a\tau\}\, , $$ where $$ C_{ba}=\frac1{\l_b(T^*T)}\left(\frac{1}{2}J_b^2+ \frac{1}{2}J_a^2-\frac{J_bJ_a}{n-1}-\frac{n(n-1)}4\right)\, , $$ $J_a$ (resp., $J_b$) is the Dirac eigenvalue on ${\cal V}_\X(j,\frac{1}{2},\cdots,\frac{1}{2},\frac{\e}{2})$ (resp., ${\cal V}_\X(j',\frac{1}{2},\cdots,\frac{1}{2},\frac{\e'}{2})$), and $\l_b(T^*T)$ is the eigenvalue of $T^*T$ on ${\cal V}_\X(j',\frac{1}{2},\cdots,\frac{1}{2},\frac{\e'}{2})$\, over $S^{n-1}$. \end{lemma}
\begin{proof}It suffices to show for the twistor operator $T$ on $S^{n-1}$
and $\w=\cos \r$ that $$
|_b\w|_aT\t=C_{ba}\cdot T(|_b\w|_a\t)\, . $$ Let $D$ be the Dirac operator on $S^{n-1}$. Then $$ \begin{array}{rl} [D^2,\w]\t & =[\N^*\N,\w]\t \; \; {\rm{\textstyle by\; Bochner\; identity}} \\ &=(\N^*\N\w)\t-2\N^k\w\N_k\t=(n-1)\w\t+2\sin\r\N_1\t\, , \\ \end{array} $$ Also $$ \begin{array}{rl} T^*(\w T\t)&=-\N^j(\w\N_j\t+\tfrac1{n-1}\w\g_j D\t) \\ &=\sin\r\N_1\t+\w\N^*\N\t +\tfrac1{n-1}\sin\r\g_1 D\t-\tfrac1{n-1}\w D^2\t \\ &=\tfrac12\left([D^2,\w]-(n-1)\w\right)\t +\w\left(D^2-\tfrac{(n-1)(n-2)}4 \right)\t+\tfrac1{n-1}[\w,D]D\t \\ &\qquad -\tfrac1{n-1}\w D^2\t \quad {\rm{\textstyle by\; the\; above\; and\; Bochner\; identity}} \\ &=\tfrac12 D^2(\w\t)+\tfrac12 \w D^2\t -\tfrac1{n-1}D(\w D\t)-\tfrac{n(n-1)}4\w\t. \end{array} $$ Therefore $$ \begin{array}{ll}
|_b\w|_aT\t&=T\left(\dfrac{1}{\l_b(T^*T)}T^*(|_b\w|_aT\t)\right)\\
&=T\left(\dfrac{1}{\l_b(T^*T)}\left(\frac{1}{2}J_b^2+\frac{1}{2}J_a^2
-\frac{1}{n-1}J_bJ_a-\frac{n(n-1)}{4}\right)|_b\w|_a\t\right)\, . \end{array} $$ \end{proof} \begin{rmk}Eigenvalues of $D$ and $T^*T$ on $S^{n-1}$ are known due to Branson (\cite{Branson:99}). \end{rmk} \noindent With the above (\ref{w}) at hand, we get
\begin{equation}\label{Dw} \begin{array}{l}
|_\b[{\cal D},\vp]|_\a\langle \theta \rangle=\HTTmx{({\cal D}_{11}^\b-{\cal D}_{11}^\a) \langle\tilde\theta\rangle} {({\cal D}_{21}^\b-C_{ba}{\cal D}_{21}^\a)\{\tilde\theta\}}{-{\cal D}_{21}^\a[\eta]}, \mbox{ where }
\left\{\htmx{\langle \tilde\theta \rangle=|_\b\vp|_\a\langle\theta\rangle}
{[\eta]=|_\b\vp|_\a\{\theta\}}\right. ,\\
|_\b[{\cal D},\vp]|_\a\{\tau\}= \HTTmx{(C_{ba}{\cal D}_{12}^\b-{\cal D}_{12}^\a)\langle\tilde\tau\rangle} {C_{ba}({\cal D}_{22}^\b-{\cal D}_{22}^\a)\{\tilde\tau\}} {({\cal D}_{33}^\b-{\cal D}_{22}^\a)[\eta]}, \mbox{ where }
\left\{\htmx{\{\tilde\tau\}=|_\b\vp|_\a\{\tau\}}{[\eta]=|_\b\vp|_\a\{\tau\}} \right. ,\mbox{ and}\\
|_\b[{\cal D},\vp]|_\a[\eta]=\HTTmx{{\cal D}_{12}^\b\langle\bar{\tau}\rangle} {({\cal D}_{22}^\b-{\cal D}_{33}^\a)\{\bar{\tau}\}} {({\cal D}_{33}^\b-{\cal D}_{33}^\a)[\tilde{\eta}]} , \mbox{ where }
\left\{\htmx{\{\bar{\tau}\}=|_\b\vp|_\a[\eta]}
{[\tilde{\eta}]=|_\b\vp|_\a[\eta]}\right. . \end{array} \end{equation} Here we use subscripts to refer to the specific entries of the ${\cal D}$ and superscripts to indicate where these entries are computed.\\ Let us now consider the compressed relation of (\ref{rel}) between neighboring $K$-types.\\ \\
{\bf Case 1: Multiplicity 2 $\leftrightarrow$ 1}
$$ \a={\cal V}_\X(f;j,\frac{1}{2},\cdots,\frac{1}{2},\frac{\e}{2}) \leftrightarrow \b={\cal V}_\X(f';j,\frac{3}{2},\frac{1}{2},\cdots, \frac{1}{2},\frac{\e}{2})\, . $$ Note that the operator $B$ in block form looks $$ B=\left(\begin{array}{ccc}B_{11}&B_{12}&0\\
B_{21}&B_{22}&0\\
0&0&B_{33}
\end{array}\right)\, . $$ With $$
|_{\a}N|_{\b}=f^2-f'^2-(n-2)\mbox{ and }
|_{\b}N|_{\a}=-|_{\a}N|_{\b} $$ and (\ref{Dw}), we get $\a\rightarrow\b$ transition quantities $$ \begin{array}{cc} \b\rightarrow\a :& \HHmx{B_{11}^{\a}}{B_{12}^{\a}}{B_{21}^{\a}}{B_{22}^{\a}} \HTmx{A_1}{E^-} =B_{33}^{\b}\HTmx{-A_1}{E^+}\, \mbox{ and}\\ \a\rightarrow\b :& \THmx{A_2}{-E^-}\HHmx{B_{11}^{\a}} {B_{12}^{\a}}{B_{21}^{\a}}{B_{22}^{\a}} =B_{33}^{\b}\THmx{-A_2}{-E^+}\, , \end{array} $$ where $$ \begin{array}{l} A_1:=\X(f-f'){\cal D}_{12}^\a\, ,\\ A_2:=-\X(f-f'){\cal D}_{21}^\a\, ,\\ E^-:=\frac{1}{2}(f^2-f'^2)-\frac{n-2}{2}-r+\X(f-f') ({\cal D}_{22}^\a-{\cal D}_{33}^\b)\, ,\\ E^+:=\frac{1}{2}(f^2-f'^2)-\frac{n-2}{2}+r-\X(f-f') ({\cal D}_{22}^\a-{\cal D}_{33}^\b)\, . \end{array} $$ In particular, we can write all $2\times 2$ entries of $B^{\a}$ in terms of $B_{21}^{\a}$ and $B_{33}^{\b}$: \begin{equation}\label{m1andm2} \begin{array}{l} B_{11}^{\a}=(E^-B_{21}^{\a}-A_2B_{33}^{\b}) /{A_2}\, ,\\ B_{12}^{\a}=-A_1B_{21}^{\a}/A_2\, , \mbox{ and}\\ B_{22}^{\a}=(-A_1B_{21}^{\a}+E^+B_{33}^{\b}) /E^-\, . \end{array} \end{equation} Thus if we can express $B_{21}^{\a}$ in terms of $B_{33}^{\b}$, we can completely determine all entries in the $2\times2$ block. \\ \\
\noindent {\bf Case 2: Multiplicity 2 $\leftrightarrow$ 2}
$$ \a={\cal V}_\X(f;j,\frac{1}{2},\cdots,\frac{1}{2},\frac{\e}{2}) \rightarrow \b={\cal V}_\X(f';j'\frac{1}{2},\cdots,\frac{1}{2},\frac{\e'}{2})\, . $$ Here we have $$
|_{\b}N|_{\a}=f'^2-f^2+J_b^2-J_a^2\, . $$ So using (\ref{Dw}), we get the transition quantities
\begin{equation}\label{m2andm2} \HHmx{B_{11}^{\b}}{B_{12}^{\b}}{B_{21}^{\b}}{B_{22}^{\b}} \HHmx{F_1^-}{G_2}{G_1}{C_{ba}F_2^-} =\HHmx{F_1^+}{-G_2}{-G_1}{C_{ba}F_2^+} \HHmx{B_{11}^{\a}}{B_{12}^{\a}}{B_{21}^{\a}}{B_{22}^{\a}}\, , \end{equation}
where $$ \begin{array}{l} F_1^-:=\frac{1}{2}(f'^2-f^2)+\frac{1}{2}(J_b^2-J_a^2) -r+\X(f'-f)({\cal D}_{11}^\b-{\cal D}_{11}^\a)\, ,\\ F_1^+:=\frac{1}{2}(f'^2-f^2)+\frac{1}{2}(J_b^2-J_a^2) +r-\X(f'-f)({\cal D}_{11}^\b-{\cal D}_{11}^\a)\, ,\\ F_2^-:=\frac{1}{2}(f'^2-f^2)+\frac{1}{2}(J_b^2-J_a^2) -r+\X(f'-f)({\cal D}_{22}^\b-{\cal D}_{22}^\a)\, ,\\ F_2^+:=\frac{1}{2}(f'^2-f^2)+\frac{1}{2}(J_b^2-J_a^2) +r-\X(f'-f)({\cal D}_{22}^\b-{\cal D}_{22}^\a)\, ,\\ G_1:=\X(f'-f)({\cal D}_{21}^\b-C_{ba}{\cal D}_{21}^\a)\, , \mbox{ and}\\ G_2:=\X(f'-f)(C_{ba}{\cal D}_{12}^\b-{\cal D}_{12}^\a)\, . \end{array} $$ Therefore we get determinant quotients of $B$ on multiplicity 2 part.\\ Note the following diagram of neighboring multiplicity 2 isotypic summands centered at ${\cal V}_\X(f;j,\frac{1}{2},\cdots,\frac{1}{2},\frac{\e}{2})$: $$ \begin{array}{lcccl} {\cal V}_\X(f-1;j+1,\frac{1}{2},\cdots,\frac{1}{2},\frac{\e}{2}) &&&& {\cal V}_\X(f+1;j+1,\frac{1}{2},\cdots,\frac{1}{2},\frac{\e}{2})\\ &\nwarrow&&\nearrow&\\ {\cal V}_\X(f-1;j,\frac{1}{2},\cdots,\frac{1}{2},-\frac{\e}{2}) &\leftarrow&\bullet&\rightarrow& {\cal V}_\X(f+1;j,\frac{1}{2},\cdots,\frac{1}{2},-\frac{\e}{2})\\ &\swarrow&&\searrow&\\ {\cal V}_\X(f-1;j-1,\frac{1}{2},\cdots,\frac{1}{2},\frac{\e}{2}) &&&& {\cal V}_\X(f+1;j-1,\frac{1}{2},\cdots,\frac{1}{2},\frac{\e}{2})\, . \end{array} $$ The determinant quotients corresponding to the above diagram are: \begin{equation}\label{det-m2} \left(\begin{array}{ll} \frac{\left(-f+J+1-\X+r+\frac{\e}{2}\X\right) \left(-f+J+1+\X+r+\frac{\e}{2}\X\right)} {\left(-f+J+1-\X-r-\frac{\e}{2}\X\right) \left(-f+J+1+\X-r-\frac{\e}{2}\X\right)}& \frac{\left(f+J+1-\X+r-\frac{\e}{2}\X\right) \left(f+J+1+\X+r-\frac{\e}{2}\X\right)} {\left(f+J+1-\X-r+\frac{\e}{2}\X\right) \left(f+J+1+\X-r+\frac{\e}{2}\X\right)}\\&\\ \frac{\left(-f+\frac{1}{2}-\X+r-\e\X J\right) \left(-f+\frac{1}{2}+\X+r-\e\X J\right)} {\left(-f+\frac{1}{2}-\X-r+\e\X J\right) \left(-f+\frac{1}{2}+\X-r+\e\X J\right)}& \frac{\left(f+\frac{1}{2}-\X+r+\e\X J\right) \left(f+\frac{1}{2}+\X+r+\e\X J\right)} {\left(f+\frac{1}{2}-\X-r-\e\X J\right) \left(f+\frac{1}{2}+\X-r-\X J\right)}\\&\\ \frac{\left(-f-J+1-\X+r-\frac{\e}{2}\X\right) \left(-f-J+1+\X+r-\frac{\e}{2}\X\right)} {\left(-f-J+1-\X-r+\frac{\e}{2}\X\right) \left(-f-J+1+\X-r+\frac{\e}{2}\X\right)}& \frac{\left(f-J+1-\X+r+\frac{\e}{2}\X\right) \left(f-J+1+\X+r+\frac{\e}{2}\X\right)} {\left(f-J+1-\X-r-\frac{\e}{2}\X\right) \left(f-J+1+\X-r-\frac{\e}{2}\X\right)} \end{array}\right)\, , \end{equation} where $J=\e J_a$. \\ And these data can be put into the following gamma function expression: $$ \frac{1}{4}\bullet \frac{\G\left(\frac12(f+J+r-\frac{\e}{2}\Xi)\right) \G\left(\frac12(-f+J+r+\frac{\e}{2}\Xi)\right)} {\G\left(\frac12(f+J-r+\frac{\e}{2}\Xi)\right) \G\left(\frac12(-f+J-r-\frac{\e}{2}\Xi)\right)} $$ $$ \bullet \frac{\G\left(\frac12(f+J+2+r-\frac{\e}{2}\Xi)\right) \G\left(\frac12(-f+J+2+r+\frac{\e}{2}\Xi)\right)} {\G\left(\frac12(f+J+2-r+\frac{\e}{2}\Xi)\right) \G\left(\frac12(-f+J+2-r-\frac{\e}{2}\Xi)\right)}\, . $$ \\ \\
{\bf Case 3: Multiplicity 1 $\leftrightarrow$ 1}
$$ \a={\cal V}_\X(f;j,\frac{3}{2},\frac{1}{2},\cdots,\frac{1}{2}, \frac{\e}{2}) \leftarrow \b={\cal V}_\X(f';j'\frac{3}{2},\frac{1}{2},\cdots,\frac{1}{2}, \frac{\e'}{2}) \, . $$ Again we have $$
|_{\a}N|_{\b}=f^2-f'^2+J_a^2-J_b^2\, . $$ And the transition quantities are \begin{equation}\label{m1andm1} B_{33}^\a P^-=P^+ B_{33}^\b\, , \end{equation} where $$ \begin{array}{l} P^-:=\frac{1}{2}(f^2-f'^2)+\frac{1}{2}(J_a^2-J_b^2) -r+\X(f-f')({\cal D}^\a_{33}-{\cal D}^\b_{33})\mbox{ and}\\ P^+:=\frac{1}{2}(f^2-f'^2)+\frac{1}{2}(J_a^2-J_b^2) +r-\X(f-f')({\cal D}^\a_{33}-{\cal D}^\b_{33})\, . \end{array} $$ The diagram of neighboring multiplicity 1 isotypic summands centered at $$ {\cal V}_\X(f;j,\frac{3}{2},\frac{1}{2},\cdots,\frac{1}{2},\frac{\e}{2}) $$ looks: $$ \begin{array}{lcccl} {\cal V}_\X(f-1;j+1,\frac{3}{2},\frac{1}{2},\cdots,\frac{1}{2},\frac{\e}{2}) &&&& {\cal V}_\X(f+1;j+1,\frac{3}{2},\frac{1}{2},\cdots,\frac{1}{2},\frac{\e}{2})\\ &\nwarrow&&\nearrow&\\ {\cal V}_\X(f-1;j,\frac{3}{2},\frac{1}{2},\cdots,\frac{1}{2},-\frac{\e}{2}) &\leftarrow&\bullet&\rightarrow& {\cal V}_\X(f+1;j,\frac{3}{2},\frac{1}{2},\cdots,\frac{1}{2},-\frac{\e}{2})\\ &\swarrow&&\searrow&\\ {\cal V}_\X(f-1;j-1,\frac{3}{2},\frac{1}{2},\cdots,\frac{1}{2},\frac{\e}{2}) &&&& {\cal V}_\X(f+1;j-1,\frac{3}{2},\frac{1}{2},\cdots,\frac{1}{2},\frac{\e}{2}) \, . \end{array} $$ And the eigenvalue quotients are: $$ \left(\begin{array}{ll} \dfrac{-f+J+1+r+\frac{\e}{2}\X}{-f+J+1-r-\frac{\e}{2}\X}& \dfrac{f+J+1+r-\frac{\e}{2}\X}{f+J+1-r+\frac{\e}{2}\X}\\&\\ \dfrac{-f+\frac{1}{2}+r-\e\X J}{-f+\frac{1}{2}-r+\e\X J}& \dfrac{f+\frac{1}{2}+r+\e\X J}{f+\frac{1}{2}-r-\e\X J}\\&\\ \dfrac{-f-J+1+r-\frac{\e}{2}\X}{-f-J+1-r+\frac{\e}{2}\X}& \dfrac{f-J+1+r+\frac{\e}{2}\X}{f-J+1-r-\frac{\e}{2}\X} \end{array}\right)\, , $$ where $J=\e J_a$.\\ Thus, following the normalization on the multiplicity 2 part, we get the spectral function on the multiplicity 1 part:
\begin{equation}\label{SpecFcn} Z(r;f,J,\Xi\e)=\frac{\e}{2}\Xi \frac{\G\left(\frac12(f+J+1+r-\frac{\e}{2}\Xi)\right) \G\left(\frac12(-f+J+1+r+\frac{\e}{2}\Xi)\right)} {\G\left(\frac12(f+J+1-r+\frac{\e}{2}\Xi)\right) \G\left(\frac12(-f+J+1-r-\frac{\e}{2}\Xi)\right)}\,. \end{equation}
In particular, $$ Z(\frac{1}{2},f,J,\X\e)=-\frac{1}{4}(f-\X\e J) =\frac{1}{4}\sqrt{-1}\mbox{ eig}(E{\cal R};f,J,\X\e)\, , $$ where $E{\cal R}$ is the exchanged Rarita-Schwinger operator.
\section{Interface between multiplicity 1 and 2 parts}
Consider the following diagram: $$ \begin{array}{ccc} \a_1={\cal V}_\X(f;j,\frac{1}{2},\cdots,\frac{1}{2},\frac{\e}{2}) &\rightarrow& \a_2={\cal V}_\X(f+1;j+1,\frac{1}{2},\cdots,\frac{1}{2},\frac{\e}{2})\\ &&\\ \updownarrow & &\updownarrow \\ &&\\ \b_1={\cal V}_\X(f+1;j,\frac{3}{2},\frac{1}{2},\cdots,\frac{1}{2},\frac{\e}{2}) &\leftarrow& \b_2={\cal V}_\X(f;j+1,\frac{3}{2},\frac{1}{2},\cdots,\frac{1}{2},\frac{\e}{2})\, . \end{array} $$ Then (\ref{m2andm2}) reads $$ B^{\a_2}M_1=M_2B^{\a_1}\, . $$ So $$ {\rm det}B^{\a_2}=\dfrac{{\rm det}M_2}{{\rm det}M_1}{\rm det}B^{\a_1}\, . $$ Note that $\dfrac{{\rm det}M_2}{{\rm det}M_1}$ is a determinant quotient computed in (\ref{det-m2}).\\ From (\ref{m1andm2}), we get a relation between $B_{12}$ and $B_{33}$: $$ \begin{array}{ll} \mbox{det}\HHmx{B_{11}}{B_{12}}{B_{21}}{B_{22}}& =B_{11}B_{22}-B_{12}B_{22}\\ &=-\dfrac{1}{A_2E^-}B_{33}\left(B_{33}A_2E^+-(E^-E^++A_1A_2)B_{21} \right)\, . \end{array} $$ We can also compare $(2,1)$ entries of both sides in (\ref{m2andm2}). Applying (\ref{m1andm2}) and (\ref{m1andm1}) to the both relations, we can finally write $B_{21}$ in terms of $B_{33}$ with a ``big'' help from computer algebra package. \par $2\times 2$ block on $$ {\cal V}_\X(f;j,\frac{1}{2},\cdots,\frac{1}{2},\frac{\e}{2}) $$ in terms of $(3,3)$ $$ {\cal V}_\X(f+1;j,\frac{3}{2},\frac{1}{2},\cdots, \frac{1}{2},\frac{\e}{2}) $$ is: \begin{equation}\label{block} \HHmx{\dfrac{4C_1C_2}{(n-1)C_3C_4}-1} {\dfrac{-2(n-2)\Xi C_5C_2}{(n-1)^2C_3C_4}} {\dfrac{8n\Xi C_2}{C_3C_4}} {\dfrac{-4C_5C_2}{(n-1)C_1C_3C_4}+\dfrac{C_6}{C_1}} \bullet Z(r;f+1,J,\X\e)\, , \end{equation} where $$ \begin{array}{l} C_1=2fn-2f-2n+1+n^2+2rn-2r-2\Xi J_a \, , \\ C_2=2fr+\Xi J_a \, ,\\ C_3=n-1+2r \, , \\ C_4=(2f+2r-\Xi+2J_a)(2f+2r+\Xi-2J_a)\, , \\ C_5=(n-1+2J_a)(n-1-2J_a) \, ,\mbox{ and} \\ C_6=2fn-2f-2n+1+n^2-2rn+2r+2\Xi J_a\, . \end{array} $$ \begin{rmk}In particular, if $r=\frac{1}{2}$ and $(3,3)$ entry $$ \sqrt{-1} f-\sqrt{-1}\X\e J $$ of the exchanged Rarita-Schwinger operator is put into the above formula, we recover the other $2\times 2$ entries $$ \left(\begin{array}{cc} -\dfrac{n-2}{n}\sqrt{-1}\left(f+\dfrac{n+1}{n-1}\X\e J\right) &-\dfrac{2\sqrt{-1}\X}{n(n-1)}\left(\dfrac{(n-1)(n-2)}{4} -\dfrac{n-2}{n-1}J^2\right)\\ 2\sqrt{-1}\X &\sqrt{-1} f-\dfrac{n-3}{n-1}\sqrt{-1}\X\e J \end{array}\right). $$ \end{rmk}
\end{document} | arXiv |
\begin{document}
\baselineskip=16pt
\title[Reaction-diffusion models with Perona--Malik diffusion]{Layered patterns in reaction-diffusion models with Perona--Malik diffusions}
\author[A. De Luca]{Alessandra De Luca} \address[Alessandra De Luca]{Dipartimento di Scienze Molecolari e Nanosistemi, Universit\`a Ca' Foscari Venezia Mestre, Campus Scientifico, Via Torino 155, 30170 Venezia Mestre (Italy)} \email{[email protected]}
\author[R. Folino]{Raffaele Folino} \address[Raffaele Folino]{Departamento de Matem\'aticas y Mec\'anica, Instituto de Investigaciones en Matem\'aticas Aplicadas y en Sistemas, Universidad Nacional Aut\'onoma de M\'exico, Circuito Escolar s/n, C.P. 04510 Cd. de M\'exico (M\'exico)} \email{[email protected]}
\author[M. Strani]{Marta Strani} \address[Marta Strani]{Dipartimento di Scienze Molecolari e Nanosistemi, Universit\`a Ca' Foscari Venezia Mestre, Campus Scientifico, Via Torino 155, 30170 Venezia Mestre (Italy)} \email{[email protected]}
\keywords{Perona--Malik diffusion; compactons; energy estimates; asymptotic behavior}
\maketitle
\begin{abstract} In this paper we deal with a reaction-diffusion equation in a bounded interval of the real line with a nonlinear diffusion of Perona--Malik's type and a balanced bistable reaction term. Under very general assumptions, we study the persistence of layered solutions, showing that it strongly depends on the behavior of the reaction term close to the stable equilibria $\pm1$, described by a parameter $\theta>1$. If $\theta\in(1,2)$, we prove existence of steady states oscillating (and touching) $\pm1$, called \emph{compactons}, while in the case $\theta=2$ we prove the presence of \emph{metastable solutions}, namely solutions with a transition layer structure which is maintained for an exponentially long time. Finally, for $\theta>2$, solutions with an unstable transition layer structure persist only for an algebraically long time. \end{abstract}
\section{Introduction}\label{sec:intro} The goal of this paper is to investigate the persistence of phase transition layer solutions to the reaction-diffusion equation \begin{equation}\label{eq:Q-model}
u_t=Q(\varepsilon^2u_x)_x-F'(u), \end{equation} where $u=u(x,t) : [a,b]\times(0,+\infty)\rightarrow \mathbb{R}$, complemented with homogeneous Neumann boundary conditions \begin{equation}\label{eq:Neu}
u_x(a,t)=u_x(b,t)=0, \qquad \qquad t>0, \end{equation} and initial datum \begin{equation}\label{eq:initial}
u(x,0)=u_0(x), \qquad \qquad x\in[a,b]. \end{equation} In \eqref{eq:Q-model} $\varepsilon>0$ is a small parameter, $Q:\mathbb{R}\to\mathbb{R}$ is a Perona--Malik's type diffusion \cite{PerMal}, while $F:\mathbb{R}\to\mathbb{R}$ is a double well potential with wells of equal depth. More precisely, we assume that $Q\in C^1(\mathbb{R})$ satisfies \begin{equation}\label{eq:Q-ass1}
\lim_{s\to\pm\infty}Q(s)=Q(0)= 0, \qquad \qquad Q(-s)=-Q(s), \end{equation} for all $s\in\mathbb{R}$ and that there exists $\kappa>0$ such that \begin{equation}\label{eq:Q-ass2}
Q'(s)>0, \quad \forall \, s \in (-\kappa,\kappa) \qquad \mbox{ and } \qquad Q'(s)<0, \quad \mbox{ if } |s|>\kappa. \end{equation} The prototype examples we have in mind are \begin{equation}\label{ex:fluxfunction}
Q(s):=\frac{s}{1+s^2} \qquad \mbox{ and } \qquad Q(s):=s e^{-s^2}, \end{equation} which satisfy assumptions \eqref{eq:Q-ass2} with $\kappa = 1$ and $\kappa= \frac{1}{\sqrt{2}}$, respectively (see the left hand picture of Figure \ref{fig1}). \begin{figure}\label{fig1}
\end{figure} Regarding the reaction term, we require that the potential $F\in C^1(\mathbb{R})$ satisfies \begin{equation}\label{ipoF1}
F(\pm1)=F'(\pm1)=0, \qquad \quad F(u)>0 \quad \forall\,u\neq\pm 1, \end{equation} and that there exist constants $0<\lambda_1\leq\lambda_2$, $\eta>0$ and $\theta>1$ such that: \begin{equation}\label{ipoF2}
\lambda_1 |1\pm u|^{\theta-2} \leq \frac{F'(u)}{u\pm1}\leq\lambda_2 |1\pm u|^{\theta-2}, \qquad\qquad \mbox{ for } \qquad |u\pm1|<\eta. \end{equation} Therefore, \eqref{ipoF1} ensures that $F$ is a double well potential with wells of equal depth in $u=\pm1$ and \eqref{ipoF2} describes the behavior of $F$ close to the minimal points. In particular, notice that by integrating \eqref{ipoF2} and using \eqref{ipoF1}, we obtain \begin{equation}\label{eq:ass-F3}
\frac{\lambda_1}{\theta}|1\pm u|^{\theta}\leq F(u)\leq \frac{\lambda_2}{\theta}|1\pm u|^{\theta},
\quad \qquad |u\pm1|<\eta. \end{equation} The simplest example of potential $F$ satisfying \eqref{ipoF1}-\eqref{ipoF2} is \begin{equation}\label{F:ex}
F(u)=\frac{1}{2\theta}|1-u^2|^\theta, \quad \theta>1, \end{equation} which is depicted in Figure \ref{fig1} for different choices of $\theta>1$. It is worth mentioning that when $\theta=2$ in \eqref{F:ex}, we obtain the classical double well potential $F(u)=\frac{1}{4}(1-u^2)^2$, which, in particular, satisfies $F''(\pm1)>0$; on the other hand, if $\theta\in(1,2)$ the second derivative of the potential \eqref{F:ex} blows up as $u\to\pm1$, while for $\theta>2$ we have the \emph{degenerate} case $F''(\pm1)=0$. Finally, notice that \eqref{ipoF1} implies that the reaction term $f=-F'$ satisfies $\displaystyle\int_{-1}^1 f(s)\,ds=0$, being the reason why we call $f$ a balanced bistable reaction term.
The competition between a balanced reaction term satisfying the additional assumption $F''(\pm1)>0$ and a classical linear diffusion is described by the celebrated Allen--Cahn equation \cite{Allen-Cahn}, which can be obtained from \eqref{eq:Q-model} by choosing $Q(s)=s$ and reads as \begin{equation}\label{eq:A-C}
u_t=\varepsilon^2u_{xx}-F'(u). \end{equation} Such model has been extensively studied since the early works \cite{Bron-Kohn,Carr-Pego,Fusco-Hale}, and it is well known that the solution of \eqref{eq:A-C} subject to \eqref{eq:Neu}-\eqref{eq:initial} exhibits a peculiar phenomenon when the diffusion coefficient $\varepsilon>0$ is very small, known in literature as \emph{metastability}: if the initial datum has a transition layer structure, that is $u_0$ is close to a step function taking values in $\{\pm1\}$ and has sharp transition layers, then the corresponding solution evolves very slowly in time and the layers move towards one another or towards the endpoints of the interval $(a,b)$ at an extremely low speed. Once two layers are close enough or one of them is sufficiently close to $a$ or $b$, they disappear quickly and, after that, again the solution enters in a slow motion regime. The latter phenomenon repeats until all the transitions disappear and the solution reaches a stable configuration, which is given by one of the two stable equilibria $u=\pm1$. A rigorous description of such metastable dynamics first appeared in the seminal work \cite{Carr-Pego} where, in particular, it is proved that the time needed for the annihilation of the closest layers is of order $\exp(Al/\varepsilon)$, where $A:=\min\left\{F''(\pm1)\right\}>0$ and $l$ is the distance between the layers. Therefore, the dynamics strongly depends on the parameter $\varepsilon>0$ and the evolution of the solution is extremely slow when $\varepsilon\to0^+$. Moreover, the assumption $F''(\pm1)>0$ is necessary to have metastability and almost 25 years later than the publication of \cite{Bron-Kohn,Carr-Pego,Fusco-Hale}, in \cite{Bet-Sme} the authors prove that in the degenerate case $F''(\pm1)=0$ the exponentially small speed of the layers is replaced by an algebraic upper bound. On the other hand, the slow motion phenomena described above appear only for potentials $F\in C^2(\mathbb{R})$, while in the case of a potential of the form \eqref{F:ex} with $\theta\in(1,2)$ there exist stationary solutions to \eqref{eq:A-C}-\eqref{eq:Neu} that attain the values $\pm1$, with an arbitrary number of layers randomly located inside the interval $(a,b)$, for details see \cite{Dra-Rob} or \cite{FPS-DCDS}. In addition, in \cite{FPS-DCDS} a more general equation than \eqref{eq:A-C} is considered:
the linear diffusion $u_{xx}$ is replaced by the (nonlinear) $p$-Laplace operator $(|u_x|^{p-2}u_x)_x$ and it is shown that the aforementioned phenomena strongly rely on the interplay between the parameters $p,\theta>1$. To be more precise, there exist stationary solutions with a transition layer structure for any $\theta\in(1,p)$; the metastable dynamics appears in the case $\theta=p$ and, finally, the solutions exhibit an algebraic slow motion for any $\theta>p>1$.
The main novelty of our work consists in considering a general function $Q$ satisfying \eqref{eq:Q-ass1}-\eqref{eq:Q-ass2}, with the purpose of extending the aforementioned results to the reaction-diffusion model \eqref{eq:Q-model}. The choice of a function $Q$ satisfying \eqref{eq:Q-ass1}-\eqref{eq:Q-ass2} is inspired by \cite{PerMal}, where the authors introduced the so-called \emph{Perona--Malik equation} (PME) \begin{equation}\label{eq:PME}
u_t=Q(u_x)_x, \end{equation} with $Q$ as in \eqref{ex:fluxfunction}, to describe noise reduction and edge detection of digitalized images. Ever since it was proposed by Perona and Malik in 1990, the nonlinear forward-backward heat equation \eqref{eq:PME} has attracted the interest of the mathematical community. Without claiming to be complete, we list some of the contributions. The Cauchy problem associated to \eqref{eq:PME} has been studied in \cite{Gobbi,Kich}, while the existence, regularity, and (non)uniqueness of solutions to \eqref{eq:PME} in a bounded interval with homogeneous Neumann boundary conditions \eqref{eq:Neu} has been investigated in \cite{Kaw-Kut}. More recently, in \cite{Guido} it is shown that PME admits a natural regularization by forward-backward diffusions possessing better analytical properties than PME itself. Many papers have also been devoted to the study of \eqref{eq:PME} with the presence of reaction and/or convection terms: for instance, we mention \cite{FS-NA} where a Burger's type equation with Perona--Malik diffusion is considered and \cite{CorMalSov}, where the authors study existence of wavefront solutions for a reaction-convection equation with Perona--Malik diffusion. Regarding the reaction-diffusion model \eqref{eq:Q-model}, we recall that the corresponding multi-dimensional version was proposed in \cite{CotGer} for edge detection and contrast enhancement in image processing. In particular, the authors prove that the mathematical model is well posed and show numerically that the processed image can be observed on the asymptotic state of its solution. The same model is studied in \cite{Morfu}, where it is shown that combining the properties of an anisotropic diffusion like the Perona--Malik's with those of bistable reaction terms provides a better processing tool which enables noise filtering, contrast enhancement and edge preserving.
Up to our knowledge, the long time behavior of phase transition layer solutions to reaction-diffusion models with a nonlinear diffusion of the form $Q(u_x)_x$ has been studied only in the case of the $p$-Laplacian \cite{FPS-DCDS} and mean curvature operators \cite{FPS,FS}. Both cases are rather different than \eqref{eq:Q-ass1}-\eqref{eq:Q-ass2}: to highlight the differences, let us expand the term $Q(u_x)_x$ as $Q'(u_x)u_{xx}$ and notice that the derivative of $Q$ plays the role of the diffusion coefficient. In the case of the $p$-Laplacian, $Q'$ is singular at $s=0$ for $p\in(1,2)$ and degenerate ($Q'(0)=0$) for $p>2$, while in the cases studied in \cite{FPS,FS} the function $Q$ is explicitly given by $$Q(s)=\frac{s}{\sqrt{1+s^2}}, \qquad \mbox{ or } \qquad Q(s)=\frac{s}{\sqrt{1-s^2}}.$$
Hence, in the first case the derivative $Q'$ is strictly positive for any $s\in\mathbb{R}$ and since $Q'(s)\to0$ as $|s|\to\infty$, one has a degenerate diffusion coefficient for large values of $|u_x|$, while in the latter case $Q'(s)\to\pm\infty$ as $s\to\pm1$ and, as a consequence, the diffusion coefficient is strictly positive but singular at $\pm1$. In the case considered in this article, \eqref{eq:Q-ass1}-\eqref{eq:Q-ass2} imply that not only the diffusion coefficient is degenerate at $\pm\infty$ and $\pm\kappa$,
but the diffusion coefficient is even strictly negative when $|u_x|>\kappa$. Thus, our work provides the first investigation of long time behavior of phase transition layer solutions in the case of a degenerate and negative diffusion. Moreover, we mention that in \cite{FPS,FPS-DCDS,FS} an explicit formula for $Q$ is considered, while here $Q$ is not explicit but it is a generic function satisfying \eqref{eq:Q-ass1}-\eqref{eq:Q-ass2}. Actually, the interested reader can check that even if considering specific examples as in \eqref{ex:fluxfunction}, one cannot obtain explicit formulas for the involved functions, so that the computations become much more complicated; for details, see Remark \ref{rem:difficile}.
The main contribution of this paper is to show that the well know results about the classic model \eqref{eq:A-C} described above can be extended to \eqref{eq:Q-model}, even if $Q$ is a non-monotone function; roughly speaking, the condition $Q'>0$ in $(-\kappa,\kappa)$ for some generic $\kappa$ is enough to obtain existence of \emph{compactons}, which are stationary solutions (hence, invariant under the dynamics of \eqref{eq:Q-model}), with a transition layer structure, in the case $F$ satisfy \eqref{ipoF1}-\eqref{ipoF2} with $\theta\in(1,2)$, existence of metastable patterns if $\theta=2$ and existence of algebraically slowly moving structures when $\theta>2$. The dynamical (in)stability of the compactons remains an interesting open problem; it is not clear whether small perturbations of compactons generate a slow motion dynamics similar to the case $\theta=2$ or the transition layer structure is maintained for all times $t>0$.
\subsection*{Plan of the paper} We close the Introduction with a short plan of the paper; Section \ref{stationary} is devoted to the stationary problem associated to \eqref{eq:Q-model}. We will consider steady states both in the whole real line and in bounded intervals, showing also that there are substantial differences depending of the value of the power $\theta$ appearing in \eqref{ipoF2}. Indeed, the existence of the aforementioned compactons is a peculiarity of the case $\theta\in(1,2)$ and it is established in Proposition \ref{prop:comp}. On the contrary, in the case $\theta\geq2$ we focus our attention on the existence of periodic solutions in the real line and their restriction on a bounded interval $[a,b]$ (see Propositions \eqref{periodicR} and \ref{periodic:bounded} respectively) that oscillate among values $\pm\bar s$, with $\bar s\approx1$ (strictly less that one). In Section \ref{sec:energy} we prove some variational results and lower bounds on the energy associated to \eqref{eq:Q-model} (for its definition, we refer to \eqref{eq:energy}) which will be crucial in order to prove the results of Section \ref{sec:slow}; we here focus on the asymptotic behavior of the solutions to \eqref{eq:Q-model}-\eqref{eq:Neu}-\eqref{eq:initial}, showing that if the dynamics starts from an initial datum with $N$ transition layers inside the interval $[a,b]$, then such configuration will be maintained for extremely long times; as it was previously mentioned, the time taken for the solution to annihilate the \emph{unstable} structure of the initial datum strongly depends on the choice of the parameter $\theta>1$ appearing in \eqref{ipoF2} and we have either metastable dynamics (exponentially slow motion) in the critical case $\theta=2$ or algebraic slow motion in the degenerate case $\theta>2$. We underline again that in the case $\theta \in (1,2)$ solutions with $N$ transition layers are either stationary solutions (compactons) or close to them. Numerical simulations, which illustrate the analytical results, are provided at the end of Section \ref{sec:slow}.
\section{Stationary solutions}\label{stationary} The aim of this section is to analyze the stationary problem associated to \eqref{eq:Q-model} and to prove the existence of some \emph{special} solutions to \begin{equation}\label{eq:staz}
Q(\varepsilon^2\varphi')'-F'(\varphi)=0, \end{equation} with $\varepsilon>0$, $Q$ satisfying assumptions \eqref{eq:Q-ass1}-\eqref{eq:Q-ass2} and $F$ as in \eqref{ipoF1}-\eqref{ipoF2}, both in the whole real line and in a bounded interval complemented with homogeneous Neumann boundary conditions \eqref{eq:Neu}. In order to prove the results of this section we actually have to require more regularity on the diffusion flux $Q$, that is $Q \in C^3(\mathbb{R})$; however, we underline that the basic examples \eqref{ex:fluxfunction} we have in mind satisfy such additional assumption as well.
\subsection{Standing waves} We start by considering problem \eqref{eq:staz} in $\mathbb{R}$, and we focus the attention on standing waves, that can be defined as follows: an increasing standing wave $\Phi_\varepsilon:=\Phi_\varepsilon(x)$ is a solution to \eqref{eq:staz} in the whole real line satisfying either \begin{equation}\label{eq:stand-teta<2}
\begin{cases}
\Phi_\varepsilon(x)=1, \qquad & x\geq x_+,\\
\Phi_\varepsilon(x)=-1, & x\leq x_-,
\end{cases} \qquad \qquad
\Phi'_\varepsilon(x)> 0, \qquad \mbox{ for any } x\in(x_-,x_+), \end{equation} for some $x_\pm \in \mathbb{R}$ with $x_-<x_+$ or \begin{equation}\label{eq:stand-teta>2}
\lim_{x\to\pm\infty}\Phi_\varepsilon(x)=\pm1, \qquad \qquad \Phi'_\varepsilon(x)> 0, \qquad \mbox{ for any } x\in\mathbb{R}. \end{equation} Similarly, a decreasing standing wave $\Psi_\varepsilon:=\Psi_\varepsilon(x)$ satisfies \eqref{eq:staz} and either \begin{equation*}
\begin{cases}
\Psi_\varepsilon(x)=-1, \qquad & x\geq x_+,\\
\Psi_\varepsilon(x)=1, & x\leq x_-,
\end{cases} \qquad \qquad
\Psi'_\varepsilon(x)<0, \qquad \mbox{ for any } x\in(x_-,x_+), \end{equation*} for some $x_\pm \in \mathbb{R}$ with $x_-<x_+$ or \begin{equation*}
\lim_{x\to\pm\infty}\Psi_\varepsilon(x)=\mp1, \qquad \qquad \Psi'_\varepsilon(x)<0, \qquad \mbox{ for any } x\in\mathbb{R}. \end{equation*} It is easy to check that solutions to \eqref{eq:staz}-\eqref{eq:stand-teta<2} and \eqref{eq:staz}-\eqref{eq:stand-teta>2} are invariant by translation; thus, in order to deduce a unique solution we add the further assumption $\Phi_\varepsilon(0)=0$ and we rewrite the problems \eqref{eq:staz}-\eqref{eq:stand-teta<2} and \eqref{eq:staz}-\eqref{eq:stand-teta>2} as \begin{equation}\label{eq:Fi}
\varepsilon^2Q'(\varepsilon^2\Phi'_\varepsilon)\Phi''_\varepsilon-F'(\Phi_\varepsilon)=0, \qquad \lim_{x\to\pm\infty}\Phi_\varepsilon(x)=\pm1, \qquad \Phi_\varepsilon(0)=0, \qquad \Phi'_\varepsilon(x)\geq 0, \end{equation} for any $x\in\mathbb{R}$. Analogously, in the decreasing case we have \begin{equation}\label{eq:Fi-dec}
\varepsilon^2Q'(\varepsilon^2\Psi'_\varepsilon)\Psi''_\varepsilon-F'(\Psi_\varepsilon)=0, \qquad \lim_{x\to\pm\infty}\Psi_\varepsilon(x)=\mp1, \qquad \Psi_\varepsilon(0)=0, \qquad \Psi'_\varepsilon(x)\leq 0, \end{equation} for any $x\in\mathbb{R}$.
As we will see below, there is a fundamental difference whether $F$ satisfies \eqref{ipoF2} with $\theta\in(1,2)$ or with $\theta>2$: in the first case, the standing waves touch the values $\pm1$, namely the increasing standing waves satisfy \eqref{eq:stand-teta<2}. Conversely, if $\theta>2$ the standing waves reach the values $\pm1$ only in the limit: for instance, in the increasing case, they satisfy \eqref{eq:stand-teta>2}. In order to prove such claim, as well as the existence of a unique solution to \eqref{eq:Fi} (or, alternatively, of \eqref{eq:Fi-dec}), we need to premise the following technical result.
\begin{lem}\label{lemma:new} Let $Q \in C^1(\mathbb{R})$ satisfying \eqref{eq:Q-ass1}-\eqref{eq:Q-ass2}. Denote by \begin{equation}\label{def:ell}
\ell := \kappa Q(\kappa)-\tilde Q(\kappa), \qquad \mbox{ where } \qquad \tilde Q(s) := \int_0^s Q(t) \, dt, \end{equation} and \begin{equation}\label{def:P} \begin{aligned}
P_\varepsilon(s):= \int_{0}^{s}\varepsilon^2z\,Q'(\varepsilon^2 z)\,dz.
\end{aligned} \end{equation} Then, there exists a unique (strictly positive) function $J_\varepsilon$ which inverts the equation $P_\varepsilon(s)=\xi$ in $[0,\kappa \varepsilon^{-2}]$: for any $s\in[0,\kappa \varepsilon^{-2}]$ and $\xi\in[0,\ell\varepsilon^{-2}]$ there holds \begin{equation}\label{eq:J_e}
P_\varepsilon(s)=\xi \qquad \qquad \mbox{ if and only if } \qquad \qquad s=J_\varepsilon(\xi). \end{equation} Moreover, the following expansion holds true \begin{equation}\label{eq:J-important}
J_\varepsilon(\xi) = \sqrt{\frac{2}{\varepsilon^2Q'(0)}\xi}+\varepsilon^{-2}\rho(\varepsilon^2\xi),
\qquad \mbox{ where } \qquad \rho(\xi)=o(\xi). \end{equation} \end{lem} \begin{proof} In order to study the invertibility of the equation $P_\varepsilon(s)=\xi$, we observe that \begin{equation}\label{defalternativa} \begin{aligned}
P_\varepsilon(s)& =\int_{0}^{s}\varepsilon^2z\,Q'(\varepsilon^2 z)\,dz =\frac{1}{\varepsilon^2} \int_0^{\varepsilon^2 s } \tau Q'(\tau) \, d\tau \\
&= \frac{1}{\varepsilon^2} \left[ \tau \, Q(\tau) \Big|^{\varepsilon^2 s}_0- \int_0^{\varepsilon^2 s} Q(\tau) \, dt\right] = s Q(\varepsilon^2 s) - \frac{1}{\varepsilon^2}\tilde Q (\varepsilon^2 s).
\end{aligned} \end{equation} Hence $P_\varepsilon$ is an even function satisfying $P_\varepsilon'(0)=P_\varepsilon'(\pm\kappa\varepsilon^{-2})=0$ and \begin{equation*}
P_\varepsilon'(s)s= \varepsilon^2 Q'(\varepsilon^2 s) \, s^2 \geq 0, \qquad \mbox{ for any } \; s \in[-\kappa \varepsilon^{-2},\kappa \varepsilon^{-2}], \end{equation*} because of the definition \eqref{def:P} and the assumptions on $Q$ \eqref{eq:Q-ass1}-\eqref{eq:Q-ass2}. Thus, the equation $P_\varepsilon(s)=\xi$ has exactly two solutions in $[-\kappa \varepsilon^{-2},\kappa \varepsilon^{-2}]$, provided that $\xi\in[P_\varepsilon(0),P_\varepsilon(\kappa \varepsilon^{-2})]$. Going further, one has $P_\varepsilon(0)=0$ while \begin{equation*}
P_\varepsilon(\kappa \varepsilon^{-2})= \kappa \varepsilon^{-2} Q(\kappa)-\varepsilon^{-2} \tilde Q(\kappa) = \frac{\ell}{\varepsilon^2}, \end{equation*} where the constant $\ell$, defined in \eqref{def:ell}, is strictly positive because $\kappa \, Q(\kappa)$ is indeed greater than $\tilde Q(\kappa)$, which represents the area underneath the function $Q$ in the interval $[0,\kappa]$. Hence, \eqref{eq:J_e} holds true and it remains to prove \eqref{eq:J-important}. Using \eqref{defalternativa}, we deduce \begin{equation}\label{eq:defH}
P_\varepsilon(s)=\varepsilon^{-2}H(\varepsilon^2s), \qquad \mbox{ where} \qquad H(s):=sQ(s)-\tilde Q(s). \end{equation} Since the function $H$ in \eqref{eq:defH} satisfies $H(0)=H'(0)=0$ and $H''(0)=Q'(0)>0$, for $s \sim 0$ we have $$P_\varepsilon(s) =\varepsilon^{-2}\left[\frac{Q'(0)}2 \,\varepsilon^4s^2 +o((\varepsilon^2s)^2)\right],$$ and the latter equality gives a hint that $J_\varepsilon$ behaves like the square root of $s$ for $s \sim 0$. To prove it, let us study the behavior of $(H^{-1})^2$ close to the origin. We have \begin{equation*}
\left[(H^{-1})^2 \right]'(s)= 2 H^{-1}(s) \left[H^{-1}\right]' (s)= \frac{2H^{-1}(s)}{H'\left(H^{-1}(s)\right)}, \end{equation*} so that \begin{align*}
\left[(H^{-1})^2 \right]'(0) &= 2\lim_{s \to 0} \frac{H^{-1}(s)}{H'\left(H^{-1}(s)\right)} = 2 \lim_{s \to 0} \frac{H^{-1}(s)}{H''(0)H^{-1}(s) + o\left(H^{-1}(s)^2\right)} \\
&= 2\lim_{s \to 0} \frac{1}{H''(0)+ \frac{o\left(H^{-1}(s)^2\right)}{H^{-1}(s)}} = \frac{2}{H''(0)} =\frac{2}{Q'(0)}>0. \end{align*} Thus, \begin{equation*}
(H^{-1})^2(s) = 0 +\frac{2}{Q'(0)} s + R(s) \quad \Longrightarrow \quad H^{-1}(s) = \sqrt{\frac{2}{Q'(0)} s + R(s)}, \end{equation*} where $R(s)=\mathcal{O}(s^2)$. Hence, we can state that \begin{equation*}
H^{-1}(s) = \sqrt{\frac{2}{Q'(0)} s} + \rho(s), \qquad \mbox{ with } \qquad \rho(s)= o(s). \end{equation*} Indeed, \begin{align*}
\lim_{s \to 0^+}\frac{\rho(s)}{s}&=\lim_{s \to 0^+}s^{-1}\left[\sqrt{\frac{2}{Q''(0)}s+R(s)}-\sqrt{\frac{2}{Q''(0)}s} \, \right]=
\lim_{s \to 0^+}\frac{s^{-1}R(s)}{\sqrt{\frac{2}{Q''(0)}s+R(s)}+\sqrt{\frac{2}{Q''(0)}s}} \\
&=\lim_{s \to 0^+}\frac{s^{-\frac{3}{2}}R(s)}{\sqrt{\frac{2}{Q''(0)}+s^{-1}{R(s)}}+\sqrt{\frac{2}{Q''(0)}}} =0. \end{align*} By using \eqref{eq:J_e}-\eqref{eq:defH}, one obtains \begin{equation*}
J_\varepsilon(\xi) =\varepsilon^{-2}H^{-1}(\varepsilon^2\xi)=\sqrt{\frac{2}{\varepsilon^2Q'(0)}\xi} +\varepsilon^{-2}\rho(\varepsilon^2\xi), \end{equation*} that is \eqref{eq:J-important} and the proof is complete. \end{proof} \begin{rem}\label{rem:difficile} It is interesting to notice that, even if we consider the explicit examples in \eqref{ex:fluxfunction}, it is not possible to give an explicit formula for the function $J_\varepsilon$ in \eqref{eq:J_e}. As we will see in the rest of the paper, not having an explicit formula for $J_\varepsilon$ considerably complicates the proof of our results, in which the expansion \eqref{eq:J-important} plays a crucial role. \end{rem}
We have now all the tools to prove the following existence result.
\begin{prop}\label{ESW} Let $Q$ satisfying \eqref{eq:Q-ass1}-\eqref{eq:Q-ass2} and $F$ satisfying \eqref{ipoF1}-\eqref{ipoF2}. Then, there exists $\varepsilon_0>0$ such that problem \eqref{eq:Fi} admits a unique solution $\Phi_\varepsilon \in C^2(\mathbb{R})$ for any $\varepsilon \in (0,\varepsilon_0)$. Moreover we have the following alternatives: \begin{itemize} \item[(i)] If $\theta\in(1,2)$, then the profile $\Phi_\varepsilon$ satisfies \eqref{eq:stand-teta<2}; more precisely one has \begin{equation}\label{ugualea1e-1}
\Phi_\varepsilon(x^\varepsilon_1)=1, \qquad \qquad \Phi_\varepsilon(x^\varepsilon_2)=-1, \end{equation} where \begin{equation}\label{eq:x-eps}
x^\varepsilon_1=\varepsilon \bar x_1+o(\varepsilon), \qquad \qquad x^\varepsilon_2=-\varepsilon \bar x_2+o(\varepsilon), \end{equation} for some $\bar x_i>0$, for $i=1,2$ which do not depend on $\varepsilon$. \item[(ii)] If $\theta=2$, then $\Phi_\varepsilon$ satisfies \eqref{eq:stand-teta>2} with the following {\it exponential decay}: \begin{equation*}
\begin{aligned}
&1-\Phi_\varepsilon(x)\leq c_1 e^{-c_2x}, \qquad &\mbox{as } x\to+\infty,\\
&\Phi_\varepsilon(x)+1\leq c_1 e^{c_2x}, &\mbox{as } x\to-\infty,
\end{aligned} \end{equation*} for some $c_1,c_2>0$. \item[(iii)] If $\theta>2$, then $\Phi_\varepsilon$ satisfies \eqref{eq:stand-teta>2} with {\it algebraic decay}: \begin{equation*}
\begin{aligned}
&1-\Phi_\varepsilon(x)\leq d_1 x^{-d_2}, \qquad &\mbox{as } x\to+\infty,\\
&\Phi_\varepsilon(x)+1\leq d_1x^{-d_2}, &\mbox{as } x\to-\infty,
\end{aligned} \end{equation*} for some $d_1,d_2>0$. \end{itemize} \end{prop}
\begin{proof} In order to prove the existence of a solution to \eqref{eq:Fi}, we multiply the ordinary differential equation by $\Phi'_\varepsilon=\Phi'_\varepsilon(x)$, deducing \begin{equation*}
\varepsilon^2Q'(\varepsilon^2\Phi'_\varepsilon)\Phi'_\varepsilon\Phi''_\varepsilon-F'(\Phi_\varepsilon)\Phi'_\varepsilon=0, \qquad\qquad \mbox{in }\, (-\infty,+\infty). \end{equation*} As a consequence, \begin{equation}\label{eq:SW}
\begin{cases}
P_\varepsilon(\Phi'_\varepsilon)=F(\Phi_\varepsilon), \qquad \qquad \mbox{in }\, (-\infty,+\infty),\\
\Phi_\varepsilon(0)=0,
\end{cases} \end{equation} where $P_\varepsilon$ is defined in \eqref{def:P}. In order to solve the Cauchy problem \eqref{eq:SW}, we apply Lemma \ref{lemma:new}; hence, we need to require $F(\Phi_\varepsilon) \leq \ell \varepsilon^{-2}$, namely we choose \begin{equation}\label{eq:maxF}
\varepsilon\in(0,\varepsilon_0), \qquad \mbox{ with } \qquad \varepsilon_0=\sqrt{\frac{\ell}{\displaystyle\max_{\Phi\in[-1,1]}F(\Phi)}}. \end{equation} Condition \eqref{eq:maxF} ensures that we can find monotone solutions to \eqref{eq:SW} applying the standard method of separation of variables; in particular, we obtain a solution $\Phi_\varepsilon$, satisfying $\Phi'_\varepsilon \in [0,\kappa \varepsilon^{-2}]$, which is implicitly defined by
\begin{equation*}
\int_0^{\Phi_\varepsilon(x)} \frac{du}{J_\varepsilon\left(F(u)\right) }= x,
\end{equation*} where $J_\varepsilon$ is defined in \eqref{eq:J_e}. Since $J_\varepsilon\left(F(u)\right)=0$ if and only if $F(u)=0$ (that is, $u= \pm 1$), in order to prove the uniqueness of $\Phi_\varepsilon$ and its behavior described in the properties (i), (ii) and (iii) of the statement, we need to study the convergence of the following improper integrals \begin{equation}\label{int_SW}
\int_{0}^1 \frac{du}{J_\varepsilon\left( F(u)\right)} \qquad \mbox{ and } \qquad \int_{-1}^0 \frac{du}{J_\varepsilon\left( F(u)\right)}. \end{equation} Substituting \eqref{eq:J-important} in the first integral of \eqref{int_SW}, we end up with \begin{equation}\label{int_SW2}
\int_{0}^1 \frac{du}{J_\varepsilon\left( F(u)\right)}=\varepsilon\bar x_1+I(\varepsilon), \end{equation} where \begin{align*}
\bar x_1&:=\sqrt{\frac{Q'(0)}{2}}\int_0^1 \frac{du}{\sqrt{F(u)}}\\
I(\varepsilon)&:=-\varepsilon^{-1}\sqrt{\frac{Q'(0)}{2}}\int_0^1\frac{\rho(\varepsilon^2F(u))}{\left(\sqrt{\frac{2}{\varepsilon^2Q'(0)} F(u)} +\varepsilon^{-2}\rho(\varepsilon^2F(u))\right)\sqrt{F(u)}}\,du. \end{align*} The crucial point is that the character of the integral in \eqref{int_SW2} is simply given by $\bar x_1$,
since there exists $\varepsilon_0>0$ such that $|I(\varepsilon)|<\infty$ for any $\varepsilon\in(0,\varepsilon_0)$.
Indeed, using the estimate $|\rho(\varepsilon^2F(u))| \leq C \varepsilon^2 F(u)$, one gets \begin{align*}
\int_0^1 \frac{|\rho(\varepsilon^2F(u))|}{\left(\sqrt{\frac{2}{\varepsilon^2Q'(0)} F(u)} +\varepsilon^{-2}\rho(\varepsilon^2F(u))\right)\sqrt{F(u)}}\,du &\leq
\int_0^1 \frac{C \varepsilon^2\sqrt{F(u)}}{\left(\sqrt{\frac{2}{\varepsilon^2Q'(0)} F(u)} +\varepsilon^{-2}\rho(\varepsilon^2F(u))\right)}\,du \\
& \leq \int_0^1 \frac{C \varepsilon^3}{\left(\sqrt{\frac{2}{Q'(0)} }+\frac{\rho(\varepsilon^2F(u))}{\varepsilon\sqrt{F(u)}}\right)}\,du, \end{align*} and we can choose $\varepsilon>0$ sufficiently small such that \begin{equation*}
\sqrt{\frac{2}{Q'(0)} } +\frac{\rho(\varepsilon^2F(u))}{\varepsilon\sqrt{F(u)}} \geq \sqrt{\frac{2}{Q'(0)} } -C\varepsilon\sqrt{F(u)} >0. \end{equation*} Moreover, notice that $\bar x_1$ does not depend on $\varepsilon$, while $I(\varepsilon)=o(\varepsilon)$. By using \eqref{eq:ass-F3}, we obtain \begin{equation*}
\bar x_1 \sim \int_0^1 \frac{du}{(1-u)^{\frac{\theta}{2}}}. \end{equation*} Hence, $\bar x_1 < +\infty$ if and only if $\theta< 2$, and the point (i) of the thesis follows; the first equality in \eqref{eq:x-eps} is a consequence of \eqref{int_SW2}. Going further, points (ii)-(iii) of the statement are a consequence of the standard theory of ODE applied to \eqref{eq:SW}, together with the fact that
$$J_\varepsilon(F(s)) \approx |1-s|^{\frac{\theta}{2}}, \qquad s \approx 1.$$ The computations are completely similar if considering the second integral in \eqref{int_SW}, and we thus proved the existence of a unique solution of \eqref{eq:Fi} satisfying properties (i), (ii) and (iii). Precisely, if $\theta\geq2$ there exists a unique solution of \eqref{eq:SW}, while if $\theta\in(1,2)$, \eqref{eq:SW} has infinitely many solutions, but the additional requirement $\Phi_\varepsilon'(x)\geq0$, for any $x\in\mathbb{R}$, guarantees that there is a unique solution of \eqref{eq:Fi}. \end{proof}
\begin{rem}We notice that the condition $\Phi'_\varepsilon \in [0,\kappa \varepsilon^{-2}]$ allows also for high values of the first derivative; to be more precise, one has \begin{equation}\label{eq:Fi-firstderivative}
|\Phi_\varepsilon'(x)|\leq \kappa\varepsilon^{-2}, \qquad \qquad \mbox{ for any } x\in\mathbb{R}. \end{equation} \end{rem}
As a corollary of Proposition \ref{ESW}, we can prove existence of a unique solution to \eqref{eq:Fi-dec}, sharing similar properties to (i)-(ii) and (iii). \begin{cor}\label{cor:standwavedecr} Under the same assumptions of Proposition \ref{ESW}, there exists $\varepsilon_0>0$ such that problem \eqref{eq:Fi-dec} admits a unique solution $\Psi_\varepsilon$ for any $\varepsilon \in (0,\varepsilon_0)$. Moreover, if $\theta\in(1,2)$, then \begin{equation*}
\Psi_\varepsilon(-x^\varepsilon_1)=1, \qquad \qquad \Psi_\varepsilon(-x^\varepsilon_2)=-1, \end{equation*} where $x^\varepsilon_i$, $i=1,2$ are defined in \eqref{eq:x-eps}. On the other hand, if $\theta=2$ ($\theta>2$) the profile $\Psi_\varepsilon$ has an exponential (algebraic) decay towards the states $\mp1$. \end{cor} \begin{proof} Using the symmetry of $Q$ and, in particular, the fact that $Q'$ is an even function, it is a simple exercise to verify that $\Psi_\varepsilon(x):=\Phi_\varepsilon(-x)$, with $\Phi_\varepsilon$ given by Proposition \ref{ESW}, is the unique solution to \eqref{eq:Fi-dec}. \end{proof}
\begin{rem} The existence of a unique solution $\Psi_\varepsilon$ to \eqref{eq:Fi-dec} can be proven independently on the one of $\Phi_\varepsilon$: indeed, it is enough to adapt the proof of Proposition \ref{ESW} by inverting the equation $P_\varepsilon(s)=\xi$ in the interval $[-\kappa\varepsilon^{-2},0]$ (in this case the inverse is $-J_\varepsilon$, see \eqref{eq:J_e}), and obtaining the existence of a unique solution $\Psi_\varepsilon$ with negative derivative $\Psi'_\varepsilon \in [-\kappa \varepsilon^{-2},0]$. \end{rem}
The previous results are instrumental to prove the existence of a {\it special} class of stationary solutions on a bounded interval in the case $\theta \in (1,2)$, as we will see in the next section. Indeed, in such a case the standing waves reaches $\pm 1$ for a finite value of the $x$-variable and with zero derivative, so that we are able to construct infinitely many steady states oscillating between $\pm 1$ and satisfying the boundary conditions \eqref{eq:Neu}. On the contrary, if $\theta \geq 2$, all the standing waves satisfy \eqref{eq:stand-teta>2}, so that they never satisfy the homogeneous Neumann boundary conditions and can never solve \eqref{eq:staz}-\eqref{eq:Neu} in any bounded interval.
\subsection{Compactons} We here consider the so-called \emph{compactons}, which are by definition stationary solutions connecting two phases on a finite interval. More explicitly, we prove the existence of infinite solutions to the stationary problem associated to \eqref{eq:Q-model}-\eqref{eq:Neu}, namely \begin{equation}\label{eq:comp}
Q(\varepsilon^2 \varphi')'-F'(\varphi)=0, \qquad \qquad \varphi'(a)=\varphi'(b)=0, \end{equation} oscillating between $-1$ and $+1$ (touching them), provided that $\varepsilon$ is sufficiently small. The existence of such solutions is shown by proving that, given an arbitrary set of real numbers in $[a,b]$, for sufficiently small $\varepsilon$, there are two solutions $\varphi_1$ and $\varphi_2$ to \eqref{eq:comp} having such numbers as zeros, satisfying \begin{equation}\label{varphi12}
\varphi_1(a)=-1 \qquad\quad \mbox{ and } \quad \qquad \varphi_2(a)=+1, \end{equation} and oscillating between $-1$ and $+1$ ( $+1$ and $-1$, respectively).
\begin{prop}\label{prop:comp} Let $1<\theta<2$, $N\in \mathbb{N}$ and let $h_1,h_2,\dots, h_N$ be any $N$ real numbers such that $a<h_1<h_2<\cdots <h_N<b$. Then, for any $\varepsilon\in(0,\bar\varepsilon)$ with $\bar\varepsilon>0$ sufficiently small, there exist two solutions $\varphi_1$ and $\varphi_2$ to \eqref{eq:comp} satisfying \eqref{varphi12}, oscillating between $-1$ and $+1$ and between $+1$ and $-1$ respectively, and having precisely $N$ zeros at $h_1,h_2,\dots, h_N$. \end{prop}
\begin{proof} We start by proving the existence of the solution $\varphi_1$ to \eqref{eq:comp} on the interval $[a,b]$ satisfying the first condition in \eqref{varphi12}, oscillating between $-1$ and $+1$ and having $h_1,h_2,\dots, h_N$ as zeros. To this aim, we consider the function $$\Phi_{\varepsilon}^1(x):=\Phi_\varepsilon(x-h_1), \qquad \, x\in\mathbb{R},$$ where $\Phi_\varepsilon$ is the increasing standing wave solution of Proposition \ref{ESW}. Then, the function $\Phi_{\varepsilon}^1$ has a zero at $h_1$. Furthermore, by \eqref{eq:Fi} and \eqref{ugualea1e-1}, recalling that \begin{equation*}
x_1^\varepsilon =\varepsilon \bar{x}_1+o(\varepsilon) \quad \text{and}\quad x_2^\varepsilon =-\varepsilon \bar{x}_2+o(\varepsilon), \end{equation*} for some $\bar{x}_1, \bar{x}_2>0$, we can conclude that $\Phi_{\varepsilon}^1$ takes the values $-1$ on $(-\infty, h_1+x_2^\varepsilon]$ and $+1$ on $[h_1+x_1^\varepsilon, +\infty)$; we notice that if $\varepsilon$ is sufficiently small, then $h_1+x_2^\varepsilon <h_1$. Let us now fix $\varepsilon>0$ small enough so that $h_1+x_2^\varepsilon >a$; the restriction of $\Phi_{\varepsilon}^1$ to the interval $[a,h_1+x_1^\varepsilon]$, denoted with the same symbol, turns out to be equal to $-1 $ for every $x\in [a,h_1+x_2^\varepsilon]$, touching $+1$ at $h_1+x_1^\varepsilon$. Let us introduce the notation \begin{equation*} y_j^\varepsilon:= \begin{cases} x_1^\varepsilon \qquad & \text{if $j$ is odd}, \\ -x_2^\varepsilon & \text{if $j$ is even}, \\ \end{cases} \end{equation*} for every $j=1,\dots, N$, and for every $i=1,\dots, N-1$ let us define the function \begin{equation*}
\Phi_{\varepsilon}^{i+1}(x):=\Phi_\varepsilon((-1)^{i}(x-h_{i+1})), \qquad x\in \mathbb{R}. \end{equation*} For each fixed $i=1,\dots, N-1$, the function $\Phi_{\varepsilon}^{i+1}$ has a zero at $h_{i+1}$, and takes the values $-1$ for every $x \geq h_{i+1}+y_{i+1}^\varepsilon$ and $+1$ for every $x\leq h_{i+1}-y_{i}^\varepsilon$ as long as $i$ is odd, otherwise it takes the values $-1$ for every $x \leq h_{i+1}-y_{i}^\varepsilon$ and $+1$ for every $x\geq h_{i+1}+y_{i+1}^\varepsilon$. Up to choosing $\varepsilon$ possibly smaller in order to have $h_i+ y_i^\varepsilon \leq h_{i+1} -y_i^\varepsilon$, for every $i=1,\dots, N-1$ (namely $2y_i^\varepsilon \leq h_{i+1} -h_i $), the restriction of the function $\Phi_{\varepsilon}^{i+1}(x)$ to the interval $[h_i+y_i^\varepsilon, h_{i+1}+y_{i+1}^\varepsilon ]$ (still using the same notation), takes the value $(-1)^{i+1}$ for every $x\in [h_i+y_i^\varepsilon, h_{i+1}-y_i^\varepsilon]$, touching $(-1)^i$ at $h_{i+1}+y_{i+1}^\varepsilon$. Selecting $\varepsilon>0$ sufficiently small so that $h_N+y_N^\varepsilon< b$, we end up defining $\varphi_1$ in the following way \begin{equation*} \varphi_1(x):= \begin{cases} \Phi_\varepsilon^1(x), \qquad & x\in [a,h_1+x_1^\varepsilon], \\ \Phi_\varepsilon^{i+1}(x), & x\in [h_i+y_i^\varepsilon, h_{i+1}+y_{i+1}^\varepsilon]\quad \text{ and }i=1,\dots,N-1, \\ (-1)^{N-1}, & x\in [h_N+y_N^\varepsilon,b]. \end{cases} \end{equation*} {By construction}, the resulting map {$\varphi_1\in C^2([a,b])$} solves \eqref{eq:comp} and has exactly $N$ zeros at points $h_1,h_2,\dots, h_N$.
Arguing as above, we can construct the compacton $\varphi_2$ which satisfies the second condition in \eqref{varphi12} in the following way
\begin{equation*} \varphi_2(x):= \begin{cases} \Phi_\varepsilon^1(-x), \qquad & x\in [a,h_1-x_2^\varepsilon], \\ \Phi_\varepsilon^{i+1}((-1)^{i+1}x), & x\in [h_i+y_{i+1}^\varepsilon, h_{i+1}+y_i^\varepsilon]\quad \text{ and } i=1,\dots,N-1, \\ (-1)^{N}, & x\in [h_N+y_{N-1}^\varepsilon,b]. \end{cases} \end{equation*} The proof is thereby complete. \end{proof}
We, again, point out that since the integrals in \eqref{int_SW} are finite only if $\theta \in(1,2)$, the compactons solutions constructed in Proposition \ref{prop:comp} exist only in such a case.
\subsection{Periodic solutions for $\theta \geq 2$} As already mentioned in the previous section, when $\theta \geq 2$ the integrals in \eqref{int_SW} diverge, so that solutions to \eqref{eq:comp} cannot touch the values $\pm 1$ and compactons solutions do not exist anymore. In this case, we construct a different type of stationary solutions with a transition layer structure, which can be seen as a restriction of {\it periodic solutions} on the whole real line. The study of all periodic solutions to \eqref{eq:staz} is beyond the scope of the paper and it is strictly connected to the specific form of the potential $F$ which, in our case, is a very generic function satisfying \eqref{ipoF1}-\eqref{ipoF2} and may give raise to infinitely many kinds of periodic solutions in the whole real line, see two examples in Figures \ref{fig3}-\ref{fig4} below. Indeed, assumptions \eqref{ipoF1}-\eqref{ipoF2} only assure that $F$ is a double well potential with wells of equal depth in $\pm 1$, and describe its behavior close to these minimal points, while give no informations of the shape of $F$ between them. Here, we are interested in periodic solutions oscillating between values close to $\pm1$ and assumptions \eqref{ipoF1}-\eqref{ipoF2} are enough to prove their existence. Denote by \begin{equation}\label{eq:Gamma}
\mathcal{Z}:=\{u\in(-1,1)\, :\, F'(u)=0\} \qquad \mbox{ and } \qquad \Gamma:=\min_{\mathcal Z}F(u)>0. \end{equation} As it was mentioned before, assumptions \eqref{ipoF1}-\eqref{ipoF2} give no information on the structure of $\mathcal{Z}$, that is the set of all the critical points of $F$ inside $(-1,1)$, that could be a discrete set or even an interval. Multiplying \eqref{eq:staz} by $\varphi'$, we deduce that \begin{equation}\label{eq:periodic}
P_\varepsilon(\varphi')-F(\varphi)= C, \qquad \qquad \mbox{ in } \mathbb{R}, \end{equation} where $C$ is an appropriate integration constant. For instance, if $C=0$ we obtain the constant solutions $\pm1$ or the standing wave constructed in Proposition \ref{ESW}; in the next result, we prove that the choice $C\in\left(-\Gamma,0\right)$ gives raise to periodic (bounded) solutions.
\begin{prop}\label{periodicR} Let $\varepsilon>0$, $P_\varepsilon$ as in \eqref{def:P}, $F$ satisfying \eqref{ipoF1}-\eqref{ipoF2} with $\theta\geq2$ and $C\in\left(-\Gamma,0\right)$, where $\Gamma$ is defined in \eqref{eq:Gamma}. Denote by $\bar s\in(0,1)$ the unique number such that $-F(\bar s)=C$. Then there exists $\varepsilon_0>0$ such that, for any $\varepsilon\in(0,\varepsilon_0)$, problem \eqref{eq:periodic} admits periodic solutions $\Phi_{T_\varepsilon}$, oscillating between $-\bar s$ and $\bar s$, with fundamental period $2T_\varepsilon$, where \begin{equation}\label{period}
T_\varepsilon(\bar s):=\int_{-\bar s}^{\bar s} \frac{ds}{J_\varepsilon\left( F(s)-F(\bar s)\right)}, \end{equation} with $J_\varepsilon$ defined in \eqref{eq:J_e}. \end{prop}
\begin{proof} Fix $C\in\left(-\Gamma,0\right)$ and notice that the assumption \eqref{ipoF1} together with definition \eqref{eq:Gamma} imply the existence of a unique $s\in(0,1)$ such that $-F(\bar s)=C$. Since any periodic solution is unique up to translation we can assume, without loss of generality, that $\Phi_{T_\varepsilon}(0)=-\bar s$. By using \eqref{eq:J_e}, from equation \eqref{eq:periodic} we can infer that $\Phi_{T_\varepsilon}$ is implicitly defined as \begin{equation*}
\int_{-\bar s}^{\Phi_{T_\varepsilon(x)}} \frac{du}{J_\varepsilon\left( F(u)+C\right)}=x, \end{equation*} provided that $F(u)+C \in [0,\ell\varepsilon^{-2}])$, where $\ell$ is defined in \eqref{def:ell}. Hence, we need to require \begin{equation*}
\max_{u \in [-\bar s,\bar s]} F(u)+C \leq \ell \varepsilon^{-2}, \end{equation*} and, since $C<0$, such condition is satisfied again as soon as \eqref{eq:maxF} holds. We now have to verify the convergence of the improper integral \begin{equation}\label{eq:int-period}
\int_{-\bar s}^{\bar s} \frac{ds}{J_\varepsilon\left(F(s)-F(\bar s)\right)}. \end{equation}
By using \eqref{eq:J-important} and the Taylor expansion $F(s)-F(\bar s)= F'(\bar s)(s-\bar s)+o\left(|s-\bar s|\right)$, with $F'(\bar s)<0$ because $\bar s\in(0,1)$, we deduce \begin{equation*}
\int_{0}^{\bar s} \frac{ds}{J_\varepsilon\left( F(s)-F(\bar s)\right)}\sim
\int_{0}^{\bar s} \frac{ds}{\sqrt{F'(\bar s)(s-\bar s)}}<+\infty. \end{equation*} Similarly, one can prove that \begin{equation*}
\int_{-\bar s}^{0} \frac{ds}{J_\varepsilon\left( F(s)-F(\bar s)\right)}\sim
\int_{-\bar s}^{0} \frac{ds}{\sqrt{F'(-\bar s)(s+\bar s)}}<+\infty, \end{equation*} and, as a consequence, the improper integral \eqref{eq:int-period} is finite. Therefore, we have constructed a solution $\Phi_{T_\varepsilon}:[0, T_\varepsilon(\bar s)]$, satisfying $$\Phi_{T_\varepsilon}(0)=-\bar s, \quad \Phi_{T_\varepsilon}\left({T_\varepsilon(\bar s)}\right)=\bar s \qquad \mbox{and} \qquad \Phi'_{T_\varepsilon}(0)=\Phi'_{T_\varepsilon}\left({T_\varepsilon(\bar s)}\right)=0.$$ Let us now define $$\Phi_{T_\varepsilon}(x)=\Phi_{T_\varepsilon}\left(T_\varepsilon(\bar s)-x\right), \qquad \qquad x\in\left[{T_\varepsilon(\bar s)},2T_\varepsilon(\bar s)\right].$$ It is easy to check that $\Phi_{T_\varepsilon}$ solves \eqref{eq:periodic} in $\left[{T_\varepsilon(\bar s)},2T_\varepsilon(\bar s)\right]$ and $$\Phi_{T_\varepsilon}\left(T_\varepsilon(\bar s) \right)=-\bar s, \qquad \Phi'_{T_\varepsilon}\left(T_\varepsilon(\bar s) \right)=0.$$ We have thus extended the solution in the interval $\left[{T_\varepsilon(\bar s)},2T_\varepsilon(\bar s)\right]$, and thus constructed a solution in $\left[0,2T_\varepsilon(\bar s)\right]$; by iterating the same argument, we can extend the solution to the whole real line by $2T_\varepsilon(\bar s) - $periodicity, and the proof is complete. \end{proof}
We now make use of the solutions constructed in Proposition \ref{periodicR} to construct solutions to \eqref{eq:comp} having $N$ equidistant zeroes in $[a,b]$.
\begin{prop}\label{periodic:bounded} Let $\varepsilon>0$, $Q$ satisfying \eqref{eq:Q-ass1}-\eqref{eq:Q-ass2}, $F$ satisfying \eqref{ipoF1}-\eqref{ipoF2} with $\theta\geq2$ and let us fix $N \in \mathbb{N}$. Then, if $\varepsilon\in(0,\bar\varepsilon)$ with $\bar\varepsilon>0$ sufficiently small, there exists $\bar s$ close to (and strictly less than) $+1$, such that problem \eqref{eq:comp} admits a solution oscillating between $\pm\bar s$ and with $N$ equidistant zeroes inside the interval $[a,b]$, located at \begin{equation}\label{locationzeros}
h_1= \frac{T_\varepsilon(\bar s)}{2}+a \qquad \mbox{and} \qquad h_i= h_{i-1}+T_\varepsilon(\bar s), \quad i=2, \dots, N, \end{equation} where $T_\varepsilon(\bar s)$ is defined in \eqref{period}. \end{prop}
\begin{proof} The solution we are looking for can be constructed by shifting and modifying properly $\Phi_{T_\varepsilon}$, the periodic solution of Proposition \ref{periodicR}. Hence, in order to construct a solution $\psi$ with exactly $N$ zeroes in $[a,b]$ such that $\psi'(a)=\psi'(b)=0$, we proceed as follows: first of all, we define $\psi(x)= \Phi_{T_\varepsilon}(x-a)$ (recall that $\Phi_{T_\varepsilon}(0)=-\bar s$). In such a way $\psi(a)=-\bar s$, and we also have $\psi'(a)=0$. Moreover, the first zero $h_1$ of $\psi$ is located exactly at $T_\varepsilon/2+a$, while $h_2=T_\varepsilon + h_1$, $h_3= T_\varepsilon+h_2$ and so on, leading to \eqref{locationzeros}. Thus, in order to have $b$ located in the middle point after the last zero of $\psi$ (so that, consequently, $\psi'(b)=0$), we have to choose $\bar s$ in such a way that $b=N T_\varepsilon(\bar s)+a$, if $\varepsilon\in(0,\bar\varepsilon)$. In other words, we have to prove that, if $\varepsilon\in(0,\bar\varepsilon)$, for any $N \in \mathbb{N}$ and $a,b \in \mathbb{R}$ there exists $\bar s\approx1$ such that \begin{equation*}
T_\varepsilon(\bar s)= \frac{b-a}{N}. \end{equation*} To this purpose, by putting the expansion \eqref{eq:J-important} in the definition \eqref{period}, we infer $$T_\varepsilon(\bar s)=\varepsilon\sqrt{\frac{Q'(0)}{2}}\int_{-\bar s}^{\bar s}\frac{ds}{\sqrt{F(s)-F(\bar s)}}+o(\varepsilon).$$ Thus, we have to find $\bar s\approx1$ such that \begin{equation}\label{eq:inutile}\int_{-\bar s}^{\bar s}\frac{ds}{\sqrt{F(s)-F(\bar s)}}=\frac{\sqrt2(b-a)}{\varepsilon \sqrt{Q'(0)}N}+\frac{o(\varepsilon)}{\varepsilon}. \end{equation} Since the integral on the left hand side is a monotone function of $\bar s\approx1$ \cite{Carr-Pego} and it satisfies $$\lim_{\bar s \to 1^- }\int_{-\bar s}^{\bar s}\frac{ds}{\sqrt{F(s)-F(\bar s)}}= +\infty,$$ we can state that if $\varepsilon$ is sufficiently small, then there exists a unique $\bar s\approx1$ such that $T_\varepsilon(\bar s)= (b-a)/N$, and the proof is complete. \end{proof}
\begin{rem} We point out that in the case $\theta\in(1,2)$ we proved existence of compactons with a generic number $N\in\mathbb{N}$ of layers located at arbitrary positions $h_1,h_2,\dots,h_N$ in the interval $[a,b]$. On the other hand, in Proposition \ref{periodic:bounded}, we proved that, for $\theta\geq 2$, there exist solutions with $N$ layers, which oscillate among the values $\pm \bar s$, with $\bar s\approx1$ determined by the condition $T_\varepsilon(\bar{s})= (b-a)/N$, but the layers position is given by \eqref{locationzeros}, and so it is not random. Nevertheless, such result can be proved also when $\theta \in (1,2)$, but only if $\varepsilon$ is large enough: indeed, one has $$\lim_{\bar s \to 1^-} \int_{-\bar s}^{\bar s} \frac{ds}{\sqrt{ F(s)-F(\bar s)}} < +\infty \qquad \mbox{if} \qquad \theta <2,$$ and one can easily see that the right hand side of \eqref{eq:inutile} diverges if $\varepsilon \to 0^+$.
Finally, notice that we proved that for $\varepsilon\in(0,\bar\varepsilon)$ there exists a unique $\bar s$ (depending on $\varepsilon$) such that $T_\varepsilon(\bar s)= (b-a)/N$ and, in particular, $\displaystyle{\lim_{\varepsilon\to0^+}}\bar s=1,$ meaning that, the more $\varepsilon$ is small, the closer $\bar s$ is to $1$. \end{rem}
\subsubsection*{Particular cases of potential $F$.} In order to give a hint of what can happens for particular choices of the potential $F$ satisfying \eqref{ipoF1}-\eqref{ipoF2}, we consider two specific examples. In the first one, $F$ is given by \eqref{F:ex} with $\theta\geq2$ and, as a consequence, the admissible levels of the energy that lead to periodic (bounded) solutions are $C \in \left(-\frac{1}{2\theta},0\right)$, see Figure \ref{fig3}.
\begin{figure}\label{fig3}
\label{fig3}
\end{figure}
On the other hand, let us consider a symmetric potential $F$ with a local minimum located in $u=0$ and, as a consequence, two local maxima located in $\pm \bar u$, for some $\bar u \in (0,1)$, see Figure \ref{fig4}, where $-F$ is depicted; in this case,
the periodic solutions described in Proposition \ref{periodicR} appear for $C \in \left( -F(0), 0\right)$ (hence with $\Gamma=F(0)$ in \eqref{eq:Gamma}), see the green line in Figure \ref{fig4}. Moreover, if $C= -F(0)$, homoclinic solutions appear (see the blue line in Figure \ref{fig4}), while for $C \in \left( -F(\bar u), F(0)\right)$ one can construct new periodic solutions entirely contained either in the negative or in the positive half plane.
Of course the case of a non-symmetric potential will be even more difficult (for instance, one will have several level of the energy corresponding to different homoclinic solutions), and this study will be the object of further investigations.
\begin{figure}
\caption{\small{In blue we depicted the level of the energy corresponding to the homoclinic solution; in dark green the level of the energy corresponding to the periodic solutions oscillating, respectively, among $-\bar s_2, -\bar s_1 <0$ and $\bar s_1, \bar s_2 >0$. }}
\label{fig4}
\label{fig4}
\end{figure}
\section{Variational results}\label{sec:energy} In this section we collect and prove some variational results needed in order to show the slow motion phenomena of the solutions to \eqref{eq:Q-model}-\eqref{eq:Neu} in the case $\theta \geq 2$, whose analysis will be performed in Section \ref{sec:slow}. The idea is to apply the strategy firstly developed by Bronsard and Kohn in \cite{Bron-Kohn}, subsequently improved by Grant in \cite{Grant} and successfully used in many other models, see for instance \cite{FPS,FPS-DCDS,FS} and references therein.
\subsection{Lyapunov functional}\label{Lyapunov} We start by introducing the energy associated to \eqref{eq:Q-model}-\eqref{eq:Neu} \begin{equation}\label{eq:energy}
E_\varepsilon[u]:=\int_a^b \left[\frac{\tilde{Q}(\varepsilon^2u_x)}{\varepsilon^3}+\frac{F(u)}{\varepsilon}\right]\, dx, \end{equation} where $\tilde Q$ is defined in \eqref{def:ell}; we first prove that \eqref{eq:energy} is a Lyapunov functional for the model \eqref{eq:Q-model}-\eqref{eq:Neu}, that is a functional whose time derivative is negative if computed along the solutions to \eqref{eq:Q-model}-\eqref{eq:Neu}.
\begin{lem}\label{lem:energy} Let $u\in C([0,T],H^2(a,b))\cap C^1([0,T],H^1(a,b))$ be a solution of \eqref{eq:Q-model}-\eqref{eq:Neu}. Let $E_\varepsilon$ be the functional defined in \eqref{eq:energy}. Then \begin{equation}\label{eq:energyestimate1}
\frac{d}{dt}E_\varepsilon[u](t)=-\varepsilon^{-1}\int_a^b u^2_t(x,t)\, dx. \end{equation} and \begin{equation}\label{eq:energyestimate}
E_\varepsilon[u](0)-E_\varepsilon[u](T)=\varepsilon^{-1}\int_0^T\!\int_a^bu_t(x,t)^2\,dx \,dt. \end{equation} \end{lem} \begin{proof} \eqref{eq:energyestimate} directly follows from \eqref{eq:energyestimate1}: indeed once \eqref{eq:energyestimate1} is proved, an integration with respect to time over the interval $[0,T]$ yields \eqref{eq:energyestimate}.
As for the proof of \eqref{eq:energyestimate1}, by differentiating with respect to time the energy $E_\varepsilon$, we have \begin{equation*}
\frac{d}{dt}E_\varepsilon[u](t)=\frac1\varepsilon\int_a^b\left[Q(\varepsilon^2 u_x)u_{xt}+F'(u)u_t\right]\,dx, \end{equation*} where we have used that $\tilde Q'(\varepsilon^2 u_x)=Q(\varepsilon^2 u_x)$. Integrating by parts, exploiting the boundary conditions \eqref{eq:Neu} and that $Q(0)=0$, we deduce \begin{equation*}
\frac{d}{dt}E_\varepsilon[u](t)=\frac1\varepsilon\int_a^b\left[-Q(\varepsilon^2 u_x)_x+F'(u)\right]u_t\,dx. \end{equation*} From this, since $u$ satisfies equation \eqref{eq:Q-model}, we end up with \eqref{eq:energyestimate1}, thus completing the proof.
\end{proof} \subsection{Lower bounds.} The aim of this subsection is to prove some lower bounds for the energy $E_\varepsilon$, defined in \eqref{eq:energy}, associated to a function which is sufficiently close in $L^1$-sense to a jump function with constant values $-1$ and $+1$ (we refer the reader to Definition \ref{vstruct}). Such variational results present a different nature depending on either $\theta=2$ or $\theta>2$; moreover, we underline that in their proof the equation \eqref{eq:Q-model} does not come into play, unlike the result contained in Subsection \ref{Lyapunov}.
\subsubsection{A crucial inequality} The first tool we need to prove the aforementioned lower bounds is an inequality involving the functions $Q$ and $J_\varepsilon$. To better understand the motivation behind such a tool, we recall the equation which identifies the standing waves solutions, that is \begin{equation*}
P_\varepsilon(\Phi'_\varepsilon)= F(\Phi_\varepsilon), \end{equation*} which in turn, using \eqref{defalternativa}, can be rewritten as follows \begin{equation}\label{uguag}
\Phi'_\varepsilon Q(\varepsilon^2 \Phi'_\varepsilon) - \frac{1}{\varepsilon^2}\tilde Q (\varepsilon^2 \Phi'_\varepsilon)= F(\Phi_\varepsilon). \end{equation} We now observe that \eqref{eq:J_e} implies $\Phi'_\varepsilon=J_\varepsilon (F(\Phi_\varepsilon))$, so that substituting into \eqref{uguag}, we arrive at \begin{equation}\label{eq:impforenergy}
\frac{\tilde Q (\varepsilon^2 \Phi'_\varepsilon)}{\varepsilon^3} + \frac{F(\Phi_\varepsilon)}{\varepsilon} = \frac{\Phi'}{\varepsilon} Q(\varepsilon^2J_\varepsilon (F(\Phi_\varepsilon))). \end{equation} Therefore, we seek a suitable inequality such that in some sense the equality holds along the standing wave solutions. Inspired by the previous considerations, we state and prove the following lemma. \begin{lem} Let $\varepsilon,L>0$. If $Q\in C^1(\mathbb{R})$ satisfies \eqref{eq:Q-ass1}-\eqref{eq:Q-ass2} and $J_\varepsilon$ satisfies \eqref{eq:J_e}, then \begin{equation}\label{eq:strangeineq+}
\frac{\tilde Q (\varepsilon^2 x)}{\varepsilon^3}+\frac{y}{\varepsilon}\geq \frac{|x|}{\varepsilon} \, Q(\varepsilon^2J_\varepsilon (y)), \end{equation} for any $(x,y)\in[-\kappa \varepsilon^{-2}, \kappa \varepsilon^{-2}]\times[0,\ell \varepsilon^{-2}]$, where $\kappa$ and $\ell$ are defined in \eqref{eq:Q-ass2} and \eqref{def:ell}, respectively. \end{lem} \begin{proof} Since, by assumption, $\tilde Q$ is even, in order to prove \eqref{eq:strangeineq+} it is sufficient to study the sign of the function \begin{equation*}
g(x,y):=\tilde Q (\varepsilon^2 x)+\varepsilon^2 y -\varepsilon^2 x \, Q(\varepsilon^2 J_\varepsilon (y)), \end{equation*} for all $ x \in [0,\kappa \varepsilon^{-2}]$ and for all $ y \in [0,\ell\varepsilon^{-2}]$. For any $(x,y)$ belonging to the inside of such a rectangle we have \begin{align*}
& g_x(x,y)=\varepsilon^2 \, \tilde Q'(\varepsilon^2 x)-\varepsilon^2 \, Q(\varepsilon^2 _\varepsilon J(y)) = 0 \qquad \mbox{ if and only if } \qquad Q(\varepsilon^2 x)= Q(\varepsilon^2 J_\varepsilon (y)).
\end{align*} Since $Q$ is strictly increasing in the interval $[0,\kappa \varepsilon^{-2}]$, we thus have $g_x(x,y)=0$ if and only if $x=J_\varepsilon (y)$. Let us now evaluate $g_y$ in such points: \begin{align*}
g_y(x,y) \Big|_{x= J_\varepsilon (y)}&=\varepsilon^2 -\varepsilon^2 x \, Q'(\varepsilon^2 J_\varepsilon (y)) \, \varepsilon^2 J'_\varepsilon (y)\Big|_{x=J_\varepsilon (y)} \\
&=\varepsilon^2 -\varepsilon^4 x \, \frac{Q'(\varepsilon^2 J_\varepsilon (y))}{P'_\varepsilon\left( J_\varepsilon (y)\right)}\Big|_{x=J_\varepsilon (y)} \\
&= \varepsilon^2 -\frac{\varepsilon^2 x}{P_\varepsilon'(x)} \, Q'(\varepsilon^2 x) = \varepsilon^2 -\frac{\varepsilon^4 x}{\varepsilon^2 x \, Q'(\varepsilon^2 x)} \, Q'(\varepsilon^2 x)=0, \end{align*} where we used \eqref{def:P} and \eqref{eq:J_e}. It follows that the only {\it internal} critical points of the function $g$ are given by $x=J_\varepsilon (y)$, and we have \begin{align*}
g(x,y) \Big|_{x= J_\varepsilon (y)} &= \tilde Q(\varepsilon^2 x) + \varepsilon^2 P_\varepsilon(x) -\varepsilon^2 x \, Q(\varepsilon^2 x) \\
&=\tilde Q(\varepsilon^2 x) + \varepsilon^2 \left[ x \, Q(\varepsilon^2 x) -\frac{1}{\varepsilon^2} \tilde Q(\varepsilon^2 x)\right] -\varepsilon^2 x \, Q(\varepsilon^2 x) =0. \end{align*} Let us now study the function $g$ on the boundary of the rectangle $[0, \kappa \varepsilon^{-2}]\times[0,\ell \varepsilon^{-2}]$, which is formed by four segments; we start with \begin{equation*}
g(0,y)=\tilde Q(0)+\varepsilon^2 y \geq 0 \qquad \mbox{for all} \quad y\in[0,\ell \varepsilon^{-2}], \end{equation*} and \begin{equation*}
f(x,0)=\tilde Q(\varepsilon^2 x)-\varepsilon^2 x \, Q(\varepsilon^2J_\varepsilon (0)) =\tilde Q(\varepsilon^2 x) \geq 0 \qquad \mbox{for all} \quad x \in[0,\kappa\varepsilon^{-2}], \end{equation*} where we used the fact that $J_\varepsilon(0)=0$ and $Q(0)=0$. Next, for $y \in [0, \ell \varepsilon^{-2}]$ we consider the function \begin{equation*}
g_1(y):=g(\kappa \varepsilon^{-2},y)=\tilde Q (\kappa)+\varepsilon^2 y-\kappa Q(\varepsilon^2 J_\varepsilon (y)), \end{equation*} and we have \begin{align*}
g'_1(y)&=\varepsilon^2 -\kappa\varepsilon^2 Q'(\varepsilon^2 J_\varepsilon (y)) \frac{1}{P'_\varepsilon(J_\varepsilon (y))}\\
&=\varepsilon^2 -\frac{\kappa \varepsilon^2 Q'(\varepsilon^2 J_\varepsilon (y))}{\varepsilon^2 J_\varepsilon (y) Q'(\varepsilon^2 J_\varepsilon (y))} = \varepsilon^2 -\frac{\kappa}{J_\varepsilon (y)}. \end{align*} Recalling \eqref{def:ell}, we observe that the function $g_1$ is such that \begin{equation*}
g_1'(y) \leq 0 \quad \forall \ y \in [0, \ell \varepsilon^{-2}] \qquad \mbox{and} \qquad \lim_{y \to 0} g_1'(y) = -\infty. \end{equation*} Moreover $g_1(0)= \tilde Q(\kappa) >0$ while \begin{align*}
g_1(\ell \varepsilon^{-2}) = g_1 \left( P_\varepsilon(\kappa \varepsilon^{-2})\right) &= \tilde Q(\kappa) + \varepsilon^2 P_\varepsilon(\kappa \varepsilon^{-2}) - \kappa Q(\kappa) \\
&=\tilde Q(\kappa) + \ell - \kappa Q(\kappa) \\
&= \tilde Q(\kappa)+ \kappa Q(k) - \tilde Q(\kappa)- \kappa Q(\kappa)=0, \end{align*} implying that $g(\kappa \varepsilon^{-2},y) \geq 0$ for all $y \in [0, \ell \varepsilon^{-2}]$. Finally, for $x \in [0,\kappa \varepsilon^{-2}]$ we consider \begin{align*}
g_2(x):=g(x,\ell \varepsilon^{-2})&=\tilde Q ( \varepsilon^2 x)+\ell- \varepsilon^2 x Q(\varepsilon^2 J_\varepsilon ( \ell \varepsilon^{-2})) \\
&=\tilde Q ( \varepsilon^2 x) + \kappa Q(\kappa)-\tilde Q(\kappa)-\varepsilon^2 x Q(\kappa), \end{align*} where in the last equality we used \eqref{def:ell}-\eqref{eq:J_e}, which imply $J_\varepsilon(\ell \varepsilon^{-2}) = \kappa \varepsilon^{-2}$. Going further, we have \begin{equation*}
g_2(0)= \ell >0, \qquad \qquad
g_2(\kappa \varepsilon^{-2}) = \tilde Q ( \kappa)+ \kappa Q(\kappa)-\tilde Q(\kappa)-\kappa Q(\kappa)=0, \end{equation*} and \begin{align*}
g_2'(x)= \varepsilon^2 \tilde Q' (\varepsilon^2x )-\varepsilon^2 Q(k) = \varepsilon^2 \left( Q(\varepsilon^2 x)-Q(k)\right) \leq 0, \end{align*} since $Q$ is increasing and $x \leq k\varepsilon^{-2}$. Hence, $g(x, \ell \varepsilon^{-2}) \geq 0$ for all $x \in [0, \kappa \varepsilon^{-2}]$.
We thus proved that $g$ is non negative on the boundary; since $g=0$ at the only internal critical points, we have that $g$ is non negative for all $(x,y)\in [0,\kappa \varepsilon^{-2}]\times[0,\ell\varepsilon^{-2}]$, and the proof is complete. \end{proof}
The inequality \eqref{eq:strangeineq+} is crucial because it allows us to state that if $\bar{u}$ is a monotone function connecting the two stable points $+1$ and $-1$ and \eqref{eq:maxF} holds true, then the energy \eqref{eq:energy} satisfies \begin{equation}\label{eq:c_eps}
E_\varepsilon[\bar{u}]\geq\int_a^b\frac{|\bar{u}'|}\varepsilon Q\left(\varepsilon^2 J_\varepsilon(F(\bar{u}))\right)\,dx=\varepsilon^{-1}\int_{-1}^{+1}Q\left(\varepsilon^2 J_\varepsilon(F(s))\right)\,ds=:c_\varepsilon. \end{equation} Our next goal is to show that the positive constant $c_\varepsilon$ defined in \eqref{eq:c_eps} represents the minimum energy to have a {\it single} transition between $-1$ and $+1$; having this in mind, we fix once for all $N\in\mathbb{N}$ and a {\it piecewise constant function} $v$ with $N$ transitions as follows: \begin{equation}\label{vstruct}
\begin{aligned}
v:[a,b]\rightarrow\{-1,+1\}\ \hbox{with $N$ jumps located at $a<h_1<h_2<\cdots<h_N<b$ and}\ r>0\\
\hbox{such that}\ (h_i-r,h_i+r)\cap(h_j-r,h_j+r)=\emptyset \ \hbox{for}\ i\neq j\ \hbox{and}\ a\leq h_1-r,\ h_N+r\leq b.
\end{aligned} \end{equation} The aforementioned lower bounds will allow us to state that if $\{ u^\varepsilon\}_{\varepsilon >0}$ is a family of functions sufficiently close to $v$ in $L^1$, then $$E_\varepsilon[{u^\varepsilon}] \geq N c_\varepsilon-R_{\theta,\varepsilon},$$ where the reminder term $R_{\theta,\varepsilon}$ goes to zero as $\varepsilon \to 0^+$ with a speed rate depending on $\theta$.
\subsubsection{Lower bound in the critical case $\theta=2$.}
Let us start by proving the lower bound in the case $\theta=2$, where the reminder term $R_{\theta,\varepsilon}$ is exponentially small as $\varepsilon\to0^+$. \begin{prop}\label{prop:lower} Assume that $Q\in C^1(\mathbb{R})$ satisfies \eqref{eq:Q-ass1}-\eqref{eq:Q-ass2} and that $F\in C^1(\mathbb{R})$ satisfies \eqref{ipoF1}-\eqref{ipoF2} with $\theta=2$. Let us set \begin{equation}\label{eq:mathc-Q}
\mathcal{Q}:=\max_{s\in[-\kappa,\kappa]}Q'(s), \end{equation} where $\kappa$ is given in \eqref{eq:Q-ass2}. Moreover, let $v$ be as in \eqref{vstruct} and $A\in(0,r\sqrt{2\lambda_1 \mathcal{Q}^{-1}})$ with $\lambda_1>0$ (independent on $\varepsilon$) as in \eqref{ipoF2}. Then, there exist $\varepsilon_0,C,\delta>0$ (depending only on $Q,F,v$ and $A$) such that if $u\in H^1(a,b)$ satisfies \begin{equation}\label{eq:u-v}
\|u-v\|_{{}_{L^1}}\leq\delta, \end{equation} then for any $\varepsilon\in(0,\varepsilon_0)$, \begin{equation}\label{eq:lower}
E_\varepsilon[u]\geq Nc_\varepsilon-C\exp(-A/\varepsilon), \end{equation} where $E_\varepsilon$ and $c_\varepsilon$ are defined in \eqref{eq:energy} and \eqref{eq:c_eps}, respectively. \end{prop} \begin{proof} Fix $u\in H^1(a,b)$ satisfying \eqref{eq:u-v} and $\varepsilon$ such that \eqref{eq:maxF} holds true. Take $\hat r\in(0,r)$ so small that \begin{equation}\label{eq:nu}
A\leq(r-\hat r)\sqrt{2\mathcal Q^{-1}\lambda_1}. \end{equation} Then, choose $0<\rho <\eta$ (with $\eta$ given by \eqref{ipoF2}) sufficiently small that \begin{equation}\label{eq:forrho2} \begin{aligned}
\int_{1-\eta}^{1-\rho}Q(\varepsilon^2J_\varepsilon(F(s)))\,ds&>\int_{1-\rho}^{1}Q(\varepsilon^2J_\varepsilon(F(s)))\,ds, \\
\int_{-1+\rho}^{-1+\eta}Q(\varepsilon^2J_\varepsilon(F(s)))\,ds&> \int_{-1}^{-1+\rho}Q(\varepsilon^2J_\varepsilon(F(s)))\,ds.
\end{aligned} \end{equation} Let us focus our attention on $h_i$, one of the discontinuous points of $v$ and, to fix ideas, let $v(h_i\pm r)=\pm1$, the other case being analogous. We claim that assumption \eqref{eq:u-v} implies the existence of $r_+$ and $r_-$ in $(0,\hat r)$ such that \begin{equation}\label{2points}
|u(h_i+r_+)-1|<\rho, \qquad \quad \mbox{ and } \qquad \quad |u(h_i-r_-)+1|<\rho. \end{equation}
Indeed, assume by contradiction that $|u-1|\geq\rho$ throughout $(h_i,h_i+\hat r)$; then \begin{equation*}
\delta\geq\|u-v\|_{{}_{L^1}}\geq\int_{h_i}^{h_i+\hat r}|u-v|\,dx\geq\hat r\rho, \end{equation*}
and this leads to a contradiction if we choose $\delta\in(0,\hat r\rho)$. Similarly, one can prove the existence of $r_-\in(0,\hat r)$ such that $|u(h_i-r_-)+1|<\rho$.
Now, we consider the interval $(h_i-r,h_i+r)$ and claim that \begin{equation}\label{eq:claim}
\int_{h_i-r}^{h_i+r}\left[\frac{\tilde{Q}(\varepsilon^2u_x)}{\varepsilon^3}+\frac{F(u)}{\varepsilon}\right]\,dx\geq c_\varepsilon-\tfrac{C}N\exp(-A/\varepsilon), \end{equation} for some $C>0$ independent on $\varepsilon$. Observe that from \eqref{eq:strangeineq+}, it follows that for any $a\leq c<d\leq b$, \begin{equation}\label{eq:ineq}
\int_c^d\left[\frac{\tilde{Q}(\varepsilon^2u_x)}{\varepsilon^3}+\frac{F(u)}{\varepsilon}\right]\,dx \geq\varepsilon^{-1}\left|\int_{u(c)}^{u(d)}Q(\varepsilon^2J_\varepsilon(F(s)))\,ds\right|. \end{equation} Hence, if $u(h_i+r_+)\geq1$ and $u(h_i-r_-)\leq-1$, then from \eqref{eq:ineq} we can conclude that \begin{equation*}
\int_{h_i-r_-}^{h_i+r_+}\left[\frac{\tilde{Q}(\varepsilon^2u_x)}{\varepsilon^3}+\frac{F(u)}{\varepsilon}\right]\,dx\geq c_\varepsilon, \end{equation*} which implies \eqref{eq:claim}. On the other hand, notice that in general we have \begin{align}
\int_{h_i-r}^{h_i+r}\left[\frac{\tilde{Q}(\varepsilon^2u_x)}{\varepsilon^3}+\frac{F(u)}{\varepsilon}\right]\,dx &
\geq \int_{h_i+r_+}^{h_i+r}\left[\frac{\tilde{Q}(\varepsilon^2u_x)}{\varepsilon^3}+\frac{F(u)}{\varepsilon}\right]\,dx\notag \\
& \quad + \int_{h_i-r}^{h_i-r_-}\left[\frac{\tilde{Q}(\varepsilon^2u_x)}{\varepsilon^3}+\frac{F(u)}{\varepsilon}\right]\,dx \notag \\
& \quad +\varepsilon^{-1}\int_{-1}^{1}Q(\varepsilon^2J_\varepsilon(F(s)))\,ds\notag\\
&\quad-\varepsilon^{-1}\int_{-1}^{u(h_i-r_-)}Q(\varepsilon^2J_\varepsilon(F(s)))\,ds \notag \\
& \quad-\varepsilon^{-1}\int_{u(h_i+r_+)}^{1}Q(\varepsilon^2J_\varepsilon(F(s)))\,ds\notag \\
&=:I_1+I_2+c_\varepsilon-\alpha_\varepsilon-\beta_\varepsilon, \label{eq:Pe} \end{align} where we again used \eqref{eq:ineq}. Let us estimate the first two terms of \eqref{eq:Pe}. Regarding $I_1$, assume that $1-\rho<u(h_i+r_+)<1$ and consider the unique minimizer $z:[h_i+r_+,h_i+r]\rightarrow\mathbb{R}$ of $I_1$ subject to the boundary condition $z(h_i+r_+)=u(h_i+r_+)$. If the range of $z$ is not contained in the interval $(1-\eta,1+\eta)$, then from \eqref{eq:ineq}, it follows that \begin{equation}\label{E>fi}
\int_{h_i+r_+}^{h_i+r}\left[\frac{\tilde{Q}(\varepsilon^2z')}{\varepsilon^3}+\frac{F(z)}{\varepsilon}\right]\,dx>\varepsilon^{-1}\int_{u(h_i+r_+)}^{1}Q(\varepsilon^2J_\varepsilon(F(s)))\,ds=\beta_\varepsilon, \end{equation} by the choice of $r_+$ and $\rho$, see \eqref{eq:forrho2}. Suppose, on the other hand, that the range of $z$ is contained in the interval $(1-\eta,1+\eta)$. Then, the Euler-Lagrange equation for $z$ is \begin{align*}
&\varepsilon Q'(\varepsilon^2z'(x))z''(x)=\varepsilon^{-1}F'(z(x)), \quad \qquad x\in(h_i+r_+,h_i+r),\\
&z(h_i+r_+)=u(h_i+r_+), \quad \qquad z'(h_i+r)=0. \end{align*} Denoting by $\psi(x):=(z(x)-1)^2$, we have $\psi'=2(z-1)z'$ and \begin{equation*}
\psi''(x)=2(z(x)-1)z''(x)+2z'(x)^2\geq\frac{2}{\mathcal{Q}\varepsilon^2}(z(x)-1)F'(z(x)), \end{equation*}
where we used the fact that $\varepsilon^2|z'|\leq\kappa$ (see \eqref{eq:Fi-firstderivative}) and \eqref{eq:mathc-Q}. Since $|z(x)-1|\leq\eta$ for any $x\in[h_i+r_+,h_i+r]$, using \eqref{ipoF2} with $\theta=2$, we obtain \begin{equation*}
\psi''(x)\geq\frac{2\lambda_1}{\mathcal{Q}\varepsilon^2}(z(x)-1)^2\geq \frac{\mu^2}{\varepsilon^2}\psi(x), \end{equation*} where $\mu=A/(r-\hat r)$ and we used \eqref{eq:nu}. Thus, $\psi$ satisfies \begin{align*}
\psi''(x)-\frac{\mu^2}{\varepsilon^2}\psi(x)\geq0, \quad \qquad x\in(h_i+r_+,h_i+r),\\
\psi(h_i+r_+)=(u(h_i+r_+)-1)^2, \quad \qquad \psi'(h_i+r)=0. \end{align*} We compare $\psi$ with the solution $\hat \psi$ of \begin{align*}
\hat\psi''(x)-\frac{\mu^2}{\varepsilon^2}\hat\psi(x)=0, \quad \qquad x\in(h_i+r_+,h_i+r),\\
\hat\psi(h_i+r_+)=(u(h_i+r_+)-1)^2, \quad \qquad \hat\psi'(h_i+r)=0, \end{align*} which can be explicitly calculated to be \begin{equation*}
\hat\psi(x)=\frac{(u(h_i+r_+)-1)^2}{\cosh\left[\frac\mu\varepsilon(r-r_+)\right]}\cosh\left[\frac\mu\varepsilon(x-(h_i+r))\right]. \end{equation*} By the maximum principle, $\psi(x)\leq\hat\psi(x)$ so, in particular, \begin{equation*}
\psi(h_i+r)\leq\frac{(u(h_i+r_+)-1)^2}{\cosh\left[\frac\mu\varepsilon(r-r_+)\right]}\leq2\exp(-A/\varepsilon)(u(h_i+r_+)-1)^2. \end{equation*} Then, we have
\begin{equation}\label{|z-v+|<exp}
|z(h_i+r)-1|\leq\sqrt2\exp(-A/2\varepsilon)\rho. \end{equation} Thanks to the expansion \begin{equation}\label{eq:expQ}
Q(s)=Q'(0)s+o(s^2), \end{equation} and \eqref{eq:ass-F3}-\eqref{eq:J-important}, we can choose $\varepsilon>0$ small enough that \begin{equation}\label{eq:crucial}
\left|Q(\varepsilon^2J_\varepsilon(F(s)))\right|\leq C\varepsilon^2 J_\varepsilon(F(s))\leq C\varepsilon\sqrt{F(s)}\leq C\varepsilon|1-s|, \end{equation}
for any $s\in[z(h_i+r),1]$; as a consequence, \eqref{|z-v+|<exp} yields \begin{equation}\label{fi<exp}
\varepsilon^{-1}\left|\int_{z(h_i+r)}^{1}Q(\varepsilon^2J_\varepsilon(F(s)))\,ds\right|\leq C\exp(-A/\varepsilon). \end{equation} From \eqref{eq:ineq}-\eqref{fi<exp} it follows that, for some constant $C>0$, \begin{align}
\int_{h_i+r_+}^{h_i+r}\left[\frac{\tilde{Q}(\varepsilon^2z')}{\varepsilon^3}+\frac{F(z)}{\varepsilon}\right]\,dx &\geq\varepsilon^{-1}\left|\int_{z(h_i+r_+)}^{1}Q(\varepsilon^2J_\varepsilon(F(s)))\,ds\,-\right.\nonumber \\
&\qquad \qquad\left.\int_{z(h_i+r)}^{1}Q(\varepsilon^2J_\varepsilon(F(s)))\,ds\right| \nonumber\\
& \geq\beta_\varepsilon-\tfrac{C}{2N}\exp(-A/\varepsilon). \label{E>fi-exp} \end{align} Combining \eqref{E>fi} and \eqref{E>fi-exp}, we get that the constrained minimizer $z$ of the proposed variational problem satisfies \begin{equation*}
\int_{h_i+r_+}^{h_i+r}\left[\frac{\tilde{Q}(\varepsilon^2z')}{\varepsilon^3}+\frac{F(z)}{\varepsilon}\right]\,dx \geq\beta_\varepsilon-\tfrac{C}{2N}\exp(-A/\varepsilon). \end{equation*} The restriction of $u$ to $[h_i+r_+,h_i+r]$ is an admissible function, so it must satisfy the same estimate and we have \begin{equation}\label{eq:I1}
I_1\geq\beta_\varepsilon-\tfrac{C}{2N}\exp(-A/\varepsilon). \end{equation} The term $I_2$ on the right hand side of \eqref{eq:Pe} is estimated similarly by analyzing the interval $[h_i-r,h_i-r_-]$ and using the second condition of \eqref{eq:forrho2} to obtain the corresponding inequality \eqref{E>fi}. The obtained lower bound reads: \begin{equation}\label{eq:I2}
I_2\geq\alpha_\varepsilon-\tfrac{C}{2N}\exp(-A/\varepsilon). \end{equation} Finally, by substituting \eqref{eq:I1} and \eqref{eq:I2} in \eqref{eq:Pe}, we deduce \eqref{eq:claim}. Summing up all of these estimates for $i=1, \dots, N$, namely for all transition points, we end up with \begin{equation*}
E_\varepsilon[u]\geq\sum_{i=1}^N\int_{h_i-r}^{h_i+r}\left[\frac{\tilde{Q}(\varepsilon^2u_x)}{\varepsilon^3}+\frac{F(u)}{\varepsilon}\right]\,dx\geq Nc_\varepsilon-C\exp(-A/\varepsilon), \end{equation*} and the proof is complete. \end{proof}
\subsubsection{Lower bound in the supercritical case $\theta>2$.} We now deal with the case $\theta>2$, where we have a {\it weaker} lower bound for the energy, which is stated and proved in the following proposition. \begin{prop}\label{prop:lower_deg} Assume that $Q\in C^1(\mathbb{R})$ satisfies \eqref{eq:Q-ass1}-\eqref{eq:Q-ass2} and that $F\in C^1(\mathbb{R})$ satisfies \eqref{ipoF1}-\eqref{ipoF2} with $\theta>2$. Let $v$ as in \eqref{vstruct} and define the sequence \begin{equation}\label{eq:exp_alg}
k_j:=\sum_{m=1}^j\alpha^m,
\qquad\qquad \mbox{ where } \quad \alpha:=\displaystyle\frac{1}{2}+\frac{1}{\theta}. \end{equation} Then, for any $j\in \mathbb N$ there exist constants $\delta_j>0$ and $C>0$ such that if $u\in H^1(a,b)$ satisfies
\begin{equation}\label{|w-v|_L^1<delta_l}
\|u-v\|_{L^1}\leq\delta_j, \end{equation} and \begin{equation}\label{E_p(w)<Nc0+eps^l}
E_\varepsilon[u]\leq Nc_\varepsilon+C\varepsilon^{k_{j}}, \end{equation} with $\varepsilon$ sufficiently small, then \begin{equation}\label{E_p(w)>Nc0-eps^l}
E_\varepsilon[u]\geq Nc_\varepsilon-C_j\varepsilon^{k_{j+1}}. \end{equation} \end{prop}
\begin{proof} We prove our statement by induction on $j\geq1$. Let us begin by considering the case of only one transition $N=1$ and let $h_1$ be the only point of discontinuity of $v$ and assume, without loss of generality, that $v=-1$ on $(a,h_1)$. Also, we choose $\delta_j$ small enough such that \begin{equation*}
(h_1-2j\delta_j,h_1+2j\delta_j)\subset(a,b). \end{equation*} Our goal is to show that for any $j\in\mathbb N$ there exist $x_j\in(h_1- 2j\delta_j,h_1)$ and $y_j\in(h_1, h_{1}+2j\delta_j)$ such that \begin{equation}\label{x_k,y_k}
u(x_j)\leq-1+C\varepsilon^\frac{k_j+1}{\theta}, \qquad u(y_j)\geq 1-C\varepsilon^\frac{k_j+1}{\theta}, \end{equation} and \begin{equation}\label{E_p(w)-xk,yk}
\int_{x_j}^{y_j}\left[\frac{\tilde{Q}(\varepsilon^2u_x)}{\varepsilon^3}+\frac{F(u)}{\varepsilon}\right]\geq c_\varepsilon-C\varepsilon^{k_{j+1}}, \end{equation}
where $\{k_j\}_{{}_{j\geq1}}$ is defined in \eqref{eq:exp_alg}. We start with the base case $j=1$, and we show that hypotheses \eqref{|w-v|_L^1<delta_l} and \eqref{E_p(w)<Nc0+eps^l} imply the existence of two points $x_1{\in(h_1-2\delta_j,h_1)}$ and $y_1\in(h_1,h_1+2\delta_j)$ such that \begin{equation}\label{x_1,y_1}
u(x_1)\leq-1+C\varepsilon^\frac1{\theta}, \qquad u(y_1)\geq1-C\varepsilon^\frac1{\theta}. \end{equation} Here and throughout, $C$ represents a positive constant that is independent of $\varepsilon$, whose value may change from line to line.
From hypothesis \eqref{|w-v|_L^1<delta_l}, we have \begin{equation}\label{int su (gamma,b)}
\int_{h_1}^b|u-1|\leq\delta_j, \end{equation} so that, denoting by $S^-:=\{y:u(y)\leq0\}$ and by $S^+:=\{y:u(y)>0\}$,
\eqref{int su (gamma,b)} yields \begin{equation*} \begin{aligned} \textrm{meas}(S^-\cap(h_1,b))\leq\delta_j \qquad \mbox{and} \qquad
\textrm{meas}(S^+\cap(h_1,h_1+2\delta_j))\geq\delta_j.
\end{aligned} \end{equation*} Furthermore, from \eqref{E_p(w)<Nc0+eps^l} with $j=1$, we obtain \begin{equation*}
\int_{S^+\cap(h_1,h_1+2\delta_j)}\frac{F(u)}\varepsilon\, dx\leq c_\varepsilon+C\varepsilon^\alpha, \end{equation*} and therefore there exists $y_1\in S^+\cap(h_1,h_1+2\delta_j)$ such that \begin{equation*}
F(u(y_1))\leq \frac{c_\varepsilon+C\varepsilon^\alpha}{\delta_j}\varepsilon. \end{equation*}
Since $F$ vanishes only at $\pm1$ and $u(y_1)>0$ we can choose $\varepsilon$ so small that the latter condition implies $|u(y_1)-1|<\eta$; hence, from \eqref{eq:ass-F3}, it follows that $u(y_1)\geq 1-C\varepsilon^{\frac1{\theta}}$. The existence of $x_1{\in S^-\cap(h_1-2\delta_j,h_1)}$ such that $u(x_1)\leq-1+C\varepsilon^\frac1{\theta}$ can be proved similarly.
Now, let us prove that \eqref{x_1,y_1} implies \eqref{E_p(w)-xk,yk} in the case $j=1$, and as a trivial consequence we obtain the statement \eqref{E_p(w)>Nc0-eps^l} with $j=1$ and $N=1$. Indeed, by using \eqref{eq:strangeineq+} and \eqref{x_1,y_1} one deduces \begin{equation*}
\begin{aligned}
E_\varepsilon[u]&\geq\int_{x_1}^{y_1}\left[\frac{\tilde{Q}(\varepsilon^2u_x)}{\varepsilon^3}+\frac{F(u)}{\varepsilon}\right]
\geq\varepsilon^{-1}\int_{u(x_1)}^{u(y_1)} Q(\varepsilon^2J_\varepsilon(F(s)))\,ds\\
&\geq c_\varepsilon-\varepsilon^{-1}\int_{1-C\varepsilon^\frac1{\theta}}^1 Q(\varepsilon^2J_\varepsilon(F(s)))\,ds-\varepsilon^{-1}\int_{-1}^{-1+C\varepsilon^\frac1{\theta}}Q(\varepsilon^2J_\varepsilon(F(s)))\,ds.
\end{aligned} \end{equation*} Reasoning as in \eqref{eq:crucial} and using \eqref{eq:ass-F3}, we infer \begin{align*}
\left|\int_{1-C\varepsilon^\frac1{\theta}}^1 Q(\varepsilon^2J_\varepsilon(F(s)))\,ds\right|&\leq
C\varepsilon\int_{1-C\varepsilon^\frac1{\theta}}^1(1-s)^{\frac\theta2}\leq C\varepsilon^{\frac32+\frac1\theta},\\
\left|\int_{-1}^{-1+C\varepsilon^\frac1{\theta}} Q(\varepsilon^2J_\varepsilon(F(s)))\,ds\right|&\leq
C\varepsilon\int_{-1}^{-1+C\varepsilon^\frac1{\theta}}(s+1)^{\frac\theta2}\leq C\varepsilon^{\frac32+\frac1\theta}, \end{align*} and, as a trivial consequence, \begin{equation}\label{stima-l=1}
E_\varepsilon[u]\geq c_\varepsilon-C\varepsilon^\alpha, \end{equation} where $\alpha$ is defined in \eqref{eq:exp_alg}. This concludes the proof in the case $j=1$ with one transition $N=1$.
We now enter the core of the induction argument, proving that if \eqref{E_p(w)-xk,yk} holds true for for any $i\in\{1,\dots,j-1\}$, $j\geq2$, then \eqref{x_k,y_k} holds true. By using \eqref{|w-v|_L^1<delta_l} we have \begin{equation}\label{meas>delta_l-k}
\textrm{meas}(S^+\cap(y_{j-1},y_{j-1}+2\delta_j))\geq\delta_j. \end{equation} Furthermore, by using \eqref{E_p(w)<Nc0+eps^l} and \eqref{E_p(w)-xk,yk} in the case $j-1$, we deduce \begin{equation*}
\int_{y_{j-1}}^b\frac{F(u)}\varepsilon dx\leq C\varepsilon^{k_j}, \end{equation*} implying \begin{equation}\label{int F(w)<Ceps^k+1}
\int_{S^+\cap(y_{j-1},y_{j-1}+2\delta_j)} F(u)\,dx\leq C\varepsilon^{k_j+1}. \end{equation} Finally, from \eqref{meas>delta_l-k} and \eqref{int F(w)<Ceps^k+1} there exists $y_{j}\in S^+\cap(y_{j-1},y_{j-1}+2\delta_j)$ such that \begin{equation*}
F(u(y_{j}))\leq \frac{C}{\delta_j}\varepsilon^{k_j+1}, \end{equation*} and, as a consequence, we have the existence of $y_{j}\in(y_{j-1},y_{j-1}+2\delta_j)$ as in \eqref{x_k,y_k}. The existence of $x_{j}\in(x_{j-1}-2\delta_j,x_{j-1})$ can be proved similarly.
Proceeding as done to obtain \eqref{stima-l=1}, one can easily check that \eqref{x_k,y_k} implies \begin{equation*}
\int_{x_{j}}^{y_{j}}\left[\frac{\tilde{Q}(\varepsilon^2u_x)}{\varepsilon^3}+\frac{F(u)}{\varepsilon}\right]\,dx\geq c_\varepsilon-C\varepsilon^{\alpha(k_j+1)}. \end{equation*} Since the definition \eqref{eq:exp_alg} implies $\alpha(k_j+1)=k_{j+1}$, the induction argument is completed, as well as the proof in case $N=1$.
The previous argument can be easily adapted to the case $N>1$. Let $v$ be as in \eqref{vstruct}, and set $a=h_0, h_{N+1}=b$. We argue as in the case $N=1$ in each point of discontinuity $h_i$, by choosing the constant $\delta_j$ so that $$
{h_i}+2j\delta_j<h_{i+1}-2j\delta_j,\qquad \quad 0\leq i\leq N, $$ and by assuming, without loss of generality, that $v=-1$ on $(a,h_1)$. Proceeding as in \eqref{x_1,y_1}, one can obtain the existence of $x^i_1\in(h_i-2\delta_j,h_i)$ and $y^i_1\in(h_i,h_i+2\delta_j)$ such that \begin{align*}
u(x^i_1)&\approx (-1)^i, &u(y^i_1)&\approx(-1)^{i+1},\\
F(u(x^i_1))&\leq C\varepsilon, &F(u(y^i_1))&\leq C\varepsilon. \end{align*} On each interval $(x_1^i,y_1^i)$ we bound from below $E_\varepsilon$ as in \eqref{stima-l=1}, so that by summing one obtains $$
\sum_{i=1}^N\int_{x_1^i}^{y_1^i}\left[\frac{\tilde{Q}(\varepsilon^2u_x)}{\varepsilon^3}+\frac{F(u)}{\varepsilon}\right]\,dx\geq Nc_\varepsilon-C\varepsilon^\alpha, $$ that is \eqref{E_p(w)>Nc0-eps^l} with $j=1$. Arguing inductively as done in the case $N=1$, we obtain \eqref{E_p(w)>Nc0-eps^l} for $E_\varepsilon$ in the general case $j\geq2$. \end{proof}
\subsubsection{Some comments on the lower bounds} First of all, let us notice that in the proof of Proposition \ref{prop:lower_deg}, we simply took advantage of the behavior of $Q$ in a neighborhood of zero, and of the fact that the potential $F \in C(\mathbb{R})$ satisfies $F(u)>0$ for any $u \neq \pm 1$ and \eqref{eq:ass-F3} for some $\theta>0$. Hence, Proposition \ref{prop:lower_deg} holds also in the case $\theta\in(0,2]$; moreover, since in such a case $\alpha>1$, the increasing sequence in \eqref{eq:exp_alg} is unbounded, and we can rewrite the estimate \eqref{E_p(w)>Nc0-eps^l} as \begin{equation*}
E_\varepsilon[u]\geq Nc_\varepsilon-C_k\varepsilon^{k}, \qquad \qquad k\in\mathbb{N}, \end{equation*} provided that \begin{equation*}
E_\varepsilon[u]\leq Nc_\varepsilon+C\varepsilon^{k},\qquad \qquad k\in\mathbb{N}. \end{equation*} Nevertheless, if $\theta=2$, we underline that \eqref{eq:lower} provides a {\it stronger} lower bound, where the error is exponentially small rather than algebraically small.
On the other hand, if $\theta>2$ then $\alpha\in(0,1)$, and, consequently
\begin{equation}\label{eq:beta}
\lim_{j\to+\infty}k_j= \sum_{m=1}^{+\infty} \alpha^m =\frac{1}{1-\alpha}-1=\frac{\theta+2}{\theta-2}:=\beta. \end{equation} Therefore, in the case $\theta>2$ one only has the lower bounds \begin{equation*}
E_\varepsilon[u]\geq Nc_\varepsilon-C_j\varepsilon^{k_j}, \qquad \mbox{for any } j\in\mathbb N,\qquad \mbox{ with } \qquad \lim_{j\to+\infty}k_j=\beta. \end{equation*}
As a direct consequence of Propositions \ref{prop:lower}-\ref{prop:lower_deg} we have the following result, showing that if the family $\{ u_\varepsilon\}_{\varepsilon >0}$ makes $N$ transitions among $+1$ and $-1$ in an {\it energy efficient way} (see \eqref{TLS2} for the rigorous definition), then $E_\varepsilon[u^\varepsilon]$ converges, as $\varepsilon \to 0$, to the minimum energy in the case of the classical Ginzburg-Landau functional (see \cite{Bron-Kohn}).
\begin{cor}\label{cor:TLS} Let $v$ as in \eqref{vstruct} and let $u^\varepsilon \in H^1(a,b)$ be such that \begin{equation}\label{TLS1}
\lim_{\varepsilon\rightarrow 0^+} \| u^\varepsilon -v \|_{L^1}=0, \end{equation} and there exists a function $\nu \, : \, (0,1) \to (0,1)$ such that \begin{equation}\label{TLS2}
E_\varepsilon[u^\varepsilon] \leq N c_\varepsilon +\nu(\varepsilon), \qquad \mbox{with} \qquad \lim_{\varepsilon \to 0^+} \nu(\varepsilon) =0, \end{equation} where $c_\varepsilon$ is defined in \eqref{eq:c_eps}. Then \begin{equation*} \lim_{\varepsilon \to 0^+} E_\varepsilon[u^\varepsilon] = N c_0, \qquad \mbox{ with } \qquad c_0:= \lim_{\varepsilon\to0^+} c_\varepsilon=\sqrt{Q'(0)}\int_{-1}^{+1} \sqrt{2F(s)}\,ds. \end{equation*} \end{cor} \begin{proof} By Propositions \ref{prop:lower}-\ref{prop:lower_deg} and from \eqref{TLS2} we have: \begin{equation}\label{eq:proofTLS} Nc_\varepsilon -C R_{\theta, \varepsilon} \leq E_\varepsilon[u^\varepsilon] \leq Nc_\varepsilon + \nu(\varepsilon), \end{equation} where \begin{equation}\label{def:Rte} R_{\theta, \varepsilon}= \left\{ \begin{aligned} & \exp(-A/\varepsilon) \quad &\mbox{if} \quad &\theta=2 \\ & \varepsilon^{k_j} \quad &\mbox{if} \quad &\theta>2, \end{aligned}\right. \end{equation} with $A$ appearing in Proposition \ref{prop:lower} and $\{ k_j \}_{j \in \mathbb{N}}$ defined in \eqref{eq:exp_alg}. Moreover, by using \eqref{eq:J-important} and \eqref{eq:expQ}, we deduce that \begin{align*}
Q(\varepsilon^2J_\varepsilon(F(s)))&=Q\left(\sqrt{\frac{2\varepsilon^2}{Q'(0)}F(s)}+\rho(\varepsilon^2F(s))\right)\\
&=Q'(0)\left[\sqrt{\frac{2\varepsilon^2}{Q'(0)}F(s)}+\rho(\varepsilon^2F(s))\right]+o(\varepsilon), \end{align*} and substituting into the definition \eqref{eq:c_eps}, we end up with \begin{equation*}
\lim_{\varepsilon\to0^+} c_\varepsilon=\varepsilon^{-1}\int_{-1}^1 Q(\varepsilon^2J_\varepsilon(F(s)))\,ds
=\sqrt{Q'(0)}\int_{-1}^{+1} \sqrt{2F(s)}\,ds. \end{equation*} Hence, the thesis follows by simply passing to the limit as $\varepsilon \to 0^+$ in \eqref{eq:proofTLS}. \end{proof}
\subsection*{Example of a function satisfying the assumptions of Corollary \ref{cor:TLS}} We conclude this section by showing that there exist a family of functions satisfying assumptions \eqref{TLS1}-\eqref{TLS2}. First of all, we observe that it is easy to check that the {\it compactons} $\varphi_1$ and $\varphi_2$ constructed in Proposition \ref{prop:comp} satisfy \eqref{TLS1} and $E_\varepsilon[\varphi_{1}] =E_\varepsilon[\varphi_{2}] = N c_\varepsilon$. However, such stationary solutions exist only in the case $\theta\in(1,2)$; the idea is to use a similar construction as the one done for compactons to obtain a function satisfying \eqref{TLS1}-\eqref{TLS2} also when $\theta \geq 2$; we underline that in this case these profiles are not stationary solutions.
Let us thus consider the increasing standing wave $\Phi_\varepsilon$, solution to \eqref{eq:Fi}, and observe that \begin{equation*}
\lim_{\varepsilon\to0}\Phi_\varepsilon(x)=
\begin{cases}
-1, \qquad & x<0,\\
0, &x=0,\\
+1, & x>0.
\end{cases} \end{equation*} Now, choose $\varepsilon_0>0$ small enough so that the condition \eqref{eq:maxF} holds true for any $\varepsilon\in(0,\varepsilon_0)$ and fix $N\in\mathbb{N}$ transition points $a<h_1<h_2<\dots<h_n<b$. Denoted by \begin{equation*}
m_1:=a, \qquad \quad m_j:=\frac{h_{j-1}+h_j}{2}, \quad j=2,\dots,N-1, \qquad \quad m_N:=b, \end{equation*} the middle points, we define \begin{equation}\label{eq:translayer}
u^\varepsilon(x):=\Phi_\varepsilon\left((-1)^j(x-h_j)\right), \qquad \qquad x\in[m_j,m_{j+1}], \qquad \qquad j=1,\dots N. \end{equation} Notice that $u^\varepsilon(h_j)=0$, for $j=1,\dots,N$ and for definiteness we choose $u^\varepsilon(a)<0$ (the case $u^\varepsilon(a)>0$ is analogous). Let us now prove that $u^\varepsilon$ satisfies \eqref{TLS1}-\eqref{TLS2}. It is easy to check that $u^\varepsilon\in H^1(a,b)$ and satisfies \eqref{TLS1}; concerning \eqref{TLS2}, the definitions of $E_\varepsilon$ and $u^\varepsilon$ give \begin{equation*}
E_\varepsilon[u^\varepsilon]=\sum_{j=1}^{N}\int_{m_j}^{m_{j+1}}\left[\frac{\tilde Q(\varepsilon^2\Phi'_\varepsilon)}{\varepsilon^3}+\frac{F(\Phi_\varepsilon)}\varepsilon\right]\,dx. \end{equation*} From \eqref{eq:impforenergy}, it follows that \begin{align*}
\int_{m_j}^{m_{j+1}}\left[\frac{\tilde Q(\varepsilon^2\Phi'_\varepsilon)}{\varepsilon^3}+\frac{F(\Phi_\varepsilon)}\varepsilon\right]\,dx&=
\varepsilon^{-1}\int_{m_j}^{m_{j+1}}\left[\Phi'Q(\varepsilon^2 J_\varepsilon (F(\Phi_\varepsilon)))\right]\,dx\\
&=\varepsilon^{-1}\int_{\Phi_\varepsilon(m_j)}^{\Phi_\varepsilon(m_{j+1})}\left[Q(\varepsilon^2 J_\varepsilon (F(s)))\right]\,ds<c_\varepsilon, \end{align*} where $c_\varepsilon$ is defined in \eqref{eq:c_eps}. Summing up all the terms we end up with $E_\varepsilon[u^\varepsilon]\leq Nc_\varepsilon$ which clearly implies \eqref{TLS2}.
\section{Slow motion}\label{sec:slow}
In this last section we investigate the long time dynamics of the solutions to \eqref{eq:Q-model}-\eqref{eq:Neu}-\eqref{eq:initial} with a special focus to their different speed rate of convergence towards an asymptotic configuration, which heavily depends on the parameter $\theta$ appearing in the potential $F$ (see the assumptions \eqref{ipoF1} and \eqref{ipoF2}). Specifically, the main results of this section are contained in Theorems \ref{thm:main} and \ref{thm:main2}: in the former we prove that in the critical case $\theta=2$ the solutions to \eqref{eq:Q-model}-\eqref{eq:Neu}-\eqref{eq:initial} exhibit a metastable behaviour, namely, they maintain the same unstable structure of the initial datum for a time which is exponentially long with respect to the parameter $\varepsilon$; in the latter, concerning instead the supercritical case $\theta>2$, we prove slow motion with a speed rate which is only algebraic with respect to $\varepsilon$.
\subsection*{Preliminary assumptions} Here and in the rest of the section, we fix a function $v$ as in \eqref{vstruct} and we assume that the initial datum in \eqref{eq:initial} depends on $\varepsilon$ and satisfies \begin{equation}\label{eq:ass-u0}
\lim_{\varepsilon\rightarrow 0^+} \|u^\varepsilon_0-v\|_{{}_{L^1}}=0. \end{equation} Moreover, we assume that there exist $C, \varepsilon_0>0$ such that, for any $\varepsilon\in(0,\varepsilon_0)$, \begin{equation}\label{eq:energy-ini}
E_\varepsilon[u^\varepsilon_0]\leq Nc_\varepsilon+ C R_{\theta,\varepsilon}, \end{equation} where $R_{\theta,\varepsilon}$ is defined in \eqref{def:Rte}, that is $u^\varepsilon_0$ satisfies the assumptions of Corollary \ref{cor:TLS}.
We emphasize that an initial datum as the one satisfying the assumptions \eqref{eq:ass-u0}-\eqref{eq:energy-ini} is far from being a stationary solution if and only if $\theta \geq 2$; indeed, when $\theta \in (1,2)$, in Proposition \ref{prop:comp} we proved the existence of a particular class of stationary solutions (compactons), which have an arbitrary number of transition layers that are randomly located inside the interval $[a,b]$. Hence, an initial datum satisfying \eqref{eq:ass-u0}-\eqref{eq:energy-ini} is either a steady state or a small perturbation of it; thus, proving that the corresponding time dependent solution maintains the same structure for long times is either trivially true (indeed, the same structure is maintained for all $t>0$) or only a partial result, since we do not know if such structure will be lost at some point. In other words, in the case $\theta\in(1,2)$ transition layers do not evolve in time and could persist forever. On the contrary, if $\theta \geq 2$, then stationary solutions can only have layers that are equidistant (see Proposition \ref{periodic:bounded}), so that an initial configuration as $u^\varepsilon_0$ is in general far away from any steady state.
\subsection{Exponentially slow motion}\label{sec:exp_met} In this subsection we examine the persistence of layered solutions to \eqref{eq:Q-model}-\eqref{eq:Neu}-\eqref{eq:initial} when $\theta=2$ in the assumption \eqref{ipoF2} on the potential $F$. In particular, we prove that in this case a metastability phenomenon occurs, showing that the solutions perpetuate the same behavior of the initial datum for an $\varepsilon$-exponentially long time, that is, at least for a time equals $m\,e ^{A/\varepsilon}$ for some $A>0$ and any $m>0$, both independent on $\varepsilon$, as stated in the following theorem.
\begin{thm}[exponentially slow motion when $\theta=2$]\label{thm:main} Assume that $Q\in C^1(\mathbb{R})$ satisfies \eqref{eq:Q-ass1}-\eqref{eq:Q-ass2} and that $F\in C^1(\mathbb{R})$ satisfies \eqref{ipoF1}-\eqref{ipoF2} with $\theta=2$. Let $v$ be as in \eqref{vstruct} and $A\in(0,r\sqrt{2\lambda_1 \mathcal{Q}^{-1}})$, with $\mathcal{Q}$ defined in \eqref{eq:mathc-Q} and $\lambda_1>0$ (independent on $\varepsilon$) as in \eqref{ipoF2}. If $u^\varepsilon$ is the solution of \eqref{eq:Q-model}-\eqref{eq:Neu}-\eqref{eq:initial} with initial datum $u_0^{\varepsilon}$ satisfying \eqref{eq:ass-u0} and \eqref{eq:energy-ini} with $R_{\theta,\varepsilon}= \exp(-A/\varepsilon)$, then \begin{equation}\label{eq:limit}
\sup_{0\leq t\leq m\exp(A/\varepsilon)}\|u^\varepsilon(\cdot,t)-v\|_{{}_{L^1}}\xrightarrow[\varepsilon\rightarrow0]{}0, \end{equation} for any $m>0$. \end{thm} The proof of Theorem \ref{thm:main} strongly relies on the following result which provides an estimate from above of the $L^2$-norm of the derivative with respect to time of the solution $u^\varepsilon(\cdot,t)$ to \eqref{eq:Q-model}-\eqref{eq:Neu}-\eqref{eq:initial} under the assumptions of Theorem \ref{thm:main}. \begin{prop}\label{prop:L2-norm}
Under the same assumptions of Theorem \ref{thm:main}, there exist positive constants $\varepsilon_0, C_1, C_2>0$ (independent on $\varepsilon$) such that \begin{equation}\label{L2-norm}
\int_0^{C_1\varepsilon^{-1}\exp(A/\varepsilon)}\|u_t^\varepsilon\|^2_{{}_{L^2}}dt\leq C_2\varepsilon\exp(-A/\varepsilon), \end{equation} for all $\varepsilon\in(0,\varepsilon_0)$. \end{prop}
\begin{proof} Let $\varepsilon_0>0$ so small that for all $\varepsilon\in(0,\varepsilon_0)$, \eqref{eq:energy-ini} holds and \begin{equation}\label{1/2delta}
\|u_0^\varepsilon-v\|_{{}_{L^1}}\leq\frac12\delta, \end{equation} where $\delta$ is the constant of Proposition \ref{prop:lower}. Let $\hat T>0$; we claim that if \begin{equation}\label{claim1}
\int_0^{\hat T}\|u_t^\varepsilon\|_{{}_{L^1}}dt\leq\frac12\delta, \end{equation} then there exists $C>0$ such that \begin{equation}\label{claim2}
E_\varepsilon[u^\varepsilon](\hat T)\geq Nc_\varepsilon-C\exp(-A/\varepsilon). \end{equation}
Indeed, inequality \eqref{claim2} follows from Proposition \ref{prop:lower} if $\|u^\varepsilon(\cdot,\hat T)-v\|_{{}_{L^1}}\leq\delta$. By using triangle inequality, \eqref{1/2delta} and \eqref{claim1}, we obtain \begin{equation*}
\|u^\varepsilon(\cdot,\hat T)-v\|_{{}_{L^1}}\leq\|u^\varepsilon(\cdot,\hat T)-u_0^\varepsilon\|_{{}_{L^1}}+\|u_0^\varepsilon-v\|_{{}_{L^1}}
\leq\int_0^{\hat T}\|u_t^\varepsilon\|_{{}_{L^1}}+\frac12\delta\leq\delta. \end{equation*} Substituting \eqref{eq:energy-ini} and \eqref{claim2} in \eqref{eq:energyestimate}, one has \begin{equation}\label{L2-norm-Teps}
\int_0^{\hat T}\|u_t^\varepsilon\|^2_{{}_{L^2}}dt\leq C_2\varepsilon\exp(-A/\varepsilon). \end{equation} It remains to prove that inequality \eqref{claim1} holds for $\hat T\geq C_1\varepsilon^{-1}\exp(A/\varepsilon)$. If \begin{equation*}
\int_0^{+\infty}\|u_t^\varepsilon\|_{{}_{L^1}}dt\leq\frac12\delta, \end{equation*} there is nothing to prove. Otherwise, choose $\hat T$ such that \begin{equation*}
\int_0^{\hat T}\|u_t^\varepsilon\|_{{}_{L^1}}dt=\frac12\delta. \end{equation*} Using H\"older's inequality and \eqref{L2-norm-Teps}, we infer \begin{equation*}
\frac12\delta\leq[\hat T(b-a)]^{1/2}\biggl(\int_0^{\hat T}\|u_t^\varepsilon\|^2_{{}_{L^2}}dt\biggr)^{1/2}\leq
\left[\hat T(b-a)C_2\varepsilon\exp(-A/\varepsilon)\right]^{1/2}, \end{equation*} so that there exists $C_1>0$ such that \begin{equation*}
\hat T\geq C_1\varepsilon^{-1}\exp(A/\varepsilon), \end{equation*} and the proof is complete. \end{proof}
Now we are ready to prove Theorem \ref{thm:main}. \begin{proof}[Proof of Theorem \ref{thm:main}] Fix $m>0$. As a consequence of the triangle inequality we have that \begin{equation}\label{trianglebar}
\sup_{0\leq t\leq m\exp(A/\varepsilon)}\|u^\varepsilon(\cdot,t)-v\|_{{}_{L^1}}\leq \sup_{0\leq t\leq m\exp(A/\varepsilon)}\|u^\varepsilon(\cdot,t)-u_0^\varepsilon\|_{{}_{L^1}}+\|u_0^\varepsilon-v\|_{{}_{L^1}}. \end{equation} Hence, since the second term in the right-hand side of \eqref{trianglebar} tends to $0$ by \eqref{eq:ass-u0}, in order to prove \eqref{eq:limit}, it is sufficient to show that \begin{equation}\label{primopezzo}
\sup_{0\leq t\leq m\exp(A/\varepsilon)}\|u^\varepsilon(\cdot,t)-u_0^\varepsilon\|_{{}_{L^1}}\xrightarrow[\varepsilon\rightarrow0]{}0. \end{equation} To this aim, we first observe that up to taking $\varepsilon$ so small that $m<C_1\varepsilon^{-1}$, we can apply \eqref{L2-norm} to deduce
\begin{equation}\label{proof:usata}
\int_0^{m\exp(A/\varepsilon)}\|u_t^\varepsilon\|^2_{{}_{L^2}}dt\leq C_2\varepsilon\exp(-A/\varepsilon). \end{equation} Moreover, for all $t\in[0,m\exp(A/\varepsilon)]$ we have \begin{equation*} \begin{aligned}
\|u^\varepsilon(\cdot,t)-u^\varepsilon_0\|_{{}_{L^1}}&\leq\int_0^{m\exp(A/\varepsilon)}\|u_t^\varepsilon(\cdot,t)\|_{{}_{L^1}}\,dt \\
&\leq \sqrt{m(b-a)} \exp(A/2\varepsilon) \left( \int_0^{m\exp(A/\varepsilon)}\|u_t^\varepsilon(\cdot,t)\|^2_{{}_{L^2}}\,dt \right)^{\frac{1}{2}} \leq C\sqrt\varepsilon, \end{aligned} \end{equation*} where we applied H\"older's inequality and \eqref{proof:usata}. We thus obtained \eqref{primopezzo} and the proof is complete. \end{proof}
\begin{rem}\label{rem:Q} We stress that the constant $\mathcal Q$ defined in \eqref{eq:mathc-Q} (and appearing the first time in the constant $A$ of the lower bound \eqref{eq:lower}) plays a relevant role in the dynamics of the solution; indeed, such lower bound is needed to prove Theorem \ref{thm:main}, and, from estimate \eqref{eq:limit}, we can clearly see that the bigger is $A$ (that is the smaller is $\mathcal Q$) the slower is the dynamics. As to give a hint on what happens even with a small variation of $\mathcal Q$, if we choose $Q'$ so that its maximum is $4$ instead of $1$ (as it is for the examples \eqref{ex:fluxfunction} we considered) then the time taken for the solution to drift apart from the initial datum $u_0$ reduces from $T_\varepsilon$ to $\sqrt{T_\varepsilon}$; we will see further details with the numerical simulations of Section \ref{numerics}. \end{rem}
\subsection{Algebraic slow motion} In this subsection we consider the case in which the potential $F$ satisfies assumption \eqref{ipoF2} with $\theta>2$
and we show that the evolution of the solutions drastically changes with respect to the critical case $\theta=2$, studied in Section \ref{sec:exp_met}. Indeed, the exponentially slow motion proved in Theorem \ref{thm:main} is a peculiar phenomenon of \emph{non-degenerate potentials}, while if $\theta>2$, then the solution maintains the same unstable structure of the initial profile \emph{only} for an algebraically long time with respect to $\varepsilon$, that is, at least for a time equals $ l\varepsilon^{-\beta}$, for any $l>0$, with $\beta>0$ defined in \eqref{eq:beta}. This is a consequence of the fact that when $\theta>2$, we have no longer a lower bound like the one exhibited in Proposition \ref{prop:lower} (with an exponentially small reminder), but only a lower bound with an algebraic small reminder, see Proposition \ref{prop:lower_deg}.
Our second main result is the following one.
\begin{thm}[algebraic slow motion when $\theta>2$]\label{thm:main2} Assume that $Q\in C^1(\mathbb{R})$ satisfies \eqref{eq:Q-ass1}-\eqref{eq:Q-ass2} and that $F\in C^1(\mathbb{R})$ satisfies \eqref{ipoF1}-\eqref{ipoF2} with $\theta>2$. Moreover, let $v$ be as in \eqref{vstruct} and let $\left\{k_j\right\}_{j\in\mathbb N}$ be as in \eqref{eq:exp_alg}. If $u^\varepsilon$ is the solution to \eqref{eq:Q-model}-\eqref{eq:Neu}-\eqref{eq:initial}, with initial profile $u_0^{\varepsilon}$ satisfying \eqref{eq:ass-u0} and \eqref{eq:energy-ini} with $R_{\theta,\varepsilon}=\varepsilon^{k_j}$, then \begin{equation}\label{eq:limit-deg}
\sup_{0\leq t\leq l{\varepsilon^{-k_j}}}\|u^\varepsilon(\cdot,t)-v\|_{{}_{L^1}}\xrightarrow[\varepsilon\rightarrow0]{}0, \end{equation} for any $l>0$. \end{thm}
\begin{proof} The proof follows the same steps of the proof of Theorem \ref{thm:main} and it is obtained by using Proposition \ref{prop:lower_deg} instead of Proposition \ref{prop:lower}. In particular, proceeding as in the proof of Proposition \ref{prop:L2-norm}, one can prove that there exist $\varepsilon_0, C_1, C_2>0$ (independent on $\varepsilon$) such that \begin{equation*}
\int_0^{C_1\varepsilon^{-(k_j+1)}}\|u_t^\varepsilon\|^2_{{}_{L^2}}dt\leq C_2\varepsilon^{k_j+1}, \end{equation*} for all $\varepsilon\in(0,\varepsilon_0)$. Thanks to the latter estimate, we can prove \eqref{eq:limit-deg} in the same way we proved \eqref{eq:limit} (see \eqref{trianglebar} and the following discussion). \end{proof}
\subsection{Layer Dynamics} In this last subsection, we ultimate our investigation giving a description of the slow motion of the transition points $h_1,\ldots,h_N$. More precisely, we will incorporate the analysis in both the critical case (that is when $\theta=2$) and the subcritical one (namely $\theta>2$), showing that the transition layers evolve with a velocity which goes to zero as $\varepsilon\to0^+$, according to Theorem \ref{thm:interface}.
For this, let us fix a function $v$ as in \eqref{vstruct} and define its {\it interface} $I[v]$ as the set \begin{equation*}
I[v]:=\{h_1,h_2,\ldots,h_N\}. \end{equation*} Moreover, for any function $u:[a,b]\rightarrow\mathbb{R}$ and for any closed subset $K\subset\mathbb{R}\backslash\{\pm1\}$, the {\it interface} $I_K[u]$ is defined by \begin{equation*}
I_K[u]:=u^{-1}(K). \end{equation*} Finally, we recall the notion of {\it Hausdorff distance} between any two subsets $A$ and $B$ of $\mathbb{R}$ denoted with $d(A,B)$ and given by
\begin{equation*}
d(A,B):=\max\biggl\{\sup_{\alpha\in A}d(\alpha,B),\,\sup_{\beta\in B}d(\beta,A)\biggr\}, \end{equation*}
where $d(\beta,A):=\inf\{|\beta-\alpha|: \alpha\in A\}$, for every $\beta\in B $.
Before stating the main result of this subsection (see Theorem \ref{thm:interface}), we prove the following lemma which is merely variational, meaning that it does not take into account the equation \eqref{eq:Q-model} , and establishes that, if a function $u\in H^1([a,b])$ is close to $v$ in $L^1$ and its energy $E_\varepsilon[u]$ (defined in \eqref{eq:energy}) exceeds for a small quantity with respect to $\varepsilon$ the minimum energy to have $N$ transitions, then the distance between the interfaces $I_K[u]$ and $I_K[v]$ remains small. \begin{lem}\label{lem:interface} Assume that $Q\in C^1(\mathbb{R})$ satisfies \eqref{eq:Q-ass1}-\eqref{eq:Q-ass2}, $F\in C^1(\mathbb{R})$ satisfies \eqref{ipoF1}-\eqref{ipoF2} with $\theta\geq 2$ and let $v$ be as in \eqref{vstruct}. Given $\delta_1\in(0,r)$ and a closed subset $K\subset\mathbb{R}\backslash\{\pm1\}$, there exist positive constants $\hat\delta,\varepsilon_0$ such that, if for all $\varepsilon\in(0,\varepsilon_0)$ $u\in H^1([a,b])$ satisfies \begin{equation}\label{eq:lem-interf}
\|u-v\|_{{}_{L^1}}<\hat\delta \qquad \quad \mbox{ and } \qquad \quad E_\varepsilon[u]\leq Nc_\varepsilon+M_\varepsilon, \end{equation} for some $M_\varepsilon>0$ and with $E_\varepsilon[u]$ defined in \eqref{eq:energy}, we have \begin{equation}\label{lem:d-interfaces}
d(I_K[u], I[v])<\tfrac12\delta_1. \end{equation} \end{lem} \begin{proof} Fix $\delta_1\in(0,r)$ and choose $\rho>0$ small enough that \begin{equation*}
I_\rho:=(-1-\rho,-1+\rho)\cup(1-\rho,1+\rho)\subset\mathbb{R}\backslash K, \end{equation*} and \begin{equation*}
\inf\left\{\varepsilon^{-1}\left|\int_{\xi_1}^{\xi_2}Q\left(\varepsilon^2 J_\varepsilon(F(s))\right)\,ds\right| : \xi_1\in K, \xi_2\in I_\rho\right\}>2M_\varepsilon, \end{equation*} where \begin{equation*}
M_\varepsilon:=2N\varepsilon^{-1}\max\left\{\int_{1-\rho}^{1}Q\left(\varepsilon^2 J_\varepsilon(F(s))\right)\,ds, \, \int_{-1}^{-1+\rho}Q\left(\varepsilon^2 J_\varepsilon(F(s))\right)\,ds \right\}. \end{equation*} By using the first assumption in \eqref{eq:lem-interf} and by reasoning as in the proof of \eqref{2points} in Proposition \ref{prop:lower}, we can prove that, if we consider $h_i$ the discontinuous points of $v$, then for each $i= 1, \dots , N$ there exist \begin{equation*}
x^-_{i}\in(h_i-\delta_1/2,h_i) \qquad \textrm{and} \qquad x^+_{i}\in(h_i,h_i+\delta_1/2), \end{equation*} such that \begin{equation*}
|u(x^-_{i})-v(x^-_{i})|<\rho \qquad \textrm{and} \qquad |u(x^+_{i})-v(x^+_{i})|<\rho. \end{equation*} Now suppose by contraddicition that \eqref{lem:d-interfaces} is violated. Using \eqref{eq:strangeineq+}, we deduce \begin{align}
E_\varepsilon[u]\geq&\sum_{i=1}^N\left|\varepsilon^{-1}\int_{u(x^-_{i})}^{u(x^+_{i})}Q\left(\varepsilon^2 J_\varepsilon(F(s))\right)\,ds\right|\notag\\
& \qquad +\inf\left\{\left|\varepsilon^{-1}\int_{\xi_1}^{\xi_2}Q\left(\varepsilon^2 J_\varepsilon(F(s))\right)\,ds\right| : \xi_1\in K, \xi_2\in I_\rho\right\}. \label{diseq:E1} \end{align} On the other hand, we have \begin{align*}
\left|\varepsilon^{-1}\int_{u(x^-_{i})}^{u(x^+_{i})}Q\left(\varepsilon^2 J_\varepsilon(F(s))\right)\,ds\right|&\geq\varepsilon^{-1}\int_{-1}^{1}Q\left(\varepsilon^2 J_\varepsilon(F(s))\right)\,ds\\
&\qquad-\varepsilon^{-1}\int_{-1}^{-1+\rho}Q\left(\varepsilon^2 J_\varepsilon(F(s))\right)\,ds\\
&\qquad -\varepsilon^{-1}\int_{1-\rho}^{1}Q\left(\varepsilon^2 J_\varepsilon(F(s))\right)\,ds\\
&\geq c_\varepsilon-\frac{M_\varepsilon}{N}. \end{align*} Substituting the latter bound in \eqref{diseq:E1}, we deduce \begin{equation*}
E_\varepsilon[u]\geq Nc_\varepsilon-M_\varepsilon+\inf\left\{\left|\varepsilon^{-1}\int_{\xi_1}^{\xi_2}Q\left(\varepsilon^2 J_\varepsilon(F(s))\right)\,ds\right| : \xi_1\in K, \xi_2\in I_\rho\right\}, \end{equation*} which implies, because of the choice of $\rho$, that \begin{align*}
E_\varepsilon[u]>Nc_\varepsilon+M_\varepsilon, \end{align*} which is a contradiction with assumption \eqref{eq:lem-interf}. Hence, the bound \eqref{lem:d-interfaces} is true and the proof is completed. \end{proof}
Finally, thanks to Theorems \ref{thm:main}, \ref{thm:main2} and Lemma \ref{lem:interface}, we can prove the main result of the present subsection, which provides information about the slow motion of the transition layers as $\varepsilon$ goes to 0. In particular, we highlight that the minimum time so that the distance between the interface of the time-dependent solution $u^\varepsilon(\cdot,t)$ and that one of the initial datum becomes greater than a fixed quantity is exponentially big with respect to $\varepsilon$ in the critical case $\theta=2$, while only algebraically large in the super critical case $\theta>2$.
\begin{thm}\label{thm:interface} Assume that $Q\in C^1(\mathbb{R})$ satisfies \eqref{eq:Q-ass1}-\eqref{eq:Q-ass2} and that $F\in C^1(\mathbb{R})$ satisfies \eqref{ipoF1}-\eqref{ipoF2} with {$\theta\geq 2$}. Let $u^\varepsilon$ be the solution of \eqref{eq:Q-model}-\eqref{eq:Neu}-\eqref{eq:initial}, with initial datum $u_0^{\varepsilon}$ satisfying \eqref{eq:ass-u0} and \eqref{eq:energy-ini}. Given $\delta_1\in(0,r)$ and a closed subset $K\subset\mathbb{R}\backslash\{\pm1\}$, set \begin{equation*}
t_\varepsilon(\delta_1)=\inf\{t:\; d(I_K[u^\varepsilon(\cdot,t)],I_K[u_0^\varepsilon])>\delta_1\}. \end{equation*} Then, there exists $\varepsilon_0>0$ such that if $\varepsilon\in(0,\varepsilon_0)$ \begin{equation*}
t_\varepsilon(\delta_1)> \left\{ \begin{aligned} & \exp(A/\varepsilon) \quad &\mbox{if} \quad &\theta=2, \\ & \varepsilon^{-k_j} \quad &\mbox{if} \quad &\theta>2, \end{aligned}\right. \end{equation*} where $A$ and $k_j$ are defined as in Propositions \ref{prop:lower} and \ref{prop:lower_deg}, respectively. \end{thm} \begin{proof} First of all we notice that if $u_0^\varepsilon$ satisfies \eqref{eq:ass-u0} and \eqref{eq:energy-ini}, then there exists $\varepsilon_0>0$ so small that $u_0^\varepsilon $ automatically verifies assumption \eqref{eq:lem-interf} for all $\varepsilon\in(0,\varepsilon_0)$.
Hence, we are in the position to apply Lemma \ref{lem:interface}, obtaining that for all $\varepsilon\in(0,\varepsilon_0)$ \begin{equation}\label{interfaces-u0}
d(I_K[u_0^\varepsilon], I[v])<\tfrac12\delta_1. \end{equation} Now, for each fixed $\varepsilon\in(0,\varepsilon_0)$, we consider $u^\varepsilon(\cdot,t)$ for all time $t>0$ such that \begin{equation}\label{time} t\leq\left\{ \begin{aligned} & \exp(A/\varepsilon) \quad &\mbox{if} \quad &\theta=2, \\ & \varepsilon^{-k_j} \quad &\mbox{if} \quad &\theta>2. \end{aligned}\right. \end{equation}
Then, $u^\varepsilon(\cdot,t)$ satisfies the first condition in assumption \eqref{eq:lem-interf} either thanks to \eqref{eq:limit} if $\theta=2$ or by \eqref{eq:limit-deg} if $\theta>2$. The second condition in \eqref{eq:lem-interf} can be easily deduced observing that the energy $E_\varepsilon[u^\varepsilon](t)$ is a non-increasing function of $t$. Then, we have also that \begin{equation}\label{interfaces-u}
d(I_K[u^\varepsilon(\cdot,t)], I[v])<\tfrac12\delta_1 \end{equation} for all $t$ as in \eqref{time}. Combining \eqref{interfaces-u0} and \eqref{interfaces-u}, and using the triangle inequality, we deduce that \begin{equation*}
d(I_K[u^\varepsilon(\cdot,t)],I_K[u_0^\varepsilon])<\delta_1 \end{equation*} for all $t$ as in \eqref{time}, as desired. \end{proof}
\begin{rem} We underline, according to Theorem \ref{thm:interface}, one must wait an extremely long time (which is either exponentially or algebraically long, depending on wether $\theta=2$ or $\theta >2$) to see an appreciable change in the position of the zeros of $u^\varepsilon$. Once again, this proves a slow dynamics of the solution only because $\theta \geq 2$; indeed, in such a case we are sure that an initial datum $u_0^\varepsilon$ satisfying \eqref{eq:ass-u0}-\eqref{eq:energy-ini} is neither a stationary solution nor is close to it, implying that there exists a {\it finite} time $\bar t$ such that the solution $u^\varepsilon(\cdot, \bar t)$ will drift apart from $u_0^\varepsilon$. Hence, proving that the layers of $u^\varepsilon(\cdot, t)$ stay close to the layers of $u_0^\varepsilon$ for long times is not a trivial result. \end{rem}
\subsection{Numerical experiments}\label{numerics}
We conclude the paper with some numerical simulations showing the slow evolution of the solutions to \eqref{eq:Q-model}-\eqref{eq:Neu}-\eqref{eq:initial} rigorously described in the previous analysis. All the numerical computations, done for the sole purpose of illustrating the theoretical results, were performed using the built-in solver \texttt{pdepe} by \textsc{Matlab}$^\copyright$, which is a set of tools to solve PDEs in one space dimension. In all the examples we consider equation \eqref{eq:Q-model} with $Q$ given by one of the two explicit functions of \eqref{ex:fluxfunction}, while $F(u)=\frac{1}{2\theta}|1-u^2|^\theta$ (see \eqref{F:ex}) with different values of $\theta \geq 2$, depending on whether we aim at showing exponentially or algebraic slow motion.
\begin{figure}\label{Num1}
\end{figure}
\subsubsection{Example no. 1} We start with an example illustrating the result of Theorem \ref{thm:main}; indeed, in this case $F(u)=\frac{1}{4}(1-u^2)^2$ (hence we are in the critical case $\theta=2$), and we choose $Q(s)=\frac{s}{1+s^2}$. In the left picture of Figure \ref{Num1} we can see the solution maintains six transitions until $t=7*10^3$ and, suddenly, the closest layers collapse; after that, one has to wait up to $t=2.9*10^4$ to see another appreciable change in the solution, see the right hand picture of Figure \ref{Num1}. It is worth mentioning that the distance between the first layers which disappear (left hand picture) is $d=0.9$, while the distance of the layers disappearing in the right hand picture is $d=1$; hence, a \emph{small} variation in the distance between the layers at the initial time $t=0$ gives rise to a \emph{big} change on the time taken for the solution to annihilate them.
\begin{figure}
\caption{In this figure we consider the same problem as in Figure \ref{Num1}, with the only difference $Q(s)= \frac{\alpha s}{1+s^2}$, $\alpha>0$.
In the left hand side $\alpha=1/4$, so that $\mathcal Q=1/4$, while on the right hand side $\alpha=\mathcal Q=2$.}
\label{Num2}
\end{figure}
\subsubsection{Example no. 2} In this second numerical experiment, we emphasize which is the role of the constant $\mathcal Q=\max \left\{ Q'(s) \, : \, s \in [-\kappa, \kappa] \right\}$ in the metastable dynamics of the solutions; in particular, and as we noticed in Remark \ref{rem:Q}, the constant $\mathcal Q$ affects the evolution of the solution, since it appears in the minimum time needed for the solution to {drift apart} from its initial transition layer structure (see Theorem \ref{thm:main}). To be more precise, if $\mathcal Q=1$ as in the example of Figure \ref{Num1}, the first {\it bump} collapses at $t \approx 7190$, while the second at $t \approx 3*10^4$; such result has to be compared with Figure \ref{Num2}, where all the data are the same as Figure \ref{Num1} except for the choice of $\mathcal Q$. In the left hand side of Figure \ref{Num2} we choose $Q$ such that $\mathcal Q=1/4$, and we can see that the evolution becomes much slower, as the first bump collapses at $t=10^{10}$, instead of $t=7190$; on the contrary, if $\mathcal Q=2$ (right hand picture), the dynamics accelerates and the first two bumps disappear respectively at $t\approx200$ and $t\approx470$.
\begin{figure}\label{Num3}
\end{figure}
\subsubsection{Example no. 3} In this example we show what happens if considering a discontinuous initial datum $u_0$ which is a small perturbation of the unstable equilibrium zero. In the left hand side of Figure \ref{Num3} we can see how, in extremely short times, such configuration develops into a {\it continuous} function with two transition layers; after that (right hand picture), we have to wait until $t \approx 10^{11}$ to see the first interface to collapse. In particular, such numerical experiment shows that, even if the solution does not satisfy the assumptions \eqref{eq:ass-u0}-\eqref{eq:energy-ini} at $t=0$, at $t=9$ we already entered into the framework described by Theorem \ref{thm:main} and we witness the exponentially slow motion of the solution. This picture is not surprising, since it seems to confirm the well known behavior of the solution to the linear diffusion equation $u_t= \varepsilon^2 u_{xx} -F'(u)$; indeed, in \cite{Chen} the author rigorously proves that there are different phases in the dynamics, the first one being the generation of a metastable layered solution, which is governed by the ODE $u_t = -F'(u)$. We conjecture that the same results hold true also in the nonlinear diffusion case \eqref{eq:Q-model}, being $Q(0)=0$.
\subsubsection{Example no. 4} In this last numerical experiment, we illustrate the results of Theorem \ref{thm:main2} by choosing $F$ as in \eqref{F:ex} with $\theta >2$. In both the pictures of Figure \ref{Num4} we select $Q(s)=se^{-s^2}$, and either the initial datum of Figure \ref{Num1} (left picture) or the discontinuous one of Figure \ref{Num3} (right picture), so that we can compare the two numerical experiments. In the left hand picture, since $\theta=4$, the time taken to see two bumps disappear is $t \approx 450$ (hence much smaller if compared to the right hand side of Figure \ref{Num1}, where one has to wait until $t \approx 10^4$). Similarly, in the right side of Figure \ref{Num4} (where $\theta=3$), the time employed by the first interface to disappear is $t \approx 8*10^4 \ll 10^{11}$, that is the time exhibited in the right hand side of Figure \ref{Num3}.
\begin{figure}\label{Num4}
\end{figure}
\end{document} | arXiv |
Water reuse and growth inhibition mechanisms for cultivation of microalga Euglena gracilis
Mingcan Wu1,2,3,
Ming Du1,
Guimei Wu1,
Feimiao Lu1,
Jing Li1,
Anping Lei1,
Hui Zhu3,
Zhangli Hu1,2 &
Jiangxin Wang ORCID: orcid.org/0000-0002-9917-067X1
Biotechnology for Biofuels volume 14, Article number: 132 (2021) Cite this article
Microalgae can contribute to more than 40% of global primary biomass production and are suitable candidates for various biotechnology applications such as food, feed products, drugs, fuels, and wastewater treatment. However, the primary limitation for large-scale algae production is the fact that algae requires large amounts of fresh water for cultivation. To address this issue, scientists around the world are working on ways to reuse the water to grow microalgae so that it can be grown in successive cycles without the need for fresh water.
In this study, we present the results when we cultivate microalgae with cultivation water that is purified and reused. Specifically, we purify the cultivation water using an ultrafiltration membrane (UFM) treatment and investigate how this treatment affects: the biomass and biochemical components of the microalgae; characteristics of microalgae growth inhibitors; the mechanism whereby potential growth inhibitors are secreted (followed using metabolomics analysis); the effect of activated carbon (AC) treatment and advanced oxidation processes (AOPs) on the removal of growth inhibitors of Euglena gracilis. Firstly, the results show that E. gracilis can be only cultivated through two growth cycles with water that has been filtered and reused, and the growth of E. gracilis is significantly inhibited when the water is used a third time. Secondly, as the number of reused water cycles increases, the Cl− concentration gradually increases in the cultivation water. When the Cl− concentration accumulates to a level of fivefold higher than that of the control, growth of E. gracilis is inhibited as the osmolality tolerance range is exceeded. Interestingly, the osmolality of the reused water can be reduced by replacing NH4Cl with urea as the source of nitrogen in the cultivation water. Thirdly, E. gracilis secretes humic acid (HA)—which is produced by the metabolic pathways for valine, leucine, and isoleucine biosynthesis and by linoleic acid metabolism—into the cultivation water. Because HA contains large fluorescent functional groups, specifically extended π(pi)-systems containing C=C and C=O groups and aromatic rings, we were able to observe a positive correlation between HA concentration and the rate of inhibition of E. gracilis growth using fluorescence spectroscopy. Moreover, photosynthetic efficiency is adversely interfered by HA, thereby reductions in the synthetic efficiency of paramylon and lipid in E. gracilis. In this way, we are able to confirm that HA is the main growth inhibitor of E. gracilis. Finally, we verify that all the HA is removed or converted into nutrients efficiently by AC or UV/H2O2/O3 treatments, respectively. As a result of these treatments, growth of E. gracilis is restored (AC treatment) and the amount of biomass is promoted (UV/H2O2/O3 treatment).
These studies have important practical and theoretical significance for the cyclic cultivation of E. gracilis and for saving water resources. Our work may also provide a useful reference for other microalgae cultivation.
Unicellular eukaryotic microalgae are a diverse and ubiquitous group of plants that are a promising biomass source and a biological feedstock [1, 2]. However, the use of microalgae as a feedstock is limited because a large amount of water is needed for its cultivation, and this negatively affects economic viability and environmental sustainability [3]. Wastewater treatment is currently being used to recycle the water that is necessary for microalgae cultivation [4]. It has been reported that to produce 1 kg of algae biomass, 1564 L of water are required under pond conditions [5]. Reusing cultivation water can reduce water usage and nutrient requirements as well as the need for algal wastewater treatment [3]. Consequently, water reuse after algae harvesting is essential for the economic viability of the microalgae industry and for environmental sustainability.
The effects of water reuse on algae growth are different across algae taxa [6]. The most researched taxa are green algae, diatoms, cyanobacteria, haptophytes, eustigmatophytes, chrysophytes, and xanthophytes [3]. However, among the genus of Euglena spp. algae, especially, there have been no reports of E. gracilis cultivation with reused water. E. gracilis is a unicellular flagellated alga characterized by the absence of a cell wall. It produces a wide variety of bioactive compounds such as paramylon, carotenoids, tocopherols, euglenophycin, and lipids. It has tremendous potential for metabolic engineering and commercialization [7]. Therefore, a deeper understanding of water reuse in the cultivation of the microalga E. gracilis, and the mechanisms underlying its growth-inhibiting secretions, are urgently needed.
After the microalgae assimilate nutrients ions, unabsorbed counter ions such as Cl−, Na+, and K+ from NH4Cl, NaHCO3, and KH2PO4, respectively, can accumulate in the cultivation water. When these ions accumulate, the osmotic pressure of the cultivation water increases, thereby inhibiting microalgae growth [8, 9]. Therefore, finding a suitable medium, which can balance the osmotic pressure between microalgae and cultivation water, is important for improving the effects of water reuse.
In addition to accumulated ions, excreted metabolites, such as dissolved organic matter (DOM) from microalgae is considered to be the main cause of negative biomass growth [3, 6, 10]. For Scenedesmus sp. LX1, DOM concentrations between 6.4 and 25.8 mg/L in reused water resulted in a decrease in the maximum algae cells density and the maximum growth rate by 50–80% and 35–70%, respectively [11]. The DOM in that study was classified into two fractions: hydrophobic or hydrophilic. Each of these fractions was further classified as acids, neutrals, or bases for a total of six fractions. In that study, all six fractions showed inhibited algal growth. Moreover, Lu et al. [10] who also used this fractionation approach, reported that the DOM of Scenedesmus acuminatus in the reused water included palmitic acid and octadecanoic acid, both of which inhibited the growth of this algae species. Although many studies have attempted to characterize the growth inhibitor present in DOM, it is unclear which major metabolic pathways within microalgae cells regulate and secrete these inhibitory substances into the cultivation water.
Recently, many researchers have tried to use traditional methods of wastewater treatment on microalgae cultivation water. Zhang et al. [12] reported that the removal of DOM with activated carbon (AC) in the reused water of cultivated Nannochloropsis oceanica moderately reduced growth inhibition and lipid accumulation. Moreover, the AC treatment significantly increased the final dry weight of S. acuminatus to 2.33 ± 0.04 g/L, which was almost the same as the dry weight obtained after growth in fresh media [13]. Advanced oxidation processes (AOPs) are another type of treatment technology in which organic pollutants are destroyed by powerful oxidizing agents [14, 15]. O3, UV/H2O2 have been successfully applied in the treatment reused water for the cultivation of Scenedesmus sp. LX1 [16] and S. acuminatus GT-2 [17], respectively. To date, few studies have investigated the removal of DOM by AC or AOPs in reused water of cultivated E. gracilis.
In this study, the main objectives were: (1) to identify the effect of treatment with ultrafiltration membrane (UFM) on the biomass and biochemical components of cultivated E. gracilis; (2) to identify the characteristics of the growth inhibitors in reused water and uncover the mechanism whereby potential inhibitors are secreted by E. gracilis via metabolomics analysis; (3) and to evaluate the effect of AC and AOPs on the removal of growth inhibitors.
Effects of reused water on the growth of E. gracilis
Since the UFM can cut off substances with a molecular weight ≥ 50 kDa, viruses, bacteria, macromolecular proteins, polysaccharides, and other substances can be filtered out [10, 12], so only unconsumed ions and DOM remain in the reused water. This study found that the DW of algae cells gradually decreased in the reused water with successive cycles. The DW of algae with each cycle of cultivation decreased by 13.1% (UFM-R1, p < 0.05), 28.6% (UFM-R2, p < 0.05), and 79.2% (UFM-R3, p < 0.01) compared to the control group on the last day of cultivation (Fig. 1). By the third cycle of cultivation, the growth of algae cells had been severely inhibited. This suggests that the increase in the presence of growth inhibitors with successive cycles of water reuse reduces algal growth beyond a tolerable range. This phenomenon is similar to the growth inhibition observed for other microalgae such as S. acuminatus [10], Chlorella. SDEC-18 [18], and N. oceanica [12]. We postulate that accumulated ions and algae cells secretions of DOM in the reused water are the main factors that inhibit the growth of E. gracilis.
Effects of reused water on the growth of E. gracilis. UFM-R0, -R1, -R2, and -R3 represent the number of times water is reused and treated with UFM; The letters F, S, and T combined with 0, 2, 4, 6, 8, and 10 represent the number of days for algae cultivation under the first (F), second (S), and third (T) conditions of reused water, respectively. Asterisk represents p < 0.05; double asterisk represents p < 0.01. The values are represented by mean ± SD, where n = 3
The effect of accumulated ions on the growth of E. gracilis
Microalgae can selectively absorb some types of ions from inorganic nutrients and assimilate them into their own organic matter. But some ions such as Cl− and Na+, cannot be absorbed by microalgae. These residual ions, which accumulate in reused water, can destroy the balance of osmolality of the microalgae, thereby inhibiting their growth [8, 9]. This study found that when algae cells were cultivated in more than eightfold the concentration of PEM medium (NH4Cl as the nitrogen source) compared to the control, the relative cell density was lower, algal cytochromes were almost absent, and the algae cells were elongated (Fig. 2a, b). When the concentration of the culture medium was within fivefold of the control medium, the DW of algal cells was about 2.7 g/L, which was not significantly different from the control group (p > 0.05). However, when the concentrations of the medium were increased to above fivefold that of the control medium, the DW gradually decreased. In fact, when the concentration of the medium was increased by eight-, nine-, and tenfold, the DW of cells decreased by 81.3% (p < 0.01), 85.0% (p < 0.01), and 92.5% (p < 0.01), respectively (Fig. 2c). In addition, the Cl− concentration was increased as the medium concentration increases, the maximal concentration was reached 11,996.4 mg/L in the tenfold medium (Fig. 2d), suggesting that accumulated Cl− in the medium may be a key growth inhibitor of E. gracilis.
Effects of accumulated ions on the growth of E. gracilis. a morphological changes and relative algae cells density (NH4Cl as nitrogen source). b Cytochrome changes in algae cells (NH4Cl as nitrogen source). c The biomass of algae cells cultured under NH4Cl and urea as nitrogen sources. d The Cl− ions concentration. e The morphological changes and relative algae cells density (urea as a nitrogen source). f Cytochrome changes of algae cells (urea as a nitrogen source). g The salinity and h, osmolality of the medium with NH4Cl and urea as nitrogen sources, respectively. "Folds-medium" on the x-axis in c, d, g, and h represent different multiples of PEM medium concentration. NS, asterisk, and double asterisk represent p > 0.05, p < 0.05, and p < 0.01, respectively. The values are represented by mean ± SD, where n = 3
To prove the above hypothesis, we used urea instead of NH4Cl as the nitrogen source (equal nitrogen content) and found that as the concentration of urea increased, the DW of algae cells increased significantly. When the concentration of the culture medium was four to tenfold, the biomass of algal cells was stably maintained at about 2.7 g/L, and there was no significant difference among them (p > 0.05) (Fig. 2c). In addition, when urea was used as a nitrogen source, the cells appeared to be fuller. With an increase in the concentration of the culture medium, the relative cells density increased and the relative content of chlorophyll gradually increased as well (Fig. 2e, f). At the same time, the salinity and osmolality in the medium were much lower than those of the medium using NH4Cl as the nitrogen source. For example, the salinities at tenfold medium concentration were 29.4 psu versus 9.1 psu for NH4Cl versus urea, respectively. The osmolalities under these conditions were 727.0 mosm versus 167.0 mosm for NH4Cl versus urea, respectively. The salinity value for NH4Cl was 3.2-fold greater than urea (Fig. 2g, p < 0.01), and the osmolality for NH4Cl was 4.3-fold greater than urea (Fig. 2h, p < 0.01). These results show that the growth of algae cells was not inhibited with a fivefold increase in salinity (< 15.6 psu) and osmolality (< 361.1 mosm) of the medium, and we can confirm that during the UFM-R3 culture cycle, the growth of E. gracilis was not hindered by the accumulated ions in the reused water. This phenomenon has also been confirmed by the cultivation of S. acuminatus in reused water [10, 19]. Our work also showed that the growth of E. gracilis had a certain tolerance range to ions. If this tolerance range was exceeded, the growth of E. gracilis was inhibited. In addition, it was determined that the traditional medium PEM with NH4Cl as the nitrogen source was not suitable for the continuous recycling of cultivation water or for batch-fed cultivation of E. gracilis (such as heterotrophic batch-fed fermentation). When urea is used, it serves as an ideal nitrogen source because it reduces the salinity and osmolality in the culture medium.
Identification of growth inhibitors in E. gracilis secretions
The growth of microalgae is not affected by certain osmotic pressures for reused water, so growth inhibitors may exist in the DOM secreted by microalgae. However, some DOM can promote the growth of microalgae while some have an inhibitory effect on microalgae growth [3], so further study of these DOM characteristics is required. This study also found that E. gracilis continuously secreted DOM during the culture process. By the time UFM-R3 was reached, the DOM concentration was 189.21 mg/L, while the control group contained only 54.92 mg/L DOM, a 3.4-fold difference (p < 0.01) (Fig. 3a). This indicates that, at elevated concentrations, DOM may have an inhibitory effect on the growth of E. gracilis.
Identification of growth inhibitors in E. gracilis secretions. a The content of DOM in reused water by using ultrafiltration membrane treatment. b Three-dimensional (3D) fluorescence excitation–emission matrix (FEEM) spectra of DOM (I and II, AP, aromatic proteins; III, FA, fulvic acid-like, IV, SMBM, soluble microbial by product-like material; V, HA, humic acid). c Percent of 6 DOM fractions from reused water of E. gracilis. d UV254 of 6 DOM fractions in 80-fold medium (HoB hydrophobic bases, HiN hydrophilic neutrals, HoA hydrophobic acids, HiB hydrophilic bases, HiA hydrophilic acids, HoN hydrophobic neutrals). e The inhibition rate of growth (IG%) of E. gracilis with stress from different DOM fractions. f The relationship between UV254 and IG% (inset: trendline linear equation and R2 value). Asterisk represents p < 0.05, double asterisk represents p < 0.01. The values represent mean ± SD, where n = 3
3D-FEEM fluorescence spectroscopy is fast and has excellent selectivity and sensitivity for fluorescent substances [20]. Therefore, in this study, this technique was used to identify the types of DOM secreted by algae cells. Chen et al. [21] used 3D-FEEM fluorescence spectroscopy to identify the following substances in the DOM present in cultivation water: aromatic proteins (AP), fulvic acid-like substances (FA), soluble microbial by product-like material (SMBM), and HA. We used those assignments to determine which types of DOM were present in our cultivation water samples (they are labeled with roman numerals in the spectrum in Fig. 3b. See caption b). It can be seen from Fig. 3b that the abundance of organic compounds with fluorescent signals in the reused water from high to low as: HA, SMBM, FA, and AP. Our spectra showed that HA was the potential main type of DOM present in E. gracilis secretions.
In order to further identify the growth inhibitors, we divided the DOM into six major fractions using fractional distillation (Fig. 3c). The percentages from high to low were: HiN (32%), HoA (27%), HoN (25%), HiB (7%), HoB (6%), and HiA (3%). From this result, we know that the DOM is mainly composed of HiN, HoA, and HoN, suggesting that HA, a potential inhibitor of E. gracilis, is composed of these organic acids. We also know that the slope of peaks in a UV spectrum (at 254 nm, given in AU/cm) for organic matter represents the content of organic functional groups that contribute to fluorescence, such as C=C bonds, C=O bonds, and aromatic rings. The importance of ultraviolet spectra for detecting pollutants in the water treatment process was described by Altmann et al. [22]. In this study, we tested the UV254 of 80-fold-concentrated DOM and found that the fluorescence intensity from high to low was: HoN (0.58 AU/cm), HoA (0.53 AU/cm), HiA (0.50 AU/cm), HiB (0.44 AU/cm), HoB (0.30 AU/cm), and HiN (0.24 AU/cm) (Fig. 3d). The inhibition of growth, IG%, of E. gracilis for each of these organic substances were: 28.8%, 25.0%, 24.6%, 19.2%, 12.1%, and 5.0% (Fig. 3e), respectively. It suggests that all of these DOM fractions can inhibit the growth of E. gracilis, especially HoN, HoA, and HiA. In addition, it is obvious that the UV254 absorption value is linearly related to the IG% for E. gracilis based on the graph in Fig. 3f for which R2 = 0.9. The degree of inhibition was positively correlated with the content of luminescent functional groups in the DOM. Based on these results, we confirmed that all fractions of DOM with C=O bonds, C=C bonds, and aromatic rings have an inhibitory effect on the growth of E. gracilis. In other words, inhibiting the growth of E. gracilis mainly depended on the concentration of different fractions.
According to the above results, DOM mainly includes HA, which was mainly composed of three organic compounds: HiN, HoA, and HoN (Fig. 3b). However, when using U254 signal to characterize these organics, in addition to HoA and HoN with relatively high signal intensity, HiA, HiB, and HoB also had an inhibitory effect on the growth of E. gracilis, suggesting that inhibitors other than HA may also be present in the recycled culture media. These growth inhibitory factors may be derived from SMBM, FA, and AP (Fig. 3b). These hydrophilic/hydrophobic fractions also have inhibitory effects on the growth of microalgae, such as FA has been proven to have an inhibitory effect on Scenedesmus species [13], indicating that this fraction, as well as HA with its highly fluorescent signals, may be potential inhibitors. However, both the concentration and the UV254 signal intensity of HiA, HiB, and HoB were lower than HoA and HoN derived from HA. In addition, although the concentration of HiN derived from HA was relatively higher, it obviously reduced the inhibitory effect on the growth of E. gracilis. Therefore, this study finally confirmed that the main growth inhibition of E. gracilis was HA, and the hydrophobic HoA and HoN organics fractions with higher content and higher UV254 signal intensity played a key inhibitory role. Lu et al. [10] only fractionated HoN-containing fatty acids and showed that they have an inhibitory effect on the growth of S. acuminatus. In addition, Zhang et al. [11] showed that all of the fractions could inhibit the growth of Scenedesmus sp. LX1, especially, HiB, HoB, and HiA. However, HoN and HoA showed the strongest inhibition of E. gracilis. This suggests that different microalgae may have different tolerances to different classifications of DOM. This scientific problem requires further research.
The influence of E. gracilis secretions on its physiology and biochemistry
The Fv/Fm ratio reflects the ability of microalgae to dissipate, absorb, and transmit light energy during photosynthesis. It is a useful parameter that indicates physiological state and growth rate, and is also an internal probe of the relationship between microalgae and their environment [13, 23, 24]. The Fv/Fm ratio for algae cells was only 0.12 in water containing HA, while that of the control group was 0.64, which is 5.3-fold difference (Fig. 4a, p < 0.01). From this result, it is obvious that HA significantly reduces the algae cell's photosynthetic efficiency. Similarly, studies on S. acuminatus [13] and Arthrospira platensis [23] also showed comparable Fv/Fm reductions when cultivated in reused water, which means that HA has a negative effect on the photosynthetic system of these microalgae too. Thus, this impact has a certain universality.
The effect of self-secreted DOM on (a) Fv/Fm ratio, (b) paramylon content, and (c) TFA content of E. gracilis. TFA total fatty acid, HA humic acid. Double asterisk represents p < 0.01. The values represent mean ± SD, where n = 3
The paramylon content and TFA content of E. gracilis in the experimental group containing DOM were 7.1% and 12.2%, respectively, while the control group was 21.2% and 35.2%. Both of these values were significantly lower than the control group, which showed a decrease of 66.5% (Fig. 4b, p < 0.01), and 65.3% (Fig. 4c, p < 0.01), respectively. These results were confirmed for the TFA of Scenedesmus sp. LX1 [11]. These results indicate that the HA secreted by E. gracilis may interfere with its own photosynthesis, and that this leads to inhibition of the synthesis of organic matter in the algae cells. The mechanism behind this process is worthy of our in-depth study in the future.
Study on the mechanism of E. gracilis growth inhibition by its own secretions
When UHPLC–QTOF-MS was used to detect metabolites in E. gracilis cells and cultivation media, the range of metabolites detected in negative ion mode was greater than that in positive ion mode, so this study only analyzes metabolites that were observed in negative ion mode to describe the mechanism whereby algae cells secrete DOM. With this analysis, we observed 4130 metabolites (Additional file 2). These metabolites were analyzed by PCA and OPLS-DA, and we can see clear separation between intracellular (IEG) and extracellular (EEG) metabolites (Additional file 1: Fig. S2, S3), indicating that there were significant differences in the metabolites in these two groups. When the OPLS-DA permutation test was performed on the data, the categorical variable Y was randomly changed 1000 times (Additional file 1: Fig. S4) and the original model R2Y was equal to 1, indicating that the established model conforms to the real situation for the sample data. The original model had a Q2 value equal to 0.997, which is very close to 1. This means that if a new sample were added to the model, it would fall within the existing distribution of data points. In general, the original model is robust and can explain the difference between the two sets of samples well. No overfitting was required to fit our data to it.
This study used VIP > 1 and a P-value < 0.05 to screen metabolites, and 108 different metabolites were obtained (see Additional file 2). According to the heat map cluster analysis, the relative concentration of 69 and 39 metabolites in the EEG and IEG were up-regulated, respectively (Additional file 1: Fig. S5). After these metabolites were annotated by the KEGG database, important metabolic pathways were screened according to their position and role in the relevant metabolic pathways (Additional file 2). According to the bubble chart, there are nine main metabolic pathways that are relevant: valine, leucine, and isoleucine biosynthesis; linoleic acid metabolism; arginine biosynthesis; the TCA cycle; pyruvate metabolism; purine metabolism; tyrosine metabolism; pyrimidine metabolism; and phenylalanine metabolism (Fig. 5). Among these, the first two are the key metabolic pathways. Some of the metabolites in these metabolic pathways were highly expressed inside the cell, and some were highly expressed outside the cell, and the latter group of metabolites may be secreted from the cell into the cultivation water. Three pathways—linoleic acid metabolism, the TCA cycle, and valine, leucine, and isoleucine biosynthesis—involve C=O and C=C bonds, while purine and pyrimidine metabolism contribute aromatic rings and C=O bonds. These metabolites accumulate in the medium and gradually become HA, which contains various functional groups (Fig. 6).
The top nine KEGG pathways of groups EEG and IEG are presented in the bubble chart. Each bubble in the bubble chart represents a metabolic pathway. The X-axis of the bubble and the bubble scale indicate the influence factor of the pathway in the topology analysis. The larger the size, the greater the influence factor; the Y-axis and the color of the bubble indicate the enrichment analysis. P value (take the negative natural logarithm, namely − log10(p), the redder the color, the smaller the P value, the more significant the enrichment degree. IEG, EEG represent intracellular and extracellular metabolites, respectively
The schematic diagram of the mechanism of E. gracilis secretion of growth inhibitors. Red font indicates that the relative concentration of metabolites outside the cell is greater than that inside the cell, and blue font indicates that the relative concentration of metabolites outside the cell was less than that inside the cell. The solid line represents the direct chemical reaction, and the dashed line represents the indirect chemical reaction. HA represents humic acid
Palmitic acid was one of the metabolites secreted by S. acuminatus to inhibit growth [10]. Similarly, this study found that the palmitic acid in linoleic acid metabolism was higher in concentration in the culture medium. Therefore, it was further proved that palmitic acid was also one of the key factors that inhibit the growth of E. gracilis. 2-Isopropylmalate is an intermediate product of valine, leucine, and isoleucine biosynthesis. Excessive secretion of this intermediate product from algal cells into the cultivation medium may also inhibit the growth of E. gracilis. However, exactly how this intermediate metabolite inhibits the growth of E. gracilis is a question that requires in-depth research in the future.
HA containing multiple functional groups can complex iron ions that are essential for photosynthesis in microalgae. However, Sun et al. [25] found that the underlying mechanism of the inhibitory effect for cyanobacteria was not to reduce the bioavailability of iron, but to inhibit the oxidative damage of cells mediated by peroxidase-mediated. More and more evidence shows that HA could directly interact with certain large plants and algae through their different functional groups, thereby interfering with photosynthesis and growth. Due to their low molecular weight (< 50 kDa), these substances can easily pass through cell membranes. When these quinone-containing metabolites enter the chloroplast, they interfere with the electron transport processes of photosynthesis [26, 27]. In fact, the toxic effects of quinones on the growth and photosynthesis of Scenedesmus strains have been confirmed [28]. In addition, we have previously found that there was no significant difference between the experimental group and the control group under heterotrophic conditions containing HA (data not disclosed) and the Fv/Fm ratio was significantly reduced (Fig. 4a), which means that these inhibitors may primarily attack the photosynthetic system of the E. gracilis chloroplast. However, no metabolites related to quinones were found in the different metabolites screened in this study (Additional file 1: Fig. S5), indicating that the photosynthetic machinery of E. gracilis was not affected by quinones. Moreover, it is possible that different functional groups (e.g. C=C and C=O bonds, aromatic rings) interfere with the electron transport processes of photosynthesis. How these functional groups in the compounds secreted by different metabolic pathways interfere with the photosynthetic system of microalgae requires further in-depth study.
Removal of growth inhibitors
Markiewicz et al. [29] have confirmed that DOM in sewage is adsorbed effectively by AC. The fluorescence spectrum after AC treatment showed that the fluorescence signal was very weak (Fig. 7a), indicating that almost all of the HA that can fluoresce had been removed. In addition, the growth curve for the experimental group was almost the same as that of the control group (Fig. 7d). By the time of the last day of culture, the DWs of algal cells were 2.4 g/L (experimental group) and 2.4 g/L (control group), with no significant difference (p > 0.05). This indicates that AC is effective at completely adsorbing and removing substances that inhibit the growth of E. gracilis. Although the reused water of cultivated N. oceanica [12] and S. acuminatus [13] showed a relatively significant effect from AC treatment, the biomass obtained was slightly lower than the control group, indicating that some growth inhibitors could not be removed. However, AC effectively adsorbs growth inhibitors secreted by E. gracilis in this study. We would like to develop recyclable AC technology, such as biological AC, to increase the utilization rate so that it can be more convenient for large-scale reuse of water resources to cultivate E. gracilis.
Removal of growth inhibitors. 3D-FEEM spectra of HA after: (a) activated carbon; (b) UV254/H2O2/O3-treatment. (c) UV254/H2O2-treatment; (d) Dry weight (DW) of E. gracilis after different treatments. The regions in these spectra are assigned to various substances found in DOM in Fig. 3b and its caption. HA represents humic acid. Asterisk represents p < 0.05. The values represent mean ± SD, where n = 3
AOPs have been widely used in the field of wastewater treatment. Oxidizers create a large number of free radicals under ultraviolet catalysis, such as hydroxyl radicals. These free radicals have strong oxidizing properties and can oxidize organic acids with unsaturated bonds [15]. According to the 3D-FEEM spectra of the reused water after AOP treatment, the fluorescence signal of the UV/H2O2/O3 group was weaker (Fig. 7b), followed by UV/H2O2 (Fig. 7c), indicating that the oxidation efficiency was higher with the participation of O3. In addition, the biomasses of the UV/H2O2/O3 group and the UV/H2O2 group on the last day were 2.89 g/L and 2.59 g/L, respectively, with the UV/H2O2/O3 experimental group significantly higher than the control group (p < 0.05). These results indicate that the advanced oxidation method not only eliminates the growth inhibitors, but may also oxidize these inhibitors into small organic molecules that could be absorbed by algae cells, thereby increasing their biomass. Our results show that the growth inhibitors were mainly HAs with luminescent functional groups (C=O and C=C bonds, aromatic rings). O3 and UV/H2O2 have been shown to work well for the treatment of Scenedesmus sp. LX1 [16] and S. acuminatus GT-2 [17] reused water, respectively. This study combines these two methods and shows that both methods together are more effective at removing growth inhibitors than either O3 or UV/H2O2 alone. Therefore, we believe that UV/H2O2/O3 is an ideal and efficient method for the removal of inhibitors of E. gracilis.
According to our previous research, the free radicals in the reused water after treatment with AOPs could also inhibit the growth of microalgae. Therefore, we need to optimize the AOP treatment process in the future by optimizing the treatment time, the concentration of the oxidizing agent, and the development of indicators for online detection of the concentration of free radicals in reused water (for example, the vitamin C reducing agent neutralization method). Use of AOPs is more conducive to the wide application of water reuse for algae cultivation. In addition, through these treatments, again it is clear that HA secreted by E. gracilis is a main growth inhibitor.
Based on the above results, we have proposed a cyclic culture model for E. gracilis (Fig. 8). The conceptual model is optimal when urea replaces NH4Cl as a nitrogen source and the reused water is filtered through an UFM and then treated with UV254/H2O2/O3. This model improves the availability of reused water, reduces the cost of cultivation, and increases the biomass of microalga E. gracilis.
The cyclic culture model of E. gracilis under reused water conditions. GI represents the growth inhibitors secreted by microalga E. gracilis
Our study demonstrated that cultivation water used three times had a significant inhibitory effect on the growth of E. gracilis. We replaced NH4Cl with urea and observed a reduction in the osmotic pressure caused by Cl− accumulation in the reused water, indicating that urea is an ideal nitrogen source. In addition, HA was identified as a main growth inhibitor of E. gracilis, and its content was positively related to the rate of growth inhibition. Moreover, we found that HA interfered with the photosynthetic efficiency of the algae and reduced the efficiency of paramylon and lipid synthesis. We determined that the key metabolic pathways for secreting these HA were valine, leucine, and isoleucine biosynthesis, and linoleic acid metabolism via metabolomics analysis. All HA been efficiently removed or converted into nutrients by AC or UV/H2O2/O3 treatment, respectively. As a result, the biomass has been recovered to the same levels as the control group (AC treatment) and even enhanced E. gracilis growth (UV/H2O2/O3 treatment). This result provides further confirmation that HA was a main growth inhibitor. An effective model for the cyclic culture of E. gracilis was thus proposed. These studies have important practical and theoretical significance for cyclic cultivation of E. gracilis or even other species of microalgae and for saving precious water resources.
Microalgae strain and growth conditions
Euglena gracilis CCAP 1224/5Z was obtained from the Culture Collection of Algae and Protozoa (CCAP) and maintained in our lab at Shenzhen University [30]. This strain was cultured in a modified photoautotrophic Euglena medium (PEM) according to Cramer and Myers [31]. PEM includes 1.8 g/L NH4Cl, 0.6 g/L KH2PO4, 1.2 g/L MgSO4·7H2O, 0.02 g/L CaCl2·2H2O, 0.55 μg/L Na2EDTA·2H2O, 2.0 μg/L Fe2(SO4)3, 0.05 μg/L CuSO4·5H2O, 0.4 μg/L ZnSO4·7H2O, 1.3 μg/L Co(NO3)2·6H2O, 1.8 μg/L MnCl2·4H2O, 0.01 μg/L Vitamin B1, and 0.0005 μg/L Vitamin B12. The pH value of the PEM medium was 3.6 adjusted with 3 mol/L NaOH and 1 mol/L HCl. The microalgae cells were grown in 2-L glass column photobioreactors with a 10-cm internal column diameter. The photobioreactors contained 1.5 L PEM medium that was stirred with 0.2 μm-filtered mixed gas (2% CO2, v/v, gas flow rate = 6 L/min) and illuminated with an LED lamp at a light intensity of 150 μmol photons m−2 s−1 (24: 0 h light–dark cycle). The temperature of the cultivation was maintained at 25.0 °C.
When the algae cells were cultured on day 10, the cultivation water sample (UFM-R0) was treated with ultrafiltration membrane (UFM) harvesting equipment [32] that was built in our laboratory (Additional file 1: Fig. S1). This equipment cuts off the molecular weight ≥ 50 kDa. One equivalent volume of the PEM nutrient medium was added to the UFM-R0 water sample. After sterilization, the solution was inoculated with microalgae to a final OD750 concentration of 0.2 for the microalgae suspension. After the first cultivation, water samples were cycled through this cultivation and UFM cycle three times and these subsequent cycles were named UFM-R1, R2, R3. Samples of the microalgae were taken every other day to monitor the cells' dry weight (DW).
DW was measured according to the method described by Wu et al. [33]. Briefly, 5 mL microalgae suspension was filtered through a preheated (105 °C, 24 h), pre-weighed glass microfiber filter (Whatman GF/C, 47 mm, UK). The filters were washed twice, each with 50 mL of 0.5 mol/L NH4HCO3. The filters were weighed after drying at 105 °C for 24 h to reach a constant mass. DW was calculated using Eq. (1):
$$\mathrm{DW}\frac{\mathrm{g}}{\mathrm{L}}= \frac{{w}_{\mathrm{a}}-{w}_{\mathrm{b}}}{v},$$
where "wa" and "wb" are the mass of the filters at the end and start of cultivation, respectively, and "v" is the volume of the microalgae suspension. Finally, the reused water samples and the algae cells were kept at − 80 °C to prepare for the next study.
Algae cells were cultured in PEM with nutrient concentrations of 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0.5, and 0.2-fold, with 1.8 g/L NH4Cl or 1.0 g/L urea (both of them contained equal nitrogen content) as the nitrogen source. When the algae cells were cultured on day 10, the morphology of the algae cells was observed with an inverted microscope (Leica DMI8, Leica Microsystems, Germany). In addition, the DW, Cl− concentration, salinity, osmolality, and the pigment changes of the algae cells in the culture medium were measured. Cl− concentration was measured using a capillary ion chromatograph (ICS 5000 +, Dionex, Sunnyvale, CA, USA) [10]. The salinity of the culture medium was measured using an Orion STAR A329 multiparameter meter (G10919, Thermo Fisher Scientific, USA). Osmolality was determined by measuring the freezing point of the solution with an automatic osmometer (Osmomat 030, Gonotec, Germany) according to Hadj-Romdhane et al. [9]. The osmolality was calculated using the freezing point depression, ΔT(°C), according to the following Eq. (2):
$$\mathrm{Osmolality} \mathrm{mosm}=\frac{\Delta T(^\circ{\rm C} )}{1.858},$$
where 1.858 is the cryoscopic constant, which is equal to the freezing point depression of a solution with an osmolality of 1 osmol.
The identification of E. gracilis growth inhibitors and quantitative analysis of their effect on growth
The differences in dissolved organic matter (DOM) content between the control (UFM-R0) and the reused water samples (UFM-R1, R2, R3) were measured using a total organic carbon analyzer (Multi N/C 2100, Analytik Jena, Germany).
DOM was determined with three-dimensional fluorescence excitation–emission matrix (3D-FEEM) spectrophotometry. Briefly, 3D-FEEM spectra were obtained using a fluorescence spectrophotometer (F-4500, Hitachi, Japan). The excitation (Ex) and emission (Em) slits were set to a bandpass of 5 nm. Ex wavelengths were scanned from 200 to 450 nm, and Em wavelengths were scanned from 220 to 550 nm. All of the 3D-FEEM spectral data were analyzed with Origin Pro 2018 software (https://www.originlab.com/origin).
The method of pretreated water sample was based on Leenheer [34]. The fractional method of DOM from water sample was optimized by Imai et al. [35] and Zhang et al. [36]. Briefly, the water sample was repeatedly passed through Amberlite XAD-8 resin at a flow rate of 5 mL/min 3 times, and then 2 bed volumes (BV) of 0.1 mol/L HCl were added to reverse the elution 3 times to obtain the hydrophobic bases (HoB). The pH of the water sample was adjusted to 2.0 using 0.1 mol/L HCl and 0.1 mol/L NaOH, then the water sample was passed through 3 columns with XAD-8 resin, D001(a microporous strong acidic ion exchange resin), and D201 (a macroporous strong base ion exchange resin) orderly. This sequence was repeated 3 times. After the water sample was passed through the resin, the remaining water sample contained only the hydrophilic neutrals (HiN). The XAD-8, D001, and D201 resins were back-eluted with 0.1 mol/L NaOH to obtain hydrophobic acids (HoA), hydrophilic bases (HiB), and hydrophilic acids (HiA), respectively. After the XAD-8 resin was air dried, ethanol was added and soxhlet extraction was run for 12 h to obtain the hydrophobic neutrals (HoN). All solvents were removed using rotary-evaporation (RV3, IKA, Germany). The volume of all DOM fractions was adjusted to 50 mL and transferred to centrifuge tubes. After diluting each group of DOM to a certain concentration, the percentage of the DOM in each fraction was measured by using a total N/C analyser (Multi N/C 2100, Analytik Jena, Germany). An absorbance value of 254 nm (UV254) of DOM in each fraction was measured by using a UV–Vis spectroscopic measurements (UV2350, UNICO, China).
The fractionated organic acids were added to fresh medium according to the percentage of DOM. After culturing the algae cells with this medium for 10 days, the DWs of the control and the experimental groups were measured. The inhibition rate of growth (IG%) was calculated using the following Eq. (3):
$$\mathrm{IG\%}=\frac{c-i}{c}\times 100\mathrm{\%},$$
where "c" is the DW of the control group and "i" is the DW of algae cells under the different fractionated DOM stressed conditions. Finally, the correlation analysis of UV254 and IG% were performed by using Origin Pro 2018 software.
The effect of growth inhibitors on the Fv/Fm ratio, paramylon and total fatty acid content
After the fractionated DOM was diluted according to the DOM content in the UFM-R3 reused water, as described above, fresh medium was added, and then the algae cells were cultivated under the afore mentioned culture conditions. The Fv/Fm ratio, the paramylon and the total fatty acid (TFA) content of the algae cells were measured after the 10th day of cultivation.
The Fv/Fm ratio was determined by dividing the variable fluorescence (Fv) by the maximum fluorescence (Fm), according to the method of Sha et al. [13]. The algae cells were placed in a quartz cube and maintained in the dark for 3 min prior to measurement of Fv/Fm. The Fv/Fm ratio for the algae cells was then measured at room temperature using a PHYTO-ED fluorimeter (Walz, Effeltrich, Germany).
Paramylon content was quantified using the method of Takenaka et al. [37] and Wu et al. [30] with the following modification: 2 mL, 30 mmol/L of EDTA chelating agent was added to 15 mL the algae cells suspensions. After each cell suspension was centrifuged and freeze-dried, 10 mg freeze-dried algae powder and 5 mL of acetone were transferred to a 15 mL of centrifuge tube and shaken for 30 s, then placed in a shaker for 2 h. After the tube was centrifuged at 2000×g for 5 min, the supernatant was removed. 1.5 mL of 1% sodium dodecyl sulfate (SDS) solution was added to the tube, then the contents were transferred into a 1.5-mL centrifuge tube and heated in a water bath at 85 °C for 2 h. Again, the supernatant was removed after the centrifuge tube was centrifuged at 2000×g for 5 min. The precipitate was washed and centrifuged in 1 mL deionized water, then oven-dried at 70 °C to a constant mass. The resulting precipitate was paramylon. The paramylon content was calculated as shown in Eq. (4):
$$\mathrm{Paramylon content \%}=\frac{P}{\mathrm{DW}}\times 100\mathrm{\%}$$
where "P" and "DW" are the DWs of the paramylon and the algae powder, respectively.
TFA content was determined using the method of Wu et al. [38]. Briefly, about 10 mg lyophilized cell pellets were disrupted by grinding three times under liquid nitrogen in the presence of methanol, chloroform, and formic acid (20:10:1, v:v:v) to extract the lipids from the algal biomass. The quantity of TFA content in extracts was measured by using an Agilent 7890B gas chromatograph coupled with a 5977A mass spectrometer (GC–MS).
The metabolic pathways of growth inhibitors secreted by E. gracilis were determined using metabolomics analysis
E. gracilis cells and cultivation water from UFM-R0 were collected and a metabolomics analysis was performed. The metabolites in the sample were extracted and analyzed according to the method mentioned by Wu et al. [38]. All of the metabolites were detected using ultra-high performance liquid chromatography coupled with quadrupole time-of-flight mass spectrometry (UHPLC–QTOF-MS). In this study, some metabolite peaks were detected after relative standard deviation noise reduction. Next, the missing values were increased by half of the minimum value. An internal standard normalization method was also employed in this data analysis. The final dataset containing the peak number, sample name, and normalized peak area was imported to a SIMCA16.0.2 software package (Sartorius Stedim Data Analytics AB, Umea, Sweden) for multivariate analysis. Data was scaled and logarithmically transformed to minimize the impact of both noise and high variance of the variables. After these transformations, principal component analysis (PCA), an analysis that reduces the dimensionality of the data, was carried out to visualize the distribution and the grouping of the samples. A 95% confidence interval in the PCA score plot was used as the threshold to identify potential outliers in the dataset. In order to visualize group separation and find significantly changed metabolites, orthogonal-projections-to-latent–structures discriminate analysis (OPLS-DA) was applied. Then a sevenfold cross-validation was performed to calculate the values of R2 and Q2. R2 indicates how well the data variance fits the model, and Q2 indicates how well a variable can be predicted. To check the robustness and predictive ability of the OPLS-DA model, 1000 permutations were further conducted. Afterward, the R2 and Q2 intercept values were obtained. The intercept value of Q2 represents the robustness and reliability of the model and the risk of overfitting (for the latter, smaller values are better). Furthermore, the value of variable importance in the projection (VIP) of the first principal component in OPLS-DA analysis was obtained. It summarizes the contribution of each variable to the model. The metabolites with VIP > 1 and p < 0.05 (Student's t-test) were considered to be significantly changed. In addition, commercial databases including KEGG database (http://www.genome.jp/kegg/) and MetaboAnalyst (http://www.metaboanalyst.ca/) were used for metabolic pathway enrichment analysis. From these analyses, bubble diagrams and metabolic pathways were made.
The methods of advanced oxidation that we used (UV/H2O2/O3 and UV/H2O2) were similar to the result of Hu et al. [16] and Wang et al. [17]. Briefly, an ozone generator, which produced O3 with a flow rate of 3000 mg/h, was fed into a solution containing 189.2 mg/L DOM (found in reused water UFM-R3) and 1% H2O2 for 2 h under UV254 ultraviolet lamp irradiation. This is the UV/H2O2/O3 experimental group. In the other set of experiments, O3 was not used, but all other conditions were the same. This is the UV/H2O2 experimental group. After all the treated water samples were freeze-dried, fresh medium was added to each. The method of AC filtration was similar to that of Sha et al. [13]. Briefly, the water sample containing the same UFM-R3 DOM concentration was filtered repeatedly through a chromatography column containing saturated AC (K04, Hainan Xingguang Active Carbon Co., LTD. China) for 2 h. Then 0.45-micron membrane filters were used to recover the water sample. DOM was characterized by using 3D-FEEM spectroscopy. Finally, the microalgae cells were cultured under the experimental conditions, and the DW of the algae cells was measured every other day.
All DWs, Cl− concentration, salinity, osmolality, DOM, UV254, IG%, Fv/Fm ratio, paramylon content, and TFA content tests were performed in triplicate and the average and standard deviations was reported. All data were statistically analyzed by Student's t-test analysis to investigate the difference between the control and experimental groups. p-values of less than 0.01 (p < 0.01) were considered significantly different, p < 0.05 values were considered statistically different, and p > 0.05 values were considered not statistically different (NS) compared to the control groups.
All data generated or analyzed in the present study are included in this article and in additional information.
UFM:
Ultrafiltration membrane
AC:
AOPs:
Advanced oxidation processes
Humic acid
Dissolved organic matter
AP:
Aromatic proteins
Fulvic-acid-like substances
SMBM:
Soluble microbial by product-like material
IEG:
CCAP:
Culture Collection of Algae and Protozoa
Photoautotrophic Euglena medium
3D-FEEM:
Three-dimensional fluorescence excitation-emission matrix
Excitation (Ex) and emission
DW:
Cells' dry weight
Hydrophobic bases
BV:
Bed volumes
HiN:
Hydrophilic neutrals
HoA:
Hydrophobic acids
HiB:
Hydrophilic bases
HiA:
Hydrophilic acids
HoN:
Hydrophobic neutrals
IG%:
The inhibition rate of growth
TFA:
The total fatty acid
Fm:
The maximum fluorescence
GC–MS:
Gas chromatograph coupled with a 5977A mass spectrometer
UHPLC–QTOF-MS:
Quadrupole time-of-flight mass spectrometry
PCA:
OPLS-DA:
Orthogonal-projections-to-latent–structures discriminate analysis
VIP:
The value of variable importance in the projection
NS:
Statistically different
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We acknowledge TopEdit LLC (www.topeditsci.com) for the linguistic editing and proofreading during the preparation of this manuscript. We thank Instrument Analysis Center of Shenzhen University for the assistance with an inverted microscope (Leica DMI8, Leica Microsystems, Germany) analysis.
This work was supported by the National Natural Science Foundation of China (Grant numbers 31670116, 41876188), Guangxi Innovation Drive Development Special Fund (Gui Ke AA18242047) and Grant Plan for Demonstration Project for Marine Economic Development in Shenzhen to Dr. Zhangli Hu; the Natural Science Foundation of Guangdong Province, China (2014A030313562), National Key R&D Program of China (2018YFA0902500), the Guangdong Innovation Research Team Fund (Grant number, 2014ZT05S078), and the Shenzhen Grant Plan for Science and Technology (Grant numbers, JCYJ20160308095910917, JCYJ20170818100339597).
Shenzhen Key Laboratory of Marine Bioresource and Eco-Environmental Science, Shenzhen Engineering Laboratory for Marine Algal Biotechnology, Guangdong Provincial Key Laboratory for Plant Epigenetics, College of Life Sciences and Oceanography, Shenzhen University, Shenzhen, 518060, China
Mingcan Wu, Ming Du, Guimei Wu, Feimiao Lu, Jing Li, Anping Lei, Zhangli Hu & Jiangxin Wang
Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and Guangdong Province, College of Optoelectronic Engineering, Shenzhen University, Shenzhen, 518060, China
Mingcan Wu & Zhangli Hu
College of Food Engineering and Biotechnology, Hanshan Normal University, Chaozhou, 521041, China
Mingcan Wu & Hui Zhu
Mingcan Wu
Ming Du
Guimei Wu
Feimiao Lu
Jing Li
Anping Lei
Hui Zhu
Zhangli Hu
Jiangxin Wang
MW conceived and designed the experiments; MW and JL performed the experiments; MW analyzed the data; MW, MD, GW, FL, JW, AL, ZH and HZ wrote and revised the manuscript. All authors read and approved the final manuscript.
Correspondence to Anping Lei or Jiangxin Wang.
All authors agree with submission to Biotechnology for Biofuel.
There are no competing interests associated with this manuscript.
: Fig. S1. Microalgae harvesting equipment. a, microalgae suspension; b, ultrafiltration membrane; c, pre-concentrated microalgae; d, reused water. Fig. S2. PCA analysis for groups EEG and IEG. IEG, EEG represent intracellular and extracellular metabolites, respectively. Fig. S3. OPLS-DA analysis for group EEG and IEG. IEG, EEG represent intracellular and extracellular metabolites, respectively. Fig. S4. Permutation test for group EEG and IEG. IEG, EEG represent intracellular and extracellular metabolites, respectively. Fig. S5. Heatmap of hierarchical clustering analysis for group EEG and IEG. IEG, EEG represent intracellular and extracellular metabolites, respectively; * represents the metabolite can be annotated in KEGG database.
. The data of statistical analysis results, different metabolites, and pathway analysis from the group EEG and IEG.
Wu, M., Du, M., Wu, G. et al. Water reuse and growth inhibition mechanisms for cultivation of microalga Euglena gracilis. Biotechnol Biofuels 14, 132 (2021). https://doi.org/10.1186/s13068-021-01980-4
Received: 29 December 2020
Microalgae
Euglena gracilis
Growth inhibitor | CommonCrawl |
\begin{document}
\setlength{\baselineskip}
{15.5pt}
\title{Strong uniqueness for both Dirichlet operators and stochastic dynamics to Gibbs measures on a path space with exponential interactions}
\author{ {\Large Sergio {Albeverio}} \\ Institut f\"ur Angewandte Mathematik, HCM and SFB 611 \\ Universit\"at Bonn \\ Endenicher Allee 60, D-53115 Bonn, Germany \\ e-mail: {\tt [email protected]}
\\ {\Large Hiroshi {Kawabi}} \footnote{Corresponding author. } \\ Department of Mathematics, Faculty of Science \\ Okayama University \\ 3-1-1, Tsushima-Naka, Kita-ku, Okayama 700-8530, Japan \\ e-mail: {\tt [email protected]}
\\ and
\\ {\Large Michael {R\"ockner}} \\ Fakult\"at f\"ur Mathematik, Universit\"at Bielefeld \\ Universit\"atsstra{\ss}e 25, D-33501 Bielefeld, Germany \\ e-mail: {\tt [email protected]} }
\date{}
\maketitle
\hspace{-8mm} {\bf { Abstract}}: We prove $L^{p}$-uniqueness of Dirichlet operators for Gibbs measures on the path space $C(\mathbb R, \mathbb R^{d})$ associated with exponential type interactions in infinite volume by extending an SPDE approach presented in previous work by the last two named authors.
We also give an SPDE characterization of the corresponding dynamics.
In particular, we prove existence and uniqueness of a strong solution for the SPDE, though the self-interaction potential is not assumed to be differentiable, hence the drift is possibly discontinuous.
As examples, to which our results apply, we mention the stochastic quantization of $P(\phi)_{1}$-, ${\rm exp}(\phi)_{1}$-, and trigonometric quantum fields in infinite volume. In particular, our results imply essential self-adjointness of the generator of the stochastic dynamics for these models.
Finally, as an application of the strong uniqueness result for the SPDE, we prove some functional inequalities for diffusion semigroups generated by the above Dirichlet operators. These inequalities are improvements of previous work by the second named author.
\\ {\bf Mathematics Subject Classifications (2000):}~
35R15, 35R60, 46N50, 47D07
\\ {\bf Keywords:} Strong uniqueness, Dirichlet operator, Gibbs measure, ${\rm exp}(\phi)_{1}$-quantum fields, $L^{p}$-uniqueness, Essential self-adjointness, SPDE,
Logarithmic Sobolev inequality.
\section{Introduction}
In recent years, there has been a growing interest in the study of infinite dimensional stochastic dynamics associated with models of Euclidean quantum field theory, hydrodynamics, and statistical mechanics, see, e.g., Liskevich--R\"ockner \cite{Lisk-Rock},
Da Prato--Tubaro \cite{DT2} and Albeverio--Liang--Zegarli\'nski \cite{ALZ}, resp. Albeverio--Flandoli--Sinai \cite{AFS}, resp. Albeverio--Kondratiev--Kozitsky--R\"ockner \cite{AKKR-book}. Equilibrium states of such dynamics are described by Gibbs measures. The stochastic dynamics corresponding to these states is given by a diffusion semigroup, see, e.g., Albeverio \cite{A}.
On some minimal domain of smooth functions, the infinitesimal generator of the semigroup coincides with the Dirichlet operator defined through a classical Dirichlet form of gradient type with a Gibbs measure. From an analytic point of view, it is very important to study $L^{p}$-uniqueness of the Dirichlet operator, that is, the question whether or not the Dirichlet operator restricted to the minimal domain has a unique closed extension in the $L^{p}$-space of the Gibbs measure under consideration, which generates a $C_{0}$-semigroup. As is well known, in the case of $p=2$, this uniqueness is equivalent to essential self-adjointness.
We recall that essential self-adjointness is crucial in applications to quantum mechanics to be sure that solutions of Schr\"odinger equations are unique.
This kind of uniqueness problem on infinite dimensional state spaces has been studied intensively by many authors. In particular, we refer to the recent work \cite{KR} by the last two named authors, where essential self-adjointness was proved in the case of $P(\phi)_{1}$-quantum fields in infinite volume by using an SPDE approach based on Da Prato--R\"ockner \cite{DR}. Besides, in \cite{KR} also the relationship between the corresponding dynamics and the $P(\phi)_{1}$-time evolution, which had been constructed as the strong solution of a parabolic SPDE (\ref{GL}) in Iwata \cite{Iwa2}, is discussed.
The first objective of the present paper is to prove $L^{p}$-uniqueness of the Dirichlet operator for all $p\geq1$, under much weaker conditions on the growth rate of the potential function of the Gibbs measure by a modification of the SPDE approach presented in \cite{KR}. Important new examples are ${\rm exp}(\phi)_{1}$-quantum fields in infinite volume in the context of Euclidean quantum field theory.
These models were introduced (for the case where $\mathbb R$ occurring in (\ref{formal-Gibbs})
below is replaced by a 2-dimensional Euclidean space-time $\mathbb R^{2}$ and where $d=1$) in H\o egh-Krohn \cite{Hk}, Albeverio--H\o egh-Krohn \cite{AHk} and further studied e.g., in Simon \cite{Si}, Fr\"ohlich \cite{Fro}, Albeverio--Gallavotti--H\o egh-Krohn \cite{AGHk} and Kusuoka \cite{Ku}.
More precisely, we are concerned with Gibbs measures on an infinite volume path space $C({\mathbb R}, {\mathbb R}^{d})$ given by the following formal expression: \begin{eqnarray}
& & \hspace{-25mm} Z^{-1}\exp \Big \{-\frac{1}{2}
\int_{\mathbb R}
\big( (-\Delta_{x}+m^{2})w(x),w(x) \big)_{\mathbb R^{d}} dx \nonumber \\ & & \hspace{25mm} -\int_{\mathbb R} dx \big ( \int_{\mathbb R^{d}} e^{(w(x), \xi )_{\mathbb R^{d}}}
\nu(d\xi) \big ) \Big \} \prod_{x\in \mathbb R} dw(x). \label{formal-Gibbs}
\end{eqnarray} Here $Z$ is a normalizing constant, $m>0$ denotes mass,
$\Delta_{x}:=d^{2}/dx^{2}$, $\nu$ is a bounded positive measure on $\mathbb R^{d}$ with compact support, and $\prod_{x\in \mathbb R} dw(x)$ stands for a (heuristic) volume measure on the space of maps from $\mathbb R$ into $\mathbb R^{d}$.
This has the interpretation of a quantized $d$-dimensional vector field with an interaction of exponential type in the $1$-dimensional space-time $\mathbb R$,
a model which is known as stochastic quantization of the ${\rm exp}(\phi)_{1}$-quantum field model (with weight measure $\nu$). We should mention that essential self-adjointness of the Dirichlet operators for such ${\rm exp}(\phi)_{1}$-quantum fields was not known yet, although the corresponding stochastic dynamics was constructed by using the Dirichlet form theory in Albeverio--R\"ockner \cite{AR} (see also Hida--Kuo--Potthoff--Streit \cite{HKPS} for an approach based on white noise calculus). Another important new example we handle is the model of trigonometric interactions, defined analogously to (\ref{formal-Gibbs}), but with $e^{(w(x), \xi )_{\mathbb R^{d}}} $ replaced by $\cos \{(w(x), \xi )_{\mathbb R^{d}}+\alpha \}, \alpha \in \mathbb R$. Such a model was studied (with $\mathbb R$ replaced by a 2-dimensional space-time $\mathbb R^{2}$ and assuming $d=1$) e.g., in Albeverio--H\o egh-Krohn \cite{AHk73}, Fr\"ohlich \cite{Fro} and Albeverio--Haba--Russo \cite{AHR}. In the present paper, we, in particular, prove essential self-adjointness of the corresponding Dirichlet operator for all these models. As a consequence, the Dirichlet operator associated with the superposition of polynomial, exponential and trigonometric interactions, is also essentially self-adjoint.
The second objective of the present paper is to discuss a characterization of the stochastic dynamics corresponding to the above Dirichlet operator.
Due to general theory, the stochastic dynamics constructed through the Dirichlet form approach solves the parabolic SPDE (\ref{GL}) weakly. However, we prove something much better, namely existence and uniqueness of a strong solution. We achieve this by first proving pathwise uniqueness for SPDE (\ref{GL}) and then applying the recent work of Ondrej\'at \cite{Ondre} on the Yamada--Watanabe theorem for mild solutions of SPDE.
As a consequence, we have the existence of a unique strong solution to SPDE (\ref{GL}) by using simple and straightforward arguments which do not rely on any finite volume approximations discussed in \cite{Iwa2} in case of polynomial (i.e., smooth) self-interaction.
Here we would like to emphasize that neither of the two uniqueness statements in Theorems \ref{ES} and \ref{ES2} respectively implies the other (cf. Remark \ref{compare} below).
The organization of this paper is as follows: In Section 2, we present the framework and state our results. There, we construct Gibbs measures as (\ref{formal-Gibbs}) rigorously by using $d$-dimensional Brownian motion and the ground states of Schr\"odinger operators on $L^{2}(\mathbb R^{d}, \mathbb R)$. After introducing our Dirichlet form and the corresponding Dirichlet operator, we state our main results (Theorems \ref{ES} and \ref{ES2}).
Section 3 contains the proofs, in which, we prove our main theorems. In our proof, we regard the Dirichlet operator as a perturbation of the infinite dimensional Ornstein--Uhlenbeck operator by a possibly discontinuous and unbounded drift term. Then we implement a modification of a technique developed in {\cite{KR}} which in turn is based on beautiful results of Da Prato, Tubaro and Priola in \cite{D, DT, priola} for Lipschitz perturbations of the Ornstein--Uhlenbeck operators.
(For other works on perturbed infinite dimensional Ornstein--Uhlenbeck operators, see also, e.g., Albeverio--R\"ockle--Steblovskaya \cite{ARS} and references therein.)
To handle our quite singular drift term, the first thing to do is to check its $L^{p}$-integrability. For this purpose, we make use of the asymptotic behavior for the ground state of the Schr\"odinger operator at infinity which, through the Feynman--Kac formula, has a close connection with the growth rate of the potential function.
We introduce an approximation scheme for the potential function by combining the Moreau--Yosida approximation (\ref{Moreau-Yosida}) with a further regularization (\ref{further}) inspired by \cite{DR, KR}, and this scheme works efficiently in our proof. To show existence and uniqueness of a strong solution to SPDE (\ref{GL}), we firstly identify our diffusion process
as a weak solution to an infinite system of SDEs. Secondly, we translate the infinite dimensional SDE into the weak form of SPDE (\ref{GL}), and show pathwise uniqueness for it based on Marinelli--R\"ockner \cite{Carlo}. In Section 4, we discuss some functional inequalities including the logarithmic Sobolev inequality (\ref{LSI-ineq}) as an application of Theorem \ref{ES2}, and in Section 5, we give another proof of the logarithmic Sobolev inequality (\ref{LSI-ineq}) by using Lemmas \ref{Hirokawa} and \ref{Conv-GS} on the approximation of the ground state. These lemmas play key roles when we combine some tightness arguments with the previous work to derive inequality (\ref{LSI-ineq}).
\section{Framework and Results}
We begin by introducing some notation and objects we will be working with. We define a weight function $\rho_{r}\in C^{\infty}({\mathbb R}, {\mathbb R} ), r\in {\mathbb R}$, by $\rho_{r}(x):=e^{r\chi (x)}$, $x\in {\mathbb R}$, where $\chi \in C^{\infty}({\mathbb R}, {\mathbb R})$ is a positive symmetric convex function satisfying $\chi (x)=\vert x \vert$ for $\vert x \vert \geq 1$. We fix a positive constant $r$ sufficiently small. In particular, we take $r>0$ such that $2r^{2}<K_{1}$ if $K_{1}>0$,
where the constant $K_{1}$ appears in condition {\bf (U1)} below. We set $E:=L^{2}({\mathbb R}, {\mathbb R}^{d};\rho_{-2r}(x)dx)$. This space is a Hilbert space with its inner product defined by $$ (w, {\tilde w})_{E}:=\int_{\mathbb R} \big( w(x),{\tilde w}(x) {\big )}_{{\mathbb R}^{d}}
\rho_{-2r}(x)dx,\quad w, {\tilde w} \in E. $$ Moreover, we set $H:=L^{2}({\mathbb R},{\mathbb R}^{d})$ and denote by $\Vert \cdot \Vert_{E}$ and $\Vert \cdot \Vert_{H}$ the corresponding norms in $E$ and $H$, respectively. We regard the dual space $E^{*}$ of $E$ as $L^{2}({\mathbb R}, {\mathbb R}^{d};\rho_{2r}(x)dx)$. We endow $C({\mathbb R}, \mathbb R^{d})$ with the compact uniform topology and introduce a tempered subspace
\begin{equation} {\cal C}:=\{ w \in C({\mathbb R}, \mathbb R^{d} )
\vert~\lim_{\vert x \vert \to \infty} \vert w(x) \vert \rho_{-r}(x)<\infty \mbox{ for every } r>0 \}. \label{temper} \nonumber \end{equation} We easily see that the inclusion ${\cal C} \subset E \cap C({\mathbb R}, \mathbb R^{d})$ is dense with respect to the topology of $E$.
Let $\cal B$ be the topological $\sigma$-field on $C({\mathbb R},\mathbb R^{d})$. For $T_{1}<T_{2}\in \mathbb R$, we define by ${\cal B}_{[T_{1},T_{2}]}$ and ${\cal B}_{[T_{1},T_{2}],c}$ the sub-$\sigma$-fields of $\cal B$ generated by $\{ w(x);T_{1}\leq x \leq T_{2} \}$ and $\{w(x); x\leq T_{1}, x\geq T_{2} \}$, respectively.
For $T_{1}, T_{2} \in \mathbb R$ and $z_{1},z_{2} \in {\mathbb R}^{d}$, let ${\cal W}_{[T_{1},T_{2}]}^{z_{1},z_{2}}$ be the path space measure of the Brownian bridge such that $w(T_{1})=z_{1}, w(T_{2})=z_{2}$.
We sometimes regard this measure as a probability measure on the measurable space $(C({\mathbb R}, {\mathbb R}^{d}), {\cal B})$ by considering $w(x)=z_{1}$ for $x\leq T_{1}$ and $w(x)=z_{2}$ for $x\geq T_{2}$.
Following Simon \cite{simon} and Iwata \cite{Iwa1}, we now proceed to introduce rigorously the Gibbs measure on $C({\mathbb R}, {\mathbb R}^{d})$.
In this paper, we impose the following conditions on the potential function $U \in C(\mathbb R^{d}, \mathbb R)$:
\\ {\bf (U1)}\quad There exist a constant $K_{1}\in {\mathbb R}$ and a convex function
$V:{\mathbb R}^{d} \to \mathbb R$ such that $$ U(z)=\frac{K_{1}}{2}\vert z \vert^{2}+V(z), \qquad z\in \mathbb R^{d}. $$
\\ {\bf (U2)} \quad There exist $K_{2}>0$, $R>0$ and $\alpha>0$ such that $$ U(z)\geq K_{2}\vert z \vert^{\alpha}, \qquad \vert z \vert > R. $$
\\ {\bf (U3)}\quad There exist $K_{3}, K_{4}>0$ and $0<\beta <1+\frac{\alpha}{2}$ such that $$
\vert {\widetilde {\nabla}} U(z) \vert \leq K_{3} \exp (K_{4}\vert z \vert^{\beta}), \qquad z \in {\mathbb R}^{d},$$ where ${\widetilde {\nabla}} U(z):=K_{1}z+\partial_{0}V(z)
, z\in \mathbb R^{d}$ and $\partial_{0}V$ is the minimal section of the subdifferential $\partial V$. (The reader is referred to Showalter \cite{Show} for the definition of the subdifferential for a convex function and its minimal section. In the case where $U\in C^{1}(\mathbb R^{d}, \mathbb R)$, ${\widetilde {\nabla}}U$ coincides with the usual gradient $\nabla U$.)
Let $H_{U}:=-\frac{1}{2}\Delta_{z}+U$ be the Schr\"odinger operator on $L^{2}({\mathbb R}^{d}, {\mathbb R})$, where $\Delta_{z}:= \sum_{i=1}^{d}{\partial^{2}}/{\partial z_{i}^{2}}$ is the $d$-dimensional Laplacian. Then condition {\bf (U2)} assures that $H_{U}$ has purely discrete spectrum and a complete set of eigenfunctions (see, e.g., Reed--Simon \cite{rs}). We denote by $\lambda_{0}(>\min U)$ the minimal eigenvalue and by $\Omega$ the corresponding normalized eigenfunction in $L^{2}({\mathbb R}^{d}, {\mathbb R})$. This eigenfunction is called ground state and it can be chosen to be strictly positive. Moreover, it has exponential decay at infinity.
To be precise, there exist some positive constants $D_{1}, D_{2}$ such that \begin{equation} 0< \Omega(z) \leq D_{1} \exp \big( -D_{2}\vert z\vert \hspace{0.5mm} U_{\frac{1}{2} \vert z \vert}(z)^{1/2} \big), \quad z\in \mathbb R^{d}, \label{falloff} \end{equation} where $U_{\frac{1}{2} \vert z \vert}(z):= \inf \{ U(y)\vert~
\vert y-z \vert \leq \frac{1}{2}\vert z\vert \}$. See \cite[Corollary 25.13]{simon} for details.
We are going to define a probability measure $\mu$ on $(C({\mathbb R}, {\mathbb R}^{d}), {\cal B})$. For $T_{1}<T_{2}$, and for all $T_{1}\leq x_{1}<x_{2}<\cdots <x_{n}\leq T_{2},~A_{1},A_{2},\cdots,A_{n} \in {\cal B}({\mathbb R}^{d})$,
we define a cylinder set $A\in {\cal B}_{[T_{1},T_{2}]}$ by $A:=\{w\in C(\mathbb R, \mathbb R^{d})~\vert~w(x_{1})\in A_{1}, w(x_{2})\in A_{2}, \cdots, w(x_{n})\in A_{n} \}$.
Next, we set
\begin{eqnarray} \mu(A)&:=& \Big( \Omega,
e^{-(x_{1}-T_{1})(H_{U}-\lambda_{0})}
\big ({ \bf 1}_{A_{1}}
e^{-(x_{2}-x_{1})(H_{U}-\lambda_{0})} \big ( {\bf 1}_{A_{2}}
\cdots \nonumber \\ &\mbox{ }& \hspace{30mm}
e^{-(x_{n}-x_{n-1})(H_{U}-\lambda_{0})} \big ({\bf 1}_{A_{n}}
e^{-(T_{2}-x_{n})(H_{U}-\lambda_{0})} \Omega \big ) \big ) \big)
\Big)_{L^{2}({\mathbb R}^{d},{\mathbb R})} \nonumber \\ &=& e^{\lambda_{0}(T_{2}-T_{1})}\int_{{\mathbb R}^{d}} dz_{1} \Omega(z_{1}) \int_{{\mathbb R}^{d}} dz_{2} \Omega(z_{2})p(T_{2}-T_{1},z_{1},z_{2}) \nonumber \\ &\mbox{ }& \times \int_{C(\mathbb R, \mathbb R^{d})} {\bf 1}_{A}(w) \exp \big( -\int_{T_{1}}^{T_{2}}U(w(x))dx \big) {\cal W}_{[T_{1},T_{2}]}^{z_{1},z_{2}}(dw),
\label{Construction-Gibbs} \end{eqnarray}
where $p(t,z_{1},z_{2}), t>0, z_{1},z_{2}\in \mathbb R^{d}$,
is the transition probability density of standard Brownian motion on ${\mathbb R}^{d}$, and we used the Feynman--Kac formula for the second line.
Then by recalling that $e^{-tH_{U}}\Omega=e^{-t\lambda_{0}}\Omega, \Vert \Omega \Vert_{L^{2}(\mathbb R^{d},\mathbb R)}=1$ and by the Markov property of the $d$-dimensional Brownian motion,
(\ref{Construction-Gibbs}) defines a consistent family of probability measures, and hence $\mu$ can be extended to a probability measure on $C({\mathbb R}, {\mathbb R}^{d})$.
In the same way as \cite[Proposition 2.7]{Iwa1}, we can see that $\mu({\cal C})=1$ and
the following DLR-equations hold even if we replace the potential function with polynomial growth by the one satisfying the much weaker condition {\bf (U3)}:
\begin{eqnarray} {\mathbb E}^{\mu} \big [ {\bf 1}_{A} \vert {\cal B}_{[T_{1},T_{2}],c} \big ](\xi) \hspace{-1.5mm} &=& \hspace{-1.5mm} Z^{-1}_{[T_{1},T_{2}]}(\xi) \int_{A} \exp \Big( -\int_{T_{1}}^{T_{2}}U(w(x))dx \Big) {\cal W}_{[T_{1},T_{2}]}^{\xi(T_{1}),\xi(T_{2})}(dw), \nonumber \\ &\mbox{ }&
\mu \mbox{-a.e. }\xi \in C(\mathbb R, \mathbb R^{d}), \mbox{ for all }A\in {\cal B}_{[T_{1},T_{2}]}, T_{1}<T_{2}, \label{DLR} \end{eqnarray} where $Z_{[T_{1},T_{2}]}(\xi):={\mathbb E}^{{\cal W}_{[T_{1},T_{2}]}^{\xi(T_{1}),\xi(T_{2})} } [\exp(-\int_{T_{1}}^{T_{2}} U(w(x))dx) ] $ is a normalizing constant.
By the continuity of the inclusion map of $\cal C$ into $E$, we may regard $\mu$ as a probability measure on $E$ by identifying it with its image measure under the inclusion map, and using that,
${\cal C} \in {\cal B}(E)$ and ${\cal B}(E) \cap {\cal C} ={\cal B}({\cal C})$ by Kuratowski's theorem.
The DLR-equations (\ref{DLR}) imply that the Gibbs measure $\mu$ is $C^{\infty}_{0}({\mathbb R}, {\mathbb R}^{d})$-quasi-invariant, i.e., $\mu(\cdot +k)$ and $\mu$ are mutually equivalent, and
$ \mu(k+dw)=
\Lambda(k,w) \mu(dw) $ holds for every $k\in C^{\infty}_{0}({\mathbb R}, {\mathbb R}^{d})$. In particular by Albeverio--R\"ockner \cite[Proposition 2.7]{AR1}, $\mu(O)>0$ for every open $\emptyset \neq O \subset E$, i.e., the topological support $\rm{supp }(\mu)$ is equal to all of $E$.
The Radon-Nikodym density $\Lambda (k,w)$ is represented by \begin{eqnarray}
\Lambda(k,w)&=&
\exp \Big \{ \int_{\mathbb R} \Big ( U\big(w(x)\big)-U\big(w(x)+k(x)\big)
\nonumber \\
&\mbox{ }& \qquad \qquad \qquad \qquad
-\frac{1}{2} \big \vert \frac{dk}{dx}(x) \big \vert^{2}+ (w(x), \Delta_{x}k(x) )_{{\mathbb R}^{d}} \Big )dx \Big \}.
\label{quasi}
\end{eqnarray}
We give the following examples which satisfy our conditions {\bf (U1)}, {\bf (U2)} and {\bf (U3)}.
\begin{exm} [$P(\phi)_{1}$-quantum fields] \label{poly-potential case}
We consider the case where the potential function $U$ is written as the following potential function on $\mathbb R^{d}$: $$ U(z)=\sum_{j=0}^{2n}a_{j}\vert z \vert^{j},\quad
a_{2n}>0,~ n\in \mathbb N. $$ Especially, in the case
$U(z)=\frac{m^{2}}{2} \vert z \vert^{2}$, $m>0$, the corresponding Gibbs measure $\mu$ is the Gaussian measure on ${\cal C}$ with mean $0$ and covariance operator $(-\Delta_{x}+m^{2})^{-1}$. It is just the (space-time) free field of mass $m$ in terms of Euclidean quantum field theory.
A double-well potential $U(z)=a(\vert z \vert^{4}-\vert z \vert^{2}), a>0$, is also particularly important from the point of view of physics.
\end{exm}
\begin{exm} [${\rm exp}(\phi)_{1}$-quantum fields] \label{exp-potential case}
We consider an exponential type potential function $U: {\mathbb R}^{d} \to {\mathbb R}$ (with weight $\nu$) given by \begin{equation} U(z) =\frac{m^{2}}{2} \vert z \vert^{2}+V(z):= \frac{m^{2}}{2} \vert z \vert^{2} + \int_{{\mathbb R}^{d}} e^{(\xi, z)_{\mathbb R^{d}}}
\nu(d\xi), \quad z\in \mathbb R^{d}, \label{U-def} \nonumber \end{equation} where
$\nu$ is a bounded positive measure with ${\rm supp}(\nu) \subset \{ \xi \in \mathbb R^{d} \vert~\vert \xi \vert \leq L \}$ for some $L>0$. We note that $U$ is a smooth strictly convex function (i.e., $\nabla^{2}U \geq m^{2}$). Hence we can take
$K_{1}=m^{2}$, $K_{2}=\frac{m^{2}}{2}$ and $\alpha=2$. Moreover, \begin{equation} \vert U(z) \vert \leq \frac{m^{2}}{2} \vert z \vert^{2}+ \nu({\mathbb R}^{d}) e^{L\vert z \vert} \leq \big( \frac{m^{2}}{2L^{2}}+\nu({\mathbb R}^{d}) \big) e^{2L\vert z \vert},
\quad z\in \mathbb R^{d}, \label{U-bound} \nonumber \end{equation} and \begin{equation} \vert \nabla U(z) \vert \leq m^{2}\vert z \vert +\int_{\mathbb R^{d}} \vert \xi \vert e^{(\xi,z)_{\mathbb R^{d}}} \nu(d\xi)
\leq (\frac{m^{2}}{L}+L\nu(\mathbb R^{d}))e^{L\vert z \vert}, \quad z\in \mathbb R^{d}. \label{3-1} \nonumber \end{equation} Thus we can take $\beta=1$, which satisfies $\beta <1+\frac{\alpha}{2}$ in condition {\bf (U3)}.
\end{exm} \begin{re} \label{Betz-Hairer}
We discuss a simple example of ${\rm exp}(\phi)_{1}$-quantum fields in the case $d=1$. This example has been discussed in the $2$-dimensional space-time case in {\rm{\cite{AHk}}}. Let $\delta_{a}$ be the Dirac measure at $a\in \mathbb R$ and we consider $\nu(d\xi):=\frac{1}{2}\big ( \delta_{-a}(d\xi) +\delta_{a}(d\xi) \big ),~a>0$. Then the corresponding potential function is $U(z)=\frac{m^{2}}{2}z^{2} +\cosh (az)$,
and {\rm{(\ref{falloff})}} implies that
the Schr\"odinger operator $H_{U}$ has a ground state $\Omega$ satisfying \begin{equation} 0<\Omega(z) \leq D_{1} \exp \big( -\frac{D_{2}}{\sqrt 2} \vert z\vert \hspace{0.5mm} e^{\frac{a}{4}\vert z \vert} \big), \quad z\in \mathbb R, \label{BH-upper} \end{equation} for some $D_{1}, D_{2}>0$.
By the translation invariance of the Gibbs measure $\mu$ and {\rm{(\ref{BH-upper})}}, there exist positive constants $M_{1}$ and $M_{2}$ such that \begin{eqnarray} A_{T}&:=& \mu \big ( \{w \in C(\mathbb R, \mathbb R) \vert~\vert w(T) \vert > \frac{4}{a} \log \log T \} \big ) \nonumber \\ &=& \int_{\vert z \vert >\frac{4}{a} \log \log T} \Omega(z)^{2}dz \nonumber \\ &\leq& M_{1} \exp \big \{ - M_{2}
(\log T)(\log \log T) \big \} =M_{1} T^{-M_{2}\log \log T} \label{BH-upper2} \end{eqnarray} for $T$ large enough, and {\rm{(\ref{BH-upper2})}} implies $ \sum_{T=1}^{\infty} A_{T}<\infty$. Then the first Borel--Cantelli lemma yields $$ \mu \big ( \{w \in C(\mathbb R, \mathbb R) \vert~\limsup_{T \to \infty} \frac{ \vert w(T) \vert}{\log \log T} \leq \frac{4}{a} \} \big )=1, $$ and thus $\mu$ is supported by a much smaller subset of $C(\mathbb R, \mathbb R)$ than ${\cal C}$. \end{re}
\begin{exm} [Trigonometric quantum fields] \label{sine-potential case}
We consider a trigonometric type potential function $U: {\mathbb R}^{d} \to {\mathbb R}$ (with weight $\nu$) given by \begin{equation} U(z) =\frac{m^{2}}{2} \vert z \vert^{2}+V(z):= \frac{m^{2}}{2} \vert z \vert^{2} + \int_{{\mathbb R}^{d}} \cos \big \{ {(\xi, z)_{\mathbb R^{d}}}+\alpha \big \}
\nu(d\xi), \quad z\in \mathbb R^{d}, \nonumber \end{equation} where $\alpha\in \mathbb R$, $m>0$, and $\nu$ is a bounded signed measure with finite second absolute moment, i.e., $$ \vert \nu \vert (\mathbb R^{d})<\infty, \quad K(\nu):=\int_{\mathbb R^{d}} \vert \xi \vert^{2} \vert \nu \vert (d\xi) <\infty. $$ This potential function is smooth, and it can be regarded as a bounded perturbation of a quadratic function. Moreover, $\nabla^{2}U \geq m^{2}-K(\nu)$ and $$ \vert \nabla U(z) \vert \leq m^{2}\vert z \vert +K(\nu)^{1/2} \vert \nu \vert(\mathbb R^{d})^{1/2}. $$
This type of potential functions corresponds to quantum field models with ``trigonometric interaction" and has been discussed especially in the {\rm{2}}-dimensional space-time case (see, e.g., {\rm{\cite{AHk73, Fro, HKPS}}}).
\end{exm}
\begin{re} Further examples can be obtained by considering $U:\mathbb R^{d} \to \mathbb R$ of the form $U(z)=\lambda_{1}U_{1}(z)+\lambda_{2}U_{2}(z)+\lambda_{3}U_{3}(z)$, where $\lambda_{i}\geq 0,~i=1,2,3$, and $U_{1}$, resp. $U_{2}$, resp. $U_{3}$, is as given in Example {\rm{\ref{poly-potential case}}}, resp. Example {\rm{\ref{exp-potential case}}}, resp. Example {\rm{\ref{sine-potential case}}}. \end{re}
Now we are in a position to introduce the pre-Dirichlet form $({\cal E},{\cal FC}_{b}^{\infty})$. Let $K\subset E^{*}$ be a dense linear subspace of $E$ and let ${\cal FC}_{b}^{\infty}(K)$ be the space of all smooth cylinder functions on $E$ having the form
$$ F(w)=f(\langle w,\varphi_{1} \rangle , \ldots , \langle w,\varphi_{n} \rangle), \quad w\in E, $$ with $n\in {\mathbb N}$, $f
\in C^{\infty}_{b}({\mathbb R}^{n}, {\mathbb R})$ and $\varphi_{1}, \ldots , \varphi_{n} \in K$.
Here we set $\langle w,\varphi \rangle:=\int_{\mathbb R} (w(x), \varphi(x))_{\mathbb R^{d}}dx$ if the integral converges absolutely, and set ${\cal FC}_{b}^{\infty}:={\cal FC}_{b}^{\infty}(C^{\infty}_{0}({\mathbb R}, {\mathbb R}^{d}))$ for simplicity. Since we have supp$(\mu)=E$, two different functions in ${\cal FC}_{b}^{\infty}(K)$ represent two different $\mu$-classes.
Note that ${\cal FC}_{b}^{\infty}$ is dense in $L^{p}(\mu)$ for all $p\geq 1$.
For $F \in {\cal FC}_{b}^{\infty}$, we define the $H$-Fr\'echet derivative $D_{H}F:E\to H$ by $$ D_{H}F(w):=\sum_{j=1}^{n}\frac{\partial {f}}{\partial \alpha_{j}} (\langle w,\varphi_{1} \rangle , \ldots, \langle w,\varphi_{n} \rangle)\varphi_{j}. $$ Then we consider the pre-Dirichlet form $({\cal E},{\cal FC}_{b}^{\infty})$ which is given by $$ {\cal E}(F,G)= \frac{1}{2} \int_{E} \big( D_{H}F(w), D_{H}G(w) \big)_{H}
\mu (dw),~~F,G\in {\cal FC}_{b}^{\infty}. $$
\begin{pr} \label{IbP} \begin{equation} {\cal E}(F,G)=-\int_{E} {\cal L}_{0}F(w) G(w) \mu(dw), \quad F,G\in {\cal FC}_{b}^{\infty}, \label{IbP3} \end{equation} where ${\cal L}_{0}F\in L^{p}(\mu),~ p\geq 1,~F\in {\cal FC}^{\infty}_{b}$, is given by
\begin{eqnarray}
\hspace{-5mm}
{\cal L}_{0}F(w)
&=&\frac{1}{2} {\rm Tr}(D_{H}^{2}F(w)) +\frac{1}{2}\big \langle w, \Delta_{x}D_{H}F(w(\cdot)) \big \rangle -\frac{1}{2}\big \langle ({\widetilde {\nabla}} U)(w(\cdot)), D_{H}F(w) \big \rangle \nonumber \\
&=&
\frac{1}{2} \sum_{i,j=1}^{n}\frac{\partial^{2}f} {\partial \alpha_{i}\partial \alpha_{j}} \big(\langle w,\varphi_{1} \rangle , \ldots, \langle w,\varphi_{n} \rangle \big )\langle \varphi_{i}, \varphi_{j}\rangle \nonumber \\ &\mbox{ }&+ \frac{1}{2} \sum_{i=1}^{n} \frac{\partial f} {\partial \alpha_{i}} \big (\langle w,\varphi_{1} \rangle , \ldots, \langle w,\varphi_{n} \rangle \big ) \cdot \big\{\langle w, \Delta_{x}\varphi_{i}\rangle -\langle ({\widetilde {\nabla}}U)(w(\cdot)), \varphi_{i}\rangle \big\}. \nonumber
\end{eqnarray} for $F(w)=f(\langle w,\varphi_{1} \rangle , \ldots, \langle w,\varphi_{n} \rangle)$. \end{pr}
This proposition means that the operator ${\cal L}_{0}$ is the pre-Dirichlet operator which is associated with the pre-Dirichlet form $({\cal E},{\cal FC}_{b}^{\infty})$. In particular, $({\cal E},{\cal FC}_{b}^{\infty})$ is closable in $L^{2}(\mu)$. Let us denote by ${\cal D(E)}$ the completion of ${\cal FC}_{b}^{\infty}$ with respect to the ${\cal E}_{1}^{1/2}$-norm. (Here we use standard notations of the theory of Dirichlet forms, see, e.g., \cite{A, FOT, MR}.) By standard theory (cf. \cite{A, AR90, FOT, MR}), $({\cal E}, {\cal D(E)})$ is a Dirichlet form and
the operator ${\cal L}_{0}$ has a self-adjoint extension $({\cal L}_{\mu}, {\rm Dom}({\cal L}_{\mu}))$, called the Friedrichs extension, corresponding to the Dirichlet form $({\cal E}, {\cal D(E)})$. The semigroup $\{ e^{t{{\cal L}_{\mu}}} \}_{t\geq 0}$ generated by $({\cal L}_{\mu}, {\rm Dom}({\cal L}_{\mu}))$ is Markovian, i.e., $0\leq e^{t{{\cal L}_{\mu}}}F \leq 1$, $\mu$-a.e. whenever $0\leq F \leq 1$, $\mu$-a.e. Moreover, since $\{ e^{t{{\cal L}_{\mu}}} \}_{t\geq 0}$ is symmetric on $L^{2}(\mu)$, the Markovian property implies that $$ \int_{E} e^{t{{\cal L}_{\mu}}} F(w) \mu(dw) \leq \int_{E} F(w) \mu(dw), \quad F\in L^{2}(\mu),~F \geq 0,~ \mu \mbox{-a.e.} $$
Hence $\Vert e^{t{{\cal L}_{\mu}}}F \Vert_{L^{1}(\mu)} \leq \Vert F \Vert_{L^{1}(\mu)}$ holds for $F\in L^{2}(\mu)$, and
$\{e^{t{{\cal L}_{\mu}}} \}_{t\geq 0}$ can be extended as a family of $C_{0}$-semigroup of contractions in $L^{p}(\mu)$ for all $p\geq 1$.
See e.g., Shigekawa \cite[Proposition 2.2]{shige} for details.
On the other hand, it is a fundamental question whether the Friedrichs extension is the only closed extension generating a $C_{0}$-semigroup on $L^{p}(\mu), p\geq 1$, which for $p=2$ is equivalent to the fundamental problem of essential self-adjointness of ${\cal L}_{0}$ in quantum physics (cf. Eberle \cite{Eb}). Even if $p=2$, in general there are many lower bounded self-adjoint extensions ${\widetilde {\cal L}}$ of ${\cal L}_{0}$ in $L^{2}(\mu)$ which therefore generate different symmetric strongly continuous semigroups $\{ e^{t{\widetilde {\cal L}}} \}_{t\geq 0}$.
If, however, we have $L^{p}(\mu)$-uniqueness of ${\cal L}_{0}$ for some $p\geq 2$, there is hence only one semigroup which is strongly continuous and with generator extending ${\cal L}_{0}$. Consequently, in this case, only one such $L^{p}$-, hence only one such $L^{2}$-dynamics exists, associated with the Gibbs measure $\mu$.
The following theorems are the main results of this paper. For the notions of ``quasi-everywhere" and ``capacity", we refer to \cite{A, FOT, MR}.
\begin{tm} \label{ES} {\rm{(1)}}~The pre-Dirichlet operator $({\cal L}_{0}, {\cal FC}_{b}^{\infty})$ is $L^{p}(\mu)$-unique for all $p \geq 1$, i.e., there exists exactly one $C_{0}$-semigroup in $L^{p}(\mu)$ such that its generator extends $({\cal L}_{0}, {\cal FC}_{b}^{\infty})$.
\\ {\rm{(2)}}~There exists a diffusion process ${\mathbb M}:=(\Theta, {\cal F}, \{ {\cal F}_{t} \}_{t\geq 0}, \{ X_{t} \}_{t\geq 0}, \{ {\mathbb P}_{w}\}_{w\in E} )$ such that
the semigroup $\{P_{t}\}_{t\geq 0}$ generated by the unique (self-adjoint) extension of $({\cal L}_{0}, {\cal FC}^{\infty}_{b})$
satisfies the following identity for any bounded measurable function $F:E \to \mathbb R$, and $t>0$: \begin{equation}
P_{t}F(w)=\int_{\Theta} F(X_{t}(\omega)) {\mathbb P}_{w}(d\omega), \quad \mu\mbox{-}a.s.~w\in E. \label{suii-hangun} \end{equation}
Moreover,
$\mathbb M$ is the unique diffusion process solving
the following ``componentwise" SDE: \begin{eqnarray} \langle X_{t}, \varphi \rangle &=& \langle w, \varphi \rangle +\langle B_{t}, \varphi \rangle +\frac{1}{2}\int_{0}^{t} \big \{ \langle X_{s}, \Delta_{x} \varphi \rangle-\langle ({\widetilde {\nabla}}U)(X_{s}(\cdot)), \varphi \rangle \big \} ds, \nonumber \\ &\mbox{ }& \hspace{55mm} t>0,~\varphi \in C^{\infty}_{0}(\mathbb R, \mathbb R^{d}), ~{\mathbb P}_{w} \mbox{-a.s.}, \label{weakform} \end{eqnarray} for quasi-every $w\in E$ and such that its corresponding semigroup given by {\rm{(\ref{suii-hangun})}} consists of locally uniformly bounded (in $t$) operators on $L^{p}(\mu), p\geq 1$, where $\{B_{t}\}_{t\geq 0}$ is an $\{{\cal F}_{t}\}_{t\geq 0}$-adapted $H$-cylindrical Brownian motion starting at zero defined on $(\Theta, {\cal F}, \{ {\cal F}_{t} \}_{t\geq 0}, {\mathbb P}_{w})$ and ${\widetilde {\nabla}}U$ was defined in condition {\bf{(U3)}}. \end{tm}
\begin{tm} \label{ES2} For quasi-every $w\in E$, the parabolic SPDE \begin{eqnarray} dX_{t}(x)=\frac{1}{2} \big \{ \Delta_{x} X_{t}(x) -({\widetilde {\nabla}}U)(X_{t}(x)) \big \} dt +dB_{t}(x), \quad x \in \mathbb R, ~t>0, \label{GL} \end{eqnarray} has a unique strong solution $X=\{ X_{t}^{w}(\cdot)\}_{t\geq 0}$ living in $C([0,\infty),E)$. Namely, there exists a set $S\subset E$ with ${\rm Cap}(S)=0$ such that for any $H$-cylindrical Brownian motion $\{B_{t}\}_{t\geq 0}$ starting at zero defined on a filtered probability space $(\Theta, {\cal F}, \{ {\cal F}_{t} \}_{t\geq 0}, {\mathbb P})$ satisfying the usual conditions and an initial datum $w \in E \setminus S$, there exists a unique $\{ {\cal F}_{t} \}_{t\geq 0}$-adapted process $X=\{ X_{t}^{w}(\cdot)\}_{t\geq 0}$ living in $C([0,\infty),E)$ satisfying {\rm{(\ref{weakform})}}. \end{tm}
\begin{re} \label{compare} Obviously, the uniqueness result in Theorem {\rm{\ref{ES2}}} implies the (thus weaker) uniqueness stated for the diffusion process $\mathbb M$ in Theorem {\rm{\ref{ES}}}. However, it does not imply the $L^{p}(\mu)$-uniqueness of the Dirichlet operator. This is obvious, since a priori the latter might have extensions which generate non-Markovian semigroups which thus have no probabilistic interpretation as transition probabilities of a process.
Therefore, neither of the two uniqueness results in Theorems {\rm{\ref{ES}}} and {\rm{\ref{ES2}}}, i.e., $L^{p}(\mu)$-uniqueness of the Dirichlet operator and strong uniqueness of the corresponding SPDE respectively, implies the other. We refer to Albeverio--R\"ockner {\rm{\cite[Sections 2 and 3]{Cornell}}} and see also {\rm{\cite[Section 8]{DR}}} for a detailed discussion. \end{re}
\begin{re} If the potential function $U$ is a $C^{1}$-function with polynomial growth at infinity,
Iwata {\rm{\cite{Iwa2}}} proves that SPDE {\rm{(\ref{GL})}}
has a unique strong solution $X^{w}=\{ X_{t}^{w}(\cdot) \}_{t\geq 0}$ living in $C([0,\infty),{\cal C})$ for every initial datum $w\in {\cal C}$.
On the other hand, in the case of ${\rm exp}(\phi)_{1}$-quantum fields, since $({\nabla} U)(w(\cdot))\notin {\cal C}$ for $w\in {\cal C}$ in general, we cannot expect to solve SPDE {\rm (\ref{GL})} in $C([0,\infty),{\cal C})$ for a given initial datum $w\in {\cal C}$.
Hence if we
replace the state space ${\cal C}$ by
a much smaller tempered subspace ${\cal C}_{e}$ such that $({\nabla} U)(w(\cdot))\in {\cal C}_{e}$ holds for $w\in {\cal C}_{e}$, we might construct a unique strong solution to SPDE {\rm (\ref{GL})} living in $C([0,\infty), {\cal C}_{e})$ for every initial datum $w\in {\cal C}_{e}$.
(A possible candidate for ${\cal C}_{e}$ could be the space of all paths behaving like $$ \vert w(x) \vert \sim \log (\log(\log(\log(\cdots x)))) $$ at infinity.) We will discuss this problem in the future. \end{re}
\section{Proof of the Main Results}
At the beginning, we give the proof of Proposition \ref{IbP}. \\ {\bf Proof of Proposition \ref{IbP}:} Firstly, we aim to prove that \begin{equation} \int_{E} \Big( \int_{\mathbb R} \vert ({\widetilde{\nabla}} U)(w(x)) \vert^{2} \rho_{-2r}(x) dx \Big )^{p/2} \mu(dw) <\infty, \quad p\geq 1. \label{p-moment} \end{equation} By the translation invariance of the Gibbs measure $\mu$, for every $p\geq 2$, it holds that
\begin{eqnarray} \lefteqn{ \int_{E} \Big( \int_{\mathbb R} \vert ({\widetilde{\nabla}} U)(w(x)) \vert^{2} \rho_{-2r}(x) dx \Big )^{p/2} \mu(dw)
} \nonumber \\ & \leq & \int_{E} \Big \{ \Big ( \int_{\mathbb R} \vert ({\widetilde{\nabla}} U)(w(x)) \vert^{p} \rho_{-2r}(x) dx \Big) \big( \int_{\mathbb R} \rho_{-2r}(x) dx \big )^{\frac{p-2}{2}} \Big \} \mu(dw) \nonumber \\ &\leq & \Big( \frac{1}{r} \Big )^{\frac{p-2}{2}} \int_{\mathbb R} \Big( \int_{E} \vert ({\widetilde{\nabla}} U)(w(0)) \vert^{p} \mu(dw) \Big) \rho_{-2r}(x) dx \nonumber \\ & \leq & \Big (\frac{1}{r} \Big )^{p/2} \int_{\mathbb R^{d}} \vert {\widetilde{\nabla}} U(z) \vert^{p} \Omega(z)^{2}dz
\nonumber \\ &\leq & \Big (\frac{K_{3}^{2}}{r} \Big )^{p/2} \int_{\mathbb R^{d}} \exp (pK_{4}\vert z \vert^{\beta}) \Omega(z)^{2}dz.
\label{Omega-estimate} \end{eqnarray} On the other hand, condition {\bf (U2)} leads to a lower bound $U_{\frac{1}{2} \vert z \vert}(z)\geq \frac{K_{2}}{2^{\alpha}}\vert z \vert^{\alpha}$ for $\vert z \vert \geq 2R$. Hence we can continue to bound the integral on the right-hand side of (\ref{Omega-estimate}) as follows:
\begin{eqnarray} \lefteqn{ \int_{\vert z \vert \geq 2R} \exp (pK_{4}\vert z \vert^{\beta}) \Omega(z)^{2}dz } \nonumber \\ &\leq & D_{1}^{2} \int_{\vert z \vert \geq 2R} \exp \big \{ p (K_{4}\vert z \vert^{\beta} -D_{2} \vert z \vert U_{\frac{1}{2} \vert z \vert}(z)^{1/2} )\big \} dz \nonumber \\ & \leq & D_{1}^{2} \int_{\vert z \vert \geq 2R} \exp \Big \{ p \big( K_{4}\vert z \vert^{\beta}- \frac{D_{2} K_{2}^{1/2} }{{2}^{\alpha/2} }\vert z \vert^{1+\frac{\alpha}{2}} \big ) \Big \}dz < \infty, \label{Omega-estimate-2} \end{eqnarray} where we used the estimate (\ref{falloff}) for the second line and $\beta<1+\frac{\alpha}{2}$ for the third line.
Hence by combining (\ref{Omega-estimate}) with (\ref{Omega-estimate-2}), we see that the left-hand side of (\ref{Omega-estimate}) is finite for all $p\geq 2$. Since $\mu$ is a probability measure on $E$, we have shown that (\ref{p-moment}) holds for all $p\geq 1$. In the same way, we also have \begin{equation} \int_{E} \Vert w \Vert_{E}^{p} \hspace{1mm} \mu(dw) <\infty, \quad p\geq 1. \label{p-moment2} \end{equation}
Next, we define $$\beta_{\varphi}(w):=\langle w, \Delta_{x}\varphi \rangle -\langle ({\widetilde{ \nabla }}U)(w(\cdot)), \varphi \rangle,~\quad w \in E,~ \varphi \in C^{\infty}_{0}(\mathbb R, \mathbb R^{d}).$$ Then by noting that $\varphi$ has compact support, one has $\Vert \Delta_{x} \varphi \Vert_{E^{*}} + \Vert \varphi \Vert_{E^{*}}<\infty$, and (\ref{p-moment}) and (\ref{p-moment2}) lead to \begin{eqnarray}
\int_{E} \vert \beta_{\varphi}(w) \vert^{p} \mu(dw) &\leq& 2^{p-1} \big ( \Vert \Delta_{x} \varphi \Vert_{E^{*}}^{p} + \Vert \varphi \Vert_{E^{*}}^{p} \big ) \nonumber \\ &\mbox{ }& \hspace{-15mm} \times \int_{E} \Big \{ \Vert w \Vert_{E}^{p} +
\Big( \int_{\mathbb R} \vert ({\widetilde{\nabla}} U)(w(x)) \vert^{2} \rho_{-2r}(x) dx \Big )^{p/2} \Big \}
\mu(dw) <\infty. \label{beta-integrability} \nonumber \end{eqnarray} Thus we have shown that ${\cal L}_{0}F \in L^{p}(\mu)$ holds for all $p\geq 1$ and $F\in {\cal FC}_{b}^{\infty}$. Hence the right-hand side of (\ref{IbP3}) is well-defined and finite, and the quasi-invariance of $\mu$ yields
(\ref{IbP3}).
\qed
Before proceeding to the proofs of our main theorems, we make some preparations. We fix a positive constant $\kappa >2r^{2}$, and set $$ G_{t}w(x):= \int_{\mathbb R} \frac{1}{{\sqrt{2\pi t}}}e^{-\frac{(x-y)^{2}}{2t}} w(y)dy, \quad t>0,~x\in \mathbb R. $$ Then by \cite[Lemma 3.2]{KR}, we see that $\{ e^{-\kappa t/2}G_{t} \}_{t \geq 0}$ is a strongly continuous contraction semigroup on $E$ with $\Vert e^{-\kappa t/2}G_{t} \Vert_{L(E,E)} \leq \exp \{-(\frac{\kappa}{2}-r^{2})t\}$.
Let $A: {\rm Dom}(A)\subset E \to E$ be the infinitesimal generator of $\{ e^{-\kappa t/2}G_{t} \}_{t \geq 0}$. We set $e^{tA}:=e^{-\kappa t/2}G_{t}$ throughout this paper. By the Hille--Yosida theorem, $(A, {\rm Dom}(A))$ is $m$-dissipative and it satisfies \begin{equation} (Aw,w)_{E}\leq (r^{2}-\frac{\kappa}{2}) \Vert w \Vert_{E}^{2}, \quad w\in {\rm{Dom}}(A). \label{fukuoka} \end{equation}
\begin{lm} \label{A} {\rm (1)} $C^{\infty}_{0}(\mathbb R, \mathbb R^{d})$ is dense in ${\rm Dom}(A)$ with respect to the graph norm $\Vert w \Vert_{A}:=\Vert w \Vert_{E}+ \Vert Aw \Vert_{E},~w\in {\rm Dom}(A)$, and we have \begin{equation} A\varphi=\frac{1}{2}(\Delta_{x}-\kappa)\varphi, \quad \varphi \in C^{\infty}_{0}(\mathbb R, \mathbb R^{d}). \label{A-hyoji} \end{equation} {\rm (2)} Let $A^{*}: {\rm Dom}(A^{*}) \subset E \to E$ denote the adjoint operator of $(A,{\rm Dom}(A))$. Then ${\rm Dom}(A^{*})={\rm Dom}(A)$. Moreover, we have \begin{eqnarray} A^{*}\varphi &=& \frac{1}{2}\Delta_{x}(\rho_{-2r}\cdot \varphi ) \rho_{2r} -\frac{\kappa}{2}\varphi \nonumber \\ &=& A\varphi
-2r \frac{d \chi}{dx}\cdot \frac{d\varphi}{dx}+ \big \{2r^{2} \big ( \frac{d\chi}{dx} \big)^{2}-r \Delta_{x}\chi
\big \} \varphi,\quad \varphi \in {C}^{\infty}_{0}(\mathbb R, \mathbb R^{d}), \label{A*-hyoji} \end{eqnarray} and \begin{equation} e^{tA^{*}}w(y):=e^{-\kappa t/2} \rho_{2r}(y) \cdot G_{t}(\rho_{-2r}\cdot w)(y), \quad t>0,~ y\in \mathbb R,~w\in E. \label{eA*} \end{equation} \end{lm}
{\bf Proof:}~(1) By a straightforward computation, we can easily see that $C^{\infty}_{0}(\mathbb R, \mathbb R^{d}) \subset {\rm Dom}(A)$ and that (\ref{A-hyoji}) holds. We introduce $$ {\cal C}^{\infty}_{\infty} :=\bigcap_{k=0}^{\infty} \bigcap_{r>0} \Big \{\varphi \in C^{\infty}(\mathbb R, \mathbb R^{d})~\vert~ \sup_{x\in \mathbb R} \big \vert \frac{d^{k}\varphi}{dx^{k}}(x) \big \vert \rho_{r}(x) <\infty \Big \}. $$ Then $C^{\infty}_{0}(\mathbb R, \mathbb R^{d}) \subset {\cal C}^{\infty}_{\infty}$ and the differential operator $A$ can be naturally extended to the domain ${\cal C}^{\infty}_{\infty}$ through (\ref{A-hyoji}). By using the cut-off argument discussed in \cite[Lemma 4.7]{KR}, we can show that $C^{\infty}_{0}(\mathbb R, \mathbb R^{d})$ is dense in ${\cal C}^{\infty}_{\infty}$ with respect to the graph norm $\Vert \cdot \Vert_{A}$.
Now, we take a function $\varphi \in {\cal C}^{\infty}_{\infty}$. Then for every $k \in \mathbb N \cup \{0 \}$ and $r>0$, we can find a positive constant $C(k,r)$ such that $ \big \vert \frac{d^{k}\varphi}{dx^{k}}(x) \big \vert \leq C(k,r) \rho_{-r}(x)$ for all $x \in \mathbb R$. Here we recall that \begin{equation} \int_{\mathbb R} \frac{1}{{\sqrt{2\pi t}}}e^{-\frac{(x-y)^{2}}{2t}} \rho_{-2r}(y)dy\leq e^{2r^{2}t}\rho_{-2r}(x),\quad t>0,x\in \mathbb R. \label{heat-kihon} \end{equation} (cf. e.g., Da Prato--Zabcyzk \cite[Lemma 9.44]{DZ}.) Then for every $k \in \mathbb N \cup \{0 \}$ and $r>0$, \begin{eqnarray} \big \vert \frac{d^{k}}{dx^{k}}(G_{t}\varphi)(x) \big \vert \rho_{r}(x) & \leq & \rho_{r}(x) \int_{\mathbb R} \frac{1}{{\sqrt{2\pi t}}}e^{-\frac{(x-y)^{2}}{2t}} \big \vert \frac{d^{k} \varphi}{dx^{k}}(y) \big \vert dy \nonumber \\ & \leq & \rho_{r}(x) \int_{\mathbb R} \frac{1}{{\sqrt{2\pi t}}}e^{-\frac{(x-y)^{2}}{2t}} \big ( C(k,2r) \rho_{-2r}(y) \big ) dy \nonumber \\ & \leq & C(k, 2r) \rho_{r}(x) \big( e^{2r^{2}t} \rho_{-2r}(x) \big) \nonumber \\ & \leq & C(k, 2r) e^{2r^{2}t} <\infty, \quad x \in \mathbb R, \label{stable-check} \nonumber \end{eqnarray} where we used (\ref{heat-kihon}) for the third line and $\rho_{-2r}(x) \rho_{r}(x) = \rho_{-r}(x) \leq 1$ for the fourth line. This means that $(e^{-\kappa t/2}G_{t})({\cal C}^{\infty}_{\infty}) \subset {\cal C}^{\infty}_{\infty}$ for all $t \geq 0$, and by \cite[Theorems 1.2 and 1.3]{Eb}, we see that ${\cal C}^{\infty}_{\infty}$ is an operator core for $A$. Hence we have shown that ${C}^{\infty}_{0}(\mathbb R, \mathbb R^{d})$ is dense in ${\rm Dom}(A)$ with respect to the graph norm $\Vert \cdot \Vert_{A}$.
\\ (2)~Since (\ref{A*-hyoji}) and (\ref{eA*}) follow by straightforward computations, it is sufficient to show the equivalence of the graph norms $\Vert \varphi \Vert_{A}$ and $\Vert \varphi \Vert_{A^{*}}$ for $\varphi\in C^{\infty}_{0}(\mathbb R, \mathbb R^{d})$.
Using integration by parts, Young's inequality $2ab \leq \delta^{-2}a^{2}+\delta^{2}b^{2}$ and that $\Vert \frac{d\chi}{dx} \Vert_{\infty} \leq 1$, we obtain \begin{eqnarray} \big \Vert \frac{d\varphi}{dx} \big \Vert_{E}^{2} &=&-(\varphi, \Delta_{x}\varphi)_{E} +2r\int_{\mathbb R} \big( \varphi(x), \frac{d \varphi}{dx}(x) \big)_{\mathbb R^{d}} \frac{d\chi}{dx}(x) \rho_{-2r}(x)dx \nonumber \\ & \leq & \big( \frac{1}{2\delta^{2}} \Vert \varphi \Vert_{E}^{2}+\frac{\delta^{2}}{2} \Vert \Delta_{x}\varphi \Vert_{E}^{2} \big)+ r \big( \frac{1}{2r} \big \Vert \frac{d \varphi}{dx} \big \Vert_{E}^{2} + 2r \Vert \varphi \Vert_{E}^{2} \big), \nonumber \end{eqnarray} which in turn implies that \begin{equation} \big \Vert \frac{d \varphi}{dx} \big \Vert_{E} \leq \big( 2r+\frac{1}{\delta} \big) \Vert \varphi \Vert_{E}+\delta \Vert \Delta_{x} \varphi \Vert_{E}, \quad \delta>0, ~ \varphi \in C^{\infty}_{0}(\mathbb R, \mathbb R^{d}). \label{jyunbi} \end{equation} Recalling (\ref{A*-hyoji}), we deduce that
\begin{eqnarray} \Vert A \varphi \Vert_{E} &\leq & \Vert A^{*} \varphi \Vert_{E}+2r \Vert \dot \varphi \Vert_{E} +(2r^{2}+r\Vert \Delta_{x} \chi \Vert_{\infty})\Vert \varphi \Vert_{E} \nonumber \\ & \leq & \Vert A^{*} \varphi \Vert_{E}+2r \big \{ \big( 2r+\frac{1}{\delta} \big) \Vert \varphi \Vert_{E}+\delta \Vert \Delta_{x} \varphi \Vert_{E} \big \} +(2r^{2}+r\Vert \Delta_{x} \chi \Vert_{\infty})\Vert \varphi \Vert_{E}
\nonumber \\ & \leq & \Vert A^{*} \varphi \Vert_{E}+4r\delta \Vert A \varphi \Vert_{E} +\big (6r^{2}+\frac{2r}{\delta}+2r\delta \kappa +r\Vert \Delta_{x} \chi \Vert_{\infty} \big) \Vert \varphi \Vert_{E}, \label{jyunbi-2} \end{eqnarray} where we used $\Vert \frac{d \chi}{dx} \Vert_{\infty} \leq 1$ again for the first line and (\ref{jyunbi}) for the second line.
Now, we choose $\delta:=\frac{1}{8r}$. Then (\ref{jyunbi-2}) implies \begin{equation} \Vert A \varphi \Vert_{E} \leq 2 \Vert A^{*} \varphi \Vert_{E}+ \big (22 r^{2}+\kappa+r\Vert \Delta_{x} \chi \Vert_{\infty} \big) \Vert \varphi \Vert_{E}, \nonumber \end{equation} and by repeating a similar argument for $A^{*}\varphi$, we also have \begin{equation} \Vert A^{*} \varphi \Vert_{E} \leq \frac{3}{2} \Vert A \varphi \Vert_{E}+ \big (22 r^{2}+\kappa+r\Vert \Delta_{x} \chi \Vert_{\infty} \big) \Vert \varphi \Vert_{E}. \nonumber \end{equation} This completes the proof. \qed
\\
{\bf Proof of Theorem \ref{ES}:} (1)~Although we mostly follow the argument in \cite{KR}, which in turn is based on a modification of a technique in \cite{DR}, we give an outline of the argument for the convenience of the reader.
We define $E_{U}:=\{ w\in E;~\Vert ({\widetilde \nabla} U)(w(\cdot)) \Vert_{E}<\infty \}$. Then by (\ref{p-moment}), we see that $E_{U}\in {\cal B}(E)$ and $\mu(E_{U})=1$. We define a measurable map ${\widetilde b}:{\rm Dom}({\widetilde b}) \subset E \to E$ with
${\rm Dom}({\widetilde b})=E_{U}$ by \begin{equation} {\widetilde b}(w)(\cdot):=-\frac{1}{2}(\partial_{0}V)(w(\cdot))=-\frac{1}{2}\big \{ ({\widetilde \nabla} U)(w(\cdot)) -K_{1}w(\cdot) \big \}, \quad w\in {\rm Dom}({\widetilde b}). \label{b-def} \end{equation} We note that $\mu({\rm Dom}({\widetilde b}))=1$, and since $V$ is convex, ${\widetilde b}$ is dissipative, i.e., \begin{equation} \big ( w_{1}-w_{2}, {\widetilde b}(w_{1})-{\widetilde b}(w_{2}) \big)_{E} \leq 0, \quad w_{1},w_{2}\in {\rm Dom}({\widetilde b}). \label{tilde-b-dissipative} \end{equation}
On the other hand, we note that ${\widetilde b}$ is not continuous on $E$ in general. Thus we need to introduce the following regularization scheme.
For $\alpha >0$, we recall the Moreau--Yosida approximation of $V$ which is defined by \begin{equation} V_{\alpha}(z):=\inf_{y\in {\mathbb R}^{d}} \big \{ \frac{1}{2\alpha} \vert y-z \vert^{2} +V(y) \big \}, \quad z\in \mathbb R^{d}.
\label{Moreau-Yosida} \end{equation} Then $V_{\alpha}(z) \nearrow V(z)$ for every $z\in \mathbb R^{d}$ as $\alpha \searrow 0$. On the other hand, $ \partial_{0}V:{\mathbb R}^{d} \to {\mathbb R}^{d}$ is maximal dissipative by convexity of $V$.
For $\alpha >0$, we set $J_{\alpha}(z):= \big(I_{\mathbb R^{d}}+\alpha \partial_{0}V \big)^{-1}(z),~z\in \mathbb R^{d}$, and define the Yosida approximation $(\partial_{0}V)_{\alpha}:\mathbb R^{d} \to \mathbb R^{d}$ by $$ (\partial_{0}V)_{\alpha}(z) :=\frac{1}{\alpha}(J_{\alpha}(z)-z) =(\partial_{0}V)(J_{\alpha}(z)), \quad z\in \mathbb R^{d}. $$ Then $(\partial_{0}V)_{\alpha}$ is monotone and the following Lipschitz continuity holds: $$\big \vert (\partial_{0}V)_{\alpha}(z_{1}) -(\partial_{0}V)_{\alpha}(z_{2}) \big \vert \leq \frac{2}{\alpha} \vert z_{1}-z_{2} \vert, \quad z_{1},z_{2}\in \mathbb R^{d},$$ Furthermore, it is known that $(\partial_{0}V)_{\alpha}(z)=(\partial_{0} V_{\alpha})(z), z\in \mathbb R^{d}$ (cf. e.g., \cite[Proposition 1.8]{Show}), and
\begin{eqnarray} & & \big \vert (\partial_{0}V)_{\alpha}(z) \big \vert \leq \big \vert \partial_{0}V(z) \big \vert, \quad z\in {\mathbb R}^{d}, \label{last-1} \\ \nonumber \\ & & \lim_{\alpha \searrow 0} (\partial_{0}V)_{\alpha}(z) =\partial_{0}V(z), \quad z\in \mathbb R^{d}. \label{last-2} \end{eqnarray} See \cite[Theorem 1.1]{Show} for details.
We define ${\widetilde b}_{\alpha}:E\to E$ in the same way as ${\widetilde b}$ with $\partial_{0}V$ replaced by $(\partial_{0}V)_{\alpha}$. Then ${\widetilde b}_{\alpha}$ is Lipschitz continuous and dissipative on $E$. By (\ref{last-1}) and (\ref{last-2}), we also have \begin{equation} \lim_{\alpha \searrow 0}{\widetilde b}_{\alpha}(w)={\widetilde b}(w), \quad w\in {\rm Dom}({\widetilde b}). \label{alpha-conv} \end{equation}
However, since ${\widetilde b}_{\alpha}$ is not differentiable in general, we need to introduce a further regularization. Let $B:{\rm Dom}(B)\subset E \to E$ be a self-adjoint negative definite operator such that $B^{-1}$ is of trace class. For any $\alpha, \beta>0$, we set \begin{equation} {\widetilde b}_{\alpha, \beta}(w):=\int_{E} e^{\beta B}{\widetilde b}_{\alpha}\big( e^{\beta B}w+y\big) N_{\frac{1}{2}B^{-1}(e^{2\beta B}-1)}(dy),\quad w\in E, \label{further} \end{equation} where $N_{Q}$ is the standard centered Gaussian measure with covariance given by a trace class operator $Q$. Then by applying \cite[Theorem 9.19]{DZ}, we prove that ${\widetilde b}_{\alpha, \beta}$ is dissipative, of class $C^{\infty}$, has bounded derivatives of all orders and \begin{equation} \lim_{\beta \searrow 0}{\widetilde b}_{\alpha, \beta}(w)={\widetilde b}_{\alpha}(w), \quad \Vert {\widetilde b}_{\alpha, \beta}(w) \Vert_{E} \leq C_{\alpha}(1+\Vert w \Vert_{E}), \quad w\in E. \label{6-00} \end{equation}
We also define a measurable map $b: {\rm Dom}(b) \subset E \to E$ with ${\rm Dom}(b)=E_{U}$ by \begin{equation} b(w):=\frac{1}{2}(\kappa -K_{1})w+{\widetilde b}(w), \quad w \in {\rm Dom}(b), \label{def-b} \end{equation} and define ${b_{\alpha, \beta}}$ with ${\widetilde b}_{\alpha, \beta}$ replacing $\widetilde b$ in (\ref{def-b}).
Now, we consider the
stochastic evolution equation on $E$ given by \begin{eqnarray} dX_{t}&=&AX_{t}dt +b_{\alpha, \beta}(X_{t})dt+{\sqrt Q}d{W}_{t} \nonumber \\ &=& AX_{t}dt+\frac{1}{2}(\kappa -K_{1})X_{t}dt+ {\widetilde b}_{\alpha, \beta}(X_{t})dt+{\sqrt Q}d{W}_{t}, ~t\geq 0, \label{abstSPDE} \end{eqnarray} where $Q$ is a bounded linear operator on $E$ defined by $Qw:=\rho_{-2r}\cdot w,~w\in E$, and $\{ W_{t} \}_{t\geq 0}$ is an $E$-cylindrical Brownian motion defined on a fixed filtered probability space $({\Theta}, {\cal F}, \{ {\cal F}_{t} \}_{t\geq 0}, {\mathbb P})$. Note that $Q^{-1}$ is not bounded on $E$. This kind of equation is regarded as an abstract formulation of SPDE (\ref{GL}) in the sense of \cite{DZ}, i.e., in the mild form.
Since each $e^{tA}{\sqrt{Q}}$ is a Hilbert--Schmidt operator on $E$ and ${b}_{\alpha, \beta}$ is Lipschitz continuous on $E$, SPDE (\ref{abstSPDE}) has a unique mild solution $X=\{ X^{w}_{t}(\cdot) \}_{t\geq 0}$
living in $C([0,\infty),E)$ for every initial datum $w\in E$. Here we recall that $X$ is a mild solution to SPDE (\ref{abstSPDE}) with
$X_{0}=w\in E$ if one has \begin{equation} X_{t}=e^{tA}w+\int_{0}^{t} e^{(t-s)A} b_{\alpha, \beta}(X_{s}) ds +\int_{0}^{t} e^{(t-s)A} {\sqrt Q}dW_{s}, \quad t>0,~ {\mathbb P}\mbox{-a.s.}
\label{abstSPDE2} \end{equation}
By a standard coupling method for SPDEs applied to (\ref{abstSPDE}), we see that
\begin{equation} \big \Vert X_{t}^{w}-X_{t}^{\tilde w} \big \Vert_{E}\leq e^{\frac{(-K_{1}+2r^{2})t}{2}} \Vert w-{\tilde w} \Vert_{E}, \qquad w, {\tilde w}\in E, \label{coupling-est} \end{equation} also holds with probability one.
We can then define the transition semigroup corresponding to SPDE (\ref{abstSPDE}), denoted by $\{P^{\alpha, \beta}_{t} \}_{t\geq 0}$.
For $F\in
{\cal FC}^{\infty}_{b}$ and $\lambda >(-\frac{K_{1}}{2}+r^{2})\vee 0$, we consider the function $$ \Phi_{\alpha, \beta}(w):=\int_{0}^{\infty} e^{-\lambda t} P_{t}^{\alpha, \beta}F(w) dt, \quad w\in E. $$
Then (\ref{coupling-est}) leads us to the estimate \begin{equation} \Vert D\Phi_{\alpha, \beta}(w) \Vert_{E} \leq \frac{2}{2\lambda +K_{1}-2r^{2}} \Vert DF \Vert_{\infty}, \quad w\in E, \label{grad-est} \end{equation} where $DF:E \to E$ is the $E$-Fr\'echet derivative of $F$. We have the relation $D_{H}F={\sqrt Q}DF$. By Proposition \ref{IbP}, $({\cal L}_{0}, {\cal FC}^{\infty}_{b})$ is dissipative in $L^{p}(\mu), p\geq 1$, and then it is closable. Let $({\overline{\cal L}}_{0}, {\rm Dom}({\overline{\cal L}}_{0}) )$ denote the closure in $L^{p}(\mu)$. However, since it is not easy to consider ${\overline{\cal L}}_{0}$ directly, we need to insert a tractable space between ${\cal FC}^{\infty}_{b}$ and ${\rm Dom}({\overline{\cal L}}_{0})$. Here we recall some beautiful results on Lipschitz perturbations of Ornstein--Uhlenbeck operators
discussed in \cite{D, DT, priola}.
By modifying the results in \cite{D, DT, priola} for our use, we deduce that $\Phi_{\alpha, \beta}$ belongs to a ``nice" domain ${\cal D}(L, C^{1}_{b,2}(E))$ (see \cite{KR} for the precise definition and details) of the Ornstein--Uhlenbeck operator $L$ associated with the SPDE \begin{equation} dY_{t}=AY_{t}dt+{\sqrt Q}d{W}_{t}, ~t\geq 0. \nonumber \end{equation} Moreover, recalling (\ref{p-moment2}), we see that ${\overline {\cal L}}_{0}F=LF+(b,DF)_{E}$ for $F\in {\cal D}(L, C^{1}_{b,2}(E))$ and this identity implies the inclusion ${\cal D}(L, C^{1}_{b,2}(E)) \subset {\rm Dom}({\overline{\cal L}}_{0})$. Hence we have $\Phi_{\alpha, \beta} \in {\rm Dom}({\overline{\cal L}}_{0})\cap C^{2}_{b}(E)$ and
moreover $\Phi_{\alpha, \beta}$
satisfies \begin{equation} (\lambda -{\overline {\cal L}}_{0})\Phi_{\alpha, \beta}=F+ \big ({\widetilde b}_{\alpha, \beta}-{\widetilde b}, D\Phi_{\alpha, \beta} \big)_{E}. \label{6-1} \end{equation} By using (\ref{grad-est}), the right-hand side of (\ref{6-1}) can be estimated as follows: \begin{eqnarray} I_{\alpha, \beta}&:=&\int_{E} \big \vert
\big ({\widetilde b}_{\alpha, \beta}(w)-{\widetilde b}(w), D\Phi_{\alpha, \beta}(w) \big )_{E} \big \vert^{p} \mu(dw)
\nonumber \\ &\leq & \Big(\frac{2}{2\lambda+K_{1}-r^{2}}\Vert DF \Vert_{\infty} \Big)^{p} \int_{E}
\big \Vert {\widetilde b}_{\alpha, \beta}(w)-{\widetilde b}(w) \big \Vert_{E}^{p} \mu(dw). \label{6-2} \end{eqnarray}
Recalling (\ref{last-1}), (\ref{last-2}), (\ref{6-00}) and using Lebesgue's dominated convergence theorem, we conclude that \begin{equation} \lim_{\alpha \searrow 0}\lim_{\beta \searrow 0}I_{\alpha, \beta} =\lim_{\alpha \searrow 0}\big(\limsup_{\beta \searrow 0}I_{\alpha, \beta}\big)=0. \nonumber
\end{equation} From this and (\ref{6-1}), (\ref{6-2}), we obtain $$ \lim_{\alpha \searrow 0}\lim_{\beta \searrow 0} (\lambda -{\overline {\cal L}}_{0})\Phi_{\alpha, \beta}=F \quad \mbox{in }L^{p}(\mu). $$ This means that the closure of ${\rm Range}(\lambda -{\overline {\cal L}}_{0})$ contains ${\cal FC}^{\infty}_{b}$.
Since
${\cal FC}^{\infty}_{b}$ is dense in $L^{p}(\mu)$, ${\rm Range}(\lambda -{\overline {\cal L}}_{0})$ is also dense in $L^{p}(\mu)$. Then by the Lumer--Phillips theorem, we have that $({\overline {\cal L}}_{0}, {\rm Dom}({\overline {\cal L}}_{0}))$ generates a $C_{0}$-semigroup in $L^{p}(\mu)$, and this completes the proof of (1).
\\
(2)~Since $C^{\infty}_{0}(\mathbb R, \mathbb R^{d})$ is dense in $E^{*}$, ${\cal D(E)}$ coincides with the closure of ${\cal FC}^{\infty}_{b}(E^{*})$ with respect to the ${\cal E}_{1}^{1/2}$-norm. Thus, we can directly apply the general methods of the theory of Dirichlet forms \cite{A, MR} to prove quasi-regularity of $({\cal E}, {\cal D(E)})$ and the existence of a diffusion process $\mathbb M$ properly associated with $({\cal E}, {\cal D(E)})$.
Here, following R\"ockner \cite{Ro-JFA} and Funaki \cite{Funaki}, we introduce scaled Sobolev spaces: $$H^{m}_{r}(\mathbb R, \mathbb R^{d}):=\{ \varphi \vert~\rho_{r}\varphi \in H^{m}(\mathbb R, \mathbb R^{d}) \}, \quad m\geq 0,~r\in \mathbb R, $$ equipped with norms
$\vert \varphi \vert_{m,r}:=\Vert \rho_{r}\varphi \Vert_{H^{m}(\mathbb R, \mathbb R^{d})}$. Note that this norm is equivalent to $\Vert \varphi \Vert_{m,r} :=\sum_{k=0}^{m} \Vert \rho_{r} \big (\frac{d^{k} \varphi}{dx^{k}}\big) \Vert_{L^{2}(\mathbb R, \mathbb R^{d})}$ in the case $m\in \mathbb N \cup \{0 \}$. Let $(H^{m}_{r}(\mathbb R, \mathbb R^{d}))^{*}$ be the dual space of $H^{m}_{r}(\mathbb R, \mathbb R^{d})$. Then we have $$(H^{m}_{r}(\mathbb R, \mathbb R^{d}))^{*}=H^{-m}_{-r}(\mathbb R, \mathbb R^{d}) =\{ w \vert~\rho_{-r}w\in H^{-m}(\mathbb R, \mathbb R^{d}) \},$$ and, clearly $H=H^{0}_{0}(\mathbb R, \mathbb R^{d})$, $E=H^{0}_{-r}(\mathbb R, \mathbb R^{d})$. For our later use, we consider a separable Hilbert space ${\cal H}:=H^{-2}_{-r}(\mathbb R, \mathbb R^{d})$. Since ${\cal H}^{*} =H^{2}_{r}(\mathbb R, \mathbb R^{d})$, we have $$ C^{\infty}_{0}(\mathbb R, \mathbb R^{d}) \subset {\cal H}^{*} \subset E^{*} \subset H^{*} \equiv H \subset E \subset {\cal H} $$ and the inclusions are dense and continuous.
Let $D:=\{\varphi_{n} \}_{n=1}^{\infty} \subset C^{\infty}_{0}(\mathbb R, \mathbb R^{d})$ be the countable weakly dense $\mathbb Q$-linear subspace of ${\cal H}^{*}$ constructed on page 369 of Albeverio--R\"ockner \cite{AR}. Then by
\cite[Theorem 5.3]{AR}, for each $n\in \mathbb N$, there exists some $S_{n} \subset E$ with ${\rm Cap}(S_{n})=0$ such that the diffusion process $\mathbb M$ satisfies
\begin{equation} \langle X_{t}, \varphi_{n} \rangle = \langle w, \varphi_{n} \rangle +B^{(n)}_{t}
+\frac{1}{2}\int_{0}^{t} \beta_{\varphi_{n}}(X_{s}) ds, \quad t>0,~{\mathbb P}_{w} \mbox{-a.s.}, \label{weakform-2} \end{equation} for all $w \in E \setminus S_{n}$, where $\{B^{(n)}_{t} \}_{t\geq 0}$ is a one-dimensional $\{ {\cal F}_{t} \}$-adapted Brownian motion on $({\Theta}, {\cal F}, {\mathbb P}_{w})$ starting at zero multiplied by $\Vert \varphi_{n} \Vert_{H}$.
On the other hand, by recalling (\ref{p-moment}), (\ref{p-moment2}) and
\cite[Lemma 4.2]{AR}, there exists a set $S_{0}\subset E$ with ${\rm Cap}(S_{0})=0$ such that \begin{equation} {\mathbb P}_{w} \Big ( \int_{0}^{T}
\big( \Vert (\tilde \nabla U)(X_{s}(\cdot)) \Vert_{E} +\Vert X_{s}(\cdot) \Vert_{E} \big) ds <\infty \mbox{ for all } T>0 \Big )=1
\label{weakform-extend} \end{equation} for any $w\in E\setminus S_{0}$. Here we set $S:=\cup_{n=0}^{\infty} S_{n}$. Obviously, ${\rm Cap}(S)=0$.
By noting that the embedding map $H \hookrightarrow {\cal H}$ is a Hilbert--Schmidt operator (cf. \cite[Remark 2.1]{Funaki}), and \cite[Remark 6.3]{AR}, we can apply \cite[Lemma 6.1 and Theorem 6.2]{AR}, which implies that
there exists an $\{ {\cal F}_{t} \}_{t\geq 0}$-Brownian motion on $({\Theta}, {\cal F}, {\mathbb P}_{w})$ with values in $\cal H$
starting at zero with covariance $(\cdot, \cdot)_{H}$ (i.e., an $H$-cylindrical Brownian motion) under $\mathbb P_{w}$ for every $w \in E \setminus S$ such that
\begin{equation} \langle B_{t}, \varphi_{n} \rangle := \mbox{ }_{{\cal H}} \langle B_{t}, \varphi_{n} \rangle_{{\cal H}^{*}}=B^{(n)}_{t}, \quad n \in \mathbb N,~t \geq 0,~~{\mathbb P}_{w} \mbox{-a.s.},~~w\in E\setminus S. \label{Wiener process} \end{equation}
Since $D$ is dense in $C^{\infty}_{0}(\mathbb R, \mathbb R^{d})$ with respect to the weak topology of ${\cal H}^{*}$, for every $\varphi \in C^{\infty}_{0}(\mathbb R, \mathbb R^{d})$, we can take a subsequence $\{ \varphi_{n(j)} \}_{j=1}^{\infty} \subset D$ such that $\varphi_{n(j)} \to \varphi$ weakly in ${\cal H}^{*}$ as $j \to \infty$. Furthermore, the Banach--Saks theorem implies that, selecting another subsequence again denoted by $\{ \varphi_{n(j)} \}_{j=1}^{\infty}$, the Ces\`aro mean ${\hat \varphi}_{k}:=\frac{1}{k} \sum_{j=1}^{k} \varphi_{n(j)}, k\in \mathbb N$,
converges to $\varphi$ strongly in ${\cal H}^{*}$ as $k \to \infty$. Thus $\Vert \varphi-{\hat \varphi}_{k} \Vert_{E^{*}}+\Vert \Delta_{x}\varphi-\Delta_{x} {\hat \varphi}_{k} \Vert_{E^{*}} \to 0$ as $k \to \infty$. On the other hand, (\ref{weakform-2}) and (\ref{Wiener process}) imply \begin{equation} \langle X_{t}, \hat \varphi_{k} \rangle = \langle w, \hat \varphi_{k} \rangle +\langle B_{t}, \hat \varphi_{k} \rangle
+\frac{1}{2}\int_{0}^{t} \beta_{\hat \varphi_{k}}(X_{s}) ds, \quad t>0,~{\mathbb P}_{w} \mbox{-a.s.}, \label{weakform-3} \end{equation} for all $w \in E \setminus S$. Hence due to (\ref{weakform-extend}) we can take the limit $k \to \infty$ on both sides of (\ref{weakform-3}) to obtain SDE (\ref{weakform}) for all $w\in E \setminus S$.
Besides, the uniqueness statement for $\mathbb M$ is derived from item (1) (cf. \cite[Sections 2 and 3]{Cornell} and also \cite[Section 8]{DR}). This completes the proof.
\qed
\\ {\bf Proof of Theorem \ref{ES2}:}~By noting (\ref{weakform-extend}), the fact that $Q^{-1}(C^{\infty}_{0}(\mathbb R, \mathbb R^{d})) =C^{\infty}_{0}(\mathbb R, \mathbb R^{d})$ and Theorem \ref{ES}, we can read (\ref{weakform}) as \begin{eqnarray} (X_{t}, \varphi)_{E} &=& (w, \varphi)_{E} +\int_{0}^{t} ({\sqrt Q}\varphi, dW_{s})_{E} +\int_{0}^{t} \big \{ (X_{s},A^{*}\varphi)_{E}+(b(X_{s}), \varphi)_{E} \big \} ds, \nonumber \\ &\mbox{ }& \hspace{35mm} t>0,~\varphi \in C^{\infty}_{0}(\mathbb R, \mathbb R^{d}), ~{\mathbb P}_{w} \mbox{-a.s.},~w\in E\setminus S. \label{weakform-E} \end{eqnarray}
for all $w\in E\setminus S$, where $\{W_{t} \}_{t\geq 0}$ is an $\{ {\cal F}_{t} \}_{t\geq 0}$-adapted $E$-cylindrical Brownian motion corresponding to the $H$-cylindrical Brownian motion $\{ B_{t} \}_{t\geq 0}$ defined on $(\Theta, {\cal F}, {\mathbb P}_{w})$. (See \cite[Remark 3.5]{KR} for details.) Furthermore,
by recalling Lemma \ref{A}, we have equation (\ref{weakform-E}) for every $\varphi \in {\rm Dom}(A^{*})$. We also mention that (\ref{weakform-E}) is equivalent to the mild-form (\ref{abstSPDE2}) of SPDE (\ref{abstSPDE}) with $b_{\alpha, \beta}$ replaced by $b$. We refer to Ondrej\'at \cite[Theorem 13]{Ondre} for details.
Now, we prove pathwise uniqueness based on the argument of Marinelli--R\"ockner \cite{Carlo}. Suppose that $X=X^{w}$ and ${\widetilde X}={\widetilde X}^{w}$ are two weak solutions to SPDE (\ref{abstSPDE}) defined on the same filtered probability space $(\Theta, {\cal F}, \{ {\cal F}_{t} \}_{t\geq 0}, {\mathbb P})$ with the same $E$-cylindrical Brownian motion $\{ W_{t} \}_{t \geq 0}$ and $X_{0}={\widetilde X}_{0}=w\in E\setminus S$ such that \begin{equation} \int_{0}^{T} \Vert b(X_{s}) \Vert_{E}ds <\infty, \quad \int_{0}^{T} \Vert b({\widetilde X}_{s}) \Vert_{E}ds <\infty \quad \mbox{ for all } T>0,~{\mathbb P}\mbox{-a.s.} \label{p-integrability} \end{equation} We fix $T>0$ from now on, and set $\Psi_{t}:=X_{t}-{\widetilde X}_{t}$. Note that it enjoys an $\omega$-wise equation $$ d\Psi_{t}=A\Psi_{t}dt +(b(X_{t})-b({\widetilde X}_{t}))dt, \quad 0<t \leq T, $$ with the initial datum $\Psi_{0}=0$, again to be understood in the mild form. Since $X$ and $\widetilde X$ have continuous paths on $E$, (\ref{p-integrability}) implies that
$b(X_{\cdot})-b({\widetilde X}_{\cdot}) \in L^{1}([0,T], E)$ and $\sup_{0\leq t \leq T} \Vert \Psi_{t} \Vert_{E} <\infty$ hold for $\mathbb P$-a.s $\omega \in \Theta$.
Let $\{ \varphi_{n} \}_{n=1}^{\infty} \subset C^{\infty}_{0}(\mathbb R, \mathbb R^{d})$ be a CONS of $H$, and we set ${\widetilde \varphi}_{n}:=\rho_{r} \varphi_{n}$ and $e_{n}:=(I+\varepsilon A^{*})^{-1}{\widetilde \varphi}_{n} \in {\rm Dom}(A^{*})$ for $n \in \mathbb N$. We mention that $\{ {\widetilde \varphi}_{n} \}_{n=1}^{\infty}$ is a CONS of $E$.
Recalling (\ref{weakform-E}) and applying It\^o's formula, we have \begin{eqnarray} (e_{n},\Psi_{t})_{E}^{2} &=& 2 \int_{0}^{t} \Psi_{n}(s) d\Psi_{n}(s) \nonumber \\
& \mbox{ }& +2 \int_{0}^{t} ( e_{n}, \Psi_{s} )_{E} \big (e_{n}, b(X_{s})-b({\widetilde X}_{s}) \big )_{E} ds
\nonumber \\ &=:& 2 \big( J_{n}^{1}(t)+J_{n}^{2}(t) \big), \qquad 0\leq t \leq T. \label{Carlo-1} \end{eqnarray}
For the first term $J_{n}^{1}(t)$, Lebesgue's dominated convergence theorem leads us to \begin{eqnarray} \sum_{n=1}^{\infty} J_{n}^{1}(t) &=& \int_{0}^{t} \sum_{n=1}^{\infty} \big( (I+\varepsilon A^{*})^{-1}{\widetilde \varphi}_{n}, \Psi_{s} \big )_{E} \cdot \big ( (A(I+\varepsilon A)^{-1})^{*}{\widetilde \varphi}_{n}, {\Psi}_{s} \big )_{E} ds \nonumber \\ &=& \int_{0}^{t} \sum_{n=1}^{\infty} \big( {\widetilde \varphi}_{n}, (I+\varepsilon A)^{-1}{\Psi}_{s} \big )_{E} \cdot \big ( {\widetilde \varphi}_{n}, A(I+\varepsilon A)^{-1}{\Psi}_{s} \big )_{E} ds \nonumber \\ &=& \int_{0}^{t} \big( (I+\varepsilon A)^{-1}\Psi_{s}, A(I+\varepsilon A)^{-1}\Psi_{s} \big )_{E} ds \nonumber \\ &\leq & \big (r^{2}-\frac{\kappa}{2} \big) \int_{0}^{t} \big \Vert (I+\varepsilon A)^{-1}\Psi_{s} \big \Vert_{E}^{2}ds, \label{Carlo-2} \end{eqnarray}
where we used $(I+\varepsilon A^{*})^{-1} =( (I+\varepsilon A)^{-1})^{*}$ and the fact that $A^{*}$ and $(I+\varepsilon A^{*})^{-1}$ commute for the second line, and (\ref{fukuoka}) for the fourth line.
For the second term $J_{n}^{2}(t)$, since we have \begin{eqnarray} \lefteqn{ \int_{0}^{t} \big \Vert b(X_{s})-b({\widetilde X}_{s}) \big \Vert_{E} \Vert \Psi_{s} \Vert_{E} ds } \nonumber \\ &\leq & \big( \sup_{0\leq t \leq T} \Vert \Psi_{s} \Vert_{E} \big) \int_{0}^{T} \big \Vert b(X_{s})-b({\widetilde X}_{s}) \big \Vert_{E} ds<\infty, \quad 0\leq t \leq T, \quad {\mathbb P}\mbox{-a.s.}, \nonumber \end{eqnarray} Lebesgue's dominated convergence theorem also yields \begin{eqnarray} \sum_{n=1}^{\infty} J_{n}^{2}(t) &=& \int_{0}^{t} \sum_{n=1}^{\infty} \big( {\widetilde \varphi}_{n}, (I+\varepsilon A)^{-1}{\Psi}_{s} \big )_{E} \cdot \big ( {\widetilde \varphi}_{n}, (I+\varepsilon A)^{-1} \big (b(X_{s})-b({\widetilde X}_{s}) \big) \big )_{E} ds \nonumber \\ &=& \int_{0}^{t} \Big( (I+\varepsilon A)^{-1}\Psi_{s}, (I+\varepsilon A)^{-1} \big (b(X_{s})-b({\widetilde X}_{s}) \big) \Big )_{E} ds. \label{Carlo-3} \end{eqnarray} Then by putting (\ref{Carlo-2}) and (\ref{Carlo-3}) into (\ref{Carlo-1}),
we have \begin{eqnarray} \Vert (I+\varepsilon A)^{-1}\Psi_{t} \Vert_{E}^{2} &=&2 \sum_{n=1}^{\infty} \big( J_{n}^{1}(t)+J_{n}^{2}(t) \big) \nonumber \\ & \leq & (2r^{2}-{\kappa}) \int_{0}^{t} \big \Vert (I+\varepsilon A)^{-1}\Psi_{s} \big \Vert_{E}^{2}ds \nonumber \\ &\mbox{ }& +2 \int_{0}^{t} \Big( (I+\varepsilon A)^{-1}\Psi_{s}, (I+\varepsilon A)^{-1} \big (b(X_{s})-b({\widetilde X}_{s}) \big) \Big )_{E} ds. \nonumber \end{eqnarray} Moreover letting $\varepsilon \searrow 0$ on both sides, and recalling the dissipativity (\ref{tilde-b-dissipative}) for $\widetilde b$ and (\ref{def-b}), we obtain that $$ \Vert \Psi_{t} \Vert_{E}^{2} \leq (-K_{1}+2r^{2}) \int_{0}^{t} \Vert \Psi_{s} \Vert_{E}^{2}ds. $$ Hence, we have $\Psi_{t}=X_{t}-{\widetilde X}_{t}=0,~0\leq t \leq T$, $\mathbb P$-almost surely by an application of Gronwall's inequality, which proves the pathwise uniqueness. Then by \cite[Theorem 2]{Ondre}, a Yamada--Watanabe type argument implies that SPDE (\ref{GL}) has a unique strong solution. This completes the proof. \qed
By repeating the same argument as in the above proof, we can easily deduce the following coupling estimates (\ref{flow}) and (\ref{flow-2}) which play crucial roles in the next section.
\begin{co} \label{cor} Let $X^{w}$ and $X^{\tilde w}$ denote the strong solutions of SPDE {\rm{(\ref{GL})}} with the initial datum $X^{w}_{0}=w\in E\setminus S$ and $X^{\tilde w}_{0}=\tilde w\in E\setminus S$, respectively. Then
\begin{equation} \big \Vert X_{t}^{w}-X_{t}^{\tilde w} \big \Vert_{E}\leq e^{\frac{(-K_{1}+2r^{2})t}{2}} \Vert w-\tilde w \Vert_{E}, \quad t\geq 0,~{\mathbb P}\mbox{-a.s.} \label{flow} \end{equation}
In addition, for every $h \in H\setminus S$, we have \begin{equation} \Vert X_{t}^{w+h}-X_{t}^{w} \Vert_{H}\leq e^{-\frac{K_{1}t}{2}} \Vert h \Vert_{H}, \quad t\geq 0,~{\mathbb P}\mbox{-a.s.} \label{flow-2} \end{equation}
\end{co}
\section{Some Functional Inequalities}
In this section, as an application of Theorem \ref{ES2} and Corollary \ref{cor}, we present some functional inequalities
for the diffusion semigroup $\{P_{t} \}_{t\geq 0}$ generated by the Dirichlet operator ${\cal L}_{\mu}$. In particular, we
prove the gradient estimate for $\{P_{t} \}_{t\geq 0}$ and logarithmic Sobolev inequalities under much weaker conditions on the regularity and the growth rate of the potential function $U$ than in the previous papers \cite{kawa-POTA, Kawa-LSI} (which however already included the $P(\phi)_{1}$-case).
There are at present some approaches to derive these functional inequalities,
and it is well-known that Bakry--\'Emery's $\Gamma_{2}$-method (cf. Bakry \cite{B}) works efficiently on finite dimensional complete Riemannian manifolds. In contrast to finite dimensions,
we face a big difficulty to define the $\Gamma_{2}$-operator when we work in infinite dimensional frameworks, because it is not so easy to check the existence of a suitable core which is not only a ring but also stable under the operations both of the diffusion semigroup and its generator.
Hence, we cannot apply this method directly to the infinite dimensional model in the present paper.
On the other hand, we have the coupling estimates (\ref{flow}) and (\ref{flow-2}) which are implied by the strong uniqueness of the solution to SPDE (\ref{GL}). By making use of them, we can apply the stochastic approach presented in \cite{kawa-POTA, Kawa-LSI}.
First, we give the following gradient estimate for $\{P_{t} \}_{t\geq 0}$.
\begin{pr}[Gradient estimate] \label{thm-GE} For any $F\in {\cal D(E)}$, we have the following gradient estimate
\begin{equation} \Vert D(P_{t}F)(w) \Vert_{H} \leq e^{-\frac{K_{1}t}{2}} P_{t}\big ( \Vert DF \Vert_{H} \big)(w), \quad \mu \mbox{-a.e.}~w\in E,~t>0. \label{GE} \end{equation} \end{pr}
{\bf Proof:}~The proof is done in the same manner as the proof of \cite[Proposition 2.4]{kawa-POTA} together with the coupling estimate (\ref{flow-2}). So we omit it here. \qed \\
Now, we are in a position to state logarithmic Sobolev inequalities.
\begin{tm} [Log-Sobolev inequalities] \label{thm-LSI}
{\rm (1)}~For $F\in {\cal D(E)}$, we have the following heat kernel
log-Sobolev inequality
\begin{eqnarray} & & \hspace{-25mm} P_{t}(F^{2}\log F^{2})(w)-P_{t}(F^{2})(w)\log P_{t}(F^{2})(w) \nonumber \\ & & \leq \frac{2(1-e^{-K_{1}t})}{K_{1}}P_{t}(\Vert DF \Vert_{H}^{2})(w), \quad \mu \mbox{-a.e.}~w\in E,~t>0. \label{Heat-LSI-ineq} \end{eqnarray}
{\rm (2)}~If $K_{1}>0$, that is, $U$ is strictly convex, then the following log-Sobolev inequality \begin{equation} \int_{E} F(w)^{2} \log \Big( \frac{F(w)^{2}}{\Vert F \Vert_{L^{2}(\mu)}^{2}} \Big)\mu(dw) \leq \frac{2}{K_{1}}\int_{E} \Vert D_{H}F(w) \Vert_{H}^{2}\mu(dw), \quad F\in {\cal D(E)}
\label{LSI-ineq} \end{equation} holds.
Consequently, we have the spectral gap estimate
$ \inf \big ( \sigma(-{\cal L}_{\mu})\setminus \{0\} \big ) \geq \frac{K_{1}}{2}$. \end{tm}
{\bf Proof:}~We first sketch the proof of (1). We refer to \cite{kawa-POTA, Kawa-LSI} for all technical details. We may assume $F\in {\cal FC}_{b}^{\infty}$, i.e., $F(w)=f(\langle w, \varphi_{1} \rangle, \ldots, \langle w, \varphi_{n} \rangle)$, where $\{ \varphi_{i} \}_{i=1}^{n} \subset C^{\infty}_{0} (\mathbb R, \mathbb R^{d})$. Note that $P_{t}F$ can be extended to a function in $C_{b}(E)$ by using the coupling estimate (\ref{flow}), and the fact that ${\rm{supp}}(\mu)=E$.
We fix $\delta>0$, and introduce a function $G:[0,t] \to L^{1}(\mu)$ by $$ G(s):=P_{t-s} \big \{ (P_{s}(F^{2})+\delta) \log (P_{s}(F^{2})+\delta) \big \}(\cdot), \quad 0\leq s \leq t. $$ Then $G$ is differentiable with respect to $s$ and \begin{equation} {\dot G}(s)=-\frac{1}{2} P_{t-s} \Big \{ \frac{ \Vert DP_{s}(F^{2}) \Vert^{2}_{H}} {P_{s}(F^{2})+\delta } \Big \} (\cdot), \quad 0<s<t. \label{G-diff} \end{equation} On the other hand, Proposition \ref{thm-GE} and Schwarz's inequality imply \begin{equation} \Vert DP_{s}(F^{2}) \Vert^{2}_{H} \leq 4e^{-K_{1}s} P_{s}(F^{2}) \cdot P_{s} \big( \Vert DF \Vert_{H}^{2} \big). \label{F2-GE} \end{equation} By combining (\ref{G-diff}) with (\ref{F2-GE}), we have $$ \dot{G}(s) \geq -2e^{-K_{1}s} P_{t-s} \big \{ P_{s} \big( \Vert DF \Vert_{H}^{2} \big) \big \} =-2e^{-K_{1}s} P_{t} \big( \Vert DF \Vert_{H}^{2} \big). $$ This imply the heat kernel logarithmic Sobolev inequality (\ref{Heat-LSI-ineq}) by first integrating over $s$ from $0$ to $t$ and then by letting $\delta \searrow 0$.
Next, we prove (2). By noting that the Gibbs measure $\mu$ is the invariant measure for our stochastic dynamics $\mathbb M$,
we have the following estimate for $w\in E\setminus S$ and $t\geq 0$: \begin{eqnarray} \big \vert P_{t}F(w)-{\mathbb E}^{\mu}[F] \big \vert &\leq & \int_{E} {\mathbb E} \big [ \vert F(X_{t}^{w})-F(X_{t}^{\widetilde w}) \vert \big ]\mu(d\widetilde w) \nonumber \\ &\leq & \Vert \nabla f \Vert_{\infty} \big (\sum_{i=1}^{n} \Vert \varphi_{i} \Vert_{E^{*}}^{2} \big )^{1/2} e^{( \frac{-K_{1}+2 {r}^{2}}{2}t)} \int \Vert w-{\widetilde w} \Vert_{E}~ \mu(d{\widetilde w}) \nonumber \\ &\leq & {\sqrt 2} \Vert \nabla f \Vert_{\infty} \big (\sum_{i=1}^{n} \Vert \varphi_{i} \Vert_{E^{*}}^{2} \big )^{1/2} e^{( \frac{-K_{1}+2 {r}^{2}}{2}t)} \Big \{ \Vert w \Vert_{E}^{2}+ \int_{E} \Vert {\widetilde w} \Vert_{E}^{2}\mu(d{\widetilde w}) \Big \}^{1/2},
\nonumber \label{ergord} \\ \end{eqnarray} where we used (\ref{flow}) for the second line. Since $r>0$ satisfies $2r^{2}<K_{1}$, (\ref{ergord}) implies the following ergodic property of $\{P_{t}\}_{t\geq 0}$: \begin{equation} \lim_{t\to \infty} P_{t}F(w)={\mathbb E}^{\mu}[F], \quad w\in E\setminus S, \label{22-ergord} \end{equation}
Finally, we have the desired logarithmic Sobolev inequality (\ref{LSI-ineq}) by letting $t\to \infty$ on both sides of (\ref{Heat-LSI-ineq}) and using (\ref{22-ergord}). This completes the proof of (2). \qed
\begin{re} The logarithmic Sobolev inequality {\rm{(\ref{LSI-ineq})}} holds with $K_{1}\geq m^{2}$ in the case of exp$(\phi)_{1}$-quantum fields. \end{re}
\begin{re} We mention that many other functional inequalities including the dimension free parabolic Harnack inequality (cf. {\rm{\cite{kawa-POTA}}}) and the Littlewood--Paley--Stein inequality (cf. Kawabi--Miyokawa {\rm{\cite{KM}}}) for our infinite dimensional model can be obtained from the gradient estimate {\rm{(\ref{GE})}}. In particular, it is a fundamental and important problem in harmonic analysis and potential theory to ask for boundedness of the Riesz transform $R_{\alpha}({\cal L}_{p}):=D_{H}(\alpha -{\cal L}_{p})^{-1/2}$ on $L^{p}(\mu)$ for all $p>1$ and some $\alpha >0$, where ${\cal L}_{p}$ is the extension of $({\cal L}_{0}, {\cal FC}^{\infty}_{b})$ in $L^{p}(\mu)$, because boundedness of $R_{\alpha}({\cal L}_{p})$ yields the Meyer equivalence of first order Sobolev norms. In {\rm{\cite{shige-OJM}}}, Shigekawa studied this problem in a general framework assuming the intertwining property of the diffusion semigroup $\{P_{t} \}_{t\geq 0}$ and another semigroup $\{ \overrightarrow {P}_{t} \}_{t\geq 0}$ acting on vector-valued functions. We note that essential self-adjointness of $({\cal L}_{0}, {\cal FC}^{\infty}_{b})$ as obtained in Theorem {\rm{\ref{ES}}} plays a crucial role to prove this property for our model. (See e.g., Shigekawa {\rm{\cite{Shige2}}} and Kawabi {\rm{\cite{RIMS}}}.) We will discuss boundedness of the Riesz transform by making use of the Littlewood--Paley--Stein inequality and this intertwining property in a forthcoming paper. \end{re}
\section{ Appendix:~ Another Approach to the Log-Sobolev Inequality (\ref{LSI-ineq}) }
In this section,
we give another approach to the log-Sobolev inequality (\ref{LSI-ineq}).
First, we prepare the following lemma taken from Arai--Hirokawa \cite[Lemma 4.9]{Hirokawa}:
\begin{lm} \label{Hirokawa} Let $\{T_{n} \}_{n=1}^{\infty}$
and $T$ be self-adjoint operators on a Hilbert space ${\cal H}$ having a common core ${\cal D}$ such that, for all $\psi \in {\cal D}$, $T_{n}\psi \to T\psi$ as $n \to \infty$. Let $\psi_{n}$ be a normalized eigenvectors of $T_{n}$ with eigenvalue $E_{n}: T_{n}\psi_{n}=E_{n}\psi_{n}$. Assume that $E:=\lim_{n\to \infty}E_{n}$ exists and that the weak limit ${\mbox{w-}}\lim_{n \to \infty}\psi_{n}=\psi$ also exists and one has $\psi \neq 0$. Then $\psi$ is an eigenvector of $T$ with eigenvalue $E$. In particular, if $\psi_{n}$ is a ground state of $T_{n}$, then $\psi$ is a ground state of $T$. \end{lm}
\begin{lm} \label{Conv-GS} Let $U_{N}(z):=\frac{K_{1}}{2}\vert z \vert^{2}+V_{1/N}(z), N=1,2, \ldots$, be potential functions, where $V_{1/N}$ is the Moreau--Yosida approximation of $V$.
We consider the Schr\"odinger operator $H_{U_{N}}=-\frac{1}{2}\Delta_{z}+U_{N}$ on $L^{2}({\mathbb R}^{d}, {\mathbb R})$, and denote by $(\lambda_{0})_{N}$ and $\Omega_{N}$ the minimal eigenvalue and the (normalized) ground state of $H_{U_{N}}$, respectively. Then the following properties hold under the assumption $K_{1}>0$: \\ {\rm{(1)}} $(\lambda_{0})_{N} \nearrow \lambda_{0}$ as $N \to \infty$. \\ {\rm{(2)}} There exists a sub-sequence $\{N(k)\}_{k=1}^{\infty}$ of $N \to \infty$
such that $\Vert \Omega_{N(k)}-\Omega \Vert_{L^{2}({\mathbb R}^{d}, {\mathbb R})} \to 0$ as $k \to \infty$. \end{lm}
{\bf Proof.}~(1)~Since $U_{N} \nearrow U$ as $N\to \infty$, we have $(\lambda_{0})_{1}\leq (\lambda_{0})_{2} \leq \cdots \leq \lambda_{0}$. Moreover, recalling (\ref{falloff}) and taking into account the estimate $(U_{N})_{\frac{1}{2}\vert z\vert }(z)\geq \frac{K_{1}}{8}\vert z \vert^{2}, z\in \mathbb R^{d}$ for every $N\in \mathbb N$, we have the following uniform pointwise upper bound for $\{\Omega_{N} \}_{N=1}^{\infty}$: \begin{equation} 0<\Omega_{N}(z) \leq D_{1} \exp (-\frac{D_{2} K_{1}^{1/2}}{2\sqrt 2} \vert z \vert^{2}), \quad z\in \mathbb R^{d}. \label{Omega-N-upper} \end{equation}
On the other hand, the variational characterization of the minimal eigenvalue and the ground state implies
\begin{eqnarray} (\lambda_{0})_{N}&=& \big({\Omega}_{N},H_{U_{N}}\Omega_{N} \big)_{L^{2}({\mathbb R}^{d}, {\mathbb R})} \nonumber \\ &=& \big({\Omega}_{N},H_{U}\Omega_{N} \big)_{L^{2}({\mathbb R}^{d}, {\mathbb R})}- \big({\Omega}_{N},(V-V_{N})\Omega_{N} \big)_{L^{2}({\mathbb R}^{d}, {\mathbb R})} \nonumber \\ &\geq & \lambda_{0}- \big({\Omega}_{N},(V-V_{N})\Omega_{N} \big)_{L^{2}({\mathbb R}^{d}, {\mathbb R})}, \label{3-2-1} \end{eqnarray} and by Lebesgue's monotone convergence theorem, we also have \begin{eqnarray} 0&\leq & \lim_{N \to \infty} \big({\Omega}_{N},(V-V_{N})\Omega_{N} \big)_{L^{2}({\mathbb R}^{d}, {\mathbb R})} \nonumber \\ &\leq & D_{1}^{2} \lim_{N \to \infty} \int_{\mathbb R^{d}} (V(z)-V_{N}(z)) \exp (-\frac{D_{2}K_{1}^{1/2}}{\sqrt 2} \vert z \vert^{2}) dz \nonumber \\ &=& D_{1}^{2} \int_{\mathbb R^{d}} \lim_{N \to \infty} (V(z)-V_{N}(z)) \exp (-\frac{D_{2}K_{1}^{1/2}}{\sqrt 2} \vert z \vert^{2}) dz=0, \label{3-2-2} \end{eqnarray} where we used (\ref{Omega-N-upper}) for the second line.
Hence by combining (\ref{3-2-1}) with (\ref{3-2-2}), we have $\lim_{N \to \infty} (\lambda_{0})_{N} \geq \lambda_{0}$, which completes the proof of (1).
\\
(2) We take $C^{\infty}_{0}(\mathbb R^{d}, \mathbb R)$ as a common core of the Schr\"odinger operators $\{H_{U_{N}} \}_{N=1}^{\infty}$ and $H_{U}$ (cf. \cite[Theorem X.28]{rs}), and by Lebesgue's monotone convergence theorem, we can easily see that $H_{U_{N}}\psi \to H_{U}\psi$ for all $\psi \in C^{\infty}_{0}(\mathbb R^{d}, \mathbb R)$ as $N \to \infty$. Since $\Vert \Omega_{N} \Vert_{L^{2}({\mathbb R}^{d}, {\mathbb R})}=1$ for all $N\in \mathbb N$, there exist a sub-sequence $\{N(k) \nearrow \infty \}$ and a function $\psi \in L^{2}({\mathbb R}^{d}, {\mathbb R})$ such that $\Omega_{N(k)} \to \psi$ weakly as $k \to \infty$.
On the other hand, by \cite[Theorem 25.16]{simon},
there exist some positive constants $D_{3}, D_{4}$ independent of $N$ such that
\begin{equation} \Omega_{N}(z) \geq D_{3} \exp \big (-D_{4} \vert z \vert {{U_{N}^{(\infty)}(z)^{1/2}}} \big),
\quad z\in \mathbb R^{d}, \label{Omega-N-lower} \end{equation}
where $U_{N}^{(\infty)}(z):= \inf \{ U_{N}(y)\vert~\vert y \vert \leq 3 \vert z\vert \}$. Recalling condition {\bf (U3)}, we see that \begin{eqnarray} U_{N}^{(\infty)}(z) & \leq & \inf \{ U(y)\vert~\vert y \vert \leq 3 \vert z\vert \} \nonumber \\ & \leq & \vert U(0) \vert +3K_{3} \vert z \vert \exp (3^{\beta} K_{4} \vert z \vert^{\beta} ),
\quad z\in \mathbb R^{d}. \label{Omega-N-lower-2} \end{eqnarray}
Then combining (\ref{Omega-N-lower}) with (\ref{Omega-N-lower-2}), we deduce that \begin{eqnarray} \Omega_{N}(z) & \geq & D_{3} \exp \Big \{ -D_{4}\vert z \vert {\sqrt{\vert U(0) \vert +3K_{3} \vert z \vert \exp (3^{\beta} K_{4} \vert z \vert^{\beta} )}} \Big \} \nonumber \\ &=:&\Psi (z), \quad z\in \mathbb R^{d},
\label{Omega-N-lower-3} \end{eqnarray} and hence the uniform pointwise lower estimate (\ref{Omega-N-lower-2}) implies that $$\lim_{k \to \infty}(\Omega_{N(k)},\Psi)_{L^{2}({\mathbb R}^{d}, {\mathbb R})} \geq \Vert \Psi \Vert^{2}_{L^{2}({\mathbb R}^{d}, {\mathbb R})} >0$$ and we now see that $\psi \neq 0$ holds.
Now by item (1) and Lemma \ref{Hirokawa}, it follows that
$\psi$ is a ground state of $H_{U}$. However, since we already know the uniqueness of the ground state of $H_{U}$, $\Omega_{N(k)} \to \psi=\Omega$ weakly as $k\to \infty$.
Moreover since $$\lim_{k \to \infty} \Vert \Omega_{N(k)} \Vert_{L^{2}({\mathbb R}^{d}, {\mathbb R})} =\Vert \Omega \Vert_{L^{2}({\mathbb R}^{d}, {\mathbb R})}=1,$$ we conclude that $\lim_{k \to \infty} \Vert \Omega_{N(k)}-\Omega \Vert_{L^{2}({\mathbb R}^{d}, {\mathbb R})}=0$.
\qed
\\
{\bf Proof of the Log-Sobolev Inequality (\ref{LSI-ineq}):} By the same procedure as in Section 2, we can construct a Gibbs measure $\mu_{N}$ with $\mu_{N}({\cal C})=1$
if we replace $U$ by $U_{N}$. As we have seen in the proof of Theorem \ref{ES}, ${\widetilde \nabla} U_{1/N}(z)=K_{1}z+\partial_{0} (V_{1/N})(z), z\in {\mathbb R}^{d}$, is Lipschitz continuous. Thus, we can apply \cite[Theorem 1.2]{Kawa-LSI}, and we see that the following logarithmic Sobolev inequality holds for each $\mu_{N}$: \begin{equation} \int_{E} F(w)^{2} \log \Big( \frac{F(w)^{2}}{\Vert F \Vert_{L^{2}(\mu_{N})}^{2}} \Big)\mu_{N}(dw) \leq \frac{2}{K_{1}}\int_{E} \Vert D_{H}F(w) \Vert_{H}^{2}\mu_{N}(dw), \quad F\in {\cal FC}^{\infty}_{b}. \label{LSI-ineq-N} \end{equation}
Next, we aim to prove the tightness of the family of probability measures $\{\mu_{N} \}_{N=1}^{\infty}$ on ${\cal C}$. Due to \cite[Lemma 5.4]{Iwa2}, it suffices to verify the following two conditions:
\begin{eqnarray} & & \hspace{-8mm} {\mbox{ (i)~There exists a constant }} \gamma>0 {\mbox{ such that }} \sup_{N\in \mathbb N} \int_{\cal C} \vert w(0) \vert^{\gamma} \mu_{N}(dw)<\infty. \nonumber \\ & & \hspace{-8mm} {\mbox{ (ii)~For each }}r>0, {\mbox{there exist constants }}p, q, M>0 {\mbox{ independent of }}N {\mbox{ such that }} \nonumber \\ & &
\int_{\cal C} \vert w(x_{1})-w(x_{2}) \vert^{p} \mu_{N}(dw)\leq M \vert x_{1}-x_{2} \vert^{2+q} \rho_{r}(x_{1}) {\mbox{ for }}x_{1},x_{2} \in \mathbb R {\mbox{ with }} \vert x_{1}-x_{2} \vert \leq 1. \nonumber \end{eqnarray}
By combining the translation invariance of $\mu_{N}$ with estimate (\ref{Omega-N-upper}), we have \begin{equation} \int_{\cal C} \vert w(0) \vert^{2} \mu_{N}(dw) =\int_{\mathbb R^{d}} \vert z \vert^{2} \Omega_{N}(z)^{2}dz \leq D_{1}^{2}\int_{\mathbb R^{d}} \vert z \vert^{2} \exp (-\frac{D_{2}K_{1}^{1/2}}{\sqrt 2} \vert z \vert^{2})dz. \nonumber \end{equation} Hence we have shown that condition (i) holds with $\gamma=2$.
Besides, in a similar way to \cite{Iwa1}, we see that \begin{eqnarray} & & \hspace{-10mm} \int_{\cal C} \vert w(x_{1})-w(x_{2}) \vert^{2m} \mu_{N}(dw) \nonumber \\ & & \leq \exp \big \{ \big ( (\lambda_{0})_{N}-\inf_{z\in \mathbb R^{d}} U_{N}(z) \big) \vert x_{1}-x_{2} \vert \big \} \big( \sup_{z \in \mathbb R^{d}} \Omega_{N}(z) \big) \nonumber \\ & & \quad \times \int_{\mathbb R^{d}} \Omega_{N}(z) dz \cdot (2m-1)!! \cdot \vert x_{1}-x_{2} \vert^{m} \nonumber \\ & & \leq \exp \big \{ \big ( \lambda_{0}-\inf_{z\in \mathbb R^{d}} U_{1}(z) \big) \vert x_{1}-x_{2} \vert \big \} D_{1}^{2}
\Big (\frac{ {\sqrt 2}\pi}{D_{2}K_{1}^{1/2}} \Big)^{d/2}
(2m-1)!! \cdot \vert x_{1}-x_{2} \vert^{m} \nonumber \end{eqnarray} for every $m\in \mathbb N$, where $(2m-1)!!:=\prod_{k=1}^{m} (2k-1)$ and we used Lemma \ref{Conv-GS} and (\ref{Omega-N-upper}) for the third line. Hence we can find a positive constant $C$ independent of $N$ such that \begin{equation} \int_{\cal C} \vert w(x_{1})-w(x_{2}) \vert^{2m} \mu_{N}(dw) \leq C \vert x_{1}-x_{2} \vert^{m}, \quad {\mbox{for }}x_{1},x_{2} \in \mathbb R {\mbox{ with }} \vert x_{1}-x_{2} \vert \leq 1, \nonumber \end{equation} and hence we have proven condition (ii).
Thus we can find a sub-sequence $\{N(j) \nearrow \infty \}$ such that $\mu_{N(j)}$ converges to some probability measure $\mu_{*}$ weakly on ${\cal C}$.
On the other hand, by virtue of the Feynman--Kac formula, we have \begin{equation} \lim_{N \to \infty} \Vert e^{-tH_{U_{N}}}\psi -e^{-tH_{U}}\psi \Vert_{L^{2}(\mathbb R^{d}, \mathbb R)}=0, \quad \psi \in L^{2}(\mathbb R^{d}, \mathbb R). \label{FKsemigroup-conv} \end{equation} Then by putting (\ref{FKsemigroup-conv}) and Lemma \ref{Conv-GS} into (\ref{Construction-Gibbs}),
we see that there exists a sub-sequence $\{N(k) \nearrow \infty \}$ of $\{N(j)\}$ such that $\lim_{k \to \infty} \mu_{N(k)}(A)=\mu(A)$ for each cylinder set $A\in {\cal B}_{[T_{1},T_{2}]}, T_{1}<T_{2}$.
Hence we obtain $\mu_{*}=\mu$.
Finally, since $F\in {\cal FC}_{b}^{\infty}$ can be regarded as an element of $C_{b}({\cal C})$ in a natural way, we can take the limit $k \to \infty$ on both sides of (\ref{LSI-ineq-N}). This implies the desired inequality (\ref{LSI-ineq}). \qed
\\
{\bf Acknowledgment.} The authors are grateful to Masao Hirokawa for useful discussions on the paper {\cite{Hirokawa}}, and to Volker Betz and Martin Hairer for providing helpful comments on Remark \ref{Betz-Hairer}. They were partially supported by the DFG--JSPS joint research project \lq \lq Dirichlet Forms, Stochastic Analysis and Interacting Systems" (2007--2008), and CRC 701, as well as by the project NEST of the Provincia Autonoma di Trento, at
University of Trento and by HCM at University of Bonn. The second named author was also partially supported by Grant-in-Aid for Young Scientists (Start-up) (18840034) and
(B) (20740076) from MEXT. This work was completed while the authors were visiting Isaac Newton Institute for Mathematical Sciences at University of Cambridge. They would like to thank the institute for its warm hospitality.
\end{document} | arXiv |
Multiset
In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set,[1] allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that element in the multiset. As a consequence, an infinite number of multisets exist which contain only elements a and b, but vary in the multiplicities of their elements:
• The set {a, b} contains only elements a and b, each having multiplicity 1 when {a, b} is seen as a multiset.
• In the multiset {a, a, b}, the element a has multiplicity 2, and b has multiplicity 1.
• In the multiset {a, a, a, b, b, b}, a and b both have multiplicity 3.
These objects are all different when viewed as multisets, although they are the same set, since they all consist of the same elements. As with sets, and in contrast to tuples, the order in which elements are listed does not matter in discriminating multisets, so {a, a, b} and {a, b, a} denote the same multiset. To distinguish between sets and multisets, a notation that incorporates square brackets is sometimes used: the multiset {a, a, b} can be denoted by [a, a, b].[2]
The cardinality of a multiset is the sum of the multiplicities of all its elements. For example, in the multiset {a, a, b, b, b, c} the multiplicities of the members a, b, and c are respectively 2, 3, and 1, and therefore the cardinality of this multiset is 6.
Nicolaas Govert de Bruijn coined the word multiset in the 1970s, according to Donald Knuth.[3]: 694 However, the concept of multisets predates the coinage of the word multiset by many centuries. Knuth himself attributes the first study of multisets to the Indian mathematician Bhāskarāchārya, who described permutations of multisets around 1150. Other names have been proposed or used for this concept, including list, bunch, bag, heap, sample, weighted set, collection, and suite.[3]: 694
History
Wayne Blizard traced multisets back to the very origin of numbers, arguing that "in ancient times, the number n was often represented by a collection of n strokes, tally marks, or units."[4] These and similar collections of objects are multisets, because strokes, tally marks, or units are considered indistinguishable. This shows that people implicitly used multisets even before mathematics emerged.
Practical needs for this structure have caused multisets to be rediscovered several times, appearing in literature under different names.[5]: 323 For instance, they were important in early AI languages, such as QA4, where they were referred to as bags, a term attributed to Peter Deutsch.[6] A multiset has been also called an aggregate, heap, bunch, sample, weighted set, occurrence set, and fireset (finitely repeated element set).[5]: 320 [7]
Although multisets were used implicitly from ancient times, their explicit exploration happened much later. The first known study of multisets is attributed to the Indian mathematician Bhāskarāchārya circa 1150, who described permutations of multisets.[3]: 694 The work of Marius Nizolius (1498–1576) contains another early reference to the concept of multisets.[8] Athanasius Kircher found the number of multiset permutations when one element can be repeated.[9] Jean Prestet published a general rule for multiset permutations in 1675.[10] John Wallis explained this rule in more detail in 1685.[11]
Multisets appeared explicitly in the work of Richard Dedekind.[12][13]
Other mathematicians formalized multisets and began to study them as precise mathematical structures in the 20th century. For example, Whitney (1933) described generalized sets ("sets" whose characteristic functions may take any integer value - positive, negative or zero).[5]: 326 [14]: 405 Monro (1987) investigated the category Mul of multisets and their morphisms, defining a multiset as a set with an equivalence relation between elements "of the same sort", and a morphism between multisets as a function which respects sorts. He also introduced a multinumber : a function f (x) from a multiset to the natural numbers, giving the multiplicity of element x in the multiset. Monro argued that the concepts of multiset and multinumber are often mixed indiscriminately, though both are useful.[5]: 327–328 [15]
Examples
One of the simplest and most natural examples is the multiset of prime factors of a natural number n. Here the underlying set of elements is the set of prime factors of n. For example, the number 120 has the prime factorization
$120=2^{3}3^{1}5^{1}$
which gives the multiset {2, 2, 2, 3, 5}.
A related example is the multiset of solutions of an algebraic equation. A quadratic equation, for example, has two solutions. However, in some cases they are both the same number. Thus the multiset of solutions of the equation could be {3, 5}, or it could be {4, 4}. In the latter case it has a solution of multiplicity 2. More generally, the fundamental theorem of algebra asserts that the complex solutions of a polynomial equation of degree d always form a multiset of cardinality d.
A special case of the above are the eigenvalues of a matrix, whose multiplicity is usually defined as their multiplicity as roots of the characteristic polynomial. However two other multiplicities are naturally defined for eigenvalues, their multiplicities as roots of the minimal polynomial, and the geometric multiplicity, which is defined as the dimension of the kernel of A − λI (where λ is an eigenvalue of the matrix A). These three multiplicities define three multisets of eigenvalues, which may be all different: Let A be a n × n matrix in Jordan normal form that has a single eigenvalue. Its multiplicity is n, its multiplicity as a root of the minimal polynomial is the size of the largest Jordan block, and its geometric multiplicity is the number of Jordan blocks.
Definition
A multiset may be formally defined as an ordered pair (A, m) where A is the underlying set of the multiset, formed from its distinct elements, and $m\colon A\to \mathbb {Z} ^{+}$ is a function from A to the set of positive integers, giving the multiplicity – that is, the number of occurrences – of the element a in the multiset as the number m(a).
Representing the function m by its graph (the set of ordered pairs $\left\{\left(a,m\left(a\right)\right):a\in A\right\}$) allows for writing the multiset {a, a, b} as ({a, b}, {(a, 2), (b, 1)}), and the multiset {a, b} as ({a, b}, {(a, 1), (b, 1)}). This notation is however not commonly used and more compact notations are employed.
If $A=\{a_{1},\ldots ,a_{n}\}$ is a finite set, the multiset (A, m) is often represented as
$\left\{a_{1}^{m(a_{1})},\ldots ,a_{n}^{m(a_{n})}\right\},\quad $ sometimes simplified to $\quad a_{1}^{m(a_{1})}\cdots a_{n}^{m(a_{n})},$
where upper indices equal to 1 are omitted. For example, the multiset {a, a, b} may be written $\{a^{2},b\}$ or $a^{2}b.$ If the elements of the multiset are numbers, a confusion is possible with ordinary arithmetic operations, those normally can be excluded from the context. On the other hand, the latter notation is coherent with the fact that the prime factorization of a positive integer is a uniquely defined multiset, as asserted by the fundamental theorem of arithmetic. Also, a monomial is a multiset of indeterminates; for example, the monomial x3y2 corresponds to the multiset {x, x, x, y, y}.
A multiset corresponds to an ordinary set if the multiplicity of every element is 1. An indexed family (ai)i∈I, where i varies over some index set I, may define a multiset, sometimes written {ai}. In this view the underlying set of the multiset is given by the image of the family, and the multiplicity of any element x is the number of index values i such that $a_{i}=x$. In this article the multiplicities are considered to be finite, so that no element occurs infinitely many times in the family; even in an infinite multiset, the multiplicities are finite numbers.
It is possible to extend the definition of a multiset by allowing multiplicities of individual elements to be infinite cardinals instead of positive integers, but not all properties carry over to this generalization.
Basic properties and operations
Elements of a multiset are generally taken in a fixed set U, sometimes called a universe, which is often the set of natural numbers. An element of U that does not belong to a given multiset is said to have a multiplicity 0 in this multiset. This extends the multiplicity function of the multiset to a function from U to the set $\mathbb {N} $ of non-negative integers. This defines a one-to-one correspondence between these functions and the multisets that have their elements in U.
This extended multiplicity function is commonly called simply the multiplicity function, and suffices for defining multisets when the universe containing the elements has been fixed. This multiplicity function is a generalization of the indicator function of a subset, and shares some properties with it.
The support of a multiset $A$ in a universe U is the underlying set of the multiset. Using the multiplicity function $m$, it is characterized as
$\operatorname {Supp} (A):=\left\{x\in U\mid m_{A}(x)>0\right\}.$
A multiset is finite if its support is finite, or, equivalently, if its cardinality
$|A|=\sum _{x\in \operatorname {Supp} (A)}m_{A}(x)=\sum _{x\in U}m_{A}(x)$
is finite. The empty multiset is the unique multiset with an empty support (underlying set), and thus a cardinality 0.
The usual operations of sets may be extended to multisets by using the multiplicity function, in a similar way to using the indicator function for subsets. In the following, A and B are multisets in a given universe U, with multiplicity functions $m_{A}$ and $m_{B}.$
• Inclusion: A is included in B, denoted A ⊆ B, if
$m_{A}(x)\leq m_{B}(x)\quad \forall x\in U.$
• Union: the union (called, in some contexts, the maximum or lowest common multiple) of A and B is the multiset C with multiplicity function[13]
$m_{C}(x)=\max(m_{A}(x),m_{B}(x))\quad \forall x\in U.$
• Intersection: the intersection (called, in some contexts, the infimum or greatest common divisor) of A and B is the multiset C with multiplicity function
$m_{C}(x)=\min(m_{A}(x),m_{B}(x))\quad \forall x\in U.$
• Sum: the sum of A and B is the multiset C with multiplicity function
$m_{C}(x)=m_{A}(x)+m_{B}(x)\quad \forall x\in U.$
It may be viewed as a generalization of the disjoint union of sets. It defines a commutative monoid structure on the finite multisets in a given universe. This monoid is a free commutative monoid, with the universe as a basis.
• Difference: the difference of A and B is the multiset C with multiplicity function
$m_{C}(x)=\max(m_{A}(x)-m_{B}(x),0)\quad \forall x\in U.$
Two multisets are disjoint if their supports are disjoint sets. This is equivalent to saying that their intersection is the empty multiset or that their sum equals their union.
There is an inclusion–exclusion principle for finite multisets (similar to the one for sets), stating that a finite union of finite multisets is the difference of two sums of multisets: in the first sum we consider all possible intersections of an odd number of the given multisets, while in the second sum we consider all possible intersections of an even number of the given multisets.
Counting multisets
See also: Stars and bars (combinatorics)
The number of multisets of cardinality k, with elements taken from a finite set of cardinality n, is called the multiset coefficient or multiset number. This number is written by some authors as $\textstyle \left(\!\!{n \choose k}\!\!\right)$, a notation that is meant to resemble that of binomial coefficients; it is used for instance in (Stanley, 1997), and could be pronounced "n multichoose k" to resemble "n choose k" for ${\tbinom {n}{k}}.$ Like the binomial distribution that involves binomial coefficients, there is a negative binomial distribution in which the multiset coefficients occur. Multiset coefficients should not be confused with the unrelated multinomial coefficients that occur in the multinomial theorem.
The value of multiset coefficients can be given explicitly as
$\left(\!\!{n \choose k}\!\!\right)={n+k-1 \choose k}={\frac {(n+k-1)!}{k!\,(n-1)!}}={n(n+1)(n+2)\cdots (n+k-1) \over k!},$
where the second expression is as a binomial coefficient;[lower-alpha 1] many authors in fact avoid separate notation and just write binomial coefficients. So, the number of such multisets is the same as the number of subsets of cardinality k of a set of cardinality n + k − 1. The analogy with binomial coefficients can be stressed by writing the numerator in the above expression as a rising factorial power
$\left(\!\!{n \choose k}\!\!\right)={n^{\overline {k}} \over k!},$
to match the expression of binomial coefficients using a falling factorial power:
${n \choose k}={n^{\underline {k}} \over k!}.$
either of which is well defined even if we allow n to be an arbitrary complex number.
There are for example 4 multisets of cardinality 3 with elements taken from the set {1, 2} of cardinality 2 (n = 2, k = 3), namely {1, 1, 1}, {1, 1, 2}, {1, 2, 2}, {2, 2, 2}. There are also 4 subsets of cardinality 3 in the set {1, 2, 3, 4} of cardinality 4 (n + k − 1), namely {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}.
One simple way to prove the equality of multiset coefficients and binomial coefficients given above involves representing multisets in the following way. First, consider the notation for multisets that would represent {a, a, a, a, a, a, b, b, c, c, c, d, d, d, d, d, d, d} (6 as, 2 bs, 3 cs, 7 ds) in this form:
• • • • • • | • • | • • • | • • • • • • •
This is a multiset of cardinality k = 18 made of elements of a set of cardinality n = 4. The number of characters including both dots and vertical lines used in this notation is 18 + 4 − 1. The number of vertical lines is 4 − 1. The number of multisets of cardinality 18 is then the number of ways to arrange the 4 − 1 vertical lines among the 18 + 4 − 1 characters, and is thus the number of subsets of cardinality 4 − 1 of a set of cardinality 18 + 4 − 1. Equivalently, it is the number of ways to arrange the 18 dots among the 18 + 4 − 1 characters, which is the number of subsets of cardinality 18 of a set of cardinality 18 + 4 − 1. This is
${4+18-1 \choose 4-1}={4+18-1 \choose 18}=1330,$
thus is the value of the multiset coefficient and its equivalencies:
${\begin{aligned}\left(\!\!{4 \choose 18}\!\!\right)&={21 \choose 18}={\frac {21!}{18!\,3!}}={21 \choose 3},\\[1ex]&={\frac {{\color {red}{\mathfrak {4\cdot 5\cdot 6\cdot 7\cdot 8\cdot 9\cdot 10\cdot 11\cdot 12\cdot 13\cdot 14\cdot 15\cdot 16\cdot 17\cdot 18}}}\cdot \mathbf {19\cdot 20\cdot 21} }{\mathbf {1\cdot 2\cdot 3} \cdot {\color {red}{\mathfrak {4\cdot 5\cdot 6\cdot 7\cdot 8\cdot 9\cdot 10\cdot 11\cdot 12\cdot 13\cdot 14\cdot 15\cdot 16\cdot 17\cdot 18}}}}},\\[1ex]&={\frac {1\cdot 2\cdot 3\cdot 4\cdot 5\cdots 16\cdot 17\cdot 18\;\mathbf {\cdot \;19\cdot 20\cdot 21} }{\,1\cdot 2\cdot 3\cdot 4\cdot 5\cdots 16\cdot 17\cdot 18\,\times \,\mathbf {1\cdot 2\cdot 3\quad } }},\\[1ex]&={\frac {19\cdot 20\cdot 21}{1\cdot 2\cdot 3}}.\end{aligned}}$
From the relation between binomial coefficients and multiset coefficients, it follows that the number of multisets of cardinality k in a set of cardinality n can be written
$\left(\!\!{n \choose k}\!\!\right)=(-1)^{k}{-n \choose k}.$
Additionally,
$\left(\!\!{n \choose k}\!\!\right)=\left(\!\!{k+1 \choose n-1}\!\!\right).$
Recurrence relation
A recurrence relation for multiset coefficients may be given as
$\left(\!\!{n \choose k}\!\!\right)=\left(\!\!{n \choose k-1}\!\!\right)+\left(\!\!{n-1 \choose k}\!\!\right)\quad {\mbox{for }}n,k>0$
with
$\left(\!\!{n \choose 0}\!\!\right)=1,\quad n\in \mathbb {N} ,\quad {\mbox{and}}\quad \left(\!\!{0 \choose k}\!\!\right)=0,\quad k>0.$
The above recurrence may be interpreted as follows. Let $[n]:=\{1,\dots ,n\}$ be the source set. There is always exactly one (empty) multiset of size 0, and if n = 0 there are no larger multisets, which gives the initial conditions.
Now, consider the case in which n, k > 0. A multiset of cardinality k with elements from [n] might or might not contain any instance of the final element n. If it does appear, then by removing n once, one is left with a multiset of cardinality k − 1 of elements from [n], and every such multiset can arise, which gives a total of
$\left(\!\!{n \choose k-1}\!\!\right)$
possibilities.
If n does not appear, then our original multiset is equal to a multiset of cardinality k with elements from [n − 1], of which there are
$\left(\!\!{n-1 \choose k}\!\!\right).$
Thus,
$\left(\!\!{n \choose k}\!\!\right)=\left(\!\!{n \choose k-1}\!\!\right)+\left(\!\!{n-1 \choose k}\!\!\right).$
Generating series
The generating function of the multiset coefficients is very simple, being
$\sum _{d=0}^{\infty }\left(\!\!{n \choose d}\!\!\right)t^{d}={\frac {1}{(1-t)^{n}}}.$
As multisets are in one-to-one correspondence with monomials, $\left(\!\!{n \choose d}\!\!\right)$ is also the number of monomials of degree d in n indeterminates. Thus, the above series is also the Hilbert series of the polynomial ring $k[x_{1},\ldots ,x_{n}].$
As $\left(\!\!{n \choose d}\!\!\right)$ is a polynomial in n, it and the generating function are well defined for any complex value of n.
Generalization and connection to the negative binomial series
The multiplicative formula allows the definition of multiset coefficients to be extended by replacing n by an arbitrary number α (negative, real, or complex):
$\left(\!\!{\alpha \choose k}\!\!\right)={\frac {\alpha ^{\overline {k}}}{k!}}={\frac {\alpha (\alpha +1)(\alpha +2)\cdots (\alpha +k-1)}{k(k-1)(k-2)\cdots 1}}\quad {\text{for }}k\in \mathbb {N} {\text{ and arbitrary }}\alpha .$
With this definition one has a generalization of the negative binomial formula (with one of the variables set to 1), which justifies calling the $\left(\!\!{\alpha \choose k}\!\!\right)$ negative binomial coefficients:
$(1-X)^{-\alpha }=\sum _{k=0}^{\infty }\left(\!\!{\alpha \choose k}\!\!\right)X^{k}.$
This Taylor series formula is valid for all complex numbers α and X with |X| < 1. It can also be interpreted as an identity of formal power series in X, where it actually can serve as definition of arbitrary powers of series with constant coefficient equal to 1; the point is that with this definition all identities hold that one expects for exponentiation, notably
$(1-X)^{-\alpha }(1-X)^{-\beta }=(1-X)^{-(\alpha +\beta )}\quad {\text{and}}\quad ((1-X)^{-\alpha })^{-\beta }=(1-X)^{-(-\alpha \beta )},$
and formulas such as these can be used to prove identities for the multiset coefficients.
If α is a nonpositive integer n, then all terms with k > −n are zero, and the infinite series becomes a finite sum. However, for other values of α, including positive integers and rational numbers, the series is infinite.
Applications
Multisets have various applications.[7] They are becoming fundamental in combinatorics.[16][17][18][19] Multisets have become an important tool in the theory of relational databases, which often uses the synonym bag.[20][21][22] For instance, multisets are often used to implement relations in database systems. In particular, a table (without a primary key) works as a multiset, because it can have multiple identical records. Similarly, SQL operates on multisets and return identical records. For instance, consider "SELECT name from Student". In the case that there are multiple records with name "Sara" in the student table, all of them are shown. That means the result of an SQL query is a multiset; if the result were instead a set, the repetitive records in the result set would have been eliminated. Another application of multisets is in modeling multigraphs. In multigraphs there can be multiple edges between any two given vertices. As such, the entity that shows edges is a multiset, and not a set.
There are also other applications. For instance, Richard Rado used multisets as a device to investigate the properties of families of sets. He wrote, "The notion of a set takes no account of multiple occurrence of any one of its members, and yet it is just this kind of information which is frequently of importance. We need only think of the set of roots of a polynomial f (x) or the spectrum of a linear operator."[5]: 328–329
Generalizations
Different generalizations of multisets have been introduced, studied and applied to solving problems.
• Real-valued multisets (in which multiplicity of an element can be any real number)[23][24]
• Fuzzy multisets[25]
• Rough multisets[26]
• Hybrid sets[27]
• Multisets whose multiplicity is any real-valued step function[28]
• Soft multisets[29]
• Soft fuzzy multisets[30]
• Named sets (unification of all generalizations of sets)[31][32][33][34]
See also
• Frequency (statistics) as multiplicity analog
• Quasi-sets
• Set theory
• Learning materials related to Partitions of multisets at Wikiversity
Notes
1. The formula (
n+k −1
k
)
does not work for n = 0 (where necessarily also k = 0) if viewed as an ordinary binomial coefficient since it evaluates to (
−1
0
)
, however the formula n(n+1)(n+2)...(n+k −1)/k! does work in this case because the numerator is an empty product giving 1/0! = 1. However (
n+k −1
k
)
does make sense for n = k = 0 if interpreted as a generalized binomial coefficient; indeed (
n+k −1
k
)
seen as a generalized binomial coefficient equals the extreme right-hand side of the above equation.
References
1. Cantor, Georg; Jourdain, Philip E.B. (Translator) (1895). "beiträge zur begründung der transfiniten Mengenlehre" [contributions to the founding of the theory of transfinite numbers]. Mathematische Annalen (in German). New York Dover Publications (1954 English translation). xlvi, xlix: 481–512, 207–246. Archived from the original on 2011-06-10. By a set (Menge) we are to understand any collection into a whole (Zusammenfassung zu einem Gansen) M of definite and separate objects m (p.85)
2. Hein, James L. (2003). Discrete mathematics. Jones & Bartlett Publishers. pp. 29–30. ISBN 0-7637-2210-3.
3. Knuth, Donald E. (1998). Seminumerical Algorithms. The Art of Computer Programming. Vol. 2 (3rd ed.). Addison Wesley. ISBN 0-201-89684-2.
4. Blizard, Wayne D (1989). "Multiset theory". Notre Dame Journal of Formal Logic. 30 (1): 36–66. doi:10.1305/ndjfl/1093634995.
5. Blizard, Wayne D. (1991). "The Development of Multiset Theory". Modern Logic. 1 (4): 319–352.
6. Rulifson, J. F.; Derkson, J. A.; Waldinger, R. J. (November 1972). QA4: A Procedural Calculus for Intuitive Reasoning (Technical report). SRI International. 73.
7. Singh, D.; Ibrahim, A. M.; Yohanna, T.; Singh, J. N. (2007). "An overview of the applications of multisets". Novi Sad Journal of Mathematics. 37 (2): 73–92.
8. Angelelli, I. (1965). "Leibniz's misunderstanding of Nizolius' notion of 'multitudo'". Notre Dame Journal of Formal Logic (6): 319–322.
9. Kircher, Athanasius (1650). Musurgia Universalis. Rome: Corbelletti.
10. Prestet, Jean (1675). Elemens des Mathematiques. Paris: André Pralard.
11. Wallis, John (1685). A treatise of algebra. London: John Playford.
12. Dedekind, Richard (1888). Was sind und was sollen die Zahlen?. Braunschweig: Vieweg. p. 114.
13. Syropoulos, Apostolos (2000). "Mathematics of multisets". In Calude, Cristian; Paun, Gheorghe; Rozenberg, Grzegorz; Salomaa, Arto (eds.). Multiset Processing, Mathematical, Computer Science, and Molecular Computing Points of View [Workshop on Multiset Processing, WMP 2000, Curtea de Arges, Romania, August 21–25, 2000]. Lecture Notes in Computer Science. Vol. 2235. Springer. pp. 347–358. doi:10.1007/3-540-45523-X_17.
14. Whitney, H. (1933). "Characteristic Functions and the Algebra of Logic". Annals of Mathematics. 34 (3): 405–414. doi:10.2307/1968168. JSTOR 1968168.
15. Monro, G. P. (1987). "The Concept of Multiset". Zeitschrift für Mathematische Logik und Grundlagen der Mathematik. 33 (2): 171–178. doi:10.1002/malq.19870330212.
16. Aigner, M. (1979). Combinatorial Theory. New York/Berlin: Springer Verlag.
17. Anderson, I. (1987). Combinatorics of Finite Sets. Oxford: Clarendon Press. ISBN 978-0-19-853367-2.
18. Stanley, Richard P. (1997). Enumerative Combinatorics. Vol. 1. Cambridge University Press. ISBN 0-521-55309-1.
19. Stanley, Richard P. (1999). Enumerative Combinatorics. Vol. 2. Cambridge University Press. ISBN 0-521-56069-1.
20. Grumbach, S.; Milo, T (1996). "Towards tractable algebras for bags". Journal of Computer and System Sciences. 52 (3): 570–588. doi:10.1006/jcss.1996.0042.
21. Libkin, L.; Wong, L. (1994). "Some properties of query languages for bags". Proceedings of the Workshop on Database Programming Languages. Springer Verlag. pp. 97–114.
22. Libkin, L.; Wong, L. (1995). "On representation and querying incomplete information in databases with bags". Information Processing Letters. 56 (4): 209–214. doi:10.1016/0020-0190(95)00154-5.
23. Blizard, Wayne D. (1989). "Real-valued Multisets and Fuzzy Sets". Fuzzy Sets and Systems. 33: 77–97. doi:10.1016/0165-0114(89)90218-2.
24. Blizard, Wayne D. (1990). "Negative Membership". Notre Dame Journal of Formal Logic. 31 (1): 346–368. doi:10.1305/ndjfl/1093635499. S2CID 42766971.
25. Yager, R. R. (1986). "On the Theory of Bags". International Journal of General Systems. 13: 23–37. doi:10.1080/03081078608934952.
26. Grzymala-Busse, J. (1987). "Learning from examples based on rough multisets". Proceedings of the 2nd International Symposium on Methodologies for Intelligent Systems. Charlotte, North Carolina. pp. 325–332.{{cite book}}: CS1 maint: location missing publisher (link)
27. Loeb, D. (1992). "Sets with a negative numbers of elements". Advances in Mathematics. 91: 64–74. doi:10.1016/0001-8708(92)90011-9.
28. Miyamoto, S. (2001). "Fuzzy Multisets and their Generalizations". Multiset Processing. Lecture Notes in Computer Science. 2235: 225–235. doi:10.1007/3-540-45523-X_11. ISBN 978-3-540-43063-6.
29. Alkhazaleh, S.; Salleh, A. R.; Hassan, N. (2011). "Soft Multisets Theory". Applied Mathematical Sciences. 5 (72): 3561–3573.
30. Alkhazaleh, S.; Salleh, A. R. (2012). "Fuzzy Soft Multiset Theory". Abstract and Applied Analysis. 2012: 1–20. doi:10.1155/2012/350603.
31. Burgin, Mark (1990). "Theory of Named Sets as a Foundational Basis for Mathematics". Structures in Mathematical Theories. San Sebastian. pp. 417–420.
32. Burgin, Mark (1992). "On the concept of a multiset in cybernetics". Cybernetics and System Analysis. 3: 165–167.
33. Burgin, Mark (2004). "Unified Foundations of Mathematics". arXiv:math/0403186.
34. Burgin, Mark (2011). Theory of Named Sets. Mathematics Research Developments. Nova Science Pub Inc. ISBN 978-1-61122-788-8.
| Wikipedia |
\begin{document}
\title[Smooth covers on symplectic manifolds] {Smooth covers on symplectic manifolds}
\author[Fran\c{c}ois Lalonde]{Fran\c{c}ois Lalonde} \author[Jordan Payette]{Jordan Payette}
\address{D\'epartement de math\'ematiques et de statistique, Universit\'e de Montr\'eal; D\'{e}partement de math\'{e}matiques et de statistique, Universit\'{e} de Montr\'{e}al.} \email{[email protected]; [email protected]}
\thanks{The first author is supported by a Canada Research Chair, a NSERC grant OGP 0092913 (Canada) and a FQRNT grant ER-1199 (Qu\'ebec); the second author is supported by a Graham Bell fellowship from NSERC (Canada) }
\date{}
\begin{abstract}
In this article, we first introduce the notion of a {\it continuous cover} of a manifold parametrised by any compact manifold $T$ endowed with a mass 1 volume-form. We prove that any such cover admits a partition of unity where the sum is replaced by integrals. When the cover is smooth, we then generalize Polterovich's notion of Poisson non-commutativity to such a context in order to get a more natural definition of non-commutativity and to be in a position where one can compare various invariants of symplectic manifolds. The main theorem of this article states that the discrete Poisson bracket invariant of Polterovich is equal to our smooth version of it, as it does not depend on the nature or dimension of the parameter space $T$. As a consequence, the Poisson-bracket invariant of a symplectic manifold can be computed either in the discrete category or in the smooth one, that is to say either by summing or integrating. The latter is in general more amenable to calculations, so that, in some sense, our result is in the spirit of the De Rham theorem equating simplicial cohomology and De Rham cohomology. We finally study the Poisson-bracket invariant associated to coverings by symplectic balls of capacity $c$, exhibiting some of its properties as the capacity $c$ varies. We end with some positive and negative speculations on the relation between uncertainty phase transitions and critical values of the Poisson bracket, which was the motivation behind this article.
\end{abstract}
\maketitle
\noindent Subject classification: 53D35, 57R17, 55R20, 57S05.
\section{Introduction}
In mathematics, the notion of partition of unity is fundamental since it is the concept that distinguishes $C^{\infty}$ geometry from analytic geometry. In the first case, where partitions of unity apply, most objects can be decomposed in local parts, while in the second case where partitions of unity do not apply, most objects are intrinsically global and indecomposable.
It is therefore of some importance to push that notion as far as we can in order to make it more natural and applicable. Our first observation is that the right context in which one should consider partitions of unity is in the continuous category (or possibly in the measurable category if one were able to make sense of that concept for families of open sets). So here continuous covers by open subsets of a given smooth manifold $M$ will be parametrised by any smooth compact manifold (possibly with boundary) $T$ endowed with a volume-form $dt$ of total mass $1$; for simplicity we shall refer to those pairs $(T, dt)$ as 'probability spaces'. We will first prove that for any compact manifold $M$, any such continuously parametrised cover of $M$ admits a smooth partition of unity made of smooth functions.
Concentrating on an arbitrary symplectic manifold $(M, {\omega})$, the covers that we will consider will be made of smooth families of symplectically embedded balls of a given capacity $c= \pi r^2$ indexed by a measure space $(T, d\mu = \mathrm{d} t)$. Here is our first theorem: the level of Poisson non-commutativity, as defined by Polterovich in the discrete case of partitions of unity, can be generalised to the case of our families of covers and associated partitions of unity; morever the number that we get in this general case, which depends {\it a priori} on the probability space, actually does not, being equal to the number associated to the corresponding discrete setting.
Our second theorem is that if one considers the function $f: [0, c_{max}] \to [0, \infty]$ that assigns to $c$ the Polterovich's level of non-commutativity of the covers made of symplectically embedded balls of capacity $c$, as generalised by us in the smooth setting, then this function enjoys the following two properties:
\noindent
1) $f$ is non-increasing, and
\noindent
2) $f$ is upper semi-continuous and left-continuous.
We end this paper with a question concerning the relation, for a given symplectic manifold $(M, \omega)$, between critical values of the Poisson-bracket invariant as the capacity $c$ of the ball varies, and the critical values (or ``phase transition'') depending on $c$ of the topology of the infinite dimensional space of symplectic embeddings of the standard ball of capacity $c$ into $(M, \omega)$.
\noindent {\it Acknowledgements}. Both authors are deeply grateful to Lev Buhovsky for suggesting and proving that the $T$-parameter spaces of smooth covers can be reduced to one-dimensional families. Although we present a different proof here, that now includes the discrete case, his idea has had a significant impact on the first version of this paper. The second author would like to thank Dominique Rathel-Fournier for inspiring discussions.
\section{Continuous and smooth covers}
Throughout this article, ``smooth" means ``of class $C^r$" for some arbitrary fixed $r \ge 1$ and $T$ is a compact smooth manifold of finite dimension endowed with a measure $\mu$ of total volume $1$ coming from a volume-form $\mathrm{d} t$. The following definition is far more restrictive than the one that we have in mind, but it will be enough for the purpose of this article.
\begin{Definition}
Let $M$ be a closed smooth manifold of dimension $n$. Let $U$ be a bounded open subset of Euclidean space $\mathbb{R}^{n}$ whose boundary is smooth, so that the closure of $U$ admits an open neighbourhood smoothly diffeomorphic to $U$.
A {\it continous cover} of $M$ of type $(T,U)$ is a continuous map
$$
G: T \times U \to M $$
such that
\begin{enumerate}
\item for each $t \in T$, the map $G_t$ is a smooth embedding of $U$ to $M$ that can be extended to a smooth embedding of some ({\it a priori} $t$-dependent) collar neighbourhood of $U$ (and therefore to the closed set $\bar{U}$), and
\item the images of $U$ as $t$ runs over the parameter space $T$, cover $M$.
\end{enumerate}
\end{Definition}
Note that, in general, the topology of $U$ could change within the $T$-family. However, to simplify the presentation, we restrict ourselves to a fixed $U$ -- this is what we had in mind in the sentence preceding this definition.
A {\it smooth cover} is defined in the same way, but now requiring that $G$ be a smooth map.
\begin{Definition} A {\it partition of unity} $F$ subordinated to a continuous cover $G$ is a smooth function
$$
\tilde{F}: T \times U \to [0, \infty)
$$
such that
\begin{enumerate}
\item each $\tilde{F}_t : U \to {\bf R}$ is a smooth function with (compact) support in $U$,
\item the closure of the union $\bigcup_{t \in T} \, \mathrm{supp} (\tilde{F}_t)$ is contained in $U$, and
\item for every $x \in M$,
$$
\int_T F_t(x) dt = 1,
$$
\end{enumerate}
\noindent
where the smooth function $F_t : M \to {\bf R}$ is the pushforward of $\tilde{F}_t$ to $M$ using $G_t$, extended by zero outside the image of $G_t$; in other words, it is $F_t(x) = \tilde{F}(t, G^{-1}_t(x))$.
\noindent The notation $F < G$ expresses that $F$ is a partition of unity subordinated to the cover $G$.
\end{Definition}
\begin{Remark} Condition (2) plays a role in the proofs of a few results of this paper by allowing us to deform $U$ a little while keeping a given $F$ fixed; we were not able to come up with arguments working without this condition. Note that we recover the usual notion of partition of unity by taking $T$ to be a finite set of points with the counting measure.
\end{Remark}
\begin{Theorem} Each continuous cover admits a partition of unity.
\end{Theorem}
\proof
Let $G$ be a continuous cover of $M$ of type $(T,U)$. The general idea of the proof is to replace $G$ by a finite open cover $G'$ of $M$, to consider a partition of unity subordinated to the latter and to use it to construct a partition of unity subordinated to $G$.
Cover $U$ by open balls such that their closure is always included inside $U$. Now, push forward this cover to $M$ using each $G_{t}$; the collection of all of these images as $t$ varies in $T$ forms an open cover of $M$ by sets diffeomorphic to the ball. Since $M$ is compact, there exists a finite subcover. Each open set in this subcover comes from some $G_t$, where $t$ is an element of a finite set $T' \subset T$.
For each $t \in T'$, consider the (finite) collection $C_t$ of open balls inside $U$ whose image under $G_t$ belongs to the aforementioned subcover, so that the latter can be expressed as $G' := \{G_t(V) \, : \, t \in T', V \in C_t\}$. Since the closure of each ball $V \in C_t$ is contained in the open set $U$, by continuity of $G$ there is an open set $B_V \subset T$ centred at $t$ such that $G_t(\bar{V}) \subset G_{\tau}(U)$ for all $\tau \in B_V$, so that each $G_{\tau}^{-1} \circ G_t : \bar{V} \to U$ is defined and is a diffeomorphism onto its image. The intersection $B_t = \cap_{V \in C_t} B_V$ contains $t$ and is open since $C_t$ is finite. For each $t \in T'$, consider a smooth nonnegative bump function $\rho_t$ supported in $B_t$ whose integral over $T$ equals $1$.
There exists a smooth partition of unity $\Phi = \{ \phi_V : M \to [0,1] \, | \, V \in \cup_{t \in T'} C_t \}$ on $M$ subordinated to the finite open cover $G'$. For $t \in T'$ and $V \in C_t$, the real-valued function $\tilde{F}_V(\tau,u) := \rho_t(\tau) \phi_V(G(\tau,u))$ defined on $T \times U$ is supported in $B_t \times U$, is smooth in both $u$ and $\tau$ and satisfies $\int_T \tilde{F}_V(\tau, G_{\tau}^{-1}(x))d\tau = \phi_V(x)$. It easily follows that the function \[ \tilde{F} : T \times U \to \mathbb{R} : (\tau, u) \mapsto \sum_{t \in T'} \sum_{V \in C_t} \tilde{F}_V(\tau, u) \] is a partition of unity subordinated to $G$.
$\square$
\section{The $pb$ invariant and Poisson non-commutativity}
Leonid Polterovich \cite{P1,P2,PR} introduced recently the notion of the level of Poisson non-com\-mutativity of a given classical (i.e finite) covering of a symplectic manifold. Here is the definition:
\begin{Definition} Let $(M, {\omega})$ be a closed symplectic manifold and $\mathcal U$ a finite cover of $M$ by open subsets $U_1, \ldots, U_N$. For each partition of unity $F = (f_1, \ldots, f_N)$ subordinated to $\mathcal U$, take the supremum of $\| \{\Sigma_i a_i f_i, \Sigma_j b_j f_j\} \|$ when the $N$-tuples of coefficients $(a_i)$ and $(b_i)$ run through the $N$-cube $[-1,1]^N$, where the bracket is the Poisson bracket and the norm is the $C^0$-supremum norm. Then take the infimum over all partitions of unity subordinated to $\mathcal U$. This is by definition the $pb$ \textit{invariant} of $\mathcal U$. To summarize:
\[ \mathrm{pb}(\mathcal U) := \underset{F < \mathcal U}{\mathrm{inf}} \; \underset{(a_i), (b_i) \in [-1,1]^N \, }{\mathrm{sup}} \, \left\| \left\{ \sum_i a_i f_i \, , \, \sum_j b_j f_j \right\} \right\| \; . \]
\end{Definition}
Roughly speaking, this number is a measure of the least amount of ``symplectic interaction" that sets in a cover $\mathcal U$ can have. It is very plausible that such a number depends on the combinatorics of the cover, but also on the symplectic properties of the (intersections of the) open sets in the cover. To illustrate this point, observe that if $\mathcal U$ is an open cover made of only two open sets, then $\mathrm{pb}(\mathcal U) = 0$. A somewhat opposite result holds for covers constituted of displaceable open sets; let's recall that a subset $U$ of $M$ is {\it displaceable} if there is a Hamiltonian diffeomorphism $\phi$ such that $\phi(U) \cap U = \emptyset$. The main result of Polterovich in \cite{P1,PR} is that for such a cover, the number $\mathrm{pb}(\mathcal U)$ (multiplied by some finite number which measures the ``symplectic size" of the sets in the cover) is bounded from below by $(2N^2)^{-1}$. In particular, $\mathrm{pb}(\mathcal U) > 0$ in such a case. This result heavily relies on techniques in quantum and Floer homologies and in the theory of quasi-morphisms and quasi-states. Unfortunately, this lower bound depends on the cardinality $N$ of the open cover; as such, it does not show if one could use the $\mathrm{pb}$-invariant in order to assign to a given symplectic manifold a (strictly positive) number that might be interpreted as its level of Poisson non-commutativity. Nevertheless, Polterovich conjectured in \cite{P2} and \cite{PR} that for covers made of displaceable open sets, there should be a strictly positive lower bound for $pb$ independent of the cardinality of the cover, an extremely hard conjecture.
One way of solving this problem might come from the extension of the $\mathrm{pb}$-invariant from finite covers to continuous or smooth covers. Indeed, such covers are morally limits of finite covers as the cardinality $N$ goes to infinity, so we can expect some relation between the minimal value of $\mathrm{pb}$ on such covers and the level of Poisson non-commutativity of the symplectic manifold. This extension has the advantage that one may then compare the $\mathrm{pb}$ invariant for continuous/smooth covers to other quantities that also depend on continuous/smooth covers, such as the critical values at which families of symplectic balls undergo a ``phase transition''. We first need the following definition:
\begin{Definition} Let $(M, {\omega})$ be a closed symplectic manifold and $G$ a continuous cover of $M$ of type $(T,U)$ by open subsets $G_t(U)$. For each partition of unity $F$ subordinated to $G$, take the supremum of $\| \{\int_N a(t) F(t) dt, \int_N b(t) F(t) dt\} \|$ over all coefficients (or \textit{weights}) $a$ and $b$ that are measurable functions defined on $T$ with $dt$-almost everywhere values in $[-1,1]$. Then take the infimum over all partitions of unity subordinated to $G$. This is by definition the $pb$ \textit{invariant} of $G$. To summarize:
\[ \mathrm{pb}(G) := \underset{F < G}{\mathrm{inf}} \; \underset{a, b : \, T \to [-1,1] \, \mbox{ \tiny{measurable}}}{\mathrm{sup}} \, \left\| \left\{ \int_T a(t) F_{t} dt \, , \, \int_T b(t) F_{t} dt \right\} \right\| \; . \]
\end{Definition}
Note that we recover Polterovich's definition by replacing $T$ by a finite set of points. The following result shows that this pb-invariant is finite.
\begin{lemma}
Given a continuous cover $G$ of type $(T,U)$, there exists a partition of unity $F$ subordinated to $G$ whose pb-invariant $\mathrm{pb} \, F$ is finite\footnote{The pb-invariant $\mathrm{pb} \, F$ of a partition of unity $F$ is defined as above without the infimum over $F < G$. That is, $\mathrm{pb} \, G = \mathrm{inf}_{F < G} \, \mathrm{pb} \, F$.}.
\end{lemma}
\proof
Consider the partition of unity $\tilde{F} : T \times U \to \mathbb{R}$ constructed in Theorem 4 above. Given a measurable function $a : T \to [-1,1]$, for any $x \in M$ we compute
\[ \int_T a(\tau) F_{\tau}(x) d\tau = \sum_{t \in T'} \sum_{V \in C_t} \bar a_t \phi_V(x) \, , \]
where $\bar{a}_t := \int_T a(\tau) \rho_t(\tau)d\tau$ is a number whose value is in $[-1,1]$. It follows that
\[ \left| \left\{ \int_T a(\tau) F_{\tau}(x) d\tau \, , \, \int_T b(\tau) F_{\tau}(x) d\tau \right\}(x) \right| \le \sum_{V, W \in \cup_{t \in T'} C_t} |\{ \phi_V, \phi_W \}(x)| < \infty \]
for any measurable functions $a, b : T \to [-1,1]$, which proves the claim. When the partition is smooth with respect to $t$ also, then the value is always finite. Indeed, since $F \in C^r(T \times M)$, it follows from Lebesgue's dominated convergence theorem that the function $a \cdot F := \int_T a(t) F_t dt \in C^r(M)$ is defined and also that $\{ a \cdot F, b \cdot F \} = \int_{T \times T} a(t) b(u) \{F_t, F_u \} dtdu < \infty$ when $a$ and $b$ are weights.
\qed\goodbreak\vskip6pt
\noindent We will be now working only with smooth covers. Therefore all partitions of unity $F$ satisfy $\mathrm{pb} \, F < \infty.$
We recall a few facts taken from Polterovich's and Rosen's recent book \cite{PR}, since we will need them. Further informations are available in this book and in the references therein. The setting is the following (\cite{PR}, chapter 4):
\begin{itemize} \item $(M^{2n}, \omega)$ a compact symplectic manifold;
\item $U \subset M$ an open set;
\item $H(U)$ the image of $\widetilde{Ham}(U)$ in $H := \widetilde{Ham}(M)$ under the map induced by the inclusion $U \subset M$;
\item $\phi$ : an element of $H(U)$;
\item $c$ : a (subadditive) spectral invariant on $H(U)$ (see the definition below);
\item $q(\phi) := c(\phi) + c(\phi^{-1})$, which is (almost) a norm on $H$;
\item $w(U) := \mathrm{sup}_{\phi \in H(U)} \, q(\phi)$ the spectral width of $U$ (which may be infinite). \end{itemize}
\begin{Definition}[\cite{PR}, 4.3.1] A function $c : H \to \mathbb{R}$ is called a \textit{subadditive spectral invariant} if it satisfies the following axioms: \begin{description} \item[ Conjugation invariance ] $c(\phi \psi \phi^{-1}) = c(\psi) \; \forall \phi, \psi \in H$;
\item[ Subadditivity ] $c(\phi \psi) \le c(\phi) + c(\psi)$;
\item[ Stability ] \[ \int_{0}^1 \mathrm{min}(f_{t} - g_{t}) dt \le c(\phi) - c(\psi) \le \int_{0}^1 \mathrm{max}(f_{t} - g_{t}) dt \, , \]
\noindent provided $\phi, \psi \in H$ are generated by normalized Hamiltonians $f$ and $g$, respectively;
\item[ Spectrality ] $c(\phi) \in \mathrm{spec}(\phi)$ for all nondegenerate elements $\phi \in H$. \end{description} \end{Definition}
\begin{Remark} The first three properties of a spectral invariant are in practice the most important ones. However, from the spectrality axiom, one can show for instance that $w(U) < \infty$ whenever $U$ is displaceable; as such, the spectrality axiom is relevant in order to tie the spectral invariant with the symplectic topology of $M$. Let's mention that a spectral invariant exists on any closed symplectic manifold, as can be shown in the context of Hamiltonian Floer theory. \end{Remark}
Given a Hamiltonian function $f \in C^{\infty}(M)$ generating the (autonomous) Hamiltonian diffeomorphism $\phi_{f} = \phi^1_{f}$ and a spectral invariant $c$, we can define the number \[ \zeta(f) := \sigma(\phi_{f}) + \langle f \rangle \in \mathbb{R} \]
\noindent where $\sigma(\phi_{f}) := \lim_{n \to \infty} \frac{1}{n}c(\phi^n_{f})$ (with $\sigma$ the {\it homogeneization} of $c$) and $\langle f \rangle := V^{-1} \, \int_{M} f \omega^n$ is the mean-value of $f$, where $V = \int_M \omega^n$ is the volume of the symplectic manifold $M$. The function $\zeta : C^{\infty}(M) \to \mathbb{R}$ is called the (\textit{partial symplectic}) {\it quasi-state} associated to $c$. It has some very important properties, among which:
\begin{description} \item[ Normalization ] $\zeta(a) = a$ for any constant $a$;
\item[ Stability ] $\mathrm{min}_{M} (f-g) \le \zeta(f) - \zeta(g) \le \mathrm{max}_{M} (f-g)$;
\item[ Monotonicity ] If $f \ge g$ on $M$, then $\zeta(f) \ge \zeta(g)$;
\item[ Homogeneity ] If $s \in [0, \infty)$, then $\zeta(sf) = s \zeta(f)$;
\item[ Vanishing ] If the support of $f$ is displaceable, then $\zeta(f) = 0$ (this is a consequence of the spectrality axiom for $c$);
\item[ Quasi-subadditivity ] If $\{f, g \} = 0$, then $\zeta(f+g) \le \zeta(f) + \zeta(g)$. \end{description}
For $f, g \in C^{\infty}(M)$, define $S(f, g) = \mathrm{min} \{ w(\mathrm{supp} \, f) \; , \; w(\mathrm{supp} \, g) \} \in [0, \infty]$. It follows from Remark 8 that this number is finite whenever either $f$ or $g$ has displaceable support.
\begin{Theorem}[\cite{EPZ}, 1.4 ; \cite{PR}, 4.6.1 ; the Poisson bracket inequality] For every pair of functions $f, g \in C^{\infty}(M)$ such that $S(f,g) < \infty$,
\[ \Pi(f,g) := \left| \zeta(f+g) - \zeta(f) - \zeta(g) \right| \le \sqrt{2 S(f,g) \, \| \{ f, g \} \| } \; . \] \end{Theorem}
\noindent We see that $\Pi(f,g)$ measures the default of additivity of $\zeta$. In fact, this theorem implies:
\begin{description} \item[ Partial quasi-linearity ] If $S(f,g) < \infty$ and if $\{f, g \} = 0$, then \[ \zeta(f+g) = \zeta(f) + \zeta(g) \; \mbox{ and } \zeta(s f) = s \zeta(f) \; \forall s \in \mathbb{R} \, . \] \end{description}
It is known that some symplectic manifolds admit a spectral invariant $c$ for which $S$ takes values in $[0, \infty)$, in which case $\zeta$ is a genuine symplectic quasi-state : it is a normalized, monotone and quasi-linear functional on the Poisson algebra $(C^{\infty}(M), \{-, - \})$.
\begin{Theorem}[\cite{P1}, 3.1 ; \cite{PR}, 9.2.2] Let $(M, \omega)$ be a symplectic manifold and consider a finite cover $U = \{ U_{1}, \dots, U_{N} \}$ of $M$ by displaceable open sets. Write $w(U) := \mathrm{max}_{i} \, w(U_{i}) < \infty$. Then \[ \mathrm{pb}(U) \, w(U) \; \ge \; \frac{1}{2N^2} \; . \] \end{Theorem}
\textit{Proof} :
Let $F$ be a partition of unity subordinated to $U$. Set \[ G_{1} = F_{1}, \, G_{2} = F_{1} + F_{2}, \, \dots , \, G_{N} = F_{1} + \dots + F_{N} \, . \]
\noindent Using Theorem 1 and the vanishing property of $\zeta$, one obtains the following estimate: \begin{align}
\notag \left| \zeta(G_{k+1}) - \zeta(G_{k}) \right| &= \left| \zeta(G_{k} + F_{k+1} \, ) - \zeta(G_{k}) - \zeta(F_{k+1}) \right| \\ \notag & \le \sqrt{2 \, \mathrm{min} ( w(\mathrm{supp} \, G_{k}) \, , \, w(\mathrm{supp} \, F_{k+1}) )} \, \sqrt{\{ G_{k} , F_{k+1} \}} \, . \end{align}
\noindent Using the definitions of $\mathrm{pb} (F)$ and of $w(U)$, one gets: \begin{align}
\notag \left| \zeta(G_{k+1}) - \zeta(G_{k}) \right| & \le \sqrt{2 \, w(U)} \, \sqrt{\mathrm{pb} (F) } \, . \end{align}
\noindent This inequality holds for all $k$. Using the normalization and vanishing properties of $\zeta$ and applying the triangle inequality to a telescopic sum, one gets: \begin{align}
\notag 1 & = \left| \zeta(1) - 0 \right| = \left| \zeta(G_{N}) - \zeta(G_{1}) \right| \le \sum_{k=1}^{N-1} \left| \zeta(G_{k+1}) - \zeta(G_{k}) \right| \\ \notag & \le \sum_{k=1}^{N-1} \sqrt{2 \, w(U) \, \mathrm{pb} (F) } \le N \sqrt{2 \, w(U) \, \mathrm{pb} (F) } \, . \end{align}
\noindent Since this is true for any $F < U$, the result easily follows.
\qed\goodbreak\vskip6pt
A similar results holds in the context of smooth covers. We say that a smooth cover $G : T \times U \to (M, \omega)$ is made of displaceable sets if each set $G_t(\bar{U}) = \overline{G_t(U)} \subset (M, \omega)$ is displaceable (recall that we assume that $G_t$ extends as a smooth embedding to the closure $T \times \bar{U}$). In other words, not only is each $G_t(U)$ displaceable, but so is a small neighborhood of it too.
\begin{Theorem} For any smooth cover $G$ of type $(T,U)$ made of displaceable sets, there exists a constant $c = c(G) > 0$ such that \[ \mathrm{pb}(G) \ge \; c(G) \; . \] \end{Theorem}
\textit{Proof} :
The proof morally consists in a coarse-graining of the smooth cover to a finite cover. Let $W_{1}, \dots, W_{N}$ be any exhaustion of the compact manifold $T$ by nested open sets with the following property: the sets $V_{1} = W_{1}$, $V_{2} = W_{2}-W_{1}$, ..., $V_{N} = W_{N} - W_{N-1}$ are such that for every $j$ the open set $U_j := \cup_{t \in V_{j}} \, \mathrm{Im}(G_{t})$ in $M$ is displaceable. Assume for the moment being that such sets $W_i$ exist. Notice that the sets $U_j$ cover $M$ and let $w(G) := \mathrm{sup}_j w(U_j) < \infty$. Now let $F$ be a partition of unity subordinated to $G$ and consider the functions $\int_{V_{1}} F_{t} dt$, ...\,, $\int_{V_{N}} F_{t} dt$ which form a partition of unity on $M$ subordinated to the $U_j$'s. As in the previous theorem, one estimates:
\begin{align}
\notag 1 &= \left| \zeta(1) - 0 \right| = \left| \zeta \left(\int_{W_{N}} F_{t}dt \right) - \zeta \left( \int_{W_{1}} F_{t}dt \right) \right| \\
\notag &\le \sum_{k=1}^{N-1} \left| \zeta \left(\int_{W_{k+1}} F_{t}dt \right) - \zeta \left( \int_{W_{k}} F_{t}dt \right) - \underset{0}{\underbrace{\zeta \left( \int_{V_{k+1}} F_{t}dt \right)}}\right| \\ \notag & \le \sum_{k=1}^{N-1} \sqrt{2 \, w(G) \, \mathrm{pb}(F)} \le N \sqrt{2 \, w(G) \, \mathrm{pb}(F)} \, . \end{align}
\noindent Since this is true for all $F < G$, and since $2N^2$ depends only on $G$ (through the choice of the $W_j$'s), the result follows with $c(G) := (2N^2 w(G))^{-1}$.
The sets $W_{j}$'s exist for the following reason. The closure of each $G_{t}(U)$ is a compact displaceable set, so that some open neighborhood $O_t$ of this set is displaceable. By the continuity of the cover $G$, for any $t$ there exists an open set $\{t \} \in Y_t \subset T$ such that $G(Y_t \times U) \subset O_t$. Since $T$ is compact, only a finite number of these $Y_t$ suffices to cover $T$, say $Y_1, \dots, Y_N$. Set $W_j = \cup_{k=1}^{j} Y_j$. Since $V_j \subset Y_j$, the sets $G(V_j \times U)$ are indeed displaceable. This concludes the proof.
\qed\goodbreak\vskip6pt
It is natural to compare the $\mathrm{pb}$ invariant of different smooth covers of type $(T,U)$, especially if they are related to each other by a smooth family of smooth covers of the same type. This might help in understanding what is the 'optimal' way to cover a symplectic manifold $(M, \omega)$ by copies of a set $U$. We are led to the following definition which lies at the heart of this article:
\begin{Definition} A \textit{constraint} on smooth covers of $M$ of type $(T,U)$ is a set $C$ of such covers; the set of all smooth covers of type $(T,U)$ corresponds to the unconstrained case. Considering the $C^r$-Whitney topology on the space of smooth covers $G : T \times U \to M$, a \textit{constrained class of smooth covers of $M$ of type $(T,U)$} is defined as a connected component of the given constraint. We define the $\mathrm{pb}$ invariant of a (constrained) class $A$ as the infimum of $\mathrm{pb}(G)$ when $G$ runs over all smooth covers in $A$. \end{Definition}
As an instance of a constraint, we shall consider later on the one given by asking for each embedding $G_t : (U^{2n}, \omega_0) \hookrightarrow (M^{2n}, \omega)$ to be symplectic. The obvious difficulty with this last notion of pb invariant is that it intertwines four extrema: the supremum in the definition of the $C^0$-norm, the supremum over coefficients, the infimum over partitions of unity and the infimum over the smooth cover in the class. As a consequence of this difficulty, it is not clear if this number is strictly positive for every $M$, a problem which is related to Polterovich's conjecture; however, this number is now known to be positive for closed surfaces, as Polterovich's conjecture was recently proved valid in this context by Buhovsky, Tanny and Logunov \cite{BLT} and by the second author for genera $g \ge 1$ \cite{Pa}.
\section{Equivalence of the smooth and discrete settings}
This section is mainly devoted to the proof of Theorem 13 below which can be summarized as follows: the pb invariant of any class of $T$-covers is equal to the pb invariant of an affiliated class of discrete covers.
Fix a pair $(T,U)$. Any constraint $C$ of type $(T,U)$ determines the subset of \textit{constrained embeddings}
\[ C^* := \{ G_t : U \hookrightarrow M \, | \, t \in T, \, G \in C \} \subset \mathrm{Emb}(U, M) \, . \]
\noindent Any section of the natural map $T \to \pi_0(T)$ -- which associates to $t \in T$ the connected component to which it belongs -- induces a well-defined, \textit{i.e.} section-independent, map $p_T : \pi_0(C) \to [\pi_0(T), C^*] \simeq \pi_0(C^*)^{\pi_0(T)}$. An element $A \in \pi_0(C)$ is just a constrained class of covers, and the element $A^* = p_T(A) \in \mathrm{Im}(p_T)$ corresponds to the $|\pi_0(T)|$ (not necessarily distinct) connected components of $C^*$ from which open sets the smooth covers in $A$ are built. Denote by $B$ the subset of $\pi_0(C^*)$ which is the image of $A^*$.
Thus $B$ comprises sufficiently many open sets to cover the whole of $M$.
Let $\langle 1, n \rangle = [1,n] \cap \mathbb{N}$. Considering the natural map $q: C^* \to \pi_0(C^*)$, for $B \subset \pi_0(C^*)$ let $B' = q^{-1}(B) \subset C^*$. Assuming that $B'$ comprises enough open sets to cover $M$, define
\[ \mathrm{pb}_{\mathrm{discrete}}(B) := \inf \, \{ \, \mathrm{pb}(G) \, | \, \exists n \in \mathbb{N}, \, G : \langle 1, n \rangle \to B' \mbox{ a cover of $M$ } \} \, . \]
\noindent To simplify the notations, we will, in the sequel, denote the set $B$ by the same symbol $A^*$.
\begin{Theorem} [Equivalence smooth-discrete]
Let $M$ be a symplectic manifold of dimension $2n$, $U$ an open subset of ${\bf R}^{2n}$ as mentioned above, and $T$ a compact manifold of strictly positive dimension endowed with a Lebesgue measure $\mu$ of total mass $1$. Consider a constraint $C$ on smooth covers of $M$ of type $(T,U)$, let $A \in \pi_0(C)$ be a constrained class of such covers and write $A^* = p_T(A) \subset \pi_0(C^*)$. Then
\[ \mathrm{pb}(A) = \mathrm{pb}_{\mathrm{discrete}}(A^*) \; . \]
\end{Theorem}
\proof
We first prove $\mathrm{pb}(A) \ge \mathrm{pb}_{\mathrm{discrete}}(A^*)$. Let $G$ be a smooth cover of type $(T,U)$ in the constrained class $A$ and consider a smooth partition of unity $F < U$. By property (2) in the definition of a partition of unity and by continuity of $G$, we deduce that for each $t \in T$ there is an open set $t \in B_t \subset T$ such that $\mathrm{supp}(G_t^*(F_s)) \subset U$ for all $s \in B_t$. Since $T$ is compact, there is a finite set $T' = \{t_1, \dots, t_n\} \subset T$ such that the collection $B = \{B_{t_1}, \dots, B_{t_n}\}$ covers $T$. Consider a partition of unity $\rho = \{\rho_1, \dots, \rho_n\}$ on $T$ subordinated to $B$ and for each $t_i$ define
\[ F'_i : M \to [0, \infty) : x \mapsto F_i(x) = \int_T \rho_i(t)F(x,t) dt \; . \]
We observe that the collection $F' = \{F'_1, \dots, F'_n\}$ is a partition of unity on $M$ by smooth functions which is subordinated to the finite cover $G' := \left. G \right|_{T'}$ of $M$. We note that $\mathrm{Im}(G') \subset (A^*)'$, where we use a notation introduced just before the statement of the theorem. For $a' = \{a'_1, \dots, a'_n\} \subset [-1,1]$, the quantity $a := \sum_{i=1}^n a'_i \rho_i : T \to [-1,1]$ is a $T$-weight. For $a',b' \in [-1,1]^n$ we easily compute
\begin{align}
\notag \left\{ \int_T a(t) F_t dt \, , \, \int_T b(u) F_u du \right\} &= \left\{ \sum_{i=1}^n a'_i F'_i \, , \, \sum_{j=1}^n b'_j F'_j \right\} \; .
\end{align}
Taking the suprema over weights thus yields $\mathrm{pb}(F) \ge \mathrm{pb}(F')$, while taking the infima over partitions of unities yields $\mathrm{pb}(G) \ge \mathrm{pb}(G')$. Taking the infima over covers in classes $A$ and $A^*$ finally yields $\mathrm{pb}(A) \ge \mathrm{pb}_{\mathrm{discrete}}(A^*)$.
We now prove $\mathrm{pb}(A) \le \mathrm{pb}_{\mathrm{discrete}}(A^*)$. Let $G' : \langle 1, n \rangle \to (A^*)'$ be a finite cover of $M$ and let $F' = \{F'_1, \dots, F'_n\}$ be a partition of unity subordinated to $G'$. Since $A^* = p_T(A)$, there exists a smooth cover $G''$ of $M$ of type $(T,U)$ in the constrained class $A \in \pi_0(C)$. Interpreting $A^* = \{A^*_1, \dots, A^*_m\}$ as a collection of connected components of $C^*$, for each connected component $A^*_i$ we can associate a point $t''_i \in T$ such that $G''_{t''_i} \in A^*_i$. From this association we can get an injective map $\langle 1, n \rangle \to T$ which associates to the integer $j$ a point $t'_j$ in the same connected component as the point $t''_i$, with $A^*_i \ni G'_j$. Call the image of this map $T' \subset T$.
From these data we shall construct a smooth cover $G$ of type $(T,U)$ in the class $A$ which could act as a substitute for $G'$, in the sense that $\left. G \right|_{T'} = G'$. In fact, we shall define a smooth family $G_s$ of covers of type $(T,U)$ with $s \in [0,1]$ so that $G_0 = G''$ and $G_1 = G$, thereby illustrating that $G$ is indeed in the constrained class $A$. Fix a Riemannian metric on $T$. Observe that smoothly deforming $G''$ within $A$ if necessary, we can assume that $G''$ is constant in an $\epsilon$-neighbordhood of $T'$. If some connected component of $T$ contains none of the points $t'_j$, just set $G_s = G''$ on that component. For any other connected component of $T$, say the one containing $t''_i$, pick a Riemannian metric on it and consider disjoint embedded closed geodesic $\epsilon$-balls centred at the points $t'_j$. Outside the reunion of these balls, set again $G_s = G''$, whereas on the ball containing $t'_j$ define $G_s$ as follows. First choose a smooth path $g_j : [0, \epsilon] \to A^*_i$ such that $g_j(0) = G'_j$and $g_j(\epsilon) = G''(t'_j)$. Also pick a smooth function $\chi : [0, \epsilon] \to [0,1]$ such that $\chi(u) = 1$ if $u < \epsilon/3$ and $\chi(u) = 0$ is $u > 2\epsilon/3$. Denoting $r(p)$ the radial distance in the $j$-th ball of a point $p$ from $t'_j$, set on that ball $G_s(p) = g_j([1 - (1- \chi(s \epsilon)) \chi(r)]\epsilon)$. This completely defines the family $G_s$ in the way we desired.
We observe that $G$ is constant on an $(\epsilon/3)$-neighbourhood of each $t'_j$. For each $j$, pick a smooth positive function $\rho_j$ with support in the $(\epsilon/3)$-ball about $t'_j$ and which integrates to $1$. We define the smooth function $F : T \times M \to [0, \infty)$ as $F(t,m) = \sum_{j=1}^n \rho_j(t) F'_j(x)$. We easily verify that this is a smooth partition of unity subordinated to $G$.
For any $T$-weight $a : T \to [-1,1]$, define $a' = (a'_1, \dots, a'_n) \in [-1,1]^n$ via $a'_j = \int_T a(t) \rho_j(t) dt$. For $T$-weights $a$ and $b$ we then easily compute
\begin{align}
\notag \left\{ \sum_{i=1}^n a'_i F'_i \, , \, \sum_{j=1}^n b'_j F'_j \right\} &= \left\{ \int_T a(t) F_t dt \, , \, \int_T b(u) F_u du \right\} \; .
\end{align}
Taking the suprema over weights thus yields $\mathrm{pb}(F') \ge \mathrm{pb}(F)$, while taking the infima over partitions of unities yields $\mathrm{pb}(G') \ge \mathrm{pb}(G)$. Taking the infima over covers in classes $A^*$ and $A$ finally yields $\mathrm{pb}_{\mathrm{discrete}}(A^*) \ge \mathrm{pb}(A)$.
\qed\goodbreak\vskip6pt
\section{Independence on the probability space}
The equivalence of the smooth and of the discrete settings suggests that the pb invariants might be independent from the underlying probability space $T$ parametrising the smooth covers. The purpose of this section is make this idea precise.
\begin{proposition}
Let $M$ be a symplectic manifold of dimension $2n$, $U$ an open subset of ${\bf R}^{2n}$ as mentioned above, and $T_1$ and $T_2$ be compact manifold of strictly positive dimension each endowed with a smooth volume form of total mass $1$. Consider constraints $C_1$ and $C_2$ on smooth covers of $M$ of type $(T_1,U)$ and $(T_2, u)$, respectively. Let $A_i \in \pi_0(C_i)$, $i=1,2$, be constrained classes and assume that the corresponding sets of embeddings $(A_i^*)' \subset C^*_i \subset \mathrm{Emb}(U, M)$ coincide in the latter space. Then
\[ \mathrm{pb}(A_1) = \mathrm{pb}(A_2) \; . \]
\end{proposition}
\proof
It follows from Theorem 13 that $\mathrm{pb}(A_i) = \mathrm{pb}_{\mathrm{discrete}}(A_i^*)$, $i=1,2$. Looking at the definition, $\mathrm{pb}_{\mathrm{discrete}}(A_i^*)$ only depends on the set $(A_i^*)'$, which is itself assumed to be independent from $i$.
\qed\goodbreak\vskip6pt
Next we discuss special sorts of constraints which not only frequently appear in practice, but also for which the hypothesis in the previous proposition follows from a somewhat less stringent assumption.
\begin{Definition}
A constraint $C$ on covers of type $(T,U)$ is \textit{prime} if there exists a set $C' \subset \mathrm{Emb}(U, M)$ such that $G \in C$ if and only if $G_t \in C'$ for every $t \in T$. In other words, $C$ is prime if it is the largest constraint such that $C^* \subset C'$ (equivalently, $C^* = C'$).
\end{Definition}
\noindent We point out that $C'$ thus admits sufficiently many open sets to cover the whole of $M$. Conversely, given a set $C' \subset \mathrm{Emb}(U, M)$ which admits sufficiently many open sets to cover $M$ and a probability space $T$, it is not guaranteed that there exists a constraint $C$ of covers of type $(T,U)$ (let alone a prime one) such that $C^* = C'$; this happens if $|\pi_0(C')| > |\pi_0(T)|$ and if no reunion of $|\pi_0(T)|$ connected components of $C'$ has sufficiently many open sets to cover $M$. In comparison, as long as $|\pi_0(C')|$ is finite, we can always find a discrete cover of $M$ made of open sets in $C'$. Note that this is however the only obstacle: given a set $C' \subset \mathrm{Emb}(U, M)$ such that there exists a smooth cover $G$ of $M$ of type $(T,U)$ with $G_* : \pi_0(T) \to \pi_0(C')$ well-defined and surjective, then $C' = C^*$ for some (prime) constraint $C$ on covers of type $(T,U)$.
\begin{Definition}
A prime constraint $C$ on covers of type $(T,U)$ with $C^* = C' \subset \mathrm{Emb}(U,M)$ is \textit{filled} if there is $G \in C$ such that the map $G_* : \pi_0(T) \to \pi_0(C')$ is surjective. By extension, we say that $C'$ is \textit{filled by $T$} if the associated prime constraint $C$ of type $(T,U)$ is filled.
\end{Definition}
\begin{corollary}
Let $M$ be a symplectic manifold of dimension $2n$, $U$ an open subset of ${\bf R}^{2n}$ as mentioned above, and $T_1$ and $T_2$ be compact manifold of strictly positive dimension each endowed with a smooth volume form of total mass $1$. Consider constraints $C_1$ and $C_2$ on smooth covers of $M$ of type $(T_1,U)$ and $(T_2, u)$, respectively. Let $C' \subset \mathrm{Emb}(U,M)$ be filled by both $T_1$ and $T_2$ and consider the corresponding prime constraints $C_1$ and $C_2$. Let $A_i \in \pi_0(C_i)$, $i=1,2$, be constrained classes and assume that the corresponding sets of embeddings $(A_i^*)' \subset C^*_i \subset \mathrm{Emb}(U, M)$ coincide in the latter space. Then
\[ \mathrm{pb}(A_1) = \mathrm{pb}(A_2) \; . \]
\end{corollary}
\begin{Remark} \label{independence}
For one application of this corollary, note that $C'$ is filled by any probability space $T$ of strictly positive dimension whenever $C'$ is connected and contains sufficiently many open sets to cover $M$. In that case $(A^*)'=C'$ for any $A \in \pi_0(C)$ (where $C$ is the prime and filled constraint associated with $C'$), since in fact $|\pi_0(C)|=1$. As a consequence, when $C'$ is not necessarily connected but each of its components contains sufficiently many embeddings to cover $M$, then the restriction of $\mathrm{pb}$ to prime constrained classes of covers parametrised by \textit{connected} $T$ comes from a function on $\pi_0(C')$.
\end{Remark}
\section{The behaviour of $\mathrm{pb}$ on symplectic balls}
For the rest of this article, we only\footnote{The results of this section can however be easily adapted for star-shaped domain $U \subset \mathbb{R}^{2n}$.} consider $U = U(c) = B^{2n}(c)$, that is the standard symplectic ball capacity $c = \pi r^2$ (where $r$ is the radius). We also only consider (smooth) \textit{symplectic} covers, that is covers $G$ of type $(T,U)$ satisfying the symplectic prime constraint $C$ given as follows: $G \in C$ if $G_t \in C' = \mathrm{Emb}_{\omega}(U, M)$ for every $t \in T$. We shall write $U(c)$, $C(c)$ and $C'(c)$ when we want to stress the dependence on $c$.
Of special interest are the cases when $T = S^n$ for some $n \ge 1$. A constrained class $A$ of $C$ determines a connected component\footnote{It is still a conjecture, that we shall dub the \textit{symplectic camel conjecture}, whether $C'(c)$ is connected (whenever nonempty) when $(M, \omega)$ is compact and for any $c$.} $A' = p_T(A) \subset C'$, and determines in fact an element of the $n$-th homotopy group $\pi_n(A')$. Conversely, since $M$ is compact and using the fact that the group $\mathrm{Symp}(M, \omega)$ is $k$-transitive for all $k \in \mathbb{N}$, any element in $\pi_n(C')$ can be represented by some class $A \in \pi_0(C)$. The pb-invariants of symplectically constrained classes hence allow to probe the homotopic properties of $C'(c)$, properties which might change with $c$. Consequently, it appears important to better understand how the pb-invariants depend on the capacity $c$. This behavior of $\mathrm{pb}$ on $c$ is the main question raised in this paper.
However, invoking \Cref{independence} and again the $k$-transitivity of $\mathrm{Symp}(M, \omega)$, we deduce that for any connected probability space $T$ there is a bijective correspondence between $\pi_0(C)$ and $\pi_0(C')$. We can thus interpret the $\mathrm{pb}$ functional on smooth covers of type $(T,U(c))$ parametrised by connected spaces $T$ simply as a map $\mathrm{pb} : \pi_0(C'(c)) \to [0, \infty)$, the latter being clearly independent from $T$. It therefore appears that the $\mathrm{pb}$-invariants can only probe the homotopy type of $C'(c)$ in a crude way.
Let $c_{max}$ denote the largest capacity a symplectic (open) ball embedded in $M$ can have; that can be much smaller than the one implied by the volume constraint $\mathrm{Vol}(U(c)) \le \mathrm{Vol}(M, \omega)$, according to the Non-Squeezing Theorem. For $0 < c < c' < c_{max}$ the obvious inclusion $U(c') \subset U(c)$ induces the restriction map $C'(c) \to C'(c')$ and hence also $r_{c,c'} : \pi_0(C'(c)) \to \pi_0(C'(c'))$.
\begin{Definition} The \textit{tree of path-connected classes of symplectic embeddings of $U$ in $M$} is the set
\[ \Psi(U,M) := \bigsqcup_{c \in (0, c_{max})} \{c\} \times \pi_0(C'(c)) \; . \]
A \textit{(short) branch of $\Psi(U,M)$} is a continuous path $\beta : (0, c_{\beta}) \to \Psi(U,M) : c \mapsto (c, A^*_{\beta}(c))$ such that $r_{c,c'}(A^*_{\beta}(c)) = A^*_{\beta}(c')$.
\end{Definition}
We can therefore define a function $\mathrm{pb} : \Psi(U,M) \to [0, \infty)$ in the obvious way. Given a branch $\beta$ with domain $(0, c_{\beta})$, we can define a map $\mathrm{pb}_{\beta} = \mathrm{pb} \circ \beta : (0, c_{\beta}) \to [0, \infty)$.
\begin{Theorem} Given any branch $\beta$, the function $\mathrm{pb}_{\beta}$ is non-increasing, upper semi-continuous and left-continuous.
\end{Theorem}
\proof
(a) Let us first show that the function is non-increasing. Fix $0 < c' < c < c_{\beta}$ and let $\epsilon > 0$. From the work done above and with the interpretation of $A^*_{\beta}(c)$ as a connected component of $C'(c)$, there exists a discrete cover $G' : \langle 1, n \rangle \to A^*_{\beta}(c')$ of $M$ such that $\mathrm{pb}(G') < \mathrm{pb}_{\beta}(c') + \epsilon$. We claim that this cover refines a cover $G : \langle 1, n \rangle \to A^*_{\beta}(c)$ of $M$; assuming this for the moment, we would then have
\[ \mathrm{pb}_{\beta}(c) \le \mathrm{pb}(G) \le \mathrm{pb}(G') < \mathrm{pb}_{\beta}(c') + \epsilon \; . \]
As this holds for any $\epsilon > 0$, we get $\mathrm{pb}_{\beta}(c) \le \mathrm{pb}_{\beta}(c')$ \textit{i.e.} $\mathrm{pb}_{\beta}$ is non-increasing.
To prove the existence of $G$, consider a symplectic embedding $B \in A^*_{\beta}(c)$. Since $\beta$ is a branch, the restriction $B'$ of $B$ to $U(c')$ is an embedding in $A^*_{\beta}(c')$; the latter space being a connected component of $C'(c')$ with respect to the Whitney $C^r$-topology, for each $j \in \langle 1, n \rangle$ there is smooth path of symplectic embeddings of $U(c')$ into $M$ joining $B'$ to $G'(j)$. By the symplectic isotopy extension theorem, each of these paths extends to a global symplectic isotopy on $M$, which thus sends $B$ to an embedding $G(j)$ of $U(c)$ into $M$. Clearly $G$ is a discrete cover of $M$ refined by $G'$.
(b) Now let us show that for every $c \in (0, c_{beta})$, the function $\mathrm{pb}_{\beta}$ is upper semi-continuous at $c$, \textit{i.e.} $\limsup_{c' \to c} \mathrm{pb}_{\beta}(c') \le \mathrm{pb}_{\beta}(c)$.
On the one hand, it follows from part (a) that $\mathrm{pb}_{\beta}(c)$ is greater or equal to all limits of $f$ from the right. On the other hand, for any $\epsilon > 0$, there are a discrete cover $G$ representing $A^*_{\beta}(c)$ and a partition of unity $F < G$ such that $\mathrm{pb}(F) < \mathrm{pb}_{\beta}(c) + \epsilon$. In fact, by our definition of a partition of unity, there is a strictly smaller capacity $c' < c$ such that the support of $F$ is compact inside the open ball $U(c') \subset U(c)$. Transporting the data to the restriction of the pair $(G,F)$ to $U(c'')$ for any $c'' \in [c',c]$, one gets
$$\mathrm{pb}_{\beta}(c'') \le \mathrm{pb}_{\beta}(c) + \epsilon.$$
\noindent Since the choice of $c'$ indirectly depends on $\epsilon > 0$ through $F$, and might get as close to $c$ when $\epsilon$ approaches to zero, we do not get $\mathrm{pb}_{\beta}(c'') \le \mathrm{pb}_{\beta}(c)$ but only that $\mathrm{pb}_{\beta}$ is upper semi-continuous from the left.
(c) We wish to prove that $\mathrm{pb}_{\beta}$ is in fact left-continuous, that is to say that $\mathrm{pb}_{\beta}(c)$ is equal to the limit of $\mathrm{pb}_{\beta}(c')$ as $c'$ tends to $c$ from the left. Consider a sequence of capacities $c_i < c$ converging to $c$ with highest value $\lim \mathrm{pb}_{\beta}(c_i)$ (the value $\infty$ is not excluded). This limit cannot be smaller than $\mathrm{pb}_{\beta}(c)$ because otherwise it would contradict the non-increasing property. However, by upper semi-continuity, it cannot be greater than $\mathrm{pb}_{\beta}(c)$. Therefore, it has to be equal to $\mathrm{pb}_{\beta}(c)$.
\qed\goodbreak\vskip6pt
With regard to the continuity of the function $\mathrm{pb}_{\beta}$ associated to a branch $\beta$ it is not possible to be much more specific than the above Theorem, at least not when $\mathrm{dim} \, M = 2$. Indeed, in that case $c_{max} = \mathrm{Area}(M, \omega)$ and Moser's argument allows to prove that the space $C'(c) = \mathrm{Emb}_{\omega}(B^{2}(c), M)$ is connected whenever non-empty, so that there is only one maximal branch $\beta$. Polterovich's conjecture has recently been established in dimension two \cite{BLT}: in fact there is a universal constant $\gamma > 0$ such that $\mathrm{pb}_{\beta}(c)c > \gamma$ whenever $c \le c_{max}/2$. Using the invariance of the quantity $\mathrm{pb}_{\beta}(c)c$ upon pullback of the data under any symplectic covering map, this inequality holds even for $c > c_{max}/2$ when $M$ has genus $g \ge 1$ (\textit{c.f.} \cite{Pa}). However for $M = S^2$, by enlarging two opposite hemispheres one gets $\mathrm{pb}_{\beta}(c) = 0$ when $c > c_{max}/2$. Consequently $\mathrm{pb}_{\beta}$ is discontinuous on $S^2$, yet might be continuous on higher genus surfaces.
\section{Phase transitions and the $\mathrm{pb}$ function}
We conclude this paper by a few speculations since they disclose the main motivation behind this article.
The first ``phase transition'' discovered in Symplectic Topology is the following one:
\begin{Theorem} (Anjos-Lalonde-Pinsonnault) In any ruled symplectic 4-manifold $(M, {\omega})$, there is a unique value $c_{crit}$ such that the infinite dimensional space $Emb(c,{\omega})$ of all symplectic embeddings of the standard closed ball of capacity $c$ in $M$ undergoes the following striking property: below $c_{crit}$, the space $Emb(c,{\omega})$ is homotopy equivalent to a finite dimensional manifold, while above that value, $Emb(c,{\omega})$ does not retract onto any finite dimensional manifold (or CW-complex) since it possesses non-trivial homology groups in dimension as high as one wishes. Below and above that critical value, the homotopy type stays the same.
\end{Theorem}
\noindent The reason for the term {\it phase transition} is still debatable, but there are several physical reasons, from Thermodynamics, to adopt that terminology.
\begin{Definition} Given a closed symplectic manifold $(M, \omega)$, let us call an {\it uncertainty phase transition} any critical value $c$ at which the space of symplectic embeddings of balls of capacity $c$ into $(M, \omega)$ undergoes a change of its homotopy type. \end{Definition}
This terminology reflects the fact that a symplectically embedded ball quantifies the uncertaintly in the position and momentum of a (collection of) particles(s).
The proof of the above theorem is quite indirect: one identifies all homology classes of symplectically embedded balls through the action of two groups on them: the full group of symplectic diffeomorphisms and the subset of these that preserve a given standard ball, the latter being viewed as the group of symplectic diffeomorphisms on the blow-up. Each of theses groups is computed by their action on a stratification of all compatible almost complex structures that realise holomophically some homology classes (essentially the homology classes that cut out the symplectic manifold in simple parts). Everything boils down to the behaviour of some J-curves in the given symplectic manifold for each $J$, generic or not (the non-generic ones playing the fundamental role since only the first stratum is generic). So, for instance, some homology class of symplectically embedded balls may disappear at some capacity $c_{crit}$ because the homology class of symplectic diffeomorphisms that preseve some standard ball of capacity $c$ in $M$ vanishes when $c$ crosses $c_{crit}$. It is conceivable that the class that vanished was supporting a covering that minimized the $pb$ at that level of capacity. We know that the dimension of the homology class, i.e. the dimension of the parametrizing space $T$, plays no role by our theorem on smooth-discrete equivalence. However, it is possible, that such a class, discretized or not, contained the optimal configuration of balls for a covering in order to mimimize $pb$. Therefore the main question that drove us to study the $pb$ invariant in the smooth setting is:
\noindent {\bf Question (Poisson-Uncertainty).} Is there a relation between the critical values of the Poisson bracket and the critical values (or phase transitions) of $Emb_{\omega}(B(c), M)$ as $c$ varies ?
This is a natural question since the latter probes the topological changes in configurations of balls, while the former looks for $pb$-optimal configurations. We do not have in mind any direct sketch proving that there is a relation. So we must simply for the moment look at the facts. We have little material to work on, since the $pb$ conjecture has been proved (very recently) only for real surfaces, while the study of the topology of balls is known only in dimension $2$ and $4$ for ruled symplectic $4$-manifolds. Thus we may just examine the case of surfaces. In this case, there is no critical value for the phase transition, but there are for the pb-invariant, showing that the answer to the above question is negative in dimension $2$.
Small displaceable balls should not see the symplectic form, actually the space of (unparamatrised) symplectic balls below the uncertainty critical value retracts to the topology of the manifold itself for ruled symplectic 4-manifolds. This refines the symplectic camel conjecture for small capacities and it leads us to state the following conjecture:
\begin{Conjecture} (The Topology conjecture). The limit of the function $c \mathrm{pb}(c)$, as $c$ tends to zero, is a finite number, and depends only on the differential topology of the symplectic manifold. \end{Conjecture}
Now, while the the Poisson-uncertainty question might have a positive answer in high dimensions, we show here that the Poisson-uncertainty question has a negative answer in dimension $2$ for the sphere.
To see this, let us consider the simple situation of $(M, \omega)$ being $S^2$ with its standard symplectic form, say of area $A$. As $M$ is a surface, it satisfies the symplectic camel conjecture, which is to say that the space $\mathrm{Emb}(c, \omega)$ is connected. The Poisson bracket function is then defined for any $c \in (0, A)$. There exists on any closed symplectic manifold a spectral invariant $c$ such that $c(Id) = 0$, see Theorem 4.7.1 in \cite{PR}. It follows from that and the other properties of $c$ that the spectral width $w(U)$ of any subset $U \subset M$ satisfies $w(U) \le 4 e_H(U)$ where $e_H(U)$ is the Hofer displacement energy of $U$. For open sets in $S^2$, $e_H(U) = \mathrm{Area}(U)$ if this area is smaller than $A/2$ and $e_H(U) = \infty$ otherwise. In this context, Polterovich's conjecture (now a theorem on surfaces \cite{BLT,Pa}) states that there is a constant $C > 0$ such that for any, continuous or discrete, cover $G$ of $S^2$ by displaceable open sets, the inequality \[ \mathrm{pb}(G)w(G) \ge C \; \mbox{ holds }. \]
\noindent This implies that $\mathrm{pb}(c) e_H(U(c)) \ge C$. Thus when $c < A/2$, we have $\mathrm{pb}(c) \ge 2C/A$. However, we observe that $\mathrm{pb}(c) = 0$ whenever $c > A/2$: two symplectic balls of capacity $c > A/2$ suffice to cover $S^2$ and the $\mathrm{pb}$-invariant of such a cover vanishes. Polterovich's conjecture hence goes against any claim that the Poisson bracket function $\mathrm{pb}(c)$ only has discontinuities when $\mathrm{Emb}(c, \omega)$ undergoes a transition in its homotopy type.
As a concluding remark, we point out that our borrowings in the thermodynamical and statistical mechanical terminology are explained by our insight that tools from these subjects might play a role in the understanding of the symplectic problems we considered in this paper. The space of symplectically embedded balls can be understood as an infinite dimensional (pre)symplectic manifolds which is some sort of limit of finite dimensional ones. In this paper, continuous covers have also been understood as limits of discretes ones. It is a recurrent theme in statistical mechanics that systems with a very large number of degrees of freedom tend to behave in universal and somewhat simpler ways.
\end{document} | arXiv |
Proving $ 1+\frac{1}{4}+\frac{1}{9}+\cdots+\frac{1}{n^2}\leq 2-\frac{1}{n}$ for all $n\geq 2$ by induction
Let $P(n)$ be the statement that $1+\dfrac{1}{4}+\dfrac{1}{9}+\cdots +\dfrac{1}{n^2} <2- \dfrac{1}{n}$. Prove by mathematical induction. Use $P(2)$ for base case.
Attempt at solution:
So I plugged in $P(2)$ for the base case, providing me with $\dfrac{1}{4} < \dfrac{3}{2}$ , which is true.
I assume $P(n)$ is true, so I need to prove $P(k) \implies P(k+1)$.
So $\dfrac{1}{(k+1)^2} < 2 - \dfrac{1}{k+1}$.
I don't know where to go from here, do I assume that by the Inductive hypothesis that it's true?
discrete-mathematics inequality summation induction
Donald DangDonald Dang
$\begingroup$ In $P(n)$, do you mean that the sum is $< n - 1/n$, rather than $<1/n$? Although as I look closer, it's probably $2 - 1/n$ on the right. $\endgroup$ – pjs36 Apr 4 '15 at 19:34
$\begingroup$ Yes it is. Sorry for the typo. Its 2 - 1/n $\endgroup$ – Donald Dang Apr 4 '15 at 19:37
$\begingroup$ The logic in the line "I assume $P(n)$ is true, so I need to prove $P(k)\Rightarrow P(k+1)$." is awkward. It should read: "I need to prove $P(k)\Rightarrow P(k+1)$, so I assume that $P(n)$ is true for $n=k$. $\endgroup$ – Michael Burr Apr 4 '15 at 19:57
$\begingroup$ The line "So $\frac{1}{(k+1)^2}<2-\frac{1}{k+1}$" doesn't seem to follow from the previous steps (even though it is true). Can you show how you conclude that? $\endgroup$ – Michael Burr Apr 4 '15 at 20:05
For $n\geq 2$, let $S(n)$ denote the statement $$ S(n) : 1+\frac{1}{4}+\frac{1}{9}+\cdots+\frac{1}{n^2}\leq 2-\frac{1}{n}. $$
Base step ($n=2$): $S(2)$ says that $1+\frac{1}{4}=\frac{5}{4}\leq\frac{3}{2}= 2-\frac{1}{2}$, and this is true.
Inductive step: Fix some $k\geq 2$ and suppose that $S(k)$ is true. It remains to show that $$ S(k+1) : 1+\frac{1}{4}+\frac{1}{9}+\cdots+\frac{1}{k^2}+\frac{1}{(k+1)^2}\leq 2-\frac{1}{k+1} $$ holds. Starting with the left-hand side of $S(k+1)$, \begin{align} 1+\frac{1}{4}+\cdots+\frac{1}{k^2}+\frac{1}{(k+1)^2} &\leq 2-\frac{1}{k}+\frac{1}{(k+1)^2}\quad\text{(by $S(k)$)}\\[1em] &= 2-\frac{1}{k+1}\left(\frac{k+1}{k}-\frac{1}{k+1}\right)\\[1em] &= 2-\frac{1}{k+1}\left(\frac{k^2+k+1}{k(k+1)}\right)\tag{simplify}\\[1em] &< 2-\frac{1}{k+1}.\tag{$\dagger$} \end{align} we see that the right-hand side of $S(k+1)$ follows. Thus, $S(k+1)$ is true, thereby completing the inductive step.
By mathematical induction, for any $n\geq 2$, the statement $S(n)$ is true.
Addendum: How did I get from the "simplify" step to the $(\dagger)$ step? Well, the numerator is $k^2+k+1$ and the denominator is $k^2+k$. We note that, $k^2+k+1>k^2+k$ (this boils down to accepting that $1>0$). Since $\frac{1}{k+1}$ is being multiplied by something greater than $1$, this means that what is being subtracted from $2$ in the "simplify" step is larger than what is being subtracting from $2$ in the $(\dagger)$ step.
Note: It really was unnecessary to start your base case at $n=2$. Starting at $n=1$ would have been perfectly fine. Also, note that this exercise shows that the sum of the reciprocals of the squares converges to something at most $2$; in fact, the series converges to $\frac{\pi^2}{6}$.
Daniel W. FarlowDaniel W. Farlow
18k1111 gold badges4848 silver badges9090 bronze badges
Although OP asks for proof by induction, other answers cover it, so I will add solution through integral estimation. What we will use is integral test for convergence of series, more precisely, the last line in the proof section of Wiki.
We have estimate $$\sum_{k=1}^n\frac 1{k^2} \leq 1 + \int_1^n \frac{dx}{x^2} = 1 + \left(-\left.\frac 1x\ \right|_1^n \right) = 2 - \frac 1 n$$ (see this for visualization)
EnnarEnnar
$\begingroup$ I will add that this idea is useful also for estimations for other sums. Some nice pictures related to the harmonic series $\sum_{k=1}^n \frac1k$ can be seen here. $\endgroup$ – Martin Sleziak Oct 25 '15 at 9:31
$\begingroup$ @MartinSleziak, thank you for the link. It serves as much better visual aid than my attempt with Alpha. $\endgroup$ – Ennar Oct 25 '15 at 9:36
Hint: You assume that the statement is true for $n=k$. In other words, $$ 1+\frac{1}{4}+\frac{1}{9}+\cdots+\frac{1}{k^2}<2-\frac{1}{k}. $$ Now, add $\frac{1}{(k+1)^2}$ to both sides to get $$ 1+\frac{1}{4}+\frac{1}{9}+\cdots+\frac{1}{k^2}+\frac{1}{(k+1)^2}<2-\frac{1}{k}+\frac{1}{(k+1)^2}. $$ What you would really like is that $$ 2-\frac{1}{k}+\frac{1}{(k+1)^2}<2-\frac{1}{k+1} $$ because then, by transitivity, your result would hold. So, can you prove that?
Michael BurrMichael Burr
Your base case is wrong. You should realize it's true since $\frac{5}{4}<\frac{3}{2}$ obtained from $$1+\frac{1}{2^2} = \frac{5}{4} < \frac{3}{2} = 2-\frac{1}{2}$$ For the induction step, suppose $P(n)$ is true for all $n \in \{1,2,\ldots,k\}$. Then $$\begin{align}1+\frac{1}{2^2}+\ldots+\frac{1}{k^2}+\frac{1}{(k+1)^2} = \left(1+\frac{1}{2^2}+\ldots+\frac{1}{k^2}\right)+\frac{1}{(k+1)^2} \\ < \left(2-\frac{1}{k} \right)+\frac{1}{(k+1)^2}\end{align}$$ Now if you can show that $$\left(2-\frac{1}{k} \right)+\frac{1}{(k+1)^2}<2-\frac{1}{k+1}$$ you are done. It is possible to get to that inequality simply starting with the fact that $\frac{1}{k+1}<\frac{1}{k}$
graydadgraydad
Notice that for $k\ge2$ you have $$\frac1{k^2} \le \frac1{k(k-1)} = \frac1{k-1} - \frac{1}{k}.$$ Using this we can get $$\sum_{k=1}^{n} \frac1{k^2} = 1+\frac1{2^2}+\frac1{3^2}+\dots+\frac1{n^2} \le 1+\frac1{2\cdot 1}+\frac1{3\cdot 2}+\dots+\frac1{n(n-1)} \le\\\le 1+\left(1-\frac12\right)+\left(\frac12-\frac13\right)+\dots+\left(\frac1{n-1}-\frac1n\right).$$ Notice that many terms cancel out and in the end we get $$\sum_{k=1}^n \frac1{k^2} \le 2-\frac1n.$$
This is called telescoping series. (In fact, it is probably the best known telescoping series - it is also mentioned in the Wikipedia article I linked to.)
It is not difficult to rewrite this to induction proof. Induction step would be $$\sum_{k=1}^n \frac1{k^2}+\frac1{(n+1)^2} \le \left(2-\frac1n\right) + \frac1{(n+1)^2} \le \left(2-\frac1n\right) + \left(\frac1n -\frac1{n+1}\right) =\\= 2-\frac1{n+1}.$$ (The first inequality is from the induction hypothesis. The second one is from the equality given at the beginning of this post.)
As an exercise you might try to prove that $\sum\limits_{k=2}^n \dfrac1{k(k-1)} =1-\dfrac1n$ in a similar way. See also this post and some other posts about the same sum.
Martin SleziakMartin Sleziak
$\begingroup$ Your cancellation while telescoping isn't correct. The partial sum should be bounded by $2-\frac 1 n$. This is probably the reason why someone downvoted this answer. I will +1 after correction. $\endgroup$ – Ennar Oct 25 '15 at 9:09
$\begingroup$ @Ennar You are right. (Thanks for noticing it.) There were also some other typos (which I have edited before). I have no problem with the downvote, and I can think of plenty other reasons why the post might have been downvoted. $\endgroup$ – Martin Sleziak Oct 25 '15 at 9:26
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Help with proof using induction: $1 + \frac{1}{4} + \frac{1}{9}+\cdots+\frac{1}{n^2}\leq 2-\frac{1}{n}$
Proof verification for proving $\forall n \ge 2, 1 + \frac1{2^2} + \frac1{3^2} + \cdots + \frac1{n^2} < 2 − \frac1n$ by induction
Proof of $\sum_{k = 1}^{n} \frac{1}{k^{2}} < 2 - \frac{1}{n}$
Proving that $\sum_{i=1}^n\frac{1}{i^2}<2-\frac1n$ for $n>1$ by induction
What is the formula for $\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\cdots +\frac{1}{n(n+1)}$
Sum of series question: $S_n = 1 + 1/4 + 1/9 + 1/16 + 1/25 + … + 1/n^2 < 2$
Proving inequality for induction proof: $\frac1{(n+1)^2} + \frac1{n+1} < \frac1n$
Proof by Induction for inequality, $\sum_{k=1}^nk^{-2}\lt2-(1/n)$
Induction on inequalities: $\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\ldots+\frac1{n^2}<2$
Proving inequality $\ 1+\frac14+\frac19+\cdots+\frac1{n^2}\le 2-\frac1n$ using induction
Backwards induction to show that $x_1\cdots x_n \leq ((x_1+\cdots+x_n)/n )^n$
Proving $\frac{1}{n+1} + \frac{1}{n+2}+\cdots+\frac{1}{2n} > \frac{13}{24}$ for $n>1,n\in\Bbb N$ by Induction
Use induction on $n$ to prove that $2n+1<2^n$ for all integers $n≥3$.
Induction: show that $\sum\limits_{k=1}^n \frac{1}{\sqrt{k}} < 2 \sqrt{n}$ for all n $\in Z_+$
Weak Mathematical Induction for Modulo Arithmetic $8\mid 3^{2n}-1$
Proving by induction that $\frac{n(n + 1)(2n + 1)}{6} = 0^2 + 1^2 + 2^2 + 3^2 + … + n^2$
Proving $n! < n^n$ by induction for all $n\geq 2$.
Proving A is a subset of S by mathematical induction?
Prove that $\tfrac{a_1a_2\cdots a_n(1-a_1-a_2-\cdots-a_n)}{(a_1+a_2+\cdots+a_n)(1-a_1)\cdots(1-a_n)} \leq \frac{1}{n^{n+1}}.$
A strange inductive proof: Induction on $n$, for all positive integers $n,n\ge1$ | CommonCrawl |
Noether Lecture
The Noether Lecture is a distinguished lecture series that honors women "who have made fundamental and sustained contributions to the mathematical sciences". The Association for Women in Mathematics (AWM) established the annual lectures in 1980 as the Emmy Noether Lectures, in honor of one of the leading mathematicians of her time. In 2013 it was renamed the AWM-AMS Noether Lecture and since 2015 is sponsored jointly with the American Mathematical Society (AMS). The recipient delivers the lecture at the yearly American Joint Mathematics Meetings held in January.[1]
The ICM Emmy Noether Lecture is an additional lecture series, sponsored by the International Mathematical Union. Beginning in 1994 this lecture was delivered at the International Congress of Mathematicians, held every four years. In 2010 the lecture series was made permanent.[2]
The 2021 Noether Lecture was supposed to have been given by Andrea Bertozzi of UCLA, but it was cancelled due to Bertozzi's connections to policing. The cancellation was made during the George Floyd protests: "This decision comes as many of this nation rise up in protest over racial discrimination and brutality by police".[3]
Noether Lecturer
YearNameLecture title
1980F. Jessie MacWilliamsA Survey of Coding Theory
1981Olga Taussky-ToddThe Many Aspects of Pythagorean Triangles
1982Julia RobinsonFunctional Equations in Arithmetic
1983Cathleen S. MorawetzHow Do Perturbations of the Wave Equation Work
1984Mary Ellen RudinParacompactness
1985Jane Cronin ScanlonA Model of Cardiac Fiber: Problems in Singularly Perturbed Systems
1986Yvonne Choquet-BruhatOn Partial Differential Equations of Gauge Theories and General Relativity
1987Joan S. BirmanStudying Links via Braids
1988Karen K. UhlenbeckMoment Maps in Stable Bundles: Where Analysis Algebra and Topology Meet
1989Mary F. WheelerLarge Scale Modeling of Problems Arising in Flow in Porous Media
1990Bhama SrinivasanThe Invasion of Geometry into Finite Group Theory
1991Alexandra BellowAlmost Everywhere Convergence: The Case for the Ergodic Viewpoint
1992Nancy KopellOscillators and Networks of Them: Which Differences Make a Difference
1993Linda KeenHyperbolic Geometry and Spaces of Riemann Surfaces
1994Lesley SibnerAnalysis in Gauge Theory
1995Judith D. SallyMeasuring Noetherian Rings
1996Olga OleinikOn Some Homogenization Problems for Differential Operators
1997Linda Preiss RothschildHow Do Real Manifolds Live in Complex Space
1998Dusa McDuffSymplectic Structures - A New Approach to Geometry
1999Krystyna M. KuperbergAperiodic Dynamical Systems
2000Margaret H. WrightThe Mathematics of Optimization
2001Sun-Yung Alice ChangNonlinear Equations in Conformal Geometry
2002Lenore BlumComputing Over the Reals: Where Turing Meets Newton
2003Jean TaylorFive Little Crystals and How They Grew
2004Svetlana KatokSymbolic Dynamics for Geodesic Flows
2005Lai-Sang YoungFrom Limit Cycles to Strange Attractors
2006Ingrid DaubechiesMathematical Results and Challenges in Learning Theory
2007Karen VogtmannAutomorphisms of Groups, Outer Space, and Beyond
2008Audrey A. TerrasFun With Zeta Functions of Graphs
2009Fan Chung GrahamNew Directions in Graph Theory
2010Carolyn S. GordonYou Can't Hear the Shape of a Manifold
2011Susan MontgomeryOrthogonal Representations: From Groups to Hopf Algebras
2012Barbara KeyfitzConservation Laws - Not Exactly a la Noether
2013Raman ParimalaA Hasse principle for quadratic forms over function fields
2014Georgia BenkartWalking on Graphs the Representation Theory Way
2015Wen-Ching Winnie LiModular forms for congruence and noncongruence
2016Karen E. SmithThe Power of Noether's Ring Theory in Understanding Singularities of Complex Algebraic Varieties
2017Lisa JeffreyCohomology of Symplectic Quotients
2018Jill PipherNonsmooth Boundary Value Problems
2019Bryna KraDynamics of systems with low complexity
2020Birgit SpehBranching Laws for Representations of Non Compact Orthogonal Groups
2021Lecture cancelled in 2021 (see above[3])
2022Marianna CsörnyeiThe Kakeya needle problem for rectifiable sets
2023Laura DeMarcoRigidity and uniformity in algebraic dynamics
2024 Anne Schilling TBA
References:[4][5][6]
ICM Emmy Noether Lecturers
YearName
1994Olga Ladyzhenskaya
1998Cathleen Synge Morawetz
2002Hesheng Hu
2006Yvonne Choquet-Bruhat
2010Idun Reiten
2014Georgia Benkart
2018Sun-Yung Alice Chang
2022Marie-France VignérasZ
References: [7]
See also
• Falconer Lecture
• Kovalevsky Lecture
• List of mathematics awards
• List of things named after Emmy Noether
References
1. "Noether Lecture". Association for Women in Mathematics. Retrieved 23 December 2018.
2. "ICM Emmy Noether Lecture". International Mathematical Union. Archived from the original on 5 August 2017.
3. Re: 2021 Noether Lecture
4. "Profiles of Women in Mathematics - The Emmy Noether Lectures". Association for Women in Mathematics. Archived from the original on 21 July 2018. Retrieved 19 August 2015.
5. "Past Noether Lectures". Association for Women in Mathematics. Retrieved 23 December 2018.
6. "2017 :: Joint Mathematics Meetings :: January 4 - 7 (Wednesday - Saturday), 2017". jointmathematicsmeetings.org.
7. "ICM Emmy Noether Lecturers". International Mathematical Union. 24 July 2014. Retrieved 7 April 2019.
External links
• Official website
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Some Thoughts on Sampling
Qasim Chaudhari●November 15, 2016●2 commentsTweet
Multirate DSP
DFT_FFT
Some time ago, I came across an interesting problem. In the explanation of sampling process, a representation of impulse sampling shown in Figure 1 below is illustrated in almost every textbook on DSP and communications. The question is: how is it possible that during sampling, the frequency axis gets scaled by $1/T_s$ -- a very large number? For an ADC operating at 10 MHz for example, the amplitude of the desired spectrum and spectral replicas is $10^7$! I thought that there must be something wrong somewhere.
This article is available in PDF format for easy printing
Figure 1: Sampling in time domain creates spectral replicas in frequency domain, each of which gets scaled by $1/T_s$
I asked a few DSP experts this question. They did not know the answer as well. Slowly I started to understand the reason why it is true, and in fact, makes perfect sense. The answer is quite straightforward but I did not realize it immediately. Since this is a blog post, I can take the liberty of explaining the route I took for this simple revelation.
A Unit Impulse
My first impression was that whenever impulses are involved, everything that branches from there becomes fictitious as well. A continuous-time unit impulse is defined as shown in Figure 2.
Area under a rectangle = $\Delta \cdot \frac{1}{\Delta} = 1$, or $\int \limits_{-\infty} ^{+\infty} \delta (t) dt = 1$
Figure 2: Definition of a unit impulse
Clearly, the sampling scale factor is independent of how we define a unit impulse and its infinite height. The next step might be to look into a sequence of unit impulses: an impulse train.
An Impulse Train
An impulse train is a sequence of impulses with period $T_s$ defined as
$$ p(t) = \sum \limits _{n=-\infty} ^{+\infty} \delta (t-nT_s) $$
The standard method to derive its Fourier Transform is through expressing the above periodic signal as a Fourier series and then finding Fourier series coefficients as follows.
First, remember that a signal $x(t)$ with period $T_s$ can be expressed as
$$x(t) = \sum \limits _{k=-\infty} ^{+\infty} x_k e^{jk \Omega_s t}$$ where $\Omega_s = 2\pi/T_s$ and it can be seen as the fundamental harmonic in the sequence of frequencies $k\Omega_s$ with $k$ ranging from $-\infty$ to $+\infty$. Here, $x_k$ are the Fourier series coefficients given as
$$ x_k = \frac{1}{T_s} \int _{-T_s/2} ^{T_s/2} x(t) e^{-jk \Omega_s t} dt$$
When $p(t)$ is an impulse train, one period between $-T_s/2$ and $+T_s/2$ only contains a single impulse $\delta (t)$. Thus, $p_k$ can be found to be
$$ p_k = \frac{1}{T_s} \int _{-T_s/2} ^{T_s/2} \delta (t) e^{-jk \Omega_s t} dt = \frac{1}{T_s} \cdot e^{-jk \Omega_s \ \cdot \ 0}= \frac{1}{T_s}$$
Plugging this in definition above and using the fact that $\mathcal{F}\left\{e^{jk \Omega_s t}\right\} = 2\pi \delta \left(\Omega - k\Omega_s \right)$ and $\Omega_s = 2\pi/T_s$, we get
$$ P(\Omega) = \frac{1}{T_s} \sum \limits _{k=-\infty} ^{+\infty} \mathcal{F}\left\{ e^{jk \Omega_s t}\right\} = \frac{2\pi}{T_s} \sum \limits _{k=-\infty} ^{+\infty} \delta \left( \Omega - k \frac{2\pi}{T_s}\right)$$
Figure 3: Fourier Transform of an impulse train in time domain is another impulse train in frequency domain
The resulting time and frequency domain sequences are drawn in Figure 3 above which is a standard figure in DSP literature. However, it really comes as a surprise that an impulse train has a Fourier Transform as another impulse train, although the Fourier Transform of a unit impulse is a constant. What happened to the concept that a signal narrow in time domain has a wide spectral representation? Going through this method, it is not immediately clear why.
So I thought of deriving it through some other technique. Note that I prefer to use the frequency variable $F$ instead of $\Omega$.
$$\int _{-\infty} ^{+\infty} \sum _{n=-\infty} ^{+\infty} \delta(t - nT_s) e^{-j2\pi F t} dt = \sum _{n=-\infty} ^{+\infty} \int _{-\infty} ^{+\infty} \delta(t - nT_s) e^{-j2\pi F t} dt $$
$$= \sum _{n=-\infty} ^{+\infty} e^{-j2\pi F nT_s}$$
From here, there are many techniques to prove the final expression, and the mathematics becomes quite cumbersome. So I just take the route of visualizing it and see where it leads.
Where the Impulses Come From
First, the Fourier Transform of a time domain impulse train can be seen as the sum of frequency domain complex sinusoids with ``frequencies'' equal to $nT_s$, or periods equal to $\frac{1}{nT_s} = \frac{F_s}{n}$. Furthermore, the above signal is periodic with period of first harmonic $1/T_s=F_s$.
$$ \sum _{n=-\infty} ^{+\infty} e^{-j2\pi (F+F_s) nT_s} = \sum _{n=-\infty} ^{+\infty} e^{-j2\pi F nT_s}\cdot \underbrace{e^{-j2\pi n}}_{1} = \sum _{n=-\infty} ^{+\infty} e^{-j2\pi F nT_s}$$
Therefore, for the sake of better visualization, we limit ourselves to the range $[0,F_s)$ when drawing the figures in frequency domain. Whatever happens here gets repeated elsewhere. Figure 4 illustrates 3 frequency domain complex sinusoids $e^{-j2\pi F nT_s}$ for $n=1,2$ and $3$, with frequencies $T_s, 2T_s$ and $3T_s$, respectively. Note how within a period of $F_s$, the first completes one cycle, while the other two complete two and three cycles, respectively. Here, we deliberately refrained from drawing the result for $n=0$ which is a straight line at $1$ on $I$ axis.
Figure 4: 3 frequency domain complex sinusoids $e^{-j2\pi F nT_s}$ for $n=1,2$ and $3$, with frequencies $T_s, 2T_s$ and $3T_s$, respectively, in the range $[0,F_s)$
For a closer look, see the signal decomposed into $I$ and $Q$ axes. Clearly, the $I$ axis consists of $\cos (2\pi F nT_s)$ and the $Q$ axis consists of $-\sin(2\pi F nT_s)$ again with $n = 1,2$ and $3$. This is shown in Figure 5.
Figure 5: $I$ and $Q$ decomposition of Figure 4. The $I$ axis consists of $\cos (2\pi F nT_s)$ and the $Q$ axis consists of $-\sin(2\pi F nT_s)$ again with $n = 1,2$ and $3$, in the range $[0,F_s)$
Having understood the basic concept, we now divide the range $[0,F_s)$ into $N$ segments each with width $\nu$.
\begin{equation*}
\nu = \frac{F_s}{N}
\end{equation*}
This implies that for integer $k$,
F \approx k\nu = k \frac{F_s}{N}, \qquad fT_s = \frac{k}{N}
The result of summing the sinusoids shown in Figure 5 for a large $N$ is drawn in Figure 6.
Figure 6: Summing the $I$ and $Q$ parts from Figure 5 but for a very large $N$ in the range $[0,F_s)$. The $Q$ part is seen converging to 0 while the $I$ part approaches an impulse, half of which is visible at $0$ and half at $F_s$
The rectangle at $k=0$ has a height of $N$ since it is a sum of $1$ for all $N$ values of $n$. It can now be concluded that
$I$ arm -- the expression $\sum _{n=-\infty} ^{+\infty} \cos(2\pi nk/N)$ -- approaches $N$ at $k=0$ in each period $[0,F_s)$. As $\nu \rightarrow 0$, $N \rightarrow \infty$ and it becomes an impulse (strictly speaking, the limit does not exist as a function and is rather known as a ``distribution'').
On the other hand, the $Q$ arm -- the expression $\sum _{n=-\infty} ^{+\infty} -\sin(2\pi n k/N)$ -- vanishes.
The scaling factor here can be found again by taking the area under the first rectangle. Thus,
Area under the rectangle $= N \cdot \nu = N \cdot \frac{F_s}{N} = F_s,$
the same as found in Figure 3 (except the factor of $2\pi$). Since it is a periodic signal, it consists of impulses repeating at integer multiples of $F_s$ -- just what we saw in Figure 3 (except the $2\pi$ factor as we are dealing in $F$ now). The complete sequence is now drawn in Figure 7.
Figure 7: Displaying the $I$ branch from Figure 6 in the range $(-3F_s,+3F_s)$ for large $N$
It was then I realized my problem. All the graphs and curves I have drawn, and DSP textbooks draw for that matter, use the wrong scaling on the axes! Although the time and frequency domain impulse sequences look pretty much similar to each other, they are not. Imagine a very large number, say $L$. Now if I draw the time and frequency domain representations of an impulse train within the range $-L$ to $+L$, the signals would look like as illustrated in Figure 8 for a sample rate of $10$ MHz. As it is evident, all the confusion in the y-axis was due to incorrect scaling at the x-axis!
Figure 8: For a $10$ MHz sample rate, there are (10 million)$^2$ impulses in time domain for just 1 in frequency domain
Here, it is easier to recognize that for an ADC operating at 10 MHz, there are (10 million)$^2$ impulses on the time axis, for just one such impulse on the frequency axis. Let us express the equivalence of energy in time and frequency domains (known as Parseval's relation) in a different way.
\lim_{L \to \infty} \int \limits _{-L}^{+L} |x(t)|^2 dt = \lim_{L \to \infty} \int \limits _{-L}^{+L} |X(F)|^2 df
\end{equation*}Clearly, the scaling of the frequency axis by $F_s$ seems perfectly logical. A signal narrow in time domain is wide in frequency domain, finally! I must say that this is the same figure as Figure 3 but until I went this route, I did not figure (pun intended) this out.
This reminded me of drawing the structure of an atom along with the solar system to indicate some similarity, without mentioning their respective scales, as in Figure 9.
Figure 9: An atom and solar system
This also reinforced my concept of using the frequency variable $F$ instead of $\Omega$. While $\Omega$ has its advantages at places, the variable $F$ connects us with what is out there: the reality. The commonly used ISM band is $2.4$ GHz, not $4.8\pi$ G-radians/second.
Going into Discrete Domain
Let us turn our thoughts in discrete domain now. There are two points to consider in this context.
The main difference between a continuous-time impulse and a discrete-time impulse is that a discrete-time impulse actually has an amplitude of unity; there is no need to define the concept related to the area under the curve.
There is a factor of $N$ that gets injected into expressions due to the way we define Fourier Transforms. This is the equivalent of the factor $2\pi$ for continuous frequency domain $\omega$ in Discrete-Time Fourier Transform (DTFT).
x[n] = \frac{1}{2\pi} \int \limits _{\omega=-\pi} ^{+\pi} X(e^{j\omega})\cdot e^{j\omega n} d \omega, \qquad x[n] = \frac{1}{N} \sum \limits_{k=-N/2}^{+N/2-1} X[k]\cdot e^{j2\pi \frac{k}{N}n}
A discrete impulse train $p[n]$ is a sequence of unit impulses repeating with a period $M$ within our observation window (and owing to DFT input periodicity, outside the observation window as well). Figure 10 illustrates it in time domain for $N=15$ and $M=3$.
Figure 10: Discrete impulse train in time domain for $N=15$ and $M=3$
To compute its $N$-point DFT $P[k]$,
\begin{align*}
P[k] =& \sum \limits _{n=0} ^{N-1} p[n] e^{-j2\pi\frac{k}{N}n} = \sum \limits _{m=0} ^{N/M-1} 1 \cdot e^{-j2\pi\frac{k}{N/M}m} \\
=& \begin{cases}
\frac{1}{M}\cdot N & k = integer \cdot \frac{N}{M} \\
0 & elsewhere
\end{cases}
\end{align*}A few things to note in this derivation are as follows.
I write DFT emphasizing on the expression $\frac{k}{N}$ because it corresponds to the continuous-time frequency $\frac{F}{F_s}$, explicitly exposing $n$ as the time variable similar to its counterpart $t$ (think of $2\pi \frac{F}{F_s}n$, or $2\pi ft$).
From going to second equation from the first, I use the fact that $p[n]=0$ except where $n=mM$.
The third equation can be derived, again, in many ways such as the one described in the continuous-time case described before.
The final DFT output is shown in Figure 11 for $N=15$ and $M=3$.
Figure 11: Discrete impulse train in frequency domain for $N=15$ and $M=3$
Note how impulse train in time domain is an impulse train in frequency domain. So for $N=15$ and a discrete-time impulse train with a period of $3$, the DFT is also an impulse train with a period of $N/3 =5$. Moreover, the unit impulses in frequency domain have an amplitude of $N/3=5$ as well. Observe that a shorter period $M$ in time domain results in a longer period $N/M$ in frequency domain. Also, the amplitude of those frequency domain impulses is $N/M$ times larger as well: compare with $2\pi/T_s$ in Figure 3.
This is the same result as in continuous frequency domain with impulse train having a period of $F_s$ as well as a scaling factor of $F_s$. However, it is much clearer to recognize in discrete domain due to the finite DFT size $N$. That is why I talked about limiting the continuous frequency range within $\pm L$ for a large $L$ and wrote Parseval's relation as above.
Sampling in Discrete Domain: Downsampling
With the above findings, we want to see what happens when a discrete-time signal is sampled with a discrete-time impulse train. In DSP terminology, this process is known as downsampling. Although it is common to discuss downsampling as a separate topic from sampling a continuous-time signal, both processes are intimately related and offer valuable insights into one another.
A signal $x[n]$ can be downsampled by a factor of $D$ by retaining every $D$th sample and discarding the remaining samples. Such a process can be imagined as multiplying $x[n]$ with an impulse train $p[n]$ of period $D$ and then throwing off the intermediate zero samples (in practice, no such multiplication occurs as zero valued samples are to be discarded). As a result, there is a convolution between $X[k]$ and $P[k]$ and $D$ spectral copies of $X[k]$ arise: one at frequency bin $k=0$ and others at $\pm integer \cdot N/D$, all having a spectral amplitude of $N/D$. As $D$ increases, the spectral replicas come closer and their amplitudes decrease proportionally. This is illustrated in Figure 12 for downsampling a signal by $D=3$.
Figure 12: Downsampling a signal $x[n]$ by $D=3$ in time and frequency domains
This is the same concept as sampling a continuous-time signal. Just as we decrease the amplitudes and spread of the resulting spectral copies by reducing the sampling rate, we decrease the amplitudes and spread of the resulting spectral copies by throwing off more samples, i.e., reducing the sampling rate. In the former, this happens in proportion to $1/T_s$; in the latter, in proportion to $1/D$.
Put in this way, $1/D$ is choosing $1$ out of every $D$ samples. And $1/T_s$ is choosing $1$ out of every $T_s$ samples if there ever were. In this context, $1/T_s$ is the probability density function (pdf) of a uniform random variable in the range $[0,T_s)$.
Carrying on, now we can look into an alternative view on sampling. First, consider the scaling property of the Fourier Transform.
x(a\cdot t) \quad \xrightarrow{{\mathcal{F}}} \quad \frac{1}{|a|} X\left(\frac{F}{a} \right)
Second, instead of impulse sampling, we can just see it as a scaling of the time axis. Taking the constant $a$ as $T_s$,
x(T_s \cdot t) = x(tT_s) \quad \xrightarrow{{\mathcal{F}}} \quad \frac{1}{T_s} X\left(F = \frac{f}{T_s} \right)
Here, $x(tT_S)$ is the counterpart of $x(nT_s)$ from impulse sampling with the difference being a real number $t$ instead of an integer $n$. Also, instead of nano and micro seconds, $T_s$ has no units and we just deal it in the limiting case where the interval $[0,T_s)$ is divided into infinitesimally small segments and $T_s$ is just a positive number. In this way, this case becomes similar to downsampling where $D$ is always greater than $1$.
The spectrum x-axis also gets scaled by $T_s$ leading to the fundamental relationship between continuous-time frequency $F$ and discrete-time frequency $f$.
f = F\cdot T_s \qquad \rightarrow \qquad \omega = \Omega \cdot T_s
\end{equation*}Within this framework, it is only after discarding the intermediate ``samples'' that the spectral replicas emerge. This is a result of defining the frequency axis in relation to periodic signals. There might be an alternative definition of frequency that would make processing the samples easier.
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Comment by Tim Wescott●November 21, 2016
I've gone through similar thinking on this. It turns out, that no matter what method you choose to scale things, you'll find that (A) there are some constraints that you have to put on the axis scaling to keep the numbers consistent as you go from continuous-time to sampled-time and back, and (B) no matter what you do, it'll be klunky. Ultimately, I gave up and just stuck to what's in the published literature.
Something that's worthy of note is your question "why does the frequency axis get scaled by a very large number", and you cite 10MHz ( \( 10^7 \) ) as an example. In this, you are making two errors.
First, you are comparing a dimensional quantity to a non-dimensional quantity; in essence, you're comparing apples and oranges. Seconds are arbitrary units that roughly fit with the shortest length of time that a non-musician can easily measure. If, however, you were to consider the sampling time in Plank units, then your "big" \( \frac{1}{T_s} \) becomes \( F_s = \frac{1}{1.855\cdot10^{36} t_p} = 5.391 \cdot 10^{-37} \frac{1}{t_p} \), and your question turns into "why does the frequency axis get scaled by a very small number?"
Second, you are assuming that all DSP is carried out for the purposes of radio waves. I work on closed-loop control systems, and while I have not yet had the pleasure of working on a system that samples at less than 1Hz, there are such systems out there. These systems -- even using your chosen units, come with the question "why does the frequency axis get scaled by a (possibly very) small number?"
Comment by qasim_chaudhari●November 22, 2016
You are right Tim. Here, I just wanted to compare seconds and cycles/second. Also, I thought about sampling rates of less than 1 Hz and discarded it as a hypothetical situation. Thank you for increasing my knowledge.
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My main area of interest is signal processing for wireless communication systems. I write about it at https://wirelesspi.com.
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Forest Ecosystems
Stand-level biomass models for predicting C stock for the main Spanish pine species
Ana Aguirre ORCID: orcid.org/0000-0001-7723-20781,2,
Miren del Río2,3,
Ricardo Ruiz-Peinado2,3 &
Sonia Condés1
Forest Ecosystems volume 8, Article number: 29 (2021) Cite this article
National and international institutions periodically demand information on forest indicators that are used for global reporting. Among other aspects, the carbon accumulated in the biomass of forest species must be reported. For this purpose, one of the main sources of data is the National Forest Inventory (NFI), which together with statistical empirical approaches and updating procedures can even allow annual estimates of the requested indicators.
Stand level biomass models, relating the dry weight of the biomass with the stand volume were developed for the five main pine species in the Iberian Peninsula (Pinus sylvestris, Pinus pinea, Pinus halepensis, Pinus nigra and Pinus pinaster). The dependence of the model on aridity and/or mean tree size was explored, as well as the importance of including the stand form factor to correct model bias. Furthermore, the capability of the models to estimate forest carbon stocks, updated for a given year, was also analysed.
The strong relationship between stand dry weight biomass and stand volume was modulated by the mean tree size, although the effect varied among the five pine species. Site humidity, measured using the Martonne aridity index, increased the biomass for a given volume in the cases of Pinus sylvestris, Pinus halepensis and Pinus nigra. Models that consider both mean tree size and stand form factor were more accurate and less biased than those that do not. The models developed allow carbon stocks in the main Iberian Peninsula pine forests to be estimated at stand level with biases of less than 0.2 Mg∙ha− 1.
The results of this study reveal the importance of considering variables related with environmental conditions and stand structure when developing stand dry weight biomass models. The described methodology together with the models developed provide a precise tool that can be used for quantifying biomass and carbon stored in the Spanish pine forests in specific years when no field data are available.
Forests are fundamental in the global carbon cycle, which plays a key role in the global greenhouse gas balance (Alberdi 2015), and therefore in climate change. As part of the strategy to mitigate climate change, forest carbon sinks were included in the Kyoto Protocol in 1998 (Breidenich et al. 1998) and subsequent resolutions as the Paris Agreements in 2015. In accordance, countries are requested to estimate forest CO2 emissions and removals as one of the mechanisms for mitigating climate change. Based on the international demands, some international institutions request periodic reports on forest indicators which are used in global reports. For example, the State of Europe's Forest 2015 (SoEF 2015) or Global Forest Resources Assessment 2020 (FRA 2020) request five-yearly information on accumulated carbon in the biomass of woody species or the accumulated carbon in other sources or sinks. Since the development of these international agreements, numerous countries have made efforts to achieve the main objective of mitigating climatic change. In Spain, for example, the Spanish Ministry for Ecological Transition and Demographic Challenge is developing a data base of the national contribution to the European Monitoring and Evaluation Program (EMEP) emission inventory, which includes Land Use, Land-Use Change and Forestry (LULUCF) sector, with the aim of estimating carbon emissions and removals in each land-use category. Furthermore, annually updated greenhouse gas emission data must be provided for the UNFCCC (United Nations Framework Convention on Climate Change) Greenhouse Gas Inventory Data.
Soil and biomass are the most important forest carbon sinks. The carbon present in soils is physically and chemically protected (Davidson and Janssens 2006), although it is more or less stable depending on the type of disturbances suffered and the environmental conditions (Ruiz-Peinado et al. 2013; Achat et al. 2015; Bravo-Oviedo et al. 2015; James and Harrison 2016). Therefore, the carbon that could be returned to the atmosphere from the ecosystem after a disturbance is mainly contained in the aboveground biomass, which accounts for 70%–90% of total forest biomass (Cairns et al. 1997). Carbon stocks and carbon sequestration in tree vegetation are usually estimated thorough biomass evaluation as the amount of carbon in woody species is about 50% of their dry weight biomass (Kollmann 1959; Houghton et al. 1996). Although species-specific values can be found in the literature, this percentage is recommended by the Intergovernmental Panel on Climate Change (IPCC) if no specific data is available (Eggleston et al. 2006).
There are two main approaches to estimating forest carbon: i) using biogeochemical-mechanisms and ii) the statistical empirical approach (Neumann et al. 2016). The second method is more common in forestry since it uses inventory data such as that provided by NFI's (Tomppo et al. 2010) and the data required does not need to be as specific as for the biogeochemical-mechanism approach. Through this approach, biomass and carbon estimates can be obtained using allometric biomass functions and/or biomass expansion factors (BEFs). Biomass functions require variables for individual trees and/or stand variables (Dahlhausen et al. 2017), while BEFs convert stand volume estimates to stand dry weight biomass (Castedo-Dorado et al. 2012). The BEF method is widely used when little data is available, this being one of the methods recommended in the IPCC guidelines (Penman et al. 2003).
BEFs, including their generalization of stand biomass functions depending on stand volume, can be affected by environmental conditions and stand characteristics, such as the species composition (Lehtonen et al. 2004; Soares and Tomé 2004; Lehtonen et al. 2007; Petersson et al. 2012; Jagodziński et al. 2017). Some authors have also pointed to the dependence of the stand biomass-volume relationship on age or stand development stage (Jalkanen et al. 2005; Peichl and Arain 2007; Tobin and Nieuwenhuis 2007; Teobaldelli et al. 2009; Jagodziński et al. 2017). When age data are not available, as is the case in several NFIs, other variables expressing the development stage can be used as a surrogate of age, such as tree size (Soares and Tomé 2004; Kassa et al. 2017; Jagodziński et al. 2020). In addition, site conditions can influence the relationship between stand biomass and stand volume (Soares and Tomé 2004). These conditions can be assessed by means of indicators such as site index or dominant height (Houghton et al. 2009; Schepaschenko et al. 2018) or directly through certain environmental variables (Briggs and Knapp 1995; Stegen et al. 2011).
Most of the information on forests at national level currently comes from the National Forest Inventories (NFIs). Consequently, many countries have adapted their NFIs to fulfil international requirements (Tomppo et al. 2010; Alberdi et al. 2017). As regards carbon stock, NFIs are widely recognized as being appropriate sources of data for estimating these stocks (Brown 2002; Goodale et al. 2002; Mäkipää et al. 2008), especially at large scales (Fang et al. 1998; Guo et al. 2010). Although most NFIs are carried out periodically, the frequency does not coincide with the international requirements for data on accumulated carbon and biomass stocks (which may be annual). In the case of the Spanish National Forest Inventory (SNFI), the time between two consecutive surveys is longer than that stated in the international requirements for forest statistics reporting. Hence, the forest indicators from SNFI data should be updated annually in order to fulfill the international requirements. Moreover, the time between two consecutive SNFI is approximately 10 years, although it is carried out a province at a time, so not all the Spanish forest area is measured in the same year. Whereas other countries measure a percentage of their NFI plots each year, distributed systematically throughout the country (allowing annual national estimates to be made, albeit with greater uncertainly), the approach used in Spain is to measure all the plots within a given province, which does not allow for annual data (or indicators) to be extrapolated at national level. As a consequence, indicators must be updated in the same year for all provinces in order to estimate carbon at national level in a given year. A possible approach to updating carbon stocks indicators from SNFI data would be to estimate the stand biomass through tree allometric biomass functions (Neumann et al. 2016), although this method would require complex individual tree models to update stand information at tree level (tree growth, tree mortality and ingrowth). Given the strong relationship between stand volume and biomass (Fang et al. 1998; Lehtonen et al. 2004), estimations of biomass could be also made by updating volume stocks from the SNFI and using BEFs. This option has the advantage that stand volume can often be easily updated through growth models (Shortt and Burkhart 1996) or even by remote sensing (McRoberts and Tomppo 2007).
According to Montero and Serrada (2013), the main pine species (Pinus sylvestris L., Pinus pinea L., Pinus halepensis Mill., Pinus nigra Arn. and Pinus pinaster Ait.) occupy around of 30% of the Spanish forest area as dominant species, which is more than 5 million ha, along with almost half a million ha of pine-pine mixtures. Their distribution across the Iberian Peninsula covers a wide range of climatic conditions (Alía et al. 2009), with arid conditions being particularly prominent. Thus, aridity was found to influence the maximum stand density and productivity of these pinewoods (Aguirre et al. 2018, 2019). Furthermore, pine species were those most used in reforestation programs, so these species play a fundamental role in carbon sequestration. According to the Second and Third National Forest Inventories, the five abovementioned species alone account for a carbon stock of around 250 × 106 Mg C (del Río et al. 2017), of which more than half corresponds to two of these forest species (P. sylvestris and P. pinaster).
The main objective of this study was to develop dry weight biomass models for pine forests (monospecific and mixed stands) according to stand volume, exploring whether basic BEFs can be improved by including site conditions and stand development stage. We hypothesized that for a given stand volume the stand dry weight biomass increases as site aridity decreases and that it decreases with the stand development stage. Therefore, the specific objectives were to study the dependence of the models on these factors and to assess the biomass expansion factors when varying these variables for the main pine species studied. The biomass models developed will allow carbon estimates to be updated for a given year when no field data from SNFI surveys are available.
The data used were from two consecutive completed surveys of the SNFI in the Iberian Peninsula, the Second and Third SNFI (SNFI-2 and SNFI-3), which were carried out from 1986 to 1996, and from 1997 to 2007 respectively, except for the provinces of Navarra, Asturias and Cantabria, where the SNFI-2 surveys were carried out using a different methodology. Data from the SNFI-3 and SNFI-4 were used for these provinces, covering the periods from 1998 to 2000 and from 2008 to 2010, respectively. The initial and final surveys are referred to regardless of the provinces considered. The time elapsed between surveys ranges from 7 to 13 years depending on the province. Data from the final SNFI surveys were used to develop dry weight biomass estimates, while data from the initial surveys, together with volume growth models by Aguirre et al. (2019), were used to evaluate model assessment capability.
The SNFI consists of permanent plots located systematically at the intersections of a 1-km squared grid in forest areas. The plots are composed of four concentric circular subplots, in which all trees with breast-height diameter of at least 7.5, 12.5, 22.5 and 42.5 cm are measured in the subplots with radii of 5, 10, 15 and 25 m, respectively. Using the appropriate expansion factor for each subplot, stand variables were calculated per species and for the total plot. For further details of the SNFI, see Alberdi et al. (2010).
The target species were five native pine species in the Spanish Iberian Peninsula: Pinus sylvestris (Ps), Pinus pinea (Pp), Pinus halepensis (Ph), Pinus nigra (Pn) and Pinus pinaster (Pt). Plots located in the peninsular pine forests were used; the criterion for selection being that the density of non-target species should not exceed 5% of the maximum capacity (Aguirre et al. 2018). The plots used for each species were those in which the proportion of the species by area was greater than 0.1. Additionally, to allow the application of the results to stands where the volume was updated through growth models, only those plots in which silvicultural fellings affected less than 5% of the total basal area were considered, as this was the criterion used for developing the existing volume growth models (Aguirre et al. 2019).
Stem volume was calculated for every tree in the plot according to SNFI volume equations developed for each province, species and stem form (Villanueva 2005). The Martin (1982) criteria were used to obtain volume growth. Dry weight biomass for different tree components was calculated at tree level using equations taken from Ruiz-Peinado et al. (2011), who developed biomass models for all the studied species, using diameter at breast height and total tree height as independent variables. Total tree aboveground dry weight biomass was calculated by adding the weight of stem (stem fraction), thick branches (diameter larger than 7 cm), medium branches (diameter between 2 and 7 cm) and thin branches with needles (diameter smaller than 2 cm). Based on tree data and using the appropriate expansion factors for each SNFI subplot, the stand level volume and dry weight biomass were obtained per species and total plot.
To estimate the aridity conditions for each plot used, the annual precipitation (P, in mm) and the mean annual temperature (Tm, in °C) were obtained from raster maps with a one-kilometer resolution developed by Gonzalo Jiménez (2010). These variables were used to obtain the Martonne aridity index (De Martonne 1926), M, calculated as M = P/(Tm + 10), in mm·°C− 1. M was chosen as an aridity indicator because of its simplicity and recognized influence on volume growth (Vicente-Serrano et al. 2006; Führer et al. 2011; Aguirre et al. 2019) and maximum stand density (Aguirre et al. 2018). Hence, M was expected to have a positive influence on dry weight biomass.
Due to the lack of age information for SNFI plots, the development stage had to be estimated through specific indicators. Tree-size related variables are commonly used as surrogates for stand development stage, one such variable being the mean tree volume (vm), which could be used to correct the lack of age information. The vm was calculated as in Eq. 1, where V is the volume of the stand in m3·ha− 1, and N is the number of the trees per hectare, both referred to the target species (sp).
$$ {vm}_{sp}=\frac{V_{sp}}{N_{sp}} $$
A summary of the data used to develop the models is shown in Table 1 (note that when a target species was studied, other pine species could be included within stands). Figure 1 summarizes the methodology that is described in the following sections.
Table 1 Summary of data used to develop dry weight biomass models. Note that plots where a target species, sp, is studied, other pine species could be present
Schematic explanation about how to apply the developed model for future projections. SNFIF is the last Spanish National Forest Inventory available, ∆T is the time elapsed between SNFIF and the projection time T, M is the Martonne aridity index, Origin is the naturalness of the stand (plantation or natural stand), dg is the quadratic mean diameter (cm), Ho is the dominant height (m), RD is the relative stand density, p is the proportion of basal area of the species in the stand, VGE is the volume growth efficiency, IV is the volume increment (m3·ha− 1·year− 1), N is the number of trees per hectare, V is the volume of the stand (m3·ha− 1), vm is the mean tree volume, f is the stand form factor, W is the dry weight biomass, and C is the weight of carbon. The subscript "F" refers to the final SNFI, the last available, while "T" refers at projection time T. The variables with the subscript "sp" refer to the target species, variables without the subscript refer to the stand
Biomass estimation models by species
Basic biomass models were developed for each species from SNFIF data in accordance with the structure used by Lehtonen et al. (2004) (Eq. 2) to estimate dry weight biomass (W) from stand volume (V) for the target species. The Basic Model was modified by including the effect of aridity, thus, the Martonne aridity index (M) was added to the Basic Model to obtain the so-called Basic M Model (Eq. 3). As regards the model structure, following a preliminary study (not shown) it was decided to include the logarithm of this variable to adapt the Basic Model (Eq. 2), modifying the 'a' coefficient according to Eq. 3.
$$ \mathrm{Basic}\ \mathrm{Model}:{W}_{jk}=\left(a+{a}_k\right)\times {V}_{jk}^b+{\varepsilon}_{jk} $$
$$ \mathrm{Basic}\ \mathrm{M}\ \mathrm{M}\mathrm{odel}:{W}_{jk}=\left(a+{a}_k\right)\times {V}_{jk}^b\times \left(1+m\times \log \left({M}_{jk}\right)\right)+{\varepsilon}_{jk} $$
where, for plot j in province k, W is the dry weight biomass of the target species in Mg·ha− 1, V is the volume of the target species in m3·ha− 1, M is the Martonne aridity index, in mm·°C− 1; and Ɛ is the model error. The coefficient a is the fixed effect, while ak is the province random effect to avoid possible correlation between plots belonging to the same province, as the measurements in the different provinces were carried out in different years and by different teams. b and m are other coefficients to be estimated: if coefficient m was not significant for a given species or its inclusion did not improve the Basic Model, M was no longer included in the species model.
To determine how the stand development stage influences the relationships between volume and dry weight biomass for each species, the mean tree volume (vm) was included in the models. This variable also multiplies the coefficient ' (a + ak) ' (Eq. 4), so that if it was not significant, the final model will be equivalent to the basic one.
$$ vm\ \mathrm{Model}:{W}_{jk}=\left(a+{a}_k\right)\times {V}_{jk}^b\times \left(1+m\times \log (M)\right)\times \left(1+{c}_1\times {vm}_{jk}^{p_1}\right)+{\varepsilon}_{jk} $$
where, a, ak, b, c1, p1 and m were the coefficients to be estimated and vm is the mean tree volume, all variables referring to the target species.
When fitting the biomass models some bias linked to the stem form was detected. Hence, the next step was to test whether it was possible to correct the model bias by adding the shape of the trees by means of the stand form factor (f) (Eq. 5). This variable was also added to multiply the coefficient ' (a + ak) ', thus obtaining the Total Model (Eq. 6).
$$ f=\frac{V}{G\times H} $$
where f is the stand form factor; V is the stand volume (m3·ha− 1); G is the basal area (m2·ha− 1); and H is the mean height of the plot (m), all variables referring to the target species.
$$ \mathrm{Total}\ \mathrm{Model}:\kern0.5em {W}_{jk}=\left(a+{a}_k\right)\times {V}_{jk}^b\times \left(1+m\times \log (M)\right)\times \left(1+{c}_1\times {vm}_{jk}^{p_1}\right)\times \left(1+{c}_2\times {f}_{jk}^{p_2}\right)+{\varepsilon}_{jk.} $$
where a, ak, b, c1, c2, p1, p2 and m were the coefficients to be estimated, f is the form factor of the stand and vm is the mean tree volume, all variables referring to the target species.
The model structure was analysed in a preliminary study where each coefficient in the allometric basic model was parametrized in function of M, vm and f, considering linear and non-linear expansions. The final model structure (Eq. 6) was selected because its better goodness of fit in terms of AIC, showing also the lowest residuals.
All models (Eqs. 2 to 4 and Eq. 6) were fitted using non-linear models with the nlme package (Pinheiro et al. 2017) from the R software (Team RC 2014). The coefficients were only included if they were statistically significant (p-value < 0.05) and their inclusion improved the model in terms of Akaike Information Criterion (AIC) (Akaike 1974). Furthermore, conditional and marginal R2 (Cox and Snell 1989; Magee 1990; Nagelkerke 1991) were calculated as a goodness-of-fit statistic using MuMIn library (Barton 2020). Once selected the model with the lowest AIC, and highest marginal and conditional R2, and to check that the improvement achieved is significant, anova tests were made.
Evaluation of biomass estimation models
In order to evaluate the goodness of fit, an analysis of the four developed models (Eqs. 2 to 4 and Eq. 6) was performed. The mean errors (Eqs. 7 to 9), estimated in Mg·ha− 1, as well as mean percentage errors (Eqs. 10 to 12) in % were calculated for each model of each species.
$$ \mathrm{Mean}\ \mathrm{error}:\mathrm{ME}=\sum {e}_j/n $$
$$ \mathrm{Mean}\ \mathrm{absolute}\ \mathrm{error}:\mathrm{MAE}=\sum \left|{e}_j\right|/n $$
$$ \mathrm{Root}\ \mathrm{mean}\ \mathrm{square}\ \mathrm{error}\ \mathrm{RMSE}=\sqrt{\sum {e_j}^2/n} $$
$$ \mathrm{Mean}\ \mathrm{percentage}\ \mathrm{error}:\mathrm{MPE}=100\times \sum {ep}_j/n $$
$$ \mathrm{Mean}\ \mathrm{absolute}\ \mathrm{percentage}\ \mathrm{error}:\mathrm{MAPE}=100\times \sum \left|{ep}_j\right|/n $$
$$ \mathrm{Root}\ \mathrm{mean}\ \mathrm{square}\ \mathrm{percentage}\ \mathrm{error}:\mathrm{RMSPE}=100\times \sqrt{\sum {ep_j}^2/n} $$
where \( {e}_j={W}_j-\hat{W_j} \) and \( {ep}_j=\left({W}_j-\hat{W_j}\right)/{W}_j \); \( \hat{W_j} \) is the estimated values of dry weight biomass for each plot j, Wj the corresponding observed values for each plot j, both referring to the target species; and n is the number of plots where the species was present.
Carbon predictions at national level
The models developed (Eqs. 4 to 6 and Eq. 8) provide estimates of dry weight biomass per species, both in monospecific and mixed stands, which could be transformed to carbon stock, considering the specific data of carbon content in wood given by Ibáñez et al. (2002) for the five studied pine species (Table 2).
Table 2 Carbon content of wood for the studied species (Ibáñez et al. 2002)
To evaluate the prediction capacity of the fitted models at time T when no field data is available, a simulation from the initial SNFI survey (SNFII) was performed at a national scale, assuming that this was the last available survey.
The first step was to obtain the predicted biomass at time T, where all variables are supposed to be unknown for each species, from the four biomass models developed (Eqs. 2 to 4 and Eq. 6). To apply these models, it was necessary to obtain the values of all independent variables, updated to year T. This procedure was done as follow:
Using the annual growth volume models by Aguirre et al. (2019), the volume \( {\hat{V}}_T \) was estimated from the SNFII volume. These authors developed a volume growth efficiency (VGE) model for the five pine species considered in this study. Volume growth efficiency is a measure of stand volume growth taking into account the species proportions by area (p), which is necessary when studying mixed stands (Condés et al. 2013), as VGE = IV/p. In monospecific stands VGE = IV. So, with these estimations (IV) and the number of years elapsed since initial SNFI (∆T), the volume at time T was estimated as \( {\hat{V}}_T={V}_I+ IV\times \Delta T \).
The mean tree volume \( {\hat{vm}}_T \) was estimated assuming that there are no extractions or high mortality in plots during ∆T, that is, assuming the number of trees per hectare remains constant (\( {\hat{N}}_T={N}_I \)), so that, \( {\hat{vm}}_T={\hat{V}}_T/{\hat{N}}_T \).
Furthermore, it was assumed that the stand form factor does not vary significantly in the time elapsed between inventories, so this variable was estimated as \( {\hat{f}}_T={f}_I \).
As the predictions were made for the same plots used to develop the growth models by Aguirre et al. (2019), biomass models can be applied directly, without the need to perform calibrations, since the fixed and random effects are known. Hence, by applying the different models (Eqs. 2 to 4 and Eq. 6) and using the independent variables described (\( {\hat{V}}_T,{\hat{vm}}_T\ \mathrm{and}\ {\hat{f}}_T \)), we obtain the biomass estimated at time T (\( {\hat{W}}_T \)), which is assumed to be unknown.
Secondly, using the carbon percentages contained in the biomass weight shown in Table 2, the carbon weight estimated for each species was obtained at time T (\( {\hat{C}}_{T\_ sp} \)). Considering all species present in each plot, the total carbon weight was estimated at time T (\( {\hat{C}}_T=\sum {\hat{C}}_{T\_ sp} \)).
Finally, in order to evaluate the predictions, time T was set to be the same as the final SNFI (SNFIF), therefore the observed values were already known and could be compared with the predictions obtained. Thus, the predicted carbon (\( {\hat{C}}_T \)) was compared with the observed carbon weight for the final SNFI (CF), obtained by multiplying the observed dry weight biomass (as explained in the data section) and the carbon content (Table 2) in the final survey (SNFIF). The mean errors were then calculated from Eqs. 9 to 14.
How to estimate carbon stocks at national level when no data is available
In this section, it is explained how to apply the developed models for predicting the carbon stock at time T required, when no data is available. For this, it is necessary to use some variables of the last Spanish National Forest Inventory available (SNFIF), ∆T years before T.
The first step is to estimate the volume growth efficiency of the target species (VGEsp), which can be estimated using Aguirre et al. (2019) models. These models estimate VGE as function on:
Origin, makes reference to the naturalness of the stand. It was a dummy variable, with value 1 when the stand was a plantation and 0 when the stand comes from natural regeneration.
dgsp, is the quadratic mean diameter of the target species.
Ho, is the dominant height of the stand.
RD, is the relative stand density (Aguirre et al. 2018, Eq. S1), and RDsp is only considering the target species.
psp, is the proportion of the species.
M, is the Martonne aridity index.
With these variables it is possible to estimate VGEsp for each pine species considered, and using its proportion, also volume growth of each species (IVsp) can be estimated. Note that in monospecific stands IVsp is equal to IV total.
Having the IVsp, the time elapsed since T and SNFIF and the volume of the target species at SNFIF (VspF) the volume at T time is estimated (\( {\hat{V}}_{sp\_T} \)).
Obtained \( {\hat{V}}_{sp\_T} \), the biomass models can be applied by using some assumptions:
The number of trees per hectare remains constant at equal to the observed in SNFIF (\( {\hat{N}}_{sp\_T}={N}_{sp F} \)).
So, the mean tree volume at time T can be estimated as: \( {\hat{vm}}_{sp\_T}={\hat{V}}_{sp\_T}/{\hat{N}}_{sp\_T} \).
The stand form factor also is considered constant at equal to the observed in SNFIF \( {\hat{\Big(f}}_{sp\_T}={f}_{sp F}\Big) \).
Using these estimated variables, biomass models can be used to obtain the estimation of dry weight biomass of the target species at time T (\( {\hat{W}}_{sp\_T} \)). The appropriate percentage of the carbon content per species (Ibáñez et al. 2002) allows to transform that value in the estimated carbon of the target species at time T (\( {\hat{C}}_{sp\_T} \)). For mixed stands, the estimated carbon of the stand (\( \hat{C} \)) is the sum of the different \( {\hat{C}}_{sp\_T} \).
Biomass estimation models for each species
Table 3 shows the coefficient estimates together with the standard errors and goodness of fit for the four models developed for dry weight biomass of the five species studied (Eqs. 2 to 4 and Eq. 6). When the Basic Model (Eq.2) was compared with the Basic M Model (Eq. 3) it was observed that aridity (M) was significant in three of the five species and in all three cases it resulted in an improvement in the Basic Model, both in terms of AIC and marginal and conditional R2. The species for which M was not significant in the models were Pt and Pp. Among the species for which M was significant, Ps and Ph showed the greatest increase in conditional and marginal R2, while a slightly negative effect was only detected in the case of Pn (Table 3).
Table 3 Coefficients estimated (a, b, m, c1, p1, c2, p2) and standard error (in brackets) for models from Eqs. 2 to 4 and Eq. 6, together the standard deviation of the random variable (StdRnd), Akaike Information Criterion (AIC) and marginal and conditional R2 (M.R2 and C.R2)
The estimates obtained for the coefficients c1 and p1 in the models that include vm indicate the high importance of this variable for estimating biomass weight. Nevertheless, its influence was less in the case of Pt, as reflected by its low p1 value (Fig. 2c, Table 3). The coefficients can be significant either as exponents or by multiplying the variables, or in both ways.
The selected model (Total Model), showing the dry weight biomass estimations for the target species (W, in Mg·ha− 1) according to: a volume of the stand for the target species (V, in m3·ha− 1); b Martonne aridity index (M, in mm·°C− 1); c mean tree volume (vm, in m3 per tree); and d stand form factor (f). The variable represented in each figure on the x axis, ranges from 1% to 99% of its distribution in the data used, while the rest of the variables remain constant and equal to: V = 150 m3·ha− 1; M = 30 mm·°C− 1; f = 0.5; and vm = 0.5 m3 per tree. Species as in Table 3
The bias observed when fitting the models was corrected by including the stand form factor f. When the Total Model and vm Model were compared, the bias correction was more clearly observed in the Ph model, while for Ps and Pt the inclusion of f only had a slight effect (Table 3).
When the estimation errors were analyzed using the different models (Table 4) it was observed that the bias was always less than 0.2 Mg·ha− 1, which in relative terms is equivalent to less than 3%. In general, the models overestimated the biomass weight (negative ME), although for Ph and Pp all the fitted models overestimated the biomass, except the Total Model for Pp. In addition, Pn and Pp were the species for which the greatest reduction in RMSE was observed, comparing the Total Model and Basic Model (greater than 4.5%), while this reduction was the lowest for Pt (around 0.06%).
Table 4 Model errors calculated through Eqs. 7 to 12
Having selected the Total Model as the best model to estimate the dry weight biomass for all species, the influence of each independent variable was analyzed. In Fig. 2, the variation of dry weight biomass with each variable was presented, assuming the rest of the variables not represented on the axis remain constant. Figure 2a shows a clear positive relationship between dry weight biomass and stand volume, with Pp being the species producing the highest stand biomass for a given volume, although it was very similar to Ph and Pn. If stand volume (V) is considered constant, it is possible to analyze the variation in W with aridity (Fig. 2b), observing that for all species where M was included in the model (Ps, Ph and Pn) the relationship was positive, that is, the higher the M value (less aridity), the higher the W value for a given V. Furthermore, the effect of aridity on this biomass-volume relationship varied according to the species, with Ps being the species for which this influence was the greatest (Fig. 2b, Table 3). Analyzing the dry weight biomass variation according to vm (Fig. 2c), it was observed that the tendency of the relationship between W and vm was similar for Pp, Pn and Ps, that is, the higher the mean tree volume, the lower the W estimated for a given V. An increase in vm, for a constant V, indicates that the stand is composed of a smaller number of larger trees whereas a decrease in vm indicates that the same stand volume comprising a greater number of smaller trees. Figure 2c shows that the vm effect is more evident when trees are smaller, while the relationship tends to be more constant as the size of trees increases. Note that for Pt and Ph, the vm effect was opposite to that for the other studied species, that is, positive. Figure 2c shows this effect clearly for Ph, despite being the species with the lowest range of vm variation, while for Pt, the influence of vm was only slight, despite being one of the species with the highest range of variation of this variable. As regards the stand form factor (f), in general, W decreased as f approached the unit value (Fig. 2d), although in the case of Pt there is a very slight positive effect of f. The influence of f on W was not decisive for Ps and Pn, while it was especially important for Pp and Ph.
Biomass expansion factors
According to the fitted models, the BEF, i.e. stand biomass weight/stand volume, is not constant but rather decreases as the stand volume increases. Figure 3 represents the species BEF variation within the inter-percentile 5%–95% range of the species stand volume in monospecific stands for the mean and the extreme values of each of the independent variables in the Total Model. For all species, the estimated BEF values generally varied between 0.5 and 1.5 Mg·m− 3, and the lowest estimations were found for Pt, for which the BEF values were almost constant and around to 0.75 Mg·m− 3. In contrast, the species for which the highest BEF was obtained was Pp, when f or vm had lower values. BEF estimations for this species could reach values of more than 1.5 Mg·m− 3 for low stand volume.
Variation of biomass expansion factor (BEF), defined as dry weight biomass (W, in Mg·ha− 1) estimated from the Total Model, divided by stand volume (V, in m3·ha− 1), for different values of: Martonne aridity index (M, in mm·°C− 1); stand form factor (f); and mean tree volume (vm, in m3 per tree). The lines are drawn within the inter-percentile 5%–95% range of stand volume distribution. Solid lines represent the mean value of the variable for each species and dashed and dotted lines represent the 5% percentiles, the mean 95% of the variable distribution for each species
Figure 3 shows that the BEF of Pt was always lower than 0.9 and was not influenced by M and hardly affected by vm or f. The BEF values presented little variation in the M range distribution for any of the pine species studied, despite being a statistically significant variable. However, it can be seen in Fig. 3 that Ps was the species most affected by aridity. In contrast, the BEF variation for different vm values was evident (Fig. 3), being the variable that produced the most change in BEFs for Ps and Pn, although it also affected Pp. Highly variable BEFs values can be observed for Pp and Ph within the f range distribution of the species, while for Ps and Pt this relationship was practically insignificant. If the different species are compared, Pn shows more constant BEF values than the other species, regardless of stand volume.
The results confirmed that the Total Model was also that which gave the lowest bias when carbon predictions were update to time T in the pine stands across peninsular Spain (Fig. 4). This model allowed carbon estimates with lower errors, both in absolute and relative terms, than the rest of the models, despite all the assumptions described, that is, constant values for both the number of trees per hectare and stand form factor in the elapsed interval considered.
Mean errors for carbon estimates at plot level for the studied pine species throughout peninsular Spain according the four studied models. ME, mean error (in Mg·ha− 1 of C); MAE, mean absolute error (in Mg·ha− 1 of C); RMSE, Root mean square error (in Mg·ha− 1 of C); MPE, mean percentage error (in %); MAPE, mean absolute percentage error (in %); RMSE, Root mean square percentage error (in %)
In Fig. 4, it can be seen that all models produced overestimations of carbon stocks, except the Total Model, which produced the lowest bias, although it slightly underestimated carbon stock. Figure 4 also shows that the inclusion of the f variable scarcely modified the errors (MAE, RMSE, MAPE and RMSPE), although the bias decreased significantly. When the Total Model was used, the RMSE obtained when making carbon stock predictions for the studied pine species in the Iberian Peninsula was less than 20%, which is slightly higher than 9 Mg·ha− 1 of C. This Total Model resulted in an important reduction in the bias, reaching around 2%.
The use of BEFs to estimate biomass at stand level provides an interesting alternative for predicting biomass and carbon stocks in forest systems since stand volume (V) is the only variable required. However, the use of traditional BEFs, mainly as constant values and generally obtained for stands under specific conditions, can result in biased biomass estimates if they are applied under different conditions (Di Cosmo et al. 2016). These biases can have a significant impact on estimated carbon in the tree layer when large-scale estimates are made, as is the case of national-scale predictions (Zhou et al. 2016). In this study, stand biomass models have been developed that include other easily obtained variables as independent variables, in addition to the stand volume. The fitted models allow us to update the carbon stocks in pine forests across mainland Spain for the five species studied using SNFI data. The strong relationship between stand biomass and stand volume (Fang et al. 1998) implies that the Basic Model can provide a good first estimate of biomass. This is confirmed by the results obtained as the Basic Model yields good fit statistics. This suggests that, to a certain extent, the stand volume should absorb the effects of other variables, such as the stand age or stand density, as well as environmental conditions (Fang et al. 2001; Guo et al. 2010; Tang et al. 2016). Therefore, in the development of the different models, the structure of the Basic Model was maintained, expanding its coefficients so that if the specific coefficients corresponding to the effects of M, vm and f were not significant, the Basic Model is returned. However, the models improved for all species with the inclusion of the other variables (Tables 3 and 4), reflecting the fact that stands with the same volume can have different structures leading to different biomass. This is observed in the improvement achieved with the Total Model, both with regard to the goodness of fit of the model and the errors (Tables 3 and 4), indicating less biased and more accurate estimates when the stand characteristics and the aridity conditions (M) are included.
The positive relationship found between the aridity index M and the dry biomass W for a given stand volume supports the findings presented by Aguirre et al. (2019), who reported higher productions in less arid conditions. This positive relationship between M and W suggests greater crown development and higher crown biomass for the same volume in less arid conditions. However, it is important to highlight that the individual tree biomass equations used did not consider this type of within-tree variation in the distribution of biomass with site conditions (Ruiz-Peinado et al. 2011). Hence, the observed effect of M must be associated with changes in the stand structure. For example, the variation in vm according to the aridity conditions, that is, the stand V is distributed over more trees of smaller size or fewer larger trees according to the aridity of the site, since the proportion of crown biomass with respect to total biomass varies with tree size (Wirth et al. 2004; Menéndez-Miguélez et al. 2021). This would entail an interaction between the effect of M and the effect of vm in the models, as reflected in the case of Pn, which varies from negative in the basic model with M to positive for the vm Model and Total Model. However, in general, M is not the most important variable to explain the variation in W (Fig. 2b), as can also be observed in the small BEF variation for the studied species in relation with M (Fig. 3).
The variable vm, as surrogate of the stand development stage, has a different influence on the models for Ph and Pt than for the rest of the species (Fig. 2c). The observed pattern for Ps, Pp and Pn indicates that the relationship between W and V, or the BEF, decreases with vm, i.e. as the stage of stand development increases, as has been observed previously in other studies (Lehtonen et al. 2004; Teobaldelli et al. 2009). This behavior may be caused by differences in the relationship between the components of the trees. For example, Schepaschenko et al. (2018) observed an important decreasing effect of age on the branch and foliar biomass factors. Similarly, Menéndez-Miguélez et al. (2021) analyzed the patterns of crown biomass proportion with respect to total aboveground biomass of the tree as its size develops for the main forest tree species in Spain. These authors found that in the cases of Ps and Pp, this pattern was decreasing; while for Pn and Pt it was constant (the study did not include Ph). These within-tree biomass distributions would validate the patterns found in the Ps, Pp and Pt models, but not the Pn model. However, Ph presents a totally different BEF behavior with the variation in vm. Analyzing the modular values of the different biomass fractions for this species presented in Montero et al. (2005), it can be observed that the proportion of crown biomass in this species increases slightly with the size of the tree, which could explain the opposite pattern observed in this species. However, this difference could also be due to the equations used to calculate the biomass (Ruiz-Peinado et al. 2011), since the maximum normal diameter of the biomass sample used in that study was 44 cm, whereas for the Iberian Peninsula as a whole it was as much as 97 cm (Villanueva 2005). Schepaschenko et al. (2018) also reported that the number of branches in low productive, sparse forest is greater than in high productive, dense forests, which may be a cause for the increasing tendency of W in Ph in relation to vm.
The results indicate an improvement in the models with the inclusion of the stand form factor, although the magnitude of the effect caused by this variable, as well as the improvement in the models, were greater for Pp and Ph than for the rest of the species (Fig. 2d, Table 3). To estimate the stand volume, diameter at breast height, total height of the tree and its shape are used, according to species and province available models (Villanueva 2005). However, to estimate stand biomass, the equations applied for the different tree components only depend on the species, the diameter at breast height and the total height of the tree, without considering the shape of the tree (Ruiz-Peinado et al. 2011). This difference explains the advisability of considering the stand form factor to avoid biases in the estimates, although it also highlights the need to study the dependence of the biomass equations on the different components of the tree according to their shape. In turn, this shape depends on genetic factors, environmental conditions, and stand structure (Cameron and Watson 1999; Brüchert and Gardiner 2006; Lines et al. 2012).
The models obtained underline the importance of considering the environmental conditions and the stand structure (size and shape of trees) when expanding the volume of the stand to biomass. If constant BEF values are used for all kinds of conditions, biomass may be underestimated in younger and less productive stands, while for more mature and/or productive stands it may be overestimated (Fang et al. 1998; Goodale et al. 2002; Yu et al. 2014). These authors also highlight the need to further our understanding of the influence of these factors on the individual tree biomass equations. In this regard, Forrester et al. (2017) found that the intraspecific variation in tree biomass depends on the climatic conditions and on the age and characteristics of the stand, such as basal area or density. The components that mostly depended on these variables were leaf and branch biomass, which suggests that it would be advantageous to have more precise equations for these tree components, which would therefore modify the stand biomass estimates. However, the inclusion of other variables in the tree biomass models in order to improve the accuracy would require a large number of destructive samples from trees under different conditions (site conditions, stand characteristics, age...), which would be difficult to obtain in most cases.
The suitability of SNFI data to develop models has been questioned by several authors (Álvarez-González et al. 2014; McCullagh et al. 2017). One of the main disadvantages is the lack of control about environmental conditions, stand age or history of the stand (Vilà et al. 2013; Condés et al. 2018; Pretzsch et al. 2019). Another shortcoming is the lack of differentiation of pine subspecies in the SNFI, like the two subspecies of Pn, salzmanii and nigra, or those of Pt, atlantica and mesogeensis, which could lead to confusing results such as those obtained for Pt, which was the only species for which the Basic Model improved with the inclusion of both variables together, vm and f. This could suggest that the relationship between volume and shape of trees differs according to the subspecies considered.
Through the models developed (Fig. 4), it is possible to provide more precise responses to the international requirements in terms of biomass and carbon stocks. Since the most recent SNFI, it has become possible to update the information at a required time. For this purpose, the least favourable situation was assumed, that is, that the only information available was that obtained from the most recent SNFI. However, the main limitation of the models developed is that they are only valid for a short time period, when the assumptions made can be assumed and when both climatic conditions and stand management do not vary (Peng 2000; Condés and McRoberts 2017). If the elapsed time would be too long for assuming that there is not mortality and that the stand form factor does not vary, the basic model could be applied. Furthermore, to achieve more precise updates, natural deaths and silvicultural fellings must be considered using scenario analysis or by estimating of past fellings (Tomter et al. 2016). Besides, a proper validation with independent data was not possible due to lack of such data. When the SNFI-4 is finished for all Spanish provinces, it would be interesting to validate the models developed.
The results reveal the importance of considering both, site conditions and stand development stage when developing stand biomass models. The inclusion of site conditions in the models for Ps, Ph and Pn, indicate that aridity conditions modulate the relationship between the dry weight biomass of a stand (W) and its volume (V), while for Pp and Pt this relationship was not influenced. As hypothesized, it was observed that for a lower aridity, the biomass weight and therefore that of carbon are higher for the same stand volume.
Besides, the results reveal the importance of considering both size and form of trees for estimating dry weight biomass, and therefore to estimate carbon stock. As expected, the relationship between dry weight biomass of the stand and its volume decreases when the stand development stage (vm) increases, except for Ph whose behavior is the opposite, and Pt which is hardly affected by vm. However, the inclusion of this variable reduces the ME, MAE and RMSE for all the studied species, which indicates the importance of its consideration in the dry weight biomass estimation.
The raw datasets used and/or analyzed during the current study are available from Ministerio para la Transición Ecológica y el Reto Demográfico of the Government of Spain (https://www.mapa.gob.es/es/desarrollo-rural/temas/politica-forestal/inventario-cartografia/inventario-forestal-nacional/default.aspx).
NFI:
National Forest Inventory
SNFI:
Spanish National Forest Inventory
BEFs:
Pinus pinea
Pinus halepensis
Pinus nigra
Pinus pinaster
M :
Martonne aridity index
vm :
Mean tree volume
W :
Dry weight biomass
Stand form factor
C :
Carbon weight
sp :
Referred to the target species
T :
Any time when no field data is available
Initial NFI survey
Final NFI survey
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This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.
Department of Natural Systems and Resources, School of Forest Engineering and Natural Resources, Universidad Politécnica de Madrid, Madrid, Spain
Ana Aguirre & Sonia Condés
INIA, Forest Research Center, Department of Forest Dynamics and Management, Madrid, Spain
Ana Aguirre, Miren del Río & Ricardo Ruiz-Peinado
iuFOR, Sustainable Forest Management Research Institute, University of Valladolid and INIA, Valladolid, Spain
Miren del Río & Ricardo Ruiz-Peinado
Ana Aguirre
Miren del Río
Ricardo Ruiz-Peinado
Sonia Condés
Condés, del Río, and Ruiz-Peinado developed the idea, Aguirre and Condés developed the models, Aguirre programmed the models, and all authors wrote the document. All authors critically participated in internal review rounds, read the final manuscript, and approved it.
Correspondence to Ana Aguirre.
Aguirre, A., del Río, M., Ruiz-Peinado, R. et al. Stand-level biomass models for predicting C stock for the main Spanish pine species. For. Ecosyst. 8, 29 (2021). https://doi.org/10.1186/s40663-021-00308-w
Carbon stock
Peninsular pine forest
Biomass expansion factor | CommonCrawl |
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EJNMMI Physics
Positron range in combination with point-spread-function correction: an evaluation of different implementations for [124I]-PET imaging
Hunor Kertész ORCID: orcid.org/0000-0002-4626-930X1,
Maurizio Conti2,
Vladimir Panin2,
Jorge Cabello2,
Deepak Bharkhada2,
Thomas Beyer1,
Laszlo Papp1,
Walter Jentzen3,
Jacobo Cal-Gonzalez1,4,
Joaquín L. Herraiz5,6,
Alejandro López-Montes5 &
Ivo Rausch1
EJNMMI Physics volume 9, Article number: 56 (2022) Cite this article
To evaluate the effect of combining positron range correction (PRC) with point-spread-function (PSF) correction and to compare different methods of implementation into iterative image reconstruction for 124I-PET imaging.
Uniform PR blurring kernels of 124I were generated using the GATE (GEANT4) framework in various material environments (lung, water, and bone) and matched to a 3D matrix. The kernels size was set to 11 × 11 × 11 based on the maximum PR in water and the voxel size of the PET system. PET image reconstruction was performed using the standard OSEM algorithm, OSEM with PRC implemented before the forward projection (OSEM+PRC simplified) and OSEM with PRC implemented in both forward- and back-projection steps (full implementation) (OSEM+PRC). Reconstructions were repeated with resolution recovery, point-spread function (PSF) included. The methods and kernel variation were validated using different phantoms filled with 124I acquired on a Siemens mCT PET/CT system. The data was evaluated for contrast recovery and image noise.
Contrast recovery improved by 2–10% and 4–37% with OSEM+PRC simplified and OSEM+PRC, respectively, depending on the sphere size of the NEMA IQ phantom. Including PSF in the reconstructions further improved contrast by 4–19% and 3–16% with the PSF+PRC simplified and PSF+PRC, respectively. The benefit of PRC was more pronounced within low-density material. OSEM-PRC and OSEM-PSF as well as OSEM-PSF+PRC in its full- and simplified implementation showed comparable noise and convergence. OSEM-PRC simplified showed comparably faster convergence but at the cost of increased image noise.
The combination of the PSF and PRC leads to increased contrast recovery with reduced image noise compared to stand-alone PSF or PRC reconstruction. For OSEM-PRC reconstructions, a full implementation in the reconstruction is necessary to handle image noise. For the combination of PRC with PSF, a simplified PRC implementation can be used to reduce reconstruction times.
Positron emission tomography (PET) is based on the detection of photon radiation arising from the annihilation of positrons originating from the PET tracers with electrons within the patient [1]. To obtain a 3D representation of the tracer distribution, image reconstruction is necessary. This is done today using iterative reconstruction algorithms where the detector signals are estimated from a guess of the tracer distribution using a mathematical model of the system (system matrix) and compared to the actual measured signal in projection space. Standard reconstruction algorithms, such as OSEM, typically use simple geometric assumptions to model the system matrix [2]. However, to optimize spatial resolution a more realistic model of the system accounting for all physical processes is needed. A realistic system response is, for example, derived from point source measurements [3]. The implementation of such a realistic detector response function into the projectors of the iterative image reconstruction process is called resolution recovery or point-spread-function (PSF) correction. The benefit of including PSF in the image reconstruction by means of reduced image noise and increased contrast was shown by various studies [4,5,6,7].
However, besides the imaging systems properties, spatial resolution in PET is further limited by positron range (PR), the distance traveled by the positron from the emission position to the annihilation position [8, 9]. For the most commonly used PET isotope (18F), median PR in soft tissue or water is of the order of 0.4 mm and thus [10] can be practically ignored in clinical PET studies. However, positron emitters with high positron energies such as 68Ga and 124I, which are increasingly applied in the diagnosis of prostate cancer [11] or used for therapy planning in patients with thyroid cancer [12], respectively, present with average PRs of ~ 10 mm in soft tissues which lead to substantial degradation of spatial resolution and image quality [13].
Positron range effects can be corrected with a similar implementation to the one used to incorporate PSF into the image reconstruction algorithm. This is done by adding the PR-induced blurring to the forward projector, for example, by convolving the current image guess with the PR-based blurring kernels before projecting the current image guess into projection space. In addition, the PR blurring has to be taken into account in the back-projection step within an iterative reconstruction since the back-projection is a transpose operation of the forward projection [14]. The main challenges of PRC are deriving adequate PR kernels and extensive computational demand due to the use of non-isotropic and non-shift-invariant PR kernels. Monte Carlo simulations are usually used for defining the ground truth of tissue-dependent PR kernels [10].
However, due to the computational challenges, PR kernels are generally based on simply using isotropic PR kernels for different tissue types [15,16,17]. Only a limited number of studies used spatially variant and tissue-dependent estimations of PR kernels including effects on tissue borders [15, 18]. These studies suggest implementing the PR blurring kernels only prior to the forward projection step in standard iterative reconstructions as an image blur to reduce the computational demand [15, 19]. Only one recent study showed the value of using a full implementation of the PRC into an OSEM algorithm [14].
Therefore, our study aimed to evaluate the effect of combined PSF correction and PRC within the reconstruction on image contrast and convergence. Further, the influence of using simplified PR correction implementation methods limiting computational demand in combination with PSF was assessed.
Positron range kernel calculation
The uniform positron range distribution of 124I was simulated using GATE 9.0 (GEANT4 10.06.p02). The simulation setup consisted of a point source with a radius of 5 nm centered in a uniform phantom with a radius of 30 cm. For the phantom material, three settings were simulated bone material (mass density 1.92 g/cm3), water (1.00 g/cm3), and lung equivalent material (0.26 g/cm3). The predefined "empenelope" physics list was used. The initial activity of 124I was 10 MBq and the emission and annihilation coordinates were recorded for 20 M annihilation events. From the recorded data, the 3D PR distribution was generated by mapping the distribution to a 3D matrix. The size of the kernel (11 × 11 × 11 voxels) was chosen based on the voxel size of the Siemens mCT PET/CT system (2 × 2 × 2 mm3) (Siemens Medical Solutions Inc., Knoxville, TN, USA) [20] and the maximum PR in water for 124I (~ 10 mm), since the use of larger kernels is leading to artifacts [14].
For each voxel of the investigated object, spatially variant and tissue-dependent PR kernels based on the underlying material distribution were created as described before [14]. In short, the attenuation map was segmented into a material map containing lung, soft tissue, and bone. Regions containing air were treated as lung material. Then, for every voxel, PR kernels were generated by combining pre-calculated uniform PR kernels for lung, water, and bone from GATE Monte Carlo simulations.
Positron range correction implementations
The PR correction was implemented into the vendor-based software in combination with OSEM and PSF algorithms as an additional PR-dependent PSF applied in image space. This was done as full implementation and, in addition, in a simplified version applying the PR-dependent blurring only before the forward projector of the reconstruction algorithm, as suggested by [19] to reduce computational demand.
The standard OSEM algorithm taken as the basis can be written as:
$$f_{j}^{n + 1} = \frac{{f_{j}^{n} }}{{\sum\nolimits_{{i^{\prime} \in S^{n} }} {a_{i^{\prime}j} } }}\sum_{{i \in S^{n} }} a_{ij} \frac{{m_{i} }}{{\sum_{k} a_{ik} f_{k}^{n} }}$$
where fjn+1 is the next image estimate of voxel j based on the current image estimate fkn. The measured projection data is mi. The system matrix describing the probability that the emission from voxel j will be detected along the line of response (LOR) i is given by Aij = (aij)IxJ. Only a subset Sn of the data was used in each update. Of note: for simplification, the random and scatter are not included in this description. The system matrix can be factorized to account for the finite resolution effects, in our case the positron range effect by matrix H = (hj'j)JxJ, a matrix X = (xij)IxJ expressing the intersection length and a matrix W = (wii)IxI describing the photon attenuation and geometrical sensitivity variations. Thus, the system matrix can be written as:
$$A =WXH$$
Combining Eq. (1) and Eq. (2), the OSEM algorithm with PRC correction can be written as [14]:
$${f}_{j}^{n+1}=\frac{{f}_{j}^{n}}{{\sum }_{b}{h}_{bj}{\sum }_{i\in {S}^{n}}{w}_{ii}{x}_{ib}}{\sum }_{b}{h}_{bj}{\sum }_{i\in {S}^{n}}{x}_{ib}\frac{{m}_{i}}{{\sum }_{p}{x}_{ip}{\sum }_{v}{h}_{pv}{f}_{v}^{n}}$$
This method is referred to as "PRC" in this work. Since PRC is computationally highly expensive, a simplified version of the PRC implementation was suggested in previous studies [18, 21], where the PR kernels are applied to the current image estimate as a convolution in image space before the forward projection. In this way, Eq. (1) can be written as:
$$f_{j}^{n + 1} = \frac{{f_{j}^{n} }}{{\sum\nolimits_{{i^{\prime} \in S^{n} }} {a_{i^{\prime}j} } }}\sum_{{i \in S^{n} }} a_{ij} \frac{{m_{i} }}{{\sum_{k} a_{ik} f_{k}^{{n^{ \sim } }} }}$$
where \(f_{k}^{{n^{ \sim } }}\) is the current image estimate blurred with the spatially variant and tissue-dependent PR kernel (\(\rho\)) calculated for the given voxel [19]:
$$f_{k}^{{n^{ \sim } }} = f \otimes \rho = \frac{{\sum_{h} f_{k} - \rho_{h} }}{{\mathop \sum \nolimits_{h}^{{}} \rho_{h} }}$$
This method is referred to as "PRC simplified" in this study. Both PRC methods (PRC and PRC simplified) together with the generated PR kernels were implemented into the Siemens e7tools (Siemens Medical Solutions USA, Inc., Knoxville, TN, USA) image reconstruction framework using MATLAB R2019a (MathWorks Inc, USA). The image reconstructions with both PRC implementations were performed with both OSEM and PSF corrected OSEM (PSF): OSEM algorithm with PR implemented in the forward projection only (OSEM+PRC simplified), OSEM with PRC implemented in both forward- and back-projection (OSEM+PRC). The same naming convention was used for PSF reconstructions (PSF+PRC simplified and PSF+PRC).
All reconstructions were done with 1–10 iterations and 12 subsets, except the reconstruction for which PSF was combined with PRC in the back-projection (PSF+PRC). In this case, image reconstructions were performed with 1–20 iterations and 12 subsets. The image size was 400 × 400 x 109 voxels, with a voxel size of 2 × 2 × 2 mm3. All emission data was corrected for attenuation, normalization, scatter, and randoms. The time-of-flight (TOF) information was included and no post-reconstruction filters were applied.
Comparison of the PRC implementations
The PRC implementations were tested and compared by means of contrast, noise, and convergence properties using the standard "NEMA Image Quality (IQ)" phantom, a modified NEMQ IQ phantom with changed hot spheres referred to as "Small-tumor" phantom and a "Bone–lung" phantom. Of note, the NEMA IQ phantom simulates hot lesions embedded in soft tissue (water) only (Additional file 1: Fig. S1a). The Small-tumor phantom is a specially-modified phantom using the housing of the NEMA IQ phantom, with fillable spheres with diameters of 3.7, 4.8, 6.5, 7.7, 8.9, and 9.7 mm (Additional file 1: Fig. S1b). The dedicated Bone–lung phantom simulates hot lesions embedded in lung and bone mimicking material, and thus, allows a comparison of the PRCs in a tissue-dependent environment (Additional file 1: Fig. S1c). All phantom acquisitions were performed at the University Clinic Essen, Germany, using the Siemens mCT PET/CT system [20].
NEMA IQ
In the NEMA IQ phantom, all the spheres (diameter of 10, 13, 17, 22, 28, and 37 mm) were filled with a 124I activity concentration of 30 kBq/ml, and the background region was filled with an activity concentration of 6 kBq/ml. The emission acquisition time was 60 min. The phantom was positioned with the centers of the spheres in the center of FOV. For the evaluation, 12 volume-of-interests (VOIs) were used in the background region (each with a diameter of 37 mm) and 6 individual VOIs covering every sphere (with the diameter corresponding to the sphere size) (Additional file 1: Fig. S1d). For placing the background VOIs, the center of the FOV slice was selected, where all the spheres are centered as well. The reconstructed images were evaluated for contrast recovery calculated as:
$${\text{Contrast}}\; {\text{recovery}} = \frac{{\frac{{{\text{MEAN}}_{{{\text{signal}}}} }}{{{\text{MEAN}}_{{{\text{background}}}} }}}}{{{\text{Activity }}\;{\text{ratio}}}}$$
where the mean value in the background was calculated as the mean overall VOIs.
Image noise was defined in percentage following the guidelines of EFOMP [22]:
$${\text{Noise}} = \frac{{{\text{STDEV}}_{{{\text{background}}}} }}{{{\text{MEAN}}_{{{\text{background}}}} }}*100$$
The convergence of the investigated reconstruction algorithms in combination with the various PRC implementations was assessed by plotting the contrast recovery vs. noise and how fast contrast recovery and noise is changing with each iteration [14, 23]. For a direct comparison of the reconstructed images, the reconstruction settings were selected to match a 10% background noise as defined as a clinical acceptable noise level as gained from phantom scans [22].
Small-tumor
The Small-tumor phantom was filled with a 124I activity concentration of 25 kBq/ml in the hot lesions and a background activity of 1.2 kBq/ml leading to a signal-to-noise ratio of 20:1. The acquisition time was 30 min [7]. The reconstructed images were evaluated for contrast recovery (Eq. 5) and image noise (Eq. 6) by defining 10 VOIs in the background region (diameter of 20 mm) and individual VOIs for every sphere (with the diameter corresponding to the sphere size) (Additional file 1: Fig. S1e).
Bone–lung
The Bone–lung phantom was composed of 3 different cylinders (each with a diameter of 50 mm) made of different materials (lung with -800 HU, bone with 500 HU, and 1000 HU) that are housed within the standard NEMA IQ phantom [24]. Within every cold cylinder, two fillable spheres were inserted, one with a diameter of 8.5 mm and one with 19.4 mm. The spheres were filled with a 124I activity concentration of 30 kBq/ml and the background with 6 kBq/ml. Similar to the NEMA IQ evaluations, 6 background VOIs (diameter of 37 mm) and VOIs covering the hot spheres (diameter of 8.5 and 19.4 mm) were defined (Additional file 1: Fig. S1f). The acquisition time was 60 min, and the reconstructed images were evaluated for contrast recovery (Eq. 5) and noise (Eq. 6).
NEMA IQ phantom
Substantial differences in convergence, image contrast, and noise properties were noticed between the different PRC implementations.
In general, three different convergence patterns were observed that can be described as follows: OSEM+PRC simplified had similar convergence as OSEM with higher achievable contrast but also noise levels (Fig. 1). In both cases, 2 iterations and 12 subsets resulted in the background noise of approximately 10%.
Noise versus contrast recovery for the NEMA IQ phantom: (a) 10 mm, (b) 13 mm, (c) 17 mm, (d) 22 mm, (e) 28 mm and (f) 37 mm spheres filled with 124I and reconstructed with OSEM (1–5 iterations), OSEM+PRC simplified (1–3 iterations), OSEM+PRC (1–10 iterations), PSF (1–10 iterations), PSF+PRC simplified (1–10 iterations), and PSF+PRC (1–20 iterations). Every point corresponds to one iteration (all reconstructions done with 12 subsets). The behavior of the OSEM+PRC is similar to the stand-alone PSF reconstructions. The selected number of iterations for the comparison of the images is marked with red
OSEM+PRC and PSF showed similar convergence, contrast, and noise level with improved contrast and noise properties, albeit with a slower enhancement of noise (the 10% noise level was reached with 8 iterations and 12 subsets) compared to OSEM and OSEM+PRC simplified (Fig. 1). PRC in combination with PSF resulted in similar convergence and contract recovery for the full and simplified implementation; however, convergence was notably slower by means of gain in contrast per iteration for PSF+PRC compared to PSF+PRC simplified (Fig. 1).
The visual inspection of the reconstructed NEMA IQ phantoms is shown in Fig. 2. With OSEM+PRC and PSF-PRC simplified, Gibbs artifacts were introduced, which became more pronounced for PSF+PRC (Fig. 3). Regardless of the PRC implementation, the effect was more pronounced for the smaller spheres (10, 13, and 17 mm).
Comparison of the central axial slice of the reconstructed (top) NEMA IQ phantom using: (a) OSEM; (b) OSEM+PRC simplified; (c) OSEM+PRC; (d) PSF; (e) PSF+PRC simplified; and (f) PSF+PRC. The images with similar noise levels measured in the background are marked with dashed boxes. These noise levels were achieved using 2 iterations for OSEM and OSEM+PRC simplified, 8 iterations from OSEM+PRC and PSF, 7 iterations for PSF+PSF simplified, and 15 iterations for PSF+PRC. All reconstructions were with 12 subsets. The OSEM+PRC is producing similar images then the stand-alone PSF; however, more pronounced Gibbs artifacts are produced. (bottom) Small-tumor phantom using: (g) OSEM; (h) OSEM+PRC simplified; (i) OSEM+PRC; (j) PSF; (k) PSF+PRC simplified, and l) PSF+PRC. The images with similar noise levels measured in the background (~ 33%) are marked with dashed boxes. These noise levels were achieved using 2 iterations for OSEM and OSEM+PRC simplified, 9 iterations from OSEM+PRC, 8 iterations for PSF, 7 iterations for PSF+PSF simplified, and 18 iterations for PSF+PRC All reconstructions were with 12 subsets
Comparison of the reconstructed (a) NEMA Image Quality phantom with OSEM (2 iterations and 12 subsets), OSEM+PRC simplified (2 iterations and 12 subsets), OSEM+PRC (8 iterations and 12 subsets), PSF (8 iterations and 12 subsets), PSF+PRC simplified (7 iterations and 12 subsets), and PSF+PRC (15 iterations and 12 subsets) and (b) Small-tumor phantom with OSEM (2 iterations and 12 subsets), OSEM+PRC simplified (2 iterations and 12 subsets), OSEM+PRC (9 iterations and 12 subsets), PSF (8 iterations and 12 subsets), PSF+PRC simplified (7 iterations and 12 subsets) and PSF+PRC (18 iterations and 12 subsets). For the NEMA IQ phantom, the highest differences (%) seen were for the smaller spheres 10, 13, and 17 mm and the edge of the phantom
Table 1 summarizes the lesion contrasts with all PRC implementations for a clinically acceptable image noise of ~ 10%. For subsequent evaluations, those reconstruction settings were selected that yielded a background noise level of ~ 10% following the convergence analyses (Fig. 1).
Table 1 Recovery coefficient for the different image reconstruction and kernel combinations. The image reconstruction settings were defined to match the background noise around ~ 10% for the NEMA IQ phantom and ~ 33% for the Small-tumor phantom. Relative deviations (%) apply relative to the OSEM and for PSF reconstructions
Small-tumor phantom
For this phantom, similar convergence patterns were observed as in the case of the NEMA IQ phantom (Fig. 4). The effect of PRC was noticeable for hot lesions down to 4.8 mm. When comparing images with similar noise levels (~ 33%) (Fig. 2), OSEM+PRC improved contrast recovery by 80% and 86% for the 4.8 and 6.5 mm spheres, respectively (Table 1), which is also clearly seen visually (Fig. 3b). Table 1 summarizes the contrasts improvements with all PRC implementations for a comparable image noise level of ~ 33%.
Noise vs. contrast recovery for the Small-tumor phantom (a) 3.7 mm, (b) 4.8 mm, (c) 6.5 mm, (d) 7.7 mm, (e) 8.9 mm, 9.7 mm spheres filled with 124I and reconstructed with OSEM (1–5 iterations), OSEM+PRC simplified (1–3 iterations), OSEM+PRC (1–10 iterations), PSF (1–10 iterations), PSF+PRC simplified (1–10 iterations), and PSF+PRC (1–20 iterations). Every point corresponds to one iteration (all reconstructions done with 12 subsets). The behavior of the OSEM+PRC is similar to the stand-alone PSF reconstructions. The selected number of iterations for the comparison of the images are marked with red
Bone–lung phantom
The convergence of the reconstructions was highly affected by the surrounding medium. Within the bone inserts, almost no change in contrast improvement was achieved for either the small or big spheres with the OSEM+PRC simplified (Fig. 5). The convergence was similar for OSEM+PRC, PSF, and PSF+PRC reconstructions. Within the lung medium, the effect of PRC was clearly visible with substantially improved contrasts (Table 2). The OSEM+PRC simplified showed similar convergence as OSEM with a substantially higher contrast recovery. The trend was similar with OSEM+PRC and PSF reconstruction with reduced image noise. PSF+PRC simplified and PSF+PRC had almost identical convergence, with the simplified implementation leading to convergence substantially faster. The effect of PRC within the lung medium is also clearly visible on the difference images shown in Fig. 6. Table 2 summarizes the contrast recoveries for all reconstruction and PRC combinations at comparable image noise (~ 10%).
Contrast recovery vs. noise calculated for the small sphere (8.5 mm) in (a) lung, (b) bone (500 HU), (c) bone (1000 HU) and for the large sphere (19.4 mm) in (d) lung, (e) bone (500 HU), (f) bone (1000 HU) reconstructed with PRC methods in combination with OSEM and PSF. The changes are most pronounced within the lung insert
Table 2 Contrast recovery for every reconstruction method and kernel variation using the Bone–lung phantom. Relative deviations (%) in relation to the standard OSEM and for the PSF reconstructions
Comparison of the reconstructed Bone–lung phantom with (a) OSEM and (b) PSF the central slices are shown through the small (8.5 mm) and (b) the central slices are shown through large (19.4 mm) spheres. The recovery is slightly overestimated for the 8.5 mm sphere within the lung medium. The black dashed lines represent the activity concentrations in the hot spheres and background
In this paper, different PRC implementation methods in combination with PSF image reconstruction were evaluated for PET imaging using 124I. The implementation of the PR-dependent blurring into the PET image reconstruction was done for the OSEM algorithm and for a PSF corrected OSEM as full and simplified implementation [14]. Both PRC implementations, simplified and full implementation, lead to increased contrast recovery compared to standard OSEM. However, the full implementations also led to a slower convergence of the algorithm although with higher achievable contrasts compared to with PRC simplified. The simplified implementation demonstrated a faster convergence although with substantially higher noise levels (Fig. 1). The reason for the faster convergence and the increased noise levels are expected due to result from the strong mismatch of the forward and backward projectors [3].
The image reconstruction with OSEM+PRC showed a similar convergence to the PSF reconstruction (Fig. 1). This can be explained by the similarity of the full implementation of the PRC and the vendor-based PSF image reconstructions [3].
When adding the PRC to the PSF reconstruction, the image contrast recovery was almost identical for the full and the simplified PRC implementation. This observation may be explained by the higher similarity of the forward and backward projector in the case of the combination of the PRC with PSF. This observation is also strengthened by the results for the OSEM-PRC simplified, which indicates that the mismatch of forward and backward projector substantially influences the noise propagation within iterative PRC implementations (Figs. 2 and 3).
The effect of PRC was the most pronounced within low-density materials, such as lung, where the reconstructions with PRC outperformed the standard reconstructions (Fig. 4). In this case, also OSEM+PRC was leading to higher contrast recovery compared to PSF (Fig. 5). On the other hand, within the bone medium, relatively small changes were seen for PRC-based reconstructions. This was expected due to the high positron range of 124I within the lung (max. 30 mm) and short PR in bone. Gibbs artifacts well known from PSF reconstructions were also present in the PRC reconstructions and more prominent when PSF and PRC were combined (Figs. 3 and 6). As shown in Fig. 6 in the line profile through the small spheres, a slight overestimation can be caused by the Gibbs artifacts. The effect is more visible in the edges of the bigger spheres. However, such Gibbs artifacts as known from PSF are mainly occurring at sharp activity concentration borders as present in phantom studies and are expected to be less pronounced in patients. Further, as shown for PSF reconstructions, such artifacts can be handled using post-reconstruction filters or by applying regularization methods [25, 26].
The composition of the spatially variant and tissue-dependent kernels does not take into account the positron energy loss, in particular when a positron is emitted from a higher density material. The solution for these limitations would require a more complex PR such as based on direct MC simulations; however, due to an extensive number of kernels that have to be computed during the image reconstruction process, the full PRC application is still a challenge especially with the aim of clinical application.
The data acquisition of the NEMA IQ phantom was 60 min. For obtaining similar count statistic as for a standard 15 min NEMA IQ scan with 18F given the reduced branching ratio for positrons in 124I. In the clinical scenario, the reconstruction settings may have to be adjusted to account for the reduced count statistics due to the shorter acquisition times.
The evaluated PRC methods are based on the same basic methodology of calculating the spatially variant and tissue-dependent kernels, by analyzing the underlying material composition from the AC maps [14]. Thus, the errors in the attenuation map directly translate into artifacts caused by the PRC. This holds also true in the case of PET/MRI, were artifacts in MR-based AC might occur more frequent that for PET/CT. Furthermore, when applying the evaluated methods for the PET/MRI, the effect of the magnetic field on the PR has to be carefully considered [15, 27].
Combining PSF and PRC notably increases achievable contrast compared to PSF or PRC alone in 124I PET and appears to be preferred for formal correctness, contrast recovery, and noise level. Our study showed that the use of a simplified PRC implementation applying the PR-based blurring exclusively to the forward projector within an OSEM algorithm, resulted in a seriously increase in image noise compared to the full implementation of PRC in both, forward- and back-projection. However, a simplified implementation of PRC in the forward projector appears acceptable, when combined with a PSF correction.
The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.
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The financial support of the Austrian Science Funds (FWF) Project I3451-N32 is gratefully acknowledged. The authors acknowledge the support of Siemens Medical Solutions USA, Inc. (Knoxville, TN, USA).
QIMP Team, Center for Medical Physics and Biomedical Engineering, Medical University of Vienna, Währinger Gürtel 18-20, 1090, Vienna, Austria
Hunor Kertész, Thomas Beyer, Laszlo Papp, Jacobo Cal-Gonzalez & Ivo Rausch
Siemens Medical Solutions USA, Inc., Knoxville, TN, USA
Maurizio Conti, Vladimir Panin, Jorge Cabello & Deepak Bharkhada
Clinic for Nuclear Medicine, University Hospital Essen, Essen, Germany
Walter Jentzen
Ion Beam Applications, Protontherapy Center Quironsalud, Madrid, Spain
Jacobo Cal-Gonzalez
Nuclear Physics Group and IPARCOS, Faculty of Physical Sciences, University Complutense of Madrid, Madrid, Spain
Joaquín L. Herraiz & Alejandro López-Montes
Health Research Institute of the Hospital Clínico San Carlos, Madrid, Spain
Joaquín L. Herraiz
Hunor Kertész
Maurizio Conti
Vladimir Panin
Jorge Cabello
Deepak Bharkhada
Thomas Beyer
Laszlo Papp
Alejandro López-Montes
Ivo Rausch
All authors contributed to writing, critically reviewing and approving the manuscript. Specific author contributions are as follows: HK implemented the PRC methods, did the image reconstructions, analyzed the data, and wrote the manuscript draft. MC, VP, and JC provided expertise on the PRC implementations, DB provided the image reconstruction tools, LP optimized the PRC implementation, WJ performed the phantom measurements, JH and ALM provided with help and expertise with the kernel calculations, TB, JCG, and IR designed, guided, and supervised the project. All authors read and approved the final manuscript.
Correspondence to Hunor Kertész.
No patient information/data was used in the manuscript.
Consent to participate
Dr. Maurizio Conti, Dr. Vladimir Panin, Dr. Jorge Cabello, and Dr. Deepak Bharkhada are employees of Siemens Medical Solutions USA, Inc., (Knoxville, TN, USA) and report no competing interests with this study.
Additional file 1
. Employed phantoms: a) NEMA IQ - Image quality phantom; b) Small tumor phantom and c) Bone-lung phantom. d-f) VOI delineations for the image analysis.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Kertész, H., Conti, M., Panin, V. et al. Positron range in combination with point-spread-function correction: an evaluation of different implementations for [124I]-PET imaging. EJNMMI Phys 9, 56 (2022). https://doi.org/10.1186/s40658-022-00482-y
Positron range correction | CommonCrawl |
Why do things move? It seems such a profound question for something that happens around us all the time. In fact, it's something that puzzled scientists and philosophers for hundreds of years. For example, Aristotle believed that all objects had a 'natural' place to which objects fell towards depending on which 'element' they were made out of. However, it wasn't until Newton's laws of motion that an understanding of forces as the causes of motion was systematically laid out. In this article, we're going to look at translational dynamics which analyses forces and how they produce a type of motion known as translational motion. Before looking at translational dynamics, let's recap Newton's law of motion.
Newton's Laws of Motion
Newton's laws of motion are three fundamental laws that describe the forces acting on an object and its motion. Sir Isaac Newton published his principles of motion in Philosophiae Naturalis Principia Mathematica on July 5, 1687. These simple laws served as the foundation of classical mechanics, and Newton himself utilized them to describe a wide range of phenomena relating to the motion of physical objects.
Newton's First Law
Newton's first law is concerned with the idea of inertia. It states that an object will remain at rest or in uniform (i.e at constant velocity) motion in a straight line unless it is acted upon by a net external force. It states that objects don't just start moving, or change velocity, of their own accord, there must be a force involved. Alternatively, we can summarise the law as 'no acceleration means there is no force.
Newton's First Law - An object will remain at rest or in uniform motion in a straight line (i.e at constant velocity) unless it is acted upon by a net external force.
It's important to remember that because velocity is a vector with a direction, changing direction is a form of acceleration. This is why the law states that an object 'remains in a straight line unless acted upon by an external force. This idea prompted Newton to suggest that the planets orbiting the sun experience the force of gravity.
Fig. 1 - A box on the ground will stay stationary until an external force is applied. This is because the weight of the box and the normal force cancel each other, meaning the net force is zero.
If an object is stationary this doesn't necessarily mean that no forces are acting on it, simply that the sum of all the forces, the net force, is zero. A stationary box is affected by both its weight and the normal force of the floor pushing up on it. Because their vector sum equals zero, these two forces cancel each other out, and the net force is zero. That's why the box doesn't just fall through the floor!
Newton's Second Law
Of all the three laws of motion, Newton's second law is the one that gives a direct mathematical relationship between the motion of an object and the force it experiences. It will probably be the single law of physics that you use most in all your studies.
Newton's Second Law - The resultant force acting on an object is equal to the product of the object's mass and its acceleration.
This is expressed mathematically by the famous equation:
\[F=ma\]
where \(F\) is the resultant or net force measured in Newtons \(\mathrm{N}\), \(m\) is the mass of the object in kilograms \(\mathrm{kg}\), and \(a\) is the acceleration of the object in meters per second squared\(\frac{\mathrm{m}}{\mathrm{s}^2}\).
If a box with a mass of 5 kg is pulled with a force of 25 N, what is the subsequent acceleration of the box?
According to Newton's second law:
\[\text{Resultant Force}=\text{mass}\cdot\text{acceleration}\]Here, the box is pulled with a force of 25 N, which is our resultant force, and the mass is given to us in the question. If we plug these into the equation for Newton's second law, we can find the magnitude of the acceleration of the box.\[\begin{align}25\,\mathrm{N}&=5\,\mathrm{kg}\cdot a\\\Rightarrow a&=\frac{25\, \mathrm{N}}{5\,\mathrm{kg}}\\&=5\frac{\mathrm{m}}{\mathrm{s}^2}\end{align}\]
Newton's first law is a special case of the second law, as clearly if the acceleration of an object is zero then there is resultant force is also zero.
Whilst the definition of Newton's second law given above is more common and is almost always sufficient for solving problems there is a more rigorous and fundamental definition that is useful to know. It states that the resultant force on an object is equal to the time derivative of the object's momentum. Mathematically,\[F=\frac{\mathrm{d}p}{\mathrm{d}t}\]
Recall that the momentum of an object is \(p=mv\). This formulation can be used in situations where the mass of the object is non-constant and is more precise as mass is not necessarily a conserved quantity whereas momentum always is.If the mass of an object is a constant then we easily recover the usual equation for Newton's second law\[\begin{align}F=&\frac{\mathrm{d}p}{\mathrm{d}t}\\=&\frac{\mathrm{d}(mv)}{\mathrm{d}t}\\=&m\frac{\mathrm{d}v}{\mathrm{d}t}\\=&ma\end{align}\]The final step comes from the definition of acceleration as the time derivative of velocity.
Newton's Third Law
The third and final Newton's law of motion concerns reaction forces. Imagine that you are sitting in a wheelchair, and you push against a wall causing you to roll backward. Why is this? Well when you push against the wall, the wall exerts an equal and opposite reaction force on you causing you to roll backward. This idea is encapsulated by Newton's third law. We call this combination of action and reaction forces, action-reaction pairs.
Newton's Third Law - For every force, there is always a reaction force with the same magnitude acting in the opposite direction.
It is worth noting that the action-reaction pairs do not apply to the same object and are of the same type of force. For example, a book resting on a table experiences a gravitational force from the earth, the reaction force is the gravitational force exerted on the earth by the book.
When a ball bounces on the ground it exerts a downwards force onto the ground, the ground then exerts a force of equal magnitude onto the ball upwards in the opposite direction. This reaction force causes the ball to bounce upwards. This may seem strange as we never see the earth move downwards when we bounce a ball, however, remember that \(F=ma\), so the huge mass of the earth means that the acceleration of the earth is negligible.
Translational Dynamics Meaning
When a body's position does not change with respect to time, we say it is at rest. However, when a body's position changes with respect to time, we say it is in motion. The study of motion in physics falls into two categories dynamics and kinematics. In kinematics, we are only concerned with things like an object's position, velocity, acceleration, etc. and how it changes over time. Kinematics does not look at the causes of motion and so ignores quantities such as momentum, force, or energy. On the other hand, dynamics looks at the broader picture looking at why an object moves and analyzing motion by looking at the resultant forces on an object or the work done by the object. Newton's Laws of Motion give us an understanding of forces as the cause of motion, from these laws we can then build up a full understanding of the motion of an object. We define two kinds of dynamics, translational dynamics, and rotational dynamics. Rotational dynamics is concerned with objects moving around an axis of rotation such as spinning objects or objects undergoing circular motion.
Fig. 4 - The motion of a Ferris wheel is rotational and falls under rotational dynamics, Wikimedia Commons
However, in this article, we will concentrate on translational dynamics.
Translational dynamics concerns the motion of objects where all parts of the body travel uniformly in the same direction. We can think of it as a kind of sliding where the orientation of the boy does not change whilst the object moves.
For example, a bullet fired from a gun undergoes translational motion as does a block sliding down an inclined plane.
Translational Dynamics Model
Being able to model the motion of an object or group of objects is the central purpose of translational dynamics. In physics, we call an object, or group of objects, under consideration a system. A system can be really simple like a single non-interacting particle, or it can be as complex as a galaxy held together by gravitational interactions.
In translational dynamics, we can simplify things by considering a system's mass to be accumulated entirely within the center of mass, this way forces only act at one point of the system and we don't have to consider how the force acts on each component of the system. We can make this simplification when the properties of the constituent particles are not important to model the behavior of the macroscopic system.
The center of mass of a system is the point of a system where the weighted relative position of the distributed mass of the system is zero. For a system composed of a finite number of point masses, it can be found using the following formula\[\vec{r}_{\mathrm{cm}}=\frac{\sum{m_i\vec{r}_i}}{\sum{m_i}}\]If the distribution of mass is continuous, like in a single solid object, then we need to integrate instead
\[\vec{r}_{cm}=\frac{\int\vec{r}\mathrm{d}m}{\int \mathrm{d}m}\]
If a force acts through an object's center of mass it will experience linear translational motion, it's only if the force acts away from the center of mass that rotational motion can occur. This is why objects balance if they're held up at their center of mass.
Fig. 5 - By holding up the toy at its center of mass the toy can balance on the person's finger
In translational dynamics, one of the most commonly occurring problems is calculating the resultant force on an object to understand the direction of an object's acceleration. This involves adding up all the force vectors acting on the object and finding the resultant force vector. Keeping track of all the forces acting on a body is easiest done using a free-body force diagram. These are simple diagrams where each force acting on an object is represented by an arrow pointing in the direction the force acts and accompanied by the magnitude of the force written beside it. Such a diagram is shown below, demonstrating the forces acting on a block sliding down an inclined plane.
Fig.6-Free-body diagram of a block on an incline plane. The friction force acts opposite to the direction of the object's motion, whilst the normal force acts perpendicularly to the surface and the weight acts downwards from the centre of mass.
When multiple forces are acting on an object that are not acting along more than one axis, then we need to select a coordinate system to resolve the forces into their components. By adding the components we can find the resultant force in this coordinate system. In the diagram above, the three forces on the block are not acting in the same direction, so we need to resolve the vectors into their \(x\) and \(y\) components to find the resultant force vector.
Notice that the angle between the weight and the normal force is equal to the angle of inclination \(\theta\). Choosing to resolve the forces into components defined by a coordinate system whose axes are parallel and perpendicular to the surface of the ramp means that we only need to resolve the weight into its components. The friction and normal force are already aligned with the axes of this coordinate system simplifying the calculation. Resolving the weight into these components gives:\[\begin{align}W_x=& mg\sin(\theta)\\W_y=& mg\cos(\theta)\end{align}\]
This model of forces and resolved force components set us up to calculate the resultant force acting on the object and its acceleration.
Translation Dynamics Formulas
Once all the forces have been drawn on a free-body diagram and an appropriate coordinate system has been chosen to resolve the force vectors into, we can use Newton's laws to calculate the acceleration of an object.The formula for calculating the resultant force \(F_{\text{net}}\) can be given as
\[\begin{align}(F_{\text{net}})_x&=\sum_i (F_i)_x\\(F_{\text{net}})_y&=\sum_i (F_i)_y\end{align}\]where \(i\) is an index for the forces acting on the object.
Returning to the block on the inclined plane, labeling the friction as \(F_{\mu}\) and the normal force \(F_{\text{norm}}\) the resultant force is given by
\[\begin{align}(F_{\text{net}})_x&=F_{\mu}+mg\sin(\theta)\\(F_{\text{net}})_y&=F_{\text{norm}}+mg\cos(\theta)\end{align}\]
We can use Newton's third law of motion to find the value of the normal force acting on the block. As every action has an equal and opposite reaction force, the normal force must be the reaction force of the plane's surface equal and opposite to the component of the weight acting on the surface.
\[\begin{align}&F_{\text{norm}}=-mg\cos(\theta)\\&\Rightarrow (F_{\text{net}})_y=F_{\text{norm}}+mg\cos(\theta)=0\end{align}\]
This is as we expect as we chose our \(y\)-axis to be perpendicular to the surface of the inclined plane, and clearly, the block is neither falling through the plane nor levitating above it! If we had instead lined up our \(y\) axis to be perpendicular to the ground, there would be a force acting downwards.The block is sliding down the plane, so we need to use Newton's second law to find the acceleration of the block along the \(x\) axis. Recall that\[F_{\text{net}}=ma\]so, the acceleration along the \(x\) axis is\[\begin{align}F_{\mu}+mg\sin(\theta)&=ma_x\\\Rightarrow a_x&=\frac{F_{\mu}}{m}+g\sin(\theta)\end{align}\]
The value of \(F_{\mu}\) is determined by the surface's coefficient of friction \(\mu\)and is given by
\[F_{\mu}=\mu F_{\text{norm}}=\mu m\cos(\theta)\]
This gives a full run-through of how we can use Newton's laws of motion to determine the translational dynamics of an object, so let's apply this method to an explicit example.
Consider a wooden block held up by two cables, as shown in the diagram below, with the wooden block having a mass of \(5.00\mathrm{kg}\). If the first cable is at an angle of \(\theta_1=-25.0^{\circ}\) to the vertical, and the second is at an angle of \(\theta_2=45.0^{\circ}\) what will the tension in each cable be?
Fig.7- By resolving the forces acting on the block we can find the tension in each cable required to hold it up.
First write down all the forces acting on the block. Let the tension in cable one be \(F_1\) and in cable two \(F_2\) and obviously the weight will be \(F_{weight}=mg=5.00\cdot9.81=49.0\,\mathrm{N}\). By choosing our coordinate axes such that the \(y\) axis is the vertical axis and \(x\) is the horizontal, we know that, if the block is in equilibrium, the following equations must hold.
\[\begin{align}(F_1)_x+(F_2)_x&=0\\(F_1)_y+(F_2)_y&=F_{\mathrm{weight}}=49.0\,\mathrm{N}\end{align}\]
We can express these components using trigonometry\[\begin{align}F_1\sin(\theta_1)+F_2\sin(\theta_2)&=0\,\mathrm{N}\\F_1\cos(\theta_1)+F_2\cos(\theta_2)&=49.0\,\mathrm{N}\end{align}\]
Re-arranging the first equation and plugging in the angles gives:
\[F_1=-F_2\frac{\sin(-25^{\circ})}{\sin(45^{\circ})}=0.60F_2\]
Substituting this into the second equation
\[\begin{align}0.60&\cdot\cos(-25.0^{\circ})F_2\,\mathrm{N}+F_2\cos(45.0^{\circ})\,\mathrm{N}=49.0\,\mathrm{N}\\\Rightarrow& 0.60\cdot(0.91)F_2\,\mathrm{N}+0.71F_2\,\mathrm{N}=49.0\,\mathrm{N}\\\Rightarrow& F_2=\frac{49}{0.60\cdot0.91+0.71}\,\mathrm{N}=39.0\,\mathrm{N}\end{align}\]This means the two tensions are
\[\begin{align}F_2&=39.0\,\mathrm{N}\\F_1&=0.60F_2=23.0\,\mathrm{N}\end{align}\]
Translational Dynamics - Key takeaways
In mechanics, an object or group of objects under consideration is called a system
Kinematics looks at how a system moves over time through quantities such as position velocity and acceleration without considering the causes of motion, whereas dynamics considers analyses the causes of motion and is concerned with things like force and energy
When a body moves as a whole and every portion of the body travels the same distance in the same amount of time, then we say the body is in translational motion. On the other hand, if the body is rotating around a fixed axis, this is called rotational motion.
Newton's Laws are the fundamental laws governing the relationship between forces and motion
Newton's first law states that if an object is at rest or in uniform motion unless there is an external force, it will preserve its status.
Newton's second law, force is the product of mass and acceleration.
Newton's third law states that for every action force, there is a reaction force of the same magnitude but in the opposite direction.
We can use free body diagrams to visualize how forces act on a system and to find the resultant force acting on the system's center of mass
Fig. 1 - A box on the ground, StudySmarter Originals
Fig. 2 - Person pushing wall, StudySmarter Originals
Fig. 3 - Bouncing ball, StudySmarter Originals
Fig. 4 - Ferris Wheel (https://commons.wikimedia.org/wiki/File:Ferris_Wheel_-_2590239345.jpg), by seabamirum licensed by CC BY 2.0 (https://creativecommons.org/licenses/by/2.0/deed.en)
Fig. 5 - Bird Toy Showing Center of Gravity (https://commons.wikimedia.org/wiki/File:Bird_toy_showing_center_of_gravity.jpg) by APN MJM licensed by CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/deed.en)
Fig. 6 - Free body force diagram, StudySmarter Originals
Fig. 7 - Cable and block, StudySmarter Originals
Frequently Asked Questions about Translational Dynamics
What is dynamic translational motion?
When a body moves as a whole and every portion of the body travels in the same direction, then we say the body is in translational motion.
What is an example of translational motion?
Examples of translational motion can be a car moving in a straight line, and the path of a bullet out of a gun.
What is the difference between translational and rotational motion?
When a body moves as a whole and every portion of the body travels in the same direction, then we say the body is in translational motion. On the other hand, if the body is rotating around a fixed axis, this is called rotational motion.
What movement is called translational rotational?
Rolling with sliding is an example of both translational and rotational motion.
Is walking a translational motion?
Yes. When a body moves as a whole and every portion of the body travels the same distance in the same amount of time, then we say the body is in translational motion.
Final Translational Dynamics Quiz
Friction is not a contact force.
A car can brake and stop without the friction force.
The static friction force is applied while the object is stationary.
If the force \(\overrightarrow F\) is exerted on the box pulling it to the right, and its magnitude is increasing with time, the box stays at rest for some time. This is because even if the force is increasing, the force of static friction is increasing as well to cancel the applied force and maintain the box at rest. The force needed to move the box is equal and opposite to the maximum static friction force, \(\mu_{\mathrm s}N\).
The type of friction where there is the ___ value of static friction force and the motion is about to start is called the limiting friction.
Maximum.
The difference between static and kinetic friction force is that the static one is applied while the object is ___, and the kinetic one is applied while the object is ___.
Stationary, moving.
Fire from a lighter or match is one of the daily life examples of friction.
Climbers climbing the mountains have nothing to do with the friction.
There is friction between the ground and the wheels when driving.
The maximum value of the static friction force is always greater than the magnitude of kinetic friction force.
For an object at rest, the magnitude of the static friction force is equal to the magnitude of an applied force.
What is the formula for the frictional force acting on an object as it slides on a surface?
$$F=\mu F_{\text{normal}}$$
where \(\mu\) is the co-efficient of friction of the surface, and \(F_{\text{normal}}\) is the normal force of the object on the surface.
When can we assume that all of a system's mass is concentrated at its center of mass?
When the properties of the system's constituent particles are not important to model the motion of the system.
What are the two types branches of mechanics that study the motion of systems?
Kinematics and Dynamics.
If a force is applied at an object's center of mass, it will undergo rotational motion. True or False?
What is the formula for Newton's second law?
\(F=ma\)
If a box with a mass of 5 kg is pulled with a force of 20 N, what is the acceleration?
\(4\,\mathrm{m}/\mathrm{s}^2\).
If a box is pulled with a force of 35 N and has an acceleration of 7 m/s2, then what is the mass of the box?
The center of mass for an object with a continuous mass distribution is given as:
\(\vec{r}_{\mathrm{cm}}=\frac{\sum{m_i\vec{r}_i}}{\sum{m_i}}\).
Action and reaction forces always act on the same object. True or False?
The gravitational force acting on a ball as it falls to earth has the same magnitude as the force pulling the earth towards the ball. True or False?
What does Newton's first law state?
Newton's first law of linear motion states that a body will remain in a state of rest or uniform motion unless acted on by a net external force.
4 m/s2
The SI unit of force is kg.m/s2 (True/False)
Newton (N)
What does Newton's second law state?
Force is the product of mass and acceleration.
A non-inertial frame of reference ____.
has acceleration concerning an inertial reference frame.
Newton's first law of motion helps to describe the behavior of which property of matter?
Inertia.
The greater a body's mass, the less its inertia is. (True/False)
False. The greater a body's mass, the more the body "resists" being accelerated.
The net force and acceleration are in the opposite direction. (True/False)
Kate pushes a bottle with a mass of 2 kg to her right along a smooth, level lunch counter. The bottle leaves her hand moving at 2 m/s, then slows down as it slides because of the countertop's constant horizontal friction force. It slides for 1 m before coming to rest. What is the magnitude of the friction force acting on the bottle?
Kate pushes a bottle to her right along a smooth, level lunch counter. The bottle leaves her hand moving at 5 m/s, then slows down as it slides because of the countertop's constant horizontal friction force. It slides for 5 m before coming to rest. What is the magnitude of the acceleration of the bottle?
-2.5 m/s2
-4 m/s2
For every action force, there is a reaction force in the same magnitude acting in the opposite direction. These action-reaction forces always exist in pairs. These forces act on the same object.
Action and reaction forces have different magnitudes.
Consider a horse trying to pull a cart. Which of the following is the correct description?
The action-reaction forces are equal but opposite, and they act on the horse such that the net force on the horse is zero, and the horse is unable to move.
A small car of mass \(300\;\mathrm{kg}\) is pushing a large truck of mass \(500\;\mathrm{kg}\) due east on a level road. The car exerts a horizontal force of \(500\;\mathrm{N}\) on the truck. What is the magnitude of the force that the truck exerts on the car?
$$500\;\mathrm{N}$$.
Boxes A and B are in contact on a horizontal, frictionless surface. Box A has a mass of \(3\;\mathrm{kg}\) and box B has a mass of \(2\;\mathrm{kg}\). A horizontal force of \(30\;\mathrm{N}\) is exerted on box A. What is the magnitude of the force that box A exerts on box B?
$$30\;\mathrm{N}$$.
Which of the following are examples of Newton's Third Law?
A boat's propeller pushes the water away, and the water pushes the boat forward.
Let's say you are driving a car, and hitting a fly. The car and the fly have action-reaction pairs. Even though they exert the same magnitude of force on each other, would the effects be the same?
If a car exerts \(500\;\mathrm N\) of force on the fly, the fly exerts the same amount of force on the car as well. However, the fly would die because it is not a force that it can handle.
Kevin is riding his skateboard and pushes off the ground with his foot. This causes him to accelerate at a rate of (10\;\frac{\mathrm m}{\mathrm s^2}\). He weighs \(600\;\mathrm N\). How strong was his push off the ground? (\(g=10\;\frac{\mathrm m}{\mathrm s^2}\))
$$600\;\mathrm N$$.
Erin is riding her skateboard and pushes off the ground with her foot. This causes her to accelerate at a rate of \(5\;\frac{\mathrm m}{\mathrm s^2}\). She weighs \(400\;\mathrm N\). How strong was her push off the ground? (\(g=10\;\frac{\mathrm m}{\mathrm s^2}\))
A book exerts a force of \(5\;\mathrm N\) downward, into a chair that exerts a force of \(10\;\mathrm N\) downward to the floor it stands on. What is the force that the floor exerts upwards on the chair?
$$15\;\mathrm N$$.
You stand next to a wall on a frictionless skateboard and push the wall with a force of \(20\;\mathrm N\). How hard does the wall push on you?
An archer shoots an arrow. The action force is the bowstring against the arrow. The reaction force is:
Arrow's push against the bowstring.
A player catches a ball. The action force is the impact of the ball against the player's glove. The reaction force is:
The force the glove exerts on the ball.
A player hits a ball with a bat. The action force is the impact of the bat against the ball. The reaction force is:
The force of the ball against the bat.
More about Translational Dynamics
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More explanations about Translational Dynamics
Static Friction Learn
Centripetal Force and Velocity Learn
Kinetic Friction Learn
Gravitational Acceleration Learn
Gravitational Force Learn
Orbital Period Learn
Critical Speed Learn
Spring Force Learn
Object in Equilibrium Learn
Free Fall and Terminal Velocity Learn
Resistive Force Learn | CommonCrawl |
Common symbols
℘ or P
SI unit
In SI base units
kg⋅m2⋅s−3
Derivations from
other quantities
℘ = E/t
℘ = F·v
℘ = U·I
℘ = τ·ω
L 2 M T − 3 {\displaystyle {\mathsf {L))^{2}{\mathsf {M)){\mathsf {T))^{-3))
Electric power is transmitted by overhead lines like these, and also through underground high-voltage cables.
Electric power is the rate at which electrical energy is transferred by an electric circuit. The SI unit of power is the watt, one joule per second. Standard prefixes apply to watts as with other SI units: thousands, millions and billions of watts are called kilowatts, megawatts and gigawatts respectively.
A common misconception is that electric power is bought and sold, but actually electrical energy is bought and sold. For example, electricity is sold to consumers in kilowatt-hours (kilowatts multiplied by hours), because energy is power multiplied by time.
Electric power is usually produced by electric generators, but can also be supplied by sources such as electric batteries. It is usually supplied to businesses and homes (as domestic mains electricity) by the electric power industry through an electrical grid.
Electric power can be delivered over long distances by transmission lines and used for applications such as motion, light or heat with high efficiency.[1]
Electric power, like mechanical power, is the rate of doing work, measured in watts, and represented by the letter P. The term wattage is used colloquially to mean "electric power in watts." The electric power in watts produced by an electric current I consisting of a charge of Q coulombs every t seconds passing through an electric potential (voltage) difference of V is
Work done per unit time = ℘ = W t = W Q Q t = V I {\displaystyle {\text{Work done per unit time))=\wp ={\frac {W}{t))={\frac {W}{Q)){\frac {Q}{t))=VI}
W is work in joules
t is time in seconds
Q is electric charge in coulombs
V is electric potential or voltage in volts
I is electric current in amperes
Animation showing power source
Electric power is transformed to other forms of energy when electric charges move through an electric potential difference (voltage), which occurs in electrical components in electric circuits. From the standpoint of electric power, components in an electric circuit can be divided into two categories:
Animation showing electric load
Active devices (power sources)
If electric current (conventional current) is forced to flow through the device in the direction from the lower electric potential to the higher, so positive charges move from the negative to the positive terminal, work will be done on the charges, and energy is being converted to electric potential energy from some other type of energy, such as mechanical energy or chemical energy. Devices in which this occurs are called active devices or power sources; such as electric generators and batteries. Some devices can be either a source or a load, depending on the voltage and current through them. For example, a rechargeable battery acts as a source when it provides power to a circuit, but as a load when it is connected to a battery charger and is being recharged.
Passive devices (loads)
If conventional current flows through the device in a direction from higher potential (voltage) to lower potential, so positive charge moves from the positive (+) terminal to the negative (−) terminal, work is done by the charges on the device. The potential energy of the charges due to the voltage between the terminals is converted to kinetic energy in the device. These devices are called passive components or loads; they 'consume' electric power from the circuit, converting it to other forms of energy such as mechanical work, heat, light, etc. Examples are electrical appliances, such as light bulbs, electric motors, and electric heaters. In alternating current (AC) circuits the direction of the voltage periodically reverses, but the current always flows from the higher potential to the lower potential side.
Transmission of power through an electric circuit
Passive sign convention
Main article: Passive sign convention
Since electric power can flow either into or out of a component, a convention is needed for which direction represents positive power flow. Electric power flowing out of a circuit into a component is arbitrarily defined to have a positive sign, while power flowing into a circuit from a component is defined to have a negative sign. Thus passive components have positive power consumption, while power sources have negative power consumption. This is called the passive sign convention.
Resistive circuits
In the case of resistive (Ohmic, or linear) loads, Joule's law can be combined with Ohm's law (V = I·R) to produce alternative expressions for the amount of power that is dissipated:
℘ = I V = I 2 R = V 2 R {\displaystyle \wp =IV=I^{2}R={\frac {V^{2)){R))}
where R is the electrical resistance.
Alternating current without harmonics
Main article: AC power
In alternating current circuits, energy storage elements such as inductance and capacitance may result in periodic reversals of the direction of energy flow. The portion of energy flow (power) that, averaged over a complete cycle of the AC waveform, results in net transfer of energy in one direction is known as real power (also referred to as active power).[2] The amplitude of that portion of energy flow (power) that results in no net transfer of energy but instead oscillates between the source and load in each cycle due to stored energy, is known as the absolute value of reactive power.[2][3][4] The product of the RMS value of the voltage wave and the RMS value of the current wave is known as apparent power. The real power P in watts consumed by a device is given by
℘ = 1 2 V p I p cos θ = V r m s I r m s cos θ {\displaystyle \wp ={1 \over 2}V_{p}I_{p}\cos \theta =V_{\rm {rms))I_{\rm {rms))\cos \theta }
Vp is the peak voltage in volts
Ip is the peak current in amperes
Vrms is the root-mean-square voltage in volts
Irms is the root-mean-square current in amperes
θ = θv − θi is the phase angle by which the voltage sine wave leads the current sine wave, or equivalently the phase angle by which the current sine wave lags the voltage sine wave
Power triangle: The components of AC power
The relationship between real power, reactive power and apparent power can be expressed by representing the quantities as vectors. Real power is represented as a horizontal vector and reactive power is represented as a vertical vector. The apparent power vector is the hypotenuse of a right triangle formed by connecting the real and reactive power vectors. This representation is often called the power triangle. Using the Pythagorean Theorem, the relationship among real, reactive and apparent power is:
(apparent power) 2 = (real power) 2 + (reactive power) 2 {\displaystyle {\text{(apparent power)))^{2}={\text{(real power)))^{2}+{\text{(reactive power)))^{2))
Real and reactive powers can also be calculated directly from the apparent power, when the current and voltage are both sinusoids with a known phase angle θ between them:
(real power) = (apparent power) cos θ {\displaystyle {\text{(real power)))={\text{(apparent power)))\cos \theta }
(reactive power) = (apparent power) sin θ {\displaystyle {\text{(reactive power)))={\text{(apparent power)))\sin \theta }
The ratio of real power to apparent power is called power factor and is a number always between −1 and 1. Where the currents and voltages have non-sinusoidal forms, power factor is generalized to include the effects of distortion.
Electrical energy flows wherever electric and magnetic fields exist together and fluctuate in the same place. The simplest example of this is in electrical circuits, as the preceding section showed. In the general case, however, the simple equation P = IV may be replaced by a more complex calculation. The closed surface integral of the cross-product of the electric field intensity and magnetic field intensity vectors gives the total instantaneous power (in watts) out of the volume: [5]
℘ = ∮ area ( E × H ) ⋅ d A . {\displaystyle \wp =\oint _{\text{area))(\mathbf {E} \times \mathbf {H} )\cdot d\mathbf {A} .}
The result is a scalar since it is the surface integral of the Poynting vector.
2019 world electricity generation by source (total generation was 27 petawatt-hours)[6][7]
Coal (37%)
Natural gas (24%)
Hydro (16%)
Nuclear (10%)
Wind (5%)
Solar (3%)
Other (5%)
Main article: Electricity generation
The fundamental principles of much electricity generation were discovered during the 1820s and early 1830s by the British scientist Michael Faraday. His basic method is still used today: electric current is generated by the movement of a loop of wire, or disc of copper between the poles of a magnet.
For electric utilities, it is the first process in the delivery of electricity to consumers. The other processes, electricity transmission, distribution, and electrical energy storage and recovery using pumped-storage methods are normally carried out by the electric power industry.
Electricity is mostly generated at a power station by electromechanical generators, driven by heat engines heated by combustion, geothermal power or nuclear fission. Other generators are driven by the kinetic energy of flowing water and wind. There are many other technologies that are used to generate electricity such as photovoltaic solar panels.
A battery is a device consisting of one or more electrochemical cells that convert stored chemical energy into electrical energy.[8] Since the invention of the first battery (or "voltaic pile") in 1800 by Alessandro Volta and especially since the technically improved Daniell cell in 1836, batteries have become a common power source for many household and industrial applications. According to a 2005 estimate, the worldwide battery industry generates US$48 billion in sales each year,[9] with 6% annual growth. There are two types of batteries: primary batteries (disposable batteries), which are designed to be used once and discarded, and secondary batteries (rechargeable batteries), which are designed to be recharged and used multiple times. Batteries are available in many sizes; from miniature button cells used to power hearing aids and wristwatches to battery banks the size of rooms that provide standby power for telephone exchanges and computer data centers.
Electric power industry
Main article: Electric power industry
The electric power industry provides the production and delivery of power, in sufficient quantities to areas that need electricity, through a grid connection. The grid distributes electrical energy to customers. Electric power is generated by central power stations or by distributed generation. The electric power industry has gradually been trending towards deregulation – with emerging players offering consumers competition to the traditional public utility companies.[10]
Electric power, produced from central generating stations and distributed over an electrical transmission grid, is widely used in industrial, commercial and consumer applications. The per capita electric power consumption of a country correlates with its industrial development. [11] Electric motors power manufacturing machinery and propel subways and railway trains. Electric lighting is the most important form of artificial light. Electrical energy is used directly in processes such as extraction of aluminum from its ores and in production of steel in electric arc furnaces. Reliable electric power is essential to telecommunications and broadcasting. Electric power is used to provide air conditioning in hot climates, and in some places electric power is an economically competitive source of energy for building space heating. Use of electric power for pumping water ranges from individual household wells to irrigation projects and energy storage projects.
Energy portal
EGRID
Electric energy consumption
Electric power system
High-voltage cable
Rural electrification
^ Smith, Clare (2001). Environmental Physics. London: Routledge. ISBN 0-415-20191-8.
^ a b Thomas, Roland E.; Rosa, Albert J.; Toussaint, Gregory J. (2016). The Analysis and Design of Linear Circuits (8 ed.). Wiley. pp. 812–813. ISBN 978-1-119-23538-5.
^ Fraile Mora, Jesús (2012). Circuitos eléctricos (in Spanish). Pearson. pp. 193–196. ISBN 978-8-48-322795-4.
^ IEEE Standard Definitions for the Measurement of Electric Power Quantities Under Sinusoidal, Nonsinusoidal, Balanced, or Unbalanced Conditions. IEEE. 2010. p. 4. doi:10.1109/IEEESTD.2010.5439063. ISBN 978-0-7381-6058-0.
^ Hayt, William H.; Buck, John A. (2012). Engineering Electromagnetics (8 ed.). McGraw-Hill. p. 385. ISBN 978-0-07-338066-7.
^ "Data & Statistics". International Energy Agency. Retrieved 2021-11-25.
^ "World gross electricity production by source, 2019 – Charts – Data & Statistics". International Energy Agency. Retrieved 2021-11-25.
^ "battery" (def. 4b), Merriam-Webster Online Dictionary (2009). Retrieved 25 May 2009.
^ Power Shift: DFJ on the lookout for more power source investments Archived 2005-12-01 at the Wayback Machine. Draper Fisher Jurvetson. Retrieved 20 November 2005.
^ The Opportunity of Energy Group-Buying Archived 2017-05-25 at the Wayback Machine EnPowered, April 18, 2016,
^ Ignacio J. Pérez-Arriaga (ed), Regulation of the Power Sector, Springer Science & Business Media, 2014 ISBN 1447150341, page 8
Reports on August 2003 Blackout, North American Electric Reliability Council website
Croft, Terrell; Summers, Wilford I. (1987). American Electricians' Handbook (Eleventh ed.). New York: McGraw Hill. ISBN 0-07-013932-6.
Fink, Donald G.; Beaty, H. Wayne (1978). Standard Handbook for Electrical Engineers (Eleventh ed.). New York: McGraw Hill. ISBN 0-07-020974-X.
Wikimedia Commons has media related to Electrical power.
U.S. Department of Energy: Electric Power
GlobTek, Inc. Glossary of Electric power Power Supply Terms
Electricity delivery
Automatic generation control
Backfeeding
Base load
Demand factor
Droop speed control
Economic dispatch
Energy return on investment
Electrical fault
Home energy storage
Grid storage
Grid code
Grid strength
Load-following
Merit order
Nameplate capacity
Peak demand
Power-flow study
Utility frequency
Variability
Vehicle-to-grid
Non-renewable
Osmotic
Sustainable biofuel
Combined cycle
Induction generator
Micro CHP
Microgeneration
Rankine cycle
Three-phase electric power
Virtual power plant
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Electric power distribution
Electricity retailing
Electrical busbar system
Electric power transmission
Electrical grid
Electrical interconnector
High-voltage direct current
High-voltage shore connection
Load management
Mains electricity by country
Pumped hydro
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Super grid
Transmission system operator (TSO)
Utility pole
Failure modes
Blackout (Rolling blackout)
Brownout
Black start
Cascading failure
Arc-fault circuit interrupter
Earth-leakage circuit breaker
Generator interlock kit
Residual-current device (GFI)
Power system protection
Protective relay
Numerical relay
Sulfur hexafluoride circuit breaker
and policies
Availability factor
Cost of electricity by source
Environmental tax
Feed-in tariff
Fossil fuel phase-out
Load factor
Pigovian tax
Renewable energy payments
Renewable energy policy
Spark/Dark/Quark/Bark spread
Statistics and
List of electricity sectors
Electricity economics
Power station technology
Temporal rates | CommonCrawl |
\begin{document}
\maketitle
\begin{abstract}
The present work deals with the canonical map of smooth, compact complex surfaces of general type in a
polarization of type $(1,2,2)$ on an abelian threefold. A natural and classical question is whether the canonical system of such surfaces is very ample in the general case. In this work, we provide a positive answer
to this question. First, we describe the structure of the canonical map of those smooth ample surfaces of type $(1,2,2)$ in an abelian threefold which are bidouble cover of principal polarizations. Then, we study the general behavior of the canonical map of general ample surfaces $\mathcal{S}$, yielding a $(1,2,2)$-polarization on an abelian threefold $A$ which is isogenous to a product.
By combining these descriptions, we show that the canonical map yields a holomorphic embedding when $A$ and $\mathcal{S}$ are both sufficiently general.
It follows, in particular, a proof of the existence of canonical irregular surfaces in $\mathbb{P}^5$ with numerical
invariants $p_g = 6$, $q = 3$ and $K^2 = 24$. \end{abstract}
\section*{Introduction}
The present work aims at studying the canonical map of smooth, compact complex surfaces of general type $\mathcal{S}$ in a polarization of type $(1, 2, 2)$ on an abelian threefold $A$. A classical question to ask is whether the canonical system of such surfaces is very ample, at least when $A$ and $\mathcal{S}$ are sufficiently general. The following theorem, which is the main result of this work, provides an affirmative answer to this question. \begin{thm1intro} \label{teoremafinaleintro}
Let be $(A, \mathcal{L})$ a general $(1,2,2)$-polarized abelian threefold and let be $\mathcal{S}$ a general surface in the linear system $|\mathcal{L}|$. Then the canonical system of $\mathcal{S}$ yields a holomorphic embedding in $\mathbb{P}^5$. \end{thm1intro}
A first motivation underlying this research question is the well-known existence problem of canonically embedded surfaces in $\mathbb{P}^5$ with prescribed numerical invariants:
\begin{que1intro} \label{questionE1} Which are the possible values of $K^2$ for the smooth surfaces of general type $\mathcal{S}$ which are canonically embedded in $\mathbb{P}^5$? \end{que1intro} \noindent It's important to recall that the requirement for the canonical map to be birational onto its image leads to some numerical constraints. From the inequality of Bogomolov-Miyaoka-Yau $K^2 \leq 9 \chi$ and the Debarre version of Castelnuovo's inequality $K^2 \geq 3p_g(\mathcal{S}) + q - 7$ (see \cite{Debarre1982}), it follows that, if the canonical map of an algebraic surface $\mathcal{S}$ with $p_g = 6$ is birational, then \begin{equation*} 11 + q \leq K^2 \leq 9(7 - q) \ \ \text{.} \end{equation*}
Several construction methods have been considered in the attempt to give a satisfactory answer to Question \ref{questionE1} at least for some values of $K^2$. One first method we want to mention here relies on the existence of a very special determinantal structure for the defining equations of a canonical surface in $\mathbb{P}^5$. More precisely, Walter proved (see \cite{Walter}) that every codimension $3$ locally Gorenstein subscheme $X$ of $\mathbb{P}^{n+3}$, which is $l$-subcanonical (i.e. $\omega_X \cong \mathcal{O}_X(l))$ and such that $\chi(\mathcal{O}_X(l/2))$ is even if $n$ is divisible by $4$ and $l$ is even, is pfaffian. This means that there exists a locally free resolution \begin{equation} \label{pfaffiand} 0 \longrightarrow \mathcal{O}_{\mathbb{P}^5}(-t) \longrightarrow \mathcal{E}^{\vee}(-t)\overset{\alpha}{\longrightarrow} \mathcal{E} \longrightarrow \mathcal{I}_{\mathcal{S}}\longrightarrow 0 \ \ \text{,} \end{equation} where $\mathcal{E}$ is a vector bundle on $\mathbb{P}^5$ of odd rank $2k+1$, and $\alpha$ is an antisymmetric map such that $\mathcal{S}$ is defined by the Pfaffians of order $2k$ of $\alpha$. This implies that every smooth canonical surface $\mathcal{S}$ in $\mathbb{P}^5$ is pfaffian with a resolution as in (\ref{pfaffiand}) with $t = 7$.
Moreover, in this case, the cohomology of the vector bundle $\mathcal{E}$ (and hence $\mathcal{E}$ itself by using the Horrock's correspondence) is completely determined by the cohomology of the structure sheaf of $\mathcal{S}$ and its ideal sheaf in $\mathbb{P}^5$, and hence by the structure of the canonical ring of $\mathcal{S}$ (see also \cite{Catanese1997})
\noindent By considering a vector bundle $\mathcal{E}$ which splits into a sum of line bundles and a suitable antisymmetric map of vector bundles $\alpha$ as in Diagram $\ref{pfaffiand}$, Catanese exhibited examples of regular canonical surfaces in $\mathbb{P}^5$ with $11 \leq K^2 \leq 17$ (see \cite{Catanese1997}). Moreover, G. and M. Kapustka construct via bilinkage methods a further example of canonical regular surface $\mathcal{S}$ in $\mathbb{P}^5$ with $K^2=18$ (see \cite{GMKapustka2016}).
The same two authors conjectured in a previous work (cf. \cite{GMKapustka2015}) that this degree would be an upper bound for the existence problem \ref{questionE1}. However, this conjecture has turned out to be false:
Catanese showed (see \cite{Catanese1999}) that examples of surfaces of general type with birational canonical map of higher degree arise by considering ramified bidouble covers with prescribed branch type, and he proved that there is a family of surfaces of general type which are bidouble covers of $\mathbb{P}^1 \times \mathbb{P}^1$, branched on three smooth curves $D_1$, $D_2$, $D_3$ of bidegree respectively $(2,3)$, $(2,3)$, $(1,4)$, whose general fibers are surfaces with $K^2 = 24$, $q=0$, $p_g = 6$ and birational canonical map. Furthermore, in a recent paper (see \cite{Catanese2017}) he proved that, in this case, the canonical system is base point free and yields an embedding in $\mathbb{P}^5$.
\noindent However, it is remarkable that the given examples in \cite{Catanese1997,Catanese1999,Catanese2017,GMKapustka2015} are examples of regular surfaces, and thus they leave unanswered the existence problem \ref{questionE1} for canonical irregular surfaces of higher degree.
\noindent Examples of irregular algebraic projective varieties can be easily produced using transcendental methods. Nice examples of irregular projective varieties arise naturally by considering abelian varieties or subvarieties of abelian varieties. However, in the case of abelian varieties, the description of the equations of their projective model and the problem of determining whether an ample line bundle is very ample have been considered challenging problems (see \cite{MumfordEquation} and \cite{DebarreTor}), although their underlying analytic structure, as well as the structure of their ample line bundles, are well understood.
A first important class of subvarieties of abelian varieties is that of their ample divisors. Since the canonical sheaf of an ample divisor $\mathcal{D}$ is an abelian variety $A$ is, by adjunction, just the restriction of the polarization $\mathcal{O}_A(\mathcal{D})$ to $\mathcal{D}$, some natural questions arise:
\begin{que1intro}\label{questionPolarizationE}
Let us consider a couple $(A, \mathcal{D})$, where $A$ a $g$-dimensional $\underline{d} := (d_1, \cdots, d_g)$ abelian variety, and $\mathcal{D}$ smooth ample divisor of type $\underline{d}$. For which polarization types $\underline{d}$ the canonical map is birational, at least for the general couple $(A, \mathcal{D})$? \end{que1intro}
\begin{que1intro}\label{questionPolarizationF} In the notations of Question \ref{questionPolarizationE}, for which polarization types $\underline{d}$ the canonical map is a holomorphic embedding for the general couple $(A, \mathcal{D})$? \end{que1intro}
The results contained in a recent work (see \cite{CatCes2021}) answer to Question \ref{questionPolarizationE}: the canonical map is birational, provided that the polarization given by $\mathcal{D}$ is non-principal (i.e $d_g \neq 1$) and $\mathcal{D}$ is sufficiently general within its polarization class. \newline Concerning Question \ref{questionPolarizationF} above, in the same work, it is proven that if the canonical bundle of a smooth ample divisor $\mathcal{D}$ is very ample, then the following inequality holds \begin{equation*} \label{NECCOND} d:=d_1 \cdot d_2 \cdot \cdots \cdot d_g \geq g +1 \ \ \text{.} \end{equation*}
In consideration of the results which we present in this work, Catanese has conjectured that the necessary condition above for the very ampleness of the canonical bundle of a smooth ample divisor $\mathcal{D}$ is also sufficient, provided that the couple $(A, \mathcal{D})$ is sufficiently general.
The present paper provides the first known result in the direction of this conjecture, at least in the case of abelian threefolds when $d=4$. However, it leaves the case of a polarization of type $(1,1,4)$ open. In this case, whether the canonical system yields an embedding in $\mathbb{P}^5$ in the general case remains unknown. Nevertheless, in the case of the polarization type $(1,2,2)$, we provide very explicit examples and descriptions of the canonical map of smooth surfaces within a $(1,2,2)$-polarization class which are interesting for their own sake. \newline
This work is organized as follows.
\noindent For the sake of completeness, we focus throughout Section 1 on some theoretical background about the structure of the canonical map of an ample divisor on an abelian variety, and we give a proof that this map can be expressed in terms of Theta functions (see Proposition \ref{desccanonicalmap}).\newline
\noindent Section 2 treats the case of a smooth surface, which is a bidouble cover of a principal polarization of a general Jacobian threefold. The study of the canonical map of surfaces in a polarization of type $(1,1,2)$, which has been carried out by Catanese and Schreier in their joint work \cite{CataneseSchreyer1}, plays here a crucial role. It turns out that the canonical map of a surface $\mathcal{S}$, which is the pullback of a principal polarization of a general Jacobian threefold along an isogeny $p$ with kernel isomorphic to $\mathbb{Z}_2^2$, is a local holomorphic embedding. However, its canonical map is never injective. Indeed, we show that, on the invariant canonical curves under the action of an involution contained in the kernel of the isogeny $p$, the canonical map has degree $2$ and it factors through the involution itself.
\noindent The (apparently negative) outcome of this first attempt to prove Theorem \ref{teoremafinaleintro} suggests that the canonical map may fail to be injective only for those surfaces $\mathcal{S}$, which are invariant under the action of an involution on $A$. In such a case, on an invariant canonical curve in $\mathcal{S}$, the restriction of the canonical map should factor through the involution itself, and in particular it is not injective. \newline
\noindent Section 3 deals with the properties of the morphisms defined by ample line bundles of type $(1,2)$ and $(2,4)$ on abelian surfaces, and treats some results, which we apply to prove Theorem \ref{teoremafinaleintro}.
\noindent In Section 4, we deal with general ample surfaces in general $(1,2,2)$-abelian threefolds which are isogenous to the product of elliptic curve. Such abelian threefolds $A$ admit changing-sign involutions, and under their action all smooth surfaces $\mathcal{S}$ within the polarization on $A$ are invariant. In particular, for every such involution, we see that the canonical map factors through it precisely on some invariant canonical curves in $\mathcal{S}$. \newline
The discussion of the above cases in which $\mathcal{S}$ is a bidouble cover of a principal polarization or $A$ is isogenous to a product constitute the background for the proof of Theorem \ref{teoremafinaleintro}, which is contained in Section 5, the last of this work. The proof of it proceeds by assuming that the canonical map of a general member $\mathcal{S}$ within the $(1,2,2)$-polarization class in a general abelian threefold $A$ always fails to be injective on a couple of points which varies when $A$ moves in the moduli space of $(1,2,2)$-polarized abelian threefolds. By specializing to the different loci of abelian threefolds which are isogenous to a product, we find that the original couple of points also specialize to a couple of points which are conjugated with respect to many different involutions. This argument will show that the original assumption leads to a contradiction, and hence it will follow that the canonical map is, in general, injective and with injective differential everywhere, according to the results of the second section.
\setcounter{section}{0} \section{Preliminaries and Notations}
Throughout this work, a polarized abelian variety will be a couple $(A, \mathcal{L})$, where $\mathcal{L}$ is an ample line bundle on a complex torus $A$ and we denote by $|\mathcal{L}|$ the linear system of effective ample divisors which are zero loci of global holomorphic sections of $\mathcal{L}$. \newline
\noindent The first Chern Class $c_1(\mathcal{L}) \in H^2(A, \mathbb{Z})$ is an integral valued alternating bilinear form on the lattice $H_1(A, \mathbb{Z})$. Applying the elementary divisors theorem, we obtain that there exists a basis $\lambda_1, \cdots, \lambda_g, \mu_1, \cdots, \mu_g$ of $\Lambda$ with respect to which the matrix of $c_1(\mathcal{L})$ is \begin{equation*} \begin{pmatrix} 0 & D \\ -D & 0 \end{pmatrix} \ \ \text{,} \end{equation*} where $D$ is a diagonal matrix $diag(d_1, \cdots, d_g)$ of positive integers with the property that every integer in the sequence divides the next. We call the sequence of integers $\underline{d} = (d_1, \cdots, d_g)$ the type of the polarization on $A$ induced by $\mathcal{L}$.
\noindent Moreover, we will say that an ample divisor $\mathcal{D}$ in an abelian variety $A$ yields a polarization of type $\underline{d} = (d_1, \cdots, d_g)$ on $A$, or just that $\mathcal{D}$ is of type $\underline{d}$ on $A$ if the type of the polarization of $\mathcal{L}(\mathcal{D})$ is $\underline{d}$. \newline
Let us consider a polarized abelian variety $(A, \mathcal{L})$ of dimension $g$, where $A := \bigslant{\mathbb{C}^g}{\Lambda}$, and $\Lambda$ denote a sublattice in $\mathbb{C}^g$. Consider $\mathcal{D}$ a smooth ample divisor in the linear system $|\mathcal{L}|$. Denoted by $[\{\phi_{\lambda}\}_{\lambda}] \in H^1(\Lambda, H^0(\mathcal{O}_V^*)$ the factor of automorphy corresponding to the ample line bundle $\mathcal{L}$ according to the Appell-Humbert theorem
(see \cite[p. 32]{BLange}), the vector space $H^0(A, \mathcal{L})$ is isomorphic to the vector space of the holomorphic functions $\theta$ on $\mathbb{C}^g$ which satisfy, for every $\lambda$ in the lattice $\Lambda$, the functional equation \begin{equation*} \theta(z + \lambda) = \phi_{\lambda}(z) \theta(z) \ \ \text{.} \end{equation*}
\noindent When $\mathcal{D} = div(\theta_0)$ is a divisor which is the zero locus of a holomorphic global section $\theta_0$ of $\mathcal{L}$, we have, by the adjunction formula, that \begin{equation} \label{canbdl}
\omega_{\mathcal{D}} = (\mathcal{O}_A(\mathcal{D}) \otimes \omega_A)|_{\mathcal{D}} = \mathcal{O}_{\mathcal{D}}(\mathcal{D}) \ \ \text{.} \end{equation} The derivatives $\frac{\partial \theta_0}{\partial z_j}$ are global holomorphic sections of $\mathcal{O}_{\mathcal{D}}(\mathcal{D})$ for every $j = 1, \cdots, g$. Indeed, for every $\lambda$ in $\Lambda$ and for every $z$ on $\mathcal{D}$ we have \begin{equation*} \frac{\partial \theta_0}{\partial z_j}(z+\lambda) = \phi_{\lambda}(z) \frac{\partial \theta_0}{\partial z_j}(z) + \frac{\partial \phi_{\lambda}}{\partial z_j}(z) \theta_0(z) = \phi_{\lambda}(z) \frac{\partial \theta_0}{\partial z_j}(z) \ \ \text{.} \end{equation*} This leads naturally to a description of the canonical map of a smooth ample divisor in an abelian variety only in terms of theta functions.
\begin{prop1} \label{desccanonicalmap} Let $A=\bigslant{\mathbb{C}^g}{\Lambda}$ be an abelian variety and $\mathcal{L}$ an ample line bundle. Let $\mathcal{D}$ be a smooth divisor defined as the zero locus of a holomorphic section $\theta_0$ of $\mathcal{L}$. Moreover, let us suppose $\theta_0, \cdots, \theta_n$ is a basis for the vector space $H^0(A, \mathcal{L})$. Then $\theta_1, \cdots, \theta_n, \frac{\partial \theta_0}{\partial z_1}, \cdots \frac{\partial \theta_0}{\partial z_g}$, where $z_1, \cdots, z_g$ are the flat uniformizing coordinates of $\mathbb{C}^g$, is a basis for $H^0(\mathcal{D}, \omega_{\mathcal{D}})$. \end{prop1} \begin{proof} From now on let us consider $\mathcal{L}$ to be $\mathcal{O}_A(\mathcal{D})$. We observe first that, for instance, by the Kodaira vanishing theorem, all the cohomology groups $H^i(A, \mathcal{O}_A(\mathcal{D}))$ vanish, so this implies that $H^0(\mathcal{D}, \mathcal{O}_{\mathcal{D}}(\mathcal{D}))$ has the desired dimension $n+g$. In order to prove the assertion of Proposition \ref{desccanonicalmap}, it is then enough to prove that the connecting homomorphism $\delta^0\colon H^0(\mathcal{D}, \mathcal{O}_A(\mathcal{D})) \longrightarrow H^1(A, \mathcal{O}_A)$ maps $\frac{\partial \theta_0}{\partial z_1}, \cdots \frac{\partial \theta_0}{\partial z_g}$ to $g$ linearly independent elements. \newline
\noindent Let us denote the projection of $\mathbb{C}^g$ onto $A$ by $\pi$ and the divisor $\pi^*(\mathcal{D})$ by $\widehat{\mathcal{D}}$. We have then the short exact sequence \begin{equation*} 0 \longrightarrow \mathcal{O}_{\mathbb{C}^g} \longrightarrow \mathcal{O}_{\mathbb{C}^g}(\widehat{\mathcal{D}}) \longrightarrow \mathcal{O}_{\widehat{\mathcal{D}}}(\widehat{\mathcal{D}}) \longrightarrow 0 \ \ \text{,} \end{equation*} and we can denote the respective cohomology groups by: \begin{equation} \label{modulesMNP} \begin{split} M&:=H^0(\mathbb{C}^g, \mathcal{O}_{\mathbb{C}^g}) = H^0(\mathbb{C}^g, \pi^*\mathcal{O}_{A})\\ N&:=H^0(\mathbb{C}^g, \mathcal{O}_{\mathbb{C}^g}(\widehat{\mathcal{D}})) = H^0(\mathbb{C}^g, \pi^*\mathcal{O}_{A}(\mathcal{D})) \\ P&:= H^0(\widehat{\mathcal{D}}, \mathcal{O}_{\widehat{\mathcal{D}}}(\widehat{\mathcal{D}})) = H^0(\widehat{\mathcal{D}}, \pi^*\mathcal{O}_{\mathcal{D}}(\mathcal{D})) \ \ \text{.} \end{split} \end{equation} The cohomology groups we have just defined in (\ref{modulesMNP}) are $\Lambda$-modules with respect to the following actions: for every element $\lambda$ of $\Lambda$ and every elements $s$, $t$, $u$ respectively in $M$, $N$, and $P$, the action of $\Lambda$ is defined as follows: \begin{equation} \label{definitionlambdaaction} \begin{split} \lambda.s(z) &:= s(z+\lambda) \\ \lambda.t(z) &:= t(z+\lambda)\phi_{\lambda}(z)^{-1} \\ \lambda.u(z) &:= u(z+\lambda)\phi_{\lambda}(z)^{-1} \ \ \text{.} \end{split} \end{equation}
According to Mumford \cite[Appendix 2]{MumfordAbelian}, there exists a natural homomorphism $\psi_{\bullet}$ from the cohomology groups sequences $H^p(\Lambda, \cdot)$ and $H^p(A, \cdot)$:
\begin{diagram}
\cdots & \rTo &H^i(\Lambda, M) &\rTo &H^i(\Lambda, N) & \rTo &H^i(\Lambda, P) & \rTo & H^{i+1}(\Lambda, M) & \rTo & \cdots \\
&&\dTo_{\psi^M_i} & &\dTo_{\psi^N_i} & & \dTo_{\psi^P_i} & & \dTo_{\psi^M_{i+1}} & \\
\cdots &\rTo &H^i(A, \mathcal{O}_A) &\rTo & H^i(A, \mathcal{O}_A(\mathcal{D}))& \rTo &H^i(\mathcal{D}, \mathcal{O}_{\mathcal{D}}(\mathcal{D})) & \rTo^{\delta^i} & H^{i+1}(A, \mathcal{O}_A) & \rTo & \cdots \\
\end{diagram} The homomorphism $\psi_{\bullet}$ is actually an isomorphism (this means that all the vertical arrows are isomorphisms), because the cohomology groups $H^i(\mathbb{C}^g, \mathcal{O}_{\mathbb{C}^g}(\widehat{\mathcal{D}}))$ vanish for every $i>0$, being $\mathbb{C}^g$ a Stein manifold. Another possible method to prove that the cohomology sequences $H^p(\Lambda, \cdot)$ and $H^p(A, \cdot)$ are isomorphic, is to use the following result: if $X$ is a variety, $G$ is a group acting freely on $X$ and $\mathcal{F}$ is a $G$-linearized sheaf, then there is a spectral sequence with $E_1$ term equal to $H^p(G, H^q(X, \mathcal{F}))$ converging to $H^{p+q}(Y, \mathcal{F})^{G}$.
\noindent The natural identification of these cohomology group sequences allows us to compute $\delta^0\bigl(\frac{\partial \theta_0}{\partial z_j} \bigr)$ using the explicit expression of the connecting homomorphism $H^0(\Lambda, P) \longrightarrow H^1(\Lambda, M)$: given an element $s$ of $P^{\Lambda}$, there exists an element $t$ in $N$ such that $t|_{\widehat{\mathcal{D}}} = s$. Then, by definition of $d\colon N \longrightarrow \mathcal{C}^{1}(\Lambda; N)$, we have \begin{equation*} (dt)_{\lambda} = \lambda.t - t \ \ \text{.} \end{equation*} where $\lambda.t$ is defined according to (\ref{definitionlambdaaction}). Now, from the invariance of $s$ under the action of $\Lambda$, we get \begin{equation*}
(\lambda.t - t)|_{\widehat{\mathcal{D}}} = \lambda.s - s = 0 \ \ \text{.} \end{equation*} Hence, for every $\lambda$ there exists a constant $c_{\lambda} \in \mathbb{C}$ such that $\lambda.t - t = c_{\lambda} \theta_0$, and it follows that, by definition, \begin{equation} \label{HOMCONN} \delta^0(s)_{\lambda} = c_{\lambda} = \frac{\lambda.t - t}{\theta_0} \ \ \text{.} \end{equation} If we apply now (\ref{HOMCONN}) to the elements $\frac{\partial \theta_0}{\partial z_j}$, we obtain
\begin{equation*} \delta^0\left(\frac{\partial \theta_0}{\partial z_j} \right) = \left[ \left( \pi H(e_j, \lambda) \right)_{\lambda} \right] \in H^1(\Lambda; M) \ \ \text{,} \end{equation*} where $H$ is the positive definite hermitian form on $\mathbb{C}^g$ which corresponds to the ample line bundle $\mathcal{O}_A(\mathcal{D})$ by applying the Appell-Humbert theorem. We prove now that these images are linearly independent in $H^1(\Lambda; M)$. Assume that we have coefficients $a_1, \cdots, a_g \in \mathbb{C}$ such that: \begin{equation*} \left[\left(a_1 H(e_1, \lambda) + \cdots + a_g H(e_g, \lambda)\right)_{\lambda}\right] = 0 \ \ \text{.} \end{equation*} This means that there exists $f \in C^0(\Lambda, M)$ such that, for every $\lambda \in \Lambda$, we have: \begin{equation*} a_1 H(e_1, \lambda) + \cdots + a_g H(e_g, \lambda) = \lambda.f(z) - f(z) = f(z+\lambda) - f(z) \ \ \text{.} \end{equation*} For such $f$, the differential $df$ is a holomorphic $\Lambda$-invariant $1$-form. Hence, for some complex constant $c$ and a certain $\mathbb{C}$-linear form $L$, we can write \begin{equation*} f(z) = L(z) + c \ \ \text{.} \end{equation*} Hence, for every $\lambda \in \Lambda$ the following holds \begin{equation*} H(a_1e_1 + \cdots + a_ge_g, \lambda) = L(\lambda) \ \ \text{,} \end{equation*} so the same holds for every $z \in \mathbb{C}^g$. This allows us to conclude that $L=0$, $L$ being both complex linear and complex antilinear. But the form $H$ is non-degenerate, so we conclude that $a_1 = \cdots = a_g = 0$. The proposition is proved. \end{proof}
We can now easily compute the invariants of a smooth ample divisor $\mathcal{D}$ on an abelian variety $A$ of dimension $g$. We will denote throughout this work by: \begin{itemize}
\item $p_g := h^0(\mathcal{D}, \omega_{\mathcal{D}})$ the geometric genus of $\mathcal{D}$,
\item $q := h^1(\mathcal{D}, \mathcal{O}_{\mathcal{D}})$ the irregularity $\mathcal{D}$,
\item $K_{\mathcal{D}}$ a canonical divisor on $\mathcal{D}$. \newline \end{itemize}
\noindent The following formulas in Proposition \ref{invariantdivisors} can be easily verified by applying the Kodaira vanishing theorem and Proposition \ref{desccanonicalmap}. \begin{prop1} \label{invariantdivisors} Let $\mathcal{D}$ a smooth divisor in a polarization of type $(d_1, \cdots ,d_g)$ on an abelian variety $A$. Then the invariants of $\mathcal{D}$ are the following: \begin{align*} p_g &= \prod_{j=1}^g d_j + g - 1 \\ q &= g \\ K_{\mathcal{D}}^{g-1} &= g! \prod_{j=1}^g d_j \ \ \text{.} \end{align*} \end{prop1}
\begin{def1} (The Gauss map)
\noindent Even though the canonical map of a smooth divisor $\mathcal{D}$ in the linear system $|\mathcal{L}|$ of a (polarized) abelian variety $(A, \mathcal{L})$ can be explicitly expressed in terms of theta functions (as we saw in Proposition \ref{desccanonicalmap}), its image is not always easy to describe.
If we allow $\mathcal{D}$ to be any divisor, not necessarily reduced and irreducible, the same Proposition \ref{desccanonicalmap} provides for us a basis for the space of holomorphic sections of $\mathcal{L}|_{\mathcal{D}}$. It makes sense, however, to consider the map defined as follows: \begin{align*} \mathit{G} : \mathcal{D} &\dashrightarrow \mathbb{P} (V)^{\vee} \\ x &\mapsto \mathbb{P} (T_x \mathcal{D}) \ \ \text{.} \end{align*}
This map is called the \textbf{Gauss Map}, and it is clearly defined on the smooth part of the support of $\mathcal{D}$. In particular, if $\mathcal{D}$ is defined as the zero locus of a holomorphic non-zero section $\theta \in H^0(A, \mathcal{L})$, the map $\mathit{G}: \mathcal{D} \dashrightarrow \mathbb{P} (V)^{\vee} \cong \mathbb{P}^{g-1}$ is defined by the linear subsystem of $|\mathcal{L}|_{\mathcal{D}}|$ generated by $\frac{\partial \theta}{\partial z_1}, \cdots, \frac{\partial \theta}{\partial z_g}$. \end{def1} \begin{ex1} \label{Gaussjacob}
\noindent Let us consider the well-known case of a principal polarization $\Theta$ of the Jacobian $\mathcal{J}$ of a smooth curve $\mathcal{C}$ of genus $g$. In this case, the Gauss Map coincides with the map defined by the complete linear system $|\mathcal{\mathcal{J}}(\Theta)|_{\Theta}|$ and it can be geometrically described as follows: the Abel-Jacobi theorem induces an isomorphism $\mathcal{J} \cong Pic^{g-1}(\mathcal{C})$, so $\Theta$ can be viewed, after a suitable translation, as a divisor of $Pic^{g-1}(\mathcal{C})$. The Riemann Singularity Theorem states (see \cite[Chapter VI]{ACGH}): \begin{equation*} mult_L \Theta = h^0(\mathcal{C}, L) \ \ \text{.} \end{equation*} By a geometric interpretation of the Riemann-Roch theorem for algebraic curves, it follows that a point $L$ on the Theta divisor represented by the divisor $D = \sum_{j=1}^g{P_j}$ is smooth precisely when the linear span $\left<\phi(P_1), \cdots, \phi(P_g)\right>$ in $\mathbb{P} H^0(\mathcal{C},\mathcal{\omega_{\mathcal{C}}})$ is a hyperplane, where $\phi: \mathcal{C} \longrightarrow \mathbb{P} H^0(\mathcal{C},\mathcal{\omega_{\mathcal{C}}})^{\vee}$ denotes the canonical map of $\mathcal{C}$. Viewing now $\mathcal{J}$ as the quotient of $H^0(\mathcal{C}, \omega_{\mathcal{C}}^{\vee})$ by the lattice $H_1(\mathcal{C}, \mathbb{Z})$, the Gauss Map associates to $L$ the tangent space $\mathbb{P}(T_{L} \Theta)$, which is a hyperplane of $\mathbb{P}(T_{L} \mathcal{J}) = \mathbb{P} H^0(\mathcal{C}, \omega_{\mathcal{C}})^{\vee}$ defined as follows: \begin{align*} \mathit{G} \colon &\Theta \dashrightarrow \mathbb{P} H^0(\mathcal{C},\mathcal{\omega_{\mathcal{C}}})^{\vee} \\ \sum_{j=1}^g{P_j}& \mapsto \left<\phi(P_1), \cdots, \phi(P_g)\right> \ \ \text{.} \end{align*} It is then easy to conclude that, in this case, the Gauss map is dominant and generically finite, with degree $\binom{2g-2}{g-1}$.
\noindent Furthermore, in the particular case in which $\mathcal{C}$ is a genus $3$ non-hyperelliptic curve, which we assume to be embedded in $\mathbb{P}^2$ via the canonical map, $\Theta$ is smooth and the Gauss map is then nothing but the map which associates, to a divisor $P+Q$, the line in $\mathbb{P}^2$ spanned by $P$ and $Q$ if $P\neq Q$ and the tangent line $T_P(\mathcal{C})$ at $P$ if $P=Q$. In particular, the Gauss map $\mathit{G}$ is a covering of degree $6$ of $\mathbb{P}^2$ branched on $\mathcal{C}^{\vee}$, the dual curve of $\mathcal{C}$, which has $28$ nodes corresponding to the bitangent lines of $\mathcal{C}$, and $24$ cusps corresponding to the tangent lines passing through a Weierstrass points of $\mathcal{C}$. \end{ex1} However, we will see in Remark \ref{rmkGauss} that this good behavior of the Gauss map arises in more general situations. There is furthermore a close connection between the property for the Gauss map of a reduced and irreducible divisor $\mathcal{D}$ of being dominant, and the property for $\mathcal{D}$ of being ample and of general type. More precisely, it is known that a divisor $\mathcal{D}$ on an abelian variety is of general type if and only if there is no non-trivial abelian subvariety whose action on $A$ by translation stabilizes $\mathcal{D}$. Indeed, the following theorem holds:
\begin{thm1} \cite[Theorem 4 - Ueno's theorem]{KAWVIE} \label{Uenothm} Let $V$ a subvariety of an abelian variety $A$. Then there exist an abelian subvariety $B$ of $A$ and an algebraic variety $W$ which is a subvariety of an abelian variety such that \begin{itemize} \item $V$ is an analytic fiber bundle over $W$ whose fiber is $B$, \item $\kappa(W) = dim W = \kappa(B)$. \end{itemize} $B$ is characterized as the maximal connected subgroup of $A$ such that $B + V \subseteq V$. \end{thm1}
\begin{rmk} \label{rmkGauss} We can conclude that, for a reduced and irreducible divisor $\mathcal{D}$ on an abelian variety $A$, the following are equivalent: \begin{itemize} \item[1)] The Gauss map of $\mathcal{D}$ is dominant and hence generically finite. \item[2)] $\mathcal{D}$ is an algebraic variety of general type. \item[3)] $\mathcal{D}$ is an ample divisor. \end{itemize} Indeed, we recall that a divisor $\mathcal{D}$ on an abelian variety is ample if and only if it is not translation invariant under the action of any non-trivial abelian subvariety of $A$. (see \cite[p. 60]{MumfordAbelian}). The equivalence of $1)$ and $3)$ follows by (\cite[Proposition 4.4.2]{BLange}), and the idea is that if the Gauss map is not dominant, then $\mathcal{D}$ is not ample because it would be invariant under the action of a non-trivial abelian subvariety. The equivalence of $2)$ and $3)$ follows now easily by applying the Ueno's Theorem \ref{Uenothm}. \end{rmk} \begin{rmk}
If $\mathcal{D}$ is a smooth, ample divisor, then the Gauss map of $\mathcal{D}$ is a finite morphism. Indeed, if $\mathcal{D} = div(\theta)$ and the Gauss map $G$ contracted a curve $\mathcal{C}$, then without loss of generality we could suppose that $\frac{\partial \theta}{\partial z_j}$ is identically $0$ on $\mathcal{C}$ for every $j = 1 \cdots g-1$, and that $\frac{\partial \theta}{\partial z_g}$ has no zeros on $\mathcal{C}$. Thus, it would follow that $\omega_{\mathcal{D}}|_{\mathcal{C}} \cong \mathcal{O}_{\mathcal{C}}$, which would contradict the fact that $\omega_{\mathcal{D}}$ is ample on $\mathcal{D}$, $\omega_{\mathcal{D}}$ being the restriction of the ample line bundle $\mathcal{O}_{A}(\mathcal{D})$ to $\mathcal{D}$.
\noindent In particular, an ample smooth surface in an abelian $3$-fold $A$ is a minimal surface of general type. \end{rmk}
\begin{prop1} Let $V$ a complex vector space of dimenson $g \geq 1$ and $\Lambda$ a lattice in $V$. Let moreover $\Lambda'$ a sublattice of $\Lambda$ and $G = \bigslant{\Lambda}{\Lambda'}$ a finite abelian group. Denote: \begin{align*} A = \bigslant{V}{\Lambda'} \\ B = \bigslant{V}{\Lambda} \end{align*} and $p: A \longrightarrow B$ the natural projection. Then we have: \begin{equation*} p_*{\mathcal{O}_A} \cong \bigoplus_{\chi \in G^*} \mathcal{L}_{\chi} \end{equation*} where $\mathcal{L}_{\chi}$ is the line bundle on $B$ associated (accordingly to Appel-Humbert theorem) to the semicharacter $\chi \circ q$ on $\Lambda$, where $q$ is the projection of $\Lambda$ onto $G$. \end{prop1} \begin{proof}
We call $\pi_B: V \longrightarrow B$ and $\pi_A: V \longrightarrow A$ the projections. The stalk of $p_*\mathcal{O}_A$, which has dimension $|G|$, is a $\mathbb{C} G$-module isomorphic to the regular representation.The trivial vector bundle $\pi_B^*(p_*\mathcal{O}_A)$ can be expressed then as: \begin{equation*} \pi_B^*(p_*\mathcal{O}_A) \cong \mathbb{C}[G] \end{equation*} where $\mathbb{C}[G]$ is the group algebra of $G$ over $\mathbb{C}$. The regular action of $G$ on $\mathbb{C}[G]$ (the action by left multiplication) decomposes as the sum: \begin{equation*} \mathbb{C}[G] \cong \bigoplus_{\chi \in G^*} W_{\chi} \end{equation*} where $W_{\chi}$ is the irreducible $\mathbb{C}[G]$-module of dimension $1$ associated to the character $\chi$. To the $\mathbb{C}[G]$-module $W_{\chi}$ is associated the line bundle $\bigslant{V \times W_{\chi}}{\sim_{\chi} }$ on $B$ where: \begin{equation*} (v, t) \sim_{\chi} (v + \lambda, q(\lambda).t) = (v + \lambda, (\chi \circ q)(\lambda)t) \end{equation*} But this is exactly the line bundle on $B$ associated (accordingly to Appel-Humbert theorem) to the semicharacter $\chi \circ q$ on $\Lambda$. \end{proof}
Let us write $J = \bigslant{V}{\Lambda}$ a principally polarized abelian variety and $A = \bigslant{V}{\Lambda'}$ an abelian variety with $\Lambda'$ a sublattice of $\Lambda$. We have the projection $p: A \longrightarrow J$ and, accordingly with the previous proposition: \begin{equation*} p_*{\mathcal{O}_A} \cong \bigoplus_{\chi \in G^*} \mathcal{L}_{\chi} \end{equation*} We have then a copy of the group $G^*$ generated by the line bundles $\mathcal{L}_{\chi}$ immersed in $\widehat{J}$.
The polarization of $\widehat{J}$ is principal, and then the morphism $\phi_{\Theta}: J \longrightarrow \widehat{J}$ is an isomorphism. Then for every $\chi \in G^*$ there exists $g_{\chi} \in J$ such that: \begin{equation*}
\phi_{\Theta}(g_{\chi}) = \mathcal{O}_{J}(t_{g_\chi}^*\Theta) \otimes \mathcal{O}_{J}(\Theta)^{-1} = \mathcal{L}_{\chi} \end{equation*} this means that, then: \begin{equation} \label{projchar} \chi \circ q = e^{2 \pi i E(g_{\chi}, \cdot)} \end{equation} where $q$ is the projection of $\Lambda$ into $G$. Howewer, is clear that: \begin{equation*} g_{\chi} + g_{\chi'} = g_{\chi \chi'} \end{equation*} And in particular we obtain that: \begin{equation*}
G \cong \{g_{\chi} \ | \ \chi \in G^* \} \leq J \end{equation*} We obtain than $\Lambda_G$ a corresponding sublattice of $V$. Note, however, that for every $g \in \Lambda_G$ is defined: \begin{align*} \psi_{g}: \Lambda \longrightarrow \mathbb{S}^1 \\ \psi_{g}(\lambda) = e^{2 \pi i E(g, \lambda)} \end{align*} For the equation \ref{projchar}, all those characters on $\Lambda$ have the sublattice $\Lambda'$ in the kernel. Let's us then define: \begin{equation*}
\Lambda^G = \bigcap_{g \in \Lambda_G} Ker(\psi_g) = \{v \in V \ | \ E(v, \Lambda_G) \subseteq \mathbb{Z} \} \end{equation*} and we have that $\Lambda'$ is a sublattice of $\Lambda^G$. But the converse is also true. Infact, \begin{equation*}
G^* \cong \{\psi_g \ | g \in G\} \end{equation*} \begin{equation*} \bigslant {\Lambda}{\Lambda^G} \cong G^* \end{equation*} All the characters of $G$ are trivial on the subgroup $H = \bigslant{\Lambda^G}{\Lambda'}$ of $G = \bigslant{\Lambda}{\Lambda'}$, and then we conclude that $H = \{1\}$. \subsection{The polarization} Let us suppose as before $A$ a polarized abelian variety, with $\mathcal{L}$ an ample line bundle. Then we have an isogeny onto a principally polarized variety $p: A \longrightarrow J$, and an associated subgroup $G$ of $J$, with $\mathcal{L} = p^* \mathcal{O}_J(\Theta)$. The polarization is defined as: \begin{align*} \phi_{\mathcal{L}}(z) = p^*(\phi_{\theta}(z)) &= p^* (\mathcal{O}_J(t_z)(\Theta) \otimes \mathcal{O}(\Theta)^{-1}) \\ &= p^*\mathcal{L}(0, e^{2 \pi i E(z, \cdot)}) \\ &= \mathcal{L}(0, e^{2 \pi iE(z,\cdot)}) \end{align*} Its kernel is: \begin{align*}
Ker(\phi_{\mathcal{L}}) = \{\bar{v} \in \bigslant{V}{\Lambda^G} \ | \ e^{2 \pi i E(v, \lambda)} = 0 \ \ \forall \lambda \in \Lambda^G \} \end{align*} We have then that: \begin{equation*} Ker(\phi_{\mathcal{L}}) = \bigslant{\Lambda_G}{\Lambda^G} \end{equation*} To study the polarization and defining a suitable basis for the space of sections, I suppose: \begin{equation*} G \cong \mathbb{Z}_{s_1} \oplus \cdots \oplus \mathbb{Z}_{s_r} \end{equation*} with $s_i$ dividing the next.
\noindent The alternating bilinear form $E$ takes than on $\Lambda_G$ values in $\frac{1}{s_r^2} \mathbb{Z}$. There must then exist $\eta_1 \cdots \eta_g, \delta_1 \cdots \delta_g$ generators of $\Lambda_G$ such that: \begin{equation*} E(\eta_i, \delta_i) =\frac{a_i}{s_r^2} \end{equation*} with $a_i$ integers dividing the next.
\begin{lem1} With $u_i$ orders of $\eta_i$ in $\Lambda_G$ modulo $\Lambda$ and $v_i$ orders of $\delta_i$ in $\Lambda_G$ modulo $\Lambda$, we have that: \begin{align*} a_i = \frac{s_r^2}{u_iv_i} \end{align*} \end{lem1} \begin{proof} In particular, since $s_r \eta_i \in \Lambda$ and $s_r \delta_i \in \Lambda$, and that: \begin{equation*} E(s_r \eta_i, \frac{s_r}{a_i} \delta_i ) = 1 \end{equation*} We obtain that $\frac{s_r}{a_i} \delta_i \in \Lambda$ and $\frac{s_r}{a_i} \eta_i \in \Lambda$. From this is clear that $\frac{s_r}{a_i}$ is a natural number, and $u_i$ divides $\frac{s_r}{a_i}$. But we obtain even that: \begin{equation*} E(u_i\eta_i, v_i\delta_i) = 1 \end{equation*} and this means that: \begin{equation*} \frac{1}{u_i v_i} = \frac{a_i}{s_r^2} \end{equation*} \end{proof} The order of $\eta_i$ in $\bigslant{\Lambda_G}{\Lambda^G}$ is then the same of the order of $\delta_i$, and this is exactly $u_iv_i$, since $\chi_{\eta_i}(u_iv_i \delta_i)=1$. From this follows that the polarization of $A$ is exactly: \begin{equation*} (d_1 \cdots d_g) = (u_gv_g, \cdots u_1v_1) \end{equation*} \subsection{The basis of theta functions}
We take as a decomposition of $V$ for $\mathcal{L}$ the decomposition $V = V_1 \oplus V_2$, $V_1$ generated by $\delta_1 \cdots \delta_g$ and $V_2$ generated by $\eta_1 \cdots \eta_g$. The decomposition distinguishes a semicharacter $\chi_0$ on $\Lambda$ whose extension on $V$ is: \begin{equation*} \chi_0(v) = e^{\pi i E(v_1, v_2)} \end{equation*}
$H$ has no imaginary part on $V_1$, and so $H$ is symmetric on $V_1$. Is then defined $B$ the $\mathbb{C}$-linear extension of $H|_{V_1 \times V_1}$ to $V$, because $V_1$ generates $V$ as $\mathbb{C}$-vector space. We have then: \begin{equation*} (H-B)(v,w) = \begin{cases} 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \text{if} (v,w) \in V \times V_1 \\
2iE(v,w) \ \ \text{if} (v,w) \in V_1 \times V
\end{cases} \end{equation*} We have then that, with $v \in V_1$, $w \in V$: \begin{align*} (H-B)(v,w) &= \overline{H(w,v)} - B(w,v) \\ &= \overline{H(w,v)} - B(w,v) \\ &= \overline{H(w,v)} - H(w,v) \\ &= -2i E(w, v) = 2iE(v,w) \end{align*}
Respect to the sum, $\chi_0$ behaves in the following way. Take $v$, $w$ in $V$. Then we have: \begin{align*} \chi_0(v+w) &= e^{\pi i E(v_1 + w_1, v_2 + w_2)} \\ &= \chi_0(v) \chi_0(w)e^{\pi i E(v_1, w_2) + \pi i E(w_1, v_2) } \\ &= \chi_0(v) \chi_0(w) e^{\pi i E(v,w) - 2 \pi i E(v_2, w_1)} \end{align*} Let us suppose that $\mathcal{O}_J(\Theta) = L(\chi_0, H)$, $\chi_0$ a semicharacter on $\Lambda$, namely, satysfing: \begin{equation*} \chi_0(\lambda + \mu) = \chi_0(\lambda)\chi_0(\mu)e^{\pi i E(\lambda, \mu) } \end{equation*} Then: \begin{equation*}
\mathcal{L} = p^{*}(\mathcal{O}_J(\Theta) ) = p^{*}L(\chi_0, H) = L(\chi_0|_{\Lambda^G}, H) \end{equation*} Then I write the cocyles respect to this decomposition: with $v \in V$ we define \begin{equation*} \phi_{v}(z) = \chi_0(v) e^{\pi (H-B)(z, v) + \frac{\pi}{2}(H-B)(v, v)} \end{equation*} They satysfies, for every $u$, $v$ and $z$ in $V$ the following relation: \begin{equation*} \phi_{u+v}(z) = \phi_u(z+v)\phi_v(z)e^{\frac{\pi}{2}(H-B)(v,v) - \frac{\pi}{2}(H-B)(u,u)} \end{equation*} Indeed: \begin{align*}
& \phi_u(z+v)\phi_v(z) \\
&= \chi_0(u)\chi_0(v)e^{\pi (H-B)(z+v, u) + \frac{\pi}{2}(H-B)(u, u)} e^{\pi (H-B)(z, v) + \frac{\pi}{2}(H-B)(v, v)} \\ &= \chi_0(u)\chi_0(v)e^{\pi (H-B)(z, v+u) + \pi (H-B)(v,u) + \frac{\pi}{2}(H-B)(u, u) + \frac{\pi}{2}(H-B)(v, v)} \\ &= \chi_0(u)\chi_0(v)e^{\pi (H-B)(z, v+u) + \frac{\pi}{2} (H-B)(v+u,u) + \frac{\pi}{2}(H-B)(v, v+u)} \\ &= e^{-\pi i E(u,v) + 2 \pi i E(u_2, v_1)} \phi_{u+v}(z) e^{- \frac{\pi}{2}(H-B)(v+u,v) + \frac{\pi}{2}(H-B)(v, v+u)} \\ &= e^{-\pi i E(u_1,v_2) + -\pi i E(v_1,u_2) - \pi i E(u_1,v_2) + \pi i E(v_1, u_2)} \phi_{u+v}(z) \\ &= e^{-2 \pi i E(u_1, v_2)}\phi_{u+v}(z) \end{align*} We define the theta functions: with $\eta \in {\Lambda_{G,2}}$: \begin{equation*} \theta^*[\eta](z) := \phi_{\eta}(z)^{-1}\theta(z+\eta) \end{equation*} \begin{lem1} \label{equalityactions} For $g \in \Lambda$, we have: \begin{equation*} \theta^*[\eta](z+g) = e^{ - 2 \pi i E(g_1, \eta) } \phi_{g_2}(z) \theta^*[\eta](z) \end{equation*} \end{lem1} \begin{proof} \begin{align*} \theta^*[\eta](z+g) &= \phi_{\eta}(z+g)^{-1}\theta(z+ \eta + g) \\
&= \phi_{\eta}(z+g)^{-1} \phi_g(z+\eta) \theta(z+ \eta ) \\
&= \phi_{\eta}(z+g)^{-1} \phi_g(z+\eta) \phi_{\eta}(z) \theta^*[\eta](z) \\
&= \phi_{\eta}(z)^{-1} e^{-\pi(H-B)(g, \eta) } \phi_g(z+\eta) \phi_{\eta}(z) \theta^*[\eta](z) \\
&= e^{-\pi(H-B)(g, \eta) } \phi_g(z+\eta) \theta^*[\eta](z) \\
&= e^{\pi(H-B)(\eta, g) -\pi(H-B)(g, \eta) } \phi_g(z) \theta^*[\eta](z) \\
&= e^{2 \pi i E(\eta_1, g_2) - 2 \pi i E(g_1, \eta_2) } \phi_{g_2}(z) \theta^*[\eta](z) \\
&= e^{ - 2 \pi i E(g_1, \eta) } \phi_{g_2}(z) \theta^*[\eta](z) \end{align*} \end{proof}
This means in particular that $\theta^*[\eta]$ satisfies, for every $\mu \in \Lambda^{G}_2$: \begin{equation*} \theta^*[\eta](z+\mu) = \phi_{\mu}(z) \theta^*[\eta](z) \end{equation*}
\begin{prop1} The sections $\theta^*[\eta]$, with $\eta \in \bigslant{\Lambda_{G,2}}{\Lambda^{G}_2 \cong G}$ give a basis for the sections of $H^0(A, \mathcal{L})$. \end{prop1} \begin{proof} We observe first that, for every $\lambda \in \Lambda_1$ we have that: \begin{equation*} \phi_{\lambda}(z) \equiv 1 \end{equation*} This means, in particular, that, for every $\eta \in \bigslant{\Lambda_{G,2}}{\Lambda^{G}_2}$ and $\lambda \in \Lambda_1$, we have: \begin{align*} \theta^*[\eta](z+\lambda) &= \phi_{\eta}(z + \lambda)^{-1}\theta(z+\eta + \lambda) \\
&= \phi_{\eta}(z) \theta(z+\eta) \phi_{\lambda}(z+\eta) \theta(z+\eta) \\
&= \phi_{\eta}(z) \theta(z+\eta) \\
& =\theta^*[\eta](z) \end{align*} Thus, $\theta^*[\eta]$ is periodic respect to $\Lambda_1$. We can then write $\theta^*[\eta]$ in the following form: \begin{equation*} \theta^*[\eta](z) = \sum_{u \in \Lambda_{G,2}} \alpha_{u} e^{\pi (H-B) (z,u) } \end{equation*} If we take $\lambda \in \Lambda_2$, we obtain: \begin{align*} \phi_{\lambda}(z)\theta^*[\eta](z) = \theta^*[\eta](z + \lambda) &= \sum_{u \in \Lambda_{G,2}} \alpha_{u} e^{\pi (H-B) (z+\lambda,u) } \\ &= \sum_{u \in \Lambda_{G,2}} \alpha_{u} e^{\pi (H-B) (z ,u)}e^{\pi (H-B) (\lambda ,u) } \end{align*} Thus: \begin{align*} \theta^*[\eta](z) &= \phi_{\lambda}(z)^{-1} \sum_{u \in \Lambda_{G,2}} \alpha_{u} e^{\pi (H-B) (z ,u)}e^{\pi (H-B) (\lambda ,u) } \\ &= e^{-\pi (H-B)(z,\lambda) - \frac{\pi}{2}(H-B)(\lambda, \lambda)} \sum_{u \in \Lambda_{G,2}} \alpha_{u} e^{\pi (H-B) (z ,u)}e^{\pi (H-B) (\lambda ,u) } \\ &= \sum_{u \in \Lambda_{G,2}} (\alpha_{u} e^{- \frac{\pi}{2}(H-B)(\lambda, \lambda)}) e^{\pi (H-B) (z ,u - \lambda)}e^{\pi (H-B) (\lambda ,u) } \\ &= \sum_{u \in \Lambda_{G,2}} (\alpha_{u+\lambda} e^{- \frac{\pi}{2}(H-B)(\lambda, \lambda)}) e^{\pi (H-B) (z ,u )}e^{\pi (H-B) (\lambda ,u+\lambda) } \\ &= \sum_{u \in \Lambda_{G,2}} (\alpha_{u+\lambda} e^{\frac{\pi}{2}(H-B)(\lambda, \lambda)}) e^{\pi (H-B) (z ,u )} \end{align*} This means that the coefficient must only satisfy for every $u \in \Lambda_{G,2}$ and every $\lambda \in \Lambda_{2}$: \begin{equation*} \alpha_{u+\lambda} = \alpha_{u} e^{- \frac{\pi}{2}(H-B)(\lambda, \lambda)} \end{equation*}
\end{proof} To conclude, let's study the action of \begin{equation*} K(\mathcal{L}) = \bigslant{\Lambda_G}{\Lambda^G} = \bigslant{\Lambda_{G,1}}{\Lambda^G_1} \oplus \bigslant{\Lambda_{G,2}}{\Lambda^G_2} \cong G \oplus G \end{equation*} We have then:
\noindent For the second copy of $G$, lets take $g$ in $\Lambda_{G,2}$. \begin{align*} \theta^*[\eta](z+g) &= \phi_{\eta}(z+g)^{-1}\theta(z + \eta + g) \\ &= \phi_{\eta}(z+g)^{-1} \phi_{\eta + g}(z) \theta^{*}[\eta + g](z) \\ &= e^{\frac{\pi}{2}(H-B)(\eta,g) - \frac{\pi}{2}(H-B)(g,\eta) + \frac{\pi}{2}(H-B)(g,g) } e^{\pi(H-B)(z,g)} \theta^*[\eta+g](z) \end{align*} For every $v$,$w$ in $V_2$, we have that: \begin{equation*} (H-B)(v,w) - (H-B)(w,v) = 0 \end{equation*} Indeed, if $v = iv'$ and $w = iw'$, we have that: \begin{align*} (H-B)(v,w) - (H-B)(w,v) &= (H-B)(v,w) - (H-B)(w,v) \\
&= (H-B)(iv',iw') - (H-B)(iw',iv') \\
&=H(v',w')+ B(v',w') - H(w',v') - B(w',v') \\
&= H(v',w') - H(w',v') = H(v',w') - H(w',v') = 0 \end{align*} So we conclude that, for every $g$ in $\Lambda_{G,2}$. \begin{align*} \theta^*[\eta](z+g) &= e^{ \frac{\pi}{2}(H-B)(g,g) } e^{\pi(H-B)(z,g)} \theta^*[\eta+g](z) \end{align*}
\subsection{Base points} In this section let $A$ an abelian variety of dimension $g$ with $\mathcal{L}$ a symmetric ample line bundle of type $(d_1 \cdots d_g)$. Let $K(\mathcal{L})$ the kernel of the polarization: \begin{equation*} \phi_{\mathcal{L}} : A \longrightarrow \widehat{A} \end{equation*} The first observaton is the following: in $\zeta \in A$ is a base point for the linear system, then $-\zeta$ is again a base point (v. p. 91 Birkenhake, 6,4). However, the group $K(\mathcal{L})_2$ acts on the set of base points (see ex.2 p. 69 Birkenhake). Even the group $K(\mathcal{L})_1$ acts on this set, and th action of $K(\mathcal{L})$ is faithful on this set by translation.
Note that if $d_1 \geq 2$ then the system has no base points (the associated map is in this case a morphism), so suppose $d_1 = 1$. We have: \begin{equation*} \mathcal{L}^{n} = g! d_2 \cdot \cdots d_g \end{equation*} so, if the linear system has no fixed components, we have at most $n! d_2 \cdot \cdots d_g$ points. On the other sides: \begin{equation*} K(\mathcal{L}) \cong \mathbb{Z}_{d_2}^2 \times \cdots \mathbb{Z}_{d_g}^2 \end{equation*} So, in the case we have a linear system with base points and without fixed components, then there exist a natural $k \geq 1$ such that: \begin{equation*}
kd_2^2 \cdots d_g^2 = |\mathcal{B}| \end{equation*} and: \begin{equation*}
g! d_2 \cdots d_g \geq |\mathcal{B}| = kd_2^2 \cdots d_g^2 \end{equation*} so we conclude that with such a $k$ we have: \begin{equation*} g! \geq k d_2 \cdots d_g \end{equation*} \subsection{Etale Abelian covers and curves}
Every curve will be considered to be nonsingular and projective over the field $\mathbb{C}$. We introduce the notion of abelian cover of curves by referring to the more general definition of abelian cover of algebraic variety as introduced in the article \cite{Pardini}. \begin{def1} Let $C$ a curve. \textbf{A cover of $C$ with group $G$} is a finite map $\pi: C' \longrightarrow C$ together with a faithful action of $G$ on $X$ such that $\pi$ exhibits $C$ as the quotient of $C'$ via $G$. \end{def1}
\noindent Let us consider a curve $C$ of genus $g \geq 1$. We will denote always: \begin{align*} V &:= H^0(C, \omega_C)^{\vee} \\ \Lambda &:= H_1(C, \mathbb{Z}) \end{align*} The abelian group $\Lambda$ will be considered as included in $V$ as a lattice via the integral operator. Moreover, we denote the Jacobian variety of $C$ as: \begin{align*} J(C) = \bigslant{V}{\Lambda} \end{align*} On $\Lambda$ is defined $E: \Lambda \times \Lambda \longrightarrow \mathbb{Z}$ the bilinear alternating form given by the intersections of cycles, and with the same letter we refer us to its linear extension on $V$. $E$ is the alternating bilinear form of the principal polarization on $J(C)$ associated to the non-degenerate hermitian form $H: V \times V \longrightarrow \mathbb{C}$ defined from $E$ as following: \begin{equation*} H(v,w) = iE(v,w) + E(iv,w)
\end{equation*}
Given an Etale abelian cover $\pi: C' \longrightarrow C$ of curves with group $G$, there exists a normal subgroup $H$ of $\pi_1(C)$ with quotient group $\bigslant{\pi_1(C)}{H}$ isomorphic to $G$ and, because $G$ is abelian, this group $H$ cooresponds to a sublattice $\Lambda'$ of $\Lambda$ with quotient isomorphic to $G$. The aim is then to construct a cartesian diagram of the type: \begin{diagram}
C' &\rInto & A \\
\dTo_{\pi} & &\dTo_{p} & \\
C &\rInto &J = J(C)
\end{diagram}
and determine the polarization of $A$.
\noindent Accordingly to the results contained in the article \cite{Pardini}, every such a cover is determined by a group of torsion bundles isomorphic to $G$ itself, and then it makes sense to consider first the $G$-covers of the Jacobian variety, $G$ an abelian group embedded in $J(C)$.
\noindent Let $\Lambda_G$ be the corresponding lattice in $V$ with: \begin{align*} \Lambda \leq \Lambda_G \leq V \\ \bigslant{\Lambda_G}{\Lambda} \cong G \end{align*} We denote the isomorphism $\phi_{\Theta}: J(C) \longrightarrow Pic^0(J(C)) \cong Hom(\Lambda, \mathbb{S}^{1})$ associated to the polarization given by the Theta-divisor $\Theta$ of $C$: \begin{align*} \phi_{\Theta}: Pic^0(C) \cong &J(C) \longrightarrow Pic^0(J(C)) \cong Hom(\Lambda, \mathbb{S}^{1})\\ \phi_{\Theta}(x)&:=\mathcal{O}_{J(C)}(t_x^*\Theta) \otimes \mathcal{O}_{J(C)}(\Theta )^{-1} \end{align*} Given $x \in J(C)$, the line bundle $\phi_{\Theta}(x)$ on $J(C)$ is defined (respect to the Appel-Humbert theorem) by the semicharacter: \begin{equation*} e^{2 \pi i E(x, \cdot)}: \Lambda \longrightarrow \mathbb{S}^1 \end{equation*} We could consider the image in $Hom(\Lambda, \mathbb{S}^{1})$ of the group $G$ via $\phi_{\Theta}$: Given $g=[\mu] \in G$, the associated character $\chi_{g}: \Lambda \longrightarrow \mathbb{S}^1$ is defined by: \begin{equation*}
\chi_g(\lambda) = e^{2 \pi i E(\mu, \lambda)} \end{equation*} \noindent We can then consider moreover: \begin{equation*}
\Lambda^G = \bigcap_{g \in \Lambda_G} Ker(\chi_g) = \{v \in V \ | \ E(v, \Lambda_G) \subseteq \mathbb{Z} \} \end{equation*} This is a sublattice of $\Lambda$ with quotient group $\bigslant{\Lambda}{\Lambda^G}$ canonically isomorphic to $G^*$, with isomorphism induced by: \begin{align*} \Phi: \Lambda \longrightarrow G^* \\ \Phi(\lambda)(g) = \chi_g(\lambda) \end{align*} which is surjective with kernel $\Lambda^G$. We define then the complex torus: \begin{equation*} A^G := \bigslant{V}{\Lambda^G} \end{equation*} with the natural projection $p: A^G \longrightarrow J(C)$. In the next step we show that with the characters $\chi_g$ we can describe the decomposition of the sheaf $p_*\mathcal{O}_{A^G}$ in line bundles on $J(C)$.
\begin{prop1} \label{mainproposition} Given $G$ finite subgroup di $J(C)$ there exists up to isomorphism an unique Etale abelian abelian cover $\pi: C^{G} \longrightarrow C$ such that: \begin{equation*} \pi_*\mathcal{O}_G \cong \bigoplus_{g \in G} g \end{equation*} This is moreover a pullback of a type: \begin{diagram}
C^G &\rInto & A^G \\
\dTo_{\pi} & &\dTo_{p} & \\
C &\rInto &J = J(C)
\end{diagram} where $A^G$ is an abelian variety, with polarization depending on the generators of $G$. Conversely, every abelian Etale abelian cover $\pi: C' \longrightarrow C$ of group $G$ can be obtained in this way. \end{prop1} \begin{proof} The second assertion follow from \cite{Pardini}, proposition 2.1 p. 199. We prove then the first. Given the group $G$ in $J(C)$, and defined $A^G$ as before, we define $C^G$ as the pullback $C \times_{J} A^G$. From the previous proposition we have: \begin{equation*} p_*{\mathcal{O}_{A^G}} \cong \bigoplus_{\chi \in G^*} \mathcal{L}_{\chi} \end{equation*} where $\mathcal{L}_{\chi}$ is the line bundle on $J(C)$ associated to the semicharacter $\chi \circ q$ on $\Lambda$, where $q$ is the projection of $\Lambda$ onto $G$. But then $\chi \circ q$ belongs to the image of $\phi_{\Theta}$ in $Hom(\Lambda, \mathbb{S}^1)$. So there exists $g \in G$ such that: \begin{equation*} \mathcal{L}_{\chi} = \mathcal{O}_{J(C)}(t_g^*\Theta) \otimes \mathcal{O}_{J(C)}(\Theta )^{-1} = \phi_{\Theta}(g) \end{equation*} and we can write: \begin{equation*} p_*{\mathcal{O}_{A^G}} \cong \bigoplus_{g \in G} \phi_{\Theta}(g) \end{equation*}
The restriction $|_C: Pic^0(J(C)) \longrightarrow Pic^0(C) \cong J(C)$ is an isomorphism and we have (see Birkenhake-Lange 11.3.5 p.329): \begin{equation*}
\phi_{\Theta}(g)|_C = g^{-1} \end{equation*} From this we can conclude the first point. \end{proof} \section{On the canonical map of the bidouble cover of a principal polarization} \label{basicsectionssect}
In this section, we consider surfaces in the polarization of a general $(1,2,2)$-polarized Abelian 3-fold $(A, \mathcal{L})$, which are étale bidouble covers of the principal polarization $\Theta$ of a Jacobian threefold. More precisely, we are interested in the canonical map of a surface $\mathcal{S}$ in the polarization of a general $(1,2,2)$-polarized Abelian 3-fold $A$, such that there exists a pullback diagram \begin{align} \label{diagrampullbackTheta} \begin{diagram} \mathcal{S} & \rInto & A& \\ \dTo_{p} & & \dTo_{p} \\ \Theta & \rInto & \mathcal{J} \end{diagram} \end{align} where $p: A \longrightarrow \mathcal{J}$ denotes an isogeny onto the Jacobian variety of a smooth quartic plane curve $\mathcal{D}$, with kernel \begin{equation} \label{KERGEQ}
\mathcal{G} := Ker(p) \cong \mathbb{Z}_2 \times \mathbb{Z}_2 \ \ \text{.} \end{equation} \begin{rmk} \label{CHOICENOTGENERAL}
We remark that those surfaces are not general in their linear system. Indeed, the number of moduli for the couples $(A, \mathcal{S})$, where $(A, \mathcal{L})$ is a $(1,2,2)$-polarized abelian threefold and $\mathcal{S}$ belongs to the linear system $|\mathcal{L}|$ is $6$. On the other hand, an ètale bidouble cover as in Diagram \ref{diagrampullbackTheta} is defined by a couple of $2$-torsion line bundles on the basis variety $\mathcal{J}$. Since there are only finitely many possible choices for the couples of $2$-torsion line bundles on $\mathcal{J}$, the number of moduli for étale bidouble covers as in Diagram \ref{diagrampullbackTheta} is $3$. \end{rmk}
\noindent In the setting of Diagram \ref{diagrampullbackTheta}, it is convenient to introduce some notations. We express $\mathcal{J}$ as a quotient in the form $\mathcal{J} \cong \bigslant{\mathbb{C}^3}{\Lambda}$, where $\Lambda$ is the lattice in $\mathbb{C}^3$ defined as \begin{equation} \label{choosendecom}
\Lambda := \Lambda_{1} \oplus \Lambda_2 \ \ \text{,} \end{equation} with $\Lambda_1 := \tau \mathbb{Z}^3$, $\Lambda_2:=\mathbb{Z}^3$ and $\tau$ is a general point in the Siegel upper half-space $\mathcal{H}_3$ of $3 \times 3$ symmetric matrices with positive definite imaginary part. The polarization $\mathcal{O}_{\mathcal{J}}(\Theta)$ on $\mathcal{J}$, which can be assumed up to translation to be of characteristic $0$ with respect to the decomposition above in (\ref{choosendecom}), is defined by the hermitian form $H$ whose matrix with respect to the standard basis of $\mathbb{C}^3$ is $(\IMh{\tau})^{-1}$. In particular, since the line bundle on $\mathcal{J}$ is assumed to be of characteristic $0$, the divisor $\Theta$ is the zero locus of the Riemann theta function: \begin{equation} \label{RIEMANNTHETA} \theta_0(z, \tau) := \sum_{n \in \mathbb{Z}^3} e^{\pi i \cdot \traspose{n}\tau n + 2 \pi i \cdot \traspose{n}z} \ \ \text{.} \end{equation} By virtue of the pullback diagram in (\ref{diagrampullbackTheta}), we may assume that \begin{equation*} (A, \mathcal{L}) = \left(\bigslant{\mathbb{C}^3}{\Gamma}, p^* \mathcal{O}_{\mathcal{J}}(\Theta) \right) \ \ \text{,} \end{equation*} where $\Gamma$ is the lattice \begin{equation*} \Gamma := \SET{\lambda \in \Lambda}{\IMh{H}(\lambda, \eta_j) \in \mathbb{Z}, \ \ j = 1,2} \ \ \text{,} \end{equation*} where $\eta_1, \eta_2$ is a couple of $2$-torsion points in $\mathcal{J}[2]$. We recall that $\IMh{H}$ is an alternating bilinear form which is moreover integrally valued on the lattice $\Lambda$: for every $u$, $v$ in $\mathbb{Z}_3$ it holds: \begin{equation} \label{bHpair} \IMh{H}(\tau \cdot u, v) = \traspose{u}\cdot v
\end{equation}
\noindent The decomposition (\ref{choosendecom}) induces a real decomposition of $\mathbb{C}^3$ into the direct sum of two real subvector spaces $V_j:= \Lambda_j \otimes_{\mathbb{Z}} \mathbb{R}$ defined by extending the scalars, and
by intersection we have an analogous decomposition of each of the sublattices \begin{equation*} \Gamma \subseteq \Lambda \subseteq \Gamma(\mathcal{L}) := \Lambda + \eta_1 \mathbb{Z} + \eta_2 \mathbb{Z} \ \ \text{.} \end{equation*}
With $j = 1,2$, we denote the members of this decomposition as follows \begin{align*} \Gamma_j:= \Gamma \cap V_j \subseteq \Lambda_j \subseteq \Gamma(\mathcal{L})_j := \Gamma(\mathcal{L}) \cap V_j \ \ \text{.} \end{align*}
Without loss of generality, we can assume that both $\eta_1$ and $\eta_2$ belong to $V_1$. Moreover, since we assume the polarization of $A$ to be of type $(1,2,2)$, the elements $\eta_1$ and $\eta_2$ must be orthogonal to each other with respect to the Weil pairing: \begin{equation} \label{bpair} W \colon \mathcal{J}[2] \times \mathcal{J}[2] \longrightarrow \bigslant{\frac{1}{4} \mathbb{Z}}{\frac{1}{2} \mathbb{Z}} \cong \mathbb{Z}_2 \ \ \text{.} \end{equation} For the reader convenience, we recall that the Weil pairing in (\ref{bpair}) is the natural symplectic pairing on $\mathcal{J}[2]$ induced by the alternating bilinear form $\IMh{H}$. The latter orthogonality condition ensures indeed that the quotient $\bigslant{\Gamma(\mathcal{L})_1}{\Gamma_1}$ is isomorphic to $\mathbb{Z}_2 \times \mathbb{Z}_2$ and generated by $\eta_1$ and $\eta_2$.
The kernel of the isogeny $p$ in the pullback diagram in (\ref{diagrampullbackTheta}), which we denoted by $\mathcal{G}$, is then isomorphic to $\bigslant{\Gamma(\mathcal{L})_2}{\Gamma_2}$, and we can find two involutions $a$ and $b$ in $\mathcal{G}$ such that $W(\eta_1, a) = W(\eta_2, b) = 0$ and $W(\eta_0, b) = W(\eta_1, a) = 1$.
The Weil pairing $W$, being a perfect pairing of $\mathbb{Z}_2$-modules, allows us to identify $\bigslant{\Gamma(\mathcal{L})_1}{\Gamma_1}$ with the character group $\mathcal{G}^*$: indeed, every element $\tau \cdot \gamma$ of $\bigslant{\Gamma(\mathcal{L})_1}{\Gamma_1}$ can be identified with the character defined on $\mathcal{G}$ as follows: \begin{equation} \label{chardef} \gamma(g) := e^{\pi i W(\tau \gamma, g)} = e^{2\pi i \traspose{\gamma} \cdot g} \end{equation}
\noindent The vector space $H^0(A, \mathcal{L})$ has dimension $4$, and it is generated by the theta functions $\theta_{0}$, $\theta_{\alpha}$, $\theta_{\beta}$, $\theta_{\alpha + \beta}$ which are defined, for every element $\gamma$ of $\mathcal{G}^{*}$ as follows \begin{equation} \label{thetadef1} \theta_{\gamma}(z, \tau) := e^{\pi i \traspose{\gamma} \tau \gamma +2\pi i \traspose{\gamma}z }\theta_0(z + \gamma) \ \ \text{.} \end{equation}
In conclusion, one can easily see that, for every $\gamma \in \mathcal{G}$ and every $g$ in $\mathcal{G}$, we have \begin{equation} \label{THETAGAMMA} \theta_{\gamma}(z + g) = e^{\pi i \lambda(\gamma, g)} = \gamma(g) \theta_{\gamma}(z) \ \ \text{.} \end{equation}
\begin{not1}\label{THETAGAMMANOTATION} From the above definition in (\ref{thetadef1}), it follows that the zero locus of each section $\theta_{\gamma}$ in $A$ is just a translated of $\mathcal{S}$, where \begin{equation*} (\theta_{\gamma})_0 = \mathcal{S} + \gamma \end{equation*} We will denote this locus by $\BBF{\mathcal{S}_g}$, where $g$ is the only element of $\mathcal{G}$ whose action leaves the section $\theta_{\gamma}$ invariant according to Equation \ref{THETAGAMMA}. \end{not1}
\begin{rmk} The multiplication by $-1$ on $A$ leaves each of these surfaces invariant, since each function $\theta_{\gamma}$ is an even function. Furthermore, it can be easily seen that the base locus $\mathcal{B}(\mathcal{L})$ of the line bundle $\mathcal{L}$ is a set of cardinality $16$ which containes precisely those $2$-torsion points on $A$ on which every function $\theta_{\gamma}$ vanishes with odd multiplicity.
\noindent The following theorem shows that, on a surface $\mathcal{S}$ which is an étale bidouble cover of a theta divisor in a general principally polarized abelian threefold the differential of the canonical map is injective at every point. However, the same theorem states that the canonical map fails to be injective precisely on the canonical curves of $\mathcal{S}$ of the type $\mathcal{S} \cap \mathcal{S}_{\gamma}$.
\end{rmk} \begin{thm1} \label{injectivedifferential} Let $\mathcal{S}$ be an étale bidouble cover of a theta divisor in a general principally polarized abelian threefold, and let be \begin{align*} \begin{diagram} \mathcal{S} & \rInto & A& \\ \dTo_{p} & & \dTo_{p} \\ \Theta & \rInto & \mathcal{J} \end{diagram} \end{align*} the corresponding pullback diagram.
\begin{itemize} \item[1)] If $P$ and $Q$ are two distinct points on $\mathcal{S}$ whose image with respect to the canonical map of $\mathcal{S}$ is the same, then one of the following cases occurs:
\begin{itemize} \item[$\bullet$] $Q = -g.P$ for some non-trivial element $g$ of $\mathcal{G}$. This case arises precisely when $P$ and $Q$ belong to the canonical curve $\mathcal{S} \cap \mathcal{S}_{g}$, where $\gamma(g)=1$. \item[$\bullet$] $Q = g.P$ for some non-trivial element $g$ of $\mathcal{G}$. This case arises precisely when $P$ and $Q$ belong to the translate $\mathcal{S}_{h}$, for every $h \in \mathcal{G} - \{g\}$.
\item[$\bullet$] $P$ and $Q$ are two base points of $|\mathcal{S}|$ which belong to the same $\mathcal{G}$-orbit. \end{itemize}
\item[2)] The differential of the canonical map of $\mathcal{S}$ is injective at every point. \end{itemize} \end{thm1} To prove Theorem \ref{injectivedifferential}, we begin by fixing some notations and recalling some known facts about the behavior of the canonical map of the surfaces in a polarization of type $(1,1,2)$, which has been developed by F. Catanese and F-O Schreier in their joint work \cite{CataneseSchreyer1}. The strategy we use to prove the first claim is essentially then to consider the canonical projections which naturally arise from the isogenies onto $(1,1,2)$-polarized abelian threefolds.\newline
\noindent To this purpose, for every non-trivial element $g$ of $\mathcal{G}$, we denote by $A_g$ the $(1,1,2)$-polarized abelian threefold obtained as the quotient of $A$ by $g$, by $q_{g}$ the projection of $A$ onto $A_g$, and by $\mathcal{T}_g$ the image $q_g(\mathcal{S})$ in $A_g$. We have then the following diagram: \begin{align} \label{diagramproj} \begin{diagram}
A &\lInto&\mathcal{S} & \rTo^{\phi_{\mathcal{S}}} & \Sigma &\rInto& \mathbb{P}^5 \\
\dTo_{q_g} && \dTo_{q_g} & &\dTo &&\dDashto \\
A_{g}&\lInto&\mathcal{T}_{g} & \rTo^{\phi_{\mathcal{T}_g}} &\Sigma_g &\rInto&\mathbb{P}^3
\end{diagram}
\end{align} where $\phi_{\mathcal{S}}$ denotes the canonical map of $\mathcal{S}$ and $\Sigma$ its image in $\mathbb{P}^5$. \newline
\noindent The canonical map $\phi_{\mathcal{T}_g}$ is defined by the theta functions $[\theta_{\gamma}, \frac{\partial \theta_{0}}{\partial z_1}, \frac{\partial \theta_{0}}{\partial z_2}, \frac{\partial \theta_{0}}{\partial z_3}]$, where $\theta_{\gamma}$ is the only holomorphic section of $\mathcal{S}$ which is invariant under the action of $g$ and anti-invariant under the action of every other non-trivial element.
\noindent Under this setup, we can state the following theorem (see \cite[Theorem 6.4]{CataneseSchreyer1}) \begin{thm1} \label{CATSCHREYER} Let $\mathcal{T}$ be a smooth divisor yielding a polarization of type $(1,1,2)$ on an Abelian threefold. Then the canonical map of $\mathcal{T}$ is, in general, a birational morphism onto a surface $\Sigma$ of degree $12$ in $\mathbb{P}^3$.
\noindent In the special case where $\mathcal{T}$ is the inverse image of the theta divisor in a principally polarized Abelian threefold, the canonical map is a degree $2$ morphism onto a sextic surface $\Sigma$ in $\mathbb{P}^3$. In this case the singularities of $\Sigma$ are in general: a plane cubic $\Gamma$ which is a double curve of nodal type for $\Sigma$ and, according to \cite[Definition 2.5]{Catanese1981}, a strictly even set of $32$ nodes for $\Sigma$. Also, in this case, the normalization of $\Sigma$ is in fact the quotient of $\mathcal{T}$ by an involution $i$ on $A$ having only isolated fixed points (on $A$), of which exactly $32$ lie on $\mathcal{T}$. \end{thm1}
For the sake of exposition, it is useful to recall here the basic ideas of the proof of Theorem \ref{CATSCHREYER}. We assume that $\mathcal{T}$ is defined as the zero locus of the Riemann Theta function $\theta_0$ (as defined in \ref{thetadef1}) in a general $(1,1,2)$-polarized abelian threefold, which we shall denote by $A'$. In this case, $\mathcal{T}$ is the pullback of a divisor $\Theta$ via an isogeny of degree $2$ whose kernel is generated by an involution $\delta$.
The vector space of the global holomorphic section of the polarization $\mathcal{L}'$ on $A'$ is generated by two even functions: \begin{equation} \label{secondDEFgamma}
H^0(A', \mathcal{L}') = \left<\theta_0, \theta_{\gamma}\right> \ \ \text{,} \end{equation} where $\theta_{\gamma}$ is anti-invariant with respect to the action of $\delta$ by translations. On the other hand, the derivatives $\frac{\partial \theta_0}{\partial z_j}$ are odd functions, and by virtue of Proposition \ref{desccanonicalmap}, we conclude that every holomorphic section of the canonical bundle of $\mathcal{T}$ is invariant under the action of the involution: \begin{equation} \label{involutioniota} \iota \colon z \mapsto -z + \delta \ \ \text{.} \end{equation} Hence we can conclude that the canonical map of $\mathcal{T}$ factors through the quotient $Z := \bigslant{\mathcal{T}}{\iota}$ and it cannot be, in particular, birational.
\noindent The quotient map on the surface $Z$ is however birational, as we are going to prove. Firstly, we count the number of fixed points of $\iota$ on $\mathcal{T}$. Let us consider the commutative diagram \begin{align}\label{put} \begin{diagram} A' & \lInto & \mathcal{T} & \rTo & Z \\ \dTo & & \dTo & & \dTo \\ \mathcal{J} & \lInto & \Theta & \rTo & Y := \bigslant{\Theta}{\left<-1\right>} \end{diagram} \end{align} By our generality assumption on $A'$, we can assume that $\mathcal{J}$ is the Jacobian variety of a smooth quartic plane curve $\mathcal{D}$. \newline
\noindent We saw in Example \ref{Gaussjacob} that every point of the Theta divisor $\Theta$ of a quartic $\mathcal{D}$ can be represented by an effective divisor of degree $2$ on $\mathcal{D}$, and the Gauss map $G \colon \Theta \longrightarrow \mathbb{P}^2$ is the map which associates, to a divisor $u+v$, the line in $\mathbb{P}^2$ spanned by $u$ and $v$ if $u\neq v$ and the tangent line to $\mathcal{D}$ at $u$ if $u=v$.
\noindent Moreover, the multiplication by $-1$ on $\mathcal{J}$ corresponds to the Serre involution $\mathcal{L} \mapsto \omega_{\mathcal{D}} \otimes \mathcal{L}^{\vee}$ on $Pic^2(\mathcal{D})$, which can be expressed on $\Theta$ as the involution which associates, to the divisor $u+v$ on $\mathcal{D}$, the unique divisor $u'+v'$ such that $u+v+u+v'$ is a canonical divisor on $\mathcal{D}$.
\noindent It can be now easily seen that the projection of $\Theta$ onto $Y$ in Diagram (\ref{put}) is a covering branched on $28$ points, which correspond precisely to the $28$ bitangents of $\mathcal{D}$, and in particular we conclude that $Y$ has precisely $28$ nodes. This set corresponds to the set of odd $2$-torsion points of the Jacobian on which $\Theta$ has odd multiplicity, and we have : \begin{equation} \label{twopoints} \begin{split}
\mathcal{J}[2]^{-}(\Theta) &= \SET{z \in \mathcal{J}[2]}{mult_{z}\Theta \text{ is odd}} \\
&= \SET{z \in \mathcal{J}[2]}{E(2z_1, 2z_2) = 1 \ \ \text{$(mod \ 2)$}} \ \ \text{,}
\end{split}
\end{equation}
where $E := \IMh H$ is the alternating bilinear form associated to the polarization on $\mathcal{J}$, which is $\mathbb{Z}$-valued on the lattice $\Lambda$
Among the odd $2$-torsion points in (\ref{twopoints}) above, there are some which belong to the image in $\mathcal{J}$ of the fixed locus of the involution $\iota$ in $A$. The set containing them is precisely \begin{align*} p(\mathcal{F}ix(\iota)) \cap \mathcal{J}[2]^{-}(\Theta) &= \SET{[v ]\in \mathcal{J}[2]^{-}(\Theta)}{[2v] = \delta \ \ \ \text{in $A'$}} \\ &= \SET{z \in \mathcal{J}[2]^{-}(\Theta)}{E(2z, 2\eta) = 1 \ \ \text{$(mod \ 2)$}} \ \ \text{.} \end{align*} This latter set has cardinality equal to $16$, and thus the involution $\iota$ has exactly $32$ fixed points on $\mathcal{S}$.
Hence the singular locus of $Z$ is a strictly even set of $32$ nodes, and it holds (see \cite[Proposition 2.11]{Catanese1981}) that \begin{align} \label{equationEulercover} \chi(\mathcal{T}) = 2\chi(Z) - 8 = 2 \ \ \text{.} \end{align}
Thus, $\chi(Z) = 5$. On the other hand, since the canonical map of $Z$ factors through $\iota$, we have $p_g(Z) = 4$, and hence we conclude that $q(Z) = 0$. \noindent The map $Z \longrightarrow Y$ in Diagram \ref{put} is a double cover which is unramified except over the remaining $12$ nodes of $Y$.
\noindent We have just proved that the degree of the canonical map of $\mathcal{T}$ is at least $2$. The following lemma ensures that the degree is exactly $2$ (see also \cite{CataneseSchreyer1} for a proof).
\begin{lem1} \label{thetafunctions112} The canonical map of $Z$ is birational. Hence, the canonical map of $\mathcal{T}$ is of degree $2$, and its image is a surface $\Sigma$ of degree $6$ in $\mathbb{P}^3$. \end{lem1} \begin{proof} We recall that the canonical map $\CAN{\mathcal{T}}$ of $\mathcal{T}$ factors through the involution $\iota \colon z \mapsto -z + \delta$ as in \ref{involutioniota}, and all canonical sections of $\mathcal{T}$ are anti-invariant with respect to $\iota$.
\noindent It is easily seen that the degree of the canonical map of the quotient $Z := \bigslant{\mathcal{T}}{\iota}$ is at most $3$. Indeed, the Gauss map $\mathit{G}$ of $\mathcal{T}$ factors through $Z$ via the Gauss map on the quotient $\mathit{G}_Z: Z \longrightarrow {\mathbb{P}^2}$. The latter map is of degree $6$ and invariant with respect to the involution $(-1)_Z$ induced by the multiplication by $(-1)$ of $\mathcal{T}$, while the canonical map is not.
Moreover, $\mathit{G}$ is ramified on a locus of degree $24$ which represents the dual curve $\mathcal{D}^{\vee}$ of $\mathcal{D}$ (of degree $12$) counted with multiplicity $2$. If we denote by $\pi: \mathbb{P}^3 \dashrightarrow \mathbb{P}^2$ the projection which forgets the first coordinate, its restriction to $\Sigma$ must have a ramification locus $R$ whose degree is divisible by $12$, and we have: \begin{equation*}
deg \Sigma = deg K_{\Sigma} = -3 deg(\pi|_{\Sigma}) + deg R \ \ \text{.} \end{equation*}
However, $deg(\pi|_{\Sigma}) = deg \Sigma$, and hence $4deg\Sigma = deg R$. In particular, the canonical map of $Z$ can have degree $1$ or $2$. \newline
\noindent Let us suppose by contradiction that $\CAN{Z}$ is of degree $2$. Then $\CAN{Z}$ would be invariant under the action of an involution $j$ on $Z$, which would imply that the map $\mathit{G}_Z$ is invariant under the action of the whole group generated by $(-1)_Z$ and $j$. This latter group has a natural faithful representation as a subgroup of the monodromy group of Gauss map of $Z$, which is isomorphic to $S_3$, the symmetric group of degree $3$. Hence, \begin{equation*} H := \left< (-1)_Z , j \right> \cong S_3 \end{equation*} \noindent The ramification locus in $Y$ of $\phi_{Y}$ has two components: \begin{align*} \mathbb{T}_1 &:= \SET{[2p]}{p \in \mathcal{D}} \\ \mathbb{T}_2 &:= \SET{[p+q]}{\left< p,q \right>.\mathcal{D} = 2p+q+r \ \text{for some $q$,$r$ on $\mathcal{D}$}} \ \ \text{.} \end{align*} The component $\mathbb{T}_2$ has to be counted twice on $Y$, while $\mathbb{T}_1$ has multiplicity $1$. The components of the counterimage of $\mathbb{T}_1$ in $Z$ are both of multiplicity $1$ and the group $H$ acts on them. Clearly, the involution $(-1)_Z$ exchanges them, while both components must be pointwise fixed under the action of $j$. But the product $(-1)_Z \cdot j$, has order $3$ (because $H \cong \mathcal{S}_3$), hence it must fix both components, and we reach a contradiction. \end{proof}
\noindent The canonical models of surfaces with invariants $p_g=4$, $q=0$, $K^2 = 6$ and birational canonical map have been studied extensively by F. Catanese (see \cite{Catanese1984}). In this case, there is a symmetric homomorphism of sheaves: \begin{equation*} \alpha = \begin{bmatrix}\alpha_{00} & \alpha_{01} \\
\alpha_{01} & \alpha_{11}
\end{bmatrix}: (\mathcal{O}_{\mathbb{P}^3} \oplus \mathcal{O}_{\mathbb{P}^3}(-2))^{\vee}(-5) \longrightarrow \mathcal{O}_{\mathbb{P}^3} \oplus \mathcal{O}_{\mathbb{P}^3}(-2) \ \ \text{,} \end{equation*}
where $\alpha_{00}$ is contained in the ideal $I$ generated by $\alpha_{01}$ and $\alpha_{11}$. The canonical model $Y$ is defined by $det (\alpha)$ and the closed curve $\Gamma$ in Theorem \ref{CATSCHREYER} is a cubic defined by the ideal $I$ and contained in the projective plane $\alpha_{11} = 0$.
\begin{rmk} \label{rmk112} \noindent In our situation, the double curve $\Gamma$ in the canonical image of the quotient $Z$ is nothing but the image of the canonical curve $\mathcal{K}$ defined as the zero locus of $\theta_{\gamma}$ in $\mathcal{T}$ (refer again to (\ref{secondDEFgamma}) for the definition of $\theta_{\gamma}$). Indeed, let us denote by $\bar{\mathcal{K}}$ the image of $\mathcal{K}$ in $Z$. The curve $\mathcal{K}$ does not contain fixed points of $\iota$, whence $\mathcal{K}$ and $\bar{\mathcal{K}}$ are isomorphic. Furthermore, the curve $\bar{\mathcal{K}}$ is stable under the involution $(-1)$ on $\mathcal{T}$, and the canonical map of $Z$ is of degree $2$ on $\bar{\mathcal{K}}$. In conclusion, the image of $\bar{\mathcal{K}}$ in $\Sigma$ is a curve of nodal type, and then it must be exactly $\Gamma$. Moreover, the set $\mathcal{P}$ of pinch points on $\Gamma$ is exactly the image of the set of the $2$-torsion points of $\mathcal{T}$ which lie on the canonical curve $\mathcal{K}$. This latter set is in bijection with the set \begin{equation*}
\left\{(x,y) \in \mathbb{Z}_2^3 \times \mathbb{Z}_2^3\ \ | \ x_2y_2 + x_3y_3 = 1 \ \right\} \ \ \text{,} \end{equation*} which consists precisely of $24$ points. The set $\mathcal{P}$ is then precisely the branch locus of the map $\bar{\mathcal{K}} \longrightarrow \Gamma$, which has degree $2$ and factors with respect to the involution $(-1)_Z$. \end{rmk}
\noindent We are now in position to prove the first claim of Theorem \ref{injectivedifferential}. \begin{proof}[of Theorem \ref{injectivedifferential} point 1)] According to Diagram \ref{diagramproj}, for every involution in the kernel $\mathcal{G}$ of the isogeny $p: A \longrightarrow \mathcal{J}$ there is a diagram \begin{align} \label{diagramproj2} \begin{diagram}
A &\lInto&\mathcal{S} & \rTo^{\phi_{\mathcal{S}}} & \Sigma &\rInto& \mathbb{P}^5 \\
\dTo_{q_g} && \dTo_{q_g} & &\dTo &&\dDashto \\
A_{g}&\lInto&\mathcal{T}_{g} & \rTo^{\phi_{\mathcal{T}_g}} &\Sigma_g &\rInto&\mathbb{P}^3 \\ &&&\rdTo(1,1)^{} Z_g \ruTo(1,1)&&
\end{diagram}
\end{align} where $Z_g$ denotes the quotient of $\mathcal{T}_{g}$ by the involution $z \mapsto -z+h$ in $A_g$, and where $h \in \mathcal{G} - \{1,g\}$. By Theorem \ref{CATSCHREYER}, we obtain that, for every non-trivial element $g$ of $\mathcal{G}$ only one of the two possible cases can occur: \begin{itemize}
\item $q_g(U) = q_g(V)$. In this case, $V = g.U$, and thus both $U$ and $V$ lie on $\mathcal{S} \cap \mathcal{S}_{h} \cap \mathcal{S}_{g+h}$, where $h \in \mathcal{G} - \{1,g\}$, and the
claim of the theorem follows.
\item $q_g(U) \neq q_g(V)$ but $(\pi_{g} \circ q_g)(U) = (\pi_{g} \circ q_g)(V)$, where $\pi_g$ denotes the projection of $\mathcal{T}_{g}$ onto $Z_g$ in Diagram \ref{diagramproj2}.
In this case, we have that $V = -U + h$ for some $h \in \mathcal{G} - \{1,g\}$, and both $U$ and $V$ belong to $\mathcal{S} \cap \mathcal{S}_{h}$.
\item $(\pi_{g} \circ q_g)(U) \neq (\pi_{g} \circ q_g)(V)$. In this case, by applying Theorem \ref{CATSCHREYER} together with
Remark \ref{rmk112}, we have that $U$ and $V$ belong to $\mathcal{S}_{g}$ and $V = -U + g$, and we conclude again that $\mathcal{S} \cap \mathcal{S}_{g}$. \end{itemize} \end{proof} The proof of the second claim of Theorem \ref{injectivedifferential} requires some more efforts. The first important step toward proving it, is to describe the pullback surface $\mathcal{S}$ as a quotient of the symmetric product of a certain genus $9$ curve $\mathcal{C}$, which is an ètale bidouble cover of a general algebraic curve $\mathcal{D}$ of genus $3$.\newline
\noindent The next lemma shows that an ètale bidouble cover $\mathcal{C}$ of a general algebraic curve $\mathcal{D}$ of genus $3$ is a genus $9$ tetragonal curve with a very ample theta characteristic with four global independent sections.´
\begin{lem1} \label{fundamentallemmagenus9} Let $(A, \mathcal{L})$ be a general $(1,2,2)$-polarized Abelian 3-fold, let $p: A \longrightarrow \mathcal{J}$ be an isogeny onto the Jacobian of a general algebraic curve $\mathcal{D}$ of genus $3$, defined by two elements $\eta_1$ and $\eta_2$ belonging to $\mathcal{J}[2]$ which satisfy the condition that $W(\eta_1, \eta_2) = 0$. Let us moreover consider the algebraic curve $\mathcal{C}$ obtained by pulling back to $A$ the curve $\mathcal{D}$ via the isogeny $p$ (here we consider $\mathcal{D}$ to be embedded in its Jacobian $\mathcal{J}$).
Then, the following hold true: \begin{itemize} \item The genus $9$ curve $\mathcal{C}$ admits $\mathcal{E}$ and $\mathcal{F}$ two distinct $\mathcal{G}$-invariant $g_{4}^1$'s, with $\mathcal{E}^{2} \ncong \mathcal{F}^{2}$ and \begin{equation*}
h^0(\mathcal{C}, \mathcal{E}) = h^0(\mathcal{C}, \mathcal{F}) = 2 \ \ \text{.} \end{equation*} \item The line bundle $\mathcal{M} := \mathcal{E} \otimes \mathcal{F}$ is a very ample theta characteristic of type $g_8^3$. \item The image of $\mathcal{C}$ in $\mathbb{P}^3 = \mathbb{P}(\mathcal{M})$ is a complete intersection of the following type: \begin{equation} \label{corollaryscrolleq} \mathcal{C} : \begin{cases} X^2 + Y^2 + Z^2 + T^2 = 0 \\ q(X^2,Y^2,Z^2,T^2) = XYZT \ \ \text{,} \end{cases} \end{equation} where $q$ is a quadric, and $[X,Y,Z,T]$ are coordinates on $\mathbb{P}^3$ such that two generators $a$ and $b$ of kernel of the isogeny $\mathcal{G}:= Ker(p)$ act as follows: \begin{align*} \begin{split} a.[X,Y,Z,T] &= [X,Y,-Z,-T]\\ b.[X,Y,Z,T] &= [X,-Y,Z,-T] \ \ \text{.} \end{split} \end{align*} \item The étale bidouble cover $p: \mathcal{C} \longrightarrow \mathcal{D}$ can be identified with the restriction to $\mathcal{C}$ of the rational map $\psi: \mathbb{P}^3 \dashrightarrow \mathbb{P}^3$ defined by the squares of the coordinates: \begin{equation*} \psi: [X,Y,Z,T] \mathrel{\mapstochar\dashrightarrow} [x,y,z,t] := [X^2,Y^2,Z^2,T^2] \ \ \text{.} \end{equation*} and the equations of $\mathcal{D}$ in $\mathbb{P}^3 = \mathbb{P}[x,y,z,t]$ are, according to the defining equations of $\mathcal{C}$ in (\ref{corollaryscrolleq}), in the following form: \begin{equation*} \mathcal{D} : \begin{cases} x + y + z + t = 0 \\ q(x,y,z,t)^2 = xyzt \ \ \text{.} \end{cases} \end{equation*} \end{itemize} \end{lem1} \begin{proof}
We show that $\mathcal{C}$ is tetragonal. \noindent Let us fix a point $Q_0 \in \mathcal{C}$ whose image $q_0 \in \mathcal{D}$ with respect to $p$ does not lie on a bitangent line of $\mathcal{D}$, and let us denote by $\mathcal{A}$ the Abel map defined respect to a point different from $q_0$. \newline
\noindent We claim first that there exists a point $\zeta$ on $\Theta$ such that: \begin{itemize}
\item The point $\zeta$ does not belong to any translated of $\Theta$ with the 2-torsion elements $\eta_1$, $\eta_2$ and $\eta_1 + \eta_2$. This condition is equivalent to require that $\zeta$ does not belong to the image with respect to $p$ of the base locus of the polarization on $A$.
\item For every special divisor of degree $3$ and every $2$-torsion element $\eta$ we have that
\begin{equation*}
\zeta \neq \mathcal{A}(q_0) - \kappa - \eta - \mathcal{A}(D) \ \ \text{,}
\end{equation*} where $\kappa$ is the vector of Riemann constants (See \cite[Theorem 2 p. 100]{Nara}). \newline \end{itemize}
\noindent Indeed, if the second condition does not hold for a certain point $\zeta$ of Theta divisor, then it exists a $2$-torsion point $\eta \in \mathcal{J}[2]$ and a special divisor $D$ of degree $3$ on $\mathcal{D}$ such that: \begin{equation*} \zeta = \mathcal{A}(q_0) - \kappa - \eta - \mathcal{A}(D) \ \ \text{,} \end{equation*} and it follows that, in particular (recall that $\mathcal{A}(K) = -2\kappa$, where $K$ is a canonical divisor on $\mathcal{D}$) \begin{equation*} 0 = 2 \zeta = \mathcal{A}(2q_0 + K - 2D) - 2\eta = \mathcal{A}(2q_0 + K - 2D) \ \ \text{.} \end{equation*} Hence, by Abel's theorem, the divisor $2(D-q_0)$ is a canonical divisor. But $D$ is supposed to be a special divisor of degree $3$, hence linearly equivalent to $K-r$ where $r$ is some point on $\mathcal{D}$. ($D$ is a $g_3^1$ on the curve $\mathcal{D}$). That means, in particular, that: \begin{align*} K \equiv 2(D-q_0) \equiv 2(K - r - q_0) \ \ \text{,} \end{align*} and we conclude that $r + q_0$ is an odd theta-characteristic. However, because the odd theta characteristics correspond to the bitangent lines on $\mathcal{D}$ we assume that $q_0$ does not lie on any bitangent line of $\mathcal{D}$, we can exclude this case. This proves the first claim. \newline
\noindent Now we prove that there exists a theta characteristic $\mathcal{M}$ on $\mathcal{C}$ which yields an embedding of $\mathcal{C}$ in $\mathbb{P}^3$, such that the image as a complete intersection as in the statement of the lemma. We are going to prove that $\mathcal{M}$ is the mobile part of a line bundle on $\mathcal{C}$ naturally induced by restricting to $\mathcal{C}$ a certain translated of the polarization $\mathcal{L}$ on $A$, which has $4$ sections.
To prove the lemma, we pick now $\zeta$ which satisfy the claim we have just proved. With such a point $\zeta$, we can define: \begin{equation} \label{definitionT}
\mathcal{T} := (t_{\mathcal{A}(q_0)-\zeta}^*\mathcal{L})|_{\mathcal{C}} \end{equation} where $t_x$ denotes in general the translation by a point $x$ of $A$. By Definition \ref{definitionT}, for every $\gamma$ the following are then easily seen to be holomorphic sections of $\mathcal{T}$: \begin{equation*}
s^{q_0}_{\gamma} := t_{\mathcal{A}(q_0)-\zeta}^* \theta_{\gamma}|_{\mathcal{C}} \ \ \text{.} \end{equation*}
\noindent Our goal is then to show that $|\mathcal{T}|$ is a linear system on $\mathcal{C}$ with $4$ base points and of degree $12$, and that its mobile part defines a $g^3_8$ on $\mathcal{C}$. \noindent To this purpose, we first notice that, if we consider a point $X \in \mathcal{C}$ and its image $x$ in $\mathcal{D}$ with respect to $p$, we have that: \begin{equation} \label{EXPSECT} s^{q_0}_{\gamma}(X) = \theta_{\gamma}(\mathcal{A}(x)-\mathcal{A}(q_0)-\zeta) \ \ \text{.} \end{equation} Hence, $s^{q_0}_{\gamma}(X)$ vanishes if and only if $ \theta_{0}(\mathcal{A}(x)-\mathcal{A}(q_0)-(\zeta + \gamma))= 0$. On the other side, one can show that (see \cite[Lemma 2, p.112]{Nara}): \begin{align*} div (\theta_0(\mathcal{A}(x)-\mathcal{A}(q_0)-\zeta - \gamma) ) &= q_0 + D_{\gamma}
\end{align*} where $D_{\gamma}$ is a divisor of degree $2$ on the quartic curve $\mathcal{D}$ which is independent on $q_0$, for which: \begin{align*}
\mathcal{A}(D_{\gamma}) &= \zeta + \eta - \kappa \ \ \text{.} \end{align*} Thus: \begin{equation*} div (s^{q_0}_{\gamma}) = \mathcal{G}.Q_0 + p^*(D_{\gamma}) \ \ \text{,} \end{equation*} where $\mathcal{G}.Q_0$ is the orbit of $Q_0$ with respect to the action of $\mathcal{G}$.
\noindent This means that $\mathcal{T}$, which is of degree $12$, has a fixed part of degree $4$, and its mobile part $|\mathcal{M}|$ has degree $8$. We can conclude now that \begin{equation*} H^0(A, \mathcal{I}_{\mathcal{C}} \otimes t_{\mathcal{A}(q_0)-\zeta}^*\mathcal{L}) = \sum_{\eta} H^0(\mathcal{J}, \mathcal{I}_{\mathcal{D}} \otimes t_{\mathcal{A}(q_0)-\zeta-\gamma}^*\Theta) = 0 \ \ \text{.} \end{equation*} Indeed, if $h^0(\mathcal{J}, \mathcal{I}_{\mathcal{D}} \otimes t_{\mathcal{A}(q_0)-\zeta-\gamma}^*\Theta) = 1$ for some $\gamma$ then, for every $p \in \mathcal{D}$, we would have: \begin{equation*} \theta_0(\mathcal{A}(q_0) - \zeta - \gamma - \mathcal{A}(p)) = 0 \ \ \text{,} \end{equation*} and we would find a special divisor $D$ such that: \begin{equation*} \mathcal{A}(q_0) - \eta - \zeta - \kappa = \mathcal{A}(D) \ \ \text{.} \end{equation*} However, this would contradict the conditions on $\zeta$. We have then the following exact sequence: \begin{equation*} 0 \longrightarrow H^0(A, t_{A(q_0)-\zeta}^*\mathcal{L}) \longrightarrow H^0(\mathcal{C}, \mathcal{T} ) \longrightarrow H^1(A, \mathcal{I}_{\mathcal{C}} \otimes t_{A(q_0)-\zeta}^*\mathcal{L}) \longrightarrow 0 \ \ \text{,} \end{equation*}
from which we can conclude that $h^0(\mathcal{C}, \mathcal{M} ) = h^0(\mathcal{C}, \mathcal{T} ) \geq 4 $. We can apply the Riemann Roch theorem to conclude that the linear system $|\mathcal{M}|$ is special on $\mathcal{C}$. Moreover, its Clifford index is: \begin{equation*}
Cliff(\mathcal{M}) = 8 - 2|\mathcal{M}| \leq 2 \ \ \text{.} \end{equation*} On the other side, the Clifford index of $\mathcal{M}$ cannot be $0$ by the Clifford theorem, so $\mathcal{M}$ is of Clifford index $2$. This implies that $\mathcal{C}$ is an algebraic curve of genus $9$ with Clifford Index $2$, and we conclude that $\mathcal{C}$ is a tetragonal curve, with $\mathcal{M}$ a linear system $g_3^{8}$. Moreover, $\mathcal{M}$ is very ample. Indeed, if it were not the case, then $\mathcal{C}$ would be hyperelliptic, which would imply that $\mathcal{D}$ also is. This, however, would lead to a contradiction, since we assume $\mathcal{D}$ to be general. \newline
\noindent The sections of $\mathcal{T}$, defined in (\ref{EXPSECT}), define coordinates $[X,Y,Z,T]$ on $\mathbb{P}^3 = \mathbb{P} H^0(\mathcal{C}, \mathcal{M})$ on which $\mathcal{G}$ acts as claimed in the statement (according to Equation \ref{THETAGAMMA}, which describes how the group $\mathcal{G}$ acts on the sections of the polarization of $A$).
The image of $\mathcal{C}$ in $\mathbb{P}^3$ is the complete intersection of a quadric $\mathcal{Q}_2$ and a quartic surface $\mathcal{Q}_4$, both $\mathcal{G}$-invariant, with $\mathcal{Q}_2 = (q_2)_0$ and $\mathcal{Q}_4 = (q_4)_0$ for certain homogeneous polynomials $q_2$ and $q_4$.
Without loss of generality, we can assume our quadric to be in the form´ $q_2 = X^2 + Y^2 + Z^2 + T^2$. We claim that also the quartic $q_4$ is $\mathcal{G}$-invariant. Indeed, if it were not the case, If it were not the case, then we could suppose (using the equation of $Q_2$) that the quartic $q_4$ is of the form \begin{equation*} XYr^2 = ZTs^2 \ \ \text{,} \end{equation*} where $r$ and $s$ are polynomials in the vector space generated by the three squares $X^2, Y^2, Z^2$. This means that, considered $\mathbb{P}^3$ with coordinates $[x,y,z,t] := [X^2, Y^2, Z^2, T^2]$, we could write the equation of $\mathcal{D}$ in the following form: \begin{equation*} \mathcal{D} : \begin{cases} x + y + z + t = 0 \\ xyr^2 = zts^2 \ \ \text{,} \end{cases} \end{equation*} where $r$ and $s$ are two lines. In this case, $\mathcal{D}$ would be singular in the point in which the lines $r$ and $s$ intersect. Hence, $\mathcal{D}$ would be hyperelliptic, which would contradict our generality assumptions on $\mathcal{D}$. Hence $q_4$ is $\mathcal{G}$-invariant, and in particular of the form described in the claim of this lemma.
To conclude the proof, is now enough to observe that the two rulings of the non-singular quadric $\mathcal{Q}_2$ induce two distinct $\mathcal{G}$-invariant $g_1^4$ which we shall denote by $\mathcal{E}$ and $\mathcal{F}$, whose tensor product is clearly isomorphic to $\mathcal{M}$. This finishes the proof of the lemma. \end{proof}
We can now easily see that, for a general quartic plane curve $\mathcal{D}$, an ètale bidouble cover $\mathcal{S}$ of its Theta divisor $\Theta$, defined by two $2$-torsion line bundles on $\mathcal{D}$ which are orthogonal with respect to the Weil pairing, can be geometrically described as a quotient of the form \begin{equation*} \label{RepresentS} \mathcal{S} = \bigslant{\mathcal{C} \times \mathcal{C}}{\Delta_{\mathcal{G}} \times \mathbb{Z}_2} \ \ \text{,} \end{equation*} where $\mathcal{C}$ is a smooth curve of genus $9$ in $\mathbb{P}^3$ as in Lemma \ref{fundamentallemmagenus9}, $\mathcal{G} \cong \mathbb{Z}_2 \times \mathbb{Z}_2$ is the Galois group of the cover $p: \mathcal{S} \longrightarrow \Theta$, $\Delta_{\mathcal{G}}$ is the diagonal subgroup of $\mathcal{G} \times \mathcal{G}$ acting naturally on $\mathcal{C} \times \mathcal{C}$ and $\mathbb{Z}_2$ denotes the group acting by switching the two factors of $\mathcal{C} \times \mathcal{C}$.
\begin{not1} \label{NOTcanbdlS}
\noindent The global holomorphic sections of the canonical bundle of $\mathcal{S}$ are clearly those of $\mathcal{C} \times \mathcal{C}$ which are invariant under the action of the group $\Delta_{\mathcal{G}} \times \mathbb{Z}_2$:
\begin{align*} H^0(\mathcal{S}, \omega_{\mathcal{S}}) &= H^0(\mathcal{C} \times \mathcal{C}, \omega_{\mathcal{C}} \boxtimes \omega_{\mathcal{C}})^{\Delta_{\mathcal{G}} \times \mathbb{Z}_2}
\end{align*} These can be seen as quadrics in the projective coordinates $[X_1, Y_1, Z_1, T_1]$ and $[X_2, Y_2, Z_2, T_2]$ of $\mathbb{P}^3 \times \mathbb{P}^3$ which are invariant under the same action. Thus, the following is easily seen to be a basis for $H^0(\mathcal{S}, \omega_{\mathcal{S}})$:
\begin{equation} \label{polynomialexpressioncanonicalmap}
\begin{split} \ETA{12} &:= \begin{vmatrix} X_1^2 & X_2^2\\
Y_1^2 & Y_2^2
\end{vmatrix} \ \ \ \ \ \ \ \ \ETA{13} := \begin{vmatrix} X_1^2 & X_2^2\\
Z_1^2 & Z_2^2
\end{vmatrix} \ \ \ \ \ \ \ \ \ \ETA{23} := \begin{vmatrix} Y_1^2 & Y_2^2\\
Z_1^2 & Z_2^2
\end{vmatrix} \\ \OMEGA{45} &:= \begin{vmatrix} X_1Y_1 & X_2Y_2\\
Z_1T_1 & Z_2T_2\\
\end{vmatrix} \ \ \ \OMEGA{67} :=
\begin{vmatrix} X_1Z_1 & X_2Z_2\\
Y_1T_1 & Y_2T_2\\
\end{vmatrix} \ \ \OMEGA{89} :=
\begin{vmatrix} X_1T_1 & X_2T_2\\
Y_1Z_1 & Y_2Z_2\\
\end{vmatrix} \ \ \text{.} \end{split} \end{equation}
\end{not1} \begin{rmk}\label{rmk_base_points}[\textit{The 16 base points of the $(1,2,2)$-polarization}]
With the notation of Lemma \ref{fundamentallemmagenus9}, we consider the quartic curve $\mathcal{D}$ in $\mathbb{P}^3$ with coordinates $[x,y,z,t]=[X^2,Y^2,Z^2,T^2]$ defined by \begin{align*} \mathcal{D}: \begin{cases} x + y + z + t &= 0 \\ q(x,y,z,t)^2&= xyzt \ \ \text{.} \end{cases} \end{align*} \noindent The lines $x,y,z,t$ in the plane $H: x+y+z+t=0$ in $\mathbb{P}^3$ are clearly bitangent lines on $\mathcal{D}$, and for each bitangent line $l$ in this set we can denote by $l_1 + l_2$ the corresponding degree $2$ on $\mathcal{D}$ such that: \begin{equation*} l.\mathcal{D} = 2(l_1 + l_2) \ \ \text{,} \end{equation*} \noindent If we consider now $L_1$ and $L_2$ points of $\mathcal{C}$ in the preimage of $l_1$ and $l_2$ respectively, then it is now easily seen that $\mathcal{G}.[(L_1, L_2)]$ is a $\mathcal{G}$-orbit of base points of the linear system $\mathcal{L}$ in $A$. Indeed, by using the polynomial expression of the holomorphic canonical section of $\mathcal{S}$ in (\ref{polynomialexpressioncanonicalmap}) one can see directly that $\mathcal{G}.[(L_1, L_2)]$ is a $\mathcal{G}$-orbit of base points of the sublinear system generated by $\OMEGA{45}$, $\OMEGA{67}$, $\OMEGA{89}$ (which we defined in Notation \ref{NOTcanbdlS}). Since the set of the base points of the polarization on the $(1,2,2)$-polarized abelian threefold $A$ containing $\mathcal{S}$ has cardinality $16$, all $\mathcal{G}$-orbit of base points must be in the form $\mathcal{G}.[(L_1, L_2)]$ for some bitangent line. \end{rmk}
\noindent We are now in the position to prove the second claim of Theorem \ref{injectivedifferential}. \begin{proof}[of the Theorem \ref{injectivedifferential}, point 2)]
\noindent We consider the projections $q_g$ of $A$ onto the $(1,1,2)$-polarized abelian threefolds $A_g:=\bigslant{A}{\left<g\right>}$ associated to the non-trivial elements $g$ of $\mathcal{G}$, and Diagram \ref{diagramproj2}: \begin{align*} \begin{diagram}
A &\lInto&\mathcal{S} & \rTo^{\phi_{\mathcal{S}}} & \Sigma &\rInto& \mathbb{P}^5 \\
\dTo_{q_g} && \dTo_{q_g} & &\dTo &&\dDashto \\
A_{g}&\lInto&\mathcal{T}_{g} & \rTo^{\phi_{\mathcal{T}_g}} &\Sigma_g &\rInto&\mathbb{P}^3 \\ &&&\rdTo(1,1)^{} Z_g \ruTo(1,1)&&
\end{diagram}
\end{align*} Recall that here $\mathcal{T}_g$ denotes the image $q_g(\mathcal{S})$ in $A_g$. If the differential $d_{z}\phi_{\mathcal{S}}$ at a point $z$ on $\mathcal{S}$ is not injective, then we have that the differential at $q_g(z)$ of the canonical map of $Z_g$ is not injective for every non-trivial element $g$ of $\mathcal{G}$. By applying Theorem \ref{CATSCHREYER}, the image in $\mathbb{P}^3$ of $q_{g}(z)$ with respect to $\phi_{\mathcal{T}_g}$
must be one of the pinch points belonging to the double nodal curve $\Gamma_{g}$ in the canonical image of $Z_g$ in $\mathbb{P}^3$. By Remark \ref{rmk112}, the latters are contained in the plane $\theta_{\gamma}=0$. Hence, $z$ must be a base point of the linear system $|\mathcal{O}_A(\mathcal{S})|$ in $A$.
\noindent Thus, it is enough to prove the claim for the base points of the linear system $|\mathcal{O}_A(\mathcal{S})|$ in $\mathcal{S}$.
\noindent Let us consider in particular a base point $z_0$. We have to prove that, for every tangent vector $\nu$ to $\mathcal{S}$ in $z_0$, there exists a divisor $D$ in the canonical class $|K_\mathcal{S}|$ such that $D$ contains $z_0$, but $\nu$ is not tangent to $D$ in $z_0$. Then, to conclude the proof of the proposition it suffices to prove the following lemma. \end{proof}
\begin{lem1} Let $b$ be a base point of the polarization $\mathcal{L}$ on a general $(1,2,2)$ abelian 3-fold $A$ with flat uniformizing coordinates $(z_1, z_2, z_3)$. Moreover, let us consider $p: A \longrightarrow \mathcal{J}$ an isogeny onto the Jacobian of a quartic plane curve $\mathcal{D}$ with kernel $\mathcal{G}$ and $(\theta_{\gamma})_{\gamma \in \mathcal{G}^*}$ a basis for $H^0(A, \mathcal{L})$ (recall that $\mathcal{G}^*$ is the group generated by the $2$-torsion bundles $\eta_1$ and $\eta_2$ defining the cover $p$).
\noindent Then the four points in $\mathbb{P}^2(\mathbb{C})$ represented by the columns of following matrix are in general position: \begin{equation*} \begin{pmatrix} \frac{\partial}{\partial z_1}\theta_{0}(b) & \frac{\partial}{\partial z_1}\theta_{\eta_1}(b) & \frac{\partial}{\partial z_1}\theta_{\eta_2}(b) & \frac{\partial}{\partial z_1}\theta_{\eta_1 + \eta_2}(b) \\
\frac{\partial}{\partial z_2}\theta_{0}(b) & \frac{\partial}{\partial z_2}\theta_{\eta_1}(b) &\frac{\partial}{\partial z_2}\theta_{\eta_2}(b) & \frac{\partial}{\partial z_2}\theta_{\eta_1 + \eta_2}(b) \\
\frac{\partial}{\partial z_3}\theta_{0}(b) & \frac{\partial}{\partial z_3}\theta_{\eta_1}(b) &\frac{\partial}{\partial z_3}\theta_{\eta_2}(b) & \frac{\partial}{\partial z_3}\theta_{\eta_1 + \eta_2}(b) \end{pmatrix} \ \ \text{.} \end{equation*} \end{lem1} \begin{proof}
With the notation of Remark \ref{rmk_base_points}, to each bitangent $l \in \{x,y,z,t\}$ of $\mathcal{D}$ we fix $b_{l} := [L_1,L_2]$ a representative of a $\mathcal{G}$-orbit of base points. The unramified bidouble covering $p: \mathcal{C} \longrightarrow \mathcal{D}$ is then defined by the $2$-torsion points in the Jacobian $\mathcal{J}(\mathcal{D})$: \begin{align*} \eta_1 &= \mathcal{O}_{\mathcal{D}}(y_1 + y_2 - x_1 - x_2) \\ \eta_2 &= \mathcal{O}_{\mathcal{D}}(z_1 + z_2 - x_1 - x_2) \\ \eta_1 \otimes \eta_2 &= \mathcal{O}_{\mathcal{D}}(t_1 + t_2 - x_1 - x_2) \ \ \text{.} \end{align*} With this notation, it follows that, as projective points, \begin{align*} [\nabla \theta_0(b_y)] &= [\nabla \theta_0(b_x + \eta_1)] = [\phi_{\eta_1}(b_x) \cdot \nabla \theta_{\eta_1}(b_x)] = [ \nabla \theta_{\eta_1}(b_x)]\\ [\nabla \theta_0(b_z)] &= [\nabla \theta_0(b_x + \eta_2)] = [\phi_{\eta_2}(b_x) \cdot \nabla \theta_{\eta_2}(b_x)] = [\nabla \theta_{\eta_2}(b_x)] \\ [\nabla \theta_0(b_t)] &=[\nabla \theta_0(b_x + \eta_1 + \eta_2)] = [\phi_{\eta_1+\eta_2}(b_x) \cdot \nabla \theta_{\eta_1 + \eta_2}(b_x)] = [\nabla \theta_{\eta_1 + \eta_2}(b_x)] \ \ \text{,} \end{align*} where $\phi_{\gamma}(b_x)$ is the non-zero automorphy factor in Equation \ref{thetadef1}, which defines the section $\theta_{\gamma}$. This proves the claim, since the projective points $[\nabla \theta_0(b_l)]$ represent image in $\mathbb{P}^2$ of the divisors $l_1 + l_2$ on $\mathcal{D}$ with respect to the Gauss map of $\Theta$. The Gauss map associates to these divisors the coordinates in $\mathbb{P}^{2\vee}$ of the bitangents $l$ passing through them. Hence, since the four bitangents are in general position, also their projective coordinates representation must be in general position. \end{proof}
\newcommand{\IM}[1]{\Im \text{m} \ #1} \newcommand{\ct}[1]{\frac{\partial \y{#1}}{\partial \x{#1}}}
\section{Polarizations of type $(1,2)$ and $(2,4)$ on abelian surfaces} \input{12polarizC}
\section{Surfaces in a $(1,2,2)$-polarization which is isogenous to a product} \label{degenerationspolarizationsection} \noindent
In Section \ref{basicsectionssect}, we investigated the behavior of the canonical map of an ètale bidouble cover $\mathcal{S}$ of type $(1,2,2)$ of a general principal polarization, and in Proposition \ref{injectivedifferential}) we proved that the canonical map has everywhere injective differential and it is injective outside of some canonical curves, on which its restriction has degree $2$. However, Remark \ref{CHOICENOTGENERAL} shows that such surfaces are quite special in their linear system, and the question whether the canonical map in the general case is an embedding still remains open. \\
\noindent In this section, we look at this question by considering sufficiently general surfaces in a polarization of type $(1, 2, 2)$ on an abelian threefold $A$, which is isogenous to a polarized product of a $(2, 2)$-polarized surface $B$ and a $(2)$-polarized elliptic curve $E$. To achieve this goal we first need to consider the special case in which $B$ itself is a product of elliptic curves. We begin this section by introducing some notation.
\begin{not1} \label{productSurfEll} \noindent Let us consider a point $\tau = (\tau_{\B}, \tau_{\E})$ in the product $\mathcal{H}_2 \times \mathcal{H}_1$.
The $(2,2,2)$-polarized abelian threefold $T$ defined by $\tau$ is then the product of a $(2,2)$-abelian surface $B := \bigslant{\mathbb{C}^2}{\tau_{\B} \mathbb{Z}^2 \oplus 2\mathbb{Z}^2}$ and a $(2)$-polarized elliptic curve $E := \bigslant{\mathbb{C}}{\tau_{\E} \mathbb{Z} \oplus 2\mathbb{Z}}$. We denote by $\mathcal{N}$ the ample line bundle on $B$ of characteristic $0$ of type $(2,2)$.
As in Notation \ref{Surftwotwo} in the previous section, it is defined a projection of $B$ \begin{equation*} {\pi_{\B}} \colon B \longrightarrow S :=\bigslant{B}{\left<e_1+e_2\right>} \end{equation*} onto a $(1,2)$-polarized surface $S$, and we have the splitting ${{\pi_{\B}}}_*(\mathcal{N}) = {\mathcal{M}^{+}} \oplus {\mathcal{M}^{-}}$ into invariant and anti-invariant part. \newline
\noindent We can now consider the $(1,2,2)$-polarized abelian threefold $(A, \mathcal{L})$ obtained as the quotient of $T$ by the translation $e_1 + e_2 + e_3$: \begin{equation*} p \colon T \longrightarrow A:= \bigslant{T}{\left<e_1+e_2+e_3\right>} \ \ \text{.} \end{equation*}
The vector space of global holomorphic sections of $\mathcal{L}$ is generated by those global sections of the polarization on $T$ which are invariant under the action by $e_1 + e_2 + e_3$, and we have the following decomposition: \begin{equation} \label{sectionsDECOMP}
\begin{split} H^0(A, \mathcal{L}) &= H^0(T, \mathcal{N} \boxtimes \mathcal{O}_{E}(2\cdot O_{E}))^{e_1 + e_2 + e_3} \\ &= H^0(S \times E, {\pi_{\B}}_* \mathcal{N} \boxtimes \mathcal{O}_{E} (2\cdot O_{E}))^{e_1 + e_2 + e_3} \\ &\cong H^0(S, {\mathcal{M}^{+}})\cdot \theta^{E}_0 \oplus H^0(S, {\mathcal{M}^{-}})\cdot \theta^{E}_1
\end{split} \end{equation} By (\ref{sectionsDECOMP}) it also follows a similar decomposition for the base locus of $\mathcal{L}$: \begin{equation} \label{BASEL1}
\mathcal{B}(\mathcal{L}) = \mathcal{B}({\mathcal{M}^{+}}) \times div(\theta^{E}_1) \cup \mathcal{B}({\mathcal{M}^{-}}) \times div(\theta^{E}_0) \end{equation} \end{not1}
The following lemma shows that the images of two distinct base points of $\mathcal{L}$ with respect to the canonical map of a general surface in $|\mathcal{L}|$ are different. This happens also in the case in which our $(1,2,2)$ abelian variety $A$, as in Notation \ref{productSurfEll}, is the quotient of three elliptic curves. \begin{lem1} \label{generalbehavior0}
Let us consider a $(1,2,2)$-polarized abelian variety $(A, \mathcal{L})$ defined as the quotient of the product of three $(2)$-polarized elliptic curves $E_j := \bigslant{\mathbb{C}}{\tau_{jj}\mathbb{Z} \oplus 2\cdot \mathbb{Z}}$ by the involution $e_1 + e_2 + e_3$. Then the canonical map of a general surface $\mathcal{S}$ in the polarization $|\mathcal{L}|$ on $A$ in injective on the set $\mathcal{B}(\mathcal{L}) \subseteq \mathcal{S}$ of base points of $|\mathcal{L}|$. \end{lem1} \begin{proof} According to the splitting of $H^0(A, \mathcal{L})$ in (\ref{sectionsDECOMP}), the vector space $H^0(A, \mathcal{L})$ is generated by four elements which we shall denote by \begin{equation} \label{THETAELLP}
\theta_{ijk}:= \theta^{E_1}_{i}\theta^{E_2}_{j}\theta^{E_3}_{k} \ \ \text{,} \end{equation} with $(ijk)$ a triplet in the set $\{(000),(011),(101),(110)\}$.
\noindent The subgroup $\mathcal{G}$ of $A$ generated by $e_1$ and $e_2$ acts on the base locus $\mathcal{B}(\mathcal{L})$ by translation, and splits $\mathcal{B}(\mathcal{L})$ as the union of four $\mathcal{G}$-orbits, each consisting of four points. We shall denote them as $\mathcal{B}_{ijk}$ for $(ijk)$ a triplet as above: \begin{equation} \label{listbasepoints122}
\mathcal{B}_{ijk} = \mathcal{G} \text{\ \textbf{.}}\left(\frac{1 + (i-1)\tau_{11}}{2}, \frac{1 + (j-1)\tau_{22}}{2}, \frac{1 + (k-1)\tau_{33}}{2} \right) \end{equation}
The projective points in $\mathbb{P}^5$ which represent the canonical images of the $\mathcal{G}$-orbits of base points only depend on their image with respect to the Gauss map of $\mathcal{S}$. on which $\mathcal{G}$ acts by changing the signs of the coordinates. Hence, it suffices to show that, with $P_{ijk}$ representatives of the $\mathcal{G}$-orbits of base points, the columns of the matrix $[\nabla \theta (P_{ijk})]_{ijk}$ represent projective points in $\mathbb{P}^3$ in general position. An easy computation shows that, if $\theta = \theta_{000} + b_{23} \theta_{011} + b_{13} \theta_{101} + b_{12} \theta_{110}$, for certain complex coefficients $b_{23}$, $b_{13}$ and $b_{12}$, then we have: \begin{equation*} (\nabla(\theta)(P_{ijk}))_{ijk} = \begin{bmatrix} \pm b_{23} \partial_{z_0} \theta_{011}(P_{000}) & \partial_{z_0} \theta_{000}(P_{011}) & \pm b_{12} \partial_{z_0} \theta_{110}(P_{101}) & \pm b_{13} \partial_{z_0} \theta_{101}(P_{110}) \\ \pm b_{13} \partial_{z_1} \theta_{101}(P_{000}) & \pm b_{12} \partial_{z_1} \theta_{110}(P_{011}) & \partial_{z_1} \theta_{000}(P_{101}) & \pm b_{23} \partial_{z_1} \theta_{011}(P_{110}) \\ \pm b_{12} \partial_{z_2} \theta_{110}(P_{000}) & \pm b_{13} \partial_{z_2} \theta_{101}(P_{011}) & \pm b_{23} \partial_{z_2} \theta_{011}(P_{101}) & \partial_{z_2} \theta_{000}(P_{110})
\end{bmatrix} \end{equation*} Here the signs $\pm$ refer to the action of $\mathcal{G}$ on the coordinates. One can see that after the specialization in some particular cases (for instance to the case in which $(b_{23}, b_{13}, b_{12}) = (1,0,0)$), one can see that the columns are in general position as claimed. This shows that this holds true also for the general choice of the coefficients $b_{ij}$. \end{proof}
When the following generality condition is satisfied,
\begin{prop1}\label{generalbehavior1}
Let $(A, \mathcal{L})$ be a $(1,2,2)$-polarized abelian variety which is isogenous to the product of three elliptic curves with flat uniformizing coordinates $(z_1, z_2, z_3)$. Let us moreover assume that $\mathcal{L}$ is of characteristic $0$ on $A$, and $\mathcal{S} := div(\theta)$ be a general surface in the polarization class $|\mathcal{L}|$.
\noindent Then we have the following: \begin{itemize}
\item For every $i=1,2,3$, the involution $\iv{i}\colon z_i \mapsto -z_i$ on $A$ which changes the sign only to the $i$-th coordinate $z_i$ leaves $\mathcal{S}$ invariant and induces an involution on every canonical divisor of the form $W_j := div \left( \frac{\partial \theta}{\partial z_j} \right)$ on $\mathcal{S}$.
\item The canonical map of $\mathcal{S}$ is injective outside of the divisors $W_j$, on which the canonical map has degree $2$ and factors respectively through the involution $\iv{j}$.
\item The differential of the canonical map is injective outside of the divisors $W_j$. For each $j$, the canonical map is not injective on the fixed points set of the involution $\iv{j}$ in $W_j$. Moreover, this latter set contains precisely $8$ points. \end{itemize}
\end{prop1}
\begin{proof} We write all global sections of $\mathcal{L}$, as in Equation \ref{THETAELLP} in Lemma \ref{generalbehavior0}, as products of the form: \begin{equation*}
\theta_{ijk}= \theta^{E_1}_{i}\theta^{E_2}_{j}\theta^{E_3}_{k} \ \ \text{.} \end{equation*} Since the factors in the latter product are even theta functions, it is straightforward to verify that $\mathcal{S}$ is invariant under every involution which only changes the sign to some of the coordinates $z_j$. In particular, the derivative $\left( \frac{\partial \theta}{\partial z_j} \right)$ is anti-invariant with respect to $\iv{i}$ precisely when $i=j$, invariant otherwise. This also shows that the restriction to $W_j$ of the canonical map of $\mathcal{S}$ factors through the involution $\iv{j}$, being the derivative $\frac{\partial \theta}{\partial z_j}$ the only anti-invariant global holomorphic section of the canonical bundle on $\mathcal{S}$. This proves the first claim. \newline
\noindent Let us prove then the second claim. We start by considering two points $P$ and $Q$ on $\mathcal{S}$ such that $\CAN{\mathcal{S}}(P) = \CAN{\mathcal{S}}(Q)$, where $\CAN{\mathcal{S}}$ denotes the canonical map of $\mathcal{S}$. By Lemma \ref{generalbehavior0} and by our generality assumption we can assume that $P$ and $Q$ are not in the base locus $\mathcal{B}(\mathcal{L}) \subseteq \mathcal{S}$. Recall that the base locus $\mathcal{B}(\mathcal{L})$ is, according with the decomposition in (\ref{BASEL1}) in the Notations (\ref{productSurfEll}): \begin{equation} \label{BASEL}
\mathcal{B}(\mathcal{L}) = \mathcal{B}({\mathcal{M}^{+}}) \times div(\theta^{E}_1) \cup \mathcal{B}({\mathcal{M}^{-}}) \times div(\theta^{E}_0) \ \ \text{.} \end{equation}
\noindent By a slight abuse of notation we shall refer to them as points $P=(x,s)$ and $Q=(y,t)$ in the product $T = B \times E$ as introduced in Notation \ref{productSurfEll}. We see that the equation of $\mathcal{S}$ can be written in the form \begin{equation}\label{product122Surfequation} \theta = \nu \theta^{E}_0 + \xi \theta^{E}_1 = 0 \ \ \text{,} \end{equation} for certain general global sections $\nu \in H^0(S, {\mathcal{M}^{+}})$ and $\xi \in H^0(S, {\mathcal{M}^{-}})$. Then, by Proposition \ref{desccanonicalmap}, the canonical map of $\mathcal{S}$ has the following structure:
\begin{equation*} \CAN{\mathcal{S}} = \left[H^0(S, {\mathcal{M}^{+}})\theta^{E}_0, H^0(S, {\mathcal{M}^{-}})\theta^{E}_1, \frac{\partial \nu}{\partial z_1} \theta^{E}_0 + \frac{\partial \xi}{\partial z_1} \theta^{E}_1, \frac{\partial \nu}{\partial z_2} \theta^{E}_0 + \frac{\partial \xi}{\partial z_2} \theta^{E}_1, \nu \theta'^{E}_0 + \xi \theta'^{E}_1 \right] \ \ \text{.} \end{equation*}
\noindent Hence, denoted by $\Cu := div(\nu)$ and $\Du := div(\xi)$ the curves in $S := \bigslant{B}{e_1 + e_2}$, respectively in the linear systems $|{\mathcal{M}^{+}}|$ and $|{\mathcal{M}^{-}}|$, we have to the following cases: \begin{itemize}
\item[a)] \underline{$\theta^{E}_0(s) = \theta^{E}_0(t) = 0$}. In this case, the surface equation in (\ref{product122Surfequation}) implies that both $x$ and $y$ represent points lying on the curve $\Du$ in $S$. Thus, by considering only the non-vanishing components of the expression of $\CAN{\mathcal{S}}$ when evaluated on $P$ and $Q$, we have:
\begin{equation}\label{CANEXP1}
\CAN{\mathcal{S}} = \left[H^0(S, {\mathcal{M}^{-}})\theta^{E}_1, \frac{\partial \xi}{\partial z_1} \theta^{E}_1, \frac{\partial \xi}{\partial z_2} \theta^{E}_1,\nu \theta'^{E}_0 + \xi \theta'^{E}_1 \right] \ \ \text{.} \end{equation} Again by Proposition \ref{desccanonicalmap}, since a set of generators for $H^0(\Du, \omega_{\Du})$ is obtained by considering the derivatives $\frac{\partial \xi}{\partial z_1}$ and $\frac{\partial \xi}{\partial z_2}$ together with the restriction of the sections of $H^0(S, {\mathcal{M}^{-}})$ to $\Du$, we can conclude that $\phi_{\omega_{\Du}}(x) = \phi_{\omega_{\Du}}(y)$, where $\phi_{\omega_{\Du}}$ is the canonical map of $\Du$. By applying Remark \ref{osservationsingularfibers}, our generality assumption implies that $\Du$ is not hyperelliptic, and we infer that $x = y$ as points in $S$. Thus, $x = y$ or $x = y + e_1 + e_2$ in $B$, and we can assume that $(x,s) = (y, -t)$ or $(x,s) = (y, -t + e_3)$ as points on $A$. This latter second subcase can be excluded, since it occurs only if $x$ or $y$ belongs to $\mathcal{B}({\mathcal{M}^{-}})$. Indeed, by the expression of the base locus of $\mathcal{L}$ in (\ref{BASEL}) it would lead to the conclusion, contrary to our hypothesis, that both $P$ and $Q$ are base points. Thus, $(x,s) = (y,-t)$ (i.e. $P = {\iota}_3(Q)$), and from the expression of the canonical map of $\mathcal{S}$ in (\ref{CANEXP1}) we conclude that both $P$ and $Q$ belong to $W_3 = div\left(\frac{\partial \theta}{\partial z_3}\right)$.
\item[b)] \underline{$\theta^{E}_1(s) = \theta^{E}_1(t) = 0$.} This case can be considered to be equivalent to the previous case. \item[c)] \underline{$\theta^{E}_0(s) = 0$, $\theta^{E}_0(t) \neq 0$ but $y \in \mathcal{B}({\mathcal{M}^{+}})$ as point on $S$.} Then, according to Equation \ref{product122Surfequation}, $x \in \Du$.
Since we are assuming that $P$ and $Q$ are ouside of the base locus of $\mathcal{L}$ (which contains $\left(\mathcal{B}({\mathcal{M}^{+}}) \times div(\theta^{E}_1)\right)$), we have that $\theta^{E}_1(t) \neq 0$. Then, since $\CAN{\mathcal{S}}(P) = \CAN{\mathcal{S}}(Q)$, we have that $y$ also belongs to $\Du$. Hence $\Du$ is a curve in the linear system $|{\mathcal{M}^{+}}|$ passing through $y$ which is a base point of $|{\mathcal{M}^{-}}|$. Since by Remark \ref{osservationsingularfibers} this implies that $\Du$ is hyperelliptic, this case can be excluded by our generality assumption on $\nu$ and $\xi$. \item[d)] \underline{$\theta^{E}_0$ and $\theta^{E}_1$ do not vanish on $s$ or $t$, and neither $x$ nor $y$ belongs to $\mathcal{B}({\mathcal{M}^{+}}) \cup \mathcal{B}({\mathcal{M}^{-}})$}. In this case, we can consider the
rational map on $S$ defined by the product of the linear systems $|{\mathcal{M}^{+}}|$ and $|{\mathcal{M}^{-}}|$ \begin{equation*} {\psi} \colon S \dashrightarrow \mathbb{P} H^0(S, {\mathcal{M}^{+}}) ´\times \mathbb{P} H^0(S, {\mathcal{M}^{-}}) \cong \mathbb{P}^1 \times \mathbb{P}^1 \end{equation*} The map ${\psi}$ is clearly defined outside the union of the base loci $\mathcal{I} = \mathcal{B}({\mathcal{M}^{+}}) \cup \mathcal{B}({\mathcal{M}^{-}})$, it is dominant and generically finite of degree $4$. Since $S$ is the quotient of a product of two elliptic curves, this map factors through the involutions ${\iota}_1$ and ${\iota}_2$. Hence, in this case we have that $y \in \left<{\iota}_1, {\iota}_2\right>.x$ as points of $S$, and $t = \pm s$. It is now straightforward to prove that there is no other possibility, and that, if $Q = {\iota}_j(P)$, then $P$ and $Q$ lie on the divisor $W_j$. This finishes the proof of the second claim. \newline \end{itemize}
\noindent We prove now the third claim of the proposition.
One can show, in general, that the complementary set of an ample divisor in an abelian variety is an affine set. Unfortunately it is not always possible to achieve an useful description of these affine sets in terms of coordinates and equations. However, our special situation suggests the idea to use affine coordinates on the factors $E_j$ and to use them to define suitable local affine coordinates on $\mathcal{S}$ and to give an expression of the canonical map of $\mathcal{S}$ in local coordinates.
As we have already observed throughout Example \ref{basicexampleEll}, the elliptic curves $E_j$ are defined in an affine open set $\EU{j} = E_j - div(\theta^{E_j}_0)$ by an equation $\Eeq{j} = 0$, where: \begin{equation}\label{LegendreNormalFormE2} \Eeq{j} := \y{j}^2 - (\x{j}^2 - 1)(\x{j}^2 - \pa{j}^2) \ \ \text{.} \end{equation}
In the affine open set $\EV{j} = E_j - div(\theta^{E_j}_1)$ around the poles $\infty_{+}$ and $\infty_{-}$ of the function $\x{j}$, we have coordinates $(\vinf{j}, \winf{j})$, with \begin{equation}\label{LegendreNormalForm2} \Eeqinf{j} := \winf{j}^2 - (1 - \vinf{j}^2)(1- \pa{j}^2\vinf{j}^{2}) \ \ \text{.} \end{equation}
On the open subset $\mathcal{U}_{000} := T - div(\theta_{000})$ of $T = E_1 \times E_2 \times E_3$ ,we have coordinates $\left( \binom{\x{1}}{\y{1}},\binom{\x{2}}{\y{2}},\binom{\x{3}}{\y{3}}\right)$, and since $\mathcal{S} = div(\theta)$ for some general holomorphic section $\theta$, we have that the pullback $\widehat{\mathcal{S}}$ of $\mathcal{S}$ in via the projection $p \colon T \longrightarrow A$ can be expressed, according to Equation \ref{LegendreRelationsEQ} as the closed subset in the open set $\mathcal{U}_{000}$ defined by \begin{equation}\label{SURFEQU}
\widehat{\mathcal{S}} \cap \mathcal{U}_{000} = \left\{\left(\binom{\x{1}}{\y{1}},\binom{\x{2}}{\y{2}},\binom{\x{3}}{\y{3}}\right) \in \mathcal{U}_{000} \ \ | \ \ f = f_{000} := 1 + b_{23} \x{2}\x{3}+b_{13} \x{1}\x{3} + b_{12} \x{1}\x{2} = 0 \right\} \ \ \text{,} \end{equation} where $b_{23}$, $b_{13}$ and $b_{12}$ are general (in particular non-zero) complex coefficients. This generality hypothesis implies that no couple of coordinates $\x{i}$ and $\x{j}$ can simultaneously vanish. Moreover, according with Equation \ref{LegendreRelationsEQ}, the action on the elliptic curve $E_j$ with a translation by $e_j$ changes the signs of both coordinates $(\x{j}, \y{j})$ (resp. $(\vinf{j}, \winf{j})$) while the involution $\iv{j}$ only inverts the sign of $\y{j}$ (resp $\winf{j}$).
\noindent By applying the coordinate changes $(\x{j}, \y{j}) \mapsto (\vinf{j}, \winf{j}) = (\x{j}^{-1}, \y{j}\x{j}^{-2})$ to the elliptic curves $E_j$, we obtain the equation of $\widehat{\mathcal{S}}$ in the other open sets of the form $\mathcal{U}_{ijk} := T - div(\theta_{ijk})$. For the reader convenience, we can express the defining equation in the open set $\mathcal{U}_{110}$. To this purpose we only need to write $\vinf{1}^{-1}$ and $\vinf{2}^{-1}$ in place of $\x{1}$ and $\x{2}$ and to multiply the resulting equality by $\vinf{1}\vinf{2}$. We obtain: \begin{equation*}
\widehat{\mathcal{S}} \cap \mathcal{U}_{110} = \left\{\left(\binom{\vinf{1}}{\winf{1}},\binom{\vinf{2}}{\winf{2}},\binom{\x{3}}{\y{3}}\right) \in \mathcal{U}_{110} \ \ | \ \ f_{110} := \vinf{1}\vinf{2} + (b_{23} + b_{13})\x{3} + b_{12} = 0 \right\} \ \ \text{.} \end{equation*}
\noindent It is easy to see that the preimage $B := p^{-1}(\mathcal{B}({\mathcal{L}}))$ in $T$ of the base locus in $A$ is the union of the orbits of the points $(\infty_{+}, \infty_{+}, \infty_{+})$, $(\infty_{+}, 0, 0)$, $(0, \infty_{+}, 0)$ and $(0,0,\infty_{+})$ with respect to the group isomorphic to $\mathbb{Z}_2^3$ which operates on each factor as the sign change $(\x{j}, \y{j}) \mapsto (-\x{j}, -\y{j})$. \newline
\noindent With this local description of $\widehat{\mathcal{S}}$ in Equation \ref{SURFEQU}, we express the canonical map in local coordinates and we determine the locus where the rank of the differential drops. \newline
\noindent For every $(ij) \in \{(12),(13),(23)\}$, the holomorphic $2$-forms $\omega_{ij}:= dz_i \wedge dz_j$ on $A$ can be looked at as non-zero elements of $H^0(\mathcal{S}, \omega_{\mathcal{S}})$ when restricted to $\mathcal{S}$. However, around a point of $E_j$ on which $\y{j}$ does not vanish (i.e. the $j$-th component is not $2$-torsion on $E_j$) the holomorphic 1-form $dz_j$ is equivalent up to a non-zero constant to $\frac{d\x{j}}{\y{j}}$, while around a point at infinity it is equivalent to $\frac{d\vinf{j}}{\winf{j}}$. The remaining holomorphic forms of the basis for $H^0(\mathcal{S}, \omega_{\mathcal{S}})$ can be expressed by applying the residue map $H^0(A, \mathcal{O}_{A}(\mathcal{S})) = H^0(A, \omega_{A}(\mathcal{S})) \longrightarrow H^0(\mathcal{S}, \omega_{\mathcal{S}})$, again by using a local equation of $\mathcal{S}$. For example, we can see how the canonical map looks like at a point $P = (\x{j}, \y{j})_j$ in $\widehat{\mathcal{S}} \cap \mathcal{U}_{000}$, for which the coordinates $\y{j}$ do not vanish. Since $\mathcal{S}$ is smooth, we can assume that, in $P$, \begin{equation} \label{temporaryassumption1proof} \frac{\partial f}{\partial \x{3}} = b_{23}\x{2}+b_{13}\x{1} \neq 0 \ \ \text{.} \end{equation} Thus, we can use $\x{1}$ and $\x{2}$ as local parameters of $\mathcal{S}$ in $P$, and we can express the global sections of the canonical bundle of $\mathcal{S}$ locally in $P$ as holomorphic forms of the type $g(x_1, x_2) d\x{1} \wedge d\x{2}$, where $g$ denotes a function around $P$ without poles. With this procedure, we can write: \begin{equation} \label{eqdiff1} \begin{split} \omega_{12} &:= dz_1 \wedge dz_2 = \frac{d\x{1}}{\y{1}} \wedge \frac{d\x{2}}{\y{2}} = \frac{1}{\y{1} \y{2}} d\x{1} \wedge d\x{2} \\ \omega_{13} &:= dz_1 \wedge dz_3 = \frac{d\x{1}}{\y{1}} \wedge \frac{d\x{3}}{\y{3}} = -\frac{b_{12} \x{1} + b_{23} \x{3}}{(b_{23} \x{2} + b_{13} \x{1})\y{1}\y{3}} d\x{1} \wedge d\x{2} \\ \omega_{23} &:= dz_2 \wedge dz_3 = \frac{d\x{2}}{\y{2}} \wedge \frac{d\x{3}}{\y{3}} = \frac{(b_{13} \x{3} + b_{12} \x{2})}{(b_{23} \x{2} + b_{13} \x{1})\y{1}\y{2}} d\x{1} \wedge d\x{2} \ \ \text{.} \end{split} \end{equation} \noindent We write down also the global holomorphic differentials which arise by the residue map $H^0(A, \mathcal{O}_{A}(\mathcal{S})) = H^0(A, \omega_{A}(\mathcal{S})) \longrightarrow H^0(\mathcal{S}, \omega_{\mathcal{S}})$. We denote, with $(ijk) \in \{(000),(011),(101),(110)\}$, \begin{align} \label{eqdiff1a} \psi_{ijk} := (\theta_{ijk} \cdot d z_{1} \wedge d z_{2} \wedge d z_{3}) \neg \left( \frac{\partial \x{3}}{\theta_{000}\frac{\partial f}{\partial \x{3}}} \right) \ \ \text{,} \end{align} where $\neg$ is the contraction operator. In conclusion, we have, up to a non-zero constant: \begin{equation}\label{eqdiff2} \begin{split} \psi_{000} &= \frac{1}{(b_{23}\x{2}+b_{13}\x{1})\y{1}\y{2}\y{3}} d \x{1} \wedge d \x{2} \\ \psi_{011} &= \frac{\x{2} \x{3}}{(b_{23}\x{2}+b_{13}\x{1})\y{1}\y{2}\y{3}} d \x{1} \wedge d \x{2} \\ \psi_{101} &= \frac{\x{1} \x{3}}{(b_{23}\x{2}+b_{13}\x{1})\y{1}\y{2}\y{3}} d \x{1} \wedge d \x{2} \\ \psi_{110} &= \frac{\x{1} \x{2}}{(b_{23}\x{2}+b_{13}\x{1})\y{1}\y{2}\y{3}} d \x{1} \wedge d \x{2} \ \ \text{.} \end{split} \end{equation} Once we have multiplied all expressions in (\ref{eqdiff1}) and (\ref{eqdiff2}) by ${(b_{23}\x{2}+b_{13}\x{1})\y{1}\y{2}\y{3}}$, we obtain the following expression of the canonical map of $\mathcal{S}$, which is defined independently on the assumption in (\ref{temporaryassumption1proof}) and every point of the affine space $\mathbb{A}^6$ with coordinates $(\x{1}, \x{2}, \x{3},\y{1}, \y{2}, \y{3})$: \begin{equation*} \phi_{\mathcal{S}} = \begin{bmatrix} (b_{23}\x{2}+b_{13}\x{1})\y{3} & (b_{23}\x{3}+b_{12}\x{1})\y{2} & (b_{12}\x{2}+b_{13}\x{3})\y{1} & 1 & \x{1}\x{2} & \x{1}\x{3} & \x{2} \x{3} \ \ \text{.} \end{bmatrix} \ \ \text{.} \end{equation*} To compute its rank, we have to consider the equations $\Eeq{j} = 0$ of $E_j$, considered in the form as in Equation \ref{LegendreNormalFormE2}, and the affine map $\Phi: \mathbb{A}^6 \longrightarrow \mathbb{A}^9$ defined by \begin{equation}\label{mapPsiaffinecase} \Phi = \begin{pmatrix} (b_{23}\x{2}+b_{13}\x{1})\y{3} & (b_{23}\x{3}+b_{12}\x{1})\y{2} & (b_{12}\x{2}+b_{13}\x{3})\y{1} & \x{1}\x{2} & \x{1}\x{3} & \x{2} \x{3} & g_1 & g_2 & g_3 \\ \end{pmatrix} \ \ \text{.} \end{equation} The differential of the canonical map of $\mathcal{S}$ is injective at the point $P$ if the matrix of the differential of $\Phi$ at $P$ has maximal rank. The matrix of the differential in $P$ is obtained, as usual, by considering the derivatives with respect to the coordinates $\x{j}$ and $\y{j}$, which is: \begin{equation} \label{matrixNequation} \mbox{\scriptsize{$
N := \begin{bmatrix}
b_{13}\y{3} & b_{12}\y{2} & 0 & 0 & \x{3} & \x{2} & 2\x{1}(\pa{1}^2-2\x{1}^2+1) & 0 & 0 \\
b_{23}\y{3} & 0 & b_{12}\y{1} & \x{3} & 0 & \x{1} & 0 & 2\x{2}(\pa{2}^2-2\x{2}^2+1) & 0 \\
0 & b_{23}\y{2} & b_{13}\y{1} & \x{2} & \x{1} & 0 & 0 & 0 & 2\x{3}(\pa{3}^2-2\x{3}^2+1) \\
0 & 0 & b_{12}\x{2}+b_{13}\x{3} & 0 & 0 & 0 & 2\y{1} & 0 & 0 \\
0 & b_{12}\x{1}+b_{23}\x{3} & 0 & 0 & 0 & 0 & 0 & 2\y{2} & 0 \\
b_{13}\x{1}+b_{23}\x{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\y{3} \\ \end{bmatrix} $}} \end{equation}
\noindent It is now straightforward to see that the determinant of the submatrix formed by the first six colums from left is: \begin{equation*}
det(N_{1,2,3,4,5,6}) := 2\x{1}\x{2}\x{3}(b_{13}\x{1}+b_{23}\x{2})(b_{12}\x{1}+b_{23}\x{3})(b_{12}\x{2}+b_{13}\x{3}) \end{equation*} On the other side, the determinant of the submatrix formed by deleting the central three colums from is: \begin{equation} \label{FIRSTDETRELATION}
det(N_{1,2,3,7,8,9}) := 16\x{1}\x{2}\x{3}\y{1}\y{2}\y{3} \end{equation} Hence, the rank of $N$ may decrease only on the divisors $div(\x{j})$ for some $j$. Indeed, by the defining equation of $\mathcal{S}$ in (\ref{SURFEQU}) and by our generality hypothesis on the coefficients in the equation, at most one of the factors involved in the previous determinant expressions in (\ref{FIRSTDETRELATION}) can vanish.
If $\x{3}=0$ (which implies that $\x{1} \neq 0$ and $\x{2} \neq 0$), we can assume, without loss of generality, that $\y{3} = \delta_{3}$.
and the differential matrix $N$ in (\ref{matrixNequation}) has the following form \begin{equation*} \mbox{\scriptsize{$ N := \begin{bmatrix}
b_{13} \delta_{3} & b_{12}\y{2} & 0 & 0 & 0 & \x{2} & 2\x{1}(\pa{1}^2-2\x{1}^2+1) & 0 & 0 \\
b_{23} \delta_{3} & 0 & b_{12}\y{1} & 0 & 0 & \x{1} & 0 & 2\x{2}(\pa{2}^2-2\x{2}^2+1) & 0 \\
0 & b_{23}\y{2} & b_{13}\y{1} & \x{2} & \x{1} & 0 & 0 & 0 & 0 \\
0 & 0 & b_{12}\x{2} & 0 & 0 & 0 & 2\y{1} & 0 & 0 \\
0 & b_{12}\x{1} & 0 & 0 & 0 & 0 & 0 & 2\y{2} & 0 \\
b_{13}\x{1}+b_{23}\x{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\delta_{3} \\ \end{bmatrix} $}} \ \ \text{.} \end{equation*}
Finally, we notice that \begin{align*}
det(N_{1,4,6,7,8,9}) &= -8b_{13} \y{1}^2\y{2}(b_{13} \x{1}-b_{23} \x{2}) \\
det(N_{2,4,6,7,8,9}) &= 8b_{12} \x{1}\y{1}\x{2}(\x{2}^2-\pa{2}^4)\\
det(N_{3,4,6,7,8,9}) &= -8b_{12} \y{2}\x{2}^2(\x{1}^2-\pa{1}^4) \ \ \text{,} \end{align*} which, according to our generality condition can not simultaneously vanish. Indeed, if this were the case and all $\y{i}$ were non-zero, then we would have that $b_{13} \x{1}-b_{23} \x{2} = 0$ and $\x{1}^2-\pa{1}^4 = \x{2}^2-\pa{2}^4 = 0$. But also this situation can be avoided if we suppose the coefficients $b_{23},b_{13},b_{12}$ to be sufficiently general. If otherwise $\y{1} = 0$, then we have clearly that $det(N_{3,4,6,7,8,9}) \neq 0$, and the conclusion follows.
On the other side, the matrix in (\ref{matrixNequation}) shows that injectivity of the differential of $\Phi$ fails at the points where $y_3 = 0 = b_{13}\x{1}+b_{23}\x{2}$, $y_2 = 0 = b_{12}\x{1}+b_{23}\x{3}$ or $\y{1} = 0 = b_{12}\x{2}+b_{13}\x{3}$. Since in local coordinates we have that $W_1 = div(b_{12}\x{2}+b_{13}\x{3})$, $W_2 = div(b_{12}\x{1}+b_{23}\x{3})$ and $W_3 = div(b_{13}\x{1}+b_{23}\x{2})$, these points are the $24$ points which belong to some divisor $W_j$ and are fixed under the corresponding involution ${\iota}_j$, as claimed. \newline
\noindent It remains to show that the differential of the canonical map has maximal rank at a base point of the polarization, when the choice of the coefficients in (\ref{SURFEQU}) is sufficiently general. \newline
\noindent Let us assume that $P = (\infty_{+}, \infty_{+}, \infty_{+})$ (the other cases can be treated similarly). Around $P$ we use the coordinates $(\vinf{j}, \winf{j})_j$ of the open set $\mathcal{U}_{111}$ the equation of $\widehat{\mathcal{S}}$ is: \begin{equation*} f_{\infty} = \vinf{1}\vinf{2}\vinf{3} + b_{23} \vinf{1} + b_{13} \vinf{2} + b_{12} \vinf{3} = 0 \end{equation*} According with the affine equations in (\ref{LegendreNormalFormE2}) and (\ref{LegendreNormalForm2}) of the elliptic curves $E_j$, we have that, in $P$, all coordinates $\vinf{j}$ vanish, while $\winf{j} = 1$ for each $j$. We have that \begin{equation*} \frac{\partial}{\partial \vinf{3}}f_{\infty} = \vinf{1}\vinf{2} + b_{12} \end{equation*} which is non-zero if we assume that $b_{12} \neq 0$. If we repeat the same procedure, we obtain a map $\mathbb{A}^6 = \mathbb{A}(\vinf{j}, \winf{j})_j \longrightarrow \mathbb{A}^9$ which represents the canonical map around $P$:
\begin{equation} \label{canonicalexpressionatinfty} \Phi_{\infty} = \begin{bmatrix} ((\vinf{1}\vinf{2} + b_{12})\winf{3} & (\vinf{1}\vinf{3} + b_{13})\winf{2} & (\vinf{2}\vinf{3} + b_{23})\winf{1} & \vinf{1} & \vinf{2} & \vinf{3} & \Eeqinf{1} & \Eeqinf{2} & \Eeqinf{3} \\ \end{bmatrix} \ \ \text{.} \end{equation} It is now easy to verify that the matrix of the differential in $P$ is of the form \begin{equation*} \begin{bmatrix}
0 & 0 & 0 & 1 & 0 & 0 & * & * & * \\ 0 & 0 & 0 & 0 & 1 & 0 & * & * & * \\ 0 & 0 & 0 & 0 & 0 & 1 & * & * & * \\ b_{12} & 0 & 0 & 0 & 0 & 0 & * & * & * \\ 0 & b_{13} & 0 & 0 & 0 & 0 & * & * & * \\ 0 & 0 & b_{23} & 0 & 0 & 0 & * & * & * \end{bmatrix} \ \ \text{,} \end{equation*} which has clearly maximal rank when the coefficients $b_{23}$, $b_{13}$ and $b_{12}$ are non-zero. This completely proves the third claim of the proposition. \end{proof}
\section{On the canonical map in the general case} \label{GENERALFINALSECTION}
Following the notations of the previous section (see Notation \ref{productSurfEll}), to every symmetric matrix $\tau = (\tau_{ij})$ in the Siegel half space ${\mathcal{H}}_3$, there is a corresponding $(2,2,2)$-polarized abelian threefold $T = \bigslant{\mathbb{C}^3}{\tau \mathbb{Z}^3 \oplus 2\mathbb{Z}^3}$ with a projection onto a $(1,2,2)$-polarized abelian threefold $A$. This latter abelian threefold is isogenous to a product of three elliptic curves if $\tau$ belongs to the diagonal subspace ${\mathcal{H}}_1^3 = {\mathcal{H}}_1 \times {\mathcal{H}}_1 \times {\mathcal{H}}_1$ of ${\mathcal{H}}_3$. In such a case, Proposition \ref{generalbehavior1} tells us which pairs of points are critical for the canonical map of a general surface $\mathcal{S} = div(\theta)$ in the $(1,2,2)$-polarization of $A$, according to the following general definition: \begin{def1}\label{CRITICALPAIRS} Let $\mathcal{D}$ be a smooth ample divisor on an abelian variety $A$. Denoted by $\CAN{\mathcal{D}}$ the canonical map of $\mathcal{D}$, a \textit{\textbf{critical pair}} for $\mathcal{D}$ is a couple of points $(P, Q)$ such that: \begin{itemize}
\item $\CAN{\mathcal{D}}(P) = \CAN{\mathcal{D}}(Q)$ if $P \neq Q$.
\item The differential of $\CAN{\mathcal{D}}$ at $P$ has not maximal rank, if $P = Q$. \end{itemize} \end{def1} Recall that, denoted by $(z_1, z_2, z_3)$ a coordinate system for an abelian threefold $A$ which is isogenous to the product of three elliptic curves, we proved in Proposition \ref{generalbehavior1} that the canonical map of $\mathcal{S} = div(\theta) \subseteq A$ fails to be an embedding precisely on those pairs of points which lie on the canonical divisors \begin{equation*} W_k := div\left(\frac{\partial \theta}{\partial z_k}\right) = div\left(dz_i \wedge dz_j \right) \ \ \text{,} \end{equation*} where $(i,j,k)$ is a permutation of the indices $(1,2,3)$. This happens because the divisors $W_k$ are invariant for the involution ${\iota}_k \colon z_k \mapsto -z_k$.
\begin{rmk} (On the generality condition in the statement of Proposition \ref{generalbehavior1}) \label{BEQUADROG} Let us consider a diagonal matrix $\tau_0 \in {\mathcal{H}}_1^3$ in the Siegel upper half space ${\mathcal{H}}_3$. As in Notation \ref{productSurfEll}, for ever couple of indices $(i,j)$ there is a $(1,2)$-polarized abelian surface $S_{ij}$ with a projection: \begin{equation*} {\pi_{\B}}_{ij} \colon B_{ij}:= E_i \times E_j \longrightarrow S_{ij} \end{equation*} For the $(1,2,2)$-abelian threefold $(A, \mathcal{L})$, which is isogenous to via a degree $2$ projection $p \colon T \longrightarrow A$, we have that the vector space of sections of the polarization decomposes, for every couple of indices $(i,j)$, as: \begin{equation} \label{sectionsDECOMP2} H^0(A, \mathcal{L}) \cong H^0(S_{ij}, \PMpij{ij})\cdot \theta^{E_k}_0 \oplus H^0(S_{ij}, \PMmij{ij})\cdot \theta^{E_k}_1 \end{equation}
In the proof of Proposition \ref{generalbehavior1}, a smooth surface $\mathcal{S} := div(\theta)$ in the polarization class $|\mathcal{L}|$ of such an abelian threefold $A$ were required to fulfill some conditions in order to be considered as \textbf{sufficiently general}. There conditions can be resumed as follows: \begin{itemize}
\item For every permutation $(i,j,k)$ of the indices $(1,2,3)$, the divisors $\Cu_{ij} := div(\nu_{ij})$ and $\Du_{ij} := div(\xi_{ij})$ of $S_{ij}$, which are defined according to the decomposition in (\ref{sectionsDECOMP2}) such that
\begin{equation*}
\theta = \nu_{ij}\cdot \theta^{E_k}_0 + \xi_{ij}\cdot \theta^{E_k}_1
\end{equation*} are smooth irreducible genus $3$ curves. \item For every point $P$ on $\mathcal{S}$ it holds that: \begin{itemize} \item If the sections $\nu_{ij}$ and $\xi_{ij}$ both vanish on $P$, then neither the $i$-th coordinate nor the $j$-th coordinate of $P$ are represented by $2$-torsion on $E_i$, resp. $E_j$. \item If the sections $\nu_{ij}$ and $\xi_{ij}$ both vanish on $P$, then $\nu_{ik}$ and $\xi_{ik}$ do not simultaneously vanish at $P$. \end{itemize} \end{itemize} If the latter generality conditions are fulfilled, then the proof of Proposition \ref{generalbehavior1} ensures that the only critical pairs on $\mathcal{S}$ are as claimed in the statement of same proposition.
\end{rmk}
In this section, we aim to prove that, when $\tau$ is sufficiently general, there are no critical pairs on a general surface $\mathcal{S}$ within the $(1,2,2)$ polarization of $A$. That means that the canonical map yields a holomorphic embedding in $\mathbb{P}^5$ in the general case, in agreement with Theorem \ref{teoremafinale}: \begin{thm1*}
Let be $(A, \mathcal{L})$ a general $(1,2,2)$-polarized abelian threefold and let be $\mathcal{S}$ a general surface in the linear system $|\mathcal{L}|$. Then the canonical map of $\mathcal{S}$ is a holomorphic embedding. \end{thm1*}
\begin{sketch} For the sake of clarity and exposition, we introduce at this place the basic ideas behind the proof of Theorem \ref{teoremafinale}, and we give a complete proof of it at the end of this section. \newline
\noindent We fix a certain general diagonal matrix $\tau_0 = diag(\tau_{ii})_i$ in the subspace ${\mathcal{H}}_1^3$ of ${\mathcal{H}}_3$ and we denote by \begin{equation}\label{HDCENTRDEF}
{\mathcal{H}_{3\Delta, \tau_0}} := \SET{\tau \in {\mathcal{H}}_3}{\text{The diagonal entries of $\tau$ coincide with those of $\tau_0$}} \ \ \text{.} \end{equation} To every open neighborhood $\mathcal{U}$ of $\tau_0$ in ${\mathcal{H}_{3\Delta, \tau_0}}$ corresponds a family of $(1,2,2)$-polarized abelian varieties $(A_{\mathcal{U}}, \mathcal{L}_{\mathcal{U}})$. The abelian threefold $(A_{\tau_0}, \mathcal{L}_{\tau_0})$ of the family is isogenous to the product of three elliptic curves as in Remark \ref{BEQUADROG}. Every non-zero holomorphic section $\theta$ of $H^0(A_{\tau_0}, \mathcal{L}_{\tau_0})$ gives rise to a family of surfaces $\mathcal{S}_{\mathcal{U}}$, each contained in the polarization class of the respective fiber of the family of abelian threefolds $A_{\mathcal{U}}$. We have, for every element $\tau$ of the basis $\mathcal{U}$, a natural identification: \begin{equation} \label{IDENTIFICATION} H^0(A_{\tau_0}, \mathcal{L}_{\tau_0}) \cong H^0(A_{\tau}, \mathcal{L}_{\tau}) \cong \mathbb{P}^3 \ \ \text{,} \end{equation} defined by the natural restriction of the holomorphic sections of $H^0(A_{\mathcal{U}}, \mathcal{L}_{\mathcal{U}})$ to the fibers of the family: \begin{equation}
H^0(A_{\mathcal{U}}, \mathcal{L}_{\mathcal{U}})|_{\tau} \cong H^0(A_{\tau}, \mathcal{L}_{\tau}) \ \ \text{.} \end{equation}
\noindent If we assume our claim to be false, then we expect to find, for every $\tau$ in a sufficiently small neighborhood $\mathcal{U}$ of $\tau_0$, a critical pair $(P_{\tau}, Q_{\tau})$ for the canonical map of $\mathcal{S}_{\tau}$.
This means that we have a whole family $\mathcal{P}$ on $\mathcal{U}$ of critical pairs $(P_{\mathcal{U}}, Q_{\mathcal{U}})$, which gives rise, by restriction to the closed loci \begin{equation} \label{LOCIEQ} {\mathcal{H}_{3\Delta, \tau_0}}^{(ij)} := \{\tau_{ik} = \tau_{jk} = 0\} \cong {\mathcal{H}}_2 \times {\mathcal{H}}_1 \ \ \text{,} \end{equation} to three different families $\mathcal{P}_{ij}$ of critical pairs, each defined respectively on $\mathcal{U}_{ij} := \mathcal{U} \cap {\mathcal{H}_{3\Delta, \tau_0}}^{(ij)}$: \begin{equation} \label{restrictionPdiagram} \begin{diagram} & &\mathcal{P}_{ij} &\rInto{}& \mathcal{P}\\ & &\dTo_{} & &\dTo\\ & &\mathcal{U}_{ij} := \mathcal{U} \cap {\mathcal{H}_{3\Delta, \tau_0}}^{(ij)} &\rInto &\mathcal{U}\\ \end{diagram} \end{equation}
It is then natural to ask whether we can classify all possible critical pairs on (general) surfaces in abelian threefolds which lie in the loci ${\mathcal{H}_{3\Delta, \tau_0}}^{(ij)}$, similarly to what we did in Proposition \ref{generalbehavior1}. Note that these loci parametrize $(1,2,2)$-abelian variety which are isogenous to products of a $(2,2)$-polarized abelian surface $B$ and a $(2)$-polarized elliptic curve $E$, according to Definition \ref{LOCIEQ} and Notation \ref{productSurfEll}.
The following proposition, which we state for simplicity for the couples of indices $(i,j) = (1,2)$, answers to this question. \end{sketch}
\begin{prop1}\label{generalbehavior2}
Let $\tau_0 = diag(\tau_{11}, \tau_{22}, \tau_{33})$ be a diagonal matrix in the Siegel upper half-space $\mathcal{H}_{3}$, let $(A_{\tau_0}, \mathcal{L}_{\tau_0})$ be the associated $(1,2,2)$-polarized abelian variety with a general divisor $S_{\tau_0} = div(\theta)$ in the polarization $|\mathcal{L}_{\tau_0}|$. Then, for a sufficiently small neighborhood $\mathcal{U}$ of $\tau_0$ in the closed locus ${\mathcal{H}_{3\Delta, \tau_0}}^{(12)}$, for every surface $\mathcal{S}_{\tau} = div(\theta)$ of the restricted family $\mathcal{U}_{12} := \mathcal{U} \cap {\mathcal{H}_{3\Delta, \tau_0}}^{(12)}$ (see Diagram \ref{restrictionPdiagram}), it holds that
\begin{itemize}
\item The involution $\iv{3} \colon (z_1, z_2, z_3) \mapsto (z_1, z_2, -z_3)$ on $A_{\tau}$ leaves $\mathcal{S}_{\tau}$ invariant and induces an involution on the canonical divisor $W_3 := div \left( \frac{\partial \theta}{\partial z_3} \right)$ on $\mathcal{S}_{\tau}$. (Recall that, in general, $W_k := div \left( \frac{\partial \theta}{\partial z_k} \right)$.)
\item The canonical map of $\mathcal{S}$ is one-to-one, except:
\begin{itemize}
\item on the divisor $W_3$, on which the canonical map has degree $2$ and factors respectively through the involution $\iv{3}$.
\item on the finite set $W_1 \cap W_2$, on which the canonical map of $\mathcal{S}$ factors through the involution $\iv{1}\cdot \iv{2}: (z_1, z_2, z_3) \mapsto (-z_1, -z_2, z_3)$. \end{itemize} \item The differential of the canonical map of $\mathcal{S}$ has everywhere maximal rank, except on those points of $W_3$ which are fixed points under the action of the involution $\iv{3}$ on $W_3$. \end{itemize} \end{prop1}
\noindent Proposition \ref{generalbehavior2} shows, in particular, how the behavior of the canonical map changes when, in the moduli space, we move away from the locus of abelian threefolds, which are isogenous to a product of elliptic curves, and we move along closed loci parametrizing abelian threefolds which are isogenous to a product of a simple $(2,2)$ abelian surface and an elliptic curve.
From Proposition \ref{generalbehavior2} also follows also that every restriction $\mathcal{P}_{ij}$ as in Diagram \ref{restrictionPdiagram} of a family $\mathcal{P}$ of critical pairs only contains pairs of points which are conjugate under the same involution. However, on the central fiber $\tau_0$, (that is, on the intersection of these loci) the three different families of critical pairs $\mathcal{P}_{ij}$ specialize to a unique pair of points $(P,Q)$ which must be conjugated under many different involutions, which would lead to a contradiction.
After this rough explaination of the basic approach to our problem, we aim to formalize the steps toward the proof of Theorem \ref{teoremafinale}.
\begin{def1}\label{generaldeformationdefinition} In the notation of Proposition \ref{generalbehavior2}, let us suppose that $(P,Q)$ is a critical pair on $\mathcal{S}_{\tau_{0}} = div(\theta)$. A \textbf{family of critical pairs around $(P,Q)$} is the datum of a couple $(\mathcal{S}_{\mathcal{U}} \longrightarrow \mathcal{U}, \mathcal{P})$, where: \begin{itemize} \item[i)] $\mathcal{U}$ is an open set of ${\mathcal{H}_{3\Delta, \tau_0}}$ (see definition of ${\mathcal{H}_{3\Delta, \tau_0}}$ in Equation \ref{HDCENTRDEF}). \item[ii)] $\mathcal{S}_{\mathcal{U}} \longrightarrow \mathcal{U}$ is a family of surfaces defined as $\mathcal{S}_{\tau} = div(\theta)$ for every $\tau$, by applying the natural identification \begin{equation} H^0(A_{\tau_0}, \mathcal{L}_{\tau_0}) \cong H^0(A_{\mathcal{U}}, \mathcal{L}_{\mathcal{U}}) \cong H^0(A_{\tau}, \mathcal{L}_{\tau}) \ \ \text{.} \end{equation}
\item[iii)] With $\Delta_{\mathcal{U}}$ the diagonal subscheme in $\mathbb{P}_{\mathcal{U}}^5 \times \mathbb{P}_{\mathcal{U}}^5$ and $\phi_{\mathcal{S}_{\mathcal{U}}}$ the map, which on each surface $\mathcal{S}_{\tau}$ of the family coincides with the canonical map $\phi_{\mathcal{S}_{\tau}}$, $\mathcal{P}$ is a closed irreducible subscheme of \begin{equation*} \mathcal{K}_{\mathcal{U}} := (\phi_{\mathcal{S}_{\mathcal{U}}} \times_{\mathcal{U}} \phi_{\mathcal{S}_{\mathcal{U}}})^{-1}(\Delta_{\mathcal{U}}) \subseteq \mathcal{S}_{\mathcal{U}} \times_{\mathcal{U}} \mathcal{S}_{\mathcal{U}}
\end{equation*} which is dominant over $\mathcal{U}$ and such that its restriction on the central fiber $\tau_0$ is $(P,Q)$. Here $\times_{\mathcal{U}}$ denotes, as usual, the cartesian product on $\mathcal{U}$ according to the pullback diagram: \begin{equation*} \begin{diagram} \mathcal{S}_{\mathcal{U}} \times_{\mathcal{U}} \mathcal{S}_{\mathcal{U}} &\rTo &\mathcal{S}_{\mathcal{U}} \\
\dTo & & \dTo \\ \mathcal{S}_{\mathcal{U}} &\rInto & \mathcal{U} \end{diagram} \end{equation*} \end{itemize} \end{def1}
\begin{rmk} \label{BEQUADROij} In the notation of the previous Definition \ref{generaldeformationdefinition}, if we restrict a family of critical points $(\mathcal{S}_{\mathcal{U}} \longrightarrow \mathcal{U}, \mathcal{P})$ to one of the loci ${\mathcal{H}_{3\Delta, \tau_0}}^{(ij)}$ (recall the definition in \ref{LOCIEQ}), we obtain a subfamily of critical points $(\mathcal{S}_{\mathcal{U}_{ij}} \longrightarrow \mathcal{U}_{ij}, \mathcal{P}_{ij})$ defined on the open set $\mathcal{U}_{ij} := \mathcal{U} \cap {\mathcal{H}_{3\Delta, \tau_0}}^{(ij)}$. Furthermore, we have in this case a family of projections \begin{equation} \label{PProjU} p_{\mathcal{U}} \colon T_{\mathcal{U}} = B_{\mathcal{U}, ij} \times E_k \longrightarrow A_{\mathcal{U}} \end{equation} and a family of $(2,2)$-polarized abelian surfaces $B_{\mathcal{U}_{ij}} \longrightarrow \mathcal{U}_{ij}$ with a projection onto a family of $(1,2)$-polarized abelian surfaces $S_{\mathcal{U}_{ij}} \longrightarrow \mathcal{U}_{ij}$, and there is a decomposition as in (\ref{sectionsDECOMP2}): \begin{equation} \label{sectionsDECOMPgen}
H^0(A_{\mathcal{U}}, \mathcal{L}_{\mathcal{U}}) \cong H^0(S_{\mathcal{U}_{ij}}, \PMpij{ij})\cdot \theta^{E_k}_0 \oplus H^0(S_{\mathcal{U}_{ij}}, \PMmij{ij})\cdot \theta^{E_k}_1 \end{equation} \end{rmk}
\begin{not1} \label{NOTNUXI}
If we write a non-zero section $\theta$ of $H^0(A_{\mathcal{U}}, \mathcal{L}_{\mathcal{U}})$ according to (\ref{sectionsDECOMPgen}), we have \begin{equation}\label{thetaDECOMPOSITIONij}
\theta = \nu_{\mathcal{U}, ij}\cdot \theta^{E_k}_0 + \xi_{\mathcal{U}, ij}\cdot \theta^{E_k}_1 \end{equation} for some $\nu_{\mathcal{U}, ij}$ and $\xi_{\mathcal{U}, ij}$ in $H^0(S_{\mathcal{U}_{ij}}, \PMpij{ij})$ and $H^0(S_{\mathcal{U}_{ij}}, \PMmij{ij})$ respectively. We denote henceforth, similarly to what we did in Remark \ref{BEQUADROG}, the following families of divisors in $S_{\mathcal{U}, ij}$ as follows \begin{align*}
\Cu_{\mathcal{U}, ij} &:= div(\nu_{\mathcal{U}, ij}) \\
\Du_{\mathcal{U}, ij} &:= div(\xi_{\mathcal{U}, ij}) \end{align*}
\end{not1}
\begin{oss1} \label{firstOssBXE} \noindent Let us consider a family of critical pairs $(\mathcal{S}_{\mathcal{U}} \longrightarrow \mathcal{U}, \mathcal{P})$ and its restriction to: \begin{equation} \label{LOCIEQ2} {\mathcal{H}_{3\Delta, \tau_0}}^{(ij)} := \{\tau_{ik} = \tau_{jk} = 0\} \cong {\mathcal{H}}_2 \times {\mathcal{H}}_1 \ \ \text{,} \end{equation} We have, following our previous discussion in Remark \ref{BEQUADROij}, a family $(\mathcal{S}_{\mathcal{U}_{ij}} \longrightarrow \mathcal{U}_{ij}, \mathcal{P}_{ij})$ on the open set $\mathcal{U}_{ij} := \mathcal{U} \cap {\mathcal{H}_{3\Delta, \tau_0}}^{(ij)}$, with $\mathcal{S}_{\mathcal{U}} = div(\theta)$ which satisfy the generality conditions stated in Remark \ref{BEQUADROij}.
Suppose that, for some $\tau$ in $\mathcal{U}_{ij}$, the point $P_{\tau}$ belongs to the canonical divisor \begin{equation*} W_k = div\left(\frac{\partial \theta}{\partial z_k}\right)_{\tau} \subseteq \mathcal{S}_{\tau} \ \ \text{.} \end{equation*} Then the set ${\pi_{\B}}_{\tau}\left(p_{\tau}^{-1}(W_k)\right)$, defined on each $\tau$ according to the following diagram \begin{equation} \label{restrictionPdiagram} \begin{diagram} & &B_{\mathcal{U}, ij} \times E_k &\rTo{p_{\mathcal{U}}}& A_{\mathcal{U}}\\ & &\dTo_{{\pi_{\B}}_{\mathcal{U}_{ij}} \times id_{E_k}} & & \\ & &S_{\mathcal{U}, ij} \times E_k & &\\ \end{diagram} \end{equation} is reducible, and we have, for every $\tau$ in $\mathcal{U}_{ij}$ (see again Remark \ref{BEQUADROij} for the notations): \begin{equation*}
({\pi_{\B}}_{\tau} \times id_{E_k})\left(p_{\tau}^{-1}(W_k)\right) = (\Cu_{\tau, ij} \cap \Du_{\tau, ij}) \times E_k \cup S_{\tau} \times E_k[2] \ \ \text{.} \end{equation*}
Indeed, if we write the section $\theta$ as in Equation \ref{thetaDECOMPOSITIONij}, then the coordinates of the points in the divisor $W_k$ must fulfill the conditions:
\begin{equation}\label{casesDEFeta} \begin{cases} \nu_{\tau, ij}\theta^{E_k}_0 &+ \ \ \xi_{\tau, ij}\theta^{E_k}_1 = 0\\ \nu_{\tau, ij}\frac{\partial}{\partial z_k} \theta^{E_k}_0 &+ \ \ \xi_{\tau, ij}\frac{\partial}{\partial z_k}\theta^{E_k}_1 = 0 \ \ \text{.} \end{cases} \end{equation} The conditions in (\ref{casesDEFeta}) can be looked at as a linear system in $\nu_{\tau, ij}$ and $\xi_{\tau, ij}$. This implies that both $\nu_{\tau}$ and $\xi_{\tau}$ vanish, or the determinant \begin{equation} \label{etadefinition} \eta_k := \begin{vmatrix}\theta^{E_k}_0 & \theta^{E_k}_1 \\ \frac{\partial}{\partial z_k} \theta^{E_k}_0 & \frac{\partial}{\partial z_k} \theta^{E_k}_1 \end{vmatrix} \in H^0(E_k, \mathcal{O}_{E_k}(4O_{E_k})) \ \ \text{.} \end{equation}
The conclusion follows now by Equation \ref{etadefinition_Y} in Example \ref{basicexampleEll} (since $div(\eta_k) = E_k[2]$ ). \end{oss1}
In the next lemma, we show that the restriction $\mathcal{P}_{ij}$ (as in Remark \ref{BEQUADROij}) of a family of critical pairs $\mathcal{P}$ around a couple of points $(P,Q)$ with $P \neq Q$ must be contained in a closed locus of $\mathcal{S}_{\mathcal{U}_{ij}} \times_{\mathcal{U}_{ij}} \mathcal{S}_{\mathcal{U}_{ij}}$ which, in \begin{equation}\label{eqproduct} \left(S_{\mathcal{U}, ij} \times E_k\right)^2 := \left(S_{\mathcal{U}, ij} \times_{\mathcal{U}_{ij}} S_{\mathcal{U}, ij} \right) \times E_k^2 \ \ \text{,} \end{equation} is defined by a holomorphic section of the $(4,4)$-polarization $\mathcal{O}_{E_k}(4O_{E_k}) \boxtimes \mathcal{O}_{E_k}(4O_{E_k})$ on $E_k \times E_k$.
\begin{lem1}\label{secondOssBXE} \noindent Let us consider a family of critical pairs $(\mathcal{S}_{\mathcal{U}} \longrightarrow \mathcal{U}, \mathcal{P})$ and its restriction $(\mathcal{S}_{\mathcal{U}_{ij}} \longrightarrow \mathcal{U}_{ij}, \mathcal{P}_{ij})$ to the open set $\mathcal{U}_{ij} := \mathcal{U} \cap {\mathcal{H}_{3\Delta, \tau_0}}^{(ij)}$ with $\mathcal{S}_{\mathcal{U}} = div(\theta)$ which satisfy the generality conditions of Remark \ref{BEQUADROij}. Let us assume that $P_{\tau} \neq Q_{\tau}$ for every $\tau$ in $\mathcal{U}_{ij}$ (i.e., $\mathcal{P}$ is a family of critical pairs of distinct points). \newline
Then, in the notations of the following diagram (cf. also Diagram \ref{restrictionPdiagram})
\begin{equation} \label{restrictionPdiagram2} \begin{diagram} & &\left(B_{\mathcal{U}, ij} \times E_k\right)^2 &\rTo{\ \ \ \ p_{\mathcal{U}} \times_{\mathcal{U}} p_{\mathcal{U}} \ \ \ \ }& A_{\mathcal{U}}^2 = A_{\mathcal{U}} \times_{\mathcal{U}} A_{\mathcal{U}}\\ & &\dTo_{({\pi_{\B}}_{\mathcal{U}_{ij}} \times id_{E_k})^2} & & \\ & &\left(S_{\mathcal{U}, ij} \times E_k \right)^2 & &\\ \end{diagram} \end{equation}
there exists a non-zero holomorphic section $v_k$ of $H^0(E_k \times E_k, \mathcal{O}_{E_k}(4O_{E_k}) \boxtimes \mathcal{O}_{E_k}(4O_{E_k}))$ such that: \begin{equation*}
({\pi_{\B}}_{\mathcal{U}_{ij}} \times id_{E_k})^2\left(\left(p_{\mathcal{U}} \times_{\mathcal{U}} p_{\mathcal{U}}\right)^{-1}(\mathcal{P}_{ij})\right) \subseteq \left[(\Cu_{\mathcal{U}, ij} \cap \Du_{\mathcal{U}, ij}) \times E_k \right]^2 \cup \left[S_{\mathcal{U}, ij}^2 \times div(v_k) \right] \end{equation*} \end{lem1} \begin{proof} It sufficies to prove the claim for a generic fiber of the family. For this purpose, we can assume $(i,j,k) = (1,2,3)$ and we pick an element $\tau \in \mathcal{U}_{12}$. The critical pair $(P,Q)$ for the canonical map of $\mathcal{S}_{\tau} = div(\theta)$ can be represented by a couple of points $(\widetilde{P}, \widetilde{Q})$ in $\left(S_{\tau, 12} \times E_3 \right)^2$ according to Diagram \ref{restrictionPdiagram2}, with: \begin{equation*} \begin{split} \widetilde{P}&=(x, z) \\ \widetilde{Q}&=(y, w) \ \ \text{.} \end{split} \end{equation*} The section $\theta$ can be written, as in Equation \ref{thetaDECOMPOSITIONij}, as: \begin{equation} \label{eqStau}
\theta = \nu_{\tau, 12}\cdot \theta^{E_3}_0 + \xi_{\tau, 12}\cdot \theta^{E_3}_1 \end{equation} To simplify the notations, the section $\nu_{\tau, 12}$ (resp. $\xi_{\tau, 12}$) in Equation \ref{eqStau} will be denoted by $\nu$ (resp. $\xi$), and the section $\theta^{E_3}_0$ (resp. $\theta^{E_3}_1$) by $\theta_0$ (resp. $\theta_1$). The condition that the canonical image of $P$ and $Q$ coincide implies that there exists $\lambda \in \mathbb{C}^*$ such that: \begin{equation*} \begin{cases} \nu(x)\theta_0(z) &= \lambda \cdot \nu(y)\theta_0(w) \\ \xi(x)\theta_1(z) &= \lambda \cdot \xi(y)\theta_1(w) \\ \nu(x)\DTH{0}(z) + \xi(x)\DTH{1}(w) &= \lambda \cdot \Bigl( \nu(y)\DTH{0}(w) + \xi(y)\DTH{1}(w) \Bigr) \ \ \text{.} \end{cases} \end{equation*} (Here $\DTH{0}$ and $\DTH{1}$ are the derivatives of $\theta_0$ and $\theta_1$). This leads to
\begin{align*} & \nu(x)\left(\frac{\DTH{0}(z)\theta_0(w) - \theta_0(z)\DTH{0}(w)}{\theta_0(w)} \right) \ \ + \\ & \xi(x)\left(\frac{\DTH{1}(z)\theta_1(w) - \theta_1(z)\DTH{1}(w)}{\theta_1(w)} \right) \ \ = 0 \ \ \text{,} \end{align*} and we can conclude that: \begin{align} \label{definitionofpsi0} \nu(x)\theta_1(w) \rhos{0}{3}(z,w) + \xi(x)\theta_0(w)\rhos{1}{3}(z,w) &= 0 \ \ \text{,} \end{align} where, with $j = 0,1$: \begin{align} \label{RHODEF} \rhos{j}{3}(z,w) := \begin{vmatrix}\theta_j(z) & \theta_j(w) \\ \DTH{j}(z) & \DTH{j}(w) \end{vmatrix} \in H^0(E_3 \times E_3, \mathcal{O}_{E_3}(2O_{E_3}) \boxtimes \mathcal{O}_{E_3}(2O_{E_3})) \end{align} Together with the defining equation of $\mathcal{S}_{\tau}$ in (\ref{eqStau}), we obtain the following linear system in $\nu(x)$ and $\xi(x)$:
\begin{align*} \label{systemfgrho} &\nu(x)\theta_0(z) & &+& &\xi(x)\theta_1(z) & &= 0 \\ &\nu(x)\theta_1(w) \rhos{0}{3}(z,w) & &+& &\xi(x)\theta_0(w)\rhos{1}{3}(z,w) & &= 0 \ \ \text{.} \end{align*}
The determinant of the matrix of the linear system above is precisely \begin{equation}\label{DEFVAB} v_3(z,w) := \begin{vmatrix}\theta_0(z)\theta_{0}(w) & \theta_1(z)\theta_1(w) \\
\rhos{0}{3}(z,w) & \rhos{1}{3}(z,w)\end{vmatrix} \in H^0(E_3 \times E_3, \mathcal{O}_{E_3}(4O_{E_3}) \boxtimes \mathcal{O}_{E_3}(4O_{E_3})) \ \ \text{,} \end{equation} and we conclude that $x$ belongs to $\left(div(\nu) \cap div(\xi)\right) = \Cu_{\tau, 12} \cap \Du_{\tau, 12}$, or \begin{equation*} v_3(z, w) = 0 \end{equation*} This proves the claim of the lemma. \end{proof}
\begin{rmk} \label{OSScomponentZ} In general, if we have an elliptic curve $E$ and $\theta_0, \theta_1$ the canonical basis for $H^0(E, \mathcal{O}_{E}(2 \cdot O_{E})$ it is easy to determine a polynomial equation, in a suitable affine open set, of the divisor $V := div(v)$ in $E \times E$ defined as in Equation \ref{DEFVAB} above as \begin{equation}\label{DEFVAB} v(z,w) := \begin{vmatrix}\theta_0(z)\theta_{0}(w) & \theta_1(z)\theta_1(w) \\
\rhos{0}{}(z,w) & \rhos{1}{}(z,w)\end{vmatrix} \ \ \text{.} \end{equation} (Here $\rhos{0}{}$ and $\rhos{1}{}$ are defined as in Equation \ref{RHODEF}.)
In the notations of Example \ref{basicexampleEll}, for every point $(z,w)$ of $E \times E$ it easily seen that: \begin{equation*} v(z,w) = \theta_0(z)\theta_1(z)\cdot \left(\theta_0(w)\DTH{1}(w) - \DTH{0}(w)\theta_1(w) \right) - \theta_0(w)\theta_1(w)\cdot \left(\theta_0(z)\DTH{1}(z) - \DTH{0}(z)\theta_1(z) \right) \end{equation*} In the affine open set $U$ where $\theta_0(z)$ and $\theta_0(w)$ do not vanish, we can divide by $\theta_0(z)^2\theta_0(w)^2$ and in local affine coordinates $(\x{1}, \y{1}, \x{2}, \y{2})$ with two equations $g_1$, $g_2$ in Legendre normal form as in Equation \ref{LegendreNormalForm} in Example \ref{basicexampleEll}, we get that: \begin{equation*} v(z,w) \sim \x{1}\y{2} - \x{2}\y{1} \end{equation*} Moreover, by using the equations $\y{j}^2 = (\x{j}^2 - 1)(\x{j}^2 - \pa{}^2)$ of $E$ on the components, we can easily recover the equations of the components of $V$ in the open set $U$: \begin{equation*}
div(V)|_U = (\x{1}^2 - \x{2}^2)(\x{1}^2\x{2}^2 - \pa{}^2) \end{equation*} In conclusion, the divisor $V$ splits into the sum of components: \begin{equation*} V = \Delta + (-1).\Delta + X + (-1).X \ \ \text{,} \end{equation*} where $\Delta$ is the diagonal locus in $E \times E$. All components are invariant under the action of the involution $(z,w) \mapsto (z + 1, w+ 1)$, and the irreducible component $X$ meets $\Delta$ in $8$ points whose coordinates are not $2$-torsion on $E$. \end{rmk}
\begin{proof}[of Proposition \ref{generalbehavior2}]
Let us consider $(A_{\tau_0}, \mathcal{L}_{\tau_0})$ the $(1,2,2)$-polarized abelian variety associated to a diagonal matrix $\tau$ in $\mathcal{H}_1^3$, with a divisor $\mathcal{S}_{\tau_0} = div(\theta)$ which is sufficiently general in the polarization $|\mathcal{L}_{\tau_0}|$, in accordance with the conditions of Remark \ref{BEQUADROG}.
Since these conditions are open, they still hold for a sufficiently small neighborhood $\mathcal{U}$ of $\tau_0$ in the closed locus ${\mathcal{H}_{3\Delta, \tau_0}}^{(12)}$. Hence, for a suitable $\mathcal{U}$, there is a family $(A_{\mathcal{U}}, \mathcal{L}_{\mathcal{U}})$ of $(1,2,2)$-polarized abelian threefolds with $\mathcal{S}_{\mathcal{U}} = div(\theta)$, such that, considered the decomposition \begin{equation*}
\theta = \nu_{\mathcal{U}}\cdot \theta_0 + \xi_{\mathcal{U}}\cdot \theta_1 \ \ \text{,} \end{equation*}
the fibers of the induced families $\Cu_{\mathcal{U}} =div \left(\nu_{\mathcal{U}}\right)$ and $\Du_{\mathcal{U}} = div\left(\xi_{\mathcal{U}}\right)$ defined as in Notation \ref{NOTNUXI} (for the sake of readability we omit the indices $1,2$ as in the proof of Lemma \ref{secondOssBXE}), are smooth genus $3$ non-hyperelliptic curves. \newline
By Proposition \ref{generalbehavior1}, the possibile critical pairs on the central fiber $\mathcal{S}_{\tau_0}$ are only of one of the following types: \newline
$\bullet$ \textbf{type (a)}: The pair belongs to the divisor $W_j = div \left(\frac{\partial \theta}{\partial z_j}\right)$ for some $j$. In this case, the points are conjugated under the the sign reversing involution $\iota_j$, which changes the sign on the $j$-th coordinate $z_j \mapsto -z_j$. \newline
$\bullet$ \textbf{type (b)}: The pair belongs to the intersection of two divisors $W_i$ and and $W_j$ for some couple of indices $i,j$. In this case, the points are conjugated under the involution $\iota_i \iota_j$. \newline
Hence, to prove the proposition, it is enough to prove the following claim. \newline
\noindent \textbf{Claim}: Let us consider a family of critical pairs $(\mathcal{S}_{\mathcal{U}} \longrightarrow \mathcal{U}, \mathcal{P}_{12})$ on $\mathcal{U}$ (up to shrinking $\mathcal{U}$). Then the family $\mathcal{P}_{12}$ is one of the following types: \begin{itemize}
\item[1)] On every fiber (up to shrinking $\mathcal{U}$), the critical pair is of type (a) with $j=3$. In particular, the pair always belongs to the divisor $W_3$.
\item[2)] On every fiber (up to shrinking $\mathcal{U}$), the critical pair is of type (b) with $(i,j)=(1,2$. In particular, the pair always belongs to the divisor $W_1 \cap W_2$. \end{itemize}
To this purpose, let us pick a fiber on a point $\tau$ of $\mathcal{U}$ and a critical pair $(P_{\tau},Q_{\tau})$ on $\mathcal{S}_{\tau}$. This critical pair can be represented (see Diagram \ref{restrictionPdiagram2}) as a couple of points $(\widetilde{P}_{\tau}, \widetilde{Q}_{\tau})$ in $\left(S_{\tau, 12} \times E_3 \right)^2$ with \begin{equation*} \begin{split} \widetilde{P}_{\tau}&=(x, z) \\ \widetilde{Q}_{\tau}&=(y, w) \ \ \text{,} \end{split} \end{equation*} where the coordinates $\widetilde{P}_{\tau}$ and $\widetilde{Q}_{\tau}$ are functions of $\tau$. By Lemma \ref{secondOssBXE}, we have that $(x,y)$ belongs to $\Cu_{\tau} \cap \Du_{\tau}$, or $(z, w)$ belongs to $ div(v_3)$.
In the first case, already the condition that $(x,y)$ belongs to $\Cu_{\tau}$ implies that $x = y$, since the curve $\Cu_{\tau}$ is non-hyperelliptic for every $\tau$, up to shrinking $\mathcal{U}$: indeed, recall that (again by Proposition \ref{desccanonicalmap}) the canonical map of $\Cu_{\tau}$ is obtained by considering the derivatives $\frac{\partial \xi}{\partial z_1}$ and $\frac{\partial \xi}{\partial z_2}$ together with the restriction of the sections of $H^0(S, {\mathcal{M}^{-}})$ to $\Du$, and we can conclude that $\phi_{\Du_{\tau}}(x) = \phi_{\Du_{\tau}}(y)$, where $\phi_{\Du_{\tau}}$ denotes the canonical map of $\Du_{\tau}$. \newline
\noindent Let us assume that $(z, w)$ belongs to $V = div(v_3)$.
\noindent The affine equations in Remark \ref{OSScomponentZ} show that $V$ splits into a sum of irreducible components: \begin{equation*} V = \Delta + (-1).\Delta + X + (-1).X \ \ \text{,} \end{equation*}
\noindent For the central fiber $\tau_0$, we described these components in Remark \ref{OSScomponentZ} and we determined their equation on an affine open set. Since the components $(z, w)$ of the points $\widetilde{P}_{\tau}$ and $\widetilde{Q}_{\tau}$ are bound to move on $V$, we have to deal with two cases: \begin{itemize}
\item $(z, w)$ is contained in $\Delta \cup (-1).\Delta$ but not in $X \cup (-1).X'$.
\item $(z, w)$ is contained in $X \cup (-1).X'$. \newline \end{itemize}
\noindent Let us assume that $(z, w)$ is contained in $\Delta \cup (-1).\Delta$ but not in $X \cup (-1).X'$. Then, because this latter condition is open, we can conclude that the same condition holds true on the neighborhood $\mathcal{U}$ up to shrinking it. Hence, on every fiber on $\tau$, we have that for the coordinates $(z, w) = (z, \pmz)$ or $(z, w) = (z, \pmz + 1)$. Let us assume that we are in the case $(z, w) = (z, z)$ (the other cases can be treated similarly). The ratio of the values $[s,t]:=[\theta_0(z), \theta_1(z)]$ of the $(2)$-polarization on $E_3$ define, according to the equation of $\mathcal{S}_{\tau}$, a curve \begin{equation} \label{EQFCICLE} \mathcal{F} \colon s \cdot \nu + t \cdot \xi = 0 \end{equation} in the $(2,2)$-polarization class of $B_{\tau}$,
on which the points $x$ and $y$ belong.
On the other side, the expression of the canonical map of $\mathcal{S}_{\tau}$ is \begin{equation}\label{CANEXP2}
\CAN{\mathcal{S}_{\tau}} = \left[H^0(S_{\tau}, {\mathcal{M}^{+}})\theta_0, H^0(S_{\tau}, {\mathcal{M}^{-}})\theta_1, \frac{\partial \nu}{\partial z_1} \theta_0 + \frac{\partial \xi}{\partial z_1} \theta_1, \frac{\partial \nu}{\partial z_2} \theta_0 + \frac{\partial \xi}{\partial z_2} \theta_1,\nu \DTH{0} + \xi \DTH{1} \right] \ \ \text{.} \end{equation} This expression can be easily compared with the one of the canonical map of $\mathcal{F}$, which is \begin{equation}\label{CANEXP3}
\CAN{\mathcal{F}} = \left[s\cdot H^0(S_{\tau}, {\mathcal{M}^{+}}), t \cdot H^0(S_{\tau}, {\mathcal{M}^{-}}), s\frac{\partial \nu}{\partial z_1} + t\frac{\partial \xi}{\partial z_1} , s\frac{\partial \nu}{\partial z_2} + t\frac{\partial \xi}{\partial z_2}\right] \ \ \text{,} \end{equation} and follows that $\CAN{\mathcal{F}}(x) = \CAN{\mathcal{F}}(y)$ .
\noindent If $\mathcal{F}$ is smooth, then it is non-hyperelliptic (because in a simple $(2,2)$-polarization there are no irreducible hyperelliptic curves). Hence, in this case, we conclude that $x=y$ and $P=Q$. \newline
\noindent If otherwise $\mathcal{F}$ is not smooth, then we can assume that $\mathcal{F}$ is a nodal curve. Indeed, since the $(1,2)$-polarization class on a simple abelian surface is a pencil only containing irreducible curves (cf. Remark \ref{osservationsingularfibers}), we may assume, by generality, that the singular curves in the pencil generated by $\nu$ and $\xi$ in the $(2,2)$-polarization class of $B$ only contains nodal curves, which also are non-hyperelliptic since they are irreducible.
\noindent This implies that $x=y$ if $x$ is not a node on $\mathcal{F}$, otherwise $x =-y$, and the second possibility occurs precisely when $(P,Q)$ is a critical pair which belong to $W_1 \cap W_2$ with $Q = \iota_1 \iota_2 (P)$.
\noindent Let us suppose now that we are in the second case, namely the case that $(z, w)$ is contained in $X \cup (-1).X$ and $w \neq \pmz$ when $\tau$ varies away from the central point $\tau_0$ in a suitable base neighborhood. Our goal is to show by contradiction that this case actually does not occur. \newline
\noindent By Proposition \ref{generalbehavior1}, on the central fiber $\tau_0$ the point $(z, w)$ must also belong on $\Delta \cup (-1).\Delta$. Hence, without loss of generality (since the other cases are similar), we may assume by contradiction that on the central fiber $(z, w)$ belongs to the intersection $\Delta \cap X$, with $z$ and $w$ not $2$-torsion on $E_3$, and on them neither $\theta_0$ nor $\theta_1$ vanish. (cf. Remark \ref{OSScomponentZ}). In this case, we can also assume that, on $\tau_0$, \begin{equation*} y = (y_1, y_2) = (x_1, -x_2) = {\iota}_2(x) \end{equation*} and both $P$ and $Q$ belong to $W_2$. Since all the sections of the canonical bundle of $\mathcal{S}_{\tau}$ are invariant with the only exception of $\DER{z_2}{\theta}$, we have that all the family of critical pairs $(P,Q)_{\tau}$ also belongs to the family of divisors $W_2$ when $\tau$ varies in a sufficiently small neighborhood of $\tau_0$, with \begin{equation}\label{KEYCONDITIONsign} x \neq \pmy \end{equation} for every $\tau$ in the base neighborhood. \newline However, $x$ and $y$ are supposed here not to remain costant with respect to $\tau$, and in particular they move outside of the preimage in $B$ of the union of the base loci $\mathcal{B}(\mathcal{M}^+) \cup \mathcal{B}(\mathcal{M}^-)$ in $S$. By the expression of the canonical map in (\ref{CANEXP2}), it follows that: \begin{equation} \label{DERF}
\phi_{|\mathcal{M}^+| \times |\mathcal{M}^-|}(x) = \phi_{|\mathcal{M}^+| \times |\mathcal{M}^-|}(y) \end{equation}
where $\phi_{|\mathcal{M}^+| \times |\mathcal{M}^-|}$ denotes the rational map defined on $S$ in $\mathbb{P}^1 \times \mathbb{P}^1$ from the pencils $|\mathcal{M}^+|$ and $|\mathcal{M}^-|$ on each factor.
On the other side, the family $(P,Q)_{\tau}$ moves inside the divisor $W_2$. Thus $x$ and $y$ must be contained, for every $\tau$, in the divisor defined by the equation:
\begin{equation*} \nu \DER{z_2}{\xi} \cdot \DER{z_2}{\nu}\xi = \SKMULT{z_2}{}(\nu \wedge \xi) = 0 \end{equation*} However, by Proposition \ref{TWOFOURTH}, for the general choice of $\nu$ and $\xi$, a couple $(x,y)$ of points in the zero locus of $\SKMULT{z_2}{}(\nu \wedge \xi)$ which also fulfills the condition in (\ref{DERF}) must be of the form $(x, x)$ or $(x, -x)$. But this contradicts Equation (\ref{KEYCONDITIONsign}) above, and the claim is proved.
\end{proof}
In conclusion, we can prove the main result of this paper: \begin{thm1} \label{teoremafinale}
Let be $(A, \mathcal{L})$ a general $(1,2,2)$-polarized abelian threefold and let be $\mathcal{S}$ a general surface in the linear system $|\mathcal{L}|$. Then the canonical map of $\mathcal{S}$ is a holomorphic embedding. \end{thm1} \begin{proof}
\noindent By Proposition \ref{injectivedifferential}, the canonical map is a local embedding in the general case. Hence, it sufficies to prove that the canonical map is injective in the general case. \newline
\noindent By way of contradiction, let us assume that the latter claim is false. \newline
\noindent Our assumption implies that, for every $(1,2,2)$-polarized abelian threefold $A$ and every smooth surface $\mathcal{S}$ within its polarization, the canonical map of $\mathcal{S}$ is not injective. Following Notation \ref{productSurfEll}, our assumption will also hold true for all abelian threefolds in a neighborhood $\mathcal{U}$ in the moduli space around a certain fixed threefold which is isogenous to a product of elliptic curves. Let us consider a $(1,2,2)$-polarized abelian threefold $A_{\tau_0}$, where $\tau_0 = (\tau_{jj})_{j=1,2,3}$ is a diagonal matrix in ${\mathcal{H}}_1^3$. We have then a isogeny: \begin{equation*} p \colon T \longrightarrow A_{\tau_0}:= \bigslant{T_{\tau_0}}{\left<e_1+e_2+e_3\right>} \ \ \text{.} \end{equation*} where $T: = E_1 \times E_2 \times E_3$ is the product of elliptic curves, each defined as a quotient $E_j = \bigslant{\mathbb{C}}{\tau_{jj}\mathbb{Z} \oplus 2\mathbb{Z}}$ carrying a natural $(2)$-polarization.
According to Proposition \ref{generalbehavior1}, on a general surface $\mathcal{S}_{\tau_{0}} = div(\theta)$ in the $(1,2,2)$-polarization class of $A_{\tau_0}$, there exists a critical pair $(P,Q)$ on $\mathcal{S}_{\tau_{0}}$ which, by our assumption, admits the existence of a family of critical pair $(\mathcal{S}_{\mathcal{U}} \longrightarrow \mathcal{U}, \mathcal{P})$. According to Definition \ref{generaldeformationdefinition}: \begin{itemize} \item[i)] $\mathcal{U}$ is an open set of ${\mathcal{H}_{3\Delta, \tau_0}}$ of matrices whose diagonal is fixed and equal to the entries of $\tau_0$ \item[ii)] $\mathcal{S}_{\mathcal{U}} \longrightarrow \mathcal{U}$ is a family of surfaces defined by $\mathcal{S}_{\tau} = div(\theta)$ for every $\tau$. \item[iii)] $\mathcal{P}$ is a closed irreducible subscheme of \begin{equation*} \mathcal{K}_{\mathcal{U}} := (\phi_{\mathcal{S}_{\mathcal{U}}} \times_{\mathcal{U}} \phi_{\mathcal{S}_{\mathcal{U}}})^{-1}(\Delta_{\mathcal{U}}) \subseteq \mathcal{S}_{\mathcal{U}} \times_{\mathcal{U}} \mathcal{S}_{\mathcal{U}}
\end{equation*} which is dominant over $\mathcal{U}$ and such that its restriction on the central fiber $\tau_0$ coincides with $(P,Q)$. \newline \end{itemize}
\noindent The geometric points of the intersection of $\mathcal{P}$ with the diagonal locus in $\mathcal{S}_{\mathcal{U}} \times_{\mathcal{U}} \mathcal{S}_{\mathcal{U}}$ represent critical pairs of infinitely near couples of points on a certain surface of the family. If $P = Q$, without loss of generality we can assume, by Proposition \ref{generalbehavior1}, that the third component of $P$ is represented by a $2$-torsion point of $E_3[2]$, and $P$ belongs to the divisor $W_{3} = div \left(\DER{z_3}{\theta}\right)$. However, by Proposition \ref{generalbehavior2}, the restriction of the family of critical pairs $\mathcal{P}$ to the close locus $\mathcal{U}_{13}$ defines a subfamily of critical pairs whose general element $(P_{\tau}, Q_{\tau})$ belongs to $W_{2}$ with $Q_{\tau} = \iota_2(P_{\tau})$, or to the intersection $W_{1} \cap W_{3}$ with $Q_{\tau} = \iota_1\iota_3(P_{\tau})$ for every $\tau$ in $\mathcal{U}_{13}$ (recall that $\iota_{h}: \mathbb{C}^3 \longrightarrow \mathbb{C}^3$ denotes the sign-changing involution of to the $h$-th coordinate, and by definition in Equation \ref{LOCIEQ} and Diagram \ref{restrictionPdiagram}, $\mathcal{U}_{ij} = \mathcal{U} \cap {\mathcal{H}_{3\Delta, \tau_0}}^{(ij)}$).
In both cases we can conclude that $P_{\tau_0}$ has $2$ coordinates which are $2$-torsion on the respective elliptic curve factors of $T$, and we would reach a contradiction with the generality conditions in Remark \ref{BEQUADROG}. \newline
From now on, we can assume (up to shrinking $\mathcal{U}$) that $\mathcal{P}$ does not intersect the diagonal locus in $\mathcal{S}_{\mathcal{U}} \times_{\mathcal{U}} \mathcal{S}_{\mathcal{U}}$. For every couple of indices $(ij)$ we can fix an irreducible component $\mathcal{P}_{ij}$ of the restriction $\mathcal{P}$ to the closed locus $\mathcal{U}_{ij}$. By our assumption on $\mathcal{P}$, the component $\mathcal{P}_{ij}$ also has empty intersection with the diagonal subscheme of $\mathcal{S}_{\mathcal{U}_{ij}} \times_{\mathcal{U}_{ij}} \mathcal{S}_{\mathcal{U}_{ij}}$.
\noindent By applying Proposition \ref{generalbehavior2}, we have that every fiber $\mathcal{P}_{\tau, ij}$ of the family $\mathcal{P}_{ij}$ must be contained in
\begin{align*} \mathcal{X}_{\tau,ij} &:= \SET{(P, \iota_{i}\iota_{j}(P)) \in \mathcal{S}_{\tau} \times \mathcal{S}_{\tau}}{P \in W_i \cap W_j} \\ \mathcal{W}_{\tau,k} &:= \SET{(P, \iota_{k}(P)) \in \mathcal{S}_{\tau} \times \mathcal{S}_{\tau}}{P \in W_k} \ \ \text{.}
\end{align*} By definition, we have that $\mathcal{U}_{12} \cap \mathcal{U}_{13} \cap \mathcal{U}_{23} = \{\tau_0\}$, hence on the central fiber we must have that $(P,Q)$ belongs to the intersections of of the different $\mathcal{X}_{\tau_0, ij}$ or to $\mathcal{Y}_{\tau_0, k}$, in accordance with the behavior of $\mathcal{P}_{ij}$.
\noindent Because all surfaces of the family $\mathcal{S}_{\mathcal{U}}$ are supposed to be smooth, then $(P,Q)$ belongs to an intersection of the form $\mathcal{X}_{\tau_0, ij} \cap \mathcal{W}_{\tau_0, i} \cap \mathcal{W}_{\tau_0, j}$ for a couple of indices $i,j$, with $i \neq j$.
On the other hand, every point of $\mathcal{X}_{\tau_0, ij} \cap \mathcal{W}_{\tau_0, i} \cap \mathcal{W}_{\tau_0, j}$ is of the form $(P,Q)$ such that $Q = \iota_{i}\iota_{j}(P)$, $Q = \iota_{i}(P)$ and $Q = \iota_{j}(P)$. But this implies that $P=Q$, and we reach a contradiction with the assumption that $\mathcal{P}$ does not intersect the diagonal sublocus.
\noindent The proof of the theorem is complete.
\end{proof}
\section{On the canonical map in the general case} \label{GENERALFINALSECTION}
Following the notations of the previous section (see Notation \ref{productSurfEll}), to every symmetric matrix $\tau = (\tau_{ij})$ in the Siegel half space ${\mathcal{H}}_3$, there is a corresponding $(2,2,2)$-polarized abelian threefold $T = \bigslant{\mathbb{C}^3}{\tau \mathbb{Z}^3 \oplus 2\mathbb{Z}^3}$ with a projection onto a $(1,2,2)$-polarized abelian threefold $A$. This latter abelian threefold is isogenous to a product of three elliptic curves if $\tau$ belongs to the diagonal subspace ${\mathcal{H}}_1^3 = {\mathcal{H}}_1 \times {\mathcal{H}}_1 \times {\mathcal{H}}_1$ of ${\mathcal{H}}_3$. In such a case, Proposition \ref{generalbehavior1} tells us which pairs of points are critical for the canonical map of a general surface $\mathcal{S} = div(\theta)$ in the $(1,2,2)$-polarization of $A$, according to the following general definition: \begin{def1}\label{CRITICALPAIRS} Let $\mathcal{D}$ be a smooth ample divisor on an abelian variety $A$. Denoted by $\CAN{\mathcal{D}}$ the canonical map of $\mathcal{D}$, a \textit{\textbf{critical pair}} for $\mathcal{D}$ is a couple of points $(P, Q)$ such that: \begin{itemize}
\item $\CAN{\mathcal{D}}(P) = \CAN{\mathcal{D}}(Q)$ if $P \neq Q$.
\item The differential of $\CAN{\mathcal{D}}$ at $P$ has not maximal rank, if $P = Q$. \end{itemize} \end{def1} Recall that, denoted by $(z_1, z_2, z_3)$ a coordinate system for an abelian threefold $A$ which is isogenous to the product of three elliptic curves, we proved in Proposition \ref{generalbehavior1} that the canonical map of $\mathcal{S} = div(\theta) \subseteq A$ fails to be an embedding precisely on those pairs of points which lie on the canonical divisors \begin{equation*} W_k := div\left(\frac{\partial \theta}{\partial z_k}\right) = div\left(dz_i \wedge dz_j \right) \ \ \text{,} \end{equation*} where $(i,j,k)$ is a permutation of the indices $(1,2,3)$. This happens because the divisors $W_k$ are invariant for the involution ${\iota}_k \colon z_k \mapsto -z_k$.
\begin{rmk} (On the generality condition in the statement of Proposition \ref{generalbehavior1}) \label{BEQUADROG} Let us consider a diagonal matrix $\tau_0 \in {\mathcal{H}}_1^3$ in the Siegel upper half space ${\mathcal{H}}_3$. As in Notation \ref{productSurfEll}, for ever couple of indices $(i,j)$ there is a $(1,2)$-polarized abelian surface $S_{ij}$ with a projection: \begin{equation*} {\pi_{\B}}_{ij} \colon B_{ij}:= E_i \times E_j \longrightarrow S_{ij} \end{equation*} For the $(1,2,2)$-abelian threefold $(A, \mathcal{L})$, which is isogenous to via a degree $2$ projection $p \colon T \longrightarrow A$, we have that the vector space of sections of the polarization decomposes, for every couple of indices $(i,j)$, as: \begin{equation} \label{sectionsDECOMP2} H^0(A, \mathcal{L}) \cong H^0(S_{ij}, \PMpij{ij})\cdot \theta^{E_k}_0 \oplus H^0(S_{ij}, \PMmij{ij})\cdot \theta^{E_k}_1 \end{equation}
In the proof of Proposition \ref{generalbehavior1}, a smooth surface $\mathcal{S} := div(\theta)$ in the polarization class $|\mathcal{L}|$ of such an abelian threefold $A$ were required to fulfill some conditions in order to be considered as \textbf{sufficiently general}. There conditions can be resumed as follows: \begin{itemize}
\item For every permutation $(i,j,k)$ of the indices $(1,2,3)$, the divisors $\Cu_{ij} := div(\nu_{ij})$ and $\Du_{ij} := div(\xi_{ij})$ of $S_{ij}$, which are defined according to the decomposition in (\ref{sectionsDECOMP2}) such that
\begin{equation*}
\theta = \nu_{ij}\cdot \theta^{E_k}_0 + \xi_{ij}\cdot \theta^{E_k}_1
\end{equation*} are smooth irreducible genus $3$ curves. \item For every point $P$ on $\mathcal{S}$ it holds that: \begin{itemize} \item If the sections $\nu_{ij}$ and $\xi_{ij}$ both vanish on $P$, then neither the $i$-th coordinate nor the $j$-th coordinate of $P$ are represented by $2$-torsion on $E_i$, resp. $E_j$. \item If the sections $\nu_{ij}$ and $\xi_{ij}$ both vanish on $P$, then $\nu_{ik}$ and $\xi_{ik}$ do not simultaneously vanish at $P$. \end{itemize} \end{itemize} If the latter generality conditions are fulfilled, then the proof of Proposition \ref{generalbehavior1} ensures that the only critical pairs on $\mathcal{S}$ are as claimed in the statement of same proposition.
\end{rmk}
In this section, we aim to prove that, when $\tau$ is sufficiently general, there are no critical pairs on a general surface $\mathcal{S}$ within the $(1,2,2)$ polarization of $A$. That means that the canonical map yields a holomorphic embedding in $\mathbb{P}^5$ in the general case, in agreement with Theorem \ref{teoremafinale}: \begin{thm1*}
Let be $(A, \mathcal{L})$ a general $(1,2,2)$-polarized abelian threefold and let be $\mathcal{S}$ a general surface in the linear system $|\mathcal{L}|$. Then the canonical map of $\mathcal{S}$ is a holomorphic embedding. \end{thm1*}
\begin{sketch} For the sake of clarity and exposition, we introduce at this place the basic ideas behind the proof of Theorem \ref{teoremafinale}, and we give a complete proof of it at the end of this section. \newline
\noindent We fix a certain general diagonal matrix $\tau_0 = diag(\tau_{ii})_i$ in the subspace ${\mathcal{H}}_1^3$ of ${\mathcal{H}}_3$ and we denote by \begin{equation}\label{HDCENTRDEF}
{\mathcal{H}_{3\Delta, \tau_0}} := \SET{\tau \in {\mathcal{H}}_3}{\text{The diagonal entries of $\tau$ coincide with those of $\tau_0$}} \ \ \text{.} \end{equation} To every open neighborhood $\mathcal{U}$ of $\tau_0$ in ${\mathcal{H}_{3\Delta, \tau_0}}$ corresponds a family of $(1,2,2)$-polarized abelian varieties $(A_{\mathcal{U}}, \mathcal{L}_{\mathcal{U}})$. The abelian threefold $(A_{\tau_0}, \mathcal{L}_{\tau_0})$ of the family is isogenous to the product of three elliptic curves as in Remark \ref{BEQUADROG}. Every non-zero holomorphic section $\theta$ of $H^0(A_{\tau_0}, \mathcal{L}_{\tau_0})$ gives rise to a family of surfaces $\mathcal{S}_{\mathcal{U}}$, each contained in the polarization class of the respective fiber of the family of abelian threefolds $A_{\mathcal{U}}$. We have, for every element $\tau$ of the basis $\mathcal{U}$, a natural identification: \begin{equation} \label{IDENTIFICATION} H^0(A_{\tau_0}, \mathcal{L}_{\tau_0}) \cong H^0(A_{\tau}, \mathcal{L}_{\tau}) \cong \mathbb{P}^3 \ \ \text{,} \end{equation} defined by the natural restriction of the holomorphic sections of $H^0(A_{\mathcal{U}}, \mathcal{L}_{\mathcal{U}})$ to the fibers of the family: \begin{equation}
H^0(A_{\mathcal{U}}, \mathcal{L}_{\mathcal{U}})|_{\tau} \cong H^0(A_{\tau}, \mathcal{L}_{\tau}) \ \ \text{.} \end{equation}
\noindent If we assume our claim to be false, then we expect to find, for every $\tau$ in a sufficiently small neighborhood $\mathcal{U}$ of $\tau_0$, a critical pair $(P_{\tau}, Q_{\tau})$ for the canonical map of $\mathcal{S}_{\tau}$.
This means that we have a whole family $\mathcal{P}$ on $\mathcal{U}$ of critical pairs $(P_{\mathcal{U}}, Q_{\mathcal{U}})$, which gives rise, by restriction to the closed loci \begin{equation} \label{LOCIEQ} {\mathcal{H}_{3\Delta, \tau_0}}^{(ij)} := \{\tau_{ik} = \tau_{jk} = 0\} \cong {\mathcal{H}}_2 \times {\mathcal{H}}_1 \ \ \text{,} \end{equation} to three different families $\mathcal{P}_{ij}$ of critical pairs, each defined respectively on $\mathcal{U}_{ij} := \mathcal{U} \cap {\mathcal{H}_{3\Delta, \tau_0}}^{(ij)}$: \begin{equation} \label{restrictionPdiagram} \begin{diagram} & &\mathcal{P}_{ij} &\rInto{}& \mathcal{P}\\ & &\dTo_{} & &\dTo\\ & &\mathcal{U}_{ij} := \mathcal{U} \cap {\mathcal{H}_{3\Delta, \tau_0}}^{(ij)} &\rInto &\mathcal{U}\\ \end{diagram} \end{equation}
It is then natural to ask whether we can classify all possible critical pairs on (general) surfaces in abelian threefolds which lie in the loci ${\mathcal{H}_{3\Delta, \tau_0}}^{(ij)}$, similarly to what we did in Proposition \ref{generalbehavior1}. Note that these loci parametrize $(1,2,2)$-abelian variety which are isogenous to products of a $(2,2)$-polarized abelian surface $B$ and a $(2)$-polarized elliptic curve $E$, according to Definition \ref{LOCIEQ} and Notation \ref{productSurfEll}.
The following proposition, which we state for simplicity for the couples of indices $(i,j) = (1,2)$, answers to this question. \end{sketch}
\begin{prop1}\label{generalbehavior2}
Let $\tau_0 = diag(\tau_{11}, \tau_{22}, \tau_{33})$ be a diagonal matrix in the Siegel upper half-space $\mathcal{H}_{3}$, let $(A_{\tau_0}, \mathcal{L}_{\tau_0})$ be the associated $(1,2,2)$-polarized abelian variety with a general divisor $S_{\tau_0} = div(\theta)$ in the polarization $|\mathcal{L}_{\tau_0}|$. Then, for a sufficiently small neighborhood $\mathcal{U}$ of $\tau_0$ in the closed locus ${\mathcal{H}_{3\Delta, \tau_0}}^{(12)}$, for every surface $\mathcal{S}_{\tau} = div(\theta)$ of the restricted family $\mathcal{U}_{12} := \mathcal{U} \cap {\mathcal{H}_{3\Delta, \tau_0}}^{(12)}$ (see Diagram \ref{restrictionPdiagram}), it holds that
\begin{itemize}
\item The involution $\iv{3} \colon (z_1, z_2, z_3) \mapsto (z_1, z_2, -z_3)$ on $A_{\tau}$ leaves $\mathcal{S}_{\tau}$ invariant and induces an involution on the canonical divisor $W_3 := div \left( \frac{\partial \theta}{\partial z_3} \right)$ on $\mathcal{S}_{\tau}$. (Recall that, in general, $W_k := div \left( \frac{\partial \theta}{\partial z_k} \right)$.)
\item The canonical map of $\mathcal{S}$ is one-to-one, except:
\begin{itemize}
\item on the divisor $W_3$, on which the canonical map has degree $2$ and factors respectively through the involution $\iv{3}$.
\item on the finite set $W_1 \cap W_2$, on which the canonical map of $\mathcal{S}$ factors through the involution $\iv{1}\cdot \iv{2}: (z_1, z_2, z_3) \mapsto (-z_1, -z_2, z_3)$. \end{itemize} \item The differential of the canonical map of $\mathcal{S}$ has everywhere maximal rank, except on those points of $W_3$ which are fixed points under the action of the involution $\iv{3}$ on $W_3$. \end{itemize} \end{prop1}
\noindent Proposition \ref{generalbehavior2} shows, in particular, how the behavior of the canonical map changes when, in the moduli space, we move away from the locus of abelian threefolds, which are isogenous to a product of elliptic curves, and we move along closed loci parametrizing abelian threefolds which are isogenous to a product of a simple $(2,2)$ abelian surface and an elliptic curve.
From Proposition \ref{generalbehavior2} also follows also that every restriction $\mathcal{P}_{ij}$ as in Diagram \ref{restrictionPdiagram} of a family $\mathcal{P}$ of critical pairs only contains pairs of points which are conjugate under the same involution. However, on the central fiber $\tau_0$, (that is, on the intersection of these loci) the three different families of critical pairs $\mathcal{P}_{ij}$ specialize to a unique pair of points $(P,Q)$ which must be conjugated under many different involutions, which would lead to a contradiction.
After this rough explaination of the basic approach to our problem, we aim to formalize the steps toward the proof of Theorem \ref{teoremafinale}.
\begin{def1}\label{generaldeformationdefinition} In the notation of Proposition \ref{generalbehavior2}, let us suppose that $(P,Q)$ is a critical pair on $\mathcal{S}_{\tau_{0}} = div(\theta)$. A \textbf{family of critical pairs around $(P,Q)$} is the datum of a couple $(\mathcal{S}_{\mathcal{U}} \longrightarrow \mathcal{U}, \mathcal{P})$, where: \begin{itemize} \item[i)] $\mathcal{U}$ is an open set of ${\mathcal{H}_{3\Delta, \tau_0}}$ (see definition of ${\mathcal{H}_{3\Delta, \tau_0}}$ in Equation \ref{HDCENTRDEF}). \item[ii)] $\mathcal{S}_{\mathcal{U}} \longrightarrow \mathcal{U}$ is a family of surfaces defined as $\mathcal{S}_{\tau} = div(\theta)$ for every $\tau$, by applying the natural identification \begin{equation} H^0(A_{\tau_0}, \mathcal{L}_{\tau_0}) \cong H^0(A_{\mathcal{U}}, \mathcal{L}_{\mathcal{U}}) \cong H^0(A_{\tau}, \mathcal{L}_{\tau}) \ \ \text{.} \end{equation}
\item[iii)] With $\Delta_{\mathcal{U}}$ the diagonal subscheme in $\mathbb{P}_{\mathcal{U}}^5 \times \mathbb{P}_{\mathcal{U}}^5$ and $\phi_{\mathcal{S}_{\mathcal{U}}}$ the map, which on each surface $\mathcal{S}_{\tau}$ of the family coincides with the canonical map $\phi_{\mathcal{S}_{\tau}}$, $\mathcal{P}$ is a closed irreducible subscheme of \begin{equation*} \mathcal{K}_{\mathcal{U}} := (\phi_{\mathcal{S}_{\mathcal{U}}} \times_{\mathcal{U}} \phi_{\mathcal{S}_{\mathcal{U}}})^{-1}(\Delta_{\mathcal{U}}) \subseteq \mathcal{S}_{\mathcal{U}} \times_{\mathcal{U}} \mathcal{S}_{\mathcal{U}}
\end{equation*} which is dominant over $\mathcal{U}$ and such that its restriction on the central fiber $\tau_0$ is $(P,Q)$. Here $\times_{\mathcal{U}}$ denotes, as usual, the cartesian product on $\mathcal{U}$ according to the pullback diagram: \begin{equation*} \begin{diagram} \mathcal{S}_{\mathcal{U}} \times_{\mathcal{U}} \mathcal{S}_{\mathcal{U}} &\rTo &\mathcal{S}_{\mathcal{U}} \\
\dTo & & \dTo \\ \mathcal{S}_{\mathcal{U}} &\rInto & \mathcal{U} \end{diagram} \end{equation*} \end{itemize} \end{def1}
\begin{rmk} \label{BEQUADROij} In the notation of the previous Definition \ref{generaldeformationdefinition}, if we restrict a family of critical points $(\mathcal{S}_{\mathcal{U}} \longrightarrow \mathcal{U}, \mathcal{P})$ to one of the loci ${\mathcal{H}_{3\Delta, \tau_0}}^{(ij)}$ (recall the definition in \ref{LOCIEQ}), we obtain a subfamily of critical points $(\mathcal{S}_{\mathcal{U}_{ij}} \longrightarrow \mathcal{U}_{ij}, \mathcal{P}_{ij})$ defined on the open set $\mathcal{U}_{ij} := \mathcal{U} \cap {\mathcal{H}_{3\Delta, \tau_0}}^{(ij)}$. Furthermore, we have in this case a family of projections \begin{equation} \label{PProjU} p_{\mathcal{U}} \colon T_{\mathcal{U}} = B_{\mathcal{U}, ij} \times E_k \longrightarrow A_{\mathcal{U}} \end{equation} and a family of $(2,2)$-polarized abelian surfaces $B_{\mathcal{U}_{ij}} \longrightarrow \mathcal{U}_{ij}$ with a projection onto a family of $(1,2)$-polarized abelian surfaces $S_{\mathcal{U}_{ij}} \longrightarrow \mathcal{U}_{ij}$, and there is a decomposition as in (\ref{sectionsDECOMP2}): \begin{equation} \label{sectionsDECOMPgen}
H^0(A_{\mathcal{U}}, \mathcal{L}_{\mathcal{U}}) \cong H^0(S_{\mathcal{U}_{ij}}, \PMpij{ij})\cdot \theta^{E_k}_0 \oplus H^0(S_{\mathcal{U}_{ij}}, \PMmij{ij})\cdot \theta^{E_k}_1 \end{equation} \end{rmk}
\begin{not1} \label{NOTNUXI}
If we write a non-zero section $\theta$ of $H^0(A_{\mathcal{U}}, \mathcal{L}_{\mathcal{U}})$ according to (\ref{sectionsDECOMPgen}), we have \begin{equation}\label{thetaDECOMPOSITIONij}
\theta = \nu_{\mathcal{U}, ij}\cdot \theta^{E_k}_0 + \xi_{\mathcal{U}, ij}\cdot \theta^{E_k}_1 \end{equation} for some $\nu_{\mathcal{U}, ij}$ and $\xi_{\mathcal{U}, ij}$ in $H^0(S_{\mathcal{U}_{ij}}, \PMpij{ij})$ and $H^0(S_{\mathcal{U}_{ij}}, \PMmij{ij})$ respectively. We denote henceforth, similarly to what we did in Remark \ref{BEQUADROG}, the following families of divisors in $S_{\mathcal{U}, ij}$ as follows \begin{align*}
\Cu_{\mathcal{U}, ij} &:= div(\nu_{\mathcal{U}, ij}) \\
\Du_{\mathcal{U}, ij} &:= div(\xi_{\mathcal{U}, ij}) \end{align*}
\end{not1}
\begin{oss1} \label{firstOssBXE} \noindent Let us consider a family of critical pairs $(\mathcal{S}_{\mathcal{U}} \longrightarrow \mathcal{U}, \mathcal{P})$ and its restriction to: \begin{equation} \label{LOCIEQ2} {\mathcal{H}_{3\Delta, \tau_0}}^{(ij)} := \{\tau_{ik} = \tau_{jk} = 0\} \cong {\mathcal{H}}_2 \times {\mathcal{H}}_1 \ \ \text{,} \end{equation} We have, following our previous discussion in Remark \ref{BEQUADROij}, a family $(\mathcal{S}_{\mathcal{U}_{ij}} \longrightarrow \mathcal{U}_{ij}, \mathcal{P}_{ij})$ on the open set $\mathcal{U}_{ij} := \mathcal{U} \cap {\mathcal{H}_{3\Delta, \tau_0}}^{(ij)}$, with $\mathcal{S}_{\mathcal{U}} = div(\theta)$ which satisfy the generality conditions stated in Remark \ref{BEQUADROij}.
Suppose that, for some $\tau$ in $\mathcal{U}_{ij}$, the point $P_{\tau}$ belongs to the canonical divisor \begin{equation*} W_k = div\left(\frac{\partial \theta}{\partial z_k}\right)_{\tau} \subseteq \mathcal{S}_{\tau} \ \ \text{.} \end{equation*} Then the set ${\pi_{\B}}_{\tau}\left(p_{\tau}^{-1}(W_k)\right)$, defined on each $\tau$ according to the following diagram \begin{equation} \label{restrictionPdiagram} \begin{diagram} & &B_{\mathcal{U}, ij} \times E_k &\rTo{p_{\mathcal{U}}}& A_{\mathcal{U}}\\ & &\dTo_{{\pi_{\B}}_{\mathcal{U}_{ij}} \times id_{E_k}} & & \\ & &S_{\mathcal{U}, ij} \times E_k & &\\ \end{diagram} \end{equation} is reducible, and we have, for every $\tau$ in $\mathcal{U}_{ij}$ (see again Remark \ref{BEQUADROij} for the notations): \begin{equation*}
({\pi_{\B}}_{\tau} \times id_{E_k})\left(p_{\tau}^{-1}(W_k)\right) = (\Cu_{\tau, ij} \cap \Du_{\tau, ij}) \times E_k \cup S_{\tau} \times E_k[2] \ \ \text{.} \end{equation*}
Indeed, if we write the section $\theta$ as in Equation \ref{thetaDECOMPOSITIONij}, then the coordinates of the points in the divisor $W_k$ must fulfill the conditions:
\begin{equation}\label{casesDEFeta} \begin{cases} \nu_{\tau, ij}\theta^{E_k}_0 &+ \ \ \xi_{\tau, ij}\theta^{E_k}_1 = 0\\ \nu_{\tau, ij}\frac{\partial}{\partial z_k} \theta^{E_k}_0 &+ \ \ \xi_{\tau, ij}\frac{\partial}{\partial z_k}\theta^{E_k}_1 = 0 \ \ \text{.} \end{cases} \end{equation} The conditions in (\ref{casesDEFeta}) can be looked at as a linear system in $\nu_{\tau, ij}$ and $\xi_{\tau, ij}$. This implies that both $\nu_{\tau}$ and $\xi_{\tau}$ vanish, or the determinant \begin{equation} \label{etadefinition} \eta_k := \begin{vmatrix}\theta^{E_k}_0 & \theta^{E_k}_1 \\ \frac{\partial}{\partial z_k} \theta^{E_k}_0 & \frac{\partial}{\partial z_k} \theta^{E_k}_1 \end{vmatrix} \in H^0(E_k, \mathcal{O}_{E_k}(4O_{E_k})) \ \ \text{.} \end{equation}
The conclusion follows now by Equation \ref{etadefinition_Y} in Example \ref{basicexampleEll} (since $div(\eta_k) = E_k[2]$ ). \end{oss1}
In the next lemma, we show that the restriction $\mathcal{P}_{ij}$ (as in Remark \ref{BEQUADROij}) of a family of critical pairs $\mathcal{P}$ around a couple of points $(P,Q)$ with $P \neq Q$ must be contained in a closed locus of $\mathcal{S}_{\mathcal{U}_{ij}} \times_{\mathcal{U}_{ij}} \mathcal{S}_{\mathcal{U}_{ij}}$ which, in \begin{equation}\label{eqproduct} \left(S_{\mathcal{U}, ij} \times E_k\right)^2 := \left(S_{\mathcal{U}, ij} \times_{\mathcal{U}_{ij}} S_{\mathcal{U}, ij} \right) \times E_k^2 \ \ \text{,} \end{equation} is defined by a holomorphic section of the $(4,4)$-polarization $\mathcal{O}_{E_k}(4O_{E_k}) \boxtimes \mathcal{O}_{E_k}(4O_{E_k})$ on $E_k \times E_k$.
\begin{lem1}\label{secondOssBXE} \noindent Let us consider a family of critical pairs $(\mathcal{S}_{\mathcal{U}} \longrightarrow \mathcal{U}, \mathcal{P})$ and its restriction $(\mathcal{S}_{\mathcal{U}_{ij}} \longrightarrow \mathcal{U}_{ij}, \mathcal{P}_{ij})$ to the open set $\mathcal{U}_{ij} := \mathcal{U} \cap {\mathcal{H}_{3\Delta, \tau_0}}^{(ij)}$ with $\mathcal{S}_{\mathcal{U}} = div(\theta)$ which satisfy the generality conditions of Remark \ref{BEQUADROij}. Let us assume that $P_{\tau} \neq Q_{\tau}$ for every $\tau$ in $\mathcal{U}_{ij}$ (i.e., $\mathcal{P}$ is a family of critical pairs of distinct points). \newline
Then, in the notations of the following diagram (cf. also Diagram \ref{restrictionPdiagram})
\begin{equation} \label{restrictionPdiagram2} \begin{diagram} & &\left(B_{\mathcal{U}, ij} \times E_k\right)^2 &\rTo{\ \ \ \ p_{\mathcal{U}} \times_{\mathcal{U}} p_{\mathcal{U}} \ \ \ \ }& A_{\mathcal{U}}^2 = A_{\mathcal{U}} \times_{\mathcal{U}} A_{\mathcal{U}}\\ & &\dTo_{({\pi_{\B}}_{\mathcal{U}_{ij}} \times id_{E_k})^2} & & \\ & &\left(S_{\mathcal{U}, ij} \times E_k \right)^2 & &\\ \end{diagram} \end{equation}
there exists a non-zero holomorphic section $v_k$ of $H^0(E_k \times E_k, \mathcal{O}_{E_k}(4O_{E_k}) \boxtimes \mathcal{O}_{E_k}(4O_{E_k}))$ such that: \begin{equation*}
({\pi_{\B}}_{\mathcal{U}_{ij}} \times id_{E_k})^2\left(\left(p_{\mathcal{U}} \times_{\mathcal{U}} p_{\mathcal{U}}\right)^{-1}(\mathcal{P}_{ij})\right) \subseteq \left[(\Cu_{\mathcal{U}, ij} \cap \Du_{\mathcal{U}, ij}) \times E_k \right]^2 \cup \left[S_{\mathcal{U}, ij}^2 \times div(v_k) \right] \end{equation*} \end{lem1} \begin{proof} It sufficies to prove the claim for a generic fiber of the family. For this purpose, we can assume $(i,j,k) = (1,2,3)$ and we pick an element $\tau \in \mathcal{U}_{12}$. The critical pair $(P,Q)$ for the canonical map of $\mathcal{S}_{\tau} = div(\theta)$ can be represented by a couple of points $(\widetilde{P}, \widetilde{Q})$ in $\left(S_{\tau, 12} \times E_3 \right)^2$ according to Diagram \ref{restrictionPdiagram2}, with: \begin{equation*} \begin{split} \widetilde{P}&=(x, z) \\ \widetilde{Q}&=(y, w) \ \ \text{.} \end{split} \end{equation*} The section $\theta$ can be written, as in Equation \ref{thetaDECOMPOSITIONij}, as: \begin{equation} \label{eqStau}
\theta = \nu_{\tau, 12}\cdot \theta^{E_3}_0 + \xi_{\tau, 12}\cdot \theta^{E_3}_1 \end{equation} To simplify the notations, the section $\nu_{\tau, 12}$ (resp. $\xi_{\tau, 12}$) in Equation \ref{eqStau} will be denoted by $\nu$ (resp. $\xi$), and the section $\theta^{E_3}_0$ (resp. $\theta^{E_3}_1$) by $\theta_0$ (resp. $\theta_1$). The condition that the canonical image of $P$ and $Q$ coincide implies that there exists $\lambda \in \mathbb{C}^*$ such that: \begin{equation*} \begin{cases} \nu(x)\theta_0(z) &= \lambda \cdot \nu(y)\theta_0(w) \\ \xi(x)\theta_1(z) &= \lambda \cdot \xi(y)\theta_1(w) \\ \nu(x)\DTH{0}(z) + \xi(x)\DTH{1}(w) &= \lambda \cdot \Bigl( \nu(y)\DTH{0}(w) + \xi(y)\DTH{1}(w) \Bigr) \ \ \text{.} \end{cases} \end{equation*} (Here $\DTH{0}$ and $\DTH{1}$ are the derivatives of $\theta_0$ and $\theta_1$). This leads to
\begin{align*} & \nu(x)\left(\frac{\DTH{0}(z)\theta_0(w) - \theta_0(z)\DTH{0}(w)}{\theta_0(w)} \right) \ \ + \\ & \xi(x)\left(\frac{\DTH{1}(z)\theta_1(w) - \theta_1(z)\DTH{1}(w)}{\theta_1(w)} \right) \ \ = 0 \ \ \text{,} \end{align*} and we can conclude that: \begin{align} \label{definitionofpsi0} \nu(x)\theta_1(w) \rhos{0}{3}(z,w) + \xi(x)\theta_0(w)\rhos{1}{3}(z,w) &= 0 \ \ \text{,} \end{align} where, with $j = 0,1$: \begin{align} \label{RHODEF} \rhos{j}{3}(z,w) := \begin{vmatrix}\theta_j(z) & \theta_j(w) \\ \DTH{j}(z) & \DTH{j}(w) \end{vmatrix} \in H^0(E_3 \times E_3, \mathcal{O}_{E_3}(2O_{E_3}) \boxtimes \mathcal{O}_{E_3}(2O_{E_3})) \end{align} Together with the defining equation of $\mathcal{S}_{\tau}$ in (\ref{eqStau}), we obtain the following linear system in $\nu(x)$ and $\xi(x)$:
\begin{align*} \label{systemfgrho} &\nu(x)\theta_0(z) & &+& &\xi(x)\theta_1(z) & &= 0 \\ &\nu(x)\theta_1(w) \rhos{0}{3}(z,w) & &+& &\xi(x)\theta_0(w)\rhos{1}{3}(z,w) & &= 0 \ \ \text{.} \end{align*}
The determinant of the matrix of the linear system above is precisely \begin{equation}\label{DEFVAB} v_3(z,w) := \begin{vmatrix}\theta_0(z)\theta_{0}(w) & \theta_1(z)\theta_1(w) \\
\rhos{0}{3}(z,w) & \rhos{1}{3}(z,w)\end{vmatrix} \in H^0(E_3 \times E_3, \mathcal{O}_{E_3}(4O_{E_3}) \boxtimes \mathcal{O}_{E_3}(4O_{E_3})) \ \ \text{,} \end{equation} and we conclude that $x$ belongs to $\left(div(\nu) \cap div(\xi)\right) = \Cu_{\tau, 12} \cap \Du_{\tau, 12}$, or \begin{equation*} v_3(z, w) = 0 \end{equation*} This proves the claim of the lemma. \end{proof}
\begin{rmk} \label{OSScomponentZ} In general, if we have an elliptic curve $E$ and $\theta_0, \theta_1$ the canonical basis for $H^0(E, \mathcal{O}_{E}(2 \cdot O_{E})$ it is easy to determine a polynomial equation, in a suitable affine open set, of the divisor $V := div(v)$ in $E \times E$ defined as in Equation \ref{DEFVAB} above as \begin{equation}\label{DEFVAB} v(z,w) := \begin{vmatrix}\theta_0(z)\theta_{0}(w) & \theta_1(z)\theta_1(w) \\
\rhos{0}{}(z,w) & \rhos{1}{}(z,w)\end{vmatrix} \ \ \text{.} \end{equation} (Here $\rhos{0}{}$ and $\rhos{1}{}$ are defined as in Equation \ref{RHODEF}.)
In the notations of Example \ref{basicexampleEll}, for every point $(z,w)$ of $E \times E$ it easily seen that: \begin{equation*} v(z,w) = \theta_0(z)\theta_1(z)\cdot \left(\theta_0(w)\DTH{1}(w) - \DTH{0}(w)\theta_1(w) \right) - \theta_0(w)\theta_1(w)\cdot \left(\theta_0(z)\DTH{1}(z) - \DTH{0}(z)\theta_1(z) \right) \end{equation*} In the affine open set $U$ where $\theta_0(z)$ and $\theta_0(w)$ do not vanish, we can divide by $\theta_0(z)^2\theta_0(w)^2$ and in local affine coordinates $(\x{1}, \y{1}, \x{2}, \y{2})$ with two equations $g_1$, $g_2$ in Legendre normal form as in Equation \ref{LegendreNormalForm} in Example \ref{basicexampleEll}, we get that: \begin{equation*} v(z,w) \sim \x{1}\y{2} - \x{2}\y{1} \end{equation*} Moreover, by using the equations $\y{j}^2 = (\x{j}^2 - 1)(\x{j}^2 - \pa{}^2)$ of $E$ on the components, we can easily recover the equations of the components of $V$ in the open set $U$: \begin{equation*}
div(V)|_U = (\x{1}^2 - \x{2}^2)(\x{1}^2\x{2}^2 - \pa{}^2) \end{equation*} In conclusion, the divisor $V$ splits into the sum of components: \begin{equation*} V = \Delta + (-1).\Delta + X + (-1).X \ \ \text{,} \end{equation*} where $\Delta$ is the diagonal locus in $E \times E$. All components are invariant under the action of the involution $(z,w) \mapsto (z + 1, w+ 1)$, and the irreducible component $X$ meets $\Delta$ in $8$ points whose coordinates are not $2$-torsion on $E$. \end{rmk}
\begin{proof}[of Proposition \ref{generalbehavior2}]
Let us consider $(A_{\tau_0}, \mathcal{L}_{\tau_0})$ the $(1,2,2)$-polarized abelian variety associated to a diagonal matrix $\tau$ in $\mathcal{H}_1^3$, with a divisor $\mathcal{S}_{\tau_0} = div(\theta)$ which is sufficiently general in the polarization $|\mathcal{L}_{\tau_0}|$, in accordance with the conditions of Remark \ref{BEQUADROG}.
Since these conditions are open, they still hold for a sufficiently small neighborhood $\mathcal{U}$ of $\tau_0$ in the closed locus ${\mathcal{H}_{3\Delta, \tau_0}}^{(12)}$. Hence, for a suitable $\mathcal{U}$, there is a family $(A_{\mathcal{U}}, \mathcal{L}_{\mathcal{U}})$ of $(1,2,2)$-polarized abelian threefolds with $\mathcal{S}_{\mathcal{U}} = div(\theta)$, such that, considered the decomposition \begin{equation*}
\theta = \nu_{\mathcal{U}}\cdot \theta_0 + \xi_{\mathcal{U}}\cdot \theta_1 \ \ \text{,} \end{equation*}
the fibers of the induced families $\Cu_{\mathcal{U}} =div \left(\nu_{\mathcal{U}}\right)$ and $\Du_{\mathcal{U}} = div\left(\xi_{\mathcal{U}}\right)$ defined as in Notation \ref{NOTNUXI} (for the sake of readability we omit the indices $1,2$ as in the proof of Lemma \ref{secondOssBXE}), are smooth genus $3$ non-hyperelliptic curves. \newline
By Proposition \ref{generalbehavior1}, the possibile critical pairs on the central fiber $\mathcal{S}_{\tau_0}$ are only of one of the following types: \newline
$\bullet$ \textbf{type (a)}: The pair belongs to the divisor $W_j = div \left(\frac{\partial \theta}{\partial z_j}\right)$ for some $j$. In this case, the points are conjugated under the the sign reversing involution $\iota_j$, which changes the sign on the $j$-th coordinate $z_j \mapsto -z_j$. \newline
$\bullet$ \textbf{type (b)}: The pair belongs to the intersection of two divisors $W_i$ and and $W_j$ for some couple of indices $i,j$. In this case, the points are conjugated under the involution $\iota_i \iota_j$. \newline
Hence, to prove the proposition, it is enough to prove the following claim. \newline
\noindent \textbf{Claim}: Let us consider a family of critical pairs $(\mathcal{S}_{\mathcal{U}} \longrightarrow \mathcal{U}, \mathcal{P}_{12})$ on $\mathcal{U}$ (up to shrinking $\mathcal{U}$). Then the family $\mathcal{P}_{12}$ is one of the following types: \begin{itemize}
\item[1)] On every fiber (up to shrinking $\mathcal{U}$), the critical pair is of type (a) with $j=3$. In particular, the pair always belongs to the divisor $W_3$.
\item[2)] On every fiber (up to shrinking $\mathcal{U}$), the critical pair is of type (b) with $(i,j)=(1,2$. In particular, the pair always belongs to the divisor $W_1 \cap W_2$. \end{itemize}
To this purpose, let us pick a fiber on a point $\tau$ of $\mathcal{U}$ and a critical pair $(P_{\tau},Q_{\tau})$ on $\mathcal{S}_{\tau}$. This critical pair can be represented (see Diagram \ref{restrictionPdiagram2}) as a couple of points $(\widetilde{P}_{\tau}, \widetilde{Q}_{\tau})$ in $\left(S_{\tau, 12} \times E_3 \right)^2$ with \begin{equation*} \begin{split} \widetilde{P}_{\tau}&=(x, z) \\ \widetilde{Q}_{\tau}&=(y, w) \ \ \text{,} \end{split} \end{equation*} where the coordinates $\widetilde{P}_{\tau}$ and $\widetilde{Q}_{\tau}$ are functions of $\tau$. By Lemma \ref{secondOssBXE}, we have that $(x,y)$ belongs to $\Cu_{\tau} \cap \Du_{\tau}$, or $(z, w)$ belongs to $ div(v_3)$.
In the first case, already the condition that $(x,y)$ belongs to $\Cu_{\tau}$ implies that $x = y$, since the curve $\Cu_{\tau}$ is non-hyperelliptic for every $\tau$, up to shrinking $\mathcal{U}$: indeed, recall that (again by Proposition \ref{desccanonicalmap}) the canonical map of $\Cu_{\tau}$ is obtained by considering the derivatives $\frac{\partial \xi}{\partial z_1}$ and $\frac{\partial \xi}{\partial z_2}$ together with the restriction of the sections of $H^0(S, {\mathcal{M}^{-}})$ to $\Du$, and we can conclude that $\phi_{\Du_{\tau}}(x) = \phi_{\Du_{\tau}}(y)$, where $\phi_{\Du_{\tau}}$ denotes the canonical map of $\Du_{\tau}$. \newline
\noindent Let us assume that $(z, w)$ belongs to $V = div(v_3)$.
\noindent The affine equations in Remark \ref{OSScomponentZ} show that $V$ splits into a sum of irreducible components: \begin{equation*} V = \Delta + (-1).\Delta + X + (-1).X \ \ \text{,} \end{equation*}
\noindent For the central fiber $\tau_0$, we described these components in Remark \ref{OSScomponentZ} and we determined their equation on an affine open set. Since the components $(z, w)$ of the points $\widetilde{P}_{\tau}$ and $\widetilde{Q}_{\tau}$ are bound to move on $V$, we have to deal with two cases: \begin{itemize}
\item $(z, w)$ is contained in $\Delta \cup (-1).\Delta$ but not in $X \cup (-1).X'$.
\item $(z, w)$ is contained in $X \cup (-1).X'$. \newline \end{itemize}
\noindent Let us assume that $(z, w)$ is contained in $\Delta \cup (-1).\Delta$ but not in $X \cup (-1).X'$. Then, because this latter condition is open, we can conclude that the same condition holds true on the neighborhood $\mathcal{U}$ up to shrinking it. Hence, on every fiber on $\tau$, we have that for the coordinates $(z, w) = (z, \pmz)$ or $(z, w) = (z, \pmz + 1)$. Let us assume that we are in the case $(z, w) = (z, z)$ (the other cases can be treated similarly). The ratio of the values $[s,t]:=[\theta_0(z), \theta_1(z)]$ of the $(2)$-polarization on $E_3$ define, according to the equation of $\mathcal{S}_{\tau}$, a curve \begin{equation} \label{EQFCICLE} \mathcal{F} \colon s \cdot \nu + t \cdot \xi = 0 \end{equation} in the $(2,2)$-polarization class of $B_{\tau}$,
on which the points $x$ and $y$ belong.
On the other side, the expression of the canonical map of $\mathcal{S}_{\tau}$ is \begin{equation}\label{CANEXP2}
\CAN{\mathcal{S}_{\tau}} = \left[H^0(S_{\tau}, {\mathcal{M}^{+}})\theta_0, H^0(S_{\tau}, {\mathcal{M}^{-}})\theta_1, \frac{\partial \nu}{\partial z_1} \theta_0 + \frac{\partial \xi}{\partial z_1} \theta_1, \frac{\partial \nu}{\partial z_2} \theta_0 + \frac{\partial \xi}{\partial z_2} \theta_1,\nu \DTH{0} + \xi \DTH{1} \right] \ \ \text{.} \end{equation} This expression can be easily compared with the one of the canonical map of $\mathcal{F}$, which is \begin{equation}\label{CANEXP3}
\CAN{\mathcal{F}} = \left[s\cdot H^0(S_{\tau}, {\mathcal{M}^{+}}), t \cdot H^0(S_{\tau}, {\mathcal{M}^{-}}), s\frac{\partial \nu}{\partial z_1} + t\frac{\partial \xi}{\partial z_1} , s\frac{\partial \nu}{\partial z_2} + t\frac{\partial \xi}{\partial z_2}\right] \ \ \text{,} \end{equation} and follows that $\CAN{\mathcal{F}}(x) = \CAN{\mathcal{F}}(y)$ .
\noindent If $\mathcal{F}$ is smooth, then it is non-hyperelliptic (because in a simple $(2,2)$-polarization there are no irreducible hyperelliptic curves). Hence, in this case, we conclude that $x=y$ and $P=Q$. \newline
\noindent If otherwise $\mathcal{F}$ is not smooth, then we can assume that $\mathcal{F}$ is a nodal curve. Indeed, since the $(1,2)$-polarization class on a simple abelian surface is a pencil only containing irreducible curves (cf. Remark \ref{osservationsingularfibers}), we may assume, by generality, that the singular curves in the pencil generated by $\nu$ and $\xi$ in the $(2,2)$-polarization class of $B$ only contains nodal curves, which also are non-hyperelliptic since they are irreducible.
\noindent This implies that $x=y$ if $x$ is not a node on $\mathcal{F}$, otherwise $x =-y$, and the second possibility occurs precisely when $(P,Q)$ is a critical pair which belong to $W_1 \cap W_2$ with $Q = \iota_1 \iota_2 (P)$.
\noindent Let us suppose now that we are in the second case, namely the case that $(z, w)$ is contained in $X \cup (-1).X$ and $w \neq \pmz$ when $\tau$ varies away from the central point $\tau_0$ in a suitable base neighborhood. Our goal is to show by contradiction that this case actually does not occur. \newline
\noindent By Proposition \ref{generalbehavior1}, on the central fiber $\tau_0$ the point $(z, w)$ must also belong on $\Delta \cup (-1).\Delta$. Hence, without loss of generality (since the other cases are similar), we may assume by contradiction that on the central fiber $(z, w)$ belongs to the intersection $\Delta \cap X$, with $z$ and $w$ not $2$-torsion on $E_3$, and on them neither $\theta_0$ nor $\theta_1$ vanish. (cf. Remark \ref{OSScomponentZ}). In this case, we can also assume that, on $\tau_0$, \begin{equation*} y = (y_1, y_2) = (x_1, -x_2) = {\iota}_2(x) \end{equation*} and both $P$ and $Q$ belong to $W_2$. Since all the sections of the canonical bundle of $\mathcal{S}_{\tau}$ are invariant with the only exception of $\DER{z_2}{\theta}$, we have that all the family of critical pairs $(P,Q)_{\tau}$ also belongs to the family of divisors $W_2$ when $\tau$ varies in a sufficiently small neighborhood of $\tau_0$, with \begin{equation}\label{KEYCONDITIONsign} x \neq \pmy \end{equation} for every $\tau$ in the base neighborhood. \newline However, $x$ and $y$ are supposed here not to remain costant with respect to $\tau$, and in particular they move outside of the preimage in $B$ of the union of the base loci $\mathcal{B}(\mathcal{M}^+) \cup \mathcal{B}(\mathcal{M}^-)$ in $S$. By the expression of the canonical map in (\ref{CANEXP2}), it follows that: \begin{equation} \label{DERF}
\phi_{|\mathcal{M}^+| \times |\mathcal{M}^-|}(x) = \phi_{|\mathcal{M}^+| \times |\mathcal{M}^-|}(y) \end{equation}
where $\phi_{|\mathcal{M}^+| \times |\mathcal{M}^-|}$ denotes the rational map defined on $S$ in $\mathbb{P}^1 \times \mathbb{P}^1$ from the pencils $|\mathcal{M}^+|$ and $|\mathcal{M}^-|$ on each factor.
On the other side, the family $(P,Q)_{\tau}$ moves inside the divisor $W_2$. Thus $x$ and $y$ must be contained, for every $\tau$, in the divisor defined by the equation:
\begin{equation*} \nu \DER{z_2}{\xi} \cdot \DER{z_2}{\nu}\xi = \SKMULT{z_2}{}(\nu \wedge \xi) = 0 \end{equation*} However, by Proposition \ref{TWOFOURTH}, for the general choice of $\nu$ and $\xi$, a couple $(x,y)$ of points in the zero locus of $\SKMULT{z_2}{}(\nu \wedge \xi)$ which also fulfills the condition in (\ref{DERF}) must be of the form $(x, x)$ or $(x, -x)$. But this contradicts Equation (\ref{KEYCONDITIONsign}) above, and the claim is proved.
\end{proof}
In conclusion, we can prove the main result of this paper: \begin{thm1} \label{teoremafinale}
Let be $(A, \mathcal{L})$ a general $(1,2,2)$-polarized abelian threefold and let be $\mathcal{S}$ a general surface in the linear system $|\mathcal{L}|$. Then the canonical map of $\mathcal{S}$ is a holomorphic embedding. \end{thm1} \begin{proof}
\noindent By Proposition \ref{injectivedifferential}, the canonical map is a local embedding in the general case. Hence, it sufficies to prove that the canonical map is injective in the general case. \newline
\noindent By way of contradiction, let us assume that the latter claim is false. \newline
\noindent Our assumption implies that, for every $(1,2,2)$-polarized abelian threefold $A$ and every smooth surface $\mathcal{S}$ within its polarization, the canonical map of $\mathcal{S}$ is not injective. Following Notation \ref{productSurfEll}, our assumption will also hold true for all abelian threefolds in a neighborhood $\mathcal{U}$ in the moduli space around a certain fixed threefold which is isogenous to a product of elliptic curves. Let us consider a $(1,2,2)$-polarized abelian threefold $A_{\tau_0}$, where $\tau_0 = (\tau_{jj})_{j=1,2,3}$ is a diagonal matrix in ${\mathcal{H}}_1^3$. We have then a isogeny: \begin{equation*} p \colon T \longrightarrow A_{\tau_0}:= \bigslant{T_{\tau_0}}{\left<e_1+e_2+e_3\right>} \ \ \text{.} \end{equation*} where $T: = E_1 \times E_2 \times E_3$ is the product of elliptic curves, each defined as a quotient $E_j = \bigslant{\mathbb{C}}{\tau_{jj}\mathbb{Z} \oplus 2\mathbb{Z}}$ carrying a natural $(2)$-polarization.
According to Proposition \ref{generalbehavior1}, on a general surface $\mathcal{S}_{\tau_{0}} = div(\theta)$ in the $(1,2,2)$-polarization class of $A_{\tau_0}$, there exists a critical pair $(P,Q)$ on $\mathcal{S}_{\tau_{0}}$ which, by our assumption, admits the existence of a family of critical pair $(\mathcal{S}_{\mathcal{U}} \longrightarrow \mathcal{U}, \mathcal{P})$. According to Definition \ref{generaldeformationdefinition}: \begin{itemize} \item[i)] $\mathcal{U}$ is an open set of ${\mathcal{H}_{3\Delta, \tau_0}}$ of matrices whose diagonal is fixed and equal to the entries of $\tau_0$ \item[ii)] $\mathcal{S}_{\mathcal{U}} \longrightarrow \mathcal{U}$ is a family of surfaces defined by $\mathcal{S}_{\tau} = div(\theta)$ for every $\tau$. \item[iii)] $\mathcal{P}$ is a closed irreducible subscheme of \begin{equation*} \mathcal{K}_{\mathcal{U}} := (\phi_{\mathcal{S}_{\mathcal{U}}} \times_{\mathcal{U}} \phi_{\mathcal{S}_{\mathcal{U}}})^{-1}(\Delta_{\mathcal{U}}) \subseteq \mathcal{S}_{\mathcal{U}} \times_{\mathcal{U}} \mathcal{S}_{\mathcal{U}}
\end{equation*} which is dominant over $\mathcal{U}$ and such that its restriction on the central fiber $\tau_0$ coincides with $(P,Q)$. \newline \end{itemize}
\noindent The geometric points of the intersection of $\mathcal{P}$ with the diagonal locus in $\mathcal{S}_{\mathcal{U}} \times_{\mathcal{U}} \mathcal{S}_{\mathcal{U}}$ represent critical pairs of infinitely near couples of points on a certain surface of the family. If $P = Q$, without loss of generality we can assume, by Proposition \ref{generalbehavior1}, that the third component of $P$ is represented by a $2$-torsion point of $E_3[2]$, and $P$ belongs to the divisor $W_{3} = div \left(\DER{z_3}{\theta}\right)$. However, by Proposition \ref{generalbehavior2}, the restriction of the family of critical pairs $\mathcal{P}$ to the close locus $\mathcal{U}_{13}$ defines a subfamily of critical pairs whose general element $(P_{\tau}, Q_{\tau})$ belongs to $W_{2}$ with $Q_{\tau} = \iota_2(P_{\tau})$, or to the intersection $W_{1} \cap W_{3}$ with $Q_{\tau} = \iota_1\iota_3(P_{\tau})$ for every $\tau$ in $\mathcal{U}_{13}$ (recall that $\iota_{h}: \mathbb{C}^3 \longrightarrow \mathbb{C}^3$ denotes the sign-changing involution of to the $h$-th coordinate, and by definition in Equation \ref{LOCIEQ} and Diagram \ref{restrictionPdiagram}, $\mathcal{U}_{ij} = \mathcal{U} \cap {\mathcal{H}_{3\Delta, \tau_0}}^{(ij)}$).
In both cases we can conclude that $P_{\tau_0}$ has $2$ coordinates which are $2$-torsion on the respective elliptic curve factors of $T$, and we would reach a contradiction with the generality conditions in Remark \ref{BEQUADROG}. \newline
From now on, we can assume (up to shrinking $\mathcal{U}$) that $\mathcal{P}$ does not intersect the diagonal locus in $\mathcal{S}_{\mathcal{U}} \times_{\mathcal{U}} \mathcal{S}_{\mathcal{U}}$. For every couple of indices $(ij)$ we can fix an irreducible component $\mathcal{P}_{ij}$ of the restriction $\mathcal{P}$ to the closed locus $\mathcal{U}_{ij}$. By our assumption on $\mathcal{P}$, the component $\mathcal{P}_{ij}$ also has empty intersection with the diagonal subscheme of $\mathcal{S}_{\mathcal{U}_{ij}} \times_{\mathcal{U}_{ij}} \mathcal{S}_{\mathcal{U}_{ij}}$.
\noindent By applying Proposition \ref{generalbehavior2}, we have that every fiber $\mathcal{P}_{\tau, ij}$ of the family $\mathcal{P}_{ij}$ must be contained in
\begin{align*} \mathcal{X}_{\tau,ij} &:= \SET{(P, \iota_{i}\iota_{j}(P)) \in \mathcal{S}_{\tau} \times \mathcal{S}_{\tau}}{P \in W_i \cap W_j} \\ \mathcal{W}_{\tau,k} &:= \SET{(P, \iota_{k}(P)) \in \mathcal{S}_{\tau} \times \mathcal{S}_{\tau}}{P \in W_k} \ \ \text{.}
\end{align*} By definition, we have that $\mathcal{U}_{12} \cap \mathcal{U}_{13} \cap \mathcal{U}_{23} = \{\tau_0\}$, hence on the central fiber we must have that $(P,Q)$ belongs to the intersections of of the different $\mathcal{X}_{\tau_0, ij}$ or to $\mathcal{Y}_{\tau_0, k}$, in accordance with the behavior of $\mathcal{P}_{ij}$.
\noindent Because all surfaces of the family $\mathcal{S}_{\mathcal{U}}$ are supposed to be smooth, then $(P,Q)$ belongs to an intersection of the form $\mathcal{X}_{\tau_0, ij} \cap \mathcal{W}_{\tau_0, i} \cap \mathcal{W}_{\tau_0, j}$ for a couple of indices $i,j$, with $i \neq j$.
On the other hand, every point of $\mathcal{X}_{\tau_0, ij} \cap \mathcal{W}_{\tau_0, i} \cap \mathcal{W}_{\tau_0, j}$ is of the form $(P,Q)$ such that $Q = \iota_{i}\iota_{j}(P)$, $Q = \iota_{i}(P)$ and $Q = \iota_{j}(P)$. But this implies that $P=Q$, and we reach a contradiction with the assumption that $\mathcal{P}$ does not intersect the diagonal sublocus.
\noindent The proof of the theorem is complete.
\end{proof}
\affiliationone{}
\end{document} | arXiv |
\begin{definition}[Definition:Reduced Ring]
Let $\struct {R, +, \circ}$ be a ring.
Then $R$ is a '''reduced ring''' {{iff}} it contains no nilpotent elements except the zero.
\end{definition} | ProofWiki |
\begin{document}
\title{Quantum Symbolic Execution} \author[1,2]{Jiang Nan} \author[1]{Wang Zichen} \author*[3,4]{Wang Jian}\email{[email protected]} \affil[1]{The Faculty of Information Technology, Beijing University of Technology, Beijing 100124, China} \affil[2]{Beijing Key Laboratory of Trusted Computing, Beijing 100124, China} \affil[3]{School of Computer and Information Technology, Beijing Jiaotong University, Beijing
100044, China} \affil[4]{Beijing Key Laboratory of Security and Privacy in Intelligent Transportation, Beijing Jiaotong University, Beijing 100044, China}
\abstract{With advances in quantum computing, researchers can now write and run many quantum programs. However, there is still a lack of effective methods for debugging quantum programs. In this paper, quantum symbolic execution (QSE) is proposed to generate test cases, which helps to finding bugs in quantum programs. The main idea of quantum symbolic execution is to find the suitable test cases from all possible ones (i.e. test case space). It is different from the way of classical symbol execution, which gets test cases by calculating instead of searching. QSE utilizes quantum superposition and parallelism to store the test case space with only a few qubits. According to the conditional statements in the debugged program, the test case space is continuously divided into subsets, subsubsets and so on. Elements in the same subset are suitable test cases that can test the corresponding branch in the code to be tested. QSE not only provides a possible way to debug quantum programs, but also avoids the difficult problem of solving constraints in classical symbolic execution.}
\keywords{quantum symbolic execution, test cases, quantum program testing, quantum program, quantum computing}
\maketitle
\section{Introduction}\label{sec1} Quantum computing has attracted much attention, because quantum superposition, entanglement and other properties can greatly improve the efficiency of computing \cite{nielsen2002quantum,jiang2021programmable}. In recent years, with the development of quantum computer hardware \cite{zhong2020quantum,arute2019quantum}, quantum software and quantum programming \cite{broughton2020tensorflow,cross2018ibm,paolini2019qpcf,selinger2004towards} has also been greatly developed. Researchers can write and run many quantum algorithms that have been proposed before but cannot be implemented due to limitations, such as Grover's algorithm \cite{adedoyin2018quantum}, quantum principal component analysis algorithm \cite{he2020exact}, quantum phase estimation \cite{o2019quantum}, and ${etc}$. In the process of writing quantum programs, some errors will inevitably occur \cite{paltenghi2021bugs,wang2018quanfuzz,miranskyy2020your}. For example, Zhao \cite{zhao2021identifying} defined a few bugs that focus on misuses of features of the quantum programming language --- Qiskit \cite{cross2018ibm}. Huang \cite{huang2019statistical} also recorded some bugs in the Scaffold compiler \cite{javadiabhari2015scaffcc}. For quantum programs we still need to take corresponding measures to find these errors and fix them. Due to the characteristics of quantum computing, we cannot debug programs as in the classical environment. This difficulty in debugging quantum programs hinders the development of quantum computing. An effective quantum program debugging scheme is needed.
Researchers have proposed some methods for debugging quantum programs, including quantum unit tests \cite{bright2017microsoft}, quantum assertions \cite{huang2019statistical,liu2020quantum,li2019proq,liu2021systematic}, and ${etc}$. Unit tests are used to determine whether a specific function is correct under a specific condition. The role of the assertion is that when the program executes to the assertion, the corresponding assertion should be true, and if the assertion is not true, the program should terminate execution. These methods have corresponding quantum versions. However, these methods are not very good to meet the needs. Currently, assertions in the quantum environment include statistical assertions \cite{huang2019statistical} based on classical observations, dynamic runtime assertions \cite{liu2020quantum} that use auxiliary qubits to obtain information indirectly, a projection-based runtime assertion \cite{li2019proq}, and dynamic assertion \cite{liu2021systematic} that extend dynamic runtime assertions \cite{liu2020quantum}. These assertions have two main shortcomings. Firstly, they are mostly used when an error has occurred during the running of the program or when the programmer suspects that there is an error somewhere in the program. Just like people do not directly set breakpoints on the entire program, but often set breakpoints only when the output is not as expected. Secondly, the use of assertions relies on the prediction of results. They need to compare the actual output with the expected result to judge whether the program is error. This is not simple for quantum programs. Microsoft's ${Q\#}$ \cite{bright2017microsoft} provides a method for unit testing of quantum programs, which tests a unit of a quantum program individually to verify whether it meets expectations, and internally still uses assertions to achieve this goal. There is another method Quito (quantum input output coverage) \cite{ali2021assessing}. The biggest contribution of this paper is to define three coverage criteria for the input and output of quantum program debugging. But the biggest flaw of this method is that it still uses statistical analysis to determine test pass and fail, which certainly does not reduce the complexity of quantum program debugging. Therefore, they cannot meet the programmer's needs for quantum program debugging very well.
Only unit tests and assertion cannot meet the needs of program debugging. In classical program debugging field, symbolic execution is another important debug method and it has appeared much earlier \cite{king1976symbolic}. With the development of constraint solving technology, symbolic execution has become an effective technology for generating high-coverage test cases \cite{cadar2013symbolic} and been widely used in different areas such as software testing, analysis and verification \cite{zhao40smart,yang2019cache,wang2017cached}.
This paper proposes a quantum symbolic execution (QSE) method, which focuses on generating high-coverage test cases for quantum programs. QSE uses quantum superposition and parallel characteristics to store the test case space with only a few qubits. According to the conditional statements in the debugged program, the test case space is continuously divided into subsets. Elements in the same subset are suitable test cases that can test the corresponding branch in the code to be tested. QSE not only provides a possible way to debug quantum programs, but also avoids the difficult problem of solving constraints in classical symbolic execution.
\section{Related Works} In this section, we briefly introduce the classical symbolic execution and some existing quantum modules that will be used in QSE. \subsection{classical symbolic execution (CSE)} \label{sec21} Programs often have conditional statements, and each branch represents an execution path to the program. In software testing, symbolic execution is a way to generate test cases that cover each execution path. Symbolic execution works by two steps: \begin{enumerate}[(1)]
\item creating execution paths, and
\item using a constraint solver to calculate the answers to the execution paths, i.e., generating test cases. \end{enumerate}
To formally accomplish this task, symbolic execution maintains two states globally: a symbolic state ${\sigma}$, which maps variables to symbolic expressions, and symbolic path constraints ${PC}$s, which are quantifier-free first-order logical formulas over symbolic expressions. At the beginning of a symbolic execution, ${\sigma}$ is initialized to an empty map and ${PC}$ is initialized to ${true}$. Both ${\sigma}$ and ${PC}$ are populated during the course of symbolic execution. The update rule of ${\sigma}$ is: \begin{enumerate}
\item[$\bullet$] At every read statement ${var=sym\_input()}$ that receives program input, symbolic execution adds the mapping ${var \mapsto s}$ to ${\sigma}$, where ${s}$ is a fresh symbolic value.
\item[$\bullet$] At every assignment ${v=e}$, symbolic execution updates ${\sigma}$ by mapping ${v}$ to ${\sigma(e)}$, where ${\sigma(e)}$ is the mapping of the symbolic state ${\sigma}$ to the expression ${e}$. \end{enumerate} The update rule of ${PC}$ is: \begin{enumerate}
\item[$\bullet$] At every conditional statement ${if \; (e) \; S1 \; else \; S2}$, ${PC}$ is updated to
${PC_1=PC\land \sigma(e)}$ (``then'' branch) and ${PC_2={PC} \land \lnot \sigma(e)}$ (``else'' branch). \end{enumerate}
For example, the symbolic execution of the code in Fig. \ref{fig1} starts with an empty symbolic state $\sigma$ and a symbolic path constraint ${true}$. After Line 03, ${\sigma=\{x \mapsto x_0, y \mapsto y_0\}}$; after Line 05, a path constraint ${(x_0+y_0<4)\land(x_0>y_0)}$ is created; and after Line 09, a path constraint ${(x_0+y_0\geq4)\land(y_0>1)}$ is created. Finally, there are 4 path constrains: $PC_{11}$, $PC_{12}$, $PC_{21}$, and $PC_{22}$. Each path constraint is solved with a constraint solver to obtain test cases. $\{x=2,y=1\}$, $\{x=1,y=2\}$, $\{x=3,y=2\}$, and $\{x=4,y=1\}$ are the possible outputs of the constraint solver for $PC_{11}$, $PC_{12}$, $PC_{21}$, and $PC_{22}$ respectively, i.e., they are suitable test cases.
All the execution paths of a program can be represented using a tree, called the execution tree. For example, Fig. \ref{fig2} gives the execution tree of the code in Fig. \ref{fig1}. The 4 branches correspond to the 4 path constrains.
\begin{figure}
\caption{An example to illustrate symbolic execution}
\label{fig1}
\end{figure}
\begin{figure}
\caption{The execution tree for the example in Fig. \ref{fig1}}
\label{fig2}
\end{figure}
\subsection{related quantum modules}\label{sec24} Suppose ${a}$ and ${b}$ are two ${n}$-qubit binary numbers, quantum adder \cite{chang2019design} ``${A}$'' implements addition of two qubits: \begin{equation*}
A(|ab \rangle |0 \rangle ^{\otimes n+1})= |ab \rangle |a+b \rangle. \end{equation*} The quantum module is shown in Fig. \ref{fig3}.
Quantum multiplier \cite{2019Quantum} ``${M}$'' implements multiplication of two qubits: \begin{equation*}
M(|ab\rangle |0\rangle^{\otimes 2n})= |ab\rangle |a\times b\rangle. \end{equation*} The quantum module is shown in Fig. \ref{fig4}.
The quantum comparator \cite{wang2012design} ``${C}$'' is used to compare two binary numbers. ${c_1}$ and ${c_2}$ are two 1-qubit outputs to record the comparison: \begin{equation*}
C(|ab\rangle|00\rangle)=|ab\rangle|c_1c_2\rangle. \end{equation*}
When ${a>b}$, ${\left|c_1c_2\right\rangle=\left|10\right\rangle}$; when ${a<b}$, ${\left|c_1c_2\right\rangle=\left|01\right\rangle}$; and when ${a=b}$, ${\left|c_1c_2\right\rangle=\left|00\right\rangle}$. The module is shown in Fig. \ref{fig5}. \begin{figure}
\caption{Three quantum modules}
\label{fig3}
\label{fig4}
\label{fig5}
\end{figure}
\section{Quantum symbolic execution} In this section, we first give the workflow of quantum symbolic execution. Then we explain how to prepare the initial test case space and use relational operators, logical operators to delineate subspaces. Then we give the overall framework of QSE. Finally give an example to illustrate.
\subsection{main idea} \label{sec31} In Section \ref{sec21}, we briefly describe the process of symbolic execution in the classical environment. Generally speaking, it first traverses the program to collect the path constraints, and then uses the constraint solver to calculate a set of inputs that meet the path constraints.
Quantum symbolic execution is completely different, which works by two steps: \begin{enumerate}[(1)]
\item generating a test case space that includes all possible test cases, and
\item according to the conditional statements in the code to be tested, partitioning the test case space into subspaces, and each subspace contains all the test cases that fit into a path constraint. \end{enumerate}
Fig. \ref{fig16} contrasts classical symbolic execution and quantum symbolic execution. \begin{figure}
\caption{The contrast between classical symbolic execution and quantum symbolic execution.}
\label{fig16}
\end{figure}
QSE uses two quantum registers: $|s\rangle$ and $|c\rangle$, where \begin{equation}
|q\rangle=\frac{1}{\sqrt{2}^n}\sum_{i=0}^{2^{n}-1}|s_i\rangle\otimes|c_i\rangle \end{equation}
$|s\rangle=|s^{n-1}s^{n-2}\cdots s^0\rangle$ consists of $n$ qubits and $s_i$ is a value used to represent a test case. $|c\rangle=|c^{m-1}c^{m-2}\cdots c^0\rangle$ consists of $m$ qubits and is the flag to subspace. $|s\rangle$ and $|c\rangle$ entangle together to realize the partition of $|s\rangle$: $s_i$ with the same $c_i$ belongs to the same subset, i.e. test cases for the same branch. $|s\rangle$ and $|c\rangle$ are collectively referred to as $|q\rangle$.
The flag $|c\rangle$ plays an important role in QSE, and it is gradually modified as the conditional statements in the code to be tested. Different conditions correspond to different ways to modify $|c\rangle$. Therefore, it is necessary to know how many types of conditions there are when programming. According to \cite{prata2014c, bruce2006java, eric2015python}, the conditions mainly include relational operation in Table \ref{table1} and logical operation in Table \ref{table2}. \begin{table}
\caption{relational operation}
\begin{center}
\begin{tabular}{cc}
\hline
relational operators & meaning \\
\hline
${<}$ & less than \\
${<=}$ & less than or equal to \\
${>}$ & greater than \\
${>=}$ & greater than or equal to \\
${==}$ & equal to \\
${!=}$ & not equal to \\
\hline
\end{tabular} \label{table1}
\end{center} \end{table}
\begin{table}
\caption{logical operation}
\begin{center}
\begin{tabular}{cc}
\hline
logical operators & meaning \\
\hline
${\&\&}$ & AND \\
${||}$ & OR \\
${!}$ & NOT \\
\hline
\end{tabular} \label{table2}
\end{center} \end{table}
The effects of relational and logical operations on $|c\rangle$ will be described in detail in Sections \ref{relationaloperator} and \ref{logicaloperator}, respectively.
\subsection{Preparation of the test case space}
Prepare $m+n$ qubits and set all of them to $|0\rangle$. The initial state of $|q\rangle$ is \begin{equation}
|q\rangle_0 =|0\rangle^{\otimes n}\otimes|0\rangle^{\otimes m} \end{equation}
i.e., $|s\rangle_0=|0\rangle^{\otimes n}$ and $|c\rangle_0=|0\rangle^{\otimes m}$.
$n$ $H$ quantum gates and $m$ $I$ quantum gates are used to transform the initial state $|q\rangle_0$ to state $|q\rangle_1$, where $$ H=\frac{1}{\sqrt{2}}\left[ \begin{array}{cc}
1 & 1 \\
1 & -1 \end{array} \right],\ \ I=\left[ \begin{array}{cc}
1 & 0 \\
0 & 1 \end{array} \right] $$
The quantum preparation of the test case space can be expressed as $U_1$: \begin{equation}
U_1 = H^{\otimes n}\otimes I^{\otimes m} \end{equation}
$U_1$ changes the initial state $|q\rangle_0$ to the test case space: \begin{equation}\label{U1}
\begin{split}
|q\rangle_1=&U_1(|q\rangle_0)\\
=&H^{\otimes n}(|s\rangle_0)\otimes I^{\otimes m}(|c\rangle_0)\\
=&(H|0\rangle)^{\otimes n}\otimes (I|0\rangle)^{\otimes m}\\
=&\frac{1}{\sqrt 2}(|0\rangle+|1\rangle) \otimes \frac{1}{\sqrt 2}(|0\rangle+|1\rangle) \otimes \cdots \otimes \frac{1}{\sqrt 2 }(|0\rangle+|1\rangle) \otimes|0\rangle^m\\
=&\frac{1}{{\sqrt 2 }^n}\left( |0 \cdots 00\rangle + |0 \cdots 01\rangle +\cdots+|1 \cdots 11\rangle\right)\otimes|0\rangle^m\\
=&\frac{1}{{\sqrt 2 }^n}\left( |0\rangle + |1\rangle+\cdots +|2^n-1\rangle\right)\otimes|0\rangle^m\\
=&\frac{1}{\sqrt{2}^n}\sum_{i=0}^{2^n-1}|i\rangle\otimes|0\rangle^m\\
=&|s\rangle\otimes|0\rangle^m
\end{split} \end{equation}
where $|s\rangle=\frac{1}{\sqrt{2}^n}\sum_{i=0}^{2^n-1}|i\rangle$. The quantum circuit is shown in Fig. \ref{fig6}.
\begin{figure}
\caption{The preparation of the test case space}
\label{fig6}
\end{figure}
Eq. (\ref{U1}) shows that the test case space $|s\rangle$ stores all integers from 0 to $2^n-1$, which are all the possible test cases. If the code to be tested contains $l$ ($l>1$) variables $x_1,x_2,\cdots,x_l$, $|s\rangle$ is still able to store all possible test cases. Divide the $n$ qubits of $|s\rangle$ into $l$ parts and each part stores all the possible value of a variable. The $i$th part contains $n_i$ qubits $|s_{x_i}\rangle=|s_{x_i}^{n_i-1}s_{x_i}^{n_i-2}\cdots s_{x_i}^{0}\rangle$, where $n=\sum_{i=1}^{l}n_i$. For example, the code in Fig. \ref{fig1} has two variables: $x$ and $y$. They contain 3 and 2 qubits respectively. Hence, $$
|s\rangle=|s_xs_y\rangle=\frac{1}{\sqrt{2}^3}\sum_{i=0}^{7}|i\rangle\otimes\frac{1}{\sqrt{2}^2}\sum_{i=0}^{3}|i\rangle=\frac{1}{\sqrt{2}^5}\sum_{i=0}^{31}|i\rangle
$$
\subsection{Relational operator} \label{relationaloperator}
Relational operators compare two numbers. Therefore, QSE uses the quantum comparator to divide the test case space. Section 2.2 shows that the quantum comparator has two output qubits: $|c_1c_2\rangle$. Suppose they correspond to some two adjacent qubits in $|c\rangle=|c^{m-1}c^{m-2}\cdots c^0\rangle$, and mark them as $|c^{i}c^{i-1}\rangle$. Combining Table \ref{table1}, we can get the relationship between the relational operators and the state of the output qubits as shown in Table \ref{table3}. In this table, ``$*$'' indicates that there is no requirement for the state of that qubit.
\begin{table}
\caption{The output of rational operation}
\begin{center}
\begin{tabular}{cc}
\hline
relational operator & $|c^{i}c^{i-1}\rangle$ \\
\hline
${<}$ & $|01\rangle$ \\
${<=}$ & $|0*\rangle$ \\
${>}$ & $|10\rangle$ \\
${>=}$ & $|*0\rangle$ \\
${==}$ & $|00\rangle$ \\
${!=}$ & $|01\rangle$ or $|10\rangle$ \\
\hline
\end{tabular} \label{table3}
\end{center} \end{table}
Sometimes, instead of directly comparing two variables, the code to be tested compares the values of two expressions. Suppose the two expressions are $e_1$ and $e_2$, and their outputs are $|\varphi_1\rangle$ and $|\varphi_2\rangle$ respectively. A quantum comparator is used to compare $|\varphi_1\rangle$ and $|\varphi_2\rangle$. $|c^{i}\rangle$ and $|c^{i-1}\rangle$ record the results of the comparison, i.e., they are the flags to segment the test case space. The segmentation of the test case space by a relational operator is expressed as $U_r$: \begin{equation}
U_r = C\otimes e_1\otimes e_2 \end{equation}
$U_r$ can segment the test case space by modifying the state of $|c^ic^{i-1}\rangle$. \begin{equation}\label{Ur}
\begin{split}
&U_r(|s\rangle|0\rangle^{\otimes k}|0\rangle^{\otimes t}|00\rangle)\\
=&C(e_1(|s\rangle|0\rangle^{\otimes k}) e_2(|s\rangle|0\rangle^{\otimes t}) |00\rangle)\\
=&C(|s\rangle|\varphi_1\rangle|\varphi_2\rangle|00\rangle)\\
=&|s\rangle\otimes C(|\varphi_1\rangle|\varphi_2\rangle|00\rangle)\\
=&|s\rangle|\varphi_1\rangle|\varphi_2\rangle|c^ic^{i-1}\rangle
\end{split} \end{equation}
In $|s\rangle\otimes|c^ic^{i-1}\rangle$, due to the entanglement between $|s\rangle$ and $|c^ic^{i-1}\rangle$, different states of $|c^ic^{i-1}\rangle$ correspond to different subspaces of $|s\rangle$. The circuit is shown in Fig. \ref{fig7}.
\begin{figure}
\caption{The segmentation of the test case space by relational operations.}
\label{fig7}
\end{figure}
In the following, we use $|c^ic^{i-1}\rangle_e$ to indicate that $|c^ic^{i-1}\rangle$ is in the output state of $e$, and $|c^ic^{i-1}\rangle_{\overline{e}}$ to indicate that $|c^ic^{i-1}\rangle$ is not in the output state of $e$, where $e=(e_1\circ e_2)$ and $\circ\in\{<,\leq,>,\geq,=,\neq\}$. For example, if $e=(e_1<e_2)$, $|c^ic^{i-1}\rangle_e=|01\rangle$, and $|c^ic^{i-1}\rangle_{\overline{e}}=|10\rangle$ or $|11\rangle$ or other non-$|01\rangle$ states.
\subsection{Logical operators} \label{logicaloperator} \subsubsection{$T$ module}
Usually, the inputs to a logical operator are the outputs of rational operator(s). A rational operator has two outputs $|c^ic^{i-1}\rangle$. Hence, Module $T$ is defined firstly to facilitate later descriptions.
$T$ is a control module that acts on two qubits $|c^ic^{i-1}\rangle$. According to Table \ref{table3}, $|c^ic^{i-1}\rangle$ have 6 states. Therefore, there are also 6 cases of $T=\{T_<,T_{\leq},T_>,T_{\geq},T_=,T_{\neq}\}$. Their circuits are shown in Fig. \ref{fig21}.
\begin{figure}
\caption{Six cases of Module $T$}
\label{figT1}
\label{figT2}
\label{figT3}
\label{figT4}
\label{figT5}
\label{figT6}
\label{fig21}
\end{figure}
For example, in Fig. \ref{figT1}, because it is $T_<$, the state of $|c^ic^{i-1}\rangle$ is $|01\rangle$. Hence, we place a 0-control on qubit $|c^i\rangle$ and a 1-control on $|c^{i-1}\rangle$. Thus, these two control qubits represent that the result of the previous relational operation is ``less than''.
\subsubsection{Logical operators} There are 3 logical operators. We will give their quantum circuits one by one.
\begin{enumerate}[(1)]
\item AND \end{enumerate}
Suppose there is an expression ${e_1\&\&e_2}$, where $e_1$ and $e_2$ are two rational operations. The logical AND in QSE is shown in Fig. \ref{fig91}, where $T_{e_1},T_{e_2}\in T$, $|c_1^i c_1^{i-1}\rangle$ are the flags of ${e_1}$, and $|c_2^i c_2^{i-1}\rangle$ are the flags of ${e_2}$. The output of logical AND is $|c_{A}\rangle$: if and only if both $e_1$ and $e_1$ are satisfied, $|c_{A}\rangle$ becomes $|1\rangle$; otherwise, it remains unchanged in $|0\rangle$ state. That is to say, $|c_{A}\rangle$ becomes a flag of logical AND.
\begin{figure}
\caption{logical AND for QSE}
\label{fig91}
\label{fig92}
\label{fig9}
\end{figure}
Define \begin{equation}
U_{Ar} = T_{e_1}\text{-}T_{e_2}\text{-NOT} \end{equation} Then, \begin{equation}\label{UAr}
\begin{split}
&U_{Ar}(|c_1^i c_1^{i-1}\rangle\otimes|c_2^i c_2^{i-1}\rangle\otimes|0\rangle)\\
=&T_{e_1}\text{-}T_{e_2}\text{-NOT}(|c_1^i c_1^{i-1}\rangle\otimes|c_2^i c_2^{i-1}\rangle\otimes|0\rangle)\\
=&|c_1^i c_1^{i-1}\rangle_{e_1}\otimes|c_2^i c_2^{i-1}\rangle_{e_2}\otimes|1\rangle+|c_1^i c_1^{i-1}\rangle_{\overline{e_1}}\otimes|c_2^i c_2^{i-1}\rangle_{{e_2}}\otimes|0\rangle\\
&+|c_1^i c_1^{i-1}\rangle_{{e_1}}\otimes|c_2^i c_2^{i-1}\rangle_{\overline{e_2}}\otimes|0\rangle+|c_1^i c_1^{i-1}\rangle_{\overline{e_1}}\otimes|c_2^i c_2^{i-1}\rangle_{\overline{e_2}}\otimes|0\rangle
\end{split} \end{equation}
If $e_1$ and $e_2$ are two logical operations, it is only necessary to replace $|c_1^i c_1^{i-1}\rangle$ with $|c_1\rangle$, $|c_2^i c_2^{i-1}\rangle$ with $|c_2\rangle$, and $ T_{e_1}$ and $T_{e_2}$ with 1-control, as shown in Fig. \ref{fig92}, where $|c_1\rangle$ and $|c_2\rangle$ are the outputs of $e_1$ and $e_2$ respectively. Now \begin{equation}
U_{Al} = \text{CC-NOT} \end{equation} and \begin{equation}\label{UAl}
\begin{split}
&U_{Al}(|c_1\rangle\otimes|c_2\rangle\otimes|0\rangle)\\
=&\text{CC-NOT}(|c_1\rangle\otimes|c_2\rangle\otimes|0\rangle)\\
=&|1\rangle\otimes|1\rangle\otimes|1\rangle+|0\rangle\otimes|1\rangle\otimes|0\rangle+|1\rangle\otimes|0\rangle\otimes|0\rangle+|0\rangle\otimes|0\rangle\otimes|0\rangle
\end{split} \end{equation}
\begin{enumerate}[(2)]
\item OR \end{enumerate}
For logical OR, there is an expression ${e_1||e_2}$. Fig. \ref{fig101} shows the logical OR in QSE if $e_1$ and $e_2$ are two rational operations. The output of logical OR is $|c_{O}\rangle$: as long as one of $e_1$ and $e_1$ is satisfied, $|c_{O}\rangle$ becomes $|1\rangle$; otherwise, it remains unchanged in $|0\rangle$ state. That is to say, $|c_{O}\rangle$ becomes a flag of logical OR.
\begin{figure}
\caption{logical OR for QSE}
\label{fig101}
\label{fig102}
\label{fig10}
\end{figure}
Define \begin{equation}
U_{Or} = T_{e_1}\text{-}T_{e_2}\text{-NOT}\otimes T_{e_2}\text{-NOT}\otimes T_{e_1}\text{-NOT} \end{equation} Then, \begin{equation}\label{UA}
\begin{split}
&U_{Or}(|c_1^i c_1^{i-1}\rangle\otimes|c_2^i c_2^{i-1}\rangle\otimes|0\rangle)\\
=&T_{e_1}\text{-}T_{e_2}\text{-NOT}\otimes T_{e_2}\text{-NOT}(T_{e_1}\text{-NOT}(|c_1^i c_1^{i-1}\rangle\otimes|c_2^i c_2^{i-1}\rangle\otimes|0\rangle))\\
=&T_{e_1}\text{-}T_{e_2}\text{-NOT}\otimes T_{e_2}\text{-NOT}(|c_1^i c_1^{i-1}\rangle_{e_1}\otimes|c_2^i c_2^{i-1}\rangle\otimes|1\rangle+|c_1^i c_1^{i-1}\rangle_{\overline{e_1}}\otimes|c_2^i c_2^{i-1}\rangle\otimes|0\rangle)\\
=&T_{e_1}\text{-}T_{e_2}\text{-NOT}(|c_1^i c_1^{i-1}\rangle_{e_1}\otimes|c_2^i c_2^{i-1}\rangle_{e_2}\otimes|0\rangle
+|c_1^i c_1^{i-1}\rangle_{e_1}\otimes|c_2^i c_2^{i-1}\rangle_{\overline{e_1}}\otimes|1\rangle\\
&+|c_1^i c_1^{i-1}\rangle_{\overline{e_1}}\otimes|c_2^i c_2^{i-1}\rangle_{e_2}\otimes|1\rangle
+|c_1^i c_1^{i-1}\rangle_{\overline{e_1}}\otimes|c_2^i c_2^{i-1}\rangle_{\overline{e_1}}\otimes|0\rangle)\\
=&|c_1^i c_1^{i-1}\rangle_{e_1}\otimes|c_2^i c_2^{i-1}\rangle_{e_2}\otimes|1\rangle
+|c_1^i c_1^{i-1}\rangle_{e_1}\otimes|c_2^i c_2^{i-1}\rangle_{\overline{e_1}}\otimes|1\rangle\\
&+|c_1^i c_1^{i-1}\rangle_{\overline{e_1}}\otimes|c_2^i c_2^{i-1}\rangle_{e_2}\otimes|1\rangle
+|c_1^i c_1^{i-1}\rangle_{\overline{e_1}}\otimes|c_2^i c_2^{i-1}\rangle_{\overline{e_1}}\otimes|0\rangle
\end{split} \end{equation}
If $e_1$ and $e_2$ are two logical operations, the quantum circuit is shown in Fig. \ref{fig102} and represented as $U_{Ol}$. The migration principle is the same as in Fig. \ref{fig9} and will not be repeated.
\begin{enumerate}[(3)]
\item NOT \end{enumerate}
NOT does not need to be implemented with any quantum circuits. For $!e$, not matter $e$ is a rational operation or a logical operation, $e$ divides $|s\rangle$ into two subsets: one satisfies $e$ and the other does not. $!e$ just reverses the satisfiability and does not affect the division of the two subsets. Therefore, there is no need for quantum circuits to change the division of the subsets or to divide the subsets further.
\subsection{Divide the test case space} Programs often have complex $e$ or the branch statements are nested. Therefore, multiple quantum operations are needed to be connected to continuously divide the test case space.
Define \begin{equation}
U_2 = U^{\otimes k} \end{equation}
where $U\in\{U_r,U_{Ar},U_{Al},U_{Or},U_{Ol}\}$ and $k$ is a positive integer. Act $U_2$ on $|q\rangle_1$: \begin{equation}\label{U2}
\begin{split}
|q\rangle_2=&U_2(|q\rangle_1)\\
=&U^{\otimes k}(|s\rangle\otimes|0\rangle^m)\\
=&\frac{1}{\sqrt{2}^n}\sum_{i=0}^{2^{n}-1}|s_i\rangle\otimes|c_i\rangle
\end{split} \end{equation}
According to the definitions of $U_r,U_{Ar},U_{Al},U_{Or},U_{Ol}$ in Section \ref{relationaloperator} and Section \ref{logicaloperator}, the qubits $|0\rangle^m$ in $|q\rangle_1$ is gradually modified based on the relational and the logical operators in the program to be tested. Eventually, through the entanglement of $|s\rangle$ and $|c\rangle$, the test case space is divided into multiple subsets. The values belonging to the same subset are test cases that can cover the same branch.
\section{Experiments} \subsection{An example} \subsubsection{The division of the test space} The program shown in Fig. \ref{fig1} is used as an example to further illustrate how QSE works. There are 3 branch statements in the program. Coupled with the process of preparing the test case space, the quantum circuit consists of 4 parts as shown in Fig. \ref{fig13}.
\begin{figure}
\caption{QSE circuit for example in Fig. \ref{fig1}}
\label{fig13}
\end{figure}
\begin{enumerate}[(1)]
\item Prepare the test case space \end{enumerate}
3 and 2 qubits are used to represent variables $x$ and $y$ respectively. Hence, 5 $H$ quantum gates transform the initial state $|0\rangle^{\otimes 5}$ to state $|s_x\rangle\otimes|s_y\rangle$, i.e., \begin{equation*}
\begin{split}
&H^{\otimes 5}(|0\rangle^{\otimes 5})=(H|0\rangle)^{\otimes 3}\otimes (H|0\rangle)^{\otimes 2}\\
=&\frac{1}{\sqrt{2}^3}\sum_{i=0}^{7}|i\rangle\otimes\frac{1}{\sqrt{2}^2}\sum_{i=0}^{3}|i\rangle=|s_x\rangle\otimes|s_y\rangle
\end{split} \end{equation*}
That is to say, $|s_x\rangle$ stores $0\sim7$ and $|s_y\rangle$ stores $0\sim3$. This is the test case space.
\begin{enumerate}[(2)]
\item $x+y<4$? \end{enumerate}
The outermost branch statement is to determine whether $x+y$ is less than 4. The quantum adder ``$A$'' is used to get the sum of $x$ and $y$. We add a $|0\rangle$ qubit as the highest bit of $|s_y\rangle$ to make $|s_y\rangle$ and $|s_x\rangle$ both have 3 qubits. The quantum comparator ``$C$'' is used to compare $|s_x+s_y\rangle$ and $|4\rangle$, and the output is $|c^1c^0\rangle$. If $x+y<4$, $|c^1c^0\rangle=|01\rangle$; otherwise, $|c^1c^0\rangle=|*0\rangle$. The whole process can be described with the following equation. \begin{equation*}
\begin{split}
&(C\otimes A)(|s_x\rangle|s_y\rangle\otimes|0\rangle|4\rangle|0\rangle|0\rangle)=C(A(|s_x\rangle|s_y\rangle|0\rangle)\otimes|4\rangle|0\rangle|0\rangle)\\
=&C((|0\rangle|0\rangle|0\rangle+|0\rangle|1\rangle|1\rangle+|0\rangle|2\rangle|2\rangle+|0\rangle|3\rangle|3\rangle\\
&+|1\rangle|0\rangle|1\rangle+|1\rangle|1\rangle|2\rangle+|1\rangle|2\rangle|3\rangle+|1\rangle|3\rangle|4\rangle\\
&+|2\rangle|0\rangle|2\rangle+|2\rangle|1\rangle|3\rangle+|2\rangle|2\rangle|4\rangle+|2\rangle|3\rangle|5\rangle\\
&+|3\rangle|0\rangle|3\rangle+|3\rangle|1\rangle|4\rangle+|3\rangle|2\rangle|5\rangle+|3\rangle|3\rangle|6\rangle\\
&+|4\rangle|0\rangle|4\rangle+|4\rangle|1\rangle|5\rangle+|4\rangle|2\rangle|6\rangle+|4\rangle|3\rangle|7\rangle\\
&+|5\rangle|0\rangle|5\rangle+|5\rangle|1\rangle|6\rangle+|5\rangle|2\rangle|7\rangle+|5\rangle|3\rangle|8\rangle\\
&+|6\rangle|0\rangle|6\rangle+|6\rangle|1\rangle|7\rangle+|6\rangle|2\rangle|8\rangle+|6\rangle|3\rangle|9\rangle\\
&+|7\rangle|0\rangle|7\rangle+|7\rangle|1\rangle|8\rangle+|7\rangle|2\rangle|9\rangle+|7\rangle|3\rangle|10\rangle)\otimes|4\rangle|0\rangle|0\rangle)\\
=&|0\rangle|0\rangle C(|0\rangle|4\rangle|0\rangle|0\rangle)+|0\rangle|1\rangle C(|1\rangle|4\rangle|0\rangle|0\rangle)+|0\rangle|2\rangle C(|2\rangle|4\rangle|0\rangle|0\rangle)\\
&+|0\rangle|3\rangle C(|3\rangle|4\rangle|0\rangle|0\rangle)+|1\rangle|0\rangle C(|1\rangle|4\rangle|0\rangle|0\rangle)+|1\rangle|1\rangle C(|2\rangle|4\rangle|0\rangle|0\rangle)\\
&+|1\rangle|2\rangle C(|3\rangle|4\rangle|0\rangle|0\rangle)+|1\rangle|3\rangle C(|4\rangle|4\rangle|0\rangle|0\rangle)+|2\rangle|0\rangle C(|2\rangle|4\rangle|0\rangle|0\rangle)\\
&+|2\rangle|1\rangle C(|3\rangle|4\rangle|0\rangle|0\rangle)+|2\rangle|2\rangle C(|4\rangle|4\rangle|0\rangle|0\rangle)+|2\rangle|3\rangle C(|5\rangle|4\rangle|0\rangle|0\rangle)\\
&+|3\rangle|0\rangle C(|3\rangle|4\rangle|0\rangle|0\rangle)+|3\rangle|1\rangle C(|4\rangle|4\rangle|0\rangle|0\rangle)+|3\rangle|2\rangle C(|5\rangle|4\rangle|0\rangle|0\rangle)\\
&+|3\rangle|3\rangle C(|6\rangle|4\rangle|0\rangle|0\rangle)+|4\rangle|0\rangle C(|4\rangle|4\rangle|0\rangle|0\rangle)+|4\rangle|1\rangle C(|5\rangle|4\rangle|0\rangle|0\rangle)\\
&+|4\rangle|2\rangle C(|6\rangle|4\rangle|0\rangle|0\rangle)+|4\rangle|3\rangle C(|7\rangle|4\rangle|0\rangle|0\rangle)+|5\rangle|0\rangle C(|5\rangle|4\rangle|0\rangle|0\rangle)\\
&+|5\rangle|1\rangle C(|6\rangle|4\rangle|0\rangle|0\rangle)+|5\rangle|2\rangle C(|7\rangle|4\rangle|0\rangle|0\rangle)+|5\rangle|3\rangle C(|8\rangle|4\rangle|0\rangle|0\rangle)\\
&+|6\rangle|0\rangle C(|6\rangle|4\rangle|0\rangle|0\rangle)+|6\rangle|1\rangle C(|7\rangle|4\rangle|0\rangle|0\rangle)+|6\rangle|2\rangle C(|8\rangle|4\rangle|0\rangle|0\rangle)\\
&+|6\rangle|3\rangle C(|9\rangle|4\rangle|0\rangle|0\rangle)+|7\rangle|0\rangle C(|7\rangle|4\rangle|0\rangle|0\rangle)+|7\rangle|1\rangle C(|8\rangle|4\rangle|0\rangle|0\rangle)\\
&+|7\rangle|2\rangle C(|9\rangle|4\rangle|0\rangle|0\rangle)+|7\rangle|3\rangle C(|10\rangle|4\rangle|0\rangle|0\rangle)\\
=&|0\rangle|0\rangle |0\rangle|4\rangle|0\rangle|1\rangle+|0\rangle|1\rangle |1\rangle|4\rangle|0\rangle|1\rangle+|0\rangle|2\rangle |2\rangle|4\rangle|0\rangle|1\rangle+|0\rangle|3\rangle |3\rangle|4\rangle|0\rangle|1\rangle\\
&+|1\rangle|0\rangle |1\rangle|4\rangle|0\rangle|1\rangle+|1\rangle|1\rangle |2\rangle|4\rangle|0\rangle|1\rangle+|1\rangle|2\rangle |3\rangle|4\rangle|0\rangle|1\rangle+|1\rangle|3\rangle |4\rangle|4\rangle|0\rangle|0\rangle\\
&+|2\rangle|0\rangle |2\rangle|4\rangle|0\rangle|1\rangle+|2\rangle|1\rangle |3\rangle|4\rangle|0\rangle|1\rangle+|2\rangle|2\rangle |4\rangle|4\rangle|0\rangle|0\rangle+|2\rangle|3\rangle |5\rangle|4\rangle|1\rangle|0\rangle\\
&+|3\rangle|0\rangle |3\rangle|4\rangle|0\rangle|1\rangle+|3\rangle|1\rangle |4\rangle|4\rangle|0\rangle|0\rangle+|3\rangle|2\rangle |5\rangle|4\rangle|1\rangle|0\rangle+|3\rangle|3\rangle |6\rangle|4\rangle|1\rangle|0\rangle\\
&+|4\rangle|0\rangle |4\rangle|4\rangle|0\rangle|0\rangle+|4\rangle|1\rangle |5\rangle|4\rangle|1\rangle|0\rangle+|4\rangle|2\rangle |6\rangle|4\rangle|1\rangle|0\rangle+|4\rangle|3\rangle |7\rangle|4\rangle|1\rangle|0\rangle\\
&+|5\rangle|0\rangle |5\rangle|4\rangle|1\rangle|0\rangle+|5\rangle|1\rangle |6\rangle|4\rangle|1\rangle|0\rangle+|5\rangle|2\rangle |7\rangle|4\rangle|1\rangle|0\rangle+|5\rangle|3\rangle |8\rangle|4\rangle|1\rangle|0\rangle\\
&+|6\rangle|0\rangle |6\rangle|4\rangle|1\rangle|0\rangle+|6\rangle|1\rangle |7\rangle|4\rangle|1\rangle|0\rangle+|6\rangle|2\rangle |8\rangle|4\rangle|1\rangle|0\rangle+|6\rangle|3\rangle |9\rangle|4\rangle|1\rangle|0\rangle\\
&+|7\rangle|0\rangle |7\rangle|4\rangle|1\rangle|0\rangle+|7\rangle|1\rangle |8\rangle|4\rangle|1\rangle|0\rangle+|7\rangle|2\rangle |9\rangle|4\rangle|1\rangle|0\rangle+|7\rangle|3\rangle |10\rangle|4\rangle|1\rangle|0\rangle
\end{split} \end{equation*}
\begin{enumerate}[(3)]
\item $x>y$? \end{enumerate}
If $x+y<4$, it needs to be further judged whether $x$ is greater than $y$. Hence, a $T_<$-$C$ module acts on the subspace $|s_x\rangle|s_y\rangle\otimes|c^3c^2c^1c^0\rangle$. \begin{equation*}
\begin{split}
&T_<\text{-}C(|0\rangle|0\rangle |0001\rangle+|0\rangle|1\rangle |0001\rangle+|0\rangle|2\rangle |0001\rangle+|0\rangle|3\rangle |0001\rangle\\
&+|1\rangle|0\rangle |0001\rangle+|1\rangle|1\rangle |0001\rangle+|1\rangle|2\rangle |0001\rangle+|1\rangle|3\rangle |0000\rangle\\
&+|2\rangle|0\rangle |0001\rangle+|2\rangle|1\rangle |0001\rangle+|2\rangle|2\rangle |0000\rangle+|2\rangle|3\rangle |0010\rangle\\
&+|3\rangle|0\rangle |0001\rangle+|3\rangle|1\rangle |0000\rangle+|3\rangle|2\rangle |0010\rangle+|3\rangle|3\rangle |0010\rangle\\
&+|4\rangle|0\rangle |0000\rangle+|4\rangle|1\rangle |0010\rangle+|4\rangle|2\rangle |0010\rangle+|4\rangle|3\rangle |0010\rangle\\
&+|5\rangle|0\rangle |0010\rangle+|5\rangle|1\rangle |0010\rangle+|5\rangle|2\rangle |0010\rangle+|5\rangle|3\rangle |0010\rangle\\
&+|6\rangle|0\rangle |0010\rangle+|6\rangle|1\rangle |0010\rangle+|6\rangle|2\rangle |0010\rangle+|6\rangle|3\rangle |0010\rangle\\
&+|7\rangle|0\rangle |0010\rangle+|7\rangle|1\rangle |0010\rangle+|7\rangle|2\rangle |0010\rangle+|7\rangle|3\rangle |0010\rangle)\\
=&|0\rangle|0\rangle |0001\rangle+|0\rangle|1\rangle |0101\rangle+|0\rangle|2\rangle |0101\rangle+|0\rangle|3\rangle |0101\rangle\\
&+|1\rangle|0\rangle |1001\rangle+|1\rangle|1\rangle |0001\rangle+|1\rangle|2\rangle |0101\rangle+|1\rangle|3\rangle |0000\rangle\\
&+|2\rangle|0\rangle |1001\rangle+|2\rangle|1\rangle |1001\rangle+|2\rangle|2\rangle |0000\rangle+|2\rangle|3\rangle |0010\rangle\\
&+|3\rangle|0\rangle |1001\rangle+|3\rangle|1\rangle |0000\rangle+|3\rangle|2\rangle |0010\rangle+|3\rangle|3\rangle |0010\rangle\\
&+|4\rangle|0\rangle |0000\rangle+|4\rangle|1\rangle |0010\rangle+|4\rangle|2\rangle |0010\rangle+|4\rangle|3\rangle |0010\rangle\\
&+|5\rangle|0\rangle |0010\rangle+|5\rangle|1\rangle |0010\rangle+|5\rangle|2\rangle |0010\rangle+|5\rangle|3\rangle |0010\rangle\\
&+|6\rangle|0\rangle |0010\rangle+|6\rangle|1\rangle |0010\rangle+|6\rangle|2\rangle |0010\rangle+|6\rangle|3\rangle |0010\rangle\\
&+|7\rangle|0\rangle |0010\rangle+|7\rangle|1\rangle |0010\rangle+|7\rangle|2\rangle |0010\rangle+|7\rangle|3\rangle |0010\rangle
\end{split} \end{equation*}
If and only if $|c^1c^0\rangle=|01\rangle$, $|s_x\rangle$ and $|s_y\rangle$ need to be compared, i.e., $|c^3c^2\rangle$ is changed according to $|s_x\rangle$ and $|s_y\rangle$: if $|s_x\rangle>|s_y\rangle$, $|c^3c^2\rangle=|10\rangle$; otherwise, $|c^3c^2\rangle=|0*\rangle$. As long as $|c^1c^0\rangle\neq|01\rangle$, $|c^3c^2\rangle$ remains unchanged at state $|00\rangle$.
\begin{enumerate}[(4)]
\item $y>1$? \end{enumerate}
If $x+y\geq4$, it needs to be further judged whether $y$ is greater than $1$. Hence, a $T_\geq$-$C$ module acts on the subspace $|s_y\rangle|1\rangle\otimes|c^3c^2c^1c^0\rangle$. \begin{equation*}
\begin{split}
&T_\geq\text{-}C(|0\rangle|1\rangle |0001\rangle+|1\rangle|1\rangle |0101\rangle+|2\rangle|1\rangle |0101\rangle+|3\rangle|1\rangle |0101\rangle\\
&+|0\rangle|1\rangle |1001\rangle+|1\rangle|1\rangle |0001\rangle+|2\rangle|1\rangle |0101\rangle+|3\rangle|1\rangle |0000\rangle\\
&+|0\rangle|1\rangle |1001\rangle+|1\rangle|1\rangle |1001\rangle+|2\rangle|1\rangle |0000\rangle+|3\rangle|1\rangle |0010\rangle\\
&+|0\rangle|1\rangle |1001\rangle+|1\rangle|1\rangle |0000\rangle+|2\rangle|1\rangle |0010\rangle+|3\rangle|1\rangle |0010\rangle\\
&+|0\rangle|1\rangle |0000\rangle+|1\rangle|1\rangle |0010\rangle+|2\rangle|1\rangle |0010\rangle+|3\rangle|1\rangle |0010\rangle\\
&+|0\rangle|1\rangle |0010\rangle+|1\rangle|1\rangle |0010\rangle+|2\rangle|1\rangle |0010\rangle+|3\rangle|1\rangle |0010\rangle\\
&+|0\rangle|1\rangle |0010\rangle+|1\rangle|1\rangle |0010\rangle+|2\rangle|1\rangle |0010\rangle+|3\rangle|1\rangle |0010\rangle\\
&+|0\rangle|1\rangle |0010\rangle+|1\rangle|1\rangle |0010\rangle+|2\rangle|1\rangle |0010\rangle+|3\rangle|1\rangle |0010\rangle)\\
=&|0\rangle|1\rangle |0001\rangle+|1\rangle|1\rangle |0101\rangle+|2\rangle|1\rangle |0101\rangle+|3\rangle|1\rangle |0101\rangle\\
&+|0\rangle|1\rangle |1001\rangle+|1\rangle|1\rangle |0001\rangle+|2\rangle|1\rangle |0101\rangle+|3\rangle|1\rangle |1000\rangle\\
&+|0\rangle|1\rangle |1001\rangle+|1\rangle|1\rangle |1001\rangle+|2\rangle|1\rangle |1000\rangle+|3\rangle|1\rangle |1010\rangle\\
&+|0\rangle|1\rangle |1001\rangle+|1\rangle|1\rangle |0000\rangle+|2\rangle|1\rangle |1010\rangle+|3\rangle|1\rangle |1010\rangle\\
&+|0\rangle|1\rangle |0100\rangle+|1\rangle|1\rangle |0010\rangle+|2\rangle|1\rangle |1010\rangle+|3\rangle|1\rangle |1010\rangle\\
&+|0\rangle|1\rangle |0110\rangle+|1\rangle|1\rangle |0010\rangle+|2\rangle|1\rangle |1010\rangle+|3\rangle|1\rangle |1010\rangle\\
&+|0\rangle|1\rangle |0110\rangle+|1\rangle|1\rangle |0010\rangle+|2\rangle|1\rangle |1010\rangle+|3\rangle|1\rangle |1010\rangle\\
&+|0\rangle|1\rangle |0110\rangle+|1\rangle|1\rangle |0010\rangle+|2\rangle|1\rangle |1010\rangle+|3\rangle|1\rangle |1010\rangle
\end{split} \end{equation*}
If and only if $|c^0\rangle=|0\rangle$, $|s_y\rangle$ and $|1\rangle$ need to be compared, i.e., $|c^3c^2\rangle$ is changed according to $|s_y\rangle$ and $|1\rangle$: if $|s_y\rangle>|1\rangle$, $|c^3c^2\rangle=|10\rangle$; otherwise, $|c^3c^2\rangle=|0*\rangle$. As long as $|c^0\rangle\neq|0\rangle$, $|c^3c^2\rangle$ remains unchanged.
Finally, the state of the subspace $|s_x\rangle|s_y\rangle\otimes|c^3c^2c^1c^0\rangle$ is \begin{equation}\label{example}
\begin{split}
&{|0\rangle|0\rangle |0001\rangle}+{|0\rangle|1\rangle |0101\rangle}+{|0\rangle|2\rangle |0101\rangle}+{|0\rangle|3\rangle |0101\rangle}\\
+&{|1\rangle|0\rangle |1001\rangle}+{|1\rangle|1\rangle |0001\rangle}+{|1\rangle|2\rangle |0101\rangle}+{|1\rangle|3\rangle |1000\rangle}\\
+&{|2\rangle|0\rangle |1001\rangle}+{|2\rangle|1\rangle |1001\rangle}+{|2\rangle|2\rangle |1000\rangle}+{|2\rangle|3\rangle |1010\rangle}\\
+&{|3\rangle|0\rangle |1001\rangle}+{|3\rangle|1\rangle |0000\rangle}+{|3\rangle|2\rangle |1010\rangle}+{|3\rangle|3\rangle |1010\rangle}\\
+&{|4\rangle|0\rangle |0100\rangle}+{|4\rangle|1\rangle |0010\rangle}+{|4\rangle|2\rangle |1010\rangle}+{|4\rangle|3\rangle |1010\rangle}\\
+&{|5\rangle|0\rangle |0110\rangle}+{|5\rangle|1\rangle |0010\rangle}+{|5\rangle|2\rangle |1010\rangle}+{|5\rangle|3\rangle |1010\rangle}\\
+&{|6\rangle|0\rangle |0110\rangle}+{|6\rangle|1\rangle |0010\rangle}+{|6\rangle|2\rangle |1010\rangle}+{|6\rangle|3\rangle |1010\rangle}\\
+&{|7\rangle|0\rangle |0110\rangle}+{|7\rangle|1\rangle |0010\rangle}+{|7\rangle|2\rangle |1010\rangle}+{|7\rangle|3\rangle |1010\rangle}
\end{split} \end{equation}
There are 4 cases of the state $|c^3c^2c^1c^0\rangle$: \begin{itemize}
\item {$|1001\rangle$}: $|c^1c^0\rangle=|01\rangle$ indicates $x+y<4$ and $|c^3c^2\rangle=|10\rangle$ indicates $x>y$. Hence, $|1001\rangle$ indicates $x+y<4 \ \&\&\ x>y$, which corresponds to $PC_{11}$ in classical symbolic execution.
\item {$|0*01\rangle$}: $|c^1c^0\rangle=|01\rangle$ indicates $x+y<4$ and $|c^3c^2\rangle=|0*\rangle$ indicates $x\leq y$. Hence, $|0*01\rangle$ indicates $x+y<4 \ \&\&\ x\leq y$, which corresponds to $PC_{12}$ in classical symbolic execution.
\item {$|10*0\rangle$}: $|c^1c^0\rangle=|*0\rangle$ indicates $x+y\geq 4$ and $|c^3c^2\rangle=|10\rangle$ indicates $y>1$. Hence, $|10*0\rangle$ indicates $x+y\geq 4 \ \&\&\ y>1$, which corresponds to $PC_{21}$ in classical symbolic execution.
\item {$|0**0\rangle$}: $|c^1c^0\rangle=|*0\rangle$ indicates $x+y\geq 4$ and $|c^3c^2\rangle=|0*\rangle$ indicates $y\leq1$. Hence, $|0**0\rangle$ indicates $x+y\geq 4 \ \&\&\ y\leq1$, which corresponds to $PC_{22}$ in classical symbolic execution. \end{itemize}
These 4 states of $|c^3c^2c^1c^0\rangle$ divide $|s_x\rangle|s_y\rangle$ into 4 subsets. As shown in Eq. \ref{example}, \begin{itemize}
\item Subset $\{|1\rangle|0\rangle,|2\rangle|0\rangle,|3\rangle|0\rangle,|2\rangle|1\rangle\}$ contains all the test cases that can test the branch $x+y<4 \ \&\&\ x>y$.
\item Subset $\{|0\rangle|0\rangle,|0\rangle|1\rangle,|0\rangle|2\rangle,|0\rangle|3\rangle,|1\rangle|1\rangle,|1\rangle|2\rangle\}$ contains all the test cases that can test the branch $x+y<4 \ \&\&\ x\leq y$.
\item Subset $\{|2\rangle|2\rangle,|3\rangle|2\rangle,|4\rangle|2\rangle,|5\rangle|2\rangle,|6\rangle|2\rangle,|7\rangle|2\rangle,|1\rangle|3\rangle,|2\rangle|3\rangle,|3\rangle|3\rangle,|4\rangle|3\rangle, \\
|5\rangle|3\rangle,|6\rangle|3\rangle,|7\rangle|3\rangle\}$ contains all the test cases that can test the branch $x+y\geq 4 \ \&\&\ y>1$.
\item Subset $\{|4\rangle|0\rangle,|5\rangle|0\rangle,|6\rangle|0\rangle,|7\rangle|0\rangle,|3\rangle|1\rangle,|4\rangle|1\rangle,|5\rangle|1\rangle,|6\rangle|1\rangle,|7\rangle|1\rangle\}$ contains all the test cases that can test the branch $x+y\geq 4 \ \&\&\ y\leq1$. \end{itemize}
\subsubsection{Running on a quantum computer}\label{sec412} We use the ${ibmq \_ qasm \_ simulator}$ quantum computer on the ${IBM \; Quantum}$ platform to perform the example. The circuit is shown in Fig. \ref{fig14}. This experiment uses 28 qubits, with $q_0$ as the lowest bit and $q_{25}$ as the highest bit: \begin{itemize}
\item $q_2q_1q_0$ represent $|s_x\rangle$;
\item $q_5q_4q_3$ represent $|s_y\rangle$;
\item $q_9q_8q_7q_6$ represent $|s_x+s_y\rangle$;
\item $q_{12}q_{11}q_{10}$ are the auxiliary qubits of the quantum adder ``$A$'';
\item $q_{16}q_{15}q_{14}q_{13}$ are used to represent constant $\left|4\right\rangle$ and $q_{17}$ is used to represent constant $\left|1\right\rangle$;
\item $q_{23}q_{22}q_{21}q_{20}q_{19}q_{18}$ are the auxiliary qubits of the quantum comparator ``$C$'';
\item $q_{24}q_{25}$ are the flags $|c^3c^2\rangle$ and $q_{26}q_{27}$ are the flags $|c^1c^0\rangle$. \end{itemize}
\begin{figure}
\caption{circuit implementation of QSE}
\label{fig14}
\end{figure}
The three purple bars in the figure are three quantum comparators. At the end of the circuit, $q_0q_1q_2q_3q_4q_5$ and $q_{24}q_{25}q_{26}q_{27}$ are measured and they have 32 results as shown in Fig. \ref{fig15}. The abscissa displays all the results and the default state of qubits that are not measured is 0. The ordinate represents the probability of each state in a total of 8192 measurements.
\begin{figure}
\caption{measurement results for the circuit in Fig. \ref{fig14}}
\label{fig15}
\end{figure}
The 32 results can be divided into four test case spaces. Fig. \ref{fig30}(a) gives the measurement results whose $|c_3c_2c_1c_0\rangle=|1001\rangle$, i.e., $x+y<4 \ \&\&\ x>y$. Fig. \ref{fig30}(b) gives the measurement results whose $|c_3c_2c_1c_0\rangle=|0*01\rangle$, i.e., $x+y<4 \ \&\&\ x\leq y$. Fig. \ref{fig30}(c) gives the measurement results whose $|c_3c_2c_1c_0\rangle=|10*0\rangle$, i.e., $x+y \geq 4 \ \&\&\ y>1$. Fig. \ref{fig30}(d) gives the measurement results whose $|c_3c_2c_1c_0\rangle=|0**0\rangle$, i.e., $x+y \geq 4 \ \&\&\ y \leq 1$.
\begin{figure}
\caption{Four test case spaces}
\label{fig30}
\end{figure}
\subsection{Experiment data}
8 real programs are used to evaluate the performance of QSE. They come from 2 references: \cite{2014Solving} and \cite{2018VulDeePecker} as shown in Table \ref{table4}. The ``Operations'' column describes the type of operations appearing in the path conditions. The ``Line of code'' column lists the number of source code lines in the program, excluding comments and empty lines.
\begin{center}
\begin{table}
\centering
\caption{Programs for the experiments}
\begin{tabular}{cccc}
\hline
Program & Operations & Line of code & From \\
\hline
dart & Polynomials & 11 & \cite{2014Solving}\\
power & Exponential function & 20 & \cite{2014Solving}\\
stat & Mean and std. dev. computation & 62 & \cite{2014Solving}\\
tcas & Constant equality checks & 82 & \cite{2014Solving}\\
early & Polynomials & 14 & \cite{2014Solving}\\
basic00181 & Constant equality checks & 30 & \cite{2018VulDeePecker}\\
snp3-ok & Constant equality checks & 24 & \cite{2018VulDeePecker}\\
CWE789 & Integer computation & 141 & \cite{2018VulDeePecker}\\
\hline
\end{tabular}\label{table4}
\end{table} \end{center}
Firstly we compare the complexity and the time consumption of CSE and QSE. The comparison results are shown in Table \ref{table5}. The main factor that affects the complexity of CSE are the number of path constraints. The main factor that affects the complexity of QSE is the number of subspace divisions. Table \ref{table5} shows that the complexity of QSE is less than that of CSE. We also compare the actual time consumption of CSE and QSE. The tool to realize CSE is JDart \cite{2016JDart}, which supports the z3 constraint solver \cite{2012Solving}. In most cases, the time consumption of QSE is also smaller than that of CSE.
\begin{center}
\begin{table}
\centering
\caption{The comparison of complexity and time consumption of CSE and QSE.}
\begin{tabular}{ccccc}\hline
\multicolumn{1}{c}{\multirow{3}{*}{Program}} & \multicolumn{2}{c}{CSE} & \multicolumn{2}{c}{QSE} \\
\cline{2-5}
& number of & \multicolumn{1}{c}{\multirow{2}{*}{time/s}} & number of & \multicolumn{1}{c}{\multirow{2}{*}{time/s}} \\
& path constraints & & subspace divisions & \\
\hline
dart & 4 & 0.48 & 3 & 0.45 \\
power & 11 & 1.32 & 7 & 1.05 \\
stat & 3 & 0.36 & 2 & 0.3 \\
tcas & 5 & 0.6 & 4 & 0.6 \\
early & 2 & 0.24 & 1 & 0.15 \\
basic00181 & 3 & 0.36 & 2 & 0.3 \\
snp3-ok & 1 & 0.12 & 1 & 0.15 \\
CWE789 & 6 & 0.72 & 3 & 0.45 \\
\hline
\end{tabular}\label{table5}
\end{table} \end{center}
We also show the impact of test case space on program branch coverage. In the example given in Section \ref{sec412}, three qubits are used for each variable. In fact, more or fewer qubits can affect the performance of QSE. Too few qubits make it impossible for QSE to cover all branches. Consider the more extreme case: there are 4 branches in the program, but only 1 qubit is used to store variables, i.e., there are only 2 test cases in the test case space. Such a test case space is unlikely to cover all branches. Isn't the more qubits used, the better? No. Too many qubits will increase the difficulty of QSE, and lead to the waste of quantum resources. Therefore, the smallest number of qubits that can cover all branches is the best choice. Fig \ref{fig35} shows the relationship between the number of qubits used by variables in the three programs in Table \ref{table5} and the program branch coverage. The best numbers of qubits for the three programs are 2, 4 and 5 respectively.
\begin{figure}
\caption{The relationship between the number of qubits and branch coverage}
\label{fig35}
\end{figure}
\section{Conclusion} This paper proposes a quantum symbolic execution for the first time to generate high-coverage test cases. It is completely different from not only classical symbolic executions, but also quantum debugging schemes. QSE divides the test case space into subsets according to the conditional statements in the debugged program, and a subset contains all test cases that can test the same program branch. QSE not only provides a possible way to debug quantum programs, but also avoids the difficult problem of solving constraints in classical symbolic execution, which obviously reduces the difficulty and improves the efficiency of the work.\\
\noindent\textbf{Funding} This work is supported by the National Natural Science Foundation of China under Grants No.61502016.
\noindent\textbf{Data availability} All data generated or analysed during this study are included in this article.
\end{document} | arXiv |
\begin{document}
\title{Towards a Singular Value Decomposition and spectral theory for all rings} \author{Ran Gutin\footnote{Department of Computer Science, Imperial College London}} \date{\today}
\maketitle
\begin{abstract}
We propose definitions of SVD, spectral decomposition (for self-adjoint matrices) and Jordan decomposition which make sense for all rings. For many rings, these decompositions can be shown to exist. For some specific rings, these decompositions are complicated to describe in full and prove the existence of. These decompositions have occurred piecemeal in the literature. We conjecture that they exist for many rings, including all Clifford algebras over the real numbers and complex numbers. The origin of this programme is not directly in module theory or linear algebra. \end{abstract}
\section{Motivation before giving the definitions}\label{motivation-before-giving-the-definitions}
\subsection{Objective}\label{objective}
The objective of this paper is to propose a general definition for three matrix decompositions, which can be shown to be satisfiable (or not) over any given ring:
\begin{itemize} \item
SVD (Singular Value Decomposition) \item
Spectral decomposition. In this paper, we see this as a decomposition
of self-adjoint matrices. \item
Jordan decomposition. \end{itemize}
The motivation for this is that we have discovered analogues of those three decompositions outside the obvious linear algebra or module theory context. These analogues present canonical forms which are not as simple as in the complex case, because they might fail to be diagonal. This phenomenon is already familiar for the Jordan decomposition (in the complex case), but is not present for the other two decompositions when working over the complex numbers. In general, the canonical forms can be complicated, and their existence is not easy to prove on a case-by-case basis.
\subsection{Summary of origins}\label{summary-of-origins}
I give an account of how I stumbled upon SVDs over different rings. Admittedly, this is somewhat subjective. It is written in the first person.
There is a 19th century ``non-Euclidean'' geometry (in the Kleinian sense) called \emph{Laguerre geometry} which admits a relationship to the algebra of \(2 \times 2\) matrices over the dual numbers. I found that based on the correspondence between the \emph{congruence transformations} of this geometry, and the \(2 \times 2\) matrices over the dual numbers, there was a strong hint towards the existence of an analogue of the Singular Value Decomposition over the ring of dual numbers instead of the usual ring of complex numbers. An account of this matrix/transformation correspondence can be found in Yaglom's book \emph{Complex numbers in geometry}. The SVD interpretation and subsequent result was inspired by a classification of the Laguerre transformations which can be found in Yaglom's book. This analogue of the SVD turned out to exist for matrices over the dual numbers, where the matrices could be of any possible dimension (not just the \(2 \times 2\) case that Yaglom considered, and not just the square matrices), even in the case where the matrices were singular, and satisfied certain uniqueness properties similar to the complex SVD.
I will now state exactly the result I obtained \cite{dualsvd}. The ring of dual numbers is denoted formally as \(\mathbb R[\varepsilon]/(\varepsilon^2)\). \footnote{More informally, a dual number is a number of the form $a + b \varepsilon$ where $\varepsilon^2 = 0$ while $\varepsilon \neq 0$. The dual numbers form an associative and commutative unital algebra over the real numbers. }The dual numbers admit an involution \(*\) for which \((a + b\varepsilon)^* = a - b \varepsilon\). This defines a conjugate-transpose or adjoint operator on matrices \(M^*\) such that \((M^*)_{ij} = (M_{ji})^*\). Analogously, we get a notion of unitary matrix \(U\) for which \(UU^* = U^* U = I\). Given a matrix \(M\) over the dual numbers (of any dimension), I obtained a result which states that \(M\) can be factorised into \(USV^*\) where \(U\) and \(V\) are unitary, and \(S\) is a direct sum of matrices in the set
\[
\begin{aligned}
&\{[x] : x \in \mathbb R; x > 0\}\\& \cup \left\{\begin{bmatrix} x & -y\varepsilon \\ y\varepsilon & x\end{bmatrix} : x \in \mathbb R, y \in \mathbb R; x > 0, y > 0\right\}\\& \cup \{[y\varepsilon] : y \in \mathbb R; y > 0\}\\&\cup \{0_{1,0}, 0_{0,1}\}.
\end{aligned} \]
This set is quite complicated, and we won't dwell on it. Notice though that the matrix \(0_{1,0}\) (and \(0_{0,1}\)) is missing in the usual account of the SVD over \(\mathbb C\), but we will show that it is implicitly there as well! It's the unique \(1 \times 0\) matrix.
To explain the occurrence of this strange matrix, let's consider the complex matrix \(M = \begin{bmatrix}1 & 2 & 0 \\ 2 & 1 & 0 \end{bmatrix}\). Ignoring the exact values of \(U\) and \(V\), but focussing our attention on \(S\), we have that \(M = USV^*\) where \(S = [3] \oplus [-1] \oplus 0_{0,1}\). The reader should verify that given a matrix \(K\), the value of \(K \oplus 0_{0,1}\) is the same matrix as \(K\) but padded with a zero column. Likewise, the value of \(K \oplus 0_{1,0}\) is the same matrix as \(K\) but padded with a zero row. The value of \([3] \oplus [-1] \oplus 0_{0,1}\) is thus \(\begin{bmatrix}3 & 0 & 0 \\ 0 & -1 & 0 \end{bmatrix}\) as expected. We conclude that the ``singular values'' of a complex matrix form a multiset (a similar object to a \emph{set} but the elements are allowed to repeat) whose elements belong to the set \(\{[x] : x \in \mathbb R; x > 0\} \cup \{0_{1,0}, 0_{0,1}\}\). This is actually a set of \emph{matrices} rather than scalar ``values''. This suggests that the notion of a singular value as a scalar value is questionable.
Given a matrix \(M\) over \(\mathbb C\) or \(\mathbb R[\varepsilon]/(\varepsilon^2)\), the direct summands that make up \(S\) in \(M = USV^*\) are unique up to permutation. In the dual number case, a direct summand can be of dimension \(2 \times 2\), and non-square direct summands can be found over both rings (and in fact all rings).
Upon discovering the above dual number SVD, I found that a dual number SVD of sorts had already been considered in the literature, and it wasn't the one above. In that case, the involution over the dual numbers is the trivial one: \(z^* := z\), and the notion of unitary matrix degenerates into \(U^T U = UU^T = I\). We then have that the direct summands that make up \(S\) are of the form:
\[\{[x + y\varepsilon] : x \in \mathbb R, y \in \mathbb R; x > 0\} \cup \{[y \varepsilon] : y \in \mathbb R; y > 0\}\cup \{0_{1,0}, 0_{0,1}\}.\]
This shows that the correct setting is not a \emph{ring} as such, but a \emph{\(*\)-ring}, which is a ring equipped with an involution. The motivation for considering matrix algebra over this particular \(*\)-ring came from the mechanics literature where this SVD was suggested for solving kinematic synthesis problems.
Over the \(*\)-ring of \emph{double numbers}, which are defined as \(\mathbb R \oplus \mathbb R\), equipped with the involution \((a,b)^* := (b,a)\), a matrix decomposition that constitutes an SVD over the double numbers has been given \cite{contragredient}. I attempted to come up with such a decomposition myself, which I called the Jordan SVD, but it had the problem that it didn't exist for all matrices over the double numbers, and therefore didn't satisfy the general criteria I give here. The paper in which the \emph{actual} double-number SVD is introduced has some limitations, chief among them is that the paper is unaware of the connection to the double numbers, or that the decomposition constitutes \emph{the} analogue of the SVD over the \(*\)-ring of double numbers. A lot of terminology is introduced there, like \emph{contragredient equivalence}, which I view as needless.
Finally, over the \emph{quaternions} an analogue of SVD exists which satisfies my definition. This has recently been extended to the case of the \emph{dual quaternions} (an algebra equal to \(\mathbb H[\varepsilon]/(\varepsilon^2)\)). These algebras are non-commutative, so our definitions don't assume commutativity.
There is clearly a need for a general result which states precisely when an analogue of the SVD, spectral decomposition for self-adjoint matrices, or Jordan decomposition, exists for a given \(*\)-ring. Such a result has already been obtained for the analogue of the Jordan decomposition which we discuss here, but this appears at the moment to be the least general of the three decompositions. We supply general definitions here and formulate conjectures.
\subsection{Rough desired criteria}\label{rough-desired-criteria}
If \(M = USV^*\) is the ``SVD'' of M, then \(S\) should be a block diagonal matrix which is unique up to permutation of the blocks. This is similar to, and somewhat related to, the fact that the Jordan decomposition of a square matrix \(M\), expressed as \(PJP^{-1}\), should be unique up to permutation of the Jordan blocks. The objective is to decide whether or not a set of such blocks can be exhibited for a given ring.
The formal definitions which we give below express in a rigorous way the desired existence and uniqueness properties which we've alluded to.
\section{Definitions}\label{definitions}
A \(*\)-ring is a ring \(R\) equipped with an automorphism denoted \(*:R \to R\) that has order \(2\). Such an automorphism is called an \emph{involution}. Every ring can be made into a \(*\)-ring in at least one way by defining \(z^* = z\) for all \(z \in R\).
We define the following monoids:
\begin{itemize} \item
\(\mat(R,*)\) is the monoid of all matrices over a \(*\)-ring \(R\) where
the monoid operation is \(\oplus\), denoting direct sum of matrices.
The matrices can be of any possible dimensions, and they don't have to
be square. The matrices can also have \(0\) rows or \(0\) columns. \item
\(\herm(R,*)\) is the monoid of all self-adjoint matrices over a
\(*\)-ring \(R\) where the monoid operation is \(\oplus\). \item
\(\sq(R)\) is the monoid of all square matrices over a ring \(R\), where
\(R\) is merely a ring and not a \(*\)-ring. \end{itemize}
We then define the following equivalence relations on \(\mat(R,*)\) and its various submonoids above:
\begin{itemize} \item
\({\sim_{\text{UE}}}\) means \emph{unitary equivalence}. In other
words, we have that \(A {\sim_{\text{UE}}} B\) is true whenever there
exist unitary matrices \(U\) and \(V\) such that \(A = UBV^*\). \item
\({\sim_{\text{US}}}\) means \emph{unitary similarity}. In other
words, we have that \(A {\sim_{\text{US}}} B\) is true whenever there
exists a unitary matrix \(V\) such that \(A = VBV^*\). \item
\({\sim_{\text{S}}}\) means \emph{similarity}. In other words, we have
that \(A {\sim_{\text{S}}} B\) is true whenever there an invertible
matrix \(P\) such that \(A = PBP^{-1}\). \end{itemize}
We aim to study the three monoids:
\begin{itemize} \item
\(\matbar(R,*) := \mat(R,*) / {\sim_{\text{UE}}}\) with the intention of generalising the
\emph{singular value decomposition}, \item
\(\hermbar(R,*) := \herm(R,*) / {\sim_{\text{US}}}\) with the intention of generalising the
\emph{spectral theorem} (on self-adjoint matrices). \item
\(\sqbar(R,*) := \sq(R) / {\sim_{\text{S}}}\) with the intention of generalising the
\emph{Jordan decomposition}. \end{itemize}
To do this, notice that all three monoids are abelian, and for some \(*\)-rings \(R\), they are even \emph{free} as abelian monoids. A \emph{free abelian monoid} consists of all finite multisets whose elements belong to some set \(S\). An isomorphism between each of the three monoids above and a free abelian monoid produces an analogue of the SVD, spectral theorem, and Jordan decompositions respectively.
As an aside: Note that all six parametrised families of monoids can be thought of as \emph{functors} between two categories, if this fact can ever be useful. The functors are all from the category of \(*\)-rings to the category of monoids.
\section{Discussion of the relationship of monoid algebra to the SVD, spectral decomposition and Jordan decompositions}\label{discussion-of-the-relationship-of-monoid-algebra-to-the-svd-spectral-decomposition-and-jordan-decompositions}
When an abelian monoid \((M,+,0)\) is free, there is a \emph{unique} subset (which we will here denote \(P\)) of \(M\) (called the generators of \(M\)) such that each element of \(M\) is a unique sum elements of the generators \(P\). It can be argued that many ``unique factorisation'' type results in mathematics merely state the fact that some abelian monoid with a complicated construction is actually free.
An example of the above is the abelian monoid of \emph{positive integers} under integer multiplication, in which the generators are the prime numbers. The fact that for every integer \emph{there exists} a prime factorisation, and moreover it is \emph{unique}, is equivalent to the fact that the abelian monoid of positive integers is free.
The abelian monoid \(\matbar(R,*)\) for a given \(*\)-ring \((R,*)\) is constructed in a complicated way. In spite of that fact, if it is free, then it is actually quite simple. Its freeness captures the existence and uniqueness of the SVD.
\section{Summary of existing results}\label{summary-of-existing-results}
\textbf{Theorem 1}: \(\sqbar(R)\) is a free abelian monoid for every Artinian ring \(R\).
\emph{Proof}. See \cite{ringjordan}.
The above theorem in particular shows that something approaching a Jordan decomposition exists for the dual numbers, which are an Artinian ring. All finite dimensional associative algebras over a field, like the quaternions and dual numbers, are Artinian, so this generalises an important aspect of the Jordan decomposition.
Let \(\operatorname{swap}(a,b) := (b,a)\). This is a natural choice of involution for \(R \oplus R\) where \(R\) is any ring. An illustrative special case is when \(R\) is \(\mathbb R\), in which case \(R \oplus R\) is commonly called either the \emph{split-complex numbers} or the \emph{double numbers}. This is an interesting hypercomplex number system considered in \cite{doublenumbers}. When \(R = \mathbb C\), the name we give to \(R \oplus R\) is the \emph{double complex numbers}.
\textbf{Theorem 2}: \(\hermbar(R\oplus R,\operatorname{swap})\cong \sqbar(R)\) for every ring \(R\).
\textbf{Remark}: When both sides are treated as functors of \(R\), the isomorphism is natural. We don't attempt to verify this fact.
\emph{Proof}. Let \(M\) be an element of \(\herm(R \oplus R, \operatorname{swap})\). We have that \(M = (A,B^T)\) because \(M\) in particular must belong to \(J(R \oplus R)\). We furthermore have that \(M\) is Hermitian, so \((A,B^T) = M = M^* = (B,A^T)\); therefore \(A = B\). In summary, we have that \(M = (A,A^T)\). Let \(\phi_R : \sq(R) \,\to \herm(R \oplus R, \operatorname{swap})\) be given by \(\phi_R(A) = (A,A^T)\). The mapping is clearly an isomorphism of monoids. But we're not done yet because we would like an isomorphism \(\psi_R : \sqbar(R) \to \hermbar(R \oplus R, \operatorname{swap})\) instead. This isomorphism is obtained by noticing that if \(M \sim_\text{S} K\) then \(\phi_R(M) \sim_\text{US} \phi_R(K)\). We show that this is indeed true: If \(M {\sim_{\text{S}}} K\) then \(M = PKP^{-1}\), so \(\phi_R(M) = (M,M^T) = (PKP^{-1},(P^T)^{-1} K^T P^T) = (P,(P^{-1} )^T) (K,K^T) (P^{-1},P^T) {\sim_{\text{US}}} (K,K^T) = \phi_R(K)\). We are done.
The above result is significant because it shows that the monoid family \(\sqbar(R)\) is somewhat redundant. The study of these monoids is subsumed by the study of monoids in the family \(\hermbar(R,*)\).
\textbf{Theorem 3}: \(\matbar(R,*)\) admits an isomorphism to a free abelian monoid with the following generators when \((R,*)\) is any of the following \(*\)-rings:
\begin{longtable}{|l|l|l|l|}
\hline
& \(R\) & \(*:R \to R\) & Generators of \(\matbar(R,*)\)\\
\hline
\hline
1 & \(0\) & \(z^* := z\) & \(\{0_{1,0}, 0_{0,1}\}\) \\
\hline
2 & \(\mathbb R\) & \(z^* := z\) &
\(\begin{aligned}&\{[x] : x \in \mathbb R; x > 0\}\\ &\cup \{0_{1,0}, 0_{0,1}\}\end{aligned}\) \\
\hline \\
3 & \(\mathbb C\) & \((a + bi)^* := a - bi\) &
\(\begin{aligned}&\{[x] : x \in \mathbb R; x > 0\} \\&\cup \{0_{1,0}, 0_{0,1}\}\end{aligned}\) \\
\hline \\
4 & \(\mathbb R[\varepsilon] / (\varepsilon^2)\) & \(z^* := z\) &
\(\begin{aligned}&\{[x + y\varepsilon] : x \in \mathbb R, y \in \mathbb R; x > 0\}\\ &\cup \{[y \varepsilon] : y \in \mathbb R; y > 0\}\cup \{0_{1,0}, 0_{0,1}\}\end{aligned}\) \\
\hline \\
5 & \(\mathbb R[\varepsilon] / (\varepsilon^2)\) &
\((a + b\varepsilon)^* := a - b\varepsilon\) &
\(\begin{aligned}&\{[x] : x \in \mathbb R; x > 0\}\\ &\cup \left\{\begin{bmatrix} x & -y\varepsilon \\ y\varepsilon & x\end{bmatrix} : x \in \mathbb R, y \in \mathbb R; x > 0, y > 0\right\}\\ &\cup \{[y\varepsilon] : y \in \mathbb R; y > 0\}\cup \{0_{1,0}, 0_{0,1}\}\end{aligned}\) \\
\hline \\
6 & \(\mathbb C \oplus \mathbb C\) & \(\operatorname{swap}\) &
\(\begin{aligned}&\{(J_m(re^{i\theta}),J_m(re^{i\theta})^T) : r > 0, \theta \in [0,\pi)\} \\&\cup \left\{\left(\begin{pmatrix}I_m & 0\end{pmatrix},\begin{pmatrix}0\\ I_m\end{pmatrix}^T\right) : m \in \mathbb N\right\}\\ &\cup \left\{\left(\begin{pmatrix}0\\ I_m\end{pmatrix},\begin{pmatrix}I_m & 0\end{pmatrix}^T\right) : m \in \mathbb N\right\}\\&\cup \{(J_m(0), I_m^T) : m \in \mathbb N\}\\& \cup \{(I_m, J_m(0)^T) : m \in \mathbb N\}\cup\{0_{1,0}, 0_{0,1}\}\end{aligned}\) \\
\hline
7 & \(\mathbb H\) & \(\begin{aligned}&(a + bi + cj + dk)^* :=\\& a - bi - cj - dk \end{aligned}\) &
\(\{[x] : x \in \mathbb R; x > 0\} \cup \{0_{1,0}, 0_{0,1}\}\) \\
\hline
8 &
\(\begin{aligned}&\operatorname{span}(1, i, \varepsilon j, \varepsilon k)\\&\subset \mathbb H[\varepsilon]/(\varepsilon^2)\end{aligned}\)
&
\(\begin{aligned}&(a + bi + c\varepsilon j + d \varepsilon k)^* :=\\& a - bi - c\varepsilon j - d \varepsilon k\end{aligned}\)
&
\(\begin{aligned}&\{[x] : x \in \mathbb R; x > 0\}\\& \cup \left\{\begin{bmatrix} x & -\delta \\ \delta & x\end{bmatrix} : x \in \mathbb R; x > 0, \delta^2 = 0\right\}\\& \cup \{[y\varepsilon j]: y \in \mathbb R; y > 0\}\cup \{0_{1,0}, 0_{0,1}\}\end{aligned}\) \\
\hline \end{longtable} \emph{Proof}. For each example in turn:
\begin{enumerate} \item
The ring here is the \(0\) ring, whose set of elements is \(\{0\}\).
The definition of the operations \(\{+,-,\times\}\) is immediate. The
generators are the two degenerate matrices \(0_{1,0}\) and
\(0_{0,1}\). The first matrix has \(1\) row and \(0\) columns. The
second matrix has \(0\) rows and \(1\) column. Matrices with \(0\)
rows or columns appear strange but they are an inevitable part of
matrix algebra because they represent linear maps which map to or from
\(0\)-dimensional vector spaces. \item
A proof of this fact can be found in any undergraduate textbook on
linear algebra for mathematics students, as long as that textbook
covers the spectral theorem and SVD. The only unusual generators are
\(0_{1,0}\) and \(0_{0,1}\), which are \(1 \times 0\) and
\(0 \times 1\) matrices respectively. While it is strange to have
matrices with these dimensions, they are an inevitable part of the
matrix formalism because they represent linear maps that go to or from
\(0\)-dimensional vector spaces. \item
A proof of this fact can be found in any undergraduate textbook on
linear algebra for mathematics students, as long as that textbook
covers the spectral theorem and SVD. The only unusual generators are
\(0_{1,0}\) and \(0_{0,1}\), which are \(1 \times 0\) and
\(0 \times 1\) matrices respectively. While it is strange to have
matrices with these dimensions, they are an inevitable part of the
matrix formalism because they represent linear maps that go to or from
\(0\)-dimensional vector spaces. \item
See \cite{dualsvd}. The only thing missing from
the paper is a complete proof of uniqueness. This can be achieved by
using the uniqueness of the eigendecomposition for dual-number
matrices (a fact proved in the reference) and observing that the
diagonal form of the block matrix
\(\begin{pmatrix}0 & M^* \\ M & 0\end{pmatrix}\) is
\(\Sigma \oplus (-\Sigma)\) where \(\Sigma\) is the normal form of
\(M\) under SVD. This implies that \(\Sigma\) is unique. \item
See \cite{dualsvd}. Uniqueness can be reduced to
the case of the involution \(z^* := z\). \item
This follows from \cite{contragredient}. \item
This follows from Theorem 7.2 of \cite{quaternions}. \item
See \cite{dualcomplex}. \end{enumerate}
\textbf{Theorem 4}: \(\hermbar(R,*)\) admits an isomorphism to a free abelian monoid with the following generators when \((R,*)\) is any of the following \(*\)-rings:
\begin{longtable}{|l|l|l|l|}
\hline
& \(R\) & \(*:R \to R\) & Generators of \(\hermbar(R,*)\)\\
\hline \hline 1 & \(0\) & \(z^* := z\) & \(\{[0]\}\) \\ \hline 2 & \(\mathbb R\) & \(z^* := z\) & \(\{[x] : x \in \mathbb R; x > 0\} \cup \{[0]\}\) \\ \hline 3 & \(\mathbb C\) & \((a + bi)^* := a - bi\) & \(\{[x] : x \in \mathbb R; x > 0\} \cup \{[0]\}\) \\ \hline 4 & \(\mathbb R[\varepsilon] / (\varepsilon^2)\) & \(z^* := z\) & \(\begin{aligned}&\{[x + y\varepsilon] : x \in \mathbb R, y \in \mathbb R; x > 0\} \\&\cup \{[y \varepsilon] : y \in \mathbb R; y > 0\}\\&\cup \{[0]\}\end{aligned}\) \\ \hline 5 & \(\mathbb R[\varepsilon] / (\varepsilon^2)\) & \((a + b\varepsilon)^* := a - b\varepsilon\) & \(\begin{aligned}&\{[x] : x \in \mathbb R; x > 0\} \\&\cup \left\{\begin{bmatrix} x & -y\varepsilon \\ y\varepsilon & x\end{bmatrix} : x \in \mathbb R, y \in \mathbb R; x > 0, y > 0\right\}\\& \cup \{y\varepsilon : y \in \mathbb R; y > 0\}\\&\cup \{[0]\}\end{aligned}\) \\ \hline 6 & \(\mathbb C \oplus \mathbb C\) & \(\operatorname{swap}\) & \(\{(J_m(re^{i\theta}),J_m(re^{i\theta})^T) : r \geq 0, \theta \in [0,\pi)\}\) \\ \hline 7 & \(\mathbb H\) & \(\begin{aligned}&(a + bi + cj + dk)^* :=\\& a - bi - cj - dk\end{aligned}\) & \(\{[x] : x \in \mathbb R; x > 0\} \cup \{[0]\}\) \\ \hline 8 & \(\begin{aligned}&\operatorname{span}(1, i, \varepsilon j, \varepsilon k)\\&\subset \mathbb H[\varepsilon]/(\varepsilon^2)\end{aligned}\) & \(\begin{aligned}&(a + bi + c\varepsilon j + d \varepsilon k)^* :=\\& a - bi - c\varepsilon j - d \varepsilon k\end{aligned}\) & \(\begin{aligned}&\{[x] : x \in \mathbb R; x > 0\} \\&\cup \left\{\begin{bmatrix} x & -\delta \\ \delta & x\end{bmatrix} : x \in \mathbb R; x > 0, \delta^2 = 0\right\}\\&\cup \{[0]\}\end{aligned}\) \\ \hline 9 & \(\mathbb Z\) & \(z^* := z\) & \begin{tabular}[t]{@{}l@{}}The generators are related to the set of \\adjacency matrices of all connected graphs. \\We don't give an explicit description.\end{tabular}\\ \hline 10 & \(\mathbb C\) & \(z^* := z\) & Very complicated. Given in \cite{complexsymmetric}. \\ \hline \end{longtable} \emph{Proof}. For each example in turn:
\begin{enumerate} \item
The ring here is the \(0\) ring, whose set of elements is \(\{0\}\).
The definition of the operations \(\{+,-,\times\}\) is immediate. \item
A proof of this fact can be found in any undergraduate textbook on
linear algebra for mathematics students, as long as that textbook
covers the spectral theorem and SVD. \item
A proof of this fact can be found in any undergraduate textbook on
linear algebra for mathematics students, as long as that textbook
covers the spectral theorem and SVD. \item
See \cite{dualsvd}. \item
See \cite{dualsvd}. \item
Consequence of Theorem 2. \item
This follows from Theorem 7.2 of \cite{quaternions}. \item
See \cite{dualcomplex}. \item
In this case, we haven't stated the result clearly enough that we can
prove it. If we wished to, we \emph{could} state it rigorously and
prove it. Note that a unitary matrix \(U\) that is also an integer
matrix is precisely a signed permutation matrix. Two undirected
graphs, represented as adjacency matrix \(M\) and \(K\), are
isomorphic iff there exists a permutation matrix \(P\) such that
\(M = PKP^{-1}\). The generators are therefore connected graphs which
are combined by direct sum to form unconnected graphs. \item
See \cite{complexsymmetric}. \end{enumerate}
\section{Meta-conjectures}\label{meta-conjectures}
By a meta-conjecture, we mean a conjecture which is true for a large class of rings, but not necessarily for all rings. We suspect that the class of rings for which these conjectures are true is large.
\textbf{Conjecture 1}: For any ring \(*\)-ring \((R,*)\), \(\hermbar(R,*)\) is free.
\textbf{Conjecture 2}: For any ring \(*\)-ring \((R,*)\), \(\matbar(R,*)\) is free.
We know from theorem 1 that \(\sqbar(R)\) is free whenever \(R\) is Artinian. \(\sqbar(R)\) may be free for some non-Artinian rings as well. Therefore we do not pose this as one of the meta-conjectures.
A proof of those two meta-conjectures may end up being non-constructive, in the sense that it might fail to give the generators of the corresponding free abelian monoids explicitly. We consider the search for a constructive proof in some special cases to be worthwhile.
{}
\end{document} | arXiv |
October 2011 , Volume 16 , Issue 3
Navier--Stokes equations on the $\beta$-plane
Mustafa A. H. Al-Jaboori and D. Wirosoetisno
We show that, given a sufficiently regular forcing, the solution of the two-dimensional Navier--Stokes equations on the periodic $\beta$-plane (i.e. with the Coriolis force varying as $f_0+\beta y$) will become nearly zonal: with the vorticity $\omega(x,y,t)=\bar\omega(y,t)+\tilde\omega(x,y,t),$ one has $|\tilde\omega|_{H^s}^2 \le \beta^{-1} M_s(\cdots)$ as $t\to\infty$. We use this show that, for sufficiently large $\beta$, the global attractor of this system reduces to a point.
Mustafa A. H. Al-Jaboori, D. Wirosoetisno. Navier--Stokes equations on the $\\beta$-plane. Discrete & Continuous Dynamical Systems - B, 2011, 16(3): 687-701. doi: 10.3934/dcdsb.2011.16.687.
Almost periodic and asymptotically almost periodic solutions of Liénard equations
Tomás Caraballo and David Cheban
The aim of this paper is to study the almost periodic and asymptotically almost periodic solutions on $(0,+\infty)$ of the Liénard equation
$ x''+f(x)x'+g(x)=F(t), $
where $F: T\to R$ ($ T= R_+$ or $R$) is an almost periodic or asymptotically almost periodic function and $g:(a,b)\to R$ is a strictly decreasing function. We study also this problem for the vectorial Liénard equation.
We analyze this problem in the framework of general non-autonomous dynamical systems (cocycles). We apply the general results obtained in our early papers [3, 7] to prove the existence of almost periodic (almost automorphic, recurrent, pseudo recurrent) and asymptotically almost periodic (asymptotically almost automorphic, asymptotically recurrent, asymptotically pseudo recurrent) solutions of Liénard equations (both scalar and vectorial).
Tom\u00E1s Caraballo, David Cheban. Almost periodicand asymptotically almost periodic solutions of Li\u00E9nard equations. Discrete & Continuous Dynamical Systems - B, 2011, 16(3): 703-717. doi: 10.3934/dcdsb.2011.16.703.
Dynamical behavior of a ratio dependent predator-prey system with distributed delay
Canan Çelik
In this paper, we consider a predator-prey system with distributed time delay where the predator dynamics is logistic with the carrying capacity proportional to prey population. In [1] and [2], we studied the impact of the discrete time delay on the stability of the model, however in this paper, we investigate the effect of the distributed delay for the same model. By choosing the delay time $\tau $ as a bifurcation parameter, we show that Hopf bifurcation can occur as the delay time $\tau $ passes some critical values. Using normal form theory and central manifold argument, we establish the direction and the stability of Hopf bifurcation. Some numerical simulations for justifying the theoretical analysis are also presented.
Canan \u00C7elik. Dynamical behavior of a ratio dependent predator-prey system with distributed delay. Discrete & Continuous Dynamical Systems - B, 2011, 16(3): 719-738. doi: 10.3934/dcdsb.2011.16.719.
Existence of radial stationary solutions for a system in combustion theory
Jérôme Coville and Juan Dávila
In this paper, we construct radially symmetric solutions of a nonlinear non-cooperative elliptic system derived from a model for flame balls with radiation losses. This model is based on a one step kinetic reaction and our system is obtained by approximating the standard Arrehnius law by an ignition nonlinearity, and by simplifying the term that models radiation. We prove the existence of 2 solutions using degree theory.
J\u00E9r\u00F4me Coville, Juan D\u00E1vila. Existence of radial stationary solutions for a system in combustion theory. Discrete & Continuous Dynamical Systems - B, 2011, 16(3): 739-766. doi: 10.3934/dcdsb.2011.16.739.
Shape minimization of the dissipated energy in dyadic trees
Xavier Dubois de La Sablonière, Benjamin Mauroy and Yannick Privat
2011, 16(3): 767-799 doi: 10.3934/dcdsb.2011.16.767 +[Abstract](1607) +[PDF](1298.0KB)
In this paper, we study the role of boundary conditions on the optimal shape of a dyadic tree in which flows a Newtonian fluid. Our optimization problem consists in finding the shape of the tree that minimizes the viscous energy dissipated by the fluid with a constrained volume, under the assumption that the total flow of the fluid is conserved throughout the structure. These hypotheses model situations where a fluid is transported from a source towards a 3D domain into which the transport network also spans. Such situations could be encountered in organs like for instance the lungs and the vascular networks.
Two fluid regimes are studied: (i) low flow regime (Poiseuille) in trees with an arbitrary number of generations using a matricial approach and (ii) non linear flow regime (Navier-Stokes, moderate regime with a Reynolds number $100$) in trees of two generations using shape derivatives in an augmented Lagrangian algorithm coupled with a 2D/3D finite elements code to solve Navier-Stokes equations. It relies on the study of a finite dimensional optimization problem in the case (i) and on a standard shape optimization problem in the case (ii). We show that the behaviours of both regimes are very similar and that the optimal shape is highly dependent on the boundary conditions of the fluid applied at the leaves of the tree.
Xavier Dubois de La Sabloni\u00E8re, Benjamin Mauroy, Yannick Privat. Shape minimization of the dissipated energy in dyadic trees. Discrete & Continuous Dynamical Systems - B, 2011, 16(3): 767-799. doi: 10.3934/dcdsb.2011.16.767.
Thermalization time in a model of neutron star
Bernard Ducomet and Šárka Nečasová
We consider an initial boundary value problem for the equation describing heat conduction in a spherical model of neutron star considered by Lattimer et al. We estimate the asymptotic decay of the solution, which provides a plausible estimate for a "thermalization time" for the system.
Bernard Ducomet, \u0160\u00E1rka Ne\u010Dasov\u00E1. Thermalization time in a model of neutron star. Discrete & Continuous Dynamical Systems - B, 2011, 16(3): 801-818. doi: 10.3934/dcdsb.2011.16.801.
Front propagation in diffusion-aggregation models with bi-stable reaction
Mikhail Kuzmin and Stefano Ruggerini
In this paper, necessary and sufficient conditions are given for the existence of travelling wave solutions of the reaction-diffusion-aggregation equation
$v_\tau=(D(v)v_x)_{x}+f(v), $
where the diffusivity $D$ changes sign twice in the interval $(0,1)$ (from positive to negative and again to positive) and the reaction $f$ is bi-stable. We show that classical travelling waves with decreasing profile do exist for a single admissible value of their speed of propagation which can be either positive or negative, according to the behavior of $f$ and $D$. An example is given, illustrating the employed techniques. The results are then generalized to a diffusivity $D$ with $2n$ sign changes.
Mikhail Kuzmin, Stefano Ruggerini. Front propagation in diffusion-aggregation models with bi-stable reaction. Discrete & Continuous Dynamical Systems - B, 2011, 16(3): 819-833. doi: 10.3934/dcdsb.2011.16.819.
Robustness of signaling gradient in drosophila wing imaginal disc
Jinzhi Lei, Frederic Y. M. Wan, Arthur D. Lander and Qing Nie
Quasi-stable gradients of signaling protein molecules (known as morphogens or ligands) bound to cell receptors are known to be responsible for differential cell signaling and gene expressions. From these follow different stable cell fates and visually patterned tissues in biological development. Recent studies have shown that the relevant basic biological processes yield gradients that are sensitive to small changes in system characteristics (such as expression level of morphogens or receptors) or environmental conditions (such as temperature changes). Additional biological activities must play an important role in the high level of robustness observed in embryonic patterning for example. It is natural to attribute observed robustness to various type of feedback control mechanisms. However, our own simulation studies have shown that feedback control is neither necessary nor sufficient for robustness of the morphogen decapentaplegic (Dpp) gradient in wing imaginal disc of Drosophilas. Furthermore, robustness can be achieved by substantial binding of the signaling morphogen Dpp with nonsignaling cell surface bound molecules (such as heparan sulfate proteoglygans) and degrading the resulting complexes at a sufficiently rapid rate. The present work provides a theoretical basis for the results of our numerical simulation studies.
Jinzhi Lei, Frederic Y. M. Wan, Arthur D. Lander, Qing Nie. Robustness of signaling gradient in drosophila wing imaginal disc. Discrete & Continuous Dynamical Systems - B, 2011, 16(3): 835-866. doi: 10.3934/dcdsb.2011.16.835.
Long time behavior of some epidemic models
Fang Li and Nung Kwan Yip
In this paper, we prove two results concerning the long time behavior of two systems of reaction diffusion equations motivated by the S-I-R model in epidemic modeling. The results generalize and simplify previous approaches. In particular, we consider the presence of directed diffusions between the two species. The new system contains an ill-posed region for arbitrary parameters. Our result is established under the assumption of small initial data.
Fang Li, Nung Kwan Yip. Long time behavior of some epidemic models. Discrete & Continuous Dynamical Systems - B, 2011, 16(3): 867-881. doi: 10.3934/dcdsb.2011.16.867.
Phase transitions of a phase field model
Honghu Liu
We consider a phase field model for the mixture of two viscous incompressible uids with the same density. The model leads to a coupled Navier-Stokes/Cahn-Hilliard system. We explore the dynamics of the system near the critical point via a dynamic phase transition theory developed recently by Ma and Wang [7, 8]. Our analysis shows qualitatively the same phase transition result as the purely dissipative Cahn-Hilliard equation, which implies that the hydrodynamics does not play a role in the phase transition process of binary systems. This is different from the sharp interface situation, where numerical studies (see e.g. [3, 6]) suggest quite different behaviors between these two models.
Honghu Liu. Phase transitions of a phase field model. Discrete & Continuous Dynamical Systems - B, 2011, 16(3): 883-894. doi: 10.3934/dcdsb.2011.16.883.
Stability of traveling waves for systems of nonlinear integral recursions in spatial population biology
Judith R. Miller and Huihui Zeng
We use spectral methods to prove a general stability theorem for traveling wave solutions to the systems of integrodifference equations arising in spatial population biology. We show that non-minimum-speed waves are exponentially asymptotically stable to small perturbations in appropriately weighted $L^\infty$ spaces, under assumptions which apply to examples including a Laplace or Gaussian dispersal kernel a monotone (or non-monotone) growth function behaving qualitatively like the Beverton-Holt function (or Ricker function with overcompensation), and a constant probability $p\in [0,1)$ (or $p=0$) of remaining sedentary for a single population; as well as to a system of two populations exhibiting non-cooperation (in particular, Hassell and Comins' model [6]) with $p=0$ and Laplace or Gaussian dispersal kernels which can be different for the two populations.
Judith R. Miller, Huihui Zeng. Stability of traveling waves for systems of nonlinear integral recursions in spatial population biology. Discrete & Continuous Dynamical Systems - B, 2011, 16(3): 895-925. doi: 10.3934/dcdsb.2011.16.895.
The logistic map of matrices
Zenonas Navickas, Rasa Smidtaite, Alfonsas Vainoras and Minvydas Ragulskis
The standard iterative logistic map is extended by replacing the scalar variable by a square matrix of variables. Dynamical properties of such an iterative map are explored in detail when the order of matrices is 2. It is shown that the evolution of the logistic map depends not only on the control parameter but also on the eigenvalues of the matrix of initial conditions. Several computational examples are used to demonstrate the convergence to periodic attractors and the sensitivity of chaotic processes to initials conditions.
Zenonas Navickas, Rasa Smidtaite, Alfonsas Vainoras, Minvydas Ragulskis. The logistic map of matrices. Discrete & Continuous Dynamical Systems - B, 2011, 16(3): 927-944. doi: 10.3934/dcdsb.2011.16.927.
Tikhonov's theorem and quasi-steady state
Lena Noethen and Sebastian Walcher
There exists a systematic approach to asymptotic properties for quasi-steady state phenomena via the classical theory of Tikhonov and Fenichel. This observation allows, on the one hand, to settle convergence issues, which are far from trivial in asymptotic expansions. On the other hand, even if one takes convergence for granted, the approach yields a natural way to compute a reduced system on the slow manifold, with a reduced equation that is frequently simpler than the one obtained by the ad hoc approach. In particular, the reduced system is always rational. The paper includes a discussion of necessary and sufficient conditions for applicability of Tikhonov's and Fenichel's theorems, computational issues and a direct determination of the reduced system. The results are applied to several relevant examples.
Lena Noethen, Sebastian Walcher. Tikhonov\'s theorem andquasi-steady state. Discrete & Continuous Dynamical Systems - B, 2011, 16(3): 945-961. doi: 10.3934/dcdsb.2011.16.945.
The flashing ratchet and unidirectional transport of matter
Dmitry Vorotnikov
We study the flashing ratchet model of a Brownian motor, which consists in cyclical switching between the Fokker-Planck equation with an asymmetric ratchet-like potential and the pure diffusion equation. We show that the motor indeed performs unidirectional transport of mass, for proper parameters of the model, by analyzing the attractor of the problem and the stationary vector of a related Markov chain.
Dmitry Vorotnikov. The flashing ratchet and unidirectional transport of matter. Discrete & Continuous Dynamical Systems - B, 2011, 16(3): 963-971. doi: 10.3934/dcdsb.2011.16.963.
Existence of traveling wavefront for discrete bistable competition model
Chin-Chin Wu
We study traveling wavefront solutions for a two-component competition system on a one-dimensional lattice. We combine the monotonic iteration method with a truncation to obtain the existence of the traveling wavefront solution.
Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete & Continuous Dynamical Systems - B, 2011, 16(3): 973-984. doi: 10.3934/dcdsb.2011.16.973.
Attractors for autonomous and nonautonomous 3D Navier-Stokes-Voight equations
Gaocheng Yue and Chengkui Zhong
2011, 16(3): 985-1002 doi: 10.3934/dcdsb.2011.16.985 +[Abstract](1895) +[PDF](457.0KB)
In this paper we study the long time behavior of the three dimensional Navier-Stokes-Voight model of viscoelastic incompressible fluid for the autonomous and nonautonomous cases. A useful decomposition method is introduced to overcome the difficulties in proving the asymptotical regularity of the 3D Navier-Stokes-Voight equations. For the autonomous case, we prove the existence of global attractor when the external forcing belongs to $V'.$ For the nonautonomous case, we only assume that $f(x,t)$ is translation bounded instead of translation compact, where $f=Pg$ and $P$ is the Helmholz-Leray orthogonal projection. By means of this useful decomposition methods, we prove the asymptotic regularity of solutions of 3D Navier-Stokes-Voight equations and also obtain the existence of the uniform attractor. Finally, we describe the structure of the uniform attractor and its regularity.
Gaocheng Yue, Chengkui Zhong. Attractors for autonomous and nonautonomous 3D Navier-Stokes-Voight equations. Discrete & Continuous Dynamical Systems - B, 2011, 16(3): 985-1002. doi: 10.3934/dcdsb.2011.16.985.
Influence of neurobiological mechanisms on speeds of traveling wave fronts in mathematical neuroscience
Linghai Zhang, Ping-Shi Wu and Melissa Anne Stoner
2011, 16(3): 1003-1037 doi: 10.3934/dcdsb.2011.16.1003 +[Abstract](1448) +[PDF](1670.2KB)
We study speeds of traveling wave fronts of the following integral differential equation
$ \frac{\partial u}{\partial t}+f(u)\hspace{6cm} $
$=(\alpha-au)\int^{\infty}_0\xi(c)[\int_R K(x-y) H(u(y,t-\frac{1}{c}|x-y|)-\theta)dy]dc $
$ +(\beta-bu)\int^{\infty}_0\eta(\tau)[\int_RW(x-y) H(u(y,t-\tau)-\Theta)dy]d\tau. $
This model equation is motivated by previous models which arise from synaptically coupled neuronal networks. In this equation, $f(u)$ is a smooth function of $u$, usually representing sodium current in the neuronal networks. Typical examples include $f(u)=u$ and $f(u)=u(u-1)(Du-1)$, where $D>1$ is a constant. The transmission speed distribution $\xi$ and the feedback delay distribution $\eta$ are probability density functions. The kernel functions $K$ and $W$ represent synaptic couplings between neurons in the neuronal networks. The function $H$ stands for the Heaviside step function: $H(u-\theta)=0$ for all $u<\theta$, $H(0)=\frac{1}{2}$ and $H(u-\theta)=1$ for all $u>\theta$. Here $H$ represents the gain function. The parameters $a \geq 0$, $b \geq 0$, $ \alpha \geq 0$, $\beta \geq 0$, $\theta > 0$ and $\Theta > 0$ represent biological mechanisms in the neuronal networks.
We will use mathematical analysis to investigate the influence of neurobiological mechanisms on the speeds of the traveling wave fronts. We will derive new estimates for the wave speeds. These results are quite different from the results obtained before, complementing the estimates obtained in many previous papers [11], [14], [15], and [16].
We will also use MATLAB to perform numerical simulations to investigate how the neurobiological mechanisms $a$, $b$, $\alpha$, $\beta$, $\theta$ and $\Theta$ influence the wave speeds.
Linghai Zhang, Ping-Shi Wu, Melissa Anne Stoner. Influence of neurobiological mechanisms on speeds of traveling wave fronts in mathematical neuroscience. Discrete & Continuous Dynamical Systems - B, 2011, 16(3): 1003-1037. doi: 10.3934/dcdsb.2011.16.1003. | CommonCrawl |
January 2020, 25(1): 99-115. doi: 10.3934/dcdsb.2019174
Noémi Nagy , and Péter L. Simon
Institute of Mathematics, Eötvös Loránd University Budapest, Numerical Analysis and Large Networks Research Group, Hungarian Academy of Sciences, Pázmány Péter sétány 1/C, H-1117 Budapest, Hungary
* Corresponding author: Noémi Nagy
Received November 2018 Published July 2019
The global behaviour of the compact pairwise approximation of SIS epidemic propagation on networks is studied. It is shown that the system can be reduced to two equations enabling us to carry out a detailed study of the dynamic properties of the solutions. It is proved that transcritical bifurcation occurs in the system at $ \tau = \tau _c = \frac{\gamma n}{\langle n^{2}\rangle-n} $, where $ \tau $ and $ \gamma $ are infection and recovery rates, respectively, $ n $ is the average degree of the network and $ \langle n^{2}\rangle $ is the second moment of the degree distribution. For subcritical values of $ \tau $ the disease-free steady state is stable, while for supercritical values a unique stable endemic equilibrium appears. We also prove that for subcritical values of $ \tau $ the disease-free steady state is globally stable under certain assumptions on the graph that cover a wide class of networks.
Keywords: SIS epidemic, pairwise approximation, transcritical bifurcation, global stability, network process.
Mathematics Subject Classification: Primary: 34C23, 34D23, 92C42.
Citation: Noémi Nagy, Péter L. Simon. Detailed analytic study of the compact pairwise model for SIS epidemic propagation on networks. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 99-115. doi: 10.3934/dcdsb.2019174
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Figure 1. Case of the globally stable disease-free equilibrium: Time evolution of the expected number of the infected nodes $ [I_1] $, $ [I_2] $, $ [I_3] $ of degree $ n_1 = 2 $, $ n_2 = 3 $, $ n_3 = 4 $ respectively, started with $ 900 $, $ 500 $ randomly chosen infected nodes (i.e. firstly $ 765 $, $ 90 $, $ 45 $ infected nodes of degree 2, 3, 4 respectively (continuous curves), secondly $ 425 $, $ 50 $, $ 25 $ infected nodes of degree $ 2 $, $ 3 $, $ 4 $ respectively (dashed curves)). The parameters are: $ N = 1000 $, $ N_1 = 850 $, $ N_2 = 100 $, $ N_3 = 50 $, $ \gamma = 1 $, $ \tau = 0.5 $, $ \tau_c = 0.7586 $
Figure 2. Case of the globally stable endemic equilibrium: Time evolution of the expected number of the infected nodes $ [I_1] $, $ [I_2] $, $ [I_3] $ of degree $ n_1 = 2 $, $ n_2 = 3 $, $ n_3 = 4 $ respectively, started with $ 900 $, $ 500 $ randomly chosen infected nodes (i.e. firstly $ 765 $, $ 90 $, $ 45 $ infected nodes of degree 2, 3, 4 respectively (continuous curves), secondly $ 425 $, $ 50 $, $ 25 $ infected nodes of degree $ 2 $, $ 3 $, $ 4 $ respectively (dashed curves)). The parameters are: $ N = 1000 $, $ N_1 = 850 $, $ N_2 = 100 $, $ N_3 = 50 $, $ \gamma = 1 $, $ \tau = 1 $, $ \tau_c = 0.7586 $
Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344
Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003
Ziang Long, Penghang Yin, Jack Xin. Global convergence and geometric characterization of slow to fast weight evolution in neural network training for classifying linearly non-separable data. Inverse Problems & Imaging, 2021, 15 (1) : 41-62. doi: 10.3934/ipi.2020077
Yangjian Sun, Changjian Liu. The Poincaré bifurcation of a SD oscillator. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1565-1577. doi: 10.3934/dcdsb.2020173
Qiang Fu, Yanlong Zhang, Yushu Zhu, Ting Li. Network centralities, demographic disparities, and voluntary participation. Mathematical Foundations of Computing, 2020, 3 (4) : 249-262. doi: 10.3934/mfc.2020011
Bo Tan, Qinglong Zhou. Approximation properties of Lüroth expansions. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020389
Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296
Bilal Al Taki, Khawla Msheik, Jacques Sainte-Marie. On the rigid-lid approximation of shallow water Bingham. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 875-905. doi: 10.3934/dcdsb.2020146
P. K. Jha, R. Lipton. Finite element approximation of nonlocal dynamic fracture models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1675-1710. doi: 10.3934/dcdsb.2020178
Simone Fagioli, Emanuela Radici. Opinion formation systems via deterministic particles approximation. Kinetic & Related Models, 2021, 14 (1) : 45-76. doi: 10.3934/krm.2020048
Shipra Singh, Aviv Gibali, Xiaolong Qin. Cooperation in traffic network problems via evolutionary split variational inequalities. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020170
Yicheng Liu, Yipeng Chen, Jun Wu, Xiao Wang. Periodic consensus in network systems with general distributed processing delays. Networks & Heterogeneous Media, 2020 doi: 10.3934/nhm.2021002
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Noémi Nagy Péter L. Simon | CommonCrawl |
Mingarelli identity
In the field of ordinary differential equations, the Mingarelli identity[1] is a theorem that provides criteria for the oscillation and non-oscillation of solutions of some linear differential equations in the real domain. It extends the Picone identity from two to three or more differential equations of the second order.
The identity
Consider the n solutions of the following (uncoupled) system of second order linear differential equations over the t–interval [a, b]:
$(p_{i}(t)x_{i}^{\prime })^{\prime }+q_{i}(t)x_{i}=0,\,\,\,\,\,\,\,\,\,\,x_{i}(a)=1,\,\,x_{i}^{\prime }(a)=R_{i}$ where $i=1,2,\ldots ,n$.
Let $\Delta $ denote the forward difference operator, i.e.
$\Delta x_{i}=x_{i+1}-x_{i}.$
The second order difference operator is found by iterating the first order operator as in
$\Delta ^{2}(x_{i})=\Delta (\Delta x_{i})=x_{i+2}-2x_{i+1}+x_{i},$,
with a similar definition for the higher iterates. Leaving out the independent variable t for convenience, and assuming the xi(t) ≠ 0 on (a, b], there holds the identity,[2]
${\begin{aligned}x_{n-1}^{2}\Delta ^{n-1}(p_{1}r_{1})]_{a}^{b}=&\int _{a}^{b}(x_{n-1}^{\prime })^{2}\Delta ^{n-1}(p_{1})-\int _{a}^{b}x_{n-1}^{2}\Delta ^{n-1}(q_{1})\\&-\sum _{k=0}^{n-1}C(n-1,k)(-1)^{n-k-1}\int _{a}^{b}p_{k+1}W^{2}(x_{k+1},x_{n-1})/x_{k+1}^{2},\end{aligned}}$
where
• $r_{i}=x_{i}^{\prime }/x_{i}$ is the logarithmic derivative,
• $W(x_{i},x_{j})=x_{i}^{\prime }x_{j}-x_{i}x_{j}^{\prime }$, is the Wronskian determinant,
• $C(n-1,k)$ are binomial coefficients.
When n = 2 this equality reduces to the Picone identity.
An application
The above identity leads quickly to the following comparison theorem for three linear differential equations,[3] which extends the classical Sturm–Picone comparison theorem.
Let pi, qi i = 1, 2, 3, be real-valued continuous functions on the interval [a, b] and let
1. $(p_{1}(t)x_{1}^{\prime })^{\prime }+q_{1}(t)x_{1}=0,\,\,\,\,\,\,\,\,\,\,x_{1}(a)=1,\,\,x_{1}^{\prime }(a)=R_{1}$
2. $(p_{2}(t)x_{2}^{\prime })^{\prime }+q_{2}(t)x_{2}=0,\,\,\,\,\,\,\,\,\,\,x_{2}(a)=1,\,\,x_{2}^{\prime }(a)=R_{2}$
3. $(p_{3}(t)x_{3}^{\prime })^{\prime }+q_{3}(t)x_{3}=0,\,\,\,\,\,\,\,\,\,\,x_{3}(a)=1,\,\,x_{3}^{\prime }(a)=R_{3}$
be three homogeneous linear second order differential equations in self-adjoint form, where
• pi(t) > 0 for each i and for all t in [a, b] , and
• the Ri are arbitrary real numbers.
Assume that for all t in [a, b] we have,
$\Delta ^{2}(q_{1})\geq 0$,
$\Delta ^{2}(p_{1})\leq 0$,
$\Delta ^{2}(p_{1}(a)R_{1})\leq 0$.
Then, if x1(t) > 0 on [a, b] and x2(b) = 0, then any solution x3(t) has at least one zero in [a, b].
Notes
1. The locution was coined by Philip Hartman, according to Clark D.N., G. Pecelli & R. Sacksteder (1981)
2. (Mingarelli 1979, p. 223).
3. (Mingarelli 1979, Theorem 2).
References
• Clark D.N.; G. Pecelli & R. Sacksteder (1981). Contributions to Analysis and Geometry. Baltimore, USA: Johns Hopkins University Press. pp. ix+357. ISBN 0-80182-779-5.
• Mingarelli, Angelo B. (1979). "Some extensions of the Sturm–Picone theorem". Comptes Rendus Mathématique. Toronto, Ontario, Canada: The Royal Society of Canada. 1 (4): 223–226.
| Wikipedia |
\begin{document}
\title{Space-time imaging, magnification and time reversal of matter waves}
\author{Brian H.\ Kolner} \email[]{[email protected]}
\affiliation{Electrical and Computer Engineering Department,
University of California,
One Shields Avenue, Davis, CA 95616, USA }
\date{\today}
\begin{abstract} An imaging system is proposed for matter-wave functions that is based on producing a quadratic phase modulation on the wavefunction of a charged particle, analogous to that produced by a space or time lens. The modulation is produced by co-propagating the wavepacket within an extremum of the harmonic vector and scalar potentials associated with a slow-wave electromagnetic structure. By preceding and following this interaction with appropriate dispersion, characteristic of a solution to the time-dependent Schr\"odinger\ equation, a system results that is capable of magnifying ({\it i.e.}, stretching or compressing the space- and time-scales) and time-reversing an arbitrary quantum wavefunction. \end{abstract}
\pacs{03.65.-w, 42.79.-e, 42.25.-p }
\maketitle
As originally proposed by Aharanov and Bohm, the presence of a static potential in the absence of an electric or magnetic field would be detectable owing to its effect on the phase of the wavefunction of a charged particle\cite{Aharonov_Bohm:59}. This idea was soon demonstrated\cite{Chambers:60} and the principle has spawned a large amount of activity over the decades
\cite{Peshkin_Tonomura:89,Hamilton:97,Batelaan_Tonomura:09}. At the heart of the Aharanov-Bohm effect is the notion that the scalar and vector potentials of electromagnetic theory are real, measurable, entities and their effects distinct from the fields to which they are related. However, an essential feature of the mechanism and proof is that the fields and therefore the potentials are static, for if they were not, Faraday's law and the Amp\`ere-Maxwell relation would require the existence of electric and magnetic fields coincident with the potentials which would contaminate the desired effect with forces\cite{Boyer:73}.
Time-varying potentials, on the other hand, have seldom been considered in terms of their influence on wavefunction phase\cite{Lee:92}. The purpose of this Letter is to suggest a mechanism whereby a quadratically-varying potential (scalar and/or vector) can produce an accumulated quadratic phase on the wavefunction of a charged particle in a manner consistent with the action of a lens. By preceding and following this interaction with the normal dispersion inherent in wavefunction propagation, an {\sl imaging system for matter waves} might be realized that would produce magnified and time-reversed replicas of the original wavefunction (Fig.\ \ref{Matter_wave_imaging_figure}). Note that this mechanism is distinctly different from, and more general than, systems that image the position of charged particles in space\cite{Peshkin_Tonomura:89,Hamilton:97,Batelaan_Tonomura:09} or map their evolution by time of flight\cite{Szriftgiser:96}.
\begin{figure}
\caption{Space-time duality between spatial and matter-wave imaging. (a) Conventional spatial imaging system. Input and output diffraction produce quadratic phase filtering in Fourier spectra of the transverse coordinates. Lens produces quadratic phase modulation directly on transverse coordinates. (b) Imaging system for quantum wavefunctions. Input and output dispersion resulting from free-space propagation distances $L_1$ and $L_2$ correspond to the object and image distances in the spatial imaging system. An electromagnetic slow-wave structure provides lens action by producing a quadratic phase on the dispersed wavefunction due to the traveling-wave interaction with the scalar and vector potentials. (Phase relationship shown for a positive charge). The output wavefunction is a rescaled and time-reversed version of the input with magnification $M=-L_2/L_1$.}
\label{Matter_wave_imaging_figure}
\end{figure}
The key component in the proposed system is an electromagnetic slow-wave structure with a longitudinal traveling-wave electric field that co-propagates with the wavepacket under consideration. If the wavepacket coincides with a zero-crossing of the field (Fig.\ \ref{Matter_wave_imaging_figure}b) it is at an extremum of the potential(s) and, as we will show, it will acquire a predominantly quadratic phase shift essential for lens action, with minimal force imparted by the field.
Central to the three seemingly disparate phenomena of Fresnel diffraction, narrowband dispersion and the quantum-mechanical description of free-particle propagation is the complex diffusion equation, which in general form is given by
\begin{equation}
\frac{\partial\psi}{\partial t} = i \alpha \frac{\partial^2 \psi}{\partial z^2} ,
\label{Sch_Eq}
\end{equation}
\noindent where the diffusivity term $\alpha$ depends on the particular problem at hand. One can consider the independent variable $t$ as guiding the evolution of the wavefunction $\psi$ while $z$ maps its profile. In quantum mechanics the diffusivity $\alpha=\hbar/2m$ while in diffraction $\alpha=1/2k$ and in optical dispersion $\alpha=(1/2)d^2\beta/d\omega^2$. (Here we use the symbols common to diffraction and dispersion analyses; $k=2\pi/\lambda$ is the wavenumber and $\beta=\omega n(\omega)/c$ is the phase constant. Both are phase shifts per-unit-length.)
Based on the similarity of their governing equations, Fresnel diffraction and narrowband dispersion have been successfully united in the concept of temporal imaging which, as its name suggests, is a system for stretching or compressing electromagnetic waveforms of carrier-envelope form while maintaining the integrity of the envelope profile \cite{Tournois:64,Caputi:65,Tournois:68,Akhmanov:69,Kolner:94c}.
A feature common to both diffraction and dispersion is the quadratic phase that is introduced into the frequency spectrum of the envelope, either of the cross-sectional profile of a beam in diffraction or the longitudinal pulse profile in dispersion. Although quadratic in frequency, this effect
also acquires strength linearly with the evolution variable. The same must be true, of course, for the Schr\"odinger\ equation.
To realize imaging in both space and time, it is also necessary to produce quadratic phase in the real-space domain of the profile variable. Thus a space lens, which produces a quadratic phase transformation in the transverse (profile) variable has its counterpart in a time lens which produces quadratic phase in the local time coordinate.\cite{Kolner:11a} To obtain corresponding lens action for the wavefunction of a particle, we look to the governing p.d.e.\ \eqref{Sch_Eq} and recognize that a quadratic phase imparted to the $z$-coordinate would suffice. As we will see, this can be accomplished by interaction between the particle and an electromagnetic potential, either vector, scalar or both.
The general solution to \eqref{Sch_Eq} as an initial-value problem in one dimension, assuming a carrier-envelope type of wavepacket and a relatively narrow momentum spectrum centered about $k_0$, can be written for the quantum-mechanical case as
\begin{multline}
\psi(z,t) = e^{i(k_0 z -\omega_0 t)} \int_{-\infty}^\infty \psi_0 (k ,0) \\
\times \exp \left[ -i \left( k v_g
+ k^2 \dfrac{\hbar}{2m} \right) t \right] e^{i k z} \, \dfrac{dk}{2\pi},
\label{General_solution}
\end{multline}
\noindent where $\psi_0 (k,0)$ is the initial spectrum of the wavefunction $\psi(z,0)$ shifted to baseband ({\it i.e.\ }$\psi_0 (k,0)=\psi(k+k_0, 0)$) and $v_g$ is the group velocity.
It will be especially useful in the context of this exposition to convert the solution \eqref{General_solution} to a traveling-wave coordinate system moving at the group velocity of the wavepacket; $ \xi \equiv z-v_g t$ and $\tau\equiv t$. Introducing these variables transforms \eqref{General_solution} into
\begin{multline}
\psi(\xi,\tau) = e^{i(k_0 \xi + \omega_0 \tau)} \int_{-\infty}^\infty \psi_0 (k ,0) \\[1ex]
\times \exp \left( -i \dfrac{\hbar \tau}{2m}k^2 \right) e^{i k\xi} \, \dfrac{dk}{2\pi},
\label{General_solution_TW_coordinates} \end{multline}
\noindent which represents a stationary envelope in the $(\xi,\tau)$ reference frame superimposed upon a backward-propagating carrier with phase velocity $\omega_0/k_0$. This interpretation can also be seen from the nature of the dispersion relation for \eqref{Sch_Eq}. For a spectrum centered at $k_0$,
\begin{equation*}
\omega(k)= \dfrac{\hbar k^2}{2m}, \>\>
v_g = \dfrac{d\omega}{dk}\biggr|_{k=k_0} \hskip -2ex = \dfrac{\hbar k_0}{m}, \>
\quad v_0=\dfrac{\omega_0}{k_0} = \dfrac{\hbar k_0}{2m},
\end{equation*}
\noindent and thus $v_g/v_0 = 2$. The quadratic phase term in \eqref{General_solution_TW_coordinates} will be seen to be an essential feature of the space-time imaging process.
To produce lens action on the quantum-mechanical wavefunction, we must generate a quadratic phase over the dispersed envelope. One mechanism that will accomplish this is interaction with an electromagnetic potential. Consider a slow-wave structure designed to interact with charged particles which has an electric field component along the axis of particle motion. Such structures have long been studied and developed for use in electron accelerators and vacuum tubes\cite{Hutter:60, Bevensee:64}.
A transverse magnetic (TM) field mode in a slow-wave guide can have an electric field on axis that is directed exclusively along the axis of the guide. Assuming a monochromatic wave propagating in such a slow-wave structure we can write the longitudinal component of the vector potential and the scalar potential as traveling waves
\begin{equation}
A_z(z,t) = A_0 e^{i(k_mz-\omega_m t)}, \>\>
\Phi(z,t) = \Phi_0 e^{i(k_m z-\omega_m t)}
\label{Traveling_wave_potentials_1} \end{equation}
where the subscripts ``$m$'' on the wavenumber and angular frequency indicate that these are associated with a modulating field.
\noindent The two peak potentials can be related using the Lorentz gauge
\begin{equation}
\nabla\cdot \mathbf{A} = - \dfrac{1}{c^2} \dfrac{\partial \Phi}{\partial t} ,
\hskip 5ex \dfrac{k_m c^2}{\omega_m} A_0 = \Phi_0
\end{equation}
\noindent so that the potentials \eqref{Traveling_wave_potentials_1} become
\begin{align}
A_z(z,t) & = A_0 e^{i(k_m z-\omega_m t)}, \label{Traveling_wave_potential_A}\\
\Phi(z,t) & = \dfrac{k_m c^2}{\omega_m} A_0 e^{i(k_m z-\omega_m t)}
\label{Traveling_wave_potential_phi} \end{align}
From a practical standpoint, guided wave structures are generally analyzed in terms of fields rather than potentials and thus it will be useful to relate the peak potential $A_0$ to the peak electric field from the defining relation (using real-valued functions)
\begin{equation*}
\mathbf{E} (z,t) = -\nabla \Phi (z,t) - \dfrac{\partial \mathbf{A}}{\partial t}
= E_0 \sin (k_m z-\omega_m t) \> \mbox{\boldmath${\hat{\mathrm{z}}}$} \notag
\end{equation*}
\noindent and thus
\begin{equation}
A_0 = \dfrac {E_0}{\omega_m \left(c^2/v_p^2 -1 \right) }
\label{E_0_Phi_0} \end{equation}
where $v_p=\omega_m/k_m$ is the phase velocity of the fields and potentials within the slow-wave structure.
Now, assume a charged particle of mass $m$ moving at the group velocity $v_g=\hbar k_0/m$ co-propagates with the electromagnetic field in the slow-wave structure and, furthermore, assume that it is synchronized with a peak of the potentials (Fig.\ \ref{Matter_wave_imaging_figure} ) where the variation is essentially quadratic. Then, in a manner similar to the Aharonov-Bohm effect for stationary fields, the wavefunction will accumulate an additional phase due to the scalar (electric) and vector (magnetic) potentials given by
\begin{equation*}
\Delta\phi_{\lower2pt\hbox{$\scriptstyle E$}} = -\dfrac{q}{\hbar}\int \Phi \, dt, \quad
\Delta\phi_{\lower1pt\hbox{$\scriptstyle M$}} = \dfrac{q}{\hbar}\int \mathbf{A}\cdot d\mathbf{s}
\end{equation*}
The combined effect of interacting with both potentials can be written as one integral over the time coordinate using $z^\prime = v_g t^\prime$. Assuming an interaction length $L$ and traveling-wave potentials given by \eqref{Traveling_wave_potential_A} and \eqref{Traveling_wave_potential_phi} the accumulated phase shift in the absence of dispersion becomes, using real-valued functions,
\begin{multline}
\Gamma(z,L/v_g) = \dfrac{q A_0}{\hbar} \left( v_g- \dfrac{c^2}{v_p} \right) \\
\times \int_0^{L/v_g} \cos (k_m z - \omega_m t + \theta)\, dt
\label{Accumulated_phase1} \end{multline}
\noindent where $\theta$ is an initial phase offset between the potentials and the wavepacket and determines whether the matter-wave lens has a positive or negative focal length.
Since the wavepacket is co-propagating with the modulating potentials, it will be useful to transform this into a traveling-wave coordinate system moving with the group velocity of the wavepacket as was done for the dispersion problem. Integral \eqref{Accumulated_phase1} is readily evaluated,
\begin{multline}
\Gamma(\xi,L/v_g) = \dfrac{q A_0 L}{\hbar} \left(1- \dfrac{c^2}{v_p v_g} \right) \\[1ex]
\times \dfrac{\sin \Delta\phi/2}{\Delta\phi/2}
\cos\left( k_m \xi + \Delta\phi/2 + \theta \right)
\label{Accumulated_phase3} \end{multline}
where
\begin{equation}
\Delta\phi \equiv \omega_m L \left(\dfrac{1}{v_p} - \dfrac{1}{v_g} \right)
\end{equation}
\noindent is the phase slip produced by the walkoff between the wavepacket and the modulating potentials. We now substitute \eqref{E_0_Phi_0} for the peak vector potential in terms of the electric field,
\begin{multline}
\Gamma(\xi,L/v_g) = - \dfrac{q E_0 L}{\hbar \omega_m} \dfrac{\left(c^2/v_p v_g-1 \right)}
{\left(c^2/v_p^2-1 \right)} \\
\times \dfrac{\sin \Delta\phi/2}{\Delta\phi/2}
\cos\left( k_m \xi + \Delta\phi/2 + \theta \right)
\label{Accumulated_phase4} \end{multline}
\noindent Note that for the case of perfect velocity matching ({\it i.e.}, $v_p=v_g$; $\Delta\phi=0$), \eqref{Accumulated_phase4} simplifies considerably,
\begin{align}
\Gamma(\xi,L/v_g) & = - \dfrac{q E_0 L}{\hbar \omega_m} \cos\left( k_m \xi +\theta \right) \\
& = - \Gamma_0 \cos\left( k_m \xi + \theta \right)
\label{Accumulated_phase_ideal} \end{align}
where
\begin{equation}
\Gamma_0\equiv q E_0 L/\hbar\omega_m
\end{equation}
is the peak phase deviation. The initial phase offset $\theta$ is chosen according to the sign of the charge $q$ in order that $\Gamma(\xi,L/v_g)>0$ in the vicinity of $\xi=0$, thus ensuring a positive focal length (see below),
\begin{equation}
\theta =
\begin{cases} \pi,& q =+|q|, \\
0 ,& q =-|q|.
\end{cases}
\label{Phase_theta} \end{equation}
The process of quadratic phase modulation by co-propagating the wavefunction with a sinusoidal potential is but a specific example of any general modulation scheme which may result in a quadratic term. As with space and time lenses, it will be very useful to define an equivalent focal length and focal-length-to-aperture ratio, or $f^\#$ of a matter-wave lens\cite{Kolner:94d}. We can draw an analogy between the focal length of a space lens \cite{Goodman:68} and our matter-wave lens by comparing the phase variation with respect to the profile variables,
\@fleqntrue\@mathmargin0pt \begin{align}
\text{\small SPACE:} && \exp\bigl[-i\phi_{\lower 0.4ex\hbox{\begin{tiny}S\end{tiny}}} (x,y)\bigr]
& = \exp \biggl[ -i \dfrac{k}{2 f_{\lower 0.4ex\hbox{\begin{scriptsize}S\end{scriptsize}}}}(x^2 + y^2) \biggr]\notag\\
\text{\small MATTER:} && \exp\bigl[-i\phi_{\lower 0.4ex\hbox{\begin{tiny}M\end{tiny}}} (\xi)\bigr]
& = \exp \biggl[ -i \dfrac{k_0}{2 f_{\lower 0.4ex\hbox{\begin{scriptsize}M\end{scriptsize}}}} \xi^2 \biggr] \label{Matter_wave_phase}
\end{align} \@fleqnfalse
To assign the concept of focal length to an arbitrary phase modulation process we expand the phase induced by that process
in a Taylor series and equate the coefficient on the second-order term to the form in \eqref{Matter_wave_phase}. For the matter-wave lens phase described by \eqref{Accumulated_phase_ideal} we find
\begin{equation}
f_{\lower 0.4ex\hbox{\begin{scriptsize}M\end{scriptsize}}} \equiv \dfrac{k_{\lower 0.2ex\hbox{\kern 0.02em$\scriptstyle 0$}}}{\Gamma_{\lower 0.2ex\hbox{\kern 0.02em$\scriptstyle 0$}} k_m^2}
\label{Focal_length_def} \end{equation}
The assumption of an accumulated phase that is generally quadratic will be valid if the extent of the wavefunction is confined predominantly to a maximum of the propagating cosine potential, perhaps with a time gate or shutter. We may consider this region as defining an effective aperture for the matter-wave lens and this suggests the concept of $f^\#$. A reasonable estimate is approximately $1/2\pi$ of the period. In space, in the traveling-wave coordinate system, this corresponds to a window $\Delta\xi=\lambda_m/2\pi=1/k_m$. Let us then define
\begin{equation}
f_{\lower 0.4ex\hbox{\begin{scriptsize}M\end{scriptsize}}}^\# \equiv \dfrac{f_{\lower 0.4ex\hbox{\begin{scriptsize}M\end{scriptsize}}}}{\Delta\xi} = \dfrac{f_{\lower 0.4ex\hbox{\begin{scriptsize}M\end{scriptsize}}}}{1/k_m} = \dfrac{k_0}{\Gamma_0 k_m}
\label{F_number_matter_1} \end{equation}
If we now equate the classical to the quantum-mechanical momentum, $k_0=mv_g/\hbar$, and substitute into \eqref{F_number_matter_1} along with the definition for $\Gamma_0$ we find
\begin{equation}
f_{\lower 0.4ex\hbox{\begin{scriptsize}M\end{scriptsize}}}^\# = \dfrac{m v_g v_p}{qE_0L} = \dfrac{mc^2}{n^2 qE_0L}
\end{equation}
where, owing to the assumption of perfect velocity matching, $v_g v_p=(c/n)^2$ and $n$ is the slowing factor for the electromagnetic slow wave. We see that the numerator is the rest mass energy of the particle, while in the denominator, $qE_0L$ would be the kinetic energy acquired by the particle in a uniform field $E_0$ accelerated from rest through a distance $L$. However, in the traveling-wave frame of the particle, the electric field is in phase quadrature with the potentials, going through a zero-crossing where the potentials are maximized, and thus the particle feels no net force if its center of mass is aligned with the zero-field point.
As an example, let's assume an electron is propagating in a slow-wave structure where $n=10$ (corresponding to a kinetic energy of 3 keV) and the interaction length $L=1$ cm. We find $ f_{\lower 0.4ex\hbox{\begin{scriptsize}M\end{scriptsize}}}^\# = 5 \times 10^5/E_0 = 5$ for an electric field of $E_{\lower 0.2ex\hbox{\kern 0.02em$\scriptstyle 0$}}=10^5$ V/m (1 kV/cm).
Imaging of matter waves is accomplished by the concatenation of dispersion, lens action and more dispersion (Fig.\ \ref{Matter_wave_imaging_figure}). Mathematically we see from \eqref{General_solution_TW_coordinates} and \eqref{Matter_wave_phase} that this corresponds to quadratic phase filtering in wavenumber space for dispersion and quadratic phase modulation in coordinate space for the matter-wave lens. To simplify expressing this three-step process, we introduce the following functions
\@fleqntrue\@mathmargin0pt \begin{align}
\text{\footnotesize INPUT DISPERSION:}
&& \mathscr{G}_1 (k,\tau_1) & = \exp\left[ -i a k^2 \right] \\
\text{\footnotesize MATTER-WAVE LENS:}
&& H(\xi) & = \exp \left[-i \xi^2 / 4c \right] \\
\text{\footnotesize OUTPUT DISPERSION:}
&& \mathscr{G}_2 (k,\tau_2) & = \exp\left[ -i b k^2 \right]
\end{align} \@fleqnfalse
\@fleqntrue\@mathmargin0pt \begin{align}
\text{where:} \qquad a =\dfrac{\hbar\tau_1}{2m}, \qquad b =\dfrac{\hbar\tau_2}{2m},
\qquad c =\dfrac{f_{\lower 0.4ex\hbox{\begin{scriptsize}M\end{scriptsize}}}}{2k_{\lower 0.2ex\hbox{\kern 0.02em$\scriptstyle 0$}}},
\label{ABC} \end{align} \@fleqnfalse
$\tau_{1,2}$ are the propagation times in the input and output dispersive regions, respectively, and $f_{\lower 0.4ex\hbox{\begin{scriptsize}M\end{scriptsize}}}$ is given by \eqref{Focal_length_def}.
Carrying out the forward and inverse Fourier transforms (indicated by $\mathscr{F}$ and $\mathscr{F}^{-1}$, respectively) of the three-step process of imaging gives the wavefunction following the second (output) dispersive region,
\begin{multline}
\psi(\xi, \tau_2) = e^{i[k_0 \xi + \omega_0(\tau_{\lower 0.3ex\hbox{$\scriptscriptstyle 1$}}+\tau_{\lower 0.3ex\hbox{$\scriptscriptstyle 2$}} + \tau_{\lower 0.3ex\hbox{$\scriptscriptstyle l$}}) + \Gamma_0 ]} \\
\times
\mathscr{F}^{-1} \biggl\lbrace \mathscr{F} \biggl\lbrace \mathscr{F}^{-1} \biggl\lbrace
\psi_0 (k,0) \mathscr{G}_1 (k,\tau_1) \biggr\rbrace \\
\times H(\xi) \biggr\rbrace
\mathscr{G}_2 (k,\tau_2) \biggr\rbrace
\label{Output_wavefunction_1} \end{multline}
where $\tau_l$ is the propagation time through the matter-wave lens. Neglecting multiplicative phase and amplitude constants, the final inverse Fourier transform contains the essential features of the imaging problem and expresses the wavefunction envelope in the local traveling-wave system,
\begin{multline}
\psi(\xi, \tau_2) \propto \int_{-\infty}^\infty \psi_0 (k,0) \\
\exp\biggl[ i \biggl( \dfrac{1}{1/c -1/b} -a \biggr) {k}^2
+ i \dfrac{c\,\xi}{c-b} k \biggr] \, dk .
\label{Output_wavefunction_3} \end{multline}
\noindent This integral represents the initial envelope spectrum $\psi_0 (k,0)$ multiplied by a quadratic spectral phase and then inverse Fourier-transformed to a re-scaled space-time coordinate. In order for this wavefunction to be a replica, or ``image'', of the input waveform, we must eliminate the quadratic phase. Thus we set $1/c -1/b = 1/a$ and find
\begin{equation}
\dfrac{m}{\hbar\tau_1} + \dfrac{m}{\hbar\tau_2} = \dfrac{k_{\lower 0.2ex\hbox{\kern 0.02em$\scriptstyle 0$}}}{f_{\lower 0.4ex\hbox{\begin{scriptsize}M\end{scriptsize}}}}
\end{equation}
or, setting $p=\hbar k_0=mv_g$ and noting that the propagation distances in the input and output dispersive regions are $L_1 = v_g \tau_1$ and $L_2 = v_g \tau_2$, this becomes
\begin{equation}
\dfrac{1}{L_1} + \dfrac{1}{L_2} = \dfrac{1}{f_{\lower 0.4ex\hbox{\begin{scriptsize}M\end{scriptsize}}}}
\label{Imaging_condition} \end{equation}
which is the {\sl imaging condition}, familiar from classical optics.
The re-scaled space coordinate in the Fourier transform kernel takes on an equally significant and familiar form. Substituting from \eqref{ABC} and the imaging condition yields
\begin{equation}
\dfrac{c-b}{c} = -\dfrac{b}{a} = -\dfrac{\tau_2}{\tau_1} = -\dfrac{L_2}{L_1} \equiv M
\label{Def_magnification} \end{equation}
which defines the {\sl magnification.} Thus, when the imaging condition is satisfied, the relationship between the input and output wavefunction envelopes is
\begin{equation}
\psi(\xi, \tau_2) \propto \psi(\xi/M, 0)
\end{equation}
Notice that the magnification $M$ \eqref{Def_magnification} takes on a negative value and therefore produces a space- and time-reversed image of the wavefunction. This implies no violation of causality as it is merely a consequence of the quadratic phase modulation and
frequency-domain filtering of the wavefunction spectrum. Indeed, there is no mechanism producing output prior to input.
To this point we have not included the concept of resolution, or how fine the structure of a wavefunction can be resolved. As in the case of conventional optical imaging systems, limitations will arise due to the aperturing effects at the lens. We hinted at this by assuming that the dispersed wavefunction entered the matter-wave lens and only interacted with $\lambda_m/2\pi$ of the guided electromagnetic wave in the slow-wave guide. Let's assume we have a mechanism that creates a shutter admitting only a portion of the dispersed wavepacket over this duration. In concert with similar analyses in spatial and temporal imaging systems, the impulse response of the system will be given by the Fourier transform of this aperture function. Then, to a good approximation, the resolution referred to the input space scale for large magnifications can be shown to be\cite{Kolner:94c}
\begin{equation}
\delta \xi_{in} \approx \lambda_0 f_{\lower 0.4ex\hbox{\begin{scriptsize}M\end{scriptsize}}}^{\#}
\end{equation}
where $\lambda_0=2\pi/k_0$ is the de Broglie wavelength.
An interesting effect with time apertures that will have an impact on a stream of particles in certain situations is the matter-wave equivalent of edge diffraction due to a semi-infinite opaque screen\cite{Goodman:68}. This was first pointed out by Moshinsky in a seminal paper in 1952\cite{Moshinsky:52} which he described as ``diffraction in time'' and is discussed, along with many other interesting transient phenomena, in the comprehensive review by del Campo, {\sl et al.}\cite{Del_Campo:09} The effect of temporal slits (equivalent to the shutter discussed here) has also been studied experimentally\cite{Szriftgiser:96} and a full three-dimensional analysis of diffraction and dispersion from a shutter has been presented by Beau and Dorlas\cite{Beau:15}. It should be noted, however, that when the imaging condition \eqref{Imaging_condition} is satisfied, the quadratic phase term in \eqref{Output_wavefunction_3} is eliminated and the aperture effect is changed from Fresnel to Fraunhofer diffraction.
The assumption that no dispersion occurs within the slow-wave structure is not necessarily valid in all cases. There will be a trade-off between interaction length $L$ and peak potential $A_0$ (or peak field amplitude $E_0$) to maintain a low $f$-number and minimize dispersion. This argues for high fields and short interaction lengths but if the latter cannot be attained, then incorporating dispersion within the slow-wave structure can be accommodated with the definitions of the net input and output dispersions.
Placement of the wavepacket on a cusp, or extremum, of the potential also places it at the zero-crossing of the electric field (Fig.\ \ref{Matter_wave_imaging_figure}). If the group velocity of the particle is not perfectly matched to the phase velocity of the potential wave then it will begin to move and experience a nonzero force $F_z = qE_z$. However, the sign of the the electric field is such as to produce a {\sl restoring force} on the charged particle, regardless of the sign of the charge as long as \eqref{Phase_theta} is satisfied.
In summary I have proposed a system for space-time transformation and imaging of matter-wave functions in a fashion entirely analogous to spatial or temporal imaging. It relies on the introduction of a quadratic phase modulation on the envelope of the wavefunction of a charged particle following and preceding regions of normal dispersive spreading. Recent work in ultra-compact electron-optical accelerators \cite{Peralta:13} and coherent control of electrons \cite{Kfir:20,Wang:20} may make ideal candidates for testing some of these ideas.
\begin{acknowledgments} This work was funded in part by the National Science Foundation under grants ECS-9110678, ECS-9521604 and ECS-9900414, and also by the David and Lucile Packard Foundation. \end{acknowledgments}
\vskip 2ex \noindent {\sffamily \bfseries\small DATA AVAILABILITY STATEMENT}
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
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\end{document} | arXiv |
Liar's Paradox & Tarksi — Does Tarski's theorem truly resolve Liar's Paradox (in Peano Arithmetic [and possibly outside of it])?
I was looking in the literature, and in my textbook, it was concluding Tarski's theorem after showing: $$\mathbf{PA} \vdash \varphi \; \longleftrightarrow \; \lnot \text{truth}(\ulcorner{\varphi}\urcorner)$$ Then it tells that in order to find a model $\mathbb{N}$ such that it models $\varphi$, we would need to resolve Liar's paradox which is a contradiction.
More formally, Tarski's theorem states that: \begin{gather*} \text{There is no}~\mathcal{L}_{PA}\text{-formula truth(}x)~\text{with one free variable}~x~\text{such that}~\mathbb{N} \models \text{truth(}\#\varphi) \leftrightarrow \varphi. \end{gather*} By the Diagonalisation Lemma there exists an $\mathcal{L}_{PA}$-sentence $\sigma$ such that: $$\mathbf{PA} \vdash \sigma \; \longleftrightarrow \; \lnot \text{truth}(\ulcorner{\sigma}\urcorner)$$ Also, then: \begin{equation}\mathbb{N} \models \text{truth}(\#\sigma) \; \iff \; \mathbb{N} \models \sigma \; \iff \; \mathbb{N} \models \lnot \text{truth}(\#\sigma) \tag{$\bot$}\end{equation} This would disallow for a statement such as: "This sentence is false" from existing as a truth in $\mathbf{PA}$. My question now is, does this resolve Liar's paradox truly or are there objections to this? Also, is this the only form of Liar's paradox, and if so, how general is Tarski's solution here (in terms of applicability to others forms of Liar sentences)?
Edit: I've also seen comments on the problem being undecidable, and thus incomplete in terms of knowing the "truth value" of it. If a solution (in a perhaps multi-valued logic) is proposed, then doesn't that counter the fact that it is incomplete or are the results inconsistent? What follows from what exactly? (Also, did Tarski make some helpful comments on it?)
logic set-theory first-order-logic paradoxes
Math3147
Math3147Math3147
$\begingroup$ I wouldn't necessarily say that Tarski "resolves" the Liar paradox. Rather, I think it's better to view the Liar itself as a very broad theorem - along the lines of "There is no consistent formal system capable of both self-reference and defining truth" (although making this precise takes serious work!). This Liar Theorem isn't something which needs resolution, it's simply true. Tarski's Undefinability Theorem is then a corollary of the Liar Theorem and the Diagonalization Lemma. (In general I think it's helpful to recast "paradoxes" as theorems and then look for concrete corollaries.) $\endgroup$
– Noah Schweber
$\begingroup$ @NoahSchweber Can you elaborate on what you mean by "simply true"? The resolution as far as I understand it, does not allow for the truth of the statement to even exist within the hierarchy (more precisely, that it cannot be expressed), and Tarski's theorem is a corollary of the diagonal lemma & self-reference. Saying in $\mathbf{PA}$ something along the lines of: "this sentence is false" would be, according to a Tarskian system, impossible (as written above). Regardless, I still think this doesn't quite help me here as I want to understand, if & why, it "resolves" it, and also its generality. $\endgroup$
– Math3147
$\begingroup$ I meant that the result saying (something like) "There is no consistent formal system capable of both self-reference and defining truth" is simply true. The Liar paradox is the thing we consider in the course of proving this theorem. $\endgroup$
$\begingroup$ @NoahSchweber Oh I see. I was already aware of this but I appreciate your recommended insight. It is a corner stone of the first incompleteness theorem to understand what you said between quotes. Regardless, I'm still baffled as to why it would be true in every system... the Tarskian system seems to probe Peano Arithmetic more so than any system; I could see this being generalized to any system capable of arithmetic as is already done but to any formal system? I could see the intuition but I've not seen the proof at least (I've tried to look it up but no one's discussing it with generality.) $\endgroup$
$\begingroup$ Edit: A certain level of arithmetic not all arithmetic. Also, in every formal system not just any system (I don't want to include paracomplete systems in here; they already accept those trivially.) $\endgroup$
Tarski's famous semantic theory of truth asserts a truth-predicate for the sentences of a given formal language cannot be defined within that language, see here. There're other solutions for Liar.
To avoid self-contradiction, it is necessary when discussing truth values to envision levels of languages, each of which can predicate truth (or falsehood) only of languages at a lower level. So, when one sentence refers to the truth-value of another, it is semantically higher. The sentence referred to is part of the "object language", while the referring sentence is considered to be a part of a "meta-language" with respect to the object language. It is legitimate for sentences in "languages" higher on the semantic hierarchy to refer to sentences lower in the "language" hierarchy, but not the other way around. This prevents a system from becoming self-referential.
However, this system is incomplete. One would like to be able to make statements such as "For every statement in level α of the hierarchy, there is a statement at level α+1 which asserts that the first statement is false." This is a true, meaningful statement about the hierarchy that Tarski defines, but it refers to statements at every level of the hierarchy, so it must be above every level of the hierarchy, and is therefore not possible within the hierarchy (although bounded versions of the sentence are possible). Saul Kripke is credited with identifying this incompleteness in Tarski's hierarchy in his highly cited paper "Outline of a theory of truth," and it is recognized as a general problem in hierarchical languages.
Smullyan (1991, 2001) has argued forcefully that Tarski's undefinability theorem deserves much of the attention garnered by Gödel's incompleteness theorems. That the latter theorems have much to say about all of mathematics and more controversially, about a range of philosophical issues (e.g., Lucas 1961) is less than evident. Tarski's theorem, on the other hand, is not directly about mathematics but about the inherent limitations of any formal language sufficiently expressive to be of real interest. Such languages are necessarily capable of enough self-reference for the diagonal lemma to apply to them. The broader philosophical import of Tarski's theorem is more strikingly evident.
An interpreted language is strongly-semantically-self-representational exactly when the language contains predicates and function symbols defining all the semantic concepts specific to the language. Hence the required functions include the "semantic valuation function" mapping a formula A to its truth value ||A||, and the "semantic denotation function" mapping a term t to the object it denotes. Tarski's theorem then generalizes as follows: No sufficiently powerful language is strongly-semantically-self-representational.
To prove the first incompleteness theorem, Gödel represented statements by numbers. Then the theory at hand, which is assumed to prove certain facts about numbers, also proves facts about its own statements. Questions about the provability of statements are represented as questions about the properties of numbers, which would be decidable by the theory if it were complete. In these terms, the Gödel sentence states that no natural number exists with a certain, strange property. A number with this property would encode a proof of the inconsistency of the theory. If there were such a number then the theory would be inconsistent, contrary to the consistency hypothesis. So, under the assumption that the theory is consistent, there is no such number. It is not possible to replace "not provable" with "false" in a Gödel sentence because the predicate "Q is the Gödel number of a false formula" cannot be represented as a formula of arithmetic. This result, known as Tarski's undefinability theorem, was discovered independently by Gödel (when he was working on the proof of the incompleteness theorem) and by Alfred Tarski.
So Tarski's undefinability theorem is a powerful and philosophically fresh way to look at semantic expressiveness limitation about all conceivable truth predicates of any formal language. Of course Tarski's theorem was developed in a bivalent system, if you're using another truth theory with multivalued-logic such as fuzzy logic, the same reference mentions it can resolve the liar paradox as having truth-value=0.5. Kripke also circumvented the consequences of Tarski's theorem by using three-valued logic as referenced here which discussed all your concerns in detail including revenge paradox. For an axiomatic formal theory of truth see SEP article here.
mohottnadmohottnad
2,51011 gold badge11 silver badge1010 bronze badges
$\begingroup$ I understand and already know this. But the generality of it is not clear, and the proof addresses Peano Arithmetic mainly, and also there are paradoxes (check out revenge paradoxes, this gets a bit philosophical unfortunately) with Tarski's truth predicate, and does this mean it is the only predicate? What about mutli-valued logics? Etc. ? $\endgroup$
$\begingroup$ @Math3147 I don't know if below linked content from same wiki reference helpful to you. "Kripke proposes a solution in the following manner. If a statement's truth value is ultimately tied up in some evaluable fact about the world, that statement is "grounded". If not, that statement is "ungrounded". Ungrounded statements do not have a truth value. Liar statements and liar-like statements are ungrounded, and therefore have no truth value." This is called correspondence theory of truth in philosophy jargon, so Kripke treats Tarski's semantic theory as correspondence, not deflationary... $\endgroup$
– mohottnad
$\begingroup$ Can you link something about Kriple's comments? Although they're more philosophy related than related to arithmetic or decidability. $\endgroup$
$\begingroup$ @Math3147 it's in the second link in my answer above (just below Tarski's solution section). IMHO philosophy is all about awareness not some niffty-gritty technical or formal details, and most people need to be aware of something first then dive into details later, then spiral (not cyclic hopefully). $\endgroup$
$\begingroup$ I meant a more comprehensive read. Philosophy to me sounds like speculative theories that don't actually prove what their talking about. I'm all for intuition... one that is grounded. Plus, this site mainly discusses mathematics too, and so it is slightly off-topic. $\endgroup$
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