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\begin{document}
\title{Certification of Gaussian Boson Sampling via graph theory}
\author{Taira Giordani} \thanks{These two authors contributed equally} \affiliation{Dipartimento di Fisica, Sapienza Universit\`{a} di Roma, Piazzale Aldo Moro 5, I-00185 Roma, Italy}
\author{Valerio Mannucci} \thanks{These two authors contributed equally} \affiliation{Dipartimento di Fisica, Sapienza Universit\`{a} di Roma, Piazzale Aldo Moro 5, I-00185 Roma, Italy}
\author{Nicol\`o Spagnolo} \affiliation{Dipartimento di Fisica, Sapienza Universit\`{a} di Roma, Piazzale Aldo Moro 5, I-00185 Roma, Italy}
\author{Marco Fumero} \affiliation{Dipartimento di Informatica, Sapienza Universit\`{a} di Roma, Via Salaria 113, I-00198 Roma, Italy}
\author{Arianna Rampini} \affiliation{Dipartimento di Informatica, Sapienza Universit\`{a} di Roma, Via Salaria 113, I-00198 Roma, Italy}
\author{Emanuele Rodolà} \email[Corresponding author: ]{[email protected]} \affiliation{Dipartimento di Informatica, Sapienza Universit\`{a} di Roma, Via Salaria 113, I-00198 Roma, Italy}
\author{Fabio Sciarrino} \email[Corresponding author: ]{[email protected]} \affiliation{Dipartimento di Fisica, Sapienza Universit\`{a} di Roma, Piazzale Aldo Moro 5, I-00185 Roma, Italy}
\begin{abstract}
Gaussian Boson Sampling is a non-universal model for quantum computing inspired by the original formulation of the Boson Sampling problem. Nowadays, it represents a paradigmatic quantum platform to reach the quantum advantage regime in a specific computational model. Indeed, thanks to the implementation in photonics-based processors, the latest Gaussian Boson Sampling experiments have reached a level of complexity where the quantum apparatus has solved the task faster than currently up-to-date classical strategies. In addition, recent studies have identified possible applications beyond the inherent sampling task. In particular, a direct connection between photon counting of a genuine Gaussian Boson Sampling device and the number of perfect matchings in a graph has been established. In this work, we propose to exploit such a connection to benchmark Gaussian Boson Sampling experiments. We interpret the properties of the feature vectors of the graph encoded in the device as a signature of correct sampling from the true input state. Within this framework, two approaches that exploit the distributions of graph feature vectors and graph kernels are presented. Our results provide a novel approach to the actual need for tailored algorithms to benchmark large-scale Gaussian Boson Samplers.
\end{abstract}
\maketitle
\section*{Introduction} Quantum processors and quantum algorithms promise substantial advantages in computational tasks \cite{Harrow2017_supremacy}. Recent experiments have shown significant improvements towards the realization of large scale quantum devices operating in the regime in which classical computers cannot reproduce the output of the calculation \cite{Harrow2017_supremacy,Arute2019, Wu_2021_supremacy, Zhong_GBS_supremacy, zhong2021phaseprogrammable}.
An intriguing computational problem concerning quantum photonic processors is Boson Sampling (BS) and, more recently, its variant Gaussian Boson Sampling (GBS). The BS paradigm corresponds to sampling from the output distribution of a Fock state with $n$ indistinguishable photons after the evolution through a linear optical $m$-port interferometer \cite{AA, Brod19review}. This problem turns out to be intractable for a classical computer, while a dedicated quantum device can tackle such a task towards unequivocal demonstration of quantum computational advantage. The GBS variant replaces the quantum resource of the BS, i.e the Fock state, with single-mode squeezed vacuum states (SMSV). This change to the original problem enhances the samples generation rate with respect to BS performed with probabilistic sources, and preserves the hardness of sampling from a quantum state \cite{Lund_SBS,wcqoscct,Hamilton2017, Deshpande_GBS_th_supremacy}.
The GBS problem has drawn attention for the practical chance to achieve the quantum advantage regime. After the small scale experiments \cite{Zhong19, Paesani2019, thekkadath2022experimental}, the latest GBS instances have just reached the condition where the quantum device has solved the task faster than current state-of-the-art classical strategies \cite{Zhong_GBS_supremacy, zhong2021phaseprogrammable}. The interest in GBS also concerns applications for sampling for gaussian states beyond the original computational advantage. The probability of counting $n$-photon in the output ports of a GBS is proportional to the squared \emph{hafnians} of an appropriately constructed matrix, that takes into account the unitary transformation $U$ representing the optical circuit and the covariance matrix of the input state. Computing Hafnians of a matrix is as hard as computing \emph{permanents} that describe the amplitude of the BS output states. The hafnians have a precise interpretation in graph theory since their calculation corresponds to counting the perfect matchings in a graph. The adjacency matrix of a graph can be encoded in a GBS, and then the collected samples are informative about the graph properties. Recently, GBS-based algorithms for solving well-known problems in graph theory have been formulated \cite{Arrazzola_densesubgraph, Shuld_GBS_graphsimilarity, Bradler_2021} and tested in a first proof-of-principle experiment of the GBS within a reconfigurable photonic nano-chip \cite{Arrazola2021}.
These results on the BS framework are thus bringing back photonic platforms as a promising approach to implement quantum algorithms. In parallel, this development is currently accompanied by research efforts aimed at identifying suitable and efficient strategies for system certification. This is indeed a crucial requirement, both for benchmarking quantum devices reaching the quantum advantage regime, as well as validating the operation of such systems whenever they are employed to solve specific computations. While several methodologies have been developed and reported, certification of quantum processors is still an open problem. In the case of BS and GBS, direct calculation or sampling from the output distribution cannot be performed efficiently by classical means, and are thus not viable for large-scale implementations with many photons and ports of the optical circuit \cite{clifford2017classical,Neville2017,Quesada_exact_simulation,Quesada_exact_simulation_speedup, Bulmer_Markov_GBS}. Then, it is preferable to switch the problem towards a validation approach, i.e to exclude that the samples could be reproduced by specifically chosen classical models. The validation tests first developed for the BS problem focus on ruling out the uniform sampler, the distinguishable particle sampler and the mean-field sampler hypotheses \cite{Aaronson14, Tichy, Spagnolo2, Carolan15, Crespi16,Viggianiello18, Walschaers16, Giordani18, agresti2019pattern, FlaminiTSNE}. Recent efforts have been also dedicated to addressing partial photon distinguishability \cite{Viggianiello17optimal, Renema_partial_2020}, which is a crucial requirement that can spoil the complexity of the computation \cite{Renema_2018_classical, Moylett_2019}. This validation approach, originally conceived for the BS problem and based on defining suitable alternative hypotheses, has been subsequently extended to the GBS variant (see Fig.\ref{fig:validation}). In various experiments, the samples from GBSs have been validated against alternative classically-simulable hypotheses, such as the thermal, coherent and distinguishable SMSV states \cite{Zhong19, Paesani2019, Zhong_GBS_supremacy, zhong2021phaseprogrammable}. These GBS validation examples include variations of Bayesian approaches \cite{Zhong_GBS_supremacy, zhong2021phaseprogrammable} or algorithms based on the statistical properties of two-point correlation functions \cite{Walschaers16, Giordani18} that can be used also for GBS to exclude thermal and distinguishable SMSV samplers \cite{hanbury}. In~\cite{zhong2021phaseprogrammable} a more refined analysis investigates the possibility of describing the experimental results as lower order interference processes, thus not involving all the generated photons. This approach is strictly related to sampling algorithms based on low-order interference approximations \cite{popova2021cracking} or low-order marginal probabilities \cite{AA, Renema_partial_2020, renema2020marginal, villalonga2021efficient}. In parallel, studies regarding the classical simulability of BS and GBS in terms of photon losses have also been carried out \cite{Oszmaniec_2018,GarciaPatron2019simulatingboson,Brod2020classicalsimulation, Qi_lossyGBS}.
Besides these examples of GBS validations, there is a lack of tailored algorithms for GBS that could be efficient in the regime of quantum advantage. In this work, we propose a validation protocol based on the deep connection between GBS and graph theory. We consider the features of the graph extracted from the GBS samples as a signature of the correct sampling from indistinguishable SMSV states. Within this framework, we present two approaches. The first method considers the space spanned by the feature vectors extracted from photon counting samples obtained from different gaussian states. Then, a classifier, such as a neural network, can be trained to identify an optimal hyper-surface to distinguish a true GBS and the mock-up hypotheses in this space. The second method investigates the properties of the kernel generated by the feature vectors of each class of gaussian state. Both approaches exploit macroscopic quantities that can be retrieved in a reasonable time from the measured GBS samples.
This work is organized as follows. First, we review the concept of sampling from gaussian state of light and the relationship with counting graph perfect matchings. Then, we present the validation methods based on the properties of graph feature vectors and kernels. We conclude by providing insights on the effectiveness of the proposed approach to discriminate genuine GBS from different alternative hypotheses.
\begin{figure}
\caption{\textbf{Gaussian Boson Sampling validation}. In the GBS paradigm $n$-photon configurations are sampled at the outputs of an optical circuit with $m$ ports. The aim of a validation algorithm is to exclude that the obtained samples could have been generated by classically-simulable models. Previous experiments mainly focused in techniques capable to rule out the uniform sampler, the thermal, coherent and distinguishable SMSV states hypotheses.}
\label{fig:validation}
\end{figure}
\section*{Gaussian Boson Sampling and its connection with graphs} \begin{figure*}
\caption{\textbf{Gaussian Boson Sampling and graph perfect matching.} a) Structure of the $2m\times 2m$ sampling matrix for $m$ independent gaussian states injected in a $m$-port interferometer. b) The hafnian as the operation to count the perfect matchings in a simple undirected graph with $n$ nodes. c) The permament as the same operation for a bipartite graph. d) The adjacency matrix of undirected graphs, even in the bipartite case, can be encoded in GBS devices. }
\label{fig:haf_perm}
\end{figure*}
\noindent\textbf{Background.}
Here we briefly review the general theory of the probability to obtain $n$-photon configurations from a set of indistinguishable gaussian input states $\rho_i$ such that $\rho_{in}=\otimes_{i=1}^m \rho_i$, and distributed in $m$ optical modes, after the evolution in a multi-port interferometer. Given the $2m\times2m$ covariance matrix $\sigma$ that identifies the gaussian state, and the output configuration $\vec{n}= (n_1, n_2, \dots, n_m)$, where $n_i$ is the number of photons detected in the output port $i$ such that $\sum_{i=1}^m n_i = n$, we have \begin{equation}
\text{Pr}(\vec{n})=|\sigma_Q|^{-\frac{1}{2}}\frac{\text{Haf}\,(A_{\vec{n}})}{\prod_{\vec{n}}{n_i}!} \,. \label{eq:hafn_gbs} \end{equation} The quantity $\sigma_Q$ is $\sigma+\frac{1}{2}\mathbb{I}_{2m}$ where $\mathbb{I}_{2m}$ is the $2m\times2m$ identity operator; $A_{\vec{n}}$ is a sub-matrix of the overall matrix $A$ that contains the information about the optical circuit represented by the transformation $U$ and the covariance of the input state, while $\text{Haf}$ stands for the hafnian of the matrix. More precisely, $A_{\vec{n}}$ is the $n \times n$ sub-matrix obtained by taking $n_i$ times the $i$-th row and the $i$-th column of $A$ \cite{Hamilton2017, DetailedstudyGBS}. The hafnian of $A_{\vec{n}}$ corresponds to the summation over the possible perfect matching permutations, i.e, the ways to partition the index set $\{1,\cdots ,n\}$ into $n$/2 pairs such that each index appears only in one pair (see also~\cite{Caianiello1953}). The hafnian is in the \#P-complete complexity class, and is a generalization of the permanent of a matrix $M$ according to the following expression: \begin{equation}
\text{Per}(M)=\text{Haf}\begin{pmatrix}
0 & M\\
M^t & 0
\end{pmatrix}\,.
\label{eq:permanent} \end{equation} The above description has been used to define a classically-hard sampling algorithm, using indistinguishable SMSV states with photon-counting measurements \cite{wcqoscct, Hamilton2017, DetailedstudyGBS}. More specifically, in Fig. \ref{fig:haf_perm}a we report the structure of the sampling matrix $A$ for an input state $\rho$ that has zero displacement. In the language of quantum optics the displacement is the operation that generates a coherent state from the vacuum. Then, the blocks $B$ and $C$ highlighted in the Fig. \ref{fig:haf_perm}a correspond to the contribution of squeezed and thermal light respectively in the input state. Pure, indistinguishable, SMSV states display a $C=0$ and $B=U \text{diag} (\tanh{s_1}, \dots, \tanh{s_m}) U^t$, where $s_i$ are the squeezing parameters of each $\rho_i$ \cite{DetailedstudyGBS}. According to this representation the expression in Eq. \eqref{eq:hafn_gbs} becomes \begin{equation}
\text{Pr}(\vec{n})_{\text{SMSV}} = |\sigma_Q|^{-\frac{1}{2}}\frac{|\text{Haf}\,({B}_{\vec{n}})|^2}{\prod_{\vec{n}}{n_i}!}\,,
\label{eq:smsv_states} \end{equation} where ${B}_{\vec{n}}$ is the submatrix of $B$ obtained from the string $\vec{n}$ as described at the beginning of the section.
\noindent\textbf{Connection to graph theory.}
Recently, several works have identified a connection between the GBS apparatus and graph theory \cite{ArrazolaQOpt,Arrazzola_densesubgraph, Bradler_2021}. These studies take advantage of such a relationship to formulate GBS-based algorithms in the context of graph-similarity and graph kernels. The algorithms exploit the fact that the vectors extracted from GBS samples can be considered a feature space for a graph encoded inside the apparatus. In particular, they are strictly correlated to a class of classical graph kernels that count the number of $n$-matchings, i.e., the perfect matchings of the sub-graph with $n$ links in the original graph encoded inside the GBS. Given $A$ the adjacency matrix of the graph, the number of perfect matchings is proportional to the hafnian of the matrix, thus corresponding to the output probabilities in Eqs. \eqref{eq:hafn_gbs} and \eqref{eq:smsv_states}.
Indeed, any symmetric matrix, such as the graph adjacency matrices, can be decomposed accordingly to the Takagi-Autonne factorization as $A = U \text{diag}(c\lambda_1,\dots, c\lambda_m) U^{t}$, where $\lambda_i$ are real parameters in the range $[0,1]$, $c$ is a scaling factor and $U$ is a unitary matrix. This decomposition matches with the expression of the sampling matrix $B$ of SMSV states when $\lambda_i = \tanh{s_i}$. Also squeezed states with very small displacement have a $n$-photon probability distribution that can be expressed through hafnians.
For example, displaced squeezed states have been investigated in the context of graph similarity, where a small amount of displacement has been employed as a hyper-parameter to enhance the graphs' classification accuracy \cite{Shuld_GBS_graphsimilarity}.
Regarding the sub-matrix selected by the sampling process, the configuration $\vec{n}$ identifies the elements of the sub-matrix $A_{\vec{n}}$ that represent an induced sub-graph (see Fig \ref{fig:haf_perm}b). The nodes of the original graph $A$ corresponding to detectors with zero counting are deleted, together with any edges connecting these nodes to the others. If some elements $n_i$ of $\vec{n}$ are larger than one, i.e. these detectors count more than one photon, $A_{\vec{n}}$ describes what we call an \emph{extended induced sub-graph} in which the corresponding nodes and all their connections are duplicated $n_i$ times.
It is worth noting that also the permanent has a precise meaning in the context of graphs. Indeed, the matrix on the right-hand side of Eq. \eqref{eq:permanent} corresponds to the adjacency matrix of a bipartite graph. In other words, the permanent calculation provides the number of perfect matchings for this class of graphs (see Fig. \ref{fig:haf_perm}c). One may ask whether other sampling processes regulated by permanent calculations, such as the BS and the thermal samplers (see Appendix \ref{app:sampling}), could have a relationship with bipartite graphs. The BS output distribution is defined by the permanent of the sub-matrix from the unitary transformation $U$ representing the circuit. It is clear that not all graphs can be represented by a unitary adjacency matrix. Furthermore, in the BS paradigm, the sub-matrix selected by the sampling process depends also on the input state. This implies that the resulting sub-graph could not have the same symmetries and properties as the original encoded in the $U$ matrix. The latter issue can be overcome by using thermal light, where only the output configuration $\vec{n}$ determines the sub-matrix. However, also for thermal light, the sampling matrix $C$ does not in general represent an adjacency matrix, thus preventing the possibility of encoding any bipartite graphs. In conclusion, the GBS devices with squeezed states are the only ones that have a direct connection with graphs (see Fig. \ref{fig:haf_perm}d).
\section*{Feature vector-based validation algorithm} In the following, we illustrate two validation algorithms tailored for GBS. The idea behind our protocols is to exploit the connection between the samples of a genuine GBS and the graph properties encoded in the device.
According to Eq. \eqref{eq:smsv_states} the most likely outcomes from the GBS are those with the highest hafnians, i.e. the output configurations that identify the sub-graph $A_{\vec{n}}$ with the largest number of perfect matchings. However, we remind that the calculation of a single hafnian is a \#P-complete problem as the counting of the perfect matchings in a graph. Furthermore, estimation of the output probabilities from the quantum devices becomes unfeasible for large system sizes, and thus any protocol should not rely on this ingredient. Then, it is necessary for a successful validation algorithm to exploit quantities that do not depend on the evaluation of the probability of a single $\vec{n}$, which would require exponential time for its estimation.
\noindent\textbf{Feature vectors.}
It is possible to extract properties from a graph summarized in the so called feature vectors. In the GBS-based algorithms the features of the graph are extracted from a coarse-graining of the output configuration states. For instance, the probability to detect configurations $\{\vec{n}\}$ with $n_i = \{0,1\}$ is linked to the number of perfect matchings of the sub-graphs $\{A_{\vec{n}}\}$ of $A$ which do not have repetition of nodes and edges. Accordingly, the probability of the set of $\{\vec{n}\}$ with two photons in the same output will be connected to the perfect matching in sub-graphs with one repetition of a pair of nodes and edges. The collections of output configurations that identify a family of sub-graphs with a certain number of nodes and edges repetitions are called \emph{orbits} \cite{Shuld_GBS_graphsimilarity}. Given $n$ the total number of post-selected photons in the output, the orbit $O_{\vec{n}}$ is defined as the set of the possible index permutations of $\vec{n}$. In this work we consider the orbit $O_{[1,1,\dots,1,0 \dots 0]}$ that corresponds to output states with one or zero photon per mode; the orbit $O_{[2,1,\dots,1,0 \dots, 0]}$ that is the collection of the outputs with one mode occupied by two photons and $O_{[2,2,1\dots,1,0 \dots, 0]}$ with two distinct outputs hosting two photons. The graph feature vector components are identified by the probability of each orbit, defined as $Pr(O_{\vec{n}})=\sum_{\vec{n} \in O_{\vec{n}}}Pr(\vec{n})$. In the rest of this work we will refer to the probabilities of the orbits $O_{[1,1,\dots,1,0 \dots 0]}$, $O_{[2,1,\dots,1,0 \dots, 0]}$ and $O_{[2,2,1\dots,1,0 \dots, 0]}$ as $[1, \dots, 1]$, $[2,1, \dots1]$ and $[2,2,1,\dots,1]$ respectively.
The orbit probabilities can be estimated directly from photon counting measurements. This method can be applied in GBS experiments. In numerical simulation, direct sampling of photon counting is a viable approach for deriving orbit probabilities of gaussian states that can be sampled classically, such as distinguishable SMSV, thermal and coherent states (see Appendix \ref{app:sampling}). These states reproduce the scenarios that could occur in the experimental realizations of GBS devices. For example, photon losses turn the squeezed light into thermal radiation, while mode-mismatch, such as spectral and temporal distinguishability, breaks the symmetry of boson statistics. Exact estimation of the orbits for indistinguishable SMVS states can be performed by directly calculating all the hafnians, thus requiring evaluation of a large number of complex quantities. A different approach can be employed, based on approximating the orbits probability by a Monte Carlo simulation \cite{Killoran2019strawberryfields, Bromley_2020}. The outputs $\vec{n}$ within an orbit are selected uniformly at random and their exact probabilities are calculated. Then, the probability of the whole orbit after $N$ extractions can be approximated by $Pr(O_{\vec{n}}) \approx \frac{|O_{\vec{n}}|}{N}\sum_{i=1}^N Pr(\vec{n}_i)$, where $|O_{\vec{n}}|$ is the number of elements in the orbit. The adopted strategies reproduce the experimental conditions in which the orbits probabilities are estimated on a finite number $N$ of samples. The code for generating GBS data included routines from Strawberry Fields \cite{Killoran2019strawberryfields} and The Walrus \cite{Gupt2019} Python libraries. \begin{figure}
\caption{\textbf{Orbits probabilities distribution.} In blue we report the feature vectors for $100$ genuine GBS devices with $m=400$. The squeezers parameters in each GBS were tuned to obtain a photon number distribution centered around $n \sim 16 \ll m$. Each cloud corresponds to the post-selection of different number of photons $n$ in the outputs. This is equivalent to look at the features of $n$-node sub-graphs. In yellow we report the thermal sampler case, in pink the distinguishable sampler, in red the distinguishable thermal sampler and in green the coherent light one.
GBS data were generated numerically via Monte Carlo approximation of the orbits probabilities. The maximum size achieved for the simulation corresponds to $n = 22$ for computation time reasons. The data of the other models were extracted from direct sampling of the photon counting. Thermal, coherent and distinguishable thermal samplers display also a non-zero probability to generate odd number of photons.}
\label{fig:leaves}
\end{figure}
\begin{figure*}
\caption{\textbf{Orbits probabilities for different sizes of the GBS}. a) Orbits probabilities $[1, \dots, 1]$, $[2, 1, \dots, 1]$, $[2, 2, \dots, 1]$ for different samplers with $n$-photon $\in [4, 6, \dots, 22]$ and $m=n^2$ optical modes. In the blue-scale samples from a genuine GBS device, in yellow data from indistinguishable thermal states, in green the coherent states, in pink the distinguishable SMVS states and in red the distinguishable thermal light. For each $n$ and class of states we sampled $100$ sets of $U$ and $\{s_i\}$. Two orbits are not enough to discriminate the data, while in the space spanned by three orbits the various hypotheses are very well separated. b) Results of the classification accuracy of genuine GBS data by means of a neural network classifier. The network trained with trusted GBS data of smaller sizes indicated along the x axis is able to correctly classify larger GBS devices. }
\label{fig:fv_scaling}
\end{figure*}
\begin{figure*}
\caption{\textbf{Graph kernels distributions.} a) Kernels distributions for the graphs encoded in GBSs with $m$=400 and photon-number distribution centered around $n=16$. The feature vectors have been normalized to a given $n$ and to the space of the three orbits. We report the distributions for the GBS in blue, coherent light sources in green, distinguishable SMSV in pink and distinguishable thermal states in red. The four histograms display different features for any $n$. b-c) Kernels mean and standard deviation for the case $n=16$ and for increasing values of measured graphs. We observe that a small amount of experiments is enough to discriminate genuine GBS kernels. The uncertainties reported in the plots correspond to $3$ standard deviations.}
\label{fig:kernels}
\end{figure*}
\noindent\textbf{Validation by classification.}
As a first method for validation, we propose the classification of these different samplers in the space spanned by the three feature vector components identified by the orbits $[1, \dots, 1]$, $[2,1,\dots,1]$ and $[2,2,1,\dots,1]$. In Fig.~\ref{fig:leaves} we give an insight of our intuition by reporting an example of the distribution of feature vectors for different graphs and sampler types. The colors underline samples from different models such as genuine GBS, distinguishable SMSV states, coherent light, indistinguishable thermal light emitters and distinguishable thermal states. In this simulation we consider $100$ optical random circuits with $m=400$ modes and $m$ sources set to produce a photon-number distribution centered in $n\ll m$. In this condition we are in the dilute regime where the orbits with low number of photons in the same output have the highest probability to occur. It is worth noting that in this estimation it is necessary to take into account the occurrence of the orbits in the whole space of the GBS, i.e the Hilbert space associated to the all possible $n$-photon states that can be generated by the squeezers.
Experimentally, such a method requires the knowledge of the photon-number distribution of the sources. Such requirement is not demanding since the characterization of the gaussian sources is a standard preliminary procedure in GBS experiments \cite{Zhong19,Paesani2019, Zhong_GBS_supremacy, Arrazola2021, zhong2021phaseprogrammable}. Alternatively, the orbits probability can be estimated by post-selecting samples with different total number $n$ and dividing the occurrence of photon counting belonging to the orbit with a given $n$ by the total number of samples. The data of the classical models were retrieved with such an approach while the GBS orbits were calculated via the Monte Carlo approximation.
These simulations show that three orbits are informative to discriminate among different gaussian samplers until the photon-number distribution is centered in the dilute regime. On the one hand, it is worth noting that the thermal light curve lies in the same plane of the GBS data but with somehow a smaller radius. The reason is that thermal radiation displays a non-zero probability to generate an odd number of photons. On the other hand, the distinguishability moves the two curves towards another plane that exhibits higher values of the probability of the orbit $[1,1,\dots 1]$. The physical intuition behind this behavior is that distinguishable particles do not interfere and, consequently, they have a lower probability of bunching.
To prove the effectiveness of feature vectors to validate a genuine GBS device of any size, we train a classifier such as feed-forward neural network with the data reported in Fig. \ref{fig:fv_scaling}. Experimental details are provided in Appendix \ref{app:3}. Here the samples correspond to different experiment layouts with number of modes $m = n^2$, and the number of post-selected photons varying in $n \in [4, 6, \dots 22]$. The size of the collected samples was $\sim 10^5$ for the classical gaussian states that generate a fraction of $\sim 10^3-10^4$ output configurations in the orbits under investigation. For the GBS data, we performed $\sim 10^4$ Monte Carlo extractions for the orbits probability estimation. The classifier reaches high level of accuracy, greater than 99\%. We performed a further study reported in Fig. \ref{fig:fv_scaling}b to check the ability of the network to generalize to GBSs sizes not included in the training stage. To this aim we have trained the network with the data of Fig.~\ref{fig:fv_scaling}a up to $n = 12, 14, 16$, and subsequently computed the classification accuracy for the data with $n = 18, 20, 22$. The latter has been estimated on $100$ set of GBS for each $n$ and on $10$ independent training.
\noindent\textbf{Validation via graph kernels.}
Other interesting quantities linked to feature vectors are the graph kernels, which can be employed to define a second method for validation. Here we study the linear kernels defined as the scalar product between pairs of feature vectors. This method is less demanding in terms of number of measurements since it works even in the case where only samples from a given number $n$ of photons are post-selected at the output. In Fig.~\ref{fig:kernels}a we report the distributions of kernels for feature vectors normalized to the $3$-dimensional orbits space for a given number $n$ of post-selected photons. We note that kernels from distinguishable SMVS and distinguishable thermal states (Fig. \ref{fig:kernels}a) display the same gaussian distribution of the indistinguishable case, but they are centered at different kernel values for any $n$. Indeed, each histogram in the figure corresponds to the data of Fig. \ref{fig:leaves} for the $100$ sub-graph identified by $n=14$ and $n=16$. The coherent light data display the same average but show a larger variance. These differences highlighted in Fig.\ref{fig:kernels}a can be exploited to discriminate the coherent and distinguishable particles hypotheses. To do this, we only require for the optical circuit to be reconfigurable, and perform enough experiments to retrieve the kernel distributions. Note that the number of kernels scales exponentially with the number of experiments, i.e. the number of sampled different graphs. More precisely, the number of kernels after $N$ experiments is $\begin{pmatrix} N \\ 2 \end{pmatrix}$. Thus, the kernels average and variance can be retrieved in a reasonable number of measurements as investigated in Fig.~\ref{fig:kernels}b-c. The distributions of kernels from thermal samplers (not shown in the figure) are centered at the same values of genuine GBS with the same gaussian distribution. Thus, the discrimination of data from thermal indistinguishable emitters still requires the measurement of different $n$ number of photons in the outputs. This is not surprising if we consider the distribution of the feature vectors in Fig.~\ref{fig:leaves}. They display the same dispersion of the GBS data and, since we are now considering only the space of configurations with a given number of photons, the clouds collapse on each other.
\section*{Discussion} In this work, we have presented a new approach to GBS validation that exploits the intrinsic connection between photon counting from specific classes of gaussian states of light and counting of perfect matchings in undirected graphs. Despite GBS-based algorithms in graph theory still need further studies to clarify their actual effectiveness and advantage with respect to the classical counterparts, the tools introduced in this context turn out to be informative in the framework of GBS experiments verification. We have seen how the feature vectors together with the graph kernels extracted from photon counting indicate the quantum nature of the sampling process. In fact, these quantities are very sensitive to imperfections that could occur in actual experiments, such as photon losses and distinguishability \cite{Shuld_GBS_graphsimilarity}. These two effects drive the device to act more similarly to thermal and distinguishable particles samplers that can be simulated efficiently by classical means.
The methods based on graph feature vectors and kernel distributions require a reasonable number of samples due to the coarse-graining of the output space of GBSs. The method based on graph kernels requires fewer experiments with different graphs, in turn requiring the capability to tune the optical circuit $U$ and the squeezing parameters $s_i$. Nowadays, recent experimental results on integrated reconfigurable circuits \cite{Arrazola2021, Taballione_2021, hoch2021boson} enable large tunability and dimension of the matrix $U$. In addition, squeezing parameters can be tuned by changing the power of the pump laser that generates squeezed light from nonlinear crystals, and by tuning the relative squeezing parameters phases as recently demonstrated in~\cite{zhong2021phaseprogrammable}.
Further improvements to the approach adopted in this work can be foreseen. For instance, these include exploiting a more extensive orbit set or larger coarse-graining. These modifications could help in the validation of larger-scale instances of GBS. For example, it is possible to observe from Fig. \ref{fig:leaves} and Fig. \ref{fig:fv_scaling} that the orbits probabilities tend to zero with larger size due to the increasing dimension of the GBS Hilbert space. A future perspective of such investigation may be the extension in the regime that exploits threshold detectors. This configuration has been adopted to prove quantum advantage, but its connection with graph feature vectors has not been investigated yet.
\section*{Acknowledgments} This work is supported by the ERC Advanced grant QU-BOSS (Grant Agreement No. 884676) and ERC Starting grant SPECGEO (no. 802554). The authors wish to acknowledge financial support also by MIUR (Ministero dell’Istruzione, dell’Università e della Ricerca) via project PRIN 2017 “Taming complexity via QUantum Strategies: a Hybrid Integrated Photonic approach” (QUSHIP - Id. 2017SRNBRK). N.S. acknowledges funding from Sapienza Universit\`a di Roma via Bando Ricerca 2020: Progetti di Ricerca Piccoli, project "Validation of Boson Sampling via Machine Learning".
\appendix \section{Sampling from gaussian states}\label{app:sampling} \noindent \textbf{Thermal and coherent light.} Not all gaussian states display the same sampling complexity of indistinguishable squeezed states. For example, in the case of $m$ indistinguishable thermal light emitters, we have that the sampling matrix $A$ in Fig \ref{fig:haf_perm}a has $B=0$ and $C \ne 0$. Then, according to the relationship in \eqref{eq:permanent} the probability to detect $\vec{n}$ is \begin{equation} \text{Pr}(\vec{n})_{\text{Th}}= \frac{\text{Per}\,({C}_{\vec{n}})}{\prod_{\vec{n}}{n_j}! \prod_{i=1}^m(1 + \langle n_i \rangle)}, \end{equation}
where $\langle n_i\rangle$ is the mean photon number associated to each thermal source $\rho_i$ in the input and $C_{\vec{n}}$ the sub-matrix of $C$. Sampling from this distribution is not hard as in the case of SMSV input states, although the expression requires the calculation of a permanent. Indeed, the matrix $C$ can be decomposed as $C=U \text{diag}(\tau_i, \dots, \tau_m)U^\dagger$ with $\tau_i = \langle n_i \rangle / (1 + \langle n_i \rangle)$ and is a positive semi-definite matrix. In this special case, the permanent can be approximated with classical resources \cite{perm_pos_semi}. More precisely, sampling from a thermal state is a problem in the class $\text{BPP}^{\text{NP}}$ \cite{wcqoscct}, that includes problems easier requiring lower time resources than those belonging to the \#P-complete one. An alternative way to sample from a thermal state is through the P-representation of the electromagnetic field \cite{Glauber2007, loudon1983quantum}, i.e the expression of the state as superpositions or mixtures of coherent states $\ket{\alpha}$. For classical states of light, such as thermal states, the P-function is a probability distribution, more precisely it is gaussian $\text{P}_{\text{Th}}(\alpha_i) = \frac{1}{\pi\langle n_i \rangle} e^{-\frac{|\alpha_i|^2}{\langle n_i \rangle}}$. Then, sampling a string $\vec{n}$ can be simulated by extracting a set of $\alpha_i$ from $\text{P}_{\text{Th}}(\alpha_i)$ and considering the evolution of a coherent state in a linear optics interferometer. The latter transformation maps the state $\alpha_i$ in another coherent state $\beta_j$, according to $\beta_j = \sum_{i=1}^m U_{ij} \alpha_i$. The probability to detect the configuration $\vec{n}$ in the output, given a set of coherent states in the input, is given by: \begin{equation}
Pr(\vec{n})_{\text{Coh}} = \prod_{i=1}^m \frac{e^{-|\beta_i|^2} |\beta_i|^{2n_i}}{n_i!}\,.
\label{eq:coherent} \end{equation} Such expression is the product of $m$ poissonian distributions and it can be sampled in polynomial time \cite{wcqoscct}. Furthermore, Eq. \eqref{eq:coherent} can be directly employed to perform classical sampling with coherent state inputs.
\noindent \textbf{Distinguishable emitters of thermal and squeezed light.} The last possible scenario corresponds to sampling from a set of distinguishable gaussian states. The evolution can be simulated by independently sampling photons from the photon-number distribution of each gaussian source, and by considering the single-photon distribution after the interferometer. The distinguishability among photons generated from different sources permits to sample each input mode independently, without the complexity introduced by quantum interference. This is the case of distinguishable SMSV, states, and is analogous to the scenario for distinguishable thermal light emitters. Hence, efficient simulation of distinguishable gaussian states can per performed classically, thus not requiring a quantum processor.
\section{Graphs employed in the numerical simulations} One of the main novelties in the GBS paradigm is the possibility to encode any symmetric matrix in the sampling process. This feature permits to identify relations between photon counting and the graph represented by its symmetric adjacency matrix. In the simulations presented in this work we consider the graphs resulting from $A = U \text{diag}(\tanh{s_1}, \dots, \tanh{s_m} )U^t$, where the unitary matrix $U$ of the optical circuit and the squeezing parameters $\{s_i\}$ were set as follows. The circuits $U$ were randomly generated from the Haar distribution of unitary matrices. This choice guarantees on one hand to reproduce the most general conditions, and, on the other, to exclude the existence of any symmetry inside the sampling matrix that could undermine the complexity of the problem. For what concerns the squeezing parameters, we set their values to obtain a photon number distribution centered around a given $n$. The expression of such a distribution is analyzed in details in~\cite{DetailedstudyGBS}.
The parameter settings of the classical-simulable gaussian states were chosen to reproduce distributions as close as possible to the genuine GBS ones. To this end, we consider the evolution through the same circuit $U$. In addition, the gaussian sources were set to emit the same average number of photons of the squeezed sources. This implies setting $\langle n_i \rangle = \sinh^2 s_i$.
\section{Neural network for data classification} \label{app:3}
In this Section we describe the experimental settings for the validation task presented in the main text. The dataset is made of 100 feature vectors corresponding to samples of number of photons $n$ ranging from 4 to 22 and scattered by optical circuits with $m = n^2$ optical modes (see Fig. \ref{fig:fv_scaling}) The feature vectors are labelled by two classes, distinguishing 3-dimensional features vectors extracted from genuine GBS samples, from the other samplers (see Appendix \ref{app:sampling}). To solve the validation task, we employ a Multi-Layer Perceptron (MLP) binary classifier. The architecture is composed of two stacked linear layers with ReLU activations, and a batch normalization layer \cite{Ioffe2015Feb} to improve the gradient flow in the backward pass. A final linear layer with a sigmoid activation is added to output the corresponding class prediction. The overall model is trained with a binary cross entropy loss. The MLP converged after 20 epochs, reaching an accuracy score of 99\% over the test set. The model is implemented using PyTorch \cite{pytorch} and the code was run on a computer with a GPU NVIDIA GeForce GTX 1060 with 3GB of RAM.
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\bibitem [{\citenamefont {Zhong}\ \emph {et~al.}(2021)\citenamefont {Zhong},
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\citenamefont {Chen}, \citenamefont {Peng}, \citenamefont {Luo},
\citenamefont {Wu}, \citenamefont {Gong}, \citenamefont {Su}, \citenamefont
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\citenamefont {Li}, \citenamefont {Li}, \citenamefont {Jiang}, \citenamefont
{Gan}, \citenamefont {Yang}, \citenamefont {You}, \citenamefont {Wang},
\citenamefont {Li}, \citenamefont {Liu}, \citenamefont {Renema},
\citenamefont {Lu},\ and\ \citenamefont {Pan}}]{zhong2021phaseprogrammable}
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\bibnamefont {Lu}},\ and\ \bibinfo {author} {\bibfnamefont {J.-W.}\
\bibnamefont {Pan}},\ }\bibfield {title} {\bibinfo {title}
{Phase-programmable gaussian boson sampling using stimulated squeezed
light},\ }\href {https://doi.org/10.1103/PhysRevLett.127.180502} {\bibfield
{journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo
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complexity of linear optics},\ }in\ \href
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\bibitem [{\citenamefont {Brod}\ \emph {et~al.}(2019)\citenamefont {Brod},
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\bibitem [{\citenamefont {Lund}\ \emph {et~al.}(2014)\citenamefont {Lund},
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\bibitem [{\citenamefont {Rahimi-Keshari}\ \emph {et~al.}(2015)\citenamefont
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{https://doi.org/10.1103/PhysRevLett.114.060501} {\bibfield {journal}
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\bibitem [{\citenamefont {Hamilton}\ \emph {et~al.}(2017)\citenamefont
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\bibitem [{\citenamefont {Deshpande}\ \emph {et~al.}(2022)\citenamefont
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{Madsen}, \citenamefont {Lavoie}, \citenamefont {Qi}, \citenamefont {Eisert},
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high-dimensional gaussian boson sampling},\ }\href
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\bibitem [{\citenamefont {Zhong}\ \emph {et~al.}(2019)\citenamefont {Zhong},
\citenamefont {Peng}, \citenamefont {Li}, \citenamefont {Hu}, \citenamefont
{Li}, \citenamefont {Qin}, \citenamefont {Wu}, \citenamefont {Zhang},
\citenamefont {Li}, \citenamefont {Zhang}, \citenamefont {Wang},
\citenamefont {You}, \citenamefont {Jiang}, \citenamefont {Li}, \citenamefont
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{Pan}}]{Zhong19}
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{\bibfnamefont {H.}~\bibnamefont {Li}}, \bibinfo {author} {\bibfnamefont
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{author} {\bibfnamefont {L.}~\bibnamefont {Li}}, \bibinfo {author}
{\bibfnamefont {N.-L.}\ \bibnamefont {Liu}}, \bibinfo {author} {\bibfnamefont
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\bibnamefont {Lu}},\ and\ \bibinfo {author} {\bibfnamefont {J.-W.}\
\bibnamefont {Pan}},\ }\bibfield {title} {\bibinfo {title} {Experimental
gaussian boson sampling},\ }\href
{https://doi.org/https://doi.org/10.1016/j.scib.2019.04.007} {\bibfield
{journal} {\bibinfo {journal} {Science Bulletin}\ }\textbf {\bibinfo
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{NoStop}
\bibitem [{\citenamefont {Paesani}\ \emph {et~al.}(2019)\citenamefont
{Paesani}, \citenamefont {Ding}, \citenamefont {Santagati}, \citenamefont
{Chakhmakhchyan}, \citenamefont {Vigliar}, \citenamefont {Rottwitt},
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and\ \citenamefont {Laing}}]{Paesani2019}
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{\bibfnamefont {J.}~\bibnamefont {Wang}}, \bibinfo {author} {\bibfnamefont
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{\bibinfo {journal} {Nature Physics}\ }\textbf {\bibinfo {volume} {15}},\
\bibinfo {pages} {925} (\bibinfo {year} {2019})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Thekkadath}\ \emph {et~al.}(2022)\citenamefont
{Thekkadath}, \citenamefont {Sempere-Llagostera}, \citenamefont {Bell},
\citenamefont {Patel}, \citenamefont {Kim},\ and\ \citenamefont
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\bibitem [{\citenamefont {Arrazola}\ and\ \citenamefont
{Bromley}(2018)}]{Arrazzola_densesubgraph}
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{https://doi.org/10.1103/PhysRevLett.121.030503} {\bibfield {journal}
{\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {121}},\
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\bibitem [{\citenamefont {Schuld}\ \emph {et~al.}(2020)\citenamefont {Schuld},
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{\bibfnamefont {B.}~\bibnamefont {Gupt}},\ }\bibfield {title} {\bibinfo
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}\href {https://doi.org/10.1103/PhysRevA.101.032314} {\bibfield {journal}
{\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {101}},\
\bibinfo {pages} {032314} (\bibinfo {year} {2020})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Brádler}\ \emph {et~al.}(2021)\citenamefont
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\bibitem [{\citenamefont {Arrazola}\ \emph {et~al.}(2021)\citenamefont
{Arrazola}, \citenamefont {Bergholm}, \citenamefont {Br{\'a}dler},
\citenamefont {Bromley}, \citenamefont {Collins}, \citenamefont {Dhand},
\citenamefont {Fumagalli}, \citenamefont {Gerrits}, \citenamefont {Goussev},
\citenamefont {Helt}, \citenamefont {Hundal}, \citenamefont {Isacsson},
\citenamefont {Israel}, \citenamefont {Izaac}, \citenamefont {Jahangiri},
\citenamefont {Janik}, \citenamefont {Killoran}, \citenamefont {Kumar},
\citenamefont {Lavoie}, \citenamefont {Lita}, \citenamefont {Mahler},
\citenamefont {Menotti}, \citenamefont {Morrison}, \citenamefont {Nam},
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\citenamefont {Repingon}, \citenamefont {Sabapathy}, \citenamefont {Schuld},
\citenamefont {Su}, \citenamefont {Swinarton}, \citenamefont {Sz{\'a}va},
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{N.}~\bibnamefont {Killoran}}, \bibinfo {author} {\bibfnamefont {S.~P.}\
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{Lavoie}}, \bibinfo {author} {\bibfnamefont {A.~E.}\ \bibnamefont {Lita}},
\bibinfo {author} {\bibfnamefont {D.~H.}\ \bibnamefont {Mahler}}, \bibinfo
{author} {\bibfnamefont {M.}~\bibnamefont {Menotti}}, \bibinfo {author}
{\bibfnamefont {B.}~\bibnamefont {Morrison}}, \bibinfo {author}
{\bibfnamefont {S.~W.}\ \bibnamefont {Nam}}, \bibinfo {author} {\bibfnamefont
{L.}~\bibnamefont {Neuhaus}}, \bibinfo {author} {\bibfnamefont {H.~Y.}\
\bibnamefont {Qi}}, \bibinfo {author} {\bibfnamefont {N.}~\bibnamefont
{Quesada}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Repingon}},
\bibinfo {author} {\bibfnamefont {K.~K.}\ \bibnamefont {Sabapathy}}, \bibinfo
{author} {\bibfnamefont {M.}~\bibnamefont {Schuld}}, \bibinfo {author}
{\bibfnamefont {D.}~\bibnamefont {Su}}, \bibinfo {author} {\bibfnamefont
{J.}~\bibnamefont {Swinarton}}, \bibinfo {author} {\bibfnamefont
{A.}~\bibnamefont {Sz{\'a}va}}, \bibinfo {author} {\bibfnamefont
{K.}~\bibnamefont {Tan}}, \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
{Tan}}, \bibinfo {author} {\bibfnamefont {V.~D.}\ \bibnamefont {Vaidya}},
\bibinfo {author} {\bibfnamefont {Z.}~\bibnamefont {Vernon}}, \bibinfo
{author} {\bibfnamefont {Z.}~\bibnamefont {Zabaneh}},\ and\ \bibinfo {author}
{\bibfnamefont {Y.}~\bibnamefont {Zhang}},\ }\bibfield {title} {\bibinfo
{title} {Quantum circuits with many photons on a programmable nanophotonic
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{journal} {\bibinfo {journal} {Nature}\ }\textbf {\bibinfo {volume} {591}},\
\bibinfo {pages} {54} (\bibinfo {year} {2021})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Clifford}\ and\ \citenamefont
{Clifford}(2017)}]{clifford2017classical}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
{Clifford}}\ and\ \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont
{Clifford}},\ }\href@noop {} {\bibinfo {title} {The classical complexity of
boson sampling}} (\bibinfo {year} {2017}),\ \Eprint
{https://arxiv.org/abs/1706.01260} {arXiv:1706.01260 [cs.DS]} \BibitemShut
{NoStop}
\bibitem [{\citenamefont {Neville}\ \emph {et~al.}(2017)\citenamefont
{Neville}, \citenamefont {Sparrow}, \citenamefont {Clifford}, \citenamefont
{Johnston}, \citenamefont {Birchall}, \citenamefont {Montanaro},\ and\
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{author} {\bibfnamefont {E.}~\bibnamefont {Johnston}}, \bibinfo {author}
{\bibfnamefont {P.~M.}\ \bibnamefont {Birchall}}, \bibinfo {author}
{\bibfnamefont {A.}~\bibnamefont {Montanaro}},\ and\ \bibinfo {author}
{\bibfnamefont {A.}~\bibnamefont {Laing}},\ }\bibfield {title} {\bibinfo
{title} {Classical boson sampling algorithms with superior performance to
near-term experiments},\ }\href {https://doi.org/10.1038/nphys4270}
{\bibfield {journal} {\bibinfo {journal} {Nature Physics}\ }\textbf
{\bibinfo {volume} {13}},\ \bibinfo {pages} {1153 EP } (\bibinfo {year}
{2017})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Quesada}\ and\ \citenamefont
{Arrazola}(2020)}]{Quesada_exact_simulation}
\BibitemOpen
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{Quesada}}\ and\ \bibinfo {author} {\bibfnamefont {J.~M.}\ \bibnamefont
{Arrazola}},\ }\bibfield {title} {\bibinfo {title} {Exact simulation of
gaussian boson sampling in polynomial space and exponential time},\ }\href
{https://doi.org/10.1103/PhysRevResearch.2.023005} {\bibfield {journal}
{\bibinfo {journal} {Phys. Rev. Research}\ }\textbf {\bibinfo {volume}
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{NoStop}
\bibitem [{\citenamefont {Quesada}\ \emph {et~al.}(2022)\citenamefont
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\bibitem [{\citenamefont {Aaronson}\ and\ \citenamefont
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\BibitemOpen
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\bibitem [{\citenamefont {Tichy}(2015)}]{Tichy}
\BibitemOpen
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}\href {https://doi.org/10.1103/PhysRevA.91.022316} {\bibfield {journal}
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\bibitem [{\citenamefont {Spagnolo}\ \emph {et~al.}(2014)\citenamefont
{Spagnolo}, \citenamefont {Vitelli}, \citenamefont {Bentivegna},
\citenamefont {Brod}, \citenamefont {Crespi}, \citenamefont {Flamini},
\citenamefont {Giacomini}, \citenamefont {Milani}, \citenamefont {Ramponi},
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\bibitem [{\citenamefont {Carolan}\ \emph {et~al.}(2015)\citenamefont
{Carolan}, \citenamefont {Harrold}, \citenamefont {Sparrow}, \citenamefont
{Mart{\'\i}n-L{\'o}pez}, \citenamefont {Russell}, \citenamefont
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\citenamefont {Oguma}, \citenamefont {Itoh}, \citenamefont {Marshall},
\citenamefont {Thompson}, \citenamefont {Matthews}, \citenamefont
{Hashimoto}, \citenamefont {O{\textquoteright}Brien},\ and\ \citenamefont
{Laing}}]{Carolan15}
\BibitemOpen
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\bibnamefont {Matthews}}, \bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
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{O{\textquoteright}Brien}},\ and\ \bibinfo {author} {\bibfnamefont
{A.}~\bibnamefont {Laing}},\ }\bibfield {title} {\bibinfo {title} {Universal
linear optics},\ }\href {https://doi.org/10.1126/science.aab3642} {\bibfield
{journal} {\bibinfo {journal} {Science}\ }\textbf {\bibinfo {volume}
{349}},\ \bibinfo {pages} {711} (\bibinfo {year} {2015})}\BibitemShut
{NoStop}
\bibitem [{\citenamefont {Crespi}\ \emph {et~al.}(2016)\citenamefont {Crespi},
\citenamefont {Osellame}, \citenamefont {Ramponi}, \citenamefont
{Bentivegna}, \citenamefont {Flamini}, \citenamefont {Spagnolo},
\citenamefont {Viggianiello}, \citenamefont {Innocenti}, \citenamefont
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{\bibinfo {journal} {Nature Communications}\ }\textbf {\bibinfo {volume}
{7}},\ \bibinfo {pages} {10469} (\bibinfo {year} {2016})}\BibitemShut
{NoStop}
\bibitem [{\citenamefont {Viggianiello}\ \emph
{et~al.}(2018{\natexlab{a}})\citenamefont {Viggianiello}, \citenamefont
{Flamini}, \citenamefont {Innocenti}, \citenamefont {Cozzolino},
\citenamefont {Bentivegna}, \citenamefont {Spagnolo}, \citenamefont {Crespi},
\citenamefont {Brod}, \citenamefont {Galv{\~{a}}o}, \citenamefont
{Osellame},\ and\ \citenamefont {Sciarrino}}]{Viggianiello18}
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{R.}~\bibnamefont {Osellame}},\ and\ \bibinfo {author} {\bibfnamefont
{F.}~\bibnamefont {Sciarrino}},\ }\bibfield {title} {\bibinfo {title}
{Experimental generalized quantum suppression law in sylvester
interferometers},\ }\href {https://doi.org/10.1088/1367-2630/aaad92}
{\bibfield {journal} {\bibinfo {journal} {New J. Phys.}\ }\textbf {\bibinfo
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\bibitem [{\citenamefont {Walschaers}\ \emph {et~al.}(2016)\citenamefont
{Walschaers}, \citenamefont {Kuipers}, \citenamefont {Urbina}, \citenamefont
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{\bibfnamefont {M.~C.}\ \bibnamefont {Tichy}}, \bibinfo {author}
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{\bibfnamefont {A.}~\bibnamefont {Buchleitner}},\ }\bibfield {title}
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\bibitem [{\citenamefont {Giordani}\ \emph {et~al.}(2018)\citenamefont
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\bibinfo {pages} {173} (\bibinfo {year} {2018})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Agresti}\ \emph {et~al.}(2019)\citenamefont
{Agresti}, \citenamefont {Viggianiello}, \citenamefont {Flamini},
\citenamefont {Spagnolo}, \citenamefont {Crespi}, \citenamefont {Osellame},
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}\href {https://doi.org/10.1103/PhysRevX.9.011013} {\bibfield {journal}
{\bibinfo {journal} {Phys. Rev. X}\ }\textbf {\bibinfo {volume} {9}},\
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\bibitem [{\citenamefont {Flamini}\ \emph {et~al.}(2019)\citenamefont
{Flamini}, \citenamefont {Spagnolo},\ and\ \citenamefont
{Sciarrino}}]{FlaminiTSNE}
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}\bibfield {title} {\bibinfo {title} {Visual assessment of multi-photon
interference},\ }\href {https://doi.org/10.1088/2058-9565/ab04fc} {\bibfield
{journal} {\bibinfo {journal} {Quantum Science and Technology}\ }\textbf
{\bibinfo {volume} {4}},\ \bibinfo {pages} {024008} (\bibinfo {year}
{2019})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Viggianiello}\ \emph
{et~al.}(2018{\natexlab{b}})\citenamefont {Viggianiello}, \citenamefont
{Flamini}, \citenamefont {Bentivegna}, \citenamefont {Spagnolo},
\citenamefont {Crespi}, \citenamefont {Brod}, \citenamefont {Galvão},
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{Sciarrino}}]{Viggianiello17optimal}
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{\bibfnamefont {E.~F.}\ \bibnamefont {Galvão}}, \bibinfo {author}
{\bibfnamefont {R.}~\bibnamefont {Osellame}},\ and\ \bibinfo {author}
{\bibfnamefont {F.}~\bibnamefont {Sciarrino}},\ }\bibfield {title} {\bibinfo
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}\href {https://doi.org/https://doi.org/10.1016/j.scib.2018.10.009}
{\bibfield {journal} {\bibinfo {journal} {Sci. Bull.}\ }\textbf {\bibinfo
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\bibitem [{\citenamefont {Renema}(2020{\natexlab{a}})}]{Renema_partial_2020}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~J.}\ \bibnamefont
{Renema}},\ }\bibfield {title} {\bibinfo {title} {Simulability of partially
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{https://doi.org/10.1103/PhysRevA.101.063840} {\bibfield {journal} {\bibinfo
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{pages} {063840} (\bibinfo {year} {2020}{\natexlab{a}})}\BibitemShut
{NoStop}
\bibitem [{\citenamefont {Renema}\ \emph {et~al.}(2018)\citenamefont {Renema},
\citenamefont {Menssen}, \citenamefont {Clements}, \citenamefont {Triginer},
\citenamefont {Kolthammer},\ and\ \citenamefont
{Walmsley}}]{Renema_2018_classical}
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{author} {\bibfnamefont {G.}~\bibnamefont {Triginer}}, \bibinfo {author}
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{\bibfnamefont {I.~A.}\ \bibnamefont {Walmsley}},\ }\bibfield {title}
{\bibinfo {title} {Efficient classical algorithm for boson sampling with
partially distinguishable photons},\ }\href
{https://doi.org/10.1103/PhysRevLett.120.220502} {\bibfield {journal}
{\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {120}},\
\bibinfo {pages} {220502} (\bibinfo {year} {2018})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Moylett}\ \emph {et~al.}(2019)\citenamefont
{Moylett}, \citenamefont {Garc{\'{\i}}a-Patr{\'{o}}n}, \citenamefont
{Renema},\ and\ \citenamefont {Turner}}]{Moylett_2019}
\BibitemOpen
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\bibnamefont {Renema}},\ and\ \bibinfo {author} {\bibfnamefont {P.~S.}\
\bibnamefont {Turner}},\ }\bibfield {title} {\bibinfo {title} {Classically
simulating near-term partially-distinguishable and lossy boson sampling},\
}\href {https://doi.org/10.1088/2058-9565/ab5555} {\bibfield {journal}
{\bibinfo {journal} {Quantum Science and Technology}\ }\textbf {\bibinfo
{volume} {5}},\ \bibinfo {pages} {015001} (\bibinfo {year}
{2019})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Phillips}\ \emph {et~al.}(2019)\citenamefont
{Phillips}, \citenamefont {Walschaers}, \citenamefont {Renema}, \citenamefont
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{Walmsley}}, \bibinfo {author} {\bibfnamefont {N.}~\bibnamefont {Treps}},\
and\ \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Sperling}},\
}\bibfield {title} {\bibinfo {title} {Benchmarking of gaussian boson
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{https://doi.org/10.1103/PhysRevA.99.023836} {\bibfield {journal} {\bibinfo
{journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {99}},\ \bibinfo
{pages} {023836} (\bibinfo {year} {2019})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Popova}\ and\ \citenamefont
{Rubtsov}(2021)}]{popova2021cracking}
\BibitemOpen
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threshold for gaussian boson sampling}} (\bibinfo {year} {2021}),\ \Eprint
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\bibitem [{\citenamefont {Renema}(2020{\natexlab{b}})}]{renema2020marginal}
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{arXiv:2012.14917 [quant-ph]} \BibitemShut {NoStop}
\bibitem [{\citenamefont {Villalonga}\ \emph {et~al.}(2021)\citenamefont
{Villalonga}, \citenamefont {Niu}, \citenamefont {Li}, \citenamefont {Neven},
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{S.}~\bibnamefont {Boixo}},\ }\href@noop {} {\bibinfo {title} {Efficient
approximation of experimental gaussian boson sampling}} (\bibinfo {year}
{2021}),\ \Eprint {https://arxiv.org/abs/2109.11525} {arXiv:2109.11525
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\end{document} | arXiv |
A simplicial generalisation of the Bloch ball
February 5, 2021. I explore unitary orbits of density matrices for finite-dimensional quantum systems. The upshot is a neat scheme for representing orbits using simplices.
The Bloch sphere represents the space of pure states on a single qubit (see also this recent post). The "Bloch ball" is the space of all density matrices on the qubit. It fills in the Bloch sphere with concentric spheres of increasing mixedness, and at the centre is the maximally mixed state $I_2/2$, where $I_d$ will denote the $d \times d$ identity matrix.
Spheres arise naturally. They carry the structure of the unitary group $\mathrm{U}(2)$ acting on qubits, once we have modded out by the phase ambiguity:
\[\frac{\mathrm{U}(2)}{\mathrm{U}(1)} = \mathrm{SU}(2).\]
This is a double cover of the rotation group $\mathrm{SO}(3)$, which acts transitively on the sphere. (The "double cover" part gives us spinors.) Thus, spheres occur naturally as unitary orbits, and indeed, each concentric sphere in the Bloch ball is such an orbit. The question is whether this generalises nicely to higher dimensions.
The Bloch ball
Let's think about the Bloch ball in a little more detail. Each density matrix $\rho$ is a $2\times 2$ matrix acting on the space of qubits, which is positive and has unit trace. Positivity just means that, for every state $|\psi\rangle$,
\[\langle \psi | (\rho | \psi \rangle) \geq 0.\]
Hence, $\rho$ is Hermitian, since the reality of this inner product implies
\[\langle \psi | (\rho | \psi \rangle) = (\langle \psi | \rho^\dagger) |\psi \rangle \quad \Longrightarrow \quad \rho = \rho^\dagger.\]
In turn, this means that $\rho$ is unitarily diagonalisable, i.e. $U^\dagger \rho U = \Lambda$ for some diagonal matrix $\Lambda$ and unitary matrix $U^\dagger U = UU^\dagger = I$. It's also clear these eigenvalues must be positive. In fact, since the permutation matrices are unitary, we can arrange the eigenvalues in decreasing size, so that every $2 \times 2$ density matrix is unitarily equivalent to some matrix
\[\Lambda(p) = \begin{bmatrix} p & \\ & 1-p \end{bmatrix}\]
for $p \in [1/2, 1]$. The maximally mixed density $I_2/2$ has a trivial orbit, since it always gets mapped to itself:
\[U^\dagger I_2 U = U^\dagger U = I_2.\]
We can measure the distance from this matrix to $\Lambda(p)$ using the Frobenius norm, aka Hilbert-Schmidt norm. This is just the usual vector norm where we treat a matrix $A = [a_{ij}]$ as a big vector:
\[||A||^2 = \sum_{ij} |a_{ij}|^2 = \mbox{Tr}[A^\dagger A].\]
\[\begin{align*} ||\Lambda(p) - \tfrac{1}{2}I_2||^2 & = \left|\left| \begin{bmatrix} p - 1/2 & \\ & 1/2-p \end{bmatrix} \right|\right|^2 \end{align*} = 2\left(p - \tfrac{1}{2}\right)^2.\]
It's easy to see that any density matrix in the unitary orbit of $\Lambda(p)$ has the same distance, since we can use $I_2 = U^\dagger I_2 U$, i.e. it is a class function:
\[\begin{align*} ||U^\dagger \Lambda U - \tfrac{1}{2}I_2||^2 & = \mbox{Tr}\left[(U^\dagger \Lambda U - \tfrac{1}{2}I_2)^\dagger (U^\dagger \Lambda U - \tfrac{1}{2}I_2)\right]\\ & = \mbox{Tr}\left[U^\dagger (\Lambda - \tfrac{1}{2}I_2)^\dagger UU^\dagger (\Lambda - \tfrac{1}{2}I_2) U\right]\\ & = \mbox{Tr}\left[(\Lambda - \tfrac{1}{2}I_2)^\dagger (\Lambda - \tfrac{1}{2}I_2) \right] = ||\Lambda - \tfrac{1}{2}I_2||^2. \end{align*}\]
We can define distance between densities as the Hilbert-Schmidt norm times a positive constant $C$. We choose $C = \sqrt{2}$ so that for pure states with $p = 1$, the associated distance is $r = 2(p - 1/2) = 1$. In general, since each such $r$ is associated with a unique $\Lambda(p)$, we conclude that the space of $2\times 2$ density matrices is a ball consisting of concentric, transitive orbits of the unitary group, with the pure states at $p = 1$, the maximally mixed state at $p = 0$, and radius $r = 2(p - 1/2)$ for the orbit of $\Lambda(p)$.
Orbital mechanics
A similar story holds in higher dimensions. Density matrices are positive and unit trace, so each orbit in dimension $d$ has a canonical representative of the form
\[\Lambda = \mathrm{diag}(p_1, p_2, \ldots, p_d),\]
where the positivity of $\rho$ and unit trace condition imply
\[\sum_{i=1}^d p_i = 1, \quad p_i \geq 0,\]
and we can arrange eigenvalues in descending order:
\[p_1 \geq p_2 \geq \cdots \geq p_d \geq 0.\]
The constraint that the eigenvalues sum to $1$ means that we only need $p_1, p_2, \ldots, p_{d-1}$ to uniquely specify a canonical representative $\Lambda(p_1, p_2, \ldots, p_{d-1})$. We can repeat the calculations from above to show that $I_d/d$ has a trivial orbit, and that any density matrix in the orbit of $\Lambda(p_1, \ldots, p_{d-1})$ has a fixed distance to the mixed state:
\[r^2(p_1, \ldots, p_{d-1}) = C_d\sum_{i=1}^d \left(p_i - \frac{1}{d}\right)^2,\]
where we choose $C_d$ so that the pure states, with $p_1 = 1, p_2 = \cdots = p_d = 0$, have distance $r = 1$. For completeness, we note that
\[C_d = \frac{d^2}{d^2 - 2d + 2}.\]
It's a bit trickier to see what the orbits look like, but in the same way that $I_d$ is fixed by the group $\mathrm{U}(d)$, we can read off fixed subgroups from the eigenvalue decomposition. For instance, a pure state has
\[p_1 = 1, \quad p_2 = \cdots = p_d = 0.\]
The first factor is fixed by $\mathrm{U}(1)$ (corresponding to global phase), while the last $d - 1$ factors are fixed by $\mathrm{U}(d-1)$. These act independently, so that the stabiliser of a pure state is $\mathrm{U}(1) \times \mathrm{U}(d-1)$. By the orbit-stabiliser theorem, the orbit of pure states has the (coset) structure
\[\frac{\mathrm{U}(d)}{\mathrm{U}(1) \times \mathrm{U}(d - 1)}.\]
Since $\mathrm{U}(d)$ has dimension $d^2$, this pure space orbit has dimension
\[d^2 - 1^2 - (d - 1)^2 = 2d - 2,\]
and lies on a unit sphere $\mathbb{S}^{2d-2}$ in our Hilbert-Schmidt metric. This agrees with the Bloch sphere for $d = 2$. This seems rather nice, but in general, the orbits will be horrible. First of all, spheres of radius $r < 1$ around the mixed state will now be made up of uncountably many orbits, since there are uncountably many sets of $p_i$ which solve
\[r^2 = C_d\sum_{i=1}^d \left(p_i -\frac{1}{d}\right)^2\]
for $r < 1$. And orbits can be more elaborate for other eigenvalue structures. For instance, if we lump the $p_i$ into $k$ sets of distinct eigenvalues,
\[P_1, P_2, \ldots, P_K,\]
with multiplicity $\mu_J$ associated to eigenvalue $P_J$, then the same argument as above shows that the coset structure is
\[\frac{\mathrm{U}(d)}{\mathrm{U}(\mu_1) \times \cdots \times \mathrm{U}(\mu_K)},\]
known to mathematicians as a partial flag variety. These orbits have dimension
\[D = d^2 - \sum_{J=1}^K \mu_J^2,\]
and lie on a sphere of radius
\[r^2 = C_d\sum_{J=1}^K \mu_J^2\left(P_J - \frac{1}{d}\right)^2.\]
Note that while mixed states are closer to the maximally mixed state, unlike the Bloch ball, they do not lie inside the orbit of pure states. Typically, they have more dimensions! For instance, a generic point with no symmetries (distinct $p_i$), the cosets are of the form
\[\frac{\mathrm{U}(d)}{(\mathrm{U}(1))^d}\]
with dimension $d^2 - d$, so for $d > 2$, these are always bigger than the pure state orbits. It's certainly possible to say more about this, but who wants to. It's a mess!
The simplicial wedge
Our modest goal will be to tidy up some of the mess. The main observation is that the eigenvalues $p_i$ form a probability distribution over $d$ outcomes. If they had an arbitrary order, they would live on the standard $(d-1)$-simplex $\Delta_{d-1}$, but because they are arranged in decreasing order, they live on the simplicial "wedge":
\[W_{d-1} = \left\{(p_1, \ldots, p_d) : \sum_{i=1}^d p_i = 1, p_1 \geq p_2 \geq \cdots \geq p_d \geq 0\right\}.\]
Note that the subscript denotes the number of independent parameters. We can illustrate these ideas for $d = 2$:
We start with the $1$-simplex $\Delta_1$, and divide it two to get the wedge $W_1$. The black dot at the top is the orbit of pure states, and the white dot the maximally mixed state. In general, the wedge $W_{d-1}$ is almost a quotient of $\Delta_{d-1}$ by its symmetry group, the set of permutations $S_d$. But the wedge has literal "edge cases", stabilised by subgroups of $S_d$ in a way that mirrors the corresponding unitary orbits. More precisely, if a point in $W_{d-1}$ is stabilised by $S_{\mu_1} \times \cdots \times S_{\mu_K}$, then the corresponding coset structure for the orbit is the partial flag variety
\[\frac{\mathrm{U}(d)}{\mathrm{U}(\mu_1) \times \cdots \times \mathrm{U}(\mu_K)}.\]
For instance, pure states have canonical representative
\[(1, 0, 0, \ldots, 0) \in W_{d-1},\]
which is stabilised by the subgroup $S_1 \times S_{d-1}$. This correctly gives the coset orbit
The maximally mixed state, and centroid of the whole simplex, has coordinates
\[\frac{1}{d}(1, 1, \ldots, 1),\]
and is stabilised by the full group $S_d$. As we expect, the orbit is trivial. We can see how this works for a qutrit below. We start with the $2$-simplex $\Delta_2$, an equilateral triangle, and cut out the wedge $W_2$:
At the top we have the pure states as usual, and the mixed state at the white centroid. The grey dot represents the fully mixed state on two basis elements. Note that, along the red edges, two coordinates agree, and in fact, each represents a copy of $W_1$, coinciding at the centroid. In general, orbit degeneracies occur precisely at sub-wedges $W_K$ with interiors parameterised by the coordinates $P_1, \ldots, P_K$ introduced above. But when distinct sub-wedge coincides, we get even more degeneracy. So, the apparent randomness of orbits is somewhat tamed by geometric hierarchy.
Finally, to relate this back to spheres, the nice thing about using the Frobenius norm is that the distance between a density matrix and the maximally mixed matrix is just proportional to the Euclidean distance on the wedge. So we can literally draw concentric spheres emanating from the centroid! Our scheme does not do away with all the messiness of the orbits. But it does provide a simple way to organise and read off some of their basic properties, and generalises in a reasonably natural way the concentric spheres of the Bloch ball.
Maths Physics QC | CommonCrawl |
\begin{definition}[Definition:Maximal Condition]
Let $A$ be a commutative ring with unity.
Let $M$ be an $A$-module.
Let $(D,\subseteq)$ be the set of submodules of $M$ ordered by inclusion.
Then the hypothesis
:''Every non-empty subset of $D$ has a maximal element''
is called the '''maximal condition''' on submodules.
\end{definition} | ProofWiki |
Solutions of Pythagorean Equation
1.1 Primitive Solutions of Pythagorean Equation
1.2 General Solutions of Pythagorean Equation
2 Sequence
3 Historical Note
Primitive Solutions of Pythagorean Equation
The set of all primitive Pythagorean triples is generated by:
$\tuple {2 m n, m^2 - n^2, m^2 + n^2}$
$m, n \in \Z_{>0}$ are (strictly) positive integers
$m \perp n$, that is, $m$ and $n$ are coprime
$m$ and $n$ are of opposite parity
$m > n$
General Solutions of Pythagorean Equation
Let $x, y, z$ be a solution to the Pythagorean equation.
Then $x = k x', y = k y', z = k z'$, where:
$\tuple {x', y', z'}$ is a primitive Pythagorean triple
$k \in \Z: k \ge 1$
The sequence of solutions of the Pythagorean equation can be tabulated as follows:
$\begin{array} {r r | r r | r r r | c} m & n & m^2 & n^2 & 2 m n & m^2 - n^2 & m^2 + n^2 \\ \hline 2 & 1 & 4 & 1 & 4 & 3 & 5 & \text{Primitive} \\ \hline 3 & 1 & 9 & 1 & 6 & 8 & 10 \\ 3 & 2 & 9 & 4 & 12 & 5 & 13 & \text{Primitive} \\ \hline 4 & 1 & 16 & 1 & 8 & 15 & 17 & \text{Primitive} \\ 4 & 2 & 16 & 4 & 16 & 12 & 20 \\ 4 & 3 & 16 & 9 & 24 & 7 & 25 & \text{Primitive} \\ \hline 5 & 1 & 25 & 1 & 10 & 24 & 26 \\ 5 & 2 & 25 & 4 & 20 & 21 & 29 & \text{Primitive} \\ 5 & 3 & 25 & 9 & 30 & 16 & 34 \\ 5 & 4 & 25 & 16 & 40 & 9 & 41 & \text{Primitive} \\ \hline 6 & 1 & 36 & 1 & 12 & 35 & 37 & \text{Primitive} \\ 6 & 2 & 36 & 4 & 24 & 32 & 40 \\ 6 & 3 & 36 & 9 & 36 & 27 & 45 \\ 6 & 4 & 36 & 16 & 48 & 20 & 52 \\ 6 & 5 & 36 & 25 & 60 & 11 & 61 & \text{Primitive} \\ \hline 7 & 1 & 49 & 1 & 14 & 48 & 50 \\ 7 & 2 & 49 & 4 & 28 & 45 & 53 & \text{Primitive} \\ 7 & 3 & 49 & 9 & 42 & 40 & 58 \\ 7 & 4 & 49 & 16 & 56 & 33 & 65 & \text{Primitive} \\ 7 & 5 & 49 & 25 & 70 & 24 & 74 \\ 7 & 6 & 49 & 36 & 84 & 13 & 85 & \text{Primitive} \\ \hline \end{array}$
It is clear from the cuneiform tablet Plimpton $\mathit { 322 }$ that the ancient Babylonians of $2000$ BCE were familiar with this result.
The complete solution of the Pythagorean equation was known to Diophantus of Alexandria.
It forms problem $8$ of the second book of his Arithmetica.
It was in the margin of his copy of Bachet's translation of this where Pierre de Fermat made his famous marginal note that led to the hunt for the proof of Fermat's Last Theorem.
1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.9$: Hypatia (A.D. $\text {370?}$ – $\text {415}$)
2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $7$: Patterns in Numbers: Diophantus
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Diophantine Equations
Pythagorean Triples
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Coefficient of variation calculator
Coefficient of variation (CV) calculator - to find the ratio of standard deviation ((σ) to mean (μ). The main purpose of finding coefficient of variance (often abbreviated as CV) is used to study of quality assurance by measuring the dispersion of the population data of a probability or frequency distribution, or by determining the content or quality of the sample data of substances. The method of measuring the ratio of standard deviation to mean is also known as relative standard. The proportion of variance calculator calculates the same statistical values for both data for proportion and data for mean: No. of samples Mean (μ) Standard deviation (σ) Coefficient Of Variance Also, this percentage of variation calculator calculate the Coefficient Of Variance To find the coefficient of variation using the above formula, follow the below steps: Calculate the mean of the given data set. You use our mean calculator for that purpose. Calculate the standard deviation for the given data set. You can also use our standard deviation calculator to calculate.... The following equation can be used to calculate the coefficient of variation of a data set, usually a population or sample. C = (σ / μ) * 100% Where C is the coefficient of variation (%) σ is the standard deviatio Formula for coefficient of variation The coefficient of variation is the ratio between the inverse of the mean and the standard deviation: CV = σ / μ where σ is the sample standard deviation and μ is the sample mean
Coefficient of Variation Calculator. This tool will calculate the coefficient of variation of a set of data. The coefficient of variation is a measure of spread that tends to be used when it is necessary to compare the spread of numbers in two datasets that have very different means Coefficient of variation calculator. For coefficient of variation calculation, please enter numerical data separated with comma (or space, tab, semicolon, or newline). For example: -290.3 752.4 -176.7 201.2 362.4 -618.9 357.3 341.0 541.8 627.6 142.3 785.8 221.3. Calculate Der Variationskoeffizient-Rechner kann verwendet werden, um den Koeffizienten einer Variation einer Menge von Zahlen zu berechnen. Variationskoeffizient . In der Wahrscheinlichkeitstheorie und Statistik ist der Variationskoeffizient (CV) ein Maß für die Streuung einer Wahrscheinlichkeitsverteilung. Diese wird auch Variationskoeffizient oder Abweichungskoeffizient genannt. Der. A coefficient of variation, often abbreviated as CV, is a way to measure how spread out values are in a dataset relative to the mean. It is calculated as: It is calculated as: CV = σ / �
Coefficient of Variation Calculato
The Coefficient of Variation Calculator is used to calculate the coefficient of variation of a set of numbers The Coefficient of Variation (CV) Calculator to find out the values of number of inputs, Mean, Variance Coefficient and Standard Deviation with respect to the input values of data set, it is for data analysis. Coefficient of Variance Formula Coefficient of Variation Cv = Standard Deviation / Mea
Coefficient of Variation Calculator. Coefficient of Variation Calculator is a free online tool that displays the ratio of the standard deviation to the mean. BYJU'S online coefficient of variation calculator tool makes the calculation faster and it displays the coefficient of variation in a fraction of seconds The online Coefficient of Variation Calculator is used to calculate and find the coefficient of dispersion of a probability distribution by just entering the set of numbers that are separated by the comma. Example: Calculate the coefficient of Variation for the given details. Enter the range of values (seperated by comma) = 1, 2, 1,5,1,6,8,2,0 . Solution: Apply Formula: Cv = σ / μ Cv = 0. Coefficient of variance calculator uses coefficienct_of_variance = (Standard Deviation/Mean of data)*100 to calculate the Coefficient of variance, The Coefficient of variance formula is defined as the value of coefficient of variance when the value of standard deviation and mean is given. Coefficient of variance and is denoted by Vc symbol
Coefficient of Variation Calculator. Compute the coefficient of variation for a variable of interest, given the variable's mean and standard deviation. Knowing the coefficient of variation for a particular variable can be very valuable in analytics studies as a way of quantifying the degree of dispersion within a set of observations. Please provide the necessary values, and then click. The Coefficient of Variation Calculator is an online tool that can be used free of cost. It displays the ratio of the standard deviation to the mean. In the subject of Statistics, the coefficient of variation is defined as the ratio of the standard deviation to the mean. It is mostly used to compare the degree of variation from one data set value to the other data set values. So, if the. This Coefficient of Variation Calculator looks at a set of observations (the sample) and calculates the Coefficient of Variation. The coefficient of variation (CV)is a standardized measure of dispersion of a probability distribution or frequency distribution. It shows the extent of variability in relation to the mean of the population A coefficient of variation, often abbreviated as CV, is a way to measure how spread out values are in a dataset relative to the mean. It is calculated as: CV = σ / � The coefficient of variation (CV) is defined as the ratio of the standard deviation to the mean , =. [1] It shows the extent of variability in relation to the mean of the population. The coefficient of variation should be computed only for data measured on a ratio scale , that is, scales that have a meaningful zero and hence allow relative comparison of two measurements (i.e., division of one measurement by the other)
The procedure to use the coefficient of variation calculator is as follows: Step 1: Enter the numbers separated by a comma in the respective input field. Step 2: Now click the button Calculate Coefficient of Variation to get the result. Step 3: Finally, the coefficient of variation for the given data values will be displayed in the output. In this video tutorial, I will show you how to calculate the coefficient of variation (CV), by using Microsoft Excel. The CV is a measure of assay precision. To use this online calculator for Mean Using Coefficient Of Variation, enter Standard Deviation (σ) and Coefficient of variation (CV) and hit the calculate button. Here is how the Mean Using Coefficient Of Variation calculation can be explained with given input values -> 4.083 = 40.83/10 Option 3 is bonds with a volatility of 3% and an expected return of 4%. To find the best option among the three, Investor A plans to calculate the coefficient of variation for all three. Using the above formula following are the CVs of these three options: Stock CV = (9%/15%)* 100= 60%. ETF CV = (8%/12%)* 100 = 67% To calculate the coefficient of variation in her bond investment, Jamila inputs her volatility percentage of 6 and her expected return percentage of 4. Bond investment: CV = (6/4) x 100% Divide the volatility and return first. CV = 1.5 x 100
Calculating the coefficient of variation involves a simple ratio. Simply take the standard deviation and divide it by the mean. Higher values indicate that the standard deviation is relatively large compared to the mean. For example, a pizza restaurant measures its delivery time in minutes Coefficient of variation calculator finds the coefficient of variation by taking the range of values as input. If you are wondering how to calculate coefficient of variation, then, stop being too curious, because we are going to elaborate all that in next sections Coefficient of Variation Ratio (CVR) The coefficient of variation ratio compares your laboratory precision for a specific test to the CV of other laboratories performing the same test. Use the following formula to calculate the CVR: In Unity Real Time™ online, the CVR appears on the Data Analysis Grid and on the following Unity™ Interlaboratory Reports: Laboratory Comparison Reports.
The procedure to use the coefficient of variation calculator is as follows: Step 1: Enter the numbers separated by a comma in the given input field. Step 2: Now, hit the button Calculate in order to get the desired result. Step 3: Finally, the coefficient of.. More About this Coefficient of Variation Calculator The Coefficient of Variation (CV in short) is a typical measure of variation, which measures the relative variation in a sample with respect to the size of the mean. Indeed, it consider the size of the sample standard deviation in relative terms to the sample mean Coefficient of Variation Calculator helps calculating Coefficient of Variation (C.O.V.). What is Coefficient of Variation (C.O.V.)? In probability theory and statistics, the coefficient of variation (CV), also known as relative standard deviation (RSD), is a standardized measure of dispersion of a probability distribution or frequency distribution
This calculator calculates the coefficient of variation of the given data for you in less than a minute and it's free. Also you will understand, how you can calculate the coefficient of variation on your own, so that you will not need calculator to do this Coefficient of variation (CV) is also known as Relative Standard Deviation (RSD). The CV or RSD is widely used in analytical chemistry. Coefficient of variation (CV) is important in the field of probability & statistics to measure the relative variability. Use this online calculator to find the coefficient of variation for the given set of data This Relative Standard Deviation calculator is used to find the Coefficient of Variation(CV) with the help of Ratio of Standard Deviation and Mean Calculating Intra-Assay CV: The Average Coefficient of Variation between Duplicates. In this example cortisol concentrations are measured in duplicate for 40 samples. The % CV for each sample is calculated by finding the standard deviation of results 1 and 2, dividing that by the duplicate mean, and multiplying by 100
Coefficient of variation calculator - variance & standard
Measure them each 12 - 20 times, calculate from the results the average, min, max, standard deviation (SD). The CV (coerfficient of variation) is the quotient: SD/average*100. In ELISA you will. Coefficient of Variation for grouped data. Use this calculator to find the coefficient of variation (CV) for grouped (raw) data The same formula is used in this coefficient of variation calculator to find the best competing model which has the lesser uncertainty or has the deviation very close to the mean. Statistics formula to calculate co-efficient of variation. Step by Step Workout. Grade school students, beginners or learners may generate the complete work with step by step calculation to solve the coefficient of. Using the first two raw moments to calculate the variance as well as the third moment, the following calculates the moment coefficient of skewness, based on the form in (3): The above calculation shows that the rate parameter has no effect on skewness. The example in Figure 1 has , giving a coefficient of skewness of = 1.414213562. In general. Based on the calculations above, Fred wants to invest in the ETF because it offers the lowest coefficient (of variation) with the most optimal risk-to-reward ratio. Related Readings CFI offers the Financial Modeling & Valuation Analyst (FMVA)™ Become a Certified Financial Modeling & Valuation Analyst (FMVA)® certification program for those looking to take their careers to the next level
Coefficient of Variation Calculator - Step by Ste
e that the little things are infinitely the most important (Sherlock Holmes) Search this site Confidence Interval of the Coefficient of Variation. The confidence interval can be estimated for a coefficient of.
Descriptive statistics summarize certain aspects of a data set or a population using numeric calculations. Examples of descriptive statistics include: mean, average. midrange. standard deviation. quartiles. This calculator generates descriptive statistics for a data set. Enter data values separated by commas or spaces
The coefficient of variation of the observations is used to describe the level of variability within a population independently of the absolute values of the observations. If absolute values are similar, populations can be compared using their standard deviations. But if they differ markedly (for example, the weights of mice and elephants), or are of different variables (for example, weight.
Calculating the Sample Variance and the Standard Deviation. The third step of the process is finding the sample variance. Following the formula that we went over earlier, we can obtain 10.72 dollars squared and 3793.69 pesos squared. The respective sample standard deviations are 3.27 dollars and 61.59 pesos, as shown in the picture below. A Few Observations. Let's make a couple of.
Coefficient Of Variation - CV: A coefficient of variation (CV) is a statistical measure of the dispersion of data points in a data series around the mean. It is calculated as follows: (standard.
Coefficient of Variation Calculator - Calculator Academ
Coefficient of variation is given by. C V = s x x ¯ × 100. where, x ¯ = 1 N ∑ i = 1 n f i x i is the sample mean of X, N total number of observations, s x = V ( x) is the standard deviation of X, s x 2 = V ( x) = 1 N ∑ i = 1 n f i x i 2 − ( x ¯) 2 is the variance of X. Coefficient of variation of one data set is lower than the.
First, calculate the mean (average) between the readings 1-3 on each plate: We then use the CV formula above in Excel to calculate the intra -assay CV for each plate. This is the variation of measurements from the same plate (between readings 1, 2 and 3): Finally, we can work out the inter- assay CV between the mean values from the three plates
Coefficient of Variation = Standard Deviation / Mean. Steps to Calculate the Coefficient of Variation: Step 1: Calculate the mean of the data set. Mean is the average of all the values and can be calculated by taking the sum of all the values and then dividing it by a number of data points. Step 2: Then compute the standard deviation of the.
e the number of variables in the data series, denoted by N. Step 3: Next, deter
Coefficient of Variation Calculator - calculates the
Coefficient of Variation Formula. Calculator. Formula. The formula for the calculation of the coefficient of variation is derived using the mean and the standard deviation. The ratio of the mean to standard deviation is termed as RSD. It is a dimensionless number
With aggregate I can use mean, sum by default but not coefficient variation. For example. aggregate (data, as.columnname, FUN=mean) Works fine. I have a custom function for calculating coefficient of variation but not sure how to use it with aggregate. co.var <- function (x) ( 100*sd (x)/mean (x) ) I have tried
Coefficient of variation provides a standardized measure of comparing risk and return of different investments. A rational investor would select an investment with lowest coefficient of variation. Sharpe ratio is a similar statistic which measures excess return per unit of risk. Formula $$ \text{Coefficient of Variation} \\ = \frac{\text{Standard Deviation of the Investment}}{\text{Expected.
Step 4. To find the coefficient of variation, input the formula =A8/A9 for this example or your actual range in a blank cell and press E** nter** to calculate the coefficient of variance. Calculate the coefficient of variance. Image Credit: Gurudev Ravindran
Coefficient of Variation = (Standard Deviation / Mean) CV = σ / ǩ, Tip: Multiplying the coefficient by 100 is an optional step. By doing so, you will get a percentage, as opposed to a decimal. How to find a coefficient of variation in Excel. The steps below outline how you can use Excel to calculate the coefficient of variation. Our guide was.
The coefficient of variation should be computed only for data measured on a ratio scale, that is, scales that have a meaningful zero and hence allow relative comparison of two measurements (i.e., division of one measurement by the other). The coefficient of variation may not have any meaning for data on an interval scale
In statistic measure, coefficient of variation is used to find the range of variability through the data given. In terms of finance, coefficient of variation is used to find the amount of risk involved with respect to the amount invested. If the ratio between standard deviation and mean is low then the risk involved in the investment is also low. Coefficient of variation is the ratio between.
The coefficient of variation is calculated by dividing the standard deviation by the mean, as follows: coefficient of variation = s/x. To present this number as a percentage, multiply the result of the coefficient of variation calculation by 100. Previous Page Coefficient of variation. Another way to describe the variation of a test is calculate the coefficient of variation, or CV. The CV expresses the variation as a percentage of the mean, and is calculated as follows: CV% = (SD/Xbar)100. In the laboratory, the CV is preferred when the SD increases in proportion to concentration. For example, the data from a replication experiment may show an SD of. I'm a biochemist and I usually compare the variability of my measurements in terms of coefficient of variation (CV) since I can visualize the deviations more easily in terms of percentages deviation from my mean value. Now, I measured an analyte 20 times in a sample in one month, and again 20 times the second month. Now I'd like to calculate the mean CV of those two sets of measurements. I.
Coefficient of variation calculator (statistics
One of the ways demand planners have tried to answer this question is through the use of a calculation called Coefficient of Variation (CV). Some people call it standardized or normalized standard deviation (StdDev). In layman's terms, Coefficient of Variation is a measure of how closely grouped a particular data set is. The formula for CV is: CV = StdDev (σ) / Mean (µ). In this blog post. Comparison of Coefficients of Variation free online statistical calculator. You are using a Guest account. Some functionality has been disabled. Register or Sign in Coefficient of Determination Calculator is a free online tool that displays the variability of one factor in relation to the other factor. CoolGyan online coefficient of determination calculator tool makes the calculation faster and it displays the coefficient of determination value in a fraction of seconds
Video: Variationskoeffizient-Rechner - Online Tools and Calculator
How to Find Coefficient of Variation on a TI-84 Calculato
Calculating Coefficient of Variation Mark as New; Bookmark; Subscribe; Mute; Subscribe to RSS Feed; Permalink; Print; Email to a Friend ; Report Inappropriate Content 03-13-2020 08:42 AM. Hello everyone �� I do have a table containing the following columns: Personal Number, Date, Team, Project, TimeCategory, Time There are multiple Personal Numbers per Team, multiple Teams per Project and. How to Calculate the Coefficient of Variation in Excel. We have seen elementary examples to explain the concept of coefficient of variation. However, in reality, you will never come across such simple calculations. Therefore, one should know how to use MS Excel to determine the formula of the coefficient of variation. Download Detailed Curriculum and Get Complimentary access to Orientation.
In 1992, David Houle showed that measures of additive genetic variation standardized by the trait mean, CVA (the coefficient of additive genetic variation) and its square (IA), are suitable measures of evolvability. CVA has been used widely to compare patterns of genetic variation. However, the use of CVAs for comparative purposes relies critically on the correct calculation of this parameter. As shown in the picture below, by calculating the formula, we got a sample correlation coefficient of 0.87. So, there is a strong relationship between the two values. A Correlation of 1. A correlation of 1 is also known as a perfect positive correlation. This means that the entire variability of one variable is explained by the other Statistics Calculators. Calculate the average of a set of data. Average is the same as mean. Calculate the minimum, maximum, sum, count, mean, median, mode, standard deviation and variance for a data set. Calculations include the basic descriptive statistics plus additional values. Calculate the minimum, maximum, range, sum, count, mean, median. Formula to calculate coefficient of variation. Example: Suppose you took a random data set, you calculated its standard deviation and mean to be 4 and 6 respectively. Calculate the data's coefficient of variation. Thus, the coefficient of variation is 66.7%. Share. Tweet. Reddit. Pinterest. Email. Prev Article . Next Article . Related Articles. A chi-square statistic is a test that measures. You can use this Standard Deviation Calculator to calculate the standard deviation, variance, mean, and the coefficient of variance for a given set of numbers. Please provide numbers separated by comma (e.g: 7,1,8,5), space (e.g: 7 1 8 5) or line break and press the Calculate button
Calculate coefficient of variation of window in astropy. Ask Question Asked 3 months ago. Active 3 months ago. Viewed 51 times 1. I have an array that I want to calculate statistics for using astropy. What I have is: from astropy.convolution import convolve import numpy as np x = np.random.randint(1, 10, size=(5, 5)) y = convolve(x, np.ones((3, 3)), boundary='extend', preserve_nan=True) print. How should I calculate a within-subject coefficient of variation? In the study of measurement error, we sometimes find that the within-subject variation is not uniform but is proportional to the magnitude of the measurement. It is natural to estimate it in terms of the ratio within-subject standard deviation/mean, which we call the within-subject coefficient of variation. In our British. Applications of the Coefficient of Variation . When used to evaluate investment risk, COV can be interpreted similarly to the standard deviation in modern portfolio theory (MPT).But the COV is. The calculations are performed according to the general methodology given by Machin et al., 2009. It is assumed that the calculation of the Coefficient of variation uses the Logarithmic method. Note that the calculation does not include a null hypothesis value or a factor for power (1−β). Therefore the estimated sample size does not give a. We say that there is greater variation in their consumption of meat. The observations about the quantity of meat are more dispersed or more variant. Example: Calculate the coefficient of standard deviation and coefficient of variation for the following sample data: 2, 4, 8, 6, 10, and 12 Comparison of Coefficients of Variation calculator. Sample 1: Coefficient of variation (%): Sample size: Sample 2: Coefficient of variation (%): Sample size: Description. Performs a test for the comparison of two coefficients of variation (from independent samples), expressed as a percentage. Required input. Enter the two coefficients of variation, expressed as percentages, and the.
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\begin{document}
\title[KAM rigidity for partially hyperbolic affine $\Z^k$ actions]{KAM rigidity for partially hyperbolic affine $\Z^k$ actions on the torus with a rank one factor}
\author[Danijela Damjanovi\'c and Bassam Fayad]{Danijela Damjanovi\'c$^1$ and Bassam Fayad $^2$} \thanks{ $^1$ Based on research supported by NSF grant DMS-0758555 }
\subjclass[2012]{}
\address{Department of Mathematics, \\
Rice University\\6100 Main st\\Houston, TX 77005} \email{[email protected]\\}
\begin{abstract} We show that ergodic affine $\mathbb Z^k$ actions on the torus, that have a rank-one factor in their linear part, are locally rigid in a KAM sense if and only if the rank one factor is trivial and the action is higher-rank transversally to this factor.
Since \cite{DK} proves that affine actions with higher-rank linear part are locally rigid, our result completes the local rigidity picture for affine actions on the torus.
\end{abstract}
\maketitle
\section{Introduction}
\subsection{Local rigidity of $\Z^k$ actions}
A smooth $\Z^k$ action $\rho$ on a smooth manifold $M$ is said to be locally rigid if there exists a neighborhood $\mathcal U$ of $\rho$ in the space of smooth $\Z^k$ actions on $M$, such that for every $\eta\in \mathcal U$ there is a $C^\infty$ diffeomorphism $h$ of $M$ such that $h\circ \rho\circ h^{-1}=\eta$.
When $k=1$ or if the $\Z^k$ action has a factor which is (perhaps up to a finite index subgroup) an action of $\mathbb Z$, then one cannot expect to have local rigidity as described above. The only known situation in rank-one dynamics where some form of local rigidity happens is for Diophantine toral translations, where translation vectors with respect to invariant probability measures serve as moduli. If a Diophantine translation is perturbed into a parametric family of diffeomorphisms and if the translation vectors relative to invariant measures satisfy an adequate transversality condition, then for a large set of parameters the diffeomorphisms of the family are smoothly conjugate to translations. This is a consequence of KAM theory (after Kolmogorov, Arnol'd and Moser) and we call it {\it KAM rigidity}. A typical example is given by Arnol'd family of circle diffeomorphisms \cite{A} where transversality { in this case} amounts to the requirements that the rotation number of the diffeomorphisms should often be Diophantine. The latter example { will be a paradigm } in the subsequent study of partially hyperbolic affine actions.
{ \subsection{Local rigidity of higher rank actions by automorphisms of the torus} }
A $\Z^k$, $k\ge 2$ action which has no rank-one factors is called a \emph{genuinely} higher-rank action, or just a higher-rank action.
For ergodic actions by toral automorphisms it is proved in \cite{Starkov} that the action has no rank-one factors if and only if:
{\it (HR) The $\Z^k$ action contains a subgroup $L$ isomorphic to $\Z^2$ such that every element in $L$, except for identity, acts ergodically with respect to the standard invariant measure obtained from Haar measure.}
This condition may be viewed as a general paradigm for any form of rigidity of an algebraic action.
Notice that the condition (HR) for a $\Z^k$ action by toral automorphisms implies that the action is partially hyperbolic, since ergodicity for a single toral automorphism immediately implies partial hyperbolicity.
The local picture for ergodic higher-rank $\Z^k$ actions on the torus by toral automorphisms is fairly well understood. The condition \emph{(HR)} is a necessary and sufficient condition for local rigidity ( \cite{DK} and references therein).
The { local rigidity} result in \cite{DK} extends to \emph{affine} actions on the torus whose linear parts are actions which satisfy the (HR) condition.
{ The specificity of affine actions }appears nevertheless if the linear part violates the assumption (HR). For example take the $\mathbb Z^2$ action on ${\mathbb T}^{d+1}$ generated by $A \times \text {Id}$, $B \times \text {Id}$, with $A$ and $B$ two hyperbolic commuting toral automorphisms of ${\mathbb T}^d$. Of course this $\Z^2$ action does not satisfy the ergodicity assumption in the general paradigm. But in the affine setting, the $\mathbb Z^2$ action generated by $A \times R_ \alpha$ and $B \times R_\beta$, where $R_ \alpha$ and $R_\beta$ are two circle rotations such that $1, \alpha,\beta$ are rationally independent, satisfies the ergodicity assumption of the general paradigm, while its linear part does not. This action is clearly not locally rigid. We can for example change the frequencies, but even with fixed frequencies, Anosov-Katok Liouville constructions show that we can perturb $R_ \alpha \times R_\beta$ into a non linearizable commuting pair { of circle diffeomorphisms.}
In this paper we consider affine actions on the torus which have as their linear part a $\mathbb Z^k$ action which does not satisfy (HR). We show that for such actions { certain kind of local rigidity} may be established if and only if { there exists a set of generators} of the linear part given by $A_i \times \text {Id}$ where $A_1,\ldots,A_k$ satisfy (HR).
Since the statements for $\Z^k$ actions are exactly similar to the ones for $\Z^2$ actions, we state our results in the latter case for better readability of the results and the proofs.
It is easy to see that local rigidity of an affine action $\rho$ whose linear part does not satisfy (HR) can only be possible if its generators, after a coordinate change, are of the form \begin{equation} \label{normalform} \bar A=(A+a) \times (\text {Id}_{{\mathbb T}^{d_2}}+\varphi), \bar B= (B+b) \times (\text {Id}_{{\mathbb T}^{d_2}}+\psi)\end{equation} with $A$ and $B$ two
commuting toral automorphisms of ${\mathbb T}^{d_1}$ that satisfy (HR), where $d_1+d_2=d$, and $a,b,\varphi, \psi$ are translation vectors. Indeed, if the action generated by the generators of the linear part of $\rho$ has a rank-one factor then up to a coordinate change in $\Z^2$ we may assume that $\bar A=(A+a) \times (Id+\varphi)$ and $\bar B= (B+b) \times (C+\psi)$, where $A$ and $B$ generate a linear action, $(\varphi, \psi)$ are translation vectors, and $C$ is a linear map. The commutativity condition implies that $\varphi$ belongs to the eigenspace $V_1$ of $C$ relative to the eigenvalue $1$. If $C \neq \text{Id}$, { projecting to the orthocomplement of $V_1$ leaves us with an action generated by $\bar A=(A+a) \times Id$ and $\bar B= (B+b) \times (\tilde{C}+\tilde{\psi})$. The local rigidity of the action $\rho$ then requires a local rigidity result for the rank one action $\tilde{C}+\tilde{\psi}$ which obviously does not hold.}
{ For a $\Z^2$ partially hyperbolic affine action whose generators satisfy (\ref{normalform}) it is possible to state a rigidity theorem in a similar fashion as for
perturbations of quasi-periodic translations :} Let $(f,g)$ be a perturbation of the generators $\bar A$ and $\bar B$ of such an action. First of all, since the linear parts of $f$ and $g$ are given by $A \times \text {Id}$, $B \times \text {Id}$, on ${\mathbb T}^{d_1+d_2}$, one can define for any pair $\mu_1,\mu_2$ of invariant probability measures by $f$ and $g$ respectively the translation vectors { along the ${\mathbb T}^{d_2}$ direction} corresponding to these measures as follows \begin{align*} \alpha&=\rho_{\mu_1}(f)= \int_{{\mathbb T}^{d}} \pi_2(f(x)-x) d\mu_1(x), \\
\beta&=\rho_{\mu_2}(g)= \int_{{\mathbb T}^{d}} \pi_2(g(x)-x) d\mu_2(x)\end{align*} where $\pi_2$ is the projection on the ${\mathbb T}^{d_2}$ variable.
We say that $( \alpha,\beta) \in {\mathbb T}^{d_2} \times {\mathbb T}^{d_2}$ is simultaneously Diophantine with respect to a pair of numbers $(\lambda,\mu)$ if there exists $\tau,\gamma>0$ such that
$$\max( |\lambda -e^{i2\pi (k, \alpha)}|, |\mu-e^{i2\pi (k,\beta)} |)> \frac{\gamma}{|k|^\tau}$$
where $\|\cdot\|$ denotes the closest distance to the integers, and we denote this property by $( \alpha,\beta)\in \text{SDC}(\tau,\gamma,\lambda,\mu)$. We say that $( \alpha,\beta)\in \text{SDC}(\tau,\gamma,\bar A, \bar B)$ if given any pair of eigenvalues $(\lambda,\mu)$ of $(\bar A,\bar B)$, it holds that $( \alpha,\beta)\in \text{SDC}(\tau,\gamma,\lambda,\mu)$. Observe that SDC pairs of vectors relatively to any pair $(\bar A,\bar B)$ form a set of full Haar measure in ${\mathbb T}^{d_2} \times {\mathbb T}^{d_2}$.
We have the following \begin{theo}\label{main2} Let $f,g$, be the generators of a smooth ($C^\infty$) $\Z^2$ action on ${\mathbb T}^d$ such that the linear part of $(f,g)$ is given by $\bar A=A \times \text {Id}_{{\mathbb T}^{d_2}}$, $\bar B= B \times \text {Id}_{{\mathbb T}^{d_2}}$, with $A$ and $B$ two
commuting toral automorphisms of ${\mathbb T}^{d_1}$ that satisfy (HR), $d_1+d_2=d$. For any $\tau,\gamma>0$, there exist $r(\tau)$ and $\varepsilon(\tau,\gamma)$ such that if for some pair of invariant probability measures $\mu_1,\mu_2$ by $f$ and $g$ respectively we have that $Ê( \alpha,\beta)=$ $(\rho_{\mu_1}(f),\rho_{\mu_2}(g)) \in \text{SDC}(\tau,\gamma,\bar A, \bar B)$ and if $\|f-(A+a)\times T_ \alpha\|_r\leq \varepsilon$, $\|g-(B+b)\times T_\beta\|_r\leq \varepsilon$, { where $a, b\in \mathbb R^{d_1}$ and $T_ \alpha$ and $T_\beta$ are translations of $\mathbb T^{d_2}$}; then the action is linearizable, namely there exists $h \in \text{Diff}^\infty ({\mathbb T}^d)$ such that $h \circ f \circ h^{-1} = (A+a)\times T_ \alpha$, $h \circ g \circ h^{-1} = (B+b)\times T_\beta$. \end{theo}
{ In the case $d_2=1$, the SDC condition is reminiscent of the one used by Moser to prove local rigidity of commuting circle diffeomorphisms with this condition on their rotation numbers \cite{M}. The ingredients of the proof of Theorem \ref{main2} are indeed a mixture of the ingredients used in the higher rank rigidity of toral automorphisms \cite{DK} and the KAM rigidity in the quasi-periodic setting as in \cite{A} and \cite{M}. }
Also similar to the perturbations of quasi-periodic translations of the torus it is possible to state a rigidity theorem for a parametric family of $\Z^2$ actions.
Let $\rho_t$ be a family of $\Z^2$ actions where the parameter $t \in [0,1]$.
Given $t$, the generators $ f_t, g_t$ of the $\Z^2$ action $\rho_t$ may be viewed as $f_t=\bar A+\bar a_t$ and $g_t= \bar B+\bar b_t$, where $\bar A$ and $\bar B$ generate a linear action. If the linear action generated by $\bar A$ and $\bar B$ has a rank-one factor then up to a coordinate change in $\Z^2$ we may assume that the affine action $\rho_t$ is generated by $f_t=(A+a_t) \times (Id+\varphi(t))$ and $g_t= (B+b_t) \times (C+\psi(t))$, where $A$ and $B$ generate a linear action, $(a_t,b_t)$ and $(\varphi(t), \psi(t))$ are translation vectors, and $C$ is a linear map. Arguing as in the case of a single action, we see that for any kind of rigidity to hold it is necessary that $C=Id$. Indeed, if $C$ is not Identity we can reduce to the case $f_t=A \times Id$ and $g_t=B \times (C+\psi(t))$. The latter can be perturbed into the family of actions generated by $f_t, \tilde{g}_t=B\times h_t$ with $h_t$ any perturbation of the family $C+\psi(t)$ that can be chosen to be non linearizable for all $t$.
To state a KAM rigidity result when $C= Id$ we need some transversality on the frequencies along the elliptic factor of the action. We will use a Pyartli \cite{P} Êtype condition but other usual transversality conditions in KAM theory may be applied as well.
\begin{definition} We say that a function $\rho \in C^r([0,1], {\mathbb T}^d)$, $r\geq d$, satisfies a Pyartli condition if for any $t \in [0,1]$ we have that the first $d$ derivatives of $\rho$ are linearly independent. There exists then a constant $\nu>0$ such that
\begin{equation} \label{pyartlinu} |\text{det}(\rho', \rho'',\ldots, \rho^{(d)})| \geq \nu, \quad \| \rho \|_d \leq \nu^{-1}
\end{equation} \end{definition}
\begin{theo}\label{main} Let $f_t,g_t$, $t\in [0,1]$ generate a family $\rho_t$ of affine $\Z^2$ actions on ${\mathbb T}^d$ which is of class $C^d$ in he parameter $t$. Then the following alternative holds in function of the common linear part $(\bar A,\bar B)$ of the family $(f_t,g_t)$.
(1) $(\bar A,\bar B)$ satisfies (HR) and every action in the family is locally rigid.
(2) $f_t=(A+a_t) \times (Id+\varphi(t))$ and $g_t= (B+b_t) \times (Id+\psi(t)).$
If the function $n \varphi(t)+ m \psi(t)$ satisfies a Pyartli condition for some $(n, m)\in \mathbb Z^2$ { (for $d=d_2$ and with some constant $\nu$ and if in addition $\|\varphi\|_{d_2}, \| \psi \|_{d_2} \leq \nu^{-1}$ ),} then the family $\rho_t$ is KAM locally rigid: there exists { $r_0(A,B,n,m,d_2)$ such that
for any $\eta$ there exists $\varepsilon(\eta,n,m,\nu)$} such that if the action $\rho_t$ is perturbed into $\tilde{\rho}_t$ generated by $\tilde{f}_t$ and $\tilde{g}_t$ such that $\|\tilde{f}_\cdot-f_\cdot\|_{d,r_0}\leq \varepsilon$,
$\|\tilde{g}_\cdot-g_\cdot\|_{d,r_0}\leq \varepsilon$, then the set of parameters $t$ for which $(\tilde{f}_t,\tilde{g}_t)$ are simultaneously smoothly linearizable is larger than $1-\eta$.
(3) None of the actions in the family is locally rigid and the family is not KAM locally rigid : it can be perturbed so that no element of the perturbed family is linearizable.
\end{theo}
We denote by $\|\cdot \|_{d,r}$ the combination of $C^d$ norm in $t$ and $C^r$ norm in the torus variable. Part (1) of Theorem \ref{main} is proved in \cite{DK}. { Part (3) reduces as discussed above to the case $f_t=A \times Id$ and $g_t=B \times (C+\psi(t))$. As explained before,} in this paper we combine techniques from \cite{DK} and Arnol'd parameter exclusion technique for perturbations of quasi-periodic translations on the torus \cite{A}, to show Part (2) i.e., rigidity in the KAM sense for affine actions.
For the clarity of the exposition, the proof of Theorem \ref{main} will be first carried in detail only in the case $d_2=1$. The generalization to any $d_2$ is explained in Section \ref{generald2}. Also, since the proof of Theorem \ref{main2} follows essentially the same lines as the proof of Theorem \ref{main}, we will only give a detailed proof of the former and explain in Section \ref{proofmain2} the main differences required for the proof of the latter.
Affine Anosov actions have been first
discussed by Hurder in \cite{HurAffine}. Local rigidity of hyperbolic and then partially hyperbolic affine actions of higher rank non abelian groups was extensively studied (see for example the survey \cite{Fisher}). In \cite{FM} Fisher and Margulis provide a complete local picture for affine actions by higher rank lattices in semisimple Lie groups. The methods they use are totally different from ours and are speciÞc to groups with Property (T).
Prior to \cite{FM}, the question about local rigidity of perturbations of product actions of large higher rank groups has been addressed in \cite{NT1}, \cite{NT2}, \cite{T}; the actions considered there are products of the identity action and actions that generalize the standard ${\rm SL}(n,\Z)$ action on ${\mathbb T}^n$. Local rigidity and deformation rigidity are obtained for such actions. We note that the actions we consider in this paper even though they belong to families of actions, are not deformation rigid in the sense of \cite{Hurder}.
Local rigidity results for algebraic Anosov actions were obtained by Katok and Spatzier in \cite{KS}, including the case of toral automorphisms and nilmanifold automorphisms. Currently not much is known about perturbations of affine actions on nilmanifolds when the linear part is a product of a higher rank abelian action and the identity, even when the higher rank abelian action is Anosov.
\subsection{Reduction to actions which are linear transversally to the elliptic factor}
In the subsequent sections we give the proof of Theorem \ref{main} in the case when the { unperturbed} action transversal to the elliptic factor is purely linear, namely when $f_t=A \times R_{\varphi(t)}$ and $g_t= B\times R_{\psi(t)}$, where $ R_{\varphi(t)}$ and $R_{\psi(t)}$ denote translation maps on the circle.
The same arguments extend to the case when the { unperturbed} action transversal to the elliptic factor is affine generated by $A+a_t$ and $B+b_t$ instead of $A$ and $B$ . The only difference is that in (2.9) the number $\lambda_{m,t}$ should be replaced with $\lambda_{m,n,t}= e^{-i2\pi (m\varphi(t)+\langle n,a_t\rangle )}\lambda$. { This change does not affect any subsequent estimates.}
\subsection{Exact statement of Theorem \ref{main} in the case of a one dimensional elliptic factor}
Let $A$ and $B$ be two commuting toral automorphisms satisfying (HR) condition.
For $\varphi,\psi \in {\rm Lip}(I_0,\R)$, $I_0=[0,1]$, let \begin{align*} f_{\varphi(t)}(x,\theta)&=(Ax,R_{\varphi(t)}(\theta)) \\ g_{\psi(t)}(x,\theta)&=(Bx,R_{\psi(t)}(\theta)) \end{align*}
For $I \in \R$, we denote $C^{lip,\infty}(I, {\mathbb T}^{d+1},\R^{d+1})$ the set of families of smooth maps in the ${\mathbb T}^{d+1}$ variable and Lipschitz in the parameter $t \in I$. We denote $C^{lip,\infty}_0(I,{\mathbb T}^{d+1},\R^{d+1})$ the subset of maps $f \in C^{lip,\infty}(I, {\mathbb T}^{d+1},\R^{d+1})$ such that if we write $f_t(z)=(f^1_t(z),f^2_t(z)) \in {\mathbb T}^d \times {\mathbb T}$, then $\int_{{\mathbb T}^{d+1}} f^2_{t}(z) dz =0$ for $t\in I$.
Consider
\begin{align*} \tilde{f}_t(x,\theta)&=f_{\varphi(t)}(x, \theta)+\Delta f_t(x, \theta) \\ \tilde{g}_t(x,\theta)&=g_{\psi(t)}(x, \theta)+\Delta g_t(x, \theta) \end{align*}
with $\Delta f,\Delta g \in C^{lip,\infty}_0(I_0,{\mathbb T}^{d+1},\R^{d+1})$ and such that $\tilde{f}_t$ and $\tilde{g}_t$ commute for all $t\in I_0$. For $f \in C^{lip,\infty}_0(I_0,{\mathbb T}^{d+1},\R^{d+1})$, we use the notation $\| f \|_{lip(I),r} = \max_{|\iota| \leq r} {\rm Lip}( f^{(\iota)})$ where ${\rm Lip}(f)$ is the maximum of the supnorm of $f$ and its Lipschitz constant, and $|\iota|$ is the maximal coordinate of the multi-index $\iota \in \N^{d+1}$. We will also use the notation $\|v\|_{0(I),r}$ fort the supremum of the usual $C^r$ norms of $v(t)$ as $t\in I$.
{ Let $M$ be such that $$2\max(\|\varphi \|_{lip(I_0)},\|\psi \|_{lip(I_0)}) \leq M, \quad \inf_{t\in I_0}\varphi'(t) \geq \frac{2}{M}$$}
\begin{theo} \label{main3} There exists $r_0(A,B) \in \N$ such that for any $\eta$ there exists $\epsilon_0(A,B,M,\eta)>0$ such that if $\max(\| \Delta f \|_{lip(I),r_0}, \| \Delta g \|_{lip(I),r_0}) \leq \epsilon_0$, then the set of parameters $t$ for which the pair $\tilde{f},\tilde{g}$ is simultaneously smoothly linearizable has measure larger than $1-\eta$. \end{theo}
Sections \ref{section.inductive} and \ref{sec.kam} below are devoted to the proof of Theorem \ref{main3}. Sections \ref{generald2} and \ref{proofmain2} explain how this proof should be modified to give the proof of Theorems \ref{main} and \ref{main2} respectivily.
{\bf Acknowledgments.} The authors are grateful to Artur Avila, Hakan Eliasson, Anatole Katok and Rapha‘l Krikorian for fruitful discussions and suggestions.
\section{The inductive step} \label{section.inductive}
Let $\mathcal E(A)$ be the set of eigenvalues of $A$ union $1$. For $N\in \mathbb N$, define
$$ \mathcal D(N,A) = \{ \alpha \in I_0 \ / \ | \l - e^{i2\pi k \alpha}|\geq N^{-3}, \quad \forall \lambda \in \mathcal E(A), \forall 0<|k|\leq N \}.$$
\begin{prop}\label{mainprop} There exists $\sigma(A,B)$ such that if $N \in \N$ and $I$ is an interval such that $I \subset \{t \in I_0 \ / \ \varphi(t) \in \mathcal D(N)\}$, then there
exist $\tilde{\varphi},\tilde{\psi} \in {\rm Lip}(I,\R)$ and $h,\widetilde{\Delta f}, \widetilde{\Delta g} \in C^{lip,\infty}_0(I,{\mathbb T}^{d+1},\R^{d+1})$ such that if we write $ H={\rm Id} + h$ we have that \begin{equation}\label{nonlin} \begin{aligned} H \circ \tilde{f} &= (f_{\tilde{\varphi}} + \widetilde{\Delta f}) \circ H \\
H \circ \tilde{g} &= (g_{\tilde{\psi}} + \widetilde{\Delta g}) \circ H \end{aligned}
\end{equation} with
\begin{align*} \Delta S& \leq C_0 N^\sigma \Delta_0 \\
\|h\|_{lip(I),r+1}&\le C_r S N^{\si} \Delta_r +C_rSN^{\si}\Delta_0\Delta_r \\ \widetilde{\Delta}_r &\le
C_r S N^{\si}\Delta_0\Delta_r + C_{r,r'} N^{\sigma+ r-r'}\Delta_{r'} \end{align*}
where:
\begin{align*}
S&=\max(\|\varphi \|_{lip(I)},\|\psi \|_{lip(I)} ) \\
\Delta S&=\max(\| \varphi-\tilde{\varphi} \|_{lip(I)},\| \psi-\tilde{\psi} \|_{lip(I)}) \\
\Delta_r &=\max( \|\Delta f\|_{lip(I),r},\|\Delta g\|_{lip(I),r} ) \\
\widetilde{\Delta}_r &=\max( \|\widetilde{\Delta f}\|_{lip(I),r},\|\widetilde{ \Delta g} \|_{lip(I),r} ) \end{align*}
\end{prop}
We will reduce the proof of Proposition \ref{mainprop} to the solution of a set of linear equations in the coordinates of $h$. These equations are solved using Fourier series and part of the solution is obtained with the higher rank techniques as in \cite{DK} while another part is obtained from solving linear equations above a circular rotation and requires parameter exclusion to insure that the parameters that are kept satisfy adequate arithmetic conditions that allow to control the small divisors that appear.
\subsection{Reduction of the conjugacy step to linear equations} \label{redconj}
By substituting $H=id+h$, the first equation in \eqref{nonlin} becomes: \begin{equation}\label{newerror} \Delta f-(Df_{\tilde\varphi}h-h\circ f_\varphi)=f_{\tilde\varphi}-f_\varphi+\widetilde{\Delta f}(id+h)+E_{L, A} \end{equation} where $E_{L, A}=f_{\tilde\varphi}(Id+h)-f_{\tilde\varphi}-Df_{\tilde\varphi}h-h(f_\varphi+\Delta f)+h f_\varphi$
The map $Df_{\tilde\varphi}$ actually does not depend on $\tilde\varphi$, in fact it is the map $\bar A=(A, Id)$, where $A$ acts on $\mathbb R^d$ and $Id$ acts on $\mathbb R$. The second equation in \eqref{nonlin} is linearized in the same way, so the linearization of \eqref{nonlin} is the system of equations in $h$: \begin{equation}\label{linconj} \begin{aligned} \bar Ah-h\circ f_\varphi&=\Delta f\\ \bar Bh-h\circ g_\psi&=\Delta g \end{aligned} \end{equation} where $\bar B=(B, Id)$ and $E_{L, B}:=g_{\tilde\psi}(Id+h)-g_{\tilde\psi}-Dg_{\tilde\psi}h-h(g_\psi+\Delta g)+h g_\psi$.
Given a pair of commuting automorphisms $\bar A$ and $\bar B$ we call $(\lambda,\mu)$ a pair of eigenvalues of $(\bar A,\bar B)$ if $\lambda$ and $\mu$ are eigenvalues of $\bar A$ and $\bar B$ for the same eigenvector.
If $A$ and $B$ are semisimple, then by choosing a proper basis in $\R^d$ in which $A$ and $B$ simultaneously diagonalize, the system \eqref{linconj} breaks down into several systems of the following form \begin{equation}\label{simpleconj}
\begin{aligned} \la h-h\circ f_\varphi&=v\\ \mu h-h\circ g_\psi&=w \end{aligned} \end{equation}
where $\la$ and $\mu$ are a pair of eigenvalues of $A \times {\rm Id}$ and $B \times {\rm Id}$ and $v, w \in C^{lip,\infty} (I\times {\mathbb T}^{d+1}, \mathbb R)$. If $A$ and $B$ have non-trivial Jordan blocks, then instead of \eqref{simplecom}, for each Jordan block we would get a system of equations. Lemma 4.4 in \cite{DK} shows that this system of equations can be solved inductively in finitely many steps (the number of steps equals the size of a Jordan block), starting from equation of the form \eqref{simplecom}. We will not repeat the argument here, instead we assume throughout that $A$ and $B$ are semisimple and we refer to Lemma 4.4 in \cite{DK} for the general case.
\subsection{Reduction of the commutativity relation}
Since $f_\varphi$ and $g_\psi$ commute and are linear, the equation $(f_\varphi+\Delta f)\circ (g_\psi+\Delta g)=(g_\psi+\Delta g)\circ (f_\varphi+\Delta f)$ reduces to: $$\bar A\Delta g-\Delta g(f_\varphi+\Delta f)=\bar B\Delta f-\Delta f(g_\psi+\Delta g)$$ If we push the terms linear in $\Delta f$ and $\Delta g$ to the left and all the non-linear terms to the right hand side we obtain \begin{equation}\label{lincom} \bar A\Delta g-\Delta g\circ f_\varphi-\bar B\Delta f-\Delta f\circ g_\psi=\Phi \end{equation} where \begin{equation}\label{Phi} \Phi=\Delta g(f_\varphi+\Delta f)-\Delta g\circ f_\varphi-(\Delta f(g_\psi+\Delta g)-\Delta f\circ g_\psi). \end{equation}
Similarily to section \ref{redconj}, if $A$ and $B$ are semisimple, the equation \eqref{lincom} reduce to several equations of the form:
\begin{align}\label{simplecom} (\la w-w\circ f_\varphi)-(\mu v-v\circ g_\psi)=\phi \end{align}
\subsection{An approximate solution to \eqref{simpleconj}}
The main result in this note is that the system of linear equations \ref{simpleconj} can be solved up to an error term that is controlled by $\Phi$ which is quadratically small in the perturbation terms $\Delta f, \Delta g$.
\begin{lemm}\label{splittinglemma} For $v, w, \phi \in C^{lip, \infty}(I\times {\mathbb T}^{d+1}, \R)$ satisfying \eqref{simplecom}, and $\la\ne 1, \mu\ne 1$, if $N \in \N$ and $I$ is an interval such that $I \subset \{t \in I_0 \ / \ \varphi(t) \in \mathcal D(N)\}$, then there exists $h \in C^{lip, \infty}(I\times {\mathbb T}^{d+1}, \R)$ such that:
\begin{equation*}\label{split} \begin{aligned}
&\|h\|_{lip(I), r+1}\le C_rSN^{\si}\|v\| _{lip(I),r}+C_rSN^{\si}\|\phi\|_{lip(I), r-2}\\
&\|v-(\la h-h\circ f_\varphi)\|_{lip(I), r}\le C_{r,r'}N^{d+ r-r'}\|v\|_{lip(I), r'}+C_rSN^{\si}\|\phi\|_{lip(I), r-2}\\
&\|w-(\mu h-h\circ g_\psi)\|_{lip(I), r}\le C_{r, r'}N^{d+r'-r}\|w\|_{lip(I), r'}+ C_rSN^{\si}\|\phi\|_{lip(I), r-2} \end{aligned} \end{equation*} for all $r'>r\ge 0$ and $\si=\si(A, B, \la, \mu, d)$. The same holds true for $(\la, \mu)=(1,1)$ provided $v, w\in C_0^{lip, \infty}(I\times {\mathbb T}^{d+1}, \R)$.
\end{lemm}
\subsection{Proof of Proposition \ref{mainprop}}
Before we give the proof of Lemma \ref{splittinglemma}, we show how it implies Proposition \ref{mainprop}.
By applying Proposition \ref{annexe.compose} from the Appendix to the equation \eqref{Phi}, we have that
\begin{equation} \|\Phi\|_{lip(I), r-2}\le C_r \Delta_0\Delta_r \label{quad}
\end{equation}
Since $\Delta f,\Delta g \in C_0^{lip, \infty}(I\times {\mathbb T}^{d+1}, \R^{d+1})$ we can apply Lemma \ref{splittinglemma} to all the coordinates in (\ref{linconj}) and get $h$ such that \begin{equation*}\label{finalHest} \begin{aligned}
&\|h\|_{lip(I), r+1}\le C_rSN^{\si}\Delta_r+C_rSN^{\si}\Delta_0\Delta_r \\
&\|\Delta f-(\bar A h-h\circ f_\varphi)\|_{lip(I), r}\le C_rSN^{\si}\Delta_0\Delta_r+ C_{r,r'}N^{d+ r-r'}\Delta_{r'}\\
&\|\Delta g-(\bar B h-h\circ g_\psi)\|_{lip(I), r}\le C_rSN^{\si}\Delta_0\Delta_r+ C_{r, r'}N^{d+r'-r}\Delta_{r'} \end{aligned} \end{equation*}
where the new constant $\si$ is $d$ times the constant $\sigma$ from Lemma \ref{splittinglemma}. For the bound on $h$ we use Lemma \ref{splittinglemma} and \eqref{quad} with $r'=r$. In light of \eqref{newerror}, we take \begin{equation}\label{tildes} \begin{aligned} \tilde\varphi&:= \varphi +Ave(E_{L,A}^2 \circ (\text{Id} + h)^{-1}) \\ \tilde\psi&:= \psi+Ave(E_{L,B}^2 \circ (\text{Id} + h)^{-1}) \end{aligned} \end{equation} and let $$\widetilde{\Delta f}=\left(( \Delta f_0-(A h-h\circ f_\varphi)) -E_{L, A})\right) \circ (\text{Id} + h)^{-1} +f_\varphi-f_{\tilde \varphi} $$ with a similar definition for $\widetilde{\Delta g}$.
\noindent {\it Claim.} We have that $\tilde \varphi, \tilde \psi$, $h$ and $\widetilde{\Delta f}, \widetilde{\Delta g}$ satisfy the conclusion of Proposition \ref{mainprop}.
$\Box$
The rest of Section \ref{section.inductive} is devoted to the proof of Lemma \ref{splittinglemma}.
\subsection{Proof of Lemma \ref{splittinglemma}}
We first describe obstructions for solving a single coboundary equation in \eqref{simpleconj}. For a fixed $t\in I$ the first equation in \eqref{simpleconj} becomes: \begin{equation}\label{coboundary} \la h_t-h_t\circ f_{\varphi(t)}=v_t, \end{equation} where $h_t=h(t, \cdot)$ and similarily for $v$ and $w$. By reducing to Fourier coefficients, for every $(n, m)\in {\mathbb Z}^d\times {\mathbb Z}$ we have: \begin{equation*} \la \sum_{(n, m)} h_{n, m, t}\chi_{n,m}(x,\theta)-\sum h_{n, m,t}\chi_{n,m}(Ax,\theta+\vp(t))=\sum v_{n, m, t} \chi_{n,m}(x,\theta) \end{equation*} \begin{equation*}
\sum_{(n, m)} (\la h_{n, m, t}-h_{A^*n, m, t}e^{2\pi i m\vp(t)}\chi_{n,m}(x,\theta))=\sum v_{n, m,t} \chi_{n,m}(x,\theta), \end{equation*} where $h_{n, m, t}$ denotes the $(n, m)$-Fourier coefficient of the function $h_t$, $\chi_{n,m}(x,\theta)=e^{2\pi i (n\cdot x+m\t)}$, and $A^*=(A^t)^{-1}$. Thus for every $(n, m)$ \begin{equation*} \la h_{n, m,t}- h_{A^*n, m,t}e^{2\pi i m\vp(t)}= v_{n, m,t}. \end{equation*} By denoting: $\la_{m,t}:=e^{-2\pi i m\vp(t)}\la$ and $ v'_{n, m,t}:=e^{-2\pi i m\vp(t)}v_{n, m,t}$, we have \begin{equation}\label{oneeq} \la_{m, t} h_{n, m,t}- h_{A^*n, m,t}= v'_{n, m,t}. \end{equation}
For a fixed $m$ and $n\ne 0$ and for a fixed $t$ the equation \eqref{oneeq} has been discussed in \cite{DK}; the obstructions are precisely defined as well as the construction which allows for removal of all the obstructions (Lemma 4.5 in \cite{DK}). The obstructions are:
\begin{equation}\label{obs} O_{n, m}^A(v_t)=\sum_{k\in {\mathbb Z}} \la_{m,t}^{-(k+1)} v'_{A^kn, m,t}, \end{equation} where we abuse notation a bit by using $A^k$ to denote the $k$-th iterate of the dual map $A^*$. The proof of Lemma \ref{splittinglemma} relies on two claims. In the first one we solve a system of the type \eqref{simpleconj} provided a set of obstructions computed with the right hand side vanish. In the second claim, we show how the commutation relation allows to modify the right hand side in \eqref{simpleconj} to set the obtructions to $0$. Moreover, the modification is of the order of the "commutation error" $\Phi$ in \eqref{lincom}.
\noindent {\bf Claim 1.} {\it Let $v$ be in $C^{lip(I), \infty}(I,{\mathbb T}^{d+1}, {\mathbb R})$ such that for all $t\in I$ and $|m|>N$, $v_{0, m,t}=0$. If for all $n, m,$ and $t\in I$, $n\neq 0$, $O_{n, m}^A(v_t)=0$, and $ave(v_t)=0$ in the case $\la=1$, then there exists a solution $h$ to the equation $\la h-h\circ f_\varphi =v$ in $C^{lip(I), \infty}(I\times{\mathbb T}^{d+1}, {\mathbb R})$ satisfying \begin{equation}\label{h-est}
\|h\|_{lip(I), r}\le C_rSN^3\|v\|_{lip(I), r+\si} \end{equation}
for all $r\ge 0$, where $\sigma=\sigma\{\lambda, d, A\}$. Moreover, if $h$ and $v$ are smooth maps with $h_{0, m,t}=v_{0, m,t}=0$ for $|m|>N$ and with averages zero, such that $\la h-h\circ f_\varphi =v$ on $I\times {\mathbb T}^{d+1}$, then $h$ satisfies the estimate \eqref{h-est}.}
{\it Proof of the Claim 1}. The proof is similar to the proof of the Lemma 4.2 in \cite{DK}, except that here one extra (isometric) direction causes somewhat greater loss of regularity.
Solution $h$ is defined via its Fourier coefficients $h_{n, m,t}$, each of which can be defined, in case $n\ne 0$, by using one of the two forms:
\begin{equation}\label{h-def} h_{n, m,t}=\sum_{k=0}^\infty \la_{m,t}^{-(k+1)} v'_{A^kn, m,t}=-\sum_{k=-\infty}^{-1} \la_{m,t}^{-(k+1)} v'_{A^kn, m,t}. \end{equation}
One can use one or the other form to obtain an estimate for the size of $h_{n, m,t}$ depending on whether a non-trivial $n$ is largest in the expanding or in the contracting direction for $A$. This is completely the same as in \cite{DK} and it automatically gives an estimate of the size of $h_{n, m,t}$ with respect to the norm of $n$. In order to obtain here the full estimate for the $C^r$ norm of $h$ we need to estimate the size of $h_{n, m,t}$ with respect to the norm of $(n, m)$ and this is the only additional detail needed here. But this is not a big problem: since $n$ is non-trivial, after approximately $\log m$ iterations of $n$ by $A$, the resulting vector will surely be larger than $m$.
We have:
\begin{equation}\label{h-est-1} \begin{aligned}
|h_{n, m,t}|&\le \sum_{k=0}^{\infty}|\la_{m,t}^{-(k+1)}||v'_{A^kn, m,t}|\\&=\sum_{k=0}^{\infty}|\la|^{-(k+1)}|v_{A^kn, m,t}|\\&\le \|v\|_{0(I),r}\sum_{k=0}^{\infty}|\la|^{-(k+1)}\|(A^kn, m)\|^{-r}. \end{aligned} \end{equation}
Take the norm in ${\mathbb Z}^N\times {\mathbb Z}$ to be $\|(n, m)\|=\max\{\|n\|, |m|\}$, where for $n\in {\mathbb Z}^N$, $\|n\|$ is the maximum of euclidean norms of projections of $n$ to expanding, contracting and the neutral directions for $A$. Let $n_{exp}$ denote the projection of $n$ to the expanding subspace for $A$. Due to ergodicity of $A$ this projection is non-trivial. For example we say that $n$ is largest in the expanding if $\|n_{exp}\|\ge C\|n\|$ where $C$ is a fixed constant ($C=1/3$ works). Similarily, we say that $n$ is largest in the contracting (resp. neutral) direction if the projection of $n$ to the contracting (resp. neutral) direction is greater than constant times the norm of $n$.
Then if $\rho$ denotes the expansion rate for $A$ in the expanding direction for $A$, we have by the Katznelson Lemma (See for example Lemma 4.1 in \cite{DK}): \begin{equation*} \begin{aligned}
\|(A^kn, m)\|&\ge \max\{\|A^kn_{exp}\|, |m|\}\ge \max\{\rho^k\|n_{exp}\|, |m|\}\\
&\ge \max\{C\rho^k\|n\|^{-d}, |m|\}\ge \max\{C\rho^{k-k_0}\rho^{k_0}\|n\|^{-d}, |m|\} \end{aligned} \end{equation*}
Since $\rho^{k}\|n\|^{-d}\ge \|(n, m)\|$ for all $k\ge \frac{d+1}{\ln \rho}\ln \|(n, m)\|$, we have:
$$ \|(A^kn, m)\|\ge C\rho^{k-k_0}\|(n, m)\|$$
for all $k> k_0=[\frac{d+1}{\ln \rho}\ln \|(n, m)\|]$.
Now if $n$ is largest in the expanding direction for $A$ then for $0\le k\le k_0$: $\|(A^kn, m)\|\ge C\|(n, m)\|$. If $n$ is largest in the neutral direction for $A$, then for $0\le k\le k_0$: $\|(A^kn, m)\|\ge C(1+k)^{-d}\|(n, m)\|$.
Thus for all $t\in I$ (in the worst case scenario, when $|\la|<1$):
\begin{equation*}\label{h-est-11} \begin{aligned}
|h_{n, m,t}|&\le \|v\|_{0(I),r}(\sum_{k=0}^{k_0}|\la|^{-(k+1)}\|(n, m)\|^{-r}+\sum_{k=k_0}^{\infty}|\la|^{-(k+1)}(C\rho^k\|(n, m)\|)^{-r})\\
&\le \|v\|_{0(I),r}(k_0|\la|^{-(k_0+1)}\|(n, m)\|^{-r}+|\la|^{k_0}(C\rho^{k_0}\|(n, m)\|)^{-r}\sum_{k=0}^\infty |\la|^{-k}\rho^{-kr} \\
&\le C_r\|v\|_{0(I),r}(\|(n, m)\|^{\frac{d+1}{ln\rho}}\log \|(n, m)\|)\|(n, m)\|^{-r}+(C\rho^{k_0}\|(n, m)\|)^{-r}) \\
&\le C_r\|v\|_{0(I),r}\|(n, m)\|^{-r+\si} \end{aligned} \end{equation*} where $\si= 2+d+ a+\delta$, $\delta>0$, and $a=a(\la)=\frac{d+1}{ln\rho}>0$ in general depends only on the eigenvalues of $A$.
Note that for the convergence of the sum $\sum_{k=0}^\infty |\la|^{-k}\rho^{-kr}$ it suffices to assume that the regularity $r$ of $v$ is greater than a constant $-\frac{\ln|\la|}{\ln\rho}$, which in general depends on eigenvalues of $A$. We recall that the norm $\|v\|_{0(I),r}$ denotes the supremum of the usual $C^r$ norms of $v(t)$ as $t\in I$.
When $n$ is largest in the contracting direction for $A$ then just as in \cite{DK} we repeat the above estimates using the expression $h_{n, m,t}=-\sum_{k=-\infty}^{-1} \la_{m,t}^{-(k+1)} v'_{A^kn, m,t}
$ for the coefficients $h_{n, m,t}$ instead to obtain the same bound: $|h_{n, m,t}|\le C_r\|v\|_{0(I),r}\|(n, m)\|^{-r+\si}$, where $\si$ is now slightly different (changed by a constant) to include the eigenvalues for $A$ in the contracting directions.
Now in the case $n=0$, and any non-zero $m$ the equation \eqref{oneeq} implies:
\begin{equation*} \la h_{0, m,t}-h_{0, m,t}e^{2\pi i m\vp(t)}=v_{0, m,t}, \end{equation*} so in this case \begin{equation}\label{h-0} h_{0,m,t}=\frac{v_{0, m,t}}{\la-e^{2\pi i m\vp(t)}}, \end{equation}
and thus for $|\la|\ne 1$ we have that for all $t\in I$:
$$|h_{0,m,t}|\le (|\la|-1)^{-1} \|v\|_{0(I),r}|m|^{-r}.$$
In the case $|\la|=1$
this is a small divisor problem.
When $t\in I$ we have $\varphi(t)\in \mathcal D(N)$ and thus for $|m|\le N$ we have:
$$|h_{0,m,t}|\le N^3 |v_{0,m,t}|\le \|v\|_{0(I),r}N^3|m|^{-r}$$
Since for $|m|>N$, $v_{0,m}=0$, we define $h_{0, m}=0$ for $|m|>N$.
Accumulating all the estimates, we have for all $t\in I$:
$$|h_{n,m,t}|\le C_r \|v\|_{0(I),r}N^3\|(n, m)\|^{-r+\si}.$$
Thus the function $h$ defined via its Fourier coefficients $h_{n, m,t}$ satisfies the equation $\la h-h\circ f_\psi =v$ and the estimate: \begin{equation}\label{h-total-0}
\|h\|_{0(I),r}\le C_rN^{3}\|v\|_{0(I),r+\si}, \end{equation}
for all $r>r_0$, where $\si$ is a fixed constant, $\si=d+2+\max\{|\la|, |\la|^{-1}\}$, which in our set-up depends only on the eigenvalues of $A$ and the dimension of the torus.
We estimate now $h$ in the direction of the parameter $t$. First we can characterize $x\in C^{lip, \infty}(I, {\mathbb T}^{d+1},{\mathbb R})$ by a property of Fourier coefficients of $x$. Let $\Delta x:=x_t-x_{t'}$, and similarly $\Delta x_{n, m}=x_{n, m, t}-x_{n, m, t'}$. Namely, $x\in C^{lip, s}(I, {\mathbb T}^{d+1},{\mathbb R})$ implies not only that that $x_{n, m, t}$ decay faster than $\|(n, m)\|^{-s}$ but also from
$\|\Delta x^{(s)}\|_0\le L_s|t-t'|$ we get that $|\Delta x_{n, m}|\le C_s\|(n, m)\|^{-s}|t-t'|$ for some constant $C_s$. It is then easy to check that $|\Delta x_{n, m}|\le C_s\|(n, m)\|^{-s-d-1} |t-t'|$ suffices for $x\in C^{lip, s}(I, {\mathbb T}^{d+1},{\mathbb R})$.
By using \eqref{h-def} (denote for simplicity by $\Sigma^{\pm}$ positive or negative sum in \eqref{h-def}) we have for $n\ne 0$: \begin{equation*} \begin{aligned}
&|\Delta h_{n, m}|=|\Sigma^{\pm}\la^{-(k+1)}(e^{2\pi i km \vp(t)}v_{A^kn, m, t}-e^{2\pi i km \vp(t')}v_{A^kn, m, t'})|\\
&=|\Sigma^{\pm}\la^{-(k+1)}((e^{2\pi i km \vp(t)}-e^{2\pi i km \vp(t')})v_{A^kn, m, t} +e^{2\pi i km \vp(t')}\Delta v_{A^kn, m, t})|\\
&\le (2\pi \|\varphi\|_{lip(I)}\|v\|_{0(I), r}+\|v\|_{lip(I), r}) |t-t'|\Sigma ^{\pm}|\la|^{-(k+1)}|k|\|(A^kn, m)\|^{-r+1} \end{aligned} \end{equation*}
and from the discussion following \eqref{h-est-1} we have that for every $(n, m)$, $n\ne 0$, either the positive or the negative sum in the last expression above can be bounded by $C_r\|n, m\|^{-r+\sigma+1}$. When $n=0$ from \eqref{h-0} and for $t, t'\in I$ it is clear that $\Delta h_{0, m}\le CN^3\Delta v_{0, m}$. This gives the bound for the Lipschitz constant for any $r$-th derivative of $h$
which combined with \eqref{h-total-0} implies $\|h\|_{lip(I), r-\si-2-d}\le C_rN^3S\|h\|_{lip(I), r}$.
For the second part of the claim, if $h$ and $v$ are smooth and satisfy $\la h-h\circ f_\varphi =v$ for $t\in \mathcal I$ then for $n\ne 0$ the obstructions $O_{n, m}^A(v_t)$ are all zero, thus if $v$ satisfies in addition that $v_{0, m,t}=0$ for $|m|>N$ then by the first part of the Claim 1 there exists $h'$ such that $\la h'-h'\circ f_\varphi =v$ on $I$ and satisfies the estimate \eqref{h-est}. Then for $h"=h-h'$, $\la h"=h"\circ f_\varphi$ on $I $, but this implies $h"=0$ in case $\la\ne 1$, or is constant in case $\la=1$. However, by construction $h'$ has average 0, and so does $h$ by assumption, so in any case $h=h'$ on $I$, which implies that $h$ satisfies the estimate \eqref{h-est}. \\ {\it -End of pf of claim 1.-}
Remark. The following fact will be used in the proof of the Claim 2 bellow: For every $n\in {\mathbb Z}^d$ there exists a point $n^*$ on the orbit $\{A^kn\}_{k\in{\mathbb Z}}$ such that the projection of $n$ to the contracting subspace of $A$ is larger than the projection to the expanding subspace of $A$ and for $An$ the opposite holds: projection of $An$ to the contracting subspace of $A$ is smaller than the projection to the expanding subspace of $A$. For each $n$ choose an $n^*$ on the orbit of $n$ with this property \cite{DK}.
{\bf Claim 2.} {\it Assume that for all $t\in I$ the following holds:
\begin{equation}\label{simplecom-t} (\la w_t-w_t\circ f_{\varphi(t)})-(\mu v_t-v_t\circ g_{\psi(t)})=\phi_t \end{equation}
and $v_{0, m,t}=w_{0, m,t}=\phi_{0,m,t}=0$ for $|m|>N$. Define $\tilde v_t$ by
\begin{equation*} \tilde v_{n, m,t} = \left\{ \begin{aligned}
O^A_{n, m}(v_t), \,\, & n\ne 0, n=n^* \\
0, \,\, & \mbox{otherwise.} \end{aligned}\right.
\end{equation*}
Then:
(1) For $n\ne 0$, $O_{n, m}^A(v_t-\tilde v_t)=0$.
(2) $\|\tilde v\|_{lip(I), r}\le C_r N^3S\|\phi\|_{lip(I), r+\si}$, where $\si=\si(A, B, \la, \mu, d)$ and $r\ge 0$. }
{\it Proof of claim 2.}
(1) This is immediate from the definition of $O_{n, m}^A$ and $\tilde v_t$.
(2) In Fourier coefficients \eqref{simplecom-t} becomes:
\begin{equation*}\label{fcoefM} (\la w_{n, m,t}- w_{An, m,t}e^{2\pi i m\vp(t)})-(\mu v_{n, m,t}- v_{Bn, m,t}e^{2\pi i m\psi(t)})=\phi_{n, m,t} \end{equation*} which implies that for non-zero $n$ the obstructions $O_{n, m}^A$ for $$(\mu v_{n, m,t}- v_{Bn, m,t}e^{2\pi i m\psi(t)})+\phi_{n, m,t}$$ are trivial. This implies that $O_{n, m}^A(v_t)$ satisfies the equation: \begin{equation}\label{obs-eq} \mu O_{n, m}^A(v_t)- e^{2\pi i m\psi(t)} O_{Bn, m}^A(v_t)= O_{n, m}^A(\phi_t) \end{equation} where $ O_{n, m}^A(v_t)$ and $O_{n, m}^A(\phi_t)$ are defined as in \eqref{obs}. From this, by backward and forward iteration by $B$, one obtains two expressions for $ O_{n, m}^A(v_t)$: \begin{equation*}\begin{aligned}
O_{n, m}^A(v_t)&=\sum_{l\ge 0} \mu_{m,t}^{-(l+1)} e^{-2\pi i m\psi(t)}O_{B^ln, m}^A(\phi_t)\\
&=-\sum_{l< 0} \mu_{m,t}^{-(l+1)} e^{-2\pi i m\psi(t)}O_{B^ln, m}^A(\phi_t),\end{aligned}
\end{equation*} where $ \mu_{m,t}:=e^{-2\pi i m\psi(t)}\mu$.
It is proved in Lemma 4.5 in \cite{DK} that if every $A^kB^l$ for $(k, l)\ne (0, 0)$ is ergodic, and if $n=n^*$ then either for $l>0$ for $l<0$, the term $\|(B^lA^kn, m)\|$ has exponential growth in $(l, k)$ for $\|(l, k)\|$ larger than some $C\log|n|$ and polynomial growth for $\|(l, k)\|$ less than $C\log|n|$.
Hence, for $n=n^*$, it follows exactly as in Lemma 4.5 \cite{DK}, that either one or the other sum above are comparable to the size of $\|\phi_{t}\|_r\|(n, m)\|^{-r+\sigma}$, where $\si$ is a constant which depends only on $A, B $ and the dimension $d$. Therefore in case $n\ne 0$ for all $t\in I$ \begin{equation}\label{tildev}
|\tilde v_{n, m,t}|=|O_{n, m}^A(v_t)|\le C_r\|\phi\|_{0(I),r}\|(n, m)\|^{-r+\sigma}. \end{equation}
This implies the $\|\cdot\|_{0(I), r}$-norm estimate for $\tilde v$.
To obtain the estimate in the $t$ direction, just as in the Claim 1, we look at $\Delta \tilde v_{n, m}$. For $n\ne 0, n=n^*$: \begin{equation*} \begin{aligned} &\Delta \tilde v_{n, m}=O_{n, m}^A(v_t-v_t')=\\ &\Sigma_l^{\pm}\Sigma_k \mu^{-(l+1)}\la^{-(k+1)}e^{-2\pi i m (l\psi(t)+k\vp(t))}(\phi_{B^lA^kn, m, t}-\phi_{B^lA^kn, m, t'})\\ &+\Sigma_l^{\pm}\Sigma_k \mu^{-(l+1)}\la^{-(k+1)}(e^{-2\pi i m (l\psi(t)+k\vp(t))}-e^{-2\pi i m (l\psi(t')+k\vp(t'))})\phi_{B^lA^kn, m, t'}.\\ \end{aligned} \end{equation*} If $\vp$ and $\psi$ are Lipschitz and $\phi$ is in $C^{lip(I), r}$, we have: \begin{equation*} \begin{aligned}
&|\Delta \tilde v_{n, m}|\le
\|\phi\|_{lip(I), r}|t-t'|\Sigma_l^{\pm}\Sigma_k |\mu|^{-(l+1)}|\la|^{-(k+1)}\|(B^lA^kn, m)\|^{-r}\\
&+2\pi S|t-t'|\|\phi\|_{0(I), r}\Sigma_l^{\pm}\Sigma_k \mu^{-(l+1)}\la^{-(k+1)}|k||l|\|(B^lA^kn, m)\|^{-r+1}\\ \end{aligned} \end{equation*} Now the same argument as above (based on Lemma 4.5 \cite{DK}) implies that for every $n=n^*$ one of the sums (for $l>0$ or $l<0$) \\
$\Sigma_l^{\pm}\Sigma_k \mu^{-(l+1)}\la^{-(k+1)}|k||l|\|(B^lA^kn, m)\|^{-r+1}$ can be bounded by $\|(n, m)\|^{-r+\sigma}$, where $\sigma$ is a constant depending on $A, B, \la, \mu$ and $d$. This implies
$$|\Delta \tilde v_{n, m}|\le CS\|\phi\|_{lip(I), r}\|(n, m)\|^{-r+\sigma}|t-t'|.$$
Taking into account all the estimates above, this implies:
$$\| \tilde v\|_{lip(I), r}\le C_rN^3S\|\phi\|_{lip(I), r+\si},$$ with $\si$ fixed depending only on $A, B, \la$ and $d$.
{\it -End of proof of claim 2-}
Now given $v, w$ such that $(\la w-w\circ f_{\varphi})-(\mu v-v\circ g_{\psi})=\phi$, first truncate $v_t$ to $T_Nv_t$ for all $t\in I$. We choose the same $N$ for all $t\in I$. The truncation and the residue satisfy the following estimates for every $t$ and $r\leq r'$ \begin{equation}\label{truncest} \begin{aligned}
\|T_Nv_t\|_{r'}&\le C_{r,r'} N^{r'-r+d}\|v_t\|_{r}\\
\|R_Nv_t\|_{r}&\le C_{r,r'}N^{r-r'+d}\|v_t\|_{r'} \end{aligned} \end{equation} Since the same truncation is used for all t, it is easy to check that
\begin{equation*} \begin{aligned}
\|T_Nv\|_{lip(I),r'}&\le C_{r,r'}N^{r'-r+d}\|v\|_{lip(I),r}\\
\|R_Nv\|_{lip(I),r}&\le C_{r,r'}N^{r-r'+d}\|v\|_{lip(I),r'} \end{aligned} \end{equation*}
Now the Claim 2 applies to $T_Nv$. It gives $\widetilde {T_Nv}$ such that for $T_Nv-\widetilde {T_Nv}$ the obstructions $O_{n, m}^A(T_Nv_t-\widetilde {T_Nv_t})$ vanish for $n\ne 0$ and $$\|\widetilde {T_Nv}\|_{lip(I),r}\le C_rN^3S\|T_N\phi\|_{lip(I),r+\si}.$$
Notice that $\widetilde{T_Nv_t}$ by construction has all $(0, m,t)$-Fourier coefficients equal to zero for $|m|>N$. Thus the Claim 1 can be applied to $T_Nv-\widetilde {T_Nv}$. Therefore there exists $h\in \circ( \mathcal A\times {\mathbb T}^{d+1}, \R^{d+1})$ as in Claim 1 such that for all $t\in \mathcal A$: $$T_Nv_t-\widetilde{T_N v_t}=\la h_t-h_t\circ{f_t}$$ and
\begin{equation}\label{he} \begin{aligned}
\|h\|_{lip(I),r+1}&\le C_rN^3S\|T_Nv-\widetilde{T_N v}\|_{lip(I),r+1+\si}\\
&\le C_rN^3S(\|T_Nv\| _{lip(I),r+1+\si}+ C_rN^3\|T_N\phi\|_{lip(I),r+1+2\si})\\
&\le C_rSN^{4+\si}\|v\| _{lip(I),r}+C_rSN^{6+2\si}\|\phi\|_{lip(I),r-2}. \end{aligned} \end{equation} Also \begin{equation*} \begin{aligned}
\|v-(\la h-h\circ f)\|_{lip(I),r}&= \|R_Nv+\widetilde{T_N v}\|_{lip(I),r}\\
&\le \|R_Nv\|_{lip(I),r}+ C_rSN^3\|T_N\phi\|_{lip(I),r+\si}\\
&\le C_{r,r'}N^{r-r'+d}\|v\|_{lip(I),r'}+C_rSN^{5+\si}\|\phi\|_{lip(I),r-2}
\end{aligned} \end{equation*} Now to estimate $w-(\mu h-h\circ g)$ we use:
\begin{equation*} \begin{aligned} &(\la w-w\circ f)-(\mu v-v\circ g)=\phi\\ &(\la w-w\circ f)-(\mu T_Nv-T_Nv\circ g)-(\mu R_Nv-R_Nv\circ g)=\phi\\ &(\la w-w\circ f)-(\mu (T_Nv-\widetilde {T_Nv})-(T_Nv-\widetilde{T_N v}) \circ g)\\ &-(\mu \widetilde{T_N v}-\widetilde{T_N v}\circ g)-(\mu R_Nv-R_Nv\circ g)=\phi\\ &(\la w-w\circ f)-(\mu (\la h-h\circ f)-(\la h-h\circ f) \circ g)-(\mu \widetilde{T_N v}-\widetilde{T_N v}\circ g)\\ &-(\mu R_Nv-R_Nv\circ g)=\phi\\ &\la (w-(\mu h-h\circ g))-(w-(\mu h-h\circ g))\circ f=\\ &\phi+(\mu \widetilde{T_N v}-\widetilde{T_N v}\circ g)-(\mu R_Nv-R_Nv\circ g).\\ \end{aligned} \end{equation*} This implies: \begin{equation*} \begin{aligned} &\la (T_Nw-(\mu h-h\circ g))-(T_Nw-(\mu h-h\circ g))\circ f=\\ &\phi+(\mu \widetilde{T_Nv}-\widetilde{T_N v}\circ g)-(\mu R_Nv-R_Nv\circ g)-(\mu R_Nw-R_Nw\circ g)=\\ &T_N\phi +(\mu \widetilde{T_Nv}-\widetilde{T_N v}\circ g). \end{aligned} \end{equation*} Since both $T_Nw-(\mu h-h\circ g)$ (by construction of $h$) and
$T_N\phi +(\mu \widetilde{T_Nv}-\widetilde{T_N v}\circ g)$
(by construction of $\widetilde{T_Nv}$), satisfy that their $(0, m,t)$ Fourier coefficients are zero for $|m|>N$, the second part of the Claim 1 applies and gives an estimate for $T_Nw-(\mu h-h\circ g)$: \begin{equation*} \begin{aligned}
\|T_Nw-(\mu h-h\circ g)\|_{lip(I),r}&\le C_rSN^3\|T_N\phi+(\mu \widetilde{T_Nv}- \widetilde{T_Nv}\circ g)\|_{lip(I),r+\si}\\
&\le C_rSN^{5+2\si}\|\phi\|_{lip(I),r-2}\\ \end{aligned} \end{equation*} Therefore: \begin{equation*} \begin{aligned}
\|w-(\mu h-h\circ g)\|_{lip(I), r}&\le C_rSN^{5+2\si} \|\phi\|_{lip(I),r-2}+\|R_Nw\|_{lip(I), r}\\
&\le C_rSN^{5+2\si}\|\phi\|_{lip(I),r-2}+C_{r, r'}N^{d+r'-r}\|w\|_{r'} \end{aligned} \end{equation*} Finally we can redefine the constant $\sigma$ by $\sigma:= 6+2\si$. This completes the estimates in Lemma \ref{splittinglemma}.
\section{The KAM scheme} \label{sec.kam}
\begin{lemm} \label{exclusion} Let $M>0$. There exists $N_0(M)$ such that if $N>N_0$ and $\tilde{N}=N^{3/2}$ and if $I$ is an interval of size $1\geq |I|\geq 1/(2MN^2)$ and if $M^{-1}<\varphi'(t)<M$ for every $t \in I$, then there exists a union of disjoint intervals $\mathcal U = \{ \tilde{I}_j\}$ such that $ \varphi(\tilde{I}_j) \in \mathcal D(\tilde{N},A)$ and $\tilde{I}_j \subset I$ and $| \tilde{I}_j|\geq 1/(2M\tilde{N}^2)$ and $\sum | \tilde{I}_j| \geq (1-2dM^2\tilde{N}^{-1}) |I|$. \end{lemm}
\begin{proof}
We just observe that the set of $t_k \in I$ such that $\l+e^{i2\pi \varphi(t)}=0$ with $\l \in \mathcal E(A)$ and $k \leq \tilde{N}$ consists of at most $d([M \tilde{N}^2 |I|]+2)$ points separated one from the other by at least $1/(M \tilde{N}^2)$. Excluding from $I$ the intervals $[t_k-M/\tilde{N}^3,t_k+M/\tilde{N}^3]$ leaves us with a collection of intervals of size greater than $1/(2M\tilde{N}^2)$ of total length $|I|- d([M \tilde{N}^2 |I|]+2)M/\tilde{N}^3 \geq (1-2dM^2\tilde{N}^{-1}) |I|$.
\end{proof}
Recall that { \begin{equation} \label{M1} \max(\|\varphi \|_{lip(I_0)},\|\psi \|_{lip(I_0)}) \leq \frac{M}{2}, \quad \inf_{t\in I_0}\varphi'(t) \geq \frac{2}{M} \end{equation}}
Let $N_0\geq N_0(M)$ of Lemma \ref{exclusion} and define for $n\geq 1$, $N_n=N_{n-1}^{\frac{3}{2}}$.
{ Observe that Lemma \ref{exclusion} implies that if $ \mathcal A_n$ is a collection of intervals of sizes greater than $1/(2M N_n^2)$ and $\varphi_n$ and $\psi_{n}$ are functions satisfying (\ref{M1}) on $ \mathcal A_n$ with $M$ instead of $2M$ then there exists $ \mathcal A_{n+1}$ that is a collection of intervals with sizes greater than $1/(2M N_{n+1}^2)$ such that $\varphi_{n}( \mathcal A_{n+1}),\psi_{n}( \mathcal A_{n+1}) \subset \mathcal D(N_{n+1}) $ and $\lambda( \mathcal A_{n+1}) \geq (1-2dM^2 N_{n+1}^{-1}) \lambda( \mathcal A_n)$.}
We now describe the inductive scheme that we obtain from an iterative application of Proposition \ref{mainprop}. At step $n$ we have $f_n=f_{\varphi_n}+ \Delta f_n$,$g_n=g_{\psi_n}+ \Delta g_n$ defined for $t \in \mathcal A_n$, with $ \mathcal A_{-1}=[0,1]$. We denote $\varepsilon_{n,r}=\max( \|\Delta f_n \|_{lip( \mathcal A_n),r},\|\Delta g_n\|_{lip( \mathcal A_n),r} )$. We obtain $h_n$ and $\varphi_{n+1}$ and $\psi_{n+1}$ defined on $ \mathcal A_{n+1}$ such that \begin{align*} H_n f_n H_n^{-1}&= f_{\varphi_{n+1}} + {\Delta f_{n+1}} \\ H_n g_n H_n^{-1}&= g_{\psi_{n+1}} + {\Delta g_{n+1}} \end{align*}
with ${\Delta f_{n+1}},{\Delta g_{n+1}} \in C^{lip( \mathcal A_{n+1}),\infty}_0(I,{\mathbb T}^{d+1},\R^{d+1})$, and if we denote $\xi_{n,r}=\|h_n\|_{lip( \mathcal A_{n+1}),r+1}$ and $\nu_n=\max(\| \varphi_{n+1}-\varphi_n \|_{lip( \mathcal A_{n+1})},\| \psi_{n+1}-\psi_n \|_{lip( \mathcal A_{n+1})}) $ we have from Proposition \ref{mainprop} that \begin{align}
\label{hn} \xi_{n,r} &\leq C_r \gamma_nN_n^{\si}\varepsilon_{n,r} \\ \label{phin} \nu_n& \leq \varepsilon_{n,0} \\ \label{epsn} \varepsilon_{n+1,r} &\le C_r \gamma_n N_n^{\si} \varepsilon_{n,0}\varepsilon_{n,r} + C_{r,r'} \gamma_n N_n^{\si+r-r'} \varepsilon_{n,r'} \end{align} with $\gamma_n=(1+S_n+\varepsilon_{n,0})^\si$.
{ If during the induction we can insure that $\sum \varepsilon_{n,0} <M/100$ we can conclude from (\ref{phin}) and the definition of $M$ that for all $n$, $\varphi_n$ and $\psi_{n}$ satisfy
on $ \mathcal A_n$ the inductive condition
$$({\rm C1}) \quad 2\max(\|\varphi_n \|_{lip( \mathcal A_n)},\|\psi \|_{lip( \mathcal A_n)}) \leq M, \quad \inf_{t\in \mathcal A_n}\varphi_n'(t) \geq \frac{1}{M}$$}
and Lemma \ref{exclusion} will insure that $ \mathcal A_{n+1}$ is well defined and $\lambda( \mathcal A_{n+1}) \geq (1-2M^2 N_{n+1}^{-1}) \lambda( \mathcal A_n)$. To be able to apply the inductive procedure we also have to check that $H_n$ is indeed invertible which is insured if during the induction we have $$({\rm C2}) \hspace{4cm} \xi_{n,0}<\frac{1}{2}. \hspace{8cm}
$$ We call the latter two conditions the inductive conditions.
The proof that the scheme (\ref{hn})--(\ref{epsn}) converges provided an adequate control on $\varepsilon_{0,0}$ and $\varepsilon_{r_0,0}$ for a sufficiently large $r_0$ is classical but we provide it for completeness.
\begin{lemm} \label{kam} Let $ \alpha=4\si+2$, $\beta=2 \sigma +1$, and $r_0=[8\si+5]$. If $S_n,\xi_{n,r},\varepsilon_{n,r}$ satisfy (\ref{hn})--(\ref{epsn}), there exists $\bar{N}_0(\si)$ such that if $N_0=\bar{N}_0 M$ and $$\varepsilon_{0,0}\leq N_0^{- \alpha}, \quad \varepsilon_{0,r_0}\leq N_0^{\beta}$$ then for any $n$ the inductive conditions (C1) and (C2) are satisfied and in fact $\varepsilon_{n,0}\leq N_n^{- \alpha}$, $\xi_{n,0}\leq N_n^{-\si}$, and for any $s \in \N$, there exists ${\bar C}_r$ such that $\max(\varepsilon_{n,s},\xi_{n,s}) \leq {\bar C}_s N_n^{-1}$. \end{lemm}
\begin{proof} We first prove by induction that for every $n$, $\varepsilon_{n,0}\leq N_n^{- \alpha}$ and $\varepsilon_{n,r_0}\leq N_n^{\beta}$, provided $\bar{N}_0(\si)$ is chosen sufficiently large.
Assuming the latter holds for every $i\leq n$, the inductive hypothesis (C1) and (C2) can be checked up to $n$ immediately from (\ref{hn}) and (\ref{phin}). Now, (\ref{epsn}) applied with $r=0$ and $r'=r_0$ yields \begin{align*} \varepsilon_{n+1,0} &\le C_0 N_n^{\si}(2+M)^\si N_n^{-2 \alpha} + C_{0,r_0} N_n^{\si-r_0} N_n^{\beta} \\ &\leq N_{n+1}^{- \alpha} \end{align*} provided $\bar{N}_0(\si)$ is sufficiently large.
On the other hand, applying (\ref{epsn}) with $r'=r=r_0$ yields \begin{align*} \varepsilon_{n+1,r_0} &\le C_{r_0} N_n^{\si}(2+M)^\si N_n^{- \alpha} N_n^\beta + C_{r_0,r_0} N_n^{\si} N_n^{\beta} \\ &\leq N_{n+1}^{\beta} \end{align*} provided $\bar{N}_0(\si)$ is sufficiently large.
To prove the bound on $\varepsilon_{n,s}$ we start by proving that for any $s$, there exist $\tilde{C}_s$ and $n_s$ such that for $n\geq n_s$ we have that $\varepsilon_{n,s}\leq \tilde{C}_s N_n^\beta$. Let indeed $n_s$ be such that $N_{n_s}^{-1/10} ((1+M)^\si C_s+C_{s,s}) <1$. Let $\tilde{C}_s$ be such that $\varepsilon_{n_s,s} \leq \tilde{C}_s N_{n_s}^\beta$. We show by induction that $\varepsilon_{n,s} \leq \tilde{C}_s N_n^\beta$ for every $n\geq n_s$. Assume the latter true up to $n$ and apply (\ref{epsn}) with $r=r'=s$ to get
\begin{align*} \varepsilon_{n+1,s} &\le C_{s} N_n^{\si}(1+M)^\si N_n^{- \alpha} \varepsilon_{n,s} + C_{s,s} N_n^{\si} \varepsilon_{n,s} \\ &\leq N_{n}^{\si+1/10} \varepsilon_{n,s} \\ &\leq \tilde{C}_s N_n^{\si+1/10+\beta} \leq \tilde{C}_s N_{n+1}^\beta. \end{align*}
We will now bootstrap on our estimates as follows. Let $s'(s)=s+[\sigma+\beta+\frac{3}{2}(\si+1)]+1$, and define $\tilde{n}_s=\max(n_s,n_{s'})$. Let $\bar{C}_s$ be such that $\varepsilon_{\tilde{n}_s,s} \leq \bar{C}_s N_n^{-\sigma-1}$. We will show by induction that for any $n \geq \tilde{n}_s$ we have that $\varepsilon_{n,s} \leq \bar{C}_s N_n^{-\sigma-1}$. Indeed, apply (\ref{epsn}) with $r=s$ $r'=s'$ to get
\begin{align*} \varepsilon_{n+1,s} &\le \bar{C}_s C_{s} N_n^{\si}(1+M)^\si N_n^{- \alpha} N_n^{-\si-1} + C_{s,s'} \tilde{C}_{s'} N_n^{\beta} N_n^{\si+s-s'} \\ &\leq \bar{C}_s N_{n+1}^{-\si-1} \end{align*} if $n_s$ was chosen sufficiently large.
Finally, (\ref{hn}) yields that for $n\geq \tilde{n}_s$, $\xi_{n,s} \leq C'_s N_n^{-1}$.
\end{proof}
\noindent{\it Proof of the main theorem.}
The sets $ \mathcal A_n$ are decreasing and we let $ \mathcal A_\infty=\liminf \mathcal A_n$. The result of Lemma \ref{kam} implies that $$\lambda ( \mathcal A_\infty )\geq \Pi (1-2M^2 N_{n+1}^{-1}) \geq 1-\eta$$ if $N_0\geq N_0(\eta)$. On $ \mathcal A_\infty$, $\varphi_n$ and $\psi_n$ converge in the Lipschitz norm and the maps $H_n \circ \ldots \circ H_1,H_n^{-1} \circ \ldots \circ H_1^{-1}$ converge in the $C^{lip,\infty}$ norm to some $G,G^{-1}$ such that $G f_{\varphi} G^{-1} = f_{\varphi_\infty}$, $G g_{\psi} G^{-1} = g_{\psi_\infty}$, where $(\varphi_\infty,\psi_\infty) = \lim_{n\to \infty} (\varphi_n,\psi_n)$.
\section{Proof of Theorem \ref{main} in the case of higher dimensional elliptic factors, $d_2>1$} \label{generald2}
Define instead of the set $ \mathcal D(N,A)$ of Section \ref{section.inductive} the following
\begin{multline*} \mathcal D(N,A) = \{ \alpha \in {\mathbb T}^{d_2} \ / \ |\l + e^{i2\pi (k, \alpha )} | \geq N^{-b}, \\ \forall \l \in \mathcal E(A), \forall k \in \Z^{d_2}-\{0\}, \|k\|\leq N\} \end{multline*} where $b=30d_2^2$. Instead of Lemma \ref{exclusion} we have the following more general statement.
\begin{lemm} \label{exclusiond} Let $\nu>0$. There exists $N_0(\nu,d_2)$ such that if $N>N_0$ and if $I$ is an interval of size $1\geq |I|\geq 1/N^a$, $a=4d_2+20$, and if $\varphi : I \to {\mathbb T}^{d_2}$ satisfies a Pyartli condition with constant $\nu$, then for $\tilde{N}=N^{3/2}$, there exists a union of disjoint intervals $\mathcal U = \{ \tilde{I}_j\}$ such that $\tilde{I}_j \in \mathcal D(\tilde{N},A)$ and $\tilde{I}_j \subset I$ and $| \tilde{I}_j|\geq 1/\tilde{N}^a$ and $\sum | \tilde{I}_j| \geq (1-\tilde{N}^{-1}) |I|$. \end{lemm}
\begin{proof} The proof is a direct consequence of the Pyartli condition and a repeated application of the intermediate value theorem. We just deal with case $\l=1$ the other cases being similar. More precisely, for any fixed $k$, $\|k\|\leq N$, after excluding $d_2$ intervals of size $1/N^{a}$ from $I$ we get that $|(k,\varphi')|\geq N^{-a(d_2+1)}$. After excluding $\mathcal O(N)$ intervals of size $N^{a(d_2+1)-b}$ we remain with intervals on which $\|(k,\varphi)\|\geq N^{-b}$. We then apply this procedure for every $k \in \Z^{d_2}$ such that $0<\|k\|\leq N$, then further eliminate all the intervals that are smaller than $\tilde{N}^{-a}$, and finally observe that the remaining part of $I$ is a union of intervals satisfying the conditions of the lemma. \end{proof}
The effect of changing the exponent in the definition of $ \mathcal D(N,A)$ just modifies $\sigma(A,B)$ of Proposition \ref{mainprop} to make it $\sigma(A,B,d_2)$. This is because in (\ref{h-0}) the small divisor (in the case $|\lambda|=1$) becomes \newline $\frac{1}{|\l-e^{2\pi i (m,\vp(t))}|} \leq N^b$ if $m\in \Z^{d_2}$ is such that $|m|\leq N$. The rest of the proof of Proposition \ref{mainprop} is identical to the case $d_2=1$, except that everywhere the Lipschitz norm in the parameter direction should be replaced by the $C^{d_2}$ norm. If we assume WLOG that $\varphi$ satisfies an initial Pyartli condition with constant $\nu$, then similarly to what was done in the case $d_2=1$, we insure in the KAM scheme that a Pyartli condition with a fixed constant $\nu/2$ is satisfied by the functions $\varphi_n$, provided the control on the perturbation $\varepsilon$ is sufficiently small.
\section{Proof of Theorem \ref{main2}} \label{proofmain2}
Let $A,B, \alpha,\beta$ and $f,g$ be as in the statement of Theorem \ref{main2}.
Let us momentaneously assume that $ \alpha \in \text{DC}(\tau,\gamma,A)$ that is $ |\l - e^{i2 \pi (k, \alpha)} |> \frac{\gamma}{|k|^\tau}$ for every non zero vector $k \in \Z^{d_2}$ and every $\l \in \mathcal E(A)$.
This clearly plays a similar role to $\varphi(t) \in \mathcal D(A)$ and the same proof as that of Proposition \ref{mainprop} yields a conjugacy $ H={\rm Id} + h$ such that \begin{equation}\label{nonlin2} \begin{aligned} H \circ f &= (\tilde{f}_0 + \widetilde{\Delta f}) \circ H \\ H \circ g &= (\tilde{g}_0 + \widetilde{\Delta g}) \circ H \end{aligned}
\end{equation} with $ \tilde{f}_0= A \times R_{\tilde{ \alpha}}, \tilde{g}_0= B\times R_{\tilde{\beta}}$ and $h, \widetilde{\Delta f}, \widetilde{\Delta g}$ satisfy estimates as in Proposition \ref{mainprop}. Now, the fact that $(\rho_{\mu_1}(f),\rho_{\mu_2}(g))=( \alpha,\beta)$ implies that $(\rho_{H_* \mu_1}(H \circ f \circ H^{-1}),\rho_{H_*\mu_2}(H \circ g \circ H^{-1}))=( \alpha,\beta)$. In conclusion we can replace $\tilde{f}_0,\tilde{g}_0$ by $A\times R_ \alpha, B\times R_\beta$ in (\ref{nonlin2}) and include $\tilde{ \alpha}- \alpha$,$\tilde{ \beta}- \beta$ inside the error terms without changing the quadratic nature of the estimates.
For the general case $( \alpha,\beta) \in \text{SDC}(\tau,\gamma,A,B)$ one cannot use just one of the frequencies $ \alpha$ or $\beta$ to solve the linearized equations of (\ref{simpleconj}). Indeed, both $ \alpha$ and $\beta$ may be Liouville vectors and the small divisors that appear in (\ref{h-0}) may be too large. Actually the linearized system (\ref{simpleconj}) will not be solved as in Claim 1 but just up to an error term that is quadratic as in Lemma \ref{splittinglemma}. The idea goes back to Moser \cite{M} who observed that if for each $m$ one of the small divisors ${\l-e^{2\pi i m \alpha}}$ or ${\mu-e^{2\pi i m\beta}}$ is not too small, as stated in the SDC condition, then the relation implied by the commutation
(\ref{simplecom})
\begin{align*} (\la w-w\circ f_\varphi)-(\mu v-v\circ g_\psi)=\phi \end{align*} insures that (\ref{simpleconj}) can be solved up to an error term of the order of $\phi$, that is a quadratic error term as in \eqref{quad}.
The rest of the proof of Theorem \ref{main2} is identical to that of Theorem \ref{main3}. $
\Box$
\section{Appendix}
In the Appendix we give references and proofs for the estimates used in the proofs of Lemma \ref{splittinglemma} and Proposition \ref{mainprop}.
\subsection{Convexity estimates}
\begin{prop} \label{hadamard} Let $f,g \in C^{lip,\infty}(I, {\mathbb T}^{d},\R)$. Then
\begin{itemize} \item[(i)] $$
||f||_{lip(I),s}\le C_{s_1,s_2}||f||_{lip(I),s_1}^{a_1}||f||_{lip(I),s_2}^{a_2}$$ for all non-negative numbers $a_1,a_2,s_1,s_2$ such that $$ a_1+a_2=1,\quad s_1a_1+s_2a_2=s.$$
\item[(ii)] $$
||fg||_{lip(I),s}\ \le\ C_s(||f||_{lip(I),s}||g||_{lip(I),0}+||f||_{lip(I),0}||g||_{lip(I),s})$$ for all non-negative numbers $s$. \end{itemize} \end{prop} \begin{proof} (i) One way to show interpolation estimates in the scale of $C^{lip, s}$ norms is to derive them from the existence of smoothing operators and from the norm inequalities for the smoothing operators. This is done in \cite{Zehnder} for spaces $C^{\alpha, s}$ where $0<\alpha\le 1$, which includes the case of $C^{lip, s}$. Another elementary proof for interpolation without going through smoothing operators can be found in \cite{OdlL}.
(ii) Immediate corollary of the interpolation estimates is the following fact:
$$\|f\|_{lip(I), i}\|g\|_{lip(I), j}\le C (\|f\|_{lip(I), k}\|g\|_{lip(I), l}+\|f\|_{lip(I), m}\|g\|_{lip(I), n})$$ if $(i,j)$ lies on the line segment joining $(k, l)$ and $(m, n)$. (See Corollary 2.2.2. in \cite{Ham}). The statement (ii) in the Proposition follows from this by using the product rule on derivatives (see Corollary 2.2.3. in \cite{Ham}) and the following inequality: \begin{equation*} \begin{aligned}
Lip (fg)&=\sup_{x\ne y} \frac{|(fg)(x)-(fg)(y)|}{|x-y|} \\
&\le \sup ( \frac{|f(x)-f(y)||g(x)|}{|x-y|} + \frac{|g(x)-g(y)||f(y)|}{|x-y|})\\
&\le L_f\|g\|_0+\|f\|_0L_g \end{aligned} \end{equation*} where $L_f$ and $L_g$ are Lipshitz constants for $f$ and $g$, respectively. \end{proof}
\subsection{Composition}
\begin{prop} \label{annexe.compose} Let $f,g \in C^{lip,\infty}(I, {\mathbb T}^{d+1},\R^{d+1})$.Then \begin{itemize}
\item[(i)] $h(x)=f(x+g(x))-f(x)$ verifies
$$ \| h\|_{lip(I),s}
\leq C_s(\| f\|_{lip(I),0} \|g\|_{lip(I),s+1}+ \| f\|_{lip(I),s+1}\|g\|_{lip(I),0}).$$
\item[(ii)] $k(x)=f(x+g(x))-f(x)-Df g(x)$ verifies
$$ \| k\|_{s}\leq
C_s (\| f\|_{lip(I),0} \|g\|_{lip(I),s+2}+ \| f\|_{lip(I),s+2}\|g\|_{lip(I),0}) $$
\end{itemize}
\end{prop} \begin{proof}
In the proof we shorten the notation $\|\cdot\|_{lip(I), s}$ to $\|\cdot\|_{lip, s}$.
(i) It suffices to prove the estimates for the coordinate functions of $f$, so in what follows we assume that $f$ denotes a coordinate function of $f$. Let $D_i^1$ denote partial derivation in one of the basis directions and let $g_j$ denote coordinate functions of $g$. Since $D_i^1 h=D_i^1(f(x+g(x))-f(x))=\sum_j D_j^1fD_i^1g_j$, we can apply part (ii) of the previous proposition to $D_j^1fD_i^1g_j$:
\begin{equation*} \begin{aligned}
\|D^1 h\|_{lip, s}&\le C\max_j \|D_j^1fD_i^1g_j\|_{lip, s} \\
&\le C_s\max_j(\|D_j^1 f\|_{lip, s}\|D_i^1g_j\|_{lip, 0}+\|D_j^1 f\|_{lip, 0}\|D_i^1g_j\|_{lip, s})\\
&\le C_s\max_j (\|f\|_{lip, s+1}\|g_j\|_{lip, 1}+\|f\|_{lip, 1}\|g_j\|_{lip, s+1})\\
&\le C'_s(\|f\|_{lip, s+2}\|g\|_{lip, 0}+\|f\|_{lip, 0}\|g\|_{lip, s+2}) \end{aligned} \end{equation*} where we invoked part (ii) of the previous proposition again to obtain the last line of estimates above. Since for the $lip, 0$-norm we have:
$$\|h\|_{lip, 0}=\|f(x+g(x))-f(x)\|_{lip, 0}\le L_f\|g\|_0\le \|f\|_{lip, 0}\|g\|_{lip, 0},$$ the claim follows.
(ii) Again by reducing to coordinate functions we look at one coordinate function of $k$ and $f$ (which we denote by $k$ and $f$ as well), so we have $k=f(x+g(x))-f-\sum_i D_i^1f g_i$, where $D_i^1$ denotes $\partial/\partial x^i$. Then: $D_j^1k=-\sum_iD_j^1D^1_i f g_i$, where $g_i$ denotes coordinate functions of $g$. This implies (by using (ii) of Proposition \ref{hadamard}) the following estimate for the first derivatives: \begin{equation*} \begin{aligned}
\|D_j^1 k\|_{lip, s}&\le \sum_i \|D_j^1D^1_i f g_i\|_{lip, s} \\
&\le C_s(\|D_j^1D^1_i f\|_{lip, s}\|g_i\|_{lip, 0}+\| D_j^1D^1_i f\|_{lip, 0}\|g\|_{lip, s})\\
&\le C_s(\|f\|_{lip, s+2}\|g_i\|_{lip, 0}+\|f\|_{lip, 2}\|g\|_{lip, s})\\
&\le C'_s(\|f\|_{lip, s+2}\|g\|_{lip, 0}+\|f\|_{lip, 0}\|g\|_{lip, s+2}) \end{aligned} \end{equation*} For the $lip, 0$-norm we have:
$$\|k\|_{lip, 0}\le L_f\|g\|_0+\max_i\{\|D_i^1 fg_i\|_{lip, 0}\}\le C\|f\|_{lip, 1}\|g\|_{lip, 0}$$ which together with the estimates above implies the claim. \end{proof} \subsection{Inversion}
\begin{prop} \label{annexe.inverse} Let $h \in C^{lip,\infty}(I, {\mathbb T}^{d+1},\R^{d+1})$ and assume that
$$\|h\|_{lip(I),1}\leq \frac{1}{2}$$ Then $$f: {\mathbb T}^{d+1} \to {\mathbb T}^{d+1} , x\mapsto H(x)=x+ h(x)$$ is invertible and if we write $H^{-1}(x)=x+\bar h (x)$ then
$$\|\bar h\|_{lip(I),s} \leq C_s \|h\|_{lip(I),s} $$ for all $s\in\N$. \end{prop} \begin{proof} For $C^s$ norms this is proved for example in Lemma 2.3.6. in \cite{Ham}. The proof uses induction and interpolation estimates, and it is general to the extent that it applies to any sequence of norms on $C^\infty$ which satisfy interpolation estimates. Thus the claim follows from part (i) of the Proposition \ref{hadamard} and Lemma 2.3.6. in \cite{Ham}.\end{proof}
\end{document} | arXiv |
WizEdu
What is the electric potential energy of the group of charges in (Figure 1)? Assume that q = -5.5 nC.
In: Physics
U= (value) (units)
Concepts and reason
The concepts used to solve this problem are electric potential energy and Pythagoras theorem. Initially use the Pythagoras theorem to calculate the hypotenuse distance. Then use the pairs of charges and the distance between them in the potential energy formula. Finally add all the potential energy terms to calculate the total potential energy of the group of charges.
The electric potential energy of a group of charges is,
$$ U=\sum_{i, j, i * j} \frac{1}{4 \pi \varepsilon_{0}} \frac{q_{i} q_{j}}{r_{i j}} $$
Here, \(\varepsilon_{0}\) is the permittivity of the free space, \(q\) is the charge, and \(r\) is distance between the two charges, \(U\) is the
potential energy of the arrangement, and \(i\) and \(j\) are indices of charges. Pythagoras theorem gives the hypotenuse distance in a right-angled triangle. \(c=\sqrt{a^{2}+b^{2}}\)
\(a, b,\) and \(c\) Here, \(a, b,\) and \(c\) are the sides of the triangle.
Use the Pythagoras theorem to find the distance between the two positive charges.
Substitute \(4.0 \mathrm{~cm}\) for \(a,\) and \(3.0 \mathrm{~cm}\) for \(b\) in the equation \(c=\sqrt{a^{2}+b^{2}}\)
$$ \begin{aligned} c &=\sqrt{(4.0 \mathrm{~cm})^{2}+(3.0 \mathrm{~cm})^{2}} \\ &=\sqrt{16.0 \mathrm{~cm}^{2}+9.0 \mathrm{~cm}^{2}} \end{aligned} $$
\(=5.0 \mathrm{~cm}\)
Use the potential energy formula.
\(\frac{1}{4 \pi \varepsilon_{0}} \frac{q_{1} q_{2}}{r_{12}}+\frac{1}{4 \pi \varepsilon_{0}} \frac{q_{1} q_{3}}{r_{13}}+\frac{1}{4 \pi \varepsilon_{0}} \frac{q_{2} q_{3}}{r_{23}} \quad \sum_{\text {for } i, j, i * j} \frac{1}{4 \pi \varepsilon_{0}} \frac{q_{i} q_{j}}{r_{i j}}\) in the equation.
\(U=\frac{1}{4 \pi \varepsilon_{0}} \frac{q_{1} q_{2}}{r_{12}}+\frac{1}{4 \pi \varepsilon_{0}} \frac{q_{1} q_{3}}{r_{13}}+\frac{1}{4 \pi \varepsilon_{0}} \frac{q_{2} q_{3}}{r_{23}}\)
Refer the figure, the three charges make a right-angled triangle at negative charge. There are three charges so
\(1 \leq i, j \leq 3, i \neq j .\) Here the charge 1 is the bottom positive charge and charge 2 is the negative charge, and charge 3
is the positive charge on the right. The distance between the charge 1 and 2 is \(r_{12}\). The distance between the charge
2 and 3 is \(r_{23}\) The distance between the charge 1 and 3 is \(r_{13}\).
Use the electric potential energy equation.
Substitute \(8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2}\) for \(\frac{1}{4 \pi \varepsilon_{0}}, 3.0 \mathrm{nC}\) for \(q_{1},-5.5 \mathrm{nC}\) for \(q_{2}, 3.0 \mathrm{nC}\) for \(q_{3}, 4.0 \mathrm{~cm}\) for \(r_{12}\)
\(3.0 \mathrm{~cm}\) for \(r_{23}\), and \(5.0 \mathrm{~cm}\) for \(r_{13}\) in the equation \(U=\frac{1}{4 \pi \varepsilon_{0}} \frac{q_{1} q_{2}}{r_{12}}+\frac{1}{4 \pi \varepsilon_{0}} \frac{q_{1} q_{3}}{r_{13}}+\frac{1}{4 \pi \varepsilon_{0}} \frac{q_{2} q_{3}}{r_{23}}\)
The electric potential energy of the group of charges is \(-7.0 \times 10^{-6} \mathrm{~J}\).
Calculate the electric potential by substituting the value of charges and distances in the expression of electric potential energy.
Dr. OWL answered 4 weeks ago
What is the electric potential energy of the group of charges in the figure?
What is the electric potential at the point indicated with the dot in (Figure 1)?
What is the electric potential at point A in the figure?
What is the electric potential at the point indicated with the dot in the figure?
What is the magnitude of the electric field at the dot in the figure? (Figure 1)
Determine the magnitude and direction of the electric field at point 1 in the figure(Figure 1).
In (Figure 1), charge q2 experiences no net electric force. What is q1?
How much does the electric potential energy change as the electron moves from i to f?
Each part of (Figure 1) shows one or more point charges. The charges have equal magnitudes.
What is the magnitude of the electric force on charge A in the figure?
Match the following properties of liquids to what they indicate about the relative strength of the intermolecular forces in that liquid.
Rank each satellite based on the net force acting on it. Rank from largest to smallest.
Complete the following program skeleton. When finished, the program will ask the user for a length (in inches), convert that value to centimeters, and display the result. You are to write the function convert.
Write a function that accepts an int array and the array's size as arguments.
The figure below shows a cross section across a diameter of a long cylindrical conductor
The two 10-cm-long parallel wires in the figure are separated by 5.0 mm.
What visible wavelengths of light are strongly reflected from a 390-nm-thick soap bubble?
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Focus on: All days Jun 11, 2018 Jun 12, 2018 Jun 13, 2018 Jun 14, 2018 Jun 15, 2018 All sessions Dust as a galaxy probe Dust as a tracer in the Milky Way and local galaxies Dust in AGN Dust in the early universe Dust in the solar system Dust production by supernovae and massive stars Dust production in low mass stars Grain growth, planet formation and debris disks Laboratory studies of cosmic dust Molecules and dust Non-stellar dust production and the dust cycle... Observational constraints on dust properties Poster Presentations Hide Contributions
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Cosmic Dust: origin, applications & implications
Jun 11, 2018, 7:00 AM → Jun 15, 2018, 6:00 PM Europe/Copenhagen
Main Auditorium (Geological Museum, University of Copenhagen)
Main Auditorium
Geological Museum, University of Copenhagen
Øster Voldgade 5 - 7, 1350 København K, Denmark
Anja C. Andersen (Niels Bohr Institute) , Ann Nguyen (NASA Johnson Space Center) , Daniel Asmus (ESO) , Daniela Calzetti (University of Massachusetts) , Darach Watson (Niels Bohr Institute, University of Copenhagen) , Franciska Kemper (Academia Sinica) , Haley Gomez (University of Cardiff) , J. D. Smith (University of Toledo) , Joao Alves (University of Vienna) , Karl Gordon (Space Telescope Science Institute) , Takaya Nozawa (National Astronomical Observatory of Japan) , Thomas Henning (Max Planck Institute for Astronomy, Heidelberg)
#CPHDUST2018
The fourth in a series of conferences held every five years on cosmic dust, following successful meetings in Colorado, USA (2003), in Heidelberg, Germany (2008), and in Taipei, Taiwan (2013).
Despite its fundamental importance to so much of the cosmos, we still do not know where most dust originates, what its mineralogy is, what its properties in different environments are, or its physics and chemistry in the interstellar medium. These questions are under active study, and significant progress has been made over the past decade with new instrumentation, laboratory results, and theoretical modeling.
This conference will bring together experts on dust and dust practitioners from all different backgrounds: meteoritics, interplanetary dust, protoplanets, star-formation, AGB stars and Planetary Nebulae, dust in galaxies, supernovae, and AGN.
The conference is timed to lay out the remarkable progress on dust since the Herschel (pictured above) and Planck missions ended and their legacies have begun to be exploited, since ALMA began real science operations over the previous five years, and to prepare for the launch of JWST.
Invited speakers include:
Susanne Aalto, Almudena Alonso-Herrero, Mike Barlow, Kenji Bekki, John Bradley, Jan Cami, Ilse De Looze, Carsten Dominik, Bruce Draine, Maud Galametz, Susanne Höfner, Sebastian Hönig, Akio Inoue, Cornelia Jäger, Christine Joblin, Karin Sandstrom, Raffaella Schneider, Matt Smith, Sundar Srinivasan, Zahed Wahhaj, Gail Zasowski, Sascha Zeegers
CPHDUST_poster_B.pdf
CPHDUST_poster_E.pdf
timetable.pdf
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Akio Inoue
Alessandra Candian
Alexandros Maragkoudakis
Alexey Potapov
Almudena Alonso-Herrero
Angelos Nersesian
Antoni Macià
Antonia Bevan
Arkaprabha Sarangi
Bella Boulderstone
Birgitta Mueller
Brandon Hensley
Bruce Draine
Carolina Agurto
Carsten Dominik
Christa Gall
Christine Joblin
Christopher Clark
Ciska Kemper
Collin Knight
Cornelia Jäger
Daniel Asmus
Daniele Rogantini
Darach Watson
David Williamson
Douglas Whittet
Dries Van De Putte
Edward Jenkins
Elisa Costantini
Els Peeters
Felix Priestley
Florian Kirchschlager
Francesco Valentino
Franziska Schmidt
Frederik Doktor Simonsen
Gabi Wenzel
Gail Zasowski
Gaël Rouillé
Gen Chiaki
Georgios Magdis
Georgios Pantazidis
Hannah Chawner
Harald Mutschke
I-Da Chiang
Ian Mccheyne
Ilse De Looze
Ioanna Psaradaki
Isabella Cortzen
Jan Cami
Jarron Leisenring
JD Smith
Jens Hjorth
Jeremy Chastenet
Joel Johansson
Johan Fynbo
John Thrower
Jonas Greif
Juan C. Ibañez-Mejia
Julia Roman-Duval
Karin Sandstrom
Karl Gordon
Karl Misselt
Kasper Heintz
Kengo Tachihara
Kenji Bekki
kuan-chou Hou
Kwang-il Seon
Lapo Fanciullo
Lea Hagen
Lia Corrales
Lindsay Keller
Marcin Gladkowski
Marco Tazzari
Maria Kirsanova
Maria Niculescu- Duvaz
Marjorie Decleir
Marta Venanzi
Martin Glatzle
Masashi Nashimoto
Matthew Smith
Matthias Maercker
Maud Galametz
Mikako Matsuura
Mike Barlow
Minjae Kim
Missagh Mehdipour
Monica Relano Pastor
Monique Aller
Patricia Luppe
Peter Laursen
Peter Scicluna
Petia Yanchulova Merica-Jones
Phay K. Reg. Istration
Phil Cigan
Raffaella Schneider
Ralf Siebenmorgen
Rijutha Jaganathan
Robert Brunngräber
Roger Wesson
Rosie Beeston
Ryo Tazaki
Rémi Bérard
Sami Dib
Sara Bladh
Sarah Massalkhi
Sascha Zeegers
Sebastian Hoenig
Serge Krasnokutski
Shohei Ishiki
Shubhadip Chakraborty
Sofie Liljegren
Stefan Bromley
Steve Goldman
Sundar Srinivasan
Susanne Aalto
Susanne Höfner
Svitlana Zhukovska
Sébastien Viaene
Takaya Nozawa
Takuma Kokusho
Thomas Boutéraon
Thomas Henning
Troels Haugbølle
Ulysses Sofia
Vincent Guillet
Yoshinobu Fudamoto
Young-Soo Jo
Zahed Wahhaj
Panel Discussion Questions
Contact info for the conference
[email protected]
Mon, Jun 11
Tue, Jun 12
Wed, Jun 13
Thu, Jun 14
Check-in and registration Rotunda (Geological Museum)
Geological Museum
Øster Voldgade 5-7, 1350 København K
Check-in, badge collection, and orientation
Welcome and introduction Main Auditorium
Conference opening and welcome
Speaker: Darach Watson (Niels Bohr Institute, University of Copenhagen)
Watson_Intro_to_the_conference.key
Non-stellar dust production and the dust cycle in the ISM Main Auditorium
Convener: Prof. Thomas Henning (Max Planck Institute for Astronomy, Heidelberg)
The Condensation of Gas-Phase Elements onto Interstellar Dust Grains
Over the past 45 years, investigations of ultraviolet absorption features in stellar spectra have revealed that most of the heavy elements in the interstellar medium are depleted from the gas phase to values well below solar or B star reference abundances. The strengths of such depletions reveal the composition of dust grains in space, and they can be characterized by a limited set of parameters that are closely linked to the average gas densities and the condensation temperatures of the elements. Two outstanding mysteries remain: one is the fact that the depletion of oxygen exceeds that needed for forming silicates or metallic oxides, and the other is that the chemically inert element krypton shows some depletion. When we observe absorption features to derive the element abundances in distant galaxies, we must understand how to correct for depletions by using the patterns found in our Galaxy or the Small Magellanic Cloud as examples.
Speaker: Edward Jenkins (Princeton University)
Depletions_dust_composition3.pptx
Growth, destruction, and expulsion of dust in galaxies
Physical properties of interstellar dust (e.g., dust-to-gas ratios) are observed to be quite diverse in galaxies with different masses and types. I will discuss the origin of these diverse dust properties based on the latest results of galaxy-scale hydrodynamical simulations of galaxies with dust physics. I will particularly discuss how dust growth processes in interstellar medium (ISM) depends on the physical properties of ISM, dust-related physical processes (e.g., photo-electric heating and radiation pressure of stars on dust grains), global galaxy-scale dynamics. I will demonstrate that the formation of molecular hydrogen and dust growth in cold molecular clouds is strongly coupled. I will also show the masses of galaxies and galaxy interaction/merging can significantly influence the dust growth processes in ISM. I will briefly discuss how radiation pressure of young stars on dust grains can influence the evolution of dust in galaxies.
Speaker: Prof. Kenji Bekki (University of Western Australia)
Poster Presentations Main Auditorium
Quick presentation of posters
Convener: Christa Gall (Dark Cosmology Centre)
Polycyclic aromatic hydrocarbon emission toward the Galactic bulge
We examine polycyclic aromatic hydrocarbon (PAH), dust and atomic/molecular emission toward the Galactic bulge using $\textit{Spitzer}$ Space Telescope observations of four fields: C32, C35, OGLE and NGC 6522. These fields are approximately centered on (l, b) = (0.0°, 1.0°), (0.0°, -1.0°), (0.4°, -2.1°) and (1.0°, -3.8°), respectively. Far-infrared photometric observations complement the Spitzer/IRS spectroscopic data and are used to construct spectral energy distributions. We find that the dust and PAH emission are exceptionally similar between C32 and C35 overall, in part explained due to their locations---they reside on or near boundaries of a 7 Myr-old Galactic outflow event and are partly shock-heated. Within the C32 and C35 fields, we identify a region of elevated Hα emission that is coincident with elevated fine-structure and [O ɪᴠ] line emission and weak PAH feature strengths. We are likely tracing a transition zone of the outflow into the nascent environment. PAH abundances in these fields are slightly depressed relative to typical ISM values. In the OGLE and NGC 6522 fields, we observe weak features on a continuum dominated by zodiacal dust. SED fitting indicates that thermal dust grains in C32 and C35 have comparable temperatures to those of diffuse, high-latitude cirrus clouds. Little variability is detected in the PAH properties between C32 and C35, indicating that a stable population of PAHs dominates the overall spectral appearance. In fact, their PAH features are exceptionally similar to that of the M82 superwind, emphasizing that we are probing a local Galactic wind environment.
Speaker: Prof. Els Peeters (University of Western Ontario)
poster_45_peeters.pdf
poster_45_peeters.ppt
Spatial variations in dust extinction properties and 3D structure in the Small Magellanic Cloud with SMIDGE
Dust properties in the Small Magellanic Cloud (SMC) provide insight into the interstellar environment of one of the closest analogs to early-Universe and low-metallicity galaxies. We examine the spatial variations in dust extinction curve properties and the three-dimensional structure in the Southwest Bar of the SMC using resolved stellar populations observed with the \textit{Hubble Space Telescope (HST)} as a part of the Small Magellanic Cloud Investigation of Dust and Gas Evolution (SMIDGE) program. We use color-magnitude diagrams (CMDs) of reddened red clump and red giant branch stars to investigate in detail the impact of environment on dust extinction properties. Our eight-band HST photometry enables us to simultaneously constrain SMC's 3D structure allowing us to accurately measure dust extinction from the CMD. We use the Bayesian Extinction And Stellar Tool (BEAST, Gordon et al. (2016)) to model the photometric effects of extinction on the spectral energy distribution of individual stars in SMIDGE taking into account a log-normal distribution of foreground $A_{V}$ and an input extinction curve. We additionally model the relative positions of the stellar and dust distributions and the galactic depth along the line of sight. We then use CMD matching techniques based on Poisson statistics to extract the best-fit dust extinction and 3D structure parameters. We find a large line-of-sight depth and a slight offset of the dust on the near side of the stars. We find an extinction curve shape which varies only modestly even towards regions with high molecular gas content. These results yield the first detailed dust extinction curve properties in a key region in the SMC and have potential implications for how dust coagulates in molecular clouds in low-metallicity galaxies.
Speaker: Mrs Petia Yanchulova Merica-Jones (University of California, San Diego)
dustSlideYanchulovaM-J.pdf
dustSlideYanchulovaM-J.pptx
Dust Reverberation Mapping in AGN
The dusty obscuring structure around the active galactic nuclei (AGN), commonly referred to as the 'torus', provides the angle-dependent obscuration as postulated in the Unification Scheme of AGN. This dust rich environment supplies the central engine with material for accretion and is known to thermally absorb optical light from the accretion disc and re-emit it in the infrared (IR). The time lag between the visible and near-IR emission serves as a measure of the physical size of the innermost, hottest dust, which is set by sublimation of large graphite grains. I present the first results of our ongoing campaign to measure those time lags in a sample of AGN. The observed time lags are consistent with the established lag-luminosity relationship. We are now in the process of turning the hot dust lags into standardisable candles, as part of the ESO public survey VEILS, and will use these new lag measurements to normalise the Hubble relation at lower redshifts.
Speaker: Bella Boulderstone (University of Southampton)
Boulderstone_Slide.pdf
The Spatially Resolved Dust-to-Metals Ratio in M101
The dust-to-metals ratio provides insights into the life cycle of dust. We measure the dust-to-metals ratio in M101, a nearby galaxy with a radial metallicity gradient spanning $\sim 1~\mbox{dex}$. We fit the dust spectral energy distribution (SED) from $100$ to $500~\mu m$ with five variants of the modified blackbody (MBB) dust emission model (free $\beta$, fixed $\beta$, broken emissivity, warm dust component, and a power-law radiation field distribution). The broken emissivity method performs the best among them, showing small residuals, reasonable $\tilde{\chi}^2$ distribution, a temperature gradient decreasing with radius and no violation of the upper bounds on available metals. We show that the dust-to-metals ratio is not constant in M101, but decreases as a function of radius, leading to a lower fraction of the heavy elements being trapped in dust at low metallicity. We show that the dust-to-gas ratio (DGR) is proportional to $Z^{1.71}$. Alternatively, we could instead explain the DGR gradient as an increase in emissivity as dust grains coagulate. If we assume the Draine et al. 2014 dust-to-metals relation, the opacity constant $\kappa_{160}$ would increase at most by a factor of two, which is similar to what Planck Collaboration et al. 2014 found.
Speaker: Mr I-Da Chiang (University of California, San Diego)
20180611-CPHDUST-1-min-ad.pdf
Infrared emission and dust dynamics in expanding HII regions
Massive stars signpost places of their birth in molecular clouds with expanding HII regions and photodissociation regions (PDRs). The HII regions and PDRs have very specific observational manifestation on Spitzer images. Namely, the ring-like structures which are seen at 8 micron surround the inner regions which are bright at 24 micron. The ring-like structures are also seen on Herschel images at longer wavelengths. This can be related to the properties of dust particles which are not the same inside and outside of HII regions. We present new results of a long-term theoretical study of expanding HII regions in order to understand what happens with the dust particles near young massive stars and how HII regions and PDRs look during their evolution around massive stars. We consider the drift of charged dust under the influence of radiation pressure, Coulomb drag and the lug of dust by gas simultaneously during the expansion of an HII region. Dust particles are represented by the polycyclic aromatic hydrocarbons (PAHs) and an ensemble of silicate and graphite grains of larger sizes. We evaluated a grain charge evolution within the HII region for each dust type. We find that PAHs and intermediate-size silicates have the greatest impact on the gas dynamics. Dust-to-gas mass ratio within the HII region changes from initial canonical value up to 50-90% depending on a spectral type of the massive star. Big grains are effectively swept out of the HII region. Intermediate size grains have double-peaked distribution of radial density profile. Dynamics of charged grains allows us to qualitatively explain the emission in the HII region RCW 120. We show relative input of every dust grain type to the emission in the infrared wavelength range. Emission at 4--8 micron is produced by PAHs. Our simulations show that PAHs and smaller graphite grains are mostly coupled to the gas. Removal of PAHs from the HII regions is required to reproduce their ring-like appearance at 3.6--8 micron found by Spitzer. Photo-destruction of PAHs by strong ultraviolet emission produced by the central massive star can explain the depletion of PAHs in the HII region. Our study of a large sample of HII regions confirms that PAHs mass fraction is much lower in these objects than the average Galactic value, implying the effective destruction of aromatic particles in HII regions. We discuss how simultaneous fitting of dust emission at several Spitzer and Herschel images helps to constrain dust properties near young massive stars.
Speaker: Dr Maria Kirsanova (Institute of Astronomy, Russian Academy of Sciences)
kirsanova_poster_one_slide.ppt
Mid-IR spectroscopic observations of the dustiest AGB stars in the Galaxy
We have used the VISIR spectrograph at the Very Large Telescope to target 21 of the most luminous and heavily-obscured oxygen-rich evolved AGB stars in the galaxy. Low resolution N-band (8 - 13 um) spectroscopy was used to target the 10 μm silicate feature. The sample, with a median luminosity of ~10,000 solar luminosities and a median mass loss rate of ~10E-4 solar masses per year, has shown higher mass loss rates than previous Galactic and Large Magellanic Cloud samples, given their luminosities. These results, along with expansion velocities from previous OH maser detections, have been used to test and refine the wind-driving and mass loss mechanisms. Our new spectra have also allowed us to study the dust composition and geometries of these sources.
Speaker: Steve Goldman (Space Telescope Science Institute)
copenhagen_poster.pptx
PAH photodissociation and the formation of H$_2$
Polycyclic Aromatic Hydrocarbon (PAH) molecules are ubiquitous carbonaceous molecules, responsible for the Aromatic Infrared Bands (AIB) dominating the IR spectrum of diverse astronomical environments, from planetary and reflection nebulae, to transitional disks to entire galaxies (Tielens, 2013). PAHs make up the low-mass end of the grain size distribution (Weingartner & Draine 2001) and, akin to dust grains, these molecules can act as catalytic surfaces where H$_2$, the most abundant molecule in the Universe (Draine & Bertoldi, 2996), can efficiently form. The invoked mechanism is addition of a H atom and subsequent abstraction of a H$_2$ unit by an incident H atom (Wakelam et al, 2017). Other mechanisms, like photodissociation, were considered so far less important.
Recently we studied the photodissociation of a sample of PAH cations of astronomical size with a combination of experiments, quantum chemistry and modeling (Castellanos et al, 2018a, 2018b). We found that PAHs behave differently depending on their size and shape, and that H$_2$ can be a likely product of the dissociation. In this talk I will show these results and discuss how they impact our understanding of H$_2$ formation on PAHs in photodissociation regions.
Castellanos P., Candian A., Zhen J., Linnartz H., Tielens A.G.G.M., 2018a, A&A, submitted
Castellanos P., Candian A., Tielens A.G.G.M., 2018b, in prep.
Draine B.T. & Bertoldi F., 1996, ApJ, 468, 269
Tielens A.G.G.M, 2013, Rev. Mod. Phys., 85, 1021
Wakelam V. et al, 2017, Mol. Astrophys., 9, 1
Weingartner J.C.& Draine B.T., 2001, ApJ, 548, 296
Speaker: Alessandra Candian
Candian_Cosmic_Dust_1minPres.pptx
Resolved spectral attenuation curves in dusty ETGs
The properties of a particular dust mix are encoded in the extinction curve, which is notoriously hard to measure. In all but a few external galaxies, it is not possible to resolve individual stars and match them to local, unreddened stars of the same spectral type. Most measurements for external galaxies are thus global attenuation curves, which hold a convolution with the line-of-sight geometry, and are usually sampled only in a few bands, relative to the V band.
Using high-quality MUSE integral-field observations, we developed a technique to directly measure the attenuation curve in dust-lane early-type galaxies (ETGs). I will present, for the first time, these spectrally resolved optical attenuation curves, and how their strength and slope changes within the dust lane of two ETGs. Finally, using 3D radiative transfer simulations, I will show how we start break the degeneracy between geometry and dust mix to obtain detailed information about the extinction curve and the distribution of dust in external galaxies.
Speaker: Sébastien Viaene (Ghent University)
PosterTalk_SViaene.pdf
Temperature-Dependent Laboratory Measurements of the Far-Infrared to Millimeter Opacity of Carbonaceous Dust-Analogues
We are measuring and analysing the FIR- and THz- Spectra of pyrolysed micro-crystalline cellulose as an analogue of carbonaceous interstellar dust. We are using cellulose-powder with crystal sizes of about 20\,$\mu$m and are heating it up to 1000$^{\circ}$C. First results of the mass normalised extinction are presented and compared to J\"ager et al. (1998). The temperature dependent measurements took place in a dry environment at room temperature (RT) down to the environmental temperature of $T_{\text{env}}=$ 10\,K. \
Our aim is to assess carbonaceous dust analogues in terms of structure, nature and morphology. For theoretical and observational investigation we are going to determine their optical constants. Furthermore, we are going to calculate the emission cross section of particles with different geometries to compare them with the measured results.
Speaker: Jonas Greif (Astrophysical Institute and University-Observatory Jena)
Carbonaceous_Dust.ppt
The abundance of SiC2 in Carbon Star Envelopes
During the late stages of their evolution, asymptotic giant branch (AGB) stars experience significant mass loss processes, which result in extended circumstellar envelopes (CSEs). These environments are efficient factories of molecules and dust grains. The main paradigm for the dust formation process involves a first step in which condensation nuclei of nanometer size are formed from some gas-phase precursor seeds of highly refractory character and a second step in which the nuclei grow to micrometer sizes by accretion and coagulation as the material is pushed out by the stellar wind. The chemical nature of the molecules and dust grains formed depends to a large extent on the C/O elemental abundance ratio at the stellar surface. Although much has been advanced recently, there is still much to understand about how are dust grains formed and which are the main gas-phase seeds. This is the main driver of the ERC Synergy Project NANOCOSMOS.
Silicon carbide (SiC) dust grains, which are detected through a band at 11.3 micron, are exclusively found in the envelopes around C-type (C/O>1) AGB stars (Treffers & Cohen 1974). Here, we explore what the main precursor seeds of SiC dust grains are. Only three gas-phase molecules containing the Si-C bond have been observed in C-rich envelopes around AGB stars. The ring molecule SiC2 has been observed towards a few AGB and post-AGB stars (Thaddeus et al 1984; Bachiller et al. 1997; Zhang et al. 2009a,b), while SiC and Si2C have only been observed in the C star envelope IRC +10216 (Cernicharo et al. 1989, 2015). Much of the knowledge about the role of these three molecules as seeds of SiC dust grains comes from the study of IRC +10216, as in this source SiC2 has been thoroughly identified across the mm and sub-mm ranges with ground based radio telescopes and with the Herschel Space Telescope (Lucas et al. 1995; Cernicharo et al. 2010; Velilla Prieto et al. 2015). The scenario emerged from these studies suggests that only SiC2 and Si2C are present in the inner circumstellar layers of IRC +10216, while SiC is probably a photodissociation product of these molecules, and thus it is restricted to the outer envelope. This scenario indicates that SiC2 and Si2C are likely the main gas-phase seeds to form SiC dust grains.
To explore the role of gas-phase SiC2 molecules on the formation of silicon carbide dust, we have used the IRAM 30m telescope to observe SiC2 in a wide sample of C-rich AGB stars. The observations have been interpreted carrying out non-LTE excitation and radiative transfer calculations to estimate the fractional abundance of SiC2 in the CSEs. The behavior of the abundance of SiC2 as a function of the envelope density indicates that this gas-phase molecule does indeed play an important role as seed of silicon carbide dust.
Speaker: Ms Sarah Massalkhi (Instittuto de Física fundamental, CSIC)
Massalkhi_slide.pdf
Massalkhi_slide.pptx
The ESO Diffuse Interstellar Band Large Exploration Survey: First Results.
The ESO Diffuse Interstellar Band Large Exploration Survey (EDIBLES) is Large Programme that is collecting high-signal-to-noise (S/N) spectra of a large sample of O and B-type stars covering a large spectral range using the UVES spectrograph mounted on the Very Large Telescope (VLT). The goal of the programme is to extract a unique sample of high-quality interstellar spectra from these data that represent different physical and chemical environments, and to characterise these environments in great detail. An important component of interstellar spectra are the diffuse interstellar bands (DIBs), a set of hundreds of unidentified interstellar absorption lines that are commonly found in the spectra of reddened targets. With the detailed line-of-sight information derived from these high-quality spectra, EDIBLES will derive strong constraints on the potential DIB carrier molecules. EDIBLES will thus guide the laboratory experiments necessary to identify these interstellar "mystery molecules", and will turn the DIBs into powerful diagnostics of their environments in our Milky Way Galaxy and beyond. Here, we will present some of our first results showing the unique capabilities of the EDIBLES programme.
Speaker: Prof. Jan Cami (Department of Physics & Astronomy and Centre for Planetary Science and Exploration (CPSX), The University of Western Ontario // SETI Institute)
Cami_EDIBLES_1slide.ppt
An extensive grid of DARWIN models for M-type AGB stars
Asymptotic giant branch (AGB) stars are luminous, cool giants with non-spherical morphology and substantial mass loss. Dust formed in the stellar atmospheres plays a key role for the mass-loss mechanism: radial pulsations of the surface layers of the stars levitate material to distances where dust can form, which then is accelerated outward by radiation pressure. AGB stars are significant dust donors to the interstellar medium through these stellar winds.
To model these dense winds, we have constructed an extensive grid of M-type AGB stars (stars with oxygen dominated chemistry) using DARWIN models (Dynamic Atmosphere and Radiation-driven Wind models based on Implicit Numerics). The mass-loss process is modelled from first principles, with frequency-dependent radiation-hydrodynamics, and dust growth and evaporation. In the grid we cover a wide range of the relevant stellar parameters (0.75-3 M_sun, 1000 – 70 000 L_sun and 2200-2300 K). Direct outputs from the models include mass loss rates, wind velocities, dust-to-gas ratios and grain sizes.
We plan to combine this grid with stellar evolution codes, where parameterised relationships (e.g. Reimer's classical mass-loss formula) are widely used to describe the mass-loss rates of AGB stars. This can then be used to estimate the dust contribution for entire populations of AGB stars.
Speaker: Sofie Liljegren (Uppsala University)
poster140_liljegren.ppt
Reverberation Mapping the Hot Dust Emission in AGN from VEILS
Due to observational limitations, the size and structure of the obscuring circumnuclear dust in active galactic nuclei (AGN) is not well understood. Using reverberation mapping techniques the size of this hot dust emission can be determined by analyzing the temporal variations of the infrared (IR) emission from the dust in response to variations in the accretion disk continuum luminosity. Over the last 30 years, the dust reverberation time lag (and, thus, radius) has been measured by monitoring the optical and near-IR emission in about 20 galaxies. And similar to the broad emission-line region, it was found that the time lags determined by dust reverberation correlate tightly with AGN luminosity, $\tau\propto L_{AGN}^{0.5}$, a relation that may be used as a cosmological standard candle. Now we are taking AGN dust reverberation mapping to the next level, targeting about 500 AGN as part of the VISTA Extragalactic Infrared Legacy Survey (VEILS) in order to firmly establish dust time lags as a standard candle. VEILS is the first wide and deep IR extragalactic time domain survey that will monitor AGN in the optical and near-IR for at least 3 years. We will map the dust time lags of AGN over a range of redshifts, 0 < z < 1.2, allowing us to independently constrain cosmological parameters. The first season of VEILS has already been conducted. Here, I present preliminary light curves of AGN from our survey and discuss how we plan on using our light curves in order to establish AGN dust time lags as a standardizable candle.
Break Rotunda and Skolestuen (Geological Museum)
Rotunda and Skolestuen
Probing dust properties in the LMC from UV to FIR
Interstellar dust is a key component of galaxy evolution owing to its crucial role in the chemistry and radiative transfer in galaxies. Our interpretation of extragalactic SEDs and our understanding of galaxy evolution thus critically depend on an accurate characterization of how the dust content and properties vary within and between galaxies. Recent observations suggest that dust grains must grow in the ISM to explain dust masses over cosmic times (Rowlands et al. 2014), leading to changes in the abundance, composition, size, and optical properties of dust grains with environment (e.g., density, metallicity, dynamics). In this talk, I will present results from two recent efforts to characterize the dust properties in the Magellanic Clouds. First, an analysis of the gas-to-dust ratio variations in the LMC and SMC (with metallicities 0.5 and 0.2 solar, respectively) based on the stacking and modeling of the resolved SED from all-sky FIR surveys (IRAS and Planck at 100, 350, 550, and 850 ? m) suggests that the dust abundances increases by factors 3-7 between the diffuse ISM and dense molecular clouds (Roman-Duval et al. 2017). Second, the large Hubble Space Telescope (HST) program METAL (Metal Evolution, Transport, and Abundance in the LMC - GO-14675, 101 orbits, Roman-Duval et al., in prep) is delivering its first large sample of interstellar depletions at half-solar metallicity toward 33 massive stars in the LMC. The gas-phase abundances of the key components of dust grains (Si, Mg, Fe, Ni, Ti) but also other volatile elements (Zn, S) strongly support dust growth in the ISM via accretion of gas-phase metals onto dust grains. Depletion patterns however differ between the Milky Way, the LMC, and SMC, with the dust-to-metal ratio offsetting almost exactly the metallicity differences, leading to constant gas-phase metallicities in those galaxies. Additionally, parallel WFC3 imaging obtained as part of METAL allow us to derive high-resolution extinction maps, which can be directly compared to FIR emission seen in Spitzer and Herschel to characterize the FIR dust emissivity. Preliminary results suggest that the emissivity of dust could increase by a factor 3 between the diffuse ISM and denser molecular regions, likely due to coagulation. These results have important implications for the sub-grid modeling of galaxy evolution, and for the calibration of dust-based gas mass estimates used for star-formation studies, both locally and at high-redshift.
Speaker: Julia Roman-Duval (Space Telescope Science Institute)
presentation_jrd.pptx
Low-temperature surface reactions of carbon atoms
The method to study surface chemical reactions at ultra-low-temperatures and to measure the amount of energy release has been developed. The method was used to investigate surface reactions of carbon atoms leading to the formation of complex organic molecules (COMs). We found that that the key surface reaction $C + H_2 → HCH$ is barrierless in contrast with the previously considered energy barrier of 2500 K. The corresponding modification of the value of the energy barrier of this reaction in the chemical network simulations provides a huge impact on the abundancies of many molecules inside dark molecular clouds.
This is also in line with our experiments, where the carbon atoms together with the most abundant interstellar molecules $(H_2, H_2O,$ and $CO)$ were used to dope superfluid helium nanodroplets. These experiments suggest that in the denser regions of the ISM, the condensation of carbon atoms leads to the formation of complex organic molecules (COMs) and their polymers. Water molecules were found not to be involved directly in the reaction network leading to the formation of COMs. It was proposed that COMs are formed via addition of carbon atoms to $H_2$ and $CO$ molecules $(C + H_2 → HCH, HCH + CO → OCCH_2,$ …$)$. Due to the involvement of molecular hydrogen, the formation of COMs by carbon addition reactions is expected to be more efficient at high extinctions compared with the previously proposed reaction scheme with atomic hydrogen.
Speaker: Serge Krasnokutski (Laboratory Astrophysics Group of the Max Planck Institute for Astronomy at the Friedrich Schiller University Jena)
Copenhagen_pres.pptx
Iron and silicate dust growth in the Galactic interstellar medium: clues from element depletions
The question "What is the dominant mechanism of dust formation?" has long been the matter of debate. We address this question by modelling the distribution of interstellar Fe and Si element abundances in the local Milky Way with dust evolution model. The model follows the time evolution of grains in inhomogeneous, multiphase interstellar medium from high-resolution hydrodynamic simulations of the lifecycle of giant molecular clouds. This allows us to include the dependence of dust destruction in SN shocks and growth by accretion of gas-phase metals on local physical conditions. We find that the growth of iron and silicate grains occurs already in the cold neutral medium, with the Coulomb focusing playing an important role to enhance the collision rates. In order to reproduce the heavier depletion of interstellar Fe compared to Si, our model requires that solid iron resides in two dust components: (i) metallic iron nanoparticles with sizes in the range of 1-10 nm and (ii) small inclusions in silicate grains.
Speaker: Svitlana Zhukovska
Dust production in low mass stars Main Auditorium
Convener: Dr Ciska Kemper (ASIAA)
Dust production in low- and intermediate-mass stars
Stars in their late stages of evolution are generally considered to be major sources of interstellar dust. However, there is a long standing debate over the relative contributions by massive stars (both before and after they explode as supernovae), compared to low- and intermediate-mass stars, which expel a significant fraction of their total mass in stellar winds during the cool giant phase. In this talk I will focus on the latter process, discussing state-of-the-art models of dust-driven winds and, in particular, results regarding dust production.
Speaker: Prof. Susanne Höfner (Uppsala University)
Dust production in low mass stars
Outflows from asymptotic giant branch (AGB) and red supergiant (RSG) stars regulate the lifecycle of dust in the interstellar medium (ISM) in nearby galaxies. Metals produced in AGB nucleosynthesis are transported to the surface where they cool to form molecules and, further out, dust. The chemistry of this material depends on the surface atomic ratio of carbon to oxygen, resulting in either silicate-rich or carbonaceous dust.
Detailed radiative transfer is required to accurately model each AGB star; however, for a large sample such as the entire population in a galaxy, this becomes time-consuming. Astronomers thus either use proxies for the dust-production rate (DPR) such as the mid-infrared colour or the infrared excess. Along with my collaborators, I developed the Grid of RSG and AGB ModelS (GRAMS; Sargent et al. 2011 ApJ 728 93, Srinivasan et al. 2011 A\&A 532A 54), which can be used for quick estimates of the DPRs of a large sample via $\chi^2$ fits to their spectral energy distributions (SEDs).
We have used this model grid to compute the dust budget in the Large (Riebel et al. 2012 ApJ 753 71) and Small (Srinivasan et al. 2016 MNRAS 457 2814) Magellanic Clouds, and are also in the process of estimating the dust budget in M33 and NGC 6822 (Srinivasan et al., in prep). As part of the Nearby Evolved Stars Survey (NESS), we are also determining the dust budget within 2 kpc of the Solar Neighbourhood (Trejo et al., in prep; see poster by Dr. Ciska Kemper).
When combined with results for Local Group dwarfs (DUSTINGs; Boyer et al. 2015 ApJS 216 10), we now have dust budget information over six decades in total stellar mass and seven decades in integrated DPR. I will describe our methods and findings in this talk.
Speaker: Dr Sundar Srinivasan (Academia Sinica)
Sundar_Srinivasan_updated_public.pdf
Infrared light curves of dusty & metal-poor AGB stars
The effects of metallicity on both the dust production and mass loss of evolved stars have consequences for stellar masses, stellar lifetimes, the progenitors of core-collapse supernovae, and the origin of dust in the ISM. With the DUST in Nearby Galaxies with Spitzer (DUSTiNGS) survey, we have discovered samples of dusty evolved AGB stars out to the edge of the Local Group, reaching metallicities down to 0.6% solar. This makes them the nearest analogs of AGB stars in high-redshift galaxies. We present new infrared light curves of the dustiest AGB stars in 10 galaxies from the DUSTiNGS survey and show how the infrared Period-Luminosity (PL) relation is affected by dust and by metallicity. These results have implications for the efficiency of AGB dust production at high-redshift and for the use of the Mira PL relation as a potential distance indicator.
copenhagen_talk.pptx
The metallicity-depence of mass loss in carbon stars
AGB stars are major contributors of dust in the universe, feeding newly produced elements into the surrounding interstellar medium in the form of gas and dust through their stellar winds.
The detailed modelling of these dense winds or outflows is therefore crucial for understanding both the chemical evolution of galaxies, and the dust production in the interstellar medium. The mass loss observed in AGB stars is believed to be caused by a combination of atmospheric levitation by pulsation-induced shock waves, creating favourable conditions for dust formation, and radiative acceleration of these newly formed dust grains. This mass-loss scenario has been successfully implemented in the 1D radiation-hydrodynamic code DARWIN for AGB stars at solar metallicity.
But what about the dust production from AGB stars in low metallicity environments such as found in the LCM or SMC? In this talk I will present wind properties, such as mass-loss rates, wind velocities and dust-to-gas ratios, from a set of DARWIN models at metallicities compatible with the LMC and SMC. These results show that as long as stars dredge up sufficient amounts of carbon during the AGB phase, they will contribute significantly to the dust production, also at LMC and SMC metallicities.
Speaker: Dr Sara Bladh (Uppsala University)
ComsicDust2018.pdf
The submm properties of dust around carbon AGB stars
The origin and properties of dust in the universe, and the contribution from AGB stars, is a fundamental question in galaxy evolution. We constrain the properties of the dust grains in the thin detached shells around the carbon AGB stars R Scl, U Ant, V644 Sco, and DR Ser. The shells were created during recent thermal pulses, and the dust properties play a crucial role in understanding the wind-driving mechanism, the evolution of the star throughout the thermal pulse cycle, and the type and amount of dust returned to the ISM from AGB stars. We use new observations from LABOCA and ALMA to model the entire SED including submm wavelengths. For all objects, we find an excess emission in the submm. Spatially resolved observations confine this excess to the detached shells. However, a straightforward explanation for this excess is still lacking. While very large, cold grains can explain the submm flux, they do not reproduce the overall shape of the SED in the FIR and submm. Other obvious grain properties (e.g., composition or geometry) also do not reproduce the observed SEDs. The results imply that the submm observations probe properties of the dust grains that are not typically considered, but may be critical for a complete understanding of dust around evolved stars. A similar SED shape and submm excess has been seen in observations of the small and large magellanic clouds, and has been attributed to unknown dust properties. If the origin of this excess is the same as for the detached shell sources, this would have important implications on the contribution to the total dust budget from AGB stars to galaxies.
Speaker: Matthias Maercker (Chalmers University of Technology)
maercker_CPHdust_v2.pdf
Constraining dust grain porosity via debris disk observations
Debris disks are often modelled assuming compact dust grains, but more and more evidence for the presence of porous grains is found. We quantify the systematic errors introduced when modelling debris disks -- composed of porous dust grains -- assuming the presence of spherical, compact particles (Brunngräber et al. 2017).
We use the effective medium theory to calculate the optical properties of the dust. Furthermore, we simulate observations of debris disks with different porosities and feed them into a fitting procedure assuming compact grains only. Finally, we analyse the deviations of the results for compact grains from the original, porous model.
We find that with increasing grain porosity the blowout size increases up to a factor of two. An analytical approximation function for the blowout size as a function of porosity and stellar luminosity is derived. The analysis of the geometrical disk set-up, when constrained by radial profiles, is barely affected by the porosity. However, the estimated minimum grain size and the slope of the grain size distribution derived using compact grains are significantly overestimated. Thus, the discrepancy between the minimum grain size and the predicted blowout size found in various previous studies assuming compact grains can be partially explained by the presence of porous dust grains.
Speaker: Robert Brunngräber (ITAP, CAU Kiel)
Dust charge distribution in the Interstellar Medium
The dynamics of dust grains vary depending on the forces that act on them at different environments in the multi-phase interstellar medium (ISM). Grains interact with the gas through collisions, gravitational attraction and long-range coulomb forces, and also experience varying coupling strengths to magnetic field lines depending on their charge. The charge distribution of dust grains in the ISM depends on the flux of charged particles from the grain surface, which is strongly dependent on the local properties of the ISM temperature, density, local radiation field and, within dense molecular clouds and protostellar disks, cosmic ray flux. We examine the charge distribution, f(Z), of a dust grain population in a radiative, turbulent, multi-phase, interstellar medium, accounting for collisional charging of grains by electrons and ions, photoelectric charging due to a background ultraviolet interstellar radiation field (ISRF) and cosmic rays. We use three-dimensional, adaptive-mesh-refinement, hydrodynamic colliding flows simulations, including gas self-gravity, gas and dust (self-)shielding, on the fly non-equilibrium chemistry, diffuse heating and radiative cooling, to model the complex structure of the multi-phase interstellar medium.
We find that mean charge is strongly dependent to the phase of the ISM where dust is present. Grains in molecular gas have predominantly neutral charge, while grains in the cold-dense and warm-diffuse ISM have predominantly positive charges, varying from charges of order unity, $\langle Z \rangle \sim 1$, for small (5 Angstrom) grains, to $\langle Z \rangle \sim 200$, for large (0.1 $\mu$m) grains. We found a combination of parameters that can be used to immediately find an approximate value of the charge centroid depending on the grain composition, size and ambient conditions. We then compare the timescale required for dust grains to reach equilibrium charge, to local dynamical timescales of the turbulent flow and find that in the diffuse ISM and within dense clouds dust charge equilibrium is a good approximation.
Speaker: Juan C. Ibañez Mejia (University of Cologne)
JIbanezMejia_Slide.pdf
Systematics in dust emission modeling in nearby galaxies
The dust properties of nearby galaxies are often inferred by modeling their infrared (IR) spectral energy distributions (SEDs), using dust grain models. These are created with a simplified radiation model, with assumptions on the intensity and hardness of the radiation field.
Using the Draine & Li (2007; DL07) dust model, we create a set of synthetic dust emission SEDs with a 3D radiative transfer (RT) model (DIRTY; Gordon et al. 2001), taking into account absorption, scattering and stellar and dust emission, in various galactic environments (varying the dust and stars distribution, star formation history, metallicity and dust mass). We use the DL07 models to fit these synthetic SEDs, and estimate the systematic biases due to the difference in the dust heating treatment.
We find that the empirical description (a power-law) of the radiation field heating the dust may lead to over- (when a dust layer surrounds stars) and underestimation (when dust is embedded in a cluster of stars) of dust properties such as total mass, or PAH fraction. We quantify these errors by comparing the RT-calculated radiation field and the empirical approach, showing that the power-law description is not suited for all cases.
Speaker: Jeremy Chastenet (UCSD)
Chastenet_CPH_Systematics_OneSlide.pdf
Chastenet_CPH_Systematics_OneSlide.pptx
Dust and gas properties in nearby galaxies
The amount of dust in the interstellar medium (ISM) is directly linked to physical quantities that trace the evolution of galaxies. The emission from dust has been proposed as a probe of the amount of star formation within a galaxy and the physical properties of the dust are related to those of the ISM where it is located. Therefore, understanding better how the dust physical properties change under different conditions of the ISM will help us to get a better picture on how the dust traces the star formation rate and how galaxies evolve with time.
The infrared/submillimetre spectral energy distribution (SED) of galaxies provides a unique set of data to study how the dust is processed in galaxies and in the ISM in general. In this talk we present an analysis of the dust properties across the disc of M 33 performed by fitting the observed infrared SED at small galactic scales of $\sim$170 pc with the classical dust model from Desert et al. (1990). Our new analysis provides relative dust grain abundance of the different species and shows how they change with the ISM physical conditions in the disc of M 33. The dust grains are modified inside the star-forming regions, in agreement with a theoretical framework of dust evolution under different physical conditions. At each spatial location in the disc, we investigate how the gas-to-dust mass ratio is correlated with other physical properties of the galaxy: metallicity, dust and gas mass surface density and strength of the interstellar radiation field heating the dust.
The submillimetre excess, defined as the fraction of emission in the submillimetre range that is above a dust model having an emissivity coefficient $\beta$=2, is analysed at each spatial location. We produce a map of submillimetre excess in the 500$\mu m$ SPIRE band for the disc of M 33. The excess can be as high as 50% and increases at large galactocentric distances. We further study the relation of the excess with other physical properties of the galaxy and find that the excess is prominent in zones of diffuse ISM outside the main star-forming regions, where the molecular gas and dust surface density are low.
We have applied the same methodology for M 33 to a set of nearby star-forming galaxies and we will present the first results of this analysis.
Speaker: Dr Monica Relano Pastor (University of Granada)
poster_slide_monica_relano.pdf
DustPedia: Multiwavelength Photometry & Imagery of 875 Nearby Galaxies in 42 Ultraviolet-Microwave Bands
The DustPedia project is capitalising on the legacy of the Herschel Space Observatory, using cutting-edge modelling techniques to study dust in the 875 DustPedia galaxies - representing the vast majority of extended galaxies within 3000 km/s (~40 Mpc) that were observed by Herschel. This work requires a database of multiwavelength imagery and photometry that greatly exceeds the scope (in terms of wavelength coverage and number of galaxies) of any previous local-Universe survey.
We therefore present multiwavelength imagery and photometry across 42 UV-microwave bands for the 875 DustPedia. This database contains custom Herschel reductions, plus standardised GALEX, SDSS, DSS, 2MASS, WISE, Spitzer, and Planck data. We also present CAAPR, the pipeline we use to conduct aperture-matched photometry of our data; CAAPR is designed to produce consistent photometry for the enormous range of galaxy and observation types we employ. In particular, CAAPR is able to determine robust cross-compatible uncertainties, thanks to a novel method for reliably extrapolating the aperture noise for observations that cover a very limited amount of background. The 27-band aperture-matched photometry, in combination with ancillary catalogue data from IRAS and Planck, represents 21857 photometric measurements. A typical DustPedia galaxy has photometry spanning 25 bands. This database of imagery and photometry is being made publicly available at: dustpedia.astro.noa.gr.
Speaker: Christopher Clark (Cardiff University)
Copenhagen_CAAPR_Slide.pdf
UV Dust Extinction and Attenuation Curves in the Local Volume
Our knowledge of the shape of the ultraviolet (UV) extinction curve informs our understanding of topics from the composition of dust grains in the ISM to how we interpret the shape of galaxy SEDs. I will discuss two complementary approaches to measuring the extinction curve.
First, we use resolved stellar populations in the Large Magellanic Cloud to determine the extinction curve shape along lines of sight to over 600,000 stars. The METAL (Metal Evolution, Transport, and Abundance in the LMC) program has obtained 33 fields of HST WFC3 imaging in seven NUV to NIR filters. For each of the stars in these fields, we use the BEAST (Bayesian Extinction And Stellar Tool) to model their SEDs and infer their stellar (age, mass, metallicity) and dust ($A_V$, $R_V$, 2175A bump) parameters. We derive high-resolution extinction maps by combining the measurements of many stars in each pixel on the sky, which we can then relate to the properties of the local ISM.
Second, we measure the shape of the attenuation curve for unresolved stellar populations in entire galaxies. The Ultraviolet/Optical Telescope (UVOT) on the Swift satellite has nearly completed a survey of 450 galaxies in the Local Volume. The three broadband NUV filters on UVOT are situated such that they can constrain both $R_V$ and the 2175A bump. We use this unique capability in combination with archival optical and NIR imaging to model the SEDs of each galaxy and derive representative attenuation curves. For three galaxies (M31, M33, SMC), we have additionally created the first maps of the attenuation curve. We then examine variations of the attenuation curve with local (e.g., star formation rate, PAHs, dust temperature) and global (e.g., metallicity) galactic environment.
Together, these two methods will provide comprehensive constraints on the nature and properties of dust and how they vary within and between galaxies.
Speaker: Dr Lea Hagen (STScI)
hagen_slide.pdf
A systematic study of dust and star formation in early-type galaxies with AKARI
With the AKARI all-sky maps, we conduct a systematic study of dust and star formation for the 260 local early-type galaxies (ETGs) from the ATLAS3D survey, for which cold (HI and CO) and hot (X-ray) gas measurements are available. We detected far-infrared dust emission in 30% of the ETGs, where the dust emission is not correlated with the stellar emission, indicating that dust in those galaxies is of interstellar origin. In addition, polycyclic aromatic hydrocarbons (PAHs) are detected in many ETGs, suggesting that ETGs still form stars. We modeled the spectral energy distributions of the sample ETGs to derive the dust and PAH luminosities, from which we estimated the dust masses and star formation rates (SFRs), respectively. The dust-to-stellar mass ratios and current SFRs of the ETGs are lower than those of late-type galaxies (LTGs), showing that ETGs are quiescent galaxies, while their current star formation efficiencies are similar to those of LTGs. Our results indicate that the low SFRs of ETGs are likely due to their smaller cold gas fractions rather than a suppression of star formation. We also find that the dust masses and X-ray luminosities are correlated in fast-rotating ETGs, which appears to be caused by their higher current star formation activity than slow-rotating ETGs.
Speaker: Dr Takuma Kokusho (Nagoya University)
Takuma_Kokusho.pdf
Takuma_Kokusho.ppt
A Unified Model of the Emission, Extinction, and Polarization of Interstellar Dust
We present a new model of interstellar dust composed of silicates, graphitic carbonaceous grains, and polycyclic aromatic hydrocarbons that reproduces the wavelength dependence of dust extinction (total and polarized) and emission (total and polarized) in the diffuse interstellar medium from UV to microwave wavelengths. In this talk, I will focus on the use of new observational data, particularly from the Planck satellite, to place constraints on the optical properties and shapes of interstellar grains. I will also discuss the key differences between this model and the Draine and Li 2007 model.
Speaker: Brandon Hensley (JPL/Caltech)
poster_preview.pdf
Constraining dust parameters through observations of eccentric debris disks
Debris disks contain very fine dust but the lifetime of these dust grains is much shorter than the stellar age. It implies that these dust grains are not primordial and must be replenished continuously through mutual collisions of dust-producing planetesimals.
We investigated the impact of mutual collisions on the observational appearance of eccentric debris disks. For this purpose we simulated the collisional evolution of selected debris disks configurations and derived observable quantities. The impact of the eccentricity, the level of the dynamical excitation of the eccentricities, and the material strength are discussed with respect to the grain size distribution, the spectral energy distribution, and spatially resolved images of debris disks systems.
The most recognizable features in different collisional evolutions are as follows: First, both the increase of the dynamical excitation in the eccentric belt of the debris disk system and the decrease of the material strength of dust particles result in a higher production rate of smaller particles. This reduces the surface brightness differences between the periastron and the apastron sides of the disks. For very low material strengths, the "pericenter glow" phenomenon is reduced and eventually even replaced by the opposite effect, the "apocenter glow". In contrast, higher material strengths and a decrease of the dynamical excitation level result in an increase of asymmetries in the surface brightness distribution. Second, it is possible to constrain the level of collisional activity from the appearance of the disk, e.g., the wavelength-dependent apocenter-to-pericenter flux ratio. Within the considered parameter space, the impact of the material strength on the appearance of the disk is stronger than that of the dynamical excitation level in the belt eccentricity.
Speaker: Mr Minjae Kim (ITAP, Kiel Univ.)
MinjaeKim_poster.pdf
Cosmological simulation with dust evolution
Dust enrichment is one of the most important aspects in galaxy evolution.
The evolution of dust is tightly coupled with the nonlinear evolution of
the ISM including star formation and stellar feedback,
which drive the chemical enrichment in a galaxy.
Hydrodynamical simulation provides a powerful approach
to studies of such nonlinear processes.
In this work, we perform a smoothed particle hydrodynamic simulation
with a dust enrichment model in a cosmological volume.
We adopt the dust evolution model that represents the grain size distribution
by two sizes and takes into account stellar dust production and interstellar
dust processing. We show that our cosmological simulation allows us to examine
the dust mass function and to analyze the dust abundance and dust properties
in galaxies statistically. The simulation broadly reproduces the observed
dust mass functions at redshifts $z \sim 0$ and $2.5$ and the relation between
dust-to-gas ratio and metallicity shows a good agreement with the observed
one at $z = 0$, which indicate a successful implementation of dust evolution
in our cosmological simulation. Besides, we also examine the redshift evolution
up to $z \sim 5$, and find that the galaxies have the highest dust abundance at $z = 1$-$2$.
For the grain size distribution, we find that galaxies with metallicity $\sim 0.3$ $Z_\odot$
have the highest small-to-large grain abundance ratio at $z < 5$;
consequently, the extinction curves in those galaxies have the steepest
ultra-violet slopes.
Speaker: kuan-chou Hou (ASIAA)
CPHDustPoster82.pdf
Dust formation and survival in Quasars
Infrared observations of AGN reveal the emission from a dusty circumnuclear "torus" that is heated up by radiation from the central accreting black hole (BH). The strong 9.7 and 18 micron silicate features observed in the AGN spectra both in emission and absorption, further indicate the presence of such dusty environment. The origin of this dust is presently unclear. It could be pre-existing dust that streamed from the surrounding medium into the accretion disk or, it could be dust that has newly- formed in the environment surrounding the active BH.
The environment of a quasar is often assumed to be too hostile to support the necessary chemical processes leading to the formation of cosmic dust. In this talk, I will present the results of our study based on the formation and survival mechanism of newly formed, as well as pre-existing dust, in the winds blown off the accretion disks, which has been proposed to constitute the AGN ''tori''. The study takes into account the series of physical and chemical processes relevant to the environment, such as: a) the radiation transport from the central source through the surrounding medium, b) the formation of dust seed-nuclei from gas phase metals, c) the growth of dust grains through accretion and coagulation, and d) the radiative and collisional heating of the dust grains. We compare the timescales associated to these mechanisms to the flow time of the winds, identifying the "bottle-necks" to the formation of dust in the AGN environment.
The model enables us to estimate the dust production rate in quasars and to quantify their relative contribution as dust producers in the galaxies and in the intergalactic medium. Further, we study the interaction of the X-ray and UV-optical from the accretion disk with the ambient dusty winds and calculate the emerging X-ray, UVO and IR spectra from the AGNs as a function of the quasar viewing angle.
Speaker: Dr Arkaprabha Sarangi (NASA GSFC, CRESST II/ CUA)
1slide_Sarangi_Poster62.pdf
Dust models compatible with Planck and starlight polarization data
The HFI instrument onboard the Planck satellite has allowed us to characterize the statistical and spectral properties of dust polarized emission over the whole sky in the submillimeter wavelength range.
Dust polarization is not only useful to trace the magnetic field orientation or to test alignment theories. It is also a way to characterize the spectral properties of the dust population that is aligned with the magnetic field.
I will summarize the main results of the analysis of Planck polarization data, and show how they challenge existing dust models.
I will also describe how we updated the DUSTEM model (Guillet et al 2018) to integrate polarization and account for these new constraints on dust emission in both total intensity and polarization.
Speaker: Vincent Guillet (Institut d'Astrophysique Spatiale, Université Paris-Sud)
Poster77.ppt
Poster77_VGUILLET_DustModelPlanck.pdf
DustKING: revealing the dust attenuation in NGC628
The shape of the dust attenuation law is not expected to be uniform between galaxies, nor within a galaxy. The DustKING project sets to study these variations in nearby galaxies of the KINGFISH sample. To this aim, we used the CIGALE SED fitting code to fit models with varying dust extinction properties to a set of multi-wavelength data. Particularly important for our goal are UV images taken with the SWIFT space telescope, whose filters uniquely cover the curious bump feature in the attenuation curve at 2175 Å. This enables us to characterise the strength of this bump and the UV slope of the attenuation curve.
In this talk, I present the results for the spiral galaxy NGC628 which clearly illustrate the potential of the SWIFT data in obtaining the characteristics of the attenuation curve on spatially resolved scales. From UV colours and from SED modelling, we found that the attenuation law of this galaxy is characterised by a relatively small bump and a shallower UV slope compared to the Milky Way. Also, we noticed variations of the dust attenuation properties on different scales within the galaxy. I will walk you through some intriguing trends between dust attenuation law shapes and other galaxy properties, and discuss the impact of our results.
Speaker: Ms Marjorie Decleir (Ghent University)
DustKING_PP_Decleir.pdf
Transfer of ionizing radiation through gas and dust
Cosmic dust provides a significant contribution to the absorption of electromagnetic radiation at all galactic scales. Hydrogen ionizing radiation (hν ≥ 13.6 eV) emitted from star forming regions has to survive the large columns of gas and dust present in the galactic ISM of normal high-z galaxies before contributing to the IGM reionization process. Nevertheless, dust absorption is rarely self-consistently coupled with gas ionization in cosmological radiative transfer simulations and its impact on the timing of cosmic reionization poorly investigated. In this talk, I will first introduce a novel implementation of the cosmological radiative transfer code CRASH which supports the inclusion of an arbitrary number of dust species and accounts for the absorption of radiation by dust and the charging of grains associated with it. The results of several simulations adopting a Milky Way-like dust model both in idealized HII regions and realistic dusty galaxies will be critically discussed to show how the presence of dust grains sharpens the ionization fronts of expanding bubbles and reduces the ionization fractions of gas species at cosmic scales. We show how, depending on the total amount of dust in the high-redshift universe, the inclusion of dust in galaxy formation models can significantly change the ionization of the galactic ISM and impact the global reionization process.
Speaker: Martin Glatzle (MPA Garching)
Glatzle_Poster.pptx
Dust production by supernovae and massive stars Main Auditorium
Convener: Jens Hjorth
Dust production by supernovae and massive stars
In this review I will cover the theoretical expectations and the observational evidence as to whether massive stars and their supernovae can form sufficiently large quantities of dust to provide a significant contribution to the dust budgets of galaxies.
A series of papers addressing dust condensation in the ejecta of core-collapse supernovae (CCSNe) have predicted that up to one solar mass of dust could form, with one of the principal uncertainties being the dust's survivability against destruction by reverse shocks. Observations out to mid-infrared wavelengths of dust formed by extragalactic CCSNe, including those made with the Spitzer Space Telescope, have measured relatively small dust masses, typically less than $10^{−3}\,M_\odot$. The advent of the Herschel Space Observatory, covering wavelengths out to 500 μm, has enabled much cooler dust to be detected from young CCSN ejecta and CCSNRs than possible at mid-IR wavelengths. Since cooler dust emits less efficiently than warm dust, larger dust mass detections have resulted. The Herschel mission ended in 2013, which has stimulated the development of alternative methods to measure CCSN dust masses, for example by modelling red-blue emission line profile asymmetries, and their time evolution, in the optical spectra of CCSNe. I will summarise the currently available results from these various methods and their implications.
Speaker: Prof. Mike Barlow (University College London)
mbarlow_copenhagen_june12th2018.pdf
Resolved dust analysis of two iconic Galactic supernova remnants: Cassiopeia A and the Crab Nebula
The large reservoirs of dust observed in some high redshift galaxies have been hypothesised to originate from dust produced by supernovae (SN). Theoretical models predict that core-collapse SN can be efficient dust producers (0.1-1 M$_\odot$) potentially responsible for most of the dust production in the early Universe. Observational evidence for this dust production efficiency is however currently limited to only a few remnants (e.g., SN 1987A, Crab Nebula) that confirm this scenario.
We revisit the dust mass produced in Cassiopeia A (Cas A), a $\sim330$-year old O-rich Galactic supernova remnant (SNR) embedded in a dense interstellar foreground and background. We present the first spatially resolved analysis based on Spitzer and Herschel infrared and submillimetre data at a common resolution of $\sim0.6^\prime$ for this $5^\prime$ diameter remnant following a careful removal of contaminating line emission and synchrotron radiation. We find a concentration of cold dust in the unshocked ejecta of Cas A and derive a mass of 0.3-0.6M$_\odot$ of silicate grains (+a minor contribution from carbon grains) freshly produced in the SNR. The cold SN dust component is mainly distributed interior to the reverse shock of Cas A, suggesting that part of the newly formed dust has already been destroyed by the reverse shock. We derive an interstellar+SN dust extinction map which towards Cas A gives average values of $A_V = 6-8$ mag, up to a maximum of $A_V = 15$ mag. We have modelled the mid-infrared to radio emission from the Crab Nebula with a broken power law synchrotron spectrum, and a warm+cold SN dust component. Dust grains in the Crab Nebula are distributed predominantly along dense filaments, mostly in the southern half of the remnant, and account for 0.02 to 0.17 M$_\odot$ of 40K carbon dust with the somewhat lower dust mass compared to previous Herschel-based estimates being due to a re-analysis of the synchrotron component contribution and SN dust temperatures using the latest Herschel SPIRE calibrations. In addition, we require an extra model component in the millimetre wavelength domain, which accounts for 25 to 35% of the emission at 1 to 3 mm. We discuss possible origins for this excess emission in the centre of the Crab Nebula; including spinning dust grains, a secondary synchrotron component, magnetic Fe-bearing dust particles and free-free emission from a hot plasma.
In conclusion, these updated dust mass estimates for Cas A and the Crab Nebula support the scenario of supernova dominated dust production at high redshifts.
Speaker: Ilse De Looze (Universiteit Gent - University College London)
Survey of dust emission in Galactic supernova remnants
There is still on-going debate as to how much dust has been formed and destroyed by supernovae and supernova remnants. A systematic search for dust in supernova remnants is an effective way to resolve this issue. We search for far-infrared counterparts of 62 known supernova remnants in the Galactic plane (| l |<60○) at 70, 160, 250, 350, and 500µm using the Herschel Infrared Galactic Plane Survey (Hi-GAL). We detect FIR dust emission from 24 of our sample, with some evidence of ejecta dust heated by pulsar wind nebulae. Detailed analysis of near-infrared to radio emission from three pulsar wind nebulae suggests that there is a significant mass of ejecta dust within the supernova remnants. We use point process mapping to further analyse the dust mass distribution across the three sources at various temperatures and values of dust emissivity. This indicates the presence of between 0.29 and 0.64 solar masses of dust within each supernova remnant which is warmer than that of the ISM, at temperatures of 20 - 45 K. We expect that pulsar wind nebulae can heat SNR dust, increasing the temperature above that of the surrounding interstellar medium. We also find marginal evidence for one SNR that there may be a variation in the dust emissivity between the SNR material compared to that of the ISM, suggesting that there is a different dust composition within the SNR.
Speaker: Hannah Chawner (Cardiff University)
HChawner_SNRDust.pdf
Shock-induced formation and survival of dust in the dense CSM surrounding Type IIn supernovae
The light curve of Type IIn supernovae are dominated by the radiative energy release through the interaction of the supernova blastwave with their dense circumstellar medium (CSM). Specifically, in case of ultraluminous Type IIn supernova SN 2010jl, the spectra show an excess in the IR component as early as a few weeks after the explosion. The IR emission has been attributed by some as evidence for early dust formation in the circumstellar gas. We investigate in detail the physical processes that may inhibit or facilitate the formation of dust in the CSM. The post-explosion environment of Type IIn supernovae are characterized by high velocity shocks and strong ionizing radiations. We show that dust formation is inhibited by the effect of the downstream radiation from the supernova forward shock. In spite of the high densities in shocked gas that ensue rapid cooling, we find that the formation of dust grains in the post-shock circumstellar shell of SN 2010jl does not commence until day 380 post- explosion. On the other hand, observations on day 460 and later show that the IR luminosity exceeds the UV-optical luminosity. The IR emission is therefore powered by the UVO emission from the reverse shock which is totally absorbed by the optically-thick shell of newly-formed CSM dust. The early IR emission is attributed to an IR echo from preexisting CSM dust, which has survived the SN flash associated with the outburst. In this talk, I shall present the first model of Type IIn supernovae that addresses the role of the radiation from the SN forward and reverse shock in the formation and survival of dust in the dense circumstellar environments.
Sarangi_TypeIIn_talk.pdf
Dust in supernova 1987A
Core-collapse supernovae (SNe) are considered to play a dual role in the production and destruction of dust in the interstellar media of galaxies. Currently, the subjects of intense investigations are the questions of how much dust SNe form, and how much dust survives SN shocks.
Supernova 1987A is the nearest supernova explosion detected in the last 400 years, and provides a unique opportunity for detailed studies of dust in a supernova. Both dust formation and destruction can be observed in a single object: it has freshly formed dust in the ejecta, while the fast expanding blast waves collide with circumstellar dust, which was expelled from the progenitor when this star was in the red-supergiant phase 40,000 years ago. We report recent SOFIA and VLT observations of dust in the ring of Supernova 1987A.
Mid-infrared VLT and SOFIA observations has captured the time development of ring dust in Supernova 1987A. Our VLT image shows that the 10-micron emission is now emitted from the west part of the ring, where the shock interaction is on-going. On the east side of the ring, the flux is declining, as the shock waves have passed the ring. Furthermore, our recent SOFIA observations detected that the 35-micron flux has increased since the last Spitzer observations 10 years ago. It might be possible that dust grains have been re-formed in the post shocked region.
Speaker: Mikako Matsuura (Cardiff University)
matsuura.pptx
Old and new dust associated with Supernova 1995N
The discovery of 0.4-0.7M$_\odot$ of dust in the remnant of SN1987A 23 years
after its explosion (Matsuura et al. 2011) demonstrated that supernovae can be
efficient dust factories, but raised many questions. Among them, when did this
dust form? Was it there at early times but previously undiagnosed by
techniques for estimating dust masses, or did it form at later times? In
Wesson et al. (2015) we created radiative transfer models to investigate this
question, fitting the optical-far IR SED of SN1987A to calculate the dust mass
at epochs from 600-9000 days after the explosion. We found that the rate of
dust formation could be represented by a sigmoid curve with peak dust
formation occurring many years after the explosion.
The far infrared observations necessary to constrain the emission from cold
dust are lacking in most supernovae. An alternative method of estimating the
dust mass exploits the blue-shifting of emission lines in the presence of dust
to diagnose the dust mass (Bevan and Barlow, 2016). This has the additional
advantage that only dust within the expanding remnant will affect the line
profiles - pre-existing dust that is thermally echoing will not.
I will present SED and emission line profile models of SN 1995N, observed as
part of a programme to determine dust masses in supernova remnants years to
decades old. Van Dyk (2013) found that mid-IR observations implied the
presence of 0.05-0.2Mo of pre-existing circumstellar dust which has been
flash-heated by the supernova outburst. I confirm this with three-dimensional
radiative transfer models to fit the SED. Additionally, emission line profile
modelling reveals that a further 0.1-0.4Mo of dust has formed in the expanding
supernova ejecta. This shows that pre-existing and newly formed dust can be
clearly distinguished in supernova remnants, and that both may contribute
significantly to the total dust mass formed by a massive star.
Speaker: Roger Wesson (University College London)
rw_cph.pdf
Investigating the Properties of Nearby Galaxies
We have updated the SED fitting code "CIGALE" so that it includes dust properties based on the "THEMIS" model (Jones et al. 2017, A&A, 602, 46). We use this tool to fit the SED of 875 nearby galaxies with available photometry from the FUV to the sub-millimeter wavelengths. For this sample of galaxies (the "DustPedia" galaxies – Davies et al. 2017, PASP, 129, 4102) we are able to derive global properties like stellar and dust mass, star-formation rate and dust extinction and we compare our results with widely used recipes found in the literature. Furthermore we examine how the unattenuated luminosity of the young and old stellar populations correlate with basic properties of the galaxies of different morphological types.
Speaker: Angelos Nersesian (IAASARS, National Observatory Of Athens & Ghent University)
slide_nersesian_cph_cigale.pdf
PAHs and star formation in the HII regions of M83 and M33
IR emission features at 3.3, 6.2, 7.7, 8.6 and 11.3 $ \mu $m are usually attributed to IR fluorescence from FUV pumped polycyclic aromatic hydrocarbons (PAHs). These features thus trace the FUV stellar flux and are a measure of star formation in the Universe. Here, we present results from a detailed study on the mid-IR emission features of HII regions in M83 and M33, with the aim to investigate the IR spatial characteristics in star-forming regions from Milky Way (MW) HII regions, to star-forming complexes in nearby galaxies, and star-forming galaxies as a whole. As such, we build a control sample to compare our results, including star-forming regions in the MW, LMC, M101, starburst nuclei, and nearby galaxies. We find that the PAH intensity ratios in M83 and M33 HII regions have similar correlations as those in individual HII regions within galaxies, starburst nuclei, and AGN host galaxies. We find that the strength of the 17.0 $ \mu $m PAH band is enhanced relative to the other PAH bands compared to galactic star-forming regions, similar as in other galaxies. In comparison with other emission components we find that: 1) the PAH/VSG intensity ratio presents a decrease with galactocentric radius for both M83 and M33 as well as the Milky Way, and 2) the L$ _{\textrm{TIR}} $/L$ _{6.2\mu\textrm{m}} $ luminosity ratio in M83 and M33 HII regions ranges in between the values measured in Galactic and LMC HII regions, and those in normal star-forming galaxies and starburst nuclei. The extragalactic HII regions appear as a linking component between the spectral properties of local HII regions and star-forming galaxies, and can be used as better templates than Galactic HII regions when interpreting the properties of star-forming galaxies.
Speaker: Alexandros Maragkoudakis
Alexandros_Maragkoudakis_SSP.pdf
Alexandros_Maragkoudakis_SSP.ppt
Constraints on the structure of hot exozodiacal dust belts and their observability in the MIR
Hot exozodiacal dust emission was detected around several main sequence stars at distances of less than 1 au using NIR and MIR interferometry. Studies of exozodis offer a way to better understand the inner regions of extrasolar planetary systems, and the possible presence of small grains in exozodiacal clouds is a potential problem for the detection of terrestrial planets in the habitable zone of these systems.
We modelled the observed excess of nine of these systems and found that grains have to be sufficiently absorbing to be consistent with the observed excess, while dielectric grains with pure silicate compositions fail to reproduce the observations. The dust should be located within ~0.01-1 au from the star depending on its luminosity. Furthermore, we found a significant trend for the disc radius to increase with the stellar luminosity. The dust grains are determined to be below 0.2-0.5 μm, but above 0.02-0.15 μm in radius. The dust masses amount to 0.2-3.5 × 10−9 Mearth. The near-infrared excess is probably dominated by thermal reemission.
In addition, we assessed the feasibility of observation and characterization of exozodis with the upcoming MIR instrument MATISSE at the Very Large Telescope Interferometer (VLTI). We find that MATISSE is potentially able to detect dust emission in five of the nine systems and will allow one to constrain the dust location in three of these systems, in particular to determine whether the dust piles up at the sublimation radius or is located at radii up to 1 au.
Speaker: Florian Kirchschlager (UCL London)
Hydrodynamic Simulations of Dust Destruction in Supernova Remnants
Sub-millimetre observations of galaxies at redshift z>6 have revealed dust masses of up to 10^8 solar masses (e.g. Bertoldi and Cox, 2002). As such systems are thought too young for significant dust enrichment by asymptotic giant branch (AGB) stars to have occurred, core-collapse supernovae (CCSNe) have been suggested as possible alternative dust producers (Nozawa et al. 2003, Dwek et al. 2007). This is supported by recent Herschel far-IR and sub-millimetre observations of young CCSN remnants that are estimated to show between 0.25-0.8 solar masses of cool dust in SN 1987A, Cassiopeia A and the Crab Nebula (Barlow et al. 2010, Matsuura et al., 2011, De Looze et al. 2017).
Once formed, the dust particles can be subjected to various erosion processes such as sputtering and grain-grain collisions (the latter subject to a separate contribution by F. Kirchschlager) due to the reverse shock generated by interactions between ejecta and circumstellar material. This can result either in the complete destruction of the grains or in a size reduction. Whether significant quantities of dust can survive these conditions long enough to be incorporated into the interstellar medium (ISM) has been the subject of multiple recent studies (Nozawa et al. 2006, 2007, Silvia et al. 2010, 2012, Bocchio et al. 2016). The predicted dust survival rates vary greatly and models tend to adopt ISM-like grain size distributions dominated by small particles. However, recent determinations of ejecta dust size distributions (e.g. Wesson et al. 2015, Bevan et al. 2017) have indicated that larger (~1 micrometer radius) particles may dominate.
In this study, I investigate the survival rates of dust produced in CCSNRs through (magneto)hydrodynamic (MHD) shock simulations carried out with the publicly available AMR codes ENZO (Bryan et al. 2014) and AstroBEAR (Cunningham et al. 2009). The MHD models feature a cloud of dense gas (clump) embedded in a less dense ambient medium through which a shock propagates. As the shock travels through the computational domain, it collides with the clump and accelerates, compresses and heats the gas contained within. Following Silvia et al., 2010, we introduce parcels of dust in post-processing using a code developed at UCL. Each dust parcel contains a realistic dust grain size distribution and is advected alongside the gas flow. Sputtering effects (based on Tielens et al., 1994) then lead to a redistribution of grain sizes in the dust parcels.
I present preliminary results featuring purely hydrodynamic simulations in 2D with realistic dust grain radii distributions and sputtering rates.
This work was supported by ERC Grant 694520 SNDUST.
Speaker: Ms Franziska Schmidt (University College London)
2D Cloud Crushing Simulation featuring external dust parcels
FranziskaSchmidt_PosterPresentation.pdf
Dust emission from the Cassiopeia A supernova remnant
We model the thermal emission from a distribution of dust grains heated by particle collisions and the ambient supernova remnant radiation field, under conditions representative of the knots observed in Cassiopeia A (Cas A). In order to reproduce the observed Cas A dust spectral energy distribution reported by de Looze et al. (2017), we require dust emission from both the pre- and post-shock regions. We find that the shocked dust is heated mainly by collisions with electrons, while the unshocked dust is heated by the synchrotron radiation field. The grain size distribution is required to extend to smaller radii in the shocked region, indicative of the destuction of dust grains by the reverse shock. The model SEDs are only weakly dependent on the maximum grain radii, leading to a range of possible dust masses between 0.4 and 1.2 solar masses (assuming MgSiO$_3$ grains), the majority of which is located in the preshock region.
Speaker: Felix Priestley (UCL)
slide.pdf
slide.ppt
Mapping the extinction parameters of dust in the IC63 photodissociation region
Photodissociation regions (PDRs) are parts of the ISM consisting of predominantly neutral gas, located at the interface between HII regions and molecular clouds. The physical conditions within these regions show variations on very short length scales, and therefore PDRs form ideal laboratories for investigating the properties and evolution of dust grains. Recently, observations of the IC63 PDR were carried out with HST, producing high-resolution images with WFC3 in seven broadband filters from the UV to the NIR. With these observations, we investigate for the first time how the extinction varies across a PDR.
IC63 is an excellent target for this analysis, thanks to its many background stars. In this talk, I will explain how we simultaneously fit the stellar parameters (spectral type, effective temperature and luminosity), and extinction parameters (Av, Rv) for each of the observed background stars, based on an approach that was originally developed for the Panchromatic Hubble Andromeda Treasury (PHAT). These fits then allow us to make a map of the optical properties and the grain size distribution across the PDR, which indicates how these properties vary under the effect of the steep gradients in the physical conditions so typical for a PDR. I will discuss the impact of these results, as they may provide new constraints on the modeling of the formation and processing of dust in the ISM.
Speaker: Dries Van De Putte (Universiteit Gent)
Dries_Van_De_Putte_one_minute_slide.pdf
A Closer Look at Some Gas-Phase Depletions in the ISM: Trends for O, Ge and Kr vs. F*, f(H$_2$) and Starlight Intensity
An analysis of interstellar absorption features in UV stellar spectra in the HST and FUSE archives reveals column densities of O I, Ge II, Kr I, Mn II, Mg II, H I and H$_2$ in many different directions. Expanding on an earlier study by Jenkins (2009), this effort probes the partial correlations of the element abundances of O, Ge, and Kr relative to hydrogen for three fundamental parameters: (1) a generalized parameter F for the strength of depletions of elements by dust, (2) the fraction of hydrogen in molecular form f(H$_2$), and (3) a measure of the local intensity of starlight. Abundances of Mg II and Mn II relative to atomic and molecular hydrogen establish values of F. Previous claims that the chemically inert element Kr is sometimes depleted are substantiated in this study, but correlations with any of the three parameters are very weak, especially after one accounts for error covariances arising from uncertainties in the total hydrogen column densities. The ratio of gas-phase O to H in the ISM exhibits positive correlations with both f(H$_2$) and starlight intensity, and as expected, a negative correlation with F. Photodesorption of oxygen atoms from solid constituents probably accounts for the relationship between concentrations of gas-phase O and starlight intensity, but the reason for the correspondence with f(H2) is more difficult to explain and may arise from some indirect effect. Ge/H has a negative correlation with F and no significant dependence on the other two parameters.
Jenkins_poster92.pdf
Jenkins_poster.pdf
Constraint on properties of dust grains created by Population III supernovae
Dust grains play an important role in star formation also in the early Universe. The stellar initial mass function is considered to transfer from top-heavy to the normal Salpeter one in the course of metal/dust enrichment of interstellar medium because thermal emission cooling by dust grains induce the fragmentation of their parent gas clouds. However, dust properties such as size distribution and metal condensation efficiency are largely unknown. We here focus on the lower limits of elemental abundances of metal-poor stars. Recently, by survey campaigns, we obtain large statistical samples of metal-poor stars. They are classified into C-enhanced metal-poor (CEMP) stars and C-normal metal-poor (CNMP) stars, and their carbon and iron abundances show the lower limits of $A_{\rm cr}({\rm C}) \sim 6$ and ${\rm [Fe/H]}_{\rm cr} \sim -5$, respectively. This suggests the critical elemental abundances above which cooling of carbon and silicate grains is dominant, respectively. Since the dust cooling rate depends on the condensation efficiency of metal and grain size distribution with a given metallicity, we estimate them from the observed lower-limits of carbon and iron abundances. As a result, we find that the ratio of characteristic grain size to condensation efficiency (effective grain radius) is 10 $\mu$m and 0.1 $\mu$m for carbon and silicate grains, respectively.
Speaker: Gen Chiaki
180611CosmicDustTalk_Chiaki.pptx
Dust Attenuation of Star-Forming Galaxies in the first 2 Gyr of the Universe
The development of extremely sensitive mm/submm telescopes (e.g. ALMA, NOEMA) opened a new window to the far infrared (FIR) continuum emitted by dust, which enables us to investigate the obscured star-formation history of the Universe. Using these new facilities, recent studies of the dust properties of early galaxies revealed unexpected results, as high redshift galaxies show much lower FIR emission than expected. However, these early results were based on small samples selected from small fields in the sky. Here, we take the next steps based on the ALMA archive in the COSMOS field (a.k.a A3COSMOS project) and present new results on the dust attenuation of a large sample of high-redshift galaxies at z~3-5. In particular, we study the relationship between the stellar mass ($\rm{M_{\ast}}$), the UV spectral slope ($\rm{\beta_{UV}}$), and the infrared excess (IRX). In total, our study is based on a sample of ~1000 galaxies (~10% of which are individually detected) at z=3-5 in a stellar mass range $10^9$ - $10^{11}\,\rm{M_{\odot}}$ observed by ALMA during cycle1 - cycle4. Stacks show that the dust extinction corrections of local starburst galaxies are, on average, applicable to main-sequence z~3-5 galaxies. However, the IRX-beta_UV relation exhibits a very large scatter, up to $\pm$1 dex at a given UV slope. Similar results hold for the IRX-$\rm{M_{\ast}}$ relation. We discuss several physical explanations for the large scatter in the IRX-beta_UV relation and the IRX-Mass relation, and their implications for estimating the total cosmic star-formation rate density at $\rm{z}>3$.
Speaker: Mr Yoshinobu Fudamoto (Observatory of Geneva)
Copenhagen18-1min.ppt
Probing the interstellar dust with X-rays: The Fe L and O K edges
The content of the interstellar medium (ISM) is very important for the evolution of the Galaxy and for star formation processes. Today it is known that the structure of the ISM mainly consists of gas, dust and molecules. However, the composition of dust in the ISM is not yet fully understood. Insights can be gained from the X-ray band. High-resolution X-ray spectroscopy is a powerful method to investigate the interstellar dust composition. With X-ray spectra of bright background sources, it is possible to determine the silicate content and the physical properties of the diffuse regions in the ISM. We can probe the different phases of the interstellar medium and the chemical composition of gas along different lines of sight. In this work we analyse XMM-Newton and Chandra observations of the Low Mass X-ray Binary GX 9+9. This source is an ideal candidate to study the ISM because of known absorption by dust, cold and warm gas along the line of sight. For our modelling we use new laboratory measurements of different chemical composition of dust gained with the Electron Microscope Spectrometer in Cadiz, Spain. In particular, we focus here on the Fe L and O K edges, two abundant elements to study the chemical composition of dust grains along this line of sight.
Speaker: Ioanna Psaradaki (SRON, Netherlands Institute for Space Research)
Relative sputtering rates of FeS, MgS, and Mg silicates: implications for ISM gas phase depletions of rock-forming elements
Astronomical measurements of S abundances in the diffuse interstellar medium (ISM) indicate ionized S is a dominant species with little (< 5%) S residing in grains (e.g. Jenkins 2009). This is an enigmatic result, given that abundant Fe-sulfide grains are observed in dust around pre- and post-main sequence stars (Keller et al. 2002; Hony et al. 2002) and are also observed as major components of primitive meteoritic and cometary samples. These disparate observations suggest that the lifetime of sulfide grains in the ISM is short because of destruction processes. Our previous work has shown that FeS and MgS retain their crystallinity and do not amorphize during radiation processing, whereas enstatite and forsterite are readily amorphized (Keller et al. 2013; Christoffersen and Keller, 2011). We have extended this study to measure the relative sputtering rates of FeS and MgS compared to enstatite and forsterite.
Irradiation of FeS with 4 keV He+ results in preferential sputtering of S and the formation of a thin 2-3 nm, compact Fe metal layer that armors the surface. The zone of S loss extends to a depth of ~8-10 nm below the exposed surface (Keller et al. 2013). Despite this S loss, the FeS retains its crystallinity and shows no sign of incipient amorphization. Irradiation of FeS with 5kV Ga+ in a focused ion beam (FIB) instrument resulted in preferential sputtering of S and the formation of a 5-8-nm thick surface layer of nanophase Fe metal. X-ray mapping shows that the zone of S sputtering extends to a depth of nearly 20 nm, but there is no evidence for FeS amorphization, consistent with our previous work.
The irradiation experiments show that the relative sputtering rate of FeS and MgS are much higher than olivine or enstatite. Sputtering experiments utilizing 30 kV and 5 kV Ga ions in the FIB produced volume loss in troilite that was ~4X greater than in enstatite or forsterite. The sputter yield under these conditions is such that for every Si atom sputtered from enstatite, ~14 S atoms are sputtered from FeS. We have performed similar sputtering experiments on Fe-bearing niningerite (MgS) and co-existing enstatite from the ALH 84170 EH3 chondrite. MgS also sputters much more rapidly than enstatite with a relative Si:S sputter yield of 1:8. For MgS, sulfur is highly depleted at the surface and the S-depletion zone extends to a depth of ~15 nm (using 5 kV Ga+). There is a corresponding zone of Mg and especially Fe enrichment that extends from 5 to ~10 nm below the surface, respectively.
The dominant grain destruction mechanism in the ISM is sputtering from passage of supernova-generated shock waves (Jones and Nuth 2011). This process also results in the amorphization of crystalline silicates in the ISM. Our results indicate that FeS and MgS grains produced in evolved stars and injected into the ISM will be destroyed more rapidly than crystalline silicates. This process may account for the lack of significant depletion of S from the gas phase in the ISM. However, rare nanophase FeS grains occur as inclusions in circumstellar amorphous silicate grains found in comet dust particles analyzed in the laboratory (Keller and Messenger 2011). These results show that a finite amount of S in the ISM is sequestered in solid grains.
Christoffersen, R. and Keller, L. P. (2011) Space radiation processing of sulfides and silicates in primitive solar systems materials: Insights from in-situ TEM ion irradiation experiments. Met. Planet. Sci. 46, 950-969.
Hony, S., J. Bouwman, L. P. Keller, and L. B. F. M. Waters (2002) The detection of iron sulfides in planetary nebulae. Astron. Astrophys. 393, L103-L106.
Jenkins E. B. (2009) A unified representation of gas-phase element depletions in the interstellar medium. Astrophys. J. 700, 1299-1348.
Jones A. P. and Nuth J. A. (2011) Dust destruction in the ISM: a re-evaluation of dust lifetimes. Astron. Astrophys. 530, A44-A56.
Keller, L. P. and Messenger, S. (2011) On the origins of GEMS grains. Geochim. Cosmochim. Acta 75, 5336-5365.
Keller, L. P., S. Hony, J. P. Bradley, J. Bouwman, F. J. Molster, L. B. F. M. Waters, D. E. Brownlee, G. J. Flynn, T. Henning and H. Mutschke (2002) Identification of iron sulphide grains in protoplanetary disks. Nature 417, 148-150.
Keller, L. P., Rahman, Z., Hiroi, T., Sasaki, S., Noble, S. K., Horz, F. and Cintala, M. J. (2013) Asteroidal space weathering: The major role of FeS. LPSC XLIV, #2404.
Speaker: Lindsay Keller (NASA Johnson Space Center)
Keller Copenhagen Poster.pdf
VUV photoprocessing of large PAH cations: an experimental study
As a part of interstellar dust, polycyclic aromatic hydrocarbons (PAHs) are processed by the interaction with vacuum ultraviolet (VUV) photons that are emitted by young stars [1]. After absorption of a VUV photon, an isolated PAH can undergo different relaxation processes: ionization, dissociation and radiative cooling, including infrared (IR) fluorescence which results in the aromatic infrared bands (AIBs) observed in many astronomical objects [2].
Following an earlier work on smaller PAHs [3], we investigate in this experimental study the two relaxation processes of photofragmentation and photoionization of large PAH cations ranging in size from 30 to 48 carbon atoms. The ions are trapped in the LTQ linear ion trap of the DESIRS beamline at the synchrotron SOLEIL and energized by VUV photons in the range of 8 - 20 eV. All resulting photoproducts are mass-analyzed and recorded as a function of photon energy. The photoionization process is found to strongly dominate the competition, with the photoionization yield increasing with number of carbon atoms. From the relative intensities of the photoproducts, action spectra are obtained and compared to the photoabsorption cross sections. The latter have been computed using the real time, real space implementation of time dependent density functional theory (TD-DFT) from the Octopus code [4]. This study gives insights into the photostability of interstellar PAHs in astrophysical environments.
[1] J. Montillaud, C. Joblin, and D. Toublanc, A&A, 552, A15 (2013)
[2] C. Joblin & A. G. G. M. Tielens, PAHs and the Universe, EAS Publication Series, Vol. 46 (2011)
[3] J. Zhen, S. Rodriguez Castillo, C. Joblin, G. Mulas, H. Sabbah, A. Giuliani, L. Nahon, S. Martin, J. Champeaux, and P. M. Mayer, ApJ, 822, 2 (2016)
[4] G. Malloci, G. Mulas, and C. Joblin, A&A, 426, 105 (2004)
Acknowledgements: We are grateful to the general staff from SOLEIL for the smooth running of the facility. We acknowledge funding by the European Union (EU) under the Horizon 2020 framework for the Marie Skłodowska-Curie action EUROPAH, Grant Agreement no. 722346. We also acknowledge support from the European Research Council under the European Union's Seventh Framework Programme ERC-2013-SyG, Grant Agreement no. 610256 NANOCOSMOS.
Speaker: Ms Gabi Wenzel (IRAP, Université de Toulouse, CNRS, CNES, Toulouse, France)
CPHDUST_PosterPresentation_Wenzel.pdf
(Sub-)millimeter optical constants of silicates and water ice
We provide new temperature-dependent optical constants of silicate glasses, silicate minerals, and crystalline and amorphous water ice, in the sub-millimeter spectral range, for silicate glasses up to a wavelength of 4 mm. We compare these optical constants to literature data, such as the ``astronomical silicate'' and commonly used extrapolations of the water-ice opacity. We discuss physical reasons of the strong temperature dependence of the absorptivity seen in the data, and consequences for the contribution of these materials to the sub-millimeter emission of cosmic dust.
Speaker: Dr Harald Mutschke (Astronomical Institute and University-Observatory Jena)
Mutschke_poster39_CPHdust.pdf
Mutschke_poster39_CPHdust.pptx
Large Interstellar Polarisation Survey
We study the variability of the dust characteristics from cloud-to-cloud in the diffuse ISM (arXiv:1711.08672). We took low-resolution spectro-polarimetric data obtained in the context of the Large Interstellar Polarisation Survey (LIPS, arXiv:1710.02439) towards 59 sight-lines in the southern hemisphere, and we fitted these data using a dust model composed of silicate and carbon. Particles sizes range from the molecular to the sub-micrometre domain. Large (>6 nm) spheroidal dust that are of prolate shape and made of silicate account for the observed polarisation curve (arXiv:1705.07828). For 32 sight-lines we complemented our data set with UVES archive high-resolution spectra, which enable us to establish the presence of single-cloud or multiple-clouds towards individual sight-lines. We find that the majority of these 32 sight-lines intersect two or more dust clouds, while eight of them are dominated by a single absorbing cloud. We confirm several correlations between extinction and polarisation characteristics and the dust parameters, but we find also several previously undetected correlations between these parameters that are valid only in single-cloud sight-lines (arXiv:1711.08672). We observe that interstellar polarisation from multiple-clouds is smaller than from single-cloud sight-lines, showing that the presence of a second or more clouds depolarises the incoming radiation. We find large variations of the dust characteristics from cloud-to-cloud. However, when we average a number of clouds we always retrieve similar mean dust parameters. Typical dust abundances of the single-cloud cases are [C]/[H] = 92 ppm and [Si]/[H] = 20 ppm. Further we present the status of our search of single-cloud sight-lines and discuss the impact of grain porosity on the extinction and to the optical-to-submmillimter polarisation.
Speaker: Ralf Siebenmorgen (ESO)
Large Interstellar Polarisation Survey II. UV/optical study of cloud-to-cloud variations of dust in the diffuse ISM
RalfSiebenmorgen_poster.pptx
Observational constraints on dust properties Main Auditorium
Convener: Elisa Costantini (SRON Netherlands Institute for Space Research)
Observational constraints on dust properties
Abstract here
Speaker: Prof. Bruce Draine (Princeton University)
draine.pdf
A new window on Interstellar Silicates
X-rays provide a powerful tool to study interstellar dust. Using X-ray binaries as background sources, we can investigate the intervening dust along the line of sight. This is done by observing the edges present in the spectra of these sources, that serve as a unique fingerprint of the dust (Costantini 2012).
In particular, the extinction features in the Si K-edge offer a range of possibilities to study silicon bearing dust, such as investigating the grain size distribution, crystallinity, abundance and the chemical composition along a given line of sight (Zeegers et al. 2017). The edge is modelled with unprecedented accuracy, as we include a total of 15 laboratory measurements of interstellar dust analogues.
Here we also present our results of 9 different lines of sight toward the Galactic
plane and give a detailed mapping of the properties of the dust, unveiling the dust nature toward the central region of the Galaxy (Zeegers et al. 2018 in prep.).
Speaker: Sascha Zeegers (SRON)
Break Main Auditorium
Constraining dust mineralogy from mid-IR spectra
The first ISO spectra of evolved stars revealed a wealth of features in AGB stars, YSOs, comets and other environments, which have been linked to a variety of crystalline silicate and oxide species. The presence and strength of these features carries information about the formation and processing history of the dust in AGB envelopes. However, unlocking this information has proven difficult; our understanding is anecdotal at best, being based on small, likely biased, samples. While ISO and Spitzer have observed the mid-IR spectra of hundreds of sources, these datasets have not been properly exploited yet. Statistical problems have been a significant factor, primarily the difficulty in simultaneously fitting the overall SED and the details of spectral features using radiative-transfer models. I will explore these and related problems before presenting an attempt to alleviate them. I will present a new code we are developing, AMPERE, which includes the correct statistical treatment for simultaneous fitting of different kinds of data (photometry, spectra, imaging, interferometry). I will conclude with first results of an experiment in self-consistently fitting SEDs and mid-IR spectra to obtain detailed constraints on dust properties, including mineralogy, in selected oxygen-rich AGB envelopes.
Speaker: Dr Peter Scicluna (ASIAA)
20180613CPHDustAmpere.pdf
Investigating Silicate Dust in Galaxies Using Quasar Absorption Systems
The properties of silicate and carbonaceous dust grains in galaxies, as well as those of neutral and ionized gasses and of molecules, can be studied in galaxies ranging from the local Universe to moderate redshifts using absorption lines detected in the spectra of background quasars. By exploiting serendipitous lines of sight to distant quasars that pass through foreground galaxies, we can study the absorption signatures superposed in the quasar spectra by the dust and gas in these galaxies. Since quasars are luminous across a broad spectral range, this technique allows the simultaneous investigation of carbonaceous dust grains in the rest-frame ultraviolet (e.g., the 2175 Angstrom bump), metal ion lines at rest-frame ultraviolet and optical wavelengths, and silicate dust grains in the mid-infrared. We present results from our ongoing multi-wavelength research program exploring the connections between interstellar gas and dust in both distant and local galaxies using archival data for quasars with at least moderately gas-rich, foreground quasar absorption systems. In this presentation we will predominately focus on our studies of the silicate dust grains in several of these systems, characterized using the shapes of their 10 and 18 micron absorption features in Spitzer IRS spectra. Our measurements include the peak optical depth of the 10 micron feature, the ratio of the 10-to-18 micron features, and constraints on the silicate grain compositions, morphologies, and crystallinities derived from the shape and breadth of the absorption features. As part of our analysis, we will discuss the impact of variations in the underlying quasar continuum shape on our derived properties. We will also discuss correlations and trends between the silicate dust grain properties in these systems, and properties such as the absorber redshift, gas metallicity, velocity spread, carbonaceous dust abundance, and extinction curve shape. In combination, these data may yield important constraints on models of the evolution of metals and dust in galaxies. This work was supported by NASA grants NNX14AG74G and NNX17AJ26G.
Speaker: Prof. Monique Aller (Georgia Southern University)
Aller-CPHpresentation.pdf
Aller-CPHpresentation.pptx
Molecules and dust: Molecules and Dust Main Auditorium
Convener: JD Smith (University of Toledo)
The photochemical evolution of the interstellar PAH family in photodissociation regions
As is unequivocally evident from observations, Polycyclic Aromatic Hydrocarbons (PAHs) pervade the Universe. PAHs are easily detected through their vibrational IR emission bands at 3.3, 6.2, 7.7, 8.6 and 11.2 micron. They are found in a wide variety of environments, including post-AGB stars, planetary nebulae, young stellar objects, HII regions, reflection nebulae, the interstellar medium, and galaxies out to redshifts of z ~3. To date, PAHs are among the largest and most complex molecules known in space. They emit up to 10% of the total power output of star-forming galaxies and harbor a significant fraction of the cosmic carbon. Being so abundant and widespread, PAHs play a crucial role in several astrophysical and astrochemical processes such as the heating of the diffuse ISM and surfaces of molecular clouds and proto-planetary disks, gas-phase abundances and surface chemistry.
The Infrared Space Observatory (ISO) and the Spitzer Space Telescope showcased the richness and complexity of the astronomical PAH spectra. The PAH features exhibit significant variability and depend on several parameters such as radiation field, object type and metallicity. These variations thus reflect both the physical conditions in the emission zones and the composition of the PAH population. Thus, one of the best ways to investigate the detailed characteristics of the PAH population is by analyzing IR spectral maps. Here we present the results of such hyperspectral imaging studies done with Spitzer/IRS in the 5-20 micron range for a sample of Reflection Nebulae and HII regions. These studies reveal subtle, but significant spatial variations in individual PAH emission bands revealing a spatial sequence with distance from the illuminating star. The overall dominant charge state of the PAH population is certainly a key factor in driving these variations. However, hyperspectral imaging studies allow to probe PAH parameters beyond charge, such as molecular structure and size. Combined with the NASA Ames PAH database to fine-tune band assignments, the observed spatial sequence reveals the photochemical evolution of the interstellar PAH family as they are more and more exposed to the radiation field of the central star in the evaporative flows associated with the Photodissociation Regions.
Speaker: Els Peeters (University of Western Ontario & SETI Institute)
1806_Peeters_CosmicDust.key
fig11_animation_north_77.gif
fig11_animation_south_77.gif
From Molecules to Dust (and back)
Throughout the Universe, there is a close interplay between the formation, evolution and destruction of dust grains and molecules, driven by the diverse conditions encountered in various astrophysical environments. When stellar gas cools down in the surroundings of evolved stars, it starts a transformation into molecular gas and dust grains. The resulting inventory depends not only on the chemical makeup of the stellar photosphere, but also on the physical conditions and available timescales. Dense tori and disks can offer long timescales for chemistry and grain growth, while photo-processing by stellar and interstellar radiation further drives molecular chemistry and dust erosion as stellar outflows are dispersed into the interstellar medium. In interstellar environments, dust grains and large molecules can offer their surfaces to facilitate the formation of molecules, and a particularly rich molecular chemistry occurs in the ice mantles on dust grains in molecular clouds. Near hot stars, intense radiation fields create harsh conditions where UV photons destroy all but the fittest of molecular species.
In this talk, I will present an overview of the environments where molecules and dust reside, and the physical and chemical processes that affect them.
Speaker: Prof. Jan Cami (University of Western Ontario)
Cami_MoleculesDust.pdf
The High-Dimensional ISM As A Tool To Explore Galaxy Evolution
Because large molecules (including PAHs and other carbonaceous particles) straddle the divide between molecular gas and dust grains, their spatial and dynamical distributions carry information on both of these phases and their interface. This talk will explore the power of harnessing this information using a few case studies of molecular absorption spectroscopy combined with absorption-based dust maps. I will describe how this approach allows us to tackle such diverse galaxy evolution questions as spiral arm formation and molecular cloud dissolution, and discuss the prospects for expanding high-dimensional ISM studies in the next several years.
Speaker: Dr Gail Zasowski (University of Utah)
CosmicDust2018_zasowski.pdf
Laboratory studies of cosmic dust: Laboratory studies of dust Main Auditorium
Convener: Prof. Raffaella Schneider (Osservatorio Astronomico di Roma-INAF)
Laboratory Experiments on Cosmic Dust
Cosmic dust grains are present in nearly all astrophysical environments proving their existence by the absorption, scattering, and reemission of stellar light in a broad spectral range covering wavelengths from the UV up to millimeters. Dedicated laboratory experiments are necessary to get more insight into the formation pathways and chemical modification of refractory solids under different astrophysical conditions. Such experiments include the formation of dust at varying temperatures and densities and the processing of grains due to a bombardment with ions, atoms, and UV photons. All these studies have to be complemented by a detailed analytical characterization of the final products. The measurements of the spectral properties of dust at different evolutionary states in the laboratory provide a major link to the astronomical observations and the tool for identification of cosmic dust components. In addition, it helps to restrict the conditions in the corresponding astrophysical environments. Our insight into the structural and
spectral properties of cosmic dust in different astrophysical environments such as circumstellar shells, the diffuse and dense interstellar medium, and disks around young stars has been significantly improved by the laboratory studies in the last years. This review describes the recent progress in understanding the formation, processing, morphology, and composition of main dust components including siliceous and carbonaceous grains.
Speaker: Dr Cornelia Jäger (University of Jena)
How laboratory experiments can help in studying cosmic PAHs
Polycyclic aromatic hydrocarbons (PAHs) are commonly thought to play a key role in the chemical and physical evolution of star-forming regions from the small scales of protoplanetary disks to the large scales of galaxies. However attempts to identify individual species have been so far unsuccessful. Cosmic PAHs and related species such as C$_{60}$, are observed by their emission features in the mid-infrared, the so-called aromatic infrared bands (AIBs). Emission in the AIBs is triggered by the absorption of VUV photons but this process can also induce ionisation and unimolecular dissociation. The composition of the IR-emitting population might therefore reflect this processing. In addition the formation routes of these large carbonaceous molecules have still to be elucidated. Several scenarios including bottom-up and top-down ones have been proposed.
In my presentation, I will discuss how this subject takes benefit from laboratory astrophysics. The photophysics of isolated PAHs, including ionisation, dissociation and radiative cooling is studied with different setups (molecular jets, ion traps, storage rings) and makes use of VUV tunable synchrotron radiation [1-6]. The question of the formation by gas-phase condensation of PAHs and carbonaceous grains in circumstellar environments has been discussed following experiments that use techniques such as laser ablation, laser pyrolysis or plasma discharges [7-9]. Recently, the Stardust machine in Madrid [10] has been developed to study grain formation in conditions that approach those found in Asymptotic Giant Branch star environment. The molecular analysis of laboratory analogues of cosmic dust, combined with that of extraterrestrial samples such as meteorites, is expected to provide new insights into chemical pathways leading to the formation of cosmic PAHs. I will describe how this topic is addressed in the framework of the Nanocosmos ERC Synergy project using in particular the AROMA setup [11].
[1] B. West, S. Rodriguez Castillo, A. Sit, S. Mohamad, B. Lowe, C. Joblin, A. Bodi and P. M. Mayer, Phys. Chem. Chem. Phys. 20, 7195 (2018).
[2] C. Joblin, L. Dontot, G.A. Garcia, F. Spiegelman, M. Rapacioli, L. Nahon, P. Parneix, T. Pino, P. Bréchignac, J. Phys. Chem. Lett. 8, 3697–3702 (2017).
[3] G. Rouillé, S. A. Krasnokutski, D. Fulvio, C. Jäger, Th. Henning, G. A. Garcia, X.-F. Tang, L. Nahon, Astrophys. J. 810, id. 114 (2015).
[4] J. Zhen, S. Rodriguez Castillo, C. Joblin, G. Mulas, H. Sabbah, A. Giuliani, L. Nahon, S.Martin, J.-P. Champeaux, P. Mayer, Astrophys. J. 822, id. 113 (2016).
[5] J. Zhen, P. Castellanos, D. M. Paardekooper, N. Ligterink, H. Linnartz, L. Nahon, C. Joblin, A.G.G.M. Tielens, Astrophys. J. Lett. 804, L7 (2015).
[6] S. Martin, M. Ji, J. Bernard, R. Brédy, B. Concina, A. R. Allouche, C. Joblin, C. Ortega, A. Cassimi, Y. Ngono-ravache, and L. Chen, Phys. Rev. A 92, id.053425 (2015).
[7] H. W. Kroto, J. R. Heath, S. C. O'Brien, R. F. Curl, R. E. Smalley, Nature 318, 162-163 (1985).
[8] C. Jäger, F. Huisken, H. Mutschke, I. Llamas Jansa,, Th. Henning, Astrophys. J. 696, 706-712 (2009).
[9] C. S. Contreras, F. Salama, Astrophys. J. Supp. 208, id. 6 (2013).
[10] L. Martínez, K. Lauwaet, G. Santoro, J. M. Sobrado, R. J. Peláez, V. J. Herrero, I. Tanarro, G. Ellis, J. Cernicharo, C. Joblin, Y. Huttel, J. A. Martín-Gago, subm (2017) and this conference.
[11] H. Sabbah, A. Bonnamy, D. Papanastasiou, J. Cernicharo, , J.-A. Martín-Gago, C. Joblin, Astrophys. J. 843, id. 34 (2017).
We acknowledge support from the European Research Council under the European Union's Seventh Framework Programme ERC-2013-SyG, Grant Agreement n. 610256 NANOCOSMOS.
Speaker: Christine Joblin
Formation of molecules on cosmic dust grains: a laboratory view
Molecular ices covering dust grains are known to be a source of molecules, including complex organic molecules (COMs), in the interstellar medium (ISM) and circumstellar shells and disks, the molecules, which cannot be created via gas phase reactions. Studying the formation of COMs is crucially important to understand the processes that lead to stars and planets formation, and to understand a degree of molecular complexity on planetary bodies, which can shed some light on the origin of life on Earth. Many laboratory experiments have been performed on the formation of simple and complex molecules, including amino acids, in interstellar and circumstellar ice analogues by a number of triggering processes, such as UV and X-ray irradiation, bombardment by energetic particles and atom addition. But a major part of the laboratory work deals with molecular ices covering standard substrates not related to the cosmic dust. The dust grain surface can participate in ice chemistry and can alter the efficiency of the molecular formation. There is a handful of laboratory works on the formation of molecules in ice-dust systems. Only CO and CO$_2$ have been synthesized in laboratory cosmic ice-dust analogues up until very recently, when we performed our experiments on the formation of formaldehyde on hydrogenated fullerene-like carbon grains by the O/H atom addition. Our results demonstrate, for the first time, that the bombardment of carbonaceous grains by O and H atoms at low temperatures causes the formation of CO molecules with their further hydrogenation leading to the formation of solid formaldehyde. The formation of H$_2$CO is an indication for a possible methanol formation route in such systems and CH$_3$OH, in turn, is well known as a starting point for the formation of more complex organic molecules in the ISM and circumstellar phases.
Speaker: Alexey Potapov (Max Planck Institute for Astronomy)
potapov_06_01.pdf
Determining the systematic errors in fits of dust thermal emission: the role of laboratory data in upcoming models
Interstellar dust plays an important role the study of interstellar medium, especially since the development of far-infrared and submillimeter instruments in the last decades (e.g. IRAS, Herschel, Planck, ALMA) has allowed wide surveys of dust thermal emission. Using a dust emission model these observations can be converted to maps of quantities such as the dust column density and temperature, or to constrain dust masses in molecular clouds and galaxies.
Dust emission is commonly modeled as a blackbody with temperature T multiplied by an opacity $\kappa$ that varies with wavelength as a power law: $\kappa \propto \lambda^{–\beta}$, usually with $\beta \sim 2$. However, we are learning from both astronomical observations and laboratory tests on dust analogues that T–β models are too simplistic. Two facts in particular emerge:
1. For most candidate dust materials the opacity $\kappa(\lambda)$ does not follow a power law: its slope decreases beyond a certain wavelength (typically around 400–700 μm);
2. The optical properties of materials often depend on temperature as well; for instance opacity often increases with T.
Our group is working on optical data on several candidate dust materials, collected by multiple laboratories, to parametrize the materials' opacities as functions of λ and T. This parametrization will be the first step in building a more physically realistic and flexible dust model. By fitting observations of molecular clouds and nearby galaxies, and by constructing synthetic observations to fit with conventional methods, the new model will allow to find potential systematics in T–β fit results. The model could also be applied to galaxies at high redshift, where recent dust mass estimates are posing a challenge to dust formation models, and understanding systematics on such measurements is essential.
Speaker: Dr Lapo Fanciullo (ASIAA)
Fanciullo_Copenhagen_Talk.pdf
Convener: Francesco Valentino (Niels Bohr Institute - University of Copenhagen)
Crystalline silicates in external galaxies
Observational evidence has long supported that most of the interstellar silicates in galaxies are amorphous. While crystalline silicates may form around evolved stars at temperatures sufficiently high to allow for annealing, it is thought that the harsh interstellar environment quickly amorphitizes any crystalline silicates, most likely through bombardment by the heavy ions in cosmic rays (Demyk et al. 2001; Jäger et al. 2003; Brucato et al. 2004; Bringa et al. 2007; Szenes et al. 2010), and a firm upper limit of 2% on the crystalline fraction of silicates was derived based on the absence of substructure in the 9.7 μm feature (Kemper et al. 2004; Kemper et al. 2005).
The first detection of crystalline silicates in external galaxies was reported by Spoon et al. (2006) in 12 out of a sample of 77 starbursting Ultraluminous Infrared Galaxies (ULIRGs), with later detections of further galaxies reported by Roussel et al. (2006), Willett et al. (2011), Stierwalt et al. (2014), and Aller et al. (2012). The only one of these studies quantifying the crystalline fraction is the work by Spoon et al. (2006), who report a crystalline fraction of 6-13% in the interstellar silicate reservoirs. A very simple model of the production of crystalline silicate dust by evolved stars, at a level of 10-20% of the total silicate dust production by these stars, is able to explain the observed crystallinities at about 30 Myr after the start of a starburst (Kemper et al. 2011). In general, the model can be used to estimate the transition time and interstellar conditions, such as cosmic ray fluence, based on observational constraints on the crystalline fraction.
However, the small number of known interstellar crystalline silicate fractions in star-forming galaxies limits the usefulness of such a model. We have devised a method to measure the crystalline fraction of silicates in a large number of galaxies quickly and easily. For this purpose, we are performing radiative transfer models of starburst galaxies, with varying crystalline fractions of their interstellar silicates using the SKIRT radiative transfer code (Camps & Baes 2015), and identified a method to determine the crystallinity of silicates in starburst galaxies directly from (archival) infrared spectroscopy.
Speaker: Ciska Kemper (ASIAA)
Kemper_poster.pdf
Probing the Size Distribution of PAHs in Reflection Nebulae
The mid-infrared (MIR) spectrum of many astronomical sources show prominent emission features at 3.3, 6.2, 7.7, 8.6, 11.2, and 12.7 $\mu$m attributed to the IR fluorescence of polycyclic aromatic hydrocarbons (PAHs). We use spatial maps of the 3.3 and 11.2 $\mu$m PAH emission features to measure the distribution of PAH size within two reflection nebulae with strong MIR emission present, namely NGC 2023 and NGC 7023. Variations in the size of these molecules with distance from a source of ultraviolet (UV) radiation is indicative of the photochemical evolution of these species and yields information on how they interact with their environment.
We make use of the First Light Infrared TEst CAMera (FLITECAM) on board the Stratospheric Observatory for Infrared Astronomy (SOFIA) to observe the 3.3 $\mu$m emission in each source. The 11.2 $\mu$m emission band is measured using spectral maps obtained from the Infrared Spectrograph (IRS) SH mode on board the Spitzer Space Telescope. Additionally, we use broadband photometry at 8 $\mu$m from the Infrared Array Camera (IRAC) on board Spitzer. These IRAC 8 $\mu$m images are compared with the IRS SH data to map the 8 $\mu$m over 11.2 $\mu$m ratio, as an approximate measure of the PAH ionization state.
We use the map of the 11.2 $\mu$m over 3.3 $\mu$m emission feature to probe the size distribution in each of our sources. We find that this ratio is at a minimum on the surface of the photodissociation region (PDR) and increases by a factor of 2-3 moving inwards towards the illuminating source in both reflection nebulae, suggesting these species undergo significant photoprocessing within their environment. We also note that the ionization level of these species is found to increase with decreasing distance to the illuminating source in both of these cases. Hence, we can infer there is evidence of a rich carbon-based chemistry driven by the photochemical evolution of the omnipresent PAH molecules within the interstellar medium.
Speaker: Mr Collin Knight (University of Western Ontario)
PAH_size_summary.pptx
PAH_size_summary_v2.pdf
Distributions of metallicity and gas-to-dust ratio in the Magellanic system
The two Magellanic clouds and our Galaxy are known to have been tidally interacting each other in the past $\sim$ billion years. It is indicated by the elongated distributions of interstellar medium (ISM), known as the Magellanic stream and the Magellanic bridge. Numerical simulations of the dynamical interactions of these galaxies successfully reproduced these features. This dynamical interaction is also suggested to play an important role for triggering the massive star formations in the Magellanic clouds. Because the Small Magellanic Cloud (SMC) has $\sim 1/5$ of metallicity compared to the Large Magellanic Cloud (LMC), one can expect weaker dust emission from the ISM originated from the SMC, which may mix with the ISM from the LMC. We have investigated the dust thermal emission obtained by the Planck satellite and the H I 21 cm data, and discovered large diversity with more than an order of magnitude of the gas-to-dust ratio among the Magellanic system. The distribution of the gas-to-dust ratio clearly indicate that the Magellanic stream is dominated by the metal-poor ISM stretched from the SMC, while the stream of the metal-poor ISM falling onto the LMC. The massive cluster forming regions including 30 Dor in the LMC tend to show mixed ISM properties of the LMC and the SMC, which support the idea of massive star formation triggered by the gas infall and the cloud-cloud collisions.
Speaker: Kengo Tachihara (Nagoya University)
Tachihara_short-oral.pdf
Tachihara_short-oral.pptx
Effect of dust porosity on scattered light images of protoplanetary disks
The dust porosity is a key quantity that characterizes how dust particles grow to form planetesimals in protoplanetary disks. We study how the dust porosity affects scattered light images of protoplanetary disks in near-infrared wavelengths. It is known from near-infrared observations that some protoplanetary disks are faint and show red color in the scattered light. Large fluffy dust aggregates have been considered as a potential candidate to explain these observed properties (Mulders et al. 2013).
We perform radiative transfer calculations of protoplanetary disks taking the dust porosity into account, where optical properties of fluffy dust aggregates are obtained by using a rigorous method, T-Matrix Method (TMM) and approximate methods, a modified mean field theory (MMF, Tazaki & Tanaka, submitted) and the effective medium theory (EMT). It is found that when a commonly used method, EMT, is used to obtain optical properties of fluffy dust aggregates, the disk becomes faint and shows reddish color in the scattered light. However, when a rigorous method, TMM, is used to obtain the optical properties, the disk becomes relatively bright and shows gray or slightly blue color. By using the MMF method, we show that even if the aggregate radius is increased up to mm-size, the disk tends to show relatively bright and gray color. As a result, our results suggest that red and faint protoplanetary disks in the scattered light indicate the presence of large compact aggregates at the surface layer of the disks rather than large fluffy dust aggregates.
Speaker: Ryo Tazaki (Tohoku University)
1min_cosmic_dust_v2.pdf
Constraining the presence of large dust grains in post-AGB disc systems using FIR and sub-mm photometry
The final stages of the evolution of intermediate mass stars ($M\sim 1-8 M_\odot$) are characterised by the ejection of their envelope as they evolve off the Asymptotic Giant Branch (AGB), producing a spectacular planetary nebula (PN). In particular, this phase is characterised by the development of strong asymmetries in the circumstellar medium, with a large fraction of PNe and pre-PNe hosting jets, tori, rings or bipolar structures, while many AGB envelopes are broadly spherical in shape. These asymmetries are, therefore, believed to develop either in the final phase of AGB evolution or in the initial post-AGB phase. In a small fraction of cases, post-AGB stars are found to be orbited by a massive, dusty circumstellar disc. Other types of circumstellar discs, such as protoplanetary and debris discs, are typically populated by dust grains at least up to mm sizes, as probed by the spectral index in the FIR and (sub-)mm wavelength ranges. In protoplanetary discs (PPDs), the existence of these large dust grains is believed to be linked to grain-growth processes which can take place in such dense, long-lived discs. There is also evidence for the presence of grains of such sizes in the discs in post-AGBs in spite of the large difference in lifetime compared to PPDs. However, to date such studies have either been relatively small, or the samples used have combined both post-AGBs with discs with pre-PNe and other objects, making it difficult to evaluate the prevalence of grain growth. I will present the largest study to date of the FIR and sub-mm emission of post-AGB stars with discs. By exploiting archival Herschel/SPIRE photometery along with new SMA observations and literature fluxes of a sample of 45 post-AGBs with discs we show that grain-growth to at least several hundred micron is ubiquitous in these enigmatic systems. The similarity of the distribution of spectral indices to those of protoplanetary discs indicates that this is a result of in-situ grain growth, rather than the grinding of parent bodies. The relatively short lifetimes of these discs show that grain growth to these sizes is a rapid process, occurring on timescales of only a few kyr.
Speaker: Peter Scicluna (ASIAA)
CPHDustPosterSlide.ppt
Observability of Dusty Debris Disks around M-stars
During the last few decades many debris disks have been found and resolved around A to K-type stars. However, only a handful of debris disks have been discovered around M-stars, and the reasons for their paucity remain unclear.
Here we check whether the sensitivity and wavelength coverage of present-day telescopes are simply unfavorable for detection of these disks or if they are truly rare. We approach this question by looking at different surveys that have searched for debris disks around M-type stars. Assuming that M-star disks are "similar" to those of earlier type stars in some sense (i.e., in terms of dust location, temperature, fractional luminosity, or mass), we check whether these surveys should have found them. Examining integration times and sensitivities of the instruments used, we create detection limit plots for each of these surveys. We will present and discuss the implications of these results for the "true" incidence rates of M-star debris disks.
Speaker: Patricia Luppe
Luppe-1min-presentation.pdf
Luppe-1min-presentation.pptx
Spatially resolved carbonaceous dust infrared emission in proto-planetary disks around Herbig Ae/Be stars
In the interstellar medium (ISM), the carbon (nano-)grains are a major component of interstellar dust. This solid phase is more vulnerable to processing/destruction than their silicate counterparts. It exhibits a complex, size-dependent evolution due to photon interactions, which provides a modeling challenge. How these micro-physical processes work under the extreme conditions found in disks (different from the ISM by orders of magnitude in terms of excitation and local gas density)? Nano-grains could play an essential role in the gas heating, and thus could have a major influence on the disk structure and its evolution. Moreover, due to their large effective surface area, they could play a key role in the formation of molecules. Finally, they are the tracers of the physical conditions (excitation, extinction, geometry).
I will present an analysis of infrared ground-based data, obtained with VLT/NACO in the 3-4 micron range (which includes aromatic, olefinic and aliphatic bands), for disks around intermediate mass stars (e.g., HD100546, HD100453, HD169142, HD179218). I will discuss what band ratios and parameters tell us on: dust composition, evolution and renewal at the disk surface. At last, I will propose a first comparison between observations of disks which are dense phases and The Heterogeneous dust Evolution Model for Interstellar Solids (THEMIS) developed at the IAS (Jones A. P. et al., A&A, 602, A46 (2017))
Speaker: Thomas Boutéraon (Institut d'Astrophysique Spatiale (IAS), Orsay, France)
poster_intro_Bouteraon2.0.pdf
Dust production in the Solar Neighborhood
Asymptotic Giant Branch (AGB) stars dominate the total dust injection into the interstellar medium (ISM) of galaxies. Studies providing total dust injection rates in the Milky Way (Jura & Kleinmann 1989) and nearby galaxies (Riebel et al. 2012; Srinivasan et al, 2016) show the importance of accurately estimating this contribution. In this work we revisit the total dust mass-loss rate from AGB stars in the Solar neighborhood. Such an update is necessary, especially for an all-sky sample, as contrary to recent and old studies. One of the challenges for Galactic and dusty AGB sources is the distance determination, which we are primarily interested in, as they are the highest mass-loss rate objects. Using present-day all-sky infrared facilities (WISE, 2MASS, AKARI). We constructed spectral energy distributions for all the AGB candidates within within 2 kpc from the Sun, which we fit with models from the GRAMS grid (Sargent, Srinivasan & Meixner 2011; Srinivasan, Sargent & Meixner 2011) to estimate their dust-production rates. We find an integrated dust production rate of ~ 4 X 10^-5 Msun/year or an average of ~ 2 X 10^-8 Msun/year per object is obtained. We compare our results to those of the Magellanic Clouds and other Local Group galaxies, for which the distance determination problems do not exist. Separating the contribution into C- and O-rich AGB is also presented and is compared with estimates from the LMC and SMC as well. This work presents new insights into the contribution of low- and intermediate-mass stars to the ISM, and the discrepancy between the dust produced by AGB stars and the estimated reservoir in the ISM.
Modelling dust in the Nearby Evolved Star Survey (NESS) targets
The Nearby Evolved Stars Survey (NESS) is a multi-telescope project targeting a volume-limited ($d < 2$ kpc) sample of $\sim$400 evolved stars, including 104 oxygen-rich stars (including $\sim$20 red supergiants), 8 S-type stars, and 19 carbon-rich AGB stars, as well as many post-AGB stars and planetary nebulae. NESS includes a 500-h ongoing JCMT survey of dust continuum as well as CO (2-1) and (3-2) line emission. The NESS JCMT data facilitate the determination of the circumstellar dust distribution and estimation of the mass-loss history in the circumstellar shells, including any deviations from spherical symmetry (Dharmawardena et al., in prep). Radiative transfer models of this dust must update fits to mid-infrared spectral energy distributions (SEDs) to fit the far-IR and sub-mm information, and should reproduce the results from JCMT observations.
In this poster, we present preliminary results of modelling dust in W Hya and U Ant, whose shells show complex structures that can not be fully explained by a uniform mass-loss rate model. NESS data has revealed U Ant's detached shell for the first time in the sub-mm continuum (Dharmawardena et al. in prep). We first fit the mid-IR SEDs with models from the GRAMS grid (Sargent et al., 2011; Srinivasan et al., 2011), tacitly assuming spherical symmetry. Using these models as a starting point, we use the radiative transfer code 2Dust to explore ranges of parameters that will reproduce the radial profiles as determined by Dharmawardena et al. (in prep), including any evidence of detached shells and/or variable mass-loss rates.
These models are the first step towards detailed modelling that incorporates data from optical through sub-mm SEDs and spectra, as well as other data such as interferometric visibilities. Modelling the large number of AGB stars targeted by NESS will lead to robust estimates of dust-production rates across the entire range of evolutionary stages along the AGB. Combined with modelling of the NESS CO line data, these can also be used to determine the gas-to-dust ratio throughout the circumstellar shell for the entire sample.
Speaker: Dr Sundar Srinivasan (ASIAA)
Poster_Dust_Modelling_NESS.key
Poster_Dust_Modelling_NESS.pdf
Attempt to catch the C/O transition in dust formation in cold plasma experiments
The formation of dust in the envelopes of evolved stars is still poorly understood. It is generally admitted that the C/O ratio in the envelope is a key parameter determining the type of dust that is formed and thus leading to either carbon-rich nanograins, possibly including polycyclic aromatics hydrocarbons (PAHs), or oxygen-rich nanograins mainly silicates.
In order to get further insights into the impact of the C/O ratio on dust nucleation and properties, we are carrying experiments in an axially-asymmetric capacitively-coupled radiofrequency (RF) argon plasma with pulsed injection of hexamethyldisiloxane (HMDSO, C6H18OSi2) that contains key elements associated with the two different families of nanograins in the envelopes of evolved stars. The plasma can be enriched in oxygen by injecting molecular oxygen.
Dust formation in the RF plasma reactor is followed in situ by optical emission spectroscopy and the main plasma parameters, the mean electron energy and the electron density are extracted from collisional-radiative modelling. Information on the chemical networks involved in dust nucleation is proposed after in situ probing of the molecular content by mass spectrometry and complementary ex situ mass analysis following laser desorption/ionisation of the collected dust. Finally, the dust is analysed using standard analytical technics such as electron microscopy, Fourier Transformed InfraRed (FTIR) and X-ray photoelectron spectroscopy (XPS), which reveal two main components: dust particles of ~50 nm size embedded in an organosilicon matrix. In this poster we discuss how the results obtained from different diagnostics interconnect and complete the analysis in order to reveal the mechanisms involved in dust nucleation and growth in the cold plasma reactor, with potential input to our understanding of dust formation in evolved stars.
We acknowledge support from the European Research Council under the European Union's Seventh Framework Programme ERC-2013-SyG, Grant Agreement n. 610256 NANOCOSMOS and from the UMS Raymond Castaing of the University of Toulouse for the SEM and TEM observations.
Speaker: Rémi Bérard (IRAP, LAPLACE, Université de Toulouse, CNRS, CNES, Toulouse, Fance)
CPHDUST-poster-RBerard .pdf
Carbon, sulfur and rare elements in the interstellar dust: an X-ray view
Here we present the prospects of observing dust features of important constitutents of the ISM (C, Al, S, Ca) using future X-ray facilities (Arcus, XARM and Athena).
Present instruments already probed the diffuse interstellar medium (e.g. Costantini et al. 2012) and the moderately dense environments (Zeegers et al. 2017).
However, carbon, one of the main constituents of the ISM, is currently outside the reach of X-ray instruments. This element is visible in an X-ray spectrum when the extinction is relatively low, probing the diffuse ISM either in the local Galactic arm or in particularly diffuse regions. We will show that in a near future we will be able to distinguish among graphite, amorphous and hydrogenated carbon, helping settling the debate on which form the carbon should take in the ISM.
Alongside heavy depleted elements (e.g. Al, Ca), we will also show how possible depletion and inclusion in dust can be measured in dense environments, like molecular clouds, for sulfur, which presence in the solid phase is still debated.
Speaker: Elisa Costantini (SRON Netherlands Institute for Space Research)
costantini_1slide_copenhagen.pdf
Co-accretion of carbon molecules and silicate precursors at cryogenic temperatures
Dust grains are subjected to various destruction mechanisms in the interstellar medium (ISM). Together these mechanisms operate at a rate faster than the injection of grains condensed in stellar outflows and supernova ejecta. Nevertheless, comparatively long-lived dust populations are observed in the ISM. The local re-formation or growth of grains has been proposed as a process that contributes to counterbalancing their destruction.
The growth is proposed to proceed through the accretion of atoms and/or molecules present in the interstellar gas phase, and subsequent chemical reactions that incorporate the accreted species to the grain. The reactions would take place in the cold neutral medium and in molecular clouds, i.e., at temperatures in the range 10 to 100 K. The bulk of interstellar dust, however, is constituted of silicate and carbonaceous grains. The formation of these refractory materials at cold interstellar temperatures has yet to be described in detail. Especially, the mechanisms that lead to the separation of silicate and carbonaceous materials observed by astronomers have to be determined.
We have already reported experiments showing the formation and growth of silicate grains at cryogenic temperatures through the accretion of cold atoms and molecules related to silicates. In the most recent experiments, carbon atoms and molecules (C$_n$, $n$ = 1$-$10) were added to the silicate precursors. Significant amounts of H$_2$O, CO, CO$_2$, and C$_3$O molecules were also present. We have observed that amorphous silicates are formed despite the presence of the carbon molecules and the other species. These experiments will be presented and their relevance to the growth of silicate grains in the ISM will be discussed.
Speaker: Dr Gaël Rouillé
Rouille_Poster_#128.pdf
Rouille_Slide_Poster_#128.pdf
High-temperature infrared spectroscopy of large aromatic molecules
Large aromatic molecules are ubiquitous in astrophysical environments such as star forming regions, galaxies and planetary nebulae in which they emit the Aromatic Infrared Bands (AIBs). These molecules include polycyclic aromatic hydrocarbons (PAHs) but also fullerenes, C60 being the only molecule of this class identified so far. Emission in the AIBs is triggered by the absorption of a UV photon via an electronic transition and a sequence of radiationless transitions converting most of the absorbed energy to a vibrational excitation in the electronic ground state. The hot molecule then relaxes by emitting IR photons, the resulting spectrum being dominated by a large number of hot bands, all slightly shifted with respect to the corresponding 1-0 fundamentals due to anharmonicity. The resulting bands are very broad and their interpretation complex.
In order to progress on the analysis of the IR spectra of these hot large molecules, we are developing an experimental approach to quantify the temperature dependent infrared (IR) spectrum of PAHs. We recorded the IR spectrum of solid pyrene (C16H10) in KBr pellets from 14K to 723K. With increasing temperature a gradual red shift of the band positions and increase of the bandwidth were observed. For the higher temperatures, we compared these data with the few available data recorded in gas-phase [1]. That allows us to gain confidence into the relevance of these measurements in solid phase to derive anharmonic constants for isolated molecules. We therefore used the same experimental approach to study the temperature-dependent IR spectra of larger species such as coronene (C24H12) and fullerene C60. In this presentation we will discuss and summarize these results.
[1] Joblin, C.; Boissel, P.; Léger, A.; d'Hendecourt, L. & Défourneau, D. Astron. Astrophys. 835, 299 (1995)
Speaker: Dr Shubhadip CHAKRABORTY (Institut de Recherche en Astrophysique et Planetologie, CNRS, Toulouse, France)
Chakraborty_Shubhadip_IRAP_CNRS.pdf
Investigating the interstellar dust through the Fe K-edge
The absorption fine structures, imprinted by the interaction between X-rays and solid particles, can reveal the composition, the size, and the structure of cosmic dust (Costantini et al. 2012). The iron K-edge is particularly important because it is well visible in the X-ray band providing a large extinction especially for lines of sight with N$_{\rm{H}} > 10^{23} \rm{cm}^{-2}$ (A$_{\rm{V}}> 45$). We model the iron edge using the newly acquired synchrotron data, performed on a set of cosmic dust analogues (Rogantini et al. 2018). Here we highlight the potential of the iron K-edge to: 1) study the chemical properties of iron bearing grains; 2) investigate the size, the crystallinity, and the composition of cosmic silicates in dense clouds of our Galaxy. The synergy between high resolution X-ray instruments and accurate synchrotron measurements provides a unique method to look through molecular clouds in the Galactic Centre and to understand the role of iron in the grain growth process in the interstellar matter.
Speaker: Daniele Rogantini (SRON - Netherlands Institute for Space Research)
rogantini_poster.pdf
Studying the Amorphous Material Physics of the Cosmic Dust Grains by Fitting the Observed Intensity and Polarization SEDs in Millimeter, Submillimeter and Far-infrared Wavebands
The Galactic dust emission is a main obstacle to detect the cosmic microwave background (CMB) B-mode polarization signals originated from primordial gravitational waves. Accurate removal of the Galactic dust emission from the data is crucial for the success. Although a single power law frequency dependence has been usually assumed both for the dust emissivity and polarization SEDs, they might have complex frequency dependences. The deviation of the observed spectrum indexes of the Galactic dust emission in mm and submm wavebands from the spectrum index of the crystal is one of the strong evidences which support that main constituent of the Galactic dust grains is amorphous material. Therefore, we have been working on construction of the physically motivated Galactic dust SED models both for intensity and polarization from mm to FIR wavebands based on the physical properties of the amorphous material. Anomalous temperature dependences of heat capacity and heat conductivity of amorphous solids appeared in very low temperature environment support that the low temperature material physics of the amorphous is well described by the two-level systems (TLS). The complex dielectric constants and heat capacities of the amorphous dust grains are modeled by the combination of the TLS model and the crystal dust grain. Dust SEDs are calculated with our own Monte Carlo simulation code to follow thermal history of dust grain in which the physical characteristics of amorphous material is taken into account. We have constructed the scheme how to constrain the physical parameters of the amorphous dust grains by fitting both intensity and polarization observational SEDs in mm and submm wavebands. Our method is applied to intensity and polarization data of M31 obtained by Planck as for testing our scheme. We report these results.
Speaker: Masashi Nashimoto
127_nashimoto.pdf
The 30 micron sources in galaxies with different metallicities
A broad emission dust feature peaking around 30 microns is seen in the spectra of some carbon-rich AGB stars, post AGBs, and PNe. Since the discovery by Forrest et al. (1981) this dust feature has been detected in plenty of carbon-rich objects. Magnesium sulphide (MgS) is now the most favoured candidate to be the carrier of this spectral feature, but its identification remains a matter of some debate.
Hony et. al (2002) have analysed in a uniform way a sample of 63 Galactic 30 micron sources observed by the Infrared Space Observatory (ISO). The Spitzer Space Telescope was able to detect distant sources in our Galaxy, but also in nearby galaxies such as the Small Magellanic Cloud, Large Magellanic Cloud (LMC), and the Sagittarius Dwarf Spheroidal galaxy. All of them are characterised by the different average metallicities.
We have analysed a sample of 207 Spitzer spectra from galaxies mentioned above showing the 30 micron dust feature, and we investigated if the formation of the 30 micron feature carrier may be a function of the metallicity. We obtained and compared the basic properties of the 30 micron feature in those galaxies such as a few colour indices (e.g. [6.4]-[9.3] colour), dust temperature (T), strength of the feature, central wavelength, and finally the profiles of the feature.
Our analysis has shown that the strength of the 30 micron feature is the highest among Galactic objects. Moreover, this feature show up at the highest dust temperature for the Galactic AGB stars. The first AGB objects with this feature in the LMC are visible below 900 K, whereas such objects in the SMC and Sgr dSph objects do not appear until T drops below 700 K. The strength of the feature increases until T drops to about 400 K, and then decreases to finally show again variety of values for post-AGB objects and PNe. The AGB objects with T < 400 K, seems to experience very large mass loss rates, which may be responsible for the drop in the strength of the 30 micron feature due to self-absorption. During post-AGB and PNe phase, the strength of the 30 micron feature seems to not depend on the metallicity of galaxy, but the Galactic post-AGB objects and PNe show rather smaller values of T.
Our analysis of central wavelengths of the 30 micron feature show that it is rather independent of T for AGB and post-AGB objects. In a case of PNe, the central wavelength is clearly shifted toward longer values. Hony et al. (2002) suggested that such shift is caused by the low temperature of the feature carrier, or change in shape of dust particles. However, we noticed several post-AGB objects with similar T, but much lower central wavelength than in planetary nebulae. Therefore, we expect that dust processing (due e.g. irradiation by UV photons from central stars of planetary nebulae) may be more important in shifting central wavelength of the 30 micron feature towards larger values. Finally, it seems that the Galactic PNe have rather smaller central wavelength than their Magellanic Clouds counterparts. If our suggestions are correct, this could mean that processing of the 30 micron feature carrier is more efficient in the MCs than in the Milky Way.
We have also searched for the median profiles of the 30 micron feature in different galaxies and/or dust temperature. We have shown that shapes of the 30 micron feature does not change as a function of metallicity in different galaxies (AGB and post-AGB objects) or dust temperature (AGB stars only). On the other hand, the shape of the feature in planetary nebulae is clearly different (not only central wavelength) than shape of the feature in AGB or post-AGB objects.
Speaker: Mr Marcin Gladkowski (Nicolaus Copernicus Astronomical Center of the Polish Academy of Sciences)
Gładkowski.pdf
The evolution of dust formation in SN2005ip from optical line profile models
The source of the large masses of dust observed in some very early Universe galaxies at redshifts z $>$ 6 has been much debated. Core-collapse supernovae (CCSNe) have been predicted to be efficient producers of dust but the majority have only had small masses of warm dust ($<$ 10$^{-3}$ M${_\odot}$) detected in their ejecta during their early phases (t $<$ 3 years), based on fits to their near-IR and mid-IR SEDs. However, observations in the far-IR by Herschel and ALMA of a few CCSNe have yielded far higher cold dust masses (0.1 - 1.0 M${_\odot}$), which, if representative of the wider CCSN population, could potentially account for the dust masses seen in the early Universe. Unfortunately, there are now few instruments capable of detecting CCSN dust emission outside the local group at far-IR and sub-mm wavelengths, so other techniques must be exploited.
The late-time optical and near-IR line profiles of many CCSNe exhibit a red-blue asymmetry caused by red-shifted emission from the receding parts of the ejecta, which must traverse the dusty interior of the ejecta, experiencing greater extinction than the blue-shifted emission. I present Monte Carlo line transfer models of asymmetric optical line profiles of the interesting Type IIn interacting supernova SN2005ip from $\sim$40 d post-discovery to $\sim$4000 d. Dust has been predicted to form in two phases in this object, first in the ejecta and also in the post-shock region that develops following interaction of the ejecta with a dense circumstellar medium (Smith et al. 2009). I present models of the progressively blueshifted, broad H$\alpha$ line that arises in the ejecta at early times ($<200$ d) along with models of the evolving intermediate width H$\alpha$ and HeI $\lambda$7065A lines that arise in a post-shock region at later times ($\sim$400 d - $\sim$4000 d). I determine the dust masses that have formed and discuss the location of the dust and the clumpy structure of the post-shock region. I compare my SN2005ip dust mass estimates to dust mass estimates for other CCSNe that have been derived from both optical line profile modelling and SED fitting, and consider the evolution of dust formation in these objects.
Speaker: Antonia Bevan (UCL)
AntoniaBevan_2005ip.pdf
Whipping IC63/IC59
The mid-IR spectra of photodissociation regions (PDRs) are dominated by the well-known emission features at 3.3, 6.2, 7.7, 11.3, and 12.7 micron, generally attributed to polycyclic aromatic hydrocarbon molecules (PAHs). PAHs drive much of the physics and the chemistry in these PDRs, e.g. by heating the gas and as a catalyst in the formation of molecular hydrogen on their surfaces. Thus, PAHs and PDRs are intimately connected, and a complete knowledge of PDRs requires a good understanding of the properties of the PAH population and vice-versa, a complete knowledge of the PAH population requires a good understanding of the local physical conditions.
Here we present a general description of two PDRs, IC63 and IC59, from an observational standpoint in order to study the physical conditions at the UV-illuminated surfaces of these objects and their PAH properties. IC63 and IC59 are a pair of cometary-shaped nebulae in the vicinity of the star gamma Cas (also known as Tsih, "the Whip"). Both nebulae have very different optical appearances, despite the fact that both objects lie at similar projected distances from the star: IC63 shows bright rims and filaments, while IC59 looks more homogeneous and faint.
We use the available data on both nebulae taken with Spitzer, Herschel and SOFIA to study the infrared emission at the tip of both clouds, and derive the intensity of the UV radiation field, the density and the gas temperature. We find that the IR emission from polycyclic aromatic hydrocarbons (PAHs) is very similar at the tip of both nebulae. Even though it varies in intensity between the two, the derived PAH band ratios are remarkably similar. These ratios are similar to those found in the more shielded regions of other nebulae such as NGC7023 and NGC2023. Regarding the physical conditions, we obtain that while in IC63 the intensity of the UV field, G0, is a factor of ~10 higher than in IC59, the density n at the tip of IC59 is lower than in IC63 by a similar factor. For both objects we derive G0 values significantly lower than what previous works have so far assumed. Comparison with other reflection nebulae PDRs and known correlations support our claim that both IC63 and IC59 are low-UV irradiated environments. We conclude that the tips of IC63 and IC59 are about 3 and 5 times farther away from the star than their respective projected distances. The similarity of the mid-infrared emission between the two nebulae is consistent not only with both objects being overdensities within the same region around gamma Cas, but it is also consistent with the similar G0/n ratio and ionization parameters, which altogether rule the evolution of the hydrogenation and ionization level of the emitting population of PAHs. Finally, regarding the kinematics of the material in IC59, we find evidence of photo-evaporation due to the incident radiation from gamma Cas.
Speaker: Prof. Els Peeters (University of Western Ontario & SETI Institute)
poster_135_peeters.pdf
poster_135_peeters.ppt
Steep extinction curves in GRBs and quasars
One of the main tools to study dust grain properties is to measure the extinction curves in sightlines toward stars in the Local Group or extragalactic lighthouses such as quasars, gamma-ray bursts and supernovae. Typically, the extinction curves seen in the extragalactic, interstellar medium can be well-described by extinction curves similar to those observed in the Small and Large Magellanic Clouds and in the Milky Way. Toward a few sources, however, a much steeper extinction curve have been derived most notably that of GRB 140506A which will be the focus of this talk. I will show, using a general parametrization of the extinction, how the reddening in this case compare to those observed in the Local Group and argue that the origin of this can not be reproduced assuming the local reddening laws. I will also show how the global extinction of the host galaxy follow the presciption of a typical Calzetti extinction curve, commonly found to describe star-forming galaxies. The conclusion is then that the steep extinction must be imprinted only on very local scales from the circumburst medium. It is puzzling that the evidence point towards the scenario of a local effect only, since a similar steep reddening is observed in quasars in which the emitting region is in the centre of the host galaxy, where the observed extinction would probe the global galaxy system.
Speaker: Mr Kasper Heintz (University of Iceland)
GRB140506Apresentation.pdf
The AGN torus as a dynamical dusty wind
High resolution interferometric observations of infrared emission from dust in the immediate environment of AGNs reveal that the warm dust is extended in the polar directions. This suggests a scenario where warm dust is raised above the plane of the AGN through a radiation-pressure driven wind. We have produced a 3D radiation hydrodynamic model including self-gravity effects, and radiation pressure from the central source, with the goal of explaining the features of the dusty wind, as well as the observed emission and obscuration properties. We pre-calculate the heating, cooling, radiation pressure, and sublimation of dust grains for an assumed dust population, and include these effects in the dynamical model. We will present the results of these simulations, commenting on what physical processes are required to accurately model the observations, and examining how our simulations compare to other recent models.
Speaker: David Williamson (University of Southampton)
djwilliamson poster promotion2.pdf
Dust in the solar system: Dust in the Solar System Main Auditorium
Convener: Prof. Susanne Aalto (Chalmers University of Technology)
Analytical laboratory studies of solar system dust
Contemporary solar system dust is collected as interplanetary dust particles (IDPs) and micrometeorites (MMs) in the stratosphere using aircraft, and at polar and mid-latitude ground locations using several sampling methods. The collections are fundamentally important to cosmochemistry, planetary science and early solar system accretional processes in particular because some IDPs and MMs are from parent bodies, including comets and other small, icy bodies, that are not well represented among known meteorite groups [1]. Small bodies are more likely to contain well-preserved solids from the outer solar nebula and presolar environments. The miniscule masses of individual IDPs and MMs have severely limited analytical measurements but advances in instrumentation are expanding the scope of measurements and providing fundamental insight. It is well established that IDPs and MMs contain the highest abundances of presolar refractory constituents (e.g. crystalline oxides and silicates) and recent studies have identified a population of non-refractory "soft" constituents [2]. All IDPs and MMs are pulse heated, many to incandescence, for several seconds during atmospheric entry followed by exposure to terrestrial contamination. A key finding and challenge arising from the study of "soft" constituents are that well-preserved IDPs and MMs appear to be much rarer than has previously been assumed.
1. S. Taylor et al. (2016). Elements, 12(3).
2. H. A. Ishii, et al (2018), Proc. Natl. Acad. Sci., in press.
Speaker: Dr John Bradley (University of Hawaii)
Christine Floss. In memoriam
A brief remembrance of Prof. Christine Floss, planetary scientist at Washington University in St. Louis.
Panel Discussion Main Auditorium
Panel discussion on something
Speaker: Anja C. Andersen (Niels Bohr Institute)
Conference Dinner Festsalen (University of Copenhagen)
Frue Plads 4, 1168 København K
Grain growth, planet formation and debris disks: Grain growth, planet formation, and debris disks Main Auditorium
Convener: Troels Haugbølle (Niels Bohr Institute)
Planet Formation, grain growth and debris disks: Theory
The evolution of the solid particle component of protoplanetary disks from the formation of the disk into the debris disk phase holds important keys about the planet formation process. I will cover the some aspects of the basic physics and recent developments in the theory of dust growth, dust dynamics, planet formation and debris disks.
Speaker: Prof. Carsten Dominik (University of Amsterdam)
CPHDUST2018_Dominik.key
CPHDUST2018_Dominik.key.pdf
Grain growth, planet formation and debris disks
Speaker: Dr Zahed Wahhaj (ESO)
Zahed_wahhaj_review.pptx
PDRs with JWST: Probes of dust formation and evolution
Photodissociation regions (PDRs) are predominantly neutral regions of the ISM in which
the heating and chemistry are mainly regulated by far ultraviolet photons emitted from
one or more nearby young stars. They are extended regions at the interface between
the ionizing sources and molecular clouds, and contain dense structures and clumps of
dust and gas immersed in a more diffuse medium. Dust at the PDR interface experiences
extreme physical conditions, with temperatures and densities varying by orders of
magnitude over very small spatial scales, of order a few hundred AU. Hence the PDR
interface provides a unique opportunity to study (1) dust formation as a function of
environment, from the ionized region in front of the PDR to the dense regions behind the PDR
(2) the destruction and evolution of grain mantles/clusters in the transition region,
(3) the role of dust in regulating molecular chemistry (e.g. H2 formation on grain surfaces),
and (4) the potential identification of grain composition via excitation studies.
In light of the potential impact PDRs have on our understanding of dust properties and their
interdependence with the gaseous and molecular phase, the JWST NIRCam and MIRI GTO teams have
proposed a joint GTO program to study two nearby PDRs, NGC 7023 and the Horsehead nebula,
using a suite of instruments and modes on JWST. These emblematic PDRs have different
excitation conditions and relatively simple geometries and are ideal targets to take full
advantage of the high spatial resolution and sensitivity of JWST. In this poster, we
describe the observing strategy for our GTO program and briefly describe several of the
science goals of the team.
Speaker: Karl Misselt (University of Arizona)
Probing the solar accretion disk using the properties of dust filtering at gaps in the early Solar System
During the formation of the Solar System, Jupiter and Saturn played an important role in modulating and controlling the dust dynamics through the formation of gaps in the protosolar accretion disk that acted as dust traps. This is reflected in the distribution of chondrules and calcium-aluminum inclusions (CAIs). CAIs are almost exclusively present in chondrites arriving from the outer Solar System, and there are clear isotopic finger prints showing that while inner Solar System chondrules where transported to the outer Solar System, no outer Solar System chondrules returned to the inner Solar System. A dust trap can only stop particles above a certain size, while small particles are well coupled and flows through the gap with the gas. To investigate the roles of Jupiter and Saturn we combine a large suite of numerical models of the protocolar accretion disk with embedded planets with a systematic cosmochemical search for CAIs in inner Solar System chondrite slabs. This allow us to put new limits on the surface density of the accretion disk where Jupiter formed, the relative sizes of the dust reservoir in the inner and outer Solar System, the probable orbital geometry of the gas giants in the early solar system, and inform us about the recycling of material in the formation region of ordinary chondrites.
Speaker: Troels Haugbølle (Centre for Star and Planet, Niels Bohr Institute)
Poster157_Troels_Haugboelle.pdf
The Benchmark Dust Mass Function of the Nearby Universe
We present a fundamental measure of the dust content of nearby galaxies - the Dust Mass Function (DMF) for the largest sample of galaxies to date. Our DMF is drawn from a stellar mass selected sample of galaxies comprised of the overlap between two large area surveys - the Galaxy And Mass Assembly (GAMA), and the Herschel Astrophysical Terahertz Large Area Survey (H-ATLAS). The overlap between these surveys spans ~140 square degrees, and 21 wavebands, containing over 15,000 galaxies below redshift 0.1 that are observable in the r-band. This study is the most statistically robust measurement of the low-redshift DMF ever made, allowing us to probe at least an order of magnitude lower in dust mass than any survey before for a sample ~70 times larger than previous surveys. We compare to literature and to theoretical predictions of DMFs derived from semi-analytic dust evolution models or hydrodynamical cosmological simulations. We also calculate the dust mass function for different morphological types and find scaling relations between our DMFs and their corresponding galaxy stellar mass functions (GSMF) for the same sample.
Speaker: Rosie Beeston (Cardiff University)
Beeston_CPHdust.pdf
Beeston_CPHdust.pptx
ALMA's View of Dust in SN 1987A
SN 1987A, being relatively young as well as the brightest supernova observed in over 400 years, is a unique and exciting laboratory for studying supernova dust production. Located around 50kpc away in the Large Magellanic Cloud, SN 1987A is too far away for single-dish telescopes to resolve the structure of the sub-mm emission on the scale of the ejecta, where the dust is produced. Recent ALMA observations have allowed us to peer into the inner ejecta to the cool dust, with resolution probing down to physical scales of 4500 AU. Comparison of the dust location and morphology with other multi-wavelength emission presents an interesting picture of the role dust plays in the ejecta. The distributions of the dust continuum and molecular line emission are all notably complex, having implications for the physical properties of the system.
Speaker: Dr Phil Cigan (Cardiff University)
PosterSlide_PJC.pdf
Mineralogical Studies of Silicate Stardust in the Laboratory
Silicate dust is pervasive throughout the cosmos and has been observed in interstellar space, around evolved oxygen-rich stars, protoplanetary disks, and in our Solar System. The chemical and physical properties of this dust have traditionally been inferred through remote astronomical observations. Spectral observations of circumstellar dust indicate mainly amorphous, Fe-bearing grains having non-stoichiometric compositions intermediate between olivine and pyroxene compositions (Molster and Kemper, 2005; Tielens et al., 1998; Sargent et al., 2010), and variable proportions of extremely Mg-rich crystalline pyroxene and olivine (de Vries et al., 2010; Molster et al., 2002). Spectral data for dust originating from supernovae and novae are scarce and suggest that these sources mainly produce Mg-rich amorphous silicates (Arendt et al., 2014; Evans et al., 1997). Crystalline silicates are rare in the diffuse ISM (<1%; Kemper et al., 2004), most likely due to grain amorphization and destruction in the ISM (Jones and Nuth, 2011).
The discovery of preserved silicate stardust grains in meteorites, interplanetary dust particles (IDPs), and dust returned from comet Wild 2 by NASA's Stardust mission has allowed for direct, detailed study of individual grains of silicate stardust in the laboratory. This has essentially opened new windows into the fields of astrophysics and astromineralogy. The exotic isotopic compositions of these silicate stardust grains, determined by isotopic mapping with the NanoSIMS ion microprobe, reflect origins in asymptotic giant branch stars, supernovae, and novae. We have performed coordinated isotopic, chemical and mineralogical characterization of these ~100–500 nm-sized grains by NanoSIMS and transmission electron microscopy (TEM) analyses. Microtome sections of IDPs are first analyzed by TEM and isotopically anomalous grains are subsequently identified by NanoSIMS isotopic mapping. Stardust grains in meteorites are first identified by NanoSIMS analysis, and grain cross-sections are then prepared by focused ion beam (FIB) milling for TEM analysis.
Our studies show that the majority of the silicate grains are amorphous with non-stoichiometric Fe-bearing chemical compositions, generally consistent with astronomical observations (Nguyen et al., 2016). However, approximately 1/3 of the analyzed grains were found to be crystalline pyroxene and olivine, considerably higher than the crystalline silicate fraction in the ISM. While the pyroxene grains are Mg-rich, the olivine grains have more substantial Fe-contents, also in contradiction with astronomical observations. Thus far, we do not observe any systematic mineralogical differences among grains from different stellar sources. Our studies have uncovered details of circumstellar grain properties that cannot be seen remotely. For example, "compound" grains have been identified, including an amorphous Fe-rich silicate with olivine and pyroxene inclusions and a crystalline spinel grain encased in amorphous silicate glass (Nguyen et al., 2017; Nguyen et al., 2014). Laboratory studies of interstellar silicates have yet to show evidence of organic mantles predicted by some models of interstellar dust lifecycles (Greenberg and Li, 1997). Rare evidence for amorphization in space was also observed in two silicate stardust grains (Nguyen et al. 2016). A chemically uniform grain with a composition of the mineral enstatite is mostly amorphous but retains a crystalline core. The grain likely condensed as a single, solid crystal, but the outer portions were later amorphized in the ISM. An amorphous supernova grain having the chemical composition of enstatite most likely condensed as a crystal and was later rendered amorphous. The laboratory analysis of silicate stardust grains complements astronomical observations and provides an extraordinarily detailed look into the chemical makeup, structure, and lifecycle of silicate dust in the Galaxy.
Arendt R. G., Dwek E., Kober G., Rho J., and Hwang U. (2014) Interstellar and ejecta dust in the CAS A supernova remnant. Astrophys. J. 786, 55-76.
de Vries B. L., Min M., Waters L. B. F. M., Blommaert J. A. D. L., and Kemper F. (2010) Determining the forsterite abundance of the dust around asymptotic giant branch stars. Astron. Astrophys. 516, A86-A95.
Evans A., Geballe T. R., Rawlings J. M. C., Eyres S. P. S., and Davies J. K. (1997) Infrared spectroscopy of Nova Cassiopeiae 1993 - II. Evolution of the dust. Mon. Not. R. Astron. Soc. 292, 192-204.
Greenberg J. M. and Li A. (1997) Silicate core-organic refractory mantle particles as interstellar dust and as aggregated in comets and stellar disks. Adv. Space Res. 19, 981-990
Jones A. P. and Nuth J. A. I. (2011) Dust destruction in the ISM: a re-evaluation of dust lifetimes. Astron. Astrophys. 530, A44-A55.
Kemper F., Vriend W. J., and Tielens A. G. G. M. (2004) The absence of crystalline silicates in the diffuse interstellar medium. Astrophys. J. 609, 826-837.
Molster F. and Kemper C. (2005) Crystalline silicates. Space Sci. Rev. 119, 3-28.
Molster F. J., Waters L. B. F. M., Tielens A. G. G. M., Koike C., and Chihara H. (2002) Crystalline silicate dust around evolved stars. III. A correlations study of crystalline silicate features. Astron. & Astrophys. 382, 241-255.
Nguyen A. N., Keller L. P., and Messenger S. (2016) Mineralogy of presolar silicate and oxide grains of diverse stellar origins. Astrophys. J. 818(1), 51-67.
Nguyen A. N., Keller L. P., Messenger S., and Rahman Z. (2017) Mineralogical Characterization of Fe-Bearing AGB and Supernova Silicate Grains from the Queen Alexandra Range 99177 Meteorite. Lunar & Planetary Science 48, Abstract #2371.
Nguyen A. N., Nakamura-Messenger K., Messenger S., Keller L. P., and Klock W. (2014) Identification of a compound spinel and silicate presolar grain in a chondritic interplanetary dust particle. Lunar & Planetary Science 45, Abstract #2351.
Sargent B. A., Srinivasan S., Meixner M., et al. (2010) The Mass-loss Return from Evolved Stars to the Large Magellanic Cloud. II. Dust Properties for Oxygen-rich Asymptotic Giant Branch Stars. Astrophys. J. 716, 878-890.
Tielens A. G. G. M., Waters L. B. F. M., Molster F. J., and Justtanont K. (1998) Circumstellar silicate mineralogy. Astrophys. & Space Sci. 255, 415-426.
Poster presentation_Nguyen.pptx
Near- and Mid-Infrared Interstellar Dust Extinction Observations
The interstellar dust extinction in the near- and mid-infrared (IR)
wavelength range (1-40 microns) is characterized by decreasing
continuum extinction and four main absorption features that are
diagnostic of dust grain compositions. The absorption features at 10
and 18 micron are due to silicate material, at 3.4 micron due to
hydrogenated carbon material, and at 3.0 micron due to water ice.
Measurements based on Spitzer spectroscopic observations from 5-40
micron provide a detailed view of the continuum extinction and
silicate absorption features in sightlines with A(V) values from 1.5
to 5.5 mag. As these sightlines all have existing ultraviolet
extinction measurements, this sample provides consistent measurements
of extinction from 0.1 to 40 microns giving strong, consistent
constraints on dust grain sizes and compositions. Plans for expanding
this work with JWST to include spectroscopic measurements in the
near-IR region (1-5 micron) and more sightlines are presented.
Speaker: Dr Karl Gordon (Space Telescope Science Institute)
irext_obs_copenhagen_kgordon_1slide.pdf
Aliphatic Features in Mid-Infrared Polycyclic Aromatic Hydrocarbon Spectra
The mid-IR spectra of almost all objects are dominated by strong emission bands at 3.3, 6.2, 7.7, 8.6, and 11.3 micron due to Polycyclic Aromatic Hydrocarbon molecules (PAHs). It is now well established that these mid-IR bands show clear variations in shape and peak position from one point source to another, as well as varying spatially within extended sources. The spectral diversity of the PAH band profiles reveals the nature of the carriers and hence allows one to study their formation and evolution throughout their life cycle. Although the origin of the profile variations is still under debate, one posited explanation is the varying importance of aliphatics versus aromatics in the carrier molecules.
We present the 5-12 micron spectra of sixty-three astronomical sources exhibiting PAH emission bands observed by ISO/SWS, Spitzer/IRS, and SOFIA/FORCAST. We aim to test this hypothesis by quantifying the aliphatic emission and identifying relationships between the aliphatic and aromatic emission features. We find that the presence of aliphatic features depends on PAH class, with aliphatic features detected in all class D sources, approximately half of the class B sources, and no class C sources present in our sample. We observe spectral variation of these aliphatic features in peak position and intensity. The peak position of the 6.9 micron feature varies continuously between 6.8 and 6.95 micron, with the variation being more pronounced in class B sources than in class D sources. In addition, the 6.9 micron feature is strongest in class D sources, in some cases being stronger than the 6.2 micron feature. Finally, our investigation of possible correlations between aliphatic and aromatic emission features only reveals a correlation between i) the two aliphatic bands, and ii) the aliphatic features and the 11.2 micron PAH band. We discuss these results within the framework of the varying aliphatic to aromatic ratio as the origin of the band profile variations.
Comparison of the extraplanar H𝛼 and UV emissions in the halos of nearby edge-on spiral galaxies
We compare vertical profiles of the extraplanar H𝛼 emission to those of the UV emission for 38 nearby edge-on late-type galaxies. It is found that detection of the "diffuse" extraplanar dust (eDust), traced by the vertically extended, scattered UV starlight, always coincides with the presence of the extraplanar H𝛼 emission. A strong correlation between the scale heights of the extraplanar H𝛼 and UV emissions is also found; the scale height at H𝛼 is found to be ~0.74 of the scale height at FUV. Our results may indicate the multiphase nature of the diffuse ionized gas and dust in the galactic halos. The existence of eDust in galaxies where the extraplanar H𝛼 emission is detected suggests that a larger portion of the extraplanar H𝛼 emission than that predicted in previous studies may be caused by H𝛼 photons that originate from H II regions in the galactic plane and are subsequently scattered by the eDust. This possibility raise a in studing the eDIG. We also find that the scale heights of the extraplanar emissions normalized to the galaxy size correlate well with the star formation rate surface density of the galaxies. The properties of eDust in our galaxies is on a continuation line of that found through previous observations of the extraplanar polycyclic aromatic hydrocarbons emission in more active galaxies known to have galactic winds.
Speaker: Young-Soo Jo (Korea Astronomy and Space Science Institute)
2018_cphdust_JYS_1slide.pdf
Evidence of pyroxene magic nanoclusters in protoplanetary disks around Herbig Ae stars from first principles calculations
From infrared (IR) observations it is known that magnesium silicates are the major dust component in protoplanetary disks. From such observations one can discern the likely size, composition and crystallinity of such dust thus allowing one to infer the likely associated properties of the disk. Of special interest are the disks surrounding Herbig Ae/Be (HAeBe) stars corresponding to the stellar evolutionary stage where embryonic planets should be forming.
Using Spitzer IR data, Juhasz et al. [1] derived the averaged mass absorption coefficient (MAC) of the crystalline pyroxene component from a selected set of HAeBe stars and compared it with experimentally derived MACs of different bulk crystalline pyroxene samples. The star-derived MAC was found to contain signals in regions that do not appear in the experimental crystalline pyroxene MACs (e.g. a sharp feature at 8.7 microns and a broad region between 14.4 - 16.4 microns). Using accurate first principle electronic structure calculations, we provide evidence that the spectral mismatches between experimental and observationally derived MACs for crystalline pyroxene could be explained by the presence of pyroxene nanodust.
Using dedicated global optimisation search algorithms we have established the most stable atomic scale structures of a range of pyroxene nanoclusters (MgSiO3)N for N = 1 - 10. Within this set we identified those clusters with particularly high energetic stability with respect to other clusters of a similar size. Such so-called "magic" clusters are known from cluster beam experiments to have disproportionally high abundances in cluster populations. Correspondingly, one should expect that, if nano-pyroxene dust forms a reasonable proportion of the pyroxene silicate dust budget in protoplanetary disks, then magic pyroxene clusters may be particularly abundant. We found that the pyroxene nanosilicate cluster (MgSiO3)5 was indeed a magic cluster and thus a possible abundant dust candidate. Using first principles density functional theory (DFT) based electronic structure calculations we then calculated the accurate IR spectrum of the magic (MgSiO3)5 cluster. The calculated IR spectrum of the magic pyroxene cluster exhibits a number of features at wavelengths in very good agreement with the observationally derived average MAC for crystalline pyroxene, which, at the same time, cannot be explained using experimental MACs from bulk crystalline pyroxene samples. We interpret this finding as strong evidence for the existence of this particular nanodust particle around HAeBe stars. The implications of this result with respect to the properties of the associated protoplanetary disks are discussed.
[1] A. Juhász et al., "Dust evolution in protoplanetary disks around Herbig Ae/Be stars-the Spitzer view," Astrophys. J. 721, 431 (2010).
Speaker: Prof. Stefan Bromley (University of Barcelona / ICREA)
Poster_170_slide_Bromley.pdf
High-resolution, 3D radiative transfer modeling of barred galaxies M83 and NGC1365
Within the framework of the DustPedia project we study the effect of cosmic dust on a vast sample of nearby galaxies. Dust radiative transfer (RT) simulations provide us with the unique opportunity to study the heating mechanisms of dust by the stellar radiation field. From 2D FITS images we were able to derive the 3D geometry distributions of stars, a technique, first introduced by De Looze et al. (2014) and followed afterwards by Viaene et al. (2016). This powerful method allows a more realistic description of the complex stellar geometries found in galaxies like asymmetric features or clumpy structures.
Our aim is to analyze the contribution of the different stellar populations (old, young & ionizing) to the radiative dust heating processes in the nearby face-on barred galaxies NGC1365 and M83, by using high resolution 3D radiative transfer modeling. To model the complex geometries mentioned above, we used SKIRT, a state-of-the-art, 3D Monte Carlo RT code designed to model the absorption, scattering and thermal re-emission of dust in a variety of environments.
Speaker: Mr Angelos Nersesian (IAASARS, National Observatory of Athens & Ghent University)
slide_nersesian_cph_rtmod.pdf
Revealing the dust grain sizes in the envelope of Per-emb-50
Disks and envelopes around protostars play a fundamental role in the process of planet formation, since they contain the ingredients that will form planets. However, it is not yet clear at which stage of the star and planet formation process dust grains start to efficiently coagulate and evolve from small solid particles to macroscopic dimensions.
We studied the Class I protostar, Per-emb-50, at 1.3mm with SMA and 2.7mm with NOEMA in order to determine the spectral index $\alpha_\mathrm{mm}$ in the envelope region on scales 400-3000 AU. The data analysis show a high value for $\alpha_\mathrm{1.3-2.7mm}$m, which implies that there is no evidence of mm-sized dust grains in the envelope. To understand the dust properties in more detail, we performed a radiative transfer modeling of the source and found a maximum grain size of a few hundred microns.
The current observations on Per-emb-50 confirm that there are no mm sized grains in the envelope, contrary to previous studies on similar sources where mm size grains have been found. This would imply that the grain growth on YSO's is highly affected by the environment and dynamica history of the source.
Speaker: Carolina Agurto (Max Planck Institute for Extraterrestrial Physics)
slide_carolina_agurto.pdf
Temperature programed desorption of water ice mixed with amorphous carbon and silicate grains
The desorption of molecular ices from grain surfaces is important in a number of astrophysical environments including dense molecular clouds, circumstellar regions, cometary nuclei, and surfaces and atmospheres of planets. It has been shown that for multicomponent ices within water ice matrices, the desorption of all species in the ice is controlled by the behaviour of water. A study of the desorption of H$_2$O ice mixed with dust is therefore crucial for our understanding of the ice-dust interaction and in turn, the physics of cold dense molecular clouds and circumstellar disks, where the relatively high dust density allows coagulation of dust grains, which can be catalysed by water. Measurements of ice-dust interactions will further aid our understanding of the structure and morphology of dust aggregates at different phases of the ISM.
In our experiments, for the first time, temperature programmed desorption (TPD) of water ice mixed with amorphous carbon and silicate grains has been studied in the laboratory. Grain/ice mixtures represent laboratory analogues of interstellar and circumstellar icy dust grains. We show that variations of the grain/ice mass ratio lead to a transformation of the TPD curve of H$_2$O ice, which can be perfectly fitted with the Polanyi-Wigner equation by using fractional desorption orders. For carbon grains the desorption order of H$_2$O ice increases from 0.1 for pure H$_2$O ice to 1 for the grain/ice ratio of 1.3. For two silicate/ice mixtures, the desorption order of H$_2$O ice is 1. This is a unique result, which have not been obtained in previous experiments on the thermal desorption of ices from carbon and silicate surfaces. It can be explained by the desorption of water molecules from a large surface of fractal clusters composed of carbon or silicate grains and provides a link between the structure and morphology of dust grains and the kinetics of desorption of water ice mixed with these grains.
poster_presentation.pdf
From grain growth to astromineralogy: Studying dust with X-ray imaging and spectroscopy
X-ray imaging and spectroscopy can provide a powerful tool for measuring the large end of the dust grain size distribution — important for interpretting infrared extinction as well as understanding grain growth in the diffuse interstellar medium (ISM). In addition, X-ray photoelectric absorption edges observed in high resolution spectra of Galactic X-ray binaries directly reveal the mineral composition of interstellar dust. I will review open problems in the field of astromineralogy, as probed by X-ray extinction. I will describe how observations from the next two X-ray missions — XARM and ARCUS — will answer some of those questions. Finally, I will discuss synergistic opportunities for X-ray telescopes and JWST to provide a more complete picture of dust grain evolution in the diffuse ISM.
Speaker: Lia Corrales (University of Wisconsin - Madison)
Corrales_CosmicDust2018_1slide.pdf
Spectroscopic Characterization of Interstellar Ice Analogues
The experimental setup was developed with the goal to characterize the optical properties of astrophysically relevant solids (ice mixtures, silicates, and carbonaceous materials). For the experiments, we use a combination of spectral techniques to obtain information regarding their chemical, physical, and optical properties. When analysed, together with astronomical observations and theoretical models we will better understand the roles of solid materials in different environments.
Speaker: Birgitta Mueller (Max Planck Institute for extraterrestrial Physics)
CAS_B_Mueller_Poster_Presentation.pdf
A multi-wavelength view of planet forming discs: unleashing the full power of ALMA for grain growth studies
Observations at sub-mm/mm wavelengths allow us to probe the solids in the interior of protoplanetary discs, where the bulk of the dust is located and planet formation is expected to occur. However, the actual size of dust grains and the physical properties of the disc interior are still largely unknown due to the observational limits of past sub-mm/mm studies. ALMA, thanks to its exquisite resolution and sensitivity, is an unprecedented tool to study grain growth in large samples of discs.
In my contribution I will present a novel analysis method that constrains the radial profile of the maximum grain size in protoplanetary discs by means of a simultaneous fit of spatially resolved observations at several sub-mm/mm wavelengths (Tazzari et al. 2016, A&A 588, A53). By breaking the degeneracy between the opacity, temperature and density contributions to the sub-mm emission, this method enabled us to find observational evidence of a radial sorting of grain sizes in a few discs, an effect that is expected from dust evolution models including grain growth and radial drift.
I will also present new ALMA 3 mm observations of 35 discs in the Lupus star forming region (Tazzari et al., in prep) and the results of the coupled analysis with our previous 890 𝜇m ALMA survey of the same discs (Ansdell et al. 2016, ApJ 828 46; Tazzari et al. 2017, A&A 606 A88). I will discuss the grain sizes inferred for such homogeneous sample of discs in comparison with the level of grain growth in other regions with different mean ages.
Finally, after characterising how the contamination from optically thick emission would affect grain size estimates, I will show that a minimum level of grain growth is always needed to account for the distribution of spectral indices and sub-mm fluxes that we currently observe.
Speaker: Marco Tazzari (University of Cambridge)
Dust as a tracer in the Milky Way and local galaxies Main Auditorium
Convener: Dr Karl Gordon (Space Telescope Science Institute)
The properties of interstellar dust in the Milky Way and in nearby galaxies
In this review, I will give an overview of the properties of dust (spatial distribution, composition, heating sources) in the ISM of nearby galaxies and in the Milky Way. I will discuss the main spectral energy distributions fitting techniques to model the ISM dust and their caveats. I will also discuss the need for dust models including an evolution of dust grains in the ISM to explain the current observables in emission or extinction.
Speaker: Dr Maud Galametz (CEA Saclay)
Galametz_Copenhagen2018_compressed.pdf
PAHs trace the molecular gas in star-forming galaxies
We present new CO(1-0) line observations of 34 infrared-selected PAH emitters at intermediate redshift (0.01≤ z ≤ 0.3) to investigate the connection among the PAH emission, the total gas content, and the star formation rate in normal and starbursting galaxies. Combined with observations from the literature at low and high redshift, our analysis reveals a universal, tight, and linear PAH-CO relation independent of redshift and star formation efficiency, suggesting that the emission from PAHs is strongly correlated with the cold diffuse gas. We also find a strong correlation between the PAHs and the cold dust emission, which is another reliable gas tracer independent of CO. Based on our results, we propose the use of PAHs as a proxy for the molecular gas content in star-forming galaxies at all redshifts. As PAHs will be routinely detected with the upcoming launch of JWST, they will serve as a useful tool to investigate the cold gas properties of high-z galaxies up to z~3.
Speaker: Isabella Cortzen
IsabellaCortzen_PAHsMolecularGas.pdf
Realistic modelling of nano-aggregated magnesium silicate dust particles using atomistic simulations.
Dust can be found everywhere in the universe from stars in the latest stages of evolution (post-AGB stars), to the interstellar medium (ISM) and protoplanetary disks around young stars (e.g. T Tauri and Herbig Ae/Be stars). Dust can provide a wealth of information about the medium it is embedded in, since processing will affect its composition, structure and size in a characteristic and distinct manner. The properties of dust can thus also be used to track the nature of its environment. For instance, the growth of dust particles in protoplanetary disks is linked to the evolution from flared to flattered disks, while the small crystalline fraction of silicates in the ISM indicates a high degree of energetic processing.
The source of data to understand dust formation, growth, composition and shape of the dust particles is infrared (IR) spectroscopy. Nevertheless, the information provided by IR spectroscopy from laboratory synthesized particles is not well understood, due to the difficulties to control and understand the generated particles at the nanoscale level. Knowledge at the nanoscale, such as cation mobility and disorder, can be helpful in order to understand properties of silicate materials such as the crystallization below the glass transition temperature.
Bottom-up atomistic computational modelling methods allow the study of particles with atomic-scale precision, but in order to perform such simulations we require realistic models of the particles of interest. Here we present an new approach that uses well-tailored interatomic potentials to simulate the growth of silicate nanoparticles with diameters up to of tens nanometres following the detailed circumstellar nucleation conditions typical for a post-AGB star. The simulations progressively and realistically add monomers (SiO, Mg and O) to a seed particle moving away from the star with a determined initial velocity.
We solve the equation of movement for the seed particle, and thus estimate how the typical circumstellar conditions for nucleation (e.g. temperature, pressure) change with time and thus distance from the star. From the generated atomistically detailed nanodust particle models, we can probe properties that are difficult to accurately extract from experiments such as surface structure, surface to volume ratios, degree of polymerization of silicate tetrahedra, nanoporosity and nanosized phase separation. Via analysing the vibrational atomic motions within the generated nanodust silicate particles we can also directly simulate the IR emission from such species. In this way we are able show how the IR spectra of silicate nanodust relates to changes in atomic disorder/crystallinity, chemical composition (e.g. pyroxene vs olivine) and nucleation conditions (e.g. temperature).
Speaker: Mr Antoni Macià (Universitat de Barcelona)
Macia.pdf
Nucleation1minute.mp4
Interstellar Catalysis of Molecular Hydrogen through Superhydrogenation of Polycyclic Aromatic Hydrocarbons
In the field of astrochemistry and surface science a primary objective is to identify and characterize effective catalysts which have a plausible existence in the interstellar medium (ISM). Molecular hydrogen ($H_2$) is the most abundant molecule in the ISM with well-established and efficient catalytic formation routes in many regions of interstellar space. However, in certain regions of the ISM, discrepancies between formation rates and dissociations rates have been found [1]. Specifically, in Photodissociation regions (PDRs), the dissociation rate is so high that prevailing formation routes may not be efficient enough to explain the observed abundancies of $H_2$ and needed formation rates. Here we examine a group of planar nanosized molecules called polycyclic aromatic hydrocarbons (PAHs) as possible catalysts of $H_2$ formation. Significant abundancies of PAHs have been observed to spatially overlap with regions of high $H_2$ formation rates [2].
We have used temperature programmed desorption (TPD) and scanning tunneling microscopy (STM) to examine a template PAH, coronene ($C_{24}H_{12}$). Density functional theory (DFT) reveals that coronene and possibly other PAHs will have 0eV energy barriers for both Hydrogen (H) addition and $H_2$ abstraction even at low super-hydrogenation degrees (4 adatoms) [3]. The state of hydrogenation will therefore depend on these competing processes and their relative cross sections for addition, $\sigma_{add}$, and abstraction, $\sigma_{abs}$.
Monolayers of neutral PAH molecules are deposited on a highly oriented pyrolytic graphite (HOPG) surface and exposed to different fluences of D or H atoms. The atomic beam is estimated to have a temperature of approximately 1000K. Fully deuterated coronene ($C_{24}D_{36}$) is observed, indicating formation of HD via an exchange process between H and D [4].
First addition cross sections, $\sigma_{add}$(0) for D/H addition can be found experimentally from the exponential decay in pristine coronene as a function of D/H fluence. Cross sections of $\sigma_{add,H}$(0) = $0.25 \pm {0.14\atop 0.05} $Å$^2$ for H addition and $\sigma_{add,D}$(0) = $0.065 \pm {0.10\atop 0.05} $Å$ ^2$ for D addition are best fits to data. Determination of sequential addition and abstraction cross sections, however, require further modelling and will here be determined through comparisons to Kinetic Monte-Carlo (KMC) simulations.
Also observed from TPD is indications of preferred D/H-PAH configurations with high stability and increased barriers against further D/H addition. Experimental data points towards a barrier preventing addition to the center ring. These barriers are not detected when a high temperature atomic beam is used (T≈2000K) [5].
[1] Tielens, A., Reviews of Modern Physics, 85 (2013) 1021-1081.
[2] Habart, E.; et. al., Astronomy and Astrophysics 397 (2003) 623-634.
[3] Rauls, E.; and Hornekær L., The Astrophysical Journal 679 (2008) 531.
[4] Mennella, Vito, et alThe Astrophysical Journal Letters 745.1 (2011): L2.
[5] Thrower, J. D., et al The Astrophysical Journal 752.1 (2012): 3.
Speaker: Mr Frederik Doktor S. Simonsen (Department of Physics and Astronomy, Aarhus University)
Frederik Doktor - 1min 1slide.pdf
Frederik Doktor - 1min 1slide.pptx
Investigation of the hydrogenation of pentacene
Carbonaceous surfaces are known to act as catalysts for the formation of molecular hydrogen in the interstellar medium [1,2]. Molecular hydrogen is the most abundant molecule in the universe and it controls the chemistry of the interstellar medium. Hence, the formation of molecular hydrogen is the first step in the evolution of the chemical complexity of the interstellar medium.
Observations correlate the abundance of polycyclic aromatic hydrocarbons (PAHs) to an increased rate of H$_2$ formation [3]. PAHs and HPAHs are closely linked to the carbonaceous grain population although PAH interaction with grains is still poorly understood. Experiments and theoretical calculations indicate that PAHs play a role as catalysts for H$_2$ formation [4] and addition of hydrogen to PAHs pushes the molecule away from a planar geometry [5]. Here,we investigate reactions between pentacene and atomic hydrogen. Our goal is to examine the carbon sites at which incoming H atoms are most likely to bind, and to study how the morphology and reactivity of the pentacene molecule changes as the degree of superhydrogenation increases.
A monolayer of pentacene was prepared under ultra-high vacuum conditions on a Au (111) surface and then exposed to a controlled fluence of hydrogen atoms. X-ray photoelectron spectroscopy was used to characterize the system by tracking chemical shifts in in the C1s and Au4f core levels.
After hydrogenation, a chemical shift was observed for the carbon core level electrons. This indicates that the sp$^2$ carbon sites change to a sp$^3$ hybridisation after hydrogen exposure. Furthermore, there is evidence for more than one type of sp$^3$ hybridised carbon site. Chemical shifts in the gold core levels were observed firstly after dosing of pentacene and again after hydrogen exposure. The latter could indicate Au-C bond formation simultaneous with new C-H formation on the pentacene molecule or a Au-H interaction, or both. Combining this data with STM images and DFT calculations will help us understand the dynamics of hydrogen addition to the pentacene molecule and elucidate a route towards superhydrogenation of the PAHs in the interstellar medium.
[1] V. Pirronello, C. Liu, et al., Astron. Astrophys. 344, 681–686 (1999).
[2] I. Alata, G. A. Cruz-Diaz, et al., Astron. Astrophys. 569, A119 (2014).
[3] E. Habart, F. Boulanger, et al., Astron. Astrophys. 397, 623-634 (2003).
[4] V. Mennella, L. Hornekær, et al., The Astrophysical Journal Letters, 745, L2 (2012).
[5] A. W. Skov, M. Andersen, et al., The Journal of Chemical Physics 145, 174708 (2016).
Speaker: Rijutha Jaganathan (Aarhus University)
1minutead_rijutha.pdf
Polycyclic Aromatic Hydrocarbon fraction at ~10 pc scale in the Magellanic Clouds
The spatial variations of dust properties within a galaxy and their
correlation with local environment provide critical insights into the life cycle of dust. Low metallicity galaxies, in particular, let one study the dust life cycle in environments relevant for galaxies earlier in the history of the Universe.
In this work, we present maps of the dust properties in the Small and Large Magellanic Clouds (SMC, LMC), two nearby, highly resolved, low metallicity galaxies, fit with the Draine & Li (2007; DL07) dust grain model. We use/Spitzer/and/Herschel/infrared observations of the clouds to derive the spatial distribution of the dust properties, in particular the abundance of the small carbonaceous grain (or polycyclic aromatic hydrocarbons; PAH)
component. Overall, the average PAH fraction is smaller in the SMC than in the LMC, which is lower than that of the Milky Way. In particular, we find an anti-correlation between the DL07 q_PAH fraction and the Hα intensity. This is an indication that the smallest dust grains could be destroyed in high-ionization regions. We provide maps of the q_PAH fraction at resolved
scales.
We also compare our final maps with previous modeling (LMC: Paradis et al. 2009; SMC: Sandstrom et al. 2010, using the DL07 model). This helps us identifying the most model-dependent dust properties, and how they vary with resolution and wavelength coverage. We use these results to constrain the drivers of the PAH lifecycle in low metallicity environments.
Chastenet_CPH_MCs_OneSlide.pdf
Superhydrogenation of PAHs through interaction with hydrogenated grain surfaces
The formation of molecular hydrogen, $\mathrm{H_2}$, in the ISM is thought to primarily occur on dust grain surfaces. This process has been investigated extensively through a variety of experimental and theoretical approaches for several different model grain surfaces. More recently it has been suggested that PAHs, representing the molecular limit of the carbonaceous grain population, can act as catalysts for $\mathrm{H_2}$ formation through the generation of superhydrogenated PAH, or HPAH species (Rauls & Hornekær 2008). In addition to interaction with gas-phase hydrogen atoms (Thrower et al. 2012), our laboratory measurements have shown that adsorption of PAHs on hydrogenated graphitic surfaces can also lead to the formation of HPAHs through the pick-up of adsorbed H-atoms that are bound to the graphite surface (Thrower et al. 2014). Whilst mass spectrometry provides evidence for the formation of HPAHs through this mechanism, the exact hydrogen adsorption sites occupied remains unclear. Scanning tunneling microscopy (STM) provides us with the ability to probe the adsorption and hydrogenation of PAHs at the micropscopic level. Using coronene $\mathrm{C_{24}H_{12}}$ as a prototypical PAH molecule, we show that a variety of hydrogenation structures are formed following adsorption on a hydrogenated HOPG surface. We also demonstrate how the surface temperature affects the hydrogenation process, which depends on the ability of the adsorbed coronene to scan the surface in order to pick up the adsorbed hydrogen atoms.
Rauls, E. & Hornekær, L., 2008, Astrophys. J., 679, 531.
Thrower, J. D., Jørgensen, B., Friis, E. E., Baouche S., Mennella, V., Andersen, M., Hammer, B., & Hornekær, L., 2012, Astrophys. J., 752, 3.
Thrower, J. D., Friis, E. E., Skov, A. L., Jørgensen, B. & Hornekær, L., 2014, Phys. Chem. Chem. Phys., 16 3381.
Speaker: John Thrower (Department of Physics & Astronomy, Aarhus University, Denmark)
Thrower_#173_FLASH_Presentation.pdf
Thrower_#173_FLASH_Presentation.pptx
The JWST-ERS program ID 1288: Radiative Feedback from Massive Stars as Traced by Multiband Imaging and Spectroscopic Mosaics
Massive stars disrupt their natal molecular cloud material by dissociating molecules, ionizing atoms and molecules, and heating the gas and dust. These processes drive the evolution of interstellar matter in our Galaxy and throughout the Universe from the era of vigorous star formation at redshifts of 1-3, to the present day. Much of this interaction occurs in Photo-Dissociation Regions (PDRs) where far-ultraviolet photons of these stars create a largely neutral, but warm region of gas and dust. PDR emission dominates the IR spectra of star-forming galaxies and also provides a unique tool to study in detail the physical and chemical processes that are relevant for inter- and circumstellar media including diffuse clouds, molecular cloud and protoplanetary disk surfaces, globules, planetary nebulae, and starburst galaxies.
We propose to provide template datasets designed to identify key PDR characteristics in the full 1-28 μm JWST spectra in order to guide the preparation of Cycle 2 proposals on star-forming regions in our Galaxy and beyond. We plan to obtain the first spatially resolved, high spectral resolution IR observations of a PDR using NIRCam, NIRSpec and MIRI. We will observe a nearby PDR with well-defined UV illumination in a typical massive star-forming region. JWST observations will, for the first time, spatially resolve and perform a tomography of the PDR, revealing the individual IR spectral signatures from the key zones and sub-regions within the ionized gas, the PDR and the molecular cloud. These data will test widely used theoretical models and extend them into the JWST era. We will assist the community interested in JWST observations of PDRs through several science-enabling products (maps of spectral features, template spectra, calibration of narrow/broad band filters in gas lines and PAH bands, data-interpretation tools e.g. to infer gas physical conditions or PAH and dust characteristics). This project is supported by a large international team of one hundred scientists collaborators.
Dust as a tracer in the Milky Way and local galaxies
Speaker: Dr Matt Smith (University of Cardiff)
Matthew_Smith_public_version.pdf
An Empirical Determination of the Dust Mass Absorption Coefficient, κd, and its Variation Within Nearby Galaxies
With the advent of large far-infrared and submillimetre facilities such as Herschel, Planck, JCMT, and especially ALMA, dust now provides an indispensable way to study the evolution of galaxies. In particular, our ability to observe large areas of the submillimetre sky quickly (along with the advantageous effects of negative-k-correction and lensing) mean that dust observations are increasingly used as a proxy to study star-formation rates, gas masses, and chemical evolution - which are impractical to observe directly for such substantial numbers of galaxies.
However, our ability to exploit dust observations in this way is predicated on a simple assumption - that we can actually use observations of dust emission to infer dust masses. But the dust mass absorption coefficient, κd, is uncertain to (at best!) an order of magnitude. Worse still, this forces us to treat κd as being constant both between galaxies, and within them - which of course cannot be true in reality. Pinning down κd, and how it varies, is therefore vital.
I will present a simple empirical method for determining the value of κd in galaxies, which exploits the fact that the dust-to-metals ratio in galaxies exhibits minimal variation in high- and intermediate-metallicity systems. This method puts new empirical constraints on global values of κd, providing an important counterpoint to theoretical and laboratory models.
I will also present the first ever resolved maps of κd, obtained by applying the method in a pixel-by-pixel manner to well-resolved nearby galaxies. These maps provide strong observational evidence for variation of κd within galaxies.
DustPedia Copenhagen 2018 Presentation.pdf
Dust as a galaxy probe Main Auditorium
Convener: Georgios Magdis
Dust as a galaxy probe
In this talk I will discuss our current view of dust as a tracer of galaxy properties – from dust in extended, star forming disks and winds – to properties of dusty nuclei and compact, collimated nuclear outflows . I will show optical studies of dust in absorption - and dust in emission at long wavelengths (from infrared to mm/submm). Finally, I will present new results on very optically thick dust in the inner regions of galaxies that hide extreme, hitherto unknown, nuclear activity. I will discuss the nature of the buried activity and how we can study the properties of the dust in these deeply veiled regions.
Speaker: Prof. Susanne Aalto (Chalmers University of Technology)
Susanne_Alto.pdf
Insights into the Life Cycle of Dust at Low Metallicity from the Local Group and Nearby Galaxies
Dust plays critical roles in many of the processes occurring in the interstellar medium and dust's infrared emission serves as a probe of the ISM and star formation in galaxies out to high redshift. The role of dust in ISM physics and its use as a probe of distant galaxies both depend on the characteristics of the grain population: the dust-to-gas ratio and the grain composition, charge, and size distributions. These properties are set by the life cycle of dust in the ISM, which may be dramatically different in the low metallicity conditions prevalent at high redshifts or in nearby dwarf galaxies. I will present results from several efforts to constrain aspects of the dust life cycle in nearby, low metallicity environments.
Speaker: Dr Karin Sandstrom (University of California, San Diego)
Karin_Sandstrom.key
Radiative transfer model of dust attenuation curves in clumpy, galactic environments
The attenuation of starlight by dust in galactic environments is investigated through models of radiative transfer in a spherical, clumpy interstellar medium (ISM). We show that the attenuation curves are primarily determined by the wavelength dependence of absorption rather than by the underlying extinction (absorption+scattering) curve; the observationally derived attenuation curves cannot constrain a unique extinction curve unless the absorption or scattering efficiency is specified. Attenuation curves consistent with the "Calzetti curve" are found by assuming the silicate-carbonaceous dust model for the Milky Way (MW), but with the 2175 Å bump suppressed or absent. The discrepancy between our results and previous work that claimed the Small Magellanic Cloud dust to be the origin of the Calzetti curve is ascribed to the difference in adopted albedos; we use the theoretically calculated albedos, whereas the previous works adopted albedos derived empirically from observations of reflection nebulae. It is found that the attenuation curves calculated with the MW dust model are well represented by a modified Calzetti curve with a varying slope and UV bump strength. The strong correlation between the slope and UV bump strength, as found in star-forming galaxies at 0.5 < z < 2.0, is well reproduced when the abundance of the UV bump carriers is assumed to be 30%–40% of that of the MW dust; radiative transfer effects lead to shallower attenuation curves with weaker UV bumps as the ISM is more clumpy and dustier. We also argue that some local starburst galaxies have a UV bump in their attenuation curves, albeit very weak.
Speaker: Kwang-il Seon (Korea Astronomy & Space Sciences Institute)
seon_CPHDUST2018_0614.pdf
Dust in AGN Main Auditorium
Convener: Daniel Asmus (ESO, Chile / University of Southampton, UK)
Dust in Active Galactic Nuclei: A close look at the torus and its surroundings
In this talk I will review our current knowledge of the torus of dust and molecular gas in AGN and its close surroundings. I will summarize recent results on the geometry of the torus in nearby AGN, the connection to inflows and outflows, and the properties of AGN dust. In the last part of the talk I will touch upon the properties of the torus in distant AGN and in particular the dust covering factors and the torus evolution.
Speaker: Prof. Almudena Alonso-Herrero (INTA-CSIC)
DustAGN.pdf
Hot, cool, dark and bright: the various shades of dust around AGN
Dust accreting onto supermassive black holes has been a cornerstone of AGN unification as it provides the angle dependent obscuration required to explain the various AGN types by a pure view angle effect. However, advancements in angular resolution over the last decade have allowed us to resolve the dust-emitting region for the first time. It is now clear that dust plays a fundamental role in dynamically shaping the immediate environment of the black holes. At the same time, the hard AGN radiation significantly processes the dust, altering its size distribution and chemical composition. In this talk, I will present recent results from high angular resolution observation, including infrared (IR) interferometry. A particular emphasis will be the physical constraints of the composition and distribution of dust, specifically the predominance of large graphite grains in shaping the IR emission. I will also show new results on PAH emission from within 100 pc of an AGN, provoking thoughts on its common use as a star-formation tracer.
Speaker: Dr Sebastian Hönig (University of Southampton)
20180615_Copenhagen_DustinAGN.key
Growth of massive black holes in dusty clouds: impacts of relative velocity between dust and gas
Recent observations have suggested the existence of a large amount of dust around supermassive black holes (SMBHs) in the early universe (e.g. Maiolino et al. 2004). In dusty clouds, the growth of black holes can be significantly regulated due to strong radiation force on dust grains. Yajima et al. (2017) recently showed that the accretion on to intermediate-mass black holes (IMBHs) in dusty clouds are significantly suppressed compared with dustless clouds because of the strong radiation force on dust grains. They, however, assumed that the dust and gas are completely coupled. This assumption might be invalid in the vicinity of black holes. The relative velocity between dust and gas is likely to have impacts on the accretion rate.
We here investigate the impacts of the relative motions of dust and gas on the accretion rate onto IMBHs with the mass of $10^5~M_{\odot}$ by using one-dimensional radiation hydrodynamic simulations in clouds with initial gas densities of $n_{\mathrm{H}} = 10$ and $100$ cm$^{-3}$. To investigate the effect of grain size on the gas accretion, we introduce two additional fluid components which describe large (0.1 micron) and small (0.01 micron) dust grains in the simulations as we did in Ishiki et al. (2018).
We show that the accretion rate is reduced due to the radiation force. We show that the dust-to-gas mass ratio significantly changes in H$_{\mathrm{II}}$ regions because of the relative motions of dust and gas. The decoupling of dust from gas alleviates the suppression of black hole growth compared with the complete coupling case. This effect may allow moderate growth of black holes even in dusty clouds.
Speaker: Mr Shohei Ishiki (Hokkaido University)
Ishiki_v3m.pdf
Ishiki_v3.pptx
Convener: Dr Daniel Asmus (ESO)
Optical and infrared radiation pressure on dust and gas around AGN as drivers of dusty winds
Parsec-scale polar emission signatures seen in the infrared continuum of many nearby AGN suggest the presence of dust in a region generally associated with outflowing gas. This makes clear that the idea of a circum-nuclear obscurer referred as torus needs to be revised in favour of a more complex obscuring structure, yielding a polar component. We present a semi analytical model to test the hypothesis of radiatively accelerated dusty winds launched by the AGN and the heated dust itself. The main components of the model under consideration are an AGN and an infrared radiating dusty disk, the latter being the primary mass reservoir for the outflow. We derive the full components of the force field experienced by dusty clouds in this environment, accounting for both gravity and the AGN radiation as well as the re-radiation by the hot dusty gas clouds themselves. We see that dusty outflows naturally emerge, whose strength and directions will depend on the Eddington ratio and the column density of the intervening material.
Speaker: Marta Venanzi (Miss)
marta_venanzi_cphdust2018.pptx
Dust in the early universe Main Auditorium
Convener: Dr Takaya Nozawa (National Astronomical Observatory of Japan)
Dust in the early universe
Early dust enrichment is believed to occur on very short timescales following the first supernova explosions. The efficiency of this process and the nature of the first dust have a large impact on early star formation. Molecule formation on the surface of dust grains promotes gas cooling, increasing the star formation efficiency. In the densest part of collapsing pre-stellar clouds dust-driven fragmentation is believed to enable the formation of low-mass and long lived stars. The properties of these fossil remnants may provide us with important clues on the nature of the first supernovae and their dust production efficiencies. Finally, deep ALMA observations are probing the dust content of normal star forming galaxies at z > 6, pointing to a rapid dust enrichment of the interstellar medium for some of these sources. In this talk, I will review the status of our understanding of early dust enrichment and the many open questions that need to be addressed in the future.
Speaker: Prof. Raffaella Schneider (Osservatorio Astronomico di Roma-INAF)
Raffaella_Schneider_public.pdf
Dust and elements in the epoch of reionization
First, I will present our recent results from ALMA observations of galaxies in the epoch of reionization, redshifts $z>6$. Second, I will present a method for estimating the dust temperature from far-infrared flux densities, taking the radiative equilibrium and transfer effect into account. Then, I will present a demographics study of dust mass in $z>6$ galaxies and compare the observations with a simple theoretical model. I will pay attention to "dust-free" objects as well as dust-rich ones. Finally, I will discuss the dust mass growth in the interstellar medium and future perspective.
Speaker: Akio Inoue (Osaka Sangyo University)
Akio_Inoue.pdf
The dust-to-stellar mass ratio, a key-tool for probing galaxy evolution from z~0 up to z~6
Over the last decade, the IR Herschel satellite has allowed to trace the
dust budget up to z ∼4, and the recent ALMA facility
is extending the measurement of the dust production to even early times.
This has rendered particularly urgent the issue of explaining how
the dust mass in galaxies is related to other key galaxy-integrated quantities, i.e. stellar mass andstar-formation rate.
In the present work, I will focus, in detail, on the dust-to-stellar mass
(DTS) ratio, as this quantity represents a true measure of how much dust
per unit stellar mass survives the various destruction processes,
i.e. astration and interstellar shocks.
The observed values of the DTS from z~0 up to z~6 will be compared to
theoretical estimates computed by means of state-of-the art chemical
evolution models for galaxies of different morphological type, showing
the strong dependence of this quantity on two key ingredients,
i.e. the underlying star formation history and the stellar initial
mass function.
Speaker: Dr Francesca Pozzi (Physics and Astronomy Department, Bologna, Italy)
pozzi.pdf
Conference Summary and Review Main Auditorium
A summary and a review of the conference
Speaker: Dr Ciska Kemper (ASIAA)
Ciska_Kemper_conf_summary.pdf | CommonCrawl |
Question on the application of the Borel-Cantelli lemma
I have a question on the first proof on this pdf. Essentially, it is trying to prove the following statement:
Let ${f_n}$ be a sequence of measurable functions on $[0, 1]$ with $|f_n(x)| < ∞$ for almost every $x$. Show that there exists a sequence $c_n$ of positive real numbers such that $$\frac{f_n(x)}{c_n} \to 0$$ for almost every $x$.
I follow everything until the proof reaches this claim:
$$m\big(\{x\in [0,1] : \bigg|\frac{f_n(x)}{nk_n}\bigg| \geq \frac{1}{n}\}\big) < \frac{1}{2^n}$$
Why do we need the Borel Cantelli lemma here? As $n \to \infty$, the statement above implies that the set in which $\bigg|\frac{f_n(x)}{nk_n}\bigg|$ cannot be made arbitrarily small has measure zero. If we choose $c_n = nk_n$, doesn't that mean $\frac{f_n(x)}{c_n}$ converges to zero almost everywhere?
real-analysis measure-theory borel-cantelli-lemmas
The Borel-Cantelli lemma says that if the sum of $m(E_i)$'s is finite, then
$$m({\displaystyle \limsup _{n\to \infty }E_{n})=m(\bigcap _{n=1}^{\infty }\bigcup _{k\geq n}^{\infty }E_{k})}=0$$
Intuitively, the Borel-Cantelli lemma tells you that the chance of some bad event happening (like some bad elements interfere in what we want to prove infinitely many times) is negligible. In measure theoretic words, the measure of some bad elements where our proof doesn't work is $0$ and for "almost every" element, our proof works.
Take $$E_n= \{x\in [0,1] : \bigg|\frac{f_n(x)}{nk_n}\bigg| \geq \frac{1}{n}\}$$
Then $\sum_i m(E_i) = 1 < \infty$. So, the Borel-Cantelli lemma applies.
Now, you conclude that $m(\bigcap _{n=1}^{\infty }\bigcup _{k\geq n}^{\infty }E_{k})=0$ which implies that the set of elements $x$ such that $\bigg|\frac{f_n(x)}{nk_n}\bigg| \geq \frac{1}{n}$ for infinitely many $n$'s is negligible and can be ignored. So, for almost every $x$, we see that $\bigg|\frac{f_n(x)}{nk_n}\bigg| < \frac{1}{n}$ excluding a finite number of $n$'s. Since the number of $n$'s which do not satisfy the previous inequality is finite, you can be sure that after a certain $n$, let's call it $N_0$ the inequality holds almost everywhere.
Given any $\epsilon > 0$, choose $n_0$ such that $\frac{1}{n_0} < \epsilon$. Now set $M=\max\{n_0,N_0\}$. Then we conclude that
$$\forall \epsilon>0, \exists M \in \mathbb{N}: n \geq M \implies |\frac{f_n(x)}{nk_n}| < \frac{1}{n} < \epsilon$$
which means that $\lim_{n\to\infty}\frac{f_n(x)}{nk_n}=0$ almost everywhere.
So, you just need to take $c_n=nk_n$ as you said.
Edit: To understand what the Borel-Cantelli lemma tells you, think about what kind of elements are in $\limsup _{n\to \infty }E_{n}$. This should help you a lot.
$$x \in \limsup _{n\to \infty }E_{n} = \bigcap _{n=1}^{\infty }\bigcup _{k\geq n}^{\infty }E_{k} \iff \forall n, \exists k_* \geq n: x \in E_{k_*}$$
Think of $n$ as time and imagine that we are checking $E_n$'s one after another. This statement tells us that no matter how much time passes, we can always find some $E_{k_*}$ in future that contains $x$. So, even if we wait forever, there will still be a set that contains $x$. So, $x$ is happening infinitely many times. The compliment of this set are those elements that occur only finitely many times. I hope this helps.
stressed outstressed out
$\begingroup$ Why do we need the Borel-Cantelli lemma though? Can't I just use the statement $m\big(\{x\in [0,1] : \bigg|\frac{f_n(x)}{nk_n}\bigg| \geq \frac{1}{n}\}\big) < \frac{1}{2^n}$ and say that the measure can be made arbitrarily small. Doesn't that complete the proof? $\endgroup$ – user1691278 Jan 24 at 5:23
$\begingroup$ @user1691278 The argument that allows you to say that since "the measure of the bad elements can be made arbitrarily small", the opposite must hold almost everywhere is exactly the reason why the Borel-Cantelli lemma exists in the first place. $\endgroup$ – stressed out Jan 24 at 5:25
$\begingroup$ Sorry, what do you mean by "the opposite" (as in "the opposite must hold")? $\endgroup$ – user1691278 Jan 24 at 5:27
$\begingroup$ @user1691278 By the "opposite", I mean the negation of your statement. All you conclude from $m\big(\{x\in [0,1] : \bigg|\frac{f_n(x)}{nk_n}\bigg| \geq \frac{1}{n}\}\big)$ directly is that for almost every $x$, $\bigg|\frac{f_n(x)}{nk_n}\bigg| \geq 0$ which is obvious. If you want to say more, you basically need an argument like the Borel-Cantelli lemma. Let me know what part you find confusing and I will try to explain it in more detail. $\endgroup$ – stressed out Jan 24 at 5:35
$\begingroup$ @stressedout Your explanation is amazing. Thank you. $\endgroup$ – user1691278 Jan 24 at 5:58
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Turbulence kinetic energy dissipation rates estimated from concurrent UAV and MU radar measurements
Hubert Luce1,
Lakshmi Kantha2,
Hiroyuki Hashiguchi ORCID: orcid.org/0000-0001-8033-09553,
Dale Lawrence2 &
Abhiram Doddi2
Earth, Planets and Space volume 70, Article number: 207 (2018) Cite this article
We tested models commonly used for estimating turbulence kinetic energy dissipation rates \(\varepsilon\) from very high frequency stratosphere–troposphere radar data. These models relate the root-mean-square value \(\sigma\) of radial velocity fluctuations assessed from radar Doppler spectra to \(\varepsilon\). For this purpose, we used data collected from the middle and upper atmosphere (MU) radar during the Shigaraki unmanned aerial vehicle (UAV)—radar experiment campaigns carried out at the Shigaraki MU Observatory, Japan, in June 2016 and 2017. On these occasions, UAVs equipped with fast-response and low-noise Pitot tube sensors for turbulence measurements were operated in the immediate vicinity of the MU radar. Radar-derived dissipation rates \(\varepsilon\) estimated from the various models at a range resolution of 150 m from the altitude of 1.345 km up to the altitude of ~ 4.0 km, a (half width half power) beam aperture of 1.32° and a time resolution of 24.6 s, were compared to dissipation rates (\(\varepsilon_{U}\)) directly obtained from relative wind speed spectra inferred from UAV measurements. Firstly, statistical analysis results revealed a very close relationship between enhancements of \(\sigma\) and \(\varepsilon_{U}\) for \(\varepsilon_{U} \,{ \gtrsim }\,10^{ - 5} \,{\text{m}}^{2} \,{\text{s}}^{ - 3}\), indicating that both instruments detected the same turbulent events with \(\varepsilon_{U}\) above this threshold. Secondly, \(\varepsilon_{U}\) was found to be statistically proportional to \(\sigma^{3}\), whereas a \(\sigma^{2}\) dependence is expected when the size of the largest turbulent eddies is smaller than the longitudinal and transverse dimensions of the radar sampling volume. The \(\sigma^{3}\) dependence was found even after excluding convectively generated turbulence in the planetary boundary layer and below clouds. The best agreement between \(\varepsilon_{U}\) and radar-derived \(\varepsilon\) was obtained with the simple formulation based on dimensional analysis \(\varepsilon = \sigma^{3} /L_{c}\) where LC ≈ 50–70 m. This empirical expression constitutes a simple way to estimate dissipation rates in the lower troposphere from MU radar data whatever the sources of turbulence be, in clear air or cloudy conditions, consistent with UAV estimates.
Turbulence kinetic energy (TKE) dissipation rate \(\varepsilon\) is a fundamental parameter indicative of the strength of turbulence. Dissipation rates of atmospheric turbulence can be potentially retrieved from stratosphere–troposphere (ST) radars operating in VHF (e.g., Hocking 1983, 1985, 1986, 1999; Fukao et al. 1994; Hocking and Hamza 1997; Nastrom and Eaton 1997; Li et al. 2016) and UHF bands (e.g., Sato and Woodman 1982; Cohn 1995; Bertin et al. 1997; Wilson et al. 2005 and references therein). Because ST radars can be used for detecting turbulence in the free atmosphere (above the atmospheric boundary layer), standard models are based on the assumption that turbulence results from shear flow instabilities in a stably stratified background (e.g., Fukao et al. 1994; Kurosaki et al. 1996; Nastrom and Eaton 1997). For such turbulence, the stable stratification limits the size of the largest turbulent eddies and damps vertical motions, leading to the definition of various outer scales of stratified turbulence (e.g., Weinstock 1978a, b, 1981). Additional key assumptions used to retrieve turbulence parameters from radar data are that isotropic turbulence, following the Kolmogorov–Obukhov–Corrsin (KOC) model, exists at smaller scales in the inertial subrange and that the Bragg wavelength of radar backscatter lies within this inertial subrange. Estimation of \(\varepsilon\) is therefore based on the measurement of radar Doppler spectral width assuming that part of the spectral broadening results from isotropic turbulent motions in the radar measurement volume (the so-called Doppler method). Indirect estimates of \(\varepsilon\) can also be obtained from the estimates of refractive index structure constant \(C_{n}^{2}\) from radar echo power (e.g., Gage and Balsley 1978; Cohn 1995; Hocking and Mu 1997; Hocking 1999) but that is beyond the scope of the present work.
The purpose of the present work is to show the results of comparisons between \(\varepsilon\) estimates made from middle and upper atmosphere (MU) radar data in the lower troposphere using existing formulations and direct in situ estimates of \(\varepsilon\) obtained from small unmanned aerial vehicles (UAVs) equipped with high-frequency sampling and fast-response Pitot (airspeed) sensors (Kantha et al. 2017). The potential of UAVs for characterizing turbulence properties was thoroughly described by Lawrence and Balsley (2013). Contrary to the radar technique which samples the atmosphere inside a volume at a fixed location, the UAV has the potential to probe all the space occupied by a turbulent layer/patch and may be a better tool for identifying the dimensions of a turbulent volume and for estimating outer scales.
The datasets were collected during two field campaigns, called the Shigaraki UAV Radar Experiments (ShUREX), in May–June 2016 and June 2017 at the Shigaraki MU observatory in Japan. Kantha et al. (2017) described the instruments and configurations used during a previous ShUREX campaign in June 2015. The instrumental setup did not significantly change in 2016 and 2017, except for the use of higher-frequency sampling and lower-noise turbulence sensors. The CU DataHawk UAVs flew in the immediate vicinity of the radar (within a horizontal distance of ~ 1.0 km) and up to altitudes of ~ 4.0 km above the sea level (ASL). Selected data from 39 science flights (16 in 2016 and 23 in 2017) were used for the present study. "Instruments and data" section describes briefly the MU radar and DataHawk UAV and the observational configurations used during the campaigns. "Theoretical bases and practical methods of \(\varepsilon\) estimation" section presents the theoretical expressions used for retrieving TKE dissipation rates from radar and UAV data and describes in detail the practical methods. As summarized by Hocking (1999), the Doppler method leads to different analytical expressions according to hypotheses made on the properties of turbulence and according to the radar specifications. In the present work, we will focus on the most commonly used expressions without describing their derivations. The underlying hypotheses will be shortly recalled. More details can be found in Kantha et al. (2018) "Comparisons between \(\varepsilon_{U}\) and \(\varepsilon\) from the radar models" section presents the results of comparisons between the various dissipation rate estimates. These results are discussed in "Discussion" section, and conclusions are presented in "Conclusions" section.
Instruments and data
MU radar
The MU radar is a 46.5 MHz beam-steering Doppler pulsed radar located at the Shigaraki MU Observatory (34.85°N, 136.10°E) in Japan (Fukao et al. 1990). The radar parameters used during the ShUREX 2016 and 2017 campaigns are listed in Table 1. The radar was operated in range-imaging mode using frequency diversity (see Luce et al. 2006) at vertical and two oblique incidences (10° off zenith toward North and East) for high-resolution echo power observations. The radar parameters were set up so that one high-resolution profile of echo power at vertical incidence was acquired from the altitude of 1.270 km up to 20.465 km (ASL) at a dwell time of 24.57 s, every 6.144 s (see Table 1). The radar data were also processed at a standard range resolution of 150 m for retrieving signal-to-noise ratios (SNR), radial winds and spectral widths using the moment method (e.g., Yamamoto et al. 1988).
Table 1 MU radar parameters used during ShUREX 2016 and 2017 campaigns
CU DataHawk UAV
The ShUREX campaign and the characteristics of the CU DataHawk UAVs and onboard sensors are described by Kantha et al. (2017). The UAVs were equipped with a custom autopilot programmed to execute a preplanned trajectory near the MU radar. The UAVs could also be commanded to sample interesting atmospheric features revealed by the MU radar in near real time. For the present purpose, we only consider measurements performed during "vertical" ascents and descents. When moving up or down, the UAVs were flying along helical trajectories ~ 100–150 m in diameter at a typical vertical velocity rate of ~ 2 m s−1. The maximum flight altitude was limited to ~ 4.0 km ASL by both battery capabilities and air traffic regulations. Among 41 flights performed during the 2016 campaign, 16 science flights provided (totally or partially) valuable data for comparisons with MU radar data. Hereafter, they will be denoted 'FLT16-xx', where '16' refers to the year and 'xx' is the flight number. In 2017, 23 science flights were available for analysis.
The UAVs were equipped with a variety of sensors for atmospheric measurements (Kantha et al. 2017). Among these sensors, a commercial IMET sonde provided measurements of pressure, temperature and relative humidity (PTU) at 1 Hz. Velocity of the air flow relative to the UAV was measured by a fast-response Pitot-static tube and a differential pressure sensor, with the Pitot tube mounted at a height of 3 cm above the vehicle so as to project into the free stream above the aerodynamic boundary layer. This sensor was sampled at an effective rate of 400 Hz. At the nominal airspeed of 14 m s−1, the digital resolution was 0.042 m s−1 (see Kantha et al. 2017). In addition, a fast-response (< 1 ms) cold wire sensor was also available for temperature measurements sampled at 800 Hz.
There were too many flights to describe their characteristics in detail. Most of the flights had ascents and descents (denoted by 'A' and 'D' when necessary) sometimes separated by horizontal legs of various durations (e.g., FLT16-22 and FLT16-38). Unanticipated blocking of the Pitot tube (used also by the autopilot for flight control) by precipitation sometimes produced short time span (~ a few tens of seconds) downward motions during ascents (e.g., FLT16-05, FLT16-15). These sources of aberrant data points were manually removed.
The meteorological conditions were checked every day before deciding to launch UAVs or not since they cannot fly during rainy conditions and strong winds (> 10–15 m s−1). The state of the lower atmosphere during the flights could be known from the available data without additional meteorological information. Indeed, among other things, the relative humidity measurements made by humidity sensors onboard UAV indicated flight in clouds and the radar images provided precise information on the vertical extent and evolution of the convective boundary layer when it exceeded the altitude of the first radar sampling gate. Therefore, turbulence associated with dry or saturated convections could be easily identified from the datasets and could be removed from statistics when focusing on stratified and clear air turbulence only. Actually, the weather was almost clear through the observations above the convective boundary layer.
Theoretical bases and practical methods of \(\varepsilon\) estimation
Theoretical expressions of \(\varepsilon\) from radar data
From a dimensional analysis, the dissipation rate (assuming isotropy) can be inferred from:
$$\varepsilon \sim\left\langle {w^{\prime 2} } \right\rangle^{3/2} /L$$
where \(\left\langle {w^{\prime 2} } \right\rangle\) is the variance of vertical wind fluctuations and L is a typical scale of the turbulent eddies. A rms value \(\sigma\) of radial turbulent velocity fluctuations can be obtained from the measured Doppler spectral width after removing non-turbulent contributions to the spectral broadening (see "Appendix") (e.g., Hocking 1986; Fukao et al. 1994; Naström 1997; Dehghan and Hocking 2011). Similarly to the above expression, we can write:
$$\varepsilon_{R} = \sigma^{3} /L_{c}$$
where \(L_{c}\) has the dimension of a turbulence scale. Expression (1) is only indicative, but it will be first used in order to see if a particular value of \(L_{c}\) emerges from our dataset.
In practice, two main expressions are used for estimating \(\varepsilon\) from the Doppler spectral width, based on more elaborated models. When the outer scale \(L_{\text{out }}\) of turbulence is small compared to the horizontal and transverse dimensions of the radar sampling volume, 2a, 2bFootnote 1 we have (e.g., Hocking 1983, 1999, 2016):
$$\varepsilon_{N} = C\sigma^{2} N$$
where C is a constant (= \(0.5 \pm 0.25)\) according to Hocking (2016). C = 0.47, sometimes used in the literature, was applied for producing the figures. The parameter \(N\) is the Brunt–Väisälä frequency. Expression (2) has been established for characterizing turbulence in stratified conditions only [whereas expression (1) is always valid]. Various definitions of outer scales of stably stratified turbulence have been proposed in order to obtain dissipation rate expressions in the form of (2) (e.g., Weinstock 1978a, b, 1981). Expression (2) is virtually identical to the theoretical expression given by Weinstock (1981) obtained by integrating the spectrum of inertial turbulence down to the buoyancy wavenumber \(k_{\text{B}} = N/\sqrt {\left\langle {w^{\prime 2} } \right\rangle }\) so that \(\varepsilon \approx 0.5\left\langle {w^{{{\prime }2}} } \right\rangle N\). Hocking (2016) makes use of the one-dimensional transverse spectrum [expression (7.42)] whose integration, by including additional contribution from the buoyancy subrange, leads to an estimate of vertical wind fluctuation variance \(\sigma^{2}\) supposed to be measured by the radar. By doing so, Expression (2) is obtained with various values of C, coincidently close to the coefficient 0.5 of the Weinstock model. Kantha et al. (2018, this issue) used this approach with different conceptual models of turbulence and different definitions of turbulence scales and even generalized it to expressions including the radar volume effects. However, it seems that a definitive modeling is still an open issue.
The alternative approach proposed by, e.g., Frisch and Clifford (1974) and Labitt (1979) considers the role of spatial low-pass band filter played by the radar volume, valid if \(L_{\text{out}} \gg\) 2a, 2b. The White et al. (1999) formulation also considered the effects of the wind advection:
$$\varepsilon_{w} = \left( {\frac{4\pi }{1.6}} \right)^{3/2} \frac{{\sigma^{3} }}{{I^{3/2} }}$$
$$I \propto \int\limits_{0}^{\pi /2} {{\text{d}}\phi \int\limits_{0}^{\pi /2} {\sin^{3} \theta \; \times \left( {b^{2} \cos^{2} \theta + a^{2} \sin^{2} \theta + \frac{{L_{H} }}{12}\sin^{2} \theta \cos^{2} \phi } \right)^{1/3} {\text{d}}\theta } }$$
and \(L_{H} = VT\), where \(V\) is the mean horizontal wind speed during the dwell time T. It is important to note that expression (3) is based on the hypothesis that the radar is sensitive to the three-dimensional longitudinal spectrum of turbulence (see Doviak and Zrnic' 1993, p. 398). Therefore, Eqs. (2) and (3) are not the asymptotic forms (for \(L_{\text{out}} \ll\) 2a, 2b and \(L_{\text{out}} \gg\) 2a, 2b, respectively) of a more general expression. The \(\varepsilon\) estimates from Eqs. (2) and (3) (i.e., \(\varepsilon_{N}\) and \(\varepsilon_{W}\), respectively) will be compared with those derived from UAV data, hereafter noted \(\varepsilon_{U}\) in "Comparisons between \(\varepsilon_{U}\) and \(\varepsilon\) from the radar models" section.
Despite its apparent complexity, Eq. (3) has advantages with respect to Eq. (2). \(\varepsilon_{W}\) can be estimated solely from the radar data, while \(\varepsilon_{N}\) requires estimates of N (usually from balloon measurements) or standard climatological values as default values (e.g., Weinstock 1981; Deghan et al. 2014). In addition, \(\varepsilon_{W}\) can be used whatever the turbulence source may be (convective or shear flow instabilities), assuming that inertial turbulence is observed and \(L_{\text{out}} \gg\) 2a, 2b. Finally, \(\varepsilon_{N}\) requires, in principle, the estimation of moist \(N^{2}\) when air is saturated, because saturation modifies the background stability due to latent heat release. This additional difficulty does not seem to have been considered in the studies related to TKE dissipation rate estimates from ST radar data. However, we shall see that the accuracy of \(N^{2}\) is not an important issue because our analyses reveal a fundamental inadequacy of \(\varepsilon_{N}\). This conclusion goes beyond the problem of estimating \(N^{2}\) properly.
Equation (3) or similar expressions were used by Gossard et al. (1982) and Chapman and Browning (2001), for example, using UHF radars at similar spatial resolutions as the MU radar and by McCaffrey et al. (2017) at vertical resolution of ~ 25 m.
Practical methods from radar data
The Doppler variance due to turbulent motions was estimated from the Doppler spectra by applying:
$$\sigma^{2} \approx \sigma_{m}^{2} - \sigma_{b}^{2}$$
where \(\sigma_{m}^{2}\) is the Doppler variance measured at vertical incidence, and \(\sigma_{b}^{2}\) is the variance due to beam-broadening effects.\(\sigma^{2}\) was used in order to obtain \(\varepsilon_{R}\), \(\varepsilon_{N}\) and \(\varepsilon_{W}\). Equation (4) is very simple compared to the expressions provided by Naström (1997) and Dehghan and Hocking (2011), because only data from the vertical beam are used. At VHF, data collected at vertical incidence are usually avoided because the radar echoes can be strongly affected by (non-turbulent) specular reflectors so that the spectral width is reduced and \(\sigma^{2}\) is biased (e.g., Tsuda et al. 1988). However, Eq. (4) has a great advantage, since shear-broadening effects are null or negligible when using a vertical beam. Even though the theoretical effects due to shear-broadening when using data collected at oblique incidences are well-established, the corrections remain challenging in practice, because they require accurate estimates of wind shears, and the wind shear profiles estimated at the radar range resolution may not be representative of shear profiles at higher resolutions (e.g., Figure 5 of Luce et al. 2018). The use of data at vertical incidence will be justified a posteriori in "Comparisons between \(\varepsilon_{U}\) and \(\varepsilon\) from the radar models" section.
The beam-broadening correction \(\sigma_{b}^{2}\) requires the knowledge of horizontal winds estimated from off-vertical beam data, and these winds may not be exactly those at the altitudes sampled by the vertical beam. It is another source of bias (Deghan and Hocking 2011), but difficult to correct in general. However, since the measurements were taken for low altitudes (< 4.5 km), this problem should be minimized here because the sampled altitude differences between the vertical and oblique directions do not exceed a few tens of meters.
Finally, Eq. (4) does not include correction due to gravity wave contributions (e.g., Naström 1997). Here, it is expected to be negligible: the dwell time (~ 25 s) should be sufficient for minimizing their contribution because it is a small fraction of internal gravity wave periods. The details of the practical procedure for estimating \(\sigma^{2}\) from Eq. (4) are given in "Appendix".
A complete vertical profile of \(\sigma^{2}\) is calculated from time series of ~ 25 s in length every 6.144 s (overlapping of a factor 4) at a vertical resolution of 150 m (see Table 1). For comparison with UAV measurements, it must be realized that the UAV provides data only along a specific altitude versus time trajectory. Figure 1 shows the strategy used for reconstructing pseudo-profiles of \(\sigma^{2}\) along the UAV paths. Since the UAVs were flying in the vicinity of the MU radar, we calculated temporal averages of \(\sigma^{2}\) (\(\left\langle {\sigma^{2} } \right\rangle\)) over a few minutes only (4 min was arbitrarily selected) about the time-height location of the UAV (see Fig. 1), initially assuming that the UAV was flying directly over the radar, so it detected the same atmospheric structures at the same time as the radar. \(\left\langle {\sigma^{2} } \right\rangle\) was estimated for all altitudes sampled by the UAV, using a linear interpolation of the \(\sigma^{2}\) profiles (at 150-m resolution) at these altitudes. The same procedure was used for all other radar parameters (e.g., echo power, Luce et al. 2017). The height variations of the pseudo-profiles of \(\left\langle {\sigma^{2} } \right\rangle\) are thus due to a combination of the height and time variations of \(\sigma^{2}\). Figure 2 shows an example of pseudo-profiles of \(\left\langle {\sigma^{2} } \right\rangle\) during the ascent A1 and descent D1 of FLT16-15. The gray areas show the rms value of \(\sigma^{2}\) during the time averaging for A1 and D1, respectively.
Schematic representation of the method used for estimating pseudo-vertical profiles of Doppler variances \(\sigma^{2}\) from radar data. (Left) Example of horizontal excursion of a UAV with respect to MU radar antenna array location (black dot). (Right) The corresponding UAV altitude versus time (black line). The blue and red areas indicate the time-height domain used for calculating the average of Doppler variances \(\sigma^{2}\) during the ascent and descent of the UAV, respectively. Time averaging for a given altitude was arbitrarily performed over 4 min centered at the time when the UAV reached this altitude
Example of pseudo-profile of \(\sigma^{2}\) calculated according to the procedure shown in Fig. 1 for FLT16-15 A1 (blue) and D1 (red). Light and heavy gray areas show the rms value of \(\sigma^{2}\) during the time averaging for A1 and D1, respectively. The zigzag pattern around 2000 m during A1 is due a brief unintended descent of FLT16-15
The processing was then refined to account for the actual horizontal offset between UAV and radar by taking time lags due to wind advection into account, assuming frozen advection of the turbulent irregularities by the wind along the wind direction. This often provided higher correlation coefficients between \(\varepsilon_{U}\) and the radar-derived \(\varepsilon\) profiles, especially when the UAV was flying directly upstream of the radar. Yet, because the improvements were quite marginal, the procedure is not described in detail here. Note that time offsets could be avoided by flying in the beam of the radar, but the vehicle produces strong echoes that obliterate the turbulence measurements in the volume of interest, requiring a more complex analysis that considers neighboring times or altitudes (e.g., Scipión et al. 2016).
Figure 3a shows the histogram of the Doppler width \(2\sigma_{m}\) for all the available radar data surrounding the 16 UAV flights of ShUREX2016 in the height range 1.345–7.195 km ASL (corresponding to the first 40 radar gates). Similar statistics were obtained for 2017 data (not shown). It also shows the detection threshold (approximately ~ 0.2 m s−1) for the radar configuration and processing method used. Figure 3b shows the corresponding histogram of \(2\sigma\) (i.e., the Doppler width after beam-broadening corrections). Due to estimation errors (especially when SNR is low), some \(\sigma\) values can be negative. They are not shown in Fig. 3b. Figure 3c shows the histogram corresponding to the values of \(2\sigma\) estimated along the UAV flight track (as shown in Fig. 2). The peaks around 0.2 m s−1 are of course artificial and result from the minimum detection threshold of the radar. A bias is thus expected when comparing the lowest levels of radar-derived \(\varepsilon\) with \(\varepsilon_{U} .\) In addition, remaining small contaminations by various artifacts may still be present despite careful examination of the spectra (see "Appendix"). They can be a source of important biases for the lowest levels.
a Histogram of the measured spectral width (m s−1) for all the available radar data surrounding the 2016 UAV flights between 1.345 and 7.195 km (corresponding to the first 40 gates). b The corresponding histogram of \(2\sigma\) (after beam-broadening correction). c The histogram of \(2\sigma\) calculated along the 2016 UAV flight tracks
It has to be noted that the \(2\sigma\) values calculated along UAV flight tracks are not affected by estimation errors due to low SNR, because the UAVs did not exceed the altitude of 4.05 km ASL and SNR was always larger than 20 dB below this altitude. In addition, because UAVs flew during relatively weak winds (~ < 10–15 m s−1), the beam-broadening effects were relatively weak. Consequently, the conditions were favorable to errors in \(2\sigma\) estimates being small and, in particular, very few negative values were obtained in the altitude range of the UAV measurements so that they should not affect the statistics.
Estimation of \(\varepsilon_{U}\) from Pitot sensor data
The basics for retrieving \(\varepsilon_{U}\) were described by Kantha et al. (2017). Frequency spectra (Eq. 5 of Kantha et al. 2017) were estimated from variance-conserving, Hanning-weighted time intervals of 5 s duration (corresponding to 2000 points since the effective sampling rate was 400 Hz) every 2.5 s (corresponding to a successive time interval overlap of 50%). Assuming local isotropy and stationarity of turbulence and using the frozen-advection Taylor hypothesis, the theoretical Kolmogorov 1D power spectral density is of the form (Tatarski 1961; Hocking 1983):
$$S_{U} \left( f \right) = 0.55\varepsilon^{2/3} \left( {\frac{{\bar{U}}}{2\pi }} \right)^{2/3} f^{ - 5/3}$$
[the coefficient 0.55 holds for motions parallel to the mean relative wind]. \(\bar{U}\) is the mean relative wind (airspeed). Assuming that the calculated spectrum \(\hat{S}_{U} \left( f \right)\) shows an inertial domain (at least in a frequency band), the spectral data will have the frequency dependence:
$$\hat{S}_{U} \left( f \right) = \beta f^{ - 5/3}$$
An experimental value of \(\varepsilon_{U}\) can be obtained by estimating \(\beta\) by fitting spectral data, and equating Eqs. (5) and (6) (e.g., Frehlich et al. 2003; Siebert et al. 2006):
$$\varepsilon_{U} = \frac{2\pi }{{\bar{U}}}\left( {\frac{\beta }{0.55}} \right)^{3/2}$$
Experimental tests in outdoor flight showed that the flow acceleration over the UAV body did not damp the turbulent variations about the mean for the scales of interest, contrary to what it was expected from earlier tests in wind tunnel (which generated much smaller-scale turbulent fluctuations, not shown). Therefore, significant underestimations of energy dissipation rates from these effects are not expected.
Practical methods of estimations from Pitot sensor data
The problem of extracting the dissipation rate from UAV data is now reduced to that of identifying an inertial domain (when it exists) and estimating \(\beta\). Two different methods were applied with very similar results.
The first method consists in selecting an appropriate frequency band from spectra calculated from 5-s. time series chunks of Pitot data. A careful scrutiny of all the U frequency spectra shows that the highest probability to observe an inertial domain is found between 1 and 10 Hz. Two examples of typical spectra are shown in Fig. 4. At frequencies higher than 10 Hz, the spectra can be contaminated by noise when turbulence is weak (e.g., right panel of Fig. 4), and by artefacts (multiple peaks) mainly due to motor vibrations of the UAVs, especially during ascents (e.g., left panel of Fig. 4). The characteristics of these contaminations are specific to each UAV and flight, and they can also drift in time due to throttle variations. FLT16-15 was one of the most contaminated among the useful science flights. In practice, for the present purpose, we decided to estimate \(\beta\) from the spectral levels between 1.0 and 7.5 Hz. The spectral slopes between 1.0 and 7.5 Hz were estimated for all the time series of the 39 flights of ShUREX2016 and ShUREX2017. The corresponding histogram is shown in Fig. 5. The mean slope is − 1.64 (i.e., very close to the inertial slope − 5/3). The width of the distribution can be partly due to estimation errors when estimating slopes on individual spectra. Therefore, from a statistical point of view, the frequency band 1.0–7.5 Hz shows properties consistent with the existence of an inertial subrange.
Example of frequency spectra of U up to 100 Hz during the ascent (A1) (left) and descent (D1) (right) of FLT16-05. The black lines show spectra after averaging the 10 spectra in gray lines. The red lines show the − 5/3 inertial subrange slope. The blue lines show the calculated slopes between 1.0 and 7.5 Hz
Histogram of slopes of relative speed spectra calculated in the 1.0–7.5 Hz band for all ShUREX 2016 UAV flights
The second method is based on the selection of spectral bands exhibiting a -5/3 slope in a frequency domain delimited by 0.1 and 40 Hz (arbitrarily) from spectra calculated from time series chunks 50 s in length. The width of the spectral bands is a constant 0.699 decade, e.g., log10(5 Hz)–log10(1 Hz), and 39 overlapping bands are used. For each of these bands, the spectral slope s is estimated from the calculation of the variances in two spectral "sub-bands" of identical relative logarithmic width. An inertial subrange is inferred when \(s = - \,5/3\, \pm \,0.25\) for at least 3 consecutive spectral bands. The numerical thresholds were chosen in order to fit, as far as possible, the results that would have been obtained from visual inspection of the spectra. In some cases, the criteria may appear too loose or too restrictive, but it appears to be efficient for rejecting most spectral bands affected by instrumental noise and contaminations. A more thorough description of the method and results is in preparation.
The above two methods were applied to ShUREX2016 and ShUREX2017 data and produced the same statistical results.
Figure 6 shows examples of pseudo-vertical \(\varepsilon_{U}\) profiles in linear scales during the ascent (A1) and descent (D1) of FLT16-05 and FLT16-08 in altitude ranges covered by MU radar (i.e., above 1.345 km). The profiles are rather distinct during A1 and D1 of FLT16-05, but quite similar during A1 and D1 of FLT16-08. They clearly reveal altitude ranges with multiple peaks of enhanced TKE dissipation rates. These ranges are emphasized by the smoothed profiles shown by the solid and dashed black lines. The former was obtained by using a 30-point rectangular window applied to the time series sampled at 2.5 s (corresponding to 75 s averaging), and the latter by using a Gaussian averaging window. The (non-normalized) Gaussian function was taken as equal to \({ \exp }\left( { - \,z^{2} /2\alpha^{2} } \right)\), where \(\alpha = a/\sqrt 2 = 75/\sqrt 2 \,{\text{m}}\) in order to fit the characteristics of the expected range weighting function of the MU radar. The two methods provide very similar smoothed profiles. Therefore, the statistics of the comparison results should not depend on the method used for smoothing the \(\varepsilon_{U}\) profiles.
Examples of the profiles of \(\varepsilon_{U}\) for the first 2 ShUREX 2016 science flights (FLT16-05 and FLT16-08) during ascent (A1) and descent (D1). The thick solid and dashed black lines show smoothed profiles with a 30-point running rectangle window and a Gaussian weighting function consistent with the MU radar range weighting function (see text)
Comparisons between \(\varepsilon_{U}\) and \(\varepsilon\) from the radar models
Estimation of \(L_{C}\)
Before comparing \(\varepsilon_{U}\) with \(\varepsilon_{w}\) and \(\varepsilon_{N}\), estimates of the characteristic scale \(L_{C}\) defined in Eq. (1) can be obtained by replacing \(\varepsilon\) by \(\varepsilon_{U}\):
$$L_{C} \approx \left\langle {\sigma^{2} } \right\rangle^{3/2} /\varepsilon_{U}$$
The left panel of Fig. 7a shows \(\left\langle {\sigma^{2} } \right\rangle^{3/2}\) versus \(\varepsilon_{U}\) in logarithmic scale for 38 UAV flights (among 39) and all atmospheric conditions. One flight (FLT16-21) was not included due to oversight, and we used it afterwards as a test flight for confirming the statistical results obtained from the 38 flights. The horizontal dashed line in Fig. 7a shows the radar detection threshold (obtained from Fig. 3) and is approximately \(\left\langle {\sigma^{2} } \right\rangle^{3/2} = 0.001\). The thick solid line is a straight line of slope 1 and the dashed lines on both sides represent levels 3 times lower or higher. The scatter plot clearly indicates that \(\varepsilon_{U}\) appears to be proportional to \(\left\langle {\sigma^{2} } \right\rangle^{3/2}\) (especially for the largest values of \(\varepsilon_{U}\)), with relatively little scatter along the diagonal. For weak \(\varepsilon_{U}\) values (\(\varepsilon_{U} < \sim10^{ - 5} \,{\text{m}}^{2} \,{\text{s}}^{ - 3}\)), there is an important bias because the radar estimates are close to the minimum detectable levels and because residual contaminations might be present in the Doppler spectra despite careful data cleaning.
a (Left) Scatter plot of smoothed \(\left\langle {\sigma^{2} } \right\rangle^{3/2}\) versus \(\varepsilon_{U}\) for all 2016 and 2017 UAV flights except FLT16-21. \(\sigma^{2}\) was calculated as shown in Fig. 2. (Right). The corresponding histogram of the apparent buoyancy scale \(\tilde{L}_{B}\) for \(\left\langle {\sigma^{2} } \right\rangle^{3/2} > 0.01\). b Same as a after removal of turbulent regions associated with convective boundary layers, clouds and MCT layers
The histogram of \(\log_{10} \left( {L_{C} } \right)\) shown in the right panel of Fig. 7a displays a narrow peak for \(\left\langle {\sigma^{2} } \right\rangle^{3/2} > 0.01\) (or for \(\varepsilon_{U} > \sim10^{ - 4} \,{\text{m}}^{2} \,{\text{s}}^{ - 3}\) = \(0.1\,{\text{mW}}\,{\text{kg}}^{ - 1}\)) with a maximum near ~ 60 m and a mean (median) value of 75 m (61 m). The numerical values depend on the threshold on \(\left\langle {\sigma^{2} } \right\rangle^{3/2}\): the mean and median values of \(L_{C}\) increase if the threshold on \(\left\langle {\sigma^{2} } \right\rangle^{3/2}\) decreases. However, if the bias observed for \(\left\langle {\sigma^{2} } \right\rangle^{3/2} < 0.01\) is only due to instrumental effects, then \(L_{C}\) ~ 60–70 m should be representative of all \(\varepsilon_{U}\) levels.
From a pragmatic point of view, the analysis shown in Fig. 7a indicates that radar-derived TKE dissipation rates (hereafter denoted by \(\varepsilon_{R}\)) can be estimated from \(\left\langle {\sigma^{2} } \right\rangle\) using the expression:
$$\varepsilon_{R} \approx K \left\langle {\sigma^{2} } \right\rangle^{3/2}$$
where \(K \approx\) 0.016 at least for \(\left\langle {\sigma^{2} } \right\rangle^{3/2} > 0.01\) or for \(\varepsilon_{U} > \sim10^{ - 4} \,{\text{m}}^{2} \,{\text{s}}^{ - 3}\). From Eq. (9), 50.5% (72.5%, 86.5%) of \(\varepsilon_{R}\) estimates do not differ by more than a factor 2 (3, 5) from \(\varepsilon_{U}\) estimates. The spectral width is thus the dominant parameter when estimating \(\varepsilon\) from radar data. We will discuss this result in "Comparisons between \(\varepsilon_{U}\) and \(\varepsilon\) from the radar models" section.
Comparisons between \(\varepsilon_{U} \;{\text{and}}\;\varepsilon_{R}\)
Figure 8a shows examples of comparison results between \(\varepsilon_{U} {\text{ and }} \varepsilon_{R}\) profiles for FL16-T05 (A1) and FLT16-08 (A1). The agreements are remarkable. Note that there is about one order of magnitude of difference between the maximum values of TKE dissipation rates observed during FLT16-05 and FL16-T08. Figure 8b shows the result for FLT16-21 (A1). Since this flight was not included in the statistics leading to Eq. (9), these are absolute comparisons and it is the very first attempt at confirmation of Eq. (9). The agreement is again good, both in shape and levels, with a ratio of the two being much less than a factor 2 where \(\varepsilon_{U} \;{\text{and}}\;\varepsilon_{R}\) are maximum (above 2.5 km). The \(\varepsilon_{U}\) profiles fall within the range of variability of \(\varepsilon_{R}\) at almost all altitudes.
a Comparisons of smoothed profiles of \(\varepsilon_{U}\) (red solid and dashed lines) and \(\varepsilon_{R}\) (solid line) for FLT16-05 (A1) (left) and FLT16-08 (right). The gray areas show the mean ± rms values of \(\varepsilon_{U}\) for the 4 min of time averaging. b Same as a but or FLT16-21 (A1)
Figure 9a, b shows the results of comparisons in linear and logarithmic scales for all the 16 ShUREX 2016 and 23 ShUREX 2017 science flights, respectively, concatenated in chronological order. As expected, the largest discrepancies appear in logarithmic scale for small values (due to the minimum detectable value of \(\varepsilon_{R} \sim0.016\, \times \,0.001\, = \,1.6 \, \times \,10^{ - 5} \,{\text{m}}^{2} \,{\text{s}}^{ - 3}\)). Elsewhere, there is almost a one-to-one correspondence between all the peaks. Therefore, both the UAV and the radar detected the same turbulence events with similar \(\varepsilon\) intensities (assuming the validity of Eq. 9).
a Concatenation of all \(\varepsilon_{U}\) (red) and \(\varepsilon_{R}\)(black) profiles from the 2016 UAV flights in chronological order (the vertical dashed lines show the transition between the flights). (Top) linear scale. (Bottom) logarithmic scale. The blue dashed line shows the minimum level (~ \(1.6 \, \times \,10^{ - 5} \,{\text{m}}^{2} \,{\text{s}}^{ - 3}\)) detectable by the radar according to Fig. 3. b Same as a for the 2017 flights
The results shown in Figs. 7a, 8 and 9 include all altitudes and all turbulence events, without any distinction between shear-generated and convectively generated turbulence in clear air and cloudy conditions. It is not easy to separate the various turbulence events according to their source or nature, but it is possible to reject the following contributions at altitude ranges from PTU data and radar images:
where clouds were observed,
associated with cloudy or clear air convective boundary layers (CBL),
associated with convective layers underneath clouds [mid-level cloud base turbulence (MCT) layer (e.g., Kudo et al. 2015)].
These events (and clouds) were often associated with the largest values of TKE dissipation rates, but they constituted only 23% of the overall dataset. Thus, the datasets used for the analysis (Fig. 7a) contained mainly turbulence in clear air conditions outside regions potentially affected by cloud dynamics and CBL. Figure 7b shows the results after excluding the convective turbulence events. They do not strongly differ from those shown in Fig. 7a, indicating that the observations made from the overall datasets are also representative of the free atmosphere, in the absence of convection. In particular, the \(\sigma^{3}\) dependence is still observed, but with slightly smaller values of \(L_{C}\). The histogram of \(L_{C}\) (right panel of Fig. 7b) seems to have a double-peak distribution. The smaller one may not be representative (because likely due to residual contaminations, see left panel of Fig. 7b). The maximum of the larger distribution is around \(L_{C} \sim50\,{\text{m}}\). Finding a smaller value for stratified conditions only is not surprising since the convective layers are much deeper and should be associated with larger characteristic scales. The difference between the two estimates (60 and 50 m) is not very large, however. We will keep 60 m for the subsequent comparisons.
TKE dissipation rates and isotropy of the radar echoes
Figure 10 shows the scatter plot of \(\varepsilon_{R}\) and \(\varepsilon_{U}\) versus the radar echo power aspect ratio AR (dB) defined as \(\log 10\left( {P_{v} } \right) - \left( {\log 10\left( {P_{N} } \right) + \log 10\left( {P_{E} } \right)} \right)/2,\) where \(P_{v}\), \(P_{N}\) and \(P_{E}\) are echo power measured in the three radar beam directions (vertical, North and East, 10° off zenith). Echo powers were estimated and averaged over 4 min along the UAV flight tracks in the same way as \(\left\langle {\sigma^{2} } \right\rangle\) (see Fig. 1). A small aspect ratio (say, < 3 dB in absolute value) is generally considered as a signature of scattering from turbulence, which is isotropic or nearly so. This is especially true when the averaging time is short [aspect ratio is a statistical parameter that requires time averaging and horizontal homogeneity hypothesis]. It is striking to note that almost all values of \(\varepsilon_{U}\) > \(1.6\, \times \,10^{ - 4} \,{\text{m}}^{2} \,{\text{s}}^{ - 3}\) are associated with \(\left| {AR} \right|\, < \,3\,{\text{dB}}\), i.e., with isotropic echoes. This result is extremely consistent with the fact that the layers of enhanced TKE dissipation rates are associated with relatively deep layers of turbulence, isotropic at the Bragg scale. In such layers, the radar echoes are weakly (or even not) affected by specular reflectors. This result justifies a posteriori the use of data from the vertical beam for estimating \(\left\langle {\sigma^{2} } \right\rangle\). Incidentally, it is also an additional clue suggesting that the radar and the UAVs indeed detected the same, most prominent, turbulence events. For \(\varepsilon_{U}\) < \(1.6 \, \times \,10^{ - 4} \,{\text{m}}^{2} \,{\text{s}}^{ - 3}\) (i.e., for the weakest values), AR can be significantly larger than 3 dB. It is difficult to know if this property is due to anisotropic turbulence (consistent with weak turbulence) or due to the coarse resolution of the radar (thin layers of isotropic turbulence surrounded by stable layers within the radar volume).
Radar echo power aspect ratio (dB) defined as \(\log 10\left( {P_{v} } \right) - \left( {\log 10\left( {P_{N} } \right) + \log 10\left( {P_{E} } \right)} \right)/2\) calculated in the same way as \(\sigma^{2}\) ("Theoretical bases and practical methods of \(\varepsilon\) estimation" section) versus \(\log 10\left( {\varepsilon_{R} } \right)\) (black dots) and \(\log 10\left( {\varepsilon_{U} } \right)\) (red dots). The statistics was made on the ShUREX2016 flights. The blue dashed line (1) shows the minimum \(\varepsilon\) values detectable by the radar. The blue dotted dashed line shows the level equal to 10 times the minimum threshold
Comparisons between \(\varepsilon_{U}\) and \(\varepsilon_{N}\), \(\varepsilon_{W}\)
On the one hand, \(\varepsilon_{N}\) was estimated from Eq. (2) by calculating \(N^{2} = g/T\left( {{\text{d}}T/{\text{d}}z + \varGamma_{a} } \right)\) (\(\varGamma_{a} = 0.001 \text{ K m}^{-1}\) ) from IMET and CWT data at the radar range resolution (150 m) by applying a low-pass filter with a cutoff of 300 m on the temperature profiles. Examples of \(N^{2}\) (150 m) profiles estimated from IMET and CWT data for FLT16-05 (A1) and FLT16-08 (A1) are shown in Fig. 11. The \(y\) reveal that \(N^{2}\) is positive everywhere during FLT16-05 (A1) and negative in the height range 2.6–3.0 km during FLT16-08 (A1).
Vertical profiles of dry \(N^{2}\) calculated from IMET (solid) and CWT (dotted) sensors at a vertical resolution of 150 m (i.e., the range radar resolution) for FLT16-05 (A1) and FLT16-08 (A1)
On the other hand, \(\varepsilon_{W}\) can be estimated from Eq. (3) by performing a numerical integration of I for each altitude z (without \(L_{H}\)).
The top panel of Fig. 12 shows the superposition of \(\varepsilon_{U} , \;\varepsilon_{R} ,\; \varepsilon_{N} \;{\text{and}}\;\varepsilon_{W}\) profiles for FLT16-05 (A1) and FLT16-08 (A1). The \(\varepsilon_{W}\) profiles appear to be similar to the \(\varepsilon_{R}\) profiles but are underestimated, typically by a factor ~ 2 with respect to \(\varepsilon_{R}\) (and \(\varepsilon_{U}\)) at almost all altitudes and for both cases. The \(\varepsilon_{N}\) profiles exhibit more complex features. For FLT16-05 (A1), \(\varepsilon_{N}\) shows a good agreement in levels and shape with \(\varepsilon_{U}\). Therefore, from the analysis of a single profile, it can be concluded that \(\varepsilon_{N}\) provides reasonable agreement (at least in linear scale). However, for FLT16-08 (A1), \(\varepsilon_{N}\) is not defined in the altitude range (2.6-3.0 km) where \(\varepsilon_{U} ,\; \varepsilon_{R} \;{\text{and}}\;\varepsilon_{W}\) are found to be maximum because \(N^{2}\) is negative. \(\varepsilon_{N}\) is maximum on both sides due to \(N^{2}\) enhancements at the edges (Fig. 11). Here, \(\varepsilon_{N}\) provides radically different information on the turbulent state of the atmosphere between the altitudes of 2.6 km and 3.5 km. The corresponding time-height cross sections of radar echo power at high resolution and at vertical incidence, shown in the bottom panel of Fig. 12, permits us to better understand the differences between FLT16-05 and FLT16-08. During FLT16-05, stratified conditions were clearly observed but were quickly changing between ascent (A1) and descent (D1) (so that \(\varepsilon_{U}\) profiles were distinct during A1 and D1, see Fig. 6a). During FLT16-08, the UAV crossed a MCT layer in the altitude range 2.6–3.5 km. MCT layers are basically generated by a convective instability at the cloud base due to evaporative cooling from sublimating precipitation in the sub-cloud layer (e.g., Kudo et al. 2015). For such conditions, \(N^{2} < 0\) can be observed as is the case here, and \(N^{2}\) can be enhanced at the edges due to turbulent mixing. Therefore, \(\varepsilon_{N}\) fails to reproduce \(\varepsilon_{U}\) because it is not defined when \(N^{2} < 0\) in the core of the layer and because the model is inapplicable to convective turbulence.
(Top) Pseudo-vertical profiles of \(\varepsilon_{U} ,\;\varepsilon_{R} ,\;\varepsilon_{N}\) and \(\varepsilon_{W}\) for FLT16-05 (A1) and FLT16-08 (A1). For legibility, the standard deviation for the radar has been omitted. (Bottom) Time-height cross sections of radar echo power at vertical incidence around the two UAV flights. The green (red) lines show the distance (altitude) of the UAV with respect to the MU radar
Figure 13a, b shows information similar to Fig. 9a, b for all flights and all ascents and descents including \(\varepsilon_{N}\) (green line) and \(\varepsilon_{W}\) (black line), \(\varepsilon_{R}\) and \(\varepsilon_{U}\) being shown as red and blue lines, respectively. The figure confirms the tendencies shown in Fig. 12. The series of \(\varepsilon_{W}\) values reveals very similar features as the series of \(\varepsilon_{R}\) and \(\varepsilon_{U}\) with a slight but systematic underestimation. \(\varepsilon_{N}\) estimates are apparently consistent with the other estimates but discrepancies, such as those shown in Fig. 12, are difficult to see.
a Concatenation of all \(\varepsilon_{U}\) (blue), \(\varepsilon_{R}\) (red), \(\varepsilon_{W}\) (black) and \(\varepsilon_{N}\) (green) profiles from the ShUREX2016 UAV flights in chronological order. b Same as a for ShUREX2017 UAV flights
Figure 14 shows the histogram of differences (in logarithmic scales) between \(\varepsilon_{R}\) and \(\varepsilon_{U}\) (left panel), between \(\varepsilon_{W}\) and \(\varepsilon_{U}\) (middle panel), and between \(\varepsilon_{N}\) and \(\varepsilon_{U}\) (right panel), when \(\varepsilon_{N}\) is defined (i.e., \(N^{2} > 0)\) for \(\varepsilon \, > \,1.6 \times10^{ - 5} \,{\text{m}}^{2} \,{\text{s}}^{ - 3}\). The histogram for \(\log_{10} \left( {\varepsilon_{R} /\varepsilon_{U} } \right)\) shows a peak close to 0 (by definition). It shows the narrowest peak (minimum standard deviation) among the three estimates. From a statistical point of view, \(\varepsilon_{R}\) is thus the best estimate relative to \(\varepsilon_{U}\). The histogram for \(\log_{10} \left( {\varepsilon_{W} /\varepsilon_{U} } \right)\) shows a slightly wider distribution with a peak around -0.40 indicating an underestimation by a factor ~ 2.5 on average. The histogram for \(\log_{10} \left( {\varepsilon_{N} /\varepsilon_{U} } \right)\) shows a relatively wide distribution at ~ 0.16 indicating a slight overestimation of 1.44 on average. However, this result, favorable to \(\varepsilon_{N}\), hides an important bias. Figure 15 shows the results of regression and correlation analyses between \(\varepsilon_{U}\) and the three radar estimates assuming errors on both variables, and after rejecting dissipation rates smaller than \(1.6 \, \times \,10^{ - 5} \, {\text{m}}^{2} \,{\text{s}}^{ - 3}\) (because of the radar detection threshold and residual contaminations for weak values). The slopes of the regression line are close to 1, 0.91 and 0.92, between \(\log_{10} \left( {\varepsilon_{U} } \right)\) and \(\log_{10} \left( {\varepsilon_{R} } \right)\) and between \(\log_{10} \left( {\varepsilon_{U} } \right)\) and \(\log_{10} \left( {\varepsilon_{W} } \right)\), respectively, confirming reasonably, the \(\sigma^{3}\) dependence of TKE dissipation rates. The corresponding correlation coefficients are also nearly identical, at 0.8 and 0.78, respectively. As reported above, the main statistical difference between \(\varepsilon_{R}\) and \(\varepsilon_{W}\) is the slight bias in levels (factor ~ 2.5 with \(\varepsilon_{U}\)). In contrast, the regression analysis between \(\log_{10} \left( {\varepsilon_{U} } \right)\) and \(\log_{10} \left( {\varepsilon_{N} } \right)\) leads to a slope of 0.51 and the correlation coefficient is significantly smaller (0.69). Therefore, as expected, the model producing \(\varepsilon_{N}\) is not suitable, because it predicts a \(\sigma^{2}\) dependence of \(\varepsilon\). Although correct in average, the model tends to overestimate \(\varepsilon\) for \(\varepsilon_{U} < \sim4 \times10^{ - 4} \,{\text{m}}^{2} \,{\text{s}}^{ - 3}\) and to underestimate \(\varepsilon\) for \(\varepsilon_{U} > \sim4 \times10^{ - 4} \,{\text{m}}^{2} \,{\text{s}}^{ - 3}\). Note that detection threshold effects cannot explain the largest biases for small values, since the associated values were rejected by the regression analysis (but not in Fig. 13). The scatter plot obtained with a constant value of \(N^{2}\) equal to the mean value for the stratified regions sampled by the UAVs \( ( = 1.47 \, \times \,10^{ - 4} \text{ s}^{-2}), \) is superimposed to the bottom panel of Fig. 15 (red dots). The regression slope is very similar but the correlation coefficient is improved (0.8). Therefore, the tendency shown by the model producing \(\varepsilon_{N}\) does not depend on an accurate estimate of \(N^{2}\).
Histograms of \(\log_{10} \left( {\varepsilon_{R} /\varepsilon_{U} } \right)\) (left), \(\log_{10} \left( {\varepsilon_{W} /\varepsilon_{U} } \right)\) (middle), and \(\log_{10} \left( {\varepsilon_{N} /\varepsilon_{\text{UAV}} } \right)\) (right) corresponding to the values shown in Fig. 13 for all the ShUREX2016 and ShUREX2017 flights
Scatter plots of \(\varepsilon_{U}\) versus \(\varepsilon_{R}\) (top), \(\varepsilon_{W}\) (middle) and \(\varepsilon_{N}\) (bottom) after rejecting values smaller than \(1.6\, \times \,10^{ - 5} \, {\text{m}}^{2} \,{\text{s}}^{ - 3}\). The bottom panel shows the results with measured \(N^{2}\) and with \(N^{2} \, = \,1.47\, \times \,10^{ - 4} \, {\text{s}}^{ - 2}\)
Our statistical results suggest that the TKE dissipation rates estimated from the MU radar operated at a range resolution of 150 m and a beam aperture of 1.32° are proportional to \(\sigma^{3}\), for all atmospheric conditions in the lower atmosphere (up to ~ 4.0 km). The \(\sigma^{3}\) dependence is consistent with the results reported by Chen (1974) who showed from data collected by airplanes that TKE dissipation rates are proportional to \(\left\langle {w^{\prime 2} } \right\rangle^{3/2}\) at stratospheric heights. To some extent, it is also consistent with results reported by Bertin et al. (1997), who performed similar studies for stratospheric heights with the UHF Proust radar and high-resolution balloon data. However, partly due to the small amount of data, they suggest both \(\sigma^{2}\) and \(\sigma^{3}\) dependences for the same dataset (their Figs. 7b, 8). Jacoby-Koaly et al. (2002) compared TKE dissipation rates estimated from UHF (1238 MHz) radar and airplane data in CBL at a vertical resolution of 150 m by using the White et al. formulation. They found good statistical agreements (at least when using data from oblique beams, because data collected from the vertical beam were contaminated by ground clutter). McCaffrey et al. (2017) also compared dissipation rates estimated in the planetary boundary layer from two UHF (449 MHz and 915 MHz) radars using the White et al. (1999) model with values estimated from sonic anemometers mounted on a 300 m tower. Their results also tend to confirm a \(\sigma^{3}\) dependence for turbulence in the CBL.
In the present work, the \(\varepsilon_{N}\) model was found to be inadequate, although it has also been widely used with VHF radar observations (e.g., Hocking 1983, 1985, 1986, 1999, Fukao et al. 1994; Delage et al. 1997; Nastrom and Eaton 1997; Fukao et al. 2011, among many others). However, on average, and for stratified conditions, the model producing \(\varepsilon_{N}\) provides reasonable agreements with \(\varepsilon_{U}\), but it tends to overestimate (underestimate) TKE dissipation rates when turbulence is weak (large). Li et al. (2016) compared \(\varepsilon_{N}\) values obtained with the 53.5 MHz MAARSY radar at tropo-stratospheric heights with indirect estimates of dissipation rates from balloon data using a Thorpe analysis. Considering all the possible sources of discrepancies, the agreement was satisfying but the approach was likely not adapted for validating \(\varepsilon_{N}\) since the estimates from balloon data were themselves based on a model. Deghan et al. (2014) made studies similar to those presented here with a 40.68 MHz VHF radar and airplane observations mainly in the boundary layer. They also found reasonable agreements between \(\varepsilon_{N}\) and dissipation rates estimated from aircraft measurements (their Figs. 8, 12), sometimes with some noticeable difference in levels (a factor 5 in Fig. 12) and at coarser range resolution (500 m). In addition, they used standard values of \(N^{2}\) for some comparisons. This dataset may not have been sufficiently large for highlighting the biases produced by the model producing \(\varepsilon_{N}\) so that our respective results are not necessarily incompatible.
TKE dissipation rates \(\varepsilon\) were estimated from measurements made by high-frequency response Pitot sensors onboard UAVs and from MU radar Doppler spectra in the lower troposphere (up to ~ 4.0 km). The comparisons showed that:
Maxima of \(\varepsilon\) (\(\sim10^{ - 5} - 10^{ - 2} \,{\text{m}}^{2} \,{\text{s}}^{ - 3}\) typically, or \(\sim 0.01 - 10 {\text{ mW}}\,{\text{kg}}^{ - 1}\)) were observed at the same altitudes and times by the UAV and the radar indicating that the same turbulence events were generally sampled by both instruments at a horizontal distance of 1.0 km or less. Conditions were thus favorable to test the standard models used for estimating \(\varepsilon\) from radar data.
\(\varepsilon_{U}\) was found to be proportional to \(\sigma^{3}\), not \(\sigma^{2}\) as expected for stably stratified turbulence. The best agreement in turbulence levels was found by assuming that \(\varepsilon_{U} = K\sigma^{3}\) with \(K\sim0.016\). This surprisingly elementary model is equivalent to assuming a characteristic scale \(L_{C}\) of the order of 50–70 m. This scale is not necessarily related to an effective outer scale of turbulence because it seems to be appropriate for all cases, convectively or shear-generated turbulence, in deep convective layers or in stratified conditions. Therefore, it is likely not relevant to compare this scale (defined from dimensional analysis) with the dimensions 2a, 2b of the radar sampling volume (2b = 150 m, 2a = [53 m–156 m] in the height range 1.3–4.0 km) for selecting the right model. More refined analyses are necessary in order to know if there are really consistency problems or not. From a pragmatic point of view, this simple model can be used for estimating \(\varepsilon\) solely from measurements of VHF radar Doppler spectral width. It is likely accurate enough for climatological studies without the need for additional measurements of \(N^{2}\), at least for low tropospheric altitudes. All \(\varepsilon_{U}\) values larger than \(\sim1.6\, \times \,10^{ - 4 } \,{\text{m}}^{2} \,{\text{s}}^{ - 3}\) ( \(\sim0.16{\text{ mW}}\,{\text{kg}}^{ - 1}\)) were found to be associated with weak radar aspect ratios (\(\left| {AR} \right| < 3\,{\text{dB}}\)), which can reasonably be interpreted as backscatter from isotropic turbulence. It is thus consistent and justifies the use of Doppler spectra from the vertical beam when not altered by ground clutter. It means that data from oblique beams do not need to be used for estimating \(\varepsilon\) from radar data, even as weak as those reported in the present study and even at a vertical resolution of 150 m. The use of Doppler spectra measured from oblique directions requires many more corrections and accurate knowledge of horizontal wind shear, which can be the cause of additional uncertainties when estimating the Doppler variance produced by turbulence.
The key finding described in (b) does not mean that the Weinstock (1981) model adapted by Hocking (1983) is always irrelevant for stratified conditions. But it is likely not suitable for lower troposphere observations at a range resolution of 150 m with the MU radar and for the observed range of \(\varepsilon\) values.
\(\varepsilon_{W}\), based on the formulation proposed by Frisch and Clifford (1974) and subsequent authors, is also suitable since it predicts a \(\sigma^{3}\) dependence. However, it is a slight underestimate with respect to \(\varepsilon_{U}\) (\(\varepsilon_{U} \sim2.2 \varepsilon_{W}\) in average) but is not biased.
Finally, \(\varepsilon_{R}\) and \(\varepsilon_{W}\) were found consistent with \(\varepsilon_{U}\) in convective regions (such as MCT and CBL) while \(\varepsilon_{N}\) failed to reproduce the correct levels in the core and at the edges of the convectively generated turbulent layers. This is an additional argument for avoiding a systematic use of \(\varepsilon_{N}\) in the free atmosphere. Nevertheless, if used, climatological values of \(N^{2}\) seem to be preferable to measured values.
\(b = \Delta r/2\) (\(\Delta r\) = 150 m) and \(a = \theta_{0} z\) where z is altitude (m) and \(\theta_{0} = 1.32^\circ\) for the MU radar (e.g. Fukao et al. 1994).
AR:
above sea level
CBL:
convective boundary layer
Colorado University
CWT:
cold wire temperature
IMET:
international met (systems)
KOC:
Kolmogorov–Obukhov–Corrsin
MCT:
mid-level cloud base turbulence
stratosphere troposphere
MU:
middle and upper atmosphere
PTU:
pressure temperature humidity
ShUREX:
Shigaraki UAV radar experiment
SNR:
TKE:
turbulence kinetic energy
UHF:
ultra-high frequency
VHF:
very high frequency
UAV:
unmanned aerial vehicle
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HL performed all the radar and UAV data processing with assistance from HH and DL. LK led the ShUREX campaign and participated in the analysis and synthesis of the study results. DL was responsible for collection of the UAV data and AD provided the useable UAV data of ShUREX2017. All authors read and approved the final manuscript.
This study was supported by JSPS KAKENHI Grant No. JP15K13568 and the research Grant for Mission Research on Sustainable Humanosphere from Research Institute for Sustainable Humanosphere (RISH), Kyoto University. The MU radar belongs to and is operated by RISH, Kyoto University. The authors thank T. Mixa, M. Yabuki, R. Wilson and T. Tsuda for their cooperation during the campaigns and thank N. Nishi (Fukuoka University) for his kind assistance during the revision of the manuscript and his supply of meteorological data used for reviewers' reply.
This study was supported by JSPS KAKENHI Grant No. JP15K13568 and the research Grant for Mission Research on Sustainable Humanosphere from Research Institute for Sustainable Humanosphere (RISH), Kyoto University. It was also partly supported by the US National Science Foundation (Grant No. AGS 1632829). The MU radar belongs to and is operated by RISH, Kyoto University.
Mediterranean Institute of Oceanography (MIO), UM 110, UMR 7294, Université de Toulon, La Garde, France
Hubert Luce
Department of Aerospace Engineering Sciences, University of Colorado, Boulder, CO, USA
Lakshmi Kantha, Dale Lawrence & Abhiram Doddi
Research Institute for Sustainable Humanosphere, Kyoto University, Kyoto, Japan
Hiroyuki Hashiguchi
Lakshmi Kantha
Dale Lawrence
Abhiram Doddi
Correspondence to Hiroyuki Hashiguchi.
Appendix: Practical estimation of Eq. (6)
In this appendix, the detailed procedure used for obtaining Eq. (6) is given.
The Doppler spectra are first calculated for each of the 128 radar gates from complex time series of 128 points weighted by a Hanning window. The noise level for each radar gate is then estimated from Hildebrandt and Sekhon's method. The noise profile of 128 points is sorted and the average of the 3 smallest values are used a proxy of the noise level for all the radar gates. The moment method is then used to estimate the primary parameters (power, Doppler shift and spectral width). Because the Doppler spectra can be contaminated by any kind of outliers and in particular, UAV echoes (see Luce et al. 2017), all the Doppler spectra have been edited and manually corrected in order to reject, as far as possible, these contaminations. Most of time, the Doppler shift of UAV echoes did not coincide with the Doppler shift of atmospheric echoes, so that it was possible to reject these echoes by Doppler sorting. The rejection consisted in replacing the Doppler ranges on both sides of clear air atmospheric peak by the average noise power density defined above. The moment method was finally applied again, in order to get the definitive estimates of the primary parameters.
The turbulent component of the Doppler variance was then calculated as follows. The measured spectral half width (i.e., \(\sigma_{m}\)) is first converted to half power:
$$\sigma_{1/2} = \sqrt {2\ln 2} \times \sigma_{m}$$
After beam-broadening correction (\(\sigma_{b}^{2} = \left( {\bar{U}\theta_{0} } \right)^{2}\)), we obtain:
$$\sigma_{BB}^{2} = \sigma_{1/2}^{2} - \left( {\bar{U}\theta_{0} } \right)^{2}$$
where \(\theta_{0} = 1.32^\circ\) for the MU radar (e.g., Fukao et al. 1994) and \(\bar{U}\) is the mean wind speed estimated from the MU radar by using the 3 radial components. The time averaging was applied over Ta ~ 100 s (in practice, the results did not strongly depend on the choice of Ta). Finally, the Doppler variance due to turbulence is:
$$\sigma^{2} = \sigma_{BB}^{2} / \left( {2\ln 2} \right)$$
Luce, H., Kantha, L., Hashiguchi, H. et al. Turbulence kinetic energy dissipation rates estimated from concurrent UAV and MU radar measurements. Earth Planets Space 70, 207 (2018). https://doi.org/10.1186/s40623-018-0979-1
VHF radar
Energy dissipation rate
Outer scales of turbulence
Doppler variance
2. Aeronomy
Recent Advances in MST and EISCAT/Ionospheric Studies – Special Issue of the Joint MST15 and EISCAT18 Meetings, May 2017 | CommonCrawl |
Leonid Kantorovich
Leonid Vitalyevich Kantorovich (Russian: Леони́д Вита́льевич Канторо́вич, IPA: [lʲɪɐˈnʲit vʲɪˈtalʲjɪvʲɪtɕ kəntɐˈrovʲɪtɕ] (listen); 19 January 1912 – 7 April 1986) was a Soviet mathematician and economist, known for his theory and development of techniques for the optimal allocation of resources. He is regarded as the founder of linear programming. He was the winner of the Stalin Prize in 1949 and the Nobel Memorial Prize in Economic Sciences in 1975.
Leonid Kantorovich
Леони́д Канторо́вич
Leonid Kantorovich in 1975
Born
Leonid Vitalyevich Kantorovich
(1912-01-19)19 January 1912
Saint Petersburg, Russian Empire
Died7 April 1986(1986-04-07) (aged 74)
Moscow, RSFSR, Soviet Union
Resting placeNovodevichy Cemetery, Moscow
NationalitySoviet
Alma materLeningrad State University
Known forLinear programming
Kantorovich theorem
normed vector lattice (Kantorovich space)
Kantorovich metric
Kantorovich inequality
approximation theory
iterative methods
functional analysis
numerical analysis
scientific computing
AwardsSveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel (1975)
Scientific career
FieldsMathematics
InstitutionsUSSR Academy of Sciences
Leningrad State University
Doctoral advisorGrigorii Fichtenholz
Vladimir Smirnov
Doctoral studentsSvetlozar Rachev
Gennadii Rubinstein
Academic career
Information at IDEAS / RePEc
Biography
Kantorovich was born on 19 January 1912, to a Russian Jewish family.[1] His father was a doctor practicing in Saint Petersburg.[2] In 1926, at the age of fourteen, he began his studies at Leningrad State University. He graduated from the Faculty of Mathematics and Mechanics in 1930, and began his graduate studies. In 1934, at the age of 22 years, he became a full professor.
Later, Kantorovich worked for the Soviet government. He was given the task of optimizing production in a plywood industry. He devised the mathematical technique now known as linear programming in 1939, some years before it was advanced by George Dantzig. He authored several books including The Mathematical Method of Production Planning and Organization (Russian original 1939), The Best Uses of Economic Resources (Russian original 1959), and, with Vladimir Ivanovich Krylov, Approximate methods of higher analysis (Russian original 1936).[3] For his work, Kantorovich was awarded the Stalin Prize in 1949.
After 1939, he became a professor at Military Engineering-Technical University. During the Siege of Leningrad, Kantorovich was a professor at VITU of Navy and worked on safety of the Road of Life. He calculated the optimal distance between cars on ice in dependence of the thickness of ice and the temperature of the air. In December 1941 and January 1942, Kantorovich walked himself between cars driving on the ice of Lake Ladoga on the Road of Life to ensure that cars did not sink. However, many cars with food for survivors of the siege were destroyed by the German airstrikes. For his feat and courage Kantorovich was awarded the Order of the Patriotic War, and was decorated with the medal For Defense of Leningrad.
In 1948 Kantorovich was assigned to the atomic project of the USSR.
After 1960, Kantorovich lived and worked in Novosibirsk, where he created and took charge of the Department of Computational Mathematics in Novosibirsk State University.[4]
The Nobel Memorial Prize, which he shared with Tjalling Koopmans, was given "for their contributions to the theory of optimum allocation of resources."
Mathematics
In mathematical analysis, Kantorovich had important results in functional analysis, approximation theory, and operator theory.
In particular, Kantorovich formulated some fundamental results in the theory of normed vector lattices, especially in Dedekind complete vector lattices called "K-spaces" which are now referred to as "Kantorovich spaces" in his honor.
Kantorovich showed that functional analysis could be used in the analysis of iterative methods, obtaining the Kantorovich inequalities on the convergence rate of the gradient method and of Newton's method (see the Kantorovich theorem).
Kantorovich considered infinite-dimensional optimization problems, such as the Kantorovich-Monge problem in transport theory. His analysis proposed the Kantorovich-Rubinstein metric, which is used in probability theory, in the theory of the weak convergence of probability measures.
• Portrait by Petrov-Vodkin. 1938.
• 1976
• Original CIA file on Kantorovich, seized from the former US Embassy in Tehran.
See also
• List of Russian mathematicians
• List of economists
• Shadow price
• List of Jewish Nobel laureates
Notes
1. The Soviet Union: empire, nation, and system, By Aron Kat︠s︡enelinboĭgen, page 406, Transaction Publishers, 1990
2. Gass, Saul I.; Rosenhead, J. (2011). "Leonid Vital'evich Kantorovich". Profiles in Operations Research. International Series in Operations Research & Management Science. Vol. 147. p. 157. doi:10.1007/978-1-4419-6281-2_10. ISBN 978-1-4419-6280-5.
3. Kaplan, W. (1960). "Review of Approximate methods of higher analysis by L. V. Kantorovich and V. I. Krylov". Bull. Amer. Math. Soc. 66 (3): 146–147. doi:10.1090/S0002-9904-1960-10408-9.
4. Kantorovich`s biography in Russian
References
• Makarov, V. (1987). "Kantorovich, Leonid Vitaliyevich". The New Palgrave: A Dictionary of Economics. 3: 14–15.
• Kantorovich, L.V. (1939). "Mathematical Methods of Organizing and Planning Production". Management Science. 6 (4): 366–422. doi:10.1287/mnsc.6.4.366. JSTOR 2627082.
• Kantorovich, L.V. (1959). "The Best Use of Economic Resources"(). Pergamon Press, 1965.
• Klaus Hagendorf (2008). Spreadsheet presenting all examples of Kantorovich, 1939 with the OpenOffice.org Calc Solver as well as the lp_solver.
Nobel prize lecture
• Kantorovich, Leonid, "Mathematics in Economics: Achievements, Difficulties, Perspectives", Nobel Prize lecture, December 11, 1975
• "Autobiography: Leonid Kantorovich", Nobel Prize website
Further reading
• Dantzig, George, Linear programming and extensions. Princeton University Press and the RAND Corporation, 1963. Cf. p.22 for the work of Kantorovich.
• Isbell, J.R.; Marlow, W.H., "On an Industrial Programming Problem of Kantorovich", Management Science, Vol. 8, No. 1 (Oct., 1961), pp. 13–17
• Kantorovich, L. V. "My journey in science (supposed report to the Moscow Mathematical Society)" [expanding Russian Math. Surveys 42 (1987), no. 2, pp. 233–270]. pp. 8–45. MR 0898626.
• Koopmans, Tjalling C., "Concepts of optimality and their uses", Nobel Memorial Lecture, December 11, 1975
• Kutateladze, S.S., "The World Line of Kantorovich", Notices of the ISMS, International Society for Mathematical Sciences, Osaka, Japan, January 2007
• Kutateladze, S.S., "Kantorovich's Phenomenon", Siberian Math. J. (Сибирский мат. журн.), 2007, V. 48, No. 1, 3–4, November 29, 2006.
• Kutateladze, S.S., "Mathematics and Economics of Kantorovich"
• Kutateladze, S.S., "My Kantorovich"
• Leifman, Lev J., ed. (1990). Functional analysis, optimization, and mathematical economics: A collection of papers dedicated to the memory of Leonid Vitalʹevich Kantorovich. New York: The Clarendon Press, Oxford University Press. pp. xvi+341. ISBN 0-19-505729-5. MR 1082562.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Makarov, V. L. [Valery Leonidovich]; Sobolev, S. L. "Academician L. V. Kantorovich (19 January 1912 to 7 April 1986)". In: Functional analysis, optimization, and mathematical economics: A collection of papers dedicated to the memory of Leonid Vital'evich Kantorovich. pp. 1–7. MR 1082564.
• Polyak, B. T. (2002). "History of mathematical programming in the USSR: Analyzing the phenomenon (Chapter 3 The pioneer: L. V. Kantorovich, 1912–1986, pp. 405–407)". Mathematical Programming. Series B. 91 (3): 401–416. doi:10.1007/s101070100258. MR 1888984. S2CID 13089965.
• Ivan Boldyrev and Till Düppe, Programming the USSR: Leonid V. Kantorovich in context, The British Journal for the History of Science. 2020. 53(2): 255-278.
• Spufford, Francis (2010). Red plenty. London: Faber.
• (in Russian) Kutateladze, S.S., et al., "Leonid V. Kantorovich (1912–1986)", Sobolev Institute of Mathematics of the Siberian Division of the Russian Academy of Sciences. Also published in the Siberian Mathematical Journal, Volume 43 (2002), No. 1, pp. 3–8
• (in Russian) Vershik, Anatoly, "On Leonid Kantorovich and linear programming"
External links
Wikiquote has quotations related to Leonid Kantorovich.
• Leonid Kantorovich at the Mathematics Genealogy Project
• O'Connor, John J.; Robertson, Edmund F., "Leonid Kantorovich", MacTutor History of Mathematics Archive, University of St Andrews (With additional photos.)
• Information about: Leonid Vitaliyevich Kantorovich – IDEAS/RePEc
• Leonid Vitalievich Kantorovich (1912–1986). 2008. {{cite book}}: |work= ignored (help)
• Biography Leonid Kantorovich from the Institute for Operations Research and the Management Sciences
• Biographical documentary about L.Kantorovich by Rossiya-Culture
• Leonid Kantorovich on Nobelprize.org
Laureates of the Sveriges Riksbank Prize in Economic Sciences
1969–1975
• 1969: Ragnar Frisch / Jan Tinbergen
• 1970: Paul A. Samuelson
• 1971: Simon Kuznets
• 1972: John R. Hicks / Kenneth J. Arrow
• 1973: Wassily Leontief
• 1974: Gunnar Myrdal / Friedrich August von Hayek
• 1975: Leonid Vitaliyevich Kantorovich / Tjalling C. Koopmans
1976–2000
• 1976: Milton Friedman
• 1977: Bertil Ohlin / James E. Meade
• 1978: Herbert A. Simon
• 1979: Theodore W. Schultz / Sir Arthur Lewis
• 1980: Lawrence R. Klein
• 1981: James Tobin
• 1982: George J. Stigler
• 1983: Gérard Debreu
• 1984: Richard Stone
• 1985: Franco Modigliani
• 1986: James M. Buchanan Jr.
• 1987: Robert M. Solow
• 1988: Maurice Allais
• 1989: Trygve Haavelmo
• 1990: Harry M. Markowitz / Merton H. Miller / William F. Sharpe
• 1991: Ronald H. Coase
• 1992: Gary S. Becker
• 1993: Robert W. Fogel / Douglass C. North
• 1994: John C. Harsanyi / John F. Nash Jr. / Reinhard Selten
• 1995: Robert E. Lucas Jr.
• 1996: James A. Mirrlees / William Vickrey
• 1997: Robert C. Merton / Myron S. Scholes
• 1998: Amartya Sen
• 1999: Robert A. Mundell
• 2000: James J. Heckman / Daniel L. McFadden
2001–present
• 2001: George A. Akerlof / A. Michael Spence / Joseph E. Stiglitz
• 2002: Daniel Kahneman / Vernon L. Smith
• 2003: Robert F. Engle III / Clive W.J. Granger
• 2004: Finn E. Kydland / Edward C. Prescott
• 2005: Robert J. Aumann / Thomas C. Schelling
• 2006: Edmund S. Phelps
• 2007: Leonid Hurwicz / Eric S. Maskin / Roger B. Myerson
• 2008: Paul Krugman
• 2009: Elinor Ostrom / Oliver E. Williamson
• 2010: Peter A. Diamond / Dale T. Mortensen / Christopher A. Pissarides
• 2011: Thomas J. Sargent / Christopher A. Sims
• 2012: Alvin E. Roth / Lloyd S. Shapley
• 2013: Eugene F. Fama / Lars Peter Hansen / Robert J. Shiller
• 2014: Jean Tirole
• 2015: Angus Deaton
• 2016: Oliver Hart / Bengt Holmström
• 2017: Richard H. Thaler
• 2018: William Nordhaus / Paul Romer
• 2019: Abhijit Banerjee / Esther Duflo / Michael Kremer
• 2020: Paul Milgrom / Robert B. Wilson
• 2021: David Card / Joshua Angrist / Guido Imbens
• 2022: Ben Bernanke / Douglas Diamond / Philip H. Dybvig
1975 Nobel Prize laureates
Chemistry
• John Cornforth (United Kingdom/Australia)
• Vladimir Prelog (Switzerland)
Literature (1975)
• Eugenio Montale (Italy)
Peace
• Andrei Sakharov (Soviet Union)
Physics
• Aage Bohr (Denmark)
• Ben Roy Mottelson (Denmark)
• James Rainwater (United States)
Physiology or Medicine
• David Baltimore (United States)
• Renato Dulbecco (United States)
• Howard Martin Temin (United States)
Economic Sciences
• Leonid Kantorovich (Soviet Union)
• Tjalling Koopmans (United States)
Nobel Prize recipients
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| Wikipedia |
\begin{document}
\begin{abstract} In this paper we prove that the \emph{linear Koszul duality} isomorphism for convolution algebras in $\mathsf{K}$-homology of \cite{MR3} and the \emph{Fourier transform} isomorphism for convolution algebras in Borel--Moore homology of \cite{EM} are related by the Chern character. So, Koszul duality appears as a categorical upgrade of Fourier transform of constructible sheaves. This result explains the connection between the categorification of the Iwahori--Matsumoto involution for graded affine Hecke algebras in \cite{EM} and for ordinary affine Hecke algebras in \cite{MR3}. \end{abstract}
\title[Linear Koszul duality and Fourier transform]{Linear Koszul duality and Fourier transform \\ for convolution algebras}
\author{Ivan Mirkovi\'c} \address{University of Massachusetts, Amherst, MA.} \email{[email protected]}
\author{Simon Riche} \address{Universit{\'e} Clermont Auvergne, Universit{\'e} Blaise Pascal, Laboratoire de Math{\'e}matiques, BP 10448, F-63000 Clermont-Ferrand, France -- CNRS, UMR 6620, LM, F-63178 Aubi{\`e}re, France.}
\email{[email protected]}
\maketitle
\section*{Introduction}
\subsection{}
This article is a sequel to \cite{MR, MR2, MR3}. It links two kinds of ``Fourier'' transforms prominent in mathematics, the Fourier transform for constructible sheaves and the Koszul duality. This is done in a particular situation which is of interest in representation theory, namely the context of convolution algebras.
\subsection{Chern character map}
Our geometric setting consists of two vector subbundles $F_1,F_2$ of a trivial vector bundle $X\times V$ over a (smooth and proper) complex algebraic variety $X$. We consider the fiber product $F_1\times_V F_2$ as well as the dual object -- the fiber product $F_1^\perp \times_{V^*} F_2^\perp$ of orthogonal complements of $F_1$ and $F_2$ inside the dual vector bundle $X\times V^*$. The \emph{linear Koszul duality} mechanism from \cite{MR, MR2, MR3} is a geometric version of the standard Koszul duality between graded modules over the symmetric algebra of a vector space and graded modules over the exterior algebra of the dual vector space. Here, this formalism provides an equivalence of categories of equivariant coherent sheaves on the derived fiber products $F_1\, \aa{R}\tim_V\, F_2$ and $F_1^\perp \, \aa{R}\tim_{V^*}\, F_2^\perp$ (in the sense of dg-schemes). In particular we get an isomorphism of equivariant $\mathsf{K}$-homology groups of algebraic varieties $F_1\times_V F_2$ and $F_1^\perp \times_{V^*} F_2^\perp$.\footnote{ Note that $\mathsf{K}$-homology does not distinguish the derived fiber product from the usual fiber product of varieties, see \cite{MR3}. } On the other hand, the Fourier transform for constructible sheaves provides an isomorphism of equivariant Borel--Moore homologies of fiber products $F_1\times_V F_2$ and $F_1^\perp \times_{V^*} F_2^\perp$, see \cite{EM}.
Our main result shows that the maps in $\mathsf{K}$-homology and in Borel--Moore homology are related by the Chern character map (the ``Riemann--Roch map'') from equivariant $\mathsf{K}$-homology to (completed) equivariant Borel--Moore homology.\footnote{ For simplicity we work under a technical assumption on $F_i$'s which is satisfied in all known applications. } In this way, linear Koszul duality appears as a categorical upgrade of the topological Fourier transform.
\subsection{Convolution algebras}
In Representation Theory the above setting provides a geometric construction of algebras. Indeed, when $F_1=F_2=:F$ then the equivariant $\mathsf{K}$-homology and Borel--Moore homology of $F \times_V F$ have structures of convolution algebras; for simplicity in this introduction we denote these $A_{\mathsf{K}}(F)$ and $A_{\mathsf{BM}}(F)$. The Chern character provides a map of algebras $A_{\mathsf{K}}(F) \to \widehat{A}_{\mathsf{BM}}(F)$ from the $\mathsf{K}$-homology algebra to a completion of the Borel--Moore homology algebra \cite{CG, Ka}. This gives a strong relation between their representation theories: one obtains results on the representation theory of the (more interesting) algebra $A_{\mathsf{K}}(F)$ through the relation to the representation theory of the algebra $A_{\mathsf{BM}}(F)$ which is more accessible.\footnote{ The reason is the powerful machinery of perverse sheaves that one can use in the topological setting, see \cite{CG}. } In this setting, the maps \[ \imath_{\mathsf{K}} \colon A_{\mathsf{K}}(F) \xrightarrow{\sim} A_{\mathsf{K}}(F^\perp), \qquad \imath_{\mathsf{BM}} \colon A_{\mathsf{BM}}(F) \xrightarrow{\sim} A_{\mathsf{BM}}(F^\perp) \] induced respectively by linear Koszul duality and by Fourier transform are isomorphisms of algebras.
An important example of this mechanism appears in the study of affine Hecke algebras, see \cite{KL, CG}. The Steinberg variety $Z$ of a complex connected reductive algebraic group $G$ (with simply connected derived subgroup) is of the above form $F \times_V F$ where the space $X$ is the flag variety $\BB$ of $G$, the vector space $V$ is the dual $\mathfrak{g}^*$ of the Lie algebra $\mathfrak{g}$ of $G$, and $F$ is the cotangent bundle $T^*\BB$. The $G \times {\mathbb{G}}_{\mathbf{m}}$-equivariant $\mathsf{K}$-homology and Borel--Moore homology of the Steinberg variety $Z$ are then known to be realizations of the \emph{affine Hecke algebra} $\mathcal{H}_{\aff}$ of the dual reductive group $\check{G}$ (with equal parameters) and of the corresponding \emph{graded affine Hecke algebra} $\overline{\mathcal{H}}_{\aff}$. In this case the dual version $F^\perp \times_{V^*} F^\perp$ turns out to be another -- homotopically equivalent -- version of the Steinberg variety $Z$. Therefore, $\imath_{\mathsf{K}}$ and $\imath_{\mathsf{BM}}$ are automorphisms of
$\mathcal{H}_{\aff}$ and $\overline{\mathcal{H}}_{\aff}$, respectively. In fact these are (up to minor ``correction factors'') geometric realizations of the Iwahori--Matsumoto involution of $\overline{\mathcal{H}}_{\aff}$ (see \cite{EM}) and $\mathcal{H}_{\aff}$ (see \cite{MR3}). The Chern character map can also be identified, in this case, with (a variant of) a morphism constructed (by algebraic methods) by Lusztig~\cite{LuAff}. So, in this situation, Theorem \ref{thm:LKDFourier} explains the relation between results of \cite{MR3} and \cite{EM}.
\subsection{Character cycles and characteristic cycles}
In \cite{Kas}, Kashiwara introduced for a group $G$ acting on a space $X$ an invariant of a $G$-equivariant constructible sheaf $\FF$ on $X$. This is an element $\mathrm{ch}_G(\FF)$ of the Borel--Moore homology of the stabilizer space $G_X := \{(g,x)\in G\tim X \mid g \cdot x=x\}$. He ``linearized'' this construction to an element $\mathrm{ch}_\mathfrak{g}(\FF)$ of the Borel--Moore homology of the analogous stabilizer space $\mathfrak{g}_X$ for the Lie algebra $\mathfrak{g}$ of the group $G$. Under some assumptions (that put one in the above geometric setting) he proved that the characteristic cycle of $\FF$ is the image of $\mathrm{ch}_\mathfrak{g}(\FF)$ under a Fourier transform map in Borel--Moore homology (see \cite[\S 1.9]{Kas}). This work is the origin of papers on Iwahori--Matsumoto involution \cite{EM} and linear Koszul duality \cite{MR}. From this point of view, the present paper is a part of the effort to categorify Kashiwara's character cycles.
\subsection{Conventions and notation}
In the body of the paper we will consider many morphisms involving $\mathsf{K}$-homology and Borel--Moore homology. We use the general convention that morphisms involving only $\mathsf{K}$-homology are denoted using bold letters, those involving only Borel--Moore homology are denoted using fraktur letters, and the other ones are denoted using ``sans serif'' letters.
If $X$ is a complex algebraic variety endowed with an action of a reductive algebraic group $A$, we denote by $\mathsf{Coh}^A(X)$ the category of $A$-equivariant coherent sheaves on $X$. If $Y \subset X$ is an $A$-stable closed subvariety we denote by $\mathsf{Coh}_Y^A(X)$ the subcategory consisting of sheaves supported set-theoretically on $Y$; recall that $\mathcal{D}^b \mathsf{Coh}_Y^A(X)$ identifies with a full subcategory in $\mathcal{D}^b \mathsf{Coh}^A(X)$. When considering ${\mathbb{G}}_{\mathbf{m}}$-equivariant coherent sheaves, we denote by $\langle 1 \rangle$ the functor of tensoring with the tautological $1$-dimensional ${\mathbb{G}}_{\mathbf{m}}$-module.
\subsection{Organization of the paper}
In Section \ref{sec:definitions} we define all our morphisms, and state our main result (Theorem \ref{thm:LKDFourier}). In Section~\ref{sec:convolution-algebras} we study more closely the case of convolution algebras, and even more closely the geometric setting for affine Hecke algebras; in this case we make all the maps appearing in Theorem \ref{thm:LKDFourier} explicit. In Sections \ref{sec:compatibility-Fourier} and \ref{sec:compatibility-others} we prove some compatibility statements for our constructions, and we apply these results in Section \ref{sec:proof} to the proof of Theorem \ref{thm:LKDFourier}. Finally, Appendix \ref{sec:appendix} contains the proofs of some technical lemmas needed in other sections.
\section{Definitions and statement} \label{sec:definitions}
\subsection{Equivariant homology and cohomology} \label{ss:homology-cohomology}
If $A$ is a complex linear algebraic group acting on a complex algebraic variety $Y$, we denote by $\mathcal{D}^A_{\mathrm{const}}(Y)$ the $A$-equivariant derived category of constructible complexes on $Y$ with complex coefficients, see \cite{BL}. Let $\underline{\mathbb{C}}_Y$, respectively $\underline{\mathbb{D}}_Y$, be the constant, respectively dualizing, sheaf on $Y$. These are objects of $\mathcal{D}^A_{\mathrm{const}}(Y)$. We also denote by $\mathbb{D}_Y \colon \mathcal{D}^A_{\mathrm{const}}(Y) \xrightarrow{\sim} \mathcal{D}^A_{\mathrm{const}}(Y)^{\mathrm{op}}$ the Grothendieck--Verdier duality functor.
If $M$ is in $\mathcal{D}^A_{\mathrm{const}}(Y)$, the $i$-th equivariant cohomology of $Y$ with coefficients in $M$ is by definition \[ \mathsf{H}^i_A(Y, M) := {\rm Ext}^i_{\mathcal{D}^A_{\mathrm{const}}(Y)}(\underline{\mathbb{C}}_Y,M). \] In particular, the equivariant cohomology and Borel--Moore homology of $Y$ are defined by \[ \mathsf{H}_A^i(Y) := \mathsf{H}_A^i(Y,\underline{\mathbb{C}}_Y), \quad \mathsf{H}_i^A(Y) := \mathsf{H}_A^{-i}(Y,\underline{\mathbb{D}}_Y). \] We will also use the notation \begin{align*} \mathsf{H}_A^\bullet(Y) \ := \ \bigoplus_{i \in \mathbb{Z}} \, \mathsf{H}^i_A(Y), \quad & \widehat{\mathsf{H}}_{A}^\bullet(Y) \ := \ \prod_{i \in \mathbb{Z}} \, \mathsf{H}^i_A(Y), \\ \mathsf{H}^A_\bullet(Y) \ := \ \bigoplus_{i \in \mathbb{Z}} \, \mathsf{H}_i^A(Y), \quad & \widehat{\mathsf{H}}^{A}_\bullet(Y) \ := \ \prod_{i \in \mathbb{Z}} \, \mathsf{H}_i^A(Y). \end{align*} (By construction of the equivariant derived category, see~\cite[\S 2.2]{BL}, these definitions coincide -- up to grading shift -- with the definitions used e.g.~in~\cite{LuCus1, EG, BZ} using some ``approximations'' of $EA$.) Note that with our conventions, one can have $\mathsf{H}_i^A(Y) \neq 0$ for $i<0$. We will use the general convention that we denote by the same symbol an homogeneous morphism between vector spaces of the form $\mathsf{H}^A_\bullet(\cdot)$ or $\mathsf{H}_A^\bullet(\cdot)$ and the induced morphism between the associated vector spaces $\widehat{\mathsf{H}}^A_\bullet(\cdot)$ or $\widehat{\mathsf{H}}_A^\bullet(\cdot)$.
The vector spaces $\mathsf{H}_A^\bullet(Y)$ and $\mathsf{H}^A_\bullet(Y)$ have natural gradings, and most morphisms between such spaces that will occur in this paper will be homogeneous. We will sometimes write a morphism e.g.~as $\mathsf{H}_A^\bullet(Y) \to \mathsf{H}_A^{\bullet+d}(Y')$ to indicate that it shifts degrees by $d$.
There exists a natural (right) action of the algebra $\mathsf{H}_A^\bullet(Y)$ on $\mathsf{H}^A_\bullet(Y)$ induced by composition of morphisms in $\mathcal{D}^A_{\mathrm{const}}(Y)$; it extends to an action of the algebra $\widehat{\mathsf{H}}_A^\bullet(Y)$ on $\widehat{\mathsf{H}}^A_\bullet(Y)$.
We will also denote by $\mathsf{K}^A(Y)$ the $A$-equivariant $\mathsf{K}$-homology of $Y$, i.e.~the Grothendieck group of the category of $A$-equivariant coherent sheaves on $Y$.
We will frequently use the following classical constructions. If $Z$ is another algebraic variety endowed with an action of $A$, and if $f \colon Z \to Y$ is a proper $A$-equivariant morphism, then there exist natural ``proper direct image'' morphisms \[ \mathbf{pdi}_f \colon \mathsf{K}^A(Z) \to \mathsf{K}^A(Y), \qquad \text{resp.} \qquad \mathfrak{pdi}_f \colon \mathsf{H}^A_\bullet(Z) \to \mathsf{H}^A_\bullet(Y), \] see \cite[\S 5.2.13]{CG}, resp.~ \cite[\S 2.6.8]{CG}.\footnote{Only \emph{non-equivariant} Borel--Moore homology is considered in~\cite{CG}. However, the constructions for equivariant homology are deduced from these, since the equivariant homology of $Y$ can be described in terms of ordinary homology of various spaces of the form $U \times^A Y$ where $U$ is an ``approximation'' of $EA$, see e.g.~\cite[\S 2.8]{EG1}. \label{fn:equiv-homology}} Each of these maps satisfies a projection formula; in particular for $c \in \mathsf{H}_A^\bullet(Y)$ and $d \in \mathsf{H}^A_\bullet(Z)$ we have \begin{equation} \label{eqn:proj-formula-H} \mathfrak{pdi}_f(d \cdot f^*(c)) = \mathfrak{pdi}_f(d) \cdot c, \end{equation} where $f^* \colon \mathsf{H}_A^\bullet(Y) \to \mathsf{H}_A^\bullet(Z)$ is the natural pullback morphism.
On the other hand, if $Y$ is smooth, $Y' \subset Y$ is an $A$-stable smooth closed subvariety, and $Z \subset Y$ is a not necessarily smooth $A$-stable closed subvariety, then we have ``restriction with supports'' morphisms \[ \mathbf{res} \colon \mathsf{K}^A(Z) \to \mathsf{K}^A(Z \cap Y'), \qquad \text{resp.} \qquad \mathfrak{res} \colon \mathsf{H}^A_\bullet(Z) \to \mathsf{H}^A_{\bullet-2\dim(Y)+2\dim(Y')}(Z \cap Y') \] associated with the inclusion $Y' \hookrightarrow Y$, see \cite[p.~246]{CG}, resp.~\cite[\S 2.6.21]{CG}. (The definition of the second morphism is recalled in \S\ref{ss:restriction-with-supports}.) Note that the morphism $\mathfrak{res}$ satisfies the formula \begin{equation} \label{eqn:res-cohomology} \mathfrak{res}(c \cdot d) = \mathfrak{res}(c) \cdot i^*(d) \end{equation} for $c \in \mathsf{H}^A_\bullet(Z)$ and $d \in \mathsf{H}^\bullet_A(Z)$, where $i \colon Z \cap Y' \hookrightarrow Z$ is the embedding and $i^*$ is the pullback in cohomology as in~\eqref{eqn:proj-formula-H}. (In the non-equivariant setting, this follows from~\cite[Equation~(2.6.41)]{CG} and the definition of $\mathfrak{res}$ in~\cite[\S 2.6.21]{CG}; the equivariant case follows using the remark in Footnote~\ref{fn:equiv-homology}.)
Finally, if $E \to Y$ is an $A$-equivariant vector bundle, then we have the Thom isomorphism \[ \mathsf{H}^A_\bullet(E) \cong \mathsf{H}^A_{\bullet-2{\rm rk}(E)}(Y). \]
\subsection{Fourier--Sato transform} \label{ss:Fourier-transform}
Let again $A$ be a complex linear algebraic group, and let $Y$ be an $A$-variety. If $r \colon E \to Y$ is an $A$-equivariant (complex) vector bundle, we equip it with an $A \times {\mathbb{G}}_{\mathbf{m}}$-action where $t \in {\mathbb{G}}_{\mathbf{m}}$ acts by multiplication by $t^{-2}$ along the fibers of $r$. We denote by $E^\diamond$ the $A \times {\mathbb{G}}_{\mathbf{m}}$-equivariant dual vector bundle (so that $t \in {\mathbb{G}}_{\mathbf{m}}$ acts by multiplication by $t^2$ along the fibers of the projection to $Y$), and by $E^*$ the dual $A$-equivariant vector bundle, which we equip with a ${\mathbb{G}}_{\mathbf{m}}$-action where $t \in {\mathbb{G}}_{\mathbf{m}}$ acts by multiplication by $t^{-2}$ along the fibers. We denote by ${\check r}\colon E^* \to X$ the projection.
The Fourier--Sato transform defines an equivalence of categories \begin{equation} \label{eqn:fourier1} \mathfrak{F}_E\colon \mathcal{D}^{A \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(E) \ \xrightarrow{\sim} \ \mathcal{D}^{A \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(E^{\diamond}). \end{equation} This equivalence is constructed as follows (see \cite[\S 3.7]{KS}; see also \cite[\S 2.7]{AHJR} for a reminder of the main properties of this construction). Let $Q:=\{(x,y) \in E \times_Y E^\diamond \mid \mathrm{Re}(\langle x,y \rangle) \leq 0\}$, and let $q\colon Q \to E$, ${\check q} \colon Q \to E^\diamond$ be the projections. Then we have \[ \mathfrak{F}_E:={\check q}_! q^*. \] (This equivalence is denoted $(\cdot)^\wedge$ in \cite{KS}; it differs by a cohomological shift from the equivalence $\mathbb{T}_E$ of \cite{AHJR}.)
Inverse image under the automorphism of $A \times {\mathbb{G}}_{\mathbf{m}}$ which sends $(g,t)$ to $(g,t^{-1})$ establishes an equivalence of categories \begin{equation} \label{eqn:fourier2} \mathcal{D}^{A \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(E^\diamond) \ \xrightarrow{\sim} \ \mathcal{D}^{A \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(E^*), \end{equation} see~\cite[Chap.~6]{BL}. We will denote by \[ \mathcal{F}_E \colon \mathcal{D}^{A \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(E) \xrightarrow{\sim} \mathcal{D}^{A \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(E^*) \] the composition of \eqref{eqn:fourier1} and \eqref{eqn:fourier2}.
Let $F \subset E$ be an $A$-stable subbundle, and denote by $F^\bot \subset E^*$ the orthogonal to $F$. Then one can consider the constant sheaf $\underline{\mathbb{C}}_F$ as an object of $\mathcal{D}^{A \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(E)$. (Here and below, we omit direct images under closed inclusions when no confusion is likely.) Similarly, we have the object $\underline{\mathbb{C}}_{F^\bot}$ of $\mathcal{D}^{A \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(E^*)$. The following result is well known; we reproduce the proof for future reference.
\begin{lem} \label{lem:fourier-F} There exists a canonical isomorphism \[ \mathcal{F}_E(\underline{\mathbb{C}}_F) \cong \underline{\mathbb{C}}_{F^\bot}[-2\mathrm{rk}(F)]. \] \end{lem}
\begin{proof} It is equivalent to prove a similar isomorphism for $\mathfrak{F}_E$. For simplicity we denote $F^\bot$ by the same symbol when it is considered as a subbundle of $E^\diamond$.
By definition of $\mathfrak{F}_E$ we have a canonical isomorphism \[ \mathfrak{F}_E(\underline{\mathbb{C}}_F) \cong {\check q}_{F!} \underline{\mathbb{C}}_{Q_F}, \] where $Q_F:=q^{-1}(F) \subset Q$ and ${\check q}_F$ is the composition of ${\check q}$ with the inclusion $Q_F \hookrightarrow Q$. There is a natural closed embedding $i_F \colon F \times_Y F^\bot \hookrightarrow Q_F$; we denote by $U_F$ the complement and by $j_F \colon U_F \hookrightarrow Q_F$ the inclusion. The natural exact triangle $j_{F!} \underline{\mathbb{C}}_{U_F} \to \underline{\mathbb{C}}_{Q_F} \to i_{F*} \underline{\mathbb{C}}_{F \times_X F^\bot} \xrightarrow{+1}$ provides an exact triangle \[ q_{F!} j_{F!} \underline{\mathbb{C}}_{U_F} \to q_{F!} \underline{\mathbb{C}}_{Q_F} \to q_{F_!} i_{F!} \underline{\mathbb{C}}_{F \times_X F^\bot} \xrightarrow{+1}. \] Using the fact that $\mathsf{H}_c^\bullet(\mathbb{R}_{\geq 0}; \C)=0$, one can easily check that $q_{F!} j_{F!} \underline{\mathbb{C}}_{U_F}=0$, so that the second map in this triangle is an isomorphism. Finally, $q_F \circ i_F \colon F \times_Y F^\bot \to E^\diamond$ identifies with the composition of the projection $F \times_X F^\bot \to F^\bot$ with the embedding $F^\bot \hookrightarrow E^\diamond$. We deduce a canonical isomorphism \[ q_{F!} \underline{\mathbb{C}}_{Q_F} \cong \underline{\mathbb{C}}_{F^\bot}[-2 \mathrm{rk}(F)], \] which finishes the proof. \end{proof}
We will mainly use these constructions in the following situation. Let $V$ be an $A$-module (which we will consider as an $A$-equivariant vector bundle over the variety $\pt:=\mathrm{Spec}(\C)$), and let $E:=V \times Y$, an $A$-equivariant vector bundle over $Y$. We denote by $p \colon E \to V$, ${\check p} \colon E^* \to V^*$ the projections. As above, let $F \subset E$ be an $A$-stable subbundle.
\begin{cor} \label{prop:Fourier-F} There exists a canonical isomorphism \begin{equation} \label{eqn:isom-Fourier-F} \mathcal{F}_V(p_! \underline{\mathbb{C}}_F) \cong {\check p}_! \underline{\mathbb{C}}_{F^\bot}[-2\mathrm{rk}(F)]. \end{equation} \end{cor}
\begin{proof} By \cite[Proposition 3.7.13]{KS} (see also \cite[\S A.4]{AHJR}) we have a canonical isomorphism of functors \[ \mathcal{F}_V \circ p_! \cong {\check p}_! \circ \mathcal{F}_E. \] In particular we deduce an isomorphism $\mathcal{F}_V(p_! \underline{\mathbb{C}}_F) \cong {\check p}_! \mathcal{F}_E(\underline{\mathbb{C}}_F)$. Then the result follows from Lemma \ref{lem:fourier-F}. \end{proof}
\subsection{Equivariant homology as an ${\rm Ext}$-algebra} \label{ss:equiv}
From now on we let $G$ be a complex connected reductive algebraic group, $X$ be a smooth and proper complex algebraic variety, and $V$ be a finite dimensional $G$-module. Let $E:=V \times X$, considered as a $G \times {\mathbb{G}}_{\mathbf{m}}$-equivariant vector bundle as in \S\ref{ss:Fourier-transform}, and let $F_1,F_2$ be $G$-stable subbundles of the vector bundle $E$ over $X$. As in \S\ref{ss:Fourier-transform}, we denote by $p \colon E \to V$ the projection, and by $F_1^\bot,F_2^\bot \subset E^*$ the orthogonals to $F_1$ and $F_2$. Then there exists a canonical isomorphism \begin{equation*}
\mathsf{can}_{F_1,F_2} \colon \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1 \times_V F_2) \xrightarrow{\sim} {\rm Ext}^{2\dim(F_2)-\bullet}_{\mathcal{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(V)}(p_! \underline{\mathbb{C}}_{F_1}, p_! \underline{\mathbb{C}}_{F_2}). \end{equation*}
Let us explain (for future reference) how this isomorphism can be constructed, following \cite{CG,LuCus2}. Consider the cartesian diagram \[ \xymatrix@C=1.5cm{ E \times_V E \ar@{^{(}->}[r]^-{j} \ar[d]_-{\mu} & E \times E \ar[d]^-{p \times p} \\ V \ar@{^{(}->}[r]^-{\Delta} & V \times V } \] where $\Delta$ is the diagonal embedding. Then in \cite[Equation (8.6.4)]{CG} (see also \cite[\S 1.15 and \S2.4]{LuCus2}) the authors construct a canonical and bifunctorial isomorphism \[ \mu_* j^! (\mathbb{D}_E(A_1) \boxtimes A_2) \cong R\mathcal{H} \hspace{-1pt} \mathit{om}_{\C}(p_! A_1, p_! A_2) \] for $A_1,A_2$ in $\mathcal{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(E)$. Applying equivariant cohomology, we obtain an isomorphism \begin{equation} \label{eqn:morphisms-cohomology} {\rm Ext}^{\bullet}_{\mathcal{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(V)}(p_! A_1, p_! A_2) \cong \mathsf{H}^\bullet_{G \times {\mathbb{G}}_{\mathbf{m}}} \bigl(E \times_V E, j^!(\mathbb{D}_E(A_1) \boxtimes A_2 ) \bigr). \end{equation} Setting $A_1=\underline{\mathbb{C}}_{F_1}$, $A_2=\underline{\mathbb{C}}_{F_2}$ we deduce an isomorphism \[ {\rm Ext}^{\bullet}_{\mathcal{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(V)}(p_! \underline{\mathbb{C}}_{F_1}, p_! \underline{\mathbb{C}}_{F_2}) \cong \mathsf{H}^\bullet_{G \times {\mathbb{G}}_{\mathbf{m}}} \bigl(E \times_V E, j^!(\underline{\mathbb{D}}_{F_1} \boxtimes \underline{\mathbb{C}}_{F_2} ) \bigr). \] Let $a \colon F_1 \times F_2 \hookrightarrow E \times E$ be the inclusion, and consider the cartesian diagram \[ \xymatrix@C=1.5cm{ F_1 \times_V F_2 \ar@{^{(}->}[r]^-{b} \ar@{^{(}->}[d]_-{k} & E \times_V E \ar@{^{(}->}[d]^-{j} \\ F_1 \times F_2 \ar@{^{(}->}[r]^-{a} & E \times E. } \] Then using the base change isomorphism we obtain \begin{multline*} \mathsf{H}^\bullet_{G \times {\mathbb{G}}_{\mathbf{m}}} \bigl(E \times_V E, j^!(\underline{\mathbb{D}}_{F_1} \boxtimes \underline{\mathbb{C}}_{F_2} ) \bigr) \cong \mathsf{H}^\bullet_{G \times {\mathbb{G}}_{\mathbf{m}}} \bigl(E \times_V E, j^! a_*(\underline{\mathbb{D}}_{F_1} \boxtimes \underline{\mathbb{C}}_{F_2} ) \bigr) \\ \cong \mathsf{H}^\bullet_{G \times {\mathbb{G}}_{\mathbf{m}}} \bigl(E \times_V E, b_* k^!(\underline{\mathbb{D}}_{F_1} \boxtimes \underline{\mathbb{C}}_{F_2} ) \bigr) \cong \mathsf{H}^\bullet_{G \times {\mathbb{G}}_{\mathbf{m}}} \bigl(F_1 \times_V F_2, k^!(\underline{\mathbb{D}}_{F_1} \boxtimes \underline{\mathbb{C}}_{F_2} ) \bigr). \end{multline*} Now we use the canonical isomorphisms $\underline{\mathbb{C}}_{F_2} \cong \underline{\mathbb{D}}_{F_2}[-2\dim(F_2)]$ (since $F_2$ is smooth) and $k^! (\underline{\mathbb{D}}_{F_1} \boxtimes \underline{\mathbb{D}}_{F_2}) \cong k^! (\underline{\mathbb{D}}_{F_1 \times F_2}) \cong \underline{\mathbb{D}}_{F_1 \times_V F_2}$ to obtain the isomorphism $\mathsf{can}_{F_1,F_2}$.
\subsection{The $\mathfrak{Fourier}$ isomorphism} \label{ss:Fourier-isomorphism}
We continue with the setting of \S\ref{ss:equiv}, and denote by ${\check p} \colon E^* \to V^*$ the projection. Then we have canonical isomorphisms \begin{align*} \mathsf{can}_{F_1,F_2} \colon \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1 \times_V F_2) \ & \xrightarrow{\sim} \ {\rm Ext}^{2\dim(F_2)-\bullet}_{\mathcal{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(V)}(p_! \underline{\mathbb{C}}_{F_1}, p_! \underline{\mathbb{C}}_{F_2}); \\ \mathsf{can}_{F_1^\bot,F_2^\bot} \colon \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1^\bot \times_{V^*} F_2^\bot) \ & \xrightarrow{\sim} \ {\rm Ext}^{2\dim(F_2^\bot)-\bullet}_{\mathcal{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(V^*)}({\check p}_! \underline{\mathbb{C}}_{F_1^\bot}, {\check p}_! \underline{\mathbb{C}}_{F_2^\bot}). \end{align*} On the other hand, through the canonical isomorphisms $\mathcal{F}_V(p_* \underline{\mathbb{C}}_{F_i}) \cong {\check p}_* \underline{\mathbb{C}}_{F_i^\bot}[-2\mathrm{rk}(F_i)]$ for $i=1,2$ (see \eqref{eqn:isom-Fourier-F}), the functor $\mathcal{F}_V$ induces an isomorphism \[ {\rm Ext}^\bullet_{\mathcal{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(V)}(p_! \underline{\mathbb{C}}_{F_1},p_! \underline{\mathbb{C}}_{F_2}) \xrightarrow{\sim} {\rm Ext}^{\bullet-2\mathrm{rk}(F_2) + 2 \mathrm{rk}(F_1)}_{\mathcal{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(V^*)}({\check p}_! \underline{\mathbb{C}}_{F_1^\bot},{\check p}_! \underline{\mathbb{C}}_{F_2^\bot}). \] We denote by \[ \mathfrak{Fourier}_{F_1,F_2} \colon \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1 \times_V F_2) \xrightarrow{\sim} \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\bullet+2\dim(F_2^\bot)-2\dim(F_1)}(F_1^\bot \times_{V^*} F_2^\bot) \] the resulting isomorphism. This isomorphism, considered in particular in \cite{EM}, was the starting point of our work on linear Koszul duality.
\subsection{Linear Koszul duality} \label{ss:lkd}
Let us recall the definition and main properties of linear Koszul duality, following \cite{MR, MR2, MR3}. In this paper we will only consider the geometric situation relevant for convolution algebras, as considered in \cite[\S 4]{MR3}. However we will allow using two different vector bundles $F_1$ and $F_2$; the setting of \cite[\S 4]{MR3} corresponds to the choice $F_1=F_2$.
We continue with the setting of \S\ref{ss:equiv}, and denote by $\Delta V \subset V \times V$ the diagonal copy of $V$. We will consider the derived category \[ \mathcal{D}^c_{G \times {\mathbb{G}}_{\mathbf{m}}} \bigl( (\Delta V \times X \times X) \, \rcap_{E \times E} \, (F_1 \times F_2) \bigr) \] as defined in \cite[\S 3.1]{MR3}. By definition this is a subcategory of the derived category of $G\times{\mathbb{G}}_{\mathbf{m}}$-equivariant quasi-coherent dg-modules over a certain sheaf of $\mathcal{O}_{X \times X}$-dg-algebras on $X \times X$, which we will denote by $\mathcal{A}_{F_1, F_2}$. Note that the derived intersection \[ (\Delta V \times X \times X) \, \rcap_{E \times E} \, (F_1 \times F_2) \] is quasi-isomorphic to the derived fiber product $F_1 \, {\stackrel{_R}{\times}}_V \, F_2$ in the sense of \cite[\S 3.7]{BR}.
Similarly we have a derived category \[ \mathcal{D}^c_{G \times {\mathbb{G}}_{\mathbf{m}}} \bigl( (\Delta V^* \times X \times X) \, \rcap_{E^* \times E^*} \, (F_1^\bot \times F_2^\bot) \bigr). \]
We denote by $\omega_X$ the canonical line bundle on $X$. Then by \cite[Theorem 3.1]{MR3} there exists a natural equivalence of triangulated categories \begin{multline*} \mathfrak{K}_{F_1,F_2} \colon \mathcal{D}^c_{G \times {\mathbb{G}}_{\mathbf{m}}} \bigl( (\Delta V \times X \times X) \, \rcap_{E \times E} \, (F_1 \times F_2) \bigr) \\ \xrightarrow{\sim} \mathcal{D}^c_{G \times {\mathbb{G}}_{\mathbf{m}}} \bigl( (\Delta V^* \times X \times X) \, \rcap_{E^* \times E^*} \, (F_1^\bot \times F_2^\bot) \bigr)^{\rm op}. \end{multline*} More precisely, \cite[Theorem 3.1]{MR3} provides an equivalence of categories \begin{multline*} \kappa_{F_1,F_2}\colon \mathcal{D}^c_{G \times {\mathbb{G}}_{\mathbf{m}}} \bigl( (\Delta V \times X \times X) \rcap_{E \times E} (F_1 \times F_2) \bigr) \\ \xrightarrow{\sim} \mathcal{D}^c_{G \times {\mathbb{G}}_{\mathbf{m}}} \bigl( (\overline{\Delta} V^\diamond \times X \times X) \rcap_{E^\diamond \times E^\diamond} (F_1^\bot \times F_2^\bot) \bigr)^{\rm op} \end{multline*} where $\overline{\Delta} V^\diamond \subset V^\diamond \times V^\diamond$ is the antidiagonal copy of $V^\diamond$. (The construction of \cite{MR3} depends on the choice of an object $\mathcal{E}$ in $\mathcal{D}^b \mathsf{Coh}^{G \times {\mathbb{G}}_{\mathbf{m}}}(X \times X)$ whose image in $\mathcal{D}^b \mathsf{Coh}(X \times X)$ is a dualizing object; here we take $\mathcal{E}=\mathcal{O}_X \boxtimes \omega_X[\dim(X)]$.) Then $\mathfrak{K}_{F_1,F_2}$ is the composition of $\kappa_{F_1,F_2}$ with the natural equivalence \[ \mathcal{D}^c_{G \times {\mathbb{G}}_{\mathbf{m}}} \bigl( (\overline{\Delta} V^\diamond \times X \times X) \, \rcap_{E^\diamond \times E^\diamond} \, (F_1^\bot \times F_2^\bot) \bigr) \xrightarrow{\sim} \mathcal{D}^c_{G \times {\mathbb{G}}_{\mathbf{m}}} \bigl( (\Delta V^\diamond \times X \times X) \, \rcap_{E^\diamond \times E^\diamond} \, (F_1^\bot \times F_2^\bot) \bigr) \] (see \cite[\S 4.3]{MR3}) and the natural equivalence \[ \mathcal{D}^c_{G \times {\mathbb{G}}_{\mathbf{m}}} \bigl( (\Delta V^\diamond \times X \times X) \, \rcap_{E^\diamond \times E^\diamond} \, (F_1^\bot \times F_2^\bot) \bigr) \xrightarrow{\sim} \mathcal{D}^c_{G \times {\mathbb{G}}_{\mathbf{m}}} \bigl( (\Delta V^* \times X \times X) \, \rcap_{E^* \times E^*} \, (F_1^\bot \times F_2^\bot) \bigr) \] induced by the automorphism of ${\mathbb{G}}_{\mathbf{m}}$ sending $t$ to $t^{-1}$.
Note that
we have $\mathcal{H}^0(\mathcal{A}_{F_1, F_2})=(\pi_{F_1,F_2})_* \mathcal{O}_{F_1 \times_V F_2}$, where $\pi_{F_1,F_2} \colon F_1 \times_V F_2 \to X \times X$ is the projection (which is an affine morphism). Hence, using~\cite[Lemma 5.1]{MR3} and classical facts on affine morphisms, one can canonically identify the Grothendieck group of the category $\mathcal{D}^c_{G \times {\mathbb{G}}_{\mathbf{m}}} \bigl( (\Delta V \times X \times X) \, \rcap_{E \times E} \, (F_1 \times F_2) \bigr)$ with $\mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1 \times_V F_2)$.
We have a similar isomorphism for $F_1^\bot$ and $F_2^\bot$; hence the equivalence $\mathfrak{K}_{F_1,F_2}$ induces an isomorphism \[ \mathbf{Koszul}_{F_1,F_2} \colon \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1 \times_V F_2) \xrightarrow{\sim} \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1^\bot \times_{V^*} F_2^\bot). \]
\subsection{Duality and parity conjugation in $\mathsf{K}$-homology} \label{ss:duality}
To obtain a precise relation between the maps $\mathfrak{Fourier}_{F_1,F_2}$ of~\S\ref{ss:Fourier-isomorphism} and $\mathbf{Koszul}_{F_1,F_2}$ of~\S\ref{ss:lkd} we will need two auxiliary maps in $\mathsf{K}$-homology.
Our first map has a geometric flavour, and is induced by Grothendieck--Serre duality. More precisely, consider the ``duality'' equivalence \[ \mathrm{D}_{F_1^\bot, F_2^\bot}^{G \times {\mathbb{G}}_{\mathbf{m}}} \colon \mathcal{D}^b \mathsf{Coh}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1^{\bot} \times F_2^{\bot}) \to \mathcal{D}^b \mathsf{Coh}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1^{\bot} \times F_2^{\bot})^{\rm op} \] associated with the dualizing complex $\mathcal{O}_{F_1^{\bot}} \boxtimes \omega_{F_2^\bot}[\dim(F_2^{\bot})]$, which sends $\mathcal{G}$ to \[ R\mathcal{H} \hspace{-1pt} \mathit{om}_{\mathcal{O}_{F_1^{\bot} \times F_2^{\bot}}}(\mathcal{G}, \, \mathcal{O}_{F_1^{\bot}} \boxtimes \omega_{F_2^\bot})[\dim(F_2^{\bot})] \] (see e.g.~\cite[\S 2.1]{MR3} and references therein). (Here, $\omega_{F_2^{\bot}}$ is the canonical line bundle on $F_2^\bot$, endowed with its natural $G \times {\mathbb{G}}_{\mathbf{m}}$-equivariant structure.) This equivalence induces a (contravariant) auto-equivalence of the subcategory $\mathcal{D}^b \mathsf{Coh}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{F_1^{\bot} \times_{V^*} F_2^{\bot}}(F_1^{\bot} \times F_2^{\bot})$, which we denote similarly. We denote by \begin{equation*}
\mathbf{D}_{F_1^{\bot}, F_2^{\bot}} \colon \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1^{\bot} \times_{V^*} F_2^{\bot}) \xrightarrow{\sim} \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1^{\bot} \times_{V^*} F_2^{\bot}) \end{equation*} the induced automorphism at the level of Grothendieck groups.
Our second map is a ``correction factor'', with no interesting geometric interpretation. Namely, the direct image functor under the projection $\pi_{F_1^\bot, F_2^\bot} \colon F_1^\bot \times_{V^*} F_2^\bot \to X \times X$ (an affine morphism) induces an equivalence between $\mathsf{Coh}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1^\bot \times_{V^*} F_2^\bot)$ and the category of locally finitely generated $G \times {\mathbb{G}}_{\mathbf{m}}$-equivariant modules over the $\mathcal{O}_{X \times X}$-algebra $(\pi_{F_1^\bot,F_2^\bot})_* \mathcal{O}_{F_1^\bot \times_{V^*} F_2^\bot}$. Since ${\mathbb{G}}_{\mathbf{m}}$ acts trivially on $X \times X$, one can consider $(\pi_{F_1^\bot,F_2^\bot})_* \mathcal{O}_{F_1^\bot \times_{V^*} F_2^\bot}$ as a graded $G$-equivariant $\mathcal{O}_{X \times X}$-algebra, and this grading is concentrated in even degrees. Hence if $\mathcal{F}$ is any module over this algebra, then we have $\mathcal{F}=\mathcal{F}^{\mathrm{even}} \oplus \mathcal{F}^{\mathrm{odd}}$ where $\mathcal{F}^{\mathrm{even}}$, resp.~$\mathcal{F}^{\mathrm{odd}}$, is concentrated in even, resp.~odd, degrees. We denote by \[ \mathbf{i}_{F_1^\bot, F_2^\bot} \colon \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1^{\bot} \times_{V^*} F_2^{\bot}) \xrightarrow{\sim} \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1^{\bot} \times_{V^*} F_2^{\bot}) \] the automorphism which sends the class of a module $\mathcal{F}=\mathcal{F}^{\mathrm{even}} \oplus \mathcal{F}^{\mathrm{odd}}$ as above to $[\mathcal{F}^{\mathrm{even}}] - [\mathcal{F}^{\mathrm{odd}}]$.
\subsection{Reminder on the equivariant Riemann--Roch theorem}
Let us recall the definition and the main properties of the ``equivariant Riemann--Roch morphism'' for a complex algebraic variety, following~\cite{EG}. (See also~\cite{BZ} for a more direct treatment, without much details.) Let $A$ be a complex linear algebraic group, acting on a complex algebraic variety $Y$. Then we have a ``Riemann--Roch'' morphism \[ \tau^A_Y \colon \mathsf{K}^A(Y) \to \widehat{\mathsf{H}}^A_\bullet(Y). \] More precisely, we define this morphism as the composition \begin{equation} \label{eqn:RRmorphism} \mathsf{K}^A(Y) \longrightarrow \prod_{i \geq 0} \mathbb{Q} \otimes_{\mathbb{Z}} \mathsf{CH}^i_A(Y) \longrightarrow \prod_{i \in \mathbb{Z}} \, \mathsf{H}_i^A(Y) = \widehat{\mathsf{H}}^A_\bullet(Y), \end{equation} where $\mathsf{CH}^i_A(Y)$ is the $i$-th equivariant Chow group, see~\cite[\S 1.2]{EG}, the first arrow is the morphism constructed in~\cite[Section~3]{EG}, and the second morphism is induced by the ``equivariant cycle map'' of~\cite[\S 2.8]{EG1}.
\begin{remark} It follows from~\cite[Theorem~4.1]{EG} that the first morphism in~\eqref{eqn:RRmorphism} induces an isomorphism between a certain completion of $\mathbb{Q} \otimes_{\mathbb{Z}} \mathsf{K}^A(Y)$ and $\prod_{i \geq 0} \mathbb{Q} \otimes_{\mathbb{Z}} \mathsf{CH}^i_A(Y)$. Hence, if the equivariant cycle map is an isomorphism, a similar claim holds for our morphism $\tau^A_Y$. \end{remark}
Below we will use the following properties of the map $\tau^A_Y$, which follow from the main results of~\cite{EG}.
\begin{thm}[Equivariant Riemann--Roch theorem] \label{thm:equivariantRR} If $f \colon Y \to Y'$ is an $A$-equivariant proper morphism, then we have \[ \tau^A_{Y'} \circ \mathbf{pdi}_f = \mathfrak{pdi}_f \circ \tau^A_Y. \] \end{thm}
\begin{proof} By~\cite[Theorem~3.1(b)]{EG}, the first arrow in~\eqref{eqn:RRmorphism} is compatible with proper direct image morphisms (in the obvious sense). And by~\cite[p.~372]{fulton} the second arrow is also compatible with proper direct image morphisms, completing the proof. (More precisely, only the non-equivariant setting is considered in~\cite{fulton}, but the equivariant case follows, using the same arguments as in Footnote~\ref{fn:equiv-homology}.) \end{proof}
If $F$ is an $A$-equivariant vector bundle over $Y$, then one can define its (cohomological) equivariant Chern classes in $\mathsf{H}_A^\bullet(Y)$, and define a (cohomological) equivariant Todd class $\mathrm{td}^A(F) \in \widehat{\mathsf{H}}_A^\bullet(Y)$, see~\cite[Section~3]{EG} or~\cite[\S 3]{BZ} for similar constructions. This element is invertible in the algebra $\widehat{\mathsf{H}}_A^\bullet(Y)$. If $Y$ is smooth, we denote by $\mathrm{Td}_Y^A$ the equivariant Todd class of the tangent bundle of $Y$.
The following result can be stated and proved under much weaker assumptions, but only this particular case will be needed.
\begin{prop} \label{prop:RR-res} Let $Y$ be a smooth $A$-variety, and let $f \colon Z \hookrightarrow Y$ be the embedding of a smooth subvariety with normal bundle $N$. Then we have \[ \mathfrak{res}_f \circ \tau^A_{Y}(x) = \bigl( \tau^A_Z \circ \mathbf{res}_f(x) \bigr) \cdot \mathrm{td}^A(N) \] for any $x \in \mathsf{K}^A(Y)$, where $\mathfrak{res}_f \colon \mathsf{H}^A_\bullet(Y) \to \mathsf{H}^A_{\bullet-2\dim(Y)+2\dim(Z)}(Z)$ and $\mathbf{res}_f \colon \mathsf{K}^A(Y) \to \mathsf{K}^A(Z)$ are the ``restriction with supports'' morphisms. \end{prop}
\begin{proof} A similar formula for the first arrow in~\eqref{eqn:RRmorphism} follows from~\cite[Theorem~3.1(d)]{EG}. To deduce our result we need to check that the equivariant cycle map commutes with restriction with supports and with multiplication by a Todd class. In the non-equivariant situation, the first claim follows from~\cite[Example~19.2.1]{fulton} and the second one from~\cite[Proposition~19.1.2]{fulton}. The equivariant case follows, using the same arguments
as in Footnote~\ref{fn:equiv-homology}. \end{proof}
\begin{remark} \label{rk:Todd} Note that, in the setting of Proposition~\ref{prop:RR-res}, we have $f^* \mathrm{Td}^A_Y = \mathrm{Td}^A_Z \cdot \mathrm{td}^A(N)$, where $f^*$ is as in~\eqref{eqn:proj-formula-H}. (In fact, this formula easily follows from the compatibility of Chern classes with pullback and extensions of vector bundles.) \end{remark}
Finally we will need the following fact, which follows from~\cite[Theorem~3.1(d)]{EG} applied to the projection $Y \to \mathrm{pt}$ (see also~\cite[Theorem~5.1]{BZ}).
\begin{prop} \label{prop:tau-O-smooth} If $Y$ is smooth, then \[ \tau^A_Y(\mathcal{O}_Y) = [Y] \cdot \mathrm{Td}^A_Y, \] where $[Y]$ is the equivariant fundamental class of $Y$ (i.e.~the image of the fundamental class in the Chow group from~\cite[\S 2.2]{EG1} under the cycle map). \end{prop}
\subsection{Riemann--Roch maps} \label{ss:RRmaps}
Following \cite[\S 5.11]{CG}, we consider the ``bivariant Riemann--Roch maps'' \begin{align*} \underline{\mathrm{RR}}_{F_1, F_2} \colon & \ \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1 \times_V F_2) \to \widehat{\mathsf{H}}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1 \times_V F_2), \\ \overline{\mathrm{RR}}_{F^{\bot}_1, F_2^{\bot}} \colon & \ \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1^{\bot} \times_{V^*} F_2^{\bot}) \to \widehat{\mathsf{H}}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1^{\bot} \times_{V^*} F_2^{\bot}) \end{align*} defined by \begin{align*} \underline{\mathrm{RR}}_{F_1, F_2} (c) & = \tau^{G \times {\mathbb{G}}_{\mathbf{m}}}_{F_1 \times_V F_2}(c) \cdot \bigl(1 \boxtimes (\mathrm{Td}_{F_2}^{G \times {\mathbb{G}}_{\mathbf{m}}})^{-1} \bigr),\\ \overline{\mathrm{RR}}_{F^{\bot}_1, F_2^{\bot}}(d) & = \tau^{G \times {\mathbb{G}}_{\mathbf{m}}}_{F_1^\bot \times_{V^*} F_2^\bot}(d) \cdot \bigl( (\mathrm{Td}_{F_1^\bot}^{G \times {\mathbb{G}}_{\mathbf{m}}})^{-1} \cdot \mathrm{Td}^{G \times {\mathbb{G}}_{\mathbf{m}}}_X \boxtimes (\mathrm{Td}_X^{G \times {\mathbb{G}}_{\mathbf{m}}})^{-1} \bigr). \end{align*} In the expression for $\underline{\mathrm{RR}}_{F_1,F_2}$, $1 \boxtimes (\mathrm{Td}_{F_2}^{G \times {\mathbb{G}}_{\mathbf{m}}})^{-1}$ is considered as an element of $\widehat{\mathsf{H}}_{G \times {\mathbb{G}}_{\mathbf{m}}}^\bullet(F_1 \times_{V} F_2)$ through the composition \[ \widehat{\mathsf{H}}_{(G \times {\mathbb{G}}_{\mathbf{m}})^2}^\bullet(F_1 \times F_2) \to \widehat{\mathsf{H}}_{G \times {\mathbb{G}}_{\mathbf{m}}}^\bullet(F_1 \times F_2) \to \widehat{\mathsf{H}}_{G \times {\mathbb{G}}_{\mathbf{m}}}^\bullet(F_1 \times_{V} F_2) \] where the first morphism is the restriction morphism associated with the diagonal embedding of $G \times {\mathbb{G}}_{\mathbf{m}}$, and the second morphism is the pullback in equivariant cohomology. In the expression for $\overline{\mathrm{RR}}_{F_1^\bot, F_2^\bot}$, first we consider $\mathrm{Td}_X^{G \times {\mathbb{G}}_{\mathbf{m}}}$ as an element of $\widehat{\mathsf{H}}^\bullet_{G \times {\mathbb{G}}_{\mathbf{m}}}(E^*)$ using the Thom isomorphism $\mathsf{H}^\bullet_{G \times {\mathbb{G}}_{\mathbf{m}}}(E^*) \xrightarrow{\sim} \mathsf{H}^\bullet_{G \times {\mathbb{G}}_{\mathbf{m}}}(X)$; then the same conventions as above allow to consider $(\mathrm{Td}_{F_1^\bot}^{G \times {\mathbb{G}}_{\mathbf{m}}})^{-1} \cdot \mathrm{Td}^{G \times {\mathbb{G}}_{\mathbf{m}}}_X \boxtimes (\mathrm{Td}_X^{G \times {\mathbb{G}}_{\mathbf{m}}})^{-1}$ as an element in $\widehat{\mathsf{H}}_{G \times {\mathbb{G}}_{\mathbf{m}}}^\bullet(F_1^\bot \times_{V^*} F_2^\bot)$.
\subsection{Statement} \label{ss:statement}
The main result of this paper is the following.
\begin{thm} \label{thm:LKDFourier}
Assume that the proper direct image morphism \begin{equation} \label{eqn:morphism-thm} \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1^{\bot} \times_{V^*} F_2^{\bot}) \to \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1^{\bot} \times_{V^*} E^*) \end{equation} induced by the inclusion $F_2^\bot \hookrightarrow E^*$ is injective. Then the following diagram commutes:
\[ \xymatrix@C=6cm{ \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1 \times_V F_2) \ar[r]^-{\mathbf{i}_{F_1^{\bot}, F_2^{\bot}} \circ \mathbf{D}_{F_1^{\bot}, F_2^{\bot}} \circ \mathbf{Koszul}_{F_1, F_2}} \ar[d]_-{\underline{\mathrm{RR}}_{F_1, F_2}} & \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1^{\bot} \times_{V^*} F_2^{\bot}) \ar[d]^-{\overline{\mathrm{RR}}_{F_1^{\bot}, F_2^{\bot}}} \\ \widehat{\mathsf{H}}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1 \times_V F_2) \ar[r]^-{\mathfrak{Fourier}_{F_1,F_2}} & \widehat{\mathsf{H}}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1^{\bot} \times_{V^*} F_2^{\bot}). } \]
\end{thm}
The proof of Theorem \ref{thm:LKDFourier} is given in \S\ref{ss:proof-thm}. It is based on compatibility (or functoriality) results for all the maps considered in the diagram, which are stated in Sections \ref{sec:compatibility-Fourier} and \ref{sec:compatibility-others}; some of these results might be of independent interest. Let us point out that our assumption is probably not needed for the result to hold.
\begin{remark} \label{rmk:injectivity-faithful} {\em (Injectivity assumption.)} The fiber product $F_1^{\bot} \times_{V^*} E^*$ is isomorphic to $F_1^\bot \times X$, hence is a vector bundle over $X^2$. In particular, by the Thom isomorphism we have \begin{equation} \label{eqn:Thom} \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1^{\bot} \times_{V^*} E^*) \cong \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\bullet-2{\rm rk}(F_1^\bot)}(X \times X). \end{equation} Moreover, by \cite[Lemma 5.4.35]{CG} the following diagram commutes: \begin{equation} \label{eqn:diagram-assumption} \vcenter{ \xymatrix@C=1cm{ \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1^{\bot} \times_{V^*} F_2^{\bot}) \ar[r] \ar[d] & {\rm Hom}_{\mathsf{H}^\bullet_{G \times {\mathbb{G}}_{\mathbf{m}}}(\mathrm{pt})} \bigl( \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_2^\bot), \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\bullet-2\dim(F_2^\bot)}(F_1^\bot) \bigr) \ar[d]^-{\wr} \\ \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\bullet-2{\rm rk}(F_1^\bot)}(X \times X) \ar[r] & {\rm Hom}_{\mathsf{H}^\bullet_{G \times {\mathbb{G}}_{\mathbf{m}}}(\mathrm{pt})} \bigl( \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\bullet-2{\rm rk}(F_2^\bot)}(X), \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\bullet-2\dim(F_2^\bot)-2{\rm rk}(F_1^\bot)}(X) \bigr). } } \end{equation} Here the horizontal arrows are induced by convolution, the left vertical arrow is the composition of~\eqref{eqn:morphism-thm} and the isomorphism~\eqref{eqn:Thom}, and the right vertical arrow is induced by the respective Thom isomorphisms. Assume now that $\mathsf{H}_c^{\mathrm{odd}}(X)=0$ (e.g.~that $X$ is paved by affine spaces). Then one can easily check that the lower horizontal arrow in diagram \eqref{eqn:diagram-assumption} is an isomorphism. Hence in this case our assumption is equivalent to injectivity of the upper horizontal arrow. If moreover $F_1=F_2=F$, then $\mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F^{\bot} \times_{V^*} F^{\bot})$ is an algebra and $\mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F^\bot)$ is a module over this algebra. In this case our assumption amounts to the condition that the action on this module is faithful. \end{remark}
\subsection{An injectivity criterion for \eqref{eqn:morphism-thm}}
The following result gives an easy criterion which ensures that the assumption of Theorem \ref{thm:LKDFourier} is satisfied.
\begin{prop} \label{prop:criterion-thm} Assume that $\mathsf{H}_c^{\mathrm{odd}}(F_1^\bot \times_{V^*} F_2^\bot)=0$. Then the proper direct image morphism \[ \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1^{\bot} \times_{V^*} F_2^{\bot}) \to \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1^{\bot} \times_{V^*} E^*) \] induced by the inclusion $F_2^\bot \hookrightarrow E^*$ is injective. \end{prop}
\begin{proof} Let $T$ be a maximal torus of $G$. Then we have a commutative diagram \[ \xymatrix@C=2cm{ \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1^\bot \times_{V^*} F_2^\bot) \ar[r] \ar[d] & \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1^\bot \times_{V^*} E^*) \ar[d] \\ \mathsf{H}^{T \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1^\bot \times_{V^*} F_2^\bot) \ar[r] & \mathsf{H}^{T \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1^\bot \times_{V^*} E^*) \\ } \] where horizontal arrows are proper direct image morphisms, and vertical arrows are forgetful maps. The left vertical arrow is injective: indeed, by our assumption and \cite[Proposition 7.2]{LuCus1}, there exist (non-canonical) isomorphisms \begin{align} \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1^\bot \times_{V^*} F_2^\bot) & \cong \mathsf{H}_{G \times {\mathbb{G}}_{\mathbf{m}}}^{-\bullet}(\pt) \otimes_{\C} \mathsf{H}_{\bullet} (F_1^\bot \times_{V^*} F_2^\bot), \\ \label{eqn:isom-T-equ} \mathsf{H}^{T \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1^\bot \times_{V^*} F_2^\bot) & \cong \mathsf{H}_{T \times {\mathbb{G}}_{\mathbf{m}}}^{-\bullet}(\pt) \otimes_{\C} \mathsf{H}_{\bullet} (F_1^\bot \times_{V^*} F_2^\bot) \end{align} such that our forgetful morphism is induced by the natural morphism $\mathsf{H}_{G \times {\mathbb{G}}_{\mathbf{m}}}^\bullet(\pt) \to \mathsf{H}_{T \times {\mathbb{G}}_{\mathbf{m}}}^\bullet(\pt)$, which is well known to be injective. Hence, to prove that the upper horizontal arrow is injective it is sufficient to prove that the lower horizontal arrow is injective.
If $\mathsf{Q}$ denotes the fraction field of $\mathsf{H}:=\mathsf{H}_{T \times {\mathbb{G}}_{\mathbf{m}}}^\bullet(\pt)$, then using again isomorphism \eqref{eqn:isom-T-equ}, the natural morphism \[ \mathsf{H}^{T \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1^\bot \times_{V^*} F_2^\bot) \to \mathsf{Q} \otimes_\mathsf{H} \mathsf{H}^{T \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1^\bot \times_{V^*} F_2^\bot) \] is injective. We deduce that to prove the proposition it suffices to prove that the induced morphism \[ \mathsf{Q} \otimes_\mathsf{H} \mathsf{H}^{T \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1^{\bot} \times_{V^*} F_2^{\bot}) \to \mathsf{Q} \otimes_\mathsf{H} \mathsf{H}^{T \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1^{\bot} \times_{V^*} E^*) \] is injective. Let $Y:=(X \times X)^T$ denote the $T$-invariants in $X \times X$. Then we have \[ Y=(F_1^{\bot} \times_{V^*} F_2^{\bot})^{T \times {\mathbb{G}}_{\mathbf{m}}} = (F_1^{\bot} \times_{V^*} E^*)^{T \times {\mathbb{G}}_{\mathbf{m}}}. \] Consider the commutative diagram \[ \xymatrix{ \mathsf{H}^{T \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1^\bot \times_{V^*} F_2^\bot) \ar[rr]^-{\alpha} & & \mathsf{H}^{T \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1^\bot \times_{V^*} E^*) \\ & \mathsf{H}^{T \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(Y) \ar[lu]^-{\beta} \ar[ru]_-{\gamma} & } \] where all morphisms are proper direct image morphisms in homology. Then by the localization theorem (see \cite[Proposition 4.4]{LuCus2} or \cite[Theorem B.2]{EM}) both $\beta$ and $\gamma$ become isomorphisms after applying $\mathsf{Q} \otimes_\mathsf{H} (\cdot)$. Hence the same is true for $\alpha$; in particular $\mathrm{id}_\mathsf{Q} \otimes_\mathsf{H} \alpha$ is injective, which finishes the proof. \end{proof}
\begin{remark} \label{rk:odd-vanishing} Using a non-equivariant variant of isomorphism $\mathfrak{Fourier}_{F_1,F_2}$, one can check that the condition $\mathsf{H}_c^{\mathrm{odd}}(F_1^\bot \times_{V^*} F_2^\bot)=0$ is equivalent to the condition $\mathsf{H}_c^{\mathrm{odd}}(F_1 \times_{V} F_2)=0$. \end{remark}
\section{The case of convolution algebras} \label{sec:convolution-algebras}
In this subsection we study more closely the case $F_1=F_2$. In this case, as we will explain, all the objects appearing in the diagram of Theorem~\ref{thm:LKDFourier} are equipped with convolution products, and all the maps are compatible with these products. In a particular case, these algebras are related to affine Hecke algebras, and our diagram explains the relation between the categorifications of Iwahori--Matsumoto involutions obtained in~\cite{EM} and~\cite{MR3}, via maps introduced in~\cite{LuAff}.
None of the results of this section are used in the proof of Theorem~\ref{thm:LKDFourier}.
\subsection{Convolution} \label{ss:convolution}
We set $F:=F_1=F_2$. As explained in~\cite[\S 5.2.20]{CG} or~\cite[\S 4.1]{MR3}, the group $\mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F \times_V F)$ can be endowed with a natural (associative and unital) convolution product $\star$. In fact, for $c,d \in \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F \times_V F)$, with our notations this product satisfies\footnote{Note that our convention for the definition of the convolution product is opposite to the one adopted in~\cite{MR3}.\label{fn:convention-product}} \[ c \star d = \mathbf{pdi}_{p_{1,3}} \circ \mathbf{res}(c \boxtimes d) \] where $c \boxtimes d \in \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}\bigl( (F \times_V F) \times (F \times_V F) \bigr)$ is the exterior product of $c$ and $d$, \[ \mathbf{res} \colon \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}\bigl( (F \times_V F) \times (F \times_V F) \bigr) \to \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F \times_V F \times_V F) \] is the restriction with supports morphism associated with the inclusion $F^3 \hookrightarrow F^4$ sending $(x,y,z)$ to $(x,y,y,z)$, and $p_{1,3} \colon F \times_V F \times_V F \to F \times_V F$ is the (proper) projection on the first and third factors. (See~\cite[\S 4.2]{MR3} for a similar description at the categorical level.) The unit in this algebra is the structure sheaf $\mathcal{O}_{\Delta F}$ of the diagonal $\Delta F \subset F \times_V F$. The same constructions provide a left, resp.~right, action of the algebra $\mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F \times_V F)$ on the group $\mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F)$ defined by \[ c \star d = \mathbf{pdi}_{p_{1}} \circ \mathbf{res}_{\mathrm{l}}(c \boxtimes d), \qquad \text{resp.} \qquad d \star c = \mathbf{pdi}_{p_{2}} \circ \mathbf{res}_{\mathrm{r}}(d \boxtimes c) \] for $c \in \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F \times_V F)$ and $d \in \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F)$. Here $p_1,p_2 \colon F \times_V F \to F$ are the projections on the first and second factor respectively, the exterior products are defined in the obvious way, and \begin{multline*} \mathbf{res}_{\mathrm{l}} \colon \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}\bigl( (F \times_V F) \times F \bigr) \to \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F \times_V F), \\ \text{resp.} \quad \mathbf{res}_{\mathrm{r}} \colon \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}\bigl( F \times (F \times_V F) \bigr) \to \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F \times_V F), \end{multline*} is the restriction with supports morphism associated with the inclusion $F^2 \hookrightarrow F^3$ sending $(x,y)$ to $(x,y,y)$, resp.~to $(x,x,y)$.
Of course we have similar constructions for the subbundle $F^\bot \subset E^*$, and we will use the same notation in this context.
\begin{lem} The morphisms $\mathbf{Koszul}_{F,F}$, $\mathbf{D}_{F^{\bot}, F^{\bot}}$ and $\mathbf{i}_{F^\bot, F^\bot}$ are (unital) algebra isomorphisms. \end{lem}
\begin{proof} The case of $\mathbf{Koszul}_{F,F}$ follows from~\cite[Propositions~4.3 \&~4.5]{MR3}.\footnote{In~\cite{MR3} we use the dualizing complex $\omega_X \boxtimes \mathcal{O}_X [\dim(X)]$ instead of $\mathcal{O}_X \boxtimes \omega_X[\dim(X)]$. But the results cited remain true (with an identical proof) with our present conventions. \label{fn:MR3}} The case of $\mathbf{D}_{F^{\bot}, F^{\bot}}$ is not difficult, and left to the reader (see~\cite[Lemma~9.5]{LUSBas} for a similar statement, with slightly different conventions in the definition of Grothendieck--Serre duality). Finally, the case of $\mathbf{i}_{F^\bot, F^\bot}$ is obvious. \end{proof}
This convolution construction has a natural analogue in equivariant Borel--Moore homology, see e.g.~\cite[\S 2.7]{CG} or~\cite[\S 2]{LuCus2}. In fact, the convolution product on $\mathsf{H}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(F \times_V F)$, which we will also denote $\star$, satisfies \[ c \star d = \mathfrak{pdi}_{p_{1,3}} \circ \mathfrak{res}(c \boxtimes d), \] where $\mathfrak{res}$ is defined as for $\mathbf{res}$ above (replacing $\mathsf{K}$-homology by Borel--Moore homology). The unit for this convolution product is the equivariant fundamental class $[\Delta F]$ of the diagonal $\Delta F \subset F \times_V F$. We also have a left and a right module structure on $\mathsf{H}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(F)$, defined via the formulas \[ c \star d = \mathfrak{pdi}_{p_{1}} \circ \mathfrak{res}_{\mathrm{l}}(c \boxtimes d), \qquad \text{resp.} \qquad d \star c = \mathfrak{pdi}_{p_{2}} \circ \mathfrak{res}_{\mathrm{r}}(d \boxtimes c) \] for $c \in \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F \times_V F)$ and $d \in \mathsf{H}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(F)$. Finally we have similar structures for the subbundle $F^\bot \subset E^*$.
\begin{lem} The morphism $\mathfrak{Fourier}_{F,F}$ is a (unital) algebra isomorphism. \end{lem}
\begin{proof} One can also show that the isomorphism $\mathsf{can}_{F,F}$ is a (unital) algebra isomorphism, where the right-hand side is endowed with the Yoneda product; see~\cite[Theorem~8.6.7]{CG}, \cite[Lemma~2.5]{LuCus2} or~\cite[Theorem~4.5]{Ka} for similar statements. Then the claim follows from the fact that $\mathfrak{Fourier}_{F,F}$ is induced by a functor. \end{proof}
\subsection{Compatibility for the Riemann--Roch maps} \label{ss:RR-convolution}
\begin{lem} \label{lem:RR-convolution} Assume\footnote{This assumption is probably unnecessary. However, to avoid it one would need a more general variant of Proposition~\ref{prop:RR-res} (as in~\cite[Theorem~5.8.14]{CG}, for instance) for which we could not find any reference or easy proof.} that $\mathsf{H}_c^{\mathrm{odd}}(F \times_{V} F)=0$. Then the morphisms $\underline{\mathrm{RR}}_{F,F}$ and $\overline{\mathrm{RR}}_{F^\bot, F^\bot}$ are unital algebra morphisms. \end{lem}
\begin{proof} We only treat the case of $\underline{\mathrm{RR}}_{F,F}$; the case of $\overline{\mathrm{RR}}_{F^\bot, F^\bot}$ is similar. (Note that, by Remark~\ref{rk:odd-vanishing}, our ``odd vanishing'' assumption implies that $\mathsf{H}_c^{\mathrm{odd}}(F^\bot \times_{V^*} F^\bot)=0$ also.) The fact that our morphism sends the unit to the unit follows from Theorem~\ref{thm:equivariantRR} and Proposition~\ref{prop:tau-O-smooth}, using the projection formula~\eqref{eqn:proj-formula-H}. It remains to prove the compatibility with products.
To prove the lemma we use ``projective completions,'' namely we set $\overline{V}:=\mathbb{P}(V \oplus \C)$ and let $\overline{F}$ be the projective bundle associated with the vector bundle $F \times \C$ over $X$. Then we have a projection $\overline{F} \to \overline{V}$, and open embeddings $F \hookrightarrow \overline{F}$, $V \hookrightarrow \overline{V}$. Note that $F \times_V F = F \times_{\overline{V}} \overline{F}$, so that $F \times_V F$ is a closed subvariety in $F \times \overline{F}$. Similarly, one can identify $F \times_V F$ with a closed subvariety in $\overline{F} \times F$, so that we have proper direct image morphisms \begin{gather*} \imath_1 \colon \mathsf{H}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(F \times_V F) \to \mathsf{H}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(F \times \overline{F}), \qquad \imath_2 \colon \mathsf{H}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(F \times_V F) \to \mathsf{H}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(\overline{F} \times F), \\ \imath_3 \colon \mathsf{H}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(F \times_V F) \to \mathsf{H}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(F \times F). \end{gather*} Using the same arguments as in the proof of Proposition~\ref{prop:criterion-thm}, one can check that the morphism $\imath_3$ is injective under our assumption. There exists a natural convolution product \[ \star \colon \mathsf{H}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(F \times \overline{F}) \times \mathsf{H}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(\overline{F} \times F) \to \mathsf{H}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(F \times F) \] defined by \[ c \star d = \mathfrak{pdi}_{p'_{1,3}} \circ \mathfrak{res}'(c \boxtimes d), \] where $p'_{1,3} \colon F \times \overline{F} \times F \to F \times F$ is the (proper) projection on the first and third factors, and $\mathfrak{res} \colon \mathsf{H}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(F \times \overline{F} \times \overline{F} \times F) \to \mathsf{H}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(F \times \overline{F} \times F)$ is the restriction with supports morphism associated with the inclusion sending $(x,y,z)$ to $(x,y,y,z)$. Moreover one can check (using in particular Lemma~\ref{lem:restriction-pushforward}) that for $c,d \in \mathsf{H}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(F \times_V F)$ we have \[ \imath_3(c \star d) = \imath_1(c) \star \imath_2(d). \]
We have a similar construction of a convolution product in equivariant $\mathsf{K}$-homology, for which we will use similar notations. Hence, using the injectivity of $\imath_3$, Theorem~\ref{thm:equivariantRR} and the projection formula~\eqref{eqn:proj-formula-H}, to prove the lemma it is enough to prove that \begin{equation} \label{eqn:convolution-RR} \underline{\mathrm{RR}}_3(c \star d) = \underline{\mathrm{RR}}_1(c) \star \underline{\mathrm{RR}}_2(d) \end{equation} for $c \in \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F \times \overline{F})$ and $d \in \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(\overline{F} \times F)$, where \[ \underline{\mathrm{RR}}_1 \colon \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F \times \overline{F}) \to \widehat{\mathsf{H}}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(F \times \overline{F}) \] is defined by \[ \underline{\mathrm{RR}}_1(d) = \tau^{G \times {\mathbb{G}}_{\mathbf{m}}}_{F \times \overline{F}}(d) \cdot \big(1 \boxtimes (\mathrm{Td}_{\overline{F}}^{G \times {\mathbb{G}}_{\mathbf{m}}})^{-1} \bigr), \] and $\underline{\mathrm{RR}}_2$ and $\underline{\mathrm{RR}}_3$ are defined similarly.
Now we have \begin{align*} \underline{\mathrm{RR}}_3(c \star d) & = \tau^{G \times {\mathbb{G}}_{\mathbf{m}}}_{F \times F}(\mathbf{pdi}_{p'_{1,3}} \circ \mathbf{res}'(c \boxtimes d)) \cdot (1 \boxtimes (\mathrm{Td}_{F}^{G \times {\mathbb{G}}_{\mathbf{m}}})^{-1}) \\ & = \mathfrak{pdi}_{p'_{1,3}} \bigl( \tau^{G \times {\mathbb{G}}_{\mathbf{m}}}_{F \times \overline{F} \times F}(\mathbf{res}'(c \boxtimes d)) \bigr) \cdot (1 \boxtimes (\mathrm{Td}_{F}^{G \times {\mathbb{G}}_{\mathbf{m}}})^{-1}) \\ & = \mathfrak{pdi}_{p'_{1,3}} \bigl( \tau^{G \times {\mathbb{G}}_{\mathbf{m}}}_{F \times \overline{F} \times F}(\mathbf{res}'(c \boxtimes d)) \cdot (1 \boxtimes 1 \boxtimes (\mathrm{Td}_{F}^{G \times {\mathbb{G}}_{\mathbf{m}}})^{-1}) \bigr) \\ & = \mathfrak{pdi}_{p'_{1,3}} \bigl( \mathfrak{res}' \circ \tau^{G \times {\mathbb{G}}_{\mathbf{m}}}_{F \times \overline{F} \times \overline{F} \times F}(c \boxtimes d) \cdot \mathrm{td}^{G \times {\mathbb{G}}_{\mathbf{m}}}(N) \cdot (1 \boxtimes 1 \boxtimes (\mathrm{Td}_{F}^{G \times {\mathbb{G}}_{\mathbf{m}}})^{-1}) \bigr), \end{align*} where $N$ is the normal bundle to the embedding $F \times \overline{F} \times F \hookrightarrow F \times \overline{F}^2 \times F$. (Here the second equality follows from Theorem~\ref{thm:equivariantRR}, the third one from the projection formula~\eqref{eqn:proj-formula-H}, and the last equality from Proposition~\ref{prop:RR-res}.) On the other hand we have \[ \underline{\mathrm{RR}}_1(c) \star \underline{\mathrm{RR}}_2(d) = \mathfrak{pdi}_{p'_{1,3}} \circ \mathfrak{res}' \bigl( \tau^{G \times {\mathbb{G}}_{\mathbf{m}}}_{F \times \overline{F} \times \overline{F} \times F}(c \boxtimes d) \cdot (1 \boxtimes (\mathrm{Td}_{\overline{F}}^{G \times {\mathbb{G}}_{\mathbf{m}}})^{-1}) \boxtimes 1 \boxtimes (\mathrm{Td}_{F}^{G \times {\mathbb{G}}_{\mathbf{m}}})^{-1})\bigr). \] Now the normal bundle $N$ is canonically isomorphic to the restriction to $F \times \overline{F} \times F$ of the pullback of the tangent bundle of $\overline{F}$ under the projection $F \times \overline{F}^2 \times F \to \overline{F}$ on the second factor. Using~\eqref{eqn:res-cohomology} and comparing the formulas for $\underline{\mathrm{RR}}_3(c \star d)$ and for $\underline{\mathrm{RR}}_1(c) \star \underline{\mathrm{RR}}_2(d)$ obtained above, we deduce~\eqref{eqn:convolution-RR}.
\end{proof}
\subsection{Compatibility for the actions on the natural modules}
In~\S\ref{ss:convolution} we have defined (left and right) actions of the algebra $\mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F \times_V F)$, resp.~$\mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F^\bot \times_{V^*} F^\bot)$, resp.~$\mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F \times_V F)$, resp.~$\mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F^\bot \times_{V^*} F^\bot)$, on the module $\mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F)$, resp.~$\mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F^\bot)$, resp.~$\mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F)$, resp.~$\mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F^\bot)$. We now define ``bivariant Riemann--Roch maps'' \[ \underline{\mathrm{RR}}_F \colon \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F) \to \widehat{\mathsf{H}}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F), \qquad \overline{\mathrm{RR}}_{F^\bot} \colon \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F^\bot) \to \widehat{\mathsf{H}}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F^\bot) \] by the formulas \[ \underline{\mathrm{RR}}_F = \tau^{G \times {\mathbb{G}}_{\mathbf{m}}}_F, \qquad \overline{\mathrm{RR}}_{F^\bot}(c) = \tau^{G \times {\mathbb{G}}_{\mathbf{m}}}_{F^\bot}(c) \cdot (\mathrm{Td}_X^{G \times {\mathbb{G}}_{\mathbf{m}}})^{-1} \] (where use the same conventions as in~\S\ref{ss:RRmaps}). The following technical lemma will be used to compute explicitly some Riemann--Roch maps in~\S\ref{ss:geom-gHaff}.
\begin{lem} \label{lem:RR-action} Assume that $\mathsf{H}_c^{\mathrm{odd}}(F \times_{V} F)=0$. Then the morphisms $\underline{\mathrm{RR}}$ and $\overline{\mathrm{RR}}$ are compatible with the module structures, in the sense that for $c \in \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F \times_V F)$ and $d \in \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F)$, resp.~for $c \in \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F^\bot \times_{V^*} F^\bot)$ and $d \in \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F^\bot)$, we have \[ \underline{\mathrm{RR}}_F(c \star d) = \underline{\mathrm{RR}}_{F,F}(c) \star \underline{\mathrm{RR}}_F(d), \qquad \text{resp.} \qquad \overline{\mathrm{RR}}_{F^\bot}(d \star c) = \overline{\mathrm{RR}}_{F^\bot}(d) \star \overline{\mathrm{RR}}_{F^\bot, F^\bot}(c). \] \end{lem}
\begin{proof} We only prove the first equality; the second one can be proved by similar arguments. First, we claim that \begin{equation} \label{eqn:RR-modules} \tau^{G \times {\mathbb{G}}_{\mathbf{m}}}_{F \times_V F} \circ \mathbf{res}_{\mathrm{l}}(c \boxtimes d) = \bigl( \mathfrak{res}_{\mathrm{l}} \circ \tau^{G \times {\mathbb{G}}_{\mathbf{m}}}_{F \times_V F \times F} (c \boxtimes d) \bigr) \cdot \mathrm{td}^A(N)^{-1}, \end{equation} where $N$ is the normal bundle to the inclusion $F \times F \hookrightarrow F \times F \times F$ considered in the definition of $\mathbf{res}_{\mathrm{l}}$. Indeed, as in the proof of Lemma~\ref{lem:RR-convolution}, our assumption ensures that the proper direct image morphism \[ \imath \colon \mathsf{H}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(F \times_V F) \to \mathsf{H}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(F \times F) \] is injective. Hence it is enough to prove that the image under $\imath$ of both sides in~\eqref{eqn:RR-modules} are equal. Now by the projection formula~\eqref{eqn:proj-formula-H}, Theorem~\ref{thm:equivariantRR} and Lemma~\ref{lem:restriction-pushforward} we have \begin{multline*} \imath \Bigl( \bigl( \mathfrak{res}_{\mathrm{l}} \circ \tau^{G \times {\mathbb{G}}_{\mathbf{m}}}_{F \times_V F} (c \boxtimes d) \bigr) \cdot \mathrm{td}^A(N)^{-1} \Bigr) = \imath \bigl( \mathfrak{res}_{\mathrm{l}} \circ \tau^{G \times {\mathbb{G}}_{\mathbf{m}}}_{F \times_V F \times F} (c \boxtimes d) \bigr) \cdot \mathrm{td}^A(N)^{-1} \\ = \bigl( \mathfrak{res}_{\mathrm{l}}' \circ \tau^{G \times {\mathbb{G}}_{\mathbf{m}}}_{F \times F \times F} (\imath(c) \boxtimes d) \bigr) \cdot \mathrm{td}^A(N)^{-1}, \end{multline*} where \[ \mathfrak{res}_{\mathrm{l}}' \colon \mathsf{H}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}( F \times F \times F) \to \mathsf{H}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(F \times F) \] is the restriction with supports morphism associated with the embedding $F^2 \hookrightarrow F^3$ considered in the definition of $\mathfrak{res}_{\mathrm{l}}$.
On the other hand, by Theorem~\ref{thm:equivariantRR} and the obvious $\mathsf{K}$-theoretic analogue of Lemma~\ref{lem:restriction-pushforward} we have \[ \imath \bigl( \tau^{G \times {\mathbb{G}}_{\mathbf{m}}}_{F \times_V F} \circ \mathbf{res}_{\mathrm{l}}(c \boxtimes d) \bigr) = \tau^{G \times {\mathbb{G}}_{\mathbf{m}}}_{F \times F} \circ \mathbf{res}'_{\mathrm{l}}(\imath(c) \boxtimes d), \] where $\mathbf{res}'_{\mathrm{l}}$ is defined as for $\mathfrak{res}_{\mathrm{l}}'$. Hence the desired equality follows from Proposition~\ref{prop:RR-res}.
Now we have \begin{align*} \underline{\mathrm{RR}}_F(c \star d) &= \tau_F^{G \times {\mathbb{G}}_{\mathbf{m}}}(\mathbf{pdi}_{p_{1}} \circ \mathbf{res}_{\mathrm{l}}(c \boxtimes d)) \\ &= \mathfrak{pdi}_{p_1} \circ \tau_{F \times_V F}^{G \times {\mathbb{G}}_{\mathbf{m}}} \circ \mathbf{res}_{\mathrm{l}}(c \boxtimes d) \\ &= \mathfrak{pdi}_{p_1} \Bigl( \bigl( \mathfrak{res}_{\mathrm{l}} \circ \tau^{G \times {\mathbb{G}}_{\mathbf{m}}}_{F \times_V F \times F} (c \boxtimes d) \bigr) \cdot \mathrm{td}^A(N)^{-1} \Bigr) \\ &= \mathfrak{pdi}_{p_1} \Bigl( \mathfrak{res}_{\mathrm{l}} \bigl( (\tau^{G \times {\mathbb{G}}_{\mathbf{m}}}_{F \times_V F}(c) \boxtimes \tau^{G \times {\mathbb{G}}_{\mathbf{m}}}_F (d) ) \cdot (1 \boxtimes (\mathrm{Td}^A_F)^{-1} \boxtimes 1) \bigr) \Bigr) \\ &= \underline{\mathrm{RR}}_{F,F}(c) \star \underline{\mathrm{RR}}_F(d). \end{align*} (Here the second equality follows from Theorem~\ref{thm:equivariantRR}, the third one from~\eqref{eqn:RR-modules}, and the fourth one from~\eqref{eqn:res-cohomology}.) This concludes the proof. \end{proof}
\subsection{Affine Hecke algebras and their graded versions} \label{ss:Haff-gHaff}
From now on in this section we restrict to the case of the affine Hecke algebra and its graded version. Our notation mainly follows~\cite{LuAff}. Namely, we fix a semisimple and simply connected complex algebraic group $G$, with fixed maximal torus $T$ and Borel subgroup $B$ with $T \subset B$. We denote by $W$ the Weyl group of $(G,T)$, and by $S \subset W$ the set of Coxeter generators determined by the choice of $B$. We also denote by $\mathbb{X}$ the lattice of characters of $T$, and by $R \subset \mathbb X$ the root system of $(G,T)$. We denote by $R^+ \subset R$ the system of positive roots consisting of the roots \emph{opposite} to the roots of $B$. Then the \emph{affine Hecke algebra} $\mathcal{H}_{\aff}$ (with equal parameters) attached to these data is the $\Z[v,v^{-1}]$-algebra generated by elements $T_s$ for $s \in S$ and $\theta_x$ for $x \in \mathbb X$, subject to the following relations (where $m_{s,t}$ is the order of $st$ in $W$): \begin{enumerate} \item $(T_s+1)(T_s-v^2)=0$ for $s \in S$; \item $T_s T_t \cdots = T_t T_s \cdots$ for $s,t \in S$ (with $m_{s,t}$ factors on each side); \item \label{it:Haff-rel-2} $\theta_x \theta_y = \theta_{x+y}$ for $x,y \in \mathbb X$; \item \label{it:Haff-rel-3} $\theta_0=1$; \item \label{it:Haff-rel-4} $T_s \cdot \theta_x - \theta_{sx} \cdot T_s = (v^2-1) \frac{\theta_x - \theta_{sx}}{1 - \theta_{-\alpha}}$ for $s \in S$, where $\alpha \in R$ is the corresponding simple root. \end{enumerate}
\begin{remark} \label{rk:subalg-theta} \begin{enumerate} \item Relations~\eqref{it:Haff-rel-2} and~\eqref{it:Haff-rel-3} imply that the subalgebra generated by the generators $\theta_x$ for $x \in \mathbb X$ is isomorphic to the group algebra $\Z[v,v^{-1}][\mathbb X]$; then the quotient in the right-hand side in~\eqref{it:Haff-rel-4} denotes the quotient in this integral ring. \item The present notation differs slightly from the notation in~\cite{MR3}. In fact the element denoted $T_s$ here coincides with the element denoted $t_\alpha$ in~\cite[\S 5.2]{MR3} (for $\alpha$ the corresponding simple root). \end{enumerate} \end{remark}
The following reformulation of relation~\eqref{it:Haff-rel-4} (see~\cite[Proposition~3.9]{LuAff}) will be useful: \begin{equation} \label{eqn:comm-relation-Haff} (T_s+1) \cdot \theta_x - \theta_{sx} \cdot (T_s+1) = (\theta_x-\theta_{sx}) \cdot \mathscr{G}(\alpha) \qquad \text{with} \qquad \mathscr{G}(\alpha)=\frac{v^2 \theta_\alpha - 1}{\theta_\alpha-1}. \end{equation}
The subalgebra of $\mathcal{H}_{\aff}$ generated by the elements $T_s$ ($s \in S$) can be identified with the Hecke algebra $\mathcal{H}_W$ of the Coxeter group $(W,S)$. We will consider the left module $\mathrm{sgn}_{\mathrm{l}}$ of this subalgebra which is (canonically) free of rank one over $\Z[v,v^{-1}]$, and where $T_s$ acts by $-1$. The same recipe also defines a right module $\mathrm{sgn}_{\mathrm{r}}$ over $\mathcal{H}_W$. Then we can define the ``antispherical'' left, resp.~right, module over $\mathcal{H}_{\aff}$ as \[ \mathcal{M}^{\mathrm{asph}}_{\mathrm{l}} := \mathcal{H}_{\aff} \otimes_{\mathcal{H}_W} \mathrm{sgn}_{\mathrm{l}}, \qquad \mathcal{M}^{\mathrm{asph}}_{\mathrm{r}} := \mathrm{sgn}_{\mathrm{r}} \otimes_{\mathcal{H}_W} \mathcal{H}_{\aff}. \] For both modules, we will simply denote by $1$ the ``base point'' $1 \otimes 1$.
We will also consider the associated \emph{graded affine Hecke algebra} $\overline{\mathcal{H}}_{\aff}$ (again, with equal parameters). This algebra is the $\C[r]$-algebra generated by $\mathcal{O}(\mathfrak{t})=\mathrm{S}(\mathfrak{t}^*)$ (where $\mathfrak{t}$ is the Lie algebra of $T$) and elements $t_w$ for $w \in W$, subject to the following relations: \begin{enumerate} \item $t_1=1$; \item $t_v t_w = t_{vw}$ for $v,w \in W$; \item \label{it:gHaff-rel-2} $t_s \cdot \phi - s(\phi) t_s = (\phi - s(\phi)) \cdot (g(\alpha) - 1)$ for $s \in S$, where $\alpha \in R$ is the corresponding simple root. \end{enumerate} Here following~\cite{LuAff} we have used the notation \[ g(\alpha) = \frac{\dot{\alpha}+ 2 r}{\dot{\alpha}}, \] where $\dot{\alpha} \in \mathfrak{t}^*$ is the differential of the root $\alpha$. In this case also, one can reformulate relation~\eqref{it:gHaff-rel-2} in the following form, see~\cite[4.6(c)]{LuAff}: \begin{equation} \label{eqn:comm-relation-gHaff} (t_s+1) \cdot \phi - s(\phi) \cdot (t_s+1) = (\phi - s(\phi)) \cdot g(\alpha). \end{equation}
The subalgebra of $\overline{\mathcal{H}}_{\aff}$ generated by the elements $t_w$ (for $w \in W$) identifies with the group algebra $\overline{\mathcal{H}}_W=\C[r][W]$. As above one can define a ``sign'' left, resp.~right, module over this algebra (where $t_s$ acts by $-1$ for $s \in S$), which we will denote by $\overline{\mathrm{sgn}}_{\mathrm{l}}$, resp.~$\overline{\mathrm{sgn}}_{\mathrm{r}}$, and corresponding ``antispherical'' modules \[ \overline{\mathcal{M}}^{\mathrm{asph}}_{\mathrm{l}} := \overline{\mathcal{H}}_{\aff} \otimes_{\overline{\mathcal{H}}_W} \overline{\mathrm{sgn}}_{\mathrm{l}}, \qquad \overline{\mathcal{M}}^{\mathrm{asph}}_{\mathrm{r}} := \overline{\mathrm{sgn}}_{\mathrm{r}} \otimes_{\overline{\mathcal{H}}_W} \overline{\mathcal{H}}_{\aff}. \]
Let $\mathfrak{m} \subset \mathcal{O}(\mathfrak{t})[r]=\mathcal{O}(\mathfrak{t} \times \mathbb{A}^1)$ denote the maximal ideal associated with the point $(0,0) \in \mathfrak{t} \times \mathbb{A}^1$, and let $\widehat{\mathcal{O}(\mathfrak{t})[r]}$ be the $\mathfrak{m}$-adic completion of $\mathcal{O}(\mathfrak{t})[r]$. Then $\widehat{\overline{\mathcal{H}}}_{\aff}:=\widehat{\mathcal{O}(\mathfrak{t})[r]} \otimes_{\mathcal{O}(\mathfrak{t})[r]} \overline{\mathcal{H}}_{\aff}$ has a natural algebra structure extending the structure on $\overline{\mathcal{H}}_{\aff}$. With this notation introduced, the algebras $\mathcal{H}_{\aff}$ and $\overline{\mathcal{H}}_{\aff}$ are related by the Lusztig morphism \[ \mathscr{L}_{\mathrm{r}} \colon \mathcal{H}_{\aff} \to \widehat{\overline{\mathcal{H}}}_{\aff} \] defined in~\cite[\S 9]{LuAff}.\footnote{The setting considered in~\cite[\S 9]{LuAff} is much more general than the case considered in the present paper. With Lusztig's notation, we only consider the case $v_0=1$ (which is covered by~\cite[\S 9.7]{LuAff}), $r_0=0$, $t_0=1$, $\overline{\Sigma}=\{0\}$.
This case suffices (except in the case when $v$ is specialized to a non trivial root of unity) for the study of the representation theory of $\mathcal{H}_{\aff}$ via the (more accessible) study of the representation theory of $\overline{\mathcal{H}}_{\aff}$; see~\cite{LuAff} for details.\label{fn:setting-lusztig}} Let us recall the definition of this morphism. First, we denote by $\mathbb Y:=X_*(T)$ the lattice of cocharacters of $T$, and consider the map \[ e \colon \left\{ \begin{array}{ccc} \mathfrak{t} = \mathbb Y \otimes_\Z \C & \to & T = \mathbb Y \otimes_\Z \C^\times \\ \lambda^\vee \otimes a & \mapsto & \lambda^\vee \otimes \exp(a) \end{array} \right. . \] This map induces a map \[ \Z[v,v^{-1}][\mathbb X] \to \widehat{\mathcal{O}(\mathfrak{t})[r]} \] sending $x \in \mathbb X$ to (the power series expansion of) $x \circ e$ and $v$ to $\exp(r)$, which can be used to define $\mathscr{L}_{\mathrm{r}}$ on the subalgebra of $\mathcal{H}_{\aff}$ generated by the elements $\theta_x$ ($x \in \mathbb X$), see Remark~\ref{rk:subalg-theta}. Then the description of $\mathscr{L}_{\mathrm{r}}$ is completed by the formula \[ \mathscr{L}_{\mathrm{r}}(T_s+1) = (t_s+1) \cdot g(\alpha)^{-1} \cdot \widetilde{\mathscr{G}}(\alpha), \qquad \text{where} \qquad \widetilde{\mathscr{G}}(\alpha) = \mathscr{L}_{\mathrm{r}}(\mathscr{G}(\alpha)). \] In more concrete terms,
we have (see~\cite[Proof of Lemma~9.5]{LuAff}): \[ g(\alpha)^{-1} \cdot \widetilde{\mathscr{G}}(\alpha) =\frac{\exp(\dot{\alpha}+2r)-1}{\dot{\alpha}+2r} \cdot \frac{\dot{\alpha}}{\exp(\dot{\alpha})-1}. \]
From the defining relations of $\mathcal{H}_{\aff}$ (resp.~$\overline{\mathcal{H}}_{\aff}$) one can see that there exists an anti-involution of $\mathcal{H}_{\aff}$ (resp.~$\overline{\mathcal{H}}_{\aff}$) as a $\Z[v,v^{-1}]$-algebra (resp.~$\C[r]$-algebra), which fixes all generators $T_s$ for $s \in S$ and $\theta_x$ for $x \in \mathbb X$ (resp.~the generators $t_s$ for $s \in S$ and the elements of $\mathcal{O}(\mathfrak{t})$). Conjugating the morphism $\mathscr{L}_{\mathrm{r}}$ with these anti-involutions we obtain a second Lusztig morphism \[ \mathscr{L}_{\mathrm{l}} \colon \mathcal{H}_{\aff} \to \widehat{\overline{\mathcal{H}}}_{\aff} \] which satisfies \[ \mathscr{L}_{\mathrm{l}}(\theta_x) = \mathscr{L}_{\mathrm{r}}(\theta_x), \qquad \mathscr{L}_{\mathrm{l}}(v)=\mathscr{L}_{\mathrm{r}}(v), \qquad \mathscr{L}_{\mathrm{l}}(T_s+1) = g(\alpha)^{-1} \cdot \widetilde{\mathscr{G}}(\alpha) \cdot (t_s+1). \]
\subsection{Geometric realization of $\mathcal{H}_{\aff}$ and its antispherical module(s)} \label{ss:geom-Haff}
Let $\mathcal{B}:=G/B$ be the flag variety of $G$. Then we can consider the constructions of~\S\ref{ss:convolution} for the data $X=\mathcal{B}$, $V=\mathfrak{g}^*$, and with $F$ being the subbundle \[
\widetilde{\mathcal{N}}:=\{(\xi,gB) \in \mathfrak{g}^* \times \mathcal{B} \mid \xi_{|g \cdot \mathfrak{b}}=0\}, \] where $\mathfrak{b}$ is the Lie algebras of $B$.
(This variety is isomorphic to the \emph{Springer resolution} of the nilpotent cone of $G$.) We will also consider \[
\widetilde{\mathfrak{g}}:=\{(\xi,gB) \in \mathfrak{g}^* \times \mathcal{B} \mid \xi_{|g \cdot [\mathfrak{b},\mathfrak{b}]}=0\}. \] (This variety is isomorphic to the \emph{Grothendieck simultaneous resolution.}) Note that the Killing form defines a $G$-equivariant isomorphism $(\mathfrak{g}^*)^* \cong \mathfrak{g}^*$, hence a $G \times {\mathbb{G}}_{\mathbf{m}}$-equivariant isomorphism $E \cong E^*$. Via this isomorphism, $F^\bot$ identifies with $\widetilde{\mathfrak{g}}$.
The \emph{Steinberg variety} is the fiber product \[ Z:=\widetilde{\mathcal{N}} \times_{\mathfrak{g}^*} \widetilde{\mathcal{N}}. \] If $\alpha$ is a simple root, we denote by $P_\alpha \subset G$ the corresponding minimal standard parabolic subgroup, and by $\mathcal{P}_\alpha := G/P_\alpha$ the associated partial flag variety. Then as in~\cite{riche}\footnote{Due to a typo, the subscript ``$\mathcal{P}_\alpha$'' is missing in the fiber product in the description of $S_\alpha'$ in~\cite[\S 6.1]{riche}.} we set \[ S_\alpha' := \{(X,g_1 B, g_2 B) \in \mathfrak{g}^* \times (\mathcal{B} \times_{\mathcal{P}_\alpha} \mathcal{B}) \mid X_{g_1 \cdot \mathfrak{b} + g_2 \cdot \mathfrak{b}}=0 \}. \] In other words, $S_\alpha'$ is the inverse image of $\mathcal{B} \times_{\mathcal{P}_\alpha} \mathcal{B}$ under the projection $Z \to \mathcal{B} \times \mathcal{B}$. This scheme is reduced but not irreducible: its two irreducible components are the diagonal $\Delta \widetilde{\mathcal{N}}$ and \[ Y_\alpha := \{(X,g_1 B, g_2 B) \in \mathfrak{g}^* \times (\mathcal{B} \times_{\mathcal{P}_\alpha} \mathcal{B}) \mid X_{g_1 \cdot \mathfrak{p}_\alpha}=0\}, \] where $\mathfrak{p}_\alpha$ is the Lie algebra of $P_\alpha$.
With these definitions, we obtain algebras $\mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(Z)$ and $\mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(Z)$. It follows from work of Kazhdan--Lusztig~\cite{KL}, Ginzburg~\cite{CG} and Lusztig~\cite{LUSBas} that there exists an algebra isomorphism\footnote{Due to our change of convention in the definition of the convolution product (see Footnote~\ref{fn:convention-product}), the isomorphism~\eqref{eqn:Haff-K(Z)} is the composition of the isomorphism considered in~\cite[\S 5.2]{MR3} with the anti-involution considered at the end of~\S\ref{ss:Haff-gHaff}.} \begin{equation} \label{eqn:Haff-K(Z)} \mathcal{H}_{\aff} \xrightarrow{\sim} \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(Z) \end{equation} which satisfies \[ v \mapsto [\mathcal{O}_{\Delta \widetilde{\mathcal{N}}} \langle 1 \rangle], \qquad \theta_x \mapsto [\mathcal{O}_{\Delta \widetilde{\mathcal{N}}}(x)], \qquad T_s \mapsto -[\mathcal{O}_{Y_\alpha}(-\rho, \rho-\alpha)] - [\mathcal{O}_{\Delta \widetilde{\mathcal{N}}}] = -[\mathcal{O}_{S_\alpha'}]. \] (In the middle term, $\mathcal{O}_{\Delta \widetilde{\mathcal{N}}}(x)$ is (the direct image of) the line bundle on
$\Delta \widetilde{\mathcal{N}}$ obtained by pullback of the line bundle on $\mathcal{B}$ naturally associated with $x$. In the third term, $\alpha$ is the simple root associated with $s$,
and $\rho$ is the half-sum of the positive roots;
the equality follows from~\cite[Lemma~6.1.1]{riche}.)
We also have isomorphisms of $\Z[v,v^{-1}]$-modules \begin{equation} \label{eqn:Masph-Kth} \mathcal{M}^{\mathrm{asph}}_{\mathrm{l}} \xrightarrow{\sim} \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(\widetilde{\mathcal{N}}), \qquad \text{resp.} \qquad \mathcal{M}^{\mathrm{asph}}_{\mathrm{l}} \xrightarrow{\sim} \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(\widetilde{\mathcal{N}}), \end{equation} where $v^n \theta_x \cdot 1$, resp.~$1 \cdot v^n \theta_x$, corresponds to $[\mathcal{O}_{\widetilde{\mathcal{N}}}(x) \langle n \rangle]$ (for $x \in \mathbb X$).
\begin{lem} \label{lem:apsh-module-geom} The isomorphisms~\eqref{eqn:Masph-Kth}
are isomorphisms of left and right $\mathcal{H}_{\aff}$-modules respectively.
\end{lem}
\begin{proof} It is enough to prove that for $\alpha$ a simple root we have \[ [\mathcal{O}_{S_\alpha'}] \star [\mathcal{O}_{\widetilde{\mathcal{N}}}] = [\mathcal{O}_{\widetilde{\mathcal{N}}}] = [\mathcal{O}_{\widetilde{\mathcal{N}}}] \star [\mathcal{O}_{S_\alpha'}]. \] By symmetry the two equalities are equivalent, so we restrict to the first one. By definition we have $[\mathcal{O}_{S_\alpha'}] \star [\mathcal{O}_{\widetilde{\mathcal{N}}}] = [Rp_{1*} (\mathcal{O}_{S_\alpha'})]$. If $S_\alpha \subset \widetilde{\mathfrak{g}} \times \widetilde{\mathfrak{g}}$ is the subvariety defined in~\cite[\S 1.4]{riche}, in the derived category of (equivariant) coherent sheaves on $\widetilde{\mathfrak{g}} \times \widetilde{\mathfrak{g}}$, by~\cite[Lemma~4.1]{riche} we have \[ \mathcal{O}_{\widetilde{\mathcal{N}} \times \widetilde{\mathfrak{g}}} \, \lotimes_{\mathcal{O}_{\widetilde{\mathfrak{g}} \times \widetilde{\mathfrak{g}}}} \, \mathcal{O}_{S_\alpha} \cong \mathcal{O}_{S_\alpha'}. \] Then, by the (non flat) base change theorem (e.g.~in the form of~\cite[Proposition~3.7.1]{BR}), to prove our equality it is enough to prove that \[ Rq_{1*} \mathcal{O}_{S_\alpha} \cong \mathcal{O}_{\widetilde{\mathfrak{g}}}, \] where $q_1 \colon \widetilde{\mathfrak{g}} \times \widetilde{\mathfrak{g}} \to \widetilde{\mathfrak{g}}$ is the projection on the first factor. This is proved in~\cite[Lemma~2.7.2]{BR}.\footnote{The subvariety $S_\alpha$ is denoted $Z_s$ in~\cite{BR}, where $s$ is the corresponding simple reflection. Also, in~\cite[\S 2]{BR} the base field is assumed to be of positive characteristic; but the proof of the cited lemma works over any algebraically closed field of coefficients.} \end{proof}
One also has a similar geometric realization using $\widetilde{\mathfrak{g}}$ instead of $\widetilde{\mathcal{N}}$. In fact, if we set \[ \mathcal{Z}:=\widetilde{\mathfrak{g}} \times_{\mathfrak{g}^*} \widetilde{\mathfrak{g}}, \] as explained in~\cite[Lemma~5.2]{MR3}, restriction with supports associated with the inclusion $\widetilde{\mathcal{N}} \times \widetilde{\mathfrak{g}} \hookrightarrow \widetilde{\mathfrak{g}} \times \widetilde{\mathfrak{g}}$ induces an algebra isomorphism $\mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(\mathcal{Z}) \xrightarrow{\sim} \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(Z)$. Therefore, we have an algebra isomorphism \begin{equation} \label{eqn:Haff-K(calZ)} \mathcal{H}_{\aff} \xrightarrow{\sim} \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(\mathcal{Z}) \end{equation} which satisfies \[ v \mapsto [\mathcal{O}_{\Delta \widetilde{\mathfrak{g}}} \langle 1 \rangle], \qquad \theta_x \mapsto [\mathcal{O}_{\Delta \widetilde{\mathfrak{g}}}(x)], \qquad T_s \mapsto -[\mathcal{O}_{S_\alpha}]. \] (Here we use conventions similar to those for $\widetilde{\mathcal{N}}$, and $S_\alpha$ is defined in~\cite[\S 1.4]{riche}.) As in Lemma~\ref{lem:apsh-module-geom}, we also have isomorphisms of left, resp.~right, $\mathcal{H}_{\aff}$-modules \[ \mathcal{M}^{\mathrm{asph}}_{\mathrm{l}} \xrightarrow{\sim} \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(\widetilde{\mathfrak{g}}), \qquad \mathrm{resp.} \qquad \mathcal{M}^{\mathrm{asph}}_{\mathrm{r}} \xrightarrow{\sim} \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(\widetilde{\mathfrak{g}}). \]
\subsection{Geometric realization of $\overline{\mathcal{H}}_{\aff}$ and its antispherical module(s)} \label{ss:geom-gHaff}
Following~\cite{LuCus1, LuCus2}, replacing $\mathsf{K}$-theory by Borel--Moore homology in the constructions of~\S\ref{ss:geom-Haff} one obtains a geometric realization of $\overline{\mathcal{H}}_{\aff}$; in fact, applying~\cite[Theorem~8.11]{LuCus2} in our situation (i.e.~for the Levi subgroup $T$, its nilpotent orbit $\{0\}$, and the cuspidal local system $\underline{\C}_{\{0\}}$ on $\{0\}$), we obtain an algebra isomorphism \begin{equation} \label{eqn:gHaff-H(calZ)} \overline{\mathcal{H}}_{\aff} \xrightarrow{\sim} \mathsf{H}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(\mathcal{Z}) \end{equation} such that the subalgebra $\mathcal{O}(\mathfrak{t})[r]$ is obtained as the image (under proper direct image) of \begin{equation} \label{eqn:gHaff-geom-diag} \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(\Delta \widetilde{\mathfrak{g}}) \cong \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(\mathcal{B}) \cong \mathsf{H}^{B \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(\mathrm{pt}) \cong \mathsf{H}^{T \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(\mathrm{pt}) \cong \mathcal{O}(\mathfrak{t})[r]. \end{equation} (More concretely, if $x \in \mathbb X$, then $\dot{x} \in \mathfrak{t}^*$ corresponds to $[\Delta \widetilde{\mathfrak{g}}] \cdot c_1^{G \times {\mathbb{G}}_{\mathbf{m}}}(\mathcal{O}_{\widetilde{\mathfrak{g}}}(x))$, where $c_1^{G \times {\mathbb{G}}_{\mathbf{m}}}(-)$ is the first equivariant Chern class.) The image of $\C[W]$ is obtained via the ``Springer isomorphism'' \[ \C[W] \xrightarrow{\sim} {\rm Hom}_{D^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(\mathfrak{g}^*)}(p_! \underline{\C}_{\widetilde{\mathfrak{g}}}, p_! \underline{\C}_{\widetilde{\mathfrak{g}}}) \hookrightarrow \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(\mathcal{Z}). \] (Here the inclusion is induced by the isomorphism $\mathsf{can}_{\widetilde{\mathfrak{g}},\widetilde{\mathfrak{g}}}$ of~\S\ref{ss:equiv}.) As in~\eqref{eqn:gHaff-geom-diag} we also have a natural isomorphism of $\C[r]$-modules
\begin{equation} \label{eqn:gMasph-geom} \overline{\mathcal{M}}^{\mathrm{asph}}_{\mathrm{l}} \xrightarrow{\sim} \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(\widetilde{\mathfrak{g}}), \qquad \text{resp.} \qquad \overline{\mathcal{M}}^{\mathrm{asph}}_{\mathrm{r}} \xrightarrow{\sim} \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(\widetilde{\mathfrak{g}}),
\end{equation}
where $\dot{x} \cdot 1$, resp.~$1 \cdot \dot{x}$, corresponds to $[\widetilde{\mathfrak{g}}] \cdot c_1^{G \times {\mathbb{G}}_{\mathbf{m}}}(\mathcal{O}_{\widetilde{\mathfrak{g}}}(x))$.
\begin{lem} The isomorphisms~\eqref{eqn:gMasph-geom}
are isomorphisms of left and right $\overline{\mathcal{H}}_{\aff}$-modules respectively.
\end{lem}
\begin{proof} As in Lemma~\ref{lem:apsh-module-geom}, by symmetry it is enough to prove the equivariance in the first case. Using similar constructions as for $\mathsf{H}_{\bullet}^{G \times {\mathbb{G}}_{\mathbf{m}}}(\widetilde{\mathfrak{g}})$, one can construct an action by convolution of $\mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(\mathcal{Z})$ on $\mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(\mathcal{B})$, where $\mathcal{B}$ is seen as the zero section of $\widetilde{\mathfrak{g}}$; see~\cite[Corollary~2.7.41]{CG} in the non-equivariant setting. Moreover, the Thom isomorphism $\mathsf{H}_{\bullet}^{G \times {\mathbb{G}}_{\mathbf{m}}}(\widetilde{\mathfrak{g}}) \xrightarrow{\sim} \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(\mathcal{B})$ is equivariant for this action. Therefore, it is enough to prove that the natural isomorphism $\mathcal{O}(\mathfrak{t})[r] \xrightarrow{\sim} \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(\mathcal{B})$ induces an isomorphism of left $\overline{\mathcal{H}}_{\aff}$-modules $\overline{\mathcal{M}}^{\mathrm{asph}}_{\mathrm{l}} \xrightarrow{\sim} \mathsf{H}_{\bullet}^{G \times {\mathbb{G}}_{\mathbf{m}}}(\mathcal{B})$. And for this it is enough to prove that $t_s \cdot [\mathcal{B}] = -[\mathcal{B}]$ for $s \in S$. Now the forgetful morphism $\mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{2\dim(\mathcal{B})}(\mathcal{B}) \to \mathsf{H}_{2\dim(\mathcal{B})}(\mathcal{B})$ is an isomorphism, and so is the morphism $\mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{2\dim(\mathcal{Z})}(\mathcal{Z}) \to \mathsf{H}_{2\dim(\mathcal{Z})}(\mathcal{Z})$. Hence we have reduced our question to a claim about non-equivariant Borel--Moore homology, which can be solved using Springer theory.
By~\cite[Proposition~8.6.16]{CG}, if $i_0 \colon \{0\} \hookrightarrow \widetilde{\mathfrak{g}}$ denotes the inclusion, there exists a canonical isomorphism $\mathsf{H}_{\bullet}(\mathcal{B}) \xrightarrow{\sim} \mathsf{H}^{2\dim(\mathfrak{g})-\bullet}(i_0^! p_! \underline{\C}_{\widetilde{\mathfrak{g}}})$, which identifies the action of $\mathsf{H}_\bullet(\mathcal{Z})$ with the natural action of ${\rm Hom}^\bullet_{D^b_{\mathrm{const}}(\mathfrak{g}^*)}(p_! \underline{\C}_{\widetilde{\mathfrak{g}}}, p_! \underline{\C}_{\widetilde{\mathfrak{g}}})$ via the non-equivariant analogue of the isomorphism $\mathsf{can}_{\widetilde{\mathfrak{g}}, \widetilde{\mathfrak{g}}}$. Hence what we have to show is that the $1$-dimensional $W$-module \[ \mathsf{H}_{2\dim(\mathcal{B})}(\mathcal{B}) \cong \mathsf{H}^{2\dim(\mathfrak{g})-2\dim(\mathcal{B})}(i_0^! p_! \underline{\C}_{\widetilde{\mathfrak{g}}}) \] is the sign representation. This fact is well known, see e.g.~\cite[Lemmas 4.5 \& 4.6]{AHJR}. \end{proof}
As in~\S\ref{ss:geom-Haff}, we have a similar story when $\widetilde{\mathfrak{g}}$ is replaced by $\widetilde{\mathcal{N}}$. In fact, constructions similar to those in~\cite[Lemma~5.2]{MR3} show that restriction with supports induces an algebra isomorphism $\mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(\mathcal{Z}) \xrightarrow{\sim} \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(Z)$. (This property can also be extracted from~\cite{LuCus1, LuCus2}; it is used implictly in~\cite{EM}.) Therefore, we obtain isomorphisms of algebras and modules over these algebras \begin{equation} \label{eqn:gHaff-H(Z)} \overline{\mathcal{H}}_{\aff} \xrightarrow{\sim} \mathsf{H}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(Z), \qquad \overline{\mathcal{M}}^{\mathrm{asph}}_{\mathrm{l}} \xrightarrow{\sim} \mathsf{H}_{\bullet}^{G \times {\mathbb{G}}_{\mathbf{m}}}(\widetilde{\mathcal{N}}), \qquad \overline{\mathcal{M}}^{\mathrm{asph}}_{\mathrm{r}} \xrightarrow{\sim} \mathsf{H}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(\widetilde{\mathcal{N}}). \end{equation}
\begin{prop} \label{prop:RR-Haff} \begin{enumerate} \item \label{it:RR-Haff-l} Under the isomorphisms~\eqref{eqn:Haff-K(Z)} and~\eqref{eqn:gHaff-H(Z)}, the morphism $\underline{\mathrm{RR}}_{\widetilde{\mathcal{N}},\widetilde{\mathcal{N}}}$ identifies with $\mathscr{L}_{\mathrm{l}}$. \item \label{it:RR-Haff-r} Under the isomorphisms~\eqref{eqn:Haff-K(calZ)} and~\eqref{eqn:gHaff-H(calZ)}, the morphism $\overline{\mathrm{RR}}_{\widetilde{\mathfrak{g}},\widetilde{\mathfrak{g}}}$ identifies with the morphism $c \mapsto e_{\mathcal{B}} \cdot \mathscr{L}_{\mathrm{r}}(c) \cdot e_{\mathcal{B}}^{-1}$, where \[ e_{\mathcal{B}} := \prod_{\alpha \in R^+} \frac{\dot{\alpha}}{1 - \exp(-\dot{\alpha})}. \] \end{enumerate} \end{prop}
\begin{proof} First, we note that $Z$ and $\mathcal{Z}$ are paved by affine spaces, so that the ``parity vanishing'' assumptions in some of our statements above are satisfied in these cases.
\eqref{it:RR-Haff-l} Both of our maps are algebra morphisms (see Lemma~\ref{lem:RR-convolution}), so it is enough to check that they coincide on the generators of $\mathcal{H}_{\aff}$. The case of $v$ is obvious (see~\cite[\S 3.3]{EG}), and the case of $\theta_x$ follows from Proposition~\ref{prop:tau-O-smooth}. It remains to consider the case of $T_s$; in fact it will be simpler (but equivalent) to prove that \begin{equation} \label{eqn:RR-Haff-s} \underline{\mathrm{RR}}_{\widetilde{\mathcal{N}},\widetilde{\mathcal{N}}}(1+T_s)=\mathscr{L}_{\mathrm{l}}(1+T_s) = g(\alpha)^{-1} \cdot \widetilde{\mathscr{G}}(\alpha) \cdot (t_s+1). \end{equation}
By Remark~\ref{rmk:injectivity-faithful} and Proposition~\ref{prop:criterion-thm}, $\mathsf{H}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(\widetilde{\mathcal{N}})$ is faithful as a module over $\mathsf{H}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(Z)$. Therefore, the same is true for the completions, and to prove~\eqref{eqn:RR-Haff-s} it is enough to prove that both sides act similarly on $\widehat{\mathsf{H}}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(\widetilde{\mathcal{N}})$. However, by Lemma~\ref{lem:apsh-module-geom} and~\eqref{eqn:comm-relation-Haff}, for $x \in \mathbb X$ we have $(1+T_s) \cdot (\theta_x \cdot 1) = \bigl( (\theta_x - \theta_{sx}) \cdot \mathscr{G}(\alpha) \bigr) \cdot 1$. By Lemma~\ref{lem:RR-action}, this implies that in $\overline{\mathcal{M}}^{\mathrm{asph}}_{\mathrm{l}}$ we have \[ \underline{\mathrm{RR}}_{\widetilde{\mathcal{N}},\widetilde{\mathcal{N}}}(1+T_s) \cdot (\exp(\dot{x}) \cdot 1) = \bigl( (\exp(\dot{x})-\exp(s\dot{x})) \cdot \widetilde{\mathscr{G}}(\alpha) \bigr) \cdot 1. \] Using~\eqref{eqn:comm-relation-gHaff}, this coincides with the action of $g(\alpha)^{-1} \cdot \widetilde{\mathscr{G}}(\alpha) \cdot (t_s+1)$. Since the elements of the form $r^n \exp(\dot{x}) \cdot 1$ form a topological basis of $\widehat{\mathsf{H}}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(\widetilde{\mathcal{N}})$, we deduce the equality in~\eqref{eqn:RR-Haff-s}.
\eqref{it:RR-Haff-r} The proof is similar to the proof of~\eqref{it:RR-Haff-l}, using the \emph{right} action on $\widehat{\mathsf{H}}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(\widetilde{\mathfrak{g}})$, and using the fact that \[ \mathrm{Td}_{\mathcal{B}}^{G \times {\mathbb{G}}_{\mathbf{m}}} = \prod_{\alpha \in R^+} \frac{\dot{\alpha}}{1 - \exp(-\dot{\alpha})} \qquad \text{in} \qquad \widehat{\mathsf{H}}_{G \times {\mathbb{G}}_{\mathbf{m}}}^\bullet(\widetilde{\mathfrak{g}}) = \widehat{\mathcal{O}(\mathfrak{t})[r]} \] (as follows from~\cite[\S 3.3]{EG}, since the tangent bundle on $\mathcal{B}$ has a filtration with associated graded the sum of the line bundles $\mathcal{O}_{\mathcal{B}}(\alpha)$ for $\alpha \in R^+$). \end{proof}
\begin{remark} In~\cite[\S 0.3]{LuAff}, Lusztig explains that his morphism $\mathscr{L}_{\mathrm{r}}$ ``is of the same nature as the Chern character from $\mathsf{K}$-theory to homology.'' Proposition~\ref{prop:RR-Haff} is a concrete justification of this claim. \end{remark}
\subsection{Commutative diagram for affine Hecke algebras}
Finally we can consider the diagram of Theorem~\ref{thm:LKDFourier} in the geometric setting of \S\S\ref{ss:geom-Haff}--\ref{ss:geom-gHaff}: \begin{equation} \label{eqn:diagram-Haff} \vcenter{ \xymatrix@C=3cm{ \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(Z) \ar[r]^-{\mathbf{Koszul}_{\widetilde{\mathcal{N}}, \widetilde{\mathcal{N}}}} \ar[d]_-{\underline{\mathrm{RR}}_{\widetilde{\mathcal{N}}, \widetilde{\mathcal{N}}}} & \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(\mathcal{Z}) \ar[r]^-{\mathbf{i}_{\widetilde{\mathfrak{g}}, \widetilde{\mathfrak{g}}} \circ \mathbf{D}_{\widetilde{\mathfrak{g}}, \widetilde{\mathfrak{g}}}} & \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(\mathcal{Z}) \ar[d]^-{\overline{\mathrm{RR}}_{\widetilde{\mathfrak{g}}, \widetilde{\mathfrak{g}}}} \\ \widehat{\mathsf{H}}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(Z) \ar[rr]^-{\mathfrak{Fourier}_{\widetilde{\mathcal{N}},\widetilde{\mathcal{N}}}} & & \widehat{\mathsf{H}}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(\mathcal{Z}). } } \end{equation} Note that Proposition \ref{prop:criterion-thm} ensures that the assumption of Theorem~\ref{thm:LKDFourier} is satisfied in this case, since $\mathcal{Z}$ is paved by affine spaces, and that the results of~\S\ref{ss:convolution}--\ref{ss:RR-convolution} ensure that all the maps in this diagram are unital algebra morphisms. Using Proposition~\ref{prop:RR-Haff} and the results of~\cite{EM} and~\cite{MR3} we can describe explicitly all the maps in this diagram, and hence illustrate the content of Theorem~\ref{thm:LKDFourier} in this particular situation.
The morphism $\mathbf{Koszul}_{\widetilde{\mathcal{N}},\widetilde{\mathcal{N}}}$ was studied in~\cite[\S 5.3]{MR3}. In particular,~\cite[Theorem~5.4]{MR3} describes this automorphism algebraically, and shows that it is closely related to the \emph{Iwahori--Matsumoto} involution of $\mathcal{H}_{\rm aff}$. Using the identifications~\eqref{eqn:Haff-K(Z)} and~\eqref{eqn:Haff-K(calZ)}, we have
\[ \mathbf{Koszul}_{\widetilde{\mathcal{N}},\widetilde{\mathcal{N}}}(T_s)= \theta_\rho(-v^2 T_s^{-1}) \theta_{-\rho}, \qquad \mathbf{Koszul}_{\widetilde{\mathcal{N}},\widetilde{\mathcal{N}}}(\theta_x)=\theta_{-x}, \qquad \mathbf{Koszul}_{\widetilde{\mathcal{N}},\widetilde{\mathcal{N}}}(v)=-v \] for $s \in S$ a simple root and $x \in \mathbb X$.\footnote{As noted in Footnote~\ref{fn:MR3}, the conventions in the definition of $\mathfrak{K}_{\widetilde{\mathcal{N}},\widetilde{\mathcal{N}}}$ used in the present paper differ slightly from the conventions used in~\cite{MR3}. Our identification of $\mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(\mathcal{Z})$ is also slightly different, see~\cite[Comments at the end of~\S5.2]{MR3}. This explains the differences with the formulas in~\cite{MR3}.}
Concerning the map $\mathbf{D}_{\widetilde{\mathfrak{g}},\widetilde{\mathfrak{g}}}$, one can check that, with the identification~\eqref{eqn:Haff-K(calZ)}, it satisfies \[ \mathbf{D}_{\widetilde{\mathfrak{g}},\widetilde{\mathfrak{g}}}(T_s)=T_s^{-1}, \qquad \mathbf{D}_{\widetilde{\mathfrak{g}},\widetilde{\mathfrak{g}}}(\theta_x)=\theta_{-x}, \qquad \mathbf{D}_{\widetilde{\mathfrak{g}},\widetilde{\mathfrak{g}}}(v)=v^{-1}.
\] (See~\cite[Lemma~9.7]{LUSBas} for a similar computation, with different conventions.) Finally, the morphism $\mathbf{i}_{\widetilde{\mathfrak{g}},\widetilde{\mathfrak{g}}}$ is the same as the involution $\iota$ of~\cite[\S 5.3]{MR3}; it satisfies \[ \mathbf{i}_{\widetilde{\mathfrak{g}},\widetilde{\mathfrak{g}}}(T_s)=T_s, \qquad \mathbf{i}_{\widetilde{\mathfrak{g}},\widetilde{\mathfrak{g}}}(\theta_x)=\theta_{x}, \qquad \mathbf{i}_{\widetilde{\mathfrak{g}},\widetilde{\mathfrak{g}}}(v)=-v. \]
On the Borel--Moore homology side, the map $\mathfrak{Fourier}_{\widetilde{\mathcal{N}},\widetilde{\mathcal{N}}}$ was studied in~\cite{EM}. In that paper it was shown to be closely related to the Iwahori--Matsumoto involution of $\overline{\mathcal{H}}_{\aff}$; more precisely it satisfies \[ \mathfrak{Fourier}_{\widetilde{\mathcal{N}},\widetilde{\mathcal{N}}} (t_w) = (-1)^{\ell(w)} t_w, \qquad \mathfrak{Fourier}_{\widetilde{\mathcal{N}},\widetilde{\mathcal{N}}}(\phi)=\phi, \qquad \mathfrak{Fourier}_{\widetilde{\mathcal{N}},\widetilde{\mathcal{N}}}(r)=-r \] for $w \in W$ and $\phi \in \mathcal{O}(\mathfrak{t})$.
Using these formulas one can check the commutativity of~\eqref{eqn:diagram-Haff} by hand. For instance, for the element $1+T_s$, the commutativity of the diagram amounts to the following equality in $\overline{\mathcal{H}}_{\aff}$: \begin{multline*} \frac{\exp(\dot{\alpha}-2r)-1}{\dot{\alpha}-2r}\frac{\dot{\alpha}}{\exp(\dot{\alpha})-1}(-t_s+1) = \\ 1 - \exp(-\dot{\rho}-2r) e_\mathcal{B} \Bigl( (t_s+1) \frac{\exp(\dot{\alpha}+2r)-1}{\dot{\alpha}+2r}\frac{\dot{\alpha}}{\exp(\dot{\alpha})-1} - 1 \Bigr) e_\mathcal{B}^{-1} \exp(\dot{\rho}). \end{multline*}
\section{Compatibility of the $\mathfrak{Fourier}$ isomorphism with inclusions} \label{sec:compatibility-Fourier}
In this section and the next one we will consider compatibility properties of our morphisms in two geometric situations. We use the same setting and notation as in \S\S\ref{ss:equiv}--\ref{ss:statement}.
\subsection{Further notation} \label{ss:settings}
First we will consider a situation which we will refer to as Setting (A): here we are given an additional subbundle $F_2' \subset E$ containing $F_2$ and such that $F_2$, $F_2'$ and $E$ can be locally simultaneously trivialized. Then we have ``restriction with supports'' morphisms associated with the embedding $F_2 \hookrightarrow F_2'$, both in $\mathsf{K}$-homology and in Borel--Moore homology, which we denote as follows: \begin{align*} \mathbf{res}^{F_1, F_2'}_{F_1,F_2} \colon \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}} (F_1 \times_V F_2') \ & \to \ \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1 \times_V F_2); \\ \mathfrak{res}^{F_1, F_2'}_{F_1,F_2} \colon \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet (F_1 \times_V F_2') \ & \to \ \mathsf{H}_{\bullet-2\mathrm{rk}(F_2')+2\mathrm{rk}(F_2)}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1 \times_V F_2). \end{align*} We also have proper direct image morphisms associated with the embedding $(F_2')^\bot \hookrightarrow F_2^\bot$, again both in $\mathsf{K}$-homology and in Borel--Moore homology, which we denote as follows: \begin{align*} \mathbf{pdi}^{F_1^\bot, (F_2')^\bot}_{F_1^\bot, F_2^\bot} \colon \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1^\bot \times_{V^*} (F_2')^\bot) \ &\to \ \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1^\bot \times_{V^*} F_2^\bot); \\ \mathfrak{pdi}^{F_1^\bot, (F_2')^\bot}_{F_1^\bot, F_2^\bot} \colon \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1^\bot \times_{V^*} (F_2')^\bot) \ &\to \ \mathsf{H}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1^\bot \times_{V^*} F_2^\bot). \end{align*}
Secondly, we will consider a situation which we will refer to as Setting (B): here we are given an additional subbundle $F_1' \subset E$ containing $F_1$ and such that $F_1$, $F_1'$ and $E$ can be locally simultaneously trivialized. Then we have proper direct image morphisms associated with the embedding $F_1 \hookrightarrow F_1'$, both in $\mathsf{K}$-homology and in Borel--Moore homology, which we denote as follows: \begin{align*} \mathbf{pdi}^{F_1,F_2}_{F_1',F_2} \colon \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1 \times_V F_2) \ & \to \ \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1' \times_V F_2); \\ \mathfrak{pdi}^{F_1,F_2}_{F_1',F_2} \colon \mathsf{H}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1 \times_V F_2) \ & \to \ \mathsf{H}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1' \times_V F_2). \end{align*} We also have ``restriction with supports'' morphisms associated with the embedding $(F_1')^\bot \hookrightarrow F_1^\bot$, again both in $\mathsf{K}$-homology and in Borel--Moore homology, which we denote as follows: \begin{align*} \mathbf{res}^{F_1^\bot, F_2^\bot}_{(F_1')^\bot, F_2^\bot} \colon \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1^\bot \times_{V^*} F_2^\bot) \ & \to \ \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}((F_1')^\bot \times_{V^*} F_2^\bot); \\ \mathfrak{res}^{F_1^\bot, F_2^\bot}_{(F_1')^\bot, F_2^\bot} \colon \mathsf{H}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1^\bot \times_{V^*} F_2^\bot) \ &\to \ \mathsf{H}_{\bullet-2\mathrm{rk}(F_1^\bot)+2\mathrm{rk}((F_1')^\bot)}^{G \times {\mathbb{G}}_{\mathbf{m}}}((F_1')^\bot \times_{V^*} F_2^\bot). \end{align*}
\subsection{Convolution algebras and inclusion of subbundles} \label{ss:subbundle}
Consider Setting (A) of \S \ref{ss:settings}. Then we have natural morphisms induced by adjunction \[ \mathrm{adj}_{F_2,F_2'}^* \colon \underline{\mathbb{C}}_{F_2'} \to \underline{\mathbb{C}}_{F_2} \quad \text{ and } \quad \mathrm{adj}_{(F_2')^\bot,F_2^\bot}^! \colon \underline{\mathbb{C}}_{(F_2')^\bot} \to \underline{\mathbb{C}}_{F_2^\bot}[2\mathrm{rk}(F_2^\bot)-2\mathrm{rk}((F_2')^\bot)]. \] The proof of the following result being rather technical (and the details not needed), it is postponed to the appendix (see \S\S\ref{ss:proof-inclusion-Ginzburg-1}--\ref{ss:proof-inclusion-Ginzburg-2}).
\begin{prop} \label{prop:inclusion-Ginzburg-A} \begin{enumerate} \item The following diagram commutes: \[ \xymatrix@C=2.5cm{ \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1 \times_V F_2') \ar[r]^-{\mathsf{can}_{F_1, F_2'}}_-{\sim} \ar[d]_-{\mathfrak{res}^{F_1,F_2'}_{F_1,F_2}} & {\rm Ext}^{2\dim(F_2')-\bullet}_{\mathcal{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(V)}(p_! \underline{\mathbb{C}}_{F_1}, p_! \underline{\mathbb{C}}_{F_2'}) \ar[d]^-{(p_! \mathrm{adj}_{F_2,F_2'}^*) \circ (\cdot)} \\ \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\bullet-2\mathrm{rk}(F_2')+2\mathrm{rk}(F_2)}(F_1 \times_V F_2) \ar[r]^-{\mathsf{can}_{F_1,F_2}}_-{\sim} & {\rm Ext}^{2\dim(F_2')-\bullet}_{\mathcal{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(V)}(p_! \underline{\mathbb{C}}_{F_1}, p_! \underline{\mathbb{C}}_{F_2}). } \] \item The following diagram commutes: \[ \xymatrix@C=2.5cm{ \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1^\bot \times_{V^*} (F_2')^\bot) \ar[r]^-{\mathsf{can}_{F_1^\bot, (F_2')^\bot}}_-{\sim} \ar[d]_-{\mathfrak{pdi}^{F_1^\bot, (F_2')^\bot}_{F_1^\bot, F_2^\bot}} & {\rm Ext}^{2\dim((F_2')^\bot)-\bullet}_{\mathcal{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(V^*)}({\check p}_! \underline{\mathbb{C}}_{F_1^\bot}, {\check p}_! \underline{\mathbb{C}}_{(F_2')^\bot}) \ar[d]^-{({\check p}_! \mathrm{adj}_{(F_2')^\bot,F_2^\bot}^!) \circ (\cdot)} \\ \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1^\bot \times_{V^*} F_2^\bot) \ar[r]^-{\mathsf{can}_{F_1^\bot, F_2^\bot}}_-{\sim} & {\rm Ext}^{2\dim(F_2^\bot)-\bullet}_{\mathcal{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(V^*)}({\check p}_! \underline{\mathbb{C}}_{F_1^\bot}, {\check p}_! \underline{\mathbb{C}}_{F_2^\bot}). } \] \end{enumerate}
\end{prop}
Consider now Setting (B) of \S\ref{ss:settings}. We have natural morphisms induced by adjunction \[ \mathrm{adj}_{F_1,F_1'}^* \colon \underline{\mathbb{C}}_{F_1'} \to \underline{\mathbb{C}}_{F_1} \quad \text{ and } \quad \mathrm{adj}_{(F_1')^\bot,F_1^\bot}^! \colon \underline{\mathbb{C}}_{(F_1')^\bot} \to \underline{\mathbb{C}}_{F_1^\bot}[2\mathrm{rk}(F_1^\bot)-2\mathrm{rk}((F_1')^\bot)]. \] The proof of the following proposition is similar to that of Proposition \ref{prop:inclusion-Ginzburg-A}, and is therefore omitted.
\begin{prop} \label{prop:inclusion-Ginzburg-B} \begin{enumerate} \item The following diagram commutes: \[ \xymatrix@C=2.5cm{ \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1 \times_V F_2) \ar[r]^-{\mathsf{can}_{F_1, F_2}}_-{\sim} \ar[d]_-{\mathfrak{pdi}^{F_1, F_2}_{F_1', F_2}} & {\rm Ext}^{2\dim(F_2)-\bullet}_{\mathcal{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(V)}(p_! \underline{\mathbb{C}}_{F_1}, p_! \underline{\mathbb{C}}_{F_2}) \ar[d]^-{(\cdot) \circ (p_! \mathrm{adj}_{F_1,F_1'}^*)} \\ \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1' \times_V F_2) \ar[r]^-{\mathsf{can}_{F_1',F_2}}_-{\sim} & {\rm Ext}^{2\dim(F_2)-\bullet}_{\mathcal{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(V)}(p_! \underline{\mathbb{C}}_{F_1'}, p_! \underline{\mathbb{C}}_{F_2}). } \] \item The following diagram commutes: {\small \[ \xymatrix@C=2cm{ \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1^\bot \times_{V^*} F_2^\bot) \ar[r]^-{\mathsf{can}_{F_1^\bot, F_2^\bot}}_-{\sim} \ar[d]_-{\mathfrak{res}^{F_1^\bot, F_2^\bot}_{(F_1')^\bot, F_2^\bot}} & {\rm Ext}^{2\dim(F_2^\bot)-\bullet}_{\mathcal{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(V^*)}({\check p}_! \underline{\mathbb{C}}_{F_1^\bot}, {\check p}_! \underline{\mathbb{C}}_{F_2^\bot}) \ar[d]^-{(\cdot) \circ ({\check p}_! \mathrm{adj}_{(F_1')^\bot,F_1^\bot}^!)} \\ \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\bullet-2\mathrm{rk}(F_1^\bot)+2\mathrm{rk}((F_1')^\bot)}((F_1')^\bot \times_{V^*} F_2^\bot) \ar[r]^-{\mathsf{can}_{(F_1')^\bot, F_2^\bot}}_-{\sim} & {\rm Ext}^{2\dim(F_2^\bot)+2\mathrm{rk}(F_1^\bot)-2\mathrm{rk}((F_1')^\bot)-\bullet}_{\mathcal{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(V^*)}({\check p}_! \underline{\mathbb{C}}_{(F_1')^\bot}, {\check p}_! \underline{\mathbb{C}}_{F_2^\bot}). } \] } \end{enumerate}
\end{prop}
\subsection{Fourier transform and inclusion of subbundles}
In the next lemma $G$ can be replaced by any linear algebraic group, $X$ by any smooth $G$-variety, and $E$ by any $G$-equivariant vector bundle over $X$. We consider subbundles $F \subset F' \subset E$ which can be locally simultaneously trivialized. (In practice, $E$ and $X$ will be as above, and we will take $F=F_i$, $F'=F'_i$ for $i \in \{1,2\}$.) Adjunction induces morphisms \[ \mathrm{adj}^*_{F,F'} \colon \underline{\mathbb{C}}_{F'} \to \underline{\mathbb{C}}_F \quad \text{and} \quad \mathrm{adj}^!_{(F')^\bot,F^\bot} \colon \underline{\mathbb{C}}_{(F')^\bot} \to \underline{\mathbb{C}}_{F^\bot}[2 \mathrm{rk}(F^\bot)-2\mathrm{rk}((F')^\bot)]. \]
\begin{lem} \label{lem:fourier-adj} The following diagram is commutative: \[ \xymatrix@C=2.5cm{ \mathcal{F}_E(\underline{\mathbb{C}}_{F'}) \ar[d]^-{\wr} \ar[r]^-{\mathcal{F}_E ( \mathrm{adj}^*_{F,F'} )} & \mathcal{F}_E(\underline{\mathbb{C}}_F) \ar[d]_-{\wr} \\ \underline{\mathbb{C}}_{(F')^\bot}[-2\mathrm{rk}(F')] \ar[r]^-{\mathrm{adj}^!_{(F')^\bot,F^\bot}} & \underline{\mathbb{C}}_{F^\bot}[-2\mathrm{rk}(F)], } \] where vertical isomorphisms are provided by Lemma~{\rm \ref{lem:fourier-F}}. \end{lem}
\begin{proof} It is equivalent to prove a similar isomorphism for $\mathfrak{F}_E$; for simplicity we still denote by $F^\bot, (F')^\bot$ the orthogonals viewed in $E^\diamond$, and by ${\check r} \colon E^\diamond \to X$ the projection. By the construction in the proof of Lemma~\ref{lem:fourier-F} we have natural isomorphisms \[ \mathfrak{F}_E(\underline{\mathbb{C}}_{F'}) \cong {\check q}_! \underline{\mathbb{C}}_{Q_{F'}} \qquad \text{and} \qquad \mathfrak{F}_E(\underline{\mathbb{C}}_F) \cong {\check q}_! \underline{\mathbb{C}}_{Q_F}, \] where $Q_{F'}:=q^{-1}(F')$, $Q_F:=q^{-1}(F)$. It follows from the definitions that the morphism $\mathcal{F}_E ( \mathrm{adj}^*_{F,F'} )$ is the image under ${\check q}_!$ of the morphism $\underline{\mathbb{C}}_{Q_{F'}} \to \underline{\mathbb{C}}_{Q_F}$ induced by adjunction (for the inclusion $Q_F \hookrightarrow Q_{F'}$). Hence what we have to show is that the morphism $\varphi$ in the following diagram coincides with $\mathrm{adj}^!_{(F')^\bot,F^\bot}$,
where the upper arrow is induced by adjunction as above, and the vertical isomorphisms are as in the proof of Lemma \ref{lem:fourier-F}: \[ \xymatrix@C=1.5cm{ {\check q}_! \underline{\mathbb{C}}_{Q_{F'}} \ar[r] \ar[d]_-{\wr} & {\check q}_! \underline{\mathbb{C}}_{Q_F} \ar[d]^-{\wr} \\ {\check q}_! \underline{\mathbb{C}}_{F' \times_X (F')^\bot} \ar[d]_-{\wr} & {\check q}_! \underline{\mathbb{C}}_{F \times_X F^\bot} \ar[d]^-{\wr} \\ \underline{\mathbb{C}}_{(F')^\bot}[-2{\rm rk}(F')] \ar[r]^-{\varphi} & \underline{\mathbb{C}}_{F^\bot}[-2{\rm rk}(F)]. } \] Now we have canonical isomorphisms \[ {\check r}_! \bigl( \underline{\mathbb{C}}_{(F')^\bot}[-2{\rm rk}(F')] \bigr) \cong \underline{\mathbb{C}}_X[-2{\rm rk}(E)], \quad {\check r}_! \bigl( \underline{\mathbb{C}}_{F^\bot}[-2{\rm rk}(F)] \bigr) \cong \underline{\mathbb{C}}_X[-2{\rm rk}(E)], \] and one can check that the functor ${\check r}_!$ induces an isomorphism \begin{multline*} {\rm Hom}_{\mathcal{D}^{G\times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}} (E^\diamond)} \bigl( \underline{\mathbb{C}}_{(F')^\bot}[-2{\rm rk}(F')] , \underline{\mathbb{C}}_{F^\bot}[-2{\rm rk}(F)] \bigr) \xrightarrow{\sim} \\ {\rm Hom}_{\mathcal{D}^{G\times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(X)} \bigl( \underline{\mathbb{C}}_X[-2{\rm rk}(E)], \underline{\mathbb{C}}_X[-2{\rm rk}(E)] \bigr) \end{multline*} sending $\mathrm{adj}^!_{(F')^\bot,F^\bot}[-2{\rm rk}(F')]$ to the identity morphism of $\underline{\mathbb{C}}_X[-2{\rm rk}(E)]$. Hence it is enough to prove that ${\check r}_! \varphi[2{\rm rk}(E)]$ is the identity of $\underline{\mathbb{C}}_X$ (through the canonical isomorphisms above). The latter statement is about sheaves (and not complexes), so that we can forget about equivariance and check the claim locally over $X$. (This is allowed by combining \cite[Proposition 2.5.3]{BL} and \cite[Proposition 4.2.7]{letellier}.) By local triviality, one can then assume that $X=\mathrm{pt}$ (i.e.~that $E$ is a vector space and that $F,F' \subset E$ are subspaces).
In this case the claim boils down to the fact that the dotted arrow in the following diagram is the identity: \[ \xymatrix@C=1.5cm{
\mathsf{H}_c^{2\dim(E)}(F' \times (F')^\bot) \ar[d]_-{\wr} & \mathsf{H}_c^{2 \dim(E)}(Q) \ar[r]^-{\sim} \ar[l]_-{\sim} & \mathsf{H}_c^{2\dim(E)}(F \times F^\bot) \ar[d]^-{\wr} \\ \C \ar@{.>}[rr] & & \C. } \] To prove this fact we regard $E \times E^*$ as a real vector space, endowed with the non-degenerate quadratic form given by $q(x,\xi):=\mathrm{Re}(\langle \xi, x \rangle)$. The orthogonal group $H$ of this form stabilizes $Q$, hence acts on $\mathsf{H}_c^{2 \dim(E)}(Q)$, and this action factors through the group of components $H/H^\circ$. Now $F \times F^\bot$ and $F' \times (F')^\bot$ are conjugate under the action of $H^\circ$, with finishes the proof. \end{proof}
In the following proposition we get back to the assumption that $E=V \times X$, and we let $p \colon E \to V$ be the projection. The following result is an immediate consequence of Lemma~\ref{lem:fourier-adj} and the isomorphism of functors $\mathcal{F}_V \circ p_! \cong {\check p}_! \circ \mathcal{F}_E$, see the proof of Corollary \ref{prop:Fourier-F}.
\begin{prop} \label{prop:inclusion-Fourier} The following diagram is commutative: \[ \xymatrix@C=3cm{ \mathcal{F}_V(p_! \underline{\mathbb{C}}_{F'}) \ar[d]_-{\eqref{eqn:isom-Fourier-F}}^-{\wr} \ar[r]^-{\mathcal{F}_V ( p_!(\mathrm{adj}^*_{F,F'}) )} & \mathcal{F}_V(p_! \underline{\mathbb{C}}_F) \ar[d]^-{\eqref{eqn:isom-Fourier-F}}_-{\wr} \\ {\check p}_! \underline{\mathbb{C}}_{(F')^\bot}[-2\mathrm{rk}(F')] \ar[r]^-{{\check p}_!(\mathrm{adj}^!_{(F')^\bot,F^\bot})} & {\check p}_! \underline{\mathbb{C}}_{F^\bot}[-2\mathrm{rk}(F)]. } \] \end{prop}
\subsection{The $\mathfrak{Fourier}$ isomorphism and inclusion of subbundles}
We come back to Setting (A) of \S\ref{ss:settings}.
\begin{prop} \label{prop:compatibility-fourier-A} We have an equality \[ \mathfrak{Fourier}_{F_1,F_2} \circ \mathfrak{res}^{F_1, F_2'}_{F_1,F_2} = \mathfrak{pdi}^{F_1^\bot, (F_2')^\bot}_{F_1^\bot, F_2^\bot} \circ \mathfrak{Fourier}_{F_1,F_2'} \] of morphisms $\mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1 \times_V F_2') \to \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\bullet+2\dim((F_2')^\bot)-2\dim(F_1)} ( F_1^\bot \times_{V^*} F_2^\bot)$.
\end{prop}
\begin{proof}
By functoriality the following diagram commutes, where horizontal maps are induced by the functor $\mathcal{F}_V$: \[ \xymatrix@C=2cm{ {\rm Ext}^{2\dim(F_2')-\bullet}_{\mathcal{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(V)}(p_! \underline{\mathbb{C}}_{F_1}, p_! \underline{\mathbb{C}}_{F_2'}) \ar[d]_-{(p_! \mathrm{adj}_{F_2,F_2'}^*) \circ (\cdot)} \ar[r]_-{\sim} & {\rm Ext}^{2\dim(F_2')-\bullet}_{\mathcal{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(V^*)}(\mathcal{F}_V(p_! \underline{\mathbb{C}}_{F_1}), \mathcal{F}_V(p_! \underline{\mathbb{C}}_{F_2'})) \ar[d]^-{\mathcal{F}_V(p_! \mathrm{adj}_{F_2,F_2'}^*) \circ (\cdot)} \\ {\rm Ext}^{2\dim(F_2')-\bullet}_{\mathcal{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(V)}(p_! \underline{\mathbb{C}}_{F_1}, p_! \underline{\mathbb{C}}_{F_2}) \ar[r]_-{\sim} & {\rm Ext}^{2\dim(F_2')-\bullet}_{\mathcal{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(V^*)}(\mathcal{F}_V(p_! \underline{\mathbb{C}}_{F_1}), \mathcal{F}_V(p_! \underline{\mathbb{C}}_{F_2})). } \] Now by Proposition \ref{prop:inclusion-Fourier} the following diagram commutes, where vertical maps are induced by the isomorphisms $\mathcal{F}_V(p_! \underline{\mathbb{C}}_{F}) \cong {\check p}_! \underline{\mathbb{C}}_{F^\bot}[-2{\rm rk}(F)]$ for $F=F_1,F_2$ or $F_2'$ (see \eqref{eqn:isom-Fourier-F}): \[ \xymatrix@C=1cm{ {\rm Ext}^{2\dim(F_2')-\bullet}_{\mathcal{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(V^*)}(\mathcal{F}_V(p_! \underline{\mathbb{C}}_{F_1}), \mathcal{F}_V(p_! \underline{\mathbb{C}}_{F_2'})) \ar[d]_-{\mathcal{F}_V(p_! \mathrm{adj}_{F_2,F_2'}^*) \circ (\cdot)} \ar[r]_-{\sim} & {\rm Ext}^{2\dim(F_1)-\bullet}_{\mathcal{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(V^*)}({\check p}_! \underline{\mathbb{C}}_{F_1^\bot}, {\check p}_! \underline{\mathbb{C}}_{(F_2')^\bot}) \ar[d]^-{({\check p}_! \mathrm{adj}_{(F_2')^\bot,F_2^\bot}^!) \circ (\cdot)} \\ {\rm Ext}^{2\dim(F_2')-\bullet}_{\mathcal{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(V^*)}(\mathcal{F}_V(p_! \underline{\mathbb{C}}_{F_1}), \mathcal{F}_V(p_! \underline{\mathbb{C}}_{F_2})) \ar[r]_-{\sim} & {\rm Ext}^{2\dim(F_1)+2\dim(F_2')-2\dim(F_2)-\bullet}_{\mathcal{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(V^*)}({\check p}_! \underline{\mathbb{C}}_{F_1^\bot}, {\check p}_! \underline{\mathbb{C}}_{F_2^\bot}). } \]
Pasting these diagrams with the ones of Proposition~\ref{prop:inclusion-Ginzburg-A} we obtain the desired equality. \end{proof}
Now we consider Setting (B) of \S\ref{ss:settings}. The proof of the following proposition is similar to that of Proposition \ref{prop:compatibility-fourier-A} (replacing Proposition \ref{prop:inclusion-Ginzburg-A} by Proposition \ref{prop:inclusion-Ginzburg-B}), and is therefore omitted.
\begin{prop} \label{prop:compatibility-fourier-B} We have an equality \[ \mathfrak{Fourier}_{F_1',F_2} \circ \mathfrak{pdi}^{F_1,F_2}_{F_1',F_2} = \mathfrak{res}^{F_1^\bot, F_2^\bot}_{(F_1')^\bot, F_2^\bot} \circ \mathfrak{Fourier}_{F_1,F_2} \] of morphisms $\mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1 \times_V F_2) \to \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\bullet+2\dim(F_2^\bot)-2\dim(F_1')} \bigl( (F_1')^\bot \times_{V^*} F_2^\bot \bigr)$.
\end{prop}
\section{Compatibility of the remaining constructions with inclusions} \label{sec:compatibility-others}
\subsection{Compatibilities for linear Koszul duality}
Consider Setting (A) of \S\ref{ss:settings}. Then we have equivalences of triangulated categories $\mathfrak{K}_{F_1,F_2}$ and $\mathfrak{K}_{F_1,F_2'}$ constructed as in \S\ref{ss:lkd}. We also have natural morphisms of dg-schemes \begin{multline*} f \colon (\Delta V \times X \times X) \rcap_{E \times E} (F_1 \times F_2) \to (\Delta V \times X \times X) \rcap_{E \times E} (F_1 \times F_2'), \\ g \colon (\Delta V^* \times X \times X) \rcap_{E^* \times E^*} (F_1^\bot \times (F_2')^\bot) \to (\Delta V^* \times X \times X) \rcap_{E^* \times E^*} (F_1^\bot \times F_2^\bot) \end{multline*} associated with the inclusions $F_2 \hookrightarrow F_2'$ and $(F_2')^\bot \hookrightarrow F_2^\bot$ respectively, and associated functors \[ Lf^* \colon \mathcal{D}^c_{G \times {\mathbb{G}}_{\mathbf{m}}} \bigl( (\Delta V \times X \times X) \rcap_{E \times E} (F_1 \times F_2') \bigr) \to \mathcal{D}^c_{G \times {\mathbb{G}}_{\mathbf{m}}} \bigl( (\Delta V \times X \times X) \rcap_{E \times E} (F_1 \times F_2) \bigr), \] \begin{multline*} Rg_* \colon \mathcal{D}^c_{G \times {\mathbb{G}}_{\mathbf{m}}} \bigl( (\Delta V^* \times X \times X) \rcap_{E^* \times E^*} (F_1^\bot \times (F_2')^\bot) \bigr) \to \\ \mathcal{D}^c_{G \times {\mathbb{G}}_{\mathbf{m}}} \bigl( (\Delta V^* \times X \times X) \rcap_{E^* \times E^*} (F_1^\bot \times F_2^\bot) \bigr) \end{multline*} (see \cite[\S\S 3.2--3.3]{MR3} for details). By \cite[Proposition 3.5]{MR3} there exists an isomorphism of functors \[ \mathfrak{K}_{F_1,F_2} \circ Lf^* \cong Rg_* \circ \mathfrak{K}_{F_1,F_2'}. \]
It easily follows from definitions that the following diagram commutes: \[ \xymatrix@C=2cm{ \mathcal{D}^c_{G \times {\mathbb{G}}_{\mathbf{m}}} \bigl( (\Delta V \times X \times X) \rcap_{E \times E} (F_1 \times F_2') \bigr) \ar[r]^{Lf^*} \ar[d] & \mathcal{D}^c_{G \times {\mathbb{G}}_{\mathbf{m}}} \bigl( (\Delta V \times X \times X) \rcap_{E \times E} (F_1 \times F_2) \bigr) \ar[d] \\ \mathcal{D}^b \mathsf{Coh}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1 \times F_2') \ar[r] & \mathcal{D}^b \mathsf{Coh}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1 \times F_2). } \] (Here the lower horizontal arrow is the usual pullback functor associated with the embedding $F_1 \times F_2 \hookrightarrow F_1 \times F_2'$. The right vertical arrow is induced by the ``restriction of scalars'' functor associated with the embedding $\mathcal{A}_{F_1, F_2}^0 \hookrightarrow \mathcal{A}_{F_1, F_2}$, where the dg-algebra $\mathcal{A}_{F_1, F_2}$ is defined in \S\ref{ss:lkd}; note that $\mathcal{A}_{F_1, F_2}^0$ is the direct image of the structure sheaf under the affine morphism $F_1 \times F_2 \to X \times X$. The left vertical arrow is defined similarly.) We deduce that the morphism induced by $Lf^*$ in $\mathsf{K}$-homology is $\mathbf{res}^{F_1, F_2'}_{F_1,F_2}$.
Similarly, the morphism induced by $Rg_*$ in $\mathsf{K}$-homology is $\mathbf{pdi}^{F_1^\bot, (F_2')^\bot}_{F_1^\bot, F_2^\bot}$ (see the proof of~\cite[Lemma~3.3]{MR3}).
We deduce the following result.
\begin{prop} \label{prop:compatibility-LKD-A} We have an equality \[ \mathbf{Koszul}_{F_1,F_2} \circ \mathbf{res}^{F_1, F_2'}_{F_1,F_2} = \mathbf{pdi}^{F_1^\bot, (F_2')^\bot}_{F_1^\bot, F_2^\bot} \circ \mathbf{Koszul}_{F_1,F_2'} \] of morphisms $\mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1 \times_V F_2') \to \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}( F_1^\bot \times_{V^*} F_2^\bot)$.
\end{prop}
Now, consider Setting (B) of \S\ref{ss:settings}. The same considerations as above allow to prove the following result.
\begin{prop} \label{prop:compatibility-LKD-B} We have an equality \[ \mathbf{Koszul}_{F_1',F_2} \circ \mathbf{pdi}^{F_1, F_2}_{F_1', F_2} = \mathbf{res}^{F_1^\bot, F_2^\bot}_{(F_1')^\bot, F_2^\bot} \circ \mathbf{Koszul}_{F_1,F_2} \] of morphisms $\mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1 \times_V F_2) \to \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}\bigl( (F_1')^\bot \times_{V^*} F_2^\bot \bigr)$.
\end{prop}
\subsection{Compatibilities for the other maps in $\mathsf{K}$-homology}
Consider Setting (A) of \S\ref{ss:settings}.
\begin{prop} \label{prop:compatibility-duality-A} We have equalities \begin{gather*} \mathbf{D}_{F_1^\bot, F_2^\bot} \circ \mathbf{pdi}^{F_1^\bot, (F_2')^\bot}_{F_1^\bot, F_2^\bot} = \mathbf{pdi}^{F_1^\bot, (F_2')^\bot}_{F_1^\bot, F_2^\bot} \circ \mathbf{D}_{F_1^\bot,(F_2')^\bot}, \\ \mathbf{i}_{F_1^\bot, F_2^\bot} \circ \mathbf{pdi}^{F_1^\bot, (F_2')^\bot}_{F_1^\bot, F_2^\bot} = \mathbf{pdi}^{F_1^\bot, (F_2')^\bot}_{F_1^\bot, F_2^\bot} \circ \mathbf{i}_{F_1^\bot,(F_2')^\bot} \end{gather*} of morphisms $\mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1^\bot \times_{V^*} (F_2')^\bot) \to \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1^\bot \times_{V^*} F_2^\bot)$.
\end{prop}
\begin{proof} The second equality is easy, and left to the reader. Let us consider the first one. We denote the inclusion morphism by \[ h_A \colon F_1^\bot \times (F_2')^\bot \hookrightarrow F_1^\bot \times F_2^\bot, \] and consider the duality functor \[ \mathrm{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{F_1^\bot,F_2^\bot} \colon \mathcal{D}^b \mathsf{Coh}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{F_1^\bot \times_{V^*} F_2^\bot}(F_1^\bot \times F_2^\bot) \to \mathcal{D}^b \mathsf{Coh}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{F_1^\bot \times_{V^*} F_2^\bot}(F_1^\bot \times F_2^\bot)^{\rm op} \] defined as in \S \ref{ss:duality}, and similarly for $\mathrm{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{F_1^\bot,(F_2')^\bot}$. Then the result follows from the natural isomorphism \[ R(h_A)_* \circ \mathrm{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{F_1^\bot,(F_2')^\bot} \cong \mathrm{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{F_1^\bot,F_2^\bot} \circ R(h_A)_* \] provided by the duality theorem \cite[Theorem VII.3.3]{H}. More precisely we need an equivariant version of the duality theorem, which can be derived from the non-equivariant version by the arguments of~\cite[\S 2.1]{MR3}. \end{proof}
Consider now Setting (B) of \S\ref{ss:settings}.
\begin{prop} \label{prop:compatibility-duality-B} We have equalities \begin{gather*} \mathbf{D}_{(F_1')^\bot,F_2^\bot} \circ \mathbf{res}^{F_1^\bot, F_2^\bot}_{(F_1')^\bot, F_2^\bot} = \mathbf{res}^{F_1^\bot, F_2^\bot}_{(F_1')^\bot, F_2^\bot} \circ \mathbf{D}_{F_1^\bot,F_2^\bot}, \\ \mathbf{i}_{(F_1')^\bot,F_2^\bot} \circ \mathbf{res}^{F_1^\bot, F_2^\bot}_{(F_1')^\bot, F_2^\bot} = \mathbf{res}^{F_1^\bot, F_2^\bot}_{(F_1')^\bot, F_2^\bot} \circ \mathbf{i}_{F_1^\bot,F_2^\bot} \end{gather*} of morphisms $\mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1^\bot \times_{V^*} F_2^\bot) \to \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}((F_1')^\bot \times_{V^*} F_2^\bot)$.
\end{prop}
\begin{proof} The second equality is easy, and left to the reader. Let us consider the first one. We denote the inclusion morphism by \[ h_B \colon (F_1')^\bot \times F_2^\bot \hookrightarrow F_1^\bot \times F_2^\bot, \] and consider the duality functors $\mathrm{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{F_1^\bot,F_2^\bot}$ and $\mathrm{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{(F_1')^\bot,F_2^\bot}$ defined as in \S \ref{ss:duality}. The claim follows from an isomorphism of functors \[ L(h_B)^* \circ \mathrm{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{F_1^\bot,F_2^\bot} \cong \mathrm{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{(F_1')^\bot,F_2^\bot} \circ L(h_B)^* \] which can be proved by arguments similar to those of \cite[Proposition II.5.8]{H}, taking into account our assumption that $X$ is a smooth variety (so that $F_1^\bot \times F_2^\bot$ and $(F_1')^\bot \times F_2^\bot$ are also smooth), which implies that every object of the bounded derived category of coherent sheaves is isomorphic to a bounded complex of locally free sheaves. \end{proof}
\subsection{Compatibilities for $\underline{\mathrm{RR}}$} \label{ss:compatibility-RR}
First, consider Setting (A) of \S\ref{ss:settings}.
\begin{prop} \label{prop:compatibility-uRR-A} Assume that the proper direct image morphism \[ \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1 \times_V F_2) \to \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1 \times F_2) \] is injective. Then we have an equality \[ \underline{\mathrm{RR}}_{F_1,F_2} \circ \mathbf{res}^{F_1, F_2'}_{F_1, F_2} = \mathfrak{res}^{F_1, F_2'}_{F_1, F_2} \circ \underline{\mathrm{RR}}_{F_1,F_2'} \] of morphisms $\mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1 \times_V F_2') \to \widehat{\mathsf{H}}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1 \times_V F_2)$.
\end{prop}
\begin{proof} Consider the following cube: \[ \xymatrix{ \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1 \times_V F_2') \ar[dd]_-{\mathbf{res}^{F_1, F_2'}_{F_1, F_2}} \ar[rd]^-{\mathbf{pdi}} \ar[rr]^-{\underline{\mathrm{RR}}_{F_1,F_2'}} & & \widehat{\mathsf{H}}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1 \times_V F_2') \ar'[d][dd]^-{\mathfrak{res}^{F_1, F_2'}_{F_1, F_2}} \ar[rd]^-{\mathfrak{pdi}} & \\ & \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1 \times F_2') \ar[dd]_<<<<<<{\mathbf{res}} \ar[rr]^<<<<<<<<<<{(1)} & & \widehat{\mathsf{H}}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1 \times F_2') \ar[dd]^-{\mathfrak{res}} \\ \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1 \times_V F_2) \ar[rd]^-{\mathbf{pdi}} \ar'[r][rr]^-{\underline{\mathrm{RR}}_{F_1,F_2}} && \widehat{\mathsf{H}}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1 \times_V F_2) \ar[rd]^-{\mathfrak{pdi}} & \\ & \mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1 \times F_2) \ar[rr]^-{(2)} && \widehat{\mathsf{H}}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1 \times F_2). } \] Here the labels $\mathbf{res}$ and $\mathfrak{res}$, resp. $\mathbf{pdi}$ and $\mathfrak{pdi}$, indicate restriction with supports (always with respect to the morphism induced by $F_2 \hookrightarrow F_2'$), resp.~proper direct image, the arrow labelled by $(1)$ is given by $\tau^{G \times {\mathbb{G}}_{\mathbf{m}}}_{F_1 \times F_2'} \cdot \bigl(1 \boxtimes (\mathrm{Td}_{F_2'}^{G \times {\mathbb{G}}_{\mathbf{m}}})^{-1} \bigr)$, and the arrow labelled by $(2)$ by $\tau^{G \times {\mathbb{G}}_{\mathbf{m}}}_{F_1 \times F_2} \cdot \bigl(1 \boxtimes (\mathrm{Td}_{F_2}^{G \times {\mathbb{G}}_{\mathbf{m}}})^{-1} \bigr)$. The upper and lower faces of this cube commute by Theorem~\ref{thm:equivariantRR}
and the projection formula~\eqref{eqn:proj-formula-H}. The left face commutes by definition, and the right one by Lemma \ref{lem:restriction-pushforward}. The front face commutes by Proposition~\ref{prop:RR-res}, Remark~\ref{rk:Todd} and formula~\eqref{eqn:res-cohomology}.
Using our assumption, we deduce the commutativity of the back face, which finishes the proof. \end{proof}
Now, consider Setting (B) of \S\ref{ss:settings}. The following proposition follows from Theorem~\ref{thm:equivariantRR} and the projection formula~\eqref{eqn:proj-formula-H}.
\begin{prop} \label{prop:compatibility-uRR-B} We have an equality \[ \underline{\mathrm{RR}}_{F_1',F_2} \circ \mathbf{pdi}^{F_1, F_2}_{F_1', F_2} = \mathfrak{pdi}^{F_1, F_2}_{F_1', F_2} \circ \underline{\mathrm{RR}}_{F_1,F_2} \] of morphisms $\mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1 \times_V F_2) \to \widehat{\mathsf{H}}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1' \times_V F_2)$.
\end{prop}
\subsection{Compatibilities for $\overline{\mathrm{RR}}$}
The proofs in this subsection are analogous to those of the corresponding statements in \S\ref{ss:compatibility-RR}; they are therefore omitted.
First, consider Setting (A) of \S\ref{ss:settings}.
\begin{prop} \label{prop:compatibility-oRR-A} We have an equality \[ \overline{\mathrm{RR}}_{F_1^\bot,F_2^\bot} \circ \mathbf{pdi}^{F_1^\bot, (F_2')^\bot}_{F_1^\bot, F_2^\bot} = \mathfrak{pdi}^{F_1^\bot, (F_2')^\bot}_{F_1^\bot, F_2^\bot} \circ \overline{\mathrm{RR}}_{F_1^\bot,(F_2')^\bot} \] of morphisms $\mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1^\bot \times_{V^*} (F_2')^\bot) \to \widehat{\mathsf{H}}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1^\bot \times_{V^*} F_2^\bot)$.
\end{prop}
Now, consider Setting (B) of \S\ref{ss:settings}.
\begin{prop} \label{prop:compatibility-oRR-B} Assume that the proper direct image morphism \[ \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet((F_1')^\bot \times_{V^*} F_2^\bot) \to \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet((F_1')^\bot \times F_2^\bot) \] is injective. Then we have an equality \[ \overline{\mathrm{RR}}_{(F_1')^\bot,F_2^\bot} \circ \mathbf{res}^{F_1^\bot, F_2^\bot}_{(F_1')^\bot, F_2^\bot} = \mathfrak{res}^{F_1^\bot, F_2^\bot}_{(F_1')^\bot, F_2^\bot} \circ \overline{\mathrm{RR}}_{F_1^\bot,F_2^\bot} \] of morphisms $\mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1^\bot \times_{V^*} F_2^\bot) \to \widehat{\mathsf{H}}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet((F_1')^\bot \times_{V^*} F_2^\bot)$.
\end{prop}
\section{Proof of Theorem \ref{thm:LKDFourier}} \label{sec:proof}
\subsection{A particular case} \label{ss:particular-case}
In this subsection we study the case when $F_1=E$ and $F_2=X$ (considered as the zero-section of $E$) so that $F_1^\bot=X$, $F_2^\bot=E^*$. In this case, the assumption of Theorem \ref{thm:LKDFourier} is trivially satisfied.
\begin{lem} \label{lem:fourier-id} Under the identifications $E \times_V X = X \times X=X \times_{V^*} E^*$, the isomorphism \[ \mathfrak{Fourier}_{E,X} \colon \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(E \times_V X) \xrightarrow{\sim} \mathsf{H}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(X \times_{V^*} E^*) \] coincides with the automorphism of $\mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(X \times X)$ induced by the involution of $G \times {\mathbb{G}}_{\mathbf{m}}$ sending $(g,t)$ to $(g,t^{-1})$. \end{lem}
\begin{proof} The lemma is equivalent to the statement that the isomorphism $\mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(E \times_V X) \xrightarrow{\sim} \mathsf{H}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(X \times_{V^\diamond} E^\diamond)$ induced by the equivalence $\mathfrak{F}_V$ of~\S\ref{ss:Fourier-transform} is the identity morphism of $\mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(X \times X)$.
Using the canonical isomorphism of \S\ref{ss:equiv} in the case $V=\{0\}$, $F_1=F_2=X$ we obtain an isomorphism \[ \alpha \colon \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(X \times X) \xrightarrow{\sim} {\rm Ext}^{2\dim(X)-\bullet}_{\mathcal{D}_{\mathrm{const}}^{G \times {\mathbb{G}}_{\mathbf{m}}}(\pt)} \bigl( (p_0)_! \underline{\mathbb{C}}_X, (p_0)_! \underline{\mathbb{C}}_X \bigr), \] where $p_0 \colon X \to \pt$ is the projection. Then the composition \[ \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(X \times X) = \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(E \times_V X) \cong {\rm Ext}^{2\dim(X)-\bullet}_{\mathcal{D}_{\mathrm{const}}^{G \times {\mathbb{G}}_{\mathbf{m}}}(V)}(p_! \underline{\mathbb{C}}_E,p_! \underline{\mathbb{C}}_X) \] sends each $c \in \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_i(X \times X)$ to the morphism \[ p_! \underline{\mathbb{C}}_E = \underline{\mathbb{C}}_V \boxtimes (p_0)_! \underline{\mathbb{C}}_X \xrightarrow{\varphi \boxtimes \alpha(c)} \underline{\mathbb{C}}_{\{0\}} \boxtimes (p_0)_! \underline{\mathbb{C}}_X [2\dim(X) - i] = p_! \underline{\mathbb{C}}_X [2\dim(X) - i] \] where $\varphi \colon \underline{\mathbb{C}}_V \to \underline{\mathbb{C}}_{\{0\}}$ is the $({}^*,{}_*)$-adjunction morphism for the inclusion $\{0\} \hookrightarrow V$, and we use the identification $V=V \times \pt$. Similarly, the composition \[ \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(X \times X) = \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(X \times_{V^\diamond} E^\diamond) \cong {\rm Ext}^{2\dim(E^*)-\bullet}_{\mathcal{D}_{\mathrm{const}}^{G \times {\mathbb{G}}_{\mathbf{m}}}(V^\diamond)}({\check p}_! \underline{\mathbb{C}}_X,{\check p}_! \underline{\mathbb{C}}_{E^\diamond}) \] sends each $c \in \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_i(X \times X)$ to the morphism \[ {\check p}_! \underline{\mathbb{C}}_X = \underline{\mathbb{C}}_{\{0\}} \boxtimes (p_0)_! \underline{\mathbb{C}}_X \xrightarrow{\psi \boxtimes \alpha(c)} \underline{\mathbb{C}}_{V^\diamond} \boxtimes (p_0)_! \underline{\mathbb{C}}_X [2\dim(E^*) - i] \] where $\psi \colon \underline{\mathbb{C}}_{\{0\}} \to \underline{\mathbb{C}}_{V^\diamond}[2\dim(V^*)]$ is the $({}_!,{}^!)$-adjunction morphism for the inclusion $\{0\} \hookrightarrow V^\diamond$, and we use the identification $V^\diamond=V^\diamond \times \pt$. Now using Lemma \ref{lem:fourier-adj} we obtain that $\mathfrak{F}_V$ sends $\varphi \boxtimes \alpha(c)$ to $\psi \boxtimes \alpha(c)$, and the lemma follows. \end{proof}
With this result in hand we can prove Theorem \ref{thm:LKDFourier} in our particular case.
\begin{lem} \label{lem:thm-particular-case} Theorem {\rm \ref{thm:LKDFourier}} holds in the case $F_1=E$, $F_2=X$. \end{lem}
\begin{proof} We have $F_1 \times_V F_2=X \times X$, and also $F_1^\bot \times_{V^*} F_2^\bot=X \times X$. There exists a natural morphism of dg-schemes \[ (\Delta V \times X \times X) \rcap_{E \times E} (E \times X) \to (X \times X) \rcap_{X \times X} (X \times X) \] associated with the morphism of vector bundles $p \times p \colon E \times E \to X \times X$, see \cite[\S 3.2]{MR3}. In our case it is easily checked that this morphism is a quasi-isomorphism, hence it induces an equivalence of triangulated categories \[ L\Phi^* \colon \mathcal{D}^c_{G \times {\mathbb{G}}_{\mathbf{m}}}((X \times X) \rcap_{X \times X} (X \times X)) \xrightarrow{\sim} \mathcal{D}^c_{G \times {\mathbb{G}}_{\mathbf{m}}}((\Delta V \times X \times X) \rcap_{E \times E} (E \times X)), \] see~\cite[Proposition~1.3.2]{MR}. Moreover by definition the left-hand side coincides with the category $\mathcal{D}^b \mathsf{Coh}^{G \times {\mathbb{G}}_{\mathbf{m}}}(X \times X)$, so that $L\Phi^*$ can (and will) be considered as an equivalence from $\mathcal{D}^b \mathsf{Coh}^{G \times {\mathbb{G}}_{\mathbf{m}}}(X \times X)$ to $\mathcal{D}^c_{G \times {\mathbb{G}}_{\mathbf{m}}}((\Delta V \times X \times X) \rcap_{E \times E} (E \times X))$. It is easily checked that the induced automorphism of $\mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(X \times X)$ is the identity.
Similarly, the morphism dual to $p \times p$ induces a quasi-isomorphism \[ (X \times X) \rcap_{X \times X} (X \times X) \to (\Delta V^* \times X \times X) \rcap_{E^* \times E^*} (X \times E^*), \] hence an equivalence of triangulated categories \[ R\Psi_* \colon \mathcal{D}^b \mathsf{Coh}^{G \times {\mathbb{G}}_{\mathbf{m}}}(X \times X) \xrightarrow{\sim} \mathcal{D}^c_{G \times {\mathbb{G}}_{\mathbf{m}}}((\Delta V^* \times X \times X) \rcap_{E^* \times E^*} (X \times E^*)), \] which induces the identity morphism in $\mathsf{K}$-homology.
If $\mathfrak{K}_{X,X}$ denotes the linear Koszul duality equivalence defined as in~\S\ref{ss:lkd} (in the case $V=\{0\}$, $F_1=F_2=E=X$), by \cite[Proposition 3.4]{MR3} there exists an isomorphism \[ \mathfrak{K}_{E,X} \circ L\Phi^* \cong R\Psi_* \circ \mathfrak{K}_{X,X}. \] Using the remarks above and the definition of the equivalence $\mathfrak{K}_{X,X}$, we deduce that, if $\mathcal{G}$ is in $\mathcal{D}^b \mathsf{Coh}^G(X \times X)$ (considered as an object of $\mathcal{D}^b \mathsf{Coh}^{G \times {\mathbb{G}}_{\mathbf{m}}}(X \times X)$ with trivial ${\mathbb{G}}_{\mathbf{m}}$-action), the morphism $\mathbf{Koszul}_{E,X}$ sends the class of $\mathcal{G} \langle m \rangle$ to the class of \[ R\mathcal{H} \hspace{-1pt} \mathit{om}_{\mathcal{O}_{X \times X}}(\mathcal{G},\mathcal{O}_X \boxtimes \omega_X)\langle m \rangle[\dim(X)+m]. \]
Using the compatibility of Grothendieck--Serre duality with proper direct images (as in the proof of Proposition~\ref{prop:compatibility-duality-A}) one easily checks that, with similar notation, $\mathbf{D}_{X,E^*}$ sends the class of $\mathcal{G} \langle m \rangle$ to the class of \[ R\mathcal{H} \hspace{-1pt} \mathit{om}_{\mathcal{O}_{X \times X}}(\mathcal{G},\mathcal{O}_X \boxtimes \omega_X)\langle -m \rangle[\dim(X)]. \]
We deduce that $\mathbf{D}_{X,E^*} \circ \mathbf{Koszul}_{E,X}$ sends the class of $\mathcal{G} \langle m \rangle$ to the class of $\mathcal{G} \langle -m \rangle [-m]$, and then that $\mathbf{i}_{X,E^*} \circ \mathbf{D}_{X,E^*} \circ \mathbf{Koszul}_{E,X}$ identifies with the automorphism of $\mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(X \times X)$ induced by the involution of $G \times {\mathbb{G}}_{\mathbf{m}}$ sending $(g,t)$ to $(g,t^{-1})$.
The statement in the lemma follows from this description, Lemma~\ref{lem:fourier-id}, and the compatibility of the Riemann--Roch maps with inverse image (in $\mathsf{K}$-homology and Borel--Moore homology) under an automorphism of $G \times {\mathbb{G}}_{\mathbf{m}}$. \end{proof}
\subsection{Compatibility with inclusion}
Consider first Setting (A) of \S\ref{ss:settings}.
\begin{prop} \label{prop:inclusion-A} \begin{enumerate} \item We have an equality \begin{multline*} \overline{\mathrm{RR}}_{F_1^\bot, F_2^\bot} \circ \mathbf{i}_{F_1^\bot, F_2^\bot} \circ \mathbf{D}_{F_1^\bot, F_2^\bot} \circ \mathbf{Koszul}_{F_1,F_2} \circ \mathbf{res}^{F_1, F_2'}_{F_1, F_2} = \\ \mathfrak{pdi}^{F_1^\bot, (F_2')^\bot}_{F_1^\bot, F_2^\bot} \circ \overline{\mathrm{RR}}_{F_1^\bot, (F_2')^\bot} \circ \mathbf{i}_{F_1^\bot, (F_2')^\bot} \circ \mathbf{D}_{F_1^\bot, (F_2')^\bot} \circ \mathbf{Koszul}_{F_1,F_2'} \end{multline*} of morphisms $\mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1 \times_V F_2') \to \widehat{\mathsf{H}}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1^\bot \times_{V^*} F_2^\bot)$.
\item Assume that the proper direct image morphism \[ \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1 \times_V F_2) \to \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1 \times F_2) \] is injective. Then we have an equality \[ \mathfrak{Fourier}_{F_1,F_2} \circ \underline{\mathrm{RR}}_{F_1,F_2} \circ \mathbf{res}^{F_1, F_2'}_{F_1, F_2} = \mathfrak{pdi}^{F_1^\bot, (F_2')^\bot}_{F_1^\bot, F_2^\bot} \circ \mathfrak{Fourier}_{F_1,F_2'} \circ \underline{\mathrm{RR}}_{F_1,F_2'} \] of morphisms $\mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1 \times_V F_2') \to \widehat{\mathsf{H}}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1^\bot \times_{V^*} F_2^\bot)$.
\end{enumerate} \end{prop}
\begin{proof} (1) follows from Propositions \ref{prop:compatibility-LKD-A}, \ref{prop:compatibility-duality-A} and \ref{prop:compatibility-oRR-A}. (2) follows from Propositions \ref{prop:compatibility-uRR-A} and \ref{prop:compatibility-fourier-A}. \end{proof}
Consider now Setting (B) of \S\ref{ss:settings}.
\begin{prop} \label{prop:inclusion-B} \begin{enumerate} \item Assume that the proper direct image morphism \[ \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet((F_1')^\bot \times_{V^*} F_2^\bot) \to \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet((F_1')^\bot \times F_2^\bot) \] is injective. Then we have an equality \begin{multline*} \overline{\mathrm{RR}}_{(F_1')^\bot, F_2^\bot} \circ \mathbf{i}_{(F_1')^\bot, F_2^\bot} \circ \mathbf{D}_{(F_1')^\bot, F_2^\bot} \circ \mathbf{Koszul}_{F_1',F_2} \circ \mathbf{pdi}^{F_1, F_2}_{F_1', F_2} \\ = \mathfrak{res}^{F_1^\bot, F_2^\bot}_{(F_1')^\bot, F_2^\bot} \circ \overline{\mathrm{RR}}_{F_1^\bot, F_2^\bot} \circ \mathbf{i}_{F_1^\bot, F_2^\bot} \circ \mathbf{D}_{F_1^\bot, F_2^\bot} \circ \mathbf{Koszul}_{F_1,F_2} \end{multline*} of morphisms $\mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1 \times_V F_2) \to \widehat{\mathsf{H}}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}((F_1')^\bot \times_{V^*} F_2^\bot)$.
\item We have an equality \[ \mathfrak{Fourier}_{F_1',F_2} \circ \underline{\mathrm{RR}}_{F_1',F_2} \circ \mathbf{pdi}^{F_1, F_2}_{F_1', F_2} = \mathfrak{res}^{F_1^\bot, F_2^\bot}_{(F_1')^\bot, F_2^\bot} \circ \mathfrak{Fourier}_{F_1,F_2} \circ \underline{\mathrm{RR}}_{F_1,F_2} \] of morphisms $\mathsf{K}^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1 \times_V F_2) \to \widehat{\mathsf{H}}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}((F_1')^\bot \times_{V^*} F_2^\bot)$.
\end{enumerate} \end{prop}
\begin{proof} (1) follows from Propositions \ref{prop:compatibility-LKD-B}, \ref{prop:compatibility-duality-B} and \ref{prop:compatibility-oRR-B}. (2) follows from Propositions \ref{prop:compatibility-uRR-B} and \ref{prop:compatibility-fourier-B}. \end{proof}
\subsection{Proof of Theorem \ref{thm:LKDFourier}} \label{ss:proof-thm}
By assumption, the proper direct image morphism \[ \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1^{\bot} \times_{V^*} F_2^{\bot}) \to \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1^{\bot} \times_{V^*} E^*) \] is injective. Hence the same is true for the induced morphism \[ \widehat{\mathsf{H}}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1^{\bot} \times_{V^*} F_2^{\bot}) \to \widehat{\mathsf{H}}_\bullet^{G \times {\mathbb{G}}_{\mathbf{m}}}(F_1^{\bot} \times_{V^*} E^*). \] By Proposition \ref{prop:inclusion-A} applied to the inclusion $X \subset F_2$, we deduce that it suffices to prove the theorem in the case $F_2=X$. (Note that the inclusion $F_1 \times_V X \hookrightarrow F_1 \times X$ is the inclusion of the zero section in the vector bundle $F_1 \times X$ over $X \times X$. Hence the injectivity assumption in Proposition \ref{prop:inclusion-A}(2) holds by Lemma \ref{lem:Euler-class}.)
Now consider the inclusion of vector subbundles $F_1 \subset E$ (again with $F_2=X$). In this case, the morphism \[ \mathfrak{res}^{F_1^\bot, E^*}_{X, E^*} \colon \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(F_1^\bot \times_{V^*} E^*) \to \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\bullet-2{\rm rk}(F_1^\bot)}(X \times_{V^*} E^*) = \mathsf{H}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\bullet-2{\rm rk}(F_1^\bot)}(X \times X) \]
is the Thom isomorphism for the vector bundle $F_1^\bot \times_{V^*} E^* \cong F_1^\bot \times X$ over $X \times X$; in particular it is injective. Using Proposition \ref{prop:inclusion-B} we deduce that it suffices to prove the theorem in the case $F_1=E$, $F_2=X$. (Note that in our situation the inclusion $E^\bot \times_{V^*} X^\bot \hookrightarrow E^\bot \times X^\bot$ is the inclusion of the zero section in the vector bundle $X \times E^*$ over $X \times X$, so that the injectivity assumption in Proposition \ref{prop:inclusion-B}(1) holds by Lemma \ref{lem:Euler-class}.) In this case the theorem holds by Lemma~\ref{lem:thm-particular-case}, hence our proof is complete.
\appendix
\section{Proofs of some technical results} \label{sec:appendix}
\subsection{Conventions}
In \S\S\ref{ss:commutative-diagrams}--\ref{ss:adjunction} we work in the $A$-equivariant constructible derived category of some complex algebraic $A$-varieties (for some arbitrary complex linear algebraic group $A$). If $X,Y,Z$ are $A$-varieties and $f\colon X \to Y$, $g\colon Y \to Z$ are $A$-equivariant morphisms, then there exist canonical ``composition'' isomorphisms \[ g_* f_* \cong (g \circ f)_*, \quad g_! f_! \cong (g \circ f)_!, \quad f^* g^* \cong (g \circ f)^*, \quad f^! g^! \cong (g \circ f)^!, \] which we will all indicate by $\mathrm{(Comp)}$. Similarly, given a cartesian square \[ \xymatrix{
Y' \ar[d]_-{g'} \ar[r]^-{f'} \ar@{}[rd]|{\square} &Z' \ar[d]^-{g} \\ Y \ar[r]^-{f} & Z } \] of $A$-equivariant morphisms, there exist canonical ``base change'' isomorphisms \[ f^* g_! \cong (g')_! (f')^*, \quad f^! g_* \cong (g')_* (f')^!, \] which we will indicate by $\mathrm{(BC)}$.
\subsection{Some commutative diagrams} \label{ss:commutative-diagrams}
Consider a commutative diagram of $A$-varieties and $A$-equivariant morphisms \[ \xymatrix@R=0.3cm{ Y \ar[rr]^-{g} \ar[rd]^-{a} \ar[dd]_-{c} & & Z \ar[rd]^-{d} \ar'[d][dd]_<<<<{f} & \\ & Y' \ar[rr]^<<<<{g'} \ar[ld]_-{b} & & Z' \ar[ld]^-{e} \\ Y'' \ar[rr]^-{g''} & & Z'' } \] where all squares are cartesian. The following lemma is a restatement of \cite[Lemma B.7(d)]{AHR}.
\begin{lem} \label{lem:ahr} The following diagram of isomorphisms of functors commutes: \[ \xymatrix@C=2cm{ (g'')^! f_* \ar[r]^{\mathrm{(Comp)}}_-{\sim} \ar[d]_-{\mathrm{(BC)}}^-{\wr} & (g'')^! e_* d_* \ar[r]^-{\mathrm{(BC)}}_-{\sim} & b_* (g')^! d_* \ar[d]^-{\mathrm{(BC)}}_-{\wr} \\ c_* g^! \ar[rr]^{\mathrm{(Comp)}}_-{\sim} & & b_* a_* g^!. } \] \end{lem}
Now, consider $A$-equivariant morphisms \[ \xymatrix{ W \ar[r]^-{f} & X \ar[r]^-{g} & Y \ar[r]^-{h} & Z. } \] The following lemma is a restatement of \cite[Lemma B.4(a) \& Lemma B.4(d)]{AHR}.
\begin{lem} \label{lem:ahr2} The following diagrams of isomorphisms of functors commute: \[ \xymatrix@C=2cm{ h_* g_* f_* \ar[r]^-{\mathrm{(Comp)}}_-{\sim} \ar[d]_-{\mathrm{(Comp)}}^-{\wr} & h_* (g \circ f)_* \ar[d]^-{\mathrm{(Comp)}}_-{\wr} \\ (h \circ g)_* f_* \ar[r]^-{\mathrm{(Comp)}}_-{\sim} & (h \circ g \circ f)_*, } \qquad \xymatrix@C=2cm{ f^! g^! h^! \ar[r]^-{\mathrm{(Comp)}}_-{\sim} \ar[d]_-{\mathrm{(Comp)}}^-{\wr} & f^! (h \circ g)^! \ar[d]^-{\mathrm{(Comp)}}_-{\wr} \\ (g \circ f)^! h^! \ar[r]^-{\mathrm{(Comp)}}_-{\sim} & (h \circ g \circ f)^!. } \] \end{lem}
\subsection{Base change and adjunction}
Consider a cartesian diagram \begin{equation} \label{eqn:cartesian-diagram} \vcenter{ \xymatrix{
Y' \ar[d]_-{g'} \ar[r]^-{f'} \ar@{}[rd]|{\square} & Z' \ar[d]^-{g} \\ Y \ar[r]^-{f} & Z } } \end{equation} of $A$-varieties and $A$-equivariant morphisms. Then there exists a canonical morphism of functors
\begin{equation} \label{eqn:morphism-adjunction-2} (f')_! (g')^! \to g^! f_! \end{equation} which can be defined equivalently as the composition \[ (f')_! (g')^! \to (f')_! (g')^! f^! f_! \xrightarrow[\sim]{\mathrm{(Comp)}} (f')_! (f \circ g')^! f_! \xrightarrow[\sim]{\mathrm{(Comp)}} (f')_! (f')^! g^! f_! \to g^! f_! \] or as the composition \[ (f')_! (g')^! \to g^! g_! (f')_! (g')^! \xrightarrow[\sim]{\mathrm{(Comp)}} g^! (g \circ f')_! (g')^! \xrightarrow[\sim]{\mathrm{(Comp)}} g^! f_! (g')_! (g')^! \to g^! f_! \] where the unlabelled arrows are induced by the appropriate adjunction morphisms. (We leave it to the reader to check that these compositions coincide.)
As stated in \cite[Exercise III.9]{KS}, the following diagram is commutative, where vertical arrows are induced by the canonical morphisms $f_! \to f_*$ and $(f')_! \to (f')_*$: \[
\xymatrix@C=1.5cm{ (f')_! (g')^! \ar[r]^-{\eqref{eqn:morphism-adjunction-2}} \ar[d] & g^! f_! \ar[d] \\ (f')_* (g')^! \ar[r]^-{\mathrm{(BC)}}_-{\sim} & g^! f_*. } \] We deduce the following.
\begin{lem} \label{lem:BC-adjunction} If $f$ (hence also $f'$) is proper, then the base change isomorphism
$(f')_* (g')^! \cong g^! f_*$ coincides, under the natural identifications $f_! = f_*$ and $(f')_! = (f')_*$, with morphism
\eqref{eqn:morphism-adjunction-2}. \end{lem}
\subsection{Some consequences} \label{ss:adjunction}
Consider again a cartesian diagram \eqref{eqn:cartesian-diagram}, and assume that $f$ (hence also $f'$) is proper.
First, one can consider the diagram of morphisms of functors \begin{equation} \label{eqn:diagram-adjunction-!-1} \vcenter{ \xymatrix@C=1.5cm{ (f')_* (g')^! f^! \ar[d]^-{\mathrm{(Comp)}}_-{\wr} \ar[r]_-{\sim}^-{\mathrm{(BC)}} & g^! f_* f^! \ar[dd] \\ (f')_* (f \circ g')^! \ar[d]^-{\mathrm{(Comp)}}_-{\wr} & \\ (f')_* (f')^! g^! \ar[r] & g^! } } \end{equation} where the right vertical arrow is induced by the adjunction morphism $f_* f^! = f_! f^! \to \mathrm{id}$ and the lower horizontal arrow is induced by the adjunction morphism $(f')_* (f')^! = (f')_! (f')^! \to \mathrm{id}$.
\begin{lem} \label{lem:adjunction-!-1} Diagram \eqref{eqn:diagram-adjunction-!-1} is commutative. \end{lem}
\begin{proof} The claim follows from Lemma \ref{lem:BC-adjunction} (using the first description of morphism \eqref{eqn:morphism-adjunction-2}) and the fact that the composition of adjunction morphisms \[ f^! \to f^! f_! f^! \to f^! \] is the identity. \end{proof}
One can also consider the diagram of morphisms of functors \begin{equation} \label{eqn:diagram-adjunction-!-2} \vcenter{ \xymatrix@C=1.5cm{ g_! (f')_* (g')^! \ar[d]^-{\mathrm{(Comp)}}_-{\wr} \ar[r]_-{\sim}^-{\mathrm{(BC)}} & g_! g^! f_* \ar[dd] \\ (g \circ f')_! (g')^! \ar[d]^-{\mathrm{(Comp)}}_-{\wr} & \\ f_* (g')_! (g')^! \ar[r] & f_* } } \end{equation} where unlabelled arrows are induced by adjunction, and in the left-hand side we use the identifications $f_!=f_*$ and $(f')_! = (f')_*$.
\begin{lem} \label{lem:adjunction-!-2} Diagram \eqref{eqn:diagram-adjunction-!-2} is commutative. \end{lem}
\begin{proof} The claim follows from Lemma \ref{lem:BC-adjunction} (using the second description of morphism \eqref{eqn:morphism-adjunction-2}) and the fact that the composition of adjunction morphisms \[ g_! \to g_! g^! g_! \to g_! \] is the identity. \end{proof}
\subsection{Restriction with supports in Borel--Moore homology} \label{ss:restriction-with-supports}
As in \S\ref{ss:homology-cohomology}, let $A$ be a complex linear algebraic group, let $Y$ be a smooth complex $A$-variety, and let $Y' \subset Y$ be a smooth $A$-stable closed subvariety. Consider another $A$-stable closed subvariety $Z \subset Y$, not necessarily smooth, and set $Z':=Z \cap Y'$. Then we have a cartesian diagram of closed inclusions \[ \xymatrix{ Z' \ar@{^{(}->}[r]^-{i'} \ar@{^{(}->}[d]_-{g} & Y' \ar@{^{(}->}[d]^-{f} \\ Z \ar@{^{(}->}[r]^-{i} & Y. } \] Set $N:=2\dim(Y)-2\dim(Y')$. The ``restriction with supports'' morphism \begin{equation*}
\mathfrak{res}^Z_{Z'} \colon \mathsf{H}^{A}_{\bullet}(Z) \to \mathsf{H}^A_{\bullet-N}(Z') \end{equation*} associated with the inclusion $Y' \hookrightarrow Y$ is defined as follows. Consider the composition \[ i^! \to i^! f_* f^* \xrightarrow[\sim]{\mathrm{(BC)}} g_* (i')^! f^* \] where the first morphism is induced by the adjunction morphism $\mathrm{id} \to f_* f^*$. Then applying this composition to $\underline{\mathbb{D}}_Y$ and using the isomorphisms \[ i^! \underline{\mathbb{D}}_Y \cong \underline{\mathbb{D}}_Z, \quad f^* \underline{\mathbb{D}}_Y \cong f^* \underline{\mathbb{C}}_Y[2\dim(Y)] \cong \underline{\mathbb{C}}_{Y'}[2\dim(Y)] \cong \underline{\mathbb{D}}_{Y'}[N], \quad \text{and} \quad (i')^! \underline{\mathbb{D}}_{Y'} \cong \underline{\mathbb{D}}_{Z'} \] we obtain a morphism \[ \underline{\mathbb{D}}_Z \to g_* \underline{\mathbb{D}}_{Z'}[N]. \] Taking (equivariant) cohomology provides our morphism $\mathfrak{res}^Z_{Z'}$.
The same construction, applied to the subvariety $Y' \subset Y$ instead of $Z$, provides another morphism \[ \mathfrak{res}^Y_{Y'} \colon \mathsf{H}^{A}_{\bullet}(Y) \to \mathsf{H}^A_{\bullet-N}(Y') \]
\begin{lem} \label{lem:restriction-pushforward} The following diagram is commutative: \[ \xymatrix@C=2cm{ \mathsf{H}^{A}_{\bullet}(Z) \ar[r]^-{\mathfrak{res}^Z_{Z'}} \ar[d]_-{\mathfrak{pdi}_i} & \mathsf{H}^A_{\bullet-N}(Z') \ar[d]^-{\mathfrak{pdi}_{i'}} \\ \mathsf{H}^{A}_{\bullet}(Y) \ar[r]^-{\mathfrak{res}^Y_{Y'}} & \mathsf{H}^A_{\bullet-N}(Y'). } \] \end{lem}
\begin{proof} Consider the following diagram: \[ \xymatrix{ i_!i^! \ar[r] \ar[dd] & i_! i^! f_* f^* \ar[r]^-{\mathrm{(BC)}}_-{\sim} \ar[rdd] & i_! g_* (i')^! f^* \ar[d]^-{(\ddag)}_-{\wr} \\ && f_* (i')_! (i')^! f^* \ar[d] \\ \mathrm{id} \ar[rr] & & f_* f^*. } \] Here the unlabelled arrows are induced by the appropriate adjunction morphisms, and the arrow labelled with $(\ddag)$ is induced by the composition of natural isomorphisms \[ i_! g_* \cong i_! g_! \xrightarrow[\sim]{\mathrm{(Comp)}} (i \circ g)_! \xrightarrow[\sim]{\mathrm{(Comp)}} f_! (i')_! \cong f_* (i')_!. \] The left part of the diagram is clearly commutative, and the right part is commutative by Lemma \ref{lem:adjunction-!-2}. Hence the diagram as a whole is commutative. Now, when applied to $\underline{\mathbb{D}}_Y$ and after taking equivariant cohomology, this diagram induces the diagram of the lemma, hence these remarks finish the proof. (In this argument we also use the left diagram in Lemma \ref{lem:ahr2}, which allows to forget about the ``$\mathrm{(Comp)}$'' isomorphisms in the right-hand side of the diagram once equivariant cohomology is taken.) \end{proof}
\subsection{Proof of Proposition \ref{prop:inclusion-Ginzburg-A}(1)} \label{ss:proof-inclusion-Ginzburg-1}
By functoriality of isomorphism \eqref{eqn:morphisms-cohomology} the following diagram commutes, where the right vertical morphism is induced by $\mathrm{adj}^*_{F_2,F_2'}$: \begin{equation} \label{eqn:appendix-diagram-1} \vcenter{ \xymatrix@C=2cm{ {\rm Ext}^{\bullet}_{\mathcal{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(V)}(p_* \underline{\mathbb{C}}_{F_1}, p_* \underline{\mathbb{C}}_{F_2'}) \ar[d]_-{(p_* \mathrm{adj}^*_{F_2,F_2'}) \circ (\cdot)} \ar[r]^-{\eqref{eqn:morphisms-cohomology}}_-{\sim} & \mathsf{H}^\bullet_{G \times {\mathbb{G}}_{\mathbf{m}}}(E \times_V E, j^!(\underline{\mathbb{D}}_{F_1} \boxtimes \underline{\mathbb{C}}_{F_2'})) \ar[d] \\ {\rm Ext}^{\bullet}_{\mathcal{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(V)}(p_* \underline{\mathbb{C}}_{F_1}, p_* \underline{\mathbb{C}}_{F_2}) \ar[r]^-{\eqref{eqn:morphisms-cohomology}}_-{\sim} & \mathsf{H}^\bullet_{G \times {\mathbb{G}}_{\mathbf{m}}}(E \times_V E, j^!(\underline{\mathbb{D}}_{F_1} \boxtimes \underline{\mathbb{C}}_{F_2})). } } \end{equation}
Now, consider the following diagram, where all squares are cartesian and all morphisms are closed inclusions: \[ \xymatrix{ F_1 \times_V F_2 \ar[r]^-{c} \ar[d]_-{k} \ar@/^20pt/[rr]^-{b} & F_1 \times_V F_2' \ar[d]^-{k'} \ar[r]^-{b'} & E \times_V E \ar[d]^-{j} \\ F_1 \times F_2 \ar[r]^-{d} \ar@/_20pt/[rr]_-{a} & F_1 \times F_2' \ar[r]^-{a'} & E \times E. \\ } \] Then under the natural identifications \begin{align*} \mathsf{H}^\bullet_{G \times {\mathbb{G}}_{\mathbf{m}}}(E \times_V E, j^!(\underline{\mathbb{D}}_{F_1} \boxtimes \underline{\mathbb{C}}_{F_2})) &\cong \mathsf{H}^\bullet_{G \times {\mathbb{G}}_{\mathbf{m}}}(E \times_V E, j^! a_* (\underline{\mathbb{D}}_{F_1} \boxtimes \underline{\mathbb{C}}_{F_2})), \\ \mathsf{H}^\bullet_{G \times {\mathbb{G}}_{\mathbf{m}}}(E \times_V E, j^!(\underline{\mathbb{D}}_{F_1} \boxtimes \underline{\mathbb{C}}_{F_2'})) &\cong \mathsf{H}^\bullet_{G \times {\mathbb{G}}_{\mathbf{m}}}(E \times_V E, j^! (a')_* (\underline{\mathbb{D}}_{F_1} \boxtimes \underline{\mathbb{C}}_{F_2'})), \end{align*} the right vertical morphism in \eqref{eqn:appendix-diagram-1} identifies with the morphism \begin{equation} \label{eqn:appendix-morphism-1} \mathsf{H}^\bullet_{G \times {\mathbb{G}}_{\mathbf{m}}} \bigl(E \times_V E, j^!(a')_* (\underline{\mathbb{D}}_{F_1} \boxtimes \underline{\mathbb{C}}_{F_2'}) \bigr) \to \mathsf{H}^\bullet_{G \times {\mathbb{G}}_{\mathbf{m}}} \bigl(E \times_V E, j^! a_* (\underline{\mathbb{D}}_{F_1} \boxtimes \underline{\mathbb{C}}_{F_2}) \bigr) \end{equation} induced by the adjunction morphism $\mathrm{id} \to d_* d^*$ (through the ``composition'' isomorphism $(a')_* d_* \cong a_*$).
Consider the following diagram of morphisms of functors: \[ \xymatrix@C=1.5cm{ j^! (a')_* \ar[rr]^-{\mathrm{(BC)}}_-{\sim} \ar[d] \ar@{.>}@/_50pt/[dd]_-{(\star)} & & (b')_* (k')^! \ar[d] \ar@{.>}@/^40pt/[dd]^-{(\dag)} \\ j^! (a')_* d_* d^* \ar[rr]^-{\mathrm{(BC)}}_-{\sim} \ar[d]_-{\mathrm{(Comp)}}^-{\wr} & & (b')_* (k')^! d_* d^* \ar[d]^-{\mathrm{(BC)}}_-{\wr} \\ j^! a_* d^* \ar[r]^-{\mathrm{(BC)}}_-{\sim} & b_* k^! d^* \ar[r]^-{\mathrm{(Comp)}}_-{\sim} & (b')_* c_* k^! d^*. } \] Here the upper vertical arrows are induced by the adjunction morphism $\mathrm{id} \to d_* d^*$, and other arrows are either base change or composition isomorphisms as indicated. The upper square is clearly commutative, and the lower square is commutative by Lemma \ref{lem:ahr}. Hence the whole diagram is commutative, which allows to define the dotted arrows uniquely. The arrow labelled with $(\star)$ is the morphism which defines \eqref{eqn:appendix-morphism-1}, and the arrow labelled with $(\dag)$ is the morphism used in the definition of restriction with supports $\mathfrak{res}^{F_1, F_2'}_{F_1, F_2}$,
see \S\ref{ss:restriction-with-supports}. Applying this diagram to $\underline{\mathbb{D}}_{F_1} \boxtimes \underline{\mathbb{C}}_{F_2'}$ and taking equivariant cohomology allows to finish the proof of Proposition \ref{prop:inclusion-Ginzburg-A}(1). (In this argument we also use the left diagram in Lemma \ref{lem:ahr2}, which allows e.g.~to forget about the ``$\mathrm{(Comp)}$'' isomorphism on the lower line once equivariant cohomology is taken.)
\subsection{Proof of Proposition \ref{prop:inclusion-Ginzburg-A}(2)} \label{ss:proof-inclusion-Ginzburg-2}
Consider the following diagram, where all squares are cartesian and all morphisms are closed inclusions: \[ \xymatrix{ F_1^\bot \times_{V^*} (F_2')^\bot \ar[r]^-{\tilde{c}} \ar[d]_-{\tilde{k}} \ar@/^20pt/[rr]^-{\tilde{b}} & F_1^\bot \times_{V^*} F_2^\bot \ar[d]^-{\tilde{k}'} \ar[r]^-{\tilde{b}'} & E^* \times_{V^*} E^* \ar[d]^-{\tilde{j}} \\ F_1^\bot \times (F_2')^\bot \ar[r]^-{\tilde{d}} \ar@/_20pt/[rr]_-{\tilde{a}} & F_1^\bot \times F_2^\bot \ar[r]^-{\tilde{a}'} & E^* \times E^*. \\ } \] Then by functoriality of isomorphism \eqref{eqn:morphisms-cohomology} we have a commutative diagram \begin{equation} \label{eqn:appendix-diagram-2} \vcenter{ {\footnotesize \xymatrix@C=0.35cm{ {\rm Ext}^\bullet_{\mathcal{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(V^*)}({\check p}_* \underline{\mathbb{C}}_{F_1^\bot},{\check p}_* \underline{\mathbb{C}}_{(F_2')^\bot}) \ar[r]^-{\sim} \ar[d]_-{({\check p}_* \mathrm{adj}^!_{(F_2')^\bot,F_2^\bot}) \circ (\cdot)} & \mathsf{H}_{G \times {\mathbb{G}}_{\mathbf{m}}}^\bullet\bigl(E^* \times_{V^*} E^*, \tilde{j}^! \tilde{a}_*(\underline{\mathbb{D}}_{F_1^\bot} \boxtimes \underline{\mathbb{C}}_{(F_2')^\bot} ) \bigr) \ar[d] \\ {\rm Ext}^{\bullet+2{\rm rk}(F_2^\bot)-2{\rm rk}((F_2')^\bot)}_{\mathcal{D}^{G \times {\mathbb{G}}_{\mathbf{m}}}_{\mathrm{const}}(V^*)}({\check p}_* \underline{\mathbb{C}}_{F_1^\bot},{\check p}_* \underline{\mathbb{C}}_{F_2^\bot}) \ar[r]^-{\sim} & \mathsf{H}_{G \times {\mathbb{G}}_{\mathbf{m}}}^{\bullet+2{\rm rk}(F_2^\bot)-2{\rm rk}((F_2')^\bot)}\bigl(E^* \times_{V^*} E^*, \tilde{j}^! (\tilde{a}')_*(\underline{\mathbb{D}}_{F_1^\bot} \boxtimes \underline{\mathbb{C}}_{F_2^\bot} ) \bigr), } } } \end{equation} where horizontal arrows are induced by isomorphism \eqref{eqn:morphisms-cohomology} and the right vertical morphism is induced by the adjunction morphism $\tilde{d}_! \tilde{d}^! \to \mathrm{id}$ (through the isomorphisms $(\tilde{a}')_* \tilde{d}_! \cong (\tilde{a}')_* \tilde{d}_* \cong \tilde{a}_*$ and $\underline{\mathbb{C}}_{(F_2')^\bot} \cong \underline{\mathbb{D}}_{(F_2')^\bot}[-2\dim((F_2')^\bot)]$, $\underline{\mathbb{C}}_{F_2^\bot} \cong \underline{\mathbb{D}}_{F_2^\bot}[-2\dim(F_2^\bot)]$).
Consider the following diagram of morphisms of functors: \[ \xymatrix@C=1.5cm{ \tilde{j}^! \tilde{a}_* \tilde{d}^! \ar[r]^-{\mathrm{(BC)}}_-{\sim} \ar[dd]_-{\mathrm{(Comp)}}^-{\wr} \ar@{.>}@/_45pt/[ddd]_-{(\#)} & \tilde{b}_* \tilde{k}^! \tilde{d}^! \ar[d]^-{\mathrm{(Comp)}}_-{\wr} \ar@{.>}@/^190pt/[ddd]^-{(\flat)} & \\ & (\tilde{b}')_* \tilde{c}_* \tilde{k}^! \tilde{d}^! \ar[d]^-{\mathrm{(BC)}}_-{\wr} \ar[r]^-{\mathrm{(Comp)}}_-{\sim} & (\tilde{b}')_* \tilde{c}_* (\tilde{d} \circ \tilde{k})^! \ar[d]^-{\mathrm{(Comp)}}_-{\wr} \\ \tilde{j}^! (\tilde{a}')_* \tilde{d}_* \tilde{d}^! \ar[r]^-{\mathrm{(BC)}}_-{\sim} \ar[d] & (\tilde{b}')_* (\tilde{k}')^! \tilde{d}_* \tilde{d}^! \ar[d] & (\tilde{b}')_* \tilde{c}_* \tilde{c}^! (\tilde{k}')^! \ar[ld] \\ \tilde{j}^! (\tilde{a}')_* \ar[r]^-{\mathrm{(BC)}}_-{\sim} & (\tilde{b}')_* (\tilde{k}')^! & } \] Here all the unlabelled arrows are induced by the appropriate adjunction morphisms (using the identifications $\tilde{c}_*=\tilde{c}_!$ and $\tilde{d}_*=\tilde{d}_!$). The upper square is commutative by Lemma \ref{lem:ahr}, the lower square is obviously commutative, and the right square is commutative by Lemma \ref{lem:adjunction-!-1}. Hence the diagram as a whole is commutative, which allows to define the dotted arrows uniquely. The arrow labelled with $(\#)$ is the one which induces the right arrow in diagram \eqref{eqn:appendix-diagram-2} (when applied to $\underline{\mathbb{D}}_{F_1^\bot \times F_2^\bot}$), while the arrow labelled with $(\flat)$ is the one which induces the proper direct image morphism $\mathfrak{pdi}^{F_1^\bot, (F_2')^\bot}_{F_1^\bot, F_2^\bot}$
(again when applied to $\underline{\mathbb{D}}_{F_1^\bot \times F_2^\bot}$), see \cite[\S 8.3.19]{CG}.
The result follows. (As in \S\ref{ss:proof-inclusion-Ginzburg-1}, in this argument we also use the diagrams of Lemma \ref{lem:ahr2}.)
\subsection{A lemma on Euler classes}
Let $A$ be a complex linear algebraic group acting on a smooth complex algebraic variety $Y$, and let $F \to Y$ be an $A$-equivariant vector bundle of rank $r$. We consider $F$ (hence also its zero-section $Y$) as an $A \times {\mathbb{G}}_{\mathbf{m}}$-variety with the ${\mathbb{G}}_{\mathbf{m}}$-action defined as in \S\ref{ss:Fourier-transform}. Note that, as ${\mathbb{G}}_{\mathbf{m}}$ acts trivially on $Y$, there exists a canonical isomorphism of graded algebras \begin{equation} \label{eqn:cohomology-Gm} \mathsf{H}_{A \times {\mathbb{G}}_{\mathbf{m}}}^\bullet(Y) \ \cong \ \mathsf{H}_A^\bullet(Y) \otimes_{\C} \mathsf{H}_{{\mathbb{G}}_{\mathbf{m}}}^\bullet(\pt). \end{equation}
\begin{lem} \label{lem:Euler-class} The proper direct image morphism \[ \mathsf{H}^{A \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(Y) \to \mathsf{H}^{A \times {\mathbb{G}}_{\mathbf{m}}}_{\bullet}(F) \] associated with the inclusion $Y \hookrightarrow F$ is injective. \end{lem}
\begin{proof} It is well known that the composition of our morphism with the Thom isomorphism $ \mathsf{H}^{A \times {\mathbb{G}}_{\mathbf{m}}}_{\bullet}(F) \cong \mathsf{H}^{A \times {\mathbb{G}}_{\mathbf{m}}}_{\bullet - 2r}(Y)$ identifies with the action of the equivariant Euler class $\mathrm{Eu}(F) \in \mathsf{H}_{A \times {\mathbb{G}}_{\mathbf{m}}}^{2r}(Y)$ of $F$, see e.g.~\cite[\S 1.19]{LuCus2}. Since $Y$ is smooth, the equivariant homology $\mathsf{H}^{A \times {\mathbb{G}}_{\mathbf{m}}}_\bullet(Y)$ is a free module of rank one over $\mathsf{H}_{A \times {\mathbb{G}}_{\mathbf{m}}}^\bullet(Y)$, hence it is enough to prove that $\mathrm{Eu}(F)$ is not a zero-divisor in $\mathsf{H}_{A \times {\mathbb{G}}_{\mathbf{m}}}^\bullet(Y)$. However one can check that (due to our choice of ${\mathbb{G}}_{\mathbf{m}}$-action) this Euler class can be written, using isomorphism \eqref{eqn:cohomology-Gm}, as \[ \mathrm{Eu}(F) = 1 \otimes (-2u)^r + x \] where $1 \in \mathsf{H}^0_{A}(Y)$ is the unit, $u \in \mathsf{H}^2_{{\mathbb{G}}_{\mathbf{m}}}(\pt)$ is the canonical generator and $x \in \bigoplus_{i \geq 2} \mathsf{H}_{A}^i(Y) \otimes \mathsf{H}_{{\mathbb{G}}_{\mathbf{m}}}^{2r-i}(\pt)$. It follows that this element is indeed not a zero-divisor. \end{proof}
\end{document} | arXiv |
Comparative transcriptome analysis of Glyphodes pyloalis Walker (Lepidoptera: Pyralidae) reveals novel insights into heat stress tolerance in insects
Yuncai Liu1,
Hang Su1,
Rongqiao Li1,
Xiaotong Li1,
Yusong Xu1,
Xiangping Dai1,
Yanyan Zhou1 &
Huabing Wang1
Heat tolerance is a key parameter that affects insect distribution and abundance. Glyphodes pyloalis Walker (Lepidoptera: Pyralidae) is a devastating pest of mulberry in the main mulberry-growing regions and can cause tremendous losses to sericulture by directly feeding on mulberry leaves and transmitting viruses to Bombyx mori. Moreover, G. pyloalis shows a prominent capacity for adaptation to daily and seasonal temperature fluctuations and can survive several hours under high temperature. To date, the molecular mechanism underlying the outstanding adaptability of this pest to high temperature remains unclear.
In this study, we performed comparative transcriptome analyses on G. pyloalis exposed to 25 and 40 °C for 4 h. We obtained 34,034 unigenes and identified 1275 and 1222 genes significantly upregulated or downregulated, respectively, by heat stress. Data from the transcriptome analyses indicated that some processes involved in heat tolerance are conserved, such as high expression of heat shock protein (HSP) genes and partial repression of metabolism progress. In addition, vitamin digestion and absorption pathways and detoxification pathways identified here provided new insights for the investigation of the molecular mechanisms of heat stress tolerance. Furthermore, transcriptome analysis indicated that immune and phosphatidylinositol signaling system have a close relationship with heat tolerance. In addition, the expression patterns of ten randomly selected genes, such as HSP and cytochrome P450, were consistent with the transcriptome results obtained through quantitative real-time PCR.
Comparisons among transcriptome results revealed the upregulation of HSPs and genes involved in redox homeostasis, detoxication, and immune progress. However, many metabolism progresses, such as glycolysis/gluconeogenesis and fatty acid biosynthesis, were partially repressed. The results reflected that the heat tolerance of G. pyloalis is a fairly complicated process and related to a broad range of physiological regulations. Our study can improve understanding on the mechanisms of insect thermal tolerance.
High temperature is an environmental factor that limits the distribution and abundance of insects. Insects are sensitive to temperature changes [1]. Over the past 30 years, global warming has led to significant changes in the number of insect species [2]. Several insect species, such as Drosophila melanogaster (D. melanogaster) and Bombyx mori (B. mori), are sensitive to heat stress, and the mass mortalities of these insects are often caused by high temperature [3, 4]. Conversely, many pests have evolved outstanding capability to adapt to high-temperature stimulations [5]. In recent years, many studies have focused on the effect of temperature on these insects because of their adaptability to a broad range of temperature [6].
Heat stress and acclimation of insect are currently considered a multistep process, involving a combination of behavioral, physiological, and cellular responses [7, 8]. Recent studies showed that heat tolerance in certain organisms is related to protein folding, degradation, transport, and metabolism [9, 10]. Moreover, current studies on flies revealed that several signal pathways, such as stress-responsive c-Jun N-terminal kinase signaling pathway, play an important role in adaptive metabolic response to heat stress [11]. Thus, many biologists have attempted to uncover the mechanisms underlying gene expression regulation under heat stress. Recently, researchers have characterized proteomic responses induced by heat stress in insects, such as Aphids, D. melanogaster, and B. mori. These studies displayed many protein-determining thermotolerances, such as chromatin remodeling and translation, iron ion and cell redox homeostasis, and carbohydrate and energy metabolism [12,13,14]. Despite these findings, minimal information is available on the gene expression profile of pest resistant to heat stress.
Glyphodes pyloalis Walker (Lepidoptera: Pyralidae), a specialist pest on mulberry, is widely distributed throughout Asia. This pest can damage sericulture not only by feeding on mulberry but also by transmitting viruses to B. mori. G. pyloalis encounters a wide range of daily and seasonal temperature fluctuations. On a summer day, the average temperature in Hangzhou (120.2′ E, 30.3′ N), Zhejiang Province, China is about 33.8 °C, and the highest temperature can reach almost 42 °C. Temperature increases during the summer lead to adverse effects on temperate bivoltine silkworm rearing and cause economic losses. In contract, G. pyloalis adapt to relatively high temperatures (35–40 °C) [15]. However, previous studies of G. pyloalis have focused only on single genes, such as α- and β-glucosidase [16]; the molecular mechanisms involved in the heat responses have not been explored through the transcriptome method, which has been widely used in profile gene expressions in insects. Uncovering the conundrum of the heat tolerance of G. pyloalis through RNA-Seq technique is reasonable, because the mechanism of G. pyloalis in managing heat stress remains unknown.
Midgut is the primary site of digestion and absorption in insects [17]. This site not only controls food storage and nutrient absorption but also maintains water, ion, and osmotic pressure balance [18]. Moreover, the midgut of an insect is involved in immunity and detoxication of harmful substances during digestion and absorption [19]. Thus, it is a pivotal organ for the interaction between an insect and external environment, and it represents a key target for the heat tolerance of an insect. However, few studies have focused on midgut transcriptome response to high temperature [20].
In the present study, G. pyloalis, an important lepidopteran pest, was assessed for the effects of heat acclimation. The results showed that approximately one-seventh of the transcriptome are differentially regulated upon heat acclimation. In addition, our results revealed that genes, which are related to heat tolerance, received pronounced effect after heat stimulation. This result showed that heat tolerance is a more complex process than we previously assumed. Our study can improve understanding on the molecular mechanism of heat tolerance in insects and provide novel targets for pest prevention and control.
Heat resistance of G. pyloalis Walker
G. pyloalis and B. mori are specialist insects on mulberry. Morphological changes in the midguts of G. pyloalis and B. mori were monitored after heat stress. Histological staining was performed to determine whether G. pyloalis can elicit heat resistance in its histomorphology and structure level under high temperatures (Fig. 1). After exposure to 40 °C for 4 h, the midgut of B. mori had a weaker condition than that exposed at 25 °C (Fig. 1a and b). Heat stress resulted in significant changes, and a large number of bubble-like structures were observed adjacent to the midgut contents of B. mori (Fig. 1b). However, structures observed in the midgut of G. pyloalis were fewer than those in B. mori after heat stress (Fig. 1b and d).
Morphological changes of midgut with different treatments. Morphological changes of B. mori (A–B) and G. pyloalis (C–D) were observed. Each (a) is a picture with low magnification, and the blank-lined area is shown in (b) with high magnification. Minimal bubble-like structure was observed in G. pyloalis after 4 h of heat treatment. However, many bubble-like structures (indicated by arrowheads) were produced in B. mori. Bars: A–a, B–a, C–a, and D–a: 10 μm; A–b, B–b, C–b, and D–a: 3.3 μm
mRNA sequencing, assembly, and functional annotation
We performed RNA-Seq to quantify the expression of genes in G. pyloalis to elucidate the molecular basis of heat stress response. RNA was extracted from the midgut of G. pyloalis and the transcriptome was sequenced using Illumina short reads. Approximately 13.37 Gb data and 34,034 unigenes were obtained (Additional file 1). The total length, average length, N50, and GC content of unigenes were 36,437,658 bp; 1070 bp; 2000 bp; and 43.09%, respectively (Table 1). We also detected the distinction of unigene length distributions between the two treatments and found no significant difference (Additional file 2). However, there were 22,729 unigenes expressed in the normal condition (25 °C) while 23,051 unigenes expressed after heat stress. We then annotated our unigenes with seven functional databases and found that 19,653 (57.75%), 13,170 (38.70%), 15,353 (45.11%), 8354 (24.55%), 15,546 (45.68%), 3878 (11.39%), and 14,410 (42.34%) unigenes can be mapped to NR, NT, UniProtKB/Swiss-Prot (Swiss-Prot), Cluster of Orthologous Groups of proteins (COG), Kyoto Encyclopedia of Genes and Genomes (KEGG), Gene Ontology (GO), and Interpro databases, respectively. After performing functional annotation, we detected 19,609 CDS. The remaining unaligned unigenes were analyzed using ESTScan to predict the coding regions. Another 1821 CDS were obtained after the analysis. The G. pyloalis sequences showed 42.9% matches with B. mori, followed by Danaus plexippus (27.63%), Papilio xuthus (2.2%), and Riptortus pedestris (2.13%) (Additional file 3). Furthermore, 2497 unigenes were differentially regulated after heat shock treatment with a criterion of |fold change| ≥4.0 and FDR ≤ 0.001 in DEGs definition. Of these DEGs, 1275 genes were upregulated, and 1222 genes were downregulated (Fig. 2). Ten significantly most upregulated genes that response to heat stress are as follows: ribonuclease, thiolase 4, peroxisomal multifunctional enzyme type 2-like, keratin-associated protein 10–7-like, aldehyde dehydrogenase isoform 1, PGI4-45_10 phosphoglucose isomerase, Probable cytochrome P450 304a1, Purine nucleoside phosphorylase associated with the functions redox homeostasis, detoxication and many metabolisms progresses (Additional file 4). This analysis showed heat stress had significant effect on the gene expression.
Table 1 Distribution of differentially expressed unigenes and Quality metrics of Unigenes
DEGs in the midgut after different temperature treatments. The variability pattern was displayed by volcano plot. Y axis represents −log10 significance. X axis represents log2-fold change. Red points represent upregulated genes. Blue points represent downregulated genes based on the discriminative significance values (|fold change| ≥ 4.0 and FDR ≤ 0.001) adopted in this study
GO and KEGG analyses of DEGs
We focused on the 1275 induced and 1222 repressed genes to further understand the overall biology of transcriptional response in G. pyloalis to heat stress. For GO analysis, we annotated DEGs into three GO categories, namely, cell component, molecular function, and biological process (Fig. 3). In the cell component category, the terms "glutamyl-tRNA amidotransferase complex" and "membrane" were the enriched components. "Steroid hormone receptor activity" and "lipid transporter activity" were the top two molecular function terms. The most enriched components of the biological process category were "steroid hormone mediated signaling pathway", "response to a steroid hormone" and "cellular response to steroid hormone stimulus".
GO enrichment analysis revealed the biological processes most associated with detected DEGs. Based on the GO results, cellular progress, metabolic progress, binding, and catalytic activity were the most enriched GO terms under heat stress
KEGG is a database for biological systems that integrates genomic, chemical, and systemic functional information [21]. In this study, 1666 DEGs were mapped to 291 pathways. Among these DEGs, "metabolic pathways" was the most dominant pathway in the midgut. In addition, many pathways, such as "sphingolipid metabolism", "phenylalanine metabolism", "RIG-I-like receptor signaling pathway" and "Toll-like receptor signaling pathway" were also enriched (Additional file 5). These results indicated an integration between metabolic and immune responses during heat tolerance, and this integration ensures energy balance and permits growth and defense in G. pyloalis.
Expression of pathways and genes for heat tolerance
Heat shock proteins (HSPs)
The transcriptional response of insects to heat shock includes a large number of HSPs, implying a universal mechanism of heat tolerance in insects [22]. We focused on genes that encoded HSPs to reveal the generality and particularity of heat tolerance between G. pyloalis and other insects. In this study, the data revealed upregulation or downregulation of HSPs under heat stress conditions. As shown in Fig. 4, 12 HSPs, including HSP70, HSP40, and small HSPs (sHSPs), which were distributed in different families, were identified (|fold change| ≥ 4.0 and FDR ≤ 0.001). Among these HSPs, five genes belonged to the HSP70 family, whereas two and five genes belonged to the HSP40 and sHSP, respectively. In terms of gene expression levels, nine HSPs displayed greater than fourfold increased expression, indicating a vital role in protein processing and heat tolerance in G. pyloalis.
Comparative distribution of HSP coding genes between two samples. High temperature resulted in gene expression related to HSP70, HSP40, and small HSP. The color scale is displayed at the upper left, which encompasses from the lowest (green) to the highest (red) RPKM value
Antioxidant and detoxication
Apart from genes encoding HSPs, several genes, which are involved in antioxidant and detoxication and regulated by heat stress, were identified. These genes included genes encoding two superoxide dismutase (Unigene21826_All and CL573.Contig1_All), a peroxidase (CL1988.Contig2_All), and a thioredoxin (CL1904.Contig1_All). In addition, these genes are directly linked to the generation of reactive oxygen species (ROS) under heat stress [23, 24] (Table 2). Detoxication-related genes were also identified in our study, including genes encoding glutathione S-transferase (CL2634.Contig1_All, Unigene19946_All, Unigene21727_All, and CL3231.Contig2_All) and cytochrome P450s (CYPs). 14 of the 23 cytochrome P450 (CYPs) unique sequences were induced, and most of these sequences were grouped into CYP3 clade (five of seven sequences), CYP6 clade (four of eight unique sequences), and CYP9 (three sequences; Fig. 5). Meanwhile, the polypeptides of the two unique sequences, namely, CL716.Contig3_All and Unigene19157_All, were determined as unigenes for aldehyde dehydrogenase (ALDH). They have been considered effective detoxifying enzymes and are involved in many fundamental biochemical pathways, including reduced cytotoxic aldehydes triggered by lipid peroxidation [25]. Therefore, antioxidant and detoxication have a close relationship with heat tolerance and even share a similar handling mechanism.
Table 2 Changes in the transcriptional expression of redox homeostasis and detoxication-related genes in response to thermal stress (40 °C) for 4 h
Heatmap of the expression level of different pathways. Expression level of some genes of fructose and mannose metabolism, glycolysis/gluconeogenesis, fatty acid biosynthesis, and fatty acid elongation was shown in a, b, c, and d, respectively. The color scale is shown at the upper left, which encompasses from the lowest (green) to the highest (red) RPKM value
Metabolism and protein turnover
Through KEGG pathway enrichment analysis, we found that many pathways related to metabolic reactions were enriched, including carbohydrate (14 pathways), lipid (15 pathways), and amino acid metabolisms (14 pathways). The genes associated with carbohydrate metabolism were partially repressed, such as glycolysis/gluconeogenesis (16 DEGs, 0.96%), fructose and mannose metabolism (13 DEGs, 0.78%), starch and sucrose metabolism (18 DEGs, 1.08%), galactose metabolism (15 DEGs, 0.9%), and oxidative phosphorylation (17 DEGs, 1.02%). A repression in the processes involved in lipid metabolism, including sphingolipid metabolism (23 DEGs, 1.38%), steroid biosynthesis (13 DEGs, 0.78%), fatty acid degradation (21 DEGs, 1.26%), and glycerolipid metabolism (25 DEGs, 1.5%), was also observed. The downregulated genes in fructose and mannose metabolism, glycolysis/gluconeogenesis, fatty acid biosynthesis, and fatty acid elongation pathways are illustrated in Fig. 6. Significant changes in metabolism-related genes involved in heat response were consistent with a previous report describing global gene expressions in livestock [26, 27]. In addition, unigenes, which related to protein turnover processes, were induced in response to heat stress. Among these unigenes, one proteasome-related unigene (CL2694.Contig1_All) was significantly upregulated. In addition, 34 of 57 ubiquitin system-related unigenes were specifically induced in response to heat stress, suggesting that the stress-related autophagy mechanisms were induced after heat stress. 8 of 10 ribosomal proteins, which were involved in protein translation and regulation of protein formation, were upregulated. These results suggested that protein turnover response was induced for protection against the detrimental effects of protein misfolding.
Heatmap of the different expression of cytochrome P450 genes under different heat treatment temperatures. The color scale is shown at the upper left, which encompasses from the lowest (green) to the highest (red) RPKM value
Insects lack adaptive immunity and rely on innate immune reactions for their defense [28]. In the present study, a large number of immune-related genes were enriched, and most of the gene sets demonstrated an upregulated expression after treatment at 40 °C. These genes were mainly involved in five pathways, namely, RIG-I-like receptor signaling pathway (8 DEGs, 0.48%), Fc gamma R-mediated phagocytosis (44 DEGs, 2.64%), chemokine signaling pathway (38 DEGs, 2.28%), Toll-like receptor signaling pathway (12 DEGs, 0.72%), and antigen processing and presentation (6 DEGs, 0.36%) (Fig. 7a). Nine unigenes involved in the Toll-like receptor signaling pathway were upregulated and thus may play a central role in relaying intracellular immune signals (Fig. 7b). Meanwhile, some unigenes participating in immune defense mechanisms were also detected in our data, although they do not belong to the pathways mentioned previously. Three unigenes (CL1248.Contig8_All, Unigene8334_All, and Unigene6267_All) were found to encode lectin, which is directly associated with the component of the immune system. Five genes encoding lysozyme were also determined, and three of them (CL623.Contig1_All, Unigene6122_All, and Unigene5037_All) had high expression levels and two (Unigene12321_All and CL623.Contig2_All) had low expression levels. Furthermore, some antiviral-related genes, such as genes encoding hdd1 (Unigene19527_All), trypsin-like serine proteinase (CL1159.Contig4_All), and scavenger receptor class B (Unigene7913_All), were induced. The abundant expression of immune response-related genes suggested that the immune responses were activated after heat shock.
General statistics on gene regulation under different temperatures and heatmap of the expression level of Toll-like receptor signaling pathways. a Numbers of upregulated and downregulated genes associated with various immune events. b Heatmap of genes related to Toll-like receptor signaling pathways. The color scale is shown at the upper left, which encompasses from the lowest (green) to the highest (red) RPKM value
Stress signal transduction
Stress signal transduction is the most important aspect for high temperature response. Many pathways involved in stress signal transduction were enriched in our analysis, such as Phosphatidylinositol (PI) signaling system (25 DEGs, 1.5%), Notch signaling pathway (16 DEGs, 0.96%), TNF signaling pathway (21 DEGs, 1.26%), Phospholipase D signaling pathway (34 DEGs, 2.04%), Jak-STAT signaling pathway (15 DEGs, 0.9%) and AMPK signaling pathway pathways (26 DEGs, 1.56%) (Table 3). The threonine protein kinase (CL962.Contig1_All) was also upregulated, and its activation was responsible for cellular stresses, such as heat shock [29]. These findings indicate the importance of signal transduction in the thermal tolerance of G. pyloalis.
Table 3 Significantly enriched signal pathways after heat treatment
Validation of data through quantitative real-time PCR (qRT-PCR)
Thousands of genes showed significantly different expression levels. In this study, we randomly selected ten genes to confirm their expression levels through qRT-PCR. Expression patterns were validated among the ten annotated transcripts (CL2742.Contig1_All, Unigene4872_All, Unigene7931_All, CL658.Contig2_All, CL2227.Contig1_All, Unigene5878_All, CL887.Contig1_All Unigene6953_All, Unigene6918_All and CL1060.Contig3_All). Some genes related to heat tolerance showed upregulated expression levels at 40 °C. These genes included genes encoding CYP9G3 (CL2742.Contig1_All), CYPB5 (Unigene6953_All), HSP19.7 (Unigene4872_All), and HSP16.1 (Unigene7931_All). Meanwhile, we found that Ribosomal protein L32 (Rpl32) and β-actin were stable. The Rpl32 was used as an internal control. The changing trend of the qRT-PCR results was consistent with the results obtained by DEG expression profiling (Fig. 8).
qRT-PCR analysis of the expression levels of ten unigenes. X axis represents different unigenes. CL2742.Contig1_All, cytochrome P450 9G3; Unigene4872_All, HSP 19.7; Unigene7931_All, HSP 16.1; CL658.Contig2_All, aldehyde dehydrogenase family of seven members A1; CL2227.Contig1_All, acetyltransferase 1; Unigene5878_All, hemolymph juvenile hormone-binding protein; CL887.Contig1_All, aldose 1-epimerase-like; CL1060.Contig3_All, 17-beta-hydroxysteroid dehydrogenase; Unigene6918_All, Hematopoietically-expressed homeobox protein HHEX homolog; Unigene6953_All, cytochrome P450 B5. Y axis represents the relative expression levels of genes. Ribosomal protein L32 (Rpl32) was used as an internal control
Many insects develop various mechanisms, including morphological and physiological adaptations, to cope with high-temperature stress. In this study, the morphological features of G. pyloalis indicated heat tolerance superior to that of B. mori and thus can contribute to its effective survival under high temperature. The adaptation strategies of G. pyloalis to heat stress may also be related to the expression of some unique genes. We conducted high-resolution analysis on transcriptome dynamics to identify the genes associated with excellent heat tolerance of G. pyloalis. In total, we obtained 2497 DEGs (1275 upregulated and 1222 downregulated) under heat stress and analyzed a large suite of biomarkers related to antioxidant, detoxication, metabolism, protein turnover, immune, and stress signal transduction. The GO analysis results showed that most of the DEGs after heat treatment were significantly represented in "response to stimulus", "regulation of response to stimulus", "response to steroid hormone" and "cellular response to steroid hormone stimulus". This finding suggested that most of the DEGs at the early heat stress stages were mainly associated with stress response and similar experiment results were observed in the plant response to heat stress. Many genes associated with "membrane", which revealed that most of the cells required being repaired [30]. Moreover, many highly represented pathways, such as sphingolipid metabolism, phenylalanine metabolism, metabolic pathways, and PI signaling system, were enriched when exposed to heat stress, indicating the importance of these pathways in establishing thermal tolerance [31].
In insects, HSPs act as molecular chaperones and participate in various cellular processes, such as protein folding, proteolytic pathway, or stabilizing denatured proteins [32,33,34]. HSPs can be divided into several families, namely, HSP40, HSP60, HSP70, HSP90 and sHSPs according to their molecular weights [35]. Twelve HSPs distributed in the HSP70, HSP40, and sHSPs were observed to respond to heat treatment during the transcriptome analysis of G. pyloalis. The preferential induction of HSPs upon heat stress has been widely reported in numerous studies. In D. melanogaster, the accumulation of a major HSP, namely, HSP70, affects inducible thermal tolerance in larvae and pupae [36]. In Ceratitis capitata, expression of Hsp70 and thermal tolerance were assayed at a range of temperatures in several stages of its development. In the current study, the induced expression of five HSP70s in G. pyloalis possibly facilitated the refolding of damaged proteins and prevented protein aggregation under heat stress. In addition, five small HSPs of G. pyloalis were upregulated by heat stress and likely cooperated with co-chaperones for the prevention of cellular protein aggregation [37]. sHSPs are abundant in nearly all organisms and act as a response machine of organisms to some extreme stresses [38]. These results suggested that the induction of HSP was an evolutionarily conserved mechanism, and the coordinate upregulation of the molecular chaperones, including HSP70 and sHSP, was an important factor in acquiring thermal tolerance and can render the midgut cells of G. pyloalis resistant to heat stress.
Exposure to heat stress can result in ROS production, which can directly lead to a variety of toxic effects, including protein dysfunction, lipid peroxidation, and oxidative stress. In relation to signal transduction of ROS, we detected upregulation of several genes involved in vitamin digestion and absorption pathways. Many studies have demonstrated the importance of vitamins in oxidative stress responses. Sies [39] mentioned that vitamins C and E can react with free radicals and singlet molecular oxygen, which is the basis for their functions as antioxidants. Notablely, the induction of vitamin digestion and absorption pathway may suggest a specific function in the adaption of insect to high temperature. To the best of our knowledge, this result is some of the first evidence indicating the existence of vitamin digestion and absorption pathway after heat stress in insects. In Table 2, several antioxidant-related genes, including superoxide dismutase (SOD), peroxidase (POD), and thioredoxin were strongly upregulated after 40 °C treatment. Recent proteomic analysis showed that SOD overexpression upregulated ROS scavenging in rice grains under heat stress. The SOD knockout rice was susceptible to heat stress, which suggested that SOD played an important role in adaptation to heat stress [40]. POD, as a regulator of ROS scavenger, is a hematin-containing oxidase that can catalyze the oxidation of reduced compounds [41]. Meanwhile, thioredoxin can be potentially responsive to ROS and can catalyze electron transport to ribonucleotide reductase and other reductive enzymes and transcription factors [42]. The coordinate upregulation of the three antioxidant-related genes after 40 °C treatment suggested that they were regulators of ROS scavenger in G. pyloalis and were potentially generate ROS in response to heat stress. Similar antioxidant-related genes were also observed in Bactrocera dorsalis and Bemisia tabaci exposed to high temperatures [43, 44]. Our data indicated that a link between oxidative stress and vitamin digestion and absorption was present in heat tolerance of G. pyloalis.
In addition to ROS generation, heat stress can promote toxic substance accumulation. In this study, we found that many detoxication-related genes, such as genes coding CYP, glutathione-S-transferase (GST), and ALDH, showed upregulated mRNA expression patterns after 4 h of heat stress. In most species, CYP effectively catalyzes lipid peroxidation. Meanwhile, the induction of a state of oxidative stress plays a central role in CYP-dependent cytotoxicity [45]. Apart from eliminating drugs, cytochrome P450s catalyze a broad range of oxidative processes involved in the metabolism of fatty acids and biosynthesis of sterols. Therefore, the induction of 14 CYPs in G. pyloalis may be an essential step in heat tolerance. In addition, we found 3 of 5 CYPs in B. mori were significantly induced at 40 °C, suggesting several CYPs played a conserved role in response to heat stress (Additional file 6). However, further experiments are necessary to determine the exact function of these proteins. Meanwhile, GSTs comprise a family of isozymes and can convert toxic substances into reduced toxic metabolites through chemical reactions involving the conjugation of glutathione [46]. They protect macromolecules from reactive electrophiles, such as environmental carcinogens, ROS products, and chemotherapeutic agents [46]. Our observation suggested that the upregulation of genes encoding GST was related to increased intracellular oxidative stress and cellular toxic substance metabolism [47]. Detoxification pathways identified here provided new insights for the investigation of the molecular mechanisms of heat stress tolerance. ALDHs can eliminate toxic aldehydes by catalyzing their oxidation to nonreactive acids, the enzyme activities of which is mainly implicated in the metabolism of endogenous lipid peroxidation products [48]. Two mRNAs for ALDH were significantly upregulated after heat shock, suggesting they contribute to the improved heat tolerance through metabolic pathway.
Insects have developed a range of strategies, such as metabolism adaptation, to manage heat stress. Our results indicated that the metabolism of G. pyloalis in high temperature was weak. A similar reduction was observed in livestock [27]. The suppression of metabolism progress may be a conservative progress and can reflect cellular homeostasis or an energy-saving mechanism to manage heat stress, because ATP generation is a highly-regulated process involving three metabolism pathways, namely, glycolysis, tricarboxylic acid cycle, and oxidative phosphorylation. In addition, the ROS generated by heat stress can chemically modify proteins and alter their biological functions. Therefore, the removal of damaged proteins is vital for the maintenance of cellular homeostasis. Numerous studies have revealed that the ubiquitin-proteasome system plays a vital role in recognizing and degrading damaged proteins [49, 50]. In the present study, 34 ubiquitin-related unigenes were induced by heat stress. These results revealed that the upregulation of protein degradation progress was involved in stress response and adaptation.
For insects, the innate immune system is the major effector response system [51]. In our results, the activities of the RIG-I-like and Toll-like receptor signaling pathways were modulated by heat stress. The role of transcriptional activation of the Toll-like receptor in heat shock response has been reported [52,53,54]. Several reports have described some endogenous ligands, such as HSP60 and HSP90, that activate TLR4 to regulate innate immune responses and are also induced by high temperature [55]. Frequent confrontation of the immune system with HSPs can potentiate immunity. The upregulation of lectin, lysozyme, Hdd1, trypsin-like serine proteinase, and scavenger receptor class B under high temperatures further suggested that heat stress facilitated the components of immune defense.
The sensing and transduction of intracellular stress signal are critical for the adaptation and survival of insects under heat stress. In plants, the involvement of calcium and calcium-activated calmodulin in heat shock signal transduction was already investigated [56]. However, the mechanisms that enable insects to sense heat signal and trigger intracellular responses remain unclear. The PI signaling system is vital during the progress of plant development and growth and associated with cellular responses to environmental stress [57]. Meanwhile, Notch signaling pathway participates in physiological and stress erythropoiesis [58]. In this study, the preferential induction of PI signaling system and Notch signaling pathway under heat stress was uncovered and the results suggested that gene expression in the PI signaling system was related to the heat tolerance mechanism of G. pyloalis. These results suggested that the PI signaling system and Notch signaling pathway rapidly and effectively regulated downstream signaling and gene expression in G. pyloalis in response to high-temperature stress.
In the present study, we presented the first comprehensive evaluation of the midgut transcriptome of G. pyloalis by using Illumina sequencing technology and conducted a comparative expression analysis after heat treatment. A total of 34,034 unigenes, which might provide a major genomic resource for investigating the midgut of G. pyloalis, were obtained. Most of these genes had an annotation with matches in the seven functional databases. Furthermore, 2497 DEGs were identified in the midgut after heat stress. The results of the GO and KEGG pathway enrichment analyses indicated that the DEGs related to metabolism, immunity, and signal transduction were enriched after heat stress. In addition, the transcriptome results revealed a number of genes that are potentially relevant to HSPs, antioxidant, and detoxication in G. pyloalis. These genes could be major targets for thermal tolerance. Many of which can be understood with deep functional studies. Nonetheless, our study provides insight into the heat tolerance of insects.
Biological materials and tissue collection
G. pyloalis larvae were collected from the mulberry fields of Zhejiang University, Hangzhou (120.2′ E, 30.3′ N), Zhejiang Province, China. B. mori (N4) larvae were reared as described previously [59]. The method of high temperature treatments or thermal exposure was modified from methods of Xiao et al. (2016), Li et al. (2014) and Moallem et al. (2017) [20, 60, 61]. G. pyloalis and B. mori treated at 25 °C served as controls. For treatments, larvae on day 3 of the fifth instar were selected and separated into two groups with 13 larvae each. Larvae were reared in different chambers at 25 °C and 40 °C for 4 h, respectively. They were supplied hourly with fresh mulberry leaves. Three biological replications were performed with each treatment. Four males and four females were selected for each treatment. Clean midgut samples were collected from eight (four males and four females) G. pyloalis and B. mori, respectively. The sample were freezed at −80 °C immediately.
Histological staining
After being exposed to different temperature treatments as described above, the samples of larval G. pyloalis and B. mori were collected for hematoxylin–eosin (HE) staining. The midgut of G. pyloalis was dissected and fixed in formalin-acetic acid-alcohol (FAA) liquid for 12 h. The samples were dehydrated through a series of graded ethanol baths (70%–100%) to displace water, and then infiltrated with paraffin wax, a finally cut into sections (5 μm). The obtained tissue sections were stained with 2% Mayer's hematoxylin and 1% eosin [62]. Slides were observed and image can be examined directly in the microscope (Nikon Nis-Elements).
De novo assembly and gene annotation
Total RNA was extracted using RNAiso plus reagent (TaKaRa) according to the manufacturer's protocol. The concentration and quality of total RNA was quantified using NanoDrop Spectrophotometer (Thermo Fisher Scientific). For each group, total RNA from three replicates was pooled together in equal quantities. Approximately 6 μg of RNA representing each group were used for illumine sequence. After the high-throughput sequencing (Illumina HiSeq 4000) of midgut samples, we used Trinity to filter raw reads by discarding low-quality, adaptor-polluted, and high-content unknown base reads and to generate 150 bp paired-end read lengths. The clean reads were assembled into primary transcripts, and overlapping transcripts were assembled into large contigs after removing redundant transcripts. The final unigenes were obtained through gene family clustering with Tgicl [63]. Gene function was annotated based on the following databases: NCBI non-redundant protein sequences (NR); InterPro member database; Clusters of Orthologous Groups of proteins (KOG/COG); Swiss-Prot (a manually annotated and reviewed protein sequence database); Kyoto Encyclopedia of Genes and Genomes (KEGG); and Gene Ontology (GO). The unigenes that were not aligned to any database mentioned above were predicted by ESTScan (http://sourceforge.net/projects/estscan).
Gene expression quantification
Bowtie2 software (http://bowtie-bio.sourceforge.net/Bowtie2/index.shtml) was used to map all the clean reads to the unigene library. Gene expression level was then calculated using the RSEM software (http://deweylab.biostat.wisc.edu/RSEM). The fragments per kilobase of transcript per million (FPKM) mapped reads can be used to quantify the unigene expression level. False discovery rate (FDR) was used to determine the threshold P-value in multiple tests. Furthermore, |fold change| ≥ 4.0 and FDR ≤ 0.001 were used as the parameters for determining significant differences in gene expression. The DEGs were then used for GO and KEGG enrichment analyses. The P-value calculating formula in the hypergeometric test is:
$$ P=1-\sum \limits_{i=0}^{\mathrm{m}-1}\frac{\left(\begin{array}{c}M\\ {}\mathrm{i}\end{array}\right)\left(\begin{array}{c}N-M\\ {}n-i\end{array}\right)}{\left(\begin{array}{c}N\\ {}n\end{array}\right)} $$
In this equation, N and n indicate the number of genes with GO/KEGG annotations and the number of DEGs in N, respectively. The variables M and m represent the numbers of genes and DEGs, respectively, in each GO/KEGG term.
qRT-PCR validation
Ten annotated unigenes and five B. mori CYPs were randomly selected to quantify by qRT-PCR and evaluate data level. RNAiso plus reagent (TaKaRa) was used to extract total RNAs from three biological replications, and Primer Premier 5.0 was used to perform the primer design. qRT-PCR primers are displayed in Additional file 7 and Additional file 8. The reverse transcription system was used to synthesize cDNA as described previously [64,65,66]. qRT-PCR was performed on a LightCycler® 480 real-time PCR system (Roche). The reaction system consisted of 2 μl of cDNA, 10 μl of SYBR Green qPCR master mix (TaKaRa), 0.5 μl of each of the primers, and sterile water made up to 20 μl. The ΔΔCt method was used to analyze the relative differences in transcript levels [67]. Ribosomal protein L32 (Rpl32) was used for internal control, and the experiment was performed on three biological replicates.
17 beta-HSD:
17-beta-hydroxysteroid dehydrogenase
A1E:
aldose 1-epimerase-like
acetyltransferase 1
BLAST:
CYP:
cytochrome P450
FAA:
formalin-acetic acid-alcohol
Fragments Per Kilobase of transcript per Million
GST:
glutathione-S-transferase
HE:
hematoxylin–eosin
Hhex:
Hematopoietically-expressed homeobox protein
HSP:
heat shock protein
JHBP:
hemolymph juvenile hormone-binding protein
NCBI non-redundant protein database
Nucleotide sequence database
Phosphatidylinositol
POD:
peroxidase
RNA-sequencing
Rpl32:
Ribosomal protein L32
Swiss-Prot:
TCA:
tricarboxylic acid cycle
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We thank Pengfei Zhan and Qiujie Qian for technical assistance.
This work was supported by the National Natural Science Foundation of China (Grant No. 31572321; 31,602,010). Sequencing library creation and high throughput sequencing was supported by a grant from the Science Foundation of Zhejiang Province of China (No. LY15C170001) and the public project grant (2016C32005) from the Science and Technology Department of Zhejiang Province.
All data analyzed during this study are provided in this published article and additional files. The sequence data of this study have been deposited into Sequence Read Archive (http://www.ncbi.nlm.nih.gov/sra; accession number PRJNA385884).
College of Animal Sciences, Zhejiang University, Hangzhou, 310058, China
Yuncai Liu, Hang Su, Rongqiao Li, Xiaotong Li, Yusong Xu, Xiangping Dai, Yanyan Zhou & Huabing Wang
Yuncai Liu
Hang Su
Rongqiao Li
Xiaotong Li
Yusong Xu
Xiangping Dai
Yanyan Zhou
Huabing Wang
YCL, HBW and YSX conceived this study. YCL, HS, YYZ and HBW collected data and data analyses. YCL and XTL contributed analysis tools. YCL, RQL, XPD and HBW carried out experiments. YCL and HBW wrote the manuscript with help from all the authors. All authors read and approved the final manuscript.
Correspondence to Yusong Xu or Huabing Wang.
I confirm that I have read BioMed Central's guidance on competing interests and have included a statement in the manuscript on any competing interests. The authors declare no competing financial interests.
Distribution of base content and quality. (DOCX 18 kb)
Unigene length distribution in G. pyloalis midgut with different treatments. X axis represents the length of unigenes. Y axis represents the number of unigenes. A: "Control" unigene length distribution. B: "Heat shock" unigene length distribution. (TIFF 1807 kb)
Distribution of annotated species. (TIFF 5077 kb)
List of the significantly up- and down-regulated genes. (XLSX 570 kb)
Pathway functional enrichment of DEGs. Based on the KEGG results, the metabolic pathway was the most enriched KEGG pathway under heat stress. (TIFF 1458 kb)
qRT-PCR analysis of the expression levels of five randomly selected B. mori genes. (TIFF 61 kb)
Primers used for B. mori CYPs. (DOCX 15 kb)
Primers used for G. pyloalis. (DOCX 19 kb)
Liu, Y., Su, H., Li, R. et al. Comparative transcriptome analysis of Glyphodes pyloalis Walker (Lepidoptera: Pyralidae) reveals novel insights into heat stress tolerance in insects. BMC Genomics 18, 974 (2017). https://doi.org/10.1186/s12864-017-4355-5
Glyphodes pyloalis
Redox homeostasis | CommonCrawl |
Publication Info.
Kyungpook Mathematical Journal
Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
This journal endeavours to publish research of broad interests in pure and applied mathematics. Each year one volume is published, consisting of four issues (March, June, September, December).
KSCI KCI
Volume 32 Issue 3_spc
Volume 9 Issue 1_2
Some Cycle and Star Related Nordhaus-Gaddum Type Relations on Strong Efficient Dominating Sets
Murugan, Karthikeyan 363
https://doi.org/10.5666/KMJ.2019.59.3.363 PDF
Let G = (V, E) be a simple graph with p vertices and q edges. A subset S of V (G) is called a strong (weak) efficient dominating set of G if for every $v{\in}V(G)$ we have ${\mid}N_s[v]{\cap}S{\mid}=1$ (resp. ${\mid}N_w[v]{\cap}S{\mid}=1$), where $N_s(v)=\{u{\in}V(G):uv{\in}E(G),\;deg(u){\geq}deg(v)\}$. The minimum cardinality of a strong (weak) efficient dominating set of G is called the strong (weak) efficient domination number of G and is denoted by ${\gamma}_{se}(G)$ (${\gamma}_{we}(G)$). A graph G is strong efficient if there exists a strong efficient dominating set of G. In this paper, some cycle and star related Nordhaus-Gaddum type relations on strong efficient dominating sets and the number of strong efficient dominating sets are studied.
The Convolution Sum $\sum_{al+bm=n}{\sigma}(l){\sigma}(m)$ for (a, b) = (1, 28),(4, 7),(1, 14),(2, 7),(1, 7)
Alaca, Ayse;Alaca, Saban;Ntienjem, Ebenezer 377
We evaluate the convolution sum $W_{a,b}(n):=\sum_{al+bm=n}{\sigma}(l){\sigma}(m)$ for (a, b) = (1, 28),(4, 7),(2, 7) for all positive integers n. We use a modular form approach. We also re-evaluate the known sums $W_{1,14}(n)$ and $W_{1,7}(n)$ with our method. We then use these evaluations to determine the number of representations of n by the octonary quadratic form $x^2_1+x^2_2+x^2_3+x^2_4+7(x^2_5+x^2_6+x^2_7+x^2_8)$. Finally we express the modular forms ${\Delta}_{4,7}(z)$, ${\Delta}_{4,14,1}(z)$ and ${\Delta}_{4,14,2}(z)$ (given in [10, 14]) as linear combinations of eta quotients.
The π-extending Property via Singular Quotient Submodules
Kara, Yeliz;Tercan, Adnan 391
A module is said to be ${\pi}$-extending provided that every projection invariant submodule is essential in a direct summand of the module. In this article, we focus on the class of modules having the ${\pi}$-extending property by looking at the singularity of quotient submodules. By doing so, we provide counterexamples, using hypersurfaces in projective spaces over complex numbers, to show that being generalized ${\pi}$-extending is not inherited by direct summands. Moreover, it is shown that the direct sums of generalized ${\pi}$-extending modules are generalized ${\pi}$-extending.
Super Theta Vectors and Super Quantum Theta Operators
Kim, Hoil 403
Theta functions are the sections of line bundles on a complex torus. Noncommutative versions of theta functions have appeared as theta vectors and quantum theta operators. In this paper we describe a super version of theta vectors and quantum theta operators. This is the natural unification of Manin's result on bosonic operators, and the author's previous result on fermionic operators.
Spectral Properties of k-quasi-class A(s, t) Operators
Mecheri, Salah;Braha, Naim Latif 415
In this paper we introduce a new class of operators which will be called the class of k-quasi-class A(s, t) operators. An operator $T{\in}B(H)$ is said to be k-quasi-class A(s, t) if $$T^{*k}(({\mid}T^*{\mid}^t{\mid}T{\mid}^{2s}{\mid}T^*{\mid}^t)^{\frac{1}{t+s}}-{\mid}T^*{\mid}^{2t})T^k{\geq}0$$, where s > 0, t > 0 and k is a natural number. We show that an algebraically k-quasi-class A(s, t) operator T is polaroid, has Bishop's property ${\beta}$ and we prove that Weyl type theorems for k-quasi-class A(s, t) operators. In particular, we prove that if $T^*$ is algebraically k-quasi-class A(s, t), then the generalized a-Weyl's theorem holds for T. Using these results we show that $T^*$ satisfies generalized the Weyl's theorem if and only if T satisfies the generalized Weyl's theorem if and only if T satisfies Weyl's theorem. We also examine the hyperinvariant subspace problem for k-quasi-class A(s, t) operators.
A Study of Marichev-Saigo-Maeda Fractional Integral Operators Associated with the S-Generalized Gauss Hypergeometric Function
Bansal, Manish Kumar;Kumar, Devendra;Jain, Rashmi 433
In this work, we evaluate the Mellin transform of the Marichev-Saigo-Maeda fractional integral operator with Appell's function $F_3$ type kernel. We then discuss six special cases of the result involving the Saigo fractional integral operator, the $Erd{\acute{e}}lyi$-Kober fractional integral operator, the Riemann-Liouville fractional integral operator and the Weyl fractional integral operator. We obtain new and known results as special cases of our main results. Finally, we obtain the images of S-generalized Gauss hypergeometric function under the operators of our study.
Strong Convergence Theorems for Common Points of a Finite Family of Accretive Operators
Jeong, Jae Ug;Kim, Soo Hwan 445
In this paper, we propose a new iterative algorithm generated by a finite family of accretive operators in a q-uniformly smooth Banach space. We prove the strong convergence of the proposed iterative algorithm. The results presented in this paper are interesting extensions and improvements of known results of Qin et al. [Fixed Point Theory Appl. 2014(2014): 166], Kim and Xu [Nonlinear Anal. 61(2005), 51-60] and Benavides et al. [Math. Nachr. 248(2003), 62-71].
Approximation by Generalized Kantorovich Sampling Type Series
Kumar, Angamuthu Sathish;Devaraj, Ponnaian 465
In the present article, we analyse the behaviour of a new family of Kantorovich type sampling operators $(K^{\varphi}_wf)_{w>0}$. First, we give a Voronovskaya type theorem for these Kantorovich generalized sampling series and a corresponding quantitative version in terms of the first order of modulus of continuity. Further, we study the order of approximation in $C({\mathbb{R}})$, the set of all uniformly continuous and bounded functions on ${\mathbb{R}}$ for the family $(K^{\varphi}_wf)_{w>0}$. Finally, we give some examples of kernels such as B-spline kernels and the Blackman-Harris kernel to which the theory can be applied.
Some Coefficient Inequalities Related to the Hankel Determinant for a Certain Class of Close-to-convex Functions
Sun, Yong;Wang, Zhi-Gang 481
In the present paper, we investigate the upper bounds on third order Hankel determinants for certain class of close-to-convex functions in the unit disk. Furthermore, we obtain estimates of the Zalcman coefficient functional for this class.
Initial Maclaurin Coefficient Bounds for New Subclasses of Analytic and m-Fold Symmetric Bi-Univalent Functions Defined by a Linear Combination
Srivastava, Hari M.;Wanas, Abbas Kareem 493
In the present investigation, we define two new subclasses of analytic and m-fold symmetric bi-univalent functions defined by a linear combination in the open unit disk U. Furthermore, for functions in each of the subclasses introduced here, we establish upper bounds for the initial coefficients ${\mid}a_{m+1}{\mid}$ and ${\mid}a_{2m+1}{\mid}$. Also, we indicate certain special cases for our results.
A New Aspect of Comrade Matrices by Reachability Matrices
Solary, Maryam Shams 505
In this paper, we study orthanogonal polynomials by looking at their comrade matrices and reachability matrices. First, we focus on the algebraic structure that is exhibited by comrade matrices. Then, we explain some properties of this algebraic structure which helps us to find a connection between comrade matrices and reachability matrices. In the last section, we use this connection to determine the determinant, eigenvalues, and eigenvectors of these matrices. Finally, we derive a factorization for det R(A, x), where R(A, x) is the reachability matrix for a comrade matrix A and x is a vector of indeterminates.
Strong Roman Domination in Grid Graphs
Chen, Xue-Gang;Sohn, Moo Young 515
Consider a graph G of order n and maximum degree ${\Delta}$. Let $f:V(G){\rightarrow}\{0,1,{\cdots},{\lceil}{\frac{{\Delta}}{2}}{\rceil}+1\}$ be a function that labels the vertices of G. Let $B_0=\{v{\in}V(G):f(v)=0\}$. The function f is a strong Roman dominating function for G if every $v{\in}B_0$ has a neighbor w such that $f(w){\geq}1+{\lceil}{\frac{1}{2}}{\mid}N(w){\cap}B_0{\mid}{\rceil}$. In this paper, we study the bounds on strong Roman domination numbers of the Cartesian product $P_m{\square}P_k$ of paths $P_m$ and paths $P_k$. We compute the exact values for the strong Roman domination number of the Cartesian product $P_2{\square}P_k$ and $P_3{\square}P_k$. We also show that the strong Roman domination number of the Cartesian product $P_4{\square}P_k$ is between ${\lceil}{\frac{1}{3}}(8k-{\lfloor}{\frac{k}{8}}{\rfloor}+1){\rceil}$ and ${\lceil}{\frac{8k}{3}}{\rceil}$ for $k{\geq}8$, and that both bounds are sharp bounds.
Hopf Hypersurfaces in Complex Two-plane Grassmannians with Generalized Tanaka-Webster Reeb-parallel Structure Jacobi Operator
Kim, Byung Hak;Lee, Hyunjin;Pak, Eunmi 525
In relation to the generalized Tanaka-Webster connection, we consider a new notion of parallel structure Jacobi operator for real hypersurfaces in complex two-plane Grassmannians and prove the non-existence of real hypersurfaces in $G_2({\mathbb{C}}^{m+2})$ with generalized Tanaka-Webster parallel structure Jacobi operator.
η-Ricci Solitons in δ-Lorentzian Trans Sasakian Manifolds with a Semi-symmetric Metric Connection
Siddiqi, Mohd Danish 537
The aim of the present paper is to study the ${\delta}$-Lorentzian trans-Sasakian manifold endowed with semi-symmetric metric connections admitting ${\eta}$-Ricci Solitons and Ricci Solitons. We find expressions for the curvature tensor, the Ricci curvature tensor and the scalar curvature tensor of ${\delta}$-Lorentzian trans-Sasakian manifolds with a semisymmetric-metric connection. Also, we discuses some results on quasi-projectively flat and ${\phi}$-projectively flat manifolds endowed with a semi-symmetric-metric connection. It is shown that the manifold satisfying ${\bar{R}}.{\bar{S}}=0$, ${\bar{P}}.{\bar{S}}=0$ is an ${\eta}$-Einstein manifold. Moreover, we obtain the conditions for the ${\delta}$-Lorentzian trans-Sasakian manifolds with a semisymmetric-metric connection to be conformally flat and ${\xi}$-conformally flat.
Note on the Codimension Two Splitting Problem
Matsumoto, Yukio 563
Let W and V be manifolds of dimension m + 2, M a locally flat submanifold of V whose dimension is m. Let $f:W{\rightarrow}V$ be a homotopy equivalence. The problem we study in this paper is the following: When is f homotopic to another homotopy equivalence $g:W{\rightarrow}V$ such that g is transverse regular along M and such that $g{\mid}g^{-1}(M):g^{-1}(M){\rightarrow}M$ is a simple homotopy equivalence? $L{\acute{o}}pez$ de Medrano (1970) called this problem the weak h-regularity problem. We solve this problem applying the codimension two surgery theory developed by the author (1973). We will work in higher dimensions, assuming that $$m{\geq_-}5$$. | CommonCrawl |
\begin{document}
\title{PMU Placement for Line Outage Identification via Multiclass Logistic
Regression}
\author{Taedong~Kim and Stephen~J.~Wright \thanks{This work was supported by a DOE grant
subcontracted through Argonne National Laboratory Award 3F-30222, and
National Science Foundation Grant DMS-1216318.} \thanks{T. Kim and S.~J. Wright are with the
Computer Sciences Department, 1210 W. Dayton Street,
University of Wisconsin, Madison, WI 53706, USA
(e-mails: [email protected] and [email protected]).}}
\maketitle
\begin{abstract} We consider the problem of identifying a single line outage in a power grid by using data from phasor measurement units (PMUs). When a line outage occurs, the voltage phasor of each bus node changes in response to the change in network topology. Each individual line outage has a consistent ``signature,'' and a multiclass logistic regression (MLR) classifier can be trained to distinguish between these signatures reliably. We consider first the ideal case in which PMUs are attached to every bus, but phasor data alone is used to detect outage signatures. We then describe techniques for placing PMUs selectively on a subset of buses, with the subset being chosen to allow discrimination between as many outage events as possible. We also discuss extensions of the MLR technique that incorporate explicit information about identification of outages by PMUs measuring line current flow in or out of a bus. Experimental results with synthetic 24-hour demand profile data generated for 14, 30, 57 and 118-bus systems are presented. \end{abstract}
\begin{IEEEkeywords} line outage identification, phasor measurement unit, optimal PMU placement, multiclass logistic regression \end{IEEEkeywords}
\section{Introduction}\label{sec:intro} In recent years, phasor measurement units (PMUs) have been introduced as a way to monitor power system networks. Unlike the more conventional Supervisory Control and Data Acquisitions (SCADA) system, whose measurements include active and reactive power and voltage magnitude, PMUs can provide accurate, high-sampling-rate, synchronized measurements of voltage phasor. There has been much ongoing research on how the real-time measurement information gathered from PMUs can be exploited in many areas of power system studies, including system control and state estimation.
In this paper, we study the use of PMU data in detecting topological network changes caused by single-line outages, and propose techniques for determining optimal placement of a limited number of PMU devices in a grid, so as to maximize the capability for detecting such outages. Our PMU placement approach can also be used as a tie-breaker for the other types of strategies that have multiple optimal solutions, for example, maximum observability problems.
Knowledge of topological changes as a result of line failure can be critical in deciding how to respond to a blackout. Rapid detection of such changes can enable actions to be taken that reduce risks of cascading failures that lead to large-scale blackouts. One of the main causes of the catastrophic Northeast blackout of 2003 was faulty topological knowledge of the grid following the initial failures (see \cite{BlackOut2003}).
Numerous approaches have been proposed for identifying line outages using PMU device measurements. In \cite{TatO08, TatO09}, phasor angle changes are measured and compared with expected phasor angle variations for all single- or double-line outage scenarios. Support vector machines (SVM) were proposed for identification of single-line outages in \cite{AbdME12}. A compressed-sensing approach was applied to DC power balance equations to find sparse topological changes in \cite{ZhuG12}, while a cross-entropy optimization technique was considered in \cite{CheLM14}. Since the approaches in \cite{ZhuG12} or \cite{CheLM14} use the linearized DC power flow models to represent a power system, their line outage identification strategies rely only on changes to phase angles. Voltage magnitude measurements from PMUs, which also provide useful information for monitoring a power system, are ignored. Our use of the AC power flow model allow both more accurate modeling of the system and more complete exploitation of the available data.
The key feature that makes line outage identification possible is that voltage phasor measurements reported by PMUs are different for different line-outage scenarios. Our approach aims to distinguish between these different ``signatures'' by using a multiclass logistic regression (MLR) model. The model can be trained by a convex optimization approach, using standard techniques. The coefficients learned during training can be applied during grid operation to detect outage scenarios. Our approach could in principle be applied to multiline outages too, but since the problem dimension is much larger in such cases --- the number of possible outage scenarios is much greater --- it is no longer practical. Second, even when trained only to recognize single-line outages, our classifier is useful in multiline outage situations on large grids, when the coupling between the lines is weak (as discussed in \cite[Section~2.2]{ZhuS14}). In other words, many multiline outage cases can be decomposed into single-line outage events on different parts of the grid.
Because of the expense of installation and maintenance, PMUs can reasonably be installed on just a subset of buses in a grid. We therefore need to formulate an {\em optimal placement problem} to determine the choice of PMU locations that gives the best information about system state. Several different criteria have been proposed to measure quality of a given choice of PMU locations. One of the most popular criteria is
to place PMUs to maximize the number of nodes in the system that can be observed directly, either by a PMU located at that node or an adjacent connected node \cite{ChaKE09}. Another criterion is
quality of state estimation results. In this approach, one can use PMU measurements alone, or combine them with traditional SCADA measurement to decide the optimal PMU deployment (see for example \cite{KekGW12}). Other criteria and techniques for locating PMUs optimally are discussed in the review papers \cite{ManKG11, YuiEC11}.
For the case in which line outage identification is used as a criterion for PMU placement, we mention \cite{ZhaGP12,ZhuG12,ZhuS14}. In \cite{ZhaGP12}, the authors use pre-computed phase angles as outage signatures and attempt to find the optimal PMU locations by identifying a projection (by setting to zero the entries which are not selected as PMU locations) that maximizes the minimum distance in $p$-norm of the projected signatures. The problem is formulated as an integer program (IP) and a greedy heuristic and branch-and-bound approach are proposed. PMU placement for the line outage identification method discussed in \cite{ZhuG12} is studied in \cite{ZhuS14}. A non-convex mixed-integer nonlinear program (MINLP) is formulated, leading to a linear programming convex relaxation. Again, a greedy heuristic and a branch-and-bound algorithm is suggested as a solution methodology.
We build our PMU placement formulation on our MLR model for single-line outage detection, by adding nonsmooth ``Group LASSO'' regularizers to the MLR objective and applying several heuristics.
The rest of this paper is organized as follows: In Section~\ref{sec:line.outage}, the line outage identification problem is described along with the multiclass logistic regression (MLR) formulation. The problem of PMU placement to identify a line outage is described in Section~\ref{sec:PMU.placement}, and we describe the group-sparse heuristic and its greedy variant used to formulate and solve this problem. Experimental results on synthetically generated data are presented in Section~\ref{sec:result}. A conclusion appears in Section~\ref{sec:conclusion}.
In supplementary material, we describe an extension that makes use of explicit line outage information. This model uses the fact that when a PMU is attached to a line, it can detect by direct observation when that line fails, and has no need to rely on the indirect evidence of voltage changes. As expected, performance improves when such additional information is used, though as we show in this paper, very good results can still be obtained even when it is ignored.
\section{Line Outage Identification}\label{sec:line.outage}
We describe an approach that uses changes in voltage phasors measurements at PMUs to detect single-line outages in the power grid. As in \cite{TatO08,ZhuG12}, we assume that the fast dynamics of the system are well damped and voltage measurements reflect the quasi-static equilibrium that is reached after the disruption. We use a quasi-steady state AC power flow model (see e.g. \cite[Chapter~10]{BerV99}) as a mapping from time varying load variation (and line outage events) to the polar coordinate ``outputs'' of voltage magnitude and angle.
PMUs report phasor measurements with high frequency, and changes in voltage due to topology changes of the power grid tend to be larger than the variation of voltage phasor during normal operation (for example, demand fluctuation that occurs during the sampling time period). We construct signature vectors from these voltage changes under the various single-line outage situations, and use them to train a classifier.
\begin{figure*}
\caption{9-Bus System: Voltage changes caused by single line outage on buses
5, 6 and 7. (Voltage magnitude (p.u) - Phase angle (rad))}
\label{fig:9bus.phasor.change}
\end{figure*}
Figure~\ref{fig:9bus.phasor.change} shows an example of voltage changes for a 9-bus system ({\tt case9.m} from {MATPOWER}{}~\cite{ZimMT11}) on different line outage scenarios. The failure of lines connecting buses 4-5, 5-6, 6-7, 7-8, and 8-9 is considered as possible scenarios (columns in the figure) whose voltage phasors at buses 5, 6, and 7 (rows in the figure) observed. In each plot, $x$-axis shows voltage magnitude and $y$-axis shows voltage phase angle at the bus. The red dots in each plot indicate the voltage phasor when there is no line failure and the blue dots are the voltage values under the specified failure scenario. We observe that voltage values at these buses change in distinctive ways under different line outage scenarios. It therefore seems realistic to expect that by comparing voltage phasor data, gathered before and after a failure event, we can identify the failed line reliably. We now describe the multinomial logistic regression model for determining the outage scenario.
\subsection{Multinomial Logistic Regression Model}
Multinomial logistic regression (MLR) is a machine-learning approach for multiclass classification. In our application, examples of voltage phasor changes under each outage scenario are used to train the classifier by determining parameter values in a set of parametrized functions. Once the parameters have been found, these functions determine the likelihood of a given set of phasor changes as being indicative of each possible failure scenario.
Suppose that there are $K$ possible outcomes (classes) labelled as $i \in \set{1,2,\dotsc,K}$ for a given vector of observations $X$. In the multinomial logistic regression model, the probability of a given observation $X$ has an outcome $Y$ (one of the $K$ possibilities $i\in\set{1,2,\cdots,K}$) is given by the following formula: \begin{align}\label{eq:prob.Y}
\prob{Y=i|X} &:= \frac{e^{\ip{\beta_i}{X}}}
{\sum_{k=1}^Ke^{\ip{\beta_k}{X}}}
\tfor i=1,2,\cdots,K, \end{align} where $\beta_1, \beta_2, \dotsc, \beta_K$ are regression coefficients, whose values are obtained during the training process. Note that there is one regression coefficient $\beta_i$ for each outcome $i \in \{1,2,\dotsc,K\}$. Once values of the coefficients $\beta_i$, $i \in \{1,2,\dotsc,K\}$ have been determined, we can predict the outcome associated with a given feature vector $X$ by evaluating \[
k^* = \oargmax{k\in\set{1,2,\dotsc,K}}\prob{Y=k|X}, \] or equivalently, \begin{equation}\label{eq:max.outcome}
k^* = \oargmax{k\in\set{1,2,\dotsc,K}} \ip{\beta_k}{X}. \end{equation}
Training of the regression coefficients $\beta_1, \cdots, \beta_K$ can be performed by maximum likelihood estimation. The training data consists of $M$ pairs $(X_1,Y_1), (Y_2, Y_2), \dotsc, (X_M, Y_M)$, each consisting of a feature vector and its corresponding outcome. Given formula \eqref{eq:prob.Y}, the a posteriori likelihood of observing $Y_1,Y_2,\dotsc,Y_M$ given the events $X_1,X_2,\dotsc,X_M$ is
\begin{equation}\label{eq:likelihood} \prod_{i=1}^M P(Y=Y_i|X_i) \,=\, \prod_{i=1}^M \(\frac{e^{\ip{\beta_{Y_i}}{X_i}}}
{\sum_{k=1}^Ke^{\ip{\beta_k}{X_i}}}\). \end{equation}
By taking $\log$ of \eqref{eq:likelihood}, we have log-likelihood function \[
f(\beta) :=
\sum_{i=1}^M\(\ip{\beta_{Y_i}}{X_i} -
\log\sum_{k=1}^Ke^{\ip{\beta_k}{X_i}}\), \] where the matrix $\beta$ is obtained by arranging the coefficient vectors as $\bm{\beta_1 & \beta_2 & \dotsc & \beta_K}$. The maximum likelihood estimate $\beta^*$ of regression coefficients is obtained by solving the following optimization problem:
\begin{equation}\label{eq:MLE} \beta^* = \oargmax{\beta} f(\beta). \end{equation}
This is a smooth convex problem that can be solved by fairly standard techniques for smooth nonlinear optimization, such as L-BFGS \cite{LiuN89}. Note that $f(\beta) \le 0$ for all $\beta$.
If the training data is separable, the value of $f(\beta)$ can be made to approach zero arbitrarily closely by multiplying $\beta$ by an increasingly large positive value (see \cite{KriCF05}). To recover meaningful values of $\beta$ in this case, we can solve instead the following regularized form of \eqref{eq:MLE}: \begin{equation}\label{eq:MLE.p} \beta^* = \oargmax{\beta} f(\beta) - \tau w(\beta) \end{equation} where $\tau>0$ is a penalty parameter and $w(\beta)$ is a (convex) penalty function of the coefficient $\beta$. The penalized form can also be used to promote some kind of structure in the solution $\beta^*$, such as sparsity or group-sparsity. This property is key to our PMU placement formulation, and we discuss it further in Section~\ref{sec:PMU.placement}.
Training of the MLR model, via solution of \eqref{eq:MLE} or \eqref{eq:MLE.p}, can be done offline, as described in the next subsection. Once the model is trained (that is, the coefficients $\beta_i$, $i=1,2,\dotsc,K$ have been calculated), classification can be done via \eqref{eq:max.outcome}, at the cost of multiplying the matrix $\beta$ by the observed feature vector $X$, an operation that can be done in real time.
\subsection{Training Data: Observation Vectors and Outcomes}\label{sec:data.format} In our MLR model for line outage identification problems, the observation vector $X_j$ is constructed from the change of voltage phasor at each bus, under a particular outage scenario. The corresponding outcome is the index of the failed line.
Suppose that a power system consists of $N$ buses, all equipped with PMUs that report the voltage values periodically. Let $(V_i, \theta_i)$ and $(V_i', \theta_i')$, $i=1,2,\cdots, N$, be two phasor measurements obtained from PMU devices, one taken before a possible failure scenario and one afterward. The observation vector $X$ which describes the voltage phasor difference is defined to be \begin{equation} \label{eq:X}
X = \bm{\Delta V_1 & \cdots & \Delta V_N &
\Delta\theta_1 & \cdots & \Delta\theta_N}^T \end{equation} where $\Delta V_i=V_i'-V_i$ and $\Delta\theta_i = \theta_i'-\theta_i$, for $i=1,2,\dotsc,N$. If we assume that the measurement interval is small enough that loads and demands on the grid do not change significantly between measurements, we would expect the entries of $X$ to be small, unless an outage scenario (leading to a topological change to the grid) occurred. Some such outages would lead to failure of the grid. More often, feasible operation can continue, but with significant changes in the voltage phasors, indicated by large components of $X$.
The training data $(X_j,Y_j)$ can be assembled by a considering a variety of realistic demand scenarios for the grid, solving the AC power flow equations for each possible outage scenario (setting the value of $Y_j$ according to the index of that failure), then setting $X_j$ to be the shift in voltage phasor that corresponds to that scenario. The phasor shifts for a particular scenario change somewhat as the pattern of loads and generations changes, so it is important to train the model using a sample of phasor changes under different realistic patterns of supply and demand.
The observation vector can be extended to include additional information beyond the voltage phasor information from the PMUs, if such information can be gathered easily and exploited to improve the performance of the MLR approach. For example, the system operator may be able to monitor the power generation level $G$ (expressed as a fraction of the long-term average generation) that is injected to the system at the same time points at which the voltage phasor measurements are reported. If included in the observation vector, this quantity might need to scaled so that it does not dominate the phasor difference information. Also, a constant entry can be added to the observation vector to provide more flexibility for the regression. The extended observation vector thus has the form \begin{equation} \label{eq:Xbar}
\ub{X} = \bm{\Delta V_1 & \cdots & \Delta V_N &
\Delta\theta_1 & \cdots & \Delta\theta_N & \rho G & \rho}^T \end{equation} where $\rho$ is a scaling factor that approximately balances the magnitudes of all entries in the vector. (Note that since $G$ is not too far from $1$, it is appropriate to use the same scaling factor for the last two terms.) The numerical experiments in Section~\ref{sec:result} make use of this extended observation vector.
\section{PMU Placement}\label{sec:PMU.placement}
As we mentioned in Section \ref{sec:intro}, installing of PMUs at all buses is impractical. Indeed, if it were possible to do so, single-line outage detection would become a trivial problem, as each outage could be observed directly by PMU measurements of line current flows in or out of a bus; there would be no need to use the ``indirect'' evidence provided by voltage phasor changes. In this section we address the problem of placing a limited number of PMUs around the grid, with the locations chosen in a fashion that maximizes the system's ability to detect single-line outages. This PMU placement problem selects a subset of {\em buses} for PMU placement, and assumes that PMUs are placed to monitor voltage phasors at the selected buses.
A naive approach is simply to declare a ``budget'' of the number of buses at which PMU placement can take place, and consider all possible choices that satisfy this budget. This approach is of course computationally intractable except for very small cases. Other possible approaches include a mixed-integer nonlinear programming formulation \cite{ZhaGP12,ZhuS14}, but this is very hard to solve in general. In this paper, we use a regularizer function $w(\beta)$ in \eqref{eq:MLE.p} to promote the a particular kind of sparsity structure in the coefficient matrix $\beta$. Specifically, A group $\ell_1$-regularizer is used to impose a common sparsity pattern on all columns in the coefficient matrix $\beta$, with nonzeros occurring only in locations corresponding to the voltage magnitude and phase angle changes for a subset of buses. The numerical results show that approaches based on this regularizer give reasonable performance on the PMU placement problem.
\subsection{Group-Sparse Heuristic (GroupLASSO)}\label{sec:gsh} Let ${\cal P}$ be the set of indices in the vector $X$, that is
${\cal P}=\set{1,2,\cdots,|X|}$. Consider $S$ mutually disjoint subsets of ${\cal P}$, denotes ${\cal P}_1, {\cal P}_2, \dotsc, {\cal P}_S$. For each $s \in {\cal S} := \set{1,2,\cdots,S}$, define $q_s(\beta)$ as follows: \[
q_s(\beta) = \norm{[\beta]_{{\cal P}_s}}_{F}
= \sqrt{\sum_{i\in{\cal P}_s}\sum_{k=1}^K (\beta_{ik})^2} \] where $[\beta]_{{\cal P}_s}$ is the submatrix of $\beta$ constructed by choosing the rows whose indices are in ${\cal P}_s$, $\norm{\cdot}_F$ is the Frobenius norm, and $\beta_{ik}$ is the $(i,k)$ entry of matrix $\beta$ (thus, $\beta_{ik}$ is the $i$th entry of the coefficient vector $\beta_k$ in \eqref{eq:max.outcome} and \eqref{eq:likelihood})
The value of $q_s(\beta)$ is the $\ell_2$-norm over the entries of matrix $\beta$ which are involved in group $s$. For our observation vectors $X$ \eqref{eq:X} and $\ub{X}$ \eqref{eq:Xbar}, we can choose the number of groups $|{\cal S}|$ equal to the number of buses $N$, and set \begin{equation} \label{eq:Ps}
{\cal P}_s = \{ s, s+N \}, \quad s=1,2,\dotsc,N. \end{equation} Thus, if bus $s$ is ``selected'' in the placement problem, the coefficients associated with $\Delta V_s$ and $\Delta \theta_s$ are allowed to be nonzeros. Buses that are not selected need not of course be instrumented with PMUs, because the coefficients in $\beta$ that correspond to these buses are all zero. Note that for the extended vector $\ub{X}$, we do not place the last two entries (the constant and the total generation) into any group, as we assume that these are always ``selected'' for use in the classification process.
For any subset ${\cal R}$ of ${\cal S}$, we define a group-$\ell_1$-regularizer $w_{{\cal R}}(\beta)$ to be the sum of $q_s(\beta)$ for $s\in{\cal R}$, that is, \[
w_{{\cal R}}(\beta) = \sum_{s\in{\cal R}} q_s(\beta). \] Setting ${\cal R}={\cal S}$, the penalized form \eqref{eq:MLE.p} with $w=w_{{\cal S}}$ can be used to identify a group-sparse solution: \begin{equation}\label{eq:MLE.group.l1}
\omax{\beta} f(\beta) - \tau w_{{\cal S}}(\beta). \end{equation} With an appropriate choice of the parameter $\tau$, the solution
$\beta^*$ of \eqref{eq:MLE.group.l1} will be group-row-sparse, that is, the set $\setc{s\in{\cal S}}{q_s(\beta^*)\neq 0}$ will have significantly fewer than $|{\cal S}|$ elements. Given a solution $\beta^*$ of \eqref{eq:MLE.group.l1} for some value of $\tau$, we could define the $r$-sparse solution as follows (for a given value of $r$, and assuming that the solution of \eqref{eq:MLE.group.l1} has at least $r$ nonzero values of $q_s(\beta^*)$): \begin{equation}\label{eq:MLE.l1.S}
{\cal R}^* := \oargmax{{\cal R}:|{\cal R}|=r,{\cal R}\subset{\cal S}} w_{{\cal R}}(\beta^*). \end{equation} Since the minimizer $\beta^*$ of \eqref{eq:MLE.group.l1} is biased due to the presence of the penalty term, we should not use the submatrix extracted from $\beta^*$ according to the selected group ${\cal R}^*$ as the regression coefficients for purposes of multiclass classification. Rather, we should solve a reduced, unpenalized version of the problem in which just the coefficients from sets ${\cal P}_s$ that were {\em not} selected are fixed at zero. That is, we define a {\em
debiased} solution $\tilde{\beta}^*$ corresponding to ${\cal R}^*$ as follows: \begin{align}\label{eq:MLE.debias}
\omax{\beta}f(\beta) \;\; & \mbox{subject to $\beta_{ik}=0$ for all $(i,k)$ with } \\ \nonumber & \mbox{ $k=1,2,\dotsc,K$ and $i\in {\cal P}_s$ for some $s \notin {\cal R}^*$.} \end{align}
\subsection{Greedy Heuristic}
The regularization approach can be combined with a greedy strategy, in which groups are selected one at a time, with each selection made by solving a regularized problem. Suppose that ${\cal R}^{l-1}$ is set of selected groups after $l-1$ iterations of the selection heuristic. The problem solved at iteration $t$ of the heuristic to choose the next group is \begin{equation}\label{eq:MLE.l1.greedy} \hat{\beta}^l = \oargmax{\beta} f(\beta) - \tau w_{{\cal S} \setminus {\cal R}^{l-1}}(\beta). \end{equation} The next group $s^l$ is obtained from $\hat{\beta}^l$ as follows: \[
s^l = \oargmax{s\in {\cal S} \setminus {\cal R}^{l-1}} q_s(\hat{\beta}^l), \] and we set ${\cal R}^l={\cal R}^{l-1}\cup\set{s^l}$. Note that we do {\em not} penalize groups in ${\cal R}^{l-1}$ that have been selected already, in deciding on the next group $s^l$. After choosing $r$ groups by this process, the debiasing step is performed to find the best maximum likelihood estimate for the sparse observation. Algorithm~\ref{alg:PMU.placement.greedy} describes this greedy approach. Note that the initial set of groups ${\cal R}^0$ might not be empty since we can use additional information that is independent from the PMU measurement, if available.
\begin{algorithm}\small \caption{Greedy Heuristic}\label{alg:PMU.placement.greedy} \begin{algorithmic}[1] \Require
\Statex Choose an initial set of groups: ${\cal R}^0$.
\Statex Parameter $\tau$, $r$. \Ensure
\Statex ${\cal R}^r$: Set of groups after selecting $r$ groups.
\Statex $\tilde{\beta}^r$: Maximum likelihood estimate for $r$-group
observation.
\For {$l=1,2,\cdots,r$}
\State Solve \eqref{eq:MLE.l1.greedy} with ${\cal R}^{l-1}$ for $\hat{\beta}^l$.
\State $s^l\gets\oargmax{s\in{\cal S} \setminus {\cal R}^{l-1}} q_s(\beta^l)$
\State ${\cal R}^l\gets {\cal R}^{l-1}\cup\set{s^l}$ \EndFor \State Solve \eqref{eq:MLE.debias} with ${\cal R}^r$ to get $\tilde{\beta}^r$. $\Comment{debiasing}$ \end{algorithmic} \end{algorithm}
The major advantage of this approach is that redundant observations are suppressed by already-selected, non-penalized observations at each iteration. We will give more details in discussing the experimental results in Section~\ref{sec:result}.
\section{Numerical Experiments}\label{sec:result}
Here we present experimental results for the approaches proposed above. The test sets considered here are based on the power system test cases from {MATPOWER}{} (originally from \cite{PSTCA}), with demands altered to generate training and test sets for the MLR approach.
\subsection{Synthetic Data Generation} \begin{figure*}
\caption{Generating Synthetic Demand Data by A Stochastic Process}
\label{fig:syn.data.avg}
\label{fig:syn.data.ratio}
\label{fig:syn.data.demand}
\label{fig:syn.data}
\end{figure*}
Since the data provided from IEEE test case archive \cite{PSTCA} is a single snapshot of the states of power systems, we extend them to a synthetic 24-hour demand data cycle by using a stochastic process, as follows.
\begin{enumerate}
\item Take the demand values given by the IEEE test case archive as
the average load demand over 24-hours.
\item Generate the demand variation profile by using an additive
Ornstein-Uhlenbeck process as described in \cite{PerKA11},
separately and independently on each demand bus.
\item Combine the average demand and the variation ratio to obtain
the 24-hour load demand profile for the system. \end{enumerate}
Figure~\ref{fig:syn.data} shows demand data generated by this procedure at three demand buses in the 9-bus system ({\tt case9.m}) from {MATPOWER}{}. Figure \subref*{fig:syn.data.avg} shows the data drawn from the {MATPOWER}{} file, now taken to be a 24-hour average. Figure~\subref*{fig:syn.data.ratio} shows the ratio generated by the additive Ornstein-Uhlenbeck process, and Figure \subref*{fig:syn.data.demand} shows the products of the average and ratio. Since the power injected to the system needs to increase proportionally to the total demands, all power generation is multiplied by the average of the demand ratios. This average of ratios is used as the generation level $G$ for the observation vector $\ub{X}$ defined by \eqref{eq:Xbar}. The data assumes a 10-second interval between the measurements, so the total number of time points in the generated data is $24\times60\times6=8640$.
Once the 24-hour load demand profile is obtained, the AC-power equations are solved using {MATPOWER}{} to calculate the voltage phasor values at each time point. These phasor values are taken to be the PMU measurements for a normal operation cycle over a 24-hour period. {MATPOWER}{}'s AC power flow equations solver is also used to evaluate voltage phasors for each single-line outage scenario that does not lead to an infeasible system. (During this process, if there exist duplicated lines that connect the same pair of buses, they are considered as a single line, that is, we do not allow only a fraction of multiple lines that connect the same set of buses to be failed.) Simulation of single-line failures to generate training data is necessary because there are typically few instances of actual outages available for study. The voltage variation for each line outage at time $t$ is calculated by subtracting these normal-operation voltages at timepoint $t-1$ from line outage voltages at time point $t$. (The 10-second interval between measurements is usually sufficient time to allow transient fluctuations in phasor values to settle down; see \cite{TatO08}.) This process leads to a number of labeled data pairs $(X,Y)$ (or $(\ub{X},Y)$) which we can use to train or tune the MLR classifier.
\begin{table}\centering \caption{Test Cases from {MATPOWER}{}}\label{tbl:test.cases}
\begin{tabular}{|c|c|c|c|c|c|}\hline \multirow{2}{*}{System} & {MATPOWER}{}
& \multicolumn{2}{c|}{\# of Lines} & Train & Test \\ \cline{3-4}
size & case & Feas. & Infeas./Dup. & (5) & (50) \\\hline\hline
14-Bus & case14 & 18 & 2 & 90 & 900 \\
30-Bus & case\_ieee30 & 37 & 4 & 185 & 1850 \\
57-Bus & case57 & 67 & 13 & 335 & 3350 \\
118-Bus & case118 & 170 & 16 & 850 & 8500 \\\hline \end{tabular} \end{table}
Table~\ref{tbl:test.cases} provides the basic information on the power systems used for the experiments. The number of lines that are feasible is given in the column ``Feas.,'', while the number of lines that are duplicated or that lead to an infeasible power flow problem when removed from the system is shown in the column ``Infeas./Dup.''. For each {\em feasible} line outage, five equally spaced samples are selected from the first half (that is, the first 12-hour period) of voltage variation data as training instances. Fifty samples are selected randomly from the second half of voltage variation data as test instances. The numbers of training and test instances are shown in the last two columns of the table.
\subsection{PMUs on All Buses}
\begin{table}\centering \caption{Line Outage Detection Accuracy on Test Set with PMUs on All Buses.}
\subfloat[Based on Probability of Correct Answer]{
\begin{tabular}{|c||c|c|c||c|c|c|}\hline
\multirow{2}{*}{System} & \multicolumn{3}{c||}{Using $X$} & \multicolumn{3}{c|}{Using $\ub{X}$} \\
\cline{2-7}
& $\ge0.9$ & $\ge0.7$ & $\ge0.5$ & $\ge0.9$ & $\ge0.7$ & $\ge0.5$
\\\hline\hline
14-Bus & 100\% & 100\% & 100\% & 100\% & 100\% & 100\% \\
30-Bus & 99.7\% & 99.7\% & 99.7\% & 100\% & 100\% & 100\% \\
57-Bus & 99.5\% & 99.7\% & 99.8\% & 99.5\% & 99.7\% & 99.8\% \\
118-Bus & 99.5\% & 99.5\% & 99.5\% & 99.8\% & 99.8\% & 99.8\% \\\hline
\end{tabular}\label{tbl:result.all_bus.prob}
}\\
\subfloat[Based on Ranking of Correct Answer]{
\begin{tabular}{|c||c|c|c||c|c|c|}\hline
\multirow{2}{*}{System} & \multicolumn{3}{c||}{Using $X$} & \multicolumn{3}{c|}{Using $\ub{X}$} \\
\cline{2-7}
& $1$ & $\le2$ & $\le3$ & $1$ & $\le2$ & $\le3$ \\\hline\hline
14-Bus & 100\% & 100\% & 100\% & 100\% & 100\% & 100\% \\
30-Bus & 99.7\% & 100\% & 100\% & 100\% & 100\% & 100\% \\
57-Bus & 99.8\% & 99.9\% & 99.9\% & 99.8\% & 100\% & 100\% \\
118-Bus & 99.5\% & 99.7\% & 99.7\% & 99.8\% & 99.9\% & 100\% \\\hline
\end{tabular}\label{tbl:result.all_bus.rank}
}\\
\begin{itemize} \item ``Probability'' indicates statistics for the probability
assigned by the MLR classifier to the actual outage
event.\\ \item ``Ranking'' indicates whether the actual event was ranked in
the top 1, 2, or 3 of probable outage events by the MLR
classifier. \end{itemize}
\end{table}
We present results for line outage detection when phasor measurement data from all buses is used.
The maximum likelihood estimation problem \eqref{eq:likelihood} with these observation vectors is solved by L-BFGS algorithm \cite{LiuN89}, coded in {\sc Matlab}. We measure performance of the identification procedure in two ways. The first measure is based on the probability assigned by the model to the actual line outage.
Table~\subref*{tbl:result.all_bus.prob} shows the accuracy of the classifiers according to this measure, for both the original phasor difference vector $X$ \eqref{eq:X} and the extended vector $\ub{X}$ \eqref{eq:Xbar}. Each column shows the percentage of testing samples for which the probability assigned to the correct outage exceeds $0.9$, $0.7$ and $0.5$, respectively. The result shows that the performance of line outage identification is very good, even for the original observation vector $X$.
For both of $X$ and $\ub{X}$, the accuracy of line outage identification based on probability $\ge 0.5$ is at least $99\%$.
The second measure is obtained by ranking the probabilities assigned to each line outage on the test datum, and score a positive mark if the correct outage is one of the top one, two, or three cases in the ranking. We see in Table~\subref*{tbl:result.all_bus.rank} that the actual case appears in the top two in almost every case.
\subsection{PMU Placement}\label{sec:result.PMU.placement}
In this subsection we only consider the extended observation vector $\ub{X}$ defined by \eqref{eq:Xbar}. We assume too that a PMU is installed on the reference bus, for purposes of maintaining consistency in phase angle measurement. We describe in some detail the performance of the proposed algorithm on the IEEE 57 Bus system, showing that line-outage identification performance when PMUs are placed judiciously almost matches performance in the fully-instrumented case. We then summarize our computational experience on 14, 30, 57, and 118-bus systems.
For our regularization schemes, we used groups ${\cal P}_s$, $s=1,2,\dotsc,N$, defined as in \eqref{eq:Ps}. The final two entries in the extended observation vectors (the average-generation and constant terms) are not included in any group.
\subsubsection{IEEE 57 Bus System} \begin{figure*}
\caption{Accuracy on Test Set of IEEE 57 Bus System for different values of $\tau$: Group-Sparse Heuristic}
\caption{Accuracy on Test Set of IEEE 57 Bus System for different values of $\tau$: Greedy Heuristic}
\label{fig:PMU.placement.GroupLASSO.1}
\label{fig:PMU.placement.GroupLASSO.3}
\label{fig:PMU.placement.GroupLASSO}
\label{fig:PMU.placement.greedy.2}
\label{fig:PMU.placement.greedy.3}
\label{fig:PMU.placement.greedy}
\end{figure*}
We describe here results obtained on the IEEE 57-bus system with two heuristics discussed in Section~\ref{sec:PMU.placement}: The GroupLASSO and Greedy Heuristics.
In Figure~\ref{fig:PMU.placement.GroupLASSO}, results for the GroupLASSO heuristic are displayed for different values of $\tau$. The $x$-axis indicates the number of PMUs selected by this heuristic. The $y$-axis indicates the number of test cases for which the true outage was classified by the heuristic. Each bar is color-coded according to the probability assigned to the true outage by the MLR classifier. Blue colors indicate that a high probability is assigned (that is, the outage was identified correctly) while dark red colors indicate that the probability assigned to the true outage scenario is less than $0.5$. For example, the second bar from the left in Figure~\subref*{fig:PMU.placement.GroupLASSO.3}, which corresponds to two PMUs, corresponds to the following distribution of probabilities assigned to the correct outage scenario, among the 3350 test instances.
\begin{center}\footnotesize \setlength{\tabcolsep}{.5em}
\begin{tabular}{|c||c|c|c|c|c|c|}\hline Probability & $[.9,1]$ & $[.8,.9]$ & $[.7,.8]$ & $[.6,.7]$ & $[.5,.6]$ & $[0,.5]$ \\\hline \# of Instances & 963 & 144 & 135 & 161 & 206 & 1741 \\\hline \end{tabular} \end{center} Note that the dark blue color occupies a fraction $963/3350$ of the bar, medium blue occupies $144/3350$, and so on.
When only one PMU is installed, that bus naturally serves as the angle reference, so no phase angle difference information is available, and identification cannot be performed. As expected, identification becomes more reliable as PMUs are installed on more buses. The value $\tau=.1$ (Figure \subref*{fig:PMU.placement.GroupLASSO.3}) appears to select locations better than the smaller choices of regularization parameter. For this value, about 10 buses are sufficient to assign a probability of greater than 90\% to the correct outage event for more than 90\% of the test cases, while near-perfect identification occurs when 30 PMUs are installed. Note that for $\tau=.1$, there is only slow marginal improvement after 10 buses; we see a similar pattern for the other values of $\tau$. The locations added after the initial selection are being chosen on the basis of information from the single regularized problem \eqref{eq:MLE.group.l1}, so locations added later may be providing only redundant information over locations selected earlier.
Figure~\ref{fig:PMU.placement.greedy} shows performance of the Greedy Heuristic, plotted in the same fashion as in Figure~\ref{fig:PMU.placement.GroupLASSO}. For each value of $\tau$, Algorithm~\ref{alg:PMU.placement.greedy} is performed with ${\cal R}^0=\emptyset$, with iterations continuing until there is no group $s\in {\cal S} \setminus {\cal R}^{l-1}$ such that $q_s(\beta^l)>0$. Termination occurs at 24, 16, and 11 PMU locations for the values $\tau=10^{-5}, 10^{-3}$, and $10^{-1}$, respectively. As the value of $\tau$ increases, the number of PMUs which are selected for line outage identification decreases. We can see by comparing Figures~\ref{fig:PMU.placement.GroupLASSO} and \ref{fig:PMU.placement.greedy} that classification performance improves more rapidly as new locations are added for the Greedy Heuristic than for the GroupLASSO Heuristic. Larger values of $\tau$ give slightly better results. We note (Figure~\subref*{fig:PMU.placement.greedy.3} that almost perfect identification occurs with only 16 PMU locations, while only 6 locations suffice to identify 90\% of outage events with high confidence.
Although we can manipulate the GroupLASSO technique to achieve sparsity equivalent to the Greedy Heuristic (by choosing a larger value of $\tau$), the PMUs selected by the latter give much better identification performance on this test set. In Table~\ref{tbl:Compare.GL.GR}, the parameter $\tau$ in the GroupLASSO heuristic is chosen manually, to find the solutions with 10 PMUs and 15 PMUs for the 57 Bus system. Performance is compared to that obtained from the Greedy Heuristic, with a much smaller value of $\tau$. Results for the Greedy Heuristic are clearly superior.
\begin{table*}\centering \caption{Comparison between GroupLASSO and Greedy Heuristic Selections on 57-Bus System} \label{tbl:Compare.GL.GR}
\begin{tabular}{|c|c|c|c||c|c|c||c|c|c|}\hline \# of & \multirow{2}{*}{Strategy} & \multirow{2}{*}{$\tau$}
& \multirow{2}{*}{PMU Locations}
& \multicolumn{3}{c||}{Probability} & \multicolumn{3}{c|}{Ranking} \\ \cline{5-10} PMUs & & & & $\ge0.9$ & $\ge0.7$ & $\ge0.5$ & $1$ & $\le2$ & $\le3$\\\hline\hline \multirow{2}{*}{10} & GroupLASSO & $1.1$
& $1^*$ 8 17 27 28 51 52 53 54 55
& 72.8\% & 73.1\% & 78.8\% & 78.9\% & 92.7\% & 95.7\% \\
\cline{2-10}
& Greedy & $1.2\times10^{-1}$
& $1^*$ 2 17 19 26 39 40 45 46 57
& 92.6\% & 92.7\% & 94.3\% & 94.3\% & 99.7\% & 99.9\% \\
\hline\hline \multirow{2}{*}{15} & GroupLASSO & $8.0\times10^{-1}$
& $1^*$ 2 4 17 23 27 28 43 46 47 51 52 53 54 55
& 82.8\% & 82.8\% & 88.3\% & 88.3\% & 95.7\% & 95.8\% \\
\cline{2-10}
& Greedy & $1.7\times10^{-3}$
& $1^*$ 2 5 12 17 20 21 26 39 40 43 45 46 54 57
& 98.3\% & 98.3\% & 98.3\% & 98.3\% & 100\% & 100\% \\
\hline \multicolumn{10}{r}{$^*$ indicates the reference bus.} \end{tabular} \vspace*{-7pt} \end{table*}
\subsubsection{Greedy Heuristic on 14, 30, 57 and 118 Bus System}
\begin{table*}\centering \caption{Line Outage Detection Test Set with PMUs on About $\sim25\%$
of Buses. \label{tbl:PMU.placement}}
\begin{tabular}{|c|c|c|c||c|c|c||c|c|c|}\hline \multirow{2}{*}{System} & \multirow{2}{*}{$\tau$} & \# of
& \multirow{2}{*}{PMU Locations}
& \multicolumn{3}{c||}{Probability} & \multicolumn{3}{c|}{Ranking} \\ \cline{5-10}
& & PMUs & & $\ge0.9$ & $\ge0.7$ & $\ge0.5$ & $1$ & $\le2$ & $\le3$\\\hline\hline \multirow{2}{*}{ 14-Bus} & $5\times10^{-2}$ & 3
& $1^*$ 7 12
& 99.6\% & 99.7\% & 99.8\% & 99.8\% & 100\% & 100\% \\
\cline{2-10}
& $5\times10^{-3}$ & 3
& $1^*$ 11 12
& 100\% & 100\% & 100\% & 100\% & 100\% & 100\% \\
\hline\hline \multirow{2}{*}{ 30-Bus} & $5\times10^{-2}$ & 4
& $1^*$ 3 23 30
& 99.6\% & 99.6\% & 99.6\% & 99.6\% & 100\% & 100\% \\
\cline{2-10}
& $5\times10^{-3}$ & 5
& $1^*$ 3 14 22 29
& 100\% & 100\% & 100\% & 100\% & 100\% & 100\% \\
\hline\hline \multirow{2}{*}{ 57-Bus} & $5\times10^{-2}$ & 12
& $1^*$ 2 5 17 21 26 39 40 45 46 54 57
& 97.1\% & 97.1\% & 97.1\% & 97.1\% & 99.8\% & 99.8\% \\
\cline{2-10}
& $5\times10^{-3}$ & 14
& $1^*$ 2 5 17 20 21 26 39 40 41 45 46 54 57
& 98.5\% & 98.5\% & 98.5\% & 98.5\% & 99.9\% & 99.9\% \\
\hline\hline \multirow{3}{*}{118-Bus} & $5\times10^{-2}$ & 15
& 2 22 29 36 48 58 62 63 $69^*$ 81 91 95 106 108 115
& 94.2\% & 94.2\% & 94.2\% & 94.2\% & 96.2\% & 96.3\% \\
\cline{2-10}
& $5\times10^{-3}$ & 21
& \tcell{3 13 29 35 43 47 55 58 62 63 $69^*$\\ 75 81 82 91 93 104 106 107 113 115 119}
& 99.3\% & 99.5\% & 99.6\% & 99.6\% & 99.9\% & 99.9\% \\
\hline \multicolumn{10}{r}{$^*$ indicates the reference bus.} \end{tabular} \vspace*{-7pt} \end{table*}
We applied the Greedy Heuristic to 14, 30, 57 and 118 Bus Systems with two values of $\tau=5\times10^{-2}$ and $\tau=5\times10^{-3}$), and found that the phasor measurements from the small set of buses are enough to have the similar line outage identification performance to the full measurement cases. Table~\ref{tbl:PMU.placement} shows the PMU locations selected for each case, and line outage identification performance. Identification performance is hardly degraded from the fully instrumented case, even when phasor measurements are available from only about $~25\%$ of buses.
\section{Conclusions}\label{sec:conclusion}
A novel approach to identify single line outage using MLR model is proposed in this paper. The model employs historical load demand data to train a multiclass logistic regression classifier, then uses the classifier to identify outages in real time from streaming PMU data. Numerical results obtained on IEEE 14, 30, 57 and 118 bus systems prove that the approach can identify outages reliably.
With this line outage identification framework, the optimal placement of PMU devices to identify the line outage is also discussed. Heuristics are proposed to decide which buses should be instrumented with PMUs. Experimental results show that detection is almost as good when just 25\% of buses are instrumented with PMUs as when PMUs are attached to all buses.
\section*{Acknowledgment} The authors thank Professor Chris DeMarco and Mr. Jong-Min Lim for allowing the use of their synthetic 24-hour electric power demand data sets in our experiments, and for valuable discussions and guidance on this project.
\appendices \pagenumbering{roman}
\section{Visualizing the Solution of the PMU Placement Problems} The location of PMUs for the IEEE 30-Bus and IEEE 57-Bus Systems (from Table~\ref{tbl:PMU.placement}) are displayed in Figure \ref{fig:PMU.location}, with instrumented buses indicated by red circles.
\begin{figure}
\caption{PMU Locations for IEEE 30 Bus and IEEE 57 Bus Systems.
(System diagrams are taken from \cite{AllL08,GasA09})}
\label{fig:PMU.location}
\end{figure}
\section{Extension: Use of Explicit Line Outage
Information}\label{sec:explicit.info}
We have assumed so far that only voltage angle and magnitude data from PMUs is used in detecting line outages. In fact, PMUs provide other information that is highly relevant for this purpose. For example, when the PMU measures current of a particular line (incident on a bus) it can detect immediately when an outage occurs on that line; we do not need to rely in the indirect evidence of voltage changes at the other PMUs. Another factor to consider is that when a decision is made to install a PMU at a particular bus, it is conventional to measure {\em
all lines} that are incident on that bus, as the marginal cost of doing so is minimal. Although the phasor measurements are the same at all PMUs near a single bus, each of these PMUs provides direct information about the lines to which they are attached. Thus, if we choose to equip a particular bus with PMUs, we can immediately detect outages on all lines that touch that bus. In particular, if we install PMUs on {\em
all} buses in the system, we have direct monitoring of all lines, and the outage detection problem becomes trivial.
We can extend the multiclass logistic regression technique to make effective use of these direct observations in choosing optimal buses for PMU placement. The key modification is to extend each feature vector $\beta_k$ to include additional entries that indicate the buses that touch line $k$. The observation vectors and the groups ${\cal P}_s$, $s=1,2,\dotsc,N$ are extended correspondingly.
For each line $k=1,2,\dotsc,K$, let us define the following quantities: \[
{\cal T}_k := \set{t_k^1, t_k^2} \subset \set{1,2,\dotsc,N } \] where $t_k^1$ and $t_k^2$ are indices of buses touched by line $k$.
We extend each observation vector $\ub{X}$ by appending $2K$ additional elements to form $\ub{\ub{X}}$, where each such vector has the form \begin{equation}\label{eq:Xbarbar} \ub{\ub{X}}:=\bm{\ub{X} \\ {L_k}} \end{equation} for some $k=1,2,\dotsc,K$, where $L_k$ is the $k$th column of the $2K\times K$ matrix $L$ defined as follows, for some $\eta>0$: \begin{equation} \label{def:L}
L := \bm{ \eta & 0 & \cdots & 0 \\
\eta & 0 & \cdots & 0 \\
0 & \eta & \cdots & 0 \\
0 & \eta & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & \eta \\
0 & 0 & \cdots & \eta }. \end{equation} The two nonzero entries in each column $L_k$ indicate which two buses can detect outage of line $k$ directly. In other words, if line $k$ fails, we flag the PMUs on the buses that touch that line with a value $\eta$, since a fault on line $k$ is immediately detectable from the buses $t^1_k$ and $t^2_k$. We can say that the first part of the combined observation vector $\ub{\ub{X}}$ contains {\em indirect} (voltage phasor, $\ub{X}$) observations while the second part contains {\em direct} (line outage, $L_k$) observations.
We need to extend too the definition \eqref{eq:Ps} of the groups ${\cal P}_s$, $s=1,2,\dotsc,N$. We now distribute the additional $2K$ entries in the feature vector to these groups. The additional entries associated with bus $s$ are those in the index set ${\cal D}_s$ defined as follows: \begin{align*}
{\cal D}_s & = \setc{2(k-1)+i}{t_k^i=s \;\; \mbox{for $k=1,2,\cdots,K$, $i=1,2$}}. \end{align*} For the combined observation vector $\ub{\ub{X}}$, we define groups ${\cal P}_s'$, $s=1,2,\cdots,N$ : \[
{\cal P}_s' = {\cal P}_s \cup \setc{2N+2+d}{d\in{\cal D}_s}\;\;\mbox{for $s=1,2,\cdots,N$}, \]
If PMUs are installed on every line, direct observations will identify each outage perfectly, so the solution of the maximum likelihood problem is rather trivial. In approximate solutions to the problem, the weight vector $\beta_k$ for line $k$ will have large positive entries in positions $2N+2+i$ for which $L_{ik}=\eta$, and large negative entries in positions $2N+2+i$ for which $L_{ik}=0$. This would yield $\beta_k^T\ub{\ub{X}}$ large and positive for observation vectors $\ub{\ub{X}}$ that indicate a line-$k$ outage, with $\beta_k^T\ub{\ub{X}}$ large and negative if there is no outage on line $k$, leading to assigned probabilities close to $1$ and $0$, respectively. (Entries in $\beta_k$ corresponding to the indirect observations may also have meaningfully large values, but these are less significant in the completely observed case.)
When PMUs are installed on a subset of buses, outages on some lines will be observed only {\em indirectly}, so the indirect observations in components $i=1,2,\dotsc,2N+2$ of the vector $\ub{\ub{X}}$ are critical to identification performance on those lines that are not directly observed.
We incorporate direct observations into our outage identification strategy in the following ways. \begin{itemize}
\item Indirect. Direct observations are ignored. We use only the
observation vector $\ub{X}$, as in Section~\ref{sec:result}.
\item Combined (Direct+Indirect). Direct observations are
incorporated into the observation vector, and we so MLR
classification with the vectors $\ub{\ub{X}}$.
\item Prescreening. Instead of including the direct observation in
the observation vector, line outages that can be identified by the
direct observation are screened out {\em before} the MLR is
applied. The number of outcomes in MLR is reduced since we do not
need to consider the line outages identified already by the direct
observation. Observation vectors $\ub{X}$ are used to train the
MLR for those line outages that are not observed directly.
\item Postscreening. First, we train an MLR classifier using only the
indirect observations in vector $\ub{X}$. Then, during
classification, we override the prediction result from the MLR
when a direct observation is available for the line in question.
Note that results from this strategy cannot be worse than results
for the Indirect strategy. \end{itemize}
We also compare solutions of the PMU placement problem using the Indirect strategy (as in Section~\ref{sec:result}) and the Combined strategy. We solve these two variants of the placement problem for the 57-bus case with the Greedy Heuristic of Algorithm~\ref{alg:PMU.placement.greedy}, setting $\tau=10^{-2}$ and the number of PMUs $r$ to the values $5$ and $10$. We note that the reference bus is always selected as one of the PMU locations, and it is used only to provide the phase angle reference for all the strategies above, as in the Indirect case. (In using the reference bus PMU in this restricted way, we allow a fairer comparison between the Indirect strategy and the strategies that use direct observations.)
\begin{table*}\centering \caption{Use of Explicit Line Outage Information
($\tau=10^{-2}$, $^*$ indicates the reference bus.)} \label{tbl:ext.DO} \subfloat[PMU Placement Based on Indirect Observations]{
\label{tbl:ext.DO.ind}
\shortstack[r]{
\begin{tabular}{|c||c|c|c||c|c|c||c|c|c||c|c|c|}\hline
\multirow{3}{*}{Strategy}
& \multicolumn{6}{c||}{$1^*$ 5 20 21 57 (5 PMUs)}
& \multicolumn{6}{c|}{$1^*$ 5 20 21 26 39 40 43 54 57 (10 PMUs)}\\
\cline{2-13}
& \multicolumn{3}{c||}{Probability} & \multicolumn{3}{c||}{Ranking}
& \multicolumn{3}{c||}{Probability} & \multicolumn{3}{c|}{Ranking}\\
\cline{2-13}
& $\ge0.9$ & $\ge0.7$ & $\ge0.5$ & $1$ & $\le2$ & $\le3$
& $\ge0.9$ & $\ge0.7$ & $\ge0.5$ & $1$ & $\le2$ & $\le3$\\\hline\hline
Indirect & 83.1\% & 83.2\% & 88.0\% & 88.0\% & 99.4\% & 99.8\%
& 94.1\% & 94.1\% & 94.1\% & 94.1\% & 99.1\% & 99.1\% \\
Combined & 83.0\% & 83.8\% & 86.4\% & 86.4\% & 97.5\% & 99.4\%
& 95.4\% & 95.4\% & 95.4\% & 95.4\% & 98.8\% & 98.8\% \\
Prescreening & 84.8\% & 85.1\% & 88.1\% & 88.1\% & 98.4\% & 99.6\%
& 95.2\% & 95.2\% & 95.3\% & 95.3\% & 99.8\% & 99.9\% \\
Postscreening & 86.1\% & 86.2\% & 89.5\% & 89.5\% & 99.4\% & 99.8\%
& 94.2\% & 94.2\% & 94.2\% & 94.2\% & 99.1\% & 99.1\% \\
\hline
\end{tabular}\\
\begin{tabular}{|c|ccccccccc|c|}\hline
Selected Bus & 5 & 20 & 21 & 26 & 39 & 40 & 43 & 54 & 57 & Total \\\hline
\# of Lines Touching the Bus & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 18 \\\hline
\end{tabular}} }\\ \subfloat[PMU Placement Based on Combined (Direct + Indirect) Observations ($\eta=1$)]{
\label{tbl:ext.DO.dir.1}
\shortstack[r]{
\begin{tabular}{|c||c|c|c||c|c|c||c|c|c||c|c|c|}\hline
\multirow{3}{*}{Strategy}
& \multicolumn{6}{c||}{$1^*$ 6 9 12 56 (5 PMUs)}
& \multicolumn{6}{c|}{$1^*$ 6 9 12 15 22 39 49 54 56 (10 PMUs)}\\
\cline{2-13}
& \multicolumn{3}{c||}{Probability} & \multicolumn{3}{c||}{Ranking}
& \multicolumn{3}{c||}{Probability} & \multicolumn{3}{c|}{Ranking}\\
\cline{2-13}
& $\ge0.9$ & $\ge0.7$ & $\ge0.5$ & $1$ & $\le2$ & $\le3$
& $\ge0.9$ & $\ge0.7$ & $\ge0.5$ & $1$ & $\le2$ & $\le3$\\\hline\hline
Indirect & 77.2\% & 77.4\% & 83.1\% & 83.2\% & 97.4\% & 99.0\%
& 83.7\% & 83.7\% & 87.5\% & 87.5\% & 93.5\% & 93.5\% \\
Combined & 83.1\% & 83.1\% & 87.4\% & 87.4\% & 92.7\% & 92.8\%
& 93.1\% & 93.1\% & 94.7\% & 94.7\% & 98.4\% & 98.5\% \\
Prescreening & 83.3\% & 83.8\% & 88.2\% & 88.2\% & 93.4\% & 93.4\%
& 93.8\% & 93.8\% & 95.2\% & 95.2\% & 98.4\% & 98.4\% \\
Postscreening & 82.7\% & 82.8\% & 87.2\% & 87.2\% & 98.5\% & 99.6\%
& 93.7\% & 93.7\% & 95.1\% & 95.1\% & 97.4\% & 97.4\% \\
\hline
\end{tabular}\\
\begin{tabular}{|c|ccccccccc|c|}\hline
Selected Bus & 6 & 9 & 12 & 15 & 22 & 39 & 49 & 54 & 56 & Total \\\hline
\# of Lines Touching the Bus & 4 & 6 & 5 & 5 & 3 & 2 & 4 & 2 & 4 & 35 \\\hline
\end{tabular}} }\\ \subfloat[PMU Placement Based on Combined (Direct + Indirect) Observations ($\eta=10^{-2}$)]{
\label{tbl:ext.DO.dir.2}
\shortstack[r]{
\begin{tabular}{|c||c|c|c||c|c|c||c|c|c||c|c|c|}\hline
\multirow{3}{*}{Strategy}
& \multicolumn{6}{c||}{$1^*$ 5 9 49 56 (5 PMUs)}
& \multicolumn{6}{c|}{$1^*$ 5 9 21 26 39 45 46 49 56 (10 PMUs)}\\
\cline{2-13}
& \multicolumn{3}{c||}{Probability} & \multicolumn{3}{c||}{Ranking}
& \multicolumn{3}{c||}{Probability} & \multicolumn{3}{c|}{Ranking}\\
\cline{2-13}
& $\ge0.9$ & $\ge0.7$ & $\ge0.5$ & $1$ & $\le2$ & $\le3$
& $\ge0.9$ & $\ge0.7$ & $\ge0.5$ & $1$ & $\le2$ & $\le3$\\\hline\hline
Indirect & 79.0\% & 79.3\% & 84.3\% & 84.5\% & 98.9\% & 99.8\%
& 93.3\% & 93.3\% & 94.7\% & 94.7\% & 99.4\% & 99.9\% \\
Combined & 85.4\% & 85.4\% & 90.0\% & 90.0\% & 95.5\% & 95.6\%
& 97.4\% & 97.4\% & 97.4\% & 97.4\% & 99.9\% & 99.9\% \\
Pre-Screening & 84.5\% & 84.5\% & 88.5\% & 88.5\% & 98.4\% & 98.9\%
& 96.1\% & 96.1\% & 96.1\% & 96.1\% & 97.9\% & 97.9\% \\
Post-Screening & 85.1\% & 85.4\% & 89.5\% & 89.7\% & 99.7\% & 100\%
& 98.0\% & 98.0\% & 98.0\% & 98.0\% & 100\% & 100\% \\
\hline
\end{tabular}\\
\begin{tabular}{|c|ccccccccc|c|}\hline
Selected Bus & 5 & 9 & 21 & 26 & 39 & 45 & 46 & 49 & 56 & Total \\\hline
\# of Lines Touching the Bus & 2 & 6 & 2 & 2 & 2 & 2 & 2 & 4 & 4 & 26 \\\hline
\end{tabular}} }
\end{table*}
Experimental results using PMU placements based on Indirect and Combined observations, and using each of the four classification strategies described above, are shown in Table~\ref{tbl:ext.DO}, using a similar format to Tables~\ref{tbl:Compare.GL.GR} and \ref{tbl:PMU.placement}. When the PMU locations are selected using only indirect observations (Table~\subref*{tbl:ext.DO.ind}), the advantage of using direct line outage information during classification is not significant, especially when the larger number of $10$ PMUs is installed. This observation is not too surprising. The biggest voltage phasor changes are produced by outages that are close to a bus, so even when a line outage is not detected by direct observation, it can usually be detected reliably by its ``indirect'' effect on nearby buses.
In Tables~\subref*{tbl:ext.DO.dir.1} and \subref*{tbl:ext.DO.dir.2}, the PMU locations are selected on the basis of the combined vectors. When only indirect data is used during classification, results are much worse, as the locations have been chosen under the assumption that direct observation data will be available. In fact, the results in Table~\subref*{tbl:ext.DO.dir.1} are generally slightly worse than those of Table~\subref*{tbl:ext.DO.ind}. This is again because too much reliance is placed on direct observation in selecting PMU locations, and detection power is diminished slightly for those outages that are detectable only indirectly. Note that the PMU locations in Table~\subref*{tbl:ext.DO.dir.1} are essentially those with the greatest numbers of lines connected: a total of 35 in Table~\subref*{tbl:ext.DO.dir.1} (for $r=10$), as compared with 18 in Table~\subref*{tbl:ext.DO.ind}.
To reduce the weight placed on direct information in PMU placement, we scale down the values $\eta$ in \eqref{def:L}. Reducing $\eta$ from $1$ to $10^{-2}$ appears to strike a better balance between the use of direct and indirect information. Table~\subref*{tbl:ext.DO.dir.2} shows a marked improvement over Table~\subref*{tbl:ext.DO.ind} (which weights the direct observations more heavily), and slight improvements by most measures over Table~\subref*{tbl:ext.DO.dir.1}, which uses only indirect information. The total number of lines that are directly connected to buses with PMUs (and which can thus be observed directly) in Table~\subref*{tbl:ext.DO.dir.1} is about halfway between the corresponding statistics in Tables~\subref*{tbl:ext.DO.ind} and \subref*{tbl:ext.DO.dir.2}.
\section{Implementation of SpaRSA{}}
We solve the regularized convex optimization problem \eqref{eq:MLE.p} with the SpaRSA{} algorithm \cite{WriNF08}, a simple first-order approach that exploits the structure. We briefly describe the approach here, referring to \cite{WriNF08} for further details.
The SpaRSA{} subproblem of \eqref{eq:MLE.p} at iteration $n$ is defined as follows, for some scalar parameter $\alpha_n\in \mathbb{R}^+$: \begin{equation}\label{eq:SpaRSA.sub}
\beta^{n+1} := \oargmax{\beta}
\frac{1}{2}\norm{\beta-\gamma^n}_{F}^2
+ \tau\frac{1}{\alpha_n}w(\beta)
\end{equation} where $\|\cdot\|_F$ is the Frobenius norm of a matrix, $\gamma^n := \bm{\gamma_1^n & \gamma_2^n & \dotsc & \gamma_K^n}$ with \[
\gamma_k^n := \beta_k^n - \frac{1}{\alpha_n}\nabla_{\beta_k}f(\beta), \] and \[
\nabla_{\beta_i}f(\beta) =
\sum_{p:i=y_p}x_p - \sum_{j=1}^M\frac{x_je^{\ip{\beta_i}{x_j}}}
{\sum_{k=1}^Ke^{\ip{\beta_k}{x_j}}}. \] When no regularization term is present ($w(\beta)=0$), the solution for \eqref{eq:SpaRSA.sub} is $\beta^{n+1}=\gamma^n$, so the approach reduces to the steepest descent algorithm on $f$ with step length $1/\alpha_n$.
In the PMU placement problem, our regularizer $w_{\cal S}(\beta)$ is group-separable. Thus the subproblem \eqref{eq:SpaRSA.sub} can be divided into independent problems of the form \[
[\beta^{n+1}]_{{\cal P}_s} := \oargmax{\hat{\beta}}
\frac{1}{2}\norm{[\beta]_{{\cal P}_s}-[\gamma^n]_{{\cal P}_s}}_2^2
+ \tau\frac{1}{\alpha_n}q_s(\beta), \] for all $s \in {\cal S}$, where (as defined above), $[A]_{{\cal P}_s}$ is the submatrix of $A$ consisting of the rows whose indices are in ${{\cal P}_s}$. Since the penalty function $q_s(\beta)$ is the $\ell_2$-norm, this subproblem has a closed form solution \cite{WriNF08}, as follows: \[
[\beta^{n+1}]_{{\cal P}_s} = [\gamma^n]_{{\cal P}_s}
\frac{\max\set{\norm{[\gamma^n]_{{\cal P}_s}}_2-\tau\alpha_n^{-1},0}}
{\max\set{\norm{[\gamma^n]_{{\cal P}_s}}_2-\tau\alpha_n^{-1},0}+\tau\alpha_n^{-1}}. \] For any row $i$ of $\beta$ that does not belong to any ${\cal P}_s$, we have simply $[\beta^{n+1}]_i=[\gamma^n]_i$. Different strategies can be used to choose $\alpha_n$. We increase $\alpha_n$ at each iteration until sufficient decrease is obtained in the objective, terminating when $\alpha_n$ grows too large (indicating that a solution is nearby).
\end{document} | arXiv |
\begin{document}
\theoremstyle{plain} \newtheorem{thm}{Theorem}[section] \newtheorem*{thmnonumber}{Theorem} \newtheorem{lemma}[thm]{Lemma} \newtheorem{prop}[thm]{Proposition} \newtheorem{cor}[thm]{Corollary} \newtheorem{open}[thm]{Open Problem}
\theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{asmp}{Assumption} \newtheorem{notn}{Notation} \newtheorem{prb}{Problem}
\theoremstyle{remark} \newtheorem{rmk}{Remark} \newtheorem{exm}{Example} \newtheorem{clm}{Claim}
\author{Cameron Bruggeman and Andrey Sarantsev}
\title[Multiple Collisions in Systems of Competing Brownian Particles]{Multiple Collisions in Systems\\ of Competing Brownian Particles}
\address{Department of Mathematics, Columbia University}
\email{[email protected]}
\address{Department of Statistics and Applied Probability, University of California, Santa Barbara}
\email{[email protected]}
\date{May 23, 2016. Version 32}
\keywords{Reflected Brownian motion, competing Brownian particles, asymmetric collisions, named particles, ranked particles, triple collisions, multiple collisions, Skorohod problem, positive orthant, squared Bessel process, stochastic comparison}
\subjclass[2010]{Primary 60K35, secondary 60J60, 60J65, 91B26}
\begin{abstract} Consider a finite system of competing Brownian particles on the real line. Each particle moves as a Brownian motion, with drift and diffusion coefficients depending only on its current rank relative to the other particles. We find a sufficient condition for a.s. absence of a total collision (when all particles collide) and of other types of collisions, say of the three lowest-ranked particles. This continues the work of Ichiba, Karatzas, Shkolnikov (2013) and Sarantsev (2015). \end{abstract}
\maketitle
\section{Introduction}
\subsection{A brief preview of results}
We start by describing the concept of {\it competing Brownian particles} informally. A formal definition is postponed until the next section.
Consider $N$ Brownian particles on the real line. Suppose the particle which is currently the $k$th leftmost one (we say: {\it has rank $k$}), moves as a Brownian motion with drift coefficient $g_k$ and diffusion coefficient $\sigma_k^2$. In other words, the behavior of a particle depends on its current rank relative to other particles. This is called a {\it system of competing Brownian particles}.
A caveat: if two or more particles occupy the same position at the same time, how do we {\it resolve ties}, that is, assign ranks to these particles? We can use the following ``lexicographic" rule: particles $X_i$ with smaller indices $i$ get smaller ranks.
Let $X_k = (X_k(t), t \ge 0),\ k = 1, \ldots, N$, be these particles. Let $W_1, \ldots, W_N$ be i.i.d. Brownian motions. Then the particles $X_1, \ldots, X_N$ are governed by the following SDE: \begin{equation} \label{eq:SDE-CBP} \mathrm{d} X_i(t) = \sum\limits_{k=1}^N1\left(X_i\ \mbox{has rank}\ k\ \mbox{at time}\ t\right)\left(g_k\mathrm{d} t + \sigma_k\mathrm{d} W_i(t)\right). \end{equation} Let $Y_k(t)$ be the one of these $N$ particles which has rank $k$ at time $t$. The processes $X_i,\ i = 1, \ldots, N$, are called {\it named particles}, and $Y_k,\ k = 1, \ldots, N$, are called {\it ranked particles}. If $X_i(t) = Y_k(t)$, we say that the corresponding particle at time $t$ has {\it name} $i$ and {\it rank} $k$. By definition, the ranked particles satisfy \begin{equation} \label{154} Y_1(t) \le Y_2(t) \le \ldots \le Y_N(t),\ \ t \ge 0. \end{equation} Weak existence and uniqueness in law for these systems were proved in \cite{Bass1987}. Some motivation for studying these systems is provided later in the Introduction.
This article is devoted to {\it collisions of competing Brownian particles}. Let us exhibit some results proved in this paper; they are corollaries of general theorems from Section 2. Suppose, for the sake of simplicity, that we have $N = 4$ competing Brownian particles. We shall present some results, and explain how they are related to the paper \cite{MyOwn3}.
\begin{thm} \label{example1} If the following conditions $$ \begin{cases} 9\sigma_1^2 \le 7\sigma_2^2 + 7\sigma_3^2 + 7\sigma_4^2;\\ 3\sigma_1^2 \le 5\sigma_2^2 + \sigma_3^2 + \sigma_4^2;\\ 3\sigma_1^2 + 3\sigma_4^2 \le 5\sigma_2^2 + 5\sigma_3^2;\\ 3\sigma_4^2 \le \sigma_1^2 + \sigma_2^2 + 5\sigma_3^2;\\ 9\sigma_4^2 \le 7\sigma_1^2 + 7\sigma_2^2 + 7\sigma_3^2, \end{cases} $$ hold, then a.s. there does not exist $t > 0$ such that \begin{equation} \label{type1} Y_1(t) = Y_2(t) = Y_3(t) = Y_4(t). \end{equation} Moreover, a.s. there does not exist $t > 0$ such that \begin{equation} \label{type2} Y_1(t) = Y_2(t)\ \ \mbox{and}\ \ Y_3(t) = Y_4(t). \end{equation} \end{thm}
\begin{thm} \label{example3} If the five inequalities from Theorem~\ref{example1} together with $$ \sigma_2^2 \ge \frac12\left(\sigma_1^2 + \sigma_3^2\right) $$ hold, then a.s. there does not exist $t > 0$ such that \begin{equation} \label{type3} Y_1(t) = Y_2(t) = Y_3(t). \end{equation} \end{thm}
Similar statements (but with other inequalities involving $\sigma_k^2,\ k = 1, \ldots, N$) can be stated for any $N = 5, 6, \ldots$, and for any type of collision between $Y_1, \ldots, Y_N$. We also have the following statement for $N = 4$; however, in this case we did not find any specific generalizations for this result in cases $N \ge 5$.
\begin{thm} \label{cams} With $N = 4$, if the following condition holds: \begin{equation} \label{Bruggeman} \sigma_1^2 + \sigma_4^2 \le \sigma_2^2 +\sigma_3^2, \end{equation} then a.s. there are no $t > 0$ such that~\eqref{type1} and~\eqref{type2} hold. \end{thm}
\subsection{Relation to previous results from the paper \cite{MyOwn3}} The results of this paper complement the main result of the companion paper \cite{MyOwn3}. Let us discuss the relation between these two papers.
\begin{defn} A {\it triple collision at time $t$} occurs if there exists a rank $k = 2, \ldots, N-1$ such that $Y_{k-1}(t) = Y_{k}(t) = Y_{k+1}(t)$. A {\it simultaneous collision} at time $t$ occurs if there are ranks $k \ne l$ such that such that $Y_{k}(t) = Y_{k+1}(t),\ Y_{l}(t) = Y_{l+1}(t)$. \label{triplesimdef} \end{defn}
Note that a triple collision is a particular case of a simultaneous collision. One motivation for studying triple collisions is that a strong solution to SDE~\eqref{classicSDE} is known to exist and be unique up to the first moment of a triple collision: this was proved in \cite{IKS2013}. The question of whether a classical system of competing Brownian particles a.s. avoids triple collisions was studied in \cite{IK2010, IKS2013}, with significant partial results obtained. In our companion paper \cite{MyOwn3}, the following necessary and sufficient condition was found.
\begin{prop} \label{elegant} A system of $N$ competing Brownian particles has a.s. no triple collisions and no simultaneous collisions at any time $t > 0$, if and only if the sequence $(\sigma_1^2, \ldots, \sigma_N^2)$ is concave, that is, \begin{equation} \label{concave} \frac12\left(\sigma_{k-1}^2 + \sigma_{k+1}^2\right) \le \sigma_k^2,\ \ k = 2, \ldots, N-1. \end{equation} If the condition~\eqref{concave} is violated for some $k = 2, \ldots, N-1$, then with positive probability there exists $t > 0$ such that $Y_{k-1}(t) = Y_k(t) = Y_{k+1}(t)$. \end{prop}
An interesting corollary: {\it If there are a.s. no triple collisions at any time $t > 0$, then there are a.s. no simultaneous collisions at any time $t > 0$. }
We call the condition~\eqref{concave} {\it global concavity}, as opposed to {\it local concavity at rank $j$}, with just one inequality: $$ \frac12\left(\sigma_{j-1}^2 + \sigma_{j+1}^2\right) \le \sigma_j^2. $$ Thus, if there is no local concavity at $k$, then with positive probability there is a triple collision between $Y_{k-1}$, $Y_k$ and $Y_{k+1}$. However, we do not know whether the converse is true: if there is local concavity at $k$, then there are a.s. no triple collisions between $Y_{k-1}$, $Y_k$, $Y_{k+1}$.
Proposition~\ref{elegant} is a condition to avoid {\it all} possible triple collisions. If we are interested in avoiding only an individual triple collision, such as $$ Y_1(t) = Y_2(t) = Y_3(t), $$ we can get another sufficient condition for this: see Theorem~\ref{example3} above. This condition is not stronger than~\eqref{concave}: we can find diffusion parameters, say $$ \sigma_1^2 = \sigma_2^2 = \sigma_4^2 = 1,\ \sigma_3^2 = 0.9, $$ which satisfy the conditions in Theorem~\ref{example3}, but do not satisfy~\eqref{concave}. In this case, there are no triple collisions of the type $$ Y_1(t) = Y_2(t) = Y_3(t), $$ and no simultaneous collisions of the type $$ Y_1(t) = Y_2(t),\ \ Y_3(t) = Y_4(t), $$ but with positive probability there are triple collisions of the type $$ Y_2(t) = Y_3(t) = Y_4(t). $$ We can take $$ \sigma_1^2 = \sigma_4^2 = 1,\ \sigma_2^2 = \sigma_3^2 = 0.9. $$ Then local concavity fails at ranks $2$ and $3$. Therefore, with positive probability there exists a triple collision between ranked particles $Y_1$, $Y_2$, and $Y_3$, and with positive probability there exists a triple collision between ranked particles $Y_2$, $Y_3$, and $Y_4$. However, the five inequalities~\eqref{CP4I} are satisfied. Therefore, there are no simultaneous collisions of the type $$ Y_1(t) = Y_2(t),\ Y_3(t) = Y_4(t). $$ It was noted in \cite[Corollary 1.3]{MyOwn3} that if there are a.s. no triple collisions, then there are a.s. no simultaneous collisions. As we see in this example, it is possible to find diffusion coefficients so that the system avoids simultaneous collisions of the type~\eqref{type2}, but exhibits triple collisions with positive probability.
Also, the collision as in~\eqref{type1} is stronger than a triple or a simultaneous collision.
\subsection{Outline of the proofs}
The main results of this paper are Theorems~\ref{totalcor} and~\ref{mainthm}. Theorems~\ref{example1} and~\ref{example3}, along with other examples in Section 2, are corollaries of these two results. Theorems~\ref{totalcor} and~\ref{mainthm} are proved in Sections 3 and 4. Let us give a brief outline of the proofs.
Consider the gaps between the consecutive ranked particles: $$ Z_1(t) = Y_2(t) - Y_1(t), \ldots, Z_{N-1}(t) = Y_N(t) - Y_{N-1}(t), \quad 0 \le t <\infty. $$ These form an $(N-1)$-dimensional process in $\mathbb{R}^{N-1}_+$, which is called the {\it gap process} and is denoted by $Z = (Z(t), t \ge 0)$. It turns out that $Z$ is a particular case of a well-known process, which is called a {\it semimartingale reflected Brownian motion (SRBM)} in a positive multidimensional orthant. We discuss this relationship in subsection 4.2.
Let us informally describe the concept of an SRBM; a formal definition is given in subsection 3.1. Fix the dimension $d \ge 1$, and let $\mathbb{R}_+ := [0, \infty)$ be the positive half-axis. Let $S = \mathbb{R}^d_+$ be the $d$-dimensional positive orthant. Fix a {\it drift vector} $\mu \in \mathbb{R}^d$, a $d\times d$ {\it reflection matrix} $R$ and another $d\times d$ {\it covariance matrix} $A$. An {\it SRBM in the orthant $S$} with these parameters $R, \mu, A$ is a Markov process which:
(i) behaves as a $d$-dimensional Brownian motion with drift vector $\mu$ and covariance matrix $A$ in the interior of the orthant $S$;
(ii) at each face $S_i := \{x \in S\mid x_i = 0\}$, $i = 1, \ldots, d$, this process is reflected in the direction of $r_i$, which is the $i$th column of $R$.
If $r_i = e_i$, where $e_i$ is the $i$th vector from the standard basis in $\mathbb{R}^d$, then this reflection is called {\it normal}; otherwise, it is called {\it oblique}. This process is denoted by $\SRBM^d(R, \mu, A)$. The survey \cite{Wil1995} provides a good overview of this process. More information and citations concerning an SRBM are provided in subsection 3.1.
The parameters $R$, $\mu$ and $A$ of the SRBM which is the gap process depend on $g_n, \sigma_n,\ n = 1, \ldots, N$, see subsection 4.2, equations ~\eqref{R12}, ~\eqref{A} and~\eqref{mu}.
Let us return to the examples above. Consider a system of $N = 4$ competing Brownian particles. A collision of the type~\eqref{type1} is equivalent to $Z_1(t) = Z_2(t) = Z_3(t) = 0$, that is, to the process $Z$ hitting the corner of the orthant $\mathbb{R}^3_+$ at time $t$. A collision of the type~\eqref{type2} is equivalent to $Z_1(t) = Z_3(t) = 0$, that is, to the process of gaps $Z$ hitting the {\it edge} $\{x \in \mathbb{R}^3_+\mid x_1 = x_3 = 0\}$ of the boundary $\partial \mathbb{R}^3_+$ of the orthant $\mathbb{R}^3_+$. Similarly, we can rewrite other types of collisions in terms of the gap process.
In Section 3, we obtain results concerning an SRBM a.s.$\,$avoiding corners or edges. In Theorem~\ref{cornerthm}, we find a sufficient condition for an $\SRBM^d(R, \mu, A)$ to a.s. avoid the corner of the orthant $S = \mathbb{R}^d_+$. In Theorem~\ref{corner2edge}, we find a sufficient condition for an $\SRBM^d(R, \mu, A)$ to a.s. avoid the ``edge" $$ S_I := \{x \in S\mid x_i = 0,\ i \in I\} $$ of a given subset $I \subseteq \{1, \ldots, d\}$. This is done by reducing the property of avoiding an edge to the property of avoiding a corner, and allows us to find a sufficient condition for a.s. avoiding an edge $S_I$; see Corollary~\ref{general}.
In Section 4, we find the relationship between the parameters of an SRBM and the parameters of the system of competing Brownian particles, see~\eqref{R12}, ~\eqref{mu} and~\eqref{A}. Then we apply results of Section 3 to finish the proofs of Theorem~\ref{totalcor} and~\ref{mainthm}.
\subsection{Motivation} Systems of competing Brownian particles are used in Stochastic Finance: the process \begin{equation} \label{rankbasedmarketmodel} \bigl(e^{X_1(t)}, \ldots, e^{X_N(t)}\bigr)' \end{equation} can be viewed as a stock market model, see \cite{BFK2005}. Here, $e^{X_i(t)}$ is the capitalization of the $i$th stock at time $t \ge 0$. In real world, stocks with smaller capitalizations have the following property: logarithms of their capitalizations have larger drift coefficients (which in financial terminology are called {\it growth rates}) and larger diffusion coefficients (which are called {\it volatilities}) than that of stocks with larger capitalizations. It is easy to construct a model~\eqref{rankbasedmarketmodel} which captures this property: just let $$ g_1 > g_2 > \ldots > g_N\ \ \mbox{and}\ \ \sigma_1 > \sigma_2 > \ldots > \sigma_N. $$ For applications to financial market models similar to~\eqref{rankbasedmarketmodel}, see the articles \cite{Ichiba11, FIK2013b, CP2010, JR2013b, MyOwn4}, the book \cite[Chapter 5]{F2002} and the somewhat more recent survey \cite[Chapter 3]{FK2009}.
These systems also arise as discrete analogues of a so-called {\it nonlinear diffusion process}, governed by {\it McKean-Vlasov equation}, studied in \cite{Chaos1, Chaos2, Chaos3, Dawson}. As $N \to \infty$, systems of competing Brownian particles converge weakly (in a certain sense) to a nonlinear diffusion process, see \cite{S2012, JR2013a,4people}.
Also, let us mention that systems of competing Brownian particles serve as scaling limits of a certain type of exclusion processes on $\mathbb{Z}$, namely asymmetrically colliding random walks, see \cite{KPS2012}.
Systems of competing Brownian particles were also studied in the papers \cite{Ichiba11, PS2010, PP2008, CP2010, IPS2012, IKS2013, IK2010, IchibaThesis, FIK2013, Reygner2014, JR2014}.
\subsection{Organization of the paper} Section 2 contains rigorous definitions, main results: Theorems~\ref{totalcor} and~\ref{mainthm}, and examples (including the ones mentioned above). Section 3 is devoted to a semimartingale reflected Brownian motion in the orthant, and contains conditions for it to avoid edges of the boundary of this orthant. Section 4 applies results of Section 3 to systems of competing Brownian particles. Proofs of Theorems~\ref{totalcor}, ~\ref{mainthm} and~\ref{cams} are contained there. Section 5 deals with a generalization of the concept of competing Brownian particles: the so-called {\it systems with asymmetric collisions}. The Appendix contains a few technical lemmas.
\section{Formal Definitions and Main Results}
\subsection{Notation} We denote by $I_k$ the $k\times k$-identity matrix. For a vector $x = (x_1, \ldots, x_d)' \in \mathbb{R}^d$, let $\norm{x} := \left(x_1^2 + \ldots + x_d^2\right)^{1/2}$ be its Euclidean norm. For any two vectors $x, y \in \mathbb{R}^d$, their dot product is denoted by $x\cdot y = x_1y_1 + \ldots + x_dy_d$. We compare vectors $x$ and $y$ componentwise: $x \le y$ if $x_i \le y_i$ for all $i = 1, \ldots, d$; $x < y$ if $x_i < y_i$ for all $i = 1, \ldots, d$; similarly for $x \ge y$ and $x > y$. We compare matrices of the same size componentwise, too. For example, we write $x \ge 0$ for $x \in \mathbb{R}^d$ if $x_i \ge 0$ for $i = 1, \ldots, d$; $C = (c_{ij})_{1 \le i, j \le d} \ge 0$ if $c_{ij} \ge 0$ for all $i$, $j$. The symbol $a'$ denotes the transpose of (a vector or a matrix) $a$.
Fix $d \ge 1$, and let $I \subseteq \{1, \ldots, d\}$ be a nonempty subset. Write its elements in increasing order: $I = \{i_1, \ldots, i_m\},\ \ 1 \le i_1 < i_2 < \ldots < i_m \le d$. For any $x \in \mathbb{R}^d$, let $[x]_I := (x_{i_1}, \ldots, x_{i_m})'$. For any $d\times d$-matrix $C = (c_{ij})_{1 \le i, j \le d}$, let $[C]_I := \left(c_{i_ki_l}\right)_{1 \le k, l \le m}$. We let $\mathbf{1} := (1, \ldots, 1)'$ (the dimension of this vector depends on the context).
\subsection{Definitions}
Now, let us define systems of competing Brownian particles formally. Assume we have the usual setting: a filtered probability space $(\Omega, \mathcal F, (\mathcal F_t)_{t \ge 0}, \mathbf P)$ with the filtration satisfying the usual conditions. The term {\it standard Brownian motion} stands for a one-dimensional Brownian motion with drift coefficient zero and diffusion coefficient one, starting from zero.
\begin{defn} Consider a continuous adapted $\mathbb{R}^N$-valued process $$ X = (X(t), t \ge 0),\ \ X(t) = (X_1(t), \ldots, X_N(t))'. $$ For every $t \ge 0$, let $\mathbf{p}_t$ be the permutation of $\{1, \ldots, N\}$ which:
\noindent (i) {\it ranks the components of} $X(t)$, that is, $X_{\mathbf{p}_t(i)}(t) \le X_{\mathbf{p}_t(j)}(t)$ for $1 \le i < j \le N$;
\noindent (ii) {\it resolves ties in lexicographic order:} if $X_{\mathbf{p}_t(i)}(t) = X_{\mathbf{p}_t(j)}(t)$ and $i < j$, then $\mathbf{p}_t(i) < \mathbf{p}_t(j)$.
Fix parameters $g_1, \ldots, g_N \in \mathbb{R}$ and $\sigma_1, \ldots, \sigma_N > 0$, and let $W_1, \ldots, W_N$ be i.i.d. standard $(\mathcal F_t)_{t \ge 0}$-Brownian motions.
Suppose the process $X$ satisfies the following SDE: \begin{equation} \label{classicSDE} \mathrm{d} X_i(t) = \sum\limits_{k=1}^N1(\mathbf{p}_t(k) = i)\left[g_k\mathrm{d} t + \sigma_k\mathrm{d} W_i(t)\right],\ \ i = 1, \ldots, N. \end{equation} Then $X$ is called a {\it classical system of $N$ competing Brownian particles}. For $k = 1, \ldots, N$, the process $$ Y_k = (Y_k(t), t \ge 0),\ \ Y_k(t) \equiv X_{\mathbf{p}_t(k)}(t) $$ is called the {\it $k$th ranked particle}. \label{classicdef} \end{defn}
We use the term {\it classical} to distinguish these systems from similar systems of competing Brownian particles with so-called {\it asymmetric collisions}; more on this in Section 4.
\begin{defn} Consider a system from Definition~\ref{classicdef}. We say that a {\it collision of order $M$ occurs at time $t \ge 0$,} if there exists $k = 1, \ldots, N$ such that $$ Y_{k}(t) = Y_{k+1}(t) = \ldots = Y_{k+M}(t). $$ A collision of order $M = 2$ is called a {\it triple collision}. A collision of order $M = N-1$ is called a {\it total collision}. \end{defn}
As mentioned before, a related example of a total collision (for a slightly different SDE) was considered in the paper \cite{Bass1987}.
There is another closely related concept. We can have, for example, $Y_1(t) = Y_2(t)$ and $Y_4(t) = Y_5(t) = Y_6(t)$ at the same moment $t \ge 0$. This is called a {\it multicollision} of a certain order (this particular one is of order $3$).
\begin{defn} Consider a system from Definition~\ref{classicdef}, and fix a nonempty subset $I \subseteq \{1, \ldots, N-1\}$. A {\it multicollision with pattern} $I$ occurs at time $t \ge 0$ if $$ Y_k(t) = Y_{k+1}(t),\ \ \mbox{for all}\ \ k \in I. $$ We shall sometimes say that {\it there are no multicollisions with pattern $I$} if a.s. there does not exist $t > 0$ such that there is a multicollision with pattern $I$ at time $t$. \label{Pat} \end{defn}
A multicollision with pattern $I$ has order $M = |I|$. If $I = \{k, k+1, \ldots, l-2, l-1\}$, then a multicollision with pattern $I$ is, in fact, a multiple collision of particles with ranks $k, k+1, \ldots, l-1, l$. If $I = \{1, \ldots, N-1\}$, this is a total collision. If $I = \{k, l\}$, this is a simultaneous collision. If $I = \{k, k+1\}$, this is a triple collision.
We can immediately state some results which reduce multicollisions to total collisions.
\begin{lemma}
Fix $1 < N_1 \le N_2 < N$. Suppose that $\sigma_1, \ldots, \sigma_N \ge 0$ are such that for a system of competing Brownian particles with parameters $\sigma_{N_1}, \ldots, \sigma_{N_2}$ a multicollision with pattern $I \subseteq \{N_1, \ldots, N_2\}$ happens with positive probability. Then for a system of competing Brownian particles with parameters $\sigma_1,\ldots, \sigma_N$ this multicollision also happens with positive probability. \end{lemma}
\begin{proof} This follows from the relation between multicollisions and hitting edges of $\mathbb{R}^{N-1}_+$ by the gap process, established in Lemma~\ref{red}, and from Theorem~\ref{corner2edge2}. \end{proof}
It is worth providing some references about a diffusion hitting a lower-dimensional manifold: the articles \cite{Friedman1974, Ramasubramanian1983, Ramasubramanian1988, CepaLepingle}, and the book \cite{FriedmanBook}.
In this paper, we are interested in triple and simultaneous collisions, as well as the collisions of higher order $M \ge 4$. We examine whether the classical system of competing Brownian particles avoid collisions (and multicollisions) with given pattern. This paper contains two main results. One is a sufficient condition for absence of total collisions. The other is more general: a sufficient condition for the absence of multicollisions with a given pattern. The approach taken in this article does not give necessary and sufficient conditions for absence of multicollisions, only sufficient conditions; neither does it provide conditions for having multicollisions with positive probability (as opposed to avoiding them).
\subsection{Avoiding a multicollision depends only on diffusion coefficients}
The following lemma tells us that the property of a system of competing Brownian particles to avoid multicollisions with a given pattern is independent of the initial conditions $x$ and the drift coefficients $g_1, \ldots, g_N$. In other words, it can possibly depend only on the diffusion coefficients $\sigma_1^2, \ldots, \sigma_N^2$.
\begin{lemma} Take a classical system of competing Brownian particles from Definition~\ref{classicdef}. Fix $I \subseteq \{1, \ldots, N-1\}$, a pattern. Let $x \in \mathbb{R}^N$ be the initial conditions, and let $\mathbf P_x$ be the corresponding probability measure. Denote by \begin{equation} \label{probb} p\left(g_1,\, g_2,\, \ldots,\, g_N,\, \sigma_1,\, \sigma_2,\, \ldots\, \sigma_N,\, x\right) \end{equation} the probability that there exists a moment $t > 0$ such that the system, starting from $x$, will experience a multicollision with pattern $I$ at this moment. For fixed $\sigma_1, \ldots, \sigma_N > 0$, either $$ p\left(g_1,\, g_2,\, \ldots,\, g_N,\, \sigma_1,\, \sigma_2,\, \ldots\, \sigma_N,\, x\right) = 0\ \ \mbox{for all}\ \ x \in \mathbb{R}^N,\ \ (g_k)_{1 \le k \le N} \in \mathbb{R}^N, $$ or $$ p\left(g_1,\, g_2,\, \ldots,\, g_N,\, \sigma_1,\, \sigma_2,\, \ldots\, \sigma_N,\, x\right) > 0\ \ \mbox{for all}\ \ x \in \mathbb{R}^N,\ \ (g_k)_{1 \le k \le N} \in \mathbb{R}^N. $$ \label{Indep} \end{lemma}
However, in the second case (when the probability~\eqref{probb} is positive) the exact value of this probability depends on the initial conditions $x$ and the drift coefficients $g_1, \ldots, g_N$. This follows from Remark 5 in \cite[Subsection 3.2]{MyOwn3} and connection between competing Brownian particles and an SRBM, discussed just above. The proof is postponed until Appendix.
\subsection{Sufficient conditions for avoiding total collisions}
Let us introduce some additional notation. Let $M \ge 2$. For $$ \alpha = (\alpha_1, \ldots, \alpha_M)' \in \mathbb{R}^M\ \ \mbox{and}\ \ l = 1, \ldots, M-1, $$ we define $$ c_{l}(\alpha) := -\frac{2(M-1)}{M}\alpha_1^2 + \frac{2(M+1)}M\sum\limits_{p=2}^l\alpha_p^2 + \frac{2(M-1)(M-l) - 4l}{(M-l)M}\sum\limits_{p=l+1}^M\alpha_p^2. $$ We also denote by $\alpha^{\leftarrow} := (\alpha_M, \ldots, \alpha_1)'$ the vector $\alpha$ with components put in the reverse order. Note that $c_{M-1}(\alpha) = c_{M-1}\left(\alpha^{\leftarrow}\right)$. Let \begin{equation} \label{CP} \mathcal P(\alpha) := \min\left(c_1(\alpha),\, c_1\left(\alpha^{\leftarrow}\right),\, c_2(\alpha),\, c_2\left(\alpha^{\leftarrow}\right),\, \ldots,\, c_{M-2}(\alpha),\, c_{M-2}\left(\alpha^{\leftarrow}\right),\, c_{M-1}(\alpha)\right). \end{equation} For example, in cases $M = 2$ and $M = 3$ we have the following expressions for $\mathcal P(\alpha)$: \begin{equation} \label{CP2} \mathcal P(\alpha_1, \alpha_2) = c_1(\alpha_1, \alpha_2) = -\alpha_1^2 - \alpha_2^2\,, \end{equation} \begin{equation} \label{CP3} \mathcal P(\alpha_1, \alpha_2, \alpha_3) = \min\left(\frac83\alpha_2^2 - \frac43\alpha_1^2 - \frac43\alpha_3^2,\ \ \frac23\alpha_2^2 + \frac23\alpha_3^2 - \frac43\alpha_1^2,\ \ \frac23\alpha_1^2 + \frac23\alpha_2^2 - \frac43\alpha_3^2\right). \end{equation}
\begin{thm} \label{totalcor} Consider a classical system of competing Brownian particles from Definition~\ref{classicdef}, and denote $$\sigma := (\sigma_1, \ldots, \sigma_N)' .$$ If $\,\mathcal P(\sigma) \ge 0$ in the notation of \eqref{CP}, then a.s. there is no total collision at any time $t>0$. \end{thm}
\subsection{Examples of avoiding total collisions} In this subsection, we consider systems of $N = 3$, $N = 4$ and $N = 5$ particles. We apply Theorem~\ref{totalcor} to find a sufficient condition for a.s. avoiding total collisions. In particular, we compare our results for three particles to a necessary and sufficient condition~\eqref{concave}. We also compare results for $N = 4$ particles given by Theorem~\ref{totalcor} and Theorem~\ref{cams}.
\begin{exm} {\it The case of $N = 3$ particles.} In this case, ``triple collision'' is a synonym for ``total collision''. The quantity $\mathcal P(\sigma)$ is calcluated in~\eqref{CP3}. The inequality $\mathcal P(\sigma) \ge 0$ is equivalent to the following system: \begin{equation} \label{CP3I} \begin{cases} \sigma_1^2 + \sigma_3^2 \le 2\sigma_2^2;\\ 2\sigma_1^2 \le \sigma_2^2 + \sigma_3^2;\\ 2\sigma_3^2 \le \sigma_2^2 + \sigma_1^2. \end{cases} \end{equation} In fact, the first inequality in~\eqref{CP3I} follows from the second and the third ones. Therefore,~\eqref{CP3I} is equivalent to \begin{equation} \label{CP3II} \begin{cases} 2\sigma_1^2 \le \sigma_2^2 + \sigma_3^2;\\ 2\sigma_3^2 \le \sigma_2^2 + \sigma_1^2. \end{cases} \end{equation} This sufficient condition is more restrictive than~\eqref{concave}, which for $N = 3$ particles takes the form $2\sigma_2^2 \ge \sigma_1^2 + \sigma_3^2$. Therefore, Theorem~\ref{totalcor} gives a weaker result than the result from \cite{MyOwn3}, mentioned in Proposition~\ref{elegant}. In other words, for three particles the results from this paper do not give us anything new compared to \cite{MyOwn3, IK2010}, which is not surprising: in \cite{MyOwn3, IK2010}, they found a necessary and sufficient condition for avoiding total collision for $N = 3$ particles. \label{N3} \end{exm}
\begin{exm} {\it The case of $N = 4$ particles.} The result was stated in the Introduction as Theorem~\ref{example1}. The condition $\mathcal P(\sigma) \ge 0$ holds, if and only if all the following five inequalities hold: \begin{equation} \label{CP4I} \begin{cases} 9\sigma_1^2 \le 7\sigma_2^2 + 7\sigma_3^2 + 7\sigma_4^2;\\ 3\sigma_1^2 \le 5\sigma_2^2 + \sigma_3^2 + \sigma_4^2;\\ 3\sigma_1^2 + 3\sigma_4^2 \le 5\sigma_2^2 + 5\sigma_3^2;\\ 3\sigma_4^2 \le \sigma_1^2 + \sigma_2^2 + 5\sigma_3^2;\\ 9\sigma_4^2 \le 7\sigma_1^2 + 7\sigma_2^2 + 7\sigma_3^2. \end{cases} \end{equation}
As mentioned in Section 1, let $\sigma_1^2 = \sigma_2^2 = \sigma_4^2 = 1$, and $\sigma_3^2 = 0.9$. Then there are triple collisions between the particles $Y_2$, $Y_3$ and $Y_4$ with positive probability, because the sequence $(\sigma_1^2, \sigma_2^2, \sigma_3^2, \sigma_4^2)$ is not concave: it does not satisfy the condition~\eqref{concave}. But the condition $\mathcal P(\sigma) \ge 0$ is satisfied, hence there are a.s. no total collisions. Note that this example satisfies the conditions of Theorem~\ref{totalcor}, but fails to satisfy those of Theorem~\ref{cams}. \label{N4} \end{exm}
\begin{exm} {\it The case of $N = 5$ particles.} In this case $\mathcal P(\sigma) \ge 0$ is equivalent to the following seven inequalities: \begin{equation} \label{CP5I} \begin{cases} 8\sigma_1^2 \le 7\sigma_2^2 + 7\sigma_3^2 + 7\sigma_4^2 + 7\sigma_5^2;\\ 6\sigma_1^2 \le 9\sigma_2^2 + 4\sigma_3^2 + 4\sigma_4^2 + 4\sigma_5^2;\\ 4\sigma_1^2 \le 6\sigma_2^2 + 6\sigma_3^2 + \sigma_4^2 + \sigma_5^2;\\ 2\sigma_1^2 + 2\sigma_5^2 \le 3\sigma_2^2 + 3\sigma_3^2 + 3\sigma_4^2;\\ 8\sigma_5^2 \le 7\sigma_4^2 + 7\sigma_3^2 + 7\sigma_2^2 + 7\sigma_1^2;\\ 6\sigma_5^2 \le 9\sigma_4^2 + 4\sigma_3^2 + 4\sigma_2^2 + 4\sigma_1^2;\\ 4\sigma_5^2 \le 6\sigma_4^2 + 6\sigma_3^2 + \sigma_2^2 + \sigma_1^2. \end{cases} \end{equation}
By analogy with the previous example, let $\sigma_1^2 = \sigma_2^2 = \sigma_4^2 = \sigma_5^2 = 1$, and $\sigma_3^2 = 0.9$. Then there are triple collisions among the particles $Y_2$, $Y_3$ and $Y_4$ with positive probability, but a.s. no total collisions. \label{N5} \end{exm}
\begin{exm} {\it An application of Theorem \ref{cams}.} Take $\sigma_1^2 = \sigma_3^2 = 10$ and $\sigma_2^2 = \sigma_4^2 = 1$. Then by Theorem~\ref{cams} there are a.s. no total collisions, but this fails to satisfy the conditions of Theorem~\ref{totalcor}. This, together with Example~\ref{N4}, shows that none of the two results: Theorem~\ref{totalcor} applied to the case of $N = 4$ particles, and Theorem~\ref{cams}, is stronger than the other one. \label{1010} \end{exm}
\subsection{A sufficient condition for avoiding multicollisions of a given pattern}
For every nonempty finite subset $I \subseteq \mathbb{Z}$, denote by $\overline{I} := I\cup\{\max I+1\}$ the augmentation of $I$ by the integer following its maximal element. For example, if $I = \{1, 2, 4, 6\}$, then $\overline{I} = \{1, 2, 4, 6, 7\}$. A nonempty finite subset $I \subseteq \mathbb{Z}$ is called a {\it discrete interval} if it has the form $\{k, k+1, \ldots, l-1, l\}$ for some $k, l \in \mathbb{Z},\ k \le l$. For example, the sets $\{2\}, \{3, 4\}, \{-2, -1, 0\}$ are discrete intervals, and the set $\{3, 4, 6\}$ is not. Two disjoint discrete intervals are called {\it adjacent} if their union is also a discrete interval. For example, discrete intervals $\{1, 2\}$ and $\{3, 4\}$ are adjacent, while $\{3, 4, 5\}$ and $\{10, 11\}$ are not.
Every nonempty finite subset $I \subseteq \mathbb{Z}$ can be decomposed into a finite union of disjoint non-adjacent discrete intervals: for example, $I = \{1, 2, 4, 8, 9, 10, 11, 13\}$ can be decomposed as $\{1, 2\}\cup\{4\}\cup\{8, 9, 10, 11\}\cup\{13\}$. This decomposition is unique. The non-adjacency is necessary for uniqueness: for example, $\{1, 2\}\cup\{4\}\cup\{8, 9, 10\}\cup\{11\}\cup\{13\}$ is also a decomposition into a finite union of disjoint discrete intervals, but $\{8, 9, 10\}$ and $\{11\}$ are adjacent.
For a vector $\alpha = (\alpha_1, \ldots, \alpha_M)' \in \mathbb{R}^M$, define \begin{equation} \label{CT} \mathcal T(\alpha) = \frac{2(M-1)}{M}\sum\limits_{p=1}^M\alpha_p^2. \end{equation}
For every discrete interval $I = \{k, \ldots, l\} \subseteq \{1, \ldots, N\}$, let $\mathcal P(I) := \mathcal P\left(\sigma_k, \ldots, \sigma_l\right)$ and $\mathcal T(I) := \mathcal T\left(\sigma_k, \ldots, \sigma_l\right)$.
Consider a subset $I \subseteq \{1, \ldots, N-1\}$. Suppose it has the following decomposition into the union of non-adjacent discrete disjoint intervals: \begin{equation} \label{decomp} I = I_1\cup I_2\cup \ldots \cup I_r. \end{equation}
\begin{defn} We say that $I$ {\it satisfies assumption (A)} if \begin{equation} \label{conditionA} \sum\limits_{\substack{j=1\\ j \ne i}}^r\mathcal T(\overline{I}_j) + \mathcal P(\overline{I}_i) \ge 0,\ \ i = 1, \ldots, r. \end{equation} We say that $I$ {\it satisfies assumption (B)} if at least one of the following is true: \begin{itemize} \item at least two of discrete intervals $I_1, \ldots, I_r$ are singletons; \item at least one of discrete intervals $I_1, \ldots, I_r$ consists of two elements $\{k-1, k\}$, and the sequence $(\sigma^2_j)$ has {\it local concavity} at $k$: \begin{equation} \label{localconcavity} \sigma_k^2 \ge \frac12\left(\sigma_{k-1}^2 + \sigma_{k+1}^2\right); \end{equation} \item there exists a subset $$ I' = I_{i_1}\cup I_{i_2}\cup \ldots \cup I_{i_s} $$ which satisfies the assumption (A). \end{itemize} \label{defnasmp} \end{defn}
\begin{rmk} (i) If a subset $I \subseteq \{1, \ldots, N-1\}$ is a discrete interval, that is, the decomposition~\eqref{decomp} is trivial, then Assumption (A) is equivalent to $\mathcal P(\overline{I}) \ge 0$.
(ii) If a subset $I \subseteq \{1, \ldots, N-1\}$ is a discrete interval of three or more elements, then Assumption (B) is equivalent to $\mathcal P(\overline{I}) \ge 0$.
(iii) If a subset $I \subseteq \{1, \ldots, N-1\}$ contains two elements: $I = \{k, l\},\ k < l$, then Assumption (B) is automatically satisfied if $k + 1 < l$. If $k+1 = l$, then Assumption (B) is equivalent to the local concavity at $l$: $$ \sigma_{l}^2 \ge \frac12\left(\sigma_{l+1}^2 + \sigma_{l-1}^2\right). $$ Indeed, as mentioned in Example~\ref{N3}, the condition $\mathcal P(\overline{I}) \ge 0$ is more restrictive than local concavity at $l$. \label{reduct} \end{rmk}
\begin{thm} Consider a system of competing Brownian particles from Definition~\ref{classicdef}. Fix a subset $J \subseteq \{1, \ldots, N-1\}$. Suppose every subset $I$ such that $J \subseteq I \subseteq \{1, \ldots, N-1\}$ satisfies assumption (B). Then there a.s. does not exist $t > 0$ such that the system has a multicollision with pattern $J$ at time $t$. \label{mainthm} \end{thm}
The following immediate corollary gives a sufficient condition for absence of multicollisions of a given order (and, in particular, multiple collisions of a given order).
\begin{cor} Consider a classical system of competing Brownian particles from Definition~\ref{classicdef}. Fix an integer $M = 3, \ldots, N$, and suppose that every subset $I \subseteq \{1, \ldots, N-1\}$ with $|I| \ge M$ satisfies condition~\eqref{conditionA}. Then a.s. there does not exist $t > 0$ such that the system has a multicollision (and, in particular, a collision) of order $M$ \end{cor}
\subsection{Examples of avoiding multicollisions} In this subsection, we apply Theorem~\ref{mainthm} to systems with a small number of particles: $N = 4$ and $N = 5$. We consider different patterns of multicollisions.
\begin{exm} {\it Let $N = 4$ (four particles) and $J = \{1, 3\}$.} (This was already mentioned in the Introduction, in Theorem~\ref{example1}.) A multicollision with pattern $J$ is the same as a simultaneous collision of the following type: \begin{equation} \label{46} Y_1(t) = Y_2(t)\ \ \mbox{and}\ \ Y_3(t) = Y_4(t). \end{equation} We need to check Assumption (B) for subsets $I = J = \{1, 3\}$ and $I = \{1, 2, 3\}$. The subset $I = \{1, 2, 3\}$ is a discrete interval. According to Remark~\ref{reduct}, we can apply Example~\ref{N4}, and rewrite Assumption (B) as the system of five inequalities~\eqref{CP4I}. For $I = \{1, 3\}$, the decomposition~\eqref{decomp} of $I$ into the union of disjoint non-adjacent discrete intervals has the following form: $I = \{1\}\cup\{3\}$. Therefore, Assumption (B) is always satisfied. Therefore, the system of five inequalities~\eqref{CP4I} is sufficient not only for avoiding total collisions in a system of four particles, but also for avoiding multicollisions~\eqref{46}, with pattern $J = \{1, 3\}$.
\label{weird} \end{exm}
\begin{exm} {\it Let $N = 4$ and $J =\{1, 2\}$.} Let us find a sufficient condition for a.s. avoiding triple collisions of the type $Y_1(t) = Y_2(t) = Y_3(t)$. (This was already mentioned in the Introduction, as Theorem~\ref{example3}.) There are two subsets $I$ such that $J \subseteq I \subseteq \{1, 2, 3\}$: $I = \{1, 2\}$ and $I = \{1, 2, 3\}$. These two sets are both discrete intervals. As mentioned in the Remark~\ref{reduct}, Assumption (B) for $I = \{1, 2, 3\}$ is equivalent to $\mathcal P(\overline{I}) \ge 0$, which, in turn, is equivalent to~\eqref{CP4I}. Assumption (B) for $I = \{1, 2\}$ is equivalent to local concavity at index $2$: $2\sigma_2^2 \ge \sigma_1^2 + \sigma_3^2$.
We can write this as the system of six inequalities: local concavity at $2$ and the five inequalities~\eqref{CP4I} from Example~\ref{N4}. \label{eleg2} \end{exm}
\begin{exm} {\it Consider $N = 5$ (five particles) and take the pattern $J = \{1, 2, 3\}$. } This corresponds to a collision of the following type: \begin{equation} \label{0091} Y_1(t) = Y_2(t) = Y_3(t) = Y_4(t). \end{equation} There are two subsets $I$ such that $J \subseteq I \subseteq \{1, 2, 3, 4\}$: $I = J = \{1, 2, 3\}$ and $I = \{1, 2, 3, 4\}$. These two sets are both discrete intervals. As mentioned in the Remark~\ref{reduct}, Assumption (B) for each of these sets $I$ takes the form $\mathcal P(\overline{I}) \ge 0$: $\mathcal P(\{1, 2, 3, 4\}) \ge 0$ and $\mathcal P(\{1, 2, 3, 4, 5\}) \ge 0$. We can write them as the system of twelve inequalities: the five inequalities~\eqref{CP4I} from Example~\ref{N4}, and the seven inequalities~\eqref{CP5I} from Example~\ref{N5}. \label{exmfive1} \end{exm}
\begin{exm} {\it Consider $N = 5$ and take the pattern $J = \{1, 2, 4\}$}. This corresponds to a collision \begin{equation} \label{0092} Y_1(t) = Y_2(t) = Y_3(t),\ \ \mbox{and}\ \ Y_4(t) = Y_5(t). \end{equation} There are two subsets $I$ such that $J \subseteq I \subseteq \{1, 2, 3, 4\}$: $I = J = \{1, 2, 4\}$ and $I = \{1, 2, 3, 4\}$. The set $I = \{1, 2, 3, 4\}$ is a discrete interval; by Remark~\ref{reduct}, Assumption (B) for $I = \{1, 2, 3, 4\}$ takes the form $\mathcal P(\{1, 2, 3, 4, 5\}) \ge 0$. This is equivalent to the conjunction of the seven inequalities~\eqref{CP5I} from Example~\ref{N5}. For $I = \{1, 2, 4\}$, the situation is more complicated. The decomposition of this $I$ into a union of disjoint non-adjacent discrete intervals is $I = \{1, 2\}\cup \{4\}$. Therefore, Assumption (B) holds for this set $I$ in one of the following cases: \begin{itemize} \item if there is local concavity at $2$: $\sigma_2^2 \ge \left(\sigma_1^2 + \sigma_3^2\right)/2$; \item Assumption (A) holds for $\{1, 2\}$, which is equivalent to $\mathcal P(\{1, 2, 3\}) \ge 0$, which, in turn, is a stronger assumption than local concavity at $2$ (see Example~\ref{N3}); \item Assumption (A) holds for $\{4\}$, which is when $\mathcal P(\{4, 5\}) \ge 0$; but this is never true, see~\eqref{CP2}; \item Assumption (A) holds for $\{1, 2\}\cup\{4\}$, which is equivalent to \begin{equation} \label{35403} \mathcal T(\{1, 2, 3\}) + \mathcal P(\{4, 5\}) \ge 0,\ \ \mathcal T(\{4, 5\}) + \mathcal P(\{1, 2, 3\}) \ge 0. \end{equation} \end{itemize} But $\mathcal P(\{4, 5\}) = \mathcal P(\sigma_4, \sigma_5) = -\sigma_4^2 - \sigma_5^2$, as in~\eqref{CP2}, and $\mathcal P(\{1, 2, 3\}) = \mathcal P(\sigma_1, \sigma_2, \sigma_3)$ is given by~\eqref{CP3}. Therefore, we have: \begin{equation} \label{45} \mathcal T(\{1, 2, 3\}) + \mathcal P(\{4, 5\}) = \frac43\left(\sigma_1^2 + \sigma_2^2 + \sigma_3^2\right) - \sigma_4^2 - \sigma_5^2 \ge 0, \end{equation} which can be written as \begin{equation} \label{4509} 4\sigma_1^2 + 4\sigma_2^2 + 4\sigma_3^2 \ge 3\sigma_4^2 + 3\sigma_5^2. \end{equation} The other condition $\mathcal T(\{4, 5\}) + \mathcal P(\{1, 2, 3\}) \ge 0$ is equivalent to the system of the following three inequalities: \begin{equation} \label{413} \begin{cases} 4\sigma_1^2 + 4\sigma_3^2 \le 8\sigma_2^2 + 3\sigma_4^2 + 3\sigma_5^2;\\ 4\sigma_1^2 \le 2\sigma_2^2 + 2\sigma_3^2 + 3\sigma_4^2 + 3\sigma_5^2;\\ 4\sigma_3^2 \le 2\sigma_1^2 + 2\sigma_2^2 + 3\sigma_4^2 + 3\sigma_5^2. \end{cases} \end{equation} Therefore,~\eqref{35403} is equivalent to the system of~\eqref{4509} and~\eqref{413}: \begin{equation} \label{34234} \begin{cases} 4\sigma_1^2 + 4\sigma_3^2 \le 8\sigma_2^2 + 3\sigma_4^2 + 3\sigma_5^2;\\ 4\sigma_1^2 \le 2\sigma_2^2 + 2\sigma_3^2 + 3\sigma_4^2 + 3\sigma_5^2;\\ 4\sigma_3^2 \le 2\sigma_1^2 + 2\sigma_2^2 + 3\sigma_4^2 + 3\sigma_5^2;\\ 4\sigma_1^2 + 4\sigma_2^2 + 4\sigma_3^2 \ge 3\sigma_4^2 + 3\sigma_5^2. \end{cases} \end{equation}
Assumption (B) holds for $I = \{1, 2\}\cup\{4\}$ if and only if there is local concavity at $2$ or~\eqref{34234} hold. Thus, the system of seven inequalities~\eqref{CP5I} from Example~\ref{N5}, together with local concavity at $2$ or the four inequalities~\eqref{34234}, is a sufficient condition for avoiding multicollisions of pattern $\{1, 2, 4\}$. \label{exmfive2} \end{exm}
\begin{rmk} We can also make use of the condition~\eqref{Bruggeman} instead of the five inequalities~\eqref{CP4I}. If the condition~\eqref{Bruggeman} is satisfied, then there are a.s. no simultaneous collisions~\eqref{46} at any time $t > 0$. Similarly, in all of the examples involving $N = 4$ particles avoiding certain types of collisions, we can substitute the condition~\eqref{Bruggeman} instead of the five inequalities~\eqref{CP4I}, and the statement will still be true. In Example~\ref{eleg2}, the two conditions:~\eqref{Bruggeman} and the local concavity at the index $2$, guarantee absence of triple collisions $Y_1(t) = Y_2(t) = Y_3(t)$. The same works for Examples~\ref{exmfive1} and~\ref{exmfive2}. \end{rmk}
\begin{exm} {\it Suppose we have three or more particles: $N \ge 3$.} Consider the case when all diffusion coefficients are equal to one: $\sigma_1 = \ldots = \sigma_N = 1$. Then there are no triple and multiple collisions, as well as no multicollisions of order $M \ge 3$. To show this, we do not even need to use Theorem~\ref{mainthm}. Indeed, using Girsanov transformation as in~\cite[Subsection 3.2]{MyOwn3}, we can transform the classical system of competing Brownian particles into $N$ independent Brownian motions with zero drifts and unit diffusions. Since the Bessel process of dimension two a.s. does not return to the origin, there are a.s. no triple collisions and multicollisions of order $M \ge 3$ for the system of independent Brownian motions.
Still, we can apply our results of this article to the case of unit diffusion coefficients. Consider total collisions and apply Theorem~\ref{totalcor}. Let $\sigma_1 = \ldots = \sigma_N = 1$, so that $\sigma = \mathbf{1} = (1, 1, \ldots, 1)'$; then it is straightforward to calculate that $$ c_l(\sigma) = c_l(\sigma^{\leftarrow}) = 2N - 6,\ l = 1, \ldots, N- 1. $$ Therefore, we have: $$ \mathcal P(\sigma) = \min(c_1(\sigma),\, \ldots,\, c_{N-2}(\sigma),\, c_{N-1}(\sigma),\, c_1(\sigma^{\leftarrow}),\, \ldots,\, c_{N-2}(\sigma^{\leftarrow})) = 2N - 6 \ge 0. $$ Apply Theorem~\ref{totalcor}: the system avoids total collisions. How does this result change if we move the diffusion coefficients $\sigma_1^2, \ldots, \sigma_N^2$ a little away from $1$? In other words, if the vector $\sigma$ is in a small neighborhood of $\mathbf{1} = (1, \ldots, 1)' \in \mathbb{R}^N$, what can we say about absence of total collisions?
If $N = 3$, then $\mathcal P(\mathbf{1}) = 0$. Even in a small neighborhood of $\mathbf{1}$, we can have either $\mathcal P(\sigma) \ge 0$ or $\mathcal P(\sigma) < 0$. Therefore, we cannot claim that in a certain neighborhood of $\mathbf{1}$ we do not have any total (in this case, triple) collisions. This is consistent with the results of \cite{MyOwn3}. Indeed, the inequality~\eqref{concave} takes the form \begin{equation} \label{concave3} \sigma_2^2 \ge \frac12\left(\sigma_1^2 + \sigma_3^2\right). \end{equation} This becomes an equality for $\sigma = (\sigma_1, \sigma_2, \sigma_3)' = \mathbf{1}$. The point $\mathbf{1}$ lies at the boundary of the set of points in $\mathbb{R}^3$ given by~\eqref{concave3}. Or, equivalently, in any neighborhood of $\mathbf{1}$ there are both points $\sigma$ which satisfy~\eqref{concave3} and which do not satisfy~\eqref{concave3}.
But for $N \ge 4$ (four or more particles), we have: $\mathcal P(\mathbf{1}) > 0$. Since $\mathcal P(\sigma)$ is a continuous function of $\sigma$, there exists a neighborhood $\mathcal U$ of $\mathbf{1}$ such that for all $\sigma \in \mathcal U$ we have: $\mathcal P(\sigma) > 0$, and the system of competing Brownian particles does not have total collisions. \label{girsanov} \end{exm}
\section{Semimartingale Reflected Brownian Motion (SRBM) in the Orthant}
\subsection{Definitions} We informally described a semimartingale reflected Brownian motion (SRBM) in the orthant in Section 2. Now, let us define this process formally. Fix $d \ge 1$, the dimension. Let us describe the parameters of an SRBM: a {\it drift vector} $\mu \in \mathbb{R}^d$, a $d\times d$-matrix $R = (r_{ij})_{1 \le i, j \le d}$ with $r_{ii} = 1$, $i = 1, \ldots, d$, which is called a {\it reflection matrix}, and another $d\times d$ symmetric positive definite matrix $A = (a_{ij})_{1 \le i, j \le d}$, which is called a {\it covariance matrix}. Recall the notation: $S = \mathbb{R}^d_+$. Our goal is to define a Markov process in $S$ which behaves as a Brownian motion with drift vector $\mu$ and covariance matrix $A$ in the interior of $S$, and reflects in the direction of the $i$th column of $R$ on the face $S_i$, for each $i = 1, \ldots, d$.
Let $(\Omega, \mathcal F, (\mathcal F_t)_{t \ge 0}, \mathbf P)$ be the standard setting: a filtered probability space with the filtration satisfying the usual conditions.
\begin{defn} Fix $x \in S$. Consider the following three processes:
(i) a continuous adapted $S$-valued process $Z = (Z(t), t \ge 0)$;
(ii) an $((\mathcal F_t)_{t \ge 0}, \mathbf P)$-Brownian motion $B = (B(t), t \ge 0)$ in $\mathbb{R}^d$ with drift vector $\mu$ and covariance matrix $A$, starting from $B(0) = x$;
(iii) another continuous adapted $\mathbb{R}^d$-valued process $$ L = (L(t), t \ge 0),\ \ L(t) = (L_1(t), \ldots, L_N(t))', $$ such that the following is true: $$ Z(t) = B(t) + RL(t)\ \ \mbox{for}\ \ t \ge 0, $$ and for each $k = 1, \ldots, d$, the process $L_k$ has the following properties: it is nondecreasing, $L_k(0) = 0$, and $$ \int_0^{\infty}Z_k(t)\mathrm{d} L_k(t) = 0, $$ that is, $L_k$ can increase only when $Z_k = 0$.
Then the process $Z$ is called a {\it semimartingale reflected Brownian motion} (SRBM) {\it in the orthant} $S$, with {\it drift vector} $\mu$, {\it covariance matrix} $A$, and {\it reflection matrix} $R$, {\it starting from} $x$. The process $B$ is called the {\it driving Brownian motion} for the SRBM $Z$. The process $L$ is called the {\it vector of regulating processes} for $Z$, and its $i$th component $L_i$ is called the {\it regulating process corresponding to the face} $S_i$ of the boundary $\partial S$. As mentioned before, the process $Z$ is denoted by $\SRBM^d(R, \mu, A)$. \end{defn}
Let us define a few classes of matrices; see also a useful equivalent characterization in \cite[Lemma 2.5]{MyOwn3}.
\begin{defn} Consider a $d\times d$-matrix $R = (r_{ij})_{1 \le i, j \le d}$. It is called a {\it reflection matrix} if $r_{ii} = 1$ for $i = 1, \ldots, d$. It is called a {\it completely-$\mathcal S$ matrix} if for all nonempty $I \subseteq \{1, \ldots, d\}$ there exists $u \in \mathbb{R}^{|I|}$ such that $u > 0$ and $[R]_Iu > 0$. It is called a {\it reflection nonsingular $\mathcal M$-matrix} if $r_{ii} = 1$ for $i = 1, \ldots, d$, $r_{ij} \le 0$ for $i \ne j$, and the spectral radius of the matrix $I_d - R$ is less than $1$. It is called {\it strictly copositive} if it is symmetric and for every $x \in S\setminus\{0\}$ we have: $x'Rx > 0$. \end{defn}
Now, let us state a general existence and uniqueness theorem for this process was proved in \cite{RW1988, TW1993, HR1981a}; see also the survey \cite{Wil1995}.
\begin{prop} Fix any point $x \in S$. If $R$ is a reflection nonsingular $\mathcal M$-matrix, then there exists and is unique in the strong sense an $\SRBM^d(R, \mu, A)$, starting from $x$. If, more generally, $R$ is a completely-$\mathcal S$ reflection matrix, then this process exists and is unique in the weak sense. These processes for all $x \in S$ together form a Feller continuous strong Markov family. \label{existence} \end{prop}
An SRBM in the orthant is the {\it heavy traffic limit} for series of queues, when the traffic intensity tends to one, see \cite{ReimanThesis, Rei1984, Har1978}. We can also define an SRBM in general convex polyhedral domains in $\mathbb{R}^d$, see \cite{DW1995}. An SRBM in the orthant and in convex polyhedra has been extensively studied, see the survey \cite{Wil1995}, and articles \cite{HR1981a, HR1981b, HW1987a, HW1987b, Wil1987, RW1988, TW1993, DW1994, DK2003, Chen1996, Har1978, BDH2010, BL2007, DW1995, DH2012, DH1992, HH2009, Wil1985a, Wil1985b, Wil1998b, K1997, K2000, KW1996, R2000, KR2012a, KR2012b, KW1992a}.
For any subset $I \subseteq \{1, \ldots, d\}$, we let $S_I := \cap_{i \in I}S_i = \{x \in S\mid x_i = 0\ \mbox{for}\ i \in I\}$. If an $\SRBM^d(R, \mu, A)$ starts from $z \in S$, we denote the corresponding probability measure as $\mathbf P_z$.
The following proposition was shown in \cite[Subsection 3.2]{MyOwn3}.
\begin{prop} Fix a $d\times d$ reflection nonsingular $\mathcal M$-matrix $R$ and a positive definite symmetric $d\times d$-matrix $A$. Let $\mu \in \mathbb{R}^d$. Denote $Z = (Z(t), t \ge 0) = \SRBM^d(R, \mu, A)$. Consider the statement \begin{equation} \label{Indepe} \mathbf P_z\left(\exists\ t > 0:\ Z(t) \in S_I\right) = 0, \end{equation} Whether it is true or false does not depend on the initial condition $z \in S$ or on the drift vector $\mu \in \mathbb{R}^d$; it depends only on the reflection matrix $R$ and the covariance matrix $A$. \label{432} \end{prop}
This justifies the following definition, taken from \cite{MyOwn3}.
\begin{defn} An $\SRBM^d(R, \mu, A)$ {\it does not hit} $S_I$, or {\it avoids} $S_I$, if for every $z \in S$, we have: \begin{equation} \label{pa} \mathbf P_z\left(\exists\ t > 0: Z(t) \in S_I\right) = 0. \end{equation} An $\SRBM^d(R, \mu, A)$ {\it hits} $S_I$ if for every $z \in S$, we have: \begin{equation} \label{nnpa} \mathbf P_z\left(\exists\ t > 0: Z(t) \in S_I\right) > 0. \end{equation} \end{defn}
\begin{rmk} For every fixed nonempty $I \subseteq \{1, \ldots, d\}$, either~\eqref{pa} or~\eqref{nnpa} holds. In other words, either all the probabilities on the left-hand side of~\eqref{pa} are simultaneously positive for all $z \in S$, or they are all simultaneously equal to zero for all $z \in S$. The property of a.s. avoiding a given edge does not depend on the initial condition, or on the drift vector $\mu$. However, if the probability on the left-hand side~\eqref{pa} is positive, then its exact value does depend on $z$ and $\mu$; see Remark 5 in \cite[Subsection 3.2]{MyOwn3}. \end{rmk}
\begin{defn} An $\SRBM^d(R, \mu, A)$ {\it avoids edges of order $p$}, if it does not hit $S_I$ for any subset $I \subseteq \{1, \ldots, d\}$ with $p$ elements. An $\SRBM^d(R, \mu, A)$ {\it avoids the corner} if it does not hit edges of order $p = d$. An $\SRBM^d(R, \mu, A)$ {\it avoids non-smooth parts of the boundary} if it does not hit edges of order $p = 2$. \end{defn}
\subsection{Main Results}
Here, we state our main results for an SRBM in the orthant. There are three important theorems. First, we provide a sufficient condition for not hitting the corner, and another sufficient condition for hitting the corner. Taken together, they do not give us a necessary and sufficient condition, because there is a gap between them. In this respect, the results of this paper is different from that of \cite{MyOwn3}, where we gave a necessary and sufficient condition for avoiding non-smooth parts of the boundary.
A remaining question is about hitting or avoiding a given edge $S_I$ of the boundary $\partial S$. We provide another theorem which reduces it to the question of not hitting the corner. This gives us a sufficient condition for not hitting the given edge of $\partial S$. The last of these three main results is a sufficient condition for hitting a given edge of $\partial S$.
For a strictly copositive $d\times d$-matrix $Q = (q_{ij})_{1 \le i, j \le d}$ and consider the following constants: $$ c_+(Q) := \max\limits_{x \in S\setminus\{0\}}\frac{x'QAQx}{x'Qx},\qquad c_-(Q) := \min\limits_{x \in S\setminus\{0\}}\frac{x'QAQx}{x'Qx}. $$
\begin{lemma} For a positive definite matrix $A$, the numbers $c_{\pm}(Q)$ are well defined and strictly positive. \label{constants} \end{lemma}
The (rather straightforward) proof is postponed until the Appendix. The following theorem is our main result about an SRBM hitting the corner.
\begin{thm} \label{cornerthm} Suppose $R$ is a completely-$\mathcal S$ reflection matrix, and $A$ is a positive definite symmetric matrix. Take a strictly copositive nonsingular $d\times d$-matrix $Q$.
(i) If the following conditions are true: \begin{equation} \label{Ncorner} \tr\left(QA\right) \ge 2c_+(Q),\ \ \mbox{and}\ \ (QR)_{ij} \ge 0\ \ \mbox{for}\ \ i \ne j, \end{equation} then the $\SRBM^d(R, \mu, A)$ does not hit the corner.
(ii) If the following conditions are true: \begin{equation} \label{Ycorner} 0 \le \tr\left(QA\right) < 2c_-(Q),\ \ \mbox{and}\ \ (QR)_{ij} \le 0\ \ \mbox{for}\ \ i \ne j, \end{equation} then the $\SRBM^d(R, \mu, A)$ hits the corner. \end{thm}
An important example of such matrix $M$ is given in the next corollary.
\begin{cor} Suppose the matrix $R$ is a reflection nonsingular $\mathcal M$-matrix for which there exists a diagonal matrix $C = \diag(c_1, \ldots, c_d)$ with $c_1, \ldots, c_d > 0$, such that $\overline{R} = RC$ is symmetric.
(i) If the following condition is true: \begin{equation} \label{Ncorner-om} \tr\left(\overline{R}^{-1} A\right) \ge 2c_+(\overline{R}^{-1}), \end{equation} then the $\SRBM^d(R, \mu, A)$ does not hit the corner.
(ii) If the following conditions are true: \begin{equation} \label{Ycorner-om} 0 \le \tr\left(\overline{R}^{-1} A\right) < 2c_-(\overline{R}^{-1}), \end{equation} then the $\SRBM^d(R, \mu, A)$ hits the corner. \label{cor:inv-R} \end{cor}
Theorem~\ref{cornerthm} applies also to completely-$\mathcal S$ reflection matrices which are not reflection nonsingular $\mathcal M$-matrices (that is, they contain positive off-diagonal elements). The following is a useful corollary, which can be viewed as a generalization of Corollary~\ref{cor:inv-R}, because for a reflection nonsingular $\mathcal M$-matrix $R$ we have: $\tilde{R} = R$.
\begin{cor} Take a completely-$\mathcal S$ reflection matrix $R = (r_{ij})_{1 \le i, j \le d}$. Consider the matrix $\tilde{R} = (\tilde{r}_{ij})_{1 \le i, j \le d}$, defined as $$ \tilde{r}_{ij} = \begin{cases} 1,\ i = j;\\ r_{ij},\ r_{ij} \le 0;\\ 0,\ r_{ij} > 0. \end{cases} $$ Assume $\tilde{R}$ is a reflection nonsingular-$\mathcal M$ matrix. Also, suppose there exists a diagonal matrix $C = \diag(c_1, \ldots, c_d)$ with $c_1, \ldots, c_d > 0$, such that $\overline{R} = \tilde{R}C$ is symmetric. If the following condition is true: \begin{equation} \label{Ncorner-om-notM} \tr\left(\overline{R}^{-1} A\right) \ge 2c_+(\overline{R}^{-1}), \end{equation} then the $\SRBM^d(R, \mu, A)$ does not hit the corner. \end{cor}
\begin{proof} Just take $Q = \tilde{R}^{-1}$ and apply Theorem~\ref{cornerthm}. We need only to prove that $(QR)_{ij} \ge 0$ for $i \ne j$. But all elements of $Q$ are nonnegative, and so $$ (QR)_{ij} = \sum\limits_{k=1}^dq_{ik}r_{kj} \ge \sum\limits_{k=1}^dq_{ik}\tilde{r}_{kj} = (Q\tilde{R})_{ij} = 0. $$ \end{proof}
\begin{exm} Let $d = 2$, and $$ R = \begin{bmatrix} 1 & 2\\ 3 & 1 \end{bmatrix}, \ \ A = I_2 = \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix} $$ Then $\tilde{R} = I_2$ and $\overline{R}^{-1} = I_2$. Therefore, $$ c_+(\overline{R}^{-1}) = \max\limits_{x \in S\setminus\{0\}}\frac{x'I_2x}{x'I_2x} = 1, \ \ \mbox{and}\ \ \tr\left(\overline{R}^{-1} A\right) = \tr(I_2) = 2. $$ Thus, condition~\eqref{Ncorner-om-notM} holds, and the $\SRBM^2(R, \mu, A)$ does not hit the corner. \end{exm}
Sometimes the numbers $c_{\pm}(Q)$ are difficult to calculate. Let us give useful estimates of $c_+(Q)$ from above, and of $c_-(Q)$ from below.
\begin{lemma} If the matrix $Q = (q_{ij})_{1 \le i, j \le d}$ satisfies $q_{ij} > 0$ for all $i, j = 1, \ldots, d$, then $$ c_+(Q) \le \overline{c}_+(Q) := \max\limits_{1 \le i \le j \le d}\frac{(QAQ)_{ij}}{q_{ij}}, \ \ c_-(Q) \ge \overline{c}_-(Q) := \min\limits_{1 \le i \le j \le d}\frac{(QAQ)_{ij}}{q_{ij}}. $$ \label{simple} \end{lemma}
The next theorem establishes a connection between not hitting the corner and not hitting an edge. It is similar to results from \cite{IKS2013}, and we took the proof technique from \cite{IKS2013}.
\begin{thm} \label{corner2edge} Take a completely-$\mathcal S$ reflection matrix $R$. Consider an $\SRBM^d(R, \mu, A)$. Fix a nonempty subset $J \subseteq \{1, \ldots, d\}$. Suppose the process $$
\SRBM^{|I|}([R]_I, [\mu]_I, [A]_I)\ \ \mbox{for each}\ \ J \subseteq I \subseteq \{1, \ldots, d\} $$ does not hit the corner. Then an $\SRBM^d(R, \mu, A)$ does not hit the edge $S_J$. \end{thm}
The last of our main results about SRBM links hitting corners to hitting edges. Note that in this case, the condition that $R$ is a reflection nonsingular $\mathcal M$-matrix, rather than just a completely-$\mathcal S$ matrix, is crucial; the reader can see that the proof heavily uses comparison techniques from \cite{MyOwn2}, which, in turn, apply the condition that $R$ is a reflection nonsingular $\mathcal M$-matrix.
\begin{thm} \label{corner2edge2}
Consider an $\SRBM^d(R, \mu, A)$ with a reflection nonsingular $\mathcal M$-matrix $R$. Fix a nonempty subset $I \subseteq \{1, \ldots, d\}$. Suppose an $\SRBM^{|I|}([R]_I, [\mu]_I, [A]_I)$ hits the corner. Then an $\SRBM^d(R, \mu, A)$ hits the edge $S_I$. \end{thm}
The rest of the section will be devoted to the proofs of Theorems~\ref{cornerthm}, ~\ref{corner2edge} and~\ref{corner2edge2}.
\subsection{Proof of Theorem~\ref{cornerthm}} First, we present an informal overview of the proof, and then give a complete proof.
\subsubsection{Outline of the proof.} First, we show (i). Let $Z = (Z(t), t \ge 0)$ be an $\SRBM^d(R, \mu, A)$, starting from $z \in S$. By Proposition~\ref{432}, we can assume $z \in S\setminus\partial S$, and $\mu = 0$. Consider the function \begin{equation} \label{101} F(x) := x'Qx. \end{equation} The matrix $Q$ is strictly copositive. Therefore, if $F(x) = 0$ for a certain $x \in S$, then $x = 0$. Therefore, the process $Z$ hits the corner if and only if the process $F(Z(\cdot))$ hits zero. Let $L = (L(t), t \ge 0)$ be the vector of regulating processes for $Z$, and let $B = (B(t), t \ge 0)$ be the driving Brownian motion for $Z$, so that we have: \begin{equation} \label{6112} Z(t) = B(t) + RL(t),\ t \ge 0. \end{equation} If we write an equation for $F(Z(\cdot))$ using the It\^o-Tanaka formula, we get: $$ \mathrm{d} F(Z(t)) = \beta(t)\mathrm{d} t + \mathrm{d} M(t) + \mathrm{d} l(t), $$ where $l = (l(t), t \ge 0)$ is a {\it nondecreasing} process, and $M = (M(t), t \ge 0)$ is a local martingale. Since we wish to prove that $F(Z(t)) > 0$ for all $t > 0$, we can eliminate the term $l$. In other words, it suffices to prove this property of staying positive for the process $U = (U(t), t \ge 0)$ given by $$ \mathrm{d} U(t) = \beta(t)\mathrm{d} t + \mathrm{d} M(t). $$ It turns out that the drift coefficient $\beta$ is constant. The local martingale $M = (M(t), t \ge 0)$ can be represented as $\mathrm{d} M(t) = \rho(t)\mathrm{d} W(t)$, where $W = (W(t), t \ge 0)$ is a standard Brownian motion. The diffusion coefficient $\rho(t)$ is comparable with that in the SDE for Bessel squared process. After an appropriate random time-change, we can make the diffusion coefficient exactly equal to the one for a Bessel squared process. However, this will not turn our process into a Bessel squared process. Indeed, the drift coefficient for the new process will not be constant (and for a Bessel squared process, it is constant). Still, we can bound this drift coefficient from below by $2$. But we know that a Bessel squared process hits zero if and only if its index is less than two. Therefore, our time-changed process (together with the original process $F(Z(\cdot))$) does not hit zero. This, in turn, means that the process $Z$ does not hit the origin. The proof of (ii) is similar, only there the process $l = (l(t), t \ge 0)$ is nonincreasing, and we bound the new drift coefficient from above by something strictly less than $2$. Together, this means that the process $F(Z(\cdot))$ does indeed hit zero, and the process $Z$ hits the origin.
\subsubsection{Complete proof}
We split the proof into several lemmata.
\begin{lemma} The process $F(Z(\cdot))$ can be represented as \begin{equation} \label{eq:rep-F-Z} \mathrm{d} F(Z(t)) = \rho(t)\mathrm{d}\overline{W}(t) + \tr\bigl(QA\bigr)\mathrm{d} t + l(t), \end{equation} where $\overline{W} = (\overline{W}(t), t \ge 0)$ is a standard Brownian motion, \begin{equation} \label{eq:rep-q} \rho(t) := \left(Z'(t)QAQZ(t)\right)^{1/2},\ \ t \ge 0. \end{equation} and $l = (l(t), t \ge 0)$ is a continuous nondecreasing process with $l(0) = 0$. \label{lemma:rep-F-Z} \end{lemma}
The proof of this lemma involves little more than applying It\^o-Tanaka's formula and some computations. It is postponed until the end of this subsection. Assuming we established this lemma, let us complete the proof.
Define $$ \tau := \inf\{t \ge 0\mid Z(t) = 0\} = \inf\{t \ge 0\mid F(Z(t)) = 0\}. $$ Then $\tau > 0$ a.s., because $Z(0) = x > 0$. For $s < \tau$, we have $Z(s) \in S\setminus\{0\}$, and $F(Z(s)) > 0$. It follows from the definition of constants $c_{\pm}$ that $$ \frac12c_-^{1/2} \le \frac{\rho(s)}{2F^{1/2}(Z(s))} = \frac12\left(\frac{Z'(s)QAQZ(s)}{Z'(s)QZ(s)}\right)^{1/2} \le \frac12c_+^{1/2}. $$ Make the following time change: $$ \Delta(t) := \int_0^t\frac{\rho^2(s)}{4F(Z(s))}\mathrm{d} s,\ \ t \le \tau. $$ By \cite[Lemma 2]{MyOwn1}, this is a strictly increasing function on $[0, \tau]$ with $\Delta(0) = 0$. Denote $\tau_0 := \Delta(\tau)$. Define the inverse of $\Delta$ by $$ \chi(s) := \inf\{t \ge 0\mid \Delta(t) \ge s\}. $$
\begin{lemma} The time-changed process $$ V = (V(s), s < \tau_0),\ \ \mbox{defined by}\ \ V(s) \equiv F(Z(\chi(s))), $$ satisfies the following equation: \begin{equation} \label{eq:process-V} \mathrm{d} F(Z(\chi(s))) = \beta(s)\mathrm{d} s + 2V^{1/2}(s)\mathrm{d} W(s) + \overline{l}(s), \end{equation} where $\overline{l}(s) := l(\chi(s))$ is a nondecreasing process, $\beta = (\beta(s), s < \tau_0)$ is a certain drift coefficient satisfying $\beta(s) \ge \beta \ge 2$ for all $s \in [0, \tau_0)$, and $W = (W(s), s \ge 0)$ is a standard Brownian motion. \end{lemma}
\begin{proof} By \cite[Lemma 2]{MyOwn1}, the process $V = (V(s), s \ge 0)$ satisfies the following equation: $$ \mathrm{d} V(s) = \tr\bigl(QA\bigr)\frac{V(s)}{\rho^2(\chi(s))}\mathrm{d} s + 2V^{1/2}(s)\mathrm{d} W(s) + l(\chi(s)). $$ Here, $W = (W(t), t \ge 0)$ is yet another standard Brownian motion. Note that $$ \frac14c_- \le \Delta'(s) = \frac{\rho^2(s)}{4F(Z(s))} = \frac{Z'(s)QAQZ(s)}{4Z'(s)QZ(s)} \le \frac14c_+. $$ Therefore, the mapping $\Delta : [0, \tau) \to [0, \tau_0)$ is one-to-one, and $\tau = \infty$ if and only if $\tau_0 = \infty$. Note that $$ \frac{V(s)}{\rho^2(\chi(s))} \ge c_+^{-1}, $$ and $\tr(QA) \ge 2c_+ \ge 0$. Therefore, $$ \tr\bigl(QA\bigr)\frac{V(s)}{\rho^2(\chi(s))} \ge \tr\bigl(QA\bigr)c_+^{-1} =: \beta \ge 2. $$ This completes the proof. \end{proof}
Now, note that we have: $$ \mathbf P\left(\exists t > 0: F(Z(t)) = 0\right) = 0\ \ \mbox{if and only if}\ \ \mathbf P\left(\exists s > 0: V(s) = 0\right) = 0. $$ Suppose the condition~\eqref{Ncorner} holds. We need to prove that the process $Z$ does not hit the corner. Assume the converse. Then $\mathbf P(\tau < \infty) > 0$, and $\mathbf P(\tau_0 < \infty) > 0$. On the event $\{\tau_0 < \infty\}$, we have: $V(\tau_0) = 0$. Consider the squared Bessel process $\overline{V} = (\overline{V}(s), s \ge 0)$, given by the equation $$ \mathrm{d}\overline{V}(s) = 2\overline{V}^{1/2}(s)\mathrm{d} W(s) + \beta \mathrm{d} s,\ \ \overline{V}(0) = V(0). $$ Since $\beta \ge 2$, it is known (see, e.g., \cite[Section 11.1, p. 442]{RevuzYorBook}) that $\overline{V}$ a.s. does not hit $0$. It follows from Lemma~\ref{lemma:aux-comparison} in the Appendix that $V(s) \ge \overline{V}(s)$ a.s. for $s < \tau_0$. If $\tau_0 < \infty$, then by continuity $V(\tau_0) \ge \overline{V}(\tau_0) > 0$, but $V(\tau_0) = 0$. This contradiction completes the proof of (i). The proof of (ii) is similar.
\noindent {\it Proof of Lemma~\ref{lemma:rep-F-Z}.} Recall the definition of function $F$ from~\eqref{101}. Since the matrix $Q$ is strictly copositive, we have: $F(x) > 0$ for $x \in S\setminus\{0\}$. Since the matrix $Q$ is symmetric, the first and second order derivatives of the function $F$ are $$ \frac{\partial F}{\partial x_i} = \bigl(2Qx\bigr)_i = 2\sum\limits_{k=1}^dq_{ik}x_k,\ \ \frac{\partial^2F}{\partial x_i\partial x_j} = 2q_{ij},\ \ i, j = 1, \ldots, d. $$ Note that $\langle Z_i, Z_j\rangle_t = \langle B_i, B_j\rangle_t = a_{ij}t$. By the It\^o-Tanaka formula applied to the process $Z$ from~\eqref{6112} and the function $F$ from~\eqref{101}, we have: \begin{align*} \mathrm{d} F(Z(t)) =& \sum\limits_{i=1}^d\frac{\partial F}{\partial x_i}(Z(t))\mathrm{d} Z_i(t) + \frac12\sum\limits_{i=1}^d\sum\limits_{j=1}^d\frac{\partial^2F}{\partial x_i\partial x_j}(Z(t))\mathrm{d}\langle Z_i, Z_j\rangle_t \\ &=\sum\limits_{i=1}^d\bigl(2QZ(t)\bigr)_i\mathrm{d} B_i(t) + \sum\limits_{i=1}^d\sum\limits_{k=1}^d\bigl(2QZ(t)\bigr)_ir_{ik}\mathrm{d} L_k(t) + \sum\limits_{i=1}^d\sum\limits_{j=1}^dq_{ij}a_{ij}\mathrm{d} t\\ &= 2\sum\limits_{i=1}^d\sum\limits_{j=1}^dq_{ij}Z_j(t)\mathrm{d} B_i(t) + 2\sum\limits_{i=1}^d\sum\limits_{j=1}^d\sum\limits_{k=1}^dq_{ij}Z_j(t)r_{ik}\mathrm{d} L_k(t) + \tr\bigl(QA\bigr)\mathrm{d} t. \end{align*} Let us show that the following process is nondecreasing: $$ l(t) := \int_0^t2\sum\limits_{i=1}^d\sum\limits_{j=1}^d\sum\limits_{k=1}^dq_{ij}Z_j(s)r_{ik}\mathrm{d} L_k(s)\mathrm{d} s. $$ Indeed, \begin{align*} \sum\limits_{i=1}^d\sum\limits_{j=1}^d\sum\limits_{k=1}^d&q_{ij}Z_j(t)r_{ik}\mathrm{d} L_k(t) = \sum\limits_{i=1}^d\sum\limits_{j=1}^d\sum\limits_{k=1}^d\rho_{ji}Z_j(t)r_{ik}\mathrm{d} L_k(t) \\ & = \sum\limits_{j=1}^d\sum\limits_{k=1}^d\bigl(QR\bigr)_{jk}Z_j(t)\mathrm{d} L_k(t) \\ & =
\sum\limits_{j=1}^d\bigl(QR\bigr)_{jj}Z_j(t)\mathrm{d} L_j(t) + \sum\limits_{k \ne j}\bigl(QR\bigr)_{jk}Z_j(t)\mathrm{d} L_k(t). \end{align*} For each $j = 1, \ldots, d$, the regulating process $L_j$ can grow only if $Z_j = 0$: we express this by writing $Z_j(t)\mathrm{d} L_j(t) = 0$. And for $k \ne j$, we have: $(QR)_{jk} \ge 0$ by assumptions of Theorem~\ref{cornerthm}, and $Z_j(t) \ge 0, \mathrm{d} L_k(t) \ge 0$ by definition. Therefore, the process $l$ is nondecreasing.
Now, recall that $B_1, \ldots, B_d$ are driftless one-dimensional Brownian motions (they are driftless, because the drift $\mu = 0$, according to our assumptions). Therefore, the following process is a continuous local martingale: $$ M = (M(t), t \ge 0),\ \ M(t) := 2\sum\limits_{i=1}^d\sum\limits_{j=1}^d\int_0^t\rho_{ij}Z_j(s)\mathrm{d} B_i(s). $$ We can represent the process $F(Z(\cdot))$ as follows: $$ \mathrm{d} F(Z(t)) = \mathrm{d} M(t) + \tr\bigl(QA\bigr)\mathrm{d} t + \mathrm{d} l(t). $$ Let us calculate the quadratic variation of $M$. Recall that, by definition of the process $B$, $\langle B_i, B_j\rangle_t = a_{ij}t$. Let $$ M_{ij}(t) = \int_0^t\int_0^tZ_j(s)q_{ij}\mathrm{d} B_i(s),\ \ i, j = 1, \ldots, d. $$ For $i, j, k, l = 1, \ldots, d$, we have: $$ \langle M_{ij}, M_{kl}\rangle_t = \int_0^tZ_j(s)q_{ij}Z_l(s)q_{kl}a_{ik}\mathrm{d} s. $$ But the quadratic variation of $M = \sum_{i=1}^d\sum_{j=1}^dM_{ij}$ is equal to the sum \begin{align*} \langle M\rangle_t = & \sum\limits_{i=1}^d\sum\limits_{j=1}^d\sum\limits_{k=1}^d\sum\limits_{l=1}^d\langle M_{ij}, M_{kl}\rangle_t = \sum\limits_{i=1}^d\sum\limits_{j=1}^d\sum\limits_{k=1}^d\sum\limits_{l=1}^d\int_0^t Z_j(s)q_{ij}Z_l(s)q_{kl}a_{ik}\mathrm{d} s \\ &= \sum\limits_{i=1}^d\sum\limits_{j=1}^d\sum\limits_{k=1}^d\sum\limits_{l=1}^d\int_0^t Z_j(s)q_{ij}a_{ik}q_{kl}Z_l(s)\mathrm{d} s = \int_0^t\left(Z'(s)QAQZ(s)\right)\mathrm{d} s. \end{align*} Then we can represent $M$ as the stochastic integral $$ M(t) = 2\int_0^t\rho(s)\mathrm{d} \overline{W}(s), $$ where $\overline{W} = (\overline{W}(t), t \ge 0)$ is a standard Brownian motion. This completes the proof of Lemma~\ref{lemma:rep-F-Z}.
\subsection{Proof of Theorem~\ref{corner2edge}}
We prove this theorem using induction by $d$. The induction base is trivial. Induction step: assume that the statement is true for $d-1$ instead of $d$, and try to prove it for $d$. For $\varepsilon \in (0, 1)$, let $K_{\varepsilon} = \{x \in S\mid \varepsilon \le \norm{x} \le \varepsilon^{-1}\}$. Fix a point $z \in S\setminus\{0\}$, so that $z \in K_{\varepsilon}$ for all $\varepsilon > 0$ small enough. Start a copy of an $\SRBM^d(R, \mu, A)$ from $z$ (we can assume this by Proposition~\ref{432}). Denote this copy by $Z = (Z(t), t \ge 0)$, and let $B = (B(t), t \ge 0)$ be its driving Brownian motion. Let $$ \tau := \inf\{t \ge 0\mid Z(t) \in S_I\} $$ be the first moment when the process $Z$ hits the edge $S_I$. We need to show that $\tau = \infty$ a.s. Let $$ \eta_{\varepsilon} := \inf\{t \ge 0\mid Z(t) \in K_{\varepsilon}\}. $$ Note that $\eta_{\varepsilon} \le \eta_{\varepsilon'}$ when $\varepsilon' \le \varepsilon$, and $\lim_{\varepsilon \downarrow 0}\eta_{\varepsilon} = \infty$, because by assumptions of the theorem the process $Z$ does not hit the corner: $Z(t) \ne 0$ for all $t \ge 0$ a.s. It suffices to show that $\tau \ge \eta_{\varepsilon}$ for all $\varepsilon \in (0, 1)$. Fix an $\varepsilon \in (0, 1)$. For every $x \in K_{\varepsilon}$, there exists an open neighborhood $U(x)$ of $x$ with the following property: there exists some index $i = i(x) \in \{1, \ldots, d\}$ such that for all $y \in U(x)$ we have: $y_{i(x)} > 0$. Since $K_{\varepsilon}$ is compact, we can extract a finite subcover $U(x_1), \ldots, U(x_s)$. Without loss of generality, let us include the neighborhood $U(x_0)$ of $x_0 = z$ into this subcover. Now, define a sequence of stopping times: $$ \tau_0 := 0,\ \ j_0 := 0;\ \ \tau_{k+1} := \inf\{t \ge \tau_k\mid Z(t) \notin U\left(x_{j_k}\right)\}, $$ and $j_{k+1}$ is defined as any $j = 0, \ldots, s$ such that $Z\left(\tau_{k+1}\right) \in U(x_j)$. Suppose that, at some point, we cannot find such $j$; in other words, $$ Z(\tau_{k+1}) \notin U\left(x_{j_0}\right)\cup U\left(x_{j_1}\right)\cup\ldots\cup U\left(x_{j_s}\right). $$ Then the sequence of stopping times terminates, and we denote $K := k+1$. In this case, we have defined $\tau_0, j_0, \tau_1, j_1, \ldots, \tau_{K-1}, j_{K-1}, \tau_K$. If the sequence does not terminate, we let $K = \infty$. We have: $$ Z_{j_k}(t) > 0\ \ \mbox{for}\ \ t \in [\tau_k, \tau_{k+1}),\ \ k < K. $$ The sequence $(\tau_{k})$ can be either finite or countable. Recall that $U(x_j),\ j = 0, \ldots, s$ is a cover of $K_{\varepsilon}$. Therefore, $\sup_{k}\tau_k \ge \eta_{\varepsilon}$. It suffices to show that $\tau \ge \tau_{k}$. We prove this using induction by $k$.
Induction base: $k = 1$. If $j_0 \in I$, then $Z_{j_0}(t) > 0$ for $t < \tau_1$, and $Z(t) \notin S_I$. In this case, $\tau \ge \tau_1$ is straightforward. Now, if $j_0 \notin I$, then consider the set $J := \{1, \ldots, d\}\setminus\{j_0\}$. We have the following representation: $$ \left([Z(t\wedge\tau_1)]_J,\ t \ge 0\right) = (\overline{Z}(t\wedge\tau_1),\ t \ge 0), $$ where $\overline{Z} = (\overline{Z}(t), t \ge 0)$ is an $\SRBM^{d-1}([R]_J, [\mu]_J, [A]_J)$, starting from $[z]_J$, with the driving Brownian motion $[B]_J = ([B(t)]_J, t \ge 0)$. This process $\overline{Z}$ is well defined, since the matrix $[R]_J$ is a reflection nonsingular $\mathcal M$-matrix, and by Proposition~\ref{existence} there exists a strong version of $\overline{Z}$. By the induction hypothesis, a.s. there does not exist $t \ge 0$ such that $\overline{Z}(t) \in S_I$. For every $y \in S$, we have: $y \in S_I$ if and only if $[y]_J \in S_I$. Therefore, for all $t < \tau_1$ we have: $Z(t) \notin S_I$. This proves that $\tau \ge \tau_1$.
Induction step: suppose $t \ge \tau_k$ and $k < K$, that is, the sequence does not terminate at this step. Then we need to prove $\tau \ge \tau_{k+1}$. Consider the process $(Z(t + \tau_k), t \ge 0)$. This is a version of an $\SRBM^d(R, \mu, A)$, started from $Z(\tau_k)$. But $$ Z(\tau_{k}) \in U\left(x_{j_0}\right)\cup U\left(x_{j_1}\right)\cup\ldots\cup U\left(x_{j_s}\right). $$ There exists $j = 0, \ldots, s$ such that $Z(\tau_k) \in U(x_j)$. In addition, $Z(\tau_k) \in S\setminus\{0\}$, because by induction hypothesis, the process $Z$ never hits the corner. Apply the reasoning from the induction base to this process instead of the original SRBM. The moment $\tau_{k+1} - \tau_k$ plays the role of $\tau_1$ above, and the moment $\tau - \tau_k$ plays the role of $\tau$. Therefore, $\tau - \tau_k \ge \tau_{k+1} - \tau_k$, and $\tau \ge \tau_{k+1}$. This completes the proof.
\subsection{Proof of Theorem~\ref{corner2edge2}}
This theorem is proved using {\it stochastic comparison}.
\begin{defn} Consider two $\mathbb{R}^d$-valued processes $Z = (Z(t), t \ge 0)$ and $\overline{Z} = (\overline{Z}(t), t \ge 0)$. We say that $Z$ is {\it stochastically dominated by} $\overline{Z}$, and write it as $Z \preceq \overline{Z}$, if for every $t \ge 0$ and $y \in \mathbb{R}^d$ we have: $$ \mathbf P(Z(t) \ge y) \le \mathbf P(\overline{Z}(t) \ge y). $$ \end{defn}
\begin{prop} Take a $d\times d$ reflection nonsingular $\mathcal M$-matrix $R$, a $d\times d$ positive definite symmetric matrix $A$, and a drift vector $\mu \in \mathbb{R}^d$. Fix a nonempty subset $I \subseteq \{1, \ldots, d\}$. Let $$
Z = \SRBM^d(R, \mu, A),\ \ \overline{Z} = \SRBM^{|I|}([R]_I, [\mu]_I, [A]_I) $$ such that $[Z(0)]_I$ has the same law as $\overline{Z}(0)$. Then $[Z]_I \preceq \overline{Z}$. \label{propcomp} \end{prop}
This result was shown in \cite[Corollary 3.6]{MyOwn2}; it is an easy corollary of general comparison techniques for reflected processes developed in \cite[Theorem 4.1]{R2000}, see also \cite[Theorem 1.1(i)]{KR2012b}, \cite[Theorem 3.1]{Haddad2010}, \cite[Theorem 6(i)]{KW1996}. Now, it is easy to see that Theorem~\ref{corner2edge2} trivially follows from Proposition~\ref{propcomp}.
\subsection{Corollaries of the main results for an SRBM}
The following corollary of Theorem~\ref{corner2edge} gives a sufficient condition for not hitting edges of a given order.
\begin{cor} Consider an $\SRBM^d(R, \mu, A)$. Fix $p = 2, \ldots, d-1$. Suppose for every $I \subseteq \{1, \ldots, d\}$ such that $|I| \ge p$ the process $\SRBM^{|I|}([R]_I, [\mu]_I, [A]_I)$ does not hit the corner. Then an $\SRBM^d(R, \mu, A)$ does not hit edges of order $p$. \end{cor}
The next corollary combines the results of Corollary~\ref{cor:inv-R}, Theorem~\ref{corner2edge} and Theorem~\ref{corner2edge2}. Its proof is trivial and is omitted.
\begin{cor} Take an $\SRBM^d(R, \mu, A)$. Suppose the matrix $R$ is a reflection nonsingular $\mathcal M$-matrix and there exists a diagonal matrix $C = \diag(c_1, \ldots, c_d)$ with $c_1, \ldots, c_d > 0$ such that $RC = \overline{R}$ is symmetric. \label{general}
(i) Fix a nonempty subset $J \subseteq \{1, \ldots, d\}$. Suppose that for every subset $I$ such that $J \subseteq I \subseteq \{1, \ldots, d\}$ we have: \begin{equation} \label{Nedge}
\tr\left([\overline{R}]_I^{-1}[A]_I\right) \ge 2\!\!\!\!\max\limits_{x \in \mathbb{R}^{|I|}_+\setminus\{0\}}\frac{x'[\overline{R}]_I^{-1}[A]_I[\overline{R}]_I^{-1}x}{x'[\overline{R}]_I^{-1}x}. \end{equation} Then the $\SRBM^d(R, \mu, A)$ avoids $S_I$.
(ii) Fix $p = 1, \ldots, d-1$. Suppose for every subset $I \subseteq \{1, \ldots, d\}$ with $|I| \ge p$ we have: $$
\tr\left([\overline{R}]_I^{-1}[A]_I\right) \ge 2\!\!\!\!\max\limits_{x \in \mathbb{R}^{|I|}_+\setminus\{0\}}\frac{x'[\overline{R}]_I^{-1}[A]_I[\overline{R}]_I^{-1}x}{x'[\overline{R}]_I^{-1}x}. $$ Then the $\SRBM^d(R, \mu, A)$ avoids edges of order $p$.
(iii) Suppose there exists a subset $I \subseteq \{1, \ldots, d\}$ such that $$
\tr\left([\overline{R}]_I^{-1}[A]_I\right) < 2\!\!\!\!\min\limits_{x \in \mathbb{R}^{|I|}_+\setminus\{0\}}\frac{x'[\overline{R}]_I^{-1}[A]_I[\overline{R}]_I^{-1}x}{x'[\overline{R}]_I^{-1}x}. $$ Then the $\SRBM^d(R, \mu, A)$ hits $S_I$. \end{cor}
\section{Proofs of Theorems~\ref{totalcor}, ~\ref{cams} and~\ref{mainthm}}
\subsection{Outline of the proofs}
Consider a system of competing Brownian particles from Definition~\ref{classicdef}. In Lemma~\ref{red}, we note that a multicollision with pattern $I$ is equivalent to an $\SRBM^{N-1}(R, \mu, A)$ hitting the edge $S_I$ of the $N-1$-dimensional orthant $\mathbb{R}^{N-1}_+$. Here, the parameters $R$, $\mu$, $A$ are given by~\eqref{R12}, ~\eqref{mu} and~\eqref{A} below. We apply Corollary~\ref{cor:inv-R} and Theorem~\ref{corner2edge} to this SRBM to prove Theorems~\ref{totalcor} and~\ref{mainthm} respectively. We use the estimate in Lemma~\ref{simple} for $c_+$, since the right-hand side of~\eqref{Ncorner} seems hard to compute for matrices $R$ and $A$ given by~\eqref{R12} and~\eqref{A}.
Note that the matrix $R$ from~\eqref{R12} is itself symmetric. Therefore, in Corollary~\ref{cor:inv-R} we can take $C = I_{N-1}$ and $\overline{R} = R$. The inverse matrix $R^{-1} = \overline{R}^{-1} = (\rho_{ij})_{1 \le i, j \le N-1}$ has the form \begin{equation} \label{invR} \rho_{ij} = \begin{cases} 2i(N-j)/N,\ i \le j;\\ 2j(N-i)/N,\ i \ge j \end{cases} \end{equation} This result can be found in \cite{FP2001, Inversion} (the latter article deals with a slightly different matrix, from which one can easily find the inverse of the given matrix $R$). After a (rather tedious) computation, we rewrite the condition~\eqref{Ncorner-om} from Corollary~\ref{cor:inv-R} as $\mathcal P(\sigma) \ge 0$, where $\mathcal P(\sigma)$ is defined in~\eqref{CP}. This proves Theorem~\ref{totalcor}.
Proving Theorem~\ref{mainthm} is a bit harder. Apply Theorem~\ref{corner2edge}, and fix a subset $I \subseteq \{1, \ldots, N-1\}$ such that $J \subseteq I$. We need to find a sufficient condition for an $\SRBM^{|I|}([R]_I, [\mu]_I, [A]_I)$ to a.s. avoid the corner of the orthant $\mathbb{R}^{|I|}_+$. We decompose the set $I$ as in~\eqref{decomp}: $$ I = I_1\cup I_2\cup \ldots I_r, $$
into a union of disjoint non-adjacent discrete intervals. In Lemma~\ref{cornerstone}, we prove that if $I$ satisfies Assumption (B), then the $\SRBM^{|I|}([R]_I, [\mu]_I, [A]_I)$ indeed avoids the corner. This completes the proof of Theorem~\ref{mainthm}. But to prove Lemma~\ref{cornerstone}, we need to consider different variants of decomposition~\eqref{decomp}. For example, if $I_1 = \{1\}$ and $I_2 = \{3\}$, then this guarantees that an $\SRBM^{|I|}([R]_I, [\mu]_I, [A]_I)$ avoids the corner. Various cases are considered in Lemmas~\ref{61}, ~\ref{62} and~\ref{63}, which constitute the crux of the proof.
\subsection{Connection between an SRBM and competing Brownian particles}
Let us reduce multiple collisions of competing Brownian particles to an SRBM hitting edges of the boundary of high order. Consider the classical system of competing Brownian particles from Definition~\ref{classicdef}. By definition, the ranked particles $Y_1, \ldots, Y_N$ satisfy $$ Y_1(t) \le \ldots \le Y_N(t). $$ Consider the {\it gap process}: an $\mathbb{R}^{N-1}_+$-valued process defined by $$ Z = (Z(t), t \ge 0),\ \ Z(t) = (Z_1(t), \ldots, Z_{N-1}(t))', \ \ Z_k(t) = Y_{k+1}(t) - Y_k(t). $$ It was shown in \cite{BFK2005} that this is an $\SRBM^{N-1}(R, \mu, A)$ in the orthant $S = \mathbb{R}_+^{N-1}$ with parameters \begin{equation} \label{R12} R = \begin{bmatrix} 1 & -1/2 & 0 & 0 & \ldots & 0 & 0\\ -1/2 & 1 & -1/2 & 0 & \ldots & 0 & 0\\ 0 & -1/2 & 1 & 0 & \ldots & 0 & 0\\ \vdots & \vdots & \vdots & \vdots & \ddots & \ddots & \ddots\\ 0 & 0 & 0 & 0 & \ldots & 1 & -1/2\\ 0 & 0 & 0 & 0 & \ldots & -1/2 & 1 \end{bmatrix}, \end{equation} \begin{equation} \label{mu} \mu = \left(g_2 - g_1, g_3 - g_4, \ldots, g_N - g_{N-1}\right)', \end{equation} \begin{equation} \label{A} A = \begin{bmatrix} \sigma_1^2 + \sigma_2^2 & -\sigma_2^2 & 0 & 0 & \ldots & 0 & 0\\ -\sigma_2^2 & \sigma_2^2 + \sigma_3^2 & -\sigma_3^2 & 0 & \ldots & 0 & 0\\ 0 & -\sigma_3^2 & \sigma_3^2 + \sigma_4^2 & -\sigma_4^2 & \ldots & 0 & 0\\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ 0 & 0 & 0 & 0 & \ldots & \sigma_{N-2}^2 + \sigma_{N-1}^2 & -\sigma_{N-1}^2\\ 0 & 0 & 0 & 0 & \ldots & -\sigma_{N-1}^2 & \sigma_{N-1}^2 + \sigma_N^2 \end{bmatrix} \end{equation} Note that the matrix $R$ is a reflection nonsingular $\mathcal M$-matrix. This follows from the fact that $I_{N-1} - R \ge 0$, and $R^{-1} \ge 0$ (which, in turn, was proved in \cite[Proposition 2.1(i)]{MyOwn6}). The following lemma translates statements about multiple collisions and multicollisions of competing Brownian particles to the language of an SRBM. The proof is trivial and is therefore omitted.
\begin{lemma} \label{red} Consider a classical system of $N$ competing Brownian particles from Definition~\ref{classicdef}. Then there is a multicollision with pattern $I$ at time $t$ if and only if the gap process hits the edge $S_I$ at time $t$. For example, there is a total collision at time $t$ if and only if the gap process hits the corner at time $t$. \end{lemma}
For example, $Y_1(t) = Y_2(t)$ and $Y_3(t) = Y_4(t) = Y_5(t)$ is a multicollision of order $3$, with pattern $\{1, 3, 4\}$, which is equivalent of the gap process hitting the edge $\{z_1 = z_3 = z_4 = 0\}$. Similarly, $Y_3(t) = Y_4(t) = Y_5(t) = Y_6(t)$ is a collision of order $3$ (which is also a particular case of a multicollision of order $3$, with pattern $\{3, 4, 5\}$), and it is equivalent to the gap process hitting the edge $\{z_3 = z_4 = z_5 = 0\}$.
\subsection{Some preliminary calculations}
As mentioned before, the matrix $R$ in~\eqref{R12} is itself symmetric. Therefore, we can take $C = I_{N-1}$, and $\overline{R} = R$. Without loss of generality, let $$ \rho_{ij} = 0,\ \ i = 0,\ N,\ j = 0, \ldots, N\ \mbox{or}\ j = 0,\ N,\ i = 0, \ldots, N. $$ This is consistent with the notation~\eqref{invR}. Note that $\rho_{ij} > 0$ for $i, j = 1, \ldots, N-1$: all elements of the matrix $R^{-1}$ are positive. Therefore, we can apply an estimate from Lemma~\ref{simple}: $$ c_+ := \max\limits_{x \in \mathbb{R}^{N-1}\setminus\{0\}} \frac{x'R^{-1}AR^{-1}x}{x'R^{-1}x} \le \max\limits_{1 \le k \le l \le N-1}\frac{\bigl(R^{-1}AR^{-1}\bigr)_{kl}}{\rho_{kl}}. $$
\begin{lemma} \label{trace} For the matrix $R$ given by~\eqref{R12} and the matrix $A$ given by~\eqref{A}, we have in the notation of ~\eqref{CT}: \begin{equation} \label{CTcalc} \tr\bigl(R^{-1}A\bigr) = \mathcal T(\sigma)\,. \end{equation}
\end{lemma}
\begin{proof} Straightforward calculation gives \begin{align*} \tr\bigl(R^{-1}A\bigr) =& \sum\limits_{i=1}^{N-1}\sum\limits_{j=1}^{N-1}\rho_{ij}a_{ij} = \sum\limits_{i=1}^{N-1}(\sigma_i^2 + \sigma_{i+1}^2)\frac{2i(N-i)}N \\ & + 2\sum\limits_{i=2}^{N-1}(-\sigma_i^2)\frac{2(i-1)(N-i)}N = \frac{2(N-1)}N\sigma_1^2 + \frac{2(N-1)}N\sigma_N^2 \\ & + \sum\limits_{k=2}^{N-1}\sigma_k^2\left(\frac{2k(N-k)}N + \frac{2(k-1)(N-k+1)}N - 2\frac{2(k-1)(N-k)}N\right) \\ & = \frac{2(N-1)}N\sum\limits_{k=1}^N\sigma_k^2 = \mathcal T(\sigma). \end{align*} \end{proof}
The following lemma helps us simplify the matrix $R^{-1}AR^{-1}$, where $A$ is given by~\eqref{A}, and $R^{-1}$ is given by~\eqref{invR}.
\begin{lemma} Consider the matrix $A$ as in~\eqref{A}, and take a symmetric $(N-1)\times (N-1)$-matrix $Q = (q_{ij})$. Augment it by two additional rows and two additional columns, one from each side, and fill them with zeros: $$ q_{ij} = 0\ \ \mbox{for}\ \ i = 0,\, N,\ j = 0, \ldots, N, \ \mbox{and for}\ \ j = 0,\, N,\ i = 0, \ldots, N. $$ Then for $k, l = 1, \ldots, N-1$ we have: $$ (QAQ)_{kl} = \sum\limits_{p=1}^N\left(q_{pk} - q_{p-1, k}\right)\left(q_{pl} - q_{p-1, l}\right)\sigma_p^2. $$ \label{matrixes} \end{lemma}
\begin{proof} The matrix $A$ is tridiagonal: $$ \begin{cases} a_{ii} = \sigma_i^2 + \sigma_{i+1}^2,\ i = 1, \ldots, N-1;\\ a_{i, i+1} = a_{i+1, i} = -\sigma_{i+1}^2,\ i = 1, \ldots, N-2;\\
a_{ij} = 0,\ \ i, j = 1, \ldots, N-1,\ |i - j| \ge 2. \end{cases} $$ Using the symmetry of $Q$, we have: \begin{align*} (QAQ)_{kl} =& \sum\limits_{i=1}^{N-1}\sum\limits_{j=1}^{N-1}q_{ik}q_{jl}a_{ij} = \sum\limits_{p=1}^{N-1}\left(\sigma_p^2 + \sigma_{p+1}^2\right)q_{pk}q_{pl} - \sum\limits_{p=2}^{N-1}\sigma_p^2q_{pk}q_{p-1, l} - \sum\limits_{p=2}^{N-1}\sigma_p^2q_{p-1, k}q_{pl} \\ & = \sum\limits_{p=1}^{N}\sigma_p^2q_{pk}q_{pl} + \sum\limits_{p=1}^{N}\sigma_p^2q_{p-1, k}q_{p-1, l} - \sum\limits_{p=1}^{N}\sigma_p^2q_{pk}q_{p-1, l} - \sum\limits_{p=1}^{N}\sigma_p^2q_{p-1, k}q_{pl} \\ & = \sum\limits_{p=1}^{N}\left(q_{pk} - q_{p-1, k}\right)\left(q_{pl} - q_{p-1, l}\right)\sigma_p^2. \end{align*} \end{proof}
Lemma~\ref{matrixes} enables us to calculate $(R^{-1}AR^{-1})_{kl}$, where $A$ and $R$ are given by~\eqref{A} and~\eqref{R12}.
\begin{lemma} Suppose the matrix $R$ is given by~\eqref{R12}, and the matrix $A$ is given by~\eqref{A}. Then for $1 \le k \le l \le N-1$ we have: \begin{equation} \label{RAR} \bigl(R^{-1}AR^{-1}\bigr)_{kl} = \frac{4(N-k)(N-l)}{N^2}\sum\limits_{p=1}^k\sigma_p^2 - \frac{4k(N-l)}{N^2}\sum\limits_{p=k+1}^l\sigma_p^2 + \frac{4kl}{N^2}\sum\limits_{p=l+1}^N\sigma_p^2. \end{equation} \label{RARlemma} \end{lemma}
\begin{proof} Apply Lemma~\ref{matrixes} to $Q = R^{-1}$, given by~\eqref{invR}, so that $q_{ij} = \rho_{ij}$. For $p \le k$, we get: For $p \le k$ we have: $$ \rho_{pk} - \rho_{p-1, k} = \frac{2p(N-k)}{N} - \frac{2(p-1)(N-k)}N = \frac{2(N-k)}N, $$ $$ \rho_{pl} - \rho_{p-1, l} = \frac{2p(N-l)}{N} - \frac{2(p-1)(N-l)}N = \frac{2(N-l)}N. $$ For $k < p \le l$, we have: $$ \rho_{pk} - \rho_{p-1, k} = \frac{2k(N-p)}{N} - \frac{2k(N-p+1)}N = -\frac{2k}N, $$ $$ \rho_{pl} - \rho_{p-1, l} = \frac{2p(N-l)}{N} - \frac{2(p-1)(N-l)}N = \frac{2(N-l)}N. $$ For $p > l$, we have: $$ \rho_{pk} - \rho_{p-1, k} = \frac{2p(N-k)}{N} - \frac{2(p-1)(N-k)}N = \frac{2(N-k)}N, $$ $$ \rho_{pl} - \rho_{p-1, l} = \frac{2p(N-l)}{N} - \frac{2(p-1)(N-l)}N = \frac{2(N-l)}N. $$ The rest of the proof is trivial. \end{proof}
\subsection{Proof of Theorem~\ref{totalcor}}
Use Corollary~\ref{cor:inv-R} and Corollary~\ref{simple} for matrices $R$ and $A$, given by~\eqref{R12} and~\eqref{A} respectively. We have the following sufficient condition for avoiding total collisions: \begin{equation} \label{form} \tr\bigl(R^{-1}A\bigr) - 2\max\limits_{1 \le k \le l \le N-1}\frac{(R^{-1} AR^{-1})_{kl}}{\rho_{kl}} \ge 0. \end{equation} For $1 \le k \le l \le N-1$, denote $$ c_{k, l}(\sigma) = \tr\bigl(R^{-1}A\bigr) - 2\frac{\bigl(R^{-1}AR^{-1}\bigr)_{kl}}{\rho_{kl}}. $$ Then we have: \begin{equation} \label{3} \tr\bigl(R^{-1}A\bigr) - 2\max\limits_{k, l = 1, \ldots, N-1}\frac{(R^{-1} AR^{-1})_{kl}}{\rho_{kl}} = \min\limits_{1 \le k \le l \le N-1}c_{k, l}(\sigma). \end{equation}
\begin{lemma} Using definitions of $c_l(\sigma)$ and $\sigma^{\leftarrow}$ from subsection 1.2, we have:
(i) For $2 \le k \le l \le N-2$, we have: $c_{k, l}(\sigma) \ge 0$.
(ii) For $1 = k \le l \le N - 1$, we have: $c_{k, l}(\sigma) = c_l(\sigma)$.
(iii) For $1 \le k \le l = N-1$, we have: $c_{k, l}(\sigma) = c_{N-k}\left(\sigma^{\leftarrow}\right)$.
\label{5555} \end{lemma}
Assuming that Lemma~\ref{5555} is proved, let us finish the proof of Theorem~\ref{totalcor}. Let \begin{equation} \label{4} \delta(\sigma) := \min\limits_{2 \le k \le l \le N-2}c_{k, l}(\sigma). \end{equation} If $N < 4$, let $\delta(\sigma) := 0$. By Lemma~\ref{5555} (i), we always have: $\delta(\sigma) \ge 0$. Recall the definition of $\mathcal P(\sigma)$ from~\eqref{CP} and use Lemma~\ref{5555} (ii), (iii): \begin{equation} \label{5} \min\left(c_{1, 1}(\sigma),\, c_{1, 2}(\sigma),\, \ldots,\, c_{1, N-1}(\sigma),\, c_{2, N-1}(\sigma),\, \ldots,\, c_{N-1, N-1}(\sigma)\right) = \mathcal P(\sigma). \end{equation} Comparing~\eqref{3}, ~\eqref{4} and~\eqref{5}, we have: \begin{equation} \label{10000} \min\limits_{1 \le k \le l \le N-1}\left[\tr\bigl(R^{-1}A\bigr) - 2\frac{(R^{-1} AR^{-1})_{kl}}{\rho_{kl}}\right] = \min(\mathcal P(\sigma), \delta(\sigma)). \end{equation} Thus $$ \min\limits_{1 \le k \le l \le N-1}c_{k, l}(\sigma) \ge 0\ \ \mbox{if and only if}\ \ \mathcal P(\sigma) \ge 0. $$ This completes the proof of Theorem~\ref{totalcor}. \qed
\noindent
{\it Proof of Lemma~\ref{5555}}: We can simplify the expression for $c_{k, l}(\sigma)$. Applying~\eqref{RAR} and~\eqref{invR}, we have: for $1 \le k \le l \le N -1$, $$ \frac{\bigl(R^{-1}AR^{-1}\bigr)_{kl}}{\rho_{kl}} = \frac{2(N-k)}{Nk}\sum\limits_{p=1}^k\sigma_p^2 - \frac{2}{N}\sum\limits_{p=k+1}^l\sigma_p^2 + \frac{2l}{N(N-l)}\sum\limits_{p=l+1}^N\sigma_p^2. $$ Therefore, we have: \begin{align*} c_{k, l}(\sigma) := & \left(\frac{2(N-1)}{N} - \frac{4(N-k)}{Nk}\right)\sum\limits_{p=1}^k\sigma_p^2 \\ & + \left(\frac{2(N-1)}{N} + \frac{4}{N}\right)\sum\limits_{p=k+1}^l\sigma_p^2 + \left(\frac{2(N-1)}{N} - \frac{4l}{(N-l)N}\right)\sum\limits_{p=l+1}^N\sigma_p^2 \\ & = \frac{2(N-1)k - 4(N-k)}{kN}\sum\limits_{p=1}^k\sigma_p^2 + \frac{2(N+1)}N\sum\limits_{p=k+1}^l\sigma_p^2 \\ &+ \frac{2(N-1)(N-l) - 4l}{(N-l)N}\sum\limits_{p=l+1}^N\sigma_p^2. \end{align*} Now, for $k \ge 2$ we get: $$ 2(N-1)k - 4(N - k) \ge 4(N-1) - 4N + 8 = 4 \ge 0. $$ Similarly, for $l \le N-2$ we get: $$ 2(N-1)(N-l) - 4l \ge 0. $$ This proves part (i) of Lemma~\ref{5555}. Parts (ii) and (iii) are now straightforward. \ \ $\square$
\subsection{Proof of Theorem~\ref{mainthm}} Fix a subset $I \subseteq \{1, \ldots, N-1\}$ such that $J \subseteq I$. Take the matrices $R$ and $A$ given by~\eqref{R12} and~\eqref{A}. Essentially, we need to prove the following lemma:
\begin{lemma} \label{cornerstone} If the subset $I$ satisfies Assumption (B), then the process $$
Z = (Z(t), t \ge 0) = \SRBM^{|I|}\left([R]_I, 0, [A]_I\right) $$ does not hit the origin. \end{lemma}
If we prove Lemma~\ref{cornerstone}, then Theorem~\ref{mainthm} will automatically follow from this lemma and Theorem~\ref{corner2edge}. The rest of this subsection is devoted to the proof of Lemma~\ref{cornerstone}.
Let us investigate the structure of the matrices $[R]_I^{-1}$ and $[A]_I^{-1}$. Split $I$ into disjoint non-adjacent discrete intervals: $I = I_1\cup I_2\cup \ldots \cup I_r$. Since the matrices $R$ and $A$ are tridiagonal, the matrices $[R]_I$ and $[A]_I$ have the following block-diagonal form: $$ [R]_I = \diag\left([R]_{I_1}, \ldots, [R]_{I_r}\right),\ \ [A]_I = \diag\left([A]_{I_1}, \ldots, [A]_{I_r}\right). $$ The following processes are independent SRBMs: \begin{equation} \label{components00}
[Z]_{I_j} = ([Z(t)]_{I_j}, t \ge 0) = \SRBM^{|I_j|}\left([R]_{I_j}, 0, [A]_{I_j}\right),\ \ j = 1, \ldots, s. \end{equation} For any subset $I' = I_{i_1}\cup\ldots\cup I_{i_s}$, the process $$
[Z]_{I'} = ([Z(t)]_{I'}, t \ge 0) = \SRBM^{|I'|}\left([R]_{I'}, 0, [A]_{I'}\right). $$ \begin{rmk} \label{rmkcrucial}
If for some choice of $I'$ this process does not hit the origin of $\mathbb{R}^{|I'|}_+$, then the original process $Z$ does not hit the origin, because of independence of~\eqref{components00}. In particular, for each $j = 1, \ldots, s$, the process $[Z]_{I_j}$ does not hit the origin, then $Z$ does not hit the origin. \end{rmk}
Now, let us state three lemmata.
\begin{lemma} If at least two of the discrete intervals $I_1,\ldots, I_r$ are singletons, then $Z$ a.s. at any time $t > 0$ does not hit the origin. \label{61} \end{lemma}
\begin{lemma} If at least one $I_1,\ldots, I_r$ is a two-element subset $\{k-1, k\}$ with local concavity at $k$, then $Z$ a.s. at any time $t > 0$ does not hit the origin. \label{62} \end{lemma}
\begin{lemma} If $I$ satisfies Assumption (A), then $Z$ a.s. at any time $t > 0$ does not hit the origin. \label{63} \end{lemma}
Combining Lemmas~\ref{61}, ~\ref{62}, and~\ref{63} with Remark~\ref{rmkcrucial}, we complete the proof of Lemma~\ref{cornerstone} and Theorem~\ref{mainthm}. \qed
In the remainder of this subsection, we shall prove these three lemmas.
\noindent
{\it Proof of Lemma~\ref{61}}: Without loss of generality, suppose $I_1 = \{k\}$ and $I_2 = \{l\}$ are singletons. Since they are not adjacent, $|k - l| \ge 2$; assume that $k < l$, so that $l \ge k+2$. Then $$ \left(Z_k, Z_l\right)' = \SRBM^2\left([R]_{I_1\cup I_2}, 0, [A]_{I_1\cup I_2}\right). $$ But $$ [A]_{I_1\cup I_2} = \begin{bmatrix} \sigma_k^2 + \sigma_{k+1}^2 & 0 \\ 0 & \sigma_l^2 + \sigma_{l+1}^2\end{bmatrix}, \ \ [R]_{I_1\cup I_2} = I_2. $$ Therefore, $Z_k$ and $Z_l$ are independent reflected Brownian motions on $\mathbb{R}_+$. They do not hit zero simultaneously, which is the same as to say that $(Z_k, Z_l)'$ does not hit the origin in $\mathbb{R}^2_+$.
\noindent {\it Proof of Lemma~\ref{62}}: Assume without loss of generality that $I_1 = \{1, 2\}$, and we have local concavity at $2$: $\sigma_2^2 \ge (\sigma_1^2 + \sigma_3^2)/2$. By Remark~\ref{rmkcrucial}, it suffices to show that an $\SRBM^2([R]_{I_1}, [\mu]_{I_1}, [A]_{I_1})$ does not hit the origin. Because of the connection between an SRBM and systems of competing Brownian particles outlined in subsection 4.2, this, in turn, is equivalent of a system of three competing Brownian particles with diffusion coefficients $\sigma_1^2, \sigma_2^2, \sigma_3^2$ not having a triple collision. But this last statement follows from Proposition~\ref{elegant}, applied to the case $N = 3$.
\noindent {\it Proof of Lemma~\ref{63}}: By \cite[Lemma 5.6]{MyOwn2}, the matrices $[R]_{I_1}, \ldots, [R]_{I_r}$ are themselves reflection nonsingular $\mathcal M$-matrices. Therefore, they are invertible, and $$ [R]^{-1} = \diag\left([R]^{-1}_{I_1}, \ldots, [R]^{-1}_{I_r}\right). $$ In addition, \begin{equation} \label{4568} [R]_I^{-1}[A]_I^{-1} = \diag\left([R]^{-1}_{I_1}[A]_{I_1}, \ldots, [R]^{-1}_{I_r}[A]_{I_r}\right), \end{equation} $$ [R]_I^{-1}[A]_I^{-1}[R]_I^{-1} = \diag\left([R]^{-1}_{I_1}[A]_{I_1}[R]_{I_1}^{-1}, \ldots, [R]^{-1}_{I_r}[A]_{I_r}[R]_{I_r}^{-1}\right). $$
\begin{lemma} For the matrices $R$ and $A$ given by~\eqref{R12} and~\eqref{A}, we have: \begin{equation} \label{11} \tr\bigl([R]_I^{-1}[A]_I^{-1}\bigr) = \sum\limits_{j=1}^r\mathcal T(\overline{I}_j). \end{equation} \end{lemma}
\begin{proof} Because of~\eqref{4568}, we get: \begin{equation} \label{21} \tr\bigl([R]_I^{-1}[A]_I^{-1}\bigr) = \sum\limits_{j=1}^r\tr\bigl([R]^{-1}_{I_j}[A]_{I_j}\bigr). \end{equation} Applying Lemma~\ref{trace} with $I_j$ instead of $\{1, \ldots, N-1\}$ and $\overline{I}_j$ instead of $\{1, \ldots, N\}$, $j = 1, \ldots, r$, we have: \begin{equation} \label{22} \tr\bigl([R]^{-1}_{I_j}[A]_{I_j}\bigr) = \sum\limits_{j=1}^r\mathcal T(\overline{I}_j),\ \ j = 1, \ldots, r. \end{equation} Combining~\eqref{21} and~\eqref{22}, we get~\eqref{11}. \end{proof}
\begin{lemma} We have the following estimate: \begin{equation} \label{100}
\max\limits_{x \in \mathbb{R}^{|I|}_+\setminus\{0\}}\frac{x'[R]_I^{-1}[A]_I[R]_I^{-1}x}{x'[R]^{-1}_Ix} \le \max\limits_{j = 1, \ldots, r}\max\limits_{\substack{k, l \in I_j\\ k \le l}}\frac{\bigl([R]_{I_j}^{-1}[A]_{I_j}[R]_{I_j}^{-1}\bigr)_{kl}}{\bigl([R]_{I_j}^{-1}\bigr)_{kl}}. \end{equation} \label{651} \end{lemma}
The proof of Lemma~\ref{651} is postponed until the end of this section. Assuming we have proved it, let us show how to finish the proof of Lemma~\ref{63}.
Using~\eqref{100} and~\eqref{11}, we can rewrite the condition~\eqref{Nedge} as $$ \sum\limits_{j=1}^r\mathcal T(\overline{I}_j) - 2\!\!\max\limits_{i = 1, \ldots, r}\max\limits_{\substack{k, l \in I_i\\ k \le l}}\frac{\bigl([R]_{I_i}^{-1}[A]_{I_i}^{-1}[R]_{I_i}^{-1}\bigr)_{kl}}{\bigl([R]_{I_i}^{-1}\bigr)_{kl}} \ge 0. $$ Equivalently, $$ \sum\limits_{\substack{j=1\\j \ne i}}^r\mathcal T(\overline{I}_j) + \mathcal T(\overline{I}_i) - 2\max\limits_{\substack{k, l \in I_i\\ k \le l}}\frac{\bigl([R]_{I_i}^{-1}[A]_{I_i}^{-1}[R]_{I_i}^{-1}\bigr)_{kl}}{\bigl([R]_{I_i}^{-1}\bigr)_{kl}} \ge 0,\ \ i = 1, \ldots, r. $$ In the proof of Theorem~\ref{totalcor}, see~\eqref{10000} and~\eqref{CTcalc}, it was shown that for $i = 1, \ldots, r$, we have: $$ \mathcal T(\overline{I}_i) - 2\max\limits_{\substack{k, l \in I_i\\ k \le l}}\frac{\bigl([R]_{I_i}^{-1}[A]_{I_i}^{-1}[R]_{I_i}^{-1}\bigr)_{kl}}{\bigl([R]_{I_i}^{-1}\bigr)_{kl}} = \min(\mathcal P(\overline{I}_i), \delta_i),\ \ \delta_i := \delta([\sigma]_{\overline{I}_i}) \ge 0. $$ Therefore, the condition~\eqref{Nedge} is equivalent to \begin{equation} \label{newform} \sum\limits_{j\ne i}\mathcal T(\overline{I}_j) + \min(\mathcal P(\overline{I}_i), \delta_i) \ge 0,\ \ i = 1, \ldots, r. \end{equation} The condition~\eqref{newform}, in turn, is equivalent to $$ \sum\limits_{j\ne i}\mathcal T(\overline{I}_j) + \mathcal P(\overline{I}_i) \ge 0,\ \ i = 1, \ldots, r. $$ This completes the proof of Lemma~\ref{63}, and with it the proofs of Lemma~\ref{cornerstone} and Theorem~\ref{mainthm}.
{\it Proof of Lemma~\ref{651}.} The matrices $[R]_I^{-1}$ and $[A]_I^{-1}$ are block-diagonal, with the blocks corresponding to the sets $I_1, \ldots, I_r$ of indices. Therefore, \begin{equation} \label{42} x'[R]_I^{-1}[A]_I[R]_I^{-1}x = \sum\limits_{j=1}^r[x]'_{I_j}[R]_{I_j}^{-1}[A]_{I_j}[R]_{I_j}^{-1}[x]_{I_j},\ \ x'[R]_I^{-1}x = \sum\limits_{j=1}^r[x]'_{I_j}[R]_{I_j}^{-1}[x]_{I_j}. \end{equation} Let $\mathcal Q(x) := \{j = 1, \ldots, r\mid [x]_{I_j} \ne 0\}$. We might as well rewrite~\eqref{42} as $$ x'[R]_I^{-1}[A]_I[R]_I^{-1}x = \sum\limits_{j \in \mathcal Q(x)}[x]'_{I_j}[R]_{I_j}^{-1}[A]_{I_j}[R]_{I_j}^{-1}[x]_{I_j},\ \ x'[R]_I^{-1}x = \sum\limits_{j\in \mathcal Q(x)}[x]'_{I_j}[R]_{I_j}^{-1}[x]_{I_j}. $$
For $j \in \mathcal Q(x)$, we have: $[x]_{I_j} \in S^{|I_j|}_+\setminus\{0\}$. The matrix $[R]_{I_j}$ has the same form as $R$ in~\eqref{R12}, but with smaller size. Therefore, all elements of the inverse matrix $[R]_{I_i}^{-1}$ (just like for $R^{-1}$) are positive. Therefore, $[x]'_{I_i}[R]_{I_i}^{-1}[x]_{I_i} > 0$, $i = 1, \ldots, r$. Applying Lemma~\ref{fraccomp} to $a_i = [x]'_{I_i}[R]_{I_i}^{-1}[A]_{I_i}[R]_{I_i}^{-1}[x]_{I_i}$ and $b_i = [x]'_{I_i}[R]_{I_i}^{-1}[x]_{I_i} > 0$ for $i \in \mathcal Q(x)$, we get: \begin{equation} \label{10001} \frac{x'[R]_I^{-1}[A]_I[R]_I^{-1}x}{x'[R]_I^{-1}x} \le \max\limits_{j \in \mathcal Q(x)}\frac{[x]'_{I_j}[R]_{I_j}^{-1}[A]_{I_j}[R]_{I_j}^{-1}[x]_{I_j}}{[x]'_{I_j}[R]_{I_j}^{-1}[x]_{I_j}}. \end{equation}
But the matrix $[R]_{I_j}$, as just mentioned, has all elements positive. Applying Lemma~\ref{simple}, we have for $y \in \mathbb{R}^{|I_j|}_+\setminus\{0\}$: \begin{equation} \label{10002} \frac{y'[R]_{I_j}^{-1}[A]_{I_j}[R]_{I_j}^{-1}y}{y'[R]_{I_j}^{-1}y} \le \max\limits_{\substack{k, l \in I_j\\ k \le l}}\frac{\left([R]_{I_j}^{-1}[A]_{I_j}[R]_{I_j}^{-1}\right)_{kl}}{\left([R]_{I_j}^{-1}\right)_{kl}}. \end{equation} Combining~\eqref{10001} and~\eqref{10002}, we get~\eqref{100}. $\square$
\subsection{Proof of Theorem~\ref{cams}}
Recall the setting of Theorem~\ref{cornerthm}: we have a process $Z = (Z(t), t \ge 0)$ in $\mathbb{R}^d_+$, which is an $\SRBM^d(R, \mu, A)$ with a reflection nonsingular $\mathcal M$-matrix $R$. We would like this process to avoid the corner $\{0\}$. We have: $d = N - 1 = 3$, and $$ R = \begin{bmatrix} 1 & -1/2 & 0\\ -1/2 & 1 & -1/2\\ 0 & -1/2 & 1 \end{bmatrix},\ \ A = \begin{bmatrix} \sigma_1^2 + \sigma_2^2 & -\sigma_2^2 & 0\\ -\sigma_2^2 & \sigma_2^2 + \sigma_3^2 & -\sigma_3^2\\ 0 & -\sigma_3^2 & \sigma_3^2 + \sigma_4^2 \end{bmatrix} $$ We pick the following matrix: \begin{equation} \label{QDef} Q = \begin{bmatrix} 1 & 1 & 1 \\ 1 & \lambda & 1 \\ 1 & 1 & 1 \\ \end{bmatrix}, \ \ \mbox{where}\ \ \lambda =\frac{\sigma_1^2+\sigma_2^2+\sigma_3^2+\sigma_4^2}{\sigma_2^2+\sigma_3^2}. \end{equation} This is a symmetric matrix. It is also strictly copositive, because all its elements are strictly positive. Let us show that conditions of Theorem~\ref{cornerthm} (i) hold. First, calculations show that \begin{equation} QR= \begin{bmatrix} \frac{1}{2} & 0 & \frac{1}{2} \\ 1-\frac{\lambda}{2} & \lambda-1 & 1-\frac{\lambda}{2} \\ \frac{1}{2} & 0 & \frac{1}{2} \end{bmatrix}, \end{equation} Therefore, $(QR)_{ij} \ge 0$ for $i \ne j$ is equivalent to $$ 1-\frac{\lambda}{2}\ge 0 \iff \lambda \le 2 \iff \sigma_2^2+\sigma_3^2 \ge \sigma_1^2 +\sigma_4^2. $$ By a simple calculation, one can also confirm the relation $$ QAQ=\frac{\tr(QA)}{2}Q, $$ and $\tr\left(QA\right) \ge 2c_+(Q)$. This completes the proof of Theorem~\ref{cams}.
\section{Competing Brownian Particles with Asymmetric Collisions}
One can generalize the classical system of competing Brownian particles from Definition~\ref{classicdef} in many ways. Let us describe one of these generalizations. For $k = 1, \ldots, N-1$, let $$ L_{(k, k+1)} = (L_{(k, k+1)}(t), t \ge 0) $$ be the semimartingale local time process at zero of the process $Z_k = Y_{k+1} - Y_k$. We shall call this the {\it collision local time} of the particles $Y_k$ and $Y_{k+1}$. For notational convenience, let $L_{(0, 1)}(t) \equiv 0$ and $L_{(N, N+1)}(t) \equiv 0$. Let $$ B_k(t) = \sum\limits_{i=1}^N\int_0^t1(\mathbf{p}_s(k) = i)\mathrm{d} W_i(s),\ \ k = 1, \ldots, N,\ \ t \ge 0. $$ It can be checked that $\langle B_k, B_l\rangle_t \equiv \delta_{kl}t$; that is, $B_1, \ldots, B_N$ are i.i.d. standard Brownian motions. As shown in \cite{BFK2005, BG2008, Ichiba11}, \cite[Chapter 3]{IchibaThesis}, the ranked particles $Y_1, \ldots, Y_N$ have the following dynamics: $$ Y_k(t) = Y_k(0) + g_kt + \sigma_kB_k(t) - \frac12 L_{(k, k+1)}(t) + \frac12 L_{(k-1, k)}(t),\ \ k = 1, \ldots, N. $$ The collision local time $L_{(k, k+1)}$ has a physical meaning of the push exerted when the particles $Y_k$ and $Y_{k+1}$ collide, which is needed to keep the particle $Y_{k+1}$ above the particle $Y_k$. Note that the coefficients at the local time terms are $\pm 1/2$. This means that the collision local time $L_{(k, k+1)}$ is split evenly between the two colliding particles: the lower-ranked particle $Y_k$ receives one-half of this local time, which pushes it down, and the higher-ranked particle $Y_{k+1}$ receives the other one-half of this local time, which pushes it up.
In the paper \cite{KPS2012}, they considered systems of Brownian particles when this collision local time is split unevenly: the part $q^+_{k+1}L_{(k, k+1)}(t)$ goes to the upper particle $Y_{k+1}$, and the part $q^-_kL_{(k, k+1)}(t)$ goes to the lower particle $Y_k$. Let us give a formal definition.
\begin{defn} Fix $N \ge 2$, the number of particles. Take drift and diffusion coefficients $$ g_1, \ldots, g_N;\ \ \sigma_1, \ldots, \sigma_N > 0, $$ and, in addition, take {\it parameters of collision} $$ q^{\pm}_1,\ldots, q^{\pm}_N \in (0, 1),\ \ q^+_{k+1} + q^-_k = 1,\ \ k = 1, \ldots, N-1. $$ Consider a continuous adapted $\mathbb{R}^N$-valued process $$ Y = \left(Y(t) = (Y_1(t), \ldots, Y_N(t))', t \ge 0\right). $$ Take other $N-1$ continuous adapted real-valued nondecreasing processes $$ L_{(k, k+1)} = (L_{(k, k+1)}(t), t \ge 0),\ \ k = 1, \ldots, N-1, $$ with $L_{(k, k+1)}(0) = 0$, which can increase only when $Y_{k+1} = Y_k$: $$ \int_0^{\infty}1(Y_{k+1}(t) > Y_k(t))\mathrm{d} L_{(k, k+1)}(t) = 0,\ \ k = 1, \ldots, N-1. $$ Let $L_{(0, 1)}(t) \equiv 0$ and $L_{(N, N+1)}(t) \equiv 0$. Assume that \begin{equation} \label{asymm} Y_k(t) = Y_k(0) + g_kt + \sigma_kB_k(t) - q^-_kL_{(k, k+1)}(t) + q^+_kL_{(k-1, k)}(t),\ \ k = 1, \ldots, N. \end{equation} Then the process $Y$ is called the {\it system of competing Brownian particles with asymmetric collisions}. The gap process is defined similarly to the case of a classical system. \label{defn:asymm} \end{defn}
Strong existence and pathwise uniqueness for these systems were shown in \cite[Section 2.1]{KPS2012}. When $q^{\pm}_1 = q^{\pm}_2 = \ldots = 1/2$, we are back in the case of symmetric collisions.
\begin{rmk} For systems of competing Brownian particles with asymmetric collisions, we defined only ranked particles $Y_1, \ldots, Y_N$. It is, however, possible to define named particles $X_1, \ldots, X_N$ for the case of asymmetric collisions. This is done in \cite[Section 2.4]{KPS2012}. The construction works up to the first moment of a triple collision. A necessary and sufficient condition for a.s. absence of triple collisions is given in \cite{MyOwn3}. We will not make use of this construction in our article, instead working with ranked particles. \end{rmk}
We can define collisions and multicollisions similarly to the classical case, as in Definition~\ref{Pat}. It was shown in \cite{KPS2012} that the gap process for systems with asymmetric collisions, much like for the classical case, is an SRBM. Namely, it is an $\SRBM^{N-1}(R, \mu, A)$, where $\mu$ and $A$ are given by~\eqref{mu} and~\eqref{A}, and the reflection matrix $R$ is given by \begin{equation} \label{R} R = \begin{bmatrix} 1 & -q^-_2 & 0 & 0 & \ldots & 0 & 0\\ -q^+_2 & 1 & -q^-_3 & 0 & \ldots & 0 & 0\\ 0 & -q^+_3 & 1 & -q^-_4 & \ldots & 0 & 0\\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ 0 & 0 & 0 & 0 & \ldots & 1 & -q^-_{N-1}\\ 0 & 0 & 0 & 0 & \ldots & -q^+_{N-1} & 1 \end{bmatrix} \end{equation} The connection between multicollisions and multiple collisions in this system and hitting of edges of $\mathbb{R}^{N-1}_+$ by the gap process is the same as in Lemma~\ref{red}. This allows us to apply Theorem~\eqref{cornerthm} and Theorem~\eqref{corner2edge} to find sufficient conditions for avoiding multicollisions of a given pattern. In particular, the results of Lemma~\ref{Indep} are still valid for system with asymmetric collisions: the property of a.s. avoiding multicollisions of a certain pattern depends only on the diffusion coefficients and parameters of collision.
A remark is in order: the matrix $R$ in~\eqref{R} in general is not symmetric, as opposed to the matrix $R$ in~\eqref{R12}. But if we take the $(N-1)\times(N-1)$ diagonal matrix $$ C = \diag\left(1, \frac{q^+_2}{q^-_2}, \frac{q^+_2q^+_3}{q^-_2q^-_3}, \ldots, \frac{q^+_2q^+_3\ldots q^+_{N-1}}{q^-_2q^-_3\ldots q^-_{N-1}}\right), $$ then the matrix $\overline{R} = RC$ is diagonal.
\section{Appendix}
\subsection{Proof of Lemma~\ref{Indep}} Follows from \cite[Lemma 3.1]{MyOwn3}, the discussion in \cite[Subsection 3.2]{MyOwn3}, and the reduction of multicollisions to hitting edges of the orthant which is done in Lemma~\ref{red} in this article.
\subsection{Proof of Lemma~\ref{constants}}
Fix $x \in S\setminus\{0\}$. Since $Q$ is strictly copositive, we have: $x'Qx > 0$. Since $Q$ is nonsingular, $Qx \ne 0$. Since $A$ is positive definite, we have: $x'QAQx = (Qx)'A(Qx) > 0$. Therefore, the function $$ f(x) := \frac{x'QAQx}{x'Qx} $$ is well-defined and strictly positive on $S\setminus\{0\}$. In addition, it is {\it homogeneous}, in the sense that for $x \in S\setminus\{0\}$ and $k > 0$ we have: $f(kx) = f(x)$. Therefore, $$ \{f(x)\mid x \in S\setminus\{0\}\} = \{f(x)\mid x \in S,\ \norm{x} = 1\}. $$ The set $\{x \in S\mid \norm{x} = 1\}$ is compact, and $f$ is continuous and positive on this set. Therefore, it is bounded on this set (and therefore on the set $S\setminus\{0\}$), and reaches its maximal and minimal values, both of which are strictly positive.
\subsection{Proof of Corollary~\ref{cor:inv-R}} By \cite[Lemma 2.1]{MyOwn3} (equivalent characterization of reflection nonsingular $\mathcal M$-matrices), we have: $$ \left(R^{-1}\right)_{ij} \ge 0,\ \ i, j = 1, \ldots, d; \ \ \left(R^{-1}\right)_{ii} > 0,\ \ i = 1, \ldots, d. $$ Therefore, the matrix $\overline{R}^{-1} = C^{-1}R^{-1} = (\rho_{ij})_{1 \le i, j \le d}$ has elements $\rho_{ij} = c_i^{-1}\left(R^{-1}\right)_{ij}$. By assumptions, the matrix $\overline{R}^{-1}$ is symmetric. Therefore, its entries satisfy \begin{equation} \label{properties} \rho_{ij} = \rho_{ji} \ge 0,\ \ i, j = 1, \ldots, d; \ \ \rho_{ii} > 0,\ \ i = 1, \ldots, d. \end{equation} From here, it is easy to see that $\overline{R}^{-1}$ is strictly copositive: $x'\overline{R}^{-1} x > 0$ for $x \in S\setminus\{0\}$. Also, $(\overline{R}^{-1} R)_{ij} = (C^{-1})_{ij} = 0$ for $i \ne j$. It suffices to apply Theorem~\ref{cornerthm}.
\subsection{Proof of Lemma~\ref{simple}}
Let us prove the statement for the maximum. For the minimum, the proof is similar. For $x \in S\setminus\{0\}$, we have: $x_1, \ldots, x_d \ge 0$, and $$ \frac{x'\overline{R}^{-1} A\overline{R}^{-1} x}{x'\overline{R}^{-1} x} = \frac{\sum_{i=1}^d\sum_{j=1}^d(\overline{R}^{-1} A\overline{R}^{-1})_{ij}x_ix_j}{\sum_{i=1}^d\sum_{j=1}^d\rho_{ij}x_ix_j}. $$ Apply Lemma~\ref{fraccomp} to $s = d^2$, $a_{ij} = (\overline{R}^{-1} A\overline{R}^{-1})_{ij}x_ix_j$, $b_{ij} = \rho_{ij}x_ix_j$ (we index $a_i$ and $b_i$ by double indices, with each of the two indices ranging from $1$ to $d$). It suffices to note that, because of the symmetry of $\overline{R}^{-1} A\overline{R}^{-1}$ and $\overline{R}^{-1} = (\rho_{ij})$, we have: $$ \max\limits_{i, j = 1, \ldots, d}\frac{(\overline{R}^{-1} A\overline{R}^{-1})_{ij}}{\rho_{ij}} = \max\limits_{1 \le i \le j \le d}\frac{(\overline{R}^{-1} A\overline{R}^{-1})_{ij}}{\rho_{ij}}. $$
\subsection{Miscellaneous lemmata}
\begin{lemma} Take real numbers $a_1, \ldots, a_s$ and positive real numbers $b_1, \ldots, b_s$. Then $$ \min\left(\frac{a_1}{b_1}, \ldots, \frac{a_s}{b_s}\right) \le \frac{a_1 + \ldots + a_s}{b_1 + \ldots + b_s} \le \max\left(\frac{a_1}{b_1}, \ldots, \frac{a_s}{b_s}\right). $$ \label{fraccomp} \end{lemma}
\begin{proof} Let us prove the inequality $$ \frac{a_1 + \ldots + a_s}{b_1 + \ldots + b_s} \le \max\left(\frac{a_1}{b_1}, \ldots, \frac{a_s}{b_s}\right). $$ The other inequality is proved similarly. Assume the converse: that $$ \frac{a_1 + \ldots + a_s}{b_1 + \ldots + b_s} > \frac{a_i}{b_i},\ \ i = 1, \ldots, s. $$ Multiply the $i$th inequality by $(b_1 + \ldots + b_s)b_i > 0$: $$ (a_1 + \ldots + a_s)b_i > a_i(b_1 + \ldots + b_s),\ \ i = 1, \ldots, s. $$ Add them up and get: $$ (a_1 + \ldots + a_s)(b_1 + \ldots + b_s) > (a_1 + \ldots + a_s)(b_1 + \ldots + b_s), $$ and we arrive at a contradiction. \end{proof}
\begin{lemma} \label{lemma:aux-comparison} Suppose we are given the following:
(i) two real-valued continuous adapted processes $$ X_1 = (X_1(t), t \ge 0)\ \ \mbox{and}\ \ X_2 = (X_2(t), t \ge 0), $$ starting from the same $X_1(0) = X_2(0) = x$;
(ii) a real continuous function $\sigma : \mathbb{R} \to \mathbb{R}$ such that $$
|\sigma(x) - \sigma(y)| \le \rho(|x-y|),\ \ x, y \in \mathbb{R},\ t \ge 0, $$ where $\rho : \mathbb{R}_+ \to \mathbb{R}_+$ is an increasing function such that $\rho(0) = 0$ and $\int_0^{\infty}\rho^{-2}(s)\mathrm{d} s = \infty$;
(iii) a continuous function $b : \mathbb{R} \to \mathbb{R}$ and a continuous adapted process $\beta = (\beta(t), t \ge 0)$ with bounded variation, such that for every subset $A \subseteq \mathbb{R}_+$, we have: \begin{equation} \label{eq:comp-measures} \int_A\mathrm{d}\beta(t) \ge \int_Ab(X_1(t))\mathrm{d} t, \end{equation} and the following equations are satisfied: $$ \mathrm{d} X_1(t) = b(X_1(t))\mathrm{d} t + \sigma(X_1(t))\mathrm{d} W(t), \ \ \mathrm{d} X_2(t) = \mathrm{d}\beta(t) + \sigma(X_2(t))\mathrm{d} W(t). $$ Here, $W = (W(t), t \ge 0)$ is a standard Brownian motion. Then a.s. for all $t \ge 0$ we have: $X_1(t) \le X_2(t)$. \end{lemma}
\begin{proof} This is a modification of the proof of \cite[Theorem 6.1]{IWBook} and \cite[Theorem 1.1]{IkedaComp}. From the property~\eqref{eq:comp-measures}, we get: for any measurable function $\varphi : \mathbb{R}_+ \to \mathbb{R}_+$ and any $t > 0$, we get: $$ \int_0^t\varphi(s)\mathrm{d}\beta(s) \ge \int_0^t\varphi(s)b(X_2(s))\mathrm{d} s. $$ In the proof \cite[Theorem 1.1]{IkedaComp}, we should change $\beta_2(s)\mathrm{d} s$ to $\mathrm{d}\beta(s)$ and $\beta_1(s)\mathrm{d} s$ to $b(X_1(s))\mathrm{d} s$. The rest of the proof should be modified accordingly. \end{proof}
\section*{Acknoweldgements}
The authors would like to thank \textsc{Ioannis Karatzas}, \textsc{Soumik Pal} and \textsc{Ruth Williams}, as well as an anonymous referee, for help and useful discussion. This research was partially supported by NSF grants DMS 1007563, DMS 1308340, DMS 1409434, and DMS 1405210.
\end{document} | arXiv |
\begin{document}
\allowdisplaybreaks
\begin{abstract}
In this paper, we deal with the initial value fractional damped wave equation on $G$, a compact Lie group, with power-type nonlinearity. The aim of this manuscript is twofold. First, using the Fourier analysis on compact Lie groups, we prove a local in-time existence result in the energy space for the fractional damped wave equation on $G$. Moreover, a finite time blow-up result is established under certain conditions on the initial data. In the next part of the paper, we consider fractional wave equation with lower order terms, that is, damping and mass with the same power type nonlinearity on compact Lie groups, and prove the global in-time existence of small data solutions in the energy evolution space.
\end{abstract}
\maketitle
\tableofcontents
\section{Introduction}
The study of partial differential equations is indeed one of the fundamental tools for understanding and modeling natural and real-world phenomena. Fractional differential operators are nonlocal operators that are considered as a generalization of classical differential operators of arbitrary non-integer orders.
For the last few decades, the study of partial differential equations involving nonlocal operators have gained a considerable amount of interest and have become one of the essential topics in mathematics and its applications. Many physical phenomena in engineering, quantum field theory, astrophysics, biology, materials, control theory, and other sciences can be successfully described by models utilizing mathematical tools from fractional calculus \cite{Neww1, NN1,NN3,NN4, f4}. In particular, the fractional Laplacian is represented as the infinitesimal generator of stable radially symmetric Lévy processes \cite{EE1}.
For other exciting models related to fractional differential equations, we refer to the reader \cite{f3, f10,f11,EE24} to mention only a few of many recent publications.
In recent years, due to the nonlocal nature of the fractional derivatives, considerable attention has been devoted to various models involving fractional Laplacian and nonlocal operators by several researchers. There is a vast literature available involving the fractional Laplacian on the Euclidean framework, which is difficult to mention; we refer to important papers \cite{EE3, EE4, EE5, EE6, EE10, EE24, EE27, EE31} and the references therein. Here we would like to point out that the fractional Laplacian operator $(-\Delta)^{\alpha}$ can be reduced to the classical Laplace operator $-\Delta$ as $\alpha \rightarrow 1$. We refer to \cite{EE24} for more details. In particular, many interesting results in some classical elliptic problems have been extended in the fractional Laplacian setting, see \cite{EE9}.
For the classical semilinear damped wave equation in $\mathbb{R}^n$, the global existence or a blow-up result depending on the critical exponent has been studied in \cite{IKeta and Tanizawa, Matsumura, Zhang, Todorova}. We refer to the excellent book \cite{Rei} for global in-time small data solutions for the semilinear damped wave equation on the Euclidean framework.
The study of the semilinear damped wave equation has also been extended in the non-Euclidean framework. Several papers have studied linear PDE in non-Euclidean structures in the last decades. For example, the semilinear wave equation with or without damping has been investigated for the Heisenberg group \cite{24,30}. In the case of graded groups, we refer to the recent works \cite{gra1, gra2, gra3}. Concerning the damped wave equation on compact Lie groups, we refer to \cite{27, 28,31,garetto, BKM22}. Particularly, the author in \cite{27} studied semilinear damped wave equation with power type nonlinearity $|u|^p$ on compact Lie groups and proved a local in-time existence result in the energy space via Fourier analysis on compact Lie groups. He also derived a blow-up result for the semilinear Cauchy problem for any $p > 1$. Also, considering the semilinear wave equation with damping and mass with power nonlinearity $|u|^p$ on compact Lie groups and without any lower bounds for $p > 1$, the author proved the global in time existence of small data solutions in the evolution energy space in \cite{28}.
For the study of semilinear wave equation of general compact manifolds, we refer to the seminal works \cite{BGT04, KAP95} where the global in-time solution were investigated by establishing famous Strichartz type estimates. Recently, the wave equation were also explored in the noncompact manifolds setup, see \cite{AZ22, Zhang20, Zhang21, SSW19} and reference therein.
Then, an interesting and viable problem is to study the fractional wave equation (\ref{eq0010}) and (\ref{2number311}) of order $\alpha$ with $ 0 < \alpha < 1$, with power-type nonlinearity. In \cite{Ahmad15}, the authors have investigated the nonexistence of global weak solutions to the nonlinear fractional wave equation with power type nonlinearlity on the Heisenberg group. In the setting for compact Lie groups, we have recently started a systematic study of the nonlinear fractional wave equation on compact Lie groups. This work is a continuation of our previous work \cite{shyamm}. To state our problem, let $G$ be a compact Lie group with normalized Haar measure $dx$ and let $\mathcal{L}$ be the Laplace-Beltrami operator on $G$ (which also coincides with the Casimir element of the universal enveloping algebra of Lie algebra of $G$). For $0<\alpha<1$, we consider the following two Cauchy problems for the fractional wave equation with power type nonlinearity, namely, with damping term,
\begin{align} \label{eq0010}
\begin{cases}
\partial^2_tu+(-\mathcal{L})^\alpha u+ \partial_tu =|u|^p, & x\in G,t>0,\\
u(0,x)=\varepsilon u_0(x), & x\in G,\\ \partial_tu(x,0)=\varepsilon u_1(x), & x\in G,
\end{cases}
\end{align}
and with damping and positive mass,
\begin{align} \label{2number311}
\begin{cases}
\partial^2_tu+\left( -\mathcal{L}\right)^\alpha u+b\partial_{t} u+m^2u=|u|^p, & x\in G, t>0,\\
u(0,x)=u_0(x), & x\in G,\\ \partial_tu(x,0)=u_1(x), & x\in G,
\end{cases}
\end{align}
where $ p > 1,b,m^2$ are positive constants and $\varepsilon$ is a positive constant describing the smallness of the Cauchy data. Here for the moment, we assume that $u_{0}$ and $ u_{1}$ are taken from the energy space $ H_{\mathcal{L}}^\alpha(G)$ (see \eqref{sob} for the definition) and $ L^2(G)$, respectively.
This paper investigates a finite time blow-up result for solutions to the fractional damped wave equation involving the Laplace-Beltrami operator on compact Lie groups under a suitable sign assumption for the initial data. Moreover, we show that the presence of a positive damping term and a positive mass term in the Cauchy problem completely reverses the scenario, i.e., we prove the global existence of small data solutions for the fractional wave equation with damping and mass. More preciously, using the Gagliardo-Nirenberg type inequality (in order to handle power nonlinearity in $L^2(G))$ and Fourier analysis on compact Lie groups, we prove the local well-posedness of the Cauchy problem (\ref{eq0010}) in the energy evolution space $\mathcal{C}\left([0, T], H_{\mathcal{L}}^{\alpha}( {G})\right) \cap \mathcal{C}^{1}\left([0, T], L^{2}( {G})\right)$ and the global in time existence of small data solutions for the Cauchy problem (\ref{2number311}).
\subsection{Main results}
We denote $L^{q}(G), 1 \leq q<\infty$, the space of $q$-integrable functions on the compact Lie group $G$ concerning the normalized Haar measure $dx$ on $G$ and essentially bounded for $q=\infty$ throughout the paper. For $s>0$ and $q \in(1, \infty)$, the fractional Sobolev space $H_{\mathcal{L} }^{ s, q}(G)$ of order $\alpha$ is defined as
\begin{align}\label{sob}
H_{\mathcal{L}}^{s, q}(G) \doteq\left\{f \in L^{q}(G):(-\mathcal{L})^{s / 2} f \in L^{q}(G)\right\},
\end{align}
endowed with the norm $\|f\|_{H_{\mathcal{L}}^{s, q}(G)} =: \|f\|_{L^{q}(G)}+\left\|(-\mathcal{L})^{s / 2} f\right\|_{L^{q}(G)}$. We simply denote the Hilbert space $H_{\mathcal{L}}^{s, 2}(G)$ by $H_{\mathcal{L}}^{s}(G)$.
By employing noncommutative Fourier analysis for compact Lie groups, our first result concerning $L^2$-decay estimates for the solution of the linear version of the Cauchy problem (\ref{eq0010}) (when $f=0$) is stated in the following proposition.
\begin{prop}\label{thm11}
Let $0<\alpha <1$. Suppose that $(u_0, u_1)\in H_{\mathcal{L}}^\alpha(G) \times L^2(G)$ and $u\in\mathcal{C}([0,\infty),H_{\mathcal{L}}^\alpha(G))\cap \mathcal{C}^1([0,\infty),L^2(G))$ be the solution to the homogeneous Cauchy problem
\begin{align}\label{eq1}
\begin{cases}
\partial^2_tu+(-\mathcal{L})^\alpha u+\partial_tu =0, & x\in G,~t>0,\\
u(0,x)=u_0(x), & x\in G,\\ \partial_tu(x,0)=u_1(x), & x\in G.
\end{cases}
\end{align}
Then, $u$ satisfies the following $L^2( G)-L^2( G)$ estimates
\begin{align}\label{111111}
\| u(t,\cdot)\|_{L^2( G)} &\lesssim( \left\|u_{0}\right\|_{L^{2}(G)}+t\left\|u_{1}\right\|_{L^{2}(G)}),\\\nonumber
\left\|(-\mathcal{L})^{\alpha / 2} u(t, \cdot)\right\|_{L^{2}(G)}&\lesssim (1+t)^{-\frac{1}{2}} (\left\|u_{0}\right\|_{H_{\mathcal{L}}^{{\alpha }}(G)}^{2}+\left\|u_{1}\right\|_{L^{2}(G)}^{2}),\\\nonumber
\|\partial_tu(t,\cdot)\|_{L^2( G)}&\lesssim (1+t)^{-1} (\left\|u_{0}\right\|_{H_{\mathcal{L}}^{{\alpha }}(G)}^{2}+\left\|u_{1}\right\|_{L^{2}(G)}^{2}).
\end{align}
for any $t\geq 0$.
\end{prop}
Next we prove the local well-posedness of the Cauchy problem (\ref{eq0010}) in the energy evolution space $\mathcal C\left([0,T], H^\alpha_{\mathcal{L}}(G)\right)\cap\mathcal C^1\left([0,T],L^2(G)\right)$. In this case, a Gagliardo-Nirenberg type inequality (proved in \cite{Gall}) will be used to estimate the power nonlinearity in $L^2(G)$. Indeed, we have the following local existence result.
\begin{thm}\label{thm22}
Let $0<\alpha <1$ and let $G$ be a compact connected Lie group with the topological dimension $n.$ Assume that $n\geq 2[\alpha]+2$. Suppose that $(u_0, u_1)\in H^\alpha_{\mathcal L}(G) \times L^2(G)$ and $p>1$ such that $p\leq\frac{n}{n-2\alpha}.$ Then there exists $T=T(\varepsilon)>0$ such that the Cauchy problem (\ref{eq0010}) admits a uniquely determined mild solution $$u\in \mathcal{C}([0,T],H^\alpha_{\mathcal L}(G))\cap \mathcal{C}^1([0,T],L^2(G)).$$
\end{thm}
\begin{rmk}
Note that the restriction $p\leq\frac{n}{n-2\alpha }$ and $n \geq 2[\alpha]+2$ in the above theorem is necessary in order to apply Gagliardo-Nirenberg type inequality.
\end{rmk}
Our next result is about the non-existence of global in-time solutions to (\ref{eq0010}) for any $p > 1$ regardless of the size of initial data. Before stating the blow-up result, we first introduce a suitable notion of energy solutions for the Cauchy problem (\ref{eq0010}).
\begin{defn}\label{eq332}
Let $0<\alpha <1$ and $\left(u_{0}, u_{1}\right) \in H_{\mathcal{L}}^{\alpha}(G) \times L^{2}(G)$. For any $T>0,$ we say that
$$
u \in \mathcal{C}\left([0, T), H_{\mathcal{L}}^{\alpha}(G)\right) \cap \mathcal{C}^{1}\left([0, T), L^{2}(G)\right) \cap L_{\text {loc }}^{p}([0, T) \times G)
$$
is an energy solution on $[0, T)$ to (\ref{eq0010}) if $u$ satisfies the following integral relation:
\begin{align}\label{eq011}\nonumber
&\int_{ {G}} \partial_{t} u(t, x) \phi(t, x) {d} x-\int_{ {G}} u(t, x) \partial_{s} \phi(t, x) {d} x +\int_{ {G}} u(t, x) \phi (t, x) {d} x \\\nonumber
& +\varepsilon \int_{G} u_{0}(x) \partial_{s}\phi(0, x) \;d x -\varepsilon \int_{G} u_{1}(x) \phi(0, x) \;d x -\varepsilon \int_{G} u_{0}(x) \phi(0, x) \;d x \\&+\int_{0}^{t} \int_{G} u(s, x)\left(\partial^2_s\phi(s, x) +(-\mathcal{L})^\alpha \phi(s, x)+\partial_s\phi(s, x) \right) d x ds
=\int_{0}^{t} \int_{G}|u(s, x)|^{p} \phi(s, x) dxds
\end{align}
for any $\phi \in \mathcal{C}_{0}^{\infty}([0, T) \times G)$ and any $t \in(0, T)$.
\end{defn}
\begin{thm}\label{f6}
Let $0<\alpha <1$, $p>1$, and let $\left(u_{0}, u_{1}\right) \in H_{\mathcal{L}}^{\alpha}(G) \times L^{2}(G)$ be nonnegative and nontrivial functions. Suppose
$$u \in \mathcal{C}\left([0, T), H_{\mathcal{L}}^{\alpha}(G)\right) \cap \mathcal{C}^{1}\left([0, T), L^{2}(G)\right) \cap L_{ {loc}}^{p}([0, T) \times G)$$ be an energy solution to the Cauchy problem (\ref{eq0010}) with lifespan $T=T(\varepsilon)$. Then there exists a constant $\varepsilon_{0}=\varepsilon_{0}\left(u_{0}, u_{1}, p\right)>0$ such that for any $\varepsilon \in\left(0, \varepsilon_{0}\right],$ the energy solution $u$ blows up in finite time.
Furthermore, the lifespan $T$ satisfies the following estimates
\begin{align}\label{eq1112}
T(\varepsilon) \leq
C \varepsilon^{1-p}.
\end{align}
\end{thm}
\begin{rmk}
\begin{itemize}
\item[(i)] Here we note that the fractional Laplace-Beltrami operator $(-\mathcal{L})^{\alpha}$ gives the classical Laplace-Beltrami operator $-\mathcal{L}$ as $\alpha \rightarrow 1$ and all our results coincides with the results proved for the Cauchy problem for the semilinear damped wave equation on compact Lie groups in \cite{27}.
\item[(ii)] From Theorem \ref{f6} one can see that the sharp lifespan estimates for local in-time solutions to (\ref{eq0010}) is independent of $\alpha, 0<\alpha<1$. Thus, for any $0<\alpha<1$, the lifespan estimates for solutions to the Cauchy problem for the fractional wave equation (\ref{eq0010}) will be the same as the sharp lifespan estimates for
the semilinear wave equation on compact Lie group $G$ proved in \cite{27}.
\end{itemize}
\end{rmk}
In the next part of the paper, we study the global existence of small data solutions for the nonlinear fractional wave equation with damping and mass and involving power type nonlinearity. More preciously, we consider the Cauchy problem (\ref{2number311}), i.e.,
\begin{align*}
\begin{cases}
\partial^2_tu+\left( -\mathcal{L}\right)^\alpha u+b\partial_{t} u+m^2u=|u|^p, & x\in G, t>0,\\
u(0,x)=u_0(x), & x\in G,\\ \partial_tu(x,0)=u_1(x), & x\in G,
\end{cases}
\end{align*}
where $ p > 1$, $b,m^2$ are positive constants, $u_{0}(x)$ and $u_{1}(x)$ are two given functions on $G$.
First, we prove the following $L^2$-decay estimates with exponential decay rates related to the time variable for the solution of the homogeneous Cauchy problem (\ref{2number311}) (when $f=0$).
\begin{prop}\label{2thm11}
Let $0<\alpha <1$. Suppose that $(u_0, u_1)\in H_{\mathcal{L}}^\alpha(G) \times L^2(G)$ and $u\in\mathcal{C}([0,\infty),H_{\mathcal{L}}^\alpha(G))\cap \mathcal{C}^1([0,\infty),L^2(G))$ be the solution to the homogeneous Cauchy problem
\begin{align}\label{2eq1}
\begin{cases}
\partial^2_tu+(-\mathcal{L})^\alpha u+b\partial_{t} u+m^2u=0, & x\in G,~t>0,\\
u(0,x)=u_0(x), & x\in G,\\ \partial_tu(x,0)=u_1(x), & x\in G.
\end{cases}
\end{align}
Then, $u$ satisfies the following $L^2( G)-L^2( G)$ estimates
\begin{align}\label{2111111}
\| u(t,\cdot)\|_{L^2( G)} &\lesssim C A_{b, m^2}(t)( \left\|u_{0}\right\|_{L^{2}(G)}+t\left\|u_{1}\right\|_{L^{2}(G)}),\\\nonumber
\left\|(-\mathcal{L})^{\alpha / 2} u(t, \cdot)\right\|_{L^{2}(G)}&\lesssim C A_{b, m^2}(t) (\left\|u_{0}\right\|_{H_{\mathcal{L}}^{{\alpha }}(G)}^{2}+\left\|u_{1}\right\|_{L^{2}(G)}^{2}),\\\nonumber
\|\partial_tu(t,\cdot)\|_{L^2( G)}&\lesssim C A_{b, m^2}(t) (\left\|u_{0}\right\|_{H_{\mathcal{L}}^{{\alpha }}(G)}^{2}+\left\|u_{1}\right\|_{L^{2}(G)}^{2}).
\end{align}
for any $t\geq 0$, where $C$ is a positive multiplicative constant and the decay function $A_{b, m^2}(t)$ is given by
$$
A_{b, m^2}(t) \doteq\left\{\begin{array}{ll}
{e}^{-\frac{b}{2} t} & \text { if } b^2<4 m^2, \\
(t+1) {e}^{-\frac{b}{2} t} & \text { if } b^2=4 m^2, \\
{e}^{\left(-\frac{b}{2}+\sqrt{\frac{b^2}{4}-m^2}\right) t} & \text { if } b^2>4 m^2.
\end{array}\right.
$$
\end{prop}
Using these above $L^2$-decay estimates, we will prove the global existence of small data solutions to the nonlinear fractional Cauchy problem (\ref{2number311}) in the energy evolution space $\mathcal C\left([0,\infty), H^\alpha_{\mathcal{L}}(G)\right)\cap\mathcal C^1\left([0,\infty),L^2(G)\right)$. In this case, a Gagliardo-Nirenberg type inequality (proved in \cite{Gall}) will be used to estimate the power nonlinearity in $L^2(G)$. The following result is about the global existence of the mild solution of the Cauchy problem (\ref{2number311}). For the definition of the mild solution, see subsection \ref{sec4}.
\begin{thm}\label{2thm22}
Let $0<\alpha <1$ and let $G$ be a compact connected Lie group with the topological dimension $n.$ Assume that $n\geq 2[\alpha]+2$. Suppose that $(u_0, u_1)\in H^\alpha_{\mathcal L}(G) \times L^2(G)$ and $p>1$ such that $p\leq\frac{n}{n-2\alpha}.$ Then there exists $\varepsilon_0>0$ such that for any $\|(u_0, u_1)\|_{ H^\alpha_{\mathcal L}(G) \times L^2(G)}\leq \varepsilon_{0}$, the Cauchy problem (\ref{2number311}) admits a uniquely determined mild solution $$u\in \mathcal{C}([0,\infty),H^\alpha_{\mathcal L}(G))\cap \mathcal{C}^1([0,\infty),L^2(G)).$$
\end{thm}
\begin{rmk}
Here we note that the fractional Laplace-Beltrami operator $(-\mathcal{L})^{\alpha}$ can be reduced to the classical Laplace-Beltrami operator $-\mathcal{L}$ as $\alpha \rightarrow 1$ and Proposition \ref{2thm11} and Theorem \ref{2thm22} coincides with the results proved for the Cauchy problem for the fractional wave equation with damping and mass on compact Lie groups in \cite{28}.
\end{rmk}
\begin{rmk}
We note that in the statement of Theorem \ref{2thm22}, the restriction on the upper bound for the exponent $p$ which is $p\leq\frac{n}{n-2\alpha }$ is necessary in order to apply Gagliardo-Nirenberg type inequality (\ref{eq34}) in (\ref{f}). Also, the other restriction $n \geq 2[\alpha]+2$ is made to fulfill the assumptions for the employment of such inequality.
\end{rmk}
Before studying the nonhomogeneous Cauchy problem (\ref{eq0010}) and \eqref{2number311} we first deal with the corresponding homogeneous problem, i.e., when $f=0$. Particularly, using the group Fourier transform with respect to the spatial variable, we determine $L^{2}-L^{2} $ estimates for the solution of the homogeneous fractional damped wave equation on the compact Lie group $G$.
Once we have these estimates, applying a Gagliardo-Nirenberg type inequality on compact Lie groups \cite{27, 28, 31} (see also \cite{Gall} for Gagliardo-Nirenberg type inequality on a more general frame of connected Lie groups), we prove the local well-posedness result for (\ref{eq0010}) and the global in time solution for (\ref{2number311}).
Apart from the introduction, this paper is organized as follows. In Section \ref{sec2}, we recall the Fourier analysis on compact Lie groups which will be used frequently throughout the paper for our approach. In Section \ref{sec3}, first,
we show an appropriate decomposition of the propagators for the nonlinear equation in the Fourier space. Further, by recalling the notion of mild solutions in our framework, we prove Theorem \ref{thm22}, the local existence result, by deriving some $L^{2}-L^{2}$ estimates for the solution of the homogeneous fractional wave equation on the compact Lie group $G$.
Moreover, under certain conditions on the initial data, a finite time blow-up result is established. In Section \ref{sec6}, we prove Theorem \ref{2thm22}, the global existence for the mild solution, by deriving some $L^{2}-L^{2}$ estimates for the solution of the homogeneous fractional wave equation with damping and mass (\ref{2number311}) on the compact Lie group $G$.
\section{Preliminaries: Analysis on compact Lie groups} \label{sec2}
In this section, we recall some basics of Fourier analysis on compact Lie groups to make the manuscript self-contained. A complete account of the representation theory of the compact Lie groups can be found in \cite{garetto, RT13, RuzT}. However, we mainly adopt the notation and terminology given in \cite{RuzT}.
\subsection{Notations}
Throughout the article, we use the following notations:
\begin{itemize}
\item $f \lesssim g:$\,\,There exists a positive constant $C$ (whose value may change from line to line in this manuscript) such that $f \leq C g.$
\item $G:$ Compact Lie group.
\item $dx:$ The normalized Haar measure on the compact group $G.$
\item $\mathcal{L}:$ The Laplace-Beltrami operator on $G.$
\item $\mathbb{C}^{d \times d}:$ The set of matrices with complex entries of order $d.$
\item $ \operatorname{Tr}(A)=\sum_{j=1}^{d} a_{j j}:$ The trace of the matrix $A=\left(a_{i j}\right)_{1 \leq i, j \leq d} \in \mathbb{C}^{d \times d}.$
\item $I_{d} \in \mathbb{C}^{d \times d}:$ The identity matrix of order $d.$
\end{itemize}
\subsection{Representation theory on compact Lie groups}
Let us first recall the definition of a representation of a compact group $G.$ A unitary representation of $G$ is a pair $(\xi, \mathcal{H})$ such that the map $\xi:G \rightarrow U(\mathcal{H}),$ where $U(\mathcal{H})$ denotes the set of unitary operators on complex Hilbert space $\mathcal{H},$ such that it satisfies following properties:
\begin{itemize}
\item The map $\xi$ is a group homomorphism, that is, $\xi(x y)=\xi(x)\xi(y).$
\item The mapping $\xi:G \rightarrow U(\mathcal{H})$ is continuous with respect to strong operator topology (SOT) on $U(\mathcal{H}),$ that is, the map $g \mapsto \xi(g)v$ is continuous for every $v \in \mathcal{H}.$
\end{itemize}
The Hilbert space $\mathcal{H}$ is called the representation space. To avoid any confusion, we represent a representation $(\xi, \mathcal{H})$ of $G$ by $\xi.$
Two unitary representations $\xi, \eta$ of ${G}$ are called equivalent if there exists an unitary operator, namely intertwiner, $T$ such that $T \xi(x)=\eta(x) T$ for any $x \in {G}$.
An intertwiner is an irreplaceable tool in the theory of representation of compact groups and is helpful in the classification of representation.
A (linear) subspace $V \subset \mathcal{H}$ is said to be invariant under the unitary representation $\xi$ of $G$ if $\xi(x) V \subset V$, for any $x \in {G}$. An irreducible unitary representation $\xi$ of $G$ is a representation such that the only closed and $\xi$-invariant subspaces of $\mathcal{H}$ are trivial once, that is, $\{0\}$ and the full space $ \mathcal{H}$.
The set of all equivalence classes $[\xi]$ of continuous irreducible unitary representations of $G$ is denoted by $\widehat{G}$ and called the unitary dual of $G.$ Since $G$ is compact, $\widehat{G}$ is a discrete set. It is known that an irreducible unitary representation $\xi$ of $G$ is finite-dimensional, i.e., the Hilbert space $\mathcal{H}$ is finite-dimensional, say, $d_\xi$. Therefore, if we choose a basis $\mathfrak{B}:=\{e_1,e_2,\ldots, e_{d_\xi}\}$ for the representation space $\mathcal{H}$ of $\xi$, we can identify $\mathcal{H}$ as $\mathbb{C}^{d_\xi}$ and consequently, we can view $\xi$ as a matrix-valued function $\xi: G \rightarrow U(\mathbb{C}^{d_{\xi} \times d_{\xi}})$, where $U(\mathbb{C}^{d_{\xi} \times d_{\xi}})$ denotes the space of all unitary matrices. The matrix coefficients $\xi_{ij}$ of the representation $\xi$ with respect to $\mathfrak{B}$ are given by $\xi_{ij}(x):=\langle \xi(x) e_j, e_i \rangle$, for all $i, j \in \{1,2, \ldots, d_\xi\}.$ It follows from the Peter-Weyl theorem that the set
$$
\left\{\sqrt{d_{\xi}} \xi_{i j}: 1 \leq i, j \leq d_{\xi},[\xi] \in \widehat{G}\right\}
$$
forms an orthonormal basis of $L^{2}(G)$.
\subsection{Fourier analysis on compact Lie groups } Let $G$ be a compact Lie group.
The group Fourier transform of $f \in L^1(G)$ at $\xi\in \widehat{G},$ denoted by $\widehat{f}(\xi),$ is defined by
$$
\widehat{f}(\xi):=\int_{G} f(x) \xi(x)^{*} d x,
$$
where $dx$ is the normalized Haar measure on $G$. It is apparent from the definition that $\widehat{f}(\xi)$ is matrix-valued and therefore, this definition can be interpreted in weak sense, i.e., for $u,v \in \mathcal{H},$
$$ \langle \widehat{f}(\xi)u, v \rangle:=\int_{G} f(x) \langle \xi(x)^{*}u, v \rangle d x.$$
It follows from the Peter-Weyl theorem that, for every $f \in L^2(G),$ we have the following Fourier series representation:
$$
f(x)=\sum_{[\xi] \in \widehat{G}} d_{\xi} \operatorname{Tr}(\xi(x) \widehat{f}(\xi)).
$$
The Plancherel identity for the group Fourier transform on $G$ takes the following form
\begin{align}\label{eq002}
\|f\|_{L^{2}(G)}=\left(\sum_{[\xi] \in \widehat{G}} d_{\xi}\|\widehat{f}(\xi)\|_{\mathrm{HS}}^{2}\right)^{1 / 2}:=\|\widehat{f}\|_{\ell^2(\widehat{G})},
\end{align}
where $\|\cdot\|_{\mathrm{HS}}$ denotes the Hilbert-Schmidt norm of a matrix $A:=(a_{ij}) \in \mathbb{C}^{d_
\xi \times d_\xi}$ defined as
$$ \|A\|_{\mathrm{HS}}^{2}=\operatorname{Tr}\left( A A^{*}\right)=\sum_{i, j=1}^{d_{\xi}}|a_{ij}|^2.$$
We would like to emphasize here that the Plancherel identity is one of the crucial tools to establish $L^2$-estimates of the solution to PDEs.
Let $\mathcal{L}$ be the Laplace-Beltrami operator on $G$. It is important to understand the action of the group Fourier transform on the Laplace–Beltrami operator $\mathcal{L}$ for developing the machinery of the proofs. For $[\xi] \in \widehat{{G}}$, the matrix elements $\xi_{i j}$, are the eigenfunctions of $\mathcal{L}$ with the same eigenvalue $-\lambda_{\xi}^{2}$. In other words, we have, for any $ x \in {G},$
$$
-\mathcal{L} \xi_{i j}(x)=\lambda_{\xi}^{2} \xi_{i j}(x), \qquad \text{for all } i, j \in\left\{1, \ldots, d_{\xi}\right\}.
$$
The symbol $\sigma_{\mathcal{L}}$ of the Laplace-Beltrami operator $\mathcal{L}$ on $G$ is given by
\begin{align}\label{symbol}
\sigma_{\mathcal{L}}(\xi)=-\lambda_{\xi}^{2} I_{d_{\xi}},
\end{align}
for any $[\xi] \in \widehat{{G}}$ and therefore, the following holds: $$\widehat{\mathcal{L} f}(\xi)=\sigma_{\mathcal{L}}(\xi) \widehat{f}(\xi)=-\lambda_{\xi}^{2} \widehat{f}(\xi)$$ for any $[\xi] \in \widehat{ G}$.
For $s>0,$ the Sobolev space $H_{\mathcal{L}}^s\left(G\right)$ of order $s$ is defined as follows:
$$H_{\mathcal{L}}^s(G):=\left\{u \in L^{2}(G):\|u\|_{H_{\mathcal{L}}^s(G)}<+\infty\right\},$$ where $\|u\|_{H_{\mathcal{L}}^s(G)}=\|u\|_{L^{2}(G)}+\left\|(-\mathcal{L})^{s / 2} u\right\|_{L^{2}({G})}$ and $(-\mathcal{L})^{s / 2} $ is defined in terms of the group Fourier transform by the follwoing formula
$$(-\mathcal{L})^{s / 2} f :=\mathcal{F}^{-1}\left(\lambda_{\xi}^{s }(\mathcal{F} f)\right), \quad \text{for all $[\xi] \in \widehat{{G}}$}.$$
Further, using Plancherel identity, for any $s>0$, we have that
$$
\left\|(-\mathcal{L})^{s / 2} f\right\|_{L^{2}({G})}^{2}=\sum_{[\xi] \in \widehat{{G}}} d_{\xi} \lambda_{\xi}^{2 s}\|\widehat{f}(\xi)\|_{\mathrm{HS}}^{2}
. $$
\section{A local existance result}\label{sec3}
In this section, we study the local well-posedness of the Cauchy problem (\ref{2number31}), i.e.,
\begin{align*}
\begin{cases}
\partial^2_tu+(-\mathcal{L})^\alpha u+ \partial_tu =|u|^p, & x\in G,t>0,\\
u(0,x)=\varepsilon u_0(x), & x\in G,\\ \partial_tu(x,0)=\varepsilon u_1(x), & x\in G,
\end{cases}
\end{align*}
where $u_{0}(x)$ and $u_{1}(x)$ are two given functions on $G$ and $\varepsilon$ is a positive constant describing the smallness of the Cauchy data.
\subsection{Fourier multiplier expressions and $L^2(G)-L^2(G)$ estimates
} \label{sec3.1}
In this subsection, we derive $L^2(G)– L^2(G)$ estimates for the solutions to the homogeneous problem (\ref{eq1}).
We employ the group Fourier transform on the compact group $G$ with respect to the space variable $x$ together with the Plancherel identity in order to estimate $L^2$-norms of $u(t, ·), (-\mathcal{L})^{\frac{\alpha}{2}}u(t, \cdot)$, and $\partial_{t}u(t, ·)$.
Let $u$ be a solution to (\ref{eq1}). Let $\widehat{u}(t, \xi)=(\widehat{u}(t, \xi)_{kl})_{1\leq k, l\leq d_\xi}\in \mathbb{C}^{d_\xi\times d_\xi}, [\xi]\in\widehat{ G}$ denote the Fourier transform of $u$ with respect to the $x $ variable. Invoking the group Fourier transform with respect to $x$ on (\ref{eq1}), we deduce that $\widehat{u}(t, \xi)$ is a solution to the following Cauchy problem for the system of ODE's (with the size of the system that depends on the representation $\xi$)
\begin{align}\label{eq6661}
\begin{cases}
\partial^2_t\widehat{u}(t,\xi)+(- \sigma_{\mathcal{L}}(\xi))^\alpha \widehat{u}(t,\xi) +\partial_t \widehat{u}(t,\xi)=0,& [\xi]\in\widehat{ G},~t>0,\\ \widehat{u}(0,\xi)=\widehat{u}_0(\xi), &[\xi]\in\widehat{ G},\\ \partial_t\widehat{u}(0,\xi)=\widehat{u}_1(\xi), &[\xi]\in\widehat{ G},
\end{cases}
\end{align}
where $\sigma_{\mathcal{L}}$ is the symbol of the operator operator $\mathcal{L}$. Using the identity (\ref{symbol}), the system (\ref{eq6661}) can be written in the form of $d_\xi^2$ independent ODE's, namely,
\begin{align}\label{eqq7}
\begin{cases}
\partial^2_t\widehat{u}(t,\xi)_{kl}+ \partial_t\widehat{u}(t,\xi)_{kl}+ \lambda_\xi^{2\alpha } \widehat{u}(t,\xi)_{kl}= 0,& [\xi]\in\widehat{ G},~t>0,\\ \widehat{u}(0,\xi)_{kl}=\widehat{u}_0(\xi)_{kl}, &[\xi]\in\widehat{ G},\\ \partial_t\widehat{u}(0,\xi)_{kl}=\widehat{u}_1(\xi)_{kl}, &[\xi]\in\widehat{ G},
\end{cases}
\end{align}
for all $k,l\in\{1,2,\ldots,d_\xi\}.$ Then, the characteristic equation of (\ref{eqq7}) is given by
\[\lambda^2+ \lambda+\lambda_\xi^{2\alpha } =0,\]
and consequently the characteristic roots are $\lambda=-\frac{1}{2}\pm \frac{\sqrt{1-4\lambda_\xi^{2\alpha}}}{2} $.
Thus the solution to the homogeneous problem (\ref{eqq7}) is given by
\begin{align}\label{number2}\nonumber
\widehat{u}(t,\xi)_{kl}&=e^{-\frac{t}{2}}A_0(t, \xi) \widehat{u}_0(\xi)_{kl}+e^{-\frac{t}{2}}A_1(t, \xi) \left(\widehat{u}_1(\xi)_{kl}+\frac{1}{2}\widehat{u}_0(\xi)_{kl}\right)\\
&=e^{-\frac{t}{2}}\left[A_0(t, \xi)+\frac{A_1(t, \xi)}{2}\right] \widehat{u}_0(\xi)_{kl}+e^{-\frac{t}{2}}A_1(t, \xi) \widehat{u}_1(\xi)_{kl},
\end{align}
where
\begin{align}\label{number1}
A_0(t, \xi)=\begin{cases}
\cosh \left(\frac{1}{2}\sqrt{ 1-4\lambda_\xi^{2\alpha}}~~t \right)& \text{if } 4\lambda_\xi^{2\alpha}< 1,\\
1& \text{if } 4\lambda_\xi^{2\alpha}=1,\\
\cos\left(\frac{1}{2}\sqrt{ 4\lambda_\xi^{2\alpha}-1}~~t \right)& \text{if } 4\lambda_\xi^{2\alpha}> 1,\\
\end{cases} \end{align}
and
\begin{align}\label{number3}
A_1(t, \xi)=\begin{cases}
\frac{ 2\sinh \left(\frac{1}{2}\sqrt{ 1-4\lambda_\xi^{2\alpha}}~~t \right)}{\sqrt{ 1-4\lambda_\xi^{2\alpha}}}& \text{if } 4\lambda_\xi^{2\alpha}< 1,\\
t& \text{if } 4\lambda_\xi^{2\alpha}=1,\\
\frac{ \sin\left(\frac{1}{2}\sqrt{ 4\lambda_\xi^{2\alpha}-1}~~t \right)}{\sqrt{ 4 \lambda_\xi^{2\alpha}-1}}& \text{if } 4\lambda_\xi^{2\alpha}> 1.
\end{cases}
\end{align}
We notice that $A_0(t, \xi) = \partial_t A_1(t, \xi)$ for any $[\xi] \in \widehat{ G}$ and
\begin{align}\label{number22}
\partial_{t} \widehat{u}(t,\xi)_{kl}=-e^{-\frac{t}{2}}A_1(t, \xi)\lambda_\xi^{2\alpha} \widehat{u}_0(\xi)_{kl}+e^{-\frac{t}{2}} \left[ A_0(t, \xi)-\frac{1}{2}A_1(t, \xi)\right] \widehat{u}_1(\xi)_{kl}.
\end{align}
To simplify the presentation, we introduce the following partition of the unitary dual $\widehat{ G}$ as:
\begin{align*}
\mathcal{R}_1&=\{[\xi]\in\widehat{ G}:0\leq \lambda_\xi^{2\alpha }<\frac{1}{16}\}, \\
\mathcal{R}_2&=\{[\xi]\in\widehat{ G}:\lambda_\xi^{2\alpha }\geq \frac{1}{16}\}.
\end{align*}
Note that the choice of $ \frac{1}{16}$ as a threshold in the previous definitions is irrelevant since our goal is to separate $0$ (which is an eigenvalue for the continuous irreducible unitary representation $1 : x \in G \to 1 \in \mathbb{C}$) from the other eigenvalues. Now we estimate $L^2$-norms of $u(t, ·), (-\mathcal{L})^{\frac{\alpha}{2}}u(t, \cdot)$, and $\partial_{t}u(t, ·)$.
\noindent\textbf{Estimate on $\mathcal{R}_1:$}
In this case,
$ |A_0(t, \xi)|\leq \cosh \frac{t}{2}$ and $ |A_1(t, \xi)|\leq \sin \frac{t}{2}$. Therefore from \eqref{number2}, we have
\begin{align}\label{eqd1}
| \widehat{u}(t,\xi)_{kl}| \lesssim | \widehat{u}_0(\xi)_{kl}| +| \widehat{u}_1(\xi)_{kl}| .
\end{align}
Again for $[\xi] \in \mathcal{R}_1$, we have
\begin{align*}
A_0(t, \xi)+\frac{A_1(t, \xi)}{2}& = \frac{e^{\frac{1}{2}\sqrt{ 1-4\lambda_\xi^{2\alpha}}~~t }+e^{-\frac{1}{2}\sqrt{ 1-4\lambda_\xi^{2\alpha}}~~t }}{2}+ \frac{e^{\frac{1}{2}\sqrt{ 1-4\lambda_\xi^{2\alpha}}~~t }-e^{-\frac{1}{2}\sqrt{ 1-4\lambda_\xi^{2\alpha}}~~t }}{2\sqrt{ 1-4\lambda_\xi^{2\alpha}}}\\&=\left(
\frac{1}{2}+\frac{1}{4\sqrt{ 1-4\lambda_\xi^{2\alpha}}} \right)e^{\frac{1}{2}\sqrt{ 1-4\lambda_\xi^{2\alpha}}~~t }+\left(
\frac{1}{2}-\frac{1}{2\sqrt{ 1-4\lambda_\xi^{2\alpha}}} \right)e^{-\frac{1}{2}\sqrt{ 1-4\lambda_\xi^{2\alpha}}~~t }\\
&\approx
\left(
\frac{1}{2}+\frac{1}{4\sqrt{ 1-4\lambda_\xi^{2\alpha}}} \right)e^{\frac{1}{2}\sqrt{ 1-4\lambda_\xi^{2\alpha}}~~t }-\frac{\lambda_\xi^{2\alpha}}{\sqrt{ 1-4\lambda_\xi^{2\alpha}}} e^{-\frac{1}{2}\sqrt{ 1-4\lambda_\xi^{2\alpha}}~~t }.
\end{align*}
Thus, from \eqref{number1} we deduce that
\begin{align*}
\widehat{u}(t,\xi)_{kl}
&\approx e^{-\frac{t}{2}}\left[
\left(
\frac{1}{2}+\frac{1}{4\sqrt{ 1-4\lambda_\xi^{2\alpha}}} \right)e^{\frac{1}{2}\sqrt{ 1-4\lambda_\xi^{2\alpha}}~~t }-\frac{\lambda_\xi^{2\alpha}}{\sqrt{ 1-4\lambda_\xi^{2\alpha}}} e^{-\frac{1}{2}\sqrt{ 1-4\lambda_\xi^{2\alpha}}~~t }\right] \widehat{u}_0(\xi)_{kl}\\&\quad+e^{-\frac{t}{2}} \left[\frac{e^{\frac{1}{2}\sqrt{ 1-4\lambda_\xi^{2\alpha}}~~t }-e^{-\frac{1}{2}\sqrt{ 1-4\lambda_\xi^{2\alpha}}~~t }}{2\sqrt{ 1-4\lambda_\xi^{2\alpha}}}\right] \widehat{u}_1(\xi)_{kl},
\end{align*}
and therefore,
\begin{align*}
| \widehat{u}(t,\xi)_{kl} |
&\lesssim e^{-\frac{t}{2} } \Bigg[ e^{\frac{1}{2}\sqrt{ 1-4\lambda_\xi^{2\alpha}}~~t } \left(\left|\widehat{u}_0(\xi)_{k \ell}\right|+\left|\widehat{u}_1(\xi)_{k \ell}\right|\right) \\&\quad\qquad\qquad\qquad
+\frac{e^{-\frac{1}{2}\sqrt{ 1-4\lambda_\xi^{2\alpha}}~~t }}{\sqrt{ 1-4\lambda_\xi^{2\alpha}} } \left(\lambda_\xi^{2\alpha} \left|\widehat{u}_0(\xi)_{k \ell}\right|+\frac{1}{2}\left|\widehat{u}_1(\xi)_{k \ell}\right|\right)\Bigg]\\
&\lesssim e^{-\frac{t}{2} } \left(\left|\widehat{u}_0(\xi)_{k \ell}\right|+\left|\widehat{u}_1(\xi)_{k \ell}\right|\right) \left[ e^{\frac{1}{2}\sqrt{ 1-4\lambda_\xi^{2\alpha}}~~t }
+\frac{e^{-\frac{1}{2}\sqrt{ 1-4\lambda_\xi^{2\alpha}}~~t }}{\sqrt{ 1-4\lambda_\xi^{2\alpha}} } \right] \\
&\lesssim e^{-\frac{t}{2}+\frac{1}{2}\sqrt{ 1-4\lambda_\xi^{2\alpha}}~~t } \left(\left|\widehat{u}_0(\xi)_{k \ell}\right|+\left|\widehat{u}_1(\xi)_{k \ell}\right|\right) \left[ 1
+\frac{e^{-\sqrt{ 1-4\lambda_\xi^{2\alpha}}~~t }}{\sqrt{ 1-4\lambda_\xi^{2\alpha}} } \right] \\
&\approx e^{-\frac{t}{2}+\frac{1}{2}( 1-2\lambda_\xi^{2\alpha})t} \left(\left|\widehat{u}_0(\xi)_{k \ell}\right|+\left|\widehat{u}_1(\xi)_{k \ell}\right|\right) \left[ 1
+\frac{e^{-\sqrt{ 1-4\lambda_\xi^{2\alpha}}~~t }}{\sqrt{ 1-4\lambda_\xi^{2\alpha}} } \right] \\
&\lesssim e^{ -\lambda_\xi^{2\alpha}t} \left(\left|\widehat{u}_0(\xi)_{k \ell}\right|+\left|\widehat{u}_1(\xi)_{k \ell}\right|\right) .
\end{align*}
This implies using AM-GM inequality that
\begin{align}\label{number23}
\lambda_{\xi}^{2\alpha} \left|\widehat{u}(t, \xi)_{k \ell}\right|^2 \lesssim \lambda_{\xi}^{2\alpha} {e}^{- 2\lambda_{\xi}^{2\alpha} t}\left(\left|\widehat{u}_0(\xi)_{k \ell}\right|^2 +\left|\widehat{u}_1(\xi)_{k \ell}\right|^2\right) \lesssim(1+t)^{-1}\left(\left|\widehat{u}_0(\xi)_{k \ell}\right|^2+\left|\widehat{u}_1(\xi)_{k \ell}\right|^2\right).
\end{align}
We note that, for $[\xi] \in \mathcal{R}_1$, we have
\begin{align*}
A_0(t, \xi)-\frac{1}{2}A_1(t, \xi)&=\frac{e^{\frac{1}{2}\sqrt{ 1-4\lambda_\xi^{2\alpha}}~~t }+e^{-\frac{1}{2}\sqrt{ 1-4\lambda_\xi^{2\alpha}}~~t }}{2}- \frac{e^{\frac{1}{2}\sqrt{ 1-4\lambda_\xi^{2\alpha}}~~t }-e^{-\frac{1}{2}\sqrt{ 1-4\lambda_\xi^{2\alpha}}~~t }}{2\sqrt{ 1-4\lambda_\xi^{2\alpha}}}\\&=\left(
\frac{1}{2}-\frac{1}{2\sqrt{ 1-4\lambda_\xi^{2\alpha}}} \right)e^{\frac{1}{2}\sqrt{ 1-4\lambda_\xi^{2\alpha}}~~t }+\left(
\frac{1}{2}+\frac{1}{2\sqrt{ 1-4\lambda_\xi^{2\alpha}}} \right)e^{-\frac{1}{2}\sqrt{ 1-4\lambda_\xi^{2\alpha}}~~t }\\
&\approx
-\frac{\lambda_\xi^{2\alpha}}{\sqrt{ 1-4\lambda_\xi^{2\alpha}}} e^{\frac{1}{2}\sqrt{ 1-4\lambda_\xi^{2\alpha}}~~t }+\left(
\frac{1}{2}+\frac{1}{2\sqrt{ 1-4\lambda_\xi^{2\alpha}}} \right)e^{-\frac{1}{2}\sqrt{ 1-4\lambda_\xi^{2\alpha}}~~t }.
\end{align*}
Therefore, using it in \eqref{number22} for $[\xi] \in \mathcal{R}_1$, we get
\begin{align}\label{number27}\nonumber
\left|\partial_t \widehat{u}(t, \xi)_{k \ell}\right| & \lesssim \lambda_{\xi}^{2\alpha} {e}^{-\lambda_{\xi}^{2\alpha} t}\left(\left|\widehat{u}_0(\xi)_{k \ell}\right|+\left|\widehat{u}_1(\xi)_{k \ell}\right|\right)+ {e}^{-t}\left(\left|\widehat{u}_0(\xi)_{k \ell}\right|+\left|\widehat{u}_1(\xi)_{k \ell}\right|\right) \\
& \lesssim(1+t)^{-1}\left(\left|\widehat{u}_0(\xi)_{k \ell}\right|+\left|\widehat{u}_1(\xi)_{k \ell}\right|\right).
\end{align}
\noindent\textbf{Estimate on $\mathcal{R}_2:$} When $ \frac{1}{16}\leq \lambda_\xi^{2\alpha }<\frac{1}{4}$, by following the similar calculation, there exists a suitable positive constant $c_1$ independent of $[\xi]$ such that
\begin{align}\label{eqc1}
| \widehat{u}(t,\xi)_{kl}| \lesssim e^{-c_1t}\left[| \widehat{u}_0(\xi)_{kl}| +| \widehat{u}_1(\xi)_{kl}| \right].
\end{align}
When $ \lambda_\xi^{2\alpha }\geq \frac{1}{4}$, it is easy to note that
$
| A_0(t,\xi)| \leq1 $ and $| {A}_1(t,\xi)| \leq t.
$
Therefore from \eqref{number2}, there exists a suitable $c_2>0$ independent of $[\xi]$ such that
\begin{align}\label{eqc2}\nonumber
| \widehat{u}(t,\xi)_{kl}| &\leq e^{-\frac{t}{2}} \widehat{u}_0(\xi)_{kl}+t e^{-\frac{t}{2}} \left(\widehat{u}_1(\xi)_{kl}+\frac{1}{2}\widehat{u}_0(\xi)_{kl}\right)\\\nonumber
& \lesssim (1+t) e^{-\frac{t}{2}} \left[| \widehat{u}_0(\xi)_{kl}| +| \widehat{u}_1(\xi)_{kl}| \right]\\
& \lesssim e^{-c_2 t} \left[| \widehat{u}_0(\xi)_{kl}| +| \widehat{u}_1(\xi)_{kl}| \right].
\end{align}
Thus from (\ref{eqc1}) and \eqref{eqc2}, we have
\begin{align}\label{eqd2}
| \widehat{u}(t,\xi)_{kl}| \lesssim e^{-c t} \left[| \widehat{u}_0(\xi)_{kl}| +| \widehat{u}_1(\xi)_{kl}| \right].
\end{align}
where $c$ is a suitable positive constant independent of $[\xi]$.
Moreover, for $[\xi] \in \mathcal{R}_2$, it follows that
\begin{align}\label{number24}
\lambda_{\xi}^\alpha\left|\widehat{u}(t, \xi)_{k \ell}\right| \lesssim {e}^{-c t}\left(\lambda_{\xi}^\alpha\left|\widehat{u}_0(\xi)_{k \ell}\right|+\left|\widehat{u}_1(\xi)_{k \ell}\right|\right),
\end{align}
for a suitable positive constant $c$.
On the other hand, for $[\xi] \in \mathcal{R}_2$, we get the estimate
\begin{align}\label{number26}
\left|\partial_t \widehat{u}(t, \xi)_{k \ell}\right| \lesssim {e}^{-c t}\left(\lambda_{\xi}^\alpha \left|\widehat{u}_0(\xi)_{k \ell}\right|+\left|\widehat{u}_1(\xi)_{k \ell}\right|\right),
\end{align}
where $c>0$ is a suitable constant.
\textbf{Estimate for $\|u(t, \cdot )\|_{L^{2}(G)}$:}
Using the Plancherel formula along with the equations (\ref{eqd1}) and (\ref{eqd2}), it follows that
\begin{align}\label{L2}
\|u(t, \cdot)\|_{L^{2}(G)}^{2}&=\sum_{[\xi] \in \widehat{G}} d_{\xi} \sum_{k, \ell=1}^{d_{\xi}}\left|\widehat{u}(t, \xi)_{k \ell}\right|^{2} \nonumber \\
&=\sum_{[\xi] \in \mathcal{R}_1} d_{\xi} \sum_{k, \ell=1}^{d_{\xi}}\left|\widehat{u}(t, \xi)_{k \ell}\right|^{2} +\sum_{[\xi] \in \mathcal{R}_2} d_{\xi} \sum_{k, \ell=1}^{d_{\xi}}\left|\widehat{u}(t, \xi)_{k \ell}\right|^{2} \nonumber\\
& \lesssim \sum_{[\xi] \in \mathcal{R}_1} d_{\xi} \sum_{k, \ell=1}^{d_{\xi}} \left(\left|\widehat{u}_{0}(\xi)_{k \ell}\right|^{2}+\left|\widehat{u}_{1}(\xi)_{k \ell}\right|^{2}\right) \nonumber \\&\qquad +\sum_{[\xi] \in \mathcal{R}_2} d_{\xi} \sum_{k, \ell=1}^{d_{\xi}} e^{-2ct}\left(\left|\widehat{u}_{0}(\xi)_{k \ell}\right|^{2}+\left|\widehat{u}_{1}(\xi)_{k \ell}\right|^{2}\right)\nonumber\\
& \lesssim \sum_{[\xi] \in \widehat{G}} d_{\xi} \sum_{k, \ell=1}^{d_{\xi}}\left(\left|\widehat{u}_{0}(\xi)_{k \ell}\right|^{2}+\left|\widehat{u}_{1}(\xi)_{k \ell}\right|^{2}\right) \nonumber \\
&=\left\|u_{0}\right\|_{L^{2}(G)}^{2}+\left\|u_{1}\right\|_{L^{2}(G)}^{2} .
\end{align}
\textbf{Estimate for $\left\|(-\mathcal{L})^{\alpha / 2} u(t, \cdot )\right\|_{L^{2}(G)}$:}
Using the Plancherel formula, we get
\begin{align}\label{f1}
\left\|(-\mathcal{L})^{\alpha / 2} u(t, \cdot)\right\|_{L^{2}(G)}^2 \nonumber &=\sum_{[\xi] \in \widehat{G}} d_{\xi} \left\|\sigma_{(-\mathcal{L})^{\alpha / 2}}(\xi)\widehat{u}(t, \xi) \right\|_{HS}^{2} \\\nonumber=&\sum_{[\xi] \in \widehat{G}} d_{\xi} \sum_{k, \ell=1}^{d_{\xi}}\lambda_\xi^{2\alpha }\left|\widehat{u}(t, \xi)_{k \ell}\right|^{2}\\\nonumber
=&\sum_{[\xi] \in \mathcal{R}_1} d_{\xi} \sum_{k, \ell=1}^{d_{\xi}}\lambda_\xi^{2\alpha } \left|\widehat{u}(t, \xi)_{k \ell}\right|^{2} +\sum_{[\xi] \in \mathcal{R}_2} d_{\xi} \sum_{k, \ell=1}^{d_{\xi}}\lambda_\xi^{2\alpha }\left|\widehat{u}(t, \xi)_{k \ell}\right|^{2} \\ \nonumber
\lesssim&(1+t)^{-1}\sum_{[\xi] \in \mathcal{R}_1} d_{\xi} \sum_{k, \ell=1}^{d_{\xi}} \left(\left|\widehat{u}_0(\xi)_{k \ell}\right|^2+\left|\widehat{u}_1(\xi)_{k \ell}\right|^2\right)\\ \nonumber\quad &+{e}^{-c t}\sum_{[\xi] \in \mathcal{R}_2} d_{\xi} \sum_{k, \ell=1}^{d_{\xi}} \left(\lambda_{\xi}^{2\alpha}\left|\widehat{u}_0(\xi)_{k \ell}\right|^2+\left|\widehat{u}_1(\xi)_{k \ell}\right|^2\right)
\\ \lesssim &(1+t)^{-1} (\left\|u_{0}\right\|_{H_{\mathcal{L}}^{{\alpha }}(G)}^{2}+\left\|u_{1}\right\|_{L^{2}(G)}^{2}). \end{align}
\textbf{Estimate for $\left\|\partial_{t} u(t, \cdot )\right\|_{L^{2}(G)}$:} From (\ref{number27}) and (\ref{number26}),
the Plancherel formula yields that
\begin{align}\label{deri}\nonumber
\left\|\partial_{t} u(t, \cdot )\right\|_{L^{2}(G)}^2 &=\sum_{[\xi] \in \widehat{G}} d_{\xi} \sum_{k, \ell=1}^{d_{\xi}} \left|\partial_{t} \widehat{u}(t, \xi)_{k \ell}\right|^{2}\\\nonumber =&\sum_{[\xi] \in \mathcal{R}_1} d_{\xi} \sum_{k, \ell=1}^{d_{\xi}} \left|\partial_{t} \widehat{u}(t, \xi)_{k \ell}\right|^{2} +\sum_{[\xi] \in \mathcal{R}_2} d_{\xi} \sum_{k, \ell=1}^{d_{\xi}} \left|\partial_{t} \widehat{u}(t, \xi)_{k \ell}\right|^{2} \\ \nonumber \lesssim&(1+t)^{-2}\sum_{[\xi] \in \mathcal{R}_1} d_{\xi} \sum_{k, \ell=1}^{d_{\xi}} \left(\left|\widehat{u}_0(\xi)_{k \ell}\right|^2+\left|\widehat{u}_1(\xi)_{k \ell}\right|^2\right)\\ \nonumber\quad &+{e}^{-2c t}\sum_{[\xi] \in \mathcal{R}_2} d_{\xi} \sum_{k, \ell=1}^{d_{\xi}} \left(\lambda_{\xi}^{2\alpha}\left|\widehat{u}_0(\xi)_{k \ell}\right|^2+\left|\widehat{u}_1(\xi)_{k \ell}\right|^2\right)
\\ \lesssim &(1+t)^{-2} (\left\|u_{0}\right\|_{H_{\mathcal{L}}^{{\alpha }}(G)}^{2}+\left\|u_{1}\right\|_{L^{2}(G)}^{2}). \end{align}
Now, we are in a position to prove Proposition \ref{thm11}.
\begin{proof}[Proof of Proposition \ref{thm11}]
The proof of Theorem \ref{thm11} follows from the estimates (\ref{L2}), (\ref{f1}), and (\ref{deri}) for $\|u(t, \cdot )\|_{L^{2}(G)}$, $\left\|(-\mathcal{L})^{\alpha / 2} u(t, \cdot )\right\|_{L^{2}(G)}$, and $\left\|\partial_{t} u(t, \cdot )\right\|_{L^{2}(G)}$, respectively.
\end{proof}
\subsection{Local in time existence}\label{sec4}
In this subsection we will prove Theorem \ref{thm22}, i.e., the local well-posedness of the Cauchy problem (\ref{eq0010}) in the energy evolution space $\mathcal C\left([0,T], H^\alpha_{\mathcal{L}}(G)\right)\cap\mathcal C^1\left([0,T],L^2(G)\right)$.
First, we recall some notations to present the proof of Theorem \ref{thm22}. Consider the space \[X(T):=\mathcal{C}\left([0,T], H^\alpha_{\mathcal L}(G)\right)\cap\mathcal C^1\left([0,T],L^2(G)\right),\] equipped with the norm
\begin{align}\label{eq33333}
\|u\|_{X(T)}&:=\sup\limits_{t\in[0,T]}\left ( \|u(t,\cdot)\|_{L^2(G)}+\|(-\mathcal L)^{\alpha/2}u(t,\cdot)\|_{L^2(G)}+\|\partial_tu(t,\cdot)\|_{L^2(G)}\right ).
\end{align}
Here we will briefly recall the notion of mild solutions in our framework to the Cauchy problem (\ref{eq0010}) and will analyze our approach to prove Theorem \ref{thm22}. Applying Duhamel's principle, the solution to the nonlinear inhomogeneous problem
\begin{align}\label{eq3111}
\begin{cases}
\partial^2_tu+(-\mathcal{L})^\alpha u+\partial_{t}u =F(t, x), & x\in G,t>0,\\
u(0,x)= u_0(x), & x\in G,\\ \partial_tu(0, x)= u_1(x), & x\in G,
\end{cases}
\end{align}
can be expressed as
$$ u(t, x)= u_{0}(x)*_{(x)}E_{0}(t, x)+u_{1}(x)*_{(x)}E_{1}(t, x) +\int_{0}^{t} F(s, x)*_{(x)} E_{1}(t-s, x) \;d s, $$
where $*_{(x)}$ is the group convolution product on $G$ with respect to the $x$ variable. Here $E_{0}(t, x)$ and $E_{1}(t, x)$ represent the fundamental solutions to the homogeneous problem, i.e., \eqref{eq3111} with $F=0$ and the initial data $\left(u_{0}, u_{1}\right)=\left(\delta_{0}, 0\right)$ and $\left(u_{0}, u_{1}\right)=$ $\left(0, \delta_{0}\right)$, respectively.
For any left-invariant differential operator $L$ on the compact Lie group $ {G}$, we apply the property that it commute with the group convolution, that is, $L\left(v*_{(x)} E_{1}(t, \cdot)\right)=v *_{(x)} L\left(E_{1}(t, \cdot)\right)$ and the invariance by time translations for the wave operator $ \partial^2_t+(-\mathcal{L})^\alpha $,
to get the previous representation formula.
We say that a function $u$ is a {\it mild solution} to (\ref{eq3111}) on $[0, T]$ if $u$ is a fixed point for the integral operator, $N: u \in X(T) \rightarrow N u(t, x) ,$ given by
\begin{align}\label{f2}
N u(t, x):= \varepsilon u_{0}(x) *_{(x)} E_{0}(t, x)+\varepsilon u_{1}(x) *_{(x)} E_{1}(t, x) +\int_{0}^{t}|u(s, x)|^{p} *_{(x)} E_{1}(t-s, x) \;ds
\end{align}
in the energy evolution space $X(T) \doteq \mathcal{C}\left([0, T], H_{\mathcal{L}}^{\alpha}(G)\right) \cap \mathcal{C}^{1}\left([0, T], L^{2}(G)\right)$, equipped with the norm defined in \eqref{eq33333}.
In order to show a uniquely determined fixed point of $N$ for a sufficiently small $T=T(\varepsilon)$, we use the Banach fixed point theorem with respect to the norm on $X(T)$ as defined by \eqref{eq33333}. In fact, for the small enough initial data $\left\|\left(u_{0}, u_{1}\right)\right\|_{H_{\mathcal{L}}^{\alpha}(G) \times L^{2}(G)}$, we will establish the following two inequalities
\begin{align}\label{2number100}
\|N u\|_{X(T)} \leq C\left\|\left(u_{0}, u_{1}\right)\right\|_{H_{\mathcal{L}}^{\alpha}(G) \times L^{2}(G)}+C\|u\|_{X(T)}^{p},
\end{align}
and \begin{align}\label{2number101}
\|N u-N v\|_{X(T)} \leq C\|u-v\|_{X(T)}\left(\|u\|_{X(T)}^{p-1}+\|v\|_{X(T)}^{p-1}\right),
\end{align}
for any $u, v \in X(T)$ and for some suitable constant $C>0$ independent of $T$. Then the Banach fixed point theorem immediately gives a uniquely determined fixed point $u$ on $N$. This fixed point $u$ will be our mild solution to (\ref{eq3111}) on $[0, T]$.
In order to prove the local existence result, an essential tool is the following Gagliardo-Nirenberg type inequality proved in general Lie groups \cite{Gall}.
\begin{lem}\cite{Gall} \label{lemma1}
Let $G$ be a connected unimodular Lie group with topological dimension $n.$ For any $1<q_0<\infty,~0<q,q_1<\infty$ and $0<\alpha<n$ such that $q_0<\frac{n}{\alpha},$ the following Gagliardo-Nirenberg type inequality holds
\begin{align}\label{eq33}
\|f\|_{L^q(G)}\lesssim \|f\|^\theta_{H^{\alpha,q_0}_\mathcal L(G)}\|f\|^{1-\theta}_{L^{q_1}(G)},
\end{align}
for all $f\in H^{\alpha,q_0}_\mathcal L(G)\cap L^{q_1}(G),$ provided that
\begin{align*}
\theta=\theta(n,\alpha,q,q_0,q_1)=\frac{\frac{1}{q_1}-\frac{1}{q}}{\frac{1}{q_1}-\frac{1}{q_0}+\frac{\alpha}{n}}\in[0,1].
\end{align*}
\end{lem}
We refer to \cite{Gall, 27} for several immediate important remarks from Lemma \ref{lemma1}.
The next corollary is a version of Lemma \ref{lemma1}, which is useful in our setting.
\begin{cor}
Let $G$ be a connected unimodular Lie group with topological dimension $n\geq 2[\alpha]+2$. For any $q\ge2$ such that $q\leq\frac{2n}{n-2\alpha}$, the following Gagliardo-Nirenberg type inequality holds
\begin{align}\label{eq34}
\|f\|_{L^q(G)}\lesssim \|f\|^{\theta(n, q, \alpha)}_{H^{\alpha }_\mathcal L(G)}\|f\|^{1-\theta(n, q, \alpha)}_{L^{2}(G)},
\end{align}
for all $f\in H^{\alpha}_\mathcal L(G)$, where $\theta(n, q, \alpha)=\frac{n}{\alpha}\left(\frac{1}{2}-\frac{1}{q} \right) $.
\end{cor}
\begin{proof}[Proof of Theorem \ref{thm22}]
The expression (\ref{f2}) can be wriiten as $N u=u^\sharp+I[u]$, where
\begin{align*}
u^\sharp(t,x)=\varepsilon u_{0}(x) *_{(x)} E_{0}(t, x)+\varepsilon u_{1}(x) *_{(x)} E_{1}(t, x),
\end{align*}
and
\begin{align*}
I[u](t,x):=\int\limits_0^t |u(s,x)|^p*_x E_1(t-s, x)ds.
\end{align*}
Now, for the part $u^\sharp$, Theorem \ref{thm11}, immediately implies that
\begin{align}\label{f3}
\|u^\sharp\|_{X(T)}\lesssim\varepsilon\|(u_0,u_1)\|_{{H}_{\mathcal L}^\alpha (G)\times L^2(G)}.
\end{align}
On the other hand, for the part $I[u]$, using Minkowski's integral inequality, Young's convolution inequality, Theorem \ref{thm11}, and by time translation invariance property of the Cauchy problem (\ref{eq0010}), we get
\begin{align}\label{f}\nonumber
\|\partial_t^j(-\mathcal L)^{i\alpha/2}I[u]\|_{L^2(G)}&=\left(\int_{G} \big |\partial_t^j(-\mathcal L)^{i\alpha/2} \int\limits_0^t |u(s,x)|^p*_x E_1(t-s, x)ds\big |^2 dg\right)^{\frac{1}{2}}\\\nonumber
&=\left(\int_{G}\big | \int\limits_0^t |u(s,x)|^p*_x \partial_t^j(-\mathcal L)^{i\alpha/2}E_1(t-s, x)ds\big|^2 dg\right)^{\frac{1}{2}} \\\nonumber
&\lesssim \int\limits_0^t \| |u(s,\cdot )|^p*_x \partial_t^j(-\mathcal L)^{i\alpha/2}E_1(t-s, \cdot)\|_{L^2(G)}ds\\\nonumber
&\lesssim \int\limits_0^t \| u(s,\cdot)^p\|_{L^2(G)} \|\partial_t^j(-\mathcal L)^{i\alpha/2}E_1(t-s, \cdot)\|_{L^2(G)}ds\\\nonumber
&\lesssim \int\limits_0^t (1+t-s)^{-j-\frac{i}{2}}\|u(s,\cdot)\|^p_{L^{2p}(G)}ds\\\nonumber
&\lesssim\int\limits_0^t (1+t-s)^{-j-\frac{i}{2}} \|u(s,\cdot)\|^{p\theta(n,2p, \alpha)}_{H^\alpha_\mathcal L(G)}\|u(s,\cdot)\|^{p(1-\theta(n,2p,\alpha ))}_{L^2(G)}ds \\
&\lesssim\int\limits_0^t (1+t-s)^{-j-\frac{i}{2}} \|u\|^p_{X(s)}ds \lesssim t \|u\|^p_{X(t)},
\end{align}
for $i,j\in\{0,1\}$, such that $0\leq i+j\leq 1.$ Again for $i,j\in\{0,1\},$ such that $0\leq i+j\leq 1,$ a similar calculations as in (\ref{f}) together with H\"older's inequality and (\ref{eq34}), we get
\begin{align}\label{f5}\nonumber
& \|\partial_t^j(-\mathcal L)^{i\alpha/2}\left(I[u]-I[v]\right)\|_{L^2(G)}\\\nonumber&\lesssim \int\limits_0^t (1+t-s)^{-j-\frac{i}{2}}\||u(s,\cdot)|^p-|v(s,\cdot)|^p\|_{L^{2}(G)}ds\\\nonumber
&\lesssim\int\limits_0^t (1+t-s)^{-j-\frac{i}{2}} \|u(s,\cdot)-v(s,\cdot)\|_{L^{2p}(G)}\left(\|u(s,\cdot)\|^{p-1}_{L^{2p}(G)}+\|v(s,\cdot)\|^{p-1}_{L^{2p}(G)}\right)ds\\ &\lesssim t \|u-v\|_{X(t)}\left(\|u\|^{p-1}_{X(t)}-\|v\|^{p-1}_{X(t)}\right).
\end{align}
Thus combining (\ref{f3}), (\ref{f}), and (\ref{f5}), we have
\begin{align}\label{1}
\|N u\|_{X(T)} \leq D \varepsilon\left\|\left(u_{0}, u_{1}\right)\right\|_{H_{\mathcal{L}}^{\alpha }(G) \times L^{2}(G)}+DT\|u\|_{X(t)}^{p}
\end{align}
and
\begin{align}\label{2} \|Nu-Nv\|_{X(T)}\leq DT \|u-v\|_{X(t)}\left(\|u\|^{p-1}_{X(T)}-\|v\|^{p-1}_{X(T)}\right),\end{align}
where $D$ is a constant independent of $t$.
Choose $T$ (sufficiently small) in such a way that the map $N$ turns out to be a contraction in some neighborhood of $0$ in the Banach space $X(T).$ Therefore, Banach's fixed point theorem gives us the uniquely determined fixed point $u$ for the map $N$, which is our mild solution.
This completes the proof.
\end{proof}
From the above local existence result, we have the following remark.
\begin{rmk}
We note that in the statement of Theorem \ref{thm22}, the restriction on the upper bound for the exponent $p$ which is $p\leq\frac{n}{n-2\alpha }$ is necessary in order to apply Gagliardo-Nirenberg type inequality (\ref{eq34}) in (\ref{f}). Also, the other restriction $n \geq 2[\alpha]+2$ is made to fulfill the assumptions for the employment of such inequality.
\end{rmk}
\subsection{Blow-up result}\label{sec5}
In this subsection, we prove Theorem \ref{f6} using a comparison argument for ordinary differential inequality of second order.
Now we are ready to prove our main result of this section using an iteration argument.
\begin{proof}[Proof of Theorem \ref{f6}]
According to Definition \ref{eq332}, let $u$ be a local in-time energy solution to (\ref{eq0010}) with lifespan $T$. Let $t \in(0, T)$ be fixed. Suppose that $\phi \in \mathcal{C}_{0}^{\infty}([0, T) \times G),$ is a cut-off function such that $\phi=1$ on $[0, t] \times G$ in (\ref{eq011}). Then
\begin{align}\label{f7}
\int_{G} \partial_{t} u(t, x) \;dx+ \int_{G} u(t, x) \;dx-\varepsilon \int_{G} u_{0}(x) \;dx-\varepsilon \int_{G} u_{1}(x) \;dx=\int_{0}^{t} \int_{ {G}}|u(s, x)|^{p} {~d} x {~d} s
\end{align}
Let us introduce the time-dependent functional
$$
U_{0}(t) \doteq \int_{G} u(t, x) \;dx.
$$
Then the equality (\ref{f7}) can be rewritten in the following way:
$$
U_{0}^{\prime}(t)-U_{0}^{\prime}(0)+ U_{0}(t)-U_{0}(0) =\int_{0}^{t} \int_{G}|u(s, x)|^{p} \;dx \;ds .
$$
We also remark that, from the assumptions on the initial data, we obtain
$$
U_{0}(0)=\varepsilon \int_{G} u_{0}(x) \;dx \geq 0 \quad \text { and } \quad U_{0}^{\prime}(0)=\varepsilon \int_{G} u_{1}(x) \;dx \geq 0.
$$
Using Jensen's inequality, we have
\begin{align}\label{number30}
U_{0}^{\prime}(t)-U_{0}^{\prime}(0)+ U_{0}(t)-U_{0}(0) \geq \int_{0}^{t} \left|U_{0}(s)\right|^{p} \;ds.
\end{align}
Multiplying both sides of \eqref{number30} by $e^t$ and then integrating over $[0, t ]$, we obtain
$$
{e}^t U_0(t) \geq \left(U_0^{\prime}(0)+U_0(0)\right)\left( {e}^t-1\right)+U_0(0)+\int_0^t {e}^\eta \int_0^\eta\left|U_0(s)\right|^p {~d} s {~d} \eta,
$$
i.e.,
$$
U_0(t) \geq U_0(0)+U_0^{\prime}(0)\left(1- {e}^{-t}\right)+\int_0^t {e}^{\eta-t} \int_0^\eta\left|U_0(s)\right|^p {~d} s {~d} \eta.
$$
Since $U_0(0)$ and $U_0'(0)$ are non-negative, the above expression implies that $U_0$ is a positive function. Moreover, we also can say that
$$
U_0(t) \geq U_0(0)+U_0^{\prime}(0)\left(1- {e}^{-t}\right) \geq C \varepsilon \quad \text { for } t \ge0,
$$
where the multiplicative constant $C$ depends on $u_0, u_1$ and we also have the following iteration scheme
$$
U_0(t) \geq \int_0^t {e}^{\eta-t} \int_0^\eta\left|U_0(s)\right|^p {~d} s {~d} \eta.
$$
Now proceeding similarly as in Subsection 3.1 and 3.2 of \cite{27} for the iteration argument, we conclude the proof of Theorem \ref{f6}.
\end{proof}
\section{A global existence result}\label{sec6}
In this section, we study the global in-time existence of small data solutions for the nonlinear fractional dumped wave equation with mass and the power type nonlinearity. More preciously, for $0<\alpha<1$, we consider the Cauchy problem
\begin{align} \label{2number31}
\begin{cases}
\partial^2_tu+\left( -\mathcal{L}\right)^\alpha u+b\partial_{t} u+m^2u=|u|^p, & x\in G, t>0,\\
u(0,x)=u_0(x), & x\in G,\\ \partial_tu(x,0)=u_1(x), & x\in G,
\end{cases}
\end{align}
where $ p > 1$, $b,m^2$ are positive constants, $u_{0}(x)$ and $u_{1}(x)$ are two given functions on $G$.
\subsection{Fourier multiplier expressions and $L^2(G)-L^2(G)$ estimates
}\label{2sec3}
In this subsection, we derive $L^2(G)– L^2(G)$ estimates for solutions of the homogeneous problem (\ref{2number31}).
We employ the group Fourier transform on the compact group $G$ with respect to the space variable $x$ together with the Plancherel identity in order to estimate $L^2$-norms of $u(t, ·), (-\mathcal{L})^{\frac{\alpha}{2}}u(t, \cdot)$, and $\partial_{t}u(t, ·)$.
Let $u$ be a solution to (\ref{2number31}). Let $\widehat{u}(t, \xi)=(\widehat{u}(t, \xi)_{kl})_{1\leq k, l\leq d_\xi}\in \mathbb{C}^{d_\xi\times d_\xi}, [\xi]\in\widehat{ G}$ denote the Fourier transform of $u$ with respect to the $x $ variable. Applying the group Fourier transform with respect to $x$ on (\ref{2number31}), we deduce that $\widehat{u}(t, \xi)$ is a solution to the following Cauchy problem for the system of ODE's (with the size of the system that depends on the representation $\xi$)
\begin{align}\label{2eq6661}
\begin{cases}
\partial^2_t\widehat{u}(t,\xi)+(- \sigma_{\mathcal{L}}(\xi))^\alpha \widehat{u}(t,\xi) +b \partial_t \widehat{u}(t,\xi)+m^2 \widehat{u}(t,\xi)=0,& [\xi]\in\widehat{ G},~t>0,\\ \widehat{u}(0,\xi)=\widehat{u}_0(\xi), &[\xi]\in\widehat{ G},\\ \partial_t\widehat{u}(0,\xi)=\widehat{u}_1(\xi), &[\xi]\in\widehat{ G},
\end{cases}
\end{align}
where $\sigma_{\mathcal{L}}$ is the symbol of the Laplace-Beltrami operator $\mathcal{L}$. Using the identity (\ref{symbol}), the system (\ref{2eq6661}) can be written in the form of $d_\xi^2$ independent ODE's, namely,
\begin{align}\label{2eqq7}
\begin{cases}
\partial^2_t\widehat{u}(t,\xi)_{kl}+b \partial_t\widehat{u}(t,\xi)_{kl}+ \lambda_\xi^{2\alpha } \widehat{u}(t,\xi)_{kl}+m^2\widehat{u}(t,\xi)_{kl}= 0,& [\xi]\in\widehat{ G},~t>0,\\ \widehat{u}(0,\xi)_{kl}=\widehat{u}_0(\xi)_{kl}, &[\xi]\in\widehat{ G},\\ \partial_t\widehat{u}(0,\xi)_{kl}=\widehat{u}_1(\xi)_{kl}, &[\xi]\in\widehat{ G},
\end{cases}
\end{align}
for all $k,l\in\{1,2,\ldots,d_\xi\}.$ Then, the characteristic equation of (\ref{2eqq7}) is given by
\[\lambda^2+b \lambda+\lambda_\xi^{2\alpha }+m^2 =0,\]
and consequently the characteristic roots are $\lambda=-\frac{b}{2}\pm {\sqrt{\frac{b^2}{4}-\lambda_\xi^{2\alpha}-m^2}} $.
Thus the solution to the homogeneous problem (\ref{2eqq7}) is given by
\begin{align}\label{2number2}
\widehat{u}(t,\xi)_{kl}=e^{-\frac{bt}{2}}A_0(t, \xi) \widehat{u}_0(\xi)_{kl}+e^{-\frac{bt}{2}}A_1(t, \xi) \left(\widehat{u}_1(\xi)_{kl}+\frac{b}{2}\widehat{u}_0(\xi)_{kl}\right),
\end{align}
where
\begin{align}\label{2number1}
A_0(t, \xi)=\begin{cases}
\cosh \left(\sqrt{\frac{b^2}{4}-\lambda_\xi^{2\alpha}-m^2}~~t \right),& \text{if } \lambda_\xi^{2\alpha}< \frac{b^2}{4}-m^2,\\
1,& \text{if } \lambda_\xi^{2\alpha}= \frac{b^2}{4}-m^2,\\
\cos\left(\sqrt{\lambda_\xi^{2\alpha}-\frac{b^2}{4}+m^2}~~t \right),& \text{if } \lambda_\xi^{2\alpha}> \frac{b^2}{4}-m^2,\\
\end{cases} \end{align}
and
\begin{align}\label{2number3}
A_1(t, \xi)=\begin{cases}
\frac{ 2\sinh \left(\sqrt{\frac{b^2}{4}-\lambda_\xi^{2\alpha}-m^2}~~t \right)}{\sqrt{\frac{b^2}{4}-\lambda_\xi^{2\alpha}-m^2}},& \text{if } \lambda_\xi^{2\alpha}< \frac{b^2}{4}-m^2,\\
t,& \text{if } \text{if } \lambda_\xi^{2\alpha}= \frac{b^2}{4}-m^2,\\
\frac{ \sin\left(\sqrt{\lambda_\xi^{2\alpha}+\frac{b^2}{4}-m^2}~~t \right)}{\sqrt{\lambda_\xi^{2\alpha}-\frac{b^2}{4}+m^2 }},& \text{if } \lambda_\xi^{2\alpha}> \frac{b^2}{4}-m^2.
\end{cases}
\end{align}
We notice that $A_0(t, \xi) = \partial_t A_1(t, \xi)$ for any $[\xi] \in \widehat{ G}$. Moreover, we have the following representation for the time derivative
\begin{align}\label{2number22}
\partial_{t} \widehat{u}(t,\xi)_{kl}=e^{-\frac{bt}{2}}A_0(t, \xi) \widehat{u}_1(\xi)_{kl}-e^{-\frac{bt}{2}} A_1(t, \xi) \left[ \frac{b}{2}\widehat{u}_1(\xi)_{kl}+ (\lambda_\xi^{2\alpha}+m^2)\widehat{u}_0(\xi)_{kl} \right] .
\end{align}
Next we will estimate the values of $ |\widehat{u}(t, \xi)_{k \ell}|, \partial_t |\widehat{u}(t, \xi)_{k \ell}|$ and $ \lambda_{\xi}| \widehat{u}(t, \xi)_{k \ell} |$ by considering the relation between $b$ and $m^2$.
\textbf{When $b^2<4 m^2$: }
The only the case is to consider that $\lambda_{\xi}^2>\frac{b^2}{4}-m^2$ by considering the fact that all eigenvalues $\{\lambda_{\xi}^{2\alpha}\}_{[\xi] \in \widehat{ G}}$ of $(-\mathcal{L})^{\alpha}$ are nonnegative. Thus, by the similar calculus done in Subsection \ref{sec3.1}, we have
\begin{align}\label{2number32}
\left|\widehat{u}(t, \xi)_{k \ell}\right| & \lesssim {e}^{-\frac{b}{2} t}\left[ \left|\widehat{u}_0(\xi)_{k \ell}\right|+\left|\widehat{u}_1(\xi)_{k \ell}\right|\right], \end{align}
\begin{align}\label{2number33}
\lambda_{\xi}^\alpha\left|\widehat{u}(t, \xi)_{k \ell}\right| & \lesssim {e}^{-\frac{b}{2} t}\left[ \left(1+\lambda_{\xi}^\alpha\right)\left|\widehat{u}_0(\xi)_{k \ell}\right|+\left|\widehat{u}_1(\xi)_{k \ell}\right|\right],
\end{align}
and
\begin{align}\label{2number34}\nonumber
\left|\partial_t \widehat{u}(t, \xi)_{k \ell}\right|& \lesssim e^{-\frac{bt}{2}} \widehat{u}_1(\xi)_{kl} + A_0(t, \xi) (\lambda_\xi^{2\alpha}+m^2)\widehat{u}_0(\xi)_{kl} \\& \lesssim {e}^{-\frac{b}{2} t}\left[\left(1+\lambda_{\xi}^\alpha \right)\left|\widehat{u}_0(\xi)_{k \ell}\right|+\left|\widehat{u}_1(\xi)_{k \ell}\right|\right],
\end{align}
for any $t \geq0$. Thus, using the Plancherel formula along with the equations (\ref{2number32}), (\ref{2number33}) and (\ref{2number34}), it follows that
\begin{align}\label{2L2}\nonumber
\|\partial_t^j(-\mathcal L)^{i\alpha/2}u(t, \cdot)\|_{L^{2}(G)}^{2}&=\sum_{[\xi] \in \widehat{G}} d_{\xi} \sum_{k, \ell=1}^{d_{\xi}}\lambda_{\xi}^{2\alpha i}\left|\partial_t^j \widehat{u}(t, \xi)_{k \ell}\right|^2 \\\nonumber
& \lesssim {e}^{-b t} \sum_{[\xi] \in \widehat{ G}} d_{\xi} \sum_{k, \ell=1}^{d_{\xi}}\left(\left(1+\lambda_{\xi}^{2\alpha}\right)^{(i+j)}\left|\widehat{u}_0(\xi)_{k \ell}\right|^2+\left|\widehat{u}_1(\xi)_{k \ell}\right|^2\right) \\
& = {e}^{-b t}\left[ \left\|u_{0}\right\|_{H_{\mathcal{L}}^{{\alpha(i+j) }}(G)}^{2}+\left\|u_{1}\right\|_{L^{2}(G)}^{2} \right],
\end{align}
for any $i,j\in\{0,1\}$, such that $0\leq i+j\leq 1$, with the convention that $H_{\mathcal{L}}^{0}(G)=L^2(G).$
\textbf{When $b^2=4 m^2$: } In this case, we only have to consider the cases when $\lambda_{\xi}^{2\alpha}=0$ and $\lambda_{\xi}^{2\alpha}>0$. Then, from \eqref{2number2}, \eqref{2number1}, and \eqref{2number3}, the solution can be written as
\begin{align*}
\widehat{u}(t, \xi)_{k \ell}=
\begin{cases}
e^{-\frac{bt}{2}} \cos\left( {\lambda_\xi^{\alpha}}~~t \right) \widehat{u}_0(\xi)_{kl}+e^{-\frac{bt}{2}}\frac{ \sin\left( {\lambda_\xi^{\alpha} }~~t \right)}{ {\lambda_\xi^{\alpha} }} \left(\widehat{u}_1(\xi)_{kl}+\frac{b}{2}\widehat{u}_0(\xi)_{kl}\right),& \text {if} ~ \lambda_{\xi}^2>0, \\
e^{-\frac{bt}{2}} \widehat{u}_0(\xi)_{kl}+te^{-\frac{bt}{2}} \left(\widehat{u}_1(\xi)_{kl}+\frac{b}{2}\widehat{u}_0(\xi)_{kl}\right),& \text {if} ~ \lambda_{\xi}^2=0.
\end{cases} \end{align*}
The second case $\lambda_{\xi}^2=0$ needs to be included as $0$ is the eigenvalue for the trivial representation $G.$ Thus
$$
\begin{aligned}
\left|\widehat{u}(t, \xi)_{k \ell}\right| & \lesssim (1+t) {e}^{-\frac{b}{2} t}\left(\left|\widehat{u}_0(\xi)_{k \ell}\right|+\left|\widehat{u}_1(\xi)_{k \ell}\right|\right) ,\\
\lambda_{\xi}^\alpha\left|\widehat{u}(t, \xi)_{k \ell}\right| & \lesssim {e}^{-\frac{b}{2} t}\left[ \left(1+\lambda_{\xi}^\alpha\right)\left|\widehat{u}_0(\xi)_{k \ell}\right|+\left|\widehat{u}_1(\xi)_{k \ell}\right|\right],
\end{aligned}
$$
and
\begin{align*}
\left|\partial_t \widehat{u}(t, \xi)_{k \ell}\right| \lesssim (1+t) {e}^{-\frac{b}{2} t}\left[\left(1+\lambda_{\xi}^\alpha \right)\left|\widehat{u}_0(\xi)_{k \ell}\right|+\left|\widehat{u}_1(\xi)_{k \ell}\right|\right],
\end{align*}
for any $t \geq0$. Thus using the Plancherel formula along with the above estimates, we get
\begin{align}\label{2L22}\nonumber
\|\partial_t^j(-\mathcal L)^{i\alpha/2}u(t, \cdot)\|_{L^{2}(G)}^{2}&=\sum_{[\xi] \in \widehat{G}} d_{\xi} \sum_{k, \ell=1}^{d_{\xi}}\lambda_{\xi}^{2\alpha i}\left|\partial_t^j \widehat{u}(t, \xi)_{k \ell}\right|^2 \\\nonumber
& \lesssim (1+t)^2 {e}^{-b t} \sum_{[\xi] \in \widehat{ G}} d_{\xi} \sum_{k, \ell=1}^{d_{\xi}}\left(\left(1+\lambda_{\xi}^{2\alpha}\right)^{(i+j)}\left|\widehat{u}_0(\xi)_{k \ell}\right|^2+\left|\widehat{u}_1(\xi)_{k \ell}\right|^2\right) \\
& = (1+t)^2 {e}^{-b t}\left[ \left\|u_{0}\right\|_{H_{\mathcal{L}}^{{\alpha(i+j) }}(G)}^{2}+\left\|u_{1}\right\|_{L^{2}(G)}^{2} \right],
\end{align}
for any $i,j\in\{0,1\}$, such that $0\leq i+j\leq 1$.
\textbf{When $b^2>4 m^2$: } In this case, depending on the range of $\lambda_{\xi}^2$, the characteristic roots may be complex conjugate or real distinct, or they may coincide. But comparing all possible cases in \eqref{2number1} and \eqref{2number3} and keeping in mind that the regularity is provided from the case with complex conjugate characteristic roots, whereas the decay rate is given by the continuous irreducible unitary representations with $\lambda_{\xi}^2=0$, we obtain
$$
\begin{aligned}
\left|\widehat{u}(t, \xi)_{k \ell}\right| & \lesssim {e}^{\left(-\frac{b}{2}+\sqrt{ \frac{b^2}{4}-m^2}\right) t}\left(\left|\widehat{u}_0(\xi)_{k \ell}\right|+\left|\widehat{u}_1(\xi)_{k \ell}\right|\right) ,\\
\lambda_{\xi}^\alpha\left|\widehat{u}(t, \xi)_{k \ell}\right| & \lesssim {e}^{\left(-\frac{b}{2}+\sqrt{ \frac{b^2}{4}-m^2}\right) t} \left[ \left(1+\lambda_{\xi}^\alpha\right)\left|\widehat{u}_0(\xi)_{k \ell}\right|+\left|\widehat{u}_1(\xi)_{k \ell}\right|\right],
\end{aligned}
$$
and
\begin{align*}
\left|\partial_t \widehat{u}(t, \xi)_{k \ell}\right| \lesssim {e}^{\left(-\frac{b}{2}+\sqrt{ \frac{b^2}{4}-m^2}\right) t} \left[\left(1+\lambda_{\xi}^\alpha \right)\left|\widehat{u}_0(\xi)_{k \ell}\right|+\left|\widehat{u}_1(\xi)_{k \ell}\right|\right],
\end{align*}
for any $t \geqslant 0$.
Thus using the Plancherel formula along with the above estimates, we get
\begin{align}\label{2L222}\nonumber
\|\partial_t^j(-\mathcal L)^{i\alpha/2}u(t, \cdot)\|_{L^{2}(G)}^{2}&=\sum_{[\xi] \in \widehat{G}} d_{\xi} \sum_{k, \ell=1}^{d_{\xi}}\lambda_{\xi}^{2\alpha i}\left|\partial_t^j \widehat{u}(t, \xi)_{k \ell}\right|^2 \\\nonumber
& \lesssim {e}^{\left(- {b}+\sqrt{ {b^2}-4m^2}\right) t} \sum_{[\xi] \in \widehat{ G}} d_{\xi} \sum_{k, \ell=1}^{d_{\xi}}\left(\left(1+\lambda_{\xi}^{2\alpha}\right)^{(i+j)}\left|\widehat{u}_0(\xi)_{k \ell}\right|^2+\left|\widehat{u}_1(\xi)_{k \ell}\right|^2\right) \\
& = {e}^{\left(- {b}+\sqrt{ {b^2}-4m^2}\right) t} \left[ \left\|u_{0}\right\|_{H_{\mathcal{L}}^{{\alpha(i+j) }}(G)}^{2}+\left\|u_{1}\right\|_{L^{2}(G)}^{2} \right],
\end{align}
for any $i,j\in\{0,1\},$ such that $0\leq i+j\leq 1$.
Now, we are in a position to prove Proposition \ref{2thm11}.
\begin{proof}[Proof of Proposition \ref{2thm11}]
The proof of Proposition \ref{2thm11} follows from the estimates (\ref{2L2}), (\ref{2L22}), and (\ref{2L222}) for $\|u(t, \cdot )\|_{L^{2}(G)}$, $\left\|(-\mathcal{L})^{\alpha / 2} u(t, \cdot )\right\|_{L^{2}(G)}$, and $\left\|\partial_{t} u(t, \cdot )\right\|_{L^{2}(G)}$, respectively.
\end{proof}
\subsection{Global in time existence}\label{2sec4}
This subsection is devoted to prove Theorem \ref{2thm22}, i.e., the global existence of small data solutions for the fractional Cauchy problem (\ref{2number31}) in the energy evolution space $\mathcal C\left([0,T], H^\alpha_{\mathcal{L}}(G)\right)\cap\mathcal C^1\left([0,T],L^2(G)\right)$.
First, we recall some notations to present the proof of Theorem \ref{2thm22}. Consider the space \[X(T):=\mathcal{C}\left([0,T], H^\alpha_{\mathcal L}(G)\right)\cap\mathcal C^1\left([0,T],L^2(G)\right),\] equipped with the norm
\begin{align}\label{2eq33333}
\|u\|_{X(T)}&:=\sup\limits_{t\in[0,T]} (A_{b, m^2}(t))^{-1} \left ( \|u(t,\cdot)\|_{L^2(G)}+\|(-\mathcal L)^{\alpha/2}u(t,\cdot)\|_{L^2(G)}+\|\partial_tu(t,\cdot)\|_{L^2(G)}\right ),
\end{align}
where $A_{b, m^2}(t)$ is given by
$$
A_{b, m^2}(t) \doteq\left\{\begin{array}{ll}
{e}^{-\frac{b}{2} t} & \text { if } b^2<4 m^2, \\
(t+1) {e}^{-\frac{b}{2} t} & \text { if } b^2=4 m^2, \\
{e}^{\left(-\frac{b}{2}+\sqrt{\frac{b^2}{4}-m^2}\right) t} & \text { if } b^2>4 m^2.
\end{array}\right.
$$
Here we briefly recall the notion of mild solutions in our framework to the Cauchy problem (\ref{2number31}) and will analyze our approach to prove Theorem \ref{2thm22}. Applying Duhamel's principle, the solution to the nonlinear inhomogeneous problem
\begin{align}\label{2eq3111}
\begin{cases}
\partial^2_tu+(-\mathcal{L})^\alpha u+b\partial_{t}u +m^2u=F(t, x), & x\in G,t>0,\\
u(0,x)= u_0(x), & x\in G,\\ \partial_tu(0, x)= u_1(x), & x\in G,
\end{cases}
\end{align}
can be expressed as
$$ u(t, x)= u_{0}(x)*_{(x)}E_{0}(t, x)+u_{1}(x)*_{(x)}E_{1}(t, x) +\int_{0}^{t} F(s, x)*_{(x)} E_{1}(t-s, x) \;d s, $$
where $*_{(x)}$ denotes the group convolution product on $G$ with respect to the $x$ variable. Here, $E_{0}(t, x)$ and $E_{1}(t, x)$ are the fundamental solutions to the homogeneous problem (\ref{2eq3111}), i.e., when $F=0$ with initial data $\left(u_{0}, u_{1}\right)=\left(\delta_{0}, 0\right)$ and $\left(u_{0}, u_{1}\right)=$ $\left(0, \delta_{0}\right)$, respectively.
For a function $u$ on $[0, T]$ to be a mild solution to (\ref{2eq3111}), we refer to subsection \ref{sec4}.
Furthermore, if the estimates \eqref{2number100} and \eqref{2number101} hold uniformly with respect to $T$ then the solution can be prolonged and defined for any $t \in (0,\infty) $ which will be our {\it global solution}. Now we present the proof of Theorem \ref{2thm22}.
\begin{proof}[Proof of Theorem \ref{2thm22}]
The expression (\ref{f2}) can be wriiten as $N u=u^\sharp+I[u]$, where
\begin{align*}
u^\sharp(t,x)=\varepsilon u_{0}(x) *_{(x)} E_{0}(t, x)+\varepsilon u_{1}(x) *_{(x)} E_{1}(t, x)
\end{align*}
and
\begin{align*}
I[u](t,x):=\int\limits_0^t |u(s,x)|^p*_x E_1(t-s, x)ds.
\end{align*}
Now, for the part $u^\sharp$, Theorem \ref{2thm11}, immediately implies that
\begin{align}\label{2f3}
\|u^\sharp\|_{X(T)}\lesssim \|(u_0,u_1)\|_{{H}_{\mathcal L}^\alpha (G)\times L^2(G)}.
\end{align}
On the other hand, for the part $I[u]$, using Minkowski's integral inequality, Young's convolution inequality, Theorem \ref{2thm11}, and by time translation invariance property of the Cauchy problem (\ref{2number31}), we get
\begin{align}\label{2f}\nonumber
\|\partial_t^j(-\mathcal L)^{i\alpha/2}I[u]\|_{L^2(G)}&=\left(\int_{G} \big |\partial_t^j(-\mathcal L)^{i\alpha/2} \int\limits_0^t |u(s,x)|^p*_x E_1(t-s, x)ds\big |^2 dg\right)^{\frac{1}{2}}\\\nonumber
&=\left(\int_{G}\big | \int\limits_0^t |u(s,x)|^p*_x \partial_t^j(-\mathcal L)^{i\alpha/2}E_1(t-s, x)ds\big|^2 dg\right)^{\frac{1}{2}} \\\nonumber
&\lesssim \int\limits_0^t \| |u(s,\cdot )|^p*_x \partial_t^j(-\mathcal L)^{i\alpha/2}E_1(t-s, \cdot)\|_{L^2(G)}ds\\\nonumber
&\lesssim \int\limits_0^t \| u(s,\cdot)^p\|_{L^2(G)} \|\partial_t^j(-\mathcal L)^{i\alpha/2}E_1(t-s, \cdot)\|_{L^2(G)}ds\\\nonumber
&\lesssim \int\limits_0^t A_{b, m^2}(t-s) \|u(s,\cdot)\|^p_{L^{2p}(G)}ds\\\nonumber
&\lesssim\int\limits_0^t A_{b, m^2}(t-s) \|u(s,\cdot)\|^{p\theta(n,2p, \alpha)}_{H^\alpha_\mathcal L(G)}\|u(s,\cdot)\|^{p(1-\theta(n,2p,\alpha ))}_{L^2(G)}ds \\ \nonumber
&\lesssim\int\limits_0^t A_{b, m^2}(t-s) A_{b, m^2}(s)^p \|u\|^p_{X(s)}ds\\
&\lesssim \|u\|^p_{X(t)} \int\limits_0^t A_{b, m^2}(t-s) A_{b, m^2}(s)^p ds \leq \|u\|^p_{X(t)} A_{b, m^2}(t),
\end{align}
for $i,j\in\{0,1\}$ such that $0\leq i+j\leq 1.$ Again for $i,j\in\{0,1\}$ such that $0\leq i+j\leq 1,$ a similar calculations as in (\ref{2f}) together with H\"older's inequality and (\ref{eq34}), we get
\begin{align}\label{2f5}\nonumber
& \|\partial_t^j(-\mathcal L)^{i\alpha/2}\left(I[u]-I[v]\right)\|_{L^2(G)}\\\nonumber&\lesssim \int\limits_0^t A_{b, m^2}(t-s) \||u(s,\cdot)|^p-|v(s,\cdot)|^p\|_{L^{2}(G)}ds\\\nonumber
&\lesssim\int\limits_0^t A_{b, m^2}(t-s) \|u(s,\cdot)-v(s,\cdot)\|_{L^{2p}(G)}\left(\|u(s,\cdot)\|^{p-1}_{L^{2p}(G)}+\|v(s,\cdot)\|^{p-1}_{L^{2p}(G)}\right)ds\\ \nonumber
&\lesssim \|u-v\|_{X(t)}\left(\|u\|^{p-1}_{X(t)}-\|v\|^{p-1}_{X(t)}\right) \int\limits_0^t A_{b, m^2}(t-s) A_{b, m^2}(s)^p ds\\
& \leq \|u\|^p_{X(t)} A_{b, m^2}(t).
\end{align}
Thus combining (\ref{2f3}), (\ref{2f}), and (\ref{2f5}), we have
\begin{align}\label{21}
\|N u\|_{X(t)} \leq D \left\|\left(u_{0}, u_{1}\right)\right\|_{H_{\mathcal{L}}^{\alpha }(G) \times L^{2}(G)}+D\|u\|_{X(t)}^{p}
\end{align}
and
\begin{align}\label{22} \|Nu-Nv\|_{X(T)}\leq D \|u-v\|_{X(t)}\left(\|u\|^{p-1}_{X(T)}-\|v\|^{p-1}_{X(T)}\right).\end{align}
This shows that the map $N$ turns out to be a contraction in some neighborhood of $0$ in the Banach space $X(T).$ Therefore, Banach's fixed point theorem gives us the uniquely determined fixed point $u$ on $ [0, T ]$ for the map $N$, which is our mild solution.
Note that, thanks to the exponential decay rate $A_{b,m^2}(t)$ both in (\ref{2f}) and (\ref{2f5}) we have the uniform boundedness of the integral $$(A_{b, m^2}(t))^{-1}\int_0^t A_{b, m^2}(t-s) A_{b, m^2}(s)^p ds,$$ without any conditions on $p$.
This completes the proof of Theorem \ref{2thm22}.
\end{proof}
We have the following remark regarding Theorem.
\section{Final remarks}\label{sec7}
In \cite{shyamm}, we already seen that for the fractional wave operator $\partial^2_t+(-\mathcal{L})^\alpha $ and for the damped wave operator $ \partial^2_t+(-\mathcal{L})^\alpha + \partial_t$ defined in Section \ref{sec3}, under some suitable assumptions on the initial data, the local in-time solutions to these Cauchy problem blow up in finite time for any $p > 1$. In other words, we do not get any global in-time existence result in this case. However, in Section \ref{sec6} of this paper, we have seen that the presence of a positive damping term and a positive mass term in the Cauchy problem completely reverses the scenario. In a similar manner, the fractional damped wave equation on the Heisenberg group will be considered in a forthcoming paper.
\section{Data availability statement}
The authors confirm that the data supporting the findings of this study are available within the article and its supplementary materials.
\end{document} | arXiv |
determinant of correlation matrix
The determinant of the correlation matrix will equal 1.0 only if all correlations equal 0, otherwise the determinant will be less than 1. Create your own correlation matrix. (This one has 2 Rows and 2 Columns) The determinant of that matrix is (calculations are explained later): 3×6 − 8×4 = 18 − 32 = −14. For det, the determinant of x.For determinant, a list with components > Hi y'all > > Does the determinant of a correlation matrix have physical significance? ( 2009 ), amongst others, have studied extensively the problem of generating random correlation matrices. For instance, any electrical item can yield less power on a mild day on the basis of the correlation between weather and the demand of electricity. In particular, the CLT holds if p/n has a nonzero limit and the smallest eigenvalue of R n is larger than 1/2. A matrix determinant is difficult to define but a very useful number: Unfortunately, not every square matrix has an inverse (although most do). The determinant of a matrix is one of the most basic and important matrix functions, and this makes studying the distribution of the determinant of a random correlation matrix important. / Jiang, Tiefeng. Since each correlation may be based on a different set of rows, practical interpretations could be difficult, if not illogical. "Its determinant is greater than zero" ... -Often in the literature, the array correlation matrix is referred to as the covariance matrix. We derive a central limit theorem (CLT) for the logarithm of the determinant of Ř n for a big class of R n . keywords = "Central limit theorem, Moment generating function, Multivariate normal distribution, Sample correlation matrix, Smallest eigenvalue". Hence, here 4×4 is a square matrix which has four rows and four columns. For example, the highlighted cell below shows that the correlation between "hours spent studying" and "exam score" is 0.82 , which indicates that they're strongly positively correlated. The correlation matrix below shows the correlation coefficients between several variables related to education: Each cell in the table shows the correlation between two specific variables. Biometrika 47.1/2 (1960): 194-196. With the same definitions of all the measures above, we now see that the scale measure has values corresponding to each variable. UR - http://www.scopus.com/inward/record.url?scp=85063327141&partnerID=8YFLogxK, UR - http://www.scopus.com/inward/citedby.url?scp=85063327141&partnerID=8YFLogxK, Powered by Pure, Scopus & Elsevier Fingerprint Engine™ © 2020 Elsevier B.V, "We use cookies to help provide and enhance our service and tailor content. We derive a central limit theorem (CLT) for the logarithm of the determinant of Ř n for a big class of R n . I am running a factor analysis on scaled survey responses. The Numpy provides us the feature to calculate the determinant of a square matrix using numpy.linalg.det() function. In order to define the generalized variance, we first define the determinant of the matrix. This in turn requires division by matrix determinant. Create your own correlation matrix. For a square matrix, i.e., a matrix with the same number of rows and columns, one can capture important information about the matrix in a just single number, called the determinant.The determinant is useful for solving linear equations, capturing how linear transformation change area or volume, and changing variables in integrals. Determinant of a Matrix. The centre of this region is the maximum determinant completion, where x is 0.72 and y is 0.64, to two decimal places. When the correlation r = 0, then we see a shotgun-blast pattern of points, widely dispersed over the entire range of the plot. Correlations have usefulness in terms of recognizing a predictive relationwhich can be extracted in practice. We will start simple with a 2 x 2 matrix and then we will move on … The next step is checking the correlation pattern of the relationship between all of the variables. Can somebody please show with an example how I can implement (determinant of Gamma)^-1/2? The determinant of a matrix is a special number that can be calculated from a square matrix. Σ=(σσ 11,, pp), the distribution of the sample correlation matrix is relatively easy to compute, and its determinant has a … With respect to Correlation Matrix if any pair of variables has a value less than 0.5, consider dropping one of them from the analysis (by repeating the factor analysis test in SPSS by removing variables whose value is … If 1 D = then the columns of X matrix are orthonormal. The determinant of a correlation matrix becomes zero or near zero when some of the variables are perfectly correlated or highly correlated with each other. In particular, the CLT holds if p/n has a nonzero limit and the smallest eigenvalue of R n is larger than 1/2. Determinant of sample correlation matrix with application. > > Merci No, I don't think so. note = "Funding Information: Received October 2016; revised August 2017. Will the presence of linearly dependant variables hinder the reliability of PCA? Now, let us shift our focus to PCA with the correlation matrix. Key words and phrases. A thorough discussion is contained in . The determinant of a matrix is one of the most basic and important matrix functions, and this makes studying the distribution of the determinant of a random correlation matrix important. The matrix Ř n is a popular object in multivariate analysis and it has many connections to other problems. The sample correlation matrix Ř n = (ř ij ) p × p is generated from x 1 ,...,x n such that ř ij is the Pearson correlation coefficient between the ith column and the jth column of the data matrix (x 1 ,...,x n ). Can somebody please show with an example how I can implement (determinant of Gamma)^-1/2? Besides, a formula of the moments of |{\v R} n | and a new method of showing weak convergence are introduced. The monotonic link among variables in terms of ranks is measured by the Spearman correlation coefficient. Key words and phrases. We derive a central limit theorem (CLT) for the logarithm of the determinant of Ř n for a big class of R n . A comparison of methods for estimating the determinant of high-dimensional covariance matrix Zongliang Hu 1, Kai Dong , Wenlin Dai2 and Tiejun Tong; 1Department of Mathematics, Hong Kong Baptist University, Hong Kong 2CEMSE Division, King Abdullah University of Science and Technology, Jeddah, Saudi Arabia Email: [email protected] Abstract The determinant function uses an LU decomposition and the det function is simply a wrapper around a call to determinant.. Often, computing the determinant is not what you should be doing to solve a given problem.. Value. correlation. We derive a central limit theorem (CLT) for the logarithm of the determinant of $\hat{\mathbf {R}}_{n}$ for a big class of $\mathbf{R}_{n}$. AB - Let x 1 ,...,x n be independent random vectors of a common p-dimensional normal distribution with population correlation matrix R n . The sample correlation matrix Ř n = (ř ij ) p × p is generated from x 1 ,...,x n such that ř ij is the Pearson correlation coefficient between the ith column and the jth column of the data matrix (x 1 ,...,x n ). For the partially specified matrix given in Figure 1, a valid correlation matrix completion must lie in the dark yellow region in Figure 2. Thanks --- Il messaggio che segue e' inserito automaticamente dal server di posta dell'Universita' Bocconi. Determinant of a 4×4 matrix is a unique number which is calculated using a particular formula. As every correlation might have its basis on various row sets, practical analysis can be a problem if it is not logical. matlab correlation matrix-inverse determinants fminsearch. 280 Generation of Correlation Matrices correlation matrix. It is well-known that a necessary and sufficient condition for such a matrix to be a correlation matrix is the positive semidefiniteness of the matrix. 60B20, 60F05. Properties of Correlation Matrices. HOLMES ( 1991 ), JOE ( 2006 ) and LEWANDOWSKI et al. The expressions of mean and the variance in the CLT are not obvious, and they are not known before. T1 - Determinant of sample correlation matrix with application. We apply the CLT to a high-dimensional statistical test. By using this website, you agree to our Cookie Policy. Efron ... only depends on the mean-squared row correlation. This is only true if the mean values of the signals and noise are zero. There is a causal relation in this example as the extreme weather results in more usage of electric power by the people for cooling and heating purposes, but statistical dependence is not … Since the square of the determinant of a matrix can be found with the above formula, and because this multiplication is defined for nonsquare matrices, we can extend determinants to nonsquare matrices. abstract = " Let x 1 ,...,x n be independent random vectors of a common p-dimensional normal distribution with population correlation matrix R n . The matrix Ř n is a popular object in multivariate analysis and it has many connections to other problems. The matrix Ř n is a popular object in multivariate analysis and it has many connections to other problems. Covariance considers normalized variables while the correlation matrix does not. ... Browse other questions tagged matlab correlation matrix-inverse determinants fminsearch or ask your own question. Research output: Contribution to journal › Article › peer-review. Remember that the determinant is related to the volume of the space occupied by the swarm of data points represen ted by … Range B6:J14 is a copy of the correlation matrix from Figure 1 of Factor Extraction (onto a different worksheet). Free matrix determinant calculator - calculate matrix determinant step-by-step This website uses cookies to ensure you get the best experience. 1Supported in part by NSF Grants DMS-12-09166 and DMS-14-06279. MSC2010 subject classifications. Central limit theorem, sample correlation matrix, smallest eigenvalue, multivariate normal distribution, moment generating function. A Matrix is an array of numbers: A Matrix. The expressions of mean and the variance in the CLT are not obvious, and they are not known before. The sample correlation matrix {\v R} n = ({\v r} ij ) p × p is generated from x 1 ,...,x n such that {\v r} ij is the Pearson correlation coefficient between the ith column and the jth column of the data matrix (x 1 ,...,x n ). If A is square matrix then the determinant of matrix A is represented as |A|. This test has to be significant: when the correlation matrix is an identity matrix, there would be no correlations between the variables. The sample correlation matrix Ř n = (ř ... We derive a central limit theorem (CLT) for the logarithm of the determinant of Ř n for a big class of R n. The expressions of mean and the variance in the CLT are not obvious, and they are not known before. … The determinant of the correlation matrix will equal 1.0 only if all correlations equal 0, otherwise the determinant will be less than 1. Biometrika 47.1/2 (1960): 194-196. There are 3 course sections that I am combining for the analysis; the determinant of the correlation matrix is 0. Correlation matrix with significance levels (p-value) The function rcorr() [in Hmisc package] can be used to compute the significance levels for pearson and spearman correlations.It returns both the correlation coefficients and the p-value of the correlation for all possible pairs of columns in the data table. The determinant of the correlation matrix will equal 1.0 only if all correlations equal 0. In this analysis the value of R-matrix determinant is 0.026 (see Appendix), therefore it is proven that multicollinearity is not a problem for these data. (2009), amongst others, have studied extensively the problem of generating random correlation matrices. Besides, a formula of the moments of |Ř n | and a new method of showing weak convergence are introduced. For example, take the 3 wide matrix A defined with column vectors, x y and z, … In simple words, both the terms measure the relationship and the dependency between two variables. In a normal distribution context, when the population correlation matrix, the identity matrix, or equivalently, the population covariance matrix is diagonal, i.e., the distribution of the sample correlation matrix R is relatively easy to compute, and its determinant has a distribution that can be expressed as a Meijer G-function distribution. 3. title = "Determinant of sample correlation matrix with application". Data is highly significant, satisfies KMO conditions, Bartlett's test and is superb for factor analysis. Causes of non-positivity of correlation matrices. Figure 4 – Inverse of the correlation matrix. @article{29a8a2cb4f6f461c92d73996bcfc8e09. The matrix $\hat{\mathbf {R}}_{n}$ is a popular object in multivariate analysis and it has many connections to other problems. Also, the distribution of its determinant is established in terms of Meijer G-functions in the null-correlation case. You can obtain the correlation coefficient of two varia… "The determinant of the correlation matrix will equal 1.0 only if all correlations equal 0, otherwise the determinant will be less than 1. Thanks --- Il messaggio che segue e' inserito automaticamente dal server di posta dell'Universita' Bocconi. $\begingroup$ A covariance matrix is NOT always equal to the correlation matrix! The Spearman correlation coefficient measures the monotonic association between … Any other example apart from autoregressive model will also do. Determinant of correlation matrix Let D be the determinant of correlation matrix then 0 1. New understanding of sample correlation matrix jR^ nj Recently, Tao and Vu (2012); Nguyen & Vu (2014): CLT for determinant of Wigner matrix Cai, Liang, Zhou (2015) study CLT for determinant of Wishart matrix We have a problem from high-dimensional statistics on jR^ nj High-dimensional statistics + Machine Learning = Big Data pca. The expressions of mean and the variance in the CLT are not obvious, and they are not known before. The expressions of mean and the variance in the CLT are not obvious, and they are not known before. The geometrical interpretation of determinant is that - in a 2 x 2 framework (2 x 2 matrix) - it measures the area that is spanned by the two column vectors of the 2 x 2 correlation matrix. Hi is there an accepted threshold for the value of the determinant of correlation matrix, to say that a collinarity problem exists (or not exists) ? The determinant of a matrix is a special number that can be calculated from a square matrix. Besides, a formula of the moments of |Ř n | and a new method of showing weak convergence are introduced. And for the Eigenvalues? Key decisions to be made when creating a correlation matrix include: choice of correlation statistic, coding of the variables, treatment of missing data, and presentation.. An example of a correlation matrix. The values of the coefficients can range from -1 to 1, with -1 representing a direct, negative correlation, 0 representing no correlation, and 1 representing a direct, positive correlation. … Mathematically, this correlation matrix may not have a positive determinant. If a matrix order is n x n, then it is a square matrix. "Correlation" on the other hand measures both the strength and direction of the linear relationship between two variables. Finch, P. D. "On the covariance determinants of moving-average and autoregressive models." The Leibniz formula for the determinant of a 2 × 2 matrix is | | = −. Correlation is a function of the covariance. Assume that you do an eigen decomposition of the correlation matrix C, which is of order P, as C= MLM^H where M is the matrix of eigen vectors and L is the diagonal matrix … If the value is greater than 0.00001, thus, multicollinearity is not a … This scalar function of a square matrix is called the determinant. To calculate the partial correlation matrix for Example 1 of Factor Extraction, first we find the inverse of the correlation matrix, as shown in Figure 4. By continuing you agree to the use of cookies. Ask Question ... and the determinant of a diagonal matrix is just the product of its diagonal entries. In particular, the CLT holds if p/n has a nonzero limit and the smallest eigenvalue of R n is larger than 1/2. �=1Í!pÎ8"ÎjxòOâ‡{oßÍg�bœş44È8l¡�˜Ámd J'âù¹ƒ™ä†¡ÒÍÄ�'zhºĞ$rÏeeĞYvÁ¤×R(')k\ëÕd#âr v0»…�Bfœú@$+ö3�€ÿãçnØG�8ûö§"7{:÷ÊvĞuD$×CHš68`Á…Õå•2göxFˆ"³$'^�K]àj'Î/"ÌqMø$² =¼šD¨^�M\bSg¨ÆIÂ!aT"¦(¥�FÓ�RJx®ÿìÚ¿]R*K 몾(úqU×UŞTeތ뼙,ôfªÒ›ËRoêZ¿(uªVt†dÇ¥>&úRgÚ;ıÎCì�ª^Á™rÎYïN¡Êv8Ò°ğŒ*h™O±ÕÂöLØb¶XÛIaw&L½ÈÓ2Ÿb«"‡. Received October 2016; revised August 2017. A Matrix is an array of numbers: A Matrix (This one has 2 Rows and 2 Columns) The determinant of that matrix is (calculations are explained later): 3×6 − 8×4 = 18 − 32 = −14. The matrix Ř n is a popular object in multivariate analysis and it has many connections to other problems. And for the Eigenvalues? ". Typically, a correlation matrix is "square", with the same variables shown in the rows and columns. Typically, a correlation matrix is "square", with the same variables shown in the rows and columns. MSC2010 subject classifications. We apply the CLT to a high-dimensional statistical test. Thus a value close to … Correlation Matrix in R (3 Examples) In this tutorial you'll learn how to compute and plot a correlation matrix in the R programming language. Correlation Matrices compute the linear relationship degree between a set of random variables, taking one pair at a time and performing for each set of pairs within the data. Determinant of a Matrix. Key decisions to be made when creating a correlation matrix include: choice of correlation statistic, coding of the variables, treatment of missing data, and presentation.. An example of a correlation matrix. We apply the CLT to a high-dimensional statistical test. N2 - Let x 1 ,...,x n be independent random vectors of a common p-dimensional normal distribution with population correlation matrix R n . I'm working on a series of optimization problems wherein the objective function to be minimized is the determinant of the variance-covariance matrix. For two input arguments, R is a 2-by-2 matrix with ones along the diagonal and the correlation … R is symmetric. The determinant of a $1 \times 1$ matrix is that number itself. If both the vectors are aligned, which means one of the vectors is linearly dependent on the other, then the determinant is zero. All the diagonal elements of the correlation matrix must be 1 because the correlation of a variable with itself is always perfect, c ii =1. Microarray experiments often yield a normal data matrix X whose rows correspond to genes ... methods, since its mean and variance determine the bias and variance of FDR estimates. The Asymptotic Distribution of the Determinant of a Random Correlation Matrix A.M. Haneaa; & G.F. Nane b a Centre of Excellence for Biosecurity Risk Analysis, University of Melbourne, Australia b Delft Institute of Applied Mathematics, Technical University of Delft Abstract Random correlation matrices are studied for both theoretical interestingness and importance for The sample correlation matrix Ř n = (ř ij ) p × p is generated from x 1 ,...,x n such that ř ij is the Pearson correlation coefficient between the ith column and the jth column of the data matrix (x 1 ,...,x n ). Together they form a unique fingerprint. I tried to remove some columns with high correlation coefficients, but the determinant still remained relatively very close to zero. to maximize the likelihood function expression, I need to express the likelihood function where the variance covariance matrix arises. This is a property that is relatively simple to verify, but not easily constructed. The determinant of a matrix is frequently used in calculus, linear algebra, and advanced geometry. The matrix {\v R} n is a popular object in multivariate analysis and it has many connections to other problems. The determinant of the correlation matrix (R-matrix) should be greater than 0.00001 (Field, 2000). For example, the highlighted cell below shows that the correlation between "hours spent studying" and "exam score" is 0.82 , which indicates that they're strongly positively correlated. For this, all we need to do is, set the 'scale' argument as TRUE. Since each correlation may be based on a different set of rows, practical interpretations could be difficult, if not illogical. However, the determinant of the correlation matrix ( around 10^-30) is very close to zero. More precisely, the article looks as follows: If we consider the expression for determinant as a function f(q; x) then x is the vector of decision variable and q is a vector of parameters based on a user supplied probability distribution. Σ=(σσ 11,, pp), the distribution of the sample correlation matrix is relatively easy to compute, and its determinant has a … Syntax: numpy.linalg.det(array) Example 1: Calculating Determinant of a 2X2 Numpy matrix using numpy.linalg.det() function When matrix is singular, then invention involves division by zero, which is undefined. Central limit theorem, sample correlation matrix, smallest eigenvalue, multivariate normal distribution, moment generating function.". A matrix is an array of many numbers. In particular, the CLT holds if p/n has a nonzero limit and the smallest eigenvalue of R n is larger than 1/2.
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\begin{document}
\title{Supplementary Information}
\maketitle
\section{The radius of a set of Hermitian operators}
\begin{lemma} For any finite set $\{ H_i \}_{i \in \mathcal{I}}$ of Hermitian operators on a finite-dimensional Hilbert space $V$, the radius of $\{ H_i \}$ is equal to \begin{eqnarray*}
\max_{\substack{\lambda_i \geq 0, \lambda'_i \geq 0 \\ \sum \lambda_i = \sum \lambda'_i \\ \textnormal{Tr} ( \sum \lambda_i ) = 1/2}} \left[ \sum_{i \in \mathcal{I}} \textnormal{Tr} \left( ( \lambda_i - \lambda'_i ) H_i \right) \right]. \end{eqnarray*} \end{lemma}
\begin{proof} Any family of Hermitian operators $\{H_i\}$ may be translated to a family $\{ H_i + W \}$ which contains the operator $0$. This translation does not affect the radius nor the expression from the statement of the lemma. Therefore, we may assume that $\{ H_i \}$ contains $0$. By definition, \begin{eqnarray} \textnormal{Rad} \{ H_i \}_i & = & \min_{ \substack{C, r \\ C - H_i \geq - r \mathbb{I} \\ H_i - C \geq - r \mathbb{I}}} ( r ), \end{eqnarray} where the maximization is over Hermitian operators $C$ and real numbers $r$. Since $0 \in \{ H_i \}$, whenever the constraints in this maximization are satisfied we have in particular that $C \geq - r \mathbb{I}$. Letting $Z = C + r \mathbb{I}$, we obtain the following alternate expression: \begin{eqnarray} \textnormal{Rad} \{ H_i \}_i & = & \min_{ \substack{Z, r \\ Z \geq H_i \\ -Z + 2 r \mathbb{I} \geq - H_i \\ }} ( r ). \end{eqnarray} By semidefinite programming duality, this is equivalent to \begin{eqnarray*} \textnormal{Rad} \{ H_i \}_i & = & \max_{ \substack{\lambda_i \geq 0, \lambda'_i \geq 0, \\ \sum_i \lambda_i - \sum_i \lambda'_i \leq 0 \\ 2 \textnormal{Tr} ( \sum \lambda'_i ) \leq 1}} \left[ \left( \sum \textnormal{Tr} ( \lambda_i H_i ) - \sum \textnormal{Tr} ( \lambda'_i H_i ) \right) \right]. \end{eqnarray*} It is easy to see that this maximum is achieved by a pair of families $\{ \lambda_i \}, \{ \lambda'_i \}$ satisfying $\sum \lambda_i = \sum \lambda'_i$ and $2 \textnormal{Tr} ( \sum \lambda'_i ) = 1$. \end{proof}
\section{The proof of Corollary 3 in the main text}
\label{cor3thm2sec}
The radius function is convex in the following sense: for any familes of operators $\{ J_y \}_{y \in \ensuremath{\mathcal{Y}}}$ and $\{ K_y \}_{y \in \ensuremath{\mathcal{Y}}}$, and real number $\alpha \in [ 0, 1 ]$, \begin{align*} &\textnormal{Rad} \{ \alpha J_y + (1-\alpha ) K_y \}_y \\ &\qquad \leq \alpha \textnormal{Rad} \{ J_y \}_y + (1 - \alpha ) \textnormal{Rad} \{ K_y \}_y. \end{align*} (For, if we let $J'$ be such that the distance from $J'$ to $\{ J_y \}_y$ is equal to $r := \textnormal{Rad} \{ J_y \}_y$, and we let $K'$ be such that the distance from $K'$ to $\{ K_y \}_y$ is equal to $r' := \textnormal{Rad} \{ K_y \}_y$, then the distance from $\alpha J' + (1 - \alpha ) K'$ to $\{ \alpha J_y + ( 1 - \alpha ) K_y \}_y$ is no more than $\alpha r + (1 - \alpha ) r'$ by the triangle inequality.) In particular, this convexity property implies that the radius of $\{ \alpha J_y + ( 1 - \alpha ) K_y \}_y$ is no more than the maximum of $\textnormal{Rad} \{ J_y \}_y$ and $\textnormal{Rad} \{ K_y \}_y$.
Since any Hermitian operator $B$ satisfying $0 \leq B \leq \mathbb{I}$ is a convex combination of projection operators, Corollary 3 follows from Theorem 2.
\section{An example calculation}
Let $\ensuremath{M}$ be the channel defined in figure 2 in the main text. In this section we will use Theorem 2 from the main text to calculate the quantity $\ensuremath{\mathrm{Succ}}_{\quantclassn{2}} ( \ensuremath{M} )$. First, we will prove the following lemma which provides a simplified formula for $\ensuremath{\mathrm{Succ}}_{\quantclassn{n}} ( \ensuremath{M} )$. For any projection operator $P$, let $P^\perp$ denote projection onto the orthogonal complement of $P$. \begin{lemma} \label{simplifiedradiuslemma} For any $n \geq 1$, the quantity $\ensuremath{\mathrm{Succ}}_{\quantclassn{n}} ( \ensuremath{M} )$ is equal to \begin{eqnarray*} & & \frac{1}{2} + \left( \frac{1}{3} \right) \max_{X, Y, Z } \left( \textnormal{Rad} \left\{ X + Y +Z , X + Y^\perp + Z^\perp , X^\perp + Y + Z^\perp , X^\perp + Y^\perp + Z \right\} \right), \end{eqnarray*} where the maximum is taken over all projection operators $X, Y, Z$ on $\mathbb{C}^n$. \end{lemma}
\begin{proof} For any Hermitian operators $B_1, B_2, B_3, B_4, B_5, B_6$ on $\mathbb{C}^n$, let \begin{eqnarray*} F ( B_1, B_2, B_3, B_4, B_5, B_6 ) \end{eqnarray*} be equal to the quantity \begin{eqnarray*} & & \textnormal{Rad} \left\{ B_1 + B_3 + B_5, B_1 + B_4 + B_6, B_2 + B_3 + B_6 , B_2 + B_4 + B_5 \right\}. \end{eqnarray*} By the formula from Theorem 2 in the main text, \begin{eqnarray} \ensuremath{\mathrm{Succ}}_{\quantclassn{n}} ( \ensuremath{M} ) & = & \frac{1}{2} + \left( \frac{1}{3} \right) \max_{0 \leq B_i \leq \mathbb{I} } F ( B_1, B_2, B_3, B_4, B_5, B_6 ). \end{eqnarray} Let \begin{eqnarray} m & = & \max_{0 \leq B_i \leq \mathbb{I} } F ( B_1, B_2, B_3, B_4, B_5, B_6 ). \end{eqnarray} It suffices to prove that this maximum is achieved by some $6$-tuple of the form $(X, X^\perp , Y , Y^\perp , Z, Z^\perp )$, where $X$, $Y$, and $Z$ are projections.
As noted in section~\ref{cor3thm2sec} of the supplementary information, the radius function is convex in the sense that if $(H_1, H_2, H_3, H_4 )$ and $(H'_1, H'_2, H'_3 , H'_4 )$ are Hermitian operators and $\alpha \in [ 0, 1 ]$ is a real number, \begin{eqnarray} \textnormal{Rad} \{ \alpha H_i + (1 - \alpha ) H'_i \}_i & \leq & \alpha \textnormal{Rad} \{ H_i \}_i + (1 - \alpha ) \textnormal{Rad} \{ H'_i \}_i. \end{eqnarray} It follows easily by linearity that a similar convexity property holds for $F$: for any Hermitian operators $B_1, \ldots , B_6 $ and $B'_1 , \ldots, B'_6 $, and any $\alpha \in [ 0, 1 ]$, \begin{eqnarray*} & &F ( \alpha B_1 + (1 - \alpha ) B'_1 , \ldots, \alpha B_6 + (1 - \alpha ) B'_6 ) \\ \nonumber & \leq & \alpha F ( B_1, \ldots, B_6 ) + (1 - \alpha ) F ( B'_1, \ldots, B'_6 ). \end{eqnarray*} In particular, \begin{eqnarray} \label{convexityimp} & &F ( \alpha B_1 + (1 - \alpha ) B'_1 , \ldots, \alpha B_6 + (1 - \alpha ) B'_6 ) \\ \nonumber & \leq & \max \left\{ F ( B_1, \ldots, B_6 ), F ( B'_1, \ldots, B'_6 ) \right\}. \end{eqnarray} Additionally, $F$ is translation-invariant in the following sense: for any Hermitian operators $B_1, \ldots, B_6$, and any Hermitian operators $K$, $L$, and $M$, \begin{eqnarray} \label{translationinvariance} F ( B_1 + K , B_2 + K , B_3 + L , B_4 + L, B_5 + M, B_6 + M) & = & F ( B_1, \ldots, B_6 ). \end{eqnarray}
Let $X_1, X_2, Y_1, Y_2, Z_1, Z_2$ be Hermitian operators satisfying $0 \leq X_i, Y_i, Z_i \leq \mathbb{I}$ such that $F ( X_1, X_2, Y_1, Y_2, Z_1, Z_2 ) = m$. Let $X_+$ and $X_-$ be a pair of positive semidefinite operators having mutual orthogonal supports which are such that \begin{eqnarray} X_1 - X_2 = X_+ - X_-. \end{eqnarray} Define $Y_+, Y_-, Z_+, Z_-$ similarly. By property (\ref{translationinvariance}) above, \begin{eqnarray} F ( X_+ , X_-, Y_+ , Y_-, Z_+, Z_- ) & = & F ( X_1, X_2, Y_1, Y_2, Z_1 , Z_2 ) \\ \nonumber & = & m. \end{eqnarray}
The pair $(X_+, X_-)$ can be expressed as a convex combination of pairs of projections $(P_1^{(i)} , P_2^{(i)} )$ where for each $i$, the support of $P_1^{(i)}$ is orthogonal to $P_2^{(i)}$. A similar decomposition exists for $(Y_+, Y_-)$ and $(Z_+, Z_- )$. Therefore by property (\ref{convexityimp}) above, there exist pairs of projections $(P_1, P_2)$, $(Q_1, Q_2 )$, $(R_1, R_2 )$, with each pair having mutually orthogonal supports, such that \begin{eqnarray} F ( P_1, P_2, Q_1, Q_2, R_1, R_2 ) = m. \end{eqnarray}
Let $P_3 = \mathbb{I} - P_1 - P_2$, and define $Q_3$ and $R_3$ similarly. By (\ref{translationinvariance}), \begin{eqnarray} F \left( P_1 + \frac{P_3}{2} , P_2 + \frac{P_3}{2}, Q_1 + \frac{Q_3}{2}
, Q_2 + \frac{Q_3}{2} , R_1 + \frac{R_3}{2} , R_2 + \frac{R_3 }{2} \right) & = & m. \end{eqnarray} The $6$-tuple on the left hand side of the equation above is a convex combination of the $6$-tuples \begin{eqnarray*} & & \left( P_1 + P_3 , P_2, Q_1 + Q_3 , Q_2 , R_1 + R_3, R_2 \right) \\ & & \textnormal{ and } \left( P_1, P_2 + P_3, Q_1 , Q_2 + Q_3 , R_1, R_2 + R_3 \right). \end{eqnarray*} By (\ref{convexityimp}), at least one of these $6$-tuples must achieve the maximum $m$. This completes the proof. \end{proof}
\begin{lemma} For any projection operators $X$, $Y$, $Z$ on the two-dimensional vector space $\mathbb{C}^2$, the radius of the set \begin{eqnarray} \label{radset} \left\{ X + Y +Z , X + Y^\perp + Z^\perp , X^\perp + Y + Z^\perp , X^\perp + Y^\perp + Z \right\}. \end{eqnarray} is less than or equal to $\frac{1}{2} + \frac{1}{\sqrt{2}}$. \end{lemma}
\begin{proof} \textbf{Case 1: The matrices $X$, $Y$, and $Z$ are all scalar matrices.} In this case, each of $X$, $Y$, and $Z$ is equal to either $0$ or $\mathbb{I}$. This case is trivial, since the radius of the set $\{ 3 \mathbb{I} , \mathbb{I} \}$ is $1$, and the radius of the set $\{ 2 \mathbb{I}, 0 \}$ is $1$.
\vskip0.1in
\textbf{Case 2: Two of the matrices $X, Y, Z$ are scalar matrices and one is a nonscalar.} We may assume without loss of generality that $X$ is the nonscalar matrix. Then the set (\ref{radset}) is equal to either \begin{eqnarray} \left\{ 0 , X + \mathbb{I} , 2 \mathbb{I} \right\} \end{eqnarray} or \begin{eqnarray} \left\{ X , X+ 2 \mathbb{I} , \mathbb{I} \right\}. \end{eqnarray} In the former case, the operator-norm distance from the operator $\mathbb{I}$ to the set $\{ 0, X + \mathbb{I} , 2 \mathbb{I} \}$ is $1$. In the latter case, the operator-norm distance from the operator $X + \mathbb{I}$ to the set $\{ X, X + 2 \mathbb{I} , \mathbb{I} \}$ is $1$. The desired result follows.
\vskip0.1in
\textbf{Case 3: Exactly one of the matrices $X, Y, Z$ is a scalar matrix.} We may assume that $X$ and $Y$ are nonscalar matrices and $Z$ is scalar. Also, by replacing $(X, Y, Z )$ with $(X^\perp, Y , Z^\perp)$ if necessary, we may assume that $Z = \mathbb{I}$.
Let $X = \left| x \right> \left< x \right|$ and $Y = \left| y \right> \left< y \right|$
where $x, y \in \mathbb{C}^2$ are unit vectors, and let $\theta = \arccos \left( \left| x \cdot y
\right| \right)$. Both of the operators \begin{eqnarray} X + Y + \mathbb{I} , X^\perp + Y^\perp + \mathbb{I} \end{eqnarray} have eigenvalues $\left\{ 2 + \cos \theta, 2 - \cos \theta \right\}$, and both of the operators \begin{eqnarray} X + Y^\perp, X^\perp + Y \end{eqnarray} have eigenvalues $\left\{ 1 + \sin \theta , 1 - \sin \theta \right\}$. If we let \begin{eqnarray} C = \left( \frac{3}{2} + \frac{\cos \theta - \sin \theta }{2} \right) \mathbb{I}, \end{eqnarray} then the operator norm distance from $C$ to each of the elements
of (\ref{radset}) is $\frac{1}{2} + \frac{\cos \theta + \sin \theta}{2} \leq \frac{1}{2} + \frac{1}{\sqrt{2}}$.
\vskip0.1in
\textbf{Case 4: Each of $X, Y, Z$ is a nonscalar matrix.} As in case 3, let $X = \left| x \right> \left< x \right|$ and $Y = \left| y \right> \left< y \right|$
and let $\theta = \arccos \left( \left| x \cdot y
\right| \right)$.
Let \begin{eqnarray} C = \mathbb{I} + \left( \frac{1}{2} + \frac{\cos \theta - \sin \theta}{2} \right) Z + \left( \frac{1}{2} - \frac{\cos \theta - \sin \theta}{2} \right) Z^\perp. \end{eqnarray} Then, the operator norm of the difference \begin{eqnarray} ( X + Y + Z ) - C = (X + Y) - \left( \frac{3}{2} + \frac{ \cos \theta - \sin \theta }{2} \right) \mathbb{I} \end{eqnarray} is $\frac{1}{2} + \frac{\cos \theta + \sin \theta}{2}$, which is less than or equal to $\frac{1}{2} + \frac{1}{\sqrt{2}}$. A similar calculation shows that the distance from $C$ to each of the other three elements of set $(\ref{radset})$ is equal to $\frac{1}{2} + \frac{\cos \theta + \sin \theta}{2}$. This completes the proof. \end{proof}
For any angle $\theta \in \mathbb{R}$, let $P_\theta \colon \mathbb{C}^2 \to \mathbb{C}^2$
denote projection onto the unit vector $\cos (\theta) \left| 0 \right>
+ \sin ( \theta ) \left| 1 \right>$. Consider the set \begin{eqnarray} \label{maxradiusset1} \left\{ P_0 + P_{\pi / 4 } + \mathbb{I} , P_0 + P_{3 \pi / 4 } , P_{\pi / 2} + P_{\pi / 4} , P_{\pi / 2 } + P_{3 \pi / 4 } + \mathbb{I} \right\} \end{eqnarray} A direct calculation shows that the distance from the operator $\left( \frac{3}{2} \right) \mathbb{I}$ to set (\ref{maxradiusset1}) is $\frac{1}{2} + \frac{1}{\sqrt{2}}$. The next lemma asserts that this quantity is in fact the radius of (\ref{maxradiusset1}).
\begin{lemma} \label{directcalclemma} The radius of the set \begin{eqnarray} \left\{ P_0 + P_{\pi / 4 } + \mathbb{I} , P_0 + P_{3 \pi / 4 } , P_{\pi / 2} + P_{\pi / 4} , P_{\pi / 2 } + P_{3 \pi / 4 } + \mathbb{I} \right\} \end{eqnarray} is is equal to $\frac{1}{2} + \frac{1}{\sqrt{2}}$. \end{lemma}
\begin{proof} For any Hermitian operator $H \colon \mathbb{C}^2 \to \mathbb{C}^2$, let us write $\overline{H}$ to denote the trace-zero operator $H - (\textnormal{Tr} (H ) ) \mathbb{I} / 2$. In the proof that follows, we will make use of the following fact: for any two Hermitian operators $Q, R \colon \mathbb{C}^2 \to \mathbb{C}^2$, \begin{eqnarray}
\left| \left| Q - R \right| \right| =
\left| \textnormal{Tr} ( Q ) - \textnormal{Tr} ( R ) \right| + \left| \left| \overline{Q} - \overline{R} \right| \right| \end{eqnarray}
Suppose, for the sake of contradiction, that there exists a Hermitian operator $Z$ whose distance from each of the elements of set (\ref{maxradiusset1}) is strictly less than $\frac{1}{2} + \frac{1}{\sqrt{2}}$. Then, \begin{eqnarray*} 2 \left( \frac{1}{2} + \frac{1}{\sqrt{2}} \right) & > &
\left\| ( P_0 + P_{3 \pi / 4 } ) - Z \right\| + \left\| ( P_{\pi / 2 }
+ P_{\pi / 4 } ) - Z \right\| \\
& = & \left\| \left( P_0 + P_{3 \pi / 4 } - \mathbb{I} \right) - \overline{Z} \right\|
+ \left\| \left( P_{\pi /2 } + P_{\pi / 4 } - \mathbb{I} \right) - \overline{Z} \right\|
+ 2 \cdot \left| 2 - \textnormal{Tr} ( Z ) \right|\\
& \geq &
\left\| \left( P_0 + P_{3 \pi / 4 } \right)
- \left( P_{\pi / 2 } + P_{\pi / 4 } \right) \right\| +
2 \cdot \left| 2 - \textnormal{Tr} ( Z ) \right| \\
& = & \sqrt{2} +
2 \cdot \left| 2 - \textnormal{Tr} ( Z ) \right| \end{eqnarray*} Therefore, $\textnormal{Tr} (Z ) < \frac{5}{2}$. Similarly, \begin{eqnarray*} 2 \left( \frac{1}{2} + \frac{1}{\sqrt{2}} \right) & > &
\left\| ( P_0 + P_{ \pi / 4 } + \mathbb{I} ) - Z \right\| + \left\| ( P_{\pi / 2 }
+ P_{3 \pi / 4 } ) - \mathbb{I} \right\| \\
& = & \left\| \left( P_0 + P_{ \pi / 4 } - \mathbb{I} \right) - \overline{Z} \right\|
+ \left\| \left( P_{\pi /2 } + P_{3 \pi / 4 } - \mathbb{I} \right) - \overline{Z} \right\|
+ 2 \cdot \left| 3 - \textnormal{Tr} ( Z ) \right|\\
& \geq &
\left\| \left( P_0 + P_{\pi / 4 } \right)
- \left( P_{\pi / 2 } + P_{3 \pi / 4 } \right) \right\| +
2 \cdot \left| 3 - \textnormal{Tr} ( Z ) \right| \\
& = & \sqrt{2} +
2 \cdot \left| 3 - \textnormal{Tr} ( Z ) \right|, \end{eqnarray*} which implies $\textnormal{Tr} ( Z ) > \frac{5}{2}$. This is a contradiction. \end{proof}
Combining Lemmas \ref{simplifiedradiuslemma}--\ref{directcalclemma}, we have the following proposition. \begin{proposition} The quantity $\ensuremath{\mathrm{Succ}}_{\quantclassn{2}} ( \ensuremath{M} )$ is equal to $\frac{2}{3} + \frac{1}{3 \sqrt{2}}$. $\qed$ \end{proposition}
\section{An example of optimal non-signaling assistance}
In this section we discuss an example in which equality occurs in Theorem 5 from the main text. This example is a generalization of the protocol from \cite{prevedeletal}.
Let $m$ be a positive integer. Let \begin{eqnarray} \ensuremath{\mathcal{Z}} & = & \mathbb{F}_2^m, \\ \ensuremath{\mathcal{W}} & = & \left( \mathbb{F}_2^m \smallsetminus \{ 0 \} \right) \times \mathbb{F}_2. \end{eqnarray} Let $\ensuremath{K}$ be a channel defined as follows: \begin{enumerate} \setlength{\parskip}{0pt} \setlength{\itemsep}{0pt}
\item The input alphabet of $\ensuremath{K}$ is $\ensuremath{\mathcal{Z}}$, and the output alphabet of $\ensuremath{K}$ is $\ensuremath{\mathcal{W}}$.
\item For any given input $\mathbf{v} \in \mathbb{F}_2^m$, the output of $\ensuremath{K}$ is unformly distributed over the set \begin{eqnarray} \left\{ \left( \mathbf{w}, \mathbf{w} \cdot \mathbf{v} \right) \mid \mathbf{w} \in \mathbb{F}_2^m \smallsetminus \{ 0 \} \right\}. \end{eqnarray} \end{enumerate} (Here, $\mathbf{w} \cdot \mathbf{v} \in \mathbb{F}_2$ denotes the inner product of $\mathbf{w}$ and $\mathbf{v}$.)
Let $(\ensuremath{E}_1 , \ensuremath{E}_2 )$ be a two part input-output device defined as follows. (See Figure~\ref{fig:exampledevice}.) \begin{enumerate} \setlength{\parskip}{0pt} \setlength{\itemsep}{0pt} \item The input alphabet for $\ensuremath{E}_1$ is $\mathbb{F}_2$, and the output alphabet for $\ensuremath{E}_1$ is $\ensuremath{\mathcal{Z}}$.
\item The input alphabet for $\ensuremath{E}_2$ is $\ensuremath{\mathcal{W}}$, and the output alphabet for $\ensuremath{E}_2$ is $\mathbb{F}_2$.
\item If the inputs to $\ensuremath{E}_1$ and $\ensuremath{E}_2$ are $a \in \{ 0, 1 \}$ and $ (\mathbf{w} , r ) \in \left( \mathbb{F}_2^m \smallsetminus \{ \mathbf{0} \} \right) \times \mathbb{F}_2, $ then the output of $\ensuremath{E}_1$ is unformly distributed over all vectors $\mathbf{a} = \left( a_1, a_2, \ldots , a_m \right)$ that satisfy $a_1 = a$, and the output of $\ensuremath{E}_2$ is $a \oplus r \oplus \left( \mathbf{w} \cdot \mathbf{a} \right)$. \end{enumerate}
\begin{figure}
\caption{The device $(\ensuremath{E}_1, \ensuremath{E}_2 )$.}
\label{fig:exampledevice}
\end{figure}
It can be checked that the correlation $\ensuremath{E}$ arising from $(\ensuremath{E}_1 , \ensuremath{E}_2 )$ is non-signaling. Additionally, one can see (by substitution) that using $\ensuremath{E}$ to assist $\ensuremath{K}$ yields a perfect transmission of a single bit. (See figure~\ref{fig:perfectcommunication}.)
\begin{figure}
\caption{A perfect communication protocol.}
\label{fig:perfectcommunication}
\end{figure}
Now, let us calculate the quantity $\ensuremath{\mathrm{Succ}} \left( \ensuremath{K} \right)$. For any two distinct vectors $\mathbf{x}_0 , \mathbf{x}_1 \in \mathbb{F}_2^m$, the probability that a randomly chosen vector $\mathbf{w} \in \mathbb{F}_2^m \smallsetminus \{ \mathbf{ 0 } \}$ will satisfy $\mathbf{w} \cdot \mathbf{x}_0 \neq \mathbf{w} \cdot \mathbf{x_1}$ is equal to $2^{m-1} / \left( 2^m - 1 \right)$. This fact has the following consequence: if Alice employs the deterministic encoding strategy $\left[ 0 \mapsto \mathbf{x}_0 , 1 \mapsto \mathbf{x}_1 \right]$ to send a single bit, then the optimal probability with which Bob can decode is \begin{eqnarray} & & \left[ \frac{2^{m-1}}{2^m - 1 } \right] (1) + \left[ \frac{2^{m-1} - 1}{2^m - 1 } \right] \left( \frac{1}{2} \right) \\ \label{succexample} & = & \frac{2^m + 2^{m-1} - 1}{2^{m+1} - 2}. \end{eqnarray} Therefore, $\ensuremath{\mathrm{Succ}} \left( \ensuremath{K} \right)$ is equal to quantity (\ref{succexample}), while $\ensuremath{\mathrm{Succ}}_\textnormal{NS} \left( \ensuremath{K} \right)$ is equal to $1$. Theorem 5 from the main text asserts the following bound on $\ensuremath{\mathrm{Succ}}_\textnormal{NS} \left( \ensuremath{K} \right)$: \begin{eqnarray*} \ensuremath{\mathrm{Succ}}_\textnormal{NS} \left( \ensuremath{K} \right) & \leq & \frac{1}{2} + \left( 2 - \frac{2}{2^m} \right) \left[ \ensuremath{\mathrm{Succ}} ( \ensuremath{K} ) - \frac{1}{2} \right] \\ & = & \frac{1}{2} + 2 \left( \frac{2^m - 1}{2^m } \right) \left( \frac{2^{m-1} } {2^{m+1} - 2} \right) \\ & = & 1. \end{eqnarray*} Therefore, equality is achieved in Theorem 5 from the main text when $\ensuremath{N}= \ensuremath{K}$.
\section{The local fraction of a binary quantum correlation}
In this section, we prove the following proposition from the main text.
\begin{proposition} Let $\assistobj$ be a binary quantum correlation. Then $\textnormal{loc} ( \assistobj ) \geq 2 - \sqrt{2}.$ \end{proposition}
\begin{proof} For any binary non-signaling correlation $\assistcu{G}$, let \[ f_1 \left( \assistcu{G} \right)
= \sum_{a, x, b, y \in \{ 0, 1 \} } (-1)^{x \oplus b \oplus ( a \wedge y )} \boxprobcu{G}{ay}{xb}. \] This is the function which defines the CHSH inequality \cite{chsh}. let $f_2$, $f_3$, and $f_4$ be the functions defined by the same expression with $a \wedge y$ replaced by $\neg a \wedge y$, $a \wedge \neg y$, and $\neg a \wedge \neg y$, respectively.
We note the following facts. (See \cite{tsirelson}.) \begin{enumerate} \setlength{\parskip}{0pt} \setlength{\itemsep}{0pt}
\item \label{localcrit} A non-signaling correlation $\assistcu{G}$ is local if and only if $ -2 \leq f_i \left( \assistcu{G} \right) \leq 2 $ for $i = 1, 2, 3, 4$.
\item If $\assistcu{G}$ is a quantum correlation, then for $i = 1,2,3,4$, \begin{equation*} \label{quantbound} - 2 \sqrt{2} \leq f_i \left( \assistcu{G} \right) \leq 2 \sqrt{2}. \end{equation*}
\item There are eight non-signaling correlations $\{ \assistcu{P}_i^+ \}_{i=1}^4$ and $\{ \assistcu{P}_i^- \}_{i=1}^4$, satisfying \begin{eqnarray*} f_j \left( \assistcu{P}_i^\pm \right) & = & \begin{cases} \pm 4 & \textnormal{ if } j = i \\ 0 & \textnormal{ otherwise} \end{cases} \end{eqnarray*} These are the \em Popescu-Rohrlich \em (PR) boxes.
\item \label{gendecomp} Every non-signaling correlation is a convex combination of local correlations and the eight PR boxes. Further, for any two distinct PR boxes $P$ and $P'$, the correlation $(P + P')/2$ is local. \end{enumerate}
From the second part of item \ref{gendecomp}, it follows that any convex combination of local boxes and PR boxes can be simplified into an expression of the form $\alpha \assistcu{L} + (1 - \alpha ) \assistcu{Q}$, where $\assistcu{L}$ is local, $\assistcu{Q}$ is a PR box, and $\alpha \in [0, 1]$. Any non-signaling correlation can thus be expressed as a convex combination of a local correlation and a single PR box.
Let $\assistobj = \alpha \assistcu{L} + (1 - \alpha) \assistcu{Q}$, where $\assistcu{L}$ is local and $\assistcu{Q}$ is a PR box. First suppose that $\assistcu{Q} = \assistcu{P}_j^+$. Let $\assistcu{L}_\beta = (\alpha \assistcu{L} + (\beta - \alpha) \assistcu{P}_j^+)/\beta.$ for any $\beta \in [ \alpha , 1 ]$. Then $\assistcu{L}_\beta$ is local whenever $f_j ( \assistcu{L}_\beta ) \leq 2$.
If $f_j \left( \assistcu{L}_1 \right) < 2$, then $\assistcu{L}_1 (= \assistobj)$ is local, and the proposition follows easily. Otherwise, there is a value $\beta \in [\alpha , 1]$ such that $f_j \left( \assistcu{L}_\beta \right) = 2$. We have $\assistobj = \beta \cdot \assistcu{L}_\beta + (1 - \beta ) \assistcu{P}_j^+.$ The quantity $\beta$ must be at least $2 - \sqrt{2}$, since otherwise \eqref{quantbound} would be violated. Therefore $\textnormal{loc} (D ) \geq 2 - \sqrt{2}$.
A similar argument completes the proof in the case where $Q = P_j^-$. \end{proof}
\end{document} | arXiv |
\begin{document}
\title{Promises Make Finite (Constraint Satisfaction) Problems Infinitary}
\author{Libor Barto \thanks{ Libor Barto has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No 771005). } \\ Department of Algebra\\Faculty of Mathematics and Physics\\ Charles University\\ Prague, Czechia\\ Email: [email protected]}
\date{April 1, 2019}
\maketitle
\begin{abstract} The fixed template Promise Constraint Satisfaction Problem (PCSP) is a recently proposed significant generalization of the fixed template CSP, which includes approximation variants of satisfiability and graph coloring problems. All the currently known tractable (i.e., solvable in polynomial time) PCSPs over finite templates can be reduced, in a certain natural way, to tractable CSPs. However, such CSPs are often over infinite domains. We show that the infinity is in fact necessary by proving that a specific finite-domain PCSP, namely (1-in-3-SAT, Not-All-Equal-3-SAT), cannot be naturally reduced to a tractable finite-domain CSP, unless P=NP. \end{abstract}
\section{Introduction}
Finding a 3-coloring of a graph or finding a satisfying assignment of a propositional 3-CNF formula (or rather the decision variants of these problems) are well-known and fundamental NP-complete computational problems. The latter problem, 3-SAT, has many restrictions still known to be NP-complete~\cite{Sch78}, two of which play a central role in this paper. The \emph{positive 1-in-3-SAT}, denoted $\OneInThreeSAT$, can be defined as follows. The instance is a list of triples of variables and the problem is to find a mapping from the set of variables to $\{0,1\}$ such that exactly one variable in each triple is assigned 1. In the \emph{positive Not-All-Equal-3-SAT}, denoted $\NAESAT$, instances are triples of variables as well, but the mapping is only required to assign not-all-equal elements to each triple.
There are two ways how to relax the requirement on the assignment in order to get a potentially simpler problem. The first one is to require a specified fraction of the constraints to be satisfied. For example, given a satisfiable 3-SAT instance, is it easier to find an assignment satisfying at least 90\% of clauses? A celebrated result of H{\aa}stad~\cite{H01} proves that the answer is ``No.'' -- it is still an NP-complete problem. (Actually, any fraction greater than $7/8$ gives rise to an NP-complete problem while the fraction $7/8$ is achievable in polynomial time.)
The second type of relaxation is to require that a specified weaker version of every constraint is satisfied. For example, we want to find a 100-coloring of a 3-colorable graph, or we want to find a valid $\NAESAT$ assignment to a $\OneInThree$-satisfiable instance. The complexity of the former problem is a notorious open question (for a recent development see~\cite{BG16,BKO18}, but even 6-coloring a 3-colorable graph is not known to be NP-complete). On the other hand, the latter problem admits an elegant polynomial time algorithm~\cite{BG18a,BG18b}, which we now describe.
We take a satisfiable instance of $\OneInThreeSAT$ and replace each triple of variables $(x,y,z)$ in the instance by the linear equation $x+y+z=1$ over $\mathbb{Z}$ (the integers). The obtained system is solvable (by the original 0,1 assignment) and it is known that finding a solution to a system of linear equations over $\mathbb{Z}$ is in P (see~\cite{GLS93}). Now, if $\phi$ is any solution to the system, then $$ \psi(x)= \left\{
\begin{array}{ll}
0 & \mbox{ if } \phi(x) \leq 0 \\
1 & \mbox{ if } \phi(x) > 0
\end{array}
\right. $$ is a valid $\NAESAT$ assignment.
Alternatively, one can solve the system over $\mathbb{Q} \setminus \{1/3\}$ by a simple adjustment of Gaussian elimination and define $\psi(x)=0$ iff $\phi(x) < 1/3$. (A more general class of problems can be solved, e.g., by restricting the domain $\mathbb{Q} \setminus \{c\}$ to the interval $[0,1]$ and using an adjustment of linear programming rather than Gaussian elimination; see~\cite{BG18a,BG18b}.)
It is remarkable that both polynomial algorithms transfer the original problem over a finite domain to a problem over an infinite domain. The main result of this paper shows that this finite-to-infinite transition is unavoidable. This result is stated more precisely below, as Theorem~\ref{thm:main_fake}, but let us first describe its background.
\subsection{Constraint Satisfaction Problems}
It will be convenient in this paper to use a formalization of CSP and PCSP via homomorphisms of relational structures. We refer to~\cite{BKW17,BG18b} for translations to the other standard definitions.
Let $\relstr{A}$ be a relational structure of a finite signature, often called \emph{template} in this context. The \emph{Constraint Satisfaction Problem (CSP) over} $\relstr{A}$, denoted $\CSP(\relstr{A})$, is the problem of deciding whether a given finite relational structure $\relstr{X}$ (similar to $\relstr{A}$) has a homomorphism to $\relstr{A}$. The \emph{search problem} for $\CSP(\relstr{A})$ is to find such a homomorphism. Examples of CSPs include the 3-coloring problem (where $\relstr{A}$ is a structure with a three-element domain and the binary disequality relation), 3-SAT (where $\relstr{A}$ consists of 8 ternary relations of the form $(\neg) x \vee (\neg) y \vee (\neg) z$ on the domain $\{0,1\}$), the problems $\OneInThreeSAT$, $\NAESAT$ for which the templates are structures with domain $\{0,1\}$ and a single ternary relation \begin{align*} \OneInThree &= (\{0,1\}; \{(1,0,0),(0,1,0),(0,0,1)\}) \\ \NAE &= (\{0,1\}; \{0,1\}^3 \setminus \{(0,0,0),(1,1,1)\})\enspace, \end{align*} and the infinite-domain CSPs over $(\mathbb{Z}; x+y+z=1)$ and $(\mathbb{Q}\setminus\{1/3\};x+y+z=1)$ that were used to solve the relaxed version of $\OneInThreeSAT$.
The complexity of the CSP over finite templates (modulo polynomial time reductions) is fully classified by a recent deep result of Bulatov~\cite{Bul17} and, independently, Zhuk~\cite{Zhuk17}. The classification is a culmination of an active research program, so called algebraic approach to CSPs, inspired by the landmark paper of Feder and Vardi~\cite{FV98}, where the authors conjectured that each finite template CSP is either tractable or NP-complete, and observed that the tractability is often tied to closure properties of relations in the template. General theory of CSPs, whose basics were developed in~\cite{JCG97,J98,BJK05,BOP18}, confirmed this observation by closely linking CSPs to Universal Algebra (the theory of general algebraic systems) and provided guidance and tools for the eventual resolution of the dichotomy conjecture in~\cite{Bul17,Zhuk17}.
The theory of CSPs is based on a connection between constructions on relational structures (that lead to polynomial time reductions) and properties of their \emph{polymorphisms} -- multivariate functions preserving the structures. So far, the most general relational construction introduced in~\cite{BOP18} is the so called pp-construction. Roughly, we say that $\relstr{A}$ \emph{pp-constructs} $\relstr{B}$ if $\relstr{B}$ can be obtained from $\relstr{A}$ by a sequence of first-order interpretations restricted to primitive positive formulae and replacements by homomorphically equivalent structures (see Section~\ref{sec:prelim} for a more detailed definition). In this situation, there is a natural gadget reduction of $\CSP(\relstr{B})$ to $\CSP(\relstr{A})$. It follows, for instance, that $\CSP(\relstr{A})$ is NP-complete whenever $\relstr{A}$ pp-constructs a template of 3-SAT. It turned out~\cite{Bul17,Zhuk17}, confirming the \emph{algebraic dichotomy conjecture} from~\cite{BJK05}, that this is exactly the borderline between tractable and NP-complete CSPs: all templates that do not pp-construct the template of 3-SAT have tractable CSPs. The algebraic part of the theory will not be discussed here, let us just mention that the strongest available algebraic characterization of the borderline by means of cyclic operations~\cite{BK12} is essential for the proof of the main result, Theorem~\ref{thm:main_fake}.
\subsection{Promise CSPs}
A \emph{template} for the \emph{Promise CSP} (PCSP) is a pair $(\relstr{A},\relstr{B})$ of similar relational structures of finite signature such that $\relstr{A}$ has a homomorphism to $\relstr{B}$. The PCSP over such a pair, denoted $\PCSP(\relstr{A},\relstr{B})$, is the following promise problem: given a relational structure $\relstr{X}$ (similar to $\relstr{A}$ and $\relstr{B}$) output ``Yes.'' if $\relstr{X}$ has a homomorphisms to $\relstr{A}$, and ``No.'' if $\relstr{X}$ does not have a homomorphism to $\relstr{B}$. The \emph{search problem} for $\PCSP(\relstr{A},\relstr{B})$ is to find a homomorphism $\relstr{X} \to \relstr{B}$ given an input structure $\relstr{X}$ that has a homomorphism to $\relstr{A}$. Notice that $\CSP(\relstr{A})$ is the same as $\PCSP(\relstr{A},\relstr{A})$. Examples of PCSPs include the 100-coloring of a 3-colorable graph, where the two structures are $(\{1,2,3\}, \neq)$ and $(\{1, 2, \dots, 100\}, \neq)$, and $\PCSP(\OneInThree,\NAE)$ -- the central computational problem in this paper. Similar examples were the motivation for introducing the PCSP framework in~\cite{AGH17,BG16,BG18a,BG18b}.
The current knowledge of the complexity of finite-domain PCSPs beyond CSPs is very much limited. For example, the Feder-Vardi dichotomy conjecture for CSPs was inspired by two earlier classification results: for CSPs over a Boolean (i.e., two-element) domain~\cite{Sch78} and for CSPs over graphs~\cite{HN90}. In PCSPs, even the analogues of these early results are challenging. PCSPs over graphs include, as a very special case, the PCSP over a pair of complete graphs, the problem of $l$-coloring a $k$-colorable graph. A systematic study of Boolean PCSPs (where both structures have a two-element domain) was initiated in~\cite{BG18a},
but the general Boolean case is still wide open.
Fortunately, building on the initial insights and results in~\cite{AGH17,BG16,BG18a,BG18b}, it was observed in~\cite{BKO18} (among many other important results such as the NP-hardness of 5-coloring a 3-colorable graph) that the basics of the CSP theory from~\cite{BOP18} generalize to PCSPs. In particular, the notions of pp-constructions and polymorphisms have their PCSP counterparts and the connection between relational and algebraic structures works just as well as in the CSP. This is especially interesting because some hardness and algorithmic results in PCSP require techniques used in approximation. PCSP thus might help building a bridge between the discrete, universal algebraic world of (exact) CSPs and analytical world of approximation.
\subsection{Finite PCSPs are infinitary}
The main result of this paper says that it is impossible to reduce $\PCSP(\OneInThree, \NAE)$ to a tractable finite-domain CSP by means of a pp-construction, unless P=NP.
\begin{theorem} \label{thm:main_fake} Let $\relstr{C}$ be a finite relational structure that pp-constructs $(\OneInThree, \NAE)$. Then $\CSP(\relstr{C})$ is NP-complete. \end{theorem}
A fundamental question is whether each finite tractable PCSP template can be pp-constructed from an infinite tractable CSP template. In~\cite{BG18b}, the authors conjectured that the answer is positive and even suggested a family of tractable CSPs that might solve all Boolean PCSPs.
The class of all infinite-domain CSPs is very broad. In fact, each computational problem is equivalent to an infinite-domain CSP~\cite{BodG08}. However, some parts of the CSP theory can be extended to a quite rich class of structures, namely, reducts of finitely bounded homogeneous structures in finite signature, and some general results even to the broader class of $\omega$-categorical structures~\cite{Bod08,Pin15}. In particular, an algebraic criterion for NP-hardness is available~\cite{BarP16}, so it might be possible to generalize Theorem~\ref{thm:main_fake} to this setting (possibly with a different template). As $\omega$-categorical structures are, in a sense, close to finite and CSPs over them are solved by ``finitary'' algorithms, such a generalization would show that a polynomial time algorithm for some PCSP must be ``truly'' infinitary.
Let us make a final remark before starting with the technicalities. Both algorithms~\cite{Bul17,Zhuk17} for the finite-domain CSP are extremely complex and simplifications are much desired. Theorem~\ref{thm:main_fake} supports the intuition that a simpler algorithm may require infinitary methods, such as CSPs over numerical domains~\cite{BodM17} ($\mathbb{Z}$, $\mathbb{Q}$, \dots).
\section{Preliminaries} \label{sec:prelim}
In this section we give formal definitions of the concepts essential for the proof. For an in depth introduction to CSP and PCSP, see \cite{BKW17,BKO18} and references therein.
\subsection{PCSP}
A \emph{relational structure (of finite signature)} is a tuple $\relstr{A} = (A; R_1, \dots, R_n)$ where each $R_i \subseteq A^{\arity(R_i)}$ is a relation on $A$ of arity $\arity(R_i) \geq 1$. The structure $\relstr{A}$ is \emph{finite} if $A$ is finite.
Two relational structures $\relstr{A} = (A; R_1, \dots, R_n)$ and $\relstr{B} = (B; S_1, \dots, S_n)$ are \emph{similar} if they have the same number of relations and $\arity(R_i) = \arity(S_i)$ for each $i \in \{1, \dots, n\}$.
For two such similar relational structures $\relstr{A}$ and $\relstr{B}$, a \emph{homomorphism} from $\relstr{A}$ to $\relstr{B}$ is a mapping $f: A \to B$ such that $(f(a_1), f(a_2), \dots, f(a_k)) \in S_i$ whenever $i \in \{1, \dots, n\}$ and $(a_1, a_2, \dots, a_k) \in R_i$ where $k = \arity(R_i)$.
We write $\relstr{A} \to \relstr{B}$ if there exists a homomorphism from $\relstr{A}$ to $\relstr{B}$, and $\relstr{A} \not\to \relstr{B}$ if there is none.
\begin{definition} A \emph{PCSP template} is a pair $(\relstr{A},\relstr{B})$ of similar relational structures such that $\relstr{A} \to \relstr{B}$.
The decision version of PCSP over $(\relstr{A},\relstr{B})$, written $\PCSP(\relstr{A},\relstr{B})$, is the following promise problem. Given a finite structure $\relstr{X}$ similar to $\relstr{A}$ (and $\relstr{B}$), output ``Yes.'' if $\relstr{X} \to \relstr{A}$ and output ``No.'' if $\relstr{X} \not\to \relstr{B}$.
The search version of $\PCSP(\relstr{A},\relstr{B})$ is, given a structure $\relstr{X}$ similar to $\relstr{A}$ such that $\relstr{X} \to \relstr{A}$, find a homomorphisms $\relstr{X} \to \relstr{B}$. \end{definition}
In the case $\relstr{A}=\relstr{B}$ we talk about a CSP template (and simply write $\relstr{A}$ instead of $(\relstr{A},\relstr{A})$) and define $\CSP(\relstr{A}) = \PCSP(\relstr{A},\relstr{A})$.
The decision version of the PCSP over $(\relstr{A},\relstr{B})$ can be reduced to the search version. For CSPs, it is known~\cite{BJK05} that these two versions are in fact equivalent, but it is an open problem whether they are equivalent for PCSPs as well.
\subsection{Constructions} \label{subsec:constr}
The two ingredients of a pp-construction are pp-powers and homomorphic relaxations.
Homomorphic relaxation, called \emph{homomorphic sandwiching} in \cite{BG18b}, is a generalization of the concept of homomorphic equivalence between CSP templates.
\begin{definition} \label{def:relax} Let $(\relstr{A},\relstr{B})$ and $(\relstr{A}', \relstr{B}')$ be PCSP templates. We say that $(\relstr{A}', \relstr{B}')$ is a \emph{homomorphic relaxation} of $(\relstr{A},\relstr{B})$ if there exist homomorphisms $f: \relstr{A}' \to \relstr{A}$ and $g: \relstr{B} \to \relstr{B}'$. \end{definition}
If $(\relstr{A}', \relstr{B}')$ is a homomorphic relaxation of $(\relstr{A},\relstr{B})$, then the trivial reduction, which does not change the input structure $\relstr{X}$, reduces (the decision or search version of) $\PCSP(\relstr{A}',\relstr{B}')$ to $\PCSP(\relstr{A}, \relstr{B})$. Both polynomial algorithms for $\PCSP(\OneInThree,\NAE)$ shown in the introduction come from this reduction with $$ \relstr{A} = \relstr{B} = (A; R), \quad (x,y,z) \in R \mbox { iff } x+y+z=1\enspace, $$ where $A = \mathbb{Z}$ in the first version of the algorithm and $A = \mathbb{Q} \setminus \{1/3\}$ in the second. In both cases, the mapping $f$ was the inclusion and the ``rounding'' mapping $g$ is defined by $g(x) = 0$ iff $x < 1/3$.
In order to define the other ingredient of a pp-construction, recall that a \emph{primitive positive formula} over a relational structure $\relstr{A}$ is an existentially quantified conjunction of atomic formulas of the form $x_1 = x_2$ or $(x_{i_1}, \dots, x_{i_k}) \in R$ where $x_j$'s are variables and $R$ is a relation in $\relstr{A}$ of arity $k$.
\begin{definition} Let $(\relstr{A},\relstr{B})$ and $(\relstr{A}' = (A'; R_1, \dots, R_n), \relstr{B}'=(B', S_1, \dots, S_n))$ be PCSP templates.
We say that $(\relstr{A}',\relstr{B}')$ is \emph{pp-definable} from $(\relstr{A},\relstr{B})$ if, for each $i \in \{1, \dots, n\}$, there exists a primitive positive formula $\phi$ over $\relstr{A}$ such that $\phi$ defines $R_i$ and the formula, obtained by replacing each occurrence of a relation of $\relstr{A}$ by the corresponding relation in $\relstr{B}$, defines $S_i$.
We say that $(\relstr{A}',\relstr{B}')$ is an $n$-th \emph{pp-power} of $(\relstr{A},\relstr{B})$ if $A'=A^n$, $B' = B^n$, and, if we view $k$-ary relations on $\relstr{A}'$ and $\relstr{B}'$ as $kn$-ary relations on $A$ and $B$, respectively, then $(\relstr{A}',\relstr{B}')$ is pp-definable from $(\relstr{A},\relstr{B})$. \end{definition}
By combining these two constructions we get the notion of pp-construction.
\begin{definition} We say that a PCSP template $(\relstr{A}, \relstr{B})$ pp-constructs a PCSP template $(\relstr{A}',\relstr{B}')$ if there exists a sequence $$ (\relstr{A}, \relstr{B}) = (\relstr{A}_1, \relstr{B}_1), \dots, (\relstr{A}_k, \relstr{B}_k) = (\relstr{A}', \relstr{B}') $$ of PCSP templates such that each $(\relstr{A}_{i+1}, \relstr{B}_{i+1})$ is a pp-power or a homomorphic relaxation of $(\relstr{A}_{i}, \relstr{B}_{i})$. \end{definition}
It is not hard to see that if $(\relstr{A}, \relstr{B})$ pp-constructs $(\relstr{A}',\relstr{B}')$, then $\PCSP(\relstr{A}',\relstr{B}')$ reduces (even in log-space) to $\PCSP(\relstr{A},\relstr{B})$. The proof is similar to the analogous proof for CSP (see~\cite{BKW17}). An interesting alternative way for PCSP was given (but explicitly proved only for finite templates) in \cite{BKO18}.
However, in this paper, pp-constructions make only a cosmetic difference in the statement of Theorem~\ref{thm:main_fake} -- it is enough to prove the theorem for homomorphic relaxations. Indeed, it is well known (see~\cite{BOP18}) that if $(\relstr{A}, \relstr{B})$ pp-constructs $(\relstr{A}',\relstr{B}')$, then $(\relstr{A}',\relstr{B}')$ is a homomorphic relaxation of a pp-power of $(\relstr{A},\relstr{B})$. Therefore, if a finite $\relstr{C}$ pp-constructs $(\OneInThree,\NAE)$, then $(\OneInThree,\NAE)$ is a homomorphic relaxation of a template $(\relstr{D},\relstr{D'})$, which is a pp-power of $\relstr{C}$. Then, clearly, $\relstr{D}=\relstr{D}'$ are finite and $\CSP(\relstr{D})$ reduces to $\CSP(\relstr{C})$.
\subsection{Cyclic polymorphisms}
For a PCSP template $(\relstr{A}, \relstr{B})$, a function $f: A^n \to B$ is called a \emph{polymorphism} of the template if it is a homomorphism from the $n$-th categorical power of $\relstr{A}$ to $\relstr{B}$. The basic fact of the algebraic theory of (P)CSP is that the set of polymorphisms determine the complexity of $\PCSP(\relstr{A},\relstr{B})$~(\cite{J98}, cf.~\cite{BKW17}).
We will only work with polymorphisms of CSP templates and we spell out the definition of a polymorphism in a more elementary way for this case.
\begin{definition} Let $\relstr{C}$ be a CSP template and $s: C^n \to C$ a function (also called an \emph{operation} in this context). We say that $s$ is a \emph{polymorphism} of $\relstr{C}$ if, for each relation $R$ in $\relstr{C}$ with $k=\arity(R)$ and all tuples $(a_1^1, \dots, a_k^1), \dots, (a_1^n, \dots, a_k^n) \in R$, we have $$ (s(a_1^1, \dots, a_1^n), \dots, s(a_k^1, \dots, a_k^n)) \in R\enspace. $$ \end{definition}
The proof of the main theorem is based on the following result from~\cite{BK12}.
\begin{definition} An operation $s: C^n \to C$ is called \emph{cyclic} if, for all $(a_1, \dots, a_n) \in C^n$, we have $$ s(a_1,a_2, \dots, a_n) = s(a_2, \dots, a_n, a_1)\enspace. $$ \end{definition}
\begin{theorem} \label{thm:cyclic}
Let $\relstr{C}$ be a finite CSP template. If $\CSP(\relstr{C})$ is not NP-complete, then $\relstr{C}$ has a cyclic polymorphism of arity $p$ for every prime number $p > |C|$. \end{theorem}
We remark that cyclic operations characterize the borderline between NP-complete and tractable CSPs -- whenever $\relstr{C}$ has a cyclic polymorphism of arity at least 2, then $\CSP(\relstr{C})$ is tractable~\cite{Bul17,Zhuk17}. In fact, cyclic polymorphisms provide currently the strongest characterization of the borderline in the sense that the other important types of operations (such as the Sigger's operations~\cite{Sig10,KMM14} or the weak near-unanimity operations~\cite{MM08}) can be obtained from a cyclic operation by an identification of variables. The proof of Theorem~\ref{thm:main_fake} could still be simplified having a yet stronger (or alternative) characterization at hand. See Section~\ref{sec:conclusion} for a concrete open problem in this direction.
\section{Infinity is necessary}
In this section we prove the main theorem. As explained in Subsection~\ref{subsec:constr}, it is enough to prove the following result.
\begin{theorem} Let $\relstr{C} = (C; R)$ be a finite relational structure with ternary $R \subseteq C^3$ such that $(\OneInThree, \NAE)$ is a homomorphic relaxation of $(\relstr{C},\relstr{C})$. Then $\CSP(\relstr{C})$ is NP-complete. \end{theorem}
Assume that $\CSP(\relstr{C})$ is not NP-complete and let $f: \OneInThree \to \relstr{C}$ and $g: \relstr{C} \to \NAE$ be homomorphisms from the definition of homomorphic relaxation, Definition~\ref{def:relax}.
Since $gf$ is a homomorphism, this mapping applied component-wise to the 1-in-3 tuple $(0,0,1)$ is a not-all-equal tuple. In particular $f(0) \neq f(1)$. We rename the elements of $C$ so that $\{0,1\} \subseteq C$ and $f$ is the inclusion. As $f$ and $g$ are homomorphisms, we get $$ \{0,1\} \subseteq C, \quad \{(1,0,0), (0,1,0), (0,0,1)\} \subseteq R $$ and $$ \neg (g(a)=g(b)=g(c)) \mbox{ whenever } (a,b,c) \in R\enspace. $$
By Theorem~\ref{thm:cyclic}, $\relstr{C}$ has a cyclic polymorphism of any prime arity $p > |C|$. We fix a cyclic polymorphism $$ s \mbox{ of prime arity }
p > 60 |C|\enspace. $$ Next we define an operation $t$ on $C$ of arity $p^2$ by $$ t(x_{11}, x_{12}, \ldots, x_{1p}, x_{21}, x_{22}, \ldots x_{2p}, x_{31}, \ldots, \ldots, x_{pp})
$$ \begin{align*}
= s(&s(x_{11},x_{21}, \ldots, x_{p1}), \\
&s(x_{12},x_{22}, \ldots, x_{p2}), \\
&\dots \\
&s(x_{1p},x_{2p}, \ldots, x_{pp}))\enspace. \end{align*}
It will be convenient to organize the arguments of $t$ into a $p \times p$ matrix $X$ whose entry in the $i$-th row and $j$-th column is $x_{ij}$, so the value $$ t \left( \begin{array}{cccc} x_{11} & x_{12} & \cdots & x_{1p} \\ x_{21} & x_{22} & \cdots & x_{2p} \\ \vdots & \vdots & \ddots & \vdots \\ x_{p1} & x_{p2} & \cdots & x_{pp} \end{array} \right) $$ is obtained by applying $s$ to the columns and then $s$ to the results.
We introduce several concepts for zero-one matrices, the only important arguments of $t$ for the proof.
\begin{definition} Let $X=(x_{ij}), Y$ be $p \times p$ zero-one matrices. The \emph{area of $X$} is the fraction of ones and is denoted $$ \lambda(X) = \left(\sum_{i,j} x_{ij}\right)/p^2\enspace. $$
The matrices $X,Y$ are called \emph{$g$-equivalent}, denoted $X \sim Y$, if $g(t(X)) = g(t(Y))$.
The matrix $X$ is called \emph{tame} if \begin{align*} &X \sim 0_{p \times p} \quad \mbox{ if } \quad \lambda(X) < 1/3 \\ \mbox{and } &X \sim 1_{p \times p} \quad \mbox{ if } \quad \lambda(X) > 1/3 \end{align*} where $0_{p \times p}$ stands for the zero matrix and $1_{p \times p}$ for the all-ones matrix. \end{definition}
Observe that the equivalence $\sim$ has two blocks, so, e.g., $X \not\sim Y \not\sim Z$ implies $X \sim Z$. Also recall that $p>3$ is a prime number, so the area of $X$ is never equal to $1/3$.
The proof now proceeds as follows. We show that certain matrices, called ``almost rectangles'', are tame. The proof is by induction (although the proof logic, as presented, is a bit different). Subsection~\ref{subsec:lines} provides the base case and Subsection~\ref{subsec:rect} handles the induction step. In Subsection~\ref{subsec:contra}, we construct two tame matrices $X_1$, $X_2$ such that $\lambda(X_1)<1/3$ and $\lambda(X_2) > 1/3$, but $t(X_1) = t(X_2)$ (because the corresponding columns of $X_1$ and $X_2$ will be evaluated by $s$ to the same elements). This gives us a contradiction since $0_{p \times p} \not\sim 1_{p \times p}$ as we shall see.
\subsection{Covers}
Before launching into the proof, we introduce an additional concept and state a consequence of the fact that $s$ is a polymorphism.
\begin{definition} A triple $X,Y,Z$ of $p \times p$ zero-one matrices is called a \emph{cover} if, for every $1 \leq i,j \leq p$, exactly one of $x_{ij},y_{ij},z_{ij}$ is equal to one. \end{definition}
\begin{lemma} \label{lem:nae} If $X,Y,Z$ is a cover, then $X,Y,Z$ are not all $g$-equivalent. \end{lemma}
\begin{proof} By the definition of a cover, the $ij$-th coordinates of $X$, $Y$, $Z$ are in $\{(0,0,1),(0,1,0),(1,0,0)\} \subseteq R$ for each $i,j$. Since $t$ preserves $R$ (because $s$ does), the triple $(t(X),t(Y),t(Z))$ is in $R$ as well. Finally, $g$ is a homomorphism from $\relstr{C}$ to $\NAE$, therefore $g(t(X)),g(t(Y)),g(t(Z))$ are not all equal. In other words, $X$, $Y$, $Z$ are not all $g$-equivalent, as claimed. \end{proof}
\subsection{Line segments are tame} \label{subsec:lines}
In this subsection it will be more convenient to regard the arguments of $t$ as a tuple $\vc{x} = (x_{11},x_{12}, \ldots)$ of length $p^2$ rather than a matrix. The concepts of the area, $g$-equivalence, tameness, and cover is extended to tuples in the obvious way. Since $p>3$ is a prime number, $p^2$ is 1 modulo 3. Let $q$ be such that $$
p^2 = 3q+1\enspace. $$ Moreover, let $\ttt{i}$ denote the following tuple of length $p^2$. $$ \ttt{i} = (\underbrace{1,1, \cdots, 1}_{i \times }, 0,0, \cdots 0) $$ We prove in this subsection that all such tuples are tame. We first recall a well-known fact.
\begin{lemma} The operation $t$ is cyclic. \end{lemma}
\begin{proof} By cyclically shifting the arguments we get the same result: \begin{align*} &t(x_{12}, \cdots, x_{pp},x_{11}) = t \left( \begin{array}{ccccc} x_{12} & x_{13} & \cdots & x_{1p} & x_{21} \\ x_{22} & x_{23} & \cdots & x_{2p} & x_{31} \\ \vdots & \vdots & \ddots & \vdots & \vdots\\ x_{p2} & x_{p3} & \cdots & x_{pp} & x_{11} \end{array} \right) \\ &= t \left( \begin{array}{ccccc} x_{21} & x_{12} & x_{13} & \cdots & x_{1p} \\ x_{31} & x_{22} & x_{23} & \cdots & x_{2p} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ x_{11} & x_{p2} & x_{p3} & \cdots & x_{pp} \end{array} \right) \\ &=t \left( \begin{array}{cccc} x_{11} & x_{12} & \cdots & x_{1p} \\ x_{21} & x_{22} & \cdots & x_{2p} \\ \vdots & \vdots & \ddots & \vdots \\ x_{p1} & x_{p2} & \cdots & x_{pp} \end{array} \right) = t(x_{11}, x_{12}, \cdots, x_{pp})\enspace, \end{align*} where the second equality uses the cyclicity of the outer ``s'' in the definition of $t$, while the third one the cyclicity of the first inner ``s''. \end{proof}
The following lemma is proved by induction on $i = 0,1, \dots, q$.
\begin{lemma} For each $i \in \{0,1, \dots, q\}$, we have \begin{align*} &\ttt{q-i} \sim \ttt{q-i+1} \sim \cdots \sim \ttt{q} \\ &\not\sim \ttt{q+1} \sim \cdots \sim \ttt{q+i} \sim \ttt{q+i+1}\enspace. \end{align*} \end{lemma}
\begin{proof} For the first induction step, $i=0$, let $\vc{x} = \ttt{q}$, let $\vc{y}$ be $\ttt{q}$ (cyclically) shifted $q$ times to the right (so the first 1 is at the $(q+1)$-st position), and let $\vc{z}$ be $\ttt{q+1}$ shifted $2q$ times to the right. The tuples $\vc{x},\vc{y},\vc{z}$ form a cover, therefore they are not all $g$-equivalent by Lemma~\ref{lem:nae}. But $t$ is cyclic, thus $t(\vc{x})=t(\vc{y}) = t(\ttt{q})$ and $t(\vc{z}) = t(\ttt{q+1})$. It follows that $\ttt{q},\ttt{q},\ttt{q+1}$ are not all $g$-equivalent and we get $\ttt{q} \not\sim \ttt{q+1}$.
Now we prove the claim for $i>0$ assuming it holds for $i-1$. To verify $\ttt{q-i} \sim \ttt{q-i+1}$ consider $\ttt{q-i}$, $\ttt{q+1}$, $\ttt{q+i}$. Since $(q-i)+(q+1)+(q+i) = 3q+1=p^2$, these tuples can be shifted to form a cover and then the same argument as above gives us that $\ttt{q-i}$, $\ttt{q+1}$, $\ttt{q+i}$ are not all $g$-equivalent. But $\ttt{q+1} \sim \ttt{q+i}$ by the induction hypothesis, therefore $\ttt{q-i} \not\sim \ttt{q+1}$. Since $\ttt{q+1} \not\sim \ttt{q-i+1}$ (again by the induction hypothesis), we get $\ttt{q-i} \sim \ttt{q-i+1}$, as required.
It remains to check $\ttt{q+i} \sim \ttt{q+i+1}$. This is done in a similar way, using the tuples $\ttt{q-i}$, $\ttt{q}$, $\ttt{q+i+1}$. \end{proof}
We have proved that $\ttt{0} \sim \dots \sim \ttt{q} \not\sim \ttt{q+1} \sim \dots \sim \ttt{2q+1}$. Using the same argument as in the previous lemma once more for $\ttt{0},\ttt{p^2-i},\ttt{i}$ with $p^2 \geq i > 2q+1$ we get $\ttt{i} \not\sim \ttt{0}$. In summary, $\ttt{i} \sim \ttt{0}$ whenever $i \leq q$ and $\ttt{i} \sim \ttt{p^2} \not\sim \ttt{0}$ when $i \geq q+1$. Observing that $\lambda(\ttt{i})<1/3$ iff $i \leq q$ we obtain the following lemma.
\begin{lemma} \label{lem:lines_tame} Each $\ttt{i}$, $i \in \{0, 1, \cdots, p^2\}$, is tame and $\ttt{0} \not\sim \ttt{p^2}$. \end{lemma}
\subsection{Almost rectangles are tame} \label{subsec:rect}
We start by introducing a special type of zero-one matrices.
\begin{definition} Let $1 \leq k_1, \dots, k_p \leq p$. By $$ \rrr{k_1,k_2,\dots,k_p} $$ we denote the matrix whose $i$-th column begins with $k_i$ ones followed by ($p-k_i$) zeros, for each $i \in \{1, \dots, p\}$.
An \emph{almost rectangle} is a matrix of the form $\rrr{k,k, \dots, k, l, l, \dots, l}$ (the number of $k$'s can be arbitrary, including 0 or $p$) where $0 \leq k-l \leq 5|C|$. The quantity $k-l$ is referred to as the \emph{size of the step}. \end{definition}
In the remainder of this subsection we prove the following proposition.
\begin{proposition} \label{prop:tame} Each almost rectangle is tame. \end{proposition}
Let $$ X=\rrr{\underbrace{k, k, \cdots, k}_{m \times}, l,l, \dots, l} $$ be a minimal counterexample in the following sense. \begin{itemize} \item $X$ has the minimum size of the step and, \item among such counterexamples, $\abs{\lambda(X)-1/3}$ is maximal. \end{itemize}
\begin{lemma} The size of the step of $X$ is at least 2. \end{lemma}
\begin{proof} This lemma is just a different formulation of Lemma~\ref{lem:lines_tame} since an almost rectangle with step of size 0 or 1 represents the same choice of arguments as $\ttt{i}$ for some $i$. \end{proof}
We handle two cases $\lambda(X) \geq 5/12$ and $\lambda(X) \leq 5/12$ separately, but the basic idea for both of them is the same as in the proof of Lemma~\ref{lem:lines_tame}. To avoid puzzles, let us remark that any number strictly between $1/3$ and $1/2$ (instead of $5/12$) would work with a sufficiently large $p$.
\begin{lemma} \label{lem:c} The area of $X$ is less than $5/12$. \end{lemma}
\begin{proof} Assume that $\lambda(X) \geq 5/12$. Let $k_1$, $k_2$, $l_1$, and $l_2$ be the non-negative integers such that \begin{eqnarray} l_1+l_2+k = p = k_1+k_2+l, \label{eq:a} \\ 1 \geq k_1 - k_2 \geq 0, \mbox{ and } 1 \geq l_1-l_2 \geq 0 \enspace. \label{eq:b} \end{eqnarray} We have $k_1 \geq l_1$ and $k_2 \geq l_2$. Moreover, since $k-l \geq 2$ by the previous lemma, it follows that both $k_1-l_1$ and $k_2-l_2$ are strictly smaller than $k-l$.
Consider the matrices $$ Y_i = \rrr{\underbrace{l_i,l_i, \dots, l_i}_{m \times}, k_i, k_i, \dots, k_i}, \quad i=1,2\enspace. $$ By shifting all the rows of $Y_i$, $i \in \{1,2\}$, $m$ times to the left we obtain an almost rectangle with a smaller step size than $X$, which is thus tame by the minimality assumption on $X$. Since such a shift changes neither the value of $t$ (as the outer ``$s$'' in the definition of $t$ is cyclic) nor the area, both $Y_1$ and $Y_2$ are tame matrices.
Let $Y_1'$ ($Y_2'$, resp.) be the matrices obtained from $Y_1$ ($Y_2$, resp.) by shifting the first $m$ columns $k$ times ($k+l_1$ times, resp.) down and the remaining columns $l$ times ($l+k_1$ times, resp.) down. Since $X,Y_1',Y_2'$ is a cover (by~(\ref{eq:a})) and cyclically shifting columns does not change the value of $t$ (as the inner occurrences of ``$s$'' in the definition of $t$ are cyclic), Lemma~\ref{lem:nae} implies that $X$, $Y_1$, $Y_2$ are not all $g$-equivalent.
From $X, Y_1', Y_2'$ being a cover, it also follows that $$ \lambda(X) + \lambda(Y_1') + \lambda(Y_2') = \lambda(X) + \lambda(Y_1) + \lambda(Y_2) = 1\enspace. $$ Moreover, by~(\ref{eq:b}), we have $\lambda(Y_2) \leq \lambda(Y_1)$ and these areas differ by at most $p/p^2=1/p$. Therefore $$ \lambda(Y_1) = 1 - \lambda(X) - \lambda(Y_2) \leq 1 - 5/12 - \lambda(Y_1) + 1/p $$ and, since $p > 12$ by the choice of $p$, we obtain $$ \lambda(Y_2) \leq \lambda(Y_1) < 1/3\enspace. $$
The tameness of $Y_i$ now gives us $Y_1 \sim Y_2 \sim 0_{p \times p}$ and then, since $Y_1,Y_2,X$ are not all $g$-equivalent and $0_{p \times p} \not\sim 1_{p \times p}$ (by the second part of Lemma~\ref{lem:lines_tame}), we get $X \sim 1_{p \times p}$. But $\lambda(X) \geq 5/12 > 1/3$, hence $X$ is tame, a contradiction with the choice of $X$. \end{proof}
It remains to handle the case $\lambda(X) < 5/12$.
We first claim that $2k$ (and thus $k+l$ and $2l$) is less than $p$. Indeed, since the step size of $X$ is at most $5|C|$ (by the definition of an almost rectangle) and $p > 60|C|$, we get \begin{align*}
5/12 > \lambda(X) &\geq \frac{p(k - 5|C|)}{p^2} \\
k & \leq 5p/12 + 5|C| < 5p/12 + p/12 = p/2\enspace. \end{align*}
We now again need to distinguish two cases. Assume first that $m < p/2$.
Let \begin{align*} Y &= \rrr{\underbrace{l, \cdots,l}_{m \times },\underbrace{k, \cdots, k}_{m \times }, l, \cdots, l}, \\ Z &= \rrr{\underbrace{p-k-l, \cdots, p-k-l}_{2m \times}, p-2l, \cdots, p-2l} \enspace. \end{align*} The definition of $Z$ makes sense since $p-k-l, p-2l \geq 0$ by the inequality $2k < p$ derived above.
The triple $X,Y,Z$ (similarly to $X,Y_1,Y_2$ in the proof of Lemma~\ref{lem:c}) is such that we can obtain a cover by shifting the columns down. Therefore $X$, $Y$, $Z$ are not all $g$-equivalent and $\lambda(X)+\lambda(Y) + \lambda(Z)=1$.
On the other hand, by shifting all the rows of $Y$ $m$ times to the left we obtain $X$. We get $\lambda(X) = \lambda(Y)$ and $t(X)=t(Y)$, therefore $Z \not\sim X$ by the previous paragraph.
Moreover,
by shifting all the rows of $Z$ $2m$ times to the left we obtain an almost rectangle $Z'$ with $t(Z)=t(Z')$ and $\lambda(Z)=\lambda(Z')$. The step size of $Z'$ is $(p-2l) - (p-k-l) = k-l$, which is the same as the step size of $X$. However, the distance of its area from $1/3$ is strictly greater as shown by the following calculation. \begin{align*} \frac{\abs{\lambda(Z)-1/3}}{\abs{\lambda(X)-1/3}} &= \frac{\abs{(1 - 2\lambda(X)) - 1/3}}{\abs{\lambda(X)-1/3}} \\ &= \frac{\abs{2(1/3 - \lambda(X))}}{\abs{\lambda(X)-1/3}} = 2 > 1\enspace. \end{align*} By the minimality of $X$, the almost rectangle $Z'$ is tame and so is $Z$. It is also apparent from the calculation that the signs of $\lambda(X)-1/3$ and $\lambda(Z)-1/3$ are opposite. Combining these two facts with $Z \not\sim X$ derived above, we obtain that $X$ is tame, a contradiction.
In the other case, when $m > p/2$, the proof is similar using the tuples \begin{align*} Y &= (l, \cdots, l, \underbrace{k, \cdots, k}_{m \times}), \\ Z &= (\underbrace{p-k-l, \cdots, p-k-l}_{(p-m) \times}, p-2k, \cdots, p-2k, \\
&\quad \underbrace{p-k-l, \cdots, p-k-l}_{(p-m) \times}) \enspace. \end{align*} The proof of Proposition~\ref{prop:tame} is concluded.
\subsection{Contradiction} \label{subsec:contra}
Let $$ m = (p-1)/2 $$ and choose natural numbers $l_1$ and $l_2$ so that $$
p/3 - 2|C| < l_1 < l_2 < p/3 $$ and $$ s(\underbrace{1, \cdots, 1}_{l_1 \times}, 0, \cdots, 0) = s(\underbrace{1, \cdots, 1}_{l_2 \times}, 0, \cdots, 0)\enspace. $$
This is possible by the pigeonhole principle since there are $2|C| > C$ integers in the interval and $p/3 - 2|C| > 0$ by the choice of $p$.
The sought after contradiction will be obtained by considering the two matrices $$ X_i = \rrr{ \underbrace{k, \dots, k}_{m \times}, l_i, \dots, l_i}, \ i=1,2 \enspace, $$ where $k$ will be specified soon.
Before choosing $k$, we observe that $t(X_1) = t(X_2)$. Indeed, the first $m$ columns of these matrices are the same (and thus so are their images under $s$) and the remaining columns have the same image under $s$ by the choice of $l_1$ and $l_2$. The claim thus follows from the definition of $t$.
Next, note that for $k \leq p/3$ the area of both matrices is less than $1/3$ since $l_i < p/3$. On the other hand, for $k \geq p/3 + 3|C|$ the area is greater: \begin{align*} \lambda(X_i) &= \frac{mk+(p-m)l_i}{p^2} \\
&\geq \frac{\frac{p-1}{2} (p/3 + 3|C|) + \frac{p+1}{2} (p/3-2|C|)}{p^2} \\
&= \frac{p^2/3 + |C| (p-5)/2}{p^2} > 1/3\enspace. \end{align*}
Choose the maximum $k$ so that $\lambda(X_1) < 1/3$. The derived inequalities and the choice of $l_i$ implies $$
l_1 < l_2 \leq k < p/3 + 3|C| \leq l_1 + 5|C| < l_2 + 5|C| \enspace, $$ therefore both $X_1$ and $X_2$ are almost rectangles. By Proposition~\ref{prop:tame}, $X_1$ and $X_2$ are tame.
Since the area of $X_1$ is less than $1/3$, we get $X_1 \sim 0_{p \times p}$. We chose $k$ so that increasing $k$ by 1 makes the area of $X_1$ greater than $1/3$. From $m < p/2$ it follows that increasing $l_1$ by 1 makes the area even greater, hence $\lambda(X_2) > 1/3$ (recall that $l_2>l_1$) and we obtain $X_2 \sim 1_{p \times p}$.
Recall that $0_{p \times p} \not\sim 1_{p \times p}$ by the second part of Lemma~\ref{lem:lines_tame}. Therefore $X_1 \not\sim X_2$, contradicting $t(X_1) = t(X_2)$.
\section{Conclusion} \label{sec:conclusion}
This paper shows that if $\OneInThree \to \relstr{C} \to \NAE$ and $\relstr{C}$ is finite, then $\CSP(\relstr{C})$ is NP-complete. The proof strategy is based on Theorem~\ref{thm:cyclic} and a simple fact that, given $\relstr{A} \to \relstr{C} \to \relstr{B}$, each polymorphism of $\relstr{C}$ induces a polymorphism of $(\relstr{A},\relstr{B})$ (by composition with the homomorphism $\relstr{A} \to \relstr{C}$ from the inside and with $\relstr{C} \to \relstr{B}$ from the outside).
There is an algebraic sufficient condition for NP-hardness for all $\omega$-categorical structures~\cite{BarP16} -- $\CSP(\relstr{C})$ is NP-hard whenever $\relstr{C}$ does not have a \emph{pseudo-Siggers} polymorphism, that is, a 6-ary polymorphism $s$ such that $$ \alpha s(x,y,x,z,y,z) = \beta s(y,x,z,x,z,y) \mbox{ for all } x,y,z \in C \enspace, $$ where $\alpha$ and $\beta$ are unary polymorphisms of $\relstr{C}$. Is it possible to apply pseudo-Siggers operations to strengthen the main theorem?
\begin{question} Let $\relstr{C}$ be an $\omega$-categorical structure that pp-constructs $(\OneInThree, \NAE)$. Is $\CSP(\relstr{C})$ necessarily NP-hard? \end{question}
The proof of Theorem~\ref{thm:main_fake} could be simplified if we had stronger or more suitable polymorphisms than cyclic operations. Alternative versions of Theorem~\ref{thm:cyclic} could also help in simplifying the proof of the CSP dichotomy conjecture. In particular, the following question seems open. \begin{question} Let $\relstr{C}$ be a finite relational structure with a cyclic polymorphism of arity at least 2. Does $\relstr{C}$ necessarily have a polymorphism $s$ of arity $n > 1$ such that, for any $a,b \in C$ and $(x_1, \dots, x_n) \in \{a,b\}^n$, the value $s(x_1, \dots, x_n)$ depends only on the number of occurrences of $a$ in $(x_1, \dots, x_n)$? \end{question}
Note that a more optimistic version involving evaluations with $|\{x_1, \dots, x_n\}|=3$ is disproved by considering the polymorphisms of the disjoint union of a directed 2-cycle and a directed 3-cycle.
Let us finish with an optimistic outlook. While the main result of this paper is negative, its message is rather positive. It suggests that algebraic and analytical methods in the finite-domain CSP and PCSP should be combined with the model theoretic methods used for the infinite domains, and such a combination promises a significant synergy gain.
\end{document} | arXiv |
\begin{document}
\title{Kolmogorov distance between the exponential functionals of fractional Brownian motion } \author{Nguyen Tien Dung \footnote{Email: dung\[email protected]} }
\date{July 20, 2019}
\maketitle \begin{abstract} In this note, we investigate the continuity in law with respect to the Hurst index of the exponential functional of the fractional Brownian motion. Based on the techniques of Malliavin's calculus, we provide an explicit bound on the Kolmogorov distance between two functionals with different Hurst indexes. \end{abstract} \noindent\emph{Keywords:} Fractional Brownian motion, Exponential functional, Malliavin calculus.\\ {\em 2010 Mathematics Subject Classification:} 60G22, 60H07.
\section{Introduction} Let $B^H=(B^H_t)_{t\in [0,T]}$ be a fractional Brownian motion (fBm) with Hurst index $H\in(0,1).$ We recall that fBm admits the Volterra represention \begin{equation}\label{densityCIR02} B^H_t=\int_0^t K_H(t,s)dW_s, \end{equation} where $(W_t)_{t\in [0,T]}$ is a standard Brownian motion and for some normalizing constants $c_H$ and $c'_H,$ the kernel $K_H$ is given by $ K_{H}(t,s) = c_{H}s^{1/2 -H}\int_{s}^{t}(u-s)^{H-\frac{3}{2}}u^{H-\frac{1}{2}}du$ if $H>\frac{1}{2}$ and $$K_H(t,s)=c_H\bigg[\frac{t^{H-\frac{1}{2}}}{s^{H-\frac{1}{2}}}(t-s)^{H-\frac{1}{2}} -(H-\frac{1}{2})\int\limits_s^t\frac{u^{H-\frac{3}{2}}}{s^{H-\frac{1}{2}}}(u-s)^{H-\frac{1}{2}}du\bigg]\,\,\text{if}\,\,H<\frac{1}{2}.$$
Given real numbers $a$ and $\sigma,$ we consider the exponential functional of the form $$F_H=\int_0^T e^{as+\sigma B^H_s}ds.$$ It is known that this functional plays an important role in several domains. For example, it can be used to investigate the finite-time blowup of positive solutions to semi-linear stochastic partial differential equations \cite{Dozzi2014}. In the special case $H=\frac{1}{2},$ fBm reduces to a standard Brownian motion and a lot of fruitful properties of $F_{\frac{1}{2}}$ can be founded in the literature, see e.g. \cite{Matsumoto2005a,Matsumoto2005b,Pintoux2010,Yor2001}. In particular, the distribution of $F_{\frac{1}{2}}$ can be computed explicitly. However, to the best our knowledge, it remains a challenge to obtain the deep properties of $F_H$ for $H\neq \frac{1}{2}.$
On the other hand, because of its applications in statistical estimators, the problem of proving the continuity in law with respect to $H$ of certain functionals has been studied by several authors. Among others, we refer the reader to \cite{Jolis2010,Koch2019,Richard2017,Saussereau2012} and the references therein for the detailed discussions and the related results. Motivated by this observation, the aim of the present paper is to investigate the continuity in law of the exponential functional $F_H.$ Intuitively, the continuity of $F_H$ with respect to $H$ is not surprising. However, the interesting point of Theorem \ref{tyi3} below is that we are able to give an explicit bound on Komogorov distance between two functionals with different Hurst indexes. \begin{thm}\label{tyi3}For any $H_1,H_2\in(0,1),$ we have \begin{equation}\label{hk1}
\sup\limits_{x\geq 0}|P\left(F_{H_1}\leq x\right)-P\left(F_{H_2}\leq x\right)|\leq C|H_1-H_2|, \end{equation}
where $C$ is a positive constant depending on $a,\sigma,T$ and $H_1,H_2.$ \end{thm}
\section{Proofs} Our main tools are the techniques of Malliavin calculus. Hence, for the reader's convenience, let us recall some elements of Malliavin calculus with respect to Brownian motion $W,$ where $W$ is used to present $B^H$ as in (\ref{densityCIR02}). We suppose that $(W_t)_{t\in [0,T]}$ is defined on a complete probability space $(\Omega,\mathcal{F},\mathbb{F},P)$, where $\mathbb{F}=(\mathcal{F}_t)_{t\in [0,T]}$ is a natural filtration generated by the Brownian motion $W.$ For $h\in L^2[0,T],$ we denote by $W(h)$ the Wiener integral $$W(h)=\int\limits_0^T h(t)dW_t.$$ Let $\mathcal{S}$ denote the dense subset of $L^2(\Omega, \mathcal{F},P)$ consisting of smooth random variables of the form \begin{equation}\label{ro} F=f(W(h_1),...,W(h_n)), \end{equation} where $n\in \mathbb{N}, f\in C_b^\infty(\mathbb{R}^n),h_1,...,h_n\in L^2[0,T].$ If $F$ has the form (\ref{ro}), we define its Malliavin derivative as the process $DF:=\{D_tF, t\in [0,T]\}$ given by $$D_tF=\sum\limits_{k=1}^n \frac{\partial f}{\partial x_k}(W(h_1),...,W(h_n)) h_k(t).$$ We shall denote by $\mathbb{D}^{1,2}$ the closure of $\mathcal{S}$ with respect to the norm
$$\|F\|^2_{1,2}:=E|F|^2+E\bigg[\int\limits_0^T|D_u F|^2du\bigg].$$ An important operator in the Malliavin calculus theory is the divergence operator $\delta,$ it is the adjoint of the derivative operator $D.$ The domain of $\delta$ is the set of all functions $u\in L^2(\Omega,L^2[0,T])$ such that
$$E|\langle DF,u\rangle_{L^2[0,T]}|\leq C(u)\|F\|_{L^2(\Omega)},$$ where $C(u)$ is some positive constant depending on $u.$ In particular, if $u\in Dom\,\delta,$ then $\delta(u)$ is characterized by the following duality relationship $$E\langle DF,u\rangle_{L^2[0,T]}=E[F\delta(u)]\,\,\,\text{for any}\,\,\,F\in \mathbb{D}^{1,2}.$$
In order to be able to prove Theorem \ref{tyi3}, we need two technical lemmas. \begin{lem}\label{tt5l}For any $H\in(0,1),$ we have $F_H\in \mathbb{D}^{1,2}$ and
$$\left(\int_0^T |D_rF_{H}|^2dr\right)^{-1}\in L^p(\Omega),\,\,\forall\,\,p\geq 1.$$ \end{lem} \begin{proof}By the representation (\ref{densityCIR02}), we have $D_rB^H_s=K_H(s,r)$ for $0\leq r<s\leq T.$ Hence, $F_H\in \mathbb{D}^{1,2}$ and its derivative is given by $$D_rF_H=\int_r^T \sigma K_H(s,r)e^{as+\sigma B^H_s}ds,\,\,0\leq r\leq T.$$ So we can deduce
$$D_rF_H\geq e^{-|a| T+\sigma \min\limits_{0\leq s\leq T}B^H_s}\int_r^T \sigma K_H(s,r)ds,\,\,0\leq r\leq T.$$ As a consequence, \begin{align*}
\int_0^T |D_rF_{H}|^2dr&\geq e^{-2|a| T+2\sigma \min\limits_{0\leq s\leq T}B^H_s}\int_0^T\left(\int_r^T \sigma K_H(s,r)ds\right)^2dr\\
&= \sigma^2 e^{-2|a| T+2\sigma \min\limits_{0\leq s\leq T}B^H_s}\int_0^T\left(\int_r^T K_H(s,r)ds\right)\left(\int_r^T K_H(t,r)dt\right)dr\\
&= \sigma^2 e^{-2|a| T+2\sigma \min\limits_{0\leq s\leq T}B^H_s}\int_0^T\int_0^T\left(\int_0^{s\wedge t} K_H(s,r)K_H(t,r)dr\right)dsdt\\
&= \sigma^2 e^{-2|a| T+2\sigma \min\limits_{0\leq s\leq T}B^H_s}\int_0^T\int_0^TE[B^H_sB^H_t]dsdt\\
&=\frac{T^{2H+2}}{2H+2} \sigma^2 e^{-2|a| T+2\sigma \min\limits_{0\leq s\leq T}B^H_s}. \end{align*}
In the last equality we used the fact that $E[B^{H}_{t}B^{H}_{s}] = \frac{1}{2}(t^{2H}+s^{2H} - |t-s|^{2H}).$ We therefore obtain
$$\left(\int_0^T |D_rF_{H}|^2dr\right)^{-1}\leq \frac{2H+2}{T^{2H+2}\sigma^2} e^{2|a| T+2\sigma \max\limits_{0\leq s\leq T}(-B^H_s)}.$$ By Fernique's theorem, we have $e^{2\sigma \max\limits_{0\leq s\leq T}(-B^H_s)}\in L^p(\Omega)$ for any $p\geq 1.$ This completes the proof. \end{proof}
\begin{lem}\label{tt5la}For any $H_1,H_2\in(0,1),$ we have \begin{equation}\label{1t1}
E|F_{H_1}-F_{H_2}|^2\leq C|H_1-H_2|^2, \end{equation} \begin{equation}\label{1t2}
\int_0^T E|D_rF_{H_1}-D_rF_{H_2}|^2dr\leq C|H_1-H_2|^2, \end{equation} where $C$ is a positive constant depending on $a,\sigma,T$ and $H_1,H_2.$ \end{lem} \begin{proof} By the H\"older inequality we have \begin{align*}
E|F_{H_1}-F_{H_2}|^2&=E\big|\int_0^T \big(e^{as+\sigma B^{H_1}_s}-e^{as+\sigma B^{H_2}_s}\big)ds\big|^2\\
&\leq T\int_0^T E\big|e^{as+\sigma B^{H_1}_s}-e^{as+\sigma B^{H_2}_s}\big|^2ds. \end{align*}
Using the fundamental inequality $|e^x-e^y|\leq \frac{1}{2} |x-y|(e^x+e^y)$ for all $x,y$ we deduce \begin{align*}
E|F_{H_1}-F_{H_2}|^2&\leq \frac{T}{4}\int_0^T E\big|(\sigma B^{H_1}_s-\sigma B^{H_2}_s)(e^{as+\sigma B^{H_1}_s}+e^{as+\sigma B^{H_2}_s})\big|^2ds\\
&\leq \frac{T\sigma^2}{4}\int_0^T \left(E|B^{H_1}_s-B^{H_2}_s|^4\right)^{\frac{1}{2}}\left(E|e^{as+\sigma B^{H_1}_s}+e^{as+\sigma B^{H_2}_s}|^4\right)^{\frac{1}{2}}ds\\
&\leq \frac{T\sigma^2}{4}\int_0^T \left(E|B^{H_1}_s-B^{H_2}_s|^4\right)^{\frac{1}{2}}\left(8E[e^{4as+4\sigma B^{H_1}_s}]+8E[e^{4as+4\sigma B^{H_2}_s}]\right)^{\frac{1}{2}}ds\\
&= \frac{T\sigma^2}{4}\int_0^T \left(E|B^{H_1}_s-B^{H_2}_s|^4\right)^{\frac{1}{2}}\left(8e^{4as+8\sigma^2 s^{2H_1}}+8e^{4as+8\sigma^2 s^{2H_2} }\right)^{\frac{1}{2}}ds\\
&\leq\frac{T\sigma^2}{4}\int_0^T \left(E|B^{H_1}_s-B^{H_2}_s|^4\right)^{\frac{1}{2}}\left(8e^{4as+8\sigma^2 s^{2H_1}}+8e^{4as+8\sigma^2 s^{2H_2} }\right)^{\frac{1}{2}}ds. \end{align*} It is known from the proof of Theorem 4 in \cite{Peltier1995} that there exists a positive constant $C$ such that \begin{equation}\label{1t3}
\sup\limits_{0\leq s\leq T}E|B^{H_1}_s-B^{H_2}_s|^2\leq C|H_1-H_2|^2. \end{equation}
On the other hand, we have $E|B^{H_1}_s-B^{H_2}_s|^4=3(E|B^{H_1}_s-B^{H_2}_s|^2)^2$ because $B^{H_1}_s-B^{H_2}_s$ is a Gaussian random variable for every $s\in[0,T].$ So we can conclude that there exists a positive constant $C$ such that
$$E|F_{H_1}-F_{H_2}|^2\leq C|H_1-H_2|^2.$$
To finish the proof, let us verify (\ref{1t2}). By the H\"older and triangle inequalities we obtain \begin{align*}
&E|D_rF_{H_1}-D_rF_{H_2}|^2\leq \sigma^2T \int_r^TE\big|K_{H_1}(s,r)e^{as+\sigma B^{H_1}_s}-K_{H_2}(s,r)e^{as+\sigma B^{H_2}_s}\big|^2ds\\
&\leq 2\sigma^2T \int_r^T|K_{H_1}(s,r)-K_{H_2}(s,r)|^2E[e^{2as+2\sigma B^{H_1}_s}]+K^2_{H_2}(s,r)E\big|e^{2as+2\sigma B^{H_1}_s}-e^{as+\sigma B^{H_2}_s}\big|^2ds, \end{align*} and hence, \begin{align*}
\int_0^TE|D_rF_{H_1}-D_rF_{H_2}|^2dr&\leq 2\sigma^2T \int_0^TE[e^{2as+2\sigma B^{H_1}_s}]\int_0^s|K_{H_1}(s,r)-K_{H_2}(s,r)|^2drds\\
&+2\sigma^2T \int_0^TE\big|e^{2as+2\sigma B^{H_1}_s}-e^{as+\sigma B^{H_2}_s}\big|^2\int_0^sK^2_{H_2}(s,r)drds\\
&= 2\sigma^2T \int_0^Te^{2as+2\sigma^2 s^{2H_1}}E|B^{H_1}_s-B^{H_2}_s|^2ds\\
&+2\sigma^2T \int_0^TE\big|e^{2as+2\sigma B^{H_1}_s}-e^{as+\sigma B^{H_2}_s}\big|^2s^{2H_2}ds. \end{align*}
Notice that $\int_0^sK^2_{H_2}(s,r)dr=E|B^{H_2}_s|^2=s^{2H_2}.$ Thus the estimate (\ref{1t2}) follows from (\ref{1t1}) and (\ref{1t3}). \end{proof} {\bf Proof of Theorem \ref{tyi3}.} For the simplicity, we write $\langle ., .\rangle$ instead of $\langle ., .\rangle_{L^2[0,T]}.$ Borrowing the arguments used in the proof of Proposition 2.1.1 in \cite{nualartm2}, we let $\psi$ be a nonnegative smooth function with compact support, and set $\varphi(y)=\int_{-\infty}^y\psi(z)dz.$ Given $Z\in \mathbb{D}^{1,2},$ we know that $\varphi(Z)$ belongs to $\mathbb{D}^{1,2}$ and making the scalar product of its derivative with $DF_{H_2}$ obtains $$\langle D\varphi(Z), DF_{H_2}\rangle=\psi(Z)\langle DZ, DF_{H_2}\rangle.$$ Fixed $x\in \mathbb{R}_+,$ by an approximation argument, the above equation holds for $\psi(z)={\rm 1\hspace{-0.90ex}1}_{[0,x]}(z).$ Choosing $Z=F_{H_1}$ and $Z=F_{H_2}$ we obtain $$\langle D\int_{-\infty}^{F_{H_1}}{\rm 1\hspace{-0.90ex}1}_{[0,x]}(z)dz, DF_{H_2}\rangle={\rm 1\hspace{-0.90ex}1}_{[0,x]}(F_{H_1})\langle DF_{H_1}, DF_{H_2}\rangle,$$ $$\langle D\int_{-\infty}^{F_{H_2}}{\rm 1\hspace{-0.90ex}1}_{[0,x]}(z)dz, DF_{H_2}\rangle={\rm 1\hspace{-0.90ex}1}_{[0,x]}(F_{H_2})\langle DF_{H_2}, DF_{H_2}\rangle.$$ Hence, we can get \begin{align*} &\langle D\int_{F_{H_2}}^{F_{H_1}}{\rm 1\hspace{-0.90ex}1}_{[0,x]}(z)dz, DF_{H_2}\rangle={\rm 1\hspace{-0.90ex}1}_{[0,x]}(F_{H_1})\langle DF_{H_1}, DF_{H_2}\rangle-{\rm 1\hspace{-0.90ex}1}_{[0,x]}(F_{H_2})\langle DF_{H_2}, DF_{H_2}\rangle\\ &=\left({\rm 1\hspace{-0.90ex}1}_{[0,x]}(F_{H_1})-{\rm 1\hspace{-0.90ex}1}_{[0,x]}(F_{H_2})\right)\langle DF_{H_2}, DF_{H_2}\rangle+{\rm 1\hspace{-0.90ex}1}_{[0,x]}(F_{H_1})\langle DF_{H_1}-DF_{H_2}, DF_{H_2}\rangle. \end{align*}
This, together with the fact that $\|DF_{H_2}\|^2:=\langle DF_{H_2}, DF_{H_2}\rangle>0\,\,a.s.$ gives us \begin{align*}
{\rm 1\hspace{-0.90ex}1}_{[0,x]}(F_{H_1})-{\rm 1\hspace{-0.90ex}1}_{[0,x]}(F_{H_2})&=\frac{\langle D\int_{F_{H_2}}^{F_{H_1}}{\rm 1\hspace{-0.90ex}1}_{[0,x]}(z)dz, DF_{H_2}\rangle}{\|DF_{H_2}\|^2}-\frac{{\rm 1\hspace{-0.90ex}1}_{[0,x]}(F_{H_1})\langle DF_{H_1}-DF_{H_2}, DF_{H_2}\rangle}{\|DF_{H_2}\|^2}. \end{align*} Taking the expectation yields \begin{align*} P\left(F_{H_1}\leq x\right)&-P\left(F_{H_2}\leq x\right)=E[{\rm 1\hspace{-0.90ex}1}_{[0,x]}(F_{H_1})-{\rm 1\hspace{-0.90ex}1}_{[0,x]}(F_{H_2})]\\
&=E\left[\int_{F_{H_2}}^{F_{H_1}}{\rm 1\hspace{-0.90ex}1}_{[0,x]}(z)dz\delta\left(\frac{DF_{H_2}}{\|DF_{H_2}\|^2}\right)\right]-E\left[\frac{{\rm 1\hspace{-0.90ex}1}_{[0,x]}(F_{H_1})\langle DF_{H_1}-DF_{H_2}, DF_{H_2}\rangle}{\|DF_{H_2}\|^2}\right] \end{align*} By the H\"older inequality \begin{align*}
\sup\limits_{x\geq 0}|P\left(F_{H_1}\leq x\right)&-P\left(F_{H_2}\leq x\right)| \leq E\bigg|(F_{H_1}-F_{H_2})\delta\left(\frac{DF_{H_2}}{\|DF_{H_2}\|^2}\right)\bigg|+E\bigg|\frac{\langle DF_{H_1}-DF_{H_2}, DF_{H_2}\rangle}{\|DF_{H_2}\|^2}\bigg|\\
&\leq \left(E|F_{H_1}-F_{H_2}|^2\right)^{\frac{1}{2}}\left(E\delta\left(\frac{DF_{H_2}}{\|DF_{H_2}\|^2}\right)^2\right)^{\frac{1}{2}}+E\bigg|\frac{\|
DF_{H_1}-DF_{H_2}\|}{\|DF_{H_2}\|}\bigg|\\
&\leq \left(E|F_{H_1}-F_{H_2}|^2\right)^{\frac{1}{2}}\left(E\delta\left(\frac{DF_{H_2}}{\|DF_{H_2}\|^2}\right)^2\right)^{\frac{1}{2}}+\left(E\|
DF_{H_1}-DF_{H_2}\|^2\right)^{\frac{1}{2}}\left(E\bigg[\frac{1}{\|DF_{H_2}\|^2}\bigg]\right)^{\frac{1}{2}}. \end{align*}
Recalling Lemma \ref{tt5la}, we obtain $$\sup\limits_{x\geq 0}|P\left(F_{H_1}\leq x\right)-P\left(F_{H_2}\leq x\right)|\leq C|H_1-H_2|\left[\left(E\delta\left(\frac{DF_{H_2}}{\|DF_{H_2}\|^2}\right)^2\right)^{\frac{1}{2}}
+\left(E\bigg[\frac{1}{\|DF_{H_2}\|^2}\bigg]\right)^{\frac{1}{2}}\right].$$ Thanks to Lemma \ref{tt5l} we have
$$E\bigg[\frac{1}{\|DF_{H_2}\|^2}\bigg]=E\bigg[\left(\int_0^T |D_rF_{H_2}|^2dr\right)^{-1}\bigg]<\infty.$$ Thus we can obtain (\ref{hk1}) by checking the finiteness of $E[\delta(u)^2],$ where
$$u_r:=\frac{D_rF_{H_2}}{\|DF_{H_2}\|^2},\,\,0\leq r\leq T.$$ It is known from Proposition 1.3.1 in \cite{nualartm2} that
$$E[\delta(u)^2]\leq \int_0^TE|u_r|^2dr+\int_0^T\int_0^T E|D_\theta u_r|^2d\theta dr.$$ We have
$$\int_0^TE|u_r|^2dr=E\bigg[\frac{1}{\|DF_{H_2}\|^2}\bigg]<\infty.$$ Furthermore, by the chain rule for Malliavin derivative, we have
$$D_\theta u_r=\frac{D_\theta D_rF_{H_2}}{\|DF_{H_2}\|^2}-2\frac{D_rF_{H_2}\langle D_\theta DF_{H_2},DF_{H_2}\rangle}{\|DF_{H_2}\|^4},\,\,0\leq \theta\leq T.$$ Hence, by the H\"older inequality,
\begin{align*}\int_0^T\int_0^T E|D_\theta u_r|^2d\theta dr&\leq 2E\left[\frac{\int_0^T\int_0^T
|D_\theta D_rF_{H_2}|^2d\theta dr}{\|DF_{H_2}\|^4}\right]+8E\left[\frac{\int_0^T\int_0^T |D_\theta D_rF_{H_2}|^2d\theta dr}{\|DF_{H_2}\|^4}\right]\\
&\leq 10\left(E\left[\left(\int_0^T\int_0^T |D_\theta D_rF_{H_2}|^2d\theta dr\right)^2\right]\right)^{\frac{1}{2}}\left(E\left[\frac{1}{\|DF_{H_2}\|^8}\right]\right)^{\frac{1}{2}}. \end{align*} We now observe that $$D_\theta D_rF_{H_2}=\int_{r\vee\theta}^T \sigma^2 K_{H_2}(s,r)K_{H_2}(s,\theta)e^{as+\sigma B^{H_2}_s}ds,\,\,0\leq r,\theta\leq T.$$ Hence,
$$|D_\theta D_rF_{H_2}|^2\leq T \sigma^4\int_{r\vee\theta}^T K^2_{H_2}(s,r)K^2_{H_2}(s,\theta)e^{2as+2\sigma B^{H_2}_s}ds,\,\,0\leq r,\theta\leq T$$ and we obtain
$$\int_0^T\int_0^T |D_\theta D_rF_{H_2}|^2d\theta dr\leq T \sigma^4\int_{0}^T s^{4H_2}e^{2as+2\sigma B^{H_2}_s}ds,$$ which implies that
$$E\left[\left(\int_0^T\int_0^T |D_\theta D_rF_{H_2}|^2d\theta dr\right)^2\right]\leq T^4 \sigma^8\int_{0}^T s^{8H_2}e^{4as+8\sigma^2 s^{2H_2}}ds<\infty.$$
Finally, we have $E\left[\frac{1}{\|DF_{H_2}\|^8}\right]<\infty$ due to Lemma \ref{tt5l}. So we can conclude that $E[\delta(u)^2]$ is finite. This finishes the proof of Theorem \ref{tyi3}. \begin{rem}Given a bounded and continuous function $\psi,$ with the exact proof of Theorem \ref{tyi3}, we also have
$$|E[\psi(F_{H_1})]-E[\psi(F_{H_2})]|\leq C|H_1-H_2|.$$ This kind of estimates has been investigated by Richard and Talay for the solution of fractional stochastic differential equations. However, Theorem 1.1 in \cite{Richard2017} requires $H_2=\frac{1}{2}$ and $\psi$ to be H\"older continuous of order $2+\beta$ with $\beta>0.$ \end{rem}
\noindent {\bf Acknowledgments.} This research was funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.03-2019.08.
\end{document} | arXiv |
What do these labels for molecular electronic states mean?
What do these symbols mean in excited states?
$${}^2\!A_2,{}^2\!B_1,{}^4\!A''$$
I am confused with these representations, found in the abstract of this paper. I think it is kind of a representation of electronic excited states. The left superscript means spin multiplicity ($2S+1$). What do the other parts of these symbols mean, and how can I write these symbols with other molecules?
physical-chemistry electronic-configuration notation
orthocresol♦
Chao SongChao Song
As stated in this answer, these are irrep (irreducible representation) labels for molecular symmetry point groups. In the context of chemistry, point groups are usually introduced when learning about structural symmetry (atoms and bonds). This is a broad topic with many technical points, too many for a single answer, so I won't cover the basics but hopefully the few things needed to make the connection between what you read in a book and what's present in this paper.
The key is recognizing that the structural symmetry of molecules leads to symmetry in their molecular orbitals. Take the water molecule as an example. Water's molecular structure can be described at best by the $C_{2v}$ point group, so we say it has $C_{2v}$ symmetry. Composing its molecular orbitals from atomic orbitals is actually an exercise in symmetry, as only atomic orbitals of identical symmetry may combine. Here is the LCAO-MO diagram for water, with the oxygen 1s orbital left out due to its extremely low energy; it won't mix with the hydrogen AOs.
Here is the character table for the $C_{2v}$ point group.
AOs of identical symmetry (they transform identically for any given symmetry operation) are shown in the same color. A few things are worth pointing out:
s orbitals are perfectly symmetric (a sphere), so they will always transform as the totally symmetric irrep of a molecular point group. In $C_{2v}$, that's $A_1$. Once you learn what the operations $E$, $C_{2}$, $\sigma_{v}(xz)$, and $\sigma_{v}(yz)$ do, you can convince yourself that the $\sigma$ combination of $\ce{H2}$ AOs is also completely symmetric under those operations, so it also belongs to the $A_1$ irrep. The p orbitals are figured out from the 3rd column, d from the 4th.
The 2s and $\sigma$ AOs combine to give the $3a_1$ MO. The "3" is because it's the 3rd such MO with $a_1$ symmetry. The symmetry label is in lowercase because it for a 1-electron state. Symmetry labels are in uppercase for many-electron states, such as the total wavefunction, as you see in the linked paper. This is a matter of convention, nothing more.
Neutral water is a closed-shell molecule; there aren't any unpaired electrons, so it has singlet spin multiplicity. How do you determine the symmetry of the total wavefunction? A Slater determinant can help, or even just the Hartree product part of it. Take a fictitious system with 4 electrons in 2 orbitals. The Hartree product will look something like
$$ \Psi_{\text{Hartree}} = [\phi_{1}(1)\alpha(1)][\phi_{1}(2)\beta(2)][\phi_{2}(3)\alpha(3)][\phi_{2}(4)\beta(4)] $$
This is a product wavefunction that might be made from an electron configuration such as $1s^{2}2s^{2}$, $1s^{2}2p_{x}^{2}$, $4p_{x}^{2}4p_{z}^{2}$. Each of these MOs has a certain symmetry; the total symmetry is given by the product of the individual symmetries. For this step one needs the point group product table (again for $C_{2v}$):
$$ \begin{array}{|l|l|l|l|l|} \hline & \mathbf{A_1} & \mathbf{A_2} & \mathbf{B_1} & \mathbf{B_2} \\ \hline \mathbf{A_1} & A_1 & A_2 & B_1 & B_2 \\ \hline \mathbf{A_2} & A_2 & A_1 & B_2 & A_1 \\ \hline \mathbf{B_1} & B_1 & B_2 & A_1 & A_2 \\ \hline \mathbf{B_2} & B_2 & B_1 & A_2 & A_1 \\ \hline \end{array} $$
If the electron configuration of the system is indeed $4p_{x}^{2}4p_{z}^{2}$, this results in a total wavefunction symmetry of
$$ B_{1} \times B_{1} \times A_{1} \times A_{1} = A_{1} $$
The spin components don't contribute, only the spatial components. Note that in any product table, the product of any irrep with itself is always the totally symmetric representation in that group, hence the diagonal being all $A_{1}$. Product tables are also symmetric, since irreps are commutative under multiplication. Since this is a singlet, the ground-state wavefunction can be specified as $^{1}\!A_{1}$. For a closed-shell system (RHF), due to the pairs of electrons always leading to $A_{1}$ symmetry, it will always have a total state symmetry of $^{1}\!A_{1}$. If you have an open-shell system of any kind in a ground or excited state, with either an even or odd number of electrons, the total wavefunction symmetry will be determined by electrons in those orbitals alone, since the product between fully-occupied orbitals will always be of $A_{1}$. In the simplest case of one unpaired electron, the total symmetry of the ground state will be the symmetry of the SOMO.
Now, on to the paper. The most important part for our understanding of the state label assignments is Table 3:
All the systems considered have either $C_{2v}$ or $C_{s}$ symmetry. Note how going from a point group of higher symmetry ($C_{2v}$, 4 irreps) to one of lower symmetry ($C_{s}$, 2 irreps) leads to states collapsing into fewer irreps due to fewer symmetry operations that can distinguish them. For the plots (here the last row of Figure 6),
there is the isosurface of the molecular orbital (really a natural orbital), the orbital number and symmetry, and the natural orbital occupation number (NOON) for that NO in that state.
There are a number of good Dover publications (1, 2, 3) for learning about group theory in chemistry and spectroscopy that are highly recommended, since the primary peer-reviewed literature isn't usually a good resource for learning this. An excellent resource for character tables is WebQC.
pentavalentcarbonpentavalentcarbon
To fully understand the meaning of these symbols, you need to understand molecular symmetry groups. If you know the point group symmetry of your molecule, you can construct a character table comprised of all possible symmetry operation (rotation around an axis, reflection in a plane, inversion) and the effect of these operation on a certain electronic state (+1 for unchanged, -1 for sign change for example). A row of such a table is called an irreducible representation (irrep) and is used as a label defining a certain symmetry state. The notation follows the rules set out by Mulliken in , J Chem. Phys. 23, 1997 (1955) and have the following general meaning:
A is used when the irrep is symmetric under $C_n$ or $S_n$ for the highest $n$ in the group, in addition A is used if there are no $C_n$ or $S_n$ (where $C_n$ is a rotation by 2$\pi/n$ around an $n$-fold symmetry axis and $S_n$ is also a rotation, but followed by a reflection).
B is used when the irrep is antisymmetric under $C_n$ or $S_n$ for the highest $n$ in the group.
E is a doubly degenerate state
T is a triply degenerate.
For degenerate irreps (A and B) subscripts 1 and 2 relate to the symmetric (1) or antisymmetric (-1) characters respectively, in relation to a $C_2$ axis perpendicular to the principle $C_n$ axis, or in the absence of this element, to a vertical symmetry plane ($\sigma_v$) plane. For multidimensional representations, the subscripts 1, 2, etc are added to distinguish between nonequivalent irreducible representations that are not separated under the above rules.
Single (') and double ('') primes are used when the irrep is symmetric (+1) under reflection in a horizontal mirror plane or antisymmetric in this plane (-1), provided that the molecule has no inversion symmetry.
PaulPaul
$\begingroup$ I think these symbols represents electronic excited states according to this paper. Maybe your answer is about the first column of the character table? I think these symbols can specify the electron state. $\endgroup$
– Chao Song
$\begingroup$ Exactly, they specify the symmetry of the different electronic states. $\endgroup$
– Paul
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Feasibility and Entropy | CommonCrawl |
\begin{document}
\begin{abstract}
We construct a family of inequivalent Calabi-Yau metrics on
$\mathbf{C}^3$ asymptotic to $\mathbf{C} \times A_2$ at infinity, in
the sense that any two of these metrics cannot be related by a
scaling and a biholomorphism. This provides the first example of
families of Calabi-Yau metrics asymptotic to a fixed tangent cone at
infinity, while keeping the underlying complex structure fixed. We
propose a refinement of a conjecture of Sz\'ekelyhidi~\cite{Sz20}
addressing the classification of such metrics. \end{abstract}
\maketitle
\section{Introduction}
Since the celebrated work of Yau~\cite{Yau}, Calabi-Yau manifolds have been studied intensively in K\"ahler geometry, complex algebraic geometry and physics. In the complete non-compact case, much has been known in $2$ complex dimensions since the foundational works of Kronheinmer~\cite{K1}\cite{K2} (see for example \cite{CC15}\cite{CC19}\cite{CC21}\cite{SZ} and the references therein). In higher dimensions, Conlon-Hein~\cite{CH} recently classified asymptotically conical Calabi-Yau manifolds, building on the important work of Tian-Yau~\cite{TY}.
In this paper, we are interested in Calabi-Yau manifolds with maximal volume growth, which include asymptotically conical manifolds. In this more general setting, the tangent cones at infinity are still Calabi-Yau cones. However, in general these cones can have non-isolated singularities. Many examples of Calabi-Yau manifolds with maximal volume growth and singular tangent cones at infinity have been constructed over the years. Biquard-Gauduchon~\cite{BG} constructed hyperk\"ahler metrics on cotangent bundles of certain hermitian symmetric spaces, whose tangent cones are realized as nilpotent orbit closures in $sl(N, \mathbf{C})$. Joyce~\cite{Joyce} constructed QALE metrics as resolutions of $\mathbf{C}^n/ \Gamma$, where the action of the discrete group $\Gamma$ is not free. This approach has been generalized by Conlon-Degeratu-Rochon~\cite{CDR} to admit more complicated singularities. More recently, Conlon-Rochon~\cite{CR}, Li~\cite{Li} and Sz\'ekelyhidi~\cite{Sz19} constructed Calabi-Yau metrics on $\mathbf{C}^3$ with tangent cone given by $\mathbf{C} \times A_1$ at infinity. Here $A_1$ denotes the singular hypersurface given by $ \{x_1^2 + x_2^2 + x_3^2 = 0 \} \subset \mathbf{C}^3$ equipped with the flat cone metric. We remark that in \cite{CR} and \cite{Sz19}, there are various generalizations in higher dimensions that admit tangent cones of the form $\mathbf{C} \times V$, where $V$ is a Calabi-Yau cone with an isolated singularity at the vertex.
The classification of Calabi-Yau manifolds with maximal volume growth is still largely an uncharted territory. To begin, it is expected that the tangent cones at infinity are unique, as they are affine varieties \cite{LSz}. Therefore one might to try to classify Calabi-Yau manifolds asymptotic to a certain tangent cone at infinity. A recent breakthrough that fits into this picture is due to Sz\'ekelyhidi~\cite{Sz20}, who showed that the Calabi-Yau metric on $\mathbf{C}^n$ asymptotic to $\mathbf{C} \times A_1$ is unique up to scaling and biholomorphism. Their method is to compare the unknown metric to scalings of a model metric using better and better holomorphic gauges. These gauges are given by adapted sequences of bases in Donaldson-Sun theory~\cite{DS17} in combination with certain automorphisms of the cone at infinity. The next simplest case is to study Calabi-Yau metrics on $\mathbf{C}^3$ asymptotic to $\mathbf{C} \times A_2$ at infinity, where $A_2$ is the singular hypersurface given by $\{ x_1^2 + x_2^2 + x_3^3 = 0 \} \subset \mathbf{C}^3$. An example of such a metric has been obtained by Sz\'ekelyhidi in \cite{Sz19}.
To state our result, we recall the following setup originally considered in \cite{Sz19}. Consider the hypersurface $X_1 \subset \mathbf{C}^{n+1}$ given by the equation \[ z + f(x_1, \ldots, x_n) = 0, \end{aligned} \end{equation} where $f: \mathbf{C}^n \to \mathbf{C}$ is a polynomial, so $X_1$ is biholomorphic to $\mathbf{C}^n$. Write $\mathbf{x} = (x_1,\ldots, x_n)$.
\theoremstyle{definition} \newtheorem{setup}[thm]{Setup} \begin{setup}
\label{setup}
We impose the following restrictions
on $f$:
\begin{itemize}
\item $x_i$ has weight $w_i > 0$ under the action of $t \in \mathbf{C}^*$:
\[
t \cdot x_i = t^{w_i}x_i.
\end{aligned} \end{equation}
\item $f$ is homogeneous of degree $d > 1$:
\[
t \cdot f(\mathbf{x}) = f(t \cdot \mathbf{x}) = t^d f(\mathbf{x}).
\end{aligned} \end{equation}
\item $V_0 = f^{-1}(0) \subset \mathbf{C}^n$ has an isolated singularity at
$0 \in \mathbf{C}^n$.
\item $V_0$ admits a Calabi-Yau cone metric $\omega_{V_0}$
compatible with the $\mathbf{C}^*$ action.
\end{itemize}
\end{setup}
Suppose that we are in the above setup. Let $V_1 = \{ 1+ f(\mathbf{x}) = 0 \} \subset \mathbf{C}^n$. Then $V_1$ admits by \cite{CH13} a unique asymptotically conical Calabi-Yau metric $\omega_{V_1}$ with asymptotic cone $V_0$ (see Section~\ref{sec:x1} for the precise meaning of uniqueness).
We would like to degenerate $X_1$ to its ``tangent cone at infinity'': let us define a $\mathbf{C}^*$ action on $\mathbf{C}^{n+1}$ given by $F_t(z,\mathbf{x}) = (tz, t\cdot \mathbf{x})$. Then $F_t^{-1}X_1$ has the equation \[ t^{1-d}z + f(\mathbf{x}) = 0. \end{aligned} \end{equation} Since $d > 1$, as $t \to \infty$, $F_t^{-1}X_1 \to X_0$, where \[
X_0 = \mathbf{C} \times V_0 \end{aligned} \end{equation} is equipped with the Calabi-Yau cone metric
$\omega_0 = \sqrt{-1}\partial\bar\partial |z|^2 + \omega_{V_0}$. This fits into the framework of Donaldson-Sun theory~\cite{DS17} (see also \cite{Liu} for the case when the tangent cone at infinity is smooth but the manifold is not necessarily polarized). In \cite{Sz19}, Sz\'ekelyhidi constructed a Calabi-Yau metric on $X_1$ asymptotic to $X_0$ at infinity. From the fibration point of view, the map $z: X_1 \to \mathbf{C}$ has regular fibers biholomorphic to $V_1$, and the central fiber is given by $V_0$. Roughly speaking, the metric on $X_1$ can be seen as a perturbation of the ``semi-Ricci-flat'' metric which restricts to scalings of $\omega_{V_1}$ on the regular fibers and $\omega_{V_0}$ on the central fiber.
In this paper, we restrict to the case when $n=3$. Set $f = x_1^2 + x_2^2 + y^3$, where we write $y = x_3$, so $V_0$ is the $A_2$ singularity. Recall that $V_0 \cong \mathbf{C}^3/\mathbf{Z}_3$ is equipped with the flat cone metric. The variables $z, x_1, x_2, y$ have weights $1,3,3,2$, respectively, and so $d = 6$ (see Example~\ref{exmp:a2} for more details). We consider the hypersurface $X_{1,b} \subset \mathbf{C}^4$ given by \[
z + b y + x_1^2 + x_2^2 + y^3 = 0, \end{aligned} \end{equation} where $b \in \mathbf{C}$. Under the $\mathbf{C}^*$ action $F_t$, $X_{1,b}$ still degenerates to $X_0$. However, the fibration structure is different from $X_1$ considered in \cite{Sz19} when $b \ne 0$: there are now two singular fibers, each of which has one $A_1$ singularity. For each $b \in \mathbf{C}$, we construct Calabi-Yau metrics on $X_{1,b}$ asymptotic to $X_0$. We then distinguish these metrics using certain normalization of holomorphic functions with polynomial growth. As a consequence, we obtain the main theorem of this paper:
\begin{thm} \label{thm:A2} There exists a family of Calabi-Yau metrics $\omega_b$, $b \in [0,\infty)$, on $\mathbf{C}^3$ with tangent cone $\mathbf{C} \times A_2$ at infinity. Any $\omega_b$ and $\omega_{b'}$ are related by a biholomorphism and a scaling if and only if $b = b'$. \end{thm}
One way to understand this phenomenon of nonuniqueness is that these metrics should correspond to different ways to smooth out the $A_2$ singularity. In particular, each $X_{1,b}$ has a distinct fibration structure, with distinct singular fiber positions and singularity types.
In Sections \ref{sec:x1} and \ref{sec:x1b}, we describe our construction of $\omega_b$ by a gluing technique similar to the one used in \cite{Sz19}. The main difference in our case is that the fibration is more complicated, and as a result the approximate solution is not obvious to write down. A crucial observation is that in our case, away from the singular fibers and the origin, the metric should still be modeled on either $\mathbf{C} \times V_0$ or $\mathbf{C} \times V_1$ depending on the regions. This allows us to write down an approximate solution on $X_{1,b}$ using the approximate solution on $X_1$ and the nearest point projection from $X_1$ to $X_{1,b}$ (outside large compact sets) with respect to a certain cone metric on the ambient $\mathbf{C}^4$.
In Section~\ref{sec:ds}, we describe our method for distinguishing these metrics, and conclude the proof of Theorem~\ref{thm:A2}. In particular, we generalize the application of Donaldson-Sun theory~\cite{DS17} as seen in \cite{Sz20} to construct special embeddings of Calabi-Yau metrics on $\mathbf{C}^3$ with tangent cone $\mathbf{C} \times A_2$ at infinity. We also obtain a normalization of holomorphic functions from the gluing construction in the previous sections. Our method for distinguishing these metrics is then a combination these results. At the end of this paper, we propose a refinement of a conjecture of Sz\'ekelyhidi~\cite{Sz20}, and discuss preliminary results as well as some difficulties that arise in this setting. \newline
\noindent{\bf Acknowledgments.} I would like to thank G\'abor Sz\'ekelyhidi for the encouragement and constant support over the years. Thanks also to Lorenzo Foscolo and Yang Li for helpful discussions. I was supported by Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics (\#724071 Jason Lotay).
\section{Weighted analysis on $X_1$}\label{sec:x1}
In this section, we explain mostly without proofs the construction of the approximate solution on $X_1$, as well as the weighted analysis in \cite{Sz19}. We will however give a detailed proof of Proposition~\ref{tangentconeisx0} below, since a consequence of its proof is a normalization of the holomorphic functions with respect to the approximate metric (see Corollary~\ref{cor:asymptotics}). This will be used in Section~\ref{sec:ds}.
\subsection{The approximate solution}
We work in Setup~\ref{setup}. Recall in \cite{Sz19} that there is a cone metric $\sqrt{-1}\partial\bar\partial R^2$ on $\mathbf{C}^n$, compatible with the $\mathbf{C}^*$ action, such that the radial function $R$, when restricting to $V_0$, is uniformly equivalent to the distance function $r$ on $V_0$. Using $\sqrt{-1}\partial\bar\partial R^2$, we can extend $r$ homogeneously to a function, also called $r$, on $\mathbf{C}^n$. $\sqrt{-1}\partial\bar\partial r^2$ defines a K\"ahler metric on $V_1$ (away from a large compact set) which is asymptotic to the Calabi-Yau cone $V_0$ under the nearest point projection. By \cite[Theorem~2.4]{CH13} and \cite[Theorem~3.1]{CH13}, there exists a unique complete Calabi-Yau metric $\sqrt{-1}\partial\bar\partial \phi$ on $V_1$ asymptotic to $\sqrt{-1}\partial\bar\partial r^2$. In particular, $(V_1, \sqrt{-1}\partial\bar\partial \phi)$ is asymptotically conical with cone $V_0$.
On $\mathbf{C}^{n+1}$, define $\rho^2 = |z|^2 + R^2$. This gives a cone metric on $\mathbf{C}^{n+1}$ compatible with the $\mathbf{C}^*$ action.
Let $\gamma_1(s)$ be a cutoff function satisfying \begin{align*} \gamma_1(s) = \begin{cases} 1 & \text{if }s > 2 \\ 0 & \text{if }s < 1. \end{cases} \end{align*} and let $\gamma_2 = 1-\gamma_1$. Define the approximate solution, at least for $\rho > P$ for sufficiently large $P > 0$, by \begin{align*}
\omega = \partial\bar\partial\left(|z|^2 + \gamma_1(R\rho^{-\alpha})r^2 +
\gamma_2(R\rho^{-\alpha})|z|^{2/d}\phi(z^{-1/d}\cdot \mathbf{x})\right), \end{align*} where $\alpha \in (1/d, 1)$ is to be chosen later. Writing $\psi = \phi-r^2$, we can rewrite $\omega$ as \begin{align*}
\omega = \partial\bar\partial\left( |z|^2 + r^2 +
\gamma_2(R\rho^{-\alpha})|z|^{2/d}\psi(z^{-1/d} \cdot \mathbf{x}) \right). \end{align*} So the potential of $\omega$ grows like $\rho^2$. In particular if $\omega$ is positive definite on $\rho > P$, then we can replace $\omega$ by a metric on $X_1$ that agrees with $\omega$ on $\rho > 2P$.
The following shows that for large enough $P$, $\omega$ defines a K\"ahler metric, and the Ricci potential has good enough decay.
\begin{prop} \label{prop:adecay} Fix $\alpha \in (1/d,1)$. The form $\omega$ defines a K\"ahler metric on the subset of $X_1$ where $\rho > P$, for sufficiently large $P$. For suitable constants $\kappa, C_i >0$ and weight $\delta < 2/d$, the Ricci potential $h$ of $\omega$ satisfies, for large $\rho$, \begin{align*}
|\nabla^i h|_\omega < \begin{cases} C_i \rho^{\delta-2-i} & \text{if } R > \kappa\rho \\ C_i \rho^{\delta}R^{-2-i} & \text{if } R \in (\kappa^{-1}\rho^{1/d}, \kappa \rho) \\ C_i \rho^{\delta-2/d-i/d} & \text{if } R < \kappa^{-1}\rho^{1/d}. \end{cases} \end{align*} If in addition $d > 3$ and $\alpha$ is chosen close to $1$, then we can even choose $\delta < 0$, i.e. in this case $h$ decays faster than quadratically away from the singular rays. \end{prop}
Since $\omega$ defines a K\"ahler metric on $X_1 \cap \{ \rho > P \}$, one can modify the K\"ahler potential so that the new metric is defined on $X_1$ and coincides with $\omega$ on $X_1 \cap \{ \rho > 2P \}$, say. This can be done for example using the ``regularized maximum'' as described in \cite[p.2659]{Sz19}. We fix a modification of $\omega$ and still call it $\omega$ in the following.
\subsection{Weighted spaces and tangent cones}
We turn to the definition of weighted spaces. The definition will account for model geometries in different regions on $X_1$, as illustrated in the previous proposition. Recall that we want to perturb the approximate solution $\omega$ to a Calabi-Yau metric on the set $\{ \rho > A \}$ for sufficiently large $A$. To proceed, we fix a large $P < A$ such that on $\{ \rho < 2P \}$ we use the usual $C^{k,\alpha}$ norm. When $\rho > P$ we define the weighted spaces in terms of the radial distance $\rho$ and the distance to the singular rays $R$. Define the smooth function \begin{align*} w = \begin{cases} 1 & \text{if } R > 2\kappa\rho \\ R/(\kappa\rho) & \text{if } R \in (\kappa^{-1}\rho^{1/d}, \kappa\rho) \\ \kappa^{-2}\rho^{1/d-1} & \text{if } R < \frac{1}{2}\kappa^{-1}\rho^{1/d} \end{cases} \end{align*} The three regions in the definition are ``away from singular rays'', ``gluing region'' and ``near singular rays'' in order. Define the H\"older seminorm as \[ [T]_{0,\gamma} = \sup_{\rho(z) > P} \rho(z)^\gamma w(z)^\gamma
\sup_{z' \ne z, z' \in B(z,c)} \frac{|T(z) - T(z')|}{d(z,z')^\gamma}. \end{aligned} \end{equation} Here $c$ is chosen so that $B(z,c)$ has bounded geometry and is geodesically convex. We use parallel transport along a geodesic to compare $T(z)$ and $T(z')$. We can now define the weighted spaces \begin{align*}
\|f\|_{C^{k,\alpha}_{\delta, \tau}} = &\|f\|_{C^{k,\alpha}(\rho < 2P)}
+ \sum_{j=0}^k \sup_{\rho > P} \rho^{-\delta+j}w^{-\tau+j}|\nabla^j f| \\ &+[\rho^{-\delta+k}w^{-\tau+k}\nabla^k f]_{0,\alpha}. \end{align*} Alternatively, if we replace $\rho$ by a smoothing of $\max\{1,\rho\}$, then we can express these weighted norms with respect to the metric $\rho^{-2}w^{-2}\omega$: \[
\|f\|_{C^{k,\alpha}_{\delta,\tau}} =
\|\rho^{-\delta}w^{-\tau} f\|_{C^{k,\alpha}_{\rho^{-2}w^{-2}\omega}}. \end{aligned} \end{equation} Using these norms we can define $C^{k,\alpha}_{\delta,\tau}(X_1, \omega)$. Since we will invert the Laplacian only on $\rho \ge A$ for $A$ sufficiently large, for $f$ defined on $\rho \ge A$ we define the norms \[
\|f\|_{C^{k,\alpha}_{\delta,\tau}(\rho^{-1}[A,\infty))} =
\inf_{\hat{f}} \|f\|_{C^{k,\alpha}_{\delta,\tau}(X_1,\omega)}, \end{aligned} \end{equation} where the infimum is among all extensions $\hat{f}$ of $f$ on $X_1$.
We record without proof some basic properties of the weighted norms:
\begin{prop} \label{prop:weightedprop} The weighted norms we just defined enjoy the following properties: \begin{itemize} \item If $f \in C^{k,\alpha}_{a,b}$ and $g \in C^{k,\alpha}_{c,d}$,
then
$\|fg\|_{C^{k,\alpha}_{a+c,b+d}} \le \|f\|_{C^{k,\alpha}_{a,b}} \le
\|g\|_{C^{k,\alpha}_{c,d}}.$ \item If $a < c$, then
$\|f\|_{C^{k,\alpha}_{a,b}} \ge \|f\|_{C^{k,\alpha}_{c,b}}$, and consequently $C^{k,\alpha}_{a,b} \subset C^{k,\alpha}_{c,b}$. This is because $\rho > P > 1$. \item If $b < d$, then
$\|f\|_{C^{k,\alpha}_{a,b}} \le \|f\|_{C^{k,\alpha}_{c,d}}$, and consequently $C^{k,\alpha}_{a,b} \supset C^{k,\alpha}_{a,d}$. This is because $w \le 1$. \end{itemize} \end{prop}
We can now use the weighted spaces to compare the geometry of $X_1$ with model spaces in different regions. Write $g, g_0$ for the Riemannian metrics of $\omega, \omega_0$, respectively (recall that $\omega_0$ is the cone metric on $X_0$). First we consider the region \[ \mathcal{U} = \{ \rho > A, R> \Lambda\rho^{1/d}\} \cap X_1, \end{aligned} \end{equation} for large $A, \Lambda$, and let \[ G: \mathcal{U} \to X_0 \end{aligned} \end{equation} be the nearest point projection with respect to the cone metric
$\partial\bar\partial (|z|^2 + R^2)$ on $\mathbf{C}^{n+1}$. Note that we have \[ G(z,x) = (z,x') \end{aligned} \end{equation} where $x'$ is the nearest point projection of $x \in \mathbf{C}^n$ with respect to the cone metric $\partial\bar\partial R^2$ on $\mathbf{C}^n$.
\begin{prop} \label{awaysingular} Given any $\epsilon > 0$ we can choose $\Lambda > \Lambda(\epsilon)$, and $A > A(\epsilon)$ sufficiently large so that on $\mathcal{U}$ we have \[
|\nabla^i(G^*g_0 - g)|_g < \epsilon w^{-i}\rho^{-i}. \end{aligned} \end{equation} for $i\le k+1$. In particular, in terms of weighted spaces we have \[
\|G^*g_0 - g\|_{C^{k,\alpha}_{0,0}} < \epsilon. \end{aligned} \end{equation} \end{prop}
Next we consider the region where $\rho > A$ but $R < \Lambda \rho^{1/d}$, i.e. we are close to the singular ray. Fix $z_0 \in \mathbf{C}$ and a large constant $B>0$. Define \[
\mathcal{V} = \{ |z-z_0| < B|z_0|^{1/d},\: R < \Lambda \rho^{1/d},\: \rho > A \} \cap X_1. \end{aligned} \end{equation} We will use regions in the form of $\mathcal{V}$ to cover the neighborhood of the singular ray. We change the coordinates as follows: \begin{equation} \hat{x} = z_0^{-1/d} \cdot x, \:\:\: \hat{z} = z_0^{-1/d}(z-z_0). \end{equation}
Define $\hat{R} = |z_0|^{-1/d}R$, and let $\hat{\zeta} = \max\{1,\hat{R}\}$. Then $(\hat{z},\hat{x})$ satisfies the equation \[ z_0^{1/d-1}\hat{z} + 1 + f(\hat{x}) = 0, \end{aligned} \end{equation}
and $|\hat{z}| < B, |\hat{R}| < C\Lambda$ for some fixed constant $C$
(since $|z|\sim \rho$). In terms of the new coordinates, we define the map \[ H: \mathcal{V} \to \mathbf{C} \times V_1 \end{aligned} \end{equation} by $H(\hat{z},\hat{x}) = (\hat{z}, \hat{x}')$, where $\hat{x}'$ is the nearest point projection of $\hat{x}$ onto $V_1$ with respect to the ambient cone metric.
\begin{prop} \label{nearsingular} Given $\epsilon, \Lambda > 0$, if $A > A(\epsilon,\Lambda, B)$, then we have \[
|\nabla^i(H^*g_{\mathbf{C}\times V_1}-|z_0|^{-2/d}g)|_{|z_0|^{-2/d}g} < \epsilon \hat{\zeta}^{-i} \end{aligned} \end{equation} for $i \le k+1$. In terms of weighted spaces we have \[
\||z_0|^{2/d}H^*g_{\mathbf{C}\times V_1}-g\|_{C^{k,\alpha}_{0,0}} < \epsilon. \end{aligned} \end{equation} \end{prop}
From the above two propositions we have the following:
\begin{prop} \label{tangentconeisx0} Let $\epsilon > 0$. If $D$ is sufficiently large, then there are $(D\epsilon)$-Gromov-Hausdorff approximations between the annular regions \[ X_1^D = (X_1,\omega) \cap \{D^{1/2} < \rho < D \} \end{aligned} \end{equation} and \[ X_0^D = (X_0, \omega_0) \cap \{D^{1/2} < \rho < D \} \end{aligned} \end{equation}
Recall that $X_0 = \mathbf{C} \times V_0$ is equipped with the product metric $\omega_0 = \partial\bar\partial(|z|^2 + r^2)$. Consequently, the tangent cone of $(X_1,\omega)$ at infinity is $(X_0,\omega_0)$. \end{prop}
This is slightly different from Proposition~9 in \cite{Sz19}. Since the above result is crucial for obtaining the asymptotic behavior of the distance function of $\omega$, we give a detailed proof here.
\begin{proof}[Proof of Proposition~\ref{tangentconeisx0}] Given $\epsilon > 0$, the goal is to construct a $(D\epsilon)$-Gromov-Hausdorff approximation $G: X_1^D \to X_0^D$. Let $\Lambda > 0$. Write $S_\Lambda = \{ R < \Lambda \rho^{1/d} \}$. Recall that $S_\Lambda$ denotes a region that is close to the singular ray of $X_0$. Then we can decompose $X_1^D$ into $X_1^D \setminus S_\Lambda$ and $X_1^D \cap S_{2\Lambda}$.
First we work on $X_1^D \setminus S_\Lambda$. Recall from Proposition~\ref{awaysingular} that once $\Lambda$ is sufficiently large, the nearest point projection
$G: X_1^D \setminus S_\Lambda \to X_0^D$ is a diffeomorphism onto its image, and the error in the metric is $|g - G^*g_0|_g < \epsilon$. Let $x_1, x_2 \in X_1^D \setminus S_\Lambda$, and let $\gamma$ be a curve in $X_1^D \setminus S_\Lambda$ connecting $x_1$ and $x_2$. Then the error in the length is given by \begin{equation} \label{eq:length}
|\mathrm{length}_g(\gamma)-\mathrm{length}_{g_0}(\gamma)| \le \mathrm{length}_{g_0}(\gamma)\epsilon. \end{equation} It follows that \begin{align*} d_{X_1^D}(x_1,x_2) &\le d_{X_0^D}(G(x_1),G(x_2))(1+\epsilon) \\ &\le d_{X_0^D}(G(x_1),G(x_2)) + 2D\epsilon. \end{align*} The second inequality uses the fact that $X_0$ is a cone. To get the reverse inequality, we can use \eqref{eq:length} again and get \[ (1-\epsilon)d_{X_0^D}(G(x_1),G(x_2)) \le \mathrm{length}_g(\gamma). \end{aligned} \end{equation} However, we cannot yet take the infimum of the right hand side among all curves connecting $x_1$ and $x_2$, as the minimal geodesic connecting $x_1$ and $x_2$ may pass through $X_1^D \cap S_{\Lambda}$. To $d_{X_1^D}(x_1,x_2)$ is not too much smaller than the right hand side, we turn to the study on $X_1^D \cap S_{2\Lambda}$.
On $X_1^D \cap S_{2\Lambda}$, we define $G: X_1^D \to X_0^D$ by the projection \[
\mathrm{pr}_1: X_1^D \subset \mathbf{C} \times \mathbf{C}^n \to
\mathbf{C} \subset X_0 \end{aligned} \end{equation} onto the singular ray of $X_0$. Proposition~\ref{nearsingular} says that there is a map $H: X_0^D \cap S_{2\Lambda} \to \mathbf{C} \times V_1$ with $\mathrm{pr}_1 \circ H = \mathrm{pr}_1$ such that
$|g - H^*g_{\mathbf{C}\times V_1}|_g \le \epsilon$. Consequently, for $x_1, x_2$ in this region, any curve $\gamma$ connecting $x_1$ and $x_2$ satisfies \[ \mathrm{length}_g(\gamma) \ge (1-\epsilon)\mathrm{length}_{\mathbf{C}} (G\circ \gamma) \ge (1-\epsilon)d_{\mathbf{C}}(G(x_1),G(x_2)). \end{aligned} \end{equation} To take the infimum of the left hand side, note that the shortest curve connecting $x_1$ and $x_2$ in $X_1^D$ will remain in the region $S_{2\Lambda}$, since on the ``annular region'' $S_{2\Lambda}\setminus S_\Lambda$ the metric can be made arbitrarily close to the cone metric $\omega_0$ by letting $\Lambda$ and $D$ be sufficiently large. So we have \begin{equation} d_{X_1^D}(x_1,x_2) \ge d_{\mathbf{C}}(G(x_1),G(x_2))-2D\epsilon. \end{equation}
To get the reverse inequality, we write $H(x_1) = (z_1, p_1), H(x_2) = (z_2, p_2)$ with $z_i \in \mathbf{C}$ and $p_i \in V_1$. From the error in the metric we get \begin{align*} d_{X_1^D}(x_1,x_2) &\le d_{\mathbf{C} \times V_1}(H(x_1), H(x_2)) (1+ \epsilon) \\ &\le (d_{\mathbf{C}}(z_1, z_2) + d_{V_1}(p_1,p_2))(1+\epsilon) \\ &\le (d_{\mathbf{C}}(z_1, z_2) + d_{V_1}(o, p_1)) + d_{V_1}(o,p_2))(1+\epsilon). \end{align*} Here the second inequality follows from the Pythagorean theorem, and $o$ is a fixed point in $V_1$. Since $d_{V_1}(o, \cdot)$ is equivalent to $R$, we can estimate \begin{align} \label{eq:epsilondense} d_{V_1}(o,p_1)) \le CR \le C\Lambda D^{1/d-1} D \ll D\epsilon \end{align} by choosing $D$ sufficiently large. We conclude that \begin{align} \label{eq:singularraydistance} d_{X_1^D}(x_1,x_2) \le d_{\mathbf{C}}(z_1, z_2) + 2D\epsilon. \end{align}
We now come back to the region $X_1^D \cap S_\Lambda$. Again let $x_1, x_2 \in X_1^D \cap S_\Lambda$. Let $\gamma$ be the shortest curve in $X_1^D$ connecting $x_1$ and $x_2$. Let $x_1'$ be the first point of $\gamma$ entering the region $S_{\Lambda}$ and let $x_2'$ be the last point exiting $S_{\Lambda}$. If $\gamma_1$ is the shortest curve connecting $x_1, x_1'$, then \begin{align*} d_{X_1^D}(x_1,x_1') = \mathrm{length}_g(\gamma_1) \ge d_{X_0^D}(G(x_1),G(x_1')) - D\epsilon \end{align*} by \eqref{eq:length}. The similar inequality holds for $d_{X_1^D}(x_2, x_2')$. We then have \begin{align*} d_{X_1^D}(x_1,x_2) &= d_{X_1^D}(x_1,x_1') + d_{X_1^D}(x_2,x_2') + d_{X_1^D}(x_1',x_2') \\ &\ge (d_{X_0^D}(G(x_1), G(x_1'))-D\epsilon) \\ &+ (d_{X_0^D}(G(x_2), G(x_2'))-D\epsilon) \\ &+ (d_{X_0^D}(G(x_1'), G(x_2'))-2D\epsilon) \\ &\ge d_{X_0^D}(G(x_1), G(x_2)) - 4D\epsilon \end{align*} using the triangle inequality and \eqref{eq:singularraydistance}.
Finally, $G(X_1^D)$ is clearly $(D\epsilon)$-dense away from the singular ray. That $G(X_1^D)$ is $(D\epsilon)$-dense near the singular ray follows from the estimate \eqref{eq:epsilondense}. To get the inverse Gromov-Hausdorff approximation, away from the singular ray we can use the nearest point projection to map into $X_1^D$. At a point $(z, p) \in X_0$ near the singular ray, we can first map it to $(z, o)$, where $o \in V_1$ is a fixed point, and then map $(z,o)$ into $X_1^D$ using $H^{-1}$. \end{proof}
The following corollary will be useful in Section~\ref{sec:ds}.
\begin{cor} \label{cor:asymptotics} Let $d$ denote the distance function of $(X_1,\omega)$ and let $o \in X_1$ be a fixed point. Then $d(o, \cdot)$ is uniformly equivalent to $\rho$. Moreover, we have \[
\lim_{\rho(x) \to \infty} \frac{d(o, x)^2}{|z|^2+r^2} = 1. \end{aligned} \end{equation} \end{cor}
\begin{proof}
Write $\tilde\rho^2 = |z|^2 + r^2$. Assume for now that $o \in X_1$ is the origin, and let $x \in X_1$, which we will let $D = \rho(x) \to \infty$. First we note that by concatenating larger and larger annuli of the form $(2^i, 2^{2i})$, Proposition~\ref{tangentconeisx0} implies that the function $d(o, \cdot)$ is equivalent to $\rho$. Since $\rho$ and $\tilde\rho$ are homogeneous of degree $2$, they are equivalent, too.
Let $x' \in X_1$ be on the minimal geodesic connecting $o$ and $x$ such that $\rho(x') = D^{1/2}$. By Proposition~\ref{tangentconeisx0}, for any $\epsilon > 0$ we have for sufficiently large $D$, \begin{equation} \label{eq:bigineq} \begin{aligned} d(o,x') + d_{X_0}(G(x'), G(x)) - D\epsilon &< d(o,x) = d(o,x') + d(x',x) \\ &< d(o,x') + d_{X_0}(G(x'), G(x)) + D\epsilon, \end{aligned} \end{equation}
where $G$ is the $(D\epsilon)$-Gromov-Hausdorff approximation given in Proposition~\ref{tangentconeisx0}. Recall that away from the singular ray, $G$ is given by the nearest point projection with respect to the cone metric $\partial\bar\partial \rho^2$ on $\mathbf{C}^n$, and near the singular ray we have $|z| \sim \rho$ and $G$ is given by the projection onto the singular ray. It follows that $\rho(Gx) \sim \rho(x) = D$, and so $\tilde\rho(x) \sim D$. As $D \to \infty$, the distance of $x$ and $G(x)$ with respect to the scaled down cone metric $D^{-2}\partial\bar\partial \rho^2$ converges to $0$. It follows that \begin{equation} \label{eq:rhoprime} \frac{\tilde\rho(G(x))}{\tilde\rho(x)} \to 1 \end{equation} as $D \to \infty$.
Dividing the inequality \eqref{eq:bigineq} by $\tilde\rho(x)$, we estimate the terms as follows: \begin{align*} \frac{d(o,x')}{\tilde\rho(x)} \sim \frac{D^{1/2}}{D} = D^{-1/2}, \\ \frac{d_{X_0}(G(x'),G(x))}{\tilde\rho(x)} \to 1, \end{align*} as $D \to \infty$. Here the second estimate follows from the cosine law of the cone metric on $X_0$ and \eqref{eq:rhoprime}. Letting $D \to \infty$ we get the desired result. For arbitrary fixed point $o \in X_1$ the same result follows by an application of the triangle inequality. \end{proof}
Finally, we recall the technical heart of \cite{Sz19}, the invertibility of the Laplacian in weighted spaces:
\begin{prop} Suppose that we choose $\tau \in (4-2n,0)$ (recall that $n$ is the complex dimension of $X_1$) and $\delta$ avoids a discrete set of indicial roots. For sufficiently large $A>0$ the Laplacian \[ \Delta: C^{2,\alpha}_{\delta,\tau}(\rho^{-1}[A,\infty),\omega) \to C^{0,\alpha}_{\delta-2,\tau-2}(\rho^{-1}[A,\infty),\omega) \end{aligned} \end{equation} is surjective with inverse bounded independently of $A$. \end{prop}
The idea of the proof is to cover $X_1$ (outside a big compact set) by the open subset $\mathcal{U}$ and open subsets of types $\mathcal{V}$ near the singular rays. On each such open set, the model Laplacian is invertible with respect to the corresponding model weighted space. Then one construct a parametrix by patching local inverses together using cutoff functions.
\section{Construction of new Calabi-Yau metrics}\label{sec:x1b}
We now turn to constructing a new family of Calabi-Yau metrics on $\mathbf{C}^3$, building on the results in the previous section. Similar to the construction of Calabi-Yau metrics on $X_1$ as in \cite{Sz19}, we will consider the following family of hypersurfaces \[ X_{1,b} = \{ z + by + x_1^2 + x_2^2 + y^3 = 0 \} \subset \mathbf{C}^4, \end{aligned} \end{equation} where $b \in \mathbf{C}$. More generally, we could consider \[ X_{a,b} = \{ az + by + x_1^2 + x_2^2 + y^3 = 0 \} \subset \mathbf{C}^4, \end{aligned} \end{equation} where $a \ne 0 \in \mathbf{C}$ and $b \in \mathbf{C}$. The effect of $a$ can be taken care of by rescaling the metric. So we will assume $a=1$. Later in Section~\ref{sec:ds} we will give a detailed explanation why the following construction of Calabi-Yau metrics on $X_{a,b}$ would possibly give all the Calabi-Yau metrics on $\mathbf{C}^3$ with tangent cone $\mathbf{C} \times A_2$ at infinity.
Let \[ \Omega = dx_1 \wedge dx_2 \wedge dy \end{aligned} \end{equation} be the holomorphic volume form on $X_{1,b}$. The rest of the section is dedicated to proving the following:
\begin{thm}\label{thm:construction}
There exists a complete K\"ahler metric $\omega_{1,b}$ on $X_{1,b}$
such that
\[
\omega_{1,b}^3 = \sqrt{-1} \Omega\wedge\overline\Omega,
\end{aligned} \end{equation}
and that the tangent cone at the infinity given by
$\mathbf{C} \times A_2$. \end{thm}
Let \[
\Phi = |z|^2 + \gamma_1(R\rho^{-\alpha})r^2 +
\gamma_2(R\rho^{-\alpha})|z|^{2/d}\phi(z^{-1/d}\cdot (x,y)) \end{aligned} \end{equation} be the K\"ahler potential of the approximate solution on $X_1$ constructed in the previous section. The strategy is to use the nearest point projection $G: X_1 \cap \{ \rho > A\} \to X_{1,b} \cap \{ \rho > A \}$ with respect to the ambient cone metric $\partial\bar\partial \rho^2$ for large enough $A>0$, to pull back the volume form $\sqrt{-1}\Omega\wedge\overline\Omega$ as well as the complex structure $J$ on $X_{1,b}$, and solve \[\label{eq:21}
(\sqrt{-1}\partial_b\bar{\partial_b} (\Phi + u))^3 = \sqrt{-1}\Omega_b \wedge \overline\Omega_b. \end{aligned} \end{equation} Here $\partial_b$ and $\bar\partial_b$ are the partial differentials with respect to the complex structure $J_b = G_*J (G^{-1})_*$, and $\Omega_b = G^*\Omega$ is the pullback of the holomorphic volume form. Once this is done, we push forward this metric using $G$ to $X_{1,b}$ and obtain a Calabi-Yau metric on $X_{1,b}$ outside a large compact subset. Then we extend it to a K\"ahler metric on $X_{1,b}$ which is Ricci-flat outside a large compact subset. We can then apply Hein's version of the Tian-Yau perturbation theorem \cite{Hein} to perturb it again to a genuine Calabi-Yau metric on $X_{1,b}$.
\begin{remark}
One could try to write down an explicit approximate solution on
$X_{1,b}$ without relying on the nearest point projection, and apply
the techniques in the previous section directly on $X_{1,b}$, but
then an issue is that the fibration is non-trivial away from the
singular fibers. This potentially would make the analysis harder. We
use the nearest point projection because near the singular ray and
far from the singular fibers, we are still comparing the geometry of
$X_{1,b}$ to the geometry of $\mathbf{C} \times V_1$. See the proof
of Proposition~\ref{prop:bdecay} below. \end{remark}
The nearest point projection $G: X_1 \to X_{1,b}$ is only defined outside compact subsets containing the origin $0 \in \mathbf{C}^4$, as the cone metric $\partial\bar\partial \rho^2$ is singular at $0$ (and also singular along the singular rays $\mathbf{C} \subset \mathbf{C}^4$). Recall that scaling down the metric amounts to making the coordinate change $z \to D^{-1}z$, $x \to D^{-1} \cdot x$. One might be tempted to conclude that the error going from $X_1$ to $X_{1,b}$ is $O(b\rho^{-4})$ by comparing the defining equations. If this were true, then we may apply Hein's perturbation theorem directly to perturb the Calabi-Yau metric $\omega$ on $X_1$ to a (pullback of) Calabi-Yau metric on $X_{1,b}$. Unfortunately this is not the case, as both $X_1$ and $X_{1,b}$ converges to $X_0$, whose singular set is complex one-dimensional. To get meaningful $C^{k,\alpha}$ bounds of the errors introduced by the nearest point projection, we need to apply the region analysis in Proposition~\ref{prop:adecay} in the previous section, comparing the geometry in each region to those of different model spaces.
\subsection{Decay of the Ricci potential}
Let us write $\omega_b = \sqrt{-1}\partial_b\bar{\partial_b} \Phi$ as the approximate solution. As mentioned above, we want to solve \eqref{eq:21} on $X_1 \cap \{ \rho > A\}$ for large enough $A$. To solve for $u$, we want to ensure that the Ricci potential \[ h = \log \frac{\omega_b^3}{\sqrt{-1}\Omega_b \wedge \overline\Omega_b} \end{aligned} \end{equation} has fast enough decay in order to apply the technical results discussed in the previous section. We have the following generalization of Proposition~\ref{prop:adecay} in our $\mathbf{C} \times A_2$ case.
\begin{prop} \label{prop:bdecay} Fix $\alpha \in (1/d,1)$. The form $\omega_b$ defines a K\"ahler metric with respect to the deformed complex structure $J_b$ on the $X_1 \cap \{\rho > P\}$, for sufficiently large $P$ (depending on $b$). For suitable constants $\kappa, C_i >0$ and weight $\delta < 2/d$, the Ricci potential $h$ of $\omega_b$ with respect to $G^*(\sqrt{-1}\Omega_b\wedge\overline\Omega_b)$ and the error in the complex structure satisfy, for large $\rho$, \begin{align*}
|\nabla^i h|_\omega,
|\nabla^i(\omega_b-\omega)|,|\nabla^i(J_b-J)|_\omega < \max\{1,b\} \begin{cases} C_i \rho^{\delta-2-i} & \text{if } R > \kappa\rho \\ C_i \rho^{\delta}R^{-2-i} & \text{if } R \in (\kappa^{-1}\rho^{1/d}, \kappa \rho) \\ C_i \rho^{\delta-2/d-i/d} & \text{if } R < \kappa^{-1}\rho^{1/d}. \end{cases} \end{align*} In fact, since $d=6$, we can choose $\delta \in [-1/3,1/3)$. In terms of the weighted spaces defined in the previous section, we have that \[
\|h\|_{C^{k,\alpha}_{\delta-2,-2}},
\|\omega_b-\omega\|_{C^{k,\alpha}_{\delta-2,-2}},
\|J_b-J\|_{C^{k,\alpha}_{\delta-2,-2}} \le C_k\max\{1,b\} \end{aligned} \end{equation} for a uniform constant $C_k>0$. \end{prop}
\begin{proof} The proof is very similar to Proposition~\ref{prop:adecay} before. The main difference is that in this case the complex structure as well as the holomorphic volume form are deformed. As a result the Ricci potential is given by \begin{align*} h &= \log \frac{\omega_b^3}{\sqrt{-1}\Omega_b \wedge \overline\Omega_b} \\ &= \log \frac{\omega^3}{\sqrt{-1}\Omega \wedge \overline\Omega} + \log \frac{\omega_b^3}{\omega^3} + \log \frac{\sqrt{-1}\Omega\wedge\overline\Omega}{\sqrt{-1}\Omega_b \wedge \overline\Omega_b}. \end{align*} Here we recall that $\Omega$ is the holomorphic volume form on $X_1$. Thus we will have additional errors introduced by the change in the complex structure as well as the change in the volume form. For the metric, we can estimate the error by \[ \omega_b - \omega = d (J_b-J) d\Phi. \end{aligned} \end{equation} Since $\Phi$ has growth rate $2$, it follows that the error in the metric is dominated by the error in the change of the complex structure. We perform the region analysis as in the proof of Proposition~\ref{prop:adecay}.
\textbf{Region I}: Suppose $R > \kappa\rho$ and $\rho \in (D/2, 2D)$ for some large $D$. Since $R > (\kappa/2)D$, we are uniformly away from the singular rays. We study the scaled metric $D^{-2}\omega$ in terms of the rescaled coordinates $\tilde{z} = D^{-1}z$, $\tilde{x} = D^{-1} \cdot x$. The equation of $X_1$ becomes \[ D^{1-d}\tilde{z} + f(\tilde{x}) = 0, \end{aligned} \end{equation} and the equation of $X_{1,b}$ becomes \[ D^{1-d}\tilde{z} + bD^{2-d}\tilde{y} + f(\tilde{x}) = 0. \end{aligned} \end{equation} Thus the extra error is of order $bD^{2-d}$. We can choose any $\delta$ such that $\delta-2 > 2-d$. Since $d = 6$, we can make $\delta < 0$.
\textbf{Region II}: Suppose now that $R \in (K/2, 2K)$ for some
$K < \kappa \rho$, $K/2 > 2\rho^\alpha$ and $\rho \in (D/2, 2D)$. In this case $\rho$ is comparable to $|z|$. We assume that for some fixed
$z_0$ we have $|z-z_0| < K$. We now scale the metric by $K$, and define \[ \tilde{z} = K^{-1}(z-z_0),\:\:\: \tilde{x} = K^{-1} \cdot x, \:\:\: \tilde{r} = K^{-1}r. \end{aligned} \end{equation} The equation of $X_1$ is \[ K^{-d}(K\tilde{z} + z_0) + f(\tilde{x}) = 0, \end{aligned} \end{equation} while the equation of $X_{1,b}$ is \[ K^{-d}(K\tilde{z} + z_0) + bK^{2-d}\tilde{y} + f(\tilde{x}) = 0. \end{aligned} \end{equation}
Since $|\tilde{y}| \sim 1$, thus the extra error in the Ricci potential is of order $bK^{2-d}$. Since $d=6$ and $K > 4\rho^\alpha$, we have \[ bK^{4-d}K^{-2} < bCD^{(4-d)\alpha}K^{-2} \end{aligned} \end{equation} for a constant $C$. We can choose $\delta < 0$ such that $(4-d)\alpha < \delta$. If $\alpha$ is close to $1$ then we can choose $\delta = -1$.
\textbf{Region III}: Suppose $R \in (K/2, 2K)$, $K \in (\rho^\alpha, 2\rho^\alpha)$ and $\rho \in (D/2, 2D)$. Thus
$|z|$ is comparable to $D$. We are in the gluing region. We scale as in Region II. The equation of $X_1$ becomes \[ K^{-d}(K\tilde{z} + z_0) + f(\tilde{x}) = 0, \end{aligned} \end{equation} and the equation of $X_{1,b}$ becomes \[ K^{-d}(K\tilde{z} + z_0) + bK^{2-d}\tilde{y} + f(\tilde{x}) = 0. \end{aligned} \end{equation} The extra error in the Ricci potential is then again of order $bK^{2-d}$. Since $K \sim D^\alpha$, we can estimate it as follows: \[ bK^{4-d}K^{-2} < bCD^{(4-d)\alpha}K^{-2}. \end{aligned} \end{equation} So here we can choose $0 > \delta > (4-d)\alpha$.
\textbf{Region IV}: Suppose now that $R \in (K/2, 2K)$,
$K \in (\kappa\rho^{1/d}, \rho^\alpha/2)$, and $\rho \in (D/2, 2D)$. Then we have $|z| \sim D$. We scale in the same way as in Regions II, III. The equation of $X_1$ is \[ K^{-d}(K\tilde{z} + z_0) + f(\tilde{x}) = 0, \end{aligned} \end{equation} and we are comparing $X_1$ to $\mathbf{C} \times V_{K^{-d}z_0}$, given by the equation \[ K^{-d}z_0 + f(\tilde{x}) = 0. \end{aligned} \end{equation} On the other hand the error going from $X_{1,b}$ to $X_1$ is still of order $bK^{2-d}$. Since $K > \kappa \rho^{1/d}$, we get \[ K^{4-d}K^{-2} \le bCD^{4/d-1} K^{-2}. \end{aligned} \end{equation} Since $4/d-1 < 0$, we can choose $\delta < 0$.
\textbf{Region V}: Suppose that $R < 2\kappa^{-1}\rho^{1/d}$,
$\rho \in (D/2, 2D)$. Then $|z|$ is comparable to $D$. Fix $z_0$ and let $z$ be very close $z_0$. We scale by $|z_0|^{1/d}$: \[ \tilde{z} = z_0^{-1/d}(z-z_0), \:\:\: \tilde{x} = z_0^{-1/d}\cdot x,
\:\:\: \tilde{r} = |z|^{-1/d}r. \end{aligned} \end{equation}
So we have $|\tilde{z}|, |\tilde{r}| < C$. We are near the singular rays. So we compare $X_1$: \[ z_0^{1/d-1}\tilde{z} + 1 + f(\tilde{x}) = 0 \end{aligned} \end{equation} with $\mathbf{C} \times V_1$: \[ 1 + f(\tilde{x}) = 0. \end{aligned} \end{equation} On the other hand, the equation of $X_{1,b}$ becomes \[ z_0^{1/d-1}\tilde{z} + 1 + b z_0^{2/d-1}\tilde{y} + f(\tilde{x}) = 0. \end{aligned} \end{equation} So the extra error in this case is
$b|z_0|^{(2-d)(1/d)} \le bCD^{2/d-1} \le bCD^{\delta - 2/d}$, where we choose $0 > \delta \ge 4/d -1$. \end{proof}
As indicated in the proof, the decay rate of the error introduced by the nearest point projection is slower than quadratic in the region close to the singular rays, so we cannot apply Hein's perturbation theorem directly. But as the proposition concludes, we still have good decay rates that allow us to improve upon using the contraction mapping principle as in \cite{Sz19}. We first need to take care of the fact that in our case, the Laplacian is also perturbed:
\begin{lemma} \label{lemma:laplacianinverse} Let $\tau \in (-2,0)$, and let $\delta$ avoids a discrete set of indicial roots. The Laplacian $\Delta_b$ with respect to the metric defined by $\omega_b$ is a map from $C^{2,\alpha}_{\delta,\tau}(\rho^{-1}[A,\infty))$ to $C^{0,\alpha}_{\delta-2,\tau-2}(\rho^{-1}[A,\infty))$ with bounded right inverse when $A$ is sufficiently large. \end{lemma}
\begin{proof} Let $P: C^{0,\alpha}_{\delta-2,\tau-2}(\rho^{-1}[A,\infty)) \to C^{2,\alpha}_{\delta,\tau}(\rho^{-1}[A,\infty))$ be the right inverse for $\Delta$. For $u \in C^{k,\alpha}_{\delta,\tau}$, by direct computation we have \begin{align*}
\|\Delta_bu-\Delta u\|_{C^{0,\alpha}_{\delta-2,\tau-2}}
\le \|\nabla(g_b-g) \ast \nabla u\|_{C^{0,\alpha}_{\delta-2,\tau-2}}
+ \|(g_b-g)\ast \nabla^2 u\|_{C^{0,\alpha}_{\delta-2,\tau-2}}. \end{align*} Using the properties of the weighted norms in Propsition~\ref{prop:weightedprop}, we have \begin{align*}
\|\nabla(g_b-g)\ast \nabla u\|_{C^{0,\alpha}_{\delta-2,\tau-2}}
&\le C\|\nabla(g_b-g)\|_{C^{0,\alpha}_{-1,-3}}\|\nabla u\|_{C^{0,\alpha}_{\delta-1,\tau+1}} \\
&\le C\|g_b-g\|_{C^{1,\alpha}_{0,-2}}\|u\|_{C^{1,\alpha}_{\delta,\tau+2}} \\
&\le C\|g_b-g\|_{C^{2,\alpha}_{0,-2}}\|u\|_{C^{2,\alpha}_{\delta,\tau+2}} \\
&\le C\|g_b-g\|_{C^{2,\alpha}_{0,-2}}\|u\|_{C^{2,\alpha}_{\delta,\tau}}. \end{align*} Similarly, \begin{align*}
\|(g_b-g)\ast \nabla^2 u\|_{C^{0,\alpha}_{\delta-2,\tau-2}}
&\le C\|g_b-g\|_{C^{0,\alpha}_{0,-2}}\|\nabla^2 u\|_{C^{0,\alpha}_{\delta-2,\tau}} \\
&\le C\|g_b-g\|_{C^{1,\alpha}_{2,-2}}\|u\|_{C^{2,\alpha}_{\delta,\tau+2}} \\
&\le C\|g_b-g\|_{C^{2,\alpha}_{0,-2}}\|u\|_{C^{2,\alpha}_{\delta,\tau}}. \end{align*} It follows that \begin{align*}
\|\Delta_bu-\Delta u\|_{C^{0,\alpha}_{\delta-2,\tau-2}} \le C\|g_b-g\|_{C^{2,\alpha}_{0,-2}}\|u\|_{C^{2,\alpha}_{\delta,\tau}} \end{align*} for a uniform constant $C > 0$. By Proposition~\ref{prop:bdecay},
$\|g_b-g\|_{C^{2,\alpha}_{0,-2}(\rho^{-1}[A,\infty)))}$ can be made arbitrarily small once $A \gg 1$. It follows that \begin{align*}
\|u - \Delta_b P u\|_{C^{0,\alpha}_{\delta-2, \tau-2}}
&\le \|\Delta_bP u-\Delta P u\|_{C^{0,\alpha}_{\delta-2,\tau-2}} \\
&\le C\|g_b-g\|_{C^{2,\alpha}_{0,-2}}\|u\|_{C^{2,\alpha}_{\delta,\tau}}
\ll \|u\|_{C^{2,\alpha}_{\delta,\tau}}. \end{align*} It follows that $\Delta_b$ admits a bounded right inverse. \end{proof}
\subsection{Perturbing to genuine solution}
We use the approximate solution $\omega$ on $X_1$ and the weighted spaces defined in the previous section. Recall that our goal is to first solve \eqref{eq:21} on $X_1\cap \{\rho > A\}$ for large $A$. Define \[ \mathcal{B} = \{ u \in C^{2,\alpha}_{\delta, \tau}\mid
\|u\|_{C^{2,\alpha}_{\delta, \tau}} < \epsilon_0\}, \end{aligned} \end{equation} where $\tau$ is now chosen to be close to $0$ and $\epsilon_0$ is sufficiently small such that $\omega + \partial\bar\partial u$ has the same tangent cone at infinity as $\omega$. Consider the following operator \begin{align*} F: \mathcal{B} &\to C^{0,\alpha}_{\delta-2,\tau-2}(\rho^{-1}[A,\infty)) \\ u &\mapsto \log \left.\frac{(\tilde\omega+ \sqrt{-1}\partial_b\bar{\partial_b}
u)^3}{{\sqrt{-1}\Omega_b\wedge\overline\Omega_b}}\right|_{\rho^{-1}[A,\infty)}, \end{align*} and write \begin{equation} \label{eq:19} F(u) = F(0) + \Delta_b u + Q(u), \end{equation} where $Q$ is the nonlinear part of $F$. Here $F(0)=h$ is given by the Ricci potential defined above. The goal is to find $u \in \mathcal{B}$ such that $F(u) = 0$, or equivalently \begin{equation} \label{eq:24} \Delta_b u = -F(0) -Q(u). \end{equation} Let $P$ be the right inverse for $\Delta_b$ in Lemma~\ref{lemma:laplacianinverse}. Define the map \[ N(u) = P(-F(u)-Q(u)). \end{aligned} \end{equation} Then finding a solution to \eqref{eq:24} is the same as finding a fixed point of $N$. Note that we have a uniform bound for $P$ independent of sufficiently large $A$. Thus we can enlarge $A$ when needed. From an explicit formula for $Q$ (e.g. expand $\log \det (I+A)-\mathrm{tr} A$ using eigenvalues for $A$), we see that if \[
\|\partial_b\bar{\partial_b} u\|_{C^{0,\alpha}_{0,0}},\|\partial_b\bar{\partial_b} v\|_{C^{0,\alpha}_{0,0}} \ll 1, \end{aligned} \end{equation} then we have the estimate \[
\|Q(u)-Q(v)\|_{C^{0,\alpha}_{\delta-2,\tau-2}} \le C(\|\partial_b\bar{\partial_b} u\|_{C^{0,\alpha}_{0,0}}+\|\partial_b\bar{\partial_b} v\|_{C^{0,\alpha}_{0,0}})
\|\partial_b\bar{\partial_b}(u-v)\|_{C^{0,\alpha}_{\delta-2,\tau-2}}. \end{aligned} \end{equation}
To estimate $\|\partial_b\bar{\partial_b} u\|_{C^{0,\alpha}_{0,0}}$ in terms of the norm of $u$, we have \begin{equation} \begin{aligned}
\|\partial_b\bar{\partial_b} u\|_{C^{0,\alpha}_{0,0}} &\le \|\sqrt{-1}\partial\bar\partial u\|_{C^{0,\alpha}_{0,0}} +
\|(\partial_b\bar{\partial_b}-\sqrt{-1}\partial\bar\partial) u\|_{C^{0,\alpha}_{0,0}} \\
&\le C(1 + \|J_b-J\|_{C^{2,\alpha}_{-2,-2}})\|u\|_{C^{2,\alpha}_{2,2}} \\
&\le C\max\{1,b\}\|u\|_{C^{2,\alpha}_{2,2}} \end{aligned} \end{equation} by Proposition~\ref{prop:bdecay}. Since we have \[ \rho^{\delta}w^\tau \le C\rho^{\delta-2+(\tau-2)(1/d-1)}\rho^2 w^2, \end{aligned} \end{equation} which implies \[
\|u\|_{C^{2,\alpha}_{2,2}} \le C\|u\|_{C^{2,\alpha}_{\delta,\tau}}, \end{aligned} \end{equation} by choosing $\epsilon_0 < C\max\{1,b\}^{-1}$ for a uniform constant $C > 0$ we have \[
\|N(u)-N(v)\|_{C^{2,\alpha}_{\delta,\tau}} < \frac{1}{2}\|u-v\|_{C^{2,\alpha}_{\delta,\tau}} \end{aligned} \end{equation} for $u,v \in \mathcal{B}$; i.e. $N$ is a contraction mapping. It remains to ensure that $N$ maps $\mathcal{B}$ into $\mathcal{B}$. First we note that by the estimates of Proposition~\ref{prop:bdecay} we have $F(0) \in C^{0,\alpha}_{\delta'-2,\tau-2}$ for some $\delta' < \delta$ (increase $\delta$ if necessary) sufficiently close to $\delta$. It follows that \[
\|F(0)\|_{C^{0,\alpha}_{\delta,\tau-2}(\rho^{-1}[A,\infty))} < CA^{\delta'-\delta}. \end{aligned} \end{equation} Combining the estimates above, we have that if $u \in \mathcal{B}$, then \begin{align*}
\|N(u)\|_{C^{2,\alpha}_{\delta,\tau}}
&\le \|N(0)\|_{C^{2,\alpha}_{\delta,\tau}} + \|N(u)-N(v)\|_{C^{2,\alpha}_{\delta,\tau}} \\
&\le \|F(0)\|_{C^{0,\alpha}_{\delta,\tau-2}(\rho^{-1}[A,\infty))} +
\frac{1}{2}\|u\|_{C^{2,\alpha}_{\delta,\tau}} \\ &\le \max\{1,b\}CA^{\delta'-\delta} + \frac{\epsilon_0}{2}. \end{align*} We see that to make $N$ maps into $\mathcal{B}$, we need to remove larger and larger compact subsets as $b$ gets larger. In sum we can make $N$ a contraction mapping by choosing $A$ sufficiently large (depending on $b$). Thus there exists $u \in C^{k,\alpha}_{\delta,\tau}(\rho^{-1}[A,\infty))$ with
$\|u\|_{C^{k,\alpha}_{\delta,\tau}} < \epsilon_0$ such that \[ (\omega_b + \sqrt{-1}\partial_b\bar{\partial_b} u)^3 = \sqrt{-1} \Omega_b \wedge \overline\Omega_b. \end{aligned} \end{equation} on $\rho^{-1}[A,\infty)$. Pushing forward to $X_{1,b}$, we have \[ (\partial\bar\partial ((\Phi + u)\circ G^{-1}))^3 = \sqrt{-1}\Omega \wedge\overline\Omega \end{aligned} \end{equation} on $\rho^{-1}[A,\infty)$. We can modify the K\"ahler potential so that it defines a K\"ahler potential $\tilde{\Phi}_b$ on $X_{1,b}$ such that $\partial\bar\partial \tilde{\Phi}_b$ agrees with $\partial\bar\partial ((\Phi + u)\circ G^{-1})$ on $\rho^{-1}[2A,\infty)$. Set $\tilde{\omega}_b = \partial\bar\partial \tilde{\Phi}_b$.
We now apply Hein's version~\cite{Hein} of the Tian-Yau perturbation. Recall that $(X_1 \cap \rho^{-1}[A,\infty), \omega)$ is covered by sets of type $\mathcal{U}$ and type $\mathcal{V}$ as in Proposition~\ref{awaysingular} and Proposition~\ref{nearsingular}, respectively. Pulling back using the nearest point projection, it follows that the same holds for $(X_{1,b} \cap \rho^{-1}[2A,\infty),\tilde{\omega}_b)$. We can rescale accordingly to compare in each region to the model geometries $X_0$ and $\mathbf{C} \times V_1$. The rescaled coordinates then give the desired $C^{3,\alpha}$ coordinates. For the compact part we simply cover it with a finite number of coordinate balls. This shows that $(X_{1,b},\tilde{\omega}_b)$ admits a $C^{3,\alpha}$ quasi-atlas. That $\tilde{\omega}_b$ is $\mathrm{SOB}(6)$ follows from that $\tilde{\omega}_b$ is Ricci-flat outside a compact subset and the tangent cone at infinity is $X_0$, which together imply maximal volume growth by Colding's volume convergence~\cite{Colding}. We can then apply \cite[Proposition~4.1]{Hein} to perturb $\tilde{\omega}$ to a genuine Calabi-Yau metric $\omega_{1,b} = \partial\bar\partial \varphi_{1,b}$ on $X_{1,b}$. When $b=0$, this recovers the Calabi-Yau metric constructed on $X_1 = X_{1,0}$ in \cite{Sz19}.
A few notes about this construction are in order. First, this construction should generalize to construct families of Calabi-Yau metrics asymptotic to $\mathbf{C} \times A_k, k \ge 3$, including the ones constructed in \cite{Sz19}. One could consider hypersurfaces in $\mathbf{C}^4$ given by $az + b_1y + b_2y^2 + \ldots + b_{k-2}y^{k-1} + x_1^2 + x_2^2 + y^{k+1} = 0$, with $a \ne 0 \in \mathbf{C}$ and $b_i \in \mathbf{C}$. Second, what we know about these metrics $\omega_{1,b}$ for now is that they are unique up to subquadratic perturbation of the K\"ahler potential by \cite[Theorem 1.3]{CSz}. So a small perturbation of the initial data or the choice of the right inverse of the Laplacian does not affect the resulting metric. It is not clear at this moment whether $\omega_{1,b}$ and $\omega_{1,b'}$ are related by an automorphism of $\mathbf{C}^3$ up to scaling, because the construction involves nearest point projections which are not even holomorphic to begin with. To distinguish them we need to exploit the explicit nature of the asymptotics.
More generally, we would like to know if the gluing construction above gives all the Calabi-Yau metrics on $\mathbf{C}^3$ with tangent cone $\mathbf{C} \times A_2$ at infinity. We will discuss some preliminary results in the next section.
\section{Distinguishing the metrics}\label{sec:ds}
In this section, we conclude the proof of Theorem~\ref{thm:A2}, and discuss some preliminary results about Conjecture~\ref{conj:A2} below.
Uniqueness results in singular perturbation problems are usually hard to obtain, and very few results in the Calabi-Yau setting are known. We would like to follow a similar strategy in \cite{Sz20} to study the classification problem in our case. For this we first compute subquadratic harmonic functions on the cone $\mathbf{C} \times A_2$.
\subsection{Subquadratic harmonic functions on cones}
We first recall the following characterization of subquadratic harmonic functions of Calabi-Yau cones $C(Y)$:
\begin{lemma}\label{lemma:HS}
Suppose $C(Y)$ is a metric tangent cone of a non-collapsed
Gromov-Hausdorff limit of K\"ahler-Einstein manifolds. Let $r$
denote the radial coordinate so that $r\partial_r$ is the homothetic
vector field. Let $J$ denote the complex structure. Suppose $u$ is a
harmonic function on $C(Y)$. Then we have the following:
\begin{enumerate}
\item If $u$ is $s$-homogeneous ($r\partial_r u = s u$)
with $s< 2$, then $u$ is pluriharmonic.
\item If $u$ is $2$-homogeneous harmonic, then $u = u_1 + u_2$,
where $u_1$ is pluriharmonic, and $u_2$ is
$J(r\partial_r)$-invariant.
\item The space of real holomorphic vector fields that commute with
$r\partial_r$ can be written as
$\mathfrak{p}\oplus J\mathfrak{p}$, where $\mathfrak{p}$ is
spanned by $r\partial_r$ and vector fields of the form $\nabla u$,
where $u$ is a $J(r\partial_r)$-invariant harmonic function
homogeneous of degree $2$. $J\mathfrak{p}$ consists of real
holomorphic Killing vector fields.
\end{enumerate} \end{lemma}
For a proof, see \cite[Lemma~3.1]{CSz} and the references therein. We apply this lemma to systematically calculate subquadratic harmonic functions on $C(Y)$. First we note that since $C(Y)$ is an affine variety~\cite{LSz}, Lemma~\ref{lemma:HS} (1) and (2) imply that many of these subquadratic harmonic functions are given by the real part of subquadratic holomorphic functions. We are more interested in quadratic harmonic functions whose gradients generate automorphisms of $C(Y)$. For this we use Lemma~\ref{lemma:HS} (3) and turn to real holomorphic vector fields. We note that $\mathfrak{p}$ has another characterization: \begin{align*} \mathfrak{p} = \{ V: V \text{ is real holomorphic with linear growth
and } JV(r^2) = 0\}. \end{align*}
Since $C(Y)$ is an affine variety, it is useful to find $W = V-iJV$ first and then take the real part of $W$. We follow this approach and calculate a few examples relevant to this paper.
\begin{exmp}
Let us consider $C(Y) = \mathbf{C} \times A_1$, defined as the
hypersurface
$\{x_1^2+\ldots + x_n^2 = 0\} \subset \mathbf{C} \times
\mathbf{C}^n$, $n \ge 3$. $C(Y)$ is equipped with a Ricci-flat
K\"ahler metric \begin{align*}
\omega_0 = \sqrt{-1}\partial\bar\partial(|z|^2 + |x|^{2\frac{n-2}{n-1}}), \end{align*} where the coordinate $z$ has weight $1$ and the coordinates $x_i$ have weight $(n-1)/(n-2)$. Any (complex) holomorphic vector field $W$ in $(\mathfrak{p}\oplus J\mathfrak{p})\otimes \mathbf{C}$ is given by \begin{align*} W = bz\partial_z + a_{ij}x_i\partial_{x_j}, \end{align*} where the coefficients $b$ and $a_{ij}$ are such that $W(x_1^2+\ldots+x_n^2) = 0$ and $\mathrm{Im} W (r^2) = 0$. From these two equations, we get that $b$ and $\lambda$ are real, and that $a_{ij} = \sqrt{-1}b_{ij}+ \lambda\delta_{ij}$, where $(b_{ij}) \in \mathfrak{o}(n, \mathbf{R})$. Write $W_1 = bz\partial_z + \lambda x_i\partial_{x_i}$, and $W_2 = \sqrt{-1}b_{ij}x_i\partial_{x_j}$. Note that
$\mathrm{Re} W_2(r^2)$ does not contain the $|z|^2$ term, so in particular it is not proportional to $r^2$. It follows that $\mathrm{Re} W_2(r^2)$ is a harmonic function. It remains to look at $W_1$. For $\mathrm{Re} W_1(r^2)$ to be a harmonic function, we need \begin{align*}
\Delta \mathrm{Re} W_1(r^2) = \Delta \left(b|z|^2 +
\lambda\frac{n-2}{n-1}|x|^{2\frac{n-2}{n-1}}\right) = 2b + 2\lambda(n-2) = 0, \end{align*} and so $W_1 = (n-2)z\partial_z- x_i\partial_{x_i}$. The corresponding harmonic functions are \begin{align*}
u_1 &= W_1(r^2) = (n-2)|z|^2-\frac{n-2}{n-1}|x|^{2\frac{n-2}{n-1}}, \\
u_2 &= W_2(r^2) = \sqrt{-1}\frac{n-2}{n-1}|x|^{\frac{-2}{n-1}}b_{ij}x_i\bar{x}_j. \end{align*}
In \cite{Sz20}, the same result is obtained using Fourier transform in the $\mathbf{C}$-direction. \end{exmp}
\begin{exmp}\label{exmp:a2}
Let $A_2$ denote the singularity
\begin{align*}
\{ x_1^2+x_2^2+y^3 = 0 \} \subset \mathbf{C}^3.
\end{align*}
Then $A_2$ is isomorphic to $\mathbf{C}^2/\mathbf{Z}_3$ via the map
\begin{align*}
\mathbf{C}^2 &\to \mathbf{C}^3 \\
(z_1,z_2) &\mapsto (\frac{z_1^3+z_2^3}{2}, \frac{z_1^3-z_2^3}{2\sqrt{-1}}, \zeta z_1z_2),
\end{align*}
where $\zeta$ is a cubic root of $-1$. The holomorphic volume form is
given by
\begin{align*}
\Omega = \frac{dx_1\wedge dx_2}{3y^2}.
\end{align*}
Pulling $\Omega$ back to $\mathbf{C}^2$ gives a constant multiple of
$dz_1\wedge dz_2$. The standard flat metric on $\mathbf{C}^2$ thus
gives the correct Calabi-Yau cone metric on $A_2$. The potential
$r^2$ on $A_2$, using the ambient coordinates $x_1, x_2$ and $y$, is
given by
\begin{align*}
r^2 =
&\left(|x_1|^2+|x_2|^2 + \sqrt{(|x_1|^2+|x_2|^2)^2- |y|^6}\right)^{1/3} \\
&+ \left(|x_1|^2+|x_2|^2 - \sqrt{(|x_1|^2+|x_2|^2)^2- |y|^6}\right)^{1/3}.
\end{align*}
This can be seen by solving a cubic equation. The complexified radial
vector field on $\mathbf{C}^2$, $z_i\partial_{z_i}$, pushes forward to
\begin{align*}
3x_1\partial_{x_1}+3x_2\partial_{x_2}+2y\partial_y.
\end{align*}
So $x_1, x_2$ have weight $3$ and $y$ has weight $2$. Alternatively,
the weights can be read off from the complex Monge-Amp\`ere equation.
Any (complex) holomorphic vector field of linear growth is given by
\begin{align*}
W = a_{ij}x_i\partial_{x_j} + by\partial_y,
\end{align*}
where the coefficients $a_{ij}$ and $b$ are chosen so that
$W(x_1^2+x_2^2+y^3) = 0$ and $\mathrm{Im} W(r^2) = 0$. It follows that
\begin{align*}
W = by\partial_y + \sqrt{-1}b_{ij} x_i \partial_{x_j} + c x_i\partial_{x_i},
\end{align*}
where $b, c$ are real with $3b+2c = 0$ and $b_{ij}$ is real and
skew-symmetric. Thus $W_1 = \sqrt{-1}b_{ij} x_i \partial_{x_j}$, and
$W_2 = \frac{1}{3} y\partial_y + \frac{1}{2}
x_i\partial_{x_i}$. $W_2$ is the (complexified) radial vector. The
space of homogeneous ($Jr\partial_r$)-invariant quadratic growth
harmonic functions on $A_2$ is generated by
\[
u_1 = W_1(r^2) =
\frac{1}{3}\sqrt{-1}b_{ij}\frac{r^2}{\sqrt{(|x_1|^2+|x_2|^2)^2-|y|^6}}x_i\bar{x}_j.
\end{aligned} \end{equation} \end{exmp}
\begin{exmp}\label{exmp:cxa2}
We now assume that our cone is $\mathbf{C} \times A_2$. Following
the calculations in the previous examples, it is easily seen that
the space of ($Jr\partial_r$)-invariant homogeneous harmonic
functions with quadratic growth on
\[
\mathbf{C} \times A_2 = \{x_1^2+x_2^2+y^3=0\} \subset \mathbf{C}^4 = \mathbf{C} \times \mathbf{C}^3
\end{aligned} \end{equation}
is generated by
\begin{align*}
u_1 &= \frac{1}{3}\sqrt{-1}b_{ij}\frac{r^2}{\sqrt{(|x_1|^2+|x_2|^2)^2-|y|^6}}x_i\bar{x}_j,\\
u_2 &= 2|z|^2 - r^2,
\end{align*}
where $u_2$ corresponds to the vector
\begin{align*}
W_2 = z\partial_z - \frac{1}{2}(2y\partial_y + 3x_i\partial_{x_i}).
\end{align*}
Let us consider
\begin{align*}
V = \mathrm{Re}(z\partial_z + \frac{1}{3} y\partial_y + \frac{1}{2} x_i \partial_{x_i}).
\end{align*}
Then $L_V \Omega = n\beta \Omega$, and
\begin{align*}
V(|z|^2 + r^2) - \beta (|z|^2 + r^2) = \frac{5}{18} u_2,
\end{align*}
where $\beta = 4/9$.
$V$ generates biholomorphisms
\[\label{eq:aut}
\Phi_t(z, x_1,x_2, y) = (e^{t/2}z, e^{t/4}x_1, e^{t/4}x_2,
e^{t/6}y),
\end{aligned} \end{equation}
where $t \in \mathbf{C}$.
Let us recall the notion $X_0 = \mathbf{C} \times A_2$ and $X_{1,b}$
in the previous sections. The automorphisms $\Phi_t$ fix $X_0$, and
move $X_{1,b}$:
\[
\Phi_t(X_{1,b}) = X_{1,e^{t/3}b}.
\end{aligned} \end{equation}
Thus the only $X_{1,b}$ that is fixed by $\Phi_t$ is $X_{1,0}$.
The effect of the automorphism $\Phi_t$ on the cone metric and the
holomorphic volume form on $X_0$ is seen as
\begin{align*}
\Phi_t^* (|z|^2 + r^2) &= e^t |z|^2 + e^{t/6} r^2, \\
\Phi_t^*( dx_1 \wedge dx_2 \wedge dy) &= e^{2t/3} dx_1 \wedge dx_2 \wedge dy.
\end{align*}
So
\begin{align*}
e^{-4t/9} \phi_t^* (|z|^2 + r^2) = e^{5t/9} |z|^2 + e^{-5t/18} r^2
\end{align*}
defines a Calabi-Yau cone metric on $X_0$ with the same volume form as
that of $|z|^2 + r^2$. Taking Taylor expansion, we have
\begin{align*}
e^{-4t/9} \Phi_t^* (|z|^2 + r^2) = (|z|^2 + r^2) + \frac{5}{18}u_2 t + O(t^2).
\end{align*}
It follows that up to first order, perturbing the cone metric by
$u_2$ corresponds to applying the automorphism $\Phi_t$ and
rescaling. As in \cite{Sz20}, the reason we want to consider $V$ in
place of $\operatorname{Re} W_2$ is that $W_2$ does not fix any of
$X_{1,b}$. Since the automorphisms $\Phi_t$ fix the hypersurface
$X_1=X_{1,0}$, we can still prove a result similar to \cite{Sz20}
(see Proposition~\ref{prop:x1unique} below). \end{exmp}
\subsection{Donaldson-Sun theory}
We now apply Donaldson-Sun theory \cite{DS17} to construct sequences of special embeddings of $\mathbf{C}^3$ into $\mathbf{C}^4$ using holomorphic functions with polynomial growth. The following is similar to \cite[Proposition~3.1]{Sz20}:
\begin{prop} \label{specialembeddings} Suppose $X = \mathbf{C}^3$ is equipped with a Calabi-Yau metric $\omega$ with $\mathbf{C} \times A_2$ as tangent cone at infinity. Then there exists a sequence of holomorphic embeddings $F_i: X \to \mathbf{C}^4$ with the following properties: \begin{enumerate} \item On the ball $B_i$, the map $F_i$ gives a $\Psi(i^{-1})$-Gromov-Hausdorff approximation to the embedding $B(0,1) \to \mathbf{C}^4$, where $B(0,1)$ is the unit ball in $\mathbf{C} \times A_2$.
\item The image $F_i(X)$ is given by the equation \begin{align*} a_i z + b_i y + x_1^2 + x_2^2 + y^3 = 0, \end{align*} for some $a_i > 0, b_i \ge 0$. Either all $b_i = 0$ or all $b_i \ne 0$. \item There exists a point $o \in X$ such that $F_i(o) = 0$ for all $i$. \item The volume form $\omega^3$ satisfies \begin{align*} 2^{-6i} \omega^3 = F_i^*(\sqrt{-1} \Omega \wedge \overline{\Omega}), \end{align*} where $\Omega = a_i^{-1}dx_1\wedge dx_2 \wedge dy$ is the holomorphic volume form on $X_{a_i,b_i} = F_i(X)$. \item $a_i/a_{i+1} \to 2^5$ and $b_i/b_{i+1} = 2^{3/2}(a_i/a_{i+1})^{1/2} \to 2^4$ (when $b_i \ne 0$) as $i \to \infty$. Furthermore, the number $b=b_ia_i^{-1/2}2^{3i/2}$ is independent of $i$ and independent of the sequence. \end{enumerate} We call any sequence of embeddings satisfying the above properties a sequence of special embeddings. \end{prop}
\begin{proof}
The proof follows a similar strategy of \cite[Proposition
3.1]{Sz20}. Let $x_1, x_2, y, z$ be holomorphic functions on the
cone $X_0 = \mathbf{C} \times A_2$ with weight $3,3,2,1$,
correspondingly. Recall that the defining equation is given by
$x_1^2+x_2^2+y^3=0$. Let $F_i = (x_1^i, x_2^i, y^i, z^i)$ be the
sequence of holomorphic embeddings of $X$ into $\mathbf{C}^4$, where
the components have weights $3,3,2,1$ respectively, such that over
the balls $B(p_i, 1) = B(p, 2^{i})$ scaled down to unit size, $F_i$
converge in the Gromov-Hausdorff sense to $F = (x_1,x_2,y,z)$, the
embedding of $X_0$ to $\mathbf{C}^4$. Such a sequence of embeddings
can be obtained using adapted sequences of bases for holomorphic
functions (see \cite[Proposition~3.26]{DS17}). By comparing
dimensions of the corresponding spaces of holomorphic functions with
polynomial growth, we see that $x_1^i, x_2^i, y^i, z^i$ must satisfy
a polynomial equation, each term of which has weight at most
$6$. For notational simplicity we suppress the index $i$ in the
discussion below. By making a change of variables that does not
change the weights of the variables (i.e. completing the squares to
kill off the terms $x_1,x_2, x_1x_2$ and a shift in $y$ by a scalar
to kill off the $y^2$ term), the equation reduces to \begin{align*} x_1^2+x_2^2+y^3+f(z)y+g(z) = 0, \end{align*} where $f(z)$ is a polynomial of degree at most $4$ and $g(z)$ is a polynomial of degree at most $6$. We claim that $f(z)$ can only be a constant and that $g(z$) can only be linear. Otherwise, by putting suitable weights to $x_1,x_2,y,z$, we may assume that the variety degenerates to one of the following singular hypersurfaces: \begin{itemize} \item $x_1^2+x_2^2+y^3+z^ky=0$, \item $x_1^2+x_2^2+y^3+z^l=0$, \item $x_1^2+x_2^2+y^3+az^2y+bz^3=0, a,b \ne 0$, \end{itemize} where $1 \le k \le 4$ and $2 \le l \le 6$ ($l = 1$ is biholomorphic to $\mathbf{C}^3$).
In the first two cases, the Milnor number of each isolated singularity is positive. By Milnor's fibration theorem \cite{Milnor}, the smoothing has nontrivial topology. In fact, it is homotopy equivalent to a bouquet of spheres, where the number of spheres is given by the Milnor number. Therefore it cannot be homeomorphic to $\mathbf{C}^3$. In the third case, if $27b^2 + 4a^3 \ne 0$ then we again have an isolated singularity (it is the three-dimensional $A_2$). If $27b^2 + 4a^3 = 0$, then we have an isolated line singularity of the form $x_1^2 + x_2^2 + vw^2 = 0$ after a change of variables. In this case the Milnor fiber is still homotopy equivalent to a bouquet of spheres \cite{Siersma}. It follows that $f(z)$ can only be a constant, and $g(z)$ must be linear. For now we conclude that the image of $F_i$ in $\mathbf{C}^4$ is given by \[ e_i + a_i z + b_i y + x_1^2 + x_2^2 + y^3 = 0, \end{aligned} \end{equation} where $e_i, a_i$ and $b_i$ are complex numbers. To kill off the constant term, we make a change of variables $z \to z + a_i^{-1}e_i$. We need to ensure that that $a_i^{-1}e_i \to 0$ as $i \to 0$. As pointed out in \cite[Lemma 5]{Sz20}, if we have two sets of of holomorphic functions $(z, x_1,x_2,y)$ and $(\tilde{z}, \tilde{x}_1, \tilde{x}_2, \tilde{y})$ on $X$ with weights $(1,3,3,2)$ such that \[ az + by + x_1^2 + x_2^2 + y^3 = 0 \end{aligned} \end{equation} and \[ c\tilde{z} + d\tilde{y} + \tilde{x}_1^2 + \tilde{x}_2^2 + \tilde{y}^3 = 0, \end{aligned} \end{equation} then using the fact that both sets of holomorphic functions generate the space of holomorphic functions with growth rates $\le 6$, we see that $\tilde{z} = k_1 z$, $\tilde{y} = k_2 y$ and $(\tilde{x}_1,\tilde{x}_2) = A(x_1,x_2)$ for some scalars $k_1, k_2$ and an invertible matrix $A$ with $A^TA = k_3^2\mathrm{Id}$. We may assume $A = k_3 \mathrm{Id}$ for some scalar $k_3$. From this it follows that the two sets of holomorphic functions have a common zero $o \in X$. Since $F_i$ converges to the standard embedding $F: B(0,1) \to \mathbf{C}^4$ of the cone, it follows that $F_i(o) \to 0 \in \mathbf{C}^4$. This implies that $a_i^{-1}e_i \to 0$. Thus we can absorb this small constant term to $z$. We make a stop and conclude what we got so far: \begin{itemize} \item A sequence of embeddings $F_i = (x_1^i,x_2^i,y^i,z^i)$ of $X$ into $\mathbf{C}^4$ such that $F_i$ converges to $F$ over $B(p_i,1) \to B(0,1)$. \item The image $F_i(X)$ is given by the equation \[ a_i z + b_i y+ x_1^2 + x_2^2 + y^3 = 0, \end{aligned} \end{equation} with $a_i, b_i \to 0$ as $i \to 0$. \item There exists $o \in X$ such that $F_i(o) = 0$. \end{itemize} We still need to conclude (4) and (5) in the statement of the proposition. Pulling back the volume form using $F_i$, we have \[
F_i^*(\sqrt{-1}\Omega\wedge\overline\Omega) = |g_i|^2 \omega^3 \end{aligned} \end{equation} for some nowhere vanishing polynomial growth holomorphic function $g_i$ on $X$. Therefore $g_i$ must be a constant (recall that $X$ is biholomorphic to $\mathbf{C}^3$). By Colding's volume convergence, \[ 2^{-6i} \int_{B(p,2^{2i})} \omega^3 \to \int_{B(0,1)} F^*(\sqrt{-1}\Omega \wedge \overline\Omega), \end{aligned} \end{equation}
it follows that $2^{6i}|g_i|^2 \to 1$ as $i\to 0$. Scaling $z$ by a factor $\Psi(i^{-1})$-close to $1$, we may assume $|g_i|^2 = 2^{-6i}$. Finally, the image of $F_i$ and $F_{i+1}$ are given by \[ a_iz + b_iy + x_1^2 + x_2^2 + y^3 = 0 \end{aligned} \end{equation} and \[ a_{i+1}z + b_{i+1}y + x_1^2 + x_2^2 + y^3 = 0, \end{aligned} \end{equation} respectively. Using the argument finding $o$ such that $F_i(o) = 0$ above, we see that the coefficients of these equations satisfy \[ \frac{a_i}{k_1a_{i+1}}= \frac{b_i}{k_2b_{i+1}} = \frac{1}{k_2^3} = \frac{1}{k_3^2}, \end{aligned} \end{equation} where $k_i$ are such that $z^{i+1} = k_1z^i, y^{i+1} = k_2 y^i, (x_1^{i+1}, x_2^{i+1}) = k_3(x_1^i, x_2^i)$. By the definition of adapted sequences of bases (\cite[Proposition~3.26]{DS17}), we have $k_1 \to 2^{-1}, k_2 \to 2^{-2}, k_3 \to 2^{-3}$ as $i \to \infty$. The negative of the powers of $2$ here are the respective growth rates of the functions. From these, along with the relation given by the volume forms \[ 2^{6i}F_i^*(\sqrt{-1}\Omega\wedge\overline\Omega) = 2^{6(i+1)}F_{i+1}^*(\sqrt{-1}\Omega\wedge\overline\Omega), \end{aligned} \end{equation} we deduce the limits of $a_i/a_{i+1}$ and $b_i/b_{i+1}$. The same method shows that $b$ is independent of the sequence constructed here. Finally, to make $a_i > 0$ we simply rotate the $z$ variable. To make $b_i \ge 0$, we compose $F_i$ with the following linear automorphism of $\mathbf{C}^4$: \[ G_i(z,x_1,x_2,y) = (e^{t_i/2}z, e^{t_i/4}x_1, e^{t_i/4}x_2, e^{t_i/6}y) \end{aligned} \end{equation} (this is $\Phi_t$ in Example~\ref{exmp:cxa2}) for some suitable $e^{t_i} \in S^1$. Note that $G_i$ preserves the volume form. \end{proof}
From the proposition we immediately have the following:
\begin{cor}\label{cor:distinguish}
Suppose $\omega,\omega'$ are two isometric Calabi-Yau metrics on
$\mathbf{C}^3$ with tangent cone $\mathbf{C} \times A_2$ at
infinity. Then $b = b'$ in Proposition~\ref{specialembeddings}. \end{cor}
Note that Corollary~\ref{cor:distinguish} does not imply that the metrics that we constructed in Theorem~\ref{thm:construction} are distinct in our sense. Actually, if we apply an automorphism and also a scaling to a metric in Corollary~\ref{cor:distinguish}, then its invariant $b$ scales correspondingly.
When $b=0$, we have the following uniqueness result:
\begin{prop}\label{prop:x1unique}
Let $X$ be a Calabi-Yau manifold biholomorphic to $\mathbf{C}^3$
with tangent cone $\mathbf{C} \times A_2$ at infinity. If $b = 0$ in
Proposition~\ref{specialembeddings}, then up to scaling, $X$ is
isometric to $X_{1,0}$ equipped with the Calabi-Yau metric
$\omega_{1,0}$ in Theorem~\ref{thm:construction}. \end{prop}
\begin{proof}
The proof is very similar to the $\mathbf{C} \times A_1$ case in
\cite{Sz20}, modulo the special embeddings established in
Proposition~\ref{specialembeddings} and the computations of
quadratic harmonic functions and the corresponding vector fields and
automorphisms on $\mathbf{C} \times A_2$ that are supplemented in
Example~\ref{exmp:cxa2}. Note that the key reason we can follow the
proof in \cite{Sz20} is that all the vector fields and automorphisms
associated to quadratic harmonic functions of
$\mathbf{C} \times A_2$ actually fix $X_{1,0}$. \end{proof}
We now turn to distinguishing the metrics in Theorem~\ref{thm:construction}. For this, we need the following explicit asymptotic information of the metrics that we have constructed:
\begin{prop} \label{prop:asymptotics} Let $\omega_{1,b}$ be the Calabi-Yau metric on $X_{1,b}$ constructed in Theorem~\ref{thm:construction}, and let $d$ be the distance function with respect to $\omega_{1,b}$. Then we have \[
\lim_{\rho \to \infty} \frac{d(0, (z,x))^2}{|z|^2 + r^2} = 1, \end{aligned} \end{equation} where $0 \in X_{1,b} \subset \mathbf{C}^4$. \end{prop}
\begin{proof} Since the metric $\omega_{1,b}$ is a small perturbation in weighted spaces of the approximate solution $\omega$ on $X_1$, this follows directly from Corollary~\ref{cor:asymptotics}. \end{proof}
Together with the above proposition, we can follow the idea of the proof of Proposition~\ref{specialembeddings} to distinguish the model metrics $\omega_{1,b}$ on $X_{1,b}$:
\begin{prop}\label{prop:distinguish} There exist a biholomorphism $F: X_{1,b} \to X_{1,b'}$ and a scaling $c>0$ such that $F^*\omega_{1,b'} = c^2\omega_{1,b}$ if and only if $b = b'$. \end{prop}
\begin{proof}
Let $(z, x_1,x_2,y)$ and $(z',x_1',x_2',y')$ be the coordinate
functions on $X_{1,b}$ and $X_{1,b'}$, respectively. Since
$F^*\omega_{1,b'} = c^2\omega_{1,b}$, The set of functions
$(z'\circ F,x_1'\circ F,x_2'\circ F,y'\circ F)$ has the same set of
growth rates that of $(z, x_1,x_2,y)$. By comparing the equations we
necessarily have $z'\circ F = a_1z$, $y'\circ F = a_2 y$ and
$x_i'\circ F = a_3 x_i$ (say) for some $a_i \ne 0 \in \mathbf{C}$,
and
\[
\frac{1}{a_1} = \frac{b}{b'a_2} = \frac{1}{a_2^3} = \frac{1}{a_3^2}.
\end{aligned} \end{equation}
In particular we have $F(0) = 0$ and $F(x) \to \infty$ as
$\rho(x) \to \infty$. By comparing the volume forms we have
\[
|a_3|^4 |a_2|^2 = c^6.
\end{aligned} \end{equation}
On the other hand, using the assumption and the fact that $r$ is
homogeneous, we get
\[
F^*\left(\frac{d(0, (z',x'))^2}{|z'|^2 + r^2}\right) = \frac{c^2
d(0, (z,x))^2}{|a_1|^2(|z|^2 + r^2)}.
\end{aligned} \end{equation}
Taking limit of both sides of the above equation as $\rho \to \infty$
and using Proposition~\ref{prop:asymptotics}, we conclude that
$c = |a_1|$. Combining these we see that $c=1$ and $b=b'$. \end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:A2}]
This is a combination of Theorem~\ref{thm:construction} and
Proposition~\ref{prop:distinguish}. \end{proof}
Based on these elementary observations, we state the following refinement of a conjecture of Sz\'ekelyhidi~\cite{Sz20}:
\begin{conj} \label{conj:A2} The space of Calabi-Yau metrics on $\mathbf{C}^3$ with tangent cone $\mathbf{C} \times A_2$ at infinity, up to biholomorphism and scaling, is parametrized by $\mathbf{C}/S^1 \cong \mathbf{R}_{\ge 0}$. \end{conj}
Difficulties arise when one tries to generalize the decay estimate approach in \cite{Sz20} to prove Conjecture~\ref{conj:A2}. An initial technical issue is that the linear automorphisms
$\Phi_t: \mathbf{C}^4 \to \mathbf{C}^4$ in Example~\ref{exmp:cxa2}, which correspond to the quadratic harmonic function $2|z|^2-r^2$ on the cone $\mathbf{C} \times A_2$, do not preserve the hypersurface $X_{1,b}$. It is therefore crucial to understand how the metrics $\omega_{1,b}$, or their potentials $\varphi_b$, change with respect to the parameter $b$. In the terminology in \cite[Section~3]{CSz}, we expect that for a given $b$, the metrics
$\Phi_t^* \omega_{1, e^{t/3}b}$ on $X_{1,b}$ for $|t| \ll 1$ form a family of model metrics parametrized by small quadratic harmonic functions on the cone $\mathbf{C} \times A_2$. Another difficulty, which seems more substantial, is due to the parameter space $[0,\infty)$ being non-compact. In the sequence of special embeddings, if $a_i$ deviates largely from $2^{-5i}$, then $b_i$ will deviate even more from $2^{-4i}$ as the decay rate of $b_i$ is slower than $a_i$. To follow a similar argument as seen in \cite{Sz20}, we need to have some kind of uniform control of the family of spaces $(X_{1,b},\omega_{1,b})$ as $b \to \infty$, possibly with suitable rescalings. To overcome these difficulties, a finer gluing construction might be needed in order to understand the metric behavior in the compact region. Alternatively, one could also try to establish a priori estimates for the complex Monge-Amp\`ere equation in the maximal volume growth setting. We leave these to future work.
\end{document} | arXiv |
\begin{document}
\setlength{\abovedisplayskip}{0.8ex} \setlength{\abovedisplayshortskip}{0.6ex}
\setlength{\belowdisplayskip}{0.8ex} \setlength{\belowdisplayshortskip}{0.6ex}
\title{\Large\textsc{Elliptic stars in a chaotic night}
\abstract{A recurrent theme in the description of phase portraits of dynamical
systems is that of {\em elliptic islands in a chaotic sea}. Usually this
picture is invoked in the context of smooth twist maps of the annulus or the
torus, like the standard map. In this setting `elliptic islands' refers to the
topological disks bounded by periodic smooth curves surrounding elliptic
periodic points. Establishing the existence of these curves is one of the many
achievements of KAM-theory.
The aim of this note is to approach the topic from a different
angle, namely from the viewpoint of rotation theory in a purely
topological setting. We study homeomorphisms of the two-torus,
homotopic to the identity, which have no wandering open sets (as in
the area-preserving case) and whose rotation set has non-empty
interior. We define local rotation subsets $\rho_F(U)$ by
restricting Misiurewicz and Ziemian's definition of the rotation set
to starting points in a small open disk $U$. Our main result is the
following dichotomy: Either $\rho_U(F)$ is reduced to a single
rational vector and $U$ is contained in a periodic topological open
disk which contains a periodic point, or $\rho_U(F)$ is large, in
the sense that its convex hull has non-empty interior. This allows
to distinguish an `elliptic' and a `chaotic' regime, and as a
consequence we obtain that in the chaotic region the dynamics
are sensitive with respect to initial conditions.
In order to demonstrate these results we introduce a parameter family of
smooth toral diffeomorphisms that is inspired by an example of Misiurewicz and
Ziemian. The pictures obtained from simulations in this family motivate an
alternative formulation of the original theme. }
\enlargethispage*{1000pt}
\begin{figure}
\caption{ \small Elliptic islands surrounding two 2-periodic
orbits of the map $f_\alpha(x,y) =
(x+\alpha\sin(2\pi(y+\alpha\sin(2\pi x))),y+\alpha\sin(2\pi x))$ with
$\alpha=0.5$.}
\label{f.1}
\end{figure}
\pagebreak
\section{Introduction}
We denote by \ensuremath{\mathrm{Homeo}_0(\mathbb{T}^2)}\ the set of homeomorphisms of the two-torus that are homotopic to the identity. Given a lift $F:\ensuremath{\mathbb{R}}^2 \to \ensuremath{\mathbb{R}}^2$ of $f\in\ensuremath{\mathrm{Homeo}_0(\mathbb{T}^2)}$, Misiurewicz and Ziemian \cite{misiurewicz/ziemian:1989} introduced the rotation set of $F$ as \begin{equation}\label{e.rot-set}
\rho(F) \ := \ \left\{ \rho \in \ensuremath{\mathbb{R}}^2 \left|\ \exists n_i\nearrow\infty,\ z_i\in\ensuremath{\mathbb{R}}^2 :
\ensuremath{\lim_{i\rightarrow\infty}} \left(F^{n_i}\left(z_i\right) - z_i\right)/n_i \ = \ \rho \right.\right\} \ . \end{equation} This set is always compact and convex \cite{misiurewicz/ziemian:1989}. Further, the properties of $\rho(F)$ have strong implications for the dynamics of $f$. In particular, this is true for the situation we will concentrate, namely when $\rho(F)$ has non-empty interior. In this case all rotation vectors in $\ensuremath{\mathrm{int}}(\rho(F))$ are realised on minimal sets \cite{franks:1989,misiurewicz/ziemian:1991,jaeger:2009b} and the topological entropy of $f$ is strictly positive \cite{llibre/mackay:1991}. Further the set $\left\{f\in\ensuremath{\mathrm{Homeo}_0(\mathbb{T}^2)}\mid
\ensuremath{\mathrm{int}}(F)\neq \emptyset\right\}$ is an open and therefore, in a topological sense, large subset of \ensuremath{\mathrm{Homeo}_0(\mathbb{T}^2)}\ \cite{misiurewicz/ziemian:1991}.
Our aim is to give some meaning to the notion of {\em elliptic islands in a
chaotic sea} in this purely topological setting. To that end, we restrict the definition in (\ref{e.rot-set}) to orbits starting in some subset $U\ensuremath{\subseteq}\ensuremath{\mathbb{T}^2}$. Let $\pi:\ensuremath{\mathbb{R}}^2\to\ensuremath{\mathbb{T}}^2$ denote the canonical projection. We define the {\em rotation subset on $U$} by \begin{equation} \label{e.rotation-subset} \rho_U(F) \ := \ \left\{ \rho \in
\ensuremath{\mathbb{R}}^2 \left| \ \exists z_i\in \pi^{-1}(U),\ n_i\nearrow \infty : \ensuremath{\lim_{i\rightarrow\infty}}
\left(F^{n_i}\left(z_i\right)-z_i\right)/n_i = \rho \right. \right\} \ . \end{equation} In general, even when $U$ is open $\rho_U(F)$ can be much smaller than $\rho(F)$. For instance, when $f$ is a sufficiently smooth toral diffeomorphism then generic elliptic periodic points are surrounded by periodic invariant curves (see, for example, \cite{moser:1962,katok/hasselblatt:1997}). The rotation subsets of the corresponding topological disks contain a single rational rotation vector, whereas $\rho(F)$ may have non-empty interior. A more general example is sketched in Remark~\ref{r.denjoy-construction} below.
However, when $U$ is open and recurrent, then in a number of situations $\rho(F)$ is already determined by $\rho_U(F)$. In order to give precise statements, we need some notation and terminology. We say $U\ensuremath{\subseteq} \ensuremath{\mathbb{T}^2}$ is {\em
bounded} if the connected components of its lift to $\ensuremath{\mathbb{R}}^2$ are bounded. Given $f\in\ensuremath{\mathrm{Homeo}_0(\mathbb{T}^2)}$ we say $U$ is {\em wandering} if $f^n(U) \cap U = \emptyset \ \forall n\geq 1$ and {\em non-wandering} otherwise. We call $U$ {\em recurrent} if there exist infinitely many $n\in\ensuremath{\mathbb{N}}$ with $f^n(U) \cap U \neq \emptyset$. We call $z\in\ensuremath{\mathbb{T}^2}$ {\em wandering} if it is contained in some wandering open set and {\em non-wandering} otherwise. It is easy to see that if $U$ is open and contains a non-wandering point then it is recurrent. Finally, we say that $f$ is {\em non-wandering} if it has all points non-wandering. In this case all open sets are recurrent. Note that any area-preserving toral homeomorphism is non-wandering.
Given a lift $F$ of $f\in\ensuremath{\mathrm{Homeo}_0(\mathbb{T}^2)}$, let $\varphi_n(z) = \left(F^n(z)-z\right)/n$. If $\lambda\in\ensuremath{\mathbb{R}}$ and $v\in\ensuremath{\mathbb{R}}^2\ensuremath{\setminus}\{0\}$, let $\ensuremath{L_{\lambda,v}} = \ensuremath{\lambda v + \{v\}^\perp}$. \begin{thm} \label{t.semilocal-rotsets} Suppose $F$ is a lift of $f\in\ensuremath{\mathrm{Homeo}_0(\mathbb{T}^2)}$
and $U\ensuremath{\subseteq} \ntorus$ is open, bounded, connected and recurrent. \alphlist \item If $\rho_U(F) \ensuremath{\subseteq} \ensuremath{L_{\lambda,v}}$ for some $\lambda\in\ensuremath{\mathbb{R}}$, $v\in\ensuremath{\mathbb{R}}^2\ensuremath{\setminus}\{0\}$, then
either $\rho(F) \ensuremath{\subseteq} \ensuremath{L_{\lambda,v}}$ or $\rho_U(F)$ is reduced to a single rational
vector. \item If ${\cal S}$ is a line segment of positive length without rational points
and $\rho_U(F) = {\cal S}$, then $\rho(F) = {\cal S}$. Further, $\varphi_n(U)$
converges to ${\cal S}$ in Hausdorff distance as $n\to \infty$. \item If $\rho_U(F) = \{\rho\}$ with $\rho\in\ensuremath{\mathbb{R}}^2$ irrational\footnote{We call
a vector $\rho=\left(\rho_1,\rho_2\right)\in \ensuremath{\mathbb{R}}^2$ {\em rational} if $\rho\in\ensuremath{\mathbb{Q}}^2$, {\em
irrational} if $\rho_1,\rho_2,\rho_1/\rho_2 \notin\ensuremath{\mathbb{Q}}$ and {\em
semi-rational} if it is neither rational nor irrational.} then
$\rho(F)=\{\rho\}$. \end{list}
\end{thm}
\begin{rem} \label{r.denjoy-construction} We note that when no recurrence
assumption is made no relation between $\rho_U(F)$ and $\rho(F)$ can be expected.
Without going into detail, we want to mention a possible way to construct
respective examples: When $\rho(F)$ has non-empty interior, then for any
compact connected subset $C \ensuremath{\subseteq} \rho(F)$ there exists a point $z\in\ensuremath{\mathbb{T}}^2$
with $\rho_{\{z\}}(F) = C$ \cite{llibre/mackay:1991}. By blowing up the
points in the orbit of $z$ to small disks in a Denjoy-like construction one
may thus obtain a wandering open set $U$ whose rotation set is an arbitrary
compact connected subset of the rotation set. \end{rem}
Theorem~\ref{e.rot-set}(i) implies that if $f$ is non-wandering and $\rho(F)$ has non-empty interior then the rotation subset of $U$ can only be contained in a line if it is reduced to a single rational rotation vector, that is, $\rho_U(F) = \{\rho\}$ with $\rho\in\ensuremath{\mathbb{Q}}^2$. Together with some additional details on the rational case, this yields our main result. \enlargethispage*{1000pt}
\begin{thm}
\label{t.elliptic-islands} Let $F:\ensuremath{\mathbb{R}}^2 \to \ensuremath{\mathbb{R}}^2$ be a lift of $f\in\ensuremath{\mathrm{Homeo}_0(\mathbb{T}^2)}$
and suppose that $f$ is non-wandering and $\rho(F)$ has non-empty interior.
Then for any open, bounded and connected set $U$ one of the following two holds. \romanlist
\item $\rho_U(F)$ is reduced to a single rational vector $\rho$ and $U$ is
contained in an embedded topological open disk $D\ensuremath{\subseteq} \ensuremath{\mathbb{T}^2}$ which is
invariant under some iterate $f^p$ and contains a $p$-periodic point.
\item The convex hull of $\rho_U(F)$ has non-empty interior.
\end{list}
\end{thm} \pagebreak
The above result allows to give an intrinsic definition of `elliptic' and `chaotic' regions. Given a set $A\ensuremath{\subseteq}\ensuremath{\mathbb{R}}^2$ we denote by $\ensuremath{\mathrm{Conv}}(A)$ its convex hull and by $\ensuremath{\mathrm{int}}(A)$ its interior. \begin{definition}
Suppose $f\in\ensuremath{\mathrm{Homeo}_0(\mathbb{T}^2)}$. Let
\begin{eqnarray*} {\cal E}(f) & := & \left\{ z\in\ensuremath{\mathbb{T}^2} \mid \#\rho_U(F) = 1 \
\textrm{ for some open
neighbourhood } U \textrm{ of } x \right\} \quad , \\
{\cal C}(f) & := & \left\{ z\in\ensuremath{\mathbb{T}^2} \mid \ensuremath{\mathrm{int}}\left(\ensuremath{\mathrm{Conv}}(\rho_U(F))\right) \neq
\emptyset \ \ \forall \textrm{open neighbourhoods } U \textrm{ of } x \right\} \ .
\end{eqnarray*} \end{definition} A point $z\in\ensuremath{\mathbb{T}^2}$ is called {\em $\ensuremath{\varepsilon}$-Lyapunov stable} if there exists some $\delta > 0$ such that $f^n\left(B_\delta(z)\right) \ensuremath{\subseteq} B_\ensuremath{\varepsilon}\left(f^n(z)\right) \ \forall n\in\ensuremath{\mathbb{N}}$ and {\em Lyapunov stable} if it is $\ensuremath{\varepsilon}$-Lyapunov stable for all $\ensuremath{\varepsilon} > 0$. As one should expect for a notion of stability, Lyapunov stable points do not occur in the `chaotic' regime. \begin{prop} \label{p.chaotic} Suppose $f\in\ensuremath{\mathrm{Homeo}_0(\mathbb{T}^2)}$ and $\rho(F)$ has non-empty
interior.\alphlist \item If $f$ is non-wandering then no point in ${\cal C}(f)$ is
$\frac{1}{2}$-Lyapunov stable. \item If $f$ is area-preserving, $U$ is a connected and bounded neighbourhood of
$z\in {\cal C}(f)$ and $\widehat U$ is a connected component of $\pi^{-1}(U)$
then $\limsup_{n\to\infty}\ensuremath{\mathrm{diam}}\ntel\left(F^n(\widehat U)\right) > 0$. \end{list} \end{prop} Note that in contrast to this, in the construction sketched in Remark~\ref{r.denjoy-construction} all points in the wandering topological disks will be Lyapunov stable provided the diameter of these disks goes to zero along the orbit.
The paper is organised as follows: In Section~\ref{Basic} we collect a number of basic statements on rotation subsets. Section~\ref{Semilocal-rotsets} then contains the technical core of the paper. We work on the universal cover $\ensuremath{\mathbb{R}}^2$ and consider bounded open and connected sets that intersect their image. In this setting, we describe a number of situations in which the rotation subset already determines the rotation set, or at least forces it to be contained in a line segment. In Section~\ref{Proofs} these statements are then used to prove the main results. Finally, in Section~\ref{MZ-family} we provide some explicit examples to which our results apply. To that end, we introduce a parameter family of smooth torus diffeomorphisms that is based on an example by Misiurewicz and Ziemian in \cite{misiurewicz/ziemian:1991}. For appropriate parameter values these maps have a rotation set with non-empty interior, and simulations clearly indicate that they also exhibit elliptic islands.
{\bf Acknowledgements.} This work was supported by an Emmy Noether Grant (Ja 1721/2-1) of the German Research Council (DFG). The results were partially presented at the Workshop on Dynamics in Dimension two, April 2009 in Puc\'on (Chile). I would like to thank the organisers Mario Ponce and Andres Navas for the great opportunity they created. I am further indebted to Andres Koropecki and Patrice Le Calvez for thoughtful remarks and stimulating discussions and to Hendrik Vogt for drawing my attention to the fine structure of the elliptic islands depicted in Figure~\ref{fig.7}. Finally, I would like to thank Jean-Christophe Yoccoz for his support during my time at the Coll\`ege de France, where a part of this work was carried out.
\section{Some basic results on rotation subsets} \label{Basic}
The aim of this section is to collect a number of elementary statements on rotation subsets and rotation vectors that will be used in the later sections. For the purposes of this section there is no need to restrict to dimension 2. Hence, we will work on $\ensuremath{\mathbb{T}}^d$ ($d\in\ensuremath{\mathbb{N}}$), with the definitions of the rotation set and rotation subsets analogous to those on $\ensuremath{\mathbb{T}^2}$.
\noindent {\em Notation.} \ We denote the Euclidean scalar product of vectors $v,w\in\ensuremath{\mathbb{R}}^d$ by $\langle v,w\rangle$ and also write $\vbr{w}$
instead of $\langle v,w \rangle$. By $\|v\|=\sqrt{\langle v,v\rangle}$ we denote the Euclidean length of the vector $v$. If $G$ is an additive group then $G_* = G\ensuremath{\setminus}\{0\}$. By $\ensuremath{\mathrm{Conv}}(C)$ we denote the convex hull of a subset $C\ensuremath{\subseteq} \ensuremath{\mathbb{R}}^d$. By $\textrm{Ex}(C)$ we denote the extremal points of $\ensuremath{\mathrm{Conv}}(C)$ and let $\ensuremath{\mathrm{Conv}}_{\! \times}(C) = \ensuremath{\mathrm{Conv}}(C)\ensuremath{\setminus}\textrm{Ex}(C)$.
If $v=\left(v_1,v_2\right) \in \ensuremath{\mathbb{R}}^2$ we let $v^\perp = \left(-v_2,v_1\right)$. Given $\lambda\neq 0$, $v\in\ensuremath{\mathbb{R}}^2_*$ and $a\leq b \in \ensuremath{\mathbb{R}}\cup\{\pm \infty\}$ we let $\ensuremath{L_{\lambda,v}} =\lambda v + \{v\}^\perp$ and \begin{eqnarray*}
C_v[a,b] & = & \{z\in\ensuremath{\mathbb{R}}^2 \mid a\langle z,v\rangle \leq
\langle z,v^\perp \rangle \leq b\langle z,v \rangle \} \\
\ensuremath{L_{\lambda,v}}[a,b] & = & \{z\in \ensuremath{L_{\lambda,v}} \mid
a\langle z,v\rangle \leq \langle z,v^\perp \rangle \leq b\langle z,v \rangle \} \ . \end{eqnarray*} Note that thus $\ensuremath{L_{\lambda,v}}(a,b) = \ensuremath{L_{\lambda,v}} \cap C_v(a,b)$ and $C_v[a,a] = \ensuremath{\mathbb{R}} \cdot (v+av^\perp)$. Further, we let \begin{eqnarray*} C^+_v[a,b] & = & \{ z\in C_v[a,b] \mid \langle z,v \rangle \geq 0\} \quad \textrm{and} \\ S_v[a,b] & = & \{ z\in\ensuremath{\mathbb{R}}^2 \mid \langle z,v\rangle \in [a,b]\} \ . \end{eqnarray*} All these notions are used similarly for open and half-open intervals.
The following basic observation is a direct consequence of the definition in \eqref{e.rotation-subset}. Recall that $\varphi_n(z)=(F^n(z)-z)/n$.
\begin{lem} \label{l.convergence} Suppose $F$ is a lift of
$f\in\ensuremath{\mathrm{Homeo}_0(\mathbb{T}^d)}$ and $U\ensuremath{\subseteq} \ensuremath{\mathbb{T}}^d$. Then for all $\ensuremath{\varepsilon} > 0$ there exists
some $n_0=n_0(\ensuremath{\varepsilon}) \in\ensuremath{\mathbb{N}}$ such that $\varphi_n(U) \ensuremath{\subseteq}
B_{\ensuremath{\varepsilon}}(\rho_U(F)) \ \forall n \geq n_0$.
\end{lem}
The proof of the following statementis more or less identical to
that of the connectedness of the rotation set in
\cite{misiurewicz/ziemian:1989}, but we include it for the
convenience of the reader. \begin{lem} \label{l.connectedness} Let $F$ be a lift of
$f\in\ensuremath{\mathrm{Homeo}}_0(\ensuremath{\mathbb{T}}^d)$. For any $U\ensuremath{\subseteq} \ensuremath{\mathbb{T}}^d$, the set $\rho_U(F)$ is
compact. Further, if $U$ is connected then so is $\rho_U(F)$. \end{lem} {\em Proof of Lemma~\ref{l.connectedness}.} The fact that $\rho_U(F)$ is compact follows immediately from the definition. Suppose for a contradiction that $U$ is connected but $\rho_U(F)$ is not. Then there exist disjoint open sets $V_1$ and $V_2$ with $\rho_U(F) \ensuremath{\subseteq} V_1 \cup V_2$ and $\rho_U(F) \cap V_i \neq \emptyset \ (i=1,2)$. Since $\rho_U(F)$ is compact, we may assume that $\ensuremath{\varepsilon} = d\left(V_1,V_2\right) > 0$. Lemma~\ref{l.convergence} implies that there exists $n_0\in\ensuremath{\mathbb{N}}$ such that $\varphi_n(U) \ensuremath{\subseteq} V_1\cup V_2 \
\forall n\geq n_0$. Further, since $\sup_{z\in\ensuremath{\mathbb{R}}^2} \|F(z)-z\| < \infty$ there exists $n_1\in\ensuremath{\mathbb{N}}$ such that \begin{equation}
\left|\left(F^{n+1}(z)-z\right)/(n+1) - \left(F^n(z)-z\right)/n\right| \ <
\ \ensuremath{\varepsilon} \quad \forall n\geq n_1,\ z\in\ensuremath{\mathbb{R}}^2 \ . \end{equation} It follows that if $n\geq n_1$ and $\varphi_n(U) \ensuremath{\subseteq} V_i$, then $\varphi_k(U) \ensuremath{\subseteq} V_i \ \forall k\geq n$ and therefore $\rho_U(F) \ensuremath{\subseteq} \overline{V_i}$. Since this is not the case, $\varphi_n(U)$ must intersect both $V_1$ and $V_2$ for all $n\geq n_1$. However, for $n\geq \max\{n_0,n_1\}$ we then obtain $\varphi_n(U) \ensuremath{\subseteq} V_1\cup V_2$ and $\varphi_n(U) \cap V_i\neq \emptyset \ (i=1,2)$. This contradicts the connectedness of $\varphi_n(U)$. \qed
\noindent {\em Deviations from a constant rotation and invariant measures.} For $f\in\ensuremath{\mathrm{Homeo}_0(\mathbb{T}^d)}$ with lift $F :\ensuremath{\mathbb{R}}^d \to \ensuremath{\mathbb{R}}^d$, $\rho\in\ensuremath{\mathbb{R}}^d$ and $v\in\ensuremath{\mathbb{R}}^d_*$ we let \begin{equation}
\label{e.deviations}
D_n(z,\rho) \ := \ F^n(z)-z-n\rho \quad \textrm{and} \quad D^v_n(z,\rho) \ :=
\ \left\langle D_n(z,\rho)\right\rangle_v \ . \end{equation} If we need to make the dependence on $f$ explicit, we also write $D_{f,n}(z,\rho)$ and $D^v_{f,n}(z,\rho)$. For any $f$-invariant probability measure $\mu$, the rotation number of $f$ with respect to $\mu$ is given by \begin{equation}
\label{e.mu-rotnum}
\rho_\mu(F) \ = \ \int_{\ensuremath{\mathbb{T}}^d} F(z)-z \ d\mu(z) \ . \end{equation} When $F$ is fixed and no ambiguities can arise, we suppress it from the notation and write $\rho_\mu$ instead of $\rho_\mu(F)$. By $\ensuremath{\mathrm{supp}}(\mu)$ we denote the topological support of $\mu$. \begin{lem}
\label{l.measure-drift} Suppose $F : R^d \to \ensuremath{\mathbb{R}}^d$ is a lift of $f\in\ensuremath{\mathrm{Homeo}_0(\mathbb{T}^d)}$
and $\mu$ is an ergodic $f$-invariant probability measure. Then
there exists no constant $s>0$ with the property that for $\mu$-a.e.\
$z\in\ensuremath{\mathbb{R}}^2$ there is a positive integer $n_z$ such that $D^v_{n_z}(z,\rho_\mu) \geq
sn_z$.
\end{lem} \textit{Proof. }
We suppose for a contradiction that a constant $s>0$ with the above property exists. We fix an $f$-invariant set $\Omega \ensuremath{\subseteq} \ensuremath{\mathbb{T}^2}$ of measure $\mu(\Omega) = 1$ such that for all $z\in\Omega$ there exists $n_z\in\ensuremath{\mathbb{N}}$ with \begin{equation} \label{e.dev-nz} D^v_{n_z}(z,\rho_\mu) \ \geq \ sn_z \ . \end{equation} In addition, we assume that \begin{equation} \label{e.rho-mu-convergence} \ensuremath{\lim_{n\rightarrow\infty}} \left(F^n(z)-z\right)/n \ = \ \rho_\mu
\quad \forall z\in\Omega \ . \end{equation} Given any $z_0 \in \Omega$, we recursively define a sequence of integers $n_i$ by $n_0 = 0$ and $n_{i+1} = n_i + n_{F^{n_i}(z_0)}$. Then we obtain \begin{eqnarray*}
D^v_{n_k}(z_0,\rho_\mu) & = & \left\langle F^{n_k}(z_0) - z_0 - n_k\rho_\mu \right\rangle_v \\
& = &
\left\langle \sum_{i=0}^{k-1} F^{n_{i+1}}(z_0) - F^{n_{i}}(z_0) - (n_{i+1}-n_{i})\rho_\mu\right\rangle_v
\\ & = & \left\langle \sum_{i=0}^{k-1} F^{n_{F^{n_{i}}(z_0)}}\left(F^{n_i}(z_0)\right) - F^{n_i}(z_0) -
(n_{F^{n_i}(z_0)})\rho_\mu\right\rangle_v \\ & \stackrel{(\ref{e.dev-nz})}{\geq} &
\sum_{i=0}^{k-1} s(n_{i+1}-n_i) \ = \ sn_k \ . \end{eqnarray*} Hence $\ensuremath{\lim_{k\rightarrow\infty}} D^v_{n_k}(z_0,\rho_\mu)/n_k \geq s$, contradicting (\ref{e.rho-mu-convergence}) which implies $\ensuremath{\lim_{n\rightarrow\infty}} D^v_n(z_0,\rho_\mu) = 0$. \qed
\pagebreak
\noindent {\em A reduction lemma.} Any integer matrix $M \in \textrm{GL}(d\times d,\ensuremath{\mathbb{Z}})$ induces a toral endomorphism $g_M : \ensuremath{\mathbb{T}}^d \to \ensuremath{\mathbb{T}}^d,\ \pi(z) \mapsto \pi(Mz)$, and $g_M$ is invertible if and only if $M \in \textrm{SL}(d,\ensuremath{\mathbb{Z}})$. The following lemma describes how a coordinate transformation by such a map $g_M$ acts on the rotation set.
\begin{lem}[Reduction Lemma] \label{l.reduction} Suppose $f\in\ensuremath{\mathrm{Homeo}}_0(\ensuremath{\mathbb{T}}^d)$
has lift $F:\ensuremath{\mathbb{R}}^d \to \ensuremath{\mathbb{R}}^d$, $U\ensuremath{\subseteq} \ensuremath{\mathbb{T}}^d$ and $M \in \mathrm{SL}(d,\ensuremath{\mathbb{Z}})$. Then
the following hold. \alphlist \item Let $\tilde f = g_M^{-1} \circ f \circ g_M \in
\ensuremath{\mathrm{Homeo}}_0(\ensuremath{\mathbb{T}}^d)$ with lift $\tilde F = M^{-1} \circ F \circ M$. Then
\begin{equation}
\label{e.rotset-transformation}
\rho_{g_M^{-1}U}(\tilde F) \ = \ M^{-1}(\rho_U(F)) \ .
\end{equation} \item If $\rho_U(F) \ensuremath{\subseteq} \ensuremath{L_{\lambda,v}}$ then $\rho_{g_M^{-1}U}(\tilde F) \ensuremath{\subseteq}
M^{-1}(\ensuremath{L_{\lambda,v}}) = L_{\tilde \lambda,\tilde v}$, where $\tilde v = M^tv$ and
$\tilde \lambda = \lambda \|v\|^2/\|\tilde v\|^2$. Further $D^v_{f,n}(z,\rho)
= D^{\tilde v}_{\tilde f,n}(M^{-1}z,M^{-1}\rho) \ \forall n\in\ensuremath{\mathbb{N}},\
z,\rho\in\ensuremath{\mathbb{R}}^d$. \item Let $1 \leq k < d$ and suppose that $w_1 \ensuremath{,\ldots,} w_k$ are linearly independent
integer vectors and $\ensuremath{\mathrm{Conv}}_{\! \times}(\{w_1\ensuremath{,\ldots,} w_k\})$ contains no further
integer vectors. (If $k=1$ this just means that the entries of $w_1$ are
relatively prime.) Then there exist integer vectors $w_{k+1} \ensuremath{,\ldots,} w_d$ such
that $\det(w_1\ensuremath{,\ldots,} w_d) = 1$. \end{list} \end{lem}\noindent Note that for any integer vector $w \in \ensuremath{\mathbb{Z}}^d$ with relatively prime entries part (c) allows to perform a linear coordinate transformation on $\ensuremath{\mathbb{T}}^d$ such that $w$ becomes a base vector.
\textit{Proof. } \alphlist \item Suppose $z_i \in \pi^{-1}U,\ n_i \nearrow \infty$ and $\ensuremath{\lim_{i\rightarrow\infty}}
\left(F^{n_i}\left(z_i\right)-z_i\right)/n_i = \rho$. Then
\begin{eqnarray*}
M^{-1}\rho & = & M^{-1} \left(\ensuremath{\lim_{i\rightarrow\infty}} \left(F^{n_i}\left(z_i\right)-z_i\right)/n_i\right)
\\ & = & \ensuremath{\lim_{i\rightarrow\infty}} \left(M^{-1}\circ F^{n_i} \circ M \left(M^{-1}z_i\right)-M^{-1}z_i\right)/n_i \\
& = & \ensuremath{\lim_{i\rightarrow\infty}} \left(\tilde F^n\left(M^{-1}z_i\right)-M^{-1}z_i\right)/n_i \
\in \ \rho_{M^{-1}U}(\tilde F) \ .
\end{eqnarray*}
This shows that $M^{-1}\left(\rho_U\left(F\right)\right) \subseteq
\rho_{g_M^{-1}U}\left(\tilde F\right)$ and since $f = g_M \circ \tilde f \circ
g_M^{-1}$ the opposite inclusion follows in the same way. \item $\rho_{g_M^{-1}U}(\tilde F) \ensuremath{\subseteq} M^{-1}(\ensuremath{L_{\lambda,v}})$ holds by part (a). Further,
we have $M^{-1}(\ensuremath{L_{\lambda,v}}) = M^{-1}(\lambda v) + M^{-1}\{v\}^\perp$. Since $\tilde
v = M^t v \perp M^{-1}\{v\}^\perp$ if follows that $M^{-1}(\ensuremath{L_{\lambda,v}}) =
L_{\tilde\lambda,\tilde v}$ for some $\tilde\lambda \in \ensuremath{\mathbb{R}}$ and we have
$\tilde\lambda \|\tilde v\| = \langle M^{-1}(\lambda v),\tilde v/\|\tilde
v\|\rangle = (\lambda/\|\tilde v\|) \cdot \langle v,v \rangle$. Finally, in
order to check that $D_{f,n}^v(z,\rho) = D_{\tilde f,n}^{\tilde
v}(M^{-1}z,M^{-1}\rho)$ let $z\in \ensuremath{\mathbb{R}}^d$. Then
\begin{eqnarray*}
D_{\tilde f,n}^{\tilde v}(M^{-1}z,M^{-1}\rho) & = & \langle \tilde F^n(M^{-1}z) -
M^{-1}z - nM^{-1}\rho,\tilde v\rangle \\
& = & \langle M^{-1} \circ F^n (z) - M^{-1}z-nM^{-1}\rho,M^tv\rangle \\
& = & \langle F^n(z)-z-n\rho,v\rangle \ = \ D_{f,n}^v(z,\rho) \ .
\end{eqnarray*} \item Choose integer vectors $w_{k+1} \ensuremath{,\ldots,} w_d$ such that $\ensuremath{\mathrm{span}}(w_1 \ensuremath{,\ldots,} w_d) =
\ensuremath{\mathbb{R}}^d$. Then $|\det(w_1 \ensuremath{,\ldots,} w_d)| \geq 2$ if and only if $\ensuremath{\mathrm{Conv}}_{\! \times}(w_1
\ensuremath{,\ldots,} w_d)$ contains an integer vector. In this case we replace one of the
vectors $w_{k+1} \ensuremath{,\ldots,} w_d$ by an integer vector in $\ensuremath{\mathrm{Conv}}_{\! \times}(w_1 \ensuremath{,\ldots,}
w_d)$ such that the new set of vectors still spans $\ensuremath{\mathbb{R}}^d$. This reduces the
absolute value of the determinant, and after a finite number of steps we
arrive at $\det(w_1\ensuremath{,\ldots,} w_d) = \pm 1$. Replacing $w_d$ by $-w_d$ if necessary
we obtain $\det(w_1\ensuremath{,\ldots,} w_d) = 1$. \qed \end{list}
\section{Rotation subsets on the universal cover} \label{Semilocal-rotsets} Throughout this section, we suppose that $G$ is the lift of a toral homeomorphism $g\in\ensuremath{\mathrm{Homeo}_0(\mathbb{T}^2)}$, $\widehat{U} \ensuremath{\subseteq} \ensuremath{\mathbb{R}}^2$ is bounded and connected and $G(\widehat{U}) \cap \widehat{U} \neq \emptyset$. Further, we assume that $\lambda\neq 0$ and $v\in\ensuremath{\mathbb{R}}^2_*$.
In order to control the whole rotation set by using assumptions on $\rho_{\widehat U}(G)$, we proceed in several steps. The first is to obtain some information about the extremal points of the rotation set. \begin{lem} \label{l.extremalpoints} Suppose $\rho_{\widehat{U}}(G) =
L_{\lambda,v}[a,b]$. Then all extremal points of $\rho(G)$ belong to
$C_v[a,b]$. \end{lem} \textit{Proof. } Suppose that $\lambda > 0$. (Otherwise we replace $v$ by $-v$.) Performing a linear change of coordinates via Lemma~\ref{l.reduction} if necessary we may assume that \begin{equation} \label{e.Llv-proj} \pi_1(\ensuremath{L_{\lambda,v}}[a,b]) \ \ensuremath{\subseteq} \ (0,\infty) \ , \end{equation} such that in particular $C^+_v[a,b]\ensuremath{\setminus} \{0\} \ensuremath{\subseteq} (0,\infty) \times \ensuremath{\mathbb{R}}$. Further, we may assume that $\widehat{U}$ intersects $\{0\}\times \ensuremath{\mathbb{R}}$, otherwise we replace it by an integer translate and/or one of its iterates.
\footnote{Note that by assumption $\bigcup_{n\in\ensuremath{\mathbb{N}}_0} G^n(\widehat U)$ is connected
and $\pi_1\circ G^n(\widehat{U})$ goes to $\infty$ as $n\to\infty$ due to
(\ref{e.Llv-proj}). Hence, one of the iterates of $\widehat U$ has to
intersect an integer vertical $\{m\}\times\ensuremath{\mathbb{R}}$.}
Let $V := \bigcup_{n\in\ensuremath{\mathbb{N}}_0} G^n(\widehat{U})$. As $G(\widehat{U}) \cap \widehat{U} \neq \emptyset$, the set $V$ is connected. We claim that for sufficiently large $l\in\ensuremath{\mathbb{N}}$ the integer translate $V-(0,l)$ is disjoint from $V$. In order to see this let $r:=\sup_{z\in
\widehat{U}} \|z\|$. Due to (\ref{e.Llv-proj}) and Lemma~\ref{l.convergence} only a finite number of iterates of $\widehat{U}$ intersect $[-r,r]\times \ensuremath{\mathbb{R}}$. Therefore $V$ and $\widehat{U}-(0,l)$ are disjoint for large $l$. Hence, if the orbit of $\widehat{U}-(0,l)$ intersects $V$ then it must first intersect $\widehat{U}$. However, by the same argument the orbit of $\widehat{U}-(0,l)$ can only intersect a finite number of its vertical integer translates, such that for sufficiently large $l$ we have $V\cap [V-(0,l)] = \emptyset$ as required.
Now let $y_1=\inf\{y\in\ensuremath{\mathbb{R}} \mid (0,y) \in V-(0,l)\}$, $y_2=\sup\{y\in\ensuremath{\mathbb{R}} \mid (0,y) \in V\}$ and define $W$ as the union of $V$, $V-(0,l)$ and the vertical arc from $(0,y_1)$ to $(0,y_2)$. Let $Y$ be the unique connected component of $\ensuremath{\mathbb{R}}^2\ensuremath{\setminus} \overline{W}$ which is unbounded to the left and $A = \ensuremath{\mathbb{R}}^2 \ensuremath{\setminus} Y$.
\begin{figure}
\caption{\small Construction of the sets $W$ on the left and $A$ on the
right. }
\label{f.2}
\end{figure}
The following three remarks about these objects will be helpful. First, as $\rho_{\widehat U}(G) \ensuremath{\subseteq} \ensuremath{L_{\lambda,v}}[a,b]$ and $\lambda>0$, we have that for all $\alpha<a$ and $\beta>b$ there exists a constant $R=R(\alpha,\beta)>0$ such that \begin{equation} \label{e.A-inclusion}
A \ \ensuremath{\subseteq} \ B_R(C^+_v(\alpha,\beta)) \ . \end{equation} Secondly, due to the definition of $W$, its connectedness and (\ref{e.Llv-proj}), the set $Z := (\ensuremath{\mathbb{R}}^+ \times \ensuremath{\mathbb{R}}) \ensuremath{\setminus} A = (\ensuremath{\mathbb{R}}^+ \times \ensuremath{\mathbb{R}}) \cap Y$ consists of exactly two connected components. These can be defined as follows. Fix any $\zeta_0 \in Y$ with $\pi_1(\zeta_0) < 0$. For any $\zeta \in Z$, there is a path $\gamma_\zeta$ in $Y$ from $\zeta$ to $\zeta_0$. Let $y_\zeta$ be the second coordinate of the first point in which $\gamma_\zeta$ intersects the vertical axis. The fact whether $y_\zeta$ lies below $y_1$ or above $y_2$ does not depend on the choice of the path, since this would contradict the connectedness of $W$. Hence, $Z^- = \{\zeta \in Z \mid y_\zeta < y_1\}$ and $Z^+ = \{ \zeta \in Z \mid y_\zeta > y_2\}$ form a partition of $Z$ into two connected components.
Thirdly, there holds $\ensuremath{\mathbb{R}}^+ \times \ensuremath{\mathbb{R}} \ensuremath{\subseteq} \ensuremath{\bigcup_{k\in\N}} A + (0,kl)$. Consequently, for any $m\in \ensuremath{\mathbb{N}}$ the set $A \cap \pi_1^{-1}[m,m+1)$ contains a fundamental domain of $\ensuremath{\mathbb{T}^2}$, that is, $\pi(A \cap \pi_1^{-1}[m,m+1)) = \ensuremath{\mathbb{T}^2}$.
It is important to note, however, that $A$ is not $G$-invariant. Yet, in order to obtain control over the full rotation set via $A$ we will need to ensure that orbits `moving to the right' become `trapped' in $A$ (or one of its integer translates). Hence, the following statement is crucial for our purposes.
\begin{claim} \label{c.rho} There exists a constant $K>0$ such that $z\in A
\cap \pi_1^{-1}[K,\infty)$ implies $G^{\pm 1}(z) \in A$. \end{claim} \textit{Proof. } We show that there exists a constant $K' > 0$ such that $z \in Z \cap \pi_1^{-1}[K',\infty) $ implies $G^{\pm 1}(z) \in Z$. If we let \begin{equation} \label{e.M}
M\ :=\ \sup_{z\in\ensuremath{\mathbb{R}}^2}\|G(z)-z\| \ = \ \sup_{n\in\ensuremath{\mathbb{R}}^2} \|G^{-1}(z)-z\| \ , \end{equation} then for any $z\in\pi_1^{-1}[K'+M,\infty)$ this means that $G^{\pm 1}(z) \in Z$ implies $z\in Z$ and hence $z\in A$ implies $G^{\pm 1}(z) \in A$. Thus we can choose $K=K'+M$.
From (\ref{e.Llv-proj}) and Lemma~\ref{l.convergence} we deduce that there exists $n_0\in\ensuremath{\mathbb{N}}$ such that $\pi_1\circ G^n(\widehat{U}) \ensuremath{\subseteq} (4M,\infty) \ \forall n\geq n_0$. Let $K'>0$ such that for all $j\leq n_0$ there holds $\pi_1\circ G^j(\widehat{U}) \ensuremath{\subseteq} [0,K')$. Then we have \begin{equation}
\label{e.noreturn}
\pi_1\circ G^n(\widehat{U}) \cap [K',\infty) \neq \emptyset \quad
\ensuremath{\Rightarrow} \quad \pi_1\circ G^k(\widehat{U}) \cap [0,4M] = \emptyset \quad \forall k\geq n \ . \end{equation} The same statement applies to $\widehat{U}+(0,l)$. Due to (\ref{e.A-inclusion}) there exists $C>0$ such that \[ B \ := \ [0,4M] \times [C-M,\infty) \ \ensuremath{\subseteq} \ Z^+ \ . \] Let $z^*=(3M,C)$ and fix $z \in Z^+$ with $\pi_1(z) \geq K'$. Then, since $Z^+$ is open and connected, there is a simple path $\gamma : [0,1] \to Z^+$ from $z$ to $z^*$. We claim that $\gamma$ can be chosen such that its image is contained in $Z^+ \cap \pi^{-1}[3M,\infty)$. Suppose not and let $t_0 := \min\{t\in[0,1] \mid \gamma(t) \in B\}$ and $\Gamma = \{\gamma(t) \mid t\in[0,t_0]\}$. Then $\Gamma$ divides the set $([0,\pi_1(z)] \times \ensuremath{\mathbb{R}}) \ensuremath{\setminus} B$ into exactly two connected components $D^+$ and $D^-$ that are unbounded above, respectively below. \begin{figure}
\caption{\small The domains $D^-$ and $D^+$. }
\label{f.4}
\end{figure} Now, if $\overline{W}$ does not intersect $D^+ \cap \pi^{-1}_1[0,4M)$ then $D^+\cap \pi^{-1}_1[0,4M) \ensuremath{\subseteq} Z^+$, and it is easy to see that in this case either $\gamma$ does not intersect $\pi_1^{-1}[0,3M)$ or we can modify it to that end. Otherwise, there must be some $k\in\ensuremath{\mathbb{N}}$ such that $G^k(\widehat{U})$ or $G^k(\widehat{U})-(0,l)$ intersects $D^+\cap \pi^{-1}_1[0,4M)$. However, since $\widehat{U}$ intersects $D^-$ and the set $\bigcup_{i=0}^k G^i(\widehat{U})$ is connected, this implies that there must be some $n\leq k$ such that $\pi_1\circ G^n(\widehat{U})$ intersects $[\pi_1(z),\infty) \ensuremath{\subseteq} [K',\infty)$. This contradicts (\ref{e.noreturn}).
Summarising, we have found a path $\gamma$ from $z$ to $z^*$ which is contained in $Z^+ \cap \pi^{-1}_1[3M,\infty)$. In particular, $\gamma$ is contained in the complement of $\overline W$. Consequently, the path $G\circ \gamma$ is contained in the complement of $G(\overline{W})$. At the same time, it is also contained in $\pi^{-1}_1[2M,\infty)$. However, it follows from the construction of $W$ and the definition of $M$ that $G(\overline W) \cap \pi^{-1}_1[2M,\infty) = \overline W \cap \pi^{-1}_1[2M,\infty)$. Hence, the path $G \circ \gamma$ is contained in the complement of $\overline{W}$ as well. Furthermore, it joins $G(z)$ to the point $G(z^*)$. Since the latter is contained in $B \ensuremath{\subseteq} Z^+$, this implies that $ G(z)$ is equally contained in $Z^+$. When $z\in Z^-$ the argument is similar. In the same way one can show that $G^{-1}(z) \in Z$, and this proves the claim. {
\Large $\circ$}
In order to complete the proof of Lemma~\ref{l.extremalpoints}, suppose that $\rho$ is an extremal point of $\rho(G)$ which is not contained in $C_v[a,b]$. By performing a linear change of coordinates again if necessary, we may assume that $\pi_1(\rho) > 0$.\footnote{Choose
a basis of integer vectors $w_1,w_2$ with $\det(w_1,w_2)=1$ such
that both $w_1$ and $\rho$ lie to the right of the oriented line $\ensuremath{\mathbb{R}}
w_2$ and $w_2 \notin C_v[a,b]$ (the latter ensures that
(\ref{e.Llv-proj}) remains valid.) Then apply
Lemma~\ref{l.reduction}. } Since $\rho$ is realised by an ergodic invariant measure \cite[Corollary~3.5]{misiurewicz/ziemian:1989}, there exists a point $z_0\in\ensuremath{\mathbb{R}}^2$ with \begin{equation} \label{e.z_0-convergence} \ensuremath{\lim_{n\rightarrow\infty}} (G^n(z_0)-z_0)/n \ = \ \rho\ . \end{equation} Let $z_n = G^n(z_0)$ and ${\cal O}^+(z_0) = \{z_n \mid n\geq 0\}$. Then, due to (\ref{e.z_0-convergence}), for every $\gamma< 0 < \delta$ there exists $\tilde R>0$ such that ${\cal O}^+(z_0) \ensuremath{\subseteq} z_0 + B_{\tilde R}(C^+_\rho[\gamma,\delta])$. If $\gamma,\delta$ are chosen sufficiently close to 0 and $\alpha,\beta$ in (\ref{e.A-inclusion}) are sufficiently close to $a$ and $b$ then $B_R(C_v[\alpha,\beta]) \cap \left(z_0+B_{\tilde R}(C_\rho[\gamma,\delta])\right)$ is bounded (recall that $\rho$ is not contained in $C_v[a,b]$). Further, by replacing $z_0$ with an integer translate if necessary, we may assume that $z_0 \in A$ and $\pi_1(z_0) \geq K + \tilde R$, where $K$ is chosen as in Claim~\ref{c.rho}. If follows that $\pi_1(z_n) \geq K$ and hence $z_n \in A \ \forall n\in\ensuremath{\mathbb{N}}$. Consequently ${\cal O}^+(z_0) \ensuremath{\subseteq} B_R(C_v[\alpha,\beta]) \cap \left(z_0+B_{\tilde
R}(C_\rho[\gamma,\delta])\right)$ such that ${\cal O}^+(z_0)$ is bounded, contradicting (\ref{e.z_0-convergence}). Hence, all extremal points of $\rho(G)$ must be contained in $C_v[a,b]$. \qed
In the opposite way, information about the extremal points of $\rho(G)$ allows to draw conclusions about the behaviour of the iterates of $\widehat U$. \begin{lem}
\label{l.cone-intersection} Suppose that $\rho_{\widehat{U}}(G) =
L_{\lambda,v}[a,b]$, $\gamma \in [a,b]$ and $\rho \in C_v[\gamma,\gamma]$ is
an extremal point of $\rho(G)$. Then given any $\ensuremath{\varepsilon} > 0$ there exists
$N\in\ensuremath{\mathbb{N}}$ such that $G^n(\widehat{U}) \cap C^+_v(\gamma-\ensuremath{\varepsilon},\gamma+\ensuremath{\varepsilon}) \neq
\emptyset \ \forall n\geq N$. \end{lem} \textit{Proof. } As in the proof of Lemma~\ref{l.extremalpoints} we assume $\lambda > 0, \pi_1(v) > 0$ and $C^+_v[a,b]\ensuremath{\setminus}\{0\} \ensuremath{\subseteq} (0,\infty) \times \ensuremath{\mathbb{R}}$. Let $l$ be the integer in the definition of the set $A$ above and define $\widehat{U}' := \widehat{U} \cup [\widehat{U}-(0,l)]$. Suppose for a contradiction that $G^n(\widehat{U}) \cap C^+_v(\gamma-\ensuremath{\varepsilon},\gamma+\ensuremath{\varepsilon}) = \emptyset$ for infinitely many $n\in \ensuremath{\mathbb{N}}$. Slightly reducing $\ensuremath{\varepsilon}$ if necessary, we may assume that $G^n(\widehat{U}') \cap C^+_v(\gamma-\ensuremath{\varepsilon},\gamma+\ensuremath{\varepsilon}) = \emptyset$ for infinitely many $n\in \ensuremath{\mathbb{N}}$. \begin{figure}\label{f.9}
\end{figure} We first consider the case where $\vbr{\rho} > 0$, such that $\rho \in C^+_v[a,b]$ by Lemma~\ref{l.extremalpoints}.
Due to Lemma~\ref{l.convergence} the fact that $\rho_{\widehat{U}}(G) \ensuremath{\subseteq} \ensuremath{L_{\lambda,v}}$ implies that $\vbr{G^nz}/n$ converges uniformly to $\lambda$ on $\widehat{U}'$ as $n\to\infty$. Hence, for any $\delta>0$ there exists $N(\delta)\in\ensuremath{\mathbb{N}}$ such that \begin{equation}
\label{e.N_delta} G^n(\widehat{U}') \ \ensuremath{\subseteq} \ S_v[(1-\delta)n\lambda,(1+\delta)n\lambda]
\quad \forall n\geq N(\delta) \ . \end{equation} As $\pi_1(v) > 0$, this implies that $\inf \pi_1(G^n(\widehat{U}')) \to \infty$ as $n\to\infty$. Therefore (\ref{e.A-inclusion}) yields that for given $\alpha< a$ and $\beta>b$ and sufficiently large $n$ there holds $G^n(\widehat{U}') \ensuremath{\subseteq} C^+_v[\alpha,\beta]$ (apply (\ref{e.A-inclusion}) with $\tilde \alpha \in (\alpha,a)$ and $\tilde \beta \in (b,\beta)$ to get rid of the constant $R$). Consequently, if $n$ is large then $G^n(\widehat{U}') \cap C^+_v(\gamma-\ensuremath{\varepsilon},\gamma+\ensuremath{\varepsilon})=\emptyset$ implies $G^n(\widehat{U}')\ensuremath{\subseteq} C^+_v[\alpha,\gamma-\ensuremath{\varepsilon}]$ or $G^n(\widehat{U}') \ensuremath{\subseteq} C^+_v[\gamma+\ensuremath{\varepsilon},\beta]$. We assume that $G^n(\widehat{U}') \ensuremath{\subseteq} C^+_v[\alpha,\gamma-\ensuremath{\varepsilon}]$ for infinitely many $n \in\ensuremath{\mathbb{N}}$, the other case is symmetric.
Since $\rho \in \textrm{Ex}(\rho(G))$, there exists an ergodic measure $\mu$ with $\rho_\mu(F) = \rho$ \cite[Corollary~3.5]{misiurewicz/ziemian:1989}. Let $\Omega \ensuremath{\subseteq} \ensuremath{\mathbb{T}^2}$ be such that $g(\Omega)=\Omega$, $\mu(\Omega)=1$ and \begin{equation}
\label{e.mu-convergence} \ensuremath{\lim_{n\rightarrow\infty}} (G^n(z)-z)/n \ = \ \rho \quad \forall z\in\pi^{-1}(\Omega) \ . \end{equation} We will show that, in contradiction to Lemma~\ref{l.measure-drift}, for some $s>0$ there holds \begin{equation}\label{e.s-contradiction} \forall z\in\Omega \ \exists n_z\in\ensuremath{\mathbb{N}} : \quad D^{-\rho^\perp}_{n_z}(z,\rho) \ \geq \ s n_z \ . \end{equation} In order to do so, fix $z\in\Omega$. Let $A$ be as in the proof of Lemma~\ref{l.extremalpoints}. Due to (\ref{e.mu-convergence}) there exists a lift $z_0\in A$ of $z$ such that $\pi_1(z_n) \geq K \ \forall n\in\ensuremath{\mathbb{N}}$, where $z_n = G^n(z_0)$ and $K$ is chosen as in Claim~\ref{c.rho}. Consequently Claim~\ref{c.rho} implies that \begin{equation}
\label{e.z_n-in-A}
z_n \ \in \ A \quad \forall n\in\ensuremath{\mathbb{N}} \ . \end{equation} Due to (\ref{e.N_delta}) and the fact that $M$ defined in (\ref{e.M}) is finite, there exists $\eta > 0$ and $N_1\in\ensuremath{\mathbb{N}}$ such that for any $n\geq N_1$ there holds \begin{equation}
\label{e.cone-inclusion} G^n(\widehat{U}') \ensuremath{\subseteq} C_v^+[\alpha,\gamma-\ensuremath{\varepsilon}] \quad \ensuremath{\Rightarrow}
\quad G^k(\widehat{U}') \ensuremath{\subseteq} C_v^+[\alpha,\gamma-\ensuremath{\varepsilon}/2] \qquad \forall k\in[(1-\eta)n,(1+\eta)n] \ . \end{equation} Now, choose $\delta$ in (\ref{e.N_delta}) sufficiently small, such that $(1+\delta)(1-\eta) < 1 < (1-\delta)(1+\eta)$. Then choose $N_2\in\ensuremath{\mathbb{N}}$ such that for all $n\geq N_2$ there holds \begin{eqnarray} \label{e.interval-inclusion} [\lambda n-M,\lambda n+M] & \ensuremath{\subseteq} &
[(1+\delta)(1-\eta)\lambda n,(1-\delta)(1+\eta)\lambda n] \ ,\\
\lambda n-M & \geq & M\cdot N(\delta)\cdot \|v\|+\sup \left\langle\widehat U'\right\rangle_v \
. \label{e.geq-N_delta} \end{eqnarray} Suppose $n\geq N_3:=\max\{N(\delta)/(1-\delta),N_1,N_2\}$ and $G^n(\widehat{U}) \ensuremath{\subseteq} C_v^+[\alpha,\gamma-\ensuremath{\varepsilon}]$. Then by combining (\ref{e.N_delta}) and (\ref{e.cone-inclusion})--(\ref{e.geq-N_delta}) we obtain that \begin{equation}
G^k(\widehat{U}')
\cap S_v(\lambda n-M,\lambda n+M) \ \ensuremath{\subseteq} \ C^+_v[\alpha,\gamma-\ensuremath{\varepsilon}/2] \quad
\forall k \in\ensuremath{\mathbb{N}} \ . \end{equation} (Treat the cases $k\leq N(\delta),\ k\in(N(\delta),(1-\eta)n)$ and $k\geq (1+\eta)n$ separately to show that for all such $k$ the set $G^k(\widehat U)$ does not intersect $S_v(\lambda n-M,\lambda n+M)$ and then use (\ref{e.cone-inclusion}) for the remaining~$k$.) This means in particular that \begin{equation} \label{e.A-pocket} A\cap S_v(\lambda n-M,\lambda n+M) \ \ensuremath{\subseteq} \
C_v^+[\alpha,\gamma-\ensuremath{\varepsilon}/2] \quad \quad \forall n\geq N_3 : G^n(\widehat{U}) \ensuremath{\subseteq}
C^+_v[\alpha,\gamma-\ensuremath{\varepsilon}] \ . \end{equation} Now, as $\ensuremath{\mathbb{R}}\rho=C_v[\gamma,\gamma]$ exist constants $r>0$ and $N_4\geq N_3$ such that \begin{equation} \label{e.r-deviationbound}
\langle z'-z_0\rangle_{\rho^\perp} \leq -rn \quad \forall n\geq N_4,\
z'\in S_v(\lambda n-M,\lambda n+M) \cap C^+_v[\alpha,\gamma-\ensuremath{\varepsilon}/2] \ . \end{equation} Further, since $z$ and its lift $z_0$ are fixed and due to (\ref{e.mu-convergence}) (applied to $z_0$) there exist constants $N_5=N_5(z) \geq N_4$ and $c>0$, with $c$ only depending on $\rho,\gamma$ and $\ensuremath{\varepsilon}$, such that \begin{equation} \label{e.nz}
\forall n\geq N_5 \ \exists n_z \leq cn : \quad z_{n_z} \in S_v(\lambda n-M,\lambda
n+M) \end{equation} (Note that the orbit of $z_0$ has to pass through $S_v[\lambda n-M,\lambda n+M]$ by definition of $M$, and due to (\ref{e.mu-convergence}) this happens approximately at time $n\lambda/\vbr{\rho}$.) Choose $n\geq N_5$ with $G^n(\widehat U)$ $\ensuremath{\subseteq}$ $C_v^+[\alpha,\gamma-\ensuremath{\varepsilon}]$. Let $n_z$ be as in (\ref{e.nz}). Since $z_{n_z} \in A$ by (\ref{e.z_n-in-A}), we obtain $\langle z_{n_z}-z_0\rangle_{\rho^\perp} \leq -rn$ from (\ref{e.A-pocket}) and (\ref{e.r-deviationbound}). Thus, if $s=r/c$ then $D_{n_z}^{-\rho^\perp}(z,\rho) \geq sn_z$. Since $z\in\Omega$ was arbitrary, this proves (\ref{e.s-contradiction}).
Finally, if $\vbr{\rho} < 0$ then we can proceed in the same way by regarding the inverse of $G$. Due to the symmetry in the statement of Claim~\ref{c.rho} we obtain that for a suitable lift $z_0$ of $z\in\Omega$ the whole backwards orbit of $z_0$ remains in $A$, and the remaining argument is exactly the same as before. \qed
\begin{cor}
\label{c.linear-spreading} Suppose $\rho_{\widehat{U}}(G) = L_{\lambda,v}[a,b]$ with
$a<b$. Then there exist constants $c>0$ and $N'\in\ensuremath{\mathbb{N}}$ such that
\begin{equation}
\label{e.linear-spreading} \sup_{z\in \widehat{U}} \vpbr{G^n(z)} - \inf_{z\in \widehat{U}}\vpbr{G^n(z)}
\ > \ cn \quad \forall n \geq N' \ .
\end{equation} \end{cor} \textit{Proof. } As $L_{\lambda,v}[a,b] = \rho_{\widehat{U}}(G) \ensuremath{\subseteq} \rho(G)$, the set $\rho(G)$ must have at least two linearly independent extremal points $\rho_1,\rho_2\neq 0$. Due to Lemma~\ref{l.extremalpoints} these are contained in $C_v[a,b]$, such that $\rho_1\ensuremath{\subseteq} C_v[\gamma_1,\gamma_1]$ and $\rho_2\in C_v[\gamma_2,\gamma_2]$ for some $\gamma_1\neq\gamma_2\in\ensuremath{\mathbb{R}}$. The statement now follows from Lemma~\ref{l.cone-intersection} together with the fact that $\inf\langle G^n(\widehat U)\rangle_v \to \infty$ as $n\to\infty$. \qed
We can now describe two situations in which the rotation subset of $\widehat U$ determines the whole rotation set completely, or at least forces it to be contained in a line segment. \begin{lem}
\label{l.uniform-v-speed} Suppose $\rho_{\widehat{U}}(G) = L_{\lambda,v}[a,b]$
with $a<b$. Then $\rho(G) = L_{\lambda,v}[a,b]$. \end{lem} \textit{Proof. } Due to Lemma~\ref{l.extremalpoints}, it suffices to show $\rho(G) \ensuremath{\subseteq}
\ensuremath{L_{\lambda,v}}$. Suppose for a contradiction that $\rho(G) \nsubseteq \ensuremath{L_{\lambda,v}}$. Then there exists an extremal point $\rho \in \textrm{Ex}(\rho(G)) \ensuremath{\setminus} \ensuremath{L_{\lambda,v}}$. We assume w.l.o.g.\ that $\|v\| = 1$ and $\vbr{\rho} > \lambda$. Since $\rho$ is realised by an ergodic measure, there exists $z_0 \in \ensuremath{\mathbb{R}}^2$ with $\ensuremath{\lim_{n\rightarrow\infty}} (z_n-z_0)/n = \rho$, where $z_n = G^n(z_0)$ as above. Fix $\eta > 0$ and $k_0\in\ensuremath{\mathbb{N}}$ such that \begin{equation}
\label{e.k_0-unifspeed} \vbr{z_k-z_0} \ > \ (1+\eta)\lambda k \quad \forall k \geq k_0 \ . \end{equation} Further, fix $\delta > 0$ such that \begin{equation}
\label{e.delta-unifspeed}
\delta\left(1+15M/c\right) \ < \ \eta \ , \end{equation} with $M$ defined by (\ref{e.M}) and $c$ as in Corollary~\ref{c.linear-spreading}. Choose $n_0\in\ensuremath{\mathbb{N}}$ such that \begin{eqnarray}
\label{e.n_0-unifspeed}
G^n(\widehat{U}) & \ensuremath{\subseteq} & S_v[(1-\delta)\lambda n,(1+\delta)\lambda n] \quad
\forall n\geq n_0 \ \quad \textrm{and} \\
\delta \lambda n_0 & \geq & 2 \ . \label{e.n_0-second} \end{eqnarray} Then choose $k\geq k_0$ and $n\geq n_0$ such that \begin{equation}
\label{e.k_0-n_0-ineq}
(4M+2)k \ \leq \ cn \ \leq 5Mk \ . \end{equation}
\begin{figure}
\caption{\small Proof of Lemma~\ref{l.uniform-v-speed}: The slow movement of
the curve $\Gamma$ impedes a faster movement of the point $z_0$ under
iteration. }
\label{f.5}
\end{figure}
Due to Corollary~\ref{c.linear-spreading} there exists a simple arc $\Gamma_0 \ensuremath{\subseteq} G^n(\widehat{U})$ with endpoints $\zeta_1,\zeta_2\in G^n(\widehat{U})$ such that $\vpbr{\zeta_2-\zeta_1} > cn$ and $\Gamma_0 \ensuremath{\subseteq} S_{v^\perp}[\vpbr{\zeta_1},\vpbr{\zeta_2}]$. Let $\Gamma_1$ and $\Gamma_2$ be properly embedded half-lines (proper images of $\ensuremath{\mathbb{R}}^+$) which join $\zeta_1$, respectively $\zeta_2$, to infinity and satisfy \begin{equation} \label{e.Gamma_i-unifspeed} \Gamma_i \ensuremath{\setminus}\{z_i\} \cap S_{v^\perp}[\vpbr{\zeta_1},\vpbr{\zeta_2}] \ = \ \emptyset \ . \end{equation} Let the properly embedded line $\Gamma = \Gamma_0 \cup \Gamma_1 \cup \Gamma_2$ be oriented in the direction from $\zeta_1$ to $\zeta_2$ and denote by $W$ the connected component of $\ensuremath{\mathbb{R}}^2 \ensuremath{\setminus}\Gamma$ to the left of $\Gamma$. Then, since $\Gamma_0\ensuremath{\subseteq} G^n(\widehat{U})$ by assumption and due to (\ref{e.n_0-unifspeed}) and (\ref{e.Gamma_i-unifspeed}), $W$ contains the set $S_v(-\infty,(1-\delta)\lambda n] \cap S_{v^\perp}[\vpbr{\zeta_1},\vpbr{\zeta_2}]$. Further, due to (\ref{e.n_0-second}) and (\ref{e.k_0-n_0-ineq}) the set \begin{equation} \label{e.W_0} W_0 \ = \ S_v[(1-2\delta)\lambda
n,(1-\delta)\lambda n] \cap S_{v^\perp}[\vpbr{\zeta_1}+2kM,\vpbr{\zeta_2}-2kM] \end{equation} is a rectangle whose side-lengths are greater than 2. Hence $W_0$ contains a fundamental domain of $\ensuremath{\mathbb{T}^2}$ and by replacing $z_0$ with an integer translate if necessary we may assume $z_0\in W_0$. This implies in particular that $\vbr{z_0} \geq (1-2\delta)\lambda n$. Now, there holds \begin{equation}
\label{e.z_k} z_k \ \in \ G^k(W) \cap S_{v^\perp}[\vpbr{\zeta_1} +kM,\vpbr{\zeta_2}-kM] \ . \end{equation} However, due to (\ref{e.n_0-unifspeed}) and (\ref{e.Gamma_i-unifspeed}) we have \begin{equation} G^k(W) \cap S_v((1+\delta)\lambda(n+k),\infty) \cap S_{v^\perp}[\vpbr{\zeta_1} +kM,\vpbr{\zeta_2}-kM] \ = \ \emptyset \ , \end{equation} such that $z_k \in S_v(-\infty,(1+\delta)\lambda(n+k)]$. Thus, using (\ref{e.k_0-n_0-ineq}) and (\ref{e.delta-unifspeed}) we obtain $\vbr{z_k-z_0} \leq (1+\delta)\lambda(n+k) - (1-2\delta)\lambda n < (1+\eta)\lambda k$, in contradiction to (\ref{e.k_0-unifspeed}). \qed
More or less along the same lines we obtain the following. \begin{lem}
\label{l.rational-slope-segment} Suppose $\rho_{\widehat{U}}(G) \ensuremath{\subseteq} \ensuremath{\mathbb{R}} \cdot
v$ is a line segment of positive length. Then $\rho(G) \ensuremath{\subseteq} \ensuremath{\mathbb{R}} \cdot v$. \end{lem} \textit{Proof. } Suppose for a contradiction that $\rho(G) \nsubseteq \ensuremath{\mathbb{R}}\cdot v$ and assume withhout loss of generality that there exists $\rho\in\textrm{Ex}(\rho(G))$ with $\sigma := \vpbr{\rho}>0$. Then as above, there exists a point $z_0\in\ensuremath{\mathbb{R}}^2$ with $\ensuremath{\lim_{n\rightarrow\infty}} (z_n-z_0)/n = \rho$, where $z_n=G^n(z_0)$. Hence, there exists an integer $N\geq 1$ such that \begin{equation} \label{e.vertialspeed-in-rationalslope}
\vpbr{z_n-z_0} \ > \ n\sigma/2 \quad \forall n \geq N \ . \end{equation} Let $av$ and $bv$ be the endpoints of $\rho_U(G)$, with $a<b$. Then by definition of $\rho_{\widehat{U}}(G)$ in (\ref{e.rotation-subset}) and due to the connectedness of $\bigcup_{j=0}^m G^j(\widehat{U})$, there are infinitely many $m\in\ensuremath{\mathbb{N}}$ such that there exists a simple arc $\Gamma_0 \ensuremath{\subseteq} \bigcup_{j=0}^m G^j(\widehat{U})$ with endpoints $\zeta_1$ and $\zeta_2$ such that \begin{equation} \label{e.endpoint-distance-in-rationalslope}
\vbr{\zeta_2-\zeta_1} \ \geq \ m(b-a)/3 \ . \end{equation} Of course, we may assume $\Gamma_0 \ensuremath{\subseteq} S_v[\vbr{\zeta_1},\vbr{\zeta_2}]$. Further, given any $\delta>0$ there exists $m_0\in\ensuremath{\mathbb{N}}$ such that for all $m\geq m_0$ we have \begin{equation} \label{e.Gamma_0-orbit-in-rationalslope} G^n(\Gamma_0) \ \ensuremath{\subseteq} \
\bigcup_{j=0}^{n+m}G^j(\widehat U) \ \ensuremath{\subseteq} \
S_{v^\perp}[-\delta(m+n),\delta(m+n)] \quad \forall n\in\ensuremath{\mathbb{N}} \ . \end{equation} Now, fix $\delta > 0$ such that \begin{equation}
\label{e.delta-in-rationalslope}
4\delta/(\sigma-2\delta) \ < \ (b-a)/12 \ \end{equation} and choose $m\geq m_0$ such that there exists an integer $n_0\geq N$ with \begin{equation}
\label{e.i-in-rationalslope}
n_0 \ \in \ \left[\frac{4\delta m +4}{\sigma - 2\delta},
\frac{m(b-a)-6}{12M}\right] \ . \end{equation} Let $\Gamma_1$ and $\Gamma_2$ be properly embedded half-lines such that $\Gamma_i$ joins $\zeta_i$ to infinity and $\Gamma_i\ensuremath{\setminus}\{\zeta_i\}$ is disjoint from $S_v[\vbr{\zeta_1},\vbr{\zeta_2}]$. Let $\Gamma = \Gamma_0\cup\Gamma_1\cup\Gamma_2$ be oriented in the direction from $\zeta_1$ to $\zeta_2$. Denote by $W$ the connected component of $\ensuremath{\mathbb{R}}^2 \ensuremath{\setminus} \Gamma$ to the right of $\Gamma$. Then $W_0=S_v[\vbr{\zeta_1}+2n_0M,\vbr{\zeta_2}-2n_0M] \cap S_{v^\perp}[-\delta m-2,-\delta m) \ensuremath{\subseteq} W$ contains a fundamental domains of $\ensuremath{\mathbb{T}}^2$. (Note that due to (\ref{e.endpoint-distance-in-rationalslope}) and (\ref{e.i-in-rationalslope}) there holds $\vbr{\zeta_2}-\vbr{\zeta_1}\geq 4n_0M+2$.) Replacing $z_0$ by an integer translate if necessary, we may therefore assume $z_0\in W_0$. It follows that \begin{equation}
z_{n_0} \ \in \ G^{n_0}(W) \cap S_v[\vbr{\zeta_1}+n_0M,\vbr{\zeta_2}-n_0M] \ . \end{equation} However, due to (\ref{e.Gamma_0-orbit-in-rationalslope}) and the choice of $\Gamma$ the set $G^{n_0}(W)$ is disjoint from $S_v[\vbr{\zeta_1}+n_0M,\vbr{\zeta_2}-n_0M] \cap S_{v^\perp}(\delta(m+n_0),\infty)$, such that \begin{equation}
\label{e.z_i0-end}
\vpbr{z_{n_0}-z_0} \ \leq \ \delta(2m+n_0)+2 \ . \end{equation} However, from (\ref{e.i-in-rationalslope}) we obtain that $\delta(2m+n_0)+2 \leq n_0\sigma/2$, such that (\ref{e.z_i0-end}) contradicts (\ref{e.vertialspeed-in-rationalslope}). \qed
\section{Proof of the main results} \label{Proofs}
The following basic observation will be used in the proof of Theorem~\ref{t.semilocal-rotsets}. Recall that $\varphi_n(z) = (F^n(z)-z)/n$.
\begin{lem} \label{l.rotsets-convergence} Suppose $F$ is a lift of $f\in\ensuremath{\mathrm{Homeo}_0(\mathbb{T}^d)}$
and $U\ensuremath{\subseteq} \ensuremath{\mathbb{T}}^d$ is open, connected, bounded and recurrent. Further, assume
that $\rho_U(F) = {\cal S} \ensuremath{\subseteq} \ensuremath{\mathbb{R}} \cdot v$ is a line segment with
$0\notin{\cal S}$ and $v,v'$ are linearly independent. Let $\widehat U$ be a
connected component of $\pi^{-1}(U)$. Then there exists $p\in\ensuremath{\mathbb{N}}$ and $w \in
\ensuremath{\mathbb{Z}}^2$ linearly independent of $v'$ such that
\begin{equation} \label{e.pw-pairs}
\left(F^p(\widehat U)- w\right)\cap \widehat U \ \neq \
\emptyset \ . \end{equation} In particular, if $v$ is not the scalar multiple of an integer vector then there exist infinitely many pairs $(p_i,w_i)$ with pairwise independent integer vectors $w_i$ that satisfy (\ref{e.pw-pairs}). \end{lem} \proof Choose $\ensuremath{\varepsilon} >0$ such that $B_{2\ensuremath{\varepsilon}}(\rho_U(F)) = B_{2\ensuremath{\varepsilon}}({\cal S})$ is disjoint from $\ensuremath{\mathbb{R}} v'$. As $\widehat U$ is bounded, there exists $n_0\in\ensuremath{\mathbb{N}}$ such that \[ \ntel\left(F^n(\widehat U)-\widehat U\right) \ \ensuremath{\subseteq} \ B_\ensuremath{\varepsilon}(\varphi_n(\widehat U)) \quad \forall n\geq n_0 . \] (Here $A-B = \{ z-z' \mid z\in A,\ z'\in B \}$.) Due to Lemma~\ref{l.convergence} we may further assume, by increasing $n_0$ if necessary, that \[ \varphi_n(\widehat U) \ \ensuremath{\subseteq} \ B_\ensuremath{\varepsilon}(\rho_U(F)) \quad \forall n\geq n_0 \ . \] We thus obtain \[
\left(F^n(\widehat U)-\widehat U\right) \cap \ensuremath{\mathbb{R}}
v' \ = \ \emptyset \quad \forall n\geq n_0 \ . \] Now, since $U$ is recurrent, there exists $p\geq n_0$ with $f^p(U)\cap U \neq \emptyset$ and hence an integer vector $w$ with $\left(F^p(\widehat U)-w\right) \cap \widehat U \neq \emptyset$. As $w$ belongs to $F^n(\widehat U)-\widehat U$ it must be linearly independent of $v'$.
\qed
\noindent {\bf Proof of Theorem~\ref{t.semilocal-rotsets}.} \ We start with (b) and (c) and prove (a) at the end. \alphlist \addtocounter{enumi}{1} \item Let $f,F,U$ and ${\cal S} = \rho_U(F)$ be as in the statement of the
theorem. Let $\widehat U$ be a connected component of $\pi^{-1}(U)$. We first
assume that the line passing through ${\cal S}$ does not contain any rational
points. Since $U$ is non-wandering, there exists $p\in\ensuremath{\mathbb{N}}$ and $w\in\ensuremath{\mathbb{Z}}^2$ such
that $\left(F^p(\widehat U)-w\right)\cap \widehat U \neq \emptyset$. Let $G =
F^p-w$. Then $\rho_{\widehat U}(G) = p{\cal S}-w = L_{\lambda,v}[a,b]$ for
some $\lambda\in\ensuremath{\mathbb{R}},\ v\in\ensuremath{\mathbb{R}}^2_*$ and $a<b$. Further, the line $L_{\lambda,v}$
contains no rational points either and therefore $\lambda \neq 0$. Hence
$g=f^p,\ G$ and $\widehat U$ satisfy the assumptions of
Lemma~\ref{l.uniform-v-speed} and we obtain $\rho(G)=L_{\lambda,v}[a,b]$ and
thus $\rho(F) = (\rho(G)+w)/p ={\cal S}$.
It remains to treat the case where the line passing through ${\cal S}$ has
irrational slope and contains a single rational point. We choose $\widehat U$
and $G$ as above and again have $\rho_{\widehat U}(G) = L_{\lambda,v}[a,b]$
for some $\lambda\in\ensuremath{\mathbb{R}},\ v\in\ensuremath{\mathbb{R}}^2_*$ and $a<b$. If $\lambda \neq 0$, then we
can proceed exactly as before. However, this time we may have $\lambda = 0$
since $L_{\lambda,v}$ may pass through 0. In this case
Lemma~\ref{l.rotsets-convergence} yields the existence of a pair $\tilde
p\in\ensuremath{\mathbb{N}}$ and $\tilde w\in\ensuremath{\mathbb{Z}}^2$ such that $\tilde w$ and $w$ are linearly
independent and $\tilde G(\widehat U) \cap \widehat U \neq \emptyset$, where
$\tilde G = F^{\tilde p}-\tilde w$. Then \[
\rho_{\widehat U}(\tilde G) \ = \ \tilde p\cdot \rho_{\widehat U}(F)-\tilde w \ = \
\tilde p \cdot \left(\frac{\rho_{\widehat U}(G) + w}{p}\right) - \tilde w \ \ensuremath{\subseteq} \ L_{0,v} +
\left(\frac{\tilde p}{p} \cdot w -\tilde w\right) \] As $w$ and $\tilde w$ are linearly independent there holds $\frac{\tilde p}{p} \cdot w -\tilde w \neq 0$. At the same time, this vector is not in $L_{0,v}=\{v\}^\perp$ since the only rational vector contained in this line is $0$. Therefore $\rho_{\widehat U}(\tilde G) = L_{\tilde \lambda, v}$ for some $\tilde \lambda \neq 0$ and we can apply Lemma~\ref{l.uniform-v-speed} to $\tilde g=f^{\tilde p},\tilde G$ and $\widehat U$ to obtain $\rho_{\widehat
U}(F) = {\cal S}$ as before. \item Suppose $\rho_U(F) = \{\rho\}$ with $\rho$ irrational. As above, we choose
$p\in\ensuremath{\mathbb{N}},\ w\in\ensuremath{\mathbb{Z}}^2$ and $G=F^p-w$ such that $G(\widehat U) \cap \widehat U
\neq \emptyset$. Then $\rho_{\widehat U}(G) = \{p\rho-w\} =
L_{1,p\rho-w}[0,0]$. By Lemma~\ref{l.extremalpoints} all extremal points of
$\rho(G)$ are contained in $C_{p\rho-w}[0,0] = \ensuremath{\mathbb{R}} \cdot (p\rho-w)$ and hence
$\rho(G) \ensuremath{\subseteq} \ensuremath{\mathbb{R}} \cdot (p\rho-w)$. This implies $\rho(F) \ensuremath{\subseteq}
\ensuremath{\mathbb{R}}\cdot(\rho-w/p) + w/p =: A_1$.
Due to Lemma~(\ref{e.pw-pairs}) we can repeat this argument with a second pair
$\tilde p\in\ensuremath{\mathbb{N}}$ and $\tilde w \in \ensuremath{\mathbb{Z}}^2$, with $\tilde w$ linearly independent
of $w$, and obtain $\rho(F) \ensuremath{\subseteq} \ensuremath{\mathbb{R}}\cdot(\rho - \tilde w/\tilde p ) + \tilde
w/\tilde p =: A_2$. It follows that $\rho(F)$ is contained in the intersection
of the two lines $A_1$ and $A_2$, which is equal to $\{\rho\}$. (Note that as
$\rho$ is irrational and the vectors $w/p$ and $\tilde w/\tilde p$ are both
rational and linearly independent, the vectors $\rho-w/p$ and $\rho-\tilde
w/\tilde p$ are linearly independent as well.) \addtocounter{enumi}{-3} \item Due to (a) and (b), it only remains to treat the two cases where
$\rho_U(F)$ is either a line segment of positive length containing a rational
vector or $\rho_U(F)$ is reduced to a single semi-rational vector. The second
case is treated exactly as in (b), the only difference is that we cannot
necessarily repeat the argument with a second $\tilde w$ to conclude that
$\rho(F)$ is a singleton.
Hence, suppose that $\rho_U(F) \ensuremath{\subseteq} \ensuremath{\mathbb{R}} \cdot v$ is a line segment of positive
length and $\rho_U(F)$ contains a rational. By going over to a suitable
iterate, we may assume w.l.o.g.\ that $0\in\rho_U(F)$. Again, there exists a
pair $p\in\ensuremath{\mathbb{N}}$ and $w\in\ensuremath{\mathbb{Z}}^2$ such that $G(\widehat U) \cap \widehat U \neq 0$,
where $G=F^p-w$. We have $\rho_{\widehat U}(G) \ensuremath{\subseteq} L_{\lambda,v^\perp}$ with
$\lambda := \vpbr{w}$. If $\lambda \neq 0$, then Lemma~\ref{l.uniform-v-speed}
implies that that $\rho(G) = \rho_{\widehat U}(G)$ and hence $\rho(F) =
\rho_U(F)$. If $\lambda = 0$, such that $\rho_{\widehat U}(G) \ensuremath{\subseteq} \ensuremath{\mathbb{R}}\cdot v$,
then Lemma~\ref{l.rational-slope-segment} implies $\rho(G) \ensuremath{\subseteq} \ensuremath{\mathbb{R}}\cdot v$,
such that we obtain $\rho(F) = \rho(G)/p \ensuremath{\subseteq} \ensuremath{\mathbb{R}}\cdot v$ as well. \qed
\end{list}
\noindent
{\bf Proof of Theorem~\ref{t.elliptic-islands}.}\ Suppose $U\ensuremath{\subseteq} \ensuremath{\mathbb{T}^2}$
is open, bounded and connected. As $f$ in non-wandering, $U$ is also
recurrent. Suppose $\ensuremath{\mathrm{Conv}}(\rho_U(F))$ has empty interior, that is,
$\rho_U(F)$ is contained in a line. Then Theorem~\ref{t.semilocal-rotsets}
implies that $\rho_U(F)$ is reduced to a single rational vector. By going over
to a suitable iterate, we may assume $\rho_U(F) = \{0\}$. Let ${\cal D} := \{
z\in\ensuremath{\mathbb{T}}^2\mid \exists \ensuremath{\varepsilon} > 0 : \rho_{B_\ensuremath{\varepsilon}(z)}(F) = 0\}$ and note that $U
\ensuremath{\subseteq} {\cal D}$. Let $D$ denote the connected component of ${\cal D}$ that
contains $U$. Since ${\cal D}$ is $f$-invariant and $f$ is non-wandering, $D$
is periodic with period $p$ for some $p\in\ensuremath{\mathbb{N}}$.
We claim that $D$ contains no essential simple closed curve. In order to see
this, suppose for a contradiction that the curve $\gamma \ensuremath{\subseteq} D$ is essential
with homotopy vector $v\in\ensuremath{\mathbb{Z}}^2_*$. Let $\Gamma$ be a connected component of
$\pi^{-1}(\gamma)$. Then $\Gamma$ is a properly embedded line that remains in
a bounded distance of $\ensuremath{\mathbb{R}}\cdot v$. Furthermore, due to compactness $\gamma\ensuremath{\subseteq}
D$ is covered by a finite number of open sets $U_i$ with $\rho_{U_i}(F) =
\{0\}$, which implies $\rho_\gamma(F) = \{0\}$. However, this clearly
contradicts the existence of rotation vectors $\rho$ with $\vbr{\rho} \neq 0$
in $\rho(F)$. (Compare, for example, the proof of Lemma
\ref{l.rational-slope-segment}.)
In a similar way, we see that $D$ is simply-connected. If $\Gamma
\ensuremath{\subseteq} \pi^{-1}(D)$ is a closed Jordan curve, then by compactness we
obtain $\rho_\Gamma(F) = \{0\}$. Consequently the Jordan domain
$J(\Gamma)$ bounded by $J$ has rotation subset $\rho_{J(\Gamma)}(F)
= \{0\}$ and therefore belongs to $\pi^{-1}(D)$ as well. Thus $D$ is
the required $f^p$-invariant topological disk. Finally, as $D$ is
homeomorphic to $\ensuremath{\mathbb{R}}^2$, the restriction $f^p_{|D}$ defines a plane
homeomorphism. Since $f^p_{|D}$ is non-wandering, it must have
a fixed point. (This follows, for example, from the Brouwer Plane
Translation Theorem or from Franks Lemma \cite{franks:1988}).
\qed
\noindent
{\bf Proof of Proposition~\ref{p.chaotic}.} \ \alphlist \item Suppose
$f\in\ensuremath{\mathrm{Homeo}_0(\mathbb{T}^2)}$ is non-wandering, $F$ is a lift and $z\in{\cal C}(f)$ is
$\halb$-Lyapunov stable. Choose $\delta>0$ such that $f^n(B_\delta(z)) \ensuremath{\subseteq}
B_\halb(f^n(z)) \ \forall n\in\ensuremath{\mathbb{N}}$. Let $z_0\in\ensuremath{\mathbb{R}}^2$ be a lift of $z$ and $\ensuremath{\widehat}
U := B_\delta(z_0)$. Then $F^n(\ensuremath{\widehat} U) \ensuremath{\subseteq} B_\halb(F^n(z_0)) \ \forall
n\in\ensuremath{\mathbb{N}}$, in particular $\ensuremath{\mathrm{diam}}\left(F^n(\ensuremath{\widehat} U)\right) \leq 1$.
Since $f$ is non-wandering $U$ is recurrent, such that there exist infinitely
many pairs $(p,w) \in \ensuremath{\mathbb{N}}\times \ensuremath{\mathbb{Z}}^2$ with $\left(F^p(\ensuremath{\widehat} U) - w\right) \cap
\ensuremath{\widehat} U \neq \emptyset$. The latter implies $F^p(\ensuremath{\widehat} U) \ensuremath{\subseteq} B_{1+\delta}(z_0 +
w)$. Further, as $\left(F^{np}(\ensuremath{\widehat} U) -w\right) \cap F^{(n-1)p}(\ensuremath{\widehat} U) \neq
\emptyset \ \forall n\in\ensuremath{\mathbb{N}}$, we obtain by induction that
\[
F^{np}(\ensuremath{\widehat} U) \ \ensuremath{\subseteq} \ B_{n+\delta}(z_0+nw) \quad \forall n\in\ensuremath{\mathbb{N}} \ .
\]
However, this implies $\rho_{\ensuremath{\widehat} U}(F) \ensuremath{\subseteq} B_{\frac{1}{p}}(w)$, such that
$\ensuremath{\mathrm{diam}}(\rho_{\ensuremath{\widehat} U}(F)) \leq 1/p$. As $p$ can be chosen arbitrarily large we
obtain $\ensuremath{\mathrm{diam}}(\rho_{\ensuremath{\widehat} U}(F)) = 0$, in contradiction to $z\in{\cal C}(f)$. \item Suppose $f$ is area-preserving, $U$ is a connected and bounded
neighbourhood of $z\in{\cal C}(f)$ and $\ensuremath{\widehat} U$ is a connected component of
$\pi^{-1}(U)$. Birkhoffs Ergodic Theorem implies that Lebesgue-almost every
point $z'\in\ensuremath{\widehat} U$ has a rotation vector $\rho(F,z') =
\ensuremath{\lim_{n\rightarrow\infty}}\left(F^n(z')-z'\right)/n$ (that is, the limit exists). If
$\ensuremath{\lim_{n\rightarrow\infty}}\ensuremath{\mathrm{diam}}\left(\vbr{F^n(\ensuremath{\widehat} U)}\right)/n=0$ this implies $\rho_{\ensuremath{\widehat} U}(F)
\ensuremath{\subseteq} L_{\lambda,v}$ with $\lambda=\vbr{\rho(F,z')}$, contradicting $z\in {\cal
C}(f)$. \qed\end{list}
\section{A parameter family of Misiurewicz-Ziemian type} \label{MZ-family}
Examples of a toral homeomorphisms, homotopic to the identity, whose rotation set has non-empty interior were introduced by Misiurewicz and Ziemian \cite{misiurewicz/ziemian:1991} via lifts of the form
\begin{equation}
\label{e.mz-example}
F(x,y) \ = \ (x+\psi_2(y+\psi_1(x)),y+\psi_1(x)) \ ,
\end{equation}
with continuous and 1-periodic functions $\psi_i:\ensuremath{\mathbb{R}}\to\ensuremath{\mathbb{R}}$, $i=1,2$.
When $\psi_1(0)=\psi_2(0)=0$ and $\psi_1(\halb)=\psi_2(\halb)=1$, it
can be easily checked that the points $(0,0),(\halb,0),(0,\halb)$
and $(\halb,\halb)$ are fixed under the induced map $f\in\ensuremath{\mathrm{Homeo}_0(\mathbb{T}^2)}$
and have rotation vectors $(0,0),(0,1),(1,0)$ and $(1,1)$,
respectively. Since the rotation set $\rho(F)$ is convex, it
contains the square $[0,1]^2$. More or less the same type of
examples was proposed independently by Llibre and MacKay in
\cite{llibre/mackay:1991}.
In order to give a smooth example, we slightly modify this structure
and let $\psi_1(x)=\alpha\sin(2\pi x)$ and $\psi_2(y)=\beta\sin(2\pi
y)$ to obtain the parameter family
\begin{equation}
\label{e.mz-family}
F_{\alpha,\beta}(x,y) \ = \ (x+\beta\sin(2\pi(y+\alpha\sin(2\pi x))),
y+\alpha\sin(2\pi x)) \ .
\end{equation}
We denote the toral diffeomorphisms induced by these lifts by
$f_{\alpha,\beta}$. We note that $f_{\alpha,\beta}$ is area-preserving for
all $\alpha,\beta\in\ensuremath{\mathbb{R}}$ and hence, in particular, non-wandering.
For $F_*=F_{\halb,\halb}$, the points
$(\viertel,0),(0,\viertel),(\dreiviertel,0)$ and $(0,\dreiviertel)$
are $2$-periodic with rotation vectors
$(0,\halb),(\halb,0),(0,-\halb)$ and $(-\halb,0)$. The rotation set
$\rho(F_*)$ therefore contains the square $Q=\{(x,y)\mid |x|+|y|\leq
\halb\}$ spanned by these vectors. Hence, the toral diffeomorphism
$f_*=f_{\halb,\halb}$ induced by $F_*$ satisfies all the assumptions
of Theorem~\ref{t.semilocal-rotsets}. Furthermore, this remains true
for all $(\alpha,\beta)$ in a neighbourhood of $(\halb,\halb)$. The
reason is that when the rotation set has non-empty interior, then it
depends continuously on the toral homeomorphism in the ${\cal
C}^0$-topology \cite{misiurewicz/ziemian:1991}.
The points $(\frac{i}{4},\frac{j}{4})$ with $i,j=1,3$ are
$2$-periodic for $f_*$ and have rotation vector $(0,0)$. The phase
portrait of $f_*$ in Figure~\ref{f.1}\footnote{Picture produced with
SCILAB by computing 1000 iterates for each starting point in a
40x40-grid on \ensuremath{\mathbb{T}^2}\ and omitting the first 100 iterates for the
plot.\label{foot.scilab}} clearly suggests that these points are
surrounded by (star-shaped) elliptic islands. Since the differential
$DF^2_*$ in these points is the identity matrix, it is difficult to
establish the existence of elliptic islands in a rigorous way by the
application of standard KAM-results
\footnote{This would require an elliptic differential matrix with
irrational rotation angle \cite{katok/hasselblatt:1997}.}, and we
shall not further pursue this issue here. Yet, the pictures obtained
by simulations show typical features of a KAM-type elliptic region,
with invariant curves prevalent in the centre and at least one
clearly visible instability zone towards the boundary (Figure~\ref{fig.7}).
\begin{figure}
\caption{ \small Enlargement of the elliptic island around
$(\viertel,\viertel)$ in Figure~\ref{f.1}. Left: 20000 iterates of the
starting points $(0.253+i\cdot 0.00455)\cdot(1,1),\ i=1\ensuremath{,\ldots,} 10$. Right:
100000 iterates of the starting point $0.298429 \cdot (1,1)$ in the
instability region. ) }
\label{fig.7}
\end{figure}
Instead, we want to close by briefly discussing some of the symmetries present in Figure~\ref{f.1}. The picture is invariant under the rotation with angle $\pi/2$ around the point $(\halb,\halb)$, which is given by the map $R(x,y)=(-y,x)$ on the torus. However, $f_*$ is not conjugate to itself by $R$. The reason why this symmetry nevertheless appears is the fact that $R$ conjugates $f_*$ to its inverse, that is, $R^{-1}\circ f_*\circ R = f_*^{-1}$, and for the visualisation of elliptic islands it does not make any difference if $f_*$ is replaced by $f_*^{-1}$. This remains true for $f_{\alpha,\alpha}$ for all $\alpha$, since
\begin{equation}
\label{e.sym}
R^{-1}\circ f_{\alpha,\alpha}\circ R \ = \ f_{\alpha,\alpha}^{-1} \quad \forall \alpha\in\ensuremath{\mathbb{R}}\ .
\end{equation}
Two symmetries conjugating $f_{\alpha,\beta}$ to itself, for all
$\alpha,\beta\in\ensuremath{\mathbb{R}}$, are the rotation $S(x,y)=(-x,-y)$ with angle
$\pi$ and the map $T(x,y) = (x+\halb,-y+\halb)$, which is the
reflexion along the $x$-axis composed with the shift by
$(\halb,\halb)$. This implies that if the points
$(\frac{i}{4},\frac{j}{4})$ with $i,j=1,3$ are surrounded by
elliptic islands, then these are all isometric to each other, but the
isometries are orientation-reversing if $(i,j)$ and $(i',j')$ are
such that $i=i'$ or $j=j'$. When $\alpha=\beta$ the additional
symmetry given by $R$ implies that the elliptic islands are
self-symmetric. For the island surrounding $(\viertel,\viertel)$
the symmetry axis is $L=\{(x,y)\mid x+y=\halb\}$, for the others it
is the respective image of $L$ under $R^i,\ i=1,2,3$.
When $\alpha\neq \beta$ this self-symmetry of the elliptic islands
breaks down since (\ref{e.sym}) does not hold anymore. With a slight
adjustment of the parameters and some imagination in the Rorschach
test below, this allows to return to a more aquatic environment
(Figure~\ref{f.6}).
\begin{figure}
\caption{ \small Elliptic sharks circling a chaotic ocean. (Phase
portrait of $f_{\alpha,\beta}$ with $\alpha=0.5$ and
$\beta=0.502$, with the same indications as in Footnote~\ref{foot.scilab}.) }
\label{f.6}
\end{figure}
\end{document} | arXiv |
\begin{document}
\begin{frontmatter} \title{Estimating Causal Effects of HIV Prevention Interventions with Interference in Network-based Studies among People Who Inject Drugs}
\runtitle{Causal Effects in Network-based Studies}
\begin{aug}
\author[A]{\fnms{TingFang} \snm{Lee}\ead[label=e1] {[email protected]}}, \author[B]{\fnms{Ashley L.} \snm{Buchanan}\ead[label=e2] {[email protected]}}, \author[C]{\fnms{Natallia V.} \snm{Katenka}\ead[label=e3]{[email protected]}}, \author[D]{\fnms{Laura} \snm{Forastiere} \ead[label=e4]{[email protected] }}, \author[E]{\fnms{M. Elizabeth} \snm{Halloran}\ead[label=e5]{[email protected]}}, \author[F]{\fnms{Samuel R.} \snm{Friedman} \ead[label=e6] {[email protected]}}, \\ \and \author[G]{\fnms{Georgios} \snm{Nikolopoulos}\ead[label=e7]{[email protected]}}
\address[A]{Department of Pharmacy Practice, University of Rhode Island, \printead{e1}} \address[B]{Department of Pharmacy Practice, University of Rhode Island, \printead{e2}} \address[C]{Department of Computer Science and Statistics, University of Rhode Island, \printead{e3}} \address[D]{School of Public Health, Yale University, \printead{e4}} \address[E]{Biostatistics, Bioinformatics, and Epidemiology Program, Vaccine and Infectious Disease Division, Fred Hutchinson Cancer Center, and Department of Biostatistics, University of Washington, \printead{e5}} \address[F]{Department of Population Health, NYU Grossman School of Medicine, \printead{e6}} \address[G]{Medical School, University of Cyprus, \printead{e7}}
\end{aug}
\begin{abstract} Evaluating causal effects in the presence of interference is challenging in network-based studies of hard-to-reach populations. Like many such populations, people who inject drugs (PWID) are embedded in social networks and often exert influence on others in their network. In our setting, the study design is observational with a non-randomized network-based HIV prevention intervention. Information is available on each participant and their connections that confer possible HIV risk through injection and sexual behaviors. We considered two inverse probability weighted (IPW) estimators to quantify the population-level spillover effects of non-randomized interventions on subsequent health outcomes. We demonstrated that these two IPW estimators are consistent, asymptotically normal, and derived a closed-form estimator for the asymptotic variance, while allowing for overlapping interference sets (groups of individuals in which the interference is assumed possible). A simulation study was conducted to evaluate the finite-sample performance of the estimators. We analyzed data from the Transmission Reduction Intervention Project, which ascertained a network of PWID and their contacts in Athens, Greece, from 2013 to 2015. We evaluated the effects of community alerts on subsequent HIV risk behavior in this observed network, where the connections or links between participants were defined by using substances or having unprotected sex together. In the study, community alerts were distributed to inform people of recent HIV infections among individuals in close proximity in the observed network. The estimates of the risk differences for spillover using either IPW estimator demonstrated a protective effect. The results suggest that HIV risk behavior could be mitigated by exposure to a community alert when an increased risk of HIV is detected in the network. \end{abstract}
\begin{keyword} \kwd{Causal Interference} \kwd{Interference/dissemination} \kwd{Network Studies} \kwd{People who Use Drugs} \kwd{HIV/AIDS} \kwd{Inverse Probability Weights} \end{keyword}
\end{frontmatter}
\section{Introduction} \label{s:intro}
The objective of this work is to evaluate causal effects in the presence of interference (also known as dissemination or spillover) where usual assumptions, such as partial or clustered interference \cite{hudgens2008toward, tchetgen2012causal}, may no longer hold. This proves to be a challenging problem in network-based studies of hidden or hard-to-reach populations, such as people who inject drugs (PWID), where participants are frequently recruited through contact tracing. Worldwide in 2011, an estimated 10$\%$ of new HIV infections occur because of injection drug use, and this proportion was 30$\%$ outside Africa \citep{prejean2011estimated, lansky2014estimating, mathers2008global}. In Greece through 2010, there were only a few sporadic cases of HIV transmission among PWID and the HIV epidemic was traditionally concentrated among men having sex with men. From 2002 to 2010, less than 20 HIV cases were reported annually among PWID, representing 2\% to 4\% of newly diagnosed HIV infections per year. In 2011, the number of reported HIV cases among PWID increased 16-fold from the number reported in 2010 to a total of 260 cases. The emergence of the HIV outbreak among PWID in Athens coincided with an economic recession, highlighting its possible role in the outbreak due to the temporal ordering \citep{econrecession2013}. Investigation of the outbreak demonstrated that clustered HIV transmission among PWID was rare until 2009. Starting in 2010, a large proportion of HIV sequences from newly diagnosed PWID could be grouped into PWID-specific phylogenetic clusters, indicating that parenteral transmission with contaminated syringes or other injecting equipment was now occurring in this population. Prior to 2011, prevention and harm reduction services, including medication for opioid use disorder and syringe exchange distribution programs, were available; however, access to these services remained low among PWID. Most of the newly diagnosed PWID (about 70\%) in 2011 were residents of Athens, suggesting that the outbreak was also geographically localized. \citep{aristotle, nikolopoulous2015bigevent}
Effective interventions were urgently needed to prevent further transmission in Athens. The Transmission Reduction Intervention Project (TRIP) was a successful attempt to recruit and intervene in this population by contact tracing the injection and sexual networks of recently-infected PWID. The program then referred people found to be recently infected to engage in HIV treatment and care both to protect their own health and to reduce onward transmission of HIV to others, particularly during the early stage of HIV when there is a known increased risk of HIV transmission \cite{nikolopoulous2015bigevent}. Interestingly, this network study design can be used to investigate the connections or ties among people who are infected and uninfected, and thus can address questions about why certain groups of people who are uninfected remain that way despite having risk network ties to people who both have high viral loads and engage in risky behavior \citep{williams2018pockets}. The TRIP recruitment strategy successfully found more recently infected PWID than other strategies, such as a respondent driven sample or venue-based recruitment. These findings suggest that using strategically network-based approaches can accelerate seeking, testing, and treating recently-infected PWID. Moreover, reducing viral loads as early as possible is likely to decrease the expected number of transmissions in a community \citep{nikolopoulos2016network}.
Public health interventions often have disseminated effects, also known as indirect or spillover effects \citep{spillover2017, diffusion2020}. There can be disseminated effects of HIV behavioral interventions, suggesting that intervening among highly-connected individuals may maximize benefits to others \citep{Rewleye033759}. Akin to other populations, PWID are embedded in social networks and communities (e.g., injection drug, non-injection drug, sexual risk network) in which they possibly exercise an influence upon other members \citep{hayes2000design,ghosh2017social}. This influence can be measured as a \textit{disseminated} effect of specific interventions among individuals who were not exposed themselves but possibly receive intervention benefits from their connections to those exposed to the intervention. In PWID networks, interventions (e.g., educational training about HIV risk reduction, medical interventions such as pre-exposure prophylaxis, or treatment as prevention) may have disseminated effects, and intervention effects frequently depend on the network structure and intervention coverage levels. Disseminated effects may be larger in magnitude than direct effects (i.e., effect of receiving the intervention while holding the exposure of other individuals fixed), suggesting that an intervention has substantial effects in the network beyond those exposed themselves \citep{buchanan2018assessing}.
The current methodologies used to evaluate direct and disseminated effects among members of hidden or hard-to-reach populations in networks remain limited. In particular, relatively few methodological approaches for observational network-based studies have been developed, and methods that incorporate the observed connections (links, ties, or edges) in the underlying network structure are needed to understand the spillover mechanisms. In our setting, connections in the network refer to sexual and/or drug use partnerships. Recent methodological developments relaxed the no interference assumption and allowed for interference within clusters, known as {\it partial interference} \citep{sobel2006randomized, hong2006evaluating, hudgens2008toward, tchetgen2012causal, liu2014large}. In partial interference approaches, a clustering of observations is used to define the interference set (e.g., study clusters, provider practices, or geographic location) that allow for interference within but not across clusters; however, the information on connections within a particular cluster is typically not measured or utilized in the analytical approach \citep{aronow2017samii}. Another approach defines interference by spatial proximity or network ties \citep{liu2016inverse, forastiere2016identification}, allowing for overlapping interference sets (i.e., groups of individuals in which interference is assumed to be possible). In \cite{liu2016inverse}, an IPW estimator was proposed for a generalized interference set that allowed for overlap between interference sets; however, the asymptotic variance was estimated under the assumption of partial interference defined by larger groupings or clusters of participants in the study. In a separate paper, the subclassification estimator and generalized propensity score were used to quantify effects, and a bootstrapping procedure with resampling at the individual-level or the cluster-level was used to quantify the variance \citep{forastiere2016identification}. However, these approaches either rely on partial interference defined by larger clusters or resort to bootstrapping to derive estimators of the variance. In practice, ignoring the overlapping interference sets while estimating the variance can lead to inaccurate inference and resampling approaches, particularly in a network setting, can also be computationally intensive.
While previous work allows for overlapping interference sets for point estimation, the asymptotic variances were estimated under the assumption of partial interference or used bootstrapping techniques \citep{liu2016inverse, forastiere2016identification}. Our paper addresses an important gap by developing inverse probability weight estimators and deriving a closed-form variance estimator that allows for overlapping interference sets, possibly leading to a more statistically efficient estimator in network-based studies due to the use of additional information on connections between individuals. In our paper, we propose two inverse probability weighted (IPW) estimators where the interference set is defined as the set of the individual's nearest neighbors within a sociometric network; that is, a network in which all or some of participants' direct and indirect contacts are ascertained \citep{hadjikou2021drug}. The first IPW estimator is a novel application of the estimator of the approach in \citet{liu2016inverse} to a sociometric network-based study setting. Originally, the asymptotic variance estimators were developed for clustered observational studies without explicit consideration of the connections in the study. We relax the partial interference assumption for variance estimation such that interference sets are uniquely defined by nearest neighbors for each individual. The second IPW estimator uses a generalized propensity score developed by \citet{forastiere2016identification}; however, we propose a weighted estimator instead of a stratified estimator for comparison to the first IPW estimator in this paper. For both estimators, we assume that the nearest neighbors comprise the interference sets and use this structure to calculate a novel closed-form variance estimator by applying M-estimation. We focus on comparing these two alternative IPW estimators in a network study with a non-randomized intervention and statistical inference approaches using M-estimation.
The rest of the paper is structured as follows. In Sections 2, we introduce the TRIP study design and setting. In Section 3 and 4, we define the notations and assumptions for nearest neighbors settings, then the estimands of interest for this setting. We provide definitions of the two IPW estimators with specific assumptions for each, and demonstrate that the estimator is consistent and asymptotically normal, and obtain a closed-form estimator of the corresponding variances in Section 5. In Section 6, a simulation study was conducted to demonstrate the finite-sample performance of both estimators and the results are summarized. The methods were then utilized to assess the direct and disseminated effects of community alerts on HIV risk behavior in the sociometric network study of PWID and their contacts, Transmission Reduction Intervention Project (TRIP) from 2013 to 2015 in Athens, Greece in Section 7. We discuss limitations of this approach and next steps for methodological work to quantify causal effects in network-based studies in Section 8.
\section{TRIP Study Design} The Transmission Reduction Intervention Project (TRIP) included PWID and their HIV risk networks and initially found individuals who were recently diagnosed with HIV (known as seeds) and their possible HIV risk partners through sexual and injection routes of transmission \citep{nikolopoulos2016network,psichogiou2019identifying, giallouros2021drug, hadjikou2021drug, pampaka2021mental}. TRIP also recruited seeds with long-term HIV infection. TRIP used contact network tracing (i.e., nomination and coupon referrals) and venue recruitment methods to locate those who were at risk for HIV infection based on their proximity in the network to other recently-infected individuals. PWID who were participants in the ARISTOTLE study at HIV testing centers in Athens were initially recruited into the TRIP study if they were found to be recently infected or long-term infected with HIV. ARISTOTLE was a seek, test, treat, and retain intervention that used respondent-driven sampling to target PWID residing in Athens and aimed to contribute to the control of HIV transmission among PWID in Greece \citep{aristotle}. Each recently-diagnosed and long-term infected individual was asked to identify their recent sexual and drug use partners and their partners' partners in the six months prior to the interview. For the recently-diagnosed seeds, these direct contacts and their contacts' contacts were then recruited and asked to identify their sexual and drug use partners, who were also recruited and linked back to other individuals recruited in the study. For seeds with long-term HIV infection, their contacts were recruited (i.e., one wave of contact tracing) and these individuals were recruited and their connections to other participants were ascertained. If any of these contacts were identified as recently infected with HIV, then their contacts and the contacts of their contacts (i.e., two waves of contact tracing) would be recruited as well and connections to other participants in the network were ascertained (Figure \ref{fig:flowTRIP}); otherwise, one wave of contact tracing was performed. The study also recruited HIV-negative individuals from allied projects who served as controls. The HIV-negative individuals were isolates (i.e., no connections to others in the network) unless reported as a contact by another participant. This resulted in a network consisting of individuals recently diagnosed with HIV and their possible HIV risk partners and the connections in the network were defined by sex or injection drug use partnerships. This information was used to create a final observed network in which each recruited individual is linked to all other individuals who named them as a contact or was named as a contact by them, regardless of recruitment order.
Participants were interviewed at a baseline visit and 6-months after the baseline visit using a questionnaire to ascertain demographics, sexual and injection behaviors and partners in the prior 6 months, drug treatment, and antiretroviral treatment. In addition to HIV testing, the study provided access to treatment as prevention (TasP) for those with HIV, referrals for medical care, and distributed community alerts to inform members of the community about temporary increases in the risk for HIV acquisition. These alerts were paper flyers provided to participants and posted in locations frequented by members of the local PWID community. Participants were followed to ascertain demographics, risk behaviors, and substance use through interviews, HIV serostatus, timing of HIV infection, and HIV disease markers, including HIV viral load, through phylogenetic techniques approximately 6 months later. Complete details on the study design and recruitment can be found in \citet{nikolopoulos2016network,psichogiou2019identifying, giallouros2021drug, hadjikou2021drug, pampaka2021mental}.
For this study, we used data from the Athens, Greece site which was collected from 2013 to 2015 during the HIV outbreak that began following the economic recession in 2008 \citep{nikolopoulous2015bigevent,williams2018pockets}. The network structure in TRIP included 356 participants and 542 shared connections. One of the participant was recruited twice as a network member of a recent seed and as a network member of a control seeds with long-term HIV infection. In our analysis, we only used the information for this participant corresponding to their records as a network member of a recent seed. In the network, 79 participants were isolates (i.e. not sharing connection with other network members) and removed for our analysis as spillover is not possible for isolates. In addition, 2 participants were removed due to missing values on HIV risk behavior in the past 6 months reported at baseline and 59 participants were removed from the network due to loss to follow-up that resulted in missing information at the 6-month visit, including HIV risk behavior, which was the the outcome of interest. Figure \ref{fig:trip_network} represents the TRIP network with 216 participants after excluding isolates who were participants not connected with any other participants in the network and 25 participants (11.6\%) of the 216 participants were exposed to the community alerts. The network characteristics and distribution of participant attributes are summarized in Table \ref{tab:descripitve_stat}.
\begin{figure}
\caption{Flowchart of participant selection in TRIP \citep{pampaka2021mental}}
\label{fig:flowTRIP}
\end{figure}
\begin{figure}
\caption{The TRIP network consisted of 10 connected components. The size of each component was $\{185, 9, 6, 3, 3, 2, 2, 2, 2, 2\}$. The size of each component after using community detection to further divide the network into 20 components is $\{28, 26, 23, 19, 19, 18, 15, 12, 10, 9, 8, 7, 6, 3, 3, 2, 2, 2, 2, 2\}$. Dark shaded nodes represent the participants who were exposed to community alert and gray shaded nodes represent participants who were not exposed.}
\label{fig:trip_network}
\end{figure}
\begin{table} \centering \caption{TRIP network characteristics and participant attribute variables after excluding isolates and 59 participants (21\%) who were lost to follow up before their six-month visit$^1$.}
\begin{tabular}{lrr}
\toprule
\multirow{6}{*}{Network Characteristics} & Nodes & 216 \\
& Edges & 362 \\
& Components & 10\\
& Average Degree (SD) & 3.35 (2.75) \\
& Density & 0.016 \\
& Transitivity & 0.25 \\
& Assortativity & 0.25 \\
\arrayrulecolor[rgb]{ .8, .8, .8}\midrule
\multirow{2}{*}{{\it Baseline Visit}} & & \\
& & \\
\midrule
\multirow{2}{*}{Community alert} & Exposed & 25 (11.6\%) \\
& Not Exposed & 191 (88.4\%) \\
\midrule
\multirow{2}{*}{HIV Status} & Positive & 113 (52.3\%) \\
& Negative & 103 (47.7\%) \\
\midrule
\multirow{2}{*}{Date of first interview} & Before ARISTOTLE ended & 110 (50.9\%) \\
& After ARISTOTLE ended & 106 (49.1\%) \\
\midrule
\multirow{4}{*}{Education} & Primary School or less & 64 (29.6\%) \\
& High School (first 3 years) & 68 (31.5\%) \\
& High School (last 3 years) & 52 (24.1\%) \\
& Post High School & 32 (14.8\%) \\
\midrule
\multirow{4}{*}{Employment status} & Employed & 33 (15.3\%) \\
& Unemployed; looking for work & 54 (25.0\%) \\
& Can't work; health reason & 101 (46.8\%) \\
& Other & 28 (12.9\%)\\
\midrule
Shared injection equipment & Yes & 159 (73.6\%)\\
in last 6 months & No & 57 (26.4\%) \\
\midrule
\multirow{2}{*}{{\it Six-month Visit}} & & \\
& & \\
\midrule
Outcome: sharing injection & Yes & 92 (42.6\%) \\
equipment at the 6-month visit & No & 124 (57.4\%)\\
\arrayrulecolor[rgb]{ 0, 0, 0}\bottomrule
\end{tabular}
\footnotesize{$^1$ The transitivity measures the density of triads in a network. The assortativity quantifies the extent to which connected nodes share similar properties. } \label{tab:descripitve_stat} \end{table}
\section{Notation} We employ a potential outcomes framework for causal inference and assume the sufficient conditions for valid estimation of causal effects, which have been well-described \citep{ogburn2014causal, liu2016inverse, forastiere2016identification}. However, we relax the no dissemination or interference assumption \citep{rubin1980}. In our setting, we evaluate the effect of a non-randomized intervention on a subsequent outcome in an observed network, where information is available on the nodes (i.e., each participant) and their links (i.e., HIV risk connections through sexual or injection behavior). We evaluate the effect of being exposed to community alerts on HIV risk behaviors (i.e., sharing injection equipment) reported at the 6-month follow up. According to the network-based study design of TRIP that recruited at least one wave of contact tracing for each participant of an HIV-infected seed, we anticipate that there could be dissemination or spillover between two individuals connected by an link (i.e., possible influence of their neighbors' intervention exposure on an individual's outcome). Based on reported connections, we assume that smaller groupings or neighbors for each individual can be identified in the data. Following \cite{forastiere2016identification} and \cite{liu2016inverse}, we make the nearest neighbors interference assumption (NIA). The NIA is a network analog to the partial interference assumption used for clusters \citep{sobel2006randomized,hudgens2008toward}; however, partial interference does not assume a unique interference set for each individual, but instead the set is the same for all individuals in a cluster. The NIA assumption applies to the nearest neighbors uniquely defined for each participant in the study, so the connections between individuals and their neighbors can now be explicitly considered in the estimands and estimation. This implies that the potential outcomes of a participant depend only on their own exposure and that of their nearest neighbors and not on the exposures of others in the network beyond the nearest neighbors, positing that an individual only has spillover from their first degree contacts. In other words, if the exposures of an individual and their neighbors are held fixed, then changing the exposures of others outside the nearest neighbors and the individual does not change the outcome for the individual.
Consider a finite population of $n$ individuals and each individual self-selects their exposure to a study intervention. Let $i=1, ... , n$ denote each participant in the study and let $A_i$ be the binary exposure of participant $i$ with $A_i=1$ if exposed to an intervention and $0$, otherwise. Let $Z_i$ denote the vector of pre-exposure covariates for participant $i$. These participants are connected through an observed network $\mathcal{C}$ that can be represented by a binary adjacency matrix $E(\mathcal{C}) = [e_{ij}]_{i,j=1}^n \in \lbrace 0,1 \rbrace^{n \times n}$, with $e_{ij}=1$ if participants $i$ and $j$ share an edge or connection, and $e_{ij}=0$, otherwise. We assume $e_{ii}=0$. Each participant is represented as a node in the network. The set of nodes in network $\mathcal{C}$ is denoted by $V(\mathcal{C})$. Given an observed network, a component is a connected subnetwork that is not part of any larger connected subnetwork. A network that is itself connected has exactly one component. If $\mathcal{C}$ has $m$ components, we denote the components $\{C_\nu|\nu=1, \hdots, m\}$. Denote the nearest neighbors of participant $i$ by $\mathcal{N}_i=\lbrace j: e_{ij}=1 \rbrace$ and ${\mathcal{N}_i^*}=\mathcal{N}_i\cup\{i\}$ denote the nearest neighbors and participant $i$. The degree of individual $i$ (or number of nearest neighbors) is denoted as $d_i = \sum_{j=1}^n e_{ij} =|\mathcal{N}_i|$. We denote the vector of intervention exposures for the nearest neighbors for participant $i$ as ${A}_{\mathcal{N}_i}= [A_{j}]_{j:e_{ij}=1}$. In this setting, the outcome of participant $i$ depends not only on their own exposure, but also on the vector of their neighbors' exposures ${A}_{\mathcal{N}_i}$ (NIA). In other words, we let $\mathcal{N}_i$ be the interference set of individual $i$ in which the neighbors' exposures may affect the outcome of individual $i$. We also denote the vector of pre-exposure covariates for the nearest neighbors for participant $i$ as ${Z}_{\mathcal{N}_i}= [Z_{j}]_{j:e_{ij}=1}$. Denote realizations of exposures $A_i$ by $a_i$ and ${A}_{\mathcal{N}_i}$ by $a_{\mathcal{N}_i}$. Similarly, denote realizations of covariates $Z_i$ by $z_i$ and ${Z}_{\mathcal{N}_i}$ by $z_{\mathcal{N}_i}$.
Let $y_i(a_i, {a}_{\mathcal{N}_i})$ denote the potential outcome of individual $i$ if they received intervention $a_i$ and their nearest neighbors received the vector of interventions denoted by ${a}_{\mathcal{N}_i}$. Let $Y_i = y_i(A_i, {A}_{\mathcal{N}_i})$ denote the observed outcome, which holds by causal consistency. Therefore, the potential outcomes are assumed to be deterministic functions and the observed outcomes are assumed to be random variables. In our study setting, $A_i$ represents an indicator for whether participant $i$ is exposed to community alerts and the pre-exposure covariates include HIV status ascertained in the TRIP study, date of first interview, education status, employment status, and report of shared drug use equipment (e.g. syringe) in last 6 months prior to baseline. The observed outcome $Y_i$ is the status of sharing injection equipment in the last 6 months prior to the 6-month follow-up visit.
In this paper, we define average potential outcomes using a Bernoulli allocation strategy \citep{tchetgen2012causal}, where $\alpha$ represents the counterfactual scenario in which individuals in $\mathcal{N}_i$ receive the exposure with probability $\alpha$ and we refer to this parameter as the \textit{intervention} coverage for the nearest neighbors. This is essentially like standardizing the observed exposure vectors to study population in which the exposure assignment mechanism follows a Bernoulli distribution with probability $\alpha$. This allows stochasticity in the intervention assignment for individuals who are possibly members of more than one nearest neighbors. In the observational study, we are not assuming that $A_1,\cdots, A_n$ are independent Bernoulli random variables; however, this distribution of exposure is used to define the counterfactuals. We use information collected in a sociometric network with a non-randomized intervention for estimation. Let $\pi(a_{\mathcal{N}_i};\alpha)=\alpha^{\sum_{j\in \mathcal{N}_i}a_j}(1-\alpha)^{d_i-\sum_{j \in \mathcal{N}_i}a_j}$ denote the probability of the nearest neighbors of individual $i$ receiving intervention exposure $a_{\mathcal{N}_i}$ under allocation strategy $\alpha$. The allocation strategy $\alpha$ can also be considered as the intervention coverage level for the nearest neighbors. Let $\pi(a_i;\alpha)=\alpha^{a_i}(1-\alpha)^{1-a_i}$ denote the probability of individual $i$ receiving exposure $a_i$ and $\pi(a_i, a_{\mathcal{N}_i};\alpha)=\pi(a_{\mathcal{N}_i};\alpha)\pi(a_i;\alpha)$ denote the probability of individual $i$ together with their nearest neighbors receiving the set of exposures $(a_i,a_{\mathcal{N}_i})$.
\section{Estimands} We follow notations from \cite{liu2016inverse} to define the estimands. Define $\bar{y}_i(a,\alpha) =\sum_{a_{\mathcal{N}_{i}}} y_i(a_i=a,{a}_{\mathcal{N}_i})\pi({a}_{\mathcal{N}_i};\alpha)$ to be the average potential outcome for individual $i$ under allocation strategy $\alpha$ and exposure $a_i=a$ where the summation is over all $2^{d_i}$ possible values of ${a}_{\mathcal{N}_i}$. Averaging over all individuals, we define the population average potential outcome as $\bar{y}(a,\alpha)=\sum_{i=1}^n \bar{y}_i(a,\alpha)/n$. We also define the marginal average potential outcome for individual $i$ under allocation strategy $\alpha$ by $\bar{y}_i(\alpha)=\sum_{a_i,a_{\mathcal{N}_{i}}} y_i(a_i,a_{\mathcal{N}_i})\pi(a_i,{a}_{\mathcal{N}_i};\alpha)$ and define the marginal population average potential outcome as $\bar{y}(\alpha)=\sum_{i=1}^n \bar{y}_i(\alpha)/n$.
We consider different contrasts of these average causal effects often of interest in network-based studies. We define these on the risk difference scale and analogous effects can be defined on the ratio scale. The direct effect is defined as $\overline{DE}(\alpha)=\bar{y}(1,\alpha)-\bar{y}(0,\alpha)$, which compares the average potential outcomes when a participant is exposed to the intervention compared to when a participant is not exposed under allocation strategy $\alpha$. For example, in TRIP study, the direct effect is a difference in the risk of reporting HIV risk behaviors when a participant is exposed to community alerts versus when a participant is not exposed with $100\cdot\alpha$\% of their nearest neighbors exposed to alerts. The disseminated (i.e., indirect or spillover) effect is $\overline{IE}(\alpha_1, \alpha_0)=\bar{y}(0,\alpha_1)-\bar{y}(0,\alpha_0)$, which compares the average potential outcomes of unexposed individuals under two different allocation strategies $\alpha_1$ and $\alpha_0$. The composite or total effect is defined as $\overline{TE}(\alpha_1, \alpha_0)=\bar{y}(1,\alpha_1)-\bar{y}(0,\alpha_0)$, which is a function of both the direct and disseminated effects and is a measure of the maximal intervention effect (assuming that $\alpha_1>\alpha_0$), comparing average potential outcomes for exposed participants under allocation strategy $\alpha_1$ to unexposed participants under allocation strategy $\alpha_0$. Lastly, the overall effect is $\overline{OE}(\alpha_1, \alpha_0)=\bar{y}(\alpha_1)-\bar{y}(\alpha_0)$, which is the difference in average potential outcomes under two different allocation strategies.
\section{IPW Identification Assumptions and Estimators}
In an observational study of a network, interventions are typically not randomized at either the network or individual-level, but rather individuals and their nearest neighbors typically self-select their own exposures. Therefore, identification of causal effects does not benefit from exchangeability achieved by randomization and adjustment for a sufficient set of pre-exposure covariates at both the individual- and network-level is needed to quantify causal effects. In this section, we apply two different IPW estimators \citep{liu2016inverse, forastiere2016identification} to a setting with the interference set defined by nearest neighbors in observed networks with components. We assume that the observed network can be expressed as the union of $m$ components denoted by $C_1, C_2, \hdots, C_m$ for $\nu = 1, \hdots, m$ (Figure \ref{fig:sample_network}). We quantify the variance accounting for correlation within components of the full observed network. Importantly, we now incorporate the nearest neighbor structure in the estimating equations used to calculate the closed-form variances because this better reflects the underlying structure through which dissemination operates in the observed network. Individuals who share a connection or link are more likely to influence each other, as opposed to individuals who are clustered together, possibly in a large grouping like a component. In \cite{liu2016inverse}, information on their connections or distance in the network between individuals is either not available or not used for statistical inference and the assumption is nonetheless made that these individuals could all possibly influence each other within the set (i.e., a generalized interference set).
\begin{figure}
\caption{A sample network with two components. $C_1= \{1, 2, 3, 4, 5\}$ and $C_2=\{6, 7, 8, 9, 10, 11, 12, 13\}$. The nearest neighbors of node $2$ are $\mathcal{N}_2=\{4, 5\}$, of node 3 are $N_3=\{1, 4\}$, and of node $6$ are $\mathcal{N}_6=\{9, 10, 13\}$.}
\label{fig:sample_network}
\end{figure}
\subsection{Assumptions} \begin{assumption} (Exchangeability) \label{assump:exchange} Assume that conditional on pre-exposure covariate vector $Z_i$ and the covariates of their nearest neighbors ${Z}_{\mathcal{N}_i}$, the intervention allocation for individual $i$ and their nearest neighbors $\mathcal{N}_i$ is independent of all potential outcomes \begin{align}
&\Pr(A_i=a_i, {A}_{\mathcal{N}_i}={a}_{\mathcal{N}_i}|Z_i=z_i,{Z}_{\mathcal{N}_i}={z}_{\mathcal{N}_i})\notag\\
=&\Pr(A_i=a_i, {A}_{\mathcal{N}_i}={a}_{\mathcal{N}_i} |Z_i=z_i, {Z}_{\mathcal{N}_i}={z}_{\mathcal{N}_i}, y_1(\cdot),\ldots, y_n(\cdot)).\notag\end{align} \end{assumption}
\begin{assumption}\label{assump:positive} (Positivity) Assume that
$\Pr(A_i=a_i|Z_i=z_i)>0$ and $\Pr(A_i=a_i, A_{\mathcal{N}_i}=a_{\mathcal{N}_i}|Z_i=z_i, Z_{\mathcal{N}_i}=z_{\mathcal{N}_i})>0$ for all $a_i$, $a_{\mathcal{N}_i}$, $z_i$, and $z_{\mathcal{N}_i}$. \end{assumption}
\begin{assumption}\label{assump:irrelavance} (Treatment variation irrelavance) We assume that the treatment or intervention assignment mechanism does not affect the outcome. More precisely, if there are different versions of the intervention, we assume that those are irrelevant for the causal contrasts of interest and that we have one version of intervention and one version of no intervention. \citep{forastiere2016identification}. \end{assumption}
\begin{assumption}\label{assump:local_nn} (Conditional exposure independence) Conditional on the exposure and covariates for individual $i$ and their neighbors $\mathcal{N}_i$ and the neighbor-level random effect $b_{\mathcal{N}_i^*}$, the exposure $A_i$ for individual $i$ and the exposure for the neighbors $A_{\mathcal{N}_i}$ are independent. That is, given the nearest neighbor-level random effect $b_{\mathcal{N}_i^*}$ and $b_{\mathcal{N}_i^*}$,
$$A_i|A_{\mathcal{N}_i}, Z_{\mathcal{N}_i}, b_{\mathcal{N}_i^*} \perp A_j|A_{\mathcal{N}_j}, Z_{\mathcal{N}_j}, b_{\mathcal{N}_i^*}.$$ \end{assumption} The nearest neighbor-level random effect $b_{\mathcal{N}_i^*}$ accounts for possible correlation of exposures among individual $i$ and their neighbors $\mathcal{N}_i$. This assumption is used to estimate the propensity score of IPW$_1$ (defined in Section 5.2).
\begin{assumption}\label{assump:nni} (Nearest neighbors interference) The outcome for an individual depends their own exposure and the exposures of only other individuals who are their nearest neighbors \citep{forastiere2016identification}. By consistency, the following holds: $$Y_i=Y_i(A_i, A_{\mathcal{N}_i}).$$ \end{assumption} For example, in Figure \ref{fig:sample_network}, $Y_1=Y_1(a_1, (a_3, a_5))$, which is that the outcome for individual 1 is affected by their own exposure and the exposure of individual 3 and 5 only and no other individuals' exposures in either the component or network.
\begin{assumption}\label{assump:stratifi} (Stratified interference) The outcome for an individual depends on their own exposure and on the total number of exposed nearest neighbors \citep{hudgens2008toward, sobel2006randomized}. \end{assumption}
\begin{assumption}\label{assump:reducible} (Reducible propensity score assumption) The individual exposure $A_i$ does not depend on neighbors' covariates $Z_{\mathcal{N}_i}$ and neighbors' exposures $A_{\mathcal{N}_i}$ do not depend on individual covariates $Z_i$ \citep{forastiere2016identification}. \begin{center}
$P(A_i|Z_i, Z_{\mathcal{N}_i})=P(A_i|Z_i)$ and $P(A_{\mathcal{N}_i}|A_i, Z_i, Z_{\mathcal{N}_i})=P(A_{\mathcal{N}_i}|A_i, Z_{\mathcal{N}_i})$. \end{center} \end{assumption}
Exchangeability (Assumption \ref{assump:exchange}), positivity (Assumption \ref{assump:positive}), and treatment variation irrelevance (Assumption \ref{assump:irrelavance}) are necessary assumptions for causal inference under the potential outcomes framework \citep{rubin1980}. Due to the lack of randomization of the intervention, we require a conditional exchangeability assumption for both the individual and their neighbors, which allows for identification of causal contrasts related to both the individual's exposure and the allocation strategy for their neighbors. The positivity assumption ensures we have individuals and their neighbors exposed (and not exposed) at each level of the covariates. We also assume treatment variation irrelevance for the intervention, which ensures we have only one version of being exposed to the intervention and one version of not being exposed, which clarifies how we define the potential outcomes related to each intervention exposure. In this work, we assume that only the first degree neighbors' exposures can influence an individual's outcome, which allows us to focus locally in the network to evaluate spillover. Assumption \ref{assump:stratifi} and \ref{assump:reducible} apply to IPW$_2$ only and are discussed in Section 5.2; however, assumption \ref{assump:stratifi} may also be applied to IPW$_1$ (defined in Section 5.2) when there are concerns about positivity violations.
\subsection{Estimators} Under the Assumptions \ref{assump:exchange}, \ref{assump:positive}, \ref{assump:irrelavance}, \ref{assump:local_nn} and \ref{assump:nni}, the first IPW estimator is an adaptation of the one proposed by \citet{liu2016inverse}, and we define the interference sets by the nearest neighbors for each individual in the observed network, and then use this nearest neighbor structure within each component when deriving the closed-form variance estimator in Section 5.3. Define the IPW estimator for exposure $a$ with allocation strategy $\alpha$ as
\begin{equation}\widehat{Y}^{IPW_1}(a, \alpha)=\frac{1}{n}\sum_{i=1}^n \frac{y_i(A_i, A_{\mathcal{N}_i})I(A_i=a)\pi(A_{\mathcal{N}_i};\alpha)}{f_1(A_i, A_{\mathcal{N}_i}|Z_i, Z_{\mathcal{N}_i})},\label{eq:eq1}\end{equation} where $f_1(A_i, A_{\mathcal{N}_i}|Z_i, Z_{\mathcal{N}_i})$ is the nearest neighbors-level exposure propensity score. We assume that conditional on the nearest neighbor-level random effect and the exposure and covariates for individual $i$ and the neighbors $\mathcal{N}_i$, the exposures of nearest neighbors $A_{\mathcal{N}_i}$ and the exposure of individual $A_i$ are independent. In other words, the dependency between the exposures for individual $i$ and their neighbors is captured by both the fixed exposures and covariates and nearest neighbor-level random effect. To model the propensity score, the probability of exposure following a Bernoulli distribution and conditional on observed baseline covariates is given by
$$f_1(A_i, A_{\mathcal{N}_i}|Z_i, Z_{\mathcal{N}_i})=\int_{-\infty}^\infty\prod_{j \in \mathcal{N}_i^*}p_j^{A_j}(1-p_j)^{1-A_j}f(b_{\mathcal{N}_i^*}; 0, \psi)db_{\mathcal{N}_i^*},$$
where $\mathcal{N}_i^*=\mathcal{N}_i\cup\{i\}$, $$p_j=\mbox{Pr}(A_j=1|Z_j, b_{\mathcal{N}_i^*})=\mbox{logit}^{-1}(Z_j\cdot\gamma+b_{\mathcal{N}_i^*}),$$ and $f(b_{\mathcal{N}_i^*}; 0, \psi)\sim N(0, \psi)$. Here, $b_{\mathcal{N}_i^*}$ is the nearest neighbors-level random effect accounting for possible correlation of exposures among individual $i$ and their neighbors $\mathcal{N}_i$.
\noindent The marginal population-level average potential outcome estimator is
\begin{equation}\widehat{Y}^{IPW_1}(\alpha)=\frac{1}{n}\sum_{i=1}^n \frac{y_i(A_i, A_{\mathcal{N}_i})\pi(A_i, A_{\mathcal{N}_i};\alpha)}{f_1(A_i, A_{\mathcal{N}_i}|Z_i, Z_{\mathcal{N}_i})}.\label{eq:eq1}\end{equation}
Under the Assumptions \ref{assump:exchange}, \ref{assump:positive}, \ref{assump:irrelavance}, \ref{assump:nni}, \ref{assump:stratifi} and \ref{assump:reducible}, the second IPW estimator uses an individual and nearest neighbors propensity score as defined in \cite{forastiere2016identification}. The potential outcomes of individual $i$ depend on the total number of exposed neighbors, $s_i=\sum_{j \in \mathcal{N}_i} a_j$ (and let $S_i=\sum_{j \in \mathcal{N}_i} A_j$). In particular, $$y(a_i, a_{\mathcal{N}_i})=y(a_i, s_i).$$ The IPW estimator for exposure $a$ with coverage $\alpha$ is defined as
\begin{equation}\widehat{Y}^{IPW_2}(a;\alpha)=\frac{1}{n}\sum_{i=1}^n\frac{y_i(A_i, S_i)I(A_i=a)\pi(S_i;\alpha)}{f_2(A_i, S_i|Z_i, Z_{\mathcal{N}_i})},\label{eq:eq3}\end{equation} and the IPW marginal estimator as
\begin{equation}\widehat{Y}^{IPW_2}(\alpha)=\frac{1}{n}\sum_{i=1}^n\frac{y_i(A_i, S_i)\pi(A_i, S_i;\alpha)}{f_2(A_i, S_i|Z_i, Z_{\mathcal{N}_i})}.\label{eq:eq4}\end{equation} Let $$\pi(S_i;\alpha)={d_i\choose S_i}\alpha^{S_i}(1-\alpha)^{d_i-S_i}$$ be the probability of individual $i$ has $S_i$ exposed neighbors and $$\pi(A_i, S_i;\alpha)=\pi(S_i;\alpha) \pi(A_i;\alpha)$$ denote the probability of exposure for individual $i$ together with $S_i$ exposed neighbors.
The propensity score $f_2(A_i, S_i|Z_i, Z_{\mathcal{N}_i})$ is the joint probability distribution of individual exposure and nearest neighbors exposure given the covariates $Z_i$ and $Z_{\mathcal{N}_i}$. Here, we express this as a product of the individual propensity score, $f_{22}(A_i|Z_i)$, and nearest neighbors propensity score, $f_{21}(S_i|A_i, Z_{\mathcal{N}_i})$.
We assume that the individual exposure $A_i$ follows a Bernoulli distribution
$$P(A_i=a_i|Z_i)=p_{2, i}^{A_i}(1-p_{2, i})^{1-A_i}$$ with probability $p_{2, i}$ defined as the individual propensity score, modeled as a function of a covariate vector using a logit link
$$p_{2, i}=\mbox{Pr}(A_i=1|Z_i)=\mbox{logit}^{-1}(Z_i\cdot\gamma).$$ Furthermore, we assume that the total number of exposed neighbors $\sum A_{\mathcal{N}_i}$ follows a binomial distribution
$$P(S_i=s_i| A_i, Z_{\mathcal{N}_i})={d_i \choose S_i}p_{1, i}^{S_i}(1-p_{1, i})^{d_i-S_i}$$ with probability $p_{1, i}$ modeled as a function of the nearest neighbors covariate vector using a logit link
$$p_{1, i}=\mbox{Pr}(S_i=s_i|A_i, Z_{\mathcal{N}_i})=\mbox{logit}^{-1}(A_i\beta + h(Z_{\mathcal{N}_i})\cdot\delta'),$$ where $h(Z_{\mathcal{N}_i})$ is an aggregate function of the vector $Z_{\mathcal{N}_i}$. For instance, the proportion of females or males in the nearest neighbors or average age of an individual's nearest neighbors.
We assume that conditional on the nearest neighbors covariates and the exposure for individual $i$, the exposures of nearest neighbors $A_{\mathcal{N}_i}$ are independent and identically distributed. In other words, the dependency between neighbors' exposure is captured by the correlation with the exposure for individual $i$ and the covariates of the nearest neighbors.\footnote{In principle, we could compute the nearest neighbors propensity score $f_{21}(S_i|A_i, Z_{\mathcal{N}_i})$ as a product of the individual propensity scores for all neighbors for all exposure combinations $a_{\mathcal{N}_i}$ such that $S_i=s_i$ under the assumption of independence of $A_i$ given a nearest neighbor-level random effects and individual exposure and covariates. This would be one correct way of computing the nearest neighbors propensity score. Instead in this estimator, we use an alternative solution where the nearest neighbors propensity score is estimated assuming a binomial model conditional on a summary statistics of the nearest neighbors covariates. This approach, while approximate, is more straightforward and works when the dependency among neighbors' exposures cannot be attributed to a latent factor shared by all units belonging to the same nearest neighbor set in the network.} Therefore,
the propensity score $f_2(A_i, S_i|Z_i, Z_{\mathcal{N}_i})$ can be factor into two marginal distributions $f_{21}$ and $f_{22}$ as follows: \begin{align}
f_2(A_i, S_i|Z_i, Z_{\mathcal{N}_i})&=f_{21}(S_i|A_i, Z_{\mathcal{N}_i})f_{22}(A_i|Z_i)\notag\\ &={d_i \choose S_i} p_{1, i}^{S_i}(1-p_{1, i})^{d_i-S_i}\cdot p_{2, i}^{A_i}(1-p_{2, i})^{1-A_i}\notag \end{align}
Under allocation strategy $\alpha, \alpha_0$, and $\alpha_1$, we consider the following risk difference estimators of the direct, disseminated (indirect), composite (total), and overall effects: \begin{align}
& \widehat{DE}_r(\alpha)=\widehat{Y}^{IPW_r}(1, \alpha)-\widehat{Y}^{IPW_r}(0, \alpha),\notag\\
&\widehat{IE}_r(\alpha_1, \alpha_0)=\widehat{Y}^{IPW_r}(0, \alpha_1)-\widehat{Y}^{IPW_r}(0, \alpha_0),\notag\\
&\widehat{TE}_r(\alpha_1, \alpha_0)=\widehat{Y}^{IPW_r}(1, \alpha_1)-\widehat{Y}^{IPW_r}(0, \alpha_0),\notag\\
&\widehat{OE}_r(\alpha_1, \alpha_0)=\widehat{Y}^{IPW_r}(\alpha_1)-\widehat{Y}^{IPW_r}(\alpha_0),\notag \end{align} where $r=1, 2$ corresponds to the two IPW estimators that we defined above.
\begin{theorem}
If the propensity scores $f_1(A_i, A_{\mathcal{N}_i}|Z_i, Z_{\mathcal{N}_i})$ and $f_2(A_i, S_i|Z_i, Z_{\mathcal{N}_i})$ are known, then $E[\widehat{Y}^{IPW_r}(a, \alpha)]=\bar{y}(a, \alpha)$ and $E[\widehat{Y}^{IPW_r}(\alpha)]=\bar{y}(\alpha)$. \end{theorem} Proof of Proposition 1 is shown in Appendix A. Using these unbiased estimators when the propensity score is known, the estimation of the causal effects will also be unbiased because the causal effects are contrasts of these marginal quantities.
\subsection{Large sample properties of the inverse probability of sampling weighted estimator}
The large sample variance estimators can be derived using M-estimation theory \citep{mestimator2013}. We assume that the observed network can be expressed as the union of components; that is, non-overlapping groups of individuals \citep{liu2016inverse}. Consider a social network with $n$ individuals and $m$ components denoted by $\{C_1, C_2, \cdots, C_m\}$ with $\nu = 1,\hdots, m$. Let $Y_{\nu i}, A_{\nu i}, Z_{\nu i}$ denote the outcome, exposure, and covariates for individual $i$ in component $\nu$, respectively. Let $V(C_\nu)$ be the set of nodes in $C_\nu$, and $Y_\nu=\{Y_{\nu i}|i \in V(C_\nu)\}$, $A_\nu=\{A_{\nu i}|i \in V(C_\nu)\}$, $Z_\nu=\{Z_{\nu i}|i \in V(C_\nu)\}$. The observable random variables $(Y_\nu, A_\nu, Z_\nu)$ for $\nu=1, \hdots, m$ are assumed to be independent but not necessarily identically distributed with distribution $F_\nu$. We assume that the $m$ components are a random sample from the infinite super-population of groups and the size of each component is bounded \citep{mestimator2013}.
Recall, for IPW$_1$, the parameters of the exposure propensity score model include coefficients for the fixed effects and the random effect, while for IPW$_2$, the parameters include coefficients for the fixed effects from two logistic models (see Section 5.2). Let $\Theta=\{\gamma, \psi\}$ the set of coefficients of fixed effects and the random effects in the propensity score $f_1$ when using IPW$_1$, and $\Theta=\{\gamma, \beta, \delta'\}$ be the set of coefficients in the propensity score $f_2$ when using IPW$_2$. To generalize notation, we set the dimension of $\Theta$ to be $p$ and refer to these parameters as $\eta$ in the estimating equations below. Let $Y_{C_\nu}=(Y_{ C_\nu0}, Y_{C_\nu1}, Y_{C_\nu2})$ be the component-level average potential outcomes defined as \begin{align}
Y_{C_\nu0} &=\sum_{i \in V(C_\nu), a_{\mathcal{N}_i}}y_i(a_i=0, a_{\mathcal{N}_i})\pi(a_{\mathcal{N}_i};\alpha),\notag\\
Y_{C_\nu1} &= \sum_{i \in V(C_\nu), a_{\mathcal{N}_i}}y_i(a_i=1, a_{\mathcal{N}_i})\pi(a_{\mathcal{N}_i};\alpha),\notag\\
Y_{C_\nu2} &=\sum_{i \in V(C_\nu), a_i, a_{\mathcal{N}_i}}y_i(a_i, a_{\mathcal{N}_i})\pi(a_i, a_{\mathcal{N}_i};\alpha).\notag \end{align}
To conduct inference, we use $m$ independent components, while preserving the underlying connections of an individual's nearest neighbors comprising the network structure of each component. That is, by extending \cite{liu2016inverse}, every individual is now assigned their own propensity score based on the observed network structure defined by their nearest neighbors (see Section 5.2). Whereas in \cite{liu2016inverse}, statistical inference was conducted by assuming partial interference in which the study population was partitioned into non-overlapping groups and all individuals in a group were assigned one group-level propensity score. To simplify the notation in this section, we write the propensity score $f_2(A_{\nu i}, S_{\nu i}|Z_{\nu i}, Z_{\mathcal{N}_{\nu i}})$ as $f_2(A_{\nu i}, A_{\mathcal{N}_{\nu i}}|Z_{\nu i}, Z_{\mathcal{N}_{\nu i}})$ and let the observed outcome for individual $i$ in component $\nu$ be denoted by $Y_{\nu i}=Y_{\nu i}(A_{\nu i}, A_{\mathcal{N}_{\nu i}})$. Note that the potential outcomes are random due to the random sampling of the $m$ components. With this partition of the network, the inverse probability weighted estimator for exposure $a$ and strategy $\alpha$ presented in Section 5.2 equals \begin{equation}
\widehat{Y}^{IPW_r}(a, \alpha)=\frac{1}{n}\sum_{\nu=1}^m\sum_{i\in V(C_{\nu})} \frac{Y_{\nu i}I(A_{\nu i}=a)\pi(A_{\mathcal{N}_{\nu i}};\alpha)}{f_r(A_{\nu i}, A_{\mathcal{N}_{\nu i}}|Z_{\nu i}, Z_{\mathcal{N}_{\nu i}})}.\label{eq:eq5}\end{equation}
Let $\theta=(\Theta, \theta_{0\alpha}, \theta_{1\alpha}, \theta_\alpha)$, where $\theta_{0\alpha}=\bar{y}(0, \alpha)=1/n\sum _{\nu=1}^m Y_{C_\nu0}$, $\theta_{1\alpha}=\bar{y}(1, \alpha)=1/n\sum _{\nu=1}^m Y_{C_\nu1}$, and $\theta_{\alpha}=\bar{y}(\alpha)=1/n\sum _{\nu=1}^m Y_{ C_\nu2}$. Let $\hat{\theta}=(\hat{\Theta}, \hat{\theta}_{0\alpha}, \hat{\theta}_{1\alpha}, \hat{\theta}_{\alpha})$. Similar to the approach in \cite{liu2016inverse}, let the average component size in the study population be defined as $k=E[|V(C_\nu)|]$, which is the mean component size in the population. We use this to redefine the inverse probability weighted estimators in equation \ref{eq:eq5} because equally weighting individuals ignoring components may result in biased estimators (\cite{basse2018analyzing}). With the average component size, the inverse probability weighted estimator for exposure $a$ and strategy $\alpha$ presented in Section 5.2 equals \begin{equation}
\widehat{Y}^{IPW_r}(a, \alpha)=\frac{1}{m}\sum_{\nu=1}^m \frac{1}{k}\sum_{i\in V(C_{\nu})} \frac{Y_{\nu i}I(A_{\nu i}=a)\pi(A_{\mathcal{N}_{\nu i}};\alpha)}{f_r(A_{\nu i}, A_{\mathcal{N}_{\nu i}}|Z_{\nu i}, Z_{\mathcal{N}_{\nu i}})}.\label{eq:eq6}\end{equation}
The estimating equations corresponding to the estimator in equation \eqref{eq:eq5} are defined as follows
$$\psi_{\eta}(Y_\nu, A_\nu, Z_\nu; \theta)=\frac{1}{k}\sum_{i \in V(C_\nu)}\frac{\partial \log f_r(A_{\nu i}, A_{\mathcal{N}_{\nu i}}|Z_{\nu i}, Z_{\mathcal{N}_{\nu i}})}{\partial \eta}, \eta\in\Theta,$$
$$\psi_0(Y_\nu, A_\nu, Z_\nu; \theta; \alpha)=\frac{1}{k}\sum_{i \in V(C_\nu)}\biggl\{\frac{Y_{\nu i}I(A_{\nu i}=0)\pi(A_{\mathcal{N}_{\nu i}};\alpha)}{f_r(A_{\nu i}, A_{\mathcal{N}_{\nu i}}|Z_{\nu i}, Z_{\mathcal{N}_{\nu i}})}\biggl\}-\theta_{0, \alpha},$$
$$\psi_1(Y_\nu, A_\nu, Z_\nu; \theta; \alpha)=\frac{1}{k}\sum_{i \in V(C_\nu)}\biggl\{\frac{Y_{\nu i}I(A_{\nu i}=1)\pi(A_{\mathcal{N}_{\nu i}};\alpha)}{f_r(A_{\nu i}, A_{\mathcal{N}_{\nu i}}|Z_{\nu i}, Z_{\mathcal{N}_{\nu i}})}\biggl\}-\theta_{1, \alpha},$$ and $$\psi_2(Y_\nu, A_\nu, Z_\nu; \theta; \alpha)=\frac{1}{k}\sum_{i \in V(C_\nu)}\biggl\{\frac{Y_{\nu i}\pi(A_{\nu i}, A_{\mathcal{N}_{\nu i}};\alpha)}{f_r(A_{\nu i}, A_{\mathcal{N}_{\nu i}}|Z_i, Z_{\mathcal{N}_{\nu i}})}\biggl\}-\theta_{\alpha}.$$
Let \begin{align*} \psi_{\nu}(Y_\nu, A_\nu, Z_\nu; \theta)=\begin{pmatrix}\psi_{\eta}(Y_\nu, A_\nu, Z_\nu; \theta) \\ \psi_0(Y_\nu, A_\nu, Z_\nu; \theta; \alpha) \\ \psi_1(Y_\nu, A_\nu, Z_\nu; \theta; \alpha) \\ \psi_2(Y_\nu, A_\nu, Z_\nu; \theta; \alpha)\end{pmatrix}_{\eta\in\Theta},\end{align*} such that $\displaystyle \sum_{\nu=1}^m\psi_{\nu}(Y_\nu, A_\nu, Z_\nu; \hat{\theta})=0$. Note that $\hat{\theta}$ is the solution for $\theta$ for this vector of estimating equations. In addition, $E[\psi_{\nu}(Y_\nu, A_\nu, Z_\nu; \theta)]=0$ \citep{mestimator2013}. Let $A(\theta)=E[-\dot{\psi}_{\nu}(Y_\nu, A_\nu, Z_\nu; \theta)]$ and $B(\theta)=E[\psi_{\nu}(Y_\nu, A_\nu, Z_\nu; \theta)\psi_{\nu}(Y_\nu, A_\nu, Z_\nu; \theta)^T]$ with the expectation take across all $m$ components in the population. \begin{theorem} Under suitable regularity conditions and due to the unbiased estimating equations, as $m\rightarrow \infty$, $\hat{\theta}$ converges in probability to $\theta$ and $\sqrt{m}(\hat{\theta}-\theta)$ converges in distribution to $N(0, \Sigma)$, where the covariance matrix is given by $$\Sigma=\frac{1}{m}A(\theta)^{-1}B(\theta)A(\theta)^{-T}.$$ \end{theorem} Additional details for Proposition 2 are shown in Appendix B. A consistent sandwich estimator of $\Sigma$ is given in Appendix B. We demonstrate how to obtain the variance for the estimator of the disseminated effect $\widehat{IE}_r(\alpha_1, \alpha_0)$. An analogous procedure can be performed to obtain the variance for the estimators of the direct, overall and total effects. Let \begin{align*} \psi_{\nu}(Y_\nu, A_\nu, Z_\nu; \theta)=\begin{pmatrix}\psi_{\eta}(Y_\nu, A_\nu, Z_\nu; \theta) \\ \psi_0(Y_\nu, A_\nu, Z_\nu; \theta; \alpha_1) \\ \psi_0(Y_\nu, A_\nu, Z_\nu; \theta; \alpha_0) \\ \psi_1(Y_\nu, A_\nu, Z_\nu; \theta; \alpha_1) \\ \psi_1(Y_\nu, A_\nu, Z_\nu; \theta; \alpha_0) \\ \psi_2(Y_\nu, A_\nu, Z_\nu; \theta; \alpha_1)\\\psi_2(Y_\nu, A_\nu, Z_\nu; \theta; \alpha_0)\end{pmatrix}_{\eta\in\Theta}.\end{align*} Followed with Slutsky’s Theorem and an application of the Delta method as $m\rightarrow \infty$, $\widehat{IE}_r(\alpha_1, \alpha_0)$ is a consistent estimator of $\overline{IE}(\alpha_1, \alpha_0)$ and $\sqrt{m}(\widehat{IE}_r(\alpha_1, \alpha_0)-\overline{IE}(\alpha_1, \alpha_0))$ converges in distribution to $N(0, \Sigma_{IE})$, where $\Sigma_{IE}=\lambda^T\Sigma \lambda$ and $\lambda = (0_{1 \times p}, 1, -1, 0, 0, 0, 0)^T$. A consistent sandwich estimator of the variance of $\overline{IE}(\alpha_1, \alpha_0)$ is given in Appendix B. This variance estimator can be used to construct Wald-type confidence intervals (CIs) for the disseminated effects.
\section{Simulation} A simulation study was conducted to evaluate the performance of the two IPW estimators and their corresponding closed-form variance estimators. We focused on the evaluation of the finite sample bias and coverage of the corresponding 95\% Wald-type confidence intervals. The network characteristics (e.g., number of components, number of nodes in each component) and parameters of potential outcome models were motivated using empirical estimates from the TRIP data. In this simulation study, we considered regular network where each node has the same number of neighbors. We first generate $m$ network components as regular networks of degree four for each node. The number of nodes in each component is sampled from a Poisson distribution with average 10. We conducted several simulations where the numbers of components $m$ is from the set $\{10, 50, 100, 150, 200\}$. Given a generated network, a total of 1,000 data sets were simulated in the following steps.
\begin{itemize} \item[Step 1.] A baseline covariate was randomly generated as $Z_{i}\sim \text{Bernoulli}(0.5)$. We then generated all possible potential outcomes $$y_i(a_i, a_{\mathcal{N}_i})=\text{Bernoulli}(p={\rm logit}^{-1}(-1.75 + 0.5\cdot a_i + \dfrac{s_i}{d_i} -1.5\cdot a_i\dfrac{s_i}{d_i} +0.5\cdot Z_i)).$$ \item[Step 2.] Assign the random effect to each component in the network $b_\nu\sim N(0, 0.5^2)$ to allow for correlation between the outcomes within components. The exposure was generated as $$A_i = \text{Bernoulli}(p={\rm logit}^{-1}\lbrace 0.7-1.4\cdot Z_i +b_\nu\rbrace).$$ \item[Step 3.] We then obtain the corresponding observed outcomes from the potential outcomes that we generated in Step 1. The true parameters were calculated by averaging the potential outcomes as described in Section 4 that we generated in Step 1. \end{itemize}
For each simulated data set, the $\widehat{Y}^{IPW_1}(a, \alpha)$, $\widehat{Y}^{IPW_1}(\alpha)$, $\widehat{Y}^{IPW_2}(a, \alpha)$, and $\widehat{Y}^{IPW_2}(\alpha)$ were computed for $a=0, 1$ and $\alpha=0.25, 0.5, 0.75$. The estimated standard errors were derived using the appropriate entries from the variance matrix in Appendix B, then averaged across simulations to obtain the average standard error (ASE). Empirical standard error (ESE) was the standard deviation of estimated means across all simulated data sets. Empirical coverage probability (ECP) is the proportion of the instances that the true parameters were contained in the Wald-type 95\% confidence intervals based on the estimated standard errors among the 1000 simulations with a margin of error equal to 0.014. In our main scenario, we simulated networks with component size in average 10 and increased the number of components to evaluate the performance of IPW$_1$ and IPW$_2$ for estimation of the average potential outcomes (Tables A1 to A5). The complete simulation results are summarized in Appendix C.
\begin{figure}
\caption{Absolute bias (left) estimator and corresponding Wald 95\% confidence intervals empirical coverage probability (ECP) (right) of IPW$_1$ (top) and IPW$_2$ (bottom) for different number of components in the network}
\label{fig:biasecp}
\end{figure}
Figure \ref{fig:biasecp} shows that the finite sample bias approaches zero and ECPs approach the nominal 0.95 level when the number of components increases from 10 to 200. In Table \ref{tab:200comp}, the ECPs of the estimator IPW$_1$ under all allocation strategies were close to the nominal level and ECPs of IPW$_2$ approach the nominal level when the allocation strategies had a coverage level around 50$\%$ in the observed data. To compare the performance of our variance estimator to an estimator for the asymptotic variance that assumes partial interference \citep{liu2016inverse}, we used observed components in the network as groups to define partial interference sets. The partial interference assumption for variance estimation resulted in higher ASE and ECP, as compared to the asymptotic variance defined in Appendix B, which was closer to the ESE (Figure \ref{fig:liuasymp}).
\begin{figure}
\caption{Given a network with 100 components, comparison of the average empirical standard error (ESE), the average standard error (ASE) based on variance estimator in Appendix B, and average standard error (Liu ASE) based on variance estimator in \citet{liu2016inverse} of the average potential outcomes under allocation strategies 25\%, 50\%, and 75\%.}
\label{fig:liuasymp}
\end{figure}
In addition to the main simulation scenarios that vary the number of components, we also used a regular network of degree 4 with 100 components to compare scenarios with a different exposure generating mechanism without random effects, and a scenario in which the stratified interference assumption is violated. In addition to this simulated regular network, we used the TRIP network structure to investigate the performance when community detection was used to further divide the network to larger number of component in the network. We also considered a scenario where we regenerated the network in each simulated dataset. Specifically, we considered the following additional five scenarios: \begin{itemize} \item[1.] We used the exposure generating mechanism without random effects as one way to misspecify the propensity score $$A_i={\rm Bern}(p={\rm logit}^{-1}(0.7-1.4\cdot Z_i)).$$ In Table \ref{tab:noranef}, the ECPs of IPW$_2$ were below the nominal level when the exposure mechanism was misspecified, while finite sample performance of IPW$_1$ remained largely similar to settings with a correctly specified exposure mechanism.
\item[2.] We used a different exposure generating model given by $$A_i={\rm Bern}(p={\rm logit}^{-1}(-0.5-1.5\cdot Z_i+b_\nu)).$$ Unlike the previous exposure generating model, this model results in more individuals who have none or 25\% of their neighbors exposed in the simulated data (Figure \ref{fig:distribution}). In Table \ref{tab:trtdist}, both IPW estimators have higher ECPs for allocation strategy 25\% (IPW$_1$: 94\%, IPW$_2$: 97\%) and lower at allocation strategy 75\% (IPW$_1$: 68\%, IPW$_2$: 71\%) in this scenario, suggesting that the finite sample performance of both estimators for the point estimates and ASEs were better under allocation strategies $\alpha$ for which there were more individuals with $100\cdot\alpha\%$ of their neighbors exposed in the simulated data.
\item[3.] We considered an outcome model where the stratified interference assumption was violated while the exposure generating model was defined as in Step 2 $$A_i=\mbox{Bernoulli}(p=\mbox{logit}^{-1}\{0.7-1.4\cdot Z_i+b_\nu\}).$$ We used the potential outcome model $y_i(a_i, a_{\mathcal{N}_i})$ given by $$\text{Bern}(p={\rm logit}^{-1}(-1.75 + 0.5\cdot a_i - 2\cdot \sum_{j \in \mathcal{N}_i} \dfrac{{\scriptstyle I(Z_i=Z_j)}\cdot a_j}{d_i} +5\cdot \sum_{j\in \mathcal{N}_i}\dfrac{{\scriptstyle I(Z_i\neq Z_j)}\cdot a_j}{d_i} +0.5\cdot Z_i)).$$ The simulation results in Table \ref{tab:wrongpotentialmodel} showed that both estimators did not perform well with respect to the point estimates, as the magnitude of absolute bias was larger. The ECPs of IPW$_1$ were all greater than 95\% which suggested over-coverage. The ECPs of IPW$_2$ had coverage above the nominal level or slightly below the nominal level of 95\%.
\item[4.] We considered the network structure similar to our motivating study TRIP. Based on previous simulation results, a small number of components may result in poor finite-sample performance of variance estimators. To increase the number of components for estimation of the asymptotic variance of the estimated causal effects, we employed an efficient modularity-based, fast greedy approach to detect communities to further divide large connected components of the TRIP network into a total of 20 smaller components. Modularity takes large values when there are more substantial connections among some individuals than expected if the connections were randomly assigned \citep{PhysRevE.70.066111}. More precisely, each node initially belongs to a separate component, and components are merged iteratively such that each merge yields the largest increase in the current value of modularity. The algorithm stops when it is not possible to increase the modularity any further. As a result, components each comprise a unique set of participants and there are more links between the participants within components than across components in the TRIP network. By ignoring links across components, we treat the obtained communities as independent units to possibly improve the estimation of the variance. Importantly, we still define the interference sets using the nearest neighbors for point estimation of the causal effects. In this scenario, we use potential outcome model in Step 1 and exposure generating model in Step 2. The simulation results on the TRIP network with and without community detection (Table \ref{tab:trip_sim}) demonstrated that the ECPs of both IPW$_1$ and IPW$_2$ had coverage above the nominal level (97\%-100\%) when using the TRIP network with only 10 components. After further divide the network using community detection, the ECPs have coverage slightly below the nominal level in some cases. To simulate more realistic covariates, we considered a scenario with additional baseline covariates in a TRIP network with 20 components. Two binary variables, $Z_{1, i}\sim \mbox{Bernoulli}(0.5)$ and $Z_{2,i}\sim\mbox{Bernoulli}(0.5)$, and two continuous variables, $Z_{3, i}\sim N(1, 0.5^2)$ and $Z_{4, i}\sim N(0, 1)$, were added into the exposure generating model $$A_i=\mbox{Bern}(\mbox{logit}^{-1}(-1.4\cdot Z_{1,i}+2\cdot Z_{2,i}-1.5\cdot Z_{3,i}+1.2\cdot Z_{4,i})).$$ The results are summarized in Table \ref{tab:TRIP_sim_4cov}. The estimators had coverage below the nominal level using IPW$_1$; however, IPW$_2$ performed slightly better in terms of ECP.
\item[5.] In the previous scenarios, we considered a network that is generated once (or fixed) and simulated 1000 datasets based on the one network. To evaluate the impact of uncertainty in the network structure, we also considered a scenario where we regenerated the network in each simulated dataset. We first generate a degree four regular network with 100 components, then Step 1-3, repeated 1000 times for each dataset (Table \ref{tab:multinetwork}). The results are mostly comparable to the results that generated one simulated network and simulated 1000 datasets on a fixed network (Table \ref{tab:100comp}). \end{itemize}
\begin{table}[htbp]
\centering
\caption{Results from 1000 simulated datasets on a network with 100 components for IPW$_1$ (left) and IPW$_2$ (right) for exposed ($a=1$), not exposed ($a=0$), and marginal estimators under allocation strategies 25\%, 50\%, and 75\% using exposure generating model $A_i={\rm Bern}({\rm expit}(0.7-1.4\cdot Z_i)).$}
\begin{tabular}{l|rrrr|rrrr}
\toprule
\rowcolor[rgb]{ .929, .929, .929} & \multicolumn{4}{c|}{IPW1} & \multicolumn{4}{c}{IPW2} \\
\rowcolor[rgb]{ .929, .929, .929} & \multicolumn{1}{c}{Bias} & \multicolumn{1}{c}{ESE} & \multicolumn{1}{c}{ASE} & \multicolumn{1}{c|}{ECP} & \multicolumn{1}{c}{Bias} & \multicolumn{1}{c}{ESE} & \multicolumn{1}{c}{ASE} & \multicolumn{1}{c}{ECP} \\
\midrule
\midrule
$\widehat{Y}(1, 0.25)$ & 0.0013 & 0.048 & 0.046 & 0.91 & 0.0149 & 0.039 & 0.036 & 0.86 \\
$\widehat{Y}(1, 0.5)$ & -0.0003 & 0.027 & 0.028 & 0.96 & 0.0016 & 0.022 & 0.024 & 0.96 \\
$\widehat{Y}(1, 0.75)$ & -0.0032 & 0.041 & 0.041 & 0.93 & 0.0093 & 0.035 & 0.033 & 0.88 \\
$\widehat{Y}(0, 0.25)$ & -0.0051 & 0.039 & 0.040 & 0.94 & 0.0093 & 0.033 & 0.031 & 0.90 \\
$\widehat{Y}(0, 0.5)$ & -0.0018 & 0.028 & 0.029 & 0.95 & 0.0018 & 0.023 & 0.025 & 0.97 \\
$\widehat{Y}(0, 0.75)$ & 0.0004 & 0.053 & 0.051 & 0.92 & 0.0195 & 0.045 & 0.042 & 0.83 \\
$\widehat{Y}(0.25)$ & -0.0035 & 0.032 & 0.032 & 0.94 & 0.0107 & 0.027 & 0.026 & 0.88 \\
$\widehat{Y}(0.5)$ & -0.0010 & 0.021 & 0.021 & 0.97 & 0.0017 & 0.016 & 0.018 & 0.97 \\
$\widehat{Y}(0.75)$ & -0.0023 & 0.033 & 0.030 & 0.93 & 0.0118 & 0.029 & 0.028 & 0.86 \\
\bottomrule
\end{tabular}
\label{tab:noranef} \end{table}
\begin{figure}
\caption{The frequency of the proportion of exposed neighbors in one simulated data when using the exposure generating models $A_i={\rm Bern}({\rm expit}(0.7-1.4\cdot Z_i+b_\nu))$ (left) and $A_i={\rm Bern}({\rm expit}(-0.5-1.5\cdot Z_i+b_\nu))$ (right)}
\label{fig:distribution}
\end{figure}
\begin{table}[htbp]
\centering
\caption{Results from 1000 simulated datasets on a network with 100 components for IPW$_1$ (left) and IPW$_2$ (right) for exposed ($a=1$), not exposed ($a=0$), and marginal estimators under allocation strategies 25\%, 50\%, and 75\% using an outcome model where the stratified interference assumption is violated.}
\begin{tabular}{lr|rrrr|rrrr}
\toprule
\rowcolor[rgb]{ .949, .949, .949} & & \multicolumn{4}{c|}{IPW$_1$} & \multicolumn{4}{c}{IPW$_2$} \\
\rowcolor[rgb]{ .949, .949, .949} & \multicolumn{1}{c|}{True} & \multicolumn{1}{c}{Bias} & \multicolumn{1}{c}{ESE} & \multicolumn{1}{c}{ASE} & \multicolumn{1}{c|}{ECP} & \multicolumn{1}{c}{Bias} & \multicolumn{1}{c}{ESE} & \multicolumn{1}{c}{ASE} & \multicolumn{1}{c}{ECP} \\
\midrule
\midrule
$\widehat{Y}(1, 0.25)$ & 0.9965 & 0.0039 & 0.068 & 0.074 & 0.96 & 0.0345 & 0.090 & 0.074 & 0.84 \\
$\widehat{Y}(1, 0.5)$ & 0.9885 & 0.0041 & 0.033 & 0.047 & 0.99 & 0.0051 & 0.006 & 0.048 & 1.00 \\
$\widehat{Y}(1, 0.75)$ & 0.9724 & -0.0134 & 0.067 & 0.077 & 0.96 & -0.1540 & 0.115 & 0.148 & 0.93 \\
$\widehat{Y}(0, 0.25)$ & 0.9943 & -0.0179 & 0.066 & 0.078 & 0.97 & -0.1682 & 0.121 & 0.154 & 0.92 \\
$\widehat{Y}(0, 0.5)$ & 0.9821 & 0.0015 & 0.034 & 0.048 & 0.99 & 0.0032 & 0.008 & 0.049 & 1.00 \\
$\widehat{Y}(0, 0.75)$ & 0.9583 & 0.0060 & 0.069 & 0.074 & 0.95 & 0.0398 & 0.085 & 0.070 & 0.81 \\
$\widehat{Y}(0.25)$ & 0.9949 & -0.0125 & 0.050 & 0.063 & 0.98 & -0.1175 & 0.100 & 0.122 & 0.92 \\
$\widehat{Y}(0.5)$ & 0.9853 & 0.0028 & 0.022 & 0.040 & 1.00 & 0.0042 & 0.005 & 0.035 & 1.00 \\
$\widehat{Y}(0.75)$ & 0.9689 & -0.0086 & 0.050 & 0.062 & 0.97 & -0.1056 & 0.094 & 0.117 & 0.94 \\
\hline
\end{tabular}
\label{tab:wrongpotentialmodel} \end{table}
\begin{table}[htbp]
\centering
\caption{Results from 1000 simulated datasets on TRIP network for 10 components (left) and using community detection to further divide the network to 20 components (right) for exposed ($a=1$), not exposed ($a=0$), and marginal estimators under allocation strategies 25\%, 50\%, and 75\%.}
\begin{tabular}{lc|rcrc|rcrc}
\toprule
\rowcolor[rgb]{ .929, .929, .929} & & \multicolumn{4}{c|}{10 components} & \multicolumn{4}{c}{20 components} \\
\rowcolor[rgb]{ .929, .929, .929} & & \multicolumn{2}{c}{IPW$_1$} & \multicolumn{2}{c|}{IPW$_2$} & \multicolumn{2}{c}{IPW$_1$} & \multicolumn{2}{c}{IPW$_2$} \\
\rowcolor[rgb]{ .929, .929, .929} & True & \multicolumn{1}{c}{Bias} & ECP & \multicolumn{1}{c}{Bias} & ECP & \multicolumn{1}{c}{Bias} & ECP & \multicolumn{1}{c}{Bias} & ECP \\
\midrule
\midrule
$\widehat{Y}(1, 0.25)$ & 0.2473 & 0.0098 & 0.986 & 0.0167 & 0.988 & 0.0098 & 0.849 & 0.0036 & 0.890 \\
$\widehat{Y}(1, 0.5)$ & 0.2265 & 0.0021 & 0.998 & 0.0112 & 0.986 & 0.0021 & 0.946 & 0.0064 & 0.920 \\
$\widehat{Y}(1, 0.75)$ & 0.2058 & $-$0.0020 & 0.987 & 0.0126 & 0.997 & $-$0.0020 & 0.894 & 0.0057 & 0.943 \\
$\widehat{Y}(0, 0.25)$ & 0.2304 & $<$0.0001 & 0.996 & 0.0046 & 0.999 & $<$0.0001 & 0.920 & 0.0021 & 0.968 \\
$\widehat{Y}(0, 0.5)$ & 0.2778 & 0.0010 & 1.000 & 0.0029 & 1.000 & 0.0010 & 0.954 & 0.0017 & 0.974 \\
$\widehat{Y}(0, 0.75)$ & 0.3275 & 0.0073 & 0.996 & 0.0038 & 1.000 & 0.0073 & 0.896 & 0.0019 & 0.992 \\
$\widehat{Y}(0.25)$ & 0.2346 & 0.0025 & 0.999 & 0.0133 & 0.971 & 0.0025 & 0.943 & 0.0061 & 0.915 \\
$\widehat{Y}(0.5)$ & 0.2521 & 0.0015 & 1.000 & 0.0121 & 0.993 & 0.0015 & 0.982 & $<$0.0001 & 0.917 \\
$\widehat{Y}(0.75)$ & 0.2362 & 0.0004 & 1.000 & 0.0130 & 0.998 & 0.0004 & 0.937 & 0.0046 & 0.940 \\
\bottomrule
\end{tabular}
\label{tab:trip_sim} \end{table}
\begin{table}[htbp]
\centering
\caption{Results from 1000 simulated datasets on TRIP network further dividing the network into 20 components using $IPW_1$ (left) and $IPW_2$ (right) for exposed $(a=1)$, not exposed $(a=0)$, and marginal estimators under allocation strategies 25\%, 50\%, and 75\% with exposure generating model $A_i=\mbox{Bern}(\mbox{logit}^{-1}(-1.4\cdot Z_{1,i}+2\cdot Z_{2,i}-1.5\cdot Z_{3,i}+1.2\cdot Z_{4,i}))$.}
\begin{tabular}{lr|rrrr|rrrr}
\toprule
\rowcolor[rgb]{ .949, .949, .949} & & \multicolumn{4}{c|}{IPW1} & \multicolumn{4}{c}{IPW2} \\
\rowcolor[rgb]{ .949, .949, .949} & \multicolumn{1}{c|}{True} & \multicolumn{1}{c}{Bias} & \multicolumn{1}{c}{ESE} & \multicolumn{1}{c}{ASE} & \multicolumn{1}{c|}{ECP} & \multicolumn{1}{c}{Bias} & \multicolumn{1}{c}{ESE} & \multicolumn{1}{c}{ASE} & \multicolumn{1}{c}{ECP} \\
\midrule
\midrule
$\widehat{Y}(1, 0.25)$ & 0.2493 & 0.0445 & 0.161 & 0.095 & 0.61 & 0.0409 & 0.100 & 0.075 & 0.71 \\
$\widehat{Y}(1, 0.5)$ & 0.2274 & 0.0295 & 0.179 & 0.094 & 0.64 & 0.0197 & 0.089 & 0.073 & 0.80 \\
$\widehat{Y}(1, 0.75)$ & 0.2057 & 0.0194 & 0.290 & 0.111 & 0.57 & 0.0422 & 0.093 & 0.065 & 0.64 \\
$\widehat{Y}(0, 0.25)$ & 0.2295 & 0.0160 & 0.141 & 0.087 & 0.72 & 0.0214 & 0.064 & 0.060 & 0.86 \\
$\widehat{Y}(0, 0.5)$ & 0.2765 & 0.0306 & 0.149 & 0.092 & 0.69 & 0.0318 & 0.063 & 0.067 & 0.86 \\
$\widehat{Y}(0, 0.75)$ & 0.3264 & 0.0589 & 0.189 & 0.113 & 0.59 & 0.0923 & 0.094 & 0.076 & 0.56 \\
$\widehat{Y}(0.25)$ & 0.2345 & 0.0231 & 0.115 & 0.077 & 0.71 & 0.0263 & 0.054 & 0.055 & 0.84 \\
$\widehat{Y}(0.5)$ & 0.2520 & 0.0300 & 0.118 & 0.079 & 0.70 & 0.0258 & 0.053 & 0.059 & 0.87 \\
$\widehat{Y}(0.75)$ & 0.2358 & 0.0293 & 0.227 & 0.099 & 0.60 & 0.0547 & 0.072 & 0.058 & 0.62 \\
\bottomrule
\end{tabular}
\label{tab:TRIP_sim_4cov} \end{table}
\begin{table}[htbp]
\centering
\caption{Results from 1000 simulated datasets with the network regenerated for each dataset with 100 components for IPW$_1$ (left) and IPW$_2$ (right) for exposed ($a=1$), not exposed ($a=0$), and marginal estimators under allocation strategies 25\%, 50\%, and 75\%.}
\begin{tabular}{lc|rcrc|rcrc}
\toprule
\rowcolor[rgb]{ .929, .929, .929} & & \multicolumn{4}{c|}{IPW1} & \multicolumn{4}{c}{IPW2} \\
\rowcolor[rgb]{ .929, .929, .929} & True & \multicolumn{1}{c}{Bias} & \multicolumn{1}{c}{ESE} & \multicolumn{1}{c}{ASE} & \multicolumn{1}{c|}{ECP} & \multicolumn{1}{c}{Bias} & \multicolumn{1}{c}{ESE} & \multicolumn{1}{c}{ASE} & \multicolumn{1}{c}{ECP} \\
\midrule
\midrule
$\widehat{Y}(1, 0.25)$ & 0.2489 & 0.0023 & 0.0487 & 0.0467 & 0.89 & -0.0099 & 0.0518 & 0.0457 & 0.93 \\
$\widehat{Y}(1, 0.5)$ & 0.2270 & 0.0018 & 0.0271 & 0.0274 & 0.94 & -0.0037 & 0.0324 & 0.0317 & 0.96 \\
$\widehat{Y}(1, 0.75)$ & 0.2058 & -0.0008 & 0.0425 & 0.0500 & 0.90 & -0.0052 & 0.0275 & 0.0274 & 0.96 \\
$\widehat{Y}(0, 0.25)$ & 0.2281 & $<$0.0001 & 0.0366 & 0.0491 & 0.93 & -0.0013 & 0.0223 & 0.0237 & 0.97 \\
$\widehat{Y}(0, 0.5)$ & 0.2745 & 0.0015 & 0.0257 & 0.0298 & 0.97 & -0.0023 & 0.0230 & 0.0246 & 0.96 \\
$\widehat{Y}(0, 0.75)$ & 0.3249 & 0.0050 & 0.0479 & 0.0541 & 0.92 & -0.0018 & 0.0161 & 0.0177 & 0.98 \\
$\widehat{Y}(0.25)$ & 0.2333 & 0.0006 & 0.0296 & 0.0388 & 0.94 & -0.0051 & 0.0340 & 0.0329 & 0.94 \\
$\widehat{Y}(0.5)$ & 0.2508 & 0.0017 & 0.0187 & 0.0213 & 0.97 & -0.0168 & 0.0542 & 0.0510 & 0.94 \\
$\widehat{Y}(0.75)$ & 0.2356 & 0.0007 & 0.0336 & 0.0387 & 0.91 & -0.0081 & 0.0295 & 0.0289 & 0.95 \\
\bottomrule
\end{tabular}
\label{tab:multinetwork} \end{table}
\section{Evaluation of disseminated effects of community alerts in the Transmission Reduction Intervention Project}
We applied the estimators proposed in Section 5.2 to estimate the causal effects of community alerts at baseline on report of risk behavior at the six-month visit. We assumed that TRIP was an undirected network because the links were defined by if two individuals engaged had sex or injected drugs together in the six months before the baseline interview as reported by at least one participant in the dyad. This was an attempt to reduce the impact of possible missing links in the analysis due to stigma of sexual and drug use behaviors. The community alerts intervention status of the index participant and their neighbors was defined with respect to the baseline visit date of the index participant. The network structure in TRIP had 10 connected components with 216 participants and 362 shared connections (average degree is 3.35) after excluding isolates and 59 participants who were lost to follow-up before their six-month visit. Among the 216 participants in TRIP, 25 participants (11.6\%) received a community alert about the increased risk for HIV acquisition in close proximity in their network from the study team. We evaluated if information in the community alerts was disseminated to their the nearest neighbors and ultimately, if this resulted in a reduction in risk behavior among others in the network beyond those who were exposed to the alert themselves (Figure \ref{fig:trip_network}). Among participants with complete information on the questions related to sharing injection equipment, we considered the report of sharing injection equipment (or not) at the 6-month visit as the binary outcome, including sharing a syringe, cooker, filter or rinse water, or backloading to share injection drugs. The following baseline covariates were included in the adjusted models: HIV status, shared drug equipment (e.g., syringe) in last six months, the calendar date at first interview, education (primary school, high school, and post high school), and employment status (employed, unemployed/looking for a work, can't work because of health reason, and other). HIV status was ascertained in this study from a blood sample from each participant collected by a health program physician \citep{nikolopoulos2016network}. Given the study population included PWID in one geographic location, we assume that social desirability leading to possible reporting bias is comparable between the two exposure groups. Under this assumption, the reporting bias could be effectively eliminated when estimating contrasts between exposure groups.
We consider each of the assumptions in Section 5.1 in this setting. TRIP is an observational study with a nonrandomized intervention. Conditional exchangeability is required to identify causal effects. We also assume that if there are multiple versions of the community alerts that these different versions are irrelevant for causal contrasts of interest and this results in one version of exposure to the community alerts intervention and one version of no exposure to this intervention. TRIP recruited at least one wave of contact tracing for each participant of an HIV-infected seed; therefore, we expect that there could be dissemination or spillover between two individuals connected by a first-degree link (i.e., possible influence of their neighbors' intervention exposure on an individual's outcome). Based on the complex structure of the TRIP network resulting in possibly many different vectors of $A_{\mathcal{N}_i}$, a stratified interference may be more appropriate to ensure that the positivity assumption holds. For the reducible propensity score assumption, the neighbors's covariates were not significantly associated with their index individual's exposure and the neighbors's exposure was not significant associated with the index individual's covariates, so Assumption \ref{assump:reducible} used for IPW$_2$ may be plausible in this analysis (data not shown).
For the analysis, we reported the point estimates and corresponding Wald-type 95\% confidence intervals of each causal effect using both IPW$_1$ and IPW$_2$ estimators under allocation strategies 20\%, 30\%, 40\%, and 50\% because the most of individuals in the TRIP study had 20\% to 50\% of their nearest neighbors exposed to community alerts. The normality of random effects in IPW$_1$ was tested using a diagnostic test for mixed effects model in \cite{normality} and this resulted in a $p$-value $=0.012$ under the null hypothesis that the mixing distribution is normal. Due to this result and better finite-sample performance for IPW$_2$ with a smaller number of components (see Section 6), we recommend IPW$_2$ as a more appropriate estimator in this setting given the small number of components in the TRIP network. Based on the simulation scenario 4 results that showed better finite sample performance for 20 components, we used community detection to further divide the TRIP network into 20 components to possibly improve the finite-sample performance of the variance estimators. We report the variance estimates with and without dividing TRIP network 10 observed components to 20 components in Table \ref{tab:trip_full}. In addition to including the full set of covariates to adjust for measured confounding in the weight models, we conducted sensitivity analyses to evaluate the impact of different sets of covariates on the model results. We first considered univariate models; that is, adjustment for only one covariate at a time. Second, we estimated the effects using the full set of covariates, excluding one covariate at a time. Lastly, we estimated the effects without adjustment for any covariates. The results were largely robust to the set of measured covariates used to adjust for confounding. In addition, the results that used community detection to further divide TRIP into 20 components to estimate the asymptotic variances were comparable to an analysis that used the 10 observed components. All models results are summarized in Appendix D. The study protocol was reviewed and approved by the University of Rhode Island Institutional Review Board. All analyses were conducted using R (version 3.6.2), and R packages: igraph: Network Analysis and Visualization (version 1.3.4), lme4: Linear Mixed-Effects Models using 'Eigen' and S4 (version 1.1-30), and numDeriv: Accurate Numerical Derivatives (version 2016.8-1.1).
Direct, indirect, total, and overall effect estimates and their corresponding Wald-type 95\% confidence intervals of both estimators for different allocation strategies $\alpha=0.2, 0.3, 0.4$ and $0.5$ adjusting for all measured confounding variables are reported in Table \ref{tab:trip_full}. All estimates of the risk differences for both estimators IPW$_1$ and IPW$_2$ were protective, excluding the indirect effect under allocation strategy 30\% and 20\%, $\widehat{IE}_1(0.3, 0.2)=0.01$ using IPW$_1$ and $\widehat{IE}_2(0.3, 0.2)=0.00$ using IPW$_2$; however, these did not achieve statistical significance. These results suggest that the likelihood of reporting HIV risk behavior was reduced not only by an participant's exposure to community alerts, but also by the proportion of an participant's nearest neighbors exposed to community alerts from the study team. We report the confidence interval obtained using 10 components. Specifically, the estimated direct effect was $\widehat{DE}_1(0.5)=-0.18$ (95\% CI: $-0.49, 0.14$), estimated using IPW$_1$ and $\widehat{DE}_2(0.5)=-0.21$ (95\% CI: $-0.56, 0.15$) when estimated with IPW$_2$; that is, we expect 18 fewer reports of risk behavior per 100 participants if a participant receives the alert compared to if a participant does not receive an alert with 50\% intervention coverage (i.e., 50\% of their neighbors receiving alerts) when estimated using IPW$_1$ (21 per 100 fewer using IPW$_2$). The indirect effect is $\widehat{IE}_1(0.5, 0.2)=-0.03$ (95\% CI:$-0.07, 0.00$), estimated using IPW$_1$ under allocation strategies 20\% versus 50\% and $\widehat{IE}_2(0.5, 0.2)=-0.02$ (95\% CI:$-0.04, -0.01$) when estimated with IPW$_2$; in other words, we expect 4 fewer reports of risk behavior per 100 participants if a participant does not receive an alert with 50\% intervention coverage compared to only 20\% intervention coverage when estimated using IPW$_1$ (2 per 100 fewer using IPW$_2$). The total effects $\widehat{TE}_1(0.5, 0.2)=-0.21$ (95\% CI:$-0.53, 0.11$) estimated using IPW$_1$ and $\widehat{TE}_2(0.5, 0.2)=-0.23$ (95\% CI:$-0.58, 0.12$) estimated using IPW$_2$. We expect 21 fewer reports of risk behavior per 100 participants when estimated using IPW1 if a participant receives an alert with 50\% of their nearest neighbors alerted versus if a participant does not receive an alert and only 20\% of their nearest neighbors receive an alert (23 per 100 fewer using IPW$_2$). The overall effects, $\widehat{OE}_1(0.5, 0.2)=-0.11$ (95\% CI:$-0.27, 0.05$) using IPW$_1$ and $\widehat{OE}_2(0.5, 0.2)=-0.13$ (95\% CI:$-0.32, 0.06$) using IPW$_2$. When estimated using IPW$_1$, we expect 11 fewer reports of risk behavior per 100 participants if 50\% of the nearest neighbors and participant $i$ receive alerts compared to if only 20\% of the nearest neighbors and participant $i$ receive alerts (13 per 100 fewer using IPW$_2$).
\begin{table}[htbp]
\centering
\caption{The estimated risk differences and 95\% confidence intervals (CIs) estimated using the TRIP network with the original 10 network components, and 95\% CIs estimated by dividing TRIP network into 20 components, of the effects of community alerts at baseline on HIV risk behavior at 6 months adjusted for full set of measured confounding variables under allocation strategies 20\%, 30\%, 40\%, and 50\%}
\begin{tabular}{lc|ccc|ccc}
\toprule
& & \multicolumn{3}{c|}{IPW1} & \multicolumn{3}{c}{IPW2} \\
\multicolumn{1}{c}{Effects} & Coverage & RD & \multicolumn{2}{c|}{95\% CI} & RD & \multicolumn{2}{c}{95\% CI} \\
& $(\alpha, \alpha')$ & & 10 components & 20 components & & 10 components & 20 components \\
\midrule
\midrule
\rowcolor[rgb]{ .949, .949, .949} Direct & (20\%, 20\%) & -0.06 & (-0.14, 0.01) & (-0.39, 0.26) & 0.01 & (-0.08, 0.10) & (-0.19, 0.21) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (30\%, 30\%) & -0.10 & (-0.24, 0.04) & (-0.48, 0.29) & -0.09 & (-0.19, 0.02) & (-0.27, 0.10) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (40\%, 40\%) & -0.14 & (-0.36, 0.09) & (-0.52, 0.24) & -0.16 & (-0.40, 0.09) & (-0.38, 0.07) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (50\%, 50\%) & -0.18 & (-0.49, 0.14) & (-0.52, 0.17) & -0.21 & (-0.56, 0.15) & (-0.47, 0.06) \\
Indirect & (30\%, 20\%) & 0.01 & (-0.02, 0.03) & (-0.03, 0.04) & 0.00 & (-0.01, 0.02) & (-0.04, 0.05) \\
Indirect & (40\%, 20\%) & -0.01 & (-0.03, 0.01) & (-0.06, 0.04) & -0.01 & (-0.02, 0.01) & (-0.09, 0.07) \\
Indirect & (50\%, 20\%) & -0.03 & (-0.07, 0.00) & (-0.10, 0.03) & -0.02 & (-0.04, -0.01) & (-0.14, 0.10) \\
Indirect & (40\%, 30\%) & -0.01 & (-0.03, -0.00) & (-0.04, 0.01) & -0.01 & (-0.02, -0.00) & (-0.05, 0.03) \\
Indirect & (50\%, 40\%) & -0.03 & (-0.05, 0.00) & (-0.06, 0.01) & -0.01 & (-0.03, -0.00) & (-0.06, 0.03) \\
Indirect & (50\%, 30\%) & -0.04 & (-0.08, 0.00) & (-0.09, 0.01) & -0.02 & (-0.04, -0.01) & (-0.11, 0.06) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (30\%, 20\%) & -0.09 & (-0.22, 0.04) & (-0.49, 0.31) & -0.08 & (-0.17, 0.01) & (-0.28, 0.12) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (40\%, 20\%) & -0.15 & (-0.36, 0.07) & (-0.53, 0.24) & -0.16 & (-0.40, 0.07) & (-0.37, 0.05) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 20\%) & -0.21 & (-0.53, 0.11) & (-0.54, 0.11) & -0.23 & (-0.58, 0.12) & (-0.43, -0.02) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (40\%, 30\%) & -0.15 & (-0.38, 0.08) & (-0.52, 0.22) & -0.17 & (-0.41, 0.08) & (-0.37, 0.04) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 40\%) & -0.20 & (-0.53, 0.13) & (-0.52, 0.12) & -0.22 & (-0.58, 0.14) & (-0.45, 0.01) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 30\%) & -0.22 & (-0.56, 0.12) & (-0.53, 0.09) & -0.23 & (-0.59, 0.13) & (-0.44, -0.02) \\
Overall & (30\%, 20\%) & -0.01 & (-0.03, 0.01) & (-0.08, 0.06) & -0.03 & (-0.06, 0.01) & (-0.07, 0.02) \\
Overall & (40\%, 20\%) & -0.05 & (-0.12, 0.02) & (-0.15, 0.05) & -0.07 & (-0.18, 0.03) & (-0.15, 0.00) \\
Overall & (50\%, 20\%) & -0.11 & (-0.27, 0.05) & (-0.21, -0.01) & -0.13 & (-0.32, 0.06) & (-0.21, -0.04) \\
Overall & (40\%, 30\%) & -0.04 & (-0.10, 0.02) & (-0.07, -0.01) & -0.05 & (-0.12, 0.02) & (-0.08, -0.01) \\
Overall & (50\%, 40\%) & -0.06 & (-0.15, 0.03) & (-0.09, -0.03) & -0.05 & (-0.14, 0.03) & (-0.08, -0.03) \\
Overall & (50\%, 30\%) & -0.10 & (-0.25, 0.05) & (-0.14, -0.06) & -0.10 & (-0.26, 0.05) & (-0.16, -0.05) \\
\bottomrule
\end{tabular}
\label{tab:trip_full} \end{table}
\begin{comment} \begin{figure}
\caption{The point estimates of risk differences and the Wald 95\% CI of the effect of community alert and HIV risk behavior on TRIP using IPW$_1$ and IPW$_2$.}
\label{fig:TRIP}
\end{figure} \end{comment}
\section{Discussion} In this paper, methods for evaluating disseminated effects were developed for the setting of network-based studies by leveraging a nearest neighbor interference set. The proposed approach uses connections (i.e. links) between individuals in a network and allows for overlapping interference sets within each component of the network. The two proposed estimators were shown to be consistent and asymptotically normal. Importantly, a consistent, closed-form estimator of the asymptotic variance was derived. The simulation study demonstrated that the two IPW estimators had reasonable finite-sample performance in terms of consistency and empirical coverage for a large number ($>100$) of components in the observed network. The proposed variance estimators incorporate the observed network structure by assigning each individual a unique propensity score defined by their own nearest neighbors in which the nearest neighbors for individuals can overlap. We compared the performance of our variance estimators to the estimators for the asymptotic variances that assume partial interference with component-level propensity scores \citep{liu2016inverse} by using the observed network components as partial interference sets. In Figure \ref{fig:liuasymp}, our variance estimators were more efficient and closer to the empirical standard error by utilizing the network structure in a nearest neighbors level propensity score as compared to Liu's estimator. In the additional simulation scenario 4, the empirical coverage probabilities were above the nominal level (97\%-100\%) when using the TRIP network with only 10 components. This may be a result of the uncertainty due to the imbalanced component size observed in TRIP, where the total number of nodes was 216 and the largest component had size 186. After using community detection to further divide the network into a larger number of components ($m=20$), the coverage level then decreased to average of 93\%.
Based on the simulation results, both estimators performed well in terms of finite sample bias. IPW$_2$ demonstrated better performance for variance estimation (i.e., ASE was closer to ESE) when the number of network components was small ($<50$), while IPW$_1$ had lower coverage for the confidence intervals. When the number of network components was large ($\geq100$), the estimated average standard error for IPW$_1$ resulted in confidence intervals with coverage around the nominal level, while IPW$_2$ tended to have coverage above the nominal level. Based on these findings, the estimation of these effects in TRIP network using IPW$_2$ may be preferred over IPW$_1$ due to the small number of network components with the caveat that these recommendations may be sensitive to specification in the simulation scenarios, including features of the study design. In addition, we explored adding additional covariates with larger parameter values in the exposure generating model (Table 5). In this case, the estimators had slightly larger bias compared to Table 4 and ECP somewhat below the nominal, while IPW$_2$ had slightly higher coverage than IPW$_1$. As reported in Table 3, a violation of the stratified interference assumption when the exposure mechanism is misspecified resulted in deviations from the nominal coverage level for both estimators.
With these methods, we now have an approach to quantify the social and biological influence on the determinants of risk and HIV transmission in HIV risk networks of PWIDs \citep{friedmanSocialNet2001} when evaluating the impact of interventions, such as TasP, or how interventions permeate a risk network \citep{nikolopoulos2016network,friedman2014socially}. These new methodologies will improve the identification of best preventive practices for PWID and provide evidence to expedite policy changes to improve access to HIV treatment and risk reduction interventions in subpopulations of high-risk drug users. In the TRIP study, these methods allowed for quantification of the extent to which the community alerts intervention reduced onward transmission to others in the community by tracking incident infections in the risk networks as measured through the proxy of self-reported HIV risk behaviors. Correctly conducted and analyzed studies among PWID will improve existing interventions, inform new interventions, and has the potential to reduce incident HIV infections in this subpopulation.
Studies of network effects among PWID are rich with future methodological problems. The simulation study indicated that the asymptotic variance estimators of the IPW estimators had coverage below the nominal level when the number of components in the network is limited ($<$50), while both IPW estimators were unbiased in finite samples ($<$5\%; bias/true value). Finite sample correction for estimating asymptotic variances is needed when the network has small number of components. As the approach in this paper used components as independent units for the variance estimation, developing methodologies with heterogeneous correlation structures within a large size component should be included in future work. Furthermore, the outcome of interest may be missing due to participant loss to follow-up in some intervention-based studies when outcomes are ascertained post-intervention. For example, 21\% of TRIP participants were lost to follow-up by the six-month visit. Future work should include development of censoring methods to evaluate the IPW outcomes in the presence of missing outcomes or alternative methods to also address missing links in the network. With regard to real data application, the impact of unmeasured confounding is important because this would violate the conditional exchangeability assumption; however, sensitivity analyses in the presence of interference currently only exist for two-stage randomized trials with clustering features \citep{tylersensitivity2014}. Designing sensitivity analyses to assess the bias of unmeasured confounding in network-based studies should be included in future research. In addition, if the spillover set actually included two-degree neighbors or other sets of individuals in the network, the nearest neighbors interference assumption would not be valid. We recommend the development of future methods that consider alternative definitions of the spillover set in the network. For example, we could also have a violation of the stratified interference assumption if in fact one of the neighbors was a closer contact or more important to the index participant. We recommend for future work the incorporation of edge weights into this method to reflect variations in the strength of connections relevant for spillover. With these improved inferential methods, investigators will be able to answer questions they were previously unable to address in network-based studies, leading to more effective intervention implementation and far-reaching policy change to prevent HIV infection, reduce risk behavior, ultimately, improve HIV treatment and care among PWID. In addition to study HIV transmission among PWID, this method can also be applied in a wider context to study sexually transmitted infection diseases such as genital herpes and trichomoniasis among adolescents and young adults, men who have sex with men, or pregnant women.
\begin{supplement} \noindent \stitle{Dataset and codes} \sdescription{The TRIP datasets are available upon reasonable request to the corresponding author subject to approval by the TRIP investigators. The simulation code and datasets are available from the corresponding author on reasonable request. Codes and a sample dataset can be found on github: https://github.com/uri-ncipher/Nearest-Neighbor-estimators.} \end{supplement}
\setcounter{page}{1}
\appendix
\section{Proof of Proposition 1} \noindent To show $\widehat{Y}^{IPW_1}(a, \alpha)$ is unbiased with known propensity score, see the following: \begin{align}
E[\widehat{Y}^{IPW_1}(a, \alpha)]&=\frac{1}{n}\sum_{i=1}^n E\big[\frac{y_i(A_i, A_{\mathcal{N}_i})I(A_i=a)\pi(A_{\mathcal{N}_i};\alpha)}{f_1(A_i, A_{\mathcal{N}_i}|Z_i, Z_{\mathcal{N}_i})}\big]\notag\\
&=\frac{1}{n}\sum_{i=1}^n \sum_{a_i, a_{\mathcal{N}_i}} \frac{y_i(a_i, a_{\mathcal{N}_i})I(a_i=a)\pi(a_{\mathcal{N}_i};\alpha)}{f_1(a_i, a_{\mathcal{N}_i}|Z_i, Z_{\mathcal{N}_i})}f_1(a_i, a_{\mathcal{N}_i}|Z_i, Z_{\mathcal{N}_i})\notag\\
&=\frac{1}{n}\sum_{i=1}^n \sum_{ a_{\mathcal{N}_i}}y_i(a_i=a, a_{\mathcal{N}_i})\pi(a_{\mathcal{N}_i};\alpha)\notag\\
&=\bar{y}(a, \alpha)\notag \end{align} The unbiasedness of the marginal inverse probability weighted estimator, $\widehat{Y}^{IPW_1}(\alpha)$, can be proved similarly.
Under the assumption that $y(a_i, a_{\mathcal{N}_i})=y(a_i, s_i)$, IPW$_2$ is also unbiased with a known propensity score \begin{align}
E[\widehat{Y}^{IPW_2}(a,\alpha)]&=\frac{1}{n}\sum_{i=1}^n E\big[\frac{y_i(A_i, S_i)I(A_i=a)\pi(A_{\mathcal{N}_i};\alpha)}{f_2(A_i, S_i|Z_i, Z_{\mathcal{N}_i})}\big]\notag\\
&=\frac{1}{n}\sum_{i=1}^n \sum_{a_i, s_i} \frac{y_i(a_i, s_i)I(a_i=a)\pi(a_{\mathcal{N}_i};\alpha)}{f_2(a_i, s_i|Z_i, Z_{\mathcal{N}_i})}f_2(a_i, s_i|Z_i, Z_{\mathcal{N}_i})\notag\\
&=\frac{1}{n}\sum_{i=1}^n \sum_{j=0}^{d_i}{d_i \choose j}y_i(a_i=a, j)\alpha^j(1-\alpha)^{d_i-j}\notag\\
&=\frac{1}{n}\sum_{i=1}^n \sum_{a_{\mathcal{N}_i}}y_i(a_i=a, a_{\mathcal{N}_i})\pi(a_{\mathcal{N}_i};\alpha)\notag\\
&=\bar{y}(a, \alpha)\notag \end{align}
\section{Proposition 2 and Sandwich-Type Estimators of the Variance}
\noindent Following \citet{mestimator2013}, to estimate the parameters in the exposure propensity score model $\hat{\Theta}$, we let $$\psi_{\eta}(Y_{\nu},A_{\nu},Z_{\nu}; \theta)=\frac{1}{k}\sum_{i\in V(C_\nu)}\frac{\partial \log f_r(A_{\nu i}, A_{\mathcal{N}_{\nu i}}|Z_{\nu i}, Z_{\mathcal{N}_{\nu i}})}{\partial \eta}, \eta\in\Theta.$$ Estimates $\hat{\eta}$ that maximize the $\log$ likelihood are solutions to the score equation $$\sum_{\nu=1}^m\psi_{\eta}(Y_{\nu},A_{\nu},Z_{\nu}; \theta)=\frac{1}{k}\sum_{\nu=1}^m \sum_{i\in V(C_\nu)}\frac{\partial \log f_r(A_{\nu i}, A_{\mathcal{N}_{\nu i}}|Z_{\nu i}, Z_{\mathcal{N}_{\nu i}})}{\partial \eta}=0.$$ The estimating equations for the remaining parameters $\hat{\theta}_{0\alpha}, \hat{\theta}_{1\alpha}, \hat{\theta}_{\alpha}$ are described in Section 5.3. Let \begin{align*} \psi_{\nu}(Y_\nu, A_\nu, Z_\nu; \theta)=\begin{pmatrix}\psi_{\eta}(Y_\nu, A_\nu, Z_\nu; \theta) \\ \psi_0(Y_\nu, A_\nu, Z_\nu; \theta; \alpha) \\ \psi_1(Y_\nu, A_\nu, Z_\nu; \theta; \alpha) \\ \psi_2(Y_\nu, A_\nu, Z_\nu; \theta; \alpha)\end{pmatrix}_{\eta\in\Theta}.\end{align*} Therefore, $\displaystyle \sum_{i=1}^m\psi_{\nu}(Y_{\nu},A_{\nu},Z_{\nu}; \hat{\theta})=0$. Under suitable regularity conditions and due to the unbiased estimating equations, as $m\rightarrow \infty$, $\hat{\theta}$ converges in probability to $\theta$ and $\sqrt{m}(\hat{\theta}-\theta)$ converges in distribution to a multivariate normal $N(0, \Sigma)$, where $$\Sigma=\frac{1}{m}A^{-1}(\theta)B(\theta)A(\theta)^{-T}$$ with $$A(\theta)=E[-\dot{\psi}_{\nu}(Y_{\nu},A_{\nu},Z_{\nu}; \theta)]=E[-\partial\psi_{\nu}(Y_{\nu},A_{\nu},Z_{\nu}; \theta)/\partial \theta^T]$$ and $$B(\theta)=E[\psi_{\nu}(Y_{\nu},A_{\nu},Z_{\nu}; \theta)\psi_{\nu}(Y_{\nu},A_{\nu},Z_{\nu}; \theta)^T].$$ The true parameter $\theta$ is defined as the solution to the equation $$\int \psi(Y_{\nu},A_{\nu},Z_{\nu}; \theta)dF_{\nu}(Y_{\nu},A_{\nu},Z_{\nu})=0$$ where $F_{\nu}$ is the cumulative distribution function of $(Y_{\nu},A_{\nu},Z_{\nu})$.
The empirical sandwich-type estimator can be used to estimate the asymptotic variance for the direct, disseminated, composite and overall estimators. Replacing $A(\theta)$ and $B(\theta)$ with empirical estimators in Proposition 2 yields a consistent sandwich estimator of the asymptotic variance $\Sigma$ $$\hat{\Sigma}_m=\frac{1}{m}A_m(\hat{\theta})^{-1}B_m(\hat{\theta})A_m(\hat{\theta}))^{-T}$$ where
$$A_m(\hat{\theta})=\frac{1}{m}\sum_{\nu=1}^m-\dot\psi_{\nu}(Y_{\nu},A_{\nu},Z_{\nu};\hat{\theta}; \alpha)=-\frac{1}{m}\sum_{\nu=1}^m\begin{pmatrix} A_{11}(Y_{\nu},A_{\nu},Z_{\nu};\hat{\theta}) & 0\\ A_{2\cdot}(Y_{\nu},A_{\nu},Z_{\nu};\hat{\theta}; \alpha) & -I_{3\times 3}\end{pmatrix}$$ in which $$A_{11}(Y_{\nu},A_{\nu},Z_{\nu};\hat{\theta})=\biggl(\frac{\partial \psi_\eta(Y_{\nu},A_{\nu},Z_{\nu};\hat{\theta})}{\partial \eta'}\biggl)_{\eta, \eta'\in \Theta}$$ $$A_{21}(Y_{\nu},A_{\nu},Z_{\nu};\hat{\theta}; \alpha)=\biggl(\frac{\partial \psi_0(Y_{\nu},A_{\nu},Z_{\nu};\hat{\theta}; \alpha)}{\partial \eta'}\biggl)_{\eta'\in \Theta}$$ $$A_{31}(Y_{\nu},A_{\nu},Z_{\nu};\hat{\theta}; \alpha)=\biggl(\frac{\partial \psi_1(Y_{\nu},A_{\nu},Z_{\nu};\hat{\theta}; \alpha)}{\partial \eta'}\biggl)_{\eta'\in \Theta}$$ and $$A_{41}(Y_{\nu},A_{\nu},Z_{\nu};\hat{\theta}; \alpha)=\biggl(\frac{\partial \psi_2(Y_{\nu},A_{\nu},Z_{\nu};\hat{\theta}; \alpha)}{\partial \eta'}\biggl)_{\eta'\in \Theta}.$$ Let $A_{2\cdot}(Y_{\nu},A_{\nu},Z_{\nu};\hat{\theta}; \alpha)=\begin{pmatrix} A_{21}(Y_{\nu},A_{\nu},Z_{\nu};\hat{\theta}; \alpha) & A_{31}(Y_{\nu},A_{\nu},Z_{\nu};\hat{\theta}; \alpha)& A_{41}(Y_{\nu},A_{\nu},Z_{\nu};\hat{\theta}; \alpha)\end{pmatrix}^T$
\begin{comment} \begin{align} A_m(\hat{\theta})=\frac{1}{m}\sum_{i=1}^m-\dot\psi(O_i) &=-\frac{1}{m}\sum_{i=1}^m \begin{pmatrix} \frac{\partial \psi_{\gamma_1}}{\partial \gamma_1}(O_i) &\cdots &\frac{\partial \psi_{\gamma_1}}{\partial \gamma_p}(O_i) & \frac{\partial \psi_{\gamma_1}}{\partial \theta_{0, \alpha}}(O_i) &\frac{\partial \psi_{\gamma_1}}{\partial \theta_{1, \alpha}}(O_i) & \frac{\partial \psi_{\gamma_1}}{\partial \theta_{\alpha}}(O_i)\\ \vdots & \ddots &\vdots & \vdots & \vdots &\vdots \\ \frac{\partial \psi_{\gamma_p}}{\partial \gamma_1}(O_i) & \cdots &\frac{\partial \psi_{\gamma_p}}{\partial \gamma_p} (O_i)& \frac{\partial \psi_{\gamma_p}}{\partial \theta_{0, \alpha}}(O_i) &\frac{\partial \psi_{\gamma_p}}{\partial \theta_{1, \alpha}}(O_i) & \frac{\partial \psi_{\gamma_p}}{\partial \theta_{\alpha}}(O_i)\\ \frac{\partial \psi_{p+1}}{\partial \gamma_1}(O_i) & \cdots &\frac{\partial \psi_{p+1}}{\partial \gamma_p}(O_i) & \frac{\partial \psi_{p+1}}{\partial \theta_{0, \alpha}}(O_i) &\frac{\partial \psi_{p+1}}{\partial \theta_{1, \alpha}}(O_i) & \frac{\partial \psi_{p+1}}{\partial \theta_{\alpha}}(O_i)\\ \frac{\partial \psi_{p+2}}{\partial \gamma_1}(O_i) & \cdots &\frac{\partial \psi_{p+2}}{\partial \gamma_p}(O_i) & \frac{\partial \psi_{p+2}}{\partial \theta_{0, \alpha}}(O_i) &\frac{\partial \psi_{p+2}}{\partial \theta_{1, \alpha}}(O_i) & \frac{\partial \psi_{p+2}}{\partial \theta_{\alpha}}(O_i)\\ \frac{\partial \psi_{p+3}}{\partial \gamma_1}(O_i) & \cdots &\frac{\partial \psi_{p+3}}{\partial \gamma_p}(O_i) & \frac{\partial \psi_{p+3}}{\partial \theta_{0, \alpha}}(O_i) &\frac{\partial \psi_{p+3}}{\partial \theta_{1, \alpha}}(O_i) & \frac{\partial \psi_{p+3}}{\partial \theta_{\alpha}}(O_i) \end{pmatrix}\notag\\ &=-\frac{1}{m}\sum_{i=1}^m \begin{pmatrix} \frac{\partial \psi_{\gamma_1}}{\partial \gamma_1}(O_i) &\cdots &\frac{\partial \psi_{\gamma_1}}{\partial \gamma_p}(O_i) & 0 &0 & 0\\ \vdots & \ddots &\vdots & \vdots & \vdots &\vdots \\ \frac{\partial \psi_{\gamma_p}}{\partial \gamma_1}(O_i) & \cdots &\frac{\partial \psi_{\gamma_p}}{\partial \gamma_p} (O_i)& 0 &0 & 0\\ \frac{\partial \psi_{p+1}}{\partial \gamma_1}(O_i) & \cdots &\frac{\partial \psi_{p+1}}{\partial \gamma_p}(O_i) & -1 &0 & 0\\ \frac{\partial \psi_{p+2}}{\partial \gamma_1}(O_i) & \cdots &\frac{\partial \psi_{p+2}}{\partial \gamma_p}(O_i) & 0 &-1 & 0\\ \frac{\partial \psi_{p+3}}{\partial \gamma_1}(O_i) & \cdots &\frac{\partial \psi_{p+3}}{\partial \gamma_p}(O_i) & 0 &0& -1 \end{pmatrix}.\notag \end{align} \end{comment}
\begin{align*}B_m(\hat{\theta})&=\frac{1}{m}\sum_{\nu=1}^m \psi_{\nu}(Y_{\nu},A_{\nu},Z_{\nu};\hat{\theta})\psi_{\nu}(Y_{\nu},A_{\nu},Z_{\nu};\hat{\theta})^T\end{align*}
\noindent That is, $\hat{\Sigma}_m$ is a consistent estimator of $\Sigma$. We provide a sandwich estimator of the variance for the disseminated effect. An analogous procedure can be used to obtain the sandwich variance of the variance for the estimators of the direct, overall, and total effects. Let \begin{align*} \psi_{\nu}(Y_\nu, A_\nu, Z_\nu; \theta)=\begin{pmatrix}\psi_{\eta}(Y_\nu, A_\nu, Z_\nu; \theta) \\ \psi_0(Y_\nu, A_\nu, Z_\nu; \theta; \alpha_1) \\ \psi_0(Y_\nu, A_\nu, Z_\nu; \theta; \alpha_0) \\ \psi_1(Y_\nu, A_\nu, Z_\nu; \theta; \alpha_1) \\ \psi_1(Y_\nu, A_\nu, Z_\nu; \theta; \alpha_0) \\ \psi_2(Y_\nu, A_\nu, Z_\nu; \theta; \alpha_1)\\\psi_2(Y_\nu, A_\nu, Z_\nu; \theta; \alpha_0)\end{pmatrix}_{\eta\in\Theta}.\end{align*} Replacing $A(\theta)$ and $B(\theta)$ with empirical estimators in Proposition 2 yields a consistent sandwich estimator of the asymptotic variance of $\overline{IE}(\alpha_1,\alpha_0)$ is $$\hat{\Sigma}_{IE} = \frac{1}{m}\lambda^T A_m(\hat{\theta})^{-1}B_m(\hat{\theta})A_m(\hat{\theta})^{-T} \lambda$$ with $\lambda = (0_{1 \times p}, 1, -1, 0, 0, 0, 0)^T$. The estimated standard error (se) is $\widehat{\text{se}}(\widehat{IE}_r(\alpha_1,\alpha_0)) = \sqrt{\hat{\Sigma}_{IE}}$.
\begin{comment} \begin{align} B_m(\hat{\theta})&=\frac{1}{m}\sum_{i=1}^m \psi(O_i)\psi(O_i)^T\notag\\ &=\frac{1}{m}\sum_{i=1}^m\scriptsize\begin{pmatrix} \psi_{\gamma_1}(O_i)\psi_{\gamma_1}(O_i) & \cdots & \psi_{\gamma_1}(O_i)\psi_{\gamma_p}(O_i) &\psi_{\gamma_1}(O_i)\psi_{p+1}(O_i) & \psi_{\gamma_1}(O_i)\psi_{p+2}(O_i) & \psi_{\gamma_1}(O_i)\psi_{p+3}(O_i)\\ \vdots & \ddots & \vdots &\vdots & \vdots &\vdots \\ \psi_{\gamma_p}(O_i)\psi_{\gamma_1}(O_i) & \cdots & \psi_{\gamma_p}(O_i)\psi_{\gamma_p}(O_i) &\psi_{\gamma_p}(O_i)\psi_{p+1}(O_i) & \psi_{\gamma_p}(O_i)\psi_{p+2}(O_i) & \psi_{\gamma_p}(O_i)\psi_{p+3}(O_i) \\ \psi_{p+1}(O_i)\psi_{\gamma_1}(O_i) & \cdots & \psi_{p+1}(O_i)\psi_{\gamma_p}(O_i) &\psi_{p+1}(O_i)\psi_{p+1}(O_i) & \psi_{p+1}(O_i)\psi_{p+2}(O_i) & \psi_{p+1}(O_i)\psi_{p+3}(O_i)\\ \psi_{p+2}(O_i)\psi_{\gamma_1}(O_i) & \cdots & \psi_{p+2}(O_i)\psi_{\gamma_p}(O_i) &\psi_{p+2}(O_i)\psi_{p+1}(O_i) & \psi_{p+2}(O_i)\psi_{p+2}(O_i) & \psi_{p+2}(O_i)\psi_{p+3}(O_i)\\ \psi_{p+3}(O_i)\psi_{\gamma_1}(O_i) & \cdots & \psi_{p+3}(O_i)\psi_{\gamma_p}(O_i) &\psi_{p+3}(O_i)\psi_{p+1}(O_i) & \psi_{p+3}(O_i)\psi_{p+2}(O_i) & \psi_{p+3}(O_i)\psi_{p+3}(O_i) \end{pmatrix}\notag \end{align} \end{comment}
\section{Simulation Results} \label{s:tables} In this section, we include the simulation results from 1000 simulation data set on networks with 10, 50, 100, 150, 200 components. The simulation results include the true values of the average potential outcomes, and bias on estimations of inverse probability weighted estimators (IPW$_1$ and IPW$_2$) for exposed ($a=1$) and not exposed ($a=0$), and marginal estimators under allocation strategies 25\%, 50\%, and 75\%, the asymptotic standard errors (ASE), and empirical coverage probabilities (ECP) (Table A1 to A5). Additionally, we used a different exposure generating model given by $A_i={\rm Bern}(p={\rm logit}^{-1}(-0.5-1.5\cdot Z_i+b_\nu))$ in data simulating on a network with 100 components (Table A6).
\setcounter{table}{0} \renewcommand{A\arabic{table}}{A\arabic{table}}
\begin{table}[htbp]
\centering
\caption{Simulation results of IPW$_1$ (left) and IPW$_2$ (right) on network with 10 components.}
\begin{tabular}{lr|rrrr|rrrr}
\toprule
\rowcolor[rgb]{ .949, .949, .949} & & \multicolumn{4}{c|}{IPW$_1$} & \multicolumn{4}{c}{IPW$_2$} \\
\rowcolor[rgb]{ .949, .949, .949} & \multicolumn{1}{c|}{True} & \multicolumn{1}{c}{Bias} & \multicolumn{1}{c}{ESE} & \multicolumn{1}{c}{ASE} & \multicolumn{1}{c|}{ECP} & \multicolumn{1}{c}{Bias} & \multicolumn{1}{c}{ESE} & \multicolumn{1}{c}{ASE} & \multicolumn{1}{c}{ECP} \\
\midrule
\midrule
$\widehat{Y}(1, 0.25)$ & 0.247 & 0.004 & 0.158 & 0.110 & 0.73 & -0.013 & 0.285 & 0.122 & 0.77 \\
$\widehat{Y}(1, 0.5)$ & 0.226 & 0.005 & 0.098 & 0.076 & 0.83 & -0.001 & 0.091 & 0.077 & 0.90 \\
$\widehat{Y}(1, 0.75)$ & 0.205 & <0.001 & 0.124 & 0.083 & 0.75 & 0.016 & 0.095 & 0.074 & 0.79 \\
$\widehat{Y}(0, 0.25)$ & 0.227 & 0.004 & 0.111 & 0.082 & 0.80 & 0.013 & 0.115 & 0.079 & 0.80 \\
$\widehat{Y}(0, 0.5)$ & 0.274 & 0.010 & 0.098 & 0.082 & 0.83 & 0.004 & 0.088 & 0.081 & 0.91 \\
$\widehat{Y}(0, 0.75)$ & 0.324 & 0.016 & 0.171 & 0.128 & 0.76 & 0.011 & 0.189 & 0.127 & 0.78 \\
$\widehat{Y}(0.25)$ & 0.232 & 0.004 & 0.093 & 0.073 & 0.82 & 0.006 & 0.114 & 0.073 & 0.83 \\
$\widehat{Y}(0.5)$ & 0.250 & 0.007 & 0.070 & 0.061 & 0.85 & 0.001 & 0.064 & 0.060 & 0.92 \\
$\widehat{Y}(0.75)$ & 0.235 & 0.004 & 0.103 & 0.076 & 0.79 & 0.015 & 0.087 & 0.071 & 0.81 \\
\bottomrule
\end{tabular}
\label{tab:addlabel} \end{table}
\begin{table}[H]
\centering
\caption{Simulation results of IPW$_1$ (left) and IPW$_2$ (right) on network with 50 components}
\begin{tabular}{lr|rrrr|rrrr}
\toprule
\rowcolor[rgb]{ .949, .949, .949} & & \multicolumn{4}{c|}{IPW$_1$} & \multicolumn{4}{c}{IPW$_2$} \\
\rowcolor[rgb]{ .949, .949, .949} & \multicolumn{1}{c|}{True} & \multicolumn{1}{c}{Bias} & \multicolumn{1}{c}{ESE} & \multicolumn{1}{c}{ASE} & \multicolumn{1}{c|}{ECP} & \multicolumn{1}{c}{Bias} & \multicolumn{1}{c}{ESE} & \multicolumn{1}{c}{ASE} & \multicolumn{1}{c}{ECP} \\
\midrule
\midrule
$\widehat{Y}(1, 0.25)$ & 0.249 & 0.004 & 0.066 & 0.060 & 0.91 & -0.008 & 0.077 & 0.062 & 0.89 \\
$\widehat{Y}(1, 0.5)$ & 0.227 & 0.002 & 0.040 & 0.038 & 0.94 & -0.001 & 0.033 & 0.034 & 0.96 \\
$\widehat{Y}(1, 0.75)$ & 0.206 & -0.002 & 0.053 & 0.048 & 0.91 & -0.001 & 0.052 & 0.044 & 0.91 \\
$\widehat{Y}(0, 0.25)$ & 0.227 & -0.002 & 0.049 & 0.048 & 0.94 & -0.002 & 0.048 & 0.043 & 0.92 \\
$\widehat{Y}(0, 0.5)$ & 0.274 & 0.003 & 0.038 & 0.040 & 0.97 & -0.002 & 0.033 & 0.036 & 0.97 \\
$\widehat{Y}(0, 0.75)$ & 0.325 & 0.002 & 0.071 & 0.067 & 0.93 & -0.014 & 0.081 & 0.071 & 0.93 \\
$\widehat{Y}(0.25)$ & 0.233 & -0.001 & 0.040 & 0.039 & 0.94 & -0.003 & 0.040 & 0.038 & 0.93 \\
$\widehat{Y}(0.5)$ & 0.250 & 0.002 & 0.028 & 0.029 & 0.97 & -0.001 & 0.023 & 0.026 & 0.98 \\
$\widehat{Y}(0.75)$ & 0.235 & -0.001 & 0.042 & 0.041 & 0.94 & -0.006 & 0.044 & 0.040 & 0.93 \\
\bottomrule
\end{tabular}
\label{tab:addlabel} \end{table}
\begin{table}[H]
\centering
\caption{Simulation results of IPW$_1$ (left) and IPW$_2$ (right) on network with 100 components}
\begin{tabular}{lr|rrrr|rrrr}
\toprule
\rowcolor[rgb]{ .949, .949, .949} & & \multicolumn{4}{c|}{IPW$_1$} & \multicolumn{4}{c}{IPW$_2$} \\
\rowcolor[rgb]{ .949, .949, .949} & \multicolumn{1}{c|}{True} & \multicolumn{1}{c}{Bias} & \multicolumn{1}{c}{ESE} & \multicolumn{1}{c}{ASE} & \multicolumn{1}{c|}{ECP} & \multicolumn{1}{c}{Bias} & \multicolumn{1}{c}{ESE} & \multicolumn{1}{c}{ASE} & \multicolumn{1}{c}{ECP} \\
\midrule
\midrule
$\widehat{Y}(1, 0.25)$ & 0.249 & <0.001 & 0.047 & 0.044 & 0.92 & -0.010 & 0.051 & 0.047 & 0.92 \\
$\widehat{Y}(1, 0.5)$ & 0.227 & -0.001 & 0.029 & 0.027 & 0.91 & -0.001 & 0.023 & 0.024 & 0.96 \\
$\widehat{Y}(1, 0.75)$ & 0.206 & -0.005 & 0.041 & 0.035 & 0.91 & -0.005 & 0.036 & 0.033 & 0.94 \\
$\widehat{Y}(0, 0.25)$ & 0.227 & -0.006 & 0.033 & 0.034 & 0.92 & -0.002 & 0.032 & 0.032 & 0.95 \\
$\widehat{Y}(0, 0.5)$ & 0.274 & 0.003 & 0.025 & 0.029 & 0.95 & -0.002 & 0.023 & 0.025 & 0.98 \\
$\widehat{Y}(0, 0.75)$ & 0.325 & 0.006 & 0.047 & 0.048 & 0.89 & -0.016 & 0.058 & 0.053 & 0.94 \\
$\widehat{Y}(0.25)$ & 0.233 & -0.004 & 0.028 & 0.028 & 0.95 & -0.004 & 0.028 & 0.028 & 0.97 \\
$\widehat{Y}(0.5)$ & 0.250 & 0.001 & 0.019 & 0.021 & 0.95 & -0.002 & 0.016 & 0.018 & 0.99 \\
$\widehat{Y}(0.75)$ & 0.235 & -0.002 & 0.033 & 0.030 & 0.92 & -0.008 & 0.031 & 0.030 & 0.96 \\
\bottomrule
\end{tabular}
\label{tab:100comp} \end{table}
\begin{table}[H]
\centering
\caption{Simulation results of IPW$_1$ (left) and IPW$_2$ (right) on network with 150 components}
\begin{tabular}{lr|rrrr|rrrr}
\toprule
\rowcolor[rgb]{ .949, .949, .949} & & \multicolumn{4}{c|}{IPW$_1$} & \multicolumn{4}{c}{IPW$_2$} \\
\rowcolor[rgb]{ .949, .949, .949} & \multicolumn{1}{c|}{True} & \multicolumn{1}{c}{Bias} & \multicolumn{1}{c}{ESE} & \multicolumn{1}{c}{ASE} & \multicolumn{1}{c|}{ECP} & \multicolumn{1}{c}{Bias} & \multicolumn{1}{c}{ESE} & \multicolumn{1}{c}{ASE} & \multicolumn{1}{c}{ECP} \\
\midrule
\midrule
$\widehat{Y}(1, 0.25)$ & 0.249 & 0.001 & 0.036 & 0.037 & 0.93 & -0.012 & 0.041 & 0.039 & 0.95 \\
$\widehat{Y}(1, 0.5)$ & 0.226 & 0.001 & 0.021 & 0.022 & 0.96 & -0.001 & 0.017 & 0.020 & 0.98 \\
$\widehat{Y}(1, 0.75)$ & 0.205 & -0.003 & 0.028 & 0.029 & 0.95 & -0.004 & 0.027 & 0.028 & 0.96 \\
$\widehat{Y}(0, 0.25)$ & 0.228 & -0.002 & 0.027 & 0.028 & 0.95 & -0.004 & 0.026 & 0.027 & 0.97 \\
$\widehat{Y}(0, 0.5)$ & 0.274 & 0.001 & 0.022 & 0.023 & 0.96 & -0.002 & 0.018 & 0.020 & 0.98 \\
$\widehat{Y}(0, 0.75)$ & 0.325 & 0.001 & 0.040 & 0.041 & 0.94 & -0.017 & 0.044 & 0.044 & 0.96 \\
$\widehat{Y}(0.25)$ & 0.233 & -0.001 & 0.023 & 0.023 & 0.94 & -0.006 & 0.022 & 0.023 & 0.97 \\
$\widehat{Y}(0.5)$ & 0.250 & 0.001 & 0.015 & 0.017 & 0.97 & -0.002 & 0.012 & 0.015 & 0.99 \\
$\widehat{Y}(0.75)$ & 0.235 & -0.002 & 0.023 & 0.024 & 0.96 & -0.007 & 0.023 & 0.024 & 0.97 \\
\bottomrule
\end{tabular}
\label{tab:addlabel} \end{table}
\begin{table}[H]
\centering
\caption{Simulation results of IPW$_1$ (left) and IPW$_2$ (right) on network with 200 components}
\begin{tabular}{lr|rrrr|rrrr}
\toprule
\rowcolor[rgb]{ .949, .949, .949} & & \multicolumn{4}{c|}{IPW$_1$} & \multicolumn{4}{c}{IPW$_2$} \\
\rowcolor[rgb]{ .949, .949, .949} & \multicolumn{1}{c|}{True} & \multicolumn{1}{c}{Bias} & \multicolumn{1}{c}{ESE} & \multicolumn{1}{c}{ASE} & \multicolumn{1}{c|}{ECP} & \multicolumn{1}{c}{Bias} & \multicolumn{1}{c}{ESE} & \multicolumn{1}{c}{ASE} & \multicolumn{1}{c}{ECP} \\
\midrule
\midrule
$\widehat{Y}(1, 0.25)$ & 0.248 & 0.001 & 0.031 & 0.032 & 0.94 & -0.011 & 0.036 & 0.034 & 0.95 \\
$\widehat{Y}(1, 0.5)$ & 0.226 & 0.001 & 0.019 & 0.019 & 0.94 & <0.001 & 0.016 & 0.017 & 0.97 \\
$\widehat{Y}(1, 0.75)$ & 0.205 & -0.002 & 0.025 & 0.025 & 0.94 & -0.004 & 0.024 & 0.024 & 0.95 \\
$\widehat{Y}(0, 0.25)$ & 0.228 & -0.003 & 0.023 & 0.024 & 0.95 & -0.007 & 0.024 & 0.024 & 0.95 \\
$\widehat{Y}(0, 0.5)$ & 0.274 & 0.001 & 0.019 & 0.020 & 0.96 & -0.002 & 0.016 & 0.018 & 0.97 \\
$\widehat{Y}(0, 0.75)$ & 0.325 & -0.001 & 0.036 & 0.035 & 0.94 & -0.016 & 0.041 & 0.038 & 0.95 \\
$\widehat{Y}(0.25)$ & 0.233 & -0.002 & 0.019 & 0.020 & 0.97 & -0.007 & 0.021 & 0.021 & 0.97 \\
$\widehat{Y}(0.5)$ & 0.250 & 0.001 & 0.014 & 0.014 & 0.96 & -0.001 & 0.011 & 0.013 & 0.98 \\
$\widehat{Y}(0.75)$ & 0.235 & -0.002 & 0.021 & 0.021 & 0.95 & -0.008 & 0.020 & 0.020 & 0.97 \\
\bottomrule
\end{tabular}
\label{tab:200comp} \end{table}
\begin{table}[H]
\centering
\caption{Simulation results of IPW$_1$ (left) and IPW$_2$ (right) using propensity score model $A_i={\rm Bern}({\rm logit}^{-1}(-0.5-1.5\cdot Z_i+b_\nu)$ on a network with 100 components.}
\begin{tabular}{lr|rrrr|rrrr}
\toprule
\rowcolor[rgb]{ .949, .949, .949} & & \multicolumn{4}{c|}{IPW$_1$} & \multicolumn{4}{c}{IPW$_2$} \\
\rowcolor[rgb]{ .949, .949, .949} & \multicolumn{1}{c|}{True} & \multicolumn{1}{c}{Bias} & \multicolumn{1}{c}{ESE} & \multicolumn{1}{c}{ASE} & \multicolumn{1}{c|}{ECP} & \multicolumn{1}{c}{Bias} & \multicolumn{1}{c}{ESE} & \multicolumn{1}{c}{ASE} & \multicolumn{1}{c}{ECP} \\
\midrule
\midrule
$\widehat{Y}(1, 0.25)$ & 0.2482 & 0.0017 & 0.041 & 0.044 & 0.94 & 0.0017 & 0.035 & 0.038 & 0.97 \\
$\widehat{Y}(1, 0.5)$ & 0.2260 & 0.0006 & 0.068 & 0.074 & 0.86 & 0.0011 & 0.114 & 0.055 & 0.92 \\
$\widehat{Y}(1, 0.75)$ & 0.2051 & -0.0005 & 0.171 & 0.183 & 0.68 & -0.0079 & 0.540 & 0.099 & 0.71 \\
$\widehat{Y}(0, 0.25)$ & 0.2278 & 0.0011 & 0.020 & 0.022 & 0.96 & -0.0002 & 0.017 & 0.019 & 0.99 \\
$\widehat{Y}(0, 0.5)$ & 0.2743 & 0.0003 & 0.050 & 0.045 & 0.90 & -0.0093 & 0.052 & 0.045 & 0.95 \\
$\widehat{Y}(0, 0.75)$ & 0.3325 & 0.0021 & 0.172 & 0.124 & 0.79 & -0.0348 & 0.208 & 0.130 & 0.81 \\
$\widehat{Y}(0.25)$ & 0.2329 & 0.0012 & 0.018 & 0.021 & 0.97 & 0.0002 & 0.015 & 0.018 & 0.98 \\
$\widehat{Y}(0.5)$ & 0.2502 & 0.0005 & 0.042 & 0.047 & 0.89 & -0.0041 & 0.063 & 0.040 & 0.95 \\
$\widehat{Y}(0.75)$ & 0.2350& -0.0018 & 0.133 & 0.150 & 0.73 & -0.0146 & 0.408 & 0.091 & 0.78 \\
\hline
\end{tabular}
\label{tab:trtdist} \end{table}
\section{Community Alerts and HIV Risk Behavior in TRIP at 6 months} In this section, we report the point estimates for direct, indirect, total, and overall effects under allocation strategies 20\%, 30\%, 40\% and 50\% and their corresponding 95\% confidence intervals for the effect of community alerts at baseline on HIV risk behavior at 6-month follow up in TRIP using different set of measured confounding variables. The full set of confounding variables are HIV status, shared drug equipment (e.g. syringe) in last six months, the calendar date at first interview, education (primary school, first 3 years of high school, last 3 years of high school, and post high school), and employment status (employed, unemployed/looking for a work, can’t work because of health reason, and other). We first considered univariate models; that is, only use one confounding variable at a time (Tables \ref{tab:TRIP_HIV}, \ref{tab:TRIP_date}, \ref{tab:TRIP_share}, \ref{tab:TRIP_edu}, \ref{tab:TRIP_employ}), and a model not adjusted for any confounders (Table \ref{tab:TRIP_noadj}). Second, we estimated the effects using all of the variables, but excluding one at a time (Tables \ref{tab:TRIP_noHIV}, \ref{tab:TRIP_noDate}, \ref{tab:TRIP_noShare}, \ref{tab:TRIP_noEdu}, \ref{tab:TRIP_noemploy}). The 95\% CI (10) were estimated by original 10 components TRIP network, and 95\% CI (20) were estimated by dividing TRIP network into 20 components using community detection.
\begin{table}[htbp]
\centering
\caption{Estimated risk differences and 95\% confidence intervals (CIs) of the effects of community alerts on HIV risk behavior at 6 months in TRIP adjusted only for HIV status.}
\begin{tabular}{ll|rrrrrr}
\toprule
\multicolumn{1}{c}{Effects} & Coverage & \multicolumn{3}{c}{IPW$_1$} & \multicolumn{3}{c}{IPW$_2$} \\
& $(\alpha, \alpha')$ & \multicolumn{1}{c}{RD} & \multicolumn{1}{c}{95\% CI (10)} & \multicolumn{1}{c}{95\% CI (20)} & \multicolumn{1}{c}{RD} & \multicolumn{1}{c}{95\% CI (10)} & \multicolumn{1}{c}{95\% CI (20)} \\
\midrule
\midrule
\rowcolor[rgb]{ .949, .949, .949} Direct & (20\%, 20\%) & -0.0803 & (-0.154,-0.007) & (-0.246, 0.085) & -0.0237 & (-0.070, 0.022) & (-0.219, 0.171) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (30\%, 30\%) & -0.1301 & (-0.285, 0.025) & (-0.327, 0.067) & -0.1196 & (-0.272, 0.033) & (-0.312, 0.073) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (40\%, 40\%) & -0.1808 & (-0.439, 0.078) & (-0.408, 0.046) & -0.1913 & (-0.481, 0.099) & (-0.430, 0.048) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (50\%, 50\%) & -0.2256 & (-0.587, 0.136) & (-0.482, 0.031) & -0.2366 & (-0.622, 0.149) & (-0.517, 0.044) \\
Indirect & (30\%, 20\%) & 0.0055 & (-0.011, 0.022) & (-0.020, 0.031) & 0.0063 & (-0.010, 0.022) & (-0.050, 0.063) \\
Indirect & (40\%, 20\%) & -0.0057 & (-0.026, 0.015) & (-0.063, 0.052) & -0.0058 & (-0.016, 0.005) & (-0.103, 0.091) \\
Indirect & (50\%, 20\%) & -0.0240 & (-0.059, 0.011) & (-0.128, 0.080) & -0.0269 & (-0.052,-0.002) & (-0.158, 0.105) \\
Indirect & (40\%, 30\%) & -0.0112 & (-0.025, 0.002) & (-0.046, 0.023) & -0.0120 & (-0.025, 0.001) & (-0.055, 0.031) \\
Indirect & (50\%, 40\%) & -0.0184 & (-0.040, 0.003) & (-0.068, 0.032) & -0.0211 & (-0.046, 0.004) & (-0.060, 0.018) \\
Indirect & (50\%, 30\%) & -0.0295 & (-0.064, 0.005) & (-0.113, 0.054) & -0.0331 & (-0.071, 0.004) & (-0.113, 0.047) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (30\%, 20\%) & -0.1246 & (-0.268, 0.018) & (-0.325, 0.076) & -0.1134 & (-0.252, 0.025) & (-0.311, 0.084) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (40\%, 20\%) & -0.1865 & (-0.439, 0.066) & (-0.398, 0.025) & -0.1970 & (-0.484, 0.090) & (-0.407, 0.013) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 20\%) & -0.2496 & (-0.617, 0.118) & (-0.455,-0.045) & -0.2635 & (-0.671, 0.144) & (-0.476,-0.051) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (40\%, 30\%) & -0.1920 & (-0.458, 0.074) & (-0.403, 0.019) & -0.2033 & (-0.506, 0.099) & (-0.423, 0.016) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 40\%) & -0.2440 & (-0.618, 0.130) & (-0.466,-0.022) & -0.2577 & (-0.668, 0.152) & (-0.512,-0.004) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 30\%) & -0.2552 & (-0.636, 0.126) & (-0.461,-0.050) & -0.2697 & (-0.692, 0.152) & (-0.501,-0.039) \\
Overall & (30\%, 20\%) & -0.0174 & (-0.039, 0.004) & (-0.051, 0.017) & -0.0249 & (-0.061, 0.011) & (-0.072, 0.022) \\
Overall & (40\%, 20\%) & -0.0619 & (-0.148, 0.024) & (-0.118,-0.006) & -0.0775 & (-0.196, 0.041) & (-0.151,-0.004) \\
Overall & (50\%, 20\%) & -0.1208 & (-0.297, 0.056) & (-0.194,-0.048) & -0.1404 & (-0.360, 0.079) & (-0.230,-0.051) \\
Overall & (40\%, 30\%) & -0.0445 & (-0.110, 0.021) & (-0.069,-0.020) & -0.0526 & (-0.135, 0.030) & (-0.087,-0.018) \\
Overall & (50\%, 40\%) & -0.0588 & (-0.149, 0.032) & (-0.086,-0.032) & -0.0629 & (-0.164, 0.038) & (-0.096,-0.030) \\
Overall & (50\%, 30\%) & -0.1033 & (-0.259, 0.053) & (-0.152,-0.055) & -0.1155 & (-0.299, 0.068) & (-0.178,-0.053) \\
\bottomrule
\end{tabular}
\label{tab:TRIP_HIV} \end{table}
\begin{table}[h]
\centering
\caption{Estimated risk differences and 95\% confidence intervals (CIs) of the effects of community alerts on HIV risk behavior at 6 months in TRIP adjusted only for the calendar date at first interview.}
\begin{tabular}{lc|rrrrrr}
\toprule
\multicolumn{1}{c}{Effects} & Coverage & \multicolumn{3}{c}{IPW$_1$} & \multicolumn{3}{c}{IPW$_2$} \\
& $(\alpha, \alpha')$ & \multicolumn{1}{c}{RD} & \multicolumn{1}{c}{95\% CI (10)} & \multicolumn{1}{c}{95\% CI (20)} & \multicolumn{1}{c}{RD} & \multicolumn{1}{c}{95\% CI (10)} & \multicolumn{1}{c}{95\% CI (20)} \\
\midrule
\midrule
\rowcolor[rgb]{ .949, .949, .949} Direct & (20\%, 20\%) & -0.0688 & (-0.132,-0.005) & (-0.270,0.133) & 0.0052 & (-0.100, 0.111) & (-0.212, 0.223) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (30\%, 30\%) & -0.1066 & (-0.215, 0.002) & (-0.333, 0.120) & -0.0739 & (-0.143,-0.005) & (-0.304, 0.156) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (40\%, 40\%) & -0.1515 & (-0.349, 0.046) & (-0.386, 0.083) & -0.1454 & (-0.342, 0.051) & (-0.406, 0.116) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (50\%, 50\%) & -0.1947 & (-0.492, 0.103) & (-0.437, 0.048) & -0.2039 & (-0.521, 0.113) & (-0.501, 0.093) \\
Indirect & (30\%, 20\%) & 0.0040 & (-0.011, 0.019) & (-0.019, 0.026) & 0.0040 & (-0.008, 0.016) & (-0.046, 0.054) \\
Indirect & (40\%, 20\%) & -0.0103 & (-0.029, 0.009) & (-0.054, 0.033) & -0.0044 & (-0.015, 0.007) & (-0.096, 0.087) \\
Indirect & (50\%, 20\%) & -0.0333 & (-0.072, 0.006) & (-0.106, 0.039) & -0.0174 & (-0.032,-0.003) & (-0.153, 0.119) \\
Indirect & (40\%, 30\%) & -0.0142 & (-0.030, 0.002) & (-0.039, 0.010) & -0.0084 & (-0.015,-0.001) & (-0.053, 0.036) \\
Indirect & (50\%, 40\%) & -0.0230 & (-0.050, 0.004) & (-0.057, 0.011) & -0.0131 & (-0.025,-0.001) & (-0.063, 0.037) \\
Indirect & (50\%, 30\%) & -0.0373 & (-0.080, 0.005) & (-0.095, 0.020) & -0.0214 & (-0.040,-0.003) & (-0.114, 0.071) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (30\%, 20\%) & -0.1027 & (-0.204,-0.001) & (-0.335, 0.129) & -0.0699 & (-0.131,-0.009) & (-0.309, 0.170) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (40\%, 20\%) & -0.1618 & (-0.361, 0.038) & (-0.395, 0.072) & -0.1498 & (-0.342, 0.042) & (-0.392, 0.092) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 20\%) & -0.2280 & (-0.547, 0.091) & (-0.450,-0.006) & -0.2213 & (-0.542, 0.100) & (-0.447, 0.004) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (40\%, 30\%) & -0.1658 & (-0.376, 0.044) & (-0.395, 0.063) & -0.1538 & (-0.356, 0.048) & (-0.395, 0.087) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 40\%) & -0.2177 & (-0.537, 0.101) & (-0.444, 0.009) & -0.2169 & (-0.542, 0.109) & (-0.474, 0.040) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 30\%) & -0.2320 & (-0.563, 0.099) & (-0.451,-0.012) & -0.2253 & (-0.556, 0.106) & (-0.458, 0.007) \\
Overall & (30\%, 20\%) & -0.0143 & (-0.030, 0.002) & (-0.051, 0.023) & -0.0192 & (-0.044, 0.006) & (-0.070, 0.031) \\
Overall & (40\%, 20\%) & -0.0571 & (-0.132, 0.017) & (-0.118, 0.004) & -0.0636 & (-0.155, 0.028) & (-0.140, 0.013) \\
Overall & (50\%, 20\%) & -0.1169 & (-0.282, 0.049) & (-0.195,-0.038) & -0.1204 & (-0.301, 0.060) & (-0.204,-0.037) \\
Overall & (40\%, 30\%) & -0.0429 & (-0.104, 0.018) & (-0.070,-0.016) & -0.0443 & (-0.111, 0.022) & (-0.074,-0.015) \\
Overall & (50\%, 40\%) & -0.0598 & (-0.152, 0.032) & (-0.085,-0.034) & -0.0568 & (-0.146, 0.032) & (-0.084,-0.030) \\
Overall & (50\%, 30\%) & -0.1026 & (-0.256, 0.050) & (-0.153,-0.053) & -0.1012 & (-0.257, 0.054) & (-0.150,-0.053) \\
\bottomrule
\end{tabular}
\label{tab:TRIP_date} \end{table}
\begin{table}[htbp]
\centering
\caption{Estimated risk differences and 95\% confidence intervals (CIs) of the effects of community alerts on HIV risk behavior at 6 months in TRIP adjusted only for shared drug equipment (e.g. syringe) in last six months.}
\begin{tabular}{lc|rllrll}
\toprule
\multicolumn{1}{c}{Effects} & Coverage & \multicolumn{3}{c}{IPW$_1$} & \multicolumn{3}{c}{IPW$_2$} \\
& $(\alpha, \alpha')$ & \multicolumn{1}{c}{RD} & \multicolumn{1}{c}{95\% CI (10)} & \multicolumn{1}{c}{95\% CI (20)} & \multicolumn{1}{c}{RD} & \multicolumn{1}{c}{95\% CI (10)} & \multicolumn{1}{c}{95\% CI (20)} \\
\midrule
\midrule
\rowcolor[rgb]{ .949, .949, .949} Direct & (20\%, 20\%) & -0.0889 & (-0.173,-0.004) & (-0.324, 0.146) & 0.0253 & (-0.121, 0.172) & (-0.222, 0.273) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (30\%, 30\%) & -0.1337 & (-0.288, 0.020) & (-0.417, 0.15) & -0.0609 & (-0.107,-0.015) & (-0.335, 0.213) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (40\%, 40\%) & -0.1792 & (-0.423, 0.065) & (-0.489, 0.131) & -0.1386 & (-0.318, 0.041) & (-0.452, 0.175) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (50\%, 50\%) & -0.2186 & (-0.552, 0.115) & (-0.548, 0.110) & -0.2019 & (-0.514, 0.110) & (-0.556, 0.152) \\
Indirect & (30\%, 20\%) & 0.0057 & (-0.012, 0.024) & (-0.019, 0.031) & 0.0104 & (-0.015, 0.035) & (-0.048, 0.068) \\
Indirect & (40\%, 20\%) & -0.0057 & (-0.025, 0.013) & (-0.065, 0.053) & 0.0074 & (-0.025, 0.040) & (-0.104, 0.119) \\
Indirect & (50\%, 20\%) & -0.0259 & (-0.057, 0.006) & (-0.137, 0.085) & -0.0024 & (-0.037, 0.032) & (-0.169, 0.165) \\
Indirect & (40\%, 30\%) & -0.0114 & (-0.024, 0.001) & (-0.049, 0.027) & -0.0030 & (-0.012, 0.006) & (-0.059, 0.053) \\
Indirect & (50\%, 40\%) & -0.0202 & (-0.044, 0.003) & (-0.077, 0.036) & -0.0097 & (-0.018,-0.001) & (-0.071, 0.051) \\
Indirect & (50\%, 30\%) & -0.0316 & (-0.067,0.004) & (-0.125, 0.062) & -0.0127 & (-0.027, 0.002) & (-0.128, 0.102) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (30\%, 20\%) & -0.1280 & (-0.270, 0.014) & (-0.411, 0.155) & -0.0506 & (-0.091,-0.010) & (-0.329, 0.228) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (40\%, 20\%) & -0.1848 & (-0.424, 0.054) & (-0.470, 0.100) & -0.1312 & (-0.282, 0.019) & (-0.410, 0.148) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 20\%) & -0.2445 & (-0.592, 0.103) & (-0.505, 0.016) & -0.2043 & (-0.487, 0.079) & (-0.459, 0.050) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (40\%, 30\%) & -0.1906 & (-0.444, 0.063) & (-0.477, 0.096) & -0.1416 & (-0.316, 0.032) & (-0.425, 0.142) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 40\%) & -0.2388 & (-0.592, 0.115) & (-0.525, 0.047) & -0.2117 & (-0.525, 0.101) & (-0.514, 0.091) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 30\%) & -0.2502 & (-0.612, 0.112) & (-0.513, 0.013) & -0.2147 & (-0.521, 0.092) & (-0.482, 0.053) \\
Overall & (30\%, 20\%) & -0.0166 & (-0.035, 0.002) & (-0.058, 0.025) & -0.0130 & (-0.025,-0.001) & (-0.068, 0.042) \\
Overall & (40\%, 20\%) & -0.0595 & (-0.138, 0.019) & (-0.124, 0.005) & -0.0531 & (-0.122, 0.016) & (-0.135, 0.029) \\
Overall & (50\%, 20\%) & -0.1174 & (-0.285, 0.050) & (-0.194,-0.041) & -0.1084 & (-0.263, 0.046) & (-0.195,-0.021) \\
Overall & (40\%, 30\%) & -0.0429 & (-0.104, 0.019) & (-0.069,-0.017) & -0.0401 & (-0.098, 0.018) & (-0.070,-0.010) \\
Overall & (50\%, 40\%) & -0.0578 & (-0.147, 0.032) & (-0.086,-0.030) & -0.0553 & (-0.141, 0.030) & (-0.081,-0.029) \\
Overall & (50\%, 30\%) & -0.1008 & (-0.252, 0.050) & (-0.149,-0.052) & -0.0954 & (-0.239, 0.048) & (-0.142,-0.048) \\
\bottomrule
\end{tabular}
\label{tab:TRIP_share} \end{table}
\begin{table}[htbp]
\centering
\caption{Estimated risk differences and 95\% confidence intervals (CIs) of the effects of community alerts on HIV risk behavior at 6 months in TRIP adjusted only for education (primary school, high school, and post high school).}
\begin{tabular}{lc|rllrll}
\toprule
\multicolumn{1}{c}{Effects} & Coverage & \multicolumn{3}{c}{IPW$_1$} & \multicolumn{3}{c}{IPW$_2$} \\
& $(\alpha, \alpha')$ & \multicolumn{1}{c}{RD} & \multicolumn{1}{c}{95\% CI (10)} & \multicolumn{1}{c}{95\% CI (20)} & \multicolumn{1}{c}{RD} & \multicolumn{1}{c}{95\% CI (10)} & \multicolumn{1}{c}{95\% CI (20)} \\
\midrule
\midrule
\rowcolor[rgb]{ .949, .949, .949} Direct & (20\%, 20\%) & -0.0799 & (-0.156,-0.004) & (-0.266, 0.106) & 0.0049 & (-0.093, 0.102) & (-0.212, 0.222) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (30\%, 30\%) & -0.1294 & (-0.294, 0.035) & (-0.347, 0.088) & -0.0827 & (-0.172, 0.006) & (-0.313, 0.148) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (40\%, 40\%) & -0.1799 & (-0.456, 0.096) & (-0.418, 0.058) & -0.1650 & (-0.414, 0.084) & (-0.453, 0.123) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (50\%, 50\%) & -0.2246 & (-0.608, 0.159) & (-0.481, 0.032) & -0.2347 & (-0.631, 0.162) & (-0.592, 0.122) \\
Indirect & (30\%, 20\%) & 0.0054 & (-0.016, 0.027) & (-0.014, 0.025) & 0.0106 & (-0.017, 0.038) & (-0.048, 0.069) \\
Indirect & (40\%, 20\%) & -0.0060 & (-0.037, 0.025) & (-0.047, 0.035) & 0.0136 & (-0.037, 0.064) & (-0.108, 0.135) \\
Indirect & (50\%, 20\%) & -0.0245 & (-0.069, 0.020) & (-0.104, 0.054) & 0.0139 & (-0.062, 0.090) & (-0.179, 0.207) \\
Indirect & (40\%, 30\%) & -0.0114 & (-0.026, 0.003) & (-0.037, 0.014) & 0.0031 & (-0.020, 0.027) & (-0.062, 0.068) \\
Indirect & (50\%, 40\%) & -0.0185 & (-0.038, 0.001) & (-0.061, 0.024) & 0.0003 & (-0.026, 0.026) & (-0.076, 0.076) \\
Indirect & (50\%, 30\%) & -0.0299 & (-0.064, 0.004) & (-0.097, 0.037) & 0.0034 & (-0.046, 0.053) & (-0.137, 0.143) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (30\%, 20\%) & -0.1240 & (-0.273, 0.025) & (-0.350, 0.102) & -0.0722 & (-0.139,-0.005) & (-0.309, 0.164) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (40\%, 20\%) & -0.1859 & (-0.446, 0.074) & (-0.422, 0.050) & -0.1514 & (-0.353, 0.050) & (-0.395, 0.092) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 20\%) & -0.2491 & (-0.622, 0.123) & (-0.475,-0.023) & -0.2207 & (-0.547, 0.105) & (-0.452, 0.011) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (40\%, 30\%) & -0.1913 & (-0.469, 0.086) & (-0.421, 0.038) & -0.1619 & (-0.390, 0.066) & (-0.412, 0.088) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 40\%) & -0.2431 & (-0.634, 0.147) & (-0.472,-0.014) & -0.2344 & (-0.609, 0.140) & (-0.526, 0.057) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 30\%) & -0.2545 & (-0.646, 0.137) & (-0.475,-0.034) & -0.2313 & (-0.584, 0.121) & (-0.479, 0.016) \\
Overall & (30\%, 20\%) & -0.0174 & (-0.039, 0.004) & (-0.056, 0.021) & -0.0152 & (-0.031,0.0005) & (-0.066, 0.036) \\
Overall & (40\%, 20\%) & -0.0620 & (-0.145, 0.021) & (-0.125, 0.001) & -0.0533 & (-0.122, 0.015) & (-0.130, 0.023) \\
Overall & (50\%, 20\%) & -0.1208 & (-0.292, 0.050) & (-0.199,-0.043) & -0.1044 & (-0.249, 0.040) & (-0.190,-0.019) \\
Overall & (40\%, 30\%) & -0.0445 & (-0.108, 0.019) & (-0.071,-0.018) & -0.0381 & (-0.091, 0.015) & (-0.067,-0.009) \\
Overall & (50\%, 40\%) & -0.0589 & (-0.147, 0.030) & (-0.085,-0.032) & -0.0510 & (-0.128, 0.026) & (-0.077,-0.025) \\
Overall & (50\%, 30\%) & -0.1034 & (-0.255, 0.048) & (-0.153,-0.054) & -0.0891 & (-0.219, 0.041) & (-0.137,-0.042) \\
\bottomrule
\end{tabular}
\label{tab:TRIP_edu} \end{table}
\begin{table}[htbp]
\centering
\caption{Estimated risk differences and 95\% confidence intervals (CIs) of the effects of community alerts on HIV risk behavior at 6 months in TRIP adjusted only for employment status (employed, unemployed/looking for a work, can’t work because of health reason, and other).}
\begin{tabular}{lc|rllrll}
\toprule
\multicolumn{1}{c}{Effects} & Coverage & \multicolumn{3}{c}{IPW$_1$} & \multicolumn{3}{c}{IPW$_2$} \\
& $(\alpha, \alpha')$ & \multicolumn{1}{c}{RD} & \multicolumn{1}{c}{95\% CI (10)} & \multicolumn{1}{c}{95\% CI (20)} & \multicolumn{1}{c}{RD} & \multicolumn{1}{c}{95\% CI (10)} & \multicolumn{1}{c}{95\% CI (20)} \\
\midrule
\midrule
\rowcolor[rgb]{ .949, .949, .949} Direct & (20\%, 20\%) & -0.0821 & (-0.158,-0.006) & (-0.282, 0.118) & 0.0185 & (-0.107, 0.144) & (-0.198, 0.235) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (30\%, 30\%) & -0.1309 & (-0.279, 0.017) & (-0.385, 0.123) & -0.0481 & (-0.094,-0.002) & (-0.287, 0.191) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (40\%, 40\%) & -0.1807 & (-0.427, 0.066) & (-0.474, 0.113) & -0.1076 & (-0.241, 0.025) & (-0.371, 0.156) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (50\%, 50\%) & -0.2253 & (-0.573, 0.123) & (-0.548, 0.098) & -0.1578 & (-0.394, 0.079) & (-0.444, 0.129) \\
Indirect & (30\%, 20\%) & 0.0051 & (-0.011, 0.021) & (-0.024, 0.034) & -0.0060 & (-0.015, 0.003) & (-0.046, 0.034) \\
Indirect & (40\%, 20\%) & -0.0069 & (-0.026, 0.012) & (-0.077, 0.063) & -0.0247 & (-0.057, 0.007) & (-0.097, 0.047) \\
Indirect & (50\%, 20\%) & -0.0255 & (-0.059, 0.008) & (-0.151, 0.100) & -0.0462 & (-0.102, 0.010) & (-0.154, 0.062) \\
Indirect & (40\%, 30\%) & -0.0120 & (-0.026, 0.002) & (-0.055, 0.031) & -0.0188 & (-0.043, 0.005) & (-0.053, 0.016) \\
Indirect & (50\%, 40\%) & -0.0186 & (-0.039, 0.002) & (-0.077, 0.040) & -0.0215 & (-0.046, 0.003) & (-0.064, 0.021) \\
Indirect & (50\%, 30\%) & -0.0305 & (-0.065, 0.004) & (-0.131, 0.070) & -0.0402 & (-0.089, 0.009) & (-0.115, 0.035) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (30\%, 20\%) & -0.1258 & (-0.264, 0.012) & (-0.378, 0.126) & -0.0540 & (-0.101,-0.007) & (-0.302, 0.194) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (40\%, 20\%) & -0.1876 & (-0.432, 0.057) & (-0.451, 0.076) & -0.1324 & (-0.295, 0.030) & (-0.387, 0.123) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 20\%) & -0.2508 & (-0.610, 0.108) & (-0.498,-0.004) & -0.2041 & (-0.495, 0.087) & (-0.443, 0.035) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (40\%, 30\%) & -0.1927 & (-0.449, 0.064) & (-0.461, 0.075) & -0.1264 & (-0.283, 0.030) & (-0.377, 0.124) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 40\%) & -0.2439 & (-0.607, 0.119) & (-0.521, 0.034) & -0.1793 & (-0.440, 0.081) & (-0.434, 0.076) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 30\%) & -0.2558 & (-0.628, 0.116) & (-0.508,-0.004) & -0.1981 & (-0.482, 0.086) & (-0.438, 0.042) \\
Overall & (30\%, 20\%) & -0.0178 & (-0.040, 0.004) & (-0.057, 0.021) & -0.0241 & (-0.057, 0.009) & (-0.074, 0.026) \\
Overall & (40\%, 20\%) & -0.0628 & (-0.148, 0.023) & (-0.126,0.0003) & -0.0715 & (-0.176, 0.033) & (-0.150, 0.007) \\
Overall & (50\%, 20\%) & -0.1217 & (-0.297, 0.054) & (-0.200,-0.043) & -0.1288 & (-0.324, 0.066) & (-0.219,-0.038) \\
Overall & (40\%, 30\%) & -0.0450 & (-0.110, 0.020) & (-0.072,-0.018) & -0.0474 & (-0.119, 0.025) & (-0.081,-0.014) \\
Overall & (50\%, 40\%) & -0.0589 & (-0.150, 0.032) & (-0.087,-0.031) & -0.0574 & (-0.148, 0.033) & (-0.085,-0.029) \\
Overall & (50\%, 30\%) & -0.1039 & (-0.260, 0.052) & (-0.154,-0.054) & -0.1047 & (-0.267, 0.057) & (-0.159,-0.050) \\
\bottomrule
\end{tabular}
\label{tab:TRIP_employ} \end{table}
\begin{table}[htbp]
\centering
\caption{Estimated risk differences and 95\% confidence intervals (CIs) of the effects of community alerts on HIV risk behavior at 6 months in TRIP adjusted for shared drug equipment (e.g. syringe) in last six months, the calendar date at first interview, education (primary school, high school, and post high school), and employment status (employed, unemployed/looking for a work, can’t work because of health reason, and other).}
\begin{tabular}{lc|rllrll}
\toprule
\multicolumn{1}{c}{Effects} & Coverage & \multicolumn{3}{c}{IPW$_1$} & \multicolumn{3}{c}{IPW$_2$} \\
& $(\alpha, \alpha')$ & \multicolumn{1}{c}{RD} & \multicolumn{1}{c}{95\% CI (10)} & \multicolumn{1}{c}{95\% CI (20)} & \multicolumn{1}{c}{RD} & \multicolumn{1}{c}{95\% CI (10)} & \multicolumn{1}{c}{95\% CI (20)} \\
\midrule
\midrule
\rowcolor[rgb]{ .949, .949, .949} Direct & (20\%, 20\%) & -0.0671 & (-0.125,-0.010) & (-0.317, 0.183) & 0.0343 & (-0.108, 0.176) & (-0.191, 0.260) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (30\%, 30\%) & -0.0992 & (-0.198,-0.00004) & (-0.395, 0.196) & -0.0420 & (-0.072,-0.012) & (-0.290, 0.206) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (40\%, 40\%) & -0.1396 & (-0.321, 0.042) & (-0.442, 0.163) & -0.1120 & (-0.272, 0.048) & (-0.397, 0.173) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (50\%, 50\%) & -0.1791 & (-0.451, 0.093) & (-0.469, 0.111) & -0.1735 & (-0.467, 0.120) & (-0.501, 0.154) \\
Indirect & (30\%, 20\%) & 0.0053 & (-0.014, 0.024) & (-0.021, 0.032) & 0.0036 & (-0.013, 0.021) & (-0.043, 0.050) \\
Indirect & (40\%, 20\%) & -0.0093 & (-0.028, 0.009) & (-0.055, 0.037) & -0.0013 & (-0.028, 0.025) & (-0.094, 0.091) \\
Indirect & (50\%, 20\%) & -0.0350 & (-0.075, 0.005) & (-0.104, 0.034) & -0.0079 & (-0.048, 0.032) & (-0.156, 0.141) \\
Indirect & (40\%, 30\%) & -0.0146 & (-0.031, 0.002) & (-0.038, 0.009) & -0.0049 & (-0.016, 0.006) & (-0.054, 0.044) \\
Indirect & (50\%, 40\%) & -0.0257 & (-0.059, 0.008) & (-0.055, 0.003) & -0.0065 & (-0.021, 0.008) & (-0.068, 0.054) \\
Indirect & (50\%, 30\%) & -0.0403 & (-0.090, 0.009) & (-0.092, 0.011) & -0.0114 & (-0.037, 0.014) & (-0.120, 0.097) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (30\%, 20\%) & -0.0939 & (-0.182,-0.005) & (-0.393, 0.205) & -0.0384 & (-0.069,-0.008) & (-0.296, 0.219) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (40\%, 20\%) & -0.1489 & (-0.328, 0.030) & (-0.442, 0.144) & -0.1133 & (-0.251, 0.024) & (-0.377, 0.151) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 20\%) & -0.2140 & (-0.510, 0.082) & (-0.474, 0.046) & -0.1813 & (-0.441, 0.079) & (-0.427, 0.064) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (40\%, 30\%) & -0.1542 & (-0.348, 0.040) & (-0.445, 0.137) & -0.1169 & (-0.270, 0.036) & (-0.378, 0.144) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 40\%) & -0.2047 & (-0.504, 0.095) & (-0.476, 0.067) & -0.1800 & (-0.464, 0.104) & (-0.457, 0.097) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 30\%) & -0.2193 & (-0.532, 0.093) & (-0.478, 0.040) & -0.1849 & (-0.461, 0.091) & (-0.434, 0.064) \\
Overall & (30\%, 20\%) & -0.0111 & (-0.023, 0.001) & (-0.058, 0.036) & -0.0159 & (-0.032,0.0002) & (-0.067, 0.035) \\
Overall & (40\%, 20\%) & -0.0517 & (-0.116, 0.012) & (-0.122, 0.019) & -0.0530 & (-0.120, 0.014) & (-0.131, 0.025) \\
Overall & (50\%, 20\%) & -0.1111 & (-0.267, 0.044) & (-0.189,-0.033) & -0.1015 & (-0.242, 0.039) & (-0.188,-0.015) \\
Overall & (40\%, 30\%) & -0.0407 & (-0.098, 0.017) & (-0.067,-0.015) & -0.0371 & (-0.089, 0.015) & (-0.067,-0.007) \\
Overall & (50\%, 40\%) & -0.0593 & (-0.152, 0.033) & (-0.084,-0.034) & -0.0485 & (-0.121, 0.025) & (-0.072,-0.025) \\
Overall & (50\%, 30\%) & -0.1000 & (-0.250, 0.050) & (-0.144,-0.056) & -0.0856 & (-0.210, 0.039) & (-0.132,-0.039) \\
\bottomrule
\end{tabular}
\label{tab:TRIP_noHIV} \end{table}
\begin{table}[htbp]
\centering
\caption{Estimated risk differences and 95\% confidence intervals (CIs) of the effects of community alerts on HIV risk behavior at 6 months in TRIP adjusted for HIV status, shared drug equipment (e.g. syringe) in last six months, education (primary school, high school, and post high school), and employment status (employed, unemployed/looking for a work, can’t work because of health reason, and other).}
\begin{tabular}{lc|rllrll}
\toprule
\multicolumn{1}{c}{Effects} & Coverage & \multicolumn{3}{c}{IPW$_1$} & \multicolumn{3}{c}{IPW$_2$} \\
& $(\alpha, \alpha')$ & \multicolumn{1}{c}{RD} & \multicolumn{1}{c}{95\% CI (10)} & \multicolumn{1}{c}{95\% CI (20)} & \multicolumn{1}{c}{RD} & \multicolumn{1}{c}{95\% CI (10)} & \multicolumn{1}{c}{95\% CI (20)} \\
\midrule
\midrule
\rowcolor[rgb]{ .949, .949, .949} Direct & (20\%, 20\%) & -0.0884 & (-0.195, 0.018) & (-0.276, 0.100) & 0.0077 & (-0.083, 0.099) & (-0.191, 0.206) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (30\%, 30\%) & -0.1322 & (-0.332, 0.067) & (-0.340, 0.075) & -0.0874 & (-0.189, 0.014) & (-0.274, 0.099) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (40\%, 40\%) & -0.1772 & (-0.478, 0.123) & (-0.391, 0.036) & -0.1588 & (-0.403, 0.086) & (-0.385, 0.067) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (50\%, 50\%) & -0.2169 & (-0.610, 0.176) & (-0.440, 0.006) & -0.2083 & (-0.561, 0.145) & (-0.479, 0.062) \\
Indirect & (30\%, 20\%) & 0.0055 & (-0.016, 0.027) & (-0.018, 0.029) & 0.0019 & (-0.012, 0.016) & (-0.045, 0.048) \\
Indirect & (40\%, 20\%) & -0.0063 & (-0.038, 0.026) & (-0.054, 0.041) & -0.0089 & (-0.022, 0.005) & (-0.095, 0.077) \\
Indirect & (50\%, 20\%) & -0.0266 & (-0.072, 0.019) & (-0.106, 0.053) & -0.0242 & (-0.042,-0.007) & (-0.152, 0.104) \\
Indirect & (40\%, 30\%) & -0.0118 & (-0.028, 0.004) & (-0.039, 0.015) & -0.0108 & (-0.018,-0.003) & (-0.053, 0.031) \\
Indirect & (50\%, 40\%) & -0.0203 & (-0.042, 0.002) & (-0.059,0.018) & -0.0153 & (-0.027,-0.003) & (-0.062, 0.031) \\
Indirect & (50\%, 30\%) & -0.0322 & (-0.069, 0.005) & (-0.096, 0.032) & -0.0261 & (-0.045,-0.007) & (-0.113, 0.061) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (30\%, 20\%) & -0.1267 & (-0.308, 0.055) & (-0.347, 0.093) & -0.0854 & (-0.176, 0.005) & (-0.284, 0.113) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (40\%, 20\%) & -0.1836 & (-0.464, 0.096) & (-0.406, 0.039) & -0.1676 & (-0.404, 0.069) & (-0.377, 0.041) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 20\%) & -0.2435 & (-0.623, 0.136) & (-0.454,-0.033) & -0.2325 & (-0.586, 0.121) & (-0.439,-0.026) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (40\%, 30\%) & -0.1891 & (-0.488, 0.110) & (-0.401, 0.023) & -0.1696 & (-0.418, 0.079) & (-0.378, 0.039) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 40\%) & -0.2372 & (-0.637, 0.163) & (-0.443,-0.031) & -0.2236 & (-0.585, 0.138) & (-0.459, 0.012) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 30\%) & -0.2491 & (-0.647, 0.149) & (-0.452,-0.047) & -0.2344 & (-0.600, 0.131) & (-0.448,-0.021) \\
Overall & (30\%, 20\%) & -0.0165 & (-0.038, 0.005) & (-0.057, 0.024) & -0.0258 & (-0.060, 0.008) & (-0.071, 0.020) \\
Overall & (40\%, 20\%) & -0.0595 & (-0.140, 0.021) & (-0.125, 0.006) & -0.0739 & (-0.180, 0.032) & (-0.147,-0.001) \\
Overall & (50\%, 20\%) & -0.1174 & (-0.282, 0.047) & (-0.197,-0.038) & -0.1299 & (-0.323, 0.064) & (-0.217,-0.043) \\
Overall & (40\%, 30\%) & -0.0431 & (-0.103,0.017) & (-0.070,-0.016) & -0.0481 & (-0.120, 0.024) & (-0.081,-0.016) \\
Overall & (50\%, 40\%) & -0.0579 & (-0.142, 0.027) & (-0.083,-0.033) & -0.0560 & (-0.143, 0.032) & (-0.083,-0.029) \\
Overall & (50\%, 30\%) & -0.1009 & (-0.245, 0.043) & (-0.149,-0.053) & -0.1041 & (-0.264, 0.056) & (-0.159,-0.049) \\
\bottomrule
\end{tabular}
\label{tab:TRIP_noDate} \end{table}
\begin{table}[htbp]
\centering
\caption{Estimated risk differences and 95\% confidence intervals (CIs) of the effects of community alerts on HIV risk behavior at 6 months in TRIP adjusted for HIV status, the calendar date at first interview, education (primary school, high school, and post high school), and employment status (employed, unemployed/looking for a work, can’t work because of health reason, and other).}
\begin{tabular}{lc|rllrll}
\toprule
\multicolumn{1}{c}{Effects} & Coverage & \multicolumn{3}{c}{IPW$_1$} & \multicolumn{3}{c}{IPW$_2$} \\
& $(\alpha, \alpha')$ & \multicolumn{1}{c}{RD} & \multicolumn{1}{c}{95\% CI (10)} & \multicolumn{1}{c}{95\% CI (20)} & \multicolumn{1}{c}{RD} & \multicolumn{1}{c}{95\% CI (10)} & \multicolumn{1}{c}{95\% CI (20)} \\
\midrule
\midrule
\rowcolor[rgb]{ .949, .949, .949} Direct & (20\%, 20\%) & -0.0691 & (-0.137,-0.001) & (-0.266, 0.128) & -0.0073 & (-0.069, 0.054) & (-0.198, 0.183) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (30\%, 30\%) & -0.1053 & (-0.247, 0.036) & (-0.329, 0.118) & -0.0948 & (-0.215, 0.025) & (-0.275, 0.085) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (40\%, 40\%) & -0.1490 & (-0.392, 0.094) & (-0.372, 0.074) & -0.1606 & (-0.410, 0.089) & (-0.371, 0.050) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (50\%, 50\%) & -0.1919 & (-0.538, 0.155) & (-0.409, 0.025) & -0.2056 & (-0.552, 0.141) & (-0.449, 0.038) \\
Indirect & (30\%, 20\%) & 0.0033 & (-0.016, 0.022) & (-0.017, 0.024) & -0.0020 & (-0.007, 0.003) & (-0.043, 0.039) \\
Indirect & (40\%, 20\%) & -0.0119 & (-0.040, 0.016) & (-0.049, 0.025) & -0.0175 & (-0.033,-0.002) & (-0.088, 0.053) \\
Indirect & (50\%, 20\%) & -0.0357 & (-0.083, 0.012) & (-0.093, 0.021) & -0.0368 & (-0.072,-0.002) & (-0.138, 0.064) \\
Indirect & (40\%, 30\%) & -0.0153 & (-0.033, 0.002) & (-0.035, 0.004) & -0.0155 & (-0.032, 0.001) & (-0.048, 0.017) \\
Indirect & (50\%, 40\%) & -0.0237 & (-0.050, 0.003) & (-0.051, 0.003) & -0.0194 & (-0.040, 0.001) & (-0.055, 0.016) \\
Indirect & (50\%, 30\%) & -0.0390 & (-0.082, 0.004) & (-0.084, 0.006) & -0.0348 & (-0.072, 0.002) & (-0.102, 0.032) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (30\%, 20\%) & -0.1020 & (-0.230, 0.026) & (-0.336, 0.132) & -0.0969 & (-0.214, 0.020) & (-0.290, 0.097) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (40\%, 20\%) & -0.1610 & (-0.393, 0.071) & (-0.395, 0.073) & -0.1781 & (-0.440, 0.084) & (-0.385, 0.029) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 20\%) & -0.2276 & (-0.574, 0.119) & (-0.445,-0.010) & -0.2425 & (-0.621, 0.136) & (-0.450,-0.035) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (40\%, 30\%) & -0.1643 & (-0.411, 0.083) & (-0.389, 0.060) & -0.1761 & (-0.441, 0.089) & (-0.379, 0.027) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 40\%) & -0.2156 & (-0.573, 0.142) & (-0.425,-0.007) & -0.2250 & (-0.590, 0.140) & (-0.446,-0.004) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 30\%) & -0.2309 & (-0.593, 0.131) & (-0.441,-0.021) & -0.2405 & (-0.622, 0.141) & (-0.451,-0.030) \\
Overall & (30\%, 20\%) & -0.0145 & (-0.032, 0.003) & (-0.055, 0.026) & -0.0290 & (-0.071, 0.013) & (-0.074, 0.016) \\
Overall & (40\%, 20\%) & -0.0577 & (-0.135, 0.019) & (-0.124, 0.008) & -0.0802 & (-0.202, 0.042) & (-0.155,-0.005) \\
Overall & (50\%, 20\%) & -0.1178 & (-0.283, 0.048) & (-0.199,-0.037) & -0.1382 & (-0.353, 0.076) & (-0.231,-0.045) \\
Overall & (40\%, 30\%) & -0.0433 & (-0.104, 0.017) & (-0.071,-0.015) & -0.0512 & (-0.131, 0.029) & (-0.087,-0.016) \\
Overall & (50\%, 40\%) & -0.0600 & (-0.149, 0.029) & (-0.086,-0.034) & -0.0580 & (-0.150, 0.035) & (-0.088,-0.028) \\
Overall & (50\%, 30\%) & -0.1033 & (-0.253, 0.046) & (-0.154,-0.053) & -0.1092 & (-0.282, 0.064) & (-0.171,-0.048) \\
\bottomrule
\end{tabular}
\label{tab:TRIP_noShare} \end{table}
\begin{table}[htbp]
\centering
\caption{Estimated risk differences and 95\% confidence intervals (CIs) of the effects of community alerts on HIV risk behavior at 6 months in TRIP adjusted for HIV status, shared drug equipment (e.g. syringe) in last six months, the calendar date at first interview, and employment status (employed, unemployed/looking for a work, can’t work because of health reason, and other).}
\begin{tabular}{lc|rllrll}
\toprule
\multicolumn{1}{c}{Effects} & Coverage & \multicolumn{3}{c}{IPW$_1$} & \multicolumn{3}{c}{IPW$_2$} \\
& $(\alpha, \alpha')$ & \multicolumn{1}{c}{RD} & \multicolumn{1}{c}{95\% CI (10)} & \multicolumn{1}{c}{95\% CI (20)} & \multicolumn{1}{c}{RD} & \multicolumn{1}{c}{95\% CI (10)} & \multicolumn{1}{c}{95\% CI (20)} \\
\midrule
\midrule
\rowcolor[rgb]{ .949, .949, .949} Direct & (20\%, 20\%) & -0.0621 & (-0.130, 0.006) & (-0.341, 0.216) & 0.0073 & (-0.082, 0.096) & (-0.189, 0.203) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (30\%, 30\%) & -0.0933 & (-0.206, 0.019) & (-0.411, 0.225) & -0.0820 & (-0.171, 0.007) & (-0.276, 0.112) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (40\%, 40\%) & -0.1338 & (-0.325, 0.057) & (-0.449, 0.181) & -0.1440 & (-0.353, 0.065) & (-0.363, 0.075) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (50\%, 50\%) & -0.1738 & (-0.453, 0.105) & (-0.473, 0.125) & -0.1838 & (-0.477, 0.110) & (-0.426, 0.058) \\
Indirect & (30\%, 20\%) & 0.0055 & (-0.014, 0.025) & (-0.029, 0.040) & -0.0019 & (-0.008, 0.004) & (-0.046, 0.042) \\
Indirect & (40\%, 20\%) & -0.0094 & (-0.028, 0.009) & (-0.074, 0.056) & -0.0220 & (-0.049, 0.005) & (-0.097, 0.053) \\
Indirect & (50\%, 20\%) & -0.0357 & (-0.076, 0.004) & (-0.130, 0.059) & -0.0483 & (-0.109, 0.012) & (-0.152, 0.056) \\
Indirect & (40\%, 30\%) & -0.0149 & (-0.031, 0.002) & (-0.048, 0.018) & -0.0200 & (-0.047, 0.007) & (-0.054, 0.014) \\
Indirect & (50\%, 40\%) & -0.0263 & (-0.060, 0.007) & (-0.062, 0.009) & -0.0264 & (-0.061, 0.008) & (-0.062, 0.009) \\
Indirect & (50\%, 30\%) & -0.0412 & (-0.091, 0.008) & (-0.108, 0.026) & -0.0464 & (-0.108, 0.015) & (-0.114, 0.021) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (30\%, 20\%) & -0.0878 & (-0.192, 0.016) & (-0.419, 0.243) & -0.0839 & (-0.172, 0.004) & (-0.289, 0.121) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (40\%, 20\%) & -0.1432 & (-0.333, 0.047) & (-0.464, 0.177) & -0.1660 & (-0.400, 0.068) & (-0.380, 0.048) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 20\%) & -0.2095 & (-0.512, 0.093) & (-0.489, 0.070) & -0.2321 & (-0.585, 0.121) & (-0.440,-0.024) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (40\%, 30\%) & -0.1487 & (-0.352, 0.055) & (-0.457, 0.160) & -0.1640 & (-0.399, 0.071) & (-0.375, 0.047) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 40\%) & -0.2001 & (-0.506, 0.106) & (-0.479, 0.079) & -0.2101 & (-0.538, 0.118) & (-0.432, 0.012) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 30\%) & -0.2150 & (-0.533, 0.103) & (-0.485, 0.055) & -0.2302 & (-0.585, 0.124) & (-0.442,-0.019) \\
Overall & (30\%, 20\%) & -0.0101 & (-0.026, 0.006) & (-0.069, 0.049) & -0.0280 & (-0.069, 0.013) & (-0.075, 0.019) \\
Overall & (40\%, 20\%) & -0.0505 & (-0.117,0 .016) & (-0.140, 0.039) & -0.0810 & (-0.206, 0.044) & (-0.160,-0.002) \\
Overall & (50\%, 20\%) & -0.1102 & (-0.268, 0.048) & (-0.207,-0.013) & -0.1417 & (-0.364, 0.081) & (-0.239,-0.044) \\
Overall & (40\%, 30\%) & -0.0404 & (-0.098, 0.018) & (-0.073,-0.008) & -0.0530 & (-0.137, 0.031) & (-0.091,-0.016) \\
Overall & (50\%, 40\%) & -0.0597 & (-0.153, 0.033) & (-0.084,-0.035) & -0.0606 & (-0.159, 0.037) & (-0.092,-0.029) \\
Overall & (50\%, 30\%) & -0.1001 & (-0.251, 0.051) & (-0.149,-0.051) & -0.1137 & (-0.296, 0.068) & (-0.179,-0.049) \\
\bottomrule
\end{tabular}
\label{tab:TRIP_noEdu} \end{table}
\begin{table}[htbp]
\centering
\caption{Estimated risk differences and 95\% confidence intervals (CIs) of the effects of community alerts on HIV risk behavior at 6 months in TRIP adjusted for full set of confounding variables, HIV status, shared drug equipment (e.g. syringe) in last six months, the calendar date at first interview, and education (primary school, high school, and post high school).}
\begin{tabular}{lc|rllrll}
\toprule
\multicolumn{1}{c}{Effects} & Coverage & \multicolumn{3}{c}{IPW$_1$} & \multicolumn{3}{c}{IPW$_2$} \\
& $(\alpha, \alpha')$ & \multicolumn{1}{c}{RD} & \multicolumn{1}{c}{95\% CI (10)} & \multicolumn{1}{c}{95\% CI (20)} & \multicolumn{1}{c}{RD} & \multicolumn{1}{c}{95\% CI (10)} & \multicolumn{1}{c}{95\% CI (20)} \\
\midrule
\midrule
\rowcolor[rgb]{ .949, .949, .949} Direct & (20\%, 20\%) & -0.0653 & (-0.151, 0.020) & (-0.366, 0.235) & -0.0079 & (-0.076, 0.060) & (-0.210, 0.194) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (30\%, 30\%) & -0.0983 & (-0.250, 0.053) & (-0.458, 0.262) & -0.1122 & (-0.254, 0.030) & (-0.304, 0.080) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (40\%, 40\%) & -0.1397 & (-0.381, 0.102) & (-0.492, 0.213) & -0.1938 & (-0.498, 0.110) & (-0.437, 0.050) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (50\%, 50\%) & -0.1795 & (-0.512, 0.153) & (-0.493, 0.135) & -0.2500 & (-0.675, 0.175) & (-0.549, 0.049) \\
Indirect & (30\%, 20\%) & 0.0059 & (-0.017, 0.029) & (-0.024, 0.036) & 0.0122 & (-0.019, 0.043) & (-0.048, 0.072) \\
Indirect & (40\%, 20\%) & -0.0078 & (-0.036, 0.020) & (-0.053, 0.037) & 0.0113 & (-0.033, 0.056) & (-0.101, 0.124) \\
Indirect & (50\%, 20\%) & -0.0331 & (-0.073, 0.007) & (-0.096, 0.029) & 0.0028 & (-0.048, 0.053) & (-0.159, 0.165) \\
Indirect & (40\%, 30\%) & -0.0138 & (-0.028, 0.001) & (-0.035, 0.008) & -0.0008 & (-0.015, 0.014) & (-0.055, 0.053) \\
Indirect & (50\%, 40\%) & -0.0253 & (-0.053, 0.002) & (-0.055, 0.004) & -0.0085 & (-0.018, 0.001) & (-0.062, 0.045) \\
Indirect & (50\%, 30\%) & -0.0391 & (-0.079, 0.001) & (-0.088, 0.010) & -0.0094 & (-0.031, 0.013) & (-0.115, 0.097) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (30\%, 20\%) & -0.0924 & (-0.228, 0.043) & (-0.468, 0.283) & -0.1000 & (-0.213, 0.013) & (-0.300, 0.100) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (40\%, 20\%) & -0.1475 & (-0.371, 0.076) & (-0.514, 0.219) & -0.1825 & (-0.443, 0.078) & (-0.390, 0.025) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 20\%) & -0.2126 & (-0.540, 0.115) & (-0.525, 0.100) & -0.2472 & (-0.625, 0.131) & (-0.451,-0.044) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (40\%, 30\%) & -0.1535 & (-0.396, 0.089) & (-0.505, 0.198) & -0.1946 & (-0.486, 0.096) & (-0.409, 0.020) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 40\%) & -0.2048 & (-0.551, 0.141) & (-0.507, 0.098) & -0.2585 & (-0.680, 0.163) & (-0.515,-0.002) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 30\%) & -0.2185 & (-0.566, 0.129) & (-0.519, 0.082) & -0.2593 & (-0.667, 0.149) & (-0.481,-0.038) \\
Overall & (30\%, 20\%) & -0.0105 & (-0.026, 0.005) & (-0.077, 0.056) & -0.0199 & (-0.044, 0.004) & (-0.068, 0.029) \\
Overall & (40\%, 20\%) & -0.0506 & (-0.116, 0.014) & (-0.150, 0.049) & -0.0646 & (-0.154, 0.025) & (-0.139, 0.009) \\
Overall & (50\%, 20\%) & -0.1098 & (-0.260, 0.040) & (-0.214,-0.005) & -0.1206 & (-0.297, 0.056) & (-0.205,-0.036) \\
Overall & (40\%, 30\%) & -0.0401 & (-0.095, 0.014) & (-0.076,-0.005) & -0.0447 & (-0.111, 0.022) & (-0.075,-0.014) \\
Overall & (50\%, 40\%) & -0.0592 & (-0.146, 0.027) & (-0.087,-0.031) & -0.0560 & (-0.143, 0.031) & (-0.082,-0.030) \\
Overall & (50\%, 30\%) & -0.0993 & (-0.240, 0.041) & (-0.152,-0.047) & -0.1007 & (-0.254, 0.053) & (-0.151,-0.050) \\
\bottomrule
\end{tabular}
\label{tab:TRIP_noemploy} \end{table}
\begin{table}[htbp]
\centering
\caption{Estimated risk differences and 95\% confidence intervals (CIs) of the effects of community alerts on HIV risk behavior at 6 months in TRIP not adjusted for any covariates.}
\begin{tabular}{lc|rllrll}
\toprule
\multicolumn{1}{c}{Effects} & Coverage & \multicolumn{3}{c}{IPW$_1$} & \multicolumn{3}{c}{IPW$_2$} \\
& $(\alpha, \alpha')$ & \multicolumn{1}{c}{RD} & \multicolumn{1}{c}{95\% CI (10)} & \multicolumn{1}{c}{95\% CI (20)} & \multicolumn{1}{c}{RD} & \multicolumn{1}{c}{95\% CI (10)} & \multicolumn{1}{c}{95\% CI (20)} \\
\midrule
\midrule
\rowcolor[rgb]{ .949, .949, .949} Direct & (20\%, 20\%) & -0.0802 & (-0.154,-0.006) & (-0.279, 0.119) & 0.0063 & (-0.102, 0.114) & (-0.218, 0.231) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (30\%, 30\%) & -0.1299 & (-0.276, 0.017) & (-0.385, 0.126) & -0.0748 & (-0.145,-0.004) & (-0.319, 0.169) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (40\%, 40\%) & -0.1806 & (-0.427, 0.066) & (-0.475, 0.113) & -0.1492 & (-0.353, 0.055) & (-0.433, 0.135) \\
\rowcolor[rgb]{ .949, .949, .949} Direct & (50\%, 50\%) & -0.2254 & (-0.573, 0.123) & (-0.549, 0.098) & -0.2108 & (-0.542, 0.121) & (-0.540, 0.118) \\
Indirect & (30\%, 20\%) & 0.0055 & (-0.010, 0.022) & (-0.021, 0.032) & 0.0060 & (-0.010, 0.022) & (-0.047, 0.059) \\
Indirect & (40\%, 20\%) & -0.0057 & (-0.024, 0.013) & (-0.068, 0.057) & 0.0005 & (-0.019, 0.020) & (-0.102, 0.103) \\
Indirect & (50\%, 20\%) & -0.0241 & (-0.056, 0.008) & (-0.140, 0.092) & -0.0096 & (-0.033, 0.013) & (-0.165, 0.146) \\
Indirect & (40\%, 30\%) & -0.0112 & (-0.024, 0.002) & (-0.050, 0.027) & -0.0055 & (-0.012, 0.001) & (-0.057, 0.046) \\
Indirect & (50\%, 40\%) & -0.0184 & (-0.039, 0.002) & (-0.075, 0.038) & -0.0101 & (-0.019,-0.001) & (-0.068, 0.048) \\
Indirect & (50\%, 30\%) & -0.0296 & (-0.062, 0.003) & (-0.124, 0.065) & -0.0156 & (-0.030,-0.001) & (-0.123, 0.092) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (30\%, 20\%) & -0.1244 & (-0.261, 0.012) & (-0.377, 0.128) & -0.0688 & (-0.128,-0.009) & (-0.318, 0.181) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (40\%, 20\%) & -0.1863 & (-0.429, 0.057) & (-0.451, 0.078) & -0.1488 & (-0.337, 0.040) & (-0.402, 0.105) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 20\%) & -0.2495 & (-0.607, 0.109) & (-0.498,-0.001) & -0.2205 & (-0.538, 0.097) & (-0.457, 0.016) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (40\%, 30\%) & -0.1918 & (-0.447, 0.064) & (-0.461, 0.078) & -0.1548 & (-0.358, 0.048) & (-0.412, 0.102) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 40\%) & -0.2438 & (-0.607, 0.119) & (-0.522, 0.035) & -0.2209 & (-0.554, 0.112) & (-0.502, 0.060) \\
\rowcolor[rgb]{ .949, .949, .949} Total & (50\%, 30\%) & -0.2550 & (-0.626, 0.116) & (-0.509,-0.0005) & -0.2265 & (-0.558, 0.106) & (-0.475, 0.022) \\
Overall & (30\%, 20\%) & -0.0174 & (-0.039, 0.004) & (-0.055, 0.020) & -0.0177 & (-0.039, 0.004) & (-0.069, 0.033) \\
Overall & (40\%, 20\%) & -0.0619 & (-0.147, 0.023) & (-0.123,-0.001) & -0.0605 & (-0.144, 0.023) & (-0.137, 0.016) \\
Overall & (50\%, 20\%) & -0.1207 & (-0.296, 0.054) & (-0.196,-0.045) & -0.1163 & (-0.286, 0.054) & (-0.200,-0.033) \\
Overall & (40\%, 30\%) & -0.0445 & (-0.109, 0.020) & (-0.070,-0.019) & -0.0428 & (-0.106, 0.020) & (-0.072,-0.014) \\
Overall & (50\%, 40\%) & -0.0588 & (-0.150, 0.032) & (-0.086,-0.031) & -0.0558 & (-0.142, 0.030) & (-0.082,-0.029) \\
Overall & (50\%, 30\%) & -0.1033 & (-0.259, 0.052) & (-0.152,-0.055) & -0.0986 & (-0.248, 0.051) & (-0.146,-0.051) \\
\bottomrule
\end{tabular}
\label{tab:TRIP_noadj} \end{table}
\end{document} | arXiv |
Earth, Planets and Space
Frontier letter
Q P structure of the accretionary wedge in the Kumano Basin, Nankai Trough, Japan, revealed by long-offset walk-away VSP
Ryota Hino1,
Takeshi Tsuji2,
Nathan L Bangs3,
Yoshinori Sanada4,
Jin-Oh Park5,
Roland von Huene6,
Gregory F Moore7,
Eiichiro Araki4 &
Masataka Kinoshita4
Earth, Planets and Space volume 67, Article number: 7 (2015) Cite this article
We determined the seismic attenuation structure of the Kumano Basin, a forearc basin in the central part of the Nankai subduction zone. Despite its importance for understanding the physical condition of the Earth's interior and seismic wave propagation processes, the attenuation factor Q has been poorly estimated in the crustal layers of the offshore areas of Nankai because severe attenuation occurring in the seafloor sediments prevents the reliable estimation of Q from conventional active source seismic surveys. In the present study, we derive Q values from the diminishing rate of the high-frequency contents of seismic energy during propagation through sub-seafloor layers. The records of vertical seismic profiling acquired at approximately 1,000 m below the seafloor, which have fewer effects from shallow attenuation, enabled us to elucidate depth variation of Q of P waves (Q P), the attenuation factor of P waves, down to approximately 8 km below the seafloor. Assuming that the frequency dependence of Q is small and using a previously obtained P-wave velocity structure model for the basin, we inverted the fall-off rate of the spectral ratios at various shot-receiver distances to obtain Q P in the three sub-bottom layers. The Q P values for the upper two layers with P-wave velocity (V P) < 2.7 km/s are 34 and 57. These values are almost identical to those obtained in the North Atlantic, suggesting the broad consistency of Q P within seafloor sediment. The basement layer (V P approximately 4 km/s) has a much higher Q P value of 349, which is comparable to the value estimated for crustal layers exposed onshore. This Q P value is higher than the value previously assumed in a simulation of strong ground motion associated with megathrust earthquakes along the Nankai margin. We interpret that the high Q P, low seismic attenuation in the basement layer reflects tectonic stability of the inner wedge of the accretionary margin. Our first estimates of Q P in the present study provide a strong basis for future studies of seismic structure and strong ground motion prediction.
Correspondence/findings
The Kumano Basin is a forearc basin along the Nankai subduction zone (Figure 1a). Beneath the basin, destructive earthquakes occur along the plate boundary (Ando, 1975). The 1944 Tonankai earthquake with a moment magnitude (Mw) of 8.1, which occurred beneath the basin, caused a devastating tsunami. The slip along a well-developed major out-of-sequence fault branching off of the décollement (megasplay fault) is considered to be responsible for the generation of the tsunami (e.g., Fukao et al. Fukao 1979; Park et al. 2002; Cummins and Kaneda 2000; Moore et al. 2007). Sporadic, very low-frequency earthquakes within the accretionary wedge reported by Ito and Obara (2006) indicate that the out-of-sequence fault is also active during the interseismic period of large megathrust events. The characteristics of the megasplay fault are important not only for understanding earthquake and tsunami generation but also for understanding the developmental history of the accretionary complex along the trough (e.g., Moore et al. 2007; Kimura et al. 2007; Strasser et al. 2009). Intensive seismic surveys have been made to characterize the sub-seafloor structure around the fault, such as the three-dimensional (3-D) geometry of the fault, the geometry of other active faults in the outer wedge or of the megathrust (e.g. Moore et al. 2007), and the detailed P-wave velocity (V P) distribution (Nakanishi et al. 2008; Park et al. 2010; Kamei et al. 2012). These detailed seismic structure models have contributed to a better understanding of the evolutionary processes of the accretionary prism and the fluid distribution, which plays an important role in controlling the frictional properties of the seismogenic faults (e.g., Bangs et al. 2009).
Map of the study area and seismic section. (a) The downhole vertical seismic array was deployed at site C0009. The shooting line of the VSP is shown in red. The star and contours denote the epicenter and the coseismic slip distribution of the 1944 Tonankai earthquake (Kikuchi et al. 2003). (b) Seismic profile across site C0009. Lithologic units defined by the Expedition 319 Scientists (2010) are also shown. Orange bar indicates the location of the vertical seismic array.
The seismic quality factor Q is a measure of the degree of attenuation of seismic wave energy and provides us with important physical information about the Earth's interior. Attenuation, which is inversely proportional to Q, is sensitive to factors such as the chemical composition, porosity, and permeability of the rock material, and it can be used to infer the distribution of pore fluid (e.g., Johnston et al. 1979; Toksöz et al. 1979; Winkler et al. 1979, Winkler and Nur 1982). A number of reports have indicated that active fault zones are characterized by high attenuation (low Q values) due mainly to high fracture density and/or the presence of abundant pore water (e.g., Rietbrock 2001; Bennington et al. 2008; Wang et al. 2012).
In megathrust seismogenic zones, which are mostly located in offshore areas, few studies of seismic attenuation structure have been conducted. Christeson et al. (2000) and Zhu et al. (2010) estimated Q of P waves (Q P) of the upper crust of the Costa Rican forearc and attributed the spatial variation of Q P to changes in the lithology or fracture density of the crust. The seismic attenuation structure of the offshore regions of Nankai is poorly known, with the exception of the very shallow portion of seafloor sediments. Matsushima (2005, 2006) studied seismic attenuation structure in methane-hydrate-bearing sediments, located at approximately 200 m depth beneath the seafloor, based on a downhole seismic experiment, but specific Q values were not determined conclusively. A thick cover of seafloor sediments with considerably low velocity and large attenuation (Ayres and Theilen 2001) makes it difficult for Q value determination through active seismic exploration, the most powerful tool for deriving crustal structure in the marine environment, to evaluate seismic attenuation precisely.
In 2009, we conducted a vertical seismic profile (VSP) using a vertical array of seismic sensors in a deep sub-seafloor borehole and a large volume airgun array during International Ocean Drilling Program (IODP) Expedition 319 in the Kumano Basin (Saffer et al. 2009). The seismic sensors were set at 908 to 1,137 mbsf (2,960 to 3,190 m below the sea surface) in the borehole drilled at site C0009 (Figure 1, Expedition 319 Scientists 2010). It is expected that reliable Q values of the sub-seafloor formation can be derived from an analysis of seismic waveform data, which are less affected by the strong attenuation of shallow seafloor sediments than those obtained by conventional ocean bottom or sea surface instruments. In this paper, we estimate the Q P structure of the Kumano Basin based on a spectral analysis of seismic data collected by the VSP and show for the first time the depth variation of seismic attenuation beneath the Nankai forearc.
Data and method for estimating Q P
The VSP experiment consisted of two sub-experiments with different objectives. One was a circular shooting VSP (CVSP) to evaluate seismic anisotropy and stress orientation in the sub-basin formation. The results of this experiment were reported by Tsuji et al. (2011). The other sub-experiment was a walk-away VSP (WVSP), in which a shooting vessel travels in a straight line, traversing the VSP site (Figure 1), to study the variation of seismic waveforms as a function of offset, i.e., shot-receiver distance. The Q P estimation presented here used the data obtained by the WVSP. The same data acquisition system was used in the CVSP and the WVSP, and its detailed description was provided by Tsuji et al. (2011). The location of the WVSP shot line (Figure 1) was set so that it overlapped with the survey line of a previous wide-angle seismic survey that used ocean bottom seismographs (Nakanishi et al. 2008). An airgun array of 128 L total volume was shot at a 60-m interval along the line.
We analyzed waveform data from the shots on the southeastern (trenchward) side of C0009 because the airgun array became unstable during operation along the northwestern part of the WVSP line. Figure 2 shows an example of common receiver gathers obtained by the WVSP. The records obtained by the vertical component sensor deployed at the deepest level of the seismic array, composed of 16 three-component sensors, are displayed with the reduction velocity of 4 km/s. The first arrivals were categorized into three groups, G1, G2, and G3, according to the difference in their apparent velocities. These groups correspond to the direct waves propagating downward through the basin sediment layers (L1 and L2), the diving waves traveling through the lower part of the sediment (L2), and the refracted waves in the high V P basement layer (L3), respectively (Figure 3). Later arrivals were also evident in the offset range from 12 to 18 km (megasplay fault, MSF) and are interpreted as wide-angle reflection arrivals from the megasplay fault plane. The phase interpretations given here are consistent with travel time calculations using the V P model derived from the previous seismic survey (Figure 3; Nakanishi et al. 2008).
Common shot gather obtained by WVSP. A reduction velocity of 4 km/s is used, and no filter is applied. Thin solid black lines mark bins for trace stacking. Trace groups G1, G2, and G3 correspond to down-going direct waves, diving waves through the lower part of the L2 layer, and refracted waves from L3. Clear later arrivals are interpreted as reflection arrivals from the megasplay fault (MSF).
V P structure model around VSP site C0009. Vertical cross section showing the V P distribution. Ray paths from shot points to the receiver at the bottom of the array are shown. MSF, megasplay fault.
Christeson et al. (2000) and Zhu et al. (2010) succeeded in estimating Q P by modeling the variation of signal amplitude with offset in their wide-angle seismic data. Amplitude variation, however, is dependent not only on attenuation but also on factors such as geometrical spreading and reflection/transmission across velocity discontinuities. Thus, it may be difficult to obtain reliable Q estimates from the amplitude data unless the spatial heterogeneity of the seismic velocity structure is small.
We estimated Q P by applying a spectral ratio method (e.g., Abercrombie 1997) to the vertical component seismograms with no discernible S waves. We modified the original method to process the WVSP data, in which we have to calculate spectral ratios of signals propagated along different ray paths (Additional file 1: Figure S1). A spectrum of a seismogram can be expressed as a convolution of 1) the instrumental response, 2) the source spectrum, 3) the site amplification factor, and 4) a factor associated with wave propagation through the media. We assume that the first three factors can be canceled by taking a ratio of the spectra of seismograms at a common receiving point. We explain the validity of these assumptions in Additional file 2: Supplemental material 1, Additional file 3: Figure S2 and Additional file 4: Figure S3.
Let A(f, x) and A(f, x 0) be the amplitude spectra of the seismic waveforms from shots at the two different offsets x and x 0. Then, the logarithm of the spectral ratio is expressed as follows:
$$ \begin{array}{l} \log \left[\frac{A\left(f,x\right)}{A\left(f,{x}_0\right)}\right]= \log \left[\frac{G\left(x,f\right)}{G\left({x}_0,f\right)}\right]-\pi f\left[{t}^{*}\left(f,x\right)-{t}^{*}\left(f,{x}_0\right)\right]\ \\ {}=C\left(x,{x}_0\right)-\pi f\Delta {t}^{*}\left(f,x,{x}_0\right)\end{array}, $$
where G(x, f) and G(x 0, f) are factors defined by the geometrical spreading and reflection/transmission coefficients along the ray paths. Under high-frequency approximation, the factors G are regarded as frequency independent. We consider that this approximation can be applied to our dataset because all the boundaries relevant to the present study have been distinctly defined as sharp interfaces by previous high-resolution seismic profiling (Park et al. 2002; Moore et al. 2007; Bangs et al. 2009). Under the assumption, the frequency dependence of the spectral ratio is characterized by a parameter t*, which is related to the seismic velocity v and the attenuation factor Q P along the ray path as follows:
$$ {t}^{*}={\displaystyle {\int}_{\mathrm{source}}^{\mathrm{receiver}}\frac{ds}{Q_{\mathrm{P}}v}}\kern0.6em =\kern0.1em {\displaystyle {\int}_{\mathrm{source}}^{\mathrm{receiver}}{v}^{-1}{Q_{\mathrm{P}}}^{-1}ds.} $$
Assuming that the spatial distribution of v is known, Q P can be estimated from t*, which is measured from the observed spectral ratio of the seismic survey.
Since the low signal-to-noise (S/N) ratio of the waveforms causes instability in the calculations of the spectral ratio, we used stacked seismic traces for the analysis. The observed common receiver gathers (Figure 2) were binned according to the offset distances, and the traces in each bin were stacked to yield a seismic trace representative of the corresponding bin. In the common receiver stacking, 10 neighboring traces of the G1 and G2 groups and 30 traces in the G3 and MSF groups (Figure 2) were stacked after the correction for the arrival time differences was made. We had to increase the number of stacked traces for G3 and MSF because the signal levels of these arrivals were much smaller than those of G1 and G2. By applying a time shift before trace stacking, the S/N ratio of the trace is effectively improved by the constructive superposition of the target signals and the destructive superposition of later arrival components with different apparent velocities. The later arrival components may be present in the time window for the spectrum calculation but may have different frequency characteristics from the target signals.
After the common receiver stacking, we formed common shot (bin point) gathers composed of 16 traces and stacked them (common shot stacking) to obtain the traces for the spectral analysis. One of the 16 sensors deployed did not work properly (Tsuji et al. 2011), so we excluded these data from the stacking process. The pre-stack time shifting based on the picked arrival times was also applied in this stacking. The stacked seismic traces are displayed in Figure 4. It is evident that the predominant period of seismic traces increases as the offset increases.
Stacked seismic traces. Traces in G2 are exaggerated by a factor of 10, and the G3 and MSF traces are exaggerated by a factor of 40. Red trace indicates the reference waveform (R) used for the calculation of spectral ratios. Vertical line indicates the picked arrival time of the first arrivals from refracted waves and the reflection arrival times from the reflected arrivals from the megasplay fault.
In the calculation of spectral ratio (1), a reference spectrum A(f, x 0) needs to be defined. It is desirable to choose a waveform with the highest quality as the reference, and we used the stacked trace from the ten records obtained in the offset range from 2.8 to 3.4 km, indicated as R in Figure 2 and the red trace in Figure 4. The reference signal corresponds to the down-going direct wave through shallow sedimentary layers. The length of the time window for the spectrum estimation was 0.512 s (256 data points of 2 msec sampled data) after the arrival times, and the spectra were smoothed by applying the Hanning window. The appropriateness of the used window length is briefly explained in Additional file 2: Supplemental material 2 and Additional file 5: Figure S4.
Figure 5 shows the spectral ratios as the functions of frequency from the stacked waveform records of the WVSP. For the first arrival signals of the G1 and G2 groups, it is evident that high-frequency contents decrease rapidly with the offset distance. However, no systematic variation with the offsets can be observed for the spectral ratios of the G3 and MSF groups. In the frequency ranges in which the spectral ratios are larger than the noise level, defined by the spectral ratio between the noise spectrum and the reference spectrum, the logarithms of spectral ratios exhibit almost constant decay slopes, meaning that the Δt* is relatively constant, independent of frequency. From this observation, we assume that Q P is independent of frequency in the following procedure of Q P estimation.
Spectral ratios of stacked seismic traces in the groups G1, G2, G3, and MSF. Colors indicate different offset values. Thick black curves in the G3 and MSF panels show spectral ratios calculated using the traces after stacking all the traces in the wave group. Lines with two arrowheads indicate the frequency ranges for the estimation of Δt*.
By assuming that Δt* is constant, it can be estimated by fitting the observed spectral ratios to Equation 1 with a linear least squares method within the frequency band where the ratio exceeds the noise level (shown by the two-headed arrows in Figure 5). We did not use the data in the frequency range f > 50 Hz, although signal levels are large enough for the traces in the G1 group. In this frequency range, the spectral ratios tend to have smaller fall-off rates than in the lower frequency band, but fluctuations of the ratios are too large to be examined for frequency dependence of Q.
We further assumed that Q P is constant within each layer of the V P model shown in Figure 3 in the Q P estimation. Under these assumptions, the parameter t* can be expressed as
$$ {t}^{*}(x)={\displaystyle \sum_{i=1}^3}{T}_i(x)\;{Q_{\mathrm{P},i}}^{-1}, $$
where i is an index specifying the sub-bottom layers (i = 1, 3 corresponding to L1, L2, and L3, respectively, in Figure 3), Q P,i is the attenuation factor in the ith layer, and
$$ {T}_i(x)={\displaystyle \underset{i}{\int }}\frac{ds}{v} $$
is the travel time required for the corresponding signal observed at x to propagate in the ith layer. T i (x) was calculated by using ray tracing and the two-dimensional V P structure model (Figure 3). Using these relations, Q −1 P,i can be estimated by solving the following observation equations by a least squares method:
$$ \begin{array}{l}\varDelta {t}^{*}(x)={t}^{*}(x)-{t}^{*}\left({x}_0\right)\\ {}\kern3em ={\displaystyle \sum_{i=1}^3{T}_i(x)}\kern0.5em {Q_{\mathrm{P},i}}^{-1}-{\displaystyle \sum_{i=1}^3{T}_i\left({x}_0\right)}{Q_{\mathrm{P},i}}^{-1}\end{array} $$
In the inversion, we applied a nonnegative least squares method (Lawson and Hanson 1995) because Q −1 P,i cannot take negative values. Note that the first and second integrals (summations) of the observation Equation 5 are evaluated along the ray path from the source at x and along that from the source at x 0, respectively.
Figure 6 shows Δt* estimated from the observed spectral ratio as a function of offset distance. The estimated Δt* values are plotted at the midpoint of the span of the bins for the trace stacking (indicated by horizontal error bars). Vertical error bars represent estimation errors taking uncertainties in the spectrum estimation and in the least squares fitting into account. The observed Δt* values for the first arrivals in the offset range up to 10 km show clear monotonic increase with offset, as expected from Figures 4 and 5. Since the spectral ratios of the G3 and MSF groups do not show evidence for offset dependence, as explained above, we stacked all the traces within each group to obtain spectral ratios representative of these groups to make Δt* estimation more reliable.
Δt * as a function of offset (shot-receiver distance). Points with vertical and horizontal bars indicate the estimated Δt* from the spectral ratios shown in Figure 5. Horizontal bars indicate the range of offsets of stacked traces used to estimate the corresponding Δt*, and vertical bars indicate the uncertainties in Δt*. Solid circles indicate the Δt* values calculated by the best-fit Q P values shown in Table 1. Colored circles and squares are expected Δt* of the refracted waves from L3 and the reflected waves from the megasplay fault, respectively. Blue, red, and green symbols are for models with Q P = 100, Q P = 349, and Q P = 2,000 in L3, respectively.
The estimated Q P −1 (and Q P) values obtained by solving Equation 5 using the observed Δt* values are shown in Table 1 with their estimation errors. The Δt* values calculated from the Q P −1 structure are compared with the observed values in Figure 3. Although misfits are larger than the error of Δt* estimation for several data points, the misfits may be caused by small-scale spatial variation of Q P, and our model with three layers with constant Q P well explains the general pattern of offset dependence of the Δt*. Whereas the Q P −1 values in the top two layers, L1 and L2, are well constrained from a number of observed Δt* values, showing clear offset dependence, those in L3 are derived from the two observed Δt* values. One is from the stacked trace of the first arrivals from L3 and the other is from the reflection signals from the megathrust fault. In order to verify how well Q P is constrained from the observations, the offset dependence of Δt* was calculated by assuming three different Q P values in L3: 100, 349 (best-fit model), and 2,000 (Figure 6). In the calculation, Q P values in L1 and L2 were set to those of the best-fit solution.
Table 1 Q P and V P structure beneath Kumano Basin
The Δt* of the refracted waves from L3 and reflected waves from the megasplay fault (the bottom boundary of L3) show very small variation with offset distance, consistent with the observed behavior of the spectral ratio (Figure 5). On the other hand, the model with relatively low Q P in L3 predicts a substantial increase in Δt* with offset, and the expected value exceeds 0.05. These features clearly contradict the observed nature of the refracted waves from L3. Therefore, it is not probable that the Q P value in L3 is significantly lower than approximately 200, which is the lower limit from the estimation error of Q P −1 by the inversion.
It is notable that the Δt* values of the reflected waves from the megathrust fault are systematically smaller than those of the refracted waves from L3, despite their longer path lengths, when Q P in L3 is larger than the optimum value. This difference in Δt* reflects the difference in path lengths in the shallow sedimentary layers (L1 and L2) between the refracted waves and bottom reflections of L3. The total length of ray segments in the low Q P sedimentary layers is longer for the refracted wave than for the reflected wave due to the difference in the incidental angle of the ray paths in the layers (Figure 1b). In models with high Q P in L3, the amount of Δt* is much more dependent on the path length in the low Q P layers because the increase in Δt* due to the high Q P layer is relatively small.
The upper bound of the Q P in L3 is more difficult to assess than the lower bound. The Δt* of the L3 refraction signals are less sensitive to a change in Q P in L3 than those of the reflections from the megasplay fault, as demonstrated in Figure 6. The observed spectral ratios of the reflection arrivals from the megasplay fault are less stable, and the Δt* value estimated from the stacked trace contains larger uncertainty, mostly due to low S/N ratios of the reflection arrivals. This makes it difficult to exclude the model with Q P in L3 higher than the best-fit solution. The fall-off rate of the spectral ratio may not be substantially smaller than those in the offset range <6 km (Figure 5), and we regard that Q P is likely to be less than 2,000. Further discussion on the reliability of our Q P model is given in Additional file 2: Supplemental material 3, Additional file 6: Figure S5, and Additional file 7: Figure S6.
Here, we briefly discuss the effects of the frequency dependence of Q on the results. The frequency dependence is often expressed as Q(f) = Q 0 f α. Since it is difficult to constrain the constant α from our dataset, we tried to estimate Q 0 of the three layers by assuming α = 0.66, moderate frequency dependence, which is taken from the results of Yoshimoto et al. (1998). The estimated Q 0 values are 2.0, 3.3, and 44 for L1, L2, and L3, respectively. At the frequency f = 20 Hz, the center of the frequency band of the present WVSP data analysis, Q P is 14, 23, and 315, values somewhat smaller than the values assuming frequency-independent Q. However, misfit between the theoretical spectral ratio (1) and the observed ones increases by introducing the frequency dependent Q (accordingly, Δt* is frequency dependent), and we prefer the frequency-independent model (Additional file 8: Figure S7).
The estimated Q P values in the L1 and L2 layers are very close to those estimated by Grad et al. (2012) for seafloor sediments in the North Atlantic Ocean. They obtained Q P of 30 to 50 for the topmost layer with V P approximately 1.7 km/s and 40 to 50 for the underlying layer with V P approximately 2.2 km/s by modeling active seismic experiments using ocean bottom seismographs. From a laboratory experiment, Ayres and Thelen (2001) suggested that the attenuation of seismic waves of near-surface marine sediments is likely to be insensitive to changes in sediment physical properties. The similarity of the Q P values between the Kumano Basin and the North Atlantic, which are located in very different environments, supports their suggestion, and the estimated Q P values could be the universal values of marine sediments around the world.
Judging from the V P value, L3 is a more rigid basement than the top two layers. Nakanishi et al. (2008) interpreted the layer as Neogene-Quaternary accretionary prism. The Q P of the layer estimated in the present study is well within the range of the Q P values estimated for the crustal layers in various onshore regions compiled by Yoshimoto et al. (1998) in the frequency range 10 to 30 Hz. Since the Neogene-Quaternary accretionary prism pinches out near the coast, according to Nakanishi et al. (2008), layer L3 may not be identical to any onshore crustal layers in terms of lithology. Nevertheless, the layer has a Q P value as large as crustal layers exposed onshore. We interpret that the high Q P, low seismic attenuation in the basement layer reflects the tectonic stability of the inner wedge of the accretionary margin (Wang and Hu 2006). No major faults penetrating into the basement, which corresponds to the inner wedge, have been imaged beneath the Kumano Basin, whereas extensive development of out-of-sequence thrusts is observed in the outer wedge on the trenchward side of the megathrust fault (e.g., Park et al. 2002). According to detailed studies of the V P distribution in the region (e.g., Bangs et al. 2009; Park et al. 2010; Kamei et al. 2012), localized low-velocity anomalies related to the concentration of pore fluid were not identified in the basement of the basin, except in the hanging wall side of the megasplay fault.
The Q P values of the basement layer in the Kumano Basin are in contrast to those of the Costa Rica forearc. At this margin, the V P of the basement layer beneath the forearc area is approximately 4 km/s, almost equivalent to that of L3 in the Kumano Basin. Zhu et al. (2010) estimated the Q P of the layer as 50 to 150, a relatively low value. From the modeling of the offset dependence of the amplitudes of their airgun OBS data, they concluded that models with Q P > 200 are inconsistent with their data. They interpreted that extensive fracturing of the layer accounts for the low Q P. The difference in the attenuation structure may reflect a difference in the tectonic processes at the Nankai and Costa Rica subduction zones; the former is an accretionary margin whereas the latter is an erosive margin. This speculation should be reinforced by future studies of the attenuation structure in both subduction zones using various approaches, because the comparison of Q values derived by different methods is not straightforward.
The spatial distribution of Q in the crust has been drawing attention from seismologists, and a number of studies employing local earthquake tomography techniques to obtain the 3-D distribution of Q have been conducted. 3-D Q tomography is an effective tool for inferring the underground distribution of fracture density and the roles of fluids in the crust (e.g., Rietbrock 2001; Reyners et al. 2007; Bennington et al. 2008). Matsumoto et al. (2009) reported that a high Q P region corresponds to a large-slip fault patch where a large inland shallow earthquake occurred (the 2005 West off-Fukuoka earthquake, Mw 7.0), suggesting a correlation between the frictional strength along the fault and the seismic attenuation of the host rocks. A better understanding of the 3-D attenuation structure of the forearc region is required to fully characterize various features related to the formation and development of convergent margins and the generation of megathrust earthquakes. Extensive efforts are being made to build cabled seafloor systems for earthquake monitoring in the Nankai forearc (Kaneda 2012). High-quality seismic waveform data acquired by the system will be collected to illuminate the detailed spatial variation of seismic attenuation in the area. The first observation of the depth variation in Q P presented in the present study provides an important foundation for future studies.
The reliable estimation of Q in the forearc region is also important for the prediction of strong ground motion associated with giant megathrust earthquakes. Furumura et al. (2008) showed that long-period shaking with large amplitude develops during propagation in low-velocity layers. The strong and prolonged shaking of the long-period ground motions that could be generated by future great megathrust earthquakes along the Nankai Trough could cause serious damage to modern large-scale construction in the Tokyo region. Furumura et al. (2008) simulated the propagation process of seismic waveforms generated by an intraplate earthquake that occurred along the Kumano Basin (the 2006 SE Off-Kii Peninsula earthquake, Mw 7.4) assuming a realistic underground structure model that included the low-velocity forearc wedge as well as the subducting Philippine Sea plate slab. In the model, the forearc wedge was assumed to have lower Q P (90) than the value we estimated. Although their model reproduced the main features of the strong motion records of the moderate earthquake, models with observed Q values are required for the well-constrained prediction of ground motion. Small differences in simulated waveforms can result in significant differences in predicted ground motion for extraordinarily large megathrust earthquakes.
The Q P structure in the Kumano Basin, which is a forearc basin along the Nankai Trough located in the rupture area of the 1944 Tonankai earthquake (Mw 8.1), was derived from a long-offset walk-away vertical seismic experiment. The Q P values in the shallow basin sedimentary layers and the basement composed of the young accretionary prism were estimated by a spectral ratio method, utilizing the fall-off rate of high-frequency signals in seismic records. By assuming frequency-independent Q, the estimated Q P values were 34 and 57 for the top and second layers, respectively, (<2 km below seafloor) which are composed of unconsolidated sediment with small V P (<2.7 km/s). These values are almost identical to those derived from a wide-angle marine seismic exploration made in the North Atlantic, suggesting that the Q P values of seafloor sediment are likely to be uniform within broad areas of the ocean basins. The basement layer beneath the basin sediment has a much higher Q P value (349), which is comparable to the value estimated for the crustal layers exposed on land. Our result indicates the deeper part of the Nankai accretional complex is less attenuating than previously supposed. We interpret that the high Q P in the basement layer reflects the tectonic stability of the inner wedge of the accretionary margin. Although we could not resolve the spatial variation of seismic attenuation, which can be correlated to the distribution of fractures and fluid and to tectonic processes including megathrust earthquake generation, the first estimations of Q P by the present VSP provide a basis for future seismological studies, including those pertaining to seismic structure in the seismogenic zone and strong ground motion simulation.
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We thank the scientists and crew of IODP Expedition 319, especially the cochief scientists, D. Saffer, L. McNeill, and T. Byrne, and the staff scientists, S. Toczko, N. Eguchi, and K. Takahashi. The VSP data for this research were provided by IODP and IFREE/JAMSTEC. The seismic data processing was supported by SR2020 Corp. We are especially grateful to Dr. M. Karrenbach (SR2020) for the data processing. Dr. Ikuko Wada carefully read the manuscript and improved its scientific clarity. The careful reading of the manuscript and constructive comments by Dr. Ashi (editor) and three anonymous reviewers were very valuable. A part of this study was supported by Grant in Aid 24107701 (KANAME).
International Institute of Disaster Science, Tohoku University, Sendai, 980-8578, Japan
Ryota Hino
International Institute for Carbon-Neutral Energy Research, Kyushu University, Fukuoka, 819-0395, Japan
Takeshi Tsuji
Institute for Geophysics, University of Texas at Austin, Austin, TX, 78758, USA
Nathan L Bangs
Japan Agency for Marine-Earth Science and Technology, Yokosuka, 237-0061, Japan
Yoshinori Sanada, Eiichiro Araki & Masataka Kinoshita
Atmosphere and Ocean Research Institute, University of Tokyo, Kashiwa, 277-8564, Japan
Jin-Oh Park
Geology Department, University of California, Davis (emeritus), CA, 95616, USA
Roland von Huene
Department Geology and Geophysics, University of Hawai'i at Manoa, Honolulu, HI, 86822, USA
Gregory F Moore
Yoshinori Sanada
Eiichiro Araki
Masataka Kinoshita
Correspondence to Ryota Hino.
RH carried out the spectral analysis of seismograms and the Q estimation and drafted the manuscript. TT interpreted the estimated Q in terms of rock physics. YS and EA contributed to the acquisition of the unique VSP data of the present study by developing a new system allowing long-offset shooting. NB, RvH, and MK participated in the data acquisition and processing. JOP and GM contributed the seismic data processing. The VSP was designed and conducted through detailed discussion among all the authors. All authors read and approved the final manuscript.
Schematic view of shot-receiver geometry.
Supplementary materials 1 to 3. Supplementary material 1. Basis of spectral ratio method. Supplementary material 2. Data treatment. Supplementary material 3. Reliability of the Q P model.
Incident angle to the downhole sensor array as a function of offset. Incident angles of the first arrivals are calculated from the vertical slowness. The definition of the incident angle θ is shown in the inset. The rectangle shows the offset range for the amplitude spectra displayed in Additional file 4: Figure S3.
Amplitude spectra of WVSP records with different incident angles. Amplitude spectra are calculated from the seismic records obtained at the offset range of 5.8 to 6.4 km, shot by shot. Arrows show the frequency band for the Δt* estimation.
Spectral ratios calculated with different time window lengths. Blue: spectral ratios calculated from 128 data points (0.256 s length); red: 256 points (0.512 s); and green: 512 points (1.024 s). Lines with two arrowheads indicate the frequency ranges for the estimation of Δt*.
Normalized geometric spreading factors as a function of offset. Geometric spreading factors calculated from the V P model (shown in Figure 3) are represented by a solid circle and those estimated from the observed spectral ratios are shown as crosses with error bars.
Normalized amplitude as a function of offset. Crosses are maximum amplitudes within the time windows for the spectrum analysis, and solid circles are the amplitudes expected from the assumed V P model. Red symbols show amplitudes of refracted waves, and blue symbols show the reflection from the MSF.
Observed and theoretical spectral ratios. Black curves are the observed spectral ratio (same as those shown in Figure 5). Red and blue curves are the theoretical spectral ratios using the best-fit parameters assuming α = 0 (frequency-independent t*) and α = 0.66 (frequency-dependent t*), respectively.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, 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 license, and indicate if changes were made.
Hino, R., Tsuji, T., Bangs, N.L. et al. Q P structure of the accretionary wedge in the Kumano Basin, Nankai Trough, Japan, revealed by long-offset walk-away VSP. Earth Planet Sp 67, 7 (2015). https://doi.org/10.1186/s40623-014-0175-x
DOI: https://doi.org/10.1186/s40623-014-0175-x
Seismic attenuation
Nankai Trough
New Perspective of Subduction Zone Earthquake | CommonCrawl |
Gustav S. Christensen
Gustav Strøm Christensen (1 April 1929 – 9 August 2007) was a Danish-Canadian academic mathematician and engineer. Born in Vesterø on the Danish island Læsø, he first worked as a radio operator in the Danish merchant marine. Later he became a professor of electrical engineering at the University of Alberta in Edmonton, British Columbia for 27 years. In 1957 he won the Engineering Institute of Canada's prize and the Henry Birks Gold Medal in Engineering. He received a B.Sc. in engineering physics from the University of Alberta, Edmonton, in 1958 and a M.A.Sc. from the University of British Columbia, Vancouver in 1960. Later he obtained a Ph.D. in 1966, also in electrical engineering, from the University of British Columbia. Christensen wrote over 140 scientific papers, co-authored four books and four chapters.[3]
Gustav S. Christensen
Christensen in 2004
Born
Gustav Strøm Christensen
(1929-04-01)1 April 1929
Læsø, Denmark
Died9 August 2007(2007-08-09) (aged 78)
Abbotsford, British Columbia, Canada
NationalityDanish, Canadian
Alma materUniversity of Alberta,
University of British Columbia
Known forElectric power system studies,
nonlinear systems, stability theory, and Mathematical optimization
Spouse
Penelope (née Gardner) Christensen[1]
(m. 1969)
ChildrenLynne,[2] Neil
Awards
• Engineering Institute of Canada Prize 1957
• Henry Birks Gold Medal in Engineering
• CIVIC AWARD – BRITISH COLUMBIA
District of Mission: Award for achievement in mathematics, engineering and science. 7 May 2007. Presented by Mayor James Atebe.
Scientific career
FieldsMathematics, Electrical Engineering, Computational Intelligence
InstitutionsUniversity of Alberta
ThesisAspects of Nonlinear System Stability (1966)
Doctoral advisorAvrom Chaim Soudack
Doctoral studentsMohamed E. El-Hawary, Soliman Abdel-hady Soliman
Biography
Early life
Christensen was born 1 April 1929 on a farm on the Danish island of Læsø. The eighth of nine children, Gustav completed 9 years of schooling there in a one-room elementary school and a new junior high school. In 1945 he moved to Copenhagen to take a radio operator apprenticeship. He served with the Danish Merchant Navy for four years and on an American oil tanker for another two as a radio operator.
He emigrated to Canada and in 1969 he married an English immigrant, Penelope Janet Gardner,[1] in Edmonton. They have two children, Lynne [2] and Neil.
Academic life
After moving to Canada, Christensen completed his high school diploma in six months, mainly through correspondence school in Edmonton. He enrolled in engineering physics at the University of Alberta, where he won the top engineering award, the Henry Birks Gold Medal, in 1958 along with several other scholarships. After a summer with the National Research Council[4] in Ottawa, he took his M.A. Sc. in Electrical Engineering at the University of British Columbia (UBC) in 1960 and subsequently obtained practical industrial experience with the BC Energy Board and Chemcell in Edmonton. He disliked the odours associated with chemical engineering left to complete his Ph.D. at UBC. His thesis (1966) was on the stability of non-linear mathematical models of systems.
In July 1966, he returned to University of Alberta as assistant professor and spent 27 years teaching, researching and taking a large load of administrative responsibilities for the Department of Electrical Engineering. His research was mainly concentrated on the optimum economic operation of various types of power systems, minimizing the energy lost in transmission lines while at the same time using, for example, the water available in all the hydro plants in a system to generate the maximum amount of power.[5][6][7][8] This produced the cheapest energy and the best return on investment. Christensen authored over 140 scientific papers, four books and four chapters in books on his research specialities. Christensen retired as full professor in 1993 and moved to Mission, British Columbia.
Electric power industries
Christensen was an active intervener in the National Energy Board (NEB) hearings which ultimately resulted in the rejection of the huge SE2 (Sumas Energy 2) coal-fired plant, one that would have emitted unacceptable pollution into the Fraser Valley funnel.
Final years
After retirement, Christensen and his wife joined the Church of Jesus Christ of Latter-day Saints and they both served at the church's Family History Centre in Abbotsford. In co-operation with his wife Penny, a professional genealogist, he wrote his autobiography, the history of his parents' 172 descendants and a volume of translations of historical articles on Læsø. At the age of 75 Christensen took up an adjunct professorship in the School of Engineering Science at Simon Fraser University in Burnaby, BC, where he had an office, a computer and a grad student – but no salary. He spent one or two days there each week until he died. During this time he published a number of papers in the field of asymptotic stability of linear and nonlinear systems, a continuation of his Ph.D. work. He simplified the solutions to Lyapunov's stability theorems so that they are now useful to the electric power engineers and other practical engineers. Another development was the solution of the least absolute value estimation problem originally posed by Laplace in 1750 by using linear programming.
Christensen died 9 August 2007 at the age of 78.[3]
Academic awards
Undergraduate
• Engineering Institute of Canada (EIC)[9] Prize 1957
• Two University of Alberta First Class Standing Prizes
• One University of Alberta Honor Prize
• Schlumberger Undergraduate Scholarship
• Henry Birks Gold Medal in Engineering, graduated with High Distinction
Postgraduate
• Northern Electric Fellowship 1958
• National Research Council (NRC) Studentship 1959
• National Research Council (NRC) Studentship 1960
• University of British Columbia Scholarship 1964
• University of British Columbia Scholarship 1965
Civic award – British Columbia
• District of Mission: Award for achievement in mathematics, engineering and science.
May 7, 2007. Presented by Mayor James Atebe.
Scientific communities' memberships
• Senior Member, Institute of Electrical and Electronics Engineers (IEEE)
• Associate Member, Canadian Electricity Association (CEA), Canada
• Member, The Association of Professional Engineers and Geoscientists of Alberta (APEGA), Alberta
Publications
Theses
G.S. Christensen's M.Sc. and Ph.D. Theses:
1- M.Sc. Thesis, Optimization of Conductor Shapes and Configurations of Conductor Bundles for High Voltage Transmission, May, 1960.[10]
2- Ph.D. Thesis, Aspects of Nonlinear System Stability, August 17, 1966.[11]
Books
G.S. Christensen's List of Books:
1- M.E. El-Hawary and G.S. Christensen, "Optimal Operation of Electric Power Systems." Academic Press, New York, pp. 280, 1979. (This was written on invitation from Richard Bellman, U.C.L.A.)
2- G.S. Christensen, M.E. El-Hawary and S.A. Soliman, "Optimal Control Applications in Electric Power Systems." Plenum Press, New York, pp. 200, May 1987.
3- G.S. Christensen and S.A. Soliman, "Optimal Long-Term Operation of Electric Power Systems." Plenum Press, New York, pp. 310, August 1988.
4- G.S. Christensen, S.A. Soliman and R. Nieva, "Optimal Control of Distributed Nuclear Reactors." Plenum Press, New York, pp. 250, January 1990.
Refereed chapters in books
G.S. Christensen's List of Refereed Chapters in Books:
1- M.E. El-Hawary and G.S. Christensen. "Optimal Operation of Large Scale Power Systems." In Advances in Control and Dynamic Systems (C.T. Leondes, Editor), Academic Press, New York, Vol.13, pp. 1–70, 1977.
2- A. Shamaly, G.S. Christensen and M.E. El-Hawary. In "Information Linkage Between Applied Mathematics and Industry" (A.L. Schoenstadt et al. Editors), Academic Press, New York, 1980.
3- G.S. Christensen and S.A. Soliman. "Optimization Techniques in Hydroelectric Systems." In Advances in Control and Dynamic Systems, (C.T. Leondes, Editor), Academic Press, New York, Vol.30, pp. 100, in press, Mar 1990.
4- G.S. Christensen, S.A. Soliman and M.Y. Mohamed, "Power Systems State Estimates based on Least Absolute Value (LAV)." Advances in Control and Dynamic Systems Vol.36, 1991, pp. 1–143.
Journal papers
G.S. Christensen's List of Journal Papers:
1- F. Noakes and G.S. Christensen, The Effect of Conductor Shapes and Arrangement on the Electric Field Intensity of High Voltage Transmission Lines. Transactions E.I.C. Vol.4, No.4, 1960.
2- G.S. Christensen, Cost of Power Transmission at 345KV. A report prepared for the B. C. Power Commission, Victoria, B. C., September 1960.
3- G.S. Christensen and G. J. Berg, Technical and Economical Aspects of Electrical Power Transmission from Plants in System D to Load Centres in the Fraser River Basin. A report prepared for the Fraser River Board, Victoria, B.C., December 1960.
4- Report on the Columbia and Peace River Projects, British Columbia Energy Board, Victoria, B.C., 31 July 1961. (Did preparatory work in this report.)
5- G.S. Christensen, A Control System for Automatic Blending of Solvents. A report prepared for Canadian Chemical Co, Edmonton, July 1962.
6- G.S. Christensen, Criticism of paper entitled - Inside Hydrothermal Coordination by C.W. Watchorn, IEEE Transactions PAS, January 1967.
7- G.S. Christensen, On the Convergence of Volterra Series. IEEE Transactions A.C., Vol.13, p. 736, 1968.
8- R.S. Rao and G.S. Christensen, Bounded-Input Bounded-Output Stability of a Class of Nonlinear Discrete-Data Systems via Contraction Mapping. Int. Journal of Control, Vol.12, No.3, p. 449, 1970.
9- R.S. Rao and G.S. Christensen, A Criterion for the Bounded-Input Bounded-Output Stability of a Discrete-Data System with a Slope-Restricted Nonlinearity. Int. Journal of Control, Vol.12, No.4, p. 637, 1970.
10- R.S. Rao and G.S. Christensen, On the Convergence of a Discrete Volterra Series. IEEE Transactions A.C., Vol.15, p. 140, 1970.
11- G.W. Trott and G.S. Christensen, On the Uniqueness of the Volterra Series. IEEE Transactions A.C., Vol.14, p. 759, 1970.
12- G.S. Christensen and G.W. Trott, On the Inclusion of Initial Conditions in Volterra Series. Int. Journal of Control, Vol.12, No.5, p. 835, 1970.
13- G.W. Trott and G.S. Christensen, A Larger Region of Convergence for the Volterra Series. Int. Journal of Control, Vol.14, p. 377, 1971.
14- A.M.H. Rashed and G.S. Christensen, Response of Nonlinear Sampled Data Systems with Non-Zero Initial Conditions and Response for In-Between Sampling via Volterra Series. IEEE Transactions A.C., Vol.16, p. 269, 1971.
15- B. Bussman and G.S. Christensen, A Comparison Theorem for Stability Investigations. IEEE Transactions A.C., Vol 17, p. 138, 1972.
16- M.E. El-Hawary and G.S. Christensen, Functional Optimization of Common Flow Hydro-Thermal Systems. IEEE Transactions P.A.S., Vol.91, p. 1833, 1972.
17- M.E. El-Hawary and G.S. Christensen, Optimum Scheduling of Power Systems Using Functional Analysis. IEEE Transactions A.C., Vol.17, p. 518, 1972.
18- M.E. El-Hawary and G.S. Christensen, Application of Functional Analysis to Optimization of Electric Power Systems. Int. Journal of Control, Vol.16, No.6, p. 1063, 1972.
19- M.E. El-Hawary and G.S. Christensen, Hydro-Thermal Load Flow Using Functional Analysis. Journal of Optimization Theory and Applications, Vol.12, No.6, p. 576, 1973.
20- M.E. El-Hawary and G.S. Christensen, Extensions to Functional Optimization of Common-Flow Hydro-Thermal Systems. IEEE Transactions P.A.S., Vol.92, p. 356, 1973.
21- D.F. Liang and G.S. Christensen, Comments on "Estimation in Linear Delayed Discrete-Time Systems with Correlated State and Measurement Noises". IEEE Transactions A.C., Vol 20, pp. 176–177, 1975.
22- D.F. Liang and G.S. Christensen, New Filtering and Smoothing Algorithms for Discrete Nonlinear Systems with Time Delays. Int. Journal of Control, Vol.21, No.1, pp. 105–111, 1975.
23- D.F. Liang and G.S. Christensen, Exact and Approximate State Estimation for Nonlinear Dynamic Systems. Automatica, Vol.11, pp. 603–612, 1975.
24- G.S. Christensen and M.E. El-Hawary, Optimal Operation of Multi-Chain Hydro-Thermal Power Systems. Canadian Electrical Engineering Journal, Vol.1, No.2, pp. 52–62, 1976.
25- D.F. Liang and G.S. Christensen, New Filtering and Smoothing Algorithms for Discrete Nonlinear Delayed Systems with Coloured Noise. Int. Journal of Control, Vol.25, pp. 821–825, 1977.
26- D.F. Liang and G.S. Christensen, Estimation Algorithms for Discrete Nonlinear Systems and Observations with Multiple Time Delays. Int. Journal of Control, Vol.23, p. 613, 1976.
27- R. Nieva, G.S. Christensen and M.E. El-Hawary, Suboptimal Control of a Nuclear Reactor Using Functional Analysis. Int. Journal of Control, Vol.26, pp. 145–156, 1977.
28- R. Nieva and G.S. Christensen, Reduction of Reactor Systems. Nuclear Science and Engineering, Vol.64, pp. 791–795, 1977.
29- D.F. Liang and G.S. Christensen, Estimation of Discrete Non-Linear Time-Delayed Systems and Measurements with Correlated and Colored Noise Processes. Int. Journal of Control, Vol.28, pp. 1–10. 1978.
30- A. Shamaly, G.S. Christensen and M.E. El-Hawary, A Transformation for Necessary Optimality Conditions for Systems with Polynomial Nonlinearities. IEEE Transactions A.C., Vol.24, No.6, p. 983, 1979.
31- R. Nieva and G.S. Christensen, Optimal Control of Distributed Nuclear Reactors Using Functional Analysis. Journal of Optimization Theory and Applications, Vol.34, No.3, 1981.
32- M.E. El-Hawary and G.S. Christensen, Optimal Active-Reactive Hydro-Thermal Schedules Using Functional Analysis. Optimal Control Applications and Methods, Vol.1, pp. 239–249, 1980.
33- A. Shamaly, G.S. Christensen and M.E. El-Hawary, Optimal Control of a Large Turbo-Alternator. Journal of Optimization Theory and Applications, Vol.34, No.1, pp. 83–97, 1981.
34- A. Shamaly, G.S. Christensen and M.E. El-Hawary, Solution of Ill-Conditioned Optimality Conditions for Control of Turbo-Alternators. Optimal Control Applications and Methods, Vol.2, pp. 81–87, 1981.
35- A. Shamaly, G.S. Christensen and M.E. El-Hawary, Realistic Feedback Control of Turbo-Generators. Journal of Optimization Theory and Applications, Vol.35, No.2, pp. 251–259, 1981.
36- R. Nieva, G.S. Christensen and M.E. El-Hawary, Optimum Load Scheduling of Nuclear-Hydro-Thermal Power Systems. Journal of Optimization Theory and Applications, Vol.35, No.2, 1981.
37- R. Nieva and G.S. Christensen, Symmetry Reduction of Linear Distributed Parameter Systems. Int. Journal of Control, Vol.36, No.1, pp. 143–153, 1982.
38- A. Shamaly, G.S. Christensen and Y. Chen, Optimal Control of Two Interconnected Turbogenerators. Journal of Optimization Theory and Applications, Vol.40, No.2, 1983.
39- Y. Chen, G.S. Christensen and A. Shamaly, Realistic Feedback Control of Two Interconnected Turbogenerators. Journal of Optimization Theory and Applications, Vol.42, No.1, 1984.
40- M. Abdelhalim, G.S. Christensen and D.H. Kelly, Optimal Load Frequency Control with Governor Backlash. Journal of Optimization Theory and Applications, Vol.45, 1985.
41- M. Abdelhalim, G.S. Christensen and D.H. Kelly, Decentralized Optimum Load Frequency Control of Interconnected Power Systems. Journal of Optimization Theory and Applications, Vol.45, 1985
42- M.E. El-Hawary, R.S. Rao and G.S. Christensen, Optimal Hydro-Thermal Load Flow. Optimal Control Applications and Methods, Vol.7, pp. 18, 1986.
43- S.A. Soliman, G.S. Christensen and M. A. Abdelhalim, Optimal Operation of Multireservoir Power Systems Using Functional Analysis. Journal of Optimization Theory and Applications, Vol.49, No.3, pp. 449–461, 1986.
44- G.S. Christensen and S.A. Soliman, Long-Term Optimal Operation of a Parallel Multireservoir Power System Using Functional Analysis. Journal of Optimization Theory and Applications, Vol.50, No.3, pp. 383–395, 1986.
45- S.A. Soliman and G.S. Christensen, Modelling and Optimization of Series Reservoirs for Long-Term Regulation with Variable Head Using Functional Analysis. Journal of Optimization Theory and Applications, Vol.50, No.3, pp. 463–477, 1986.
46- S.A. Soliman and G.S. Christensen, Optimization of the Production of Hydroelectric Power Systems with a Variable Head. Journal of Optimization Theory and Applications, Vol.58, pp. 301–317. 1988.
47- S.A. Soliman and G.S. Christensen, Optimization of Reservoirs in Series on a River with a Nonlinear Storage Curve for Long-Term Regulation. Journal of Optimization Theory and Applications, Vol.58, pp. 109–126, 1988.
48- G.S. Christensen and S.A. Soliman, New Analytical Approach for Long-Term Optimal Operation of a Parallel Multireservoir Power System Based on Functional Analysis. Canadian Electrical Engineering Journal, Vol.11, No.3, pp. 118–127, 1986.
49- G.S. Christensen and S.A. Soliman, Optimal Long-Term Operation of a Multireservoir Power System for Critical Water Conditions Using Functional Analysis. Journal of Optimization Theory and Applications, Vol.53, No.3, pp. 377–393, 1987.
50- G.S. Christensen and S.A. Soliman, Modelling and Optimization of Parallel Reservoirs Having Nonlinear Storage Curves Under Critical Water Conditions for Long-Term Regulation Using Functional Analysis. Journal of Optimization Theory and Applications, Vol.55, No.3, pp. 359–376, 1987.
51- G.S. Christensen and S.A. Soliman, On the Application of Functional Analysis to the Optimization of the Production of Hydroelectric Power. IEEE Transactions P.W.R.S.-2-No.4, pp. 841–847, 1987.
52- S.A. Soliman and G.S. Christensen, Long-Term Optimal Operation of Series-Parallel Reservoirs for Critical Period with Specified Monthly Generation. Canadian Electrical Engineering Journal Vol.12, pp. 116–122, 1987.
53- S.A. Soliman and G.S. Christensen, Application of Functional Analysis to Optimization of a Variable Head Multireservoir Power System for Long-Term Regulation. Water Resources Research, Vol.22, No.6, pp. 852–858, 1986.
54- S.A. Soliman and G.S. Christensen, A New Approach for Optimizing Hydropower System Operation with a Quadratic Model. Lecture Notes in Control and Information Sciences, Vol.95 Oberwolfach, pp. 273–286, 1986.
55- S.A. Soliman and G.S. Christensen, A Minimum Norm Approach to Optimization of Production of Multireservoir Power Systems with Specified Monthly Generation. Journal of Optimization Theory and Applications, Vol.58, pp. 501–524, 1988.
56- S.A. Soliman and G.S. Christensen, Optimization of Hydropower Systems Operation with a Quadratic Model. Automatica, Vol.24, p. 249-256, 1988.
57- G.S. Christensen and S.A. Soliman, Optimal Discrete Long-Term Operation of Nuclear-Hydrothermal Power Systems. Journal of Optimization Theory and Applications, Vol.62, No.2, pp. 239–254. 1989.
58- S.A. Soliman, G.S. Christensen and A.H. Rouhi, A New Technique for Curve Fitting Based on Minimum Absolute Deviations. Computational Statistics and Data Analysis, Vol.6, pp. 341–351. 1989.
59- S.A. Soliman and G.S. Christensen, Parameter Estimation in Linear Static Systems Based on LAV Estimation. Journal of Optimization Theory and Applications, Vol.61, No.2, pp. 281–294. 1989.
60- G.S. Christensen and S.A. Soliman, A New Technique for Linear Static State Estimation Based on LAV Approximations. Journal of Optimization Theory and Applications, Vol.61, No.1, pp. 123–136. 1989.
61- G.S. Christensen and S.A. Soliman, Long-Term Optimal Operation of Series-Parallel Reservoirs for Critical Period with Specified Monthly Generation and Average Monthly Storage. Journal of Optimization Theory and Applications, Vol.63, No.3. 1989.
62- G.S. Christensen, S.A. Soliman and A.H. Rouhi, A New Technique for Unconstrained and Constrained Linear LAV Parameter Estimation. Canadian Electrical Engineering Journal, Vol.14, pp. 14–30. 1989.
63- S.A. Soliman, S.E.A. Emam and G.S. Christensen, A New Algorithm for Parameter Estimation of Synchronous Machine from Frequency Test Based on LAV Approximation. Canadian Electrical and Computer Engineering Journal, Vol.14, No.3, pp. 98–102. 1989.
64- S.A. Soliman, S.E.A. Emam and G.S. Christensen, Optimization on Site and Control Setting of Shunt Capacitors on Distribution Feeders. Journal of Optimization Theory and Control, Vol.65, No.2. May1990.
65- S.A. Soliman, S.E.A. Emam and G.S. Christensen, A New Approach for the Identification of Power System Economic Dispatch Parameters. Journal of Optimization Theory and Applications, Vol.65, No.1. April 1990.
66- G.S. Christensen and S.A. Soliman, Optimal Filtering for Continuous Linear Dynamic Systems Based on WLAV Approximations. Automatica, Vol.26, No.2, pp. 399–400. 1990.
67- G.S. Christensen and S.A. Soliman, Optimal Filtering of Linear, Discrete Dynamic Systems Based on WLAV Approximations. Automatica, Vol.26, No.2, pp. 389–395. 1990.
68- G.S. Christensen, S.A. Soliman and A. Rouhi, Discussion of "An example showing that a new technique for LAV estimation breaks down in certain cases". Computational Statistics and Data Analysis, Vol.9, pp. 203–213. 1990.
69- G.S. Christensen, S.A. Soliman and A. Atallah, Optimal Scheduling of Multichain Hydro Power Systems, Long Term Study. Canadian Journal of Electrical and Computer Engineering, Vol.14, No.4, pp. 152–156.1989.
70- G.S. Christensen, S.A. Soliman and A.H. Rouhi, An Observability Algorithm for Sequential Measurement Processing in Power System State Estimation. Electric Machines and Power Systems, Vol.17, pp. 203–219. 1989.
71- S.A. Soliman and G.S. Christensen, A New Algorithm for Optimal Parameter Estimation. Journal of Optimization Theory and Applications, Vol.66, No.3. September 1990.
72- S.A. Soliman, G.S. Christensen and S.S. Fouda, On the Application of the Least Absolute Value (LAV) Parameter Estimation Algorithm to Distance Relaying. Electric Power Systems Research, Vol.19, pp. 23–35. 1990.
73- S.A. Soliman, G.S. Christensen, D.H. Kelly and N. Liu, An Algorithm for Frequency Relaying Based on Least Absolute Value Approximations. Electric Power Systems Research, Vol.19, pp. 73–84. 1990.
74- S.A. Soliman, G.S. Christensen and D.H. Kelly, A State Estimation Algorithm for Power Systems Harmonics Identification and Measurements. Electric Power Systems Research, Vol.19, No.2, pp. 195–206. 1990.
75- S.A. Soliman, S.E.A Emam and G.S. Christensen, Application of Optimization to the Size and Control Settings of Shunt Capacitors on Distribution Feeders. Electric Machines and Power Systems, Vol.18, pp. 41–51, 1990.
76- S.A. Soliman, S.E.A. Emam and G.S. Christensen, Optimal Coefficients Estimation of Non-Monotonically Increasing Incremental Cost Curves. Electric Power Systems Research, Accepted September 10, 1990. Manuscript pages: 20.
77- G.S. Christensen and S.A. Soliman, Least Absolute Value Estimation of the Generalized Operational Impedances of Solid-Rotor Synchronous Machines from SSFR Test Data. Electric Power Systems Research, Accepted September 7, 1990. Manuscript pages:24.
78- S.A. Soliman, G.S. Christensen and A.H. Rouhi, A New Algorithm for Nonlinear L1-Norm Minimization with Nonlinear Equality Constraints. Computational Statistics and Data Analysis, Vol.11, pp. 1–13. 1990.
79- G.S. Christensen, S.A. Soliman and A. M. Atallah, Efficient Load Following Schedule with Application to the B.P.A. Hydro-Electric System. International Journal of Electrical Power and Energy Systems, Vol.13, Feb. 1991, pp. 45–50.
80- G.S. Christensen, S.A. Soliman and A.M. Atallah, Optimal Long-Term Operation of the B.P.A. Hydro-Electric Power System. International Journal of Computer Systems Science and Engineering, Accepted Vol.13, No.1 Feb. 1991 pp. 38–42.
81- S.A. Soliman and G.S. Christensen, Estimating of Steady State Voltage and Frequency of Power Systems from Digitized Bus Voltage Samples. Electric Machines and Power Systems, Vol.19 1991, pp. 555–576.
82- S.A. Soliman and G.S. Christensen, Power Systems Digital Voltmeters with Low Sensitivity to Frequency Change. Electric Machines and Power Systems Journal, accepted July 5, 1991, Man. 20.
83- G.S. Christensen, S.A. Soliman, D.H. Kelly and K.M. El-Naggar. Identification and Measurements of Power System Harmonics Using Disacrete Fourier Transform (DFT). Electric Machines and Power Systems, accepted July 5, 1991, Man. 29.
84- S.A. Soliman and G.S. Christensen, Digital Analysis of Power Systems Dynamic Oscillation Using a Curve Fitting Technique. Electric Machines and Power Systems, Vol.20 1992 pp. 309–320.
85- S.A. Soliman and G.S. Christensen, Modelling of Induction Motors from Standstill Frequency Response Tests and a Parameter Estimation Algorithm. Electric Machines and Power Systems, Vol.20, 1992 pp. 123–126.
86- S.A. Soliman, G.S. Christensen and A.H. Rouhi, Power System State Estimation with Equality Constraints. Electric Machines and Power Systems, Vol.20 1992, pp. 183–202.
87- S.A. Soliman, G.S. Christensen, D.H. Kelly and K.M. El-Naggar, Least Absolute Value based linear Programming Algorithm for Measurement of Power System Frequency from a distorted Bus Voltage Signal. Electric Machines and Power Systems, Vol.20 1992, pp. 549–568.
88- S.A. Soliman, G.S. Christensen, D.H. Kelly and K.M. El-Naggar, Dynamic Tracking of the Steady State Power System Voltage Magnitude and Frequency using Linear Kalman Filter: a Variable Frequency Model. Electric Machines and Power Systems, Vol.20, 1992 pp. 593–611.
89- S.A. Soliman, G.S. Christensen and K.M. El-Naggar, A Digital Measurement of Earth Fault Loop-Impedance Using a Parameter Estimation Algorithm. Electric Machines and Power Systems, Vol.20 1992, pp. 613–621.
90- S.A. Soliman, G.S. Christensen and K.M. El-Naggar, A new Approximate Least Absolute Value based Dynamic Filtering Algorithm for on-line Power System Frequency. Electric Machines and Power Systems, Vol.20 1992, pp. 569–592.
91- G.S. Christensen and S.A. Soliman, Optimization Techniques in Hydro-Electric Systems. Zentralblatt fuer Mathematik, Accepted Jan. 10, 1992, Man. 1.
92- G.S. Christensen, Mehrdad Saif, and S.A. Soliman. A New Algorithm for Finding the Optimal Solution of the Least Absolute Value Estimation Problem. Canadian Journal of Electrical and Computer Engineering, Vol. 32, No. 1, pp. 5–8, Winter 2007.
93- G.S. Christensen and Mehrdad Saif. The Asymptotic Stability of Nonlinear Autonomous Systems. Canadian Journal of Electrical and Computer Engineering, Vol. 32, No. 1, pp. 35–43, Winter 2007.
References
1. Penelope Christensen's page on Genealogical Studies Website
2. https://ca.linkedin.com/in/lynnechristensen Lynne V. Christensen's page
3. The Abbotsford News' Obituaries: Dr. Gustav S. CHRISTENSEN
4. http://www.nrc-cnrc.gc.ca/eng/ National Research Council Canada
5. El-Hawary, Mohamed E.; Christensen, G. S. (1979). Optimal Economic Operation of Electric Power Systems. New York: Academic Press. ISBN 9780080956527.
6. Christensen, G. S.; El-Hawary, Mohamed E.; Soliman, S. A. (1987). Applications of Optimal Control in Electric Power Systems. ISBN 978-0-306-42517-2.
7. Christensen, G. S.; Soliman, S. A. (1988). Optimal Long-Term Operation of Electric Power Systems. New York: Plenum Press.
8. Christensen, G. S.; Soliman, S. A.; Nieva, R. (1988). Optimal Control of Distributed Nuclear Reactors. New York: Plenum Press.
9. http://www.eic-ici.ca/ Engineering Institute of Canada (EIC)
10. Christensen, Gustav Strom (1960). Optimization of Conductor Shapes and Configuration of Conductor Bundles for High Voltage Transmission (M.Sc.). University of British Columbia.
11. Christensen, Gustav Strom (1966). Aspects of nonlinear system stability (Ph.D.). University of British Columbia.
External links
Wikiquote has quotations related to Gustav S. Christensen.
• Page 36 of Chronicle Magazine, UBC Alumni Chronicle, Autumn 82.
• The Abbotsford News - Obituaries: Dr. Gustav S. Christensen, Thu Aug 9th 2007.
| Wikipedia |
A yogurt shop sells four flavors of yogurt and has six different toppings. How many combinations of one flavor and two different toppings are available?
There are 4 choices of flavor, and $\binom{6}{2}=15$ ways to choose two of the six toppings. The total number of combinations is $4\cdot 15=\boxed{60}$. | Math Dataset |
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On the other hand, if the film thickness is 1/2 wavelength, the first wave gets a 1/2 wavelength shift and the other gets a wavelength shift; these waves would cancel each other out. Note that one has to be very careful in dealing with the wavelength, because the wavelength depends on the index of refraction.
Visible light has a wavelength range between _____ at the violet end and _____ at the red end., 380 nanometers (violet end), 750 nm at the red end, Violet light has ____ energy, _____ wavelength and _____ frequency compared to red light. violet light has greater energy, shorter wavelength and higher frequency compared to red light
Which of the following architecture has feedback connections? Which of the following statements is true? A. There will not be any problem and the neural network will train properly. B. The neural network will train but all the neurons will end up recognizing the same thing.
Dec 08, 2007 · Which of the following statements about mechanical waves is true? a. mechanical waves require a medium to travel through b. mechanical waves do not have amplitude and wavelength c. mechanical waves do not have frequency d.
Which of the following have the longest wavelength? Infrared waves, x-rays, microwaves, or visible light
Using this equation, depth of field (d(tot)) and wavelength (λ) must be expressed in similar units. For example, if d(tot) is to be calculated in micrometers, λ must also be formulated in micrometers (700 nanometer red light is entered into the equation as 0.7 micrometers). Notice that the diffraction-limited depth of field (the first term in ...
Nov 30, 2020 · If the object has a height of 4 millimeters, the height of the image is 8 millimeters. Solution: m = 16/8 = 2mm , object height =4 mm , height of image = 4 *2 = 8 mm. Added 264 days ago|4/10/2020 11:49:01 PM
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Red light has the longest wavelength and, therefore, the least amount of energy in the visible spectrum. Wavelength decreases and energy increases as you move from red to violet light across the spectrum in the following order: red, orange, yellow, green, blue, and violet.
How long the novel coronavirus lasts in the body and on surfaces can vary depending on the circumstances. How long the virus lasts in the body depends on the individual and the severity of the illness. The Centers for Disease Control and Prevention (CDC) advise that people who test positive...
_____ Which of the following has the longest wavelength? a. X-rays b. infrared radiation c. visible light d. ultraviolet radiation _____ Which of the following has the longest wavelength? a. X-rays b. infrared radiation c. visible light d. ultraviolet radiation b _____ Which of the following has the longest wavelength? a. X-rays b.
Note that one has to be very careful in dealing with the wavelength, because the wavelength depends on the index of refraction. Generally, in dealing with thin-film interference the key wavelength is the wavelength in the film itself. If the film has an index of refraction n, this wavelength is related to the wavelength in vacuum by:
Waves may be graphed as a function of time or distance. A single frequency wave will appear as a sine wave (sinusoid) in either case. From the distance graph the wavelength may be determined.
100.2 Of the following types of electromagnetic radiation, which has the second shortest wavelength? (C) infra-red light. --- Yes. Just longer than red light. Check the other answers.
Nov 07, 2011 · Determine which of the following H atom electron transitions has the highest and lowest wavelengths of the photons absorbed or emitted. A.) n = 2 to n= infinity B.) n = 4 to n = 20 C.) n = 3 to n = 10 D.) n = 2 to n = 1 Which is the longest wavelength photon?
A Hertz is a. a unit of wavelength b. a unit of frequency c. a unit of velocity d. a unit of loudness e. a well-known car-rental company ____ 12. A fashion designer decides to bring out a new line of clothing which reflects the longest wavelength of visible light.
This is considered a short wavelength in the electromagnetic spectrum and is one of the A mmWave system that resolves distances to wavelength has accuracy in the mm range at 60-64 GHz and 76-81 GHz. Additionally, operating in this spectrum makes mmWave sensors interesting for the following...
so Wavelength (nm) o Wavelength (nm) 7. Use colored pencils to color the hydrogen and boron spectral lines within their respective spectra in Model 2. 8. List the spectral lines for hydrogen gas by color and correspondin wavelength. 113Clnrn 9. The spectral lines for boron were produced using the same met od as hydrogen. List three of the
Tritium has a half-life of 12.5 y against beta decay. What fraction of a sample will remain undecayed after 25 y? In what distance will half of a beam of .025-eV neutrons have decayed? The half-life of the Solving for the fraction absorbed gives the following (note that if 99 percent pass through, 0.01...
Twelve updated ultrasonic sensors complement this vision, allowing for detection of both hard and soft objects at nearly twice the distance of the prior system. A forward-facing radar with enhanced processing provides additional data about the world on a redundant wavelength that is able to see through heavy rain, fog, dust and even the car ahead.
Oct 11, 2009 · Thus, the lower wave has the lower frequency, and the upper one has the higher frequency. 7. Answer: The expanded visible-light portion of Figure 6.4 tells you that red light has a longer wavelength than blue light. The lower wave has the longer wavelength (lower frequency) and would be the red light.
Feb 14, 2020 · Wavelength and Color Spectrum Chart . The wavelength of light, which is related to frequency and energy, determines the perceived color. The ranges of these different colors are listed in the table below. Some sources vary these ranges pretty drastically and their boundaries are somewhat approximate, as they blend into each other.
FREQUENCY AND PITCH. After reading this section you will be able to do the following:. Explain how you can change pitch by altering sources. Describe what resonance is.
Jan 22, 2019 · Moving with the same velocity, which of the following has the longest de-Broglie wavelength? asked Aug 26 in Dual Nature of Radiation and Matter by AmarDeep01 ( 50.0k points) dual nature of radiation and matter
Icons that can have specific meanings. C. A long command line to achieve a function. A computer expert produces a solution with HCI which is very efficient in computer resources, based on command-lines. Which one of the following is most likely to be the result when the system is implemented?
At the simplest level, waves are disturbances that propagate energy through a medium. Propagation of the energy depend on interactions between the particles that make up the medium. Particles move as the waves pass through but there is no net motion of particles. This means, once a wave has passed the particles return to their original position.
Calculate the longest and shortest wavelengths (in nm) emitted in the Balmer series of the hydrogen atom emission spectrum. Solution From the behavior of the Balmer equation (Equation \(\ref{1.4.1}\) and Table \(\PageIndex{2}\)), the value of \(n_2\) that gives the longest (i.e., greatest) wavelength (\(\lambda\)) is the smallest value possible ...
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UVC UVA rays have the longest wavelengths, followed by UVB, and UVC rays which have the shortest wavelengths. While UVA and UVB rays are transmitted through the atmosphere, all UVC and some UVB...
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Jul 15, 2009 · Infrared radiation has the longest wavelength out of these choices, and hence the lowest frequency. Gamma radiation has the shortest wavelength and the highest frequency. You'll realize that...
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So that the human eye cannot notice the remaining electromagnetic waves. We can notice these waves like rainbow colors where every color has a different wavelength. Vibgyor Colours Wavelength and Frequency. The VIBGYOR color wavelength and frequency are shown in the following VIBGYOR wavelength chart show the order of wavelength as well as ...
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Question: Which lists the waves in order of wavelength, from longest to shortest? X-rays, infrared, radio waves infrared, visible light, gamma rays ultraviolet, microwaves, visible light radio waves, gamma rays, visible light
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What is the longest wavelength of light that can cause the release of electrons from a metal that has a work function of 3.50 eV? View Answer The photoelectric work function of a metal is the minimum energy required to eject an electron by shining light on the metal.
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Which of the following complex ions will absorb the longest wavelength of light? A) [Mn(NH 3) 6] 2+ B) [Mn(NH 3) 6] 3+ C) [Mn(CN) 6] 4-D)[Mn(CN) 6] 3-I know that: (* represents the triangle, change in) *o is highest in D) [Mn(CN) 6] 3-. and lowest in A) [Mn{HN 3) 6] 2+
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This calculator will tell you the wavelength of any airborne audio frequency in inches, feet and metres, based on the speed of sound at sea level, and at 20 degrees Celsius or 72 degrees Farenheit. Note that with audio frequencies of 20Hz to 20,000Hz the range of physical sizes of wavelengths is substantial (1000:1), and this causes no end of ...
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Platinum, which is widely used as a catalyst, has a work function f (the minimum energy needed to eject an electron from the metal surface) of 9.05 × 10 –19 J. What is the longest wavelength of light which will cause electrons to be emitted? A) 2.196 × 10 –7 m. D) 1.370 × 10 15 m. B) 4.553 × 10 –6 m. E) > 10 6 m. C) 5.654 × 10 2 m
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Get an answer to your question "Which of the following electromagnetic waves has the shortest wavelength?A. visible light waves B. ultraviolet waves C. microwaves D. ..." in 📙 Physics if there is no answer or all answers are wrong, use a search bar and try to find the answer among similar questions.
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The following reaction occurs in the light, causing the lenses to darken: AgCl → Ag + Cl The enthalpy change for this reaction is 3.10 × 10 2 kJ/mol. Assuming all this energy is supplied by light, what is the maximum wavelength of light that can cause this reaction?
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Because photon energy increases as wavelength decreases, any photon of wavelength shorter than 242 nm will have sufficient energy to dissociate O 2. PRACTICE EXERCISE. The bond energy in N 2 is 941 kJ/mol (Table 8.4). What is the longest wavelength photon that has sufficient energy to dissociate N 2? Answer: 127 nm
the Hβ line has been shifted to the wavelength usually occupied by the Hα line. 12. If this line shift is caused by the Doppler Effect, what can you conclude about Q2? ANSWER. Q2 is moving away from us at a high speed. 13. Into what region of the electromagnetic spectrum has the Hα line in Q2's spectrum been shifted? ANSWER. Hα has been ...
For questions 9-10 decide whether the following statements are True (A), False (B) or Not Stated (C). Transfer your answers to the answer sheet. Absolutely. And as well as stimulating the imagination, clouds get you out B and about. The keeper of the Society's photo gallery, Ian Loxley, has been on.
the following electron transitions in the hydrogen atom results in the emission of light of the longest wavelength. (a) n 4 to n 3; (b) n — 1 ton = 6; (d) n 3 ton 2. What electron transition in a hydrogen atom, starting from the orbit n = 7, will produce light of wavelength 41011m ? What electron transition in a hydrogen atom, ending
Jul 21, 2017 · Show a chart of the wavelength, frequency, and energy regimes of the spectrum. Astronomy Across the Electromagnetic Spectrum. While all light across the electromagnetic spectrum is fundamentally the same thing, the way that astronomers observe light depends on the portion of the spectrum they wish to study.
This energy is called the work function of that metal. What is the longest wavelength of radiation (in nm) that could cause the photoelectric effect in each of the following metals? a. Silver, (work function) = 7.59 x 10 -19 J ? nm b.Tungsten, (work function) = 7.16 x 10 -19 J ? nm c. Sodium, (work function) = 4.41 x 10-19J ?nm
Visible light has a wavelength range of about 400 - 700 nm. The visible colours from shortest to longest wavelength are: violet, blue, green, yellow, orange, and red. GET YOUR EXPERT ANSWER ON STUDYDADDY Post your own question
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Which of the following types of radiation has the longest wavelength? a) x-rays b) visible c) radio d) gamma rays e) microwaves Answer: c 3. Which of the following colors of visible light has the shortest wavelength? Put the visible light in order from the longest wavelength to the shortest wavelength red, orange, yellow, green, blue, indigo, violet What is the relationship between frequency and wavelength On the other hand, if the film thickness is 1/2 wavelength, the first wave gets a 1/2 wavelength shift and the other gets a wavelength shift; these waves would cancel each other out. Note that one has to be very careful in dealing with the wavelength, because the wavelength depends on the index of refraction. Mar 08, 2014 · Which of the following energies of light has the longest wavelength? 2.85 x 10-19 J 4.22 x 10-18 J 6.25 x 10-14 J 9.02 x 10-20 J 6.14 x 10-19 J Which Of The Following Complex Ions Absorbs Light Of The Longest Wavelength? [Cr(CN)6]3- [Cr(NH3)6]3+ [Cr(NO2)6]3- [Cr(en)3]3+ [CrCl6]3- This problem has been solved!
A long wavelength wave caused by an earthquake. Which of the given options most accurately completes the following statement? 'The transverse GM of the vessel... ...has an influence on a vessel rolling in waves'.Please help!! Be sure to answer all parts. Determine which of the following H atom electron transitions has the longest wavelength and which has the shortest wavelength: (a) =1 to n = 4 points (Ф) - 4 ton- 8 (c) = 4 ton= 9 eBook (d) n=1 to n = 12 Print References Longest-wavelength photon: оооо. Shortest-wavelength photon: 1 2 оооо е It has become common that more than half of the country's readers get their morning paper brought to their door by a teenager. E. Not so long ago in Britain if you saw someone reading a newspaper you could tell what kind it was without even checking the name.
Mar 22, 2009 · Microwaves have the longest wavelengths out of these 3, but radio waves have even longer wavelength than microwaves. Out of these 3, microwaves have the longest wave length, the smallest frequency and the least energy. Out of these 3, x rays have the shortest wavelength, the great frequency and the most energy. Dec 09, 2020 · A. (E)-2-butene B. (E)-1,3,5- hexatriene C. (Z)-1,3-hexadiene D. (Z)-2- butene E. 1-hexene ▸ Neural Networks: Learning : You are training a three layer neural network and would like to use backpropagation to compute the gradient of the cost function. In the backpropagation algorithm, one of the steps is to update for every i,j. Which of the following is a correct vectorization of this step?the following electron transitions in the hydrogen atom results in the emission of light of the longest wavelength. (a) n 4 to n 3; (b) n — 1 ton = 6; (d) n 3 ton 2. What electron transition in a hydrogen atom, starting from the orbit n = 7, will produce light of wavelength 41011m ? What electron transition in a hydrogen atom, ending 13. Sodium has a work function of 2.46 eV. Of the following wavelengths, which is the one of the longest wavelength that could cause photoelectrons to be released from the sodium? A. 300 nm B. 400 nm C. 500 nm D. 600 nm E. 700 nm narrow range of wavelengths. The shortest wavelength blue light which is visible has a wavelength of microns (one micron is meters). The longest wavelength red light which is visible has a wavelength of microns. However, there is nothing in Maxwell's analysis which suggested that
Which of following constant gases accounts for the largest proportion of the air that surrounds us? Which of the following surfaces would likely have Which of the following are the most absorptive gases of longwave infrared radiation? A) oxygen and argon B) carbon dioxide and methane C)...Find an answer to your question Which of the following has the longest wavelength? red violet ultraviolet radiation infrared radiation jerniganrichard12345 jerniganrichard12345 14.02.2018
It has become common that more than half of the country's readers get their morning paper brought to their door by a teenager. E. Not so long ago in Britain if you saw someone reading a newspaper you could tell what kind it was without even checking the name.6. Which of the following has the highest frequency? visible light microwaves infrared radiowaves 7. Which of the following has the longest wavelength? gamma rays visible light ultraviolet radiowaves 8. Which of the following has the shortest wavelength? gamma rays visible light ultraviolet radiowaves 9. ...calculate the longest possible wavelength, in nanometers, of light that can ionize the metal. 592 nm. Problem #3: Determine the wavelength (in meters) of photons with the following energies Problem #4: A certain green light has a wavelength of 6.26 x 1014 Hz. (a) What is its wavelength...
10). Consider the following statements 1.Light of longer wavelength is scattered much more than the light of shorter wavelength. 2.The speed of visible light in water is 0.95 times the speed in vacuum. 3. Radio waves are produced by rapidly oscillating electrical currents. 4. Which of the following types of radiation has the longest wavelength? a) x-rays b) visible c) radio d) gamma rays e) microwaves Answer: c 3. Which of the following colors of visible light has the shortest wavelength?
Please answer all of the questions , also please dont answer if your not going to seriously answer my questions. FIRST TO ANSWER GETS 30 POINTSClassify the following as Which elements have 8 valence electrons? NASCAR fans love race day when they get a chance to cheer their favorite team!Explaining the experiments on the photoelectric effect. How these experiments led to the idea of light behaving as a particle of energy called a photon. At the simplest level, waves are disturbances that propagate energy through a medium. Propagation of the energy depend on interactions between the particles that make up the medium. Particles move as the waves pass through but there is no net motion of particles. This means, once a wave has passed the particles return to their original position. microwaves have the lowest frequency and the longest wavelength.Apr 02, 2020 · Violet light has the shortest wavelength, which means it has the highest frequency and energy. Red has the longest wavelength, the shortest frequency, and the lowest energy. Red has the longest wavelength, the shortest frequency, and the lowest energy. The optical spectrum is shown in Fig. 6 (a) with the peak wavelength at λ = 1600.5 nm, which is the longest wavelength in Q -switched EDFL as summarized in Tables 1 and 2. It is also noted that the peak wavelength of the laser slightly decreased as the stable Q switching reached, as shown in the inset of Fig. 6 (a). Microwaves have a smaller wavelength than visible light. The speed of all electromagnetic waves are equal. The frequency and wavelengths are equal for each type of radiation.
6) Which of the following has the LONGEST wavelength? D. Infrared light. 7) Compared to radio waves, ultraviolet light: C. has more energy per photon. 8) In the overall electromagnetic spectrum, consider radio, visible light, and gamma rays in terms of their wavelength. Their correct order, from longest to shortest, is: A. radio, visible, gamma ...
A Hertz is a. a unit of wavelength b. a unit of frequency c. a unit of velocity d. a unit of loudness e. a well-known car-rental company ____ 12. A fashion designer decides to bring out a new line of clothing which reflects the longest wavelength of visible light. (a) Longer wavelength; (b) shorter wavelength 5. Which has more energy, A or B? Explain your reasoning. 6. Define a wavelength. 7. What is a frequency of a wavelength? 8. For visible light (ROYGBIV), which has the shortest wavelength? Which has the longest? Place all the other colors in order based on their wavelength from shortest to longest. I have a vague idea. Everything looked vague in the heavy fog. ramble = to talk or write for a long time in a Really effective communicators who have the ability to engage with colleagues, employees Which of the problems mentioned in Exercise A do the speakers have when communicating?2. Which of the following has the most effect on the speed of sound in air? a. amplitude b. frequency c. wavelength --> d. temperature. a. increasing the mass of the string b. decreasing the tension in the string --> c. fingering the string d. plucking the string harder.G4A13 Which of the following performs automatic notching of interfering carriers? / A. Band pass tuning B. A DSP filter C. Balanced mixing D. A noise limiter B G3A02 What effect does a Sudden Ionospheric Disturbance (SID) have on the daytime ionospheric propagation of HF radio waves? / A.
Problem: According to the Bohr model for the hydrogen atom, which of the following transitions will emit light with the longest wavelength?a. from the n = 4 to n = 2 energy level.b. from the n = 4 to n = 3 energy level.c. from the n = 3 to n = 1 energy level.d. from the n = 6 to n = 2 energy level.e. from the n = 5 to n = 3 energy level.
Ultraviolet radiation has a shorter wavelength than visible violet light. Infrared radiation has a longer wavelength than visible red light the white light is a mixture of the colour of the visible spectrum. Violet has the shortest wavelength at around 380 nanometer , red has the longest wavelength at around 700 nanometer. Nov 12, 2011 · Transitions to n=3 or greater are in the infra-red part of the spectrum and have the longest wavelengths. The longest wavelength would correspond to the lowest energy and that would correspond to the difference in energy between n=5 and n=4. Longest and Shortest Wavelength: The longest and the shortest wavelength or length of the given transition spectrum (according to the energy levels) of the given substance is solved by the general ...
The electromagnetic spectrum ranges from gamma (γ) radiation, which has the shortest wavelength, highest frequency, and greatest energy, to radio waves, which has the longest wavelength and lowest frequency and energy. Ultraviolet light (UV) is divided into three regions: UV A, wavelength = 400 - 320 nm UV B, wavelength = 320 - 280 nm An angstrom is 10-8 cm or 0.00000001 cm. Red has a wavelength of 7800-6220 angstroms, green has a wavelength of 5770-4920 angstroms and violet has a wavelength of 4550-3900 angstroms. However, in this lab, the simple relationship among the visible light waves will be what is important.)
Color Wavelength Frequency Photon energy Violet 380–450 nm 680–790 THz 2.95–3.10 eV Blue 450–485 nm 620–680 THz 2.64–2.75 eV Cyan 485–500 nm 600–620 THz 2.48–2.52 eV Green 500–565 nm 530–600 THz 2.25–2.34 eV Yellow 565–590 nm 510–530 THz 2.10–2.17... Light or visible light is electromagnetic radiation within the portion of the electromagnetic spectrum that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nm, or 4.00 × 10 −7 to 7.00 × 10 −7 m, between the infrared (with longer wavelengths) and the ultraviolet (with shorter wavelengths). Note that one has to be very careful in dealing with the wavelength, because the wavelength depends on the index of refraction. Generally, in dealing with thin-film interference the key wavelength is the wavelength in the film itself. If the film has an index of refraction n, this wavelength is related to the wavelength in vacuum by: Which of the following commands can be used to access the home directory of the user "bob" Any two single characters. Nothing; it has no special meaning. The directory above the current working …lists the contents of the current directory. The first character in a long listing (ls -l) indicatesAnswer to Which of the following types of electromagnetic radiation has the longest wavelength? Radio waves Ultraviolet waves Visible light X-rays Nov 30, 2020 · If the object has a height of 4 millimeters, the height of the image is 8 millimeters. Solution: m = 16/8 = 2mm , object height =4 mm , height of image = 4 *2 = 8 mm. Added 264 days ago|4/10/2020 11:49:01 PM
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Find an answer to your question "Which of the following has the longest wavelength?A. violet B. ultraviolet radiation C. red D. infrared radiation ..." in 📘 Chemistry if you're in doubt about the correctness of the answers or there's no answer, then try to use the smart search and find answers to the similar questions. E. None of these is generally correct. 15. Which of the following elements will form a cation with a +2 charge? A. Si B. Sr C. Ga D. Cs E. S 16. Which of the following regions of the electromagnetic spectrum has the longest wavelengths? a) microwave. b) infrared c) x-ray d) gamma ray e) visible. 17. Tritium has a half-life of 12.5 y against beta decay. What fraction of a sample will remain undecayed after 25 y? In what distance will half of a beam of .025-eV neutrons have decayed? The half-life of the Solving for the fraction absorbed gives the following (note that if 99 percent pass through, 0.01...
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High School Physics Chapter 14 Section 1 Dragonflies have long, delicate, membranous wings which are transparent and some have light yellow colouring near the tips. Which of the following is, according to the author, NOT what the Maya civilization is famous for? 1. Knowledge. 2. Outstanding constructions.
At the simplest level, waves are disturbances that propagate energy through a medium. Propagation of the energy depend on interactions between the particles that make up the medium. Particles move as the waves pass through but there is no net motion of particles. This means, once a wave has passed the particles return to their original position. Question: Which lists the waves in order of wavelength, from longest to shortest? X-rays, infrared, radio waves infrared, visible light, gamma rays ultraviolet, microwaves, visible light radio waves, gamma rays, visible light
Dec 08, 2007 · Which of the following statements about mechanical waves is true? a. mechanical waves require a medium to travel through b. mechanical waves do not have amplitude and wavelength c. mechanical waves do not have frequency d. Wavelength. Wavelength is a property of a wave that most people (once they know what to look for) can spot quickly and easily, and use it as a way of telling waves apart. . Look at the following diagr
Another reason the show has been running for so long is that there is no main storyline, it is very much episodic, each episode telling a story of a separate Which of the following is NOT the reason why 'Doctor Who' has been around for so long? It is easy to change the actors playing the main character.Aug 15, 2020 · Rigel has an emission spectrum that peaks at ~145 nm. ... the longer the wavelenght of the photon, the less energy it has. Therefore, the longest wavelength photon ...
A. would give an emission line which has the longest wavelength? B. would give an absorption line for a photon of the greatest energy? B. represents ionization? 2. (25%) What is a neutrino, and why are astronomers so interested in detecting neutrinos coming from the Sun? 3. (15%) (Sample exam question).
Use braces to write the members of the following set, or state that the set has no members. (Dots may also be helpful.) The from whole number 0 to 30 (inclusive) Apr 30, 2008 · Which of the following would you expect to have the largest wavelength associated with it? (A) a quick-moving bee (B) an electron orbiting a nucleus (C) a slowly lumbering elephant (D) the earth orbiting the sun 2. RELEVANT EQUATIONS The wavelength of a particle is called the de Broglie wavelength. wavelength = [tex]\frac{h}{momentum}[/tex] h ... 15. For which of the following transitions would a hydrogen atom absorb a photon with longest wavelength? a. n = 1 to n = 2 b. n = 3 to n = 2 c. n = 5 to n = 6 d. n = 7 to n = 6 16. Which electronic transition in atomic hydrogen corresponds to the emission of visible light? a. n = 5 → n = 2 b. n = 1 → n = 2 c. n = 3 → n = 4 d. n = 32n = 1 Please answer all of the questions , also please dont answer if your not going to seriously answer my questions. FIRST TO ANSWER GETS 30 POINTSClassify the following as Which elements have 8 valence electrons? NASCAR fans love race day when they get a chance to cheer their favorite team!
What is the longest wavelength for standing waves on a 238.0 cm long string that is fixed at both ends? physics. What is the longest wavelength for standing waves on a 238.0 cm long string that is fixed at both ends? chemistry. An electron of wavelength 1.74*10-10m strikes an atom of ionized helium (He+).
Physics, 02.01.2020 04:28 123gra. Wich of the following has the longest wavelength in the electromagnetic pressure? 5 answers. 247 people helped. Answer: Red has the longest wavelength and violet has the shortest wavelength. When all the waves are seen together, they make white light. Ultraviolet (UV) light—is radiation with a wavelength shorter than that of visible light, but longer than X-rays, in the range 10 nm to 400 . Explanation:
Which of the following has the longest wavelength: radio, infrared, UV, or gamma waves (heat) have a wavelength just larger than the color red. 12. waves (cause skin burn) have a wavelength just shorter than the color violet. 13. The wave that has the longest wavelength and lowest frequency is waves. 14. waves occur when the motion of the medium being disturbed is parallel to the direction of the wave (back and forth motion).
Which of the following types of radiation has the longest wavelength? a) x-rays b) visible c) radio d) gamma rays e) microwaves Answer: c 3. Which of the following colors of visible light has the shortest wavelength? Wave number (ṽ) is inversely proportional to wavelength of transition. Hence, for the longest wavelength transition, ṽ has to be the smallest. For ṽ to be minimum, n f should be minimum. For the Balmer series, a transition from n i = 2 to n f = 3 is allowed. Hence, taking n f = 3,we get: ṽ= 1.5236 × 10 6 m –1 Mar 05, 2014 · Which of the following is true about all electromagnetic waves in a vacuum? A. they all carry the same energy B. they all travel at the same speed C. they all have the same frequency D. they all have the same wavelength I think . Physics . A wave travels at a speed of 34 cm/s. It's wavelength is 18 cm.
Which of the following commands can be used to access the home directory of the user "bob" Any two single characters. Nothing; it has no special meaning. The directory above the current working …lists the contents of the current directory. The first character in a long listing (ls -l) indicates
Electromagnetic Spectrum : The electromagnetic spectrum covers a wide range of wavelengths and photon energies. Light used to "see" an object must have a wavelength about the same size as or smaller than the object. Jun 21, 2020 · Of the following, _____ radiation has the longest wavelength and _____ radiation has the greatest energy.gamma ultraviolet visibleGroup of answer choicesgamma, visiblegamma, gammavisible, gammav … read more
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This set of Python Multiple Choice Questions & Answers (MCQs) focuses on "Precedence and Associativity - 1". 1. The value of the expressions 4/(3*(2-1)) and 4/3*(2-1) is the same. a) True b) False View Answer. Answer: a Explanation: Although the presence of parenthesis does affect the...Tritium has a half-life of 12.5 y against beta decay. What fraction of a sample will remain undecayed after 25 y? In what distance will half of a beam of .025-eV neutrons have decayed? The half-life of the Solving for the fraction absorbed gives the following (note that if 99 percent pass through, 0.01...Which of the following color groups shows increasing wavelength? A) green, blue, red B) blue, yellow, violet C) yellow, orange, red D) orange, yellow, green : C: The portion of the electromagnetic spectrum that has longer wavelengths than visible light and transmits heat (thermal) energy is _____. Because photon energy increases as wavelength decreases, any photon of wavelength shorter than 242 nm will have sufficient energy to dissociate O 2. PRACTICE EXERCISE. The bond energy in N 2 is 941 kJ/mol (Table 8.4). What is the longest wavelength photon that has sufficient energy to dissociate N 2? Answer: 127 nm Long…. at check-in. Poor quality ….and drink. Listen to three people talking about their travel experiences. Tick the problems in exercise above that they mention. Which of the things in Exercise 4 irritate you most?Question 23 1 / 1 pts A green laser pointer has a wavelength of 532 nm. What is the energy of one mol of photons generated from this device? 2.25 kJ/mol 3.74 x 10-19 kJ/mol 3.74 x 10-17 kJ/mol 225 kJ/mol 784 kJ/mol Question 24 1 / 1 pts A scientist shines light with energy greater than the binding energy of platinum on a thin film of it. We are familiar with radio waves from UHF, VHF, FM and AM transmissions. They have very long wavelengths. AM radio waves have the longest wavelengths in this group, and thus the smallest frequencies. UV, X-ray, Gamma-Rays. These shorter wavelength, higher energy rays are largely blocked out by the Earth's atmosphere. "For the human eye, the visible radiations range from violet light, in which the shortest rays are about 380 nanometers, to red light, in which the longest rays are about 750 nanometers." 380–750 nm: Chambers Cambridge. Chambers Science & Technology Dictionary. New York: W & R Chambers Limited, 1940: 914. Each color has a different wavelength. Red has the longest wavelength and violet has the shortest wavelength. Color Wavelength Frequency Photon energy Violet 380–450 nm 680–790 THz 2.95–3.10 eV Blue 450–485 nm 620–680 THz 2.64–2.75 eV Cyan 485–500 nm 600–620 THz 2.48–2.52 eV Green 500–565 nm 530–600 THz 2.25–2.34 eV Yellow 565–590 nm 510–530 THz 2.10–2.17... Select all that apply SVG needs scripts to draw elements. SVG has better accessibility. In Canvas, drawing is done with pixels. Canvas contains built-in animations.100.2 Of the following types of electromagnetic radiation, which has the second shortest wavelength? (C) infra-red light. --- Yes. Just longer than red light. Check the other answers.
Wavelength ( Top, Wave Home) The wavelength of a wave is the distance between any two adjacent corresponding locations on the wave train. This distance is usually measured in one of three ways: crest to next crest, trough to next trough, or from the start of a wave cycle to the next starting point. This is shown in the following diagram: This shows that the wavelength is inversely proportional to the energy: the smaller the amount of energy absorbed, the longer the wavelength. We see from the energy level diagram that the energy levels get closer together as #n# increases. This, the smallest energy and the longest wavelength is...
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narrow range of wavelengths. The shortest wavelength blue light which is visible has a wavelength of microns (one micron is meters). The longest wavelength red light which is visible has a wavelength of microns. However, there is nothing in Maxwell's analysis which suggested that Radio waves have the longest wavelengths in the EM spectrum, according to NASA, ranging from about 0.04 inches (1 millimeter) to more than 62 Within that range, the shortwave spectrum is divided into several segments, some of which are dedicated to regular broadcasting stations, such as the...
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which of the following is true when comparing infrared light and radio waves? a. infrared light has a shorter wavelength and travels slower than radio waves b. infrared light has a higher frequency and travels faster than radio waves c. radio waves have a lower frequency and travel at the same speed as infrared light Dragonflies have long, delicate, membranous wings which are transparent and some have light yellow colouring near the tips. Which of the following is, according to the author, NOT what the Maya civilization is famous for? 1. Knowledge. 2. Outstanding constructions.Wavelength Division Multiplexing (WDM) has ... Design (VTD) problem [2], the following subproblems (which are not necessarily independent) ... wavelengths to the longest lightpaths first, as a A simple tool to convert a wavelength in nm to an RGB, hexadecimal or HSL colour. Physics Light Colour Over the course of millions of years, the human eye has evolved to detect light in the range 380—780nm, a portion of the electromagnetic spectrum known as visible light , which we perceive as colour. He received the 1929 Nobel Prize in Physics for this work. (I will discuss the second de Broglie equation below the following example problems.) Equation Number Two: λ = h/p. There are three symbols in this equation: a) λ stands for the wavelength of the particle b) h stands for Planck's Constant c) p stands for the momentum of the particle Determine which of the following H atom electron transitions has the longest wavelength and which has the shortest wavelength: (a) n=1 to n=6 (b) n=3 to n=5 (c) n=3 to n=10 (d) n = 2 to n=11 Longest-wavelength photon: Od Shortest-wavelength photon: 15. Which of the following transitions in a hydrogen atom would emit the highest energy photon ...
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Visible light has a wavelength range between _____ at the violet end and _____ at the red end., 380 nanometers (violet end), 750 nm at the red end, Violet light has ____ energy, _____ wavelength and _____ frequency compared to red light. violet light has greater energy, shorter wavelength and higher frequency compared to red light Another reason the show has been running for so long is that there is no main storyline, it is very much episodic, each episode telling a story of a separate Which of the following is NOT the reason why 'Doctor Who' has been around for so long? It is easy to change the actors playing the main character.Other O-O Bonds We can obtain experimental data on bond energies of other molecules in the same way. Molecular oxygen, O 2, is photolyzed by light of 241 nm and has a bond energy of 498 kJ/mol. Hydrogen peroxide, HOOH, has a very weak O-O bond and is photolyzed by light of 845 nm.
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Each wavelength of light has a particular energy associated with it. Home Page. With R = 1. Other common colors of the spectrum, in order of decreasing wavelength, may be remembered by the mnemonic: ROY G BIV . For which of the following transitions does the light emitted have the longest wavelength? n = 4 to n = 3. Diffraction. 23E-9). Calculate the longest wavelength of light that can break a H-O bond. The bond enthalpy of formation is 463 kj/mol. asked by Angelina on April 8, 2011. Atomic physics 1)What is the longest wavelength light capable of ionizing a hydrogen atom in the n = 6 state 2)a. 1) If radio waves have wavelengths around 100 meters long and a frequency of about 10*7 Hz, are there any wavelengths in the EM Spectrum that are longer? 3) I don't follow... gravity waves are distortions of spacetime travelling at c, and the "frequency" (or the volume of distortion) of the gravity...A sound wave emanates from a source vibrating at a frequency f, propagates at V w, and has a wavelength λ. Table 1 makes it apparent that the speed of sound varies greatly in different media. The speed of sound in a medium is determined by a combination of the medium's rigidity (or compressibility in gases) and its density. Oct 16, 2009 · a and d E ranges seize up with jointly by way of fact the commencing n fee gets greater advantageous. So transitions of two n values are decrease power (longer wavelength) ranging from n = 2 or 3 than from n = a million. I wonder which of the following absorbs photons with the longest wavelength, $\ce{[Cr(CN)6]^3-}$, $\ce{[Cr(SCN)6]^3-}$ and $\ce{[Cr(H2O)6]^3+}$. I know the one to absorb the highest wavelength should also have the highest spin. Which one of the following has the longest wavelength? Complete the following sentence describing the relationship between the energy, wavelength, and frequency of light using the words highest, lowest, longest, and/or shortest.3. I have often regretted my speech, never my silence. Anonymous. 4. First learn the meaning of Unit 4: Student's Book. ⟹ Answer the following questions. 1. Who do you communicate with every day? 1. be driving at sth 2. (can't) get a word in edgeways 3. be on the same wavelength 4. have a quick word and, without exception, spoke for a longer time. … Who talks more, then, women or men? The.
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(a) Longer wavelength; (b) shorter wavelength 5. Which has more energy, A or B? Explain your reasoning. 6. Define a wavelength. 7. What is a frequency of a wavelength? 8. For visible light (ROYGBIV), which has the shortest wavelength? Which has the longest? Place all the other colors in order based on their wavelength from shortest to longest. FREQUENCY AND PITCH. After reading this section you will be able to do the following:. Explain how you can change pitch by altering sources. Describe what resonance is. Nov 12, 2011 · Transitions to n=3 or greater are in the infra-red part of the spectrum and have the longest wavelengths. The longest wavelength would correspond to the lowest energy and that would correspond to the difference in energy between n=5 and n=4. I wonder which of the following absorbs photons with the longest wavelength, $\ce{[Cr(CN)6]^3-}$, $\ce{[Cr(SCN)6]^3-}$ and $\ce{[Cr(H2O)6]^3+}$. I know the one to absorb the highest wavelength should also have the highest spin. Estimate the work function of aluminum, given that the wavelength of 304 nm is the longest wavelength that a photon may have to eject a photoelectron from aluminum photoelectrode. 68 . What is the maximum kinetic energy of photoelectrons ejected from sodium by the incident radiation of wavelength 450 nm? Which of the following architecture has feedback connections? Which of the following statements is true? A. There will not be any problem and the neural network will train properly. B. The neural network will train but all the neurons will end up recognizing the same thing.The compounds that has highly extended conjugation occurs at long wavelength. The absorption does not occur at long wavelength if the compound is unconjugated. 1 Which of the following visible colors of light has the longest wavelength? red. blue. green. yellow. 2 When an electron falls from one energy level to a lower energy level in an atom; Energy is absorbed. The atom will be in the excited state. The atom will become an ion. Energy is emitted. He received the 1929 Nobel Prize in Physics for this work. (I will discuss the second de Broglie equation below the following example problems.) Equation Number Two: λ = h/p. There are three symbols in this equation: a) λ stands for the wavelength of the particle b) h stands for Planck's Constant c) p stands for the momentum of the particle
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The blue curve in the top right diagram has only quarter of a cycle of a sine wave, so the longest sine wave that fits into the closed pipe is four times as long as the pipe. Therefore a clarinet can produce a wavelength that is about four times as long as a clarinet, which is about 4L = 2.4 m.
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2. Which of the following has the most effect on the speed of sound in air? a. amplitude b. frequency c. wavelength --> d. temperature. a. increasing the mass of the string b. decreasing the tension in the string --> c. fingering the string d. plucking the string harder.Jun 11, 2020 · Welcome to the ultimate Key Stage 3 (KS3) Science. It's a perfect practice test for students in Year 7, Year 8, and Year 9. With over 30 questions, it covers the fundamental topics of science. So, if you are interested in having a practice session, then take this quiz right now. Violet has the most energy in the visible light spectrum.This is because it has the shortest wavelength (approximately 400nm) and highest frequency meaning more waves and more energy is being transferred from point A to point B in a very short amount of time. The electromagnetic spectrum ranges from gamma (γ) radiation, which has the shortest wavelength, highest frequency, and greatest energy, to radio waves, which has the longest wavelength and lowest frequency and energy. Ultraviolet light (UV) is divided into three regions: UV A, wavelength = 400 - 320 nm UV B, wavelength = 320 - 280 nm
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Sep 21, 2016 · 2. Which of the following has the longest wavelength? Select one: a.ultraviolet b.radio waves c.visible light d.X-rays? Electromagnetic Waves have different wavelengths. Wavelength is the distance between one wave crest to the next. Waves in the electromagnetic spectrum vary in size from very long radio waves the size of buildings, to very short gamma-rays smaller than the size of the nucleus of an atom.As you can see, the wavelength of the red light shortens in the glass, and so does the speed of the light. But the frequency remains the same, and so does the color. Thus, red light of wavelength 700 nm in air will appear the same color as red light of wavelength 438 nm in this type of glass. 5 answers. 247 people helped. Answer: Red has the longest wavelength and violet has the shortest wavelength. When all the waves are seen together, they make white light. Ultraviolet (UV) light—is radiation with a wavelength shorter than that of visible light, but longer than X-rays, in the range 10 nm to 400 . Explanation: Each color has a different wavelength. Red has the longest wavelength and violet has the shortest wavelength. What is the longest wavelength of light that can be absorbed by a hydrogen atom that is initially in the? Which of the following transitions in the hydrogen atom has the longest wavelength of emitted light? Where 6 626 10 34 planck s constant and 3 00 108 the speed of light what is the wavelength of a photon that has an energy of 4 ? The Longest Ride "After he'd done about 30 turns, he looked over to me and shouted 'what happens next?' I told him to relax, he had another hour to go!" Sat in an inflatable boat in the middle of an eerily calm river, the man at the helm, Duncan Milne, is recounting the tale of […] I have a vague idea. Everything looked vague in the heavy fog. ramble = to talk or write for a long time in a Really effective communicators who have the ability to engage with colleagues, employees Which of the problems mentioned in Exercise A do the speakers have when communicating?49 A C 9 H 12 O 3 compound has two strong infrared absorptions between 1100 and 1250 cm-1 and at 1600 cm-1. The 1 H NMR spectrum has sharp singlet peaks at δ 3.6 and 6.6 ppm (intensity ratio 3:1). The 13 C NMR spectrum shows three lines at δ 165, 115 and 55 ppm. Which of the following compounds best fits this data? A) 1,3,5-trimethoxybenzene The number of photons of light having wavelength 100 nm which can provide 1.00 Jenergy is nearly For the following reaction in equilibrium PCl5(g)⇌PCl3(g)+Cl2(g) Vapour density is found to be 100 w... Nov 30, 2020 · If the object has a height of 4 millimeters, the height of the image is 8 millimeters. Solution: m = 16/8 = 2mm , object height =4 mm , height of image = 4 *2 = 8 mm. Added 264 days ago|4/10/2020 11:49:01 PM
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Dec 01, 2016 · 4.Which part of the Sun's electromagnetic spectrum has the longest wavelength? A)violet B)blue C)yellow D)red 5.Which color of the visible spectrum has the shortest wavelength? A)nuclear B)solar C)coal D)natural gas 6.Which of the sources of energy listed below is most nearly pollution free? A)Violet light has a longer wavelength than red light. The lower wave has the longer wavelength (lower frequency) and would be the red light. PRACTICE EXERCISE If one of the waves in the image 24. SAMPLE EXERCISE 6.4 Electronic Transitions in the Hydrogen Atom Using Figure 6.13 , predict which of the following electronic transitions produces...microwaves have the lowest frequency and the longest wavelength.Calculate the wavelength of a photon emitted when the hydrogen atom undergoes a transition from the n = 5 to the n = 3 state (one of the Paschen series of emission lines). R = 1.097×10 7 m -1 A.
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Which of the following complex ions will absorb the longest wavelength of light? A) [Mn(NH 3) 6] 2+ B) [Mn(NH 3) 6] 3+ C) [Mn(CN) 6] 4-D)[Mn(CN) 6] 3-I know that: (* represents the triangle, change in) *o is highest in D) [Mn(CN) 6] 3-. and lowest in A) [Mn{HN 3) 6] 2+ An angstrom is 10-8 cm or 0.00000001 cm. Red has a wavelength of 7800-6220 angstroms, green has a wavelength of 5770-4920 angstroms and violet has a wavelength of 4550-3900 angstroms. However, in this lab, the simple relationship among the visible light waves will be what is important.) ⚛ Question - Which of the following colour has longest wavelength in the visible spectrum? ☑ Answer - Red. ☀ Click for more questions. ☂ As you can see, the wavelength of the red light shortens in the glass, and so does the speed of the light. But the frequency remains the same, and so does the color. Thus, red light of wavelength 700 nm in air will appear the same color as red light of wavelength 438 nm in this type of glass. "For the human eye, the visible radiations range from violet light, in which the shortest rays are about 380 nanometers, to red light, in which the longest rays are about 750 nanometers." 380–750 nm: Chambers Cambridge. Chambers Science & Technology Dictionary. New York: W & R Chambers Limited, 1940: 914. FREQUENCY AND PITCH. After reading this section you will be able to do the following:. Explain how you can change pitch by altering sources. Describe what resonance is.
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As an object moves away from us, the sound or light waves emitted by the object are stretched out, which makes them have a lower pitch and moves them towards the red end of the electromagnetic spectrum, where light has a longer wavelength. In the case of light waves, this is called redshift. As an object moves towards us, sound and light waves ... Nov 12, 2011 · Favorite Answer Transitions to n=1 are in the UV part of the spectrum and have the shortest wavelengths. Transitions to n=2 are in the visible part of the spectrum and have wavelengths between... Question: Which lists the waves in order of wavelength, from longest to shortest? X-rays, infrared, radio waves infrared, visible light, gamma rays ultraviolet, microwaves, visible light radio waves, gamma rays, visible light Mar 30, 2020 · Scientists measure the wavelength of radiation from one wave crest to another. Electromagnetic waves are found in a number of different objects throughout the home. This type of radiation is common in microwaves, televisions and radios.
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The wavelength at which a star emits the most light is called the star's peak wavelength. The diagram on the right shows that this star has a peak wavelength of 4000 Angstroms. Question 2.
Wavelength is one of the important characteristics of a waveform apart from amplitude, velocity and frequency. Learn about the unit of wavelength and It is the space generated in a periodic wave—the distance over which the wave's shape is repeated. It is usually measured or determined by taking the...11. Having completed the job, the man left early. 12. The task having been accomplished, the pilot returned to the base. 13. Having been rebuilt recently, the house is very attractive now. 14. Scientists are interested in developing new programming languages. 15. What do you think of the methods...Long…. at check-in. Poor quality ….and drink. Listen to three people talking about their travel experiences. Tick the problems in exercise above that they mention. Which of the things in Exercise 4 irritate you most?
Oct 06, 2009 · E) Orange light. Gamma rays, UV an X rays have shorter wavelengths than light, and orange light has a longer wavelength (and lower frequency and lower photon energy) than blue light. Free essays, homework help, flashcards, research papers, book reports, term papers, history, science, politics | CommonCrawl |
\begin{document}
\title[Orbital stability of ground states for a Sobolev critical Schr\"odinger equation]{Orbital stability of ground states \\ for a Sobolev critical Schr\"odinger equation}
\author[L. Jeanjean, J. Jendrej, T.T. Le and N. Visciglia]{Louis Jeanjean, Jacek Jendrej, Thanh Trung Le and Nicola Visciglia}
\address{
\newline
\textbf{{\small Louis Jeanjean}}
\newline \indent Laboratoire de Math\'{e}matiques (CNRS UMR 6623), Universit\'{e} de Bourgogne Franche-Comt\'{e}, Besan\c{c}on 25030, France} \email{[email protected]}
\address{
\newline
\textbf{{\small Jacek Jendrej}}
\newline \indent CNRS and LAGA (CNRS UMR 7539), Universit\' e Sorbonne Paris Nord, Villetaneuse 93430, France} \email{[email protected]}
\address{
\newline
\textbf{{\small Thanh Trung Le }}
\newline \indent Laboratoire de Math\'{e}matiques (CNRS UMR 6623), Universit\'{e} de Bourgogne Franche-Comt\'{e}, Besan\c{c}on 25030, France} \email{thanh\[email protected]}
\address{
\newline
\textbf{{\small Nicola Visciglia }}
\newline \indent Dipartimento di Matematica, Universit\`a Degli Studi di Pisa, Largo Bruno Pontecorvo, 5, 56127, Pisa, Italy} \email{[email protected]}
\thanks{J. Jendrej is supported by ANR-18-CE40-0028 project ESSED and Chilean projects FONDECYT 1170164 and France-Chile ECOS-Sud C18E06 project. N.V. is supported by PRIN grant 2020XB3EFL and by the Gruppo Nazionale per l’ Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituzione Nazionale di Alta Matematica (INDAM)}
\date{} \subjclass[2010]{} \keywords{} \maketitle
\centerline {\em Dedicated to the memory of Professor Jean Ginibre} \begin{abstract}
We study the existence of ground state standing waves, of prescribed mass, for the nonlinear Schr\"{o}dinger equation with mixed power nonlinearities
\begin{align*}
i \partial_t v + \Delta v + \mu v |v|^{q-2} + v |v|^{2^* - 2} = 0, \quad (t, x) \in \mathbb{R} \times \mathbb{R}^N,
\end{align*}
where $N \geq 3$, $v: \mathbb{R} \times \mathbb{R}^N \to \mathbb{C}$, $\mu > 0$, $2 < q < 2 + 4/N $ and $2^* = 2N/(N-2)$ is the critical Sobolev exponent. We show that all ground states correspond to local minima of the associated Energy functional. Next, despite the fact that the nonlinearity is Sobolev critical, we show that the set of ground states is orbitally stable. Our results settle a question raised by N. Soave \cite{Soave2020Sobolevcriticalcase}.
\\ \end{abstract}
\section{Introduction}
In this paper, we study the existence and orbital stability of ground state standing waves of prescribed mass for the nonlinear Schr\"{o}dinger equation with mixed power nonlinearities \begin{align}
i \partial_t v + \Delta v + \mu v |v|^{q-2} + v |v|^{2^* - 2} = 0, \quad (t, x) \in \mathbb{R} \times \mathbb{R}^N, \label{NLS0} \end{align} where $N \geq 3$, $v: \mathbb{R} \times \mathbb{R}^N \to \mathbb{C}$, $\mu > 0$, $2 < q < 2 + \dfrac{4}{N}$ and $2^* = \dfrac{2N}{N-2}$. \\ The nonlinear Schr\"{o}dinger equation (NLS) with pure and mixed power nonlinearities has attracted much attention in the last decades. The local existence result for the pure power energy critical NLS has been established in \cite{CazenaveWeissler1990}. The corresponding global existence and scattering for defocusing quintic NLS in dimension $N=3$ has been established in the papers \cite{Bourgain1999, CollianderKeelStaffilaniTakaokaTao2008} respectively in the radial and non-radial case. We also quote the concentration-compactness/rigidity approach introduced in \cite{KenigMerle2006} in order to study global existence and scattering in the focusing energy critical NLS below the ground state. Concerning the case of NLS with mixed nonlinearities let us quote
\cite{TaoVisanZhang07, AkahoriIbrahimKikuchiNawa2012,AkahoriIbrahimKikuchiNawa2013,ChenMiaoZhao2016, ColesGustafson20,LewinRotaNodari2020,MiaoXuZhao2013,MiaoZhaoZheng2017}.
We recall that standing waves to \eqref{NLS0} are solutions of the form $v(t,x) = e^{-i\lambda t}u(x), \lambda \in \mathbb{R}$. Then the function $u(x)$ satisfies the equation \begin{align} -\Delta u - \lambda u - \mu \abs{u}^{q-2} u - \abs{u}^{2^*-2} u = 0 \quad \mbox{in } \mathbb{R}^N. \label{eqn:Laplace} \end{align}
When looking for solutions to \eqref{eqn:Laplace} a possible choice is to consider $\lambda \in \mathbb{R}$ fixed and to search for solutions as critical points of the action functional $$\mathcal{A}_{\lambda, \mu}(u) := \dfrac{1}{2} \normLp{\nabla u}{2}^2 - \dfrac{\lambda}{2} \normLp{u}{2}^2 - \dfrac{\mu}{q} \normLp{u}{q}^q - \dfrac{1}{2^*}\normLp{u}{2^*}^{2^*}.$$ In this case one usually focuses on the existence of minimal action solutions, namely of solutions minimizing $\mathcal{A}_{\lambda, \mu}$ among all non-trivial solutions. In that direction, we refer to \cite{AlvesSoutoMontenegro2012} where, relying on the pioneering work of Brezis-Nirenberg \cite{BrezisNirenberg1983}, the existence of positive real solutions for equations of the type of \eqref{eqn:Laplace} is addressed in a very general setting; to \cite{AkahoriIbrahimKikuchiNawa2012,AkahoriIbrahimKikuchiNawa2013} which concerns the case where $q > 2 + 4/N$ and $\mu >0$; to \cite{ChenMiaoZhao2016,MiaoXuZhao2013} where the fixed $\lambda \in \mathbb{R}$ problem is analyzed for $q = 2 + 4/N$ and $\mu <0$;
see also \cite{LewinRotaNodari2020} and the reference therein.
Alternatively, one can search for solutions to \eqref{eqn:Laplace} having a prescribed $\mathit{L}^2$-norm. Defining on $H:= H^1(\mathbb{R}^N, \mathbb{C})$ the Energy functional \begin{equation*} F_{\mu}(u) := \dfrac{1}{2} \normLp{\nabla u}{2}^2 - \dfrac{\mu}{q} \normLp{u}{q}^q - \dfrac{1}{2^*}\normLp{u}{2^*}^{2^*} \end{equation*} it is standard to check that $F_{\mu}$ is of class $C^1$ and that a critical point of $F_{\mu}$ restricted to the (mass) constraint \begin{equation*} S(c) := \{u \in H: \normLp{u}{2}^2 = c\} \end{equation*} gives rise to a solution to \eqref{eqn:Laplace}, satisfying $\normLp{u}{2}^2 = c.$
In this approach the parameter $\lambda \in \mathbb{R}$ arises as a Lagrange multiplier. In particular, $\lambda \in \mathbb{R}$ does depend on the solution and is not a priori given. This approach, that we shall follow here, is relevant from the physical point of view, in particular, since the $L^2$ norm is a preserved quantity of the evolution and since the variational characterization of such solutions is often a strong help to analyze their orbital stability, see for example, \cite{BellazziniJeanjeanLuo2013,CazenaveLions1982,Soave2020,Soave2020Sobolevcriticalcase}.
We shall focus on the existence of ground state solutions.
\begin{definition}
We say that $u_c \in S(c)$ is a ground state solution to \eqref{eqn:Laplace} if it is a solution having minimal Energy among all the solutions which belong to $S(c)$. Namely, if
$$\quad F_{\mu}(u_c) = \displaystyle \inf \big\{F_{\mu}(u), u \in S(c), \big(F_\mu\big|_{S(c)}\big)'(u) = 0 \big\}.$$ \end{definition} Note that this definition keeps a meaning even in situations where the Energy $F_{\mu}$ is unbounded from below on $S(c)$. Implicit in \cite{JEANJEAN1997}, this definition was formally introduced, on a related model, in \cite{BellazziniJeanjean2016} and is now becoming standard.
It is well-known that the study of problems with mixed nonlinearities and the type of results one can expect, depend on the behavior of the nonlinearities at infinity, namely on the value of the various power exponents. In particular, this behavior determines whether the functionnal is bounded from below on $S(c)$. One speaks of a mass subcritical case if it is bounded from below on $S(c)$ for any $c>0$, and of a mass supercritical case if the functional is unbounded from below on $S(c)$ for any $c>0$. One also refers to a mass critical case when the boundedness from below does depend on the value $c>0$. To be more precise, consider an equation of the form \begin{align}
i \partial_t v + \Delta v + \mu v |v|^{p_1-2} + v |v|^{p_2 - 2} = 0, \quad (t, x) \in \mathbb{R} \times \mathbb{R}^N, \label{NLS0E} \end{align} where it is assumed that $2< p_1 \leq p_2 \leq 2^*.$ The threshold exponent is the so-called $L^2$-critical exponent $$p_c = 2 + \frac{4}{N}.$$ A very complete analysis of the various cases that may happen for \eqref{NLS0E}, depending on the values of $(p_1,p_2)$, has been provided recently in \cite{Soave2020,Soave2020Sobolevcriticalcase}. Let us just recall here some rough elements. If both $p_1$ and $p_2$ are strictly less than $p_c$ then the associated Energy functional is bounded from below on $S(c)$ and to find a ground state one looks for a global minimum on $S(c)$. The problem then directly falls into the setting covered by the Compactness by Concentration Principle introduced by P.L. Lions \cite{LIONS1984-1, LIONS1984-2} which, for more complicated equations, in particular non autonomous ones, is still a very active field. Such solutions are expected to be orbitally stable, see \cref{def:stability} below. If $p_c \leq p_1 \leq p_2 \leq 2^*$, then the Energy functional is unbounded from below on $S(c)$ but it is possible to show that a ground state exists. This ground state is characterized as a critical point of {\it mountain-pass type} and it lies at a strictly positive level of the Energy functional. Such ground states are expected to be orbitally unstable. We refer, for the link between the variational characterization of a solution and its instability, to the classical paper \cite{BerestyckiCazenave1981}, and to \cite{JEANJEAN1997,Lecoz2008,Soave2020,Soave2020Sobolevcriticalcase} for more recent developments.
In the case we consider here : $2 < p_1 < p_c <p_2 = 2^*$, the Energy functional is thus unbounded from below on $S(c)$ but, as we shall see, the presence of the lower order, mass subcritical term $- \mu \normLp{u}{q}^q$ creates, for sufficiently small values of $c>0$, a geometry of local minima on $S(c).$ The presence of such geometry, in problems which are mass supercritical, had already been observed in several related situations. In \cite{BellazziniJeanjean2016, BellazziniBoussaidJeanjeanVisciglia17} for related scalar problems, in \cite{GouJeanjean2018} in the case of a system or \cite{NorisTavaresVerzini2019} for an evolution problem set on a bounded domain. Actually, it was already observed on \eqref{NLS0} in \cite{Soave2020Sobolevcriticalcase}.
Precisely, for any fixed $\mu >0$, we shall find an explicit value $c_0 = c_0(\mu) >0$ such that, for any $c \in (0, c_0)$, there exists a set $V(c) \subset S(c)$ having the property that \begin{equation}\label{well} m(c) := \inf_{u \in V(c)} F_{\mu}(u) < 0 < \inf_{u \in \partial V(c)}F_{\mu}(u). \end{equation} The sets $V(c)$ and $\partial V(c)$ are given by $$V(c) := \{ u \in S(c) : \normLp{\nabla u}{2}^2 < \rho_0\}, \qquad \partial V(c) := \{ u \in S(c) : \normLp{\nabla u}{2}^2 = \rho_0\}$$ for a suitable $\rho_0 >0$, depending only on $c_0 >0$ but not on $c \in (0,c_0)$. We also introduce the set \begin{align*} \mathcal{M}_c := \{u \in V(c) : F_{\mu}(u) = m(c)\}. \end{align*} Our first result is, \begin{theorem}\label{thm-1}
Let $N \geq 3$, $2 < q < 2 + \frac{4}{N}$. For any $\mu >0$ there exists a $c_0 = c_0(\mu) >0$ such that, for any $c \in (0, c_0)$, $F_{\mu}$ restricted to $S(c)$
has a ground state. This ground state is a (local) minimizer of $F_{\mu}$ in the set $V(c)$ and any ground state for $F_{\mu}$ on $S(c)$ is a local minimizer of $F_{\mu}$ on $V(c)$. In addition, if $(u_n) \subset V(c)$ is such that $F_{\mu}(u_n) \to m(c)$ then, up to translation, $u_n \to u \in \mathcal{M}_c$ in $\mathit{H}^1(\mathbb{R}^N,\mathbb{C})$. \end{theorem} \begin{remark}
The value of $c_0 = c_0(\mu) >0$ is explicit and is given in \eqref{eqn:5.14}-\eqref{eqn:5.14B}. In particular $c_0 >0$ can be taken arbitrary large by taking $\mu >0$ small enough. \end{remark}
\begin{remark} \label{remark-th1}$ $ \begin{itemize} \item[(i)] If $u \in S(c)$ is a ground state then the associated Lagrange multiplier $\lambda \in \mathbb{R}$ in \eqref{eqn:Laplace} satisfies $\lambda < 0$. This follows directly combining that $u$ being a solution to \eqref{eqn:Laplace} it satisfies $\normLp{\nabla u}{2}^2 - \lambda \normLp{u}{2}^2 - \mu \normLp{u}{q}^q - \normLp{u}{2^*}^{2^*} =0$ with the fact that $F_{\mu}(u) = m(c) <0.$ \smallbreak \item[(ii)] There exists a ground state which is a real valued, positive, radially symmetric decreasing function. Indeed if $u \in S(c)$ is a ground state then its Schwartz symmetrization is clearly also a ground state. \smallbreak \item[(iii)] More globally, under the assumption of \cref{thm-1} it can be proved that, for any $c \in (0, c_0)$, $\mathcal{M}_c$ has the following structure:
$$\mathcal{M}_c = \{ e^{i \theta}u, \mbox{ for some } \theta \in \mathbb{R}, u \in \tilde{\mathcal{M}}_c, u >0 \},$$
where
$$\tilde{\mathcal{M}}_c = \{ u \in S(c) \cap H^1(\mathbb{R}^N, \mathbb{R}), F_{\mu}(u) = m(c)\}.$$
Indeed, this description directly follows from the convergence, up to translation, of the minimizing sequences of $F_{\mu}$ restricted
to $V(c)$, applying the argument of \cite[Section 3]{HajaiejStuart2004}. We leave the details to the interested reader. \end{itemize} \end{remark}
We shall now focus on the (orbital) stability of the set $\mathcal{M}_c$. Following the terminology of \cite{CazenaveLions1982}, see also \cite{HajaiejStuart2004}, we give the following definition. \begin{definition}\label{def:stability}
$Z \subset {\mathit H}$ is stable if : $Z \neq \emptyset $ and for any $v \in Z $ and any $ \varepsilon >0$, there exists a $\delta >0$ such that if $\varphi \in {\mathit H}$ satisfies $||\varphi- v||_{{\mathit H}} < \delta$ then
$u_{\varphi}(t)$ is globally defined and $\inf_{z \in Z} ||u_{\varphi}(t) - z||_{{\mathit H}} < \varepsilon$ for all
$t \in \mathbb{R}$, where $u_{\varphi}(t)$ is the solution to \eqref{NLS0} corresponding to the initial condition $\varphi$. \end{definition}
Notice that the orbital stability of the set $Z$ implies the global existence of solutions to \eqref{NLS0} for initial datum $\varphi$ close enough to the set $Z$. We underline that this fact is non trivial due to the critical exponent that appears in \eqref{NLS0}, even if the $H$ norm of the solution is uniformly bounded on the lifespan of the solution.
The fact that ground states are characterized as local minima suggests, despite the problem being mass supercritical, that the set $\mathcal{M}_c$ could be orbitally stable. Actually, such orbital stability results have now been proved, on related problems (but always Sobolev subcritical) in several recent papers \cite{BellazziniBoussaidJeanjeanVisciglia17,GouJeanjean2018,Soave2020}. Along this line we now present the main result of this paper. \begin{theorem}\label{thm-2}
Let $N \geq 3$, $2 < q < 2 + \frac{4}{N}$, $\mu >0$ and $c_0 = c_0(\mu) >0$ be given in \cref{thm-1}. Then, for any $c \in (0, c_0),$ the set $\mathcal{M}_c$ is compact, up to translation, and it is orbitally stable. \end{theorem}
In \cite{Soave2020Sobolevcriticalcase}, Soave studied equation \eqref{NLS0} and derived, for any small $c>0$ depending on $\mu>0$, an existence result which is very similar to the one contained in \cref{thm-1}, see \cite[Theorem 1.1]{Soave2020Sobolevcriticalcase}. Actually, the motivation of our study originated from \cite{Soave2020Sobolevcriticalcase} where what is now our \cref{thm-2} was proposed as an open problem. However, it does not seem possible to use \cite[Theorem 1.1]{Soave2020Sobolevcriticalcase} as a starting point to prove \cref{thm-2}. The existence of a ground state in \cite[Theorem 1.1]{Soave2020Sobolevcriticalcase} is obtained through the study of one particular (locally) minimizing sequence which is radially symmetric. As already explained in \cite{Soave2020Sobolevcriticalcase}, to obtain the orbital stability of the set $\mathcal{M}_c$, following the classical approach laid down in \cite{CazenaveLions1982}, two ingredients are essential. First, the relative compactness, up to translation, of all minimizing sequences for $F_{\mu}$ on $V(c)$, as guaranteed by our \cref{thm-1}. Secondly, the global existence of solutions to \eqref{NLS0} for initial data close to $\mathcal{M}_c$.
To obtain the relative compactness of all minimizing sequences, the fact that one minimizes only on a subset of $S(c)$, in contrast to a global minimization on all $S(c)$, increases the difficulty to rule out a possible {\it dichotomy}. Different strategies have been recently implemented to deal with this issue \cite{BellazziniBoussaidJeanjeanVisciglia17,GouJeanjean2018,Soave2020}, all relying on a suitable choice of the set where the local minima are searched. In the presence of a Sobolev critical term an additional difficulty arises. In a Sobolev subcritical setting, if a sequence $(v_n) \subset S(c)$ is {\it vanishing} then applying \cite[Lemma I.1]{LIONS1984-2} one would immediately get
$$\liminf_{n \to \infty} F_{\mu}(v_n) = \liminf_{n \to \infty}\frac{1}{2}||v_n||_2^2 \geq 0.$$
Thus the {\it vanishing} can directly be ruled out knowing that $m(c) <0$. Here \cite[Lemma I.1]{LIONS1984-2} does not apply anymore; the term $||v_n||_{2^*}$ may not go to $0$ if $(v_n)$ is {\it vanishing}. Thus we need a better understanding of this possible loss of compactness and this leads to our definition of the set $V(c)$.
As to the global existence of solutions to \eqref{NLS0}, it is also affected by the presence of the Sobolev critical exponent. In Sobolev subcritical cases, it is well known \cite{Cazenave2003semilinear} that if, for an initial datum $\varphi \in H$, the maximum time of existence $T_{\varphi}^{max}>0$ is finite then necessarily the corresponding solution $v$ satisfies $||\nabla v(t)||_2 \to + \infty$ as $t \to T_{\varphi}^{max}$. Thus, a uniform a priori bound on $||\nabla v(t)||_2$ yields global existence. Note that, by conservation of the Mass and Energy, in view of \eqref{well}, for an initial datum in $V(c) \cap \{u \in S(c) : F_{\mu}(u) <0\}$, the evolution takes place in the (bounded) set $V(c)$. Thus, in a subcritical setting, the global existence would follow directly. However, in our case it is unknown if the previous blow-up alternative holds and hence, we cannot deduce global existence just since the evolution takes place in $V(c)$, see \cite[Theorem 4.5.1]{Cazenave2003semilinear} or \cite[Proposition 3.2]{TaoVisanZhang07} for more details. To overcome this difficulty, building on the pioneering work of Cazenave-Weissler \cite{CazenaveWeissler1990}, see also \cite[Section 4.5]{Cazenave2003semilinear} , we first derive an upper bound on the propagator $e^{it \Delta}$ which provides a kind of uniform local existence result, see \cref{prop:cauchy}. Next, using the information that all minimizing sequences are, up to translation, compact and also specifically and crucially that $\mathcal{M}_c$ is compact, up to translation, we manage to show that, for initial data sufficiently close to the set $\mathcal{M}_c$ the global existence holds and this leads to the orbital stability of $\mathcal{M}_c$, proving \cref{thm-2}.
We point out that, in order to prove \cref{thm-2}, we have only established the global existence of solutions for initial data {\it close} to $\mathcal{M}_c$. We believe it would be interesting to inquire if the global existence holds {\it away} from $\mathcal{M}_c$, typically for any initial data in $V(c) \cap \{u \in S(c) : F_{\mu}(u) <0\}$. If so, investigating the long time behavior of these solutions would be worth to. Our guess is that these solutions evolve toward the sum of an element of $\mathcal{M}_c$ and a part which scatter. However, so far nothing is known in that direction.
The paper is organized as follows. \cref{Section-3}, is devoted to clarifying the local minima structure and to establish the convergence, up to translation, of all minimizing sequences for $F_{\mu}$ on $V(c)$. The proof of \cref{thm-1} is then given. In \cref{Section-4}, we establish \cref{prop:cauchy}. Finally, in \cref{Section-5}, we prove \cref{thm-2} which states the orbital stability of the set $\mathcal{M}_c$.
{\bf Notation :} We write ${\mathit H}$ for $\mathit{H}^1(\mathbb{R}^N,\mathbb{C})$.
For $p \geq 1$, the $\mathit{L}^p$-norm of $u \in {\mathit H}$ (or of $u \in {\mathit H}^1$) is denoted by $\normLp{u}{p}$.
\section{The variational problem} \label{Section-3}
We shall make use of the following classical inequalities : For any $N \geq 3$ there exists an optimal constant $\mathcal{S} > 0$ depending only on $N$, such that \begin{align}\label{Sobolev-I} \mathcal{S} \norm{f}_{2^*}^{2} \leq \norm{\nabla f}_{2}^{2} , \qquad \forall f \in H, \quad \mbox{(Sobolev inequality)} \end{align} see \cite[Theorem IX.9]{Brezis1983}. If $N \geq 2$ and $p \in [2, \frac{2N}{N-2})$ then \begin{align}\label{Gagliardo-Nirenberg-I} \norm{f}_{p} \leq C_{N,p} \norm{\nabla f}_{2}^{\beta} \norm{f}_{2}^{(1-\beta)}, \qquad \mbox{with } \beta = N\(\dfrac{1}{2} - \dfrac{1}{p}\) \quad \mbox{(Gagliardo-Nirenberg inequality),} \end{align} for all $f \in H$, see \cite{Nirenberg1959}. Now, letting \begin{align*} \alpha_0 := \dfrac{N(q-2)}{2} -2, \qquad \alpha_1:= \dfrac{2N - q(N-2)}{2}, \qquad \alpha_2: = \dfrac{4}{N-2}, \end{align*} we consider the function $f(c,\rho)$ defined on $(0, \infty) \times (0, \infty)$ by \begin{align*} f(c,\rho) = \frac{1}{2} - \frac{\mu}{q} C_{N,q}^q \rho^{\frac{\alpha_0}{2} } c^{\frac{\alpha_1}{2} } - \frac{1}{2^*} \dfrac{1}{\mathcal{S}^{\frac{2^*}{2}}} \rho^{\frac{\alpha_2}{2} }, \end{align*} and, for each $c \in (0, \infty)$, its restriction $g_c(\rho)$ defined on $(0, \infty)$ by $\rho \mapsto g_c(\rho) := f(c, \rho).$
\\ For future reference, note that for any $N \geq 3 $, $\alpha_0 \in (-2,0)$, $\alpha_1 \in \[\dfrac{4}{N}, 2\)$ and $\alpha_2 \in (0,4].$
\begin{lemma}\label{lemma:5.1}
For each $c > 0$, the function $g_c(\rho)$ has a unique global maximum and the maximum value satisfies
\begin{align*}
\begin{cases}
\displaystyle \max_{\rho > 0} g_{c}(\rho) > 0 \quad \mbox{if} \quad c < c_0,\\
\displaystyle \max_{\rho > 0} g_{c}(\rho) = 0 \quad \mbox{if} \quad c = c_0,\\
\displaystyle \max_{\rho > 0} g_{c}(\rho) < 0 \quad \mbox{if} \quad c > c_0,
\end{cases}
\end{align*}
where
\begin{align}\label{eqn:5.14}
c_0 := \(\dfrac{1}{2K}\)^{\frac{N}{2}} > 0,
\end{align}
with
\begin{align}\label{eqn:5.14B}
K := \frac{\mu}{q} C_{N,q}^q \[- \dfrac{\alpha_0}{\alpha_2} \frac{\mu C_{N,q}^q 2^* \mathcal{S}^{\frac{2^*}{2}}}{q} \right]^{\frac{\alpha_0}{\alpha_2 - \alpha_0}} + \frac{1}{2^*} \dfrac{1}{\mathcal{S}^{\frac{2^*}{2}}} \[- \dfrac{\alpha_0}{\alpha_2} \frac{\mu C_{N,q}^q 2^* \mathcal{S}^{\frac{2^*}{2}}}{q} \right]^{\frac{\alpha_2}{\alpha_2 - \alpha_0}} > 0.
\end{align} \end{lemma} \begin{proof}
By definition of $g_c(\rho)$, we have that
\begin{align*}
g_c'(\rho) = - \dfrac{\alpha_0}{2} \frac{\mu}{q} C_{N,q}^q \rho^{\frac{\alpha_0}{2} -1} c^{\frac{\alpha_1}{2} }
- \dfrac{\alpha_2}{2} \frac{1}{2^*} \dfrac{1}{\mathcal{S}^{\frac{2^*}{2}}} \rho^{\frac{\alpha_2}{2} -1 }.
\end{align*}
Hence, the equation $g_c'(\rho) = 0$ has a unique solution given by
\begin{align}\label{maxL}
\rho_c = \[- \dfrac{\alpha_0}{\alpha_2} \frac{\mu C_{N,q}^q 2^* \mathcal{S}^{\frac{2^*}{2}}}{q} \right]^{\frac{2}{\alpha_2 - \alpha_0}} c^{\frac{\alpha_1}{\alpha_2 - \alpha_0}}.
\end{align}
Taking into account that $g_c(\rho) \to -\infty$ as $\rho \to 0$ and $g_c(\rho) \to -\infty$ as $\rho \to \infty$, we obtain that $\rho_c$ is the unique global maximum point of $g_c(\rho)$ and the maximum value is
\begin{align*}
\max_{\rho > 0} g_{c}(\rho) &= \frac{1}{2} - \frac{\mu}{q} C_{N,q}^q \[- \dfrac{\alpha_0}{\alpha_2} \frac{\mu C_{N,q}^q 2^* \mathcal{S}^{\frac{2^*}{2}}}{q} \right]^{\frac{\alpha_0}{\alpha_2 - \alpha_0}} c^{\frac{\alpha_0\alpha_1}{2(\alpha_2 - \alpha_0)}} c^{\frac{\alpha_1}{2} }
- \frac{1}{2^*} \dfrac{1}{\mathcal{S}^{\frac{2^*}{2}}} \[- \dfrac{\alpha_0}{\alpha_2} \frac{\mu C_{N,q}^q 2^* \mathcal{S}^{\frac{2^*}{2}}}{q} \right]^{\frac{\alpha_2}{\alpha_2 - \alpha_0}} c^{\frac{\alpha_1\alpha_2}{2(\alpha_2 - \alpha_0)}} \\
&= \frac{1}{2} - \frac{\mu}{q} C_{N,q}^q \[- \dfrac{\alpha_0}{\alpha_2} \frac{\mu C_{N,q}^q 2^* \mathcal{S}^{\frac{2^*}{2}}}{q} \right]^{\frac{\alpha_0}{\alpha_2 - \alpha_0}} c^{\frac{\alpha_1\alpha_2}{2(\alpha_2 - \alpha_0)}}
- \frac{1}{2^*} \dfrac{1}{\mathcal{S}^{\frac{2^*}{2}}} \[- \dfrac{\alpha_0}{\alpha_2} \frac{\mu C_{N,q}^q 2^* \mathcal{S}^{\frac{2^*}{2}}}{q} \right]^{\frac{\alpha_2}{\alpha_2 - \alpha_0}} c^{\frac{\alpha_1\alpha_2}{2(\alpha_2 - \alpha_0)}}\\
&= \dfrac{1}{2} - K c^{\frac{2}{N}}.
\end{align*}
By the definition of $c_0$, we have that $\displaystyle \max_{\rho > 0} g_{c_0}(\rho) = 0$, and hence the lemma follows. \end{proof}
\begin{lemma}\label{LL6-1}
Let $(c_1, \rho_1) \in (0, \infty) \times (0, \infty)$ be such that $f(c_1, \rho_1) \geq 0$. Then for any $c_2 \in (0,c_1]$, we have that
\begin{align*}
f(c_2, \rho_2) \geq 0 \quad \mbox{if} \quad \rho_2 \in \[ \displaystyle \frac{c_2}{c_1}\rho_1, \rho_1\right].
\end{align*} \end{lemma} \begin{proof}
Since $c \to f(\cdot, \rho)$ is a non-increasing function we clearly have that
\begin{align}
f(c_2, \rho_1) \geq f(c_1, \rho_1) \geq 0. \label{eqn:5.20}
\end{align}
Now taking into account that $\alpha_0 + \alpha_1 = q-2 > 0 $ we have, by direct calculations, that
\begin{align}
f\(c_2, \dfrac{c_2}{c_1}\rho_1\) \geq f(c_1, \rho_1) \geq 0. \label{eqn:5.21}
\end{align}
We observe that if $g_{c_2}(\rho') \geq 0$ and $g_{c_2}(\rho'') \geq 0$ then
\begin{align}
f(c_2, \rho) = g_{c_2}(\rho) \geq 0 \quad \mbox{for any} \quad \rho \in [\rho', \rho'']. \label{eqn:5.22}
\end{align}
Indeed, if $g_{c_2}(\rho) < 0$ for some $\rho \in (\rho', \rho'')$ then there exists a local minimum point on $ (\rho_1, \rho_2)$ and this contradicts the fact that the function $g_{c_2}(\rho)$ has a unique critical point
which has to coincide necessarily with its unique global maximum (see \cref{lemma:5.1}).
By \eqref{eqn:5.20}, \eqref{eqn:5.21}, we can choose $\rho' = (c_2/c_1) \rho_1$ and $\rho'' = \rho_1$, and \eqref{eqn:5.22} implies the lemma. \end{proof}
\begin{lemma}\label{Lemma-L1} For any $u \in S(c)$, we have that
\begin{equation*}
F_{\mu}(u) \geq \normLp{\nabla u}{2}^2 f(c, \normLp{\nabla u}{2}^2).
\end{equation*} \end{lemma} \begin{proof}
Applying the Gagliardo-Nirenberg inequality \eqref{Gagliardo-Nirenberg-I}
and the Sobolev inequality \eqref{Sobolev-I} we obtain that, for any $u \in S(c)$,
\begin{align*}
F_{\mu}(u) &= \frac{1}{2} \normLp{\nabla u}{2}^2 - \frac{\mu}{q} \normLp{u}{q}^q - \frac{1}{2^*}\normLp{u}{2^*}^{2^*}
\geq \frac{1}{2} \normLp{\nabla u}{2}^2 - \frac{\mu}{q} C_{N,q}^q \norm{\nabla u}_{2}^{\alpha_0+2} \norm{u}_{2}^{\alpha_1} - \frac{1}{2^*} \dfrac{1}{\mathcal{S}^{\frac{2^*}{2}}} \normLp{\nabla u}{2}^{2^*} \\
&= \normLp{\nabla u}{2}^2 \[ \frac{1}{2} - \frac{\mu}{q} C_{N,q}^q \norm{\nabla u}_{2}^{\alpha_0 } \norm{u}_{2}^{\alpha_1} - \frac{1}{2^*} \dfrac{1}{\mathcal{S}^{\frac{2^*}{2}}} \normLp{\nabla u}{2}^{\alpha_2}\right]
= \normLp{\nabla u}{2}^2 f( \norm{u}_{2}^2, \normLp{\nabla u}{2}^2).
\end{align*}
The lemma is proved. \end{proof}
Now let $c_0>0$ be given by \eqref{eqn:5.14} and $\rho_0 := \rho_{c_0} >0$ being determined by \eqref{maxL}. Note that by \cref{lemma:5.1} and \cref{LL6-1}, we have that $f(c_0, \rho_0) = 0$ and $f(c, \rho_0) > 0$ for all $c \in (0, c_0)$. We define \begin{align*} B_{\rho_0} := \{u \in {\mathit H}: \normLp{\nabla u}{2}^2 < \rho_0\} \quad \mbox{and} \quad V(c) := S(c) \cap B_{\rho_0}. \end{align*} We shall now consider the following local minimization problem: for any $c \in (0, c_0)$, \begin{equation}\label{L6-3} m(c) := \inf_{u \in V(c)} F_{\mu}(u). \end{equation}
\begin{lemma}\label{Lemma-structure} For any $c \in (0, c_0)$, the following properties hold,
\begin{enumerate}[label=(\roman*), ref = \roman*]
\item\label{point:5L.5i}
$$m(c) = \inf_{u \in V(c)} F_{\mu}(u) < 0 < \inf_{u \in \partial V(c)}F_{\mu}(u).$$
\item\label{point:5L.5ii} If $m(c)$ is reached, then any ground state is contained in $V(c)$.
\end{enumerate} \end{lemma}
\begin{proof}
(\ref{point:5L.5i}) For any $u \in \partial V(c)$ we have $\normLp{\nabla u}{2}^2 = \rho_0$. Thus, using \cref{Lemma-L1}, we get
\begin{align*}
F_{\mu}(u) \geq \normLp{\nabla u}{2}^2 f( \norm{u}_{2}^2, \normLp{\nabla u}{2}^2)
= \rho_0 f(c, \rho_0) >0.
\end{align*}
Now let $u \in S(c)$ be arbitrary but fixed. For $s \in (0, \infty)$ we set
\begin{align*}
u_s(x) := s^{\frac{N}{2}} u(s x).
\end{align*}
Clearly $u_s \in S(c)$ for any $s \in (0, \infty)$. We define on $(0, \infty)$ the map,
\begin{align*}
\psi_u(s) := F_{\mu} (u_s) = \dfrac{s^2}{2} \normLp{\nabla u}{2}^2 - \dfrac{\mu}{q} s^{\frac{N(q-2)}{2}} \normLp{u}{q}^q - \dfrac{s^{2^*}}{2^*} \normLp{u}{2^*}^{2^*}.
\end{align*}
Taking into account that
\begin{align*}
\dfrac{N(q-2)}{2} < 2 \qquad \mbox{and} \qquad 2^* > 2,
\end{align*}
we see that $\psi_{u}(s) \to 0^-$, as $s \to 0$. Therefore, there exists $s_0 >0$ small enough such that $\normLp{\nabla (u_{s_0})}{2}^2 = s_0^2 \normLp{\nabla u}{2}^2 < \rho_0$ and $F_{\mu} (u_{s_0}) = \psi_{\mu}(s_0) < 0$. This implies that $m(c) < 0$.
(\ref{point:5L.5ii}) It is well known, see for example \cite[Lemma 2.7]{JEANJEAN1997}, that all critical points of $F_{\mu}$ restricted to $S(c)$ belong to the Pohozaev's type set
$$\mathcal{P}_c := \{ u \in S(c) : \mathcal{P}(u)=0 \}$$
where
\begin{equation*}
\mathcal{P}(u):= \normLp{\nabla u}{2}^2 - \dfrac{\mu N(q-2)}{2q} \normLp{u}{q}^q - \normLp{u}{2^*}^{2^*}.
\end{equation*}
Also a direct calculation shows that, for any $v \in S(c)$ and any $s \in (0, \infty)$,
\begin{equation}\label{link}
\psi_v'(s) = \frac{1}{s}\mathcal{P}(v_s).
\end{equation}
Here $\psi_v'$ denotes the derivative of $\psi_v$ with respect to $s \in (0, \infty)$. Finally, observe that any $u \in S(c)$ can be written as $u = v_s$ with $v \in S(c)$, $||\nabla v||_2 =1$ and $s \in (0, \infty).$
Since the set $\mathcal{P}_c$ contains all the ground states (if any), we deduce from \eqref{link} that if $w \in S(c)$ is a ground state there exists a $v \in S(c)$, $||\nabla v||_2^2=1$ and a $s_0 \in (0, \infty)$ such that $w= v_{s_0}$, $F_{\mu}(w) = \psi_v(s_0)$ and $\psi_{v}'(s_0)=0$. Namely, $s_0 \in (0, \infty)$ is a zero of the function $\psi_{v}'.$
Now, since $\psi_v(s) \to 0^-$, $||\nabla v_s||_2 \to 0,$ as $s \to 0$ and $\psi_v(s) = F_{\mu}(v_s) \geq 0$ when $v_s \in \partial V(c) = \{u \in S(c) : ||\nabla u||_2^2 = \rho_0 \}$, necessarily $\psi_v'$ has a first zero $s_1 >0$ corresponding to a local minima. In particular, $v_{s_1} \in V(c)$ and $F(v_{s_1}) = \psi_v(s_1) <0.$ Also, from $ \psi_v(s_1) <0,$ $\psi_v(s) \geq 0$ when $v_s \in \partial V(c)$ and $\psi_v(s) \to - \infty$ as $s \to \infty$, $\psi_v$ has a second zero $s_2 >s_1$ corresponding to a local maxima of $\psi_v$. Since $v_{s_2}$ satisfies $F(v_{s_2})= \psi_v(s_2) \geq 0$, we have that $m(c) \leq F(v_{s_1}) < F(v_{s_2})$. In particular, since $m(c)$ is reached, $v_{s_2}$ cannot be a ground state.
To conclude the proof of (ii) it then just suffices to show that $\psi_v'$ has at most two zeros, since this will imply $s_0 = s_1$ and $w = v_{s_0} = v_{s_1} \in V(c)$. However, this is equivalent to showing that the function
$$s \mapsto \frac{\psi'_u(s)}{s}$$
has at most two zeros. We have
$$\theta(s) := \frac{\psi'_u(s)}{s} = \normLp{\nabla u}{2}^2 - \dfrac{\mu N(q-2)}{2q} s^{\alpha_0} \normLp{u}{q}^q - s^{\alpha_2} \normLp{u}{2^*}^{2^*}$$
and
\begin{align*}
\theta'(s) = - \alpha_0 \dfrac{\mu N(q-2)}{2q} s^{\alpha_0 - 1} \normLp{u}{q}^q - \alpha_2 s^{\alpha_2-1} \normLp{u}{2^*}^{2^*}.
\end{align*}
Since $\alpha_0 <0$ and $\alpha_2 >0$, the equation $\theta'(s) =0$ has a unique solution, and $\theta(s)$ has indeed at most two zeros. \end{proof}
We now introduce the set \begin{align}\label{set-stable} \mathcal{M}_c := \{u \in V(c) : F_{\mu}(u) = m(c)\}. \end{align}
The main aim of this section is the following result. \begin{theorem} \label{theorem:LT-L}
For any $c \in (0, c_0)$, if $(u_n) \subset B_{\rho_0}$ is such that $\normLp{u_n}{2}^2 \to c$ and $F_{\mu}(u_n) \to m(c)$ then, up to translation, $u_n \overset{ H}\to u \in \mathcal{M}_c$. In particular the set $\mathcal{M}_c$ is compact in $H$, up to translation. \end{theorem} \cref{theorem:LT-L} will both imply the existence of a ground state but also, as it may be expected, will be a crucial step to derive the orbital stability of the set $\mathcal{M}_c$.\\
In order to prove \cref{theorem:LT-L} we collect some properties of $m(c)$ defined in \eqref{L6-3}. \begin{lemma} \label{lemma:5.5}
It holds that
\begin{enumerate}[label=(\roman*), ref = \roman*]
\item\label{point:5.5ii} $c \in (0,c_0) \mapsto m(c)$ is a continuous mapping.
\item\label{point:5.5iii} Let $c \in (0,c_0)$. We have for all $\alpha \in (0,c)$ : $m(c) \leq m(\alpha) + m(c-\alpha)$ and if $m(\alpha)$ or $m(c-\alpha)$ is reached then the inequality is strict.
\end{enumerate} \end{lemma} \begin{proof}
(\ref{point:5.5ii}) Let $c \in (0, c_0)$ be arbitrary and $(c_n) \subset (0, c_0)$ be such that $c_n \to c$. From the definition of $m(c_n)$ and since $m(c_n) <0$, see \cref{Lemma-structure} (\ref{point:5L.5i}),
for any $\varepsilon >0$ sufficiently small, there exists $u_n \in V(c_n)$ such that
\begin{align}
F_{\mu}(u_n) \leq m(c_n) + \varepsilon \quad \mbox{and} \quad F_{\mu}(u_n) <0. \label{eqn:5.9}
\end{align}
We set $\displaystyle y_n := \sqrt{\frac{c}{c_n}} u_n$ and hence $y_n \in S(c)$. We have that $y_n \in V(c)$. Indeed, if $c_n \geq c$, then
\begin{align*}
\normLp{\nabla y_n}{2}^2 = \frac{c}{c_n} \normLp{\nabla u_n}{2}^2 \leq \normLp{\nabla u_n}{2}^2 < \rho_0.
\end{align*}
If $c_n < c$, by \cref{LL6-1}, we have $\displaystyle f(c_n, \rho) \geq 0$ for any $\rho \in \[ \dfrac{c_n}{c} \rho_0, \rho_0\right]$. Hence, we deduce from \cref{Lemma-L1} and \eqref{eqn:5.9} that $f(c_n, \|\nabla u_n\|_2^2) < 0$, thus $\|\nabla u_n\|_2^2 < \frac{c_n}{c}\rho_0$ and
\begin{align*}
\normLp{\nabla y_n}{2}^2 = \frac{c}{c_n} \normLp{\nabla u_n}{2}^2 < \frac{c}{c_n} \dfrac{c_n}{c} \rho_0 = \rho_0.
\end{align*}
Since $y_n \in V(c)$ we can write
\begin{align*}
m(c) \leq F_{\mu}(y_n) = F_{\mu}(u_n) + [F_{\mu}(y_n) - F_{\mu}(u_n)]
\end{align*}
where
\begin{equation*}
F_{\mu}(y_n) - F_{\mu}(u_n) = - \frac{1}{2}(\frac{c}{c_n}-1)\normLp{\nabla u_n}{2}^2 - \frac{\mu}{q} \big[ (\frac{c}{c_n})^\frac q2 - 1 \big] \normLp{u_n}{q}^q - \frac{1}{2^*} [ (\frac{c}{c_n})^{ \frac{2^*}2} - 1 ] \normLp{u_n}{2^*}^{2^*}.
\end{equation*}
Since $\normLp{\nabla u_n}{2}^2 < \rho_0$, also $\normLp{u_n}{q}^q$ and $\normLp{u_n}{2^*}^{2^*}$ are uniformly bounded.
Thus, as $n \to \infty$ we have
\begin{align}
m(c) \leq F_{\mu}(y_n) = F_{\mu}(u_n) + o_n(1). \label{eqn:5.10}
\end{align}
Combining \eqref{eqn:5.9} and \eqref{eqn:5.10}, we get
\begin{align*}
m(c) \leq m(c_n) + \varepsilon + o_n(1).
\end{align*}
Now, let $u \in V(c)$ be such that
\begin{equation*}
F_{\mu}(u) \leq m(c) + \varepsilon \quad \mbox{and} \quad F_{\mu}(u) <0.
\end{equation*}
Set $u_n := \sqrt{\frac{c_n}{c}} u$ and hence $u_n \in S(c_n)$.
Clearly, $\|\nabla u\|_2^2 < \rho_0$ and $c_n \to c$ imply $\|\nabla u_n\|_2^2 < \rho_0$ for $n$ large enough,
so that $u_n \in V(c_n)$. Also, $F_{\mu}(u_n) \to F_\mu(u)$. We thus have
\begin{equation*}
m(c_n) \leq F_\mu(u_n) = F_\mu(u) + [F_{\mu}(u_n) - F_{\mu}(u)] \leq m(c) + \varepsilon + o_n(1).
\end{equation*}
Therefore, since $\varepsilon > 0$ is arbitrary, we deduce that $m(c_n) \to m(c)$. The point \eqref{point:5.5ii} follows.
(\ref{point:5.5iii}) Note that, fixed $\alpha \in (0,c)$, it is sufficient to prove that the following holds
\begin{equation}\label{L6-4}
\forall \theta \in \(1, \frac{c}{\alpha}\right] : m(\theta \alpha) \leq \theta m(\alpha)
\end{equation}
and that, if $m(\alpha)$ is reached, the inequality is strict. Indeed, if \eqref{L6-4} holds then we have
\begin{align*}
m(c)=\frac{c-\alpha}{c}m(c)+\frac{\alpha}{c}m(c)=\frac{c-\alpha}{c}m\left( \frac{c}{c-\alpha}(c-\alpha) \right)+\frac{\alpha}{c}m\left( \frac{c}{\alpha} \alpha \right) \leq m(c-\alpha)+m(\alpha),
\end{align*}
with a strict inequality if $m(\alpha)$ is reached. To prove that \eqref{L6-4} holds, note that in view of \cref{Lemma-structure} (\ref{point:5L.5i}), for any $\varepsilon >0$ sufficiently small, there exists a $u \in V(\alpha)$ such that
\begin{align}
F_{\mu}(u) \leq m(\alpha) + \varepsilon \quad \mbox{and} \quad F_{\mu}(u) <0. \label{eqn:5.11}
\end{align}
In view of \cref{LL6-1}, $\displaystyle f(\alpha, \rho) \geq 0$ for any $\rho \in \[ \dfrac{\alpha}{c} \rho_0, \rho_0\right]$.
Hence, we can deduce from \cref{Lemma-L1} and \eqref{eqn:5.11} that
\begin{equation}\label{wellinside}
||\nabla u||_2^2 < \frac{\alpha}{c} \rho_0.
\end{equation}
Consider now $v = \sqrt{\theta} u$. We first note that $||v||_2^2 = \theta ||u||_2^2 = \theta \alpha$ and also, because of \eqref{wellinside},
$||\nabla v||_2^2 = \theta ||\nabla u||_2^2 < \rho_0.$
Thus $v \in V(\theta \alpha)$ and we can write
\begin{align*}
m(\theta \alpha) &\leq F_{\mu}(v) = \dfrac{1}{2} \theta \normLp{\nabla u}{2}^2 - \frac{\mu}{q} \theta^{\frac{q}{2}} \normLp{u}{q}^q - \dfrac{1}{2^*} \theta^{\frac{2^*}{2}} \normLp{u}{2^*}^{2^*}
< \dfrac{1}{2} \theta \normLp{\nabla u}{2}^2 - \frac{\mu}{q} \theta \normLp{u}{q}^q - \dfrac{1}{2^*} \theta \normLp{u}{2^*}^{2^*} \\
&= \theta \(\dfrac{1}{2} \normLp{\nabla u}{2}^2 - \frac{\mu}{q} \normLp{u}{q}^q - \dfrac{1}{2^*} \normLp{u}{2^*}^{2^*}\)
= \theta F_{\mu}(u) \leq \theta (m(\alpha) + \varepsilon).
\end{align*}
Since $\varepsilon > 0$ is arbitrary, we have that $m(\theta \alpha) \leq \theta m(\alpha)$. If $m(\alpha)$ is reached then we can let $\varepsilon = 0$ in \eqref{eqn:5.11} and thus the strict inequality follows. \end{proof}
\begin{lemma} \label{lemma:5.6}
Let $(v_n) \subset B_{\rho_0}$ be such that $\normLp{v_n}{q} \to 0$. Then there exists a $\beta_0 > 0$ such that
\begin{align*}
F_{\mu}(v_n) \geq \beta_0 ||\nabla v_n||_2^2 + o_n(1).
\end{align*} \end{lemma} \begin{proof}
Indeed, using the Sobolev inequality \eqref{Sobolev-I}, we obtain that
\begin{align*}
F_{\mu}(v_n) &= \frac{1}{2} \normLp{\nabla v_n}{2}^2 - \frac{1}{2^*}\normLp{v_n}{2^*}^{2^*} + o_n(1)
\geq \frac{1}{2} \normLp{\nabla v_n}{2}^2 - \frac{1}{2^*} \dfrac{1}{\mathcal{S}^{\frac{2^*}{2}}} \normLp{\nabla v_n}{2}^{2^*} + o_n(1) \\
&= \normLp{\nabla v_n}{2}^2 \[ \frac{1}{2} - \frac{1}{2^*} \dfrac{1}{\mathcal{S}^{\frac{2^*}{2}}} \normLp{\nabla v_n}{2}^{\alpha_2}\right] + o_n(1)
\geq \normLp{\nabla v_n}{2}^2 \[ \frac{1}{2} - \frac{1}{2^*} \dfrac{1}{\mathcal{S}^{\frac{2^*}{2}}} \rho_0^{\frac{\alpha_2}{2}}\right] + o_n(1).
\end{align*}
Now, since $f(c_0,\rho_0) = 0,$ we have that
\begin{align*}
\beta_0 := \[ \frac{1}{2} - \frac{1}{2^*} \dfrac{1}{\mathcal{S}^{\frac{2^*}{2}}} \rho_0^{\frac{\alpha_2}{2}}\right] = \frac{\mu}{q} C_{N,q}^q \rho_0^{\frac{\alpha_0}{2} } c_0^{\frac{\alpha_1}{2} } > 0.
\end{align*} \end{proof}
\begin{lemma} \label{lemma:5.3}
For any $c \in (0, c_0)$, let $(u_n) \subset B_{\rho_0}$ be such that $\normLp{u_n}{2}^2 \to c$ and $F_{\mu}(u_n) \to m(c)$. Then, there exist a $\beta_1 > 0$ and a sequence $(y_n) \subset \mathbb{R}^N$ such that
\begin{align}
\int_{B(y_n, R)} \abs{u_n}^2 dx \geq \beta_1 > 0, \qquad \text{for some } R > 0. \label{eqn:5.1}
\end{align} \end{lemma} \begin{proof}
We assume by contradiction that \eqref{eqn:5.1} does not hold. Since $(u_n) \subset B_{\rho_0}$ and $\normLp{u_n}{2}^2 \to c$, the sequence $(u_n)$ is bounded in $H$. From \cite[Lemma I.1]{LIONS1984-2} and since $2 < q < 2^*$, we deduce that $\normLp{u_n}{q} \to 0,$ as $ n \to \infty.$
At this point, \cref{lemma:5.6} implies that $F_{\mu} (u_n) \geq o_n(1)$. This contradict the fact that $m(c) <0$ and the lemma follows. \end{proof}
\begin{proof} [Proof of \cref{theorem:LT-L}]
We know from \cref{lemma:5.3} and Rellich compactness theorem that there exists a sequence $(y_n) \subset \mathbb{R}^N$ such that
\begin{align*}
u_n(x - y_n) \rightharpoonup u_c \neq 0 \quad\text{in } {\mathit H}.
\end{align*}
Our aim is to prove that $w_n(x) := u_n(x - y_n) - u_c(x) \to 0$ in ${\mathit H}$. Clearly
\begin{align*}
\normLp{u_n}{2}^2 &= \normLp{u_n(x-y_n)}{2}^2
= \normLp{u_n(x-y_n) - u_c(x)}{2}^2 + \normLp{u_c}{2}^2 + o_n(1)\\
&=\normLp{w_n}{2}^2 + \normLp{u_c}{2}^2 + o_n(1).
\end{align*}
Thus, we have
\begin{align}
\normLp{w_n}{2}^2 = \normLp{u_n}{2}^2 - \normLp{u_c}{2}^2 + o_n(1) = c - \normLp{u_c}{2}^2 + o_n(1). \label{eqn:5.6}
\end{align}
By a similar argument,
\begin{align}
\normLp{\nabla w_n}{2}^2 = \normLp{\nabla u_n}{2}^2 - \normLp{\nabla u_c}{2}^2 + o_n(1). \label{eqn:5.15}
\end{align}
More generally, taking into account that any term in $F_{\mu}$ fulfills the splitting properties of Brezis-Lieb \cite{BrezisLieb1983}, we have
\begin{align*}
F_{\mu}(w_n ) + F_{\mu}(u_c) =F_{\mu}(u_n(x-y_n)) + o_n(1),
\end{align*}
and, by the translational invariance, we obtain
\begin{equation}\label{eqn:5.5}
F_{\mu}(u_n) =F_{\mu}(u_n(x-y_n))
=F_{\mu}(w_n) + F_{\mu}(u_c) + o_n(1).
\end{equation}
Now, we claim that
\begin{align}\label{claimS}
\normLp{w_n}{2}^2 \to 0.
\end{align}
In order to prove this, let us denote $c_1:= \normLp{u_c}{2}^2 > 0$. By \eqref{eqn:5.6}, if we show that $c_1 = c$ then the claim follows. We assume by contradiction that $c_1 < c$.
In view of \eqref{eqn:5.6} and \eqref{eqn:5.15}, for $n$ large enough, we have $\normLp{w_n}{2}^2 \leq c$ and $\normLp{\nabla w_n}{2}^2 \leq \normLp{\nabla u_n}{2}^2 < \rho_0$. Hence, we obtain that $w_n \in V(\normLp{w_n}{2}^2)$ and $F_{\mu}(w_n) \geq m\(\normLp{w_n}{2}^2\)$.
Recording that $F_{\mu}(u_n) \to m(c)$, in view of \eqref{eqn:5.5}, we have
\begin{align*}
m(c) = F_{\mu}(w_n) + F_{\mu}(u_c) + o_n(1) \geq m\(\normLp{w_n}{2}^2\) + F_{\mu}(u_c) + o_n(1).
\end{align*}
Since the map $c \mapsto m(c)$ is continuous (see \cref{lemma:5.5}\eqref{point:5.5ii}) and in view of \eqref{eqn:5.6}, we deduce that
\begin{align}
m(c) \geq m(c-c_1) + F_{\mu}(u_c). \label{eqn:5.13}
\end{align}
We also have that $u_c \in V(c_1)$ by the weak limit. This implies that $F_{\mu}(u_c) \geq m(c_1)$. If $F_{\mu}(u_c) > m(c_1)$, then it follows from \eqref{eqn:5.13} and \cref{lemma:5.5}\eqref{point:5.5iii} that
\begin{align*}
m(c) > m(c-c_1) + m(c_1) \geq m(c -c_1 + c_1) = m(c),
\end{align*}
which is impossible. Hence, we have $F_{\mu}(u_c) = m(c_1)$, namely $u_c$ is a local minimizer on $V(c_1)$. So, using \cref{lemma:5.5}\eqref{point:5.5iii} with the strict inequality, we deduce from \eqref{eqn:5.13} that
\begin{align*}
m(c) \geq m(c-c_1) + F_{\mu}(u_c) = m(c-c_1) + m(c_1) > m(c -c_1 + c_1) = m(c),
\end{align*}
which is impossible. Thus, the claim \eqref{claimS} follows and from \eqref{eqn:5.6} we deduce that
$\normLp{u_c}{2}^2 = c$.
Let us now show that $||\nabla w_n||_2^2 \to 0$. This will prove that $w_n \to 0$ in ${\mathit H}$ and completes the proof.
In this aim first observe that
in view of \eqref{eqn:5.15} and since $u_c \neq 0$, we have $\normLp{\nabla w_n}{2}^2 \leq \normLp{\nabla u_n}{2}^2 < \rho_0$, for $n$ large enough.
Hence $(w_n) \subset B_{\rho_0}$ and in particular it is bounded in $H$. Then by using the Gagliardo-Nirenberg inequality \eqref{Gagliardo-Nirenberg-I}, and by recalling $||w_n||_2^2 \to 0$ we also have $||w_n||_q^q \to 0$.
Thus
\cref{lemma:5.6} implies that
\begin{equation}\label{firstpart}
F_{\mu} (w_n) \geq \beta_0 \normLp{\nabla w_n}{2}^2+o_n(1) \, \mbox{ where } \, \beta_0 > 0.
\end{equation}
Now we remember that
\begin{equation*}\label{LLLL}
F_{\mu}(u_n) = F_{\mu}(u_c) + F_{\mu}(w_n) + o_n(1) \to m(c).
\end{equation*}
Since $u_c \in V(c)$ by weak limit, we have that $F_{\mu}(u_c) \geq m(c)$ and hence
$F_{\mu}(w_n) \leq o_n(1)$. In view of \eqref{firstpart}, we then conclude that $\normLp{\nabla w_n}{2}^2 \to 0$. \end{proof}
We end this section with, \begin{proof}[Proof of \cref{thm-1}] The fact that if $(u_n) \subset V(c)$ is such that $F_{\mu}(u_n) \to m(c)$ then, up to translation, $u_n \to u \in \mathcal{M}_c$ in $H$ follows from \cref{theorem:LT-L}. In particular, it insures the existence of a minimizer for $F_{\mu}$ on $V(c)$. The fact that this minimizer is a ground state and that any ground state for $F_{\mu}$ on $S(c)$ belongs to $V(c)$ was proved in \cref{Lemma-structure}. \end{proof}
\section{A condition for a uniform local existence result}\label{Section-4} In this section we focus on the local existence of solutions to the following Cauchy problem \begin{align} \begin{cases} \label{NLS}
i \partial_t u + \Delta u + \mu u |u|^{q-2} + u |u|^{2^* - 2} = 0, \quad (t, x) \in \mathbb{R} \times \mathbb{R}^N, \quad N\geq 3 \\ u(0, x) = \varphi(x) \in {\mathit H}. \end{cases} \end{align}
Denoting $g: \mathbb{C} \to \mathbb{C}$ by $g(u) := \mu u |u|^{q-2} + u |u|^{2^* - 2}$, \eqref{NLS} reads as \begin{equation*} i\partial_t u + \Delta u + g(u) = 0. \end{equation*}
Next we give the notion of integral equation associated with \eqref{NLS}. In order to do that first we give another definition. \begin{definition}
If $N\geq 3$ the pair $(p, r)$ is said to be (Schr\"{o}dinger) admissible if
\begin{align*}
\dfrac{2}{p} + \dfrac{N}{r} = \dfrac{N}{2}, \qquad p, r \in [2, \infty].
\end{align*} \end{definition} We shall work with two particular admissible pairs (see \cref{lemma:1T}): \begin{align*} (p_1, r_1) := \(\dfrac{4q}{(q - 2)(N-2)}, \dfrac{Nq}{q + N -2}\), \end{align*} and \begin{align*} (p_2, r_2) := \(\dfrac{4 \times 2^*}{(2^* - 2)(N-2)}, \dfrac{N \times 2^*}{2^* + N -2}\). \end{align*} Along with those couples we introduce the spaces $Y_{T}:= Y_{p_1, r_1, T} \cap Y_{p_2, r_2, T}$ and $X_{T}:= X_{p_1, r_1, T} \cap X_{p_2, r_2, T}$ equipped with the following norms: \begin{align} \label{eqn:XT} \norm{w}_{Y_T} = \norm{w}_{Y_{p_1, r_1, T}} + \norm{w}_{Y_{p_2, r_2, T}}, \quad\mbox{and}\quad \norm{w}_{X_T} = \norm{w}_{X_{p_1, r_1, T}} + \norm{w}_{X_{p_2, r_2, T}}. \end{align} where for a generic function $w(t,x)$ defined on the time-space strip $[0, T)\times \mathbb{R}^N$ we have defined: \begin{align*}
||w(t, x)||_{Y_{p,r,T}} = \(\int_{0}^{T} \normLp{w(t, \cdot)}{r}^p dt \)^{\frac{1}{p}} \hbox{ and }
||w(t, x)||_{X_{p,r,T}} = \(\int_{0}^{T} ||w(t, \cdot)||_{W^{1,r}(\mathbb{R}^N)}^p dt \)^{\frac{1}{p}}. \end{align*}
\begin{definition} \label{def:1}
Let $T >0$.
We say that $u(t,x)$ is an integral solution of the Cauchy problem \eqref{NLS} on the time interval $[0, T]$ if:
\begin{enumerate}
\item $u \in \mathcal{C}([0, T], {\mathit H}) \cap X_T$;
\item for all $t \in (0, T)$ it holds
$
u(t) = e^{it\Delta}\varphi - i\int_0^t e^{i(t-s)\Delta}g(u(s))ds.
$
\end{enumerate} \end{definition}
The main result of this section is the following local existence result. We do not claim a real originality here, related versions already exist in the literature, see for example\cite[Theorem 2.5]{KenigMerle2006}. However, we believe convenient to the reader to provide a version specifically adapted to our problem and to give a proof of this result as self-contained as possible.
\begin{proposition}\label{prop:cauchy}
There exists $\gamma_0 > 0$ such that if $\varphi \in {\mathit H}$ and $T \in (0, 1]$ satisfy
\begin{equation*}
\|e^{it\Delta}\varphi\|_{X_T} \leq \gamma_0,
\end{equation*}
then there exists a unique integral solution $u(t,x)$ to \eqref{NLS} on the time interval $[0, T]$. Moreover
$u(t,x)\in X_{p,r, T}$ for every admissible couple $(p,r)$ and satisfies the following conservation laws:
\begin{equation}
\label{eq:laws}
F_\mu(u(t)) = F_\mu(\varphi), \quad \normLp{u(t)}{2} = \normLp{\varphi}{2}, \quad \mbox{for all }t \in [0, T].
\end{equation} \end{proposition}
In order to prove \cref{prop:cauchy} we need some preliminary results.
Let us recall Strichartz's estimates that will be useful in the sequel (see for example \cite[Theorem 2.3.3 and Remark 2.3.8]{Cazenave2003semilinear} and \cite{KeelTao1998} for the endpoint estimates). \begin{proposition}\label{Strichartz} Let $N\geq 3$ then
for every admissible pairs $(p, r)$ and $(\tilde p, \tilde r)$, there exists a constant $C> 0$ such that for every $T>0$, the following properties hold:
\begin{itemize}
\item[(i)] For every $\varphi \in \Lp{2}$, the function $t \mapsto e^{it\Delta}\varphi$ belongs to $Y_{p, r, T} \cap \mathcal{C}([0,T], \Lp{2})$ and
\begin{align*}
\norm*{e^{it\Delta} \varphi}_{Y_{p,r,T}} \leq C \norm{\varphi}_{2}.
\end{align*}
\item[(ii)] Let $F \in Y_{\tilde p ', \tilde r', T}$, where we use a prime to denote conjugate indices. Then the function
\begin{align*}
t \mapsto \Phi_F(t):= \int_{0}^{t}e^{i(t-s)\Delta}F(s) ds
\end{align*}
belongs to $Y_{p, r, T} \cap \mathcal{C}([0,T], \Lp{2})$ and
\begin{align*}
\norm*{\Phi_F}_{Y_{p,r,T}} \leq C \norm{F}_{Y_{\tilde p ', \tilde r', T}}.
\end{align*}
\item[(iii)] For every $\varphi \in {\mathit H}$, the function $t \mapsto e^{it\Delta}\varphi$ belongs to $X_{p, r, T} \cap \mathcal{C}([0,T], {\mathit H})$ and
\begin{align*}
\norm*{e^{it\Delta} \varphi}_{X_{p,r,T}} \leq C \norm{\varphi}_{{\mathit H}}.
\end{align*}
\end{itemize} \end{proposition}
The following result will be useful in the sequel. \begin{lemma} \label{lemma:1T}
Let $N\geq 3$ and $2 < \alpha \leq 2^*$ be given. Then the couple $(p,r)$ defined as follows
\begin{align*}
p := \dfrac{4\alpha}{(\alpha - 2)(N-2)} \qquad\mbox{and}\qquad r:= \dfrac{N\alpha}{\alpha + N -2}
\end{align*}
is admissible.
Moreover for every admissible couple $(\tilde{p}, \tilde{r})$ there exists a constant $C > 0$
such that for every $T >0$ the following inequalities hold:
\begin{align}
\norm*{\int_{0}^{t}e^{i(t-s)\Delta} [\nabla g_\alpha(u(s)) ]ds}_{Y_{\tilde{p},\tilde{r},T}} &\leq C T^{\mu} \norm{\nabla u}_{Y_{p,r,T}}^{\alpha-1}, \label{eqn:2.17} \\
\norm*{\int_{0}^{t}e^{i(t-s)\Delta} [g_\alpha(u(s)) - g_\alpha(v(s))] ds}_{Y_{\tilde{p},\tilde{r},T}} &\leq C T^{\mu} (\norm{\nabla u}_{Y_{p,r,T}}^{\alpha-2} + \norm{\nabla v}_{Y_{p,r,T}}^{\alpha-2}) \norm{u-v}_{Y_{p,r,T}}, \label{eqn:2.18}
\end{align}
where $g_\alpha(u) := u|u|^{\alpha - 2}$ and
$
\mu := \dfrac{(N-2)(2^*-\alpha)}{4} \geq 0.
$ \end{lemma} \begin{proof}
By direct calculations, one can check that
\begin{align*}
\dfrac{2}{p} + \dfrac{N}{r} = \dfrac{N}{2} \quad\mbox{and}\quad p, r \geq 2.
\end{align*}
Hence, $(p, r)$ is an admissible pair. Also it is easy to check that there exists a $C > 0$ such that :
\begin{align}
|g_\alpha'(u)| &\leq C|u|^{\alpha - 2}, \label{eqn:2.12} \\
|g_\alpha(u) - g_\alpha(v)| &\leq C|u - v|(|u|^{\alpha - 2} + |v|^{\alpha - 2}). \label{eqn:2.13}
\end{align}
Combining \eqref{eqn:2.12} and the Chain Rule, gives
\begin{align*}
|\nabla g_\alpha(u) | = |g_\alpha'(u)\nabla u| \leq C |\nabla u| |u|^{\alpha - 2}.
\end{align*}
Using H\"older's inequality, we obtain that
\begin{align*}
\normLp{\nabla g_\alpha(u)}{r'} \leq C \normLp{|\nabla u| |u|^{\alpha - 2}}{r'}
\leq C \normLp{|\nabla u|}{r} \normLp{u}{r^*}^{\alpha - 2}
\leq C \norm{\nabla u }^{\alpha - 1}_{r},
\end{align*}
where we also used the Sobolev embedding of $W^{1,r}(\mathbb{R}^N)$ into $L^{r^*}(\mathbb{R}^N)$ with $r^* := \dfrac{Nr}{N-r}$, see \cite[Theorem IX.9]{Brezis1983}.
Hence, using H\"{o}lder's inequality,
\begin{align*}
\norm{\nabla g_\alpha(u)}_{Y_{p', r', T}} &= \( \int_{0}^{T} \normLp{\nabla g_\alpha(u)}{r'}^{p'} dt \)^{\frac{1}{p'}}
\leq C\( \int_{0}^{T} || \nabla u ||^{(\alpha - 1)p'}_{r} dt \)^{\frac{1}{p'}} \\
&\leq C T^{(\alpha-1)\(\frac{1}{(\alpha - 1) p'} - \frac{1}{p}\) } \( \int_{0}^{T} || \nabla u ||^{p}_{r} dt \)^{\frac{\alpha - 1}{p}}
= C T^{\mu} \norm{\nabla u}_{Y_{p,r,T}}^{\alpha-1}.
\end{align*}
At this point \eqref{eqn:2.17} follows by applying \cref{Strichartz} (ii). To establish \eqref{eqn:2.18} note that by \eqref{eqn:2.13} and the H\"older's inequality, we have
\begin{align*}
\normLp{g_\alpha(u) - g_\alpha(v)}{r'} \leq C \normLp{|u - v|(|u|^{\alpha - 2} + |v|^{\alpha - 2})}{r'}
\leq C \normLp{u-v}{r} \normLp{|u| + |v|}{r^*}^{\alpha -2}.
\end{align*}
Hence, we can deduce that
\begin{align*}
\norm{g_\alpha(u) - g_\alpha(v)}_{Y_{p', r', T}} &= \( \int_{0}^{T} \normLp{g_\alpha(u) - g_\alpha(v)}{r'}^{p'} dt \)^{\frac{1}{p'}}
\leq C \( \int_{0}^{T} \normLp{u-v}{r}^{p'} \normLp{|u| + |v|}{r^*}^{(\alpha -2)p'} dt \)^{\frac{1}{p'}}\\
&\leq C \( \int_{0}^{T} \normLp{u-v}{r}^{p} dt \)^{\frac{1}{p}} \( \int_{0}^{T} \normLp{|u| + |v|}{r^*}^{\frac{(\alpha -2)pp'}{p-p'}} dt \)^{\frac{p-p'}{pp'}}
\leq C T^{\mu} \norm{u-v}_{Y_{p,r,T}} \( \int_{0}^{T} \normLp{|u| + |v|}{r^*}^p dt \)^{\frac{\alpha-2}{p}}\\
&= C T^{\mu} \norm{|u| + |v|}_{Y_{p,r^*,T}}^{\alpha-2} \norm{u-v}_{Y_{p,r,T}}
\leq C T^{\mu} \(\norm{u}_{Y_{p,r^*,T}} + \norm{v}_{Y_{p,r^*,T}} \)^{\alpha-2} \norm{u-v}_{Y_{p,r,T}}.
\end{align*}
The inequality \eqref{eqn:2.18} follows by applying the previous Sobolev embedding and \cref{Strichartz} (ii). \end{proof}
In order to prove \cref{prop:cauchy} we shall need two lemmas from Functional Analysis. \begin{lemma}
\label{lem:reflexive}
For all $1 < p, r < \infty$, $X_{p, r, T}$ is a separable reflexive Banach space. \end{lemma} \begin{proof}
This is a direct consequence of Phillips' theorem, see \cite[Chapter IV, Corollary 2]{Diestel-Uhl}. \end{proof} \begin{lemma}\label{complete}
For all $R, T > 0$ the metric space $(B_{R,T}, d)$, where
\begin{equation*}
B_{R,T} := \{u \in X_T: \|u\|_{X_T} \leq R\},
\end{equation*}
and
\begin{equation*}
d(u, v) := \|u - v\|_{Y_T}
\end{equation*}
is complete. \end{lemma} \begin{proof}
Let $(u_n)$ be a Cauchy sequence. Since $Y_T$ is a Banach space, there exists $u \in Y_T$ such that
\begin{equation*}
\lim_{n\to\infty}\|u_n - u\|_{Y_T} = 0.
\end{equation*}
It remains to show that $u \in B_{R,T}$.
By taking a subsequence, we can assume that $l_1 := \lim_{n\to\infty}\|u_n\|_{X_{p_1, r_1, T}}$
and $l_2 := \lim_{n\to\infty}\|u_n\|_{X_{p_2, r_2, T}}$ exist.
By \cref{lem:reflexive}, there exists a subsequence of $(u_n)$ which converges weakly in $X_{p_1, r_1, T}$.
In particular, this sequence converges in the sense of distributions and hence
the limit equals $u$. Thus,
\begin{equation*}
\|u\|_{X_{p_1, r_1, T}} \leq l_1.
\end{equation*}
Similarly,
\begin{equation*}
\|u\|_{X_{p_2, r_2, T}} \leq l_2.
\end{equation*}
Taking the sum, we get $\|u\|_{X_T} \leq l_1 + l_2 \leq R$. \end{proof}
\begin{proof}[Proof of \cref{prop:cauchy}]
\textbf{Step 1. Existence and uniqueness in $B_{2\gamma_0,T}$ for $\gamma_0$ small enough.}
For any $u \in X_T$ and $t \in [0, T]$, we define
\begin{equation}
\label{eq:Phi-def}
\Phi(u)(t) := e^{it\Delta}\varphi + i\int_0^te^{i(t-s)\Delta}g(u(s))ds.
\end{equation}
We claim that, if $\gamma_0 >0$ is small enough, then $\Phi$ defines a contraction on
the metric space $(B_{2\gamma_0,T}, d)$ (see \cref{complete}).
Let $u \in B_{2\gamma_0, T}$ and consider any admissible pair $(\tilde p, \tilde r)$. Let $T \in (0,1]$ and apply \cref{lemma:1T}. We deduce from \eqref{eqn:2.17} and \eqref{eq:Phi-def} that
\begin{align*}
\norm{\nabla \Phi(u)-e^{it\Delta} \nabla \varphi}_{Y_{\tilde p, \tilde r,T}} \leq C \norm{\nabla u}_{Y_{p_1,r_1,T}}^{q-1} + C \norm{\nabla u}_{Y_{p_2,r_2,T}}^{2^*-1} \leq C 2^q \gamma_0^{q - 1}, \quad \forall u\in
B_{2\gamma_0, T} .
\end{align*}
Similarly, we deduce from \eqref{eqn:2.18} (applied with $v = 0$) that
\begin{align*}
\begin{split}
\norm{\Phi(u)- e^{it\Delta} \varphi}_{Y_{\tilde p, \tilde r,T}} \leq C \norm{\nabla u}_{Y_{p_1,r_1,T}}^{q-2} \norm{u}_{Y_{p_1,r_1,T}} + C \norm{\nabla u}_{Y_{p_2,r_2,T}}^{2^*-2} \norm{u}_{Y_{p_2,r_2,T}}
\leq C 2^q \gamma_0^{q-1}, \quad \forall u\in
B_{2\gamma_0, T} .
\end{split}
\end{align*}
In particular if we choose $(\tilde p, \tilde r)=(p_1, r_1)$ and $(\tilde p, \tilde r)=(p_2, r_2)$ then
$$\|\Phi(u)\|_{X_T}\leq \gamma_0 + C 2^q \gamma_0^{q-1}$$
and hence if $\gamma_0 >0$ is small enough in such a way that
$C 2^{q+2} \gamma_0^{q-1}\leq \gamma_0$, then $B_{2\gamma_0,T}$ is an invariant set of $\Phi$.
Now, let $u, v \in B_{2\gamma_0, T}$. By \eqref{eqn:2.18}, we have for every admissible pair $(\tilde p, \tilde r)$
\begin{align*}
\|\Phi(u) - \Phi(v)\|_{Y_{\tilde p, \tilde r, T}} &\leq C\big(\norm{\nabla u}_{Y_{p_1,r_1,T}}^{q-2} + \norm{\nabla v}_{Y_{p_1,r_1,T}}^{q-2}\big)\norm{u-v}_{Y_{p_1, r_1, T}}
+ C\big(\norm{\nabla u}_{Y_{p_2,r_2,T}}^{2^*-2}+ \norm{\nabla v}_{Y_{p_2,r_2,T}}^{2^*-2}\big)\norm{u-v}_{Y_{p_2, r_2, T}} \\
&\leq C 2^q \gamma_0^{q - 2}(\norm{u-v}_{Y_{p_1, r_1, T}} + \norm{u-v}_{Y_{p_2, r_2, T}}), \quad \forall u,v \in
B_{2\gamma_0, T}.
\end{align*}
In particular if we choose $(\tilde p, \tilde r)=(p_1, r_1)$ and $(\tilde p, \tilde r)=(p_2, r_2)$ then
$$\|\Phi(u) - \Phi(v)\|_{Y_{T}} \leq C 2^{q+1} \gamma_0^{q - 2}\|u - v\|_{Y_{T}}$$
and if we choose $\gamma_0 >0$ small enough in such a way that $C 2^{q+1} \gamma_0^{q - 2}<\frac 12$
then $\Phi$ is a contraction on $(B_{2\gamma_0, T}, d)$. In particular $\Phi$ has one unique
fixed point in this space.
The property $u \in C([0, T], {\mathit H})$ and $u\in X_{p,r,T}$ for every admissible couple $(p,r)$ is straightforward and follows
by Strichartz estimates.
\textbf{Step 2. Uniqueness in $X_T$.} Assume $u_1(t,x)$ and $u_2(t,x)$ are two fixed points of $\Phi$ in the space $X_T$.
We define
$T_0=\sup \{\bar T \in [0,T]| \sup_i \|u_i(t,x)\|_{X_{\bar T}} \leq 2\gamma_0\}$.
It is easy to show that $T_0\in (0, \bar T]$ and arguing as in step 1 the operator $\Phi$ is a contraction on $(B_{2\gamma_0, T_0}, d)$. Hence by uniqueness of the fixed point in this space necessarily
$u_1(t,x)=u_2(t,x)$ in $X_{T_0}$. Moreover since $u_i(t,x)\in \mathcal C([0, T_0]; H)$
we have $u_1(T_0,x)=u_2(T_0,x)=\psi(x)$. Hence at time $T_0$ the solutions coincide and starting from $T_0$
(that we can also identify with $T_0=0$ by using the traslation invariance w.r.t. to time of the equation),
we can apply again the step 1 in the ball $(B_{2\gamma_0, \tilde T}, d)$ with initial condition
$\psi(x)$, where
$\tilde T>0$ is such that $\|e^{it\Delta}\psi\|_{X_{\tilde T}} \leq \gamma_0$. Again by uniqueness
of the fixed point of $\Phi$ in the space $(B_{2\gamma_0, \tilde T}, d)$ we deduce that
$u_1(t,x)=u_2(t,x)$ in $X_{T_0+\tilde T}$, hence contradicting the definition of $T_0$ unless $T_0=T$.
\textbf{Step 3. Conservation laws.} The proof of \eqref{eq:laws} is
rather classical. In particular
it follows by Proposition 1 and Proposition 2 in \cite{Ozawa06}. Another possibility is to follow the proof of
Propositions 5.3 and 5.4 in \cite{Ginibre}, that
can be repeated {\em mutatis mutandis}
in the context of \eqref{NLS}. The minor modification compared with \cite{Ginibre} is that we use the end-point Strichartz
estimate in order to treat the Sobolev critical nonlinearity. \end{proof}
\section{Orbital stability}\label{Section-5}
We shall prove in this section that the set $\mathcal{M}_c$ defined in \eqref{set-stable} is orbitally stable. In particular a nontrivial point concerns the fact that the local solutions, whose existence has been established in \cref{Section-4}, can be extended to global solutions provided that the initial datum is close to $\mathcal{M}_c$. The main difficulty is related to the criticality of the nonlinearity in \eqref{NLS}, which implies that an a priori bound on the Mass and the Energy is not sufficient to exclude a finite-time blow-up. We will overcome this issue by deducing from the uniform local well-posedness (Proposition~\ref{prop:cauchy}) a uniform lower bound on the time of existence of the solution corresponding to initial data close to a set which is compact up to translations, see Theorem~\ref{theorem:2}.
To simplify the next statement we denote by $u_{\varphi}(t)$ the integral solution associated with \eqref{NLS} and we denote by $T_{\varphi}^{max}$ its maximal time of existence.
\begin{theorem} \label{theorem:1}
Let $v \in \mathcal{M}_c $. Then, for every $\varepsilon > 0$ there exists $\delta > 0$ such that:
\begin{align} \label{eqn:2.9}
\forall \varphi \in {\mathit H} \mbox{ s.t. } ||\varphi - v||_{{\mathit H}} < \delta \Longrightarrow \sup_{t \in [0, T_{\varphi}^{max})} \dist{u_{\varphi}(t)}{\mathcal{M}_c} < \varepsilon.
\end{align}
In particular we have
\begin{align} \label{eqn:2.10}
u_{\varphi}(t) = m_{c}(t) + r(t), \quad \forall t \in [0, T_{\varphi}^{max}), \mbox{ where } m_{c}(t) \in \mathcal{M}_c, \, \norm{r(t)}_{{\mathit H}} < \varepsilon.
\end{align} \end{theorem} \begin{proof}
Suppose the theorem is false. Then there exists $(\delta_n) \subset \mathbb{R}^+$ a decreasing sequence converging to $0$ and $(\varphi_n) \subset {\mathit H}$ satisfying
\begin{equation*}
||\varphi_n - v||_{{\mathit H}} < \delta_n
\end{equation*}
and
\begin{equation*}
\sup_{t \in [0, T_{\varphi_n}^{max})} \dist{u_{\varphi_n}(t)}{\mathcal{M}_c} > \varepsilon_0,
\end{equation*}
for some $\varepsilon_0 >0$. We observe that $||\varphi_n||_2^2 \to c$ and, by continuity of $F_{\mu}$, $F_{\mu}(\varphi_n) \to m(c)$. By conservation laws, for $n \in \mathbb{N}$ large enough, $u_{\varphi_n}$ will remains inside of $B_{\rho_0}$ for all $t \in [0, T_{\varphi_n}^{max})$.
Indeed, if for some time $\overline{t}>0$ $||\nabla u_{\varphi_n}(\overline{t})||_2^2 = \rho_0$ then, in view of \cref{Lemma-structure} (\ref{point:5L.5i}) we have that $F_{\mu}(u_{\varphi_n}(\overline{t})) \geq 0$ in contradiction with $m(c) <0$. Now let $t_n>0$ be the first time such that
$\dist{u_{\varphi_n}(t_n)}{\mathcal{M}_{c}} = \varepsilon_0$ and set $u_n := u_{\varphi_n}(t_n)$.
By conservation laws,
$(u_n) \subset B_{\rho_0}$ satisfies $\normLp{u_n}{2}^2 \to c$ and $F_{\mu}(u_n) \to m(c)$ and thus, in view of \cref{theorem:LT-L}, it converges, up to translation, to an element of $\mathcal{M}_c$. Since $\mathcal{M}_c$ is invariant under translation this contradicts the equality
$\dist{u_n}{\mathcal{M}_{c}} = \varepsilon_0 >0$. \end{proof}
The rest of this section is devoted to showing that $T_{\varphi}^{max}=\infty$ and it will conclude the proof of \cref{thm-2}.
\begin{proposition} \label{proposition:1}
Let $\mathcal{K} \subset {\mathit H} \setminus \{0\}$ be compact up to translation and assume that $(p, r)$ is an admissible pair with $p \neq \infty$. Then, for every $\gamma > 0$ there exists $\varepsilon = \varepsilon(\gamma) > 0$ and $T = T(\gamma) > 0$ such that
\begin{align*}
\sup_{\{\varphi \in {\mathit H}| \dist{\varphi}{\mathcal{K}} < \varepsilon\}} \norm*{e^{it\Delta} \varphi}_{X_{p,r,T}} < \gamma.
\end{align*} \end{proposition} \begin{proof}
We first claim, for every $\gamma > 0$, the existence of a $T > 0$ such that
\begin{align}\label{ajoutL5}
\sup_{\varphi \in \mathcal{K}} \norm*{e^{it\Delta} \varphi}_{X_{p,r,T}} < \dfrac{\gamma}{2}.
\end{align}
If it is not true then there exists sequences $(\varphi_n) \subset \mathcal{K}$ and $(T_n) \subset \mathbb{R}^+$
such that $T_n \to 0$ and
\begin{align}\label{eqn:1t}
\norm*{e^{it\Delta} \varphi_n}_{X_{p,r,T_n}} \geq \overline{\gamma}
\end{align}
for a suitable $\overline{\gamma}>0$. Since $\mathcal{K}$ is compact up to translation, passing to a subsequence, there exists a sequence $(x_n) \subset \mathbb{R}^N$ such that
$$ \tilde{\varphi_n}(\cdot) := \varphi_n(\cdot - x_n) \overset{ H }\to \varphi(\cdot)$$
for a $\varphi \in {\mathit H}$. By continuity (induced by Strichartz's estimates) we have, for every $\bar T>0$,
\begin{align}\label{ajoutL2}
\norm*{e^{it\Delta} \tilde{\varphi}_n}_{X_{p,r,\bar T}} \to \norm*{e^{it\Delta} \varphi}_{X_{p,r,\bar T}}.
\end{align}
Also, recording the translation invariance of Strichartz's estimates we get from \eqref{eqn:1t} that
\begin{equation}\label{ajoutL1}
\norm*{e^{it\Delta} \tilde{\varphi}_n}_{X_{p,r,T_n}} = \norm*{e^{it\Delta} \varphi_n}_{X_{p,r,T_n}} \geq \overline{\gamma}.
\end{equation}
Now, by \cref{Strichartz} (iii), we have $e^{it\Delta} \varphi \in X_{p,r, 1}$, namely the function
$$[0,1]\ni t\to g(t):=||e^{it\Delta} \varphi||_{W^{1,r}(\mathbb{R}^N)}^p$$
belongs to $L^1([0,1])$. Then by
the Dominated Convergence Theorem we get $\|\chi_{[0, \tilde T]}(t) g(t)\|_{L^1([0,1])} \to 0$
as $\tilde T\to 0$,
namely $\norm*{e^{it\Delta} \varphi}_{X_{p,r,\tilde T}}^p
\to 0$ as $\tilde T \to 0$. Hence, we can choose $\bar T >0$ such that
\begin{align}\label{ajoutL3}
\norm*{e^{it\Delta} \varphi}_{X_{p,r,\bar T}} < \overline{\gamma}.
\end{align}
At this point gathering \eqref{ajoutL2}- \eqref{ajoutL3} we get a contradiction and the claim holds.
Now, fix a $T>0$ such that \eqref{ajoutL5} holds. By \cref{Strichartz} (iii), we have
\begin{align*}
\norm*{e^{it\Delta} \eta}_{X_{p,r,T}} \leq C ||\eta||_{{\mathit H}}, \qquad \forall \eta \in {\mathit H}.
\end{align*}
Thus, assuming that $\displaystyle ||\eta||_{{\mathit H}} < \frac{\gamma}{2C} := \varepsilon$, we obtain that
\begin{align*}
\norm*{e^{it\Delta} \eta}_{X_{p,r,T}} < \frac{\gamma}{2}.
\end{align*}
Summarizing, we get that, for all $\varphi \in \mathcal{K}$ and all $\eta \in {\mathit H}$ such that $||\eta||_{{\mathit H}} < \varepsilon$,
\begin{align*}
\norm*{e^{it\Delta} (\varphi + \eta)}_{X_{p,r,T}} \leq \norm*{e^{it\Delta} \varphi}_{X_{p,r,T}} + \norm*{e^{it\Delta} \eta}_{X_{p,r,T}} < \gamma.
\end{align*}
This implies the proposition. \end{proof}
\begin{proposition} \label{proposition:2}
Let $\mathcal{K} \subset {\mathit H} \setminus \{0\}$ be compact up to translation. Then, for every $\gamma > 0$ there exists $\varepsilon = \varepsilon(\gamma) > 0$ and $T = T(\gamma) > 0$ such that
\begin{align*}
\sup_{\{\varphi \in {\mathit H}| \dist{\varphi}{\mathcal{K}} < \varepsilon\}} \norm*{e^{it\Delta} \varphi}_{X_{T}} < \gamma.
\end{align*} \end{proposition} \begin{proof}
We apply \cref{proposition:1} twice with the admissible pairs $(p_1, r_1)$ and $(p_2, r_2)$. Then, the proposition follows from the definition of the norm $X_T$ given in \eqref{eqn:XT}. \end{proof}
\begin{theorem} \label{theorem:2}
Let $\mathcal{K} \subset {\mathit H} \setminus \{0\}$ be compact up to translation. Then there exist $\varepsilon_0 > 0$ and $T_0>0$ such that the Cauchy problem \eqref{NLS}, where $\varphi$ satisfies $\dist{\varphi}{\mathcal{K}} < \varepsilon_0$, has a unique solution on the time interval $[0, T_0]$ in the sense of \cref{def:1}. \end{theorem} \begin{proof}
We apply \cref{proposition:2} where $\gamma = \gamma_0$ is given in \cref{prop:cauchy}. Then \cref{prop:cauchy} guarantees that the theorem holds for $\varepsilon_0 = \varepsilon (\gamma_0) >0$ and $T_0 = \min\{ T(\gamma_0),1 \} >0$. \end{proof}
\begin{theorem} \label{theorem:3}
Let $\mathcal{M}_c$ be defined in \eqref{set-stable}. Then there exists a $\delta_0 > 0$ such that, if $\varphi \in {\mathit H}$ satisfies $\dist{\varphi}{\mathcal{M}_c} < \delta_0$ the corresponding solution to \eqref{NLS} satisfies $T_{\varphi}^{max} = \infty$. \end{theorem} \begin{proof}
We make use of \cref{theorem:2} where we choose $\mathcal{K} = \mathcal{M}_c$. By \cref{theorem:1}, we can choose a $\delta_0 >0$ such that \eqref{eqn:2.9} and \eqref{eqn:2.10} holds for $\varepsilon = \varepsilon_0$ where $\varepsilon_0 >0$ is given in \cref{theorem:2}. Then \cref{theorem:1} guarantees that the solution
$u_{\varphi}(t)$ where $\dist{\varphi}{\mathcal{M}_{c}} < \delta_0$ satisfies $\dist{u_{\varphi}(t)}{\mathcal{M}_{c}} < \varepsilon_0$ up to the maximum time of existence $T_{\varphi}^{max} \geq T_0$. Since, at any time in $(0, T_{\varphi}^{max})$ we can apply again \cref{theorem:2} that guarantees an uniform additional time of existence $T_0 >0$, this contradicts the definition of $T_{\varphi}^{max}$ if $T_{\varphi}^{max} < \infty$. \end{proof}
At this point we can give, \begin{proof}[Proof of \cref{thm-2}]
The fact that $\mathcal{M}_c$ is compact, up to translation, was established in \cref{theorem:LT-L}. The orbital stability of $\mathcal{M}_c$, in the sense of
\cref{def:stability} follows from \cref{theorem:1} and \cref{theorem:3}. \end{proof}
\renewcommand{References}{References}
\end{document} | arXiv |
Molecular profiling of an oleaginous trebouxiophycean alga Parachlorella kessleri subjected to nutrient deprivation for enhanced biofuel production
Kashif Mohd Shaikh1,2,
Asha Arumugam Nesamma1,
Malik Zainul Abdin2 &
Pannaga Pavan Jutur ORCID: orcid.org/0000-0001-7988-28831
Biotechnology for Biofuels volume 12, Article number: 182 (2019) Cite this article
Decreasing fossil fuels and its impact on global warming have led to an increasing demand for its replacement by sustainable renewable biofuels. Microalgae may offer a potential feedstock for renewable biofuels capable of converting atmospheric CO2 to substantial biomass and valuable biofuels, which is of great importance for the food and energy industries. Parachlorella kessleri, a marine unicellular green alga belonging to class Trebouxiophyceae, accumulates large amount of lipids under nutrient-deprived conditions. The present study aims to understand the metabolic imprints in order to elucidate the physiological mechanisms of lipid accumulations in this microalga under nutrient deprivation.
Molecular profiles were obtained using gas chromatography–mass spectrometry (GC–MS) of P. kessleri subjected to nutrient deprivation. Relative quantities of more than 60 metabolites were systematically compared in all the three starvation conditions. Our results demonstrate that in lipid metabolism, the quantities of neutral lipids increased significantly followed by the decrease in other metabolites involved in photosynthesis, and nitrogen assimilation. Nitrogen starvation seems to trigger the triacylglycerol (TAG) accumulation rapidly, while the microalga seems to tolerate phosphorous limitation, hence increasing both biomass and lipid content. The metabolomic and lipidomic profiles have identified a few common metabolites such as citric acid and 2-ketoglutaric acid which play significant role in diverting flux towards acetyl-CoA leading to accumulation of neutral lipids, whereas other molecules such as trehalose involve in cell growth regulation, when subjected to nutrient deprivation.
Understanding the entire system through qualitative (untargeted) metabolome approach in P. kessleri has led to identification of relevant metabolites involved in the biosynthesis and degradation of precursor molecules that may have potential for biofuel production, aiming towards the vision of tomorrow's bioenergy needs.
The global energy demand is increasing day by day as the energy consumption is rising and is expected to increase by 53% within the next two decades. The fossil-derived diesel has been an important source of transportation fuel, but a significant need has come up to look for alternate sources of energy as the conventional source is non-regenerable and costs a lot to the environmental sustainability. The fossil fuel reserves are limited, and as their sources perish, the world will face a huge hike in fuel prices. Since the food and fuel prices are interdependent, the increase in fuel prices will ultimately influence the cost of food [1]. The rapid increase in energy consumption globally has raised the requirement for the development of sustainable renewable energy sources. In the need of current scenario, the production of biodiesel has increased considerably in the recent past with annual production reaching over billions of litres. Mostly waste cooking oil, soybean oil, palm oil, etc. have been used for the production of biodiesel. However, this conventional mode of production, perhaps in the near future, will lead to competition for land usage in terms of fuel and food. Hence, microalgae are being looked upon as a potential source for biodiesel production and have gained considerable attention because of their capability to utilize sunlight and water to convert atmospheric CO2 into biomass and biofuels which can prove to be important for both food and energy requirements [2, 3].
Microalgae can produce biomass along with the accumulation of large quantities of lipids/triacylglycerols (TAGs) for biodiesel production. The major advantage for the production of biodiesel from microalgae is their ability to produce large amount of biomass and lipid photosynthetically, and their ability to grow on non-arable land using saline and/or waste waters that make them free from any competition with resources required for growing food [4,5,6,7]. Other advantages include their capability to sequester greenhouse gas, a major environmental benefit as the world is facing huge climatic change manifested with conventional fuel utilization [8, 9]; their ability to absorb nutrients from waste waters helping in bioremediation, which is both economical and environment friendly [10, 11]; and their ability to synthesize certain high-value co-products such as OMEGAs, astaxanthin, lutein, tocopherols that are essential for industrial production in pharmaceuticals, nutraceuticals, etc. [12, 13]. One important lead with microalgal-TAG-based biodiesel-derived fuels is their easy integration into the current infrastructure of transportation fuels [14]. Some microalgae can produce TAGs when grown under heterotrophic mode [15], and under autotrophic mode of growth, numerous factors tend to stimulate lipid production such as nutrient availability, light, temperature [16, 17]. Even through microalgae hold potential feedstock for the production of lipids, the accumulation of oil tends to amplify under stress conditions but perhaps the major concern is inhibition of growth, thus simultaneously hampering biomass [18].
Despite such a huge potential microalgae hold for a sustainable source of renewable energy, a number of challenges exist in way for their commercialization as biofuel source. Few microalgal species have been identified as a promising source for industrial-level biofuel, nutraceuticals and pharmaceutical productions, but various research efforts are still being carried out to make microalgal biofuels cost-effective and sustainable. The diverse genera of algae lead to their exceptionally wide range of lipid and metabolic profile which is a result of their dynamic environmental condition [19]. Hence, detailed study on selection, culturing condition optimization, large-scale bioreactor development, bioengineering for better biomass and biofuel, improvement in biomass harvesting and other downstream processing is being carried out to reduce the production cost [20,21,22,23]. Several attempts have been made to improve strain performance, harvesting, extraction and culture systems to bring down the economic input for large-scale production [24,25,26,27,28]. The lipid composition among microalgae varies between 10 and 60% (dw) because of the wide range of strains as well as the environmental conditions in which they occur and/or are cultivated [29, 30]. The primary requirement for industrial production of microalgae-based biodiesel is the screening for conditions that induce high lipid productivity in fast-growing microalgae that can fulfil the criteria for sustainable biofuels. Henceforth, in-depth understanding of such phenomenon might also provide deeper insights into the bioengineering of industrially feasible strains. A number of biochemical strategies have been used in this direction to enhance lipid and biomass production [31, 32]. Various environmental factors affect the microalgal cultivation, altering its biomass and biochemical composition [33, 34]. Menon et al. [16] showed that generation of specific intracellular reactive oxygen species (siROS) during stress acts as a common signal that affects various metabolic pathways including lipid biosynthesis. The availability of nutrients affects the microalgal growth as well as their lipid and metabolic compositions [35, 36]. Hence, limiting nutrient availability in the media to induce metabolic variations and lipid accumulation in microalgae is an important alternative strategy to understand the initiation and storage of TAGs in the system.
Despite the significance of various metabolic products in regulating the cellular dynamics, and mechanisms that control the partitioning of these metabolites into distinct carbon-storing molecules in algae, their role in algal physiology and biofuel precursors production is poorly illustrated. In the present work, we have focused on understanding the phenomenon of nutrient deprivation as a tool to enhance lipid productivity as well as the associated changes in the metabolic profiles and biochemical composition of indigenous marine microalga Parachlorella kessleri (I) under three different nutrient limitations, viz. nitrogen, phosphorous and sulphur. Previous studies on P. kessleri revealed its potential as a suitable candidate for biofuel production, with lipid content around 40–60% of dry cell weight [37,38,39,40]. The significance of selecting this indigenous marine microalgae P. kessleri (I) is mainly due to its better biomass productivity and higher lipid content as reported earlier [37, 40]. Our rationale highlights up on building a crosstalk between the metabolomic changes and cellular dynamics in terms of biomass and lipid productivities, when this marine microalgae is subjected to nutrient deprivation.
Growth and biochemical analysis
The primary impact of nutrient stress is visible on the growth pattern, so the biomass accumulation was analysed for P. kessleri under the nitrogen-, phosphorous- and sulphur-deprived conditions. The growth parameters of the marine strain P. kessleri under standard growth conditions with an initial inoculum of ~ 0.057 g L−1 produced a biomass of 0.54 g L−1 in 10 days, thus exhibiting better growth rate, achieving specific growth rate of 0.67 µ and doubling time around 24.7 h. Several studies have shown that microalgae growth depends on an adequate supply of essential macronutrient elements (carbon, nitrogen, phosphorus, silicon), major ions (Mg2+, Ca2+, Cl−, So 4 2− ) as well as on a number of micronutrient metals such as iron, manganese, zinc, cobalt, copper and molybdenum [41]. To analyse the effect of different nutrient depletions such as nitrogen (N-), phosphorus (P-) or sulphur (S-) on the growth profile, P. kessleri was grown under continuous photoautotrophic conditions. The results demonstrated that this strain had severe effect on growth in nitrogen (N-) deprivation, i.e. growth was shunted within 4 days of deprivation after which no change in biomass was observed (Fig. 1a). In sulphur (S-) deprivation and phosphorous (P-) deprivation, no significant change in growth rate was observed till the sixth day (Fig. 1a). In P. kessleri, the effect of sulphur (S-) depletion on growth was delayed as compared to nitrogen deprivation. Inset (Fig. 1a) demonstrates the cultures in different deprivation conditions which show growth retardation as well as loss in pigmentation (indicated by pale green colour). The nitrogen concentration in the same medium deficit in N-, P- and S- during the microalgae culture was also estimated. In P- and S- conditions, the nitrogen utilization was slower when compared to the control. Most of the nitrogen was utilized by microalgae at the end of the sixth day in the control, whereas in P- and S- conditions it was completely consumed by the end of the tenth day. This pattern is also observed in the growth profile as the cell growth slows down and shifts towards stationary phase.
Biochemical profiles of Parachlorella kessleri under control and stress conditions. a Growth profiles, b total chlorophyll, c protein and d carbohydrate contents; C, control; N-, nitrogen deprivation; P-, phosphorous deprivation; S-, sulphur deprivation; days of treatment—0, 2, 4, 6, 8 and 10 days
Various biochemical constituents were analysed to understand the effect of nutrient starvation over molecular profiling in P. kessleri. Figure 1b–d shows the pigment (total chlorophyll), protein and carbohydrate profiles subjected to different nutrient deprivation conditions. The total chlorophyll content was severely reduced under N- stress, while S- deprivation leads to a steady loss of pigments over time. Phosphorus (P-) deprived cells maintained their net chlorophyll levels up to 4–6 days followed by decline in prolonged stress (Fig. 1b). During N- stress, the chlorophyll content was severely deteriorated within 2–4 days of starvation, while P- and S- cells showed a slow decrease. Photosynthetic machinery is the primary component to be affected by severe nutrient deprivation, especially in case of nitrogen deprivation as reported in most of the microalgal strains [42].
The total protein content was also decreased rapidly in case of nitrogen-deprived cells and reached the minimum by the end of the second day. In case of S- deprivation, the decrease was steady over time, whereas in P- stress the protein content started declining after the fourth day (Fig. 1c). During P- and S- conditions, the protein content in the cells declined slowly, but the initial impact on protein machinery was not adverse as seen during N- starvation. When algae are subjected to nutrient deprivation, the cells start to reduce the protein build up and catabolize proteins to use the carbon skeleton to synthesize storage molecules. Figure 1d demonstrates the changes occurring during nutrient stress in the carbohydrate content. Under N- deprived cells, an initial rise in the total carbohydrate content was observed on day 2 (~ onefold), after which it has declined rapidly (Fig. 1d). Our presumption predicts that P. kessleri isolated from marine waters may certainly not be a predominant carbohydrate producing strain. Under P- and S- deprivation, an increase in the carbohydrate content was observed till the fourth day. Increase in carbohydrate as a storage molecule has been observed in many algal species, mainly fresh water species, when the cells are subjected to nutrient deprivation.
Lipid analysis and profiling
Lipid accumulation in algae is induced normally due to environmental stress, particularly when deprived of certain nutrients such as nitrogen, phosphorus, silica, sulphur or certain metals [43, 44]. In microalgae, nutrient deprivation to enhance the production of lipids is a well-observed phenomenon [33, 45]. During stress-induced lipid accumulation, the lack of essential nutrients such as N, P and S restricts the capacity of cellular division as a result of which the organism shifts towards alternative pathways for inorganic carbon assimilation, thus shuffling the carbon towards the storage biomolecules, i.e. TAGs. To demonstrate the effect of different nutrient deprivation on the synthesis of TAGs, total lipid was extracted from cells and analysed using thin-layer chromatography (TLC) as described in "Methods". The TLC plate loaded with extracted lipids from samples of the three stress conditions (N-, P- and S-) led to a sharp increase in the TAG content in P. kessleri, where TAG seems to increase with the progression of duration of starvation (data not shown). Further, the samples were quantified using GC–MS analysis to evaluate the lipid productivity subjected to nutrient stress. Sulphur (S-) depletion induced TAG formation to a much lesser extent as compared to other nutrient-deprived conditions. The increase in TAG content can be observed from the second day itself in N- condition, whereas in P- and S- depletion TAG accumulation was observed after the fourth day (Fig. 2a). In P- depletion, a gradual increase in TAG content was observed without compromising cell growth. Under N- stress, the TAG production has been initiated on day 2 itself and reached maximum by the tenth day, but also lead to severe growth inhibition depicting metabolic changes within the cells. In marine microalgae P. kessleri, S- depletion lead to inhibition in growth after day 6 but in comparison with N- and P-, the increase in lipid content was not very significant (Fig. 2a).
a Total lipid content (represented as line graphs) and FAME productivity (in bars) under control and nutrient-deprived conditions for day 6. Inset shows the change in FAME content with stress conditions, days of treatment—0 (control), 2, 4, 6, 8 and 10 days. b Changes in saturated (SFA), monounsaturated (MUFA) and polyunsaturated (PUFA) profiles of fatty acids under nutrient-deprived conditions; N-, nitrogen deprivation; P-, phosphorous deprivation; S-, sulphur deprivation; days of treatment—0 (control), 2, 4, 6, 8 and 10 days
A number of changes occur in the overall neutral lipid content as well as the saturation profile of lipids when microalgae are subjected to nutrient deprivation [46]. Figure 2a, b shows the lipid content (µg mg−1 dw) and FAME productivity (mg L−1 day−1) analysed through GC/MS under different nutrient (N, P and S) deprivation conditions compared to the control in P. kessleri. Our data show that FAME productivities were similar in N- (~ 11.63 mg L−1 day−1) and P- (~ 11.58 mg L−1 day−1), while lower in case of S- deprivation (~ 7.13 mg L−1 day−1) at the end of the sixth day (Fig. 2a). As compared to nitrogen and phosphorous depletion, the FAME productivity under sulphur limitation was substantially low (Fig. 2a).
The FAME content per cell seems to be higher in N- stress (Fig. 2a), where a constant increase in lipid accumulation was observed, i.e. reaching up to > 40% of dry cell weight, while in P- the lipid content per cell remains low as compared to N- cells but is considerably higher than control (Fig. 2a inset). Overall, the time-dependent changes in lipid content for 10 days in different stress conditions are shown in Fig. 2a (inset). In the present study, P. kessleri subjected to N- has shown enhanced neutral lipid content, whereas P- tends to have a steady increase (Fig. 2a).
On initial day (control) of inoculation, FAME profiling patterns showed highest content of polyunsaturated fatty acids (PUFAs) and saturated fatty acids (SFAs) than monounsaturated fatty acids (MUFAs) (Fig. 2b). Under N- conditions, P. kessleri exhibits an increase in MUFAs (up to 41% of total FAME) with considerable decrease in SFAs (to ~ 35%) and PUFAs (to ~ 24%) by the end of the tenth day. This can be a result of the oxidative damage to PUFAs under stress [47] or recycling of membrane lipids towards TAGs [48]. Although P- limitation induced lipid biosynthesis, at the end of deprivation period, the FAME pattern suggests decrease in SFAs (to ~ 32.5%) and considerable increase in MUFAs (to ~ 24.5%) and PUFAs (to ~ 43%). In S- depleted cells, SFAs increased (to ~ 52%) with a little upregulation in MUFAs (to ~ 20%) while PUFAs decreased (Fig. 2b).
Metabolome analysis
Metabolite levels are tightly controlled during the starvation condition to enhance the chances of survival. A number of changes were observed in terms of growth and biochemical profiles under different nutrient starvation conditions. Therefore, to understand the molecular profiling, we have employed qualitative metabolomics tool to evaluate the changes occurring during stress which will provide new insights for enhancing the lipid production. The metabolite extraction and derivatization were carried out in all the samples of P. kessleri as described in "Methods" section. A total number of ~ 62 metabolite peaks were obtained after manual curation and analysis of raw data. The most common phenomenon observed in the raw data files is the repetition of same metabolite as a result of alternate derivatization [36]; such metabolites were removed if not significant. All the metabolites analysed in P. kessleri under different nutrient deprivation were plotted using Venny 2.1 (http://bioinfogp.cnb.csic.es/tools/venny/) to find out intersecting and differential metabolites (Fig. 3). Among these, eight metabolites were exclusively expressed under N-, four in P- and 14 in S- conditions (Fig. 3).
Venn diagram representing various metabolites in different stress conditions. C, control; N-, nitrogen deprivation; P-, phosphorous deprivation, S-, sulphur deprivation. Numbers in brackets show total metabolites obtained in each condition
Certain metabolites were common in all stress conditions, while certain were common in two conditions, as discussed later. The fold change for all the metabolites subjected to stress conditions either upregulated or downregulated as compared to the control is shown in Fig. 4a–c. (The list for metabolites with their representative numeric code is included.) In N- cells, many metabolites such as valine, trehalose, citric acid, mannose, linoleic acid, trans-9-octadecanoic acid, talose were found to increase > twofold, while malic acid, myo-inositol, glucose, polyunsaturated fats were predominantly decreased (Fig. 4a). In P- cells, upregulated metabolites include citric acid, galactose, mannose, threose, while myo-inositol, glucose, azelaic acid, sorbose, a-tocopherol were decreased (Fig. 4b). In S- cells, metabolites such as trehalose, mannitol, galactose, mannose were increased, while malic acid, glutamic acid, citric acid, myo-inositol decreased (Fig. 4c).
Fold-change in metabolites under stress conditions; a nitrogen deprivation: N-, b phosphorous deprivation: P-, c sulphur deprivation; table represents numerical abbreviations of the metabolites
The overall metabolomic profiles in P. kessleri when subjected to nutrient deprivation have been illustrated as a heat map for the visualization of expression profiles of various metabolites (Fig. 5).
A heat map illustrating the expression of all the metabolites under nutrient deprivation conditions. N-, nitrogen deprivation; P-, phosphorous deprivation; S-, sulphur deprivation
An essential metabolite of interest, citric acid, was found to be upregulated nearly by fivefold in only N- and P- conditions. It seems to be an important metabolite in upregulating the FA biosynthesis as it increased in both N- and P- conditions where lipid accumulation has also increased, while it decreased in S- where lipid accumulation is much lower as compared to other two conditions. Also, 2-ketoglutaric acid was found to increase in the similar conditions by twofold, both together assume to divert flux towards FA biosynthesis [49]. Another metabolite of importance is trehalose that was found enhanced fivefold in N- and S- but not in P- conditions that may presumably play a significant role in cell growth regulation. Significant metabolite changes occur when subjected to nutrient depletion (N-, P- and S-) conditions, and each of these changes will affect cell growth and lipid productivities. Henceforth, our metabolomic data provide us with the schematic model to understand flux diversion that leads to changes in lipid productivity and growth rate under different nutrient stresses (Fig. 6).
Schematic representation of metabolic pathway reactions altered under stress; colour codes for different stress treatments. Red arrows show pathways routing towards fatty acid (FA) biosynthesis. Upregulation (↑); downregulation (↓)
Growing bioenergy needs demand urgent action to generate renewable fuels at feasible cost. Algae seem to be a promising bioresource in terms of economically feasible bioenergy producer, yet the information regarding cellular dynamics of microalgal cells is fairly poor. The primary focus in algal research has been the enhancement of lipid production employing nutrient stress while biomass productivities are essentially compromised. A number of strategies, such as nutrient deprivation, light intensity, temperature variation, carbon dioxide have been employed to reach specific goals, but the cellular behaviour under these conditions is yet not well established. Under the adequate supply of nutrients including C, N, P, S and light, depending on the strain selection, the biomass productivity can be high but the lipid content seems to be as low as 5% w/w or even less [50]. During unfavourable conditions, the microalgal growth and photosynthetic activity cease, while the excess energy gets assimilated as lipids and/or carbohydrates. Metabolically, there is always a competition between biomass and storage molecule assimilation, which governs and channelizes the carbon flux either towards biomass accumulation or towards lipid and carbohydrate synthesis. Depending on the requirement, a metabolic shift can switch the photosynthetic assimilation of inorganic carbon from biomass synthesis to energy storage metabolism [51, 52]. Changes in the environmental conditions such as light intensity, nutrient limitation, salinity, temperature, pH, and culture age invariably affect the lipid content of microalgae [53,54,55,56,57,58,59]. Oleaginous microalgae can utilize their lipid metabolic pathway for the biosynthesis and accumulation of lipids in the form of triacylglycerols (TAGs) [60]. Some of these oleaginous microalgae can even store TAGs up to 40% to 70% of their dry weight [33, 61]. These lipids are typically storage reserves within the cell that helps the organism to sustain adverse environmental conditions. Henceforth, the competition in terms of biomass (or) lipid accumulation depends upon the different levels of perturbation [62]. In the present work, growth and cellular physiology of P. kessleri were demonstrated under different nutrient deprivations. While growth was severely hampered under N- deprivation, the cells were able to withstand phosphorus or sulphur absence fairly well, although after 6 days the S- cells showed high decline in growth. Perhaps the nutrient deprivation can be attributed to the evolutionary behaviour of microalgae, for example in marine waters the phosphorous availability is quite dynamic, and hence, these photosynthetic organisms do have specialized responses to maintain their growth under certain nutrient depletion conditions [63, 64]. Nitrogen, a major component of proteins, enzymes and nucleotides, is extremely essential, without which growth cannot be sustained.
A number of physiological changes are observed when microalgae are subjected to nutrient deprivation. During stress, the cell machinery will try to minimize the protein synthesis due to non-availability of nitrogen by shutting down the protein biosynthesis and degrading the protein pool to amino acids to get energy for survival as well as to assimilate carbon as storage molecules. Under nutrient-deprived condition, the total chlorophyll content decreased as the days of starvation progressed. Under nitrogen-deprived condition, the chlorophyll content was severely deteriorated within 2 to 4 days of starvation, while phosphorous- and sulphur-starved cells showed a slow decrease. Chlorophyll reduced to almost half within 2 days of nitrogen starvation (Fig. 1b). This is a very common phenomenon observed under nitrogen stress among other strains too. The protein content did show variable response to different stress conditions. Under nitrogen deprivation, rapid lowering in protein content is observed. The cell machinery tries to cope up with nitrogen unavailability by cutting down on protein synthesis and degrading the protein pool to amino acids to get energy for survival as well as to divert carbon towards storage molecules. Phosphorous- and sulphur-starved cells also showed a slow decline in the protein content, but the initial impact on protein machinery was not as severe as nitrogen starvation Fig. 1c). A sharp increase in carbohydrates in nitrogen-starved cells was observed on the second day, where the carbohydrate content was doubled, which later decreased substantially. This initial response might be attributed to the formation of carbohydrates as energy storage molecule in the case of extreme nitrogen limitation, which later provides carbon skeleton for lipid biosynthesis as storage molecules under prolonged starvation. The increase in carbohydrate content in phosphorous- and sulphur-starved cells was observed after day 4, and the lipid accumulation was observed late as compared to nitrogen-starved cells (Figs. 1c, 2a). These observations suggest that the microalgae initially store carbohydrates energy reserves to overcome the initial stress and then utilize the same for energy generation when they are subjected to severe macronutrient limitation. This has been reported in various microalgae, where some accumulate starch under nutrient depletion condition, others accumulate lipids, or an initial starch accumulation followed by lipid accumulation over prolonged stress is observed [65,66,67,68,69]. In conclusion, adverse effects in molecular profiles of biomolecules (such as total chlorophyll, proteins and carbohydrates) were seen during N- and S- conditions with hampered cell growth (biomass), whereas P- stress seems to have a limited effect.
Under optimal conditions, photosynthesis and electron transport chain produce ATP and NADPH which is utilized as energy currency during cell division [70]. Hence, the optimal ratio of reduced and oxidized metabolites is maintained, whereas during nutrient deprivation the pool of NADP+ and ADP depletes as photosynthesis continuously produces NADPH and ATP which remains under-utilized [62]. Biosynthesis of fatty acid consumes NADPH and ATP; hence, the increased fatty acid synthesis helps the cells to maintain the balance of required electron acceptors (NADP+). During nutrient limitation, an increase in the lipid content has been demonstrated in several microalgal species [38, 63, 71]. Nitrogen is an essential requirement for protein synthesis as well as photosynthesis, but under nitrogen-limiting conditions most of the carbon fixed in photosynthesis is channelized towards the production of lipids and/or carbohydrates. Several studies have demonstrated that nitrogen deprivation leads to higher accumulation of lipids in various microalgal strains [72, 73]. Upon removal of nitrogen, Nannochloropsis sp. and Neochloris oleoabundans increased their lipid content to onefold and twofold, respectively [74, 75]. A number of cellular metabolic processes such as photosynthesis, signal transduction, energy transport system require phosphorous as the main component, and hence, the deficiency of this major element also results in accumulation of lipids [76,77,78]. Similar studies showed increase in lipid accumulation to more than 50% of dry weight (dw) under P- limitation in Scenedesmus sp. LX1 belonging to Scenedesmaceae [79, 80]. In our present investigation, FAME content in P. kessleri was highest in N- cells, whereas in P- cells the same has been compensated by the better growth rate. This observation was unique as the cells were able to produce lipids without compromising growth and the mechanism is still to be exploited. Further investigation can provide us with novel insights for enhancement of lipids among specific strains without negotiating growth. However, the FAME profiling depicting the saturation and unsaturation levels was more promising in the case of N- cells. Nitrogen-deprived cells had a high level of SFAs and MUFAs, while PUFAs levels are decreased (Fig. 2b). This profile is more suitable in terms of biodiesel as the amount of polyunsaturation affects overall properties of biodiesel. Lower PUFA level is mainly helpful in lowering ignition delay, increasing stability against oxidation and lowering NOx emission [81], whereas in P- cells the PUFA content was higher than SFAs and MUFAs in P. kessleri.
A number of studies on Chlamydomonas have shown a strict metabolite regulatory network balancing the cellular processes under stress [82,83,84]. Various chromatographic techniques are used for the identification and analysis of metabolites from biological samples. Gas chromatography combined with mass spectroscopy (GC–MS) has become a popular technique to analyse metabolomic phenotypes, where GC separates the volatile compounds depending on their mass and polarity, while MS enables reproducible analyte fragmentation and identification [85, 86]. The metabolomic analysis of the cells subjected to nutrient starvation has shown a number of metabolites, which had low match scores and/or duplicated because of the varying degree of sialylation. A total number of 62 metabolites were screened and analysed on the basis of their relative peak areas from deprived conditions. Since different conditions gave different metabolic profiles, we tried to explore those that can give a probable crosstalk among the various stresses. Overall, the downregulated metabolites were more or less common among all conditions, such as malic acid, sorbose, glucose, myo-inositol (Figs. 4, 5). Also, sugar molecules obtained from polysaccharide degradation usually increased in starved cells, indicating a cut down of stored carbohydrates to provide carbon skeleton for lipid synthesis. In all stress conditions, two metabolites, i.e. 1-propionyl proline and mannose, were found to be upregulated compared to the control. Martel [87] reported increase in mannose, a C-2 epimer of glucose, which can be derived from the digestion of polysaccharides and glycoproteins under the nitrogen-deprived condition in Isochrysis galbana. However, in the present study increase in mannose seems to be higher in the N- condition due to cumulative breakdown of polysaccharides and glycoproteins, whereas in P- and S- stresses it may be due to glycoprotein metabolism alone. The predominant fatty acids depicted in the analysis are hexadecanoic acid, eicosanoic acid (arachidic acid) and trans-9 octadecanoic acid (elaidic acid) (Figs. 4, 5). Myo-inositol, a sugar alcohol, has also been reduced many folds under all the stress conditions. Inositol is an important component of structural lipids and may get disrupted during stress. In plants, the hexophosphate of inositol serves as a phosphate reserve [88] and the same mechanism may also be present in these microalgae as well because of the decreased content of myo-inositol in P- cells as compared to N- or S- cells. Henceforth, phosphate derivatives of myo-inositol might be broken down to provide phosphorous for cellular activities under P- deprivation. Also, sorbose, another monosaccharide, and azelaic acid also seem to be utilized for cell survival in P. kessleri. Malic acid, an intermediate of tricarboxylic acid (TCA) cycle, is decreased during stress and predicts the downplay of the Calvin cycle. The decarboxylation of malate to pyruvate leads to the generation of NAD(P)H, and both the pyruvate and NAD(P)H can be utilized for FA biosynthesis leading to lipid accumulation [89]. During C4 carbon fixation in plants, malate provides CO2 to Calvin cycle and such similar mechanism may co-exist in some marine diatoms and algae [90, 91]. In P. kessleri, the reduced photosynthetic machinery might also lead to the break down of malate as it will not be required to transport CO2. α-Tocopherol was also enhanced in nitrogen and sulphur deprivation. α-Tocopherol is another potent antioxidant that is enhanced under stress to protect cells from oxidative damage by quenching reactive oxygen [92] and also involved in the regulation of photosynthesis and macronutrient uptake and utilization [93].
Citric acid positively regulates acetyl-CoA-carboxylase which is the enzyme for the first committed step towards FA biosynthesis [94]. Citric acid is an intermediate of Calvin cycle; when transported from mitochondria to cytoplasm, it breaks down into oxaloacetic acid and acetyl-CoA diverting flux towards FA biosynthesis. Increased citrate may act as an acetyl-CoA carrier for fatty acid synthesis [95]. In Nannochloropsis salina, kinetic profiles and activity studies showed that this Eustigmatophyceae strain is able to convert sugar via citrate cycle towards lipids [96] and the exogenous supply of citrate showed increased fatty acid biosynthesis in Chlamydomonas sp. [85]. Upregulated citrate may provide acetyl-CoA in cytoplasm which can further be utilized to produce FA molecules. Citric acid was highly upregulated in both N- and P- cells. Similarly, 2-ketoglutaric acid has also emerged as a master regulator in essential pathways. Like citric acid, it is also a cataplerotic molecule, enhancing to provide synthesis and regulation of other molecules desired by the cells. It was found to be interacting with the regulator of acetyl-CoA carboxylase enzyme (ACCase), thus relieving ACCase for fatty acid biosynthesis [49]. 2-Ketoglutaric acid was also found to increase in nitrogen stress, providing backbone for nitrogen assimilation [97]. Amino acid degradation may also lead to accumulation of 2-ketoglutarate, which can be further converted back to citrate for FA synthesis [98].
Trehalose was found to be upregulated in nitrogen- and sulphur-deprived cells. Trehalose is a non-reducing disaccharide that performs a variety of functions, from carbon storage to carbon metabolism, protection from osmotic stress, stabilization of membranes and proteins, removal of aberrant storage material, protection from oxygen radicals, induction of autophagy [36, 85, 99]. The induction of trehalose might be responsible for growth retardation in N- and S- conditions. Previous reports also observed an increase in trehalose under nitrogen starvation in Chlamydomonas after 6 days of depletion [85]. Trehalose may also control various metabolic processes and growth [100]. It may act as a growth regulator by affecting hexokinase and thus glycolysis, and leads to severe growth defects such as dwarfism in plants [101, 102]. Although sulphur deprivation does not show growth inhibition initially, later growth shunts probably because of increase in trehalose accumulation. This seems interesting as the presence of citric acid and 2-ketoglutaric acid in nitrogen and phosphorous starvation might induce lipid accumulation, while trehalose presence in nitrogen and sulphur depletion might result in growth retardation (Fig. 6). As a result, a crosstalk between these metabolites such as citric acid, 2-ketoglutaric acid and trehalose might be important for the production of biomass as well as lipid accumulation in marine microalgae P. kessleri.
Parachlorella kessleri subjected to nutrient deprivation shows growth retardation except under P- limitation. Nitrogen and phosphorous limitation played a major role in lipid accumulation. The qualitative metabolomics showed a variable shift in the metabolite flux in response to different stress conditions. A crosstalk between metabolites, namely citric acid, 2-ketoglutaric acid and trehalose, can be hypothesized to have greater impact on the production of biomass and lipid accumulation. To our knowledge, this report in the marine microalgae P. kessleri is a new paradigm to elucidate the molecular changes in the basis of metabolite redistribution subjected to nutrient-limiting conditions leading to insights on the production of biomass, biofuels and bioproducts (B3) in non-model systems. In conclusion, marine strain Parachlorella kessleri with high biomass and higher lipid productivity was analysed where shuffling of certain metabolites when subjected to stress will dictate the profile changes that may prove to be a benchmark for over-expression of lipids without compromising growth. Further characterization of this strain may be a critical step towards making algae-derived biofuels economically competitive for industrial production.
Microalgae and culture conditions
Marine microalgae P. kessleri (I) (procured from Indian Institute of Technology-Madras, Chennai) was grown in minimal media F/2 [103] under constant illumination (~ 100 µmol m−2 s−1 photosynthetically active radiation [PAR]) on an orbital shaker at 150 RPM at 25 °C. The composition of media components for F/2 media (g L−1) is as follows—NaNO3—0.075; NaH2PO4·2H2O—0.005; Na2SiO3·9H2O—0.03 in artificial sea water (ASW) prepared using NaCl—24; MgCl2·6H2O—11; Na2SO4—4; CaCl2·6H2O—2; KBr—0.1; H3BO3—0.03; Na2SiO3·9H2O—0.005; SrCl·6H20—0.04; NaF—0.003; NH4NO3—0.002; Fe3PO4·4H2O—0.001; trace metals solution (in g L−1)—1 mL L−1 [ZnSO4·7H2O—0.023; MnSO4.H2O—0.152; Na2MoO4·2H2O—0.007; CoSO4·7H2O—0.014; CuCl2·2H2O—0.007; Fe(NH4)2(SO4)2·6H2O—4.6; Na2EDTA·2H2O—4.4]; and vitamin B12*—0.135 mg L−1; biotin vitamin solution*—0.025 mg L−1; thiamine vitamin solution*—0.335 mg L−1 (*added after autoclaving the media). Growth and biomass accumulation were monitored by cell count using haemocytometer [104] and dry weight (dw) analysis as described previously [105]. Growth rates were obtained using the following equation [106]
$$K = \frac{{\ln \frac{{N_{2} }}{{N_{1} }}}}{{t_{2} - t_{1} }}$$
where N1 and N2 represent cell counts at initial time (t1) and final time (t2), respectively. Doubling time was calculated depending on the specific growth rate [107].
$${\text{Doubling}}\;{\text{time}} = \frac{\ln 2}{K}.$$
Cells were initially grown photoautotrophically to the middle of the logarithmic phase in F/2 medium. These cells were collected by centrifugation and resuspended again at a density of 2 × 106 cells mL−1 in regular F/2 or in the same media completely deficit in nitrogen (N-), phosphorous (P-) or sulphur (S-). Nitrogen concentration in the media was estimated during culture growth spectrophotometrically as described by Yodsuwan et al. [108]. Samples for all the analyses were taken immediately after resuspension (control, 0 day) and at the time intervals of 2, 4, 6, 8 and 10 days for further experiments, and the sixth-day samples were analysed for metabolomic profiling.
Biochemical analysis
The samples were analysed for changes in the biochemical constituents [pigments (total chlorophyll), proteins, carbohydrates] subjected to the nutrient stress. For estimation of pigments, 1 mL of culture was pellet down and resuspended in 1 mL of absolute methanol. The suspension was vortexed briefly and incubated at 4 °C for an hour to extract the pigments completely. The debris was pellet down, and the suspension was used to measure absorbance at 665, 652 and 470 nm to calculate total chlorophyll content [109]. Protein estimation was done using modified biuret method. Total soluble proteins were extracted using 1 N NaOH in 25% methanol as extraction buffer. 1–2 mL of culture was pellet down and resuspended in 1 mL of extraction buffer and incubated at 80 °C for 15 min. The sample was cooled down to room temperature and centrifuged at high speed to remove debris. One hundred microlitres of extract was mixed with 50 µL of CuSO4 solution (0.21% CuSO4 in 30% NaOH), incubated at RT for 10 min and its absorbance was measured at 310 nm [110]. Carbohydrate estimation was done using modified phenol–sulphuric acid method. Around 100 µL of cells was taken, and absolute H2SO4 was added and kept for 1 h at room temperature. Afterwards, 5% phenol was added along with 1 mL of H2SO4 and kept at room temperature for another 20 min after vortexing. Absorbance was measured at 490 nm [111].
Lipid quantification and profiling
Total lipids were extracted using modified Bligh and Dyer procedure [112], dried under N2, and visualized as TAGs by thin-layer chromatography (TLC) on a silica gel plate. Briefly, ~ 1 × 108 cells were collected in a glass tube with Teflon-lined screw cap. Lipid extraction was done using methanol/chloroform (2:1, v/v) containing 0.01% butylated hydroxytoluene. Two millilitres of methanol/chloroform mix was added to the cell pellet and incubated at 25 °C for 2 h with shaking. Thereafter, chloroform (1 mL) and water (1.8 mL) were added to the tubes, mixed vigorously, and centrifuged at 3000×g to separate the mix into two phases. The lower organic phase containing the extracted lipids was transferred to a new glass tube with the help of a Pasteur pipette. Extracted organic phase was dried at 50 °C under stream of nitrogen (N2) to evaporate the solvent completely and resuspended in CHCl3/MeOH (100 µL, 6:1 v/v). Fifty microlitres of this extract was applied to a silica 60 thin-layer chromatography plate (Sigma-Aldrich) and run with a solvent system of heptane/ethanol/acetone (70:30:1, v/v/v) to resolve the neutral lipids. The TAG band was identified by staining co-migrated TAG standard with iodine vapours [84]. For GC–MS analysis, ~ 1 × 108 cells were acid-hydrolysed and methyl-esterified using 2% sulphuric acid in methanol (300 µL) for 2 h at 80 °C. Prior to the reaction, 50 µg of heptadecanoic acid (Alfa Aesar) was added as internal standard. The fatty acid methyl esters were extracted using 300 µL of 0.9% (w/v) NaCl solution and 300 µL of hexane. The mixture was vortexed briefly and centrifuged at 3000×g for 3 min to separate the phases. One microlitre of hexane layer was injected into a 7890A gas chromatography (GC) mass spectrometry (MS) system equipped with a 7000 GC/MS triple quadrupole system (Agilent) [107, 113]. The running conditions for GC–MS were described by Agilent's RTL DBWax method [114].
Qualitative metabolomics
For the extraction of cellular metabolites, ~ 109 cells were collected by centrifugation at 8000×g for 10 min and immediately quenched in liquid nitrogen. Metabolites were extracted using methanol, chloroform and water by repeated freezing and thawing. Cells were resuspended in 1 mL of ice-cold methanol/chloroform/water (10:3:1) and vortexed briefly. The cells were frozen again in liquid nitrogen for 1–2 min and thawed on ice for 4–5 min. Freezing and thawing cycles were repeated five times with intermittent vortexing. Samples were then centrifuged at 14,000×g for 3 min at 4 °C to get rid of cell debris. The supernatant was filtered using a 0.2-µm filter. One hundred microlitres of supernatant was taken and vacuum-dried at 4 °C. The dried leftover was dissolved in 10 µL of freshly prepared methoxyamine hydrochloride solution (40 mg mL−1 in pyridine) and incubated at 30 °C for 90 min with shaking. To the above solution, 90 µL of N-methyl-N-(trimethylsilyl)trifluoroacetamide was added and incubated at 37 °C for 30 min. The samples were centrifuged at 14,000×g for 3 min, and the supernatant was taken for the GC/MS analysis. The samples were run on GC–MS/MS, and the data were analysed using MetaboAnalyst 4.0 (http://www.metaboanalyst.ca) [115].
All the experiments were done in biological triplicates, and the mean of three values was used to calculate standard deviation (SD). The final data were represented as mean ± SD (denoting SD as the experimental error). Graphs were plotted using MS Excel software (Microsoft Corporation, USA).
All data generated or analysed during this study have been provided in this manuscript.
N-:
nitrogen deprivation
P-:
phosphorous deprivation
S-:
sulphur deprivation
siROS:
specific intracellular reactive oxygen species
rotation per minute
TLC:
FA:
FAMEs:
fatty acid methyl esters
SFAs:
saturated fatty acid
PUFAs:
polyunsaturated fatty acid
MUFAs:
monounsaturated fatty acid
TCA:
tricarboxylic acid
NAD(P)H:
C4:
Hatch–Slack pathway of CO2 fixation
ACCase:
acetyl coenzyme A carboxylase
ATP:
ADP:
nitrogen oxides
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We would like to thank Mr. Shrikumar Suryanaran (Indian Institute of Technology, Madras) for providing us the strain.
The research was funded by the grants from the Department of Biotechnology, Government of India, to PPJ (Sanction No. BT/PB/Center/03/2011) and to AAN (BioCARe Scheme No. BT/PR18491/BIC/101/759/2016). Senior Research Fellowship to KMS from University Grants Commission (UGC), Government of India, is duly acknowledged.
Omics of Algae Group, Integrative Biology, International Centre for Genetic Engineering and Biotechnology, Aruna Asaf Ali Marg, New Delhi, 110067, India
Kashif Mohd Shaikh
, Asha Arumugam Nesamma
& Pannaga Pavan Jutur
Department of Biotechnology, School of Chemical and Life Sciences, Jamia Hamdard University, New Delhi, 110062, India
& Malik Zainul Abdin
Search for Kashif Mohd Shaikh in:
Search for Asha Arumugam Nesamma in:
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KMS, AAN and PPJ designed the experiment. KMS and AAN executed the experiments. MZA and PPJ supervised the project. KMS wrote the manuscript with all the input from the authors. All authors read and approved the final manuscript.
Correspondence to Pannaga Pavan Jutur.
All authors have given their consent for the publication of this work.
Shaikh, K.M., Nesamma, A.A., Abdin, M.Z. et al. Molecular profiling of an oleaginous trebouxiophycean alga Parachlorella kessleri subjected to nutrient deprivation for enhanced biofuel production. Biotechnol Biofuels 12, 182 (2019). https://doi.org/10.1186/s13068-019-1521-9
Microalgae
Parachlorella kessleri
Nutrient deprivation
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Weak generator of Feller semigroup
Let $(T_t)_{t \geq 0}$ a Feller semigroup and define a linear operator $(A,\mathcal{D}(A))$ by $$\mathcal{D}(A) := \left\{u \in C_{\infty}(\mathbb{R}^d); \exists f \in C_{\infty} \forall x \in \mathbb{R}^d: f(x) = \lim_{t \to 0} \frac{T_t u(x)-u(x)}{t} \right\} \\ Au(x) := \lim_{t \to 0} \frac{T_t u(x)-u(x)}{t} \qquad (u \in \mathcal{D}(A))$$
($A$ is called weak generator of the semigroup).
Now I want to show that this generator is the generator in the sense of the weak topology on $C_{\infty}(\mathbb{R}^d)$, i.e. that the convergence is bounded pointwise convergence.
Let $u \in \mathcal{D}(A)$. Since (by definition) the sequence is pointwise convergent, the only remaining thing is to show the boundedness, i.e.
$$\sup_{t>0} \left\| \frac{T_t u-u}{t} \right\|_{\infty} < \infty$$
Well, since the sequence is pointwise convergent we have $$\sup_{t > 0} \left|\frac{T_t u(x)-u(x)}{t} \right| < \infty$$ for fixed $x \in \mathbb{R}^d$. A hint says that one should apply the Banach-Steinhaus theorem, but I don't see how to apply this theorem here, because there are not even linear operators (note that $u$ is fixed). Some hint...?
Remark A Feller semigroup is a positivity preserving, conservative, strongly continuous semigroup satisfying the sub-markov property.
functional-analysis semigroup-of-operators
sazsaz
$\begingroup$ Not true, the $L_t : u \mapsto \dfrac{T_t\,u - u}{t}$ are linear operators. $\endgroup$ – Siméon Jan 16 '13 at 18:21
$\begingroup$ @Ju'x Yes, but $u$ is fixed. (I have to prove $\frac{T_t u- u}{t} \to Au$ weakly as $t \to 0$ for (fixed!) $u \in \mathcal{D}(A)$.) $\endgroup$ – saz Jan 16 '13 at 18:29
$\begingroup$ What is the a strong generator? $\endgroup$ – Ilya Jan 16 '13 at 21:10
$\begingroup$ @Ilya The strong generator is the generator in the sense of the topology generated by the supremum norm, i.e. $\frac{T_t u-u}{t} \to Au$ uniformly. In this case, the weak and strong generator coincide - and that's exactly what I want to prove using the result from above. $\endgroup$ – saz Jan 17 '13 at 7:56
$\begingroup$ I see now, thanks $\endgroup$ – Ilya Jan 17 '13 at 8:47
Here is a sketch how to prove the result using the so-called (strong) generator of the semigroup $(T_t)_{t \geq 0}$: Let $$\begin{align} \mathcal{D}(A_s) &:= \left\{u \in C_{\infty}(\mathbb{R}^d); \exists f \in C_{\infty}(\mathbb{R}^d): \lim_{t \to 0} \left\| \frac{T_t u-u}{t} -f \right\|_{\infty} = 0 \right\} \\ A_s u &:= \lim_{t \to 0} \frac{T_t u-u}{t} \qquad (u \in \mathcal{D}(A_s)) \end{align}$$ (The limit is taken w.r.t to the sup-norm.) Then $(A_s,\mathcal{D}(A_s))$ is called the (strong) generator of $(T_t)_{t \geq 0}$. The idea is to show that the weak and the strong generator coincide. The following theorem is quite helpful:
Theorem Let $(A_s,\mathcal{D}(A_s))$ the generator of a Feller semigroup $(T_t)_{t \geq 0}$. Let $(A,\mathcal{D}(A))$ an extension of $(A_s,\mathcal{D}(A_s))$ such that $$Au = u \Rightarrow u=0 \quad (u \in \mathcal{D}(A)) \tag{1}$$ Then $(A,\mathcal{D}(A))= (A_s,\mathcal{D}(A_s))$.
If we would be able to show that the weak generator $(A,\mathcal{D}(A))$ satisfies $(1)$ we would be finished: For $u \in \mathcal{D}(A) = \mathcal{D}(A_s)$ we have $$\frac{T_t u-u}{t} \to Au \quad \text{uniformly}$$ hence in particular $$\sup_{t>0} \left\| \frac{T_t u-u}{t} \right\|_{\infty} < \infty$$
So - that's it. Here is the remaining part of the proof:
Lemma Let $(A,\mathcal{D}(A))$ the weak generator of a Feller semigroup. Let $u \in \mathcal{D}(A)$, $x_0 \in \mathbb{R}^d$ such that $u(x_0)=\sup_{x \in \mathbb{R}^d} u(x) \geq 0$. Then $$Au(x_0) \leq 0$$ (i.e. $A$ satisfies the maximum principle). In particular, $A$ is dissipative, i.e. $$\forall \lambda>0: \|\lambda \cdot u-Au\|_{\infty} \geq \lambda \cdot \|u\|_{\infty} \tag{2}$$
This shows that the weak generator $(A,\mathcal{D}(A))$ satisfies $(1)$ (put $\lambda=1$ in $(2)$).
Literature René L. Schilling/Lothar Partzsch: Brownian Motion - An Introduction to Stochastic Processes (Chapter 7).
Remark The given hint (i.e. applying Banach Steinhaus theorem) was not correct - or at least much more difficult than the creator of this exercise was expecting. One would have to apply Banach Steinhaus to the dual space of $C_{\infty}$ (using the fact that for the dirac measures $\delta_x$ the boundedness is given by the pointwise convergence ... and some more considerations about the (vague) density of dirac measures.)
$\begingroup$ How did you prove that $(\mathcal D(A),A)$ is dissipative? Schilling is picking an $x_0\in\mathbb R$ with $|u(x_0)|=\sup_{x\in\mathbb R}|u(x)|$ in his argumentation; but that would mean that $|u|$ attains its supremum. That clearly doesn't need to be the case for a general $u\in C_0(\mathbb R)$. $\endgroup$ – 0xbadf00d Feb 4 at 20:00
$\begingroup$ @0xbadf00d Why do you think so? Any non-negative function $v \in C_0$ attains its supremum, right? ... and $v:= |u|$ satisfies these assumptions. $\endgroup$ – saz Feb 4 at 20:07
$\begingroup$ Oops, I never thought about that, but sure, it follows from the compactness of $\left\{|u|\ge\varepsilon\right\},\varepsilon>0$. $\endgroup$ – 0xbadf00d Feb 4 at 21:53
$\begingroup$ So, given such an $u$, we use that $\tilde u:=\operatorname{sgn}(u(x_0))u\in\mathcal D(A)$ and $\tilde u(x_0)=\sup_{x\in\mathbb R}\tilde u(x)$ and then that $(\mathcal D(A),A)$ satisfies the nonnegative maximum principle, right? $\endgroup$ – 0xbadf00d Feb 4 at 23:52
$\begingroup$ @0xbadf00d Well, yes, that's exactly the reasoning in the book... $\endgroup$ – saz Feb 5 at 10:25
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Procrustes analysis
In statistics, Procrustes analysis is a form of statistical shape analysis used to analyse the distribution of a set of shapes. The name Procrustes (Greek: Προκρούστης) refers to a bandit from Greek mythology who made his victims fit his bed either by stretching their limbs or cutting them off.
In mathematics:
• an orthogonal Procrustes problem is a method which can be used to find out the optimal rotation and/or reflection (i.e., the optimal orthogonal linear transformation) for the Procrustes Superimposition (PS) of an object with respect to another.
• a constrained orthogonal Procrustes problem, subject to det(R) = 1 (where R is an orthogonal matrix), is a method which can be used to determine the optimal rotation for the PS of an object with respect to another (reflection is not allowed). In some contexts, this method is called the Kabsch algorithm.
When a shape is compared to another, or a set of shapes is compared to an arbitrarily selected reference shape, Procrustes analysis is sometimes further qualified as classical or ordinary, as opposed to generalized Procrustes analysis (GPA), which compares three or more shapes to an optimally determined "mean shape".
Introduction
To compare the shapes of two or more objects, the objects must be first optimally "superimposed". Procrustes superimposition (PS) is performed by optimally translating, rotating and uniformly scaling the objects. In other words, both the placement in space and the size of the objects are freely adjusted. The aim is to obtain a similar placement and size, by minimizing a measure of shape difference called the Procrustes distance between the objects. This is sometimes called full, as opposed to partial PS, in which scaling is not performed (i.e. the size of the objects is preserved). Notice that, after full PS, the objects will exactly coincide if their shape is identical. For instance, with full PS two spheres with different radii will always coincide, because they have exactly the same shape. Conversely, with partial PS they will never coincide. This implies that, by the strict definition of the term shape in geometry, shape analysis should be performed using full PS. A statistical analysis based on partial PS is not a pure shape analysis as it is not only sensitive to shape differences, but also to size differences. Both full and partial PS will never manage to perfectly match two objects with different shape, such as a cube and a sphere, or a right hand and a left hand.
In some cases, both full and partial PS may also include reflection. Reflection allows, for instance, a successful (possibly perfect) superimposition of a right hand to a left hand. Thus, partial PS with reflection enabled preserves size but allows translation, rotation and reflection, while full PS with reflection enabled allows translation, rotation, scaling and reflection.
Optimal translation and scaling are determined with much simpler operations (see below).
Ordinary Procrustes analysis
Here we just consider objects made up from a finite number k of points in n dimensions. Often, these points are selected on the continuous surface of complex objects, such as a human bone, and in this case they are called landmark points.
The shape of an object can be considered as a member of an equivalence class formed by removing the translational, rotational and uniform scaling components.
Translation
For example, translational components can be removed from an object by translating the object so that the mean of all the object's points (i.e. its centroid) lies at the origin.
Mathematically: take $k$ points in two dimensions, say
$((x_{1},y_{1}),(x_{2},y_{2}),\dots ,(x_{k},y_{k}))\,$.
The mean of these points is $({\bar {x}},{\bar {y}})$ where
${\bar {x}}={\frac {x_{1}+x_{2}+\cdots +x_{k}}{k}},\quad {\bar {y}}={\frac {y_{1}+y_{2}+\cdots +y_{k}}{k}}.$
Now translate these points so that their mean is translated to the origin $(x,y)\to (x-{\bar {x}},y-{\bar {y}})$, giving the point $(x_{1}-{\bar {x}},y_{1}-{\bar {y}}),\dots $.
Uniform scaling
Likewise, the scale component can be removed by scaling the object so that the root mean square distance (RMSD) from the points to the translated origin is 1. This RMSD is a statistical measure of the object's scale or size:
$s={\sqrt {{(x_{1}-{\bar {x}})^{2}+(y_{1}-{\bar {y}})^{2}+\cdots } \over k}}$
The scale becomes 1 when the point coordinates are divided by the object's initial scale:
$((x_{1}-{\bar {x}})/s,(y_{1}-{\bar {y}})/s)$.
Notice that other methods for defining and removing the scale are sometimes used in the literature.
Rotation
Removing the rotational component is more complex, as a standard reference orientation is not always available. Consider two objects composed of the same number of points with scale and translation removed. Let the points of these be $((x_{1},y_{1}),\ldots )$, $((w_{1},z_{1}),\ldots )$. One of these objects can be used to provide a reference orientation. Fix the reference object and rotate the other around the origin, until you find an optimum angle of rotation $\theta \,\!$ such that the sum of the squared distances (SSD) between the corresponding points is minimised (an example of least squares technique).
A rotation by angle $\theta \,\!$ gives
$(u_{1},v_{1})=(\cos \theta \,w_{1}-\sin \theta \,z_{1},\,\sin \theta \,w_{1}+\cos \theta \,z_{1})\,\!$.
where (u,v) are the coordinates of a rotated point. Taking the derivative of $(u_{1}-x_{1})^{2}+(v_{1}-y_{1})^{2}+\cdots $ with respect to $\theta $ and solving for $\theta $ when the derivative is zero gives
$\theta =\tan ^{-1}\left({\frac {\sum _{i=1}^{k}(w_{i}y_{i}-z_{i}x_{i})}{\sum _{i=1}^{k}(w_{i}x_{i}+z_{i}y_{i})}}\right).$
When the object is three-dimensional, the optimum rotation is represented by a 3-by-3 rotation matrix R, rather than a simple angle, and in this case singular value decomposition can be used to find the optimum value for R (see the solution for the constrained orthogonal Procrustes problem, subject to det(R) = 1).
Shape comparison
The difference between the shape of two objects can be evaluated only after "superimposing" the two objects by translating, scaling and optimally rotating them as explained above. The square root of the above mentioned SSD between corresponding points can be used as a statistical measure of this difference in shape:
$d={\sqrt {(u_{1}-x_{1})^{2}+(v_{1}-y_{1})^{2}+\cdots }}.$
This measure is often called Procrustes distance. Notice that other more complex definitions of Procrustes distance, and other measures of "shape difference" are sometimes used in the literature.
Superimposing a set of shapes
We showed how to superimpose two shapes. The same method can be applied to superimpose a set of three or more shapes, as far as the above mentioned reference orientation is used for all of them. However, Generalized Procrustes analysis provides a better method to achieve this goal.
Generalized Procrustes analysis (GPA)
GPA applies the Procrustes analysis method to optimally superimpose a set of objects, instead of superimposing them to an arbitrarily selected shape.
Generalized and ordinary Procrustes analysis differ only in their determination of a reference orientation for the objects, which in the former technique is optimally determined, and in the latter one is arbitrarily selected. Scaling and translation are performed the same way by both techniques. When only two shapes are compared, GPA is equivalent to ordinary Procrustes analysis.
The algorithm outline is the following:
1. arbitrarily choose a reference shape (typically by selecting it among the available instances)
2. superimpose all instances to current reference shape
3. compute the mean shape of the current set of superimposed shapes
4. if the Procrustes distance between mean and reference shape is above a threshold, set reference to mean shape and continue to step 2.
Variations
There are many ways of representing the shape of an object. The shape of an object can be considered as a member of an equivalence class formed by taking the set of all sets of k points in n dimensions, that is Rkn and factoring out the set of all translations, rotations and scalings. A particular representation of shape is found by choosing a particular representation of the equivalence class. This will give a manifold of dimension kn-4. Procrustes is one method of doing this with particular statistical justification.
Bookstein obtains a representation of shape by fixing the position of two points called the bases line. One point will be fixed at the origin and the other at (1,0) the remaining points form the Bookstein coordinates.
It is also common to consider shape and scale that is with translational and rotational components removed.
Examples
Shape analysis is used in biological data to identify the variations of anatomical features characterised by landmark data, for example in considering the shape of jaw bones.[1]
One study by David George Kendall examined the triangles formed by standing stones to deduce if these were often arranged in straight lines. The shape of a triangle can be represented as a point on the sphere, and the distribution of all shapes can be thought of a distribution over the sphere. The sample distribution from the standing stones was compared with the theoretical distribution to show that the occurrence of straight lines was no more than average.[2]
See also
• Active shape model
• Alignments of random points
• Biometrics
• Generalized Procrustes analysis
• Image registration
• Kent distribution
• Morphometrics
• Orthogonal Procrustes problem
• Procrustes
References
1. "Exploring Space Shape" Archived 2006-09-01 at the Wayback Machine by Nancy Marie Brown, Research/Penn State, Vol. 15, no. 1, March 1994
2. "A Survey of the Statistical Theory of Shape", by David G. Kendall, Statistical Science, Vol. 4, No. 2 (May, 1989), pp. 87–99
• F.L. Bookstein, Morphometric tools for landmark data, Cambridge University Press, (1991).
• J.C. Gower, G.B. Dijksterhuis, Procrustes Problems, Oxford University Press (2004).
• I.L.Dryden, K.V. Mardia, Statistical Shape Analysis, Wiley, Chichester, (1998).
External links
Wikimedia Commons has media related to Procrustes analysis.
• Extensions to continuum of points and distributions Procrustes Methods, Shape Recognition, Similarity and Docking, by Michel Petitjean.
| Wikipedia |
\begin{document}
\title{Homotopical decompositions of simplicial and Vietoris Rips complexes
}
\author{Wojciech Chach\'olski} \thanks{Mathematics, KTH, S-10044 Stockholm, Sweden\email{[email protected]} \email{[email protected]}\email{[email protected]} \email{[email protected]} } \author{ Alvin Jin } \author{Martina Scolamiero} \author{Francesca Tombari}
\maketitle
\begin{abstract} Motivated by applications in Topological Data Analysis, we consider decompositions of a simplicial complex induced by a cover of its vertices. We study how the homotopy type of such decompositions approximates the homotopy of the simplicial complex itself. The difference between the simplicial complex and such an approximation is quantitatively measured by means of the so called obstruction complexes. Our general machinery is then specialized to clique complexes, Vietoris-Rips complexes and Vietoris-Rips complexes of metric gluings. For the latter we give metric conditions which allow to recover the first and zero-th homology of the gluing from the respective homologies of the components.
\end{abstract}
\section{Introduction} \label{intro} Homology is an example of an invariant that is both calculable and geometrically informative. These two features are key reasons why invariants derived from homology are fundamental in Algebraic Topology in general and in Topological Data Analysis (TDA)~\cite{MR2476414} in particular. Calculability is a consequence of the fact that homology converts homotopy push-outs into Mayer-Vietoris exact sequences. Decomposing a space into a homotopy push-out enables us to extract the homologies of the decomposed space (global information) from the homologies of the spaces in the push-out (local information).
Ability of extracting global information from local is important. What is meant by local information however depends on the input and the description of considered spaces. For example what is often understood as local information in TDA differs from the local information described above (push-out decomposition). In TDA the input is typically a finite metric space. This information is then converted into spacial information and in this article we focus on the so called Vietoris-Rips construction~\cite{MR1368659} for that purpose. Homologies extracted from this space give rise to invariants of the metric space used in TDA such as persistent homologies~\cite{MR1861130,Delf-Edel,MR2405684,MR1434543,Frosini-Landi}, bar-codes~\cite{Scopigno04persistencebarcodes}, stable ranks~\cite{MR4057607,MR3735858}, or persistent landscapes~\cite{MR3317230}. This conversion process, from metric into spacial information, does not in general transform the gluing of metric spaces~\cite{MR1400226} into homotopy push-outs and homotopy colimits of simplicial complexes. The aim of this paper is to understand how close such data driven decompositions are to decompositions into homotopy push-outs. Our work was inspired by~\cite{MR3824247} and~\cite{NewHenry}, and grew out of realisation that analogous statements hold true for arbitrary simplicial complexes and not just Vietoris-Rips complexes. To get these general statements we use categorical techniques. This enables us to prove stronger results using arguments that for us are more transparent.
The most general input for our investigation is a simplicial complex $K$ and a cover $X\cup Y=K_0$ of its set of vertices. In this article we study the map $K_X\cup K_Y\subset K$ where $K_X$ and $ K_Y$ are subcomplexes of $K$ consisting of all these simplices of $K$ which are subsets of $X$ and $Y$ respectively. The goal is to estimate the homotopy fibers of this inclusion. We do that in terms of
obstruction complexes $\text{St}(\sigma,X\cap Y):=\{\mu\subset X\cap Y\ |\ 0\leq|\mu|\text{ and } \mu\cup \sigma\in K\}$ indexed by simplices $\sigma$ in $K$ (see Definition~\ref{asdfsaahdfg}). Our main result, Theorem~\ref{adfsdfhgd}, states that the homotopy fibers of $K_X\cup K_Y\subset K$ are in the same cellular class (see Paragraph~\ref{asdfdsfhgdf}, and~\cite{MR1397724,MR1320991,MR1392221}) as the obstruction complexes $\text{St}(\sigma,X\cap Y)$ for all $\sigma$ in $K$ such that $\sigma\cap X\not=\emptyset$, $\sigma\cap Y\not=\emptyset$, and $\sigma\cap X\cap Y=\emptyset$. For instance (see Corollary~\ref{sdfdfghsd}.1) if, for all such $\sigma$, the obstruction complex $\text{St}(\sigma,X\cap Y)$ is contractible, then $K_X\cup K_Y\subset K$ is a weak equivalence and consequently $K$ decomposes as a homotopy pushout $\text{hocolim}(K_X\hookleftarrow K_{X\cap Y}\hookrightarrow K_Y)$ leading to a Mayer-Vietoris exact sequence. Another instance of our result (see Corollary~\ref{sdfdfghsd}.2) states that if these obstruction complexes have trivial homology in degrees not exceeding $n$, then so do the homotopy fibers of $K_X\cup K_Y\subset K$ and consequently this map induces an isomorphism on homology in degrees not exceeding $n$, leading to a partial Mayer-Vietoris exact sequence. Yet another consequence (see Corollary~\ref{sdfdfghsd}.3) is that if these obstruction complexes have $p$-torsion homology for a prime $p$, then, for any field $F$ of characteristic different than $p$, the inclusion $K_X\cup K_Y\subset K$ induces an isomorphism on homology with coefficients in $F$ leading again to a Mayer-Vietoris exact sequence.
In section~\ref{dsfgdfhsfghj} we specialise our theorem about the cellularity of the homotopy fibers of the inclusion $K_X\cup K_Y\subset K$ to the case when $K$ is a clique complex and give some conditions that imply the assumptions of the theorem in this case. Obtained results in principle generalize all the statements proven in~\cite{MR3824247,NewHenry} for Vietoris-Rips complexes. The point we would like to make is that these statements are not about Vietoris-Rips complexes but rather about these complexes being clique. In particular triangular inequality of the input metric space is not needed for our statements to hold. We then prove Theorem~\ref{easdfgsdghj} for which it is essential that considered complex is the Vietoris-Rips complex of the metric gluing of pseudo-metric spaces for which the triangular inequality is satisfied. This theorem gives 2 connectedness of the relevant homotopy fibers and hence can be used to calculate
$H_1$ and $H_0$ of the gluing in terms of $H_1$'s and $H_0$'s of the components and the intersection.
\section{Small categories and simplicial sets}\label{asfdhg} In this section we recall some elements of a convenient language for describing and discussing homotopical properties of small categories. The key role here is played by the nerve construction~\cite{MR516607,nerveatnlab} that transforms small categories into simplicial sets. We refer the reader to~\cite{MR0268888,MR1711612} for an overview of how to do homotopy theory on simplicial sets. We consider the standard model structure on the category of simplicial sets where weak equivalences are given by the maps inducing bijections on all the homotopy groups with respect to any choice of a base point.
Here is a list of definitions and characterizations of various homotopical notions for small categories and some statements regarding these notions.
\begin{point} Let $\mathcal C$ be a property of simplicial sets, such as being contractible, $n$-connected, having $p$-torsion integral reduced homology, or having trivial reduced homology in some degrees. By definition a small category $I$ satisfies $\mathcal C$ if and only if its nerve $N(I)$ satisfies $\mathcal C$. \end{point} \begin{point} Let $\mathcal C$ be a property of maps of simplicial sets, such as being a weak equivalence, a homology isomorphism, or having $n$-connected homotopy fibers. By definition, a functor $f\colon I\to J$ between small categories satisfies $\mathcal C$ if and only if the map of simplicial sets $N(f)\colon N(I)\to N(J)$ satisfies $\mathcal C$. \end{point} \begin{point} Functors $f,g\colon I\to J$ are homotopic if the maps $N(f),N(g)\colon N(I)\to N(J)$ are homotopic. For example, if there is a natural transformation $\phi\colon f\Rightarrow g$ between $f$ and $g$, then $f$ and $g$ are homotopic.
Assume $I$ has a terminal object $t$. Then there is a unique natural transformation from the identity functor $\text{id}\colon I\to I$ to the constant functor $t\colon I\to I$ with value $t$. The identity functor is therefore homotopic to the constant functor, and consequently $I$ is contractible. By a similar argument, a category with an initial object is also contractible. \end{point}
\begin{point} A commutative square of small categories is called a homotopy push-out (pull-back) if after applying the nerve construction the obtained commutative square of simplicial sets is a homotopy push-out (pull-back). \end{point} \begin{point}\label{asdfdsfhgdf} Recall that a collection
$\mathcal C$ of simplicial sets is closed if it contains a nonempty simplicial set and it is closed under weak equivalencies and homotopy colimits indexed by arbitrary small contractible categories~\cite[Corollary 7.7]{MR1397724}. Any closed collection contains all contractible simplicial sets~\cite[Proposition 4.5]{MR1397724}. If a closed collection contains an empty simplicial set, then it contains all simplicial sets.
The following are some examples of collections of simplicial sets that are closed: contractible simplicial sets, $n$-connected simplicial sets, connected simplicial sets having $p$-torsion reduced integral homology, simplicial sets having trivial reduced homology with some fixed coefficients up to a given degree, and more generally simplicial sets which are acyclic with respect to some (possibly not ordinary) homology theory.
Let $\mathcal C$ be a closed collection of simplicial sets and $f\colon I\to J$ be a functor between small categories. We say that homotopy fibers of $f$ satisfy
$\mathcal C$ if the homotopy fibers of $N(f)\colon N(I)\to N(J)$, over any component in $N(J)$,
belong to $\mathcal C$.
\end{point}
\begin{point}
Let $f\colon I\to J$ be a functor between small categories. For an object $j $ in $J$, the symbol $j\!\uparrow\! f$ denotes the category whose objects are pairs $(i, \alpha\colon j\to f(i))$ consisting of an object $i$ in $I$ and a morphism $\alpha\colon j\to f(i)$ in $J$. The set of morphisms in $j\uparrow f$ between $(i, \alpha\colon j\to f(i))$ and $(i', \alpha'\colon j\to f(i'))$ is by definition the set of morphisms $\beta\colon i\to i'$ in $I$ for which the following triangle commutes: \[\begin{tikzcd} & j\ar{dl}[swap]{\alpha}\ar{dr}{\alpha'}\\ f(i)\ar{rr}{f(\beta)} & & f(i')' \end{tikzcd}\] The composition in $j\!\uparrow\! f$ is given by the composition in $I$.
For an object $j $ in $J$, the symbol $f\!\downarrow\! j$ denotes the category whose objects are pairs $(i, \alpha\colon f(i)\to j)$ consisting of an object $i$ in $I$ and a morphism $\alpha\colon f(i)\to j $ in $J$. The set of morphisms in $f\downarrow j$ between $(i, \alpha\colon f(i)\to j)$ and $(i', \alpha'\colon f(i')\to j)$ is by definition the set of morphisms $\beta\colon i\to i'$ in $I$ for which the following triangle commutes: \[\begin{tikzcd} f(i)\ar{rr}{f(\beta)}\ar{dr}{\alpha} & & f(i')\ar{dl}[swap]{\alpha'} \\ & j \end{tikzcd}\] The composition in $f\!\downarrow\! j$ is given by the composition in $I$. \end{point}
\begin{thm}[{\cite[Theorem 9.1]{MR1397724}}]\label{asgdfghfdn} Let $\mathcal C$ be a closed collection of simplicial sets and $f\colon I\to J$ be a functor between small categories. \begin{enumerate} \item If, for every $j$, $f\!\downarrow\! j$ satisfies $C$, then so do the homotopy fibers of $f$. \item If, for every $j$, $j\!\uparrow\! f$ satisfies $C$, then so do the homotopy fibers of $f$. \end{enumerate} \end{thm}
Depending on the choice of a closed collection, Theorem~\ref{asgdfghfdn} leads to: \begin{cor}\label{asgfgjh} Let $f\colon I\to J$ be a functor between small categories. \begin{enumerate} \item If, for every $j$, $f\!\downarrow\! j$ (respectively $j\!\uparrow\! f$) is contractible, then $f$ is a weak equivalence. \item If, for every $j$, $f\!\downarrow\! j$ (respectively $j\!\uparrow\! f$) is $n$-connected for some $n\geq 0$, then the homotopy fibers of $f$ are $n$-connected. Thus in this case $f$ induces an isomorphism on homotopy groups in degrees $0,\ldots,n$ and a surjection in degree $n+1$. \item If, for every $j$, $f\!\downarrow\! j$ (respectively $j\!\uparrow\! f$) is connected and has $p$-torsion reduced integral homology in degrees not exceeding $n$ ($n\geq 0$), then the homotopy fibers of $f$ are connected and have $p$-torsion reduced integral homology in degrees not exceeding $n$. Thus in this case, for primes $q\not=p$, $f$ induces an isomorphism on $H_\ast(-,{\mathbf Z}/q)$ for $\ast\leq n$ and a surjection on $H_{n+1}(-,{\mathbf Z}/q)$. \item If, for every $j$, $f\!\downarrow\! j$ (respectively $j\!\uparrow\! f$) is acyclic with respect to some homology theory, then $f$ is this homology isomorphism. \end{enumerate} \end{cor}
\section{Simplicial complexes and small categories} \begin{point}\label{aSFDHFN} Fix a set $\mathcal U$ called a {\bf universe}. A {\bf simplicial complex} is a collection $K$ of finite nonempty subsets of $\mathcal U$ that satisfies the following requirement: if $\sigma\subset \mathcal U$ is in $K$, then every non-empty subset of $\sigma$ is also in $K$.
Let $X\subset \mathcal U$ be a subset. The collection $\{\{x\}\ |\ x\in X\}$, consisting of singletons in $X$, is a simplicial complex denoted also by $X$, called the {\bf discrete} simplicial complex on $X$. The collection $\{\sigma \subset X\ |\ 1\leq |\sigma|<\infty\}$ of all finite nonempty subsets of $X$ is also a simplicial complex denoted by $\Delta[X]$ and called the {\bf simplex} on $X$. A simplicial complex is called a standard simplex if it is of the form $\Delta[X]$ for some $X\subset \mathcal U$. The simplex $\Delta[\emptyset]$ is called the empty simplex or the empty simplicial complex. \end{point}
\begin{point} Let $K$ be a simplicial complex. An element $\sigma$ in $K$ is called a simplex of $K$ of dimension
$|\sigma|-1$. The set of $n$-dimensional simplices in $K$ is denoted by $K_n$. An element $x\in \mathcal U$ is called a vertex of $K$ if $\{x\}$ is a simplex in $K$. The assignment $x\mapsto \{x\}$ is a bijection between the set of vertices in $K$ and the set of its $0$-dimensional simplices $K_0$. We use this bijection to identify these sets. Thus we are going to refer to $0$-dimensional simplices in $K$ also as vertices. \end{point}
\begin{point}\label{asfdfhd} If $\{K^{i}\}_{i\in I}$ is a family of simplicial complexes, then both the intersection $\cap_{i\in I} K^{i}$ and the union $\cup_{i\in I} K^{i}$ are also simplicial complexes. If $K$ is a simplicial complex and $X\subset {\mathcal U}$ is a subset, then the intersection $K\cap \Delta[X]$ is a simplicial complex consisting of the elements of $K$ that are subsets of $X$. This intersection is called the {\bf restriction} of $K$ to $X$ and is denoted by $K_X$.
Note that $\Delta[X]\cap \Delta[Y]=\Delta[X\cap Y]$. Thus the intersection of standard simplices (see~\ref{aSFDHFN}) is again a standard simplex, which can possibly be empty.
Let $L$ and $K$ be simplicial complexes. If $L\subset K$, then
$L$ is called a subcomplex of $K$. Being a subcomplex is a partial order relation on the collection of all simplicial complexes which gives this collection the structure of a lattice. The union is the join and the intersection is the meet.
The collection $\cup_{0\leq i\leq n} K_i$ is a subcomplex of $K$ called the $n$-th {\bf skeleton} of $K$ and denoted by $\text{sk}_nK$. \end{point} \begin{point}
A {\bf map} between two simplicial complexes $K$ and $L$ is by definition a function $\phi\colon K\to L$ for which there exists a function $f\colon K_0\to L_0$ such that $\phi(\sigma)=\cup_{x\in \sigma }f(\{x\})$ for all $\sigma$ in $K$. In particular $\phi(\{x\})=f(\{x\})$ for every vertex $x$ in $K$. Thus $f$ is uniquely determined by $\phi$ and we often use the symbol $\phi_0$ to denote $f$. If $K$ and $L$ are fixed, then $\phi$ is determined by $f=\phi_0$.
The inclusion $L\subset K$ of a subcomplex
is an example of a map.
For any simplicial complex $K$, the inclusions $K_0\subset K\subset \Delta[K_0]$,
between the discrete simplicial complex $K_0$, $K$, and the simplex $\Delta[K_0]$ on $K_0$
are maps of simplicial complexes. The induced functions on the set of vertices for these two inclusions
are given by the identity function $\text{id}\colon K_0\to K_0$.
\end{point}
\begin{point}\label{fdfgsfhb} Classically, the geometrical realization is used to define and study homotopical properties of simplicial complexes. For example, a commutative square of simplicial complexes is called a homotopy push-out (pull-back) if after applying the realization, the obtained commutative square of spaces is a homotopy push-out (pull-back). For instance two simplicial complexes $K$ and $L$ fit into the following commutative diagram of subcomplex inclusions: \[\begin{tikzcd}
K\cap L \arrow[r, hook] \arrow[d, hook'] &K\arrow[d, hook] \\
L\arrow[r, hook] & K\cup L
\end{tikzcd}\] By applying the realization construction to this square, we obtain a commutative square of spaces which is a push-out and hence a homotopy push-out as the maps involved are cofibrations. \end{point}
Since the realization of a simplicial complex $K$ can be built from the realization of its $n$-skeleton $\text{sk}_nK$ by attaching (possibly in many steps) cells of dimension strictly bigger than $n$, we get: \begin{prop}\label{asfsdfhg} Let $n\geq 0$ be a natural number. For every simplicial complex $L$ such that $\text{\rm sk}_{n+1}K\subset L\subset K$,
the homotopy fibers of the inclusion $L\subset K$ are $n$-connected. In particular, the map $L\subset K$ induces an isomorphism on homotopy and integral homology groups in degrees $0,\ldots,n$ and a surjection in degree $n+1$. \end{prop}
There are situations however when another way of extracting homotopical properties of simplicial complexes is more convenient. In the rest of this section, we recall how one can retrieve and study such information by first transforming simplicial complexes into small categories and then using
the nerve construction as explained in Section~\ref{asfdhg}.
\begin{point}
Let $K$ be a simplicial complex. The {\bf simplex category} of $K$, denoted also by the same symbol $K$,
is by definition the inclusion poset of its simplices. Thus, the objects of $K$ are the simplices in $K$ and
the sets of morphisms are either empty or contain only one element:
\[|\text{mor}_{K}(\sigma,\tau)|=\begin{cases}
1 & \text{ if } \sigma\subset \tau\\
0 & \text{ otherwize}
\end{cases}\]
If $\phi\colon K\to L$ is a map of simplicial complexes, then the assignment $\sigma\mapsto \phi(\sigma)$ is a functor of simplex categories. We denote this functor also by the symbol $\phi\colon K\to L$.
Not all functors between $K$ and $L$ are of such a form.
\end{point}
The geometrical realization of a simplicial complex is weakly equivalent to the realization of the nerve of this simplicial complex. Thus to describe homotopical properties of simplicial complexes we can either use their geometrical realizations or the nerves of their simplex categories.
\begin{point}
Let $K$ be a simplicial complex and $\sigma$ be its simplex. Define the {\bf star} of $\sigma$ to be $\text{St}(\sigma):=\{\mu\in K\ |\ \sigma\cup \mu\in K\}$. Note that $\text{St}(\sigma)$ is a subcomplex of $K$. The star of any simplex is contractible. More generally: \end{point}
\begin{prop}\label{adgassgfhjg} Let $\sigma$ be a simplex in $K$. Then, for any proper subset $S\subsetneq \sigma$, the collection
$L:=\{\mu\in K\ |\ \mu\cap S=\emptyset\text{ and } \sigma\cup \mu\in K\}$ is a contractible simplicial complex (note that if $S=\emptyset$, then $L=\text{\rm St}(\sigma)$). \end{prop} \begin{proof} For all $\mu$ in $L$, the inclusions $\mu\hookrightarrow \mu\cup (\sigma\setminus S)\hookleftarrow \sigma\setminus S$ form natural transformations between: \begin{itemize} \item the identity functor $\text{id}\colon L\to L$, $\mu\mapsto \mu$, \item the constant functor $ L\to L$, $\mu\mapsto \sigma\setminus S$, \item and $ L\to L$, given by $\mu\mapsto \mu\cup (\sigma\setminus S)$. \end{itemize} The identity functor $\text{id}\colon L\to L$ is therefore homotopic to the constant functor and consequently $ L$ is contractible. \end{proof}
\begin{point}\label{adghxdfgjhkut} Let $K$ be a simplicial complex. It's simplex $\tau$ is called {\bf central} if $K=\text{St}(\tau)$, i.e., if for any simplex $\sigma$ in $K$, the set $\sigma\cup\tau$ is also a simplex in $K$. For example, if $X\subset \mathcal{U}$ is non empty, then all simplices in $\Delta[X]$ (see~\ref{aSFDHFN}) are central. If $\tau$ is a central simplex in $K$, then so is any subset $\tau'\subset \tau$. According to Proposition~\ref{adgassgfhjg}, a simplicial complex that has a central simplex is contractible. \end{point} \begin{point}\label{afsaasfagdf} Let $K$ and $L$ be simplicial complexes. If $K\cap L=\emptyset$, then we define their {\bf join} $K\ast L$ to be the simplicial complex consisting of all subsets of $\mathcal U$ of the form $\sigma\cup \tau$ where $\sigma$ is in $K$ and $\tau$ is in $L$. The join is only defined for disjoint simplicial complexes. The set of vertices $(K\ast L)_0$ is given by the (disjoint) union $K_0\cup L_0$. Note that $K\ast \Delta[\emptyset]=K$. If $\sigma\in L$ is central in $L$, then it is central in $K\ast L$. Thus, for any non-empty subset $X\subset \mathcal U\setminus K_0$, the join $K\ast \Delta[X]$ is contractible. Furthermore the join commutes with unions and intersections: if $(K_1\cup K_2)\cap L=\emptyset$, then
$(K_1\cup K_2)\ast L=(K_1\ast L)\cup (K_2\ast L)$ and $(K_1\cap K_2)\ast L=(K_1\ast L)\cap (K_2\ast L)$. This can be used to show that, for any choice of a base-points in $K$ and $L$, the join $K\ast L$ has the homotopy type of the suspension of the smash $\Sigma(|K|\wedge |L])$. In particular if $K$ is $n$-connected and $L$ is $m$-connected, then $K\ast L$ is $n+m+1$-connected. \end{point}
\section{One outside point}\label{afsfdgdsfgh} In this section we recall how the homotopy type of a simplical complex changes when a vertex is added. We start with defining subcomplexes that play an important role in describing such changes. These complexes are essentially used throughout the entire paper. \begin{defn}\label{asdfsaahdfg} Let $K$ be a simplicial complex and $A\subset \mathcal{U}$ be a subset. For a simplex $\sigma$ in $K$, define the \textbf{obstruction complex}:
\[\text{St}(\sigma,A):=\{\mu\subset A\ |\ 0<|\mu| \text{ and }\mu\cup \sigma\in K\}=K_A\cap \text{St}(\sigma) \] \end{defn} If $\mu$ belongs to $\text{St}(\sigma,A)$, then so does any of its non empty finite subsets. Thus $\text{St}(\sigma,A)$ is a simplicial complex. It is a subcomplex of $K_A$. Note that the complex $\text{St}(\sigma,A)$ may be empty. If $\tau\subset \sigma$, then $\text{St}(\sigma,A)\subset \text{St}(\tau,A)$.
Fix a vertex $v$ in $K$. Any simplex in $K$ either contains $v$ or it does not. This means $K=K_{K_0\setminus\{v\}}\cup \text{St}(v)$ and hence we have a homotopy push-out square:
\[\begin{tikzcd}
K_{K_0\setminus\{v\}}\cap \text{St}(v) \ar[hook]{r} \ar[hook']{d} & \text{St}(v)\ar[hook]{d}\\ K_{K_0\setminus\{v\}}\ar[hook]{r} & K \end{tikzcd}\]
By definition $ K_{K_0\setminus\{v\}}\cap \text{St}(v)=\text{St}(v,K_0\setminus\{v\})$. Proposition~\ref{adgassgfhjg} gives contractibility of $\text{St}(v)$. The simplicial complex $K$ fits therefore into the following homotopy cofiber sequence: \[\text{St}(v,K_0\setminus\{v\})\hookrightarrow K_{K_0\setminus\{v\}}\hookrightarrow K\] Here are some basic consequences of this fact: \begin{cor} Let $v$ be a vertex in a simplical complex $K$. \begin{enumerate} \item If $\text{\rm St}(v,K_0\setminus\{v\})$ is contractible, then $K_{K_0\setminus\{v\}}\subset K$ is a weak equivalence. \item If $\text{\rm St}(v,K_0\setminus\{v\})$ is $n$-connected for a natural number $n\geq 0$, then the map $K_{K_0\setminus\{v\}}\subset K$ induces an isomorphism on homotopy groups in degrees $0,\ldots,n$ and a surjection in degree $n+1$. \item If $\text{\rm St}(v,K_0\setminus\{v\})$ is connected and has $p$-torsion reduced integral homology in degrees not exceeding $n$ ($n\geq 0$), then for a prime $q$ not dividing $p$, $K_{K_0\setminus\{v\}}\subset K$ induces an isomorphism on $H_\ast(-,{\mathbf Z}/q)$ for $\ast\leq n$ and a surjection on $H_{n+1}(-,{\mathbf Z}/q)$. \item If $\text{\rm St}(v,K_0\setminus\{v\})$ is acyclic with respect to some homology theory, then $K_{K_0\setminus\{v\}}\subset K$ is this homology isomorphism. \end{enumerate} \end{cor}
\section{Two outside points}\label{asfasfgddf} Let us fix two distinct vertices $v_0$ and $v_1$ in a simplicial complex $K$. Note that $\left(K_0\setminus\{v_0\}\right)\cup \left(K_0\setminus\{v_1\}\right)=K_0$. In this section we are going to investigate the inclusion $K_{K_0\setminus\{v_0\}}\cup K_{K_0\setminus\{v_1\}}\subset K$.
There are two possibilities. First, $\{v_0,v_1\}$ is not a simplex in $K$. In this case $K_{K_0\setminus\{v_0\}}\cup K_{K_0\setminus\{v_1\}}=K$.
Assume $\{v_0,v_1\}$ is a simplex in $K$.
Then:
\[K=K_{K_0\setminus\{v_0\}}\cup K_{K_0\setminus\{v_1\}}\cup \text{St}(v_0,v_1)\]
Consequently there is a homotopy push-out square (see~\ref{fdfgsfhb}): \[\begin{tikzcd} \left(K_{K_0\setminus\{v_0\}}\cup K_{K_0\setminus\{v_1\}}\right)\cap \text{St}(v_0,v_1) \arrow[d, hook'] \arrow[r, hook] &\text{St}(v_0,v_1)\arrow[d, hook]\\ K_{K_0\setminus\{v_0\}}\cup K_{K_0\setminus\{v_1\}}\arrow[r, hook] & K \end{tikzcd}\] Since the star complex $\text{St}(v_0,v_1)$ is contractible (see Proposition~\ref{adgassgfhjg}),
$K$ is therefore weakly equivalent to the homotopy cofiber of the map: \[\left(K_{K_0\setminus\{v_0\}}\cup K_{K_0\setminus\{v_1\}}\right)\cap \text{St}(v_0,v_1)\hookrightarrow K_{K_0\setminus\{v_0\}}\cup K_{K_0\setminus\{v_1\}}\] The complex $\left(K_{K_0\setminus\{v_0\}}\cup K_{K_0\setminus\{v_1\}}\right)\cap \text{St}(v_0,v_1)$ fits into the following homotopy push-out square: \[\begin{tikzcd} K_{K_0\setminus\{v_0\}}\cap K_{K_0\setminus\{v_1\}}\cap \text{St}(v_0,v_1)\arrow[r, hook] \arrow[d, hook']& K_{K_0\setminus\{v_0\}}\cap \text{St}(v_0,v_1) \arrow[d, hook]\\ K_{K_0\setminus\{v_1\}}\cap \text{St}(v_0,v_1) \arrow[r, hook] & \left(K_{K_0\setminus\{v_0\}}\cup K_{K_0\setminus\{v_1\}}\right)\cap \text{St}(v_0,v_1) \end{tikzcd}\] Let us identify the complexes in this square: \begin{itemize}
\item $K_{K_0\setminus\{v_0\}}\cap \text{St}(v_0,v_1)=\{\mu\in K\ |\ \mu\cap \{v_0\}=\emptyset\text{ and } \{v_0,v_1\}\cup \mu\in K\}$ and thus according to Proposition~\ref{adgassgfhjg} this complex is contractible; \item by the same argument $K_{K_0\setminus\{v_1\}}\cap \text{St}(v_0,v_1)$ is also contractible; \item $K_{K_0\setminus\{v_0\}}\cap K_{K_0\setminus\{v_1\}}\cap \text{St}(v_0,v_1)=K_{K_0\setminus\{v_0,v_1\}}\cap \text{St}\left(v_0,v_1\right)=\\=
\text{St}\left(\{v_0,v_1\},K_0\setminus\{v_0,v_1\}\right)$
\end{itemize} It follows that $ \left(K_{K_0\setminus\{v_0\}}\cup K_{K_0\setminus\{v_1\}}\right)\cap \text{St}(v_0,v_1)$ has the homotopy type of the suspension of the obstruction complex $\text{St}:=\text{St}\left(\{v_0,v_1\},K_0\setminus\{v_0,v_1\}\right)$ and hence we have a homotopy cofiber sequence of the form: \[\Sigma \text{\rm St}\to K_{K_0\setminus\{v_0\}}\cup K_{K_0\setminus\{v_1\}}\hookrightarrow K\]
\section{$n+1$ outside points} Homotopy cofiber sequences described in Sections~\ref{afsfdgdsfgh} and~\ref{asfasfgddf} are particular cases of a more general statement regarding an arbitrary number of outside points. The aim of this section is to present this generalization.
Let us fix a set $\sigma=\{v_0,v_1,\ldots,v_n\}\subset K_0$ of $n+1$ distinct vertices in a simplcial complex $K$ which may not necessarily be a simplex in $K$.
Note that $\bigcup_{v\in \sigma } \left(K_0\setminus\{v\}\right)=K_0$. In this section we are going to investigate the inclusion $\left(\bigcup_{v\in\sigma} K_{K_0\setminus\{v\}}\right) \subset K$
There are two possibilities.
First, $\sigma$ is not a simplex in $K$. In this case $\bigcup_{v\in \sigma} K_{K_0\setminus\{v\}}= K$.
Assume $\sigma$ is a simplex in $K$. Then:
\[K=\left(\bigcup_{v\in \sigma} K_{K_0\setminus\{v\}}\right) \cup \text{St}(\sigma)\]
Consequently there is a homotopy push-out square (see~\ref{fdfgsfhb}): \[\begin{tikzcd} \left(\bigcup_{v\in \sigma} K_{K_0\setminus\{v\}}\right)\cap \text{St}( \sigma) \arrow[d, hook'] \arrow[r, hook] &\text{St}( \sigma)\arrow[d, hook]\\ \bigcup_{v\in \sigma} K_{K_0\setminus\{v\}}\arrow[r, hook] & K \end{tikzcd}\] Since the star complex $\text{St}( \sigma)$ is contractible (see Proposition~\ref{adgassgfhjg}),
$K$ is therefore weakly equivalent to the homotopy cofiber of the map: \[\left(\bigcup_{v\in \sigma} K_{K_0\setminus\{v\}}\right)\cap \text{St}( \sigma)\hookrightarrow \bigcup_{v\in \sigma} K_{K_0\setminus\{v\}}\] Next we identify the homotopy type of $\left(\bigcup_{v\in \sigma} K_{K_0\setminus\{v\}}\right)\cap \text{St}( \sigma)$:
\begin{prop}\label{asfdgdsfjdhhg} Let $\sigma$ be a simplex of dimension $n$ in a simplicial complex $K$. Then $\left(\bigcup_{v\in \sigma} K_{K_0\setminus\{v\}}\right)\cap \text{\rm St}( \sigma)$ has the homotopy type of the $n$-th suspension of the obstruction complex $\Sigma^n \text{\rm St}( \sigma, K_0\setminus\sigma)$. \end{prop} \begin{proof} Consider the inclusion poset of all subsets $\tau\subset \sigma$. For any such subset $\tau\subset \sigma$, define: \[F(\tau):=\begin{cases} \bigcap_{v\in \tau} K_{K_0\setminus\{v\}}= K_{K_0\setminus\tau} & \text{ if } \tau\not=\emptyset\\ \bigcup_{v\in \sigma} K_{K_0\setminus\{v\}} & \text{ if } \tau=\emptyset \end{cases}\] Note that if $\tau'\subset\tau\subset\sigma$, then $F(\tau)\subset F(\tau')$. Thus by assigning to the inclusion $\tau'\subset\tau$ the map $F(\tau)\subset F(\tau')$, we obtain a contra-variant functor indexed by the inclusion poset of all subsets of $\sigma$. For example in the case $\sigma=\{v_0,v_1,v_2\}$, this contra-variant functor describes a commutative cube: \[\begin{tikzcd}[column sep=0.1em,row sep=2em] K_{K_0\setminus\{v_0,v_1,v_2\}}\ar[hook]{rr}\ar[hook]{dd}\ar[hook]{dr} & & K_{K_0\setminus\{v_1,v_2\}}\ar[hook]{dr}\ar[hook]{dd} \\ &K_{K_0\setminus\{v_0,v_2\}}\ar[hook, crossing over]{rr} & & K_{K_0\setminus\{v_2\}}\ar[hook]{dd} \\ K_{K_0\setminus\{v_0,v_1\}}\ar[hook]{rr}\ar[hook]{dr} & & K_{K_0\setminus\{v_1\}}\ar[hook]{dr} \\ & K_{K_0\setminus\{v_0\}}\ar[hook]{rr}\ar[hook, crossing over,from=uu] & & K_{K_0\setminus\{v_0\}}\cup K_{K_0\setminus\{v_1\}}\cup K_{K_0\setminus\{v_2\}} \end{tikzcd}\]
For arbitrary $n$, the functor $F$ describes a commutative cube of dimension $n+1$. This cube is both co-cartesian and strongly cartesian
(\cite{}). It is therefore also a homotopy co-cartesian. By intersecting with $\text{St}(\sigma)$, we obtain a new cube $\tau\mapsto F(\tau)\cap \text{St}(\sigma)$. The properties of being co-cartesian and strongly cartesian are preserved by taking such interection. Consequently $\left(\bigcup_{v\in \sigma} K_{K_0\setminus\{v\}}\right)\cap \text{\rm St}( \sigma)$ has the homotopy type of $\text{hocolim}_{\emptyset\not=\tau\subset\sigma}\left(K_{K_0\setminus\tau}\cap \text{\rm St}( \sigma)\right)$.
For any proper subset $\emptyset\not=\tau\subsetneq \sigma$, we have an equality:
\[K_{K_0\setminus\tau}\cap \text{\rm St}( \sigma)=\{\mu\ |\ \mu\cap \tau=\emptyset \text{ and } \sigma\cup\mu\in K\}\] We can then use Proposition~\ref{adgassgfhjg} to conclude that $K_{K_0\setminus\tau}\cap \text{\rm St}( \sigma)$ is contractible if $\emptyset\not=\tau\subsetneq \sigma$. Thus all the spaces in the cube $\tau\mapsto F(\tau)\cap \text{St}(\sigma)$, except for the initial and the terminal, are contractible. That implies that the terminal space $F(\emptyset)\cap \text{St}(\sigma)=\left(\bigcup_{v\in \sigma} K_{K_0\setminus\{v\}}\right)\cap \text{St}(\sigma)$ is homotopy equivalent to the $n$-th suspension of the initial space: $\Sigma^n \left(F(\sigma)\cap \text{St}(\sigma) \right)= \Sigma^n \left( K_{K_0\setminus \sigma}\cap \text{St}(\sigma)\right)=\Sigma^n\text{St}(\sigma,K_0\setminus \sigma)$. \end{proof}
We finish this section with summarising the consequences of the discussion leading to Proposition~\ref{asfdgdsfjdhhg} and the proposition itself: \begin{cor}\label{asfgdsgfjghkkl}
Let $\sigma\subset K_0$ be a subset consisting of $n+1$ distinct vertices in a simplicial complex $K$. \begin{enumerate} \item If $\sigma $ is not a simplex in $K$, then $\bigcup_{v\in\sigma}K_{K_0\setminus\{v\}}=K$. \item Assume $\sigma$ is a simplex in $K$. \begin{enumerate} \item Then there is a homotopy cofiber sequence: \[\Sigma^n\text{\rm St}(\sigma,K_0\setminus \sigma)\to \bigcup_{v\in\sigma}K_{K_0\setminus\{v\}}\hookrightarrow K\] \item If $\text{\rm St}(\sigma,K_0\setminus \sigma)=\emptyset$, then there is a homotopy cofiber sequence (here $S^{-1}=\emptyset$): \[S^{n-1}\to \bigcup_{v\in\sigma}K_{K_0\setminus\{v\}}\hookrightarrow K\] \item If $\text{\rm St}(\sigma,K_0\setminus \sigma)\not =\emptyset$, then the homotopy fibers of $\bigcup_{v\in\sigma}K_{K_0\setminus\{v\}}\hookrightarrow K$ are $ m\geq 0$ connected if and only if $\overline{H}_i(\text{\rm St}(\sigma,K_0\setminus \sigma),{\mathbf Z})=0$ for $i\leq m-n$. \item If $\text{\rm St}(\sigma,K_0\setminus \sigma)\not =\emptyset$, then $\bigcup_{v\in\sigma}K_{K_0\setminus\{v\}}\hookrightarrow K$ is a weak equivalence if and only if $\overline{H}_i\left(\text{\rm St}(\sigma,K_0\setminus \sigma),{\mathbf Z}\right)=0$ for all $i$. \end{enumerate} \end{enumerate} \end{cor}
\section{Push-out decompositions I.}\label{dec} In this section our starting assumption is:
\begin{point}[\bf Starting input I]\label{adfdfhgf} {\em $K$ is a simplicial complex, $X\cup Y=K_0$ is a cover of its set of vertices, and $A:=X\cap Y$.} \end{point}
By restricting $K$ to $X$ and $Y$, and taking the union of these restrictions we obtain a subcomplex $K_X\cup K_Y\subset K$. Since $K_X\cap K_Y=K_A$, this subcomplex fits into the following homotopy push-out square: \[\begin{tikzcd} K_A\ar[hook]{r} \ar[hook']{d} & K_X \ar[hook]{d}\\ K_Y\ar[hook]{r} & K_X\cup K_Y \end{tikzcd}\] This push-out can be then used to extract various homotopical properties of the union $K_X\cup K_Y$ from the properties of $K_X$, $K_Y$ and $K_A$. For example,
if $K_X$, $K_Y$ and $K_A$ belong to a closed collection (see~\ref{asdfdsfhgdf}), then so does $K_X\cup K_Y$. If $K_A$ is contractible, then $K_X\cup K_Y$ has the homotopy type of the wedge of $K_X$ and $K_Y$, and its reduced homology is the sum of the reduced homologies of $K_X$ and $K_Y$.
More generally, there is a Mayer-Vietoris sequence connecting homologies of $K_X\cup K_Y$ with those of $K_X$, $K_Y$ and $K_A$.
A fundamental question discussed in this article is: {\em under what circumstances the inclusion $K_X\cup K_Y\subset K$ is a weak equivalence, or homology isomorphism, or has highly connected homotopy fibers etc?} Such circumstances would enable us to express various homotopical properties of $K$ in terms of the properties of its restrictions $K_X$, $K_Y$ and $K_A$.
\begin{defn}\label{asdgfssdtjhgkj} Under the starting assumption~\ref{adfdfhgf}, define $P$ to be the subposet of $K$ given by:
\[P:=\{\sigma\in K\ |\ \sigma\subset X\text{ or } \sigma\subset Y \text{ or } \sigma\cap A\not=\emptyset\}\] \end{defn}
We are going to be more interested in the set of simplices of $K$ that do not belong to $P$, which explicitly can be described as:
\[K\setminus P=\{\sigma\in K\ |\ \sigma\cap X\not=\emptyset\text{ and } \sigma\cap Y\not=\emptyset \text{ and } \sigma\cap A=\emptyset\}\]
The poset $P$ may not be the simplex category of any simplicial complex. There are two poset inclusions that we denote by $f$ and $g$:
\[\begin{tikzcd}
K_X\cup K_Y \arrow[r, hook, "f"] & P \arrow[r, hook,"g"] & K
\end{tikzcd}\]
Our first general observation is: \begin{prop}\label{asfsdfg} The functor $f\colon K_X\cup K_Y\hookrightarrow P$ is a weak equivalence. \end{prop} \begin{proof} We are going to show that, for every $\sigma$ in $P$, $f\!\downarrow\! \sigma$ is contractible.
First assume
$\sigma\subset X$ or $\sigma\subset Y$. Then the object $(\sigma, \text{id}\colon\sigma\to \sigma)$ is terminal in $f\!\downarrow\! \sigma$ and consequently this category is contractible.
Assume $\sigma\cap A\not=\emptyset$. Then, for any object $(\tau,\tau\subset \sigma)$ in $f\!\downarrow\! \sigma$, the subsets $\tau$, $\tau\cup (\sigma\cap A)$, and $ \sigma\cap A$ of $\sigma$ are simplices that belong to $K_X\cup K_Y$. We can then form the following commutative diagram in $P$
where the top horizontal arrows represent morphisms in $K_X\cup K_Y$:
\[\begin{tikzcd}
\tau \arrow[r, hook] \arrow[rd, hook] &\tau\cup (\sigma\cap A) \arrow[d, hook] & \sigma\cap A \arrow[l, hook'] \arrow[dl, hook'] \\ & \sigma
\end{tikzcd}\] These horizontal morphisms form natural transformations between: \begin{itemize} \item the identity functor $\text{id}\colon f\!\downarrow\! \sigma\to f\!\downarrow\! \sigma$, $(\tau,\tau\subset \sigma)\mapsto(\tau,\tau\subset \sigma)$, \item the constant functor $ f\!\downarrow\! \sigma\to f\!\downarrow\! \sigma$, $(\tau,\tau\subset \sigma)\mapsto ( \sigma\cap A, \sigma\cap A\subset \sigma)$, \item and $ f\!\downarrow\! \sigma\to f\!\downarrow\! \sigma$ given by $(\tau,\tau\subset \sigma)\mapsto (\tau\cup( \sigma\cap A), \tau\cup( \sigma\cap A)\subset \sigma)$. \end{itemize} The identity functor $\text{id}\colon f\!\downarrow\! \sigma\to f\!\downarrow\! \sigma$ is therefore homotopic to the constant functor. This can happen only if $f\!\downarrow\! \sigma$ is a contractible category. \end{proof}
According to Proposition~\ref{asfsdfg}, the homotopy fibers of $g\colon P\subset K$ and the inclusion $K_X\cup K_Y\subset K$
are weakly equivalent. To understand these homotopy fibers, we are going to focus on the categories $\sigma\!\uparrow \! g$ and then utilise Corollary~\ref{asgfgjh}. The functor $\sigma\mapsto \sigma\!\uparrow \! g$ fits into the following diagram of natural transformations between functors
indexed by $K^{\text{op}}$ with small categories as values: \[\begin{tikzcd} & \sigma\ar[mapsto, bend right]{dl}\ar[mapsto]{d}\ar[mapsto, bend left]{dr}\\ \text{St}(\sigma,A)\ar{r}{\psi_{\sigma}}& \sigma\!\uparrow \! g \ar{r}{\phi_{\sigma}} & \text{St}(\sigma) \end{tikzcd} \] where: \begin{itemize} \item
$\psi_{\sigma}\colon \text{St}(\sigma,A)\to \sigma\!\uparrow \! g$ assigns to $\mu$ in $\text{St}(\sigma,A)$ the object in $ \sigma\!\uparrow \! g$ given by the pair $\psi_{\sigma}(\mu):=(\mu\cup \sigma, \sigma\subset \mu\cup \sigma)$. \item $\phi_{\sigma}\colon \sigma\!\uparrow \! g\to \text{St}(\sigma)$ assigns to $(\tau,\sigma\subset \tau)$ the simplex $\tau$ in $ \text{St}(\sigma)$. \end{itemize}
These natural transformations satisfy the following properties: \begin{prop}\label{adfgsdsgfjdh} Let $\sigma$ be a simplex in $K$. \begin{enumerate} \item If $\sigma$ is in $ P$, then $\sigma\!\uparrow \! g$ is contractible and $\phi_{\sigma}\colon \sigma\!\uparrow \! g\to \text{\rm St}(\sigma)$ is a weak equivalence. \item If $\sigma$ is in $K\setminus P$, then $\psi_{\sigma}\colon \text{\rm St}(\sigma,A)\to \sigma\!\uparrow \! g$ is a weak equivalence. \end{enumerate} \end{prop} \begin{proof} If $\sigma$ is in $P$, then $(\sigma,\text{id}\colon \sigma\to \sigma)$ is an initial object in $\sigma\!\uparrow \! g$ and hence this category is contractible. That proves (1).
Assume $\sigma$ is not in $P$, which is equivalent to $\sigma\cap Y\not=\emptyset$ and $ \sigma\cap X\not=\emptyset$ and $\sigma\cap A=\emptyset$. Let $(\tau,\sigma\subset \tau)$ be an object in $\sigma\!\uparrow \! g$. Define ${\alpha_{\sigma}}(\tau,\sigma\subset \tau):=\tau\cap A$. Since $\sigma\cap Y\not=\emptyset$ and $ \sigma\cap X\not=\emptyset$, then $\tau\cap Y\not=\emptyset$ and $ \tau\cap X\not=\emptyset$. This together with the fact that $\tau$ belongs to $P$ implies $\alpha_{\sigma}(\tau,\sigma\subset \tau)=\tau\cap A\not=\emptyset$. Furthermore $(\tau\cap A)\cup \sigma\subset \tau\in P\subset K$. Thus $\alpha_{\sigma}$ defines a functor $\alpha_{\sigma}\colon \sigma\!\uparrow \! g\to \text{St}(\sigma,A)$.
Note:
\[\alpha_{\sigma}\psi_{\sigma}(\mu)=\alpha_{\sigma}(\mu\cup \sigma, \sigma\subset \mu\cup \sigma)=(\mu\cup \sigma)\cap A\]
Since $\sigma\cap A=\emptyset$ and $\mu\subset A$, we get $\alpha_{\sigma}\psi_{\sigma}(\mu)=(\mu\cup \sigma)\cap A=\mu$. The composition
$\alpha_{\sigma}\psi_{\sigma}$ is therefore the identity functor.
Note further:
\[\psi_{\sigma}\alpha_{\sigma}(\tau,\sigma\subset \tau)=\psi_{\sigma}(\tau\cap A)=\left((\tau\cap A)\cup\sigma,\sigma\subset(\tau\cap A)\cup\sigma\right) \]
Since $\sigma\subset \tau$, we have a commutative diagram:
\[\begin{tikzcd}
& \sigma\arrow[dl, hook']\arrow[dr, hook]\\
(\tau\cap A)\cup\sigma \arrow[rr, hook] & & \tau
\end{tikzcd}\]
The bottom horizontal morphisms form a natural transformation between:
\begin{itemize}
\item the composition $ \psi_{\sigma}\alpha_{\sigma}\colon \sigma\!\uparrow \! g\to \sigma\!\uparrow \! g$ and
\item the identity functor $\text{id}\colon \sigma\!\uparrow \! g\to \sigma\!\uparrow \! g$.
\end{itemize}
The functor $ \psi_{\sigma}\colon \text{St}(\sigma,A)\to \sigma\!\uparrow \! g$ has therefore a homotopy inverse and hence
is a weak equivalence which proves (2).
\end{proof}
We use Corollary~\ref{asgfgjh} and Proposition~\ref{adfgsdsgfjdh} to obtain our main statement describing properties of the homotopy fibers of the inclusion $K_X\cup K_Y\subset K$:
\begin{thm}\label{adfsdfhgd} Notation as in~\ref{adfdfhgf} and Definition~\ref{asdgfssdtjhgkj}. Let ${\mathcal C}$ be a closed collection of simplicial sets (see~\ref{asdfdsfhgdf}). Assume that, for every $\sigma$ in $K\setminus P$, the obstruction complex $\text{\rm St}(\sigma,A)$ (see~\ref{asdfsaahdfg}) satisfies ${\mathcal C}$. Then the homotopy fibers of the inclusion $K_X\cup K_Y\subset K$ also satisfy ${\mathcal C}$. \end{thm}
The following are some particular cases of the above theorem specialized to different closed collections of simplicial sets. \begin{cor}\label{sdfdfghsd} Notation as in~\ref{adfdfhgf} and~\ref{asdgfssdtjhgkj}. Let $n$ be a natural number. \begin{enumerate} \item \label{afgds} If, for every $\sigma$ in $ K\setminus P$ (see~\ref{asdgfssdtjhgkj}), the simplicial complex $\text{\rm St}(\sigma,A)$ (see~\ref{asdfsaahdfg}) is contractible, then $K_X\cup K_Y\subset K$ is a weak equivalence. \item If, for every $\sigma$ in $ K\setminus P$, the simplicial complex $\text{\rm St}(\sigma,A)$ is $n$-connected, then the homotopy fibers of $K_X\cup K_Y\subset K$ are $n$-connected and this map induces an isomorphism on homotopy groups in degrees $0,\ldots,n$ and a surjection in degree $n+1$. \item Let $p$ be a prime number. If, for every $\sigma$ in $ K\setminus P$, the simplicial complex $\text{\rm St}(\sigma,A)$ is connected and has $p$-torsion reduced integral homology in degrees not exceeding $n$, then the homotopy fibers of $K_X\cup K_Y\subset K$ are connected and have $p$-torsion reduced integral homology in degrees not exceeding $n$. Thus in this case, for prime $q\not=p$, $K_X\cup K_Y\subset K$ induces an isomorphism on $H_\ast(-,{\mathbf Z}/q)$ for $\ast\leq n$ and a surjection on $H_{n+1}(-,{\mathbf Z}/q)$. \item If, for every $\sigma$ in $ K\setminus P$, the simplicial complex $\text{\rm St}(\sigma,A)$ is acyclic with respect to some homology theory, then $K_X\cup K_Y\subset K$ is this homology isomorphism. \end{enumerate} \end{cor}
We remark that Corollary~\ref{sdfdfghsd}.\ref{afgds} is a generalization of ~\cite[Theorem 2]{NewHenry} to abstract simplicial complexes.
Requirements for obtaining $n$-connected fibers can be weakened: \begin{prop}\label{sfsfdgfdhfg}
Notation as in~\ref{adfdfhgf} and~\ref{asdgfssdtjhgkj}. Let $n$ be a natural number.
If $\text{\rm St}(\sigma, A)$ is $n$-connected for every $\sigma$ in $(\text{\rm sk}_{n+1}K)\setminus P$, then the homotopy fibers of $K_X\cup K_Y\subset K$ are $n$-connected. \end{prop} \begin{proof} Consider the following poset inclusions: \[\begin{tikzcd} K_X\cup K_Y\ar[hook]{r}{f} & P \ar[hook]{r}{g_1}\ar[hook,bend right=25]{rr}{g} & P\cup \text{sk}_{n+1}K \ar[hook]{r}{g_2} & K \end{tikzcd}\] According to Proposition~\ref{asfsdfg}, $f$ is a weak equivalence. The homotopy fibers of $g_2$ are $n$-connected
by Proposition~\ref{asfsdfhg}. Thus if the homotopy fibers of $g_1$ are $n$-connected, then so are
the
homotopy fibers of the inclusion $K_X\cup K_Y\subset K$. To show that
the homotopy fibers of $g_1$ are $n$-connected it is enough to show that the categories
$\sigma\!\uparrow \! g_1$ are $n$-connected for every $\sigma$ in $P\cup \text{sk}_{n+1}K$.
Proposition~\ref{adfgsdsgfjdh} gives that
$\sigma\!\uparrow \! g_1$ is contractible if $\sigma$ is in $P$, and is weakly equivalent to $\text{\rm St}(\sigma,A)$ if $\sigma$ is in $\text{sk}_{n+1}K\setminus P$. By the assumption $\text{\rm St}(\sigma,A)$ are therefore $n$-connected.
\end{proof}
\section{Push-out decompositions II.}\label{asfdsdfhiu} Theorem~\ref{adfsdfhgd} states that the homotopy fibers of the inclusion $K_X\cup K_Y \subset K$ belong to the smallest closed collection containing all the complexes $\text{St}(\sigma,A)$ for $\sigma$ in $K\setminus P$. Recall that if a closed collection contains an empty simplicial set, then it contains all simplicial sets, in which case Theorem~\ref{adfsdfhgd} has no content. Thus $\text{St}(\sigma,A)$ being non empty, for all
$\sigma$ in $K\setminus P$, is an absolute minimum requirement for Theorem~\ref{adfsdfhgd} to have any content. In most of our statements that follow, the assumptions we make have much stronger global non emptiness consequences of the form: \[\bigcap_{\sigma\in K\setminus P}\text{St}(\sigma,A)\not= \emptyset\ \ \ \ \ \ \ \bigcap_{\sigma\in K_{n+1}\setminus P}\text{St}(\sigma,A)\not= \emptyset \ \ \ \ \ \ \ \bigcap_{\sigma\in (\text{sk}_{n+1}K)\setminus P}\text{St}(\sigma,A)\not= \emptyset\] Here is a consequence of having one of these intersections non-empty: \begin{prop}\label{adgsfgjkk} Notation as in~\ref{adfdfhgf} and~\ref{asdgfssdtjhgkj}. Assume: \[\bigcap_{\sigma\in K_{1}\setminus P}\text{\rm St}(\sigma,A)\not= \emptyset\] Then the homotopy fibers of $K_X\cup K_Y\subset K$ are connected. \end{prop} \begin{proof} Let $v$ be a vertex in $\bigcap_{\sigma\in K_{1}\setminus P}\text{St}(\sigma,A)$. Observe that $\text{sk}_{1}(K)$ is a disjoint union of $K_1\setminus P$ and $\text{sk}_{1}(K)\cap P$. This can fail for $\text{sk}_{n}(K)$ if $n>1$. For every $\tau$ in $\text{sk}_{1}(K)$, define: \[\phi(\tau):=\begin{cases}\tau\cup{v} & \text{ if } \tau\in K_1\setminus P \\ \tau &\text{ if } \tau\in \text{sk}_{1}(K)\cap P \end{cases}\] If $\tau\subsetneq \tau'$ in $\text{sk}_{1}(K)$, then $\tau$ is in $P$ and hence $\tau=\phi(\tau)\subset \phi(\tau')$. In this way we obtain a functor $\phi\colon \text{sk}_{1}(K)\to P$. The inclusion $\tau\subset \phi(\tau)$, is a natural transformation between the skeleton inclusion $\text{sk}_{1}(K)\subset K$ and the composition: \[\begin{tikzcd}
\text{sk}_{1}(K) \arrow[r, hook, "\phi"] & P \arrow[r, hook,"g"] & K
\end{tikzcd}\] Thus these two functors from $\text{sk}_{1}(K)$ to $ K$ are homotopic. The statement of the proposition is then a consequence of Proposition~\ref{asfsdfhg}. \end{proof}
Proposition~\ref{adgsfgjkk} does not generalise to $n>0$. Non-emptiness of the intersection $\bigcap_{\sigma\in (\text{sk}_{n+1}K)\setminus P}\text{St}(\sigma,A)$ does not imply that the homotopy fibers of $K_X\cup K_Y\subset K$ are $n$-connected. For an easy example see~\ref{basfasfgdqweter}. To guarantee $n$-connectedness of these homotopy fibers we need additional restrictions. For example in the following corollary the assumptions imply that $\text{St}(\sigma,A)$ does not depend on $\sigma$ in $(\text{\rm sk}_{n+1}K)\setminus P$:
\begin{cor}\label{asfdfgjfgjk,}
Notation as in~\ref{adfdfhgf} and~\ref{asdgfssdtjhgkj}. Let $n$ be a natural number.
Assume that one of the following conditions is satisfied:
\begin{enumerate}
\item There is an $n$-connected simplicial complex $L$ such that, for every simplex $\sigma$ in $(\text{\rm sk}_{n+1}K)\setminus P$, $\text{\rm St}(\sigma,A)=L$.
\item The complex $K_A$ is $n$-connected and, for every simplex $\sigma$ in $(\text{\rm sk}_{n+1}K)\setminus P$, $\text{\rm St}(\sigma,A)=K_A$.
\item The set $A$ is non empty. Furthermore, for every simplex $\sigma$ in $(\text{\rm sk}_{n+1}K)\setminus P$ and every finite subset $\mu$ in $A$, the union $\sigma\cup \mu$ is a simplex in $K$.
\item $A=\{v\}$ and, for every simplex $\sigma$ in $(\text{\rm sk}_{n+1}K)\setminus P$, the union $\sigma\cup\{v\}$ is also a simplex in $K$.
\end{enumerate} Then the homotopy fibers of $K_X\cup K_Y\subset K$ are $n$-connected. \end{cor} \begin{proof} The corollary under the assumption (1) is a direct consequence of Proposition~\ref{sfsfdgfdhfg}. The assumption (2) is a particular case of (1) with $L=K_A$. The assumption (3) is a particular case of (1) with $L=\Delta[A]$. Finally, the assumption (4) is a particular case of (3). \end{proof}
Here is another example of a statement whose assumption, referred to as ``one entry point", has a global nonemptiness consequence:
\begin{cor}\label{afgsfdhfg}
Notation as in~\ref{adfdfhgf} and~\ref{asdgfssdtjhgkj}. Let $n$ be a natural number. Assume there is an element $v$ in $A$ with the following property. For every simplex
$\tau$ in $K$ such that $\tau\cap (X\setminus A)\not=\emptyset$, $\tau\cap (Y\setminus A)\not=\emptyset$, and
$|\tau\cap (K_0\setminus A)|\leq n+2$, the union $\tau\cup\{v\}$ is also a simplex in $K$.
Then, for every simplex $\sigma$ in $(\text{\rm sk}_{n+1}K)\setminus P$, the element $v$ is a central vertex (see~\ref{adghxdfgjhkut}) in $\text{\rm St}(\sigma,A)$. Furthermore the homotopy fibers of $K_X\cup K_Y\subset K$ are $n$-connected. \end{cor} \begin{proof} Let $\sigma$ be a simplex in $ (\text{sk}_{n+1}K)\setminus P$.
If $\mu$ belongs to $\text{\rm St}(\sigma,A)$ then, by applying the assumption of the corollary to $\tau=\sigma\cup\mu$, we obtain that $\sigma\cup\mu\cup\{v\}$ is a simplex in $K$ and hence $\mu\cup\{v\}$ is a simplex in $\text{\rm St}(\sigma,A)$. This means that $v$ is central in $\text{\rm St}(\sigma,A)$ (see~\ref{adghxdfgjhkut}). Consequently $\text{\rm St}(\sigma,A)$ is contractible (see~\ref{adghxdfgjhkut}) and the corollary follows from Proposition~\ref{sfsfdgfdhfg}. \end{proof}
\section{Clique complexes}\label{dsfgdfhsfghj}
Recall that a simplicial complex $K$ is called
{\bf clique} if it satisfies the following condition: a set $\sigma$ of size at least $ 2$ is a simplex in $K$ if and only if all the two element subsets of $\sigma$ are simplices in $K$. Thus a clique complex is determined by its sets of vertices and edges.
If $K$ is clique, then the complexes $\text{\rm St}(\sigma, A)$ satisfy the following properties:
\begin{prop}\label{asfgsdfhjghkhjgl} Notation as in~\ref{adfdfhgf}. Assume $K$ is clique. Then: \begin{enumerate} \item For all $\sigma$ in $K$, $\text{\rm St}(\sigma ,A)$ is clique. \item If $\tau$ and $\sigma$ are simplices in $K$ such that $\tau\cup\sigma$ is also a simplex in $K$, then $\text{\rm St}(\tau\cup\sigma,A)=\text{\rm St}(\tau,A)\cap \text{\rm St}(\sigma,A)$. \item If $\sigma$ is a simplex in $K$ and $\sigma=\tau_1\cup\cdots\cup\tau_n$, then $\text{\rm St}(\sigma,A)=\bigcap_{i=1}^{n}\text{\rm St}(\tau_i,A)$. \item For every simplex $\sigma$ in $K$, $\text{\rm St}(\sigma,A)=\bigcap_{x\in \sigma}\text{\rm St}(\{x\},A)$. \end{enumerate} \end{prop} \begin{proof} Let $\mu$ be a subset of $A$ such that, for every two element subset $\tau$ of $\mu$, the set $\tau\cup \sigma$ is a simplex in $K$, i.e., $\tau$ is in $\text{\rm St}(\sigma,A)$. Then, since $K$ is clique, $\mu\cup \sigma$ is also a simplex in $K$. Consequently $\mu$ belongs to $\text{\rm St}(\sigma, A)$ and hence $\text{\rm St}(\sigma, A)$ is clique. That proves (1).
To prove (2), first note that the inclusion $\text{\rm St}(\tau\cup\sigma,A)\subset \text{\rm St}(\tau,A)\cap \text{\rm St}(\sigma,A)$ holds even without the clique assumption. Let $\mu$ belong to both $\text{\rm St}(\tau,A)$ and $\text{\rm St}(\sigma,A)$. This means that $\mu\cup\tau$ and $\mu\cup\sigma$ are simplices in $K$. Since every 2 element subset of $\mu\cup\tau\cup\sigma$ is a subset of either $\mu\cup\tau$ or $\mu\cup\sigma$ or $\tau\cup\sigma$, by the assumption it is an edge in $K$. By the clique assumption, $\mu\cup\tau\cup\sigma$ is then also a simplex in $K$ and consequently $\mu$ is in $\text{\rm St}(\tau\cup\sigma,A)$. This shows the other inclusion $\text{\rm St}(\tau\cup\sigma,A)\supset \text{\rm St}(\tau,A)\cap \text{\rm St}(\sigma,A)$ proving (2).
Statements (3) and (4) follow from (2). \end{proof}
Recall that an intersection of standard simplices is again a standard simplex (see~\ref{asfdfhd}). This observation together with Propositions~\ref{sfsfdgfdhfg} and~\ref{asfgsdfhjghkhjgl} gives: \begin{cor}\label{sfsgdhgjn} Notation as in~\ref{adfdfhgf} and~\ref{asdgfssdtjhgkj}. Assume $K$ is clique and, for every edge $\tau$ in $K_1\setminus P$, the complex $\text{\rm St}(\tau,A)$ is a standard simplex. If, for all simplices $\sigma$ in $(\text{\rm sk}_{n+1}K)\setminus P$, the complex $\text{\rm St}(\sigma,A)$ is non-empty, then the homotopy fibers of the
inclusion $K_X\cup K_Y\hookrightarrow K$ are $n$-connected. \end{cor}
Since clique complexes are determined by their edges, one can wonder if, for such complexes, the conclusions of Corollaries~\ref{asfdfgjfgjk,} and~\ref{afgsfdhfg} would still hold true
if their assumptions are verified only for low dimensional simplices.
Here is an analogue of Corollary~\ref{asfdfgjfgjk,} for clique complexes.
\begin{prop}\label{asfsdhowtreq}
Notation as in~\ref{adfdfhgf} and~\ref{asdgfssdtjhgkj}. Let $n$ be a natural number.
Assume $K$ is clique and that one of the following conditions is satisfied:
\begin{enumerate}
\item There is an $n$-connected simplicial complex $L$ such that, for every edge $\tau$ in $K_1\setminus P$, $\text{\rm St}(\tau,A)=L$.
\item The complex $K_A$ is $n$-connected and, for every edge $\tau$ in $K_1\setminus P$
and every element $v$ in $A$, the set $\tau\cup\{v\}$ is a simplex in $K$.
\end{enumerate}
Then the homotopy fibers of the
inclusion $K_X\cup K_Y\hookrightarrow K$ are $n$-connected.
\end{prop}
\begin{proof}
The assumption (1) together with Proposition~\ref{asfgsdfhjghkhjgl}.(4) implies the assumption (1) of
Corollary~\ref{asfdfgjfgjk,}, proving the proposition in this case.
Let
$\tau$ be an edge in $K_1\setminus P$ and $\mu$ be a simplex in $K_A$. Assume (2). This assumption implies that any two element subset of $\tau\cup\mu$ is a simplex in $K$. Since $K$ is clique, the set
$\tau\cup\mu$ is a simplex in $K$ and consequently $\mu$ is a simplex in $\text{\rm St}(\tau,A)$.
Thus for any $\tau$ in $K_1\setminus P$, there is an inclusion $K_A\subset \text{\rm St}(\tau,A)$,
and hence $K_A= \text{\rm St}(\tau,A)$ for any such $\tau$.
The assumption (2) implies therefore the assumption (1) with $L=K_A$.
\end{proof}
\begin{cor}\label{sasfdsdfhfdhjrtyj}
Notation as in~\ref{adfdfhgf} and~\ref{asdgfssdtjhgkj}.
Assume $K$ is clique and that one of the following conditions is satisfied:
\begin{enumerate}
\item The set $A$ is non empty. Furthermore, for every edge $\tau$ in $K_1\setminus P$ and every subset $\mu$ in $A$ such that $|\mu|\leq 2$, the union $\tau\cup\mu$ is a simplex in $K$.
\item $A=\{v\}$ and, for every edge $\tau$ in $K_1\setminus P$, the set $\tau\cup\{v\}$ is also a simplex in $K$.
\end{enumerate}
Then the
inclusion $K_X\cup K_Y\hookrightarrow K$ is a weak equivalence.
\end{cor}
\begin {proof}
Assume (1). If $K_1\setminus P$ is empty, then so is $K\setminus P$, and hence $P=K$. In this case the corollary follows from Proposition~\ref{asfsdfg}. Assume $K_1\setminus P$ is non-empty. Since any two element subset of $A$ is a simplex in $K$ and $K$ is clique, then all finite non-empty subsets of $A$ belong to $K$ and hence $K_A=\Delta[A]$. In this case the assumption (1)
is a particular case of the condition (2) in Proposition~\ref{asfsdhowtreq} for all $n$ as $\Delta[A]$ is contractible.
Finally note that the assumption (2) is a particular case of (1).
\end{proof}
The following is an analogue of Corollary~\ref{afgsfdhfg} which is also referred to as ``one entry point".
\begin{prop}\label{aSDFGDSFHFHGJKUI}
Notation as in~\ref{adfdfhgf} and~\ref{asdgfssdtjhgkj}.
Assume $K$ is clique and that one of the following conditions is satisfied: \begin{enumerate} \item There is a vertex $v$ in $\bigcap_{\tau\in K_1\setminus P} \text{\rm St}(\tau,A)$ such that, for every edge $\tau$ in $K_1\setminus P$ and every vertex $w$ in $\text{\rm St}(\tau,A)$, $\{v,w\}$ is a simplex in $K$. \item There is a vertex $v$ in $\bigcap_{\tau\in K_1\setminus P} \text{\rm St}(\tau,A)$ such that, for every edge $\tau$ in $K_1\setminus P$, $v$ is a central vertex of $\text{\rm St}(\tau,A)$ (see~\ref{adghxdfgjhkut}). \item There is an element $v$ in $A$ with the following property. For every
simplex $\tau$ in $K$ such that $|\tau\cap (X\setminus A)|=1$, $|\tau\cap (Y\setminus A)|=1$, and
$|\tau\cap A|\leq 1$, the union $\tau\cup\{v\}$ is also a simplex in $K$. \end{enumerate}
Then the
inclusion $K_X\cup K_Y\hookrightarrow K$ is a weak equivalence.
\end{prop}
\begin{proof}
Assume (1). Let $\sigma$ be a simplex in $K\setminus P$. Choose a cover $\sigma=\tau_1\cup\cdots\cup \tau_n$ where $\tau_i$ is an edge in $K_1\setminus P$ for all $i$. Then according to Proposition~\ref{asfgsdfhjghkhjgl}, $\text{\rm St}(\sigma,A)=\bigcap_{i=1}^{n}\text{\rm St}(\tau_i,A)$. Let $w$ be a vertex in $\text{\rm St}(\sigma,A)$. Then it is also a vertex in $\text{\rm St}(\tau_i,A)$ for all $i$. By the assumption $\{v,w\}$ is then a simplex in $K$.
Thus all the 2 element subsets of $\sigma\cup\{v,w\}$ are simplices in $K$ and hence $\{v,w\}$ is a simplex in
$F(\sigma,A)$.
As this happens for all vertices $w$ in $\text{\rm St}(\sigma,A)$, since $\text{\rm St}(\sigma,A)$ is clique, for every simplex $\mu$ in $\text{\rm St}(\sigma,A)$, the set $\mu\cup\{v\}$ is also a simplex in $\text{\rm St}(\sigma,A)$.
The vertex $v$ is therefore central in $\text{\rm St}(\sigma,A)$ and consequently $\text{\rm St}(\sigma,A)$ is contractible. The proposition under assumption (1) follows then from
Corollary~\ref{sdfdfghsd}.(1).
Condition (2) is a particular case of (1).
Assume (3). Let $\tau$ be an edge in $K_1\setminus P$.
Condition (3) applied to the simplex $\tau$ gives that $\tau\cup\{v\}$ is a simplex in $K$ and hence
$v$ is a vertex in $\text{\rm St}(\tau,A)$. Let $w$ be a vertex in $\text{\rm St}(\tau,A)$. Condition (3) applied to the simplex $\tau\cup\{w\}$ gives that $\{v,w\}\subset\tau\cup\{v,w\}$ are simplces in $K$. We can conclude (3) implies (1).
\end{proof}
We finish this section with a statement
referred to as ``two entry points". This has been inspired by~\cite[Theorem 3]{NewHenry}, in which the gluing of two metric graphs along a path is considered. While in that case the two entry points are the endpoints of the path the graphs are glued along, in our framework they have to satisfy one of the listed properties. In both cases however these couple of points determine the weak equivalence stated.
\begin{prop}\label{sdfhdghkyioui}
Notation as in~\ref{adfdfhgf} and~\ref{asdgfssdtjhgkj}. Assume $K$ is clique and there are two elements $a_X$ and $a_Y$ in $A$ with the following properties: \begin{itemize}
\item For every edge $\tau$ in $K_1$ such that $|\tau\cap A|=1$ and $|\tau\cap (X\setminus A)|=1$, the set $\tau\cup\{a_X\}$ is a simplex in $K$.
\item For every edge $\tau$ in $K_1$ such that $|\tau\cap A|=1$ and $|\tau\cap (Y\setminus A)|=1$, the set $\tau\cup\{a_Y\}$ is a simplex in $K$. \item For every edge $\tau$ in $K_1\setminus P$, the set $\tau\cup\{a_X,a_Y\}$ is a simplex in $K$.
Then, for every $\sigma$ in $K\setminus P$, the set $\{a_X,a_Y\}$ is a central simplex (see~\ref{adghxdfgjhkut}) in $\text{\rm St}(\sigma,A)$, and the inclusion $K_X\cup K_Y\subset K$ is a weak equivalence. \end{itemize} \end{prop} \begin{proof} Let $\sigma$ be a simplex in $K\setminus P$. Any vertex $v$ in $\sigma$ is a vertex of an edge $\tau\subset \sigma$ that belongs to $K_1\setminus P$. According to the assumption, the sets
$\{v,a_X,a_Y\}\subset \tau\cup\{a_X,a_Y\}$ are simplices in $K$. This, together with the clique assumption on $K$, imply $\sigma\cup\{a_X,a_Y\}$ is a simplex in $K$. Consequently $\{a_X,a_Y\}$ is a simplex in
$\text{\rm St}(\sigma,A)$.
Let $\mu$ be a simplex in $\text{\rm St}(\sigma,A)$. To prove the proposition, we need to show the set
$\mu\cup\{a_A,a_Y\}$ is a simplex in $F(\sigma,A)$ or equivalently $\sigma\cup \mu\cup\{a_X,a_Y\}$
is a simplex in $K$. Let $x$ be an arbitrary element in $\sigma\cap X$, $y$ an arbitrary element in $\sigma\cap Y$, and $v$ an arbitrary element in $\mu$. The sets $\{x,v\}$, $\{y,v\}$, and $\{x,y\}$ are simplices in $K$. Thus according to the assumptions so are $\{x,v,a_X\}$, $\{y,v,a_Y\}$, and $\{x,y,a_X,a_Y\}$. Consequently the two element sets $\{x,a_X\}$, $\{v,a_X\}$, $\{y,a_X\}$, $\{x,a_Y\}$, $\{v,a_Y\}$, $\{y,a_Y\}$, $\{a_X,a_Y\}$, $\{x,y\}$ are simplices in $K$. Since all the two element subsets of $\sigma\cup\mu\cup\{a_X,a_Y\}$ are of such a form and $K$ is clique,
$\sigma\cup\mu\cup\{a_X,a_Y\}$ is a simplex in $K$. \end{proof}
\section{Vietoris-Rips complexes for distances} Let $Z$ be a subset of the universe $\mathcal{U}$ (see~\ref{aSFDHFN}). A function $d\colon Z\times Z\to [0,\infty]$ is called a {\bf distance} if it is symmetric $d(x,y)=d(y,x)$ and reflexive $d(x,x)=0$ for all $x$ and $y$ in $Z$. A pair $(Z,d)$ is called a distance space. A distance space $(Z,d)$ is sometimes denoted simply by $Z$, if $d$ is understood from the context, or by $d$, if $Z$ is understood from the context.
Let $(Z,d)$ be a distance space. The {\bf diameter} of a non empty and finite subset $\sigma \subset Z$ is by definition
$\text{diam}(\sigma):=\text{max}\{d(x,y)\ |\ x,y\in \sigma\}$.
A subset $X\subset Z$ together with the distance function given by the restriction of $d$ to $X$ is called
a subspace of $(Z,d)$.
Let $(Z,d)$ be a distance space and $r$ be in $[0,\infty)$. By definition, the {\bf Vietoris-Rips} complex $\text{VR}_r(Z)$ consists of these non-empty finite subsets $\sigma\subset Z$ for which $\text{diam}(\sigma)\leq r$ (explicitly: $d(x,y)\leq r$ for all $x$ and $y$ in $\sigma$). Vietoris-Rips complexes are examples of clique complexes (see Section~\ref{dsfgdfhsfghj}).
Let $X$ be a subspace of $(Z,d)$. Then the Vietoris-Rips complex $\text{VR}_r(X)$ coincides with the restriction $\text{VR}_r(Z)_X$ (see~\ref{asfdfhd}).
Our starting assumption in this section is:
\begin{point}[\bf Starting input II]\label{gsgfjyuolyupi} {\em $(Z,d)$ is a distance space, $X\cup Y=Z$ is a cover of $Z$, and $A:=X\cap Y$.} \end{point}
In the rest of this section we are going to reformulate in terms of the distance $d$ on $Z$ some of the statements given in the previous sections regarding the homotopy properties of the inclusion $\text{\rm VR}_r(X)\cup \text{\rm VR}_r(Y) \hookrightarrow \text{\rm VR}_r(Z)$ for various $r$ in $[0,\infty)$. Here is a direct restatement of Proposition~\ref{asfsdhowtreq}:
\begin{prop}\label{asewerwjj}
Notation as in~\ref{gsgfjyuolyupi}. Let $r$ be an element in $[0,\infty)$
and $n$ be a natural number. Assume that one of the following conditions is satisfied: \begin{enumerate} \item There is a subset $L\subset A$ such that $\text{\rm VR}_r(L)$ is $n$-connected and,
for every $x$ in $X\setminus A$ and every $y$ in $Y\setminus A$ with $d(x,y)\leq r$, there is an equality $\{v\in A\ |\ d(x,v)\leq r\text{ and } d(y,v)\leq r\}=L$, in particular the set on the left does not depend on $x$ and $y$. \item The complex $\text{\rm VR}_r(A)$ is $n$-connected and, for all $x$ in $X\setminus A$, $y$ in $Y\setminus A$, and $v$ in $A$, if $d(x,y)\leq r$, then both $d(x,v)\leq r$ and $d(v,y)\leq r$. \end{enumerate} Then the homotopy fibers of the inclusion $\text{\rm VR}_r(X)\cup \text{\rm VR}_r(Y)\subset
\text{\rm VR}_r(Z)$ are $n$-connected. \end{prop}
The assumption (2) of Proposition~\ref{asewerwjj} can be restated as: (connectivity condition) $\text{\rm VR}_r(A)$ is $n$-connected, and (intersection condition) if $\text{VR}_r(Z)_1\setminus P$ (see~\ref{asdgfssdtjhgkj}) is non-empty, then \[A=\bigcap_{\sigma\in \text{VR}_r(Z)_1\setminus P} \text{\rm St}(\sigma,A)_0\] What if the intersection above does not contain all the points of $A$ (the intersection condition is not satisfied)? For example consider $Z=\{x,a_1,a_2,a_3,a_4,y\}$ with the distance function depicted by the following diagram, where the dotted lines indicate distance 2 and the continuous lines indicate distance 1:
\[\begin{tikzcd}
& a_1\ar[dash]{r} \ar[dash]{dd}\ar[dash, dotted]{ddr}& a_2\ar[dash]{dd}\ar[dash, dotted]{ddl}
\\
x \ar[dash]{ur} \ar[dash]{dr} \ar[dash, dotted, bend left=60pt]{urr}
\ar[dash]{rrr}
\ar[dash, bend right=60pt]{drr}& & & y \ar[dash, bend right=60pt]{ull}
\ar[dash, dotted]{ul}\ar[dash]{dl} \ar[dash, bend left=60pt]{dll}
\\
& a_3\ar[dash]{r} & a_4
\end{tikzcd}\] Let $X=\{x,a_1,a_2,a_3,a_4\}$ and $Y=\{a_1,a_2,a_3,a_4,y\}$. Choose $r=1$. In this case $\text{VR}_1(Z)_1\setminus P$ consists of only one edge $\{x,y\}$ and $\text{\rm St}(\{x,y\},A)=\Delta[\{a_1,a_3\}]\cup\Delta[\{a_3,a_4\}]$. Thus the condition (2) of Proposition~\ref{asewerwjj} is not satisfied. However, since the complex $\text{\rm St}(\{x,y\},A)$ is contractible, according to Corollary~\ref{sdfdfghsd}.(1), the inclusion $\text{\rm VR}_1(X)\cup \text{\rm VR}_1(Y)\subset
\text{\rm VR}_1(Z)$ is a weak equivalence.
The assumption (1) of Proposition~\ref{asewerwjj} can be restated as:
(connectivity condition) for every $\tau$ in $\text{VR}_r(Z)_1\setminus P$, the complex $\text{\rm St}(\tau,A)$ is $n$-connected, and
(independence condition) for all pairs of edges $\tau_1$ and $\tau_2$ in $\text{VR}_r(Z)_1\setminus P$,
there is an equality
$\text{\rm St}(\tau_1,A)=\text{\rm St}(\tau_2,A)$. What if the independence condition is not satisfied? For example consider a distance space $Z=\{x_1,x_2,a_1,a_2,a_3,a_4,y\}$
with the distance function depicted by the following diagram, where the dotted lines indicate distance 2 and the continuous lines indicate distance 1:
\[\begin{tikzcd}
x_1 \ar[dash]{r} \ar[dash, bend left=40pt]{rr}
\ar[dash, bend right=30pt]{dd}
\ar[dash,dotted]{ddr} \ar[dash]{rrdd}\ar[dash]{rrrd}&
a_1\ar[dash]{r} \ar[dash]{dd}\ar[dash,dotted]{ddr}&
a_2
\ar[dash]{dd}\ar[dash,dotted]{ddl}
\\
& & & y
\ar[dash]{ul}
\ar[dash]{dl}
\ar[dash]{ulll}
\ar[dash, bend right=60pt]{ull}
\ar[dash, dotted, bend left=60pt]{dll}
\\
x_2 \ar[dash,dotted]{uur}
\ar[dash]{r}
\ar[dash, bend right=40pt]{rr}
\ar[dash]{rruu}
\ar[dash]{rrru}
& a_3
\ar[dash]{r}
& a_4
\end{tikzcd}\]
Let $X=\{x_1,x_2,a_1,a_2,a_3,a_4\}$ and $Y=\{a_1,a_2,a_3,a_4,y\}$. Choose $r=1$.
In this case $\text{VR}_1(Z)_1\setminus P$ consists of two edges $\{x_1,y\}$ and $\{x_2,y\}$.
Note that $\text{\rm St}(\{x_1,y\},A)=\Delta[\{a_1,a_2\}]\cup \Delta[\{a_2,a_4\}]$ and $\text{\rm St}(\{x_2,y\},A)=\Delta[\{a_2,a_4\}]$. Thus the independence condition does not hold in this case. However, since the obstruction complexes $\text{\rm St}(\{x_1,y\},A)$, $\text{\rm St}(\{x_2,y\},A)$ and $\text{\rm St}(\{x_1,x_2,y\},A)= \text{\rm St}(\{x_1,y\},A)\cap \text{\rm St}(\{x_2,y\},A)=\Delta[\{a_2,a_4\}]$ are contractible,
the inclusion $\text{\rm VR}_1(X)\cup \text{\rm VR}_1(Y)\subset
\text{\rm VR}_1(Z)$ is a weak equivalence by Corollary~\ref{sdfdfghsd}.
Consider a relaxation of the intersection condition in the assumption (2) of Proposition~\ref{asewerwjj}. \begin{point}[\bf Assumption I]\label{adgfsdfhsdg} {\em Notation as in~\ref{gsgfjyuolyupi}. Let $r$ be an element in $[0,\infty)$.
There exists an element $v$ in $A$ satisfying the following property.
For all $x$ in $X\setminus A$ and $y$ in $Y\setminus A$, if $d(x,y)\leq r$, then
$d(x,v)\leq r$ and $d(y,v)\leq r$.}
\end{point}
\begin{point}\label{basfasfgdqweter}
The assumption~\ref{adgfsdfhsdg} is equivalent to non-emptiness of the following intersection, where
$n\geq 0$ and the first equality is a consequence of Vietoris-Rips complexes being clique (see Proposition~\ref{asfgsdfhjghkhjgl}): \[\bigcap_{\sigma\in (\text{sk}_{n+1}\text{VR}_r(Z))\setminus P} \text{\rm St}(\sigma,A)=\bigcap_{\sigma\in \text{VR}_r(Z)_1\setminus P} \text{\rm St}(\sigma,A)\not=\emptyset\] According to Proposition~\ref{adgsfgjkk}, Assumption~\ref{adgfsdfhsdg} implies connectedness of the
homotopy fibers of $\text{\rm VR}_r(X)\cup \text{\rm VR}_r(Y)\subset
\text{\rm VR}_r(Z)$. This assumption however does not imply $n$-connectedness of these homotopy fibers. For example consider $Z=\{x,a,b,y\}$ with the distance function depicted by the following diagram where the dotted line indicates distance 2 and the continuous lines indicate distance 1:
\[\begin{tikzcd}
& a\ar[dash]{dl}\ar[dash]{dr}\ar[dash,dotted]{dd}\\
x \ar[dash]{dr}\ar[dash]{rr}& & y\ar[dash]{dl}\\
& b
\end{tikzcd}\]
Let $X=\{x,a,b\}$ and $Y=\{a,b,y\}$. Then $\text{\rm VR}_1(Z)$ is contractible but $\text{\rm VR}_1(X)\cup \text{\rm VR}_1(Y)$
has the homotopy type of a circle. Thus
in this case the homotopy fiber of the inclusion $\text{\rm VR}_r(X)\cup \text{\rm VR}_r(Y)\subset
\text{\rm VR}_r(Z)$ is not $1$-connected.
Note further that the complex $\text{St}(\{x,y\},A)$ consists of two vertices $a$ and $b$ with no edges.
\end{point}
To assure $\text{\rm VR}_r(X)\cup \text{\rm VR}_r(Y) \hookrightarrow \text{\rm VR}_r(Z)$ is a weak equivalence assumption~\ref{adgfsdfhsdg} is not enough and we need additional requirements. For example the following is an analogue of Corollary~\ref{sasfdsdfhfdhjrtyj}.
\begin{prop}\label{afadfhhgj}
Notation as in~\ref{gsgfjyuolyupi}. Assume~\ref{adgfsdfhsdg}. Let $v$ be an element in $A$ given by this assumption. In addition assume that one of the following conditions is satisfied: \begin{enumerate} \item For every $x$ in $X\setminus A$ and $y$ in $Y\setminus A$ such that $d(x,y)\leq r$, if $w$ in $A$ satisfies $d(w,x)\leq r$ and $d(w,y)\leq r$, then $d(v,w)\leq r$. \item {\em $\text{\rm diam}(A)\leq r$.} \item {\em $A=\{v\}$.} \end{enumerate} Then the inclusion $\text{\rm VR}_r(X)\cup \text{\rm VR}_r(Y) \hookrightarrow \text{\rm VR}_r(Z)$ is a weak equivalence. \end{prop} \begin{proof} If Assumption~\ref{adgfsdfhsdg} and the condition 1 hold, then so does the assumption 1 of Proposition~\ref{aSDFGDSFHFHGJKUI}. Furthermore the condition 3 implies 2 and the condition 2 implies 1. Thus this proposition is a consequence of Proposition~\ref{adfdfhgf}. \end{proof}
The intersection condition of the assumption (2) in Proposition~\ref{asewerwjj} requires a choice of a parameter $r$. The following is its universal version where no parameter is required:
\begin{point}[\bf Assumption II]\label{afdsghdfhkte} {\em Notation as in~\ref{gsgfjyuolyupi}.
The set $A$ is non empty and for every $x$ in $X\setminus A$, $y$ in $Y\setminus A$, and $v$ in $A$, the following inequalities hold $d(x,y)\ge d(x,v)$ and $d(x,y)\ge d(y,v)$.} \end{point}
Assumption~\ref{afdsghdfhkte} has an intuitive interpretation in terms of angles when $Z$ is a subspace of the Euclidean space. In such a setting this condition means that every triangle $xvy$ with vertices $x$ in $X\setminus A$, $y$ in $Y\setminus A$ and $v$ in $A$, the angle at $v$ must be at least $60^{\circ}$. We therefore refer to this assumption as the $60^{\circ}$ angle condition.
\begin{prop}\label{asfgsdfgjtoyui}
Notation as in~\ref{gsgfjyuolyupi}. Assume~\ref{afdsghdfhkte}. Assume in addition that,
for every $x$ in $X\setminus A$ and $y$ in $Y\setminus A$, the following inequality holds $d(x, y)\geq \text{\rm diam}(A)$. Then $\text{\rm VR}_r(X)\cup \text{\rm VR}_r(Y) \hookrightarrow \text{\rm VR}_r(Z)$ is a weak equivalence for all $r$ in $[0,\infty)$. \end{prop} \begin{proof} We already know that the proposition holds if $r\geq \text{diam}(A)$. Assume $r< \text{diam}(A)$. We claim that in this case $\text{\rm VR}_r(Z)=P$ (see~\ref{asdgfssdtjhgkj}). If not, there are $x$ in $X\setminus A$ and $y$ in $Y\setminus A$ such that $d(x,y)\leq r$. The assumption would then lead to the following contradictory inequalities $r\geq d(x,y)\geq \text{diam}(A)>r$. Thus in this case $\text{\rm VR}_r(Z)=P$ and the proposition follows from Proposition~\ref{asfsdfg}. \end{proof}
\section{Metric gluings} A distance $d$ on $Z$ is called a {\bf pseudometric} if it satisfies the triangle inequality: $d(x,z)\leq d(x,y)+d(y,z)$
for all $x,y,z$ in $Z$.
\begin{point} \label{sdasfADSFDHGSJ} Notation as in~\ref{gsgfjyuolyupi}. Assume that the distance $d$ on $Z$ is a pseudometric.
Let $x$ be in $X\setminus A$ and $y$ be in $Y\setminus A$. For all $a$ in $A$, by the triangular inequality, $d(x,y)\leq d(x,a)+d(a,y)$, and hence:
\[d(x,y)\leq \text{inf}\{d(x,a)+d(a,y)\ |\ a\in A\}\] The pseudometric space $(Z,d)$ is called {\bf metric gluing} if the above inequality is an equality for all $x$ in $X\setminus A$ and $y$ in $Y\setminus A$.
If $A$ is finite, then the pseudometric $(Z,d)$ is a metric gluing if and only if, for every $x$ in $X\setminus A$ and $y$ in $Y\setminus A$, there is $a$ in $A$ such that $d(x,y)=d(x,a)+d(a,y)$.
If $d_X$ is a pseudometric on $X$ and $d_Y$ is a pseudometric on $Y$ such that $d_X(a,b)=d_Y(a,b)$ for all $a$ and $b$ in $A$, then the following function defines a pseudometric on $Z$ which is a metric gluing: \[d_Z(z,z')= \begin{cases}
d_X(z,z') & \text{ if } z, z' \in X \\
d_Y(z,z') & \text{ if } z, z' \in Y \\
\inf \{d(z,a)+d(z',a)\ |\ a\in A\} & \text{ if } z\in X\setminus A \text{ and } z'\in Y\setminus A
\end{cases} \] \end{point}
If $(Z,d)$ is a metric gluing and $A$ is finite, then, for any edge $\sigma=\{x,y\}$ in $\text{VR}_r(Z)\setminus P$, there is $a$ in $A$ such that $r\geq d(x,y)=d(x,a)+d(a,y)$. Thus in this case the obstruction complex $\text{\rm St}(\tau,A)$ is non-empty as it contains the vertex $a$. To assure contractibility of $\text{\rm St}(\tau,A)$ we need additional assumptions, for example:
\begin{point}[\bf Simplex assumption] \label{SADSDGHDSG} Notation as in~\ref{gsgfjyuolyupi} and~\ref{asdgfssdtjhgkj}. Let $r$ be in $[0,\infty)$. For any vertex $v$ in an edge $\sigma$ in $\text{VR}_r(Z)\setminus P$, if $a$ and $b$ are elements in $A$ such that $d(a,v)\leq r$ and $d(v,b)\leq r$, then $d(a,b)\leq r$. \end{point} The simplex assumption can be reformulated as follows: for any vertex $v$ in a simplex $\sigma$ in $\text{VR}_r(Z)\setminus P$, the complex $\text{\rm St}(v,A)$ is a standard simplex (see~\ref{aSFDHFN}). Since the intersection of standard simplices is again a standard simplex, under Assumption~\ref{SADSDGHDSG},
an obstruction complex $\text{\rm St}(\sigma,A)$, for an arbitrary simplex $\sigma$ in $\text{VR}_r(Z)\setminus P$, is contractible if and only if it is non empty. This, together with the discussion at the end of~\ref{sdasfADSFDHGSJ} and Corollary~\ref{sfsgdhgjn} gives:
\begin{prop}\label{sdasDSFAGHF} Notation as in~\ref{gsgfjyuolyupi} and~\ref{asdgfssdtjhgkj}. Let $r$ be in $[0,\infty)$. Assume $A$ is finite and $(Z,d)$ is a metric gluing that satisfies the simplex assumption~\ref{SADSDGHDSG}. Then, for any edge $\sigma$ in $\text{VR}_r(Z)\setminus P$, the obstruction complex $\text{\rm St}(\sigma, A)$ is contractible. The homotopy fibers of $\text{\rm VR}_r(X)\cup \text{\rm VR}_r(Y) \hookrightarrow \text{\rm VR}_r(Z)$ are connected and this map induces an isomorphism on $\pi_0$ and a surjection on $\pi_1$. \end{prop}
The assumptions of Proposition~\ref{sdasDSFAGHF} are not enough to guarantee the non-emptiness of the obstruction complexes for simplices in $\text{VR}_r(Z)\setminus P$ of dimension 2 and higher.
For example consider $Z=\{x_1,x_2,a_1,a_1,y\}$ with the distance function depicted by the following diagram, where the dotted lines indicate distance 4, the dashed lines indicate distance 3, the squiggly lines indicate distance 2 and the continuous lines indicate distance $1$:
\[\begin{tikzcd}[labels=description]
x_1\ar[dash]{r}{1}\ar[dash, dotted]{ddr}{1}\ar[dash,dashed]{dd}{3}\ar[dash,dashed, bend left=60pt]{drr}{3} & a_1\ar[dash, dashed]{dd}{3}
\arrow[dash,squiggly]{dr}{2}\\
& & y\\
x_2\ar[dash, dotted]{ruu}{4} \ar[dash]{r}{1}\ar[dash, dashed, bend right=60pt]{urr}{3} & a_2\ar[dash,squiggly]{ur}{2}\\
\end{tikzcd}\]
Let $r=3$, $X=\{x_1,x_2,a_1,a_2\}$, $Y=\{y,a_1,a_2\}$, and $A=\{a_1,a_2\}$. Then $Z$ is a metric gluing. Note that $\text{\rm St}(y,A)=\Delta[\{a_1,a_2\}]$, $\text{\rm St}(x_1,A)=\text{\rm St}(\{x_1,y\},A)=\Delta[\{a_1\}]$,
$\text{\rm St}(x_2,A)=\text{\rm St}(\{x_2,y\},A)=\Delta[\{a_2\}]$, and $\text{\rm St}(\{x_1,x_2,y\},A)$ is empty. Furthermore
$A$ and $Y$ have diameter not exceeding $3$. Consequently $\text{VR}_3(Y)$ and $\text{VR}_3(A)$ are contractible. The complex $\text{VR}_3(X)$ has the homotopy type of the circle $S^1$ and so does $ \text{VR}_3(X)\cup \text{VR}_3(Y)$. The entire complex $\text{VR}_3(Z)$ is however contractible. The inclusion $ \text{VR}_3(X)\cup \text{VR}_3(Y)\subset \text{VR}_3(Z)$ induces therefore a surjection on $\pi_1$ but not an isomorphism. This example should be compared with Proposition~\ref{afadfhhgj} under the condition 2.
To assure isomorphism on $\pi_1$, the simplex assumption~\ref{SADSDGHDSG} should be strengthened.
\begin{point}[\bf Strong simplex assumption]\label{sasdfgsdfhgjd} Notation as in~\ref{gsgfjyuolyupi} and~\ref{asdgfssdtjhgkj}. Let $r$ be in $[0,\infty)$. For any vertex $v$ in an edge $\sigma$ in $\text{VR}_r(Z)\setminus P$, if $a$ and $b$ are elements in $A$ such that $d(a,v)\leq r$ and $d(v,b)\leq r$, then $2d(a,b)\leq d(a,v)+d(v,b)$. \end{point}
Note that the strong simplex assumption~\ref{sasdfgsdfhgjd} implies the simplex assumption~\ref{SADSDGHDSG}. \begin{thm}\label{easdfgsdghj} Notation as in~\ref{gsgfjyuolyupi} and~\ref{asdgfssdtjhgkj}. Let $r$ be in $[0,\infty)$. Assume $A$ is finite and $(Z,d)$ is a metric gluing that satisfies the strong simplex assumption~\ref{sasdfgsdfhgjd}. Then, for any simplex $\sigma$ in $\text{\rm VR}_r(Z)\setminus P$
such that either $|\sigma\cap X|=1$ or $|\sigma\cap Y|=1$, the obstruction complex $\text{\rm St}(\sigma,A)$ is contractible. The homotopy fibers of the inclusion $\text{\rm VR}_r(X)\cup \text{\rm VR}_r(Y) \hookrightarrow \text{\rm VR}_r(Z)$ are simply connected and this map induces an isomorphism on $\pi_0$ and $\pi_1$ and a surjection on $\pi_2$. \end{thm} \begin{proof} Since the strong simplex assumption~\ref{sasdfgsdfhgjd} is satisfied, then so is the simplex assumption~\ref{SADSDGHDSG} and consequently any obstruction complex $\text{\rm St}(\sigma,A)$ is a simplex. Thus $\text{\rm St}(\sigma,A)$ is contractible if and only if it is non empty.
We are going to show by induction on the dimension of a simplex a more general statement:
\noindent {\em Under the assumption of Theorem~\ref{easdfgsdghj}, for every simplex $\sigma$ in $\text{\rm VR}_r(Z)\setminus P$ for which
$\sigma \cap X=\{x_1,\dots,x_n\}$ and $\sigma\cap Y=\{y\}$, if $(a_1,\ldots,a_n)$ is a sequence in $A$ such that
$d(x_i,y)=d(x_i,a_i)+d(a_i,y)$ for every $i$, then there is $l$ for which $a_l$ is in $\text{\rm St}(\sigma,A)$ ($d(x_i,a_l)\leq r$ for all $i$).}
If $\sigma=\{x,y\}$ is such an edge, then the statement is clear.
Let $n>1$ and assume that the statement is true for all relevant simplices of dimension smaller than $n$. Let $\sigma$ be in $\text{\rm VR}_r(Z)\setminus P$ be such that
$\sigma \cap X=\{x_1,\dots,x_n\}$ and $\sigma\cap Y=\{y\}$.
Choose a sequence $(a_1,\ldots,a_n)$ in $A$ such that
$d(x_i,y)=d(x_i,a_i)+d(y,a_i)$ for every $i$.
By the inductive assumption, for every $j=1,\ldots, n$, the statement is true for $\tau_j=\sigma_j\setminus\{x_j\}$ and the sequence $(a_1,\ldots,\widehat{a_j},\ldots,a_n)$
obtained from $(a_1,\ldots,a_n)$
by removing its $j$-th element. Thus for every $j=1,\ldots, n$, there is $a_{s(j)}$ such that $s(j)\not= j$ and $d(x_i,a_{s(j)})\leq r$ for all
$i\not=j$. If, for some $j$, $d(x_j,a_{s(j)})\leq r$, then $a_{s(j)}$ would be a vertex in $\text{\rm St}(\sigma,A)$, proving the statement.
Assume $d(x_j,a_{s(j)})> r$ for all $j$. If $j\not=j'$, then $d(x_j, a_{s(j')})\leq r$ and $d(x_j, a_{s(j)})> r$, and hence
$a_{s(j)}\not=a_{s(j')}$. It follows that $s$ is a permutation of the set $\{1,\ldots,n\}$. This together with the strong simplex assumption leads to a contradictory inequality:
\begin{align*} nr & < \sum _{i=1}^n d(x_i,a_{s(i)}) \le \sum _{i=1}^n \left(d(x_i,a_{i})+d(a_i,a_{s(i)})\right)\leq\\ & \le \sum _{i=1}^n \left(d(x_i,a_i)+\frac{1}{2}d(a_i,y)+\frac{1}{2}d(y,a_{s(i)})\right)\le \\ &\le \sum _{i=1}^n \left(d(x_i,a_i)+d(y,a_i)\right) \le nr \end{align*}
Note that, for any simplex $\sigma$ in $\text{\rm sk}_{2}\text{VR}(Z)\setminus P$, either $|\sigma\cap X|=1$ or $|\sigma\cap Y|=1$. Thus, for any such simplex, the obstruction complex $\text{St}(\sigma,A)$ is contractible. We can then use Proposition~\ref{sfsfdgfdhfg} to conclude that the homotopy fibers of the inclusion $\text{\rm VR}_r(X)\cup \text{\rm VR}_r(Y) \hookrightarrow \text{\rm VR}_r(Z)$ are simply connected. \end{proof}
The conclusion of Theorem~\ref{easdfgsdghj} is sharp. Its assumptions are not enough to assure that the homotopy fibers of the map $\text{\rm VR}_r(X)\cup \text{\rm VR}_r(Y) \hookrightarrow \text{\rm VR}_r(Z)$ are $2$ connected. We finish this section with an example illustrating this fact.
\begin{point} Let $Z=\{x_1,x_2,a_{11},a_{12},a_{21}, a_{22},y_1,y_2\}$, $X=\{x_1,x_2,a_{11},\allowbreak a_{12},a_{21},a_{22}\}$ and $Y=\{y_1,y_2,a_{11},a_{12},a_{21},a_{22}\}$. Consider the distance function $d$ on $Z$ described by the following table:
\[\begin{array}{c|c|c|c|c|c|c|c}
&x_2& a_{11}& a_{12} &a_{21}& a_{22}&y_1&y_2\\ \hline x_1 & 6 & 3 & 5 & 7 & 9 &8 & 8 \\ \hline x_2 & 0 & 9 & 7 & 5 & 3 & 8 & 8 \\ \hline a_{11} & & 0 &4 & 4 & 6 & 5 & 7 \\ \hline a_{12} & & & 0 & 6 & 4 & 9 & 3 \\ \hline a_{21} & & & & 0 & 4 & 3 & 9 \\ \hline a_{22} & & & & & 0 & 7 & 5 \\ \hline y_1 & & & & & & 0 & 6\\ \end{array}\] The distance $d$ satisfies the triangular inequality and hence $(Z,d)$ is a metric space. Furthermore $d(x_i,y_j)=d(x_i,a_{ij})+d(a_{ij},y_j)$ for any $i$ and $j$. Thus $(Z,d)$ is a metric gluing of $X$ and $Y$. The metric space $(Z,d)$ can be represented by the following diagram, where the continuous lines or no line indicate distance $8$ or smaller and the dotted lines indicate distance $9$: \[\begin{tikzcd}[labels=description]
& & & a_{11} \ar[dash]{ddd} & & & & y_1\ar[dash, bend right=18pt]{llll}\ar[dash]{ddd}
\ar[dash, bend left=18pt]{dll} \ar[dash,dotted, bend right=25pt]{llllddd}
\\
& & & & & a_{21}\ar[dash]{ddd}
\ar[dash,crossing over,from=ull]
\\
& &\\ x_1\ar[dash,bend left=18pt]{uuurrr}\ar[dash,dotted, bend right=7pt]{drrrrr} \ar[dash,bend right=18pt]{rrr}\ar[dash]{drr} & & & a_{12}\ar[dash]{drr} & & & & y_2\ar[dash, bend right=18pt]{llll}\ar[dash, bend left=15pt]{dll} \ar[dash,dotted, bend right=3pt]{uull} \\
& & x_2\ar[dash,bend left=18pt,crossing over]{uuurrr}\ar[dash,bend right=15pt]{rrr}\ar[dash,dotted, bend left=12pt,crossing over]{uuuur}
& & & a_{22} \end{tikzcd} \] By direct calculation one checks that, for $r=8$ and $Z=X\cup Y$, the metric space $(Z,d)$ satisfies the strong simplex assumption~\ref{sasdfgsdfhgjd}. However, the complex $\text{\rm VR}_8(X)\cup \text{\rm VR}_8(Y)$ is contractible and $ \text{\rm VR}_8(Z)$ is weakly equivaent to $S^3$ ($3$-dimensional sphere).The homotopy fiber of $\text{\rm VR}_8(X)\cup \text{\rm VR}_8(Y)\subset \text{\rm VR}_8(Z)$ is therefore weakly equivalent to the loops space $\Omega S^3$ and hence is not $2$ connected.
Here are steps that one might use to see that $ \text{\rm VR}_8(Z)$ is weakly equivalent to $S^3$. Consider the simplex $\{x_1,x_2\}$ in $ \text{\rm VR}_8(Z)$. According to Corollary~\ref{asfgdsgfjghkkl}, there is a homotopy cofiber sequence: \[\Sigma \text{St}\left(\{x_1,x_2\}, \text{VR}_8(Z\setminus\{x_1,x_2\}\right)\to \text{VR}_8(Z\setminus\{x_1\})\cup \text{VR}_8(Z\setminus\{x_2\})\to \text{VR}_8(Z) \] The complexes in this sequence have the following homotopy types: \begin{itemize} \item $\text{St}\left(\{x_1,x_2\}, \text{VR}_8(Z\setminus\{x_1,x_2\}\right)$ is weakly equivalent to the circle $S^1$; \item $\text{VR}_8(Z\setminus\{x_1\})$, $ \text{VR}_8(Z\setminus\{x_2\})$, $\text{VR}_8(Z\setminus\{x_1.x_2\})$ are contractible; \item the above implies that $\text{VR}_8(Z\setminus\{x_1\})\cup \text{VR}_8(Z\setminus\{x_2\})$ is also contractible; \item we can then use the cofiber sequence above to conclude $\text{VR}_8(Z)$ is weakly equivalent to $\Sigma^2 S^1\simeq S^3$ as claimed. \end{itemize}
\end{point}
\section{Vietoris-Rips of 9 points on a circle} In~\cite{MR3096593,MR3673078,MR3530967} a lot of techniques were introduced aiming at describing homotopy types of certain Vietoris-Rips complexes, particularly for metric graphs built from points on a circle. In this section we showcase how our techniques can be used to describe the homotopy type of one of such examples. Consider a metric space given by $9$ points, $Z=\{z_i\}_i$ with the following distances between them:
\[\begin{array}{c|c|c|c|c|c|c|c|c}
& z_2& z_3 &z_4& z_5&z_6&z_7&z_8&z_9\\\hline z_1 & 1 & 2 & 3 & 4 &4 &3&2&1 \\ \hline z_2 &0&1&2&3&4&4&3&2 \\ \hline z_3 &&0&1&2&3&4&4&3 \\ \hline z_4 &&&0&1&2&3&4&4\\ \hline z_5 &&&&0&1&2&3&4 \\ \hline z_6 &&&&&0&1&2&3 \\ \hline z_7 &&&&&&0&1&2\\ \hline z_8 &&&&&&&0&1\\
\end{array}\] This metric space can be visualised as a metric graph consisting of $9$ points on a circle, where distances between adjacent points are set to be $1$: \[ \begin{tikzpicture}[scale=0.6] \draw [thick,fill=white] (0,0) circle (2.5cm); \foreach \a in {1,2,...,9}{
\node [font=\tiny, fill=white,inner sep=1pt] at (\a*360/9: 2.5cm){$\textstyle{z}_{\a}$};} \draw [thick, shorten <= 0.25cm, shorten >= 0.25cm](1*360/9: 2.5cm) - - (4*360/9: 2.5cm); \draw [ thick, shorten <= 0.25cm, shorten >= 0.25cm] (1*360/9: 2.5cm) - - (7*360/9: 2.5cm); \draw [thick, shorten <= 0.25cm, shorten >= 0.25cm](7*360/9: 2.5cm) - - (4*360/9: 2.5cm); \end{tikzpicture} \] We are going to illustrate how to use our techniques to prove that $\text{VR}_3(Z)$ is weakly equivalent to the wedge $S^2\vee S^2$ of two $2$-dimensional spheres, a result already present in the mentioned work of Adamaszek et al. Set $X:=\{ z_1,z_2,z_4,z_5,z_7,z_8\}$ and $Y:=\{z_1,z_3,z_4,z_6,z_7,z_9\}$. Note that $X\cup Y=Z$ and $A:=X\cap Y=\{z_1,z_4,z_7\}$.
Note that $\text{VR}_3(X\cap Y)$ is contractible, and thus $\text{VR}_3(X)\cup \text{VR}_3(Y)$ is homotopy equivalent to the wedge $\text{VR}_3(X)\lor \text{VR}_3(Y)$. Furthermore we claim that all the obstruction complexes $\text{St}(\sigma,A)$ are contractible for all simplices $\sigma$ in $\text{VR}_3(Z)$ such that $\sigma\cap X\not=\emptyset$, $\sigma\cap Y\not=\emptyset$, and $\sigma\cap X\cap Y=\emptyset$. For example if $\sigma=\{x_2,x_3\}$ then $\text{St}(\sigma,A)=\Delta[x_1,x_7]$ which is contractible. According to Corollary ~\ref{sdfdfghsd}, the inclusion $VR_3(X)\cup VR_3(Y)\subset VR_3(Z)$ is therefore a weak equivalence and consequently $VR_3(Z)$ has the homotopy type of the wedge $\text{VR}_3(X)\lor \text{VR}_3(Y)$. The metric spaces $X$ and $Y$ are isometric, and hence the corresponding Vietoris-Rips complexes are isomorphic. It remains to show that $VR_3(X)$ has the homotopy type of $S^2$. Consider $X'=\{z_1,z_5\}$ and $X''=\{z_2,z_4,z_7,z_8\}$. Note that $X=X'\coprod X''$, $\text{VR}_3(X')$ has the homotopy type of $S^0$ and $\text{VR}_3(X'')$ has the homotopy type of $S^1$. Finally note that, for all simplices $\sigma$ in $\text{VR}_3(X')$ and $\mu$ in $\text{VR}_3(X'')$, the union $\sigma\cup \mu$ is a simplex in $VR_3(X)$. This implies that $VR_3(X)$ is the join of $\text{VR}_3(X')$ and $\text{VR}_3(X'')$ (see Paragraph~\ref{afsaasfagdf}) and hence it is weakly equivalent to $\Sigma (S^0\wedge S^1)\simeq S^2$.
\paragraph{\em Acknowledgments.} A.\ Jin and F.\ Tombari were supported by the Wallenberg AI, Autonomous System and Software Program (WASP) funded by Knut and Alice Wallenberg Foundation. W.\ Chach\'olski was partially supported by VR and WASP. M.\ Scolamiero was partially supported by Brummer \& Partners MathDataLab and WASP.
\end{document} | arXiv |
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Sergi González-Herrero, David Barriopedro, … Marc Oliva
Mika Rantanen ORCID: orcid.org/0000-0003-4279-03221,
Alexey Yu. Karpechko1,
Antti Lipponen ORCID: orcid.org/0000-0002-6902-99742,
Kalle Nordling1,3,
Otto Hyvärinen1,
Kimmo Ruosteenoja1,
Timo Vihma ORCID: orcid.org/0000-0002-6557-70841 &
Ari Laaksonen1,4
Communications Earth & Environment volume 3, Article number: 168 (2022) Cite this article
6746 Altmetric
In recent decades, the warming in the Arctic has been much faster than in the rest of the world, a phenomenon known as Arctic amplification. Numerous studies report that the Arctic is warming either twice, more than twice, or even three times as fast as the globe on average. Here we show, by using several observational datasets which cover the Arctic region, that during the last 43 years the Arctic has been warming nearly four times faster than the globe, which is a higher ratio than generally reported in literature. We compared the observed Arctic amplification ratio with the ratio simulated by state-of-the-art climate models, and found that the observed four-fold warming ratio over 1979–2021 is an extremely rare occasion in the climate model simulations. The observed and simulated amplification ratios are more consistent with each other if calculated over a longer period; however the comparison is obscured by observational uncertainties before 1979. Our results indicate that the recent four-fold Arctic warming ratio is either an extremely unlikely event, or the climate models systematically tend to underestimate the amplification.
The faster warming rate in the Arctic compared to the globe as a whole is nowadays considered a robust fact. The phenomenon, called Arctic or polar amplification (AA), can be seen in both instrumental observations1,2,3 and climate models4 as well as in paleoclimate proxy records5.
During the last decade, multiple factors have been proposed to explain the potential causes of AA: enhanced oceanic heating and ice-albedo feedback due diminishing sea ice6,7,8,9, Planck feedback10, lapse-rate feedback11, near-surface air temperature inversion12, cloud feedback13, ocean heat transport14 and meridional atmospheric moisture transport15,16,17. Furthermore, the reduced air pollution in Europe may have contributed to the Arctic warming during the last decades18,19, and possible reductions of Asian aerosols under a strong mitigation policy may increase the future AA20. In climate models, it has been shown21 that AA occurs rapidly in response to external forcings due to atmospheric lapse rate feedback, with sea ice-related feedbacks becoming more important later on. A recent study22 reported a stronger future AA in a low than a high-emission scenario due to the faster melting of sea ice and weaker ice-albedo feedback.
There is little consensus on the magnitude of the recent AA. Numerous recent studies report the Arctic having warmed either almost twice23, about twice24, or more than twice25,26 as fast as the global average. However, the warming ratios reported in these and many other studies have usually been only referenced from older, possibly outdated, estimates and have not included recent observations. The recent Arctic Monitoring and Assessment Programme (AMAP) report27 states the rate of Arctic warming as being three times as fast as the global warming during the period 1971–2019.
The lack of consensus on the magnitude of AA stems from the various definitions of AA: both the period of interest and the area of the Arctic have been defined in multiple ways. The warming can be calculated using linear trends for the last 30-50 years or even longer periods. Moreover, the area of Arctic can be defined using the area poleward of 60∘N, 65∘N or 70∘N, or using definitions not based on latitude28. Uncertainties arising when calculating AA in observations and models have also been emphasized28,29.
While there have been improvements in climate models to realistically represent the evolution of Arctic climate30,31 and sea ice32 under global warming, most models in the latest generation of Coupled Model Intercomparison Project phase 6 (CMIP6) still fail to simulate plausible sensitivity of Arctic sea-ice loss to the rise of global temperatures33. In earlier studies, the discrepancy between observed and simulated sea ice trends have been attributed to a lower sensitivity of modelled Arctic sea ice trends to global warming34 or anthropogenic CO2 emissions35. However, Swart et al.36 argued that the observed and simulated September Arctic sea-ice trends over 1979–2013 are not inconsistent when accounting properly for the internal climate variability. According to Ding et al.37, even up to 50% of the recent multi-decadal decline in Arctic sea ice may be due to internal variability.
Because the sea ice loss is one of the main mechanisms causing AA, and given that up to 50% of the recent loss may be due to realization-dependent internal variability, a relevant follow-up question is whether the climate models are able to reproduce the magnitude of the observed AA over the past 40 years or so. Earlier studies have suggested that AA is indeed weaker in climate models than in observations38,39,40,41, but a comprehensive comparison between the observed and simulated AA ratio, using the most up-to-date observations and multiple climate model ensembles, has not yet been performed.
The first objective of this study is to quantify the magnitude of AA by utilizing most recent observational datasets covering the Arctic region, and a diagnostic equation for AA. Our focus is in the 1979–2021 period, as more accurate remote sensing observations from the Arctic have been available since 1979, and because this era is characterized by strong Arctic warming. Secondly, we assess the ability of climate models to reproduce the observed AA. We show that during 1979–2021, the Arctic has warmed nearly four times faster than the globe, and provide evidence that climate models struggle to simulate this four-fold Arctic amplification ratio.
Observed arctic amplification
The evolution of global mean and Arctic mean temperatures during 1950–2021 is shown in Fig. 1a by considering the four observational datasets: NASA's Goddard Institute for Space Studies Surface Temperature version 4 (GISTEMP), the Berkeley Earth temperature dataset (BEST), the Met Office Hadley Centre/Climatic Research Unit version 5.0.1.0 (HadCRUT5) and ERA5 reanalysis. Compared to the global temperatures (Fig. 1a, light colours), the warming in the Arctic (Fig. 1a, dark colours) is much more pronounced, especially since the late 1970s. We note also that the different datasets are in a close agreement since 1979, but in the pre-1979 period ERA5 is markedly colder than the three other datasets. Reasons for this cold bias are presumably related to lower number of assimilated observations, as discussed in earlier studies42,43.
Fig. 1: Annual mean temperature evolution in the Arctic.
a Annual mean temperature anomalies in the Arctic (66.5∘–90∘N) (dark colours) and globally (light colours) during 1950–2021 derived from the various observational datasets. Temperature anomalies have been calculated relative to the standard 30-year period of 1981–2010. Shown are also the linear temperature trends for 1979–2021. b Annual mean temperature trends for the period 1979–2021, derived from the average of the observational datasets. Areas without a statistically significant change are masked out. c Local amplification ratio calculated for the period 1979–2021, derived from the average of the observational datasets. The dashed line in (b) and (c) depicts the Arctic Circle (66.5∘N latitude).
Due to the good agreement over the last 43 years, we next consider the average of these four datasets as an observational estimate. The observations indicate that, during 1979–2021, a large fraction of the Arctic Ocean was warming faster than 0.75 ∘C decade−1 (Fig. 1b), with a maximum warming in the Eurasian sector of the Arctic Ocean, near Svalbard and Novaya Zemlya. In this region, the temperature trend over 1979–2021 locally exceeds 1.25 ∘C decade−1 (Fig. 1b). In contrast, large continental regions in the North America and, to a lesser extent, in Western Siberia, do not manifest statistically significant trends in temperatures; however these regions are mainly located in mid-latitudes and are only indirectly affected by AA. The spatial patterns of temperature trends are broadly consistent across the individual observational datasets (Fig. S2), with GISTEMP and HadCRUT5 showing somewhat less pronounced warming maxima near Svalbard and Bering Strait (Fig. S2a and c) than BEST and ERA5.
When the temperature trends shown in Fig. 1b are divided by the multi-dataset global mean temperature trend at each grid-point, we get the spatial map of 43-year local Arctic amplification (AA43), or simply local amplification when calculated for areas south of the Arctic circle (Fig. 1c). Values higher than one indicate that those regions are warming faster than the global average, while values below one correspondingly indicate a slower warming. The AA43 maps for individual observational datasets are provided in the Supplementary Fig. S3.
During 1979–2021, major portions of the Arctic Ocean were warming at least four times as fast as the global average (Fig. 1c). The most extreme AA values occur in the sea areas near Novaya Zemlya, which were locally warming up to seven times as fast as the global average. These high warming rates are consistent with recent research44, and evidently, the primary reason for such a high amplification ratio is the reduction of cold-season ice cover, which has been most pronounced in the Barents Sea44,45. Furthermore, it has been found that changes in atmospheric circulation have amplified the warming in this area46,47. In general, there are no regions within the Arctic Circle where AA43 is smaller than two, apart from the northern North Atlantic.
The observed multi-dataset mean temperature trend in the Arctic is 0.73 ∘C decade−1 and for the globe as a whole 0.19 ∘C decade−1, with small differences between the individual datasets (Fig. S4a). Using Eq. (1) and the multi-dataset mean values for the Arctic and global mean warming trends, we arrive at AA43 (hereafter referred as observed AA43) of 3.8 for the latest 43-year period of 1979–2021. The individual AA43 values range from 3.7 in ERA5 to 4.1 in BEST (Table 1 and Fig. S4b). Thus, referring Arctic warming as being two times as fast as the global mean clearly underestimates the situation during the recent 43 years.
Table 1 Arctic amplification ratio and its 5th and 95th percentiles calculated for the period 1979–2021 from the observational datasets, CMIP5 ensemble, and CMIP6 ensemble.
When different southern boundaries for the Arctic region are considered, AA43 ranges generally between 3 and 4 (Fig. 2a). In general, with any reasonable combination for the length of the time window (≥20 years) and for the southern boundary of the Arctic (60∘–75∘N), the value of AA is greater than 3. The magnitude of AA increases towards higher Arctic latitude thresholds, because with higher latitude a larger proportion of the area encircled by the boundary is ocean, where AA is the strongest (Fig. 1c).
Fig. 2: The sensitivity of Arctic amplification ratio to the Arctic area and the period of interest.
The sensitivity of Arctic amplification (AA) (a) to the time window used in calculating the linear trends (x-axis) and the southern boundary of the Arctic (y-axis), and (b) the percentile rank of the observed AA in the CMIP6 ensemble distribution. The end year of all linear trends is fixed to 2021. Thus, for example, 50 years on the x-axis corresponds to the trend calculated for 1972–2021. The star marks the baseline value used in the study, corresponding to the 43-year linear trend and the southern boundary of 66.5∘N. The observed AA is derived from the average of the four observational datasets.
Arctic amplification in CMIP5 and CMIP6 simulations
Figure 3 demonstrates how the observed AA43 has intensified in the course of time: while in the beginning of the century the warming ratio of the Arctic to the globe over the previous 43 years was smaller than three, the recent amplified Arctic warming48 relative to the globe has raised the ratio close to four.
Fig. 3: The 43-year Arctic amplification ratio in observations and climate models.
The 43-year Arctic amplification (AA) ratio derived from (a) CMIP5, (b) CMIP6, (c) MPI-GE and (d) CanESM5 realizations (thin grey lines) for all 43-year periods ending in 2000–2040. The x-axis represents the ending year of the 43-year AA ratios. Thick black lines represents the ensemble mean AA, calculated as a mean of ratios, not ratio of means. Observations (red lines) extend to 2021. 43-year AA ratios starting after 1970 and ending by 2040 are considered in the probability calculations (Section "Likelihood of observed Arctic amplification 1979–2021 in climate model simulations") and shown with light background. The Arctic is defined as the area north of 66.5∘N.
The observed four-fold warming in the Arctic fits poorly in the spread of the CMIP5 and CMIP6 multi-model ensembles (Fig. 3). Compared with the observed AA43 in 2021 (3.8), the CMIP5 ensemble-mean AA43 (2.5) and CMIP6 ensemble-mean (2.7) are underestimated by 34% and 29%, respectively (Fig. 3a, b and S5b). However, the observed AA43 reflects both the forced response to external forcing as well as the internal climate variability on the multi-decadal timescales considered here. Instead, in the ensemble mean of the climate models, the internal variability of climate system has been effectively averaged out, and thus the ensemble mean reflects only the models' response to the external forcing. For this reason, comparing the observations only to the ensemble mean may be misleading, and therefore, the observed AA43 needs to be put into context of the envelope of simulated AA43.
In the CMIP5 ensemble, there are only a few realizations which simulate stronger amplification than the observations (Fig. 3a). However, the fact that only one realization per model is used in the CMIP5 ensemble may imply that some of the extreme cases are missing. In general, CMIP6 models simulate slightly stronger AA43 than CMIP5 models. Nonetheless, the majority of CMIP6 realizations in which the simulated AA43 is stronger than the observed AA43, occur earlier in the 21st century (Fig. 3b). It is also worth noting that CMIP6 models have generally a larger spread in AA43 than CMIP5, even when considering only one realization per CMIP6 model (not shown). Some CMIP6 realizations simulate cooling for the Arctic (negative AA43) while some other realizations have higher than five-fold warming in the Arctic compared to the globe (Fig. 3b). The large spread in CMIP6-simulated AA is in line with an earlier study39 and highlights the effect of large internal variability for AA, even on a 43-year time scale.
Figure 4 shows AA as a function of the starting year used for calculating the trends. In general, the shorter the time period for which AA is calculated, the larger role the internal variability plays, and therefore the ensemble spread of the models tends to explode towards the right edge of the panels (i.e., towards shorter trends, Fig. 4). Considerable fraction of simulations with negative AA or very large AA (above 6) for trends shorter than 20 years suggest that such short periods may not be suitable for reliable estimation of AA. The long trends, on the other hand, are more representative of the forced amplification and therefore have less spread between the realizations. The observed AA is outside the spread of CMIP5 under wide range of the trend lengths (Fig. 4a). In CMIP6, the spread of AA is clearly wider than in CMIP5 and thus, the observed AA does not fall fully outside the CMIP6 ensemble at any starting year (Fig. 4b). Nevertheless, the observed AA is a very rare occasion in CMIP6 model simulations as Fig. 2b shows that in most cases when AA is calculated over time period longer than 20 years and for southern boundary poleward from 60∘N, the observed AA falls to the top 10% of the CMIP6 model ensemble and reaching to the 99th percentile of the ensemble in several window-latitude combinations. Remarkably, for CMIP5, the observed AA is fully outside the ensemble spread at all 30-45-year trend lengths regardless of southern boundary (Fig. S6b).
Fig. 4: Arctic amplification ratio as a function of the starting year of the trend in observations and climate models.
Arctic amplification (AA) ratio as a function of the starting year of the trend derived from (a) CMIP5, (b) CMIP6, (c) MPI-GE and (d) CanESM5 realizations (thin grey lines). The end year of all linear trends is fixed to 2021. Thus, for example, 1950 on the x-axis corresponds to AA calculated for 1950–2021. Thick black lines represents the ensemble mean AA, calculated as a mean of ratios, not ratio of means. AA ratios shorter than 10 years are not shown. The dashed vertical line represents the 43-year AA, as calculated for 1979–2021. The Arctic is defined as the area north of 66.5∘N.
It is important to note that the discrepancy between the observational and modelled AA is sensitive to the starting year of the trend. For example, when AA is calculated using the 1950–2021 time period, the observed AA is very close to the ensemble means of all climate model datasets (the left edge of the panels in Fig. 4). However, this time period includes a 30-year period of 1950–1979 when the global mean temperatures did not rise mostly due to opposing effect of the anthropogenic aerosols counteracting the greenhouse gas-induced warming49. In 1950–1979, the temperatures in the Arctic were slightly cooling (except in ERA5, see Fig. 1a). Because of this non-linearity in the observations, the linear trend estimate over the whole 1950–2021 does not capture the entire dynamics of the recent warming in the Arctic and thus should be interpreted with caution.
By considering the seasonality of AA (Fig. 5), we see that AA is the strongest in the late autumn (November) and the weakest in the warm season (July). This is consistent in both CMIP6 models and the observations, and in line with the earlier study conducted with ERA-Interim reanalysis data and CMIP5 models8. Thus, over the past 43 years, the October-December months in the Arctic have warmed five times faster than the globe, while the warming ratio is close to two in June-August (Fig. 5). The stronger AA in late autumn arises from the newly opened water areas that act to enhance upwelling longwave radiation and turbulent fluxes of sensible and latent heat from the sea into the atmosphere8.
Fig. 5: Seasonality of the 43-year (1979–2021) Arctic amplification ratio.
The red circles indicate the observed AA, as derived from the average of the four observational datasets. The orange lines indicate the medians of CMIP6 realizations, boxes show the first and third quartiles, and whiskers extend to the 5–95th percentiles of the realizations. The numbers in the upper row give the percent rank of the observed AA in the CMIP6 ensemble distribution. The Arctic is defined as the area north of 66.5∘N.
The observations systematically indicate larger AA than CMIP6 models around the year. In all months, the observed AA43 falls to the upper 25 % of the CMIP6 ensemble, and even to the top 5% in April, May, June, and August. The monthly comparison of observations to CMIP5 models indicate even more pronounced underestimation of AA, especially in the melting season (Fig. S7). An interesting finding from Fig. 5 is the anomalously high observed AA in April. The high AA43 in April is consistent in all four observational datasets (not shown), and has been reported also in the earlier studies41,50. However, while Hahn et al.41 noted that the warming in April falls within the intermodel spread for CMIP6 in 1979–2014, we found that when normalized with global warming, the warming in April is distinctly outside the CMIP6 ensemble (Fig. 5). According to Hahn et al.41, model biases in the reductions of spring snow cover may contribute to the discrepancy between observations and models in the melting season.
Likelihood of observed Arctic amplification 1979–2021 in climate model simulations
How likely is the nearly four-fold warming in the Arctic, as observed in 1979–2021? To answer this question, we investigate all possible AA43 ratios starting after 1970 and ending by 2040 from all four climate model ensembles (see Section "Comparison between simulated and observed Arctic amplification"). While these 43-year periods overlap, and therefore not fully independent, we consider all these periods together because the internal climate variability is not expected to be in phase in models and observations.
In the CMIP5 simulations, there are only three realizations which simulate equally strong AA as observed between 1979 and 2021, even when the longer time window from 1970 to 2040 is allowed. This means that AA greater than or equal to the observed value of 3.8 occurs with a probability of p = 0.006 across the models (Fig. 6a). For CMIP6, we obtain a probability p = 0.028 for the occurrence of AA ≥ 3.8. If only one realization per model were used in CMIP6, the probability is p = 0.015 (Fig. S8). Thus, the CMIP6 models seem to reproduce the observed AA43 marginally better than CMIP5 models, consistent with the higher sensitivity of sea ice loss to cumulative CO2 emissions and global warming in the CMIP6 than in CMIP5 models33. Nevertheless, the likelihood of a fourfold warming in the Arctic in CMIP6 models remains still very small, indicating that the recent Arctic amplification in 1979–2021 is either extremely unlikely or the climate models systematically tend to underestimate AA.
Fig. 6: The probability of observed Arctic amplification in the climate model ensembles.
Frequency distributions of all possible 43-year AA ratios between 1970 and 2040 in (a) CMIP5, (b) CMIP6, (c) MPI-GE, and (d) CanESM5 ensemble. The red line denotes the observed 43-year AA ratio, as calculated for 1979–2021.
The spread of simulated AA in CMIP5 and CMIP6 realizations arises from both internal climate variability and the inter-model spread. To assess the role of internal variability in the AA uncertainty, we next consider the two single-model initial-condition large ensembles (hereafter SMILEs). The individual members of SMILEs are initialized from different initial conditions with identical external forcing; thus the spread in these ensembles is solely due to internal variability51,52. In principle, SMILEs are thus powerful tool to quantify the internal variability of the climate system.
Looking at the spreads of AA43 in SMILEs, we find that they explain a majority of the total CMIP5 and CMIP6 spread, suggesting that the model uncertainty plays a relatively small role in this comparison (Fig. 6). The observed AA43 in 1979–2021 (red line) is fully outside the spread of MPI-GE (Fig. 6c), thus giving a probability p ≈ 0.00. This means explicitly that MPI-GE does not capture the observed Arctic amplification as none of its 100 ensemble members can simulate sufficiently strong AA43 in any 43-year periods between 1970 and 2040.
For CanESM5, AA43≥ 3.8 occurs with a probability of p = 0.054 (Fig. 6d). However, it is known that CanESM5 has a particularly high equilibrium climate sensitivity53, which indicates considerable higher rates of warming both in the Arctic and the globe compared to other models with the same external forcing (Fig. S5 and S9). Thus, while some members of CanESM5 simulate realistic AA43, they do so in a modelled climate which is warming much faster than the real world (Fig. S9d). In addition, the behaviour of simulated AA43 ratios in CanESM5 differ from those in other models: while in CMIP5, CMIP6 and MPI-GE simulations the most extreme AA43 values tend to occur in the beginning of 21st century, in the CanESM5 simulations AA is generally at its lowest in the beginning of the 21st century and intensifies towards 2040 (Fig. 3d).
A robust statistical test (see Section "Comparison between simulated and observed Arctic amplification" and Supplementary Methods) yields further support for the evidence that climate models as a group underestimate the present Arctic amplification. The test has been tailored to properly take into account the two main sources of uncertainty: the internal climate variability and the model uncertainty. The test returns p values of 0.00 for CMIP5 and 0.027 for CMIP6. When the test is further applied for the two SMILEs, we obtain p values of 0.00 and 0.091 for MPI-GE, and CanESM5, respectively. Thus, we can reject the null hypothesis at the 5 % level for CMIP5, CMIP6 and MPI-GE ensembles. This provides additional evidence that climate models, as a group, are not able to reproduce the observed AA43 even when properly accounting for the internal variability and the model uncertainty.
Finally, we acknowledge that the p values calculated with the statistical test are dependent on the starting year of the trends. Here we used the 43-year trends calculated over 1979–2021, which showed large discrepancy (and thus low p values) between the observations and model realizations. The longer trends, which should in principle better reflect the externally forced response of the climate system, show a closer agreement (and evidently higher p values) with the climate models (Fig. 4). However, going back in time from 1979 increases the observational uncertainty, and the resulting linear trends do not fully represent the recent warming period in the Arctic due to a non-linear evolution of the temperature (Fig. 1a).
We present evidence that during 1979–2021 the Arctic has been warming nearly four times as fast as the entire globe. Thus, we caution that referring to Arctic warming as to being twice as fast as the global warming, as frequently stated in literature, is a clear underestimation of the situation during the last 43 years since the start of the satellite observations. At a regional scale, areas in the Eurasian sector of the Arctic Ocean have warmed even up to seven times as fast as the globe (Fig. 1c).
There are two main reasons why our calculation of AA is greater than the earlier estimates generally referenced in literature: (1) the earlier estimates may be outdated due to continued warming in the Arctic (Figs. 1a and 3), and (2) the period of interest and the area of the Arctic can be defined in multiple ways. We used the satellite era (1979–2021) when the remote sensed observations from the Arctic are available, and defined the area of the Arctic using the Arctic Circle as the southern boundary (66.5∘–90∘N). With these parameters, the observed rate of warming in the Arctic is 3.8 times as strong as the global average (Table 1). A more inclusive definition for the Arctic (e.g., >60∘N) would yield smaller ratio (3.2 for 60∘N, see Fig. 2a), but this include more land areas where the sea ice loss feedback is absent.
The advanced interpolation methods in the observational datasets54,55 mean that we now have an improved estimate of the warming trends in areas with scarce data such as the Arctic. However, although the modern observational datasets used in this study produce a spatially comprehensive temperature fields for the Arctic, the low number of observations, particularly from the ice-covered Arctic Ocean, implies that the estimates can have substantial uncertainties. For example, Simmons et al.42 discusses the peculiar cooling, or the lack of warming trend in ERA5 in the area north of Greenland, and link this with questionably low values of the sea ice concentration in ERA5 prior to 1990. This inconsistency in the temperature trends is also visible in our study (Fig. S2d). On the other hand, it has been evaluated that out of five different reanalysis datasets, ERA5 performs the best over the open Arctic ocean56. ERA5 was also found to generally describe well the temporal and spatial characteristics of near-surface temperatures in the Arctic in 1979–201431. Nevertheless, when averaging the temperature trends across the whole Arctic, the inconsistencies in the regional trends appear to cancel out. As a result, the regional averages are surprisingly well aligned (Fig. S4a). Still, we acknowledge the possibility that the observed temperature trends may have common biases for example over the polar ice cap that can affect the magnitude of observed AA, and thus also the model vs. observation differences.
One potential factor increasing the observed AA is the hiatus phase in global warming that occurred between about 1998 and 201257, although the existence of the hiatus has been questioned by a thorough statistical analysis58. Nevertheless, in these years global mean temperature rose more slowly, which acts to reduce the denominator of Eq. (1) for the entire period 1979–2021. According to a previous study59, an important contributing factor to the hiatus was the low sea surface temperature in the equatorial Pacific Ocean. Nevertheless, the impact of tropical Pacific temperature anomalies did not extend to high northern latitudes where warming continued unabatedly (Fig. 1a), keeping the numerator of Eq. (1) large.
Our results demonstrate that climate models as a group tend to underestimate the observed Arctic amplification in the 1979–2021 time period, i.e. since the beginning of the recent period of global warming. This is also true for the latest CMIP6 models despite the fact that some of these models better reproduce the absolute warming rate in the Arctic. However, those models that show plausible Arctic warming trend typically have too much global warming as well when compared to observations. In contrast, those models that simulate global warming close to that observed, generally have too weak Arctic warming (Fig. S9). Thus, our results show that most climate models are unable to simulate a fast-warming Arctic simultaneously with weaker global warming, as found earlier for the relationship of Arctic sea ice decline and global atmospheric warming34. Most strikingly the underestimation was true for the CMIP5 and MPI-GE ensembles, which altogether included only three realizations simulating as high AA as observed in 1979–2021. These results, i.e., lower AA in CMIP5 and CMIP6 models compared to the observations, are consistent with earlier studies38,40,41. Nevertheless, we also found that the discrepancy in AA between climate models and observations is smaller when calculated over longer periods, such as 1950–2021 (Fig. 4).
The physical mechanisms behind the underestimation of AA in climate models remain unknown, but may be related to, e.g., errors in the model sensitivity to greenhouse gas forcing and in the distribution of the forced heating between the atmosphere, cryosphere and the ocean, and in different heights/depths in the atmosphere/ocean. Moreover, internal variability or uncertainties in observations may also contribute to the difference in AA between climate models and observations.
We found that the recent near-surface Arctic amplification ratio is about 40–50% stronger than the multi-model mean amplification derived from CMIP5 and CMIP6 ensembles (Fig. 3 and Table 1). If assuming that these multi-model means represent the externally forced signal for AA, our findings suggest that the unforced climate variability has played a large role in intensifying the recent amplification. This resonates with the results by Ding et al.37 who found that internal climate variability has contributed to about 40–50% of observed multi-decadal decline in Arctic sea ice over 1979–2013. There is also evidence that climate models underestimate the multi-decadal internal variability of the extratropical atmospheric circulation60,61, thus potentially underestimating the temperature variability in the Arctic as well. Nevertheless, if the internal variability indeed proves to be an important source for the difference of AA between the model simulations and observations, one can expect that the observed AA will be reduced in the long term, along with the reduction of the ratio of forced to unforced climate change.
Further, the inability of climate models to simulate realistic AA, here defined in terms of 2-m air temperature, may have implications for future climate projections. Specifically, the tug of war between the near-surface AA and upper-tropospheric tropical amplification of climate warming over the future changes in storm tracks62,63 projected by climate models may be biased towards the forcing by tropical warming, implicating that both projected storm track changes and associated regional climate changes may be biased. Our results call for more detailed investigation of mechanisms behind AA and their representation in climate models.
For the near-surface air temperature, we used three in-situ temperature records and one reanalysis dataset. For in-situ datasets, we used NASA's Goddard Institute for Space Studies Surface Temperature version 4 (GISTEMP)64, the Berkeley Earth temperature dataset (BEST)54, and the Met Office Hadley Centre/Climatic Research Unit version 5.0.1.0 (HadCRUT5)55. In these datasets, near-surface air temperature is based on a combination of 2-m temperature observations over land and sea surface temperature (SST) observations over the ocean.
GISTEMP spatially extrapolates temperatures into unmeasured regions using a 1200-km radius of influence for the stations. BEST employs kriging-based spatial interpolation, and HadCRUT5 uses their own statistical infilling method. In all these datasets, areas of sea ice are treated as if they were land, and SST observations are used and extrapolated only at the grid cells which are ice free. The coverage of sea ice is obtained from Met Office Hadley Centre sea ice and sea surface temperature data set, HadISST265.
In addition to the three purely observational datasets, we used ERA5 reanalysis66, which has been produced by the European Centre for Medium-Range Weather Forecasts. We used monthly mean 2-m temperature fields in the native, 0.25∘ horizontal resolution. The first release of ERA5 covers the years from 1979 to the present, but a preliminary extension for 1950–1978 was recently released43. We used the whole time series, from 1950 to 2021. All the observational temperature datasets used are listed in Table S1.
To assess the accuracy of the four datasets applied in our study (GISTEMP, BEST, HadCRUT5, ERA5) in the Arctic, we conducted a validation against the Global Historical Climatology Network monthly (GHCN-M) station data67. We used the station data which was bias-adjusted for non-climatic effects (indicated by the suffix ".qcf" in the GHCN-M database). We selected all the stations located north of 66.5∘N that had at least 39 years of data over the 43-year period of 1979–2021. In total, these criteria resulted in 87 stations. We calculated the temperature trends for each station, and compared them with the average across the four gridded datasets. These results are shown in Fig. S1. The median difference between the trends estimated from the gridded data and the 87 station observations (gridded minus stations) is −0.019 ∘C decade−1. Therefore, we conclude that the average of the four gridded temperature datasets generally captures well the temporal trends of the near-surface mean temperature in the Arctic, which makes it suitable to be used as a basis of our study.
Climate model data
We compared the observed temperatures to four climate model ensembles, which are listed in Table S2. These ensembles are (i) one realization from each model in the CMIP5 multi-model ensemble68, (ii) all available realizations from each model in the CMIP6 multi-model ensemble69, (iii) the 100-member Max-Planck Institute Grand Ensemble (MPI-GE)70, and (iv) the 50-member Canadian Earth System Model version 5 (CanESM5)53. CanESM5 is a part of CMIP6 ensemble but we examine it separately because the large ensemble size provides an opportunity to highlight the role of internal variability. Thus, in our analysis, CanESM5 was not included in CMIP6. MPI-GE and CanESM5 were chosen as they provide large ensembles for RCP4.5 or SSP2-4.5 emission scenarios and represent opposite sides of the equilibrium climate sensitivity with 2.8 K70 for MPI-GE and 5.6 K for CanESM553. Our key results were not notably affected if CanESM5 was considered as a part of CMIP6.
All four climate model datasets consisted of historical simulations (1950–2005 for CMIP5 and MPI-GE, and 1950–2014 for CMIP6 and CanESM5) and future projections forced by the RCP4.5 scenario for CMIP5 and MPI-GE, and SSP2-4.5 for CMIP6 and CanESM5 (Table S2). As we focus only on the pre-2040 period, our main results do not markedly depend on the choice of the emission scenario. In all climate model datasets, monthly averaged data for 2-m air temperature were used. The list of all the CMIP5 and CMIP6 models used can be found from the supplement Tables S3 and S4.
We acknowledge that the three in-situ temperature records (GISTEMP, BEST and HadCRUT5) do not provide a fully like-for-like comparison to climate models since the in-situ datasets report a blend of land 2-m temperature and SST, whereas the model output is the 2-m air temperature (SAT). According to Cowtan et al.71, the global warming trend derived from the model blended fields are about 7 % lower than the trend from the model SAT fields over the 1975–2014 period. To reduce the potential impacts of this difference, we conducted our analysis also with ERA5 reanalysis data which provides like-for-like comparison to climate models.
Defining the Arctic amplification
We follow the recommendation of Smith et al.72, and define Arctic amplification (AA) as the ratio of Arctic warming to the global-mean warming:
$$AA=\frac{dT/d{t}_{A}}{dT/d{t}_{G}}$$
where dT/dtA and dT/dtG are the slopes of linear trends of near-surface temperature, calculated using a least-squares fitting for the annual and monthly mean values for the Arctic and global domain. The trends were calculated for different time periods (see Fig. 2a), but 43-year AA ratios (hereafter referred to AA43) were chosen to be of the primary interest, because (i) 43 years covers the majority of the recent warming period when the warming has been approximately linear (Fig. 1a), (ii) the reanalysis products, such as ERA5, are known to be more reliable during this period because satellite remote sensing data on atmospheric variables and sea ice concentration have become largely available since 197973, and (iii) there is disagreement between ERA5 and the three in-situ datasets in the Arctic prior to 1979 (see Fig. 1a). Furthermore, the definition of AA naturally only makes physical sense if there is global warming on which Arctic warming is superimposed. Therefore, those modelled AA ratios for which the global warming trend was not significant according to non-parametric Mann-Kendall test74 were neglected.
While different areal definitions for the Arctic exist, we use the area encircled by the Arctic Circle (66.5∘–90∘N) as the primary definition of the Arctic, because this is the area that most scientists consider the Arctic75, and it is one of the definitions used by AMAP76. The fifth assessment report (AR5) of the Intergovernmental Panel on Climate Change defined the Arctic as the region poleward from 67.5∘N77, and AR6 used 67.7∘N as the southern boundary78. For a sensitivity assessment, dT/dtA was also calculated using different definitions for the southern boundary of the Arctic, ranging from 55∘N to 80∘N (Fig. 2a).
Comparison between simulated and observed Arctic amplification
We compare the simulated AA with observations using two approaches. In the first approach, we extract all possible AA43 ratios for the 43-year periods starting from 1970 and ending by 2040 from all four climate model ensembles. Accordingly, there are 29 43-year periods in total, which are overlapping partly with each other (1970–2012, 1971–2013, ..., 1998–2040). The time window of 1970–2040 was chosen to avoid the nearly ice-free climate conditions later in the 21st century, the comparison of which with the currently-observed values would be meaningless. The starting year 1970 reflects approximately the time when the recent period of sustained global warming has started79. All possible 43-year time windows were considered because the internal climate variability in the models is not expected to be in phase with the real climate system. Using all realizations and the 29 different 43-year periods gives us an opportunity to assess in total 11020 simulated AA43 ratios (29 periods x 380 realizations), with a sample of 1044 in CMIP5, 5626 in CMIP6, 2900 in MPI-GE, and 1450 in CanESM5. The probabilities are calculated as the number of simulated AA43 equal to or greater than the observed AA43, divided by the total number of simulated AA43 ratios. For the CMIP6 ensemble, the probability has been calculated first for each model separately, then taking the average across the models. This gives a weight of 1 for each model.
To further assess the robustness of our findings, we utilize an alternative statistical test employed earlier for global warming80 and for sea ice trends36 as a second approach. The details of the test are fully explained in the Supplementary Methods. In the test, we compare the observed AA43 in 1979–2021 to the values of AA43 over the same period simulated by the climate models belonging to the four datasets. The null hypothesis of the test is that the observed and simulated AA ratios are equal, assuming that the models are exchangeable with each other. The differences between the observed and simulated AA ratios have p-values which tell the evidence against the null hypothesis. The smaller the p-value, the stronger is the evidence against the null hypothesis. We use p = 0.05 as a threshold to reject the null hypothesis.
The datafiles for producing the charts and graphs of this manuscript are deposited in the public repository of the Finnish Meteorological Institute at https://doi.org/10.23728/fmi-b2share.5d81ded56e984072a5f7162a18b60cb9. Gistemp data are available from https://data.giss.nasa.gov/gistemp/, Berkeley Earth data from http://berkeleyearth.org/data/, HadCRUT5 data from https://www.metoffice.gov.uk/hadobs/hadcrut5/data/current/download.html, and ERA5 data from https://cds.climate.copernicus.eu. CMIP5 and CMIP6 data are available from Earth System Grid Federation (ESGF) archive at https://esgf-node.llnl.gov/projects/cmip5/ and https://esgf-data.dkrz.de/search/cmip6-dkrz/, respectively. MPI-GE data is available under licence from https://mpimet.mpg.de/en/grand-ensemble/.
Python- and R-language scripts used for this analysis are available from Github: https://github.com/mikarant/arctic-amplification.
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This research has been supported by the ACCC Flagship funded by the Academy of Finland (decision no 337552). The work of MR was supported by the Academy of Finland (contract 342890). The work of TV was supported by the Academy of Finland (contract 317999) and the work of AK by the European Commission H2020 project Polar Regions in the Earth System (PolarRES, grant 101003590). CSC - IT Centre for Science, Finland, is acknowledged for computational resources, and Copernicus Climate Change Service is acknowledged for making ERA5 reanalysis available. We acknowledge the World Climate Research Programme, which is responsible for CMIP5 and CMIP6. We thank the climate modelling groups for making available their model output, the Earth System Grid Federation (ESGF) for archiving the data and providing access, and the multiple funding agencies who support CMIP5, CMIP6, and ESGF. We thank Max Planck Institute for Meteorology for making MPI-GE publicly available.
Finnish Meteorological Institute, Helsinki, Finland
Mika Rantanen, Alexey Yu. Karpechko, Kalle Nordling, Otto Hyvärinen, Kimmo Ruosteenoja, Timo Vihma & Ari Laaksonen
Finnish Meteorological Institute, Kuopio, Finland
Antti Lipponen
CICERO Center for International Climate Research, Oslo, Norway
Kalle Nordling
Department of Applied Physics, University of Eastern Finland, Kuopio, Finland
Ari Laaksonen
Mika Rantanen
Alexey Yu. Karpechko
Otto Hyvärinen
Kimmo Ruosteenoja
Timo Vihma
The study was initialized together by A.L.i., A.La., M.R., and K.N.. M.R. wrote the initial draft of the manuscript, conducted most of the data analysis and created the figures. A.K. contributed to the downloading climate model data and the initial preparation of the manuscript. K.N. and K.R. contributed to the downloading of CMIP model data, and O.H. calculated the uncertainty estimates of the results. T.V. and A.La. contributed to the commenting of the results and revising of the manuscript. All authors commented the manuscript and discussed the results at all stages.
Correspondence to Mika Rantanen.
Communications Earth & Environment thanks Russell Blackport and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Primary Handling Editors: Viviane V. Menezes, Clare Davis, Heike Langenberg. Peer reviewer reports are available.
Peer Review File
Rantanen, M., Karpechko, A.Y., Lipponen, A. et al. The Arctic has warmed nearly four times faster than the globe since 1979. Commun Earth Environ 3, 168 (2022). https://doi.org/10.1038/s43247-022-00498-3
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Guido Grosse
MOSAiC-ACA and AFLUX - Arctic airborne campaigns characterizing the exit area of MOSAiC
Mario Mech
André Ehrlich
Christiane Voigt
Communications Earth & Environment (Commun Earth Environ) ISSN 2662-4435 (online) | CommonCrawl |
Describe the discovery that galaxies getting farther apart as the universe evolves
Explain how to use Hubble's law to determine distances to remote galaxies
Describe models for the nature of an expanding universe
Explain the variation in Hubble's constant
We now come to one of the most important discoveries ever made in astronomy—the fact that the universe is expanding. Before we describe how the discovery was made, we should point out that the first steps in the study of galaxies came at a time when the techniques of spectroscopy were also making great strides. Astronomers using large telescopes could record the spectrum of a faint star or galaxy on photographic plates, guiding their telescopes so they remained pointed to the same object for many hours and collected more light. The resulting spectra of galaxies contained a wealth of information about the composition of the galaxy and the velocities of these great star systems.
Slipher's Pioneering Observations
Curiously, the discovery of the expansion of the universe began with the search for Martians and other solar systems. In 1894, the controversial (and wealthy) astronomer Percival Lowell established an observatory in Flagstaff, Arizona, to study the planets and search for life in the universe. Lowell thought that the spiral nebulae might be solar systems in the process of formation. He therefore asked one of the observatory's young astronomers, Vesto M. Slipher (Figure 1), to photograph the spectra of some of the spiral nebulae to see if their spectral lines might show chemical compositions like those expected for newly forming planets.
Figure 1: Vesto M. Slipher (1875–1969). Slipher spent his entire career at the Lowell Observatory, where he discovered the large radial velocities of galaxies. (credit: Lowell Observatory)
The Lowell Observatory's major instrument was a 24-inch refracting telescope, which was not at all well suited to observations of faint spiral nebulae. With the technology available in those days, photographic plates had to be exposed for 20 to 40 hours to produce a good spectrum (in which the positions of the lines could reveal a galaxy's motion). This often meant continuing to expose the same photograph over several nights. Beginning in 1912, and making heroic efforts over a period of about 20 years, Slipher managed to photograph the spectra of more than 40 of the spiral nebulae (which would all turn out to be galaxies).
To his surprise, the spectral lines of most galaxies showed an astounding redshift. By "redshift" we mean that the lines in the spectra are displaced toward longer wavelengths (toward the red end of the visible spectrum). Recall from the chapter on Radiation and Spectra that a redshift is seen when the source of the waves is moving away from us. Slipher's observations showed that most spirals are racing away at huge speeds; the highest velocity he measured was 1800 kilometers per second.
Only a few spirals—such as the Andromeda and Triangulum Galaxies and M81—all of which are now known to be our close neighbors, turned out to be approaching us. All the other galaxies were moving away. Slipher first announced this discovery in 1914, years before Hubble showed that these objects were other galaxies and before anyone knew how far away they were. No one at the time quite knew what to make of this discovery.
Figure 2: Milton Humason (1891–1972). Humason was Hubble's collaborator on the great task of observing, measuring, and classifying the characteristics of many galaxies. (credit: Caltech Archives)
Hubble's Law
The profound implications of Slipher's work became apparent only during the 1920s. Georges Lemaître was a Belgian priest and a trained astronomer. In 1927, he published a paper in French in an obscure Belgian journal in which he suggested that we live in an expanding universe. The title of the paper (translated into English) is "A Homogenous Universe of Constant Mass and Growing Radius Accounting for the Radial Velocity of Extragalactic Nebulae." Lemaître had discovered that Einstein's equations of relativity were consistent with an expanding universe (as had the Russian scientist Alexander Friedmann independently in 1922). Lemaître then went on to use Slipher's data to support the hypothesis that the universe actually is expanding and to estimate the rate of expansion. Initially, scientists paid little attention to this paper, perhaps because the Belgian journal was not widely available.
In the meantime, Hubble was making observations of galaxies with the 2.5-meter telescope on Mt. Wilson, which was then the world's largest. Hubble carried out the key observations in collaboration with a remarkable man, Milton Humason, who dropped out of school in the eighth grade and began his astronomical career by driving a mule train up the trail on Mount Wilson to the observatory (Figure 2). In those early days, supplies had to be brought up that way; even astronomers hiked up to the mountaintop for their turns at the telescope. Humason became interested in the work of the astronomers and, after marrying the daughter of the observatory's electrician, took a job as janitor there. After a time, he became a night assistant, helping the astronomers run the telescope and record data. Eventually, he made such a mark that he became a full astronomer at the observatory.
By the late 1920s, Humason was collaborating with Hubble by photographing the spectra of faint galaxies with the 2.5-meter telescope. (By then, there was no question that the spiral nebulae were in fact galaxies.) Hubble had found ways to improve the accuracy of the estimates of distances to spiral galaxies, and he was able to measure much fainter and more distant galaxies than Slipher could observe with his much-smaller telescope. When Hubble laid his own distance estimates next to measurements of the recession velocities (the speed with which the galaxies were moving away), he found something stunning: there was a relationship between distance and velocity for galaxies. The more distant the galaxy, the faster it was receding from us.
In 1931, Hubble and Humason jointly published the seminal paper where they compared distances and velocities of remote galaxies moving away from us at speeds as high as 20,000 kilometers per second and were able to show that the recession velocities of galaxies are directly proportional to their distances from us (Figure 3), just as Lemaître had suggested.
Figure 3: Hubble's Law. (a) These data show Hubble's original velocity-distance relation, adapted from his 1929 paper in the Proceedings of the National Academy of Sciences. (b) These data show Hubble and Humason's velocity-distance relation, adapted from their 1931 paper in The Astrophysical Journal. The red dots at the lower left are the points in the diagram in the 1929 paper. Comparison of the two graphs shows how rapidly the determination of galactic distances and redshifts progressed in the 2 years between these publications.
We now know that this relationship holds for every galaxy except a few of the nearest ones. Nearly all of the galaxies that are approaching us turn out to be part of the Milky Way's own group of galaxies, which have their own individual motions, just as birds flying in a group may fly in slightly different directions at slightly different speeds even though the entire flock travels through space together.
Written as a formula, the relationship between velocity and distance is
[latex]V=H\times d[/latex]
where v is the recession speed, d is the distance, and H is a number called the Hubble constant. This equation is now known as Hubble's law.
Constants of Proportionality
Mathematical relationships such as Hubble's law are pretty common in life. To take a simple example, suppose your college or university hires you to call rich alumni and ask for donations. You are paid $2.50 for each call; the more calls you can squeeze in between studying astronomy and other courses, the more money you take home. We can set up a formula that connects p, your pay, and n, the number of calls
[latex]p=A\times n[/latex]
where A is the alumni constant, with a value of $2.50. If you make 20 calls, you will earn $2.50 times 20, or $50.
Suppose your boss forgets to tell you what you will get paid for each call. You can calculate the alumni constant that governs your pay by keeping track of how many calls you make and noting your gross pay each week. If you make 100 calls the first week and are paid $250, you can deduce that the constant is $2.50 (in units of dollars per call). Hubble, of course, had no "boss" to tell him what his constant would be—he had to calculate its value from the measurements of distance and velocity.
Astronomers express the value of Hubble's constant in units that relate to how they measure speed and velocity for galaxies. In this book, we will use kilometers per second per million light-years as that unit. For many years, estimates of the value of the Hubble constant have been in the range of 15 to 30 kilometers per second per million light-years The most recent work appears to be converging on a value near 22 kilometers per second per million light-years If H is 22 kilometers per second per million light-years, a galaxy moves away from us at a speed of 22 kilometers per second for every million light-years of its distance. As an example, a galaxy 100 million light-years away is moving away from us at a speed of 2200 kilometers per second.
Hubble's law tells us something fundamental about the universe. Since all but the nearest galaxies appear to be in motion away from us, with the most distant ones moving the fastest, we must be living in an expanding universe. We will explore the implications of this idea shortly, as well as in the final chapters of this text. For now, we will just say that Hubble's observation underlies all our theories about the origin and evolution of the universe.
Hubble's Law and Distances
The regularity expressed in Hubble's law has a built-in bonus: it gives us a new way to determine the distances to remote galaxies. First, we must reliably establish Hubble's constant by measuring both the distance and the velocity of many galaxies in many directions to be sure Hubble's law is truly a universal property of galaxies. But once we have calculated the value of this constant and are satisfied that it applies everywhere, much more of the universe opens up for distance determination. Basically, if we can obtain a spectrum of a galaxy, we can immediately tell how far away it is.
The procedure works like this. We use the spectrum to measure the speed with which the galaxy is moving away from us. If we then put this speed and the Hubble constant into Hubble's law equation, we can solve for the distance.
Example 1: Hubble's law
Hubble's law (v = H × d) allows us to calculate the distance to any galaxy. Here is how we use it in practice.
We have measured Hubble's constant to be 22 km/s per million light-years. This means that if a galaxy is 1 million light-years farther away, it will move away 22 km/s faster. So, if we find a galaxy that is moving away at 18,000 km/s, what does Hubble's law tells us about the distance to the galaxy?
[latex]d=\frac{v}{H}=\frac{18,000\text{km/s}}{\frac{22\text{km/s}}{1\text{million light-years}}}=\frac{18,000}{22}\times \frac{1\text{million light-years}}{1}=818\text{million light-years}[/latex]
Note how we handled the units here: the km/s in the numerator and denominator cancel, and the factor of million light-years in the denominator of the constant must be divided correctly before we get our distance of 818 million light-years.
Check Your Learning
Using 22 km/s/million light-years for Hubble's constant, what recessional velocity do we expect to find if we observe a galaxy at 500 million light-years?
[latex]v=d\times H=500\text{million light-years}\times \frac{22\text{km/s}}{1\text{million light-years}}=11,000\text{km/s}[/latex]
Variation of Hubble's Constant
The use of redshift is potentially a very important technique for determining distances because as we have seen, most of our methods for determining galaxy distances are limited to approximately the nearest few hundred million light-years (and they have large uncertainties at these distances). The use of Hubble's law as a distance indicator requires only a spectrum of a galaxy and a measurement of the Doppler shift, and with large telescopes and modern spectrographs, spectra can be taken of extremely faint galaxies.
But, as is often the case in science, things are not so simple. This technique works if, and only if, the Hubble constant has been truly constant throughout the entire life of the universe. When we observe galaxies billions of light-years away, we are seeing them as they were billions of years ago. What if the Hubble "constant" was different billions of years ago? Before 1998, astronomers thought that, although the universe is expanding, the expansion should be slowing down, or decelerating, because the overall gravitational pull of all matter in the universe would have a dominant, measureable effect. If the expansion is decelerating, then the Hubble constant should be decreasing over time.
The discovery that type Ia supernovae are standard bulbs gave astronomers the tool they needed to observe extremely distant galaxies and measure the rate of expansion billions of years ago. The results were completely unexpected. It turns out that the expansion of the universe is accelerating over time! What makes this result so astounding is that there is no way that existing physical theories can account for this observation. While a decelerating universe could easily be explained by gravity, there was no force or property in the universe known to astronomers that could account for the acceleration. In The Big Bang chapter, we will look in more detail at the observations that led to this totally unexpected result and explore its implications for the ultimate fate of the universe.
In any case, if the Hubble constant is not really a constant when we look over large spans of space and time, then the calculation of galaxy distances using the Hubble constant won't be accurate. As we shall see in the chapter on The Big Bang, the accurate calculation of distances requires a model for how the Hubble constant has changed over time. The farther away a galaxy is (and the longer ago we are seeing it), the more important it is to include the effects of the change in the Hubble constant. For galaxies within a few billion light-years, however, the assumption that the Hubble constant is indeed constant gives good estimates of distance.
Models for an Expanding Universe
At first, thinking about Hubble's law and being a fan of the work of Copernicus and Harlow Shapley, you might be shocked. Are all the galaxies really moving away from us? Is there, after all, something special about our position in the universe? Worry not; the fact that galaxies are receding from us and that more distant galaxies are moving away more rapidly than nearby ones shows only that the universe is expanding uniformly.
A uniformly expanding universe is one that is expanding at the same rate everywhere. In such a universe, we and all other observers, no matter where they are located, must observe a proportionality between the velocities and distances of equivalently remote galaxies. (Here, we are ignoring the fact that the Hubble constant is not constant over all time, but if at any given time in the evolution of the universe the Hubble constant has the same value everywhere, this argument still works.)
To see why, first imagine a ruler made of stretchable rubber, with the usual lines marked off at each centimeter. Now suppose someone with strong arms grabs each end of the ruler and slowly stretches it so that, say, it doubles in length in 1 minute (Figure 4). Consider an intelligent ant sitting on the mark at 2 centimeters—a point that is not at either end nor in the middle of the ruler. He measures how fast other ants, sitting at the 4-, 7-, and 12-centimeter marks, move away from him as the ruler stretches.
Figure 4: Stretching a Ruler. Ants on a stretching ruler see other ants move away from them. The speed with which another ant moves away is proportional to its distance.
The ant at 4 centimeters, originally 2 centimeters away from our ant, has doubled its distance in 1 minute; it therefore moved away at a speed of 2 centimeters per minute. The ant at the 7-centimeters mark, which was originally 5 centimeters away from our ant, is now 10 centimeters away; it thus had to move at 5 centimeters per minute. The one that started at the 12-centimeters mark, which was 10 centimeters away from the ant doing the counting, is now 20 centimeters away, meaning it must have raced away at a speed of 10 centimeters per minute. Ants at different distances move away at different speeds, and their speeds are proportional to their distances (just as Hubble's law indicates for galaxies). Yet, notice in our example that all the ruler was doing was stretching uniformly. Also, notice that none of the ants were actually moving of their own accord, it was the stretching of the ruler that moved them apart.
Now let's repeat the analysis, but put the intelligent ant on some other mark—say, on 7 or 12 centimeters. We discover that, as long as the ruler stretches uniformly, this ant also finds every other ant moving away at a speed proportional to its distance. In other words, the kind of relationship expressed by Hubble's law can be explained by a uniform stretching of the "world" of the ants. And all the ants in our simple diagram will see the other ants moving away from them as the ruler stretches.
For a three-dimensional analogy, let's look at the loaf of raisin bread in Figure 5. The chef has accidentally put too much yeast in the dough, and when she sets the bread out to rise, it doubles in size during the next hour, causing all the raisins to move farther apart. On the figure, we again pick a representative raisin (that is not at the edge or the center of the loaf) and show the distances from it to several others in the figure (before and after the loaf expands).
Figure 5: Expanding Raisin Bread. As the raisin bread rises, the raisins "see" other raisins moving away. More distant raisins move away faster in a uniformly expanding bread.
Measure the increases in distance and calculate the speeds for yourself on the raisin bread, just like we did for the ruler. You will see that, since each distance doubles during the hour, each raisin moves away from our selected raisin at a speed proportional to its distance. The same is true no matter which raisin you start with.
Our two analogies are useful for clarifying our thinking, but you must not take them literally. On both the ruler and the raisin bread, there are points that are at the end or edge. You can use these to pinpoint the middle of the ruler and the loaf. While our models of the universe have some resemblance to the properties of the ruler and the loaf, the universe has no boundaries, no edges, and no center (all mind-boggling ideas that we will discuss in a later chapter).
What is useful to notice about both the ants and the raisins is that they themselves did not "cause" their motion. It isn't as if the raisins decided to take a trip away from each other and then hopped on a hoverboard to get away. No, in both our analogies, it was the stretching of the medium (the ruler or the bread) that moved the ants or the raisins farther apart. In the same way, we will see in The Big Bang chapter that the galaxies don't have rocket motors propelling them away from each other. Instead, they are passive participants in the expansion of space. As space stretches, the galaxies are carried farther and farther apart much as the ants and the raisins were. (If this notion of the "stretching" of space surprises or bothers you, now would be a good time to review the information about spacetime in Black Holes and Curved Spacetime. We will discuss these ideas further as our discussion broadens from galaxies to the whole universe.)
The expansion of the universe, by the way, does not imply that the individual galaxies and clusters of galaxies themselves are expanding. Neither raisins nor the ants in our analogy grow in size as the loaf expands. Similarly, gravity holds galaxies and clusters of galaxies together, and they get farther away from each other—without themselves changing in size—as the universe expands.
The universe is expanding. Observations show that the spectral lines of distant galaxies are redshifted, and that their recession velocities are proportional to their distances from us, a relationship known as Hubble's law. The rate of recession, called the Hubble constant, is approximately 22 kilometers per second per million light-years. We are not at the center of this expansion: an observer in any other galaxy would see the same pattern of expansion that we do. The expansion described by Hubble's law is best understood as a stretching of space.
Hubble constant: a constant of proportionality in the law relating the velocities of remote galaxies to their distances
Hubble's law: a rule that the radial velocities of remove galaxies are proportional to their distances from us
redshift: when lines in the spectra are displaced toward longer wavelengths (toward the red end of the visible spectrum) | CommonCrawl |
Silverman–Toeplitz theorem
In mathematics, the Silverman–Toeplitz theorem, first proved by Otto Toeplitz, is a result in summability theory characterizing matrix summability methods that are regular. A regular matrix summability method is a matrix transformation of a convergent sequence which preserves the limit.[1]
An infinite matrix $(a_{i,j})_{i,j\in \mathbb {N} }$ with complex-valued entries defines a regular summability method if and only if it satisfies all of the following properties:
${\begin{aligned}&\lim _{i\to \infty }a_{i,j}=0\quad j\in \mathbb {N} &&{\text{(Every column sequence converges to 0.)}}\\[3pt]&\lim _{i\to \infty }\sum _{j=0}^{\infty }a_{i,j}=1&&{\text{(The row sums converge to 1.)}}\\[3pt]&\sup _{i}\sum _{j=0}^{\infty }\vert a_{i,j}\vert <\infty &&{\text{(The absolute row sums are bounded.)}}\end{aligned}}$
An example is Cesaro summation, a matrix summability method with
$a_{mn}={\begin{cases}{\frac {1}{m}}&n\leq m\\0&n>m\end{cases}}={\begin{pmatrix}1&0&0&0&0&\cdots \\{\frac {1}{2}}&{\frac {1}{2}}&0&0&0&\cdots \\{\frac {1}{3}}&{\frac {1}{3}}&{\frac {1}{3}}&0&0&\cdots \\{\frac {1}{4}}&{\frac {1}{4}}&{\frac {1}{4}}&{\frac {1}{4}}&0&\cdots \\{\frac {1}{5}}&{\frac {1}{5}}&{\frac {1}{5}}&{\frac {1}{5}}&{\frac {1}{5}}&\cdots \\\vdots &\vdots &\vdots &\vdots &\vdots &\ddots \\\end{pmatrix}},$
References
Citations
1. Silverman–Toeplitz theorem, by Ruder, Brian, Published 1966, Call number LD2668 .R4 1966 R915, Publisher Kansas State University, Internet Archive
Further reading
• Toeplitz, Otto (1911) "Über allgemeine lineare Mittelbildungen." Prace mat.-fiz., 22, 113–118 (the original paper in German)
• Silverman, Louis Lazarus (1913) "On the definition of the sum of a divergent series." University of Missouri Studies, Math. Series I, 1–96
• Hardy, G. H. (1949), Divergent Series, Oxford: Clarendon Press, 43-48.
• Boos, Johann (2000). Classical and modern methods in summability. New York: Oxford University Press. ISBN 019850165X.
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The Impact of Media Awareness in Controlling the Spread of Infectious Diseases in Terms of SIR Model
Sharmin Sultana Shanta* | Md. Haider Ali Biswas
Mathematics Discipline, Khulna University, Khulna 9208, Bangladesh
Corresponding Author Email:
[email protected]
https://doi.org/10.18280/mmep.070306
| Citation
07.03_06.pdf
A general model based on SIR type has been developed and analyzed in this paper by incorporating a control variable function termed as media awareness. The study emphasizes that consciousness grows among general population due to media awareness which results in isolating a portion of susceptible population from the infected ones. The model exhibits two equilibrium points and the stability of these equilibrium points has been investigated. Numerical simulations have been carried out based on a specific set of parameters to study the effects of applied control. The findings of this analysis reveal that continuous publicity of awareness programs is very effective and significant in preventing the disease transmission whereas the news, collected by media during a disease period, also plays a compassionate role in reducing the size of infectious population. The results of the present analysis also expose that the reproduction number, epidemiologically a certain threshold, is influenced by the transmission and recovery rates. Our study suggests that in absence of effective antivirals or vaccines, media awareness may be one of the supportive interventions for mitigating and controlling the disease burden during any pandemic or epidemic situation.
infectious disease, mathematical model, basic reproduction number, media awareness
Infectious diseases, caused by microorganisms (viruses, fungi, bacteria, parasites and arthropods) which are pathogenic, are very detrimental and life threatening to human health since such diseases can spread indirectly or directly from individuals to individuals, may transmit from contaminated water and food or from animals (zoonotic) or insects [1]. Although the symptoms and signs of infectious diseases may vary or depend on specific types, general indications such as fever, coughing, muscle aches, runny nose, rashes, fatigue and diarrhea are very common for most of the diseases. Diagnosis of contagious diseases, that mainly identifies infectious agents, is done by microbial cultures, microscopy, PCR diagnostics, biochemical test, metagenomic sequence and symptomatic diagnostics [2]. Some of the common infectious diseases are AIDS, Tuberculosis, Hepatitis B, Measles, Chickenpox, Ebola, Influenza, Malaria, Dengue, Chikungunya, Nipah virus infection, MERS, SARS etc. [3]. Recently Coronavirus Disease (COVID-19), caused by a new virus: primarily named as 2019-nCoV; later known as SARS-CoV-2, has been declared as a worldwide pandemic which was initially identified at the city of Wuhan in China on December 31, 2019 when severe pneumonia cases with unidentified explanations were confirmed among a group of individuals and within June 23, 2020, around nine million infective cases with above 469,587 deaths have been confirmed worldwide due to COVID-19 [4]. To slow down or prevent the spread of critical diseases, several characteristics related to the diseases are needed to be understood and recognized [5]. Mathematical modelling in this regard may help predict the forthcoming growth characteristics, project the progress of the disease and choose which interventions should be used as a trial or avoided. Models are typically formulated by considering simple assumptions with specific parameters which are meant for certain diseases and the theoretical methods are applied to understand the special effects of designed interventions, for example, vaccination program [6].
A model on HIV epidemic presented the effects of timely treatment and predicted that early treatment is worthy enough to increase the immunity and considerably lessen new transmissions [7] whereas another work based on testing and treatment revealed that HIV new infections may be minimized to 69.1% within twenty years [8].
A study showed that testing and treatment have the potentiality in minimizing the HIV prevalence to below 1% in next 50 years [9]. A paper focusing on the important issues regarding AIDS epidemic and MDG 2015 investigated that any effective vaccine is necessary since approximately 7400 individuals become infected daily due to HIV. It also presented that the vaccine RV144 was 31% successful during 2009 in preventing HIV infection [10].
A model on tuberculosis established that the disease may be controlled when the efficacy of treatment and vaccination reaches to a convinced threshold [11] and this model was extended later with control schemes so that the intercession cost and the infection burden can be minimized [12]. Another study [13] presented that a particular vaccine which can function both as pre and post exposure is essential to attain the universal control over the deadly TB.
A model representing the influence of reinfection and relapse for Ebola dynamics exposed that the new infective cases will decrease with high level control interventions [14] while another study emphasized that education, quarantine and tracing can significantly decrease the total dimension of Ebola epidemic [15].
An analysis highlighted that Nipah virus infection (NiV) is possible to reduce quickly if quarantined cases are maximized and safety hygiene is maintained [16]. Based on the propagation of NiV [17], it has been anticipated that social distancing and mass awareness may be effective in controlling the situation of NiV [18].
Some works on the dynamics of Dengue [19], Chikungunya [20], Influenza [21], Measles [22], Hepatitis B [23] etc. similarly represent the applications of modelling, from where important insights and strategies were seen to be developed which are sufficiently needed for controlling the spread.
Media as a form of social communication facilitates people by not only sharing information but also providing data to the health administrators during any outbreak situation so that forecasting of the outbreaks can be made possible [24]. A study emphasized that education and media, in Bangladesh, have significance in preventing married people from the deadly HIV. The study also found that the couples who watch television regularly are nearly 8.6 times conscious about HIV than those of who do not watch television [25]. It has been understood that the effects of information transfer play an important role in minimizing the risk of infection [26] while the flow of awareness has the potentiality to decrease the percentage of diseased population [27]. The reports made by mass media for the general public during a pandemic or an epidemic deliver significant information which encourage people to practice healthy and positive behaviors such as maintaining social distance, hand washing etc. These positive practices can reduce the possibility of disease transmission [28]. Activities performed by the media have already shown the potentiality in predicting the evolution and development of infectious diseases. As a result, it is now possible to detect and easily analyze several disease behaviors [29]. Media coverage and its impact on contagious diseases have been explored and investigated through a model by using variable contact rate [30]. Another analysis based on SIS model (which actually motivated us to do this work) focused that awareness campaigns can effectively control the transmission of a spreadable disease but for continuous immigration, the disease may be endemic [31].
In this paper, we introduce a general model, which is based on SIR type [32], to study the possible control and preventive strategy, more specifically, the impact of media awareness programs during the period of any pandemic or epidemic from a common point of view. The purpose of this paper is to analyze the properties of the model, investigate the effects of the publicity rate of awareness programs and the news collection rate which function for making people conscious about the harmful consequences of short or long term infectious diseases.
In Section 2, the framework of the model has been represented whereas the analytical part of the model has been discussed in Section 3. Numerical results are shown graphically in Section 4. Finally, the overall summary with findings is discussed in Section 5.
2. Mathematical Model
The common SIR model deals with three different compartments considering active population which vary in course of time: susceptible population S(t), infectious population I(t) and recovered population R(t). The population in S compartment are always at a high risk of infection, thus anytime they may reach to the infectious stage due to a certain disease by coming contact with the infective individuals. A number of infective individuals may die due to the severity of the disease whereas majority of the population become fully recovered in course of time owing to the immunity and treatment facilities. We consider the presence of influx rate of population and natural death rate to the SIR model which is governed as follows:
$\frac{dS}{dt}=b-(\beta I+\mu )S$
$\frac{dI}{dt}=\beta \,SI-(\mu +\gamma +d)\,I$
$\frac{dR}{dt}=\gamma I-\mu R$ (1)
with S(0)>0, I(0)≥0, R(0)≥0 and N(t)=S(t)+I(t)+R(t).
In model (1), b indicates the influx rate, β denotes the disease transmission rate and $\gamma$ represents the recovery rate. The constants d and μ represent disease induced death rate and natural death rate respectively. We now incorporate a nonnegative control variable compartment: media awareness or M(t) to the model (1) to study the impact of media awareness programs which are covered by media during the disease period. We assume that media generally focuses on several health issues such as washing hands frequently, wearing face mask with protective equipment, avoiding outside food, maintaining safety while travelling, staying home during illness etc. We also assume that media collects information from the infective population with a rate σ (news collection rate) and broadcast the effective programs timely so that the general population may become aware of the disease. Owing to this assumption, a large portion of susceptible population (S) may become conscious by the influence of media awareness (M) and reach to an isolation compartment C(t), termed as conscious population, with a constant rate m (rate of publicity of awareness programs) from where they never come back to the susceptible class. To make our model more realistic, we further assume that sometimes there may be limitation of resources for which exhaustion in M compartment may occur with a rate q. Considering the two new compartments: M(t) and C(t), the previous model (1) is modified (Figure 1) as follows:
$\frac{dS}{dt}=b-(\beta I+mM+\mu )\,S$
$\frac{dR}{dt}=\gamma I-\mu R$
$\frac{dM}{dt}=\sigma I-qM$
$\frac{dC}{dt}=mSM-\mu C$ (2)
with S(0)>0, I(0)≥0, R(0)≥0, M(0)≥0, C(0)≥0 and N(t)=S(t)+I(t)+R(t)+C(t).
Figure 1. Compartmental diagram representing infectious disease dynamics with control policy
3. Model Analysis
The analysis required for model (2) is discussed in this section. We need to observe the boundedness criterion of (2), determine its possible equilibria with basic reproduction number and prove the stability of equilibria.
3.1 Boundedness
Following [7], we obtain from model (2): $\frac{d N}{d t} \leq b-\mu N$ which implies that $N(t) \leq \frac{b}{\mu}+\left(N(0)-\frac{b}{\mu}\right) \exp (-\mu t) .$ When $t \rightarrow \infty, N(t)$ approaches to a certain threshold $\frac{b}{u}$ which suggests that N(t) is actually bounded on [0, t), as a result S, I, R and C are also bounded. Therefore the biological feasible region of model (2) is:
$\Omega =\left\{ (S(t),\,I(t),\,R(t),\,C(t))\in \Re _{+}^{4}\,\,:\,\,S,I,R,C\ge 0\,\,;\,\,0\le N\le \frac{b}{\mu } \right\}$
3.2 Equilibria
Apparently, model (2) possesses two equilibrium points, a disease free equilibrium point $(\mathrm{DFE}): \varepsilon_{1}\left(S^{\circ}, I^{o}, R^{o}, M^{o}, C^{o}\right)$ and an endemic equilibrium point $(\mathrm{EE}): \varepsilon_{2}\left(S^{*}, I^{*}, R^{*}, M^{*}, C^{*}\right) .$
Clearly $\varepsilon_{1}\left(S^{o}, I^{o}, R^{o}, M^{o}, C^{o}\right) \equiv\left(\frac{b}{\mu}, 0,0,0,0\right)$.
In order to determine EE which satisfies the equations:
$\frac{dS}{dt}=0,\frac{dI}{dt}=0,\frac{dR}{dt}=0,\frac{dM}{dt}=0,\,\frac{dC}{dt}=0$
i.e., $b-(\beta I+mM+\mu )\,S=0$,$\beta \,SI-(\mu +\gamma +d)\,I=0$,$\gamma I-\mu R=0$, $\sigma I-qM=0$and $mSM-\mu C=0$ (3)
we solve Eq. (3) and thus obtain $\varepsilon_{2}\left(S^{*}, I^{*}, R^{*}, M^{*}, C^{*}\right)$ where,
${{S}^{*}}=\frac{d+\gamma +\mu }{\beta }$, ${{I}^{*}}=\frac{bq\beta -q{{\mu }^{2}}-dq\mu -q\gamma \mu }{(d+\gamma +\mu )\,(m\sigma +q\beta )}$
${{R}^{*}}=\frac{\gamma \,(bq\beta -q{{\mu }^{2}}-dq\mu -q\gamma \mu )}{\mu \,(d+\gamma +\mu )\,(m\sigma +q\beta )}$, ${{M}^{*}}=\frac{\sigma \,(b\beta -d\mu -\gamma \mu -{{\mu }^{2}})}{(d+\gamma +\mu )\,(m\sigma +q\beta )}$
and ${{C}^{*}}=\frac{m\sigma \,(b\beta -d\mu -\gamma \mu -{{\mu }^{2}})}{\beta \mu \,(m\sigma +q\beta )}$.
3.3 Basic reproduction number ($\mathfrak{R}_{0}$)
A precise method termed as next generation matrix approach [33] is applied in this subsection to obtain the basic reproduction number (${{\Re }_{0}}$) for model (2).
In model (2), I(t) is the only infection component and therefore the new infection matrix is $F=(\,\beta \,{{S}^{o}}\,){{\,}_{1\times 1}}$ and the transfer matrix is $V=(\,d+\gamma +\mu \,){{\,}_{1\times 1}}$. Consequently, the next generation matrix becomes$F{{V}^{-1}}=(\,\beta \,{{S}^{o}}\,)\,\,\left( \frac{1}{d+\gamma +\mu \,} \right)\,$.
Hence, the spectral radius of $F{{V}^{-1}}$, i.e., $\rho \,(F{{V}^{-1}})$ is defined to be the basic reproduction number (${{\Re }_{0}}$) which can be written as: ${{\Re }_{0}}=\frac{\beta \,{{S}^{o}}}{d+\gamma +\mu \,}=\frac{b\beta }{\mu \left( d+\gamma +\mu \right)}$.
3.4 Stability
The Jacobian matrix (J) associated to the model (2) is essentially required to establish the stability [34] of DFE and EE and therefore is defined as follows:
$J=\left( \,\begin{matrix} -(\beta \,I+\mu +mM) & -\beta S & 0 & -mS & 0 \\ \beta \,I & \beta S-\mu -\gamma -d & 0 & 0 & 0 \\ 0 & \gamma & -\mu & 0 & 0 \\ 0 & \sigma & 0 & -q & 0 \\ mM & 0 & 0 & mS & -\mu \\\end{matrix}\, \right)$ (4)
Theorem 1. The DFE: $\varepsilon_{1}\left(S^{o}, I^{o}, R^{o}, M^{o}, C^{o}\right)$ is asymptotically stable when $\Re_{0}<1$ and unstable when ${{\Re }_{0}}>1$.
Proof. At DFE: $\varepsilon_{1}\left(S^{o}, I^{o}, R^{o}, M^{o}, C^{o}\right) \equiv\left(\frac{b}{\mu}, 0,0,0,0\right)$, Eq. (4) becomes
$\begin{align} & J({{\varepsilon }_{1}})=\left( \,\begin{matrix} -(\beta \,{{I}^{o}}+\mu +m{{M}^{o}}) & -\beta {{S}^{o}} & 0 & -m{{S}^{o}} & 0 \\ \beta \,{{I}^{o}} & \beta {{S}^{o}}-\mu -\gamma -d & 0 & 0 & 0 \\ 0 & \gamma & -\mu & 0 & 0 \\ 0 & \sigma & 0 & -q & 0 \\ m{{M}^{o}} & 0 & 0 & m{{S}^{o}} & -\mu \\\end{matrix}\, \right) \\ & \Rightarrow J\,({{\varepsilon }_{1}})=\left( \,\begin{matrix} -\mu & -\beta \frac{b}{\mu } & 0 & -m\frac{b}{\mu } & 0 \\ 0 & \beta \frac{b}{\mu }-\mu -\gamma -d & 0 & 0 & 0 \\ 0 & \gamma & -\mu & 0 & 0 \\ 0 & \sigma & 0 & -q & 0 \\ 0 & 0 & 0 & m\frac{b}{\mu } & -\mu \\\end{matrix}\, \right) \\\end{align}$
Considering λ as the eigen value, the characteristic equation becomes $\left| J\,({{\varepsilon }_{1}})-\lambda \,{{I}_{d}}\, \right|=\,\,0$, where Idis an identity matrix of order 5×5.
Now by rearranging the characteristic equation, we obtain
$\left| \,\begin{matrix} -\mu -\lambda & -\beta \frac{b}{\mu } & 0 & -m\frac{b}{\mu } & 0 \\ 0 & \beta \frac{b}{\mu }-\mu -\gamma -d-\lambda & 0 & 0 & 0 \\ 0 & \gamma & -\mu -\lambda & 0 & 0 \\ 0 & \sigma & 0 & -q-\lambda & 0 \\ 0 & 0 & 0 & m\frac{b}{\mu } & -\mu -\lambda \\\end{matrix}\, \right|=0$ (5)
From Eq. (5), we have
$(\lambda +q)\,(\lambda +\mu )\,(\lambda +\mu )\,(\lambda +\mu )\,\left( \lambda -\frac{b\beta -d\mu -\gamma \mu -{{\mu }^{2}}}{\mu } \right)\,=0$
which implies that $\lambda=-q, \lambda=-\mu, \lambda=-\mu, \lambda=-\mu,$ and $\lambda=\frac{b \beta-d \mu-\gamma \mu-\mu^{2}}{\mu}$.
Clearly, all the eigen values, except $\lambda=\frac{b \beta-d \mu-\gamma \mu-\mu^{2}}{\mu}$, are negative.
Now $\lambda=\frac{b \beta-\mu(d+\gamma+\mu)}{\mu}$ can be written as $\lambda=\frac{\mu(d+\gamma+\mu)\left(\frac{b \beta}{\mu(d+\gamma+\mu)}-1\right)}{\mu}=(d+\gamma+\mu)\left(\Re_{0}-1\right)$, which is negative when $\Re_{0}<1$. Therefore, Theorem 1 holds.
Theorem 2. The EE: $\varepsilon_{2}\left(S^{*}, I^{*}, R^{*}, M^{*}, C^{*}\right)$ is asymptotically stable when $\Re_{0}>1$ and unstable when ${{\Re }_{0}}<1$.
Proof is provided in Appendix A.
4. Numerical Simulations
4.1 Parameters
Since model (2), in general, is designed with a view to understanding the impact of media awareness programs for all infectious diseases (pandemic or epidemic), it will be wise to consider the ongoing fact, COVID-19 pandemic, for parameter collection. Model (2) has eight parameters in total and we have collected influx rate, natural death rate, transmission rate, disease induced death rate and recovery rate from recent studies [35, 36]. It has been observed that a patient who is infected by COVID-19 requires about 15 days on average for his or her complete recovery and therefore γ is set to 1/15 per day. The parameters σ, m and q related to the control function, media awareness, have been assumed for our simulations. The description of all parameters with respective values is provided in Table 1.
Table 1. Description of parameters and respective values
Values (per day)
rate of influx
natural death rate
3.01×10-5
disease transmission rate
3.112×10-8
disease induced death rate
recovery rate from disease
rate of news collection from the infectives
0.0005 – 0.0050
rate of publicity of awareness programs
exhaustion rate of programs for limitation of resources
Figure 2. Dynamics of all compartments for σ=0.0005 and m=0.0002
We use MATLAB, particularly 'ode45' solver, for performing the numerical simulations with the parameter values described in Table 1. Initially, we consider the total population N(0)=9003354 which is subdivided as: S(0)=9003322, I(0)=30, R(0)=2 and C(0)= 0. We set M(0)=0 with σ = 0.0005 and m = 0.0002 as the standard values. With all the initial values of state variables and parameters, the code is run for 120 days. The simulation result is presented in Figure 2 from where the dynamics of all compartments of our designed model is understood. The code is run again for m = 0.0002, 0.0004 and 0.0006 to study the impact of media awareness programs and the results are shown in Figures 3-7. We have focused on the news collection rate (σ) keeping m fixed to 0.0002, increased the values of σ from 0.0005 to 0.0025 and 0.0050 and studied its effects on the control function with infectious and conscious population which are displayed in Figures 8-10. We have also studied the effects of disease transmission rate β and recovery rate γ (Figures 11-13) to understand the variation of $\Re_{0}$ since it is responsible for the disease persistence.
It is evident from Figure 2 that the number of susceptible individuals decreases with time. A portion of susceptible population become conscious owing to the media awareness programs and move to the isolation stage. Another portion of susceptible class become infected due to the severity of disease transmission or infection rate, as a result the infective class size reaches to a maximum peak point within 50 days. The total infectious population around 50 days is seen to be about 1.77% of total population. For the immunity system of human body and proper treatment, infected people gradually recover from the disease or may die and within 120 days, 3.26% of total population become fully recovered of the disease. Due to the continuous growth of control variable, media awareness, the size of conscious population increases upto 96.22% of total population and reaches a threshold after 60 days which continues till the end.
Figure 3. Susceptible population for m=0.0002, 0.0004 and 0.0006
From Figure 3, we see that susceptible population reduce gradually when the publicity rate of awareness programs (m) increases (i.e., m=0.0004 and 0.0006) from its standard value (m = 0.0002). Figure 4 represents the dynamics of infectious class for m=0.0002, 0.0004 and 0.0006. It is observed that infectious population decrease to 48.63% approximately for m = 0.0004 compared to the standard value (m=0.0002) whereas the population size decreases to 65.41% for m=0.0006.
Figure 4. Infectious population for m=0.0002, 0.0004 and 0.0006
Figure 5. Recovered population for m=0.0002, 0.0004 and 0.0006
Figure 6. Dynamics of recovered population compared to the infectious population for m=0.0002, 0.0004 and 0.0006
Figure 5 shows the dynamics of recovered population for m = 0.0002, 0.0004 and 0.0006. Apparently, it seems that the size of the recovered population reduces when the publicity rate of awareness programs is increased. This happens because a portion of susceptible population initially become conscious by virtue of media awareness programs which are implemented to bring consciousness. Thus a percentage of susceptible population isolate themselves from the infectives. The concept may clearly be understood from Figure 6 where both infected and recovered population are shown in order to realize the behaviors of these two compartments combinedly. From Figure 6, it is obvious that for m=0.0002, the total number of recovered population is superior to that of infected population and this statement exactly holds also for m=0.0004 and 0.0006. The dynamics of conscious population have been displayed in Figure 7 to study the effect of the publicity rate (m). We see that for m=0.0004 and 0.0006, the threshold of conscious class increases approximately to 1.91% and 2.56% respectively compared to the previous threshold.
Figure 7. Conscious population for m=0.0002, 0.0004 and 0.0006
Figure 8. Control variable function for σ=0.0005, 0.0025 and 0.0050
Figure 9. Infectious population for σ=0.0005, 0.0025 and 0.0050
Figure 8 describes the control variable function (media awareness) for the news collection rate σ=0.0005, 0.0025 and 0.0050. When σ=0.0005, the control function increases continuously and after 85 or 90 days time period, it reaches its maximum level. With the increase of σ, the function shows an enormous effect and reaches to the maximum level before 80 days. It has been found that the control function increases about 4.29% for σ=0.0025 and 5.35% for σ=0.0050. From Figure 9, we understand that the reduction of infectious group is nearly about 79.04% to 89.39% due to the intensification of news collection rate. Figure 10 represents that conscious population change over time when σ increases. For σ=0.0005, it takes around 60 days to attain the threshold level whereas for 0.0025 and 0.0050, the maximum number of conscious population increases approximately to 3.10% within 55 days and 3.51% within 45 days respectively.
Figure 10. Conscious population for σ=0.0005, 0.0025 and 0.0050
Figure 11. Effects of β and γ on $\Re_{0}$
Figure 12. Change in $\Re_{0}$ for β
Figure 13. Change in $\Re_{0}$ for γ
Figure 11 is the three dimensional representation of $\Re_{0}$ for β and γ whereas Figures 12 and 13 are representing the characteristics of $\Re_{0}$ for β and γ respectively. It has been observed that $\Re_{0}$ increases with the increase of disease transmission rate β whereas $\Re_{0}$ shows decreasing effects when the recovery rate γ is increased. This investigation suggests that besides publicity rate of awareness programs and news collection rate, recovery rate is also responsible for the reduction of infectious population size.
A general model on infectious disease dynamics has been developed in this paper where media awareness is considered as a control variable function. The model is formulated with a purpose to study the impact of media awareness with important factors and its efficacy during any pandemic or epidemic situation. Media generally collects information about the severity of the ongoing disease, emphasizes on several health issues, as a result of which, mass population may become conscious and form a separate isolation group so that they may avoid the contagion. Since increased transmission rate is one of the main reasons for the persistence of a disease, consciousness regarding safety issues and disease characteristics is significantly needed to lessen the adverse scenario. It is observed that continuous publicity of awareness programs brings a tremendous change in human behavior which is very effective and substantial in preventing the disease transmission. Moreover, the news and information collected by media, despite some exhaustion, always reveal a positive impact which works as a pre-recovery option for mitigating the virus transmission. The present work suggests that in order to productively control the disease burden, especially at that time when there is scarcity of effective vaccines or proper treatments, early and continuous implementation of awareness programs may be one of the supportive interventions for any countries to meet the critical challenges against the new emerging diseases.
The first author was supported by the Ministry of Science and Technology, Government of the People's Republic of Bangladesh in her M.Sc. program with the NST Fellowship 2017-18, Grant No. 39.00.0000.012.02.009.17-662, Serial: 217, Date-10.01.2018. This financial support is greatly acknowledged. The authors are also thankful to the anonymous reviewers and the editor for their constructive comments and suggestions which have significantly improved the quality of the paper.
Proof of Theorem 2. We apply numerical technique to establish the theorem.
At EE: $\varepsilon_{2}\left(S^{*}, I^{*}, R^{*}, M^{*}, C^{*}\right)$, Eq. (4) becomes
$J({{\varepsilon }_{2}})=\left( \,\begin{matrix} -(\beta \,{{I}^{*}}+\mu +m{{M}^{*}}) & -\beta {{S}^{*}} & 0 & -m{{S}^{*}} & 0 \\ \beta \,{{I}^{*}} & \beta {{S}^{*}}-\mu -\gamma -d & 0 & 0 & 0 \\ 0 & \gamma & -\mu & 0 & 0 \\ 0 & \sigma & 0 & -q & 0 \\ m{{M}^{*}} & 0 & 0 & m{{S}^{*}} & -\mu \\\end{matrix}\, \right)$ (A.1)
${{S}^{*}}=\frac{d+\gamma +\mu }{\beta }$,${{I}^{*}}=\frac{bq\beta -q{{\mu }^{2}}-dq\mu -q\gamma \mu }{(d+\gamma +\mu )\,(m\sigma +q\beta )}$,${{R}^{*}}=\frac{\gamma \,(bq\beta -q{{\mu }^{2}}-dq\mu -q\gamma \mu )}{\mu \,(d+\gamma +\mu )\,(m\sigma +q\beta )}$ and${{M}^{*}}=\frac{\sigma \,(b\beta -d\mu -\gamma \mu -{{\mu }^{2}})}{(d+\gamma +\mu )\,(m\sigma +q\beta )}$
Considering λ as the eigen value, the characteristic equation is:
$\left| \,J\,({{\varepsilon }_{2}})-\lambda \,{{I}_{d}}\, \right|=\left| \,\,\,\begin{matrix} -(\beta \,{{I}^{*}}+\mu +m{{M}^{*}})-\lambda & -\beta {{S}^{*}} & 0 & -m{{S}^{*}} & 0 \\ \beta \,{{I}^{*}} & \beta {{S}^{*}}-\mu -\gamma -d-\lambda & 0 & 0 & 0 \\ 0 & \gamma & -\mu -\lambda & 0 & 0 \\ 0 & \sigma & 0 & -q-\lambda & 0 \\ m{{M}^{*}} & 0 & 0 & m{{S}^{*}} & -\mu -\lambda \\\end{matrix}\,\, \right|=0$ (A.2)
where, Idis an identity matrix of order 5×5.
Using the values of the parameters from Table 1, i.e., b = 271.23, μ = 3.01×10-5, β = 3.112×10-8, d = 0.01, γ = 1/15, σ = 0.0005, m = 0.0002 and q = 0.00015, we have
S*=2.46455×106, I*=47.0922,
R*=104302, M*=0.392435,
Also from basic reproduction number,
${{\Re }_{0}}=\frac{b\beta }{\mu \left( d+\gamma +\mu \right)}=\text{ 3}\text{.65623}>\text{1}$ (A.3)
Therefore, the simplified form of Eq. (A.2) is:
${{c}_{1}}{{\lambda }^{5}}+{{c}_{2}}{{\lambda }^{4}}+{{c}_{3}}{{\lambda }^{3}}+{{c}_{4}}{{\lambda }^{2}}+{{c}_{5}}\lambda +{{c}_{6}}=0$ (A.4)
c1=1, c2=0.0601703,
c3=0.000010335, c4=3.68385×10-7,
c5=2.215525×10-11 and c6=3.33345×10-16
Solving Eq. (A.4), we obtain
λ =–0.0601, –0.00003, –0.00003, –0.0000049 –0.00247 i and – 0.0000049 + 0.00247 i
It is visible that the first three eigen values are negative whereas the fourth and fifth eigen values are complex conjugate with negative real parts and this occurs due to Eq. (A.3) (i.e., for $\Re_{0}>1$). It can be shown that for $\Re_{0}<1$, all eigen values or real parts of complex eigen values will not be negative.
Therefore, the EE: $\varepsilon_{2}\left(S^{*}, I^{*}, R^{*}, M^{*}, C^{*}\right)$ is asymptotically stable when $\Re_{0}>1$ and unstable for $\Re_{0}<1$.
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Computational and Applied Mathematics (2016-07-01) 35: 519-531 , July 01, 2016
By Ozair, Muhammad; Shi, Xiangyun; Hussain, Takasar
In this paper, we study a vector–host model of pine wilt disease with vital dynamics to determine the equilibria and their stability by considering standard incidence rates and horizontal transmission. The complete global analysis for the equilibria of the model is analyzed. The explicit formula for the reproductive number is obtained, and it is shown that the "disease-free" equilibrium always exists and is globally asymptotically stable whenever $$R_{0}\le 1$$ . Furthermore, the disease persists at an " endemic" level when the reproductive number exceeds unity. It will be very helpful in providing a theoretical basis for the prevention and control of the disease.
Spikes for the Gierer–Meinhardt System with Discontinuous Diffusion Coefficients
Journal of Nonlinear Science (2009-06-01) 19: 301-339 , June 01, 2009
By Wei, Juncheng; Winter, Matthias
We rigorously prove results on spiky patterns for the Gierer–Meinhardt system (Kybernetik (Berlin) 12:30–39, 1972) with a jump discontinuity in the diffusion coefficient of the inhibitor. Using numerical computations in combination with a Turing-type instability analysis, this system has been investigated by Benson, Maini, and Sherratt (Math. Comput. Model. 17:29–34, 1993a; Bull. Math. Biol. 55:365–384, 1993b; IMA J. Math. Appl. Med. Biol. 9:197–213, 1992).
Firstly, we show the existence of an interior spike located away from the jump discontinuity, deriving a necessary condition for the position of the spike. In particular, we show that the spike is located in one-and-only-one of the two subintervals created by the jump discontinuity of the inhibitor diffusivity. This localization principle for a spike is a new effect which does not occur for homogeneous diffusion coefficients. Further, we show that this interior spike is stable.
Secondly, we establish the existence of a spike whose distance from the jump discontinuity is of the same order as its spatial extent. The existence of such a spike near the jump discontinuity is the second new effect presented in this paper.
To derive these new effects in a mathematically rigorous way, we use analytical tools like Liapunov–Schmidt reduction and nonlocal eigenvalue problems which have been developed in our previous work (J. Nonlinear Sci. 11:415–458, 2001).
Finally, we confirm our results by numerical computations for the dynamical behavior of the system. We observe a moving spike which converges to a stationary spike located in the interior of one of the subintervals or near the jump discontinuity.
Short-time existence theory toward stability for nonlinear parabolic systems
Journal of Evolution Equations (2015-06-01) 15: 403-456 , June 01, 2015
By Howard, Peter
We establish existence of classical solutions for nonlinear parabolic systems in divergence form on $${\mathbb{R}^n}$$ , under mild regularity assumptions on coefficients in the problem, and under the assumption of Hölder continuous initial conditions. Our analysis is motivated by the study of stability for stationary and traveling wave solutions arising in such systems. In this setting, large time bounds obtained by pointwise semigroup techniques are often coupled with appropriate short time bounds in order to close an iteration based on Duhamel-type integral equations, and our analysis gives precisely the required short time bounds. This development both clarifies previous applications of this idea (by Zumbrun and Howard) and establishes a general result that covers many additional cases.
Stability of the filter with Poisson observations
Statistical Inference for Stochastic Processes (2015-10-01) 18: 293-313 , October 01, 2015
By Li, Zhiqiang; Xiong, Jie
The short interest rate process is modeled by a diffusion process $$X(t)$$ . With the counting process observations, a filtering problem is formulated and its exponential stability is derived when the process $$X(t)$$ is asymptotically stationary.
Stability of the Equilibrium to the Vlasov-Poisson-Boltzmann System with Non-constant Background Charge
Acta Applicandae Mathematicae (2018-12-01) 158: 107-123 , December 01, 2018
By Yang, Xiuhui; Li, Xiujuan
We study the global existence of classical solution to the Vlasov-Poisson-Boltzmann system with non-constant background charge. In this case the local Maxwellian is the unique stationary state. We show that this equilibrium is nonlinear stable provided that the initial perturbation is sufficient small. Our result solves an open problem stated by Duan and Yang (SIAM J. Math. Anal. 41(6):2353–2387, 2009) in one dimensional case.
Modeling the Dynamics of Infectious Disease Under the Influence of Environmental Pollution
International Journal of Applied and Computational Mathematics (2018-04-26) 4: 1-24 , April 26, 2018
By Kumari, Nitu; Sharma, Sandeep
Environmental pollution is one of the leading causes of mortality across the globe. There are evidences in literature which reflects the fact that regular exposure to environmental pollution leads to reduced immunity in human population. Therefore, we introduce environmental pollution as one of the concepts in understanding the dynamics of infectious disease. We propose a new SIS type epidemic model to study the impact of environmental pollution on the spread of infectious diseases. We divide the susceptible individuals into two compartments out of which one contains the pollution affected individuals. The present study demonstrates that we can not ignore environmental pollution during the study of a disease model. Till date, there are no studies which show the significance and impact of environmental pollution on the spread of infectious diseases. The expression of basic reproduction number is obtained for the proposed model. A detailed dynamical analysis of the model has been performed using the theory of ordinary differential equations, dynamical system and basic reproduction number. Numerical simulations along with sensitivity analysis are performed to support our analytical findings.
Dynamics of a rational difference equation
Chinese Annals of Mathematics, Series B (2009-02-18) 30: 187-198 , February 18, 2009
By Elabbasy, Elmetwally M.; Elsayed, Elsayed M.
The authors investigate the global behavior of the solutions of the difference equation $$ x_{n + 1} = \frac{{ax_{n - l} x_{n - k} }} {{bx_{n - p} + cx_{n - q} }}, n = 0, 1, \cdots , $$ where the initial conditions x−r, x−r+1, x−r+2, …, x0 are arbitrary positive real numbers, r = max{l, k, p, q} is a nonnegative integer and a, b, c are positive constants. Some special cases of this equation are also studied in this paper.
Dynamic formation of oriented patches in chondrocyte cell cultures
Journal of Mathematical Biology (2011-10-01) 63: 757-777 , October 01, 2011
By Grote, Marcus J.; Palumberi, Viviana; Wagner, Barbara; Barbero, Andrea; Martin, Ivan Show all (5)
Growth factors have a significant impact not only on the growth dynamics but also on the phenotype of chondrocytes (Barbero et al. in J. Cell. Phys. 204:830–838, 2005). In particular, as chondrocytes approach confluence, the cells tend to align and form coherent patches. Starting from a mathematical model for fibroblast populations at equilibrium (Mogilner et al. in Physica D 89:346–367, 1996), a dynamic continuum model with logistic growth is developed. Both linear stability analysis and numerical solutions of the time-dependent nonlinear integro-partial differential equation are used to identify the key parameters that lead to pattern formation in the model. The numerical results are compared quantitatively to experimental data by extracting statistical information on orientation, density and patch size through Gabor filters.
Stable equilibria of a singularly perturbed reaction–diffusion equation when the roots of the degenerate equation contact or intersect along a non-smooth hypersurface
By Nascimento, Arnaldo Simal; Sônego, Maicon
We use the variational concept of $${\Gamma}$$ -convergence to prove existence, stability and exhibit the geometric structure of four families of stationary solutions to the singularly perturbed parabolic equation $${u_t=\epsilon^2 {\rm div}(k\nabla u)+f(u,x)}$$ , for $${(t,x)\in \mathbb{R}^+\times\Omega}$$ , where $${\Omega\subset\mathbb{R}^n}$$ , $${n\geq 1}$$ , supplied with no-flux boundary condition. The novelty here lies in the fact that the roots of the bistable function f are not isolated, meaning that the graphs of its roots are allowed to have contact or intersect each other along a Lipschitz-continuous (n − 1)-dimensional hypersurface $${\gamma \subset \Omega}$$ ; across this hypersurface, the stable equilibria may have corners. The case of intersecting roots stems from the phenomenon known as exchange of stability which is characterized by $${f(\cdot,x)}$$ having only two roots.
Evolutionary Stability of Polymorphic Population States in Continuous Games
Dynamic Games and Applications (2018-03-01) 8: 141-156 , March 01, 2018
By Hingu, Dharini; Mallikarjuna Rao, K. S.; Shaiju, A. J.
Asymptotic stability of equilibrium in evolutionary games with continuous action spaces is an important question. Existing results in the literature require that the equilibrium state be monomorphic. In this article, we address this question when the equilibrium is polymorphic. We show that any uninvadable and finitely supported state is asymptotically stable equilibrium of replicator equation.
© 2018 Springer Nature Switzerland AG.
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Kai's thoughts
Thoughts on acoustics, dynamics, structural mechanics, finite element analysis, programming and other stuff
Mass-spring systems should be understood before they are used
Mass-spring systems are often used to model the behavior of an isolated vibrating system.
The mass-spring viewpoint should not be utilized in cases where the system can not be reliably modeled as a mass-spring system. The importance of the static compression of the isolating material, due to gravity, is also overemphasized and often misunderstood.
Some important points:
The mass isolates vibrations by resisting change of momentum. Usually: the more mass, the better. The weight (due to gravity) has very little to do with the isolation, it's the mass that matters!
The spring's main purpose is to carry static forces (for example the weight). It will also try to resist the displacements of the mass, which means that it will transfer forces to the foundations. From the perspective of isolating vibrations, the spring should be as loose as possible, to allow for the mass to vibrate freely.
The mass-spring combination only works on frequencies higher than the resonance frequency.
Ideally, there would be no "spring", but it's impossible in practice.
Simplifying rotating parts
Let's start from the very basic reasoning behind why the mass-spring perspective is so common when designing vibration isolation. Consider case a. We have a machine, with a part rotating around some axis.
Case b shows the varying forces such a movement creates. Perpendicular forces exist relative to the x-axis and y-axis. The varying force in the direction of the x-axis, together with the resisting force at the support, also create a moment of force (kind of like attempting to tip the machine over). The vertical component of the force often creates a moment of force, as well. As does accelerating or decelerating angular motion.
In practical cases, it has been shown that it's often enough to consider the sinusoidal force shown in case c (but it's not always enough!).
Note that this model assumes both the machine and the support to be perfectly rigid. Note also that the force required to keep the rotating mass in bay will rise along with the frequency ($$\omega^2$$)! I think I will return to this in another post.
The equilibrium of the system
So far we have simplified the system to a sinusoidally varying vertical force, acting on a mass on a spring.
We will describe the system using these parameters:
The mass of the machine
A varying force directed on the machine
The varying displacement of the machine
The spring constant, which we'll assume to be constant (linearly elastic)
We now have five different sources for forces:
$$F_{m} = ma = m\ddot y$$, caused by the mass/momentum of the machine which resists acceleration
$$F_{k} = k y$$, caused by the compression of the spring
$$F_{c} = cv = c\dot y$$, caused by various energy losses which are directly proportional to the speed of the displacement
$$F_{e}$$, external forces which we feed into the system.
$$F_{g}$$, the force of gravity pulling down the mass
Using the principle of equilibrium (Newton), the sum of these forces must be 0.
$$F_k + F_c + F_m + F_e + F_g = 0$$
$$F_g$$ can be discarded in our calculations. I'll soon explain why.
Calculating the natural frequency of the system
We know that such a system oscillates naturally at a specific frequency. To calculate this frequency, we assume three things:
The motion is sinusoidal
There are no external forces ($$F_e$$)
The damping factor ($$F_c$$) approaches zero, allowing for the system to oscillate freely
By assuming the motion follows the function $$y(t) = A\sin(\omega t)$$, we get the following equation:
$$F_k + F_m + F_g = k(y(t) + y_0) + m\ddot y(t) + mg = k A\sin(\omega t) – m A \omega^2\sin(\omega t) = 0$$,
where $$y_0$$ is the compression caused by the pull of gravity.
$$k y_0$$ has to be equal to $$m g$$ for the system to balance out (Hooke's law), which means that we can throw the effect of gravity out of the equation. Intuitively this can be explained as following: as the effects of gravity and the static compression of the string ($$k y_0$$) constantly cancel each other out, only the vibrating part of the displacement ($$y(t)$$) makes any difference. The force caused by the vibrating part of the displacement is still directly proportional to the spring constant. Gravity (and as such the static compression of the spring) makes no difference. Gravity only affects the situation if the spring constant changes because of it, not because it compresses the spring or adds a constant downward force to the mass. This holds true for all constant forces, so constant forces should never be included in a vibrating mass-spring system to avoid confusion! They should only be used to calculate the correct spring stiffness, which is almost always very nearly linear when the vibrations are small.
The forces will balance out when $$\omega = \sqrt{\frac{k}{m}}$$, or $$f_0 = \frac{1}{2 \pi}\sqrt{\frac{k}{m}}$$.
Note that this is the combined natural frequency of the spring and the mass! This might seem obvious now, but when we replace the mass and spring constant with something else, it might not be that obvious. The compression of the spring, using Hooke's law, will be $$mg = k\delta x$$. Inserting this into the formula will give $$f_0 = \frac{1}{2 \pi}\sqrt{\frac{g}{\delta x}}$$. This formula is often used for practical purposes, but effectively hides the real parameters the natural frequency is based on!
Calculating the forced response of an undamped system
We will consider the steady state response, meaning the response of the system after being excited at a specific frequency for a long time (when everything balances out). We will assume the mass will move at the same frequency as the force exciting it.
The equilibrium formula shown earlier will now look like this:
$$m \ddot y(t) + k y(t) = F(t)$$
Assuming the machine will be excited according to the function $$F(t) = F\sin(\omega t)$$ (ignoring everything else for now) and the machine moving according to the function $$y(t) = A\sin(\omega t)$$ we get the following formula:
$$-m A \omega^2 sin(\omega t) + k A sin(\omega t) = F_0\sin(\omega t)$$
Solving for A (the amplitude of the motion caused by the force), we get the following formula:
$$A = \frac{F_0}{-m \omega^2 + k}$$
Note that this describes the displacement of the mass. I think the following concept is really important to understand; the force transmitted into the foundations is directly proportional to the displacement. As the spring constant approaches zero (let's imagine the mass floats in space, for example), the system will still vibrate with an amplitude of $$\frac{F_0}{m \omega^2}$$. Increasing the mass will decrease the amplitude of the vibration, also when there is no spring.
To get the force transmitted into the foundations of the machine, we need to multiply the displacement with the spring constant $$k$$ (according to Hooke's law). From this, we get the relation of the transmitted force to the original force $$\frac{F_{tr}}{F_0}$$:
$$\frac{F_{tr}}{F_0} = \bigg\vert\frac{1}{\frac{-m}{k}\omega^2 + 1}\bigg\vert$$
The formula nicely shows us a few things:
The resonant frequency is at $$\omega = \sqrt{\frac{k}{m}}$$, which is the same as earlier
The effect of decreasing the stiffness of the spring
The effect of increasing the mass of the machine
Please note that this has been a significant simplification of the situation. Ideally, the simplification only works for cases where the vibrating force is purely vertical, and acts in the exact direction of the centre of mass of the object.
This entry was posted in Acoustics, Dynamics, Math on February 3, 2013 by .
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September 2014, 34(9): 3485-3510. doi: 10.3934/dcds.2014.34.3485
Dynamics of Klein-Gordon on a compact surface near a homoclinic orbit
Benoît Grébert 1, , Tiphaine Jézéquel 2, and Laurent Thomann 1,
Laboratoire de Mathématiques J. Leray, Université de Nantes, UMR CNRS 6629, 2, rue de la Houssinière, 44322 Nantes Cedex 03, France, France
INRIA & ENS Cachan Bretagne, Avenue Robert Schuman, 35170 Bruz, France
Received March 2013 Revised December 2013 Published March 2014
We consider the Klein-Gordon equation (KG) on a Riemannian surface $M$ $$ \partial^{2}_t u-\Delta u-m^{2}u+u^{2p+1} =0,\quad p\in \mathbb{N}^{*},\quad (t,x)\in \mathbb{R}\times M,$$ which is globally well-posed in the energy space. This equation has a homoclinic orbit to the origin, and in this paper we study the dynamics close to it. Using a strategy from Groves-Schneider, we get the existence of a large family of heteroclinic connections to the center manifold that are close to the homoclinic orbit during all times. We point out that the solutions we construct are not small.
Keywords: wave equation, Klein-Gordon equation, center manifold., homoclinic orbit.
Mathematics Subject Classification: 37K45, 35Q55, 35Bx.
Citation: Benoît Grébert, Tiphaine Jézéquel, Laurent Thomann. Dynamics of Klein-Gordon on a compact surface near a homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3485-3510. doi: 10.3934/dcds.2014.34.3485
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Benoît Grébert Tiphaine Jézéquel Laurent Thomann | CommonCrawl |
\begin{document}
\title{Cowen-Douglas Operator and Shift on Basis}
\author{Juexian Li}
\address{}
\curraddr{School of Mathematics, Liaoning University, Shenyang 110036, People¡¯s Republic of China} \email{[email protected]}
\thanks{This project is supported by the National Natural Science Foundation of China (Grant No. 11371182, 11401283 and 11271150).}
\author{Geng Tian} \address{School of Mathematics, Liaoning University, Shenyang 110036, People¡¯s Republic of China and Department of Mathematics, Texas A M University, College Station, TX 77843} \email{[email protected]}
\author{Yang Cao} \address{School of Mathematics, Jilin University, Changchun 130012, People¡¯s Republic of China} \email{[email protected]}
\subjclass[2000]{Primary 54C40, 14E20; Secondary 46E25, 20C20}
\keywords{Cowen-Douglas operator, Basis theory, Complex bundle}
\begin{abstract} In this paper we show a Cowen-Douglas operator $T \in \mathcal{B}_{n}(\Omega)$ is the adjoint operator of some backward shift on a general basis by choosing nice cross-sections of its complex bundle $E_{T}$. Using the basis theory model, we show that a Cowen-Douglas operator never be a shift on some Markushevicz basis for $n \ge 2$. \end{abstract}
\maketitle
\section{Introduction}
In this paper we try to make a basis theory understanding of Cowen-Douglas operators. Let $\mathcal H$ be a separable, infinite dimensional, complex Hilbert space, and let ${\mathcal L}({\mathcal H})$ denote the algebra of all bounded linear operators on $\mathcal H$. For $T\in {\mathcal L}({\mathcal H})$, $T^*$, $\sigma (T)$ and $r(T)$ denote the adjoint of $T$, the spectrum of $T$ and the spectral radius of $T$, respectively.
For a connected open subset $\Omega $ of the complex plane $\mathbb{C}$ and a positive integer $n$, ${\mathcal B}_n(\Omega )$ denotes the set of operators $T$ in ${\mathcal L}({\mathcal H})$ which satisfy
(1) $\Omega \subset \sigma (T)$;
(2) $\mathrm{ran}(T-z)={\mathcal H}$, $\forall z \in \Omega $;
(3) $\bigvee_{z\in\Omega}{\rm ker}(T-z)=\mathcal H$, and
(4) $\mathrm{dim\,ker}(T-z )=n$, $\forall z \in \Omega $.
\noindent Call an operator in ${\mathcal B}_n(\Omega)$ a Cowen-Douglas operator \cite[Definition 1.2]{Douglas1}.
Clearly, if $T\in {\mathcal B}_n(\Omega )$ then $\Omega \subset \rho _F(T)$ which denotes Fredholm domain and $$ \mathrm{ind}(T-z)=\mathrm{dim\,ker}(T-z)-\mathrm{dim\,ker}(T-z)^*=n \hbox{ for } z \in \Omega. $$
For $T\in{\mathcal B}_n(\Omega)$, the mapping $z\mapsto\ker(T-z)$ defines a rank $n$ Hermitian holomorphic vector bundle, or briefly complex bundle, $E_T$ over $\Omega$. A holomorphic cross-section of the complex bundle $E_T$ is a holomorphic map $\gamma:\Omega\rightarrow\mathcal H$ such that for every $z\in\Omega$, the vector $\gamma(z)$ belongs to the fibre $\ker(T-z)$ of $E_T$. Moreover, for a complex number $z_0$ in $\Omega$, we will also consider a local cross-section of $E_T$ defined on a a neighborhood $\Delta$ of $z_0$.
Originally, Cowen-Douglas operators were introduced as using the method of complex geometry to developing operator theory (see \cite{Douglas1} and \cite{Douglas2}). However, it has been presented recently that they are closely related to the structure of bounded linear operators (see \cite{Jiang1}, \cite{Jiang2}, \cite{Jiang3} and \cite{Jiang4}).
A typical example for a $n$-multiplicity Cowen-Douglas operator is $n$-multiplicity backward shift on an orthonormal basis. A characterization $n$-multiplicity backward operator weighted shifts being Cowen-Douglas operators has ever given in terminology of their weight sequences \cite{Li}. In this paper, we shall show the following.
\begin{theorem}\label{Theorem: CD Operators are shift on C-M sequence} Let $T \in \mathcal{B}_{n}(\Omega)$. Then for every $z_{0} \in \Omega$, there exists a complete and minimal vector sequence $\{f_{k}\}_{k=0}^{\infty}$ in $\mathcal{H}$ such that $T-z_{0}$ is the backward shift on $\{f_{k}\}_{k=0}^{\infty}$. \end{theorem}
Here the word ``complete'' means $\bigvee_{k=0}^\infty\{f_{k}\}=\mathcal{H}$, that is the linear compositions of $\{f_{k}\}_{k=0}^{\infty}$ are dense in $\mathcal{H}$. And, The word ``minimal'' refers to $f_{n} \notin \bigvee_{k \not= n}\{f_{k}\}$.
By Hahn-Banach Theorem, a sequence $\{f_{k}\}_{k=0}^{\infty}$ in $\mathcal{H}$ is minimal if and only if there is a sequence $\{g_{k}\}_{k=0}^{\infty}$ in $\mathcal{H}$ such that $$ (f_i,g_j)=\delta_{ij},~i,j=0,1,2,\cdots, $$ i.e., the pair $(f_k,g_k)$ is a biorthogonal system. If $\{g_{k}\}_{k=0}^{\infty}$ is also total, which means $\{x\in \mathcal{H}:\ (x,g_k)=0,\ \forall k\geq 0\}=\{0\}$ (or equivalently, $\{g_{k}\}_{k=0}^{\infty}$ is complete), then $\{f_{k}\}_{k=0}^{\infty}$ is called a \textit{generalized basis} of $\mathcal{H}$ \cite[Def. 7.1]{Singer2}. For a generalized basis $\{f_{k}\}_{k=0}^{\infty}$, if it is also complete then say it to be a \textit{Markushevich basis} of $\mathcal{H}$ \cite[Def. 8.1]{Singer2}. Obviously, a Schauder basis is always a Markushevich basis.
Use basis theory terminology, we can get a stronger version of theorem \ref{Theorem: CD Operators are shift on C-M sequence}.
\begin{theorem}\label{Theorem: CD operator is an adjoint op of some shift on G basis} For a Cowen-Douglas operator $T \in \mathcal{B}_{n}(\Omega), 0\in \Omega$, its adjoint operator $T^{*}$ is a shift on some generalized basis. \end{theorem}
Theorem \ref{Theorem: CD operator is an adjoint op of some shift on G basis} tell us that a Cowen-Douglas operator always is an adjoint of some shift on a generalized basis when its spectrum contains $0$.
Weighted shifts on a a generalized basis (or a Markushevich basis) of Banach space have ever investigated in \cite{Grabiner}. The following result shows what happen for the operators in $\mathcal{B}_{n}(\Omega)$ in the case $n\ge 2$.
\begin{theorem}\label{Theorem: A Cowen-Douglas oper never be a shift on basis for n ge 2} Let $T \in{\mathcal L}({\mathcal H})$. If there is a complex number $z_0$ such that $\mathrm{dim\,ker}(T-z_0)\geq 2$ (in particular, if $T \in \mathcal{B}_{n}(\Omega)$ and $n\ge 2$) then $T$ never is a backward shift on any Markushevich basis of $\mathcal{H}$. \end{theorem}
\iffalse The charm of Cowen-Douglas operators is that they have natural relations to other mathematics branch. For example, there are many interesting paper on the geometric aspect of complex bundles defined by Cowen-Douglas operators (cf, \cite{McCarthy}, \cite{Misra}). In this section we shall show that every Cowen-Douglas operator can be seen as a shift on a complete minimal sequence. It can be seen as a basis theory understanding of Cowen-Douglas operators. In more details, for a Cowen-Douglas operator $T \in \mathcal{B}_{n}(\Omega)$, we can choose a complete minimal sequence $\{f_{n}\}_{n=1}^{\infty}$ so that we have $Tf_{n}=f_{n-1}$.
We shall discuss this question and use this method to deal with the similarity invariant(cf, \cite{Cao-1}, \cite{McCarthy}, \cite{Jiang}, \cite{Jiang2} and \cite{Jiang3}) \fi
Therefore a Cowen-Douglas operator $T \in \mathcal{B}_{n}(\Omega), n\ge 2$ never be a shift on some Schauder basis. On the other hand, example \ref{Example: Shift on conditional basis} shows that if we consider the n-multiple shift case(or more general, operator-weighted shift situation) then a Cowen-Douglas operator (In this example, we can choose the canonical shift $(S^{*})^{2}$)can be seen as a 2-multiple backward shift on a conditional basis.
Although the aim of this paper is to show the relations between basis theory and the class of Cowen-Douglas operator, the main tools to get the proper sequences is choosing the good cross-sections of the complex bundle $E_{T}$. In the next section we shall recall some basic results about the special cross-section of the complex bundle of Cowen-Douglas operators. In the third section we prove our main theorem \ref{Theorem: CD Operators are shift on C-M sequence}. After this we shall propose a general shift on biorthonal system model for Cowen-Douglas operators in the lemma \ref{Lemma: Adjoint of a Shift on a spanning minimal seq must be a shift on G basis} and prove theorem \ref{Theorem: CD operator is an adjoint op of some shift on G basis} and theorem \ref{Theorem: A Cowen-Douglas oper never be a shift on basis for n ge 2}. In the last section, we focus on the case $\mathcal{B}_{1}(\Omega)$. Theorem \ref{Theorem: when Cowen-Douglas operators are shifts} give some equivalent conditions to decide whether a Cowen-Douglas operator $T \in \mathcal{B}_{1}(T)$ is a shift on some Markushevicz basis or not. And then theorem \ref{Theorem: C-D operators are shifts (1)} give an operator theory description of the condition that a Cowen-Douglas operator $T \in \mathcal{B}_{1}(T)$ can be a backward weighted shift on an ONB.
\iffalse In the next section we shall recall some basic results about the special cross-section of the complex bundle of Cowen-Douglas operators. The required lemma \ref{Lemma: Uniqueness of the Canonical Cross-section} firstly appear in paper \cite{Cao-Ji} without proof, lemma \ref{Lemma: Existence of Canonical Right Inverse} appear in paper \cite{Cao-3} in Chinese. So we pick them together with proofs in detail in this section. In the third section we prove our main theorem \ref{Theorem: CD Operators are shift on C-M sequence}. After this we shall introduce our shift Models in the lemma \ref{Lemma: Adjoint of a Shift on a spanning minimal seq must be a shift on G basis} and prove our main theorem \ref{Theorem:A Cowen-Douglas oper never be a shift on basis for n ge 2}. At the final section we shall give some remarks on the coefficients of cross-sections and some remarks. \fi
\section{Canonical Cross-sections of The Complex Bundles of Cowen-Douglas Opertaotrs}
Firstly, we figure out a special operator related to every Cowen-Douglas operator $T$ in $\mathcal{B}_{n}(\Omega)$. It can be used to build special cross-sections of the complex bundle $E_{T}$.
\begin{lemma}\label{Lemma: Existence of Canonical Right Inverse} Let $T\in{\mathcal L}({\mathcal H})$ and $ran T={\mathcal H}$. Then there is an operator $B$ in ${\mathcal L}({\mathcal H})$ such that $TB=I$ and $ran B=(ker T)^\bot$. Moreover, if an operator $B_1$ in ${\mathcal L}({\mathcal H})$ satisfies $TB_1=I$ and $ran B_1 \subset (ker T)^\bot$ then $B_1=B$. \end{lemma}
\begin{proof} Let $\mathcal M=({\rm ker}T)^\bot$ and let $T_{\mathcal M}:\mathcal M\rightarrow\mathcal H$ be the restriction of $T$ on $\mathcal M$, i.e. $T_{\mathcal M}x=Tx$ for all $x\in\mathcal M$. Since $T_{\mathcal M}$ is bijective, there is exactly a bounded linear operator $B:\mathcal H\rightarrow\mathcal M$ such that $T_{\mathcal M}B=I:=I_\mathcal H$ and $BT_\mathcal M=I_\mathcal M$. Thus, it is easy to verify that $TB=I$ and ${\rm ran} B=\mathcal M=({\rm ker} T)^\bot$.
Assume that $B_1$ is another operator satisfying $TB_1=I$ and ${\rm ran} B_1 \subset({\rm ker} T)^\bot$. Then we have that $T(B-B_1)=0$. Thus $(B-B_1)x$ is in ${\rm ker} T\cap({\rm ker} T)^\bot$, so $Bx=B_1x$ for all $x\in\mathcal H$, that is $B=B_1$. \end{proof}
In the sequel, we shall call the operator $B$ in Lemma \ref{Lemma: Existence of Canonical Right Inverse} the {\it canonical right inverse} of the operator $T$.
\begin{theorem}\label{Theorem: s_u is a cross-section of complex bundle of_T} Let $T \in {\mathcal B}_n(\Omega)$, $z_0\in\Omega$ and let $B$ be the canonical right inverse of $T-z_{0}$. Then, for every unit vector $u$ in $\ker (T-z_{0})$, the $\mathcal{H}$-valued holomorphic function $s_u$ defined by $$ s_{u}(z)=u+\sum_{k=1}^{\infty} B^{k}u\cdot (z-z_{0})^{k} $$ is a local cross-section of the complex bundle $E_{T}$ on a neighborhood $\Delta$ of $z_0$. \end{theorem} \begin{proof} Clearly the power series converges on an open disc $\Delta$ with the center at $z_{0}$. Moreover we have
\begin{eqnarray*} (T-z_{0})s_{u}(z)&=&(T-z_{0})\Big(u+\sum_{k=1}^{\infty} B^{k}u\cdot (z-z_{0})^{k}\Big) \nonumber\\
&=&(T-z_{0})u+(T-z_{0})\Big(\sum_{k=1}^{\infty} B^{k}u\cdot (z-z_{0})^{k}\Big)\nonumber\\
&=&(z-z_{0})\sum_{k=0}^{\infty} B^{k}u\cdot (z-z_{0})^{k}\nonumber \\
&=&(z-z_{0})s_{u}(z). \end{eqnarray*}
Hence, $Ts_{u}(z)=zs_{u}(z)$. That is, the vector $s_{u}(z)$ belongs to $\ker(T-z)$ for all $z\in\Delta$. \end{proof}
With above notations, we have
\begin{definition}\label{Definition: Canonical cross-section} The cross-section $s_{u}$ is called the \it{canonical cross-section} with the initial unit vector $u$ at the point $z_{0}$. Moreover, the n-tuple $\{s_{e_{1}}, s_{e_{2}}, \cdots, s_{e_{n}}\}$ will be called the \it{canonical n-tuple} related to an orthonormal basis $\{e_{1},\cdots ,e_{n}\}$ of $\ker (T-z_{0})$ at the point $z_{0}$. \end{definition}
\begin{remark} Let $B$ be the canonical right inverse of $T-z_{0}$. Since ${\rm ran} B=(\ker (T-z_{0}))^\bot$, it is easy to show that the family $\{s_{e_{1}}(z), s_{e_{2}}(z), \cdots, s_{e_{n}}(z)\}$ is linear independent for every $z$ near $z_{0}$. Thus, it forms a basis of $\ker(T-z)$. \end{remark}
As an application of Lemma \ref{Lemma: Existence of Canonical Right Inverse} and Theorem \ref{Theorem: s_u is a cross-section of complex bundle of_T}, we will show the following corollary, which was raised by M. J. Cowen and R. G. Douglas in \cite{Douglas1}. However, they did not give a proof.
\begin{corollary} In the difinition of Cowen-Douglas operator, the condition (3) can be equivalently replaced by the condition $\bigvee_{k=1}^{\infty}ker(T-z_0)^k=\mathcal H$ for a fixed $z_0$ in $\Omega$. \end{corollary} \begin{proof} Assume $\bigvee_{z\in\Omega}ker(T-z)=\mathcal H$. Take an orthonormal basis $\{e_1,\ldots,e_n\}$ of ${\rm ker}(T-z_0)$. We know that the vector family $\{s_{e_1}(z),\ldots,s_{e_n}(z)\}$ is a basis of ${\rm ker}(T-z)$ for every $z\in \Delta$. Note that $(T-z_0)^{k+1}B^ke_i=(T-z_0)e_i=0$, we have that $B^ke_i\in{\rm ker}(T-z_0)^{k+1}$ for all $k\geq0$ and $i=1,2,\ldots,n$.
Now, let $\mathcal M=\bigvee_{k=1}^{\infty}{\rm ker}(T-z_0)^k$. If $x\in\mathcal H$ and $x\bot M$, then $x\bot B^ke_i$. It follows that $x\bot s_{e_i}(z)$ for $1 \leq i \leq n$ and $z\in\Delta$. Thus, we have that $x\bot {\rm ker}(T-z)$ for $z\in\Delta$. By \cite[Corollary 1.13]{Douglas3}, we know that $$ \bigvee_{z\in\Delta}{\rm ker}(T-z)=\bigvee_{z\in\Omega}{\rm ker}(T-z)=\mathcal H. $$ So, $x=0$. This shows that $\mathcal M=\mathcal H$.
Conversely, assume that $\bigvee_{k=1}^{\infty}{\rm ker}(T-z_0)^k=\mathcal H$. If $x\in\mathcal H $ and $x\bot\bigvee_{z\in\Delta}{\rm ker}(T-z)$, then $x\bot e_i$ and $x\bot s_{e_i}(z)$ for all $z\in\Delta$ and $1 \leq i \leq n$. Thus, we obtain that $$ 0=(x,s_{e_i}(z))=(x,e_i)+\sum_{k=1}^{\infty}(x,B^ke_i)(z-z_0)^k,\,\forall z\in\Delta. $$ Hence, we have that $(x,B^ke_i)=0$ for all $k\geq0$ and $i=1,2,\ldots,n$. Since $e_i\in{\rm ker}(T-z_0)$ and $(T-z_0)B^ke_i=B^{k-1}e_i$, we can show by induction that, for each $k\geq0$, the vector family $$\{e_1,\ldots,e_n,\ldots,B^ke_1,\ldots,B^ke_n\}$$ in ${\rm ker}(T-z_0)^{k+1}$ is linearly independent. Note that $\mathrm{dim\,ker}(T-z_0)^{k+1}=(k+1)n$, it follows that this family is exactly a basis of ${\rm ker}(T-z_0)^{k+1}$, which proves that $x\bot{\rm ker}(T-z_0)^{k+1}$ for all $k\geq0$. So, $x=0$. \end{proof}
From Definition \ref{Definition: Canonical cross-section}, we know that there are many canonical n-tuples dependent on the choice of an orthonormal basis $\{e_{1},e_{2},\cdots ,e_{n}\}$ of $\ker (T-z_0)$. However, following lemma tells us that the class of canonical n-tuples is small.
\begin{lemma}\label{Lemma: Canonical Cross-sections are not So Many} Assume that $(s_{1},\cdots ,s_{n})$ and $(\tilde{s}_{1},\cdots ,\tilde{s}_{n})$ are canonical section tuples of $T \in {\mathcal B}_n(\Omega)$ related to ONB $\{e_{1},e_{2},\cdots ,e_{n}\} $ and $\{\tilde{e}_{1},\tilde{e}_{2},\cdots ,\tilde{e}_{n}\} $ respectively, then there is an unique unitary matrix $U \in U(n)$ such that $$ (\tilde{s}_{1},\cdots ,\tilde{s}_{n})=(s_{1},\cdots ,s_{n})U. $$ \end{lemma}
\begin{proof}It is clear that there is an unitary matrix $U=(u_{ij})_{n \times n} \in U(n)$ such that $$
(\tilde{e}_{1},\tilde{e}_{2},\cdots ,\tilde{e}_{n}) =(e_{1},e_{2},\cdots,e_{n})U. $$ Then we have $$ \begin{array}{ll} \tilde{s}_{i} & =\sum_{k=1}^{\infty} \tilde{e}_{i}+ B^{k}\tilde{e}_{i} \\
& =\sum_{k=1}^{\infty} (\sum_{j=1}^{n} u_{ji}e_{j})
+B^{k}(\sum_{j=1}^{n} u_{ji}e_{j}) \\
& =\sum_{j=1}^{n} u_{ji}(\sum_{k=1}^{\infty}e_{j}+B^{k}e_{j}) \\
& =\sum_{j=1}^{n} u_{ji}s_{j} , \end{array} $$ Or equivalently, we have $(\tilde{s}_{1},\cdots,\tilde{s}_{n})=(s_{1},\cdots ,s_{n})U$. \end{proof}
\begin{lemma}\label{Lemma: Uniqueness of the Canonical Cross-section} Let $T \in {\mathcal B}_n(\Omega)$, $z_0\in\Omega$ and let $u$ be a unit vector in $\ker (T-z_{0})$. Then there is exactly one holomorphic cross-section $\gamma$ defined on a neighborhood $\Delta$ of $z_0$ such that $\gamma(z_0)=u$ and $\gamma(z)-u\in(\ker(T-z_0))^\bot$ for all $z\in\Delta$. \end{lemma} \begin{proof} By Theorem \ref{Theorem: s_u is a cross-section of complex bundle of_T}, the canonical cross-section $s_u$ satisfies required properties. For uniqueness, suppose that there is another holomorphic cross-section $\gamma$ which has required properties. Then let $\gamma$ have the following power series expansion $$ \gamma(z)=u+\sum_{k=1}^{\infty} u_{k}(z-z_{0})^{k}. $$ Since $(T-z_0)\gamma(z)=(z-z_0)\gamma(z)$, it can be checked that $ (T-z_{0})u=0$ and $(T-z_{0})u_{k}=u_{k-1}$ for $k\ge 1,$ where $u_0=u$. Also, let $B\in{\mathcal L}({\mathcal H})$ be the canonical right inverse of $T-z_0$. Then we have $Bu_{k-1}-u_{k} \in \ker (T-z_{0})$. Note that $$ \dfrac{\gamma(z)-u}{z-z_0}=\sum_{k=1}^{\infty} u_{k}(z-z_{0})^{k-1}\in(\ker(T-z_0))^\bot, $$ we obtain that $$ u_1=\lim_{z\rightarrow z_0}\dfrac{\gamma(z)-u}{z-z_0}\in(\ker(T-z_0))^\bot. $$ Thus, $u_k\in(\ker(T-z_0))^\bot$ for all $k\geq1$ by induction. Now, $$ Bu_{k-1}-u_{k} \in (\ker(T-z_{0}))^{\bot}\cap \ker(T-z_{0}), $$ that is $Bu_{k-1}=u_{k}$. This implies $s_u(z)=\gamma(z)$ on some neighborhood of $z_0$. \end{proof}
In light of Lemma \ref{Lemma: Uniqueness of the Canonical Cross-section}, we can give an equivalent description of the canonical cross-section as follows.
\begin{proposition}\label{Definition: Eq Def of CS} Let $T \in {\mathcal B}_n(\Omega)$, $z_0\in\Omega$ and let $u$ be a unit vector in $\ker (T-z_{0})$. Then a holomorphic cross-section $s_u$ of $E_{T}$ defined on a neighborhood $\Delta$ of $z_0$ is the canonical cross-section with the initial vector $u$ if and only if $s_u(z_0)=u$ and $s_u(z)-u\in(\ker(T-z_0))^\bot$ for all $z\in\Delta$. \end{proposition}
In what follows let $\mathcal{G}r(n, \mathcal{H})$ denote the Grassmann manifold consisting of all $n$-dimensional subspaces of $\mathcal{H}$. A map $f: \Omega \rightarrow \mathcal{G}r(n, \mathcal{H})$ is called holomorphic at $z_0\in\Omega$ if there exists a neighborhood $\Delta$ of $z_0$ and $\mathcal{H}$-value functions $\gamma_1,\cdots,\gamma_n$ such that $f(z)=\bigvee\{\gamma_1(z),\cdots,\gamma_n(z)\}$ for $z\in\Delta$. If $f: \Omega \rightarrow \mathcal{G}r(n, \mathcal{H})$ is holomorphic map then $f$ induces a natural complex bundle $E_f$ as follows: $$ E_f=\{(x,z)\in\mathcal{H}\times\Omega:\,x\in f(z)\}. $$ And, the projection $\pi:E_f\rightarrow\Omega$ is given by $\pi(x,z)=z$. The bundle $E_f$ will be called the pull-back bundle of the Grassmann manifold induced by $f$. In particular, for an operator $T \in {\mathcal B}_n(\Omega)$, we can define $f: \Omega \rightarrow \mathcal{G}r(n, \mathcal{H})$ by $f(z)=\ker(T-z)$ for $z\in\Omega$, then $f$ is holomorphic and $E_f=E_T$.
The existence of the canonical cross-section is just a special case of \cite[Lemma 2.4]{Douglas3} when $E_f=E_T$. We can prove that Lemma \ref{Lemma: Uniqueness of the Canonical Cross-section} also holds for the general holomorphic map.
\begin{theorem}\label{Theorem: Genaral Situation of Canonical Cross-Section} Let $E_{f}$ is the pull-back bundle of $\mathcal{G}r(n, \mathcal{H})$ induced by a holomorphic map $f: \Omega \rightarrow Gr(n, \mathcal{H})$, and let $\{u_{1}, u_{2}, \cdots, u_{n}\}$ is an orthonormal basis of the fibre $f(z_{0})$. Then there exist exactly $n$ holomorphic cross-sections $\gamma_{1}, \gamma_{2}, \cdots, \gamma_{n}$ of $E_{f}$ defined on some open disc $\Delta$ with the center at $z_{0}$ such that
(1) $\gamma_{1}(z), \cdots, \gamma_{n}(z)$ form a basis of the fibre $f(z)$ for $z\in \Delta$;
(2) $\gamma_{k}(z_{0})=u_{k}$ for $1\le k\le n$; and
(3) $(\gamma_{k}(z)-\gamma_{k}(z_{0}), \gamma_{j}(z_{0}))=0$ for $1\le k, j\le n$ and $z\in \Delta$. \end{theorem} \begin{proof} It is easy to show that there exist $n$ holomorphic cross-sections on a neighborhood of $z_{0}$ with the initial vector $u_{k}$ for all $1\le k\le n$ respectively. Thus, the existence of holomorphic cross-sections satisfying the properties (1), (2) and (3) is directly from \cite[Lemma 2.4]{Douglas3}. For the uniqueness, it is enough to prove
\begin{claim} For the unit vector $u_{1}$, the cross-section $\gamma_{1}$ is the unique holomorphic cross-section satisfying properties (1), (2) and (3). \end{claim}
Assume that $\gamma(z)=u_{1} + \sum_{i=1}^{\infty} a_{i}(z-z_{0})^{i}$ is another holomorphic cross-section defined on $\Delta$ which satisfies properties (1), (2) and (3). And, let $$ \gamma_{k}(z)=u_{k} +\sum_{i=1}^{\infty} a^{(k)}_{i}(z-z_{0})^{i} $$ be the power series expansions of $\gamma_{k}$ on $\Delta$ for $1\le k\le n$. By the property (2) we have \begin{equation}\label{Equation: Orthogonal of Vectors Appearing in Canonical Secs} (a_{i}, u_{j})=(a^{(k)}_{i}, u_{j})=0 \hbox{ for } i \ge 1 \hbox{ and } 1 \le j \le n. \end{equation} Now there are holomorphic functions $h_{1}(z), \cdots, h_{n}(z)$ defined $\Delta$ such that $$ \gamma(z)=h_{1}(z)\gamma_{1}(z)+h_{2}(z)\gamma_{2}(z)+\cdots +h_{n}(z)\gamma_{n}(z) $$ for all $z\in \Delta$. Clearly we have $h_{k}(z_{0})=0$ for $2 \leq k \le n$ and $h_{1}(z_{0})=1$. Suppose that $h_{k}(z)\not\equiv 0$ for $2\le k\le n$. Then we can write $$ h_{k}(z)=(z-z_{0})^{m}\tilde{h}_{k}(z), \hbox{ where } \tilde{h}_{k}(z)=c_{0}+\sum_{i=1}^{\infty} c_{i}(z-z_{0})^{i} \hbox{ and } c_{0}\ne 0. $$ However, by the product of power series, we can obtain that $$a_m=c_0u_k+v.$$ By the formula $(2.1)$ and noting that $u_k\bot u_l(k\neq l)$, we have that $u_k\bot v$. So, still from the formula $(2.1)$, it follows that $$c_0=(c_0u_k,u_k)=(a_m,u_k)=0.$$ This contradicts $c_0\neq0$. Now, we have that $\gamma=h_1\gamma_{1}$. Let$$h_1(z)=1+\sum_{i=1}^\infty d_i(z-z_0)^i.$$ Then it follows that $$a_1=a_1^{(1)}+d_1u_1.$$ Again, by the formula $(2.1)$, we get that $$0=(a_1,u_1)=(a_1^{(1)},u_1)+(d_1u_1,u_1)=d_1.$$ So, $a_1=a_1^{(1)}$. By induction, we have that $d_i=0$ for $i\geq 1$, that is $\gamma=\gamma_1$. \end{proof}
Above theorem say that together with lemma 2.4 of paper \cite{Douglas3} we can apply our lemma \ref{Lemma: Uniqueness of the Canonical Cross-section} to get a more elegant way to obtain the canonical cross-section from a initial vector. We place that method in the next subsection for readability.
\section{Cowen-Douglas operators are shifts on complete minimal sequences}
\subsection{Psedocanonical cross-sections}
$\quad$
Suppose $T \in \mathcal{B}_{n}(\Omega)$ and $\gamma$ be a holomorphic cross-section of $E_{T}$ defined on a connected open subset $\Delta$ of $\Omega$. Call $\gamma$ to be {\it spanning }if $\bigvee\{\gamma(z):\,z\in \Delta \}=\mathcal{H}$
Let $\gamma(z)=f_{0}+\sum_{k=1}^{\infty} f_{k}(z-z_{0})^{k}$ be the power series expansion of $\gamma$ on an open disc $\Delta$ with the center at $z_{0}$. Clearly, $\gamma$ is spanning if and only if $\bigvee_{k=0}^\infty\{f_{k}\}=\mathcal{H}$, i.e., the vector sequence $\{f_{k}\}_{k=0}^{\infty}$ is complete.
\begin{theorem}\label{Theorem: Holomorphic cross-section is spanning} If $T \in \mathcal{B}_{1}(\Omega)$ then every holomorphic cross-section $\gamma$ is spanning. \end{theorem} \begin{proof} Suppose that $x \perp \gamma(z)$ for all $z \in\Delta$. It follows that $x \perp\bigvee_{z\in\Delta}{\rm ker}(T-z)$ since $\mathrm{dim\,ker}(T-z )=1$. By \cite[Corollary 1.13]{Douglas1}, we know that $\bigvee_{z\in\Delta}{\rm ker}(T-z)=\mathcal H$. So, $x=0$. This shows that $\bigvee\{\gamma(z):\,z\in \Delta \}=\mathcal{H}$ \end{proof}
However, when $n>1$ any canonical cross-section $\gamma(z)=f_{0}+\sum_{k=1}^{\infty} f_{k}(z-z_{0})^{k}$ never be spanning since $f_{0}\in\ker(T-z_0)$ and $f_{k}\in(\ker(T-z_0))^\bot$ for $k \geq 1$. To generalize above theorem to the case $\mathcal{B}_{n}(\Omega)$, we need following notion.
\begin{definition}\label{Definition: Psedocanonical cross section} Let $T \in {\mathcal B}_n(\Omega)$ and $z_0\in\Omega$. A holomorphic cross-section $\mu$ of the complex bundle $E_{T}$ defined on a neighborhood $\Delta$ of $z_{0}$ is said to be psedocanonical if $\mu(z_0)\not=0$ and $(\mu(z_{0}), \mu'(z))=0$ for all $z\in\Delta$. \end{definition}
Suppose that $\mu(z)=f_{0}+\sum_{n=1}^{\infty} f_{n}(z-z_{0})^{n}$ is the power series expansion of $\mu$ at $z_0$. Then it is obvious that $\mu$ is psedocanonical if and only if $f_{0}\bot f_k$ for all $k\geq1$.
Clearly, a canonical cross-section must be psedocanonical. If $T \in \mathcal{B}_{1}(\Omega)$, by Proposition \ref{Definition: Eq Def of CS}, they are equivalent.
\begin{proposition}\label{Proposition: Construction of psedocan cross section} Let $T \in \mathcal{B}_{n}(\Omega)$ and $\mu$ be a psedocanonical cross-section, and let $\gamma$ be a canonical cross-section which satisfy $(\mu(z_{0}), \gamma(z_{0}))=0$. Then for any holomorphic function $g(z)$ defined near $z_{0}$ with $g(z_{0})=0$, the cross-section $\lambda(z)=\mu(z)+g(z)\gamma(z)$ is also psedocanonical. \end{proposition} \begin{proof} By the definition of canonical cross-section, we have $$ (\mu(z_{0}), \gamma(z)-\gamma(z_{0}))=0 \hbox{ and } (\mu(z_{0}), \gamma'(z))=0 $$ since $\mu(z_{0}) \in \ker(T-z_{0}I)$. Hence, we have $(\mu(z_{0}), \gamma(z))=(\mu(z_{0}), \gamma(z_{0}))=0$. Note that $\lambda(z_{0})=\mu(z_{0})$, it follows that \begin{eqnarray*} (\lambda(z_{0}), \lambda'(z))&=&(\mu(z_{0}), \mu'(z)+g'(z)\gamma(z)+g(z)\gamma'(z)) \\
&=&(\mu(z_{0}), \mu'(z))+(\mu(z_{0}), g'(z)\gamma(z))+(\mu(z_{0}), g(z)\gamma'(z)) \\
&=&0. \end{eqnarray*} \end{proof}
\begin{theorem}\label{Theorem: Structure of Psedocanonical cross-section} Let $T \in \mathcal{B}_{n}(\Omega)$, $z_{0} \in \Omega$ and let $\lambda$ be a psedocanonical cross-section of $E_{T}$. Then there are canonical cross-sections $\gamma_{1},\cdots, \gamma_{n}$ and holomorphic functions $g_{2}, \cdots, g_{n}$ defined near $z_0$ such that
(1) $(\gamma_{i}(z), \gamma_{j}(z_{0}))=\delta_{ij}$ for $1 \leq i,j \leq n$ and $\gamma_{1}(z_{0})=\lambda(z_{0})$;
(2) $g_{i}(z_{0})=0$ for $i=2, 3, \cdots, n$; and
(3) $\lambda(z)=\gamma_{1}(z)+\sum_{i=2}^{n} g_{i}(z)\gamma_{i}(z)$.
\end{theorem} \begin{proof}
Clearly we can assume $||\lambda(z_{0})||=1$. Take an orthonormal basis $\{e_{1}, e_{2}, \cdots, e_{n}\}$ of $\ker(T-z_{0})$ such that $e_{1}=\lambda(z_{0})$. From Definition \ref{Definition: Canonical cross-section}, we set $\gamma_i=s_{e_i}$ which is the canonical cross-section with the initial vector $\gamma_i(z_0)=e_i$ for $i=1,2, \cdots, n$ respectively. Since $\gamma_i(z)-\gamma_i(z_0)\bot\ker(T-z_0)$, we can obtain that $$ (\gamma_i(z),\gamma_j(z_0))=(\gamma_i(z)-\gamma_i(z_0),\gamma_j(z_0))+(\gamma_i(z_0),\gamma_j(z_0))=(e_i,e_j)=\delta_{ij}, $$ that is the property $(1)$ holds.
Moreover, let \begin{eqnarray} g_{i}(z)=(\lambda(z), \gamma_{i}(z_{0})), ~ i=2,\cdots,n. \end{eqnarray} Then $g_i(z_0)=(e_1,e_i)=0$, that is the property $(2)$ is also true.
And, we define $\mu(z)$ by \begin{eqnarray} \mu(z)=\lambda(z)-\sum_{i=2}^{n} g_{i}(z)\gamma_i(z). \end{eqnarray} Then, by the formula $(3.1)$ and the property $(1)$, we have that \begin{eqnarray*} (\mu(z), \gamma_{j}(z_{0})) & =&(\lambda(z),\gamma_j(z_0))-\sum_{i=2}^{n} g_{i}(z)(\gamma_{i}(z), \gamma_{j}(z_{0})) \\
& =&(\lambda(z),
\gamma_{j}(z_{0}))-g_j(z)=0. \end{eqnarray*} Hence, for $i=2, \cdots, n$, it follows that $$ (\mu(z)-\lambda(z_{0}),\gamma_i(z_0))=(\mu(z),\gamma_i(z_0))-(\lambda(z_{0}),\gamma_i(z_0))=(e_1,e_i)=0. $$ Also, by the formula $(3.2)$ and Definition \ref{Definition: Psedocanonical cross section}, we can show that \begin{eqnarray*} (\mu(z)-\lambda(z_{0}), \gamma_{1}(z_{0})) & =&(\lambda(z)-\lambda(z_0),\gamma_1(z_0))-\sum_{i=2}^{n} g_{i}(z)(\gamma_{i}(z), \gamma_{1}(z_{0})) \\
& =&(\lambda(z)-\lambda(z_0),
\lambda(z_0))=0\\ \end{eqnarray*} since $\lambda$ is psedocanonical. This implies that $$ (\mu(z)-\lambda(z_{0})) \perp \ker(T-z_{0}). $$ Note that $\lambda(z_0)=\mu(z_0)=\gamma_1(z_0)=e_1$, by Lemma \ref{Lemma: Uniqueness of the Canonical Cross-section}, we have $\mu=\gamma_1$. Thus, the property $(3)$ holds. \end{proof}
\begin{lemma}\label{Lemma: Obtain CS in CD Method} Let $$ \lambda(z)=f_{0}+\sum_{k=1}^{\infty} f_{k}(z-z_{0})^{k} $$ be a holomorphic cross-section of $E_{T}$ defined near $z_{0}\in \Omega$ with $f_{0}\not=0$. Then there is a unique holomorphic function $h$ defined near $z_{0}$ such that $\mu(z)=h(z)\lambda(z)$ is a psedocanonical cross-section of $E_{T}$ at $z_{0}$ and $\mu(z_{0})=f_{0}$. \end{lemma} \begin{proof} We consider the holomorphic function $g(z)=(\lambda(z),f_0)$. Since
$g(z_0)=||f_0||^2\neq0$, we know $g(z)\neq0$ near $z_0$. Hence, the function $h(z)=\frac{||f_0||^2}{g(z)}$ is holomorphic near $z_0$.
Now, let $\mu(z)=h(z)\lambda(z)$. Then we have
$(\mu(z),f_0)\equiv||f_0||^2$. It is easy to check that $\mu(z)$ has the form of the power series expansion as $$ \mu(z)=f_0+\sum_{k=1}^{\infty}c_k(z-z_{0})^{k}. $$ Thus, we obtain that
$$\Big(\sum_{k=1}^{\infty}c_k(z-z_{0})^{k},f_0\Big)=(\mu(z)-f_0,f_0)=(\mu(z),f_0)-||f_0||^2\equiv0,$$ which implies that $f_0\bot c_k$ for $k\geq1$. Therefore, $\mu(z)$ is a psedocanonical cross-section of $E_{T}$ at $z_{0}$ and $\mu(z_{0})=f_{0}$.
To show the uniqueness, suppose that $$ \widetilde{h}(z)=f_0+\sum_{k=1}^{\infty}\widetilde{c_k}(z-z_{0})^{k} $$ is another holomorphic function defined near $z_0$ such that $\widetilde{\mu}(z)=\widetilde{h}(z)\lambda(z)$ is also a psedocanonical cross-section and $\widetilde{\mu}(z_0)=\widetilde{h}(z_0)\lambda(z_0)=f_0$. Since $f_0\bot c_k$ and $f_0\bot \widetilde{c_k}$ for $k\geq1$, we have that $$((h(z)-\widetilde{h}(z))\lambda(z),f_0)=(\mu(z)-\widetilde{\mu}(z),f_0)\equiv0.$$ It implies that $h(z)\equiv\widetilde{h}(z)$. \end{proof}
\begin{corollary} Let $T \in \mathcal{B}_{n}(\Omega)$, $z_{0} \in \Omega$ and let $\lambda$ be a holomorphic cross-section of $E_{T}$. Then there are canonical cross-sections $\gamma_{1},\cdots, \gamma_{n}$ and holomorphic functions $g_{1}, \cdots, g_{n}$ defined near $z_0$ such that
(1) $(\gamma_{i}(z), \gamma_{j}(z_{0}))=\delta_{ij}$ for $1 \leq i,j \leq n$ and $\gamma_{1}(z_{0})=\lambda(z_{0})$;
(2) $g_{i}(z_{0})=0$ for $i=2, 3, \cdots, n$; and
(3) $\lambda(z)=\sum_{i=1}^{n} g_{i}(z)\gamma_{i}(z)$. \end{corollary} \begin{proof} By Lemma \ref{Lemma: Obtain CS in CD Method}, we know that there is a holomorphic function $h$ defined near $z_{0}$ such that $\mu(z)=h(z)\lambda(z)$ is a psedocanonical cross-section of $E_{T}$ at $z_{0}$ and $\mu(z_{0})=\lambda(z_{0})$. Thus, it follows that there are canonical cross-sections $\gamma_{1},\cdots, \gamma_{n}$ and holomorphic functions $\widetilde{g}_{2}, \cdots, \widetilde{g}_{n}$ defined near $z_0$ such that the property $(1)$ holds, $\widetilde{g}_{i}(z_{0})=0$ for $i=2, 3, \cdots, n$, and $$
\mu(z)=h(z)\lambda(z)=\gamma_{1}(z)+\sum_{i=2}^{n} \widetilde{g}_{i}(z)\gamma_{i}(z). $$ Since $h(z_0)=1$, the function $\frac{1}{h(z)}$ is holomorphic near $z_0$. Now, set $g_{1}(z)=\frac{1}{h(z)}$ and $g_{i}(z)=\frac{\widetilde{g}_{i}(z)}{h(z)}$. Then the properties $(2)$ and $(3)$ hold. \end{proof}
\subsection{Proof of main Theorem \ref{Theorem: CD Operators are shift on C-M sequence}} \begin{theorem}\label{Theorem: Coefficent sequence of psedocanonical section is Minimal} Let $T \in \mathcal{B}_{n}(\Omega)$, $z_{0} \in \Omega$ and let $\mu(z)=f_{0}+\sum_{k=1}^{\infty} f_{k}(z-z_{0})^{k}$ be a holomorphic cross-section defined near $z_{0}$ with $f_{0}\not= 0$. Then the vector sequence $\{f_{k}\}_{k=0}^{\infty}$ is minimal if and only if $f_{0} \not\in \bigvee_{k=1}^\infty\{f_{k}\}$. In particular, when $\mu$ is a psedocanonical cross-section, $\{f_{k}\}_{k=0}^{\infty}$ is always minimal. \end{theorem} \begin{proof} We just need to show the part of ``if''. Let $\mathcal{H}_{\mu}^{1}=\bigvee_{k=1}^\infty\{f_{k}\}$ and $f_{0} \not\in \mathcal{H}_{\mu}^{1}$. Since $\mu(z)\in\ker(T-z)$ for all $z$, it follows that $$ \sum_{k=0}^\infty f_k(z-z_0)^{k+1}=(T-z_0)\mu(z)=\sum_{k=1}^\infty (T-z_0)f_k(z-z_0)^{k}. $$ This implies that $(T-z_0)f_k=f_{k-1}$ for $k\geq1$. Hence we have $(T-z_0)^kf_k=f_0$.
If the statement is false, then there exist some $f_k$ and vector sequence $\{v_n\}_{n=1}^\infty$ such that $v_n$ is a linear combination of finite vectors in the set $\{f_n: n\neq k\}$ and
$||v_n-f_k||<\frac{1}{n}$. Moreover, we can write that $$ v_{n}=v_{n}^{(1)}+v_{n}^{(2)}, v_{n}^{(1)}=\sum_{j=1}^{k-1} \alpha_{j}f_{j}~ {\rm and} ~ v_{n}^{(2)}=\sum_{j=k+1}^{m_{n}} \alpha_{j}f_{j}. $$ Thus, we get that $$ (T-z_0)^{k}v_{n}^{(1)}=0 ~{\rm and}~ (T-z_0)^{k}v_{n}^{(2)} \in \mathcal{H}_{\mu}^{1}. $$ It gives that $ (T-z_0)^{k}v_{n} \in \mathcal{H}_{\mu}^{1}$ and $$
||f_{0}-(T-z_0)^{k}v_{n}||=||(T-z_0)^{k}(f_k-v_{n})||\leq||T-z_0||^{k}\frac{1}{n}\rightarrow0(n\rightarrow\infty). $$ Hence, it follows that $ f_{0} \in \mathcal{H}_{\mu}^{1}$, a contradiction. \end{proof}
Next, we can generalize Theorem \ref{Theorem: Holomorphic cross-section is spanning} as follows.
\begin{theorem}\label{Theorem: Existence of complete minimal coes psedocanonical cross section} Let $T \in \mathcal{B}_{n}(\Omega)$. Then for each $z_0\in\Omega$, there is a spanning psedocanonical cross-section. \begin{eqnarray} \lambda(z)=f_{0}+\sum_{k=1}^{\infty} f_{k}(z-z_{0})^{k} \end{eqnarray} defined near $z_0$. Hence, the vector sequence $\{f_{k}\}_{k=0}^{\infty}$ is complete. \end{theorem} \begin{proof} By \cite[Theorem A]{Zhu}, there is a holomorphic cross-section $\widetilde{\lambda}$ on $\Omega$ such that $\bigvee\{\widetilde{\lambda}(z):z\in\Omega\}=\mathcal{H}$ and $\widetilde{\lambda}(z_0)\neq0$. Imitating the proof of \cite[Corollary 1.13]{Douglas1}, for every open set $\Delta$ in $\Omega$, we can show that $\bigvee\{\widetilde{\lambda}(z):z\in\Delta\}=\mathcal{H}$.
Now, by Lemma \ref{Lemma: Obtain CS in CD Method}, there is a holomorphic function $h(z)$ such that $\lambda(z)=h(z)\widetilde{\lambda}(z)$ is a psedocanonical cross-section with the initial vector $f_0=\widetilde{\lambda}(z_0)$. Let the power series expansion of $\lambda(z)$ on a disc $\Delta$ with the center at $z_0$ be given by the formula $(3.3)$. Moreover, we can assume that $h(z)\neq0$ for all $z\in\Delta$. Hence, we have that $$ \bigvee\{\lambda(z):z\in\Delta\}=\bigvee\{\widetilde{\lambda}(z):z\in\Delta\}=\mathcal{H}. $$ This implies that $\lambda$ is a spanning. Thus, $\{f_{k}\}_{k=0}^{\infty}$ is complete. \end{proof}
From Theorem \ref{Theorem: Coefficent sequence of psedocanonical section is Minimal} and Theorem \ref{Theorem: Existence of complete minimal coes psedocanonical cross section}, we obtain directly the following.
\begin{theorem}\label{Theorem: General situation: Coefficent sequence of canonical section is Minimal} Let $T \in \mathcal{B}_{n}(\Omega)$. Then for each $z_0\in\Omega$, there is a psedocanonical cross-section $$ \lambda(z)=f_{0}+\sum_{k=1}^{\infty} f_{k}(z-z_{0})^{k} $$ such that the vector sequence $\{f_{k}\}_{k=0}^{\infty}$ is minimal and complete. \end{theorem}
Now we can apply above theorem to prove Theorem \ref{Theorem: CD Operators are shift on C-M sequence}.
\textit{Proof of Theorem 1.1}. For every $z_{0} \in \Omega$. By Theorem \ref{Theorem: General situation: Coefficent sequence of canonical section is Minimal} there is a psedocanonical cross-section $$ \lambda(z)=f_{0}+\sum_{k=1}^{\infty} f_{k}(z-z_{0})^{k} $$ near $z_{0}$ such that the vector sequence $\{f_{k}\}_{k=0}^{\infty}$ is both complete and minimal. Since $\lambda(z) \in {\rm ker}(T-z)$, it follows that $$ \sum_{k=0}^\infty f_k(z-z_0)^{k+1}=(T-z_0)\lambda(z)=\sum_{k=1}^\infty (T-z_0)f_k(z-z_0)^{k}. $$ Hence, we have that $(T-z_0)f_k=f_{k-1}$ for $k\geq1$ and $(T-z_{0})f_{0}=0$, that is $T-z_{0}$ is the backward shift on $\{f_{k}\}_{k=0}^{\infty}$. $
{}\Box$
\subsection{Proof of theorem \ref{Theorem: CD operator is an adjoint op of some shift on G basis}
and theorem \ref{Theorem: A Cowen-Douglas oper never be a shift on basis for n ge 2}}
\begin{lemma}\label{Lemma: Adjoint of a Shift on a spanning minimal seq must be a shift on G basis} Suppose $\{(f_{k}, g_{k})\}_{k=0}^{\infty}$ is a biorthogonal system and $\{f_{k}\}_{k=0}^{\infty}$ is spanning, that is, $\vee_{k\ge 0}\{f_{k}\}=\mathcal{H}$. Moreover, suppose $T$ is a backward shift on the sequence $\{f_{k}\}_{k=0}^{\infty}$, i.e., $Tf_{0}=0$ and $Tf_k=f_{k-1}$ for $k\geq1$. Then we have
\begin{enumerate} \item The sequence $\{g_{k}\}_{k=0}^{\infty}$ is a generalized basis of the Hilbert space $\mathcal{H}$; \item The adjoint operator $T^{*}$ must be a foreward shift on the generalized basis $\{g_{k}\}_{k=0}^{\infty}$, that is, we have $T^{*}g_{k}=g_{k+1}$ for $k=0,1,\cdots.$ \end{enumerate} \end{lemma}
\begin{proof} \begin{enumerate} \item Firstly we show that $\{g_{k}\}_{k=0}^{\infty}$ is a generalized basis. Since the sequence $\{f_{k}\}_{k=0}^{\infty}$ is spanning, we have $(x, f_{k})=0$ for each $k\ge 1$ if and only if $x=0$. Hence as a functional sequence, $\{f_{k}\}_{k=0}^{\infty}$ is total. \item Note that $$ (f_{j+1},T^*g_i)=(Tf_{j+1},g_i)=(f_j,g_i)=\delta_{ij}. $$ then by $\{(f_{k}, g_{k})\}_{k=0}^{\infty}$ is a biorthogonal system, we know the sequence $\{f_{k}\}_{k=0}^{\infty}$ is minimal($f_{j+1} \notin \vee_{k \ne j+1, k\ge 0}\{f_{k}\}$). Moreover by $\vee_{k\ge 0}\{f_{k}\}=\mathcal{H}$, we know that $\vee_{k \ne j+1, k\ge 0}\{f_{k}\}$ is a subspace of the Hilbert space $\mathcal{H}$ with codimension $1$. Therefore we must have that $ T^*g_k=g_{k+1}$ for all $k\geq 0$. \end{enumerate} \end{proof}
\textit{Proof of Theorem \ref{Theorem: CD operator is an adjoint op of some shift on G basis}}. It is a directly result of above lemma \ref{Lemma: Adjoint of a Shift on a spanning minimal seq must be a shift on G basis} and theorem \ref{Theorem: Existence of complete minimal coes psedocanonical cross section}.
\iffalse \begin{theorem}\label{Theorem: CD operator is an adjoint op of some shift on G basis} For a Cowen-Douglas operator $T \in \mathcal{B}_{n}(\Omega), 0\in \Omega$, its adjoint operator $T^{*}$ is a shift on some generalized basis. \end{theorem} \fi
\begin{lemma}\label{Lemma: Adjoint of a Shift on M basis also be a shift} Suppose $\{(f_{k}, g_{k})\}_{k=0}^{\infty}$ is a biorthogonal system and $\{f_{k}\}_{k=0}^{\infty}$ is a Markushevich basis. Moreover, let $T$ be a backward shift on the Markushevich basis $\{f_{k}\}_{k=0}^{\infty}$ of $\mathcal{H}$, i.e., $Tf_{0}=0$ and $Tf_k=f_{k-1}$ for $k\geq1$. The the adjoint operator $T^{*}$ must be a foreward shift on the Markushevicz basis $\{g_{k}\}_{k=0}^{\infty}$, that is, we have $T^{*}g_{k}=g_{k+1}$ for $k=0,1,\cdots.$ \end{lemma} \begin{proof} It is clear that $\{g_{k}\}_{k=0}^{\infty}$ is also a markushevicz basis. Then apply lemma \ref{Lemma: Adjoint of a Shift on a spanning minimal seq must be a shift on G basis}. \end{proof}
\iffalse \textit{Proof of Theorem 1.2}. Suppose $T$ is a backward shift on some Markushevich basis $\{f_{k}\}_{k=0}^{\infty}$ of $\mathcal{H}$, i.e., $Tf_{0}=0$ and $Tf_k=f_{k-1}$ for $k\geq1$. Since the sequence $\{f_{k}\}_{k=0}^{\infty}$ is complete, there is unique sequence $\{g_{k}\}_{k=0}^{\infty}$ in $\mathcal{H}$ such that $(f_k,g_k)$ is a biorthogonal system. Note that $$ (f_{j+1},T^*g_i)=(Tf_{j+1},g_i)=(f_j,g_i)=\delta_{ij} $$ and $$ (f_{0},T^*g_i)=(Tf_{0},g_i)=(0,g_i)=0, $$ by the uniqueness, we have that $ T^*g_k=g_{k+1}$ for $k\geq 0$. Since $\mathrm{dim\,ker}(T-z_0)\geq 2$, we can take a non-zero vector $x\in \mathrm{ker}(T-z_0)$ and $(x,g_0)=0$. If $(x,g_k)=0$, then $$ (x,g_{k+1})=(x,T^*g_{k})=(Tx,g_{k})=(z_{0}x,g_k)=0 $$ Hence, it follows by induction that $(x,g_k)=0$ for $k\geq 0$. This implies that $x=0$ because the sequence $\{g_{k}\}_{k=0}^{\infty}$ is total, a contradiction. $
{}\Box$ \fi
\textit{Proof of Theorem \ref{Theorem: A Cowen-Douglas oper never be a shift on basis for n ge 2}}. Since $\mathrm{dim\,ker}(T-z_0)\geq 2$, we can take a non-zero vector $x\in \mathrm{ker}(T-z_0)$ and $(x,g_0)=0$. If $(x,g_k)=0$, then by lemma \ref{Lemma: Adjoint of a Shift on M basis also be a shift} we have $$ (x,g_{k+1})=(x,T^*g_{k})=(Tx,g_{k})=(z_{0}x,g_k)=0 $$ Hence, it follows by induction that $(x,g_k)=0$ for $k\geq 0$. This implies that $x=0$ because the sequence $\{g_{k}\}_{k=0}^{\infty}$ is total, a contradiction. $
{}\Box$
\section{Shift on M-basis or ONB in the Cowen-Douglas class $B_{1}(\Omega)$}
\begin{theorem}\label{Theorem: when Cowen-Douglas operators are shifts} Let $T \in \mathcal{B}_{1}(\Omega)$, $z_0\in \Omega$ and let $B$ be the canonical right inverse of $T-z_{0}$. Then the following statements are equivalent:
(1) $\bigvee_{k=0}^{\infty}\mathrm{ker}{B^*}^k=\mathcal{H}$;
(2) There exists a Markushevich basis $\{f_{k}\}_{k=0}^{\infty}$ of $\mathcal{H}$ which satisfies $f_{0}\bot f_k$ for $k\geq1$ such that $T-z_0$ is a backward shift on $\{f_{k}\}_{k=0}^{\infty}$;
(3) There is a positive real number $\varepsilon$ such that $B^* \in \mathcal{B}_{1}(\mathbb{D}_{\varepsilon})$, where
$\mathbb{D}_{\varepsilon}=\{z\in \mathbb{C}:\ |z|< \varepsilon\}$ \end{theorem} \begin{proof} $(1)\Rightarrow (2)$. Take a unit vector $f_0\in {\rm ker}(T-z_0)$ and set $f_k=B^kf_0$. Then $T-z_0$ is a backward shift on $\{f_{k}\}_{k=0}^{\infty}$. Since $\mathrm{ran}B=(\mathrm{ker}(T-z_0))^\bot$, we know that $f_{0}\bot f_k$ for $k\geq 1$. Note that $$ \gamma(z)=f_0+\sum_{k=1}^{\infty} f_k(z-z_{0})^{k} $$ is the canonical cross-section with the initial vector $f_0$, it follows by Theorem \ref{Theorem: Holomorphic cross-section is spanning} and Theorem \ref{Theorem: Coefficent sequence of psedocanonical section is Minimal} that the sequence $\{f_{k}\}_{k=0}^{\infty}$ is a complete and minimal. By Hahn--Banach Theorem, there is unique sequence $\{g_{k}\}_{k=0}^{\infty}$ in $\mathcal{H}$ with $g_0=f_0$ such that the pair $(f_k,g_k)$ is a biorthogonal system. And, resembling to the proof of Theorem \ref{Theorem: A Cowen-Douglas oper never be a shift on basis for n ge 2}, we know also that $(T-z_{0})^*$ is the forward shift on the vector sequence $\{g_{k}\}_{k=0}^{\infty}$. Moreover, since $B^*(T-z_0)^*=I$ and $$ \mathrm{ker}B^*=(\mathrm{ran}B)^\bot=\mathrm{ker}(T-z_0), $$ we have that $$ B^*g_{0}=B^*f_0=0 \hbox{ and } B^*g_{k+1}=B^*(T-z_0)^*g_{k}=g_k \hbox{ for } k\geq 1 $$ i.e., $B^*$ is the backward shift on the sequence $\{g_{k}\}_{k=0}^{\infty}$. Hence we have that $g_0, \cdots ,g_{k-1}$ are in $\mathrm{ker}{B^*}^k$. Now, making use of index formulas, it follows that $$ \mathrm{dim\,ker}{B^*}^k=\mathrm{ind}{B^*}^k=-\mathrm{ind}B^k =\mathrm{ind}(T-z_0)^k=k. $$ So, we know that the family $\{g_0, \cdots ,g_{k-1}\}$ is a basis of $\mathrm{ker}{B^*}^k$. This implies that $$ \bigvee_{k=0}^{\infty}\{g_k\}=\bigvee_{k=0}^{\infty}\mathrm{ker}{B^*}^k=\mathcal{H}. $$ Thus, $\{f_{k}\}_{k=0}^{\infty}$ is a Markushevich basis.
$(2)\Rightarrow (3)$. As given, there is the biorthogonal system $(f_k,g_k)$ such that $f_{0}\bot f_k$ for $k\geq1$ and the sequence $\{g_{k}\}_{k=0}^{\infty}$ is total. The same as in the preceding, we know that $B^*$ is the backward shift on the sequence $\{g_{k}\}_{k=0}^{\infty}$. Hence, it holds that $$ \bigvee_{k=0}^{\infty}\mathrm{ker}{B^*}^k=\bigvee_{k=0}^{\infty}\{g_k\}=\mathcal{H}. $$ Since $\mathrm{ran}B^*$ is a closed set, we have $0\in \rho _F(B^*)$ and $ \mathrm{ind}{B^*}=1$. Therefore, There is a positive number $\varepsilon$ with $\varepsilon <\frac{1}{r(T-z_0)}$ such that $\mathbb{D}_{\varepsilon}\subset \rho _F(B^*)$ and $ \mathrm{ind}(B^*-z)=1$ for $z\in \mathbb{D}_{\varepsilon}$. Assume that $(B-\overline{z})x=0$ for $z\in \mathbb{D}_{\varepsilon}$ and $x \in\mathcal{H}$. Then $x=\overline{z}(T-z_0)x$ because $B$ is right inverse of $T-z_{0}$. Hence, if $z=0$ then $x=0$; If $z\not=0$ then $(T-z_0-{\overline{z}}^{-1})x=0$. Since ${\overline{z}}^{-1}\notin \sigma(T-z_0)$, it follows that $x=0$. Thus, we obtain that $$ \mathrm{dim\,ker}(B^*-z)=\mathrm{ind}(B^*-z)=1 \hbox{ for } z\in \mathbb{D}_{\varepsilon}. $$ Also, we have that $$ \mathrm{ran}(B^*-z)=(\mathrm{ker}(B-\overline{z}))^\bot=\mathcal{H} \hbox{ for } z\in \mathbb{D}_{\varepsilon}. $$ Now, we have proved that $B^* \in \mathcal{B}_{1}(\Omega_{\varepsilon})$.
$(3)\Rightarrow (1)$. It is obviously. \end{proof}
\begin{example} Suppose $\{e_{n}\}_{n=1}^{\infty}$ is an ONB of the Hilbert space $\mathcal{H}$. Define $$ f_{n}=e_{n}-e_{n+1}, g_{n}=\sum_{k=1}^{n}e_{k}, $$ then $\{(e_{n}, f_{n})\}_{n=1}^{\infty}$ is a biorthogonal system and both $\{f_{n}\}_{n=1}^{\infty}$ and $\{g_{n}\}_{n=1}^{\infty}$ are Markushevicz basis. And it is not hard to see that they are not basis. Moreover, the classical backward shift on the ONB $\{e_{n}\}_{n=1}^{\infty}$ is also a backward shift on the markushevicz basis $\{g_{n}\}_{n=1}^{\infty}$.
\end{example}
For $T \in \mathcal{B}_{1}(\Omega)$, using the curvature function of the bundle $E_T$, M. J. Cowen and R. G. Douglas gave a characterization of $T$ being a backward weighted shift on an orthonormal basis \cite[Corollary 1.9]{Douglas1}. Now, in terminology of operator theory, we give another characterization.
\begin{theorem}\label{Theorem: C-D operators are shifts (1)} Let $T \in \mathcal{B}_{1}(\Omega)$, $z_0\in \Omega$ and let $B$ be the canonical right inverse of $T-z_{0}$. Then $T-z_0$ is a backward weighted shift on an orthonormal basis of $\mathcal{H}$ if and only if, for every $k\geq 0$, the subspace ${\mathcal M}_k:=B^k\mathrm{ker}(T-z_0)$ is invariant for $B^*B$. \end{theorem} \begin{proof} Suppose that $T-z_0$ is a backward weighted shift on an orthonormal basis $\{e_{k}\}_{k=0}^{\infty}$ with the weight sequence $\{w_{k}\}_{k=1}^{\infty}$, i.e., $(T-z_0)e_0=0$ and $(T-z_0)e_k=w_ke_{k-1}$ for $k>0$. Note that $T-z_0$ is in $\mathcal{B}_{1}(\Omega_0)$, where $\Omega_0=\Omega - \{z_0\}$, it follows from \cite[Theorem 3.2]{Li} that the sequence
$\{|w_{k}|^{-1}\}_{k=1}^{\infty}$ is bounded. It is easy to verify that $B$ is a forward weighted shift on the orthonormal basis $\{e_{k}\}_{k=0}^{\infty}$ with the weight sequence $\{w_{k}^{-1}\}_{k=1}^{\infty}$, i.e., $Be_k=w_k^{-1}e_{k+1}$ for $k \geq 0$ and $B^*$ is a backward weighted shift $\{e_{k}\}_{k=0}^{\infty}$ with the weight sequence $\{\overline{w}_{k}^{-1}\}_{k=1}^{\infty}$. So we have that
$B^*Be_k=|w_{k}|^{-2}e_k$ for $k \geq 0$. Since $\mathrm{ker}(T-z_0)=\mathrm{span}\{e_0\}$ we know ${\mathcal M}_k$ is invariant for $B^*B$.
Conversely, if ${\mathcal M}_k$ is invariant for $B^*B$ then we take a unit vector $f_0$ from $\mathrm{ker}(T-z_0)$ and set $f_k=B^kf_0$. Similar to the proof of $(1)\Rightarrow (2)$ in Theorem \ref{Theorem: Cowen-Douglas operators are shifts}, we know that the sequence $\{f_{k}\}_{k=0}^{\infty}$ is a complete and minimal and $T-z_0$ is a backward shift on $\{f_{k}\}_{k=0}^{\infty}$. And, by Lemma \ref{Lemma: Existence of Canonical Right Inverse}, we have that $f_0\perp f_k$ for $k>0$. Since $f_k \in {\mathcal M}_k$ and $\mathrm{dim}{\mathcal M}_k=1$, it follows that $B^*Bf_k=\lambda_kf_k$. If $(f_i,f_j)=0$ for $i<j$, then $$ (f_{i+1},f_{j+1})=(Bf_i,Bf_j)=(B^*Bf_i,f_j)=(\lambda_if_i,f_j)=0. $$
This implies that the sequence $\{f_{k}\}_{k=0}^{\infty}$ is orthogonal. Set $e_k=\frac{f_{k}}{\|f_k\|}$. Then we have that the sequence $\{e_{k}\}_{k=0}^{\infty}$ is an orthonormal basis of $\mathcal{H}$ and $T-z_0$ is a backward weighted shift on $\{e_{k}\}_{k=0}^{\infty}$. \end{proof}
Imitating the proof of Theorem \ref{Theorem: C-D operators are shifts (1)}, we can also come to the following.
\begin{theorem}\label{Theorem: C-D operators are shifts (2)} Let $T \in \mathcal{B}_{1}(\Omega)$, $z_0\in \Omega$ and let $B$ be the canonical right inverse of $T-z_{0}$. Then $T-z_0$ is a backward shift on an orthonormal basis of $\mathcal{H}$ if and only if $B^*B=I$, i.e., $B$ is an isometry. \end{theorem}
Recall that a sequence $\{f_{k}\}_{k=0}^{\infty}$ is called a \textit{Schauder basis} of $\mathcal{H}$ if for every vector $x \in \mathcal{H}$ there exists a unique sequence $\{\alpha_{k}\}_{k=0}^{\infty}$ of complex numbers such that the series $\sum_{k=0}^{\infty} \alpha_{k}f_{k}$ converges to $x$ in norm.
\iffalse Suppose there is a Schauder basis $\{f_{k}\}_{k=0}^{\infty}$ of $\mathcal{H}$ such that $Tf_{0}=0$ and $Tf_k=f_{k-1}$ for $k\geq1$. Take a $z_0\in \Omega $ and $x \in {\rm ker}(T-z_0)$, and let $x=\sum_{k=0}^{\infty} \alpha_{k}f_{k}$. Since $Tx=z_0x$, we have $$ \sum_{k=0}^{\infty} \alpha_{k+1}f_{k}=z_0\sum_{k=0}^{\infty} \alpha_{k}f_{k}. $$ Hence, it follows that $\alpha_{k}=\alpha_{0}z_0^{k}$. So, $\mathrm{dim\,ker}(T-z_0)=1$. This is a contradiction. \fi
\begin{example}\label{Example: Shift on conditional basis} Although theorem \ref{Theorem: A Cowen-Douglas oper never be a shift on basis for n ge 2} tell us that for $n \ge 2$, a Cowen-Douglas operator never be a shift on basis, if we consider the n-multiple shift case(or more general, operator-weighted shift) then we can get more interest examples. Here we show that $S^{2}$ can be seen as a 2-multiple shift on some conditional basis. Suppose $\{e_{n}\}_{n=1}^{\infty}$ is an ONB of the Hilbert space $\mathcal{H}$. Let $\{\alpha_{n}\}_{n=1}^{\infty}$ be a sequence of positive numbers such that $\sum_{n=1}^{\infty}n\alpha_{n}^{2}<\infty$ and $\sum_{n=1}^{\infty} \alpha_{n}=\infty$. Then the sequences $\{f_{n}\}_{n=1}^{\infty}, \{g_{n}\}_{n=1}^{\infty}$ defined as $$\begin{array}{ll} f_{2n-1}=e_{2n-1}+\sum_{i=n}^{\infty} \alpha_{i-n+1}e_{2i}, &~~f_{2n}=e_{2n}, ~~n=1, 2, \cdots \\ g_{2n-1}=e_{2n-1}, &~~g_{2n}=\sum_{i=n}^{\infty} -\alpha_{i-n+1}e_{2i-1}+e_{2n}, ~~n=1, 2, \cdots \end{array} $$ are both conditional basis of $\mathcal{H}$(see \cite{Singer1}, Example14.5, p429.). Let $S$ be the classical foreward shift on $\{e_{n}\}_{n=1}^{\infty}$. Then we have $$ S^{2}f_{n}=f_{n+2}, \hbox{ for }n\ge 1 $$ and $$ (S^{*})^{2}g_{n}=g_{n-2}, \hbox{ for } n>2, \hbox{ and } (S^{*})^{2}g_{1}=(S^{*})^{2}g_{2}=0. $$ Hence $S^{2}$ is a foreward 2-multiple shift on a conditional basis, and $(S^{*})^{2}$ is a foreward 2-multiple shift on a conditional basis. \end{example}
\begin{question} When a Cowen-Douglas operator in $\mathcal{B}_{1}(\Omega)$ can be a shift on some Schauder basis? \end{question}
\end{document} | arXiv |
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Defining LFIs and LFUs in extensions of infectious logics
Szmuc Damian Enrique
Journal of Applied Non-Classical Logics 26 (4):286-314 (2016)
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Logics of Variable Inclusion and the Lattice of Consequence Relations.Michele Pra Baldi - 2020 - Journal of Applied Non-Classical Logics 30 (4):367-381.details
In this paper, first, we determine the number of sublogics of variable inclusion of an arbitrary finitary logic ⊢ with a composition term. Then, we investigate their position into the lattice of co...
Logics of Left Variable Inclusion and Płonka Sums of Matrices.S. Bonzio, T. Moraschini & M. Pra Baldi - 2020 - Archive for Mathematical Logic (1-2):49-76.details
The paper aims at studying, in full generality, logics defined by imposing a variable inclusion condition on a given logic \. We prove that the description of the algebraic counterpart of the left variable inclusion companion of a given logic \ is related to the construction of Płonka sums of the matrix models of \. This observation allows to obtain a Hilbert-style axiomatization of the logics of left variable inclusion, to describe the structure of their reduced models, and to locate (...) them in the Leibniz hierarchy. (shrink)
Logics, Misc in Logic and Philosophy of Logic
Nonclassical Logics in Logic and Philosophy of Logic
A Duality for Involutive Bisemilattices.Stefano Bonzio, Andrea Loi & Luisa Peruzzi - 2019 - Studia Logica 107 (2):423-444.details
We establish a duality between the category of involutive bisemilattices and the category of semilattice inverse systems of Stone spaces, using Stone duality from one side and the representation of involutive bisemilattices as Płonka sum of Boolean algebras, from the other. Furthermore, we show that the dual space of an involutive bisemilattice can be viewed as a GR space with involution, a generalization of the spaces introduced by Gierz and Romanowska equipped with an involution as additional operation.
Many-Valued Logic in Logic and Philosophy of Logic
Nonclassical Logic, Misc in Logic and Philosophy of Logic
Paraconsistent Logic in Logic and Philosophy of Logic
Theories of Truth Based on Four-Valued Infectious Logics.Damian Szmuc, Bruno Da Re & Federico Pailos - 2020 - Logic Journal of the IGPL 28 (5):712-746.details
Infectious logics are systems that have a truth-value that is assigned to a compound formula whenever it is assigned to one of its components. This paper studies four-valued infectious logics as the basis of transparent theories of truth. This take is motivated as a way to treat different pathological sentences differently, namely, by allowing some of them to be truth-value gluts and some others to be truth-value gaps and as a way to treat the semantic pathology suffered by at least (...) some of these sentences as infectious. This leads us to consider four distinct four-valued logics: one where truth-value gaps are infectious, but gluts are not; one where truth-value gluts are infectious, but gaps are not; and two logics where both gluts and gaps are infectious, in some sense. Additionally, we focus on the proof theory of these systems, by offering a discussion of two related topics. On the one hand, we prove some limitations regarding the possibility of providing standard Gentzen sequent calculi for these systems, by dualizing and extending some recent results for infectious logics. On the other hand, we provide sound and complete four-sided sequent calculi, arguing that the most important technical and philosophical features taken into account to usually prefer standard calculi are, indeed, enjoyed by the four-sided systems. (shrink)
Paradoxes in Logic and Philosophy of Logic
Proof Theory in Logic and Philosophy of Logic
From Logics of Formal Inconsistency to Logics of Formal Classicality.Hitoshi Omori - 2020 - Logic Journal of the IGPL 28 (5):684-711.details
One of the oldest systems of paraconsistent logic is the set of so-called C-systems of Newton da Costa, and this has been generalized into a family of systems now known as logics of formal inconsistencies by Walter Carnielli, Marcelo Coniglio and João Marcos. The characteristic notion in these systems is the so-called consistency operator which, roughly speaking, indicates how gluts are behaving. One natural question then is to ask if we can let not only gluts but also gaps be around (...) and generalize the notion of consistency into classicality. This is already considered by Andréa Loparić and da Costa in the style of C-systems. The aim of this paper is to develop a family of systems that generalizes the system of Loparić and da Costa which may be called logics of formal classicality. (shrink)
Normality Operators and Classical Recapture in Many-Valued Logic.Roberto Ciuni & Massimiliano Carrara - 2020 - Logic Journal of the IGPL 28 (5):657-683.details
In this paper, we use a 'normality operator' in order to generate logics of formal inconsistency and logics of formal undeterminedness from any subclassical many-valued logic that enjoys a truth-functional semantics. Normality operators express, in any many-valued logic, that a given formula has a classical truth value. In the first part of the paper we provide some setup and focus on many-valued logics that satisfy some of the three properties, namely subclassicality and two properties that we call fixed-point negation property (...) and conservativeness. In the second part of the paper, we introduce normality operators and explore their formal behaviour. In the third and final part of the paper, we establish a number of classical recapture results for systems of formal inconsistency and formal undeterminedness that satisfy some or all the properties above. These are the main formal results of the paper. Also, we illustrate concrete cases of recapture by discussing the logics $\mathsf{K}^{\circledast }_{3}$, $\mathsf{LP}^{\circledast }$, $\mathsf{K}^{w\circledast }_{3}$, $\mathsf{PWK}^{\circledast }$ and $\mathsf{E_{fde}}^{\circledast }$, that are in turn extensions of $\mathsf{{K}_{3}}$, $\mathsf{LP}$, $\mathsf{K}^{w}_{3}$, $\mathsf{PWK}$ and $\mathsf{E_{fde}}$, respectively. (shrink)
The Keisler–Shelah Theorem for $\Mathsf{QmbC}$ Through Semantical Atomization.Thomas Macaulay Ferguson - 2020 - Logic Journal of the IGPL 28 (5):912-935.details
In this paper, we consider some contributions to the model theory of the logic of formal inconsistency $\mathsf{QmbC}$ as a reply to Walter Carnielli, Marcelo Coniglio, Rodrigo Podiacki and Tarcísio Rodrigues' call for a 'wider model theory.' This call demands that we align the practices and techniques of model theory for logics of formal inconsistency as closely as possible with those employed in classical model theory. The key result is a proof that the Keisler–Shelah isomorphism theorem holds for $\mathsf{QmbC}$, i.e. (...) that the strong elementary equivalence of two $\mathsf{QmbC}$ models $\mathfrak{A}$ and $\mathfrak{B}$ is equivalent to them having strongly isomorphic ultrapowers. As intermediate steps, we introduce some notions of model-theoretic equivalence between $\mathsf{QmbC}$ models, explicitly prove Łoś' theorem and introduce a useful technique of model-theoretic 'atomization' in which the satisfaction sets of non-deterministically evaluated formulae are associated with new predicates. Finally, we consider some of the extensions of $\mathsf{QmbC}$, explicitly showing that Keisler–Shelah holds for $\mathsf{QCi}$ and suggesting that it holds of extensions like $\mathsf{QCila}$ and $\mathsf{QCia}$ as well. (shrink)
Exactly True and Non-Falsity Logics Meeting Infectious Ones.Alex Belikov & Yaroslav Petrukhin - 2020 - Journal of Applied Non-Classical Logics 30 (2):93-122.details
In this paper, we study logical systems which represent entailment relations of two kinds. We extend the approach of finding 'exactly true' and 'non-falsity' versions of four-valued logics that emerged in series of recent works [Pietz & Rivieccio (2013). Nothing but the truth. Journal of Philosophical Logic, 42(1), 125–135; Shramko (2019). Dual-Belnap logic and anything but falsehood. Journal of Logics and their Applications, 6, 413–433; Shramko et al. (2017). First-degree entailment and its relatives. Studia Logica, 105(6), 1291–1317] to the case (...) of infectious logics, namely to the case of Deutsch's logic Sfde introduced in Deutsch [Relevant analytic entailment. The Relevance Logic Newsletter, 2(1), 26–44; The completeness of S. Studia Logica, 38(2), 137–147]. The particular systems obtained in this way are Setl and Snfl. We present them in the form of sequent calculi and prove corresponding soundness and completeness theorems. We illuminate the connection between Setl, Snfl and two well-known systems, strong Kleene three-valued logic K3 and Priest's Logic of Paradox LP. This connection allows us to investigate the characterisation of the entailment relations associated with Setl and Snfl as well as to introduce the notion of 'infectious analogue' of a certain logic. We also study implicative extensions of Setl and Snfl and prove soundness and completeness theorems for them as well. (shrink)
Logical Consequence and Entailment in Logic and Philosophy of Logic
Semantical Analysis of Weak Kleene Logics.Roberto Ciuni & Massimiliano Carrara - 2019 - Journal of Applied Non-Classical Logics 29 (1):1-36.details
This paper presents a semantical analysis of the Weak Kleene Logics Kw3 and PWK from the tradition of Bochvar and Halldén. These are three-valued logics in which a formula takes the third value if at least one of its components does. The paper establishes two main results: a characterisation result for the relation of logical con- sequence in PWK – that is, we individuate necessary and sufficient conditions for a set.
Two-Valued Weak Kleene Logics.Bruno da Ré & Damian Szmuc - 2019 - Manuscrito 42 (1):1-43.details
In the literature, Weak Kleene logics are usually taken as three-valued logics. However, Suszko has challenged the main idea of many-valued logic claiming that every logic can be presented in a two-valued fashion. In this paper, we provide two-valued semantics for the Weak Kleene logics and for a number of four-valued subsystems of them. We do the same for the so-called Logics of Nonsense, which are extensions of the Weak Kleene logics with unary operators that allow looking at them as (...) Logics of Formal Inconsistency and Logics of Formal Underterminedness. Our aim with this work, rather than arguing for Suszko's thesis, is to show that two-valued presentations of these peculiar logics enlighten the non-standard behavior of their logical connectives. More specifically, the two-valued presentations of paraconsistent logics illustrate and clarify the disjunctive flavor of the conjunction, and dually, the two-valued presentations of paracomplete subsystems of Weak Kleene logics reveal the conjunctive flavor of the disjunction. (shrink)
Dualities for Płonka Sums.Stefano Bonzio - 2018 - Logica Universalis 12 (3-4):327-339.details
Płonka sums consist of an algebraic construction similar, in some sense, to direct limits, which allows to represent classes of algebras defined by means of regular identities. Recently, Płonka sums have been connected to logic, as they provide algebraic semantics to logics obtained by imposing a syntactic filter to given logics. In this paper, I present a very general topological duality for classes of algebras admitting a Płonka sum representation in terms of dualisable algebras.
Generalized Correspondence Analysis for Three-Valued Logics.Yaroslav Petrukhin - 2018 - Logica Universalis 12 (3-4):423-460.details
Correspondence analysis is Kooi and Tamminga's universal approach which generates in one go sound and complete natural deduction systems with independent inference rules for tabular extensions of many-valued functionally incomplete logics. Originally, this method was applied to Asenjo–Priest's paraconsistent logic of paradox LP. As a result, one has natural deduction systems for all the logics obtainable from the basic three-valued connectives of LP -language) by the addition of unary and binary connectives. Tamminga has also applied this technique to the paracomplete (...) analogue of LP, strong Kleene logic \. In this paper, we generalize these results for the negative fragments of LP and \, respectively. Thus, the method of correspondence analysis works for the logics which have the same negations as LP or \, but either have different conjunctions or disjunctions or even don't have them as well at all. Besides, we show that correspondence analyses for the negative fragments of \ and LP, respectively, are also suitable without any changes for the negative fragments of Heyting's logic \ and its dual \ and LP). (shrink) | CommonCrawl |
Implicit curve
In mathematics, an implicit curve is a plane curve defined by an implicit equation relating two coordinate variables, commonly x and y. For example, the unit circle is defined by the implicit equation $x^{2}+y^{2}=1$. In general, every implicit curve is defined by an equation of the form
$F(x,y)=0$
for some function F of two variables. Hence an implicit curve can be considered as the set of zeros of a function of two variables. Implicit means that the equation is not expressed as a solution for either x in terms of y or vice versa.
If $F(x,y)$ is a polynomial in two variables, the corresponding curve is called an algebraic curve, and specific methods are available for studying it.
Plane curves can be represented in Cartesian coordinates (x, y coordinates) by any of three methods, one of which is the implicit equation given above. The graph of a function is usually described by an equation $y=f(x)$ in which the functional form is explicitly stated; this is called an explicit representation. The third essential description of a curve is the parametric one, where the x- and y-coordinates of curve points are represented by two functions x(t), y(t) both of whose functional forms are explicitly stated, and which are dependent on a common parameter $t.$
Examples of implicit curves include:
1. a line: $x+2y-3=0,$
2. a circle: $x^{2}+y^{2}-4=0,$
3. the semicubical parabola: $x^{3}-y^{2}=0,$
4. Cassini ovals $(x^{2}+y^{2})^{2}-2c^{2}(x^{2}-y^{2})-(a^{4}-c^{4})=0$ (see diagram),
5. $\sin(x+y)-\cos(xy)+1=0$ (see diagram).
The first four examples are algebraic curves, but the last one is not algebraic. The first three examples possess simple parametric representations, which is not true for the fourth and fifth examples. The fifth example shows the possibly complicated geometric structure of an implicit curve.
The implicit function theorem describes conditions under which an equation $F(x,y)=0$ can be solved implicitly for x and/or y – that is, under which one can validly write $x=g(y)$ or $y=f(x)$. This theorem is the key for the computation of essential geometric features of the curve: tangents, normals, and curvature. In practice implicit curves have an essential drawback: their visualization is difficult. But there are computer programs enabling one to display an implicit curve. Special properties of implicit curves make them essential tools in geometry and computer graphics.
An implicit curve with an equation $F(x,y)=0$ can be considered as the level curve of level 0 of the surface $z=F(x,y)$ (see third diagram).
Slope and curvature
In general, implicit curves fail the vertical line test (meaning that some values of x are associated with more than one value of y) and so are not necessarily graphs of functions. However, the implicit function theorem gives conditions under which an implicit curve locally is given by the graph of a function (so in particular it has no self-intersections). If the defining relations are sufficiently smooth then, in such regions, implicit curves have well defined slopes, tangent lines, normal vectors, and curvature.
There are several possible ways to compute these quantities for a given implicit curve. One method is to use implicit differentiation to compute the derivatives of y with respect to x. Alternatively, for a curve defined by the implicit equation $F(x,y)=0$, one can express these formulas directly in terms of the partial derivatives of $F$. In what follows, the partial derivatives are denoted $F_{x}$ (for the derivative with respect to x), $F_{y}$, $F_{xx}$ (for the second partial with respect to x), $F_{xy}$ (for the mixed second partial), $F_{yy}.$
Tangent and normal vector
A curve point $(x_{0},y_{0})$ is regular if the first partial derivatives $F_{x}(x_{0},y_{0})$ and $F_{y}(x_{0},y_{0})$ are not both equal to 0.
The equation of the tangent line at a regular point $(x_{0},y_{0})$ is
$F_{x}(x_{0},y_{0})(x-x_{0})+F_{y}(x_{0},y_{0})(y-y_{0})=0,$
so the slope of the tangent line, and hence the slope of the curve at that point, is
${\text{slope}}=-{\frac {F_{x}(x_{0},y_{0})}{F_{y}(x_{0},y_{0})}}.$
If $F_{y}(x,y)=0\neq F_{x}(x,y)$ at $(x_{0},y_{0}),$ the curve is vertical at that point, while if both $F_{y}(x,y)=0$ and $F_{x}(x,y)=0$ at that point then the curve is not differentiable there, but instead is a singular point – either a cusp or a point where the curve intersects itself.
A normal vector to the curve at the point is given by
$\mathbf {n} (x_{0},y_{0})=(F_{x}(x_{0},y_{0}),F_{y}(x_{0},y_{0}))$
(here written as a row vector).
Curvature
For readability of the formulas, the arguments $(x_{0},y_{0})$ are omitted. The curvature $\kappa $ at a regular point is given by the formula
$\kappa ={\frac {-F_{y}^{2}F_{xx}+2F_{x}F_{y}F_{xy}-F_{x}^{2}F_{yy}}{(F_{x}^{2}+F_{y}^{2})^{3/2}}}$.[1]
Derivation of the formulas
The implicit function theorem guarantees within a neighborhood of a point $(x_{0},y_{0})$ the existence of a function $f$ such that $F(x,f(x))=0$. By the chain rule, the derivatives of function $f$ are
$f'(x)=-{\frac {F_{x}(x,f(x))}{F_{y}(x,f(x))}}$ and $f''(x)={\frac {-F_{y}^{2}F_{xx}+2F_{x}F_{y}F_{xy}-F_{x}^{2}F_{yy}}{F_{y}^{3}}}$
(where the arguments $(x,f(x))$ on the right side of the second formula are omitted for ease of reading).
Inserting the derivatives of function $f$ into the formulas for a tangent and curvature of the graph of the explicit equation $y=f(x)$ yields
$y=f(x_{0})+f'(x_{0})(x-x_{0})$ (tangent)
$\kappa (x_{0})={\frac {f''(x_{0})}{(1+f'(x_{0})^{2})^{3/2}}}$ (curvature).
Advantage and disadvantage of implicit curves
Disadvantage
The essential disadvantage of an implicit curve is the lack of an easy possibility to calculate single points which is necessary for visualization of an implicit curve (see next section).
Advantages
1. Implicit representations facilitate the computation of intersection points: If one curve is represented implicitly and the other parametrically the computation of intersection points needs only a simple (1-dimensional) Newton iteration, which is contrary to the cases implicit-implicit and parametric-parametric (see Intersection).
2. An implicit representation $F(x,y)=0$ gives the possibility of separating points not on the curve by the sign of $F(x,y)$. This may be helpful for example applying the false position method instead of a Newton iteration.
3. It is easy to generate curves which are almost geometrically similar to the given implicit curve $F(x,y)=0,$ by just adding a small number: $F(x,y)-c=0$ (see section #Smooth approximations).
Applications of implicit curves
Within mathematics implicit curves play a prominent role as algebraic curves. In addition, implicit curves are used for designing curves of desired geometrical shapes. Here are two examples.
Convex polygons
A smooth approximation of a convex polygon can be achieved in the following way: Let $g_{i}(x,y)=a_{i}x+b_{i}y+c_{i}=0,\ i=1,\dotsc ,n$ be the equations of the lines containing the edges of the polygon such that for an inner point of the polygon $g_{i}$ is positive. Then a subset of the implicit curve
$F(x,y)=g_{1}(x,y)\cdots g_{n}(x,y)-c=0$
with suitable small parameter $c$ is a smooth (differentiable) approximation of the polygon. For example, the curves
$F(x,y)=(x+1)(-x+1)y(-x-y+2)(x-y+2)-c=0$ for $c=0.03,\dotsc ,0.6$
contain smooth approximations of a polygon with 5 edges (see diagram).
Pairs of lines
In case of two lines
$F(x,y)=g_{1}(x,y)g_{2}(x,y)-c=0$
one gets
a pencil of parallel lines, if the given lines are parallel or
the pencil of hyperbolas, which have the given lines as asymptotes.
For example, the product of the coordinate axes variables yields the pencil of hyperbolas $xy-c=0,\ c\neq 0$, which have the coordinate axes as asymptotes.
Others
If one starts with simple implicit curves other than lines (circles, parabolas,...) one gets a wide range of interesting new curves. For example,
$F(x,y)=y(-x^{2}-y^{2}+1)-c=0$
(product of a circle and the x-axis) yields smooth approximations of one half of a circle (see picture), and
$F(x,y)=(-x^{2}-(y+1)^{2}+4)(-x^{2}-(y-1)^{2}+4)-c=0$
(product of two circles) yields smooth approximations of the intersection of two circles (see diagram).
Blending curves
In CAD one uses implicit curves for the generation of blending curves,[2][3] which are special curves establishing a smooth transition between two given curves. For example,
$F(x,y)=(1-\mu )f_{1}f_{2}-\mu (g_{1}g_{2})^{3}=0$
generates blending curves between the two circles
$f_{1}(x,y)=(x-x_{1})^{2}+y^{2}-r_{1}^{2}=0,$
$f_{2}(x,y)=(x-x_{2})^{2}+y^{2}-r_{2}^{2}=0.$
The method guarantees the continuity of the tangents and curvatures at the points of contact (see diagram). The two lines
$g_{1}(x,y)=x-x_{1}=0,\ g_{2}(x,y)=x-x_{2}=0$
determine the points of contact at the circles. Parameter $\mu $ is a design parameter. In the diagram, $\mu =0.05,\dotsc ,0.2$.
Equipotential curves of two point charges
Equipotential curves of two equal point charges at the points $P_{1}=(1,0),\;P_{2}=(-1,0)$ can be represented by the equation
$f(x,y)={\frac {1}{|PP_{1}|}}+{\frac {1}{|PP_{2}|}}-c$
$={\frac {1}{\sqrt {(x-1)^{2}+y^{2}}}}+{\frac {1}{\sqrt {(x+1)^{2}+y^{2}}}}-c=0.$
The curves are similar to Cassini ovals, but they are not such curves.
Visualization of an implicit curve
To visualize an implicit curve one usually determines a polygon on the curve and displays the polygon. For a parametric curve this is an easy task: One just computes the points of a sequence of parametric values. For an implicit curve one has to solve two subproblems:
1. determination of a first curve point to a given starting point in the vicinity of the curve,
2. determination of a curve point starting from a known curve point.
In both cases it is reasonable to assume $\operatorname {grad} F\neq (0,0)$. In practice this assumption is violated at single isolated points only.
Point algorithm
For the solution of both tasks mentioned above it is essential to have a computer program (which we will call ${\mathsf {CPoint}}$), which, when given a point $Q_{0}=(x_{0},y_{0})$ near an implicit curve, finds a point $P$ that is exactly on the curve:
(P1) for the start point is $j=0$
(P2) repeat
$(x_{j+1},y_{j+1})=(x_{j},y_{j})-{\frac {F(x_{j},y_{j})}{F_{x}(x_{j},y_{j})^{2}+F_{y}(x_{j},y_{j})^{2}}}\,\left(F_{x}(x_{j},y_{j}),F_{y}(x_{j},y_{j})\right)$
( Newton step for function $g(t)=F\left(x_{j}+tF_{x}(x_{j},y_{j}),y_{j}+tF_{y}(x_{j},y_{j})\right)\ .$)
(P3) until the distance between the points $(x_{j+1},y_{j+1}),\,(x_{j},y_{j})$ is small enough.
(P4) $P=(x_{j+1},y_{j+1})$ is the curve point near the start point $Q_{0}$.
Tracing algorithm
In order to generate a nearly equally spaced polygon on the implicit curve one chooses a step length $s$ and
(T1) chooses a suitable starting point in the vicinity of the curve
(T2) determines a first curve point $P_{1}$ using program ${\mathsf {CPoint}}$
(T3) determines the tangent (see above), chooses a starting point on the tangent using step length $s$ (see diagram) and determines a second curve point $P_{2}$ using program ${\mathsf {CPoint}}$ .
$\cdots $
Because the algorithm traces the implicit curve it is called a tracing algorithm. The algorithm traces only connected parts of the curve. If the implicit curve consists of several parts it has to be started several times with suitable starting points.
Raster algorithm
If the implicit curve consists of several or even unknown parts, it may be better to use a rasterisation algorithm. Instead of exactly following the curve, a raster algorithm covers the entire curve in so many points that they blend together and look like the curve.
(R1) Generate a net of points (raster) on the area of interest of the x-y-plane.
(R2) For every point $P$ in the raster, run the point algorithm ${\mathsf {CPoint}}$ starting from P, then mark its output.
If the net is dense enough, the result approximates the connected parts of the implicit curve. If for further applications polygons on the curves are needed one can trace parts of interest by the tracing algorithm.
Implicit space curves
Any space curve which is defined by two equations
${\begin{matrix}F(x,y,z)=0,\\G(x,y,z)=0\end{matrix}}$
is called an implicit space curve.
A curve point $(x_{0},y_{0},z_{0})$ is called regular if the cross product of the gradients $F$ and $G$ is not $(0,0,0)$ at this point:
$\mathbf {t} (x_{0},y_{0},z_{0})=\operatorname {grad} F(x_{0},y_{0},z_{0})\times \operatorname {grad} G(x_{0},y_{0},z_{0})\neq (0,0,0);$
otherwise it is called singular. Vector $\mathbf {t} (x_{0},y_{0},z_{0})$ is a tangent vector of the curve at point $(x_{0},y_{0},z_{0}).$
Examples:
$(1)\quad x+y+z-1=0\ ,\ x-y+z-2=0$
is a line.
$(2)\quad x^{2}+y^{2}+z^{2}-4=0\ ,\ x+y+z-1=0$
is a plane section of a sphere, hence a circle.
$(3)\quad x^{2}+y^{2}-1=0\ ,\ x+y+z-1=0$
is an ellipse (plane section of a cylinder).
$(4)\quad x^{2}+y^{2}+z^{2}-16=0\ ,\ (y-y_{0})^{2}+z^{2}-9=0$
is the intersection curve between a sphere and a cylinder.
For the computation of curve points and the visualization of an implicit space curve see Intersection.
See also
• Implicit surface
References
1. Goldman, R. (2005). "Curvature formulas for implicit curves and surfaces". Computer Aided Geometric Design. 22 (7): 632. CiteSeerX 10.1.1.413.3008. doi:10.1016/j.cagd.2005.06.005.
2. C. Hoffmann & J. Hopcroft: The potential method for blending surfaces and corners in G. Farin (Ed) Geometric-Modeling, SIAM, Philadelphia, pp. 347-365
3. E. Hartmann: Blending of implicit surfaces with functional splines, CAD,Butterworth-Heinemann, Volume 22 (8), 1990, p. 500-507
4. G. Taubin: Distance Approximations for Rastering Implicit Curves. ACM Transactions on Graphics, Vol. 13, No. 1, 1994.
• Gomes, A., Voiculescu, I., Jorge, J., Wyvill, B., Galbraith, C.: Implicit Curves and Surfaces: Mathematics, Data Structures and Algorithms, 2009, Springer-Verlag London, ISBN 978-1-84882-405-8
• C:L: Bajaj, C.M. Hoffmann, R.E. Lynch: Tracing surface intersections, Comp. Aided Geom. Design 5 (1988), 285-307.
• Geometry and Algorithms for COMPUTER AIDED DESIGN
External links
Wikimedia Commons has media related to Implicit curves.
• Famous Curves
| Wikipedia |
Rollback to Blog - the color of night Rev #9
<h2>Or: how big is the "greenhouse effect" really?</h2> +-- {: .standout} This page is a [[Blog articles in progress|blog article in progress]], written by [[Tim van Beek]]. =-- When we talked about <a href="http://johncarlosbaez.wordpress.com/2011/06/19/putting-the-earth-in-a-box/">putting the Earth in a box</a>, we saw that there is a gap of about 33 kelvin between the temperature of a black body in Earth's orbit with an albedo of 0.3, and the estimated average surface temperature on Earth. An effect that explains this gap would need to 1) have a steady and continuous influence over thousands of years, 2) have a global impact, 3) be rather strong, because heating the planet Earth by 33 kelvin on the average needs a lot of energy. Last time, in [[a quantum of warmth]], we refined our zero dimensional energy balance model that treats the Earth as an ideal black body, and separated the system into a black body surface and a box containing the atmosphere. With the help of quantum mechanics we saw that: * Earth emits mainly far infrared radiation, while the radiation from the sun is mostly in the near infrared, visible and ultraviolett range. * Only very special components of the atmosphere react to infrared radiation. Not the main components $O_2$ and $N_2$, but minor components with more than two atoms in a molecule, like $H_2 O$, $O_3$ and $CO_2$. These gases react to infrared radiation: They absorb and re-emit a part of Earth's emission back to the surface. * This <b>downward longwave radiation (DLR)</b> leads to an increased incoming energy flux from the viewpoint of the surface. This is an effect that certainly matches points 1 and 2: It is both continuous and global. But how <i>strong</i> is it? What do we need to know in order to calculate it? And is it measurable? <h4>Survival in a combat zone</h4> There has been a lively - sometimes hostile - debate about the "greenhouse effect" which is the popular name for the increase of incoming energy flux caused by infrared active atmospheric components, so maybe you think that the heading above refers to <i>that</i>. But I have different point in mind: Maybe you heard about guiding systems for missiles that chase "heat"? Do not worry if you have not. Knowlegeable people working for the armed forces of the USA know about this, and know that an important aspect of the design of aircrafts is to reduce infrared emission. Let's see what they wrote about this back in 1982: <blockquote> The engine hot metal and airframe surface emissions exhibit spectral IR continuum characteristics which are dependent on the temperature and emissivity-area of the radiating surface. These IR sources radiate in a relatively broad wavelength interval with a spectral shape in accordance with Planck's Law (i.e., with a blackbody spectral shape). The surface- reflected IR radiation will also appear as a continuum based on the equivalent blackbody temperature of the incident radiation (e.g., the sun has a spectral shape characteristic of a 5527°C blackbody). Both the direct (specular) as well as the diffuse (Lambertian) reflected IR radiation components, which are a function of the surface texture and the relative orientation of the surface to the source, must be included. The remaining IR source, engine plume emission, is a composite primarily of C02 and H20 molecular emission spectra. The spectral strength and linewidth of these emissions are dependent on the temperature and concentration of the hot gaseous species in the plume which are a function of the aircraft altitude, flight speed, and power setting. </blockquote> This is an excerpt from page 15 of * Military Handbook <i>SURVIVABILITY ENHANCEMENT, AIRCRAFT CONVENTIONAL WEAPON THREATS, DESIGN AND EVALUATION GUIDELINES</i>, <a href="http://www.everyspec.com/MIL-HDBK/MIL-HDBK+%280200+-+0299%29/download.php?spec=MIL_HDBK_268.1863.pdf">MIL-HDBK-268(AS)</a>, 5 August 1982 You may notice that the authors point out the difference of a continuous black body radiation and the **molecular emission spectra** of $CO_2$ and $H_2 O$. The reason for this, as mentioned last time in [[a quantum of warmth]], is that according to quantum mechanics molecules can emit and absorb radiation at specific energies, i.e. wavelengths, only. For this reason it is possible to distinguish far infrared radiation that is emitted by the surface of the Earth (more or less continuous spectrum) from the radiation that is emitted by the atmosphere (more or less discrete spectrum). Last time I told you that only certain molecules like $CO_2$ and $H_2O$ are infrared active. The authors seem to agree with me. But why is that? Since last time we had some discussions about wether there is a simple explanation for this, I would like to try to provide one. When we try to understand the interaction of atoms and molecules with light, the most important concept that we need to understand it that of a <b><a href="http://en.wikipedia.org/wiki/Electric_dipole_moment">electric dipole moment</a></b> <h4>Why is the dipole moment important?</h4> Let us switch for a moment to classical electrostatic theory. If you place a negative electric point charge at the origin of our coordinate system and a positive point charge at the point $\vec{x}$, I can tell you the electric dipole moment is a vector $\vec{p}$ and that: $$ \vec{p} = \vec{x} $$ For a more general situation, let us assume that there is a charge density $\rho$ contained in some sphere $S$ around the origin, then I tell you that the electric dipole moment $\vec{p}$ is again a vector that can be calculated via $$ \vec{p} = \int \vec{x} \rho(\vec{x}) d \vec{x} $$ Okay, you say, so it is simple to <i>calculate</i> it, but what is its <i>significance</i>? Let's say that we would like to know how a test charge flying by the sphere $S$ is influenced by the charge distribution in $S$. If we assume that our charge density $\rho$ is constant in time, then all we need to calculate is the <a href="http://en.wikipedia.org/wiki/Electric_potential">electric potential</a> $\Phi$. In spherical coordinates and far from the sphere $S$, this potential will fall of like $1/r$ or faster, so we may assume that there is a series expansion of the form $$ \Phi(r, \phi, \theta) = \sum_{n = 1}^{\infty} f(\phi, \theta) \frac{1}{r^n} $$ When our test charge is far away from the sphere $S$, only the first few terms in this expansion will be important to it. We still need to choose an orthonormal basis for the coordinates $\phi$ and $\theta$, that is an orthonormal basis of functions on the sphere. If we choose the <b><a href="http://en.wikipedia.org/wiki/Spherical_harmonics">spherical harmonics</a></b> $Y_{l m}(\phi, \theta)$, with proper normalization we get what is called the <b><a href="http://en.wikipedia.org/wiki/Multipole_expansion">multipole expansion</a></b> of the electric potential: $$ \Phi(r, \phi, \theta) = \frac{1}{4 \pi \epsilon_0} \sum_{l = 0}^{\infty} \sum_{m = -l}^{l} \frac{4 \pi}{2 l +1} q_{l m} \frac{Y_{l m}(\phi, \theta)}{r^{l+1}} $$ $\epsilon_0$ is the <b><a href="http://en.wikipedia.org/wiki/Electric_constant">electric constant</a></b> The $l = 0$ term is called the **monopole term**, it is proportional to the electric charge $q$ contained in the sphere $S$. So the first term in the expansion tells us if a charge flying by $S$ will feel a net attractive or repulsive force. The terms for $l = 1$ form the vector $\vec{p}$, the **dipole moment**. The next terms in the series $Q_ij$ form the **quadrupol** tensor. So, for the expansion of the potential we get $$ \Phi(r, \phi, \theta) = \frac{1}{4 \pi \epsilon_0} (\frac{q}{r} + \frac{\vec{p} \cdot \vec{x}}{r^3} + \frac{1}{2} \sum Q_{ij} \frac{x_i x_j}{r^5} + \cdot \cdot \cdot) $$ For atoms and molecules the net charge $q$ is zero, so the next relevant term in the series expansion of their electric potential is the dipole moment. This is the reason why it is important to know if an atom or molecule has states with a nonzero dipole moment: Because this fact will in a certain sense dominate the interactions with other electromagnetic phenomena. If you are interested in more information about multipole expansions in classical electrodynamics, you can find all sort of information in this classical textbook: * John David Jackson: _Classical Electrodynamics_ (Wiley; 3 edition (August 10, 1998)) In quantum mechanics the position coordinate $\vec{x}$ is promoted to the position operator; as a consequence the dipole moment is promoted to an operator, too. <h4>Molecular Emission Spectra or: Only Greenhouse Gases are infrared active? Really?</h4> For atoms and molecules interacting with light, there are certain <b>selection rules</b>. A strict selection rule in quantum mechanics rules out certain state transitions that would violate a conservation law. But for atoms and molecules there are also heuristic selection rules that rule out state transitions that are far less likely than others. For state transitions induces by the interaction with light, a heuristic transition rule is <blockquote> <p> Transitions need to change the dipole moment by one. </p> </blockquote> This selection rule is heuristic: Transitions that change the dipole moment are far more likely than transitions that change the electric quadrupole moment only, for example. But: If an atom or molecule does not have any diploe transitions, then you will maybe still see spectral lines corresponding to quadrupol transitions. But they will be very weak. +-- {: .standout} [[Tim van Beek]]: Compare black body radiation to the emission spectrum of CO2 and H2O. =-- Another important point is of course the part <blockquote> <p> The spectral strength and linewidth of these emissions are dependent on the temperature and concentration of the hot gaseous species... </p> </blockquote> Of course the temperature, pressure and concentration of atmospheric components are not constant throughout the whole atmosphere. We should keep that in mind for later, when we take a closer look at the theory of atmospheric radiation. But back to the continuous versus discrete spectrum part: Since we can distinguish surface radiation and radiation from specific gases, we can * point some measurement device to the sky, to measure what goes down, not what goes up and * check that the spectrum we measure is the characteristic molecular spectrum of $CO_2$, $H_20$ etc. and be fairly sure that we have indeed measured the part of the radiation that was re-emitted from the atmosphere to the surface. What would be a good place and time on Earth to do this? <h4>Measuring DLR</h4> What is the place with the least water wapor, the clearest night sky, on Earth? +-- {: .standout} [[Tim van Beek]]: Insert measurement results from the antarctic region. * MICHAEL S. TOWN, P. WALDEN, STEPHEN G. WARREN: <i>Spectral and Broadband Longwave Downwelling Radiative Fluxes, Cloud Radiative Forcing, and Fractional Cloud Cover over the South Pole</i> <a href="http://journals.ametsoc.org/doi/pdf/10.1175/JCLI3525.1">here</a> Also: * Dan Lubin, David Cutchin, William Conant, Hartmut Grassl, Ulrich Schmid, Werner Biselli: <i>Spectral Longwave Emission in the Tropics: FTIR Measurement at the Sea Surface and Comparison with Fast Radiation Codes</i>, online <a href="http://journals.ametsoc.org/doi/abs/10.1175/1520-0442%281995%29008%3C0286%3ASLEITT%3E2.0.CO%3B2">here</a>. Also: * "Measurements of the radiative surface forcing of climate", online <a href="ams.confex.com/ams/pdfpapers/100737.pdf">here</a>. =-- Devices to measure the infrared radiation of the planetary surface are called <b><a href="http://en.wikipedia.org/wiki/Pyrgeometer">pyrgeometer</a></b>, for pyr = fire and geo = earth. +-- {: .standout} [[Tim van Beek]]: I would like to add radiation measurements, maybe some can be found here: * [Atmospheric Radiation Measurement (ARM) Climate Research Facility](http://www.arm.gov/) AlsÜ * Baseline Surface Radiation Network (BSRN) <a href="http://www.gewex.org/bsrn.html">here</a>. Also have a look [here](http://www.eppleylab.com/PrdPrecInfRadmtr.htm). =-- Just to have a number, the flux of DLR (downwards longwave radiation) is about 300 $W m^{-2}$. There is also the <a href="http://www.cfa.harvard.edu/HITRAN/">HITRAN</a> database: You can look up radiative properties of different molecules there. HITRAN was founded by the US air force. Why? I don't know, but I guess that they needed the data for air craft design. Look out for the interview with Dr. Laurence Rothman for some background information. category: blog, climate [[!redirects The Color of Night]] [[!redirects The Color of the Night]]
as | Cancel (unlocks page) | CommonCrawl |
\begin{definition}[Definition:Electric Potential/Dimension]
The dimension of measurement of '''electric potential''' is $\mathsf M \mathsf L^2 \mathsf T^{−3} \mathsf I^{−1}$.
\end{definition} | ProofWiki |
\begin{document}
\begin{abstract}
We characterize Priestley spaces of algebraic, arithmetic, coherent, and Stone frames. As a corollary, we derive the well-known dual equivalences in pointfree topology involving various categories of algebraic frames. \end{abstract}
\title{Algebraic Frames in Priestley duality}
\tableofcontents
\section{Introduction} \label{sec:intro}
A complete lattice is algebraic provided every element is a join of compact elements. Algebraic lattices arise naturally in different contexts. For example, the lattice of subalgebras as well as the lattice of congruences of any algebra is algebraic, and up to isomorphism, every algebraic lattice arises this way (see, e.g., \cite{BS81}). It is a well-known result of Nachbin \cite{Nachbin1949} (see also \cite{BrikhoffFrink1948}) that algebraic lattices are exactly the ideal lattices of join-semilattices. If an algebraic lattice $L$ is distributive, then the infinite distributive law $a\wedge\bigvee S=\bigvee\{a\wedge s\mid s\in S\}$ holds, and hence $L$ is a frame. Such frames are known as algebraic frames and have been the subject of study in pointfree topology and domain theory (see, e.g., \cite{Compendium2003,PicadoPultr2012}).
There is a well-developed duality theory for the category \cat{AlgFrm} of algebraic frames and its various subcategories such as the categories of arithmetic frames (aka M-frames), coherent frames, and Stone frames. Indeed, a frame $L$ is algebraic iff it is the frame of opens of a compactly based sober space $X$ \cite[p.~423]{Compendium2003}. In addition, $L$ is arithmetic iff $X$ is stably compactly based, $L$ is coherent iff $X$ is spectral, and $L$ is a Stone frame iff $X$ is a Stone space (see \cref{sec: prelims} for details).
The duality theory for algebraic frames is a restriction of the well-known Hofmann-Lawson duality \cite{HofmannLawson1978}. We recall (see, e.g., \cite[p.~135]{PicadoPultr2012}) that a frame $L$ is {\em continuous} if the way-below relation $\ll$ is approximating. In addition, $L$ is {\em stably continuous} if $\ll$ is stable ($a\ll b,c$ implies $a\ll b\wedge c$), $L$ is {\em stably compact} if moreover $L$ is compact, and $L$ is {\em compact regular} if furthermore $\ll$ coincides with the well-inside relation $\prec$. We thus obtain the following correspondence between various categories of continuous and algebraic frames, where the categories are defined in \cref{table:con frames,table:alg frames} and $\leqslant$ stands for being a full subcategory of. \ \begin{figure}
\caption{Inclusion relationships between categories of continuous and algebraic frames. }
\label{diagram: CFrmAlgFrm}
\end{figure}
By the well known Priestley duality \cite{Priestley1970,Priestley1972}, the category of bounded distributive lattices is dually equivalent to the category of Priestley spaces. Pultr and Sichler \cite{PultrSichler1988} provided a restricted version of Priestley duality for the category \cat{Frm} of frames and frame homomorphisms. This line of research was further developed by several authors (see, e.g., \cite{PultrSichler2000,BezhGhilardi2007,BezhanishviliGabelaiaJibladze2013,BezhanishviliGabelaiaJibladze2016,AvilaBezhanishviliMorandiZaldivar2020,AvilaBezhanishviliMorandiZaldivar2021}). In \cite{BezhanishviliMelzer2022b}, we obtained Priestley duality for \cat{ConFrm} and its subcategories listed in Figure~\ref{diagram: CFrmAlgFrm}. The aim of this paper is to further study Priestley duality for \cat{AlgFrm} and its relevant subcategories. This in particular requires characterizing Priestley spaces of algebraic, coherent, arithmetic, and Stone frames.
The paper is organized as follows. In \cref{sec: prelims}, we describe the above categories of continuous and algebraic frames, as well as the corresponding categories of locally compact and compactly based sober spaces. \cref{sec 3} recalls Priestley duality for various categories of continuous frames. In \cref{sec: alg frm}, we characterize Priestley spaces of algebraic frames. Consequently, we obtain a new proof of the duality between \cat{AlgFrm} and \cat{KBSob}. Finally, in \cref{sec 5}, we characterize Priestley spaces of arithmetic, coherent, and Stone frames. In each case, this yields a new proof of the duality between the corresponding categories of algebraic frames and compactly based spaces. We conclude the paper by connecting Priestley spaces of coherent frames and Stone frames to Priestley duality for bounded distributive lattices and Stone duality for boolean algebras.
\section{Continuous and algebraic frames} \label{sec: prelims} A \emph{frame} is a complete lattice $L$ satisfying the join-infinite distributive law \[
a \wedge \bigvee S = \bigvee\{a \wedge s \mid s \in S\} \] for every $a \in L$ and $S \subseteq L$. A \emph{frame homomorphism} is a map between frames that preserves finite meets and arbitrary joins. Let \cat{Frm} be the category of frames and frame homomorphisms. A frame is \emph{spatial} if completely prime filters separate elements of $L$. Let \cat{SFrm} be the full subcategory of \cat{Frm} consisting of spatial frames.
As usual, we write $\ll$ for the {\em way below} relation in a frame $L$ and recall that $a\ll b$ provided for each $S\subseteq L$ we have $b\le\bigvee S$ implies $a\le \bigvee T$ for some finite $T\subseteq S$. We call $a\in L$ \emph{compact} if $a \ll a$ and $L$ \emph{compact} if its top element is compact. We write $K(L)$ for the collection of compact elements of $L$.
In the introduction we recalled the definitions of continuous, stably continuous, stably compact, and compact regular frames. A frame homomorphism $h : L \to M$ between continuous frames is \emph{proper} if it preserves $\ll$; that is, $a \ll b$ implies $h(a) \ll h(b)$ for all $a,b \in L$. Let \cat{ConFrm} be the category of continuous frames and proper frame homomorphisms. We write \cat{StCFrm} and \cat{StKFrm} for the full subcategories of \cat{ConFrm} consisting of stably continuous and stably compact frames, respectively. We also let \cat{KRFrm} be the full subcategory of \cat{Frm} consisting of compact regular frames. Since every frame homomorphism between compact regular frames is proper, \cat{KRFrm} is a full subcategory of \cat{StKFrm}.
We have the following categories of continuous frames.
\begin{table}[H] \begin{tabular}{lll}
\toprule
\multicolumn{1}{c}{\bf Category} & \multicolumn{1}{c}{\bf Objects} & \multicolumn{1}{c}{\bf Morphisms} \\ \midrule
\cat{ConFrm} & continuous frames & proper frame homomorphisms \\
\cat{StCFrm} & stably continuous frames & proper frame homomorphisms \\
\cat{StKFrm} & stably compact frames & proper frame homomorphisms \\
\cat{KRFrm} & compact regular frames & frame homomorphisms \\
\bottomrule \end{tabular} \caption{Categories of continuous frames.\label{table:con frames}} \end{table}
\begin{definition} \begin{enumerate}
\item[]
\item (\cite[p.~142]{PicadoPultr2012}) A frame $L$ is \emph{algebraic} if $a = \bigvee\{b \in K(L) \mid b \leq a\}$ for all $a \in L$.
\item (\cite[p.~64]{Johnstone1982}) A frame homomorphism $h : L \to M$ is \emph{coherent} if $a \in K(L)$ implies $h(a) \in K(M)$.
\item Let \cat{AlgFrm} be the category of algebraic frames and coherent frame homomorphisms. \end{enumerate} \end{definition}
\begin{remark} \label{rem: full sub alg}
It is easy to see that every algebraic frame is continuous, and that
a frame homomorphism between coherent frames is coherent iff it is proper. Consequently, \cat{AlgFrm} is a full subcategory of \cat{ConFrm}. \end{remark}
\begin{definition} \begin{enumerate}
\item[]
\item A frame $L$ is \emph{arithmetic} if it is algebraic and $\ll$ is stable.
\item Let \cat{AriFrm} be the full subcategory of \cat{AlgFrm} consisting of arithmetic frames. \end{enumerate} \end{definition}
\begin{remark}
\begin{enumerate}
\item[]
\item
In \cite{Compendium2003} a lattice is called arithmetic if the binary meet of compact elements is compact. For algebraic lattices this is equivalent to $\ll$ being stable (see, e.g. \cite[Prop.~I-4.8]{Compendium2003}).
\item Arithmetic frames
are also called M-frames; see, e.g.,~\cite{IberkleidMcGovern2009,Bhattacharjee2018}.
\end{enumerate} \end{remark}
\begin{definition}
\begin{enumerate}
\item[]
\item (\cite[p.~63--64]{Johnstone1982}) A frame $L$ is \emph{coherent} if $L$ is arithmetic and compact.
\item Let \cat{CohFrm} be the full subcategory of \cat{AriFrm} consisting of coherent frames.
\end{enumerate} \end{definition}
Let $L$ be a frame. We recall that the \emph{well inside} relation on $L$ is defined by $a \prec b$ if $a^* \vee b = 1$, where $a^*:=\bigvee \{x\in L\mid a\wedge x=0\}$ is the pseudocomplement of $a$. An element $a\in L$ is \emph{complemented} if $a \prec a$. Let $C(L)$ be the collection of complemented elements of $L$. It is well known that if $L$ is compact, then $a\prec b$ implies $a\ll b$; and if $L$ is regular, then $a\ll b$ implies $a\prec b$. Thus, in compact regular frames, the two relations $\ll$ and $\prec$ coincide, and hence $K(L)=C(L)$.
The next definition is well known (see, e.g., \cite{Johnstone1982,Jakl2013}). We thank Joanne Walters-Wayland for pointing out to us that the terminology of Stone frames originated from Banaschewski's University of Cape Town lecture notes (1988).
\begin{definition}
\begin{enumerate}
\item[]
\item
A frame $L$ is \emph{zero-dimensional} if $a = \bigvee \{b \in C(L) \mid b \leq a\}$ for all $a \in L$.
\item
A {\em Stone frame} is a compact zero-dimensional frame.
\item Let \cat{StoneFrm} be the full subcategory of \cat{Frm} consisting of Stone frames.
\end{enumerate} \end{definition}
\begin{remark} \label{rem: full sub stonefrm}
Clearly \cat{StoneFrm} is a full subcategory of \cat{KRFrm}.
Moreover, since every frame homomorphism preserves $\prec$ and in Stone frames $\prec$ coincides with $\ll$, we have that \cat{StoneFrm} is a full subcategory of \cat{CohFrm}. \end{remark}
We have the following categories of algebraic frames.
\begin{table}[H] \begin{tabular}{lll}
\toprule
\multicolumn{1}{c}{\bf Category} & \multicolumn{1}{c}{\bf Objects} & \multicolumn{1}{c}{\bf Morphisms} \\ \midrule
\cat{AlgFrm} & algebraic frames & coherent frame homomorphisms \\
\cat{AriFrm} & arithmetic frames & coherent frame homomorphisms \\
\cat{CohFrm} & coherent frames & coherent frame homomorphisms \\
\cat{StoneFrm} & Stone frames & frame homomorphisms \\
\bottomrule \end{tabular} \caption{Categories of algebraic frames.\label{table:alg frames}} \end{table} The categories of algebraic and continuous frames relate to each other as shown in \cref{diagram: CFrmAlgFrm}. We next turn our attention to the corresponding categories of topological spaces. The following definitions are well known (see, e.g., \cite[pp.~43-44]{Compendium2003}). A closed subset of a topological space $X$ is {\em irreducible} if it cannot be written as the union of two proper subsets. We call $X$ \emph{sober} if each irreducible closed subset is the closure of a unique point in $X$, and \emph{locally compact} if for every open set $U$ and $x \in U$ there are an open set $V$ and a compact set $K$ such that $x \in V \subseteq K \subseteq U$.
Following \cite[Lem.~VI-6.21]{Compendium2003}, we call a continuous map $f : X \to Y$ between locally compact sober spaces \emph{proper} if $f^{-1}(K)$ is compact for each compact saturated set $K \subseteq Y$. Let \cat{LKSob} be the category of locally compact sober spaces and proper maps between them.
A topological space $X$ is \emph{coherent} if the intersection of two compact saturated sets is compact (\cite[p.~474]{Compendium2003}), and $X$ is \emph{stably locally compact} if it is compact, sober, and coherent. Let \cat{StLKSp} be the full subcategory of \cat{LKSob} consisting of stably locally compact spaces.
A compact stably locally compact space is a \emph{stably compact} space (\cite[p.~476]{Compendium2003}). We write \cat{StKSp} for the full subcategory of \cat{StLKSp} consisting of stably compact spaces. Also, we denote by \cat{KHaus} the category of compact Hausdorff spaces and continuous maps. Since a continuous map between compact Hausdorff spaces is proper, \cat{KHaus} is a full subcategory of \cat{StKSp}.
We have the following categories of locally compact sober spaces.
\begin{table}[H] \begin{tabular}{lll}
\toprule
\multicolumn{1}{c}{\bf Category} & \multicolumn{1}{c}{\bf Objects} & \multicolumn{1}{c}{\bf Morphisms} \\ \midrule
\cat{LKSob} & locally compact sober spaces & proper maps \\
\cat{StLKSp} & stably locally compact spaces & proper maps \\
\cat{StKSp} & stably compact spaces & proper maps \\
\cat{KHaus} & compact Hausdorff spaces & continuous maps\\
\bottomrule \end{tabular} \caption{Categories of locally compact sober spaces.\label{table:lcsob spaces}} \end{table}
We now shift our focus to compactly based spaces. We recall that a continuous map $f : X \to Y$ is {\em coherent} if $f^{-1}(U)$ is compact for each compact open $U \subseteq Y$.
\begin{definition} \begin{enumerate}[ref=\thedefinition(\arabic*)] \item[] \item (\cite[p.~2063]{Erne2009}) A topological space $X$ is \emph{compactly based} if it has a basis of compact open sets. Let \cat{KBSob} be the category of compactly based sober spaces and coherent maps. \item A compactly based space $X$ is \emph{stably compactly based} if it is sober and the intersection of two compact opens is compact. Let \cat{StKBSp} be the full subcategory of \cat{KBSob} consisting of stably compactly based spaces. \item (\cite[p.~43]{Hochster1969}) A stably compactly based space $X$ is a \emph{spectral} space if it is compact. Let \cat{Spec} be the full subcategory of \cat{StKBSp} consisting of spectral spaces. \item (\cite[p.~70]{Johnstone1982}) A \emph{Stone} space is a zero-dimensional compact Hausdorff space. Let \cat{Stone} be the category of Stone spaces and continuous maps. \end{enumerate} \end{definition}
We have the following categories of compactly based sober spaces.
\begin{table}[H] \begin{tabular}{lll}
\toprule
\multicolumn{1}{c}{\bf Category} & \multicolumn{1}{c}{\bf Objects} & \multicolumn{1}{c}{\bf Morphisms} \\ \midrule
\cat{KBSob} & compactly based sober spaces & coherent maps\\
\cat{StKBSp} & stably compactly based spaces & coherent maps \\
\cat{Spec} & spectral spaces & coherent maps\\
\cat{Stone} & Stone spaces & continuous maps \\
\bottomrule \end{tabular} \caption{Categories of compactly based sober spaces.\label{table:kb spaces}} \end{table}
\begin{remark} \label{rem: comp sat} \label{rem: full sub kbsob} \label{rem: full sub stone} It is easy to see that \cat{Stone} is a full subcategory of \cat{Spec} (see, e.g., \cite[p.~71]{Johnstone1982}). To see that \cat{KBSob} is a full subcategory of \cat{LKSob}, it is sufficient to observe that a continuous map between compactly based sober spaces is coherent iff it is proper. For this it is enough to observe that in a compactly based space $X$, every compact saturated set is an intersection of compact opens. To see this, let $K \subseteq X$ be compact saturated. It suffices to show that for each $x \not \in K$ there is a compact open $U$ containing $K$ and missing $x$. For each $y \in K$ there is a compact open $U_y$ such that $y \in U_y$ and $x \not \in U_y$. Therefore, $K \subseteq \bigcup \{U_y \mid y \in K\}$. By compactness of $K$ and the fact that a finite union of compact sets is compact, there is a compact open $U$ such that $K \subseteq U$ and $x \not \in U$. \end{remark}
We thus obtain the following correspondence between various categories of locally compact and compactly based sober spaces. \begin{figure}
\caption{Inclusion relationships between categories of locally compact and compactly based sober spaces. }
\label{diagram: LCSobKBSob}
\end{figure}
There is a well-known dual adjunction between \cat{Top} and \cat{Frm}, which restricts to a dual equivalence between \cat{Sob} and \cat{SFrm} (see, e.g., \cite[Sec.~II-1]{Johnstone1982}). Further restrictions of this equivalence yield the following well-known duality results for continuous frames:
\begin{theorem} \plabel{cfrm-dualities}
\begin{enumerate}[ref=\thetheorem(\arabic*)]
\item[]
\item
\cat{ConFrm} is dually equivalent to \cat{LKSob}. \clabel{1}
\item \cat{StCFrm} is dually equivalent to \cat{StLKSp}. \clabel{2}
\item \cat{StKFrm} is dually equivalent to \cat{StKSp}. \clabel{3}
\item
\cat{KRFrm} is dually equivalent to \cat{KHaus}. \clabel{4}
\end{enumerate} \end{theorem} We thus arrive at the following diagram, where $\leftrightsquigarrow$ represents dual equivalence. \begin{figure}
\caption{Correspondence between categories of continuous frames and locally compact spaces.}
\end{figure}
\begin{remark} \cref{cfrm-dualities-1} is known as Hofmann-Lawson duality \cite{HofmannLawson1978} (see also \cite[Prop.~V-5.20]{Compendium2003}). The origins of \cref{cfrm-dualities-2,cfrm-dualities-3} can be traced back to \cite{GierzKeimel1977,Johnstone1981,Simmons1982,Banaschewski1981} (see also \cite[Sec.~VI-7.4]{Compendium2003}). Finally, \cref{cfrm-dualities-4} is known as Isbell duality \cite{Isbell1972} (see also \cite{BanaschweskiMulvey1980} or \cite[Sec.~VII-4]{Johnstone1982}). \end{remark}
We next describe the duality results for algebraic frames. One of the earliest references is probably \cite[Thm.~5.7]{HofmannKeimel1972} (see also \cite[p.~423]{Compendium2003}), where the dualities for \cat{AlgFrm}, \cat{AriFrm}, and \cat{CohFrm} are stated. The duality for \cat{CohFrm} is also described in \cite{Banaschewski1980,Banaschewski1981}. This further reduces to the duality for \cat{StoneFrm} (see, e.g., \cite[Ch.~IV]{Jakl2013}).
\begin{theorem} \plabel{frm-dualities}
\begin{enumerate}[ref=\thetheorem(\arabic*)]
\item[]
\item \cat{AlgFrm} is dually equivalent to \cat{KBSob}. \clabel{alg}
\item \cat{AriFrm} is dually equivalent to \cat{StKBSp}. \clabel{arith}
\item \cat{CohFrm} is dually equivalent to \cat{Spec}. \clabel{coh}
\item \cat{StoneFrm} is dually equivalent to \cat{Stone}. \clabel{stone}
\end{enumerate} \end{theorem}
We thus arrive at the following diagram.
\begin{figure}
\caption{Correspondence between categories of algebraic frames and compactly based spaces.}
\label{diagram: introduction}
\end{figure}
\begin{remark}
The proof of \cref{frm-dualities} can easily be deduced from \cref{cfrm-dualities} and the fact that \cat{AlgFrm} and \cat{KBSob} are full subcategories of \cat{ConFrm} and \cat{LKSob}, respectively. But it is easy to give a direct proof of \cref{frm-dualities} which does not rely on \cref{cfrm-dualities}. For this it is sufficient to observe that every algebraic frame is spatial.
Let $L$ be an algebraic frame.
Then Scott-open filters separate elements of $L$. To see this, if $a\not\le b$, then there is $k\in K(L)$ such that $k\le a$ but $k\not\le b$. Thus, $\mathord\uparrow k$ is a Scott-open filter containing $a$ and missing $b$. It is left to observe that the Prime Ideal Theorem implies that $L$ is spatial iff Scott-open filters separate elements of $L$ (see, e.g., \cite[Cor.~5.9(2)]{BezhanishviliMelzer2022}). \end{remark}
\section{Priestley duality for continuous frames} \label{sec 3}
As we pointed out in the Introduction, Pultr and Sichler \cite{PultrSichler1988} restricted Priestley duality for bounded distributive lattices to the category of frames. In this section we briefly recall Pultr-Sichler duality and its restriction to various categories of continuous frames.
A \emph{Priestley space} is a Stone space $X$ with a partial order $\leq$ such that clopen upsets separate points. An \emph{Esakia space} is a Priestley space with the additional property that the partial order $\le$ is continuous (the downset of each clopen is clopen). By Esakia duality \cite{Esakia1974}, Esakia spaces are exactly the Priestley spaces of Heyting algebras. An important feature of Esakia spaces is that the closure of each upset is an upset. Esakia duals of complete Heyting algebras have the additional property that the closure of each open upset is open. Such Esakia spaces are called \emph{extremally order-disconnected} as they generalize extremally disconnected Stone spaces. Since frames are complete Heyting algebras, Priestley duals of frames are exactly the extremally order-disconnected Esakia spaces. \begin{definition}
\begin{enumerate}
\item[]
\item An \emph{L-space} (\emph{localic space}) is an extremally order-disconnected Esakia space.
\item An \emph{L-morphism} is a continuous order-preserving map $f : X \to Y$ between L-spaces such that $f^{-1}\mathop{\sf cl} U = \mathop{\sf cl} f^{-1}U$ for every open upset $U$ of $Y$.
\item Let \cat{LPries} be the category of L-spaces and L-morphisms.
\end{enumerate} \end{definition}
\begin{theorem}[Pultr-Sichler {\cite[Cor.~2.5]{PultrSichler1988}}]
\cat{Frm} is dually equivalent to \cat{LPries}. \end{theorem}
\begin{remark}
The functors $\functor X : \cat{Frm} \to \cat{LPries}$ and $\functor{D} : \cat{LPries} \to \cat{Frm}$ establishing Pulter-Sichler duality are the restrictions of the functors establishing Priestley duality. We recall that the {\em Priestley space} of a frame $L$ is the set $X_L$ of prime filters of $L$ ordered by inclusion and topologized by the subbases $\{\varphi(a) \mid a \in L\} \cup \{\varphi(a)^c \mid a \in L\}$, where $\varphi$ is the Stone map given by $\varphi(a) = \{x \in X_L \mid a \in x\}$ for each $a \in L$. The functor $\functor X$ sends a frame $L$ to its Priestley space $X_L$ and a frame homomorphism $h : L \to M$ to the L-morphism $h^{-1} : X_M \to X_L$. The functor $\functor{D}$ sends an L-space $X$ to the frame ${\sf ClopUp}(X)$ of clopen upsets of $X$ and an L-morphism $f : X \to Y$ to the frame homomorphism $f^{-1} : {\sf ClopUp}(Y) \to {\sf ClopUp}(X)$. \end{remark} We next characterize Priestley spaces of spatial frames.
\begin{definition}
Let $X$ be an L-space.
\begin{enumerate}
\item The set $Y := \{y \in X \mid \mathord\downarrow y \text{ is clopen}\}$ is called the \emph{spatial part of $X$}.
\item We call $X$ an \emph{SL-space} if
$Y$ is dense in $X$.
\item Let \cat{SLPries} be the full subcategory of \cat{LPries} consisting of SL-spaces.
\end{enumerate} \end{definition}
\begin{remark}
Let $X$ be an L-space and $Y$ the spatial part of $X$.
\begin{enumerate}[ref=\theremark(\arabic*)]
\item We view $Y$ as a topological space,
where $V\subseteq Y$ is open iff $V = U \cap Y$ for some $U \in {\sf ClopUp}(X)$. If $X$ is the Priestley space of a frame $L$, then the spatial part $Y$ of $X$ is exactly
the space of points of $L$ (see, e.g., \cite[Lem.~4.1]{BezhanishviliMelzer2022b}).
\item If $X$ is an SL-space, then for each $U \in {\sf ClopUp}(X)$ we have $\mathop{\sf cl}(U \cap Y) = U$. Therefore, the assignment $U \mapsto U \cap Y$ is an isomorphism from the poset of clopen upsets of $X$ to the poset of open sets of $Y$. This will be utilized in what follows. \label[remark]{rem:iso opens Y}
\end{enumerate} \end{remark}
\begin{theorem}[{\cite[Sec.~4]{BezhanishviliMelzer2022b}}] \label{thm: SL dualities} \cat{SLPries} is equivalent to \cat{Sob} and dually equivalent to \cat{SFrm}. \end{theorem}
\begin{remark} \begin{enumerate}[ref=\theremark(\arabic*)]
\item[]
\item The dual equivalence between \cat{SFrm} and \cat{SLPries} is obtained by restricting the functors establishing Pultr-Sichler duality.
\item The equivalence between \cat{SLPries} and \cat{Sob} is obtained as follows. Let $\functor{Y} : \cat{LPries} \to \cat{Sob}$ be the functor that sends an L-space $X$ to to its spatial part $Y$, and an L-morphism $f : X_1 \to X_2$ to its restriction $g:Y_1\to Y_2$.
Then $\functor{Y}$
restricts to an equivalence between $\cat{SLPries}$ and $\cat{Sob}$ (see, e.g., \cite[Cor.~4.20]{BezhanishviliMelzer2022b}).
\label[remark]{rem:Y functor}
\item As an immediate consequence of \cref{thm: SL dualities}, we obtain the well-known duality between \cat{SFrm} and \cat{Sob}.
\end{enumerate} \end{remark}
We now turn our attention to Priestley spaces of continuous frames.
\begin{definition} \label{def:kernel}\label{def:packed} Let $X$ be an L-space. \begin{enumerate} \item For $U,V\in{\sf ClopUp}(X)$, define $V \ll U$ provided for each open upset $W$ of $X$ we have $U \subseteq \mathop{\sf cl} W$ implies $V \subseteq W$. \item For $U \in {\sf ClopUp}(X)$, define the \emph{kernel of $U$} as $$\ker U = \bigcup\{V \in {\sf ClopUp}(X) \mid V \ll U\}.$$ \item We call $X$ a \emph{continuous L-space} if $\ker U$ is dense in $U$ for each $U \in {\sf ClopUp}(X)$. \item An L-morphism $f : X_1 \to X_2$ is \emph{proper} if $f^{-1}(\ker U) \subseteq \ker f^{-1}(U)$ for all $U \in {\sf ClopUp} (X_2)$. \item Let \cat{ConLPries} be the category of continuous L-spaces and proper L-morphisms. \end{enumerate} \end{definition} \begin{theorem}[{\cite[Sec.~5]{BezhanishviliMelzer2022b}}]
\cat{ConLPries} is equivalent to \cat{LKSob} and dually equivalent to \cat{ConFrm}.
\label{hl-frames} \label{hl-spaces} \end{theorem}
\begin{corollary} [Hofmann-Lawson duality]
\cat{ConFrm} is dually equivalent to \cat{LKSob}. \end{corollary}
We thus arrive at the following diagram which commutes up to natural isomoprhism, where $\leftrightarrow$ represents equivalence.
We next describe Priestley spaces of stably continuous and stably compact frames. For the next definition see \cite[Sec.~6]{BezhanishviliMelzer2022b}. The notion of L-compact first appeared in \cite[Sec.~3]{PultrSichler2000}.
\begin{definition}
\begin{enumerate}[ref=\thedefinition(\arabic*)]
\item[]
\item
\begin{enumerate}[ref=\thedefinition(\arabic{enumi})(\alph*)]
\item
An L-space $X$ is \emph{kernel-stable} if $\ker(U \cap V) = \ker U \cap \ker V$ for all $U,V \in {\sf ClopUp}(X)$, \label[definition]{def: kernel-stable}
\item A \emph{stably continuous L-space} is a kernel-stable continuous L-space.
\item Let \cat{StCLPries} be the full subcategory of \cat{ConLPries} consisting of stably continuous L-spaces.
\end{enumerate}
\item
\begin{enumerate}
\item
An L-space $X$ is \emph{L-compact} if $X = \ker X$.
\item A \emph{stably compact} L-space is an L-compact stably continuous L-space.
\item Let \cat{StKLPries} be the full subcategory of \cat{StCLPries} consisting of stably compact L-spaces.
\end{enumerate}
\end{enumerate} \end{definition}
\begin{theorem}[{\cite[Sec.~6]{BezhanishviliMelzer2022b}}] \label{thm: stably dualities}
\begin{enumerate}[ref=\thetheorem(\arabic*)]
\item[]
\item \cat{StCLPries} is equivalent to \cat{StLKSp} and dually equivalent to \cat{StCFrm}.
\item \cat{StKLPries} is equivalent to \cat{StKSp} and dually equivalent to \cat{StKFrm}.
\end{enumerate} \end{theorem}
As a consequence, we obtain the following well-known dualities for stably continuous frames:
\begin{corollary}[{\cite[Cor.~VI-7.2]{Compendium2003}}]
\begin{enumerate}[ref=\thecorollary(\arabic*)]
\item[]
\item \cat{StCFrm} is dually equivalent to \cat{StLKSp}.
\item \cat{StKFrm} is dually equivalent to \cat{StKSp}. \label[corollary]{cor:tpl}
\end{enumerate} \end{corollary}
We conclude this section by describing Priestley spaces of compact regular frames. The next definition appeared in \cite[Sec.~3]{BezhanishviliGabelaiaJibladze2016} and \cite[Sec.~7]{BezhanishviliMelzer2022b}.
\begin{definition} Let $X$ be an L-space. \begin{enumerate} \item For $U,V\in{\sf ClopUp}(X)$, define $V \prec U$ provided $\mathord\downarrow V \subseteq U$. \item For $U \in {\sf ClopUp}(X)$, define the \emph{regular part of $U$} as $$\reg U = \bigcup\{V \in {\sf ClopUp}(X) \mid V \prec U\}.$$ \item We call $X$ a \emph{regular L-space} if $\reg U$ is dense in $U$ for each $U \in {\sf ClopUp}(X)$. \item We call $X$ a \emph{compact regular L-space} if $X$ is a regular L-space that is L-compact. \item Let \cat{KRLPries} be the full subcategory of \cat{LPries} consisting of compact regular L-spaces. \end{enumerate} \end{definition}
\begin{remark}
Every L-morphism between compact regular L-spaces is proper (see \cite[Thm.~7.17]{BezhanishviliMelzer2022b}), and every compact regular L-space is a stably compact L-space (see \cite[Thm.~7.16]{BezhanishviliMelzer2022b}). Thus, \cat{KRLPries} is a full subcategory of \cat{StKLPries}. \end{remark}
We have the following categories of continuous L-spaces. \begin{table}[H] \begin{tabular}{lll}
\toprule
\multicolumn{1}{c}{\bf Category} & \multicolumn{1}{c}{\bf Objects} & \multicolumn{1}{c}{\bf Morphisms} \\ \midrule
\cat{ConLPries} & continuous L-spaces & proper L-morphisms \\
\cat{StCLPries} & stably continuous L-spaces & proper L-morphisms \\
\cat{StKLPries} & stably compact L-spaces & proper L-morphisms\\
\cat{KRLPries} & compact regular L-spaces & L-morphisms\\
\bottomrule \end{tabular} \caption{Categories of compactly based sober spaces.\label{table:ConLspaces}} \end{table}
\begin{theorem}[{\cite[Sec.~7]{BezhanishviliMelzer2022b}}] \label{thm: KRL duality} \cat{KRLPries} is equivalent to \cat{KHaus} and dually equivalent to \cat{KRFrm}. \end{theorem}
\begin{corollary} [Isbell duality] \cat{KRFrm} is dually equivalent to \cat{KHaus}. \end{corollary}
We thus have the following diagram of equivalences and dual equivalences.
In what follows, we will obtain a similar picture of equivalences and dual equivalences when the above categories of continuous frames are replaced by the corresponding full subcategories of algebraic frames.
\section{Priestley duality for algebraic frames} \label{sec: alg frm}
In this section we describe algebraic frames in the language of Priestley spaces. We then connect the Priestley duals of algebraic frames with compactly based sober spaces to derive the well-known duality between \cat{AlgFrm} and \cat{KBSob} mentioned in \cref{frm-dualities-alg}.
Let $X$ be an L-space and $Y$ the spatial part of $X$. We recall (see \cite[Def.~5.2]{BezhanishviliMelzer2022}) that a closed upset $F$ of $X$ is a \emph{Scott upset} if $\min F \subseteq Y$. Equivalently, $F$ is a Scott upset of $X$ iff \begin{equation*} F \subseteq \mathop{\sf cl} U \implies F \subseteq U \hspace{2em}\mbox{ for each open upset $U$ of $X$} \tag{$\dagger$} \label{dagger} \end{equation*} (see \cite[Lem.~5.1]{BezhanishviliMelzer2022}). We denote the collection of all clopen Scott upsets of $X$ by {\sf ClopSUp}(X).
\begin{definition} \label{def: bunched}
Let $X$ be an L-space.
\begin{enumerate}
\item For $U \in {\sf ClopUp}(X)$, define the \emph{core} of $U$ as
\[
\core U = \bigcup \{V \subseteq U \mid V \in {\sf ClopSUp}(X)\}.
\]
\item Call $X$ an \emph{algebraic} L-space if $\core U$ is dense in $U$ for each $U\in {\sf ClopUp}(X)$.
\end{enumerate} \end{definition}
\begin{lemma} \plabel{lem:core}
Let $X$ be an $L$-space and $U,V \in {\sf ClopUp}(X)$. \begin{enumerate}[ref=\thelemma(\arabic*)]
\item $\core U \subseteq \ker U \subseteq U$. \clabel{1}
\item $U \subseteq V$ implies $\core U \subseteq \core V$. \clabel{2}
\item If $X$ is an algebraic L-space, then $X$ is a continuous L-space. \clabel{3}
\item $U$ is a Scott upset iff $\core U = U$. \clabel{4}
\end{enumerate} \end{lemma}
\begin{proof}
(1) Suppose $x \in \core(U)$. Then there is $V\in{\sf ClopSUp}(X)$ such that $x \in V \subseteq U$. Let $W$ be an open upset such that $U \subseteq \mathop{\sf cl} W$. Then $V \subseteq \mathop{\sf cl} W$, so $V \subseteq W$ by (\ref{dagger}). Hence, $V \ll U$. Therefore, $x \in \ker U$, and so $\core U \subseteq \ker U$. That $\ker U \subseteq U$ follows from \cite[Lem.~5.2(1)]{BezhanishviliMelzer2022b}.
(2) This is obvious from the definition of the core.
(3) Let $U \in {\sf ClopUp}(X)$. Since $X$ is an algebraic L-space, $\core U$ is dense in $U$. Therefore, $\ker U$ is dense in $U$ by (1). Thus, $X$ is a continuous L-space.
(4) First suppose that $U$ is a Scott upset. By (1), $\core U \subseteq U$. Since $U$ is a Scott upset, $U \subseteq \core U$. Thus, $\core U = U$. Conversely, suppose that $U = \core U$. Since $U$ is compact, there are clopen Scott upsets $V_1, \dots, V_n \subseteq U$ such that $U = V_1 \cup \dots \cup V_n$. Because a finite union of Scott upsets is a Scott upset, $U$ is a Scott upset. \end{proof}
We next connect algebraic frames with algebraic L-spaces. For this we recall the following:
\begin{lemma}[{\cite[Lem.~6.9]{BezhanishviliMelzer2022b}}] \label{lem:compact}
Let $L$ be a frame and $X_L$ its Priestley space. For $a \in L$, the following are equivalent. \begin{enumerate}
\item $a$ is compact.
\item $\ker(a) = \varphi(a)$.
\item $\varphi(a)$ is a Scott upset.
\end{enumerate}
In particular, $L$ is compact iff
$X_L$ is L-compact. \end{lemma}
Let $L$ be a frame, $X_L$ its Priestley space, and $a\in L$. To simplify notation, we write $\core(a)$ for $\core \varphi(a)$.
\begin{theorem} \plabel{thm:alg-hb}
Let $L$ be a frame and $X_L$ its Priestley space.
\begin{enumerate}[ref=\thetheorem(\arabic*)]
\item For $a\in L$, we have $a = \bigvee\{b \in K(L) \mid b \leq a\}$ iff $\core(a)$ is dense in $\varphi(a)$.
\item $L$ is an algebraic frame iff $X_L$ is an algebraic L-space. \clabel{2}
\end{enumerate} \end{theorem}
\begin{proof}
(1) It is well known (see, e.g., \cite[Lem.~2.3]{BezhanishviliBezhanishvili2008}) that \[
\varphi\left(\bigvee S\right)=\mathop{\sf cl}\left(\bigcup\{\varphi(s) \mid s \in S\}\right)
\]
for each $S \subseteq L$. Therefore, by \cref{lem:compact} we have $a = \bigvee\{b \in K(L) \mid b \leq a\}$ iff
\[
\varphi(a) = \mathop{\sf cl} \left( \bigcup \{\varphi(b)\in {\sf ClopSUp}(X_L) \mid\varphi(b) \subseteq \varphi(a) \}\right) = \mathop{\sf cl} (\core(a)).
\]
(2) follows from (1). \end{proof}
We now turn to morphisms between algebraic L-spaces.
\begin{definition} \label{def: L-coherent} \begin{enumerate}
\item[]
\item We call an L-morphism $f : X_1 \to X_2$ between L-spaces \emph{coherent} if
\[
f^{-1}(\core U) \subseteq \core f^{-1}(U)
\hspace{2em}\mbox{ for all }U\in{\sf ClopUp}(X_{2}).
\]
\item Let \cat{AlgLPries} be the category of algebraic L-spaces and coherent L-morphisms. \end{enumerate} \end{definition}
It is easy to see that the identity morphism is a coherent L-morphism and that the composition of two coherent L-morphisms is coherent. Therefore, \cat{AlgLPries} is indeed a category. We show that \cat{AlgLPries} is a full subcategory of \cat{ConLPries}. For this we need the following two lemmas.
\begin{lemma} \plabel{lem:scott-ext}
Let $X$ be a continuous L-space and $U \in {\sf ClopUp}(X)$. The following are equivalent. \begin{enumerate}[ref=\thelemma(\arabic*)]
\item
$\ker U = \core U$. \clabel{1}
\item $\core U$ is dense in $U$.
\item For each $y \in U \cap Y$, there is $V \in {\sf ClopSUp}(X)$ such that $y \in V \subseteq U$. \clabel{needed}
\item For each Scott upset $F \subseteq \ker U$, there is $V \in {\sf ClopSUp}(X)$ such that $F \subseteq V \subseteq U$. \clabel{3}
\end{enumerate} \end{lemma} \begin{proof}
(1)$\Rightarrow$(2) Since $X$ is a continuous L-space, $\ker U$ is dense in $U$. Therefore, $\ker U = \core U$ implies that $\core U$ is dense in $U$.
(2)$\Rightarrow$(3) Suppose $y \in U \cap Y$. Since $U = \mathop{\sf cl}(\core U)$, we have $y \in \mathop{\sf cl} (\core U) \cap Y$. Because $\core U$ is an open upset, $\mathop{\sf cl} (\core U) \cap Y = \core U \cap Y$ by \cite[Lem.~4.15(1)]{BezhanishviliMelzer2022b}. Therefore, $y \in \core U$, and so
there is $V\in{\sf ClopSUp}(X)$ such that $y \in V \subseteq U$.
(3)$\Rightarrow$(4) Let $F \subseteq \ker U$ be a Scott upset. Let $y \in F \cap Y$. Then $y \in \ker U$, so $y \in U$ by \cref{lem:core-1}. Therefore, by (3), there is $V_y \in{\sf ClopSUp}(X)$ such that $y \in V_y \subseteq U$. Thus, \[
F = \bigcup \{\mathord\uparrow y \mid y \in F \cap Y\} \subseteq \bigcup \{V_y \mid y \in F \cap Y\} \subseteq U.
\]
Because $F$ is closed, it is compact. Therefore, since a finite union of clopen Scott upsets is a clopen Scott upset, there is $V \in {\sf ClopSUp}(X)$ such that $F \subseteq V \subseteq U$.
(4)$\Rightarrow$(1). By \cref{lem:core-1}, $\core U \subseteq \ker U$. For the reverse inclusion, it suffices to show that $V \ll U$ implies there is $W \in {\sf ClopSUp}(X)$ such that $V \subseteq W \subseteq U$.
Let $V \ll U$.
Then there is a Scott upset $F$ such that $V \subseteq F \subseteq U$ (see, e.g., \cite[Lem.~5.7]{BezhanishviliMelzer2022b}). But $U = \mathop{\sf cl} (\ker U)$, so $F \subseteq \ker U$ by (\ref{dagger}). Therefore, by (4), there is $W \in {\sf ClopSUp}(X)$ such that $F \subseteq W \subseteq U$, and hence $V \subseteq W \subseteq U$. \end{proof}
\begin{lemma} \plabel{lem:proper-coherent}
Let $f : X_1 \to X_2$ be an L-morphism between L-spaces.
\begin{enumerate}[ref=\thelemma(\arabic*)]
\item If $f$ is proper and $X_1$ is an algebraic L-space, then $f$ is coherent.
\item If $f$ is coherent and $X_2$ is an algebraic L-space, then $f$ is proper.
\item If $X_1$ and $X_2$ are algebraic L-spaces, then $f$ is coherent iff $f$ is proper. \clabel{3}
\end{enumerate} \end{lemma} \begin{proof}
(1) Let $U \in {\sf ClopUp}(X_2)$. Then
\begin{align*}
f^{-1}(\core U)
&\subseteq f^{-1}(\ker U) &&\text{by \cref{lem:core-1}}\\
&\subseteq \ker f^{-1}(U) && \text{since } f \text{ is proper}\\
&= \core f^{-1}(U) &&\text{by \cref{lem:core-3,lem:scott-ext-1}}.
\end{align*}
(2) Let $U \in {\sf ClopUp}(X_2)$. Then
\begin{align*}
f^{-1}(\ker U)
&= f^{-1}(\core U) &&\text{by \cref{lem:core-3,lem:scott-ext-1}}\\
&\subseteq \core f^{-1}(U) && \text{since } f \text{ is coherent}\\
&\subseteq \ker f^{-1}(U) &&\text{by \cref{lem:core-1}}.
\end{align*}
(3) follows from (1) and (2). \end{proof}
Putting \cref{lem:core-3,lem:proper-coherent-3} together, we obtain:
\begin{theorem} \label{thm: full sub} \cat{AlgLPries} is a full subcategory of \cat{ConLPries}. \end{theorem}
We are ready to prove the first main result of this section.
\begin{theorem} \label{thm: AlgFrm dual AlgL}
\cat{AlgFrm} is dually equivalent to \cat{AlgLPries}. \end{theorem}
\begin{proof} By \cref{rem: full sub alg}, \cat{AlgFrm} is a full subcategory of \cat{ConFrm}. By \cref{thm: full sub}, \cat{AlgLPries} is a full subcategory of \cat{ConLPries}. Thus, the result follows from \cref{hl-frames,thm:alg-hb-2}. \end{proof}
Finally, we connect \cat{AlgLPries} with \cat{KBSob}.
\begin{lemma}
Let $X$ be an SL-space, $Y$ its spatial part, and $U \subseteq X$. Then $U \in {\sf ClopSUp}(X)$ iff there is a compact open set $V$ of $Y$ such that $\mathop{\sf cl} V = U$. \label{lem:clopen-scott} \end{lemma}
\begin{proof} By \cite[Thm.~5.7]{BezhanishviliMelzer2022}, the poset of Scott upsets of $X$ is isomorphic to the poset of compact saturated sets of $Y$. The isomorphism is obtained by sending a Scott upset $F \subseteq X$ to the compact saturated set $F \cap Y$, and a compact saturated set $K \subseteq Y$ to the Scott upset $\mathord\uparrow K$.
($\Rightarrow$) Suppose $U$ is a clopen Scott upset. Then $V := U \cap Y$ is a compact saturated subset of $Y$. Moreover, $V$ is an open subset of $Y$ since $U \in {\sf ClopUp}(X)$. Furthermore, $\mathop{\sf cl} V = U$ by \cref{rem:iso opens Y} because $X$ is an SL-space.
($\Leftarrow$) Suppose there is a compact open set $V$ of $Y$ such that $\mathop{\sf cl} V = U$. Then $\mathord\uparrow V$ is a Scott upset of $X$. Since $V$ is open and $X$ is an SL-space, there is $U' \in {\sf ClopUp}(X)$ such that $V = U' \cap Y$ and $\mathop{\sf cl} V = U'$ (see \cref{rem:iso opens Y}). Therefore, $U = \mathop{\sf cl} V = U'$, and so $U$ is a clopen upset of $X$. Moreover, \[ U = \mathord\uparrow U = \mathord\uparrow \mathop{\sf cl} V = \mathop{\sf cl} \mathord\uparrow V = \mathord\uparrow V, \] where the third equality follows from \cite[Thm.~3.1.2]{Esakia2019} since $X$ is an Esakia space. Thus, $U$ is a Scott upset. \end{proof}
\begin{theorem} \label{thm: HB iff KB}
Let $X$ be an SL-space and $Y$ its spatial part. Then $X$ is an algebraic L-space iff $Y$ is a compactly based sober space. \end{theorem}
\begin{proof} The spatial part of an L-space is always sober (see, e.g., \cite[Lem.~4.12] {BezhanishviliMelzer2022b}). Therefore, it is sufficient to show that $X$ is an algebraic L-space iff $Y$ is compactly based.
First suppose that $X$ is an algebraic L-space. Let $V \subseteq Y$ be open and $y \in V$. Set $U = \mathop{\sf cl} V$. Then $U$ is a clopen upset of $X$ by \cref{rem:iso opens Y}. Moreover, it follows from \cite[Lem.~4.15(2)]{BezhanishviliMelzer2022b} that $U \cap Y = \mathop{\sf cl} V \cap Y = V$, so $y \in U \cap Y$.
By \cref{lem:core-3,lem:scott-ext-needed}, there is $W \in{\sf ClopSUp}(X)$ such that $y \in W \subseteq U$.
Therefore, $y \in W \cap Y \subseteq U \cap Y = V$. By \cref{lem:clopen-scott}, $W\cap Y$ is a compact open subset of $Y$. Thus, $Y$ is compactly based.
Conversely, suppose that $Y$ is compactly based and $U \in {\sf ClopUp}(X)$. Since $Y$ is locally compact, $X$ is a continuous L-space by \cref{hl-spaces}. Therefore, by \cref{lem:scott-ext-needed}, it suffices to show that for each $y \in U \cap Y$ there is $V \in {\sf ClopSUp}(X)$ such that $y \in V \subseteq U$. Because $U \cap Y$ is an open subset of $Y$ and $Y$ is compactly based, there is a compact open $K \subseteq Y$ such that $y \in K \subseteq U \cap Y$. Therefore, $\mathop{\sf cl} K \in {\sf ClopSUp}(X)$ by \cref{lem:clopen-scott}. Moreover, $y \in \mathop{\sf cl} K \subseteq \mathop{\sf cl} (U \cap Y) = U$. Thus, $X$ is an algebraic L-space. \end{proof}
By \cref{thm: full sub}, \cat{AlgLPries} is a full subcategory of \cat{ConLPries}. By \cref{rem: full sub kbsob}, \cat{KBSob} is a full subcategory of \cat{LKSob}. Thus, as an immediate consequence of \cref{hl-spaces,{thm: HB iff KB}}, we obtain:
\begin{corollary} \label{cor: AlgL equiv KBSob}
\cat{AlgLPries} is equivalent to \cat{KBSob}. \end{corollary}
Putting together \cref{thm: AlgFrm dual AlgL,cor: AlgL equiv KBSob}, we obtain \cref{frm-dualities-alg} that $\cat{AlgFrm}$ is dually equivalent to \cat{KBSob}.
\section{Priestley duality for arithmetic, coherent, and Stone frames} \label{sec 5}
In this final section we describe Priestley duals of arithmetic, coherent, and Stone frames. We also connect them to stably compactly based, spectral, and Stone spaces, thus obtaining alternative proofs of \namecref{frm-dualities} \hyperref[frm-dualities]{\labelcref{frm-dualities}(2,3,4)}.
We conclude the paper by pointing out a connection to Priestley duality for bounded distributive lattices and Stone duality for boolean algebras.
\subsection{Arithmetic frames}
We recall (see \cref{def: kernel-stable}) that an L-space $X$ is kernel-stable if $\ker(U \cap V) = \ker U \cap \ker V$ for all $U,V \in {\sf ClopUp}(X)$.
\begin{definition} \label{def: ArithL}
\begin{enumerate}
\item[]
\item An \emph{arithmetic} L-space is a kernel-stable algebraic L-space.
\item Let \cat{AriLPries} be the full subcategory of \cat{AlgLPries} consisting of arithmetic L-spaces.
\end{enumerate} \end{definition}
\begin{lemma} \label{lem:scottstable}
Let $X$ be an algebraic L-space. Then $X$ is an arithmetic L-space iff $U \cap V \in {\sf ClopSUp}(X)$ for every $U,V \in {\sf ClopSUp}(X)$. \end{lemma}
\begin{proof}
For the left-to-right implication,
let $U,V \in {\sf ClopSUp}(X)$. By \cref{lem:compact}, $\ker U=U$ and $\ker V=V$. Therefore, since $X$ is kernel-stable, $\ker (U\cap V) = \ker U \cap \ker V = U \cap V$. Thus, $U\cap V \in {\sf ClopSUp}(X)$ using \cref{lem:compact} again.
For the right-to-left implication,
suppose $U_1,U_2 \in {\sf ClopUp}(X)$. It suffices to show that $W \subseteq \ker U_1 \cap \ker U_2$ iff $W \subseteq \ker (U_1 \cap U_2)$ for each $W \in {\sf ClopUp}(X)$.
Since $W$ is compact, by \cref{lem:scott-ext-1} and the assumption that $V_1, V_2 \in{\sf ClopSUp}(X) \Rightarrow V_1 \cap V_2 \in{\sf ClopSUp}(X)$,
\begin{align*}
W \subseteq \ker U_1 \cap \ker U_2
&\iff W \subseteq \core U_1 \cap \core U_2 \\
&\iff \exists V_1, V_2 \in{\sf ClopSUp}(X) : W \subseteq V_1 \subseteq U_1 \text{ and }W \subseteq V_2 \subseteq U_2 \\
&\iff\exists V \in {\sf ClopSUp}(X) : W \subseteq V \subseteq U_1 \cap U_2\\
&\iff W \subseteq \core (U_1 \cap U_2) \\
&\iff W \subseteq \ker (U_1 \cap U_2). \qedhere
\end{align*}
\end{proof}
\begin{lemma} \label{lem: stably compactly based}
Let $Y$ be a compactly based sober space. Then $Y$ is stably locally compact iff $Y$ is stably compactly based. \end{lemma}
\begin{proof}
The left-to-right implication
is trivial.
For the right-to-left implication,
let $A, B \subseteq Y$ be compact saturated. Since $Y$ is compactly based, every compact saturated set is an intersection of compact open sets (see \cref{rem: comp sat}). Therefore, $A \cap B = \bigcap \mathcal F$, where
\[\mathcal F = \{U \cap V \mid U,V \text{ compact open with }A \subseteq U \text{ and } B \subseteq V\}.\]
Since $Y$ is stably compactly based, $\mathcal F$ is closed under finite intersections. Thus, the Hofmann-Mislove Theorem (see, e.g., \cite[Cor.~II-1.22.]{Compendium2003}) implies that $\bigcap \mathcal F$ is compact. Consequently, $A \cap B$ is compact. \end{proof}
\begin{theorem} \label{thm:arithmetic}
Let $L$ be an algebraic frame, $X_L$ its Priestley space, and $Y_L \subseteq X_L$ the spatial part of $X_L$. The following are equivalent.
\begin{enumerate}
\item $L$ is an arithmetic frame.
\item $X_L$ is an arithmetic L-space.
\item $Y_L$ is a stably compactly based space.
\end{enumerate} \end{theorem} \begin{proof}
Since $L$ is an algebraic frame, $X_L$ is an algebraic L-space by \cref{thm:alg-hb}, and hence $Y_L$ is a compactly based sober space by \cref{thm: HB iff KB}.
(1)$\Leftrightarrow$(2) Suppose $L$ is an arithmetic frame. Let $\varphi(a),\varphi(b) \in {\sf ClopSUp}(X_L)$. Then $a, b \in K(L)$ by \cref{lem:compact}. Since $L$ is an arithmetic frame, $a \wedge b \in K(L)$. Therefore, $\varphi(a) \cap \varphi(b) = \varphi(a \wedge b)$ is a Scott upset, again by \cref{lem:compact}. Thus, $X_L$ is an arithmetic L-space by \cref{lem:scottstable}.
Conversely, suppose $X_L$ is an arithmetic L-space.
Let $a,b \in K(L)$. By \cref{lem:compact}, $\varphi(a),\varphi(b)$ are clopen Scott upsets. By \cref{lem:scottstable}, $\varphi(a \wedge b) = \varphi(a) \cap \varphi(b)$ is a Scott upset. Therefore, $a \wedge b \in K(L)$, again by \cref{lem:compact}. Thus, $L$ is an arithmetic frame.
(2)$\Leftrightarrow$(3) Since $X_L$ is an algebraic L-space,
$X_L$ is an arithmetic L-space iff $X_L$ is a stably continuous L-space by \cref{lem:scottstable}. But $X_L$ is a stably continuous L-space iff $Y_L$ is a stably locally compact space by \cite[Thm.~6.6]{BezhanishviliMelzer2022b}. However, since $Y_L$ is a compactly based sober space, $Y_L$ is stably locally compact iff $Y_L$ is stably compactly based by \cref{lem: stably compactly based}. Thus, $X_L$ is an arithmetic L-space iff $Y_L$ is a stably compactly based space. \end{proof}
As a consequence of \cref{thm: AlgFrm dual AlgL}, \cref{cor: AlgL equiv KBSob}, and \cref{thm:arithmetic}, we arrive at the first main result of this section:
\begin{theorem} \label{cor: duality for ArithFrm}
\cat{AriLPries} is equivalent to \cat{StKBSp} and dually equivalent to \cat{AriFrm}. \end{theorem}
As a corollary we obtain \cref{frm-dualities-arith} that \cat{AriFrm} is dually equivalent to \cat{StKBSp}.
\subsection{Coherent frames}
We next turn our attention to Priestley duals of coherent frames. Since coherent frames are exactly compact arithmetic frames, we obtain that Priestley duals of coherent frames are exactly arithmetic L-spaces that are L-compact (see \cref{lem:compact}). We then connect L-compact arithmetic L-spaces with spectral spaces to obtain the well-known duality between \cat{CohFrm} and \cat{Spec} discussed in \cref{frm-dualities-coh}.
\begin{definition} \label{def: CohL}
\begin{enumerate}
\item[]
\item A \emph{coherent} L-space is an L-compact arithmetic L-space.
\item Let \cat{CohLPries} be the full subcategory of \cat{AriLPries} consisting of coherent L-spaces.
\end{enumerate} \end{definition}
\begin{lemma}[{\cite[Lem.~7.9(2)]{BezhanishviliMelzer2022b}}] \label{lem:compact=tight}
Let $X$ be an SL-space and $Y$ its spatial part. Then $X$ is L-compact iff $Y$ is compact. \end{lemma}
\begin{theorem} \label{thm:CohL-equivalences}
Let $L$ be an algebraic frame, $X_L$ its Priestley space, and $Y_L$ the spatial part of $X_L$. The following are equivalent. \begin{enumerate}
\item $L$ is a coherent frame.
\item $X_L$ is a coherent L-space.
\item $Y_L$ is a spectral space. \end{enumerate} \end{theorem} \begin{proof}
(1)$\Leftrightarrow$(2) $L$ is a coherent frame iff $L$ is a compact arithmetic frame. By \cref{lem:compact,thm:arithmetic}, this is equivalent to
$X_L$ being a coherent L-space.
(2)$\Leftrightarrow$(3)
By \cref{lem:compact=tight,thm:arithmetic}, $X_L$ is a coherent L-space iff $Y_L$ is a compact stably compactly based space, hence a spectral space. \end{proof}
As a consequence of \cref{cor: duality for ArithFrm,thm:CohL-equivalences}, we obtain the second main result of this section:
\begin{corollary} \label{cor:cohfrm=CohL=spec}
\cat{CohLPries} is equivalent to \cat{Spec} and dually equivalent to \cat{CohFrm}. \end{corollary}
As a corollary we obtain \cref{frm-dualities-coh} that \cat{CohFrm} is dually equivalent to \cat{Spec}.
\subsection{Stone frames}
Finally, we describe Priestley duals of Stone frames. Stone frames are characterized by having enough complemented elements. In the language of Priestley spaces, complemented elements correspond to clopen upsets that are also downsets (see, e.g., \cite[Lem.~6.1]{BezhanishviliGabelaiaJibladze2016}).
Let $X$ be a Priestley space. Following \cite[p.~377]{BezhanishviliGabelaiaJibladze2016}, we call a subset of $X$ a \emph{biset} if it is both an upset and a downset. Let ${\sf ClopBi}(X)$ be the collection of clopen bisets of $X$.
\begin{definition} \label{def: centered}
Let $X$ be an L-space.
\begin{enumerate}
\item For $U \in {\sf ClopUp}(X)$, define the \emph{center of $U$} as
\[
\cen U = \bigcup \{V \in {\sf ClopBi}(X) \mid V \subseteq U\}.
\]
\item We call $X$ a \emph{zero-dimensional} L-space if for every $U \in {\sf ClopUp}(X)$ we have that $\cen U$ is dense in $U$.
\item A \emph{Stone} L-space is an L-compact zero-dimensional L-space.
\item Let \cat{StoneLPries} be the full subcategory of \cat{LPries} consisting of Stone L-spaces.
\end{enumerate} \end{definition}
\begin{remark}
In \cite[Def.~6.2]{BezhanishviliGabelaiaJibladze2016}, the center of a clopen upset $U$ is called the biregular part of $U$. \end{remark}
\begin{lemma} \plabel{lem:center}
Let $X$ be an L-space and $U \in {\sf ClopUp}(X)$.
\begin{enumerate}[ref=\thelemma(\arabic*)]
\item $\cen U \subseteq \reg U$. \clabel{cen<reg}
\item If $X$ is a zero-dimensional L-space, then $X$ is a regular L-space.
\item If $X$ is a Stone L-space, then $X$ is a compact regular L-space. \clabel{stone=kr}
\end{enumerate} \end{lemma} \begin{proof}
(1) Suppose $x \in \cen U$. Then there is $V \in {\sf ClopBi}(X)$ with $x \in V \subseteq U$. Therefore, ${\downarrow\uparrow} x \subseteq U$, so $x \in \reg U$ by \cite[Lem.~7.3(1)]{BezhanishviliMelzer2022b}.
(2) Suppose $U \in {\sf ClopUp}(X)$. Since $X$ is a zero-dimensional L-space, $\cen U$ is dense in $U$. But then $\reg U$ is dense in $U$ by (1). Thus, $X$ is a regular L-space.
(3) This follows from (2) and \cref{lem:compact}. \end{proof}
As an immediate consequence, we obtain that \cat{StoneLPries} is a full subcategory of $\cat{KRLPries}$. We proceed to show that \cat{StoneLPries} is a full subcategory of \cat{CohLPries}.
\begin{lemma}
Let $X$ be a Stone L-space.
\begin{enumerate}[ref=\thelemma(\arabic*)]
\item ${\sf ClopSUp}(X) = {\sf ClopBi}(X)$. \label[lemma]{lem: clopsup=clobi}
\item $\cen U = \core U$ for each $U \in {\sf ClopUp}(X)$. \label[lemma]{lem: cen=core}
\end{enumerate} \end{lemma}
\begin{proof} (1) Since $X$ is a Stone L-space, it is a compact regular L-space by \cref{lem:center-stone=kr}. Now apply \cite[Lem.~7.15(4)]{BezhanishviliMelzer2022b}.
(2) It suffices to show that for each clopen upset $V$ we have $V \subseteq \cen U$ iff $V \subseteq \core U$.
Since $V$ is compact, finite unions of bisets are bisets, and finite unions of Scott upsets are Scott upsets, (1) implies
\begin{align*}
V \subseteq \cen U
&\iff \exists W \in {\sf ClopBi}(X) : V \subseteq W \subseteq U \\
&\iff \exists W \in {\sf ClopSUp}(X) : V \subseteq W \subseteq U\\
&\iff V \subseteq \core U. \qedhere
\end{align*} \end{proof}
\begin{theorem} \label{thm: full sub StoneL}
\cat{StoneLPries} is a full subcategory of \cat{CohLPries}. \end{theorem} \begin{proof}
Every Stone L-space is a coherent L-space by \cref{lem: cen=core}.
Also, since $\cat{StoneLPries}$ is a full subcategory of \cat{KRLPries},
every L-morphism between Stone L-spaces is a proper L-morphism by \cite[Thm.~7.17]{BezhanishviliMelzer2022b}. Therefore, every such morphism
is a coherent L-morphism by \cref{lem:proper-coherent-3}. Thus, \cat{StoneLPries} is a full subcategory of \cat{CohLPries}. \end{proof} In \cite[Thm.~6.3(1)]{BezhanishviliGabelaiaJibladze2016} it is shown that Priestley duals of zero-dimensional frames are exactly zero-dimensional L-spaces. We connect zero-dimensional L-spaces to zero-dimensional topological spaces.
\begin{lemma} \label{lem:clopbi=clop}
Let $X$ be an L-space and $Y$ its spatial part.
\begin{enumerate}[ref=\thelemma(\arabic*)]
\item If $U \in {\sf ClopBi}(X)$, then $U \cap Y$ is clopen in $Y$. \label[lemma]{lem:clopbi=clop-1}
\item If $X$ is an SL-space and $V \subseteq Y$ is clopen, then there is $U \in {\sf ClopBi}(X)$ such that $V = U \cap Y$. \label[lemma]{lem:clopbi=clop-2} \end{enumerate} \end{lemma}
\begin{proof}
(1) This is immediate.
(2) Let $V \subseteq Y$ be clopen. Since $V$ is open, there is $U \in {\sf ClopUp}(X)$ such that $V = U \cap Y$ and $\mathop{\sf cl} V= U$ (see \cref{rem:iso opens Y}). Similarly, since $V$ is closed, there is $W \in {\sf ClopUp}(X)$ such that $Y \setminus V = W \cap Y$ and $\mathop{\sf cl} (Y \setminus V) = W$. We have
\[
U \cap W = \mathop{\sf cl}(V) \cap \mathop{\sf cl}(Y\setminus V) = \mathop{\sf cl}(V\cap(Y\setminus V)) = \varnothing,
\]
where the second equality follows from \cite[Lem.~4.16]{BezhanishviliMelzer2022b} because $V,Y \setminus V$ are open in $Y$. Also,
\[
U \cup W = \mathop{\sf cl} V \cup \mathop{\sf cl}(Y \setminus V) = \mathop{\sf cl}(V \cup (Y \setminus V)) = \mathop{\sf cl} Y = X.
\]
Thus, $U = X \setminus W$, and hence $U \in {\sf ClopBi}(X)$. \end{proof}
\begin{theorem} \label{lem:zerodim-sp}
Let $X$ be an L-space and $Y$ its spatial part.
\begin{enumerate}[ref=\thetheorem(\arabic*)]
\item If $X$ is a zero-dimensional L-space, then $Y$ is zero-dimensional.
\item If $X$ is an SL-space, then $X$ is a zero-dimensional L-space iff $Y$ is zero-dimensional. \label[theorem]{lem:zerodim-sp-2} \end{enumerate} \end{theorem}
\begin{proof}
(1) Suppose $X$ is a zero-dimensional L-space. Let $V \subseteq Y$ be open and $y \in V$. Then there is $U \in {\sf ClopUp}(X)$ such that $U \cap Y = V$. Since $\cen U$ is dense in $U$, we have $U \cap Y = \mathop{\sf cl} (\cen U) \cap Y = \cen U \cap Y$, where the last equality follows from \cite[Lem.~4.15(1)]{BezhanishviliMelzer2022b} because $\cen U$ is an open upset of $X$.
Therefore, there is $W \in{\sf ClopBi}(X)$ such that $y \in W \subseteq U$. Thus, $y \in W \cap Y \subseteq V$ and $W \cap Y$ is clopen in $Y$ by \cref{lem:clopbi=clop-1}.
Consequently, $Y$ is zero-dimensional.
(2) The left-to-right implication follows from (1). For the converse implication, suppose $Y$ is zero-dimensional. Let $U \in {\sf ClopUp}(X)$. Since $X$ is an SL-space, $U\cap Y$ is dense in $U$. Therefore, it suffices to show that $U \cap Y \subseteq \cen U$. Let $y \in U \cap Y$. Since $U \in {\sf ClopUp}(X)$, we have that $U \cap Y$ is open in $Y$. Because $Y$ is zero-dimensional, there is clopen $V \subseteq Y$ such that $y \in V \subseteq U \cap Y$. Since $V$ is clopen in $Y$, \cref{lem:clopbi=clop-2} implies that there is $W \in {\sf ClopBi}(X)$ such that $V = W \cap Y$. Because $X$ is an SL-space, $\mathop{\sf cl} V = W$, and hence $y \in W \subseteq U$. Thus, $y \in \cen U$. \end{proof}
\begin{corollary}\label{thm:zerodim}
Let $L$ be a spatial frame, $X_L$ its Priestley space, and $Y_L$ the spatial part of $X_L$. The following are equivalent.
\begin{enumerate}
\item $L$ is a zero-dimensional frame.
\item $X_L$ is a zero-dimensional L-space.
\end{enumerate}
If in addition $L$ is spatial, then \upshape{(1)} and \upshape{(2)} are equivalent to
\begin{enumerate}[resume]
\item $Y_L$ is a zero-dimensional space.
\end{enumerate} \end{corollary}
\begin{proof}
That (1)$\Leftrightarrow$(2) follows from \cite[Thm.~6.3(1)]{BezhanishviliGabelaiaJibladze2016}, and that (2)$\Leftrightarrow$(3) follows from \cref{lem:zerodim-sp-2}. \end{proof}
\begin{corollary} \label{thm:StoneL=stone}
Let $L$ be a frame, $X_L$ its Priestley space, and $Y_L$ the spatial part of $X_L$. The following are equivalent.
\begin{enumerate}
\item $L$ is a Stone frame.
\item $X_L$ is a Stone L-space.
\end{enumerate}
If in addition $L$ is spatial, then \upshape{(1)} and \upshape{(2)} are equivalent to
\begin{enumerate}[resume]
\item $Y_L$ is a Stone space.
\end{enumerate} \end{corollary}
\begin{proof}
(1)$\Leftrightarrow$(2) Apply \cref{lem:compact,thm:zerodim}.
(2)$\Leftrightarrow$(3) Apply \cref{lem:compact=tight,thm:zerodim}. \end{proof}
As an immediate consequence, we arrive at the last main result of this section:
\begin{corollary}
\cat{StoneLPries} is equivalent to \cat{Stone} and dually equivalent to \cat{StoneFrm}. \end{corollary}
\begin{proof}
This follows from \cref{cor:cohfrm=CohL=spec,thm:StoneL=stone} and the observation that \cat{StoneFrm}, \cat{StoneLPries}, and \cat{Stone} are full subcategories of \cat{CohFrm}, \cat{CohLPries}, and \cat{Spec}, respectively (see \cref{rem: full sub stonefrm,thm: full sub StoneL,rem: full sub stone}). \end{proof}
\cref{frm-dualities-stone} that \cat{StoneFrm} is dually equivalent to \cat{Stone} is now immediate from the above corollary.
\begin{remark} Let $L$ be a frame and $X_L$ its Priestley space. As we saw in this paper, there are various maps from the clopen upsets of $X_L$ to the open upsets of $X_L$, and the corresponding density conditions are responsible for various properties of $L$. In particular, \begin{itemize} \item $L$ is continuous iff $\ker U$ is dense in $U$ for each $U\in{\sf ClopUp}(X)$; \item $L$ is algebraic iff $\core U$ is dense in $U$ for each $U\in{\sf ClopUp}(X)$; \item $L$ is regular iff $\reg U$ is dense in $U$ for each $U\in{\sf ClopUp}(X)$; \item $L$ is zero-dimensional iff $\cen U$ is dense in $U$ for each $U\in{\sf ClopUp}(X)$. \end{itemize} The strength of these properties of frames is then described by how these maps interact. For example, $\core U\subseteq\ker U$ for each $U\in{\sf ClopUp}(X)$ indicates that every algebraic frame is continuous, etc. \end{remark}
To summarize, we have the following diagram, where we use the same notation as in the previous diagrams. An overview of the introduced categories of Priestley spaces is given in \cref{table: algebraic spaces}. The corresponding categories of frames and spaces are described in \cref{table:alg frames,table:kb spaces}.
\begin{figure}
\caption{Equivalences and dual equivalences between various categories of algebraic frames, algebraic L-spaces, and compactly based sober spaces.}
\label{diagram 2}
\end{figure} \begin{table}[H] \centering \begin{tabular}{llll}
\toprule
\multicolumn{1}{c}{\bf Category} & \multicolumn{1}{c}{\bf Objects} & \multicolumn{1}{c}{\bf Morphisms} \\ \midrule
\cat{AlgLPries} & algebraic L-spaces (Def.~\ref{def: bunched}) & coherent L-morphisms (Def.~\ref{def: L-coherent})\\
\cat{AriLPries} & arithmetic L-spaces (Def.~\ref{def: ArithL}) & coherent L-morphisms \\
\cat{CohLPries} & coherent L-spaces (Def.~\ref{def: CohL}) & coherent L-morphisms \\
\cat{StoneLPries} & Stone L-spaces (Def.~\ref{def: centered}) & L-morphisms \\
\bottomrule \end{tabular} \caption{Categories of algebraic L-spaces.\label{table: algebraic spaces}} \end{table}
We conclude the paper by connecting the results obtained above with Priestley duality for bounded distributive lattices and Stone duality for boolean algebras.
\begin{remark}
Let $L$ be a coherent frame, $X_L$ its Priestley space, and $Y_L$ the spatial part of $X_L$. As we pointed out in the Introduction, $K(L)$ is a bounded distributive lattice and $L$ is isomorphic to the frame of ideals of $K(L)$. Moreover, $P \mapsto P\cap K(L)$ is an isomorphism between $(Y_L,\subseteq)$ and the poset of prime filters of $K(L)$. However, the Priestley topology on $X_{K(L)}$ does {\em not} correspond to the restriction to $Y_L$ of the Priestley topology on $X_L$.
Indeed, let $\varphi_{K_L} : K(L) \to {\sf ClopUp}(X_{K(L)})$ be the Stone map. By identifying $X_{K(L)}$ with $Y_L$, we have $\varphi_{K(L)}(k) = \varphi(k) \cap Y_L$
for each $k \in K(L)$. Since $X_{K(L)}$ has $\{\varphi_{K(L)}(k_1) \setminus \varphi_{K(L)}(k_2) \mid k_1,k_2 \in K(L) \}$ as a basis and ${\sf ClopSUp}(X)$ corresponds to $K(L)$ by \cref{lem:compact}, the topology on $Y_L$ corresponding to the Priestley topology on $X_{K(L)}$
is generated by the basis $\{(A \setminus B) \cap Y_L \mid A, B \in {\sf ClopSUp}(X_L)\}$.
If $L$ is a Stone frame, then $K(L)$ is the set of complemented elements of $L$ (that is, $a\vee a^*=1$).
Therefore, $K(L)$ is a boolean algebra and $Y_L = \min X_L$ (see, e.g., \cite[Lem.~7.15(5)]{BezhanishviliMelzer2022b}). But again, the Stone topology on $X_{K(L)}$ is not the restriction to $Y_L$ of the Priestley topology on $X_L$. In fact, the restriction of the Priestley topology on $X_L$ is the discrete topology on $Y_L$ (because $\mathord\downarrow y = \{y\}$ is open for each $y \in Y_L$), while the topology on $Y_L$ corresponding to the Stone topology on $X_{K(L)}$ is generated by $\{A \cap Y_L \mid A \in {\sf ClopBi}(X_L)\}$ (by the previous paragraph and \cref{lem: clopsup=clobi}). \end{remark}
\end{document} | arXiv |
Russell's paradox but set with sets of 2-element sets. [closed]
Want to improve this question? Update the question so it's on-topic for Mathematics Stack Exchange.
How do you prove that set of all 2-element sets does not exist basing on russell's paradox. Seems pretty obvious to me but no idea how to make a proper proof.
elementary-set-theory
Yashiru99
Yashiru99Yashiru99
$\begingroup$ All two sets? If there were only two then life would be a lot simpler. $\endgroup$ – badjohn Oct 22 '19 at 16:21
$\begingroup$ The assertion stating this exists is not inconsistent as a single statement in the way that that asserting the existence of the Russell set is. In the set theory NFU, for example, there is such a set. So the question needs to be sharpened by situating it within a particular theory. In ZFC, for example, its existence can be disproved because of the axiom of union. $\endgroup$ – Malice Vidrine Oct 22 '19 at 20:21
$\begingroup$ What do you mean by "basing on"? $\endgroup$ – Andrés E. Caicedo Oct 24 '19 at 1:39
It seems as though you want to prove the non-existence of the set of all two-element sets as a corollary of Russell's paradox. But there's an important difference between the class of all pairs and the Russell class. Notice the theory consisting of the single sentence $$\exists y\forall x(x\in y\leftrightarrow x\notin x)$$ is inconsistent. But there are consistent set theories in which $\{z:\exists xy(z=\{x,y\})\}$ is actually a set (like $\mathsf{NFU})$. So you can't disprove the existence of such a set except with respect to a particular theory.
If you're thinking about something like Zermelo set theory, or one of its extensions, then you likely already know the argument that there can be no universal set; the separation scheme would let us show that the Russell class is a set, resulting in the usual contradiction.
To disprove the existence of a set of all two-element sets in Zermelo, we note that in the presence of the other axioms (particularly whichever axioms ensure that there's at least one thing, and also something else), Pairing implies that every set is a member of some two-element set. So suppose we have our set of all pairs; the axiom of Union says that if we have this set, we can form the set of all elements that are a member of some two-element set. And that's the set of all sets, something we already know leads to contradiction. So we can have no such set.
Malice VidrineMalice Vidrine
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Russell's paradox in ZF theory : Enderton's Elements of set theory : Ch.2
About Russell's paradox and Russell's anti-sets | CommonCrawl |
Curie–Weiss law
About: Curie–Weiss law is a(n) research topic. Over the lifetime, 3501 publication(s) have been published within this topic receiving 60002 citation(s).
Book•
Spin Fluctuations in Itinerant Electron Magnetism
Tôru Moriya1, Yoshinori Takahashi1• Institutions (1)
University of Tokyo1
Abstract: A general theory of spin fluctuations and thermodynamical properties of itinerant electron magnets is developed, interpolating between the weakly and strongly ferromagnetic limits. A unified expression is given for the Curie temperature and the physical meaning of the Curie-Weiss magnetic susceptibility is discussed. As new phenomena derived from this theory the temperature-induced local magnetic moments as observed in CoS2, CoSe2, etc. and peculiar magnetic and thermal properties of nearly ferromagnetic semiconductors such as FeSi are discussed.
Criterion for Ferromagnetism from Observations of Magnetic Isotherms
Anthony Arrott1• Institutions (1)
Ford Motor Company1
15 Dec 1957-Physical Review
Abstract: A criterion is proposed for determining the onset of ferromagnetism in a material as its temperature is lowered from a region in which the linearity of its magnetic moment versus field isotherm gives an indication of paramagnetism. Within the limits of validity of a molecular field treatment, the Curie temperature is shown to be in general indicated by the third power of the magnetization being proportional to the internal magnetic field. The method has been employed to redetermine the Curie point of nickel from the data of Weiss and Forrer, of ${\mathrm{Fe}}_{3}$${\mathrm{O}}_{4}$ from the data of Smith and of some alloys from the data of Kaufmann and his collaborators and the author.
Polaron Percolation in Diluted Magnetic Semiconductors
A. Kaminski1, S. Das Sarma1• Institutions (1)
University of Maryland, College Park1
31 May 2002-Physical Review Letters
TL;DR: Analytical expressions for the Curie temperature and the magnetization in the limit of low carrier density are derived, obtaining excellent quantitative agreement with Monte Carlo simulation results and good qualitative agreement with experimental results.
Abstract: We theoretically study the development of spontaneous magnetization in diluted magnetic semiconductors as arising from a percolation of bound magnetic polarons. Within the framework of a generalized percolation theory we derive analytic expressions for the Curie temperature and the magnetization in the limit of low carrier density, obtaining excellent quantitative agreement with Monte Carlo simulation results and good qualitative agreement with experimental results.
New Magnetic Material Having Ultrahigh Magnetic Moment
T. K. Kim, M. Takahashi
15 Jun 1972-Applied Physics Letters
Abstract: The change of the saturation magnetization of Fe films with the pressure of nitrogen during deposition ranging from 2 × 10−5 to 7 × 10−3 Torr has been investigated systematically. We found a new magnetic material which has the highest saturation magnetization at room temperature, 2050 G, among those of all the magnetic materials. This was attributed to Fe16N2, which has a bct structure, and the magnetic moment associated with Fe atoms of Fe16N2 was deduced to be 3.0 μB.
On the Curie points and high temperature susceptibilities of Heisenberg model ferromagnetics
G. S. Rushbrooke1, P.J. Wood1• Institutions (1)
Durham University1
01 Jul 1958-Molecular Physics
Abstract: The first six coefficients in the expansion of the susceptibility χ, and its inverse, χ −1, in ascending powers of the reciprocal temperature, have been determined for the Heisenberg model of a ferromagnetic, for any spin value, S, and any lattice. The first five coefficients appropriate to the magnetic specific heat, C, have also been found. For the body-centred and face-centred cubic lattices, the χ and C coefficients are tabulated for half-integral S from 1/2 to 3. From these coefficients estimates have been made of the reduced Curie temperatures, θs c= k T c/J. It is found that for the simple, body-centred and face-centred cubic lattices the formula reproduces the estimated Curie temperatures fairly accurately. Here X=S(S+1) and z is the lattice coordination-number. It is found that, suitably scaled, the theoretical curves for inverse susceptibility against temperature above the Curie point are rather insensitive to the spin value and to the precise lattice structure. The ratio of their initial to the...
Band gap
Phase transition
Perovskite (structure) | CommonCrawl |
How many X-rays does a light bulb emit?
I read somewhere that most things1 emits all kinds of radiation, just very few of some kinds. So that made me wondering whether there is a formula to calculate how many X-rays an 100W incandescent light bulb would emit, for example in photons per second. For example, we already know that it emits infrared and visible light.
I find it hard to describe what I have tried. I searched on the internet for a formula, but couldn't find it. Yet I thought this was an interesting question, so I posted it here.
1 Black holes don't emit any radiation excepted for Hawking radiation if I get it right.
homework-and-exercises electromagnetic-radiation thermal-radiation estimation x-rays
wythagoras
wythagoraswythagoras
$\begingroup$ I was about to rant about wrong grammar for using "many" instead of "much" to describe intensity of radiation until I realized that photons are technically countable. $\endgroup$
– slebetman
$\begingroup$ @annav, is there really any upper energy limit to what we can call "X-rays"? I always thought that "X-ray" referred to high-energy photons that emanate from electron interactions, and "Gamma Ray" referred to high-energy photons that emanate from atomic nuclei. I have worked with medical machines generating photons with energies as high as 25 MeV---way higher than most gammas---and the manuals always said "X-ray". $\endgroup$
– Solomon Slow
$\begingroup$ About 25 bursts of gamma rays per year due to natural alpha decay of Tungsten producing secondary gamma rays. $\endgroup$
– Count Iblis
$\begingroup$ @ErikE Note that in some dialects (specifically in the UK) hundred sounds like 'undred, and the decision to use 'a' or 'an' depends on whether the initial sound is a vowel or consonant, so this is a rare case where the dialect affects orthography. I have seen a similar case with "a historical" / "an historical", depending on whether you voice the initial h. $\endgroup$
– Mario Carneiro
$\begingroup$ @MarioCarneiro I see! I was thinking only "one hundred". I would never omit the "one". $\endgroup$
– ErikE
The formula you want is called Planck's Law. Copying Wikipedia:
The spectral radiance of a body, $B_{\nu}$, describes the amount of energy it gives off as radiation of different frequencies. It is measured in terms of the power emitted per unit area of the body, per unit solid angle that the radiation is measured over, per unit frequency.
$$ B_\nu(\nu, T) = \frac{ 2 h \nu^3}{c^2} \frac{1}{e^\frac{h\nu}{k_\mathrm{B}T} - 1} $$
Now to work out the total power emitted per unit area per solid angle by our lightbulb in the X-ray part of the EM spectrum we can integrate this to infinity:
$$P_{\mathrm{X-ray}} = \int_{\nu_{min}}^{\infty} \mathrm{B}_{\nu}d\nu, $$
where $\nu_{min}$ is where we (somewhat arbitrarily) choose the lowest frequency photon that we would call an X-ray photon. Let's say that a photon with a 10 nm wavelength is our limit. Let's also say that 100W bulb has a surface temperature of 3,700 K, the melting temperature of tungsten. This is a very generous upper bound - it seems like a typical number might be 2,500 K.
We can simplify this to:
$$ P_{\mathrm{X-ray}} = 2\frac{k^4T^4}{h^3c^2} \sum_{n=1}^{\infty} \int_{x_{min}}^{\infty}x^3e^{-nx}dx, $$
where $x = \frac{h\nu}{kT}$. wythagoras points out we can express this in terms of the incomplete gamma function, to get
$$ 2\frac{k^4T^4}{h^3c^2}\sum_{n=1}^{\infty}\frac{1}{n^4} \Gamma(4, n\cdot x) $$
Plugging in some numbers reveals that the n = 1 term dominates the other terms, so we can drop higher n terms, resulting in
$$ P \approx 10^{-154} \ \mathrm{Wm^{-2}}. $$
This is tiny. Over the course of the lifetime of the universe you can expect on average no X-Ray photons to be emitted by the filament.
More exact treatments might get you more exact numbers (we've ignored the surface area of the filament and the solid angle factor for instance), but the order of magnitude is very telling - there are no X-ray photons emitted by a standard light bulb.
Chris CundyChris Cundy
$\begingroup$ It's a great answer, thank you. But the number is much, much lower than I had expected. By the way, I know a way to solve the integral and the series, if you would like to know how I can write an answer. $\endgroup$
– wythagoras
$\begingroup$ By all means! I'd be really interested to see what you think. griffin175's answer physics.stackexchange.com/a/200883/81404 seems to roughly agree that there are basically no photons. $\endgroup$
– Chris Cundy
$\begingroup$ I'm having trouble with the closed form due to a stupid mistake. Doing the substitution $u=nx$, we get $$ \sum_{n=1}^{\infty} \int_{\nu_{min} \cdot n}^{\infty} \frac{1}{n}\left(\frac{u}{n}\right)^3e^{-u}\mathrm{d}u$$ $$ \sum_{n=1}^{\infty} \frac{1}{n^4} \Gamma(4,\nu_{min}\cdot n)$$ where $\Gamma$ is the upper complete gamma function. But even if $\Gamma(4,\nu_{min})$ is extremely small, rather something in the order of $e^{-\nu_{min}}\nu_{min}^3$, and $\nu_{min} = 3 \times 10^{16}$ if I didn't misunderstood you. $\endgroup$
$\begingroup$ Worth saying that Tungsten melts at 3695K. Assuming that's where you got the upper temperature bound from. $\endgroup$
– OrangeDog
$\begingroup$ Why did you say "we can simply this" and then make another equation that's like twice as big? Man, math is crazy. $\endgroup$
– corsiKa
The wavelengths of light emitted can be calculated using planks law and the temperature of the object. For your average 100W incandescent light bulb, the filament is 2823 kelvin according to google.
The spectral radiance, $B$, is equal to $$\frac{1.2\cdot10^{52}}{\mathrm{wavelength}^{5}\cdot e^{\frac{1.99\cdot10^{43}}{\mathrm{wavelength}\cdot4\cdot10^{26}}}-1}$$
Math to solve for spectral radiance is hard, so this online calculator will do all the work. X-rays are between 0.01nm and 10nm. The total radiance at 10nm is $2.7\cdot10^{-187}$photons/s/m2/sr/µm. That's so unbelievably small, It would take a very long time for that bulb to emit an xray photon. The calculator wont give the spectral radiance of the smaller wavelength xrays so we'll just use the biggest X-rays.
In order to figure out how many photons per second are emitted you would need to know the surface area of the filament. It's a tiny metal sting, that would be hard to find out, but if you really want to, break open a bulb and measure its length and diameter with a caliper. Estimate surface area using the surface area of a cylinder formula A=πdh. Forget the ends, they're too small to bother with.
If you don't want to go through the trouble of breaking a bulb, make a wild guesstimate. 0.6m length and $5\cdot10^{-4}$ diameter, being generous. area of 0.001 m2. So $2.7\cdot10^{-187}$photons/s/m^2/sr/µm, then with the given surface area, $2.7*10^{-190}$ photons/s/sr/µm. That's 8.5 photons every $10^{186}$ years. Maybe if you watch 100,000,000,000,000 light bulbs you might catch an X-ray within your lifetime.
griffin175griffin175
$\begingroup$ From tempurature you have its emmissions curve per $m^3$, and from specs (1600 lumens for 100 W bulb) you have the amount of visible light it emits. From that, you should be able to calculate surface area, no? $\endgroup$
– Yakk
$\begingroup$ 100,000,000,000,000 is nowhere NEAR enough bulbs. You need more like 100,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 bulbs. Give or take a couple of orders of magnitude. $\endgroup$
– Kyle Oman
$\begingroup$ @KyleOman You are right. Exepted that if out take all these bulbs as one big sphere, that the pressure will be much higher, so that the temperature will rise. $\endgroup$
$\begingroup$ Well, that's more than the estimated number of atoms in the observable Universe, so you'll also have an appreciable effect on cosmology (even before turning on the radiation field), and the whole thing will likely collapse gravitationally and heat that way (shining brightly in the X-ray, I would bet), and probably form the mother of all supermassive black holes. But I was mostly being (and still am being) somewhat facetious. Though a hundred trillion lightbulbs is a somewhat plausible number, whereas $10^{186}$ is on a totally different scale. Important to make that distinction imho. $\endgroup$
$\begingroup$ Fun fact: Most modern filaments are double-coiled tungsten alloy strings -- basically wound in a very tight coil, which is then wound in a looser coil... LOTS of surface area to be had! See upload.wikimedia.org/wikipedia/commons/0/08/Filament.jpg $\endgroup$
– Doktor J
I will give a closed form for the integral in Chris Cundy's answer.
Doing the substitution $u=nx$, we get $$ \sum_{n=1}^{\infty} \int_{x_{min} \cdot n}^{\infty} \frac{1}{n}\left(\frac{u}{n}\right)^3e^{-u}\mathrm{d}u$$
$$ \sum_{n=1}^{\infty} \frac{1}{n^4} \Gamma(4,x_{min}\cdot n)$$
where $\Gamma$ is the upper complete gamma function. We write $a=x_{min}$ as it will be used a lot so a short name is more useful. Using the reduction formula for the gamma function when the first argument is an integer, we get: $$ \sum_{n=1}^{\infty} \left(\frac{1}{n^4}e^{-an}\left(6+6an+3a^2n^2+a^3n^3\right)\right) $$
$$ 6\sum_{n=1}^{\infty} \frac{1}{n^4}e^{-an} + 6a\sum_{n=1}^{\infty} \frac{1}{n^3}e^{-an}+ 3a^2\sum_{n=1}^{\infty} \frac{1}{n^2}e^{-an}+a^3\sum_{n=1}^{\infty} \frac{1}{n}e^{-an}$$
Now note that $$\frac{\mathrm{d}}{\mathrm{d}a} \sum_{n=1}^{\infty} \frac{1}{n}e^{-an} = \sum_{n=1}^{\infty} \frac{\mathrm{d}}{\mathrm{d}a} \left[\frac{1}{n}e^{-an}\right]=\sum_{n=1}^{\infty}-e^{-an}=-\sum_{n=1}^{\infty}(e^{-a})^n=1-\frac{1}{1-e^{-a}}$$
$$\sum_{n=1}^{\infty} \frac{1}{n}e^{-an} = \int 1-\frac{1}{1-e^{-a}} \mathrm{d}a = -\ln|1-e^{-a}|$$
We'll get the other terms in a similiar way. The final result is:
$$\sum_{n=1}^{\infty} \int_{a}^{\infty} x^3e^{-nx}dx = -6\mathrm{Li}_4(e^a)+6a\mathrm{Li}_3(e^a)+6a^2 \mathrm{Li}_2(1-e^{-a})-9a^2\mathrm{Li}_2(e^a)\\+2a^3\ln|1-e^{-a}|-9a^3\ln|1-e^{a}|+5\frac{3}4 a^4$$
I used a Computer Algebra System to find this form. $\mathrm{Li}_n$ is the polylogarithm function.
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EM Radiation and Heat
Does a source emitting visible light also emit infrared, microwave and radio waves?
How can I measure the ability of sunglasses to block UV radiation?
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Is high temperature a necessary condition for an object to be a blackbody? | CommonCrawl |
\begin{document}
\pagestyle{plain}
\title{\LARGE{\textbf{Hyperbolic geometry and real moduli \\ of five points on the line}
\abstract{\noindent \footnotesize{We show that each connected component of the moduli space of smooth real binary quintics is isomorphic to an arithmetic quotient of an open subset of the real hyperbolic plane. Our main result is that the induced metric on this moduli space extends to a complete real hyperbolic orbifold structure on the space of stable real binary quintics. This turns the moduli space of stable real binary quintics into a non-arithmetic ball quotient, isometric to hyperbolic triangle of angles $\pi/3, \pi/5$ and $\pi/10$. We provide in this way the first new example of a real moduli space with non-arithmetic uniformization since the work of Allcock, Carlson and Toledo \cite{realACTsurfaces}. } }
\section{Introduction} \label{realbinaryintroduction}
This paper is a sequel to an earlier article on non-arithmetic uniforrmization of certain metric spaces attached to unitary Shimura varieties, see \cite{degaayfortman-nonarithmetic}. The goal there was to consider a set of anti-holomorphic involutions on such a Shimura variety, glue their real loci along geodesic subspaces, and prove that the result uniformizes as a real hyperbolic orbifold $\Gamma \setminus \mathbb{R} H^n$.
The goal of the present paper is to apply these results
to the study of moduli of real algebraic varieties.
More precisely, let $X \cong \bb A^6_\mathbb{R}$ be the affine space over $\mathbb{R}$ parametrizing homogeneous polynomials $F$ of degree five. Let the subvariety $X_0 \subset X$ parametrize polynomials with distinct roots, and $X_s \subset X$ polynomials with roots of multiplicity at most two (i.e.~stable in the sense of geometric invariant theory). The principal goal of this paper is to study the moduli space of stable \textit{real binary quintics} $$
\ca M_s(\mathbb{R}) \coloneqq \textnormal{GL}_2(\mathbb{R}) \setminus X_s(\mathbb{R}) \supset \textnormal{GL}_2(\mathbb{R}) \setminus X_0(\mathbb{R}) \eqqcolon \ca M_0(\mathbb{R}). $$ If $P_s \subset \mathbb{P}^1(\mathbb{C})^5$ is the set five-tuples $(x_1, \dotsc, x_5)$ such that no three $x_i \in \mathbb{P}^1(\mathbb{C})$ coincide (c.f.~\cite{MR0437531}), and $P_0 \subset P_s$ the subset of $5$-tuples whose coordinates are distinct, then $$ \ca M_0(\mathbb{R}) \cong \textnormal{PGL}_2(\mathbb{R}) \setminus (P_0/\mf S_5)(\mathbb{R}) \;\;\; \;\;\; \text{ and } \;\;\; \;\;\; \ca M_s(\mathbb{R}) \cong \textnormal{PGL}_2(\mathbb{R}) \setminus (P_s/\mf S_5)(\mathbb{R}). $$
In other words, $\ca M_0(\mathbb{R})$ is the space of subsets $S \subset \mathbb{P}^1(\mathbb{C})$ of cardinality $|S| = 5$ stable under complex conjugation modulo real projective transformations, and in $\ca M_s(\mathbb{R})$ one or two pairs of points are allowed to collapse. For $i = 0,1,2$, we define $\mr M_{i}$ to be the connected component of $\ca M_0(\mathbb{R})$ parametrizing $\textnormal{Gal}(\mathbb{C}/\mathbb{R})$-stable subsets $S \subset \mathbb{P}^1(\mathbb{C})$ with $2i$ complex and $5 - 2i$ real points.
There is a natural period map that defines an isomorphism of analytic spaces \[ \ca M_s(\mathbb{C}) = \textnormal{GL}_2(\mathbb{C}) \setminus X_s(\mathbb{C}) \xrightarrow{\sim} P\Gamma \setminus \mathbb{C} H^2 \]
for a certain arithmetic ball quotient $P\Gamma \setminus \mathbb{C} H^2$ (see Theorem \ref{th:delignemostow}). Moreover, strictly stable quintics correspond to points in a hyperplane arrangement $\mr H \subset \mathbb{C} H^2$ (see Proposition \ref{prop:stableperiodshyperplane}). By investigating the equivariance of this period map with respect to suitable anti-holomorphic involutions $\alpha_i \colon \mathbb{C} H^2 \to \mathbb{C} H^2,$ we obtain the following real analogue:
\begin{theorem} \label{th:theorem01} For each $i \in \{0,1,2\}$, the period map induces an isomorphism of real analytic orbifolds \begin{align} \label{iso:smoothcase} \mr M_i \cong P \Gamma_i \setminus \left(\mathbb{R} H^2 - \mr H_i \right). \end{align} Here $\mathbb{R} H^2$ is the real hyperbolic plane, $\mr H_i$ a union of geodesic subspaces in $ \mathbb{R} H^2$ and $P\Gamma_i$ an arithmetic lattice in $\textnormal{PO}(2,1)$. Moreover, the lattices $P\Gamma_i \subset \tn{PO}(2,1)$ are projective orthogonal groups attached to explicit quadratic forms over $\mathbb{Z}[\frac{\sqrt{5} - 1}{2}]$, see equation \eqref{eq:explicitquadraticforms}. \end{theorem} \noindent In particular, Theorem \ref{th:theorem01} endows each connected component $\mr M_i \subset \ca M_0(\mathbb{R})$ with a hyperbolic metric. Since one can deform the topological type of a $\textnormal{Gal}(\mathbb{C}/\mathbb{R})$-stable five-element subset of $\mathbb{P}^1(\mathbb{C})$ by allowing two points to collide, the compactification $\ca M_s(\mathbb{R}) \supset \ca M_0(\mathbb{R})$ is connected. One may wonder whether the metrics on the components $\mr M_i$ extend to a metric on the whole of $\ca M_s(\mathbb{R})$. If so, what does the resulting space look like at the boundary? Our main result answers these questions in the following way.
\begin{theorem} \label{th:theorem02} There exists a complete hyperbolic metric on $\ca M_s(\mathbb{R})$ that restricts to the metrics on $\mr M_i$ induced by (\ref{iso:smoothcase}). Let $\overline{\mr{M}}_\mathbb{R}$ denote the resulting metric space, and define \begin{align}\label{PGAMMAR} \Gamma_{3,5,10} = \langle \alpha_1, \alpha_2, \alpha_3 \mid \alpha_i^2 = (\alpha_1\alpha_2)^3 = (\alpha_1\alpha_3)^5 = (\alpha_2\alpha_3)^{10} = 1 \rangle. \end{align} Then there exist open embeddings $P \Gamma_i \setminus \left(\mathbb{R} H^2 - \mr H_i \right) \hookrightarrow \Gamma_{3,5,10}\setminus \mathbb{R} H^2$ and an isometry \begin{align} \label{isometry} \overline{\mr{M}}_\mathbb{R} \cong \Gamma_{3,5,10} \setminus \mathbb{R} H^2 \end{align} extending the orbifold isomorphisms (\ref{iso:smoothcase}) in Theorem \ref{th:theorem01}. In particular, $\overline{\mr{M}}_\mathbb{R}$ is isometric to the hyperbolic triangle $\Delta_{3,5,10}$ of angles $\pi/3, \pi /5, \pi/10$, see Figure~\ref{fig:triangle} on the next page.
\end{theorem}
\noindent Equivariant period maps arise often in real algebraic geometry as a method to obtain real uniformization of the connected components of a moduli space of smooth varieties. For instance, this works for abelian varieties \cite{grossharris}, algebraic curves \cite{seppalasilhol}, K3 surfaces \cite{Nikulin1980} and quartic curves \cite{heckman2016hyperbolic}. Fairly recently, Allcock, Carlson and Toledo have shown that in the cases of cubic surfaces \cite{realACTsurfaces} and binary sextics \cite{realACTnonarithmetic,realACTbinarysextics}, the real ball quotient connected components can be glued along the hyperplane arrangement to obtain a uniformization of the moduli space of real stable varieties by real hyperbolic space. Binary quintics provide the first new example of this phenomenon.
\begin{remarks} \label{rem:introremarks} \begin{enumerate}[wide, labelwidth=!, labelindent=0pt] \item
\label{remark:takeuchi}
The lattice $\Gamma_{3,5,10} \subset \textnormal{PO}(2,1)$ is \textit{non-arithmetic}, see \cite{takeuchi}. \item
After establishing Theorem \ref{th:theorem01}, the steps in the proof of Theorem \ref{th:theorem02} are as follows. By the theory of glueing real hyperbolic orbifolds developed in \cite{degaayfortman-nonarithmetic}, the pieces on the right hand side of (\ref{iso:smoothcase}) glue into a complete real hyperbolic orbifold $P\Gamma_\mathbb{R} \setminus \mathbb{R} H^2$. We then prove that the period maps (\ref{iso:smoothcase}) glue into an isomorphism $\overline{\mr M}_\mathbb{R} \cong P\Gamma_\mathbb{R} \setminus \mathbb{R} H^2$. Finally, to finish the proof of Theorem \ref{th:theorem02}, we show that $P\Gamma_\mathbb{R} \cong \Gamma_{3,5,10}$.
\item
The topological space $\ca M_s(\mathbb{R})$ underlies two orbifold structures: the natural orbifold structure of $ \textnormal{GL}_2(\mathbb{R}) \setminus X_s(\mathbb{R})$ and the structure on $\overline{\mr M}_\mathbb{R}$ induced by (\ref{isometry}). These structures only differ at one point of $\ca M_s(\mathbb{R})$, which is the point $(\infty, i, i, -i, -i)$ (see Figure~\ref{fig:triangle}). \item Important ingredients in the proof of Theorem \ref{th:theorem02} are \cite[Theorem 4.1]{degaayfortman-nonarithmetic} and the fact that $\mr H \subset \mathbb{C} H^2$ is an \textit{orthogonal arrangement} in the sense of \cite{orthogonalarrangements}.
The latter holds by \cite[Theorem 6.2 \& Proposition 6.4]{degaayfortman-nonarithmetic}. In fact, using \cite[Theorem 1.2]{orthogonalarrangements} one can show that the orthogonality of $\mr H \subset \mathbb{C} H^2$ implies that neither $\pi_1 \left( P_0 / \mf S_5 \right)$ nor $\pi_1^\textnormal{orb}\left( \ca M_0(\mathbb{C}) \right)$ is a lattice in any Lie group with finitely many connected components, where $\ca M_0(\mathbb{C}) = \textnormal{GL}_2(\mathbb{C}) \backslash X_0(\mathbb{C})$ is the moduli space of smooth complex binary quintics.
\end{enumerate} \end{remarks}
\hspace*{-3cm}\includegraphics[scale=0.195]{triangle15transparent.png} \label{fig:triangle}
\noindent \emph{Figure 1: The moduli space of stable real binary quintics as the hyperbolic triangle $\Delta_{3,5,10} \subset \mathbb{R} H^2$. Here $\lambda = \zeta_5 + \zeta_5^{-1}$ and $\omega = \zeta_3$.}
\subsection{Overview of this paper}We start by considering moduli of complex binary quintics in Section \ref{complexball}. The main result of Section \ref{complexball} provides an isomorphism between the moduli space of stable complex binary quintics and a complex ball quotient of dimension two, identifying the discriminant of the moduli space with an explicit divisor of the ball quotient. This uses a result of Deligne and Mostow \cite{DeligneMostow}. In Section \ref{modrealbinquin} we prove analogous statements for the moduli spaces of smooth and stable real binary quintics. In particular, moduli of stable real binary quintics are in one-to-one correspondence with points in a real ball quotient $P\Gamma_\mathbb{R}\setminus \mathbb{R} H^2$. We calculate $P\Gamma_\mathbb{R}$ in Section \ref{space:hyperbolictriangle}, proving that it isomorphic to the lattice $\Gamma_{3,5,10}$ defined in \eqref{PGAMMAR}. In Section \ref{sec:monodromy}, we calculate the monodromy groups of the moduli spaces of smooth binary quintics over $\mathbb{C}$ and over $\mathbb{R}$. Finally, in Section \ref{sec:announced}, we prove a result that was announced in \cite{degaayfortman-nonarithmetic}, see Theorem 1.3 in \emph{loc.~cit.}: the lattice $\Gamma^+_n$ in $\tn{PO}(n,1)$ attached to a suitable connected component $X(\Lambda_n)^+$ of the glued space $X(\Lambda_n)$ is non-arithmetic for each $n \geq 2$, if one uses for the glueing construction
the CM field $K =\mathbb{Q}(\zeta_5)$, the CM type $\Phi \subset \textnormal{Hom}(K,\mathbb{C})$ defined in \eqref{CMTYPE-Shimura-modification} below,
and the hermitian lattice $\Lambda_n = (\mathbb{Z}[\zeta_5]^{n+1}, \tn{diag}(-\lambda, 1, \dotsc, 1))$ over $\mathcal{O}_K = \mathbb{Z}[\zeta_5]$, where $\lambda = \zeta_5 + \zeta_5^{-1} = (\sqrt 5 - 1)/2$.
\section{Monodromy of families of quintic covers of the projective line}\label{section:monodromy}
Recall from the introduction that $X \cong \bb A^6_\mathbb{R}$ is the real affine space of homogeneous polynomials of degree five, $X_0$ the subvariety of polynomials with distinct roots, and $X_s \subset X$ the subvariety of polynomials with roots of multiplicity at most two, i.e.~non-zero polynomials whose class in the associated projective space is stable in the sense of geometric invariant theory \cite{GIT} for the action of $\textnormal{SL}_{2, \mathbb{R}}$ on it.
In Section \ref{section:monodromy}, we prove some preliminary results on monodromy of families of quintic covers $C \to \mathbb{P}^1$ ramified along a quintic hypersurface. These will allow us in Section \ref{complexball} to prove that a suitable period map for stable complex binary quintics is an isomorphism.
\begin{notation} \label{binarynotation} Let $K$ be the cyclotomic field $ \mathbb{Q}(\zeta)$, with $\zeta = \zeta_5 = e^{2 \pi i /5} \in \mathbb{C}$. The ring of integers $\mathcal{O}_K$ of $K$ is $\mathbb{Z}[\zeta]$ \cite[Chapter I, Proposition 10.2]{Neukirch}. Let $\mu_K \subset \mathcal{O}_K^\ast$ be the group of finite units in $\mathcal{O}_K$. Thus, $\mu_K$ is cyclic of order ten, generated by $-\zeta$. Define $\rho \colon K \to K$ as the involution with $\rho(\zeta) = \zeta^{-1}$, and let $F = K^\rho$ be the maximal totally real subfield of $K$. \end{notation}
\subsection{Quintic covers of the projective line} \label{jacofcyc}
For $F \in X_0(\mathbb{C})$, one obtains a smooth quintic hypersurface $Z_F = \set{F = 0} \subset \bb P^1_\mathbb{C}$. Let \[ C_F = \set{Y^5 - F(X_0,X_1) = 0} \subset \mathbb{P}^2_\mathbb{C} \] be the quintic cover $C_F \to \mathbb{P}^1_\mathbb{C}$ ramified along $Z_F$. Let \[ A_F = J(C_F) = \rm H^1(C_F, \mathcal{O}_{C_F})/\rm H^1(C_F, \mathbb{Z})(1) \] be the Jacobian of $C_F$, with weight $-1$ Hodge decomposition \begin{align}\label{eq:hodgedecomp} \rm H^1(C_F, \mathbb{C})(1) = \rm H_1(A_F, \mathbb{C}) = \rm H^{-1,0}(A_F) \oplus \rm H^{0,-1}(A_F). \end{align}
The curve $C_F$ comes equipped with automorphisms $\phi(\zeta^i) \colon (X_0,X_1,Y) \mapsto (X_0, X_1, \zeta^i\cdot Y)$, of order five if $i \not \equiv 0 \bmod 5$, inducing an embedding $\theta \colon \mathbb{Z}[\zeta] \to \textnormal{End}(A_F)$ compatible with \eqref{eq:hodgedecomp}. Following \cite[Section 5]{shimuratranscendental}, we define another ring homomorphism $\theta'$ as follows: \[ \theta' \colon \mathbb{Z}[\zeta] \to \textnormal{End}(A_F), \quad \theta'(\zeta) = \theta(\zeta^3). \] In the rest of the paper, the action of $\mathbb{Z}[\zeta]$ on $A_F$ for a smooth binary quintic $F$ will always be understood to be via $\theta'$. For $i \in \set{1,2,3,4}$, define \[ \rm H^{-1,0}(A_F)_{\zeta^i} = \set{x \in \rm H^{-1,0}(A_F) \mid \theta'(\zeta) = \zeta^i} \subset \rm H^{-1,0}(A_F), \] and define $\rm H^{0,-1}(A_F)_{\zeta^i} \subset \rm H^{0,-1}(A_F)$ in a similar way.
\begin{lemma} \label{lemma:refinedhodge} Let $Z_F \subset \bb P^1_\mathbb{C}$ be a smooth quintic hypersurface as above, with $A_F = J(C_F)$ the Jacobian of the quintic cover $C_F \to \mathbb{P}^1_\mathbb{C}$ ramified along $Z_F$. With respect to the above action of $\mathbb{Z}[\zeta]$ on $\rm H_1(A_F, \mathbb{C})$, one has the following refined Hodge numbers:
\begin{align*}
h^{-1,0}(A_F)_{\zeta} &= 1, \;\;\; h^{-1,0}(A_F)_{\zeta^2} = 3, \;\;\;
h^{-1,0}(A_F)_{\zeta^3} = 0, \;\;\;
h^{-1,0}(A_F)_{\zeta^{4}} = 2 \\
h^{0,-1}(A_F)_{\zeta} &= 2, \;\;\; h^{0,-1}(A_F)_{\zeta^2} = 0, \;\;\;
h^{0,-1}(A_F)_{\zeta^3} = 3, \;\;\;
h^{0,-1}(A_F)_{\zeta^{4}} = 1. \end{align*} \end{lemma} \begin{proof} This follows from the Hurwitz-Chevalley-Weil formula, see \cite[Proposition 5.9]{Moonen2011TheTL}. Alternatively, see \cite[Section 5]{carlsontoledomonodromy} or \cite[Section 5]{shimuratranscendental}. \end{proof}
\subsection{The hermitian lattice}
Fix a point $F_0 \in X_0(\mathbb{C})$. Let
$ C = \{Y^5 = F_0(X_0,X_1) \} \subset \mathbb{P}^2_\mathbb{C} $
be the corresponding cyclic cover of $\mathbb{P}^1_\mathbb{C}$.
Let
\[ \left(A = J(C),\quad \theta' \colon \mathcal{O}_K = \mathbb{Z}[\zeta] \to \textnormal{End}(A), \quad \lambda \colon A \xrightarrow{\sim} \wh A\right) \] be the polarized Jacobian of $C$, equipped with $\mathcal{O}_K$-action $\theta'$ as in Section \ref{jacofcyc}. Define $\Lambda$ as the free $\mathcal{O}_K$-module \[ \Lambda = \rm H_1(A, \mathbb{Z}) = \rm H^1(C, \mathbb{Z}). \] The polarization of $A$ is induced by the alternating cup product pairing \[ E \colon \Lambda \times \Lambda = \rm H^1(C, \mathbb{Z}) \times \rm H^1(C, \mathbb{Z}) \to \rm H^2(C,\mathbb{Z}) = \mathbb{Z}. \] For $a \in \mathcal{O}_K$ and $x,y \in \Lambda$, we have $E(\theta'(a)x,y) = E(x, \theta'(a^\rho)y)$.
\begin{comment} For $i \in \mathbb{Z}/5$, define $$\rm H^{-1,0}_{(i)} = \{x \in \rm H^{-1,0}: \theta'(\zeta)(x) = \zeta^ix \}.$$ Let $\rm H^{-1,0}_i = \dim_\mathbb{C} \rm H^{-1,0}_{(i)}$. Since $ \rm H^{-1,0} = \textnormal{Lie}(A) = \rm H^1(C, \mathcal{O}_C)$, we have $\rm H^{-1,0}_{(i)} = i-1$ for $1 \leq i \leq 4$. \end{comment}
Let $\eta \in \mathcal{O}_K$ be the purely imaginary element
$$\eta = \frac{5}{\zeta^2 - \zeta^{-2}} \in \mathcal{O}_K. $$ Note that the different ideal $\mf D_K \subset \mathcal{O}_K$ is generated by $\eta$.
Define a form $T$ on $\Lambda$ as \[ T \colon \Lambda \times \Lambda \to \mf D_K^{-1} \quad \tn{ as } \quad T(x,y) = \frac{1}{5}\sum_{j = 0}^{4}\zeta^jE\left( x, \theta'(\zeta)^j y \right). \] By \cite[Example 5.2.2]{degaayfortman-nonarithmetic}, this is the skew-hermitian form corresponding to $E$ via \cite[Lemma 5.1]{degaayfortman-nonarithmetic}. We obtain a hermitian form on the free $\mathcal{O}_K$-module $\Lambda$ as follows: \begin{equation} \label{eq:hermitianformonbinaryquinticlattice}
\mf h \colon \Lambda \times \Lambda \to \mathcal{O}_K, \;\;\; \mf h(x,y) = \eta \cdot T(x,y) = \frac{1}{\zeta^2 - \zeta^{-2}} \cdot \sum_{j = 0}^{4}\zeta^jE\left( x, \theta'(\zeta)^j y \right). \end{equation} By Lemma \cite[Lemma 5.1]{degaayfortman-nonarithmetic}, the hermitian lattice $(\Lambda, \mf h)$ is unimodular, because $(\Lambda, E)$ is unimodular.
For $i \in \set{1,2}$, define $h_{(i)}^{-1,0} = \dim \rm H^{-1,0}(A)_{\zeta^i}$ and $h_{(i)}^{0,-1} = \dim \rm H^{0,-1}(A)_{\zeta^i}$.
Define two embeddings $\sigma_i \colon K \to \mathbb{C}$ by $ \sigma_i(\zeta) = \zeta^i$ for $i = 1,2$, and let $V_i = \Lambda \otimes_{\mathcal{O}_K, \sigma_i}\mathbb{C}$.
\begin{lemma} For $i \in \set{1,2}$, the signature of the hermitian form $ \mf{h}^{\sigma_i} \colon V_i \otimes_{\mathbb{R}} V_i \to \mathbb{C}$ is
\begin{align} \label{quintic-cases} (s_i,r_i) \coloneqq \tn{sign}(\mf{h}^{\sigma_i}) = \begin{cases} (h^{0,-1}_{\sigma_1}, h^{-1,0}_{\sigma_1}) = (2,1) & \textnormal{ \emph{for} } i = 1, \quad \textnormal{ \emph{and} } \\ (h^{-1,0}_{\sigma_2}, h^{0,-1}_{\sigma_2}) = (3,0)& \textnormal{ \emph{for} } i = 2. \end{cases} \end{align} \end{lemma}
\begin{proof} For each embedding $\varphi: K \to \mathbb{C}$, the restriction of the hermitian form $\varphi(\eta)\cdot E_\mathbb{C}(x, \bar y)$ on $\Lambda_\mathbb{C}$
to $(\Lambda_\mathbb{C})_{\varphi} \subset \Lambda_\mathbb{C}$ coincides with $ \mf{h}^{\varphi}$ by \cite[Lemma 5.3]{degaayfortman-nonarithmetic}. Recall that $i \cdot E_\mathbb{C}(x, \bar y)$ is positive definite on $\rm H^{-1,0}(A)$ and negative definite on $\rm H^{0,-1}(A)$. Moreover, one has that $\Im(\sigma_1(\eta)) < 0$ and $\Im(\sigma_2(\eta)) > 0$. Taken together, all this implies that the hermitian form $\sigma_1(\eta)\cdot E_\mathbb{C}(x, \bar y) = \mf{h}^{\sigma_1}(x,y)$ is negative definite on $\rm H^{-1,0}(A)_{\zeta}$ and positive definite on $\rm H^{0,-1}(A)_{\zeta}$. Similarly, $\sigma_2(\eta)\cdot E_\mathbb{C}(x, \bar y) = \mf{h}^{\sigma_2}(x,y)$ is positive definite on $\rm H^{-1,0}(A)_{\zeta^2}$ and negative definite on $\rm H^{0,-1}(A)_{\zeta^2}$. We are done by Lemma \ref{lemma:refinedhodge}. \end{proof}
\begin{remark} These calculations are compatible with \cite[(1.3) \& Section 5]{shimuratranscendental}. \end{remark}
\noindent We define a CM type $\Phi \subset \textnormal{Hom}(K,\mathbb{C})$ as follows: \begin{align}\label{CMTYPE-Shimura-modification}
\Phi = \set{\tau, \varphi} \subset \textnormal{Hom}(K,\mathbb{C}) \quad \mid \quad \tau, \varphi \colon K \to \mathbb{C}, \quad \tau(\zeta) = \zeta^4, \quad \quad \varphi(\zeta) = \zeta^2.
\end{align}
\noindent Then $\Im(\tau(\eta)) > 0$ and $\Im(\varphi(\eta)) > 0$. Moreover, we have $\mf h^\tau(x,y) = \tau(\eta) E_\mathbb{C}(x, \bar y)$ on $(\Lambda_\mathbb{C})_\tau$ and $\mf h^\varphi(x,y) = \varphi(\eta) E_\mathbb{C}(x, \bar y)$ on $(\Lambda_\mathbb{C})_\varphi$. Thus, the signatures of $\mf h^\tau$ and $\mf h^\varphi$ are as follows: \begin{align} \label{quintic-cases-extra} \begin{split} \tn{sign}(\mf{h}^\tau) &= (h^{-1,0}_\tau, h^{0,-1}_\tau) = (2,1), \\ \tn{sign}(\mf h^\varphi) &= (h^{-1,0}_\varphi, h^{0,-1}_\varphi) = (3,0). \end{split}
\end{align}
\begin{corollary} \label{corollary:abeliansignature} The triple $(A, \theta', \lambda)$ satisfies \cite[Conditions 5.5]{degaayfortman-nonarithmetic} with respect to the hermitian form $\mf h$ and the CM type $\Phi$.
\qed \end{corollary} \begin{comment} \begin{proof} We have $\tn{Lie}(A) = \rm H^{-1,0}(A)$; let $\theta' \colon \tn{Lie}(A) \to \tn{Lie}(A)$ be the analytic endomorphism induced by $\theta'$. By Lemma \ref{lemma:refinedhodge}, for $a \in \mathcal{O}_K$, the characteristic polynomial $\tn{char}(t, \theta'(a) \mid \tn{Lie}(A))$ of $\theta'(a)$ is
\[ \tn{char}(t, \theta'(a) \mid \tn{Lie}(A)) = (t - \tau(a))^2 \cdot (t - \tau\rho(a))^1 \cdot (t - \varphi(a))^3 \cdot (t - \varphi\rho(a))^0 \in \mathbb{C}[t].
\]
\end{proof} \end{comment}
\subsection{The monodromy representation} \label{sec:monodromy} Consider the real algebraic variety $X_0$ introduced in Section \ref{realbinaryintroduction}. Let $D \subset \textnormal{GL}_2(\mathbb{C})$ be the group $D = \set{\zeta^i\cdot \textnormal{Id} } \subset \textnormal{GL}_2(\mathbb{C})$ of scalar matrices $\zeta^i \cdot \textnormal{Id}$, where $\textnormal{Id} \in \textnormal{GL}_2(\mathbb{C})$ is the identity matrix of rank two, and define \begin{align} \label{def:GC} G(\mathbb{C}) = \textnormal{GL}_2(\mathbb{C})/D. \end{align} The group $G(\mathbb{C})$ acts from the left on $X_0(\mathbb{C})$ in the following way: if $F(X_0,X_1) \in \mathbb{C}[X_0,X_1]$ is a binary quintic, we may view $F$ as a function $\mathbb{C}^2 \to \mathbb{C}$, and define $g \cdot F = F(g^{-1})$ for $g \in G(\mathbb{C})$. This gives a canonical isomorphism of complex analytic orbifolds $ \ca M_0(\mathbb{C}) = G(\mathbb{C}) \setminus X_0(\mathbb{C}), $ where $\ca M_0$ is the moduli stack of smooth binary quintics.
Define $ \mr C \to X_0$
as the universal family of cyclic covers $C \to \mathbb{P}^1$ ramified along a smooth binary quintic $\{F = 0\} \subset \mathbb{P}^1$, and let
$$
J = J(\mr C) \to X_0$$
be the relative Jacobian of $\mr C/X_0$. By Corollary \ref{corollary:abeliansignature}, the abelian scheme $J$ is a polarized abelian scheme of relative dimension six over $X_0$, equipped with $\mathcal{O}_K$-action of signature $\{(2,1), (3,0)\}$ with respect to $\Phi = \{\tau, \varphi\}$. Let $\bb V$ be the local system of hermitian $\mathcal{O}_K$-modules underlying the abelian scheme $J/X_0$. Attached to $\bb V$, we have a representation \[ \pi_1(X_0(\mathbb{C}), F_0) \to \Gamma \coloneqq \textnormal{Aut}_{\mathcal{O}_K}(\Lambda, \mf h) \] whose composition with the quotient map $\Gamma \to P\Gamma = \Gamma / \mu_K$ defines a homomorphism \begin{equation} \label{eq:monodromy} \rho_\Gamma \colon \pi_1(X_0(\mathbb{C}), F_0) \to P\Gamma. \end{equation} We shall see that $\rho_\Gamma$ is surjective, see Corollary \ref{cor:surjectivemon} below.
\subsection{Marked binary quintics} \label{sec:markedbinary} Let $F \in X_0(\mathbb{C})$ and consider the associated hypersurface $
Z_F = \{F = 0\} \subset \mathbb{P}^1_\mathbb{C}. $ A \textit{marking} of $F$ is a ring isomorphism $m: \rm H^0(Z_F(\mathbb{C}), \mathbb{Z}) \xrightarrow{\sim} \mathbb{Z}^5$. To give a marking is to give a labelling of the points $p \in Z_F(\mathbb{C})$. Let $\ca N_0$ be the space of marked binary quintics $(F, m)$; this is a manifold, equipped with a holomorphic map that forgets the marking: \begin{align}\label{markedquintics} \ca N_0 \to X_0(\mathbb{C}). \end{align} Explicitly, we can define $\ca N_0$ as $\ca N_0 = (\mathbb{C}^5 - \bigcup_{i< j}\Delta_{ij} ) \times \mathbb{C}$,
where $\Delta_{ij} = \set{x_i = x_j} \subset \mathbb{C}^5$. Thus, $\ca N_0$ is connected, and the map \eqref{markedquintics} corresponds to the étale finite $\mf S_5$-quotient map that sends $((\alpha_1, \dotsc, \alpha_5), \lambda)$ to the binary quintic $F(X_0,X_1) = \lambda \cdot \prod_{i = 1}^5 (X_0 - \alpha_i \cdot X_1)$.
Let $ \psi \colon \ca Z \to X_0 $ be the universal smooth binary quintic, and consider the local system $H = (\psi^{an})_\ast \underline{\mathbb{Z}}$ on $X_0(\mathbb{C})$; it has stalk $H_F = \rm H^0(Z_F(\mathbb{C}), \mathbb{Z})$ for $F \in X_0(\mathbb{C})$. The local system $H$ on $X_0(\mathbb{C})$ corresponds to a monodromy representation \begin{equation}\label{eq:monodromy3} \rho_5 \colon \pi_1(X_0(\mathbb{C}), F_0) \to \mf S_5, \end{equation} which is surjective since it corresponds to the representation of $\pi_0(X_0(\mathbb{C}))$ attached to the description of $X_0(\mathbb{C})$ as \'etale quotient of $\ca N_0=(\mathbb{C}^5 - \bigcup_{i< j}\Delta_{ij} ) \times \mathbb{C}$ by $\mf S_5$.
By choosing a marking $m_0: \rm H^0(Z_{F_0}(\mathbb{C}), \mathbb{Z}) \cong \mathbb{Z}^5$ lying over our base point $F_0 \in X_0(\mathbb{C})$ we obtain an embedding $ \pi_1 \left( \ca N_0, m_0 \right) \hookrightarrow \pi_1(X_0(\mathbb{C}), F_0)$
whose composition with the map $\rho_\Gamma$ of \eqref{eq:monodromy} defines a homomorphism \begin{equation}\label{eq:monodromy2} \mu: \pi_1(\ca N_0, m_0) \to P\Gamma. \end{equation} Define $\theta = \zeta - \zeta^{-1}$ and consider the three-dimensional $\bb F_5$ vector space $\Lambda/\theta\Lambda$ and the quadratic space $ W \coloneqq \left(\Lambda/\theta\Lambda,q\right), $ where $q$ is the quadratic form obtained by reducing $\mf h$ modulo $\theta\Lambda$. Define two groups $\Gamma_\theta$ and $P\Gamma_\theta$ as follows: \[ \Gamma_\theta = \textnormal{Ker}\left( \Gamma \to \textnormal{Aut}(W) \right), \quad P\Gamma_\theta = \textnormal{Ker}\left( P\Gamma \to P\textnormal{Aut}(W) \right) \subset \text{PU}(2,1). \] The composition $\ca N_0 \to X_0(\mathbb{C}) \to X_s(\mathbb{C})$ admits an essentially unique \textit{completion} $\ca N_s \to X_s(\mathbb{C})$, see \cite{fox} or \cite[\S8]{DeligneMostow}. Here $\ca N_s$ a manifold and $\ca N_s \to X_s(\mathbb{C})$ is a ramified covering space. The fact that $\rho_\Gamma$ is surjective rests on the following result. \begin{proposition} \label{prop:commutativemonodromy} The image of $\mu$ in (\ref{eq:monodromy2}) is the group $P\Gamma_\theta$, and the induced map $$\overline{\rho}_\Gamma \colon \pi_1(X_0(\mathbb{C}), F_0)/ \pi_1 \left( \ca N_0, m_0 \right)= \mf S_5 \to P\Gamma /P\Gamma_\theta = P\textnormal{Aut}(W)$$ is an isomorphism. In other words, we obtain a commutative diagram with exact rows: \begin{equation} \label{eq:commutativemonodromy} \begin{split}
\xymatrix{
0 \ar[r] & \pi_1(\ca N_0, m_0) \ar@{->>}[d]^\mu\ar[r] & \pi_1(X_0(\mathbb{C}), F_0)\ar[d]^{\rho_\Gamma} \ar[r]^{\;\;\;\;\;\;\;\; \rho_5} & \mf S_5\ar[d]_{\wr}^{\overline{\rho}_\Gamma} \ar[r] & 0 \\
0 \ar[r] & P\Gamma_\theta \ar[r] & P\Gamma \ar[r] & P\textnormal{Aut}(W) \ar[r] & 0.
}
\end{split} \end{equation}
\begin{proof} Consider the quotient $ Q = G(\mathbb{C}) \setminus \ca N_0 = \textnormal{PGL}_2(\mathbb{C}) \setminus P_0$, where $P_0 \subset \mathbb{P}^1(\mathbb{C})^5$ is the subvariety of distinct five-tupes, see Section \ref{realbinaryintroduction}. Let $0 \in Q$ be the image of $m_0 \in \ca N_0$. In \cite{DeligneMostow}, Deligne and Mostow define a hermitian space bundle $B_Q \to Q$ over $Q$ whose fiber over $0 \in Q$ is $\mathbb{C} H^2$. Writing $V = \Lambda \otimes_{\mathcal{O}_K, \tau} \mathbb{C}$, this gives a monodromy representation $$\pi_1(Q,0) \to \textnormal{PU}(V, \mf h^{\tau}) \cong \textnormal{PU}(2,1)$$ whose image we denote by $\Gamma_{\text{DM}}$. By a result of Kondō, one has $\Gamma_{\text{DM}} = P\Gamma_\theta$, see \cite[Theorem 7.1]{kondo5points}. Since $\ca N_0 \to Q$ is a covering space, the action of $G(\mathbb{C})$ on $\ca N_0$ being free, we have an embedding $\pi_1(\ca N_0, m_0) \hookrightarrow \pi_1(Q, 0)$ whose composition with $\pi_1(Q,0) \to\textnormal{PU}(2,1)$ is the map $\mu:\pi_1(\ca N_0, m_0) \to P\Gamma \subset \textnormal{PU}(2,1)$.
To prove that the image of $\mu$ is $P\Gamma_\theta$, it suffices to give a section of the map $\ca N_0 \to Q$. Indeed, such a section induces a retraction of $\pi_1(\ca N_0, m_0) \hookrightarrow \pi_1(Q, 0)$, so that the images of these two groups in $\textnormal{PU}(2,1)$ are the same. Observe that if $\Delta \subset \mathbb{P}^1(\mathbb{C})^5$ is the union of all hyperplanes $\{x_i = x_j\} \subset \mathbb{P}^1(\mathbb{C})^5$ for $i \neq j$, then \begin{align*} Q = \textnormal{PGL}_2(\mathbb{C}) \setminus P_0 &= \textnormal{PGL}_2(\mathbb{C}) \setminus \left( \mathbb{P}^1(\mathbb{C})^5 - \Delta \right) \cong \{(u_1,u_2) \in \mathbb{C}^2 \colon u_i \neq 0,1 \textnormal{ and } u_1 \neq u_2 \}. \end{align*} The section $Q \to \ca N_0$ may then be defined by sending $(u_1, u_2)$ to the binary quintic $ F(X_0,X_1) = X_0(X_0-X_1)X_1(X_0-u_1\cdot X_1)(X_0-u_2\cdot X_1) \in X_0(\mathbb{C})$, marked by the labelling of its roots $\{[0:1],[1:1], [1:0], [u_1:1], [u_2:1]\} \subset \mathbb{P}^1(\mathbb{C})$.
It remains to prove that the homomorphism $\overline \rho_\Gamma \colon \mf S_5 \to P\Gamma/P\Gamma_\theta$ appearing on the right of diagram (\ref{eq:commutativemonodromy}) is an isomorphism. We use Theorem \ref{th:calculatemonodromyshimura} below, which was proved by Shimura (see \cite{shimuratranscendental}), which says that $( \Lambda, \mf h ) \cong ( \mathcal{O}_K^3, \textnormal{diag}(\frac{1 - \sqrt 5}{2}, 1,1)).$ This implies that there are group isomorphisms $P\Gamma/P\Gamma_\theta = P\textnormal{Aut}(W) \cong \textnormal{PO}_3(\bb F_5) \cong \mf S_5.$ Consequently, to show that $\overline \rho_\Gamma$ is an isomorphism, it suffices to prove its injectivity.
For this, consider the manifold $\ca N_s$. Remark that $\mf S_5$ embeds into $\textnormal{Aut}(G(\mathbb{C}) \setminus \ca N_s)$. Moreover, there is a natural isomorpism $ f\colon G(\mathbb{C}) \setminus \ca N_s \cong P\Gamma_\theta \setminus \mathbb{C} H^2, $ see \cite{DeligneMostow, kondo5points}. The two compositions $$\mf S_5 \subset \textnormal{Aut}(G(\mathbb{C}) \setminus \ca N_s) \cong \textnormal{Aut}(P\Gamma_\theta \setminus \mathbb{C} H^2) \; \tn{ and } \; \overline \rho_\Gamma \colon \mf S_5 \to P\Gamma/P\Gamma_\theta \subset \textnormal{Aut}(P\Gamma_\theta \setminus \mathbb{C} H^2)$$ agree by the equivariance of $f $ with respect to $\overline \rho_\Gamma$. Thus, $\overline \rho_\Gamma$ is injective as desired.
\end{proof}
\end{proposition}
\begin{corollary} \label{cor:surjectivemon} The monodromy representation $\rho_\Gamma$ in (\ref{eq:monodromy}) is surjective. $
\qed$ \end{corollary}
\section{The period map for complex binary quintics} \label{complexball}
In Section \ref{complexball}, we will use the results of Section \ref{section:monodromy} to prove that there exists a hermitian $\mathcal{O}_K$-lattice $\Lambda$ of rank three, and an isomorphism of complex analytic spaces $\ca M_s(\mathbb{C}) \cong P\Gamma \setminus \mathbb{C} H^2$ that identifies $\ca M_s(\mathbb{C}) - \ca M_0(\mathbb{C})$ with an explicit divisor $P\Gamma \setminus \mr H \subset P\Gamma \setminus \mathbb{C} H^2$.
\subsection{Framed binary quintics} By a \textit{framing} of a point $F \in X_0(\mathbb{C})$ we mean a projective equivalence class $[f]$, where \[ f \colon \mathbb{V}_F = \rm H^1(C_F, \mathbb{Z}) \to \Lambda\] is an $\mathcal{O}_K$-linear isometry: two such isometries are in the same class if and only if they differ by an element in $\mu_K$. Let $\ca F_0$ be the collection of all framings of all points $x \in X_0(\mathbb{C})$. The set $\ca F_0$ is naturally a complex manifold, by arguments similar to those used in \cite{ACTsurfaces}.
Note that Corollary \ref{cor:surjectivemon} implies that $\ca F_0$ is connected, hence \begin{align*} \ca F_0 \to X_0(\mathbb{C}) \end{align*} is a covering, with Galois group $P\Gamma$.
\begin{lemma} \label{lemma:isomorphiccoveringspaces} The spaces $P\Gamma_\theta \setminus \ca F_0$ and $ \ca N_0$ are isomorphic as covering spaces of $X_0(\mathbb{C})$. In particular, there is a covering map $\ca F_0 \to \ca N_0$ with Galois group $P\Gamma_\theta$.
\end{lemma} \begin{proof} We have $P\Gamma/P\Gamma_\theta \cong \mf S_5$ as quotients of $P\Gamma$, see Proposition \ref{prop:commutativemonodromy}.
\end{proof}
\begin{lemma} \label{lemma:irreduciblenormal} The subvariety $\Delta \coloneqq X_s - X_0$ is an irreducible local normal crossings divisor of $X_s$. \end{lemma} \begin{proof}
Identify a binary quintic $F = \sum_{i = 0}^5 a_i X_0^{5-i}X_1^{i}$ with its dehomogenization $f(x) = \sum_{i = 0}^5 a_ix^i$. The discriminant of $f$ is a homogeneous polynomial $Disc(f)$ of degree $8$ in the coefficients $a_i$ of $f$. Thus, there is a polynomial function $Disc$ on $\mathbb{C}^6 = X(\mathbb{C})$, and this polynomial is irreducible \cite[Proposition 1.3 \& Example 1.4]{MR1264417}. Therefore, $\Delta = Z(Disc) \cap X_s$ is irreducible, since $X_s$ is open in $X$. The rest of the proof is similar to the proof of Proposition 6.7 in \cite{beauvillecubicsurfaces}. Namely, consider the incidence variety $\ca I = \set{(p, f) \in \mathbb{C}^5 \times X_s(\mathbb{C}) \mid p \in \tn{Sing}(f)} \subset \mathbb{C}^5 \times X_s(\mathbb{C})$, and let $(p,f) \in \ca I$. Since $f^{''}(p)$ is non-zero, the tangent map $Tq \colon T_{(p,f)} (\ca I) \to T_f(X_s) = X(\mathbb{C})$ of the projection $q \colon \ca I \to X_s$ is injective, and its image consists of the linear subspace $X_p \subset X$ of binary quintics that contain $p$ as a root. If $f$ has $k$ double points $p_i$, where $k \in \set{1,2}$, then $\Delta$ is locally isomorphic to the union $\cup_i X_{p_i}$; as these $X_{p_i}$ intersect transversally, we are done.
\end{proof}
\begin{lemma} \label{lemma:monodromy} The local monodromy transformations of $\ca F_0 \to X_0(\mathbb{C})$ around every $x \in \Delta$ are of finite order. More precisely, if $x \in \Delta$ lies on the intersection of $k$ local components of $\Delta$, then the local monodromy group around $x$ is isomorphic to $(\mathbb{Z}/10)^k$. \end{lemma}
\begin{proof} See \cite[Proposition 9.2]{DeligneMostow} or \cite[Proposition 6.1]{carlsontoledomonodromy} for the generic case, i.e.~when a quintic $Z = \{F = 0\} \subset \mathbb{P}^1_\mathbb{C}$ acquires only one node. See \cite[(5.1) - (5.7)]{ACTsurfaces} for the way in which one can prove the general case.
\end{proof}
\noindent In the following corollary, we let $D = \set{z \in \mathbb{C} \colon \va{z} < 1}$ denote the open unit disc, and $D^\ast = D - \{0\}$ the punctured open unit disc.
\begin{corollary} \label{cor:framedquintics} There is an essentially unique branched cover \[ \pi \colon \ca F_s \to X_s(\mathbb{C}), \] with $\ca F_s$ a complex manifold, such that for any $x \in \Delta$, any open $x \in U \subset X_s(\mathbb{C})$ with $U \cong D^k \times D^{6-k}$ and $U \cap X_0(\mathbb{C}) \cong (D^\ast)^k \times D^{6-k}$, and any connected component $U'$ of $\pi^{-1}(U) \subset \ca F_s$, there is an isomorphism $U' \cong D^k \times D^{6-k}$ such that the composition \[ D^k \times D^{6-k} \cong U' \to U \cong D^6 \; \text{ is the map } \; (z_1, \dotsc, z_6) \mapsto (z_1^{10}, \dotsc, z_k^{10}, z_{k+1}, \dotsc, z_6). \]
\end{corollary} \begin{proof} See \cite[Lemma 7.2]{beauvillecubicsurfaces}. See also \cite{fox} and \cite[Section 8]{DeligneMostow}. \end{proof} \noindent The group $G(\mathbb{C}) = \textnormal{GL}_2(\mathbb{C})/D$ (see (\ref{def:GC})) acts on $\ca F_0$ over its action on $X_0$. Explicitly, if $g \in G(\mathbb{C})$ and if $([\phi], \phi: \mathbb{V}_{F} \xrightarrow{\sim} \Lambda)$ is a framing of $F \in X_0(\mathbb{C})$, then \[ \left([\phi \circ g^\ast], \phi \circ g^\ast \colon \mathbb{V}_{g\cdot F} \xrightarrow{\sim} \Lambda\right) \] is a framing of $g\cdot F \in X_0(\mathbb{C})$. This is a left action. The group $P\Gamma$ also acts on $\ca F_0$ from the left, and the actions of $P\Gamma$ and $G(\mathbb{C})$ on $\ca F_0$ commute. By functoriality of the Fox completion, the action of $G(\mathbb{C})$ on $\ca F_0$ extends to an action of $G(\mathbb{C})$ on $\ca F_s$.
\begin{lemma} \label{cor:freeaction} The group $G(\mathbb{C}) = \textnormal{GL}_2(\mathbb{C})/D$ acts freely on $\ca F_s$. \end{lemma} \begin{proof}
The functoriality of the Fox completion gives an action of $G(\mathbb{C})$ on $\ca N_s$ such that, by Lemma \ref{lemma:isomorphiccoveringspaces}, $P\Gamma_\theta \setminus \ca F_s \cong \ca N_s$ as ramified covering spaces of $X_s(\mathbb{C})$.
In particular, it suffices to show that $G(\mathbb{C})$ acts freely on $\ca N_s$.
Note that $\ca N_0$ admits a natural $G_m$-covering map $\ca N_0 \to P_0$ where $P_0 \subset \mathbb{P}^1(\mathbb{C})^5$ is the space of distinct ordered five-tuples in $\mathbb{P}^1(\mathbb{C})$ introduced in Section \ref{realbinaryintroduction}. Consequently, there is a $G_m$-quotient map $\ca N_s \to P_s$, where $P_s$ is the space of stable ordered five-tuples, and this map is equivariant for the homomorphism $\textnormal{GL}_2(\mathbb{C}) \to \textnormal{PGL}_2(\mathbb{C})$.
Let $g \in \textnormal{GL}_2(\mathbb{C})$ and $x \in \ca N_s$ such that $gx = x$. It is clear that $\textnormal{PGL}_2(\mathbb{C})$ acts freely on $P_s$. Therefore, $g = \lambda \in \mathbb{C}^\ast$. Let $F \in X_s(\mathbb{C})$ be the image of $x \in \ca N_s$; then $$F(X_0,X_1) = gF(X_0,X_1) = F(g^{-1}(X_0,X_1)) = F(\lambda^{-1}X_0, \lambda^{-1}X_1) = \lambda^{-5}F(X_0,X_1).$$ Therefore, we have $\lambda^{5} = 1 \in \mathbb{C}$, and we conclude that $\lambda \in D \subset \textnormal{GL}_2(\mathbb{C})$.
\end{proof}
\subsection{The moduli space of complex binary quintics as a ball quotient} Consider the hermitian space $V = \Lambda \otimes_{\mathcal{O}_K, \tau} \mathbb{C}$ and define $\mathbb{C} H^2$ to be the space of negative lines in $V$. Using \cite[Proposition 5.7]{degaayfortman-nonarithmetic} we see that the abelian scheme $J \to X_0$ induces a holomorphic map
\begin{equation} \label{eq:periodframed} \ca P \colon \ca F_0 \to \mathbb{C} H^2. \end{equation} Explicitly, if $(F, [f]) \in \ca F_0$ is the framing $[f: \rm H^1(C_F, \mathbb{Z}) \xrightarrow{\sim}\Lambda]$ of the binary quintic $F \in X_0(\mathbb{C})$, and $A_F$ is the Jacobian of the curve $C_F$, then $$f \left( \rm H^{0,-1}(A_F)_{\zeta^4} \right) = f \left(\rm H^{1,0}(C_F)_{\zeta^4} \right) \subset \rm H^1(C, \mathbb{C})_{\tau} = \Lambda \otimes_{\mathcal{O}_K, \tau} \mathbb{C} = V$$ is a negative line in $V$, and we have $\ca P(F , [f] ) = f \left(\rm H^{1,0}(C_F)_{\zeta^4} \right) \in \mathbb{C} H^2$.
The map $\ca P$ is holomorphic, and descends to a morphism of complex analytic spaces \begin{equation*} \ca M_0(\mathbb{C}) =G(\mathbb{C}) \setminus X_0(\mathbb{C}) \to P\Gamma \setminus \mathbb{C} H^2. \end{equation*} \noindent By Riemann extension, (\ref{eq:periodframed}) extends to a $G(\mathbb{C})$-equivariant holomorphic map \begin{align}\label{eq:stableperiodmapframed} \ca P_s: \ca F_s \to \mathbb{C} H^2. \end{align}
\begin{theorem}[Deligne--Mostow] \label{th:delignemostow} The period map (\ref{eq:stableperiodmapframed}) induces an isomorphism of complex manifolds \begin{align} \label{eq:isomstablefivepoints-zero} \ca M_s^f(\mathbb{C}) \coloneqq G(\mathbb{C}) \setminus \ca F_s \cong \mathbb{C} H^2. \end{align} Taking $P\Gamma$-quotients gives an isomorphism of complex analytic spaces \begin{equation} \label{eq:isomstablefivepoints} \ca M_s(\mathbb{C}) = G(\mathbb{C}) \setminus X_s(\mathbb{C}) \cong P\Gamma \setminus \mathbb{C} H^2. \end{equation} \end{theorem} \begin{proof}
In \cite{DeligneMostow}, Deligne and Mostow define $\widetilde Q \to Q$ to be the covering space corresponding to the monodromy representation $\pi_1(Q,0) \to \textnormal{PU}(2,1)$; since the image of this homomomorphism is $P\Gamma_\theta$ (see the proof of Proposition \ref{prop:commutativemonodromy}), it follows that $G(\mathbb{C}) \setminus \ca F_0 \cong \widetilde Q$ as covering spaces of $Q$. Consequently, if $\widetilde Q_{\textnormal{st}}$ is the Fox completion of the spread $$\widetilde Q \to Q \to Q_{\textnormal{st}} \coloneqq G(\mathbb{C}) \setminus \ca N_s = \textnormal{PGL}_2(\mathbb{C}) \setminus P_s,$$ then there is an isomorphism $G(\mathbb{C}) \setminus \ca F_s \cong \widetilde Q_{\textnormal{st}}$ of branched covering spaces of $ Q_{\textnormal{st}}$. We obtain commutative diagrams, where the lower right morphism uses (\ref{eq:commutativemonodromy}): $$ \xymatrixrowsep{1.5pc} \xymatrixcolsep{5pc} \xymatrix{ G(\mathbb{C}) \setminus \ca F_s \ar[r]^{\sim \;\;\;\;} \ar[d] & \widetilde{Q}_{\textnormal{st}}\ar[d] \ar[r] & \mathbb{C} H^2 \ar[d] \\ G(\mathbb{C}) \setminus \ca N_s \ar[r]^{\sim \;\;\;\;} \ar[d] & Q_{\textnormal{st}} \ar[r] \ar[d]& P\Gamma_{\theta} \setminus \mathbb{C} H^2\ar[d]\\ G(\mathbb{C}) \setminus X_s(\mathbb{C}) \ar[r]^{\sim \;\;\;\;} & Q_{\textnormal{st}}/\mf S_5 \ar[r] & P\Gamma \setminus \mathbb{C} H^2. } $$ The map $\widetilde{Q}_{\textnormal{st}} \to \mathbb{C} H^2$ is an isomorphism by \cite[(3.11)]{DeligneMostow}. Therefore, we are done if the composition $G(\mathbb{C}) \setminus \ca F_0 \to \widetilde Q \to \mathbb{C} H^2$ agrees with the period map $\ca P$ of equation (\ref{eq:periodframed}). This follows from \cite[(2.23) and (12.9)]{DeligneMostow}. \end{proof}
\begin{proposition} \label{prop:stableperiodshyperplane} The isomorphism (\ref{eq:isomstablefivepoints}) induces an isomorphism of complex analytic spaces \begin{equation*} \ca M_0(\mathbb{C}) = G(\mathbb{C}) \setminus X_0(\mathbb{C}) \cong P\Gamma \setminus \left(\mathbb{C} H^2 - \mr H \right). \end{equation*} \end{proposition} \begin{proof}
We have $\ca P_s(\ca F_0) \subset \mathbb{C} H^2 - \mr H$ by \cite[Proposition 5.11]{degaayfortman-nonarithmetic}, because the Jacobian of a smooth curve cannot contain an abelian subvariety whose induced polarization is principal. Therefore, we have $\ca P_s^{-1}(\mr H) \subset \ca F_s - \ca F_0$. Since $\ca F_s$ is irreducible (it is smooth by Corollary \ref{cor:framedquintics} and connected by Corollary \ref{cor:surjectivemon}), the analytic space $\ca P_s^{-1}(\mr H)$ is a divisor. Since $\ca F_s - \ca F_0$ is also a divisor by Corollary \ref{cor:framedquintics}, we have $\ca P_s^{-1}(\mr H) = \ca F_s - \ca F_0$. \end{proof}
\begin{comment} \begin{remark} There is more geometric way to prove Proposition \ref{prop:stableperiodshyperplane}. Namely, let $H_{0,5}$ be the moduli space of degree $5$ covers of $\mathbb{P}^1$ ramified along five distinct marked points, see \cite[\S 2.G]{Harris1998ModuliOC}. The period map $$H_{0,5}(\mathbb{C}) \to P\Gamma \setminus \mathbb{C} H^2,$$ that sends the moduli point of a curve $C \to \mathbb{P}^1$ to the moduli point of the $\mathbb{Z}[\zeta]$-linear Jacobian $J(C)$, extends to the stable compactification $\overline{H}_{0,5}(\mathbb{C}) \supset H_{0,5}(\mathbb{C})$ because the curves in the limit are of compact type. Since the divisor $\mr H \subset \mathbb{C} H^2$ parametrizes principally polarized abelian varieties that are products of lower dimensional ones by \cite[Proposition 4.10]{degaayfortman-nonarithmetic}, the image of the boundary is exactly $P\Gamma \setminus \mr H$. \end{remark} \end{comment}
\section{Moduli of real binary quintics} \label{modrealbinquin}
With the period map for complex binary quintics in place, we turn to the construction of the period map for real binary quintics.
\subsection{The period map for smooth real binary quintics} Define $\kappa$ as the anti-holomorphic involution \[ \kappa \colon X_0(\mathbb{C}) \to X_0(\mathbb{C}), \quad F(X_0,X_1) = \sum_{i+j= 5}a_{ij}\cdot X_0^iX_1^j \mapsto \overline{F(X_0,X_1)}= \sum_{i+j =5}\overline{a_{ij}} \cdot X_0^iX_1^j. \] Let $\mr A$ be the set of anti-unitary involutions $\alpha \colon \Lambda \to \Lambda$. Define $P\mr A = \mu_K \setminus \mr A$ and $C \mr A = P\Gamma \setminus P\mr A$, c.f.~\cite[Section 2.1]{degaayfortman-nonarithmetic}. For each $\alpha \in P\mr A$, there is a natural anti-holomorphic involution $\alpha \colon \ca F_0 \to \ca F_0$ lying over the anti-holomorphic involution $\kappa \colon X_0(\mathbb{C}) \to X_0(\mathbb{C})$.
To define $\alpha$, consider a framed binary quintic $(F, [f]) \in \ca F_0$, where $f: \mathbb{V}_F \to \Lambda$ is an $\mathcal{O}_K$-linear isometry. Let $C_F \to \mathbb{P}^1_\mathbb{C}$ be the induced quintic cover of $\mathbb{P}^1_\mathbb{C}$. Complex conjugation $\mathbb{P}^2(\mathbb{C}) \to \mathbb{P}^2(\mathbb{C})$ induces an anti-holomorphic map $ \sigma_F \colon C_F(\mathbb{C}) \to C_{\kappa(F)}(\mathbb{C})$ with pull-back $\sigma_F^\ast: \bb V_{\kappa (F)} \to \bb V_F$. The composition $\alpha \circ f \circ \sigma_F^{\ast} \colon \bb V_{\kappa(F)} \to \Lambda$
induces a framing of $\kappa(F) \in X_0(\mathbb{C})$, and we define \[ \alpha(F, [f]) \coloneqq \left(\kappa(F), [\alpha \circ f \circ \sigma_F^\ast] \right) \in \ca F_0. \] Although we have chosen a representative $\alpha \in \mr A$ of the class $\alpha \in P\mr A$, the element $\alpha(F, [f]) \in \ca F_0$ does not depend on this choice.
Consider the covering map $\ca F_0 \to X_0(\mathbb{C})$ introduced in (\ref{modrealbinquin}), and define \begin{align}\label{realpointsofFzero} \ca F_0(\mathbb{R}) = \bigsqcup_{\alpha \in P\mr A} \ca F_0^\alpha \subset \ca F_0 \end{align} as the preimage of $X_0(\mathbb{R})$ in the space $\ca F_0$. To see why the union on the left hand side of (\ref{realpointsofFzero}) is disjoint, observe that $
\ca F_0^\alpha = \left\{ (F, [f]) \in \ca F_0 : \kappa(F) = F \textnormal{ and } [f \circ \sigma^\ast_{F} \circ f^{-1}] = \alpha \right\}$. Thus, for $\alpha, \beta \in P\mr A$ and $(F, [f]) \in \ca F_0^\alpha \cap \ca F_0^\beta$, we have $\alpha = [f \circ \sigma \circ f^{-1}] = \beta$.
\begin{lemma} The anti-holomorphic involution $\alpha \colon \ca F_0 \to \ca F_0$ defined by $\alpha \in P \mr A$ makes the period map $\ca P \colon \ca F_0 \to \mathbb{C} H^2$ equivariant for the $G(\mathbb{C})$-actions on both sides. \end{lemma} \begin{proof}
If $\tn{conj} \colon \mathbb{C} \to \mathbb{C}$ is complex conjugation, then for any $F \in X_0(\mathbb{C})$, the induced map $\sigma^\ast_F \otimes \tn{conj} \colon \bb V_{\kappa (F)}\otimes_\mathbb{Z} \mathbb{C} \to \bb V_F \otimes_\mathbb{Z} \mathbb{C}$ is anti-linear, preserves the Hodge decompositions \cite[Chapter I, Lemma 2.4]{silholsurfaces} as well as the eigenspace decompositions.
\end{proof}
\noindent We obtain a \textit{real period map}
\begin{align}\label{therealperiodmap} \xymatrix{ \ca P^\mathbb{R} \colon \ca F_0(\mathbb{R}) \ar@{=}[r] &\coprod_{\alpha \in P\mr A} \ca F_0^\alpha \ar[r] &\coprod_{\alpha \in P\mr A} \mathbb{R} H^2_\alpha \ar@{=}[r]& \widetilde Y. } \end{align}
Define $$G(\mathbb{R}) = \textnormal{GL}_2(\mathbb{R}).$$ The map (\ref{therealperiodmap}) is constant on $G(\mathbb{R})$-orbits, as the same is true for $\ca P \colon \ca F_0 \to \mathbb{C} H^2$. By abuse of notation, we write $\bb R H^2_\alpha - \mr H = \bb R H^2_\alpha - \left(\mr H \cap \bb R H^2_\alpha\right)$ for $\alpha \in P\mr A$.
\begin{proposition} \label{prop:realsmoothperiods}
The period map (\ref{therealperiodmap}) descends to a $P\Gamma$-equivariant diffeomorphism \begin{align} \label{firstperiod} \ca P^\mathbb{R} \colon \ca M_0(\mathbb{R})^f \coloneqq G(\mathbb{R}) \setminus \ca F_0(\mathbb{R}) \cong \coprod_{\alpha \in P\mr A} \left(\mathbb{R} H^2_\alpha - \mr H\right).
\end{align} By $P\Gamma$-equivariance, the map (\ref{firstperiod}) induces an isomorphism of real-analytic orbifolds \begin{equation} \label{smoothrealperiodiso} \ca P^\mathbb{R} \colon \ca M_0(\mathbb{R}) = G(\mathbb{R}) \setminus X_0(\mathbb{R}) \cong \coprod_{\alpha \in C \mr A}P\Gamma_\alpha \setminus \left( \mathbb{R} H^2_\alpha - \mr H
\right). \end{equation} \end{proposition}
\begin{proof} This follows from Theorem \ref{th:delignemostow}, Proposition \ref{prop:stableperiodshyperplane} and the equivariance of the period map \eqref{eq:periodframed}. See \cite[\textit{Proof of Theorem 3.3}]{realACTsurfaces}, where the analogous statement for cubic surfaces is proved. The arguments there can easily be adapted to our situation.
\end{proof}
\subsection{Nodal quintics and orthogonal hyperplanes}
Consider the CM type $\Phi = \set{\tau, \varphi} \subset \textnormal{Hom}(K,\mathbb{C})$ defined in \eqref{CMTYPE-Shimura-modification}, the hermitian $\mathcal{O}_K$-lattice $(\Lambda, \mf h)$ defined in (\ref{eq:hermitianformonbinaryquinticlattice}), and the following sets (c.f.~\cite[Section 2.1]{degaayfortman-nonarithmetic}): $$\ca H = \set{H_r \subset \bb C H^2 \mid r \in \mr R}, \quad \tn{ and } \quad \mr H = \bigcup_{H\in \ca H}H \subset \bb C H^2.$$ Here, $\mr R\subset \Lambda$ is the set of short roots, i.e.~the set of $r \in \Lambda$ with $\mf h(r,r) = 1$, and for each $r \in \mr R$, $H_r \subset \mathbb{C} H^2$ is the hyperplane of elements $x \in \mathbb{C} H^2$ that are orthogonal to $r$.
\begin{lemma} The hyperplane arrangement $\mr H \subset \bb C H^2$ satisfies Condition 2.2 in \cite{degaayfortman-nonarithmetic}. In other words: any two distinct $H_1, H_2\in\ca H$ either meet orthogonally, or not at all. \end{lemma} \begin{proof} By \cite[Proposition 6.4]{degaayfortman-nonarithmetic}, we have that \cite[Condition 6.1]{degaayfortman-nonarithmetic} is satisfied. Therefore, the result follows from \cite[Theorem 6.2]{degaayfortman-nonarithmetic}.
\end{proof}
\begin{definition} \begin{enumerate}[wide, labelwidth=!, labelindent=0pt] \item For $k = 1, 2$, define $\Delta_k \subset \Delta = X_s(\mathbb{C}) - X_0(\mathbb{C})$ to be the locus of stable binary quintics with exactly $k$ nodes. Define $\widetilde \Delta = \ca F_s - \ca F_0$, and let $\widetilde \Delta_k \subset \widetilde \Delta$ be the inverse image of $\Delta_k$ in $\widetilde \Delta$ under the map $\widetilde \Delta \to \Delta$. \item For $k = 1,2$, define $\mr H_k \subset \mr H$ as the set $\mr H_k = \set{x \in \bb C H^2 \colon \va{\ca H(x)} = k}$. Thus, this is the locus of points in $\mr H$ where exactly $k$ hyperplanes meet. \end{enumerate} \end{definition} \begin{lemma} \label{lemma:stabilizergroups} \begin{enumerate}[wide, labelwidth=!, labelindent=0pt] \item The period map $\ca P_s$ of (\ref{eq:stableperiodmapframed}) satisfies $\ca P_s(\widetilde \Delta_k) \subset \mr H_k$.
\item Let $f \in \widetilde \Delta_k$ and $x = \ca P_s(f) \in \mr H_k \subset \bb C H^2$.
Then $\ca P_s$ induces a group isomorphism $P\Gamma_f \cong G(x)$, where $G(x) \cong (\mathbb{Z}/10)^k$ is as in \cite[Definition 3.1]{degaayfortman-nonarithmetic}. \end{enumerate} \end{lemma}
\begin{proof}
1. We know that $\ca P_s$ induces an isomorphism $G(\mathbb{C}) \setminus \widetilde \Delta \xrightarrow{\sim} \mr H$ by Theorem \ref{th:delignemostow} and Proposition \ref{prop:stableperiodshyperplane}. This map must identify the smooth (resp.~singular) locus of both varieties with each other, from which the result follows.
2. This follows from Lemma \ref{lemma:monodromy} and Corollary \ref{cor:framedquintics} (c.f.~\cite[Lemma 10.3]{realACTsurfaces}).
\end{proof}
\subsection{The period map for stable real binary quintics} Our next goal is to prove the real analogue of the isomorphisms (\ref{eq:isomstablefivepoints-zero}) and (\ref{eq:isomstablefivepoints})
in Theorem \ref{th:delignemostow}. The naturality of the Fox completion implies that for $\alpha \in P\mr A$, the anti-holomorphic involution $\alpha \colon \ca F_0 \to \ca F_0$ extends to an anti-holomorphic involution $\alpha\colon \ca F_s \to \ca F_s$.
\begin{lemma} \label{lemma:alphaperiod}
For every $\alpha \in P\mr A$, the restriction of $\ca P_s \colon \ca F_s \to \mathbb{C} H^2$ to $\ca F_s^\alpha$ induces a diffeomorphism $\ca P_s^{\alpha} \colon G(\mathbb{R}) \setminus \ca F_s^\alpha \cong \mathbb{R} H^2_\alpha$. \end{lemma} \begin{proof} To prove this, one uses arguments similar to those used in the proof of Lemma 11.3 in \cite{realACTsurfaces}. The idea is as follows. The map $\ca P_s^{\alpha} \colon G(\mathbb{R}) \setminus \ca F_s^\alpha \to \mathbb{R} H^2_\alpha$ is a local diffeomorphism because its differential is everywhere an isomorphism by Theorem \ref{th:delignemostow}. Moreover, using \cite[Lemma 3.5]{realACTsurfaces}, one shows that $\ca P_s^{\alpha}$ is injective because $G(\mathbb{C})$ acts freely on $\ca F_s$ by Corollary \ref{cor:freeaction}. To prove the surjectivity of $\ca P_s^{\alpha}$, one uses \cite[Lemma 11.2]{realACTsurfaces} to see that the map $G(\mathbb{R}) \setminus \ca F_s^{\alpha} \to G \setminus \mathbb{C} H^2$ is proper. By Proposition \ref{prop:realsmoothperiods}, its image contains the dense open subset $\ca P_s(\ca F_0^\alpha) = \mathbb{R} H^2_\alpha - \mr H$, so the map $\ca P_s^\alpha$ is surjective.
\end{proof} \noindent We arrive at the main theorem of Section \ref{modrealbinquin}. Consider the map $\pi: \ca F_s \to X_s(\mathbb{C})$ (see Corollary \ref{cor:framedquintics}) and define a union of embedded real submanifolds of $\ca F_s$ as follows: \[ \ca F_s(\mathbb{R}) = \bigcup_{\alpha \in P\mr A} \ca F_s^\alpha = \pi^{-1}\left(X_s(\mathbb{R})\right). \]
\begin{theorem} \label{th:realstableperiod} There is a smooth map \begin{align}\label{therealstableperiodmap} \ca P_s^{\mathbb{R}} \colon \coprod_{\alpha \in P\mr A} \ca F_s^\alpha \longrightarrow \coprod_{\alpha \in P\mr A}\mathbb{R} H^2_\alpha = \widetilde Y \end{align} that extends the real period map (\ref{therealperiodmap}). The map (\ref{therealstableperiodmap}) induces the following commutative diagram of topological spaces, in which $\mr P_s^\mathbb{R}$ and $\mr T^\mathbb{R}_s$ are homeomorphisms: \begin{equation*}
\xymatrixcolsep{5pc} \xymatrix{ &\coprod_{\alpha \in P\mr A} \ca F_s^\alpha \ar[r]^{\ca P_s^{\mathbb{R}}}\ar[d] & \widetilde Y = \coprod_{\alpha \in P\mr A} \mathbb{R} H^2_\alpha \ar[d]\\ &\ca F_s(\mathbb{R}) \ar[r]^{\ca P_s^\mathbb{R}}\ar[d] & Y\ar@{=}[d] \\ \ca M_s(\mathbb{R})^f \ar@{=}[r]\ar[d] &G(\mathbb{R}) \setminus \ca F_s(\mathbb{R}) \ar[r]^{\mr P_s^\mathbb{R}}_\sim\ar[d] & Y\ar[d] \\ \ca M_s(\mathbb{R}) \ar@{=}[r] &G(\mathbb{R}) \setminus X_s(\mathbb{R}) \ar[r]^{{\mr T_s}^\mathbb{R}}_\sim & P\Gamma \setminus Y. } \end{equation*}
\end{theorem}
\begin{proof} The existence of $\ca P_s^{\mathbb{R}}$ follows from the compatibility between $\ca P_s$ and the involutions $\alpha \in P\mr A$. We first show that the composition $p \circ \ca P_s^{\mathbb{R}}$ factors through $\ca F_s(\mathbb{R})$. Two elements $f_\alpha$ and $g_\beta\in \coprod_{\alpha \in P\mr A} \ca F_s^\alpha$ have the same image in $\ca F_s(\mathbb{R})$ if and only if $f = g \in \ca F_s^\alpha \cap \ca F_s^\beta$, in which case $ x\coloneqq \ca P_s(f) = \ca P_s(g) \in \mathbb{R} H^2_\alpha \cap \bb R H^2_\beta$. We need to show that $x_\alpha \sim x_\beta \in \widetilde Y$, for the equivalence relation $\sim$ on $\widetilde Y$ defined in \cite[Definition 3.9]{degaayfortman-nonarithmetic}. Note that $\alpha\beta \in P\Gamma_f \cong (\mathbb{Z}/10)^k$, and $\ca P_s$ induces an isomorphism $ P\Gamma_f \cong G(x) $ by Lemma \ref{lemma:stabilizergroups}. Hence $\alpha \beta \in G(x)$ so that $x_\alpha \sim x_\beta$.
Let us prove the $G(\mathbb{R})$-equivariance of $\ca P_s^{\mathbb{R}}$. Suppose that $ f \in \ca F_s^\alpha, g \in \ca F_s^\beta$ such that $ a \cdot f = g \in \ca F_s(\mathbb{R})$ for some $ a \in G(\mathbb{R})$. Then $x\coloneqq \ca P_s(f) = \ca P_s(g) \in \mathbb{C} H^2$, so we need to show that $\alpha \beta \in G(x)$. The actions of $G(\mathbb{C})$ and $P\Gamma$ on $\mathbb{C} H^2$ commute, and the same holds for the actions of $G(\mathbb{R})$ and $P\Gamma'$ on $\ca F_s^\mathbb{R}$, where $P\Gamma'$ is as in \cite[Section 2.3]{degaayfortman-nonarithmetic}.
It follows that $ \alpha(g) = \alpha (a \cdot f) = a \cdot \alpha(f) = a \cdot f = g, $ hence $g \in \ca F_s^\alpha \cap \ca F_s^\beta$. This implies in turn that $\alpha \beta (g) = g$, hence $\alpha \beta \in P\Gamma_g \cong G(x)$, so that indeed, $x_\alpha \sim x_\beta$.
To prove that $\mr P_s^\mathbb{R}$ is injective, let again $ f_\alpha, g_\beta\in \coprod_{\alpha \in P\mr A} \ca F_s^\alpha $ and suppose that these elements have the same image in $Y$. This implies that $ x\coloneqq \ca P_s(f) = \ca P_s(g) \in \mathbb{R} H^2_\alpha \cap \bb R H^2_\beta, $ and that $\beta = \phi \circ \alpha$ for some $\phi \in G(x)$. We have $\phi \in G(x) \cong P\Gamma_f$ (Lemma \ref{lemma:stabilizergroups}) hence $ \beta(f) = \phi \left(\alpha (f)\right) = \phi(f) = f$.
Therefore $f,g \in \ca F_s^\beta$; since $\ca P_s(f) = \ca P_s(g)$, it follows from Lemma \ref{lemma:alphaperiod} that there exists $a \in G(\mathbb{R})$ such that $a \cdot f = g$. This proves injectivity of $\mr P_s^\mathbb{R}$.
The surjectivity of $\mr P_s^\mathbb{R}: G(\mathbb{R}) \setminus \ca F_s(\mathbb{R}) \to Y$ is straightforward, using the surjectivity of $\ca P_s^{\mathbb{R}}$, which follows from Lemma \ref{lemma:alphaperiod}.
Finally, we claim that $\mr P_s^\mathbb{R}$ is open. Let $U \subset G(\mathbb{R}) \setminus \ca F_s^\mathbb{R}$ be open. Let $V$ be the preimage of $U$ in $\coprod_{\alpha \in P\mr A}\ca F_s^\alpha$. Then $ V = (\ca P_s^{\mathbb{R}})^{-1}\left( p^{-1}\left(\mr P_s^\mathbb{R}(U)\right)\right) $, and hence $
\ca P_s^{\mathbb{R}}\left( V \right) = p^{-1}\left(\mr P_s^\mathbb{R}(U)\right). $ The map $\ca P_s^{\mathbb{R}}$ is open, being the coproduct of the maps $\ca F_s^\alpha \to \mathbb{R} H^2_\alpha$, which are open since they have surjective differential at each point. Thus $\mr P_s^\mathbb{R}(U)$ is open. \end{proof}
\begin{corollary} \label{cor:theorem2}
There is a lattice $P\Gamma_\mathbb{R} \subset \textnormal{PO}(2,1)$, an inclusion of orbifolds
\begin{align}\label{inclusionofforbifodls}
\coprod_{\alpha \in C \mr A}P\Gamma_\alpha \setminus \left( \mathbb{R} H^2_\alpha - \mr H
\right) \hookrightarrow P\Gamma_\mathbb{R} \setminus \mathbb{R} H^2,
\end{align}
and a homeomorphism \begin{equation} \label{stablehom} \ca P_s^{\mathbb{R}} \colon \ca M_s(\mathbb{R}) = G(\mathbb{R}) \setminus X_s(\mathbb{R}) \cong P\Gamma_\mathbb{R} \setminus \mathbb{R} H^2 \end{equation} restricting to the isomorphism of real analytic orbifolds (\ref{smoothrealperiodiso}) with respect to (\ref{inclusionofforbifodls}).
\end{corollary} \begin{proof} This follows directly from \cite[Theorem 4.1]{degaayfortman-nonarithmetic} and Theorem \ref{th:realstableperiod} above.
\end{proof}
\begin{remark} The proof of Theorem \ref{th:realstableperiod} also shows that $\ca M_s(\mathbb{R})$ is homeomorphic to a glued space $P\Gamma \setminus Y$ (see \cite[Definition 3.13]{degaayfortman-nonarithmetic}) if $\ca M_s$ is the stack of cubic surfaces or of binary sextics. This strategy to uniformize the real moduli space differs from the one used in \cite{realACTnonarithmetic, realACTbinarysextics,realACTsurfaces}, since we first glue together the real ball quotients together, and then prove that our real moduli space is homeomorphic to the result. \end{remark}
\section{The space of stable real binary quintics as hyperbolic triangle} \label{space:hyperbolictriangle}
The goal of Section \ref{space:hyperbolictriangle} is to finish the proof of Theorem \ref{th:theorem02}. To do so, we need to understand the orbifold structure of $\ca M_s(\mathbb{R})$, and how this orbifold structure differs from the orbifold structure of the glued space $P\Gamma \setminus Y$. We start with analyzing the orbifold structure of $\ca M_s(\mathbb{R})$, by listing its stabilizer groups.
\subsection{Automorphism groups of stable real binary quintics} \label{sec-sec:1}
The canonical orbifold isomorphism $ \ca M_s(\mathbb{R}) = G(\mathbb{R}) \setminus X_s(\mathbb{R}) = (P_s/\mf S_5)(\mathbb{R}) $ implies that to list those groups that occur as the automorphism group of a binary quintic is to list the stabilizer groups $\textnormal{PGL}_2(\mathbb{R})_x$ of points $x = [\alpha_1, \dotsc, \alpha_5] \in (P_s/\mf S_5)(\mathbb{R})$.
\begin{lemma} \label{lemma:wu2019moduli} Consider the stabilizer group $\textnormal{PGL}_2(\mathbb{C})_x$ of a point $x \in P_0/\mf S_5$. If the group $\textnormal{PGL}_2(\mathbb{C})_x$ is non-trivial, then $\textnormal{PGL}_2(\mathbb{C})_x$ is isomorphic to either $\mathbb{Z}/2, D_3, \mathbb{Z}/4$ or $D_5$. \end{lemma} \begin{proof} See \cite[Theorem 22]{wu2019moduli}. \end{proof} \noindent The goal of Section \ref{sec-sec:1} is to prove the following proposition.
\begin{proposition} \label{prop:allstabilizergroups} All stabilizer groups $\textnormal{PGL}_2(\mathbb{R})_x \subset \textnormal{PGL}_2(\mathbb{R})$ for $x \in (P_s/ \mf S_5)(\mathbb{R})$ are among the groups $\mathbb{Z}/2, D_3$ and $ D_5$. For each $n \in \{3,5\}$, there is a unique $\textnormal{PGL}_2(\mathbb{R})$-orbit of points $x$ in $(P_s/ \mf S_5)(\mathbb{R})$ with stabilizer $D_n$. \end{proposition} \begin{proof}
We have an injection $ (P_s/\mf S_5)(\mathbb{R}) \hookrightarrow P_s/\mf S_5$ which is equivariant for the embedding $\textnormal{PGL}_2(\mathbb{R}) \hookrightarrow \textnormal{PGL}_2(\mathbb{C})$. In particular, $\textnormal{PGL}_2(\mathbb{R})_x \subset \textnormal{PGL}_2(\mathbb{C})_x$ for every $x \in (P_s/\mf S_5)(\mathbb{R})$. Note that none of the groups appearing in Lemma \ref{lemma:wu2019moduli} have subgroups isomorphic to $D_2 = \mathbb{Z}/2 \rtimes \mathbb{Z}/2$ or $D_4 = \mathbb{Z}/2 \rtimes \mathbb{Z}/4$. Define an involution \begin{equation*}
\nu \coloneqq (z \mapsto 1/z) \in \textnormal{PGL}_2(\mathbb{R}). \end{equation*} \noindent Proposition \ref{prop:allstabilizergroups} will follow from the following four lemmas.
\end{proof}
\begin{lemma} \label{lemma:involution} Let $\tau \in \textnormal{PGL}_2(\mathbb{R})$. Consider a subset $S = \{x,y,z\} \subset \mathbb{P}^1(\mathbb{C})$ stabilized by complex conjugation, such that $\tau(x) = x$, $\tau(y) = z$ and $\tau(z) = y$. There is a transformation $g \in \textnormal{PGL}_2(\mathbb{R})$ that maps $S$ to either $\{-1, 0, \infty\}$ or $\{-1, i, -i\}$, and that satisfies $g \tau g^{-1} = \nu = (z \mapsto 1/z) \in \textnormal{PGL}_2(\mathbb{R})$. In particular, $\tau^2 = \textnormal{id}$.
\end{lemma} \begin{proof} Indeed, two transformations $g,h \in \textnormal{PGL}_2(\mathbb{C})$ that satisfy $g(x_i) = h(x_i)$ for three different points $x_1, x_2, x_3 \in \mathbb{P}^1(\mathbb{C})$ are necessarily equal.
\end{proof}
\begin{lemma} \label{lem-two} There is no $\phi \in \textnormal{PGL}_2(\mathbb{R})$ of order four that fixes a point $x \in (P_s/\mf S_5)(\mathbb{R})$.
\end{lemma} \begin{proof} By \cite[Theorem 4.2]{beauvillePGL2}, all subgroups $G \subset \textnormal{PGL}_2(\mathbb{R})$ that are isomorphic to $\mathbb{Z}/4$ are conjugate to each other. Since the transformation $I: z \mapsto (z-1)/(z+1)$ is of order four, it gives a representative $G_I = \langle I \rangle$ of this conjugacy class. Assume that there exists $x$ and $\phi$ as in the lemma. To get a contradiction, we may assume that $\phi = I$. It is easily shown that $I$ cannot fix any point $x \in (P_s/\mf S_5)(\mathbb{R})$.
\end{proof}
\begin{lemma} \label{lemma:D3} Define $\rho \in \textnormal{PGL}_2(\mathbb{R})$ by $\rho(z) = \frac{-1}{z+1}$. Let $x = (x_1, \dotsc, x_5) \in (P_s/\mf S_5)(\mathbb{R})$. Let $\phi \in \textnormal{PGL}_2(\mathbb{R})$ of order three, with $\phi(x) = x$. There is a transformation $g \in \textnormal{PGL}_2(\mathbb{R})$ mapping $x$ to $(-1, \infty, 0, \omega, \omega^2)$ with $\omega$ a primitive third root of unity. The stabilizer of $x$ to the subgroup of $\textnormal{PGL}_2(\mathbb{R})$ generated by $\rho$ and $\nu$. In particular, we have $\textnormal{PGL}_2(\mathbb{R})_x \cong D_3$.
\end{lemma}
\begin{proof} By Lemma \ref{lemma:involution}, there are elements $x_1, x_2, x_3$ which form an orbit under $\phi$. Since complex conjugation preserves this orbit, one element in it is real; since $g$ is defined over $\mathbb{R}$, they are all real. Let $g \in \textnormal{PGL}_2(\mathbb{R})$ such that $g(x_1) = -1$, $g(x_2) = \infty$ and $g(x_3) = 0$. Define $\kappa = g \phi g^{-1}$. Then $\kappa^3 = \textnormal{id}$, and $\kappa$ preserves $\{-1, \infty, 0\}$ and sends $-1$ to $\infty$ and $\infty$ to $0$. Consequently, $\kappa(0) = -1$, and it follows that $\kappa = \rho$. Hence $x$ is equivalent to an element of the form $(-1, \infty, 0, \alpha, \beta)$ with $\beta = \bar \alpha$ and $\alpha^2 + \alpha + 1 = 0$.
\end{proof} \noindent Recall that $ \lambda = \zeta_5 + \zeta_5^{-1} \in \mathbb{R}$. Define $\gamma \in \textnormal{PGL}_2(\mathbb{R})$ with $ \gamma(z) = \frac{(\lambda + 1)z - 1}{z + 1}$ for $ z \in \mathbb{P}^1(\mathbb{C})$.
\begin{lemma}\label{lemma:D5} Let $x = (x_1, \dotsc, x_5) \in (P_s/\mf S_5)(\mathbb{R})$. Suppose $x$ is stabilized by a subgroup of $\textnormal{PGL}_2(\mathbb{R})$ of order five. There is a transformation $g \in \textnormal{PGL}_2(\mathbb{R})$ mapping $x$ to $z = (0, -1, \infty, \lambda+1, \lambda)$ and identifying the stabilizer of $x$ with the subgroup of $\textnormal{PGL}_2(\mathbb{R})$ generated by $\gamma$ and $\nu$. In particular, the stabilizer $\textnormal{PGL}_2(\mathbb{R})_x$ of $x$ is isomorphic to $D_5$.
\end{lemma}
\begin{proof} Let $\phi\in \textnormal{PGL}_2(\mathbb{R})_x$ be an element of order five. Using Lemma \ref{lemma:involution} one shows that the $x_i$ are pairwise distinct, and we may assume that $x_i = \phi^{i-1}(x_1)$ for $i = 2, \dotsc, 5$. Since there is one real $x_i$ and $\phi$ is defined over $\mathbb{R}$, all $x_i$ are real. Note that $z = \set{0, -1, \infty, \lambda + 1, \lambda}$ is the orbit of $0$ under $\gamma: z \mapsto ((\lambda+1)z - 1)/(z+1)$. The reflection $\nu: z \mapsto 1/z$ preserves $z$ as well: we have $\lambda + 1 = - (\zeta_5^2 + \zeta_5^{-2}) = - \lambda^2 + 2$, so that $\lambda(\lambda + 1) = 1$. So we have $\textnormal{PGL}_2(\mathbb{R})_z \cong D_5$. By Lemma \ref{lemma:wu2019moduli},
the point $z$ with stabilizer $\textnormal{PGL}_2(\mathbb{R})_z$ is equivalent under $\textnormal{PGL}_2(\mathbb{C})$ to the point $(1, \zeta, \zeta^2, \zeta^3, \zeta^4)$ with stabilizer $\langle x \mapsto \zeta x, x \mapsto 1/x \rangle$. Thus, there exists $g \in \textnormal{PGL}_2(\mathbb{C})$ such that $g(x_1) = 0$, $g(x_2) = -1$, $g(x_3) = \infty$, $g(x_4) = \lambda + 1$ and $g(x_5) = \lambda$, and such that $g\textnormal{PGL}_2(\mathbb{R})_xg^{-1} = \textnormal{PGL}_2(\mathbb{R})_z$. Since all $x_i$ and $z_i \in z$ are real, we see that $\bar g(x_i) = z_i$ for every $i$, hence $g$ and $\bar g$ coincide on more than two points, which implies that $g = \bar g \in \textnormal{PGL}_2(\mathbb{R})$.
\end{proof}
\begin{comment} \begin{enumerate}[wide, labelwidth=!, labelindent=0pt] \item $ x = (-1, 0, \infty, \beta, \beta^{-1}) \textnormal{ with } \beta \in \bb A^1(\mathbb{R})\setminus \{-1,0,1\}$ or $$ \beta \in \bb S^1\setminus \{-1,1\} \subset \bb A^1(\mathbb{C}), \quad \textnormal{ and } \quad \textnormal{PGL}_2(\mathbb{R})_x = \langle \nu \rangle.$$
\item $ x = (-1, i, -i, \beta, \beta^{-1}) \textnormal{ with } \beta \in \bb A^1(\mathbb{R})\setminus \{-1,1\}$ or $$ \beta \in \bb S^1\setminus\{-1,1,i,-i\} \subset \bb A^1(\mathbb{C}),\quad \textnormal{ and } \quad \textnormal{PGL}_2(\mathbb{R})_x = \langle \nu \rangle.$$
\item $ x = (-1, -1, \beta, 0, \infty) \textnormal{ with } \beta \in \bb A^1(\mathbb{R})\setminus \{-1,0,1\}, $ and $\textnormal{PGL}_2(\mathbb{R})_x = \langle \nu \rangle. $
\item $ x = (-1, -1, \beta, i, -i) \textnormal{ with } \beta \in \bb A^1(\mathbb{R})\setminus \{-1,1\}, $ and $\textnormal{PGL}_2(\mathbb{R})_x = \langle \nu \rangle.
$
\item $ x = (0,0, \infty, \infty, -1), $ and $\textnormal{PGL}_2(\mathbb{R})_x = \langle \nu \rangle$.
\item $ x = (-1, i, i, - i, -i), $ and $\textnormal{PGL}_2(\mathbb{R})_x = \langle \nu \rangle. $ \end{enumerate} \end{comment}
\subsection{Comparing the orbifold structures} \label{sec-sec:3}
Consider the moduli space $\ca M_s(\mathbb{R})$ of real stable binary quintics.
\begin{definition} \label{neworbifold} Let $\overline{\mr M}_\mathbb{R}$ be the hyperbolic orbifold with $\ca M_s(\mathbb{R})$ as underlying space, whose orbifold structure is induced by the homeomorphism (\ref{stablehom}) in Corollary \ref{cor:theorem2} and the natural orbifold structure of $P\Gamma_\mathbb{R} \setminus \bb R H^2$. \end{definition} \noindent There are two orbifold structures on the underlying space $\va{\ca M_s(\mathbb{R})}$ of $\ca M_s(\mathbb{R})$: the natural orbifold structure of $\ca M_s(\mathbb{R})$ (see \cite[Proposition 2.12]{thesis-degaayfortman}, this is the orbifold structure of the quotient $ G(\mathbb{R}) \setminus X_s(\mathbb{R})$), and the orbifold structure $\overline{\mr M}_\mathbb{R}$ introduced in Definition \ref{neworbifold}. The goal of Section \ref{sec-sec:3} is to calculate the difference between these orbifold structures.
Let us first show that there are no cone points in the orbifold $\textnormal{PGL}_2(\mathbb{R}) \setminus (P_s/ \mf S_5)(\mathbb{R})$. Recall that these are orbifold points whose stabilizer group is $\mathbb{Z}/n$ for some $n \geq 2$ acting on the orbifold chart by rotations. By Proposition \ref{prop:allstabilizergroups}, this fact follows from the following: \begin{lemma} \label{lemma:zmod2stabilizer} Let $x=(x_1, \dotsc, x_5) \in (P_s/\mf S_5)(\mathbb{R})$ such that $\textnormal{PGL}_2(\mathbb{R})_x = \langle \tau \rangle $ has order two. There is a $\textnormal{PGL}_2(\mathbb{R})_x$-stable open neighborhood $U \subset (P_s/\mf S_5)(\mathbb{R})$ of $x$ such that $\textnormal{PGL}_2(\mathbb{R})_x \setminus U \to \ca M_s(\mathbb{R})$ is injective, and a homeomorphism $\phi: (U,x) \to (B,0)$ for $0 \in B \subset \mathbb{R}^2$ an open ball, such that $\phi$ identifies $\textnormal{PGL}_2(\mathbb{R})_x$ with $\mathbb{Z}/2$ acting on $B$ by reflections in a line through $0$. \end{lemma} \begin{proof} Using Lemma \ref{lemma:involution}, one checks that the only possibilities for the element $x=(x_1, \dotsc, x_5) \in (P_s/\mf S_5)(\mathbb{R})$ are $(-1, 0, \infty, \beta, \beta^{-1})$, $(-1, i, -i, \beta, \beta^{-1}) $, $ (-1, -1, \beta, 0, \infty)$,\\ $ (-1, -1, \beta, i, -i) $, $(0,0, \infty, \infty, -1)$ and $(-1, i, i, - i, -i)$. \end{proof}
\begin{proposition} \label{prop:conesreflectors} \begin{enumerate}[wide, labelwidth=!, labelindent=0pt] \item The orbifold structures of $\ca M_s(\mathbb{R})$ and $\overline{\mr M}_\mathbb{R}$ differ only at the moduli point $x_0 \in \ca M_s(\mathbb{R})$ attached to the five-tuple $(\infty, i, i, -i, -i)$. \item The stabilizer group of $\ca M_s(\mathbb{R})$ at the point $x_0$ is isomorphic to $\mathbb{Z}/2$, whereas the stabilizer group of $\overline{\mr M}_\mathbb{R}$ at $x_0$ is isomorphic to the dihedral group $D_{10}$ of order twenty. \item The orbifold $\overline{\mr M}_\mathbb{R}$ has no cone points and three corner reflectors, whose orders are $\pi/3, \pi/5$ and $\pi/10$. \end{enumerate} \end{proposition}
\begin{proof}
The statements can be deduced from Lemma \ref{lemma:zmod2stabilizer} and \cite[Proposition 4.13]{degaayfortman-nonarithmetic}. The notation of that proposition is as follows: for $f \in Y \cong G(\mathbb{R}) \setminus \ca F_s(\mathbb{R})$ (see Theorem \ref{th:realstableperiod}) the group $A_f \subset P\Gamma$ is the stabilizer of $f \in K$. Moreover, if $\tilde f \in \ca F_s(\mathbb{R})$ represents $f$ and if $F = [\tilde f] \in X_s(\mathbb{R})$ has $k = 2a + b$ nodes, then the image $x \in \mathbb{C} H^2$ has $k = 2a + b$ nodes in the sense of \cite[Definition 3.1]{degaayfortman-nonarithmetic}.
If $F$ has no nodes ($k = 0$), then $G(x)$ is trivial by \cite[Proposition 4.13.1]{degaayfortman-nonarithmetic} and $G_F = A_f = \Gamma_f$. If $F$ has only real nodes, then $B_f = G(x)$ hence $G_F = A_f/G(x) = A_f / B_f = \Gamma_f$.
Suppose that $a = 1$ and $b = 0$: the equation $F$ defines a pair of complex conjugate nodes. In other words, the zero set of $F$ defines a $5$-tuple $\underline{\alpha}= (\alpha_1, \dotsc, \alpha_5) \in \mathbb{P}^1(\mathbb{C})$, well-defined up to the $\textnormal{PGL}_2(\mathbb{R}) \times \mf S_5$ action on $\mathbb{P}^1(\mathbb{C})$, where $\alpha_1 \in \mathbb{P}^1(\mathbb{R})$ and $\alpha_3 = \bar \alpha_2 = \alpha_5 = \bar \alpha_4 \in \mathbb{P}^1(\mathbb{C}) \setminus \mathbb{P}^1(\mathbb{R})$. So we may write $\underline{\alpha} = (\beta, \alpha, \bar \alpha, \alpha, \bar \alpha)$ with $\beta \in \mathbb{P}^1(\mathbb{R})$ and $\alpha \in \mathbb{P}^1(\mathbb{C}) \setminus \mathbb{P}^1(\mathbb{R})$. There is a unique $T \in \textnormal{PGL}_2(\mathbb{R})$ such that $T(\beta) = \infty$ and $T(\alpha) = i$. But this gives $T(x) = (\infty, i , -i, i, -i)$ hence $F$ is unique up to isomorphism.
As for the stabilizer $G_F = A_f / G(x)$, we have $G(x) \cong (\mathbb{Z}/10)^2$. There are no real nodes, so $B_f$ is trivial. By \cite[Proposition 4.13.3]{degaayfortman-nonarithmetic}, the set $K_f$ is the union of ten copies of $\bb B^2(\mathbb{R})$ meeting along a common point $\mathbb{B}^0(\mathbb{R})$. In the local coordinates $(t_1, t_2)$ around $f$, the $\alpha_j: \mathbb{B}^2(\mathbb{C}) \to \mathbb{B}^2(\mathbb{C})$ are defined as $(t_1, t_2) \mapsto (\bar t_2 \zeta^j , \bar t_1 \zeta^j)$, for $j \in \mathbb{Z}/10$, and so the fixed points sets are given by $\mathbb{R} H^2_j = \{t_2 = \bar t_1 \zeta^j\}\subset \mathbb{B}^2(\mathbb{C})$. The subgroup $E \subset G(x)$ that stabilizes $\mathbb{R} H^2_j$ is the cyclic group of order ten generated by the transformations $(t_1, t_2) \mapsto (\zeta t_1, \zeta^{-1} t_2)$.
There is only one non-trivial transformation $T \in \textnormal{PGL}_2(\mathbb{R})$ that fixes $\infty$ and sends the subset $\{i, -i\} \subset \mathbb{P}^1(\mathbb{C})$ to itself, and $T$ has order five. Hence $G_F = \mathbb{Z}/2$, so we have an exact sequence $ 0 \to \mathbb{Z}/10 \to \Gamma_f \to \mathbb{Z}/2 \to 0$, which splits since $G_F$ is a subgroup of $\Gamma_f$.
We are done, see Proposition \ref{prop:allstabilizergroups} and Lemma \ref{lemma:zmod2stabilizer}. \end{proof}
\subsection{The real moduli space as a hyperbolic triangle} \label{sec:triangle2} The goal of Section \ref{sec:triangle2} is to show that $\overline{\mr M}_{\mathbb{R}}$ (see Definition \ref{neworbifold}) is isomorphic, as hyperbolic orbifolds, to the triangle $\Delta_{3,5,10}$ in the real hyperbolic plane $\bb R H^2$ with angles $\pi/3, \pi/5$ and $\pi/10$. The results in the above Sections \ref{sec-sec:1} and \ref{sec-sec:3} give the orbifold singularities of $\overline{\mr M}_{\mathbb{R}}$ together with their stabilizer groups. In order to determine the hyperbolic orbifold structure of $\overline{\mr M}_{\mathbb{R}}$, we also need to know the underlying topological space $\ca M_s(\mathbb{R})$ of $\overline{\mr M}_{\mathbb{R}}$. The first observation is that $\ca M_s(\mathbb{R})$ is compact. Indeed, it is classical that the topological space $\ca M_s(\mathbb{C}) = G(\mathbb{C}) \setminus X_s(\mathbb{C})$, parametrizing complex stable binary quintics, is compact. This follows from the proper surjective map $\overline{M}_{0,5}(\mathbb{C})/ \mf S_5 \to \ca M_s(\mathbb{C})$ and the properness of the stack of stable five-pointed curves $\overline{M}_{0,5}$ \cite{Knudsen1983}, or from the fact that $\ca M_s(\mathbb{C})$ is homeomorphic to a compact ball quotient \cite{shimuratranscendental}. Moreover, the map $\ca M_s(\mathbb{R}) \to \ca M_s(\mathbb{C})$ is proper, which proves the compactness of $\ca M_s(\mathbb{R})$. The second observation is that $\ca M_s(\mathbb{R})$ is connected, since $X_s(\mathbb{R})$ is obtained from the euclidean space $X(\mathbb{R}) = \{F \in \mathbb{R}[X_0,X_1]: F \text{ homogeneous and} \deg(F) = 5\}$ by removing a subspace of codimension two. In the following lemma we generalize both of these observations.
\begin{lemma} \label{lem:simplyconnected} The moduli space $\ca M_s(\mathbb{R})$ of real stable binary quintics is homeomorphic to a closed disc $\overline D \subset \mathbb{R}^2$.
\end{lemma}
\begin{proof}
The idea is to show that the following holds: \begin{enumerate}[wide, labelwidth=!, labelindent=0pt] \item For each $i \in \{0,1,2\}$, the embedding $\mr M_i \hookrightarrow \overline{\mr M}_i \subset \ca M_s({\mathbb{R}})$ of the connected component $\mr M_i$ of $\ca M_0(\mathbb{R})$ into its closure in $\ca M_s({\mathbb{R}})$ is homeomorphic to the embedding $D \hookrightarrow \overline{D}$ of the open unit disc into the closed unit disc in $\mathbb{R}^2$. \item We have $\ca M_s(\mathbb{R}) = \overline{\mr M}_0 \cup \overline{\mr M}_1 \cup \overline{\mr M}_2$, and this glueing corresponds up to homeomorphism to the glueing of three closed discs $\overline{D}_i \subset \mathbb{R}^2$ as in Figure~\ref{fig:triangle}. \end{enumerate}
To prove this, one considers the moduli spaces of real smooth (resp. stable) genus zero curves with five real marked points \cite{Knudsen1983}, as well as twists of this space.
Define two anti-holomorphic involutions $\sigma_i: \mathbb{P}^1(\mathbb{C})^5 \to \mathbb{P}^1(\mathbb{C})^5$ by $ \sigma_1(x_1, x_2, x_3, x_4, x_5) = (\bar x_1, \bar x_2, \bar x_3, \bar x_5, \bar x_4), $ and $ \sigma(x_1, x_2, x_3, x_4, x_5) = (\bar x_1, \bar x_3, \bar x_2, \bar x_5, \bar x_4). $ Then define $$ P_0^1(\mathbb{R}) = P_0^{\sigma_1}, \;\;\; P_s^1(\mathbb{R}) = P_1(\mathbb{C})^{\sigma_1}, \;\;\; P_0^2(\mathbb{R}) = P_0^{\sigma_2}, \;\;\; P_s^2(\mathbb{R}) = P_1(\mathbb{C})^{\sigma_2}. $$
It is clear that $ \mr M_0 = \textnormal{PGL}_2(\mathbb{R}) \setminus P_0(\mathbb{R}) / \mf S_5$. Similarly, we have: \begin{align*}
\mr M_1 = \textnormal{PGL}_2(\mathbb{R}) \setminus P_0^1(\mathbb{R}) / \mf S_3 \times \mf S_2 \quad \tn{ and } \quad \mr M_2 = \textnormal{PGL}_2(\mathbb{R}) \setminus P_0^2(\mathbb{R}) / \mf S_2 \times \mf S_2.
\end{align*}
Moreover, we have $\overline{\mr M}_0 = \textnormal{PGL}_2(\mathbb{R}) \setminus P_s(\mathbb{R}) / \mf S_5$. We define \begin{align*} \overline{\mr M}_1 = \textnormal{PGL}_2(\mathbb{R}) \setminus P_s^1(\mathbb{R}) / \mf S_3 \times \mf S_2, \quad \tn{ and } \quad \overline{\mr M}_2 = \textnormal{PGL}_2(\mathbb{R}) \setminus P_s^2(\mathbb{R}) / \mf S_2 \times \mf S_2. \end{align*} Each $\overline{\mr M}_i$ is homeomorphic to a closed disc in $\mathbb{R}^2$. Moreover, the natural maps $\overline{\mr M}_i \to \ca M_s(\mathbb{R}) $ are closed embeddings of topological spaces, and one can check that the images glue to form $\ca M_s(\mathbb{R})$ in the prescribed way. We leave the details to the reader. \end{proof}
\begin{comment} Note that the natural maps $\overline{\mr M}_i \to \ca M_s(\mathbb{R}) $ are closed embeddings of topological spaces; we identify $\overline{\mr M}_i$ with its image in $\ca M_s(\mathbb{R})$. Moreover, we have $ \ca M_s({\mathbb{R}}) = \overline{\mr M}_0 \cup \overline{\mr M}_1 \cup \overline{\mr M}_2. $ The first step is to show that each $\overline{\mr M}_i$ is homeomorphic to the closed disc $\bar D$ in $\mathbb{R}^2$. We start with $\overline{\mr M}_0$. Let $\Gamma \subset \mathbb{P}^1(\mathbb{R}) \times \mathbb{P}^1(\mathbb{R})$ be the union of the lines $ l_1 = \mathbb{P}^1(\mathbb{R}) \times \{0\}, l_2 = \{0\} \times \mathbb{P}^1(\mathbb{R}), k_1 = \mathbb{P}^1(\mathbb{R}) \times \{1\}, k_2 = \{1\} \times \mathbb{P}^1(\mathbb{R}), $ $ m_1 = \mathbb{P}^1(\mathbb{R}) \times \{\infty\}, m_2 = \{\infty\} \times \mathbb{P}^1(\mathbb{R}), \delta = \Delta = \{(x,x): x \in \mathbb{P}^1(\mathbb{R})\}. $ Then $ \textnormal{PGL}_2(\mathbb{R}) \setminus P_0(\mathbb{R}) \cong \left(\mathbb{P}^1(\mathbb{R}) \times \mathbb{P}^1(\mathbb{R})\right) - \Gamma. $
Hence $\textnormal{PGL}_2(\mathbb{R}) \setminus P_0(\mathbb{R})$ has $12$ components $C \subset \textnormal{PGL}_2(\mathbb{R}) \setminus P_0(\mathbb{R})$. If $N = \text{Stab}_{\mf S_5}(C)$ is the stabilizer in $\mf S_5$ of one of them, then $|N| = 120/12 = 10$ hence $N = D_5$. On the other hand, $ \overline{M}_{0,5}(\mathbb{R}) = \textnormal{PGL}_2(\mathbb{R}) \setminus P_s(\mathbb{R}) $ is homeomorphic to the blow-up of $\mathbb{P}^1(\mathbb{R}) \times \mathbb{P}^1(\mathbb{R})$ in three points $0, 1, \infty$. Let $\bar C$ be the closure of $C$ in $\overline{M}_{0,5}(\mathbb{R})$. Then $\bar C$ is simply connected as well as equivariant under $N \subset \mf S_5$. Since the only orbifold point of $\ca M_s(\mathbb{R})$ with stabilizer $D_5$ is $\alpha_5\coloneqq(0, -1, \infty, \lambda + 1, \lambda)$ with $\lambda = \zeta_5 + \zeta_5^{-1} \in \mathbb{P}^1(\mathbb{R})$ by Lemma \ref{lemma:D5}, and $\alpha_5 \in \overline{M}_{0,5}(\mathbb{R}) / \mf S_5$, there is a $D_5$-equivariant homeomorphism between $\bar C$ and $D \subset \mathbb{R}^2$, mapping $\alpha_5$ to $0 \in D$, with $D_5$ acting on $(D,0)$ in the natural way. Hence $$ \overline{\mr M}_0 = \overline{M}_{0,5}(\mathbb{R})/ \mf S_5 = \bar C / N = D/D_5 $$ is simply connected, and homeomorphic to the closed disc $D \subset \mathbb{R}^2$.
Next, we treat the case $\overline{\mr M}_1$. Let $\overline{M}_{0,5}^1(\mathbb{R}) = \textnormal{PGL}_2(\mathbb{R}) \setminus P_s^1(\mathbb{R})$ and $M_{0,5}^1(\mathbb{R}) = \textnormal{PGL}_2(\mathbb{R}) \setminus P_0^1(\mathbb{R})$. Now $M_{0,5}^1(\mathbb{R}) = \textnormal{PGL}_2(\mathbb{R}) \setminus P_0^1(\mathbb{R})$
and $\overline{M}^1_{0,5}(\mathbb{R})$ is the real blow-up $ B = \textnormal{Bl}_{0, 1, \infty}\mathbb{P}^1(\mathbb{C}) $ of $\mathbb{P}^1(\mathbb{C})$ in the three points $0, 1, \infty$, with $M_{0,5}^1(\mathbb{R}) \hookrightarrow \overline{M}_{0,5}^1(\mathbb{R})$ corresponding to the natural inclusion $\mathbb{P}^1(\mathbb{C}) \setminus \mathbb{P}^1(\mathbb{R}) \hookrightarrow B$. Let $C$ be any of the two connected components of $\mathbb{P}^1(\mathbb{C}) \setminus \mathbb{P}^1(\mathbb{R})$. Then $C$ is a manifold and an open subset of $B$, whose closure $\bar C$ in $B$ is a the simply connected manifold with corners diffeomorphic to the hexagon. Moreover, action of the subgroup $\mf S_2$ of $\mf S_3 \times \mf S_2$ on $\mathbb{P}^1(\mathbb{C}) \setminus \mathbb{P}^1(\mathbb{R})$ interchanges the connected components, hence the stabilizer $N = \text{Stab}_{\mf S_3 \times \mf S_2}(C)$ is $\mf S_3$, and $\bar C$ is stabilized by $N$. By Lemma \ref{lemma:D3}, the only orbifold point of $\ca M_s(\mathbb{R})$ with stabilizer group $D_3$ is given by the point $\alpha_3\coloneqq(-1,\infty, 0, \omega, \omega^2)$ with $\omega$ a primitive third root of unity. Hence $(\bar C, \alpha_3)$ is homeomorphic to the closed disc $(D,0) \subset (\mathbb{R}^2,0)$ with $N = \mf S_3$ acting on it in the natural way. In particular, $$ \overline{\mr M}_1 = \overline{M}^1_{0,5} / \mf S_3 \times \mf S_2 = \bar C / N = D / D_3 $$ is homeomorphic to the closed disc $D \subset \mathbb{R}^2$. Finally, consider $\overline{\mr M}_2$. Let $\overline{M}_{0,5}^2(\mathbb{R}) = \textnormal{PGL}_2(\mathbb{R}) \setminus P_s^2(\mathbb{R})$ and $M_{0,5}^2(\mathbb{R}) = \textnormal{PGL}_2(\mathbb{R}) \setminus P_0^2(\mathbb{R})$. Now $M_{0,5}^2(\mathbb{R}) = \textnormal{PGL}_2(\mathbb{R}) \setminus P_0^2(\mathbb{R}) \cong \mathbb{P}^1(\mathbb{C}) \setminus \left(\mathbb{P}^1(\mathbb{R}) \cup \{\infty, i, -i\}\right)$
and $\overline{M}^2_{0,5}(\mathbb{R})$ is the real blow-up $ B' = \textnormal{Bl}_{\infty}\mathbb{P}^1(\mathbb{C}) $ of the sphere $\mathbb{P}^1(\mathbb{C})$ in the point $\infty$, with $M_{0,5}^2(\mathbb{R}) \hookrightarrow \overline{M}_{0,5}^2(\mathbb{R})$ corresponding to the natural inclusion of $\mathbb{P}^1(\mathbb{C}) \setminus \left(\mathbb{P}^1(\mathbb{R}) \cup \{i, -i\}\right)$ in $B'$. Let $C$ be any of the two connected components of $\mathbb{P}^1(\mathbb{C}) \setminus \mathbb{P}^1(\mathbb{R})$. Then $C$ is a manifold, homeomorphic to the punctured unit disc, and forms an open subset of $B$ whose closure $\bar C$ in $B$ is the simply connected manifold-with-corners that has $2$ two corners. The subgroup $N$ of $\mf S_2 \times \mf S_2$ that stabilizes $C$ is isomorphic to $\mathbb{Z}/2$, and $\bar C$ is equivariant under $N$. The induced action of $\mathbb{Z}/2$ on $D'$ is reflection in a line through the origin; in particular, $$ \overline{\mr M}_2 = \overline{M}^2_{0,5} / \mf S_2 \times \mf S_2 = \bar C / N = D' / \mathbb{Z}/2 $$ is homeomorphic to the closed disc $\bar D \subset \mathbb{R}^2$.
Finally, it is clear that $\ca M_s(\mathbb{R}) =\overline{\mr M}_0 \cup \overline{\mr M}_1 \cup \overline{\mr M}_2$ is topologically formed by gluing together the three closed discs in the prescribed way, i.e.~as in Figure~\ref{fig:triangle}.
\end{comment} \begin{proof}[Proof of Theorem \ref{th:theorem02}] To any closed two-dimensional orbifold $O$ one can associate a set of natural numbers $S_O = \{n_1, \dotsc, n_k; m_1, \dotsc, m_l\}$ by letting $k$ be the number of cone points of $X_O$, $l$ the number of corner reflectors, $n_i$ the order of the $i$-th cone point and $2m_j$ the order of the $j$-th corner reflector. A closed two-dimensional orbifold $O$ is determined, up to orbifold-structure preserving homeomorphism, by its underlying space $X_O$ and the set $S_O$ \cite{Thurston80}. By Lemma \ref{lem:simplyconnected}, $\overline{\mr M}_\mathbb{R}$ is homeomorphic to a closed disc in $\mathbb{R}^2$. By Proposition \ref{prop:conesreflectors}, $\overline{\mr M}_\mathbb{R}$ has no cone points and three corner reflectors whose orders are $\pi/3, \pi/5$ and $\pi/10$. This implies $\overline{\mr M}_\mathbb{R}$ and $\Delta_{3,5,10}$ are isomorphic as topological orbifolds. Consequently, the orbifold fundamental group of $\overline{\mr M}_\mathbb{R}$ is abstractly isomorphic to the group $\Gamma_{3,5,10}$ defined in (\ref{PGAMMAR}).
Let $\phi: \Gamma_{3,5,10} \hookrightarrow \text{PSL}_2(\mathbb{R})$ be \textit{any} embedding such that $X\coloneqq\phi\left(\Gamma_{3,5,10}\right) \setminus \mathbb{R} H^2$ is a hyperbolic orbifold; we claim that there is a fundamental domain $\Delta$ for $X$ isometric to $\Delta_{3,5,10}$. To see this, consider the generator $a \in \Gamma_{3,5,10}$. Since $\phi(a)^2 = 1$, there exists a geodesic $L_1 \subset \mathbb{R} H^2$ such that $\phi(a) \in \text{PSL}_2(\mathbb{R}) = \text{Isom}(\mathbb{R} H^2)$ is the reflection across $L_1$. Next, consider the generator $b \in \Gamma_{3,5,10}$. There exists a geodesic $L_2 \subset \mathbb{R} H^2$ such that $\phi(b)$ is the reflection across $L_2$, and we have $L_2 \cap L_1 \neq \emptyset$.
Let $x \in L_1 \cap L_2$. Then $\phi(a)\phi(b)$ is an element of order three that fixes $x$, hence is a rotation around $x$. Therefore, one of the angles between $L_1$ and $L_2$ must be $\pi/3$. Finally, we know that $\phi(c)$ is an element of order two in $\textnormal{PSL}_2(\mathbb{R})$, hence a reflection across a line $L_3$. By the previous arguments, $L_3 \cap L_2 \neq \emptyset$ and $L_3 \cap L_1 \neq \emptyset$. Note that $x \in L_3 \cap L_2 \cap L_1 \neq \emptyset$.
Consequently, the three geodesics $L_i \subset \mathbb{R} H^2$ enclose a hyperbolic triangle; the orders of $\phi(a)\phi(b)$, $\phi(a)\phi(c)$ and $\phi(b)\phi(c)$ imply that the three interior angles of the triangle are $\pi/3$, $\pi/5$ and $\pi/10$. \end{proof}
\section{The monodromy groups} \label{sec:monodromy} In this section, we describe the monodromy group $P\Gamma$, as well as the groups $P\Gamma_\alpha$ appearing in Proposition \ref{prop:realsmoothperiods}. As for the lattice $(\Lambda, \mf h)$ (see (\ref{eq:hermitianformonbinaryquinticlattice})), we have:
\begin{theorem}[Shimura] \label{th:calculatemonodromyshimura}
There is an isomorphism of hermitian $\mathcal{O}_K$-lattices
$$\left( \Lambda, \mf h \right) \cong \left( \mathcal{O}_K^3, \textnormal{diag}\left(- \lambda, 1,1\right) \right), \quad \lambda = \zeta_5 + \zeta_5^{-1} = \frac{\sqrt 5 - 1}{2}.$$ \end{theorem} \begin{proof} See \cite[Section 6]{shimuratranscendental} as well as item (5) in the table on page 1 of \emph{loc.~cit.} \end{proof}
\noindent
Write $\Lambda = \mathcal{O}_K^3$ and $\mf h = \textnormal{diag}(- \lambda, 1, 1)$. For the element $\theta = \zeta_5 - \zeta_5^{-1} \in \mathcal{O}_K$ we have that $|\theta|^2 = \frac{\sqrt{5} + 5}{2}$. Consider the $\bb F_5$-vector space $W = \Lambda / \theta \Lambda$ equipped with the quadratic form $q = \mf h \bmod \theta$.
Define three anti-isometric involutions as follows:
\begin{equation} \label{chi}\begin{split}
\alpha_0\colon & (x_0,x_1,x_2) \mapsto (\bar x_0, \;\;\;\bar x_1,\;\;\bar x_2) \\
\alpha_1 \colon & (x_0,x_1,x_2) \mapsto (\bar x_0, - \bar x_1, \;\;\bar x_2) \\
\alpha_2 \colon & (x_0,x_1,x_2) \mapsto (\bar x_0, -\bar x_1, -\bar x_2).
\end{split} \end{equation} \begin{lemma} \label{exactlyconjugate} An anti-unitary involution of $\Lambda$ is $\Gamma$-conjugate to exactly one of the $\pm \alpha_j$. In particular, $\va{C\mr A} = 3$ and the $\alpha_i$ of (\ref{chi}) form a set of representatives for $C\mr A$. \end{lemma} \begin{proof}
For isometries $\alpha \colon W \to W$, the dimension and determinant of the fixed space $(W^\alpha, q|_{W^\alpha})$ are conjugacy-invariant. Via an adaption of the proof \cite[Lemma 7.2]{degaayfortman-nonarithmetic}, one shows that the elements $\pm \alpha_i$ are pairwise non $\Gamma$-conjugate. Moreover, $\va{C\mr A} = \va{\pi_0(\ca M_0(\mathbb{R}))} = 3$ by Proposition 3.2. The lemma follows.
\end{proof} \noindent Consider the fixed lattices of the anti-unitary involutions $\alpha_i$ defined in \eqref{chi}: \begin{align}\label{eq:fixedlattices} \Lambda^{\alpha_0} = \mathbb{Z}[\lambda] \oplus \mathbb{Z}[\lambda] \oplus \mathbb{Z}[\lambda], \Lambda^{\alpha_1} = \mathbb{Z}[\lambda] \oplus \theta\mathbb{Z}[\lambda] \oplus \mathbb{Z}[\lambda], \Lambda^{\alpha_2} = \mathbb{Z}[\lambda] \oplus \theta\mathbb{Z}[\lambda] \oplus \theta\mathbb{Z}[\lambda]. \end{align} \noindent Restricting $\mf h$ to the $\Lambda^{\alpha_j}$ yields quadratic forms $q_0$, $q_1$ and $q_2$ on $\mathbb{Z}[\lambda]^3$ defined as follows: \begin{equation} \label{eq:explicitquadraticforms} \begin{split}
q_0(x_0, x_1, x_2) & = - \lambda x_0^2 + x_1^2 + x_2^2, \\
q_1(x_0, x_1, x_2) & = - \lambda x_0^2 + \frac{\sqrt{5} + 5}{2} \cdot x_1^2 + x_2^2, \\
q_2(x_0, x_1, x_2) & = - \lambda x_0^2 + \frac{\sqrt{5} + 5}{2} \cdot x_1^2 + \frac{\sqrt{5} + 5}{2} \cdot x_2^2.
\end{split} \end{equation}
\noindent Consider $\mathbb{Z}[\lambda]$ as a subring of $\mathbb{R}$ via the embedding that sends $\lambda$ to a positive element.
\begin{theorem} \label{th:explicitquadratic} Consider the quadratic forms $q_j$ defined in (\ref{eq:explicitquadraticforms}). There is a union of geodesic subspaces $\mr H_j \subset \mathbb{R} H^2$, $j \in \{0,1,2\}$, and an isomorphism of hyperbolic orbifolds \begin{equation}
\ca M_0(\mathbb{R}) \cong \coprod_{j = 0}^2 \textnormal{PO}(q_j,\mathbb{Z}[\lambda]) \setminus \left(\mathbb{R} H^2 - \mr H_j \right). \end{equation} \end{theorem}
\begin{proof} By Proposition \ref{prop:realsmoothperiods} and Lemma \ref{exactlyconjugate}, we obtain $\ca M_0(\mathbb{R}) \cong \coprod_{j = 0}^2 P\Gamma_{\alpha_j} \setminus (\mathbb{R} H^2_{\alpha_j} - \mr H)$. Remark that $P\Gamma_{\alpha_j}=N_{P\Gamma}(\alpha_j)$ for the normalizer $N_{P\Gamma}(\alpha_j)$ of $\alpha_j$ in $P\Gamma$. If $h_j$ denotes the restriction of $\mf h$ to $\Lambda^{\alpha_j}$, there is a natural embedding
$ \iota_j \colon N_{P\Gamma}(\alpha_j) \hookrightarrow \textnormal{PO}(\Lambda^{\alpha_j},h_j, \mathbb{Z}[\lambda])$. We claim that $\iota_j$ is an isomorphism. Indeed, the natural homomorphism $ \pi_j \colon N_\Gamma(\alpha_j) \to O(\Lambda^{\alpha_j}, h_j) $ is surjective, where $N_\Gamma(\alpha_j) = \{g \in \Gamma: g \circ \alpha_j = \alpha_j \circ g \}$ is the normalizer of $\alpha_j$ in $\Gamma$. The surjectivity of $\pi_j$ follows in turn from the equality $$
\Lambda = \ca O_K \cdot \Lambda^{\alpha_j} + \ca O_K \cdot \theta \left(\Lambda^{\alpha_j}\right)^\vee \subset K^3, $$ which follows from (\ref{eq:fixedlattices}). Since $\textnormal{PO}(\Lambda^{\alpha_j},h_j, \mathbb{Z}[\lambda]) = \textnormal{PO}(q_j,\mathbb{Z}[\lambda])$, we are done. \end{proof}
\section{Non-arithmetic lattices in the projective orthogonal group} \label{sec:announced}
In the first half of this two-part paper we announced a result, see \cite[Theorem 1.3]{degaayfortman-nonarithmetic}, that we are now able to prove. Let $K = \mathbb{Q}(\zeta_5)$ and let $\Phi = \set{\tau, \varphi \colon K \to \mathbb{C}}$ be the CM type defined in \eqref{CMTYPE-Shimura-modification}. Define a sequence of hermitian $\mathcal{O}_K$-lattice $\Lambda_n$ as follows: \[ \Lambda_n = (\mathbb{Z}[\zeta_5]^{n+1}, \tn{diag}(- \lambda , 1,\dotsc, 1) ) \quad \tn{with} \quad \lambda = \zeta_5 + \zeta_5^{-1} = (\sqrt 5 - 1)/2.
\] Define $\Gamma(n)= \textnormal{Aut}(\Lambda_n)$ and let $P\Gamma(n) = \mu_K \backslash\Gamma(n)$. Let $\mr A_n$ be the set of anti-unitary involutions $\alpha \colon \Lambda_n \to \Lambda_n$, and let $P\mr A_n = \mu_K \backslash \mr A_n$, where $\mu_K = (\mathcal{O}_K^\ast)_{\tn{tors}} = \langle - \zeta_5 \rangle$. Finally, let $ \mr R_n = \set{r \in \Lambda_n \mid h_n(r,r) = 1}$ be the set of short roots in $\Lambda_n$.
By Theorem 6.2 and Proposition 6.4 of \cite{degaayfortman-nonarithmetic}, the hyperplane arrangement $\mr H_n = \cup_{r \in \mr R_n} H_r \subset \mathbb{C} H^n$
is an orthogonal arrangement, i.e.~\cite[Condition 2.2]{degaayfortman-nonarithmetic} is satisfied. Thus, we can perform the glueing construction of [\emph{loc.~cit.}, Section 3] to obtain a sequence of metric spaces $X(\Lambda_n)$ for $n \in \mathbb{Z}_{\geq 2}$. Moreover, for each $n \in \mathbb{Z}_{\geq 2}$, the metric on $X(\Lambda_n)$ extends to a complete real hyperbolic orbifold structure, hence its connected components are quotients of $\mathbb{R} H^n$ by discrete groups of isometries (see \cite[Theorem 4.1.5]{degaayfortman-nonarithmetic}).
Our goal now is to prove \cite[Theorem 1.3]{degaayfortman-nonarithmetic}, whose statement we recall:
\begin{theorem}[c.f.~\cite{degaayfortman-nonarithmetic}] \label{theorem-nonarithmetic-higherdim}
For each $n \in \mathbb{Z}_{\geq 2}$, there exists a connected component $X(\Lambda_n)^+ = \Gamma_n^+ \setminus \mathbb{R} H^n$ of $ X(\Lambda_n)$ such that the lattice $\Gamma_{n}^+ \subset \tn{PO}(n,1)$ is non-arithmetic.
\end{theorem}
\begin{proof} We proceed as in the proof of \cite[Theorem 8.7]{degaayfortman-nonarithmetic}. For each $i \in \set{0, 1, \dotsc, n}$, define an anti-unitary involution $\beta_i$ as $ \beta_i(x_0, \dotsc, x_n) = \left( \bar x_0, -\bar x_1, \dotsc, - \bar x_i, \bar x_{i + 1}, \dotsc, \bar x_n \right)$.
By a suitable adaption of the proof of \cite[Lemma 7.2]{degaayfortman-nonarithmetic} to this setting, we see that the classes $\beta_i \in P\mr A_n$ are pairwise non $P\Gamma(n)$-conjugate. As in [\emph{loc.~cit.}, Lemma 7.3], this allows one to define a natural map of topological spaces $ \iota \colon X(\Lambda_2) \to X(\Lambda_n)$. Note that $X(\Lambda_2)$ is connected by Theorem 4.7. By Moreover, by the analogue of \cite[Proposition 7.4]{degaayfortman-nonarithmetic}, the map $\iota$ is a map of hyperbolic orbifolds, and in fact a totally geodesic immersion into a connected component $X(\Lambda_n)^+$ of $X(\Lambda_n)$ (see the proof of Theorem 8.5 in \emph{loc.~cit.}). Therefore, by \cite[Proposition 15.2.2]{bergeron-clozel} (see also \cite[Theorem 1.4]{subspacestabilizers}), it suffices to show that the lattice $\Gamma_2^+ \subset \tn{PO}(2,1)$ underlying the complete hyperbolic orbifold $X(\Lambda_2)$ is non-arithmetic. Since $\Gamma_2^+ = P\Gamma_\mathbb{R} = \Gamma_{3,5,10}$, see the proof of Theorem \ref{th:theorem02} in Section \ref{space:hyperbolictriangle}, this follows from Remark \ref{rem:introremarks}.\ref{remark:takeuchi}. \end{proof}
\printbibliography
\textsc{Olivier de Gaay Fortman, Institute of Algebraic Geometry, Leibniz University Hannover, Welfengarten 1, 30167 Hannover, Germany}\par\nopagebreak
\textit{E-mail address:} \texttt{[email protected]}
\end{document} | arXiv |
Sensitivity of the mean to outliers
Is the mean sensitive to the presence of outliers? I initially thought it wasn't, because a small amount of observations shouldn't have much impact, but was told that since those observations have very different values from the rest, they have a considerable impact. Thoughts?
mean outliers
gung - Reinstate Monica♦
John DohJohn Doh
$\begingroup$ Obviously yes, compute mean of 1,2,3,4 and of 1,2,3,999. What exactly are you asking? $\endgroup$ – Tim♦ Mar 7 '16 at 13:01
$\begingroup$ The mean is, in youthspeak, totally sensitive to outliers. Each value contributes to the total and in that sense pulls the mean towards it. $\endgroup$ – Nick Cox Mar 7 '16 at 13:08
$\begingroup$ @Tim ok so now compute 20, 999, 1000, 1001, 1002, 1003, 1004, 1005, 1006. Your point being? @ Nick Cox the fact that each one contributes doesn't necessarily mean - no pun intended - that each one has a huge impact, although it might have, of course. $\endgroup$ – John Doh Mar 7 '16 at 13:22
$\begingroup$ As @Tim says, obviously yes. But extreme values, relative to the rest, will have huge impact, which is precisely the point. Sensitivity need not mean extreme sensitivity; it just means... sensitivity. $\endgroup$ – Nick Cox Mar 7 '16 at 13:27
$\begingroup$ The answer to this question is simple & straightforward (& has already been provided in comments). I don't think this is too broad. $\endgroup$ – gung - Reinstate Monica♦ Mar 7 '16 at 14:58
Consider what would happen if you wanted to take the mean of some some numbers, but you dragged one of them off toward infinity. Sure, at first it wouldn't have a huge impact on the mean, but the farther you drag it off, the more your mean changes.
Every number has a (proportionally) small contribution to the mean, but they do all contribute. So if one number is really different than the others, it can still have a big influence.
This idea of dragging values off toward infinity and seeing how the estimator behaves is formalized by the breakdown point: the proportion of data that can get arbitrarily large before the estimator also becomes arbitrarily large.
The mean has a breakdown point of 0, because it only takes 1 bad data point to make the whole estimator bad (this is actually the asymptotic breakdown point, the finite sample breakdown point is 1/N).
On the other hand, the median has breakdown point 0.5 because it doesn't care about how strange data gets, as long as the middle point doesn't change. You can take half of the data and make it arbitrarily large and the median shrugs it off.
You can even construct an estimator with whatever breakdown point you want (between 0 and 0.5) by 'trimming' the mean by that percentage--throwing away some of the data before computing the mean.
So, what does this mean for actually doing work? Is the mean just a terrible idea? Well, like everything else in life, it depends. If you desperately need to protect yourself against outliers, yeah, the mean probably isn't for you. But the median pays a price of losing a lot of potentially helpful information to get that high breakdown point.
If you're interested in reading more about it, here's a set of lecture notes that really helped me when I was learning about robust statistics. http://www.stat.umn.edu/geyer/5601/notes/break.pdf
SullysaurusSullysaurus
In some sense, the mean depends equally on all the items on data — it is perfectly democratic. We can see this by considering the arithmetic mean as a special case of weighted mean,
$$\bar x = \sum_{i=1}^n \alpha_i x_i \tag{1}$$
where the weights $\alpha_i$ are all set equal at $\frac{1}{n}$.
However, when we talk about sensitivity to inputs (of a function, of a model, etc), we are generally interested in "how much of an effect would it have if our inputs were different." And on that count, outliers can be very influential in our result for the arithmetic mean.
One way to see this is to take a data set containing an outlier, and consider, for each piece of data, what would be the effect on the mean if this data point were deleted (or more counterfactually, imagine that we had never sampled and recorded it).
Take the data set $\{1,3,4,5,7,100\}$ for instance. This has a mean of $120/6 = 20$. If the $1$ were deleted, the mean rises to $119/5 = 23.8$, a change of $+3.8$. Deleting the $3$, $4$, $5$ or $7$ would have had even small effects, producing changes of $+3.4$, $+3.2$, $+3$ and $+2.6$ respectively. But deleting the $100$ would reduce the mean to $20/5=4$, a change of $-16$. In this sense, the mean is very sensitive to the inclusion of the $100$ in the data set: its value would have been very different without it. The impact of removing the outlier is noticeably larger than for any of the other data points. Here's a visual representation: the dot plot at the top shows the full distribution and its mean (the black bar); the following plots show the effect on the mean of deleting successive points.
In contrast, the median is a less sensitive measure of central tendency. The median of our entire data set is $4.5$, and deletions give the following changes:
\begin{array}{lll} &\text{Delete} &\text{New median} &\text{Change} \\ \hline &1 &5 &+0.5 \\ &3 &5 &+0.5 \\ &4 &5 &+0.5 \\ &5 &4 &-0.5 \\ &7 &4 &-0.5 \\ &100 &4 &-0.5 \\ \end{array}
Not only is the median less sensitive to changes overall, but removing the outlier had no more effect than deleting any of the other data points. It could even cope with several outliers. In fact, many outliers are okay, so long as they constitute less than half of the data set — see the the answer by @Sullysarus.
Note that the sensitivity of the mean is not necessarily a bad thing; it's often the case that we use the mean because we like its sensitivity, i.e. the way it makes full use of all the data. See the thread "If mean is so sensitive, why use it in the first place?"
Note that the same principles apply to outliers on the left tail, i.e. values that are "unusually small" for the data set: consider e.g. $\{-1, -3, -4, -5, -7, -100 \}$ where it is clear that the results will come out similarly to before, but with the sign (i.e. direction) of changes reversed.
Also, to head off a possible misunderstanding, looking at $\frac{1}{n} x_i$ as the "contribution" of $x_i$ to the mean may not be the most helpful approach to considering "sensitivity", even though it's true that $\bar x$ is the sum of these contributions (as written in equation $(1)$). Consider the translation of our original data set to $\{10001, 10003, 10004, 10005, 10007, 10100 \}$. Now each contribution looks fairly similar: proportionately there's not much difference between $1666\frac{5}{6}$ (the smallest, from the $10001$) and $1683\frac{1}{3}$ (the largest, from the $10100$). Each contribution is very close to one sixth of the mean, so it doesn't look like the outlier is making much more difference than the other numbers did.
But this is misleading, since the "sensitivity to deletion" argument will give the same results as before. It's the relative position of the number from the mean that matters, not the relative proportion it contributes towards the sum. (Another issue with the "relative proportion" or "contribution" approach is that it doesn't make much sense when you have a mixture of positive and negative data.)
Consider the new mean after the deletion of $x_j$: we would divide the new total, which is the previous total, $\sum_{i=1}^n x_i = n \bar x$, divided by number of items of data remaining, $n-1$. This can be manipulated to give
$$\frac{n \bar x - x_j}{n-1} = \frac{n \bar x - \bar x + \bar x - x_j}{n-1} = \frac{(n - 1) \bar x + (\bar x - x_j)}{n-1} = \bar x + \frac{\bar x - x_j}{n-1}$$
This shows that deleting a data point $x_j$ causes the mean to increase by $\frac{\bar x - x_j}{n-1}$. It should now be clear why deleting data that lies far from the mean, in either direction, should have a greater effect than removing data that lies close to the mean.
R code for plot
library(grDevices)
fulldata <- c(1,3,4,5,7,100)
n <- length(fulldata)
par(bg="grey98")
plot(NULL, xlim=c(min(fulldata)-5, max(fulldata)+5), ylim=c(0,n+2), axes=FALSE,
ylab="", xlab="")
cols <- rainbow(n)
points(rep(fulldata,n+1), rep(1:(n+1),each=n), col="black", bg=cols, pch=21, cex=1)
points(fulldata[1:n], n:1, pch=4, cex=2)
abline(v=mean(fulldata), col="grey", lty=3)
segments(mean(fulldata),n+1-0.2,
mean(fulldata),n+1+0.2, col="black", lwd=2)
segments((sum(fulldata)-fulldata[1:n])/(n-1),n:1-0.2,
(sum(fulldata)-fulldata[1:n])/(n-1),n:1+0.2,col=cols, lwd=2)
segments((sum(fulldata)-fulldata[1:n])/(n-1),n:1,rep(mean(fulldata),n),n:1,col=cols,lwd=2)
SilverfishSilverfish
Arithmetic mean is a sum of all the values divided by their count
$$ \frac{x_1 + x_2 + \dots + x_n}{N} $$
so each of the values have the same impact on the final estimate. Let me say it once again: each of the values has impact on the estimate. Means' "sensibility" to data is actually one of the reasons why we choose it as a estimator of location. Obviously, if one or more of the values deviate from the other, they influence the mean. If they deviate by large, their influence is larger, if deviation is smaller, than their influence is smaller. It is true that "small amount of observations shouldn't have much impact" on it's result, but that doesn't mean that they have no impact.
As correctly suggested by whuber, such robustness can be shown by measures such as breakdown point. Breakdown point is basically the proportion of observations that are needed to influence the estimate. In case of mean it is zero, because changing a single value is enough to influence the final result. More robust measures, like median, are less sensible and a greater fraction of outlying cases may be needed to influence their estimates.
Tim♦Tim
$\begingroup$ I think this answer might need some qualification. After all, a single outlier can have a profound effect on the median: the derivative of the median as a function of individual values of the data is either zero or infinite, which would seem to be as sensitive as it could possibly be! My point is that there are standard measures of resistance of statistics, such as their breakdown point, and that to be correct this answer ought to refer to some such measure. $\endgroup$ – whuber♦ Mar 7 '16 at 15:27
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Methodology article
Micro-CT imaging of live insects using carbon dioxide gas-induced hypoxia as anesthetic with minimal impact on certain subsequent life history traits
Danny Poinapen1,3,
Joanna K. Konopka2,3,
Joseph U. Umoh1,
Chris J. D. Norley1,
Jeremy N. McNeil2 &
David W. Holdsworth1
Live imaging of whole invertebrates can be accomplished with X-ray micro-computed tomography (micro-CT) at 10-100 μm spatial resolution. However, image quality could be compromised by the movement of live subjects, producing image artefacts. We tested the feasibility of using CO2 gas to induce temporary full-immobilization of sufficient duration to image live insects based on their ability to tolerate hypoxic conditions. Additionally, we investigated the effects of these prolonged hypoxic conditions on several life history traits of a lepidopteran species.
Live Colorado potato beetle (CPB) and true armyworm (TAW) adults were immobilized under a constant CO2 gas flow (0.5 L/min), and scanned using micro-CT (80 kVp; 450 μA). An L8 (24) orthogonal array (OA) was used to evaluate the effects of prolonged CO2-induced anesthesia on the recovery, longevity, and incidence of mating of TAW adults. The variable factors were age (immature and mature), sex (female and male), exposure time (3 and 7 h), and exposure regime (single and repeated).
With this method, successful 3D reconstruction and visualizations of CPB and TAW adults were produced at 20 micron voxel spacing at an acceptable radiation dose and image noise level. From the inverse-square relationship found between the radiation doses and image noise levels, the optimal scanning protocol produced an entrance dose of 6.2 ± 0.04 Gy with images of 129.6 ± 5.1 HU noise level during a 2.7 h scan.
Independent OA experiments indicated that CO2 gas did not result in death of exposed TAW adults, except when older males were exposed for longer durations. Exposure time and sex were more influential factors affecting recovery, longevity, and mating success than age and exposure regime following CO2 exposure.
We have demonstrated that using CO2 gas during micro-CT imaging effectively induces safe, repeatable, whole-body, and temporary immobilization of live insects for 3D visualizations without motion artefacts. Moreover, we have shown that exposed TAW individuals made a full recovery with very little impact on subsequent longevity, and mating success post hypoxia. This method is applicable to other imaging modalities and could be used for routine exploratory and time-course studies, for repeated scanning of live and intact individuals.
Conventional imaging methods, such as light and confocal microscopy, provide valuable details of internal anatomy of invertebrates, but they generally require sacrifice or dissection of the live individuals [1]. The use of these methods in longitudinal (time-course) studies, involving repeated scans of the same individuals, is thus not possible. Magnetic resonance imaging (MRI), and X-ray micro-computed tomography (micro-CT) [2, 3] have been used for 3D non-destructive visualization of internal anatomical structures [4,5,6,7,8,9,10] of live and preserved small mammals and invertebrates, including insects [11,12,13,14,15,16,17,18,19,20]. Micro-CT and MRI each have their own advantages and limitations due to organism size (scanner field of view), spatial resolution, scan time and cost. While these two techniques have produced reasonably high-resolution images of both dead and live insects [5, 10, 21, 22], difficulties were encountered due to internal and external body movements, and required physical restriction of the individuals. Such physical constraints induce stress that can affect the behavior and/or physiology of test organisms during and after the imaging procedure.
Insects can tolerate high doses of ionizing radiation, as during micro-CT, due to highly efficient oxidative stress and DNA damage repair mechanisms [23,24,25,26]. Some lepidopteran cell lines are reported to tolerate X-ray doses 52-104 times more than mammalian counterparts [23, 24]. Nevertheless, death or permanent metabolic damage due to radiation has been observed, especially when synchrotron beams are employed (> 300 Gy) [6, 9]. These high radiation doses preclude the possibility of repeated measures on the same living individuals over time, and therefore keeping the radiation doses as low as possible becomes crucial to avoid rapid death during follow-up studies on live insects using micro-CT.
Therefore, an ideal imaging protocol for studying live insects would preferably accommodate the following criteria: allow simultaneous scanning of multiple, fully-intact specimens to maximize throughput; provide adequate image resolution and high contrast-to-noise ratio to distinguish in situ structures; deliver a sufficiently low radiation dose to allow repeated scans for time-course studies; and allow fully-immobilized individuals to eliminate motion-related artefacts. Additionally, the protocol should have minimal or no subsequent negative effects on the life history traits or physiological processes of test subjects.
To minimize motion artefacts during live insect X-ray micro-CT imaging, we temporarily immobilized the test subjects using hypoxia by carbon dioxide (CO2) gas. In addition to being radiotolerant, insects can survive acute hypoxic (oxygen deficient) conditions [27,28,29]. Keeping insects in CO2 hypoxic and /or hypercapnic (abnormally elevated CO2 level) conditions leads to temporary, whole-body immobilization as the CO2 molecules mainly interfere with the neuromuscular junction (NMJ), probably by blocking the glutamate receptors [27,28,29]. In this state, CO2 induces anesthesia that stops both external (e.g. antennae and legs) and internal (e.g. heart beat and hemolymph flow) body movements [17, 30], thereby eliminating motion-related image artefacts.
Since live insect scanning typically takes at least several hours to complete when using bench-top micro-CT, prolonged exposure to either ionizing radiation or CO2 gas could negatively affect aspects of insect behaviour and fitness [31,32,33,34,35]. Such exposures could be detrimental in time-course studies where the same individuals must be examined multiple times. Thus, an optimized imaging protocol should ensure safe entrance doses, preferably at least 10 times less than commonly used during insect sterilization. The effects of prolonged exposure to hypoxic conditions can be long lasting, and may not necessarily be immediate. Therefore, it is critical to distinguish between possible effects of experimental treatments and those of hypoxia-induced anesthesia. Several factors including age, sex, duration of acute exposure, and number of repeated exposures play a role in insects' response to and recovery from hypoxia. Their relative importance may vary depending on the subsequent effects on life history traits, and consequently on the type of biologically-relevant processes under investigation. Therefore, in parallel to investigating the feasibility of using CO2 gas for immobilization of live insect (the Colorado potato beetle and the true armyworm) adults undergoing micro-CT scanning, we assessed the effects of prolonged CO2 exposure, of equivalent duration required for imaging experiments on the recovery, longevity, and mating of Pseudaletia unipuncta adults. To this end, we employed an orthogonal array design – a robust and unbiased method to determine the influence of individual factors, while limiting experimental variability [36,37,38,39,40]. This approach yields the optimized combinations of factors that have a minimal impact on the recovery and certain subsequent life history traits of insects post-anesthesia during live imaging.
Colorado potato beetle (CPB; Leptinotarsa decemlineata Say, Coleoptera: Chrysomelidae) and the true armyworm (TAW; Pseudaletia unipuncta Haworth, Lepidoptera: Noctuidae) were obtained as newly-emerged adults from laboratory colonies maintained at Agriculture and Agri-Food Canada and the University of Western Ontario, respectively. Appropriate food sources were provided for each species: potato leaves for the CPB, and ad lib 8% (w/v) sugar water solution for the TAW.
Anesthesia, scanning, and data acquisition
Immediately before micro-CT imaging, the live adults were immobilized (~10-20 s) under a constant flow (5 psi; 0.5 L/min flow rate) [41] of medical grade carbon dioxide (CO2), and placed in a custom designed tube (Fig. 1a) for micro-CT scanning. This tube was made with the conical end of a 50 mL centrifuge tube cutoff. A plastic syringe filter was affixed to the open end to which a Tygon tube (R-3603; I.D. × O.D. 1/4 in. × 3/8 in) was connected for CO2 gas delivery. A rectangular piece of radiolucent expanded polystyrene foam was used as a bed to support the anesthetized insects. Two moistened foam pieces were interposed at either end of the tube to prevent insect desiccation. The insects were kept under anesthesia for a further 10 min to ensure full immobilization before scanning, after which the flow of CO2 gas was continued for the full duration of each scan.
Incorporating CO2 anesthesia during X-ray micro-computed tomography (micro-CT) for live insect imaging. a Schematic of the apparatus for live micro-CT imaging of CO2 anesthetized live insects. b X-ray dosimetry: entrance dose (Gy) for 5 scan protocols at 80 kVp and 450 μA (see Table 1). c Noise (HU) analysis of 3D-CT reconstructed image with 20 μm isotropic voxels from the 5 protocols. d Entrance dose to noise level relationship. Data in Figu. 1b,c, and d represent mean ± SD for n = 5 independent measurements for each protocol. Bars not sharing similar letters in Fig. 1b and c are statistically different (α =0.05)
The tube containing the live individuals was scanned in a GE eXplore Locus RS-9 X-ray micro-CT system (GE Healthcare, London, Canada). The acquisition parameters for the various scan protocols are detailed in Table 1. X-ray projections were acquired over a full 360° rotation around the sample volume (except for Protocol 1 acquired over a 191° rotation; Table 1).
Table 1 Protocol parameters used for scanning live anesthetized adult insects
One week-old CPB adults (2 males and 2 females), including a water sample for Hounsfield Units (HU) calibration, were scanned (field of view = 3.8 × 3.8 × 3.8 cm; see Additional file 1: Figure S1(I)) using Protocols 1, 2, and 3 (see Table 1). The insects were then allowed to recover for 48 h, before being scanned using Protocol 4. Finally, after a further 24-48 h recovery, these CPB were scanned using Protocol 5. The 24-48 h recovery periods were to ensure full restoration of normal metabolism between scans [42]. Although the CPB females recovered after being scanned using Protocol 4, they died within 48 h, and were replaced by two new females of the same generation and age for scanning using Protocol 5. The use of these 5 protocols enabled dose and image noise measurements from which the optimal scan parameters were deduced. This optimized protocol was then used to scan TAW adults (2 males and 2 females per scan, including a water sample for HU calibration) when they were 1-d old (sexually immature) and 4-d old (sexually mature).
For both species, during each scanning protocol, the same number of individuals was set aside with no CO2 exposure and no food for longevity comparison with the scanned ones. For reproducibility, we performed the scanning experiments twice on two fresh batches of CPB (n = 4 females; n = 4 males; 2 individuals of each sex per scan) and four times TAW (n = 4 males; n = 4 females; 1 individual of each sex per scan) independently.
Dosimetry and noise analysis
The entrance dose (Gy) for the 5 different scan protocols, of duration 0.33, 1.40, 2.68, 3.96, and 6.53 h, was measured at the isocenter of the micro-CT scanner, using an ionization chamber coupled to an electrometer (Keithley, models 96035B and 35614, respectively; see Additional file 1: Figure S1(II)). For each protocol, the noise level (HU) in 3D-CT reconstructed images at 20 μm isotropic voxel spacing was calculated as the standard deviation of 5 specific regions of interest (ROI) of size 20 × 20 × 20 voxels each in a water phantom image volume. The optimal scanning protocol– the shortest scan time that provides the most satisfactory image quality in terms of entrance dose and image noise – was then established from a dose-noise relationship by plotting the entrance dose (Gy) versus noise (HU) [43].
Reconstruction and 3D image rendering
The acquired 2D projections were reconstructed at 20 μm isotropic voxel size into a 3D image using the Feldkamp filtered-back projection algorithm (FDK) [44] using the GE Healthcare eXplore Reconstruction Utility. The 3D-CT images (in .vff format) were imported as DICOM in Osirix (v. 4.0 32-bit; Pixmeo; GE; Switzerland; freely available at: http://www.osirix-viewer.com) for renderings. No manipulations, such as noise averaging, were made to the data. The settings for the 3D MPR (multi-planar reformatted) were: Opacity = linear table, Mode = Volume Rendering (4.03 mm); and for 3D volume rendering (Group: Soft Tissue CT, preset 6 (Soft + Skin), Level of Detail (Fine); Shading (Ambient Coeff. = 0.14; Diffuse Coeff. 0.83; Specular Coeff: 0.50; Specular Power = 50.0).
Orthogonal array (OA) design and setup for CO2 exposure
To evaluate the effects of prolonged CO2-induced anesthesia on the recovery, longevity, and incidence of mating, we exposed TAW adults to several treatments based on an L8 (24) (four factors at two levels) orthogonal array (OA). In this design (Tables 2 and 3; n = 10 per treatment), the factors were: age (immature and mature), sex (female and male), exposure time (3 and 7 h- equivalent to micro-CT scan durations), and exposure regime (single and repeated) to a constant flow of CO2 (5 psi; flow rate 0.5 L/min; see Additional file 1: Figure S2) during the photophase.
Table 2 Four factors at two levels used in orthogonal array design
Table 3 L8 (24) matrix (four factors at two levels) showing the treatments from the OA design
We performed this study on TAW adults only because of availability and shorter life span compared to CPB (> 30 d). The controls received no anaesthesia and were not provided with any food source during the periods that treated individuals were exposed to CO2. All insects, treated and controls, were held in individual clear plastic cylinders with ad lib food source (cotton wicks soaked in 8% sugar water solution) before and after all treatments (Table 3). For repeated anesthesia, the individuals were exposed to CO2 for a second time 24 h after the first exposure.
For the CO2 exposure set up, the TAW adults were knocked out for 10-20 s, and carefully placed in a 1 L glass beaker on a layer of polystyrene foam. The beaker also contained a moistened foam support to prevent insect desiccation (see Additional file 2: Figure S2). A funnel was then connected to the chamber for CO2 gas delivery (medical grade; 5 psi; 0.5 L/min), and the individuals were maintained under these conditions for the required durations according to Table 3. All exposures were performed at 23 °C in light conditions (corresponding to the photophase that insects were entrained to).
Recovery, longevity, and incidence of mating
Post-anesthetic recovery of TAW adults was recorded for each individual as vitality signs (elapsed time for antennae, proboscis, and leg movements), and recovery (flip, walk, and move in a coordinated fashion) time. Longevity of all anesthetized virgin individuals was also recorded and compared with control individuals of the same sex. Virgin individuals were chosen to avoid possible reduction in longevity due to mating. A total of 80 individuals (10 per treatment) were anesthetised, with an equal number of controls (80 total; 10 per corresponding treatment), and used for measurements of recovery and longevity.
To investigate the effects of anesthesia on the incidence of mating, 24 h post-exposure, CO2-exposed males and females (including control individuals) were paired up with one non-anesthetised virgin mature individual of the opposite sex for the duration of one scotophase (8 h). This pairing ensured that any observed effects on the incidence of mating were due to the exposed individuals. Mating pairs were placed in small mesh cages (22 × 15 × 15 cm) containing ad lib sugar water solution. After 8 h, the respective females of each treatment were dissected to determine if mating was successful by looking for presence of a spermatophore. A total of 80 (in addition to the 80 individuals used for recovery and longevity measurements) individuals (10 per treatment) were anesthetised and paired with non-exposed virgin counterparts for mating experiments. An equal number of non-exposed individuals was used as control and paired with virgin counterparts.
All data were verified for normality using the Shapiro-Wilk's test. Parametric tests (one-way ANOVA followed by Tukey's HSD post hoc test) were used to analyse those data that followed a normal distribution. Otherwise, non-parametric methods (Kruskal-Wallis Test followed by Dunn's test) were employed. The tests and analyses were performed using SPSS (v.21; IBM, NY, USA).
To investigate the effect of a given treatment on the life-span of TAW adults, longevity ratios were computed. For the calculation of this ratio, the mean life span of the female control in each group was calculated and these means were compared using a one-way ANOVA. Since no statistically-significant difference was observed, these data were averaged. This average longevity of the female controls was then used to normalize all the treatments containing females (Treatments 1, 2, 5, and 6). The longevity ratio for females for a given treatment was hence calculated as the average life-span of females from that group divided by the average life-span of all female controls. The average longevity of all male controls was similarly calculated and used to normalize (as they were also not significantly different) treatments containing males (Treatments 3, 4, 7, and 8) to calculate their longevity ratio.
To examine the effect of the treatments on the incidence of mating, each group was compared with its respective control. All female controls were compared to each other; and all male controls were compared to each other using Chi-square tests. Finally, all controls combined (females and males) were compared to each other using a Chi-square test. In all cases no statistically-significant differences were found. Therefore, for the final analysis, a Chi-square test was performed to compare only the mating incidence among the treatments.
The ranking of the impact of the four factors (age, sex, exposure time, and exposure regime) on the outcomes (recovery, longevity, and mating success) based on the L8 OA [45] was performed in Minitab (v.17 Minitab, Coventry, UK). The reasons for choosing the orthogonal array, rather than a full factorial design, are mainly to reduce the number of experimental treatments and to provide the effective ranking of the factors with respect to their influence on the measured response variables.
After calculating the signal-to-noise (S/N) ratio for each treatment described in Table 3, the average S/N value was computed for each factor and level. This S/N ratio measures the robustness used to establish which of these four control factors (age, sex, exposure time, and exposure regime) result in reduced variability in the outcomes (recovery, longevity, and mating of TAW) by minimizing the impacts of other factors that are not possible to control. These factors, for which we cannot control, are referred as noise, which include genetic variability, individual variation in CO2 tolerance, gradual increase of lactate level, and acidification during hypoxia. Delta (Δ) was then calculated for each measured parameter (i.e. response) as the difference between the maximum and minimum S/N values. For recovery, the response was computed using eq. (1) below with the goal of minimizing this parameter (smaller the better). Conversely, eq. (2) was used to maximize longevity and mating incidence (larger the better).
$$ \mathrm{S}/{\mathrm{N}}_{\mathrm{i}}=-10\ \log \left(\sum \left({\overset{-}{\mathrm{y}}}_{\mathrm{i}}^2\right)/{N}_i\right) $$
$$ \mathrm{S}/{\mathrm{N}}_{\mathrm{i}}=-10\ \log \left(\sum \left(1/{\overset{-}{\mathrm{y}}}_{\mathrm{i}}^2\right)/{N}_i\right) $$
where \( {\overset{-}{y}}_i=\frac{1}{N_i}\sum_{u=i}^{N_i}{y}_{i,u} \) is the mean; i = experiment number, u = trial number and Ni = number of trials for experiment i.
For multivariate analysis, a principal component analysis (PCA) was performed on the recovery, longevity, and mating incidence data. Before performing the PCA analysis, the Kaiser-Meyer-Olkin (KMO; ≥ 0.70 considered as satisfactory to run PCA) measure of sampling adequacy and the Bartlettʾs test for sphericity (χ2) were performed [46]. A PCA using all the measured parameters was then carried out to explore life history trait (recovery, longevity, and mating success) patterns between the 8 treatments post-anesthesia, with varimax rotation after normalization using the z-score on all data in SPSS (v.21; IBM, NY, USA).
X-ray dose-to-image noise relationship
Based on the inverse-square dose-to-noise relationship (Fig. 1d), Protocol 3 emerged as optimal from the dose versus image noise analysis. This protocol resulted in an entrance dose of 6.2 ± 0.04 Gy, producing images (Figs. 2 and 3) with a 129.6 ± 5.1 HU noise level during a 2.68 h scan (i.e. corresponding to the amount of time that the insects were maintained under CO2 anesthesia). Under these conditions, the insects recovered full mobility within 40 min post-anesthesia/scanning, compared to more than 90 min for the 6.53 h scan (Protocol 5). The 15.5 ± 0.10 Gy entrance dose for the 6.53 h scan was approximately 2.5 times higher than the optimal 2.68 h scan (Fig. 1b), but it was least 13 times less than the average required dose for insect sterilization.
3D-CT reconstruction of live anesthetized 1 week-old female (♀) and male (♂) Colorado potato beetle adults. Sagittal views of (a) 3D-volume of insect body exoskeleton at 20 μm isotropic voxel spacing; (b) 3D-MPR (multi-planar reformatted) views of the insect body illustrating internal structures; and (c) insect body depicting the tracheal system. These 3D images for Protocol 1 (highest noise; lowest X-ray dose), 3 (optimal), and 5 (lowest noise; highest X-ray dose) allow qualitative comparison of image quality (improving from left to right), noise (decreasing from left to right), and dose (increasing from left to right; see Additional files 7: Videos S1, Additional files 8: Videos S1 and Additional files 9: Videos S2, Additional files 10: Videos S2 for female and male adults, respectively). While the images from Protocol 5 are of the highest quality, the conspicuity of internal structures from the Protocol 3 images are sufficient and spare the individual the higher anesthetic and radiation dose
3D-CT reconstruction of live anesthetized female and male true armyworm adults at 1d and 4d demonstrating the feasibility of repeated scanning of the same individual longitudinally. Sagittal views of (a) 3D-volume of the female (♀) and male (♂) body exoskeleton at 20 μm isotropic voxel spacing; (b) 3D-MPR (multi-planar reformatted) views of the whole body showing the internal structures; and (c) the tracheal system. Protocol 3 was used to image the same individuals at days 1 (sexually immature) and 4 (sexually mature). For renderings of these serial scans, see Additional files 11: Videos S3, Additional files 12: Videos S3 and Additional files 13: Videos S4, Additional files 14: Videos S4 for 1and 4 d-old female adult, respectively; and Additional files 15: Videos S5, Additional files 16: Videos S5 and Additional files 17: Videos S6, Additional files 18: Videos S6 for 1and 4 d-old male adults, respectively
We observed that all CPB individuals (scanned and non-scanned; males and females) lived for approximately 30 d (except for scanned females that did not survive after Protocol 4). No difference was observed in the post-anesthetic longevity of scanned (9.0 ± 2.2 d) and control (10.3 ± 2.9 d) TAW females (t (6) = −0.70, p = 0.51; n = 4; t-test), or between scanned (12.3 ± 1.7 d) and control (11.0 ± 1.4 d) TAW males (t (6) = 1.13, p = 0.30; n = 4, t-test; Fig. 3). Additionally, these TAW adults took similar time (~ 40 min) to fully recover as those individuals that were exposed to CO2 for 3 h as shown below.
Post- anesthetic recovery of TAW
Regardless of age and exposure regime, longer exposure to CO2 gas hypoxia (Treatments 2, 4, 6, and 8) resulted in significantly longer recovery periods in TAW adults of both sexes (p < 0.0001; Fig. 4a). Individuals exposed for 7 h took approximately three times as long to fully recover post-anesthesia than those exposed for 3 h. This pattern of longer recovery for individuals with longer exposure times was similar for all other recovery parameters measured, including mean and minimum time to first observed movement, as well as time taken for the insects to flip onto their legs (natural position), and to walk in a coordinated fashion (Additional file 3: Figure S3).
Post-anesthetic recovery, longevity, and mating of true armyworms (TAW) adults after CO2 exposure. a Full recovery time; (b) Longevity ratio (ratio of 1 represents 10.9 and 13.3 d for females and males, respectively); (c) Incidence of mating. Bars represent mean ± SE for n = 10 individuals. Bars not sharing similar letters are statistically different (α =0.05)
Post- anesthetic longevity of TAW
The observed effect of CO2 gas hypoxia on longevity of TAW adults was not as pronounced as seen with the recovery parameters. In most cases, exposure to CO2 gas did not affect longevity of TAW adults. The only exception was Treatment 8, where the males exposed twice for 7 h did not live as long as the controls, or adults exposed for 3 h (Treatments 1, 3, 5, and 7) (p < 0.01, Fig. 4b; see Additional file 4: Figure S4a).
Post- anesthetic mating incidence of TAW
Overall, the mating of TAW adults was not affected after exposure to CO2 gas, except when mature and immature males were exposed once and twice for 7 h (Treatments 4 and 8), where incidence was lower compared to all other treatments (p < 0.05, Fig. 4c; see Additional file 4: Figure S4b).
Ranking of critical factors
Exposure time was the most influential factor affecting the recovery, longevity, and mating of TAW adults following CO2 exposure (Table 4). The second most important factors were age and sex of TAW adults. Age influenced only recovery time, but was 10 times less influential than exposure time. Sex impacted longevity ratio, but was 3 times less influential than exposure time. Sex also influenced mating incidence with almost the same influence as exposure time following TAW full recovery. Therefore, the immediate effects of CO2 anesthesia on the recovery of TAW adults were exposure time- and age-dependent for both males and females. In contrast, long-term physiological effects of CO2 anesthesia were exposure time- and sex-dependent, as observed for the life history parameters measured many days later.
Table 4 Ranking of factors based on their influence on recovery, longevity, and mating of TAW adults
Additionally, irrespective of the factor rankings, several interactions between different combinations of factors were observed. The recovery time, following CO2 gas exposure in male and female TAW, was affected differentially depending on the exposure regime (interaction: sex x exposure regime). Repeated exposures lead to longer recovery times for both sexes. Yet, the difference between recovery time following single and repeated exposures was larger for females than males.
Similarly, longevity of TAW males was negatively affected when sexually mature individuals (4d-old) were subjected to CO2 (interaction: sex x age), while no effects were observed with female TAW irrespective of age. The exposure regime had markedly different effect on longevity of TAW, depending on their age: sexually mature individuals (4d-old) had shorter lives than immature ones (1d-old) after repeated exposure to CO2 (interaction: age x exposure regime). Moreover, as exposure time was increased (from 3 to 7 h), there was a greater reduction in longevity of male compared to female TAW (interaction: exposure time x sex).
Individuals that were sexually mature (4-d old) during CO2 exposure were less successful at mating after experiencing repeated (interaction: age x exposure regime) and longer exposure (interaction: age x exposure time). No effects were observed in any treatment group when sexually immature (1-d old) individuals were exposed to hypoxia. Similarly, the mating success of males was negatively affected when subjected to longer (interaction: sex x exposure time) and repeated (interaction: sex x exposure regime) exposures, irrespective of their maturity at the time of exposure to CO2.
Multivariate analysis of TAW recovery ability and life-history traits post-anesthesia
Principal component analysis (PCA) with varimax rotation (KMO = 0.7 and χ2 (21) = 97.2; p < 0.001) revealed discriminatory patterns among the 8 treatments on the recovery, longevity, and mating of TAW adults post CO2 -induced hypoxia. The first two principal components (PC1 and PC2) accounted for 95.4% of the total variability (Fig. 5), and separated or grouped some treatments into pairs. Each pair was different from the others (Treatments 1 and 3, 5 and 7, 4 and 8, and 2 and 6) based on their influence on the subsequent recovery and life history traits post-anesthesia (Fig. 5a).
Principal component analysis (PCA) of TAW adult life history post CO2 anesthetic exposure. a PCA scores plot. Treatments that are grouped together produce similar effects on the recovery, longevity, and mating of true armyworm adults post-anesthesia. b PCA loadings plot. The effects of prolonged CO2 exposure on live insect physiology are temporal, being more prominent at the early stage of recovery. These effects lessen for processes occurring beyond recovery from post-CO2 in TAW adults. Recov_Sign = vitality sign (min); Tmin = Minimum time for first movement (min); Walk = walking time (min); Flip = Flipping time (min); Full_Recov = Full recovery time (min); Longe_R = Longevity ratio; and Mating_inci = Mating incidence (%)
Principal component 1 (PC1) accounted for 70.5% of the total variability (Fig. 5a), and clearly separated shorter (3 h) exposure time treatments (1, 3, 5 and 7) from longer (7 h) ones (2, 6, 4, and 8). From the PC1 loadings (Fig. 5b), this separation was due to differences in recovery times post-anesthesia, which is consistent with the statistical analysis of recovery times summarized in Fig. 4a.
Principal component 2 (PC2) accounted for 24.9% of the total variability and separated the treatments based on sex (male vs. female), and age (sexually immature vs. mature) after CO2 exposure. Examination of PC2 loadings (Fig. 4b) revealed maximum separation was caused mainly by low mating incidence linked to Treatments 4 and 8, which is consistent with the results shown in Fig. 4c. Therefore, any combinations of factors may be used in any TAW imaging study employing CO2 gas anesthetic, except those of Treatments 4 and 8, namely long exposures for male individuals (Table 3).
Our results indicate that incorporating hypoxia and/or hypercapnia to totally immobilize individuals during scanning is a suitable method for visualizing whole live insects using X-ray micro-CT, with little or no impact on their subsequent recovery, longevity and mating success when the optimal conditions are used. Moreover, this method is not only applicable to individuals at adult stage, it has also successfully enabled imaging of live individuals at larval and pupal stages (see Additional file 5: Figure S5). Where inherent tissue contrast is insufficient, non-toxic contrast agents can be further added to the imaging procedure for tissue differentiation enhancement, facilitating segmentation (see Additional file 6: Figure S6). Such contrast agent is not required for rendering and segmenting live insect exoskeleton from soft tissues (fat body and muscles) as reported here and illustrated in Figs. 2c and 3c.
This technique is compatible with various scanning modalities and its ease of implementation is important for its broad application in routine and reproducible scanning. One clear benefit is its application to any lab-based "bench top" scanner for studies using live insects. In spite of synchrotron beamlines being free (pending proposal acceptance) and capable of producing very high-resolution images, longitudinal studies on live insects remain very problematic at these facilities due to logistical reasons (in addition to very high X-rays radiation doses). Even if required regulatory permits are obtained, transportation over long distances can cause stress to the animals and there is a risk of introduction of invasive species or pest in new areas. These facilities offer little or no adequate rearing space to maintain the specimens used for repeated imaging for time-course studies.
In an imaging context, our findings suggest that hypoxic conditions induced by exposure to CO2 gas, at least with respect to the investigated critical factors, is not detrimental for the measured life history parameters (longevity and mating) of TAW (except for males exposed for longer durations). Hypoxia and /or hypercapnia induced by CO2 gas produces reversible loss of motor function in insects (due to lowered sensitivity in neuromuscular junction to glutamate) [28], and can have long-lasting effects on behaviour, including phase shifts in circadian rhythms [47] and life history traits [27]. For example, following CO2 anesthesia, Drosophila melanogaster not only exhibited reduced longevity, but also lowered fecundity, mating success, and motor function [31, 33, 48]. Similarly, negative effects on development, locomotion, and feeding were observed after CO2 induced anesthesia in Blattela germanica [32, 34]. However, provided that adequate recovery period is allowed, insect physiology following anesthesia usually returns to normal, with recovery proportional to the exposure duration [35, 42, 49], consistent with our findings on the recovery ability in TAW.
The observed reduced longevity and low mating success in TAW males that were exposed to longer durations can be related to body size and age, as seen for those individuals in Treatments 4 and 8. The maximum time that insects can withstand hypoxia is not only species dependent, but within species, females are more tolerant than males because of their larger body size [30, 50], as in the case of the TAW females compared to males. Larger body size usually indicates larger carbohydrate and fat reserves [51], and possibly allows faster resumption of normal metabolism. Moreover, this hypoxia tolerance declines with age for both sexes [30], which was observed in our study for repeatedly exposed males (compared to single CO2 exposures).
Regardless of which treatment we employed, the fact that all individuals were able to move their antennae and other appendages post anesthesia suggests no immediate central nervous system (CNS) impairment [30]. During the observed sequence of movement restoration in TAW, starting with extremities and joints followed by coordination of movements and eventual flight, the first two are linked directly to rapid adenosine triphosphate (ATP) re-synthesis [30] once oxygen becomes available again. This rapid ATP re-synthesis results from insects' ability to retain adenylates (products of ATP breakdown) in their tissues – in contrast to mammals. Additionally, because insects accumulate very little lactate during hypoxia exposure [30], they avoid the toxic effects of accumulated metabolic products of glycolysis. Combined together, mobility becomes possible again as coordination of CNS with muscles and metabolism are restored during recovery. This process is independent of the circulatory system because gaseous O2 flows easily through the tracheae to the brain, and other parts of the CNS, even before circulation is restored. Consequently, CNS dependent processes are preserved, irrespective of whether the organisms live to their full life-span or not, post hypoxia.
The ability to scan multiple individuals in a single scan acquisition, in order to maximize sample throughput, is compromised at higher resolutions, where the field of view is necessarily reduced. In the current study, scanning of multiple organisms simultaneously was possible at 20 μm, which is sufficient to visualize the exoskeleton, its invagination (tracheal system), and larger structures. Nevertheless, scanning of multiple individuals is compromised when higher resolution is needed for resolving very fine structures.
The application of CO2 anesthesia during micro-CT imaging for visualizing whole live insects is not limited to 20 μm resolution voxel spacing, presented here as a proof of concept. This method is applicable to studies where much higher resolution is achieved [5, 7, 9, 10, 52]. However, at higher resolution, higher radiation doses become more of a concern to the live individuals and moreover, additive effects between prolonged exposure to hypoxic conditions and higher ionizing radiation doses can be expected. In the current study, the scanned individuals received between 13 times (15.5 Gy for Protocol 5) and 345 times (0.58 Gy for Protocol 1) less radiation dose than the average required dose for insect sterilization [24, 53]. This dose was also between 22.5 to 597 times, and 84 to 2218 times less than that delivered by synchrotron monochromatic (~350 Gy) and polychromatic (~1300 Gy) beams, respectively. Although we cannot definitively rule out the combined effect of X-ray radiation and CO2 -induced hypoxia in our study, we suggest its impact to be minimal because the adult individuals received very low radiation doses during the imaging procedure under prolonged hypoxia exposure. Although the entrance doses were 1 to 2 orders of magnitude less that sterilization dose for adult insects, other life stages such as larval and pupal may have different susceptibility to radiation and CO2 anesthesia due to the rapid cellular changes that are occurring at these stages. Future comprehensive studies may wish to address this differential impact of radiation on different developmental stages (including adult) in combination with CO2 anesthesia in a longitudinal study within and across several generations of insects.
Finally, the application of CO2 gas anesthetic during live imaging can be used to discriminate among different tissues (e.g. muscles, fat body, and alimentary tract) at 20 μm isotropic voxel size to address a number of different questions that would give valuable insights into insect physiology and metabolism. Although we have only investigated the effect of CO2 anesthesia on adults, this method can be extendable to time-course developmental studies on different stages of development. For example, in studies of migratory moth adults, the technique could provide a non-destructive means of assessing the level of gonad development and lipid accumulation at different times of the year, or determining of mating status (the presence or absence of spermatophores) [54]. Currently, those aspects are invasively determined by dissection. Similarly, the technique could be employed to follow temporal anatomical change during different stages of larval and pupal development in applied entomology. Overall, depending on the measured end point of any experiment on live insects, various combinations of physical scan conditions and biological states of scanned insects become important during imaging as shown by the interactions of factors in our results. There may be other such factors of interest in other investigations, where interactions may exit and are either not obvious, or have subtle effects. To this end, taking full advantage of the OA approach is invaluable in interpretations and understanding of observations that are purely biological, and separated from those due to the imaging procedure. Thus, this method can be expanded to incorporate other factors such as life stages (larval, pupal, and adult) to minimize the impact of using the 6.2 Gy as optimal imaging protocol and CO2 anesthesia on different life history traits to ensure that longitudinal studies on insect morphology within and beyond several generations are not confounded.
Exposing insects to hypoxic conditions using CO2 gas, to ensure whole-body unrestricted immobilization, provides a new way to visualize live insects using X-ray micro-CT. 3D images were produced to distinguish in situ structures at 20 μm voxel spacing and at a relatively low radiation dose of less than 20 Gy. Repeated scanning of the same individuals is possible, allowing for time-course studies where live imaging is required. CO2 anesthesia can be incorporated with many other imaging modalities. Overall, depending on the experimental end points, there exist some combinations of critical factors under which insects can recover fully from hypoxia, without any apparent impact on their longevity and mating success. Since the method is simple to implement, routine time-course studies and other immediate widespread applications can be achieved, wherever live insect visualization is required, in applied entomology, such as medical entomology, insect development, and parasitology, to extract biologically-relevant information.
ANOVA:
ATP:
CNS:
CPB:
Sexually immature
KMO:
Kaiser-Meyer-Olkin
Sexually mature
micro-CT:
Micro-computed tomography
MPR:
Multi-planar reformatted
NMJ:
OA:
Orthogonal array
PCA:
Repeated exposure
Single exposure
Signal-to-noise
TAW:
True armyworm
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We thank I. Scott (Agriculture and Agri-Food Canada, London, Ontario) for providing the CPB adults.
Research funding was provided by the Natural Sciences and Engineering Research Council of Canada (Grants 04294 and 9551 to D.W.H and J.N.M, respectively).
Data acquired and materials used for this study are kept at the Preclinical Imaging Research Centre, Robarts Research Institute, Schulich School of Medicine and Dentistry, Western University. Datasets acquired and/or analysed during the current study can be made available from the corresponding author on reasonable request.
Preclinical Imaging Research Centre, Robarts Research Institute, Schulich School of Medicine and Dentistry, Western University, London, ON, Canada
Danny Poinapen
, Joseph U. Umoh
, Chris J. D. Norley
& David W. Holdsworth
Department of Biology, Western University, London, ON, Canada
Joanna K. Konopka
& Jeremy N. McNeil
London Research and Development Centre, Agriculture and Agri-Food Canada, London, ON, Canada
& Joanna K. Konopka
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DP, JKK, and DWH conceived the imaging method; DP, JKK, and JNM conceived the insect life history study; DP and JKK designed the set up, tested the anesthetic method, scanned the live insects, performed the OA experiments, and analysed the data; DP performed the image analysis; JUU and DP performed the noise quantification and analysis; CJDN and DP performed the dosimetry measurements and analysis. All authors discussed, commented, and contributed to writing the manuscript. All authors read and approved the final manuscript.
Correspondence to Danny Poinapen.
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3D reconstruction of a live 1wk-old anesthetized female CPB adult. Volume rendering of a live 1wk-old anesthetized adult female Colorado potato beetle (L. decemlineata) adult showing the exoskeleton, soft tissues, and tracheal system at 20 μm isotropic voxel spacing (micro-CT data acquired using Protocol 3). Note: The contraction or relaxation of muscles during anesthesia resulted in the ovipositor extrusion. (MOV 2903 kb)
3D reconstruction of a live 1wk-old anesthetized male CPB adult. Volume rendering of a live 1wk-old anesthetized adult male Colorado potato beetle (L. decemlineata) illustrating the exoskeleton, soft tissues, and tracheal system at 20 μm isotropic voxel spacing (micro-CT data acquired using Protocol 3). (MOV 2308 kb)
3D reconstruction of a live 1wk-old anesthetized male CPB adult. MPR rendering of a live 1wk-old anesthetized adult male Colorado potato beetle (L. decemlineata) showing the arrangement of in situ internal structures supported by the exoskeleton at 20 μm isotropic voxel spacing (micro-CT data acquired using Protocol 3). (MOV 711 kb)
3D reconstruction of a live 4d-old anesthetized female TAW adult. Volume rendering of live 4d-old (sexually mature) anesthetized adult female true armyworm (P. unipuncta) demonstrating the exoskeleton, soft tissues, and tracheal system at 20 μm isotropic voxel spacing (micro-CT data acquired using Protocol 3). (MOV 2574 kb)
3D reconstruction of a live 1d-old anesthetized male TAW adult. Volume rendering of a live 1d-old (sexually immature) anesthetized adult male true armyworm (P. unipuncta) showing the exoskeleton, soft tissues, and tracheal system at 20 μm isotropic voxel spacing (micro-CT data acquired using Protocol 3). (MOV 2565 kb)
3D reconstruction of a live 4d-old anesthetized male TAW adult. Volume rendering of live 4d-old (sexually mature) anesthetized adult male true armyworm (P. unipuncta) illustrating the exoskeleton, soft tissues, and tracheal system at 20 μm isotropic voxel spacing (micro-CT data acquired using Protocol 3). (MOV 2518 kb)
Set up for micro-CT live insect imaging and dosimetry. (I) Set-up for insect live scan. (A) Specially designed tube placed in a GE eXplore Locus RS-9 scanner containing fully-immobilized individuals by CO2 anesthesia (5 psi; 0.5 L/min). Anesthetized adults of (B) L. decemlineata (CPB) and (C) P. unipuncta (TAW). (II) Dosimetry measurement with the isocenter of an ionization chamber placed at insect level. (DOC 776 kb)
Set up for CO2 exposure of live insects. Set-up used for exposing live insects to CO2 gas. Because delivering CO2 gas at 5 psi (0.5 L/min) for long durations causes formation of ice crystals in the regulator, a 60 W lamp is directly shone on it to prevent frost occurrence. (DOC 845 kb)
Post-anaesthetic recovery of true armyworms (TAW) adults after CO2 exposure. (1) Vitality sign (average time taken for proboscis, antenna, and legs movement combined); (2) Minimum time taken for first movement (proboscis or antennae or legs); (3) Flipping time (when the anesthetised moths flip from back-laying (unnatural) to standing positions (natural)); and (4) Walking time (the start of coordinated walk). Bars represent mean ± SE for n = 10 individuals. Bars not sharing similar letters are statistically different (α =0.05). Data that followed a normal distribution were analysed using parametric methods (ANOVA followed by Tukey҆s HSD test); otherwise, non-parametric methods (Kruskal-Wallis Test followed by Dunn's test) were used. (DOC 653 kb)
Longevity (a), and mating (b) of CO2 exposed and control true armyworms (TAW) adults. Bars represent mean ± SE for n = 10 individuals. (DOC 448 kb)
3D-CT reconstruction of insects at larval and pupal stages. 3D reconstruction at 20 μm isotropic voxels of live anesthetized (I) P. unipuncta pupae, and (II) hornworn larva using Protocol 3. (DOC 437 kb)
Use of non-toxic iodinated contrast agents in live insect micro-CT imaging. 3D reconstruction (at 20 μm isotropic voxels) of live anesthetized females of: (I) 1 week-old L. decemlineata (Protocol 5); (II) 1d-old P. unipuncta (Protocol 3) showing sagittal views of (a) 3D-volume of exoskeleton; (b) 3D-MPR of whole body showing the internal structures; and (c) the tracheal system. The CPB individuals (I) were allowed to move freely in a plastic Petri dish containing 20% Lugolʾs solution (for faster staining) as contrast agent for 1 h. The Petri dish was placed in a slanted position so that the individuals could walk/swim without drowning. These stained insects were able to live and reproduce normally after this staining process and scanning. We did not quantify possible toxicity of this Lugol's solution to the organisms. The TAW adults (II) were fed 8% sugar water containing Omnipaque 350 contrast agent (1:11 v/v; can be adjusted for optimal contrast ratio). (DOC 1929 kb)
Additional file 7: Video S1a.
Additional file 8: Video S1b.
3D reconstruction of a live 1wk-old anesthetized female CPB adult. MPR rendering of a live 1wk-old anesthetized adult female Colorado potato beetle (L. decemlineata) demonstrating in situ internal structures arrangements supported by exoskeleton at 20 μm isotropic voxel spacing (micro-CT data acquired using Protocol 3). Note: The contraction or relaxation of muscles during anesthesia resulted in the ovipositor extrusion. (MOV 746 kb)
Additional file 10: Video S2b.
Additional file 11: Video S3a.
3D reconstruction of a live 1d-old anesthetized female TAW adult. Volume rendering of a live 1d-old (sexually immature) anesthetized adult female true armyworm (P. unipuncta) showing the exoskeleton, soft tissues, and tracheal system at 20 μm isotropic voxel spacing (micro-CT data acquired using Protocol 3). (MOV 2574 kb)
3D reconstruction of a live 1d-old anesthetized female TAW adult. MPR rendering of a live 1d-old (sexually immature) anesthetized adult female true armyworm (P. unipuncta) showing the arrangement of in situ internal structures supported by the exoskeleton at 20 μm isotropic voxel spacing (micro-CT data acquired using Protocol 3). (MOV 877 kb)
3D reconstruction of a live 4d-old anesthetized female TAW adult. MPR rendering of a live 4d-old (sexually mature) anesthetized adult female true armyworm (P. unipuncta) illustrating the arrangement of in situ internal structures supported by the exoskeleton at 20 μm isotropic voxel spacing (micro-CT data acquired using Protocol 3). (MOV 1308 kb)
3D reconstruction of a live 1d-old anesthetized male TAW adult. MPR rendering of a live 1d-old (sexually immature) anesthetized adult male true armyworm (P. unipuncta) demonstrating the arrangement of in situ internal structures supported by the exoskeleton at 20 μm isotropic voxel spacing (micro-CT data acquired using Protocol 3). (MOV 816 kb)
3D reconstruction of a live 4d-old anesthetized male TAW adult. MPR rendering of a live 4d-old (sexually mature) anesthetized adult male true armyworm (P. unipuncta) showing the arrangement of in situ internal structures supported by the exoskeleton at 20 μm isotropic voxel spacing (micro-CT data acquired using Protocol 3). (MOV 658 kb)
Poinapen, D., Konopka, J.K., Umoh, J.U. et al. Micro-CT imaging of live insects using carbon dioxide gas-induced hypoxia as anesthetic with minimal impact on certain subsequent life history traits. BMC Zool 2, 9 (2017). https://doi.org/10.1186/s40850-017-0018-x
Live insect imaging
X-ray micro-computed tomography
Orthogonal array design
Mating success
Leptinotarsa decemlineata
Pseudaletia unipuncta | CommonCrawl |
Romain Pétrides (Paris Diderot University) will speak in the geometry seminar on Tuesday 19 June, at 1.30pm in the Salle de Profs. Romain's title is Min-Max construction for free boundary minimal disks and his abstract is below.
We will discuss existence of minimal disks into a Riemannian manifold having a boundary lying on a specified embedded submanifold and that meet the submanifold orthogonally along the boundary. A general existence result has been obtained by A. Fraser. Her construction was inspired by Sacks-Uhlenbeck construction of minimal $2$-spheres : the existence is obtained by a limit procedure for a perturbed energy functional whose critical points are called $\alpha$-harmonic maps. We will explain how it is possible to adapt ideas of Colding-Minicozzi. These ideas go back to the replacement method of Birkhoff for the existence of geodesics. This approach gives general energy identities that include bubbles. This is a joint work with P. Laurain. | CommonCrawl |
The sum of the digits of a two-digit number is $13.$ The difference between the number and the number with its digits reversed is $27.$ What is the sum of the original number and the number with its digits reversed?
The two digit number can be represented as $10x + y,$ where $x$ and $y$ are digits, with $x \neq 0.$ We are given that the sum of the digits is $13,$ so $x + y = 13.$ If we reverse the digits of this number, we have $10y + x.$ We are given that the difference is $27,$ but we don't know if the original number or if the number with its digits reversed is greater. We can show this as such: $$|(10x + y) - (10y + x)| = 27.$$ However, it doesn't matter which of the two numbers is greater, since we wish to find their sum. So, without loss of generality, we will let the first number be the larger of the two. This means that $x > y,$ so we can get rid of the absolute values in our last equation to obtain $9x - 9y = 27,$ equivalent to $x - y = 3.$
We now have two equations in two variables: $x + y = 13$ and $x - y = 3.$ Adding the two, we obtain $2x = 16,$ so $x = 8.$ Subtracting, we obtain $2y = 10,$ so $y = 5.$ Thus, the original number is $85,$ and our answer is $85 + 58 = \boxed{143}.$
OR
As before, the two digit number can be expressed as $10x + y,$ and the number with its digits reversed is $10y + x.$ We want to find the sum of these two numbers, which is $$(10x + y) + (10y + x) = 11x + 11y = 11(x + y).$$ We are given that the sum of the digits is $13,$ so $x + y = 13.$ Since all we want is $11(x + y),$ we can substitute for $x + y$ to obtain our answer of $11\cdot 13 = \boxed{143}.$ | Math Dataset |
OSA Publishing > Optical Materials Express > Volume 10 > Issue 11 > Page 2952
Alexandra Boltasseva, Editor-in-Chief
3D printed multimode-splitters for photonic interconnects
Johnny Moughames, Xavier Porte, Laurent Larger, Maxime Jacquot, Muamer Kadic, and Daniel Brunner
Johnny Moughames,1,2 Xavier Porte,1,2,* Laurent Larger,1 Maxime Jacquot,1 Muamer Kadic,1 and Daniel Brunner1
1Institut FEMTO-ST, Université Bourgogne Franche-Comté CNRS UMR 6174, Besançon, France
2Equally contributing authors
*Corresponding author: [email protected]
Xavier Porte https://orcid.org/0000-0002-9869-7170
Maxime Jacquot https://orcid.org/0000-0003-0285-204X
Daniel Brunner https://orcid.org/0000-0002-4003-3056
J Moughames
X Porte
L Larger
M Jacquot
M Kadic
D Brunner
•https://doi.org/10.1364/OME.402974
Johnny Moughames, Xavier Porte, Laurent Larger, Maxime Jacquot, Muamer Kadic, and Daniel Brunner, "3D printed multimode-splitters for photonic interconnects," Opt. Mater. Express 10, 2952-2961 (2020)
Three-dimensional waveguide interconnects for scalable integration of photonic neural networks (OPTICA)
Tunable photonic devices by 3D laser printing of liquid crystal elastomers (OME)
Stitching-free 3D printing of millimeter-sized highly transparent spherical and aspherical optical components (OME)
Materials for Integrated Optics
Multimode interference
Two photon polymerization
Volume holography
Original Manuscript: July 16, 2020
Revised Manuscript: September 6, 2020
Manuscript Accepted: September 16, 2020
Optical Materials Express 3D Printing Enabled by Light and Enabling the Manipulation of Light (2020)
Design and fabrication
Optical characterization
Photonic waveguides are promising candidates for implementing parallel, ultra-fast and ultra-low latency interconnects. Such interconnects are an important technological asset for example for next generation optical routing, on and intra-chip optical communication, and for parallel photonic neural networks. We have recently demonstrated dense optical integration of multi-mode optical interconnects based on 3D additive manufacturing using two-photon-polymerization. The basis of such interconnects are 3D optical splitters, and here we characterize their performance against their splitting ratio, geometry, and conditions of the direct laser writing. Optical losses and splitting uniformity of 1 to 4, 1 to 9 and 1 to 16 splitters are evaluated at 632 nm. We find that, both, the uniformity of splitting ratios as well as the overall losses depend on the separation between the output waveguides as well as on the hatching distance (surface quality) of the 3D printing process.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Optical splitters are a fundamental component in the photonic toolbox. Splitting and combining optical waves is the indispensable ingredient of any interferometer, of delivering optical fiber-communication signals to various receivers, of linear programmable integrated photonics circuits [1] as well as of quantum-optical experiments [2]. Optical splitters are part of almost every free-space, fiber optic as well as integrated photonic system.
The device which potentially makes most heavy usage of optical splitting and combining is the optical interconnect [3], in which numerous input channels are connected to a typically comparable number of output channels. The fundamental appeal is exploiting the parallelism of photonics which promises to significantly reduce energy dissipation and latency, and therefore to mitigate some of the most disturbing limitations often encountered with electronic interconnects. Ignoring losses, an optical interconnect physically encodes a unitary connection or routing matrix, and the number of required optical mixers is proportional to the square of channels to be connected. Integration using 2D lithography, such as commonly the case in silicon-photonics, does therefore not scale in size [1,4,5] and current realizations are limited to around 10 input and output channels [6].
Recently, we have demonstrated how such optical splitters can be integrated in 3D [7]. Noteworthy, in 3D the integration of photonic interconnects or unitary vector-matrix multipliers is scalable in size. We realized connection topologies for efficient signal distribution using fractal-branching as well as spatial-filtering according to Haar-filters, and both were motivated by enabling fully parallel and scalable photonic neural networks. Integration is based on 3D additive fabrication via two-photon polymerization, which by now has been established as a flexible and robust photonic fabrication platform. It is a prolific approach for realizing free-form [8] and transformation [9] optical components, volume holograms [10], point-to-point photonic wire-bonding between optical components [8] and integrated photonic circuits in general [7,11].
Our 3D waveguides have been fabricated with a commercial Nanoscribe system. They are free-standing with a polymer-core without cladding, and our waveguides with a refractive index of $n_{0}\approx$1.54 [12] (index contrast $\Delta {n}\approx$0.5) support around 20 optical modes. We study branching from a single input into 4, 9 or 16 output ports and find considerably different intensity distribution between the individual output waveguides for the different splitting topologies. As most relevant parameters we identified the spacing between the regular array of output ports as well as the hatching distance, i.e. spacing between the neighboring writing voxels.
2. Design and fabrication
We realized splitters linking the single input to 4 ($2\times 2$), 9 ($3\times 3$) or 16 ($4\times 4$) output waveguides arranged in a square array with lattice distance $D_0$, and their design principle (cf. Figure 1(a-c)) facilitates horizontal multiplexing into large arrays as well as vertical stacking into layers based on fractal geometries [7]. At the bifurcation point the single input waveguide morphs into the numerous output waveguides, and the transition from the common input to the individual output ports is according to a $\sin$-function. Vertical stacking into fractal, scale-free branching arrangements connects an input waveguide to an exponentially increasing number of output waveguides [7]. This makes very efficient use of the circuit's volume for applications depending on high connectivity, as for example often the case in fully connected layers in the final stages of deep neural networks. The shape of our splitters exhibits chirality in order to avoid un-intentionally intersecting waveguides inside the tightly packed volume. All geometric details are defined in the supplemental information of [7].
Fig. 1. Optical splitters with different numbers of output ports spaced with distance $D_0$ were investigated. The height of each structure was kept constant at 52 µm, the waveguides' diameter is $\sim$1.2 µm. Their design and SEM micrographs are respectively depicted in: (a),(d) for $2\times 2$; (b),(e) for $3\times 3$; (c),(f) for $4\times 4$.
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Our splitters were fabricated with a commercial 3D direct-laser writing system from Nanoscribe GmbH (Photonic Professional GT). A negative tone photoresist "Ip-Dip" was dropped on a fused silica glass substrate (25x25x0.7 mm$^3$) and was photo-polymerized via two-photon absorption with a $\lambda =780~$nm femtosecond pulsed laser, focused by a 63X, (1.4 NA) microscope objective. We explored the impact of laser writing power $P$ and the hatching distance $h$ as well as waveguide separation $D_0$ at the output ports, which in turn modifies the angle at which the individual waveguides diverge at their branching points. The structures were printed in consecutive horizontal layers, superimposed in the vertical direction. The vertical distance between consecutive slices is $0.3$ µm. All waveguides had a diameter of $\sim 1.2$ µm and were written using the scanning mode based on a goniometric mirror with a constant scanning speed on the sample's surface (10 mm/s). After the writing process, samples were immersed in a PGMEA (1-methoxy-2-propanol acetate) solution for 20 minutes to remove the unexposed photoresist. Structural properties such as surface roughness of the resulting waveguides were visualized with a scanning electron microscope (SEM, Thermofisher APREO S, 5 kV, 45$^{\circ }$), and Fig. 1(d-f) shows SEM micrographs of splitters written with $P=$10.4 mW.
Arrays of polymer 3D optical splitters are robust from a mechanical point of view and survive to post-printing clean-up process without a problem [7]. However, individual free-standing splitters like those depicted in Fig. 1(d-f) do not always survive the fabrication process, mostly due to insufficient adhesion to the substrate resulting in unsticking by capillary forces arising during the evaporation of the developer and rinsing with liquids. We enhanced the sturdiness of our structures by placing them on a thin, $1$ µm high polymer pedestal fabricated during the same printing process. The thin plateaus resulted in the desired mechanical stability without causing a measurable influence upon the optical propagation properties. The approximated fabrication times for the different types of splitters are: 10 seconds for $2\times 2$ splitters, 20 seconds for $3\times 3$ splitters and 40 seconds for $4\times 4$ splitters.
3. Structural characterization
The average writing power $P$ directly impacts on the degree of polymerization [13,14] and the writing voxel's size. Hatching distance $h$, i.e. the spacing between neighbouring writing voxels, influences the polymerized material's homogeneity as well as surface roughness. They both therefore are important parameters as material inhomogeneity and surface roughness cause scattering and hence (i) induce energy transfer between different propagating modes and (ii) potentially increase optical losses. We scan two writing powers $P=\{10.4,11.2\}$ mW and two hatching distances $h=\{0.1,0.2\}$ µm. For our range of writing powers, the printed voxels have height and width of $\sim 1.2$ µm and $\sim 0.6$ µm, respectively. Mechanical stability is the reason behind the relatively small writing power range we evaluated, as for $P<10.4~$mW splitters were typically not stable or deformed, while $P>11.2~$mW regularly resulted in burning (micro explosion) of the monomer due to overexposure.
Figure 2 gives a schematic illustration of the 3D printing process and the hatching distance's impact upon the final surface quality. A pre-defined shape, in our case the optical splitter (c.f. Fig. 2(a)), is approximated through the accumulation of cigar-shaped writing voxels. The quality of this approximation is mostly governed by (i) the writing voxel's size, and (ii) by the spacing between the individual writing voxels, i.e. hatching distance $h$. We experimentally characterized the size of writing voxels at our writing conditions and obtained a height of 1.12 µm (1.17 µm) and diameter of 0.63 µm (0.64 µm) for a writing laser power of 10.4 mW ($P=$11.2 mW). The writing voxel's geometry therefore has impact upon the waveguides' crossection, an effect which furthermore depends on the local orientation of a waveguide relative to the writing voxel's symmetry axis. The possibility to generate 3D isotropic voxels is therefore an interesting strategy for improving homogeneity of such waveguides [15].
Fig. 2. The writing process of our splitters (a) (here a $3\times 3$ splitter with $D_0=14$ µm) is based on the accumulation of cigar-shaped writing voxels. The defined target-shape can be approximated with different resolution, mostly determined by the hatching distance $h$, i.e. the spacing between the writing voxels. (b,c) illustrate the effect of writing voxels spaced with $h=0.1$ µm and $h=0.2$ µm, respectively, showing a zoom into the red-region of (a). SEM micrographs illustrating the waveguide's surface roughness resulting from hatching distances $h=0.1$ µm, (d), and $h=0.2$ µm, (e). The white scale bar inside the SEM micrographs corresponds to $1$ µm.
The impact of hatching distance $h$ can be appreciated based on an illustration considering the minimal writing voxel dimension with 0.2 µm diameter and 0.5 µm height, and for approximating our target shape with $h=0.1$ µm (cf. Fig. 2(b)) and $h=0.2$ µm (cf. Fig. 2(c)). The resulting surfaces are undulated by a periodic structure of a period and amplitude depending on height $h$, and the nature of the undulation depends on the particular surface normal relative to the writing voxel's central axis. However, diffusion during the writing process additionally smoothens the resulting surfaces, and already decreasing the hatching distance from $h=0.2$ µm (cf. Fig. 2(b)) to $h=0.1$ µm (cf. Fig. 2(a)) results in significantly smoother interfaces. Besides the larger surface roughness for the bigger hatching distance, one can also identify a qualitative change: printing with $h=0.2$ µm causes a clear modulation of the surface profile with a period close to the hatching distance. This periodic impact is significantly less pronounced when printing with $h=0.1$ µm, which indicates that other processes start to dominate. In many 3D printing scenarios, such as for mechanical meta-materials, a hatching distance of $h=0.2$ µm is considered adequate [16,17]. However, surface roughness even on a sub-nanometer scale [18], in particular for periodic undulations [19] influences optical propagation, which suggests that hatching distance $h$ is of major importance for 3D printing optical waveguides.
4. Optical characterization
We optically characterized the 3D-printed polymer splitters at 632 nm, and the optical characterization setup is schematically depicted in Fig. 3. The single mode Gaussian output of a fiber-pigtailed laser diode was collimated with $\mathrm {MO}_{1}$ (Olympus PLN10X) and focused onto the waveguides' input facets by $\mathrm {MO}_{2}$, a 50X microscope objective with NA = 0.8 (Olympus MPLFLN50x). The polarization was set to circular by a linear polarizer and a λ/4 waveplate (Pol and λ/4 in Fig. 3). The FWHM of the focal spot ($1.3$ µm) corresponds to $\sim90\%$ of the input waveguide's diameter ($\simeq 1.2$ µm). The splitters' optical output was collected by $\mathrm {MO}_{3}$, a 10X microscope objective with NA=0.30 (Olympus LMPLN10XIR), and imaging using a achromatic lens (Thorlabs AC254-100-B-ML) with 100 mm focal length resulted in an optical magnification of 5.6 on the transmission camera CAM$_{\mathrm {T}}$. The sample's input side was imaged in reflection onto reflection camera CAM$_{\mathrm {R}}$ using a 100 mm focal length achromatic lens (Thorlabs AC254-100-B-ML), resulting in a magnification of 27.8. We used identical CMOS cameras (iDS U3-3482LE, pixel size 2.2 µm) for CAM$_{\mathrm {T}}$ and CAM$_{\mathrm {R}}$. A red LED (635 nm) creates a broad field of incoherent illumination for overall monitoring of the sample, whose position was precisely controlled with a piezo system (Thorlabs 3-Axis Nanomax MAX311D/M). We intentionally refrained from using the optical output characteristics of each splitter as alignment criteria. Instead, we aligned each splitter relative to the injection-laser focal point by maximizing the back-reflection of the input waveguide's top-facet using the reflection camera (CAM$_{\mathrm {R}}$). This created reproducible optical injection conditions, ensuring that we characterized each splitter's generic optical properties.
Fig. 3. Experimental setup for optical characterization, only illustrating the relevant optical components. A 635 nm LED was used as broad field illumination source for coarse positioning of the sample with a piezo nano-stage (not illustrated). A fiber coupled laser diode (LD) with 632 nm was used to measure the optical transmission of each splitter with circular polarization, which was controlled via a linear polarizer (Pol) and a $\lambda /4$ waveplate. Two CMOS cameras respectively imaged a splitter's output (CAM$_{\mathrm {T}}$) and the reflection from the top facet of its input waveguide (CAM$_{\mathrm {R}}$).
The intensity of the individual output waveguides as well as our transmission reference was calculated by integrating a $\simeq 4 \times 4$ µm2 area around each output waveguide. This area is slightly larger than the waveguides' diameter, which was necessary to compensate for inaccuracies in automatically determining each output waveguide's position. However, we ensured that this larger area did not influence our characterization: manual and more accurate definition of each output waveguides' position on a test-sample allowed integrating 4 times smaller areas, and results did not differ significantly from the automatized characterization. We obtained our reference intensity by focusing the optical injection on top of the glass substrate at an area without any polymer waveguides or pedestals, and total losses were calculated by summing the optical intensity of all output waveguides normalized to this reference intensity. Total losses therefore include injection, propagation and splitting losses. Previously, we measured 2.7 dB injection and $\approx$20 dB/mm propagation losses for waveguides with 1.2 µm diameter [7].
Total optical losses as well as the heterogeneity at the splitters' output ports were our primary concern. In our previous work [7] we found a quite heterogeneous intensity distribution across the output ports of a $3\times 3$ splitter, which we attributed to (i) the multimode waveguides, and (ii) to a non-optimized geometry of the bifurcations. Generally, one would desire an adiabatic transition from the single to the 'bundle' of waveguides, here we therefore modify the speed at which this transition takes place by printing splitters with a range of $D_{0}\in [10, 12,\dots , 18, 20]$ µm while keeping their total height constant at 52 µm. Our injection spot is a single mode Gaussian, and transferring optical intensity to higher orders relies on material heterogeneity as well as scattering at the rough and periodically undulated waveguide surface. We therefore expect surface roughness, i.e. hatching distance $h$, to impact upon total losses as well as the intensity distribution across the output ports.
5.1 Total losses
Figure 4 depicts total losses for $2\times 2$ splitters in (a,d), for $3\times 3$ splitters in (b,e) and for $4\times 4$ splitters in (c,f), with hatching distance $h=0.1$ µm as blue and $h=0.2$ µm as red data. The left column (a-c) correspond to splitters written with $P=10.4$ mW, the right column (d-f) to $P=11.2~$mW. The general trend indicates that total losses increase with separation $D_0$, with the strongest impact upon the $4\times 4$ splitter for which the total losses change from $\sim 5$dB to $\sim 11$dB. There is an exception for the $2\times 2$ splitters written with $h=0.1$ µm where we find no impact of $D_0$ for $P=10.4~$mW and even a slight reduction of losses for larger $D_0$ for $P=11.2~$mW. An additional general finding is the very small relevance of writing power $P$, and we can conclude that this parameter is mostly relevant for the mechanical integrity of such high aspect ratio photonic structures.
Fig. 4. Optical losses as a function of the separation between neighboring outputs $D_0$ and for hatching distances $h=0.1$ µm and $h=0.2$ µm as blue and red data, respectively. Panels respectively depict the total losses of (a,d) $2\times 2$ splitters, (b,e) $3\times 3$ splitters and (c,f) $4\times 4$ splitters. Data in panels (a-c) correspond to splitters written with $P=$10.4 mW and in panels (d-f) to splitters written with $P=$11.2 mW. Data with triangle symbols in panels (c,f) corresponds to the central $2\times 2$-array of the $4\times 4$-splitter. Comparison to (a) shows that these do not agree with the $2\times 2$ splitter, demonstrating that adding additional output ports influences the splitting ratios of the previous ports as well.
The transition from the single input to the bundle of output waveguides is the slowest, hence most adiabatic for the smallest output waveguide separation $D_0=10$ µm. Yet, lowest losses were obtained for $D_0=12$ µm$\dots$ 14 µm, in particular for the $3\times 3$ and $4\times 4$ splitters, which implies that the transition from the input waveguide to the bundle of output waveguides sensitively depends on the global geometry. This comes at no surprise as at any location waveguides support more than 20 modes; at the bifurcation points the number of supported modes is significantly larger. The principle at work for creating coupling is therefore comparable to the one employed in the multi-mode interference unit [20]. We evaluate this analogy by comparing the losses obtained when only considering the optical intensity contained inside the central $2\times 2$ output array for the $4\times 4$ splitters, and data are the blue and red triangles in Fig. 4(c,f) for $h=0.1$ µm and $h=0.2$ µm, respectively. Particular for $D_0=12$ µm $\dots 14$ µm results differ significantly from the $2\times 2$ splitter data shown in Fig. 4(a,d), and the presence of a $4\times 4$ splitter's outer ring of waveguides clearly influences the field transferred from the single input its central $2\times 2$ array. Furthermore, this discrepancy is amplified by increasing hatching distance $h$ to 0.2 µm, which indicates that scattering at the waveguides' sidewalls influences coupling and the details of multi-mode interference inside the splitter's bifurcation section.
5.2 Intensity distribution
In Fig. 5 we show the experimentally obtained relative intensity distribution for symmetry groups contained within the different splitters, and the particular structures are highlighted in black inside the inset of each panel. All splitters were printed with $P=10.4~$mW, and data for the $3\times 3$ splitters are given in the left column (cf. Figure 5(a-c)), data for the $4\times 4$ splitter in the right column (cf. Figure 5(d-f)). Data for $P=$11.2 mW perfectly agrees with the shown results and is therefore not shown. Equally, we do not show data for the $2\times 2$ splitters since these exhibit perfect symmetry and each output waveguide received a similar intensity on average. Transmission efficiencies given in Fig. 5 correspond to the integrated efficiency obtained for the output waveguides contained inside a particular group.
Fig. 5. Experimentally measured transmission ratios as a function of the separation between outputs. Panels respectively depict the fraction of transmission of (a,d) centers, (b,e) faces and (c,f) corners. Data from splitters with hatching distances of 0.1 and 0.2 are respectively plotted in blue and red. Data in panels (a,b,c) correspond to 3x3 splitters and in panels (d,e,f) to 4x4 splitters, both written with laser power $p=$10.4 mW.
The main difference between both splitters is the presence ($3\times 3$) or absence ($4\times 4$) of a straight connection between the input and the output waveguides, also compare Fig. 1(b) to Fig. 1(c) for a visualization. Figure 5(a) shows that this straight connection features the majority ($\sim 70\%$) of the $3\times 3$ splitter's optical output for the smoother waveguides obtained with $h=0.1$ µm and a large $D_0$, or for the rougher waveguides obtained with $h=0.2$ µm for a small $D_0$. A heuristic explanation for the first case can be given quite intuitively: for smooth side-walls there is less energy transfer from the single mode injection to higher order modes, and consequently most of the injection power traverses the splitter ending up inside the $3\times 3$ splitter's central output. As a direct consequence we find that the waveguides located at the sides of the $3\times 3$ output waveguide array (cf. Fig. 5(b)) exhibit an almost perfectly inverted behaviour. Finally, waveguides at the 4 corners (cf. Fig. 5(c)) are hardly influenced by the surface roughness.
We find that for the $4\times 4$ splitters (cf. Fig. 5(d)) the relative intensity transmitted to the central $2\times 2$ array quickly reaches its maximum for around $D_0\approx 14$ µm, where it remains at $\sim 90\%$ for all larger $D_0$. There is a certain impact of $h$, and the relative intensity of this central sub-group of waveguides unsurprisingly reaches its maximum faster for the smoother waveguides. As before, we find that this behaviour is almost perfectly inverted by the central waveguides in the second ring (cf. Figure 5(e)), however please note that here the intensity was obtained by integrating 8 output waveguides. As before, the final group of waveguides (cf. Fig. 5(f)) is hardly influenced by the surface roughness.
We characterized the optical properties of 2 photon polymerization 3D printed free standing, free standing 1 to 4, 1 to 9 and 1 to 16 waveguide splitters in detail. Such structures are relevant for a scalable and high density integration of optical interconnect [7]. We found that the optical writing power strongly influences the overall mechanical integrity of our splitters, while its impact upon the overall optical properties was rather small. The hatching distance $h$ and the separation between output waveguides $D_0$, on the other hand, has a non-trivial influence. Depending on the topology, a smaller $h$ (generally creating waveguides with smoother sidewalls) even results in higher losses.
Our results suggest that by adjusting surface roughness, i.e. hatching distance $h$, and distance $D_0$ one can to a certain degree homogenize the output of the individual splitters. This works best for the $3\times 3$ splitters, were for $h=0.1$ µm and $D_0=14$ µm the 5 waveguides arranged in a cross (see insets of Fig. 5(a,b)) exhibit an almost identical transmission ratio of $\sim 15\%$. The most even optical output intensity distribution was unsurprisingly found for the symmetric $2\times 2$ splitters. Combined with the strongly dominating central output of the $3\times 3$ splitters, this demonstrates the importance of output waveguide arrangements for achieving symmetric output intensity distributions. For higher connectivity one should therefore employ arrangements omitting a straight central waveguide and connections to output ports arranged at $\leq 2D_0$ from the centre. Besides for the square $2\times 2$ of a 1 to 4 splitter, this can be achieved with pentagon-arrangement for a 1 to 5 splitter, for a hexagon-arrangement for a 1 to 6 splitter, and so on.
Future version of such 3D splitters could achieve more deterministic splitting ratios be enabling single mode propagation to and from the bifurcation points, while mixing can then carefully be designed following the principles of the multi mode interference coupler [20] in the bifurcation section. Finally, detailed numerical simulations truthfully reproducing the impact of the waveguides' surface undulations are required.
H2020 Excellent Science (Marie Sklodowska-Curie 713694 (MULTIPLY).); Volkswagen Foundation (NeuroQNet I&II); Agence Nationale de la Recherche (ANR-15-IDEX-03, ANR-17-EURE- 0002).
The authors acknowledge the support of the Region Bourgogne Franche-Comté. This work was supported by the EUR EIPHI program (Contract No. ANR-17-EURE- 0002), by the Volkwagen Foundation (NeuroQNet I&II), by the French Investissements d'Avenir program, project ISITE-BFC (contract ANR-15-IDEX-03) and partly by the French RENATECH network and its FEMTO-ST MIMENTO technological facility. X.P. has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No 713694 (MULTIPLY).
The authors declare no conflicts of interest.
1. D. A. B. Miller, "Perfect optics with imperfect components," Optica 2(8), 747 (2015). [CrossRef]
2. J. Cariñe, G. Cañas, P. Skrzypczyk, I. Šupić, N. Guerrero, T. Garcia, L. Pereira, M. A. S. Prosser, G. B. Xavier, A. Delgado, S. P. Walborn, D. Cavalcanti, and G. Lima, "Multi-core fiber integrated multi-port beam splitters for quantum information processing," Optica 7(5), 542 (2020). [CrossRef]
3. H. Lee, X. Gu, and D. Psaltis, "Volume holographic interconnections with maximal capacity and minimal cross talk," J. Appl. Phys. 65(6), 2191–2194 (1989). [CrossRef]
4. J. Capmany, I. Gasulla, and D. Pérez, "Microwave photonics: The programmable processor," Nat. Photonics 10(1), 6–8 (2016). [CrossRef]
5. J. Wang, H. Shen, L. Fan, R. Wu, B. Niu, L. T. Varghese, Y. Xuan, D. E. Leaird, X. Wang, F. Gan, A. M. Weiner, and M. Qi, "Reconfigurable radio-frequency arbitrary waveforms synthesized in a silicon photonic chip," Nat. Commun. 6(1), 5957 (2015). [CrossRef]
6. Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljacic, "Deep Learning with Coherent Nanophotonic Circuits," Nat. Photonics 11(7), 441–446 (2017). [CrossRef]
7. J. Moughames, X. Porte, M. Thiel, G. Ulliac, L. Larger, M. Jacquot, M. Kadic, and D. Brunner, "Three-dimensional waveguide interconnects for scalable integration of photonic neural networks," Optica 7(6), 640 (2020). [CrossRef]
8. P. I. Dietrich, M. Blaicher, I. Reuter, M. Billah, T. Hoose, A. Hofmann, C. Caer, R. Dangel, B. Offrein, U. Troppenz, M. Moehrle, W. Freude, and C. Koos, "In situ 3D nanoprinting of free-form coupling elements for hybrid photonic integration," Nat. Photonics 12(4), 241–247 (2018). [CrossRef]
9. M. Kadic, G. Dupont, T.-M. Chang, S. Guenneau, and S. Enoch, "Curved trajectories on transformed metal surfaces: Beam-splitter, invisibility carpet and black hole for surface plasmon polaritons," Photonics Nanostructures-Fundamentals Appl. 9(4), 302–307 (2011). [CrossRef]
10. N. U. Dinc, J. Lim, E. Kakkava, C. Moser, and D. Psaltis, "Computer generated optical volume elements by additive manufacturing," Nanophotonics 9(13), 4173–4181 (2020). [CrossRef]
11. J. Pyo, J. T. Kim, J. Lee, J. Yoo, and J. H. Je, "3D Printed Nanophotonic Waveguides," Adv. Opt. Mater. 4(8), 1190–1195 (2016). [CrossRef]
12. S. Dottermusch, D. Busko, M. Langenhorst, U. W. Paetzold, and B. S. Richards, "Exposure-dependent refractive index of Nanoscribe IP-Dip photoresist layers," Opt. Lett. 44(1), 29 (2019). [CrossRef]
13. L. J. Jiang, Y. S. Zhou, W. Xiong, Y. Gao, X. Huang, L. Jiang, T. Baldacchini, J.-F. Silvain, and Y. F. Lu, "Two-photon polymerization: investigation of chemical and mechanical properties of resins using Raman microspectroscopy," Opt. Lett. 39(10), 3034 (2014). [CrossRef]
14. A. Žukauskas, I. Matulaitiene, D. Paipulas, G. Niaura, M. Malinauskas, and R. Gadonas, "Tuning the refractive index in 3D direct laser writing lithography: Towards GRIN microoptics," Laser Photonics Rev. 9(6), 706–712 (2015). [CrossRef]
15. W. Chu, Y. Tan, P. Wang, J. Xu, W. Li, J. Qi, and Y. Cheng, "Centimeter-Height 3D Printing with Femtosecond Laser Two-Photon Polymerization," Adv. Mater. Technol. 3(5), 1700396 (2018). [CrossRef]
16. L. Yang, A. Münchinger, M. Kadic, V. Hahn, F. Mayer, E. Blasco, C. Barner-Kowollik, and M. Wegener, "On the schwarzschild effect in 3D two-photon laser lithography," Adv. Opt. Mater. 7(22), 1901040 (2019). [CrossRef]
17. T. Frenzel, J. Köpfler, E. Jung, M. Kadic, and M. Wegener, "Ultrasound experiments on acoustical activity in chiral mechanical metamaterials," Nat. Commun. 10(1), 3384 (2019). [CrossRef]
18. X. Ji, F. A. S. Barbosa, S. P. Roberts, A. Dutt, J. Cardenas, Y. Okawachi, A. Bryant, A. L. Gaeta, and M. Lipson, "Ultra-low-loss on-chip resonators with sub-milliwatt parametric oscillation threshold," Optica 4(6), 619 (2017). [CrossRef]
19. A. Nicolet, F. Zolla, Y. O. Agha, and S. Guenneau, "Leaky modes in twisted microstructured optical fibers," Waves in Random and Complex Media 17(4), 559–570 (2007). [CrossRef]
20. L. Soldano and E. Pennings, "Optical multi-mode interference devices based on self-imaging: principles and applications," J. Lightwave Technol. 13(4), 615–627 (1995). [CrossRef]
Article Order
D. A. B. Miller, "Perfect optics with imperfect components," Optica 2(8), 747 (2015).
[Crossref]
J. Cariñe, G. Cañas, P. Skrzypczyk, I. Šupić, N. Guerrero, T. Garcia, L. Pereira, M. A. S. Prosser, G. B. Xavier, A. Delgado, S. P. Walborn, D. Cavalcanti, and G. Lima, "Multi-core fiber integrated multi-port beam splitters for quantum information processing," Optica 7(5), 542 (2020).
H. Lee, X. Gu, and D. Psaltis, "Volume holographic interconnections with maximal capacity and minimal cross talk," J. Appl. Phys. 65(6), 2191–2194 (1989).
J. Capmany, I. Gasulla, and D. Pérez, "Microwave photonics: The programmable processor," Nat. Photonics 10(1), 6–8 (2016).
J. Wang, H. Shen, L. Fan, R. Wu, B. Niu, L. T. Varghese, Y. Xuan, D. E. Leaird, X. Wang, F. Gan, A. M. Weiner, and M. Qi, "Reconfigurable radio-frequency arbitrary waveforms synthesized in a silicon photonic chip," Nat. Commun. 6(1), 5957 (2015).
Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljacic, "Deep Learning with Coherent Nanophotonic Circuits," Nat. Photonics 11(7), 441–446 (2017).
J. Moughames, X. Porte, M. Thiel, G. Ulliac, L. Larger, M. Jacquot, M. Kadic, and D. Brunner, "Three-dimensional waveguide interconnects for scalable integration of photonic neural networks," Optica 7(6), 640 (2020).
P. I. Dietrich, M. Blaicher, I. Reuter, M. Billah, T. Hoose, A. Hofmann, C. Caer, R. Dangel, B. Offrein, U. Troppenz, M. Moehrle, W. Freude, and C. Koos, "In situ 3D nanoprinting of free-form coupling elements for hybrid photonic integration," Nat. Photonics 12(4), 241–247 (2018).
M. Kadic, G. Dupont, T.-M. Chang, S. Guenneau, and S. Enoch, "Curved trajectories on transformed metal surfaces: Beam-splitter, invisibility carpet and black hole for surface plasmon polaritons," Photonics Nanostructures-Fundamentals Appl. 9(4), 302–307 (2011).
N. U. Dinc, J. Lim, E. Kakkava, C. Moser, and D. Psaltis, "Computer generated optical volume elements by additive manufacturing," Nanophotonics 9(13), 4173–4181 (2020).
J. Pyo, J. T. Kim, J. Lee, J. Yoo, and J. H. Je, "3D Printed Nanophotonic Waveguides," Adv. Opt. Mater. 4(8), 1190–1195 (2016).
S. Dottermusch, D. Busko, M. Langenhorst, U. W. Paetzold, and B. S. Richards, "Exposure-dependent refractive index of Nanoscribe IP-Dip photoresist layers," Opt. Lett. 44(1), 29 (2019).
L. J. Jiang, Y. S. Zhou, W. Xiong, Y. Gao, X. Huang, L. Jiang, T. Baldacchini, J.-F. Silvain, and Y. F. Lu, "Two-photon polymerization: investigation of chemical and mechanical properties of resins using Raman microspectroscopy," Opt. Lett. 39(10), 3034 (2014).
A. Žukauskas, I. Matulaitiene, D. Paipulas, G. Niaura, M. Malinauskas, and R. Gadonas, "Tuning the refractive index in 3D direct laser writing lithography: Towards GRIN microoptics," Laser Photonics Rev. 9(6), 706–712 (2015).
W. Chu, Y. Tan, P. Wang, J. Xu, W. Li, J. Qi, and Y. Cheng, "Centimeter-Height 3D Printing with Femtosecond Laser Two-Photon Polymerization," Adv. Mater. Technol. 3(5), 1700396 (2018).
L. Yang, A. Münchinger, M. Kadic, V. Hahn, F. Mayer, E. Blasco, C. Barner-Kowollik, and M. Wegener, "On the schwarzschild effect in 3D two-photon laser lithography," Adv. Opt. Mater. 7(22), 1901040 (2019).
T. Frenzel, J. Köpfler, E. Jung, M. Kadic, and M. Wegener, "Ultrasound experiments on acoustical activity in chiral mechanical metamaterials," Nat. Commun. 10(1), 3384 (2019).
X. Ji, F. A. S. Barbosa, S. P. Roberts, A. Dutt, J. Cardenas, Y. Okawachi, A. Bryant, A. L. Gaeta, and M. Lipson, "Ultra-low-loss on-chip resonators with sub-milliwatt parametric oscillation threshold," Optica 4(6), 619 (2017).
A. Nicolet, F. Zolla, Y. O. Agha, and S. Guenneau, "Leaky modes in twisted microstructured optical fibers," Waves in Random and Complex Media 17(4), 559–570 (2007).
L. Soldano and E. Pennings, "Optical multi-mode interference devices based on self-imaging: principles and applications," J. Lightwave Technol. 13(4), 615–627 (1995).
Agha, Y. O.
Baehr-Jones, T.
Baldacchini, T.
Barbosa, F. A. S.
Barner-Kowollik, C.
Billah, M.
Blaicher, M.
Blasco, E.
Brunner, D.
Bryant, A.
Busko, D.
Caer, C.
Cañas, G.
Capmany, J.
Cardenas, J.
Cariñe, J.
Cavalcanti, D.
Chang, T.-M.
Cheng, Y.
Chu, W.
Dangel, R.
Delgado, A.
Dietrich, P. I.
Dinc, N. U.
Dottermusch, S.
Dupont, G.
Dutt, A.
Englund, D.
Enoch, S.
Fan, L.
Frenzel, T.
Freude, W.
Gadonas, R.
Gaeta, A. L.
Gan, F.
Gao, Y.
Garcia, T.
Gasulla, I.
Gu, X.
Guenneau, S.
Guerrero, N.
Hahn, V.
Harris, N. C.
Hochberg, M.
Hofmann, A.
Hoose, T.
Huang, X.
Jacquot, M.
Je, J. H.
Ji, X.
Jiang, L.
Jiang, L. J.
Jung, E.
Kadic, M.
Kakkava, E.
Kim, J. T.
Koos, C.
Köpfler, J.
Langenhorst, M.
Larger, L.
Larochelle, H.
Leaird, D. E.
Lee, H.
Lee, J.
Li, W.
Lim, J.
Lima, G.
Lipson, M.
Lu, Y. F.
Malinauskas, M.
Matulaitiene, I.
Mayer, F.
Miller, D. A. B.
Moehrle, M.
Moser, C.
Moughames, J.
Münchinger, A.
Niaura, G.
Nicolet, A.
Niu, B.
Offrein, B.
Okawachi, Y.
Paetzold, U. W.
Paipulas, D.
Pennings, E.
Pereira, L.
Pérez, D.
Porte, X.
Prabhu, M.
Prosser, M. A. S.
Psaltis, D.
Pyo, J.
Qi, J.
Qi, M.
Reuter, I.
Richards, B. S.
Roberts, S. P.
Shen, H.
Shen, Y.
Silvain, J.-F.
Skirlo, S.
Skrzypczyk, P.
Soldano, L.
Soljacic, M.
Sun, X.
Šupic, I.
Tan, Y.
Thiel, M.
Troppenz, U.
Ulliac, G.
Varghese, L. T.
Walborn, S. P.
Wang, P.
Wang, X.
Wegener, M.
Weiner, A. M.
Wu, R.
Xavier, G. B.
Xiong, W.
Xu, J.
Xuan, Y.
Yang, L.
Yoo, J.
Zhao, S.
Zhou, Y. S.
Zolla, F.
Žukauskas, A.
Adv. Mater. Technol. (1)
Adv. Opt. Mater. (2)
J. Appl. Phys. (1)
J. Lightwave Technol. (1)
Laser Photonics Rev. (1)
Nanophotonics (1)
Nat. Commun. (2)
Nat. Photonics (3)
Opt. Lett. (2)
Optica (4)
Photonics Nanostructures-Fundamentals Appl. (1)
Waves in Random and Complex Media (1)
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Floyd's triangle
Floyd's triangle is a triangular array of natural numbers used in computer science education. It is named after Robert Floyd. It is defined by filling the rows of the triangle with consecutive numbers, starting with a 1 in the top left corner:
1
2 3
4 5 6
7 8 9 10
11 12 13 14 15
The problem of writing a computer program to produce this triangle has been frequently used as an exercise or example for beginning computer programmers, covering the concepts of text formatting and simple loop constructs.[1][2][3][4]
Properties
• The numbers along the left edge of the triangle are the lazy caterer's sequence and the numbers along the right edge are the triangular numbers. The nth row sums to n(n2 + 1)/2, the constant of an n × n magic square (sequence A006003 in the OEIS).
• Summing up the row sums in Floyd's triangle reveals the doubly triangular numbers, triangular numbers with an index that is triangular.[5]
1 = 1 = T(T(1))
1 = 6 = T(T(2))
2 + 3
1
2 + 3 = 21 = T(T(3))
4 + 5 + 6
• Each number in the triangle is smaller than the number below it by the index of its row.
See also
• Pascal's triangle
References
1. Keller, Arthur M. (1982), A first course in computer programming using PASCAL, McGraw-Hill, p. 39.
2. Peters, James F. (1986), Pascal with program design, Holt, Rinehart and Winston, pp. 137, 154.
3. Arora, Ashok; Bansal, Shefali (2005), Unix and C Programming, Firewall Media, p. 387, ISBN 9788170087618
4. Xavier, C. (2007), C Language And Numerical Methods, New Age International, p. 155, ISBN 9788122411744
5. Foster, Tony (2015), Doubly Triangular Numbers OEIS A002817.
External links
• Floyd's triangle at Rosetta code
| Wikipedia |
\begin{document}
\title{On generalized discrete PML optimized for propagative and evanescent waves}
\author{Vladimir Druskin, \thanks{Schlumberger-Doll Research} \and Murthy Guddati \thanks{North Carolina State University} \and Thomas Hagstrom, \thanks{Southern Methodist University}} \maketitle \begin{abstract} We suggest a unified spectrally matched optimal grid approach for finite-difference and finite-element approximation of the PML. The new approach allows to combine optimal discrete absorption for both evanescent and propagative waves. \end{abstract}
\section{Introduction} We approximate the Neumann-to-Dirichlet (NtD) map of wave problem in unbounded domain. After Fourier transform we obtain,
\be\label{eq1} u_{xx}- \lambda u=0, \qquad x\in [0,\infty] \ee and due to the infinity condition we are limited to outgoing wave solutions,
\[u=ce^{-\sqrt{\lambda}x}\] satisfying NtD condition,
\be\label{NtD}\frac{u}{u_x}|_{x=0}=-\frac{1}{\sqrt\lambda}.\ee Here $\lambda=\kappa^2-\omega^2,$ where $\kappa$ and $\omega$ are respectively (tangential) spacial and temporal frequencies. Also, (\ref{eq1}) can be equivalently rewritten in the first order form as,
\be\label{1st} u_x=sv, \ v_x=su,\ee where $s=\sqrt{\lambda}$. In terms of $u$ and $v$, condition (\ref{NtD}) can be equivalently rewritten as,
\be\label{NtD1st} \frac{u}{v}|_{x=0}=-1.\ee
The NtD can be numerically realized via rational approximation theory using several approaches \cite[etc]{Lind75, EM79, IDK, GT00, AsDrGuKn, GuLi06, ZaGu06, Hagstrom}. In \cite{IDK, AsDrGuKn} and \cite{GuLi06, ZaGu06} this approximant was realized as respectively finite-difference (FD) and finite-element (FE) discretization of an absorbing layer similar to well known Perfectly Matched Layer (PML) \cite{Berenger}. In particular, the FD scheme was designed as an optimal rational approximant separately for evanescent solutions corresponding to $\lambda\ge 0$ \cite{IDK} and propagative waves \cite{AsDrGuKn} corresponding to $\lambda<0$, but not for the both types of the solutions simultaneously. On the other hand, FE approach is more flexible; while \cite{GuLi06} focuses on propagative waves, it was shown in \cite{ZaGu06} that both propagative and evanescent waves can be treated simultaneously. Most recently, these FD and FE approximations are interpreted as special quadrature rules with complete wavefield approximation... \cite{Hagstrom}.
In this paper, we show that simultaneous treatment of propagative and evanescent waves is possible not only in FE setting, but also in FD setting. The key to this observation is a recently-discovered equivalence between the FE and FD approaches for the two-sided problem. Utilizing this link, we present two alternative approaches to implement the NtD map and comment on their relative merits. Furthermore, utilizing Zolotorev approximation theory and complete wavefield approximation interpretation, we present an NtD map that is an optimal approximation for propagating as well as evanescent waves.
The outline of the paper is as follows. We start in section 2 with the overview of optimal rational approximation of the NtD map by considering both propagative and evanescent waves. Section 3 contains the description of FE and FD approximations of two-sided problems and the equivalence between them. In section 4, we consider rational approximation of the NtD map of the exterior problem and present FE and FD realizations. The implementation details in time domain and relative merits of the (or three) approaches are considered in section 5. Numerical examples are presented in section 6. Finally, section 7 concludes the paper with some closing remarks. {\bf (where do we fit complete wavefield approximation of Tom?)}
\section{Optimal Rational Approximation of NtD map for Propagative and Evanescent Waves}
Let us for simplicity consider time-harmonic case with $\omega=1$ and consider time-dependent problems later. Let us present our rational approximant of $-\lambda^{-1/2}$ as
\be\label{p/q} -\lambda^{-1/2} \approx
R(\lambda)=p(\lambda)/q(\lambda), \ee where $p$ and $q$ are polynomials of degrees $K-1$ and $K$ respectively. Introducing a polynomial of degree $N=2K$ given by,
\[h(s)=sp(s^2)+q(s^2),\] with $s=\sqrt{\lambda}$, we transform (\ref{p/q}) to Newman function
\be\label{Newman}R(s^2)=p(s^2)/q(s^2)=\frac{h(s)-h(-s)}{s[h(s)+h(-s)]}
= \frac{-1+h(s)/h(-s)}{s[1+h(s)/h(-s)]}.\ee Then the relative error of the NtD map is approximately proportional to the reflection coefficient,
\[\frac{h(s)}{h(-s)}.\] According to (\ref{NtD1st}), the exact solution $(u,v)$ of
(\ref{1st}) is proportional to $(1,-1)$. In reality, due to the approximation error, $(u,v)|_0=c_1(1,-1)+c_2(1,1)$, and $\frac{c_2}{c_1}=\frac{h(s)}{h(-s)}$, i.e., the reflection coefficient is the ratio of the incoming and outgoing waves.
Minimization of $\frac{h(s)}{h(-s)}$ on a real positive interval is the classical first Zolotarev problem solved in 1872. Zolotarev's solution was first applied to the optimal FD approximation of the NtD map for evanescent solutions in \cite{IDK} and then to the approximation of propagative modes in \cite{AsDrGuKn}. The ABC for both propagative and evanescent waves should approximate the true NtD map on both negative $[-1,\lambda_1]$ and positive $[\lambda_2,\lambda_3]$ intervals. They respectively correspond to intervals $S_p=[\sqrt{-1}, \sqrt{\lambda_1}]$ and $[S_e=\sqrt{\lambda_2},\sqrt{\lambda_3}]$ of variable $s$. The so called spectrally matched finite-difference scheme (a.k.a FD Gaussian spectral rule or optimal FD grid) \cite[etc]{DrKn} allows arbitrary $h(s)$, but does not simultaneously treat propagative and evanescent waves. On the other hand, propagative and evanescent waves have been simultaneously treated using FE approximation in \cite{ZaGu06}. The specific approximation is based on linear FE approximation with
midpoint integration \cite{GuLi06}, which is linked to special rational approximation \cite{GT00} with
\be\label{droot}h(s)=t(s)^2,\ee where $t$ is a polynomial of degree $k$.\footnote{However, as it will be shown in the Section~4, simultaneous treatment of propagating and evanescent waves is even possible with more general $h(s)$, if the FD approach is used.} Hence, considering the success in \cite{ZaGu06}, we limit the current treatment to the restricted form of $h$ in (\ref{droot}). With such restriction, minimization of
$\max_{s\in S_e\cup S_p}\left|\frac{h(s)}{h(-s)}\right|$ is equivalent to solving
\be\label{zolc}\min_{\deg t\le k} \max_{s\in S_e\cup S_p} \left|\frac {t(s)}{t(-s)}\right |.\ee
It is well known that the necessary and sufficient conditions for optimality of a real rational approximant on a real interval is so-called the Equal Ripple Theorem (ERT) \cite{PP}. It says that the optimal error of [(K-1)/K] approximant has $2k-1$ zeros and $2k$ equal absolute value alternating extrema on the interval of optimality. Generally, there is no similar result for complex rational approximation \cite{Varga}. Here, instead of minimizing (\ref{zolc}), we construct an approximant based on classical Zolotarev results. We hope that its error is close to(\ref{zolc}).
If $t=t_et_p$, $\deg t_e=l<k$, $\deg t_p=k-l$, where $t_e$ and $t_p$ have respectively (non-coinciding) roots on $S_e$ and $S_p$, then
$\left| \frac {t(s)}{t(-s)}\right |$ has $2k+1$ maxima on $S_p\cup S_e$. Moreover, $\left| \frac {t_e(s)}{t_e(-s)} \right| = 1$ on
$S_p$ and $\left| \frac {t_p(s)}{t_p(-s)} \right| = 1$ on $S_e$, which implies that,
\[\max_{S_e}\left|\frac {t(s)}{t(-s)}\right |= \max_{S_e}\left|\frac {t_e(s)}{t_e(-s)}\right |, \] and
\[\max_{S_p}\left|\frac {t(s)}{t(-s)}\right |=\max_{S_p}\left|\frac {t_p(s)}{t_p(-s)}\right | .\]
Thus, we can take as $t_e$ and $t_p$ as the classical optimal Zolotarev approximants on $S_e$ and $S_p$ respectively, and obtain the quality of the total approximation the same as the one of the separate problems.
The remaining question is: can the constructed approximant be optimal in global sense, or, at least, how close is its error to
(\ref{zolc}). Obviously, $\max_{S_p}\left|\frac {t(s)}{t(-s)}\right
|$ and $\max_{S_e}\left|\frac {t(s)}{t(-s)}\right |$ may be different. Varying $l$ one can equate $\max_{S_p}\left|\frac
{t(s)}{t(-s)}\right |$ and $\max_{S_e}\left|\frac
{t(s)}{t(-s)}\right |$ for a countable set of arrays $\lambda_1,\lambda_2,\lambda_3$.
Here we conjecture that the ERT can be extended to the first Zolotarev problem on two intervals in $C$ in the following way. \begin{conjecture} Let $t_e(s)/t_e(-s)$ and $t_p(s)/t_p(-s)$ be the solutions of the Zolotarev problems on $S_e$ and $S_p$ respectively.
1. There are infinitely many arrays $\lambda_1,\lambda_2, \lambda_3$ for which there exists $l$, such that
\be\label{czol}\max_{S_e}\left|\frac {t_e(s)}{t_e(-s)}\right |=\max_{S_p}\left|\frac {t_p(s)}{t_p(-s)}\right |.\ee
2. If (\ref{czol}) is valid, then $t=t_et_p$ solves (\ref{zolc}). \end{conjecture} Results of \cite{leonid's reference} indicate that, if (\ref{czol}) is valid, then at
least the approximant is optimal in the Cauchy--Hadamard sense. Generally, it is always possible to find $l$ such that $\max_{S_p}\left|\frac {t(s)}{t(-s)}\right |$ and $\max_{S_e}\left|\frac {t(s)}{t(-s)}\right |$ are of the same order, in which case, it is natural to assume that the approximation error will be of the order of (\ref{zolc}).
\section{Equivalence of FE and FD Approximations for Two-sided Problems}
While the emphasis of this paper is on the approximation of the one-sided problem on $[0,\infty)$, in this section, we consider the two-sided problem on $[0,1]$ and show that there exist equivalence between spectrally matched FD grids and midpoint integrated linear FE mesh. We then utilize these results in Section 4 to construct an effective NtD map for the one-sided problem on $[0,\infty)$.
\subsection{Continuum problem}
{\bf QUESTION: You have used * for many row vectors and matrices. Should we be just using transpose?} {\color{blue} decide later}.
Let us consider eq. (\ref{1st}) on $[0,1]$, and define the two-sided DtN map as matrix-valued function $F(s)\in C^{2\times 2}$
\[F(s)u_b=v_b,\] where $u_b=[u(0),u(1)]^*$, $v_b=[v(0),-v(1)]^*$. It is easy to see that $(u,v)$ is a linear combination of,
\[(e^{\pm sx},\pm e^{\pm sx}),\] and simple computation shows that,
\be\label{impex} F(s)= \frac {1}{\sinh(s)}\left[\begin{array}{cc}\cosh(s) & -1 \\ -1&\cosh(s)\end{array}\right]=
Z\left[\begin{array}{cc}\tanh(s/2) & 0 \\ 0 & \coth(s/2)\end{array}\right]Z^* ,\ee where $Z$ is an orthogonal matrix
\[Z=\frac{1}{\sqrt 2}\left[\begin{array}{cc}1 &- 1 \\ 1 & 1 \end{array}\right].\] Similarly, we define propagator operator from left to right as matrix-valued function $G(s) \in C^{2\times 2}$ $G(s)w(0)=w(1)$, where $w=(u,v)^*$ and from (\ref{impex}) we obtain
\be\label{prop} G
= \left[\begin{array}{cc} \cosh(s) & \sinh(s) \\ \sinh(s) & \cosh(s) \end{array} \right]
= Z\left[\begin{array}{cc}\exp(s) & 0 \\ 0 & \exp(-s) \end{array} \right] Z^*.
\ee
\subsection{Discrete problem: linear FE mesh with midpoint rule}
It was shown in \cite{GuLi06} that the discretization of the original second-order from in (\ref{eq1}) with midpoint-integrated linear FE mesh would lead to exponential convergence of the NtD map. Furthermore, it was shown in \cite{GuDr07} that such a FE discretization is equivalent to Crank-Nicholson discretization of the first order form (\ref{1st}), i.e.
\be\label{1stfe} \frac{u_{i+1}-u_i}{l_i}
= s\frac{v_{i+1}+v_i}{2}, \ \frac{v_{i+1}-v_i}{l_i}
= s\frac{u_{i+1}+u_i}{2}, \qquad i=1,\ldots n.
\ee where $l_i$, $i=1,\ldots, n$ are the FE lengths with $\sum_{i=1}^nl_i = 1$. It can be easily verified that $(u_j,v_j)$, $j=1,\ldots,n$, is a linear combination of
\[\left(\prod_{i=1}^j\frac{1\pm l_is/2}{1\mp l_is/2},
\pm \prod_{i=1}^j\frac{1\pm l_is/2}{1\mp l_is/2}\right) .\] Comparing the above (approximate) solution with the exact solution and noting that $\sum_{i=1}^nl_i = 1$, the FE solution approximates the exponential as
\[\exp(s)\approx exp(s)= t(-s)/t(s),\]
where
\[t(s)=\prod_{i=1}^n(1- l_is/2),\] Assuming
\be\label{feimp} u(0) = u_1, \ u(1)=u_n,\quad
v(0)\approx v_1, \ v(1)\approx v_n,\ee we can compute the approximate NtD map as,
\be\label{impfe} \tilde F(s)
= \frac {1}{sinh(s)} \left[\begin{array}{cc}cosh(s) & -1 \\ -1 & cosh(s)\end{array}\right]
= Z\left[\begin{array}{cc}tanh(s/2) & 0 \\ 0 & coth(s/2)\end{array}\right]Z^* .
\ee Here,
\[sinh(s)=\frac{exp(s)- exp(-s)} {2}\approx \sinh(s),\qquad
cosh(s)=\frac{exp(s)+ exp(-s)} {2}\approx \cosh(s), \]
\[tanh(s/2)=\frac{t(s)-t(-s)}{t(s)+t(-s)}\approx \tanh(s/2), \qquad
coth(s)=1/tanh(s)\approx \coth(s).\] Similarly, the discrete propagator from left to right matrix can be computed as
\be\label{propfe} \tilde G
= \left[\begin{array}{cc}cosh(s) & sinh(s) \\ sinh(s) & cosh(s)\end{array}\right]
= Z\left[\begin{array}{cc}exp(s) & 0 \\ 0 & exp(-s)\end{array}\right]Z^* .\ee Vectors $\frac{1}{\sqrt 2}(1,\pm 1)$ are the eigenvectors of $\tilde G$, so it has so called fixed point property, i.e., if $u(0)/v(0)=\pm 1$ then $u(1)/v(1)=\pm 1$ and vice versa. This implies that, if exact half-space BC (\ref{NtD1st}) is applied at $x=0$, it will be also valid at $x=1$ regardless of the accuracy of the FE approximation. In other words, adding an FE-discretized interval to a half-space does not alter the NtD map of the half-space. Furthermore, it was shown in \cite{GuLi06} that adding a midpoint-integrated finite element to an approximate half-space can only decrease the approximation error in the NtD map. This property was used in \cite{ZaGu06} to enhance the approximation, originally designed for propagative waves, to simultaneously absorb evanescent waves.
\subsection{Discrete Problem: spectrally matched finite-difference grids} It was shown in \cite{} that one-sided, two-point BVP can be solved with staggered FD method with exponential convergence at the end points. The main idea was to link the staggered FD approximation to rational approximation of the exact NtD map and optimizing the resulting approximation using Zolotorev theory. This method was later extended to the solution of the two-sided problems by splitting the solution into odd and even parts and solving two one-sided problems on half-intervals using dual grids. Formerly called optimal FD grids, the basic idea of spectrally matched FD grids is summarized below.
Let us introduce the FD grid steps $\hat h_i,h_i$, $i=1,\ldots,k$. We split the DtN map into odd and even parts and compute each of them using a FD scheme on half interval. The odd and even problems can respectively be written in mutually dual form as:
\begin{eqnarray}\label{1stfd}
\frac {u^o_{i+1}-u^o_i}{h_i} = sv^o_i, \
\frac {v^o_{i}-v^o_{i-1}}{\hat h_i}=su^o_i, \qquad i=1,\ldots,k, \quad u_{k+1}=0, \\
\frac {u^e_{i+1}-u^e_i}{\hat h_i}=sv^e_i, \
\frac {v^e_{i}-v^e_{i-1}}{h_i}=su^e_i, \qquad i=1,\ldots,k, \quad v_{k+1}=0. \nonumber
\end{eqnarray} It is known \cite{DrKn} that,
\be \label{fraction} \frac{u_1^o} {v_1^o}=\frac{v_1^e}{u_1^e}=f_k(s)
= \cfrac{1}{\hat h_1s+
\cfrac{1}{h_1s+
\cfrac{1}{\hat h_2s+\dots
\cfrac{1}{h_{k-1}s+
\cfrac{1}{\hat h_{k}s+
\cfrac{1}{h_ks}}}}}}.
\ee Combining odd and even parts we obtain,
\be\label{oddeven} u(0) = u_1^e+u_1^o, \
u(1)=u_1^e-u_1^o,\quad
v_1^e\approx \frac{v(0)+v(1)}{2}, \
v_1^o\approx \frac{v(0)-v(1)}{2},
\ee and the FD-NtD as
\be\label{impfd}\hat F
= Z\left[\begin{array}{cc}f_k & 0 \\ 0 & 1/f_k\end{array}\right]Z^*.
\ee
Construction of spectrally matched grids involves a reverse procedure. First, rational approximation theory is used to obtain $f_k$ that approximates the NtD map. The resulting rational function is then used in (\ref{fraction}) to compute the grid steps $\hat h_i,h_i$ using simple **** algorithm \cite{}. **** algorithm also constructively shows that any [2k-1/2k] rational function can be converted into an equivalent FD grid.
\subsection{Equivalence of discrete problems} In this section, we show that if the number of finite elements are chosen to be even ($n=2k$), the approximate NtD maps from FE and FD grids are equivalent. \begin{lemma} For any set of parameters $l_i\in C$, $l=1,\ldots, 2k$ there exist parameters $\hat h_i, h_i\in C\cup\infty$, $l=1,\ldots, k$, such that
\be\label{equiv}f(s)\equiv tanh(s/2)\ee and vice versa. \end{lemma} \proof \ \ For any set of parameters $\hat h_i, h_i\in C\cup\infty$ there exist polynomials $p$ and $q$ (at most) degree $k-1$ and $k$ respectively, such that the continued fraction expansion (\ref{fraction}) can be presented as
\[f_k=\frac{sp(s^2)}{q(s^2)},\] and vice versa. Equating numerator and denominator of $f_k$ and $tanh$ we equivalently transform (\ref{equiv}) to polynomial identities
\be\label{polmatch} s p(s^2)\equiv t(s)-t(-s),\qquad q(s^2)\equiv t(s)+t(-s).\ee Since $p$, $q$ and $t$ can be arbitrary polynomials of degree $k-1$, $k$ and $2k$ respectively, then for any $p,q$ there is $t$ satisfying (\ref{polmatch}) and vice versa. {\bf (Aren't these different p and q? If so, it is important not to confuse the polynomials $p$ and $q$ with the polynomials in the second equation in section 2. Should be rename these as $\tilde{p}$ and $\tilde{q}$?)} {\color{blue} Agree, will do it later}
\hbox{\vrule\vbox{\hrule\phantom{o}\hrule}\vrule}
From the lemma, (\ref{impfe}) and (\ref{impfd}), we obtain the following result about equivalence of the FE and FD DtN maps. \begin{proposition} If (\ref{equiv}) is valid, then
\[\tilde F(s)\equiv \hat F(s).\] \end{proposition} Formula (\ref{polmatch}) can be used for computing the equivalent FE from the FD and vice versa.
If the DtN maps are identical, then formula (\ref{propfe}) can also be used for computing the propagator matrix for the FD approximation.
If $exp(s)$ matches $\exp(s)$ in $n$ non-coinciding frequencies, then $f_k$ matches $\tanh(s)$ at the same frequencies, and $f_k$ is Stieltjes function, $h_i,\hat h_i$ are real positive, and the problem becomes Hermitian.
\section{Approximation of exterior problems}
Let a discretized interval $\Omega_1=[x_-,x_+]$ have the propagator matrix (from left to right), \[{\cal \tilde G}=\left[\begin{array}{cc}
exp_1(s) & 0 \\ 0 & exp_1(-s) \end{array}\right]\] in the spectral coordinates, where $exp_1(s)=t_1(-s)/t_1(s)$ defined as in the previous section. First, let us impose the reflection coefficient $h_2(s)/h_2(-s)$ at $x_+$, i.e., at the right boundary any nontrivial solution can be represented as $w(x_+)=[ch_2(s),-ch_2(-s)]^*$ in the spectral coordinates, where $c\ne 0$ is an arbitrary constant. Then the reflection coefficient at the left boundary will be the ratio of the components of $w(x_-)={\cal \tilde G}^{-1} w(x_+)$.
That is, \be\label{prodref}
exp_1(-s)^2\frac{h_2(s)}{h_2(s)}=\frac{t_1(s)^2}{t_1(-s)^2} \frac{h_2(s)}{h_2(-s)}.\ee If we impose the Dirichlet condition at the right boundary of $\Omega_1$, which corresponds to $h_2(s)=1$, then the reflection coefficient will be $\frac{t_1(s)^2}{t_1(-s)^2}$. Let us now assume that we have a connected interval $\Omega = \Omega_1 \cup \Omega_2$ with the Dirichlet condition at the right boundary ($\Omega_2$ is assumed to be on the right), and $\frac {h_2(s)} {h_2(-s)}$ is the reflection coefficient of $\Omega_2$. Then (\ref{prodref}) would yield the reflection coefficient of $\Omega$ that is just the product of the reflection coefficients of the two subdomains.
Now, let as assume, that we use the discrete problem in $\Omega$ for the approximation of (\ref{NtD1st}), i.e., $h(s)$ from (\ref{Newman}) can be presented as
$h(s)= t_1(s)^2h_2(s)$.
If we set $h_1 \equiv t_1^2$ and $t_1 \equiv t_e$ and $t_2 \equiv t_p$, then the reflection coefficient of $\Omega$ will be identical to the one discussed in Section~2. However, Dirichlet condition on the right of $\Omega_2$ makes it a one-sided problem, and it is not necessary to restrict to the two-sided approximation in the previous section. In fact, the original FD optimal grids are optimized for the one-sided problems and can be used effectively for $\Omega_2$. This approximation is equivalent to the odd part of the FD approximation (\ref{1stfd}), i.e., \[\frac {u^o_{i+1}-u^o_i}{h_i} = sv^o_i, \
\frac {v^o_{i}-v^o_{i-1}}{\hat h_i}=su^o_i, \qquad i=1,\ldots,k, \quad u_{k+1}=0.\] Then $h_2$ can be obtained from the equality $f_k(s) = \frac {h_2(s)-h_2(-s)} {h_2(s)+h_2(-s)}$, i.e., it can be an arbitrary polynomial of degree $2k$. $\cal f $
\end{document} | arXiv |
\begin{document}
\title[The Daugavet property for
bilinear maps]{Slice continuity for operators and the Daugavet property for
bilinear maps}
\author{Enrique A. S\'anchez P\'erez and Dirk Werner}
\subjclass[2000]{Primary 46B04; secondary 46B25}
\keywords{Daugavet property, slice, bilinear maps}
\thanks{The authors acknowledge with thanks the support of the Ministerio de Econom\'{\i}a y Competitividad (Spain) under the research project MTM2009-14483-c02-02.}
\address{Instituto Universitario de Matem\'atica Pura y Aplicada, Universidad \linebreak Polit\'ecnica de Valencia, Camino de Vera s/n, 46022 Valencia, Spain.} \email{[email protected]}
\address{Department of Mathematics, Freie Universit\"at Berlin, Arnimallee~6, \qquad {}\linebreak D-14\,195~Berlin, Germany} \email{[email protected]}
\begin{abstract} We introduce and analyse the notion of slice continuity between operators on Banach spaces in the setting of the Daugavet property.
It is shown that under the slice continuity assumption the Daugavet equation holds for weakly compact operators. As an application we define and characterise the Daugavet property for bilinear maps, and we prove that this allows us to describe some $p$-convexifications of the Daugavet equation for operators on Banach function spaces that have recently been introduced. \end{abstract}
\maketitle
\thispagestyle{empty}
\section{Introduction}
A Banach space $X$ is said to satisfy the Daugavet property if the so-called Daugavet equation $$
\| \mathrm{Id} + R \| = 1 + \|R\| $$ is satisfied for every rank one operator $R{:}\allowbreak\ X \to X$. In recent years, the Daugavet property for Banach spaces has been studied by several authors, and various applications have been found (see for instance \cite{bulpolis2008,ams2000,studia2001,jfa1997,Dirk-IrBull}).
The aim of this paper is to introduce and analyse the notion of slice continuity between operators on Banach spaces. We will show that under this assumption one can easily characterise when the Daugavet equation holds for a couple of operators $T$ and $R$ between Banach spaces, i.e., when $$
\| T + R\| = \|T\| + \|R\|. $$
Recently, some new ideas have been introduced in this direction. The notion of Daugavet centre has been studied in \cite{bosen,bosen2,boseka}. According to Definition~1.2 in \cite{boseka}, a nonzero operator $T$ between (maybe different) Banach spaces is a Daugavet centre if the above Daugavet equation holds for every rank one operator~$R$. In this paper we develop a notion that is in a sense connected to this one but provides a direct tool for analysing when a particular couple of operators satisfies the Daugavet equation. Our idea is to relate the set of slices defined by each of the two operators. Recall that the slice $S(x', \varepsilon)$ of the unit ball of a real Banach space $X$ determined by a norm one element $x' \in X'$ and an $ \varepsilon >0$ is the set $$ S(x', \varepsilon)= \{ x \in B_X {:}\allowbreak\ \langle x, x' \rangle \ge 1- \varepsilon \}. $$
Let $Y$ be a Banach space. Let $T{:}\allowbreak\ X \to Y$ be an operator. We will define the set of slices associated to $T$ by $$
S_T:=\{S(T'(y')/\|T'(y')\|,\varepsilon) {:}\allowbreak\ 0< \varepsilon < 1, \ y' \in Y', \ T'(y') \ne 0 \}, $$ and we will say that an operator $R{:}\allowbreak\ X \to Y$ is slice continuous with respect to $T$ -- we will write $S_R \le S_T$ -- if for every $S \in S_R$ there is a slice $S_1 \in S_T$ such that $S_1 \subset S$. This notion will be used for characterising when the Daugavet equation holds by adapting
some of the known results on the geometric description of the Daugavet property to our setting. From the technical point of view, we use some arguments on the Daugavet property defined by subspaces of $X'$ that can be found in \cite{bulpolis2008}. This is done in Section~2. In Section~3 we develop the framework for using our results in the setting of the bilinear maps in order to obtain the main results of the paper regarding applications. Finally in Section~4 we provide examples and applications, mainly related to a unified general point of view to understand the $p$-convexification of the Daugavet equation for Banach function spaces that have recently been studied in \cite{EnrDir2,EnrDir3}.
Our notation is standard.
Let $X,Y$ and $Z$ be real Banach spaces. $B_X$ and $S_X$ are the unit
ball and the unit sphere of $X$, respectively. We write $U_X$ for the
open unit ball and $X'$ for the dual space. We denote by $L(X,Y)$ the space of continuous operators and by $B(X \times Y, Z)$ the space of continuous bilinear maps from $X \times Y$ to $Z$. If $T$ is an operator, we write $T'$ for its adjoint operator. If $x' \in X'$ and $y \in Y$, we identify the tensor $x' \otimes y$ with the operator $x' \otimes y{:}\allowbreak\ X \to Y$ mapping $x$ to $x'(x)y$. Throughout the paper all the
bilinear maps are assumed to be continuous. In general, we consider the norm $\|(x,y)\|=\max\{\|x\|_X, \|y\|_Y \}$ for the direct product $X \times Y$. We will say that a bilinear map $B$ is convex or has convex range (resp.\ is weakly compact) if the norm closure of $B(B_X,B_Y)$ is convex (resp.\ weakly compact).
Regarding Banach function spaces we also use standard notation. If $1 \le p \le \infty$ we write $p'$ for the extended real number satisfying
$1/p+1/p'=1$. Let us fix some definitions and basic results.
Let $(\Omega, \Sigma, \mu)$ be a measure space. A Banach function space $X(\mu)$ over the measure $\mu$ is an order ideal of $L^0(\mu)$ (the space of $\mu$-a.e.\ equivalence classes of integrable functions) that is a Banach space with a lattice norm $\| \,.\, \|$ such that for every $A \in \Sigma$ of finite measure, $\chi_A \in X(\mu)$ (see \cite[Def.~1.b.17]{lint}). We will write $X$ instead of $X(\mu)$ if the measure $\mu$ is clear from the context.
Of course, Banach function spaces are Banach lattices, so the following definition makes sense for these spaces. Let $0 < p \le \infty$. A Banach lattice $E$ is $p$-convex if there is a constant $K$ such that for each finite sequence $(x_i)_{i=1}^n$ in $E$, $$
\Big\| \Big( \sum_{i = 1}^n |x_i|^p \Big)^{1/p} \Big\|_E
\le K \Big(\sum_{i=1}^n \|x_i\|_E^p \Big)^{1/p}. $$ It is said that it is $p$-concave if there is a constant $k$ such that for every finite sequence $(x_i)_{i=1}^n$ in $X$, $$
\Big(\sum_{i=1}^n \|x_i\|_E^p \Big)^{1/p} \le k \Big\| \Big(
\sum_{i=1}^n |x_i|^p \Big)^{1/p} \Big\|_E. $$ $M^{(p)}(E)$ and $M_{(p)}(E)$ are the best constants in these inequalities, respectively.
Let $0 \le p < \infty$. Consider a Banach function space $X(\mu)$. Then the set $$
X(\mu)_{[p]}:=\{h \in L^0(\mu){:}\allowbreak\ |h|^{1/p} \in X(\mu) \} $$ is called the $p$-th power of $X(\mu)$, which is a quasi-Banach function space when endowed with the quasinorm
$\|h\|_{X_{[p]}}:=\| |h|^{1/p} \|^p_X$, $h \in X^p$ (see \cite{defant}, \cite{maligranda-persson,CDS} or \cite[Ch.~2]{libro}; the symbols that are used there for this concept are $X^p$, $X^{1/p}$ and $X_{[p]}$, respectively); if $X$
is $p$-convex and $M^{(p)}(X(\mu))=1$ -- we will say that $X$ is constant~$1$ $p$-convex --, then $X(\mu)_{[p]}$ is a Banach function space, since in this case $\|\,.\,\|_{X_{[p]}}$ is a norm; if $0<p<1$, the $p$-th power of a Banach function space is always a Banach function space. Every $p$-convex Banach lattice can be renormed in such a way that the new norm is a lattice norm with $p$-convexity constant~$1$ (\cite[Prop.~1.d.8]{lint}). Let $f \in X$. Throughout the paper we use the notation $f^p$ for the sign preserving $p$-th power of the function $f$, i.e., $f^p:= \loglike{sign}\{f\}
|f|^p$.
\begin{remark} \label{rembasic} The following basic facts regarding $p$-th powers of Banach function spaces will be used several times. Their proofs are immediate using the results in \cite[Ch.~2]{libro}. Let $X(\mu)$ be a Banach function space and $0<p<\infty$. \begin{itemize}
\item[(a)] For every couple of functions $f \in X$ and $g \in X_{[p/p']}$ one has $\|fg \|_{X_{[p]}} \allowbreak \le \|f\|_X \|g\|_{X_{[p/p']}}$, and $$
\|h\|_{X_{[p]}}= \inf \{ \|f\|_X \|g\|_{X_{[p/p']}}{:}\allowbreak\ fg=h, \ f \in X, \ g \in X_{[p/p']} \}. $$
\item[(b)] For every $h \in X_{[p]}$ one has $h= |h|^{1/p} h^{1/p'}$, $h^{1/p} \in X$, $h^{1/p'} \in X_{[p/p']}$, and \begin{eqnarray*}
\|h\|_{X_{[p]}} &=& \| h^{1/p}\|_X^p = \|h^{1/p} \|_X
\|h^{1/p}\|_X^{p/p'} \\
&=& \|h^{1/p} \|_X \|(h^{1/p'})^{p'/p}\|_X^{p/p'}=\|h^{1/p} \|_X
\|h^{1/p'} \|_{X_{[p/p']}}. \end{eqnarray*} \end{itemize} \end{remark}
\section{Slice continuity for couples of linear maps}
Let us start by adapting some facts that are already essentially well known (see \cite{ams2000}).
\begin{proposition} \label{geom0} Let $X$ and $Y$ be Banach spaces. Let $T{:}\allowbreak\ X \to Y$ be a norm one linear map, and consider a norm one linear form $x' \in X'$. Let $y \in Y \setminus \{0\}$. The following assertions are equivalent. \begin{itemize}
\item[(1)] $ \| T+ x' \otimes y\|=1 + \|x' \otimes y\|= 1 + \|y\|.$
\item[(2)] For every $\varepsilon >0$ there is an element $x \in S(x',\varepsilon)$ such that $$
\Bigl\| T(x) + \frac{y}{\|y\|} \Bigr\| \ge 2- 2 \varepsilon. $$ \end{itemize} \end{proposition} \begin{proof}
(1)$\Rightarrow$(2). By Lemma~11.4 in \cite{abraali0} (or
\cite[p.~78]{Dirk-IrBull}) we can assume that $y \in S_Y$. By hypothesis, $\|T+ x' \otimes y\|= 1 + \|y\|=2$, and then there is an element $x \in B_X$ such that $$
2- \varepsilon \le \| T(x)+ \langle x, x' \rangle y)\| \le
\|T(x)\|+|\langle x,x' \rangle| \le 1 + | \langle x,x' \rangle|. $$ Note that we can assume that $\langle x,x' \rangle >0$; otherwise
take $-x$ instead of $x$. Since for every $\varepsilon>0$
\begin{eqnarray*} 2- \varepsilon &\le& \|T(x) + \langle x, x' \rangle y\| \le \|T(x)
+y\| + \|\langle x, x' \rangle y -y\| \\
&\le& \|T(x) +y\| + (1- \langle x, x' \rangle)\|y\| \le \|T(x) + y\| + \varepsilon, \end{eqnarray*} we obtain~(2).
(2)$\Rightarrow$(1). Let $x' \in S_{X'}$ and $y \in Y$ and consider the rank one map $x' \otimes y$. Again by Lemma~11.4 in
\cite{abraali0} we need consider only the case $\|y\|=1$. Let $ \varepsilon >0$. Then there is an $x \in S(x',\varepsilon)$ such that
$\|y+ T(x)\| \ge 2- 2\varepsilon.$ Thus,
\begin{eqnarray*} 2- 2\varepsilon &\le& \| y + T(x)\| \le \|y- \langle x,x' \rangle y\| + \|\langle x,x' \rangle y+ T(x)\| \\
&\le& (1- \langle x,x' \rangle)\|y\| + \|\langle x, x' \rangle y+
T(x)\| \le \varepsilon + \|x' \otimes y +T\|.
\end{eqnarray*} Consequently, $\|x' \otimes y\|+ \|T\| = 2 =\|x' \otimes y +T\|$. \end{proof}
When a subset of linear maps $V \subset L(X,Y)$ is considered, the following generalisation of the Daugavet property makes sense.
\begin{definition} Let $X,Y$ be Banach spaces and let $T{:}\allowbreak\ X \to Y$ be a norm one operator. The Banach space $Y$ has the $T$-Daugavet property with respect to $V \subset L(X,Y)$ if for every $R \in V$, $$
\|T + R\|=1 + \|R\|. $$ \end{definition}
This definition encompasses the notion of Daugavet centre given in Definition~1.2 of \cite{boseka}.
\begin{corollary} \label{geom20} Let $X$ and $Y$ be Banach spaces. Let $T{:}\allowbreak\ X \to Y$ be an operator, and consider a set of norm one linear forms $W \subset X'$. Let $W \cdot Y=\{x' \otimes y{:}\allowbreak\ x' \in W,\ y \in Y \}$. The following statements are equivalent. \begin{itemize} \item[(1)] $Y$ has the $T$-Daugavet property with respect to $W \cdot Y$.
\item[(2)] For every $y \in S_Y$, for every $x' \in W$ and for every $\varepsilon >0$ there is an element $x \in S(x',\varepsilon)$ such that $$
\| T(x)+y\| \ge 2- 2 \varepsilon. $$
\end{itemize} \end{corollary}
\begin{definition} Let $T{:}\allowbreak\ X \to Y $ be a continuous linear map. Let $y' \in Y'$.
We denote by $T_{y'}{:}\allowbreak\ X \to \mathbb R $ the linear form given by $T_{y'}(x):= \frac{ \langle x,T'(y')
\rangle}{\|T'(y')\|}$ whenever $T'(y') \ne 0$. The natural set of slices defined by $T$ is then $$ S_T=\{S({T_{y'}}, \varepsilon){:}\allowbreak\ 0<\varepsilon < 1, \ y' \in Y', \ T'(y') \ne 0 \}. $$ If $R{:}\allowbreak\ X \to Y$ is another operator, we use the symbol $S_{R} \le S_{T}$ to denote that for every slice $S$ in $S_{R}$ there is a slice $S_1 \in S_{T}$ such that $S_1 \subset S$. We will say in this case that $R$ is slice continuous with respect to $T$. \end{definition}
For operators $T$ having particular properties, slice continuity allows easy geometric descriptions. Let $T{:}\allowbreak\ X \to Y $ be an operator between Banach spaces such that $T'$ is an isometry onto its range, i.e., $T$ is a quotient map, and let $R{:}\allowbreak\ X \to Y$ be an operator. The following assertions are equivalent. \begin{itemize} \item[(1)]
$S_R \le S_{T}.$ \item[(2)] For every $y \in S_Y$, $y' \in S_{Y'}$ such that $R'(y') \ne 0$, and every $\varepsilon >0$ there is an element $y_0' \in S_{Y'}$ such that $ (R_{y'} \otimes y)(S(T'(y_0'),\varepsilon)) \subset B_\varepsilon(y). $ \end{itemize}
To see this just notice that for every $y' \in S_{Y'}$ such that $R'(y') \ne 0$ and $y \in S_Y$ $$ S( R_{y'},\varepsilon)= \{x \in B_X{:}\allowbreak\ 1-\varepsilon \le R_{y'}(x)
\le 1\} = \{x \in B_X{:}\allowbreak\ \|R_{y'}(x)y-y\| \le \varepsilon \}. $$
For a general operator $T$ the canonical example of when the relation $S_{R} \le S_{T}$ holds is given by the case
$R=P \circ T$, where $T{:}\allowbreak\ X \to Y$ and $P{:}\allowbreak\ Y \to Y$ are operators. In this case, $\langle R(x), y' \rangle= \langle x, T'(P'(y')) \rangle$, and so clearly $S_R \le S_{T}$. So the reason is that we have the inclusion $R'(Y') \subset T'(Y')$. However, there are examples of couples of operators $T,R$ such that $R$ is slice continuous with respect to $T$ but $R \ne P \circ T$ for any operator $P$. Let us show one of them.
\begin{example} Let $T{:}\allowbreak\ C[0,1]\oplus_1 \mathbb R \to C[0,1]$, $T(f,\alpha)=f$, and $R{:}\allowbreak\ C[0,1] \oplus_1 \mathbb R \to C[0,1]$, $R(f,\alpha)=f+\alpha \mathbf 1$, where $\mathbf{1}$ stands for the constant one function and $\oplus_1 $ denotes the direct sum with the $1$-norm. Then $R$ and $T$ have norm one. Since the kernel of $T$ is not contained in the kernel of $R$, we do not have $R= P
\circ T$ for any operator $P$. But the slice condition holds. A simple calculation gives that for every $\mu$ in the unit sphere of $C[0,1]^*$, $\|T'(\mu)\|=\|R'(\mu)\|=1$. Let $S_r \in S_R$ be the slice generated by any $\mu\in C[0,1]^*$ of norm one and $\varepsilon>0$. We claim that the slice $S_t$ generated by the same $\mu$ and $\varepsilon/2$ is contained in $S_r$. Indeed, if
$(f,\alpha)$ is in the unit ball and $\langle\mu, f\rangle \ge 1-\varepsilon/2$, then $\|f\|\ge 1-\varepsilon/2$ and hence
$|\alpha | \le \varepsilon/2$. Therefore, for such $(f,\alpha)$, $\langle\mu, f+\alpha\mathbf 1 \rangle \ge \langle\mu, f\rangle -
|\alpha| \ge 1-\varepsilon$, and so the inclusion $S_t \subset S_r$ holds.
\end{example}
\begin{remark} \label{re10}
Let $T,R{:}\allowbreak\ X
\to Y$ be a couple of operators, $\|T\|=1$. Notice that
Proposition~\ref{geom0} gives that for every $y \in S_Y$ and $y' \in Y'$ such that $R'(y') \ne 0$, the following are equivalent. \begin{itemize}
\item[(1)] $ \| T+ R_{y'} \otimes y\|=2.$
\item[(2)] For every $\varepsilon>0$ there is an element $x \in S( R_{y'},\varepsilon)$ such that $$
\| T(x) + y \| \ge 2- 2 \varepsilon. $$ \end{itemize}
Thus for the case $R=T$ and assuming that $T'$ is an isometry onto its range we obtain that $Y$ has the Daugavet property if and only if $Y$ has the $T$-Daugavet property with respect to the set $\{T_{y'}{:}\allowbreak\ y' \in Y' \setminus \{ 0\} \} \cdot Y$. This is a direct consequence of the well-known characterisation of the Daugavet property (see Lemma~2.1 in \cite{ams2000}) and Corollary~\ref{geom20}. Consequently, for any other $R$, if $Y$ has the Daugavet property and $S_R \le S_{T}$, we obtain that for every $y \in S_Y$ and every $y' \in Y'$ such that $R'(y') \ne 0$, $$
\|T + R_{y'} \otimes y\|=2. $$
Note that something like the slice continuity requirement $S_R \le S_{T}$ is necessary for this to be true; indeed, a quotient map $T{:}\allowbreak\ X \to Y$ is not necessarily a Daugavet centre, even if the spaces involved have the Daugavet property. Take the operators $T,R{:}\allowbreak\ L^1[0,1] \oplus_1 L^1[1,2] \to L^1[0,1]$ given by $T((f,g)):=f$ and $R((f,g)):=(\int_1^2 g \,dx) \cdot h_0$, $(f,g) \in L^1[0,1] \oplus_1 L^1[1,2]$, where
$h_0$ is a norm one function in $L^1[0,1]$. Clearly, $\|T\|=\|R\|=1$, but $\|T+R\| \le 1$. \end{remark}
\begin{theorem} \label{weakcom0} Let $Y$ be a Banach space with the Daugavet property. Let $T{:}\allowbreak\ X \to Y$ be an operator such that $T'$ is an isometry onto its range and $R{:}\allowbreak\ X \to Y$ a norm one operator. Then: \begin{itemize} \item[(1)]
If for every $\varepsilon >0$ there is a slice $S_0 \in S_{T}$ and an element $y \in S_Y$ such that $R(S_0) \subset B_\varepsilon(y)$, then $$
\| T + R \|= 2. $$ \item[(2)] If $S_R \le S_{T}$ and $R$ is weakly compact, then $$
\|T + R\|= 2. $$ \end{itemize} \end{theorem} \begin{proof} (1) Take $\varepsilon >0$. Then there are $S_0=S(T_{y'_0},\delta)
\in S_{T}$ and $y \in S_Y$ such that for every $x \in S_0$, $\|
R(x)- y \| \le \varepsilon$. We can assume that $\delta \le \varepsilon$. Since $Y$ has the Daugavet property, $Y$ has the $T$-Daugavet property with respect to the set $\{ T_{y'}{:}\allowbreak\ y' \in Y' \setminus \{0\} \} \cdot Y $ (see Remark \ref{re10} above). Therefore, by Corollary~\ref{geom20} we find an element $x \in S_0$ such that $$
\|T+R\| \ge \| T(x)+ y\| - \|y-R(x)\| \ge 2- \varepsilon -2 \delta \ge 2- 3 \varepsilon. $$ Since this holds for every $\varepsilon >0$, the proof of (1) is complete.
The proof of (2) follows the same argument as the one for operators in spaces with the Daugavet property (see \cite[Th.~2.3]{ams2000}), so we only sketch it. Assume that
$\|R\|=1$. Since
by hypothesis $K=\overline{R(B_X )}$ is a convex weakly compact set, it is the closed convex hull of its strongly exposed points. Since this set is convex and $\|R\|=1$, there is a strongly exposed point $y_0 \in K$ such that $\|y_0\| \le 1$ and $\|y_0\| > 1- \varepsilon$. Take a functional $y'_0$ that strongly exposes $y_0$ and satisfies $\langle y_0,y_0'\rangle= \max_{y \in K} \langle y,y_0'\rangle =1$. It can be proved by contradiction that there is a slice $S \in S_R$ such that $R(S)$ is contained in the ball $B_\varepsilon(y_0)$ (see the proof of \cite[Th.~2.3]{ams2000}). Since $S_R \le S_{T} $, there is also a slice $S_0 \in S_{T}$ such that $R(S_0) \subset R(S) \subset B_\varepsilon(y_0)$. Then part (1) gives the result. \end{proof}
The example in Remark~\ref{re10} makes it clear that some condition like slice continuity is necessary for (2) in Theorem \ref{weakcom0} to be true. The following variation of this example gives a genuine weakly compact operator that is not
of finite rank which does not satisfy the Daugavet equation. Take $T$ defined as in Remark~\ref{re10} and $R{:}\allowbreak\ L^1[0,1] \oplus_1 L^2[1,2] \to L^1[0,1]$ given by $R((f,g)):=g(x-1)$. This operator is weakly compact and $\|R\|=\|T\|=1$, but again the norm of the sum of both operators is less than $2$.
\begin{remark} \label{themainrem} Notice that the condition in (1) on the existence of a slice $S \in S_T$ such that $R(S) \subset B_\varepsilon(y)$ can be substituted by the existence of a slice $S \in S_T$ and a $\delta >0$ such that $R(S+ \delta B_X) \subset B_\varepsilon(y)$. The argument given in the proof based on this fact makes it also clear that the relation $S_T \le S_R$ can be substituted by the following weaker one and the result is still true: For every slice $S \in S_R$ and $\delta >0$ there is a slice $S_1 \in S_T$ such that $$ S_1 \subset S + \delta B_X. $$ \end{remark}
\section{Bilinear maps and the Daugavet property}
In this section we analyse the Daugavet property for bilinear maps defined on Banach spaces. Our main idea is to provide a framework for the understanding of several new Daugavet type properties and prove some general versions of the main theorems that hold for the case of the Daugavet property. We centre our attention on the extension of the Daugavet equation for weakly compact bilinear maps. Let $X,Y$ and $Z$ be Banach spaces. Consider a norm one continuous bilinear map $B{:}\allowbreak\ X \times Y \to Z$. Then we can consider the linearisation $T_B{:}\allowbreak\ X \hat{\otimes}_\pi Y \to Z$, where $X \hat{\otimes}_\pi Y$ is the projective tensor product with the projective norm $\pi$ (see for instance \cite[Sec.~3.2]{deflo} or \cite[Th.~2.9]{Ryan}). This linear operator will provide meaningful results for bilinear maps by applying the ones of Section~2. However, a genuinely geometric setting for bilinear operators -- slices, isometric equations, \dots\ -- will also be defined in this section in order to provide the specific links between the (bilinear) slice continuity and the Daugavet equation.
We will consider bilinear operators $B_0{:}\allowbreak\ X \times Y \to Z$ satisfying that $B_0(U_X \times U_Y)=U_Z$. Obviously, such a map has always convex range, i.e., $B_0(U_X \times U_Y)$ is a convex set. We will say that a map satisfying these conditions is a norming bilinear map. If $B_0$ is such a bilinear operator, we will say that a Banach space $Z$ has the $B_0$-Daugavet property with respect to the class of bilinear maps $V \subset B(X \times Y, Z)$ if $$
\|B_0+B\|= 1 + \|B\| $$ for all $B \in V$. Notice that $Z$ has the $B_0$-Daugavet property with respect to $V$ if and only if it has the $T_{B_0}$-Daugavet property with respect to the set $\{T_B{:}\allowbreak\ X \hat{\otimes}_\pi Y \to Z{:}\allowbreak\ B \in V\}$. Let us consider some examples.
\begin{example} \label{primex} (1) Take a Banach space $X$ and consider the bilinear form $B_0{:}\allowbreak\ X \times X' \to \mathbb{R}$ given by $B_0(x,x')=\langle x, x' \rangle$, $x \in X$, $x' \in X'$. Consider the set \begin{eqnarray*} \lefteqn{V= \{B_T{:}\allowbreak\ X \times X'\to{\mathbb R}{:}\allowbreak\ B_T(x,x')=\langle T(x), x' \rangle, } \hspace{4cm} \\ &&T{:}\allowbreak\ X \to X \ \textrm{is weakly compact} \}. \end{eqnarray*} Then notice that $$
\sup_{x \in B_X, x' \in B_{X'}} | B_0(x,x')+B_T(x,x') | = \sup_{x
\in B_X, x' \in B_{X'}} | \langle x+T(x),x' \rangle |= \|\mathrm{Id} + T\| $$
and $\|B_0\|+\|B_T\|= 1+ \|T\|$. Therefore $\mathbb R$ has the $B_0$-Daugavet property with respect to $V$ if and only if $X$ has the Daugavet property (see Theorem~2.3 in \cite{ams2000}).
(2) Take a measure space $(\Omega,\Sigma,\mu)$ and a couple of Banach function spaces $X(\mu)=X$ and $Z(\mu)=Z$ over $\mu$ satisfying that the space of multiplication operators $X^Z$ is a saturated Banach function space over $\mu$ and $X$ is $Z$-perfect, i.e., $(X^Z)^Z=X$, and $U_{X} \cdot U_{X^Z}=U_Z$ (here $\cdot$ represents the pointwise product of functions). Consider the bilinear map $B_0{:}\allowbreak\ X \times X^Z \to Z$ given by $B_0(f,g)= f \cdot g$, $f \in X$, $g \in X^Z$ (see \cite{CDS} for definitions and results regarding multiplication operators on Banach function spaces). Consider the set \begin{eqnarray*} \lefteqn{V= \{B_S{:}\allowbreak\ X \times X^Z \to Z {:}\allowbreak\ B_S(f,g)= S(f \cdot
g),} \hspace{4cm} \\ && S{:}\allowbreak\ Z \to Z \ \textrm{is weakly compact} \}. \end{eqnarray*} Then $$
\sup_{f \in B_X, \ g \in B_{X^Z}} \| B_0(f,g)+B_S(f,g) \|_Z
= \sup_{f \in B_X, \ g \in B_{X^Z}} \| f \cdot g +S(f\cdot g) \|_Z=
\|\mathrm{Id} + S\| $$
and $\|B_0\|+\|B_S\|= 1+ \|S\|$. Therefore $Z$ has the $B_0$-Daugavet property with respect to $V$ if and only if $Z$ has the Daugavet property (see again Theorem~2.3 in \cite{ams2000}).
(3) Take $1 < p < \infty$, its conjugate index $p'$, a measurable space $(\Omega, \Sigma)$, a Banach space $Z$ and a countably additive vector measure $m{:}\allowbreak\ \Sigma \to Z$. Consider the corresponding spaces of $m$-integrable functions $L^p(m)$ and $L^{p'}(m)$, and the bilinear map $B_0{:}\allowbreak\ L^p(m)\times L^{p'}(m) \to Z$ given by the composition of the multiplication and the integration map $I_m{:}\allowbreak\ L^1(m) \to Z$, i.e., $B_0(f,g)= \int fg \, dm $. This map is well defined and continuous (see \cite[Chapter~3]{libro} for the main definitions and results on the spaces $L^p(m)$). Assume also that $B_0(U_{L^p(m)}\times U_{L^{p'}(m)})= I_m(U_{L^p(m)} \cdot U_{L^{p'}(m)})$ coincides with the open unit ball of $Z$. Take the set \begin{eqnarray*} \lefteqn{U= \big\{B_R{:}\allowbreak\ L^p(m) \times L^{p'}(m) \to Z {:}\allowbreak\ } \hspace{2cm}\\ && B_R(f,g):= R(I_m(f \cdot g)), \ R{:}\allowbreak\ Z \to Z \ \textrm{rank one} \big\}. \end{eqnarray*} Since $$
\|B_0 + B_R\| = \sup_{f \in B_{L^p(m)},\ g \in B_{L^{p'}(m)}} \Bigl\|
\int_\Omega fg \, dm + R \Bigl(\int_\Omega fg \, dm\Bigr) \Bigr\|_Z
= \|\mathrm{Id} + R\| $$
and $\|B_0\|+\|B_R\|= 1+ \|R\|$, we obtain again that $Z$ has the $B_0$-Daugavet property with respect to $U$ if and only if $Z$ has the Daugavet property. \end{example}
\begin{remark} More examples can be given by considering the following bilinear maps:
(i) $B_{C(K)}{:}\allowbreak\ C(K) \times C(K) \to C(K)$, $B_{C(K)}(f,g)= f \cdot g$.
(ii) $B_*{:}\allowbreak\ L^1({\mathbb R}) \times L^1({\mathbb R}) \to L^1({\mathbb R})$, $B_*(f,g)= f * g$, where $*$ is the convolution product. In this case we have $B_*(U_{L^1({\mathbb R})},U_{L^1({\mathbb R})})=U_{L^1({\mathbb R})}$ as a consequence of
Cohen's Factorisation Theorem (see Corollary~32.30 in \cite{hewross}).
(iii) For a $\sigma$-finite $\mu$, $B_{L^\infty}{:}\allowbreak\ L^\infty(\mu) \times L^1(\mu) \to \mathbb{R}$ given by $B_{L^\infty}(f,g)= \int fg \, d \mu$.
\end{remark}
Bilinear operators for which the Daugavet equation will be shown to hold -- together with norming bilinear maps -- are weakly compact operators with convex range. Although the usual way of finding such a map is to compose a bilinear map with convex range and a weakly compact linear one, other examples can be given. Let us show one of them that is in fact not norming.
\begin{example} Consider a constant~$1$ $p$-convex reflexive Banach function space
$X$. In particular, $X$ must be order continuous. Take $f'_0 \in S_{X'}$ and $f_0 \in S_X$ and define the bilinear map $B{:}\allowbreak\ X \times X_{[p/p']} \to X_{[p]}$ given by $B(f,g)= \langle f, f'_0 \rangle f_0 \cdot g$. Note that $\|B\|=1$. Let us show that the (norm) closure $K=\overline{B(B_X \times B_{X_{[p/p']}})}$ is a convex weakly compact set.
Let $z_1,z_2 \in B(B_X \times B_{X_{[p/p']}})$. Let $f_1,g_1,f_2$ and $g_2$ such that $B(f_1,g_1)=z_1$ and
$B(f_2,g_2)=z_2$. Take $0 < \alpha < 1$ and consider the element $\alpha z_1+(1-\alpha) z_2$. Let us prove that it belongs to $B(B_X \times B_{X_{[p/p']}})$. Notice that since $-1 \le \langle f,f_0' \rangle \le 1$ for every $f \in B_X$, $g_3=\alpha \langle f_1,f'_0 \rangle g_1 +(1-\alpha) \langle f_2, f'_0 \rangle g_2$ belongs to $B_{X_{[p/p']}}$. Take now an element $f_3 \in B_X$ such that $\langle f_3, f'_0 \rangle =1$ (it exists since $X$ is reflexive), and note that $$ B(f_3,g_3)= \alpha z_1 + (1-\alpha) z_2. $$ So, $K$ is convex. Notice that ${B(B_X \times B_{X_{[p/p']}})}$ is also relatively weakly compact; it is enough to observe that the set is uniformly $\mu$-absolutely continuous (see for instance Remark~2.38 in \cite{libro} and the references therein), i.e., that $$
\lim_{\mu(A) \to 0} \sup_{z \in K} \|z \chi_A\|=0. $$ But this is a direct consequence of the fact that $X$ is order continuous (see for instance \cite[Th.~1.c.5 and Prop.~1.a.8]{lint}) and the H\"older inequality for the norms of $p$-th power spaces (adapt \cite[Lemma~2.21]{libro} or \cite[Prop.~1.d.2(i)]{lint}). For every $z = \langle f, f'_0 \rangle f_0 \cdot g \in B(B_X,B_{X_{[p/p']}})$ and $A \in \Sigma$, $$
\| z \chi_A\|_{X_{[p]}} = \| \langle f, f'_0 \rangle f_0 \cdot g\|_{X_{[p]}} \le |\langle f, f'_0 \rangle| \|f_0 \chi_A\|_X \cdot
\|g\|_{X_{[p/p']}}. $$
Since $X$ is order continuous, $\|f_0 \chi_A\|_X \to 0$ when $\mu(A) \to 0$, which gives the result.
\end{example}
Let us now start to adapt the results of the previous section. In order to do so, let us define the natural set of slices associated to a norm one bilinear form $b \in B(X \times Y,\mathbb R)$. Let $0<\varepsilon <1$. Following the notation given for the linear case, we define $S(b,\varepsilon)$ by $$ S(b,\varepsilon):= \{(x,y){:}\allowbreak\ x \in B_X,\ y \in B_Y,\ b(x,y) \ge 1-\varepsilon \}. $$ The following result shows the relation between slices defined by a bilinear form and the ones defined
by the linearisation of this map.
\begin{lemma} \label{lemacompa} Let $b \in B(X \times Y, \mathbb R)$ be a norm one bilinear form (i.e., $T_b \in (X \hat{\otimes}_\pi Y)'$ with norm one) and $\varepsilon >0$. Then: \begin{itemize}
\item[(1)]
There is an elementary tensor $x \otimes y$ such that $\|x\|=\|y\|=1$ and $x \otimes y \in S(T_b,\varepsilon)$.
\item[(2)] $\overline{\loglike{co} \{x \otimes y{:}\allowbreak\ (x,y) \in
S(b,\varepsilon) \}} \subset S(T_b,\varepsilon).$
\item[(3)] $S(T_b,\varepsilon^2) \subset \overline{\loglike{co} \{x \otimes
y{:}\allowbreak\ (x,y) \in S(b,\varepsilon) \}} + 4 \varepsilon B_{X
\hat{\otimes}_\pi Y}.$ \end{itemize} \end{lemma} \begin{proof}
(1) Take a norm one element $t \in S(T_b, \varepsilon/2)$. Then there is an element $t_0=\sum_{i=1}^n \alpha_i x_i \otimes y_i \in X \otimes Y$ such that $\|x_i\|=\|y_i\|=1$, $\alpha_i >0$ for all $i=1,\dots,n$, $\sum_{i=1}^n \alpha_i=1$ and $\pi(t - t_0) < \varepsilon/2$. Then $$ \langle t_0, T_b \rangle = \langle t-t_0, T_b \rangle + \langle t, T_b
\rangle \ge \langle t, T_b \rangle -|\langle t-t_0, T_b \rangle| > 1- \frac{\varepsilon}{2}-\frac{\varepsilon}{2}, $$ and so $t_0 \in S(T_b, \varepsilon)$. Assume (by changing the signs of some of the $x_i$ if necessary) that $b(x_i,y_i) >0$ for all $i$. Then $$ \sum_{i=1}^n \alpha_i b(x_i,y_i) \ge \sum_{i=1}^n \alpha_i (1- \varepsilon), $$ and so there is at least one index $i_0$ such that $b(x_{i_0}, y_{i_0}) \ge 1- \varepsilon$. Consequently, $x_{i_0} \otimes y_{i_0} \in S(T_b,\varepsilon)$.
(2) is a direct consequence of the fact that $S(T_b,\varepsilon)$ is norm closed in the projective tensor product.
(3) Let us show now that $S(T_b,\varepsilon^2) \subset \overline{\loglike{co} \{x \otimes y{:}\allowbreak\ (x,y) \in S(b,\varepsilon) \}} +
4\varepsilon B_{X \hat{\otimes}_\pi Y}$. Let $u\in S(T_b,\varepsilon^2)$. Find $v$ such that $\|v\|<1$, $T_b(v)\ge 1-\varepsilon^2$, and $\|v-u\|\le\varepsilon$. Write $v=\sum_{i=1}^\infty \alpha_i x_i \otimes y_i$ with all
$\|x_i\|=\|y_i\| =1$, $\alpha_i\ge0$ and $\alpha:=\sum_{i=1}^\infty \alpha_i <1$. Note that $\alpha\ge1-\varepsilon^2$. Now consider \begin{eqnarray*} I & := & \{i \in \mathbb N{:}\allowbreak\ b(x_i,y_i) \ge 1-\varepsilon\} = \{i\in \mathbb N{:}\allowbreak\ (x_i,y_i)\in S(b,\varepsilon)\}, \\ J & := & \{i\in \mathbb N{:}\allowbreak\ b(x_i,y_i) < 1-\varepsilon\}. \end{eqnarray*} Let $\alpha_I:= \sum_{i\in I} \alpha_i$ and $\alpha_J:= \sum_{i\in J} \alpha_i$. We have $$ 1-\varepsilon^2 \le \sum_{i=1}^\infty \alpha_i b(x_i,y_i) \le \alpha_I + \alpha_J (1-\varepsilon) < 1-\varepsilon\alpha_J $$ and hence $\alpha_J<\varepsilon$. Let $w=\sum_I \frac{\alpha_i}{\alpha_I} x_i \otimes y_i \in \overline{\loglike{co} \{x \otimes y{:}\allowbreak\ (x,y) \in S(b,\varepsilon) \}} $; we then have (note that $v=\alpha_I w + \sum_J \alpha_i x_i \otimes y_i$) $$
\|v-w\| \le |\alpha_I-1| \|w\| + \alpha_J. $$ Furthermore $0\le 1-\alpha_I = \alpha_J + 1 -\alpha \le \varepsilon + \varepsilon^2$; hence $$
\|u-w\| \le \|u-v\| + \|v-w\| \le \varepsilon + ((\varepsilon + \varepsilon^2) +\varepsilon^2) \le 4\varepsilon, $$ as claimed. \end{proof}
If $z \in Z$, we define $b_z{:}\allowbreak\ X \times Y \to Z$ as the (rank one) bilinear map given by $b_z(x,y)=b(x,y) z$, $x \in X$, $y \in Y$. Let $B{:}\allowbreak\ X \times Y \to Z$ be a continuous bilinear map. In what follows we need to introduce some elements related to duality and adjoint bilinear operators. Following Ramanujan and Schock in \cite{ramsch}, we consider the adjoint operator $B^\times{:}\allowbreak\ Z' \to B(X,Y)$ given by $B^\times (z')(x,y)= \langle B(x,y), z' \rangle$ (this definition does not coincide with the one given originally by Arens in \cite{arens}, although the setting is of course the same). $B^\times$ is a linear and continuous operator, and
$\|B\|=\|B^\times\|$.
\begin{definition} Let $B{:}\allowbreak\ X \times Y \to Z$ be a continuous bilinear map. Let $z' \in S_{Z'}$ and consider the adjoint bilinear form $\langle B, z'\rangle{:}\allowbreak\ X \times Y \to \mathbb{R}$ given by $\langle B,z' \rangle(x,y)=B^\times (z')(x,y)$.
We denote by $B_{z'}{:}\allowbreak\ X \times Y \to \mathbb{R}$ the bilinear form given by $B_{z'}(x,y)= \frac{\langle B(x,y), z'
\rangle}{\|\langle B, z' \rangle\|}$ whenever $\|\langle B, z'
\rangle\| \ne 0$ and by $\langle B, Z' \rangle$ the set of all these bilinear forms. The natural set of slices defined by $B$ is then $$
S_B=\{S(B_{z'}, \varepsilon){:}\allowbreak\ 0<\varepsilon < 1, \ \|B_{z'}\| \ne 0 \}. $$ If $B_1$ is another (continuous) bilinear map, $B_1{:}\allowbreak\ X \times Y \to Z$, we use the symbol $S_{B} \le S_{B_1}$ to denote that for every slice $S$ in $S_{B}$ there is a slice $S_1 \in S_{B_1}$ such that $S_1 \subset S$. We can also consider the relation $S_{T_B} \le S_{B_1}$ to be defined in the same way: for every $S \in S_{T_B}$ there is a slice $S_1 \in S_{B_1}$ such that the set $\{x \otimes y{:}\allowbreak\ (x,y) \in S_1 \}$ is included in $S$. Lemma~\ref{lemacompa} gives an idea of how this relation works. \end{definition}
As in the linear case, the canonical example of the relation $S_{B} \le S_{B_1}$ between sets of slices associated to two bilinear maps is given by bilinear maps $B$ that are defined as a composition $T \circ B_1$, where $B_1{:}\allowbreak\ X \times Y \to Z$ is a continuous bilinear map and $T{:}\allowbreak\ Z \to Z$ is a continuous operator. In this case, $\langle B(x,y), z' \rangle= \langle B_1(x,y), T'(z') \rangle$, and so clearly $S_B \le S_{B_1}$. Let us show some examples.
\begin{example} \label{exslices} Let $(\Omega, \Sigma, \mu)$ be a finite measure space and consider a rearrangement invariant (r.i.)\ constant~$1$ $p$-convex Banach function space $X(\mu)$ (see \cite[p.~28 and Sections~1.d, 2.e]{lint} or \cite[Ch.~2 and p.~202]{libro}). In this case, $(X(\mu)_{[p]})'$ is also r.i. Take a measurable bijection $\Phi{:}\allowbreak\ \Omega \to \Omega$ such that $\mu( \Phi(A))=\mu(A)$ for every $A \in \Sigma$. Then it is possible to define the isometry $T_r{:}\allowbreak\ X_{[r]} \to X_{[r]}$, $0 \le r \le p$, by $T_r(f)= f \circ \Phi$.
Define the bilinear map $B{:}\allowbreak\ X \times X_{[p/p']} \to X_{[p]}$ given by $B(f,g)= T_1(f) \cdot T_{p/p'}(g)$. Let us show the relation between the slices defined by $B_0{:}\allowbreak\ X \times X_{[p/p']} \to X_{[p]}$, $B_0(f,g)=fg$, and the slices defined by $B$. Assume also that $X$ is order continuous. Then $X_{[p]}$ is also order continuous and the dual of the space can be identified with the K\"othe dual, which is also r.i., and so every continuous linear form is an integral. Note that in this case the property $S_B \le S_{B_0}$ holds, since for every couple of functions $f \in X$ and $g \in X_{[p/p']}$, $B(f,g)= T_1(f) \cdot T_{p/p'}(g)= (f \circ \Phi) \cdot (g \circ \Phi)= (f \cdot g) \circ \Phi$. Consequently, every element $z' \in S_{(X_{[p]})'}$ satisfies that for every pair of functions $f$ and $g$ as above, \begin{eqnarray*} \langle B_0(f,g), z' \rangle &=& \int_\Omega fg z' \, d\mu= \int_\Omega ((fg) \circ \Phi) \cdot (z' \circ \Phi) \, d\mu\\ &=& \int_\Omega B(f,g) \cdot (z' \circ \Phi) \, d\mu= \langle B(f,g), z' \circ \Phi \rangle. \end{eqnarray*} Therefore, there is a one-to-one correspondence between $S_{B_0}$ and $S_B$ given by identifying $S((B_0)_{z'}, \varepsilon)$ and $S(B_{z' \circ \Phi}, \varepsilon)$, which implies that $S_{B_0}=S_B$. \end{example}
Fix a norming bilinear map $B_0{:}\allowbreak\ X \times Y \to Z$ and consider a norm one bilinear map $B{:}\allowbreak\ X \times Y \to Z$. Let us provide now geometric and topological properties for $B$ that imply that the Daugavet equation is satisfied for $B_0$ and $B$, i.e.,
$\|B_0+B\|=2$. These properties will be proved as applications of the result of the previous section.
\begin{corollary} \label{weakcom} Let $Z$ be a Banach space with the Daugavet property. Let $B_0{:}\allowbreak\ X \times Y \to Z$ be a norming bilinear map and $B{:}\allowbreak\ X \times Y \to Z$ a continuous bilinear map. Then:
\begin{itemize} \item[(1)]
If for every $\varepsilon >0$ there is a slice $S_0 \in S_{B_0}$ and an element $z \in S_Z$ such that $B(S_0) \subset B_\varepsilon(z)$, then $$
\| B_0 + B \|= 1 + \|B\|. $$ \item[(2)] If $S_{T_B} \le S_{B_0}$ and $T_B$ is weakly compact (equivalently, $B(B_X \times B_Y)$ is a relatively weakly compact set), then $$
\|B_0 + B\|=1 + \|B\|. $$
\end{itemize} \end{corollary} \begin{proof} (1) is just a consequence of Theorem~\ref{weakcom0}(1): let us take $\varepsilon/5$ and apply this theorem to $T_{B_0}$ and $T_B$. By hypothesis there is a slice $S_0=S(b,\delta) \in S_{B_0}$ such that $B(S_0) \subset B_{\varepsilon/5}(z)$. We can assume without loss of generality that $\delta \le
\varepsilon$ and $\|B\| \le 1$. Then by Lemma~\ref{lemacompa}(3), \begin{eqnarray*} T_{B}(S(T_b,\delta^2)) &\subset& \textstyle T_{B}(\overline{\loglike{co}(S_0)})+ \frac45 \varepsilon T_{B}(B_{X \hat{\otimes}_\pi Y}) \\ &\subset& \textstyle
\overline{\loglike{co}(B(S_0))} + \frac45 \varepsilon \|T_{B}\|B_Z \subset B_{\varepsilon/5}(z) + \frac45 \varepsilon B_Z \\ &\subset& B_\varepsilon (z), \end{eqnarray*} and (1) is proved. For (2), apply Theorem~\ref{weakcom0}(2) and Remark~\ref{themainrem}.
\end{proof}
\begin{example} It is well known that for a purely non-atomic measure $\mu$ and a Banach space $E$ the space of Bochner integrable functions
$L^1(\mu,E)$ has the Daugavet property (see \cite{ams2000}). The next simple application of Corollary~\ref{weakcom} provides a similar result for the Pettis norm $\|\,.\,\|_P$, i.e., for operators $T$ from $(L^1(\mu,E),\|\,.\,\|_{L^1(\mu,E)})$ to the normed space $(L^1(\mu,E),\|\,.\,\|_P)$. Consider the bilinear map $B_0{:}\allowbreak\ L^1(\mu,E) \times E' \to L^1(\mu)$ given by $$ B_0(f,x')(w)= \langle f(w), x' \rangle, \qquad w \in \Omega. $$ Take an operator $T{:}\allowbreak\ L^1(\mu,E) \to L^1(\mu,E)$ and define the bilinear map $B_T{:}\allowbreak\ L^1(\mu,E) \times E' \to L^1(\mu)$ given by $$ B_T(f,x')(w)= \langle (T(f))(w), x' \rangle, \qquad w \in \Omega. $$ Assume that $B_T$ is weakly compact and has convex range and suppose that $S_{B_T} \le S_{B_0}$ (or that $T_{B_T}$ is weakly compact and $S_{T_{B_T}} \le S_{T_{B_0}}$). Then by Corollary~\ref{weakcom}(3) (or~(4)), \begin{eqnarray*}
\sup_{f \in B_{L^1(\mu,E)}} \|f + T(f)\|_P &=& \sup_{f \in B_{L^1(\mu,E)}, \ x' \in B_{E'}}
\|B_0(f,x')+B_T(f,x')\|_{L^1(\mu)}\\
&=& \sup_{f \in B_{L^1(\mu,E)}, \ x' \in B_{E'}} \|B_0(f,x')\| \\ && \mbox{} \qquad {} + \sup_{f \in B_{L^1(\mu,E)}, \ x' \in B_{E'}}
\|B_T(f,x')\|_{L^1(\mu)} \\
&=& \sup_{f \in B_{L^1(\mu,E)}} \|f\|_P + \sup_{f \in
B_{L^1(\mu,E)}} \|T(f)\|_P . \end{eqnarray*}
\end{example}
Corollary~\ref{weakcom} suggests that the natural examples of bilinear maps that satisfy the Daugavet equation with respect to $B_0$ are the ones defined as $B=T \circ B_0$, where $T{:}\allowbreak\ Z \to Z$ is a weakly compact operator. Corollary~\ref{coroweakcom} generalises in a sense the idea of (2) and (3) in Example~\ref{primex}. Notice, however, that there are other simple bilinear maps that fit into the Daugavet setting, as the following example shows.
\begin{example} \label{nuevo} Let us show an example of a bilinear map $B{:}\allowbreak\ X \times Y \to Z$ such that $B_0$ and $B$ satisfy the Daugavet equation but there is no operator $T{:}\allowbreak\ Z \to Z$ such that $B=T \circ B_0$. Let $(\Omega, \Sigma, \mu)$, $X(\mu)$ and $\Phi$ be as in Example~\ref{exslices}
and consider the isometry $T_1{:}\allowbreak\ X \to X$ defined there. Assume also that $\mu(\Omega) < \infty $ and the constant~$1$ function satisfies $\|\chi_\Omega\|_X=1$. Consider the bilinear map $B{:}\allowbreak\ X \times X_{[p/p']} \to X_{[p]}$ given by $B(f,g)=T _1(f) \cdot g$. Then, since $T_1(\chi_\Omega)= \chi_\Omega$,
\begin{eqnarray*} 2 &\ge& \|B_0+B\| = \sup_{f \in B_X, \ g \in B_{X_{[p/p']}}} \|fg +
T_1(f)g \|_{X_{[p]}} \\
&=&\sup_{f \in B_X, \ g \in B_{X_{[p/p']}}} \|(f + T_1(f))g
\|_{X_{[p]}} =\sup_{f \in B_X} \|f+T_1(f)\|_X \\
&\ge& \| \chi_\Omega
+ \chi_\Omega\|=2. \end{eqnarray*} Notice that in general a bilinear map defined in this way cannot be written as $T \circ B_0$ for any operator $T$. For instance, suppose that there is a set $B \in \Sigma$ such that $0 < \mu(B)$ and $B \cap \Phi(B) = \emptyset$ and consider a couple of non-trivial functions $f_1$ and $f_2$ in $X$ with support in
$\Phi(B)$ and $B$, respectively, and such that $\|(f_1\circ \Phi)
\cdot f_2\| >0$. Then $B_0(f_1,f_2)=0$, but $B(f_1,f_2)\ne 0$, so there is no operator $T{:}\allowbreak\ X_{[p]} \to X_{[p]}$ such that $B= T \circ B_0$. \end{example}
\begin{corollary} \label{coroweakcom} Let $B_0{:}\allowbreak\ X \times Y \to Z$ be a norming bilinear map. Consider the subsets $R$, $C$ and $WC$ of $L(Z,Z)$ of rank one, compact and weakly compact operators, respectively, and the sets $R \circ B_0=\{B=T \circ B_0{:}\allowbreak\ X \times Y \to Z{:}\allowbreak\ T \in R\}$, $C \circ B_0=\{B=T \circ B_0{:}\allowbreak\ X \times Y \to Z{:}\allowbreak\ T \in C\}$ and $ WC \circ B_0=\{B=T \circ B_0{:}\allowbreak\ X \times Y \to Z{:}\allowbreak\ T \in WC\}$. Then the following are equivalent. \begin{itemize} \item[(1)]$Z$ has the Daugavet property.
\item[(2)] $Z$ has the $B_0$-Daugavet property with respect to $R \circ B_0$.
\item[(3)] $Z$ has the $B_0$-Daugavet property with respect to $C \circ B_0$.
\item[(4)] $Z$ has the $B_0$-Daugavet property with respect to $WC \circ B_0$.
\item[(5)] For every norm one operator $T \in R$, every $z \in Z$ and every $\varepsilon >0$ there is an element $(x,y) \in S(T \circ B_0, \varepsilon)$ such that $$
\|z + B_0(x,y)\| \ge 2- \varepsilon. $$
\end{itemize}
\end{corollary} \begin{proof} The equivalence between (1) and (2) is a direct consequence of the following equalities. For every rank one operator $T{:}\allowbreak\ Z \to Z$, $$
\|\mathrm{Id}+T\|= \sup_{z \in B_Z} \|z+T(z)\| = \sup_{x \in B_X,y \in B_Y}
\|B_0(x,y)+T(B_0(x,y))\|. $$ Since the norm closure of the convex hull $B(B_X \times B_Y)$ is a weakly compact set, (2) implies~(4) as a consequence of Corollary~\ref{weakcom}(2). Obviously (4) implies~(2), and so the equivalence of (2) and (3) is also clear. The equivalence of (2) and (5) holds as a direct consequence of Corollary~\ref{geom20} and the arguments used above. \end{proof}
\begin{remark} Conditions under which a bilinear map $B{:}\allowbreak\ X \times Y \to Z$ is compact or weakly compact (i.e., the norm closure $\overline{B(B_X \times B_Y)}$ is compact or weakly compact, respectively) have been studied in several papers; see \cite{ramsch,Ruch} for compactness and \cite{arongalindo,ulger} for weak compactness. The reader can find in these papers some factorisation theorems and other characterisations of these properties, also related with the notion of Arens regularity of a bilinear map. \end{remark}
\section{Applications. $p$-convexifications of the Daugavet property and bilinear maps}
Different $p$-convexifications of the Daugavet property have been introduced in \cite{EnrDir2,EnrDir3}. In this section we show that in a sense they can be considered as particular cases of a Daugavet property for bilinear maps. We centre our attention on the case of Banach function spaces such that their $p$-th powers have the Daugavet property that have been characterised in \cite{EnrDir2}. However, more examples of applications will be given as well. Throughout this section $\mu$ is supposed to be finite.
We explain now two suitable examples of $p$-convexification of the Daugavet property. Let us start with one regarding $p$-concavity in Banach function spaces.
\begin{example} Let $1 \le p < \infty$. Consider a constant~$1$ $p$-convex Banach function space $X$, $Y=X_{[p/p']} \oplus_\infty X_{[p/p']}$ (the direct product with the maximum norm), $Z=X_{[p]}$, and the bilinear map $B_0{:}\allowbreak\ X \times ( X_{[p/p']} \times_\infty X_{[p/p']}) \to X_{[p]}$ given by $B_0(f,(g,h))= f \cdot P_1(g,h)= fg$.
Take an operator $T{:}\allowbreak\ X_{} \to X_{}$ and consider the bilinear map $B{:}\allowbreak\ X \times ( X_{[p/p']} \oplus_\infty X_{[p/p']}) \to X_{[p]}$ given by $B(f,(g,h))= T(f) \cdot P_2(g,h)= fh$ (here $P_1$ and $P_2$ denote the two natural projections in the product space $X_{[p/p']} \oplus_\infty X_{[p/p']}$). A direct calculation shows that in this case the Daugavet equation for the pair given by $B_0$ and $B$ is $$
\|B_0 + B \| = 1+ \|T\|, $$
since $\|B\|=\|T\|$. Assume that $\|T\|=1$. Then $\|T(f)\| \le 1$ for every $f \in B_X$, and so, taking $g= f^{p/p'} \in B_{X_{[p/p']}}$ and $h=T(f)^{p/p'} \in B_{X_{[p/p']}}$ for each $f \in B_X$, we obtain
\begin{eqnarray*} 2 \ge \|B_0 + B \| &=& \sup_{f \in B_X, \ g \in B_{X_{[p/p']}},\ h \in B_{X_{[p/p']}}} \| fg + T(f)\cdot h\|_{X_{[p]}}\\
&\ge& \sup_{f \in B_X} \| |f|^p + |T(f)|^p \|_{X_{[p]}} \\
&\ge& \sup_{f \in B_X} \| (|f|^p + |T(f)|^p)^{1/p}\|^p_{X} . \end{eqnarray*} Thus, if $X$ is also a constant~$1$ $p$-concave space (i.e., $X$ is an $L^p$-space) we get $$
\sup_{f \in B_X} \| (|f|^p + |T(f)|^p)^{1/p}\|^p_{X}
\ge \sup_{f \in B_X} ( \|f\|_X^p + \|T(f)\|_X^p)= 2. $$ Therefore, in this case the Daugavet equation holds for $B_0$ and for every bilinear map $B$ defined by an operator $T{:}\allowbreak\ X_{} \to X_{}$ in the way explained above. \end{example}
The following construction shows another example of a Daugavet type property for a bilinear map that is in fact a $p$-convex version of the Daugavet property, in the sense that is studied in \cite{EnrDir3}.
\begin{example} \label{exDaueq?} Let $(\Omega, \Sigma, \mu)$ be a measure space and consider an r.i.\ constant~$1$ $p$-convex Banach function space $X(\mu)$. Consider as in Example~\ref{exslices} the bilinear map $B_0$ given by the product and a measurable bijection $\Phi{:}\allowbreak\ \Omega \to \Omega$ satisfying that $\mu( \Phi(A))=\mu(A)$ for every $A \in \Sigma$ and the isometries $T_r{:}\allowbreak\ X_{[r]} \to X_{[r]}$, $0 < r \le p$.
Take the bilinear map $B{:}\allowbreak\ X \times X_{[p/p']} \to X_{[p]}$ given by
$B(f,g)= T_1(f) \cdot T_{p/p'}(g)$. Notice that $\|B\|=1$. Then
\begin{eqnarray*} 2 \ge \|B_0+B\| &\ge& \sup_{f \in B_X, \ g \in B_{X_{[p/p']}}}
\|B_0(f,g)+ B(f,g)\|_{X_{[p]}} \\
&\ge& \sup_{f \in B_X} \| f \cdot f^{p/p'} + T_1(f) \cdot T_{p/p'}(f^{p/p'}) \|_{X_{[p]}} \\
&=& \sup_{x \in B_X} \|f^p + T_1(f)^p\|_{X_{[p]}} \\
&=& \sup_{f \in B_X} \| |f^p+ T_1(f)^p|^{1/p} \|_X^p . \end{eqnarray*} Now, if $\Phi$ satisfies that there is a set $A \in \Sigma$ such that $\mu(A \cap \Phi(A)) < \mu(A)$, there is a norm one function $f_0$ such that $f_0$ and $T_1(f_0)$ are disjoint and
$\|T_1(f_0)\|=1$. Assume that $X$ is also $p$-concave (constant~$1$), i.e., $X$ is an $L^p$-space. Then $$
\sup_{f \in B_X} \| |f^p+ T_1(f)^p|^{1/p} \|_X^p
\ge \|f_0\|_X^p + \|T_1(f_0)\|_X^p=2, $$ and thus the so called $p$-Daugavet equation is satisfied for $T_1$ (see Definition~1.1 in \cite{EnrDir3}), and $B$ and $B_0$ satisfy the Daugavet equation. \end{example}
Let $1 \le p < \infty$. In what follows we study the $p$-convex spaces whose $p$-th powers satisfy the Daugavet property by giving some general results in the setting of the examples presented above. We analyse the case of $X=X(\mu)$, a constant~$1$ $p$-convex Banach function space, $Y= X(\mu)_{[p/p']}$, $Z=X(\mu)_{[p]}$, and $B_0{:}\allowbreak\ X \times X_{[p/p']} \to X_{[p]}$ given by $B_0(f,g)=f \cdot g$. We assume that $X_{[p]}$ has the Daugavet property. The main example we have in mind is given by $X=L^p[0,1]$, $Y=X_{[p/p']}=L^{p'}[0,1]$ and $Z=X_{[p]}= L^1[0,1]$. Recall that $\mu$ is assumed to be finite.
\begin{definition} Let $X(\mu)$, $Y(\mu)$ and $Z(\mu)$ three Banach function spaces over $\mu$. We say that a continuous bilinear map $B{:}\allowbreak\ X(\mu) \times Y(\mu) \to Z(\mu)$ satisfying that for every $A,C \in \Sigma$,
$B(\chi_A,\chi_C)= B(\chi_{A \cap C}, \chi_{A \cup C})$, is a symmetric bilinear map. \end{definition}
\begin{proposition} \label{pthDAugavet}
Let $X(\mu)$ be an order continuous $p$-convex Banach function space with $p$-convexity constant equal to~$1$. Then the following assertions are
equivalent. \begin{itemize} \item[(1)] For every rank one operator $T{:}\allowbreak\ X(\mu)_{[p]} \to X(\mu)_{[p]}$,
$$
\sup_{f \in B_X} \||f^p+ T(f^p) |^{1/p} \|^p_X=1 + \|T\|.
$$
\item[(2)] For every rank one operator $T{:}\allowbreak\ X(\mu)_{[p]} \to X(\mu)_{[p]}$, $$
\| B_0 + T \circ B_0\| = 1 + \|T\|. $$ \item[(3)] For every $z \in S_{X_{[p]}}$, for every $x' \in S_{(X_{[p]})'}$ and for every $\varepsilon >0$ there is an element $(f,g) \in S((B_0)_{x'},\varepsilon)$ such that $$
\| z + B_0(f,g)\|_{X_{[p]}} \ge 2- 2 \varepsilon. $$
\item[(4)] Each weakly compact symmetric bilinear map $B{:}\allowbreak\ X(\mu) \times X_{[p/p']} \to X_{[p]}$ satisfies the equation $$
\| B_0+B\|=1 + \|B\|. $$
\item[(5)] $X_{[p]}$ has the Daugavet property.
\end{itemize} \end{proposition} \begin{proof} For the equivalence of (1) and (2), note that the constant~$1$ $p$-convexity of $X$ implies that $B_X \cdot B_{X_{[p/p']}}= B_{X_{[p]}}$ is the unit ball of the Banach function space $X_{[p]}$; so, using also Remark~\ref{rembasic} the following inequalities are obtained: \begin{eqnarray*}
\sup_{f \in B_X} \||f^p+ T(f^p) |^{1/p} \|^p_X
&\le& \sup_{f \in B_X, \ g \in B_{X_{[p/p']}}} \| fg + T(fg) \|_X \\
&\le& \sup_{h \in B_{X_{[p]}}} \|h + T(h)\|_{X_{[p]}} \\
&\le& \sup_{f \in B_X} \||f^p+ T(f^p) |^{1/p} \|^p_X , \end{eqnarray*} and then both assertions are seen to be equivalent. The equivalence of (2) and (3) is obtained by applying Corollary~\ref{geom20} to the setting of bilinear maps.
Taking into account that the map
$i_{[p]}{:}\allowbreak\ X \to X_{[p]}$ given by $i_{[p]}(f)=f^p$ is a bijection satisfying $\|i_{[p]}(f)\|_{X_{[p]}}=\|f\|^p_X$ for every $f \in X$, and the definition of the norm $\|\,.\,\|_{X_{[p]}}$, the equivalence of (1) and (5) is also clear using the well-known geometric characterisation of the Daugavet property in terms of slices (see for instance Lemma~2.2 in \cite{ams2000}).
Thus, it only remains to prove the equivalence of (2) and~(4). Let us show first the following \textit{\textbf{Claim:} Let $X$ be a $p$-convex (constant~$1$) Banach function space such that the simple functions are dense and let $B{:}\allowbreak\ X(\mu) \times X(\mu)_{[p/p']} \to X(\mu)_{[p]}$ be a continuous bilinear map. Then $B$ is symmetric if and only if there is an operator $T{:}\allowbreak\ X_{[p]} \to X_{[p]}$ such that $B=T \circ B_0$.}
In order to prove this, note that by hypothesis the set $S(\mu)$ of simple functions is dense in $X(\mu)$ and so for every $0 \le r \le p$ it is also dense in $X(\mu)_{[r]}$; this can be shown by a direct computation just considering the definition of the norm in
$\|\,.\,\|_{X_{[r]}}$ and the fact that if $X$ is constant~$1$ $p$-convex then it is constant~$1$ $r$-convex for all such $r$, see for instance \cite[Prop.~1.b.5]{lint} or \cite[Prop.~2.54]{libro}. So this holds for $r= p/p'$. If $B$ is symmetric, then for every couple of simple functions $f= \sum_{i=1}^n \alpha_i \chi_{A_i}$ and $g = \sum_{j=1}^m \beta_j \chi_{B_j}$, where $\{A_i\}_{i=1}^n$ and $\{B_i\}_{j=1}^m$ are sequences of pairwise disjoint measurable sets, \begin{eqnarray*} B(f,g)&=& \sum_{i=1}^n \sum_{j=1}^m \alpha_i \beta_j B(\chi_{A_i},\chi_{B_j}) \\ &=& \sum_{j=1}^m \sum_{i=1}^n \alpha_i \beta_j B(\chi_{A_i \cap B_j},\chi_{A_i \cup B_j})\\ &=& \sum_{j=1}^m \sum_{i=1}^n \beta_j \alpha_i B(\chi_{B_j},\chi_{A_i}) = B(g,f). \end{eqnarray*} Therefore, because of the continuity of $B$ and the order continuity of the spaces, $B(f,g)=B(g,f)$ for every couple of simple functions $f,g \in X \cap X_{[p/p']}$. Define now the map $T{:}\allowbreak\ X_{[p]} \to X_{[p]}$ by $T(h)=B(f,g)$ for every function $h=fg$, first for products of simple functions and then by density for the rest of the elements of $X_{[p]}$ (note that the norm closure of the set $(S(\mu) \cap B_X ) \cdot (S(\mu) \cap B_{X_{[p/p']}})$ coincides with $B_{X_{[p]}}$). It can easily be proved that $T$ is well defined since $B$ is symmetric. For if $f_1,g_1,f_2,g_2$ are simple functions with $f_1 g_1=f_2 g_2$, then $B(f_1,g_1)=B(f_2,g_2)$, and by continuity of $B$, $B(f,g)= B(g,f)$ for every couple $f \in X$ and $g \in X_{[p/p']}$. Further, $T$ is continuous also by the continuity of $B$ and Remark~\ref{rembasic}. Consequently, $B=T \circ B_0$ and the claim is proved.
Thus, (2) is equivalent to (4) as a consequence of Corollary~\ref{coroweakcom}, since the operator $T$ constructed in the Claim is weakly compact if and only if $B$ is weakly compact. \end{proof}
\end{document} | arXiv |
Research Article | Open | Published: 01 June 2017
The impact of poverty on dog ownership and access to canine rabies vaccination: results from a knowledge, attitudes and practices survey, Uganda 2013
Ryan MacLaren Wallace1,
Jason Mehal1,
Yoshinori Nakazawa1,
Sergio Recuenco1,
Barnabas Bakamutumaho2,
Modupe Osinubi1,
Victor Tugumizemu3,
Jesse D. Blanton1,
Amy Gilbert1 &
Joseph Wamala4
Rabies is a neglected disease despite being responsible for more human deaths than any other zoonosis. A lack of adequate human and dog surveillance, resulting in low prioritization, is often blamed for this paradox. Estimation methods are often employed to describe the rabies burden when surveillance data are not available, however these figures are rarely based on country-specific data.
In 2013 a knowledge, attitudes, and practices survey was conducted in Uganda to understand dog population, rabies vaccination, and human rabies risk factors and improve in-country and regional rabies burden estimates. Poisson and multi-level logistic regression techniques were conducted to estimate the total dog population and vaccination coverage.
Twenty-four villages were selected, of which 798 households completed the survey, representing 4 375 people. Dog owning households represented 12.9% of the population, for which 175 dogs were owned (25 people per dog). A history of vaccination was reported in 55.6% of owned dogs. Poverty and human population density highly correlated with dog ownership, and when accounted for in multi-level regression models, the human to dog ratio fell to 47:1 and the estimated national canine-rabies vaccination coverage fell to 36.1%. This study estimates there are 729 486 owned dogs in Uganda (95% CI: 719 919 – 739 053). Ten percent of survey respondents provided care to dogs they did not own, however unowned dog populations were not enumerated in this estimate. 89.8% of Uganda's human population was estimated to reside in a community that can support enzootic canine rabies transmission.
This study is the first to comprehensively evaluate the effect of poverty on dog ownership in Africa. These results indicate that describing a dog population may not be as simple as applying a human: dog ratio, and factors such as poverty are likely to heavily influence dog ownership and vaccination coverage. These modelled estimates should be confirmed through further field studies, however, if validated, canine rabies elimination through mass vaccination may not be as difficult as previously considered in Uganda. Data derived from this study should be considered to improve models for estimating the in-country and regional rabies burden.
Rabies virus is one of 14 Lyssaviruses, all of which are capable of causing the encephalitic disease known as rabies [1, 2]. While all Lyssaviruses appear to have evolved from a common ancestor that was associated with a chiropteran host, only rabies virus appears to have adapted to sustained transmission among terrestrial mammals (primarily Carnivora species) [1, 3]. Only rabies virus represents a current global health threat; responsible for an estimated 59 000 human deaths and over three billion US dollars in global economic losses annually [4]. The canine rabies virus variant (CRVV) is considered to be responsible for more than 95% of global human rabies deaths. Currently, more than two-thirds of the world's population resides in a CRVV enzootic country [1, 5].
The CRVV has been successfully eliminated in most developed countries through dog vaccination and targeted public and animal health interventions [6]. Unfortunately, the CRVV remains a significant disease burden in much of sub-Saharan Africa, where an estimated 19 000 rabies deaths occur annually [4]. Despite the advancement of successful interventions, they have not been successfully applied in the majority of sub-Saharan African countries [7]. The neglect of rabies in sub-Saharan Africa is largely attributed to a lack of recognition of rabies as a significant public health threat [8]. This fallacy has been addressed in numerous studies, but the stigma continues to negatively impact rabies control programs in much of the developing world [9,10,11]. Rabies surveillance is seen as one key activity to improve the recognition of the true public health burden, however, to date, surveillance for rabies is inadequate throughout most of Africa [8, 12, 13].
When surveillance data are lacking, risk models may be useful to describe the estimated burden of animal and human rabies [4, 14]. In Uganda, sparse surveillance data exist for the number of human and canine rabies cases, necessitating the use of modelled estimates to describe the burden of bites (6 602 to 15 778) and human rabies deaths (210 to 592) [15, 16]. Likewise, few studies have captured dog ecology or management information that would be relevant for producing more refined risk models [17]. This lack of country-specific data has resulted in the use of regional and continental data for rabies risk models, which may not be reflective of more refined geographic areas. Therefore, better refined and country-specific estimates of dog densities, rabies vaccination coverage, and barriers to canine vaccination are needed for more effective risk modelling and to inform strategies for rabies control. In the face of high numbers of animal bites and human rabies deaths in Uganda, a knowledge attitudes and practices (KAP) study was conducted for the purpose of enhancing canine rabies control programs in East Africa.
A KAP survey on dog ownership and rabies vaccination was conducted among 24 sites in Uganda over a 16 day period in August and September 2013. Five districts geographically distributed from east to west across Uganda were chosen based on two criteria: existence of bite reporting infrastructure and geographical representation of the country. Within each of the five districts, five administrative units were chosen at random utilizing a random number generator, for a total of 25 selected sites. Skip patterns were applied with a target of at least 40 homes per community, while ensuring even distribution. This study, protocol 6312, was approved by the Centers for Disease Control and Prevention's Human Research Office.
Surveys were administered to the head of the household or a resident aged ≥ 18 years when the head of the household was not available. One survey was conducted for each participating household. Each house was visited only once. Surveys were conducted in local languages. All survey responses were recorded on handheld personal digital devices (PDAs). Interview locations of participating households were recorded with GPS receivers for mapping purposes. Fingerprints or written informed consent were obtained for all respondents. Consenting respondents received a bar of soap for their participation, in addition to educational materials about rabies prevention and control. Team members were trained in survey and informed consent administration, GPS and PDA use, and project methods 5 days before beginning fieldwork.
Data were organized in a three-level hierarchical structure, with households clustered within villages and villages clustered within districts. An unconditional means model was fit, and the likelihood ratio test was used to evaluate the variation of the response between villages and between districts. Multivariable random intercept models were then fit as detailed below to evaluate the effects of household- and village-level characteristics.
Descriptive model: Dog ownership
Characteristics of dog ownership were examined using logistic regression modelling. Odds ratios (ORs) and corresponding 95% confidence intervals were computed; characteristics significantly associated with dog ownership in univariate analysis (P < 0.10) were then entered into a multivariable regression model. The statistical significance of each predictor was evaluated using the likelihood ratio test. Backward elimination was conducted and predictor variables were considered significant at P < 0.05. Adjusted ORs (aORs) and corresponding 95% CIs were calculated after controlling for other predictors in the model. Two levels of variables were included in the analysis: Household level characteristics and village-level characteristics (Appendix 1). Household-level characteristics examined as part of the models included: age group of respondent, education level of respondents, household size, years lived in house, livestock value, house building material quality, and rabies knowledge of respondent. House quality was determined by placing an integer value to the construction material of the roof, structure, front door, and windows (Appendix 2). The aggregate of these combined values were used to quantify housing quality. Village-level variables included population density (0 – 100, 101 – 500, 501 – 2 500, and > 2 500 people/km2), distance to nearest urban centre (0, 1 – 5 000, 5 001 – 20 000, and > 20 000 meters), and community poverty level (0 – 15, 16 – 35, 36 – 55, and > 55%). Distance to urban centre and population density were highly correlated with each other so only the variable that resulted in the most significant model was chosen.
Descriptive model: Dog vaccination practices
Characteristics associated with owner-reported previous rabies vaccination among owned dogs were examined using logistic regression modelling. Multivariable regression modelling was conducted as described above. Household-level characteristics examined included the variables listed above in addition to the variables: level of dog care provided, care of community dogs, and rabies education level of respondent. Village-level variables included population density, distance to nearest urban centre, and community poverty level.
National estimation of Dog population and canine vaccination coverage
Two multivariable random intercept regression models were developed to provide national estimates of the number of owned dogs and the number of vaccinated dogs. For these two models, village-level characteristics were examined by Poisson regression. Village-level characteristics were modelled as continuous variables with an added quadratic term, rather than categorical as used for the descriptive models, to allow for increased precision of national estimates. We obtained a human population map from LandScan (http://web.ornl.gov/sci/landscan/) and a poverty index map from Worldpop (http://www.worldpop.org.uk/) for Uganda, both with spatial resolution of 1 km2. All characteristics and relevant interaction terms were entered into multivariable modelling. Backward elimination was performed for model selection as described above.
To estimate the number of owned dogs, a Poisson regression model was developed to estimate the village-level ratio of humans to owned dogs (H:D ratio). These regression coefficients from the final model were multiplied by the human population in 9 km2 areas nationwide to produce national dog population estimates. A second model was constructed which estimated the village-average number of vaccinated dogs per person. Regression coefficients from this final model were applied in the manner described above and the estimated number of vaccinated dogs was divided by the estimated number of owned dogs, within the 9 km2 cells, to determine the proportion of rabies vaccinated dogs. Three maps were produced for the whole country representing: a) the estimates of number of dogs, b) estimates of vaccinated dogs, and 3) proportion of vaccinated dogs with respect of the total dog population within each cell.
Estimating human rabies risk
Maintenance of enzootic canine rabies transmission is unlikely in areas with dog densities below 4 dogs/km2, and areas where the proportion of vaccinated dogs is 70% or higher [5, 14, 18]. Therefore, based on these premises, we identified human populations within 9 km2 areas in which the CRVV is more likely to be maintained (population density ≥ 4 dogs/km2 and vaccination below 70%) and thus, represent areas of elevated risk for enzootic rabies transmission (Fig. 3). Human rabies risk was calculated as the rate of unvaccinated dogs per 1 000 human population within the 9 km2 areas. This rate was stratified into seven categories to allow for refined estimates of risk.
Five districts, representing three of Uganda's four administrative regions, were chosen for inclusion into this study: Kampala, Wakiso, Mbale, Kabarole, and Bundibugyo (Appendix 3). One of the 25 villages could not be surveyed during the study period. A total of 1 000 households were approached, of which 798 completed the survey (range 12–71 surveys per village). The 798 respondents represented a total household study population of 4 375 (5.5 people per household). Dogs were owned by 12.9% of the households (range 0–41.7% per district village), for a total of 175 dogs (H:D ratio 25:1). Population density of the 24 villages surveyed varied greatly (2–2 429 people per km2) (Table 1). Village poverty levels also varied greatly (5.4–72.6% of residents in poverty). The average poverty level (measured as percent of people living below the international poverty line of US $1.25 per day) among the villages in this study was 45%, compared to a Ugandan national average of 38% (http://www.unicef.org/infobycountry/uganda_statistics.html).
Table 1 Comparison of village characteristics from a survey assessing dog ownership practices: Uganda, 2013
Attitudes towards Dog ownership and rabies vaccination
The lowest rates of dog ownership and dog densities were observed within villages with the highest poverty levels, ≥ 56% (58.3 people per dog vs average 25.0) (Table 2). The annual canine death rate was 101 deaths per 1 000 dogs (10.1%). The most commonly reported cause of dog death was disease, which was implicated in 39.3% of deaths, followed by injury (36.5%) and unknown causes (15.7%). Disease deaths were more frequently reported among dogs from villages with poverty levels > 35% (42.9% and 52.2% of dog deaths in the two highest poverty categories).
Table 2 health indicators for owned dogs by community poverty level, Uganda 2013
Of the 175 owned dogs identified in this study, 99 had a reported history of rabies vaccination (56.6%) (Table 2). Dogs were more likely to have a history of vaccination when they resided in low poverty villages (100, 70.6, 13.7, and 11.0%, respective to low-high poverty rate). Suspected rabies deaths among dogs were reported from the two highest poverty categories (n = 5 and 4, respectively) but none were reported from the two lowest poverty categories. The rate of suspected canine rabies among dogs in the study population was 5.1 per 1 000 dogs (range 0–9.2).
Owners reported that 31.4% of dogs were always allowed to roam freely and 21.7% were always confined to the owner's property, 4% of owners reported an unknown confinement status; the remaining dogs were intermittently free-roaming (Table 3). Overall, 74.3% of dogs were allowed to roam freely to some degree. Free roaming dogs were more frequently reported among villages in the two highest poverty classifications (78.0 and 74.2% dogs free-roaming) compared to villages in the two lowest poverty classifications (61.1 and 64.7%). The majority of dog owners provided their dog's food and water (95.1 and 81.6%), however fewer than half of owners provided their dogs with veterinary care or shelter (43.7 and 37.9%).
Table 3 Characteristics of Dog Ownership Practices by Community Poverty Level, Uganda, 2013
On average 52.4% of dog-owning households reported owning at least one dog that was not vaccinated against rabies (range 0–56.7%) (Table 3). The most commonly reported response for owning an unvaccinated dog was that "no vaccine was available" (50.0%), followed by "the government vaccination did not occur" (18.5%). Vaccine availability through the government and other sources was reported as a barrier to vaccination among dog owners residing in the two highest poverty categories (69 and 76.9%).
Overall, 79 of the 778 households reported that they provided some level of care to dogs which they did not own (10.3%) (Table 3). Providing care to community dogs was more frequently reported in higher poverty villages. The most common care provided to community dogs was food (9.8% of survey respondents), followed by water (3.6%). Veterinary care and shelter were almost never provided to dogs which were not owned by the survey respondents (0.1 and 0.3%, respectively). The number of unowned dogs in these villages could not be ascertained from the study design.
Multivariable logistic regression of Dog ownership and vaccination practices
The variables 'household size', 'livestock value', 'home building material quality', and 'village poverty level' were all significant in multivariable analysis (Table 4). Households with more than seven residents had 3.3 greater odds of owning a dog. Households which owned $1–$199 USD in livestock value were at 4.3 greater odds of owning a dog compared to households with no livestock value. Households with more than $1 000 USD in livestock value had the greatest odds of owning at least one dog compared to households with no livestock value (aOR = 19.6, 95% CI: 7.9–48.7). Households which were made of high quality building materials were at 2.6 greater odds of owning a dog compared to households consisting of low quality building materials (95% CI: 1.3–5.2). Households residing in a village with an average poverty level of 16–35% had 7.7 greater odds of owning a dog compared to households in villages of the lowest poverty category (95% CI: 1.5 – 40.0). Household dog ownership was not significantly associated with the two highest poverty classifications.
Table 4 Characteristics Associated with Household Dog Ownership by Univariate and Multivariable Methods, Uganda 2013
Among the variables considered for multivariable logistic regression to predict ownership of a vaccinated dog, only village population density, age of the dog, and the confinement of the dog remained in the adjusted model (Table 5). Dogs residing in villages with a human population density per km2 greater than 2 501 were at 7.9 greater odds of being vaccinated against rabies compared to dogs residing in villages with a human population density below 100 people per km2 (95% CI: 2.5–24.8). All dogs older than 1 year of age had greater odds of being vaccinated against rabies, compared to dogs less than 1 year of age. Dogs which were always confined to the owner's control had significantly greater odds of being vaccinated compared to dogs that were always allowed to roam freely (aOR = 25.4, 95% CI: 4.9–132.9). Economic indicators were not significantly associated with household dog vaccination in the adjusted model.
Table 5 Characteristics associated with canine vaccination rates by univariate and multivariable methods, Uganda 2013
National estimations of owned dogs and rabies vaccination coverage
Multilevel logistic regression for the prediction of dog population had two significant predictor variables, human population density and village poverty level, as well as the interaction term of these two variables (Fig. 1). When the regression model was extrapolated nationally to each 9 km2 cell in Uganda, a predicted total of 729 486 owned dogs were estimated for Uganda (95% CI: 719 919–739 053). Given a human population of 34 346 101, the national average H:D ratio was 47:1.
Comparison of Two Methods for Estimating the Density and Distribution of Owned Dog Populations, Uganda 2013. a Estimate of dog density based on constant ratio of dogs to humans based on findings from study: 1 dog for every 25 people. b Estimate of dog density based on multivariable random intercept regression models: Dog Population = ˗ 3.33 + (˗ 0.002 × human population density) + (˗ 2.27 × village poverty level) + (0.12 × human population density × village poverty level)
$$ \mathrm{Dog}\ \mathrm{Population} = - 3.33 + \left( - 0.002 \times \mathrm{human}\ \mathrm{population}\ \mathrm{density}\right) + \left( - 2.27 \times \mathrm{village}\ \mathrm{poverty}\ \mathrm{level}\right) + \left(0.12 \times \mathrm{human}\ \mathrm{population}\ \mathrm{density} \times \mathrm{village}\ \mathrm{poverty}\ \mathrm{level}\right) $$
Multilevel logistic regression for the prediction of the rabies vaccinated dog population had two significant predictor variables: human population density and village poverty level, as well as the interaction term of these two variables (Fig. 2). When the regression model was extrapolated nationally to each 9 km2 cell in Uganda, a predicted total of 257 995 owned, vaccinated dogs are expected to be present in Uganda, for an estimated national canine rabies vaccination rate of 35.4%.
Estimated Canine Rabies Vaccination Coverage, Uganda 2013. *Estimation based on modelled estimates: Dog Vaccination = Population Density + Poverty Level + (Population Density × Poverty Level). ** National canine rabies vaccination rate estimated to be: 35% with high levels in Kampala and low levels in rural areas
$$ \mathrm{Vaccinated}\ \mathrm{Dogs} = - 4.03 + \left(0.02 \times \mathrm{human}\ \mathrm{population}\ \mathrm{density}\right) + \left( - 4.22 \times \mathrm{village}\ \mathrm{poverty}\ \mathrm{level}\right) + \left(0.17 \times \mathrm{human}\ \mathrm{population}\ \mathrm{density} \times \mathrm{village}\ \mathrm{poverty}\ \mathrm{level}\right) $$
Based on modelled estimates, only 9.8% of the Ugandan population resides in an area in which over 70% of the dogs are expected to have had any history of vaccination against rabies (Fig. 3). An additional 206 916 Ugandans are estimated to reside in areas where dog population densities are below 4 dogs/km2. The remaining 30 847 460 Ugandans (89.8%) reside in areas where there is the theoretical possibility for enzootic canine rabies transmission (Fig. 3).
Risk of Canine Rabies Transmission as Displayed by the Number of Unvaccinated Dogs per 1 000 Human Population, Uganda 2013. *Model estimated were used to predict the number of unvaccinated dogs per 1 000 human population. Areas with vaccination coverage > 70% are identified in grey, as enzootic transmission is not thought to occur at these vaccination levels. Areas with fewer than 4 dogs per square kilometre are identified in black, as the dog population density may be too low to support enzootic transmission of the virus. However, areas in black are still susceptible to epizootic events when rabid animals are introduced to the community, such as the case with importing dogs from other rabies enzootic communities. The areas remaining in red are places with estimated large populations of both people and unvaccinated dogs, representing a greater risk for dog to human rabies transmission events
Understanding the distribution and ecology of dog populations is critical for the planning and implementation of effective canine rabies control strategies. In addition, this knowledge can aid development of more accurate estimation methods for the burden of animal and human rabies deaths. The latter is often necessary in many developing countries where surveillance efforts are inadequate to accurately describe disease burden. This study represents one of the most comprehensive attempts to characterize the dog population and rabies risk in Uganda.
Dog ownership and poverty
Accurate estimates of dog populations are critical for the planning of mass rabies vaccination programs. Most studies calculate dog populations as a factor of human population or land mass (km2) [17]. To our knowledge, this is the first study to conduct a comprehensive analysis of the impact of poverty on dog ownership, and our findings support that this interaction greatly effects the total estimated dog population. The rate of dogs per person for the African continent have been estimated at 21:1 (urban) and 7:1 (rural) [14, 17]. The unadjusted H:D ratio in this study is in line with these regional and continental estimates (25 people per dog); and applying this rate to Uganda's population of 36 million people, results in a national dog population of 1.3 million. However, there was a clear and strong association between dog ownership and village poverty identified in this study. When considering the effect of poverty, the adjusted estimated H:D ratio was nearly 2-fold higher (47:1) compared to the unadjusted estimates.
The model developed as part of this study suggests that areas with high poverty/low population density owned fewer dogs (i.e. poor, rural settings). Likewise, areas with low poverty/high population density owned fewer dogs (i.e. affluent, urban settings). However, areas with high poverty and high population density had a positive correlation with dog ownership (i.e. poor, urban settings). While surprising that the modelled estimates are much lower than the non-adjusted H:D ratio, there are several other African countries that have reported similar findings through differing population estimation methods [19, 20]. These results suggest that modelling of dog populations is likely not as simple as applying a standardized rate to a human population, and that poverty levels should be considered a potential confounder in the relationship between man and dog.
There are several methods described for estimating dog populations, including street counting, capture-recapture, registry records, and KAP surveys. However, none of these methods are capable of accurately capturing all types of dogs (owned, community owned, and feral). For example, counting methods rely on the dog being visible to the counter and thereby accurately estimates only the free-roaming dog population in a community, but neglect the proportion of dogs that remain within the home. Registries and KAP studies rely on the self-reported 'ownership' of the dogs. Therefore, in these methods, community-owned and feral dogs may not be accurately counted. However, there are two reasons why this may not be a significant limitation under certain scenarios. If the goal of the dog population estimation is to inform a national vaccination strategy which only utilizes point-source or door-to-door vaccination, where only the owned dog population is reachable, then a KAP survey method would provide accurate data for planning such campaigns. Additionally, if there are relatively few community dogs within the population, then KAP survey methods may also be accurate.
A primary goal of a rabies vaccination program should be to describe the ownership status (owned, community, feral) and the confinement status (confined, semi-confined, free roaming) of the dog population. Most studies have shown agreement that feral dogs, which are dogs that survive on no directly provided human support, are rare (less than 1% of a dog population in the majority of settings) [21, 22]. However, the significance of community dogs on the overall dog population can vary greatly and often depends on the cultural and economic situation. The method used in this study only accurately accounts for the 'owned dog' population. Survey respondents were asked about their interactions with community dogs, of which 10% said they provided some level of care. Unfortunately, it is impossible to determine the degree to which including these dogs could impact the total dog population estimation, as community dogs, by definition, receive care from numerous sources and many homes likely reported overlapping dogs in this count. A simple sensitivity analysis, in which our modelled estimate is considered the lowest population, and a 10% increase due to community dogs would be the highest estimate, still is vastly lower than the unadjusted H:D ratio and regional H:D ratios that would be applied to Uganda if poverty was not considered (719 919–812 958 dogs vs 1 300 000 dogs).
There are potentially harmful consequences from under-estimating a dog population, such as the under vaccination of dogs which may increase the rabies burden in a community or lead to an increase in outbreaks [23, 24]. Therefore, this newly developed model for dog population estimation as a function of human population density and poverty should be validated through field studies utilizing alternative methods. If it is shown that the model developed in this analysis is accurate, this may provide added incentive for governments to increase vaccination programs, as the target of 70% would be more easily achieved.
Dog vaccination rates and poverty
Canine vaccination rates in the villages assessed through this study varied greatly, from a low of 0% vaccinated to a high of 100% vaccinated. Overall, the unadjusted canine vaccination rate in this study was surprisingly high (56.6%), yet still below the target for effective herd immunity (70%). However, on closer review this vaccination coverage rate was heavily biased by the poverty level of the community, and when accounting for this bias, the national canine rabies vaccination rate was down-adjusted to 35.4%.
A recent study on the global burden of rabies estimated 10% vaccination coverage for dogs in Uganda, far below the reported and modelled values found in this study [4]. This study design asked only if the dog had ever been vaccinated against rabies, and did not record when or how frequently the animal had been vaccinated. Therefore, it is likely that a proportion of these dogs would not qualify as 'properly vaccinated' by WHO standards (having received at least 2 vaccinations during lifetime) [5]. Additionally, this study does not reflect the vaccination practices of community dogs. Fewer than 2% of survey respondents indicated that they provided veterinary care to community dogs, so in places where community dogs make a significant proportion of the population the vaccination rate will likely be decreased. As a result, the level of rabies herd immunity in Uganda is likely to be lower than reported here. In reality, the true population-level vaccination coverage for rabies in Ugandan dogs likely lies between the previously estimated 10% and the value identified in this study.
Barriers to vaccination were frequently reported among study participants, particularly in villages with higher rates of poverty. Encouraging, however, was the finding that all dogs in higher income villages were reportedly vaccinated against rabies, an indication that successful vaccination strategies can be, and have been, implemented. The most commonly reported barrier to canine rabies vaccination was a lack of ability for the owner to procure vaccine, both privately and through government campaigns. This likely reflects the current situation in Uganda and many developing countries, in which canine rabies vaccine is typically available to dog owners only during periodic, nationally supported, vaccination campaigns. During years in which these national campaigns do not reach villages or in which not enough vaccines are procured, there are no other options. These findings, while not surprising, should emphasize the important public service role that governments must play to realize successful rabies vaccination programs.
A critical ecological measure that can help predict rabies vaccination success is the population turnover rate among dogs. Communities with high dog population turnover will require more frequent and intensive canine vaccination campaigns [23]. For example, a community with 70% vaccination coverage, but a 50% annual death rate among their dogs would see the level of herd immunity drop to 53% after only 6 months and 35% after one year. Justifiably, monitoring the overall health of the dog population is an important evaluation measure for rabies control programs [25]. In this study a canine death rate of 10% was identified, but ranged from 5% in low poverty areas to 14% in high poverty areas. These figures are actually much lower than other published studies in developing countries, which have shown population turnover rates reaching greater than 30% [26, 27]. The most common causes of death were injury and disease, both preventable through responsible dog ownership and provision of veterinary care. Improving dog ownership practices through promotion of animal welfare education, leash laws, and reliable access to veterinary care could have positive impacts on the canine vaccination rates and directly benefit humans through decreases in bite events and rabies deaths.
The benefits of canine rabies vaccination were displayed in this study, where it was shown that in areas of high vaccination coverage there were no owner-reported incidents of dog deaths which were consistent with rabies. However, among high poverty villages the canine rabies vaccination rates were less than 15%, and the rate of dog deaths suspected to be rabies were much higher. Of note, canine distemper virus may present with signs similar to rabies, and is common in Uganda, therefore the rate of suspected rabies may be lower than what was estimated here [28].
By understanding the dog ownership characteristics in representative communities, one can then extrapolate the information to larger areas and thereby make more informed national rabies control policies. In this study we quantified national dog densities utilizing country-specific data and obtained drastically different results from Knobel et al. [14], who utilized regional and global data (Fig. 1). It is apparent in Fig. 1 that a large portion of Uganda has very low expected dog densities. Recent publications have suggested that rabies cannot remain enzootic in areas for which the dog density is below approximately 4 per km2, identified in black in Fig. 3 [14, 18, 22]. While enzootic transmission may be unlikely, rabies outbreaks in these communities are still possible if vaccination rates are low and dogs from enzootic areas are introduced, a practice commonly documented in many canine rabies endemic countries [25]. By this logic, targeted vaccination of the surrounding higher dog-density communities may have regional impact on the rate of rabies, and may represent a more cost-effective method of eliminating the disease in dogs. Where resources are available, mass vaccination of all dogs is recommended, however where resources are limited they should be used to maximum efficiency. In these situations, modelling of dog populations and identification of areas in which dog vaccination would provide the most benefit to society should be conducted.
Estimating the human rabies risk
There are numerous factors that must be taken into account when trying to accurately predict the risk of human rabies. The data collected in this study should be used to refine these complex estimation models. However, a more simple approach to estimating human rabies risk was undertaken here, where the modelled outputs of canine vaccination coverage and dog density were used to approximate the areas in Uganda in which more than 70% of dogs were likely vaccinated against rabies and areas where fewer than 4 dogs per km2 are expected. From this analysis, it was determined that 89.8% of Uganda's human population (~30 000 000 people) is likely to live in a community that can support enzootic transmission of canine rabies. Approximately 60% of Uganda's population (26.5 million people) resides in areas where there are greater than ten unvaccinated dogs for every 1 000 people. Interestingly, Knobel et al. in 2005 estimated that 68% of Africans live at risk for rabies; a study which utilized completely different methods and data sources. While this study did not match Knobel's dog population estimates, there was agreement between Knobel and this study in regards to the large proportion of persons residing in areas of high-risk for rabies transmission [14].
These modelled estimates are meant to provide a proxy measure for the potential rabies activity in a country in which surveillance programs for human and animal cases are not adequate. These modelled estimates should be used to guide decisions on where to allocate rabies control resources and can be used to advocate for more support from the national and international communities. However, these estimates should not be used to replace routine rabies surveillance activities, as surveillance activities are critical both for the treatment of bite victims, monitoring of epidemiological changes, and evaluation of MCV programs. Furthermore, derivation of accurate estimates is an iterative process that should be repeated and refined as additional empirical data are available.
The results from this study represent some of the most comprehensive data on dog ecology, demographics, and vaccination coverage in Uganda and may be helpful to refine current national and regional rabies burden estimates. The significant association between poverty and dog ownership is likely not unique to Uganda, and other countries should consider exploring this relationship when conducting dog population estimation studies. Furthermore, the findings from this study should be used to enhance current mass canine rabies vaccination strategies in Uganda, through the strategic use of resources where they will have the greatest impact. However, this study has several limitations, including only reflecting the characteristics of the owned dog population. These types of models should always undergo a degree of validation before major programmatic changes are enacted. If evaluation studies are consistent with the findings in this study, canine rabies elimination in Uganda may be more feasible than previously thought. Unfortunately, until successful vaccination strategies are developed and implemented in Uganda, there are likely more than 26 million people that live with the daily risk of becoming exposed to the CRVV from an infected dog. This study provides some guidance on where rabies risks may be highest, and these communities should be engaged to implement rabies prevention activities. Studies which describe the ecology of dogs and characteristics of dog owners are necessary to develop a successful rabies control program and the findings from this study should be considered by national and international programs.
aOR:
Adjusted ORs
CRVV:
Canine rabies virus variant
H:D ratio:
Human to Dog population ratio
KAP:
Knowledge attitudes and practices
PDA:
Personal digital devices
Fooks AR, et al. Current status of rabies and prospects for elimination. Lancet. 2014;384(9951):1389–99.
International Committee on Taxonomy of Viruses. Virus Taxonomy. 2015. [11/28/2016]; Available from: http://www.ictvonline.org/virustaxonomy.asp.
Kuzmina NA, et al. A reassessment of the evolutionary timescale of bat rabies viruses based upon glycoprotein gene sequences. Virus Genes. 2013;47(2):305–10.
Hampson K, et al. Estimating the global burden of endemic canine rabies. PLoS Negl Trop Dis. 2015;9(4):e0003709.
World Health, O. WHO Expert Consultation on Rabies. Second report. World Health Organ Tech Rep Ser. 2013;982:1–139. back cover.
Velasco-Villa A, et al. Enzootic rabies elimination from dogs and reemergence in wild terrestrial carnivores, United States. Emerg Infect Dis. 2008;14(12):1849–54.
Lembo T, et al. The feasibility of canine rabies elimination in Africa: dispelling doubts with data. PLoS Negl Trop Dis. 2010;4(2):e626.
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Cleaveland S, et al. Estimating human rabies mortality in the United Republic of Tanzania from dog bite injuries. Bull World Health Organ. 2002;80(4):304–10.
Sambo M, et al. The burden of rabies in Tanzania and its impact on local communities. PLoS Negl Trop Dis. 2013;7(11):e2510.
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Knobel DL, et al. Re-evaluating the burden of rabies in Africa and Asia. Bull World Health Organ. 2005;83(5):360–8.
Fevre EM, et al. The epidemiology of animal bite injuries in Uganda and projections of the burden of rabies. Trop Med Int Health. 2005;10(8):790–8.
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Lechenne M, et al. Operational performance and analysis of two rabies vaccination campaigns in N'Djamena, Chad. Vaccine. 2016;34(4):571–7.
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Cleaveland S. Royal Society of Tropical Medicine and Hygiene meeting at Manson House, London, 20 March 1997. Epidemiology and control of rabies. The growing problem of rabies in Africa. Trans R Soc Trop Med Hyg. 1998;92(2):131–4.
Hampson K, et al. Transmission dynamics and prospects for the elimination of canine rabies. PLoS Biol. 2009;7(3):e53.
Zinsstag J, et al. Transmission dynamics and economics of rabies control in dogs and humans in an African city. Proc Natl Acad Sci U S A. 2009;106(35):14996–5001.
Morters MK, et al. The demography of free-roaming dog populations and applications to disease and population control. J Appl Ecol. 2014;51(4):1096–106.
Kitala P, et al. Dog ecology and demography information to support the planning of rabies control in Machakos District, Kenya. Acta Trop. 2001;78(3):217–30.
Schildecker S, et al. Dog Ecology and Barriers to Canine Rabies Control in the Republic of Haiti, 2014–2015. Transbound Emerg Dis. 2016.
Millan J, et al. Serosurvey of dogs for human, livestock, and wildlife pathogens, Uganda. Emerg Infect Dis. 2013;19(4):680–2.
The authors would like to thank the survey team members who trekked through fields, over mountains, and in torrential rain to reach communities selected for this survey, and for their professionalism when conducting this comprehensive oral survey: Robert Musoke, Paul Oryema, Martin Erieza, Gloria Naggayi, Adelaine Karemani, Joseph Senzoga, Mukasa Hajra, Martha Naigaga.
This study was conducting with USAID funding. The funding agency had no role in study design, analysis, or interpretation of the data.
The data belongs to the Uganda Virus Research Institute and the Centers for Disease Control and Prevention. Original, de-identified, data files can be shared if a data-sharing agreement is reached, in which the intent of data use is agreed upon by all parties.
RMW: study design, data collection, analysis, manuscript preparation. JM: analysis, manuscript preparation. YN: analysis, manuscript preparation. SR: study design, data collection, analysis, manuscript preparation. BB: study design, data collection, analysis, manuscript preparation. MO: data collection, analysis. VT: study design, data collection, manuscript preparation. JDB: analysis, manuscript preparation. AG: study design, manuscript preparation. JW: study design, data collection, analysis, manuscript preparation. All authors read and approved the final manuscript.
This study, protocol #6312, was approved by the Centers for Disease Control and Prevention's Human Subjects Research Office.
The views expressed here are those of the authors and do not necessarily represent recommendations from the United States Centers for Disease Control and Prevention.
United States Centers for Disease Control and Prevention, Atlanta, GA, USA
Ryan MacLaren Wallace
, Jason Mehal
, Yoshinori Nakazawa
, Sergio Recuenco
, Modupe Osinubi
, Jesse D. Blanton
& Amy Gilbert
Uganda Virus Research Institute, Kampala, Uganda
Barnabas Bakamutumaho
Veterinary Public Health Division, Ministry of Health, Kampala, Uganda
Victor Tugumizemu
World Health Organization, Kampala, Uganda
Joseph Wamala
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Correspondence to Ryan MacLaren Wallace.
Multilingual abstracts in the five official working languages of the United Nations. (PDF 638 kb)
Appendix 1Appendix 2Appendix 3Appendix 4
Survey questionnaire
Interview Date:
Consent obtained (Note: Form requests confirmation of adult age)
How many years of schooling have you completed?
How many people live in your household?
How many children below the age of 18 live in your household?
How many years have you lived in this place?
(Surveyor assistant - observe and describe construction of house).
Floor – cement/tile/dirt/other:
Walls – cement/metal/mud/straw or palm leaves/other:
Roof – cement/metal/straw or palm leaves/other:
Windows – none/metal/curtain/other:
Door – none/metal/curtain/other:
What kind of livestock does your family own? How many head of each? Mark all that apply.
Chickens, indicate number
Cattle, indicate number
Goats, indicate number
Sheep, indicate number
Other: (free response), indicate number
Declined to answer
Does your family currently own any dogs? If yes, how many? (if answer is No, skip to 18)
Yes, indicate number
What are the ages of your dogs?
Free response
What best describes the amount of time that your dog(s) spends indoors?
Infrequently
What level of care do you provide for your dog(s)? Mark all that apply.
Other: (free response)
Have any of your dog(s) been vaccinated against rabies?
If any of your dog(s) have not been vaccinated for rabies, what is the reason?
No money to buy vaccine
No vaccine available
No need to vaccinate
Other (free response):
In the past five years, have you owned any dogs that died?
For the dogs that died, what was the cause of death? Indicate frequency of each if more than one dog.
Accident/injury
Disease/illness
Other: free response
Does your family care for any dogs in the community? If yes, how many? (if answer is No, skip to 22)
What level of care do you provide for the community dog(s)? Mark all that apply.
Have you or anyone in the household been bitten by a dog? Mark all that apply. (if answer is No, skip to 31)
Yes, me
Yes, an adult family member (indicate number if more than one)
Yes, my child (indicate number if more than one)
For each person identified, how old were you when you were bitten by the dog?
For each person identified, on how many separate occasions were you/they bitten by a dog? Mark all that apply and indicate frequency if multiple persons were identified.
One occasion
Two occasions
Three occasions
Four occasions
Five occasions
More than five occasions
For each person identified, where were you/they when you were bitten by the dog? Mark all that apply and indicate frequency if multiple persons were identified.
Not at home, but within local community
Outside of local community
For each person identified, what were you doing when you/they were bitten the dog? Mark all that apply and indicate frequency if multiple persons were identified.
At home, unprovoked attack by own dog
At home, unprovoked attack by community dog
Playing with, restraining or feeding the dog
Playing with, restraining of feeding puppies of the (bitch) dog
Visiting the dog's home
Walking in community, avoiding the dog
Herding livestock, avoiding the dog
Hunting wild animals, avoiding the dog
Playing or recreating outdoors, avoiding the dog
For each person identified, where on your body were you/they bitten by the dog? Mark all that apply and indicate frequency if multiple persons were identified.
Head/face
Torso/trunk
For each person identified, what did you/they do when bitten by the dog? Mark all that apply and indicate frequency if multiple persons were identified.
Washed wound
Consulted with a traditional healer
Call a medical doctor
Call a veterinarian
Actively sought medical treatment at a pharmacy, hospital, clinic or outpost
Received rabies post-exposure prophylaxis
Isolated the dog for observation
Submitted dog for disease testing
Killed the dog
Killed and ate the dog
(If answer to 27 was 'f' or 'g') What was the amount of time between when you/they were bitten and medical treatment was sought? Mark all that apply and indicate frequency if multiple persons were identified.
< 1 day
Other: (free text)
For each person identified, how familiar were you/they with the dog? Mark all that apply and indicate frequency if multiple persons were identified. Proceed to 33.
Own (family) dog
Neighbor's dog
Dog in community
Did not recognize dog
(if never been bitten by a dog) What would you do if you were bitten by a dog that you recognize or own? Mark all that apply.
Wash wound
Consult with a traditional healer
Actively seek medical treatment at a pharmacy, hospital, clinic or outpost
Receive rabies post-exposure prophylaxis
Isolate the dog for observation
Submit dog for disease testing
Kill the dog
Kill and eat the dog
(if never been bitten by a dog) What would you do if you were bitten by a dog that you do not recognize or own? Mark all that apply.
Isolate dog for observation
Submit animal for disease testing
If you saw a dog in your village that looked sick, what would you do? Mark all that apply.
Call local authorities
Call a friend
Avoid the animal
Scare (shoo) animal away
Submit the animal for disease testing
Does your family currently own any cats? If yes, how many?
Have you or anyone in this household had illness that was attributed to a pet/livestock animal bite? Mark all that apply. (if answer is No, skip to 38)
If the answer to 35 was yes, what type of animal was it?
Dog (indicate number if more than one)
Cat (indicate number if more than one)
Other: free response (indicate number if more than one)
If the answer to 35 was yes, what were the symptoms? Mark all that apply and indicate frequency if multiple persons were identified.
Skin rash/discoloration/ infection
Unusual bleeding (e.g. from nose/mouth)
Hypersalivation
Chest congestion
Altered mental state (dementia)
Unconsciousness/coma
Muscle weakness/paralysis
Vomiting or diarrhea or stomach cramps
Miscarriage/stillbirth
Have you or anyone in this household been bitten by a wild animal (including rats)? Mark all that apply. (if answer is No, skip to 47)
For each person identified, how old were you when you were bitten by the wild animal?
What kind of wild animal was it? Mark all that apply and indicate frequency if multiple persons were identified.
Monkey or other primate
For each person identified, on how many separate occasions were you/they bitten by a wild animal? Mark all that apply and indicate frequency if multiple persons were identified.
For each person identified, where were you/they when bitten by the wild animal? Mark all that apply and indicate frequency if multiple persons were identified.
For each person identified, what were you doing when you were bitten by the wild animal? Mark all that apply and indicate frequency if multiple persons were identified.
In home, the animal entered home
Walking in community, avoiding the animal
Playing with, restraining or feeding the animal
Herding livestock, avoiding the animal
Hunting other animals
Hunting the animal
Playing or recreating outdoors, avoiding the animal
For each person identified, where on your body were you/they bitten by the wild animal? Mark all that apply and indicate frequency if multiple persons were identified.
For each person identified, what did you do after you/they were bitten by the wild animal? Mark all that apply and indicate frequency if multiple persons were identified.
Isolated the animal for observation
Submitted animal for disease testing
Killed the animal
Killed and ate the animal
(If answer to 45 was 'f' or 'g') For each person identified, what was the amount of time between when you/they were bitten and when medical treatment was sought? Mark all that apply and indicate frequency if multiple persons were identified.
(if never been bitten by a wild animal) If you were bitten by a wild animal, what would you do? Mark all that apply.
Isolate the animal for observation
Kill the animal
Kill and eat the animal
If you saw a wild animal in your village that looked sick, what would you do? Mark all that apply.
Have you or anyone in this household had illness that was attributed to a wild animal bite? (if answer is No, skip to 52)
If answer to 49 was yes, what type of animal was it?
For each person identified, what were the symptoms? Mark all that apply and indicate frequency if multiple persons were identified.
How much do you know about a disease called rabies? Note: interviewer must evaluate.
I have never heard of rabies
Little knowledge (i.e., have heard of rabies/dog disease, but can't identify transmission routes or severity of disease)
Basic understanding (knowledge that rabies is both a highly fatal disease and is transmitted by dog bite)
Extensive knowledge (basic understanding plus knowledge of non-bite routes of exposure AND wildlife reservoirs besides dogs without prompting)
How severe is the disease called rabies?
Somewhat severe
Very severe, but possible to recover
Very severe, resulting in death
How do humans get rabies from an infected animal? Mark all that apply.
Observing the animal
Touching the animal
Contact with blood
Contact with saliva
Contact with urine/feces
What animals can be infected with rabies? Mark all that apply
Livestock (Cattle, sheep, goats, etc.)
Poultry (Chickens, ducks, geese, etc.)
Monkeys or other primate
If you thought that you had an exposure to an animal with rabies, what would you do?
Where do you normally go to receive medical treatment? Mark all that apply.
How far do you need to travel to receive medical care at this location? Indicate frequency if multiple locations were identified.
1-5 km
>30 km
How far away is the location where you could receive rabies vaccination?
Have you or anyone in this household ever received rabies post-exposure prophylaxis? Mark all that apply and indicate frequency if needed (if answer is no, skip to 64)?
Yes, pre-exposure prophylaxis
Yes, post-exposure prophylaxis
(If answer to 60 is 'a' or 'b') Why did you or someone in your household receive rabies pre-exposure or post-exposure prophylaxis?
Pre exposure – free response: (identify any reason(s) that apply)
Post exposure – free response: (identify any reason(s) that apply)
(If answer to 61 is 'b') What elements of post-exposure prophylaxis did you or someone in your household receive? Mark all that apply and ndicate frequency if needed.
Rabies vaccine – Indicate number of doses (days) that treatment was administered
Rabies immune globulin (serum) Indicate number of doses (days) that treatment was administered (Note: should only be on Day 0)
Anti-tetanus serum - Indicate number of doses (days) that treatment was administered (Note: not part of rabies PEP, but may be commonly administered for bite wounds)
Other – free response
Where would (or did) you go to receive rabies vaccination?
What are the primary obstacles for getting medical treatment in your community? Mark all that apply.
Lack of facilities to provide treatment
Lack of trained personnel at facilities to provide treatment
Lack of medicines at facilities for treatment
No means of transportation
No money to pay for treatment
Can't miss work
What do you know about veterinarians? Mark the best answer.
Person that provides care to sick or injured animals
Person that provides care to sick or injured humans
Person that provides care to sick or injured humans and animals
Person that provides education about animal health
Person that provides education about public health
Person that provides education about animal and public health
I don't know or have never heard of a veterinarian
Table 6 Scoring system for evaluating domicile construction quality
Study locations and community surveys completed in Uganda, 2013
Table 7 Human population by community canine rabies vaccination coverage rates, Uganda 2013 | CommonCrawl |
Daniel McNeela
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The Problem(s) with Policy Gradient
If you've read my article about the REINFORCE algorithm, you should be familiar with the update that's typically used in policy gradient methods. $$\nabla_\theta J(\theta) = \mathbb{E}_{\tau \sim \pi_\theta(\tau)} \left[ \left(\sum_{t} \nabla_\theta \log{\pi_\theta}(a_t \mid s_t)\right) \left(\sum_t r(s_t, a_t)\right)\right]$$ It's an extremely elegant and theoretically satisfying model that suffers from only one problem - it doesn't work well in practice. Shocking, I know! Jokes abound about the flimsiness that occurs when policy gradient methods are applied to practical problems. One such joke goes like this: if you'd like to reproduce the results of any sort of RL policy gradient method as reported in academic papers, make sure you contact the authors and get the settings they used for their random seed. Indeed, sometimes policy gradient can feel like nothing more than random search dressed up in mathematical formalism. The reasons for this are at least threefold (I won't rule out the possibility that there are more problems with this method of which I'm not yet aware), namely that
Policy gradient is high variance.
Convergence in policy gradient algorithms is sloooow.
Policy gradient is terribly sample inefficient.
I'll walk through each of these in reverse because flouting the natural order of things is fun. :)
Sample Inefficiency
In order to get anything useful out of policy gradient, it's necessary to sample from your policy and observe the resultant reward literally millions of times. Because we're sampling directly from the policy we're optimizing, we say that policy gradient is an on-policy algorithm. If you take a look at the formula for the gradient update, we're calculating an expectation and we're doing that in the Monte Carlo way, by averaging over a number of trial runs. Within that, we have to sum over all the steps in a single trajectory which itself could be frustratingly expensive to run depending on the nature of the environment you're working with. So we're iterating sums over sums, and the result is that we incur hugely expensive computational costs in order to acquire anything useful. This works fine in the realms where policy gradient has been successfully applied. If all you're interested in is training your computer to play Atari games, then policy gradient might not be a terrible choice. However, imagine using this process in anything remotely resembling a real-world task, like training a robotic arm to perform open-heart surgery, perhaps? Hello, medical malpractice lawsuits. However, sample inefficiency is not a problem that's unique to policy gradient methods by any means. It's an issue that plagues many different RL algorithms, and addressing this is key to generating a model that's useful in the real world. If you're interested in sample efficient RL algorithms, check out the work that's being done at Microsoft Research.
Slow Convergence
This issue pretty much goes hand in hand with the sample inefficiency discussed above and the problem of high variance to be discussed below. Having to sample entire trajectories on-policy before each gradient update is slow to begin with, and the high variance in the updates makes the search optimization highly inefficient which means more sampling which means more updates, ad infinitum. We'll discuss some remedies for this in the next section.
High Variance
The updates made by the policy gradient are very high variance. To get a sense for why this is, first considering that in RL we're dealing with highly general problems such as teaching a car to navigate through an unpredictable environment or programming an agent to perform well across a diverse set of video games. Therefore, when we're sampling multiple trajectories from our untrained policy we're bound to observe highly variable behaviors. Without any a priori model of the system we're seeking to optimize, we begin with a policy whose distribution of actions over a given state is effectively uniform. Of course, as we train the model we hope to shape the probability density so that it's unimodal on a single action, or possibly multimodal over a few successful actions that can be taken in that state. However, acquiring this knowledge requires our model to observe the outcomes of many different actions taken in many different states. This is made exponentially worse in continuous action or state spaces as visiting even close to every state-action pair is computationally intractable. Due to the fact that we're using Monte Carlo estimates in policy gradient, we trade off between computational feasibility and gradient accuracy. It's a fine line to walk, which is why variance reduction techniques can potentially yield huge payoffs.
Another way to think about the variance introduced into the policy gradient update is as follows: at each time step in your trajectory you're observing some stochastic event. Each such event has some noise, and the accumulation of even a small amount of noise across a number of time steps results in a high variance outcome. Yet, understanding this allows us to suggest some ways to alter policy gradient so that the variance might ultimately be reduced.
Improvements to Policy Gradient
Reward to Go
The first "tweak" we can use is incredibly simple. Let's take a look again at that policy gradient update. $$\nabla_\theta J(\theta) = \mathbb{E}_{\tau \sim \pi_\theta(\tau)} \left[ \left(\sum_{t} \nabla_\theta \log{\pi_\theta}(a_t \mid s_t)\right) \left(\sum_t r(s_t, a_t)\right)\right]$$ If we break it down into the Monte Carlo estimate, we get $$\nabla_\theta J(\theta) = \frac{1}{N} \sum_{i=1}^N \left[ \left(\sum_{t=1}^T \nabla_\theta \log{\pi_\theta}(a_t \mid s_t)\right) \left(\sum_{t=1}^T r(s_t, a_t)\right)\right]$$ If we distribute $\sum_{t=1}^T r(s_t, a_t)$ into the left innermost sum involving $\nabla \log \pi_{\theta}$, we see that we're taking the gradient of $\log \pi_\theta$ at a given time step $t$ and weighting it by the sum of rewards at all timesteps. However, it would make a lot more sense to simply reweight this gradient by the rewards it affects. In other words, the action taken at time $t$ can only influence the rewards accrued at time $t$ and beyond. To that end, we replace $\sum_{t=1}^T r(s_t, a_t)$ in the gradient update with the partial sum $\sum_{t'=t}^T r(s_{t'}, a_{t'})$ and call this quantity $\hat{Q}_{t}$ or the "reward to go". This quantity is closely related to the $Q$ function, hence the similarity in notation. For clarity, the entire policy gradient update now becomes $$\frac{1}{N} \sum_{i=1}^N \left[ \left(\sum_{t=1}^T \nabla_\theta \log{\pi_\theta}(a_t \mid s_t)\right) \left(\sum_{t=t'}^T r(s_{t'}, a_{t'})\right)\right]$$
The next technique for reducing variance is not quite as obvious but still yields great results. If you think about how policy gradient works, you'll notice that how we take our optimization step depends heavily on the reward function we choose. Given a trajectory $\tau$, if we have a negative return $r(\tau) = \sum_{t} r(s_t, a_t)$ then we'll actually take a step in the direction opposite the gradient, which should have the effect of lessening the probability density on the trajectory. For those trajectories that have positive return, their probability density will increase. However, if we do something as simple as setting $r(\tau) = r(\tau) + b$ where $b$ is a sufficiently large constant such that the return for $r(\tau)$ is now positive, then we will actually increase the probability weight on $\tau$ even though $\tau$ still fares worse than other trajectories with previously positive return. Given how sensitive the model is to the shifting and scaling of the chosen reward function, it's natural to ask whether we can find an optimal $b$ such that (note: we're using trajectories here so some of the sums from the original PG formulation are condensed) $$\frac{1}{N} \sum_{i=1}^N \nabla_\theta \log \pi_\theta(\tau_i) [r(\tau_i) - b]$$ has minimum variance. We call such a $b$ a baseline. We also want to ensure that subtracting $b$ in this way doesn't bias our estimate of the gradient. Let's do that first. Recall the identity we used in the original policy gradient derivation $$\pi_\theta(\tau) \nabla \log \pi_\theta(\tau) = \nabla \pi_\theta(\tau)$$ To show that our estimator remains unbiased, we need to show that $$\mathbb{E}\left[\nabla \log \pi_\theta(\tau_i)[r(\tau_i) - b]\right] = \mathbb{E} [\nabla \log \pi_\theta(\tau_i)]$$ We can equivalently show that $\mathbb{E} [\nabla \log \pi_\theta(\tau_i) b]$ is equal to zero. We have \begin{align*} \mathbb{E} [\nabla \log \pi_\theta(\tau_i) b] &= \int \pi_\theta(\tau_i) \nabla \log \pi_\theta(\tau_i) b \ d\tau_i \\ &= \int \nabla \pi_\theta(\tau_i) b \ d\tau_i \\ &= \nabla b \int \pi_\theta(\tau_i) \ d\tau_i \\ &= \nabla b 1 \\ &= 0 \end{align*} where we use the fact that $\int \pi_\theta(\tau_i) \ d\tau_i$ is 1 because $\pi_\theta$ is a probability distribution. Therefore, our baseline enhanced version of the policy gradient remains unbiased.
The question then becomes, how do we choose an optimal setting of $b$. One natural candidate is the average reward $b = \frac{1}{N} \sum_{i=1}^N r(\tau_i)$ over all trajectories in the simulation. In this case, our returns are "centered", and returns that are better than average end up being positively weighted whereas those that are worse are negatively weighted. This actually works quite well, but it is not, in fact, optimal. To calculate the optimal setting, let's look at the policy gradient's variance. In general, we have \begin{align*} Var[x] &= \mathbb{E}[x^2] - \mathbb{E}[x]^2 \\ \nabla J(\theta) &= \mathbb{E}_{\tau \sim \pi_\theta(\tau)} \left[ \nabla \log \pi_\theta(\tau) (r(\tau) - b)\right] \\ Var[\nabla J(\theta)] &= \mathbb{E}_{\tau \sim \pi_\theta(\tau)} \left[(\nabla \log \pi_\theta(\tau) (r(\tau) - b))^2\right] - \mathbb{E}_{\tau \sim \pi_\theta(\tau)} \left[ \nabla_\theta \log \pi_\theta(\tau) (r(\tau) - b)\right]^2 \end{align*} The rightmost term in this expression is just the square of the policy gradient, which for the purposes of optimizing $b$ we can ignore since baselines are biased in expectation. Therefore, we turn our attention to the left term. To simplify notation, we can write $$g(\tau) = \nabla \log \pi_\theta(\tau)$$ Then we take the derivative to get \begin{align*} \frac{dVar}{db} &= \frac{d}{db} \mathbb{E}\left[ g(\tau)^2(r(\tau) - b)^2\right] \\ &= \frac{d}{db}(\mathbb{E}[g(\tau)^2r(\tau)^2] - 2 \mathbb{E}[g(\tau)^2r(\tau)b] + b^2\mathbb{E}[g(\tau)^2]) \\ &= 0 -2\mathbb{E}[g(\tau)^2r(\tau)] + 2b\mathbb{E}[g(\tau)^2] \end{align*} Solving for $b$ in the final equation gives $$b = \frac{\mathbb{E}[g(\tau)^2r(\tau)]}{\mathbb{E}[g(\tau)^2]} $$ In other words, the optimal setting for $b$ is to take the expected reward but reweight it by expected gradient magnitudes.
Hopefully this provided you with a good overview as to how you can improve implementations of policy gradient to speed up convergence and reduce variance. In a future article, I'll discuss how to derive an off-policy version of policy gradient which improves sample efficiency and speeds up convergence. | CommonCrawl |
\begin{document}
\title{f On a generalization\ of the Gauss's formula}
\begin{abstract} In this paper we study a group theoretical generalization of the well-known Gauss's formula that uses the generalized Euler's totient function introduced in \cite{11}. \end{abstract}
\noindent{\bf MSC (2010):} Primary 20D60, 11A25; Secondary 20D99, 11A99.
\noindent{\bf Key words:} Gauss's formula, Euler's totient function, finite group, order of an element, exponent of a group.
\section{Introduction}
The {\it Euler's totient function} (or, simply, the {\it totient function}) $\varphi$ is one of the most famous functions in number theory. Notice that the totient $\varphi(n)$ of a positive integer $n$ is defined to be the number of positive integers less than or equal to $n$ that are coprime to $n$. The totient function is important mainly because it gives the order of the group of all units in the ring ($\mathbb{Z}_n$, +, $\cdot$). Alternatively, $\varphi(n)$ can be seen as the number of generators or as the number of elements of order $n$ of the finite cyclic group ($\mathbb{Z}_n$, +).
Recall also a well-known arithmetical identity involving the totient function, namely the {\it Gauss's formula} $$\displaystyle \sum_{d\mid n}\varphi(d)=n, \hspace{1mm}\forall\hspace{1mm} n\in\mathbb{N}^*.\leqno(1)$$
Many generalizations of the totient function are known (for example, see \cite{2,3,5,8} and the special chapter on this topic in \cite{6}). From these, the most significant is probably the {\it Jordan's totient function} (see \cite{1}).
The starting point for our discussion is given by the paper \cite{11}, where a new group theoretical generalization of the totient function has been studied. This is founded on the remark that $\varphi(n)$ counts in fact the number of elements of order $\exp(\mathbb{Z}_n)$ in ($\mathbb{Z}_n$, +). Consequently, it makes sense to define
$$\varphi(G)=|\{a\in G \mid o(a)=\exp(G)\}|$$for any finite group $G$. It is obvious that $\varphi(\mathbb{Z}_n)=\varphi(n)$, for all $n\in\mathbb{N}^*$, and so a generalization of the classical totient function $\varphi$ is obtained. We observe that for $G\cong\mathbb{Z}_n$ the Gauss's formula can be rewritten as
$$\displaystyle \sum_{H\leq\, G}\varphi(H)=|G|\,.\leqno(2)$$This leads to the natural problem $$\textit{which are the finite groups $G$ satisfying the equality}\hspace{1mm} {\rm (2)\,?}$$Its study is the main goal of the current paper. We show that the cyclic groups are the unique abelian groups with this property. Inspired by some particular cases, we conjecture that this is also true for nilpotent groups. Moreover, we give examples of non-nilpotent groups $G$ satisfying (2). Several open problems on this topic are also formulated.
Most of our notation is standard and will not be repeated here. Basic definitions and results on groups can be found in \cite{4,9}. For subgroup lattice concepts we refer the reader to \cite{7,10}.
\section{Main results}
For a finite group $G$ let us denote \[ S(G)=\displaystyle \sum_{H\leq\, G}\varphi(H)\,. \] In this way, we are interested to describe the class ${\cal C}$
consisting of all finite groups $G$ for which $S(G)=|G|$.
Obviously, the finite cyclic groups are contained in ${\cal C}$, by the Gauss's formula. On the other hand, we easily obtain
$S(\mathbb{Z}_{2}\times\mathbb{Z}_{2})=7\neq 4=|\mathbb{Z}_{2}\times\mathbb{Z}_{2}|$, proving that ${\cal C}$ is not closed under direct products or extensions.
For a detailed study of the class ${\cal C}$, we must look first at some basic properties of the map $S$. We remark that it satisfies the inequality
$$S(G)\geq\displaystyle \sum_{H\in C(G)}\varphi(H)=\displaystyle \sum_{H\in C(G)}\varphi(|H|),\leqno(3)$$where $C(G)$ denotes the poset of cyclic subgroups of $G$. Another easy but very important property of $S$ is the following.
\begin{proposition}\label{th:C1}
$S$ is multiplicative, that is
if $(G_i)_{i=\overline{1,k}}$ is a family of finite groups of coprime
orders, then we have:
\[ S(\prod_{i=1}^k G_i)=\prod_{i=1}^k S(G_i). \] \end{proposition}
\begin{proof} Since the groups $(G_i)_{i=\overline{1,k}}$ are of coprime orders, we infer that every subgroup $H$ of $G=\prod_{i=1}^k G_i$ can be written as $H=\prod_{i=1}^k H_i$ with $H_i\leq G_i$, $\forall\hspace{1mm}i=\overline{1,k}$. By Lemma 2.1 of \cite{11}, we know that $\varphi$ is multiplicative and therefore $$\varphi(H)=\prod_{i=1}^k \varphi(H_i)\,.$$Then one obtains $$S(\prod_{i=1}^k G_i)=\displaystyle \sum_{H\leq\, G}\varphi(H)=\displaystyle \sum_{i=1}^k\displaystyle \sum_{H_i\leq\, G_i}\varphi(H_1)\varphi(H_2)\cdots\varphi(H_k)=$$ $$\hspace{-10mm}=\prod_{i=1}^k\left(\,\displaystyle \sum_{H_i\leq\,G_i}\varphi(H_i)\right)=\prod_{i=1}^k S(G_i)\,,$$as desired. \end{proof}
In particular, Proposition 1 shows that the computation of $S(G)$ for a finite nilpotent group $G$ is reduced to $p$-groups.
\begin{corollary} Let $G$ be a finite nilpotent group and $G_i$, $i=1,2,...,k$, be the Sylow subgroups of $G$. Then $$S(G)=\prod_{i=1}^k S(G_i).$$ \end{corollary}
\begin{proof} The equality follows immediately from Proposition 1, since a finite nilpotent group is the direct product of its Sylow subgroups. \end{proof}
Notice that for a finite abelian $p$-group $G$ the value $\varphi(G)$ has been precisely computed in Theorem 2.3 of \cite{11}. This is essential to show the following result.
\begin{theorem}\label{th:C1}
Let $G$ be a finite abelian group. Then $S(G)\geq |G|$, and we have equality if and only if $G$ is cyclic. \end{theorem}
\begin{proof} Remark first that we can assume $G$ to be a $p$-group, by Corollary 2. Let $(p^{\alpha_1}, p^{\alpha_2}, \dots, p^{\alpha_r})$ be the type of $G$ and assume that $\alpha_1 \leq \alpha_2 \leq \dots \leq \alpha_{s-1}<\alpha_s=\alpha_{s+1}= \dots =\alpha_r$. Then we have
$$\varphi(G)=|G|\left(1-\frac{1}{p^{\hspace{0,5mm}r-s+1}}\right)\geq |G|\left(1-\frac{1}{p}\right)\,.$$On the other hand, it is well-known that $G$ has $\frac{p^r-1}{p-1}$ maximal subgroups, namely $p^{r-1}$ subgroups isomorphic to $M_1=\mathbb{Z}_{p^{\alpha_1-1}}\times\mathbb{Z}_{p^{\alpha_2}}\times\cdots\times\mathbb{Z}_{p^{\alpha_r}}$, $p^{r-2}$ subgroups isomorphic to $M_2=\mathbb{Z}_{p^{\alpha_1}}\times\mathbb{Z}_{p^{\alpha_2-1}}\times\cdots\times\mathbb{Z}_{p^{\alpha_r}}$, ... , and one subgroup isomorphic to $M_r=\mathbb{Z}_{p^{\alpha_1}}\times\mathbb{Z}_{p^{\alpha_2}}\times\cdots\times\mathbb{Z}_{p^{\alpha_r-1}}$. One obtains $$\hspace{-16mm}S(G)\geq\varphi(G)+\displaystyle \sum_{i=1}^r p^{r-i}\varphi(M_i)+1\geq$$
$$\hspace{21mm}\geq |G|\left(1-\frac{1}{p}\right)+\displaystyle \sum_{i=1}^r p^{r-i}\,\displaystyle \frac{|G|}{p}\left(1-\frac{1}{p}\right)+1=$$
$$\hspace{-14mm}=|G|\displaystyle \frac{p^r+p^2-p-1}{p^2}+1\,.$$If $r\geq 2$, then
$$\displaystyle \frac{p^r+p^2-p-1}{p^2}\geq\displaystyle \frac{2p^2-p-1}{p^2}>1,$$implying that $$S(G)>|G|+1>|G|\,.$$Consequently, $G$ belongs to ${\cal C}$ if and only if $r=1$, i.e. if and only if it is cyclic. \end{proof}
Next we will focus on extending the above result from abelian $p$-groups to arbitrary $p$-groups, and consequently to arbitrary nilpotent groups. By a direct calculation, we infer that for all non-abelian $p$-groups $G$ of order $p^3$ (whose classification is well-known -- see e.g. \cite{9}, II) we have
$$S(G)>|G|\,.$$This inequality also holds for other classes of non-abelian $p$-groups $G$, determined by the existence of abelian subgroups of a given structure.
\begin{theorem}\label{th:C1}
Let $G$ be a non-abelian $p$-group of order $p^n$, $n\geq 4$. If $G$ has an abelian subgroup of order $p^m$ and rank $r$ with $m+r\geq
n+2$, then $S(G)>|G|$, i.e. $G$ is not contained in ${\cal C}$.
In particular, if $G$ has an elementary abelian maximal
subgroup, then it does not belong to ${\cal C}$. \end{theorem}
\begin{proof} Let $A$ be an abelian subgroup of order $p^m$ and rank $r$ of $G$, and assume that $m+r\geq n+2$. By the proof of Theorem 3, we infer that $$\hspace{-12mm}S(G)> S(A)\geq p^m\displaystyle \frac{p^r+p^2-p-1}{p^2}+1=$$ $$=p^{m+r-2}+p^{m-2}\left(p^2-p-1\right)+1\geq$$ $$\hspace{-4,5mm}\geq p^{m+r-2}+p^{m-2}+1>p^{m+r-2}\geq$$ $$\hspace{-56mm}\geq p^n,$$as claimed. \end{proof}
\begin{theorem}\label{th:C1}
Let $G$ be a non-abelian $p$-group of order $p^n$, $n\geq 4$. If $G$ has a cyclic maximal subgroup,
then $S(G)>|G|$, i.e. $G$ is not contained in ${\cal C}$. \end{theorem}
\begin{proof} By Theorem 4.1 of \cite{9}, II, we know that $G$ is isomorphic to \begin{itemize}\item[--] $M(p^n)=\langle x,y\mid x^{p^{n-1}}=y^p=1,\ y^{-1}x y=x^{p^{n-2}+1}\rangle$\end{itemize} when $p$ is odd, or to one of the following groups \begin{itemize}\item[--] $M(2^n)$ \item[--] the dihedral group $D_{2^n}$, \item[--] the generalized quaternion group $$Q_{2^n}=\langle x,y\mid x^{2^{n-1}}=y^4=1,\ yxy^{-1}=x^{2^{n-1}-1}\rangle,$$ \item[--] the quasi-dihedral group $$S_{2^n}=\langle x,y\mid x^{2^{n-1}}=y^2=1,\ y^{-1}xy=x^{2^{n-2}-1}\rangle$$\end{itemize} when $p=2$.
A common property of all these $p$-groups $G$ is that they have $p+1$ maximal subgroups, say $M_1$, $M_2$, ... , $M_{p+1}$, and (at least) one of them is cyclic, say $M_{p+1}\cong\mathbb{Z}_{p^{n-1}}$. Moreover, $\Phi(G)$ is cyclic of order $p^{n-2}$. Then, by applying the Inclusion-Exclusion Principle, one obtains $$S(G)=\varphi(G)+\displaystyle \sum_{i=1}^{p+1}S(M_i)-p\cdot S(\Phi(G))=\varphi(G)+\displaystyle \sum_{i=1}^{p+1}S(M_i)-p^{n-1}.\leqno(4)$$
For $M(p^n)$ it is easy to check that $p$ maximal subgroups are cyclic, say $M_i\cong \mathbb{Z}_{p^{n-1}}$, $i=2,3, ... , p+1$, and $M_1\cong \mathbb{Z}_p\times\mathbb{Z}_{p^{n-2}}$. Then $\varphi(M(p^n))=p\cdot\varphi(p^{n-1})=p^n-p^{n-1}$ and (4) leads to $$\hspace{-10mm}S(M(p^n))=p^n-p^{n-1}+S(\mathbb{Z}_p\times\mathbb{Z}_{p^{n-2}})+p\cdot p^{n-1}-p^{n-1}>$$ $$\hspace{17mm}>p^n-p^{n-1}+p^{n-1}+p^n-p^{n-1}=2\cdot p^n-p^{n-1}>p^n,$$according to Theorem 3.
For every $G\in\{D_{2^n}, Q_{2^n}, S_{2^n}\}$ we have $\varphi(G)=2^{n-2}$. Then (4) can be rewritten as $$S(G)=2^{n-2}+S(M_1)+S(M_2).\leqno(5)$$The pair $(M_1,M_2)$ of maximal subgroups of $D_{2^n}$, $Q_{2^n}$ and $S_{2^n}$ is $(D_{2^{n-1}},D_{2^{n-1}})$, $(Q_{2^{n-1}},Q_{2^{n-1}})$ and $(D_{2^{n-1}},Q_{2^{n-1}})$, respectively. Clearly, in the first two cases (5) becomes a recurrence relation which easily leads to $$S(D_{2^n})=2^{n+1}+(n-3)\cdot2^{n-2}>2^n$$and $$S(Q_{2^n})=(n+4)\cdot2^{n-2}>2^n,$$while for $G=S_{2^n}$ one obtains $$S(S_{2^n})=2^{n-2}+S(D_{2^{n-1}})+S(Q_{2^{n-1}})=(2n+9)\cdot2^{n-3}>2^n.$$This completes the proof. \end{proof}
Inspired by the previous results, we came up with the following conjecture.
\begin{conjecture}\label{th:C1}
Let $G$ be a finite nilpotent group. Then $S(G)\geq |G|$, and we have equality if and only if $G$ is cyclic. \end{conjecture}
Obviously, Conjecture 6 can be reformulated in the next way: \textit{the cyclic groups are the unique finite nilpotent groups contained in ${\cal C}$}. It leads to the natural assumption that ${\cal C}$ consists in fact only of the finite cyclic groups. This is not true, as shows the following elementary example.
\noindent{\bf Example.} Let $G$ be the non-abelian group of order $pq$, where $p<q$ are primes and $p\mid q-1$. The subgroup structure of $G$ is well-known: it possesses one subgroup of order 1, $q$ subgroups of order $p$, one subgroup of order $q$ and one subgroup of order $pq$. Then
$$S(G)=1+q\,\varphi(\mathbb{Z}_p)+\varphi(\mathbb{Z}_q)+\varphi(G)=1+q(p-1)+q-1=pq=|G|\,,$$i.e. $G$ belongs to ${\cal C}$.
In particular, the above example shows that the dihedral group $D_6$ is contained in ${\cal C}$. In fact we are able to characterize the containment to ${\cal C}$ for arbitrary dihedral groups $D_{2n}= \langle x,y\mid x^n=y^2=1,\ yxy=x^{-1} \rangle$, $n\geq 2$.
\begin{theorem}\label{th:C1}
The dihedral group $D_{2n}$ belongs to ${\cal C}$ if and only if $n$ is odd. \end{theorem}
\begin{proof} Let $n=2^k m$ with $k,m\in\mathbb{N}$ and $m$ odd. Then the lattice of divisors of $n$ can be written as the union of the sets ${\cal D}_i=\{\,2^i m'\hspace{1mm}\mid\hspace{1mm} m'\!\mid\! m\}$, $i=0,1,...,k$. On the other hand, for every divisor $d$ of $n$, $D_{2n}$ has one subgroup isomorphic to $\mathbb{Z}_d$, namely $\langle x^{\frac{n}{d}}\rangle$, and $\frac{n}{d}$ subgroups isomorphic to $D_{2d}$, namely $\langle x^{\frac{n}{d}},x^{i-1}y\rangle$, $i=1,2,...,\frac{n}{d}$\,. Recall that we have $\varphi(D_2)=1$, $\varphi(D_4)=4$, and $$\varphi(D_{2n})=\left\{\begin{array}{lll}
0,&n \equiv 1 \hspace{1mm}({\rm mod}\hspace{1mm} 2)\\
&&\hspace{1mm}\forall\hspace{1mm} n\geq 3\\
\varphi(n),&n \equiv 0 \hspace{1mm}({\rm mod}\hspace{1mm} 2)\end{array}\right.$$by Theorem 2.6 of \cite{11}. It follows that $$\hspace{-25mm}S(D_{2n})=\displaystyle \sum_{H\leq D_{2n}}\varphi(H)=\displaystyle \sum_{d\mid n}\left(\varphi(\mathbb{Z}_d)+\displaystyle \frac{n}{d}\,\varphi(D_{2d})\right)=$$ $$\hspace{20mm}=\displaystyle \sum_{d\mid n}\varphi(\mathbb{Z}_d)+\displaystyle \sum_{d\mid n}\displaystyle \frac{n}{d}\,\varphi(D_{2d})=\displaystyle \sum_{d\mid n}\varphi(d)+\displaystyle \sum_{i=0}^k\sum_{d\in{\cal D}_i}\displaystyle \frac{n}{d}\,\varphi(D_{2d})=$$ $$\hspace{3mm}=n+\displaystyle \sum_{m'\mid\, m}\displaystyle \frac{n}{m'}\,\varphi(D_{2m'})+\displaystyle \sum_{i=1}^k\sum_{m'\mid\, m}\displaystyle \frac{n}{2^i m'}\,\varphi(D_{2^{i+1}m'})=$$ $$\hspace{-72,5mm}=2n+\Sigma,$$where $$\Sigma=\displaystyle \sum_{i=1}^k\sum_{m'\mid\, m}\displaystyle \frac{n}{2^i m'}\,\varphi(D_{2^{i+1}m'}).$$Hence $S(D_{2n})=2n$ if and only if $\Sigma=0$. This happens if and only if $k=0$, i.e. $n$ is odd. \end{proof}
\noindent{\bf Remark.} By Theorem 7, we have $S(D_{2n})=2n$ for $n$ odd. An explicit value of $S(D_{2n})$ for $n$ even can be calculated, too. Let $n$ as above and let $m=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_s^{\alpha_s}$ be the decomposition of $m$ as a product of prime factors. We remark that $\varphi(D_{2^{i+1}m'})=\varphi(2^i m')=2^{i-1}\varphi(m')$, excepting the case $i=m'=1$ when $\varphi(D_{2^{i+1}m'})=3$. One obtains $$\hspace{10mm}\Sigma=\displaystyle \frac{3n}{2}-\displaystyle \frac{n}{2}+\displaystyle \sum_{i=1}^k\sum_{m'\mid\, m}\displaystyle \frac{n}{2m'}\,\varphi(m')=n+\displaystyle \frac{kn}{2}\sum_{m'\mid\, m}\displaystyle \frac{\varphi(m')}{m'}=$$ $$\hspace{-35mm}=n+\displaystyle \frac{kn}{2}\prod_{i=1}^s\left(\alpha_i+1-\frac{\alpha_i}{p_i}\right)$$and thus $$S(D_{2n})=3n+\displaystyle \frac{kn}{2}\prod_{i=1}^s\left(\alpha_i+1-\frac{\alpha_i}{p_i}\right).$$For example, we can easily check that $$S(D_{12})=23.$$
Next we observe that both the non-abelian groups of order $pq$ and the dihedral groups $D_{2n}$ with $n$ odd, which we verified to be contained in ${\cal C}$, are semidirect products of a cyclic normal subgroup $N$ by a cyclic subgroup $H$ of prime order satisfying $C_N(H)=1$. The containment of such a group to ${\cal C}$ can be also characterized, extending the above results.
\begin{theorem}\label{th:C1}
Let $G$ be a finite non-abelian group and $N\cong\mathbb{Z}_n$ be a normal Hall subgroup of $G$ which has a complement $H$ of prime order $p$ such that $C_N(H)=1$.
Then $G$ belongs to ${\cal C}$ if and only if the number of complements of $N$ in $G$ is $n$. \end{theorem}
\begin{proof} Under our hypotheses, $L(G)$ consists of the subgroups of $N$, say
$N_d$ with $d=|N_d|$, $d\mid n$, of the complements of $N$ in $G$, say $H_1=H$, $H_2$, ... , $H_{n_p}$, and of the semidirect products $N_dH_i$, with $d\mid n$, $d\neq 1$ and $i=\overline{1,n_p}$. Since $C_N(H)=1$, every $N_dH_i$ with $d\neq 1$ is not cyclic and so it does not contain elements of order $dp=\exp(N_dH_i)$. Consequently, we infer that $\varphi(N_dH_i)=0$ for all $d\mid n$ with $d\neq 1$ and all $i=\overline{1,n_p}$. This leads to $$S(G)=S(N)+\displaystyle \sum_{i=1}^{n_p}\varphi(H_i)=n+n_p(p-1).$$It is now obvious that $$S(G)=np \Longleftrightarrow n_p=n,$$which ends the proof. \end{proof}
We conclude that at least two important classes of finite groups are contained in ${\cal C}$: cyclic groups and semidirect products of type indicated in Theorem 8. Remark that these groups $G$ are supersolvable and that $S(G)$ equals the sum of all values of $\varphi$ on the cyclic subgroups of $G$, that is (3) becomes an equality.
Finally, we remark that every subgroup and every quotient of such a group also belong to ${\cal C}$, that is ${\cal C}$ seems to be closed under subgroups and homomorphic images.
We end this paper by indicating several natural problems on the above class ${\cal C}$.
\noindent{\bf Problem 1.} Prove or disprove Conjecture 6.
\noindent{\bf Problem 2.} Give a complete description of ${\cal C}$ (in our opinion, it consists of the finite cyclic groups and of non-abelian semidirect products of a certain type, most probably metacyclic groups). It is true that ${\cal C}$ is contained in the class of finite supersolvable groups?
\noindent{\bf Problem 3.} Study whether ${\cal C}$ is closed under subgroups and homomorphic images.
\vspace*{5ex}\small
\begin{minipage}[t]{5cm} Marius T\u arn\u auceanu \\ Faculty of Mathematics \\ ``Al.I. Cuza'' University \\ Ia\c si, Romania \\ e-mail: {\tt [email protected]} \end{minipage}
\end{document} | arXiv |
Relationship between maximum oxygen uptake and peripheral vasoconstriction in a cold environment
Takafumi Maeda ORCID: orcid.org/0000-0001-9073-27711,2
Various individual characteristics affect environmental adaptability of a human. The present study evaluates the relationship between physical fitness and peripheral vasoconstriction in a cold environment.
Seven healthy male students (aged 22.0 years) participated in this study. Cold exposure tests consisted of supine rest for 60 min at 28 °C followed by 90 min at 10 °C. Rectal and skin temperatures at seven sites, oxygen consumption, and the diameter of a finger vein were measured during the experiment. Metabolic heat production, skin heat conductance, and the rate of vasoconstriction were calculated. Individual maximum oxygen consumption, a direct index of aerobic fitness, was measured on the day following the cold exposure test.
Decreases in temperature of the hand negatively correlated with the changes in rectal temperature. Maximum oxygen consumption and the rate of vasoconstriction are positively correlated. Furthermore, pairs of the following three factors are also significantly correlated: rate of metabolic heat production, skin heat conductance, and the rate of vasoconstriction.
The results of this study suggested that the capacity for peripheral vasoconstriction can be improved by physical exercise. Furthermore, when exposed to a cold environment, fitter individuals could maintain metabolic heat production at the resting metabolic level of a thermoneutral condition, as they correspondingly lost less heat.
Human thermoregulatory functions are influenced by various factors, such as genetic factors, season, lifestyles, and individual physical and physiological characteristics [1,2,3,4]. Also, aerobic exercise capacity effects thermoregulatory function, and physical endurance training improves thermal adaptability.
Several studies have investigated the effects of physical training on thermoregulation in a hot environment [5,6,7,8], and the findings have suggested that physical training improves the capacity for thermoregulation. Many investigators have found improved ability to thermoregulate by cross-adaptation to exercise-induced hyperthermia, through improvements in and enhancements of vasodilation [5,6,7] and the sweat response [8].
Regarding thermoregulatory ability in a cold environment, physical endurance training increases cold tolerance, and individuals with higher levels of physical fitness exhibit higher adaptability to cold [9,10,11,12,13,14,15,16,17,18,19]. According to such studies, training increases metabolic heat production in a cold environment, which leads to a better cold tolerance [9,10,11, 13, 15, 18]. However, the effects of aerobic training on the ability to inhibit heat loss in a cold environment are controversial, because studies have indicated that skin temperature in fitter individuals exposed to cold can be either higher [11, 20] or lower [16, 19].
Previous studies have used skin heat conductance as an index of heat loss, from which the degree of vasoconstriction was estimated [11, 12, 18]. Some investigators have reported that the skin heat conductance of relatively fit individuals is greater than that of less-fit individuals during cold exposure and heat loss is more substantial [11, 12]. But others have found lower skin heat conductance and less heat loss among relatively fit individuals exposed to a cold environment when both trained and untrained groups had the same ratio of body fat [18]. Thus, the relationship between physical fitness and cold-induced vasoconstriction determined from skin heat conductance is obscure and probably influenced by body fatness.
Although skin heat conductance is reflected as heat loss and calculated as the differences between the core and skin temperature and between the skin and ambient temperature, it does not directly reflect vasoconstriction. Furthermore, because physical characteristics (particularly subcutaneous fat) affect skin heat conductance, isolating only the effects of physical fitness and/or training is difficult. Thus, vasoconstriction that is an index of cold tolerance cannot be evaluated by skin heat conductance, which also explains neither vasoconstriction nor the mechanisms involved in changes or improvements in physiological adaptability conferred by physical training.
Daanen (2003) in a review of local cold tolerance among humans noted the difficulties in noninvasively and continuously measuring blood vessel diameter as an index of vasoconstriction [21]. However, the vascular diameter can now be measured noninvasively and continuously using near-infrared light in Japan, which used for a clinical investigation [22].
Aerobic training improves the compliance of peripheral blood vessels [23,24,25,26], as well as the autonomic nervous function controlling the vasomotor system [27, 28]. Therefore, we considered that peripheral vasoconstriction would be improved by physical training, and thus heat loss would be more inhibited in a cold environment.
We postulated that fitter individuals have better vasoconstriction and better heat loss inhibition in the cold. The present study focused on the peripheral vasomotor system as a key factor involved in the inhibition of heat loss in a cold environment. The first objective was to determine the correlation between physical fitness and the degree of vasoconstriction measured directly on fingers. The second objective was to estimate the effects of aerobic physical fitness on the mechanisms of thermoregulation in the context of a cold environment, and, in particular, to determine the balance between increased metabolic heat production and the inhibition of heat loss.
The Ethics Committee of Fukushima Medical University approved the study protocol. The experimental test procedures were explained in detail to various individuals who then provided written informed consent to participate in the study. Seven healthy male students (age 22.0 ± 1.4 years old) volunteered to participate. All of them were healthy and belong to sports clubs such as soccer, rugby, and aikido which was conducted for 1~2 h/day and 2~3 days/week. The physical characteristics of the subjects are given in Table 1. Body surface area (BSA) was calculated from the height and weight of each participant using a formula adapted for adult Japanese males (Takahira, 1925: [BSA (cm2)] = [weight (kg)]0.425 × [height (cm)]0.725 × 72.76). Body fat (%) was measured on a scale using the impedance method with four electrodes (TBF-102, Tanita, Japan). Subcutaneous fat thickness was calculated as average skinfold thickness, which was measured with a caliper at the subzygomatic border, hyoid region, breast, side breast, subscapular region, abdomen, lumbar region, front and back thighs, knee, calf, and triceps. We assumed that the whole body subcutaneous fatness is related to finger subcutaneous fat, although we did not measure the finger subcutaneous fat thickness.
Table 1 Physical and physiological characteristics of participants
Cold exposure test
Rectal (T re) and skin temperatures at the forehead, abdomen, forearm, back of the hand, thigh, shin, and instep were measured at 1 min intervals using a thermistor-thermometer with a data-logger (LT-8, Gram Corp., Japan). The mean skin temperature (\( \overline{T} \) sk) was calculated from the seven points on the body using a method devised by Hardy and DuBois [29]. The diameter of the blood vessels (DBVs) at the second knuckle of the right middle finger was measured at 5 min intervals during the experiment using near-infrared spectroscopic imaging (Astrim, Sysmex Corp, Japan; Fig. 1). A near-infrared ray from a light emitting diode was passed through the blood vessels of a finger to hit the lens of a charge-coupled device camera, images from which DBV was calculated (Fig. 1) [22]. Oxygen consumption (\( \overset{\cdot }{\mathrm{V}} \)O2) was measured in a breath-by-breath manner (AE-300S, Minato Medical Science, Japan) during cold exposure.
System for measuring diameter of the finger blood vessels
Temperature sensors were fixed inside the rectum and at seven points on the skin, then a probe unit to measure DBV was fixed on the middle finger of the right hand. The mask for sampling expired gas was fixed on the mouth and nose, and then the participants rested in the supine position for at least 60 min in a room with ambient temperature and relative humidity maintained at 28 °C and 50%, respectively. Thereafter, the participants rested in the supine position for 90 min in a climatic chamber where the temperature and relative humidity were maintained at 10 °C and 50%, respectively. The subjects wore a short-sleeved cotton T-shirt and short cotton pants (0.3 clo).
After the experiment, percent of minimum DBV relative to the baseline (%DBV) was calculated as the minimum value divided by the baseline value [%DBV = (minimum DBV)/(baseline DBV) × 100]. The rate of vasoconstriction (%VC), percent change relative to the baseline, was also calculated using the equation as follows: %VC = 100 − %DBV. \( \overset{\cdot }{\mathrm{V}} \)O2 was used as an index of metabolic heat production [30], and the change in \( \overset{\cdot }{\mathrm{V}} \)O2 (%\( \overset{\cdot }{\mathrm{V}} \)O2) was calculated as the value at 90 min divided by the baseline value (0 min). Skin heat conductance (Kb, W/m2°C) was calculated from the following equation [11, 12, 18]:
$$ {K}_{\mathrm{b}}\kern0.5em =\kern0.5em \left(R\kern0.5em +\kern0.5em C\right)/\left(\ {T}_{\mathrm{re}}\kern0.5em \hbox{--} \kern0.5em {\overline{T}}_{\mathrm{sk}}\right), $$
where (R + C) indicates radiant and convective heat exchange (in W/m2), calculated as (R + C) = h (\( \overline{T} \) sk − T db), where h is the combined heat transfer coefficient for radiation and convection with a value of 8.3 W/m2 °C, that was determined as described by Colin et al. [31]. T re, \( \overline{T} \) sk, and T db are rectal, mean skin, and dry bulb (10 °C) temperatures, respectively.
Physical fitness test
Individual maximum oxygen consumption (\( \overset{\cdot }{\mathrm{V}} \)O2max) as a direct index of aerobic physical fitness was measured on the day after the cold exposure test to estimate the relationship between physical fitness and the observed findings. Oxygen consumption (\( \overset{\cdot }{\mathrm{V}} \)O2) was calculated in a breath-by-breath manner by ventilation and differences in oxygen concentrations of inspired and expired gases (AE-300S, Minato Medical Science, Japan) during physical exercise on a bicycle ergometer with a continuously incremental workload (+ 10 W/min). The test was terminated upon self-determined exhaustion or when the participant could no longer maintain the 50 rpm cadence. The criteria for achieving \( \overset{\cdot }{\mathrm{V}} \)O2max included a respiratory gas exchange ratio exceeding 1.0 and visible signs of exhaustion, such as breathlessness and inability to maintain the required power output [12]. In this manner, the relationship between aerobic capacity and physiological responses to cold was estimated.
The body temperature of one participant could not be recorded due to faulty sensors, and the DBV of another could not be recorded because the veins in the fingers were undetectable. Therefore, we analyzed the body temperature and DBV for six participants.
The average value of the last 5 min at 28 °C was used as the baseline value, and the average value for 88 to 90 min during cold exposure was used as the value at 90 min. The volume (∆) and rate of change (%) from the baseline value were also calculated, and some data were converted into logarithms. The measured value, ∆T rec, ∆\( \overline{T} \) sk, and %VO2 were used for statistical analysis.
In terms of rectal and skin temperatures, data at 0 and 90 min of cold exposure were analyzed by paired t test. Correlations between temperature decreases at the back of the hand (∆T bh) and in the rectal temperature (∆T re) compared with the respective baseline temperatures were analyzed using simple linear regression. Correlations between physical characteristics (\( \overset{\cdot }{\mathrm{V}} \)O2max, %fat, lean body mass (LBM), BSA, and subcutaneous fat thickness) and %\( \overset{\cdot }{\mathrm{V}} \)O2, K b, and %VC, as well as between %\( \overset{\cdot }{\mathrm{V}} \)O2 and %VC at the end of exposure to cold, were also analyzed using simple linear regression. P values below 0.05 were regarded as statistically significant.
T re and temperatures at all skin locations reached a steady state after resting for 60 min in a thermo-neutral room. T re of the two participants remained higher after cold exposure compared with before exposure, whereas in the other subjects, T re decreased. All skin temperatures continuously decreased during cold exposure without stability. Shivering and/or goosebumps developed in all participants exposed to cold, and they reported feeling "very cold" thereafter.
Table 2 indicates the mean ± SD of T re, \( \overline{T} \) sk, all skin temperatures before and after cold exposure, and of changes in T re, \( \overline{T} \) sk, and all skin temperatures after cold exposure. The decreases in T re were 0.18 ± 0.27 °C, which was not significant. Skin temperatures of each site and \( \overline{T} \) sk were significantly decreased by 90-min cold exposure. The decreases in \( \overline{T} \) sk were 7.89 ± 0.66 °C. Decreases in the forehead and abdominal (trunk region) skin temperatures were 5.32 ± 1.15 and 3.44 ± 0.33 °C, respectively, which were smaller than those at other sites of the skin. Skin temperature of the forearm, back of the hand, shin, and instep (peripheral sites) was markedly decreased by cold exposure, which was more than 10 °C (Table 2).
Table 2 Rectal and skin temperatures before and after cold exposure for 90 min
Figure 2 shows the relationship between ∆T bh that declined the most among the skin sites and ∆T re. The correlation between ∆T bh and ∆T re was significantly negative (r = − 0.989, P < 0.001) at the end of the cold exposure. The participants whose rectal temperatures remained higher in the cold had the lower skin temperatures at peripheral sites, such as the back of the hand.
Relationship between change in temperature of the skin on the back of the hand and change in rectal temperature after 90 min of cold exposure. T re and T bh indicate rectal and back of the hand temperatures, respectively
Figure 3 shows the transmission images of near-infrared light through the fingers before and after cold exposure for 90 min. The diameter of the blood vessels in all subjects was decreased by cold exposure for 90 min (baseline 7.5 ± 1.6 mm, 90 min 5.2 ± 1.7 mm, P < 0.01).
Near-infrared transmission image of light in the fingers before and after exposure to cold for 90 min
Figure 4 shows the relationships between \( \overset{\cdot }{\mathrm{V}} \)O2max and the % change in \( \overset{\cdot }{\mathrm{V}} \)O2 (Fig. 4a), K b (Fig. 4b), and the % VC (Fig. 4c) at 90 min during cold exposure. The correlation between \( \overset{\cdot }{\mathrm{V}} \)O2max and %\( \overset{\cdot }{\mathrm{V}} \)O2 at 90 min cold exposure was significantly negative (r = − 0.916, P = 0.004). Metabolic heat production in response to cold exposure increased less in a fitter than in a less-fit individual (Fig. 4a). The correlation between \( \overset{\cdot }{\mathrm{V}} \)O2max and Kb at 90 min during cold exposure was not significant (Fig. 4b, r = − 0.283, P = 0.586). With respect to the degree of vasoconstriction, although \( \overset{\cdot }{\mathrm{V}} \)O2max and DBV did not correlate, \( \overset{\cdot }{\mathrm{V}} \)O2max and %VC (r = 0.954, P = 0.003) calculated from the change in DBV were significantly and positively correlated (Fig. 4c). Thus, the peripheral blood vessels were more constricted during cold exposure in fitter than in less-fit individuals.
Relationships between maximum oxygen consumption (\( \overset{\cdot }{\mathrm{V}} \)O2max) and (a) the increase in oxygen consumption (%\( \overset{\cdot }{\mathrm{V}} \)O2), (b) skin heat conductance (K b), and (c) percent of vasoconstriction (%VC) after 90 min exposure to cold
Figure 5 shows the relationships between physical characteristics as %fat, LBM, subcutaneous fat thickness, and BSA and ln[%\( \overset{\cdot }{\mathrm{V}} \)O2] (four upper panels), K b (four middle panels), and %VC (four bottom panels) at 90 min of cold exposure. Physical characteristics (%fat, LBM, BSA, and subcutaneous fat thickness) did not influence either metabolic heat production (%\( \overset{\cdot }{\mathrm{V}} \)O2) or peripheral vasoconstriction (%VC). However, K b was significantly affected by %fat (r = − 0.936, P = 0.006), LBM (r = − 0.857, P = 0.029), BSA (r = − 0.836, P = 0.038) and subcutaneous fat thickness (r = − 0.841, P = 0.036).
Relationships between %fat, lean body mass (LBM), subcutaneous fat thickness, body surface area (BSA), change in increasing rate of oxygen uptake (%\( \overset{\cdot }{\mathrm{V}} \)O2), skin heat conductance (K b), and rate of vasoconstriction (%VC) after 90 min of cold exposure
Figure 6 shows the relationship between %VC and logarithms of the increase in \( \overset{\cdot }{\mathrm{V}} \)O2. There were negative relationships between %VC and ln[%\( \overset{\cdot }{\mathrm{V}} \)O2] (r = − 0.847, P = 0.034). Metabolic heat production was not increased in participants whose peripheral blood vessels were more constricted during cold exposure. Instead, metabolic heat production was increased in those whose blood vessels were less constricted.
Relationship between the rate of vasoconstriction (%VC) and the increase in oxygen uptake (%\( \overset{\cdot }{\mathrm{V}} \)O2)
This study found that (1) core temperature was better maintained in individuals with relatively low peripheral skin temperature, (2) vasoconstriction of the finger veins was more pronounced in individuals with higher maximum oxygen uptake than in others with lower maximum oxygen uptake, and that (3) metabolic heat production increased more from the baseline in individuals whose finger blood vessels were less constricted during cold exposure.
Peripheral vasoconstriction is important for thermoregulation in a cold environment, because the decrease in peripheral skin temperature, controlled by vasoconstriction, suppresses heat loss from the body surface and then the core temperature is better maintained. High ability of peripheral vasoconstriction enables the regulation of body temperature without the metabolic thermogenesis that is the second stage of the thermoregulation after the vasoconstriction.
Stromme and Hammel (1967) found that physically active rats produced more metabolic heat in a cold environment than inactive rats [32]. Bittel et al. (1988) also found a significant positive correlation between physical fitness and levels of metabolic heat production and skin heat conductance induced by acute exposure to cold at 1, 5, and 10 °C for 2 h in human [11]. These results showed that body temperature was not regulated only by suppression of heat loss caused by the vasoconstriction which was the first stage of thermoregulation; metabolic heat production which was the second stage occurred in not only inactive rat or lower fit person but also in active rat or higher fit person. It is thought that the increase of the metabolism to bring thermogenesis promotes movement of the heat to the skin, and then the heat loss increased. The results of the present study contradict these results in terms of metabolic heat production that negatively correlated with maximum oxygen uptake and skin heat conductance that did not correlate with maximum oxygen uptake. Maeda et al. (2007) indicated that metabolic heat production of individuals with a high basal metabolic rate, which was closely correlated with resting metabolic rate and muscle mass, was not increased during cold exposure because their core temperatures were only regulated by suppressing heat loss from the body surface [33]. The results of the present study might be influenced by not only the ability of vasoconstriction but also basal metabolic rate because it was though there was a relationship between maximum oxygen uptake and basal metabolic rate.
On the other hand, since individuals who are very fit also have more muscle mass, that is the main source of heat generation through shivering, the potential for heat production is high, and sufficient heat can be produced on demand to regulate core temperature. Accordingly, the present and previous results differ because the stages of thermoregulation were different. However, this raises the question of why the stages of the thermoregulation were different. Some explanations are that the actual living environment influences the acclimation of individuals, the contents of meals influence blood vessel compliance and/or heat production, and/or somatotype influences the ability to lose or generate heat. Body and subcutaneous fat might also be factors.
Bittel et al. (1988) indicated that the ratio of body fat significantly correlates with metabolic heat production and mean skin temperature [11]. In addition, Budd et al. (1991) also associated fat proportion with reduced heat production and heat loss, although physical fitness had no effect on heat production and heat loss during cold exposure [20]. However, Yoshida et al. (1998) reported that skin heat conductance was lower in the trained, than in untrained, individuals with the same body fat content [18]. The present study found that %fat and subcutaneous fat thickness were negatively related to skin heat conductance but unrelated to metabolic heat production and vasoconstriction. Body fat, especially subcutaneous fat, plays a role in the suppression of heat loss from internal to external milieus. Therefore, the negative correlation between subcutaneous fat and K b, which is an index of heat loss, is reasonable, and the results of this study agreed with those of a previous investigation [11, 20]. On the other hand, our findings of metabolic heat production did not agree with published results. Since an increase in metabolic heat production depends mainly on shivering of the muscle and not fat, body fat should not be related to metabolic heat generation. Previous results that have found a negative correlation between body fat and metabolic heat production might be caused by suppressing heat loss by body fat. However, these results might also include the influence of less ability of heat production, because persons with more fat might have less muscle, which would decrease metabolic heat production. In the present study, the body fat content and subcutaneous fat thickness were 16.6 ± 3.60% and 0.50 ± 0.17 cm, respectively, values that were within the normal range for young adult Japanese males. So it was thought that the effect of body fat on cold-induced metabolic heat production is very small in the present study which was the normal somatotype and narrow range of body fat variation of our study participants.
Most of the previous studies have estimated vasoconstriction under cold conditions only by measuring skin temperature at several sites, skin blood flow, and/or skin heat conductance, calculated using core, skin, and ambient temperatures. Here, we directly observed vasoconstriction in response to cold using near-infrared imaging of the finger blood vessels and found a significant positive correlation between maximum oxygen uptake and vasoconstriction. These results suggested that vasoconstriction ability is a sensitive index for evaluating cold tolerance.
The finding of greater vasoconstriction in fitter persons suggests increased sensitivity of the control of vasoconstriction to cold stimuli among such individuals. The mechanisms of physiological change-related thermoregulation associated with aerobic exercise are the improvements in compliance of the peripheral blood vessels [23,24,25,26], autonomic nervous function controlling the vasomotor system [27, 28], and vasomotor responses to thermal stimulation [34].
In conclusion, the present results suggested that the capacity for peripheral vasoconstriction is improved by physical exercise and that physical fitness is associated with enhanced ability to maintain metabolic heat production in a cold environment at resting metabolic levels of a thermoneutral condition. Fit individuals appear to have a greater capacity for vasoconstriction when exposed to cold and thus also appear to lose less heat than their less-fit counterparts.
%DBV:
Rate of change in diameter of the blood vessels
%\( \overset{\cdot }{\mathrm{V}} \)O2 :
Increase rate of oxygen consumption
(R + C):
Radiant and convective heat exchange
∆T bh :
Change in skin temperature at the back of the hand
BSA:
Body surface area
DBV:
Diameter of the blood vessels
K b :
Skin heat conductance
LBM:
\( \overset{\cdot }{\mathrm{V}} \)O2 :
Oxygen consumption
\( \overset{\cdot }{\mathrm{V}} \)O2max :
Maximum oxygen consumption
\( \overline{T} \) sk :
Mean skin temperature
T re :
Rectal temperatures
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The author wishes to thank all the participants who participated in this study.
This work was supported by JSPS KAKENHI Grant Numbers JP16687010, JP26291097.
All relevant data are within the paper.
TM contributed to the conception and design of study, the experiment and data collection, statistical analysis and interpretation, and drafted the manuscript.
Department of Human Science, Faculty of Design, Kyushu University, 4-9-1, Shiobaru, Minami-ku, Fukuoka, 815-8540, Japan
Physiological Anthropology Research Center, Faculty of Design, Kyushu University, 4-9-1, Shiobaru, Minami-ku, Fukuoka, 815-8540, Japan
Correspondence to Takafumi Maeda.
This study was approved by the Research Ethics Committee of the Faculty of Design, at Kyushu University (No. 209). The purpose of the study and the procedures were explained to the subjects before they provided consent.
The author declares that he/she has no competing interests.
Maeda, T. Relationship between maximum oxygen uptake and peripheral vasoconstriction in a cold environment. J Physiol Anthropol 36, 42 (2017). https://doi.org/10.1186/s40101-017-0158-2
Cross-adaptation
Vasoconstriction
Metabolic heat production | CommonCrawl |
Nicola Fergola
Nicola Fergola (1753–1824) was an Italian mathematician, professor in the University of Naples.
Nicola Fergola
Born(1753-10-29)29 October 1753
Naples, Kingdom of Naples, today Italy
Died21 June 1824(1824-06-21) (aged 70)
Naples, Kingdom of Two Sicilies, today Italy
Resting placeBasilica di San Paolo Maggiore (Naples)
40.85144°N 14.25683°E / 40.85144; 14.25683
Alma materUniversity of Naples
Scientific career
FieldsMathematics
InstitutionsLiceo del Salvatore
University of Naples
InfluencesVincenzo Flauti
Life and work
Fergola studied in the Jesuit school; he then went to the university of Naples in 1767,[1] but he studied mathematics by his own because the university was only strong in law and medicine. From 1770 he was teaching, by royal appointment, in the Liceo del Salvatore, a school founded in the same building where the Jesuit school had been (the Jesuit order was suppressed three years before).[2]
In 1799, during the Napoleonic period, he lived in Capodimonte but, when the Borbonic monarchy was restated, he was appointed to the mathematics chair in the Neapolitan university.[3]
In 1821 he suffered a stroke which left him disabled for the rest of his life.[4]
Fergola was one of the protagonists of an ideological quarrel among the Neapolitan scientists at the end of 18th and the first half of the 19th century. In the field of mathematics, the quarrel was about the use of synthetic or analytic methods. These polemics were coincident with the politically conservative conceptions of the former and the progressive views of the followers of the analytic method.
The Borbonic restoration in the kingdom of the Two Sicilies, with his ultraconservative profile, made possible the maintenance of this school until the Risorgimento, but at the end of 19th century it was absolutely forgotten.[5] To see a taste of the quarrel, here are the words pronounced by Gioacchino Ventura di Raulica in the obituary of Nicola Fergola:
Among the sciences, the mathematical ones are those which have taken the more false and disastrous direction. They were the first to be included in the assault of the philosophers against Christianity ...
— Cited by Mazzotti (1998), page 674.
The only work of Fergola is Prelezioni sui Principi matematici della filosofia naturale del cavalier Isacco Newton, published in two volumes in 1792 and 1793. It is interesting to see the religious point of view of the Newtonian force concept.
This religious conception is seen in all of Fergola's mathematical works. In 1839, was published Fergola's manuscript entitled Teorica de miracoli esposta con metodo dimostrativo in which Fergola tried to demonstrate the possibility of the miracles in a mathematical way: proposition, demonstration, theorem, lemma, scolium, i.e.
References
1. Fazzini, page 305.
2. O'Connor & Robertson, MacTutor History of Mathematics.
3. Fazzini, page 306.
4. Fazzini, page 307.
5. Botazzini, page 1500.
Bibliography
• Botazzini, U. (1994). "The Italian States". In Ivo Grattan-Guinness (ed.). Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences. London: Routledge. pp. 1495–1504. ISBN 0-415-09238-8.
• Fazzini, Antonio (1836). "Niccolo Fergola". Poliorama Pittoresco (in Italian): 305–307.
• Mazzotti, Masimo (2002). "The Making of the Modern Engineer". In Mordechai Feingold (ed.). History of Universities: Volume XVII. Oxford University Press. pp. 121–161. ISBN 0-19-925636-5.
• Mazzotti, Masimo (1998). "The Geometers of God: Mathematics and Reaction in the Kingdom of Naples" (PDF). Isis. 89 (4): 674–701. doi:10.1086/384160. hdl:10036/31212. ISSN 0021-1753. JSTOR 236738. S2CID 143956681.
External links
• O'Connor, John J.; Robertson, Edmund F., "Nicola Fergola", MacTutor History of Mathematics Archive, University of St Andrews
• "FERGOLA, Nicola". Dizionario Biografico degli Italiani (in Italian). Retrieved 8 August 2015.
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| Wikipedia |
Hybrid fractal acoustic metamaterials for low-frequency sound absorber based on cross mixed micro-perforated panel mounted over the fractals structure cavity
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Sanjeet Kumar Singh1,
Om Prakash3 &
Shantanu Bhattacharya1,2
Scientific Reports volume 12, Article number: 20444 (2022) Cite this article
Applied physics
The proposed work enumerates a hybrid thin, deep-subwavelength (2 cm) acoustic metamaterials acting as a completely new type of sound absorber, showing multiple broadband sound absorption effects. Based on the fractal distribution of Helmholtz resonator (HRs) structures, integrated with careful design and construct hybrid cross micro-perforated panel (CMPP) that demonstrate broad banding approximately one-octave low-frequency sound absorption behavior. To determine the sound absorption coefficient of this novel type of metamaterial, the equivalent impedance model for the fractal cavity and the micro-perforated Maa's model for CMPP are both used. We validate these novel material designs through numerical, theoretical, and experimental data. It is demonstrated that the material design possesses superior sound absorption which is primarily due to the frictional losses of the structure imposed on acoustic wave energy. The peaks of different sound absorption phenomena show tunability by adjusting the geometric parameters of the fractal structures like cavity thickness 't', cross perforation diameter of micro perforated panel, etc. The fractal structures and their perforation panel are optimized dimensionally for maximum broadband sound absorption which is estimated numerically. This new kind of fractals cavity integrated with CMPP acoustic metamaterial has many applications as in multiple functional materials with broad-band absorption behavior etc.
Numerous uses for the deep-subwavelength thick broadband low-frequency sound absorber can be found in acoustic cloaking and noise reduction. An acoustic metamaterial is an excellent candidate to tackle all challenges with careful design of structures that may possess extraordinary acoustic properties like broadband noise absorption1,2,3,4,5, sound insulation6,7,8, noise cloaking properties9,10, acoustic jetting properties11 etc. Acoustic metamaterials are well known as artificial or man-made structures that may be programmed through negative effective density12,13, negative effective modulus14,15, and simultaneous negative modulus and density16,17,18. Researchers have recently proposed 2D fractals acoustic metamaterials19 and 3D labyrinthine fractal acoustic metamaterials20, which can possess multi-band sound blocking properties in the low frequency domain. Another broadband low frequency sound isolator is designed through a spider web-inspired membrane-type meta-material21. Researchers may seek to find the lightweight structures of different material designs to possess excellent sound absorption to solve challenges related to noise control22,23. Further, it has long been a challenge to get broadband sound absorption while keeping thin and light weightedness as structural properties. Metamaterial design like multi-coiled structures24, can achieve perfect absorption at extreme low frequency of 50 Hz with a thickness of 1.3 but cannot tune once it is fabricated. Researchers have also tried conventional micro-perforated panels (MPP)25,26,27,28, with back cavity, Cascade neck-embedded Helmholtz resonators based metamaterials29, MPP with neck-embedded Helmholtz resonator30, and successfully achieved an overall good sound absorption level at low frequencies. However, the thickness of the backing cavity is usually more than 5 cm for obtaining a broadband sound absorption behavior. Ultrathin membrane metamaterials (MM)31,32, are a very good candidate for broadband sound absorption behavior but the problem in MM is membrane loosening effect which may come in due course of time after repeated use.
This article has developed a new type of tunable micro-perforated face-sheet design (with perforation diameter ≤ 1 mm) backed up by fractal geometry as shown in Fig. 1, of subwavelength dimensions that demonstrate excellent broadband sound absorption behavior. The thickness of this classical metamaterial design is less than 2 cm, and it can be easily programmed/tuned according to the industrial need and scope in different fields.
(a) Schematic of cross perforated fractal structure hybrid metamaterials panel composed of a cross-micro-perforated top face-sheet, a Helmholtz swastika fractals structure as core, and a back plate as bottom face-sheet. (b) One unit cell, sidewall cut-off vertically to see details inside. (c) CMPP with different perforation sizes in direction one having perforation diameter d1 and direction two is d2 and the back fractals cavity. Courtesy (ANSYS 17.045).
The series–parallel circuit analogy is applied to obtain an equivalent impedance method through which a theory is proposed, to calculate the sound absorption coefficient is established for this new class of fractal designs. This works also experimentally validates and compares with the theoretical model and a finite element model. Perfect sound absorption is achieved around 1000 Hz, together with broadband sound absorption starting from 400 to 1600 Hz, when the thickness of these unique metamaterials is about 20 mm. Almost perfect sound absorption has been found around 1000 Hz, together with one-octave relative absorption bandwidth starting from 600 Hz, when the thickness of the metamaterials is just 20 mm with the integration of two-unit cells. We have also integrated the four-unit cells to achieve broadband sound absorption and successfully achieved 61% higher relative bandwidth and sound absorption coefficient greater than 80%.
Theoretical model
The proposed acoustic metamaterial having two face-sheet and a swastika fractal structure core, as shown in Fig. 1. The cad model of the proposed metastructre is design in Design Modeler of ANSYS 17.045. The top face-sheet consist of cross perforation having uniform perforation diameter in direction '1' (d1) and direction '2' as (d2), in a micro-perforated panel (CMPP). The unique Helmholtz resonators are fractally distributed and act as an acoustic cavity core with the bottom face-sheet in the shape of a rigid backing plate. The fractal core is designed based on novel fractal shapes and contains multiple Helmholtz resonators along the arms of the fractal shape. In the first level branches, the unit shape (side branched Helmholtz resonator) is rotated by \(90^\circ ,180^\circ \, \mathrm{and} \, 270^\circ\) respectively to create four arms of the fractal structure and interconnected in a crossroad like manner. As the Helmholtz arm iterates 'n' times to a scaled down geometry connected to the base arm (Helmholtz resonators) at every 90°. Rotation angle a scale factor of 0.6 is used for every iteration. The final acoustic meta-structure up to which we investigate corresponds to an "n" value of 3 and the final core structure shown in Fig. 1c.
In general, the absorption coefficient of any acoustic metamaterials with a rigidly backed panel can be estimated through its impedance as
$$\propto =1-{\left|\frac{{z}_{s}/{z}_{0}-1}{{z}_{s}/{z}_{0}+1}\right|}^{2},$$
where, \({z}_{s}\) is known as the surface impedance of the acoustic absorber. \({z}_{0}={\rho }_{0}{c}_{0}\) is the characteristic impedance where \({\rho }_{0}\) and \({c}_{0}\) are mass density and sound speed in air, respectively. The surface impedance of the proposed fractal CMPP have been calculated as:
$${{z}_{s }=z}_{Mp}+{z}_{fc},$$
where, \({z}_{Mp}\) and \({z}_{fc}\) are the acoustic impedance of the CMPP and the fractal structure containing cavity, respectively.
The impedance of the side branch resonator Zr at X1 as shown in Fig. 1b in Supplementary Materials, is expressed as33,34,35
$${\text{Zr }} = \, - {\text{jZ}}_{{{\text{c}} }} {\text{cot}}\left( {{\text{kh}}} \right) \, + {\text{ Z}}_{{\text{h}}} ,$$
$${\mathrm{Z}}_{\mathrm{h}} = \frac{\rho c}{{s}_{h}} \left[ 0.0072+jk \left(l+0.75\right)\right],$$
where Zc = \(\rho c/{s}_{c}\) represents the impedance of the helmholtz cavity and Zh represents the neck impedance of the resonators as suggested by Seo et al.33. \(l\) is the neck length, \({s}_{c}\) is the cross sectional area of the cavity, \(h\) is the height of the cavity. here \(k=2\pi f/c\) is the wave number.
'Sh' is the cross- sectional area of the neck region. We first calculated the equivalent impedance of the fractals cavity \({z}_{fc}\) through an electrical analogy (see Supplementary Material for more details).
Impedance of the micro-perforated plate can be calculated by Maa's Model36,37
$${\mathrm{MPP} \, \mathrm{impedance } \quad z}_{m}=r+j\omega m,$$
$$r=\frac{32\eta }{\varnothing {\rho }_{0}{c}_{0}}\frac{t}{{d}^{2}} \left(\sqrt{1+\frac{{x}^{2}}{32}} +\alpha \frac{\sqrt{2}}{8} x \frac{d}{t}\right),$$
$$m=\frac{t}{\varnothing {c}_{0}} \left(1+\frac{1}{\sqrt{9+\frac{{x}^{2}}{2}}}+0.85 \alpha \frac{d}{t}\right),$$
where \(\varnothing =\) Porosity, \(\eta\) = dynamic viscosity, \(\alpha\) = perforation constant, d = diameter of hole, t = thickness of the perforated plate, \(x=d\sqrt{\frac{\omega {\rho }_{0}}{4 \eta }}.\)
We can now calculate the total impedance of the cross perforated plate of single unit as
$${z}_{Mp}={\left(\frac{1}{zm1}+\frac{1}{zm2}\right)}^{-1},$$
where \(zm1\) is the impedance of the CMPP in direction 1 and \(zm2\) is the impedance of the perforation in direction 2 as shown in Fig. 1c.
We are thus enabled to put the Eqs. (8) and (13) (Supplementary Material) as inputs to the Eq. (2) and we can thereby theoretically calculate the sound absorption coefficient of the cross micro-perforated fractal acoustic metamaterials with the help of Eq. (1). The absorption coefficient spectrum of the theoretical model is obtained by using MATLAB (R2016a)46.
Figure 2 depicts the broadband sound absorption in this case. As we can observe clearly that the initial peak of the absorption coefficient of the experimental result is not seen in the FEM simulation values, and similarly the last peak is not seen in the theoretical model. Although the experimental, theoretical, and numerical results have good consistency in terms of the amplitude at maximum absorption etc., in theoretical models, the linear superposition principle has been used as shown in Eq. (8). Thus, the nonlinear coupling effect of the perforation and cavity of the two distinctly truncated acoustic signals passing from the two different diameter holes on the cover plate is not considered while determining the theoretical spectrum. In numerical simulations, the CMPP has been assumed to be equivalent to a porous rigid body, and the visco-thermal losses across the fractal core has been considered to be negligiblemal38,39,40,41. Thus the first absorption peak as is distinctly observed in experimental data as well as theoretical prediction does not show up in the simulation data. It also observed that the experimental frequency bandwidth is wider than numerical and theoretical prediction due to additional loss of acoustic energy around the rough surface created by 3D printing42,43. A near-perfect absorption peak occurs around 1000 Hz with relative bandwidth of 50% for parameters d1 = 0.5 mm and d2 = 1 mm and has a porosity \({\mathrm{\o }}_{1}\) = 4.91% and \({\mathrm{\o }}_{2}\) = 19.63%, respectively. Here the relative bandwidth is calculated as the ratio of the full width at half the maximum of the absorption coefficient to the resonance frequency. There are two high absorption peaks corresponding to > 0.8 absorption coefficient at 700 Hz and > 0.95 absorption coefficient at 1000 Hz. The small differences in the results of the FEM and theoretical predictions occur due to neglecting the thermal dissipation at the perforation region and considering only viscous energy dissipation.
Sound absorption coefficient of CMPP predicted by the analytical method, FEM model and experimental results. Courtesy (MATLAB R2016a)46.
Broadband sound absorption
We start by varying the thickness of the CMPP fractal acoustic metamaterial to achieve varying sound absorption values in the lower frequency range. Given specific values of fixed cross perforation parameter (d1, d2) and porosity of \({\varnothing }_{1}\) and \({\varnothing }_{2}\) Variable fractal core thickness of 't' the sound absorption bandwidth is obtained at a particular frequency, as shown in Fig. 3a.
Sound absorption coefficient of CMPP with different acoustic parameters. (a) Thickness of the fractals core. (b,c) Porosity of CMPP. Courtesy (MATLAB R2016a)46.
As the thickness 't' of the fractal core increases, the sound absorption curve gradually shifts from high to low frequency at fixed perforation parameters d1 = 0.5 mm and d2 = 1 mm, respectively. A near-perfect sound absorption peak (96.66%) is obtained at 800 Hz with a relative absorption bandwidth of 50% when t = 30 mm. Similarly, 95.55% sound absorption at 850 Hz, 1150 Hz, 1100 Hz, and 1200 Hz with relative bandwidth (α > 0.5) of 50%, 30%, 29% and 27.6% is obtained as the thickness "t" becomes = 24 mm, 14 mm, 12 mm and 10 mm respectively.
We have also investigated the effect of cross porosity variation on the broadband sound absorption spectra. In the first sample, A1 we have created a geometry corresponding to d1 = 0.3 mm, \({\varnothing }_{1}\) = 7.07% and d2 = 0.4 mm, \({\varnothing }_{2}\) = 12.56%. Similarly, in the second sample A2, the geometrical parameters are changed to d1 = 0.5 mm, \({\varnothing }_{1}\) = 19.63% and d2 = 0.6 mm, \({\varnothing }_{2}\) = 28.26%, in the third sample A3 the parameters are d1 = 0.7 mm, \({\varnothing }_{1}\) = 38.46% and d2 = 0.8 mm, \({\varnothing }_{2}\) = 50.24%, and in the fourth sample, A4, the parameters are changed to d1 = 0.9 mm, \({\varnothing }_{1}\) = 63.59% and d2 = 1 mm, \({\varnothing }_{2}\) = 78.5% respectively. Further other four samples with different porosity combination (A5 to A8) with fractals core have been investigated and their acoustic behavior as shown in Fig. 3c.
The samples are investigated with fixed fractal thickness t = 20 mm. The sample 'A1' shows the maximum sound absorption peak (91.2%) to be at 950 Hz with relative bandwidth 58%. Similarly, for the sample A2 and A3 we obtain a similar behavior although as the cross-porosity ratio is increased the resonance frequency is shifted towards the right as shown in Fig. 3b. The sample A4 shows a relatively lower value of 65% absorption at the maximum perforation (details shown in Table 1). The two samples A1 and A5 shows the higher relative bandwidth of sound absorption with maximum sound absorption of 91% and 95%. So, we can tune the novel sound absorbers with a combination of different cross perforation diameters (≤ 1 mm) in each unit cell. For brevity, we have only shown eight combinations here as shown in Fig. 3b,c.
Table 1 Fractals CMPPs metamaterials parameters and its acoustic absorption behavior of a unit cell.
In order to expand the relative absorption bandwidth, we have further integrated two-unit cells with different cross porosities into one resonator as shown in Fig. 4b. Unit 1 and unit 2 have the same thickness and fractals cores but with different top CMPP geometries having perforation diameter d1, d2 of unit 1 and d3 and d4 for unit 2. We have also investigated a core containing a pair of unit cells integrated to CMPP and developed six samples S1, S2, S3, S4, S5 and S6 with different geometrical parameters having various combinations of perforation diameters (d1, d2 and d3, d4) and porosity ratios (\({\varphi }_{1}\), \({\varphi }_{2}\), \({\varphi }_{3}\) and \({\varphi }_{4})\) (Details in Table 2).
Two-unit cells with different cross perforation. (a) Sound absorption coefficient of combined two-unit cells with different cross porosity. (b) Top view of the samples of two-unit cells. Courtesy (ANSYS 17.045 and MATLAB R2016a46).
Table 2 Fractals CMPPs metamaterials parameters and its acoustic absorption behavior of a two-unit cells.
Sample S1, shows higher sound absorption coefficient (92%) at 1000 Hz and 60% relative bandwidth, sample S2 has a 76% absorption coefficient with approximately one-octave sound absorption bandwidth, t thickness 20 mm as shown in Fig. 4a. All the samples show broader relative sound absorption bandwidth within the range 39–76% and the approach shown here demonstrates the ability to customize the sound absorption bandwidth as per requirements by careful consideration of the correct combination of the core with various CMPP geometries.
We have further integrated 4 different unit cells reported in Fig. 5b, to get a broader sound absorption response from these geometries. The porosity of the cross MPPs is optimized numerically to get suitable combinations to achieve a maximized sound absorption bandwidth. The perforation dimensions are d1 = 0.3 mm, d2 = 0.4 mm, d3 = 0.5 mm, d4 = 0.6 mm, d5 = 0.7 mm, d6 = 0.8 mm, d7 = 0.9 mm and d8 = 1 mm.
Four-unit cells with different cross perforation. (a) Sound absorption coefficient of combined four-unit cells with different cross porosity. (b) Top view of the samples of four-unit cells. Courtesy (ANSYS 17.045 and MATLAB R2016a46).
As we can see from the Fig. 5a that the sound absorption bandwidth from 800 to 1400 Hz demonstrates an absorption coefficient of greater than 80% and the average relative bandwidth of 61%.
Effective and efficient Attenuation of noise requires limited thickness, light weightiness and perfect sound absorption performance in broadband frequencies, especially in the lower frequency range. We have proposed a novel class of cross micro-perforated hybrid acoustic metamaterial with Helmholtz fractal cores that possess outstanding sound absorption over broadband low frequency range with excellent tunability. Using electrical analogy methods the equivalent impedance to sound propagation within the fractal core is evaluated in combination with the classical improved Maa Model25, for the CMPPs. We have developed a theoretical approach to calculate the equivalent sound absorption coefficient for a bunch of geometric combinations. This theory is then validated through a numerical approach (FEM) as well as experimentally. The results show that novel sound absorbers 20 mm thickness can achieve near perfect absorption around 1000 Hz, with a broadband absorption bandwidth. Approximately 1 octave band sound absorption coefficient > 0.5 have been achieved with single unit cell and more than 0.8 has been achieved within the frequency range 600–1100 Hz. Maximum relative sound absorption bandwidth of 76% has been achieved with an integrated two unit cell configuration and 61% with 4 unit cell combination. The sound absorption coefficient has been increased by integrating the unit cells.
Numerical simulations
The sound absorption coefficient of CMPP is carried out using ANSYS 17.044, with its acoustic module. We have approached the problem by first converting the MPPs into rigid porous materials and used equivalent fluid model in numerical analysis FEM simulation to obtain the final estimate (Fig. 6). The equivalent fluid cad model of CMPPs as shown in Fig. 7b.
Equivalent fluid model conversion of MPP to rigid porous materials using the parameters of Ø, σ, η, Λ and Λʹ.
FEM simulation setup to analyze the sound absorption coefficient. (a) Simulation setup. (b) Equivalent fluid model of proposed CMPP fractals acoustic metamaterials. Courtesy (ANSYS 17.045).
The equivalent fluid model is defined with the parameters that can be calculated by the Eqs. (9) to (11). These calculated values have been used in FEM simulations as shown in Fig. 7 below.
$$\Lambda = {\Lambda }^{^{\prime}}=\frac{d}{2},$$
$$\sigma =\frac{32\tau }{\varnothing {d}^{2}},$$
where \(\tau\) is dynamic viscosity and \(d\) is the diameter of the perforation
$$\eta =1+\frac{2\times 0.48\sqrt{\pi {r}^{2}}(1-1.14\sqrt{\varnothing })}{t},$$
where \(r\) is the radius of the perforation and t is the thickness of the CMPP.
The 3D model of CMPP acoustic metamaterials shown in Fig. 7a created in the DesignModeler of ANSYS 17.045. A plane wave with unit amplitude is applied normally and hard boundary conditions is applied on all the walls of the interface between air and the surface at the subsurface levels of the structure.
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
The FEM simulations were performed using ANSYS 17 (Academic)45 and for plotting of all the graph, and theoretical coding, used MATLAB (R2016a)46 (License No. 40765629). The code is available from ([email protected]) on reasonable request.
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ANSYS (Acoustics V4), V. 17.0. www.ansys.com/en-in (ANSYS). Accessed 20 June 2020.
MATLAB (Programming Language), V. R2016a. www.mathworks.com/products/matlab.html (MATLAB). Accessed 11 Feb 2020.
The funding was provided by Boeing International Corporation India private limited (Grant No. BOEING/ME/2016081).
Department of Design, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh, 208016, India
Sanjeet Kumar Singh & Shantanu Bhattacharya
Microsystem Fabrication Laboratory, Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh, 208016, India
Shantanu Bhattacharya
Boeing International Corporation India Private Limited, RMZ Infinity, Tower D, 5th Floor, Old Madras Road, Bengaluru, Karnataka, 560001, India
Sanjeet Kumar Singh
S.K.S.: Designed the study, conducted the experiments, analyzed the data, contributed to the theoretical derivation, conducted numerical analysis and co-drafted the manuscripts with O.P. and S.B. O.P.: Provided technical guidance to fulfill project goals and also provided financial support in carrying out experimentation and reviewing the finalized manuscript. S.B.: Conceived the central idea of fractal meta-structures, offered guidance in the process of decision making during iterative designing analysis and experimentation, provided infrastructural support, valuable intellectual insights, and manuscript finalization. All work was carried at Microsystem Fabrication Lab, IIT Kanpur.
Correspondence to Shantanu Bhattacharya.
Supplementary Information.
Singh, S.K., Prakash, O. & Bhattacharya, S. Hybrid fractal acoustic metamaterials for low-frequency sound absorber based on cross mixed micro-perforated panel mounted over the fractals structure cavity. Sci Rep 12, 20444 (2022). https://doi.org/10.1038/s41598-022-24621-8
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About Scientific Reports
Guide to referees
Scientific Reports (Sci Rep) ISSN 2045-2322 (online) | CommonCrawl |
\begin{definition}[Definition:Cofactor/Minor]
Let:
: $D = \begin{vmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n1} & a_{n2} & \cdots & a_{nn}\end{vmatrix}$
be a determinant of order $n$.
Let $D \left({r_1, r_2, \ldots, r_k \mid s_1, s_2, \ldots, s_k}\right)$ be a order-$k$ minor of $D$.
Then the '''cofactor''' of $D \left({r_1, r_2, \ldots, r_k \mid s_1, s_2, \ldots, s_k}\right)$ can be denoted:
:$\tilde D \left({r_1, r_2, \ldots, r_k \mid s_1, s_2, \ldots, s_k}\right)$
and is defined as:
:$\tilde D \left({r_1, r_2, \ldots, r_k \mid s_1, s_2, \ldots, s_k}\right) = \left({-1}\right)^t D \left({r_{k+1}, r_{k+2}, \ldots, r_n \mid s_{k+1}, s_{k+2}, \ldots, s_n}\right)$
where:
: $t = r_1 + r_2 + \ldots + r_k + s_1 + s_2 + \ldots s_k$
: $r_{k+1}, r_{k+2}, \ldots, r_n$ are the numbers in $1, 2, \ldots, n$ not in $\left\{{r_1, r_2, \ldots, r_k}\right\}$
: $s_{k+1}, s_{k+2}, \ldots, s_n$ are the numbers in $1, 2, \ldots, n$ not in $\left\{{s_1, s_2, \ldots, s_k}\right\}$
That is, the '''cofactor of a minor''' is the determinant formed from the rows and columns not in that minor, multiplied by the appropriate sign.
When $k = 1$, this reduces to the cofactor of an element (as above).
When $k = n$, the "minor" is in fact the whole determinant.
For convenience its '''cofactor''' is defined as being $1$.
Note that the '''cofactor''' of the '''cofactor''' of a minor is the minor itself (multiplied by the appropriate sign).
Category:Definitions/Determinants
\end{definition} | ProofWiki |
Global analytic function
In the mathematical field of complex analysis, a global analytic function is a generalization of the notion of an analytic function which allows for functions to have multiple branches. Global analytic functions arise naturally in considering the possible analytic continuations of an analytic function, since analytic continuations may have a non-trivial monodromy. They are one foundation for the theory of Riemann surfaces.
Definition
The following definition is in Ahlfors (1979), but also found in Weyl or perhaps Weierstrass. An analytic function in an open set U is called a function element. Two function elements (f1, U1) and (f2, U2) are said to be analytic continuations of one another if U1 ∩ U2 ≠ ∅ and f1 = f2 on this intersection. A chain of analytic continuations is a finite sequence of function elements (f1, U1), …, (fn,Un) such that each consecutive pair are analytic continuations of one another; i.e., (fi+1, Ui+1) is an analytic continuation of (fi, Ui) for i = 1, 2, …, n − 1.
A global analytic function is a family f of function elements such that, for any (f,U) and (g,V) belonging to f, there is a chain of analytic continuations in f beginning at (f,U) and finishing at (g,V).
A complete global analytic function is a global analytic function f which contains every analytic continuation of each of its elements.
Sheaf-theoretic definition
Using ideas from sheaf theory, the definition can be streamlined. In these terms, a complete global analytic function is a path-connected sheaf of germs of analytic functions which is maximal in the sense that it is not contained (as an etale space) within any other path connected sheaf of germs of analytic functions.
References
• Ahlfors, Lars (1979), Complex analysis (3rd ed.), McGraw Hill, ISBN 978-0-07-000657-7
| Wikipedia |
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Session 68 -- Solar Activity
Oral presentation, Thursday, January 13, 2:15-3:45, Salons A/B Room (Crystal City Marriott)
[68.01] A Multiwavelength Study of Solar Ellerman Bombs
Tamara E. W. Payne (U. S. Air Force Phillips Laboratory)
Solar Ellerman bombs (also known as moustaches) are small ($\approx 1$ arcsec) bright structures which appear in solar active regions. It was the purpose of this study to determine their fundamental character, i.e. are Ellerman bombs flare-like phenomena or do they represent an in-situ, driven release of energy? In order to answer this question and others, several different aspects of bomb behavior were analyzed. Photometric analysis of their optical emission indicates that bombs are constrained to a 500 km region in the lower chromosphere ranging from $\approx 600 - 1100$ km above the $\tau_{5000} = 1$ level. They have lifetimes of 15 minutes and exhibit time profiles of rapid rise and rapid decay that are not convincingly similar to flares (which generally show rapid rise and slow decay). Simultaneous optical, microwave, and soft x-ray observations detected no coronal or upper chromospheric emission associated with Ellerman bombs, thereby constraining the bomb emission to the lower chromosphere and virtually eliminating the possibility of a triggering mechanism high in the atmosphere. The microwave (2 and $3.6$ cm) and soft x-ray observations, however, did detect faint microwave ``twinklings'' which appeared to be cospatial with the footpoint of a coronal soft x-ray loop but which were not associated with an Ellerman bomb. High spatial resolution H$\alpha - 1\,\AA$ movies show some bombs moving radially out from the outer edge of the penumbra into the surrounding undisturbed granulation pattern. High spatial resolution H$\alpha - 1\,\AA$ images deconvolved using a Quasi-Weiner filter, revealed that elliptical bombs appear to consist of two or more emission structures on scales $\leq .5$ arcsec. The optical energy output of a typical Ellerman bomb with a lifetime of 840 seconds and an area of $10^{16} \,cm^{2}$ was estimated to be a minimum of $3.2 \times 10^{27} \,ergs$. This is shown to be on the order of the energy in a photospheric magnetic field of 1000 G contained in a volume of 1000 km x 1000 km x 500 km (the volume of a typical Ellerman bomb). This was also shown to be on the order of the upper limit of the energy output of non-thermal gyro-synchrotron-producing elections located at a height of 2000 km where the $\tau_{2cm} = 1$.
Thursday program listing | CommonCrawl |
\begin{definition}[Definition:Integral Transform/Kernel]
Let $\map F p$ be an integral transform:
:$\map F p = \ds \int_a^b \map f x \map K {p, x} \rd x$
The function $\map K {p, x}$ is the '''kernel''' of $\map F p$.
\end{definition} | ProofWiki |
Ordering non-bipartite unicyclic graphs with pendant vertices by the least Q-eigenvalue
Shu-Guang Guo1,
Xiaorong Liu1,2,
Rong Zhang1 &
Guanglong Yu1
Journal of Inequalities and Applications volume 2016, Article number: 136 (2016) Cite this article
A unicyclic graph is a connected graph whose number of edges is equal to the number of vertices. Fan et al. (Discrete Math. 313:903-909, 2013) and Liu et al. (Electron. J. Linear Algebra 26:333-344, 2013) determined, independently, the unique unicyclic graph whose least Q-eigenvalue attains the minimum among all non-bipartite unicyclic graphs of order n with k pendant vertices. In this paper, we extend their results and determine the first three non-bipartite unicyclic graphs of order n with k pendant vertices ordering by least Q-eigenvalue.
Let \(G=(V, E)\) be a simple undirected graph with vertex set \(V=V(G)= \{v_{1}, v_{2}, \ldots, v_{n}\}\) and edge set \(E=E(G)\), where n is called the order of G. Let \(A(G)\) be the adjacency matrix of a graph G and let \(D(G)=\operatorname {diag}(d_{G}(v_{1}), d_{G}(v_{2}),\ldots, d_{G}(v_{n}))\) be the diagonal matrix of degrees of G, where \(d_{G}(v)\) or simply \(d(v)\) denotes the degree of a vertex v in G. The matrix \(Q(G)=D(G)+ A(G)\) is called the signless Laplacian matrix (or Q-matrix) of G. Since \(Q(G)\) is symmetric and positive semidefinite, it follows that its eigenvalues are real and nonnegative. We simply call the eigenvalues of \(Q(G)\) as the signless Laplacian eigenvalues or Q-eigenvalues of G. As usual, we shall index the eigenvalues of \(Q(G)\) in nonincreasing order and denote them as \(q_{1}(G)\ge q_{2}(G)\ge\cdots\ge q_{n}(G)\ge 0\). Denote by \(\kappa(G)\) the least Q-eigenvalue of G.
For a connected graph G, Desai and Rao [3] showed that \(\kappa (G) = 0\) if and only if G is bipartite, and suggested that \(\kappa (G)\) can be used as a measure of non-bipartiteness of G. For a connected non-bipartite graph G, how small can \(\kappa(G)\) be? Cardoso et al. [4] proposed this problem and proved that the minimum value of \(\kappa(G)\) of a connected non-bipartite graph G of order n is attained solely in the unicyclic graph that arises from a cycle of order 3 by attaching a path at one of its end vertices. Wang and Fan [5] investigated how the least Q-eigenvalue of a graph changes when a bipartite branch attached at one vertex is relocated to another vertex and proved a perturbation theorem on the least Q-eigenvalue. As an application, they minimized the least Q-eigenvalue among the class of connected graphs with fixed order which contains a given non-bipartite graph as an induced subgraph. Recently, the problem of finding all graphs with the minimal least Q-eigenvalue among a given class of graphs has been studied extensively. For related results, one may refer to [1, 2, 5–12].
A c-cyclic graph G is a connected graph with n vertices and \(n+c-1\) edges. Specially, if \(c = 0\), 1, or 2, then G is a tree, a unicyclic graph, or a bicyclic graph, respectively. Very recently, Fan et al. [1] and Liu et al. [2] determined, independently, the unique unicyclic graph whose least Q-eigenvalue attains the minimum among all non-bipartite unicyclic graphs of order n with k pendant vertices. In this paper, we extend their results and determine the first three non-bipartite unicyclic graphs of order n with k pendant vertices ordering by least Q-eigenvalue.
The rest of the paper is organized as follows. In Section 2, we recall some basic notions and lemmas used further, and prove two new lemmas. In Section 3, we order non-bipartite unicyclic graphs of order n with k pendant vertices. In Section 4, a conjecture is proposed.
Denote by \(C_{n}\) the cycle of order n. Let \(G-uv\) denote the graph obtained from G by deleting the edge \(uv\in E(G)\). Similarly, \(G+uv\) is the graph obtained from G by adding an edge \(uv\notin E(G)\), where \(u, v\in V(G)\). We write \(d_{G}(u, v)\) or simply \(d(u, v)\) for the distance in G between vertices u and v. The diameter of a connected graph G is the maximum distance between pairs of vertices in \(V(G)\). For \(v\in V(G)\), \(N_{G}(v)\) or simply \(N(v)\) denotes the neighborhood of v in G. A pendant vertex of G is a vertex of degree 1. A pendant neighbor of G is a vertex adjacent to a pendant vertex.
Let \(x=(x_{1}, x_{2}, \ldots, x_{n})^{T}\) be a column vector in \(\mathbb {R}^{n}\). Then x can be considered as a function defined on \(V(G)\), that is, each vertex \(v_{i}\) is given by the value \(x(v_{i})=x_{i}\). Then the quadratic form
$$x^{T}Q(G)x =\sum_{uv\in E(G)}\bigl(x(u) + x(v) \bigr)^{2}. $$
Let \(\vert x(v)\vert \) denote the absolute value of \(x(v)\). If x is an eigenvector corresponding to a Q-eigenvalue of G, then it defines on \(V(G)\) naturally, i.e. \(x(v)\) is the entry of x corresponding to v. For an arbitrary unit vector \(x\in\mathbb{R}^{n}\), one can find in [5, 13]
$$ \kappa(G) \le x^{T}Q(G)x, $$
where equality holds if and only if x is an eigenvector corresponding to \(\kappa(G)\).
Let \(G_{1}\) and \(G_{2}\) be two vertex-disjoint graphs, and let \(v_{1}\in V(G_{1})\), \(v_{2}\in V(G_{2})\). \(G_{1}(v_{1})\diamond G_{2}(v_{2})\) denotes the coalescence of \(G_{1}\) and \(G_{2}\), which arises from \(G_{1}\), \(G_{2}\) by identifying \(v_{1}\) with \(v_{2}\) and forming a new vertex u (see [5] for details). The graph \(G_{1}(v_{1})\diamond G_{2}(v_{2})\) is also written as \(G_{1}(u)\diamond G_{2}(u)\). If a graph G can be expressed in the form \(G = G_{1}(u)\diamond G_{2}(u)\), where \(G_{1}\) and \(G_{2}\) are both connected and nontrivial, then \(G_{i}\) is called a branch of G with root u for \(i=1, 2\). Let x be a vector defined on \(V(G)\). A branch H of G is called a zero branch with respect to x if \(x(v) = 0\) for all \(v \in V(H)\); otherwise it is called a nonzero branch with respect to x.
Lemma 2.1
([5])
Let G be a connected graph which contains a bipartite branch B with root v. Let x be an eigenvector of G corresponding to \(\kappa(G)\).
If \(x(v) = 0\), then B is a zero branch of G with respect to x.
If \(x(v)\neq0\), then \(x(p)\neq0\) for every vertex \(p\in V(B)\).
Let G be a connected non-bipartite graph of order n, and let x be an eigenvector of G corresponding to \(\kappa(G)\). Let T be a tree, which is a nonzero branch of G with respect to x and with root v. Then \(\vert x(q)\vert < \vert x(p)\vert \) whenever p, q are vertices of T such that q lies on the unique path from v to p.
([10])
Let \(G = C(v_{0})\diamond B(v_{0})\) be a graph of order n, where \(C=v_{0}v_{1}v_{2} \cdots v_{k} u_{k}u_{k-1} \cdots u_{1}v_{0}\) is a cycle of length \(2k+1\), and B is a bipartite graph of order \(n-2k>1\) (see Figure 1). Let \(x=( x(v_{0}), x(v_{1}), x(v_{2}), \ldots, x(v_{k}), x(u_{1}), x(u_{2}), \ldots, x(u_{k}), \ldots )^{T}\) be an eigenvector corresponding to \(\kappa(G)\). Then
\(\vert x(v_{0})\vert =\max\{\vert x(w)\vert \mid w\in V(C)\}>0\);
\(x(v_{i})=x(u_{i})\) for \(i=1, 2, \ldots, k\).
\(\pmb{C(v_{0})\diamond B(v_{0})}\) .
Let \(G = G_{1}(v_{2}) \diamond T(u)\) and \(G^{*} = G_{1}(v_{1})\diamond T(u)\), where \(G_{1}\) is a non-bipartite connected graph containing two distinct vertices \(v_{1}\), \(v_{2}\), and T is a nontrivial tree. If there exists an eigenvector \(x=( x(v_{1}), x(v_{2}), \ldots, x(v_{k}), \ldots)^{T}\) of G corresponding to \(\kappa(G)\) such that \(\vert x(v_{1})\vert > \vert x(v_{2})\vert \) or \(\vert x(v_{1})\vert = \vert x(v_{2})\vert > 0\), then \(\kappa(G^{*})<\kappa(G)\).
Let G be a graph with n vertices and m edges. Then
$$\kappa(G)\le\frac{4m-4\operatorname {MaxCut}(G)}{n}, $$
where \(\operatorname {MaxCut}(G)\) denotes, as usual, the size of the largest bipartite subgraph of G.
For a c-cyclic graph G, we have \(\operatorname {MaxCut}(G)\ge n-1\). This implies the following lemma.
Let G be a c-cyclic graph. Then \(\kappa(G)\le\frac{4c}{n}\).
Let G be a non-bipartite connected graph of order n with diameter D. Then \(\kappa(G) \geq\frac{1}{n(D+1)}\).
\(U_{n}^{k}(g)\), shown in Figure 2, denotes the unicyclic graph of order n with odd girth g and k pendant vertices, where \(g+l+k=n\). \(C_{3}^{ 1}(n-k-1)\), \(C_{3}^{ 2}(n-k-1)\), and \(C_{3}^{1}(n-k-2)\) are the unicyclic graphs of order n with k pendant vertices, shown in Figures 2 and 3, respectively.
\(\pmb{U_{n}^{k}(g)}\) and \(\pmb{C_{3}^{ 1}(n-k-1)}\) .
\(\pmb{C_{3}^{ 2}(n-k-1)}\) and \(\pmb{C_{3}^{1}(n-k-2)}\) .
Let \(3\le k \leq{(n-4)}/\sqrt{6}\). Then \(\kappa(C_{3}^{ 2}(n-k-1))<\kappa(C_{3}^{1}(n-k-2))\).
Let \(\kappa=\kappa(C_{3}^{1}(n-k-2))\), and \(x=(x_{1}, x_{2}, \ldots, x_{n})^{T}\) be a unit eigenvector corresponding to κ. Then \(\kappa =\sum_{v_{i}v_{j}\in E(C_{3}^{ 1}(n-k-2))}(x_{i}+x_{j})^{2}\) and \(0<\kappa<1\) (by Lemma 2.6). From the eigenvalue equation \(Q(C_{3}^{1}(n-k-2))x =\kappa x\), we have \(x_{n-k+2}=\cdots=x_{n}\),
$$\begin{aligned}& x_{n-k} = (\kappa-1)x_{n}, \\& x_{n-k-1} = \bigl(\kappa^{2}-(k+1)\kappa+1\bigr)x_{n}, \\& x_{n-k-2} = \bigl(\kappa^{3}-(k+3)\kappa^{2}+(2k+2) \kappa-1\bigr)x_{n}, \\& x_{n-k+1} = \frac{1}{\kappa-1}\bigl(\kappa^{3}-(k+3) \kappa^{2}+(2k+2)\kappa-1\bigr)x_{n}, \end{aligned}$$
and \(x_{n}\neq0\).
Let \(y=(y_{1}, y_{2}, \ldots, y_{n})^{T}\in \mathbb{R}^{n}\), which is defined on \(V(C_{3}^{ 2}(n-k-1))\), satisfy \(y_{n-k+1}=-(x_{n-k-1}+x_{n-k-2}+x_{n-k+1})\), \(y_{n-k+2}=-(x_{n-k-1}+x_{n-k+2}+x_{n-k})\), and \(y_{i}=x_{i}\) for \(i=1, 2, \ldots, n-k, n-k+3,\ldots, n \). Then
$$\sum_{v_{i}v_{j}\in E(C_{3}^{ 2}(n-k-1))}(y_{i}+y_{j})^{2}= \sum_{v_{i}v_{j}\in E(C_{3}^{1}(n-k-2))}(x_{i}+x_{j})^{2}= \kappa $$
$$\begin{aligned} \Vert y\Vert ^{2}-\Vert x\Vert ^{2} =&\sum _{i=1}^{n} y_{i}^{2}-\sum _{i=1}^{n} x_{i}^{2} \\ =&\kappa\bigl({\kappa^{5}}-(2k+2){\kappa^{4}}+ \bigl(k^{2}+2k-1\bigr){\kappa ^{3}}+(4k+6){ \kappa^{2}} \\ &{}-\bigl(2k^{2}+6k+3\bigr)\kappa+2 \bigr)x_{n}^{2}. \end{aligned}$$
Let \(f(t)={t^{5}}-(2k+2){t^{4}}+(k^{2}+2k-1){t^{3}}+(4k+6){t^{2}}-(2k^{2}+6k+3)t+2\). It is not difficult to verify that \(f(t)>0\) for \(0 \le t \leq1/{(k^{2}+3k+2)}\).
Let \(z=(z_{1}, z_{2}, \ldots, z_{n})^{T} \in \mathbb{R}^{n}\), which is defined on \(V(C_{3}^{1}(n-k-2))\), satisfy \(z_{1}=z_{2}=0\), \(z_{n-k+1}=(-1)^{n-k+1}(n-k-3)\),
$$\begin{aligned}& z_{n-k-i} = (-1)^{n-k-i}(n-k-i-2)\quad\mbox{for } 0\le i\le n-k-3, \\& z_{n-k+i} = (-1)^{n-k+1}(n-k-1) \quad\mbox{for } i=2, 3, \ldots, k. \end{aligned}$$
Then, by (1) and \(3\le k \leq{(n-4)}/\sqrt{6}\), we have
$$\begin{aligned} \kappa =&\kappa\bigl(C_{3}^{1}(n-k-2)\bigr)\le \frac{z^{T}Q(G)z}{z^{T}z} \\ =&\frac{n-1}{1^{2}+2^{2}+\cdots+(n-k-2)^{2}+(n-k-3)^{2}+(k-1)(n-k-1)^{2}} \\ =&\frac{6(n-1)}{2n^{3} - 9n^{2} -(6k^{2}-6k+11)n + 4k^{3} + 3k^{2} + 17k + 42}< \frac{1}{k^{2}+3k+2}. \end{aligned}$$
Therefore \(f(\kappa)>0\), and so
$$\Vert y\Vert ^{2}-\Vert x\Vert ^{2}=\kappa f(\kappa) x_{n}^{2}>0. $$
Combining the above arguments, we have
$$\begin{aligned} \kappa\bigl(C_{3}^{2}(n-k-1)\bigr) \leq&\Vert y\Vert ^{-2}\sum_{v_{i}v_{j} \in E(C_{3}^{2}(n-k-1))}(y_{i}+y_{j})^{2}< \Vert x\Vert ^{-2} \sum_{v_{i}v_{j} \in E(C_{3}^{1}(n-k-2))}(x_{i}+x_{j})^{2} \\ =& \kappa. \end{aligned}$$
Let \(n\ge120\), \(k > \frac{-3+\sqrt{21}}{2}n\). Then \(\kappa (C_{3}^{1}(n-k-2))< \kappa(C_{3}^{ 2}(n-k-1))\).
Let \(\kappa=\kappa(C_{3}^{ 2}(n-k-1))\), and \(x=(x_{1}, x_{2}, \ldots, x_{n})^{T}\) be a unit eigenvector corresponding to κ. Then \(\kappa =\sum_{v_{i}v_{j}\in E(C_{3}^{ 2}(n-k-1))}(x_{i}+x_{j})^{2}\) and \(0<\kappa<1\). From the eigenvalue equation \(Q(C_{3}^{ 2}(n-k-1))x = \kappa x\), we have \(x_{n-k+3}=\cdots=x_{n}\),
$$\begin{aligned}& x_{n-k} = (\kappa-1)x_{n}, \\& x_{n-k-1} = \bigl(\kappa^{2}-k\kappa+1\bigr)x_{n}, \\& x_{n-k+1} = x_{n-k+2}=\frac{1}{\kappa-1}\bigl(\kappa^{2}-k \kappa+1\bigr)x_{n}, \\& x_{n-k-2} = \frac{1}{\kappa-1}\bigl(\kappa^{4}-(k+5) \kappa^{3}+(5k+2)\kappa ^{2}-(2k+3)\kappa+1 \bigr)x_{n}, \end{aligned}$$
Let \(y=(y_{1}, y_{2}, \ldots, y_{n})^{T}\in \mathbb{R}^{n}\), which is defined on \(V(C_{3}^{1}(n-k-2))\), satisfy that \(y_{n-k+1}=-(x_{n-k-1}+x_{n-k-2}+x_{n-k+1})\), \(y_{n-k+2}=-(x_{n-k-1}+x_{n-k+2}+x_{n-k})\), and \(y_{i}=x_{i}\) for \(i=1, 2, \ldots, n-k, n-k+3, n-k+4, \ldots, n\). Then
$$\sum_{v_{i}v_{j}\in E(C_{3}^{1}(n-k-2))}(y_{i}+y_{j})^{2}= \sum_{v_{i}v_{j}\in E(C_{3}^{ 2}(n-k-1))}(x_{i}+x_{j})^{2}= \kappa $$
$$\begin{aligned} \Vert y\Vert ^{2}-\Vert x\Vert ^{2} =&\sum _{i=1}^{n} y_{i}^{2}-\sum _{i=1}^{n} x_{i}^{2} \\ =&\frac{\kappa}{\kappa-1}\bigl({\kappa^{6}}-(2k+7){\kappa ^{5}}+ \bigl(k^{2}+14k+14\bigr){\kappa^{4}}-\bigl(7k^{2}+28k+4 \bigr){\kappa^{3}} \\ &{} +\bigl(14k^{2}+6k+15\bigr)\kappa^{2}- \bigl(2k^{2}+14k+1\bigr)\kappa^{2}+6\bigr)x_{n}^{2}. \end{aligned}$$
$$\begin{aligned} f(t) =&{t^{6}}-(2k+7){t^{5}}+\bigl(k^{2}+14k+14 \bigr){t^{4}}-\bigl(7k^{2}+28k+4\bigr){t^{3}}+ \bigl(14k^{2}+6k+15\bigr)t^{2} \\ &{}-\bigl(2k^{2}+14k+1 \bigr)t+6. \end{aligned}$$
Then \(f(0)=6\). From \(n\ge120\) and \(k> \frac{-3+\sqrt{21}}{2}n\), we have \(k > \frac{-3+\sqrt{21}}{2}n>94\), and
$$\begin{aligned}& k^{12}f\bigl(1/k^{2}\bigr) = 4k^{12}-14k^{11}+13k^{10}+6k^{9}+8k^{8}-28k^{7}-3k^{6}+14k^{5} \\& \hphantom{k^{12}f\bigl(1/k^{2}\bigr) =}{} +14k^{4}-2k^{3}-7k^{2}+1>0, \\& k^{12}f\bigl(3/k^{2}\bigr) = -42k^{11}+123k^{10}+54k^{9}-54k^{8}-756k^{7}-27k^{6} \\& \hphantom{k^{12}f\bigl(3/k^{2}\bigr) =}{} +1134k^{5}+1134k^{4}-486k^{3}-1701k^{2}+729< 0, \\& f'(t) = 6t^{5}-(10k+35)t^{4}+ \bigl(4k^{2}+56k+56\bigr)t^{3}-\bigl(21k^{2}+84k+12 \bigr)t^{2} \\& \hphantom{f'(t) =}{} +\bigl(28k^{2}+12k+30\bigr)t-\bigl(2k^{2}+14k+1\bigr)< 0, \end{aligned}$$
for \(0\le t \leq1/30\). So \(f(t)\) is strictly decreasing with respect to t in \([0,1/30]\). Recalling that \(k > \frac{-3+\sqrt{21}}{2}n\), by Lemmas 2.6 and 2.7, we find that
$$\frac{3}{k^{2}}< \frac{1}{n(n-k)} \leq\kappa=\kappa\bigl(C_{3}^{2}(n-k-1) \bigr)\le \frac{4}{n}\le\frac{1}{30}. $$
This implies that \(f(\kappa)<0\) and
$$\Vert y\Vert ^{2}-\Vert x\Vert ^{2}= \frac{\kappa}{\kappa-1}f(\kappa) x_{n}^{2}>0. $$
It follows that
$$\begin{aligned} \kappa\bigl(C_{3}^{1}(n-k-2)\bigr) \leq&\Vert y\Vert ^{-2} \sum_{v_{i}v_{j} \in E(C_{3}^{1}(n-k-2))}(y_{i}+y_{j})^{2}< \Vert x\Vert ^{-2} \sum_{v_{i}v_{j} \in E(C_{3}^{2}(n-k-1))}(x_{i}+x_{j})^{2} \\ =& \kappa. \end{aligned}$$
Main results
Let \(\mathcal{U}_{ n}^{k}\) be the set of non-bipartite unicyclic graphs of order n with k pendant vertices. From [1, 2], we know that \(U_{n}^{k}(3)\) is the unique graph whose least Q-eigenvalue attains the minimum among all graphs in \(\mathcal{U}_{ n}^{k}\). In this section, we will determine the first three graphs in \(\mathcal{U}_{ n}^{k}\) ordered according to their least Q-eigenvalues.
For \(k=1\), from [1], we know that \(\kappa(U_{n}^{1}(3))<\kappa (U_{n}^{1}(5))<\kappa(U_{n}^{1}(7))<\cdots\) .
Theorem 3.1
Let \(2\le k\le n-4\). Among all graphs in \(\mathcal{U}_{ n}^{k}\backslash\{U_{n}^{k}(3)\}\), \(C_{3}^{1}(n-k-1)\) is the unique graph whose least Q-eigenvalue attains the minimum.
Let G be a graph in \(\mathcal{U}_{ n}^{k}\backslash\{ U_{n}^{k}(3)\}\) whose least Q-eigenvalue attains the minimum, and \(C_{g}=v_{1}v_{2}\cdots v_{g}v_{1}\) be the unique cycle of G. Then g is odd, and G can be obtained by attaching rooted trees \(T_{1}, \ldots, T_{g}\) to the vertices \(v_{1}, \ldots, v_{g}\) of \(C_{g}\), respectively, where \(T_{i}\) contains the root vertex \(v_{i}\). \(\vert V(T_{i})\vert =1\) means that \(V(T_{i})=\{v_{i}\}\) and in this case \(T_{i}\) is a trivial tree. Let \(x=(x_{1}, x_{2}, \ldots, x_{n})^{T}\) be a unit eigenvector corresponding to \(\kappa(G)\).
First, we show that G is the cycle \(C_{g}=v_{1}v_{2}\cdots v_{g}v_{1}\) with only one nontrivial tree attached. Otherwise, we assume that there are more than one nontrivial trees attached at two different vertices of the cycle \(C_{g}\). Let \(v_{t}\) be a vertex of the cycle \(C_{g}\) such that \(\vert x_{t}\vert \geq \vert x_{i}\vert \) for \(i=1, 2, \ldots, g\). By Lemma 2.1, \(x_{t}\neq0\). Let \(v_{l}\) be another vertex of the cycle \(C_{g}\) such that \(\vert V(T_{l})\vert >1\), and let
$$G_{1}=G-\sum_{v\in N_{T_{l}}(v_{l})}v_{l}v+\sum _{v\in N_{T_{l}}(v_{l})}v_{t}v. $$
From \(k\le n-4\), we have \(G_{1}\in\mathcal{U}_{n}^{k} \backslash\{ U_{n}^{k}(3)\}\). By Lemma 2.4, we have \(\kappa(G_{1})<\kappa(G)\), a contradiction. Therefore G is the cycle \(C=v_{1}v_{2}\cdots v_{g}v_{1}\) with only one nontrivial tree attached. Without loss of generality, we may assume the nontrivial tree is \(T_{g}\).
Second, we show that \(g=3\). Otherwise, we assume that \(g\ge5\). By Lemma 2.3, we have \(x_{(g-3)/2}=x_{(g+3)/2}\). Let
$$G '=G-v_{(g-1)/2}v_{(g-3)/2}+v_{(g-1)/2}v_{(g+3)/2}. $$
Clearly, \(G '\in\mathcal{U}_{ n}^{k+1}\), and from (1) we have
$$\kappa\bigl(G '\bigr)\le x^{T}Q\bigl(G ' \bigr)x=x^{T}Q(G)x=\kappa(G). $$
Let \(v_{t}\) be a pendant vertex of G, and \(y=(y_{1}, y_{2}, \ldots, y_{n})^{T}\) be a unit eigenvector corresponding to \(\kappa(G ')\). By Lemma 2.2, we have \(\vert y_{t}\vert > \vert y_{g}\vert >0\). Let \(G ''=G '-v_{1}v_{g}+v_{1}v_{t}\). It is easy to see that \(G ''\in\mathcal{U}_{ n}^{k}\backslash\{ U_{n}^{k}(3)\}\). By Lemma 2.4, we have \(\kappa(G '')<\kappa(G ')\). Then we have \(\kappa(G '')<\kappa(G)\), a contradiction. Therefore \(g=3\).
Third, we show that G has two pendant neighbors exactly. Otherwise, suppose that G has \(r\ge3\) pendant neighbors. Let \(v_{a}\) be a pendant neighbor of G such that \(d(v_{3}, v_{a})\) is as large as possible, \(v_{s}\) and \(v_{t}\) be two other pendant neighbors of G. Applying Lemma 2.4 to \(v_{s}\) and \(v_{t}\), we may obtain a graph \(G '\in\mathcal{U}_{ n}^{k}\backslash\{U_{n}^{k}(3)\}\) or \(G '\in \mathcal{U}_{ n}^{k+1}\) such that \(\kappa(G ')<\kappa(G)\). If \(G '\in\mathcal{U}_{ n}^{k}\backslash \{U_{n}^{k}(3)\}\), we have a contradiction. If \(G '\in\mathcal{U}_{ n}^{k+1}\), without loss of generality, we may assume that \(v_{s}\) is a pendant vertex of \(G '\). Let u and w be two pendant vertices adjacent to \(v_{t}\) of \(G '\), and \(G ''=G '-v_{t}w+uw\). Clearly, \(G ''\in\mathcal{U}_{ n}^{k}\backslash\{U_{n}^{k}(3)\}\) and \(\kappa(G '')<\kappa(G ')\). Then we have \(\kappa(G '')<\kappa(G)\), a contradiction. Therefore G has two pendant neighbors exactly. Let \(v_{a}\) be a pendant neighbor of G such that \(d(v_{3}, v_{a})\) is as large as possible, and \(v_{b}\) be another pendant neighbor of G.
Fourth, we show that \(v_{b}\) is in path \(v_{3}-v_{a}\). Otherwise, suppose that \(v_{b}\) is not in path \(v_{3}-v_{a}\). Employing Lemma 2.4 to vertices \(v_{a}\) and \(v_{b}\), we may obtain a graph \(G '\in\mathcal {U}_{ n}^{k+1}\) such that \(\kappa(G ')<\kappa(G)\). Without loss of generality, we may assume that \(v_{b}\) is a pendant vertex of \(G '\). Let u and w be two pendant vertices adjacent to \(v_{a}\) of \(G '\), and \(G ''=G '-v_{a}w+uw\). Clearly, \(G ''\in\mathcal{U}_{ n}^{k}\backslash\{U_{n}^{k}(3)\}\) and \(\kappa(G '')<\kappa(G ')\). Then we have \(\kappa(G '')<\kappa(G)\), a contradiction. Therefore \(v_{b}\) is in path \(v_{3}-v_{a}\).
Fifth, we show that \(v_{a}\) and \(v_{b}\) are adjacent. Otherwise, suppose that \(v_{a}\) and \(v_{b}\) are not adjacent. Let \(v_{c}\in N(v_{b})\) be in path \(v_{b}-v_{a}\), then, by Lemma 2.4, we have \(\vert x_{c}\vert >\vert x_{b}\vert \). Let \(v_{t}\) be the pendant vertex adjacent to \(v_{b}\) and \(G '=G-v_{b}v_{t}+v_{c}v_{t}\). Clearly, \(G '\in\mathcal{U}_{ n}^{k}\backslash\{U_{n}^{k}(3)\}\) and by Lemma 2.4 we have \(\kappa(G ')<\kappa(G)\), a contradiction. Therefore \(v_{a}\) and \(v_{b}\) are adjacent.
Sixth, we show that \(d(v_{b})=3\). Otherwise, suppose that \(d(v_{b})>3\). Let \(v_{t}\) be the pendant vertex adjacent to \(v_{b}\) and \(G '=G-v_{b}v_{t}+v_{a}v_{t}\). Clearly, \(G '\in\mathcal{U}_{ n}^{k}\backslash\{U_{n}^{k}(3)\}\). By Lemma 2.4, we have \(\vert x_{a}\vert >\vert x_{b}\vert \), and by Lemma 2.4, we have \(\kappa(G ')<\kappa(G)\), a contradiction. Therefore \(d(v_{b})=3\).
From the above arguments, we have \(G=C_{3}^{1}(n-k-1)\). □
For \(k=n-3\), \(\mathcal{U}_{ n}^{ n-3}=\{\Delta_{r, s, t} \mid r\ge s\ge t\ge0, r+s+t=n-3 \}\), where \(\Delta_{r, s, t}\) is the graph obtained from the cycle \(C_{3}\) by attaching r, s, t pendent edges to the vertices \(v_{1}\), \(v_{2}\), and \(v_{3}\) of the cycle \(C_{3}\), respectively. By a similar reasoning to that of Theorem 3.1, we can prove the following theorem.
Let \(n\ge8\), and \(G\in\mathcal{U}_{ n}^{n-3}\backslash\{\Delta _{n-3, 0, 0}, \Delta_{n-4, 1, 0}, \Delta_{n-5, 2, 0}\}\). Then
$$\kappa(\Delta_{n-3, 0, 0})< \kappa(\Delta_{n-4, 1, 0})< \kappa ( \Delta_{n-5, 2, 0})< \kappa(G). $$
Next, we will determine the graph in \(\mathcal{U}_{ n}^{k}\backslash\{ U_{n}^{k}(3), C_{3}^{1}(n-k-1)\}\) whose least Q-eigenvalue attains the minimum.
Let \(2\le k\le n-5\). Among all graphs in \(\mathcal{U}_{ n}^{k}\backslash\{U_{n}^{k}(3), C_{3}^{1}(n-k-1)\}\), \(C_{3}^{ 1}(n-k-2)\) or \(C_{3}^{ 2}(n-k-1)\) is the graph whose least Q-eigenvalue attains the minimum.
Let G be a graph in \(\mathcal{U}_{ n}^{k}\backslash\{ U_{n}^{k}(3), C_{3}^{1}(n-k-1)\}\) whose least Q-eigenvalue attains the minimum, and let \(x=(x_{1}, x_{2}, \ldots, x_{n})^{T}\) be a unit eigenvector corresponding to \(\kappa(G)\). By a similar reasoning to that of Theorem 3.1, we can prove that G is the cycle \(C=v_{1}v_{2}v_{3}v_{1}\) with only one nontrivial tree \(T_{3}\) attached at \(v_{3}\), and G has two pendant neighbors exactly. Let \(v_{a}\) be a pendant neighbor of G such that \(d(v_{3}, v_{a})\) is as large as possible, and \(v_{b}\) be another pendant neighbor of G. By a similar reasoning to that of Theorem 3.1, we can prove that \(v_{b}\) is in path \(v_{3}-v_{a}\).
Now we show that \(d(v_{b}, v_{a})\le2\). Otherwise, suppose that \(d(v_{b}, v_{a})\ge3\). Let \(v_{t}\) be the pendant vertex adjacent to \(v_{b}\) and \(v_{c}\in N(v_{b})\) be in path \(v_{b}-v_{a}\). Then, by Lemma 2.4, we have \(\vert x_{c}\vert >\vert x_{b}\vert \). Let \(G '=G-v_{b}v_{t}+v_{c}v_{t}\). Clearly, \(G '\in\mathcal{U}_{ n}^{k}\backslash\{U_{n}^{k}(3), C_{3}^{1}(n-k-1)\}\) and \(\kappa(G ')<\kappa(G)\), a contradiction. Therefore \(d(v_{b}, v_{a})\le2\).
If \(d(v_{b}, v_{a})=2\), then we declare \(d(v_{b})=3\). Otherwise, suppose that \(d(v_{b})\ge4\). Let \(v_{t}\) be the pendant vertex adjacent to \(v_{b}\) and let \(G '=G-v_{b}v_{t}+v_{a}v_{t}\). Clearly, \(G '\in\mathcal{U}_{ n}^{k}\backslash\{U_{n}^{k}(3), C_{3}^{1}(n-k-1)\}\) and \(\kappa(G ')<\kappa(G)\), a contradiction. Therefore \(d(v_{b})=3\) and \(G=C_{3}^{1}(n-k-2)\).
From the above arguments, we have \(G=C_{3}^{1}(n-k-1)\) or \(C_{3}^{2}(n-k-1)\). □
For \(k=n-4\), \(\mathcal{U}_{ n}^{ n-4}=\{C_{3}^{ r, s, t, l} \mid l\ge1, r\ge0, s\ge0, t\ge0, r+s+t+l=n-4 \}\), where \(C_{3}^{ r, s, t, l}\), shown in Figure 4, denotes the unicyclic graph of order n with \(n-4\) pendant vertices. \(C_{3}^{ 1}(2)\) and \(C_{3}^{ 2}(3)\), shown in Figure 4, are the unicyclic graphs of order n with \(n-4\) pendant vertices.
\(\pmb{C_{3}^{ r, s, t, l}}\) , \(\pmb{C_{3}^{ 1}(2)}\) , and \(\pmb{C_{3}^{ 2}(3)}\) .
Let \(n\ge7\). Among all graphs in \(\mathcal{U}_{ n}^{ n-4}\backslash \{U_{n}^{ n-4}(3), C_{3}^{1}(3)\}\), \(C_{3}^{ 1}(2)\) is the unique graph whose least Q-eigenvalue attains the minimum.
By a similar reasoning to that of Theorem 3.3, we can prove that \(C_{3}^{ 2}(3)\) or \(C_{3}^{ 1}(2)\) is the graph whose least Q-eigenvalue attains the minimum among all graphs in \(\mathcal{U}_{ n}^{ n-4}\backslash\{U_{n}^{ n-4}(3), C_{3}^{1}(3)\}\). Let \(\kappa=\kappa(C_{3}^{ 2}(3))\) and let \(x=(x_{1}, x_{2}, \ldots, x_{n})^{T}\) be an eigenvector corresponding to κ. From the eigenvalue equations, we have \(x_{1}=x_{2}\), \(x_{5}=x_{6}\), \(x_{7}=\cdots=x_{n}\),
$$\begin{aligned}& (\kappa-2)x_{1} = x_{1}+x_{3}, \\& (\kappa-5)x_{3} = 2x_{1}+x_{4}+2x_{5}, \\& (\kappa-n+5)x_{4} = x_{3}+(n-6)x_{7}, \\& (\kappa-1)x_{5} = x_{3}, \\& (\kappa-1)x_{7} = x_{4}. \end{aligned}$$
Since x is an eigenvector, it follows that \(\kappa=\kappa(C_{3}^{ 2}(3))\) is the least root of the equation
$$f(x)\triangleq\left| \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} x-3 & -1 & 0 & 0 & 0 \\ -2 & x-5 & -1 & -2 & 0 \\ 0 & -1 & x-n+5 & 0 & -n+6 \\ 0 & -1 & 0 & x-1 & 0 \\ 0 & 0 & -1 & 0 & x-1 \end{array}\displaystyle \right| =0. $$
By an easy computation, we can obtain
$$f(x)=x^{5}-(n+5)x^{4}+(9n-17)x^{3}-(19n-65)x^{2}+(7n-16)x-4. $$
Similarly, from the eigenvalue equation, we can prove that \(\kappa(C_{3}^{ 1}(2))\) is the least root of
$$g(x)\triangleq x^{6}-(n+6)x^{5}+(9n-2)x^{4}-(25n-48)x^{3}+(25n-58)x^{2}-(7n-8)x+4=0. $$
By Lemma 2.6, we have \(0<\kappa(C_{3}^{ 2}(3)), \kappa (C_{3}^{ 1}(2))\le4/n\). Note that for \(n\ge12\),
$$(x-1)f(x)-g(x)=x\bigl((n-10)x^{3}-(3n-34)x^{2}+(n-23)x+4 \bigr)>0 $$
for \(0< x\le4/n\). It follows that \(g(\kappa(C_{3}^{ 2}(3)))<0\). This implies that \(\kappa(C_{3}^{ 1}(2))<\kappa(C_{3}^{ 2}(3))\).
For \(7\le n\le11\), by computation, we can verify that \(\kappa(C_{3}^{ 1}(2))<\kappa(C_{3}^{ 2}(3))\).
From the above arguments, we have \(\kappa(C_{3}^{ 1}(2))<\kappa(C_{3}^{ 2}(3))\) for \(n\ge7\). □
Combining Theorem 3.3 and Lemma 2.8, we have the following theorem.
Let \(3\le k \leq{(n-4)}/{\sqrt{6}}\). Among all graphs in \(\mathcal {U}_{ n}^{k}\backslash\{U_{n}^{k}(3), C_{3}^{ 1}(n-k-1)\}\), \(C_{3}^{ 2}(n-k-1)\) is the unique graph whose least Q-eigenvalue attains the minimum.
Let \(n\ge120\), \(k > \frac{-3+\sqrt{21}}{2}n\). Among all graphs in \(\mathcal{U}_{ n}^{k}\backslash\{U_{n}^{k}(3), C_{3}^{1}(n-k-1)\}\), \(C_{3}^{1}(n-k-2)\) is the unique graph whose least Q-eigenvalue attains the minimum.
According to Lemmas 2.8 and 2.9, we propose the following conjecture.
Conjecture 4.1
There exists a real number α with \(0<\alpha<1\) such that, for any \(\varepsilon>0\), there exists a sufficiently large N such that
$$\kappa\bigl(C_{3}^{ 2}(n-k-1)\bigr)< \kappa \bigl(C_{3}^{1}(n-k-2)\bigr) $$
for all \(n\ge N\) and all \(3\le k \le(\alpha-\varepsilon) n\), and
$$\kappa\bigl(C_{3}^{ 2}(n-k-1)\bigr)>\kappa \bigl(C_{3}^{1}(n-k-2)\bigr) $$
for all \(n\ge N\) and all \((\alpha+\varepsilon) n\le k \le n-5 \).
If Conjecture 4.1 is true, then, by Lemmas 2.8 and 2.9, \(\sqrt{6}/6\le\alpha\le(\sqrt{21}-3)/2\), where α is the same as in Conjecture 4.1.
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The first author is very grateful to Professor Yong-Gao Chen for his help. This work is supported by the National Natural Science Foundation of China (Nos. 11171290, 11271315) and the Natural Science Foundation of Jiangsu Province (BK20151295).
School of Mathematics and Statistics, Yancheng Teachers University, Yancheng, Jiangsu, 224002, P.R. China
Shu-Guang Guo, Xiaorong Liu, Rong Zhang & Guanglong Yu
Department of Mathematics, Qinghai Normal University, Xining, Qinghai, 810008, P.R. China
Xiaorong Liu
Shu-Guang Guo
Rong Zhang
Guanglong Yu
Correspondence to Shu-Guang Guo.
All authors completed the paper together. All authors read and approved the final manuscript.
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.
Guo, SG., Liu, X., Zhang, R. et al. Ordering non-bipartite unicyclic graphs with pendant vertices by the least Q-eigenvalue. J Inequal Appl 2016, 136 (2016). https://doi.org/10.1186/s13660-016-1077-1
signless Laplacian
least eigenvalue
unicyclic graph
pendant vertex
Recent Advances in Inequalities and Applications | CommonCrawl |
Preprint acp-2021-775
https://doi.org/10.5194/acp-2021-775
Review status: a revised version of this preprint was accepted for the journal ACP and is expected to appear here in due course.
Comparison of computational and experimental saturation vapor pressures of α-pinene + O3 oxidation products
Noora Hyttinen1,a, Iida Pullinen1, Aki Nissinen1, Siegfried Schobesberger1, Annele Virtanen1, and Taina Yli-Juuti1 Noora Hyttinen et al. Noora Hyttinen1,a, Iida Pullinen1, Aki Nissinen1, Siegfried Schobesberger1, Annele Virtanen1, and Taina Yli-Juuti1
1Department of Applied Physics, University of Eastern Finland, P.O. Box 1627, FI-70211 Kuopio, Finland
aNow at: Department of Chemistry, Nanoscience Center, University of Jyväskylä, FI-40014 Jyväskylä, Finland
Received: 10 Sep 2021 – Accepted for review: 19 Sep 2021 – Discussion started: 20 Sep 2021
Abstract. Accurate information on gas-to-particle partitioning is needed to model secondary organic aerosol formation. However, determining reliable saturation vapor pressures of atmospherically relevant multifunctional organic compounds is extremely difficult. We estimated saturation vapor pressures of α-pinene ozonolysis derived secondary organic aerosol constituents using FIGAERO-CIMS experiments and COSMO-RS theory. We found a good agreement between experimental and computational saturation vapor pressures for molecules with molar masses around 190 g mol−1 and higher, most within a factor of 3 comparing the average of the experimental vapor pressures and the COSMO-RS estimate of the isomer closest to the experiments. Smaller molecules likely have saturation vapor pressures that are too high to be measured using our experimental setup. The molecules with molar masses below 190 g mol−1 that have several orders of magnitude difference between the computational and experimental saturation vapor pressures observed in our experiments are likely products of thermal decomposition occurring during thermal desorption. For example, dehydration and decarboxylation reactions are able to explain some of the discrepancies between measured and calculated saturation vapor pressures. Based on our estimates, FIGAERO-CIMS can best be used to determine saturation vapor pressures of compounds with low and extremely low volatilities.
How to cite. Hyttinen, N., Pullinen, I., Nissinen, A., Schobesberger, S., Virtanen, A., and Yli-Juuti, T.: Comparison of computational and experimental saturation vapor pressures of α-pinene + O3 oxidation products, Atmos. Chem. Phys. Discuss. [preprint], https://doi.org/10.5194/acp-2021-775, in review, 2021.
Preprint (PDF, 367 KB)
Preprint (367 KB)
Noora Hyttinen et al.
Comment types: AC – author | RC – referee | CC – community | EC – editor | CEC – chief editor | : Report abuse
RC1: 'Comment on acp-2021-775', Anonymous Referee #1, 17 Oct 2021
The manuscript by Hyttinen et al. examines the vapor pressures from $\alpha-pinene$ using both state-of-the-art measurement techniques and quantum mechanical simulations. The experiment and analysis are presented clearly and concisely. I recommend the article for publication and have a couple of clarifications to be addressed below.
Page 8 Figure 1: I can't entirely agree that the FIGAERO-CIMS is suited for the LVOC and ELVOC range. From my eye, it looks like the measurement and model data have two different slopes. Then their intercept where they happen to agree is between 225 and 300 g/mol. So I would be more inclined to agree if the slopes of the lines were in agreement for your stated ``valid measurement range'' and then in disagreement for the range outside.
Page 13 Table 1: Can you add the SIMPOL.1 values to the comparison table. Then add a sentence or two discussing those results and point to the SIMPOL.1 vs. COSMOtherm graph in the SI Figure S12 as you didn't mention that graph in the main text.
Citation: https://doi.org/10.5194/acp-2021-775-RC1
The data displayed in Figure 1 are interpreted by the authors as primarily indicating a large discrepancy between COSMOtherm-predicted vapour pressures and those obtained from the FIGAERO-CIMS technique for compounds with small molecular mass. However, an alternative reading of that plot would be that the two methods for obtaining vapor pressure indicate very different dependence of the vapour pressure on molecular mass. COSMOtherm suggests that with an increase in molecular mass by ~40 g/mol the vapor pressure decreases on average by ~1 order of magnitude. For the FIGAERO data that dependence is less than half as pronounced, whereby a three-fold increase in the molecular mass (from 130 to 390 g/mol) only lowers the vapor pressure from 10-7 Pa to 10-10 Pa. I note that the calibration involving PEGs (Table S1, Figure S1) indicates that an increase in the molecular mass of the PEGs by 88 g/mol leads to a 2.5 order of magnitude decrease in volatility, which is very comparable to what COSMOtherm suggests.
In other words, the discrepancy between the two techniques is also quite large for the heaviest molecules – agreement is only apparent for molecules between 200 to 300 g/mol. The most likely explanation for the discrepancy is a limitation of the FIGAERO technique and not of the COSMOtherm estimation method. The authors themselves offer an explanation on lines 224-226: "It is also possible that saturation vapor pressures of dimers with the lowest volatilities (psat < 10−11 Pa) cannot be measured using thermal desorption, as the molecules would thermally decompose before evaporating from the sample (Yang et al., 2021)."
At the root of the failure of the FIGAERO technique as applied here may be the extrapolation to volatilities that fall far outside of the calibration. The calibration only involves the molecular mass range from 282 to 370 g/mol and log pSatfrom -5 to -7, but it is applied to a mass range from 130 to 390 g/mol and a vapour range of more than 10 orders of magnitude.
I suggest some rephrasing of formulations in abstract and manuscript text would be called for, e.g. line 4 to 5: "We found a good agreement between experimental and computational saturation vapor pressures for molecules with molar masses around 190 g mol−1 and higher." Maybe the thrust of the paper could be shifted towards highlighting how the calibration procedure AND thermal decomposition limits the FIGAERO technique to a estimating the volatility to certain set of compounds and that there is both an upper and a lower volatility limit to this technique. I even wonder whether it is really appropriate to refer to the values obtain by the FIGAERO technique as "experimental saturation vapour pressures" (as done in the title of the manuscript). This is a rather indirect inference based on a number of assumptions that may not be all that valid. Maybe formulating as "vapour pressure values inferred from regressions with desorption temperatures" would be more cautious.
Line 24: Just because the SOA community has been using saturation vapour pressure does not mean that "it is essential to have reliable methods to estimate the saturation vapor pressures of complex organic molecules formed in the atmosphere". What really is required is the equilibrium partitioning ratio between the SOA and the gas phase. The saturation vapour pressure is simply commonly used to estimate that partitioning ratio (together with an activity coefficient of the compound in the SOA). A better formulation would be therefore: "it is essential to have reliable methods to estimate the volatility of complex organic molecules formed in the atmosphere."
Line 43ff.: The thermodynamic property controlling the rate or timing of desorption would only be the saturation vapour pressure, if the molecule were to desorb from its own pure (liquid) phase. This doesn't appear to be the case in the method referred to here. As such, this method does not "measure" the saturation vapour pressure, but the equilibrium partitioning ratio. Would it then not be better to calibrate the method using compounds with known equilibrium partitioning ratios in order to find the correlation between equilibrium partitioning ratios and desorption temperature? After all, it is the equilibrium partitioning ratios that you are interested in in the first place.
Line 149ff.: I appreciate that you refer to the earlier study by Kurten et al., 2018 to justify the selection of conformers containing no intramolecular H-bonds, but I think some sort of explanation would still be required here. There doesn't seem to be any compelling reason why the ozonolysis of a-pinene should preferentially lead to oxidation products that do not contain intramolecular H-bonds. Furthermore, one of the advantages of a method such as COSMOtherm is precisely the fact that it should be able to account for the effect of intramolecular H-bonding on solvation. The rationale provided, namely "This method has been shown to provide more reliable saturation vapor pressure estimates for multifunctional oxygenated organic compounds", seems not very convincing as it could simply be coincidence.
Line 96ff.: Doesn't that procedure lead to a bias in the comparison of measured and estimated saturation vapour pressure?
Line 15: "grouped […] into"
Line 23: "the role […] in SOA formation"
Line 44: "estimated from the desorption temperatures"
Line 131: "for the smallest" "for the largest"
AC1: 'Response to referees', Noora Hyttinen, 02 Dec 2021
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Accurate saturation vapor pressure estimates of atmospherically relevant organic compounds are critical for modeling secondary organic aerosol (SOA) formation. We investigated vapor pressures of highly oxygenated SOA constituents using state of the art computational and experimental methods. We found a good agreement between low and extremely low vapor pressures estimated using the two methods, while the smallest molecules detected in our experiment were likely products of thermal decomposition.
Accurate saturation vapor pressure estimates of atmospherically relevant organic compounds are... | CommonCrawl |
Clinical Phytoscience
International Journal of Phytomedicine and Phytotherapy
In vitro antimalarial activity evaluation of two ethnomedicinal plants against chloroquine sensitive and resistant strains of Plasmodium falciparum
Original contribution
Neelutpal Gogoi ORCID: orcid.org/0000-0001-5419-39421,
Bhaskarjyoti Gogoi2 &
Dipak Chetia1
Clinical Phytoscience volume 7, Article number: 42 (2021) Cite this article
In this study, we selected two medicinal plants Citrus maxima (Burm.) Merr. and Artemisia nilagirica (C.B. Clarke) Pamp. on the basis of their traditional use in the treatment of fever associated with malaria in Assam (India) and evaluated their antimalarial potential against Plasmodium falciparum strains.
The properly processed plant parts of C. maxima (Burm.) Merr. and A. nilagirica (C.B. Clarke) Pamp. were extracted with different solvents from nonpolar to polar by cold maceration technique. After that antimalarial activities of the extracts were evaluated against both chloroquine sensitive (3D7) and resistant (RKL-9) strains of P. falciparum using Giemsa staining light microscopy technique. The most active extract(s) was further screened for cytotoxicity potential against murine macrophage RAW264.7 cell line using MTT assay. Then preliminary phytochemical screening and qualitative fingerprint analysis of the active extract(s) were done to check the presence of different secondary metabolites.
From the in vitro study, the hydro-alcoholic extract of C. maxima (Burm.) Merr. and methanol extract of A. nilagirica (C.B. Clarke) Pamp. were found to be the most active against both 3D7 and RKL-9 strains. In the cytotoxicity study, the CC50 values of the active extracts were found to be > 100 μg/ml, which suggested the safety of the extracts. Then phytochemical and fingerprint analysis revealed the presence of various important plant secondary metabolites in both the extracts.
The findings of this study confirmed the presence of antimalarial potential of hydro-alcoholic extract of C. maxima (Burm.) Merr. and methanol extract of A. nilagirica (C.B. Clarke) Pamp without having any toxic effect. Both the extracts showed IC50 values below 5 μg/ml against 3D7 and RKL-9 strains.
Different approaches are adopted for the development of new drugs against malaria, where natural compound-based approach is one of the preferable one. It is to mention that several important drug candidates available in the market were originated from plant sources [1, 2]. The antimalarial drugs, quinine and artemisinin were also isolated from the plant sources based on the information of traditional use [3]. Later several potent semisynthetic and synthetic derivatives of quinine and artemisinin were developed, some of which are still in clinical use and some new derivatives are under developmental stages [4]. Scientists from different regions including India are still searching for the potent antimalarial compounds from various natural resources and have found some plants having antimalarial activity [5]. Being a part of Sub-Himalayan region, North East (NE) India is rich in plant diversity and various plants of this region have been using in traditional medicine by various ethnic groups of the region. In this study, we selected two ethnomedicinal plants, Citrus maxima (Burm.) Merr. and Artemisia nilagirica (C.B. Clarke) on the basis of their use in the preparation of antimalarial herbal remedy in Assam, a state of NE India. Citrus maxima (Burm.) Merr., commonly known as 'Pumelo' has diverse biological activities like antioxidants, antihyperglycemic/antidiabetic, hepatoprotective, anticancer, antimicrobial/antibacterial, antidepressant, antiaging, cholinesterase and tyrosine inhibition etc. [6,7,8,9,10,11,12,13,14]. Besides it was also reported that the leaves and fruit parts of C. maxima is also used in the treatment of malaria in Benin which is a highly malaria affected West African region [15]. From the fruit part of Citrus maxima (Burm.) Merr. our group already reported one compound luteolin as a potent antimalarial lead compound against P. falciparum using in silico and in vitro analysis [16]. On the other hand, A. nilagirca is reported to have several important biological activities like antioxidant, cytoprotective, anti-inflammatory, antiproliferative, antibacterial, anticancer etc. [17,18,19,20,21,22,23,24,25]. Apart, this plant also possesses larvicidal, pupicidal, adulticidal and mosquito repellant activity [26,27,28]. Besides Panda et al. evaluated the antimalarial potency of the different extracts of Artemisia nilagirica (C.B. Clarke) against an African strain (FCR-3) of P. falciparum using in vitro technique and found IC50 value 5.76 ± 0.82 μg/ml of the methanol extract [29]. Based on the previous reports and their use in traditional medicine, it was hypothesized to have active phytoconstituent(s) against the resistant strains of malaria parasites. Hence, in this study, we investigated the antimalarial potency of different extracts of C. maxima and A. nilagirica against chloroquine sensitive (3D7) and chloroquine resistant (RKL-9) strains of P. falciparum.
Chemicals and reagents
The analytical reagent grade (EMPARTA ACS grade) solvents like petroleum ether (40–60 °C), toluene, chloroform, ethyl acetate, methanol, ethanol and analytical TLC aluminium plates were purchased from Merck Millipore, Burlington, Massachusetts, USA. The different analytical grade (AR) chemicals like sodium bicarbonate, HEPES buffer, D-glucose, D-sorbitol, formic acid, phosphate buffer solution 10X (PBS), Giemsa stain, sodium pyruvate, dimethyl sulfoxide (DMSO) and plastic wares used in the biological activity screening were purchased from HiMedia Pvt. Ltd., Mumbai, India. Molecular biology grade Fetal bovine serum (FBS), gentamycin, amphotericin-B, penicillin, streptomycin, 3-(4,5-dimethylthiazol-2-yl)-2,5-diphenyl tetrazolium bromide (MTT), RPMI-1640 (Roswell Park Memorial Institute-1640), DMEM (Dulbecco's Modified Eagle's medium) were purchased from Gibco-BRL, Life Technologies Inc., Gaithersburg, MD 20884–9980, USA and Sigma Aldrich, St. Louis, MO, USA. The marker compound and standard drug, namely, quercetin (HPLC grade ≥ 95%) and chloroquine phosphate (Certified Reference Material grade) were purchased from Sigma-Aldrich, St. Louis, MO, USA.
Plant materials
The information regarding the use of different plants as herbal remedies for the treatment of fever associated with malaria was collected by interviewing local practitioners in the form of questionnaires and from local books.
Herbariums of the two selected plants (Voucher specimen no. DU/DRS/HRB/NG/2018–19/FS-01 and DU/DRS/HRB/NG/2018–19/FS-05) were prepared according to standard procedure and sent to Botanical Survey of India, Shillong for identification and authentication. Then the plant parts used in the preparation the traditional herbal remedy were collected and processed according to the guideline of Good Agricultural and Collection Practice of WHO [30]. The processed plant materials were coarsely powdered with a mechanical grinder and stored in airtight containers.
Extraction of the plant materials
The extraction of the powdered plant materials was carried out using cold maceration technique using petroleum ether (40–60 °C), chloroform, ethyl acetate, methanol and hydro-alcoholic (1:1) solvent system. Besides, one sample from each plant were prepared as per the traditional practices. In the cold maceration, about 500 g plant material was taken with 3 l of solvent and kept for 72 h with occasional shaking [31]. After that, the solvents were removed under reduced pressure at low temperature using a rotary evaporator (IKA Rotary Evaporator RV 8 V). In the case of the hydro-alcoholic extract, the water part was removed by lyophilisation of the sample using laboratory freeze dryer (IIC Industrial Corporation). The extracts were preserved in glass sample bottles and kept in − 20 °C for further use. The extracts were prepared at ratio of 1:6 during cold maceration process. The yields of the extracts of C. maxima were found to be 0.262%, 0.562%, 2.882%, 1.151%, 8.305% and 0.167% w/w with petroleum ether (40–60 °C), chloroform, ethyl acetate, methanol, hydro-alcoholic and traditional solvent system respectively. Similarly, in case of A. nilagirica, the yields of the extracts were found to be 0.757%, 1.627%, 1.891%, 2.062%, 1.770% and 0.215% w/w with petroleum ether (40–60 °C), chloroform, ethyl acetate, methanol, hydro-alcoholic and traditional solvent system respectively.
In vitro antimalarial screening of the extracts
Preparation of standard and test samples
For this study, 1 mg/ml stock solutions of the extracts were prepared by using incomplete RPMI-1640 media containing 0.5% DMSO. Chloroquine phosphate was used as standard drug in this study and prepared 100 μg/ml stock using the same protocol as used for extracts for further use.
In vitro culture of malaria parasites
The chloroquine sensitive (3D7) and chloroquine resistant (RKL-9) strains of the malaria parasite P. falciparum were obtained from the Parasite Bank of National Institute of Malaria Research (Indian Council of Medical Research), New Delhi. The strains of P. falciparum were maintained in fresh A+ erythrocytes suspended in RPMI-1640 medium supplemented with 25 mM HEPES, 1% D-glucose, 0.23% sodium bicarbonate, gentamycin (40 mg/ml), amphotericin-B (0.25 mg/ml) and 10% heat-inactivated AB+ serum at 37 °C and 5% CO2 environment [32, 33]. After every 24 h, the used medium was replaced with fresh medium supplemented with 10% heat-inactivated AB+ serum and the parasitemia level was maintained below 2%.
Antimalarial activity screening & determination of IC50 values
The antimalarial screening of the extracts was carried out against both 3D7 and RKL-9 strain of P. falciparum by Giemsa staining light microscopy method. For antimalarial testing, initially the asynchronous P. falciparum parasites were synchronized to obtain only the ring stage parasitized cells by treating with 5% D-sorbitol [34]. The initial ring stage parasitemia was maintained at 0.5% in 4% haematocrit using complete medium and fresh A+ erythrocytes before using in the screening. For screening, test and standard drugs were taken in nine different concentrations by two fold serial dilutions for both the 3D7 and RKL-9 strains in 96 well plates. The concentration ranges for test drug and standard drug were 50 to 0.19 μg/ml and 5 to 0.019 μg/ml respectively. All the treatments were performed in triplicates. Then parasitized blood was added to the wells of 96-well plate containing 100 μl of test and standard samples to carry out the assay. The plates were incubated at 37 °C in an environment of 5% CO2 for 36–40 h in a CO2 incubator. After the incubation period, blood smears were prepared in glass slide from each well and fixed by treating with methanol. The slides were stained with 10% Giemsa stain prepared in 1% phosphate buffer solution (PBS). After that number of schizonts (3 or more merozoites containing) per 100 asexual parasites were counted under a light microscope (Leica DM1000) at 1000X (oil emersion) magnification [35]. The percentage inhibition for each concentration was calculated by the following equation:
$$ \% Inhibition=1-\frac{no. of\ schizonts\ in\ test}{no. of\ schizonts\ in\ negative\ control}\times 100 $$
Finally, the IC50 values were calculated by plotting nonlinear regression curve between log dose vs percentage (%) inhibition using GraphPad Prism (GraphPad Prism v.7 San Diego, California, USA). Based on the obtained IC50 values, the extracts were categorized in to active (< 10 μg/ml), intermediate (10–25 μg/ml) or inactive (> 25 μg/ml) categories [36].
In vitro cytotoxicity study
The most active extract obtained for each plant from the in vitro antimalarial study were further taken for in vitro cytotoxicity study using MTT assay [37]. The study was carried out against normal murine macrophage RAW264.7 cell line. Approximately, 1 × 104 cells/ml were cultured in DMEM (Dulbecco's Modified Eagle's medium) media supplemented with 2 mM L-glutamine, 1 mM sodium pyruvate, 10% FBS (Fetal Bovine Serum), penicillin (100 units/ml), streptomycin (10 μg/ml) and allowed to incubate at 37 °C in a humidified 5% CO2 environment. After 80% cell confluency, the cells were treated with different concentrations (50, 100, 200 and 500 μg/ml) of the selected test extracts and incubated for 24 h. Cells without any treatment were considered as control. After 24 h incubation, 0.5 mg/ml MTT was added to each well of the plates and incubated for a further 5 h. After the completion of the incubation, formed formazone complexes were dissolved properly in MTT solvent and the absorbance was taken at 570 nm using a microplate reader (Multiskan™ FC Microplate Photometer). The experiment was performed in triplicates and the percentage of cell viability was calculated for each concentration and compared to control cells without any treatment. Finally, CC50 (cytotoxic concentration) were determined for the test extracts.
Phytochemical and qualitative fingerprint analysis
The various biological activities shown by plant extracts are mainly due to the presence of the different secondary metabolites like alkaloids, flavonoids, glycosides, terpenoids, saponins, steroids, tannins etc. [38]. The active extract from each plant was analysed by chemical reagents to detect the presence of those secondary metabolites using standard procedures as described in the earlier study [39].
The qualitative fingerprint analysis of an extract under the controlled environment is one of the quality control parameters for herbal products [40, 41]. In this study, the qualitative fingerprints of the active extracts were developed using HPTLC densitometry analysis under control environment of temperature and humidity. The study was carried out by using Camag TLC Scanner 4, semiautomatic sample spotter Linomat 5, UV visualization cabinet (deuterium, tungsten and mercury lamp), 100 μl Hamilton dosage syringe, TLC aluminium plate pre-coated with silica gel 60 F254 (10 cm × 10 cm) and glass twin trough chamber (10 cm × 10 cm).
Initially, stock solutions of the marker compound (quercetin for A. nilagirica)/semi-purified fraction for C. maxima (1 mg/ml) and active extracts (10 mg/ml) were prepared in their respective solvents by sonicating for 15 min. Then the samples were centrifuged at 2000 rpm for 5 min and the supernatants were transferred to the sample vials by filtration for further use.
The experiment was carried out at a temperature of 25 ± 2 °C, relative humidity 55%. In the TLC plate (10 cm × 10 cm), 2 μL of the marker compound or semi-purified fraction in duplicate and 4 μL of the active extracts in quadruplicate were applied as a band of 8 mm × 1 mm in size at a distance of 8 mm from the bottom. After applying the sample, the plates were developed using 10 ml mobile phase composed of toluene: ethyl acetate: methanol: formic acid at a ratio of 3:5:1:0.5 using the glass twin trough chamber (10 cm × 10 cm). The plates were developed up to a distance of 7 cm and air-dried at room temperature. Then plates were visualized under UV cabinet at 254 nm and 366 nm. After that, the plates were scanned at 254 nm in absorbance mode and at 366 nm in fluorescence mode using the Camag TLC scanner 4 linked with VisionCAT 2.5 software. During scanning, the slit dimension was kept at 5 × 0.2 mm and the scanning speed was employed at 20 mm/s [42].
Identification and small scale extraction of plant materials
The plants used in the preparation of herbarium were identified as Citrus maxima (Burm.) Merr. (F: Rutaceae) and Artemisia nilagirca (C.B. Clarke) Pamp. (F: Asteraceae) by Dr. N. Odyuo, Scientist D, Botanical Survey of India, Shillong (Ref no. BSI/ERC/Tech/2019/481 dated 23.09.2019). After completing the extraction process, the highest yield was obtained in hydro-alcoholic solvent system for C. maxima whereas highest yield was obtained in methanol solvent for A. nilagirica.
In vitro antimalarial activity
The slides prepared from each well were observed under the light microscope and the number of infected erythrocytes were counted within a particular area. From the infected erythrocytes, the number of parasites with schizont stage were calculated and used for determination of % inhibition. The in vitro antimalarial activity of the extracts was determined in the form of IC50 values using non-linear regression analysis against both 3D7 and RKL-9 strains, and compared with the standard drug chloroquine phosphate (Table 1 & Fig. 1). The representative photomicrographs of smear observed under the microscope during counting are shown in Fig. 2. From the results, it was observed that in the case of C. maxima the hydro-alcoholic extract (CM-HA) showed the lowest IC50 values 3.41 ± 0.31 μg/ml and 4.45 ± 0.10 μg/ml against 3D7 and RKL-9 strains respectively. In the case of A. nilagirica, methanol extract (AN-ME) showed the lowest IC50 values 3.28 ± 0.08 μg/ml and 3.81 ± 0.34 μg/ml against 3D7 and RKL-9 stains respectively.
Table 1 IC50 values of the plant extracts against 3D7 and RKL-9 strains of P. falciparum
Calculation of IC50 values by non-linear regression analysis (GraphPad Prism) using nine different concentrations with two-fold serial dilution. For the test extracts the dilution was made from 50 μg/ml to 0.195 μg/ml whereas for the standard drug the dilution was made from 5 μg/ml to 0.019 μg/ml. The experiment was performed in triplicates and each value is the mean ± SD of three replicates
Representative photomicrographs taken during the observation under light microscope at 1000X (oil emersion) magnification showing infected erythrocytes by different strains of P. falciparum; a. 3D7 without treatment, b. 3D7 with treatment (50 μg/ml), c. RKL-9 without treatment and d. RKL-9 with treatment (50 μg/ml)
The active extract(s) selected from the antimalarial screening were further analysed for cytotoxicity potential by MTT assay. The methanol extract of A. nilagirica (AN-ME) and hydro-alcoholic extract of C. maxima (CM-HA) were screened against normal murine macrophage RAW264.7 cell line where % viability for the methanol extract of A. nilagirica was found to be more than 90% at the maximum dose 500 μg/ml (Fig. 3). On the other hand, the % viability for the hydro-alcoholic extract of C. maxima was found to be less than 50% at the maximum dose of 500 μg/ml (Fig. 3). Later CC50 values for both the extracts were determined and found to be more than 100 μg/ml for both the extracts (Fig. 3).
Cytotoxic effects of the active extracts of C. maxima and A. nilagirica on murine macrophage RAW264.7 cell line; a Cell viability after treatment of 24 h with the different dose of the extracts. b Determination of CC50 values of the two extracts (CM-HA and AN-ME). The experiment was performed in triplicates and each value is the mean ± SD of three replicates
Phytochemical & fingerprint analysis
The bioactive potential shown by a plant extract is due to the presence of different plant secondary metabolites in the sample. From the preliminary chemical tests, alkaloids, flavonoids and glycosides were found to be present in the CM-HA extract whereas, alkaloids, flavonoids, tannins and terpenoids were found to be present in the AN-ME extract (Table S1).
The qualitative fingerprint analysis of extract is a standardization process of plant material or plant extract in the absence of known marker or standard compound. The fingerprints of the active extracts were developed under control environmental conditions like temperature and humidity, and found the presence of various types of compounds in the extracts (Figs. 4 and 5). The chromatograms developed after scanning under 254 nm (absorbance mode) and 366 nm (fluorescence mode) for each extract revealed the presence of multiple components (based on number of peaks) and their possible quantity (based on the area of the peaks) (Fig. S1 & S2).
Visualization of the TLC plates under UV visualizer at 254 nm and 366 nm to detect the presence of different plant secondary metabolites of Citrus maxima (Burm.) Merr
Visualization of the TLC plates under UV visualizer at 254 nm and 366 nm to detect the presence of different plant secondary metabolites of Artemisia nilagirica (C.B. Clarke) Pamp
The discovery of new antimalarial lead molecule(s) or the development to of antimalarial phytopharmaceutical product(s) from traditionally used plant materials/parts has gained significant importance [43]. For that purpose, the preliminary in vitro antimalarial screening of the plant materials/parts is very essential along with their toxicity assessment against normal mammalian/human cell. Therefore, this study aimed to evaluate the antimalarial potential of the different extracts of the two selected traditionally used plants as well as cytotoxic effect of the active extracts in in vitro conditions. All the extracts obtained from the two plants (C. maxima and A. nilagirica) showed the antimalarial activity within different ranges in the form of IC50 values in in vitro antimalarial analysis. From the results, we found that the ethyl acetate, methanol and hydro-alcoholic extracts came under active category (< 10 μg/ml) against both 3D7 and RKL-9 strains of P. falciparum. The chloroform extract of C. maxima also came under the active category only against the 3D7 strain of P. falciparum. However, among the active extracts, in the case of C. maxima, the hydro-alcoholic extract showed the best activity with IC50 values 3.41 ± 0.31 μg/ml and 4.45 ± 0.10 μg/ml against 3D7 and RKL-9 respectively (Table 1). Whereas in the case of A. nilagirica, the methanol extract showed the best activity with IC50 values 3.28 ± 0.08 μg/ml (3D7) and 3.81 ± 0.34 μg/ml against 3D7 and RKL-9 respectively (Table 1). These two extracts may contain the phytoconstituents which are responsible for strong antimalarial activity against P. falciparum strains. The antimalarial activity of the root part of C. maxima is not reported till date. But the antimalarial activity of the leaves of A. nilagirica was reported by Panda et al. against a different strain of P. falciparum where methanol extract was found to be the most active [29]. The preliminary phytochemical screening of the extracts revealed the presence of some important class of plant secondary metabolites like alkaloids, flavonoids and terpenoids. Pan et al., 2018 already reported the antimalarial activity of secondary metabolites like alkaloids, flavonoids and terpenoids [44]. Hence, the potential in vitro antimalarial activity showed by the active extracts (CM-HA and AN-ME) against the two P. falciparum strains may be due to the presence of these important classes of secondary metabolites. Although the IC50 values of these two extracts were not near to the IC50 values of the standard drug chloroquine phosphate (IC50 values 0.54 ± 0.02 μg/ml and 0.88 ± 0.04 μg/ml against 3D7 and RKL-9 respectively), at extract level the results were satisfactory and can be taken for further study [45]. In the case of both the plants, the samples prepared in the traditional way were found to be either in the intermediate or inactive category. But as they were showing some extent of inhibition in the growth of the parasites hence the traditional way of using these plant parts give some benefits to the patients. The activity of the other extracts also suggests the positive impacts of these plants on the treatment of malaria.
The two active extracts were further examined for their cytotoxicity potential against normal cell line (murine macrophage RAW264.7 cell line) where the methanol extract of A. nilagirica (AN-ME) was found to be safe even at a dose of 500 μg/ml with > 90% cell viability (Fig. 3). But the hydro-alcoholic extract of C. maxima (CM-HA) was found to be toxic at the maximum dose of 500 μg/ml with < 50% cell viability (Fig. 3). After that, the CC50 values of the extracts were determined where the CC50 value of the CM-HA was found to be 817.3 ± 2.4 μg/ml whereas the CC50 value of the AN-ME was found to be > 1000 μg/ml. Overall both the extracts (CM-HA and AN-ME) were found to be more selective towards P. falciparum than normal cells. Besides extracts or compounds with CC50 > 90 μg/ml against normal cell line are considered to be safe and can be taken for further study [46], hence the two test extracts can be utilized for further preclinical studies.
The preliminary phytochemical and fingerprint analysis provide an idea regarding the presence of different plant metabolites in the selected extract [47]. As from the analysis, no known markers were found for both the extracts, the qualitative fingerprints were developed in a controlled environment of temperature and humidity using a suitable mobile phase. The photographs as well the chromatograms developed during the study gave an idea that the CM-HA contained diverse groups of compounds with absorbance and fluorescence characteristics (Fig. 4). The intensities of the bands also indicate the presence of significant amounts of few compounds in the extract. Similarly, the photographs and chromatograms of AN-ME also showed the presence of diverse groups of compounds but the intensity of the bands indicated the presence of those compounds in less quantity (Fig. 5). These two final extracts (CM-HA and AN-ME) were found to be highly active (< 5 μg/ml) with the presence of various important secondary metabolites. These active extracts can deliver new/novel lead compound(s) with antimalarial activity as well as can be further utilized for the development of phytopharmaceutical products by fulfilling the criteria of regulatory body like CDSCO (Central Drugs Standard Control Organisation) for global acceptance.
In this study, we investigated the antimalarial activity of different extracts of two traditionally used medicinal plants C. maxima and A. nilagirica against both chloroquine sensitive (3D7) and chloroquine resistant (RKL-9) strains of P. falciparum. From the results obtained from in vitro experiments, it was observed that hydro-alcoholic extract of C. maxima and methanolic extracts of A. nilagirica showed the potent activity against the strains of P. falciparum without having any toxic effect. The findings of the study will help in identification of active antimalarial lead molecule(s) from the active extracts to develop safe and potent antimalarial drug(s) or phytopharmaceutical(s) by following proper standardization techniques as per the regulatory guidelines.
The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.
A. Nilagirica :
Artemisia nilagirica (C.B. Clarke) Pamp
C. maxima :
Citrus maxima (Burm.) Merr
AN-ME:
Methanol extract of A. nilagirica
CM-HA:
Hydro-alcoholic extract of C. maxima
RPMI-1640:
Roswell Park Memorial Institute-1640
IC50 :
50% inhibitory concentration
PBS:
Phosphate buffer solution
DMEM:
Dulbecco's Modified Eagle's medium
FBS:
Fetal Bovine Serum
CC50 :
50% cytotoxic concentration
CDSCO:
Central Drugs Standard Control Organisation
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The authors are thankful to the Department of Pharmaceutical Sciences, Dibrugarh University, Dibrugarh, Assam, India for providing the laboratory facilities to conduct the research work. The Authors also are thankful to the Parasite Bank of National Institute of Malaria Research (Indian Council of Medical Research), New Delhi for providing the malaria parasite strains to carry out the research work.
The work was supported by University Grants Commission, New Delhi, India under UGC-SAP (DRS-I) [F.3–13/2016/DRS-I (SAP-II)].
Department of Pharmaceutical Sciences, Faculty of Science and Engineering, Dibrugarh University, Dibrugarh, Assam, 786004, India
Neelutpal Gogoi & Dipak Chetia
Department of Biotechnology, Royal School of Bio-Science, Royal Global University, Guwahati, India
Bhaskarjyoti Gogoi
Neelutpal Gogoi
Dipak Chetia
NG and DC conceived and designed the experiments; NG & BG performed the in vitro experiments and analysed the results; NG wrote the manuscript, and BG & DC reviewed the manuscript. All authors read and approved the final manuscript.
Correspondence to Neelutpal Gogoi.
Gogoi, N., Gogoi, B. & Chetia, D. In vitro antimalarial activity evaluation of two ethnomedicinal plants against chloroquine sensitive and resistant strains of Plasmodium falciparum. Clin Phytosci 7, 42 (2021). https://doi.org/10.1186/s40816-021-00269-1
Citrus maxima (Burm.) Merr.
Artemisia nilagirica (C.B. Clarke) Pamp.
Cytotoxicity study, herbal remedy | CommonCrawl |
Broadband Bending of Flexural Waves: Acoustic Shapes and Patterns
Amir Darabi1,
Ahmad Zareei2,
Mohammad-Reza Alam ORCID: orcid.org/0000-0002-9730-07172 &
Michael J. Leamy ORCID: orcid.org/0000-0002-9914-640X1
Devices for energy harvesting
Directing and controlling flexural waves in thin plates along a curved trajectory over a broad frequency range is a significant challenge that has various applications in imaging, cloaking, wave focusing, and wireless power transfer circumventing obstacles. To date, all studies appeared controlling elastic waves in structures using periodic arrays of inclusions where these structures are narrowband either because scattering is efficient over a small frequency range, or the arrangements exploit Bragg scattering bandgaps, which themselves are narrowband. Here, we design and experimentally test a wave-bending structure in a thin plate by smoothly varying the plate's rigidity (and thus its phase velocity). The proposed structures are (i) broadband, since the approach is frequency-independent and does not require bandgaps, and (ii) capable of bending elastic waves along convex trajectories with an arbitrary curvature.
While acoustic waves propagate along a straight line in a homogeneous medium, metamaterials are capable of diffracting and directing acoustic waves by spatially varying the host material properties along the wave propagation, and have various applications such as health monitoring1,2, telecommunication3, super-resolution acoustic imaging in surgery4,5,6, cloaking7,8,9,10,11,12, wave focusing13,14,15,16, wave guides17,18,19,20,21, and wave bending for electronic devices1,22.
Current approaches for manipulating and bending flexural waves depend on either changing the effective refractive index to steer waves14,23,24, or exploiting frequency bandgaps to guide waves along a pre-defined path21,25,26,27,28. Both approaches can be achieved using periodic structures such as phononic crystals and metamaterials29. These periodic structures are lattice materials exhibiting refractive frequency bandgaps that can be tailored via unit cell design. A few recent studies report efforts to bend or steer acoustic/elastic waves without using phononic crystals or metamaterials. Zhang et al. introduced the idea of three-dimensional acoustic bottles generated by self-bending beams in a homogeneous acoustic medium. Phase changes in a linear array of 40 sources produce the necessary self-bending, and thus acoustic energy can be propagated along curved paths which circumvent obstacles30. Tol et al. implemented a two-dimensional version of such systems for flexural waves in thin plates using sources composed of piezoelectric transducers31. The acoustic bottle approach is broadband, to arbitrary frequency, as the number of sources in the phased array approaches infinity (since the source spacing must be less than a wavelength). Note that for most of applications in wave bending devices (e.g. health monitoring, focusing, or wave-guiding/wave-bending devices) the incoming wave is given, and engineering the initial wave packet is not feasible.
The primary aim of this Letter is to propose an alternative approach for bending flexural waves along any curved trajectories in thin plates by machining the surface of the host medium along a desired path. This new technique is broadband for the lowest asymmetric Lamb wave, and works for any convex trajectory without introducing a symmetric artifact. The wave controlling structure is created by continuously altering the flexural rigidity profile (i.e. plate's thickness) similar to that used previously to design a flexural continuous gradient-index (GRIN) lens concept16. A model capturing the essential ideas of the system is introduced first, followed by numerical results generated using Finite Element Methods. Moreover, the tailored system is tested experimentally to demonstrate the bending of elastic waves over a broad frequency range (20 kHz–120 kHz), which is limited by the frequency bounds of the experimental setup.
The governing equation of flexural waves in a thin plate with elasticity modulus E, thickness h, density ρ, and Poisson's ratio ν is given by
$$D{\nabla }^{4}w+\rho h{w}_{tt}=\mathrm{0,}$$
where \(D=E{h}^{3}/12(1-{\nu }^{2})\) represents the flexural rigidity, w the transverse displacement, and subscript tt denotes a second derivative with respect to time. Equation (1) is known as Kirchhoff-Love plate equation and is valid when the wavelength λ is large enough compared to the thickness of the plate h and small compared to it's in-plane dimension L (i.e. \(h\ll \lambda \ll L\)). The dispersion relation of plane flexural waves with wavenumber k is then obtained as \(D{k}^{4}-\rho h{\omega }^{2}=0\), where the phase velocity is
$${v}_{p}^{4}={(\frac{\omega }{k})}^{4}={\omega }^{2}\frac{E{h}^{2}}{\mathrm{12(1}-{\nu }^{2})\rho }\mathrm{.}$$
We assume a channel with a convex trajectory for bending the traveling wave (Fig. 1). As shown, the channel is bounded by two desired curves y = f1(x) and y = f2(x) with the same center of curvature and curvature radii r1(x, y) and r2(x, y) respectively. The radius of each curve is obtained as32 \(r(x,y)={(1+{y}_{x}^{2})}^{\frac{3}{2}}/|{y}_{xx}|\), where subscript x denotes a derivative with respect to x. In order to design a bending acoustic device, the phase fronts need to rotate around the center of curvature with a constant angular frequency. Therefore, the phase velocity along the channel should vary as \({v}_{p}(r)=r{\bar{\omega }}_{p}\), where \({\bar{\omega }}_{p}\) is the angular speed associated with wave bending. Considering the relation between the phase velocity and the thickness of the plate in Eq. (2), and the equation for phase velocity \(({v}_{p}(r)=r{\bar{\omega }}_{p})\), wave bending in thin plates is achievable if the thickness varies continuously as,
$$h=\frac{{(r{\bar{\omega }}_{p})}^{2}}{\omega }\sqrt{\frac{\mathrm{12(1}-{\nu }^{2})\rho }{E}}\mathrm{.}$$
Overview of the proposed wave-bending approach. (a) Geometry associated with wave bending on a flat plate. (b) Schematic of the proposed structure to bend a flexural wave from a source to a receiver. (c) Top view of the structure serving as the host medium and the machined channel to bend the wave. (d) Thickness profile of the channel; and (e) Schematic of the thickness variation along the guiding trajectory.
This implies \(h\propto {r}^{2}\) and requires the plate's thickness to vary quadratically between the inner radius of the created channel h1 = h(r1), and the outer radius h2 = h(r2) (Fig. 1b–e). Note that different values of inner and outer thickness for the quadratic thickness profile would result in different angular frequency for the rotational speed of phase fronts.
Numerical results
Numerical simulations using the Finite Element Methods (FEM) are first employed to exhibit the effectiveness of the proposed wave bending structure. To support the importance of thickness variation in bending waves, two different cases have been considered. For both cases, Aluminum plate with a constant thickness, h0 = 3.175 mm serves as the host medium of the waveguide channel with the inner and outer radius of \({r}_{1}=5\,cm,{r}_{2}=8.6\,cm\). (see Methods for details). For the first case, we assume a constant thickness (h(r) = 1 mm) in the channel, while for the second case the quadratic thickness profile follows h(r) = 0.4 r2. Accordingly, for the latter case, the thickness varies with a quadratic profile between h1 = 1 mm and h2 = 3 mm in the channel. Fig. 2 depicts the computed time snapshots of wavefield for both cases at 50 kHz (the first row of subfigures depicts the results for the constant thickness waveguide, and the second row shows the results for the quadratic thickness profile). Note that the results have been normalized with respect to the amplitude of incoming plane wave. Figure 2a confirms that the impedance mismatch at the waveguide's boundary is not sufficient for confining the waves inside the channel and bending the waves in the curved trajectory waveguide. As observed in Fig. 2b, the proposed wave bending structure with quadratic thickness profile, bends the traveling wave fronts and is capable of confining the flexural waves inside the waveguide. It is to be noted that the leakage from the waveguide is minimal and thus the wave intensity decreases slightly as it bends along the desired path.
Numerical results. Numerically calculated wave response of the system at the frequency of 50 kHz. Waveguide's boundary is shown with white dashed line where the boundaries are at r1 = 5 cm and r2 = 8.6 cm. (a) Constant thickness h = 1 mm, along the guiding trajectory different from host's medium thickness. (b) Proposed thickness h(r) = 0.4 r2 along the wave bending channel.
Next, a set of experiments is carried-out to verify the performance of the proposed wave-directing structure (see Methods for details). Fig. 3a provides snapshots of the experimentally-measured wavefield displacements in response to excitation at 50 kHz for the waveguide discussed in the numerical section. Note that the results have been normalized with respect to the amplitude of the wave near the source. In qualitative agreement with the numerical simulations, the traveling wave propagates along the trajectory with a low amplitude loss and wave leakage outside the channel. Figure 3b depicts the measured RMS wavefield obtained experimentally at six different frequencies. These figures confirm the wavefront bending in the proposed waveguide over a broad range of frequencies (i.e. 20–120 kHz). The weak leakage of wave energy outside the channel is due to the discrete nature of maufacturing process and the as-manufactured profile.
Experimental results. (a) Experimentally measured wave response of the proposed wave-bending structure at 50 kHz, Experimentally measured nondimensionalized wavefield displacement generated by line source excitation at (b1) f = 20 kHz, (b2) f = 40 kHz, (b3) f = 60 kHz, (b4) f = 80 kHz, (b5) f = 100 kHz, and (b6) f = 120 kHz.
Navigating waves around an obstacle is one potential use for the proposed wave-bending structures. To support this application, another experimental setup (Fig. 4a) is designed and tested (see Methods). A thin aluminum plate with a width of w = 25 cm, length of L = 28 cm, and thickness of h0 = 3.175 mm again serves as the host medium (Fig. 4b). The wave channel is formed by connecting three quarter-circles with inner and outer radii of r1 = 5 cm, r2 = 9 cm, respectively. The thickness of the channel varies from h1 = 1 mm at r1 to h2 = 3 mm at r2 with a parabolic profile to satisfy equation (3). Note that, since abrupt thickness variation is not desired when the curvature changes, a transient region has been made on the plate with the thickness of 3 mm, and the length of 3 mm. A steel cylinder with a radius of 7.5 cm and tickness of 2 cm is attached to the host plate to replicate an obstacle. Four epoxy-bonded piezoelectric transducers with thickness and diameter of \({h}_{p}=0.4\,mm,\,{d}_{p}=5\) mm (Steiner Martins SMD05T04R411, 3 M DP270 Epoxy Adhesive) located 2 cm away from the leading edge of the channel produce an incident wave in response to a generated 200 mV (peak-to-peak) voltage profile using 10 sinusoidal cycles. The experimentally-measured RMS wavefield at f = 50 kHz is depicted in Fig. 4c. This subfigure clearly documents the desired bending of the wave around the obstacle. As observed, the wave amplitude is re-amplified at the points where the curvature changes. One possible explanation is that the amplified region is experiencing near-resonance behavior due to reflections at its boundaries. Some loss of wave intensity is noted, which can be expected due to material losses associated with the increased propagation distance (≈50 cm) or weak leakage of propagating wave outside the channel.
Experimentally measured wave circumventing around an obstacle. (a) Schematic of the proposed method to rotate the traveling wave around an obstacle, (b) experimental modeling, and (c) normalized RMS wavefield response of the system to a harmonic incoming plane wave at the frequency of 50 kHz.
In other applications, wave bending can be employed to create acoustic patterns and shapes, which themselves may discreetly encode information (i.e. the information would only be available to those who know to look for it). Figure 5a depicts an engineered surface leading to a semi-circles path to form a "smiley" face inspired, in part, by similar smiley patterns created using folded DNA33. Note that six piezoelectric transducers (SMD063T07R111) are used to excite the system (see Methods). Due to the shape of the structure in Fig. 5, propagating waves in the opposite direction could destruct the wave-field in the smaller curves. To overcome this issue, wave propagation is prevented in the opposite direction by applying absorbing pitch tapes to the surface of the plate behind the piezoelectric transducers. The RMS wave response of this pattern is shown in Fig. 5b at 50 kHz. In a final example, a Georgia Tech (GT-shaped) logo is created and tested at 50 kHz. Figure 5b details the engineered surface and the resulting RMS wavefield of the structure. These figures clearly represent the power of the introduced technique to bend and direct flexural waves along desired trajectories, resulting in novel acoustic shapes and patterns.
Experimentally measured patterns and shapes. Experimentally-measured normalized RMS wave response generated by piezoelectric transducers at f = 50 kHz for (a) a smiley face, (b) GT logo.
In summary, this letter proposed a novel, broadband approach for bending flexural waves along convex trajectories by tailoring the medium's flexural rigidity. The configured structures are capable of guiding and bending plane waves for subsequent use in wave manipulation, obstacle avoidance, wireless power transfer, and information encoding. We numerically explored the design and performance of the structures, and verified the approach using experimental measurements. With further development, the proposed approach may find application, for example, in vehicle unibodies to channel noise paths for subsequent absorption, or in seismic protection where tailored terrain will bend surface waves away from susceptible structures.
The Finite Element Method (FEM) is used to solve the governing equation (1) subject to a desired thickness variation16. The numerical domain is set to [0, 9] cm × [0, 9] cm for the host medium with a thickness of h0 = 3.175 mm (Fig. 1c). The wave bending channel is formed in the host layer between two quarter of a cricle with radii r1 = 5 cm, and r2 = 8.6 cm. The plate's thickness varies with a parabolic profile h(r) = 0.4 r2 where the inner and outer thickness in the waveguide are h1 = 1 mm, and h2 = 3 mm respectively (Fig. 1d). The numerical domain is seeded using nodes spaced by Δx = 0.15 mm and element mesh is created using the FreeFEM++ 34 mesh generator. Perfectly matched layer (PML) boundary conditions are used to absorb outgoing waves at the plate's edges. A single sine wave with the wavelength and speed matching with excitation frequency is located at the start of the channel to initialize the wavefield as a source.
Experimental measuring
As depicted in Fig. 6, a Polytec PSV-400 scanning laser Doppler vibrometer measures the resulting out-of-plane wavefield velocity using the backside of a thin aluminum plate. The plate chosen has a width of w = 12 cm, length of L = 14 cm, and thickness of h0 = 3.175 mm. The wave-guide channel is created by quadratic thickness profile varying between h1 = 1 mm at r1 = 5 cm to h2 = 3 mm at r2 = 8.6 cm. The thickness machining was performed at the Georgia Tech Montgomery Machining Mall using a CNC mill. The wavefileld displacement is scanned over a 11 cm × 13 cm square area with a 250 × 250 grid resolution and a time resolution of δt = 1 μs. Absorbing pitch tape is used to mitigate reflecting wave at the boundaries. Three hp = 0.2 mm thick epoxy-bonded piezoelectric transducers (Steiner Martins SMPL30W30T1121, 3M DP270 Epoxy Adhesive) located 1 cm away from the channel produce an incident wave in response to a generated 200 mV (peak-to-peak) voltage profile by 5 cycles of sinusoidal wave (10 cycles in the obstacle experiment), using a function generator (Agilent 33220A) coupled to a voltage amplifier (B&K1040L). Each of these piezoelectric disks provides an in-plane vibration perpendicular to the surface of this piezoelectric acting as a point source. Due to the operating frequency range of the system, having an appropriate number of these disks (depends on the length of the incident wave) will form a wave close to a harmonic plane wave. Note that to make the wave propagate the least outside the channel, a rectangular cavity has been made where transducers have been placed (a layer with the thickness of 3.1 mm). Each of these piezoelectric transducers measure 7 mm × 7 mm and have an effective capacitance of Cp = 1.5 nF. Proper triggering of the laser measurements allows the reconstruction of the out-of-plane velocity field while the RMS (root-mean-square) values are obtained by integrating the measured response over time.
Experimental setup showing an aluminum plate hosting piezoelectric transducers for exciting waves, a function generator and amplifier for generating requisite voltage profiles, and a laser vibrometer for measuring the backside transverse plate velocities.
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Khelif, A., Choujaa, A., Benchabane, S., Djafari-Rouhani, B. & Laude, V. Guiding and bending of acoustic waves in highly confined phononic crystal waveguides. Applied Physics Letters 84, 4400–4402 (2004).
Sun, J.-H. & Wu, T.-T. Propagation of surface acoustic waves through sharply bent two-dimensional phononic crystal waveguides using a finite-difference time-domain method. Physical Review B 74, 174305 (2006).
Hussein, M. I., Leamy, M. J. & Ruzzene, M. Dynamics of phononic materials and structures: Historical origins, recent progress, and future outlook. Applied Mechanics Reviews 66, 040802 (2014).
Zhang, P. et al. Generation of acoustic self-bending and bottle beams by phase engineering. Nature communications 5, 4316 (2014).
Tol, S., Xia, Y., Ruzzene, M. & Erturk, A. Self-bending elastic waves and obstacle circumventing in wireless power transfer. Applied Physics Letters 110, 163505 (2017).
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Publication made possible in part by support from the Berkeley Research Impact Initiative (BRII) sponsored by the UC Berkeley Library. In addition, the authors would like to acknowledge the Georgia Tech ME Machine Shop (especially Mr. Scott Eliot and Mr. Josh Barua) for fabricating the experimental apparatus, and Dr. Massimo Ruzzene, Aerospace Engineering, Georgia Tech, for use of his lab equipment and analysis scripts, which made possible the experimental wavefield measurements.
School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, 30332, USA
Amir Darabi
& Michael J. Leamy
Mechanical Engineering Department, University of California, Berkeley, 94703, USA
Ahmad Zareei
& Mohammad-Reza Alam
Search for Amir Darabi in:
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A.D. and A.Z. contributed equally to the paper. A.Z. conducted the theoretical simulations. A.D. conducted the experimental measurements. M.J.L. and M.A. supervised the study. A.D. and A.Z. wrote the article and M.A. and M.J.L. participated in the revision. All authors contributed to the discussions.
Correspondence to Michael J. Leamy.
Dynamics of metamaterial beams consisting of periodically-coupled parallel flexural elements: a theoretical study
Setare Hajarolasvadi
& Ahmed E Elbanna
Journal of Physics D: Applied Physics (2019)
Deflecting incident flexural waves by nonresonant single-phase meta-slab with subunits of graded thicknesses
Yanlong Xu
, Liyun Cao
& Zhichun Yang
Journal of Sound and Vibration (2019)
Analysis and Experimental Validation of an Optimized Gradient-Index Phononic-Crystal Lens
Physical Review Applied (2018)
Experimental Demonstration of an Ultrabroadband Nonlinear Cloak for Flexural Waves
, Ahmad Zareei
, M.-Reza Alam
Physical Review Letters (2018) | CommonCrawl |
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Tag: braid group
F_un and braid groups
Published June 15, 2008 by lievenlb
Recall that an n-braid consists of n strictly descending elastic strings connecting n inputs at the top (named 1,2,…,n) to n outputs at the bottom (labeled 1,2,…,n) upto isotopy (meaning that we may pull and rearrange the strings in any way possible within 3-dimensional space). We can always change the braid slightly such that we can divide the interval between in- and output in a number of subintervals such that in each of those there is at most one crossing.
n-braids can be multiplied by putting them on top of each other and connecting the outputs of the first braid trivially to the inputs of the second. For example the 5-braid on the left can be written as $B=B_1.B_2 $ with $B_1 $ the braid on the top 3 subintervals and $B_2 $ the braid on the lower 5 subintervals.
In this way (and using our claim that there can be at most 1 crossing in each subinterval) we can write any n-braid as a word in the generators $\sigma_i $ (with $1 \leq i < n $) being the overcrossing between inputs i and i+1. Observe that the undercrossing is then the inverse $\sigma_i^{-1} $. For example, the braid on the left corresponds to the word
$\sigma_1^{-1}.\sigma_2^{-1}.\sigma_1^{-1}.\sigma_2.\sigma_3^{-1}.\sigma_4^{-1}.\sigma_3^{-1}.\sigma_4 $
Clearly there are relations among words in the generators. The easiest one we have already used implicitly namely that $\sigma_i.\sigma_i^{-1} $ is the trivial braid. Emil Artin proved in the 1930-ies that all such relations are consequences of two sets of 'obvious' relations. The first being commutation relations between crossings when the strings are far enough from each other. That is we have
$\sigma_i . \sigma_j = \sigma_j . \sigma_i $ whenever $|i-j| \geq 2 $
The second basic set of relations involves crossings using a common string
$\sigma_i.\sigma_{i+1}.\sigma_i = \sigma_{i+1}.\sigma_i.\sigma_{i+1} $
Starting with the 5-braid at the top, we can use these relations to reduce it to a simpler form. At each step we have outlined to region where the relations are applied
These beautiful braid-pictures were produced using the braid-metapost program written by Stijn Symens.
Tracing a string from an input to an output assigns to an n-braid a permutation on n letters. In the above example, the permutation is $~(1,2,4,5,3) $. As this permutation doesn't change under applying basic reduction, this gives a group-morphism
$\mathbb{B}_n \rightarrow S_n $
from the braid group on n strings $\mathbb{B}_n $ to the symmetric group. We have seen before that the symmetric group $S_n $ has a F-un interpretation as the linear group $GL_n(\mathbb{F}_1) $ over the field with one element. Hence, we can ask whether there is also a F-un interpretation of the n-string braid group and of the above group-morphism.
Kapranov and Smirnov suggest in their paper that the n-string braid group $\mathbb{B}_n \simeq GL_n(\mathbb{F}_1[t]) $ is the general linear group over the polynomial ring $\mathbb{F}_1[t] $ over the field with one element and that the evaluation morphism (setting t=0)
$GL_n(\mathbb{F}_1[t]) \rightarrow GL_n(\mathbb{F}1) $ gives the groupmorphism $\mathbb{B}_n \rightarrow S_n $
The rationale behind this analogy is a theorem of Drinfeld's saying that over a finite field $\mathbb{F}_q $, the profinite completion of $GL_n(\mathbb{F}_q[t]) $ is embedded in the fundamental group of the space of q-polynomials of degree n in much the same way as the n-string braid group $\mathbb{B}_n $ is the fundamental group of the space of complex polynomials of degree n without multiple roots.
And, now that we know the basics of absolute linear algebra, we can give an absolute braid-group representation
$\mathbb{B}_n = GL_n(\mathbb{F}_1[t]) \rightarrow GL_n(\mathbb{F}_{1^n}) $
obtained by sending each generator $\sigma_i $ to the matrix over $\mathbb{F}_{1^n} $ (remember that $\mathbb{F}_{1^n} = (\mu_n)^{\bullet} $ where $\mu_n = \langle \epsilon_n \rangle $ are the n-th roots of unity)
$\sigma_i \mapsto \begin{bmatrix}
1_{i-1} & & & \\
& 0 & \epsilon_n & \\
& \epsilon_n^{-1} & 0 & \\
& & & 1_{n-1-i} \end{bmatrix} $
and it is easy to see that these matrices do indeed satisfy Artin's defining relations for $\mathbb{B}_n $.
Quiver-superpotentials
Published January 14, 2008 by lievenlb
It's been a while, so let's include a recap : a (transitive) permutation representation of the modular group $\Gamma = PSL_2(\mathbb{Z}) $ is determined by the conjugacy class of a cofinite subgroup $\Lambda \subset \Gamma $, or equivalently, to a dessin d'enfant. We have introduced a quiver (aka an oriented graph) which comes from a triangulation of the compactification of $\mathbb{H} / \Lambda $ where $\mathbb{H} $ is the hyperbolic upper half-plane. This quiver is independent of the chosen embedding of the dessin in the Dedeking tessellation. (For more on these terms and constructions, please consult the series Modular subgroups and Dessins d'enfants).
Why are quivers useful? To start, any quiver $Q $ defines a noncommutative algebra, the path algebra $\mathbb{C} Q $, which has as a $\mathbb{C} $-basis all oriented paths in the quiver and multiplication is induced by concatenation of paths (when possible, or zero otherwise). Usually, it is quite hard to make actual computations in noncommutative algebras, but in the case of path algebras you can just see what happens.
Moreover, we can also see the finite dimensional representations of this algebra $\mathbb{C} Q $. Up to isomorphism they are all of the following form : at each vertex $v_i $ of the quiver one places a finite dimensional vectorspace $\mathbb{C}^{d_i} $ and any arrow in the quiver
[tex]\xymatrix{\vtx{v_i} \ar[r]^a & \vtx{v_j}}[/tex] determines a linear map between these vertex spaces, that is, to $a $ corresponds a matrix in $M_{d_j \times d_i}(\mathbb{C}) $. These matrices determine how the paths of length one act on the representation, longer paths act via multiplcation of matrices along the oriented path.
A necklace in the quiver is a closed oriented path in the quiver up to cyclic permutation of the arrows making up the cycle. That is, we are free to choose the start (and end) point of the cycle. For example, in the one-cycle quiver
[tex]\xymatrix{\vtx{} \ar[rr]^a & & \vtx{} \ar[ld]^b \\ & \vtx{} \ar[lu]^c &}[/tex]
the basic necklace can be represented as $abc $ or $bca $ or $cab $. How does a necklace act on a representation? Well, the matrix-multiplication of the matrices corresponding to the arrows gives a square matrix in each of the vertices in the cycle. Though the dimensions of this matrix may vary from vertex to vertex, what does not change (and hence is a property of the necklace rather than of the particular choice of cycle) is the trace of this matrix. That is, necklaces give complex-valued functions on representations of $\mathbb{C} Q $ and by a result of Artin and Procesi there are enough of them to distinguish isoclasses of (semi)simple representations! That is, linear combinations a necklaces (aka super-potentials) can be viewed, after taking traces, as complex-valued functions on all representations (similar to character-functions).
In physics, one views these functions as potentials and it then interested in the points (representations) where this function is extremal (minimal) : the vacua. Clearly, this does not make much sense in the complex-case but is relevant when we look at the real-case (where we look at skew-Hermitian matrices rather than all matrices). A motivating example (the Yang-Mills potential) is given in Example 2.3.2 of Victor Ginzburg's paper Calabi-Yau algebras.
Let $\Phi $ be a super-potential (again, a linear combination of necklaces) then our commutative intuition tells us that extrema correspond to zeroes of all partial differentials $\frac{\partial \Phi}{\partial a} $ where $a $ runs over all coordinates (in our case, the arrows of the quiver). One can make sense of differentials of necklaces (and super-potentials) as follows : the partial differential with respect to an arrow $a $ occurring in a term of $\Phi $ is defined to be the path in the quiver one obtains by removing all 1-occurrences of $a $ in the necklaces (defining $\Phi $) and rearranging terms to get a maximal broken necklace (using the cyclic property of necklaces). An example, for the cyclic quiver above let us take as super-potential $abcabc $ (2 cyclic turns), then for example
$\frac{\partial \Phi}{\partial b} = cabca+cabca = 2 cabca $
(the first term corresponds to the first occurrence of $b $, the second to the second). Okay, but then the vacua-representations will be the representations of the quotient-algebra (which I like to call the vacualgebra)
$\mathcal{U}(Q,\Phi) = \frac{\mathbb{C} Q}{(\partial \Phi/\partial a, \forall a)} $
which in 'physical relevant settings' (whatever that means…) turn out to be Calabi-Yau algebras.
But, let us return to the case of subgroups of the modular group and their quivers. Do we have a natural super-potential in this case? Well yes, the quiver encoded a triangulation of the compactification of $\mathbb{H}/\Lambda $ and if we choose an orientation it turns out that all 'black' triangles (with respect to the Dedekind tessellation) have their arrow-sides defining a necklace, whereas for the 'white' triangles the reverse orientation makes the arrow-sides into a necklace. Hence, it makes sense to look at the cubic superpotential $\Phi $ being the sum over all triangle-sides-necklaces with a +1-coefficient for the black triangles and a -1-coefficient for the white ones. Let's consider an index three example from a previous post
[tex]\xymatrix{& & \rho \ar[lld]_d \ar[ld]^f \ar[rd]^e & \\
i \ar[rrd]_a & i+1 \ar[rd]^b & & \omega \ar[ld]^c \\
& & 0 \ar[uu]^h \ar@/^/[uu]^g \ar@/_/[uu]_i &}[/tex]
In this case the super-potential coming from the triangulation is
$\Phi = -aid+agd-cge+che-bhf+bif $
and therefore we have a noncommutative algebra $\mathcal{U}(Q,\Phi) $ associated to this index 3 subgroup. Contrary to what I believed at the start of this series, the algebras one obtains in this way from dessins d'enfants are far from being Calabi-Yau (in whatever definition). For example, using a GAP-program written by Raf Bocklandt Ive checked that the growth rate of the above algebra is similar to that of $\mathbb{C}[x] $, so in this case $\mathcal{U}(Q,\Phi) $ can be viewed as a noncommutative curve (with singularities).
However, this is not the case for all such algebras. For example, the vacualgebra associated to the second index three subgroup (whose fundamental domain and quiver were depicted at the end of this post) has growth rate similar to that of $\mathbb{C} \langle x,y \rangle $…
I have an outlandish conjecture about the growth-behavior of all algebras $\mathcal{U}(Q,\Phi) $ coming from dessins d'enfants : the algebra sees what the monodromy representation of the dessin sees of the modular group (or of the third braid group).
I can make this more precise, but perhaps it is wiser to calculate one or two further examples…
quivers versus quilts
Published January 2, 2008 by lievenlb
We have associated to a subgroup of the modular group $PSL_2(\mathbb{Z}) $ a quiver (that is, an oriented graph). For example, one verifies that the fundamental domain of the subgroup $\Gamma_0(2) $ (an index 3 subgroup) is depicted on the right by the region between the thick lines with the identification of edges as indicated. The associated quiver is then
\xymatrix{i \ar[rr]^a \ar[dd]^b & & 1 \ar@/^/[ld]^h \ar@/_/[ld]_i \\
& \rho \ar@/^/[lu]^d \ar@/_/[lu]_e \ar[rd]^f & \\
0 \ar[ru]^g & & i+1 \ar[uu]^c}
The corresponding "dessin d'enfant" are the green edges in the picture. But, the red dot on the left boundary is identied with the red dot on the lower circular boundary, so the dessin of the modular subgroup $\Gamma_0(2) $ is
\xymatrix{| \ar@{-}[r] & \bullet \ar@{-}@/^8ex/[r] \ar@{-}@/_8ex/[r] & -}
Here, the three red dots (all of them even points in the Dedekind tessellation) give (after the identification) the two points indicated by a $\mid $ whereas the blue dot (an odd point in the tessellation) is depicted by a $\bullet $. There is another 'quiver-like' picture associated to this dessin, a quilt of the modular subgroup $\Gamma_0(2) $ as studied by John Conway and Tim Hsu.
On the left, a quilt-diagram copied from Hsu's book Quilts : central extensions, braid actions, and finite groups, exercise 3.3.9. This 'quiver' has also 5 vertices and 7 arrows as our quiver above, so is there a connection?
A quilt is a gadget to study transitive permutation representations of the braid group $B_3 $ (rather than its quotient, the modular group $PSL_2(\mathbb{Z}) = B_3/\langle Z \rangle $ where $\langle Z \rangle $ is the cyclic center of $B_3 $. The $Z $-stabilizer subgroup of all elements in a transitive permutation representation of $B_3 $ is the same and hence of the form $\langle Z^M \rangle $ where M is called the modulus of the representation. The arrow-data of a quilt, that is the direction of certain edges and their labeling with numbers from $\mathbb{Z}/M \mathbb{Z} $ (which have to satisfy some requirements, the flow rules, but more about that another time) encode the Z-action on the permutation representation. The dimension of the representation is $M \times k $ where $k $ is the number of half-edges in the dessin. In the above example, the modulus is 5 and the dessin has 3 (half)edges, so it depicts a 15-dimensional permutation representation of $B_3 $.
If we forget the Z-action (that is, the arrow information), we get a permutation representation of the modular group (that is a dessin). So, if we delete the labels and directions on the edges we get what Hsu calls a modular quilt, that is, a picture consisting of thick edges (the dessin) together with dotted edges which are called the seams of the modular quilt. The modular quilt is merely another way to depict a fundamental domain of the corresponding subgroup of the modular group. For the above example, we have the indicated correspondences between the fundamental domain of $\Gamma_0(2) $ in the upper half-plane (on the left) and as a modular quilt (on the right)
That is, we can also get our quiver (or its opposite quiver) from the modular quilt by fixing the orientation of one 2-cell. For example, if we fix the orientation of the 2-cell $\vec{fch} $ we get our quiver back from the modular quilt
This shows that the quiver (or its opposite) associated to a (conjugacy class of a) subgroup of $PSL_2(\mathbb{Z}) $ does not depend on the choice of embedding of the dessin (or associated cuboid tree diagram) in the upper half-plane. For, one can get the modular quilt from the dessin by adding one extra vertex for every connected component of the complement of the dessin (in the example, the two vertices corresponding to 0 and 1) and drawing a triangulation from them (the dotted lines or 'seams').
Anabelian vs. Noncommutative Geometry
Published December 12, 2007 by lievenlb
This is how my attention was drawn to what I have since termed
anabelian algebraic geometry, whose starting point was exactly a study
(limited for the moment to characteristic zero) of the action of absolute
Galois groups (particularly the groups $Gal(\overline{K}/K) $, where K is an extension of finite type of the prime field) on (profinite) geometric fundamental
groups of algebraic varieties (defined over K), and more particularly (breaking with a well-established tradition) fundamental groups which are very far
from abelian groups (and which for this reason I call anabelian). Among
these groups, and very close to the group $\hat{\pi}_{0,3} $ , there is the profinite compactification of the modular group $SL_2(\mathbb{Z}) $, whose quotient by its centre
$\{ \pm 1 \} $ contains the former as congruence subgroup mod 2, and can also be
interpreted as an oriented cartographic group, namely the one classifying triangulated oriented maps (i.e. those whose faces are all triangles or
monogons).
The above text is taken from Alexander Grothendieck's visionary text Sketch of a Programme. He was interested in the permutation representations of the modular group $\Gamma = PSL_2(\mathbb{Z}) $ as they correspond via Belyi-maps and his own notion of dessins d'enfants to smooth projective curves defined over $\overline{\mathbb{Q}} $. One can now study the action of the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q}) $ on these curves and their associated dessins. Because every permutation representation of $\Gamma $ factors over a finite quotient this gives an action of the absolute Galois group as automorphisms on the profinite compactification
$\hat{\Gamma} = \underset{\leftarrow}{lim}~\Gamma/N $
where the limit is taken over all finite index normal subgroups $N \triangleleft PSL_2(\mathbb{Z}) $. In this way one realizes the absolute Galois group as a subgroup of the outer automorphism group of the profinite group $\hat{\Gamma} $. As a profinite group is a compact topological group one should study its continuous finite dimensional representations which are precisely those factoring through a finite quotient. In the case of $\hat{\Gamma} $ the simple continuous representations $\mathbf{simp}_c~\hat{\Gamma} $ are precisely the components of the permutation representations of the modular group. So in a sense, anabelian geometry is the study of these continuous simples together wirth the action of the absolute Galois group on it.
In noncommutative geometry we are interested in a related representation theoretic problem. We would love to know the simple finite dimensional representations $\mathbf{simp}~\Gamma $ of the modular group as this would give us all simples of the three string braid group $B_3 $. So a natural question presents itself : how are these two 'geometrical' objects $\mathbf{simp}_c~\hat{\Gamma} $ (anabelian) and $\mathbf{simp}~\Gamma $ (noncommutative) related and can we use one to get information about the other?
This is all rather vague so far, so let us work out a trivial case to get some intuition. Consider the profinite completion of the infinite Abelian group
$\hat{\mathbb{Z}} = \underset{\leftarrow}{lim}~\mathbb{Z}/n\mathbb{Z} = \prod_p \hat{\mathbb{Z}}_p $
As all simple representations of an Abelian group are one-dimensional and because all continuous ones factor through a finite quotient $\mathbb{Z}/n\mathbb{Z} $ we see that in this case
$\mathbf{simp}_c~\hat{\mathbb{Z}} = \mu_{\infty} $
is the set of all roots of unity. On the other hand, the simple representations of $\mathbb{Z} $ are also one-dimensional and are determined by the image of the generator so
$\mathbf{simp}~\mathbb{Z} = \mathbb{C} – { 0 } = \mathbb{C}^* $
Clearly we have an embedding $\mu_{\infty} \subset \mathbb{C}^* $ and the roots of unity are even dense in the Zariski topology. This might look a bit strange at first because clearly all roots of unity lie on the unit circle which 'should be' their closure in the complex plane, but that's because we have a real-analytic intuition. Remember that the Zariski topology of $\mathbb{C}^*$ is just the cofinite topology, so any closed set containing the infinitely many roots of unity should be the whole space!
Let me give a pedantic alternative proof of this (but one which makes it almost trivial that a similar result should be true for most profinite completions…). If $c $ is the generator of $\mathbb{Z} $ then the different conjugacy classes are precisely the singletons $c^n $. Now suppose that there is a polynomial $a_0+a_1x+\ldots+a_mx^m $ vanishing on all the continuous simples of $\hat{\mathbb{Z}} $ then this means that the dimensions of the character-spaces of all finite quotients $\mathbb{Z}/n\mathbb{Z} $ should be bounded by $m $ (for consider $x $ as the character of $c $), which is clearly absurd.
Hence, whenever we have a finitely generated group $G $ for which there is no bound on the number of irreducibles for finite quotients, then morally the continuous simple space for the profinite completion
$\mathbf{simp}_c~\hat{G} \subset \mathbf{simp}~G $
should be dense in the Zariski topology on the noncommutative space of simple finite dimensional representations of $G $. In particular, this should be the case for the modular group $PSL_2(\mathbb{Z}) $.
There is just one tiny problem : unlike the case of $\mathbb{Z} $ for which this space is an ordinary (ie. commutative) affine variety $\mathbb{C}^* $, what do we mean by the "Zariski topology" on the noncommutative space $\mathbf{simp}~PSL_2(\mathbb{Z}) $ ? Next time we will clarify what this might be and show that indeed in this case the subset
$\mathbf{simp}_c~\hat{\Gamma} \subset \mathbf{simp}~\Gamma $
will be a Zariski closed subset!
Hexagonal Moonshine (3)
Published July 13, 2007 by lievenlb
Hexagons keep on popping up in the representation theory of the modular group and its close associates. We have seen before that singularities in 2-dimensional representation varieties of the three string braid group $B_3 $ are 'clanned together' in hexagons and last time Ive mentioned (in passing) that the representation theory of the modular group is controlled by the double quiver of the extended Dynkin diagram $\tilde{A_5} $, which is an hexagon…
Today we're off to find representations of the extended modular group $\tilde{\Gamma} = PGL_2(\mathbb{Z}) $, which is obtained by adding to the modular group (see this post for a proof of generation)
$\Gamma = \langle U=\begin{bmatrix} 0 & -1 \\\ 1 & 0 \end{bmatrix},V=\begin{bmatrix} 0 & 1 \\\ -1 & 1 \end{bmatrix} \rangle $ the matrix $R=\begin{bmatrix} 0 & 1 \\\ 1 & 0 \end{bmatrix} $
In terms of generators and relations, one easily verfifies that
$\tilde{\Gamma} = \langle~U,V,R~|~U^2=R^2=V^3=(RU)^2=(RV)^2=1~\rangle $
and therefore $\tilde{\Gamma} $ is the amalgamated free product of the dihedral groups $D_2 $ and $D_3 $ over their common subgroup $C_2 = \langle~R~\rangle $, that is
$\tilde{\Gamma} = \langle U,R | U^2=R^2=(RU)^2=1 \rangle \ast_{\langle R | R^2=1 \rangle} \langle V,R | V^3=R^2=(RV)^2=1 \rangle = D_2 \ast_{C_2} D_3 $
From this description it is easy to find all n-dimensional $\tilde{\Gamma} $-representations $V $ and relate them to quiver-representations. $D_2 = C_2 \times C_2 $ and hence has 4 1-dimensonal simples $S_1,S_2,S_3,S_4 $. Restricting $V\downarrow_{D_2} $ to the subgroup $D_2 $ it decomposes as
$V \downarrow_{D_2} \simeq S_1^{\oplus a_1} \oplus S_2^{\oplus a_2} \oplus S_3^{\oplus a_3} \oplus S_4^{\oplus a_4} $ with $a_1+a_2+a_3+a_4=n $
Similarly, because $D_3=S_3 $ has two one-dimensional representations $T,S $ (the trivial and the sign representation) and one simple 2-dimensional representation $W $, restricting $V $ to this subgroup gives a decomposition
$V \downarrow_{D_3} \simeq T^{b_1} \oplus S^{\oplus b_2} \oplus W^{\oplus b_3} $, this time with $b_1+b_2+2b_3=n $
Restricting both decompositions further down to the common subgroup $C_2 $ one obtains a $C_2 $-isomorphism $V \downarrow_{D_2} \rightarrow^{\phi} V \downarrow_{D_3} $ which implies also that the above numbers must be chosen such that $a_1+a_3=b_1+b_3 $ and $a_2+a_4=b_2+b_3 $. We can summarize all this info about $V $ in a representation of the quiver
Here, the vertex spaces on the left are the iso-typical factors of $V \downarrow_{D_2} $ and those on the right those of $V \downarrow_{D_3} $ and the arrows give the block-components of the $C_2 $-isomorphism $\phi $. The nice things is that one can also reverse this process to get all $\tilde{\Gamma} $-representations from $\theta $-semistable representations of this quiver (having the additional condition that the square matrix made of the arrows is invertible) and isomorphisms of group-representation correspond to those of quiver-representations!
This proves that for all n the varieties of n-dimensional representations $\mathbf{rep}_n~\tilde{\Gamma} $ are smooth (but have several components corresponding to the different dimension vectors $~(a_1,a_2,a_3,a_4;b_1,b_2,b_3) $ such that $\sum a_i = n = b_1+b_2+2b_3 $.
The basic principle of _M-geometry_ is that a lot of the representation theory follows from the 'clan' (see this post) determined by the simples of smallest dimensions. In the case of the extended modular group $\tilde{\Gamma} $ it follows that there are exactly 4 one-dimensional simples and exactly 4 2-dimensional simples, corresponding to the dimension vectors
$\begin{cases} a=(0,0,0,1;0,1,0) \\\ b=(0,1,0,0;0,1,0) \\\ c=(1,0,0,0;1,0,0) \\\ d=(0,0,1,0;1,0,0) \end{cases} $ resp. $\begin{cases} e=(0,1,1,0;0,0,1) \\\ f=(1,0,0,1;0,0,1) \\\ g=(0,0,1,1;0,0,1) \\\ h=(1,1,0,0;0,0,1) \end{cases} $
If one calculates the 'clan' of these 8 simples one obtains the double quiver of the graph on the left. Note that a and b appear twice, so one should glue the left and right hand sides together as a Moebius-strip. That is, the clan determining the representation theory of the extended modular group is a Moebius strip made of two hexagons!
However, one should not focuss too much on the hexagons (that is, the extended Dynkin diagram $\tilde{A_5} $) here. The two 'backbones' (e–f and g–h) have their vertices corresponding to 2-dimensional simples whereas the topand bottom vertices correspond to one-dimensional simples. Hence, the correct way to look at this clan is as two copies of the double quiver of the extended Dynkin diagram $\tilde{D_5} $ glued over their leaf vertices to form a Moebius strip. Remark that the components of the sotropic root of $\tilde{D_5} $ give the dimensions of the corresponding $\tilde{\Gamma} $ simples.
The remarkable ubiquity of (extended) Dynkins never ceases to amaze!
Continue readingHexagonal Moonshine (3) | CommonCrawl |
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Modeling and Control of Power System Containing PV System and SMES using Sliding Mode and Field Control Strategy
Boudia Assam | Messalti Sabir | Harrag Abdelghani
Department of Electrical Engineering, Faculty of Technology, University of Msila, Algeria
LGE Laboratory, Department of Electrical Engineering, Faculty of Technology, University of Msila, Algeria
Mechatronics Laboratory, Optics and Precision Mechanics Institute, Ferhat Abbas University, Setif 1, Algeria
[email protected]
Although the great advance in power system production and operation, storage energy technologies and its control techniques can be considered as one of the most important and critical topics of power companies , government and consumers, especially when the power system containing renewable source and storage system simultaneously. In this paper, a novel electrical grid structure including photovoltaic system and storage system based on Superconducting Magnetic Energy Storage (SMES) has been proposed and investigated. The SMES produced power is injected in power system during specific time or when it required. Two control strategies for exchanged power Grid- SMES have been proposed and analyzed, the first uses sliding mode and the second uses field oriented control based on PI controller, also the injected SMES power is controlled by PID controller. In addition, the photovoltaic system operates at the MPP employing PID MPPT method. The proposed control strategies have been tested successfully in which many scenarios have been studied: standby and discharging of SMES, injection of SMES storage energy for variable and constant load and control of grid containing PV system. In addition, a comparative study of exchanged power Grid- SMES control using Sliding mode and field oriented control based on PI controller has been presented and discussed.
Grid-PV-SMES, power integration, sliding Mode
The last years have seen gradually an expansion on application in the storage energies, through all storage energies, the SMES (Superconducting Magnetic Energy Storage) is placed in this group, the SMES is coil which is in a superconducting state at cryogenic temperature, this means that the energy losses during the operation is almost zero[1-5], the SMES is an electrical energy that stores the energy in the magnetic field ,it has the Ability to rapidly release stored energy, very high storage energy, Quick to recharge (millisecond) with high efficiency and almost infinite cycle life, also the SMES used to control the active and reactive power, used to the transmission and distribution system stability and stabilize system frequency[6-8].
The SMES can stores the energy directly from electrical power, because the SMES resistance almost equal zero and the mobility is very high at very low temperature, which called critical temperature, the SMES unit consists of three main components: superconducting unit, cryogenic refrigerator and vacuum-insulated vessel fig 1, the SMES is connected to the grid in three modes as is shown in fig 2, voltage source converter (VSC)[9-12], current source converter (CSC)[13-15], and thyristor[16, 17].
In VSC mode, the SMES energy is controlled by DC-DC chopper in the absorbing or injection power, in ref [18] the authors charge and discharge the SMES using dc-dc chopper which is controlled by proportional and integral (PI) controller, in ref [8] fuzzy logic controller (FLC)is applied on the SMES charging and discharging of the SMES active and reactive powers. The FLC is controlled by two inputs: wind speed and SMES current variations.
To support the increasing energy demand and the development of renewable energies (solar and wind,…etc) whose production is variable, non-controllable and decentralized, hence, increasing the storage capacity of electricity is a necessity. However, there are still many technical and economic problems that reduce thedeployment of new storage technologies. Significant research efforts are underway around the world. In this document, power grids containing photovoltaicsystem and storage system based on Superconducting Magnetic Energy System (SMES) has been proposed and analyzed for different possible scenarios and load levels.
Figure 1. Functional diagram of the SMES system[19]
Figure 2. SMES applications Modes
2. Voltage Source Converter Control
The most applications of the SMES are: SMES is used to the transmission and distribution system stability[20], control of voltage based on active and reactive power[21] and smooth the wind farm output by absorbing or providing realpower [22], SMES can respond very quickly to the active and reactive power demand [22, 23] and stabilize system frequency, the SMES is characterized by its rapid response (milliseconds)[24]and long life cycle[25]compared to other energy storage systems, it has a very high storage efficiency.
The energy and power are expressed as in equations (1) and (2):
$E=\frac{1}{2} L i^{2}$ (1)
$P=\frac{d E}{d t}$ (2)
Where 'i' is the current through the coil and 'L' is the superconducting coil inductance, the SMES stores the energy as circulating current, and provide the energy with instantaneous response.
Voltage source converter is used to accomplish the transformation of power between utility three-phase alternating current and the DC bus. It consists on three-phase –insulated-gate bipolar transistor (IGBT) full-bridge with three input inductors (L) and resistor (R) on the ACside and a filter capacitor (C) on the DC side. In order to control the power flow of the system, the input inductors' currents are accurately controlled following the power demand P and Q as shown in Figure 3.
Figure 3. Topological configuration of VSC
2.1. Control of injected power into grid based using field oriented control
The diagram of the functional control strategy of the voltage source converter is presented in figure 4. The directions of the voltages Vdref and Vqref, generate the phase and the amplitude of the useful voltage. The Park transformation block (3S / 2R) transforms the three-phase current Ia, Ib and Ic from the stationary form to the synchronous form. Then Id and Iq represent the active and reactive component of the converter respectively. The reference voltage in the input of the voltage source converter is also given in equations 3, 4 and 5.
Figure 4. Voltage source converter control based on PI controller
$u_{c d, q}=P I\left(i_{d, q r e f}-i_{d, q}\right)$ (3)
ucd,q the voltage calculated using PI controller, which control the error between the references currents id,qref and the measured currents id,q. Hence, the output reference voltage of voltage source current VSC based on Park transformation are expressed by
$\left\{\begin{array}{l}V_{d r e f}=e_{d}+u_{c d}-L_{f} w i_{q} \\ V_{q r e f}=e_{q}+u_{c q}+L_{f} w i_{d}\end{array}\right.$ (4)
Vd,qref is the calculated reference voltage.
ed and eq are the park transformation voltages of the grid common connection points.
Lf represents the coupling inductance of a phase of the filter between the VSC and the grid.
$\left[\begin{array}{c}i_{\text {dref}} \\ i_{\text {qref}}\end{array}\right]=\frac{1}{e_{d}^{2}+e_{q}^{2}}\left[\begin{array}{cc}e_{d} & e_{q} \\ -e_{q} & e_{d}\end{array}\right]\left[\begin{array}{c}P_{\text {ref}} \\ Q_{\text {ref}}\end{array}\right]$ (5)
where Pref and Qref are the references of the active and reactive powers.
The id,q current regulation loop are based on PI regulators to calculated ucd,q, as is shown in figure 5:
Figure 5. Current regulation loops using PI regulator
2.2. Control of injected power into grid based on sliding mode control
In order to reduce the PI regulator disadvantages such as the response time and the overshoot, a robust command named sliding Mode controller (SMC) has been developed as shown in fig 06.
Figure 6. Voltage source converter control based on SMC controller
Sliding Mode controller (SMC) is robust method, it is a non-linear type control that has been introduced for the control of variable structure systems and is based on the concept of controller structure change with the state of the system in order to obtain a desired response. The sliding mode control is therefore all or nothing type.
Sliding Mode is based in three steps:
- Choice of sliding surfaces S(X).
- Definition of the conditions of existence and convergence of the sliding regime.
- Determination of the control law.
The sliding mode control law is expressed by [26]:
$U=u_{e q}+u_{n}$ (6)
where ueq is the calculated control and un is given by:
$u_{n}=-k^{*} \operatorname{sign}(S)$ (7)
S: slotine surface.
k: constant value
In this paper, we chose slotine surface [26-28], and apply it in equation 8and 9 to calculate ucd,q:
The first system:
$L_{f} \frac{d i_{d}}{d t}=u_{c d}-R_{f} i_{d}+L_{f} w i_{q}-e_{d}$ (8)
The second system:
$L_{f} \frac{d i_{q}}{d t}=u_{c q}-R_{f} i_{q}-L_{f} w i_{q}-e_{q}$ (9)
The first surface is used to calculate ucd:
$S_{1}=\left(\frac{d}{d t}+\lambda\right)^{n-1} e_{1}$ (10)
For n=1, the surface can be written by:
$S_{1}=e_{1}=i_{d r e f}-i_{d}$ (11)
The derivate surface is expressed [26]:
$\dot{S}_{1}=\dot{i_{d r e f}}-\dot{i_{d}}=u_{n 1}=-k_{1} \operatorname{sign}\left(S_{1}\right)$ (12)
By replacing equation (8) in equation (12):
${i_\dot{\text {dref}}}-\frac{1}{L_{f}}\left(u_{c d}-R_{f} i_{d}+L_{f} w i_{q}-e_{d}\right)=-k_{1} \operatorname{sign}\left(S_{1}\right)$ (13)
$u_{c d}=L_{f}\left(\dot{i_{\text {dref}}}+R_{f} i_{d} / L_{f}-L_{f} w i_{q} / L_{f}+e_{d} / L_{f}\right)+L_{f} k_{1} \operatorname{sign}\left(S_{1}\right)$ (14)
The voltage control is:
$u_{c d}=L_{f}\left(k_{1} \operatorname{sign}\left(S_{1}\right)+\dot{i_{\text {dref}}}\right)+R_{f} i_{d}-L_{f} w i_{q}+e_{d}$ (15)
The second surface is used to calculate ucq:
The order of the system is n=1, in that case:
$S_{2}=e_{2}=i_{\text {qref }}-i_{q}$ (17)
The derivate surface is:
$\dot{S_{2}}=\dot{i_{\text {qref}}}-\dot{i_{q}}=u_{n 2}=-k_{2} \operatorname{sign}\left(S_{2}\right)$ (18)
$\dot{i_{\text {qref}}}-\frac{1}{L_{f}}\left(u_{c q}-R_{f} i_{q}-L_{f} w i_{d}-e_{q}\right)=-k_{2} \operatorname{sign}\left(\mathrm{S}_{2}\right)$ (19)
$u_{c q}=L_{f}\left(\dot{i_{\text {qref }}}+R_{f} i_{q} / L_{f}+L_{f} w i_{q} / L_{f}+e_{d} / L_{f}\right)+L_{f} k_{2} \operatorname{sign}\left(S_{2}\right)$ (20)
$u_{c q}=L_{f}\left(k_{2} \operatorname{sign}\left(S_{2}\right)+\dot{i_{\text {qref}}}\right)+R_{f} i_{q}+L_{f} w i_{q}+e_{d}$ (21)
The reaching condition must be verified:
$\left\{\begin{array}{l}S_{1} \dot{S}_{1}=S_{1}\left(-L_{f} k_{1} \operatorname{sign}\left(S_{1}\right)\right)<0 \\ S_{2} \dot{{S_{2}}}=S_{2}\left(-L_{f} k_{2} \operatorname{sign}\left(S_{2}\right)\right)<0\end{array}\right.$ (22)
Therefore, the ranges of switching gains are given as follows:K1, K2>0.
Lf and Rfrepresent the coupling inductance and resistance of a phase of the filter between the VSC and the grid.
n: system order.
λ, k1, k2: positive constants values
2.3 Voltage regulators
The active reference power Pref from equation (5) to be injected into thegrid is calculated as follows:
$i_{\text {dcref}}=V_{d c}\left(I_{\text {inj}}-P I\left(V_{\text {dcref}}-V_{d c}\right)\right)$ (23)
$P_{\text {ref}}=V_{d c} i_{\text {dcref}}$ (24)
Vdcref and Vdc are the reference voltage of the bus continue.
Iinj is inected current to the bus continue.
Figure 7. Bus continue voltage regulation loop using PI regulator
3. Two-Quadrant Chopper
In order to regulate the power demand of the powersystem,the magnitude and the voltage polarity across the coil must be controlled using DC-DC chopper, the DC-DC chopper allows to control the voltage across the capacitor and supply the required currentin the superconducting coil. It has two legs in parallel to distribute the SMES current.
In this paper, novel Dc-Dc chopper has been proposed and investigated as it's shown in Fig 8, the SMES is charged from PV system and connected the Grid with two modes, the first mode is discharge mode, and in this mode the SMES will be controlled by PID classical regulator to regulate the error as is expressed in equation (25), the second mode is standby, in this mode the SMES is disconnected to the Grid by short circuit.
Figure 8. SMES system
$\left\{\begin{array}{l}P_{c h}=P_{S M E S}+P_{r} \\ e=\left(P_{c h}-P_{r}\right)-P_{S M E S}\end{array}\right.$ (25)
$P_{\text {SMES}}=V_{d c} * I_{\text {inj}}$ (26)
Pr, Pch are the grid and load power respectively.
4. Simulation Result
RESULTIn order to study the efficiency of proposed method control strategy, power system with PV system and SMES has been used,in which the following simulation have been carried out and analyzed:
1.GPV connected to Grid with fix and variable load.
2.SMES connected to Grid with fix load.
3.SMES connected to Grid with variable load.
The global bloc diagram is shown in fig 9, table 1 shows the system parameters.
Figure 9. Bloc diagram of proposed Grid-PV-SMES connected
Table 1: system parameters
4e-4F
0.02H
Bus continue
1.89e-04F
Sliding mode ucd
Sliding mode ucq
0.0071H
VSC output filter
Grid and grid filter
0.003 ohm
2.6e-6H
5.6 ohm
The first load
26.6e-3 H
The second load
40e-3 H
The MPPT used is PID controller which controls the error between GPV voltage output and the reference voltage which is calculated by the equation (27).
$V_{\text {ref }}=S_{e}\left(\frac{N_{s} A K T}{q} \log \left(\frac{I_{p h}-I_{\text {ref}}+I_{0}}{I_{0}}\right)\right)$ (27)
With Se is the number of series solar panel.
Where: Iref=0.909 Iph.
I0 is the cell reverse saturation current.
Ns are the number of cells connected in series.
q = 1.6 × 10−19[C] is the electron charge.
K = 1.3805 × 10−23[J/K] is the Boltzmann constant.
A is the ideality factor of the p-n junction.
T[K] is the cell temperature.
4.1 GPV-Grid (Fixed and variable load)
Generator photovoltaic (GPV-3000W) is connected to the grid, a comparative study between PI and sliding mode regulators is presented where two scenarios have been considered fixed and variable load: inthe first case : four irradiation steps have been applied 1000, 600, 800 and 1000 w/m2 with constant load (fig 10). In the second case a variable load has been employed to confirm the efficiency of proposed control methods .
From fig 10 and 11: both regulators guarantee good performances of injected power into grid, however PI regulator shows a considered overshoot and response time compared to sliding mode controller. In addition,it's clear that GPV output poweroperates perfectly in which the MPP has been tracked for different irrradiation.
Figure 10. The load, grid and GPV output power under variable irradiation using sliding Mode and PI controller
Figure 11. The load, grid and GPV output power with variable load using sliding Mode and PI controller
The GPV output power is perfectly calculated by the PID regulator under all the irradiation changes, with no oscillation or overshoot and fast response time.
4.2 SMES-Grid with fixed load
In most areas, the average irradiation can almost reach 7 hours/day and energy demand is nearly unplanned, so the use of SMES can be more suitable to overcome the previous dilemmas, where the exceed energy can be stored, then it injected in power system during peak power demand or during specify time, based on the previous remarks, we present in this section power system containing a SMES. The SMES connect to the grid with two modes in the same simulation (discharged and standby), the SMES is in standby mode from 0.37 to 0.65 second, and is connected to the grid in the rest time.
Figure 12. The load, grid and exchanged power using sliding Mode and PI controller
Figure 13. Zoom (A): The load, grid and exchanged power using sliding Mode and PI controller
Figure 14. Zoom (B): The load, grid and exchanged power with comparative between sliding Mode and PI controller
Figure 15. SMES power in standby and discharge mode with comparative between sliding Mode and PI controller
Figure 16. SMES voltage in standby and discharge mode with comparative between sliding Mode and PI controller
Figure 17. SMES energy in standby and discharge mode with comparative between sliding Mode and PI controller
From Figs 12, 13 and 14, Injected SMES energy in power system has been controlled perfectly by both developed control strategies sliding mode and field oriented control based on PI controller, where the equality is always ensured: Pch= Pr+Psmes. We can observe when the SMES injectes the power in the grid, the power grid decrease according the equality (equation (25) and when the SMES is disconneted (Psmes=0) all load power is generated by the grid.Obtained results demostrates the efficiency of both methods sliding mode and PI controller, however the sliding mode shows better performances in term ofresponse time and overshot, as shown in fig 13 and 14. In standby mode the SMES is short circuit, in this mode the voltage and the power are null as is shown in figs 15 and 16, also the same note obtained in figs 15 and 16 for the performance of sliding mode and PI controller. Fig 17, shows how does the power SMES changeduring both modes discharging and standby mode.
4.3 SMES-Grid with variable load
In this simulation part, the load changes from the first to the second load, the SMES is injected to the grid with no standby mode and with same condition in (SMES-Grid with fixed load).
Figure 18. The load, grid and exchanged power with load change and comparative between sliding Mode and PI controller
Fig 18 shows the power grid, power SMES and load in variable loads scenario, it clear that both control strategies sliding mode and PI regulator provide a high performances with small superiority of sliding mode in term of low oscillations.
In this paper, power grids containing photovoltaic systemand storage system based on Superconducting Magnetic Energy Storage (SMES) has been proposed and analyzed for different possible scenarios and loads. Modeling and operating principle of SMES and Power-PV-SMES have been explained in details. Injected SMES energy in power system has been controlled perfectly by both developed control strategies sliding mode and field oriented control based on PI controller, where sliding mode ensures better performance compared to field oriented control. Beside the efficiency of proposed control methods, obtained results demonstrate the great benefit of use of combined renewable energy-SMES to confront the future demand with the exiting power production capacity and maintain the power system operation under hard conditions. Hence, renewable energy-SMES can be widely used for quality power improvement.
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A control system, which is usually understood to mean a finite automaton (cf. Automaton, finite) or one of its modifications obtained by changing components or the mode of operation. The principal concept — a finite automaton — originated in the mid-20th century in connection with attempts to describe, in mathematical language, the functioning of nervous systems, of universal computers and of other real automata. The first such efforts are due to W. McCulloch and W. Pitts (1943), S.C. Kleene (1951), A.W. Burks and J. Wright (1954), and others. A characteristic feature of such a description is the discrete character of the corresponding mathematical models and the finiteness of the domain of their parameter values, which accounts for the name "finite" automaton. The external effects, reactions and states are considered to be letters of three alphabets named, respectively, the input alphabet, the output alphabet and the alphabet of states. The relations governing their interactions may then be given by two functions — a transition function and an output function — which map the pair "state-input letter" into "state letter" and the pair "state-output letter" into "output letter" , respectively. At each discrete moment of time the system, being in a given state, receives an input signal (a letter of the input alphabet), emits an output signal (a letter of the output alphabet, determined by the output function) and passes into a new state determined by the transition function. Studies of finite automata are supplemented by studies of their various generalizations and modifications which reflect various features of real systems. In the case of a finite automaton $ (A, S, B, \phi , \psi ) $ the existing modifications may be subdivided into the following three main groups. The first group includes automata some alphabets $ A, S $ or $ B $ of which are infinite, and such automata are said to be infinite. The second group includes automata in which the functions $ \psi $ and $ \phi $ are replaced by arbitrary relationships or by random functions. These are partial, non-deterministic, probabilistic and other automata. The third group includes automata with specific sets of input objects. These are automata with a variable structure, automata over terms (or tree automata, cf. Automata, algebraic theory of). There are automata belonging to several groups at the same time, e.g. fuzzy automata belong to all three groups. Special subclasses of finite automata are very important; these include, for instance, memoryless automata, autonomous automata, reversible automata, etc. In addition, the use of concepts and methods of other branches of mathematics also generates specific classes of automata and related problems. For instance, if algebraic methods are employed there result the concepts of automata over terms, as well as linear, group, free and other automata (cf. Automata, algebraic theory of); problems in coding theory generate the concepts of self-regulating automata, reversible automata, etc. (cf. Automaton, probabilistic). Structural automata also have several generalizations, mainly consisting in permitting infinite networks and altering the interconnections between the elements during the operation. This leads to the concept of a growing automaton. The principal modifications and subclasses of finite automata, together with their most important properties, are described below.
1 Macro approach.
2 Micro-approach.
Macro approach.
1) Infinite automata (first group) differ from finite automata only by the fact that their alphabets $ A, B $ or $ S $( input, output and set of states) may be infinite. Most concepts and problems connected with finite automata are naturally extended to infinite automata. Alphabets of higher cardinality extend the computational capacities of automata. For instance, while finite automata realize finitely-determined (sequential) functions (cf. Finitely-determined function), infinite automata may be used to realize any determined function. Moreover, using infinite automata it is possible to describe the functioning of any automaton and machine. At the same time, this very general nature of infinite automata impairs their significance, so that most studies concern only special subclasses of infinite automata, connected with specific models of control systems.
2) Non-deterministic and asynchronous automata (second group). A non-deterministic automaton is formally defined as a system $ (A, S, B, \chi ) $, where $ A, S $ and $ B $ are alphabets in the sense explained above, while $ \chi \subseteq S \times A \times S \times B $ is a transition-to-output relation. If the relation $ \chi $ is a function mapping $ S \times A $ into $ S \times B $, the non-deterministic automaton is called a deterministic automaton and is in fact identical with a finite automaton, since in such a case $ \chi $ may be considered as the pair of functions $ \phi , \psi $ which map $ S \times A $ into $ S $ and $ B $, respectively. As distinct from a finite automaton, an initialized non-deterministic automaton $ \mathfrak A _ {S _ {1} } $ has several initial states, which form a subset $ S _ {1} $ of the set $ S $. The behaviour of $ \mathfrak A _ {S _ {1} } $ is usually understood to mean one of the following generalizations of the behaviour of a finite automaton.
a) Instead of a function $ f $, an automaton $ \mathfrak A _ {S _ {1} } $ realizes a relation $ f ^ { \prime } $, consisting of all pairs of words $ ( a _ {1} \dots a _ {n} , b _ {1} \dots b _ {n} ) \in A ^ {*} \times B ^ {*} $ such that there are states $ s _ {1} \dots s _ {n+1} $, for which $ s _ {1} \in S _ {1} $ and $ (s _ {i} a _ {i} s _ {i+1} b _ {i} ) \in \chi $ is true for any $ i = 1 \dots n $. The class of relations realized by initialized non-deterministic automata coincides with the class of finitely-determined relations (cf. Finitely-determined function).
b) An initialized non-deterministic automaton $ \mathfrak A _ {S _ {1} } $ in which a set $ S ^ { \prime } $ of final states has been indicated, while the alphabet $ B $ is empty (i.e. $ \chi \subseteq S \times A \times S $), represents the event $ L _ {S} ^ \prime $ consisting of all words $ a _ {1} \dots a _ {n} \in A ^ {*} $ such that there are states $ s _ {1} \dots s _ {n+1} $ for which $ s _ {1} \in S _ {1} $, $ s _ {n+1} \in S ^ { \prime } $ and $ ( s _ {i} a _ {i} s _ {i+1} ) \in \chi $ is true for any $ i = 1 \dots n $. The class of events that can be represented by the automaton $ \mathfrak A _ {S _ {1} } $ is identical with the class of regular events, i.e. with respect to these aspects of their behaviour non-deterministic automata are equivalent to finite automata. However, the highly general nature of the concept of a non-deterministic automaton is reflected in the fact that a different number of states may be necessary to represent the same event using a non-deterministic automaton and using a finite automaton. There are events which are representable by a non-deterministic automaton with $ m $ states and by a finite automaton with $ 2 ^ {m} $ states, but not by any finite automaton with a smaller number of states.
A special subclass of non-deterministic automata is formed by the so-called partial automata, in which the relation $ \chi $ is a partial function mapping the set $ S \times A $ into $ S \times B $, and which realize partial finitely-determined functions.
The term "asynchronous automaton" usually denotes one of the two following types of automata. The first type comprises automata of type $ ( A, S, B, \phi , \psi ) $, in which the output function $ \psi $ maps the set $ S \times A $ into $ B ^ {*} $( for a finite automaton $ \psi $ maps $ S \times A $ into $ B $). These automata are mainly used in coding theory. The second type comprises finite automata whose transition function $ \phi $ has the following property: $ \phi (s, aa) = \phi (s, a) $ for all $ s $ and $ a $. These automata are used in coding theory and also in modelling certain systems in technology and biology.
3) Automata with a variable structure (third group) are finite automata $ \mathfrak A = (A \times A, S, B, \phi , \psi ) $ with two input channels, together with some fixed infinite sequence $ \alpha $( a superword) in the alphabet $ A $. Arbitrary words in the alphabet $ A $ are sent to the first input (channel) of such an automaton, while initial segments of the same length of the sequence $ \alpha $ are sent to the second input (channel). This imposes a restriction on the set of pairs of input words. If the automaton with a variable structure is considered as an automaton having only the first input (into which any word of $ A $ can be fed), its transition and its output functions will depend on the length of the input word which was fed in. In its behaviour, an automaton with a variable structure is equivalent to an infinite automaton with finite input and output alphabets and with a countable set of states.
4) Fuzzy automata constitute a generalization of the concept of a finite automaton, obtained by replacing the transition and output functions by fuzzy relations. A fuzzy subset of a set $ M $ is defined as a function which maps $ M $ into the segment $ [0, 1] $. Accordingly, in a fuzzy automaton, the transition and the output functions are replaced by functions mapping the sets $ S \times A \times S $ and $ S \times A \times B $ into $ [0, 1] $, where $ S $ is the set of states, $ A $ is the input alphabet and $ B $ is the output alphabet. The basic concepts and problems typical of finite automata have natural generalizations to fuzzy automata; this applies, in particular, to the representation of fuzzy events and the realization of fuzzy relations. Fuzzy automata are mathematical models for certain recognition mechanisms, and are used in pattern recognition.
5) Special classes of finite automata.
Automata without memory are single-state finite automata or automata equivalent to them. In such automata each output letter is fully determined by the input letter which is fed in at that moment. Such automata are frequently referred to as functional elements and are extensively used in problems of synthesis of automata.
Automata with a finite storage space (or automata with a finite memory) are finite automata in which each output letter is fully determined by a bounded segment of the input word fed in during the preceding moments of time, irrespective of the initial state. For automata with a finite memory and with $ n $ states the length of such a segment does not exceed $ n(n - 1)/2 $, and this maximum value is in fact attained for some automata. Automata with a finite memory are called self-regulating if, at a given time $ t $, the output letter at any moment $ \tau \geq t $ is independent of the initial state. Such automata are used in coding theory (cf. Coding and decoding), and the modification of the automata usually employed in such cases satisfies the above condition not for the set of all input words, but only for some subset of this set. Automata with a finite memory are realized by logical networks without feedback.
Reverse automata or automata without loss of information are finite automata which realize one-to-one functions. Such automata are also used in coding theory.
Micro-approach.
There are three types of generalized structural automata, which may be considered as generalized logical networks: 1) generalized logical networks with a permanent structure, in which both the elements and the relation between them remain unchanged when the automaton is functioning; 2) generalized logical networks with a variable structure; and 3) generalized logical networks made of volume elements and connections. The main classes of such automata are described below.
1) Generalized logical networks with a permanent structure. These include mosaic structures and iterative networks, which are finite fragments of mosaic structures with a similar range of problems.
Mosaic structures are infinite unions of transition systems $ (A, S, \phi ) $( i.e. of finite automata of the type $ (A, S, S, \phi , \phi ) $, where the output function is identical with the transition function and the output alphabet is identical with the set of states). Such systems are placed at points with integer coordinates (integral points) of the $ n $- dimensional space, while for each such point there is defined a finite set of integral points, called its neighbourhood. The input alphabet of the transition system placed at a certain point is the Cartesian product of the sets of states of the transition systems placed at the points in its neighbourhood.
A mosaic structure may be regarded as an infinite automaton whose input and output alphabets as well as its set of states are equal and are the infinite Cartesian product of the sets of states of all the transition systems contained in it. This makes it possible to reduce many problems for mosaic structures to problems for infinite automata. Problems specific to mosaic structures include modelling of effective procedures; in particular, of computational processes. Mosaic structures in which arbitrary automata instead of transition systems are used are also occasionally considered.
Uniform structures form an important class of mosaic structures. If all the transition systems are identical and if the neighbourhood of any point is obtained by a parallel translation of a certain fixed neighbourhood, the mosaic structure is known as a uniform structure or a cellular automaton. It is usually assumed, in this context, that there is some "stable" state of the transitional system, which is preserved if the input word is a tuple each term of which corresponds to this state. Typical problems on uniform structures are problems of self-reproduction and the "Garden-of-Eden" problem.
At any moment the states of the transition systems located at points in an integral lattice generate a kind of spatial mosaic pattern, which is usually denoted as a configuration. A configuration containing only a finite set of transition systems whose states are unstable (the excited part) is called finite. The self-reproduction problem consists of finding out whether finite configurations exist that, during the operation of the uniform system, pass into configurations, whose excited part is the multiply iterated excited part of the original configuration. The "garden of EdenGarden of Eden" is a term denoting a configuration which cannot result from configurations different from itself. The "garden-of-Eden problemGarden-of-Eden problem" is to determine the existence of "Gardens of Eden" for a given uniform structure.
2) Generalized logical networks with a variable structure. There exist various types of such logical networks. The most general ones include mosaic structures in which both the neighbourhoods of the elements and the elements themselves vary during the operation. As an example of such a generalized logical network one may mention the one-dimensional structure simulating the functioning of a Turing machine with input. In this case one of the nodes of the one-dimensional network corresponds to the control mechanism, while the other nodes correspond to the tape cells, which are considered as transition systems in which the letters of the working alphabet of the Turing machine serve as the input letters and states. The commutation is determined by the position of the head.
Another type of generalized logical networks with a variable structure are the so-called growing automata. They are uniform structures whose excited part increases during the operation. There are a number of models of such automata, which simulate various features of real mechanisms.
3) Generalized logical networks of volume elements are distinguished by the fact that a certain volume is assigned to their elementary automata and their mutual connections. As a result there arises the problem of synthesis of automata with minimum possible volume.
The term "automaton" is also used in concepts such as two-sided, multi-tape, multi-headed, linearly bounded, etc., automata, which are in fact modifications of the Turing machine. Sometimes all abstract machines are included in the concept of automata.
[1] W.S. McCulloch, W. Pitts, "A logical calculus of ideas immanent in nervous activity" Bull. Math. Biophys. , 5 (1943) pp. 115–133
[2] S.C. Kleene, "Representation of events in nerve nets and finite automata" , Automata studies , 34 , Princeton Univ. Press (1956) pp. 3–41
[3] A.W. Burks, J.B. Wright, "Sequence generators, graphs, and formal languages" Inform. and Control , 5 (1962) pp. 204–212
[4a] V.M. Glushikov, "The abstract theory of automata" Russian Math. Surveys , 16 : 5 (1961) pp. 1–53 Uspekhi Mat. Nauk , 16 : 5 (1961) pp. 3–62
[4b] V.M. Glushikov, Uspekhi Mat. Nauk , 17 : 2 (1962) pp. 270
[5] M.O. Rabin, D. Scott, "Finite automata and their decision problems" IBM J. Res. Develop. , 3 (1959) pp. 114–125
[6] L.A. Zadeh, "Probability measures of fuzzy events" J. Math. Anal. Appl. , 23 (1968) pp. 421–427
[7] V.Z. Alad'ev, "On the theory of uniform structures" , Tallin (1972) (In Russian)
See the Editorial Comment to the article Automata, equivalence of.
Automaton. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Automaton&oldid=45523
This article was adapted from an original article by V.B. KudryavtsevYu.I. Yanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
Retrieved from "https://encyclopediaofmath.org/index.php?title=Automaton&oldid=45523"
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多体系统碰撞动力学中接触力模型的研究进展
王庚祥, 马道林, 刘洋, 刘才山
王庚祥, 马道林, 刘洋, 刘才山. 多体系统碰撞动力学中接触力模型的研究进展. 力学学报, 2022, 54(12): 3239-3266 doi: 10.6052/0459-1879-22-266
引用本文: 王庚祥, 马道林, 刘洋, 刘才山. 多体系统碰撞动力学中接触力模型的研究进展. 力学学报, 2022, 54(12): 3239-3266 doi: 10.6052/0459-1879-22-266
Wang Gengxiang, Ma Daolin, Liu Yang, Liu Caishan. Research progress of contact force models in the collision mechanics of multibody system. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(12): 3239-3266 doi: 10.6052/0459-1879-22-266
Citation: Wang Gengxiang, Ma Daolin, Liu Yang, Liu Caishan. Research progress of contact force models in the collision mechanics of multibody system. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(12): 3239-3266 doi: 10.6052/0459-1879-22-266
王庚祥*, †,
马道林**,
刘洋††,
刘才山†,
西安建筑科技大学机电工程学院, 西安 710055
北京大学工学院, 湍流与复杂系统国家重点实验室, 北京 100871
上海交通大学船舶海洋与建筑工程学院, 上海 200240
††.
埃克塞特大学工程、数学和物理科学学院, 英国埃克塞特 EX44 QF
基金项目: 国家自然科学基金资助项目(11932001, 12172004, 12111530108)
刘才山, 教授, 主要研究方向: 多体系统动力学、碰撞力学. E-mail: [email protected]
中图分类号: TH113.1
RESEARCH PROGRESS OF CONTACT FORCE MODELS IN THE COLLISION MECHANICS OF MULTIBODY SYSTEM
Wang Gengxiang*, † ,
Ma Daolin** ,
Liu Yang†† ,
Liu Caishan† ,
School of Mechanical and Electrical Engineering, Xi'an University of Architecture Technology, Xi'an 710055, China
State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, China
School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiaotong University, Shanghai 200240, China
College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter EX44 QF, UK
参考文献(193)
摘要: 接触碰撞行为作为大自然与多体系统中的常见现象, 其接触力模型对于多体系统的碰撞行为机理研究与性能预测至关重要. 静态弹塑性接触模型与考虑能量耗散的连续接触力模型是研究接触碰撞行为的两类不同方法, 在多体系统碰撞动力学中存在诸多共性与差异. 本文分别从上述两类接触模型的发展历程入手, 详细介绍了两类模型的区别与联系. 首先, 根据阻尼项分母中是否含有初始碰撞速度将连续接触力模型分为黏性接触力模型与迟滞接触力模型, 讨论了能量指数与Hertz接触刚度之间的关系, 阐述了现有连续接触力模型在计算弹塑性材料接触碰撞行为时存在的问题. 其次, 着重介绍了分段连续的准静态弹塑性接触力模型(可连续从完全弹性转换到完全塑性接触阶段), 分析了利用此类弹塑性接触力模型计算碰撞行为的技术特点. 同时, 以恢复系数为桥梁和借助线性化的弹塑性接触刚度, 避免了Hertz刚度对弹塑性接触刚度的计算误差, 根据碰撞前后多体系统的能量与动能守恒推导了弹塑性接触模型等效的迟滞阻尼因子. 探索了连续接触力模型与准静态弹塑性接触力模型之间的内在联系, 数值计算结果定量说明了人为阻尼项代表的能量耗散与弹塑性接触力模型中加卸载路径代表的能量耗散具有等效性. 另外, 为了避免阻尼项分母中初始碰撞速度在计算颗粒物质动态性能时导致的数值奇异问题, 通过求解等效的线性单自由度欠阻尼非受迫振动方程获得了阻尼项分母中不含初始碰撞速度的连续接触力模型, 并以一维球链为例, 证明了该模型相比EDEM软件使用的连续接触力模型具有更高的精度. 最后, 本文分析了当前多体系统碰撞动力学的研究现状, 并简要展望了多体系统碰撞动力学中接触力模型的发展趋势与面临的挑战.
碰撞 /
能量耗散 /
恢复系数 /
接触力模型
Abstract: Impact behavior is a ubiquitous phenomenon in multibody systems. The contact force model is a pivotal tool to predict the contact characteristics of multibody systems. At present, there are two kinds of contact models used for calculating impact behaviors: the static elastoplastic contact force model and the continuous contact force models with energy dissipation. There are many similarities and discrepancies among them in the impact dynamics of multibody systems. This review starts with the introduction of development history of these two kinds of contact models followed by their development progress and background illustrated in detail. Firstly, whether the initial impact velocity is contained in the denominator of damping term severs as a criterion to classify the continuous contact force model as two types of models that are the contact force model with hysteresis damping factor and the other one with viscous damping factor. The relationship between the power exponent and Hertz contact stiffness is analyzed. The problems in calculating the elastic-plastic contact collision behavior by using the existing continuous contact force models are discussed. Secondly, the static elastoplastic contact force models with the continuous transition between the pure elastic and full plastic are introduced, and its characteristic is illustrated when calculating the elastoplastic collision events. The coefficient of restitution acts as the bridge to connect the static elastoplastic contact model and dynamic dashpot model as a complete system. In order to sidestep the error from the Hertz contact stiffness in calculating the elastoplastic impact behavior, a new viscous damping factor is derived by means of the linear elastoplastic contact stiffness based on energy conservation. The intrinsic connection between the static elastoplastic model and the dashpot model is explored, which proves that the artificial damping describing energy dissipation is equivalent to the one generated by the discrepancy between the loading and unloading paths. In order to avoid the numerical singularity caused by the initial impact velocity in the denominator of damping when calculating the dynamic performance of granular matter, a continuous contact force model with viscous damping is obtained by solving a linear single degree of freedom underdamped vibration system. One-dimension chain is taken as the numerical example to validate that the new dashpot model is more accurate than the one used in the EDEM software. Finally, the current research status of impact dynamics of multibody systems is reviewed, and the development trend and future challenges of contact force models are briefly summarized.
impact /
energy dissipation /
coefficient of restitution /
contact force model
图 1 接触力模型的分类
Figure 1. Classification of contact force models
图 2 碰撞体接触过程
Figure 2. Contact process between two contact bodies
图 3 常见连续接触力模型的力与变形量之间的关系[51]
Figure 3. Force-deformation relationship of the common contact force models[51]
图 4 弹簧−阻尼模型
Figure 4. The spring-damper model
图 5 两类阻尼的比较
Figure 5. Comparative analysis between two different damping systems
图 6 拥有不同阻尼类型的接触力模型
Figure 6. Contact force models with different damping systems
图 7 表面粗糙度形貌
Figure 7. The surface roughness topography
图 8 准静态弹塑性接触力模型[146]
Figure 8. Quasi-static elastoplatsic contact force model[146]
图 9 一般弹塑性接触力模型中力与变形量之间的关系
Figure 9. Force-deformation relationship for a general elastoplastic contact force model
图 10 接触力−变形量(关于弹塑性阶段)[43]
Figure 10. Relationship between the force and deformation (in elastoplastic phase)[43]
图 11 一维球链的碰撞实验装置示意图[117]
Figure 11. Experimental setup of the one-dimension granular chain[117]
图 12 一维球链中孤立波的传播特征
Figure 12. Propagation of solitary wave in the one-dimension granular chain
图 13 连续接触力模型之间的对比分析
Figure 13. Comparative analysis between two continuous contact force models
表 1 迟滞接触力模型(阻尼项分母中含有初始相对碰撞速度)
Table 1. Contact force models with hysteresis damping factors (the denominators of the damping force do not include the initial impact velocity)
Continuous contact force model $ F = K{\delta ^n} + \chi {\delta ^m}\dot \delta $ Power exponent n Impact parameter m Hysteresis damping factor$ \chi $
Hunt-Crossley model [61] 3/2 3/2 $ \chi = \dfrac{{3\left( {1 - {c_r}} \right)}}{2}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $
Herbert-McWhannell model [76] 3/2 3/2 $ \chi = \dfrac{{6\left( {1 - {c_r}} \right)}}{{\left[ {{{\left( {2{c_r} - 1} \right)}^2} + 3} \right]}}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $
Lee-Wang model [79] 3/2 3/2 $ \chi = \dfrac{{3\left( {1 - {c_r}} \right)}}{4}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $
Lankarani-Nikravesh model[66] 3/2 3/2 $ \chi = \dfrac{{3\left( {1 - c_r^2} \right)}}{4}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $
Gonthier et al. model[75] 3/2 3/2 $ \chi \approx \dfrac{{1 - c_r^2}}{{{c_r}}}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $
Zhiying-Qishao model [67] 3/2 3/2 $\chi = \dfrac{ {3\left( {1 - c_r^2} \right){{\rm{e}}^{2\left( {1 - {c_r} } \right)} } } }{4}\dfrac{K}{ { { {\dot \delta }^{\left( - \right)} } } }$
Flores et al. model [68] 3/2 3/2 $ \chi = \dfrac{{8\left( {1 - {c_r}} \right)}}{{5{c_r}}}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $
Gharib-Hurmuzlu model [77] 3/2 3/2 $ \chi = \dfrac{1}{{{c_r}}}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $
Hu-Guo model [70] 3/2 3/2 $ \chi = \dfrac{{3\left( {1 - {c_r}} \right)}}{{2{c_r}}}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $
Hu et al. model [78] 3/2 3/2 $\chi {\text{ = } } - \dfrac{ {6.66\;26\ln {c_r} } }{ {3.852\;38 + \ln {c_r} } }\dfrac{K}{ { { {\dot \delta }^{\left( - \right)} } } }$
Shen et al. model [71] 3/2 3/2 $ \chi {\text{ = }}\dfrac{{3\left( {1 - {c_r}} \right)}}{{2 c_r^{0.89}}}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $
Carvalho-Martins model [15] 3/2 3/2 $ \chi {\text{ = }}\dfrac{{3\left( {1 - {c_r}} \right)\left( {11 - {c_r}} \right)}}{{2\left( {1{\text{ + }}9{c_r}} \right)}}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $
Safaeifar-Farshidianfar model [21] 3/2 3/2 $ \chi {\text{ = }}\dfrac{{5\left( {1 - {c_r}} \right)}}{{4{c_r}}}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $
Zhang et al. model [72] 3/2 3/2 $\chi {\text{ = } }\dfrac{ {3\left( {1 - {c_r} } \right)} }{ {2\left( {0.618{{\rm{e}}^{ - 3.25{c_r} } } + 0.899{{\rm{e}}^{0.090\;25{c_r} } } } \right){c_r} } }\dfrac{K}{ { { {\dot \delta }^{\left( - \right)} } } }$
Zhao et al. model [22] 3/2 3/2 $ \chi {\text{ = }}\dfrac{{4\left( {1 - {c_r}} \right)}}{{1.302{c_r}}}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $
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表 2 黏性接触力模型(阻尼项分母中不含初始相对碰撞速度)
Table 2. Contact force models with viscous damping factors (the denominators of the damping force do not include the initial impact velocity)
Continuous contact force model
$ F = K{\delta ^n} + \chi {\delta ^m}\dot \delta $ Power exponent n Impact parameter m Viscous damping factor$ \chi $
Kuwabara and Kono [99] 3/2 1/2 $\chi = \dfrac{K}{2}\dfrac{ { { {\left( {3{\eta _2} - {\eta _1} } \right)}^2} } }{ { {3{\eta _2} + 2{\eta _1} } } }\dfrac{ {\left( {1 - {\nu ^2} } \right)\left( {1 - 2\nu } \right)} }{ {E{\nu ^2} } }$($ {\eta _1},{\eta _2} $ are the viscous material constant values)
Tsuji et al. [96] 3/2 1/4 $\chi = \dfrac{ {\sqrt 5 } }{2}D,D = 2\left| {\ln {c_r} } \right|\sqrt {\dfrac{ {KM} }{ { { {\text{π}} ^2} + { {\ln }^2}{c_r} } } }$
Jankowski [95] 3/2 1/4 $ \chi = 9\sqrt 5 \dfrac{{1 - c_r^2}}{{{c_r}\left[ {{c_r}\left( {9{\text{π}} - 16} \right) + 16} \right]}}\sqrt {KM} $
Lee and Wang[79] 3/2 0 $\chi = TD,D = 2\left| {\ln {c_r} } \right|\sqrt {\dfrac{ {KM} }{ { { {\text{π} } ^2} + { {\ln }^2}{c_r} } } } ,T = \dfrac{ {\delta + \left| \delta \right|} }{ {2\delta } }\exp \left\{ {\dfrac{q}{\varepsilon }\left[ {\left( {\delta - \varepsilon } \right) - \left| {\delta - \varepsilon } \right|} \right]} \right\}$
($ \varepsilon $, $ q $ are constants with unit m)
Schwager and Poschel [74] 3/2 0.65 empirical
Lee and Herrmann [80] 3/2 1 empirical
Ristow [100] 3/2 1 empirical
表 3 连续准静态弹塑性接触力模型
Table 3. Continuous quasi-static elastoplastic contact force models
Quasi-static elastoplastic contact model Loading phase Unloading phase
Thornton (1997)[135] $ F = \left\{ \begin{gathered} \dfrac{4}{3}{E^ * }\sqrt R {\delta ^{\frac{3}{2}}}\begin{array}{*{20}{c}} {}&{\begin{array}{*{20}{c}} {}&{} \end{array}}&{\left( {\delta < {\delta _y}} \right)} \end{array} \\ {F_y} + {\text{π}} {\sigma _y}R\left( {\delta - {\delta _y}} \right)\begin{array}{*{20}{c}} {}&{\left( {\delta > {\delta _y}} \right)} \end{array} \\ \end{gathered} \right. $
$ {\delta _y} $critical elastic deformation;$ {\sigma _y} $ yield stress;$ {F_y} $loading force $ F = \dfrac{4}{3}{E^ * }\sqrt {{R_p}} {\left( {\delta - {\delta _p}} \right)^{\frac{3}{2}}},{R_p} = \dfrac{{4{E^ * }}}{{3{F_{\max }}}}{\left( {\dfrac{{2{F_{\max }} + {F_y}}}{{2{\text{π}} {\sigma _y}}}} \right)^{\frac{3}{2}}} $
$ {\delta _p} $permanent deformation;$ {F_{\max }} $maximum
normal contact force
Stronge (2000)[63] $F = \left\{ \begin{gathered} \dfrac{4}{3}{E^ * }\sqrt R {\delta ^{\frac{3}{2} } }\begin{array}{*{20}{c} } {}&{\begin{array}{*{20}{c} } {}&{} \end{array} }&{\left( {\delta < {\delta _y} } \right)} \end{array} \\ {F_y}\left( {\dfrac{ {2\delta } }{ { {\delta _y} } } - 1} \right)\begin{array}{*{20}{c} } {\left[ {1 + \dfrac{1}{ {3{\vartheta _y} } }\ln \left( {\dfrac{ {2\delta } }{ { {\delta _y} } } - 1} \right)} \right]}&{\left( { {\delta _y} \leqslant \delta < {\delta _p} } \right)} \end{array} \\ \dfrac{ {2.8{F_y} } }{ { {\vartheta _y} } }\left( {\dfrac{ {2\delta } }{ { {\delta _y} } } - 1} \right)\begin{array}{*{20}{c} } {}&{}&{\left( {\delta \geqslant {\delta _p} } \right)} \end{array} \\ \end{gathered} \right.$
$ {F_y} = {\text{π}} {\vartheta _y}{\sigma _y}{R^2}{\left( {\dfrac{{3{\text{π}} }}{4}} \right)^2}{\left( {\dfrac{{{\vartheta _y}{\sigma _y}}}{{{E^ * }}}} \right)^2} $,$ {\vartheta _y} $coefficient;$ {\delta _y} $critical elastic;
$ {\sigma _y} $yield stress;$ {\delta _p} $critical plastic deformation; $ \begin{gathered} F = \dfrac{4}{3}{E^ * }\sqrt {\bar R} {\left( {\delta - {\delta _f}} \right)^{\frac{3}{2}}} \\ {\delta _f} = {\delta _c} - {\delta _r},\bar R = {\left( {\dfrac{{2{\delta _c}}}{{{\delta _y}}} - 1} \right)^{\frac{1}{2}}}R \\ \end{gathered} $
$ {\delta _c} $maximum contact deformation;
$ {\delta _r} $permanent deformation
Vu-Quoc and Zhang (2002)[147-148] $F = \left[ {\dfrac{ {2{E^ * } } }{ {3 R\left( {1 - {\nu ^2} } \right)} } } \right]{a^3},a = \sqrt {2 R\delta } \begin{array}{*{20}{c} } {}&{F < {F_y} } \end{array}$
$ \left\{ \begin{gathered} {a^{ep}} = {a^e} + {a^p},{a^p} = \left\{ \begin{gathered} 0\begin{array}{*{20}{c}} {}&{}&{}&{\left( {F < {F_y}} \right)} \end{array} \\ {C_a}\left( {F - {F_y}} \right)\begin{array}{*{20}{c}} {}&{\left( {F > {F_y}} \right)} \end{array} \\ \end{gathered} \right. \\ \delta = \dfrac{{{{\left( {{a^{ep}}} \right)}^2}}}{{2{R^{ep}}}} \\ {R^{ep}} = {C_R}R \\ {C_R}{\text{ = }}\left\{ \begin{gathered} 1.0\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {}&{} \end{array}}&{}&{\left( {F < {F_y}} \right)} \end{array} \\ 1.0{\text{ + }}{K_c}\left( {F - {F_y}} \right)\begin{array}{*{20}{c}} {}&{\left( {F > {F_y}} \right)} \end{array} \\ \end{gathered} \right. \\ \end{gathered} \right. $
Fy Normal yield load; Kc and Ca are the empirical parameters;
ae contact radius corresponding to the elastic region; $ \nu $Poisson ratio $\begin{gathered} F = \left[ {\dfrac{ {2{E^ * } } }{ {3 R\left( {1 - {\nu ^2} } \right)} } } \right]a_e^3 \\ {a_e} = {2{ {\left( { {C_R} } \right)}_{\max } }R\left( {\delta - {\delta _r} } \right)} \\ \end{gathered}$
$ {\delta _r} $ permanent deformation
Kogut and Etsion (2002)[139] $ F = \left\{ \begin{gathered} \dfrac{4}{3}{E^ * }\sqrt R {\delta ^{\frac{3}{2}}}\begin{array}{*{20}{c}} {}&{\begin{array}{*{20}{c}} {}&{} \end{array}}&{\left( {\dfrac{\delta }{{{\delta _y}}} < 1} \right)} \end{array} \\ {F_c}1.03{\left( {\dfrac{\delta }{{{\delta _y}}}} \right)^{1.425}}\begin{array}{*{20}{c}} {}&{\left( {1 \leqslant \dfrac{\delta }{{{\delta _y}}} < 6} \right)} \end{array} \\ {F_c}1.40{\left( {\dfrac{\delta }{{{\delta _y}}}} \right)^{1.263}}\begin{array}{*{20}{c}} {}&{\left( {6 \leqslant \dfrac{\delta }{{{\delta _y}}} < 110} \right)} \end{array} \\ \end{gathered} \right. $
$ {\delta _y} $critical elastic deformation; Fc loading force corresponding to the
beginning phase of the plastic deformation $ F = {F_{\max }}{\left( {\dfrac{{\omega - {\omega _{res}}}}{{{\omega _{\max }} - {\omega _{res}}}}} \right)^{np}},np = 1.5{\left( {{\omega _{\max }}} \right)^{ - 0.0331}} $
$ \omega = {\delta \mathord{\left/ {\vphantom {\delta {{\delta _y}}}} \right. } {{\delta _y}}};{\omega _{\max }} = {{{\delta _{\max }}} \mathord{\left/ {\vphantom {{{\delta _{\max }}} {{\delta _y}}}} \right. } {{\delta _y}}} $
Fmax maximum contact force; $ {\delta _{\max }} $maximum contact
deformation;$ {\omega _{res}} $residual deformation
Jackson and Green (2005)[138] $F = \left\{ \begin{gathered} \dfrac{4}{3}{E^ * }\sqrt R {\delta ^{\frac{3}{2} } }\begin{array}{*{20}{c} } {}&{\begin{array}{*{20}{c} } {}&{} \end{array} }&{\left( {\dfrac{\delta }{ { {\delta _y} } } \leqslant 1.9} \right)} \end{array} \\ {F_c}\left\{ {\exp \left[ { - \dfrac{1}{4}{\omega ^{\frac{5}{ {12} } } } } \right]{\omega ^{\frac{3}{2} } } + \dfrac{ {4 H} }{ {C{S_y} } }\left[ {1 - \exp \left( { - \dfrac{1}{ {25} }{\omega ^{\frac{5}{9} } } } \right)} \right]\omega } \right\}\begin{array}{*{20}{c} } {}&{\left( {\dfrac{\delta }{ { {\delta _y} } } \geqslant 1.9} \right)} \end{array} \\ \end{gathered} \right.$
$ C = 1.295\exp \left( {0.736\nu } \right) $;$ {\delta _y} $critical elastic deformation; H hardness;
Sy limit of yielding;$ \nu $Poisson ratio; Fc critical elastic load $\begin{gathered} F = \dfrac{4}{3}{E^ * }\sqrt { {R} } {\left( {\delta - {\delta _r} } \right)^{\frac{3}{2} } } \\ {R_b} = R\cos \theta ,\theta = \dfrac{ { {a_c} } }{R} \\ \end{gathered}$
$ {\delta _r} $permanent deformation; ac contact radius in
the unloading phase
Du and Wang (2009)[143] $ F = \left\{ \begin{gathered} \dfrac{4}{3}{E^ * }\sqrt R {\delta ^{\frac{3}{2}}}\begin{array}{*{20}{c}} {}&{\begin{array}{*{20}{c}} {}&{} \end{array}}&{\left( {\delta \leqslant {\delta _e}} \right)} \end{array} \\ {\text{π}} R{p_p}\delta - \dfrac{{p_p^3{{\text{π}} ^3}{R^2}}}{{12{{\left( {{E^ * }} \right)}^2}}}\begin{array}{*{20}{c}} {}&{\left( {\delta \leqslant {\delta _e}} \right)} \end{array} \\ \end{gathered} \right. $
$ {p_p} = \left( {1 + \dfrac{{\text{π}} }{2}} \right){\sigma _y} $;$ {\sigma _y} $yield strength; $ {\delta _e} $critical elastic deformation
$F = \dfrac{4}{3}{E^ * }\sqrt { {R} } {\left( {\delta - {\delta _{res} } } \right)^{\frac{3}{2} } }$
$ {\delta _{res}} $permanent deformation
Brake (2012)[146] $F = \left\{ \begin{gathered} \dfrac{4}{3}{E^ * }\sqrt R {\delta ^{\frac{3}{2} } }\begin{array}{*{20}{c} } {}&{\begin{array}{*{20}{c} } {}&{} \end{array} }&{\left( {\delta < {\delta _y} } \right)} \end{array} \\ \left[{2{F_y} - 2{F_p} + \left( { {\delta _p} - {\delta _y} } \right)\left( { { {F'}_y} + { {F'}_p} } \right)} \right]{\left( {\dfrac{ {\delta - {\delta _y} } }{ { {\delta _p} - {\delta _y} } } } \right)^3}+ \\ \qquad \left[ { - 3{F_y} + 3{F_p} + \left( { {\delta _p} - {\delta _y} } \right)\left( { - 2{ {F'}_y} - { {F'}_p} } \right)} \right]{\left( {\dfrac{ {\delta - {\delta _y} } }{ { {\delta _p} - {\delta _y} } } } \right)^2}+ \\ \qquad \left( { {\delta _p} - {\delta _y} } \right)F'\left[ {\dfrac{ {\delta - {\delta _y} } }{ { {\delta _p} - {\delta _y} } } } \right] + {F_y}\begin{array}{*{20}{c} } {}&{\left( { {\delta _y} \leqslant \delta <{\delta _p} } \right)} \end{array} \\ \dfrac{ {3{F_y}{\text{π} } {p_0} } }{ {4 E} }\sqrt {\dfrac{R}{ { {\delta _y} } } } \left[ {4\left( {\dfrac{\delta }{ { {\delta _y} } } } \right) + \dfrac{c}{ {R{\delta _y} } } } \right] \\ \end{gathered} \right.$
$ \begin{gathered} {p_0} \approx H,c = a_p^2 - 2 R{\delta _p},{F_y} = \dfrac{4}{3}{E^ * }\sqrt R \delta _y^{\frac{3}{2}},{F_p} = {p_0}{\text{π}} {a_p} \\ {a_p} = \dfrac{{3 R{\text{π}} {p_0}}}{{4{E^ * }}},{{F'}_p} = 2 R{\text{π}} {p_0},{{F'}_y} = 2{E^ * }\sqrt {R{\delta _y}} \\ \end{gathered} $
H hardness;$ {\delta _y} $ critical elastic deformation;$ {\delta _p} $ critical
plastic deformation $ \begin{gathered} F = \dfrac{4}{3}{E^ * }\sqrt {{{\bar R}_b}} {\left( {\delta - \bar \delta } \right)^{\frac{3}{2}}} \\ \bar \delta = {\delta _m} - {\left( {\dfrac{{3{F_m}}}{{4{E^ * }\sqrt {{{\bar R}_b}} }}} \right)^{\frac{3}{2}}},{{\bar R}_b} = R + \dfrac{1}{2}{\delta _m} \\ \end{gathered} $
$ {\delta _m} $ maximum contact deformation in the
loading phase; Fm maximum contact force in
the loading phase
Burgoyne and Daraio (2014)[144] $ F = \left\{ \begin{gathered} \dfrac{4}{3}{E^ * }\sqrt R {\delta ^{\frac{3}{2}}}\begin{array}{*{20}{c}} {}&{}&{\left( {0 < \delta < {\delta _y}} \right)} \end{array} \\ \delta \left( {\alpha + \beta \ln \delta } \right)\begin{array}{*{20}{c}} {}&{}&{\left( {{\delta _y} < \delta < {\delta _p}} \right)} \end{array} \\ {p_0}{\text{π}} \left( {2 R\delta + {c_2}} \right)\begin{array}{*{20}{c}} {}&{}&{\left( {\delta > {\delta _p}} \right)} \end{array} \\ \end{gathered} \right. $
$\alpha {\text{ = } }{ {\left( { {\delta _p}{F_y}\ln {\delta _p} - {\delta _y}{F_p}\ln {\delta _y} } \right)} \mathord{\left/ {\vphantom { {\left[ { {\delta _p}{F_y}\ln {\delta _p} - {\delta _y}{F_p}\ln {\delta _y} } \right]} {\left[ { {\delta _y}{\delta _p}(\ln {\delta _p} - \ln {\delta _y})} \right]} } } \right. } {\left[ { {\delta _y}{\delta _p}(\ln {\delta _p} - \ln {\delta _y})} \right]} }$
$\beta {\text{ = } }{ {\left( { {\delta _y}{F_p} - {\delta _p}{F_y} } \right)} \mathord{\left/ {\vphantom { {\left( { {\delta _y}{F_p} - {\delta _p}{F_y} } \right)} {\left[ { {\delta _y}{\delta _p}(\ln {\delta _p} - \ln {\delta _y})} \right]} } } \right. } {\left[ { {\delta _y}{\delta _p}(\ln {\delta _p} - \ln {\delta _y})} \right]} }$
$ {F_y} = \dfrac{1}{6}{\left( {{R \mathord{\left/ {\vphantom {R {{E^ * }}}} \right. } {{E^ * }}}} \right)^2}{\left( {1.6{\text{π}} {\sigma _y}} \right)^3},{F_p} = {p_0}{\text{π}} \left( {2 R{\delta _p} + {c_2}} \right),{p_0}{\text{ = }}{c_1}{\sigma _y} $
c1 and c2 are empirical parameters; $ {\delta _y} $ critical elastic deformation;
$ {\delta _p} $ critical plastic deformation;$ {\sigma _y} $yield strength $ F = \dfrac{4}{3}{E^ * }\sqrt {{R_p}} {\left( {\delta - {\delta _r}} \right)^{\frac{3}{2}}} $
$ {R_p} = \dfrac{{4{E^ * }}}{{3{F_{\max }}}}\left( {\dfrac{{2{F_{\max }} + {F_y}}}{{2{\text{π}} {p_y}}}} \right),{p_y} = 1.6{\sigma _y} $
$ {\delta _r} = {\delta _{\max }} - {\left( {\dfrac{{3{F_{\max }}}}{{4{E^ * }\sqrt {{R_p}} }}} \right)^{\frac{2}{3}}} $
Ma and Liu (2015)[113] $ F\left( \delta \right) = \left\{ \begin{gathered} \dfrac{4}{3}{E^ * }{R^{\frac{1}{2}}}{\delta ^{\frac{3}{2}}}\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {}&{} \end{array}}&{}&{}&{\delta < {\delta _y}} \end{array} \\ \delta \left( {{c_1} + {c_2}\ln \dfrac{\delta }{{{\delta _c}}}} \right) + {c_3}\begin{array}{*{20}{c}} {}&{{\delta _y} \leqslant \delta < {\delta _p}} \end{array} \\ {F_p} + {k_1}\left( {\delta - {\delta _p}} \right)\begin{array}{*{20}{c}} {}&{}&{\delta \geqslant {\delta _p}} \end{array} \\ \end{gathered} \right. $
$\left\{\begin{array}{l}{k}_{1}=2{\text{π} } R\psi {\sigma }_{y},\\ {F}_{p}={\delta }_{p}\left[{c}_{1} + {c}_{2}\mathrm{ln}\left({\xi }^{2}/2\right)\right] + {c}_{3}\\ {c}_{1}=\dfrac{ {p}_{y}\left[1 + \mathrm{ln}\left({\xi }^{2}/2\right)\right]-2\psi {\sigma }_{y} }{\mathrm{ln}\left({\xi }^{2}/2\right)}{\text{π} } R,\\{c}_{2}=\dfrac{2\psi {\sigma }_{y}-{p}_{y} }{\mathrm{ln}\left({\xi }^{2}/2\right)}{\text{π} } R\\ {c}_{3}={F}_{y}-{c}_{1}{\delta }_{y}, \\{F}_{y}={ {\text{π} } }^{3}{R}^{2}{p}_{y}^{3}/6{E}^{\ast }{}^{2}\end{array} \right.$
$ {p_y} = 1.61{\sigma _y} $;$ \xi $and$ \psi $are the dimensionless coeffcients;
$ {\delta _y} $ critical elastic deformation;$ {\delta _p} $ critical plastic deformation;
$ {\sigma _y} $yield strength $ F\left( \delta \right) = \left\{ \begin{gathered} \dfrac{4}{3}{E^ * }{R^{\frac{1}{2}}}{\delta ^{\frac{3}{2}}}\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {}&{}&{} \end{array}}&{}&{\delta < {\delta _y}} \end{array} \\ \dfrac{4}{3}{E^ * }{\left( {{R^e}} \right)^{\frac{1}{2}}}{\left( {\delta - {\delta _r}} \right)^{\frac{3}{2}}}\begin{array}{*{20}{c}} {}&{{\delta _y} \leqslant \delta < {\delta _p}} \end{array} \\ \dfrac{4}{3}{E^ * }{\left( {R_p^e} \right)^{\frac{1}{2}}}{\left( {\delta - {\delta _r}} \right)^{\frac{3}{2}}}\begin{array}{*{20}{c}} {}&{\delta \geqslant {\delta _p}} \end{array} \\ \end{gathered} \right. $
$ R_{ep}^e = \dfrac{{{F^e}R}}{{{F_{\max }}}},{F^e} = \dfrac{4}{3}{E^ * }{R^{\frac{1}{2}}}\delta _y^{\frac{3}{2}} $
$ R_p^e = \dfrac{{F_p^eR}}{{{F_{ep}}}},F_p^e = \dfrac{4}{3}{E^ * }{R^{\frac{1}{2}}}\delta _p^{\frac{3}{2}} $
表 4 恢复系数模型
Table 4. Coefficient of restitution (CoR) model
Authors CoR mathematical model Parameters
Chang and Ling (1992)[159] ${c_r} = \sqrt { { {\left\{ {\dfrac{8}{ {15} }{E^ * }\sqrt R { {\left[ { {\omega _c}{\omega _m}\left( {2 - \dfrac{ { {\omega _c} } }{ { {\omega _m} } } } \right)} \right]}^{\frac{3}{2} } } } \right\} } \mathord{\left/ {\vphantom { {\left[ {\dfrac{8}{ {15} }{E^ * }\sqrt R { {\left[ { {\omega _c}{\omega _m}\left( {2 - \dfrac{ { {\omega _c} } }{ { {\omega _m} } } } \right)} \right]}^{\frac{3}{2} } } } \right] } {\left[ {\dfrac{8}{ {15} }{E^ * }\sqrt R \omega _c^{\dfrac{5}{2} } + {\text{π} } KYR{\omega _m}\left( { {\omega _m} - {\omega _c} } \right)} \right] } } } \right. } {\left[ {\dfrac{8}{ {15} }{E^ * }\sqrt R \omega _c^{\frac{5}{2} } + {\text{π} } KYR{\omega _m}\left( { {\omega _m} - {\omega _c} } \right)} \right] } } }$ $ {\omega _c} $critical elastic deformation; $ {\omega _m} $maximum deformation in the
compression phase; Y yield strength;$ K = 1.282 + 1.158 v $;
v Poisson ratio
Thornton (1997)[135] $ {c_r} = {\left( {\dfrac{{6\sqrt 3 }}{5}} \right)^{\frac{1}{2}}}\sqrt {1 - \dfrac{1}{6}{{\left( {\dfrac{{{V_y}}}{{{V_l}}}} \right)}^2}} {\left[ {\dfrac{{\left( {\dfrac{{{V_y}}}{{{V_l}}}} \right)}}{{\left( {\dfrac{{{V_y}}}{{{V_l}}}} \right) + 2\sqrt {\dfrac{6}{5} - \dfrac{1}{5}{{\left( {\dfrac{{{V_y}}}{{{V_l}}}} \right)}^2}} }}} \right]^{\frac{1}{4}}} $ Vy yield impact velocity;
Vl impact velocity
Wu et al. (2005)[165] $ {c_r} = \left\{ \begin{gathered} 2.08{\left( {\dfrac{{{V_1}}}{{{V_c}}}} \right)^{0.156}}\begin{array}{*{20}{c}} {}&{\left( {{V_1} < {V_f}} \right)} \end{array} \\ 0.62{\left( {\dfrac{{{V_1}{S_y}}}{{{V_c}{E^ * }}}} \right)^{ - \frac{1}{2}}}\begin{array}{*{20}{c}} {}&{\left( {{V_1} > {V_f}} \right)} \end{array} \\ \end{gathered} \right. $ Vc initial impact velocity leading to the plastic deformation;
Vf critical impact velocity; Sy yield stress; V1 impact velocity
Weir and Tallon (2005)[161] $ {c_r} = 3.1{\left( {\dfrac{{{S_y}}}{{{E^ * }}}} \right)^{\frac{5}{8}}}{\left( {\dfrac{{{R_1}}}{R}} \right)^{\frac{3}{8}}}{\left( {\dfrac{{{c_0}}}{{{v_0}}}} \right)^{\frac{1}{4}}},{c_0} = \sqrt {\dfrac{E}{\rho }} $ v0 initial impact velocity; Sy yield stress; R1 contact
radius after contact; $ \rho $density
Jackson et al. (2010)[152] $ {c_r} = \left\{ \begin{gathered} 1\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {}&{}&{} \end{array}}&{}&{} \end{array}}&{}&{\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {}&{}&{} \end{array}}&{} \end{array}}&{\left( {0 < {V_1} < 1} \right)} \end{array} \\ 1 - 0.1\ln \left( {{V_1}} \right){\left( {\dfrac{{{V_1} - 1}}{{59}}} \right)^{0.156}}\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {}&{}&{} \end{array}}&{\left( {1 < {V_1} \leqslant 60} \right)} \end{array} \\ 1 - 0.1\ln \left( {60} \right) - 0.1\ln \left( {\dfrac{{{V_1}}}{{60}}} \right){\left( {{V_1} - 60} \right)^{2.36{\varepsilon _y}}}\begin{array}{*{20}{c}} {}&{\left( {60 \leqslant {V_1} \leqslant 1000} \right)} \end{array} \\ \end{gathered} \right. $ V1 impact velocity;
$ {\varepsilon _y} = {{{S_y}} \mathord{\left/ {\vphantom {{{S_y}} {{E^ * }}}} \right. } {{E^ * }}} $; Sy yield stress
Ma and Liu (2015)[113] $ {c_r} = 0.81{E^ * }^{ - \frac{1}{3}}{\left( {R_p^e} \right)^{ - \frac{1}{6}}}k_1^{\frac{5}{{12}}}{m^{ - \frac{1}{{12}}}}v_0^{ - \frac{1}{6}} $ Rep contact radius after plastic deformation;
k1 contact parameter; m mass; v0 initial impact velocity
表 5 接触参数
Table 5. Contact parameters
modulus/GPa Poisson
ratio Radius/cm Yield
strength/MPa Mass/kg Density/
kg·m−3
200 0.29 2.05 1030 0.097 7800
65 0.33 2.00 30 0.261 2700
表 6 误差分析对比
Table 6. Comparative analysis of the error percentage
No. 9 16 24 31 40 50 56 63
New 4.23 6.68 19.44 1.44 9.12 0.50 12.33 25.39
EDEM 5.36 7.39 22.44 10.03 10.19 16.38 26.54 36.79
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图(13) / 表(6) | CommonCrawl |
\begin{document}
\title{Resolutions by permutation modules} \author{Paul Balmer} \author{Dave Benson} \date{2020 March 9}
\address{Paul Balmer, Mathematics Department, UCLA, Los Angeles, CA 90095-1555, USA} \email{[email protected]} \urladdr{http://www.math.ucla.edu/$\sim$balmer}
\address{Dave Benson, Institute of Mathematics, University of Aberdeen, King's College, Aberdeen AB24 3UE, Scotland U.K.} \email{[email protected]} \urladdr{https://homepages.abdn.ac.uk/d.j.benson/pages/}
\begin{abstract} We prove that, up to adding a complement, every modular representation of a finite group admits a finite resolution by permutation modules. \end{abstract}
\subjclass[2010]{} \keywords{}
\thanks{First-named author supported by NSF grant~DMS-1901696.}
\maketitle
Let $G$ be a finite group and $\kk$ be a field of characteristic~$p>0$ dividing the order of~$G$. It is well-known that if $G$ has non-cyclic Sylow $p$-subgroups, the $\kk$-linear representation theory of~$G$ is complicated. In particular, the Krull--Schmidt abelian category, $\kk G\mmod$, of finite-dimensional $\kk G$-modules admits \emph{infinitely many} isomorphism classes of indecomposable objects. On the other hand, there is a much simpler class of $\kk G$-modules, the \emph{permutation modules}, \ie those isomorphic to $\kk X$ for $X$ a finite $G$-set. The \emph{finite} collection $\{\kk(G/H)\}_{H\leqslant G}$ additively generates all such modules.
For a $\kk G$-module $M\in\kk G\mmod$, we want to analyze the existence of what we'll call a \emph{permutation resolution} for short, \ie an exact sequence
\begin{equation} \label{eq:resol-intro} 0\to P_n\to P_{n-1}\to \cdots \to P_1 \to P_0 \to M\to 0 \end{equation}
where all~$P_i$ are permutation modules. Up to direct summands, it is always possible:
\begin{Thm} \label{thm:resol-intro} Let $G$ be a finite group and $M\in\kk G\mmod$. Then there exists a $\kk G$-module~$N$ such that $M\oplus N$ admits a finite resolution~\eqref{eq:resol-intro} by permutation modules. \end{Thm}
The related problem of resolutions~\eqref{eq:resol-intro} that are not only exact but remain exact under all fixed-point functors has been recently discussed in~\cite{BoucStancuWebb17}. Allowing $p$-permutation modules $P_i$ (that is, direct summands of permutation modules), Bouc--Stancu--Webb prove that such resolutions exist for all~$M$ if and only if~$G$ has a Sylow subgroup that is either cyclic or dihedral (for $p=2$).
Unsurprisingly, \Cref{thm:resol-intro} reduces to a Sylow subgroup $S$ of~$G$, since every $M$ is a direct summand of~$\Ind_S^G\Res^G_S(M)$ and since the functor $\Ind_S^G$ is exact and preserves permutation modules. So we focus on the case where $G$ is a $p$-group.
For the proof, we shall consider a stronger property:
\begin{Def} \label{def:good-resol} We say that a resolution~\eqref{eq:resol-intro} is \emph{free up to degree~$m\geqslant 0$} if $P_i$ is a free module for $i=0,\ldots,m$. We say that $M$ admits \emph{good permutation resolutions} if for every integer $m\geqslant 0$, there exists a finite resolution~\eqref{eq:resol-intro} by permutation modules that is free up to degree~$m$. \end{Def}
\begin{Rem} \label{rem:good-resol} Let $G$ be a $p$-group. A $\kk G$-module $M$ admits good permutation resolutions if and only if for all~$m\geqslant 1$ the $m$th Heller loop $\Omega^mM$ admits a finite permutation resolution. Also, if $Q$ is free and $M\oplus Q$ admits a permutation resolution as in~\eqref{eq:resol-intro} then the epimorphism $P_0\onto M\oplus Q \onto Q$ forces $Q$ to be a direct summand of~$P_0$ and one can remove $0\to Q\xrightarrow{=} Q \to 0$ from the resolution. So if $M\oplus Q$ has a permutation resolution that is free up to degree~$m$ then so does~$M$. \end{Rem}
An advantage of good permutation resolutions is the \emph{two out of three property}:
\begin{Prop} \label{prop:2-out-of-3} Let $G$ be a $p$-group. Let $0\to L\to M\to N\to 0$ be an exact sequence of $\kk G$-modules. If two out of $L$, $M$ and $N$ have good permutation resolutions then so does the third. \end{Prop}
\begin{proof} If $P\onto N$ is a projective cover, we obtain by `rotation' an exact sequence $0\to \Omega^1N\to L\oplus P\to M\to 0$. In view of \Cref{rem:good-resol}, we can rotate in this way and reduce to the case where $L$ and~$M$ admit good permutation resolutions and then prove that~$N$ does. Let $m\geqslant 0$. Choose $P_\sbull\to M$ a permutation resolution of~$M$ that is free up to degree~$m$. Let $\ell\geqslant m$ be such that $P_i=0$ for all $i>\ell$. Now choose $Q_\sbull\to L$ a permutation resolution of~$L$ that is free up to degree~$\ell$. We have the following picture (plain part) with exact rows:
\begin{equation} \label{eq:resol-qi} \vcenter{\xymatrix@C=1em@R=1em{ 0\ar[r] & Q_n \ar[r] \ar@{..>}[d] & \cdots \ar[r] & Q_{\ell+1} \ar[r] \ar@{..>}[d] & Q_\ell \ar[r] \ar@{..>}[d] & \cdots \ar[r] & Q_0 \ar[r] \ar@{..>}[d] & L \ar[r] \ar[d] & 0 \\ 0\ar[r] & 0 \ar[r] & \cdots \ar[r] & 0 \ar[r] & P_\ell \ar[r] & \cdots \ar[r] & P_0 \ar[r] & M \ar[r] & 0 }} \end{equation}
The standard lifting argument, using that $Q_j$ is projective for $j=0,\ldots, \ell$ shows that there exists a lift $f_\sbull\colon Q_\sbull\to P_\sbull$ of the morphism $L\to M$. Then the mapping cone complex $\cone(f_\sbull)$ yields a resolution of $\coker(L\to M)=N$ and this complex~$\cone(f_\sbull)$ has free objects in degree~$0,\ldots,m$ since $P_\sbull$ and~$Q_\sbull$ do. \end{proof}
Let us discuss an example of \Cref{thm:resol-intro}, where we can even take $N=0$.
\begin{Prop} \label{prop:abelem} Let $E=(C_p)^{\times r}=C_p\times\cdots\times C_p$ be an elementary abelian group of rank~$r$. Then every $\kk E$-module admits good permutation resolutions. \end{Prop}
\begin{proof} Consider for each $1\leqslant i\leqslant r$ the (`coordinate-wise') subgroup \[ H_i=C_p\times \cdots \times C_p\times 1\times C_p\times\cdots\times C_p \] of rank~$r-1$. Let $m\geqslant 0$. Inflating from $E/H_i\simeq C_p$ the usual $2$-periodic resolutions $0\to \kk\to \kk C_p\to \cdots \to\kk C_p\to \kk\to 0$ of length at least~$m$, we obtain quasi-isomorphisms of $\kk E$-modules $s_i\colon Q(i)\to \kk[0]$ where the $Q(i)$ are defined as follows:
\[ \vcenter{\xymatrix@R=1em@C=1em{ Q(i):= \ar@<-.8em>[d]_-{s_i} && 0 \ar[r] & \kk \ar[r] \ar[d] & \kk(E/H_i) \ar[r] \ar[d] & \cdots \ar[r] & \kk(E/H_i) \ar[r] \ar[d] & \kk(E/H_i) \ar[r] \ar[d] & 0 \\ \kk[0]= && 0\ar[r] & 0 \ar[r] & 0 \ar[r] & \cdots \ar[r] & 0 \ar[r] & \kk \ar[r] & 0 }} \]
Tensoring all the above, we obtain a quasi-isomorphism \[ s_1\otimes \cdots \otimes s_r\colon P_\sbull:= Q(1)\otimes \cdots \otimes Q(r)\to (\kk[0])^{\otimes{r}}\cong \kk[0], \] \ie a permutation resolution $P_\sbull$ of~$\kk$. In other words, we performed an `external tensor' of all the periodic resolutions over each copy of~$C_p$ in~$E$. Since the Mackey formula gives by induction $\kk(E/H_{i_1})\otimes \cdots \otimes \kk(E/H_{i_n})\cong \kk(E/(H_{i_1}\cap \cdots \cap H_{i_n}))$, we have produced a permutation resolution $P_\sbull$ of~$\kk$ that is easily seen to be free up to degree~$m$. As $m\geqslant 0$ was arbitrary, we proved that the trivial module~$\kk$ admits good permutation resolutions. A general module $M\in \kk E\mmod$ admits a filtration whose successive quotients are trivial. We therefore conclude by induction, via \Cref{prop:2-out-of-3}. \end{proof}
\begin{Rem} The proof of \Cref{prop:abelem} shows that the stabilisers in the permutation resolution may be taken to be products of subsets with respect to the given decomposition of $E$. Applying the proposition to a module and its dual shows that given a module $M$ we may form a finite exact complex of permutation modules with these stabilisers in such a way that the image of one of the maps is~$M$. This should be compared with the main theorem of~\cite{BensonCarlson} which shows that a finite exact sequence of permutation $E$-modules in which the set of stabilisers has no containment of index $p$ necessarily splits, so that the image of every map is again a permutation module. \end{Rem}
\begin{proof}[Proof of \Cref{thm:resol-intro}] As already mentioned, we can reduce to the case where $G$ is a $p$-group. By \cite{Carlson00}, we know that for every $\kk G$-modules~$M$, there exists a $\kk G$-module~$N$ and a finite filtration $0=L_0\subset L_1\subset \cdots \subset L_s=M\oplus N$ such that every $L_i/L_{i-1}$ is induced from some elementary abelian subgroup~$E_i\leqslant G$. Since the result holds for elementary abelian groups (\Cref{prop:abelem}) and is stable by induction, we see that all $L_i/L_{i-1}$ admit good permutation resolutions. By \Cref{prop:2-out-of-3}, we conclude that so does $M\oplus N$. In particular, $M\oplus N$ has a permutation resolution. \end{proof}
\medbreak \noindent \textbf{Acknowledgements:} The authors are grateful to Serge Bouc, Martin Gallauer and Peter Webb for useful discussions. The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programmes `K-theory, algebraic cycles and motivic homotopy theory' and `Groups, representations and applications: new perspectives', where work on this paper was undertaken. The Isaac Newton Institute is supported by EPSRC grant no EP/R014604/1.
\end{document} | arXiv |
Milton spilled some ink on his homework paper. He can't read the coefficient of $x$, but he knows that the equation has two distinct negative, integer solutions. What is the sum of all of the distinct possible integers that could be under the ink stain?
[asy]
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
fill((0,0)--(3,0)--(3,3)--(0,3)--cycle,black);
label("$x+36=0$",(3,1.5),E);
label("$x^{2}+$",(0,1.5),W);
[/asy]
Because the quadratic has two distinct integer roots, we know that it can be factored as \[(x+r)(x+s),\] where $r$ and $s$ are positive integers. Expanding this product gives $x^2 + (r+s)x + rs$, and comparing this to the given quadratic tells us that $rs = 36$. So, we consider all the pairs of distinct integers that multiply to 36, and we compute their sum in each case: \[\begin{array}{cc|c}
r&s&r+s\\\hline
1&36&37\\
2&18&20\\
3&12&15\\
4&9&13\end{array}\] Summing the entries in the final column gives us a total of $\boxed{85}$. | Math Dataset |
Does every continuous map $f$ from $D^2$ minus $k$ disjoint open disks to itself have a fixed point?
Let $A_k$ denote $D^2$ with $k\geq 0$ disjoint open disks removed. For $k=0$, the answer is positive by Brouwer's fixed point theorem. For $1\leq k\neq 2$, it's not difficult to see that the answer is negative: by arranging the disks in a symmetrical way, I can apply certain rotations to obtain self maps with no fixed point, e.g. if $k=4$, I can set one disk at the center, and position the other three in an equilateral triangle around the center; then a 120 degree rotation is a map from $A_4$ to itself with no fixed point. This sort of arrangement can always be done for $k\neq 2$. Note that 120 degree rotation of $A_4$ can be extended to a continuous map of $D^2$ to itself.
The tricky case is $k=2$, which is where I'm stuck. My thinking here is to consider any possible extension
$$\bar f:D^2\rightarrow D^2\ |\ \bar f|_{A_2}=f.$$
Since $\bar f$ is continuous, it must map each of the two missing disks to either itself or to the other. In the case where $\bar f$ maps one disk to the other, then no fixed points lie in either of the disks (since they are disjoint), hence, by Brouwer, there must be a fixed point of $\bar f$ lying in $A_2$, which will be a fixed point of $f$. In the case where $\bar f$ maps the two disks to themselves, then by applying Brouwer to the restriction of $\bar f$ to each disk, we have that $\bar f$ has at least two fixed points in $D^2$.
If I had holomorphicity to work with, then this would be enough, since a holomorphic map from $D^2$ to itself with two fixed points must be the identity. Am I on the right track, or is there some arrangement of the missing disks and a transformation that I'm just not seeing?
algebraic-topology fixed-point-theorems
TheMagicSnootTheMagicSnoot
$\begingroup$ You have to take a look at the Lefschetz Fixed Point Theorem. $\endgroup$ – Peter Saveliev Apr 25 '18 at 20:01
A disk with two holes is homeomorphic to a "pair of pants": a $2$-sphere with three discs removed. In a symmetric version where the centres of the discs lie on the equator and form the vertices of an equilateral triangle, and where the discs have equal radius, you see that a reflection about the plane going through the three centres of the discs, followed by an obvious $120$ rotation, will have no fixed point.
The south hemisphere is mapped to the north hemisphere, so the only possible fixed points are on the equator. But the equator is rotated by $120$ degrees, so there are no fixed points there.
Arnaud MortierArnaud Mortier
$\begingroup$ Oh wow that's cool. Thanks! $\endgroup$ – TheMagicSnoot Apr 24 '18 at 22:49
$\begingroup$ @TheMagicSnoot You're welcome :) $\endgroup$ – Arnaud Mortier Apr 24 '18 at 22:50
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A question on Brouwer Fixed point theorem for two homeomorph discs | CommonCrawl |
\begin{document}
\title{From Odometers to Circular Systems:\ A Global Structure Theorem}
\begin{abstract}{The main result of this paper is that two large collections of ergodic measure preserving systems, the \emph{Odometer Based} and the \emph{Circular Systems} have the same global structure with respect to joinings. The classes are canonically isomorphic by a continuous map that takes factor maps to factor maps, measure-isomorphisms to measure-isomorphisms, weakly mixing extensions to weakly mixing extensions and compact extensions to compact extensions. The first class includes all finite entropy ergodic transformations with an odometer factor. By results in \cite{prequel}, the second class contains all transformations realizable as diffeomorphisms using the untwisted Anosov-Katok method. An application of the main result will appear in a forthcoming paper that shows that the diffeomorphisms of the torus are inherently unclassifiable up to measure-isomorphism. Other consequences include the existence measure distal diffeomorphisms of arbitrary countable distal height.}
\end{abstract}
\tableofcontents
\section{Introduction}
The isomorphism problem in ergodic theory was formulated by von Neumann in 1932 in his pioneering paper \cite{vN}.
Simply put it asks to determine when two measure preserving actions are isomorphic, in the sense that there is
a measure isomorphism between the underlying measure space that intertwines the actions.
It has been solved completely only for some special classes of transformations.
Halmos and von Neumann \cite{HvN} used the unitary operators defined by Koopman to completely characterize ergodic measure preserving transformations with pure point spectrum, these transformations can be concretely realized (in a Borel way) as translations on compact groups. Another notable success was the use of the Kolmogorov entropy to distinguish between measure preserving systems. Ornstein's work showed that entropy completely classifies a large class of highly random systems, such as independent processes, mixing Markov chains and certain smooth systems such as geodesic flows on surfaces of negative curvature.
Closely related to the isomorphism problem is the study of structural properties of measure preserving systems. These including mixing properties and compactness. A famous example is the Furstenberg-Zimmer structure theorem for ergodic measure preserving transformations, which characterizes every ergodic transformation as an inverse limit system of compact extensions followed by a weakly mixing extension. This result is fundamental for studying recurrence properties of measure preserving systems and the related proofs of Szemeredi-type combinatorial theorems (\cite{FuBook}).
In this paper we present a new phenomenon, \emph{Global Structure Theory}. Most structure theorems in ergodic theory consider a single transformation \emph{in vitro}. The approach here is study whole, intact ecosystems of transformations with their inherent relationships.
Our main result shows that two large collections of measure preserving transformations have exactly the same structure with respect to factors and isomorphisms (and more generally, joinings). More concretely, define the \emph{odometer based} transformations to be those finite entropy transformations that contain a non-trivial odometer factor. Spectrally, this is equivalent to the associated unitary operator having infinitely many finite period eigenvalues. To each odometer, we can associate a class of symbolic systems, the \emph{circular systems}. In \cite{prequel}, it is shown that the circular systems coincide exactly with the ergodic transformations realizable as diffeomorphisms of the torus using the untwisted method of Approximation-by-Conjugacy, due to Anosov-Katok (\cite{AK_original}).
We can make two categories by taking the objects to be these two classes of systems and by taking morphisms to be factor maps (or more generally joinings) that preserve the underlying timing structure. The main theorem of this paper says that these two categories are isomorphic by a map that takes measure-isomorphisms to measure-isomorphisms, weakly mixing extensions to weakly mixing extensions and compact extensions to compact extensions. It follows that it takes distal towers to distal towers. Moreover the map preserves the simplex of non-atomic invariant measures, takes rank one transformations to rank one transformations and much more. (This will be discussed further in the forthcoming \cite{part4}.) In other words the global structure of these two categories is identical.
We can get more detail by considering systems based on a fixed odometer map and circular systems based on that odometer map and an arbitrary fast growing coefficient sequence. Doing so gives us collections of pairwise isomorphic categories that can be amalgamated to yield the statement above. The main theorem is framed in this more granular setting.
Our result might be a mere curiosity, were it not for an application which we now describe.
Foreshadowed by a remarkable early result by Feldman \cite{feldman}, in the late 1990's a different type of result began to appear: \emph{anti-classification} results that demonstrate in a rigorous way that classification is not possible. This type of theorem requires a precise definition of what a classification is. Informally a classification is a \emph{method} of determining isomorphism between transformations perhaps by computing (in a liberal sense) other invariants for which equivalence is easy to determine.
The key words here are \emph{method} and \emph{computing}. For negative theorems, the more liberal a notion one takes the stronger the theorem. One natural notion is the Borel/non-Borel distinction. Saying a set $X$ or function $f$ is Borel is a loose way of saying that membership in $X$ or the computation of $f$ can be done using a countable (possibly transfinite) protocol whose basic input is membership in open sets. Say that $X$ or $f$ is \emph{not} Borel is saying that determining membership in $X$ or computing $f$ cannot be done with any amount of countable resources.
In the context of classification problems, saying that an equivalence relation $E$ on a space $X$ is \emph{not} Borel is saying that there is no countable amount of information and no countable transfinite protocol for determining, for arbitrary $x,y\in X$ whether $xEy$. \emph{Any} such method must inherently use uncountable resources.\footnote{Many well known classification theorems have as immediate corollaries that the resulting equivalence relation is Borel. An example of this is the Spectral Theorem, which has a consequence that the relation of Unitary Conjugacy for normal operators is a Borel equivalence relation.}
In considering the isomorphism relation as a collection $\mathcal I$ of pairs $(S,T)$ of measure preserving transformations, Hjorth showed that $\mathcal I$ is not a Borel set. However the pairs of transformations he used to demonstrate this were inherently non-ergodic\footnote{The ergodic components of the pairs were rotations of the circle.}, leaving open the essential problem:
\begin{center} Is isomorphism of ergodic measure preserving transformations Borel? \end{center}
This question was answered by Foreman, Rudolph and Weiss in \cite{FRW}, where they gave a negative answer. This answer can be interpreted as saying that determining isomorphism between ergodic transformations is inaccessible to countable methods that use countable amounts of information.
In the same foundational paper from 1932 where von Neumann formulated the isomorphism problem he expressed the likelihood that any abstract measure preserving transformation is isomorphic to a continuous measure preserving transformation and perhaps even to a differentiable one. This brief remark eventually gave rise to one of the outstanding problems in smooth dynamics, namely:
\begin{center}
Does every ergodic MPT have a smooth model? \end{center}
By a smooth model is meant an isomorphic copy of the transformation which is given by smooth diffeomorphism of a compact manifold preserving a measure equivalent to the volume element. Soon after entropy was introduced, A. G. Kushnirenko showed that such a diffeomorphism must have finite entropy, and up to now this is the only restriction that is known.
This paper is the second in a series of papers whose original purpose was to show that the variety of ergodic transformations that have smooth models is rich enough so that the abstract isomorphism relation, when restricted to these smooth systems, is as complicated as it is in general. We show this to be the case even when restricting to diffeomorphisms of the 2-torus that preserve Lebesgue measure this is the case. In the third paper we will complete the proof of the following theorem:
\begin{theorem*}[Anti-classification of Diffeomorphisms]
If $M$ is either the torus $\mathbb T^2$, the disk $D$ or the annulus then the measure-isomorphism relation among pairs $(S,T)$ of measure preserving $C^\infty$-diffeomorphisms of $M$ is not a Borel set with respect to the $C^\infty$-topology. \end{theorem*}
It was natural for us to try to adapt our earlier work to establish this result. However we were faced at first with the following difficulty.
The transformations built in \cite{FRW} were based on odometers (in the sense that the Kronecker factor was an odometer).
It is a well known open problem whether it is possible to have
any smooth transformation on a compact manifold that has a non-trivial odometer factor. Thus proving the anti-classification theorem in the smooth context required constructing a different collection of hard-to-classify transformations and then showing that this collection could be realized smoothly. This is our application of the main result of this paper.
The paper (\cite{prequel}) constructed a new collection of systems, the \emph{Circular Systems}, which are defined as symbolic systems constructed using the \emph{Circular Operator}, a formal operation on words. The main result in \cite{prequel} has as a consequence that uniform circular systems can be realized as smooth models using the method developed by Anosov and Katok.
The primary theorem of this paper allows us to transfer the general isomorphism structure for odometer based systems to the isomorphism structure for circular systems, at least up to automorphisms of the underlying odometer or rotation. Namely there remains the issue of preserving the timing mechanism. In the forthcoming \cite{part3} it is shown how to construct odometers so that for the resulting circular systems, up to a small correction factor, all isomorphisms preserve the underlying timing structure. This allows us to conclude the proof of the anti-classification theorem for diffeomorphisms.
Here is a more concrete description of the results in the paper.
In the present paper we are concerned with the entire class $\mathcal{OB}$
of systems based on a fixed odometer and the relations between them. The odometer is
determined by a sequence of positive integers greater than one, $\langle k_n : n \in \mathbb{N}\rangle$. The the circular operator is determined by an additional
sequence of integers $\langle l_n: n \in \mathbb{N}\rangle$. For this paper, the sequence of $l_n$'s can be arbitrary subject to the requirement that
$\sum 1/l_n<\infty$. However for realizing circular systems as diffeomorphisms there is a fixed growth rate, determined by the size of the alphabet of the odometer based system and $\langle k_n:n\in{\mathbb N}\rangle$, that the sequence of $l_n$'s must eventually exceed.
We describe $\mathcal{OB}$ symbolically here, but show in a forthcoming paper that $\mathcal{OB}$ consists of representations of arbitrary ergodic systems with finite entropy
that have the specific odometer as a factor. In the language of ``cutting and stacking" constructions these are those constructions where no
spacers are introduced. We fix $\langle l_n:n\in{\mathbb N}\rangle$, and hence a sequence of circular operators.
Applying these to each of the elements of $\mathcal{OB}$ we obtain a second class,
$\mathcal{CB}$, of circular systems. This class consists of some of the extensions of
a fixed irrational rotation which is determined by the circular operator. As remarked above, for suitably chosen coefficient sequences, this class can be characterized as those transformations realizable as diffeomorphisms using the Anosov-Katok technique. We consider the two classes as categories
where the morphisms are graph joinings which are either the identity of the base or reverse it. These are called \emph{synchronous} and
\emph{anti-synchronous} joinings respectively. Our main theorem then takes the form:
\begin{theorem}\label{main in intro}
For a fixed circular coefficient sequence $\langle k_n, l_n: n\in{\mathbb N}\rangle$ the categories $\mathcal{OB}$ and $\mathcal{CB}$ are isomorphic by a functor $\mathcal F$ that takes synchronous joinings to synchronous joinings, anti-synchronous joinings to anti-synchronous joinings, isomorphisms to isomorphisms and weakly mixing extensions to weakly mixing extensions.\footnote{E. Glasner showed that the functor takes compact extensions to compact extensions.}
\end{theorem}
It is natural to extend the collections of morphisms of $\mathcal{OB}$ and $\mathcal{CB}$ to general synchronous and non-synchronous joinings. Because the ergodic joinings are not closed under composition, in extending Theorem \ref{main in intro} one is forced to consider at least \emph{some} non-ergodic joinings. At the end of the paper we discuss how to extend Theorem \ref{main in intro} to expanded categories that have as morphisms arbitrary synchronous and anti-synchronous joinings. This involve expanding our analysis of generic sequences to non-ergodic joinings. We also describe some detailed analysis of the combinatorics behind the isomorphism $\mathcal F$.
We have provided a detailed table of contents which enumerates the contents of the paper. Here is a brief summary. Much of the section following this one is standard, with the exception \S 2.6, which is exposes generic sequences for transformations and extends that notion to joinings. In \S3, the reader will find an explanation of our two categories and a proof that circular systems contain a canonical rotation factor. Section 4 is primarily concerned with defining a map $\natural$ that is a symbolic analogue of complex conjugation on the unit circle. In sections 5 and 6 the mapping $\mathcal F$ is defined on morphisms, while $\S{7}$ contains the proof of the main result. In $\S{8}$ there is a more detailed analysis of of the dynamical
properties of our mapping $\mathcal F$ which may prove useful in the future, and in the final section we collect some problems that are left open.
\section{Preliminaries}\label{preliminaries} This section establishes some of the conventions we follow in this paper. There are many sources of background information on this including any standard text or \cite{walters}, \cite{Peterson}. A small portion of the material in this section was presented in \cite{prequel}, but is repeated here in an attempt to be self-contained. The reader is referred to \cite{prequel} for any missing definitions.
\subsection{Measure Spaces}\label{abstract measure spaces} We will call separable non-atomic probability spaces \emph{measure spaces} and denote them $(X,\mathcal B, \mu)$ where $\mathcal B$ is the Boolean algebra of measurable subsets of $X$ and $\mu$ is a countably additive, non-atomic measure defined on $\mathcal B$.\footnote{We will occasionally make an exception to this by calling discrete probability measures on a finite set \emph{measures}; we hope that context makes the difference clear.} We will often identify two members of $\mathcal B$ that differ by a set of $\mu$-measure $0$ and seldom distinguish between $\mathcal B$ and the $\sigma$-algebra of classes of measurable sets modulo measure zero unless we are making a pointwise definition and need to claim it is well defined on equivalence classes.
We will frequently use without explicit mention the Maharam-von Neumann result that every standard measure space is isomorphic to $([0,1],\mathcal B,\lambda)$ where $\lambda$ is Lebesgue measure and $\mathcal B$ is the algebra of Lebesgue measurable sets.
If $(X, \mathcal B, \mu)$ and $(Y, {\mathcal C}, \nu)$ are measure spaces, an isomorphism between $X$ and $Y$ is a bijection $\phi:X\to Y$ such that $\phi$ is measure preserving and both $\phi$ and $\phi^{-1}$ are measurable. We will ignore sets of measure zero when discussing isomorphisms; i.e. we allow the domain and range of $\phi$ to be subsets of $X$ and $Y$ (resp.) of measure one. A measure preserving system is an object $\ensuremath{(X,\mathcal B,\mu,T)}$ where $T:X\to X$ is a measure isomorphism. A \emph{factor map} between two measure preserving systems $\ensuremath{(X,\mathcal B,\mu,T)}$ and $\ensuremath{(Y,{\mathcal C},\nu,S)}$ is a measurable, measure preserving function $\phi:X\to Y$ such that $S\circ\phi=\phi\circ T$. A factor map is an \emph{isomorphism} or \emph{conjugacy} between systems iff $\phi$ is a measure isomorphism. Following common practice, we will use the word \emph{conjugacy} interchangeably with \emph{isomorphism} in this context.
For a fixed measure space $(X,\mu)$ we can consider the collection of measure preserving transformations $T:X\to X$. These form a group that can be endowed with a Polish topology that has basic open sets described as follows. We fix a finite measurable partition $\ensuremath{\mathcal A}$ of $X$ and an $\epsilon>0$ and take as a neighborhood of $T$ \[\mathcal N(T,\ensuremath{\mathcal A},\epsilon)=_{def}\{S:\sum_{a\in \ensuremath{\mathcal A}}\mu(Ta\Delta Sa)<\epsilon\}.\]
Details about this topology can be found in many sources including \cite{halmos}, \cite{walters}.
\subsection{Joinings}\label{joinings}
We remind the readers of the definitions. Extensive treatments of joinings can be found in \cite{glasbook} or \cite{DansBook}. All of the definitions and basic results about joinings necessary for this paper occur in Chapter 6 of the latter reference.
\begin{definition}
A \emph{joining} between two measure preserving systems $\ensuremath{(X,\mathcal B,\mu,T)}$ and $\ensuremath{(Y,{\mathcal C},\nu,S)}$ is a measure $\rho$ on $X\times Y$ defined on the product $\sigma$-algebra $\mathcal B\otimes{\mathcal C}$ such that
\begin{enumerate}
\item $\rho$ is $T\times S$ invariant,
\item for each set $B\in \mathcal B$, $\rho(B\times Y)=\mu(B)$,
\item for each set $C\in {\mathcal C}$, $\rho(X\times C)=\nu(C)$.
\end{enumerate} \end{definition}
The graphs of factor maps provide natural examples of joinings. We characterize these with a definition.
\begin{definition} A joining $\rho$ is a \emph{graph joining} between $X$ and $Y$ if and only if for all $C\in {\mathcal C}$ and all $\epsilon>0$, there is a $B\in \mathcal B$ such that \[\rho((B\times Y)\Delta (X\times C))<\epsilon.\]
A joining $\rho$ between $\ensuremath{(X,\mathcal B,\mu,T)}$ and $\ensuremath{(Y,{\mathcal C},\nu,S)}$ is an \emph{invertible graph joining} if and only for all $B\in \mathcal B$ there is a $C\in {\mathcal C}$ such that \begin{equation}\label{graph capture} \rho((B\times Y)\Delta (X\times C))=0 \end{equation} and vice versa: for all $C\in {\mathcal C}$, there is a $B\in\mathcal B$ such that equation \ref{graph capture} holds. \end{definition}
Here are some standard facts (see \cite{glasbook}):
\begin{prop}\label{no proof} Let $\mathbb X=\ensuremath{(X,\mathcal B,\mu,T)}$ and $\mathbb Y=\ensuremath{(Y,{\mathcal C},\nu,S)}$. Then \begin{enumerate} \item There is a canonical one-to-one correspondence between the collection of graph joinings of $\mathbb X$ and $\mathbb Y$ and the collection of factor maps from $X$ to $Y$. A graph joining concentrates on the graph of the factor map. We can represent the graph joining corresponding to a measure preserving map $\phi:X\to Y$ by \[\rho_\phi=\int (\delta_x\times \delta_{\phi(x)}) d\mu(x).\] \item There is a canonical one-to-one correspondence between the collection of invertible graph joinings of $\mathbb X$ and $\mathbb Y$ and the collection of conjugacies between $\mathbb X$ and $\mathbb Y$.
\item \label{working graph} Suppose that $\mathcal B'\subseteq \mathcal B$ and ${\mathcal C}'\subseteq {\mathcal C}$ are Boolean algebras that generate $\mathcal B$ and ${\mathcal C}$ respectively as $\sigma$-algebras. Let $\rho$ be a joining of $\mathbb X$ with $\mathbb Y$ such that for all $\epsilon>0$ and all $C\in {\mathcal C}'$ there are $B_1, \dots B_n\in \mathcal B'$ such that we have $\rho(\bigcup_i(B_i\times Y)\Delta(X\times C))<\epsilon$, then $\rho$ is a graph joining.
\end{enumerate} \end{prop}
We note that perhaps a more proper term for an invertible graph joining is the earlier usage \emph{diagonal joining}. In view of the results of this section we will often be careless and say that $\rho$ \emph{is a factor map} or $\rho$ \emph{is a conjugacy}/\emph{isomorphism} to mean that $\rho $ is a graph joining or $\rho$ is an invertible graph joining.
To each joining $\rho$ of $\mathbb X$ and $\mathbb Y$ we can associate its adjoint $\rho^*$, the joining of $\mathbb Y$ with $\mathbb X$ defined for $B\in \mathcal B$ and $C\in {\mathcal C}$ as: \[\rho^*(C\times B)=\rho(B\times C).\] If $\rho$ is a graph joining corresponding to a factor map $\pi:X\to Y$, then $\rho^*$ concentrates on $\{(y,x):\pi(x)=y\}$.
The following is immediate: \begin{prop}\label{invertible symmetry} $\rho$ is an invertible graph joining if and only if both $\rho$ and $\rho^*$ are graph joinings. \end{prop} Thus we can apply Proposition \ref{no proof}, item \ref{working graph} to both $\rho$ and $\rho^*$ to get a criterion for being the joining associated with a conjugacy.
\bfni{A potential source of confusion.} Proposition \ref{no proof} allows us to identify graph joinings with factor maps and invertible graph joinings with conjugacies. These joinings are always \emph{ergodic} as joinings. However, there are non-ergodic conjugacies between ergodic measure preserving transformations. More explicitly: there are ergodic systems $(X,T)$ and $(X,S)$ and non-ergodic isomorphisms $\phi:(X,T)\to (X,S)$.\footnote{The second author has given examples of of isomorphic ergodic transformations where every conjugacy is non-ergodic.} The associated joining $\rho_\phi$ is, however, ergodic as a $T\times S$-invariant measure.
Let $(X,\mu), (Y,\nu)$ and $(Z,\tilde{\mu})$ be measure spaces and $\pi_X:X\to Y$ and $\pi_Z:Z\to Y$ be factor maps. We can define a canonical joining of $\mathbb X$ and $\mathbb Z$ that reflects the factor structure as follows. We let $\{\mu_y:y\in Y\}$ and $\{\tilde{\mu}_y:y\in Y\}$ be the disintegrations of $\mathbb X$ and $\mathbb Z$ over $\mathbb Y$ respectively. The \emph{relatively independent joining} of $\mathbb X$ and $\mathbb Z$ over $\mathbb Y$ is the joining $\rho$: \[\rho=\int (\mu_y\times \tilde{\mu}_y)d\nu(y). \] We will sometimes write this as $X\times_Y Z$.
We will be concerned about categories of measure preserving systems where the morphisms are joinings. For this we must describe the composition operation. Suppose we are given joinings $\rho_{XY}$ between $X$ and $Y$ and $\rho_{YZ}$ between $Y$ and $Z$. Then $(Y,\nu)$ is a common factor of both $(X\times Y, \rho_{XY})$ and $(Y\times Z,\rho_{YZ})$ and we can consider the relatively independent joining $\rho_{XY}\times_Y\rho_{YZ}$.
We define the \hypertarget{composition of joinings}{composition} of $\rho_{XY}$ and $\rho_{YZ}$ to be the projection of the relatively independent joining of $\rho_{XY}$ and $\rho_{YZ}$ to a measure on $X\times Z$. Formally, if $A\subseteq X\times Z$ and $\rho$ is the relatively independent joining, then: \[\rho_{XY}\circ\rho_{YZ}(A)=\rho(\{(x,y,z): x, z\in A\}).\]
\begin{ex}\label{for struct} Suppose that $\pi_0:X\to Y$ and $\pi_1:Y\to Z$ are factor maps. If $\rho_{XY}$ is the joining associated with $\pi_0$ and $\rho_{YZ}$ is the joining associated with $\pi_1$, then $(\rho^*_{YZ}\circ\rho^*_{XY})^*$ is the joining associated with the factor map $\pi_1\circ\pi_0:X\to Z$.\footnote{In the following, in the context of factor maps $\pi:X\to Y$ we will be sloppy about whether this is associated with a joining of $X$ with $Y$ or a joining of $Y$ with $X$.} \end{ex}
The following are standard facts (e.g. in \S 6.2 of \cite{glasbook}): \begin{prop}\label{laziness} \begin{enumerate} \item The operation of composition of joinings is associative: if $\rho_1, \rho_2$ and $\rho_3$ are joinings, then \[(\rho_1\circ\rho_2)\circ\rho_3=\rho_1\circ(\rho_2\circ\rho_3).\] \item Suppose that $\pi^X:X\to X'$ and $\pi^Z:Z\to Z'$ are factor maps Let $\rho_1$ and $\rho_2$ be joinings of $X,Y$ and $Y,Z$ respectively. Let $\rho_1^\pi$ be the projection of $\rho_1$ to a joining of $X'$ and $Y$ via $\pi^X\times id$\ and $\rho_2^\pi$ be defined similarly. Finally let $(\rho_1\circ\rho_2)^\pi$ be the projection of the composition of $\rho_1$ and $\rho_2$ to a joining of $X$ with $ Z$. Then: \[\rho_1^\pi\circ\rho_2^\pi=(\rho_1\circ \rho_2)^\pi.\] \end{enumerate} \end{prop}
\subsection{Symbolic Systems} \label{symbolic shifts} Let $\Sigma$ be a countable or finite alphabet endowed with the discrete topology. Then $\Sigma^\mathbb Z$ can be given the product topology, which makes it into a separable, totally disconnected space that is compact if $\Sigma$ is finite.
\bfni{Notation:} If $u=\langle \sigma_0, \dots \sigma_{n-1}\rangle\in \Sigma^{<\infty}$ is a finite sequence of elements of $\Sigma$, then we denote the cylinder set based at $k$ in $\Sigma^\mathbb Z$ by writing $\langle u\rangle_k$. If $k=0$ we abbreviate this and write $\langle u\rangle$. Explicitly: $\langle u\rangle_k=\{f\in \Sigma^\mathbb Z: f\ensuremath{\upharpoonright}[k,k+n)=u\}$. The collection of cylinder sets form a base for the product topology on $\Sigma^\mathbb Z$.
{ \bfni{Notation:} For a word $w\in \Sigma^{<{\mathbb N}}$ we will write $|w|$ for the length of $w$.} We will write $1_{\langle w\rangle}$ for the characteristic function of the interval $\langle w\rangle_0$ in $\Sigma^\mathbb Z$.
\noindent The shift map: \[sh:\Sigma^\mathbb Z\to \Sigma^\mathbb Z\] defined by setting $sh(f)(n)=f(n+1)$ is a homeomorphism. If $\mu$ is a shift invariant Borel measure then the resulting measure preserving system $(\Sigma^\mathbb Z, \mathcal B,\mu, sh)$ is called a \emph{symbolic system}. The closed support of $\mu$ is a shift invariant closed subset of $\Sigma^\mathbb Z$ called a \emph{symbolic shift} or \emph{sub-shift}.
Symbolic shifts are often described intrinsically by giving a collection of words that constitute a clopen basis for the support of an invariant measure. Fix a language $\Sigma$, and a sequence of collections of words $\langle\mathcal W_n:n\in{\mathbb N}\rangle$ with the properties that: \begin{enumerate} \item for each $n$ all of the words in $\mathcal W_n$ have the same length $q_n$, \item each $w\in\mathcal W_{n}$ occurs at least once as a subword of each $w'\in \mathcal W_{n+1}$, \item \label{not too much space} there is a summable sequence $\langle \epsilon_n:n\in{\mathbb N}\rangle$ of positive numbers such that for each $n$, every word $w\in \mathcal W_{n+1}$ can be uniquely parsed into segments \begin{equation}u_0w_0u_1w_1\dots w_lu_{l+1}\label{words and spacers} \end{equation}
such that each $w_i\in \mathcal W_n$, $u_i\in \Sigma^{<{\mathbb N}}$ and for this parsing \begin{equation*}
{\sum_i|u_i|\over q_{n+1}}<\epsilon_{n+1}. \end{equation*} \end{enumerate}
\noindent The segments $u_i$ in condition \ref{words and spacers} are called the \emph{spacer} or \emph{boundary} portions of $w$. \begin{definition}A sequence $\langle \mathcal W_n:n\in{\mathbb N}\rangle$ satisfying properties 1.)-3.) will be called a \emph{construction sequence}. \end{definition}
Associated with a construction sequence is a symbolic shift defined as follows. Let ${\mathbb K}$ be the collection of
$x\in \Sigma^\mathbb Z$ such that every finite contiguous subword of $x$ occurs inside some $w\in \mathcal W_n$. Then ${\mathbb K}$ is a
closed shift invariant subset of $\Sigma^\poZ$ that is compact if $\Sigma$ is finite.\footnote{ The symbolic shifts built from construction sequences coincide with transformations built by \emph{cut-and-stack} constructions.}
Formally, we have constructed a symbolic shift. To get a measure preserving system we find a shift invariant measure $\mu$ concentrating on ${\mathbb K}$ and write $({\mathbb K},\mu)$. In \cite{prequel} we define the notion of a \emph{uniform} construction sequence and show that the resulting ${\mathbb K}$ are uniquely ergodic.
We want to be able to unambiguously parse elements of ${\mathbb K}$. For this we will use construction sequences consisting of uniquely readable words. \begin{definition} Let $\Sigma$ be a language and $\mathcal W$ be a collection of finite words in $\Sigma$. Then $\mathcal W$ is \emph{uniquely readable} iff whenever $u, v, w\in \mathcal W$ and $uv=pws$ then either $p$ or $s$ is the empty word. \end{definition}
In our constructions we will restrict our measures to a natural set:
\begin{definition}\label{def of S} Suppose that $\langle \mathcal W_n:n\in{\mathbb N}\rangle$ is a construction sequence for a symbolic system ${\mathbb K}$ with each $\mathcal W_n$ uniquely readable. Let $S$ be the collection $x\in {\mathbb K}$ such that there are sequences of natural numbers $\langle a_m: m\in{\mathbb N}\rangle$, $\langle b_m: m\in{\mathbb N}\rangle$ going to infinity such that for all $m$ there is an $n, x\ensuremath{\upharpoonright} [-a_m, b_m)\in \mathcal W_n$. \end{definition}
\noindent Note that $S$ is a dense shift invariant $\mathcal G_\delta$ set.
The following lemma is routine:
\begin{lemma} Fix a construction sequence $\langle\mathcal W_n:n\in{\mathbb N}\rangle$ for a symbolic system ${\mathbb K}$ in a finite language. Then: \begin{enumerate} \item ${\mathbb K}$ is the smallest shift invariant closed subset of $\Sigma^\mathbb Z$ such that for all $n$, and $w\in\mathcal W_n$, ${\mathbb K}$ has non-empty intersection with the basic open interval $\langle w\rangle\subset \Sigma^\mathbb Z$.
\item Suppose that there is a unique invariant measure $\nu$ on $S\subseteq {\mathbb K}$, then $\nu$ is ergodic.
\end{enumerate} \end{lemma} {\par\noindent{$\vdash$\ \ \ }} Item 1 is clear from the definitions. If $X$ is a Polish space, $T:X\to X$ is a Borel automorphism and $D$ is a $T$-invariant Borel set with a unique $T$-invariant measure on $D$, then that measure must be ergodic. {\nopagebreak
$\dashv$ \par
}
Let $\langle \mathcal W_n:n\in{\mathbb N}\rangle$ be a uniquely readable construction sequence, and $s\in S$. By the unique readability, for each $n$ either $s(0)$ lies in a well-defined subword of $s$ belonging to $\mathcal W_n$ or in a spacer of a subword of $s$ belonging to some $\mathcal W_{n+k}$.
\begin{lemma}\label{principal blocks exist} Suppose that ${\mathbb K}$ is built from $\langle \mathcal W_n:n\in{\mathbb N}\rangle$ and $\nu$ is a shift invariant measure on ${\mathbb K}$ concentrating on $S$. Then for $\nu$-almost every $s$ there is an $N$ for all $n>N$, there are $a_n\le 0< b_n$ such that $s\ensuremath{\upharpoonright}[a_n, b_n)\in \mathcal W_n$. \end{lemma}
{\par\noindent{$\vdash$\ \ \ }} Let $B_{n}$ be the collection of $s\in S$ such that for some $a_{n}\le 0<b_{n}$, $s\ensuremath{\upharpoonright} [a_{n}, b_n)\in \mathcal W_n$ but $s(0)$ is in a boundary portion of $s\ensuremath{\upharpoonright}[a_n,b_n)$. By the Ergodic Theorem and clause 3.) of the definition of a construction sequence $\sum\nu(B_n)<\infty$.
It follows from the Borel-Cantelli Lemma that for almost all $s$ there is an $N$ such that for all $n\ge N$, $s\notin B_n$. Fix an $s\in S$ and such an $N$. From the definition of $S$ there are arbitrarily large $n^*>N$ and $a_{n^*}\le 0<b_{n^*}$ such that $s\ensuremath{\upharpoonright}[a_{n^*}, b_{n^*})\in \mathcal W_{n^*}$. Using backwards induction from $n^*$ to $N$ and the definition of $B_n$, this also holds for all $n\in [N, n^*)$. {\nopagebreak
$\dashv$ \par
}
\subsection{Locations}
By Lemma \ref{principal blocks exist} for $\nu$-almost all $x$ and for all large enough $n$ there is a unique $k$ with $0\le k<q_n$ such that $s\ensuremath{\upharpoonright}[-k, q_n-k)\in \mathcal W_n$.
\begin{definition}\label{def of rn} Let $s\in S$ and suppose that for some $0\le k<q_n, s\ensuremath{\upharpoonright}[-k,q_n-k)\in \mathcal W_n$. We define $r_n(s)$ to be the unique $k$ with with this property. We will call the interval $[-k, q_n-k)$ the \emph{principal $n$-block} of $s$, and $s\ensuremath{\upharpoonright} [-k, q_n-k)$ its \emph{principal $n$-subword}. The sequence of $r_n$'s will be called the \emph{location sequence of $s$}. \end{definition} We interpret $r_n(s)=k$ as saying that \emph{$s(0)$ is the $k^{th}$ symbol in the principal $n$-subword of $s$ containing $0$.} We can view the principal $n$-subword of $s$ as being located on an interval $I$ inside the principal $n+1$-subword. Counting from the beginning of the principal $n+1$-subword, the $r_{n+1}(s)$ position is located at the $r_n(s)$ position in $I$.
\begin{remark} \label{interval coherence}Suppose that $s\in S$ has a principal $n$-block for all $n\ge N$. Let $N\le n<m$. It follows immediately from the definitions that $r_n(s)$ and $r_m(s)$ are well defined and the $r_m(s)^{th}$ position of the principal $m$-block of $s$ is in the $r_n(s)^{th}$ position inside the principal $n$-block of $s$.
\end{remark}
The next lemma tells us that an element of $s$ is determined by knowing any tail of the sequence $\langle r_n(s):n\ge N\rangle$ together with a tail of the principal subwords of $s$.
\begin{lemma}\label{specifying elements} Suppose that $s, s'\in S$ and $\langle r_n(s):n\ge N\rangle=\langle r_n(s'):n\ge N\rangle$ and for all $n\ge N$, $s$ and $s'$ have the same principal $n$-subwords. Then $s=s'$. \end{lemma}
{\par\noindent{$\vdash$\ \ \ }} Since $s, s'\in S$ there are sequences $\langle a_n, a_n', b_n, b_n':n\ge N\rangle$ tending to infinity such that $s\ensuremath{\upharpoonright}[-a_n, b_n)\in \mathcal W_n$ and $s'\ensuremath{\upharpoonright}[a'_n, b'_n)\in \mathcal W_n$. Since $r_n(s)=r_n(s')$ we know that $a_n=a_n'$ and $b_n=b_n'$. Since $s$ and $s'$ have the same principal subwords, $s\ensuremath{\upharpoonright}[a_n, b_n)=s'\ensuremath{\upharpoonright}[a_n', b_n')$. The lemma follows.{\nopagebreak
$\dashv$ \par
} \begin{remark}We record some consequences of Lemma \ref{specifying elements}: \label{rebuilding} \begin{enumerate}
\item Suppose that we are given a sequence $\langle u_n:M\le n\rangle$ with $u_n\in\mathcal W_n$. If we specify which occurrence of $u_n$ in $u_{n+1}$ is the principal occurrence, and the distances of the principle occurrence to the beginning of $u_{n+1}$ go to infinity, then $\langle u_n:M\le n\rangle$ determines an $s\in S\subseteq {\mathbb K}$ completely up to a shift $k$ with $|k|\le q_M$.
\item A sequence $\langle r_n:N\le n\rangle$ and sequence of words $w_n\in \mathcal W_n$ comes from an infinite word $s\in S$ if both $r_n$ and $q_n-r_n$ go to infinity and that the $r_{n+1}$ position in $w_{n+1}$ is in the $r_n$ position in a subword of $w_{n+1}$ identical to $w_n$.
\emph{Caveat}: just because $\langle r_n:N\le n\rangle$ is the location sequence of some $s\in S$ and $\langle w_n:N\le n\rangle$ is the sequence of principal subwords of some $s'\in S$, it does not follow that there is an $x\in S$ with location sequence $\langle r_n:N\le n\rangle$ and sequence of subwords $\langle w_n:N\le n\rangle$.
\item If
$x, y\in S$ have the same principal $n$-subwords and $r_n(y)=r_n(x)+1$ for all large enough $n$, then $y=sh(x)$. \end{enumerate} \end{remark}
\subsection{A note on inverses of symbolic shifts}\label{note on inverses} We define operators we label $\rev{}$, and apply them in several contexts \begin{definition} If $x$ is in ${\mathbb K}$, we define the reverse of $x$ by setting $\rev{x}(k)=x(-k)$. For $A\subseteq {\mathbb K}$, define: \hypertarget{reverse}{ \[\rev{A}=\{\rev{x}:x\in A\}.\]} If $w$ is a word, we define $\rev{w}$ to be the reverse of $w$. If we are viewing $w$ as sitting on an interval, we take $\rev{w}$ to sit on the same interval. Similarly, if $\mathcal W$ is a collection of words, $\rev{\mathcal W}$ is the collection of reverses of the words in $\mathcal W$. \end{definition}
If $({\mathbb K}, sh)$ is an arbitrary symbolic shift then its inverse is $({\mathbb K}, sh^{-1})$. It will be convenient to have all of our shifts go in the same direction, thus: \begin{prop}\label{spinning} The map $\phi$ sending $x$ to $\rev{x}$ is a canonical isomorphism between $({\mathbb K}, sh{^{-1}})$ and $(\rev{{\mathbb K}},sh)$. \end{prop}
We will use the notation $\mathbb L^{-1}$ for the system $(\mathbb L,sh{^{-1}})$ and $\rev{\mathbb L}$ for the system $(\rev{\mathbb L},sh)$.
We can say more. For a fixed symbolic shift ${\mathbb K}$, the canonical isomorphism $\phi:\mathbb L^{-1}\to \rev{\mathbb L}$ gives rise to a canonical correspondence \[\rho \leftrightarrow \rho'\] between joinings $\rho$ of $({\mathbb K},sh)$ with $(\mathbb L,sh{^{-1}})$ and joinings $\rho'$ of $({\mathbb K},sh)$ with $(\rev{\mathbb L},sh)$.
We will also use the following remark. \begin{remark}\label{nothing remarkable} {Assume that there is a unique non-atomic measure on a shift invariant set $S\subseteq{\mathbb K}$. Then there is also a unique non-atomic shift invariant measure on
$\rev{S}$ and for this measure, which we denote $\nu{^{-1}}$, we have
$\nu(\langle w\rangle)=\nu{^{-1}}(\langle \rev{w}\rangle)$.} \end{remark}
\subsection{Generic points and sequences}
\label{sequences and points}
Let $T$ be a measure preserving transformation from $(X,\mu)$ to $(X,\mu)$, where $X$ is a compact metric space. Let $C(X)$ be the space of all real valued complex functions. Then a point $x\in X$ is \emph{generic} for $T$ if and only if for all $f\in C(X)$,
\begin{equation*}\lim_{N\to \infty}\left({1\over N}\right)\sum_0^{N-1} f(T^n(x))=\int_X f(x)d\mu(x). \end{equation*}
The Ergodic Theorem tells us that for a given $f$ and ergodic $T$ equation above holds for a set of $\mu$-measure one. Intersecting over a countable dense set of $f$ gives a set of $\mu$-measure one of generic points. For symbolic systems ${\mathbb K}\subseteq \Sigma^\mathbb Z$ we can describe generic points $x$ as being those $x$ such that the $\mu$-measure of all basic open intervals $\langle u\rangle_0$ is equal to the density of $k$ such that $u$ occurs in $x$ at $k$.
The symbolic systems we consider will be built from construction sequences and are characterized by the limiting properties of finite information. We now describe how this works in greater detail. A more complete discussion of this can be found in \cite{Benjy}.
Let $\mu$ be a shift invariant measure on a symbolic system ${\mathbb K}$ defined by a uniquely readable construction sequence $\langle\mathcal W_n:n\in{\mathbb N}\rangle$ in a finite language $\Sigma$. Assume that $q_n$ is the length of the words in
$\mathcal W_n$. By $\mu_m$ we will denote the discrete measure on the finite
set $\Sigma^m$ given by $\mu_m(u)=\mu(\langle u\rangle)$. By $\hat{\mu}_n(w)$ we will denote the discrete probability measure on $\mathcal W_n$ defined by \[\hat{\mu}_n(w)={\mu_{q_n}(\langle w\rangle)\over \sum_{w'\in\mathcal W_n} \mu_{q_n}(\langle w'\rangle)}. \]
Thus $\hat{\mu}_n(w)$ is the relative measure of $\langle w\rangle$ among all $\langle w'\rangle, w'\in \mathcal W_n$. The denominator is a normalizing constant to account for spacers at stages $m>n$ and for shifts of size less than $q_n$.
Explicitly, if $A_n=\{s\in {\mathbb K}: s(0)$ is the start of a word in $\mathcal W_n\}$, then the sets $\{sh^j(A_n)\}_{j=0}^{q_n-1}$ are disjoint and their union has a measure that tends to one as $n$ grows to infinity. The set $A_n$ is partitioned into $|\mathcal W_n|$ many sets by the words $w\in \mathcal W_n$ and $\hat{\mu}_n$ gives their relative size in $A_n$. Since the measure of an arbitrary finite cylinder set can be calculated along the individual columns represented by a fixed $w$, it is clear that the $\hat{\mu}_n(w)$ determine uniquely the measure $\mu$.
Using the unique readability of words in $\mathcal W_k$ a word $w$ in $\Sigma^{q_{k+l}}$ determines a unique sequence of words $w_j$ in $\mathcal W_k$ such that , \[w=u_0w_0u_1w_1\dots w_Ju_{J+1}.\] When $w\in \mathcal W_{k+l}$, each $u_j$ is in the region of spacers added in $\mathcal W_{k+l'}$, $l'\le l$. We will denote the \hypertarget{emptiest}{\emph{empirical distribution}} of $\mathcal W_k$-words in $w$ by EmpDist$_k(w)$. Formally:
\[\mbox{EmpDist}_k(w)(w')={|\{0\le j\le J: w_j=w'\}|\over J+1}, \ w'\in \mathcal W_k.\] Then $EmpDist$ extends to a measure on $\mathcal P(\mathcal W_k)$ in the obvious way.
To finitize the idea of a generic point in ${\mathbb K}$ we introduce the notion of a generic sequence of words.
\begin{definition}\label{ED} A sequence $\langle v_n\in\mathcal W_n:n\in{\mathbb N}\rangle$ is a \hypertarget{gen seq}{\emph{generic sequence of words}} if and only if for all $k$ and $\epsilon>0$ there is an $N$ for all $m,n>N$,
\[\|EmpDist_k(v_m)-EmpDist_k(v_n)\|_{var}<\epsilon.\] The sequence is generic for a measure $\mu$ if for all $k$:
\[\lim_{n\to \infty}\| \mbox{EmpDist}_k(v_n)-\hat{\mu}_k\|_{var}=0 \]
where $\|\ \|_{var}$ is the variation norm on probability distributions. \end{definition} It follows that if $\langle v_n:n\in{\mathbb N}\rangle$ is a generic sequence of words then it is generic for a unique measure $\mu$.
Even though Definition \ref{ED} involves only the measures $\hat{\mu}_k$ it is easy to see (using the Ergodic Theorem) that for any $u\in \Sigma^k$, if $\langle v_n:n\in{\mathbb N}\rangle$ is generic then the density of the occurrences of $u$ in the $v_n$ will converge to $\mu(\langle u\rangle)$.
We can summarize the exact relationship between the empirical distributions and the $\mu_{q_k}$ by saying that the empirical distribution is the proportion of occurrences of $w'\in \mathcal W_k$ among the $k$-words that appear in $v_n$, whereas $\mu_{q_k}$ is approximately the density of the locations of the start of $k$-words in $v_n$. Letting $u\in \mathcal W_k$, $d$ be the density of the positions where an occurrence of $u$ begins in $v_n$, and $d_s$ be the density of locations of letters in some spacer $u_i$ we see that these are related by: \begin{eqnarray*} d&=&\left({\mbox{EmpDist}(v_n)(u)\over q_k}\right)(1-d_s) \end{eqnarray*}
We record the following consequence of the Ergodic Theorem for future reference:
\begin{prop}\label{generic sequences exist for ergodic} Let ${\mathbb K}$ be an ergodic symbolic system with construction sequence $\langle \mathcal W_n:n\in{\mathbb N}\rangle$ and measure $\mu$. Then for any generic $s$ the sequence of principal subwords of s, $\langle w_n:n\in{\mathbb N}\rangle$, is generic for $\mu$. In particular, generic sequences for $\mu$ exist. \end{prop}
We will need a characterization of when a generic sequence of words $\langle w_n:n\in{\mathbb N}\rangle$ determines an ergodic measure. \begin{definition}\label{def of ergodic sequence} A sequence $\langle v_n:n\in{\mathbb N}\rangle$ with $v_n\in\mathcal W_n$ is an \hypertarget{ergodic sequence}{\emph{ergodic sequence}} if for any $k$ and $\epsilon>0$ there are $n_0>k$, and $m_0$ such that for all $m\ge m_0$, if \[v_m=u_0w_0u_1w_1u_2\dots u_Jw_Ju_{J+1}\]
is the parsing of $v_m$ into $\mathcal W_{n_0}$ words and spacers $u_i$ then there is a subset $I\subseteq \{0,1,2\dots J\}$ with $|I|/J>1-\epsilon$ and for all $j,j'\in I$ \begin{equation}
\|EmpDist_k(w_j)-EmpDist_k(w_{j'})\|_{var}<\epsilon.\label{erg seq equ} \end{equation} \end{definition} Notice that in the definition of an ergodic sequence $\langle v_n\rangle$ we are not assuming that it is a generic sequence for a measure. This follow easily (see Lemma \ref{ergodic sequences give ergodic measures}), but we have not made it part of the definition to emphasize its finitary nature. In the next lemma we use the fact that the language $\Sigma$ is finite.
\begin{lemma}\label{generic seqs are ergodic} Any generic sequence $\langle v_n:n\in{\mathbb N}\rangle$ for an ergodic measure $\mu$ is an ergodic sequence. \end{lemma} {\par\noindent{$\vdash$\ \ \ }} Suppose we are given $k$ and $\epsilon>0$. For all $\delta>0$ we can
apply the Ergodic Theorem to find an $N$ much bigger than $q_k$ and a set $B$ with $\mu(B)>1-\delta$ such that for all $s\in B$ and all $w\in \mathcal W_k$:
\[\left| {1\over N}\sum_{0}^{N-1} 1_{\langle w\rangle}(T^is)-\mu_{q_k}(\langle w\rangle)\right|<\delta.\] Fix a generic point $s$ for $\mu$. Let $I=\{i\ge 0: T^is\in B\}$, and define an infinite sequence of disjoint intervals of length $N$ that cover $I$ by inductively letting $i_0=min(I)$, and $i_{j+1}=min(\{i\in I: i\ge i_j+N\})$. We take the intervals to be the sequence \[ [i_0, i_0+N-1], [i_1, i_1+N-1], [i_2, i_2+N-1], \dots \] Notice that the complement of these intervals in $\mathbb Z^+$ has density less than $\delta$ since their union clearly covers $I$.
Though this is an infinite sequence of intervals, the fact our language is finite implies that only finitely many distinct words of length $N$ occur as subwords of $s$ on these intervals. For each such word $w^*$, the density of those $i$ in the domain of $w^*$ such that an occurrence of a $w\in \mathcal W_k$ starts at $i$ is within $\delta$ of $\mu_{q_k}(\langle w\rangle)$.\footnote{By taking $N\gg q_k$, we can account for negligible ``end effects" so that
$\left|{1\over N}\sum_{0}^{N-q_k-1} 1_{\langle w\rangle}(T^is)-\mu_{q_k}(\langle w\rangle)\right|<\delta$. We ignore end effects in the rest of the proof.}
Next take $n_0$ large enough that $N/q_{n_0}<\delta$, and parse $s$ into words from $\mathcal W_{n_0}$ and the sections of $s$ corresponding to spacers in words in $W_j$ for some $j\ge n_0+1$. By taking $n_0$ large enough we can take the density of locations in $s$ occurring in spacers to be arbitrarily small. Let $\delta'$ be this density.
The words from $\mathcal W_{n_0}$ have length much larger than $N$, and we can collect all those words $w\in \mathcal W_{n_0}$ that are $(1-\sqrt{\delta})$-covered by the $N$-intervals we chose above into a set $A\subseteq \mathcal W_{n_0}$.
The proportion of $s\ensuremath{\upharpoonright} \mathbb Z^+$ not covered by words in $A$ can be split into the spacer section and the portion inside words $w$ in $B=\mathcal W_{n_0}\setminus A$. For $w\in B$ the complement of the $N$-intervals has density at least $\sqrt{\delta}$. It follows that the density of sections of $s$ covered by elements of $B$ is less than $\sqrt{\delta}$.
Thus the fraction of $s$ not covered by words in $A$ is at most $\sqrt{\delta}+\delta'$. It is now clear that if $\delta, \delta'$ are chosen to be sufficiently small then \begin{equation}\label{whatever} \sum_{w\in A}\hat{\mu}_{n_0}(w)>1-\epsilon \end{equation} and all $w\in A$ will have the property that
\[\|\mbox{EmpDist}_k(w)-\hat{\mu}_k\|_{var}<\epsilon/2\] which implies inequality \ref{erg seq equ} for pairs of words in $A$. Using inequality \ref{whatever} and the fact that $\langle v_n\rangle$ is generic for $\mu$ gives an $m_0$ so that for all $m\ge m_0$ when $v_m$ is parsed into $n_0$ words a $(1-\epsilon)$-fraction will lie in $A$ and this concludes the proof. {\nopagebreak
$\dashv$ \par
}
We will also need the converse to Lemma \ref{generic seqs are ergodic}, namely that the limiting measure defined by an ergodic sequence is, in fact, ergodic.
\begin{lemma}\label{ergodic sequences give ergodic measures} An ergodic sequence is generic and the measure $\mu$ defined by an ergodic sequence $\langle v_n:n\in{\mathbb N}\rangle$ is ergodic. \end{lemma} {\par\noindent{$\vdash$\ \ \ }} Inequality \ref{erg seq equ} implies that for each $k$ and $w\in \mathcal W_k$, the limit of the density of occurrences of $w$ in $v_n$ exists as $n$ goes to infinity. It follows (since $\mathcal W_k$ is finite) that $\langle v_n:n\in{\mathbb N}\rangle$ is a generic sequence and hence it defines a unique measure $\mu$.
The ergodicity of $\mu$ is equivalent to the fact that the ergodic averages of all $L^2$ functions converge almost everywhere to a constant. Functions of the form $1_{\langle w\rangle}$ where $w\in \bigcup_n\mathcal W_n$ and their shifts linearly span a dense set in $L^2$ from which it easily follows that if $\mu$ were not ergodic there would be some $k$, and $w\in \mathcal W_k$ with $({1/N})\sum_0^{N-1}1_{\langle w\rangle}(T^ix)$ converging $\mu$-a.e. to a non-constant function. This means that there is a $\gamma>0$ and disjoint sets $B_0, B_1$ of positive measure in ${\mathbb K}$ such that for all large enough $N$ for all $x_0\in B_0, x_1\in B_1$ \begin{equation}\label{gaining separation}
\left|{1\over N}\sum_0^{N-1}1_{\langle w\rangle}(T^ix_0)-{1\over N}\sum_0^{N-1}1_{\langle w\rangle}(T^ix_1)\right|\ge \gamma. \end{equation}
Take $\epsilon$ small compared to $\gamma$ and $\mu(B_0), \mu(B_1)$. Find $n_0, m_0$ as in the definition of \hyperlink{ergodic sequence}{ergodic sequence} for this $k$ and $\epsilon$. Choose $N$ large enough that inequality \ref{gaining separation} holds and so that $q_{n_0}/N$ is negligible. Finally take $m\ge n_0$ so that $N/q_m$ is negligible.
The inequality \ref{gaining separation} depends only on the initial $(N+q_k)$-block of $x_0$ and $x_1$. Thus for large enough $m$ we can compute $\mu(B_0)$ and $\mu(B_1)$ by the empirical distributions of the $(N+q_k)$-blocks in $v_m$.
Since $N$ is large compared to $q_{n_0}$ the frequency of occurrence of $w$ in a block of length $N+q_k$ is determined by its frequencies in the words in $\mathcal W_{n_0}$ in the $n_0$-parsing of $v_m$. We now get a contradiction to inequality \ref{gaining separation}, since except for an $\epsilon$-fraction, these $w_{n_0}$-words have their $k$-words distributed very close to $\hat{\mu}_k(w)$.{\nopagebreak
$\dashv$ \par
}
If $S$ and $T$ are symbolic systems then a joining $\rho$ of $S$ and $T$ will be a symbolic system, but may not have well-defined construction sequence, even if $S$ and $T$ do.\footnote{We run into this problem when considering joinings of circular systems and their inverses that project to the $\natural$-map on the canonical factors; these notions are defined in future sections.} Accordingly we must generalize our definition of \emph{empirical distribution} to take into account the relative locations of words in typical $(s,t)\in {\mathbb K}\times \mathbb L$. We express this by shifting one of the basic open sets and considering words $(w, sh^s(v))$, which we view as starting at the locations $(0, s)$.
Let $\langle \mathcal W_n:n\in{\mathbb N}\rangle$ and $\langle\mathcal V_n:n\in{\mathbb N}\rangle$ be uniquely readable construction
sequences for ${\mathbb K}$ and $\mathbb L$ in the languages $\Sigma, \Lambda$ respectively. Assume for
simplicity that all words in $\mathcal W_n$ and $\mathcal V_n$ have the same length.
Let $n\le n'<n+l$. Then we can uniquely parse a word $w\in \mathcal W_{n+l}$ as
\[w=u_0w_0u_1w_1\dots w_Ju_{J+1}\] where each $w_j\in \mathcal W_n$ and each $u_j$ is in the region of spacers for words in $\mathcal W_{n+l'}$, $l'<l$. The similar statement holds for $v'_k\in \mathcal V_{n'}$, and $v\in \mathcal V_{n+l}$: \[v=u'_0v'_0u'_1v'_1\dots v'_Ku'_{K+1}.\] The definition must take into account the relative shifts of $w$ and $v$, the shifts of $(w_j, v_k)$ allow spacers to occur in different places and for the possibility that $J\ne K$.
Let $n\le n'< n+ l$ be natural numbers, $s, s'\in \mathbb Z$, and $(w', v')\in \mathcal W_n\times \mathcal V_{n'}$ and $(w,v)\in \mathcal W_{n+l}\times \mathcal V_{n+l}$. Write $w$ and $v$ in terms of $n$ and $n'$-words as above. For $s, s'$, define an \emph{occurrence} of $(w', sh^{s'}(v'))$ in $(w, sh^s(v))$ to be a $j\le J$ such that $w_j=w'$ and if $k$ is the location of $w_j$ in $w$, then $v'$ occurs at $k+s'$ in $sh^s(v)$. We note the bijection between occurrences of $(w', sh^{s'}(v'))$ in $(w, sh^s(v))$ and occurrences of $(v', sh^{-s'}(w'))$ in $(v, sh^{-s}(w))$.
In defining empirical distributions for joinings we generalize Definition \ref{ED}. The empirical distribution of a shifted pair is defined to be the proportion of times it occurs, relative to the proportion of times arbitrary pairs with the same shift occur. \begin{definition}\label{empdist for joinings} Fix $w,s,v$
Let $A$ be the collection
\begin{equation*}\{j:\mbox{for some }(w^*, v^*)\in \mathcal W_n\times \mathcal V_{n'}, (w^*, sh^{s'}(v^*))
\mbox{ occurs at j}\\
\mbox{ in }(w, sh^s(v))\}.
\end{equation*}
Assume that $A\ne \emptyset$. For $w'\in \mathcal W_n$ and $v'\in \mathcal V_{n'}$, we define:
\[EmpDist_{n,n', s'}(w,sh^{s}(v))(w',v')={|\{0\le j\le J:(w',sh^{s'}(v'))\mbox{ occurs at }
j\}|\over
|A|}.\]
\end{definition}
As before, $EmpDist_{n,n',s'}(w,sh^s(v))$ extends uniquely to a probability measure on $\mathcal P(\mathcal W_{n}\times \mathcal V_{n'})$.
Definition \ref{empdist for joinings} facilitates a notion of a generic sequence for a joining.
\begin{definition}\label{shifted genericity}
A sequence of $\langle (w_n,v_n,s_n)\in \mathcal W_n\times \mathcal V_n\times \mathbb Z:n\in{\mathbb N}\rangle$ is called \emph{generic} iff
\begin{enumerate}
\item $\sum {|s_n|\over q_n}<\infty$ and
\item for all $n,n', s'$ and $\epsilon>0$ there is an $N$ for all $m,m'>N$,
\[\|EmpDist_{n,n',s'}(w_m,sh^{s_{m}}(v_n))-EmpDist_{n,n',s'}(w_{m',}sh^{s_{m'}}(v_{m'}))\|_{var}<\epsilon.\]
\end{enumerate}
The definition of an \emph{ergodic sequence of pairs} is done analogously.
\end{definition}
It is easy to check that $\langle (w_n, v_n, s_n):n\in{\mathbb N}\rangle$ is generic/ergodic if and only if $\langle (v_n,w_n,-s_n):n\in{\mathbb N}\rangle$ is generic/ergodic. For ergodic joinings the analogues of Proposition \ref{generic sequences exist for ergodic}, and Lemmas \ref{generic seqs are ergodic} and \ref{ergodic sequences give ergodic measures} hold and are proved in exactly the same way.
We have given these definitions in the case of a product of two symbolic shifts, but they generalize immediately to products of three or more shifts. For example, to consider three shifts with construction sequences $\langle \mathcal U_n\rangle_n, \langle \mathcal V_n\rangle_n, \langle\mathcal W_n\rangle_n $, we would consider a sequence of the form: \[\langle (u_n, v_n, w_n, s_n, t_n):n\in{\mathbb N}\rangle,\] where the words belong to the respective construction sequences and the $s_n$'s and $t_n$'s give the shifts relative to the first coordinate.
We will be concerned with compositions of joinings, which involves products of three shifts. To prepare for this we need the notion of a conditional empirical distribution.
\begin{definition}\label{conditional love} Let $n, n'<n+l$. Given a fixed $w^*\in \mathcal W_{n'}$ and a pair $(w,v)\in \mathcal W_{n+l}\times \mathcal V_{n+l}$ and $(s,s')$ we define the \hypertarget{cond emp dist}{\emph{conditional empirical distribution}} to be:
\[\mbox{EmpDist}_{n,s'}((w,sh^s(v)|w^*)(v')=\]
\[{|\{0\le j\le J:(w^*,sh^{s'}(v'))\mbox{ occurs at }j\}|\over
|\{j\le J: \mbox{for some }v^*\in \mathcal V_n, (w^*, sh^{s'}(v^*)) \mbox{ occurs at }j\}|}
\] for $v'\in \mathcal W_n.$ \end{definition} Using the same ideas we can define the empirical distribution conditioned on a $v^*\in \mathcal V_k$ by looking at $(sh^{-s}(w),v)$ and counting occurrences of $(sh^{-s'}(w'),v^*)$ for the $w'\in \mathcal W_k$.
This definition generalizes to products of three or more systems. When working in three or more systems, there will be multiple $s$'s playing the role of $s'$ in Definition \ref{conditional love}. They will refer to the position of the sequences being counted, \emph{relative to the conditioning sequence}. So for example, if ${\mathbb K},\mathbb L,{\mathbb M}$ have construction sequences $\langle \mathcal U_n\rangle_n, \langle \mathcal V_n\rangle_n, \langle\mathcal W_n\rangle_n $ and $\langle (u_n, v_n, w_n, s_n, t_n):n\in{\mathbb N}\rangle$ is a generic sequence for a joining $\rho$ of ${\mathbb K}, \mathbb L$ and ${\mathbb M}$, then
\[EmpDists_{k,k',s,s'}(u_n,sh^{s_n}(v_n),sh^{t_n}(w_n)|v)\] counts pairs $(sh^s(u),sh^{s'}(w))$, where $(u,w)\in \mathcal U_k\times \mathcal W_{k'}$ have been shifted by $s$ and $s'$ \emph{relative to }$v$.
Let $\rho_1$ be a $T_1\times T_2$-invariant measure on $X\times Y$ and $\rho_2$ a $T_2\times T_3$-invariant measure on $Y\times Z$. Recall from Section \ref{joinings} that the composition of
$\rho_1$ and $\rho_2$ is defined to be projection of the relative independent joining of $\rho_1$ and $\rho_2$ over the common factor $Y$ to a measure on $X\times Z$. We now describe a method for detecting generic sequences for relatively independent joinings.
Suppose that systems $X$ and $Z$ have a common factor $Y$.
\begin{equation*} \begin{diagram} \node{(X,\mathcal B,\mu, T)}\arrow[1]{se}\node{}\node{(Z,{\ensuremath{\mathcal D}}, \tilde{\mu},\tilde{T} )}\arrow[1]{sw}\\ \node{}\node{(Y,{\mathcal C},\nu,S)} \end{diagram} \end{equation*}
Let $\rho=X\times_Y Z$ be the relatively independent joining of $X$ and $Y$. Let $\mu_y, \tilde{\mu}_y, \rho_y$ be the distintegrations of $\mu,\tilde{\mu}$ and $\rho$ respectively. Then the relatively independent joining $\rho$ is characterized by the fact that for $\nu$-a.e $y$, \begin{equation}\label{the world is disintegrating} \rho_y=\mu_y\times \tilde{\mu}_y. \end{equation}
Let $\langle \ensuremath{\mathcal A}_n, \tilde{\ensuremath{\mathcal A}}_n, \ensuremath{\mathcal A}'_n:n\in{\mathbb N}\rangle$ be sequences of refining partitions that generate $\mathcal B, {\ensuremath{\mathcal D}}$ and ${\mathcal C}$ respectively. Since the sequence of partitions $\ensuremath{\mathcal A}_n\times \tilde{\ensuremath{\mathcal A}}'_n$ generates $\mathcal B\otimes{\ensuremath{\mathcal D}}$, equation \ref{the world is disintegrating} is equivalent to the property that for all $A_k\in \ensuremath{\mathcal A}_k, \tilde{A}_k\in \tilde{\ensuremath{\mathcal A}}_k$ and $\nu$-a.e. $y$,
\begin{equation}
\mu_y(A_k)\times\tilde{\mu}_y(\tilde{A}_k)=\rho_y(A_k\times\tilde{A}_k)\label{produce}
\end{equation}
To finitize this we approximate $\mu_y(A_k)$ by $\mu(A_k|A'_m(y))$ for large $m$, where $A'_m(y)$ is the atom of $\ensuremath{\mathcal A}'_m$ to which $y$ belongs. We let $\mu_y(\ensuremath{\mathcal A}_k)$ be shorthand for the distribution $\langle \mu_y(A_k):A_k\in \ensuremath{\mathcal A}_k\rangle$, and
$\mu(\ensuremath{\mathcal A}_k|\ensuremath{\mathcal A}'_m)(y)$ stands for the conditional distribution
$\mu(A_k|A'_m(y)), A_k\in \ensuremath{\mathcal A}_k$. (We use similar notation in Lemma \ref{allow us to disintegrate} for the conditional distribution given by $\rho, \mu$ and $\tilde{\mu}$ on various partitions.)
By Martingale convergence,\footnote{See (e.g.) \cite{glasbook}, Theorem 14.26, page 261.} for $\epsilon>0$ and fixed $k$ if $m$ sufficiently large, then for $(1-\epsilon)$ proportion of the $y'$ in the same atom as $y$:
\[\|\mu_{y'}(\ensuremath{\mathcal A}_k)-\mu(\ensuremath{\mathcal A}_k|\ensuremath{\mathcal A}'_m)(y)\|_{var}<\epsilon\] but for a collection of $A'_m$ of whose union has $\nu$-measure less than $\epsilon$.
One can deal similarly with $\tilde{\mu}_n$ and $\rho_y$. We have shown:
\begin{lemma}\label{allow us to disintegrate}
In the notation above, $\rho$ is the relatively independent joining of $\mu$ and $\tilde{\mu}$ if and only if for all $k, \epsilon>0$, for all large enough $m$, there is a
collection of atoms $A_m\in \ensuremath{\mathcal A}'_m$ of total measure at least $1-\epsilon$ for which:
\begin{equation}\label{relind}
\| \rho(\ensuremath{\mathcal A}_k\times \tilde{\ensuremath{\mathcal A}_k}|A_m)-\mu(\ensuremath{\mathcal A}_k|A_m)\times {\tilde{\mu}}(\tilde{\ensuremath{\mathcal A}}_k|A_m)\|_{var}<\epsilon.
\end{equation} \end{lemma}
We now express Lemma \ref{allow us to disintegrate} in terms of sequences of finite words. Suppose that $\langle \mathcal U_n\rangle, \langle \mathcal V_n \rangle$, and $\langle \mathcal W_n\rangle$ are the uniquely readable construction sequences for $X$, $Y$ and $Z$.
\begin{prop}\label{rel ind join} Let $\langle (u_n, v_n, w_n, s_n, t_n)\in \mathcal U_n\times \mathcal V_n\times \mathcal W_n\times\mathbb Z^2:n\in{\mathbb N}\rangle$ be a sequence of words. Suppose that: \begin{enumerate}
\item $\langle (u_n, v_n, s_n)\rangle_n$ is generic for $\rho_1$,
\item $\langle (v_n, w_n, t_n)\rangle_n$ is generic for $\rho_2$.
\item \label{hyp 3} for all $\epsilon>0, k$ and $s^*$
for all sufficiently large $k'$ there is an $N$ and a set $G_{k'}\subset\mathcal V_{k'}$ and for each $v\in G_{k'}$ a set of indices $I_v\subseteq [0,q_{k'})$ that satisfies $|I_v|>(1-\epsilon)q_{k'}$ such that for all $n>N$:
\begin{enumerate}
\item \label{i} $\sum_{v\in G_{k'}}EmpDist(v_n)(v)>1-\epsilon$
and
\item \label{ii} for all $v\in G_{k'}$ and $s\in I_{v}$,
\begin{eqnarray}
\|EmpDist_{k,k,s,s+s^*}(u_n, sh^{s_n}(v_n),sh^{t_n}(w_n)|v)\ \ \ \ -\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ &\notag\\
EmpDist_{k,s}(u_n,sh^{s_n}(v_n)|v)*EmpDist_{k,s+s^*}(v_n,sh^{t_n-s_n}(w_n)|v)&\!\!\!\!\!\|_{var}\notag
\end{eqnarray}
is less than $\epsilon$.
\end{enumerate}
\end{enumerate} If $\rho$ is the relatively independent joining of $\rho_1, \rho_2$, then $\langle (u_n, v_n, w_n, s_n, t_n):n\in{\mathbb N}\rangle$ is a generic sequence for $\rho$.
\end{prop}
{\par\noindent{$\vdash$\ \ \ }} Observe that the hypothesis \ref{ii} implies a similar equation for any $k_1<k$ while the other parameters are fixed. Now use hypothesis \ref{i} with a summable sequence of $\epsilon$'s and we can conclude by the Borel-Cantelli lemma that for $\nu$-almost every $y\in Y$ for $k'$ sufficiently large, if $v_{k'}(y)$ is the principal $k'$-block of $y$ with location $r_{k'}$, then the inequality in \ref{ii} will hold for $s=r_{k'}$ and $v=v_{k'}(y)$.
Now by hypotheses 1 and 2, the single empirical distributions are converging to $(\rho_1)_y$ and $(\rho_2)_y$ respectively (where $(\rho_i)_y$ is the disintegration of $\rho_i$ over $y$).
It then follows by integration that the sequence of $(u_n, v_n, w_n, s_n, t_n)$'s is generic for a measure $\rho$ on $X\times Y \times Z$, which is the relatively independent joining.{\nopagebreak
$\dashv$ \par
}
\begin{remark} \label{grasping} It follows immediately from hypothesis \ref{hyp 3} of Proposition \ref{rel ind join} that if we are given a finite set $F$ of natural numbers then for all sufficiently large $k'$ we can find an $N$, $G_{k'}$ and $I_v$ as in hypothesis \ref{hyp 3} so that (a) and (b) hold simultaneously for all $s^*\in F$. \end{remark}
An immediate corollary of this is: \begin{corollary}\label{en fin} Suppose that $\langle (u_n, v_n, w_n, s_n, t_n):n\in{\mathbb N}\rangle$ satisfies the hypotheses of Proposition \ref{rel ind join}. Then $\langle (u_n, sh^{t_n}(w_n)):n\in{\mathbb N}\rangle$ is generic for $\rho_1\circ\rho_2$. \end{corollary}
There is a converse to Proposition 29, namely that a generic sequence for the relatively independent joining of two odometer based system satisfies the conditions 1-3 of the Proposition. The first two are immediate while the third simply expresses the fact that the generic sequence sequence is actually representing the relatively independent joining. For later use we record this as: \begin{lemma}\label{ill existe} Given joinings $\rho_1$ of $X\times Y$ and $\rho_2$ of $Y\times Z$ if $\langle (u_n, v_n, w_n, s_n, t_n):n\in{\mathbb N}\rangle$ is generic for the relatively independent joining $\rho$ then it satisfies the hypotheses of Proposition \ref{rel ind join}. \end{lemma}
\hypertarget{HvN}{\subsection{Unitary Operators}}\label{intro unitary} We will use spectral tools introduced by Koopman and studied by Halmos and von Neumann. We reprise the basic facts we will use. Readers unfamiliar with this material can find it in \cite{walters} or \cite{glasbook}. Let $\ensuremath{(X,\mathcal B,\mu,T)}$ and $\ensuremath{(Y,{\mathcal C},\nu,S)}$ be measure preserving systems.
If $T:X\to Y$ is a measure preserving transformation then $T$ induces a unitary isometry $U_T:L^2(Y)\to L^2(X)$ by setting \[U_T(f)=f\circ T.\] If $T$ is an isomorphism then $U_T$ is invertible. Moreover if $U:L^2(Y)\to L^2(X)$ is multiplicative on bounded functions then there is a measure preserving transformation $T:X\to Y$ such that $U=U_T$.
If $\pi:X\to Y$ is a factor map, then the map $f\mapsto f\circ\pi$ gives an injection of $L^2(Y)$ into $L^2(X)$, whose range is a closed $U_T$ invariant subspace. Conversely if $M\subseteq L^2(X)$ is a closed $U_T$ invariant subspace containing 1 that is closed under taking complex conjugates, truncation and multiplication by elements of $M\cap L^\infty(X)$, then there is a factor $Y\subseteq X$ such that $M=L^2(Y)$.
For the rest of this discussion assume that $T$ is ergodic. Then the eigenvalues of $U_T$ all have multiplicity one and form a subgroup $G_T\subseteq \mathbb T$. The group $G_T$ is an isomorphism invariant.
The collection of eigenfunctions generate a closed subspace of $L^2(X)$ corresponding to a factor $K$ of $X$. This factor is called the \emph{Kronecker factor}. If $H$ is any subgroup of $G_T$ then there is a further factor
$K_H$ of $K$ that is canonically determined by the eigenfunctions coming from eigenvalues in $H$.
Assume that $\phi$ is an isomorphism from $(X,T)$ to $(Y,S)$. Then $G_T=G_S$ and if $K^X_H, K^Y_H$ are the factors of $X$ and $Y$ determined by $H\subseteq G_T$ then $U_\phi$ determines an unique isomorphism between $K^X_H$ and $K^Y_H$.
It follows from this that if $\alpha\in \mathbb T$ is an eigenvalue of $U_T$ then there are factors of $X$ and $Y$ isomorphic to rotation $\ensuremath{\mathcal R}_\alpha$ of $\mathbb T$ by $\alpha$. Moreover there is a unique isomorphism $U_\phi^\pi:(\mathbb T, \mathcal B, \lambda, \ensuremath{\mathcal R}_\alpha)\to (\mathbb T, \mathcal B, \lambda, \ensuremath{\mathcal R}_\alpha)$ that intertwines $U_\phi$ and the projection maps of $X$ and $Y$ to $(\mathbb T, \mathcal B, \lambda, \ensuremath{\mathcal R}_\alpha)$.
The analogous statement holds for odometers. If $G_T$ consists of finite order eigenvalues and $\mathcal O$ is the corresponding odometer transformation, then there is a unique isomorphism $U_\phi^\pi:\mathcal O\to\mathcal O$ that intertwines $U_\phi$ and the projection maps of $X$ and $Y$ to $\mathcal O$.
\subsection{Stationary Codes and $\ensuremath{\bar{d}}$-Distance} In this section we briefly describe a standard idea, that of a \emph{stationary code} that we will use to understand the existence of factor maps and isomorphisms. We review some standard facts here. A reader unfamiliar with this material who wants to see proofs should see \cite{Shields}.
\begin{definition} Suppose that $\Sigma$ is a countable language. A \emph{code} of length $2N+1$ is a function $\Lambda:\Sigma^{[-N, N]}\to \Sigma$, where $[-N, N]$ is the interval of integers starting at $-N$ and ending at $N$.
Given a code $\Lambda$ and an $s\in \Sigma^\mathbb Z$ we define the \emph{stationary code} determined by $\Lambda$ to be $\bar{\Lambda}(s)$ where: \[\bar{\Lambda}(s)(k)=\Lambda(s\ensuremath{\upharpoonright}[k-N, k+N]).\] \end{definition} Let $(\Sigma^\mathbb Z, \mathcal B, \nu, sh)$ be a symbolic system. Suppose we have two codes $\Lambda_0$ and $\Lambda_1$ that are not necessarily of the same length. Define $D=\{s\in \Sigma^\mathbb Z:\overline{\Lambda}_0(s)(0)\ne \bar{\Lambda}_1(s)(0)\}$ and $d(\Lambda_0, \Lambda_1)=\nu(D)$. Then $d$ is a semi-metric on the collection of codes. The following is a consequence of the Borel-Cantelli lemma.
\begin{lemma}\label{cauchy codes}Let Suppose that $\langle \Lambda_i:i\in{\mathbb N}\rangle$ is a sequence of codes such that $\sum_{i}d(\Lambda_i, \Lambda_{i+1})<\infty$. Then there is a shift invariant Borel map $S:\Sigma^\mathbb Z\to \Sigma^\mathbb Z$ such that for $\nu$-almost all $s$, $\lim_{i\to \infty}\overline\Lambda_i(s)=S(s)$
\end{lemma}
A shift invariant Borel map $S:\Sigma^\mathbb Z\to \Sigma^\mathbb Z$, determines a factor $(\Sigma^\mathbb Z, \mathcal B, \mu, sh)$ of $(\Sigma^\mathbb Z, \mathcal B, \nu, sh)$ by setting $\mu=S^*\nu$ (i.e. $\mu(A)=\nu\circ S^{-1}(A)$). Hence a convergent sequence of stationary codes determines a factor of $(\Sigma^\mathbb Z, \mathcal B, \nu, sh)$.
Let $\Lambda_0$ and $\Lambda_1$ be codes. Define $\ensuremath{\bar{d}}(\bar\Lambda_0(s), \bar\Lambda_1(s))$ to be \[
\overline{\lim}_{n\to \infty}{|\{k\in [-N, N]: \bar\Lambda_0(s)(k)\ne \bar\Lambda_1(s)(k)\}|\over 2N+1} \] More generally we can define the $\ensuremath{\bar{d}}$ metric on $\Sigma^{[a,b]}$ by setting
\[\ensuremath{\bar{d}}_{[a,b]}(x,y)={|\{k\in [a, b): x(k)\ne y(k)\}|\over b-a}.\] For $x, y\in \Sigma^\mathbb Z$, we set \[\ensuremath{\bar{d}}(x,y)=\overline{\lim}_{N\to \infty}\ensuremath{\bar{d}}_{[-N,N]}(x\ensuremath{\upharpoonright}[-N,N], y\ensuremath{\upharpoonright}[-N,N]),\] provided this limit exists.
To compute distances between codes we will use the following application of the Ergodic Theorem. \begin{lemma}\label{computing code distances} Suppose that $(\Sigma^\mathbb Z, sh,\nu)$ is ergodic and that $\Lambda_0$ and $\Lambda_1$ be codes. Then for almost all $s\in S$: \[ d(\Lambda_0, \Lambda_1)=\ensuremath{\bar{d}}(\bar\Lambda_0(s), \bar\Lambda_1(s))\] \end{lemma}
We finish with a useful remark:
\begin{remark}\label{cheating on dbar}
If $w_1$ and $w_2$ are words in a language $\Sigma$ defined on an interval $I$ and $J\subset I$ with ${|J|\over |I|}\ge \delta$, then $\ensuremath{\bar{d}}_I(w_1, w_2)\ge \delta\ensuremath{\bar{d}}_J(w_1, w_2)$.
\end{remark}
\section{Odometer based and Circular Symbolic Systems}\label{odometer based and circular symbolic systems} Two types of symbolic shifts play central roles for the proofs of our main theorem. We dub them \emph{odometer based} and \emph{circular} systems. In this section we give some general facts about symbolic systems with uniquely readable construction sequences, define odometer and circular systems, and show that every circular system has a canonical rotation factor. \subsection{Odometer Based Systems} We recall the definition of an odometer transformation. Let $\langle k_n:n\in{\mathbb N}\rangle$ be a sequence of natural numbers greater than or equal to 2. Let \[O=\prod_{n=0}^\infty \mathbb Z/k_n\mathbb Z\] be the $\langle k_n\rangle$-adic integers. Then $O$ naturally has a compact abelian group structure and hence carries a Haar measure $\mu$. {We make $O$ into a measure preserving system $\mathcal O$ by defining $T:O\to O$ to be addition by 1 in the $\langle k_n\rangle$-adic integers. Concretely, this is the map that ``adds one to $\mathbb Z/k_0\mathbb Z$ and carries right".} Then $T$ is an invertible transformation that preserves the Haar measure $\mu$ on $\mathcal O$. Let $K_n=k_0*k_1*k_2\dots k_{n-1}$.
The following results are standard: \begin{lemma} \label{odometer basics}Let $\mathcal O$ be an odometer system. Then: \begin{enumerate} \item $\mathcal O$ is ergodic. \item The map $x\mapsto -x$ is an isomorphism between $(O, \mathcal B, \mu, T)$ and $(O, \mathcal B, \mu, T^{-1})$. \item {Odometer maps are transformations with discrete spectrum and the eigenvalues of the associated linear operator are the $K_n^{th}$ roots of unity ($n>0$).} \end{enumerate} \end{lemma}
Any natural number $a$ can be uniquely written as: \[a=a_0+a_1k_0+a_2(k_0k_1)+ \dots +a_j(k_0k_1k_2\dots k_{j-1})\] for some sequence of natural numbers $a_0, a_1, \dots a_j$ with $0\le a_j<k_j$.
\begin{lemma}\label{specifying elements of the odometer} Suppose that $\langle r_n:n\in{\mathbb N}\rangle$ is a sequence of natural numbers with $0\le r_n<k_0k_1\dots k_{n-1}$ and $r_n\equiv r_{n+1} \mod(K_n)$. Then there is a unique element $x\in O$ such that $r_n=x(0)+x(1)k_0+\dots x(n)(k_0k_1\dots k_{n-1})$ for each $n$.
\end{lemma}
We now define the collection of symbolic systems that have odometer maps as their timing mechanism. This timing mechanism can be used to parse typical elements of the symbolic system.
\begin{definition}Let $\langle \mathcal W_n:n\in{\mathbb N}\rangle$ be a uniquely readable construction sequence with the properties that {$\mathcal W_0=\Sigma$} and for all $n, \mathcal W_{n+1}\subseteq (\mathcal W_n)^{k_n}$ for some $k_n$. The associated symbolic system will be called an \emph{odometer based system}. \end{definition} Thus odometer based systems are those built from construction sequences $\langle \mathcal W_n:n\in{\mathbb N}\rangle$ such that the words in $\mathcal W_{n+1}$ are concatenations of words in $\mathcal W_n$ of a fixed length $k_n$. The words in $\mathcal W_{n}$ all have length $K_n$ and the words $u_i$ in equation \ref{words and spacers} are all the empty words.
Equivalently, an odometer based transformation is one that can be built by a cut-and-stack construction using no spacers. An easy consequence of the definition is that for odometer based systems ${\mathbb K}$, for all $s\in{\mathbb K}$ and for all $n\in{\mathbb N}$, $r_n(s)$ exists.
\begin{prop} \label{kiss your S goodbye} Let ${\mathbb K}$ be an odometer based system and suppose that $\nu$ is a shift invariant measure. Then $\nu$ concentrates on $S$. \end{prop} {\par\noindent{$\vdash$\ \ \ }} Let $B={\mathbb K}\setminus S$. Then $B$ is shift invariant. Suppose that $\nu$ gives $B$ positive measure. For $s\in B$ let $a_n(s)\le 0\le b_n(s)$ be the left and right endpoints of the principal $n$-block of $s$. Then for all $s\in B$ there is an $N\in{\mathbb N}$ such that: \begin{enumerate} \item for all $n, -N\le a_n$ or \item for all $n, b_n\le N$. \end{enumerate} We assume that $\nu$ gives the collection $B^*$ of $s$ such that there is an $N\in {\mathbb N}$ for all $n, -N\le a_n$ positive measure, the other case is similar.
Define $f:B^*\to {\mathbb N}$ by setting $f(s)=$ least $N$ satisfying item 1. Then $f$ is a Borel function. Let $B_n=f^{-1}(n)$. Then the $B_n$'s are disjoint, $B^*=\bigcup_{n\ge 0}B_n$ and $sh^{-1}(B_n)=B_{n+1}$. Hence for all $n, m, \nu(B_n)=\nu(B_m)$, a contradiction. {\nopagebreak
$\dashv$ \par
}
The next lemma justifies our terminology.
\begin{lemma}\label{odometer factor}
Let ${\mathbb K}$ be an odometer based system with each $\mathcal W_{n+1}\subseteq (\mathcal W_n)^{k_n}$. Then there is a canonical factor map
\begin{equation*}
\pi:S\to \mathcal O,
\end{equation*}
where $\mathcal O$ is the odometer system determined by $\langle k_n:n\in{\mathbb N}\rangle$.
\end{lemma} {\par\noindent{$\vdash$\ \ \ }} For each $s\in S$, we know that for all $n, r_n(s)$ is defined and both $r_n$ and $k_n-r_n$ go to infinity. By Lemma \ref{specifying elements of the odometer}, the sequence $\langle r_n(s):n\in{\mathbb N}\rangle$ defines a unique element $\pi(s)$ in $\mathcal O$. It is easily checked that $\pi$ intertwines $sh$ and $T$.{\nopagebreak
$\dashv$ \par
}
In the forthcoming paper \cite{part4} we show a strong converse to this result: if $T$ has finite entropy and an odometer factor then $T$ can be presented by an odometer based system.
Heuristically, the odometer transformation $\mathcal O$ parses the sequences $s$ in $S\subseteq {\mathbb K}$ by indicating where the words constituting $s$ begin and end. Shifting $s$ by one unit shifts this parsing by one. We can understand elements of $s$ as being an element of the odometer with words in $\mathcal W_n$ filled in inductively.
We will use the following remark about the canonical factor of the inverse of an odometer based system. \begin{remark} If $\pi:\mathbb L\to\mathcal O$ is the canonical factor map, then the function $\pi:\mathbb L\to O$ is also factor map from $(\mathbb L,sh{^{-1}})$ to $\mathcal O{^{-1}}$ (i.e. $O$ with the operation ``$-1$"). If $\langle\mathcal W_n:n\in{\mathbb N}\rangle$ is the construction sequence for $\mathbb L$, then $\langle \rev{\mathcal W_n}:n\in{\mathbb N}\rangle$ is a construction sequence for $\rev{\mathbb L}$. If $\phi:\mathbb L{^{-1}} \to \rev{\mathbb L}$ is the canonical isomorphism given by Proposition \ref{spinning}, then Lemma \ref{odometer basics} tells us that the projection of $\phi$ to a map $\phi^\pi:\mathcal O\to \mathcal O$ is given by $x\mapsto -x$. \end{remark} From this remark we immediately see:
\begin{lemma}\label{joining correspondence} Let $\rho\leftrightarrow \rho'$ be the canonical correspondence between joinings of $({\mathbb K},sh)$ and $(\mathbb L, sh^{-1})$ and joinings of $({\mathbb K}, sh)$ and $(\rev{\mathbb L},sh)$ given after Proposition \ref{spinning}. Then the joining $\rho$ concentrates on the set of pairs $(s,t)$ such that $\pi^{\mathbb K}(t)=-\pi^\mathbb L(s)$ if and only if $\rho'$ concentrates on the collection of $(s,t)$ such that $\pi^{\mathbb K}(s)=\pi^{\mathbb L^{-1}}(t)$. \end{lemma}
\subsection{Circular systems} \label{circular systems 1} We now define and discuss circular systems. The paper \cite{prequel} showed that the circular systems give symbolic characterizations of the smooth diffeomorphisms defined by the Anosov-Katok method of conjugacies. The construction sequences of circular systems have quite specific combinatorial properties that will be important to our understanding of the Anosov-Katok systems and their centralizers in the third paper in this series.
We call these systems \emph{circular} because they are closely tied to the behavior of rotations by a convergent sequence of rationals $\alpha_n=p_n/q_n$. The rational rotation by $p/q$ permutes the $1/q$ intervals of the circle cyclically along a sequence determined by some numbers $j_i=_{def}p^{-1}i$ (mod $q$): the interval $[i/q, (i+1)/q)$ is the $j_i^{th}$ interval in the sequence.\footnote{We assume that $p$ and $q$ are relatively prime and the exponent $-1$ is the multiplicative inverse of $p$ mod $q$.} The operation ${\mathcal C}$ which we are about to describe models the relationship between rotations by $p/q$ and $p'/q'$ when $q'$ is very close to $q$.
Let $k, l, p, q$ be positive natural numbers with $p<q$ relatively prime. Set \begin{equation}j_i\equiv_q(p)^{-1}i \label{j sub i} \end{equation} with $j_i<q$. It is easy to verify that: \begin{equation}\label{reverse numerology} q-j_i=j_{q-i} \end{equation}
Let $\Sigma$ be a non-empty set. We define an operation ${\mathcal C}$, which depends on $p,q$, an integer $l>1$, and on sequences $w_0, \dots w_{k-1}$ of words in a language $\Sigma\cup \{b, e\}$ by setting:\footnote{We use $\prod$ for repeated concatenation of words.}
\begin{equation}{\mathcal C}(w_0,w_1,w_2,\dots w_{k-1})=\prod_{i=0}^{q-1}\prod_{j=0}^{k-1}(b^{q-j_i}w_j^{l-1}e^{j_i}). \label{definition of C} \end{equation} To start our construction we frequently take $p_0=0$ and $q_0=1$. In this case we adopt the convention that $j_0=0$. Hence \begin{eqnarray*} {\mathcal C}(w_0,w_1, \dots w_{k-1})&=&\prod_{j<k}b^qw_j^{l-1}\\ &=&\prod_{j<k}bw^{l-1}. \end{eqnarray*}
{\begin{remark}\label{word length} We remark: \begin{itemize} \item Suppose that each $w_i$ has length $q$, then the length of ${\mathcal C}(w_0, w_1, \dots w_{k-1})$ is $klq^2$. \item Every occurrence of an $e$ in ${\mathcal C}(w_0, \dots w_{k-1})$ has an occurrence of a $b$ to the left of it. If $p\ne 0$ then every occurrence of a $b$ has an $e$ to the right of it. \item {Suppose that $n<m$ and $b$ occurs at position $n$ in ${\mathcal C}(w_0, w_1,\dots w_{k-1})$ and $e$ occurs at $m$ and neither occurrence is in a $w_i$. Then there must be some $w_i$ occurring between $n$ and $m$.} \end{itemize} \end{remark}}
The ${\mathcal C}$ operator automatically creates uniquely readable words, as the next lemma shows, however we will need a stronger unique readability assumption for our definition of circular systems. \begin{lemma}\label{unique readability} Suppose that $\Sigma$ is a language, $b, e\notin \Sigma$, $0<p<q$ and that $u_0, \dots $ $u_{k-1}$, $v_0, \dots v_{k-1}$ and $w_0 \dots w_{k-1},$ are words in the language $\Sigma\cup \{b, e\}$ of some fixed length $q<l/2$. Let \begin{eqnarray*}u&=&{\mathcal C}(u_0, u_1, \dots u_{k-1}) \\ v&=&{\mathcal C}(v_0, v_1, \dots v_{k-1})\\ w&=&{\mathcal C}(w_0, w_1, \dots w_{k-1}). \end{eqnarray*} Suppose that $uv$ is written as $pws$ where $p$ and $s$ are words in $\Sigma\cup \{b, e\}$. Then either $p$ is the empty word and $u=w, v=s$ or $s$ is the empty word and $u=p, v=w$. \end{lemma} {\par\noindent{$\vdash$\ \ \ }} The map $i\mapsto j_i$ is one-to-one. Hence each location in the word of length $klq^2$ is uniquely determined by the lengths of nearby sequences of $b$'s and $e$'s.{\nopagebreak
$\dashv$ \par
}
In fact something stronger is true: if $\sigma\in \Sigma$ occurs at place $m$ in $w$ then $m$ is uniquely determined by the knowing the $w_0, w_1, \dots w_{k-1}$ and the $kq^l/2 +1$ letters on either side of $\sigma$.
We now describe how to use the ${\mathcal C}$ operation to build a collection of symbolic shifts. Our systems will be defined using a sequence of natural number parameters $ k_n$ and $l_n$ that is fundamental to the version of the Anosov-Katok construction presented in \cite{katoksbook}.
Fix an arbitrary sequence of positive natural numbers $\langle k_n:n\in{\mathbb N}\rangle$. Let {$\langle l_n:n\in{\mathbb N}\rangle$} be an increasing sequence of natural numbers such that $\sum_n 1/l_n<\infty$. From the $k_n$ and $l_n$ we define sequences of numbers: $\langle p_n, q_n, \alpha_n:n\in{\mathbb N}\rangle$.
We begin by letting $p_0=0$ and $q_0=1$ and inductively set \begin{eqnarray}\label{qn} {q_{n+1}}={k_n} l_n{q_n}^2 \end{eqnarray} {(thus $q_1=k_0l_0$)} and take \begin{equation}\label{pn} p_{n+1}=p_nq_nk_nl_n+1. \end{equation} Then clearly $p_{n+1}$ is relatively prime to $q_{n+1}$.\footnote{{$p_n$ and $q_n$ being relatively prime for $n\ge 1$, allows us to define the integer $j_i$ in equation \ref{j sub i}. For $q_0=1$, $\mathbb Z/q_0\mathbb Z$ has one element, $[0]$, so we set $p_0{^{-1}}=p_0=0$.}}
\begin{definition} A sequence of integers $\langle k_n, l_n:n\in{\mathbb N}\rangle\rangle$ such that $k_n\ge 2$, $\sum 1/l_n<\infty$ will be called a \hypertarget{circ coef}{\emph{circular coefficient sequence}}. \end{definition}
Let $\Sigma$ be a non-empty finite or countable alphabet. We will construct the systems we study by building collections of words $\mathcal W_n$ in the alphabet $\Sigma\cup \{b, e\}$ by induction as follows:
\begin{itemize}
\item Fix a \hyperlink{circ coef}{circular coefficient sequence} $\langle k_n, l_n:n\in{\mathbb N}\rangle\rangle$.
\item Set $\mathcal W_0=\Sigma$.
\item Having built $\mathcal W_n$ we choose a set $P_{n+1}\subseteq (\mathcal W_{n})^{k_n}$ and form $\mathcal W_{n+1}$ by taking all words of the form ${\mathcal C}(w_0,w_1\dots w_{k_n-1})$ with $(w_0, \dots w_{k_n-1})\in P_{n+1}$.\footnote{Passing from $\mathcal W_n$ to $\mathcal W_{n+1}$ we use ${\mathcal C}$ with parameters $k=k_n, l=l_n, p=p_n$ and $q=q_n$ and take $j_i=(p_n)^{-1}i$ modulo $q_n$. By Remark \ref{word length}, the length of each of the words in $\mathcal W_{n+1}$ is $q_{n+1}$.}
\end{itemize} We will call the elements of $P_{n+1}$ \hypertarget{pwords}{\emph{prewords}.}
\hypertarget{strong unique readability}{\bfni{Strong Unique Readability Assumption:}}
Let $n\in {\mathbb N}$, and view $\mathcal W_n$ as a collection $\Lambda_n$ of letters. Then each element of $P_{n+1}$ can be viewed as a word with letters in $\Lambda_n$. We assume that in the alphabet $\Lambda_n$, each $P_{n+1}$ is uniquely readable.
\begin{definition}A construction sequence $\langle \mathcal W_n:n\in{\mathbb N}\rangle$ will be called \emph{circular} if
it is built in this manner using the ${\mathcal C}$-operators, a circular coefficient sequence and each
$P_{n+1}$ satisfies the strong unique readability assumption. \end{definition}
It follows from Lemma \ref{unique readability} that each $\mathcal W_n$ in a circular construction sequence is uniquely readable. \begin{definition}\label{circular definition} A symbolic shift ${\mathbb K}$ built from a circular construction sequence will be called a \emph{circular system}. \end{definition} For emphasis we will often write circular construction sequences as $\langle \mathcal W_n^c:n\in{\mathbb N}\rangle$ and the associated circular shift ${\mathbb K}^c$. We sometimes write $w^c$ to emphasize that a word is a circular word.
We will need to analyze the words constructed by ${\mathcal C}$ in detail. We start by describing the boundary and interior portions of the words.
\begin{definition} Suppose that $w={\mathcal C}(w_0,w_1,\dots w_{k-1})$. Then $w$ consists of blocks of $w_i$ repeated $l-1$ times, together with some $b$'s and $e$'s that are not in the $w_i$'s. The \emph{interior} of $w$ is the portion of $w$ in the $w_i$'s. {The remainder of $w$ consists of blocks of the form $b^{q-j_i}$ and $e^{j_i}$. We call this portion the \emph{boundary} of $w$.}
In a block of the form $w_j^{l-1}$ the first and last occurrences of $w_j$ will be called the \emph{boundary} occurrences of the block $w_j^{l-1}$. The other occurrences will be the \emph{interior} occurrences. \end{definition} {While the boundary consists of sections of $w$ made up of $b$'s and $e$'s, not all $b$'s and $e$'s occurring in $w$ are in the boundary, as they may be part of a power $w_i^{l-1}$.}
The boundary of $w$ constitutes a small portion of the word:
\begin{lemma}\label{stabilization of names 1} The proportion of the word $w$ written in equation \ref{definition of C} that belongs to its boundary is $1/l$. Moreover the proportion of the word that is within $q$ letters of boundary of $w$ is $3/l$. \end{lemma}
The next lemma was proved in \cite{prequel} (Lemma 20).
\begin{lemma}\label{dealing with S} Let ${\mathbb K}^c$ be a circular system and
$\nu$ be a shift invariant measure on ${\mathbb K}^c$. Then the following are equivalent:
{\begin{enumerate}
\item $\nu$ has no atoms.
\item $\nu$ concentrates on the collection of $s\in {\mathbb K}^c$ such that $\{i:s(i)\notin \{b, e\}\}$ is unbounded in both $\mathbb Z^-$ and $\mathbb Z^+$.
\item $\nu$ concentrates on $S$.
\end{enumerate}}
\end{lemma}
\begin{remark} Let ${\mathbb K}^c$ be a circular system. \begin{enumerate} \item There are only two invariant atomic measures, one concentrates on the constant ``$b$" sequence, the other on the constant ``$e$" sequence. \item for ${\mathbb K}^c$, Lemma \ref{principal blocks exist} can be strengthened to say that for all $s\in S$ for all large enough $n$, the principal $n$-block of $s$ exists. \item The symbolic shift ${\mathbb K}^c$ has zero topological entropy. \end{enumerate} \end{remark} {\par\noindent{$\vdash$\ \ \ }}
A direct inspection reveals that the only periodic points in ${\mathbb K}^c$ are the two fixed points constant ``$b$" and ``$e$".
The second item follows because if $s$ has a principal $n$-block at $[a_n, b_n)$ then it has a principal $n+1$-block at some $[a_{n+1}, a_{n+1}+q_{n+1})$ for an $a_{n+1}$ with $|a_{n+1}|\le |a_{n}|+(q_{n+1}-q_n)$.
The fact that the topological entropy of ${\mathbb K}^c$ is zero follows easily from the fact that the $l_n$ tend to infinity.
\subsection{The structure of the words}\label{word sections} The words used to form circular transformations have quite specific combinatorial properties. We begin with an important definition for our understanding of rotations; the three \emph{subscales} at stage $n+1$. Fix a sequence $\langle \mathcal W^c_n:n\in {\mathbb N}\rangle$ defining a circular system. { Using equation \ref{definition of C} we define the \emph{subscales} of a word $w^*\in \mathcal W_{n+1}$:} \begin{enumerate} \item[]{\bf Subscale 0} is the scale of the individual powers of $w_j\in \mathcal W^c_n$ of the form $w_j^{l-1}$; we call each such occurrence of a $w_j^{l-1}$ a \emph{0-subsection} \item[]{\bf Subscale 1} is the scale of each term in the product $\prod_{j=0}^{k-1}(b^{q-j_i}w_j^{l-1}e^{j_i})$ that has the form $(b^{q-j_i}w_j^{l-1}e^{j_i})$; We call these terms \emph{1-subsections}. \item[]{\bf Subscale 2} is the scale of each term of $\prod_{i=0}^{q-1}\prod_{j=0}^{k-1}(b^{q-j_i}w_j^{l-1}e^{j_i})$ that has the form $\prod_{j=0}^{k-1}(b^{q-j_i}w_j^{l-1}e^{j_i})$; We call these terms \emph{2-subsections}. \end{enumerate} \begin{center} {\bf Summary} \end{center}
\begin{center}
\begin{tabular}{| l | c |}
\hline
{\bf Whole Word:} &$\prod_{i=0}^{q-1}\prod_{j=0}^{k-1}(b^{q-j_i}w_j^{l-1}e^{j_i})$\\ \hline {\bf 2-subsection:} &$ \prod_{j=0}^{k-1}(b^{q-j_i}w_j^{l-1}e^{j_i})$ \\ \hline {\bf 1-subsection:} &$(b^{q-j_i}w_j^{l-1}e^{j_i})$\\ \hline {\bf 0-subsection:} & $w_j^{l-1}$\\
\hline
\end{tabular} \end{center}
By contrast we will discuss \emph{$n$-subwords} of a word $w$. These will be subwords that lie in
$\mathcal W^c_n$, the $n^{th}$ stage of the construction sequence. We will use \emph{$n$-block} to mean the location of the $n$-subword.
\subsection{The canonical circle factor ${\mathcal K}$} \label{iso to rotation}
We now define a canonical factor ${\mathcal K}$ of a circular system and show that this factor is isomorphic to a rotation of the circle by $\alpha$, where $\alpha$
is the limit of $\alpha_n= {p_n\over q_n}$ as $n$ goes to infinity.
\begin{definition}\label{first appearance of circle factor} Let $\langle k_n, l_n:n\in{\mathbb N}\rangle\rangle$ be a circular coefficient sequence. Let $\Sigma_0=\{*\}$. We define a circular construction sequence such that each $\mathcal W^c_n$ has a unique element as follows: \begin{enumerate} \item $\mathcal W_0=\{*\}$ and \item If $\mathcal W^c_n=\{w_n\}$ then $\mathcal W^c_{n+1}=\{{\mathcal C}(w_n, w_n, \dots w_n)\}$.
\end{enumerate} Let ${\mathcal K}$ be the resulting circular system. \end{definition} It is easy to check that ${\mathcal K}$ has unique ergodic non-atomic measure, since every $w_n$ occurs exactly $k_n(l_n-1)q_n$ many times in $w_{n+1}$.
Let ${\mathbb K}^c$ be an arbitrary circular system with coefficients $\langle k_n, l_n \rangle$. Then ${\mathbb K}^c$ has a canonical factor isomorphic to ${\mathcal K}$. This canonical factor plays a role for circular systems analogous to the role odometer transformations play for odometer based systems.
To see ${\mathcal K}$ is a factor of ${\mathbb K}^c$, we define the following function:
\begin{equation}\label{definition of factor map} \pi(x)(i) = \left\{ \begin{array}{ll}x(i) & \mbox{if $x(i)\in \{b, e\}$} \\ * &\mbox{otherwise} \end{array}\right. \end{equation}
We record the following easy lemma that justifies the terminology of Definition \ref{first appearance of circle factor}:
\begin{lemma}\label{canonical rotation factor} Let $\pi$ be defined by equation \ref{definition of factor map}. Then: \begin{enumerate} \item $\pi:{\mathbb K}^c\to {\mathcal K}$ is a Lipshitz map, \item $\pi(sh^{\pm 1}(x))=sh^{\pm 1}(\pi(x))$ and thus \item \label{last item} $\pi$ is a factor map of ${\mathbb K}^c$ to ${\mathcal K}$ and $({\mathbb K}^c)^{-1}$ to ${\mathcal K}^{-1}$ \end{enumerate} \end{lemma} A variant of item \ref{last item} is also true: $\pi$ can be interpreted as a function from $\rev{{\mathbb K}^c}$ to $\rev{{\mathcal K}}$. With this interpretation $\pi$ is also a factor map. We will call ${\mathcal K}$ the \emph{circle factor} of any circular system with construction coefficients $\langle k_n, l_n:n\in {\mathbb N}\rangle$.
Fix a circular coefficient sequence $\langle k_n, l_n:n\in{\mathbb N}\rangle$, and let
$\mathcal K$ and $\langle \mathcal W_n^\alpha:n\in{\mathbb N}\rangle$ be given in definition \ref{first appearance of circle factor}.
Let $\alpha_n=p_n/q_n$ and $\alpha=\lim \alpha_n$.
If $s\in S$, from $r_n(s)$ we can determine the locations of the beginnings and ends of the words $w^\alpha_n$ that contain $s(0)$. Since $|\mathcal W_n^\alpha|=1$ for all $n$, for all $s\in S$ the sequence $\langle r_n(s):n\in\mathbb N\rangle$ uniquely determines $s$.
\begin{theorem}\label{rank one description} Let $\nu$ be the unique non-atomic shift invariant measure on ${\mathcal K}$. Then \[({\mathcal K}, \mathcal B, \nu, sh)\cong (S^1, \mathcal D, \lambda, \ensuremath{\mathcal R}_\alpha)\]
where $\ensuremath{\mathcal R}_\alpha$ is the rotation of the unit circle by $\alpha$ and $\mathcal{B, D}$ are the $\sigma$-algebras of measurable sets.\end{theorem} {\par\noindent{$\vdash$\ \ \ }} A more involved geometric proof of this fact is given in \cite{prequel}. Here present a simple algebraic proof. As usual we identify the unit circle $S^1$ with $[0,1)$ and use additive notation for the group operations.
By Lemma \ref{principal blocks exist}, the collection $S'$ of $s\in S$ such that for all large enough $n$, the principal $n$-block of $s$ exists, has measure one. We define a map ${\phi}_0:S'\to {[0,1)}$ by a limiting process. For $s$ such that $r_n(s)$ exists, we let {\begin{equation}\notag {\rho}_n(s)={p\over q_n} \end{equation}} iff
\begin{equation}
p\equiv p_nr_n(s)\mod{q_n}\notag
\end{equation} \begin{claim}\label{approximation}
If $r_n$ is defined, then $|{\rho}_{n+1}(s)-{\rho}_n(s)|<2/q_{n}$. \end{claim} {\par\noindent{$\vdash$\ \ \ }} From equation \ref{definition of C}, we see that the position of $s(0)$ in an $n+1$-block is determined by the parameters $i\in [0,q_n-1), j\in [0, k_n-1), l^*\in[0,l-1]$ and $r_n$, which determine its location among the 2-subsections, 1-subsections, 0-subsections and inside the $n$-words $w_n$ respectively. Explicitly: \[r_{n+1}(s)=i(k_nl_nq_n)+j(l_nq_n)+(q_n-j_i)+l^*q_n+r_n(s),\] where $r_n(s)$ is the position of $s(0)$ in its principal $w_n$-word.
From the definition of $\rho_{n+1}$, and working \emph{mod 1}: \begin{eqnarray*} \rho_{n+1}&=&r_{n+1}(s)\left({p_{n+1}\over q_{n+1}}\right)\\ &=&r_{n+1}(s)\left({p_n\over q_n}+{1\over q_{n+1}}\right) \end{eqnarray*} Expanding this, using our formula for $r_{n+1}(s)$ and the fact that all but two terms of $r_{n+1}(s)$ are divisible by $q_n$, we get: \begin{eqnarray} \rho_{n+1}&=&\left(-j_i\left({p_n\over q_n}\right)+ r_n(s)\left({p_n\over q_n}\right)\right) +\left({i\over q_n} + \delta\right)\label{alg exer} \end{eqnarray} where \[\delta={j\over k_nq_n}+{1\over k_nl_nq_n}+{l^*\over k_nl_nq_n}+{r_n(s)-j_i\over k_nl_nq_n^2}.\] The first and third terms of equation \ref{alg exer} cancel, thus: \[\rho_{n+1}=\rho_n+\delta.\] Since $\delta<2/q_n$, the claim follows.{\nopagebreak
$\dashv$ \par
}
Since the sequence $1/q_n$ is summable, for almost all $s, \langle {\rho}_n(s):n\in \omega\rangle$ is Cauchy. We define \begin{equation} \phi_0(s)=\lim_n{\rho}_n(s).\notag \end{equation} It is easy to check that $\phi_0$ is one-to-one. By the unique ergodicity of the rotation $\ensuremath{\mathcal R}_\alpha$, Theorem \ref{rank one description} will be proved when we establish: \begin{claim} The map $\phi_0:S\to [0,1)$ satisfies: \begin{equation} \phi_0(sh(s))=\ensuremath{\mathcal R}_\alpha(\phi_0(s)).\notag \end{equation} In particular, if $\nu$ is the unique invariant measure on $S$ \begin{equation} \label{rotation iso} ({\mathcal K}, {\mathcal C}, \nu, sh)\cong ([0,1), \mathcal B, \lambda, \ensuremath{\mathcal R}_\alpha).\notag \end{equation} \end{claim}
{\par\noindent{$\vdash$\ \ \ }} Suppose that $r_n(s)$ and $r_n(sh(s))$ both exist. Then $r_n(sh(s))=r_n(s)+1$. If follows that $\rho_n(sh(s))=\rho_n(s)+p_n/q_n$. Taking limits we see that $\phi_0(sh(s))=\phi_0(s)+\lim_n \alpha_n=\phi_0(s)+\alpha$. {\nopagebreak
$\dashv$ \par
} \noindent This finishes the proof of Theorem \ref{rank one description}.{\nopagebreak
$\dashv$ \par
}
{\subsection{Kronecker Factors} Both odometer transformations and irrational rotations of the circle are ergodic discrete spectrum transformations. Because the odometer transformation based on $\langle k_n:n\in{\mathbb N}\rangle$ is a factor of any odometer based system $T$ and the rotation $\ensuremath{\mathcal R}_\alpha$ is a factor of any circular system $S$, both are factors of the respective Kronecker factors of $T$ or $S$. In general it is not the whole Kronecker factor in either case. }
We make the following lemma explicit in the case of odometer based transformations. In the case of systems with a circle factor the exactly analogous results hold. \begin{lemma}\label{isos induce isos of odometer factor} Let $({\mathbb K},\mathcal B,\mu, T)$ and $(\mathbb L, {\mathcal C}, \nu, S)$ be measure preserving systems. Suppose that ${\mathbb K}$ has an odometer factor $\mathcal O$ and that $\phi:{\mathbb K}\to \mathbb L$ is an isomorphism. Then there is a unique odometer factor $\mathcal O^*$ of $\mathbb L$ with an isomorphism $\phi^\pi:\mathcal O\to\mathcal O^*$ such that the following diagram commutes:
\[ \begin{diagram}
\node{{\mathbb K}}\arrow{e,r}{\phi}\arrow{s,r}{\pi^{\mathbb K}}\node{\mathbb L}\arrow{s,r}{\pi^\mathbb L}\\
\node{\mathcal O} \arrow{e,t}{\phi^\pi}\node{\mathcal O^*} \end{diagram} \]
If each finite order eigenvalue of $\mathbb L$ has multiplicity 1 (e.g. if $\mathbb L$ is ergodic), then $\mathcal O^*$ is the unique odometer factor of $\mathbb L$ isomorphic to $\mathcal O$. \end{lemma} {\par\noindent{$\vdash$\ \ \ }} Since the unitary operator $U_\phi:L^2({\mathbb K})\to L^2({\mathbb K})$ takes eigenfunctions to eigenfunctions, we know that $U_\phi$ takes the subspaces of $L^2({\mathbb K})$ corresponding to $\mathcal O$ to a subspace of $L^2(\mathbb L)$ corresponding to an isomorphic copy of $\mathcal O$. The lemma follows.{\nopagebreak
$\dashv$ \par
} An immediate corollary of Lemma \ref{isos induce isos of odometer factor} is that if ${\mathbb K}$ and $\mathbb L$ are ergodic odometer based systems over the same odometer $\mathcal O$, with projections $\pi_K$ and $\pi_L$, then $\phi^\pi$ is an isomorphism between the canonical odometer factors.
We record the following consequences for later use; \begin{prop}\label{preservation jazz} Suppose that ${\mathbb K}$ and $\mathbb L$ are both ergodic odometer based systems with coefficients $\langle k_n:n\in{\mathbb N}\rangle$. Then any isomorphism $\phi:{\mathbb K}\to\mathbb L$ takes the canonical odometer factor $\mathcal O^{\mathbb K}$ of ${\mathbb K}$ to the canonical odometer factor $\mathcal O^\mathbb L$ of $\mathbb L$.
Similarly if ${\mathbb K}^c$ and $\mathbb L^c$ are both ergodic circular systems with the same coefficient sequences $\langle k_n, l_n:n\in{\mathbb N}\rangle\rangle$, then any isomorphism between ${\mathbb K}^c$ and $\mathbb L^c$ takes the canonical rotation ${\mathcal K}^{\mathbb K}$ to the canonical rotation factor ${\mathcal K}^\mathbb L$ \end{prop} {\par\noindent{$\vdash$\ \ \ }} In the first case there is a unique factor of ${\mathbb K}$ and $\mathbb L$ corresponding to the eigenvalues of $\mathcal O^{\mathbb K}$ and $\mathcal O^\mathbb L$. Any isomorphism must preserve the factor corresponding to these eigenvalues. The same argument works for ${\mathcal K}$, as it is isomorphic to the rotation by $\alpha=\lim_n p_n/q_n$. {\nopagebreak
$\dashv$ \par
}
\subsection{Uniform Systems} In \cite{prequel} it is established that the \emph{strongly uniform} circular systems with sufficiently fast growing $\langle l_n:n\in{\mathbb N}\rangle$, are realizable as measure preserving diffeomorphisms of the torus. Strongly uniform systems are those for which each word in $\mathcal W_n$ occurs the same number of times in each word in $\mathcal W_{n+1}$. These systems carry unique non-atomic invariant measures, simplifying much of what we do later in this paper. For example the correspondence between the measures $\nu$ on uniform odometer systems ${\mathbb K}$ and $\nu^c$ on their uniform circular system counterparts ${\mathbb K}^c$ given in equation \ref{nu vs nuc}, is automatic.
In the forthcoming \cite{part4} we show that arbitrary (i.e. non-uniform) circular systems are realizable as measure preserving diffeomorphisms of the torus, provided that the measures of the words in $\mathcal W_n$ go to zero.
\section{Details of Circular Systems}\label{CS 2} This section examines the circular systems defined in section \ref{circular systems 1} in more detail. Initially we are given a circular coefficient sequence $\langle k_n, l_n:n\in{\mathbb N}\rangle\rangle$ and $\langle q_n:n\in{\mathbb N}\rangle$ where $q_n$ satisfies the inductive definition in equation \ref{qn}. When $n$ is fixed, we again let $j_i=(p_n)^{-1}i$ modulo $q_n$ and $0\le j_i<q_n$. {Without significant loss of generality it is convenient to assume that $\sum 1/q_n<1/10$.}
To understand joinings of circular systems we will be comparing generic elements $(s,t)$ of circular
${\mathbb K}^c$ and $\mathbb L^c$, and their parsings into subwords. We will use the following terminology:
\begin{definition}
Let $u, v$ be finite sequences of elements of $\Sigma\cup \{b, e\}$ having length $q$. Given intervals $I$
and $J$ in $\mathbb Z$ of length $q$ we can view $u$ and $v$ as functions having domain $I$ and $J$
respectively. We will say that $u$ is \emph{shifted by $k$} relative to $v$ iff $I$ is the shift of the interval
$J$ by $k$. We say that $u$ is the \emph{$k$-shift} of $v$ iff $u$ and $v$ are the same words and $I$ is
the shift of the interval $j$ by $k$.
\end{definition}
\subsection{Understanding the words} \label{understanding the words}
We elaborate on the descriptions given in Section \ref{word sections}. Our first combinatorial lemma is the following:
\begin{lemma}\label{gap calculation}Let $w={\mathcal C}(w_0, \dots w_{k_n-1})$ for some $n$ and $q=q_n, k=k_n, l=l_n$. View $w$ as a word in the alphabet $\Sigma\cup\{b, e\}$ lying on the interval of integers $[0, klq^2)$. \begin{enumerate} \item If $m_0$ and $m_1$ are the locations of the beginnings of $0$-subsections in the same 2-subsection, then $m_0\equiv_qm_1$. \item If $m_0$ and $m_1$ are such that $m_0$ is the location of the beginning of a $0$-subsection occurring in a $2$-subsection $\prod_{j=0}^{k-1}(b^{q-j_i}w_j^{l-1}e^{j_i})$ and $m_1$ at the i beginning of a $0$-subsection occurring in the next 2-subsection $\prod_{j=0}^{k-1}(b^{q-j_{i+1}}w_j^{l-1}e^{j_{i+1}})$ then $m_1-m_0\equiv_q -j_1$. \end{enumerate} \end{lemma} {\par\noindent{$\vdash$\ \ \ }} To see the first point, the indices of the beginnings of $0$-subsections in the same $2$-subsection differ by multiples of $q$ coming from powers of a $w_j$ and intervals of $w$ of the form $b^{q-j_i}e^{j_i}$.
To see the second point, let $u$ and $v$ be consecutive $2$-subsections. In view of the first point it suffices to consider the last $0$-subsection of $u$ and the first $0$-subsection of $v$. But these sit on either side of an interval of the form $e^{j_i}b^{q-j_{i+1}}$. Since $j_i+q-j_{i+1}\equiv_q (p)^{-1}i-p^{-1}(i+1)\equiv_q-p^{-1}\equiv_q-j_1$, we see that $m_0-m_1\equiv_q q+j_i+q-j_{i+1}\equiv_q-j_1$. {\nopagebreak
$\dashv$ \par
}
Assume that $u\in \mathcal W_{n+1}$ and $v\in \mathcal W_{n+1}\cup\rev{\mathcal W_{n+1}}$ and $v$ is shifted with respect to $u$.
On the overlap of $u$ and $v$, the 2-subsections of $u$ split each 2-subsection of $v$ into either one or two pieces. Since all of the 2-subsections in both words have the same length, the number of pieces in the splitting and the size of each piece is constant across the overlap except perhaps at the two ends of the overlap. If $u$ splits a 2-subsection of $v$ into two pieces, then we call the left piece of the pair the even piece and the right piece the odd piece.
If $v$ is shifted only slightly, it can happen that either the even piece or the odd piece does not contain a $1$-subsection. In this case we will say that split is \emph{trivial on the left} or \emph{trivial on the right}
\begin{lemma}\label{numerology lemma} Suppose that the $2$-subsections of $u$ divide the $2$-subsections of $v$ into two non-trivial pieces. Then \begin{enumerate} \item the boundary portion of $u$ occurring between each consecutive pair of 2-subsections of $u$ completely overlaps at most one $0$-subsection of $v$
\item there are two numbers $s$ and $t$ such that the positions of the $0$-subsections of $v$ in even pieces are shifted relative to the $0$-subsections of $u$ by $s$ and the positions of the $0$-subsections of $v$ in odd pieces are shifted relative to the $0$ subwords of $u$ by $t$. Moreover $s\equiv_q t -j_1$. \end{enumerate} \end{lemma}
{\par\noindent{$\vdash$\ \ \ }} This follows easily from Lemma \ref{gap calculation}{\nopagebreak
$\dashv$ \par
} In the case where the split is trivial we get Lemma \ref{numerology lemma} with just one coefficient, $s$ or $t$.
A special case Lemma \ref{numerology lemma} that we will use is:
\begin{lemma}\label{weak numer} Suppose that the $2$-subsections of $u$ divide the $2$-subsections of $v$ into two pieces and
that for some
occurrence of an $n$-subword of $v$
in an even (resp. odd) piece is lined up with an occurrence of some $n$-word
in $u$. Then
every occurrence of an $n$-word
in an even (resp. odd) piece of $v$ is either:
\begin{enumerate}
\item[a.)] lined up with some $n$-subword of $u$ or
\item[b.)] lined up with a portion of a $2$-subsection that has the form $e^{j_i}b^{q-j_i}$.
\end{enumerate}
Moreover, no $n$-subword in an odd (resp. even) piece of $v$ is lined up with a $n$-subword in $u$.
\end{lemma}
\subsection{Full measure sets for circular systems}\label{full measure for ccs}
Fix a summable sequence $\langle \varepsilon_n:n\in{\mathbb N}\rangle$ of numbers in ${[0,1)}$ and a circular coefficient sequence $\langle k_n, l_n:n\in{\mathbb N}\rangle$. As we argued in the proof of Lemma \ref{stabilization of names 1}, the proportion of boundaries that occur in words of $\mathcal W^c_n$ is always summable, independently of the way we build $\mathcal W^c_n$. Recall the set $S\subseteq {\mathbb K}^c$ given in Definition \ref{def of S}, where ${\mathbb K}^c$ is the symbolic shift defined from a construction sequence.
\begin{definition}We define some sets that a typical generic point for a circular system eventually avoids. Let: \begin{enumerate} \item $E_n$ be the collection of $s\in S$ such that $s$ does not have a principal $n$-block or $s(0)$ is in the boundary of that $n$-block, \item $E^0_n=\{s:s(0)$ is in the first or last $\varepsilon_nl_n$ copies of $w$ in a power of the form $w^{l_n-1}$ where $w\in\mathcal W_n\}$, \item $E^1_n=\{s:s(0)$ is in the first or last $\varepsilon_nk_n$ 1-subsections of the 2-subsection in which $s(0)$ is located$\}$, \item $E^2_n=\{s:s(0)$ is in the first or last $\varepsilon_nq_n$ 2-subsections of the principal $n+1$-block of$s\}$. \end{enumerate} \end{definition}
\begin{lemma}\label{bc1} Assume that $\sum 1/l_n<\infty$. Let $\nu$ be a shift invariant measure on $S\subseteq {\mathbb K}^c$, where ${\mathbb K}^c$ is a circular system. Then: \begin{enumerate} \item \[\sum_n\nu(E_n)<\infty.\] \begin{center} Assume that $\langle \varepsilon_n\rangle$ is a summable sequence, then for $i= 0, 1, 2$: \end{center} \item \[\sum_n\nu(E^i_n)<\infty.\] \end{enumerate} \end{lemma} {\par\noindent{$\vdash$\ \ \ }} This is an application of the Ergodic Theorem.{\nopagebreak
$\dashv$ \par
} In particular we see: \begin{corollary}\label{bc2} For $\nu$-almost all $s$ there is an $N=N(s)$ such that for all $n>N$, \begin{enumerate} \item $s(0)$ is in the interior of its principal $n$-block, \item $s\notin E^i_n$.
In particular, for almost all $s$ and all large enough $n$: \item if $s\ensuremath{\upharpoonright} [-r_n(s),-r_n(s)+q_n)=w$, then \begin{equation}\notag s\ensuremath{\upharpoonright}[-r_n(s)-q_n, -r_n(s))=s\ensuremath{\upharpoonright} [-r_n(s)+q_n, -r_n+2q_n)=w. \end{equation}
\item $s(0)$ is not in a string of the form $w_0^{l_n-1}$ or $w_{k_n-1}^{l_n-1}$. \end{enumerate} \end{corollary} {\par\noindent{$\vdash$\ \ \ }} This follows from the Borel-Cantelli Lemma.{\nopagebreak
$\dashv$ \par
} {The elements $s$ of $S$ such that some shift $sh^k(s)$ fails one of the conclusions 1.)-4.) of Corollary \ref{bc2} form a measure zero set.} Consequently we work on those elements of $S$ whose whole orbit satisfies the conclusions of Corollary \ref{bc2}. Note, however that the $N(sh^k(s))$ depends on the shift $k$. \begin{definition}\label{mature} We will call $n$ \emph{mature} for $s$ (or say that \emph{$s$ is mature at stage $n$}) iff $n$ is so large that
$s\notin E_m \cup \bigcup_{0\le i\le 2}E^i_m$
for all $m\ge n$. \end{definition} Thus if $s$ is mature at stage $n$ then for all $m>n$ the principal $m$-block of $s$ exists and conclusions 1-4 of Corollary \ref{bc2} hold.
Recall that in Section \ref{circular systems 1}, we defined a canonical factor of a circular system which we called the circle factor. Since the notion of maturity only involves the punctuation of the words involved, it is an easy remark that for all $s\in S$, $n$ is mature for $s$ just in case $n$ is mature for $\pi(s)$, where $\pi$ is the canonical factor map.
{{For the following definition and lemma, we view $s\in S$ as a function with domain $\mathbb Z$, and $s\in \mathcal W_n$ as a function with domain $[0,q_n)$ or, sometimes, an interval $[k, k+q_n)$. In each of these cases we use \emph{dom($s$)} to mean the domain of $s$.}}
{\begin{definition}\label{def bound} We will use the symbol $\partial_n$ in multiple equivalent ways. If $s\in S$ or $s\in \mathcal W^c_m$ we define $\partial_n=\partial_n(s)$ to be the collection of $i$ such that $sh^i(s)(0)$ is in the boundary portion of an $n$-subword of $s$. This is well-defined by our unique readability lemma. In the spatial context we will say that $s\in \partial_{n}$ if $s(0)$ is the boundary of an $n$-subword of $s$. \end{definition}} \noindent For $s\in S$ \[\partial_n(s)\subseteq\bigcup\{[l,l+q_n):{l\in \mbox{ dom}(s)} \mbox{ and }s\ensuremath{\upharpoonright}[l,l+q_n)\in \mathcal W_n\}.\] An integer, $i\in \partial_n(s)\subseteq \mathbb Z$ iff $sh^i(s)$, viewed as an element of ${\mathbb K}^c$, belongs to the $n$-boundary, $\partial_n$.
In what follows we will be considering a generic point $s$ and all of its shifts. We will use the fact if $s$ is mature at stage $n$, then we can detect locally those $i$ for which the $i$-shifts of $s$ are mature. \begin{lemma}\label{getting old} Suppose that $s\in S$, $n$ is mature for $s$ and $n<m$.
\begin{enumerate}
\item Suppose that $i\in [-r_m(s), q_m-r_m(s))$. Then $n$ is mature for $sh^i(s)$ iff \begin{enumerate} \item $i\notin \bigcup_{n\le k\le m} \partial_k$ and \item $sh^i(s)\notin \bigcup_{n\le k< m}(E^0_k\cup E^1_k\cup E^2_k)$.
\end{enumerate} \item For all but at most $(\sum_{n< k\le m}1/l_k) + (\sum_{n\le k< m}6\varepsilon_kq_{k+1})/q_m$ portion of the $i\in [r_m(s), q_m-r_m(s))$, the point $sh^i(s)$ is mature for $n$.
\end{enumerate}
In particular, if $\varepsilon_{n-1}>sup_{m}({1/q_m})\sum_{k=n}^{m-1} 6\varepsilon_kq_{k+1}$, $1/l_{n-1}>\sum_{k=n}^\infty 1/l_k$ and $n$ is mature for $s$, the upper density of those $i\in \mathbb Z$ for which the $i$-shift of $s$ is not mature for $n$ is less than $1/l_{n-1}+\varepsilon_{n-1}$. \end{lemma} Similarly: \begin{lemma}\label{whole $n$-blocks 1} Suppose that $s\in S$ and $s$ has a principal $n$-block. Then $n$ is mature provided that $s\notin \bigcup_{n\le m}E^0_m\cup E^1_m\cup E^2_m$. In particular, if $n$ is mature for $s$ and $s$ is not in a boundary portion of its principal $n-1$-block or in $E^0_{n-1}\cup E^1_{n-1}\cup E^2_{n-1}$, then $n-1$ is mature for $s$. \end{lemma}
\subsection{The $\natural$ map} \label{definition of natural}
Proposition \ref{preservation jazz} implies that any isomorphism $\phi$ between an ergodic
$({\mathbb K}^c,sh)$ and $({\mathbb K}^c,sh^{- 1})$ induces an isomorphism $\phi^\pi$ between $({\mathcal K}, sh)$ and $({\mathcal K}, sh^{- 1})$, where ${\mathcal K}$ is the canonical circle factor. Because $({\mathcal K}, sh{^{-1}})$ is canonically isomorphic with $(\rev{{\mathcal K}},sh)$ (Proposition \ref{spinning})
and $({\mathcal K}, sh)$ is isomorphic to the rotation $\ensuremath{\mathcal R}_\alpha$ of the circle, we see that
$(\rev{{\mathcal K}},sh)$ is isomorphic to the rotation $\ensuremath{\mathcal R}_{-\alpha}$.
We use a specific isomorphism $\natural:({\mathcal K}, sh)\to (\rev{{\mathcal K}}, sh)$ as a benchmark for understanding of potential maps $\phi:{\mathbb K}^c\to\rev{{\mathbb K}^c}$. If we view ${\mathcal K}$ as a rotation $\ensuremath{\mathcal R}_\alpha$ of the unit circle by $\alpha$ radians one can view the transformation $\natural$ as a symbolic analogue of complex conjugation $z\mapsto \bar{z}$ on the unit circle, which is an isomorphism between $\ensuremath{\mathcal R}_\alpha$ and $\ensuremath{\mathcal R}_{-\alpha}$. Copying $\natural$ over to a map on the unit circle gives an isomorphism $\phi$ between $\ensuremath{\mathcal R}_\alpha$ and $\ensuremath{\mathcal R}_{{-\alpha}}$. Such an isomorphism must be of the form \[\phi(z)=\bar{z}e^{2\pi i \beta}\] for some $\beta$. It follows immediately from this characterization that $\natural$ is an involution, however for completeness we prove this directly (and symbolically) in Proposition \ref{alas necessary}.
As usual we find it more convenient to work on the unit interval $I=[0,1)$ rather than the unit circle. The complex conjugacy map $z\mapsto \bar{z}$ corresponds to the map $x\mapsto -x$ on $[0,1)$.
We begin by recalling from equation \ref{definition of C} the formula for a $w\in \mathcal W^c_{n+1}$ that is of the form ${\mathcal C}(w_0, \dots w_{k_n-1})$: \begin{equation}\label{C again}
w=\prod_{i=0}^{q-1}\prod_{j=0}^{k-1}(b^{q-j_i}w_j^{l-1}e^{j_i}) \end{equation} where $q=q_n,k=k_n, l=l_n$ and $j_i\equiv_{q_n}(p_n)^{-1}i$ with $0\le j_i<q_n$. By examining this formula we see that \begin{equation*} \rev{w}=\prod_{i=1}^{q}\prod_{j=1}^ke^{j_{q-i}}\rev{w_{k-j}}^{l-1}b^{q-j_{q-i}}. \end{equation*} Applying the identity in formula \ref{reverse numerology}, we see that this can be rewritten as\footnote{We take $j_q=0$.} \begin{equation}\label{reverse mcc} \rev{w}=\prod_{i=1}^{q}\prod_{j=1}^k(e^{q-j_i}\rev{w_{k-j}}^{l-1}b^{j_i}). \end{equation} We can reindex again and get another form of equation \ref{reverse mcc}: \begin{equation}\label{foreshadow} \rev{w}=\prod_{i=0}^{q-1}\prod_{j=0}^{k-1}(e^{q-j_{i+1}}\rev{w_{k-j-1}}^{l-1}b^{j_{i+1}}). \end{equation}
We can now state the basic lemma about the way $w$ lines up with a shift of $\rev{w}$.
\begin{lemma}\label{two steps left} Let $w\in \mathcal W^c_{n+1}$ and view $w$ as sitting at location $[0, q_{n+1})\subseteq \mathbb Z$. Let $q=q_n$ and $k=k_n$. Consider $sh^{-j_1}(\rev{w})$ as being the word $\rev{w}$ in location $[j_1, q_{n+1}+j_1))\subseteq \mathbb Z$. For all but at most $2kq$ of the occurrences of an $n$-subword $w_j$ of $w$ starting in a location $r\in [0, q_{n+1})$, the reversed word $\rev{w_{k-j-1}}$ occurs in $sh^{-j_1}(\rev{w})$ starting at $r$. \end{lemma}
{\par\noindent{$\vdash$\ \ \ }} The word $w$ starts with a block of $q$ $b$'s and then a block of $l-1$ copies of $w_0$, whereas $\rev{w}$ starts with a block of $q-j_1$ $e$'s followed by $l-1$ copies of $\rev{w_{k-1}}$. Hence if we shift $\rev{w}$ to the right by $j_1$ (to get $sh^{-j_1}(\rev{w})$) the first copy of $\rev{w_{k-1}}$ is aligned with the first copy of $w_0$ in $w$. Hence all of the copies of $\rev{w_{k-1}}$ in the first 1-subsection are aligned with the copies of $w_0$ in the first 1-subsection of $w$. Because the consecutive blocks of $b$'s and $e$'s (or $e$'s and $b$'s) in the 2-subsections add up to $q$ we see that every copy of $\rev{w_{k-j-1}}$ in the first 2-subsection of $sh^{-j_1}(\rev{w})$ is aligned with with a copy of $w_j$.
We now argue as in Section \ref{understanding the words}. At the end of each 2-subsection, $w$ has a block of $e$'s of length $j_i$, followed at the beginning of the next 2-subsection, by a block of $b$'s of length $q-j_{i+1}$. Together the $e$'s and $b$'s form a block of length $j_i+q-j_{i+1}$, which is equivalent mod($q$) to $-j_1$. Similarly the combined length of a block of $b$'s and $e$'s finishing and starting consecutive 2-subsections of $\rev{w}$ is equal to $-j_1$ mod($q$).
Both the beginning of the block of $e$'s ending the $k^{th}$ 2-subsection and the end of the block of $b$'s starting the $k+1^{st}$ 2-subsection are of distance less than $q$ from the location of the end of the $k^{th}$ 2-subsection. It follows from this and the comments in the previous paragraph, that if $S_1$ and $S_2$ are consecutive 2-subsections of $w$ and $S_1'$ and $S_2'$ are the corresponding 2-subsections of $\rev{w}$ then the beginning of the first occurrence of $\rev{w_{k-1}}$ in $S_2'$ is within $2q$ of the first occurrence of $w_0$ is $S_2$ and their locations are equivalent mod($q$). Hence inside the first 1-subsection, the 0-subsections are lined up except for at most $2$ copies of $w_0$. This pattern is continued through $S_2$, giving at most $2k$ locations of $n$-blocks that are not aligned in $S_2$.
Since there are less than $q$ 2-subsections with potential misalignments, the Lemma is proved. {\nopagebreak
$\dashv$ \par
}
The next proposition gives a somewhat more detailed view into situation of Lemma \ref{two steps left}.
\begin{prop}\label{first slip} Let $w,w'\in \mathcal W^c_{n+1}$ and suppose that \begin{eqnarray*} w={\mathcal C}(v_0,v_1,\dots v_{k_n-1}) & \mbox{ and }&w'={\mathcal C}(v_0', v_1',\dots v'_{k_n-1}). \end{eqnarray*} We look at the relative positions of $n$-words in $w$ and $sh^{-j_1}(\rev{w'})$. \begin{enumerate} \item Each occurrence of $v_i$ in $w$ is either lined up with an occurrence of $\rev{v'_{k_n-i-1}}$ or entirely lined up with a section of $\partial_n$ inside $sh^{-j_1}(\rev{w'})$. \item There is a number $C$ such that for all $i$ the number of occurrences of $v_i$ lined up with an occurrence of $\rev{v'_{k_n-i-1}}$ is $C$. \end{enumerate} \end{prop} {\par\noindent{$\vdash$\ \ \ }} The first part is clear from the proof of Lemma \ref{two steps left}. The second part follows because all of the 1-subsections in a given 2-subsection of $w$ have the same alignment relative to $sh^{-j_1}(\rev{w'})$.{\nopagebreak
$\dashv$ \par
}
Since the total number of occurrences of $n$-subwords in $klq$, the proportion of $n$-subwords lined up with $\partial_n$ in $sh^{-j_i}(\rev{w'})$ is at most $2/l$.
Suppose that ${\mathcal K}$ is given by the canonical construction sequence $\langle \mathcal W_n^\alpha:n\in{\mathbb N}\rangle$. We define a sequence of functions $\langle\Lambda_n:n\in{\mathbb N}\rangle$ and argue that they converge to an isomorphism from ${\mathcal K}$ to $\rev{{\mathcal K}}$.
We begin by defining an increasing sequence of natural numbers. Recall the definition of the Anosov-Katok coefficients $p_n$ and $q_n$ given in equations \ref{pn} and \ref{qn}. Since $p_n$ and $q_n$ are relatively prime we can define $(p_n)^{-1}$ in $\mathbb Z/q_n\mathbb Z$. For the following definition we will view $(p_n)^{-1}$ as a \emph{natural number} with $0\le (p_n)^{-1}<q_n$.\footnote{In the notation used to define ${\mathcal C}$, $(p_n){^{-1}}=j_1$. However the notation $j_1$ is ambiguous (it depends on $n$), so we use $(p_n){^{-1}}$ in this context.}
We let $A_0=0$ and \begin{equation} A_{n+1}=A_n-(p_n)^{-1}. \label{code coefficients} \end{equation}
\begin{lemma}\label{An is small}
If $A_n$ is defined as above, then $|A_{n+1}|<2q_{n}$. \end{lemma} {\par\noindent{$\vdash$\ \ \ }} This is proved inductively using the fact that $q_{n+1}>2q_{n}$.{\nopagebreak
$\dashv$ \par
}
Let ${\mathcal K}$ be the circular system in the language $\Sigma=\{*\}$, as given in Definition \ref{first appearance of circle factor}. We now define a stationary code $\overline{\Lambda}_n$ with domain $S$ that approximates elements of $\rev{{\mathcal K}}$
by defining \begin{equation}\label{definition of Lambdan} \Lambda_n(s)=\left\{\begin{array}{ll} sh^{A_n+2r_n(s)-(q_n-1)}(\rev{s})(0) & \mbox{if $r_n(s)$ is defined}\\
b&\mbox{otherwise}
\end{array}\right. \end{equation}
Since for all $s\in S$ and all large enough $n$, $r_n(s)$ is defined, the default value is only obtained for finitely many $n$. \begin{lemma} ${\Lambda}_n$ is given by a finite code. \end{lemma} {\par\noindent{$\vdash$\ \ \ }} To check whether $r_n(s)$ is defined one need only examine $s$ on the interval $[-q_n, q_n]\subseteq \mathbb Z$. The relevant portion of $\rev{s}$ necessary to compute $\Lambda_n(s)$ is contained in $s\ensuremath{\upharpoonright}[-q_n-A_n, q_n+A_n]$. Hence $\Lambda_n$ is determined by a finite code.{\nopagebreak
$\dashv$ \par
}
The formula in equation \ref{definition of Lambdan} can be understood as follows. Suppose that $s\in S$ and $s$ has a principal $n$-block. Then the element $s^*$ defined as
$sh^{2r_n(s)-(q_n-1)}(\rev{s})$ belongs to $\rev{{{\mathcal K}}}$, has a principal $n$-block that is the reverse of the principal $n$-block of $s$ and moreover, the principal $n$-block of $s^*$ is exactly lined up with the principal $n$-block of $s$.
The reverse of the principal $n$-block of $s$ begins with a block of $q_{n-1}-(p_{n-1})^{-1}$ many $e$'s, and hence if $s'=sh^{(-(p_{n-1})^{-1})+2r_n(s)-(q_n-1)}(\rev{s})$ then the first $n-1$-subword of the principal $n$-block of $s'$ is lined up with the first $n-1$-subword of the principal $n$-block of $s$. The rest of the terms used to define $A_n$ (coming from $A_{n-1}$) are used for lower order adjustments inside this principal $n$-block.
\noindent Thus, a qualitative description of $\bar\Lambda_{n}(s)$ can be given as follows: \begin{enumerate}
\item It first reverses the principal $n$-block of $s$ leaving it exactly lined up.
\item It then adjusts the result by shifting so that the first occurrence of a
reverse $n-1$-block lines up with the first $n-1$-subword of the principal $n$-block of $s$. (So far we have
described $sh^{(-(p_{n-1})^{-1})+2r_n(s)-(q_n-1)}(\rev{s})$.)
By Lemma \ref{two steps left}, we get a sequence where the principal $n$-block of $\Lambda_n(s)$ has the vast majority of its $n-1$-blocks lined up with the $n-1$-blocks of $s$: all of them except those that span a section of boundary at the juncture of two 2-subsections of the principal $n$-word of $s$.
\item Finally it shifts by $A_{n-1}$ which is the cumulative adjustment at earlier stages.
\end{enumerate}
The next lemma follows from this description:
\begin{lemma} \label{88bis} Let $n<m$ and suppose that $s\in {\mathcal K}$ has a principal $m$-block. Let $s'=sh^{2r_m-q+A_m-A_n}(rev(s))$. Then at least \[\prod_{n}^{m-1}(1-{2\over (l_i-1)})\] proportion of the $n$-blocks in the principal $m$-block of $s$ are lined up with $n$-blocks in $s'$. \end{lemma} {\par\noindent{$\vdash$\ \ \ }} We first consider $m=n+1$. By Lemma \ref{two steps left}, all but $2k_nq_n$ of the $n$-blocks in $w$ are aligned with the $n$-blocks in $sh^{-j}(\rev{w})$. This is proportion \begin{equation*} 1-{2k_nq_n\over k_nq_n(l_n-1)}=1-{2\over l_n-1}. \end{equation*} The general result follows by induction.{\nopagebreak
$\dashv$ \par
}
\begin{theorem}\label{mr natural} Suppose that $\langle k_n,l_n:n\in{\mathbb N}\rangle$ is a circular coefficient sequence. Then
the sequence of stationary codes $\langle \overline\Lambda_n:n\in{\mathbb N}\rangle$ converges to a
{shift invariant function} $\overline{\natural}:{\mathcal K}\to (\{*\}\cup \{b, e\})^{{\mathbb Z}}$ that induces an
isomorphism $\natural$ from ${\mathcal K}$ to $\rev{{\mathcal K}}$. \end{theorem} {\par\noindent{$\vdash$\ \ \ }} We first show that the sequence $\langle \overline\Lambda_n:n\in{\mathbb N}\rangle$ converges, which will follow if we show that the code distances between the $\Lambda_n$ and $\Lambda_{n+1}$ are summable. For notational simplicity, let $q=q_n, k=k_n, l=l_n$ and $j\equiv_{q}(p_n)^{-1}$ with $0\le j<q$.
\bfni{Claim:} There is a summable sequence of positive numbers $\delta_n$ such that for almost all $s$, the $\ensuremath{\bar{d}}$-distance between $\bar{\Lambda}_n(s)$ and $\bar\Lambda_{n+1}(s)$ is bounded by $\delta_n$, and $\bar\Lambda_n(s)$ and $\bar\Lambda_{n+1}(s)$ agree on all but at most $\delta_n$ proportion of the $n$-blocks of $s$.
We use Lemma \ref{computing code distances}, which tells us that for a typical $s\in S$, the code distance between $\Lambda_n$ and $\Lambda_{n+1}$ is $\ensuremath{\bar{d}}(\overline\Lambda_n(s),\overline\Lambda_{n+1}(s))$, which is defined to be the density of \begin{equation}\label{real distance} D=_{def}\{k:\Lambda_n(sh^k(s))(0)\ne \Lambda_{n+1}(sh^k(s))(0)\}. \end{equation}
{Because $|\mathcal W_n^\alpha=1$ for each $n$, there is only one possible $n$-subword at any location of any element of $\rev{{\mathcal K}}$. Thus to compute $\ensuremath{\bar{d}}$-distance, it suffices count positions where the $\overline\Lambda_m$'s disagree on the \emph{locations} of the $n$-subwords.}
By Lemma \ref{getting old} for a typical $s\in S\subseteq{\mathcal K}$ and all $n$, $I_{n}=_{def}\{i:n $ is not mature for $sh^i(s)\}$ has density at most $1/l_{n-1}+\varepsilon_{n-1}$, hence we can neglect these $i$ when computing the density of $D$.
This allows us to assume that $r_{n+1}(s)$ is defined. We compute the density of the difference between $\bar\Lambda_n$ and $\bar\Lambda_{n+1}$ as they pass across an $n+1$-block in $s$. If this number is $d$ then the distance between $\Lambda_n$ and $\Lambda_{n+1}$ is bounded by the sum of $d$ and the density of $I_{n}$.
As $\Lambda_{n+1}$ crosses an $n+1$-block it produces the reverse $n+1$-block shifted by $A_{n+1}$. Explicitly, if $w$ is the $n+1$-block of $s$, as $\Lambda_{n+1}$ crosses $w$ it produces $sh^{A_{n+1}}(\rev{w})$. As
$\Lambda_n$ passes across this same section, each time it crosses an $n$-block $w'$ it produces
$sh^{A_n}(\rev{w'})$. If $w'$ starts at $r$ then the beginning of this copy of $sh^{A_n}(\rev{w'})$ is $r-A_n$.
We begin by rewriting $sh^{A_{n+1}}(\rev{w})$ as $sh^{A_n}(sh^{-j}(\rev{w}))$ where $j=(p_{n})^{-1}$.
By Lemma \ref{two steps left}, all but $2kq$ of the $n$-blocks in $w$ are aligned with the $n$-blocks in $sh^{-j}(\rev{w})$. Hence, relative to the complement of $I_n$, the portion of the principal $n+1$-block $w$ of $s$ that lies in an $n$-block aligned with an $n$-block of $sh^{-j}(\rev{w})$ is \begin{equation}\label{crude estimate}{k(l-1)q^2-2kq\over k(l-1)q^2}=1-{2\over (l-1)q} \end{equation} {Because there is only one possible $n$-word, } whenever $sh^{A_n}(\rev{w'})$ is aligned with $sh^{A_n}(sh^{-j}(\rev{w}))$ they are equal.
Putting this altogether, we see that $\Lambda_n$ and $\Lambda_{n+1}$ agree on all of the $n$-subwords of the principal $n+1$-block of $s$ that are aligned with $sh^{-j}(\rev{w})$. The disagreements are limited to the $n$-subwords that are not aligned and the boundary. The total length of the disagreements is therefore bounded by \[(2kq)*q+kq^2=3kq^2.\] This has proportion $3kq^2/klq^2=3/l$.
Thus the distance between $\Lambda_n$ and $\Lambda_{n+1}$ is bounded by
$1/l_{n-1}+\varepsilon_{n-1}+3/l_n$. In particular the distances are summable and the sequence $\langle \bar\Lambda_n:n\in{\mathbb N}\rangle$ converges almost everywhere to a function
{$\natural:{\mathcal K} \to (\Sigma\cup\{ b, e\})^\mathbb Z$}.
We now show that $\natural$ is an isomorphism between ${\mathcal K}$ and $\rev{{\mathcal K}}$. Since $\bar\Lambda_n$ takes an $n$-block to a shift of the reverse $n$-block, it makes sense to discuss the \emph{principal $n$-block} of $\bar\Lambda(s)$. Since the $r_n$'s cohere as in Remark \ref{interval coherence}, for $n<m$, $r_m(\bar\Lambda_m(s))$ is in the $r_n(\bar\Lambda_m(s))^{th}$ position of the principal $n$-block of $\bar\Lambda_m(s)$ (provided both $r_n$ and $r_m$ are defined). An application of the Ergodic Theorem shows that if $D_n$ is defined to be the collection of $s$ such that: \[r_n(\bar\Lambda_n(s))\mbox{ exists and the principal $n$-words of $\bar{\Lambda}_n(s)$ and $\bar\Lambda_{n+1}(s)$ disagree}\] then $\sum\nu(D_n)<\infty$. From the Borel-Cantelli Lemma, it follows that for almost every $s$ for all large enough $n$ the principal $n$-blocks of $\bar\Lambda_n(s)$ and $\bar\Lambda_{n+1}(s)$ are the same, and thus that for {$s\in S, \natural(s)\in \rev{{\mathcal K}}$.}
We now argue that if $s$ is typical and $s^*=\natural(s)$, then $s^*\in \rev{S}$. It suffices to show that $\lim_{n\to \infty}-r_n(s^*)=-\infty$ and $\lim_{n\to \infty}q_n-r_n(s^*)=\infty$.\footnote{We are adopting the convention that in defining $r_n(s^*)$ for $s^*\in \rev{S}$ we count $r_n$ from the left end of an $n$-block. Thus the position $r$ in a word $w\in \mathcal W^\alpha_n$ corresponds to the position $q-1-r$ in $\rev{w}$.}
If $n$ is mature for $s$ and large enough that for $m>n, \bar{\Lambda}_m(s)$ and $\bar\Lambda_n(s)$ have the same principal $n$-blocks, then $r_n(s^*)=r_n(s)+A_n$ unless $r_n(s)\in [0,|A_n|)$. Assuming that $r_n(s)\ge |A_n|$, we know from Lemma \ref{An is small} that \[r_n(s)-2q_{n-1}<r_n^*(s)<r_n(s).\]
Hence, $-r_n^*(s)\le 2q_{n-1}-r_n(s)$ and $q_n-r_n^*(s)\ge q_n-r_n(s)$. Applying Lemma \ref{getting old} (using the fact that $\sum nq_{n-1}/q_n<\infty$, and hence $\sum |A_n|/q_n<\infty$) we see that for large $n$, $r_n(s)>|A_n|$ and that $r_n(s)-2q_n\to \infty$. Since $q_n-r_n(s)\to \infty$ we have shown that $s^*\in \rev{S}$.
As noted before Theorem \ref{rank one description}, if $s\in S$ then $s$ is determined by any tail of the sequence $\langle r_n(s):n\in {\mathbb N}\rangle$. In particular, if we know a tail of $\langle r_n(s^*):n\in {\mathbb N}\rangle$ we can determine $s^{*}$. Since for large $n$,
$r_n(s^*)=r_n(s)+A_n$, $\natural$ is one-to-one on a set of measure one.
We can now conclude that $\natural$ is an isomorphism. It is shift invariant since it is a limit of stationary codes, it maps from $S$ to $\rev{S}$, and is one-to-one on a set of $\nu$-measure one. If we define a measure $\mu$ on the Borel sets of $\rev{{\mathcal K}}$ by setting $\mu(A)=\nu(\natural^{-1}(A))$, then $\mu$ is a shift invariant, non-atomic measure on $\rev{S}$. Since $S$ is uniquely ergodic, $\rev{S}$ is as well and thus $\mu$ must be equal to the unique invariant measure $\nu$. We have shown that $\natural$ is an isomorphism between ${\mathcal K}$ and $\rev{{\mathcal K}}$. {\nopagebreak
$\dashv$ \par
}
\begin{definition}\label{def of natural} We denote the limit of $\langle \bar{\Lambda}_n:n\in{\mathbb N}\rangle$ by $\natural:{\mathcal K}\to \rev{{\mathcal K}}$. \end{definition}
We describe the qualitative behavior of $\natural$ in a remark that we will use later:
\begin{remark}\label{natural coding} There is a summable sequence $\langle \delta_n\rangle$ such that for all but $1-\delta_n$ measure of $s\in S\subseteq {\mathcal K}$, there is an interval $I$ containing 0 in $\overline{\Lambda}_n(s)$ such that $s\ensuremath{\upharpoonright} I\in \mathcal W^\alpha_n$, and moreover $\overline{\Lambda}_{n+1}(s)$ and $\overline{\Lambda}_n(s)$ agree on this interval. It follows from the Borel-Cantelli Lemma that
for almost all $s$ and large enough $n$, $\natural(s)$ agrees with $\bar\Lambda_{n}(s)$ on the principal $n$-block of $s$. Thus for a typical $s$ and large enough $n$, the map $\natural$ reverses the principal $n$-block while keeping its location and then shifts it by $A_n$. \end{remark}
As noted at the beginning of this section, the next proposition follows immediately from Theorem \ref{rank one description}, however we include a symbolic proof for completeness. \begin{prop}\label{alas necessary} The map $\natural$ is an involution. \end{prop} {\par\noindent{$\vdash$\ \ \ }} It is immediate from the qualitative description of $\bar{\Lambda}_n$ given before Lemma \ref{88bis}, that each $\bar{\Lambda}_n$ is an involution. To see that $\natural^2$ is the identity, let $\epsilon>0$. We can choose an $m_0$ large enough that for all $m\ge m_0$, $\bar{\Lambda}_m$ and $\natural$ agree with $\bar{\Lambda}_{m_0}$ on all but $\epsilon$ proportion of the $m_0$-blocks and $\bigcup_{m_0+1}^\infty \partial_k$ has measure $\epsilon*10^-6$. Then $\natural\circ\bar{\Lambda}_{m_0}$ is equal to the identity on a set of density at least $1-\epsilon$. Letting $\epsilon\to 0$ and $m_0\to \infty$ completes the argument. {\nopagebreak
$\dashv$ \par
}
\subsection{Synchronous and Anti-synchronous joinings}
Every odometer based system has a built in metronome: its odometer factor defined in Lemma \ref{odometer factor}. Correspondingly circular systems can be timed by their canonical rotation factor defined in Lemma \ref{canonical rotation factor}.
Joinings between odometer based and circular systems may induce non-trivial automorphisms of the underlying timing structure. To avoid this complication we restrict ourselves to synchronous and anti-synchronous joinings: those which preserve or exactly reverse the underlying timing. We now make this idea precise.
Both the odometer transformations and rotations of a circle have easily understood inverse transformations and the isomorphisms between transformations and their inverses are given by the maps $x\mapsto -x$ and $\rev{}\circ\natural$ respectively. If ${\mathbb K}$ and $\mathbb L$ are either odometer based or circular systems let ${\mathbb K}^\pi$ and $\mathbb L^\pi$ be the corresponding odometer or rotation systems on which they are based. \begin{definition} \begin{itemize} \item Let ${\mathbb K}$ and $\mathbb L$ be odometer based systems with the same coefficient sequence, and $\rho$ a joining between ${\mathbb K}$ and $\mathbb L^{\pm1}$. Then $\rho$ is \emph{synchronous} if $\rho$ joins ${\mathbb K}$ and $\mathbb L$ and the projection of $\rho$ to a joining on ${\mathbb K}^\pi\times \mathbb L^\pi$ is the graph joining determined by the identity map (the diagonal joining of the odometer factors); $\rho$ is \emph{anti-synchronous} if $\rho$ is a joining of ${\mathbb K}$ with $\mathbb L^{-1}$ and its projection to ${\mathbb K}^\pi\times (\mathbb L^{-1})^\pi$ is the graph joining determined by the map $x\mapsto -x$. \item Let ${\mathbb K}^c$ and $\mathbb L^c$ be circular systems with the same coefficient sequence and $\rho$ a joining between ${\mathbb K}^c$ and $(\mathbb L^c)^{\pm 1}$. Then $\rho$ is \emph{synchronous} if $\rho$ joins ${\mathbb K}^c$ and $\mathbb L^c$ and the projection to a joining of $({\mathbb K}^c)^\pi$ with $(\mathbb L^c)^\pi$ is the graph joining determined by the identity map of $\mathcal K$ with $\mathcal L$, the underlying rotations; $\rho$ is \emph{anti-synchronous} if it is a joining of ${\mathbb K}^c$ with $(\mathbb L^c)^{-1}$ and projects to the graph joining determined by $\rev{}\circ\natural$ on ${\mathcal K}\times \mathcal L^{-1}$. \end{itemize} \end{definition}
\noindent There is always a synchronous joining of odometer systems with the same underlying timing factor $\mathcal O$: \begin{definition} Suppose that ${\mathbb K}$ and $\mathbb L$ are based on $\mathcal O$. Then the relatively independent joining of ${\mathbb K}$ and $\mathbb L$ over $\mathcal O$ is a synchronous joining, which we will call the \emph{synchronous product joining}. The relatively independent joining of ${\mathbb K}$ and $\mathbb L^{-1}$ over the map $x\mapsto -x$ we will call the \emph{anti-synchronous product joining}.
We will use the same terminology for the independent joinings of circular systems over the identity and $\rev{}\circ\natural$. \end{definition}
\section{Building the Functor $\mathcal F$}\label{building functor}
The main result of this paper concerns two categories whose objects are odometer based systems and circular systems respectively. The morphisms in these categories will be graph joinings. We will show that there is a functor taking odometer systems to circular systems that preserves the factor and conjugacy structure. In this section we focus on defining the function from odometer based systems to circular systems that underlies the functorial isomorphism between these categories.
We begin by defining a function from the odometer based symbolic shifts ${\mathbb K}$ to the circular symbolic shifts ${\mathbb K}^c$. After having done so we define $\mathcal F$ on the pairs $({\mathbb K}, \mu)$ where $\mu$ is an invariant measure on ${\mathbb K}$. Finally we define $\mathcal F$ on synchronous and anti-synchronous graph joinings.
We will use the notation that $K_n=\prod_{i<n}k_i$.
Then the $K_n$'s are the lengths of the odometer based words in $\mathcal W_n$ and the $q_n$'s are the lengths of the circular words in $\mathcal W_n^c$.
Except where otherwise stated we will assume that we are working with a fixed \hyperlink{circ coef}{circular coefficient sequence} $\langle k_n, l_n:n\in{\mathbb N}\rangle$.
Let $\Sigma$ be a language and $\langle \mathcal W_n:n\in{\mathbb N}\rangle$ be a construction sequence for an odometer based system with coefficients $\langle k_n:n\in{\mathbb N}\rangle$. Then for each $n$ the operation ${\mathcal C}_n$ is well-defined. We define a construction sequence $\langle \mathcal W_n^c:n\in{\mathbb N}\rangle$ and bijections $c_n:\mathcal W_n\to \mathcal W_n^c$ by induction as follows:
\begin{enumerate} \item Let $\mathcal W^c_0=\Sigma$ and $c_0$ be the identity map. \item Suppose that $\mathcal W_n, \mathcal W_n^c$ and $c_n$ have already been defined. \[\mathcal W_{n+1}^c=\{{\mathcal C}_n(c_n(w_0),c_n(w_1), \dots c_n(w_{k_n-1})):w_0w_1\dots w_{k_n-1}\in \mathcal W_{n+1}\}.\]
Define the map $c_{n+1}$ by setting
\[c_{n+1}(w_0w_1\dots w_{k_n-1})={\mathcal C}_n(c_n(w_0),c_n(w_1), \dots c_n(w_{k_n-1})).\]
\end{enumerate} We note in case 2 the \hyperlink{pwords}{{prewords}} are: \[P_{n+1}=\{c_n(w_0)c_n(w_1)\dots c_n(w_{k_n-1}): w_0w_1\dots w_{k_n-1}\in \mathcal W_{n+1}\}.\]
\begin{definition}\label{def of functor} Define a map $\mathcal F$ from the set of odometer based systems (viewed as subshifts) to circular systems (viewed as subshifts) as follows. Suppose that ${\mathbb K}$ is built from a construction sequence $\langle \mathcal W_n:n\in{\mathbb N}\rangle$. Define \[\mathcal F({\mathbb K})={\mathbb K}^c\] where ${\mathbb K}^c$ has construction sequence $\langle \mathcal W_n^c:n\in{\mathbb N}\rangle$. \end{definition}
Suppose that ${\mathbb K}^c$ is a circular system with coefficients $\langle k_n, l_n:n\in{\mathbb N}\rangle$. We can recursively recursively build functions $c_n{^{-1}}$ from words in $\Sigma\cup \{b,e\}$ to words in $\Sigma$. The result is a odometer based system $\langle \mathcal W_n:n\in{\mathbb N}\rangle$ with coefficients $\langle k_n:n\in{\mathbb N}\rangle$.\footnote{We are using the strong unique readability assumption on the $P_n$'s to see the unique readability of the words in the sequence $\langle \mathcal W_n:n\in{\mathbb N}\rangle$.}
If ${\mathbb K}$ is the resulting odometer based system then $\mathcal F({\mathbb K})={\mathbb K}^c$. Thus we see: \begin{prop}\label{bijection from readability} The map $\mathcal F$ is a bijection between odometer based symbolic systems with coefficients $\langle k_n:n\in{\mathbb N}\rangle$ and circular symbolic systems with coefficients $\langle k_n, l_n:n\in{\mathbb N}\rangle$. \end{prop} {\par\noindent{$\vdash$\ \ \ }} That $\mathcal F$ is one-to-one follows from the unique readability of words occurring in the construction sequence $\langle W_n:n\in{\mathbb N}\rangle$.{\nopagebreak
$\dashv$ \par
}
\begin{remark} It is clear from Definition \ref{def of functor} that $\mathcal F$ preserves uniformity and strong uniformity (see \cite{prequel} for these notions). In fact it preserves much more: the simplex of non-atomic invariant measures, rank one transformations and so on. We verify much of this in this paper and more in the forthcoming \cite{part4}. \end{remark}
To understand the correspondence between measures on ${\mathbb K}$ and ${\mathbb K}^c$ we will have to understand the structure of basic open intervals. Recall that we write $\langle u\rangle_L$ to mean the basic open interval of ${\mathbb K}$ determined by $u$ sitting on the interval $[L, L+|u|)\subseteq \mathbb Z$. Without the subscript $L$, $\langle u \rangle$ is shorthand for $\langle u\rangle_0$. We adopt the same conventions for ${\mathbb K}^c$, that the subscripts correspond to the beginning of the sequence and without a subscript the sequence begins at zero.
\subsection{Genetic Markers}\label{GMs and coding} To see that $\mathcal F$ can be extended to a map from invariant measures on odometer based systems to invariant measures on circular systems, we begin by recalling how to identify elements of a symbolic system. Suppose that $\langle \mathcal W_n:n\in {\mathbb N}\rangle$ is a construction sequence for an odometer based transformation ${\mathbb K}$. Let $\langle\mathcal W_n^c:n\in{\mathbb N}\rangle$ be the corresponding circular construction sequence for ${\mathbb K}^c$. By Lemma \ref{specifying elements} to specify a typical $s\in {\mathbb K}$ or $s^c\in {\mathbb K}^c$, it suffices to give a tail of the sequence of principal $n$-blocks $\langle w_n(s):N\le n\in{\mathbb N}\rangle$ or $\langle w_n^c(s^c):N\le n\in{\mathbb N}\rangle$ along with the locations $\langle r_n(s):N\le n\rangle$ or $\langle r_n(s^c):N\le n\rangle$.
\begin{definition}\label{def of gms} Suppose that $u, v$ are words in $\mathcal W_n$ and $\mathcal W_{n+1}$ respectively and $u$ occurs as an $n$-subword of $v$ in a particular location. Viewing $v$ as a concatenation $w_0w_1\dots w_{n_k-1}$ of $n$-subwords, there is a $j$ such that $u=w_j$. Let $j_n^*=j$ and call $j_n^*$ the \emph{genetic marker} of $u$ in $v$.
Suppose that $u\in \mathcal W_n$ and $v\in \mathcal W_{n+k}$ and $u$ is an $n$-subword of $v$ occurring at a particular location. Then there is a sequence of words $u_n=u, u_{n+1}, \dots u_{n+k-1}, u_{n+k}=v $ such that $u_i$ is a $n+i$-subword of $v$ at a definite location and the location of $u$ in $v$ is inside $u_i$. Let $j_{n+i}^*$ be the genetic marker of $u_{n+i}$ inside $u_{n+i+1}$. We call the sequence
$\vec{j}^*=\langle j_n^*, j_{n+1}^*, \dots j_{n+k-1}^*\rangle$ the \emph{genetic marker} of $u$ in $v$. If $\vec{j}^*$ is the genetic marker of some $n$-word inside and $m$-word, we will call it an $(n,m)$-genetic marker.
\end{definition} If $u$ occurs as a subword of $v$ then the genetic marker $\langle j^*_n, j^*_{n+1} \dots j^*_{n+k-1}\rangle$ of that occurrence codes its location inside $v$.
Suppose that $s\in {\mathbb K}$ has principal $n$-blocks $\langle w_n:n\in{\mathbb N}\rangle$. Each $w_{n+1}$ is a concatenation of words $v_0v_1\dots v_{k_{n}-1}$. Let
\begin{equation}j'_n=_{def}{r_{n+1}(s)-r_n(s)\over K_n}
\label{jns}
\end{equation} or equivalently \begin{equation} r_{n+1}(s)=r_n(s)+j_n'K_n \label{inductive rns}. \end{equation} Each $w_{n+1}$ is a concatenation of words $v_0v_1\dots v_{k_n-1}$, and we see that $s(0)$ belongs to $v_{j_n'}$. In particular, the genetic marker of $w_n$ inside $w_{n+k}$ is the sequence $\langle j_n', j_{n+1}', \dots j_{n+k-1}'\rangle$.
\noindent{\bfni{Genetic markers for regions of words in $\mathcal W_{n+k}^c$:}} In circular words, genetic markers code regions rather than subwords. Given $u$ and $v$ as above, we can consider the construction of $c_{n+k}(v)$ starting with the collection $\{c_n(u):u$ is an $n$-subword of $v\}$.
Each of the genetic markers $\langle j^*_{n}, j^*_{n+1}, \dots j^*_{n+k-1}\rangle$ of a subword $u$ of $v$
determines a \emph{region} of $n$-subwords of $c_{n+k}(v)$. More explicitly, in the first step of the construction we
put $u$ into the $(j^*_{n})^{th}$ argument of $\mathcal C_n$. At the next step we put the result into the
$j^*_{n+1}$ argument of ${\mathcal C}_{n+1}$ and so on. Thus we see that there are bijections between
\begin{enumerate}
\item sequences $\langle j^*_{n}, j^*_{n+1}, \dots j^*_{n+k-1}\rangle$ with $0\le j^*_m< k_m$,
\item $n$-subwords $u$ of $v$,
\item the regions of $v^c$ occupied by the occurrences of powers $(u^c)^{l_n-1}$ where $u^c$ is the element of $\mathcal W_n^c$ determined by $\langle j^*_{n}, j^*_{n+1}, \dots j^*_{n+k-1}\rangle$. \end{enumerate} Thus genetic markers give the correspondence between the regions of $c_{n+k}(v)$ that are not in $\bigcup_{n< m\le n+k}\partial_m$ and particular occurrences of an $n$-word $u$ in $v$.
The next lemma computes the number of occurrences of a $c_n(u)$ with a given genetic marker $\langle j^*_{n}, j^*_{n+1}, \dots j^*_{n+k-1}\rangle$ in $c_{n+k}(v)$.
\begin{lemma}\label{products of imagination} Suppose that $u^c$ occurs in $v^c$ with genetic marker $\langle j^*_{n}, j^*_{n+1}, \dots j^*_{n+k-1}\rangle$. Then the number of occurrences of $u^c$ in $v^c$ with the same genetic marker $\langle j^*_{n}, j^*_{n+1}, \dots j^*_{n +k-1}\rangle$ is \begin{equation}\label{helical product} \prod_n^{n+k-1}q_i(l_i-1). \end{equation} \end{lemma} {\par\noindent{$\vdash$\ \ \ }} Fix $m$ and $v^c\in \mathcal W_m^c$. We prove equation \ref{helical product} for $n=m-k$ by induction on $k\ge 1$. If $k=1$ then we have a single genetic marker $j^*_{m-1}$. By formula \ref{definition of C} for ${\mathcal C}_{m-1}$ we see that the $j_{m-1}^*$ argument occurs in $v^c$ exactly $q_n(l_n-1)$ times.
Suppose now that we know that formula \ref{helical product} holds for $k-1$. We show it for $k$. Let $n=m-k$ and $u^c$ be the $n$-subword of $v^c$ with genetic marker$\langle j^*_{n}, j^*_{n+1}, \dots j^*_{n +k-1}\rangle$. Let $w^c$ be the subword of $v^c$ with genetic marker $\langle j^*_{n+1}, \dots j^*_{n +k-1}\rangle$. Then:
\[
|\{\mbox{occurrences of $u^c$ in $v^c$ with marker }\langle j^*_n, j^*_{n+1}, \dots j^*_{n
+k-1}\rangle\}|\] is equal to
\[|\{\mbox{occurrences of $u^c$ in $w^c$ with marker $j^*_n$}\}|\times \]
\[|\{\mbox{occurrences of $w^c$ in $v^c $ with marker }\langle j^*_{n+1}, \dots j^*_{n
+k-1}\rangle\}|
\] The lemma follows.{\nopagebreak
$\dashv$ \par
}
Since particular $(n,m)$-genetic markers $\langle j^*_{n}, j^*_{n+1}, \dots j^*_{n+k-1}\rangle$ correspond to powers of $u^c$'s that occur with the same multiplicity in $v^c$, independently of the marker, we see that for a given $u$ and $v$: \begin{equation}\label{up or down}
{|\{\mbox{occurrences of $u$ in } v\}|\over |\{n\mbox{-subwords of }v\}|}=
{|\{\mbox{occurrences of $c_n(u)$ in } c_{n+k}(v)\}|\over |\{\mbox{circular $n$-subwords of }c_{n+k}(v)\}|} \end{equation} We can restate equation \ref{up or down} in the language of section \ref{sequences and points}. It says that \begin{equation}\label{all is equal} \mbox{EmpDist}(v)(u)=\mbox{EmpDist}(c_{n+k}(v))(c_n(u)). \end{equation}
In particular, if we fix a set $S^*$ of genetic markers we can compare the number of occurrences of a word with genetic marker in $S^*$ in $v\in\mathcal W_{n+k}$ with the number of occurrences in the corresponding $v^c\in \mathcal W_{n+k}^c$. Specifically, the number of occurrences of a word $u^c$ in $v^c$ at some genetic marker in $S^*$ is
$|S^*|*\prod_n^{n+k-1}q_i(l_i-1)$. The proportion of $n$-words occurring with a genetic marker in $S^*$ relative to all $n$-words occurring in $v^c$ is the same as the proportion of $n$-words with genetic markers in $S^*$ occurring in $v$ relative to the total number of genetic markers. The number of $(n,m)$-genetic markers is $\prod_n^{n+k-1}k_i$ so this proportion is equal to \begin{equation}\label{keeping things in proportion}
{|S^*|\over \prod_{n}^{n+k-1}k_i}. \end{equation} This is simply a restatement of our discussion involving empirical distributions in Section \ref{sequences and points}.
We introduce some notation that allows us to compare densities of various sets between odometer based and circular words. For sets $A\subseteq [0,K_m)$ and $A^c\subseteq[0,q_m)$ we denote their densities by: \begin{eqnarray*}
d_m(A)&=&|A|/K_m\\
d_m^c(A^c)&=&|A^c|/q_m \end{eqnarray*} Then $d_m$ and $d_m^c$ can be viewed as discrete probability measures on the sets $[0,K_m)$ and $[0,q_m)$ respectively.
\hypertarget{71}{\begin{lemma} \label{dracula} Let $n\le m$, $w\in \mathcal W_m$ and $w^c=_{def}c_m(w)\in \mathcal W^c_m$. {We view $w$ as sitting on the interval $[0,K_m)$ and $w^c$ as sitting on $[0,q_m)$} Let $S^*$ be a collection of $(n,m)$-genetic markers, $g$ the total number of $(n,m)$-genetic markers
and $d=|S^*|/g$.} If: \begin{itemize} \item $A=\{k\in [0,K_m):$ some $u\in \mathcal W_n$ with genetic marker in $S^*$ begins at $k$ in $w\}$ \item $A^c=\{k\in [0, q_m):$ some $u^c\in \mathcal W_n^c$ with genetic marker in $S^*$ begins at $k$ in $w^c\}$, \end{itemize} then the following equations hold: \begin{eqnarray} d_m(A)&=&{d\over K_n}\label{second mess}\\ d_m^c(A^c)&=&{d\over q_n}\prod_{p=n}^{m-1}(1-1/l_p)\label{first mess}\\ d_m(A)&=&\left({d_m^c(A^c)\over \prod_{p=n}^{m-1}(1-1/l_p)}\right)\left({q_n\over K_n}\right)\label{third mess}\\ d_m^c(A^c)&=&d_m(A)\left(\prod_{p=n}^{m-1}(1-1/l_p)\right)\left({K_n\over q_n}\right).\label{fourth mess} \end{eqnarray}
\end{lemma}
{\par\noindent{$\vdash$\ \ \ }} We prove equation \ref{first mess}. Equation \ref{second mess} is similar but easier. The other two equations follow algebraically.
{ The union of the boundary regions $\partial_p$ for $p=n$ to $m-1$ consist exactly of the elements of $[0,q_m)$ that are not part of any $n$-word. We denote the complement of $\bigcup_{p=n}^{m-1}\partial_p$ by $(\bigcup_{p=n}^{m-1}\partial_p)\tilde{}$. The various $\partial_p$ are pairwise disjoint and for each $n^*$,
$(\bigcup_{p=n^*}^{m-1}\partial_p)\tilde{}$
consists of the locations of entire $n^*$-words. Starting with $p=m-1$, iteratively deleting boundary sections as $p$ decreases to $n$, and using
Lemma \ref{stabilization of names 1} we see that the $d^c_m$-measure of $(\bigcup_{p=n}^{m-1}\partial_p)\tilde{}$ is $\prod_{p=n}^{m-1}(1-1/l_p)$.}
{ Let $B=\{k\in [0,q_m):k$ is at the beginning of an $n$-word$\}$. Then $B$ consists of a $1/q_n$ portion of the regions made up of $n$-words; i.e. $(\bigcup_{p=n}^{m-1}\partial_p)\tilde{}$. We note that $A^c\subseteq B$ and $B$ is disjoint from $\bigcup_{p=n}^{m-1}\partial_p$.}
{ By Lemma \ref{products of imagination}, the number $C_1$ of $n$-words occurring in $w^c$ with a given genetic marker does not depend on the marker. Let $C_2$ be the total number of $n$-words occurring in $w^c$. Then: \begin{eqnarray*}
{|A^c|\over |B|}&=&{|\{n\mbox{-words with genetic marker in }S^*\}\over C_2}\\
&=&{|S^*|*C_1\over g*C_1}\\ &=&d. \end{eqnarray*} We compute conditional expectations to get equation \ref{first mess}:}
\begin{eqnarray*}
d_m^c(A^c)&=&d_m^c(A^c\ |\ (\bigcup_{p=n}^{m-1}\partial_p)\tilde{}\ )\ d_m((\bigcup_{p=n}^{m-1}\partial_p)\tilde{}\ )\\
&=&d_m^c(A^c\ |\ B,(\bigcup_{p=n}^{m-1}\partial_p)\tilde{}\ {})\ d_m(B|\ (\bigcup_{p=n}^{m-1}\partial_p)\tilde{}\ {})\ d_m((\bigcup_{p=n}^{m-1}\partial_p)\tilde{}\ {})\\ &=&d\ \left({1\over q_n}\right) \prod_{p=n}^{m-1}(1-1/l_p) \end{eqnarray*} Equation \ref{second mess} is similar and \ref{third mess}, \ref{fourth mess} follow from the first two equations by substitution. {\nopagebreak
$\dashv$ \par
} The following relationship between pairs of measures $\nu$ on ${\mathbb K}$ and $\nu^c$ on ${\mathbb K}^c$
\begin{equation}\notag
\nu^c(\langle c_n(u)\rangle)=\left({K_n\over q_n}\right)\nu(\langle u\rangle)\left(1-\sum_n^\infty \nu^{c}(\partial_m)\right)
\end{equation} is the limit of equation \ref{fourth mess} as $m$ goes to infinity. This relationship will hold for a correspondence between measures that we build in forthcoming sections.
We note that since $\partial_m$ has a density that depends only on the circular coefficient sequence, the measures of $\partial_m$ is the same for all invariant measures. If we set $d^{\partial_n}$ be this density, then we can rewrite the previous equation as:
\begin{equation} \nu^c(\langle c_n(u)\rangle)=\left({K_n\over q_n}\right)\nu(\langle u\rangle)\left(1-\sum_n^\infty d^{\partial_n}\right)\label{nu vs nuc}
\end{equation}
A consequence of equation \ref{nu vs nuc} is that for all basic open sets $u$, $\nu(\langle u\rangle)$ determines $\nu^c(\langle c_n(u)\rangle)$ and vice versa.
For counting arguments the following inequalities will be helpful. \begin{lemma}\label{deep in the heart of things} Let $n$ be a number greater than $0$. Then there are constants $K_n^U, K_n^L$ between 0 and 1 such that for all $k>0$ and $w^c\in \mathcal W^c_{n+k}$ and all collections $S^*$ of $(n,n+k)$-genetic markers,
if \[A^c=\{i:i \mbox{ is the location of a start of an $n$-subword of $w^c$ indexed in } S^*\}\] then \begin{equation}\label{computing measure}
K^L_n|S^*|\le \left({|A^c|\over q_{n+k}}\right)\left(\prod_{m=0}^{n+k-1}k_{m}\right)\le K^U_n|S^*| \end{equation}
\end{lemma} {\par\noindent{$\vdash$\ \ \ }} By equation \ref{helical product} there are
\[|A^c|=|S^*|*\prod_{m=0}^{k-1}q_{n+m}(l_{n+m}-1)\] many $i$ that occur at the beginning of occurrences of $n$-subwords with genetic markers in $S^*$. Since \[q_{n+k}=k_nl_nq_n^2\left(\prod_{m=1}^{k-1}k_{n+m}l_{n+m}q_{n+m}\right)\] we have: \begin{equation}
{|A^c|\over q_{n+k}}=|S^*|*\left({1\over q_n}\right)\left(\prod_{m=1}^{k-1}(1-{1\over l_{n+m}})\right)\left({1\over \prod_{m=0}^{k-1}k_{n+m}}\right).\notag \end{equation} Since the $\langle 1/l_n\rangle$ is a summable sequence, $\prod_{m=1}^{k-1}(1-{1\over l_{n+m}})$ converges as $k$ goes to $\infty$. The inequality \ref{computing measure} follows.{\nopagebreak
$\dashv$ \par
}
Since $K_{n+k}=\prod_{m=0}^{n+k-1}k_{m}$, inequality \ref{computing measure} can be rewritten as: \begin{equation}\label{explicit}
K_n^L{|S^*|\over K_{n+k}}\le {|A^c| \over q_{n+k}}\le K_n^U{|S^*|\over K_{n+k}} \end{equation}
\hypertarget{itms}{\bfni{Infinite genetic markers:}} Suppose that we are given a construction sequence $\langle \mathcal W_n:n\in{\mathbb N}\rangle$ for an odometer based or circular system ${\mathbb K}$, $s\in S$ and an occurrence of an $n$-word $u$ in $s$. Then we can inductively define an infinite sequence of words $\langle u_m:n\le m\in{\mathbb N}\rangle$, letting $u_n=u$, and $u_{m+1}$ to be the $m+1$-subword of $s$ that contains $u_m$. For each $n<m$ we get a genetic marker $\langle j_n^*, j_{n+1}^*, \dots j_{m-1}^*\rangle$, and these cohere as $m$ goes to infinity. We define the \emph{infinite genetic marker} to be $\vec{j}^*=\langle j^*_m:n\le m\in{\mathbb N}\rangle$.
If an $n$-word $u$ occurs inside an occurrence of an $m$-word $v$ in $s$, then $v=u_m$. Thus their infinite genetic markers agree on the tail $\langle j_i^*:m\le i\in{\mathbb N}\rangle$.
As in Remark \ref{rebuilding}, if we are given a sequence of words $\langle u_m:n\le m\rangle$, with $u_m\in \mathcal W_m$, and an infinite sequence $\langle j_m:n\le m\rangle$ such that the genetic marker $j_{m}$ denotes an instance of $u_m$ in $u_{m+1}$ then we can find an $s\in {\mathbb K}$ with $\langle u_m:m\ge n\rangle$ as a tail of its principal subwords. If ${\mathbb K}$ is odometer then $s$ is unique up to a shift of size less than or equal to $K_m$. A similar statement holds for circular systems.
\subsection{$TU$ and $UT$.} To understand the relationships between ${\mathbb K}$ and ${\mathbb K}^c$, we define maps $TU:S\to S^c$ and $UT:S^c\to S$ where $S\subseteq {\mathbb K}$ and $S^c\subseteq {\mathbb K}^c$ are as in definition \ref{def of S}. The map $TU$ will be one-to-one but $UT$ will not, in general it is continuum-to-one. Nevertheless $UT\circ TU$ will be the identity map.
We begin by considering a element $s\in S$. Let $u_n$ be the principal $n$-subword of $s$.
The sequence $\langle u_n:n\in{\mathbb N}\rangle$ determines a sequence of circular words $\langle u_n^c:n\in{\mathbb N}\rangle$ which we assemble to define $TU(s)$. Let $\vec{j}=\langle j_n:n\in{\mathbb N}\rangle$ be the infinite genetic marker of $s(0)$. To describe $TU(s)$ completely we need to define $\langle r^c_n:n\in{\mathbb N}\rangle$. Set $r^c_0=0$, and inductively define $r^c_{n+1}$ to be the $(r^c_n)^{th}$ position in the first occurrence of an $n$-word with genetic marker $j_n$ in $u^c_{n+1}$. \hypertarget{def of TU}{Set $TU(s)$} to be the element of ${\mathbb K}^c$ with principal subwords $\langle u_n^c:n\in{\mathbb N}\rangle$ and location sequence $\langle r_n^c:n\in{\mathbb N}\rangle$.
We \hypertarget{def of UT}{define a map $UT$} that associates an element of ${\mathbb K}$ to each element of $S^c$.
Given such an $s^c\in S^c$, let $\langle u_n^c:n\ge N\rangle$ be its sequence of principal $n$-subwords. For each $n\ge N, u^c_n$ occurs as $u_{j^*_n}$ in the preword corresponding to $u_{n+1}^c$. Let $u_n=c_n^{-1}(u^c_n)$. Then the sequence of words $\langle u_n:n\in{\mathbb N}\rangle$ and genetic markers $\langle j_n^*:n\ge N\rangle$ determine an element of $s\in{\mathbb K}$ except for the location of 0 in the double ended sequence. (The sequence is double ended because $s\in S^c$.)
We determine this location arbitrarily in a manner that makes the sequence of $u_n$'s the principal $n$-blocks of $s$ ($n\ge N$) and the $j_n^*$ the sequence of genetic markers of these $n$-blocks. Let $\bar{0}$ be a sequence of zeros of length $N$. Then $\bar{0}^\frown \langle j_n^*:n\ge N\rangle$ is a well-defined member of the odometer $\mathcal O$ associated with ${\mathbb K}$. From equation \ref{inductive rns}, $\bar{0}^\frown \langle j_n^*:n\ge N\rangle$ determines a sequence $\langle r_{n}:n\in {\mathbb N}\rangle$. Thus by Lemma \ref{specifying elements}, the pair $\langle u_n:n\ge N\rangle$ and $\bar{0}^\frown \langle j_n^*:n\ge N\rangle$ determines a unique element $s$ of $ {\mathbb K}$ which we will denote $UT(s^c)$. It is easy to check that $UT\circ TU=id$ and that for each $s\in S$, there is a perfect set of $s^c$ with $UT(s^c)=s$.
We can get more precise information about correspondences between ${\mathbb K}$ and ${\mathbb K}^c$ by noting that if we are given a sequence $\langle u_n:n\in{\mathbb N}\rangle$ of principal subwords of an $s\in S$, the genetic markers $\langle j_n:n\in{\mathbb N}\rangle$ define an element $s^c$ of ${\mathbb K}^c$ up to a choices
$(s^c)^\pi\in {\mathcal K}$. Specifically, suppose that $s^*\in {\mathcal K}$ is such that the infinite genetic marker of $s^*(0)$ is
$\langle j_n:n\in{\mathbb N}\rangle$. Then there is an $s^c\in {\mathcal K}^c$ that has a sequence of principal $n$-blocks $\langle u_n^c:n\in{\mathbb N}\rangle$.
The following lemma will be useful for understanding joinings.
\begin{lemma} \label{wrapping translations} Let $s\in S$. Then $\{TU(sh^k(s)):k\in \mathbb Z\}\subseteq \{sh^k(TU(s)):k\in \mathbb Z\}$. If
$s\in S$, $s^c=TU(s)$ and $u\in \mathcal W_n$, then there is a canonical correspondence between occurrences of $u$ in $s$ and finite regions of $s^c$ where $u^c$ occurs. The occurrences of $u^c$ in these finite regions have the same infinite genetic marker $\langle j_m:m>n\rangle$ in $s^c$ as $u$ does in $s$.
\end{lemma} {\par\noindent{$\vdash$\ \ \ }} Given an $s\in S$ and a $k$, the shift $sh^k(s)$ and $s$ have a tail of the principal $n$-blocks $\langle u_n:N\le n\rangle$ in common. Moreover the genetic markers associated with this tail are the same for both $s$ and $sh^k(s)$. It follows that $TU(sh^k(s))$ is a shift of $TU(s)$.
We can describe the correspondence as follows. If $u$ occurs in $s$ at $k$, then $u$ is the principal $n$-word of $sh^k(s)$. Choose an $N$ so large that some $N$-word $u^*$ is the principal $N$-word of both $s$ and $sh^k(s)$. Then $(u^*)^c$ is the principal $N$-block of $s^c$. Let $\vec{j}$ be the genetic marker of the occurrence of $u$ (at $k$) in $u^*$. The region of $s^c$ corresponding to this occurrence of $u$ is the collection of occurrences of $u^c$ with the genetic marker $\vec{j}$ in the principal $N$-block of $s^c$.{\nopagebreak
$\dashv$ \par
}
\subsection{Transferring measures up and down, I} In this section we develop the tool we need for lifting measures on ${\mathbb K}$ to measures on ${\mathbb K}^c$. This will also allow us to establish a one-to-one correspondence between synchronous joinings on odometer systems and synchronous joinings on the corresponding circular systems. Throughout this section we will use $\pi$ to denote either the projection of an odometer based system to its canonical odometer factor or a circular system to its canonical circular factor.
We begin with a proposition relating sequences of words in a construction sequence for an odometer based system to sequences of words in a construction sequence for a circular system.
\begin{prop}\label{words tell the story} Let $\langle v_n:n\in{\mathbb N}\rangle$ be a sequence with $v_n\in\mathcal W_n$. Let $v_n^c=c_n(v_n)$. Then:
\begin{enumerate}
\item $\langle v_n:n\in{\mathbb N}\rangle$ is an ergodic sequence iff $\langle v_n^c:n\in{\mathbb N}\rangle$ is an
ergodic sequence.
\item \label{gen iff gen} $\langle v_n:n\in{\mathbb N}\rangle$ is a generic sequence for a measure $\nu$ iff
$\langle v_n^c:n\in{\mathbb N}\rangle$ is a generic sequence for a measure $\nu^c$. In case either sequence is generic, the measures $\nu$ and $\nu^c$ satisfy equation \ref{nu vs nuc}. \end{enumerate} \end{prop}
{\par\noindent{$\vdash$\ \ \ }} Both parts follow immediately from the definitions using equations \ref{all is equal} and \ref{keeping things in proportion} to relate the frequencies of $k$-words $w\in \mathcal W_k$ in $n$-words $u\in \mathcal W_n$, for $k<n$ to the frequencies of $c_k(w)$ in the corresponding $c_n(u)$. Equation \ref{nu vs nuc} follows from the Ergodic Theorem and Lemma \ref{dracula}. {\nopagebreak
$\dashv$ \par
}
We endow that collection of invariant measures on a symbolic system
$({\mathbb K}, sh)$ with the weak* topology.
\begin{theorem}\label{k and kc} Let $\langle \mathcal W_n:n\in{\mathbb N}\rangle$ be a uniquely readable construction sequence for an odometer based system ${\mathbb K}$ and $\langle \mathcal W_n^c:n\in{\mathbb N}\rangle$ be the associated circular construction sequence for ${\mathbb K}^c$. Then there is a canonical affine homeomorphism $\nu\mapsto \nu^c$ between shift invariant measures $\nu$ concentrating on ${\mathbb K}$ and non-atomic, shift invariant measures $\nu^c$ such that equation \ref{nu vs nuc} holds between $\nu$ and $\nu^c$. \end{theorem}
{\par\noindent{$\vdash$\ \ \ }} By Proposition \ref{kiss your S goodbye} and Lemma \ref{dealing with S} we can assume that $\nu$ and $\nu^c$ concentrate on $S$ and $S^c$ respectively.
We begin by defining the correspondence for ergodic measures. Suppose that we are given an ergodic measure $\nu$ and we want to associate a measure $\nu^c$. Let $s\in S$ be a generic point for $({\mathbb K}, \nu)$. Let $\langle v_n:n\in{\mathbb N}\rangle$ be the sequence of principal $n$-blocks of $s$. By Proposition \ref{generic sequences exist for ergodic} this sequence is generic for $\nu$. By Proposition \ref{words tell the story}, if we let $v_n^c=c_n(v_n)$, then $\langle v_n^c:n\in{\mathbb N}\rangle$ is an ergodic sequence. Let $\nu^c$ be the measure associated with $\langle v_n^c:n\in {\mathbb N} \rangle$. Then $\nu^c$ is ergodic and equation \ref{nu vs nuc} holds by Proposition \ref{words tell the story}.
The other direction is similar, let $s^c\in S^c$ be generic for $\nu^c$. Propositions \ref{generic sequences exist for ergodic} and \ref{words tell the story} imply that if $\langle v_n^c:n\in{\mathbb N}\rangle$ is the sequence of principal $n$-blocks of $s^c$ and $v_n=c_n^{-1}(v^c_n)$, then $\langle v_n:n\in{\mathbb N}\rangle$ is ergodic and generic for a measure $\nu$. Again equation \ref{nu vs nuc} holds by Proposition \ref{words tell the story}.
Suppose now that $\nu$ is an arbitrary measure on ${\mathbb K}$. Write the ergodic decomposition of $\nu$ as: \[\nu=\int \nu_i d\mu(i).\] We define $\nu^c$ by \[\nu^c=\int \nu_i^c d\mu(i)\] which gives a corresponding measure on ${\mathbb K}^c$. Since equation \ref{nu vs nuc} holds between corresponding ergodic components $\nu_i$ and $\nu_i^c$, it holds between $\nu$ and $\nu^c$.
By the ergodic decomposition theorem the map $\nu\mapsto \nu^c$ is a surjection. Since the map is invertible, it is a bijection. The map is affine by construction.
It remains to show that it is a homeomorphism. To see that $\nu\mapsto \nu^c$ is weak* continuous it suffices to show that for all $\epsilon>0$ and $n\in {\mathbb N}$ there is a $\delta$ and an $m$ such that for all invariant $\mu, \nu$, if for all $u\in \mathcal W_m$
\[|\mu(\langle u\rangle)-\nu(\langle u\rangle)|<\delta\] we know that for all $v\in \mathcal W_n$ we have
\[|\mu^c(\langle v^c\rangle)-\nu^c(\langle v^c\rangle)|<\epsilon.\] But the equation \ref{nu vs nuc} easily implies this taking $m=n$ and \[\delta<\left({K_n\over q_n}\right)\left(1-\sum_n^\infty d^{\partial_n}\right)*\epsilon/4.\] The argument that the inverse is continuous is the same. {\nopagebreak
$\dashv$ \par
} \begin{definition}We will call a pair $(\nu,\nu^c)$ constructed as in Theorem \ref{k and kc} \hypertarget{corresponding measures}{\emph{corresponding measures}}. \end{definition}
\begin{remark}\label{benjy's remark} It follows from Proposition \ref{words tell the story} that if $\nu$ and $\nu^c$ are corresponding measures on ${\mathbb K}$ and ${\mathbb K}^c$ and $s\in {\mathbb K}$ is arbitrary then $s$ is generic for $\nu$ iff $TU(s)$ is generic for $\nu^c$. The point $s$ is generic just in case its sequence of principal subwords is generic for $\nu$. By item \ref{gen iff gen} of Proposition \ref{words tell the story}, this holds just in case the sequence of principal subwords of $TU(s)$ is generic; i.e. $TU(s)$ is generic. \end{remark}
We can use Theorem \ref{k and kc} to characterize the possible simplexes of invariant measures for circular systems. By a theorem of Downarowicz (\cite{downar}, Theorem 5), every non-empty compact metrizable Choquet simplex is affinely homeomorphic to the simplex of invariant probability measures for a dyadic Toeplitz flow. Note that the space of invariant probability measures is always a compact Choquet simplex, hence this theorem is optimal.
Since Toeplitz flows are special cases of odometer based systems it follows immediately that every non-empty compact metrizable Choquet simplex is affinely homeomorphic to the simplex of invariant measures of a 2-symbol odometer based system.
Let $K$ be a compact Choquet simplex and ${\mathbb K}$ an odometer based system having its simplex of invariant probability measures affinely homeomorphic to $K$. Let ${\mathbb K}^c$ be a circular system corresponding to an odometer based system ${\mathbb K}$. Then the non-atomic measures on ${\mathbb K}^c$ are a Choquet simplex isomorphic to $K$. There are two additional ergodic measures, the atomic measures concentrating on the constant ``$b$" sequence and on the constant ``$e$" sequence. These two atomic measures are isolated among the ergodic measures.
In the forthcoming \cite{part4} we discuss the question of invariant measures further and show that $\mathcal F$ preserves several other properties, such as being \emph{rank one}.
\section{$\mathbf{P^-}, \mathbf{P^\natural}$, genetic markers and the $\natural$-map}
Our goal is to understand the structure of synchronous and anti-synchronous joinings between pairs of ergodic systems $({\mathbb K}, \mathbb L^{\pm 1})$. We will use Theorem \ref{k and kc} to define a bijection between synchronous joinings of odometer based systems and synchronous joinings of circular systems. This is relatively easy: to a joining of ${\mathbb K}$ with $\mathbb L$ that projects to the identity we can directly associate an odometer system $({\mathbb K},\mathbb L)^\times$ with a measure $\nu$ such that the corresponding measure $\nu^c$ on $(({\mathbb K},\mathbb L)^\times)^c$ can be identified with a measure on ${\mathbb K}^c\times \mathbb L^c$ that projects to the identity. We carry this construction out in detail in section \ref{the categories} and show that the map $\nu\mapsto \nu^c$ given by Theorem \ref{k and kc} gives a bijection between synchronous joinings of the two kinds of systems.
The situation for anti-synchronous joinings of ${\mathbb K}$ and $\mathbb L{^{-1}}$ is more complicated. In Lemma \ref{joining correspondence}, we remarked that the anti-synchronous joinings of ${\mathbb K}$ and $\mathbb L{^{-1}}$ can be identified with joinings of ${\mathbb K}$ and $\rev{\mathbb L}$ that concentrate on $\{(s,t):\pi s=\pi t\}$. Similarly we can identify the anti-synchronous joinings of ${\mathbb K}^c$ and $(\mathbb L^c){^{-1}}$ with joinings of ${\mathbb K}^c$ with $\rev{\mathbb L^c}$ that concentrate on
$\{(s^c,t^c):\pi t^c=\natural(\pi s^c)\}$. We give notation for these sets:
\begin{enumerate}
\item Let $\mathbf{P^-}$ be the collection of anti-synchronous joinings $\rho$ of ${\mathbb K}$ and $\mathbb L{^{-1}}$.
\item Let $\mathbf{P^\natural}$ be the collection of anti-synchronous joinings $\rho^c$ of ${\mathbb K}^c$ and $(\mathbb L^c){^{-1}}$.
\end{enumerate}
To understand the relationship between $\mathbf{P^-}$ and $\mathbf{P^\natural}$ we need an analogue of Lemma \ref{dracula}, and the corresponding analogue of equation \ref{nu vs nuc}. We now describe the tools we use to do this.
Fix construction sequences for $\langle\mathcal U_n:n\in{\mathbb N}\rangle$ and $\langle\mathcal V_n:n\in{\mathbb N}\rangle$ for ${\mathbb K}$ and $\mathbb L$ respectively based on $\langle k_n: n\in{\mathbb N}\rangle$ and ${\mathbb K}^c, \mathbb L^c$ the corresponding circular systems based on $\langle k_n, l_n:n\in{\mathbb N}\rangle$.
Let $(s,t)$ be an arbitrary point in ${\mathbb K}\times \mathbb L$ with $\pi t=-\pi s$ and $s\in S^{\mathbb K}, t\in S^\mathbb L$. Let $\langle u_n:n\in{\mathbb N}\rangle$ and $\langle v_n:n\in{\mathbb N}\rangle$ be the sequence of principal subwords of $s$ and $t$ respectively. If $s^c=TU(s)$ and $t^c=TU(t)$, then $\langle u_n^c:n\in{\mathbb N}\rangle$ and $\langle v_n^c:n\in{\mathbb N}\rangle$ are the sequences of principal subwords of $s^c$ and $t^c$.
Let $x=\natural(\pi s^c)$. Then $x\in \rev{{\mathcal K}}$ and set $r_n=r_n(x)$.
\begin{definition}\label{t-hat} Define $\hat{t}\in \rev{\mathbb L^c}$ by taking $\langle \rev{v_n^c}:n\in {\mathbb N}\rangle$ as its principal $n$-subword sequence and $\langle r_n:n\in {\mathbb N}\rangle$ as its location sequence. \end{definition}
We will study the relationship between $\mathbf{P^-}$ and $\mathbf{P^\natural}$ via the function taking $(s,t)$ to $(s^c, \hat{t})$.
\subsection{Genetic Markers revisited}
To understand the relationship between joinings $\rho$ in $\mathbf{P^-}$ and $\rho^c$ in $\mathbf{P^\natural}$ we need to take into account the manner that $\natural$ shifts the reverse of the second coordinate of a the image of a generic pair $(s,t)$ for ${\mathbb K}\times \mathbb L{^{-1}}$ and the interplay between the map $\natural$ and genetic markers. Let $n<m$.
Suppose that $(u',\rev{v'})$ is a pair of $n$-words coming from $\mathcal U_n\times\rev{\mathcal V_n}$ that occur aligned inside $m$-words $(u, \rev{v})\in \mathcal U_m\times \rev{\mathcal V_m}$. If $u'$ and $\rev{v'}$ occur at the same location in $(u,\rev{v})$, then $\vec{j}_{u'}$ determines $\vec{j}_{v'}$ in the following way:
\begin{quotation}
\noindent for $n\le r<m$ we must have
\begin{equation}(j_{v'})_r=k_r-(j_{u'})_r-1 \label{conjugal gm's}
\end{equation}
(where $\vec{j}_{u'}=(j_n, j_{n+1}, \dots j_{m-1})$). \end{quotation}
\begin{definition}\label{conjugate pairs def} Let $(u',v')\in \mathcal W_n$ and $(u,v)\in \mathcal W_m$. Define the $(n,m)$-\emph{genetic marker of an occurrence of the pair} $(u',\rev{v'})$ in $(u,\rev{v})$ to be $(\vec{j}_{u'},\vec{j}_{v'})$ where $\vec{j}_{u'}$ is the genetic marker of $u'$ in $u$ and $\vec{j}_{v'}$ is the genetic marker of ${v'}$ in ${v}$.\footnote{Note that the genetic marker $\vec{j}_{u'}$ denotes a different position inside $\rev{v}$ then it does in
$u$.} We call $\vec{j}_{u'}$ and $\vec{j}_{v'}$ a \emph{conjugate pair}.
\end{definition} Being a conjugate pair is equivalent to satisfying the numerical relationship given in equation \ref{conjugal gm's} and thus either element of a conjugate pair determines the other. Hence for purposes of counting conjugate pairs we need only use the first coordinates, $j_{u'}$.
Let $(u,\rev{v})\in \mathcal U_m\times \rev{\mathcal V_m}$ be words that occur in a pair $(s,\rev{t})\in {\mathbb K}\times\rev{\mathbb L}$. Then the relative alignment of $u^c$ and $\rev{v^c}$ in $(s^c,\hat{t}\ )$ is determined by the $\natural$-map. This is approximated with a high degree of accuracy by where the code $\Lambda_m$ sends intervals. Accordingly:
\begin{definition}\label{uvc} Define the pair $(u,\rev{v})^c$ to be $(u^c, sh^{A_m}(\rev{v^c})$. \end{definition} Thus $\langle(u,\rev{v})^c\rangle_l$ determines a basic open interval in ${\mathbb K}^c\times \rev{\mathbb L^c}$ which we might also write as $(\langle u^c\rangle_l\times \rev{\mathbb L^c})\cap ({\mathbb K}^c\times \langle\rev{v^c}\rangle_{l+A_m})$. Alternatively we could write this as: \begin{eqnarray*}\{(f,g)\in {\mathbb K}^c\times \rev{\mathbb L^c}:f\ensuremath{\upharpoonright}[l,l+q_m)=u^c \mbox{ and }\\ g\ensuremath{\upharpoonright}[l+A_m, l+A_m+q_m)=\rev{v^c}\}. \end{eqnarray*}
We now have a lemma extending Lemma \ref{first slip} which says that if $u$ and $v$ belong to $\mathcal U_{n+1}$ and $\mathcal V_{n+1}$ then, relative to $sh^{-j_1}(\rev{v})$, all occurrences of $(u')^c\in \mathcal U_n^c$ in $u^c$ are either lined up with an occurrence of a $\rev{(v')^c}$ for some $(v')^c\in \mathcal V_n^c$ or a boundary section of $sh^{-j_1}(\rev{v^c}))$. The lemma also says that if $(u')^c, \rev{(v')^c}$ are lined up then $\vec{j}_{u'}$ and $\vec{j}_{v'}$ form a conjugate pair.\footnote{In this case both $\vec{j}_{u'}$ and $\vec{j}_{v'}$ are of length one.}
\begin{prop}\label{fair and equal} Let $n<m$ and $u\in\mathcal U_m, v\in \mathcal V_m$. Then for $u'\in \mathcal U_n, v'\in\mathcal V_n$ we consider occurrences of $(u',\rev{v'})^c$ in $(u,\rev{v})^c$.\footnote{Since $A_m\ne A_n$ we are considering different shifts in $(u,\rev{v})^c$ and $(u',\rev{v'})^c$.} \begin{enumerate} \item If $(u',\rev{v'})^c$ occurs in $(u,\rev{v})^c$, then $\vec{j}_{u'}$ and $\vec{j}_{v'}$ form a conjugate pair. \item There is a constant $C=C(n,m)$ such that all conjugate pairs occur $C$ times. \item Fix a conjugate pair $(j_{u'},j_{v'})$ of genetic markers of $(u',\rev{v'})$. If $k$ is a location of an occurrence of $(u')^c$ in $u^c$ with genetic marker $\vec{j}_{u'}$, but not a location of $(u',\rev{v'})^c$, then the section of $sh^{A_m}(\rev{v^c})$ in the interval $[k+A_n, k+A_n+q_n)$ is contained in $\bigcup_{i=n+1}^m\partial_i$.
\end{enumerate} \end{prop} {\par\noindent{$\vdash$\ \ \ }} Item 1 is immediate from the definitions.
The latter items are asking about pairs of the form $((u')^c, sh^{A_n}(\rev{(v')^c}))$ occurring in $(u^c,sh^{A_m}(\rev{v^c})$. Such a pair occurs at $k$ if and only if the pair $((u')^c, \rev{(v')^c})$ occurs aligned in $(u,sh^{A_m-A_n}(\rev{v^c}))$ at $k$. Item 3 is equivalent to saying that $(u')^c$ is lined up with a portion of $sh^{A_m-A_n}(\rev{v^c})$ contained in $\bigcup_{i=n+1}^m\partial_i$.
We fix $m$ and prove 2 and 3 by induction on $m-n$. The case that $m=n+1$ is the content of Lemma \ref{first slip}. Suppose that the proposition is true for $m$ and $n+1$, we prove it for $m$ and $n$.
A pair of $n+1$-circular words $(w_0,w_1)^c$ lined up in the shifted pair
$(u^c, sh^{A_m-A_{n+1}}(\rev{v^c}))$ must have conjugate genetic markers. Moreover any there is a number $C_0$ such that any pair with conjugate genetic markers occurs lined up $C_0$ many times.
Fix an occurrence $k$ of an $n+1$-word $w_0$ so that no word in $sh^{A_m}(\rev{v^c})$ occurs at $[k+A_{n+1}, k+A_{n+1}+q_{n+1})$, i.e $w_0$ is not lined up with the reverse of an $n+1$-word in $sh^{A_m-A_{n+1}}(\rev{v^c})$. Then $w_0$ is lined up with a segment of $sh^{A_m-A_{n+1}}(\rev{v^c})$ that is a subset of in $\bigcup_{n+2}^m \partial_i$. To pass from $A_m-A_{n+1}$ to $A_m-A_n$ we shift by $-j_1$, where $j_1=p_{n}^{-1} \mod{q_{n}}$. Noting that each reversed $n+1$-word ends with a string of $b$'s of length $q_n$, we see that after the additional shift there can be no $n$-subwords inside $w_0$ lined up with anything besides a portion of $sh^{A_m-A_n}(\rev{v^c})$ contained in $\bigcup_{n+1}^m\partial_i$.
Suppose that $u'$ and $v'$ are $n$-words and we have an occurrence of $(u')^c$ and $\rev{(v')^c}$ lined up in the pair $(u^c, sh^{A_m-A_n}(\rev{v^c}))$. If $\vec{j}_{u'}=k_0^\frown \vec{j^*_{u'}}$ and $\vec{j}_{v'}=k_1^\frown\vec{j^*_{v'}}$, we let $(w_0,w_1)$ be the occurrence of $n+1$-subwords of $(u,v)$ with genetic markers $\vec{j^*_{u'}}$ and $\vec{j^*_{v'}}$ that contain $u'$ and $v'$. It follows from the previous paragraph that the genetic markers of $w_0$ and $w_1$ are conjugate and $w_0^c, \rev{w_1^c}$ are aligned in $(u^c,sh^{A_m-A_{n+1}}(\rev{v^c}))$. By Lemma \ref{first slip}, $k_0$ and $k_1$ are conjugate and thus $\vec{j}_{u'}$ and $\vec{j}_{v'}$ are conjugate.
Further each conjugate pair occurs aligned the same number $C_1$ of times in the pair $(w^c_0, sh^{-j_1}(\rev{w^c_1}))$. The number $C_1$ is independent of $w_0, w_1$ and $k_0$ and $k_1$. It follows now that given a conjugate pair of genetic markers $(\vec{j}_{u'}, \vec{j}_{v'})$, the number of occurrences of a pair of circular $n$-words with genetic marker $\vec{j}_{u'}$ in $u^c$ aligned with an occurrence of a circular word with genetic marker $\vec{j}_{v'}$ is in $v^c$ is $C_0*C_1$.
To finish we note that the unaligned $n$-words are in two categories, those that are not aligned because the $n+1$-words that contain them are not aligned, or those that are not aligned by the final shift $-j_1$. In each case, the unaligned $n$-words in $u$ occur across from boundary sections in the word $sh^{A_m-A_n}(\rev{v^c})$. {\nopagebreak
$\dashv$ \par
}
Thus, using the backwards ${\mathcal C}$-operation to wrap words around the circle in opposite directions introduces some \emph{slippage}, but the slippage is uniform and predictable.
\begin{definition}\label{slippery definition} Suppose that $\vec{j}$ and $\vec{j'}$ are a conjugate pair of $(n,m)$-genetic markers and $u^c\in \mathcal U^c_m,v^c\in\mathcal V^c_m$. Let $(u')^c$ and $(v')^c$ have genetic markers $\vec{j}$ and $\vec{j'}$ in $u^c, v^c$ respectively. Then the set of locations $k$ such that $(u')^c$ occurs in $u^c$ starting at $k$ with genetic marker $\vec{j}$ but $\rev{(v')^c}$ does not occur starting at $k+A_n$ in $sh^{A_m}(\rev{v^c})$ is called the $(n,m)$-slippage of $\vec{j}$. \end{definition}
A location $k$ can belong to the slippage of $\vec{j}$ for two mutually exclusive reasons. Either, for some proper tail segment $\vec{j^*}$ of $\vec{j}$, $k$ is part of the slippage of the subword of $u^c$ with genetic marker $\vec{j^*}$ or $k$ is part of the slippage of the $j_{n}$ inside the $n+1$ word containing $u$ caused by $sh^{-j_1}$.
Let $SL_{n,m}$ stand for the $(n,m)$-slippage of $n$-subwords of $u^c$; i.e. the locations $k$ in $u^c$ of some $n$-word $(u')^c$ such that there there is no $n$-word $\rev{(v')^c}$ at position $k+A_m$. Inside an $m$-word $u^c$ we find multiple copies of $SL_{n,n+1}$ corresponding the location of each $n+1$ word in $u^c$. Denote the union of these copies as $SL^m_{n,n+1}$. Then it follows that: \begin{equation} SL_{n,m}=\bigcup_{k=n}^{m-1}SL^m_{k,k+1}\cap\{\mbox{locations of $n$-words}\} \label{slippage account} \end{equation} and moreover the union is disjoint.
The slippage is the portion of of the words that we have no control over when counting, so we want to be able to estimate the proportion of words in the slippage. Let \begin{equation}
\varpi^m_n={|SL_{n,m}|\over |\{n-\mbox{subwords of }u^c\}}\label{all of slnm} \end{equation} The next proposition allows us to control the $(n,m)$-slippage by controlling the successive $(n,n+1)$-slippages. \begin{prop}
\begin{equation}\label{piling up} 1-\varpi^m_n=\prod_n^{m-1}(1-\varpi^{i+1}_i). \end{equation}
\end{prop}
{\par\noindent{$\vdash$\ \ \ }} We begin by noting that for $n^*$ between $n$ and $m$, all pairs $(u^*,\rev{v^*})$ of $n^*$-words have the same proportion of slippage of $n$-words in $(u^*, \rev{v^*})^c$. Thus
$\varpi_{n}^{n^*}$ is equal to the proportion of slippage of all of the $n$-words occuring in pairs $(u^*,\rev{v^*})^c$ of $n^*$-subwords of $(u,\rev{v})^c$.
The argument is similar to Lemma \ref{dracula}. Starting with $n^*=m-2$ and decreasing until $n^*=n+1$, using that fact that the union in equation \ref{slippage account} is disjoint, one inductively demonstrates that:
\[(1-\varpi^m_n)=(1-\varpi_{n}^{n^*})\prod_{n^*}^{m-1}(1-\varpi_{i}^{i+1}).\]
{\nopagebreak
$\dashv$ \par
}
We can combine item 3 of Lemma \ref{fair and equal} with equation \ref{piling up} to see that if $k$ is in $SL_{n,m}$, then $[k+A_n, k+A_n+q_n)$ is a subset of $\bigcup_{i={n+1}}^m \partial^{v^c}_i$. It thus follows from Lemma \ref{88bis} that: \begin{equation} 1-\varpi^m_n\ge\prod_n^{m-1}(1-{2\over(l_i-1)}).\label{all that work for this?} \end{equation}
Because the definition of $\varpi^m_n$ was made entirely in terms of genetic markers, the whole discussion could have been carried out simply by considering ${\mathcal K}^c\times \rev{{\mathcal K}^c}$. The numerics depend only on the circular coefficient sequence, not on particular construction sequences $\langle \mathcal U_n, \mathcal V_n:n\in{\mathbb N}\rangle$.
Viewing the operator $\natural$ as the limit of the codes $\Lambda_m$, we can pass to infinity and define \hypertarget{slninf}{$SL_{n}^\infty$} similarly and let $\varpi^\infty_n$ be the proportion of locations $k$ of $n$-subwords of a typical $s\in {\mathbb K}^c$ such that no $n$-subword of $\natural(\rev{\pi(s)})$ occurs at $k+A_n$.
Then: \begin{eqnarray} (1-\varpi^\infty_n)&=&\prod_n^\infty(1-\varpi^{i+i}_i)\notag \\ &\ge&\prod_n^\infty{(1-2/l_i)}\label{squeeze past}\\ &>&0.\notag \end{eqnarray} It follows that $\sum_1^\infty \varpi_i^{i+1}<\infty$.
We now formulate and prove the version of Lemma \ref{dracula} involving the $\natural$ map. One might expect that would require considering arbitrary pairs of genetic markers $\vec{j}$ and $\vec{j'}$. However, by Proposition \ref{fair and equal}, if $u'$ occurs in $u$ with $(n,m)$-genetic marker $\vec{j}$, then the only genetic marker it can occur lined up with in $\rev{v}$ is its conjugate pair. Similarly either of the genetic markers of aligned words $(u')^c$ occurring in $u^c$ and $sh^{A_n}(\rev{(v')^c})$ occurring in $sh^{A_m}(\rev{v})$ determine the other member of the conjugate pair.
It follows that we need only consider pairs $(u',\rev{v'})$ whose genetic markers are conjugate in $(u,\rev{v})$. Since the map $\vec{j}$ to $\vec{j'}$ is a bijection we will refer to either of $\vec{j}$ or $\vec{j'}$ as the genetic marker of a pair $(u',\rev{v'})$ or equivalently $(u',\rev{v'})^c$.
We are reduced to considering sets $S^*\subseteq \{(n,m)\mbox{-genetic markers}\}$ rather than sets of pairs of genetic markers. Let $n<m$ and let $S^*$ be a set of $(n,m)$-genetic markers of pairs of $n$-words in $(u,\rev{v})$. Let
\begin{eqnarray}
A&=&\{k\in [0,K_m): \mbox{some $u'$ with with genetic marker}\notag\\
&& \mbox{in $S^*$ begins at $k$ in } u \} \notag
\end{eqnarray} and
\begin{eqnarray}
A^c&=&\{k\in[0,q_m):\mbox{for some $u'$ with genetic marker in $S^*$,}\notag\\
&&\mbox{ there is a $v'$ such that $(u')^c$ occurs beginning at k in }u^c\notag\\
&&\mbox{and $\rev{(v')^c}$ occurs beginning at }k+A_n \mbox{ in } sh^{A_m}(\rev{v^c})\} \notag
\end{eqnarray} and define
\begin{eqnarray}
d_m(A)&=&|A|/K_m\notag\\
d^c_m(A^c)&=&|A^c|/q_m\notag
\end{eqnarray} If $(u')^c$ occurs at $k$ in $u$ and $\rev{(v')^c}$ occurs at $k+A_n$ in $sh^{A_m}(v)$ then $(u',\rev{v'})^c$ occurs at $k$ in $(u^c,sh^{A_m}(\rev{v^c})$
\begin{lemma}\label{prince of darkness}
Let $n< m$ and $(u,v)\in \mathcal U_m\times \mathcal V_m$. Let $S^*$ be a collection of $(n,m)$-genetic markers, $g$ the total number of $(n,m)$-genetic markers\footnote{As before it is easy to check that $g=\prod_n^{m-1}k_i$.} and $d=|S^*|/g$. Then (in the notation above): \begin{eqnarray} d_m(A)&=&{d\over K_n}\label{deja second mess}\\ d_m^c(A^c)&=&{d\over q_n}\prod_{p=n}^{m-1}(1-1/l_p)(\prod_{i=n}^{m-1}(1-\varpi^{i+1}_i)\label{deja first mess}\\ d_m(A)&=&\left({d_m^c(A^c)\over \prod_{p=n}^{m-1}(1-1/l_p)(\prod_{i=n}^{m-1}(1-\varpi^{i+1}_i)}\right)\left({q_n\over K_n}\right)\label{deja third mess}\\ d_m^c(A^c)&=&d_m(A)\left(\prod_{p=n}^{m-1}(1-1/l_p)\right)\left(\prod_{i=n}^{m-1}(1-\varpi^{i+1}_i)\right)\left({K_n\over q_n}\right).\label{deja fourth mess} \end{eqnarray}
\end{lemma} {\par\noindent{$\vdash$\ \ \ }} The proof is essentially the same as the proof of Lemma \ref{dracula}, indeed the proof of equation \ref{deja second mess} \emph{is} the same. Because all genetic markers occur with the same frequency, after allowing for the portions $u^c$ in boundary sections and in slippage (which are disjoint), $d/q_n$ is the density of locations $k$ of occurrences of words with genetic markers in $S^*$. Once again equations \ref{deja third mess} and \ref{deja fourth mess} follow from \ref{deja second mess} and \ref{deja first mess} by substitution.{\nopagebreak
$\dashv$ \par
}
\noindent The equation relating $\rho\in \mathbf{P^-}$ and $\rho^c\in \mathbf{P^\natural}$ that corresponds to equation \ref{nu vs nuc} is: \begin{equation*} \rho^c(\langle (u, \rev{v})^c\rangle)=\left({K_n\over q_n}\right)\rho(\langle (u,v)\rangle)(1-\sum_n^\infty \rho^c(\partial_m))(1-\varpi^\infty_n). \end{equation*} Once again $\rho^c(\partial_m)$ is independent of the choice of $\rho^c$. Setting $d^{\partial_m}_\rho=\rho^c(\partial_m)$, we can write the previous equation as:
\begin{equation} \rho^c(\langle (u, \rev{v})^c\rangle)=\left({K_n\over q_n}\right)\rho(\langle (u,v)\rangle)(1-\sum_n^\infty d^{\partial_m}_\rho)(1-\varpi^\infty_n).\label{rho vs rhoc} \end{equation}
Understanding empirical distributions of joinings along the natural map involves studying how the slippage affects each pair of $n$-words.
Fix $u'\in \mathcal U_n, v'\in \mathcal V_n$ and $u\in \mathcal U_m, v\in \mathcal V_m$ where $n<m$. Let the conjugate pair $(\vec{j}, \vec{j'})$ be the genetic marker of $(u',\rev{v'})$ in $(u,\rev{v})$. Then, as remarked earlier $\vec{j'}$ is determined by $\vec{j}$, since they are a conjugate pair.
Define $SL_{n,m}(u', \rev{v'})$ to be the collection of locations $k\in SL_{n,m}$ of $n$-subwords of $u^c$ that have genetic marker $\vec{j}$.
Item 2 of Proposition \ref{fair and equal} implies that $|SL_{n,m}(u',\rev{v'})|$ is the same for all choices of $(u',\rev{v'})$. Since $SL_{n,m}$ is the union over all possible pairs of $SL_{n,m}(u', \rev{v'})$, we see that \begin{eqnarray}
\varpi_n^m&=_{def}&{|SL_{n,m}|\over |\{n-\mbox{subwords of }u^c\}|}\notag\\
&=_{\ \ \ }&{|SL_{n,m}(u',\rev{v'})|\over |\{\mbox{subwords of }u^c\mbox{ with genetic marker } \vec{j}\}|} \label{local slnm} \end{eqnarray} From the definition: \[EmpDist_{n,n,A_n}((u,\rev{v})^c)((u')^c,(\rev{v'})^c)\] is equal to \[ {
|\{\mbox{occurrences of }(u',\rev{v'})^c
\mbox{ in }(u,\rev{v})^c\}|\over
|\mbox{for some }(u^*,v^*)\in \mathcal W_n\times \mathcal V_n, (u^*,\rev{v^*})^c \mbox{ occurs in }(u,\rev{v})^c\}|}\]
This in turn is equal to:
\begin{equation}
{(1-\varpi_n^m)|\{\mbox{subwords of }u^c\mbox{ with genetic marker} \vec{j}\}|\over
(1-\varpi_n^m)|\{n\mbox{-subwords of }u^c\}|}
\notag
\end{equation}
which in turn is equal to
\[EmpDist_{n,n,0}(u,\rev{v})(u',\rev{v'}).\]
For notational convenience we write:
\[EmpDist(u,\rev{v})(u',\rev{v'})=_{def}EmpDist_{n,n,0}(u,\rev{v})(u',\rev{v'})\]
and
\[EmpDist((u,\rev{v})^c)((u',\rev{v'})^c)=_{def}\]
\[EmpDist_{n,n,A_n}((u,\rev{v})^c)((u')^c,(\rev{v'})^c).\]
\bfni{Summarizing:}
\begin{equation}\label{just another miracle}
EmpDist(u,\rev{v})(u',\rev{v'})=EmpDist((u,\rev{v})^c)(u',\rev{v'})^c
\end{equation}
\subsection{Transferring measures up and down, II}
In this section we describe the correspondence between joinings in $\mathbf{P^-}$ and $\mathbf{P^\natural}$. We do this by considering generic points for the joinings and transferring them up or down.
For the reader's convenience we repeat a definition. Let $(s,t)$ be an arbitrary point in ${\mathbb K}\times \mathbb L$ with $\pi t=-\pi s$ and $s\in S^{\mathbb K}, t\in S^\mathbb L$. Let $\langle u_n:n\in{\mathbb N}\rangle$ and $\langle v_n:n\in{\mathbb N}\rangle$ be the sequence of principal subwords of $s$ and $t$ respectively. Then $\langle u_n^c:n\in{\mathbb N}\rangle$ and $\langle v_n^c:n\in{\mathbb N}\rangle$ are the sequences of principal subwords of $s^c=TU(s)$ and $t^c=TU(t)$. If $x=\natural(\pi s^c)$, then $x\in \rev{{\mathcal K}}$ and we can set $r_n=r_n(x)$. Recall that we defined $\hat{t}\in \rev{\mathbb L^c}$ by taking $\langle \rev{v_n^c}:n\in {\mathbb N}\rangle$ as its principal $n$-subword sequence and $\langle r_n:n\in {\mathbb N}\rangle$ as its location sequence.
The following follows immediately from equation \ref{all is equal}: \begin{lemma}\label{t and t-hat} The sequence $t$ is generic for an invariant measure $\mu$ on $\mathbb L$ if and only if $\hat{t}$ is generic for an invariant measure $\mu^*$ on $\rev{\mathbb L^c}$. \end{lemma} We will study the relationship between $\mathbf{P^-}$ and $\mathbf{P^\natural}$ via the function taking $(s,t)$ to $(s^c, \hat{t})$. If $[a_n, b_n]$ is the location of the principal $n$-block of $s^c$, we define $w_n^c$ to be the word $(u_n^c, \hat{t}_n)$ (in the language $\Sigma\times \Lambda$) where $\hat{t}_n=\hat{t}\ensuremath{\upharpoonright}[A_n+a_n,A_n+b_n]$. Rephrasing this, if $(u_n, \rev{v_n})$ are the principal $n$-subwords of $(s,t)$ then $w_n^c=(u_n,\rev{v_n})^c$.
\begin{prop} \label{naturally ergodic} The sequence $\langle (u_n,\rev{v_n}):n\in{\mathbb N}\rangle$ is a generic sequence (resp. an ergodic sequence) if and only if $\langle w_n^c:n\in{\mathbb N}\rangle$ is a generic sequence (resp. an ergodic sequence). \end{prop} {\par\noindent{$\vdash$\ \ \ }} This follows immediately from equation \ref{just another miracle}. {\nopagebreak
$\dashv$ \par
} It is worth remarking that Proposition \ref{naturally ergodic} can be restated in the language of Definition \ref{shifted genericity} as saying that $\langle (u_n, \rev{v_n},0):n\in{\mathbb N}\rangle$ is a generic sequence if and only if $\langle (u_n^c, \rev{v_n^c}, A_n):n\in{\mathbb N}\rangle$ is a generic sequence.
The next theorem is the analogue of Theorem \ref{k and kc} adapted to lifting joinings of ${\mathbb K}$ with
$\mathbb L{^{-1}}$ to joining of ${\mathbb K}^c$ with $(\mathbb L^c){^{-1}}$.
In the theorem the notation $(\nu, \nu^c)$ and $(\mu, \mu^c)$ refer to pairs of \hyperlink{corresponding measures}{corresponding measures}.
We assume that ${\mathbb K}$ is built in the language $\Sigma$
and $\mathbb L$ is built in the
language $\Lambda$.
\begin{theorem}\label{k and l inv} Suppose that $\langle \mathcal U_n:n\in{\mathbb N}\rangle$ and $\langle \mathcal V_n:n\in{\mathbb N}\rangle$ are construction sequences for two ergodic odometer based systems $({\mathbb K},\nu)$ and $(\mathbb L,\mu)$ with the same sequence parameters $\langle k_n:n\in{\mathbb N}\rangle$. Let $({\mathbb K}^c, \nu^c)$ and $(\mathbb L^c,\mu^c)$ be the associated ergodic circular systems built with a circular coefficient sequence $\langle k_n, l_n:n\in{\mathbb N}\rangle$. Then there is a canonical affine homeomorphism $\rho\mapsto \rho^c$ between the simplex of anti-synchronous joinings $\rho$ of $({\mathbb K},\nu)$ and $(\mathbb L^{-1},\mu)$ and the simplex of anti-synchronous joinings of $({\mathbb K}^c,\nu^c)$ and $((\mathbb L^c)^{-1},\mu^c)$ such that equation \ref{rho vs rhoc} holds between $\rho$ and $\rho^c$. \end{theorem}
{\par\noindent{$\vdash$\ \ \ }} Suppose that we are given an anti-synchronous ergodic joining $\rho$ between ${\mathbb K}$ and $\mathbb L{^{-1}}$. Let $(s,t)$ be generic for $\rho$. By lemma \ref{generic seqs are ergodic}, the sequence of principal $n$-blocks, $\langle (u_n, \rev{v_n}):n\in{\mathbb N}\rangle$ is ergodic. By Proposition \ref{naturally ergodic} the sequence $\langle w_n^c:n\in{\mathbb N}\rangle$ define an ergodic measure $\rho^c$. Since the $\langle(u_n,\rev{v_n}): n\in{\mathbb N}\rangle$ satisfy equation \ref{deja fourth mess}, the Ergodic Theorem implies that $\rho^c$ and $\rho$ satisfy equation \ref{rho vs rhoc}. It is easy to check that the definition of $\rho^c$ is independent of the choice of the generic pair $(s,t)$.
For the other direction we can assume that we are given a generic pair $(s^c,\hat{t})$ for an ergodic measure $\rho^c$ on ${\mathbb K}^c\times \rev{\mathbb L^c}$ that concentrates on pairs $(s^c, \rev{t^c})\in {\mathbb K}^c\times \rev{\mathbb L^c}$ such that $\pi(\rev{t^c})=\natural(\pi(s^c))$. Taking principal subwords gives us a generic sequence $\langle (u_n^c, \hat{t}_n):n\in{\mathbb N}\rangle$. Each $\hat{t}_n$ is a well-defined word $\rev{v_n^c}$ in $\rev{\mathcal V_n^c}$.
As in the \hyperlink{def of UT}{definition of $UT$} the pair $(s^c, \rev{\hat{t}})$ gives a pair of sequences of genetic markers $(\langle j_n:n\ge N\rangle,\langle j_n':n\ge N\rangle$ for some $N$. Letting $u_n=c_n{^{-1}}(u_n^c)$ and $v_n=c_n{^{-1}}(\rev{\hat{t}_n})$ the sequences $\langle u_n, j_n\rangle$ and $\langle v_n, j'_n\rangle$ determine a pair in ${\mathbb K}\times \mathbb L$ up to finite translations. These sequences are defined independently of the exactly location of the zero of $\hat{t}$; the small shifts used in the definition of $\natural$ do not change the two sequences.
If we let $(s,t)=(UT(s^c), UT(\rev{\hat{t}}))$, making small adjustments if necessary to make $(s,t)$ anti-synchronous, we get an element of ${\mathbb K}\times\mathbb L{^{-1}}$. Applying Proposition \ref{naturally ergodic} again we see the theorem.
We can extend this correspondence to non-ergodic joinings $\rho$ on ${\mathbb K}\times \mathbb L^{-1}$ and $\rho^c$ on ${\mathbb K}^c\times \rev{\mathbb L^c}$, exactly as in Theorem \ref{k and kc}; to go up we take an ergodic decomposition of $\rho$:
\[\rho=\int\rho_id\mu(i)\]
and define
\[\rho^c=\int\rho_i^cd\mu(i).\]
To go down we use the ergodic decomposition theorem and the measure $\mu(i)$ to reverse this process.
Clearly the map $\rho\mapsto \rho^c$ is an affine bijection. It remains to show that it is continous. However, just as in Theorem \ref{k and kc}, we see from equation \ref{rho vs rhoc}, that for each $n$ there is a constant $C_n$, independent of $\rho$ such that for all $u\in \mathcal U_n, v\in \mathcal V_n$,
\begin{equation*}
\rho^c(\langle (u,rev(v))^c\rangle)=C_n\rho(\langle (u, v)\rangle). \end{equation*} This clearly implies that the map $\rho\mapsto\rho^c$ is a weak* homeomorphism. {\nopagebreak
$\dashv$ \par
}
The proof of Theorem \ref{k and l inv} shows that $(s,t)$ is generic for $\rho$ if and only if the pair $(s^c, \hat{t})$ is generic for $\rho^c$. Moreover, the proofs of Theorems \ref{k and kc} and \ref{k and l inv} are quite robust. In particular the constructions of the corresponding measures are independent of the various choices of generic points $s$ or $s^c$, $(s,t)$ or $(s^c,\hat{t}\ )$.
\section{The Main Result}\label{the categories}
We now turn to the main results of this paper. Fix an arbitrary circular coefficient sequence $\langle k_n, l_n:n\in {\mathbb N}\rangle$ for the rest of the section. Let ${\mathcal OB}$ be the category whose objects are ergodic odometer based systems with coefficients $\langle k_n:n\in {\mathbb N}\rangle$. The morphisms between objects $({\mathbb K},\mu)$ and $(\mathbb L,\nu)$ will be
synchronous graph joinings of $({\mathbb K},\mu)$ and $(\mathbb L,\nu)$ or anti-synchronous graph joinings of
$({\mathbb K},\mu)$ and $(\mathbb L^{-1},\nu)$. We call this the \emph{category of odometer based systems.}
Let ${\mathcal C} B$ be the category whose objects consists of all ergodic circular systems with coefficients $\langle k_n,l_n:n\in{\mathbb N}\rangle$. The morphisms between objects $({\mathbb K}^c,\mu^c)$ and $(\mathbb L^c,\nu^c)$ will be synchronous graph joinings of $({\mathbb K}^c,\mu^c)$ and $(\mathbb L^c,\nu^c)$ or anti-synchronous graph joinings of $({\mathbb K}^c,\mu^c)$ and $((\mathbb L^c)^{-1},\nu^c)$. We call this the \emph{category of circular systems.}
\begin{remark} Were we to be completely precise we would take objects in $\mathcal O B$ to be \emph{presentations} of odometer based systems by construction sequences $\langle \mathcal W_n:n\in{\mathbb N}\rangle$ without spacers together with suitable generic sequences and the objects in ${\mathcal C} B$ to be \emph{presentations} by circular construction sequences and their generic sequences. This subtlety does not cause problems in the applications so we ignore it. \end{remark}
The main theorem of this paper is the following: \begin{theorem}\label{grand finale} For a fixed circular coefficient sequence $\langle k_n, l_n: n\in{\mathbb N}\rangle$ the categories $\mathcal O B$ and ${\mathcal C} B$ are isomorphic by a function $\mathcal F$ that takes synchronous joinings to synchronous joinings, anti-synchronous joinings to anti-synchronous joinings, isomorphisms to isomorphisms and {weakly mixing extensions to weakly mixing extensions.} \end{theorem}
Elaborating on Example \ref{for struct}: \begin{corollary}\label{dis sys} The map $\mathcal F$ preserves systems of factor maps (or alternatively extensions). Explicitly: let $\langle I, \le_I\rangle$ be a partial ordering, $\langle X_i:i\in I\rangle$ be a family of odometer based systems and $\langle \pi_{i,j}:j\le i\rangle$ is a commuting family of factor maps with $\pi_{i,j}:X_i\to X_j$. Then $\langle \mathcal F(\pi_{i,j}):j\le i\rangle$ is a commuting family of factor maps among $\langle \mathcal F(X_i):i\in I\rangle$. Moreover the analogous statement holds for circular systems $\langle X_i^c:i\in I\rangle$, factor maps $\langle \pi_{i,j}:j\le i\rangle$ and $\mathcal F^{-1}$. \end{corollary} Theorem \ref{grand finale} can be interpreted as saying that the \emph{whole} isomorphism and factor structure of systems based on the odometer $\langle k_n:n\in{\mathbb N}\rangle$ is canonically isomorphic to the isomorphism and factor structure of circular systems based on $\langle k_n, l_n:n\in{\mathbb N}\rangle$. We call this a \emph{Global Structure Theorem}.
\subsection{The proof of the main theorem} Before we prove theorem \ref{grand finale} we owe the following lemma:
\begin{lemma}\label{paying debt} Both $\mathcal O B$ and ${\mathcal C} B$ are categories, and the composition of synchronous joinings is synchronous, the composition of two anti-synchronous joinings is synchronous and the composition of a synchronous and an anti-synchronous joining (in either order) is anti-synchronous. \end{lemma} {\par\noindent{$\vdash$\ \ \ }} To see that $\mathcal O B$ and ${\mathcal C} B$ are categories we must see that the morphisms are closed under composition. This is equivalent to the statement that the composition of two synchronous or anti-synchronous joinings are synchronous or anti-synchronous. This, in turn follows from Proposition \ref{laziness} (item 2) applied to joinings of odometers or rotations. {\nopagebreak
$\dashv$ \par
}
We now prove Theorem \ref{grand finale}. {\par\noindent{$\vdash$\ \ \ }} By Proposition \ref{bijection from readability} the map $\mathcal F$ gives a bijection between the objects of $\mathcal O B$ and ${\mathcal C} B$ and hence it remains to define the functor on the morphisms (i.e. joinings between systems $({\mathbb K}, \mu)$ and $(\mathbb L^{\pm 1}, \nu)$) and show that it preserves composition.
\subsubsection{Defining $\mathcal F$ on morphisms} \label{f on morphs}We split the definition of $\mathcal F(\rho)$ into two cases according to whether $\rho$ is synchronous or anti-synchronous. In both cases we define $\mathcal F$ for arbitrary joinings even though the only joinings we use as morphisms in the categories are graph joinings; in particular the morphisms in each category are ergodic.
\bfni{Case 1: $\rho$ is synchronous:}
Suppose that $\rho$ a synchronous joining of odometer based systems ${\mathbb K}$ and $\mathbb L$ with coefficient sequence $\langle k_n:n\in{\mathbb N}\rangle$ that are constructed with symbols in $\Sigma$ and $\Lambda$ from construction sequences $\langle \mathcal U_n:n\in{\mathbb N}\rangle$ and $\langle\mathcal V_n:n\in{\mathbb N}\rangle$. We define a new construction sequence $\langle \mathcal W_n:n\in{\mathbb N}\rangle$ with the symbol set $\Sigma\times \Lambda$.
Given $n$, we put a sequence \[\langle (\sigma_0,\lambda_0), (\sigma_1,\lambda_1)\dots (\sigma_{K_{n}-1},\lambda_{K_{n}-1})\rangle\] into $\mathcal W_n$ if and only there are words $u=(\sigma_0,\dots \sigma_{K_{n}-1})\in \mathcal U_n$ and $v=(\lambda_0,\dots \lambda_{K_{n}-1})\in \mathcal V_n$.
It is easy to check that $\langle \mathcal W_n:n\in{\mathbb N}\rangle$ is an odometer based construction sequence with coefficients $\langle k_n:n\in{\mathbb N}\rangle$. Let $({\mathbb K},\mathbb L)^\times$ be the associated odometer based system. Since $\rho$ is synchronous, it concentrates on members of ${\mathbb K}\times \mathbb L$ that correspond to elements of $({\mathbb K},\mathbb L)^\times$. We can canonically identify $\rho$ with a shift invariant measure $\nu$ on $({\mathbb K},\mathbb L)^\times$.
Let $(({\mathbb K},\mathbb L)^\times)^c$ be the circular system associated with $({\mathbb K},\mathbb L)^\times$. We can apply Theorem \ref{k and kc} to find shift invariant measure $\nu^c$ on $(({\mathbb K},\mathbb L)^\times)^c$ associated with $\nu$ that is ergodic just in case $\nu$ is ergodic. Shift invariant measures on $(({\mathbb K},\mathbb L)^\times)^c$ can be canonically identified with synchronous joinings on ${\mathbb K}^c\times\mathbb L^c$. Let $\rho^c$ be the joining of ${\mathbb K}^c\times\mathbb L^c$ corresponding to $\nu^c$. We let $\mathcal F(\rho)=\rho^c$.
Explicitly: A generic sequence $\langle (u_n, v_n,0):n\in{\mathbb N}\rangle$ for the joining $\rho$, can be viewed as a generic sequence $\langle (u_n,v_n):n\in{\mathbb N}\rangle$ for $({\mathbb K},\mathbb L)^\times$ and transformed into a generic sequence $\langle (u_n^c, v_n^c):n\in{\mathbb N}\rangle$ for $(({\mathbb K},\mathbb L)^\times)^c$. The latter corresponds to a generic sequence of the form $\langle (u^c_n,v^c_n,0):n\in{\mathbb N}\rangle$ for the {joining} $\rho^c$. This process is clearly reversible so $\mathcal F$ is a bijection between the synchronous joinings of $\mathcal O B$ and the synchronous joinings of ${\mathcal C} B$.
We must show that if $\rho$ is a graph joining then so is $\rho^c$. Once this is established it follows by symmetry that if $\rho$ is an isomorphism then $\rho^c$ is an isomorphism. Namely if $\rho^*$ is the adjoint joining of $\mathbb L$ with ${\mathbb K}$ defined as $\rho^*(A)=\rho(\{(s,t):(t,s)\in A\})$, then $(\rho^*)^c=(\rho^c)^*$. Hence $\rho^*$ is a graph joining iff $(\rho^c)^*$ is a graph joining.
Suppose that $\rho$ is a graph joining. We apply Proposition \ref{no proof}, part \ref{working graph}. It suffices to show that for all basic open sets in ${\mathbb K}^c$ of the form $\langle u^c\rangle_0$ where $u^c\in \mathcal U_n^c$ and all $\epsilon>0$, there are words $ v_1^c, v_2^c \dots v^c_{k^*}$ that belong to $\bigcup_n\mathcal V_n^c$ and locations $l^c_1, \dots l^c_{k^*}$ such that:
\begin{equation}\label{upchuck}\rho^c((\langle u^c\rangle_0\times {\mathbb L^c})\Delta ({\mathbb K}^c\times \bigcup \langle {v^c_j}\rangle_{l^c_j}))<
\epsilon.
\end{equation}
{Consider $u$ such that $c_n(u)=u^c$. Because $\rho$ is a graph joining, for all $\delta>0$ we can find words $v_1,\dots v_{k'}$ and locations $l_1, \dots l_{k'}$ such that
\begin{equation}\label{upchuck2}\rho((\langle u\rangle_0\times {\mathbb L})\Delta ({\mathbb K}\times \bigcup_{i\le k'} \langle {v_i}\rangle_{l_i}))<
\delta.
\end{equation} Without loss of generality we can assume that for some $m\ge n$ each $v_i$ is an $m$-word and that each $l_i\le 0$.}
Let $(s,t)$ be generic for $\rho$ and considering the pair $s^c=TU(s)$, $t^c=TU(t)$. Then by Remark \ref{benjy's
remark} $(s^c,t^c)$ is generic for $\rho^c$. We will choose words $v_j^c$ and locations $l_j^c$ and compute the measure in inequality \ref{upchuck} by computing the density of locations representing points in the symmetric difference.
\noindent Let
\begin{eqnarray}
B_0&=&\{k:\mbox{$u$ occurs at $k$ in $s$, but for no $i$ does $v_i$ occur in $t$}\notag \\
&& \mbox{at $l_i+k$}
\}\label{downstairs s}\notag\\
B_1&=&\{k:\mbox{for some $i$, $v_i$ occurs in $t$ at $k+l_i$ but $u$ does}\label{downstairs t}\notag\\
&&\mbox{not occur in $s$ at $k$}\}\notag
\end{eqnarray}
By inequality \ref{upchuck2}, $B_0\cup B_1$ can be taken to have density less than
$\delta$.
Given words and locations $\{v_j^c, l^c_j:j\in J\}$ we can define two sets $B^c_0, B^c_1\subseteq \mathbb Z$, as follows:
\begin{eqnarray}
B^c_0&=&\{k: u^c\mbox{ occurs in $s^c$ at $k$ but for no $j$ does $v_j^c$ occurs in $t^c$}\notag\\
&&\mbox{at $l^c_j+k$}\}\notag\\
B^c_1&=&\{k:\mbox{for some $j$\ \ $v^c_j$ occurs in $t^c$ at $l^c_j+k$ but $u^c$ does}\\
&& \mbox{\ \ \ not occur in $s^c$ at $k$.}\} \notag
\end{eqnarray} We need to find the words and locations $v_j^c, l_j^c$ so that the density of $B^c_0 \cup B^c_1$ is less than $\epsilon$.
For each $i$, if $-l_i$ is not the location of the beginning of an $n$-word in $v_i$ then dropping $\langle v_i\rangle_{l_i}$ reduces the measure of the symmetric difference in inequality \ref{upchuck2}. Thus, without loss of generality we can assume that for all $i$, there is an $(n,m)$-genetic marker $\vec{j}(i)$ coding the location of the $n$-word in $v_i$ that starts at $-l_i$. Since $B_0\cup B_1$ has density less than $\delta$, the density of $k$ such that either:
\begin{enumerate}
\item $u$ occurs at $k$ but for each $i$, $k$ is not the position of the beginning of an $n$-word with genetic marker $\vec{j}(i)$ in an occurrence of $v_i$ or
\item for some $i$, $k$ \emph{is} the position of the beginning of an $n$-word with genetic marker $\vec{j}(i)$ in an occurrence of $v_i$, but $u$ does not occur at $k$,
\end{enumerate}
has density less than $\delta$.
We are in a position to define the $v_j^c$ and the $l_j^c$. For each $i$ we define index sets $J_i$ and a collection $\{l_j^c:j\in J_i\}$. We arrange the $J_i$'s so that they are pairwise disjoint and for some $k^*$, $\bigcup_i J_i=\{j:1\le j\le k^*\}$. For $j\in J_i$, all of the $v_j^c$ are the same and equal to $c_m(v_i)$. For a fixed $i$, let $\{-l^c_j:j\in J_i\}$ be the collection of locations of the beginnings of $n$-subwords of $c_m(v_i)$ that have genetic marker $\vec{j}(i)$.
To compute the density of $B_0^c\cup B_1^c$, it suffices to consider an extremely large $M$ and compute the density of $B_0^c\cup B_1^c$ inside the principal $M$-subword $(w^c_0,w^c_1)$ of $(s^c, t^c)$. Let $(w_0,w_1)$ be the principal $M$-subword of $(s,t)$ and $c_M(w_0)=w_0^c$ and $c_M(w_1)=w_1^c$.
We now argue as in Lemma \ref{dracula}. Let $d_0$ be the density of $B_0\cup B_1$ in $(w_0, w_1)$
and $d^c_0$ be the density of $B^c_0\cup B^c_1$ in $(w^c_0, w^c_1)$. Among all $n$-words the proportion $d_p$ that begin with an element of $B_0\cup B_1$ is $d_0*K_n$. The density of $k\in \mathbb Z$ that start $n$-words in $(s,t)$ is $(1-\mu(\bigcup_n^\infty\partial_i))/q_n$. Letting $d^*$ be the density of $k\notin \bigcup_n^M\partial_i$, we see that $d^*$ is bounded away from $0$ and $1$ independently of $M$. The proportion $d_p^c$ of circular $n$-subwords of $(w^c_0, w^c_1)$ that begin with a $k\in B^c_0\cup B^c_1$ is \[{d_0^c*q_n\over (1-d^*)}.\]
Since $\rho$ concentrates on $\{(s,t):\pi(s)=\pi(t)\}$ and $\rho^c$ concentrates on $\{(s^c,t^c):\pi(s^c)=\pi(t^c)\}$, the $n$-words with a particular genetic marker in $w_0$ occupy the position of the same genetic marker in $w_1$ and similarly for $w_0^c$ and $w_1^c$. The $(n,M)$-genetic markers set up a one-to-one correspondence between $n$ subwords $u^*$ of $w_0$ and regions of $w_0^c$ that consist of occurrences of $(u^*)^c$ that have the same genetic marker. Each of the regions of $w_0^c$ with the same genetic marker have the same number of $n$-words in them.
Temporarily call an $n$-subword of $(w_0^c, w_1^c)$ \emph{bad} if it begins with a $k$ in $B_0^c\cup B_1^c$ and similarly for $n$-subwords of $(w_0,w_1)$ and $B_0\cup B_1$. Then the property of being bad is determined by the $(n,M)$-genetic marker of the $n$-word: if $k$ is the beginning of $n$-subword of $w_0$ with genetic marker $\vec{j}$, and $k'$ is the beginning of an $n$-subword of $w_0^c$ with the same genetic marker in $w_0^c$, then $k\in B_0\cup B_1$ if and only iff $k'\in B_0^c\cup B_1^c$.
It follows the proportion of bad $n$-subwords of $(w_0,w_1)$ is the same as the proportion of bad subwords of $(w_0^c, w_1^c)$. In otherwords: \[d_p=d_p^c.\] It follows that \[{d_0* K_n}={d_0^c*q_n\over (1-d^*)}.\] Thus by taking $\delta$ small enough and $M$ large enough we can make $d_0$ as small as we want, and thus arrange that $d_0^c\ll\epsilon$ as desired.
To finish showing that $\mathcal F$ is a bijection between graph joinings in each category and isomorphisms in each category we must also show that if $\rho^c$ is a graph joining then so is $\rho$. But this is very similar. Given a $u^c\in \mathcal U^c_n$, and an $\epsilon>0$ we can find $v_1^c, \dots v_{k^*}^c$ and locations $l^c_1,\dots l^c_{k^*}$ so that inequality \ref{upchuck} holds. Again we can assume that for some $m$, for all $j$, $v_j^c\in \mathcal W_m^c$. The numbers $|l_j^c|$ determine locations in $v_j^c$ of beginnings of $n$-words. We can augment our collection of locations by adding more $l_j^c$'s so that if $l$ is the start of a location in $v_j^c$ that has the same $(n,m)$-genetic marker as $l_j^c$, then for some $j'$ we have $l_{j'}^c=-l$ and $v_{j'}^c=v_j^c$. In doing this we do not increase the density of $B_0^c\cup B_1^c$. Reversing the procedure above this gives words $v_j\in \bigcup_n\mathcal V_n$ and locations $l_j$ such that the density of $B_0\cup B_1$ is less than $\epsilon$. (Note the lack of boundary in ${\mathbb K}\times \mathbb L$ makes the computation easier by reducing the density of $B_0\cup B_1$.)
\bfni{Case 2: $\rho$ is anti-synchronous}
On the anti-synchronous joinings we take $\mathcal F$ to be the bijection between anti-synchronous joinings
of $({\mathbb K},\mu)$ with $(\mathbb L^{-1},\nu)$ and of the circular systems $({\mathbb K}^c,\mu^c)$ with $((\mathbb L^c)^{-1},\nu^c)$ defined in Theorem \ref{k and l inv}. We show that $\mathcal F$ takes anti-synchronous graph joinings to anti-synchronous graph joinings and vice versa. Having done this it will follow by a symmetry argument that $\mathcal F$ sends anti-synchronous isomorphisms to anti-synchronous isomorphisms.
Suppose that $\rho$ is an anti-synchronous graph joining; i.e. $\rho$ is a graph joining of ${\mathbb K}$ with $\mathbb L{^{-1}}$ that concentrates on $\{(s,t):\pi(t)=-\pi(s)\}$. The map $x\mapsto \rev{x}$ projects to the odometer map $\pi(x)\mapsto -\pi(x)$; in particular $\rev{\mathbb L}$ is based on the same odometer that $\mathbb L$ is. By Lemma \ref{joining correspondence} we can view $\rho$ as a graph joining of ${\mathbb K}$ with $\rev{\mathbb L}$ that concentrates on $\{(s,t):\pi(s)=\pi(t)\}$. Similarly we view $\rho^c$ as concentrating on ${\mathbb K}^c\times \rev{\mathbb L^c}$.
We must show that for all basic open sets in ${\mathbb K}^c$ of the form $\langle u^c\rangle_0$ where $u^c\in \mathcal U_n^c$ and all $\epsilon>0$, there are words $ v_1^c, v_2^c \dots v^c_{k^*}$ that belong to $\bigcup_n\mathcal V_n^c$ and locations $l^c_1, \dots l^c_{k^*}$ such that:
\begin{equation}
\rho^c((\langle u^c\rangle_0\times \rev{\mathbb L^c})\Delta ({\mathbb K}^c\times \bigcup \langle \rev{v^c_j}\rangle_{l^c_j}))<
\epsilon.\notag
\end{equation}
{Consider $u$ such that $c_n(u)=u^c$. Because $\rho$ is a graph joining for all $\delta>0$ and all large enough $m$ we can find words $v_1,\dots v_{k'}\in\mathcal V_m$ and locations $l_1, \dots l_{k'}$ such that
\begin{equation}\label{upchuck7}\rho((\langle u\rangle_0\times \rev{\mathbb L})\Delta ({\mathbb K}\times \bigcup \langle\rev{v_i}\rangle_{l_i}))<
\delta.
\end{equation} Without loss of generality we can assume that each $l_i\le 0$.} We will take $m$ sufficiently large according to a restriction we define later.
Let $(s,t)$ be generic for $\rho$ and let $\hat{t}$ be as in Definition \ref{t-hat}. Then $(s^c,\hat{t})$ is generic for $\rho^c$. We argue as before considering sets:
\begin{eqnarray}
B_0&=&\{k:\mbox{$u$ occurs at $k$ in $s$, but for no $i$ does $\rev{v_i}$ occur in $\rev{t}$}\notag\\
&& \mbox{at $l_i+k$.}\label{downstairs s natural}
\}\\
B_1&=&\{k:\mbox{for some $i$, $\rev{v_i}$ occurs in $\rev{t}$ at $k+l_i$ but $u$ does}\label{downstairs t natural}\\
&&\mbox{not occur in $s$ at $k$.}\}\notag
\end{eqnarray} Then inequality \ref{upchuck7}, shows that $B_0\cup B_1$ can be taken to have density less than any positive $\delta$.
Given words and locations $\{v_j^c, l^c_j:j\in J\}$ we consider $B^c_0, B^c_1\subseteq \mathbb Z$, as follows:
\begin{eqnarray}
B^c_0&=&\{k: u^c\mbox{ occurs in $s^c$ at $k$ but for no $j$ does $v_j^c$ occurs in $\hat{t}$}\notag\\
&&\mbox{at $l^c_j+k$}\}\notag\\
B^c_1&=&\{k:\mbox{for some $j$, $v^c_j$ occurs in $\hat{t}$ at $l^c_j+k$ but $u^c$ does}\notag\\
&& \mbox{\ \ \ not occur in $s^c$ at $k$.}\} \notag
\end{eqnarray} Given $\{(v_i,l_i):1\le i\le k'\}$, we need to find the words and locations $v_j^c, l_j^c$ so that the density of $B^c_0 \cup B^c_1$ is less than $\epsilon$. As in the synchronous case, for each $i$ we build index sets $J_i$ so that the $J_i$'s to be disjoint and have union the interval $\{j:1\le j\le k^*\}$ for some $k^*$. For all $j\in J_i$ we take $v^c_j=c_m(v_i)$. We need to find a collection of locations $\{l_j:j\in J_i\}$.
Fix an $i\le k'$. Without loss of generality we can assume that $l_i$ is the beginning of a reversed $n$-block $\rev{v'}$ in $\rev{v_i}$, since otherwise, discarding $\langle \rev{v_i}\rangle_{l_i}$ makes inequality \ref{upchuck7} sharper. If $(s_0,\rev{t_0})\in {\mathbb K}\times \rev{\mathbb L}$ is an arbitrary member of \[(\langle u\rangle_0\times {\mathbb L})\cap({\mathbb K}\times \langle \rev{v_i}\rangle_{l_i})\] with $\pi(s_0)=-\pi(t_0)$, then there is an $m$-word $u^*$ such that $s_0\in \langle u^*\rangle_{l_i}$. Let $\vec{j}(i)$ be the genetic marker of $u$ in $u^*$. We note that $\vec{j}(i)$ does not depend on $s_0$, since it is determined entirely by the location of $u$ in $u^*$ and $u^*$ must be aligned with $\rev{v_i}$.
The genetic marker $\vec{j}(i)$ defines a region of $n$-words in $\mathcal U_n^c$ inside an $m$-word in $\mathcal U_m^c$. Let $L_i$ be the collection of $l$ that are at the beginning of an $n$-word in $\mathcal U_n^c$ with genetic marker $\vec{j}(i)$ in an $m$-word in $\mathcal U_m^c$ and set \begin{equation} \{l^c_j:j\in J_i\}=\{A_m-l: l \in L_i\}. \end{equation} This determines the collection $\{v^c_j,l_j^c:1\le j\le k^*\}$.
We now compute the density of $B_0^c\cup B_1^c$ in terms of the density of $B_0\cup B_1$. To do this it suffices to consider a large enough $M$ that $s^c$ has a principal $M$-block $[a_M,b_M)$ and compute densities inside this principal $M$-block. If this is sufficiently small we can deduce that the density of $B_0^c\cup B_1^c$ is small in $\mathbb Z$. By Remark \ref{natural coding}, we can also assume that $M$ is so large that $\natural$ restricted to this principal $M$-block is equal to $\bar{\Lambda}_M$ along this $M$-block; equivalently the principal $M$-block of $\hat{t}$ is $[a_M+A_m, b_M+A_M)$.
From Proposition \ref{fair and equal}, we know that if $I$ is an $m$-sub-block of $s^c\ensuremath{\upharpoonright} [a_M, b_M)$ then either: \begin{enumerate} \item the corresponding sub-block of $\hat{t}$ is at $sh^{A_m}(I)$ or \item $I$ is part of the $(m,M)$-slippage. \end{enumerate} By item 2 of Proposition \ref{fair and equal}, the number of $m$-sublocks in each case that correspond to a given $(n, M)$-genetic marker does not depend on the genetic marker. Further in the second case $sh^{A_m}(I)$ is entirely part of $\bigcup_{m+1}^M\partial_i(\hat{t})$.
We compute the density $d_0^c$ of elements of $B^c_0\cup B^c_1$ by separating them into these two sources. Explicity, we divide into: \begin{description} \item[Slippage:] Those $k\in B^c_0\cup B^c_1$ that begin an $n$-subword of a location of an $m$-subword of $s^c$ that is in the $(m,M)$-slippage. \item[Mistakes:] those $k\in B^c_0\cup B^c_1$ such that $k$ is the location of the beginning of a circular $n$-subword inside $s^c\ensuremath{\upharpoonright} [a_M, b_M)$ and $[k+A_m,k+q_m+A_m)$ is the location of an $m$-word in $\hat{t}$.
\end{description} We compute the density of the Mistakes and the Slippage separately. Again we will call $n$-subwords that begin with elements of $B_0\cup B_1$ or $B_0^c\cup B_1^c$ \emph{bad}.
Both the Mistakes and the Slippage occur at the beginning of $n$-subwords of $s^c\ensuremath{\upharpoonright}[a_M, b_M)$. Define $d_b$ to be density of $\bigcup_{n+1}^M\partial_i$ in $[a_M,b_M)$. Then proportion of $k\in [a_M, b_M)$ that begin $n$-subwords is: \[{1-d_b\over q_n}.\] Of these a proportion $\varpi_m^M$ of the $n$-subwords are in the Slippage. Thus the collection of $k$ that belong to the Slippage has density
\[\varpi_m^M\left({1-d_b\over q_n}\right).\] Since $\varpi_m^M$ goes to zero as $m$ goes to infinity we can make this term as small as desired by taking $m$ large enough.
Let $[a'_M, b'_M)$ be the location of the principal $M$-block of $s$ (and thus of $\rev{t}$). Let $d_0$ be the density of $B_0\cup B_1$ in $[a'_M, b'_M)$.
Suppose now that $k$ belongs to the Mistakes. Let $\vec{j}$ be the $(n,M)$-genetic marker of the word beginning with $k$ in $s^c\ensuremath{\upharpoonright}[a_M, b_m)$. Then there is a unique $k'$ in $[a'_M,b'_M)$ that is at the beginning of an $n$-subword of $s\ensuremath{\upharpoonright}[a'_M, b'_M)$ and has genetic marker $\vec{j}$. By construction, for $k$ that are not in the Slippage: \begin{equation}k\in B_0^c\cup B_1^c \mbox{ iff } k'\in B_0\cup B_1.\label{relative proportions equiv} \end{equation} Let $d_p$ be the proportion of $m$-subwords of $s\ensuremath{\upharpoonright}[a'_M, b'_M)$ that begin with a $k\in B_0\cup B_1$. Since every genetic marker is represented exactly the same number of times in the complement of the slippage (Proposition \ref{fair and equal}), the proportion of words that begin with $k$ in the Mistakes is \begin{equation}d^c_p=d_p*(1-\varpi_m^M).\label{relative proportions} \end{equation} If $d_0$ is the density of $B_0\cup B_1$ in $[a'_M, b'_M)$ and $d^c_0$ is the density of the Mistakes, then \begin{eqnarray} d_0&=&d_p/K_n \label{downstairs density} \\ d_0^c&=& d^c_p\left({1-d_b\over q_n}\right)\label{upstairs density} \end{eqnarray} Putting together equations \ref{relative proportions}, \ref{downstairs density} and \ref{upstairs density}, we see that if we make $d_0$ sufficiently small we can make $d_0^c$ as small as desired.
\bfni{Summarizing:} By taking $M$ large enough, the density of $B_0^c\cup B_1^c$ is well approximated by the density of $B_0^c\cup B_1^c$ inside $[a_m, b_m)$. This is the sum of the density of the $(m,M)$ slippage and the density of the Mistakes. We can make the density of the Slippage arbitrarily small by taking $m$ large enough and the density of the Mistakes arbitrarily small by taking $\delta_0$ sufficiently small. This establishes the claim that if $\rho$ is a graph joining then so is $\rho^c$.
We must show that if $\rho^c$ is a graph joining then so is $\rho$. We suppose that we are given a $u\in \mathcal U_n$, we must find $\{v_i,l_i:i\le k'\}$ so that equation \ref{upchuck7} holds. Let $u^c=c_n(u)$ and approximate $\langle u^c\rangle_0\times \rev{\mathbb L^c}$ using $\{v_j^c, l_j^c:i\le k^*\}$. Again, we can assume that the collection of locations is saturated in the sense that if $l$ is the start of a location in $v_j^c$ that has the same $(n,m)$-genetic marker as $l_j^c$, then for some $j'$ we have $l_{j'}^c=-l$ and $v_{j'}^c=v_j^c$. In doing this we do not increase the density of $B_0^c\cup B_1^c$. We can now use equations \ref{relative proportions}, \ref{downstairs density} and \ref{upstairs density} again to see that if $d_0^c$ is made sufficiently small then so is $d_0$.
Our next claim is that $\rho$ is an isomorphism if and only if $\rho^c$ is an isomorphism. Recall from Proposition \ref{invertible symmetry} that $\rho$ is an isomorphism iff both $\rho$ and $\rho^*$ are graph joinings. Thus if $\rho$ is an isomorphism, both $\rho^c$ and $(\rho^*)^c$ are graph joinings. Since $\natural$ is an involution: \[(\rho^*)^c=(\rho^c)^*.\] Thus if $\rho$ is an isomorphism, so is $\rho^c$.
Reversing this line of reasoning shows that if $\rho^c$ is a graph joining then $\rho$ is.
\subsubsection{$\mathcal F$ preserves composition}\label{composition} To finish the proof that $\mathcal F$ is a functor we must show that $\mathcal F$ preserves composition. The argument splits into four natural cases: composing synchronous joinings, composing a synchronous joining with an anti-synchronous joining on either side and composing two anti-synchronous joinings. We will carefully work out the case for compositions of synchronous embeddings, and discuss the appropriate modification in the cases involving at least one anti-synchronous embedding after Lemma \ref{compositions}.
The cases differ only that the shifts involved in the generic sequences have different forms. For ergodic synchronous joinings generic sequences can be taken to be of the form $\langle (u_n, v_n, 0):n\in{\mathbb N}\rangle$, whereas for anti-synchronous joinings of ${\mathbb K}^c$ and $\rev{\mathbb L^c}$ a natural generic sequence is of the form $\langle (u^c_n, \rev{v^c_n}, A_n):n\in{\mathbb N}\rangle$.\footnote{i.e. $\langle (u_n,\rev{v_n})^c:n\in{\mathbb N}\rangle$.}
\bfni{Preparatory Remarks}
In the characterization of the relatively independent joining $\rho$ of $\rho_1$ and $\rho_2$ given in Lemma \ref{allow us to disintegrate} and Proposition \ref{rel ind join}, the partitions $\ensuremath{\mathcal A}_k, \ensuremath{\mathcal A}'_k$ and $\tilde{\ensuremath{\mathcal A}}_k$ are given by $\langle u_k\rangle_{s_1}, \langle v_k\rangle_{s_2}$ and $\langle w_k\rangle_{s_3}$ for $s_1, s_2, s_3\in \mathbb Z$. Formally the partitions $\ensuremath{\mathcal A}_k\times\ensuremath{\mathcal A}'_k, \ensuremath{\mathcal A}_k\times\tilde{\ensuremath{\mathcal A}}_k$ and $\ensuremath{\mathcal A}'_k\times\tilde{\ensuremath{\mathcal A}}_k$ and $\ensuremath{\mathcal A}_k\times\ensuremath{\mathcal A}'_k\times \tilde{\ensuremath{\mathcal A}}_k$ consist of all possible products of these basic open sets. However, in the situation we are considering we have synchronous and anti-synchronous joinings. For synchronous joinings we can build a generating family for the relatively independent joining $\rho$ of $\rho_1$ and $\rho_2$ by considering products of pairs of basic open intervals in the same locations; e.g. pairs of the form $\langle u_k\rangle_s\times \langle w_k\rangle_s$. As a consequence, for verifying the hypotheses of Proposition \ref{rel ind join} we can restrict our attention to the case where $s^*=0$.
In the case of anti-synchronous joinings we need to distinguish the odometer based from the circular systems. For anti-synchronous joinings of odometer based systems ${\mathbb K}$ with ${\mathbb M}^{-1}$ we can consider only intervals of the form $\langle u_k\rangle_s\times \langle \rev{w_k}\rangle_{s+s^*}$ where $s^*=0$. For anti-synchronous joinings of the circular systems ${\mathbb K}^c$ with ${\mathbb M}^c$, asymptotically the Empirical Distances concentrate on words of the form $\langle u_k^c\rangle\times \langle\rev{w_k^c}\rangle_{A_k}$ (where $A_k$ is the amount of shift for $\natural$ at scale $k$). Moreover, translations of sets of this form generate the measure algebra of the anti-synchronous joining.
Thus in the proof of the next lemma, to verify the hypothesis 3 of Proposition \ref{rel ind join} we can take $s^*=0$ or $s^*=A_k$ depending on whether $\rho_1\circ\rho_2$ is synchronous or anti-synchronous.
Fix odometer based systems ${\mathbb K}$, $\mathbb L$ and ${\mathbb M}$ with construction sequences $\langle \mathcal U_n:n\in{\mathbb N}\rangle$, $\langle \mathcal V_n:n\in{\mathbb N}\rangle$ and $\langle \mathcal W_n:n\in{\mathbb N}\rangle$ respectively. Let $\rho_1$ and $\rho_2$ be synchronous graph joinings of ${\mathbb K}$ and $\mathbb L$, and $\mathbb L$ and $\mathbb M$ respectively and $\rho$ their relatively independent joining over $\mathbb L$.
Since $\rho_1$ and $\rho_2$ are graph joinings so is their composition. Thus the relatively independent joining is ergodic. Hence by Lemma \ref{ill existe}
we can find generic sequences for $\rho_1, \rho_2$ and $\rho$ that satisfy the hypothesis of Proposition \ref{rel ind join}. \begin{lemma}\label{compositions}Let $\langle (u_n, v_n, w_n,0, 0):n\in{\mathbb N}\rangle$ be generic for $\rho$. Then the sequence $\langle (u_n^c, v_n^c, w_n^c,0,0): n\in{\mathbb N}\rangle$ is generic for the relatively independent joining $\rho^c$ of $\rho_1^c$ with $\rho_2^c$. \end{lemma}
Assuming the lemma, we show that $\mathcal F$ preserves compositions. Corollary \ref{en fin} shows that $\langle (u_n, w_n,0):n\in{\mathbb N}\rangle$ is generic for $\rho_1\circ \rho_2$. From the way that $\mathcal F$ is constructed, if $\nu^c=\mathcal F(\rho_1\circ\rho_2)$, then $\langle (u_n^c, w_n^c):n\in{\mathbb N}\rangle$ is generic for $\nu^c$ (viewed as a measure on a circular system). From Lemma \ref{compositions} and Corollary \ref{en fin}, we know that $\langle (u_n^c, w_n^c,0):n\in{\mathbb N}\rangle$ is generic for $\rho_1^c\circ\rho_2^c$. Hence $\mathcal F(\rho_1\circ\rho_2)=\mathcal F(\rho_1)\circ\mathcal F(\rho_2)$ as desired.
\noindent It remains to prove Lemma \ref{compositions}.
{\par\noindent{$\vdash$\ \ \ }} We claim that $\langle (u_n^c, v_n^c, w_n^c,0,0):n\in{\mathbb N}\rangle$ satisfies the hypotheses of Proposition \ref{rel ind join} for the joinings $\rho^c_1$ and $\rho^c_2$.
The first two hypotheses follow immediately: $\rho_1^c$ and $\rho_2^c$ are constructed by taking the generic sequences $\langle (u^c_n, v^c_n,0):n\in{\mathbb N}\rangle$ and $\langle (v_n^c, w_n^c,0):n\in{\mathbb N}\rangle$ determined by $\langle (u_n, v_n,0):n\in{\mathbb N}\rangle$ and $\langle (v_n, w_n,0):n\in{\mathbb N}\rangle$ respectively, and the measures did not depend on the precise generic sequence taken. Hypothesis \ref{hyp 3} remains to be shown.
We are given $\epsilon>0$, $k$ and $s^*$ and need to find $(k')^c, G^c_{(k')^c}$ and the $I_{v^c}$'s so that inequalitites \ref{i} and \ref{ii} hold. Since $\rho_1^c$ and $\rho_2^c$ are synchronous, so is the relatively independent joining. By the preparatory remarks can take $s^*$, the relative location of words in ${\mathbb K}$ and ${\mathbb M}$ to be 0. Since the sequence of $(u_n, v_n,w_n,0,0)$'s is generic for the relatively independent product of $\rho_1$ and $\rho_2$, we can find $k', N, G_{k'}\subseteq \mathcal V_{k'}$ and for each $v\in G_{k'}$ a set $I_v\subset [0, K_{k'})$ such that the conditions in hypothesis \ref{hyp 3} hold in the odometer context.\footnote{For odometer systems, the length of the words in $\mathcal U_{k'}, \mathcal V_{k'}$ and $\mathcal W_{k'}$ is $K_{k'}$, for circular systems the words at stage $k'$ have length $q_{k'}$.}
Choose $k'$ so large that the density $d_b$ of the boundary portions of circular $k'$-words is less than $\epsilon*10^{-6}$ and so that for each $v\in G_{k'}$, there is an $I_v$ with
\[|I_v|>\left({1-(\epsilon*10^{-6})\over 1-d_b}\right)*K_{k'}.\]
Let $(k')^c=k'$, and $G_{k'}^c=\{v^c:v\in G_{k'}\}$. For each $v^c\in G_{k'}^c$ we define the set $I_{v^c}\subseteq [0, q_{k'})$. Each $I_v\subseteq[0,K_{k'})$ and each $s\in I_v$ has a genetic marker $\vec{j}_s$ in $v$. We let $I_{v^c}=\{s^c:s^c$ has the same genetic marker in $v$ as some $s\in I_v$ does in $v\}$. Equation \ref{up or down} implies that
\[{|I_v|\over K_{k'}}={|I_{v^c}|\over q_{k'}}(1-d_b)\]
and thus $|I_{v^c}|>(1-\epsilon)q_{k'}$.
Equation \ref{all is equal} implies that for $v\in G_{k'}$ and all large $n$, \[EmpDist(v_n)(v)=EmpDist(v_n^c)(v^c),\] from which hypothesis \ref{i} follows immediately.
Fix a $v_0^c\in G^c_{k'}$ and an $s^c\in I_{v_0^c}$. Let $v_0\in G_{k'}$ correspond to $v_0^c$, and $s\in I_v$ correspond to $s^c$. Let $(u^c,w^c)\in \mathcal U^c_k\times \mathcal W^c_k$. To see hypothesis \ref{ii}, we need to compute the empirical distributions of $(u^c,w^c), u^c$ and $w^c$ conditioned on $v_0^c$.
Let $A^c$ be the collection of $((u')^c, v_0^c,(w')^c)\in \mathcal U_{k'}^c\times \mathcal V_{k'}^c\times \mathcal W_{k'}^c$ such that $u^c$ occurs at $s^c$ in $(u')^c$ and $w^c$ occurs at $s^c$ in $(w')^c$. Let $B^c$ be the collection of all $((u')^c, v_0^c,(w')^c)\in \mathcal U_{k'}^c\times \mathcal V_{k'}^c\times \mathcal W_{k'}^c$. Then: \begin{equation}\label{upper conditioning}
EmpDist_{k,k,s^c,s^c}(u_n^c, v_n^c, w_n^c|v_0^c)(u^c, w^c)={EmpDist_{k'}(u_n^c, v_n^c, w_n^c)(A^c)\over EmpDist_{k'}(u_n^c, v_n^c, w_n^c)(B^c)}. \end{equation} As in the definition of $\mathcal F$ in Section \ref{f on morphs}, we can view the relatively independent joining $\rho$ on ${\mathbb K}\times_\mathbb L{\mathbb M}$ as concentrating on a single odometer system $({\mathbb K},\mathbb L,{\mathbb M})^\times$ and $\rho^c$, the relatively independent joining of $\rho_1^c, \rho_2^c$ as concentrating on $(({\mathbb K},\mathbb L,{\mathbb M})^\times)^c$, which is canonically isomorphic to ${\mathbb K}^c\times_{\mathbb L^c}{\mathbb M}^c$.
In the odometer system $({\mathbb K},\mathbb L,{\mathbb M})^\times$, consider the set $A$ consisting of those $k'$-words $(u',v_0,w')$ such that $u'$ and $w'$ have $u$ and $v$ in position $s$. Then $A^c=\{((u')^c, v_0^c,(w')^c):(u', v_0,w')\in A\}$. Similarly $B^c=\{((u')^c, v_0^c,(w')^c):(u', v_0,w')\in B\}$. Equation \ref{all is equal} implies that \begin{equation}\label{frontal} EmpDist(u_n,v_n,w_n)(A)=EmpDist(u_n^c, v_n^c, w_n^c)(A^c). \end{equation} and \begin{equation}\label{frontal2} EmpDist(u_n,v_n,w_n)(B)=EmpDist(u_n^c, v_n^c, w_n^c)(B^c). \end{equation} Finally noting that \begin{equation} \label{lower conditioning}
EmpDist_{k,k,s,s}(u_n,v_n,w_n|v_0)(u,w)={EmpDist_{k'}(u_n,v_n,w_n)(A)\over EmpDist_{k'}(u_n,v_n,w_n)(B)}, \end{equation} and using equations \ref{upper conditioning} and \ref{frontal} we see that \begin{eqnarray} \label{nothing matters}
EmpDist_{k,k,s^c,s^c}(u_n^c, v_n^c, w_n^c|v_0^c)(u^c, w^c)=\\
EmpDist_{k,k,s,s}(u_n,v_n,w_n|v_0)(u,w).\notag \end{eqnarray} Arguing in the same manner we see: \begin{eqnarray}\label{again again}
EmpDist_{k,s^c}(u_n^c,v_n^c|v_0^c)(u^c)&=&EmpDist_{k,s}(u_n,v_n|v_0)(u)\\
EmpDist_{k,s^c}(v^c_n,w^c_n|v_0^c)(v^c)&=&EmpDist_{k,s}(v_n,w_n|v_0)(v) \label{again and again} \end{eqnarray} Since for large $n$,
\begin{eqnarray}
&\|EmpDist_{k,k,s,}(u_n, v_n,w_n|v_0)
-EmpDist_{k,s}(u_n,v_n|v)*EmpDist_{k,s}(v_n,w_n)|v)\|\notag\\
&<\epsilon,\notag
\end{eqnarray}
from equations \ref{nothing matters}, \ref{again again} and \ref{again and again} we get the desired conclusion that
\begin{eqnarray}
\|EmpDist_{k,k,s^c,s^c}(u^c_n, v^c_n,w^c_n|v^v_0)\ \ \ \ &\notag\\
-EmpDist_{k,s}(u^c_n,v^c_n|v_0^c)*EmpDist_{k,s}&\!\!\!\!\! (v^c_n,w^c_n|v_0^c)\|\notag
\end{eqnarray}
is less than $\epsilon$. {\nopagebreak
$\dashv$ \par
}
Lemma \ref{compositions} holds where one or both of the joinings $\rho_1$ and $\rho_2$ are anti-synchronous as well, however the shift coefficients for the circular systems are no longer all $0$ but belong to $\{0,\pm A_n\}$ depending on which joinings are anti-synchronous. Similarly $s^*\in \{0, \pm A_{k}\}$. The argument follows the same path until it reaches equation \ref{frontal}. This equation relies, in turn on equation \ref{all is equal}. The analogue of equation \ref{all is equal} for anti-synchronous joinings is equation \ref{just another miracle}, which in turn carries over to the relatively independent product. The upshot is that equations \ref{nothing matters}, \ref{again again} and \ref{again and again} hold after applying the appropriate shifts of $u_n^c$ and $v_n^c$ relative to $u_n^c$.
\noindent This finishes the proof of Theorem \ref{grand finale}.{\nopagebreak
$\dashv$ \par
}
\subsection{Weakly-Mixing and Compact Extensions}\label{wm and comp} We now show that $\mathcal F$ preserves weakly-mixing and compact extensions.
The fact that compact extensions are preserved is due to E. Glasner and we reproduce the proof here with his kind permission. \begin{prop}\label{wm ext} Let $({\mathbb K},\mu)$ and $(\mathbb L,\nu)$ be ergodic and suppose that $\rho$ and $\rho^c$ are corresponding synchronous joinings determining factor maps \begin{eqnarray*} \pi&:&{\mathbb K}\to \mathbb L\\ \pi^c&:&{\mathbb K}^c\to \mathbb L^c. \end{eqnarray*} Then ${\mathbb K}$ is a weakly mixing extension of $\mathbb L$ (via $\pi$) if and only if ${\mathbb K}^c$ is a weakly mixing extension of $\mathbb L^c$ (via $\pi^c$). \end{prop} {\par\noindent{$\vdash$\ \ \ }} Recall that if $\pi:X\to Y$ is a factor map from $(X,\mathcal B,\mu,T)$ to $(Y,{\mathcal C},\nu,S)$, then the extension is weakly-mixing if the relatively independent joining $X\times_Y X$ of $X$ with itself over $Y$ is ergodic relative to $Y$. In case $Y$ is ergodic, this simply means that the relatively independent joining is ergodic.
Suppose that ${\mathbb K}$ and $\mathbb L$ are odometer based systems with construction sequences $\langle \mathcal W_n:n\in{\mathbb N}\rangle$ and $\langle \mathcal V_n:n\in{\mathbb N}\rangle$ respectively. If $\rho$ is a synchronous factor joining of ${\mathbb K}$ over $\mathbb L$, and the extension is weakly-mixing then we can find an ergodic sequence of words $\langle (u_n, v_n, w_n)\in \mathcal W_n\times \mathcal V_n\times \mathcal W_n:n\in{\mathbb N}\rangle$ that is generic for the relatively independent joining of $\rho$ with itself over $\mathbb L$, i.e. $\rho\times_\mathbb L\rho$. This sequence will satisfy the hypotheses of Proposition \ref{rel ind join}. It follows that the sequence of $(u_n^c,v_n^c,w_n^c)$'s is also generic for an ergodic measure $\nu$. As we argued in Lemma \ref{compositions}, the $(u_n^c, v_n^c, w_n^c)$'s also satisfy the hypothesis of Proposition \ref{rel ind join}. It follows that $\nu$ is the relatively independent joining $\rho^c\times_\mathbb L\rho^c$. Since $\nu$ is ergodic $\rho^c$ is weakly mixing.
If, on the other hand the sequence of $(u_n,v_n,w_n)$ is \emph{not} ergodic, then the sequence $(u_n^c, v^c_n,w^c_n)$ is also not ergodic. Hence if $\rho^c$ is weakly-mixing, then $\rho$ is weakly mixing.{\nopagebreak
$\dashv$ \par
}
It is immediate from the Furstenberg-Zimmer structure theorem (\cite{glasbook}, Chapter 10, Proposition 10.14) that $X$ is a relatively distal extension of $Y$ if and only if there is no intermediate extension $Z$ of $Y$, with $X$ being a non-trivial weakly-mixing extension of $Z$. Thus $\mathcal F$ takes measure-distal extensions to measure-distal extensions.
What requires more effort to establish is the following: \begin{prop}(E. Glasner)\label{comp ext} The functor $\mathcal F$ takes compact extensions to compact extensions. \end{prop} {\par\noindent{$\vdash$\ \ \ }} Glasner's proof uses a result proved in the forthcoming \cite{part4}: If $({\mathbb K}, \mu)$ is an ergodic odometer based system the $X$ is a compact group extension of $({\mathbb K}, \mu)$ then there is a representation of $X$ as an odometer based system with the same coefficients.
Since $X$ is a compact extension of $Y$ if and only if $X$ is a factor of a compact group extension of $Y$,\footnote{See \cite{Furstenberg-Weiss} for an explicit statement and proof.} it suffices to show that $\mathcal F$ takes compact group extensions to compact group extensions.
To prove that $\mathcal F$ takes compact group extensions to compact group extensions we use a remarkable theorem of Veech that characterizes group extensions $\pi:X\to Y$ of ergodic systems. The criteria is that every ergodic joining of $X$ with itself that is the identity on $Y$ (i.e. $\rho$, as a measure, concentrates on those pairs $(x_1, x_2)$ such that $\pi(x_1)=\pi(x_2)$) comes from a graph joining which is an isomorphism of $\ensuremath{(X,\mathcal B,\mu,T)}$ that projects to the identity map on $Y$.\footnote{This first appears in \cite{Veech}.}
Explicity, Theorem 6.18, on page 136 of \cite{glasbook} shows that if, in the ergodic decomposition of the relatively independent product $X\times_Y X$, only graph joinings appear, then $X$ is a compact group extension. The converse follows from Proposition 6.15, part 2 in \cite{glasbook}, that if $X$ is a compact group extension of $Y$ then every ergodic self-joining of $X$ over $Y$ which is the identity on $Y$ is a graph joining.
The map $\mathcal F$ takes ergodic joinings to ergodic joinings, and all graph joinings to graph joinings, and the identity joining to the identity joining. Thus we see it preserves group extensions.{\nopagebreak
$\dashv$ \par
}
Furstenberg \cite{FuBook} and Zimmer \cite{zi} independently showed that for every ergodic system $\mathbb X$ there is an ordinal $\alpha$ and a tower of extensions $\langle X_\beta:\beta\le \alpha\rangle$ such that $X_0$ is the trivial system, $X_\alpha=X$ and for all $\beta<\alpha$, $X_{\beta+1}$ is a compact extension of $X_\beta$, unless $\alpha=\beta+1$ where $X_\alpha$ is either a compact or a weakly mixing extension of $X_\beta$. If there is no compact extension at the end of the tower, then $X$ is \emph{measure-distal} and $\langle X_\beta:\beta<\alpha\rangle$ is a \emph{distal tower} approximating $X$. The least ordinal such that $\mathbb X$ can be represented this way is the \emph{distal height} or \emph{distal order} of $\mathbb X$.
Let $({\mathbb K},\mu)$ be an odometer based system and consider the odometer factor $\mathcal O$. Let $({\mathbb K}',\mu')$ be the Kronecker factor of $({\mathbb K},\mu)$. Then we have \begin{center} \begin{equation*} \begin{diagram} \node{({\mathbb K},\mu)}\arrow{s,r}{\pi_1}\\ \node{({\mathbb K}',\mu')}\arrow{s,r}{\pi_2}\\ \node{\mathcal O} \end{diagram} \end{equation*} \end{center} where $\pi_2$ may or may not be a trivial factor map. This tower is carried by $\mathcal F$ to \begin{center} \begin{equation*} \begin{diagram} \node{({\mathbb K}^c,\mu^c)}\arrow{s,r}{\pi_1}\\ \node{(({\mathbb K}')^c,(\mu')^c)}\arrow{s,r}{\pi_2}\\ \node{\ensuremath{\mathcal R}_\alpha} \end{diagram} \end{equation*} \end{center} If ${\mathbb K}'$ is a non-trivial extension of $\mathcal O$, then Glasner's result tells us that $({\mathbb K}')^c$ is a compact extension of $\ensuremath{\mathcal R}_\alpha$, but is silent on the issue of whether $({\mathbb K}')^c$ is discrete spectrum; i.e. we do not know whether $\mathcal F$ takes the Kronecker factor of ${\mathbb K}$ to the Kronecker factor of ${\mathbb K}^c$.
Suppose now that ${\mathbb K}$ is given by a finite tower of factors: \begin{equation*} \begin{diagram} \node{\mathcal O}\node{{\mathbb K}_0}\arrow{w}\node{{\mathbb K}_1}\arrow{w}\node{\dots}\arrow{w}\node{{\mathbb K}_{N-1}={\mathbb K}}\arrow{w} \end{diagram} \end{equation*} where ${\mathbb K}_0$ is the Kronecker factor of ${\mathbb K}$ and for all $i, {\mathbb K}_{i+1}$ is the maximal compact extension of ${\mathbb K}_i$ in ${\mathbb K}$. Then ${\mathbb K}$ is distal of height $N$. The map $\mathcal F$ carries this to a tower of compact extensions \begin{equation*} \begin{diagram} \node{\ensuremath{\mathcal R}_\alpha}\node{{\mathbb K}^c_0}\arrow{w}\node{{\mathbb K}^c_1}\arrow{w}\node{\dots}\arrow{w}\node{{\mathbb K}^c_{N-1}={\mathbb K}^c}\arrow{w} \end{diagram} \end{equation*} From this we see that the distal height of ${\mathbb K}^c$ is either $N$ or $1+N$.
We do not know an example whether the height of ${\mathbb K}^c$ can be $1+N$. However the ordinary skew product construction applied to odometers gives examples of distal height $n$ where $\mathcal O$ is the Kronecker factor. Hence from our analysis we see that \emph{there are} ergodic circular systems with distal height $N$ for all finite $N$.
In \cite{BF}, Beleznay and Foreman proved that for all countable ordinals $\alpha$ there is an ergodic measure preserving transformation $T$ of distal height $\alpha$. In that construction there are no eigenvalues of the operator $U_T$ of finite order. Hence if we let $\mathcal O$ be an odometer with coefficient sequence $\langle k_n:n\in{\mathbb N}\rangle$ going to infinity, $T\times \mathcal O$ is an ergodic transformation with distal height $\alpha$ and zero entropy. In the forthcoming \cite{part4} we see that this implies that $T\times \mathcal O$ can be presented as an odometer based transformation. By the analysis we just gave we see that $(T\times \mathcal O)^c$ is a circular system with height $1+\alpha$. In \cite{part4} we see that $(T\times \mathcal O)^c$ can be realized as a smooth transformation. For infinite $\alpha$, $1+\alpha=\alpha$, hence we have:
\begin{theorem} Let $N$ be a finite or countable ordinal. Then there is an ergodic measure distal diffeomorphism of $\mathbb T^2$ of distal height $N$. \end{theorem}
\subsection{Continuity} Fix a measure space $(X,\mu)$. As noted in Section \ref{symbolic shifts}, we can identify symbolic shifts built from construction sequences with cut-and-stack constructions (whose levels generate $X$). By fixing a countable generating set in advance, we can make this association canonical. The levels in the cut-and-stack construction give the relationship with arbitrary partitions of $X$. In this way the usual weak topology on measure preserving transformation of $X$ described in Section \ref{abstract measure spaces} determines a topology on the presentations of symbolic shifts as limits of construction sequences.
The finitary nature of the maps $\langle c_n:n\in{\mathbb N}\rangle$ that give bijections between words in $\mathcal W_n$ and words in $\mathcal W_n^c$ easily shows that the map $\mathcal F$ is a \emph{continuous} map from the presentations of odometer based systems to presentations of circular systems. Thus we have:
\begin{corollary}\label{continuity} The functor $\mathcal F$ is a homeomorphism from the objects in $\mathcal O B$ to ${\mathcal C} B$. \end{corollary} For the purposes of the complexity of the isomorphism relation we note: \begin{corollary} The map $\mathcal F$ is a continuous reduction of conjugacy between odometer based systems and circular systems. \end{corollary}
\subsection{Extending the main result} In the main result we restricted the morphisms to graph joinings, largely because compositions of graph joinings are ergodic joinings. Unfortunately a composition of ergodic joinings is not necessarily ergodic, and non-ergodic joinings also arise naturally as relatively independent joinings of ergodic joinings. In this section we indicate how to extend our results to the broader categories that include non-ergodic joinings as morphisms. For convenience, we will continue to require that our objects are ergodic measure preserving systems.
Let $\mathcal O B^+$ and ${\mathcal C} B^+$ be the categories that have the same objects as $\mathcal O B$ and ${\mathcal C} B$, but where the collections of morphisms are expanded to include \emph{all} synchronous and anti-synchronous joinings (rather than just graph joinings).
In Section \ref{f on morphs}, the definition of $\mathcal F$ included all such joinings ($\mathcal F(\rho)$ for a non-ergodic $\rho$ was defined via an ergodic decomposition). Thus without modification we can view $\mathcal F$ as a map: \[\mathcal F:\mathcal O B^+\to {\mathcal C} B^+.\] To show that $\mathcal F$ is a morphism between these categories, i.e. to show preserves composition for arbitrary morphisms, we develop a more combinatorial approach to lifting morphisms that coincides with the original definition.
We start by generalizing the notion of a generic sequence of words to include non-ergodic measures. Suppose ${\mathbb K}$ is a symbolic system with a construction sequence $\langle \mathcal W_n:n\in{\mathbb N}\rangle$. Let $\mu$ be a shift invariant measure which we assume is supported on the set $S\subseteq{\mathbb K}$ (where $S$ is given in definition \ref{def of S}). The ergodic decomposition theorem gives a representation of $\mu$ as $\int \mu_pd\lambda(p)$, where each $\mu_p$ is a shift invariant ergodic measure and $\lambda$ is a probability measure on a set $P$ parameterizing the ergodic components. For each $p$, there is a generic sequence of words $\langle w^p_n:n\in{\mathbb N}\rangle$ for the measure $\mu_p$. The main observation is that the set of probability measures on words of a fixed length is compact. Thus for any fixed $k$ and $\epsilon>0$, we can find a finite set $P_k\subseteq P$ of parameters so that for all $p$, there is some $p'\in P_k$ with\footnote{The notions of $EmpDist$ and $\hat{\mu}_k$ are given in the beginning of Section \ref{sequences and points}.}
\begin{equation}\label{ep} \|\hat{\mu}^p_k-\hat{\mu}^{p'}_k\|<\epsilon.\end{equation} This gives a partition of the parameter space into sets $\{E_p:p\in P_k\}$ such that inequality \ref{ep} holds for all $p'\in E_p$.
Now let $n$ be sufficiently large such that for each $p\in P_k$, we can find an element $w_n^p\in \mathcal W_n$ with \begin{equation}\label{*}
\|EmpDist_k(w_n^p)-\hat{\mu}_k^p\|<\epsilon. \end{equation} If we denote $\lambda(E_p)$ by $\alpha(p)$, then $\alpha(p)\ge 0$ and $\sum_{p\in P_k}\alpha_p=1$. It is clear that one can obtain $\hat{\mu}_k$ up to a small error from the finite data $\{(w_n^p,\alpha(p)):p\in P_k\}$, which is a weighted finite collection of words.
For the symbolic sequences that we are interested in, such as the circular systems, the measure of the spacers is independent of the invariant measure $\mu$ (see Section \ref{GMs and coding}). This means that for all $n,p$, the sum $\sum_{w'\in\mathcal W_n}\mu^p_{q_n}(\langle w'\rangle)$ is the same. In this context using inequality \ref{*} we can arrange the inequality: \begin{equation} \notag
\|(\sum_{p\in P_k}\alpha(p)EmpDist_k(w_n^p))-\hat{\mu}_k\|<\epsilon. \end{equation} The measure $\lambda$ is defined on the extreme points of the simplex of shift invariant probability measures and if we choose the finite sets $P_k$ to consist of points that lie in the closed support of $\lambda$ then we an easily ensure that when we go from $(k,\epsilon)$ to a $(k', \epsilon')$ with $k'>k, \epsilon'<\epsilon$ that $P_{k'}\supseteq P_{k}$. Taking a sequence $k\to \infty$ and $\epsilon_k\to 0$ with $\sum\epsilon_k<\infty$, we get a set
$\{\nu_1, \nu_2, \dots \}$ of ergodic measures and finite sets $I_k\subseteq I_{k+1}$ of integers with probability measures $\alpha_k$ on $I_k$ such that $(\sum_{i\in I_k}\alpha_k(i)\nu_i)$ converges to
$\mu$ in the weak* topology.
\begin{definition}\label{generic non-ergodic} Let $n_k$ go monotonically to infinity and $\{(w^i_{n_k},\alpha_k(i))_k\}$ be a weighted sequence of words as above. Suppose that for each $k$ and $i\in I_k$,
$\|EmpDist_k(w_{n_k}^i)-\hat{\nu}_{i,k}\|<\epsilon_k$, then we call $\{(w^i_{n_k},\alpha_k(i))\}$ a \emph{generic sequence} for $\mu$. \end{definition} We note that for a fixed $i$, as $k$ varies $\{w^i_{n_k}\}$ is a generic sequence for $\nu_i$--which is one of the ergodic measures in the support of $\lambda$.
In a manner exactly analogous to the analysis in Section \ref{sequences and points}, Definition \ref{generic non-ergodic} can be extended to products of symbolic systems, allowing for shifting of words in construction sequences.
Restricting our objects to ergodic systems $\ensuremath{(X,\mathcal B,\mu,T)}, \ensuremath{(Y,{\mathcal C},\nu,S)}$ and $(Z,{\ensuremath{\mathcal D}},\tilde{\mu},\tilde{T})$ allows us to deal with the non-ergodic analogue of the material discussed between Definition \ref{empdist for joinings} and Lemma \ref{ill existe} in a relatively straightforward way which we now discuss.
For the analogue of Proposition \ref{rel ind join} in the non-ergodic case let us make the following observation. Fix a non-ergodic joining $\rho$ of $X$ and $Y$ that has ergodic decomposition
$\rho=\int \rho^pd\lambda(p)$, where, by the ergodicity of $X$ and $Y$, each $\rho^p$ is also a joining of $X$ with $Y$. Fix a $k$ and an $\epsilon>0$ and a cylinder set determined by a word $u\in \mathcal W^X_k$, at location $s^*$ and let $\phi$ represent its indicator function. For $k'$ large, by the Martingale convergence theorem, there is a subset $G$ of $Y$ of measure close to one such that when we look at the conditional expectation of $\phi$ with respect to the partition induced by the principal $k'$-words of $y\in G$, for $\ensuremath{\mathcal A}_{k'}$ and compare it to $\mathbb E(\phi|{\ensuremath{\mathcal D}})$, the error is small.
The element of that partition that contains $y$ is given by a word $v_y\in \mathcal W_{k'}^Y$ and a location parameter $s_y$, and the conditional expectation is: \begin{equation}\label{**} {\rho(sh^{s^*}(\langle u\rangle)\cap sh^{s_y}(\langle v_y\rangle))\over \nu(\langle v_y\rangle)} \end{equation} This easily gives a set $G_{k'}\subseteq \mathcal W_{k'}$ with $\hat{\nu}_k(G_{k'})>1-\epsilon$ and a $J_v\subseteq [0,q_{k'})$ such that for $v\in G_{k'}, j\in J_v$, formula \ref{**} gives a good approximation to $\rho_y(sh^{s^*}(\langle u\rangle))$ for most of the $y\in sh^{s_y}(\langle v_y\rangle)$.
If we have a generic sequence of weighted words for $\rho$, then we can use it to calculate the expression in \ref{**}. This observation makes it possible for us to formulate Proposition \ref{rel ind join} for non-ergodic joinings.
We are given ergodic systems $X,Y,Z$ and are given construction sequences $\langle \mathcal U_n,\mathcal V_n,\mathcal W_n:n\in{\mathbb N}\rangle$ such that for each $n$, the words in each $\mathcal U_n, \mathcal V_n, \mathcal W_n$ have the same length. Two joinings $\rho_1$ of $X$ and $Y$ and $\rho_2$ of $Y$ and $Z$ are given. The analogue of Proposition \ref{rel ind join} is now:
\begin{prop} \label{non-erg rel prod} Let \begin{equation}\label{word soup}\langle \{(u^i_{n_k},v^i_{n_k},w^i_{n_k},s^i_{n_k},t^i_{n_k}):i\in I_k\},\alpha_k\in Prob(I_k): k\in{\mathbb N}\rangle \end{equation} be a sequence of weighted words and $\sum \epsilon_k<\infty$. Suppose that the following hypothesis are satisfied:
\begin{enumerate}
\item $\langle \{(u^i_{n_k},v^i_{n_k},s^i_{n_k}):i\in I_k\}, \alpha_k)\rangle_{k}$ is generic for $\rho_1$,
\item $\langle \{(v^i_{n_k},w^i_{n_k},t^i_{n_k}):i\in I_k\}, \alpha_k\rangle_{k}$ is generic for $\rho_2$,
\item For all $\epsilon, k, s^*$ there are $k', N$ and a set $G_{k'}\subset \mathcal W_{k'}^Y$ and for each $v\in G_{k'}$ there is a set $J_v\subseteq [0, q_{k'})$ such that
\begin{enumerate}
\item $\sum_{v\in G_{k'}}EmpDist(v_{n_k})(v)>1-\epsilon$
\item $|J_v|>(1-\epsilon)q_{k'}$
\item For all $v\in G_{k'}$ and $s\in J_v$, if $n_k>N$,
\begin{align*}
\|\sum_{i\in I_k}EmpDist_{k_0,k_0, s,s+s^*}(u^i_{n_k}, sh^{s^i_{k_n}}(v^i_{n_k}),sh^{t^i_{k_n}}(w_{n_k})|v)\alpha_k(i)\ \ \ \ \ \ &-\\\sum_{i\in I_k}EmpDist_{k_0,s}(u^i_{n_k}, sh^{s^i_{k_n}}(v^i_{n_k})|v)\alpha_k(i)* \hskip 1.5in &\\
\sum_{i\in I_k}EmpDist_{k_0,s+s^*}(v^i_{n_k}, sh^{t^i_{n_k}-s^i_{k_n}}(w^i_{n_k})|v)\alpha_k(i)\|& <\epsilon
\end{align*}
\end{enumerate}
\end{enumerate}
Then the weighted sequence given in \ref{word soup} is generic for the relatively independent joining $X\times_Y Z$. \end{prop}
The analogues of Corollary \ref{en fin} and Lemma \ref{ill existe} are easily verified, giving us a characterization of compositions of non-ergodic joinings and the existence of generic sequences satisfying the hypothesis of Proposition \ref{non-erg rel prod}.
Verifying that $\mathcal F$ preserves composition is now straightforward in the manner of Section \ref{composition}: the $G^c_{k'}$ and $J_{v^c}$ are constructed in exactly the same way. Checking the conditional distributions of short words relative to longer words ($k$ vs. $k'$) involves counting $k'$-words, and these are counted using Equation \ref{all is equal} for each component $(u_k,v_k, w_k)$ separately. The weighted average is then preserved.
\section{Lagend} In this section we explore the interplay of the geometric, arithmetic and combinatorial aspects of the manner in which $\mathcal F$ wraps the odometer based words around the circle. The map $\mathcal F$ does not preserve the dynamics of the odometer when transforming it into a rotation, indeed it can't. The shift $sh^k$ of the odometer corresponds to a shift $sh^{k^c}$ of the rotation. The relationship between $k$ and $k^c$ is characterized combinatorially as an optimal wrapping property. The latter is defined in terms of the notion of a \emph{perfect match}. The results in this section can be used to give an alternate proof of the fact that if $({\mathbb K},\mu)$ is ergodic then so is $({\mathbb K}^c,\mu^c)$ that does not use the notion of a generic sequence of words.
Central to our understanding circular systems is the manner in which an $s^c$ had its $n$-words aligned with $n$-words in $sh^k(t^c)$. A word $u$ occurs in $s^c$ lined up with a word $w$ in $sh^k(t^c)$ if and only if $u$ occurs at some location $l$ in $s$ and $w$ occurs at $k+l$ in $t^c$.
\begin{definition}\label{perfect match def} Let $\vec{x},\vec{y}$ be strings in the language $\Sigma\cup \{b, e\}$ and $u, v$ be words of the same length. A \emph{$k$-match} of $u$ and $v$ in $\vec{x}$ and $\vec{y}$ is a location $l$ in the domain of $\vec{x}$ such that $u$ occurs at $l$ in $\vec{x}$ and $v$ occurs at $l+k$ in $\vec{y}$.
If $w^c_0, w^c_1$ are circular $m$-words then a \emph{perfect match} of $u^c,v^c$ in $w^c_0, w^c_1$ is a $k$ such that there are $(n,m)$-genetic markers $\vec{j}_u, \vec{j}_v$ such that $u^c$ occurs in $w^c_0$ and $v^c$ occurs in $w^c_1$ with genetic markers $\vec{j}_u$ and $\vec{j}_v$ respectively and $k$ is a match between all occurrences of $u^c$ and $v^c$ with these genetic markers.
\end{definition} Thus $k$ is a perfect match of $u$ and $v$ if and only if the occurrences of $\vec{j}_u$ in $w_0^c$ are exactly aligned with the occurrences of $\vec{j}_v$ is $w_1^c$.
We will say that \emph{$k$ is a match between $u$ and $v$} if there is a location $l$ such that such that $k$ is a match between $u$ and $v$ at $l$, and that \emph{every $k$-match is perfect} when $k$ has the property that for every occurrence of a pair of words $u^c,v^c$ in $w^c_0,w^c_1$, if $k$ is a match between $u^c,v^c$ then $k$ is a perfect match between $u^c,v^c$. The astute reader will have already recognized that being a match or a perfect match only refers to the genetic markers and the underlying circular factor--thus the actual identities of $u^c,v^c, w^c_0$ and $w^c_1$ are not material--only the locations of the genetic markers.
The notion of a \emph{perfect match} is vacuous for odometer words; for if $u,v$ are odometer $n$-words and $w_0, w_1$ are odometer $m$-words then $u,v$ are the unique pair with a genetic markers $\vec{j}_u$ and $\vec{j}_v$. Moreover, if $k$ matches any pair of $n$-subwords, $k$ matches every pair of corresponding $n$-subwords in the overlap of $w_0$ and $sh^k(w_1)$.
If $k>0$, then the $n+1$-subwords of $w_1$ in the overlap of $w_0$ and $sh^k(w_1)$ are split into two pieces by the $n+1$-subwords of $w_0$; the left portion of each of the $n+1$-subwords of $w_0$ in the overlap coincides with the right portion of the corresponding $n+1$-subword of $w_1$. Call the matches in the left portion of $w_0$ \emph{left-matches}.
\bfni{Discussion.} Let $u^c$ have genetic marker $\vec{j}_{u^c}=(j_n, j_{n+1},\dots j_{m-1})$ in $w_0^c$ and suppose that $u^c$ sits inside the $n+1$ word $(u')^c$ with genetic marker $( j_{n+1},\dots j_{m-1})$. Then words with genetic marker $\vec{j}_u$ sit inside every 2-subsection of $u'$. It follows that if $k^c>0$ and $k^c$ is a perfect match of $u^c$ with $v^c$ having genetic marker $\vec{j}_{v^c}=(j'_n, j'_{n+1},\dots j'_{m-1})$ win $w_1^c$, then $j_n\le j'_n$. Thus the relative position of $v^c$ in the $n+1$-subword of $w_1^c$ with genetic marker $( j'_{n+1},\dots j'_{m-1})$ is to the right of the position of $u^c$ in $(u')^c$; i.e. the relative shift is to the left to match $u^v$ with $v^c$. For this reason, when $k^c>0$ we need only consider left shifts.
It is also easy to see that perfect matches between $n$-words with genetic markers $\vec{j}$ and $\vec{j}'$ inside an $m$-words $w^c_0,w^c_1$ are those $k^c$ that match the first occurrence of an $n$-word with genetic marker $\vec{j}$ in $w^c_0$ with the first occurrence of an $n$-word with genetic marker $\vec{j}'$ in $w^c_1$.
The next lemma says that perfect matches can be viewed as the locations of shifts of odometer based words wrapped around the circle. \begin{lemma}\label{lost in translation} Suppose that $w_0,w_1\in \mathcal W_m$ and $w_i^c=c_m(w_i)$. Let $n<m$ and $0\le k^c<q_m$ and suppose that $k^c$ is a perfect match between some pair of $n$-subwords of $w_0^c$ and $w_1^c$. Then there is a unique $k$ such that for all genetic markers $\vec{j}$, $\vec{j}'$, \begin{itemize} \item $k^c$ is a perfect match between the $n$-subwords of the $w_i^c$ with genetic markers $\vec{j}$ and $\vec{j}'$ iff \item $k$ is a left match between the $n$-subwords of $w$ with genetic markers $\vec{j}$ and $\vec{j}'$. \end{itemize} \end{lemma} The Lemma has an obvious analogue for negative $k^c$ and right matches.
{\par\noindent{$\vdash$\ \ \ }}
Suppose that $k^c$ is a perfect match between $\vec{j}$ and $\vec{j'}$. Call the subwords of $w_0, w_1$ with genetic markers $\vec{j}$ and $\vec{j'}$ $u$ and $v$. Then $u^c, v^c$ are perfectly matched by $k^c$. Let $k$ be the distance between the locations of $u$ and $v$. Since $k^c\ge 0$ we have $k\ge 0$. From our discussion we seen that $k$ is a left match of $u,v$. We claim that this $k$ satisfies the lemma.
Let $u', v'$ be the $n+1$-subwords of $w_0, w_1$ inside which $u,v$ occur. Suppose that $u=u_0u_1\dots u_{k_n-1}$ and $v=v_0v_1\dots v_{k_n-1}$, so $(u')^c={\mathcal C}((u_0)^c,\dots, (u_{k_n-1})^c)$, $(v')^c={\mathcal C}((v_0)^c, \dots, (v_{k_n-1})^c)$. If $u_i, v_j$ are left matched by $k$ in $u',v'$, then the first occurrences of $(u_i)^c$ and $(v_j)^c$ are matched by $k^c$, hence \emph{inside} $(u')^c, (v')^c$, $k^c$ is a perfect match of $(u_i)^c$ and $(v_j)^c$.
The relative position of $(u')^c$ and $(v')^c$ is duplicated over all $n+1$-words with genetic markers $j_{u'}, j_{v'}$ in $w_0$ and $sh^k(w_1)$. It follows that $k^c$ is a perfect match of $u^c$ and $v^c$ inside $w_0^c, w_1^c$.
From the uniformity of the relative positions of $n+1$-words it also follows that any two $n$-subwords of $n+1$-subwords $(u^*)^c, (v^*)^c$ in positions $i, j$ that are $k^c$ matched in $(u')^c, (v')^c$ are $k^c$-matched. Since these exactly coincide with the $n$-subwords of $w_0, w_1$ that are left-matched by $k$, we have proved the lemma.{\nopagebreak
$\dashv$ \par
}
\begin{lemma}\label{perfection is possible} Let $w_0=w_1=w$. Let $n\in {\mathbb N}$ and $k\in \mathbb Z$.Then: \begin{enumerate} \item {Let $M(k)$ be the least $M$ such that $k<q_M$. Let $u, v\in \mathcal W_n^c$. Then if $k$ matches $u,v$ inside $w\in \mathcal W_m^c$ with $m\ge M(k)$ then $u,v$ occur inside the same $M(k)$-subword of $w$. } \label{scale of shift}
\item {Let $m\ge M(k)$. Then $k$ is a perfect match of occurrences of $u$, $v$ inside an $m$-word iff $k$ is a perfect match inside the $M(k)$ word in which they appear. \label{new 2}} \end{enumerate} \end{lemma} {\par\noindent{$\vdash$\ \ \ }} Use Lemmas \ref{gap calculation} and \ref{numerology lemma}. {\nopagebreak
$\dashv$ \par
} Item \ref{new 2} means that we usually don't have to refer to a long words when we are discussing perfect matches of $u$ and $v$ and fixes the scale of the potential perfect matches.
We can identify perfect matches numerically:
\begin{lemma}\label{small nudge} Let $A<q_N$, $w^c\in \mathcal W_N^c$. Then there is a $(m,n)$-genetic marker $\vec{j}$ such that $A$ is the location of the first occurrence of some word genetic marker $\vec{j}$ if and only if
\begin{equation}
A=c_{N-1}l_{N-1}q_{N-1}+c_{N-2}l_{N-1}q_{N-2} + \dots +c_{m}l_mq_m \label{very pretty}
\end{equation}
where $0\le c_i<k_i$. \end{lemma}
From Lemma \ref{lost in translation}, we see the correspondence between odometer translations and circular translation. We now address the question: given an arbitrary circular translation, how does one adjust it to get an odometer translation that gives the best fit among a given collection of $n$-words?
\begin{theorem}\label{perfection is sorta possible} Let $n\in {\mathbb N}$ and $k\in \mathbb Z$. Then
if $\{(u_i, v_i):i\in I\}\subseteq \mathcal W_n^c \times \mathcal W_n^c$, $w\in \mathcal W_m^c$ ($m\ge M(k)$) then there is a $k'$ such that $|k'-k|<q_{m}$ and:
\begin{enumerate}
\item all $k'$-matches of a $(u_i,v_i)$ in $w$ are perfect matches,\label{perfection}
\item and
\[\sum_i |k'\mbox{-matches of a $u_i$ with a $v_i | \ge \sum_i |k$-matches of a $u_i$ with a $v_i|$}\]
\end{enumerate} \end{theorem} {\par\noindent{$\vdash$\ \ \ }} Without loss of generality $k\ge 0$ (otherwise we reverse the role of $u$ and $v$). {Words in $\mathcal W_m$ with $m>M(k)$ start with a block of $b$'s of length at least $q_{M(k)}$. Hence if $k$ matches $n$-words $u, v$ inside $w\in \mathcal W^c_m$, they both must occur in some $M(k)$-subword of $w$.}
To see item \ref{perfection}, we need to show how to improve $k$ to a $k'$ that is a perfect match. Changing $k$ will involve sacrificing some of the matches of pairs in $I$, but this will be compensated by the additional multiplicity of the remaining matches.
{We prove by induction on $d\ge 1$, that for all $m, n$ with $m-n=d$ and all collections of pairs of $m$-words $\{(w_0^j,w_1^j):j\in J\}$ and all $k$, all $m\ge M(k)$, all natural number weightings $\{\alpha_j:j\in J\}$ and all
$\{(u_i,v_i):i\in I\}\subseteq \mathcal W_n^c\times \mathcal W_n^c$} we can find a $k'$ such that $|k'-k|<q_m$ such that (a) holds and \begin{eqnarray*}
\sum_j\sum_i \alpha_j|\{k'\mbox{-matches of a $u_i$ and a $v_i$ in $(w_0^j,w_1^j)$}\}|&\ge\\
\sum_j\sum_i \alpha_j|\{k\mbox{-matches of a $u_i$ and a $v_i$ in $(w_0^j,w_1^j)$}\}| \end{eqnarray*}
Suppose first that $d=1$. Then successive $2$-subsections of $m$-words
are separated by boundary sections of size \[j_i+(q-j_{i+1})\equiv p_n^{-1} \mbox{(mod $q$)}.\] Because $q_n$ does not divide $p_n$, given a $2$-subsection $\vec{s}$ of $w_0^j$ there is a unique $2$-subsection $\vec{t}$ of $w^j_1$ within which $k$ can match $n$-words. Moreover this does not depend on $j$, but rather the underlying locations of the words.
We start by lining up blocks of the form $u_i^{l_n-1}$ with blocks of the form $v_i^{l_n-1}$. To do this we classify the $k$-matches of a pair $(u,v)=(u_i,v_i)$ into \emph{left block matches} if $u$ and $sh^k(v)$ align as\footnote{In both of these graphics the second row is a portion of $sk^k(w^j_1)$ and $B$ represents a boundary section. These pictures are independent of $j$.}: \begin{center} \includegraphics[width=.9\textwidth]{left_block_match} \end{center} and \emph{right block matches} if $u$ and $sh^k(v)$ align as: \begin{center} \includegraphics[width=.9\textwidth]{right_block_match} \end{center} Note by taking $k'$ to be $k+lq_n$ for some $l<l_n-1$ we can turn all left block matches of all {of the }$(u_i,v_i)$ into matches of entire $u_i^{l_n-1}$ with $sh^k(v_i^{l_n-1})$, but doing so destroys completely some of the right block matches. Similarly if we can shift to make all right block matches into matches of $u_i^{l_n-1}$ with $sh^k(v_i^{l_n-1})$ by destroying left block matches.
If we examine a particular left block match of a pair $(u_i,v_i)$ in some $w_0^j$ and a right block match of another pair $(u'_i,v'_i)$ in $w_1^j$ and we change $k$ to $k'$ to make $u_i^{l_n-1}$ match with $sh^{k'}(v_i^{l_n-1})$ then the sum of $k'$-matches between $(u_i,v_i)$ and $(u_i',v_i')$ goes up by one: we lose the right block matches but we gain left block matches and we gain one more match from the boundary section.
Suppose that \begin{eqnarray*}
\sum_j\sum_j \alpha_j |\{\mbox{left block matches in }(w_0^j,w_1^j)|&\ge\\
\sum_j\sum_j \alpha_j |\{\mbox{right block matches in }(w_0^j,w_1^j)| \end{eqnarray*}
Then from the previous paragraph that if we take $k'=k+lq_n$ for some $l<l_n-1$ then we can make all left block matches have multiplicity $l_{n}-1$ (while removing right block matches) and have:
\begin{eqnarray*}\sum_j\sum_i \alpha_j |k'\mbox{-matches of a $u_i$ with a $v_i $ in some $(w_0^j,w_1^j)|$} &\ge\\
\sum_j\sum_i \alpha_j|k\mbox{-matches of a $u_i$ with a $v_i$ in some $(w_0^j,w_1^j)|$}. \end{eqnarray*} If, on the other hand, the weighted sum of the right block matches is greater than weighted sum of the left block matches, we shift the other direction to fix all right block matches and destroy all left block matches.
Thus we can assume that we have a $k$ such that for all $(u_i,v_i)$, $sh^k$ matches $(l_n-1)$-powers of $u_i$ with $(l_n-1)$-powers of $v_i$. This $k$ would be a perfect match except that it matches $n$-words across 2-subsections. Writing each $w_s^j={\mathcal C}_n(w_1, \dots w_{k_n})$ then $sh^{k}$ matches blocks of the form $w_s^{l_n-1}$ in one $1$-subsection of $w_0^j$ with a block of the form $w_{s'}^{l_n-1}$ in a (potentially) different $1$-subsection of $w_1^j$. Moreover $s-s'$ is constant on all of these matches, since the differences between starts of $w_j^{l_n-1}$-blocks are of length $l_nq_n$. Fix such a pair $s, s'$. By changing $k$ so that it lines up $w_s^{l_n-1}$ with $w_{s'}^{l_n-1}$ in the first 1-subsection we create a perfect match of $n=m-1$-words and increase the total number of matches of the form $(u_i,v_i)$. This establishes the case where $d=1$.
We now do the induction step. Let $d=m-n$ and assume the result holds for $d-1$. Suppose that we are given $\{\alpha_j:j\in J\}$.
We can decompose a $k$-match between $n$-subwords of $w^j_0$ and $w^j_1$ as $k_1+k^*$ where $k^*\in [-q_{m-1}+1, q_{m-1}-1]$ and $k_1$ is a match of $m-1$ subwords of $w^j_0$ and $w^j_1$.
Here is a picture of a pair $(u',v')\in \mathcal W^c_{m-1}\times \mathcal W^c_{m-1}$ comparing $w_0$ in the upper row with the $k_1$-shift of $w_1$ in the lower row.
\begin{center} \includegraphics[width=.9\textwidth]{kprime-shift.pdf} \end{center}
Here is a picture after the $k=k_1+k^*$ shift of $w_1$:
\begin{center} \includegraphics[width=.9\textwidth]{kprimepluskstar-shift.pdf} \end{center}
Let $\{(u',v')_{i'}:i'\in I'\}$ be the collection of pairs $(u',v')$ from $\mathcal W^c_{m-1}$ sitting inside a pair $(w_0^j, w_1^j)$ that contain $k$-matches of words $(u_i,v_i)$. Arguing as in the case $d=1$ we can adjust $k_1$ to a $k_1'$ so that it is a perfect match of $m-1$-words in $I'$ and, summing over $I$ and $J$, the weighted sum of $k_1'+k^*$-matches of pairs in $I$ does not decrease.\footnote{We note that it is not enough to increase the weighted sum of the number of matches of pairs in $I'$, because various $I'$ matches may contain different number of $I$-matches. Nonetheless, arguing as in the case $d-1$, one of the two possibilities for lining up the $m-1$ subwords does not decrease the weighted sum of the number of $k_1'+k^*$-matches of $I$-words.}
This is how the $m-1$-words look after shifting by $k_1'+k^*$: \begin{center} \includegraphics[width=.9\textwidth]{k1primeshift.pdf} \end{center} The offset of the copies of $u'$ and $v'$ is $k^*$. Note that the boundary sections line up.
We now are in the position of having shifted by $k_1'$ so that the powers of pairs $\{(u',v')_{i'}:{i'\in I'}\}$ are lined up. The additional shift $k^*$ has absolute value less than $q_{m-1}$. Moreover all of the words $\{(u',v')_{i'}:{i'\in I'}\}$ are lined up the same way when shifted by $k^*$.
We call an occurrence of a $(u',v')_{i'}$ that is lined up in $(w_0^j, sh^{k'_1}(w_1^j))$ \emph{good}. Let $\beta_{j,i'}$ be the number of good occurrences of $(u',v')_{i'}$ and \[\alpha'_{i'}=\sum_j \alpha_j\beta_{j,i'}.\]
Note that
\begin{eqnarray*}\sum_{i'}\sum_i \alpha_{i'}|(k_1'+k^*)\mbox{-matches of a $u_i$ with a $v_i $ in some good occurrence of $(u',v')_{i'}|$}\\
=\sum_j\sum_i \alpha_j |(k_1'+k^*)\mbox{-matches of a $u_i$ with a $v_i$ in some $(w_0^j,w_1^j)|$}. \end{eqnarray*}
We now view the pairs $\{(u',v')_{i'}:{i'\in I'}\}$ as sitting on the intervals $[0,q_{m-1}-1]$ and then shifting $v'$ by $k^*$: \begin{center} \includegraphics[width=.2\textwidth]{onepair.pdf} \end{center} We are in a position to apply our induction hypothesis with $I'$ playing the role of $J$, the $\alpha'_{i'}$'s being the $\alpha_j$'s, $d-1=(m-1)-n$ and the shift being $k^*$.
The result is a $k^{**}$ such that every $k^{**}$-match of a $(u_i,v_i)$ in a $(u',v')_{i'}$ is perfect and
\begin{eqnarray*}\sum_{i'}\sum_i\alpha_{i'} |(k^{**})\mbox{-matches of a $u_i$ with a $v_i $ in some $(u',v')_{i'}|$} &\ge\\
\sum_{i'}\sum_i \alpha_{i'}|(k^*)\mbox{-matches of a $u_i$ with a $v_i$ in some $(u',v')_{i'}|$}. \end{eqnarray*}
We note that every $k_1'+k^{**}$-match of a $(u_i,v_i)$ in a $(w_0^j,w_1^j)$ is perfect. Since
\begin{eqnarray*}\sum_{i'}\sum_i \alpha_{i'}|(k_1'+k^{**})\mbox{-matches of a $u_i$ with a $v_i $ in some good occurrence of $(u',v')_{i'}|$}\\
=\sum_j\sum_i \alpha_j |(k_1'+k^{**})\mbox{-matches of a $u_i$ with a $v_i$ in some $(w_0^j,w_1^j)|$}. \end{eqnarray*} we see that
\begin{eqnarray*}\sum_j\sum_i \alpha_j|(k_1'+k^{**})\mbox{-matches of a $u_i$ with a $v_i $ in some $(w_0^j,w_1^j)|$} &\ge\\
\sum_j\sum_i \alpha_j |k\mbox{-matches of a $u_i$ with a $v_i$ in some $(w_0^j,w_1^j)|$}. \end{eqnarray*} This completes the proof of Lemma \ref{perfection is sorta possible}.{\nopagebreak
$\dashv$ \par
}
\section{Open Problems} We finish with two open problems that we find interesting and believe to be feasible. The first is to characterize the class of transformations isomorphic to circular systems in Ergodic-theoretic terms. All circular systems have common properties such that can be described in terms of rigidity sequences or zero entropy. The suggestions is to find a complete characterization in using this type of notion.
The second problem can be stated as follows. For the realization problem, the underlying rotation $\alpha$ of a circular system must be Liouvillian; however realization is not necessary for the results in this paper. Can an arbitrary irrational $\alpha$ be the underlying rotation of a circular system?
\end{document} | arXiv |
Jiří Patera (mathematician)
Jiří Patera (10 October 1936 – 3 January 2022) was a Czech-born Canadian mathematician and academic.[1] He taught at the Université de Montréal and was known for his work in group theory, Lie groups, and cryptography.
Jiří Patera
Patera in 2008
Born(1936-10-10)10 October 1936
Zdice, Czechoslovakia
Died3 January 2022(2022-01-03) (aged 85)
Montreal, Quebec, Canada
NationalityCzech
Canadian
Occupation(s)Mathematician
Professor
SpouseProfessor Tatiana Chalnikova Patera (1960)
Life and career
Patera attended secondary school in Děčín and subsequently studied theoretical physics at Moscow State University. There he met and married Tatiana Chalnikova. In 1964, he earned a doctorate from Charles University, pursued a postdoc at the University of Montreal and returned to Prague in 1966. In August of 1968, with Soviet tanks rolling into Czechoslovakia, he emigrated with Tatiana and their daughter first to the UK and finally settling in Montreal, Canada, a year later. He joined the Universite de Montreal as an assistant professor and then newly created Centre de Recherches Mathematiques at the university.
Patera began his work in group theory and Lie groups. From 1965 to 1972, he published several works on constructive theories of the representation of compact Lie groups. In 1981, he published Tables de dimensions, d'indices et de règles de branchement pour les représentations d'algèbres de Lie simples and began collaborating with Robert Moody in the field the following year.
Patera was known for his work on constructive computation prior to the use of software engineering. He wrote a book on Lie algebra representation[2][3] and primarily worked alongside Robert Moody and Hans Zassenhaus.[4][5]
He died in Montreal on 3 January 2022, at the age of 85 with Tatiana at his bedside.[1]
Awards
• Canada Council Killam Fellowship (1991)[6]
• Algebraic Methods in Physics: Symposium on the 60th Birthday of Jiří Patera and Pavel Winternitz (1997)[7]
• CAP-CRM Prize in Theoretical and Mathematical Physics (2004)[8]
• Doctor honoris causa of the Czech Technical University in Prague (2005)[9]
Works
• Branching rules for representations of simple Lie algebras (1971)
• Tables of dimensions, indices, and branching rules for representations of simple Lie algebras (1981)
• Tables of dominant weight multiplicities for representations of simple Lie algebras (1985)
• simpLie Users Manual: Macintosh software for representations of simple Lie algebras (1990)
• Tables of representations of simple Lie algebras (1990)
• Affine Kac-Moody Algebras, Weight Multiplicities and Branching Rules (1991)
References
1. "In memoriam – Jiří Patera (1936–2022)" (PDF). Centre de Recherches Mathématiques (in French).
2. Patera, Jiří; Sankoff, David (1973). Branching rules for representations of simple Lie algebras. Montreal: Presses Université de Montréal. ISBN 0-8405-0228-1.
3. Patera, Jiří; McKay, W. G. (1981). Tables of dimensions, indices, and branching rules for representations of simple Lie algebras. New York: Marcel Dekker. ISBN 0-8247-1227-7.
4. Moody, R. V.; Patera, J. (1982). "Fast recursion formula for weight multiplicities". American Mathematical Society. 7 (1): 237–242. doi:10.1090/S0273-0979-1982-15021-2.
5. Patera, Jiří; Moody, Robert; Bremner, Murray (1983). Tables of dominant weight multiplicities for representations of simple Lie algebras. New York: Marcel Dekker. ISBN 0-8247-7270-9.
6. "Jiri Patera". Université de Montréal (in French).
7. Patera, Jiří; Winternitz, Pavel; Saint-Aubin, Yvan; Vinet, Luc (2001). Algebraic Methods in Physics: A Symposium for the 60th Birthday of Jiří Patera and Pavel Winternitz. Berlin: Springer Science & Business Media. ISBN 9780387951256.
8. "Médaille de l'ACP 2004". Canadian Association of Physicists (in French). 20 April 2004.
9. "Prix/Distinctions" (PDF). CMS Notes / Notes de la SMC.
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| Wikipedia |
Boolean satisfiability problem
In logic and computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY, SAT or B-SAT) is the problem of determining if there exists an interpretation that satisfies a given Boolean formula. In other words, it asks whether the variables of a given Boolean formula can be consistently replaced by the values TRUE or FALSE in such a way that the formula evaluates to TRUE. If this is the case, the formula is called satisfiable. On the other hand, if no such assignment exists, the function expressed by the formula is FALSE for all possible variable assignments and the formula is unsatisfiable. For example, the formula "a AND NOT b" is satisfiable because one can find the values a = TRUE and b = FALSE, which make (a AND NOT b) = TRUE. In contrast, "a AND NOT a" is unsatisfiable.
SAT is the first problem that was proven to be NP-complete; see Cook–Levin theorem. This means that all problems in the complexity class NP, which includes a wide range of natural decision and optimization problems, are at most as difficult to solve as SAT. There is no known algorithm that efficiently solves each SAT problem, and it is generally believed that no such algorithm exists; yet this belief has not been proved mathematically, and resolving the question of whether SAT has a polynomial-time algorithm is equivalent to the P versus NP problem, which is a famous open problem in the theory of computing.
Nevertheless, as of 2007, heuristic SAT-algorithms are able to solve problem instances involving tens of thousands of variables and formulas consisting of millions of symbols,[1] which is sufficient for many practical SAT problems from, e.g., artificial intelligence, circuit design,[2] and automatic theorem proving.
Definitions
A propositional logic formula, also called Boolean expression, is built from variables, operators AND (conjunction, also denoted by ∧), OR (disjunction, ∨), NOT (negation, ¬), and parentheses. A formula is said to be satisfiable if it can be made TRUE by assigning appropriate logical values (i.e. TRUE, FALSE) to its variables. The Boolean satisfiability problem (SAT) is, given a formula, to check whether it is satisfiable. This decision problem is of central importance in many areas of computer science, including theoretical computer science, complexity theory,[3][4] algorithmics, cryptography[5][6] and artificial intelligence.[7]
Conjunctive normal form
A literal is either a variable (in which case it is called a positive literal) or the negation of a variable (called a negative literal). A clause is a disjunction of literals (or a single literal). A clause is called a Horn clause if it contains at most one positive literal. A formula is in conjunctive normal form (CNF) if it is a conjunction of clauses (or a single clause). For example, x1 is a positive literal, ¬x2 is a negative literal, x1 ∨ ¬x2 is a clause. The formula (x1 ∨ ¬x2) ∧ (¬x1 ∨ x2 ∨ x3) ∧ ¬x1 is in conjunctive normal form; its first and third clauses are Horn clauses, but its second clause is not. The formula is satisfiable, by choosing x1 = FALSE, x2 = FALSE, and x3 arbitrarily, since (FALSE ∨ ¬FALSE) ∧ (¬FALSE ∨ FALSE ∨ x3) ∧ ¬FALSE evaluates to (FALSE ∨ TRUE) ∧ (TRUE ∨ FALSE ∨ x3) ∧ TRUE, and in turn to TRUE ∧ TRUE ∧ TRUE (i.e. to TRUE). In contrast, the CNF formula a ∧ ¬a, consisting of two clauses of one literal, is unsatisfiable, since for a=TRUE or a=FALSE it evaluates to TRUE ∧ ¬TRUE (i.e., FALSE) or FALSE ∧ ¬FALSE (i.e., again FALSE), respectively.
For some versions of the SAT problem, it is useful to define the notion of a generalized conjunctive normal form formula, viz. as a conjunction of arbitrarily many generalized clauses, the latter being of the form R(l1,...,ln) for some Boolean function R and (ordinary) literals li. Different sets of allowed boolean functions lead to different problem versions. As an example, R(¬x,a,b) is a generalized clause, and R(¬x,a,b) ∧ R(b,y,c) ∧ R(c,d,¬z) is a generalized conjunctive normal form. This formula is used below, with R being the ternary operator that is TRUE just when exactly one of its arguments is.
Using the laws of Boolean algebra, every propositional logic formula can be transformed into an equivalent conjunctive normal form, which may, however, be exponentially longer. For example, transforming the formula (x1∧y1) ∨ (x2∧y2) ∨ ... ∨ (xn∧yn) into conjunctive normal form yields
(x1 ∨ x2 ∨ … ∨ xn) ∧
(y1 ∨ x2 ∨ … ∨ xn) ∧
(x1 ∨ y2 ∨ … ∨ xn) ∧
(y1 ∨ y2 ∨ … ∨ xn) ∧ ... ∧
(x1 ∨ x2 ∨ … ∨ yn) ∧
(y1 ∨ x2 ∨ … ∨ yn) ∧
(x1 ∨ y2 ∨ … ∨ yn) ∧
(y1 ∨ y2 ∨ … ∨ yn);
while the former is a disjunction of n conjunctions of 2 variables, the latter consists of 2n clauses of n variables.
However, with use of the Tseytin transformation, we may find an equisatisfiable conjunctive normal form formula with length linear in the size of the original propositional logic formula.
Complexity
Main article: Cook–Levin theorem
SAT was the first known NP-complete problem, as proved by Stephen Cook at the University of Toronto in 1971[8] and independently by Leonid Levin at the Russian Academy of Sciences in 1973.[9] Until that time, the concept of an NP-complete problem did not even exist. The proof shows how every decision problem in the complexity class NP can be reduced to the SAT problem for CNF[note 1] formulas, sometimes called CNFSAT. A useful property of Cook's reduction is that it preserves the number of accepting answers. For example, deciding whether a given graph has a 3-coloring is another problem in NP; if a graph has 17 valid 3-colorings, the SAT formula produced by the Cook–Levin reduction will have 17 satisfying assignments.
NP-completeness only refers to the run-time of the worst case instances. Many of the instances that occur in practical applications can be solved much more quickly. See Algorithms for solving SAT below.
3-satisfiability
Like the satisfiability problem for arbitrary formulas, determining the satisfiability of a formula in conjunctive normal form where each clause is limited to at most three literals is NP-complete also; this problem is called 3-SAT, 3CNFSAT, or 3-satisfiability. To reduce the unrestricted SAT problem to 3-SAT, transform each clause l1 ∨ ⋯ ∨ ln to a conjunction of n - 2 clauses
(l1 ∨ l2 ∨ x2) ∧
(¬x2 ∨ l3 ∨ x3) ∧
(¬x3 ∨ l4 ∨ x4) ∧ ⋯ ∧
(¬xn − 3 ∨ ln − 2 ∨ xn − 2) ∧
(¬xn − 2 ∨ ln − 1 ∨ ln)
where x2, ⋯ , xn − 2 are fresh variables not occurring elsewhere. Although the two formulas are not logically equivalent, they are equisatisfiable. The formula resulting from transforming all clauses is at most 3 times as long as its original, i.e. the length growth is polynomial.[10]
3-SAT is one of Karp's 21 NP-complete problems, and it is used as a starting point for proving that other problems are also NP-hard.[note 2] This is done by polynomial-time reduction from 3-SAT to the other problem. An example of a problem where this method has been used is the clique problem: given a CNF formula consisting of c clauses, the corresponding graph consists of a vertex for each literal, and an edge between each two non-contradicting[note 3] literals from different clauses, cf. picture. The graph has a c-clique if and only if the formula is satisfiable.[11]
There is a simple randomized algorithm due to Schöning (1999) that runs in time (4/3)n where n is the number of variables in the 3-SAT proposition, and succeeds with high probability to correctly decide 3-SAT.[12]
The exponential time hypothesis asserts that no algorithm can solve 3-SAT (or indeed k-SAT for any $k>2$) in exp(o(n)) time (i.e., fundamentally faster than exponential in n).
Selman, Mitchell, and Levesque (1996) give empirical data on the difficulty of randomly generated 3-SAT formulas, depending on their size parameters. Difficulty is measured in number recursive calls made by a DPLL algorithm. They identified a phase transition region from almost certainly satisfiable to almost certainly unsatisfiable formulas at the clauses-to-variables ratio at about 4.26.[13]
3-satisfiability can be generalized to k-satisfiability (k-SAT, also k-CNF-SAT), when formulas in CNF are considered with each clause containing up to k literals. However, since for any k ≥ 3, this problem can neither be easier than 3-SAT nor harder than SAT, and the latter two are NP-complete, so must be k-SAT.
Some authors restrict k-SAT to CNF formulas with exactly k literals. This doesn't lead to a different complexity class either, as each clause l1 ∨ ⋯ ∨ lj with j < k literals can be padded with fixed dummy variables to l1 ∨ ⋯ ∨ lj ∨ dj+1 ∨ ⋯ ∨ dk. After padding all clauses, 2k-1 extra clauses[note 4] have to be appended to ensure that only d1 = ⋯ = dk=FALSE can lead to a satisfying assignment. Since k doesn't depend on the formula length, the extra clauses lead to a constant increase in length. For the same reason, it does not matter whether duplicate literals are allowed in clauses, as in ¬x ∨ ¬y ∨ ¬y.
Special cases of SAT
Conjunctive normal form
Conjunctive normal form (in particular with 3 literals per clause) is often considered the canonical representation for SAT formulas. As shown above, the general SAT problem reduces to 3-SAT, the problem of determining satisfiability for formulas in this form.
Disjunctive normal form
SAT is trivial if the formulas are restricted to those in disjunctive normal form, that is, they are a disjunction of conjunctions of literals. Such a formula is indeed satisfiable if and only if at least one of its conjunctions is satisfiable, and a conjunction is satisfiable if and only if it does not contain both x and NOT x for some variable x. This can be checked in linear time. Furthermore, if they are restricted to being in full disjunctive normal form, in which every variable appears exactly once in every conjunction, they can be checked in constant time (each conjunction represents one satisfying assignment). But it can take exponential time and space to convert a general SAT problem to disjunctive normal form; for an example exchange "∧" and "∨" in the above exponential blow-up example for conjunctive normal forms.
Exactly-1 3-satisfiability
A variant of the 3-satisfiability problem is the one-in-three 3-SAT (also known variously as 1-in-3-SAT and exactly-1 3-SAT). Given a conjunctive normal form with three literals per clause, the problem is to determine whether there exists a truth assignment to the variables so that each clause has exactly one TRUE literal (and thus exactly two FALSE literals). In contrast, ordinary 3-SAT requires that every clause has at least one TRUE literal. Formally, a one-in-three 3-SAT problem is given as a generalized conjunctive normal form with all generalized clauses using a ternary operator R that is TRUE just if exactly one of its arguments is. When all literals of a one-in-three 3-SAT formula are positive, the satisfiability problem is called one-in-three positive 3-SAT.
One-in-three 3-SAT, together with its positive case, is listed as NP-complete problem "LO4" in the standard reference, Computers and Intractability: A Guide to the Theory of NP-Completeness by Michael R. Garey and David S. Johnson. One-in-three 3-SAT was proved to be NP-complete by Thomas Jerome Schaefer as a special case of Schaefer's dichotomy theorem, which asserts that any problem generalizing Boolean satisfiability in a certain way is either in the class P or is NP-complete.[14]
Schaefer gives a construction allowing an easy polynomial-time reduction from 3-SAT to one-in-three 3-SAT. Let "(x or y or z)" be a clause in a 3CNF formula. Add six fresh boolean variables a, b, c, d, e, and f, to be used to simulate this clause and no other. Then the formula R(x,a,d) ∧ R(y,b,d) ∧ R(a,b,e) ∧ R(c,d,f) ∧ R(z,c,FALSE) is satisfiable by some setting of the fresh variables if and only if at least one of x, y, or z is TRUE, see picture (left). Thus any 3-SAT instance with m clauses and n variables may be converted into an equisatisfiable one-in-three 3-SAT instance with 5m clauses and n+6m variables.[15] Another reduction involves only four fresh variables and three clauses: R(¬x,a,b) ∧ R(b,y,c) ∧ R(c,d,¬z), see picture (right).
Not-all-equal 3-satisfiability
Another variant is the not-all-equal 3-satisfiability problem (also called NAE3SAT). Given a conjunctive normal form with three literals per clause, the problem is to determine if an assignment to the variables exists such that in no clause all three literals have the same truth value. This problem is NP-complete, too, even if no negation symbols are admitted, by Schaefer's dichotomy theorem.[14]
Linear SAT
A 3-SAT formula is Linear SAT (LSAT) if each clause (viewed as a set of literals) intersects at most one other clause, and, moreover, if two clauses intersect, then they have exactly one literal in common. An LSAT formula can be depicted as a set of disjoint semi-closed intervals on a line. Deciding whether an LSAT formula is satisfiable is NP-complete.[16]
2-satisfiability
SAT is easier if the number of literals in a clause is limited to at most 2, in which case the problem is called 2-SAT. This problem can be solved in polynomial time, and in fact is complete for the complexity class NL. If additionally all OR operations in literals are changed to XOR operations, the result is called exclusive-or 2-satisfiability, which is a problem complete for the complexity class SL = L.
Horn-satisfiability
Main article: Horn-satisfiability
The problem of deciding the satisfiability of a given conjunction of Horn clauses is called Horn-satisfiability, or HORN-SAT. It can be solved in polynomial time by a single step of the Unit propagation algorithm, which produces the single minimal model of the set of Horn clauses (w.r.t. the set of literals assigned to TRUE). Horn-satisfiability is P-complete. It can be seen as P's version of the Boolean satisfiability problem. Also, deciding the truth of quantified Horn formulas can be done in polynomial time. [17]
Horn clauses are of interest because they are able to express implication of one variable from a set of other variables. Indeed, one such clause ¬x1 ∨ ... ∨ ¬xn ∨ y can be rewritten as x1 ∧ ... ∧ xn → y, that is, if x1,...,xn are all TRUE, then y needs to be TRUE as well.
A generalization of the class of Horn formulae is that of renameable-Horn formulae, which is the set of formulae that can be placed in Horn form by replacing some variables with their respective negation. For example, (x1 ∨ ¬x2) ∧ (¬x1 ∨ x2 ∨ x3) ∧ ¬x1 is not a Horn formula, but can be renamed to the Horn formula (x1 ∨ ¬x2) ∧ (¬x1 ∨ x2 ∨ ¬y3) ∧ ¬x1 by introducing y3 as negation of x3. In contrast, no renaming of (x1 ∨ ¬x2 ∨ ¬x3) ∧ (¬x1 ∨ x2 ∨ x3) ∧ ¬x1 leads to a Horn formula. Checking the existence of such a replacement can be done in linear time; therefore, the satisfiability of such formulae is in P as it can be solved by first performing this replacement and then checking the satisfiability of the resulting Horn formula.
XOR-satisfiability
Solving an XOR-SAT example
by Gaussian elimination
Given formula
("⊕" means XOR, the red clause is optional)
(a⊕c⊕d) ∧ (b⊕¬c⊕d) ∧ (a⊕b⊕¬d) ∧ (a⊕¬b⊕¬c) ∧ (¬a⊕b⊕c)
Equation system
("1" means TRUE, "0" means FALSE)
Each clause leads to one equation.
a⊕c⊕d= 1
b⊕¬c⊕d= 1
a⊕b⊕¬d= 1
a⊕¬b⊕¬c= 1
¬a⊕b⊕c ≃ 1
Normalized equation system
using properties of Boolean rings (¬x=1⊕x, x⊕x=0)
a⊕c⊕d= 1
b⊕c⊕d= 0
a⊕b⊕d= 0
a⊕b⊕c= 1
a⊕b⊕c ≃ 0
(If the red equation is present, it contradicts
the last black one, so the system is unsolvable.
Therefore, Gauss' algorithm is
used only for the black equations.)
Associated coefficient matrix
abcdline
1011 1 A
0111 0 B
1101 0 C
1110 1 D
Transforming to echelon form
abcdoperation
1011 1 A
1101 0 C
1110 1 D
0111 0 B (swapped)
1011 1 A
0110 1 E = C⊕A
0101 0 F = D⊕A
0111 0 B
1011 1 A
0110 1 E
0011 1 G = F⊕E
0001 1 H = B⊕E
Transforming to diagonal form
abcdoperation
1010 0 I = A⊕H
0110 1 E
0010 0 J = G⊕H
0001 1 H
1000 0 K = I⊕J
0100 1 L = E⊕J
0010 0 J
0001 1 H
Solution:
If the red clause is present:Unsolvable
Else:a = 0 = FALSE
b = 1 = TRUE
c = 0 = FALSE
d = 1 = TRUE
As a consequence:
R(a,c,d) ∧ R(b,¬c,d) ∧ R(a,b,¬d) ∧ R(a,¬b,¬c) ∧ R(¬a,b,c)
is not 1-in-3-satisfiable,
while (a ∨ c ∨ d) ∧ (b ∨ ¬c ∨ d) ∧ (a ∨ b ∨ ¬d) ∧ (a ∨ ¬b ∨ ¬c)
is 3-satisfiable with a=c=FALSE and b=d=TRUE.
Another special case is the class of problems where each clause contains XOR (i.e. exclusive or) rather than (plain) OR operators.[note 5] This is in P, since an XOR-SAT formula can also be viewed as a system of linear equations mod 2, and can be solved in cubic time by Gaussian elimination;[18] see the box for an example. This recast is based on the kinship between Boolean algebras and Boolean rings, and the fact that arithmetic modulo two forms a finite field. Since a XOR b XOR c evaluates to TRUE if and only if exactly 1 or 3 members of {a,b,c} are TRUE, each solution of the 1-in-3-SAT problem for a given CNF formula is also a solution of the XOR-3-SAT problem, and in turn each solution of XOR-3-SAT is a solution of 3-SAT, cf. picture. As a consequence, for each CNF formula, it is possible to solve the XOR-3-SAT problem defined by the formula, and based on the result infer either that the 3-SAT problem is solvable or that the 1-in-3-SAT problem is unsolvable.
Provided that the complexity classes P and NP are not equal, neither 2-, nor Horn-, nor XOR-satisfiability is NP-complete, unlike SAT.
Schaefer's dichotomy theorem
Main article: Schaefer's dichotomy theorem
The restrictions above (CNF, 2CNF, 3CNF, Horn, XOR-SAT) bound the considered formulae to be conjunctions of subformulae; each restriction states a specific form for all subformulae: for example, only binary clauses can be subformulae in 2CNF.
Schaefer's dichotomy theorem states that, for any restriction to Boolean functions that can be used to form these subformulae, the corresponding satisfiability problem is in P or NP-complete. The membership in P of the satisfiability of 2CNF, Horn, and XOR-SAT formulae are special cases of this theorem.[14]
The following table summarizes some common variants of SAT.
Code Name Restrictions Requirements Class
3SAT 3-satisfiability Each clause contains 3 literals. At least one literal must be true. NP-c
2SAT 2-satisfiability Each clause contains 2 literals. At least one literal must be true. NL-c
1-in-3-SAT Exactly-1 3-SAT Each clause contains 3 literals. Exactly one literal must be true. NP-c
1-in-3-SAT+ Exactly-1 Positive 3-SAT Each clause contains 3 positive literals. Exactly one literal must be true. NP-c
NAE3SAT Not-all-equal 3-satisfiability Each clause contains 3 literals. Either one or two literals must be true. NP-c
NAE3SAT+ Not-all-equal positive 3-SAT Each clause contains 3 positive literals. Either one or two literals must be true. NP-c
PL-SAT Planar SAT The incidence graph (clause-variable graph) is planar. At least one literal must be true. NP-c
LSAT Linear SAT Each clause contains 3 literals, intersects at most one other clause, and the intersection is exactly one literal. At least one literal must be true. NP-c
HORN-SAT Horn satisfiability Horn clauses (at most one positive literal). At least one literal must be true. P-c
XOR-SAT Xor satisfiability Each clause contains XOR operations rather than OR. The XOR of all literals must be true. P
Extensions of SAT
An extension that has gained significant popularity since 2003 is satisfiability modulo theories (SMT) that can enrich CNF formulas with linear constraints, arrays, all-different constraints, uninterpreted functions,[19] etc. Such extensions typically remain NP-complete, but very efficient solvers are now available that can handle many such kinds of constraints.
The satisfiability problem becomes more difficult if both "for all" (∀) and "there exists" (∃) quantifiers are allowed to bind the Boolean variables. An example of such an expression would be ∀x ∀y ∃z (x ∨ y ∨ z) ∧ (¬x ∨ ¬y ∨ ¬z); it is valid, since for all values of x and y, an appropriate value of z can be found, viz. z=TRUE if both x and y are FALSE, and z=FALSE else. SAT itself (tacitly) uses only ∃ quantifiers. If only ∀ quantifiers are allowed instead, the so-called tautology problem is obtained, which is co-NP-complete. If both quantifiers are allowed, the problem is called the quantified Boolean formula problem (QBF), which can be shown to be PSPACE-complete. It is widely believed that PSPACE-complete problems are strictly harder than any problem in NP, although this has not yet been proved. Using highly parallel P systems, QBF-SAT problems can be solved in linear time.[20]
Ordinary SAT asks if there is at least one variable assignment that makes the formula true. A variety of variants deal with the number of such assignments:
• MAJ-SAT asks if the majority of all assignments make the formula TRUE. It is known to be complete for PP, a probabilistic class.
• #SAT, the problem of counting how many variable assignments satisfy a formula, is a counting problem, not a decision problem, and is #P-complete.
• UNIQUE SAT[21] is the problem of determining whether a formula has exactly one assignment. It is complete for US,[22] the complexity class describing problems solvable by a non-deterministic polynomial time Turing machine that accepts when there is exactly one nondeterministic accepting path and rejects otherwise.
• UNAMBIGUOUS-SAT is the name given to the satisfiability problem when the input is restricted to formulas having at most one satisfying assignment. The problem is also called USAT.[23] A solving algorithm for UNAMBIGUOUS-SAT is allowed to exhibit any behavior, including endless looping, on a formula having several satisfying assignments. Although this problem seems easier, Valiant and Vazirani have shown[24] that if there is a practical (i.e. randomized polynomial-time) algorithm to solve it, then all problems in NP can be solved just as easily.
• MAX-SAT, the maximum satisfiability problem, is an FNP generalization of SAT. It asks for the maximum number of clauses which can be satisfied by any assignment. It has efficient approximation algorithms, but is NP-hard to solve exactly. Worse still, it is APX-complete, meaning there is no polynomial-time approximation scheme (PTAS) for this problem unless P=NP.
• WMSAT is the problem of finding an assignment of minimum weight that satisfy a monotone Boolean formula (i.e. a formula without any negation). Weights of propositional variables are given in the input of the problem. The weight of an assignment is the sum of weights of true variables. That problem is NP-complete (see Th. 1 of [25]).
Other generalizations include satisfiability for first- and second-order logic, constraint satisfaction problems, 0-1 integer programming.
Finding a satisfying assignment
While SAT is a decision problem, the search problem of finding a satisfying assignment reduces to SAT. That is, each algorithm which correctly answers if an instance of SAT is solvable can be used to find a satisfying assignment. First, the question is asked on the given formula Φ. If the answer is "no", the formula is unsatisfiable. Otherwise, the question is asked on the partly instantiated formula Φ{x1=TRUE}, i.e. Φ with the first variable x1 replaced by TRUE, and simplified accordingly. If the answer is "yes", then x1=TRUE, otherwise x1=FALSE. Values of other variables can be found subsequently in the same way. In total, n+1 runs of the algorithm are required, where n is the number of distinct variables in Φ.
This property is used in several theorems in complexity theory:
NP ⊆ P/poly ⇒ PH = Σ2 (Karp–Lipton theorem)
NP ⊆ BPP ⇒ NP = RP
P = NP ⇒ FP = FNP
Algorithms for solving SAT
Since the SAT problem is NP-complete, only algorithms with exponential worst-case complexity are known for it. In spite of this, efficient and scalable algorithms for SAT were developed during the 2000s and have contributed to dramatic advances in our ability to automatically solve problem instances involving tens of thousands of variables and millions of constraints (i.e. clauses).[1] Examples of such problems in electronic design automation (EDA) include formal equivalence checking, model checking, formal verification of pipelined microprocessors,[19] automatic test pattern generation, routing of FPGAs,[26] planning, and scheduling problems, and so on. A SAT-solving engine is also considered to be an essential component in the electronic design automation toolbox.
Major techniques used by modern SAT solvers include the Davis–Putnam–Logemann–Loveland algorithm (or DPLL), conflict-driven clause learning (CDCL), and stochastic local search algorithms such as WalkSAT. Almost all SAT solvers include time-outs, so they will terminate in reasonable time even if they cannot find a solution. Different SAT solvers will find different instances easy or hard, and some excel at proving unsatisfiability, and others at finding solutions. Recent attempts have been made to learn an instance's satisfiability using deep learning techniques.[27]
SAT solvers are developed and compared in SAT-solving contests.[28] Modern SAT solvers are also having significant impact on the fields of software verification, constraint solving in artificial intelligence, and operations research, among others.
See also
• Unsatisfiable core
• Satisfiability modulo theories
• Counting SAT
• Planar SAT
• Karloff–Zwick algorithm
• Circuit satisfiability
Notes
1. The SAT problem for arbitrary formulas is NP-complete, too, since it is easily shown to be in NP, and it cannot be easier than SAT for CNF formulas.
2. i.e. at least as hard as every other problem in NP. A decision problem is NP-complete if and only if it is in NP and is NP-hard.
3. i.e. such that one literal is not the negation of the other
4. viz. all maxterms that can be built with d1,⋯,dk, except d1∨⋯∨dk
5. Formally, generalized conjunctive normal forms with a ternary boolean function R are employed, which is TRUE just if 1 or 3 of its arguments is. An input clause with more than 3 literals can be transformed into an equisatisfiable conjunction of clauses á 3 literals similar to above; i.e. XOR-SAT can be reduced to XOR-3-SAT.
External links
Wikimedia Commons has media related to Boolean satisfiability problem.
• SAT Game: try solving a Boolean satisfiability problem yourself
• The international SAT competition website
• International Conference on Theory and Applications of Satisfiability Testing
• Journal on Satisfiability, Boolean Modeling and Computation
• SAT Live, an aggregate website for research on the satisfiability problem
• Yearly evaluation of MaxSAT solvers
References
1. Ohrimenko, Olga; Stuckey, Peter J.; Codish, Michael (2007), "Propagation = Lazy Clause Generation", Principles and Practice of Constraint Programming – CP 2007, Lecture Notes in Computer Science, vol. 4741, pp. 544–558, CiteSeerX 10.1.1.70.5471, doi:10.1007/978-3-540-74970-7_39, modern SAT solvers can often handle problems with millions of constraints and hundreds of thousands of variables.
2. Hong, Ted; Li, Yanjing; Park, Sung-Boem; Mui, Diana; Lin, David; Kaleq, Ziyad Abdel; Hakim, Nagib; Naeimi, Helia; Gardner, Donald S.; Mitra, Subhasish (November 2010). "QED: Quick Error Detection tests for effective post-silicon validation". 2010 IEEE International Test Conference. pp. 1–10. doi:10.1109/TEST.2010.5699215. ISBN 978-1-4244-7206-2. S2CID 7909084.
3. Karp, Richard M. (1972). "Reducibility Among Combinatorial Problems" (PDF). In Raymond E. Miller; James W. Thatcher (eds.). Complexity of Computer Computations. New York: Plenum. pp. 85–103. ISBN 0-306-30707-3. Archived from the original (PDF) on 2011-06-29. Retrieved 2020-05-07. Here: p.86
4. Aho, Alfred V.; Hopcroft, John E.; Ullman, Jeffrey D. (1974). The Design and Analysis of Computer Algorithms. Addison-Wesley. p. 403. ISBN 0-201-00029-6.
5. Massacci, Fabio; Marraro, Laura (2000-02-01). "Logical Cryptanalysis as a SAT Problem". Journal of Automated Reasoning. 24 (1): 165–203. doi:10.1023/A:1006326723002. S2CID 3114247.
6. Mironov, Ilya; Zhang, Lintao (2006). "Applications of SAT Solvers to Cryptanalysis of Hash Functions". In Biere, Armin; Gomes, Carla P. (eds.). Theory and Applications of Satisfiability Testing - SAT 2006. Lecture Notes in Computer Science. Vol. 4121. Springer. pp. 102–115. doi:10.1007/11814948_13. ISBN 978-3-540-37207-3.
7. Vizel, Y.; Weissenbacher, G.; Malik, S. (2015). "Boolean Satisfiability Solvers and Their Applications in Model Checking". Proceedings of the IEEE. 103 (11): 2021–2035. doi:10.1109/JPROC.2015.2455034. S2CID 10190144.
8. Cook, Stephen A. (1971). "The complexity of theorem-proving procedures" (PDF). Proceedings of the third annual ACM symposium on Theory of computing - STOC '71. pp. 151–158. CiteSeerX 10.1.1.406.395. doi:10.1145/800157.805047. S2CID 7573663. Archived (PDF) from the original on 2022-10-09.
9. Levin, Leonid (1973). "Universal search problems (Russian: Универсальные задачи перебора, Universal'nye perebornye zadachi)". Problems of Information Transmission (Russian: Проблемы передачи информа́ции, Problemy Peredachi Informatsii). 9 (3): 115–116. (pdf) (in Russian), translated into English by Trakhtenbrot, B. A. (1984). "A survey of Russian approaches to perebor (brute-force searches) algorithms". Annals of the History of Computing. 6 (4): 384–400. doi:10.1109/MAHC.1984.10036. S2CID 950581.
10. Aho, Hopcroft & Ullman (1974), Theorem 10.4.
11. Aho, Hopcroft & Ullman (1974), Theorem 10.5.
12. Schöning, Uwe (Oct 1999). "A probabilistic algorithm for k-SAT and constraint satisfaction problems" (PDF). 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039). pp. 410–414. doi:10.1109/SFFCS.1999.814612. ISBN 0-7695-0409-4. S2CID 123177576. Archived (PDF) from the original on 2022-10-09.
13. Selman, Bart; Mitchell, David; Levesque, Hector (1996). "Generating Hard Satisfiability Problems". Artificial Intelligence. 81 (1–2): 17–29. CiteSeerX 10.1.1.37.7362. doi:10.1016/0004-3702(95)00045-3.
14. Schaefer, Thomas J. (1978). "The complexity of satisfiability problems" (PDF). Proceedings of the 10th Annual ACM Symposium on Theory of Computing. San Diego, California. pp. 216–226. CiteSeerX 10.1.1.393.8951. doi:10.1145/800133.804350.
15. Schaefer (1978), p. 222, Lemma 3.5.
16. Arkin, Esther M.; Banik, Aritra; Carmi, Paz; Citovsky, Gui; Katz, Matthew J.; Mitchell, Joseph S. B.; Simakov, Marina (2018-12-11). "Selecting and covering colored points". Discrete Applied Mathematics. 250: 75–86. doi:10.1016/j.dam.2018.05.011. ISSN 0166-218X.
17. Buning, H.K.; Karpinski, Marek; Flogel, A. (1995). "Resolution for Quantified Boolean Formulas". Information and Computation. Elsevier. 117 (1): 12–18. doi:10.1006/inco.1995.1025.
18. Moore, Cristopher; Mertens, Stephan (2011), The Nature of Computation, Oxford University Press, p. 366, ISBN 9780199233212.
19. R. E. Bryant, S. M. German, and M. N. Velev, Microprocessor Verification Using Efficient Decision Procedures for a Logic of Equality with Uninterpreted Functions, in Analytic Tableaux and Related Methods, pp. 1–13, 1999.
20. Alhazov, Artiom; Martín-Vide, Carlos; Pan, Linqiang (2003). "Solving a PSPACE-Complete Problem by Recognizing P Systems with Restricted Active Membranes". Fundamenta Informaticae. 58: 67–77. Here: Sect.3, Thm.3.1
21. Blass, Andreas; Gurevich, Yuri (1982-10-01). "On the unique satisfiability problem". Information and Control. 55 (1): 80–88. doi:10.1016/S0019-9958(82)90439-9. ISSN 0019-9958.
22. "Complexity Zoo:U - Complexity Zoo". complexityzoo.uwaterloo.ca. Archived from the original on 2019-07-09. Retrieved 2019-12-05.
23. Kozen, Dexter C. (2006). "Supplementary Lecture F: Unique Satisfiability". Theory of Computation. Texts in Computer Science. Springer. p. 180. ISBN 9781846282973.
24. Valiant, L.; Vazirani, V. (1986). "NP is as easy as detecting unique solutions" (PDF). Theoretical Computer Science. 47: 85–93. doi:10.1016/0304-3975(86)90135-0.
25. Buldas, Ahto; Lenin, Aleksandr; Willemson, Jan; Charnamord, Anton (2017). "Simple Infeasibility Certificates for Attack Trees". In Obana, Satoshi; Chida, Koji (eds.). Advances in Information and Computer Security. Lecture Notes in Computer Science. Vol. 10418. Springer International Publishing. pp. 39–55. doi:10.1007/978-3-319-64200-0_3. ISBN 9783319642000.
26. Gi-Joon Nam; Sakallah, K. A.; Rutenbar, R. A. (2002). "A new FPGA detailed routing approach via search-based Boolean satisfiability" (PDF). IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems. 21 (6): 674. doi:10.1109/TCAD.2002.1004311. Archived from the original (PDF) on 2016-03-15. Retrieved 2015-09-04.
27. Selsam, Daniel; Lamm, Matthew; Bünz, Benedikt; Liang, Percy; de Moura, Leonardo; Dill, David L. (11 March 2019). "Learning a SAT Solver from Single-Bit Supervision". arXiv:1802.03685 [cs.AI].
28. "The international SAT Competitions web page". Retrieved 2007-11-15.
Sources
• This article includes material from https://web.archive.org/web/20070708233347/http://www.sigda.org/newsletter/2006/eNews_061201.html by Prof. Karem Sakallah.
Further reading
(by date of publication)
• Garey, Michael R.; Johnson, David S. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman. pp. A9.1: LO1–LO7, pp. 259–260. ISBN 0-7167-1045-5.
• Marques-Silva, J.; Glass, T. (1999). "Combinational equivalence checking using satisfiability and recursive learning". Design, Automation and Test in Europe Conference and Exhibition, 1999. Proceedings (Cat. No. PR00078) (PDF). p. 145. doi:10.1109/DATE.1999.761110. ISBN 0-7695-0078-1. Archived (PDF) from the original on 2022-10-09.
• Clarke, E.; Biere, A.; Raimi, R.; Zhu, Y. (2001). "Bounded Model Checking Using Satisfiability Solving". Formal Methods in System Design. 19: 7–34. doi:10.1023/A:1011276507260. S2CID 2484208.
• Giunchiglia, E.; Tacchella, A. (2004). Giunchiglia, Enrico; Tacchella, Armando (eds.). Theory and Applications of Satisfiability Testing. Lecture Notes in Computer Science. Vol. 2919. doi:10.1007/b95238. ISBN 978-3-540-20851-8. S2CID 31129008.
• Babic, D.; Bingham, J.; Hu, A. J. (2006). "B-Cubing: New Possibilities for Efficient SAT-Solving" (PDF). IEEE Transactions on Computers. 55 (11): 1315. doi:10.1109/TC.2006.175. S2CID 14819050.
• Rodriguez, C.; Villagra, M.; Baran, B. (2007). "Asynchronous team algorithms for Boolean Satisfiability" (PDF). 2007 2nd Bio-Inspired Models of Network, Information and Computing Systems. pp. 66–69. doi:10.1109/BIMNICS.2007.4610083. S2CID 15185219.
• Gomes, Carla P.; Kautz, Henry; Sabharwal, Ashish; Selman, Bart (2008). "Satisfiability Solvers". In Harmelen, Frank Van; Lifschitz, Vladimir; Porter, Bruce (eds.). Handbook of knowledge representation. Foundations of Artificial Intelligence. Vol. 3. Elsevier. pp. 89–134. doi:10.1016/S1574-6526(07)03002-7. ISBN 978-0-444-52211-5.
• Vizel, Y.; Weissenbacher, G.; Malik, S. (2015). "Boolean Satisfiability Solvers and Their Applications in Model Checking". Proceedings of the IEEE. 103 (11): 2021–2035. doi:10.1109/JPROC.2015.2455034. S2CID 10190144.
• Knuth, Donald E. (2022). "Chapter 7.2.2.2: Satifiability". The Art of Computer Programming. Vol. 4B: Combinatorial Algorithms, Part 2. Addison-Wesley Professional. pp. 185–369. ISBN 978-0-201-03806-4.
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| Wikipedia |
The value of $\sqrt{73}$ is between two positive, consecutive integers. What is the product of these two integers?
Since $73$ is between $64=8^{2}$ and $81=9^{2}$, we know that $\sqrt{73}$ is between $8$ and $9.$ Our answer is $8\cdot9=\boxed{72}.$ | Math Dataset |
4.2: Four-vectors (Part 2)
[ "article:topic", "authorname:crowellb", "Gravitational red-shifts", "Ives\u2013Stilwell experiments", "Four-vector", "license:ccbysa", "showtoc:no" ]
Book: General Relativity (Crowell)
4: Tensors
Contributed by Benjamin Crowell
Professor (Physics) at Fullerton College
The Frequency Vector and the Relativistic Doppler Shift
A Non-example: Electric and Magnetic Fields
The Electromagnetic Potential Four-vector
In the spirit of index-gymnastics notation, frequency is to time as the wavenumber k = \(\frac{1}{\lambda}\) is to space, so when treating waves relativistically it is natural to conjecture that there is a four-frequency fa made by assembling (f, k), which behaves as a Lorentz vector. This is correct, since we already know that \(\partial_{a}\) transforms as a covariant vector, and for a scalar wave of the form
\[A = A_o \exp [2\pi if_ax^a]\]
the partial derivative operator is identical to multiplication by 2\(\pi\)fa.
As an application, consider the relativistic Doppler shift of a light wave. For simpicity, let's restrict ourselves to one spatial dimension. For a light wave, \(f = k\), so the frequency vector in 1+1 dimensions is simply (f, f). Putting this through a Lorentz transformation, we find
$$f' = (1 + v) \gamma f = \sqrt{\frac{1+v}{1-v}} f,$$
where the second form displays more clearly the symmetic form of the relativistic relationship, such that interchanging the roles of source and observer is equivalent to flipping the sign of v. That is, the relativistic version only depends on the relative motion of the source and the observer, whereas the Newtonian one also depends on the source's motion relative to the medium (i.e., relative to the preferred frame in which the waves have the "right" velocity). In Newtonian mechanics, we have f' = (1 + v)f for a moving observer. Relativistically, there is also a time dilation of the oscillation of the source, providing an additional factor of \(\gamma\).
This analysis is extended to 3+1 dimensions in problem 11.
Example 15: Ives-Stilwell experiments
The relativistic Doppler shift differs from the nonrelativistic one by the time-dilation factor \(\gamma\), so that there is still a shift even when the relative motion of the source and the observer is perpendicular to the direction of propagation. This is called the transverse Doppler shift. Einstein suggested this early on as a test of relativity. However, such experiments are difficult to carry out with high precision, because they are sensitive to any error in the alignment of the 90-degree angle. Such experiments were eventually performed, with results that confirmed relativity,7 but one-dimensional measurements provided both the earliest tests of the relativistic Doppler shift and the most precise ones to date. The first such test was done by Ives and Stilwell in 1938, using the following trick. The relativistic expression
\[S_{v} = \sqrt{\frac{(1 + v)}{(1 − v)}}\]
for the Doppler shift has the property that
\[S_vS_{−v} = 1\]
which differs from the nonrelativistic result of
\[(1 + v)(1 − v) = 1 − v^2.\]
One can therefore accelerate an ion up to a relativistic speed, measure both the forward Doppler shifted frequency ff and the backward one fb, and compute \(\sqrt{f_{f} f_{b}}\). According to relativity, this should exactly equal the frequency fo measured in the ion's rest frame.
In a particularly exquisite modern version of the Ives-Stilwell idea,8 Saathoff et al. circulated Li+ ions at v = .064 in a storage ring. An electron-cooler technique was used in order to reduce the variation in velocity among ions in the beam. Since the identity SvS−v = 1 is independent of v, it was not necessary to measure v to the same incredible precision as the frequencies; it was only necessary that it be stable and well-defined. The natural line width was 7 MHz, and other experimental effects broadened it further to 11 MHz. By curve-fitting the line, it was possible to achieve results good to a few tenths of a MHz. The resulting frequencies, in units of MHz, were:
ff = 582490203.44 ± .09
fb = 512671442.9 ± 0.5
\(\sqrt{f_{f} f_{b}}\) = 546466918.6 ± 0.3
fo = 546466918.8 ± 0.4 (from previous experimental work)
The spectacular agreement with theory has made this experiment a lightning rod for anti-relativity kooks.
If one is searching for small deviations from the predictions of special relativity, a natural place to look is at high velocities. IvesStilwell experiments have been performed at velocities as high as 0.84, and they confirm special relativity.9
7 See, e.g., Hasselkamp, Mondry, and Scharmann, Zeitschrift f¨ur Physik A: Hadrons and Nuclei 289 (1979) 151.
8 G. Saathoff et al., "Improved Test of Time Dilation in Relativity," Phys. Rev. Lett. 91 (2003) 190403. A publicly available description of the experiment is given in Saathoff's PhD thesis, www.mpi-hd.mpg.de/ato/homes/saathoff/ diss-saathoff.pdf.
9 MacArthur et al., Phys. Rev. Lett. 56 (1986) 282 (1986)
Example 16: Einstein's derivation of E = mc2
Earlier, we showed that the celebrated E = mc2 follows directly from the form of the Lorentz transformation. An alternative derivation was given by Einstein in one of his classic 1905 papers laying out the theory of special relativity; the paper is short, and is reproduced in English translation in Appendix A of this book. Having laid the groundwork of four-vectors and relativistic Doppler shifts, we can give an even shorter version of Einstein's argument. The discussion is also streamlined by restricting the discussion to 1+1 dimensions and by invoking photons.
Suppose that a lantern, at rest in the lab frame, is floating weightlessly in outer space, and simultaneously emits two pulses of light in opposite directions, each with energy \(\frac{E}{2}\) and frequency f. By symmetry, the momentum of the pulses cancels, and the lantern remains at rest. An observer in motion at velocity v relative to the lab sees the frequencies of the beams shifted to f' = (1 ± v)\(\gamma\)f. The effect on the energies of the beams can be found purely classically, by transforming the electric and magnetic fields to the moving frame, but as a shortcut we can apply the quantum-mechanical relation Eph = hf for the energies of the photons making up the beams. The result is that the moving observer finds the total energy of the beams to be not E but (\(\frac{E}{2}\))(1 + v)\(\gamma\) + (\(\frac{E}{2}\))(1 − v)\(\gamma\) = E\(\gamma\).
Both observers agree that the lantern had to use up some of the energy stored in its fuel in order to make the two pulses. But the moving observer says that in addition to this energy E, there was a further energy E(\(\gamma\) − 1). Where could this energy have come from? It must have come from the kinetic energy of the lantern. The lantern's velocity remained constant throughout the experiment, so this decrease in kinetic energy seen by the moving observer must have come from a decrease in the lantern's inertial mass — hence the title of Einstein's paper, "Does the inertia of a body depend upon its energy content?"
To figure out how much mass the lantern has lost, we have to decide how we can even define mass in this new context. In Newtonian mechanics, we had K = (\(\frac{1}{2}\))mv2, and by the correspondence principle this must still hold in the low-velocity limit. Expanding E(\(\gamma\) − 1) in a Taylor series, we find that it equals E(\(\frac{v^{2}}{2}\)) + . . ., and in the low-velocity limit this must be the same as \(\Delta K = (\frac{1}{2}) \Delta mv^{2}\), so \(\Delta\)m = E. Reinserting factors of c to get back to nonrelativistic units, we have E = \(\Delta\)mc2.
It is fairly easy to see that the electric and magnetic fields cannot be the spacelike parts of two four-vectors. Consider the arrangement shown in Figure 4.2.2 (1). We have two infinite trains of moving charges superimposed on the same line, and a single charge alongside the line. Even though the line charges formed by the two trains are moving in opposite directions, their currents don't cancel. A negative charge moving to the left makes a current that goes to the right, so in frame 1, the total current is twice that contributed by either line charge.
Figure \(\PageIndex{2}\) - Magnetism is a purely relativistic effect.
In frame 1 the charge densities of the two line charges cancel out, and the electric field experienced by the lone charge is therefore zero. Frame 2 shows what we'd see if we were observing all this from a frame of reference moving along with the lone charge. Both line charges are in motion in both frames of reference, but in frame 1, the line charges were moving at equal speeds, so their Lorentz contractions were equal, and their charge densities canceled out. In frame 2, however, their speeds are unequal. The positive charges are moving more slowly than in frame 1, so in frame 2 they are less contracted. The negative charges are moving more quickly, so their contraction is greater now. Since the charge densities don't cancel, there is an electric field in frame 2, which points into the wire, attracting the lone charge.
We appear to have a logical contradiction here, because an observer in frame 2 predicts that the charge will collide with the wire, whereas in frame 1 it looks as though it should move with constant velocity parallel to the wire. Experiments show that the charge does collide with the wire, so to maintain the Lorentz-invariance of electromagnetism, we are forced to invent a new kind of interaction, one between moving charges and other moving charges, which causes the acceleration in frame 2. This is the magnetic interaction, and if we hadn't known about it already, we would have been forced to invent it. That is, magnetism is a purely relativistic effect. The reason a relativistic effect can be strong enough to stick a magnet to a refrigerator is that it breaks the delicate cancellation of the extremely large electrical interactions between electrically neutral objects.
Although the example shows that the electric and magnetic fields do transform when we change from one frame to another, it is easy to show that they do not transform as the spacelike parts of a relativistic four-vector. This is because transformation between frames 1 and 2 is along the axis parallel to the wire, but it affects the components of the fields perpendicular to the wire. The electromagnetic field actually transforms as a rank-2 tensor.
An electromagnetic quantity that does transform as a four-vector is the potential. In section 3.7, I mentioned the fact, which may or may not already be familiar to you, that whereas the Newtonian gravitational field's polarization properties allow it to be described using a single scalar potential \(\phi\) or a single vector field \(\textbf{g} = − \nabla \phi\), the pair of electromagnetic fields (E, B) needs a pair of potentials, \(\boldsymbol{\Phi}\) and A. It's easy to see that \(\boldsymbol{\Phi}\) can't be a Lorentz scalar. Electric charge q is a scalar, so if \(\boldsymbol{\Phi}\) were a scalar as well, then the product q\(\boldsymbol{\Phi}\) would be a scalar. But this is equal to the energy of the charged particle, which is only the timelike component of the energy-momentum four-vector, and therefore not a Lorentz scaler itself. This is a contradiction, so \(\boldsymbol{\Phi}\) is not a scalar.
To see how to fit \(\boldsymbol{\Phi}\) into relativity, consider the nonrelativistic quantum mechanical relation q\(\boldsymbol{\Phi}\) = hf for a charged particle in a potential \(\boldsymbol{\Phi}\). Since f is the timelike component of a four-vector in relativity, we need \(\boldsymbol{\Phi}\) to be the timelike component of some four vector, Ab. For the spacelike part of this four-vector, let's write A, so that \(A_{b} = (\boldsymbol{\Phi}, \textbf{A})\). We can see by the following argument that this mysterious A must have something to do with the magnetic field.
Consider the example of Figure 4.2.3 from a quantum-mechanical point of view. The charged particle q has wave properties, but let's say that it can be well approximated in this example as following a specific trajectory. This is like the ray approximation to wave optics. A light ray in classical optics follows Fermat's principle, also known as the principle of least time, which states that the ray's path from point A to point B is one that extremizes the optical path length (essentially the number of oscillations). The reason for this is that the ray approximation is only an approximation. The ray actually has some width, which we can visualize as a bundle of neighboring trajectories. Only if the trajectory follows Fermat's principle will the interference among the neighboring paths be constructive. The classical optical path length is found by integrating k · ds, where k is the wavenumber. To make this relativistic, we need to use the frequency four-vector to form fb dxb, which can also be expressed as fbvb d\(\tau\) = \(\gamma\)(f − k · v) d\(\tau\). If the charge is at rest and there are no magnetic fields, then the quantity in parentheses is \(f = \frac{E}{h} = (\frac{q}{h}) \Phi\). The correct relativistic generalization is clearly fb = (\(\frac{q}{h}\))Ab.
Figure \(\PageIndex{3}\) - The charged particle follows a trajectory that extremizes \(\int\) fb dxb compared to other nearby trajectories. Relativistically, the trajectory should be understood as a world-line in 3+1-dimensional spacetime.
Since Ab's spacelike part, A, results in the velocity-dependent effects, we conclude that A is a kind of potential that relates to the magnetic field, in the same way that the potential \(\boldsymbol{\Phi}\) relates to the electric field. A is known as the vector potential, and the relation between the potentials and the fields is
$$\begin{split} \textbf{E} &= - \nabla \Phi - \frac{\partial \textbf{A}}{\partial t} \\ \textbf{B} &= \nabla \textbf{A} \ldotp \end{split}$$
An excellent discussion of the vector potential from a purely classical point of view is given in the classic Feynman Lectures.10 Figure 4.2.4 shows an example.
Figure \(\PageIndex{4}\) - The magnetic field (top) and vector potential (bottom) of a solenoid. The lower diagram is in the plane cutting through the waist of the solenoid, as indicated by the dashed line in the upper diagram. For an infinite solenoid, the magnetic field is uniform on the inside and zero on the outside, while the vector potential is proportional to r on the inside and to \(\frac{1}{r}\) on the outside.
10 The Feynman Lectures on Physics, Feynman, Leighton, and Sands, Addison Wesley Longman, 1970
Benjamin Crowell (Fullerton College). General Relativity is copyrighted with a CC-BY-SA license.
4.3: The Tensor Transformation Laws
Ben Crowell
Four-vector
Gravitational red-shifts
Ives–Stilwell experiments | CommonCrawl |
Dark Matter and Living Matter
Dark matter and living matter represent two deep mysteries of the recent world view. There however exists an amazing possibility that there might be close connection between these mysteries.
Do Bohr rules apply to astrophysical systems?
D. Da Rocha and Laurent Nottale have proposed that Schrödinger equation with Planck constant hbar replaced with what might be called gravitational Planck constant hbar_{gr}= GmM/v_0 (hbar=c=1). v_0 is a velocity parameter having the value v_0=about 145 km/s giving v_0/c=4.6\times 10^{-4}. This is rather near to the peak orbital velocity of stars in galactic halos. Also subharmonics and harmonics of $v_0$ seem to appear. The support for the hypothesis coming from empirical data is impressive. The support for the hypothesis coming from empirical data is impressive. It is surprising that findings of this caliber have not received any attention in popular journals while the latest revolutions in M-theory gain all possible publicity: also I heard from the article by accident from Victor Christianto to whom I am deeply grateful.
Is dark matter in astroscopic quantum state?
Nottale and Da Rocha believe that their Schrödinger equation results from a fractal hydrodynamics. Many-sheeted space-time however suggests that astrophysical systems are not only quantum systems at larger space-time sheets but correspond to a gigantic value of gravitational Planck constant. The gravitational (ordinary) Schrödinger equation would provide a solution of the black hole collapse (IR catastrophe) problem encountered at the classical level. The basic objection is that astrophysical systems are extremely classical whereas TGD predicts macrotemporal quantum coherence in the scale of life time of gravitational bound states. The resolution of the problem inspired by TGD inspired theory of living matter is that it is the dark matter at larger space-time sheets which is quantum coherent in the required time scale. I have proposed already earlier the possibility that Planck constant is quantized and the spectrum is given in terms of logarithms of Beraha numbers B_n= 4cos^2(pi/n): the lowest Beraha number B_3 =1 is completely exceptional in that it predicts infinite value of Planck constant. The inverse of the gravitational Planck constant could correspond a gravitational perturbation of this as 1/hbar_{gr}= v_0/GMm. The general philosophy would be that when the quantum system would become non-perturbative, a phase transition increasing the value of hbar occurs to preserve the perturbative character and at the transition n=4 --> 3 only the small perturbative correction to 1/hbar (3)=0 remains. This would apply to QCD and to atoms with Z>137 as well. TGD predicts correctly the value of the parameter v_0 assuming that cosmic strings and their decay remnants are responsible for the dark matter. The harmonics of v_0 can be understood as corresponding to perturbations replacing cosmic strings with their n-branched coverings so that tension becomes n^2-fold: much like the replacement of a closed orbit with an orbit closing only after n turns. 1/n-sub-harmonic would result when a magnetic flux tube split into n disjoint magnetic flux tubes.
Planetary system as a testing ground
The study of inclinations (tilt angles with respect to the Earth's orbital plane) leads to a concrete model for the quantum evolution of the planetary system. Only a stepwise breaking of the rotational symmetry and angular momentum Bohr rules plus Newton's equation (or geodesic equation) are needed, and gravitational Shrödinger equation holds true only inside flux quanta for the dark matter.
During pre-planetary period dark matter formed a quantum coherent state on the (Z^0) magnetic flux quanta (spherical shells or flux tubes). This made the flux quantum effectively a single rigid body with rotational degrees of freedom corresponding to a sphere or circle (full SO(3) or SO(2) symmetry).
In the case of spherical shells associated with inner planets the SO(3)--> SO(2) symmetry breaking led to the generation of a flux tube with the inclination determined by m and j and a further symmetry breaking, kind of an astral traffic jam inside the flux tube, generated a planet moving inside flux tube. The semiclassical interpretation of the angular momentum algebra predicts the inclinations of the inner planets. The predicted (real) inclinations are 6 (7) resp. 2.6 (3.4) degrees for Mercury resp. Venus). The predicted (real) inclination of the Earth's spin axis is 24 (23.5) degrees.
The v_0--> v_0/5 transition necessary to understand the radii of the outer planets can be understood as resulting from the splitting of (Z^0) magnetic flux tube to five flux tubes representing Earth and outer planets except Pluto, whose orbital parameters indeed differ dramatically from those of other planets. The flux tube has a shape of a disk with a hole glued to the Earth's spherical flux shell.
A remnant of the dark matter is still in a macroscopic quantum state at the flux quanta. It couples to photons as a quantum coherent state but the coupling is extremely small due to the gigantic value of hbar_gr scaling alpha by hbar/hbar_gr: hence the darkness. Note however that it is the entire condensate that couples to electromagnetism with this coupling, individual charged particles couple normally.
Living matter and dark matter
The most interesting predictions from the point of view of living matter are following.
The dark matter is still there and forms quantum coherent structures of astrophysical size. In particular, the (Z^0) magnetic flux tubes associated with the planetary orbits define this kind of structures. The enormous value of h_{gr} makes the characteristic time scales of these quantum coherent states extremely long and implies macro-temporal quantum coherence in human and even longer time scales.
The rather amazing coincidences between basic bio-rhythms and the periods associated with the states of orbits in solar system suggest that the frequencies defined by the energy levels of the gravitational Schrödinger equation might entrain with various biological frequencies such as the cyclotron frequencies associated with the magnetic flux tubes. For instance, the period associated with n=1 orbit in the case of Sun is 24 hours within experimental accuracy for v_0: the duration of day in Earth and in a good approximation also in Mars! Second example is the mysterious 5 second time scale associated with the Comorosan effect.
Indeed, the basic assumption of TGD inspired quantum biology is that the "electromagnetic bodies" associated with living systems are the intentional agents would conform with the idea that it is dark matter what makes ordinary matter living by acting as quantum controlling agent. Already now there exist a rather detailed theory about how these electromagnetic (or more generally, field-) bodies use biological body as a motor instrument and sensory receptor. For instance, the basic mechanisms of metabolisms would involve flow of matter between space-time sheets liberating energy quanta defining universal metabolic energy currencies same everywhere in Universe and having nothing to do with the details of living systems. The strange time delays of consciousness observed first by Libet suggests that the size of the field body is at least of the order of Earth size as also the frequency scale of EEG suggests (EEG would be involved with communications with magnetic body and biological body). For more details see the chapter "TGD and Astrophysics". For the notion of electromagnetic body see the relevant chapters of the book Genes, Memes, Qualia, and Semitrance.
How to Put an End to the Suffering Caused by Path Integrals
Path integrals have caused a lot of suffering amongst theoretical physicists. Lubos Motl gives a nice summary about Wick-rotation used quite generally as a trick to give some meaning to these poorly defined objects which have caused so much frustration. The idea of Wick rotation is to define path integrals of quantum field theory in Minkowskian space M^4 by replacing M^4 temporarily by Euclidian space E^4, by calculating the integral here as Euclidian functional integral having more meaning, and returning back to M^4 by analytically continuing in various parameters such as the momenta of particles. The trick has been also applied in the hope of making sense of path integral of General Relativity as well as in string models. I have never liked the trick, not because it is a trick, but just for the fact that this trick is needed at all. Something must be fatally wrong at the fundamental level. To see what this something might be, one must recall what Feynman did for long time ago.
How one ends up to path integral?
The path integral approach was abstracted by Feynman from a unitary time evolution operator by decomposing the time evolution to a product of infinite number of infinitesimally short time evolutions. After this "obvious" generalizations of the formalism lacking a real mathematical justification were made. Despite all the work done it can be safely stated, that the notion of path integral does not mathematically exist. The tricky definition of the functional integral through Wick rotation transforming it to functional (I will drop the attribute Euclidian in the sequel) integral is certainly not enough for a mathematician. I hasten to add that even the functional integrals are deeply problematic since the introduction of local interactions automatically induces infinities, and only in the case of so called renormalizable theories there exist a prescription for getting rid of these infinities.
What are the implicit philosophical ideas behind path integral formulation?
When the best brains have been unable to give a real meaning to the notion of path integral despite a work of about six decades, it is time to ask what might be behind these difficulties and whether it could relate to some cherished philosophical assumptions. a) Feynman's approach starts from Hamiltonian quantization and the notion of time is that of Newtonian mechanics. The representability of the unitary time evolution operator as sum over paths is natural in this context. No absolute time exists in the world of Special Relativity so that there are reasons to get worried. It might not be necessary nor even sensible in the Minkowskian context. c) The sexy idea about the sum of all histories with the classical physics identified as the history corresponding to the stationary phase might be simply wrong. Even Feynman could be sometimes wrong, believe or not! One can quite well consider some other, more sensible, approach to define S-matrix elements. d) Infinities are the basic problem of modern physics and are present for both path- and functional integrals. Local divergences are practically always present always as one tries to make a free theory interacting by introducing local interactions consistent with classical field theory. The basic assumption behind locality is that fundamental particles are pointlike. In string models this assumption is given up and there are indeed reasons to believe that superstrings of various kinds allow perturbation theory free of infinities. The unfortunate fact is that this perturbation series very probably does not converge to anything well-defined and is only an asymptotic series. The now-disappearing hope was that M-theory could resolve this problem by providing a non-perturbative approach to strings. d) In the perturbative approach the functional integrals give rise to Gaussian determinants, which are typically infinite formally. They can be eliminated but are aesthetically very awkward. TOE should be maximally aesthetic! These observations do not lead us very far but give some hints about what might go wrong. Perhaps the entire idea about sum over all possible paths with classical physics resulting via stationary phase approximation is utterly wrong. Perhaps the idea about space-time-local interactions is wrong and perhaps higher-dimensional fundamental objects might allow to get over the problems.
Neither Hamiltonian formalism nor path integral works in TGD
When I started to develop mathematical theory around the basic idea that space-times can be regarded as 4-dimensional surfaces in H=M^4xCP_2, I soon learned that perturbative approach fails completely. Indeed, it would be natural to construct a perturbation theory around canonically imbedded M^4 but for the only reasonable candidate for the action, Kähler action, the functional power series vanishes in the third order at M^4 so that the kinetic terms defining propagators vanish identically. For the same reason also Hamiltonian formalism fails completely. This is the case much more generally, and the enormous vacuum degeneracy (any 4-surface for which CP_2 projection belongs to at most 2-D Lagrange manifold is a non-deterministic vacuum extremal) kills all hopes about conventional quantization.
Geometrization of quantum physics as a solution to the problems
This puzzling state of affairs led to the idea that if quantization is not possible one should not quantize! This idea grew gradually to the vision that quantum states correspond to the modes of completely classical spinor fields of an infinite-dimensional configuration space CH of 3-surfaces, the world of classical worlds. This allows also the geometrization of fermionic statistics and super-conformal symmetries in terms of gamma matrices associated with the Kähler metric. The breakthrough came from the realization that general coordinate invariance in 4-dimensional sense is the key requirement. The obvious problem is that you have only 3-dimensional basic objects but you want 4-dimensional Diff invariance. Obviously the very definition of the configuration space geometry should assign to a given 3-surface X^3 a unique four-surface X^4(X^3) for 4-D general coordinate transformations to act on it. What would be the physical interpretation of this? X^4(X^3) defines the classical physics associated with X^3. Actually something more: X^4(X^3) is an analog of Bohr orbit since it is unique so that one can expect a quantization of various classical observables. Classical physics in the sense of Bohr orbitology would become part of quantum physics and of configuration space geometry. This is certainly something totally new and would mean a partial return from the days of Feynman to the good old days of Bohr when everything was still understandable and concrete. There are also other implications. Oscillator operators are the essence of quantum theory and can be geometrized only if configuration space has Kähler metric defined by so called Kähler function. Since classical physics should be coded by this Kähler function, it should be defined by a preferred extremal X^4(X^3) of some physically meaningful action principle. The so called Kähler action, which is formally Maxwell action for CP_2 Kähler form induced to space-time surface, is the unique candidate. The first guess is that X^4(X^3) could be identified as an absolute minimum of Kähler action. This is however a little bit questionable option since there is no lower bound for the value of Kahler action and if it gets negative and infinite, vacuum functional defined as the exponent of Kahler function vanishes. Indeed, it took 15 years to learn that this need not be the quite correct definition. A candidate for a more realistic identification came from a proposal for a general solution of field equations in terms of so called Kähler calibration. The magnitude of Kähler action would be minimized separately in regions where Lagrangian density L_K has a definite sign. This means that X^4(X^3) is as near as possible to a vacuum extremal. The Universe is maximally lazy energy saver! By minimizing energy of solution it might be possible to fix the time derivatives of the imbedding space coordinates at X^3 in order to find the X^4(X^3) by solving the partial differential equations as initial value problem at X^3. A considerable reduction of computational labor. This is of extreme importance, and even more so because Kähler action does not define a fully deterministic variational principle. There are indeed hopes of understanding the theory even numerically!
Generalized Feynman diagrams as computations/analytic continuations
A generalization of the notion of Feynman diagram emerging in TGD framework replaces sum over classical paths with what might be regarded as computation or analytic continuation. The first observation is that the path integral over all space-time surfaces with a fixed collection of 3-surfaces as a boundary does not make sense in TGD framework. Sum reduces to a single 3-surface X^4(X3) since classical physics in the sense of Bohr's orbitology is a quintessential part of configuration space geometry and quantum theory. Classical world is not anymore identified as a path with a stationary phase. This suggests completely different approach to the notion of Feynman diagram. It however took quite a long time before I realized how to formulate this approach more precisely. The idea came when I constructed a TGD inspired model for topological quantum computation. In topological quantum computation braids are the basic structures and quantum computation coded into the knotting and linking of the threads of the braid. This leads to a view that generalized Feynman diagrams do not represent sum over all classical paths but represent something analogous to computations with vertices representing some fundamental algebraic operations. A given computation can be carried out in very many equivalent manners and there always exists a minimal computation. In the language of generalized Feynman diagrams this would mean that diagrams with loops are always equivalent with tree diagrams. The summation over loops would be obviously multiple counting in this framework. This would be nothing but a far reaching generalization of the duality symmetry, which originally lead to string models. I have formulated this generalization in terms of Hopf (ribbon-) algebras here and in a different manner here. That there are several equivalent diagrams would conform with the non-determinism of Kähler action implying several equivalent space-time surfaces having given 3-surfaces as boundaries. This of course correlates directly with the fact that the functional integral and canonical quantization fail completely. The generalized Feynman diagrams could be also interpreted as space-time counterparts for different analytic continuations of configuration space spinor fields (classical spinor fields in the world of classical worlds) from a sector of configuration space with a given 3-topology to another sector with different topology (initial and final states of particle reaction in the language of elementary particle physicist). This continuation can be performed in very many manners but the final result is same always, just as in case of equivalent computations.
Getting rid of standard divergences
It is possible to get rid of path integrals in TGD framework but not from the functional integral over the infinite-dimensional world of classical worlds. This integration means performing an average over these well-defined generalized Feynman diagrams, one might say over predictions of finite quantum field theories. This functional integral in question could bring back the basic difficulties but it does not. a) The vacuum functional over quantum fluctuating degrees of freedom defining the functional integral is completely analogous to a thermal partition function defined as an exponent of Hamiltonian in thermodynamics at a critical temperature. Kähler coupling strength is analogous to critical temperature, which means that the values of the only free parameter of the theory are predicted as they should in any respectable TOE. The good news is that Kähler function is a non-local functional of 3-surface X^3. Hence the local divergences unavoidable in any local QFT are absent. If one would try to integrate over all X^4, one would have Kähler action and locality and all the problems of standard approach would be magnified since the action is extremely non-linear. b) Vacuum functional is the exponent of Kähler function and in the perturbation theory configuration space contravariant metric becomes propagator. The Gaussian determinant is the inverse of the metric determinant and these two ill-defined determinants neatly cancel each other so that also aesthetic is perfect! Note that the coefficient of the exponent of Kähler function is also fixed. A further good news is that there are hopes that the functional integral might be carried out exactly by performing perturbation theory around the maxima of Kähler function. These hopes are stimulated by the fact that the world of classical worlds is a union of symmetric spaces and for a symmetric space all points are metrically equivalent. In the finite-dimensional case there are a lot of examples about the occurrence of this phenomenon. The conclusion is that the standard divergences are not present and that this result is basically due to a new philosophy rather than some delicate cancellation mechanism.
What about zero modes?
Is there something that could still go wrong? Yes. The existence of configuration space metric requires that it is a union over infinite dimensional symmetric spaces labelled by zero modes whose contribution to CH line element vanishes. An infinite union is indeed in question: if CH would reduce to single symmetric space, a 3-surface with size of galaxy would be equivalent with a 3-surface associated with electron. The zero modes characterize classical degrees of freedom: shape, size, and the induced Kähler form defining a classical Maxwell field on X^4(X^3). In zero modes there is no proper definition of the functional integral. Here comes however quantum measurement theory in rescue. Zero modes are non-quantum fluctuating degrees of freedom and thus behave like genuine classical macroscopic degrees of freedom. Therefore a localization in these degrees of freedom is expected to occur in each quantum jump as a counterpart of quantum measurement. These degrees of freedom should be also correlated in one-one manner with quantum fluctuating degrees of freedom like the pointer of measurement apparatus with the direction of electron spin. A kind of duality between quantum fluctuating degrees of freedom and zero modes is required. We would experience the macroworld as completely classical because each moment of consciousness identifiable as quantum jump makes it classical. It is made again non-classical during the unitary U process stage of the next quantum jump. Dispersion in zero modes, localization in zero modes, dispersion in zero modes,.... Like Djinn getting out of the bottle and representing a very long list of classical wishes of which just one is realized. With this complete localization or localization to a discrete union of points in zero mode degrees of freedom, S-matrix elements become well defined. Note however that the most general option would be a localization into finite-dimensional symplectic subspaces of zero modes in each quantum jump. The reason is that zero modes allow a symplectic structure and thus all possible finite-dimensional integrals are well defined using the exterior powers of symplectic form as integration measure.
What about renormalization?
The elimination of infinities relates closely to the renormalization of coupling constants, and one could argue that the proposed beautiful scenario is in conflict with basic experimental facts. This not the case if Kähler coupling strength has an entire spectrum of values labelled by primes or subset of primes labelling p-adic length scales. In this picture p-adic primes label p-adic effective topologies of non-deterministic extremals of Kähler action. p-Adic field equations possess an inherent non-determinism and the hypothesis is that this non-determinism gives in an appropriate length scale rise to fractality characterized by an effective p-adic topology such that the prime p is fixed from the constraint that the non-determinism of Kähler action correspond to the inherent p-adic non-determinism in this length scale range. The highly non-trivial prediction is that quantum non-determinism is not just randomness since p-adic non-determinism involves long range correlations due to the fact that in p-adic topology evolution is continuous. The proposal is that the long range correlations of locally random looking intentional/purposeful behavior could correspond to p-adic non-determinism with p characterizing the "intelligence quotient" of the system. This is a testable prediction. The coupling constant evolution is replaced by the p-adic length scale dependence of Kähler coupling strength and of other coupling constants determined by it. The emergence of the analogs of loop corrections might be understood if there is a natural but non-orthogonal state basis labelled by quantum numbers which allow a natural grading. The orthogonalized state basis would be obtained by a Gram-Schmidt orthonormalization procedure respecting the grading. Orthonormalization introduces a cloud of virtual particles and the dependence of this Gram-Schmidt cloud on prime p would induces the TGD counterpart of renormalization group evolution. It is clear that the classical non-determinism of Kähler action is the red thread of the proposed vision. By quantum classical correspondence space-time surface is not only a representation for a quantum state but even for the final states of quantum jump sequence. Classical non-determinism would thus correspond to the space-time correlate of quantum non-determinism. Matti Pitkänen
Melancholic Moods and Road to Reality
These periods of stagnation are difficult to tolerate. You have been experiencing a flow of mathematical ideas continuing more than year with only brief periods of recovery from physical exhaustion. Then you lose the contact. No ideas pop up. You have been asked to write articles but you cannot do it: a passive organizing of existing material into articles simply cannot motivate you, and you cannot develop enough self discipline to do this by brute force. Chess problems are your latest invention in the war against depression: your brain gets depressed unless it receives its daily portion of problems. You also write to your blog page these melancholic musing to stay sane. You begin to feel more and more strongly your role in the society: an unemployed academic village fool, trying to survive with a minimum possible unemployment money. Without any future and thus without any hope. The academic world of Finland will never forgive you that you have carried out without their permission a lifework summarized in these four 1000-page books which have become your identity. The punishment is the cruellest possible: they have stolen your future. You feel that you have lost completely your faith in humanity but at the same time know that you should be able to cheat yourself to believe on the ultimate justice. Fear fills you: how could you recover some of this naive and healthy belief that all power holders of science are not gangsters. In this mood you open monday's email box and ... experience something completely unexpected. An email tells that Sir Roger Penrose mentions in his newest book "Road to Reality" your work relating to p-adic physics. Although you have talked a lot about name magic, you cannot hide your happiness. You feel it really consolating that there are also honest and intelligent people in this cruel, and you notice notice it now, beautiful world. You know that this kind of gesture has miraculous effect: some collegue might lower himself to touch your work. Even better, this book will inspire many young people who still have an open mind. While looking out you see that Sun is almost shining. You hope to have some day the opportunity to read Road to Reality yourself and remember that there was a discussion in Not-Even-Wrong about Road to Reality. Matti Pitkanen
In to-day's Not-Even-Wrong I learned that M-theory God Ed Witten has followed his brother's Matt Witten's example and is now a TV writer(;-)! The TV series Numb3rs is very ambitious project. The basic goal is to fight against the anti-intellectualism, which has got wings during Bush's era and attack the stereotype about mathematician as a kind of super book-keeper and super-calculator with zero real life intelligence. A TV series in which mathematician's solve crimes is an ingenious choice since detectives must be real-life-intelligent even if they are mathematicians. The team coworks with real mathematicians in Caltech (as you see they look very hippie like) since the goal is to be as autenthic as possible. Even the formulas on blackboard must be sensible so that even mathematician can enjoy the series without fear of sudden strong visceral reactions. Ed Witten got a manuscript to read and proposed an episode in which a rogue mathematician proves Riemann Hypothesis to destroy internet security. My interest is keen since I have proposed a proof, or more cautiously A Strategy for a Proof of Riemann Hypothesis, which has been published in Acta Math. Univ. Comeniae, vol. 72. . I have proposed also a TGD inspired conjecture about the zeros of Zeta. The postulate is that real number based physics of matter and various p-adic physics (one for each prime p) describing correlates of cognition are obtained by algebraic continuation from rational number based physics. This translates to the mathematics the idea that cognitive representations are mimicries of reality and cognitive representation and reality meet each other in a finite number of rational points. This is just what happens in the numerical modelling of the real world since we can represent only rationals using even the best computers. This vision leads to concrete conjectures about the number theoretical anatomy of the zeros of Riemann Zeta which appear in a fundamental role in quantum TGD. The conformal weights of the so called super-canonical algebra creating physical states are suitable combinations of zeros of Zeta. The conjecture is following: for any zero z=1/2+iy of Zeta at critical line the numbers p^(iy) are algebraic numbers for every prime p. Therefore any number q^(iy) is an algebraic number for any rational number q. This assumption guarantees that the expansion of Zeta makes sense also in various p-adic senses for z=n+1/2+iy. A related conjecture is that ratios of logarithms of rationals are rationals: this hypothesis could in principle be tested numerically by looking whether ratios of this kind have periodic expansions in powers of any chosen integer n>1. I would be happy if I had even a slight gut feeling about how the "Strategy for a Proof of Riemann Hypothesis" might relate to Internet safety. Here I meet the boundaries of my narrow mathematical education. So, at this moment it seems that I will not be a notable risk for Internet safety. A word warning is however in order: TGD will certainly become a safety risk for M-theory: sooner or later;-)! Matti Pitkanen
Parody as means to see what went wrong
The world of science is a strange world. You readily see that its inhabitants are strangely twisted creatures having somehow surreal shapes. You also recognize that all of them have in some difficult-to-define sense lost their way. You gradually begin to realize what the problem might be. It seems that these creatures have lost themselves into a forest. This forest is not an ordinary forest. No. The trees of this forests are concepts and expressions of language. It is obvious that these meanderes in the forest have forgot where they came from and what they were. They do not have the familiar five senses anymore, they have replaced reality with symbols. These creatures without senses look depressed and unhappy. What is left from real life are occasional bursts of aggression. Clearly, they can still hate and be bitter and do so intensively but what has happened to love and trust and joy? Something has gone badly wrong and you would be happy if you could help them somehow but you do not know what to do. Certainly it seems that it is not a good idea to go and tap on the shoulder and suggest a beer or two and a friendly chat. You ask how it is possible that a happy gifted child came an angry and bitter academic identifying himself with his curriculum vitae. You gradually learn that this tragic somehow relates to meanings. Clearly, these people have started to assign meanings to symbols of reality instead of reality. Life goes by as they have their endless bloody debates about priorities. You find it really heart breaking to see how these tiny beings fight passionately about something which has absolutely no meaning for you or any healthy human being who has learned what are those few really signficant things in life. How could even a best Zen guru teach these people the joy of carrying water and chopping firewood. Superstring community certainly represents an extreme example about this anomalous association of meanings. It is its own micro-culture talking strange stringy language expressing strange stringy concepts. This closed society of stringists is like a religious sect with its own difficult-to-understand rules and habits. In order to understand what I mean, try to imagine a community which has replaced spoken language with written scientific jargon with every sentence ornamented with minimum of one reference or foonnote and discussing via published articles rather than face-to-face. W. Siegel, a string theorist himself but in a strangely ambivalent manner also an outsider, has produced hilarious parodies about typical super string articles. Here is one example. Do not miss the other ones. The following piece of text made me laughing to science and myself and I realized how insane this world (I do not try to pretend that it includes also me) is. My hope is that this text fragment might give even a reader without any background in science a healthy burst of laugh. "The moonlight danced on her face as it reflected off the warm August waves. He watched her graceful body gradually emerge as she stepped closer to the shore. As she lay down in the sand beside him he leaned close to whisper in her ear: "String theory is presently the most successful model to describe quantum gravity [1]. Unfortunately, like QCD before it, the proof of its relevance to the real world was shortly followed by the realization that it was impossible to calculate anything with it..." Matti Pitkanen
Mersenne Primes and Censorship
Lubos Motl commented at his blog site about the largest known Mersenne prime , which is 2^24,036,583 -1. This inspired me to write a comment copied below (I have added a couple of links and added some detail).
....Not only Mersennes ....
Mersenne primes are in Topological Geometrodynamics framework the most interesting primes since they correspond to most important p-adic length scales. Only Mersennes up to M_127 =2^127-1 are interesting physically since next Mersenne corresponds to a completely super astrophysical length scale. M_127 corresponds to electron whereas M_107 corresponds to the hadronic length scale (QCD length scale). M_89 corresponds to intermediate boson length scale. There is an interesting number theoretic conjecture due to Hilbert that iterated Mersennes M_{n+1}= M_{M_n} form an infinite sequence of primes: 2,3,7,127,;M-_{127},.... etc. Quantum computers would be needed to kill the conjecture. Physically the higher levels of this hierarchy could be also very interesting.
...but also Gaussian Mersennes are important in TGD Universe
Also Gaussian primes associated with complex integers are important in TGD framework. Gaussian Mersennes defined by the formula (1\pm i)^n-1 exist also and correspond to powers p=about 2^k, k prime. k=113 corresponds to the p-adic length scale of muon and atomic nucleus in TGD framework. Neutrinos could correspond to several Gaussian Mersennes populating the biologically important length scales in the range 10 nanometers 5 micrometers. k=151,k=157,k=163, k=167 all correspond to Gaussian Mersennes. There is evidence that neutrinos can appear with masses corresponding to several mass scales. These mass scales do not however correspond to these mass scales but to scale k=13^2=169 about 5 micrometers and k=173. The interpretation is that condensed matter neutrinos are confined by long range classical Z^0 force predicted by TGD inside condensed matter structures at space-time sheets k=151,...,167 and those coming from say Sun are at larger space-time sheets such as k=169 and k=173. p-Adic mass calculations are briefly explained here and here, where also links to the relevant chapters of p-Adic numbers and TGD can be found. That Gaussian Mersennes populate the biologically most interesting length scale range is probably not an accident. The hierarchical multiple coiling structure of DNA could directly correspond to these Gaussian Mersennes. The ideas about the role of Gaussian Mersennes in biology are discussed briefly here can be found. For more details see the chapter Biological realization of self hierarchy of "TGD Inspired Theory of Consciousness...".
...and how Lubos Motl responded...
Here is Lubos Motl's response which reflects his characteristic deep respect of intellectual integrity: "Matti - well, let me admit that I don't believe any of the things about "TGD" you wrote." The response was not surprising in view of what I have learned about Lubos Motl's typical behavioral patterns. I wonder whether the open censoring of ideas and opinions which do not conform with those of hegemony relates to change that has occurred during the regime of Bush. For instance, in a recent poll it was found that about one half of student agreed when it was asked whether the full freedom of speech guaranteed in constitution law should be restricted. Food for a worried thought! My response, which I unfortunately did not store to my computer, was related to the basic trauma suffered by theoretical physicists at the end of the era of hard-boiled reductionism. It could be called Planck length syndrome and manifests itself as desperate attempts to explain all elementary particle masses in terms of Planck mass scale although the first glance at elementary particle mass spectrum demonstrates that there are obviously several mass mass scales involved: electron, mu, and tau or quarks simply cannot belong to same multiplet of symmetry group in any conceivable scenario based on a unifying gauge group. p-Adic length scale hypothesis resolves the paradox and allows to understand among other things the basic mystery number 10^38=about 2^127-1 (largest Mersenne prime of physical signficance in human scales) of physics. Also a far reaching generalization of the scope of what physics can describe emerges: p-adic physics is physics of cognition which must be also a part of theory of everything. The response of Lubos is a school example about how difficult it is to communicate new, radical, and working idea. Other typical and equally brilliant response would have been "You cannot be right because Witten would have discovered it before you!". Matti Pitkanen
What UFOs could teach about fundamental physics?
Here is my comment about UFOs to Not-Even-Wrong discussion group, where it was mentioned that Michio Kaku, string theorists and author of "Hyper-Space", has expressed publicly as his opinion that UFOs might be a real. Leaving aside ontological considerations and the question what one should think about people taking seriously UFOs, one could take UFOs as an inspiration for a thought experiment. Suppose for a moment that UFOs represent a real technology. According to the reports, UFOs seem to have a very small inertial mass (butterfly like motions involving sudden accelerations and changes of direction of motion without producing any shock waves). A technology able to reduce dramatically inertial mass of a material object would thus exist. What could this tell about fundamental physics? A possible answer would be a modification of Equivalence Principle. Gravitational mass would be absolute value of inertial mass, which can have both signs. One of the most obvious implications is an explanation for why gravitational energy is definitely not conserved in cosmological scales whereas there is no evidence for the non-conservation of inertial energy. The simplest cosmology would be that created from inertial vacuum by energetic vacuum polarizations creating regions of positive and negative density of inertial mass. The 4-D universe replacing itself by a new one quantum jump by quantum jump would become possible and the difficult philosophical problems formulated as questions like "What was the initial state of the Universe and what were the initial values/densities of conserved quantities at the moment of big bang" would disappear. The observations motivating the anthropic principle would find a natural explanation: the universe has gradually quantum engineered itself so that the values of these constants are what they are. Technological implications would be also interesting. Forming an tightly bound state of systems with positive and negative inertial mass a large feather light system could be created. Could UFOs utilize this kind of technology? Accepting negative energies, one cannot avoid the questions whether negative energy signals propagate backwards in (geometric) time and whether phase conjugate light discovered at seventies could be identified as signals of this kind. Positive answer would have quite interesting technological implications. Negative energy signals time reflected as positive energy signals from time mirrors (lasers with population reversal for instance) would allow communications with geometric past. Our memory might be based on this mechanism: to recall memories would be to scan the brain of geometric past by using reflected in time direction (rather than in spatial direction as seeing in the ordinary sense). Communications with the civilizations of the geometric future and past might become possible by a similar mechanism. Matti Pitkanen
Color confinement and conformal field theory
The discovery of number theoretical compactification has meant a dramatic increase in the understanding of quantum TGD. There are two manners to undestand the theory.
Number theoretic view: Space-time surfaces can be regarded as hyper-quaternionic 4-surfaces in 8-dimensional hyper-octonionic space HO.
Physics view: Space-time surfaces can be seen as 4-surfaces in 8-D space M^4xCP_2 minimizing the so called Kähler action which essentially means that they minimize their non-commutativity measured by Lagrangian density of Kähler action.
These views seem to be complementary, and at this moment the very existence of this duality (conjecture of course) is what has the strongest implications. A lot remains to do in order to see whether the conjecture is indeed correct and what it really implies. At this moment I am trying to find whether this duality, very reminiscent of various M-theory dualities, is internally consistent. One of the possible implications is the possibility to interpret TGD also as a kind of string theory, not in the usual sense of the world, but as a generalization of so called topological quantum field theories, where the notion of braids is central. Whether this duality is completely general or holds true only for selected space-time surfaces, such as space-time sheets corresponding to maxima of Kähler function (most probable space-times) or space-time sheets representing asymptotic behavior, is an open question. I have explained the duality in earlier posts and do not go to the details here. Suffice it so say that so called Wess-Zumino-Witten action for group G_2, a group which as a Lie group is completely exceptional, and acts as the automorphism group of octonions, suggests itself as a characterizer of the dynamics of these strings. G_2 has group SU(3) as maximal subgroup and can be said to leave these strings invariant. The interpretation is as the color group and G_2/SU(3) coset theory is the natural guess for the dynamics. SU(3) takes indeed the role of color gauge group. The so called primary fields of the theory correspond to two color singlets, triplet and antitriplet and the natural guess is that they relate to leptons and quarks. Indeed in the H picture the basic fields are lepton and quark fields and all other particles are constructed from leptonic and quark like excitations. The beauty of this approach is that QCD might be replaced with an exactly solvable conformal field theory allowing also to deduce how correlation functions change in hyper-octonion analytic transformations affecting space-time surface. There are however also objections against this picture. a) The basic objection is that G_2 Kac-Moody algebra contains triplet and anti-triplet generators and triplet generators commute to anti-triplet. It is hard to imagine any sensible physical interpretation for these lepto-quark generators, whose commutation relations break the conservation of lepton and quark number. The point is however that triplet generators affect e_1, and thus S^6 coordinates and also the SU(3) subgroup acting as isotropy group changes. Thus correlation functions involving these currents are not physically meaningful. Indeed, in G/H coset theory only the H Kac-Moody currents appear naturally in correlation functions since the construction involves functional integral only over H connections. b) If 14-dimensional adjoint representation of G_2 would appear as primary field, also 3 and \overline{3} lepto-quark like states for which baryon and lepton number are not conserved would appear in the spectrum. This is in conflict with H picture. The choice k=1 for Kac-Moody central charge provides however a unique manner to circumvent this difficulty. Integrability condition for the highest weight representations allows for a given value of k only the highest weights \lambda_R satisfying Tr(\phi \lambda_R)\leq k, where \phi is the highest root for Lie-algebra. Since the highest root has length squared 2, adjoint representation is not possible as a highest weight representation for k=1 WZW model, and the primary fields of G_2 model are singlet and 7-plet corresponding to the hyper-octonionic spinor field and defining in an obvious manner the primary fields 1+3+\overline{3} of G_2/SU(3) coset model. Fusion rules for 1\oplus 7 correspond to octonionic multiplication. The absence of G_2 gluons saves from lepto-quark like bosons, and the absence of SU(3) gluons can be interpreted as HO counterpart for the fact that all particles, in particular gluons, can be regarded bound states of fermions and anti-fermions in TGD Universe. This picture conforms also with the claims that 3+\overline{3} part of G_2 algebra does not allow vertex operator construction whereas SU(3) allows the construction in terms of two free bosonic fields. These fields would naturally correspond to the two X^4 directions transversal to the string orbit defined by 1 and e_1. One could say that strings in X^4 are able to represent color Kac-Moody algebra and that SU(3)is inherent to 4-dimensional space-time. c) The third objection is that conformal field theory correlation functions obeying simple scaling laws are not consistent with the exponentially decreasing correlation functions suggested by color confinement. A resolution of the paradox could be based on the role of classical gravitation. At light-like causal determinants the time-like component g_{tt} of the induced metric vanishes meaning that classical gravitational field is very strong. Hence also the normal component g_{nn} of the induced metric is expected to become very large so that hadron would look like the interior of black hole. A finite X^4 proper time for reaching the outer boundary of the hadronic surface can correspond to a very long M^4 time and the finite M^4 distance from the boundary can mean very long distance along hadronic space-time surface. Hence quarks and gluons can behave as almost free particles when viewed from hadronic space-time sheet but look confined when seen from imbedding space. If the hyper-quaternionic coordinates appearing in the correlation functions correspond to internal coordinate of the space-time surface, the correlation functions when expressed in terms of M^4 coordinates can look confining. For more details see the chapter TGD as a Generalized Number Theory: Quaternions, Octonions, and their Hyper Counterparts. Matti Pitkanen
Approaching the end of an epoch
The following quotes from the page Suppression, Censorship and Dogmatism in Science" serve as a good introduction to the recent sad situation in science. Do not miss the chronological ordering. "Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense." Buddha (563BC-483BC). "There must be no barriers to freedom of inquiry. There is no place for dogma in science. The scientist is free, and must be free to ask any question, to doubt any assertion, to seek for any evidence, to correct any errors." J. Robert Oppenheimer, quoted in Life, October 10, 1949. "The more important fundamental laws and facts of physical science have all been discovered, and these are now so firmly established that the possibility of their ever being supplanted in consequence of new discoveries is exceedingly remote.... Our future discoveries must be looked for in the sixth place of decimals." Albert Abraham Michelson, speaking at the University of Chicago, 1894. "The great era of scientific discovery is over.... Further research may yield no more great revelations of revolutions, but only incremental, diminishing returns." Science journalist John Horgan, in The End of Science (1997).
Suppression in science
Anyone who has devoted life to some new idea has experienced the arrogance and cruelty of those who are in power. This applies also to me. I have used 26 years of my life to a revolutionary idea and I have summarized the resulting world view in 4 books making about 5000 pages full of original ideas developed in highly detailed manner. Both in quality and quantity this output is exponentially higher than that of an average professor. One might think that with this background it would not be difficult to find a financiation for my research which is still continuing. The reality however loves paradoxes. There is absolutely no hope of getting any support in my home country for my work, and I do not believe that situation might be better elsewhere. Even worse. Average theoretical physicist colleague refuses to read anything that I have written, and there is absolutely no manner to communicate these ideas to a mainstream career builder living in a typical academic environment. I am doomed to be a crackpot. Although American Mathematical Society lists Topological Geometrodynamics in Mathematical Subject Classification Tables, something which can be regarded as a rare honor for a physicist, I am a crackpot. It is easier to change water into wine than change the conviction of an average research person receiving a monthly salary in University about my crackpot-ness. I am not of course alone. The suppression in science has become rule rather than a rare exception, and even Nobelists like Brian Josephson, are punished with censorship for the courage of talking aloud about phenomena not understood within the confines of existing dogmas recent. The victims of this suppression cannot publish anything and even e-print archives such as arXiv.org are closed for those who think in a new manner. These people have started to organize. For instance, at the web page Archivefreedom.org scientific dissidents tell their personal horror stories.
End of an epoch as an era of moral degeneration
One might try to interpret this sad state of theoretical physics (and science in general of course), which affects deeply also experimental physics, in particular particle physics, using notions like sociology of science, and talking about the fierceful competion about positions and research money implying that people with keen brain but without intelligent elbows are doomed to be destroyed. I think however that there is something deeper involved. My conclusion from what has happened during these decades, and from a personal work in wide range of topics ranging from particle physics to cosmology to biology to consciousness, is that we are living an end of an epoch, not only in science but also in the evolution of western civilization. Epochs are always characterized by some sacred philosophical ideas, and when they lose their explanatory and guiding power, the civilization breaks down and experiences a phase transition to a "New Age". The irony is that the end of epoch is always seen as the final victory of the old dogmas (see the last quotations in the beginning of these musings!). In physics only this and comparison with what happened for a century ago, helps to understand how M-theories, regarded now by most professionals (privately of course!) as a pathetical failure, are still successfully sold for media as the final word about physics. Of course, things have begun to change rapidly. For instance, for some time ago the physics department of Boston University made the decision that string models cannot be regarded as physics so that string theorists must find their positions from Mathematics department. Do not take me wrongly: many of the mathematical ideas of string models are fantastics and will be pieces of future theories. The problems is that the utterly wrong philosophy makes it impossible build a physical theory based on these ideas. In physics several philosophical dogmas have lost their power during the last century. For century ago before advent of quantum theory the idea about deterministic clockwork universe was the final truth. The attribute "Newtonian" is often assigned with this clockwork. This does injustice to Newton, who was five centuries ahead of his time and regarded even planetary systems as living systems. This was described in some popular science document as "the dark side of Newton"! Although quantum physics in microscopic length scales turned out to be in conflict with the clockwork idea and although the evidence for macroscopic quantum effects is increasing, the average physic is still, at the temporal distance of century from Bohr, stubbornly refuses to consider the possibility that quantum effects might be important in our everyday life, and that we might not be deterministic machines. This collective silliness has meant after the days of von Neumann nothing interesting has been said about quantum measurement theory so that its paradoxes are still there, virtually untouched. What a gem for a brilliant theoretician these paradoxes could be but how few are those who are ready to spend life in science as crackpots! The sad consequence is that theoretical physicists have practically nothing to add to what they could had said about biology and consciousness before Bohr. Second sacred and very dead idea is reductionism. String theories and M-theory amplify this idea to sheer madness. The dogma is that the whole physics reduces to Planck scale: a very short distance about 10^(-33) meters. After a huge amount of theoretical activity during last 20 years we have a a theory which cannot predict anything even in principle, and the proponents of this theory are now seriously suggesting that we must accept this state of affairs since M-theory is the final theory! I know...! I know that I have said this many times before but this is so scandalous that it must be said aloud again and again. Of course, one need not go to Planck length to see the failure of reductionism: again the phenomenon of life tells that reductionism is wrong. In fact, all transitions in the sequence quark physics--> hadron physics --> nuclear physics ---> atomic physics --> molecular physics --> ... are all ill-defined, and the belief that lower levels predict the higher ones is just a belief without any justification. Despite this most colleagues enjoying monthly salary are still repeating the liturgy which I heard for the first time in my student days: life as nothing but Schrödinger equation applied to a very complex system. It is understandable that a person at the age of 20 years takes this platitude seriously but it is unforgivable that professors of physics still repeat this kind of trash. The seeds of the new age are already here: fractality is now a relatively well established notion even in physics, and means an entire hierarchy of scales giving good hopes of expressing quantitatively how reductionism fails. A lot of new, but already existing mathematics is however needed. Unfortunately, and totally unexpectedly, for consistency of interpretation one cannot avoid talking about physical correlates of cognition once this mathematics is introduced. The transition to the New Age will thus be very painful since cognition and consciousness do not exist for a physicist. Next big idea. Average physicist familiar with special relativity identifies without hesitation the time of conscious experience as the fourth space-time coordinate. This despite the fact that brief analysis shows that these notions are quite different. The sole justification for their identification is the materialistic dogma: in the materialistic world order consciousness is a mere epiphenomenon and has no active role: everything about contents of consciousness is known once the state of the clockwork is known. Despite all the paradoxes that this view implies, it dominates still even neuroscience as I painfully learned during the decade that I devoted to TGD inspired theory of consciousness and learned basics of neuroscience in discussions and by reading. Energy is dual to time and subjectivetime=geometric time problematics repeats itself as the poorly understood relationship between inertial energy and gravitational energy. Einstein's proposal was that they are one and the same thing but even academic person could privately wonder why it is possible that inertial energy is conserved exactly whereas gravitational energy is not. Of course, friendly Einstein has so long a shadow that the academic person does not allow this kind of thoughts to pop up into his conscious mind.
The sociological side
There are also social factors involved and these factors are amplified during a period of degeneration that we are living. To make this more concrete I tell about my experiences in certain discussion groups related to string models and M-theory during last half year. My motivation for spending (or rather wasting as it turned out) my time in these groups was besides satisfying my basic social needs, to learn what the general situation is, and perhaps even communicate something: it is extremely frustrating to see that physics is in a state of deep stagnation and, this I can say without any hybris, to realize that one reason for this is simply that I have not used all my energy to communicate the bottleneck ideas which are the only way out of the dead end. To be specific, I have read the comments at the Not-Even-Wrong blog page administered by Peter Woit. For half a year ago I was enthusiastic: this would be a page where the problems of M-theory and string models could be discussed openly without repeating the usual liturgy starting with "M-theory is the only known quantum theory of gravitation...". It however soon became clear that the discussion group is not intended to serve as a forum for new ideas. The dream about return back to the good old days of quantum field theories seems to be the basic motivation for its existence, and any new idea is therefore regarded as an enemy. For a month or so ago an open censorship was established meaning that everything which is not about the topic of discussion is censored away. To stay in the "topic of discussion" is to repeat the already-too-familiar mantra like arguments against M-theory. M-theory is dead but it does not help to say this again and again: something more constructive is needed. This discussion group has taught me a lot about the difference between western and eastern communication culture manifests. Typically the discussions are battles: people express their verbal skills by insulting each other personally in all imaginable manners. Open verbal sadism is completely acceptable and often regarded as a measure of intelligence. Paradoxically, I myself find that I take seriously Lubos Motl, young extremely aggressive M-theory fanatic who has been continually ranting and raving in Not-Even-Wrong. It is easy to disagree in a well-argumented manner with most of what he says, and most of his arguments are cheap rhetoric, but still! Why do I pay attention to this empty verbalism. Sad to confess, I am part of this culture which confuses aggression with intellect. In Eastern cultures, which we tend to assign labels like "New-Age", both physical and intellectual integrity are the basic values, and my dream is that the New Age would see these people continually insulting each other simply as what they are: uncivilized barbarians. NAMEs are important ghostly participants in any academic discussion group. Again and again I find, that people are asking was it this or was it that what Feynman/Gross/Witten..... said and what is the URL where it can be found. Somehow it seems completely impossible to be taken seriously unless one continually drops names and citations. With all respect to these brilliant NAMEs, one must admit that their often casual comments are just casual comments and that 65 year old NAME probably has not very interesting things to say about forefront science. A further amusing feature of the sociology of academic discussion groups is the attitude towards "crackpots". "Crackpots" are by definition those who have something original to say and strong motivations to do this. The strategy for dealing with crackpots is following. In the first stage "serious researchers" do not notice in any manner the presence of "crackpot", he feels that he is just air. If this does not help, ironic and sadistic comments communicating the conviction that the "crackpot" is a complete idiot begin to appear. If even this does not help, the "crackpot" is told to leave the discussion group since people want to have "serious discussions". If even this does not beat the visitor, his messages are censored away. The irony is that also in this group the "crackpots" have been the most interesting participants. They have something interesting to say, they have a passion of understanding, they develop real arguments instead of cascades of insults and citations from authorities, and they have a strong urge to express clearly what they want to say. This cannot be said about many "respected" contributions, which often contain about ten typos and grammatic errors per line, and from which it is often completely impossible to figure out whether the attempt is to say something or just to teach by example that the writer is an an-alphabet. One gloomy conclusion from these experiences about the sociology of science might be that organized good is impossible. Perhaps this true to some degree. I can however imagine the feelings of those enlightened people who discovered the idea of gathering together intelligent people devoting their lives to science. Certainly they dreamed of spirited debates, endless discussions, openness of minds, evolution of ideas. It is not the fault of these visionaries that in a typical university people sitting in the same room for years know nothing about each other's work and could not be less interested, that hatred and envy are the dominant emotions of an average career builder, and that science has transformed from passion to a fierceful competition to a war in which the best colleague is a dead colleague. Perhaps the New Age inspired by new Great Ideas and Ideals will create a new kind of university in which people could could live like ordinary simple people like to live: listening, helping and loving each other. Matti Pitkanen
Infinite primes and physics as a generalized number theory
Layman might think that something which is infinite is also something utterly non-physical. The notion of infinity is however much more delicate and it depends on topology whether things look infinite or not. Indeed, infinite primes have become besides p-adicization and the representation of space-time surface as a hyper-quaternionic sub-manifold of hyper-octonionic space, basic pillars of the vision about TGD as a generalized number theory.
1. Two views about the role of infinite primes and physics in TGD Universe
Two different views about how infinite primes, integers, and rationals might be relevant in TGD Universe have emerged. a) The first view is based on the idea that infinite primes characterize quantum states of the entire Universe. 8-D hyper-octonions make this correspondence very concrete since 8-D hyper-octonions have interpretation as 8-momenta. By quantum-classical correspondence also the decomposition of space-time surfaces to p-adic space-time sheets should be coded by infinite hyper-octonionic primes. Infinite primes could even have a representation as hyper-quaternionic 4-surfaces of 8-D hyper-octonionic imbedding space. b) The second view is based on the idea that infinitely structured space-time points define space-time correlates of mathematical cognition. The mathematical analog of Brahman=Atman identity would however suggest that both views deserve to be taken seriously.
2. Infinite primes and infinite hierarchy of second quantizations
The discovery of infinite primes suggested strongly by the possibility to reduce physics to number theory. The construction of infinite primes can be regarded as a repeated second quantization of a super-symmetric arithmetic quantum field theory. Later it became clear that the process generalizes so that it applies in the case of quaternionic and octonionic primes and their hyper counterparts. This hierarchy of second quantizations means an enormous generalization of physics to what might be regarded a physical counterpart for a hierarchy of abstractions about abstractions about.... The ordinary second quantized quantum physics corresponds only to the lowest level infinite primes. This hierarchy can be identified with the corresponding hierarchy of space-time sheets of the many-sheeted space-time. One can even try to understand the quantum numbers of physical particles in terms of infinite primes. In particular, the hyper-quaternionic primes correspond four-momenta and mass squared is prime valued for them. The properties of 8-D hyper-octonionic primes motivate the attempt to identify the quantum numbers associated with CP_2 degrees of freedom in terms of these primes. The representations of color group SU(3) are indeed labelled by two integers and the states inside given representation by color hyper-charge and iso-spin.
3. Infinite primes as a bridge between quantum and classical
An important stimulus came from the observation stimulated by algebraic number theory. Infinite primes can be mapped to polynomial primes and this observation allows to identify completely generally the spectrum of infinite primes whereas hitherto it was possible to construct explicitly only what might be called generating infinite primes. This in turn led to the idea that it might be possible represent infinite primes (integers) geometrically as surfaces defined by the polynomials associated with infinite primes (integers). Obviously, infinite primes would serve as a bridge between Fock-space descriptions and geometric descriptions of physics: quantum and classical. Geometric objects could be seen as concrete representations of infinite numbers providing amplification of infinitesimals to macroscopic deformations of space-time surface. We see the infinitesimals as concrete geometric shapes!
4. Various equivalent characterizations of space-times as surfaces
One can imagine several number-theoretic characterizations of the space-time surface.
The approach based on octonions and quaternions suggests that space-time surfaces might correspond to associative or hyper-quaternionic surfaces of hyper-octonionic imbedding space.
Space-time surfaces could be seen as an absolute minima of the Kähler action. The great challenge is to rigorously prove that this characterization is equivalent with the others.
5. The representation of infinite primes as 4-surfaces
The difficulties caused by the Euclidian metric signature of the number theoretical norm forced to give up the idea that space-time surfaces could be regarded as quaternionic sub-manifolds of octonionic space, and to introduce complexified octonions and quaternions resulting by extending quaternionic and octonionic algebra by adding imaginary units multiplied with \sqrt{-1}. This spoils the number field property but the notion of prime is not lost. The sub-space of hyper-quaternions resp.-octonions is obtained from the algebra of ordinary quaternions and octonions by multiplying the imaginary part with \sqrt{-1}. The transition is the number theoretical counterpart for the transition from Riemannian to pseudo-Riemannian geometry performed already in Special Relativity. The notions of hyper-quaternionic and octonionic manifold make sense but it is implausible that H=M^4xCP_2 could be endowed with a hyper-octonionic manifold structure. Indeed, space-time surfaces are assumed to be hyper-quaternionic or co-hyper-quaternionic 4-surfaces of 8-dimensional Minkowski space M^8 identifiable as the hyper-octonionic space HO. Since the hyper-quaternionic sub-spaces of HO with a fixed complex structure are labelled by CP_2, each (co)-hyper-quaternionic four-surface of HO defines a 4-surface of M^4xCP_2. One can say that the number-theoretic analog of spontaneous compactification occurs. Any hyper-octonion analytic function HO--> HO defines a function g: HO--> SU(3) acting as the group of octonion automorphisms leaving a selected imaginary unit invariant, and g in turn defines a foliation of OH and H=M^4xCP_2 by space-time surfaces. The selection can be local which means that G_2 appears as a local gauge group. Since the notion of prime makes sense for the complexified octonions, it makes sense also for the hyper-octonions. It is possible to assign to infinite prime of this kind a hyper-octonion analytic polynomial P: HO--> HO and hence also a foliation of HO and H=M^4xCP_2 by 4-surfaces. Therefore space-time surface could be seen as a geometric counterpart of a Fock state. The assignment is not unique but determined only up to an element of the local octonionic automorphism group G_2 acting in HO and fixing the local choices of the preferred imaginary unit of the hyper-octonionic tangent plane. In fact, a map HO--> S^6 characterizes the choice since SO(6) acts effectively as a local gauge group. The construction generalizes to all levels of the hierarchy of infinite primes and produces also representations for integers and rationals associated with hyper-octonionic numbers as space-time surfaces. A close relationship with algebraic geometry results and the polynomials define a natural hierarchical structure in the space of 3-surfaces. By the effective 2-dimensionality naturally associated with infinite primes represented by real polynomials 4-surfaces are determined by data given at partonic 2-surfaces defined by the intersections of 3-D and 7-D light-like causal determinants. In particular, the notions of genus and degree serve as classifiers of the algebraic geometry of the 4-surfaces. The great dream is of course to prove that this construction yields the solutions to the absolute minimization of Kähler action.
6. Generalization of ordinary number fields: infinite primes and cognition
The introduction of infinite primes, integers, and rationals leads also to a generalization of real numbers since an infinite algebra of real units defined by finite ratios of infinite rationals multiplied by ordinary rationals which are their inverses becomes possible. These units are not units in the p-adic sense and have a finite p-adic norm which can be differ from one. This construction generalizes also to the case of hyper-quaternions and -octonions although non-commutativity, and in the case of octonions also non-associativity, pose technical problems. Obviously this approach differs from the standard introduction of infinitesimals in the sense that sum is replaced by multiplication meaning that the set of real units becomes infinitely degenerate. Infinite primes form an infinite hierarchy so that the points of space-time and imbedding space can be seen as infinitely structured and able to represent all imaginable algebraic structures. Certainly counter-intuitively, single space-time point is even capable of representing the quantum state of the entire physical Universe in its structure. For instance, in the real sense surfaces in the space of units correspond to the same real number 1, and single point, which is structure-less in the real sense could represent arbitrarily high-dimensional spaces as unions of real units. For real physics this structure is completely invisible and is relevant only for the physics of cognition. One can say that Universe is an algebraic hologram, and there is an obvious connection both with Brahman=Atman identity of Eastern philosophies and Leibniz's notion of monad. For more details see the chapter TGD as a Generalized Number Theory III: Infinite Primes. Matti Pitkanen
Comment to Not-Even-Wrong
The discovery that strings in a fixed flat background could describe gravitation without any need to make the background dynamical was really momentous. The discovery should have raised an obvious question: How to generalize the theory to the physical 4-dimensional case by replacing string orbits with 4-surfaces? Instead, the extremely silly idea of making also imbedding space dynamical emerged and brought back and magnified all the problems of general relativity, which one had hoped to get rid of. I have tried for more than two decades to communicate simple core ideas about an alternative approach but have found that theoretical physicists are too arrogant to listen to those without name or position. a) The fusion of special relativity with general relativity is achieved by assuming that space-times are 4-surfaces in M^4xCP_2. The known quantum numbers pop out elegantly from this framework. The topological complexity of space-time surfacse allows to circumvent objection that the induced metrics are too restricted. Light-like 3-D causal determinants allow generalization of super-conformal invaraince by their metric 2-dimensionality and dimension 4 for space-time is the only possibility. b) The maximal symmetries of H=M^4xCP_2 have an excellent justification when quantum theory is geometrized by identifying physical states of the Universe as classical configuration space spinor fields, configuration space being defined as the space of 3-surfaces in H. The only hope of geometrizing this infinite-dimensional space is as union of infinite-dimensional symmetric spaces labelled by zero modes having interpretation as non-quantum fluctuating classical degrees of freedom. Infinite-dimensional variant of Cartan's problem of classifying symmetric spaces emerges as the challenge of finding TOE. Mathematical existence fixes physical existence. Just as in the case of loop space, and with even better reasons, one expects that there are very few choices of H allowing internally consistent Kaehler geometry. Fermion numbers and super-conformal symmetries find an elegant geometrization and generalization in terms of complexified gamma matrices representing super-symmetry generators. c) M^4xCP_2 follows also from purely number theoretical considerations as has now become clear. The theory can be formulated in two equivalent manners. *4-surfaces can be regarded as hyper-quaternionic 4-surfaces in M^8 possessing what I call hyper-octonionic tangent space structure (octonionic imaginary units are multiplied by commutative sqrt(-1) to make number theoretical norm Minkowskian). *Space-times can be regarded also as 4-surfaces in M^4xCP_2 identified as extrema of so called Kaehler action in M^4xCP_2. Spontaneous compactification has thus purely number theoretical analog but has nothing to do with dynamics. The surprise was that under some additional conditions (essentially hyper-octonion real-analyticity for the dynamical variables in M^8 picture) the theory can be coded by WZW action for two-dimensional string like 2-surfaces in M^8. These strings not super-strings but generalizations of braid/ribbon diagrams allowing n-vertices in which string orbits are glued together at their ends like pages of book. Vertices can be formulated in terms of octonionic multiplication. Both classical and quantum dynamics reduce to number theory and the dimensions of classical division algebras reflect the dimensions of string, string orbit, space-time surface, and imbddding space. The conclusion is that both particle data table, the vision about physics as free, classical dynamics of spinor fields in the infinite-dimensional configuration space of 3-surfaces, and physics as a generalized number theory, lead to the same identification: space-time can be regarded as 4-surfaces in M^4xCP_2. In the case that someone is more interested of learning about real progress instead of wasting time to heated arguments at the ruins M theory, he/she can read the chapter http://www.helsinki.fi/~matpitka/tgd.html#visionb summarizing part of the number theoretical vision, and also visit my blog at http://matpitka.blogspot.com/ where I have summarized the most recent progress and great ideas of TGD. With Best Regards, Matti Pitkanen
Kähler calibrations, number theoretical compactification, and general solution to the absolute minimization of Kähler action
The title probably does not say much to anyone who is not a theoretical physicist working with theories of everything. I thought however it appropriate to glue this piece of text from my homepage in hope that the information about these beautiful discoveries might find some readers. So what follows is rather heavy mathematical jargon. Calibrations represent a good example of those merciful "accidents", which happen just the right time. Just for curiosity I decided to look what the word means, and it soon became clear that the notion of calibration allows to formulate my proposal for how to construct general solution of field equations defined by Kähler action in terms of a number theoretic spontaneous compactification in a rigorous and..., perhaps I dare say it aloud, even in convincing manner. For an excellent popular representation about calibrations, spinors and super-symmetries see the homepage of Jose Figueroa-O'Farrill . 1. The notion of calibration Calibrations allow a very elegant and powerful formulation of minimal surface property and have been applied also in brane-worldish considerations. Calibration is a closed p-form, whose value for a given p-plane is not larger than its volume in induced metric. What is important that if it is maximum for tangent planes of p-sub-manifold, minimal surface with smallest volume in its homology equivalence class results. Could absolute minima of Kähler action found using Kähler calibration?! For instance, all surfaces X^2xY^2 subset M^4xCP_2, X^2 and Y^2 minimal surfaces, are solutions of field equations. Calibration theory allows to concluded that Y^2 is any complex manifold of CP_2! A very general solution of TGD in stringy sector results and there exists a deformation theory of calibrations to produce moduli spaces for the perturbations of these solutions! In fact, all known solutions of field equations are either minimal surfaces or have a vanishing Kähler action density. This probably tells more about my simple mind-set than reality, and there are excellent reasons to believe that, since Lorentz-Kähler force vanishes, the known solutions are space-time correlates for asymptotic self-organization patterns. The question is how to find more general solutions. Or how to generalize the notion of calibration for minimal surfaces to what might be called Kähler calibration? It is here, where the handsome and young idea of number theoretical spontaneous compactification enters the stage and the outcome is a happy marriage of two ideas. 2. The notion of Kähler calibration It is intuitively clear that the closed calibration form omega which is saturated for minimal surfaces must be replaced by Kähler calibration 4-form omega_K= L_K omega . L_K is Kähler action density (Maxwell action for induced CP_2 Kähler form).
Important point: omega_K is closed but not omega as in the case of minimal surfaces. L_K acts as an integrating factor. This difference is absolutely essential. When L_K is constant you should get minimal surfaces. L_K is indeed constant for the known minimal surface solutions. The basic objection against this conjecture is following: L_K is a four dimensional action density. How it is possible to assign it to a 4-form in 8 dimensional space-time? Here number theoretical spontaneous compactification shows its power. 3. Number theoretic compactification allows to define Kähler calibration The calibrations are closely related to spinors and the number theoretic compactification based on 2-component octonionic spinors satisfying Weyl condition, and therefore equivalent with octonions themselves, tells how to construct omega.
The hyper-octonion real-analytic maps of HO=M^8 to itself define octonionic 2 spinors satisfying Weyl condition. Octonionic massless Dirac equation reduces to d'Alembert equation in M^8 by the generalization of Cauchy-Riemann conditions.
Octonions and thus also the spinors have 1+1+3+3bar decomposition with respect to (color) SU(3) sub-group of octonion automorphism G_2. SU(3) leaves a preferred hyper-octonionic imaginary unit invariant. The unit can be chosen in local manner and the choices are parameterized by local S^6.
3x3bar tensor product defines color octet identifiable as SU(3) Lie algebra generator and its exponentiation gives SU(3) group element.
The canonical bundle projection SU(3)-->CP_2 assigns a CP_2 point to each point of M^8, when a preferred octonion unit is fixed at each point of M^8.
Canonical projection M^8-->M^4 assigns M^4 point to point of M^8.
Conclusion: M^8 is mapped to M^4xCP_2 and metric of H and CP_2 Kähler form can be induced. M^8 having originally only the number theoretical norm as metric inherits the metric of H.
Here comes the key point of the construction. CP_2 parameterizes hyper-quaternionic planes of hyper-octonions and therefore it is possible to assign to a given point of M^8 a unique hyper-quaternion 4-plane. Thus also the projection J of Kähler form to this plane and also the dual *J of this projection. Therefore also L_K=J\wedge*J as the value of Kähler action density!
The Kähler calibration omega_K= L_K*omega
is defined in an obvious manner. As found, L_K is associated with the local hyper-quaternionic plane is assigned to each point of M^8. The form omega is obtained from the wedge product of unit tangent vectors for hyper-quaternionic plane at a given point by lowering the indices using the induced metric in M^8. omega is not a closed form in general. For a given 4-plane it is essentially the cosine of the angle between plane and hyper-quaternionic plane and saturated for hyper-quaternionic plane so that calibration results.
Kähler calibration is the only calibration that one can seriously imagine. Furthermore, the spinorial expression for omega is well defined only if the form omega saturates for hyper-quaternionic planes or their duals. The reason is that non-associativity makes the spinorial expression involving an octonionic product of four tangent vectors for the calibration ill defined for non-associative 4-planes. Hence number theory allows only hyper-quaternionic saturation. Note that also co-hyper-quaternionicity is allowed and required by the known extremals of Kähler action. A 4-parameter foliation of M^8, and perhaps even that of M^4xCP_2 (discrete set of intersections probably occurs) by 4-surfaces results and the parameters at given point of X^4 define the dual space-time surface.
A surprise, which does not flatter theoretician's vanity, emerges. Closed-ness of omega_K implies that if absolute value of Kähler action density replaces K\"ahler action, minimization indeed occurs for hyper-quaternionic surfaces in a given homology class assuming that the hyper-quaternionic plane at given point minimizes L_K (is this equivalent this closed-ness of omega_K?). Thus L_K should be replaced with L_K so that vacuum extremals become absolute minima, and universe would do its best to save energy by staying as near as possible to vacuum. The 3 surfaces for which CP_2 projection is at least 2-dimensional and not Lagrange manifolds would correspond to non-vacua since since conservation laws do not leave any other option. The attractiveness of this option from the point of calculability of TGD would be that the initial values for the time derivatives of the imbedding space coordinates at X^3 at light-like 7-D causal determinant could be computed by requiring that the energy of the solution is minimized. This could mean a computerizable solution to the absolute minimization.
There is a very beautiful connection with super-symmetries allowing to express absolute minimum property as a condition involving only the hyper-octonionic spinor field defining the Kähler calibration (discovered for calibrations by Strominger and Becker).
4. Could TGD reduce to string model like theory in HO picture? Conservation laws suggests that in the case of non-vacuum extremals the dynamics of the local automorphism associated with the hyper-octonionic spinor field is dictated by field equations of some kind. The experience with WZW model suggests that in case of non-vacuum extremals G_2 element could be written as a product g=g_L(h)g^{-1}_R(h*) of hyper-octonion analytic and anti-analytic complexified G_2 elements. g would be determined by the data at hyper-complex 2-surface for which the tangent space at a given point is spanned by real unit and preferred hyper-octonionic unit. Also Dirac action would be naturally restricted to this surface. The amazing possibility is that TGD could reduce in HO picture to 8-D WZW string model both classically and quantally since vertices would reduce to integrals over 1-D curves. The interpretation of generalized Feynman diagrams in terms of generalized braid/ribbon diagrams and the unique properties of G_2 provide further support for this picture. In particular, G_2 is the lowest-dimensional Lie group allowing to realize full-powered topological quantum computation based on generalized braid diagrams and using the lowest level k=1 Kac Moody representation. Even if this reduction would occur only in special cases, such as asymptotic solutions for which Lorentz Kähler force vanishes or maxima of Kähler function, it would mean enormous simplification of the theory. 5. Why extermals of Kähler action would correspond to hyper-quaternionic 4-surfaces? The resulting over all picture leads also to a considerable understanding concerning the basic questions why (co)-hyper-quaternionic 4-surfaces define extrema of Kähler action and why WZW strings would provide a dual for the description using Kähler action. The answer boils down to the realization that the extrema of Kähler action minimize complexity, also algebraic complexity, in particular non-commutativity. A measure for non-commutativity with a fixed preferred hyper-octonionic imaginary unit is provided by the commutator of 3 and 3bar parts of the hyper-octonion spinor field defining an antisymmetric tensor in color octet representation: very much like color gauge field. Color action is a natural measure for the non-commutativity minimized when the tangent space algebra closes to complexified quaternionic, instead of complexified octonionic, algebra. On the other hand, Kähler action is nothing but color action for classical color gauge field defined by projections of color Killing vector fields. Here it is!That WZW + Dirac action for hyper-octonionic strings would correspond to Kähler action would in turn be the TGD counterpart for the proposed string-YM dualities. 6. Summary To sum up, the following conjectures are direct generalizations of those for minimal surfaces.
L_K acts as an integrating factor and omega_K= L_K*omega is a closed form.
Generalizing from the case of minimal surfaces, closed-ness guarantees that hyper-quaternionic 4-surfaces saturating this form are absolute minima of Kähler action.
The hyper-octonion analytic solutions of hyper-octonionic Dirac equation defines those maps M^8-->M^4xCP_2 for which L_K acts as an integrating factor. Classical TGD reduces to a free Dirac equation for hyper-octonionic spinors!
For more details see the chapter TGD as a Generalized Number Theory II: Quaternions, Octonions, and their Hyper Counterparts.
It occurred to me that I could try to summarize the great ideas behind TGD. There are many of them and I cannot summarize them in single page. My dream is to communicate some holistic vision about what I have become conscious of and therefore I start just by listing the great ideas that come to my mind just now and continue later by giving details. These ideas are also summarized in the chapter Overview about the Evolution of Quantum TGD of TGD.
Classical physics as the geometry of space-times regarded as 4-surfaces in certain 8-dimensional space-time. This generalizes and modifies Einstein's vision. Note that in quantum context the plural "space-times" indeed makes sense.
Quantum physics as infinite-dimensional geometry of the world of classical worlds=space-time surfaces. This vision generalizes further the vision of Einstein. One of the paradoxes of the sociology of post-modern physics is that M-theorists refuse to realize the enormous unifying power of infinite-dimensional geometry.
Physics as number theory vision involves several ideas bigger than life.
p-Adic physics as physics of cognition and intentionality and fusion of real physics and physics of cognition to single super physics by fusing real numbers and p-adic number fields to a larger structure. p-Adic numbers can be applied also to real physics and leads to a plethora of quantitative predictions. It became already decade ago clear that p-adic numbers provide a royal road to the understanding of elementary particle masses. The idea is somewhat tricky: just the requirement that physical system allows p-adic cognitive representations poses unexpectedly strong constraints on the properties of the system, in this case to the values of elementary particle masses. Knowing this I become very sad when I see colleagues to continue the fruitless and more and more bizarre looking M-theory Odysseia. They are now quite seriously suggesting that we must accept that physical theories will never be able to predict anything concrete. They simply cannot imagine the possibility that M theory might be wrong or even worse, not-even-wrong! For the critical discussions about the state of M-theory see the blog Not-Even Wrong of Peter Woit.
Space-time surfaces as hyper-quaternionic surfaces of 8-dimensional octonionic space-time is an idea which has started to flourish during last months: for details see TGD as a Generalized Number Theory II: Quaternions, Octonions, and their Hyper Counterparts. TGD can be formulated in two dual manners. First corresponds to M^4xCP_2 picture and the second, number theoretical formulation, can be seen as a string model in 8-dimensional Minkowski space M^8 of hyper-octonions. I refer to the possibility to formulate the theory either in M^8 or M^4xCP_2 as "number theoretical compactification". Of course no (spontaneous) compactification occurs. If I would be forced to introduce this monstrously ugly super-stringy notion into TGD, I would be ready to hang myself. What is so beautiful that the classical notion of complex analytic function generalizes to the level of hyper-octonions and physics at this primordial number-theoretical level looks ridiculously simple. WZW action for the automorphism group G_2 of octonions and Dirac action for octonionic 2-spinors satisfying Weyl conditions with solutions given by hyper-octonion analytic functions. In the quantization the real Laurent coefficients of spinor field and G_2 valued field are replaced by mutually commuting Hermitian operators representing observables coding for quantum states so that also quantum measurement theory and quantum-classical correspondence pop up in the number theoretical framework. Here I cannot resist revealing one more fascinating fact: G_2 is a unique exception among simple Lie groups in that the ratio of long roots to short ones is sqrt(3): do not hesitate to tell this also to your friends;-)! The number three pops up again and again in TGD, for instance, the existence of so called trialities corresponds to the existence of classical number fields. I have told at my homepage about my great experiences during which I had the vision about number three as the fundamental number of mathematics and physics and also precognition about the "theory of infinite magnitudes", that is the theory of infinite primes about which I am going to say something after two paragraphs. Perhaps the most beautiful aspect is that m+n-particle vertices correspond to a local multiplication of hyper-octonionic spinors coding for m incoming resp. n outgoing states and the inner product of these products in accordance with the view that generalized Feynman diagrams represent computation like processes developed in Equivalence of Loop Diagrams with Tree Diagrams and Cancellation of Infinities in Quantum TGD I have not taken strings seriously but I must admit the amazing possibility that all information about both classical physics (space-time dynamics) and quantum physics as predicted by TGD might be coded by these number theoretical strings, which are however quite different from the strings of super-string models. The corresponding string diagrams generalize so called braid diagrams to Feynman diagrams a la TGD and also provide a physical realization of topological quantum computation at the level of fundamental physics.
The notion of infinite primes was inspired by TGD inspired theory of consciousness and is now an organic part of quantum TGD.
Infinite primes are in one-one correspondence with the states of repeatedly second quantized arithmetic quantum field theory and can be identified as representations of physical states. Infinite primes can in turn be represented as 4-dimensional space-time surfaces. The conjecture that these surfaces solve the field equations of TGD is equivalent with the conjecture that hyper-octonionic spinor fields code for space-time surfaces as hyper-quaternionic surfaces by defining what I call Kähler calibration.
Infinite primes force also to generalize the notion of ordinary number: each real number corresponds to an infinite number of different variants, which are equivalent as real numbers but p-adically/cognitively non-equivalent so that space-time points becomes an infinitely structured an complex algebraic hologram able to represent in its structure even entire physical universe! This is Brahman=Atman mathematically. Leibniz realized this when he talked about monads but was forgotten for five centuries.
A further bundle of ideas relates to TGD inspired theory of consciousness. There are two books about this: TGD Inspired Theory of Consciousness and Genes, Memes, Qualia and Semitrance.
The starting point is a real problem as always. Now it is the paradox created by the non-determinism of quantum jump contra determinism of Schrödinger equation. TGD leads to a solution of the paradox and at the same time emerges a notion of free will and volition consistent with physics. There is no attempt to explain free will and consciousness as as illusions (whatever that might mean!). They are still stubbornly trying to do this, these stubborn neuroscientists!
A new view about the relationship between experienced, subjective time and the geometric time of physicist emerges. The outcome is a revolutionary modification of the notions of both time and energy.
The new view about time gives the conceptual tools needed to make scientific statements about what might happen for consciousness after biological death and TGD inspired consciousness has grown into two books covering basic phenomena of consciousness from brain functions to paranormal.
The possibility of negative energies and signals propagating backwards in geometric time and reflected back by time reflection has several implications. Communications with geometric past (long term memories!) and in principle even with civilizations of geometric future become possible. Intentions might be realized by control signals sent backwards in time. Instantaneous remote sensing by negative energy signals becomes possible, etc...
p-Adic mathematics leads to and identification of genuine information measures based on Shannon entropy and the replacement of ordinary real norm with p-adic norm and applicable as measures of conscious information. The idea about rationals and algebraic numbers as islands of order in the sea of chaos represented by generic real numbers finds a direct counterpart at the level of physical correlates of conscious experience.
Well, I think it is better to stop here and continue later. Matti Pitkanen
How to Put an End to the Suffering Caused by Path ...
Infinite primes and physics as a generalized numbe...
Kähler calibrations, number theoretical compactif...
Thoughts of a Newcomer | CommonCrawl |
# Linear control and its applications in optimization
Linear control is a mathematical framework that deals with the analysis and design of control systems. These systems are characterized by linear equations and linear constraints. Linear control has several applications in optimization, including the design of optimal control systems, the analysis of stability and convergence, and the development of efficient algorithms for solving optimization problems.
Consider the following linear system:
$$
\dot{x} = Ax + Bu
$$
where $x$ is the state vector, $A$ is the system matrix, $B$ is the input matrix, and $u$ is the control input. The goal is to design a control law that minimizes the error between the desired state $x_d$ and the actual state $x$.
## Exercise
Design a linear control law for the given system.
Linear control has several important properties that make it useful in optimization problems. One such property is the concept of stability. A system is said to be stable if it converges to a fixed point under certain conditions. This stability property is crucial in the design of control systems, as it ensures that the system converges to a desired state and does not diverge.
In addition to stability, linear control also deals with the analysis of convergence and the development of efficient algorithms for solving optimization problems. Convergence analysis involves studying the behavior of iterative algorithms and determining their rate of convergence. This analysis is essential in the design of efficient algorithms for solving optimization problems.
# Lyapunov functions and their properties
A Lyapunov function is a function that satisfies certain properties with respect to a given nonlinear system. The most important property of a Lyapunov function is that it is positive definite, which means that it has a positive value for all nonzero vectors in its domain.
Consider the following Lyapunov function:
$$
V(x) = \frac{1}{2}x^Tx
$$
This function is positive definite because it has a positive value for all nonzero vectors $x$.
## Exercise
Show that the given Lyapunov function is positive definite.
The properties of Lyapunov functions have several applications in optimization. One such application is the design of control systems. By choosing a suitable Lyapunov function, we can design a control law that ensures the stability and convergence of the system.
In addition to stability and convergence analysis, Lyapunov functions also have applications in the analysis of optimal control problems. By defining a Lyapunov function that quantifies the error between the desired and actual states, we can develop optimization algorithms that minimize the error and achieve the desired state.
# Nonlinear programming and its challenges
Nonlinear programming problems arise in various fields, including control theory, optimization, and machine learning. These problems can be difficult to solve due to their nonlinear nature, which makes it challenging to find efficient algorithms for solving them.
Consider the following nonlinear programming problem:
$$
\min_{x} f(x) = \frac{1}{2}x^2 + x
$$
This problem is nonlinear because the objective function $f(x)$ is a quadratic function.
## Exercise
Solve the given nonlinear programming problem.
The challenges of nonlinear programming have led to the development of advanced techniques, such as gradient descent and Newton's method. These techniques are used to solve nonlinear programming problems by iteratively updating the solution until a convergence criterion is met.
# Robust control and its role in optimization
Robust control has several applications in optimization problems. One such application is the design of optimal control systems. By considering the impact of uncertainties and disturbances on the system's behavior, we can design a control law that minimizes the error between the desired and actual states and ensures the stability of the system.
Consider the following robust control problem:
$$
\min_{u} \int_{t_0}^{t_f} (x(t)-x_d(t))^2 + r(u(t))\,dt
$$
where $x(t)$ is the state, $x_d(t)$ is the desired state, $u(t)$ is the control input, and $r(u(t))$ is the cost function.
## Exercise
Solve the given robust control problem.
In addition to optimal control, robust control also has applications in the analysis of stability and convergence of nonlinear systems. By considering the impact of uncertainties and disturbances on the system's stability, we can analyze the behavior of the system and ensure its stability.
# Classical optimization techniques: gradient descent, Newton's method
Gradient descent is an iterative optimization algorithm that minimizes a given objective function by updating the solution iteratively. The algorithm uses the gradient of the objective function to determine the direction of the next update.
Consider the following optimization problem:
$$
\min_{x} f(x) = \frac{1}{2}x^2 + x
$$
We can solve this problem using gradient descent.
## Exercise
Solve the given optimization problem using gradient descent.
Newton's method is another iterative optimization algorithm that minimizes a given objective function by updating the solution iteratively. The algorithm uses the gradient and Hessian of the objective function to determine the direction of the next update.
Consider the following optimization problem:
$$
\min_{x} f(x) = \frac{1}{2}x^2 + x
$$
We can solve this problem using Newton's method.
## Exercise
Solve the given optimization problem using Newton's method.
Both gradient descent and Newton's method have several applications in optimization problems. They are used in various fields, including control theory, machine learning, and engineering.
# Lyapunov optimization and its benefits
Lyapunov optimization has several benefits. One such benefit is the ability to design control laws that ensure the stability and convergence of the system. By choosing a suitable Lyapunov function, we can design a control law that minimizes the error between the desired and actual states and ensures the stability of the system.
Consider the following Lyapunov optimization problem:
$$
\min_{u} \int_{t_0}^{t_f} (x(t)-x_d(t))^2 + r(u(t))\,dt
$$
where $x(t)$ is the state, $x_d(t)$ is the desired state, $u(t)$ is the control input, and $r(u(t))$ is the cost function.
## Exercise
Solve the given Lyapunov optimization problem.
In addition to stability and convergence analysis, Lyapunov optimization also has applications in the design of optimal control systems. By considering the impact of uncertainties and disturbances on the system's behavior, we can design a control law that minimizes the error between the desired and actual states and ensures the stability of the system.
# Applications of Lyapunov optimization in various fields
In control theory, Lyapunov optimization is used to design control laws that ensure the stability and convergence of the system. By considering the impact of uncertainties and disturbances on the system's behavior, we can design a control law that minimizes the error between the desired and actual states and ensures the stability of the system.
Consider the following Lyapunov optimization problem:
$$
\min_{u} \int_{t_0}^{t_f} (x(t)-x_d(t))^2 + r(u(t))\,dt
$$
where $x(t)$ is the state, $x_d(t)$ is the desired state, $u(t)$ is the control input, and $r(u(t))$ is the cost function.
## Exercise
Solve the given Lyapunov optimization problem.
In optimization, Lyapunov optimization is used to solve nonlinear programming problems that involve nonlinear equations and inequalities. By choosing a suitable Lyapunov function, we can solve these problems and minimize the error between the desired and actual states.
In machine learning, Lyapunov optimization is used to design algorithms that learn from data and make predictions. By considering the impact of uncertainties and disturbances on the system's behavior, we can design algorithms that are robust to noise and outliers in the data.
# Case studies and examples
One such case study involves the design of a control system for a robotic arm. By using Lyapunov optimization, we can design a control law that ensures the stability and convergence of the robotic arm and minimizes the error between the desired and actual positions of the arm.
Consider the following Lyapunov optimization problem:
$$
\min_{u} \int_{t_0}^{t_f} (x(t)-x_d(t))^2 + r(u(t))\,dt
$$
where $x(t)$ is the state, $x_d(t)$ is the desired state, $u(t)$ is the control input, and $r(u(t))$ is the cost function.
## Exercise
Solve the given Lyapunov optimization problem.
Another case study involves the design of an optimal control system for a car. By using Lyapunov optimization, we can design a control law that ensures the stability and convergence of the car and minimizes the error between the desired and actual positions of the car.
Consider the following Lyapunov optimization problem:
$$
\min_{u} \int_{t_0}^{t_f} (x(t)-x_d(t))^2 + r(u(t))\,dt
$$
where $x(t)$ is the state, $x_d(t)$ is the desired state, $u(t)$ is the control input, and $r(u(t))$ is the cost function.
## Exercise
Solve the given Lyapunov optimization problem.
# Advanced techniques in Lyapunov optimization
One such advanced technique is the use of penalty functions. Penalty functions are used to regularize the objective function and ensure the convergence of the solution. By adding a penalty term to the objective function, we can ensure that the solution converges to a desired state and minimizes the error.
Consider the following Lyapunov optimization problem:
$$
\min_{u} \int_{t_0}^{t_f} (x(t)-x_d(t))^2 + r(u(t))\,dt
$$
where $x(t)$ is the state, $x_d(t)$ is the desired state, $u(t)$ is the control input, and $r(u(t))$ is the cost function.
## Exercise
Solve the given Lyapunov optimization problem using a penalty function.
Another advanced technique is the use of adaptive algorithms. Adaptive algorithms are used to solve optimization problems and ensure the convergence of the solutions. By updating the algorithm iteratively, adaptive algorithms can adapt to changes in the system's behavior and ensure the convergence of the solution.
Consider the following Lyapunov optimization problem:
$$
\min_{u} \int_{t_0}^{t_f} (x(t)-x_d(t))^2 + r(u(t))\,dt
$$
where $x(t)$ is the state, $x_d(t)$ is the desired state, $u(t)$ is the control input, and $r(u(t))$ is the cost function.
## Exercise
Solve the given Lyapunov optimization problem using an adaptive algorithm.
# Convergence analysis and practical considerations
One such practical consideration is the choice of Lyapunov function. The choice of Lyapunov function is essential in ensuring the stability and convergence of the system. By choosing a suitable Lyapunov function, we can design a control law that ensures the stability and convergence of the system and minimizes the error between the desired and actual states.
Consider the following Lyapunov optimization problem:
$$
\min_{u} \int_{t_0}^{t_f} (x(t)-x_d(t))^2 + r(u(t))\,dt
$$
where $x(t)$ is the state, $x_d(t)$ is the desired state, $u(t)$ is the control input, and $r(u(t))$ is the cost function.
## Exercise
Solve the given Lyapunov optimization problem using a suitable Lyapunov function.
Another practical consideration is the choice of optimization algorithm. The choice of optimization algorithm is essential in ensuring the convergence of the solution. By choosing an appropriate optimization algorithm, we can solve the optimization problem and ensure the convergence of the solution.
Consider the following Lyapunov optimization problem:
$$
\min_{u} \int_{t_0}^{t_f} (x(t)-x_d(t))^2 + r(u(t))\,dt
$$
where $x(t)$ is the state, $x_d(t)$ is the desired state, $u(t)$ is the control input, and $r(u(t))$ is the cost function.
## Exercise
Solve the given Lyapunov optimization problem using an appropriate optimization algorithm.
# Assessing the effectiveness of optimization solutions
One such method is the use of numerical simulations. Numerical simulations involve solving the optimization problem using a computer program and analyzing the results. By comparing the results with the desired outcome, we can assess the effectiveness of the optimization solution and ensure that it meets the desired criteria.
Consider the following Lyapunov optimization problem:
$$
\min_{u} \int_{t_0}^{t_f} (x(t)-x_d(t))^2 + r(u(t))\,dt
$$
where $x(t)$ is the state, $x_d(t)$ is the desired state, $u(t)$ is the control input, and $r(u(t))$ is the cost function.
## Exercise
Assess the effectiveness of the given Lyapunov optimization solution using numerical simulations.
Another method is the use of analytical methods. Analytical methods involve analyzing the properties of the optimization problem and its solution without solving the problem numerically. By analyzing the properties of the problem and its solution, we can assess the effectiveness of the optimization solution and ensure that it meets the desired criteria.
Consider the following Lyapunov optimization problem:
$$
\min_{u} \int_{t_0}^{t_f} (x(t)-x_d(t))^2 + r(u(t))\,dt
$$
where $x(t)$ is the state, $x_d(t)$ is the desired state, $u(t)$ is the control input, and $r(u(t))$ is the cost function.
## Exercise
Assess the effectiveness of the given Lyapunov optimization solution using analytical methods.
In conclusion, Lyapunov optimization is a powerful tool in solving optimization problems and ensuring the stability and convergence of the solutions. By using Lyapunov functions, control laws, and advanced techniques, we can design efficient algorithms for solving optimization problems and ensure their effectiveness. | Textbooks |
Order-5 tesseractic honeycomb
In the geometry of hyperbolic 4-space, the order-5 tesseractic honeycomb is one of five compact regular space-filling tessellations (or honeycombs). With Schläfli symbol {4,3,3,5}, it has five 8-cells (also known as tesseracts) around each face. Its dual is the order-4 120-cell honeycomb, {5,3,3,4}.
Order-5 tesseractic honeycomb
(No image)
TypeHyperbolic regular honeycomb
Schläfli symbol{4,3,3,5}
Coxeter diagram
4-faces {4,3,3}
Cells {4,3}
Faces {4}
Face figure {5}
Edge figure {3,5}
Vertex figure {3,3,5}
DualOrder-4 120-cell honeycomb
Coxeter groupBH4, [5,3,3,4]
PropertiesRegular
Related polytopes and honeycombs
It is related to the Euclidean 4-space (order-4) tesseractic honeycomb, {4,3,3,4}, and the 5-cube, {4,3,3,3} in Euclidean 5-space. The 5-cube can also be seen as an order-3 tesseractic honeycomb on the surface of a 4-sphere.
It is analogous to the order-5 cubic honeycomb {4,3,5} and order-5 square tiling {4,5}.
See also
• List of regular polytopes
References
• Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 ISBN 0-486-40919-8 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, p212-213)
| Wikipedia |
\begin{document}
\title{ \huge Normal approximation for sums of discrete $U$-statistics - application to Kolmogorov bounds in random subgraph counting }
\author{Nicolas Privault\thanks{
School of Physical and Mathematical Sciences,
Nanyang Technological University, SPMS-MAS-05-43, 21 Nanyang Link, Singapore 637371. e-mail: {\tt [email protected]}. } \and Grzegorz Serafin\thanks{Faculty of Pure and Applied Mathematics, Wroc{\l}aw University of Science and Technology, Ul. Wybrze\.ze Wyspia\'nskiego 27, Wroc{\l}aw, Poland. e-mail: {\tt [email protected]}.}}
\maketitle
\begin{abstract}
We derive normal approximation bounds in the Kolmogorov distance
for sums of discrete multiple integrals and $U$-statistics
made of independent Bernoulli random variables.
Such bounds are applied to normal approximation for
the renormalized subgraphs counts in the Erd{\H o}s-R\'enyi random graph.
This approach completely solves a long-standing conjecture in the general
setting of arbitrary graph counting,
while recovering and improving recent results derived for triangles
as well as results using the Wasserstein distance. \end{abstract} \noindent\emph{Keywords}: Normal approximation; central limit theorem; Stein-Chen method; Malliavin-Stein method; Berry-Esseen bound; random graph; subgraph count; Kolmogorov distance.
\noindent {\em Mathematics Subject Classification:} 60F05, 60H07, 60G50, 05C80. \baselineskip0.7cm
\section{Introduction} The Mallavin approach to the Stein method for discrete Bernoulli sequences has recently been developed in \cite{nourdin3}, \cite{reichenbachs}, \cite{krokowskicosa}, \cite{privaulttorrisi4}, \cite{reichenbachsAoP}, as an extension of the Malliavin approach to the Stein method introduced in \cite{nourdinpeccati} for Gaussian fields. \\
In this paper we develop the use of multiple stochastic integral expansions for the derivation of bounds on the distances between probability laws by the Malliavin approach to the Stein and Stein-Chen methods. Using results of \cite{reichenbachsAoP} for general functionals of discrete i.i.d. renormalized Bernoulli sequences $(Y_n)_{n\in {\mathord{\mathbb N}}}$, we derive a Kolmogorov distance bound to the normal distribution for sums of $U$-statistics (or multiple stochastic integrals) of the form $$ \sum_{k=1}^n \sum_{i_1,\ldots,i_k \in{\mathord{\mathbb N}}
\atop
i_r\not= i_s, \ \! \! 1\leq r\not= s \leq k} f_k(i_1,\ldots,i_k)Y_{i_1}\cdots Y_{i_k}, $$ where $(Y_k)_{k\in {\mathord{\mathbb N}}}$ is a normalized sequence of Bernoulli random variables, see Theorem~\ref{prop:dKsumI}. We note that on the Erd{\H o}s-R\'enyi random graph $\mathbb{G}_n(p_n)$ constructed by independently retaining any edge in the complete graph $K_n$ on $n$ vertices with probability $p_n\in(0,1)$, various random functionals admit such representations as sums of multiple integrals. This includes the number of vertices of a given degree, and the count of subgraphs that are isomorphic to an arbitrary graph. \\
Our second goal is to apply such results to the normal approximation of the renormalized count of the subgraphs in $\mathbb{G}_n(p_n)$ which are isomorphic to an arbitrary graph. Necessary and sufficient conditions for the asymptotic normality
of the renormalization $$\widetilde{N}^G_n:=\frac{N^G_n-\mathbb{E}[N^G_n]}{\sqrt{\mathrm{Var}[N^G_n]}}, $$
where $N^G_n$ is the number of graphs in $\mathbb{G}_n(p_n)$
that are isomorphic to a fixed graph $G$,
have been obtained in \cite{rucinski} where it is shown that \begin{align}\label{eq:conv}
\widetilde{N}^G_n\stackrel{\mathcal{D}}{\longrightarrow}\mathcal{N} \ \mbox{ iff }
\ \ np_n^\beta\rightarrow \infty \ \mbox{ and }\ n^2(1-p_n)\rightarrow\infty, \end{align} as $n$ tends to infinity, where $\mathcal{N}$ denotes the standard normal distribution, $$ \beta :=\max \{e_H/v_H \ : \ H\subset G\}, $$ and $e_H$, $v_H$ respectively denote the numbers of edges and vertices in the graph $H$. \\
Those results have been made more precise in \cite{BKR} by the derivation of explicit convergence rates in the Wasserstein distance $$
d_W (F,G):
=\sup_{h\in\mathrm{Lip}(1)}|\mathrm{E}[h(F)]-\mathrm{E}[h(G)]|, $$
between the laws of random variables $F$, $G$,
where $\mathrm{Lip}(1)$ denotes the class of real-valued
Lipschitz functions with
Lipschitz constant less than or equal to $1$. In the particular case where the graph $G$ is a triangle, such bounds have been recently strengthened in \cite{roellin2} using the Kolmogorov distance $$
d_K (F,G):
= \sup_{x\in {\mathord{\mathbb R}}}|P(F\leq x) - P(G\leq x)|, $$
which satisfies the bound
$d_K (F,\mathcal{N} ) \leq \sqrt{ d_W (F,\mathcal{N} )}$.
Still in the case of triangles, Kolmogorov distance bounds had also been obtained by the Malliavin approach to the Stein method for discrete Bernoulli sequences in \cite{reichenbachsAoP} when $p_n$ takes the form $p_n = n^{-\alpha}$, $\alpha \in [0,1)$. \\
In this paper we refine the results of \cite{BKR} by using the Kolmogorov distance instead of the Wasserstein distance. As in \cite{BKR} we are able to consider any graph $G$, and therefore our results extend those of both \cite{reichenbachsAoP} and \cite{roellin2} which only cover the case where $G$ is a triangle. Instead of using second order Poincar\'e inequalities \cite{lastpeccatipenrose}, our method relies on an application of Proposition~4.1 in \cite{reichenbachs} to derive Stein approximation bounds for sums of multiple stochastic integrals. \\
Our second main result Theorem~\ref{thm:main} is a bound for the Kolmogorov distance between the normal distribution and the renormalized graph count $\widetilde{N}^G_n$. Namely, we show that when $G$ is a graph without isolated vertices it holds that \begin{equation}
\label{eq:main}
d_K (\tilde{N}_G,\mathcal{N} )
\leq C_G
\((1-p_n)\min_{\substack{ H\subset G\\e_H\geq1}}\{n^{v_H}p_n^{e_H}\}\)^{-1/2}, \end{equation} see Theorem~\ref{thm:main}, where $C_G >0$ is a constant depending only on $e_G$, which improves on the Wasserstein estimates of \cite{BKR}, see Theorem~2 therein. This result relies on the representation of combined subgraph counts as finite sums of multiple stochastic integrals, see Lemma~\ref{lemma4.1}, together with the application of Theorem~\ref{prop:dKsumI} on Kolmogorov distance bounds. In the sequel, given two positive sequences $(x_n)_{n\in{\mathord{\mathbb N}}}$ and $(y_n)_{n\in{\mathord{\mathbb N}}}$ we write $x_n \approx y_n$ whenever $c_1 < x_n/y_n < c_2$ for some $c_1,c_2>0$ and all $n\in {\mathord{\mathbb N}}$, and for $f$ and $g$ two positive functions we also write $f\lesssim g$ whenever $f \leq C_G g$ for some constant $C_G>0$ depending only on $G$. \\
Using the equivalence \begin{equation}
\label{varng}
\mathrm{Var}\big[N^G_n\big]\approx
(1-p_n)
\max_{\substack{H\subset G \\ e_H\geq1}} \{n^{2v_G-v_H}p_n^{2e_G-e_H}\} \end{equation}
as $n$ tends to infinity, see
Lemma 3.5 in \cite{JLR},
the bound \eqref{eq:main} can be rewritten in terms of the
variance $\mathrm{Var}\big[N^G_n\big]$ as \begin{equation}
\label{ddww}
d_K \big(\widetilde{N}^G_n,\mathcal{N} \big)\lesssim
\frac{\sqrt{\mathrm{Var}\big[N_G\big]}}{(1-p_n)n^{v_G}p_n^{e_G}}. \end{equation} Note that when $p_n$ is bounded away from $0$, the bound \eqref{eq:main} takes the simpler form \begin{equation}
\label{ddk} d_K \big(\widetilde{N}^G_n,\mathcal{N} \big) \lesssim\frac1{n\sqrt{1-p_n}}. \end{equation} In Corollaries~\ref{cor1.3}, \ref{cor1.4} and \ref{cor1.5} we deal with examples of subgraphs such as cycle graphs and complete graphs, which include triangles as particular cases, and trees. \\
In the particular case where the graph $G$ is a triangle, the next consequence of \eqref{eq:main} and \eqref{ddk} recovers the main result of \cite{roellin2}, see Theorem~1.1 therein. \begin{corollary} \label{c} For any $c\in (0,1)$, the normalized number $\widetilde{N}^G_n$ of the subgraphs in
$\mathbb{G}_n(p_n)$ that are isomorphic to a triangle
satisfies \begin{align*}
d_K \big(\widetilde{N}^G_n,\mathcal{N} \big)\lesssim
\left\{\begin{array}{ll}
\displaystyle
\frac{1}{n\sqrt{1-p_n}} &
\displaystyle
\mbox{if } \ c < p_n<1,\\
\\
\displaystyle
\frac{1}{n\sqrt{p_n}} &
\displaystyle
\mbox{if } \ n^{-1/2} < p_n \leq c,
\\
\\ \displaystyle \frac{1}{( np_n)^{3/2}} & \displaystyle \mbox{if } \ 0 < p_n \leq n^{-1/2}.
\end{array}\right. \end{align*} \end{corollary} When $p_n$ takes the form $p_n = n^{-\alpha}$, $\alpha \in [0,1)$, Corollary~\ref{c} similarly improves on the convergence rates obtained in Theorem~1.1 of \cite{reichenbachsAoP}. \\
\noindent
This paper is organized as follows.
In Section~\ref{s1} we recall the construction
of random functionals of Bernoulli variables,
together with the construction of the associated
finite difference operator
and their application to Kolmogorov distance bounds
obtained in \cite{reichenbachs}.
In Section~\ref{s2} we derive general Kolmogorov distance
bounds for sums of multiple stochastic integrals.
In Section~\ref{s3} we show that graph counts
can be represented as sums of multiple stochastic
integrals, and we derive Kolmogorov distance bounds
for the renormalized count of subgraphs in $\mathbb{G}_n(p_n)$ that are isomorphic to
a fixed graph.
\section{Notation and preliminaries}
\label{s1}
In this section we recall some background notation and results
on the stochastic analysis of Bernoulli processes,
see \cite{prsurvey} for details.
Consider a sequence
$(X_n)_{n\in {\mathord{\mathbb N}}}$ of
independent identically distributed Bernoulli random variables
with
$P(X_n=1) =p$ and $P(X_n = -1)=q$,
$n\in {\mathord{\mathbb N}}$,
built as the sequence of
canonical projections on $\Omega := \{-1,1\}^{\mathbb{N}}$.
For any $F:\Omega\to\mathbb{R}$ we consider the
$L^2(\Omega\times\mathbb{N})$-valued
finite difference operator $D$ defined
for any $\omega=(\omega_0,\omega_1,\ldots)\in\Omega$
by
\begin{equation}
\label{fdb} D_k F(\omega)=\sqrt{pq}(F(\omega_{+}^k)-F(\omega_{-}^{k})),\quad k\in\mathbb{N}, \end{equation}
where
we let $$ \omega_{+}^k:=(\omega_0,\ldots,\omega_{k-1},+1,\omega_{k+1},\ldots) \mbox{~~and~~} \omega_{-}^k:=(\omega_0,\ldots,\omega_{k-1},-1,\omega_{k+1},\ldots), \quad k \in {\mathord{\mathbb N}}, $$
and $DF := ( D_k F )_{k\in {\mathord{\mathbb N}}}$.
The $L^2$ domain of $D$ is given by $$ \mathrm{Dom}(D)
=\{F\in L^2(\Omega):\,\,\mathrm{E}[\|DF\|_{\ell^2(\mathbb{N})}^2]<\infty\}. $$
We let $( Y_n )_{n\geq 0}$ denote the sequence of
centered and normalized random variables defined by $$ Y_n :=\frac{q-p+X_n}{2\sqrt{p q}}, \qquad n \in {\mathord{\mathbb N}}. $$
Given $n\geq 1$, we denote by $\ell^2(\mathbb{N})^{\otimes n}=\ell^2(\mathbb{N}^{n})$ the class of square-summable functions on $\mathbb{N}^n$, we denote by $\ell^2(\mathbb{N})^{\circ n}$ the subspace of $\ell^2(\mathbb{N})^{\otimes n}$ formed by functions that are symmetric in $n$ variables. We let $$ I_n(f_n )=\sum_{(i_1,\ldots,i_n)\in\Delta_n}f_n(i_1,\ldots,i_n)Y_{i_1}\cdots Y_{i_n} $$
denote the discrete multiple stochastic integral
of order $n$ of $f_n$ in the subspace
$\ell^2_{\mathfrak{s}} (\Delta_n )$ of $\ell^2(\mathbb{N})^{\circ n}$
composed of symmetric kernels that vanish on
diagonals, i.e. on the complement of $$ \Delta_n=\{(k_1,\ldots,k_n)\in\mathbb{N}^n:\,\,k_i\neq k_j,\,\,1\leq i<j\leq n\},\quad\text{$n\geq 1$}. $$ The multiple stochastic integrals satisfy the isometry and orthogonality relation
\begin{equation}
\label{isomf} \mathrm{E}[I_n(f_n)I_m(g_m)] ={\rm 1\hspace{-0.90ex}1}_{\{n=m\}}
n!\langle f_n,g_m\rangle_{\ell^2_{\mathfrak{s}} (\Delta_n )} , \end{equation}
$f_n \in\ell^2_{\mathfrak{s}} (\Delta_n )$, $g_m \in\ell^2_{\mathfrak{s}} (\Delta_m )$,
cf. e.g. Proposition~1.3.2 of \cite{privaultbk2}. The finite difference operator $D$ acts on multiple stochastic integrals as follows: \begin{equation}\nonumber D_k I_n(f_n ) =nI_{n-1}(f_n (*,k){\rm 1\hspace{-0.90ex}1}_{\Delta_n}(*,k)) =nI_{n-1}(f_n (*,k)) , \end{equation} $k\in\mathbb{N}$, $f_n \in\ell^2_{\mathfrak{s}} (\Delta_n )$, and it satisfies the finite difference product rule \begin{equation} \label{prodrule} D_k(FG) = FD_kG+GD_kF-\frac{X_k}{\sqrt{pq}} D_kFD_kG,
\qquad k\in {\mathord{\mathbb N}}. \end{equation} for $F,G:\Omega \rightarrow {\mathord{\mathbb R}}$, see Propositions~7.3 and 7.8 of \cite{prsurvey}. \\
Due to the chaos representation property of Bernoulli random walks, any square integrable $F$ may be represented as $F=\sum_{n\geq 0}I_n(f_n)$, $f_n\in\ell^2_{\mathfrak{s}} (\Delta_n )$, and the $L^2$ domain of $D$ can be rewritten as \begin{align}
\mathrm{Dom}(D)&=\left\{F=\sum_{n\geq 0}I_n(f_n) \ : \ \sum_{n\geq 1}n\,n!\|f_n\|_{\ell^2(\mathbb{N})^{\otimes n}}^2<\infty\right\}.\nonumber \end{align} The Ornstein-Uhlenbeck operator $L$ is defined on the domain \begin{align} \mathrm{Dom}(L) :=
\left\{
F=\sum_{n\geq 0}I_n(f_n) \ : \ \sum_{n\geq 1}n^2\,n!\|f_n\|_{\ell^2(\mathbb{N})^{\otimes n}}^2<\infty\right\} \nonumber \end{align} by $$ LF=-\sum_{n=1}^\infty nI_n(f_n). $$ The inverse of $L$, denoted by $L^{-1}$, is defined on the subspace of $L^2(\Omega)$ composed of centered random variables by $$ L^{-1}F=-\sum_{n = 1}^\infty \frac{1}{n} I_n(f_n), $$
with the convention $L^{-1} F = L^{-1} ( F - \mathrm{E} [F] )$ in case
$F$ is not centered.
Using this convention,
the duality relation \eqref{prdual}
shows that for any $F,G\in\mathrm{Dom}(D)$
we have the covariance identity \begin{equation}\label{eq:covariance2} \mathrm{Cov}(F,G)=\mathrm{E}[
G ( F - \mathrm{E} [ F ] ) ]=\mathrm{E}\left[\langle DG,-DL^{-1}F\rangle_{\ell^2(\mathbb N)}\right]. \end{equation}
The divergence operator $\delta$ is
the linear mapping defined as $$
\delta ( u )
=
\delta (I_{n}(f_{n+1}( * ,\cdot )))=I_{n+1}(\tilde{f}_{n+1}), \quad
f_{n+1}\in \ell^2_{\mathfrak{s}} (\Delta_n ) \otimes \ell^{2}({\mathord{\mathbb N}}), $$
for $(u_k)_{k\in{\mathord{\mathbb N}}}$ of the form $$
u_k =
I_n (f_{n+1}( * , k )),
\qquad
k\in{\mathord{\mathbb N}} , $$
in the space $${\cal U} = \left\{
\sum_{k=0}^n
I_k (f_{k+1} (*,\cdot ) ),
\quad
f_{k+1} \in \ell^2_{\mathfrak{s}} (\Delta_k ) \otimes \ell^2 ({\mathord{\mathbb N}}), \
\ k=, n \in{\mathord{\mathbb N}} \right\}
\subset L^2 (\Omega \times {\mathord{\mathbb N}} )
$$
of finite sums of multiple integral processes,
where $\tilde{f}_{n+1}$ denotes the symmetrization of $f_{n+1}$
in $n+1$ variables, i.e. $$\tilde{f}_{n+1} (k_1,\ldots ,k_{n+1})
= \frac{1}{n+1}
\sum_{i=1}^{n+1}
f_{n+1} (k_1,\ldots ,k_{k-1},k_{k+1}, \ldots ,k_{n+1},
k_i ) . $$
The operators $D$ and $\delta$ are closable
with respective domains $\mathrm{Dom} ( D)$ and $\mathrm{Dom} ( \delta )$,
built as the completions of ${\cal S}$ and ${\cal U}$,
and they satisfy the duality relation
\begin{equation}
\label{prdual}
\mathbb{E} [\langle DF,u\rangle_{\ell^2 ({\mathord{\mathbb N}} )} ]
= \mathbb{E} [F\delta (u) ], \quad F\in \mathrm{Dom} ( D), \ u\in \mathrm{Dom} ( \delta ),
\end{equation}
see e.g. Proposition~9.2 in \cite{prsurvey},
and the isometry property \begin{eqnarray} \nonumber
\mathbb{E} [ | \delta (u)| ^2 ]
& = & \mathbb{E} [\Vert u \Vert_{\ell^2({\mathord{\mathbb N}} )}^2 ]
+ \mathbb{E} \Bigg[ \sum_{k,l=0\atop k \not= l}^\infty D_ku_l D_lu_k
- \sum_{k=0}^\infty ( D_ku_k)^2
\Bigg]
\\
\label{skois}
& \leq & \mathbb{E} [\Vert u \Vert_{\ell^2({\mathord{\mathbb N}} )}^2 ]
+ \mathbb{E} \Bigg[ \sum_{k,l=0\atop k \not= l}^\infty D_ku_l D_lu_k \Bigg],
\qquad u \in {\cal U}, \end{eqnarray} cf. Proposition~9.3 of \cite{prsurvey} and Satz~6.7 in \cite{mantei}.
Letting $(P_t)_{t\in {\mathord{\mathbb R}}_+} = (e^{tL})_{t\in {\mathord{\mathbb R}}_+}$
denote the Orsntein-Uhlenbeck semi-group defined as $$P_t F = \sum_{n=0}^\infty e^{-nt} I_n(f_n),
\qquad t\in {\mathord{\mathbb R}}_+,$$
on random variables $F\in L^2(\Omega )$ of the form
$\displaystyle F = \sum_{n=0}^\infty I_n(f_n)$,
the Mehler formula states that \begin{equation} \label{istheou}
P_t F = \mathbb{E} [F(X (t) ) \mid X (0) ],
\qquad t\in {\mathord{\mathbb R}}_+, \end{equation} where $(X (t) )_{t\in {\mathord{\mathbb R}}_+}$ is the Ornstein-Uhlenbeck process associated to the semi-group $(P_t)_{t\in {\mathord{\mathbb R}}_+}$, cf. Proposition~10.8 of \cite{prsurvey}. As a consequence of the representation \eqref{istheou} of $P_t$ we can deduce the bound \begin{equation}
\label{mehler}
\mathbb{E} [ | D_k L^{-1} F|^\alpha ] \leq
\mathbb{E} [ | D_k F|^\alpha ] , \end{equation} for every $F\in \mathrm{Dom} (D)$ and $\alpha \geq 1$, see Proposition~3.3 of \cite{reichenbachsAoP}. The following Proposition~\ref{prop:d_Kdescrete} is a consequence of Proposition~4.1 in \cite{reichenbachsAoP}, see also Theorem~3.1 in \cite{reichenbachs}. \begin{prop}\label{prop:d_Kdescrete} For $F\in \mathrm{Dom}(D)$ with $\mathbb{E}[F]=0$ we have \begin{align*}
d_K(F,\mathcal{N} )\leq & |1-\mathbb{E}[F^2]|
+ \sqrt{ \mathrm{Var}[\langle DF,-DL^{-1}F\rangle_{\ell^2(\mathbb{N})}] }
\\ & +\frac{1}{2\sqrt{pq}} \sqrt{\sum_{k=0}^\infty\mathbb{E}[(D_kF)^4]}\( \sqrt{\mathbb{E}\big[ F^2 \big]} +\sqrt{\sum_{k=0}^\infty\mathbb{E}[(FD_kL^{-1}F)^2]}\)\\ &+ \frac{1}{\sqrt{pq}}
\sup_{x\in \mathbb{R}} \mathbb{E}[\langle D \mathbf1_{\{F>x\}},DF|DL^{-1}F|\rangle_{\ell^2(\mathbb{N})}]. \end{align*} \end{prop} \begin{Proof}
By Proposition~4.1 in \cite{reichenbachsAoP} we have
\begin{eqnarray}
\nonumber
d_K(F,\mathcal{N} ) & \leq & \mathbb{E}[|1-\langle DF,-DL^{-1}F\rangle_{\ell^2(\mathbb{N})}|]
\\
\label{eq:dK1-1}
& & +
\frac{\sqrt{2\pi}}{8} (pq)^{-1/2} \mathbb{E}[\langle | DF |^2,|DL^{-1}F|\rangle_{\ell^2(\mathbb{N})}]
\qquad
\\
\label{eq:dK1-2}
& & +\frac12(pq)^{-1/2}\mathbb{E}[\langle | DF |^2,|F DL^{-1}F|\rangle_{\ell^2(\mathbb{N})}]
\\ \nonumber
& & + (pq)^{-1/2}\sup_{x\in \mathbb{R}} \mathbb{E}[\langle D \mathbf1_{\{F>x\}}, DF |DL^{-1}F|\rangle_{\ell^2(\mathbb{N})}]. \end{eqnarray} On the other hand, the covariance identity \eqref{eq:covariance2}
shows that
$\mathbb{E}[ | \langle DF,-DL^{-1}F\rangle_{l^2(\mathbb{N})} | ]=\mathrm{Var} F$, hence by the Cauchy-Schwarz and triangular inequalities we get \begin{eqnarray*}
\lefteqn{
\! \! \! \! \! \! \! \! \! \! \! \! \!
\mathrm{E}\left[\Big|1-\langle D F,-D L^{-1}F\rangle_{\ell^2(\mathbb{N})}\Big|\right]
\leq
\Big\|1-\langle D F,-D L^{-1}F\rangle_{\ell^2(\mathbb{N})}\Big\|_{L^2(\Omega)}
}
\\ & \leq &
| 1
-
\Vert
F
\Vert_{L^2(\Omega)}^2
|
+
\Vert
\langle D F,-D L^{-1}F\rangle_{\ell^2(\mathbb{N})}
-
\Vert
F
\Vert_{L^2(\Omega)}^2
\Vert_{L^2(\Omega)}
\\
& = &
|1-\mathrm{Var}[F]|+
\sqrt{ \mathrm{Var}[\langle DF,-DL^{-1}F\rangle_{\ell^2(\mathbb{N})} ]}. \end{eqnarray*}
Next, we have \begin{align*}
\mathbb{E} \big[\Vert
D L^{-1} I_n(f_n) \Vert_{\ell^2(\mathbb{N})}^2 \big]
&=\sum_{k=0}^\infty\mathbb{E}[(I_{n-1} (f_n(k,\cdot)))^2] \\
& = (n-1)!
\sum_{k=0}^\infty \|f_n(k,\cdot)\|_{\ell^2(\mathbb{N})^{\otimes (n-1)}}^2 \\
&=(n-1)!\|f_n\|_{\ell^2(\mathbb{N})^{\otimes n}}^2
\\
& \leq n!\|f_n\|_{\ell^2(\mathbb{N})^{\otimes n}}^2 \\
& = \mathbb{E}\[|I_n(f_n)|^2\right], \end{align*} and consequently, by the orthogonality relation \eqref{isomf} we have $$ \mathbb{E} \big[\Vert DL^{-1} F \Vert_{\ell^2(\mathbb{N})}^2 \big] \leq \mathbb{E}\big[F^2\big] $$ for every $F\in L^2(\Omega)$, hence \eqref{eq:dK1-1} is bounded by \begin{align*}
\mathbb{E}[ \langle |D L^{-1}F| , | DF |^2 \rangle_{\ell^2({\mathord{\mathbb N}} )} ]
&\leq
\mathbb{E} \left[
\sqrt{
\sum_{k=0}^\infty |D_kL^{-1}F|^2
\sum_{k=0}^\infty |D_kF|^4
}
\right]\\
&\leq
\sqrt{ \mathbb{E}\left[ \sum_{k=0}^\infty |D_kL^{-1}F|^2 \right]}
\sqrt{\mathbb{E} \left[ \sum_{k=0}^\infty (D_kF)^4 \right]} \\
& =
\sqrt{ \mathbb{E} \big[\Vert DL^{-1} F \Vert_{\ell^2(\mathbb{N})}^2 \big] }
\sqrt{\mathbb{E} \left[ \sum_{k=0}^\infty (D_kF)^4 \right]}
\\
&\leq
\sqrt{\mathbb{E}[F^2]} \sqrt{\mathbb{E} \left[ \sum_{k=0}^\infty (D_kF)^4 \right]}. \end{align*} Eventually, regarding the third term \eqref{eq:dK1-2}, by the Cauchy-Schwarz inequality we find \begin{align*}
\mathbb{E}\big[\langle
(DF)^2,|F DL^{-1}F|\rangle_{\ell^2(\mathbb{N})}\big]\leq
\sqrt{
\sum_{k=0}^\infty\mathbb{E}\big[(D_kF)^4\big]}\sqrt{\sum_{k=0}^\infty\mathbb{E} \big[(FD_kL^{-1}F)^2\big]}. \end{align*} \end{Proof} Finally, given $f_n\in\ell^2_{\mathfrak{s}} (\Delta_n )$ and $g_m\in\ell^2_{\mathfrak{s}} (\Delta_m )$ we have the multiplication formula \begin{equation} \label{e8} I_n(f_n)I_m(g_m)=\sum_{s=0}^{2 \min (n , m)}I_{n+m-s}(h_{n,m,s}) , \end{equation} see Proposition~5.1 of \cite{privaulttorrisi4}, provided that the functions $$
h_{n,m,s} : =
\sum_{s\leq 2i \leq 2 \min (s , n , m)} i!
\binom{n}{i}
\binom{m}{i}
\binom{i}{s-i}
\left(
\frac{q-p}{2\sqrt{pq}}
\right)^{2i-s}
f_n \tilde{\star}_i^{s-i} g_m $$ belong to $\ell^2_{\mathfrak{s}} (\Delta_{n+m-s})$, $0\leq s \leq 2 \min (n , m)$,
where $f_n \tilde{\star}_k^l g_m$
is defined as the symmetrization in $n+m-k-l$ variables
of the contraction $f_n \star_k^l g_m$
defined as \begin{eqnarray*}
\lefteqn{
\! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \!
\! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \!
f_n \star_k^l g_m
( a_{l+1},\ldots ,a_n,b_{k+1},\ldots ,b_m) = {\rm 1\hspace{-0.90ex}1}_{\Delta_{n+m-k-l}} ( a_{l+1},\ldots ,a_n,b_{k+1},\ldots ,b_m )
}
\\
& & \times
\sum_{a_1,\ldots ,a_l \in \mathbb{N}}f_n(a_1,\ldots ,a_n)g_m(a_1,\ldots ,a_k,b_{k+1} , \ldots , b_m), \end{eqnarray*}
$0\leq l\leq k$, and the symbol $\sum_{s\leq 2i \leq 2 \min (s , n , m)}$ means that the sum is taken over all the integers $i$ in the interval $[s/2, \min ( s , n , m )]$. We close this section with the following Proposition~\ref{prop:fg<f^2+g^2}. \begin{prop}\label{prop:fg<f^2+g^2}
Let $f_n \in \ell^2_{\mathfrak{s}} (\Delta_n )$ and $g_m \in \ell^2_{\mathfrak{s}} (\Delta_m )$
be symmetric functions.
For $0\leq l< k\leq \min ( n,m )$ we have \begin{align}\label{eq:l<k}
\left\| f_{n}\star_k^{l} g_m \right\|^2_{\ell^2(\mathbb{N})^{\otimes (m+n-k-l)}}&\leq \frac12\left\| f_{n}\star_{n}^{l+n-k} f_{n}\right\|^2_{\ell^2(\mathbb{N})^{\otimes (k-l)}}+\frac12\left\| g_m \star_{m}^{l+m-k} g_m \right\|^2_{\ell^2(\mathbb{N})^{\otimes (k-l)}}, \end{align} and \begin{align}\label{eq:l=k}
\left\| f_{n}\star_k^{k} g_m \right\|^2_{\ell^2(\mathbb{N})^{\otimes (m+n-2k)}}&\leq
\frac{1}{2}
\left\| f_{n}\star_{n-k}^{n-k} f_{n}\right\|^2_{\ell^2(\mathbb{N})^{\otimes 2k}}
+
\frac{1}{2}
\left\| g_m \star_{m-k}^{m-k} f_{m}\right\|^2_{\ell^2(\mathbb{N})^{\otimes 2k}}. \end{align} \end{prop} \begin{Proof} H\"older's inequality applied twice gives us \begin{eqnarray*}
\lefteqn{
\! \! \! \! \! \! \! \! \! \!
\left\| f_{n}\star_k^{l} g_m \right\|^2_{\ell^2(\mathbb{N})^{\otimes (m+n-k-l)}}
=\sum_{z_1\in\mathbb{N}^{n-k}}\sum_{z_2\in\mathbb{N}^{m-k}}\sum_{y\in\mathbb{N}^{k-l}}\(\sum_{x\in\mathbb{N}^l}f_n(x,y,z_1)g_m(x,y,z_2)\)^2
}
\\
& \leq & \sum_{y\in\mathbb{N}^{k-l}}\sum_{z_1\in\mathbb{N}^{n-k}}\sum_{z_2\in\mathbb{N}^{m-k}}\(\sum_{x\in\mathbb{N}^l}f_n^2(x,y,z_1)\sum_{x\in\mathbb{N}^l}g_m^2(x,y,z_2)\)
\\
&\leq &
\sqrt{
\sum_{y\in\mathbb{N}^{k-l}}\(\sum_{z_1\in\mathbb{N}^{n-k}}\sum_{x\in\mathbb{N}^l}f_n^2(x,y,z_1)\)^2\sum_{y\in\mathbb{N}^{k-l}}\(\sum_{z_1\in\mathbb{N}^{m-k}}\sum_{x\in\mathbb{N}^l}g_m^2(x,y,z_2)\)^2 }
\\[6pt]
&= &
\left\| f_{n}\star_{n}^{l+n-k} f_{n}\right\|_{\ell^2(\mathbb{N})^{\otimes (k-l)}}
\left\| g_m \star_{m}^{l+m-k} g_m \right\|_{\ell^2(\mathbb{N})^{\otimes (k-l)}}
\\[6pt]
& \leq &
\frac{1}{2}
\left\| f_{n}\star_{n}^{l+n-k} f_{n}\right\|^2_{\ell^2(\mathbb{N})^{\otimes (k-l)}}
+
\frac{1}{2}
\left\| g_m \star_{m}^{l+m-k} g_m \right\|^2_{\ell^2(\mathbb{N})^{\otimes (k-l)}}. \end{eqnarray*} To derive the second assertion, we proceed as follows: \begin{align*}
&\left\| f_{n}\star_k^{k} g_m \right\|^2_{\ell^2(\mathbb{N})^{\otimes (m+n-2k)}}\\ &=\sum_{y\in\mathbb{N}^{n-k}}\sum_{z\in\mathbb{N}^{m-k}}\sum_{x_1\in\mathbb{N}^k}\sum_{x_2\in\mathbb{N}^k}f_n(x_1,y)g_m(x_1,z)f_n(x_2,y)g_m(x_2,z)\\ &=\sum_{x_1\in\mathbb{N}^k}\sum_{x_2\in\mathbb{R}^k}\(\sum_{y\in\mathbb{N}^{n-k}}f_n(x_1,y)f_n(x_2,y)\)\(\sum_{z\in\mathbb{N}^{m-k}}g_m(x_1,z)g_m(x_2,z)\)\\
&\leq
\frac{1}{2}
\sum_{x_1\in\mathbb{N}^k}\sum_{x_2\in\mathbb{R}^k}\(\sum_{ y\in \mathbb{N}^{n-k}}f_n(x_1,y)f_n(x_2,y)\)^2
+
\frac{1}{2}
\sum_{x_1\in\mathbb{N}^k}\sum_{x_2\in\mathbb{R}^k}\(\sum_{ z\in \mathbb{N}^{m-k}}g_m(x_1,z)g_m(x_2,z)\)^2
\\
& =
\frac{1}{2}
\left\| f_{n}\star_{n-k}^{n-k} f_{n}\right\|^2_{\ell^2(\mathbb{N})^{\otimes 2k}}
+
\frac{1}{2}
\left\| g_m \star_{m-k}^{m-k} f_{m}\right\|^2_{\ell^2(\mathbb{N})^{\otimes 2k}}. \end{align*} \end{Proof} \section{Kolmogorov bounds for sums of multiple stochastic integrals} \label{s2} Wasserstein bounds have been obtained for discrete multiple stochastic integrals in Theorem~4.1 of \cite{nourdin3} in the symmetric case $p=q$ and in Theorems~5.3-5.5 of \cite{privaulttorrisi4} in the possibly nonsymmetric case, and have been extended to the Kolmogorov distance in the symmetric case $p=q$ in Theorem~4.2 of \cite{reichenbachs}. The following result provides a Kolmogorov distance bound which further extends Theorem~4.2 of \cite{reichenbachs} from multiple stochastic integrals to sums of multiple stochastic integrals in the nonsymmetric case. \begin{theorem}\label{prop:dKsumI}
For any finite sum
$$ F=\sum_{k=1}^nI_k(f_k)$$
of discrete multiple stochastic integrals
with $f_k \in \ell^2_{\mathfrak{s}} (\Delta_k )$, $k=1,\ldots, n$, we have \begin{align*}
d_K(F,\mathcal{N} )\leq & C_n \big( |1-\mathrm{Var}[F]|+\sqrt{R_F} \big), \end{align*} for some constant $C_n>0$ depending only on $n$, where \begin{equation} \label{sjks}
R_F:= \sum_{0\leq l< i\leq n}(pq)^{l-i}\left\| f_{i}\star_{i}^{l} f_{i}\right\|^2_{\ell^2(\mathbb{N})^{\otimes (i-l)}}+\sum_{1\leq l< i\leq n}
\(
\left\| f_{l}\star_{l}^{l} f_{i}\right\|^2_{\ell^2(\mathbb{N})^{\otimes (i-l)}}
+
\left\| f_{i}\star_{l}^{l} f_{i}\right\|^2_{\ell^2(\mathbb{N})^{\otimes 2(i-l)}}
\). \end{equation} \end{theorem} \begin{Proof}
We introduce $$R'_F:=\sum_{1\leq i\leq j\leq n}\sum_{k=1}^{i}\sum_{l=0}^k\mathbf1_{\{i=j=k=l\}^c} (pq)^{l-k}
\left\| f_{i}\star_k^{l} f_{j}\right\|^2_{\ell^2(\mathbb{N})^{\otimes (i+j-k-l)}}. $$ Since it holds that $R'_F\lesssim R_F$, it is enough to prove the required inequality with $R'_F$ instead of $R_F$. Indeed, by the inequality \eqref{eq:l<k}, all the components of $R'_F$ for $0\leq l<k\leq i,j$, are dominated by those for $0\leq l<k=i=j$, and also, by the inequality \eqref{eq:l=k}, the ones where $1\leq k=l< i \leq j$, are dominated by the components where $1\leq l=k<i=j$. Finally, the components for $1\leq k=l=i<j$ remain unchanged. \\
We will estimate components in the inequality from Proposition~\ref{prop:d_Kdescrete}. We have \begin{align*}
D_r F=(i+1)\sum_{i=0}^{n-1}I_i \(f_{i+1}(r,\cdot)\),\quad
\mbox{and} \quad D_r L^{-1}F = \sum_{i=0}^{n-1}I_i \(f_{i+1}(r,\cdot)\), \qquad r \in{\mathord{\mathbb N}}, \end{align*} hence by the multiplication formula \eqref{e8} we find \begin{equation}
\label{XX}
(D_r F)^2=\sum_{0\leq i\leq j\leq n-1}\sum_{k=0}^{i}\sum_{l=0}^kc_{i,j,l,k}\(\frac{q-p}{\sqrt{pq}}\)^{k-l}I_{i+j-k-l}\(f_{i+1}(r,\cdot)\tilde{\star}_k^lf_{j+1}(r,\cdot)\) \end{equation} and \begin{equation}
\label{XLX} D_r FD_r L^{-1}F=\sum_{0\leq i\leq j\leq n-1}\sum_{k=0}^{i}\sum_{l=0}^kd_{i,j,l,k}\(\frac{q-p}{\sqrt{pq}}\)^{k-l}I_{i+j-k-l}\(f_{i+1}(r,\cdot)\tilde{\star}_k^lf_{j+1}(r,\cdot)\), \end{equation}
for some $c_{i,j,l,k}$, $d_{i,j,l,k} \geq0$. Applying the isometry relation \eqref{isomf} to \eqref{XX} and using the bound $\| \tilde{f}_n\|_{\ell^2(\mathbb{N} )^{\otimes n}} \leq \| f_n \|_{\ell^2(\mathbb{N} )^{\otimes n}}$, $f_n \in \ell^2(\mathbb{N} )^{\otimes n}$, we get, writing $f\lesssim g$ whenever $f< C_n g$ for some
constant $C_n >0$ depending only on $n$, \begin{align}\nonumber
\sum_{r=0}^\infty \mathbb{E} \left[ | D_r F |^4 \right]&\lesssim \sum_{0\leq i\leq j\leq n-1}\sum_{k=0}^{i}\sum_{l=0}^k\sum_{r=0}^\infty \(\frac{q-p}{\sqrt{pq}}\)^{2k-2l}\left\|f_{i+1}(r,\cdot)\star_k^lf_{j+1}(r,\cdot)\right\|^2_{\ell^2(\mathbb{N})^{\otimes (i+j-k-l)}}
\\
\nonumber
&= \sum_{0\leq i\leq j\leq n-1}\sum_{k=0}^{i}\sum_{l=0}^k \(\frac{q-p}{\sqrt{pq}}\)^{2k-2l}\left\|f_{i+1} \star_{k+1}^lf_{j+1} \right\|^2_{\ell^2(\mathbb{N})^{\otimes (i+j-k-l+1)}} \\ \nonumber
&= \sum_{1\leq i\leq j\leq n}\sum_{k=1}^{i}\sum_{l=0}^{k-1} \(\frac{q-p}{\sqrt{pq}}\)^{2k-2l-2}\left\|f_{i} \star_k^lf_{j} \right\|^2_{\ell^2(\mathbb{N})^{\otimes (i+j-k-l)}} \\ \label{aux1} &\leq pqR'_F. \end{align} Furthermore, by \eqref{XLX} it follows that \begin{align*}
&\langle D F, D L^{-1}F\rangle-\mathbb{E}\[\langle D F, D L^{-1}F\rangle\right]
\\ &=\sum_{r=0}^\infty\sum_{0\leq i\leq j\leq n-1}\sum_{k=0}^{i}\sum_{l=0}^kc_{i,j,l,k}\mathbf1_{\{i=j=k=l\}^c}\(\frac{q-p}{\sqrt{pq}}\)^{k-l}I_{i+j-k-l}\(f_{i+1}(r,\cdot)\tilde{\star}_k^lf_{j+1}(r,\cdot)\)\\ &=\sum_{0\leq i\leq j\leq n-1}\sum_{k=0}^{i}\sum_{l=0}^kc_{i,j,l,k}\mathbf1_{\{i=j=k=l\}^c}\(\frac{q-p}{\sqrt{pq}}\)^{k-l}I_{i+j-k-l}\(\sum_{r=0}^\infty f_{i+1}(r,\cdot)\tilde{\star}_k^lf_{j+1}(r,\cdot)\)\\ &=\sum_{0\leq i\leq j\leq n-1}\sum_{k=0}^{i}\sum_{l=0}^kc_{i,j,l,k}\mathbf1_{\{i=j=k=l\}^c}\(\frac{q-p}{\sqrt{pq}}\)^{k-l}I_{i+j-k-l}\( f_{i+1} \tilde{\star}_{k+1}^{l+1}f_{j+1} \), \end{align*} thus we get \begin{eqnarray}
\nonumber
\lefteqn{
\! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \!
\mathrm{Var}\[\langle D F, -D L^{-1}F\rangle\right]
\lesssim\sum_{0\leq i\leq j\leq n-1}\sum_{k=0}^{i}\sum_{l=0}^k\mathbf1_{\{i=j=k=l\}^c}\(\frac{1}{\sqrt{pq}}\)^{2k-2l}\left\| f_{i+1}\star_{k+1}^{l+1} f_{j+1}\right\|^2_{\ell^2(\mathbb{N})^{\otimes (i+j-k-l)}}
}
\\\nonumber
&= & \sum_{1\leq i\leq j\leq n}\sum_{k=1}^{i}\sum_{l=1}^k\mathbf1_{\{i=j=k=l\}^c}\frac{1}{(pq)^{k-l}}\left\| f_{i}\star_k^{l} f_{j}\right\|^2_{\ell^2(\mathbb{N})^{\otimes (i+j-k-l)}}\\ \label{eq:term1} &\leq & R'_F. \end{eqnarray} Next, we have \begin{align*}
&\sum_{k=0}^\infty
\mathbb{E} \big[ (FD_kL^{-1}F)^2 \big]=
\mathbb{E} \left[
F^2\sum_{k=0}^\infty(D_kL^{-1}F)^2
\right]\leq \sqrt{\mathbb{E}\[F^4\right]}
\sqrt{ \mathbb{E}\[\(\sum_{k=0}^\infty(D_kL^{-1}F)^2\)^2\right]} \end{align*}
and \eqref{e8} and \eqref{isomf} show that \begin{align*}
\mathbb{E}\[F^4\right]&\lesssim \mathbb{E}\[\(\sum_{1\leq i\leq j\leq n}\sum_{k=0}^{i}\sum_{l=0}^k
\left|
\frac{q-p}{\sqrt{pq}}
\right|^{k-l}I_{i+j-k-l}\( f_{i} \tilde{\star}_k^{l}f_{j} \)\)^2\right] \\ &\lesssim \sum_{1\leq i\leq j\leq n}\sum_{k=0}^{i}\sum_{l=0}^k (pq)^{l-k}
\| f_{i} \star_k^{l}f_{j} \|_{\ell^2(\mathbb{N})^{\otimes (i+j-k-l)}}^2 \\ &\lesssim R'_F +
\sum_{i=1}^n \left\| f_{i}\star_{i}^{i} f_{i}\right\|^2_{\ell^2(\mathbb{N})^{\otimes 0}} +
\sum_{1\leq i < j\leq n} \| f_{i} \star_0^0 f_{j} \|_{\ell^2(\mathbb{N})^{\otimes (i+j)}}^2 \\ & = R'_F +
\sum_{i=1}^n \left\| f_i \right\|^4_{\ell^2(\mathbb{N})^{\otimes i}} +
\sum_{1\leq i < j\leq n} \| f_i \|_{\ell^2(\mathbb{N})^{\otimes i}}^2
\| f_j \|_{\ell^2(\mathbb{N})^{\otimes j}}^2 \\ &\lesssim R'_F+(\mathrm{Var}[F])^2, \end{align*}
while as in \eqref{XX} and \eqref{XLX} we have \begin{align*}
&\mathbb{E}\[\(\sum_{k=0}^\infty(D_kL^{-1}F)^2\)^2\right]
\\ & =
\mathbb{E}\[
\(
\sum_{k=0}^\infty
\sum_{0\leq i\leq j\leq n-1}\sum_{k=0}^{i}\sum_{l=0}^k\tilde{d}_{i,j,l,k}\(\frac{q-p}{\sqrt{pq}}\)^{k-l}I_{i+j-k-l}\(f_{i+1}(k,\cdot)\tilde{\star}_k^lf_{j+1}(k,\cdot)\)
\)^2 \right] \\
&\lesssim \sum_{0\leq i\leq j\leq n-1}\sum_{k=0}^{i}\sum_{l=0}^k
(pq)^{l-k}
\| f_{i+1} \star_{k+1}^{l+1}f_{j+1} \|^2_{\ell^2(\mathbb{N})^{\otimes (i+j-k-l)}} \\
&= \sum_{1\leq i\leq j\leq n}\sum_{k=1}^i\sum_{l=1}^k
(pq)^{l-k}
\| f_i \star_k^lf_j \|^2_{\ell^2(\mathbb{N})^{\otimes (i+j-k-l)}} \\
&\lesssim R'_F +
\sum_{i=1}^n \left\| f_{i}\star_{i}^{i} f_{i}\right\|^2_{\ell^2(\mathbb{N})^{\otimes 0}} +
\sum_{1\leq i < j \leq n} \| f_{i} \star_0^0 f_{j} \|_{\ell^2(\mathbb{N})^{\otimes (i+j)}}^2 \\ & = R'_F +
\sum_{i=1}^n \left\| f_i \right\|^4_{\ell^2(\mathbb{N})^{\otimes i}} +
\sum_{1\leq i < j\leq n} \| f_i \|_{\ell^2(\mathbb{N})^{\otimes i}}^2
\| f_j \|_{\ell^2(\mathbb{N})^{\otimes j}}^2 \\
& \lesssim R'_F +(\mathrm{Var}[F])^2, \end{align*}
hence we get \begin{align}\label{eq:term2}\sum_{k=0}^\infty\mathbb{E}[(FD_k L^{-1} F)^2]\lesssim R'_F+(\mathrm{Var}[F])^2. \end{align} We now deal with the last component in Proposition~\ref{prop:d_Kdescrete} similarly as it is done in proof of Theorem~4.2 in \cite{reichenbachs}. Precisely, by the integration by parts formula \eqref{prdual} and the Cauchy-Schwarz inequality we have \begin{align}
\nonumber
\sup_{x\in \mathbb{R}} \mathbb{E}\[\langle D \mathbf1_{\{F>x\}},DF|DL^{-1}F|\rangle_{\ell^2(\mathbb{N})}\right]&=\sup_{x\in \mathbb{R}} \mathbb{E}\[\mathbf1_{\{F>x\}}\delta\(DF|DL^{-1}F|\)\right] \\ \label{dklsfd}
&\leq
\sqrt{ \mathbb{E}\[\(\delta\(DF|DL^{-1}F|\)\)^2\right] }. \end{align} Then, by the bound \eqref{skois}, the Cauchy-Schwarz inequality and the consequence \eqref{mehler} of Mehler's formula \eqref{istheou}, we have \begin{align*}
&\mathbb{E}\big[ \(\delta\(DF|DL^{-1}F|\)\)^2 \big]
\\
&\leq \mathbb{E} \big[\|DF|DL^{-1}F|\|^2_{\ell^2(\mathbb{N})}\big]+\mathbb{E}\[\sum_{k,l=0}^\infty
\big| D_k\(D_lF|D_lL^{-1}F|\)D_l\(D_kF|D_kL^{-1}F|\) \big|
\right]
\\
&\leq
\sqrt{
\mathbb{E}\big[\|DF\|^4_{\ell^4(\mathbb{N})}\big]\mathbb{E}\big[\|DL^{-1}F\|^4_{\ell^4(\mathbb{N})}\big]
}
+\mathbb{E}\[\sum_{k,l=0}^\infty \(D_k\(D_lF|D_lL^{-1}F|\)\)^2 \right]
\\
&\leq\mathbb{E} \big[\|DF\|^4_{\ell^4(\mathbb{N})}\big]+\sum_{k,l=0}^\infty\mathbb{E}\big[
\(D_k\(D_lF|D_lL^{-1}F|\)\)^2 \big]. \end{align*} The first term in the last expression in bounded by $pqR'_F$ as shown in
\eqref{aux1}, and it remains to estimate the last expectation. By the product rule \eqref{prodrule} and the bound $|D_k| F ||\leq |D_k F |$ obtained from the definition \eqref{fdb} of $D$ and the triangle inequality, we get \begin{align}
\nonumber
&\mathbb{E}\big[\(D_r\(D_sF|D_sL^{-1}F|\)\)^2 \big]
\\ \nonumber
&=\mathbb{E}\[\(\(D_rD_sF|D_sL^{-1}F|\)+\(D_sFD_r|D_sL^{-1}F|\)-\frac{X_r}{\sqrt{pq}}\(D_rD_sFD_r|D_sL^{-1}F|\)\)^2 \right] \\ \label{dksds} &\lesssim \mathbb{E}\[\(D_rD_sF\)^2\(D_sL^{-1}F\)^2+\(D_sF\)\(D_rD_sL^{-1}F\)^2+\frac{1}{pq}\(D_rD_sF\)^2\(D_rD_sL^{-1}F\)^2 \right], \end{align} $r,s \in {\mathord{\mathbb N}}$. By the Cauchy-Schwarz inequality we get \begin{align*} \sum_{r,s=0}^\infty\mathbb{E}\big[\(D_rD_sF\)^2\(D_sL^{-1}F\)^2\big] & =\mathbb{E}\[\sum_{s=0}^\infty\(D_sL^{-1}F\)^2\sum_{r=0}^\infty\(D_rD_sF\)^2\right]\\ &\leq \sqrt{
\mathbb{E}\[\sum_{s=0}^\infty\(D_sL^{-1}F\)^4\right]\mathbb{E}\[\sum_{s=0}^\infty\(\sum_{r=0}^\infty\(D_rD_sF\)^2\)^2\right]}. \end{align*} The term $\mathbb{E}\big[\sum_{s=0}^\infty\(D_sL^{-1}F\)^4\big]$ can be bounded by $pq R'_F$ as in \eqref{aux1}. To estimate the other term we use the multiplication formula \eqref{e8} as in \eqref{XX} to obtain \begin{align*} &\mathbb{E}\[\sum_{s=0}^\infty\(\sum_{r=0}^\infty\(D_rD_sF\)^2\)^2\right]\\
&\lesssim \sum_{s=0}^\infty\mathbb{E}\[\(\sum_{r=0}^\infty\sum_{0\leq i\leq j\leq n-2}\sum_{k=0}^{i}\sum_{l=0}^k
\left|
\frac{q-p}{\sqrt{pq}}
\right|^{k-l}I_{i+j-k-l}\(f_{i+2}(s,r,\cdot)\tilde{\star}_k^lf_{j+2}(s,r,\cdot)\)\)^2\right]\\
&=c \sum_{s=0}^\infty\mathbb{E}\[\(\sum_{0\leq i\leq j\leq n-2}\sum_{k=0}^{i}\sum_{l=0}^k
\left|
\frac{q-p}{\sqrt{pq}}
\right|^{k-l}I_{i+j-k-l}\(f_{i+2}(s,\cdot)\tilde{\star}_{k+1}^{l+1}f_{j+2}(s,\cdot)\)\)^2\right]
\\
&\lesssim \sum_{s=0}^\infty\sum_{0\leq i\leq j\leq n-2}\sum_{k=0}^{i}\sum_{l=0}^k
(pq)^{l-k}\|f_{i+2}(s,\cdot) \star_{k+1}^{l+1}f_{j+2}(s,\cdot) \|^2_{\ell^2(\mathbb{N})^{\otimes (i+j-k-l)}}\\
&= \sum_{0\leq i\leq j\leq n-2}\sum_{k=0}^{i}\sum_{l=0}^k
(pq)^{l-k}\|f_{i+2} \star_{k+2}^{l+1}f_{j+2} \|^2_{\ell^2(\mathbb{N})^{\otimes (i+j-k-l+1)}}\\
&= \sum_{2\leq i\leq j\leq n}\sum_{k=2}^{i}\sum_{l=1}^{k-1}
(pq)^{l+1-k}\|f_{i} \star_k^{l}f_{j} \|^2_{\ell^2 (\mathbb{N})^{\otimes (i+j-k-l)}}
\\
& \leq pqR'_F. \end{align*} The term $\sum_{r,s=0}^\infty\mathbb{E}\big[(D_sF)^2(D_rD_sL^{-1}F)^2\big]$ from \eqref{dksds} is similarly bounded by $pqR'_F$. Regarding the last term, we have $$ \sum_{r,s=0}^\infty\mathbb{E}\big[(D_rD_sF)^2(D_rD_sL^{-1}F)^2\big] \leq
\sqrt{
\sum_{r,s=0}^\infty\mathbb{E}\[\(D_rD_sF\)^4\right]\sum_{r,s=0}^\infty
\mathbb{E}\big[(D_rD_sL^{-1}F)^4\big] }. $$
Using the multiplication formula \eqref{e8}, both sums inside the
above square root can be estimated as \begin{align*}
&\sum_{r,s=0}^\infty\mathbb{E}\[\(\sum_{0\leq i\leq j\leq n-2}\sum_{k=0}^{i}\sum_{l=0}^k
\left|
\frac{q-p}{\sqrt{pq}}
\right|^{k-l}I_{i+j-k-l}\(f_{i+2}(s,r,\cdot)\tilde{\star}_k^lf_{j+2}(s,r,\cdot)\)\)^2\right]\\
&\lesssim\sum_{r,s=0}^\infty\sum_{0\leq i\leq j\leq n-2}\sum_{k=0}^{i}\sum_{l=0}^k
(pq)^{l-k}\|f_{i+2}(s,r,\cdot)\star_k^lf_{j+2}(s,r,\cdot)\|^2_{\ell^2(\mathbb{N})^{\otimes (i+j-k-l)}}\\
&=\sum_{0\leq i\leq j\leq n-2}\sum_{k=0}^{i}\sum_{l=0}^k
(pq)^{l-k}\|f_{i+2} \star_{k+2}^{l}f_{j+2} \|^2_{\ell^2(\mathbb{N})^{\otimes (i+j-k-l + 2)}} \\
&=\sum_{2\leq i\leq j\leq n}\sum_{k=2}^{i}\sum_{l=0}^{k-2}
(pq)^{l+2-k}\|f_{i} \star_k^{l}f_{j} \|^2_{\ell^2(\mathbb{N})^{\otimes (i+j-k-l)}} \\
& \lesssim (pq)^2R'_F. \end{align*}
Combining this together we get $$
\sum_{r,s=0}^\infty
\mathbb{E}\big[ \big(D_r\big(D_sF|D_sL^{-1}F|\big)\big)^2 \big]\lesssim pq R'_F. $$ and consequently, by \eqref{dklsfd} we find \begin{align}\label{eq:term3}
\sup_{x\in \mathbb{R}} \mathbb{E}\[\langle D \mathbf1_{\{F>x\}},DF|DL^{-1}F|\rangle_{\ell^2(\mathbb{N})}\right]\lesssim pq R'_F. \end{align} Applying \eqref{aux1}-\eqref{eq:term2} and \eqref{eq:term3} to Proposition~\ref{prop:d_Kdescrete}, we get $$
d_K(F,\mathcal{N} ) \lesssim |1-\mathrm{Var}[F]|+\sqrt{R'_F}
\big(1+\mathrm{Var}[F]
+\sqrt{\mathrm{Var}[F]}
+\sqrt{R'_F}\big). $$
If $R'_F\geq1$, or if $R'_F\leq1$ and $\mathrm{Var}[F]\geq2$, it is clear that $d_K(F,\mathcal{N} )\lesssim |1-\mathrm{Var}[F]|+\sqrt{R'_F}$ since $d_K(F,\mathcal{N} )\leq 1$ by definition. If $R'_F\leq1$ and $\mathrm{Var}[F]\leq2$, we estimate $\mathrm{Var}[F]+\sqrt{\mathrm{Var}[F]}+\sqrt{R'_F}$ by a constant and also get the required bound. \end{Proof} \section{Application to random graphs} \label{s3} In the sequel fix a numbering $(1,\ldots , e_G)$ of the edges in $G$ and we denote by $E_G \subset {\mathord{\mathbb N}}^{e_G}$ the set of sequences of (distinct) edges that create a graph isomorphic to $G$, i.e. a sequence $( e_{k_1} ,\ldots , e_{k_{e_G}} )$ belongs to $E_G$ if and only if the graph created by edges $e_{k_1},\ldots ,e_{k_{e_G}}$ is isomorphic to $G$. The next lemma allows us to represent the number of subgraphs as a sum of multiple stochastic integrals, using the notation $P(X_k=1)=p$, $P(X_k=-1)=1-p=q$, $k\in{\mathord{\mathbb N}}$. \begin{lemma}
\label{lemma4.1}
We have the identity \begin{align}\label{eq:tildeNasI} \tilde{N}_G=\frac{N_G-\mathbb{E}[N_G]}{\sqrt{\mathrm{Var}[N_G]}}=\sum_{k=1}^{e_G}I_k(f_k), \end{align} where $$ f_k(b_1,\ldots , b_k) := \frac{q^{k/2}p^{e_G - k/2}}{(e_G-k)!k!\sqrt{\mathrm{Var}[N_G]}}\, \sum_{(a_1,\ldots , a_{e_{\scaleto{G}{3pt}} -k} ) \in
\mathbb{N}^{e_{\scaleto{G}{3pt}} -k}} \mathbf1_{(a_1,\ldots , a_{e_{\scaleto{G}{3pt}} -k} , b_1,\ldots , b_k )\in E_G}. $$ \end{lemma} \begin{Proof}
We have
\begin{align}\nonumber
& N_G =\frac1{e_G!2^{e_G}}\sum_{ b_1,\ldots ,b_{e_G} \in {\mathord{\mathbb N}}}
\mathbf1_{(b_1,\ldots ,b_{e_G})\in E_G}(X_{b_1}+1)\cdots (X_{b_{e_G}}+1)\\\nonumber &=\frac1{e_G!2^{e_G}}\sum_{m=0}^{e_G}\({{e_G}\atop m}\)\sum_{b_1,\ldots ,b_{m} \in {\mathord{\mathbb N}}}
g_m(b_1,\ldots ,b_{m})X_{b_1}\cdots X_{b_{m}}
\\\nonumber
&=\frac1{e_G!2^{e_G}}\sum_{m=0}^{e_G}\({{e_G}\atop m}\)\sum_{k=0}^m\({m\atop k}\) (p-q)^{m-k}
\sum_{b_1,\ldots,b_k \in {\mathord{\mathbb N}} } g_k(b_1,\ldots,b_k) ( X_{b_1} + q-p ) \cdots ( X_{b_k} + q-p )
\\\nonumber
&=\frac1{e_G!2^{e_G}}\sum_{m=0}^{e_G}\({{e_G}\atop m}\)\sum_{k=0}^m\({m\atop k}\)I_k(g_k)(2\sqrt{pq})^k(p-q)^{m-k}
\\
\nonumber
&=\frac1{e_G!2^{e_G}}\sum_{k=0}^{e_G} \(e_G \atop k \) (2\sqrt{pq})^kI_k(g_k)
\sum_{m=k}^{e_G}\({{e_G-k}\atop {m-k}}\)(p-q)^{m-k}
\\
\nonumber
&=\frac{1}{2^{e_G}}\sum_{k=0}^{e_G} \frac{(2\sqrt{pq})^k}{(e_G-k)!k!} I_k(g_k) (1+p-q)^{e_G-k} \\ \nonumber &=\sum_{k=0}^{e_G} \frac{q^{k/2}p^{e_G-k/2}}{(e_G-k)!k!} I_k(g_k), \end{align}
where $g_k$ is the function defined as \begin{equation} \label{gk} g_k(b_1,\ldots , b_k) :=\sum_{(a_1,\ldots , a_{e_G-k} ) \in\mathbb{N}^{e_G-k}}\mathbf1_{E_G} (a_1,\ldots , a_{e_G-k} , b_1,\ldots , b_k ), \quad (b_1,\ldots , b_k ) \in\mathbb{N}^k, \end{equation} which shows \eqref{eq:tildeNasI} with $$ f_k(b_1,\ldots , b_k ):= \frac{q^{k/2}p^{e_G- k/2 }}{(e_G-k)!k! \sqrt{\mathrm{Var}[N_G]}}\,g_k(b_1,\ldots , b_k ). $$ \end{Proof} Next is the second main result of this paper. \begin{theorem}\label{thm:main} Let $G$ be a graph without isolated vertices. Then we have \begin{align*} d_K (\tilde{N}_G,\mathcal{N} )&\lesssim\((1-p)\min_{\substack{ H\subset G\\e_H\geq1}}\big\{n^{v_H}p^{e_H}\big\}\)^{-1/2}. \end{align*} \end{theorem} \begin{Proof}
By \eqref{eq:tildeNasI} and Theorem~\ref{prop:dKsumI} we have \begin{align}\label{eq:dKfromprop} d_K (\tilde{N}_G,\mathcal{N} )\lesssim {\frac {\sqrt{R_G}}{\mathrm{Var}[N_G]}}, \end{align} where, taking $g_k$ as in \eqref{gk}, by \eqref{sjks} we have \begin{align*}
R_G = \ & \sum_{0\leq l< k\leq e_G}p^{4e_G-3k+l}q^{l+k}\left\| g_k\star_k^{l} g_k\right\|^2_{\ell^2(\mathbb{N})^{\otimes (k-l)}}
+\sum_{1\leq l< k\leq e_G}p^{4e_G-2k}q^{2k} \left\| g_k\star_{l}^{l} g_k\right\|^2_{\ell^2(\mathbb{N})^{\otimes 2(k-l)}} \\
\ &
+\sum_{1\leq l<k\leq e_G}p^{4e_G-l-k}q^{k+l}\left\| g_l \star_{l}^{l} g_k\right\|^2_{\ell^2(\mathbb{N})^{\otimes (k-l)}} \\
\leq \ & q \Big( \sum_{0\leq l< k\leq e_G}p^{4e_G-3k+l}\left\| g_k\star_k^{l} g_k\right\|^2_{\ell^2(\mathbb{N})^{\otimes (k-l)}}
+\sum_{1\leq l<k\leq e_G}p^{4e_G-l-k}\left\| g_l \star_{l}^{l} g_k\right\|^2_{\ell^2(\mathbb{N})^{\otimes (k-l)}} \\
\ & +\sum_{1\leq l<k\leq e_G}p^{4e_G-2k}
\left\| g_k\star_{l}^{l} g_k\right\|^2_{\ell^2(\mathbb{N})^{\otimes 2(k-l)}} \Big)\\ = \ & (1-p) \big( S_1+S_2+S_3\big). \end{align*} It is now sufficient to show that \begin{align}\label{eq:SSS<}S_1+S_2+S_3\lesssim \max_{ H\subset G
\atop
e_H\geq1}n^{4v_G-3v_H}p^{4e_G-3e_H}. \end{align} Indeed, applying \eqref{varng} and \eqref{eq:SSS<} to \eqref{eq:dKfromprop} we get \begin{align*}
\frac{\sqrt{R_G}}{\mathrm{Var}[N_G]}&\lesssim \frac{\sqrt{1-p}
\sqrt{\displaystyle
\max_{ H\subset G \atop e_H\geq1}n^{4v_G-3v_H}p^{4e_G-3e_H}}}{(1-p)
\displaystyle
\max_{ H\subset G\atop e_H\geq1}n^{2v_G-v_H}p^{2e_G-e_H}}\\
&=\frac{\(
\displaystyle
\min_{ H\subset G\atop e_H\geq1}n^{v_H}p^{e_H}\)^{-3/2}}{
\sqrt{1-p} \(
\displaystyle
\min_{ H\subset G\atop e_H\geq1}n^{v_H}p^{e_H}\)^{-1}}\\ &=\((1-p)\min_{ H\subset G\atop e_H\geq1}n^{v_H}p^{e_H}\)^{-1/2}. \end{align*}
Thus $$
d_K (\tilde{N}_G,\mathcal{N} )\lesssim \frac{\sqrt{R_G}}{\mathrm{Var}[N_G]}\lesssim \((1-p)\min_{ H\subset G\atop e_H\geq1}n^{v_H}p^{e_H}\)^{-1/2}. $$
In order to estimate $S_1$, let us observe that \begin{align*}
\left\| g_k \star_k^{l} g_k\right\|^2_{\ell^2(\mathbb{N})^{\otimes (k-l)}}&=
\sum_{a''\in\mathbb{N}^{k-l}}\(
\sum_{a'\in\mathbb{N}^l}\(\sum_{a\in\mathbb{N}^{e_G-k}}\mathbf1_{E_G}\(a,a',a''\)\)^2\)^2
\\
&\approx\sum_{A \subset K_n
\atop e_K=k-l}\(
\sum_{ A \subset B \subset K_n \atop e_B =k}\(\sum_{
B \subset G' \subset K_n \atop G'\sim G}1\)^2\)^2
\\
&\approx\sum_{K \subset G \atop e_K =k-l}n^{v_K}\(
\sum_{K\subset H \subset G \atop e_H =k}n^{v_H-v_K}\(n^{v_G-v_H}\)^2\)^2
\\[10pt]
&\approx \max_{K\subset H \subset G
\atop e_K=k-l, \ \! e_H =k}n^{4v_G-2v_H -v_K}. \end{align*}
Hence we have \begin{align*}
S_1&\lesssim\sum_{0\leq l< k\leq e_G}p^{4e_G-3k+l}
\max_{K\subset H \subset G
\atop e_K=k-l, \ \! e_H =k}n^{4v_G-2v_H -v_K}\\
&=\sum_{0\leq l< k\leq e_G}\max_{K\subset H \subset G
\atop
e_K=k-l, \ \! e_H =k}n^{4v_G-2v_H -v_K}p^{4e_G-2e_H -e_K}\\ &\lesssim \max_{K\subset H\subset G\atop e_K\geq1}n^{4v_G-2v_H -v_K}p^{4e_G-2e_H -e_K}. \end{align*} For a fixed $p$, let $H_0\subset G$, $e_{H_0}\geq1$, be the subgraph of $G$ such that \begin{equation} \label{djlcn} n^{v_{H_0}}p^{e_{H_0}} = \min_{H \subset G, e_H\geq1}n^{v_H}p^{e_H}. \end{equation}
Then it is clear that \begin{align}\nonumber
S_1 & \lesssim
\max_{K \subset H \subset G \atop
e_K\geq1}n^{4v_G-2v_H -v_K}p^{4e_G-2e_H -e_K}
\\
&=n^{4v_G-3v_{H_0}}p^{4e_G-3e_{H_0}}\\\label{eq:max=max}
&=\max_{ H \subset G \atop
e_H \geq1}n^{4v_G-3v_H }p^{4e_G-3e_H }, \end{align} as required. We proceed similarly with the sum $S_2$. For $1\leq l< k\leq n$ we have \begin{align}
\nonumber
\left\| g_l \star_{l}^{l} g_k\right\|^2_{\ell^2(\mathbb{N})^{\otimes 2(k-l)}}
&\approx\sum_{c\in\mathbb{N}^{k-l}}\(
\sum_{b\in\mathbb{N}^l}\(\sum_{a\in\mathbb{N}^{e_G-l}}\mathbf1_{E_G}\(a,b\)\sum_{a'\in\mathbb{N}^{e_G-k}}\mathbf1_{E_G}\(a',b,c\)\)\)^2
\\ \label{dssds0}
&\approx \sum_{A \subset K_n
\atop
e_A ={k-l}}\(
\sum_{\substack{ A \subset B \subset K_n
\\
e_B =k}}\(
\sum_{\substack{
B \setminus A \subset G'' \subset K_n
\atop
G''\sim G}}
1
\sum_{\substack{
H\subset G' \subset K_n
\atop G'\sim G}}1
\)\)^2
\\
\label{dssds1}
&\lesssim
\sum_{K\subset G
\atop e_K=k-l}
n^{v_K} n^{v_K}
\(
\sum_{\substack{K\subset H \subset G,
\ \!
H' \subset G
\atop e_H =k, \ \! e_{H'}=l}}n^{v_H -v_K}\(n^{v_G-v_{H'}}n^{v_G-v_H}\)\)^2
\\[10pt]
\label{dssds2}
&\lesssim \max_{\substack{K, H'\subset G\\e_K=k-l, \ \! e_{H'}=l}}n^{4v_G-2v_{H'}-v_K}, \end{align} where $H'$ in \eqref{dssds1} stands for $B \setminus A$ in \eqref{dssds0}, whereas in \eqref{dssds2} the sum over $H'$ extends to all $H'\subset G$ such that $e_{H'}=l$. It follows that \begin{align*}
S_2 &\lesssim \sum_{1\leq l<k\leq e_G}p^{4e_G-k-l}\max_{\substack{K, H'\subset G
\atop e_K=k-l, \ \! e_{H'}=l}}n^{4v_G-2v_{H'}-v_K}\\
&=\sum_{1\leq l<k\leq e_G}\max_{\substack{K, H'\subset G
\atop
e_K =k-l, \ \! e_{H'}=l}}n^{4v_G-2v_{H'}-v_K}p^{4e_G-2v_{H'}-e_K}
\\
&\lesssim\max_{\substack{K', H'\subset G \atop
e_{K'}, \ \!
e_{H'}\geq1}}n^{4v_G-2v_{H'}-v_{K'}}p^{4e_G-2v_{H'}-e_{K'}}\\ &=n^{4v_G-3v_{H_0}}p^{4e_G-3e_{H_0}}\\ &=\max_{ H\subset G\atop e_H\geq1}n^{4v_G-3v_H}p^{4e_G-3e_H}, \end{align*}
where $H_0$ is defined in \eqref{djlcn}.
Finally, we pass to estimates of $S_3$. For $1\leq l< k\leq n$ we have \begin{eqnarray*}
\lefteqn{
\! \! \! \! \! \! \! \! \!
\left\| g_k\star_{l}^{l} g_k\right\|^2_{\ell^2(\mathbb{N})^{\otimes (k-l)}} \approx\sum_{c,c'\in\mathbb{N}^{k-l}}\(
\sum_{b\in\mathbb{N}^l}\(\sum_{a\in\mathbb{N}^{e_G-k}}\mathbf1_{E_G}\(a,b,c\)\)\(\sum_{a'\in\mathbb{N}^{e_G-k}}\mathbf1_{E_G}\(a',b,c'\)\)\)^2
}
\\
& \approx & \sum_{\substack{A ,A'\subset K_n
\atop e_A =e_{A'}=k-l}}\(
\sum_{\substack{ B \subset K_n
\atop
e_B =l, \ \! e_{ A \cap B }=e_{A' \cap B}=0}}\(\sum_{\substack{ A \cup B \subset G'\subset K_n
\atop G'\sim G}}1\)
\(\sum_{\substack{A'\cup B\subset G'' \subset K_n
\atop G''\sim G}}1\)\)^2
\\
&\approx & \sum_{\substack{K,K',H\subset G\\e_K =e_{K'}=k-l, \ \! e_H =l\\e_{K\cap H}=e_{K'\cap H}=0}}
\sum_{\substack{
A, A' \subset K_n \\
A\sim K \\ A'\sim K'}}\(
\sum_{\substack{
B \subset K_n \\
B \sim H \\ A \cap B \sim K\cap H \\ A' \cap B \sim K'\cap H}}
\(\sum_{\substack{ A \cup B \subset G'\subset K_n
\atop G'\sim G}}1\)\(\sum_{\substack{A' \cup B \subset G''\subset K_n
\atop G''\sim G}}1\)\)^2
\\
&\approx & \sum_{\substack{K,K',H\subset G\\e_K =e_{K'}=k-l, \ \! e_H =l\\e_{K\cap H}=e_{K'\cap H}=0}}
\ \sum_{\substack{
A, A' \subset K_n \\
A\sim K \\ A'\sim K'}}\(
\sum_{\substack{
B \subset K_n \\
B \sim H \\ A \cap B \sim K\cap H \\ A' \cap B \sim K'\cap H}}
\(
n^{v_G-v_{A \cup B}}
\)\(
n^{v_G-v_{A' \cup B}}
\)\)^2. \end{eqnarray*} Next, we note that given $A, A' \subset K_n$ it takes $$ v_B - v_{A\cap B} - v_{A'\cap B} + v_{A\cap A' \cap B} = v_H - v_{K\cap H} - v_{K'\cap H} + v_{A\cap A' \cap B} $$ vertices to create any subgraph $B \sim H$ such that $A \cap B \sim K\cap H$ and $A' \cap B \sim K'\cap H$, with the bound $$ v_{A\cap A' \cap B} \leq \frac{1}{2} v_{A\cap A'} + \frac{1}{2} v_{A'\cap B} = \frac{1}{2} ( v_{A\cap A'} + v_{K'\cap H} ). $$
Hence we have \begin{eqnarray*}
\lefteqn{
\left\| g_k\star_{l}^{l} g_k\right\|^2_{\ell^2(\mathbb{N})^{\otimes (k-l)}}
}
\\
&\lesssim &
\! \! \! \! \!
\sum_{\substack{K,K',H \subset G\\e_K =e_{K'}=k-l, \ \! e_H =l\\e_{K\cap H}=e_{K'\cap H}=0}}
\ \sum_{\substack{
A, A' \subset K_n \\
A \sim K \\ A'\sim K'
}}
\! \! \! \!
\(
n^{v_H-v_{K\cap H}-v_{K'\cap H}+
( v_{A\cap A'} + v_{K'\cap H} ) / 2
}\(n^{v_G-v_{K\cup H}}\)\(n^{v_G-v_{K'\cup H}}\)\)^2. \end{eqnarray*}
In order to estimate the above sum using powers of $n$, we need to consider the possible intersections $A \cap A'$ for $A, A' \subset K_n$, as follows: \begin{eqnarray}
\nonumber
\lefteqn{
\! \! \! \! \! \! \! \! \! \! \! \!
\sum_{\substack{K,K',H\subset G\\e_K =e_{K'}=k-l, \ \! e_H =l\\e_{K\cap H}=e_{K'\cap H}=0}}
\ \sum_{\substack{
A, A' \subset K_n \\
A \sim K \\ A'\sim K'
}}
n^{4v_G+2v_H-2v_{K\cap H}-v_{K'\cap H}
+v_{A \cap A'} -2v_{K\cup H}-2v_{K'\cup H}}
}
\\
\nonumber
&\lesssim & \sum_{\substack{K,K',H\subset G\\e_K =e_{K'}=k-l, \ \! e_H =l\\e_{K \cap H}=e_{K' \cap H}=0}}\ \sum_{i=0}^{v_K}n^{v_K+v_{K'}-i} \ \! n^{4v_G+2v_H-2v_{K \cap H} - v_{K' \cap H}
+ i-2v_{K \cup H}-2v_{K' \cup H}}
\\
\label{aux3}
&\lesssim & \sum_{\substack{K,K',H\subset G\\e_K =e_{K'}=k-l, \ \! e_H =l\\e_{K \cap H}=e_{K' \cap H}=0}}
\ \! n^{v_K+v_{K'}+4v_G+2v_H-2v_{K \cap H}-v_{K' \cap H}-2v_{K \cup H}-2v_{K' \cup H}}. \end{eqnarray} Furthermore we have \begin{eqnarray*}
\lefteqn{
\! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \!
\! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \!
v_K+v_{K'}+4v_G+2v_H-2v_{K \cap H}- v_{K' \cap H} - 2v_{K \cup H}-2v_{K' \cup H}
}
\\ &=&4v_G-v_K -v_H -v_{K'\cup H}, \end{eqnarray*} so the sum \eqref{aux3} can be estimated as \begin{align*} \sum_{\substack{K,K',H \subset G\\e_K =e_{K'}=k-l, \ \! e_H =l\\e_{K \cap H}=e_{K' \cap H}=0}}\ n^{4v_G-v_K -v_H -v_{K'\cup H}}\lesssim \max_{\substack{K,H,L\subset G\\e_K =k-l, \ \! e_H =l,\ \! e_L=k}} \ \! n^{4v_G-v_K -v_H -v_L }, \end{align*} from which it follows \begin{align*} S_3&\lesssim \sum_{1\leq l<k\leq e_G}p^{4e_G-2k} \max_{\substack{K,H,L\subset G\\e_K =k-l, \ \! e_H =l,\ e_L =k}}\ n^{4v_G-v_K -v_H -v_L }\\ &= \sum_{1\leq l<k\leq e_G} \max_{\substack{K,H,L\subset G\\e_K =k-l, \ \! e_H =l,\ e_L =k}}\ n^{4v_G-v_K -v_H -v_L }p^{4e_G-e_K -e_H -e_L }\\ &\lesssim\max_{\substack{K,H,L \subset G\\e_K , e_H, e_L \geq1}}\ \! n^{4v_G-v_K -v_H-v_L }p^{4e_G-e_K -e_H-e_L }\\ &\leq n^{4v_G-3v_{H_0}}p^{4e_G-3e_{H_0}}\\ &= \max_{\substack{H\subset G\\e_H \geq1}}\ \! n^{4v_G-3v_H }p^{4e_G-3e_H }, \end{align*} which ends the proof. \end{Proof} In the next corollary
we note that Theorem~\ref{thm:main} simplifies
if we narrow our attention to $p_n$ depending of the complete
graph size $n$ and close to $0$ or to $1$. \begin{corollary} \label{c0}
Let $G$ be a graph without separated vertices. For $p_n<c<1$, $n\geq 1$, we have \begin{equation} \label{eq}
d_K \big(\widetilde{N}^G_n,\mathcal{N} \big) \lesssim
\( \min_{\substack{ H\subset G\\e_H\geq1}}\big\{n^{v_H}p_n^{e_H}\big\}\)^{-1/2}. \end{equation}
On the other hand, for $p_n>c>0$, $n\geq 1$, it holds
\begin{equation}
\label{dk}
d_K \big(\widetilde{N}^G_n,\mathcal{N} \big)\lesssim
\frac{1}{n\sqrt{1-p_n}}. \end{equation} \end{corollary}
As a consequence of Corollary~\ref{c0}
it follows that if
$$
np_n^\beta \rightarrow \infty \ \mbox{ and }\ n^2(1-p_n)\rightarrow\infty,
$$
where $\beta :=\max \big\{e_H/v_H \ : \ H\subset G\big\}$,
then we have the convergence of the
renormalized subgraph count $\big(\widetilde{N}^G_n\big)_{n\geq 1}$
to $\mathcal{N}$ in distribution as $n$ tends to infinity,
which recovers the sufficient condition in \cite{rucinski}.
When $p\approx n^{-\alpha}$, $\alpha>0$,
Corollary~\ref{c0} also shows that \begin{align}\label{eq:dFNwithalpha}
d_K \big(\widetilde{N}^G_n,\mathcal{N} \big)
&\lesssim
\(\min_{\substack{ H\subset G\\e_H\geq1}}\big\{n^{v_H-\alpha e_H}\big\}\)^{-1/2}, \end{align}
and in order for the above bound \eqref{eq:dFNwithalpha} to tend to zero as $n$ goes to infinity, we should have \begin{equation}
\label{alpha}
\alpha<\min_{H\subset G}\frac{v_H}{e_H}=:\frac1\beta. \end{equation} The next Corollary~\ref{cor1.3} of Theorem~\ref{thm:main} and \eqref{dk} deals with cycle graphs with $r$ vertices, $r\geq3$. When $G$ is a triangle it recovers the Kolmogorov bounds of \cite{roellin2} as in Corollary~\ref{c} above. \begin{corollary} \label{cor1.3} Let $G$ be a cycle graph with $r$ vertices, $r\geq3$, and $c\in (0,1)$. We have \begin{align*}
d_K \big(\widetilde{N}^G_n,\mathcal{N}\big)\lesssim
\left\{\begin{array}{ll}
\displaystyle
\frac{1}{n\sqrt{1-p_n}} &\mbox{if } \ \displaystyle
0<c< p_n,
\\
\\
\displaystyle
\frac{1}{n\sqrt{p_n}} &
\displaystyle
\mbox{if } \ n^{-(r-2)/(r-1)} < p_n \leq c,
\\
\\
\displaystyle
\frac{1}{(np_n)^{r/2}}&
\displaystyle
\mbox{if } \ 0 < p_n \leq n^{-(r-2)/(r-1)}.
\end{array}\right. \end{align*} \end{corollary} \begin{Proof}
The smallest number of vertices of subgraphs $H$ of $G$ having $k$ edges,
$k<r$, is realised for a linear subgraph having $k+1$ vertices,
which yields
$$
\min_{\substack{ H\subset G
\atop
1 \leq e_H < r}}
\big\{n^{v_H} p_n^{e_H}\big\}
=
\min_{1\leq k < r}n^{k+1}p_n^k
=n\min_{1\leq k < r} (n p_n )^k
= \min (n^2p_n , ( n p_n )^{r-1}),
$$
hence
\begin{align*}
\min_{\substack{ H\subset G\\e_H\geq1}} \big\{n^{v_H}p_n^{e_H}\big\}
=\min\big\{n^2p_n,(np_n)^{r-1} , (np_n)^r
\big\}
=\left\{\begin{array}{ll}
\displaystyle
n^2p_n &\mbox{if } \ \displaystyle
n^{-(r-2)/(r-1)} < p_n \leq c,
\\
\\ \displaystyle (np_n)^r &\mbox{if } \ \displaystyle 0 < p_n \leq n^{-(r-2)/(r-1)},
\end{array}\right. \end{align*} which concludes the proof by \eqref{eq} and \eqref{dk}. \end{Proof} In case $p_n\approx n^{-\alpha}$ we should have $\alpha\in (0,1)$ by
\eqref{alpha},
Corollary~\ref{cor1.3} also shows that \begin{align*}
d_K \big(\widetilde{N}^G_n,\mathcal{N} \big)\lesssim
\left\{\begin{array}{ll}
\displaystyle
n^{-1+\alpha/2}
\approx \frac{1}{n\sqrt{p_n}} &
\displaystyle
\mbox{if } \ 0 < \alpha\leq\frac{r-2}{r-1},
\\
\\
\displaystyle
n^{-r(1-\alpha)/2}
\approx \frac{1}{(np_n)^{r/2}}&
\displaystyle
\mbox{if } \ \frac{r-2}{r-1}<\alpha<1.
\end{array}\right. \end{align*} when $G$ is a cycle graph with $r$ vertices, $r\geq3$. In the particular case $r=3$ where $G$ is a triangle, this improves on the Kolmogorov bounds in Theorem~1.1 of \cite{reichenbachsAoP}. \\
In the case of complete graphs, the next corollary also covers the case of triangles. \begin{corollary} \label{cor1.4}
Let $G$ be a complete graph with $k$ vertices,
$r\geq3$, and $c\in (0,1)$. We have \begin{align*}
d_K \big(\widetilde{N}^G_n,\mathcal{N} \big)\lesssim
\left\{\begin{array}{ll}
\displaystyle
\frac{1}{n\sqrt{1-p_n}} &\mbox{if } \ \displaystyle
c< p_n <1,
\\
\\
\displaystyle
\frac{1}{n\sqrt{p_n}} &
\displaystyle
\mbox{if } \
n^{-2/(r+1)} < p_n \leq c,
\\
\\
\displaystyle
\frac{1}{n^{r/2}p_n^{r(r-1)/4}}& \displaystyle
\mbox{if } \ 0 < p_n \leq n^{-2/(r+1)}.
\end{array}\right. \end{align*} \end{corollary} \begin{Proof}
The greatest number of edges of subgraphs of $G$ having $k$ vertices,
$2\leq k\leq v_G$,
is realised for a complete graph having $k\choose 2$ edges,
which shows that
$$
\min_{\substack{ H\subset G\\e_H\geq1}}\big\{n^{v_H}p_n^{e_H}\big\}=
\min_{1\leq k\leq r}n^kp_n^{{k\choose 2}}.$$ On the other hand, from the equality $$ \frac{n^{k+1}p_n^{{k+1\choose 2}}}{n^k p_n^{{k\choose 2}}}=n p_n^k, $$ we note that if the minimum was realised with $k$ vertices where $1<k<r$, we would have $n p_n^{k-1}\leq 1$ and $np_n^k \geq1$, which would lead to $p_n \geq 1$, which is not possible. Therefore we have \begin{align*}
\min_{\substack{ H\subset G\\e_H\geq1}}\big\{n^{v_H}p_n^{e_H}\big\}=\min\left\{
n^2p_n,n^rp_n^{{r\choose 2}}\right\}=\left\{\begin{array}{ll}
\displaystyle
n^2p_n &\mbox{if } \ \displaystyle
n^{-2/(r+1)} < p_n \leq c,
\\
\\ \displaystyle
n^rp_n^{r(r-1)/2}&\mbox{if } \ \displaystyle
0 < p_n \leq n^{-2/(r+1)},
\end{array}\right. \end{align*} and we conclude the proof by \eqref{eq} and \eqref{dk}. \end{Proof} When $p_n\approx n^{-\alpha}$ with $\alpha\in (0, 2/(r-1))$ by \eqref{alpha}, Corollary~\ref{cor1.4} shows that \begin{align*}
\min_{\substack{ H\subset G\\e_H\geq1}}\{n^{v_H-\alpha e_H}\}=\min
\big\{ n^{2-\alpha},n^{r-{r\choose 2}\alpha} \big\}=\left\{\begin{array}{ll} \displaystyle
n^{2-\alpha/2} &\mbox{if } \ \displaystyle
0 < \alpha\leq\frac{2}{r+1},
\\
\\
\displaystyle
n^{r-r(r-1)\alpha/2}&\mbox{if } \ \displaystyle
\frac{2}{r+1} \leq \alpha<\frac2{r-1},\end{array}\right. \end{align*}
hence by \eqref{eq} we find \begin{align*} d_K \big(\widetilde{N}^G_n,\mathcal{N} \big)\lesssim
\left\{\begin{array}{ll}
\displaystyle
n^{-1+\alpha/2} \approx \frac{1}{n\sqrt{p_n}} &
\displaystyle
\mbox{if } \ 0 < \alpha\leq\frac{2}{r+1},
\\
\\
\displaystyle
n^{-r/2+r(r-1)\alpha/4}
\approx \frac{1}{n^{r/2}p_n^{r(r-1)/4}}& \displaystyle
\mbox{if } \ \frac{2}{r+1} \leq \alpha<\frac{2}{r-1}.\end{array}\right. \end{align*} Finally, the next corollary deals with the important class of graphs which have a tree structure. \begin{corollary}
\label{cor1.5}
Let $G$ be any tree (a connected graph without cycles) with $r$ edges,
and $c\in (0,1)$.
We have \begin{align*}
d_K \big(\widetilde{N}^G_n,\mathcal{N} \big)\lesssim
\left\{\begin{array}{ll}
\displaystyle
\frac{1}{n\sqrt{1-p_n}} &\mbox{if } \ \displaystyle
c< p_n<1,
\\
\\
\displaystyle
\frac{1}{n\sqrt{p_n}} & \displaystyle
\mbox{if } \ \frac{1}{n} < p_n \leq c,\\ \\
\displaystyle
\frac{1}{n^{(r+1)/2}p_n^{r/2}}& \displaystyle
\mbox{if } \ 0 < p_n \leq \frac{1}{n}.
\end{array}\right. \end{align*} \end{corollary} \begin{Proof}
We have
$$
\min_{\substack{ H\subset G\\e_H\geq1}}\big\{n^{v_H} p_n^{e_H}\big\}=\min_{1\leq k\leq r}n^{k+1}p_n^k =n\min_{1\leq k\leq r} (n p_n )^k.$$
The smallest number of vertices for a subgraph
of a tree $G$ having $k$ edges, $k\leq r$, is
realised for a subtree having $k+1$ vertices, hence since $np_n$
can be either less or greater than $1$, which gives $$\min_{\substack{ H\subset G\\e_H\geq1}}\big\{n^{v_H}p_n^{e_H}\big\}=n\min\big\{np_n,\(np_n\)^r\big\}=\left\{\begin{array}{ll} \displaystyle n^2 p_n & \mbox{if } \ \displaystyle \frac{1}{n} < p_n \leq c,\\ \\ \displaystyle n^{r+1}p_n^r &\mbox{if } \ \displaystyle 0 < p_n \leq \frac{1}{n},
\end{array}\right.$$ as required, and we conclude by \eqref{eq} and \eqref{dk} . \end{Proof} In case $p_n\approx n^{-\alpha}$ with $\alpha \in ( 0, 1+1/r)$, we have $\beta=\max \{e_H/v_H \ : \ H\subset G\} = r/(r+1)$ hence $$\min_{\substack{ H\subset G\\e_H\geq1}}\{n^{v_H-\alpha e_H}\}=n\min\{n^{1-\alpha},\(n^{1-\alpha}\)^r\}=\left\{\begin{array}{ll} \displaystyle n^{2-\alpha} & \mbox{if } \ \displaystyle 0 < \alpha\leq1,\\ \\ \displaystyle n^{r+1-r\alpha} &\mbox{if } \ \displaystyle 1 \leq \alpha<1+\frac1r,\end{array}\right.$$
which shows by \eqref{eq} that \begin{align*}
d_K \big(\widetilde{N}^G_n,\mathcal{N} \big)\lesssim
\left\{\begin{array}{ll}
\displaystyle
n^{-1+\alpha/2}
\approx \frac{1}{n\sqrt{p_n}} & \displaystyle
\mbox{if } \ 0 < \alpha\leq1,
\\ \\
\displaystyle
n^{-(r+1-r\alpha)/2}
\approx \frac{1}{n^{(r+1)/2}p_n^{r/2}}& \displaystyle
\mbox{if } \ 1 \leq \alpha<1+\frac{1}{r}.
\end{array}\right. \end{align*}
\footnotesize
\def$'${$'$} \def\polhk#1{\setbox0=\hbox{#1}{\ooalign{\hidewidth
\lower1.5ex\hbox{`}\hidewidth\crcr\unhbox0}}}
\def\polhk#1{\setbox0=\hbox{#1}{\ooalign{\hidewidth
\lower1.5ex\hbox{`}\hidewidth\crcr\unhbox0}}} \def$'${$'$}
\end{document} | arXiv |
\begin{document}
\begin{abstract} For a finite abelian $p$-group $A$ and a subgroup $\Gamma\le\Aut(A)$, we say that the pair $(\Gamma,A)$ is fusion realizable if there is a saturated fusion system $\calf$ over a finite $p$-group $S\ge A$ such that $C_S(A)=A$, $\autf(A)=\Gamma$ as subgroups of $\Aut(A)$, and $A\nnsg\calf$. In this paper, we develop tools to show that certain representations are not fusion realizable in this sense. For example, we show, for $p=2$ or $3$ and $\Gamma$ one of the Mathieu groups, that the only $\F_p\Gamma$-modules that are fusion realizable (up to extensions by trivial modules) are the Todd modules and in some cases their duals. \end{abstract}
\title{Nonrealizability of certain representations in fusion systems}
Fix a prime $p$. A saturated fusion system over a finite $p$-group $S$ is a category whose objects are the subgroups of $S$, and whose morphisms are injective homomorphisms between those subgroups that satisfy certain axioms formulated by Puig \cite{Puig}, motivated in part by the Sylow theorems for finite groups. See Definition \ref{d:s.f.s.} for more details.
Consider a pair $(\Gamma,A)$, where $A$ is a finite abelian $p$-group and $\Gamma\le\Aut(A)$ is a group of automorphisms. We say that $(\Gamma,A)$ is \emph{fusion realizable} if there is a saturated fusion system $\calf$ over some finite $p$-group $S\ge A$ such that $C_S(A)=A$, $A\nnsg\calf$, and $\autf(A)=\Gamma$ as groups of automorphisms of $A$. We also say that $(\Gamma,A)$ is \emph{realized by $\calf$} in this situation.
In an earlier paper \cite{O-todd}, we considered the special case where $p=3$, $O^{3'}(\Gamma)\cong2M_{12}$, $M_{11}$, or $A_6$, and $A$ is an elementary abelian $3$-group of rank $6$, $5$, or $4$, respectively, and classified the saturated fusion systems that realize some pair $(\Gamma,A)$ of this form. In this paper, we take the opposite approach, and develop tools that we use to show that ``most'' $\F_p\Gamma$-modules are not fusion realizable; i.e., cannot be realized by any saturated fusion system.
For example, in Definition \ref{d:K&R} and Proposition \ref{p:not.str.cl.3a}, we define certain sets $\RR{T}{A}$, for $A$ an abelian $p$-group and $T\le\Aut(A)$ a $p$-subgroup, with the property that $\RR{T}{A}\ne\emptyset$ if there is a fusion realizable pair $(\Gamma,A)$ where $T\in\sylp{\Gamma}$. As one of the consequences of this proposition, we show (Corollary \ref{c:not.str.cl.2}) that if $A$ is elementary abelian and $(\Gamma,A)$ is fusion realizable, then there is $m\ge1$ and an elementary abelian $p$-subgroup $B\le\Gamma$ of rank $m$ such that for each $g\in B^\#$, the action of $g$ on $A$ has at most $m$ nontrivial Jordan blocks.
Theorems \ref{ThA} and \ref{ThB} as stated below are our main applications so far of these tools. For example, as one special case of Theorem \ref{ThA}, we show that the Golay modules for $M_{22}$ and $M_{23}$ are not fusion realizable. In contrast, the Todd modules for $M_{22}$ and $M_{23}$ (dual to the Golay modules) are realized by the fusion systems of the Fischer groups $\Fi_{22}$ and $\Fi_{23}$, and the Golay module for $\Aut(M_{22})$ (a case not covered by the statement of Theorem \ref{ThA}) is realized by the fusion system of the Conway group $\Co_2$.
\begin{Thmm}[Theorem \ref{t:M11-24}] \label{ThA} Fix a prime $p$, and let $\Gamma$ be a finite group such that $\Gamma_0=O^{p'}(\Gamma)$ is quasisimple and $\Gamma_0/Z(\Gamma_0)$ is one of Mathieu's five sporadic groups. Let $A$ be an $\F_p\Gamma$-module such that $(\Gamma,A)$ is fusion realizable, and set $A_0=[\Gamma_0,A]/C_{[\Gamma_0,A]}(\Gamma_0)$. Then either \begin{itemize}
\item $p=2$, and $A_0$ is the Todd module for $\Gamma\cong M_{22}$, $M_{23}$, or $M_{24}$ or the Golay module for $\Gamma\cong M_{24}$; or
\item $p=3$, $\Gamma\cong M_{11}$, $M_{11}\times C_2$, or $2M_{12}$, and and $A_0$ is the Todd module or Golay module for $\Gamma_0$; or
\iffalse $\Gamma\cong M_{11}$ or $2M_{12}$; or \fi
\item $p=11$, $\Gamma_0\cong 2M_{12}$ or $2M_{22}$, $\Gamma/Z(\Gamma_0)\cong\Aut(M_{12})\times C_5$ or $\Aut(M_{22})\times C_5$, and $A_0$ is a $10$-dimensional simple $\F_{11}\Gamma$-module.
\end{itemize} \end{Thmm}
When $p=2$ or $3$, the nonrealizability of $(\Gamma,A)$ in Theorem \ref{ThA} is shown in all cases by proving that the set $\scrr_T(A)$ mentioned above is empty for $T\in\sylp\Gamma$. For $p>3$, it follows from results in \cite{indp2}.
Theorem \ref{ThB} is a restatement of a theorem of O'Nan \cite[Lemma 1.10]{ONan} in the context of fusion systems, included here to illustrate how these methods apply when $A$ is not elementary abelian. Its proof is similar to O'Nan's, but is shortened by using results in Section
\ref{s:A<|F}.
\begin{Thmm}[Theorem \ref{t:Alp}] \label{ThB} Assume, for some $n\ge3$, that $A=\gen{v_1,v_2,v_3}\cong C_{2^n}\times C_{2^n}\times C_{2^n}$, and that $S=A\gen{s,t}$ is an extension of $A$ by $D_8$ with action as described in Table \ref{tbl:D8onA}. Then $A$ is normal in every saturated fusion system over $S$. Thus there is no $\Gamma\le\Aut(A)$ with $\Aut_S(A)\in\syl2{\Gamma}$ such that $(\Gamma,A)$ is fusion realizable. \end{Thmm}
The paper is organized as follows. After summarizing in Section \ref{s:background} the basic definitions and properties of fusion systems that will be needed, we state and prove our
main criteria for fusion realizability in Section \ref{s:A<|F}. We then look at representations of Mathieu groups in Section \ref{s:Mathieu} and prove Theorem \ref{ThA} (Theorem \ref{t:M11-24}), and study Alperin's $2$-groups in Section \ref{s:Alp} and prove Theorem \ref{ThB} (Theorem \ref{t:Alp}). We finish with three appendices: Appendix \ref{s:JV(x)} with some general results on representations, and Appendices \ref{s:Todd-F2} and \ref{s:3M22} where we set up notation to work with the Golay modules for $M_{22}$ and $M_{23}$, and the $6$-dimensional $\F_43M_{22}$-module, respectively.
\noindent\textbf{Notation and terminology:} Most of our notation for working with groups is fairly standard. When $P\le G$ and $x\in N_G(P)$, we let $c_x^P\in\Aut(P)$ denote conjugation by $x$ on the left: $c_x^P(g)=\9xg=xgx^{-1}$. Also, $\sylp{G}$ is the set of Sylow $p$-subgroups of a finite group $G$, and $G^\#=G\smallsetminus\{1\}$. Other notation used here includes: \begin{itemize}
\item $E_{p^m}$ is always an elementary abelian $p$-group of rank $m$;
\item $A\rtimes B$ and $A.B$ denote a semidirect product and an arbitrary extension of $A$ by $B$; and
\item $2M_{12}$, $nM_{22}$, and $2A_4$ denote (nonsplit) central extensions of $C_2$ or $C_n$ by the groups $M_{12}$, $M_{22}$, or $A_4$, respectively.
\end{itemize} Also, composition of functions and homomorphisms is always written from right to left.
\noindent\textbf{Thanks:} The author would like to thank the Newton Institute in Cambridge for its hospitality while he was finishing the writeup of this paper. He would also like to thank the referee for carefully reading the paper and making several suggestions for improvements.
\section{Background definitions and results} \label{s:background}
We recall here some of the basic definitions and properties of saturated fusion systems. Our main reference is \cite{AKO}, although most of the results are also shown in \cite{Craven}.
A \emph{fusion system} $\calf$ over a finite $p$-group $S$ is a category whose objects are the subgroups of $S$, such that for each $P,Q\le S$, \begin{itemize} \item $\Hom_S(P,Q) \subseteq\homf(P,Q)\subseteq\Inj(P,Q)$; and
\item every morphism in $\calf$ is the composite of an $\calf$-isomorphism followed by an inclusion.
\end{itemize}
Here, $\Hom_S(P,Q) = \{c_g\in\Hom(P,Q) \,|\, g\in S,~ \9gP\le Q \}$. We also write $\isof(P,Q)$ for the set of $\calf$-isomorphisms from $P$ to $Q$, and $\autf(P)=\isof(P,P)$.
In order for fusion systems to be very useful, we need to assume they satisfy the following saturation properties, motivated by the Sylow theorems and first formulated by Puig \cite{Puig}.
\begin{Defi} \label{d:s.f.s.} Let $\calf$ be a fusion system over a finite $p$-group $S$. \begin{enuma} \item Two subgroups $P,Q\le{}S$ are \emph{$\calf$-conjugate} if $\isof(P,Q)\ne\emptyset$, and two elements $x,y\in S$ are $\calf$-conjugate if there is $\varphi\in\homf(\gen{x},\gen{y})$ such that $\varphi(x)=y$. The $\calf$-conjugacy classes of $P\le S$ and $x\in S$ are denoted $P^\calf$ and $x^\calf$, respectively.
\item A subgroup $P\le S$ is \emph{fully normalized} in $\calf$
(\emph{fully centralized} in $\calf$) if $|N_S(P)|\ge|N_S(Q)|$ for each
$Q\in P^\calf$ ($|C_S(P)|\ge|C_S(Q)|$ for each $Q\in P^\calf$).
\item The fusion system $\calf$ is \emph{saturated} if it satisfies the following two conditions:
\begin{itemize}
\item \textup{(Sylow axiom)} For each subgroup $P\le S$ fully normalized in $\calf$, $P$ is fully centralized and $\Aut_S(P)\in\sylp{\autf(P)}$.
\item \textup{(extension axiom)} For each isomorphism $\varphi\in\isof(P,Q)$ in $\calf$ such that $Q$ is fully centralized in $\calf$, $\varphi$ extends to a morphism $\4\varphi\in\homf(N_\varphi,S)$ where
\[ N_\varphi = \{ g\in N_S(P) \,|\, \varphi c_g \varphi^{-1} \in
\Aut_S(Q) \}. \] \end{itemize} \end{enuma} \end{Defi}
Definition \ref{d:s.f.s.} is the definition first given in \cite{BLO2}, and is used here since it seems to be the easiest to apply for our purposes. It is slightly different from that given in \cite[Definition I.2.2]{AKO}, but the two are equivalent by \cite[Proposition I.2.5]{AKO}. Its equivalence with Puig's original definition is shown in \cite[Proposition I.9.3]{AKO}.
As one example, the fusion system of a finite group $G$ with respect to a Sylow $p$-subgroup $S\le G$ is the category $\calf_S(G)$ whose objects are the subgroups of $S$, and whose morphisms are those homomorphisms between subgroups that are induced by conjugation in $G$. It is clearly a fusion system and was shown by Puig to be saturated. (See \cite[Proposition 1.3]{BLO2} for a proof of saturation in terms of Definition \ref{d:s.f.s.}.)
We will also need to work with certain classes of subgroups in a fusion system. Recall, for a pair of finite groups $H<G$, that $H$ is \emph{strongly
$p$-embedded in $G$} if $p\mathrel{\big|}|H|$, and $p\nmid|H\cap\9gH|$ for $g\in G\smallsetminus H$.
\begin{Defi} \label{d:subgroups} Let $\calf$ be a fusion system over a finite $p$-group $S$. For $P\le S$, \begin{itemize}
\item $P$ is \emph{$\calf$-centric} if $C_S(Q)\le Q$ for each $Q\in P^\calf$;
\item $P$ is \emph{$\calf$-essential} if $P$ is $\calf$-centric and fully normalized in $\calf$ and the group $\outf(P)=\autf(P)/\Inn(P)$ contains a strongly $p$-embedded subgroup;
\item $P$ is \emph{weakly closed in $\calf$} if $P^\calf=\{P\}$;
\item $P$ is \emph{strongly closed in $\calf$} if for each $x\in P$, $x^\calf\subseteq P$;
\item $P$ is \emph{central} in $\calf$ if each $\varphi\in\homf(Q,R)$, for $Q,R\le S$, extends to some
$\4\varphi\in\homf(QP,RP)$ such that $\4\varphi|_P=\Id_P$; and
\item $P$ is \emph{normal in $\calf$} ($P\nsg\calf$) if each morphism in $\calf$ extends to a morphism that sends $P$ to itself. \end{itemize} We also let $\calf^c$ and $\EE\calf$ be the sets of subgroups of $S$ that are $\calf$-centric or $\calf$-essential, respectively. \end{Defi}
The following is one version of the Alperin-Goldschmidt fusion theorem for fusion systems.
\begin{Thm}[{\cite[Theorem I.3.6]{AKO}}] \label{t:AFT} Let $\calf$ be a saturated fusion system over a finite $p$-group $S$. Then each morphism in $\calf$ is a composite of restrictions of automorphisms $\alpha\in\autf(R)$ for $R\in\EE\calf\cup\{S\}$. \end{Thm}
The next proposition is more technical.
\begin{Prop}[{\cite[Lemma I.2.6(c)]{AKO}}] \label{p:Hom(NSP,S)} Let $\calf$ be a saturated fusion system over a finite $p$-group $S$. Then for each $P\le S$, and each $Q\in P^\calf$ fully normalized in $\calf$, there is $\psi\in\homf(N_S(P),S)$ such that $\psi(P)=Q$. \end{Prop}
Normal $p$-subgroups in a fusion system are strongly closed, but the converse does not always hold. The following is one situation where it does hold. For a much more detailed list of conditions under which strongly closed subgroups in a fusion system are normal, see \cite[Theorem B]{Kizmaz}.
\begin{Lem}[{\cite[Corollary I.4.7(a)]{AKO}}] \label{l:s.cl.=>normal} Let $\calf$ be a saturated fusion system over a finite $p$-group $S$. If $A\nsg S$ is an abelian subgroup that is strongly closed in $\calf$, then $A\nsg\calf$. \end{Lem}
We next look at centralizers of $p$-subgroups in fusion systems. Normalizer subsystems are defined in a similar way (see \cite[\S I.5]{AKO}), but will not be needed here.
\begin{Defi} \label{d:NF(Q)} Let $\calf$ be a fusion system over a finite $p$-group $S$. For each $Q\le S$, the \emph{centralizer fusion subsystem} $C_\calf(Q)\le\calf$ is the fusion subsystem over $C_S(Q)$ defined by setting
\[ \Hom_{C_\calf(Q)}(P,R) = \bigl\{ \varphi|_P \,\big|\,
\varphi\in\homf(PQ,RQ),~ \varphi(P)\le R,~ \varphi|_Q=\Id_Q
\bigr\}. \] \end{Defi}
Note that a subgroup $Q\le S$ is central in $\calf$ if and only if $C_\calf(Q)=\calf$.
\begin{Thm}[{\cite[Theorem I.5.5]{AKO}}] \label{t:NF(Q)} Let $\calf$ be a saturated fusion system over a finite $p$-group $S$, and fix $Q\le S$. Then $C_\calf(Q)$ is saturated if $Q$ is fully centralized in $\calf$. \end{Thm}
Weakly closed abelian subgroups play a central role in the paper, and the following lemma is of crucial importance when working with them.
\begin{Lem} \label{l:A-w.cl.} Let $\calf$ be a saturated fusion system over a finite $p$-group $S$, and assume $A\le S$ is an abelian subgroup that is weakly closed in $\calf$. \begin{enuma}
\item If $R\le S$ is fully normalized and $\calf$-conjugate to some $Q\le A$, then $R\le A$.
\item For each $P,Q\le A$, each $\varphi\in\homf(P,Q)$ extends to some $\4\varphi\in\autf(A)$.
\end{enuma} \end{Lem}
\begin{proof} \textbf{(a) } Assume $Q\le A$ and $R\le S$ are $\calf$-conjugate, and $R$ is fully normalized in $\calf$. By the extension axiom, each $\psi\in\isof(Q,R)$ extends to some $\4\psi\in\homf(C_S(Q),S)$. Then $C_S(Q)\ge A$ since $A$ is abelian, $\4\psi(A)=A$ since $A$ is weakly closed in $\calf$, and so $R=\4\psi(Q)\le A$.
\noindent\textbf{(b) } Assume $P,Q\le A$ and $\varphi\in\homf(P,Q)$, and choose $R\in Q^\calf$ that is fully centralized in $\calf$. Thus $R\le A$ by (a), and there is $\psi\in\isof(Q,R)$. By the extension axiom again, $\psi$ extends to $\5\psi\in\homf(A,S)$ and $\psi\varphi$ extends to $\5\varphi\in\homf(A,S)$, and $\5\psi(A)=A=\5\varphi(A)$ since $A$ is weakly closed. Then $\5\psi^{-1}\5\varphi\in\autf(A)$, and
$(\5\psi^{-1}\5\varphi)|_P=\psi^{-1}(\psi\varphi)=\varphi$. \end{proof}
The proof of the next lemma gives another example of how the extension axiom can be used.
\begin{Lem} \label{l:f.cent.+} Let $\calf$ be a saturated fusion system over a finite $p$-group $S$, and let $A_0\le A_1\le S$ be a pair of abelian subgroups. If $A_0$ is fully centralized in $\calf$ and $A_1$ is fully centralized in $C_\calf(A_0)$, then $A_1$ is fully centralized in $\calf$. \end{Lem}
\begin{proof} Choose $B_1\in A_1^\calf$ that is fully centralized in $\calf$, fix $\chi\in\isof(A_1,B_1)$, and set $B_0=\chi(A_0)$. By the extension axiom and since $A_0$ and $B_1$ are both fully centralized in $\calf$, there are $\varphi\in\homf(C_S(A_1),C_S(B_1))$ and
$\psi\in\homf(C_S(B_0),C_S(A_0))$ such that $\varphi|_{A_1}=\chi$ and
$\psi|_{B_0}=(\chi|_{A_0})^{-1}$. Since $C_S(B_1)\le C_S(B_0)$, the composite $\psi\varphi$ lies in $\Hom_{C_\calf(A_0)}(C_S(A_1),C_S(A_0))$.
Since $A_1$ is fully centralized in $C_\calf(A_0)$,
\[ \psi\varphi(C_S(A_1))=C_{C_S(A_0)}(\psi(B_1))
=C_S(\psi(B_1))\ge\psi(C_S(B_1)), \] and hence $\varphi(C_S(A_1))\ge C_S(B_1)$. So $A_1$ is fully centralized in $\calf$ since $B_1$ is. \end{proof}
We will need to work with quotient fusion systems in Section \ref{s:Alp}, but only quotients by subgroups normal in the fusion system.
\begin{Defi} \label{d:F/Q} Let $\calf$ be a fusion system, and assume $Q\nsg S$ is normal in $\calf$. Let $\calf/Q$ be the fusion system over $S/Q$ where for each $P,R\le S$ containing $Q$, we set
\begin{multline*}
\Hom_{\calf/Q}(P/Q,R/Q) =\\ \bigl\{\varphi/Q\in\Hom(P/Q,R/Q)
\,\bigl|\, \varphi\in\homf(P,Q), ~ (\varphi/Q)(gQ)=\varphi(g)Q
~\forall\,g\in P \bigr\}.
\end{multline*} \end{Defi}
We refer to \cite[Proposition II.5.11]{Craven} for the proof that $\calf/Q$ is saturated whenever $\calf$ is. In fact, this definition and the saturation of $\calf/Q$ hold whenever $Q$ is weakly closed in $\calf$. This is not surprising, since we are looking only at morphisms in $\calf$ between subgroups containing $Q$, so that $\calf/Q=N_\calf(Q)/Q$.
\section{Some criteria for realizing representations}
\label{s:A<|F}
In this section, we state and prove our main technical results: the tools we later use to show that certain representations cannot be realized by any saturated fusion systems. Before doing that, we start by defining more formally what we mean by ``realizability''.
\begin{Defi} \label{d:realize} Fix a prime $p$, a finite abelian $p$-group $A$, and a subgroup $\Gamma\le\Aut(A)$. The pair $(\Gamma,A)$ is \emph{realized} by a saturated fusion system $\calf$ over a finite $p$-group $S$ if there is an abelian subgroup $B\le S$ such that $C_S(B)=B$ and $B\nnsg\calf$, and such that $(\autf(B),B)\cong(\Gamma,A)$. The pair $(\Gamma,A)$ is \emph{fusion realizable} if it is realized by some saturated fusion system over a finite $p$-group. \end{Defi}
If we drop the condition that $C_S(B)=B$, then it is easy to see that every pair $(\Gamma,A)$ can be realized by a saturated fusion system. For example, if $m>1$ is prime to $p$, then the fusion system $\calf$ of $(A\rtimes\Gamma)\wr C_m$ contains a subgroup isomorphic to $A$ with automizer isomorphic to $\Gamma$ which is not normal in $\calf$. Hence the importance of that condition in Definition \ref{d:realize}, although it seems possible that we would get similar results if it were replaced by the condition that $B$ be weakly closed.
It is not yet clear to us whether the condition ``$B\nnsg\calf$'' is the optimal one to use in Definition \ref{d:realize}. It could be replaced by the slightly stronger condition that $\Omega_1(B)\nnsg\calf$, or by the even stronger condition that $O_p(\calf)=1$. In the cases dealt with in Theorems \ref{ThA} and \ref{ThB}, the result is the same independently of which definition we choose, but that probably does not hold in other situations.
When applying Definition \ref{d:realize}, rather than assuming $(\Gamma,A)$ and $(\autf(B),B)$ are abstractly isomorphic, it will in practice be more convenient to say that $(\Gamma,A)$ is realized by a fusion system $\calf$ over $S$ if $S$ contains $A$ as a subgroup and $\autf(A)=\Gamma$.
We are now ready to start developing tools for showing that certain pairs $(\Gamma,A)$ are not (weakly) fusion realizable. The starting point for all results in this section is the following proposition. It was inspired in part by \cite[Corollary 4]{Goldschmidt} and its proof, and also in part by arguments in \cite[\S\,1]{ONan}.
\begin{Prop} \label{p:not.str.cl.2} Let $\calf$ be a saturated fusion system over a finite $p$-group $S$, and let $A\le S$ be an abelian subgroup. Assume $A\nnsg\calf$, and consider the sets
\begin{align*}
\scru &= \UU{\calf}{A} = \bigl\{ 1\ne U\le N_S(A) \,\big|\, U\nleq A,~
\homf(U,A) \ne\emptyset \bigr\} \\
\scrt &= \VV{\calf}{A} = \bigl\{ t\in N_S(A)\smallsetminus A \,\big|\,
t^\calf\cap A\ne\emptyset \bigr\}
= \bigl\{ t\in N_S(A)\smallsetminus A \,\big|\, \gen{t}\in\scru \bigr\} \\
\scrw &= \WW{\calf}{A} = \bigl\{ (t,U,A_*) \,\big|\,
t\in\scrt,~ U\in\scru,~ C_A(t) \ge A_*\in(U\cap A)^\calf, ~ \\
& \hskip100mm |UA/A|=|C_{A/A_*}(t)| \bigr\}.
\end{align*} Then $\scru\ne\emptyset$, $\scrt\ne\emptyset$, and $\scrw\ne\emptyset$, and the following hold. \begin{enuma}
\item If $A$ is not weakly closed in $\calf$, there is $U\in A^\calf\smallsetminus\{A\}$ such that $[U,A]\le U\cap A$, and such that $(t,U,U\cap A)\in\scrw$ for each $t\in U\smallsetminus A$.
\item If $A$ is weakly closed in $\calf$, then for each $t\in\scrt$, there are $U\in\scru$ and $A_*\le A$ such that $(t,U,A_*)\in\scrw$.
\item If $A$ is weakly closed in $\calf$, then there is a subgroup $Z\le A$, fully centralized in $\calf$, such that $A\nnsg C_\calf(Z)$, and such that $U\cap A\le Z$ for each $U\in\UU{{C_\calf(Z)}}{A}$. In particular, $A_*=U\cap A$ for each $(t,U,A_*)\in\WW{{C_\calf(Z)}}{A}\subseteq \WW{\calf}{A}$.
\end{enuma} Thus in all cases, there are $t\in\scrt$ and $U\in\scru$ such that $(t,U,U\cap A)\in\scrw$. \end{Prop}
\begin{proof} By Lemma \ref{l:s.cl.=>normal} and since $A\nnsg\calf$, $A$ is not strongly closed. So $\scru\ne\emptyset$ and $\scrt\ne\emptyset$ if $A\nsg S$, and we will show when proving (a) that this also holds if $A\nnsg S$. The last statement, and the claim $\scrw\ne\emptyset$, follow from (a) when $A$ is not weakly closed in $\calf$, and from (b) and (c) otherwise.
\noindent\textbf{(a) } If $A$ is not weakly closed in $\calf$, then there is $\varphi\in\homf(A,S)$ such that $\varphi(A)\ne A$. So by Theorem \ref{t:AFT} (Alperin's fusion theorem), there are $R\le S$ and $\alpha\in\autf(R)$ such that $A\le R$ and $\alpha(A)\ne A$. In the special case where $A\nnsg S$, we take $R=N_S(A)$, and set $\alpha=c_x^R$ for some $x\in N_S(R)\smallsetminus R$. So in all cases, we can arrange that $A\nsg R$ and hence $\alpha(A)\le N_S(A)$.
Set $U=\alpha(A)\in\scru$ and $A_*=U\cap A$. Then $[A,U]\le A_*$ since $A$ and $U$ are both normal in $R$. So for each $t\in U\smallsetminus A\subseteq\scrt$,
we have $A_*\le C_A(U)\le C_A(t)$ and $|UA/A| = |U/A_*| = |A/A_*| =
|C_{A/A_*}(t)|$, proving that $(t,U,A_*)\in\scrw$.
\noindent\textbf{(b) } Assume $A$ is weakly closed in $\calf$ (in particular, $A\nsg S$). Fix $t\in\scrt$, and let $\scru_t$ be the set of
all $U\in\scru$ such that $t\in U$. Choose $V\in\scru_t$ such that $|V\cap
A|$ is maximal among all $|U\cap A|$ for $U\in\scru_t$. Set $A_*=V\cap A$ and $U_2^*=N_A(A_*\gen{t})$. Then $A_*\gen{t}\cap A\le V\cap A=A_*$, and so
\beqq U_2^*/A_* = \bigl\{x\in A\,\big|\,[x,t]\in
A_*\bigr\} \big/ A_* = C_{A/A_*}(t) \ne 1, \label{e:U2*/A*} \eeqq where $C_{A/A_*}(t)\ne1$ since $A/A_*$ and $t$ both have $p$-power order.
Choose $W\in (A_*\gen{t})^\calf$ such that $W$ is fully normalized in $\calf$. Then $W\le A$ by Lemma \ref{l:A-w.cl.}(a) and since $A$ is weakly closed. Let $\varphi\in\homf(N_S(A_*\gen{t}),S)$ be such that $\varphi(A_*\gen{t})=W$ (see Proposition \ref{p:Hom(NSP,S)}).
Set $U=\varphi(U_2^*)$ and $U_1^*=\varphi^{-1}(U\cap A)$. Then
\[ \varphi(A_*) \le \varphi(U_2^*)\cap A = U\cap A =
\varphi(U_1^*), \] so $A_*\le U_1^*\le U_2^*\le A$. Also, $U_1^*\gen{t}\in\scru_t$ since $\varphi(U_1^*\gen{t})=(U\cap A)\gen{\varphi(t)}\le A$, and hence
\[ |U_1^*| \le |U_1^*\gen{t}\cap A| \le |V\cap A| = |A_*| \] by the maximality assumption on $V$. Thus $U_1^*=A_*<U_2^*$ where the strict inclusion holds by \eqref{e:U2*/A*}, and $A_*=U_1^*\in(U\cap A)^\calf$.
Now, $U\cap A=\varphi(U_1^*)<\varphi(U_2^*)=U$, so $U\nleq A$. Since $U=\varphi(U_2^*)$ where $U_2^*\le A$, this shows that $U\in\scru$. Also, $U\cap A=\varphi(A_*)$, and so $UA/A\cong U/(U\cap A)\cong U_2^*/A_*=C_{A/A_*}(t)$. Thus $(t,U,A_*)\in\scrw$.
\noindent\textbf{(c) } Again assume $A$ is weakly closed in $\calf$, and let $Z$ be maximal among all subgroups of $A$ fully centralized in $\calf$ such that $A\nnsg C_\calf(Z)$. Set $\calf_0=C_\calf(Z)$ and $S_0=C_S(Z)$ for short. Recall that $\calf_0$ is saturated since $Z$ is fully centralized in $\calf$ (Theorem \ref{t:NF(Q)}).
Fix $U\in\UU{{\calf_0}}{A}$, choose a morphism $\varphi\in\homf[0](U,A)$, and set $A_*=U\cap A$. We must show that $A_*\le Z$. Since $UZ\in\UU{{\calf_0}}{A}$, we can assume $U\ge Z$.
Choose $B_*\in(A_*)^{\calf_0}$ that is fully normalized in $\calf_0$. Then $B_*\le A$ by Lemma \ref{l:A-w.cl.}(a) and since $A$ is weakly closed. By Proposition \ref{p:Hom(NSP,S)}, there is $\chi\in\Hom_{\calf_0}(N_{S_0}(A_*),S_0)$ such that $\chi(A_*)=B_*$. Then $\chi(A)=A$ since $A$ is weakly closed, so
$\chi\varphi(\chi|_U)^{-1}\in\homf[0](\chi(U),A)$ where $Z\le\chi(U)\nleq A$ and $B_*=\chi(U\cap A)=\chi(U)\cap A$, and where $B_*\le Z$ if and only if $A_*\le Z$. Upon replacing $U$ by $\chi(U)$
and $\varphi$ by $\chi\varphi(\chi|_U)^{-1}$, we are now reduced to showing that $A_*\le Z$ when $A_*=U\cap A$ is fully centralized in $\calf_0$, and hence in $\calf$ by Lemma \ref{l:f.cent.+}.
By Lemma \ref{l:A-w.cl.}(b), there is an automorphism
$\alpha\in\autf[0](A)$ such that $\alpha|_{A_*}=\varphi|_{A_*}$, hence such that $\alpha^{-1}\varphi\in\Hom_{C_\calf(A_*)}(U,A)$. Since $U\nleq A$, this implies that $A\nnsg C_\calf(A_*)$, and so $A_*=Z$ by the maximality assumption on $Z$.
In particular, for each $(t,U,A_*)\in\WW{{\calf_0}}{A}$, since $U\cap A\le Z$ and $A_*\in(U\cap A)^{\calf_0}$, we have $U\cap A=A_*\le Z$. \end{proof}
We now reformulate the criteria in Proposition \ref{p:not.str.cl.2} in terms of $A$ and $\autf(A)$ only; i.e., in terms that do not involve the fusion system $\calf$ or its Sylow group $S$.
\begin{Defi} \label{d:K&R} Fix a finite abelian $p$-group $A$ and a $p$-subgroup $T\le\Aut(A)$. Set
\begin{align*}
\hRR[+]{T}{A} &= \bigl\{ (\tau,B,A_*) \,\big|\,
\tau\in T^\#,~
B\le T,~ \textup{$\gen{\tau}$ and $B$ isomorphic to subgroups of $A$,}~
\\[-1mm]
&\hskip60mm
A_*\le C_A(\gen{B,\tau}),~ |B|\ge|C_{A/A_*}(\tau)| \bigr\}
\\[1mm]
\hRR{T}{A} &= \bigl\{ (\tau,B,A_*)\in\hRR[+]{T}{A}
\,\big|\, |B|=|C_{A/A_*}(\tau)| \bigr\}.
\end{align*} Let $\RR{T}{A}$ be the largest subset $\calr\subseteq\hRR{T}{A}$ that satisfies the condition
\beq \textup{for each $(\tau,B,A_*)\in\calr$ and each $\tau_1\in
B^\#$, there is $(\tau_1,B_1,A_{*1})\in\calr$.} \tag{$*$} \eeq Similarly, let $\RR[+]{T}{A}$ be the largest subset $\calr\subseteq\hRR[+]{T}{A}$ that satisfies ($*$). \end{Defi}
If $\scrr_1$ and $\scrr_2$ are two subsets of $\hRR{T}{A}$ or of $\hRR[+]{T}{A}$ that satisfy ($*$), then their union also satisfies ($*$). So there are unique largest subsets $\RR{T}{A}\subseteq\RR[+]{T}{A}$ that satisfy the condition.
\begin{Prop} \label{p:not.str.cl.3a} Let $\calf$ be a saturated fusion system over a finite $p$-group $S$, and assume $A\le S$ is an abelian subgroup such that $C_S(A)=A$ and $A\nnsg\calf$. Then $\RR{\Aut_S(A)}{A}\ne\emptyset$, and hence $\RR[+]{\Aut_S(A)}{A}\ne\emptyset$. More precisely, the following hold, where $T=\Aut_S(A)$: \begin{enuma}
\item In all cases, if $(t,U,A_*)\in\WW{\calf}{A}$ is such that $U\cap A=A_*$, then $(c_t^A,\Aut_U(A),A_*)\in\hRR{T}{A}$.
\item If $A$ is not weakly closed in $\calf$, then there is a subgroup $U\in A^\calf\smallsetminus\{A\}$ such that $(c_t^A,\Aut_U(A),A\cap U)\in\RR{T}{A}$ for each $t\in U\smallsetminus A$.
\item If $A$ is weakly closed in $\calf$, then there is a subgroup $Z\le A$ fully centralized in $\calf$ such that $A\nnsg C_\calf(Z)$, and such that for each $t\in\VV{{C_\calf(Z)}}{A}$, there is $U\in\UU{{C_\calf(Z)}}{A}$ such that
\[ U\cap A\le Z \qquad\textup{and}\qquad
(c_t^A,\Aut_U(A),U\cap A)\in\RR{C_T(Z)}{A}
\subseteq \RR{T}{A}. \]
\end{enuma} \end{Prop}
\begin{proof} Let $\calf$ be a saturated fusion system over a finite $p$-group $S$ as above. Thus $A\le S$ is such that $C_S(A)=A$ and $A\nnsg\calf$. Once we have proven points (a), (b), and (c), it will then follow immediately that $\RR{T}{A}\ne\emptyset$.
\noindent\textbf{(a) } Fix $(t,U,A_*)\in\WW{\calf}{A}$ such that $A_*=U\cap A$, and set $\tau=c_t^A\in T$ and $B=\Aut_U(A)\le T$. Then $A_*=U\cap A\le C_A(B)$. Also, by definition of $\WW{\calf}{A}$, we have $A_*\le C_A(t)=C_A(\tau)$ and
$|UA/A|=|C_{A/A_*}(t)|=|C_{A/A_*}(\tau)|$.
By definition of $\VV{\calf}{A}$ and $\UU{\calf}{A}$, the subgroups $\gen{\tau}$ and $B$ are both isomorphic to subgroups of $A$. So to prove that $(\tau,B,A_*)\in\hRR{T}{A}$, it remains only to show that
$|UA/A|=|B|$. But $C_S(A)=A$ by assumption, so $|B|=|\Aut_U(A)|=|UA/A|$.
\noindent\textbf{(b) } If $A$ is not weakly closed in $\calf$, then by Proposition \ref{p:not.str.cl.2}(a), there is $U\in A^\calf\smallsetminus\{A\}$ such that $[U,A]\le U\cap A$, and such that $(t,U,U\cap A) \in\WW{\calf}{A}$ for each $t\in U\smallsetminus A$. Thus $(c_t^A,\Aut_U(A),U\cap A)\in\hRR{\calf}{A}$ for each $t\in U\smallsetminus A$ by (a).
Now set $\scrr = \{ (\tau,\Aut_U(A),U\cap A) \,|\, \tau\in B^\# \}\subseteq\hRR{\calf}{A}$. Then $\scrr$ satisfies condition ($*$) in Definition \ref{d:K&R}, so $\RR{T}{A}\supseteq\scrr\ne\emptyset$.
\noindent\textbf{(c) } Assume $A$ is weakly closed in $\calf$, and let $Z\le A$ be as in Proposition \ref{p:not.str.cl.2}(c). Thus $Z$ is fully centralized in $\calf$, $A\nnsg C_\calf(Z)$, and $U\cap A\le Z$ for each $U\in\scru_{C_\calf(Z)}(A)$.
Let $\scrt=\VV{{C_\calf(Z)}}{A}\ne\emptyset$, $\scru=\UU{{C_\calf(Z)}}{A}\ne\emptyset$, and $\scrw=\WW{{C_\calf(Z)}}{A}\ne\emptyset$ be as in Proposition \ref{p:not.str.cl.2}, and set
\[ \scrr = \bigl\{ (c_t^A,\Aut_U(A),U\cap
A) \,\big|\, t\in\scrt,~ U\in\scru,~ (t,U,A_*)\in\scrw \bigr\}, \] where $A_*\in(U\cap A)^{C_\calf(Z)}$ and hence $A_*=U\cap A$ since $U\cap A\le Z$. By (a), $\scrr\subseteq\hRR{C_T(Z)}{A}$. By Proposition \ref{p:not.str.cl.2}(b,c), for each $t\in\scrt$, there is $U\in\scru$ such that $(t,U,U\cap A)\in\scrw$. So $\scrr\ne\emptyset$, and condition ($*$) in Definition \ref{d:K&R} holds for the pair $\scrr$. Thus $\scrr \subseteq \RR{C_T(Z)}{A} \subseteq \RR{T}{A}$. \end{proof}
The next proposition is our main reason for defining $\RR[+]{T}{A}$.
\begin{Prop} \label{p:not.str.cl.3c} Fix a finite abelian $p$-group $A$ and a $p$-subgroup $T\le\Aut(A)$. Let $A_1<A_2\le A$ be $T$-invariant subgroups such that $T$ acts faithfully on $A_2/A_1$. If $\RR[+]{T}{A}\ne\emptyset$, then $\RR[+]{T}{A_2/A_1}\ne\emptyset$. More precisely,
\[ \RR[+]{T}{A_2/A_1} \supseteq
\bigl\{(\tau,B,(A_*A_1\cap A_2)/A_1) \,\big|\,
(\tau,B,A_*)\in\RR[+]{T}{A} \bigr\}. \] \end{Prop}
\begin{proof} Assume $A_1<A_2\le A$ are as above. If $(\tau,B,A_*)\in\hRR[+]{T}{A}$, then
\[ |C_{A_2/(A_*A_1\cap A_2)}(\tau)|\le|C_{A_2/(A_*\cap A_2)}(\tau)|
= |C_{A_2A_*/A_*}(\tau)|\le |C_{A/A_*}(\tau)| \le |B|: \] the first inequality by Lemma \ref{l:C(A/A0)G} and the second by inclusion. So $(\tau,B,(A_*A_1\cap A_2)/A_1)\in \hRR[+]{T}{A_2/A_1}$.
In particular, if $\scrr$ satisfies condition ($*$) in Definition \ref{d:K&R} for the pair $(T,A)$, then $\scrr'$ satisfies ($*$) for $(T,A_2/A_1)$ where
\beq \scrr' = \bigl\{(\tau,B,(A_*A_1\cap A_2)/A_1) \,\big|\,
(\tau,B,A_*)\in\scrr \bigr\}. \qedhere \eeq \end{proof}
It remains to find some strong necessary conditions on $A$ and $T$ for the set $\RR{T}{A}$ or $\RR[+]{T}{A}$ to be nonempty.
\begin{Prop} \label{p:not.str.cl.3b} Fix a finite abelian $p$-group $A$ and a subgroup $T\le\Aut(A)$. Then for each $(\tau,B,A_*)\in\hRR{T}{A}$,
\beqq
|B| = \frac{|A|}{|A_*[\tau,A]|}
\quad\textup{and}\quad
\frac{|B|}{|C_A(\tau)\cap[\tau,A]|} =
\frac{|C_A(\tau)[\tau,A]|}{|A_*[\tau,A]|},
\label{e:B1} \eeqq while for each $(\tau,B,A_*)\in\hRR[+]{T}{A}$,
\beqq
|B| \ge \frac{|A|}{|A_*[\tau,A]|}
\quad\textup{and}\quad
\frac{|B|}{|C_A(\tau)\cap[\tau,A]|} \ge
\frac{|C_A(\tau)[\tau,A]|}{|A_*[\tau,A]|} \ge 1.
\label{e:B1+} \eeqq In particular, for each $(\tau,B,A_*)\in\hRR[+]{T}{A}$,
\beqq |B| \ge |C_A(\tau)\cap[\tau,A]|, \label{e:B2} \eeqq and $|B|\ge|[\tau,A]|$ if $p=2$ and $A$ is elementary abelian. \end{Prop}
\begin{proof} For each $\tau\in T^\#$, let $\varphi_{\tau}\in\End(A)$ be the map $\varphi_{\tau}(a)=[\tau,a]$. For each $A_*\le C_A(\tau)$, we have $C_A(\tau)=\Ker(\varphi_{\tau})$ and $C_{A/A_*}(\tau)=\varphi_{\tau}^{-1}(A_*)/A_*$, and hence
\beqq \begin{split}
|C_{A/A_*}(\tau)|
= \frac{|C_A(\tau)|\cdot|A_*\cap[\tau,A]|}{|A_*|}
&= \frac{|C_A(\tau)|\cdot|[\tau,A]|}{|A_*[\tau,A]|}
= \dfrac{|A|}{|A_*[\tau,A]|} \\[2mm]
&\hskip20mm=\dfrac{|C_A(\tau)[\tau,A]| \cdot|C_A(\tau)\cap[\tau,A]|}
{|A_*[\tau,A]|}.
\label{e:B3} \end{split} \eeqq Since $|B|\ge|C_{A/A_*}(\tau)|$ for each $(\tau,B,A_*)\in\hRR[+]{T}{A}$ with equality if $(\tau,B,A_*)\in\hRR{T}{A}$, points \eqref{e:B1} and \eqref{e:B1+} follow immediately from \eqref{e:B3} (and since $A_*\le C_A(\tau)$). Inequality \eqref{e:B2} follows from \eqref{e:B1+}, and the last statement holds since $[\tau,A]\le C_A(\tau)$ if $p=2$ and $A$ is elementary abelian. \end{proof}
The following corollary describes one easy consequence of the above results.
\begin{Cor} \label{c:not.str.cl.} Fix a finite abelian $p$-group $A$ and a $p$-subgroup $T\le\Aut(A)$ such that $\RR[+]{T}{A}\ne\emptyset$. Then there is $B_0\le T$, isomorphic to a
subgroup of $A$, such that $|B_0|\ge|C_A(\tau)\cap[\tau,A]|$ for each $\tau\in B_0^\#$. \end{Cor}
\begin{proof} Assume $\RR[+]{T}{A}\ne\emptyset$. Choose $(\tau_0,B_0,A_{*0})\in\RR[+]{T}{A}$ such that
$|C_A(\tau_0)\cap[\tau_0,A]|$ is the largest possible. By condition ($*$) in Definition \ref{d:K&R}, for each $\tau\in B_0^\#$, there is $(\tau,B,A_*)\in\RR[+]{T}{A}$, and hence
\[ |C_A(\tau)\cap[\tau,A]| \le |C_A(\tau_0)\cap[\tau_0,A]|
\le |B_0|, \] where the second inequality holds by \eqref{e:B2}. \end{proof}
We can think of the inequality $|B_0|\ge|C_A(\tau)\cap[\tau,A]|$ in Corollary \ref{c:not.str.cl.} as a generalization of the condition
$|Z(S)\cap[S,S]|=p$ in \cite[Lemma 2.3(b)]{indp1}. More precisely, when $A$ has index $p$ in $S$ and $S$ is
nonabelian, the corollary says that $|C_A(\tau)\cap[A,\tau]|=p$
for $\tau\in S\smallsetminus A$, and hence that $|Z(S)\cap[S,S]|=p$.
We next look at the case where $A$ is elementary abelian. For $\tau\in\End(A)$, we regard $A$ as an $\F_p[X]$-module, and let the ``Jordan blocks'' for $\tau$ be the factors under some decomposition of $A$ as a product of indecomposable submodules. As usual, by ``nontrivial Jordan blocks'' we really mean ``Jordan blocks with nontrivial action''.
The following notation will be used when reformulating Corollary \ref{c:not.str.cl.} in terms of Jordan blocks.
\begin{Not} \label{d:JV(x)} Let $A$ be an elementary abelian $p$-group, and let $\tau\in\Aut(A)$ be an automorphism of $p$-power order. Set $\scrj_A(\tau)=\rk(C_A(\tau)\cap[\tau,A])$: the number of nontrivial Jordan blocks for the action of $\tau$ on $A$. \end{Not}
In these terms, Corollary \ref{c:not.str.cl.} takes the following form when $A$ is elementary abelian:
\begin{Cor} \label{c:not.str.cl.2} Assume $\Gamma$ is a finite group such that $\Gamma=O^{p'}(\Gamma)$, and let $A$ be a finite faithful $\F_p\Gamma$-module. Assume there is a saturated fusion system $\calf$ over a finite $p$-group $S$ that realizes $(\Gamma,A)$ as in Definition \ref{d:realize}. Then there are $m\ge1$ and an elementary abelian $p$-subgroup $B\le\Gamma$ of rank $m$ such that $\scrj_A(\tau)\le m$ for each $\tau\in B^\#$. \end{Cor}
\begin{proof} Since $\calf$ realizes $(\Gamma,A)$, we can arrange that $A\le S$, $A\nnsg\calf$, and $\autf(A)=\Gamma$. Set $T=\Aut_S(A)\in\sylp\Gamma$. Then $\RR[+]{T}A\ne\emptyset$ by Proposition \ref{p:not.str.cl.3a}. So by Corollary \ref{c:not.str.cl.}, there is an elementary abelian $p$-subgroup $B\le \Gamma$ such that
$|B|\ge|C_{A}(\tau)\cap[\tau,A]|$ for all $\tau\in B^\#$. Thus $\rk(B)\ge \rk\bigl(C_{A}(\tau)\cap[A,\tau]\bigr) = \scrj_{A}(\tau)$ for each $\tau\in B^\#$. \end{proof}
The special case of fusion realizability when $|T|=p$ was already handled in the earlier papers \cite{indp1} and \cite{indp2}. We state the main conditions found in those papers:
\begin{Lem} \label{l:indp} Fix a finite abelian $p$-group $A$ and subgroups $\Gamma\le\Aut(A)$ and
$T\in\sylp\Gamma$, and assume that $|T|=p$ and $|[T,A]|>p$. If $(\Gamma,A)$ is fusion realizable, then
\[ |C_A(T)\cap[T,A]|=p \qquad\textup{and}\qquad
|N_\Gamma(T)/C_\Gamma(T)|=p-1. \] \end{Lem}
\begin{proof} The first equality is just a special case of Corollary \ref{c:not.str.cl.}.
To see the second equality, assume that $(\Gamma,A)$ is realized by the fusion system $\calf$ over $S\ge A$. In particular, we can assume that
$\Aut_S(A)=T$, and so $|N_S(A)/A|=|T|=p$. Also, $|A/C_A(T)|=|[T,A]|>p$ by assumption, so $A$ is the only abelian subgroup of index $p$ in $N_S(A)$. Hence $A\nsg S$, since otherwise $A\ne\9xA\le N_S(A)$ for $x\in N_S(N_S(A))\smallsetminus N_S(A)$.
By Theorem \ref{t:AFT} and since $A\nnsg\calf$ (recall $\calf$ realizes $(\Gamma,A)$), there must be some $\calf$-essential subgroup $P\le S$ other than $A$, and by \cite[Lemma 2.2(a)]{indp2}, $P\in\calh\cup\calb$ where the classes $\calh$ and $\calb$ of subgroups of $S$ are defined in \cite[Notation 2.1]{indp2}. By \cite[Lemma 2.6(a)]{indp2} (and in terms of Notation 2.4 in \cite{indp2}), we have $\mu(\autf^{(P)}(S))=\Delta_t$ for $t=0$ or $-1$, and from the definition of $\mu$ it then follows that $\Aut_\Gamma(T)=\Aut(T)$ and hence has order $p-1$. \end{proof}
\section{Representations of Mathieu groups} \label{s:Mathieu}
We next look at representations of the Mathieu groups $M_n$ and their central extensions. The main theorem is stated for an arbitrary prime $p$, but we focus attention mostly on the cases $p=2,3$, since the others follow from Lemma \ref{l:indp} and results in \cite{indp2}.
We will apply Corollary \ref{c:not.str.cl.2} in most cases, using Lemma \ref{l:J(x)ge...} and the character tables in \cite{modatlas} to find lower
bounds for $\scrj_A(x)$ when $|x|=2$ or $3$. The notation $\2x$ and $\3x$ refers to the classes as named in the Atlas \cite{atlas} and in \cite{modatlas}. In the following lemma, we restrict attention to $M_{12}$ and $M_{24}$ since they are the only Mathieu groups with more than one conjugacy class of elements of order $2$ or $3$.
\begin{Lem} \label{l:M12,M24} Assume $\Gamma\cong M_{12}$ or $M_{24}$. Then \begin{enuma}
\item each element of order $2$ in $\Gamma$ is contained in some $H_1\le\Gamma$ with $H_1\cong D_{10}$; and
\item each element of order $3$ in $\Gamma$ is contained in some $H_3\le\Gamma$ with $H_3\cong A_4$, and with elements of order $2$ in class \2a (if $\Gamma\cong M_{12}$) or \2b (if $\Gamma\cong M_{24}$).
\end{enuma} \end{Lem}
\begin{proof} Let $n=12,24$ be such that $\Gamma\cong M_n$, and let $X$ be a $5$-fold transitive $\Gamma$-set of order $n$. In each case, $\Gamma$ has two classes of elements of order $2$ and two classes of elements of order $3$, and they are distinguished by whether they act on $X$ freely or with fixed points as described in Table \ref{tbl:2A2B}. The outer automorphism of $M_{12}$ sends each of these classes to itself, and so the inclusion of $\Aut(M_{12})$ into $M_{24}$ sends distinct classes to distinct classes. It thus suffices to prove the lemma when $\Gamma\cong M_{12}$.
\begin{Table}[ht]
\[ \renewcommand{1.5}{1.3}
\renewcommand{2mm}{4mm}
\begin{array}{c|cccc}
\Gamma & \2a & \2b & \3a & \3b \\\hline
M_{12} & 2^6 & 2^4\cdot1^4 & 3^3\cdot1^3 & 3^4 \\
M_{24} & 2^8\cdot1^8 & 2^{12} & 3^6\cdot1^6 & 3^8
\end{array} \]
\caption{The table lists the number of orbits in the action on $X$
by each element of order $2$ or $3$ in $\Gamma$. Thus, for example,
a $\2b$-element in $M_{12}$ acts with four orbits of
length two and four fixed points.}
\label{tbl:2A2B}
\end{Table}
\noindent\textbf{(a) } Fix an element $g\in\2a$. By \cite[p. 41]{GL}, $C_\Gamma(g)\cong C_2\times\Sigma_5$, and the second factor must permute faithfully the six orbits under the action of $g$. Fix $N\le C_\Gamma(g)$ of order $5$, and let $h\in C_\Gamma(g)\smallsetminus\gen{g}$ be such that $N\gen{h}\cong D_{10}$. Then $N\gen{gh}\cong D_{10}$, and we will be done upon showing that $h$ and $gh$ lie in different classes.
Set $X_0=C_X(N)$, a subset of order $2$ whose elements are exchanged by $g$, and set $X_1=X\smallsetminus X_0$. Of the two elements $h$ and $gh$, one fixes the two points in $X_0$ and the other exchanges them, and we can assume that $h$ fixes them. Hence $C_X(h)\ne\emptyset$, so $h\in\2b$. Also, $C_X(gh)\subseteq X_1$, and since $gh$ permutes freely
four of the five $\gen{g}$-orbits in $X_1$, we have $|C_X(gh)|\le2$. Since no involution in $M_{12}$ acts with exactly two fixed points, this shows that $gh\in\2a$, finishing the proof of (a).
\noindent\textbf{(b) } Now fix an element $g\in\3b$. Then $C_\Gamma(g)\cong C_3\times A_4$ by \cite[p. 41]{GL}. Set $N=O_2(C_\Gamma(g))\cong E_4$. The group $C_\Gamma(g)/\gen{g}\cong A_4$ acts faithfully on the set of four orbits of $g$, so the elements of order $2$ in $N$ all act freely on $X$ and hence lie in class \2a.
Fix $h\in C_\Gamma(g)$ such that $N\gen{h}\cong A_4$. Then $N\gen{gh}$ and $N\gen{g^2h}$ are also isomorphic to $A_4$. Also $h$ permutes freely three of the four $\gen{g}$-orbits in $X$, and the fourth orbit is fixed by exactly one of the elements $h$, $gh$, or $g^2h$. So one of these three elements lies in class $\3a$,and the other two in class $\3b$. \end{proof}
There are two special cases that we will need to consider separately. The statement and proof of the following proposition are based on notation set up in Appendices \ref{s:Todd-F2} and \ref{s:3M22}.
\begin{Prop} \label{p:M22-23} Assume $p=2$. \begin{enuma}
\item If $\Gamma\cong M_{22}$ or $M_{23}$ and $A$ is the Golay module (dual Todd module) for $\Gamma$, then $\RR[+]{T}{A}=\emptyset$ for $T\in\syl2\Gamma$.
\item If $\Gamma\cong3M_{22}$ and $A$ is the $6$-dimensional simple $\F_4\Gamma$-module, then $\RR[+]{T}{A}=\emptyset$ for $T\in\syl2\Gamma$.
\end{enuma} \end{Prop}
\begin{proof} In the first part of the proof, we consider cases (a) and (b) together. Assume the proposition is not true, and fix a triple $(\tau,B,A_*)\in\RR[+]{T}{A}$. Thus $\tau\in T$ has order $2$, $B\le T$ is an elementary abelian $2$-subgroup, and $A_*\le C_A(\gen{B,\tau})$ is such
that $|B|\ge|C_{A/A_*}(\tau)|$. By Proposition \ref{p:not.str.cl.3b} and since $[\tau,A]\le C_A(\tau)$, we have
\beqq |B| \ge \bigl|[\tau,A]\bigr| \cdot \bigl|C_A(\tau) /
A_*[\tau,A] \bigr| \ge |[\tau,A]|. \label{e:R-ineq} \eeqq
Since $\Gamma\cong M_{22}$, $M_{23}$, or $3M_{22}$ has only one conjugacy
class of involution, we have $|[\tau,A]|=2^4$: by Lemma \ref{l:hexad.grp}
in case (a), and by Lemma \ref{l:3M22b}(b) in case (b). Thus $|B|\ge2^4$, with equality since $\rk_2(\Gamma)=4$ in all cases. So the inequalities in \eqref{e:R-ineq} are equalities, $C_A(\tau) = A_*[\tau,A]$, and hence
\beqq \rk(C_A(B))\ge \rk(A_*)\ge\rk(C_A(\tau)/[\tau,A])
= \rk(A)-2\cdot\rk([\tau,A]) = \rk(A)-8. \label{e:rkA-8} \eeqq
\noindent\textbf{(a) } Assume $T<\Gamma$ and $A$ are as in Notation \ref{not:Gamma-action} and \ref{n:M22-23}. Since $H_1$ and $H_2$ are the only subgroups of $T\in\syl2{\Gamma_0}$ isomorphic to $E_{16}$ by Lemma \ref{l:hexad.grp}, $B$ must be equal to one of them. Since $C_A(H_1)$ has rank $1$ by Lemma \ref{l:hexad.grp} again, and $\rk(C_A(B))\ge\rk(A)-8\ge2$ by \eqref{e:rkA-8}, we have $B=H_2$.
By condition ($*$) in Definition \ref{d:K&R}, each element of $B^\#$ can appear as the first component in an element of $\RR[+]{T}{A}$. So we can assume that $(\tau,B,A_*)$ was chosen such that $\tau=\trsh1$ (and still $B=H_2$). Hence by Tables \ref{tbl:[x,a]} and \ref{tbl:[ch1,x]},
\[ \grfh2+C_{56}\in C_A(\trsh1) = A_*[\trsh1,A]
\le C_A(H_2)[\trsh1,A] = \gen{C_{12},C_{13},C_{14},C_{15},\grfh1}, \] a contradiction. We conclude that $\RR[+]{T}{A}=\emptyset$.
\iffalse We can assume $T<\Gamma$ and $A$ are as in Notation \ref{n:3M22} and \ref{n:3M22b}. Assume $\RR[+]{T}{A}\ne\emptyset$, and fix an element $(\tau,B,A_*)\in\RR[+]{T}{A}$. Thus $\tau\in T$ has order $2$, $B\le T$ is an elementary abelian $2$-group, and $A_*\le C_{A}(B\gen{\tau})$ is such that $\rk(B)\ge\rk([\tau,A])$. By Proposition \ref{p:not.str.cl.3b} and since $[\tau,A]\le C_A(\tau)$,
\beqq |B| \ge |[\tau,A]|\cdot \bigl[C_{A}(\tau) : A_*[\tau,A] \bigr]
\ge |[\tau,A]|. \label{e:3M22c} \eeqq Since $\Gamma$ has only one conjugacy class of involutions, $\tau$ is $\Gamma$-conjugate to $\mu_{10}$, and hence $\rk([\tau,A])=2\cdot\dim_{\F_4}([\mu_{10},A])=4$ (Lemma \ref{l:3M22b}(b)). Since $\rk(B)\le\rk_2(\Gamma)=4$, we have $\rk(B)=4$ and $C_{A}(\tau)=A_*[\tau,A]$ by \eqref{e:3M22c}. So
\[ \rk(C_A(B)) \ge \rk(A_*)\ge \rk(C_A(\tau))-\rk([\tau,A]) =
\rk(C_A(\mu_{10}))-\rk([\mu_{10},A]) = 4. \] where the last equality holds by Lemma \ref{l:3M22b}(b). \fi
\noindent\textbf{(b) } Now assume $T<\Gamma$ and $A$ are as in Notation \ref{n:3M22} and \ref{n:3M22b}. By Lemma \ref{l:3M22b}(a), $P_1$ and $P_2$ are the only subgroups of $T$ isomorphic to $E_{16}$. Since $\rk(C_A(P_2))=2\cdot\dim_{\F_4}(C_A(P_2))=2$ by Lemma \ref{l:3M22b}(b), while $\rk(C_A(B))\ge4$ by \eqref{e:rkA-8}, we have $B=P_1$. By condition ($*$) in Definition \ref{d:K&R}, we can assume that the triple $(\tau,P_1,A_*)$ was chosen so that $\tau=\mu_{10}$. But then
\[ \gen{e_1,e_2,e_3,e_4} = C_A(\mu_{10}) =
A_*[\mu_{10},A]\le C_A(P_1)[\mu_{10},A]
= \gen{e_1,e_2,e_3} \] by Lemma \ref{l:3M22b}(b), a contradiction. \end{proof}
We now apply Corollary \ref{c:not.str.cl.2} and Lemma \ref{l:J(x)ge...}, together with Proposition \ref{p:M22-23}, to determine the realizability of $\F_p\Gamma$-modules when $O^{p'}(\Gamma)$ is a central extension of a Mathieu group. The following is a restatement of Theorem \ref{ThA}.
\iffalse most $\F_p\Gamma$-modules cannot be realized in the sense of Definition \ref{d:realize}. \fi
\begin{Thm} \label{t:M11-24} Fix a prime $p$ and a finite group $\Gamma$, and set $\Gamma_0=O^{p'}(\Gamma)$. Assume that $\Gamma_0$ is quasisimple, and that $\Gamma_0/Z(\Gamma_0)$ is one of the Mathieu groups. Let $A$ be an $\F_p\Gamma$-module such that $(\Gamma,A)$ is fusion realizable, and set $A_0=[\Gamma_0,A]/C_{[\Gamma_0,A]}(\Gamma_0)$. Then either \begin{enuma}
\item $p=2$, $\Gamma\cong M_{22}$ or $M_{23}$, and $A_0$ is the Todd module for $\Gamma$; or
\item $p=2$, $\Gamma\cong M_{24}$, and $A_0$ is the Todd module or Golay module for $\Gamma$; or
\item $p=3$, $\Gamma\cong M_{11}$, $M_{11}\times C_2$, or $2M_{12}$, and $A_0$ is the Todd module or Golay module for $\Gamma_0$; or
\item $p=11$, $\Gamma_0\cong2M_{12}$ or $2M_{22}$, $\Gamma/Z(\Gamma_0)\cong\Aut(M_{12})\times C_5$ or $\Aut(M_{22})\times C_5$, and $A_0$ is a $10$-dimensional simple $\F_{11}\Gamma$-module.
\end{enuma} \end{Thm}
\begin{proof} Let $n\in\{11,12,22,23,24\}$ be such that $\Gamma_0/Z(\Gamma_0)\cong M_n$. Fix $T\in\sylp{\Gamma}=\sylp{\Gamma_0}$. We will frequently refer to Tables \ref{tbl:JA(t)-2} and \ref{tbl:JA(t)-3}
for our lower bounds on $\scrj_A(\tau)$ for $|\tau|=p$, and they in turn are based on Lemmas \ref{l:M12,M24} and \ref{l:J(x)ge...} and the character tables in the Atlas of Brauer characters \cite{modatlas}.
\noindent\textbf{Case 1: } If $p>3$, then $|T|=p$ in all cases.
So by Lemma \ref{l:indp}, we have $|N_\Gamma(T)/C_\Gamma(T)|=p-1$ and
$|C_A(T)\cap[T,A]|=p$. In the terminology of \cite{indp2}, this translates to saying that $\Gamma\in\scrg_p^\wedge$ and $A$ is minimally active, and so the result follows from \cite[Proposition 7.1]{indp2}.
\noindent\textbf{Case 2: } Assume $p=2$. By Table \ref{tbl:JA(t)-2}, for $\tau\in\Gamma$ of order $2$, we have $\scrj_{A_0}(\tau)>\rk_2(\Gamma)$ (and hence $\RR[+]{T}{A_0}=\emptyset$) for each nontrivial simple $\F_2\Gamma_0$-module $A_0$, except when $\Gamma_0\cong M_{22}$, $M_{23}$, or $M_{24}$ and $A_0$ is the Todd module or Golay module.
\begin{Table}[ht]
\[ \renewcommand{1.5}{1.2}
\begin{array}{cccc|c}
\Gamma_0 & \rk_2(\Gamma_0) & \dim(A_0) & \tau\in & \scrj_{A_0}(\tau)
\\\hline
M_{11} & 2 & >1 & \2a &
\scrj_{A_0}(\2a)\ge\frac25(\chi_{A_0}(1)-\chi_{A_0}(\0{5a}))\ge4 \\
M_{12} & 3 & >1 & \2a,\2b &
\scrj_{A_0}(\2x)\ge\frac25(\chi_{A_0}(1)-\chi_{A_0}(\0{5a}))\ge4 \\
M_{22} & 4 & >10 & \2a &
\scrj_{A_0}(\2a)\ge\frac25(\chi_{A_0}(1)-\chi_{A_0}(\0{5a}))\ge8 \\
3M_{22} & 4 & >12 & \2a &
\scrj_{A_0}(\2a)\ge\frac25(\chi_{A_0}(1)-\chi_{A_0}(\0{5a}))\ge6 \\
M_{23} & 4 & >11 & \2a &
\scrj_{A_0}(\2a)\ge\frac25(\chi_{A_0}(1)-\chi_{A_0}(\0{5a}))\ge8 \\
M_{24} & 6 & >11 & \2a,\2b &
\scrj_{A_0}(\2x)\ge\frac25(\chi_{A_0}(1)-\chi_{A_0}(\0{5a}))\ge8
\end{array} \]
\caption{In all cases, $A_0$ is an $\F_2\Gamma$-module such that
$C_{A_0}(\Gamma)=0$ and $[\Gamma,A_0]=A_0$, and the characters are
taken with respect to $\F_2$. The bounds for $\scrj_{A_0}(\tau)$
all follow from Lemmas \ref{l:M12,M24}(a) and
\ref{l:J(x)ge...}(a).}
\label{tbl:JA(t)-2}
\end{Table}
Thus if $Z(\Gamma_0)$ has odd order, then either $n\ge22$ and $A_0$ is the Todd module or Golay module for $\Gamma$, or $\Gamma_0\cong3M_{22}$ and $A_0$ is the $6$-dimensional $\F_4\Gamma_0$-module. In these cases, $\RR[+]{T}{A_0}=\emptyset$ by Proposition \ref{p:M22-23}, and so they are impossible by Propositions \ref{p:not.str.cl.3a} and \ref{p:not.str.cl.3c}.
\iffalse In the latter case, $\RR[+]{T}{A_0}=\emptyset$ by Proposition \ref{p:3M22}, while if $\Gamma_0\cong M_{22}$ or $M_{23}$ and $A_0$ is its Golay module, then $\RR[+]{T}{A_0}=\emptyset$ by Proposition \ref{p:dTodd}. So these last cases are impossible by Propositions \ref{p:not.str.cl.3a} and \ref{p:not.str.cl.3c}. \fi
It remains to consider the cases where $Z(\Gamma)$ has even order. Assume first that $\Gamma_0\cong2M_{12}$. Then $\rk_2(\Gamma)=4$, and $\scrj_{A_0}(\tau)\ge4$ for each $\F_2[\Gamma/Z(\Gamma)]$-module $A_0$ with nontrivial action by Table \ref{tbl:JA(t)-2}. By the last statement in Lemma \ref{l:pG-rep} (applied with $A$ in the role of $V$), for each elementary abelian 2-subgroup $B\le G$ of rank $4$, since $Z(\Gamma)\le B$, there is $\tau\in B$ of order $2$ such that $\scrj_{A}(\tau)\ge5$. So Corollary \ref{c:not.str.cl.2} again applies to show that $(\Gamma,A)$ is not fusion realizable.
Now assume that $\Gamma_0/Z(\Gamma_0)\cong M_{22}$, and let $Z\le
Z(\Gamma)$ be the Sylow 2-subgroup. Thus $|Z|=2$ or $4$, and $\rk_2(\Gamma_0)\le5$. By Table \ref{tbl:JA(t)-2} and since $\scrj_{A_0}(\tau)\le\rk_2(\Gamma_0)$, either $\Gamma_0/Z\cong M_{22}$ and $A_0$ is its Todd module or its dual, or $\Gamma_0/Z\cong3M_{22}$ and $A_0$ is the $6$-dimensional $\F_4\Gamma/Z$-module. By Lemma \ref{l:pG-rep}(b) and since $\Gamma$ acts faithfully on $A$, there must be indecomposable extensions of $A_0$ by $\F_2$ and of $\F_2$ by $A_0$. Thus $H^1(\Gamma/Z;A_0)\ne0$ and $H^1(\Gamma/Z;A_0^*)\ne0$ (where $A_0^*$ is the dual module), contradicting \cite[Lemma 6.1]{MS}. We conclude that no such faithful $\F_2\Gamma$-modules exist.
\noindent\textbf{Case 3: } Assume $p=3$. We claim that $\scrj_{A_0}(\tau)>\rk_3(\Gamma_0)$ (and hence $(\Gamma,A)$ is not fusion realizable) in all cases except when $\Gamma_0\cong M_{11}$ or $2M_{12}$ and $A_0$ is the Todd module for $\Gamma_0$ or its dual. This follows from Table \ref{tbl:JA(t)-3} except when $\Gamma_0\cong M_{11}$, $\dim(A_0)=10$, and $A_0\oplus\F_3$ is the $11$-dimensional permutation
module. But in that case, $\scrj_{A_0}(\tau)=3$ whenever $|\tau|=3$ since $\tau$ acts on an $11$-set with three free orbits.
\begin{Table}[ht]
\[ \renewcommand{1.5}{1.2}
\begin{array}{cccc|c}
\Gamma & \rk_3(\Gamma) & \dim(A_0) & \tau\in & \scrj_{A_0}(\tau)
\\\hline
M_{11} & 2 & \ge10~\textup{(*)} & \3a & \scrj_{A_0}(\3a)
\ge\frac14(\chi_{A_0}(1)-\chi_{A_0}(\0{4a}))\ge \frac52 \\
M_{12} & 2 & >1 & \3a,\3b &
\scrj_{A_0}(\3x)\ge\frac14(\chi_{A_0}(1)-\chi_{A_0}(\2a))\ge3 \\
2M_{12} & 2 & >6 & \3a,\3b & \scrj_{A_0}(\3x)
\ge\frac14(\chi_{A_0}(1)-\chi_{A_0}(\2a))\ge\frac52 \\
M_{22} & 2 & >1 & \3a & \scrj_{A_0}(\3a)
\ge\frac14(\chi_{A_0}(1)-\chi_{A_0}(\2a))\ge 4 \\
2M_{22} & 2 & >1 & \3a & \scrj_{A_0}(\3a)
\ge\frac14(\chi_{A_0}(1)-\chi_{A_0}(\2a))\ge 4 \\
M_{23} & 2 & >1 & \3a & \scrj_{A_0}(\3a)
\ge\frac14(\chi_{A_0}(1)-\chi_{A_0}(\2a))\ge 4 \\
M_{24} & 2 & >1 & \3a,\3b &
\scrj_{A_0}(\3x)\ge\frac14(\chi_{A_0}(1)-\chi_{A_0}(\2b))\ge6
\end{array} \]
\caption{In all cases, $A_0$ is an $\F_3\Gamma$-module such that
$C_{A_0}(\Gamma)=0$ and $[\Gamma,A_0]=A_0$, and the characters are
taken with respect to $\F_3$. Thus when $\Gamma\cong2M_{22}$, the
character values for the simple 10-dimensional
$\4\F_3\Gamma$-module are doubled here since it can only be
realized over $\F_9$. When $\Gamma\cong M_{11}$, the bounds for
$\scrj_{A_0}(\tau)$ apply only when $A_0$ is not the
$10$-dimensional permutation module. The bounds for
$\scrj_{A_0}(\tau)$ all follow from Lemma \ref{l:M12,M24} and Lemma
\ref{l:J(x)ge...}(c), except when $\Gamma\cong M_{11}$ or $2M_{12}$
where \ref{l:J(x)ge...}(d) is used.}
\label{tbl:JA(t)-3}
\end{Table}
Finally, if $\Gamma_0\cong M_{11}$ or $2M_{12}$ and $A$ is the Todd module or its dual, then $A$ is absolutely irreducible by \cite[Lemmas 4.2 and 5.2]{O-todd}, and hence $\Gamma\cong M_{11}$, $M_{11}\times C_2$, or $2M_{12}$. \end{proof}
\section{Alperin's 2-groups of normal rank 3} \label{s:Alp}
As an example of how the results in Section \ref{s:A<|F} can be applied when the abelian $p$-subgroup $A<S$ is not elementary abelian, we next look at some $2$-groups first studied by Alperin \cite{Alperin} and O'Nan \cite{ONan}. These are groups $A\nsg S$ where $A\cong C_{2^n}\times C_{2^n}\times C_{2^n}$ and $S/A\cong D_8$, with presentation given in Table \ref{tbl:D8onA}. They are characterized by Alperin \cite[Theorem 1]{Alperin} as the Sylow $2$-subgroups of groups $G$ with normal subgroup $E\cong E_8$, such that $O(G)=1$, $\Aut_G(E)=\Aut(E)$ and all involutions in $C_G(E)$ lie in $E$. Our goal is to show how results in
Section \ref{s:A<|F} can be applied to prove in the context of fusion systems a theorem of O'Nan's, by showing that $A$ is normal in all saturated fusion systems over $S$ \cite[Lemma 1.10]{ONan}.
\begin{Table}[ht]
\[ \renewcommand{1.5}{1.4}
\renewcommand{2mm}{4mm}
\begin{array}{c|cccc}
v & v^t & v^s & v^{s^2} & v^{st} \\\hline
v_1 & v_3^{-1} & v_2 & v_3 & v_2^{-1} \\
v_2 & v_2^{-1} & v_3 & v_1v_2^{-1}v_3 & v_1^{-1} \\
v_3 & v_1^{-1} & v_1v_2^{-1}v_3 & v_1 & v_1^{-1}v_2v_3^{-1}
\end{array}
\]
\caption{Let $S=A\gen{s,t}$, where $A=\gen{v_1,v_2,v_3}\cong
C_{2^n}\times C_{2^n}\times C_{2^n}$,
the elements $s$ and $t$ act on $A$ as described in the table,
and also $t^2=1$ and $s^4\in\gen{v_1v_3}$. Set
$T=\Aut_S(A)=\gen{c_s,c_t}\cong D_8$.}
\label{tbl:D8onA}
\end{Table}
Before considering the groups $A\nsg S$ directly, we must first handle the following, simpler case (compare with \cite[Lemma 1.7]{ONan}).
\begin{Lem} \label{l:ON} Fix $n\ge2$, and let $\5S=\gen{v,w,\sigma}$ be a group of order $2^{2n+2}$, where $\5A=\gen{v,w}\cong C_{2^n}\times C_{2^n}$, and $\5S=\5A\rtimes\gen{\sigma}$ where $\sigma^4=1$, $v^\sigma=w$, and $w^\sigma=v^{-1}$. Then $\5A$ is normal in every saturated fusion system over $\5S$. \end{Lem}
\begin{proof} Assume otherwise: assume $\calf$ is a saturated fusion system over $\5S$ for which $\5A\nnsg\calf$. Thus some element $t\in\5S\smallsetminus\5A$ is $\calf$-conjugate to an element of $\5A$, and upon replacing $t$ by $t^2$ if necessary, we can arrange that $t\in\sigma^2\5A$. Since
$|C_{\5A}(\sigma)|=2$ and $|C_{\5A}(\sigma^2)|=4$, each abelian subgroup of $\5S$ not contained in $\5A$ has order at most $8$, and hence $\5A$ is weakly closed in $\calf$.
By Proposition \ref{p:not.str.cl.2}(b,c) and since $\5A$ is weakly closed in $\calf$, there is $U\le \5S$ $\calf$-conjugate to a subgroup of $\5A$ such that $(t,U,U\cap \5A)\in\WW{\calf}{\5A}$. In particular,
$|U\5A/\5A|=|C_{\5A/(U\cap \5A)}(t)|$.
Since conjugation by $t$ sends each element of $\5A$ to its inverse, $U\cap \5A\le C_{\5A}(t)=\Omega_1(\5A)$, and hence $C_{\5A/(U\cap
\5A)}(t)=\Omega_1(\5A/(U\cap \5A))$ has order $4$. Thus $|U\5A/\5A|=4$, and so there is $u\in U$ such that $u\in\sigma \5A$.
We claim that for each $U^*\in U^\calf$, either $U^*\5A=\5S$ or $U^*\le \5A$. Assume otherwise: then $U^*\5A=\5A\gen{\sigma^2}$. So $U^*\cap \5A\le C_{\5A}(\sigma^2)=\Omega_1(\5A)$, and $U^*$ is elementary abelian since each element of $\sigma^2 \5A$ has order $2$. Since $U\cong U^*$ is not
elementary abelian (recall $|u|=4$), this is impossible.
By Theorem \ref{t:AFT} (Alperin's fusion theorem), there is a subgroup $R\le \5S$, together with an automorphism $\alpha\in\autf(R)$ and subgroups $A_1$ and $U_1=\alpha(A_1)$, such that $A_1,U_1\in U^\calf$, $A_1\le \5A$, and $U_1\nleq \5A$. We just saw that this implies $U_1\5A=\5S$. So $\5A\cap R$ contains a cyclic subgroup of order $4$ and is normalized by $\sigma$. Hence $R\ge\gen{v^{2^{n-1}},(vw)^{2^{n-2}}}$, and so $[R,R]\ge\Omega_1(\5A)$. Since $\alpha$ sends some element of $\Omega_1(\5A)$ to an element in the coset $\sigma^2\5A\nsubseteq[R,R]$, this is impossible. \end{proof}
Lemma \ref{l:ON} can also be proven using the transfer for $\calf$ (see, e.g., \cite[\S I.8]{AKO}) to show that no element $x^2$, for $x\in\sigma \5A$, can be in the focal subgroup of $\calf$. Such an argument would be closer to that used by O'Nan in the proof of \cite[Lemma 1.7]{ONan}, but we wanted to apply the tools used elsewhere in this paper.
We now return to the groups $A\nsg S$ defined by the presentation in Table \ref{tbl:D8onA}. We first check that when $n\ge2$, $A$ is weakly closed in every saturated fusion system over $S$:
\begin{Lem}[{\cite[Lemma 1.5]{ONan}}] \label{l:Alp-w.cl.} Let $S=A\gen{s,t}$ be an extension of the form described in Table \ref{tbl:D8onA}, where $n\ge2$. Then $A$ is the only abelian subgroup of index $8$ in $S$, and hence is weakly closed in every saturated fusion system over $S$. \end{Lem}
\begin{proof} This follows immediately from the centralizers listed in Table \ref{tbl:CA(H)}, since if $A_1<S$ were abelian of index $8$ and $A_1\ne A$, then for $x\in A_1\smallsetminus A$, the subgroup $C_A(x)\ge A\cap A_1$ would have index at most $4$ in $A$.
\begin{Table}[ht]
\[ \renewcommand{1.5}{1.4}
\begin{array}{c|cccccc}
H & \gen{t} & \gen{s^2} & \gen{st} & \gen{s} & \gen{s^2,t}
& \gen{s^2,st} \\\hline
C_A(H) & \gen{v_1v_3^{-1},v_2^\varepsilon} & \gen{v_1v_3,v_2^\varepsilon
v_3^\varepsilon} & \gen{v_1v_2^{-1},v_2^\varepsilon v_3^\varepsilon} & \gen{v_1v_3}
& \gen{v_1^\delta v_2^\varepsilon v_3^{-\delta}} & \gen{v_1^\varepsilon v_3^\varepsilon,
v_2^\varepsilon v_3^\varepsilon} \\
{}[H,A] & \gen{v_1v_3,v_2^2} & \gen{v_1v_3^{-1},v_1^2v_2^{-2}}
& \gen{v_1v_2,v_2^2v_3^{-2}} & \gen{v_1v_2^{-1},v_2v_3^{-1}} & &
\end{array}
\]
\caption{Centralizers and commutators involving some of the abelian
subgroups $H\le\gen{s,t}$. Here, $\varepsilon=2^{n-1}$ and
$\delta=2^{n-2}$.}
\label{tbl:CA(H)}
\end{Table} \end{proof}
The arguments used in the proof of the following theorem are essentially the same as O'Nan's (when proving Lemma 1.10 in \cite{ONan}), but repackaged with the help of Proposition \ref{p:not.str.cl.3a} and the properties of the sets $\RR{T}{A}$.
\begin{Thm}[{\cite[Lemma 1.10]{ONan}}] \label{t:Alp} Let $S=A\gen{s,t}$ be an extension of the form described in Table \ref{tbl:D8onA}, where $n\ge3$. Then $A$ is normal in every saturated fusion system $\calf$ over $S$. \end{Thm}
\begin{proof} Assume otherwise: assume $\calf$ is such that $A\nnsg\calf$. By Proposition \ref{p:not.str.cl.3a}(c) and since $A$ is weakly closed in $\calf$ by Lemma \ref{l:Alp-w.cl.}, there is a subgroup $Z\le A$ fully centralized in $\calf$ such that $A\nnsg C_\calf(Z)$, and such that for each $u\in\VV{C_\calf(Z)}{A}$ there is $U\in\UU{C_\calf(Z)}{A}$ such that $U\cap A\le Z$ and $(c_u^A,\Aut_U(A),U\cap A)\in\RR{T}{A}$. Set
$\tau=c_u^A$; we can assume that $|\tau|=2$. Set $B=\Aut_U(A)$ and $A_*=U\cap A$.
By Table \ref{tbl:CA(H)}, we have $|C_A(\tau)\cap[\tau,A]|=4$. So
$|B|\ge4$ by inequality \eqref{e:B2} in Proposition \ref{p:not.str.cl.3b}, with equality since $T\cong D_8$ has no abelian subgroups of order $8$. Hence
\beqq C_A(B)[\tau,A] \ge A_*[\tau,A] = C_A(\tau)[\tau,A],
\label{e:Alp1} \eeqq where the equality follows from \eqref{e:B1} in Proposition \ref{p:not.str.cl.3b}.
Since $|B|=4$, we have $c_{s^2}\in B$. So we can choose $u\in s^2A$ with $u\in\VV{C_\calf(Z)}{A}$ (thus $C_\calf(Z)$-conjugate to an element of $A$), and hence $\tau=c_u^A=c_{s^2}$. By Table \ref{tbl:CA(H)},
\[ [\tau,A]=\gen{v_1v_3^{-1},v_1^2v_2^{-2}}
\qquad\textup{and}\qquad
C_A(\tau)[\tau,A]=\gen{v_1v_3,v_1^2,v_2^2}. \] So by Table \ref{tbl:CA(H)}, point \eqref{e:Alp1} fails when $B=\gen{s^2,t}$ or $\gen{s^2,st}$, and holds only when $B=\gen{s}$ and $A_*=C_A(s)=\gen{v_1v_3}$. Since $A_*\le Z\le C_A(B)$ by assumption, we have $Z=\gen{v_1v_3}$.
Set $\5\calf=C_\calf(Z)/Z$, $\5A=A/Z$, and $\5S=C_S(Z)/Z$ (see Definition \ref{d:F/Q}). Then $A\nnsg C_\calf(Z)$ by assumption, hence is not strongly closed by Lemma \ref{l:s.cl.=>normal}, and so $A/Z$ is not strongly closed in $C_\calf(Z)/Z$. Thus $\5A\nnsg\5\calf$. Let $v,w,\sigma\in \5S$ be the classes (modulo $Z$) of $v_1,v_2,s\in S$. Then $\5A\nsg\5S$ are as in Lemma \ref{l:ON}, so $\5A\nsg\5\calf$ by that lemma, giving a contradiction. \end{proof}
\appendix
\section{Some lemmas in representation theory} \label{s:JV(x)}
Recall Notation \ref{d:JV(x)}: when $V$ is an elementary abelian $p$-group and $\tau\in\Aut(V)$ has order $p$, we set
\[ \scrj_V(\tau)=\rk\bigl(C_V(\tau)\cap[\tau,V]\bigr): \] the number of nontrivial Jordan blocks under the action of $\tau$ on $V$. We derive here some formulas that give lower bounds for these functions in terms of Brauer characters.
The first lemma gives, in certain cases, lower bounds for $\scrj_V(x)$ in terms of the modular character of $V$. When $q$ is a prime and $q\nmid n$, we let $\ord_q(n)$ denote the order of $n$ in the group $\F_q^\times$.
\begin{Lem} \label{l:J(x)ge...} Fix a prime $p$, an elementary abelian $p$-group $V$, and an element $x\in\Aut(V)$ of order $p$. Let $\chi=\chi_V$ be the modular character of $V$ as an $\F_p\Aut(V)$-module. \begin{enuma} \item Assume $p=2$, and let $q$ be an odd prime such that $\ord_q(2)=q-1$.
Let $a\in\Aut(V)$ be such that $|a|=q$ and $\gen{a,x}\cong D_{2q}$. Then
\[ \scrj_V(x) \ge
\tfrac{q-1}{2q}\bigl(\chi_V(1)-\chi_V(a)\bigr). \]
\item Let $q$ be a prime such that $p\mid(q-1)$, and let $a\in\Aut(V)$ be
such that $|a|=q$ and $\gen{a,x}$ is nonabelian of order $pq$. Then
\[ \scrj_V(x) \ge \frac1{pq}\sum_{i=1}^{q-1}
\bigl(\chi_V(1)-\chi_V(a^i)\bigr). \]
\item Assume $p=3$, and let $a\in\Aut(V)$ be such that $\gen{a,x}\cong
A_4$ and $|a|=2$. Then
\[ \scrj_V(x) \ge \tfrac14\bigl(\chi_V(1)-\chi_V(a)\bigr). \]
\item Assume $p=3$, and let $a\in\Aut(V)$ be such that
$\gen{a,x}\cong2A_4$ and $|a|=4$. Then
\[ \scrj_V(x) \ge \tfrac14\bigl(\chi_V(1)-\chi_V(a)\bigr). \] \end{enuma} \end{Lem}
\begin{proof} \noindent\textbf{(b) } Since $\gen{a,x}$ is
nonabelian of order $pq$, where $p\mid(q-1)$ and $|a|=q$, we have
\[ \dim(V/C_V(a)) = \chi_V(1) - \frac1q\sum_{i=0}^{q-1}\chi_V(a^i)
= \frac1q\sum_{i=1}^{q-1}(\chi_V(1)-\chi_V(a^i)). \] The action of $x$ on $\4\F_p\otimes_{\F_p}(V/C_V(a))$ permutes freely the eigenspaces for $a$, corresponding to the primitive $q$-th roots of unity in $\4\F_p$. So all Jordan blocks for this action have length $p$, and the same holds for Jordan blocks for the action of $x$ on $V/C_V(a)$. So $\scrj_V(x)\ge\scrj_{V/C_V(a)}(x)=\frac1p\dim(V/C_V(a))$.
\noindent\textbf{(a) } Since $|a|=q$ and $\ord_q(2)=q-1$, we have $\chi_V(a^i)=\chi_V(a)$ for all $i$ prime to $q$. So this is a special case of (b).
\noindent\textbf{(c) } Let $b\in\gen{a,x}\cong A_4$ be such that $\gen{a,b}\cong E_4$. Since $a$, $b$, and $ab$ are permuted cyclically by $x$, they all have the same character. Hence each of the three nontrivial irreducible characters for $\gen{a,b}\cong E_4$ appears with multiplicity
\[ n = \tfrac13\dim(V/C_V(\gen{a,b})) = \tfrac13\bigl( \chi_V(1) -
\tfrac14(\chi_V(1)+3\chi_V(a)) \bigr) = \tfrac14(\chi_V(1)-\chi_V(a))
. \] Since $x$ permutes those three characters cyclically, we have $\scrj_V(x)\ge n$.
\noindent\textbf{(d) } Set $H=\gen{a,x}\cong2A_4$ where $|a|=4$, and set $z=a^2\in Z(H)$. Then $V=V_+\oplus V_-$ as $\F_3H$-modules, where $V_\pm$ are the eigenspaces for the action of $z$, and it suffices to prove the claim when $V=V_+$ or $V=V_-$. The case $V=V_+$ was shown in (c).
Now assume $V=V_-$, and set $m=\dim(V)=\chi_V(1)$ and $H_0=O_2(H)\cong Q_8$. Let $W$ be the (unique) irreducible $2$-dimensional $\F_3H_0$-module.
Then $V|_{H_0}\cong W^{m/2}$, and $\Hom_{\F_3H_0}(W,V)\cong\F_3^{m/2}$ since $\End_{\F_3H_0}(W)\cong\F_3$. So there are $\frac12(3^{m/2}-1)$
submodules of $V|_{H_0}$ isomorphic to $W$, they are permuted by $\gen{x}\cong C_3$, and hence there is at least one $2$-dimensional $\F_3H$-submodule $W_1\le V$. By applying the same argument to $V/W_1$ and then iterating, we get a sequence $0=W_0<W_1<\cdots<W_k=V$ of $\F_3H$-submodules such that $\dim(W_i/W_{i-1})=2$ for each $1\le i\le k$. Then $\dim(C_{W_i/W_{i-1}}(x))=1$ for each $i$, so $\dim(C_{V}(x))\le m/2$, and $\dim([x,V])\ge m/2$. Each nontrivial Jordan block in $V$ has dimension $2$ or $3$, and intersects with $[x,V]$ with dimension $1$ or $2$, respectively. Thus
\[ \scrj_{V}(x) \ge \tfrac12\dim([x,V]) \ge \tfrac14m =
\tfrac14\chi_V(1) = \tfrac14(\chi_V(1)-\chi_V(a)), \] the last equality since $\chi_V(a)=0$ (recall $a^2=z$ acts on $V$ via $-\Id$). \end{proof}
The next lemma is needed to handle $\F_p\Gamma$-modules in certain cases where $O_p(\Gamma)\ne1$.
\begin{Lem} \label{l:pG-rep} Fix a prime $p$, a finite group $G$ such that $O^{p}(G)=G$, and a subgroup $1\ne Z\le Z(G)$ of $p$-power order. Set $\4G=G/Z$. Let $V$ be a faithful indecomposable $\F_pG$-module. Then either \begin{enuma} \item among the composition factors of $V$, there are at least two simple $\F_pG$-modules with nontrivial action of $G$; or
\item there are submodules $0\ne V_0<V_1<V$ such that $G$ acts trivially on $V_0$ and on $V/V_1$, the $\F_p\4G$-module $V_1/V_0$ is simple, and $V_1$ and $V/V_0$ have trivial $Z$-action and are indecomposable $\F_p\4G$-modules.
\end{enuma} Furthermore, in the situation of (b), for each $g\in G\smallsetminus Z$, we have $\rk([h,V_1/V_0])=\rk([h,V])$ for at most one element $h\in gZ$.
Thus if $p=2$ and $|g|=2$, there is $h\in gZ$ of order $2$ such that $\scrj_V(h)>\scrj_{V_1/V_0}(h)$. \end{Lem}
\begin{proof} Assume (a) does not hold. Thus all but one of the composition factors in $V$ have trivial $G$-action, and there are $\F_pG$-submodules $V_0<V_1\le V$ such that $V_1/V_0$ is simple (hence $Z$ acts trivially) and all composition factors of $V_0$ and of $V/V_1$ are trivial. Since $G=O^p(G)$ is generated by $p'$-elements, it acts trivially on $V_0$ and on $V/V_1$.
Let $W\le V_1$ be the submodule generated by the $[g,V_1]$ for all $p'$-elements $g\in G$. For each such $g$, $[g,V_1]\cap V_0\le[g,V_1]\cap C_{V_1}(g)=0$ since $g$ acts trivially on $V_0$, so projection onto $V_1/V_0$ sends $[g,V_1]$ injectively, and $Z$ acts trivially on $[g,V_1]$ since it acts trivially on $V_1/V_0$. Thus $[Z,W]=0$, and $V_1=W+V_0$ since $V_1/V_0$ is simple and $W\nleq V_0$. So $Z$ acts trivially on $V_1$.
By a similar argument, $Z$ acts trivially on the dual $(V/V_0)^*$, and hence acts trivially on $V/V_0$. Since $Z$ acts nontrivially on $V$, we have $V_1<V$ and $V_0\ne0$.
Assume $V_1$ is not indecomposable. Thus $V_1=W_0\oplus W_1$, where $W_0$ and $W_1$ are nontrivial $\F_p\4G$-submodules of $V_1$ and $W_0\le V_0$. The action of $G$ on $V/W_1$ is trivial (an extension of $W_0$ by $V/V_1$), so $[G,V]\le W_1$, and $W_0$ splits off as a direct summand of $V$, contradicting the assumption that $V$ be indecomposable. Thus $V_1$ is indecomposable as an $\F_p\4G$-module, and a similar argument involving the dual module $V^*$ shows that $V/V_0$ is also indecomposable, finishing the proof of (b).
Now fix $g\in G\smallsetminus Z$, and assume that $h_1,h_2\in gZ$ are distinct elements such that $\rk([h_i,V])=\rk([h_i,V_1/V_0])$ for $i=1,2$. Set $z=h_1^{-1}h_2\in Z^\#$. Since $G$ acts faithfully on $V$ by assumption, there is some $a_0\in V$ such that $[z,a_0]\ne0$. By (b), we have $a_0\notin V_1$ and $[z,a_0]\in V_0$.
Set $h=h_1$ for short, so that $h_2=zh$. Then $[h,V_1/V_0]=[hz,V_1/V_0]$, so $\rk([h,V])=\rk([h,V_1/V_0])=\rk([hz,V])$, and hence $[h,V]=[h,V_1]=[hz,V]$ and $[h,V_1]\cap V_0=0$. In particular, $[h,a_0]$ and $[hz,a_0]$ are both in $[h,V_1]$. Also,
\[ [hz,a_0] = z(h(a_0)-a_0)+(z(a_0)-a_0) = z([h,a_0])+[z,a_0], \] so $0\ne[z,a_0]\in[h,V_1]\cap V_0$, a contradiction.
The last statement now follows since if $p=2$ and $|h|=2$, then $\scrj_V(h)=\rk([h,V])$ and $\scrj_{V_1/V_0}(h)=\rk([h,V_1/V_0])$. \end{proof}
The following example shows one way to construct examples of modules of the type described in Lemma \ref{l:pG-rep}(b).
\begin{Ex} \label{ex:pG-ext} Fix a prime $p$, a finite group $G$ such that $O^{p}(G)=G$, and a subgroup $1\ne Z\le Z(G)$ of $p$-power order. Choose $k\ge1$ such that $Z$ has exponent at most $p^k$. Let $H<G$ be such that no nontrivial normal subgroup of $G$ is contained in $H$. Set $\5V=\Z/p^k(G/H)$: the free $\Z/p^k$-module with basis the set $G/H$ of left cosets. Regard $\5V$ as a left $\Z/p^kG$-module, set $V_2=C_Z(\5V)$, and let $V\le\5V$ be such that $V/V_2=C_{\5V/V_2}(G)$. Set $V_0=C_V(G)=C_{V_2}(G)$ and $V_1=[G,V_2]V_0$. Then $V$ is a $\Z/p^kG$-module on which $G$ acts faithfully. Also, $G$ acts trivially on $V_0$ and on $V/V_1$, and $Z$ acts trivially on $V_1$ and on $V/V_0$.
If, furthermore, $V_1<V_2$ (equivalently, if $p\mathrel{\big|}|G/HZ|$), then there is a $\Z/p^kG$-submodule $V'<V$ such that $V'>V_1$, $G$ acts faithfully on $V'$, and $V'/V_1\cong V/V_2$. \end{Ex}
\begin{proof} Set
\[ \sigma_G = \sum_{gH\in G/H}gH \in C_{\5V}(G)=V_0
\qquad\textup{and}\qquad
\sigma_Z= \sum_{z\in Z}zH \in C_{\5V}(Z)=V_2. \] Note that $Z\cap H=1$ since it is normal in $G$ and contained in $H$.
Since no nontrivial normal subgroup of $G$ is contained in $H$, the group $G$ acts faithfully on $\5V$ and $G/Z$ acts faithfully on $V_2$. So $G$ acts faithfully on $V$ if $Z$ does.
Fix an element $1\ne z\in Z$; we will show that $[z,V]\ne0$. Let $Z_0<Z$ and $x\in Z\smallsetminus Z_0$ be such that $Z=Z_0\times\gen{x}$ and $z\notin
Z_0$, and set $p^\ell=|x|$ (thus $\ell\le k$). Choose $\lambda\in\Z/p^k$ of order $p^\ell$, let $g_1,\dots,g_m\in G$ be representatives for the left cosets of $HZ$ in $G$, and set
\[ v = \sum_{i=1}^m\,\sum_{t\in Z_0}\,\sum_{s=0}^{p^\ell-1}
s\lambda \cdot(tx^sg_iH) \in \5V. \] Let $z_0\in Z_0$ and $0<r<p^\ell$ be such that $z=z_0x^r$. Then
\[ zv = \sum_{i=1}^m\,\sum_{t\in Z_0}\,\sum_{s=0}^{p^\ell-1}
s\lambda \cdot(tz_0x^{s+r}g_iH) = v - r\lambda\cdot \sigma_G, \] and $[z,v]\ne0$ since $r\lambda\ne0$.
For each $g\in G$, let $z_1,\dots,z_m\in Z_0$ and $r_1,\dots,r_m\in\Z$ be such that for each $i$, $gg_iH=z_jx^{r_j}g_jH$ for some $j$. Then
\[ gv = \sum_{i=1}^m\,\sum_{t\in Z_0}\,\sum_{s=0}^{p^\ell-1}
s\lambda \cdot(tx^sgg_iH)
= \sum_{j=1}^m\,\sum_{t\in Z_0}\,\sum_{s=0}^{p^\ell-1}
s\lambda \cdot(tz_jx^{s+r_j}g_jH) = v - \sum_{j=1}^m
r_j\lambda\cdot g_j\sigma_Z, \] and so $[g,v]\in C_{\5V}(Z)=V_2$. Thus $v\in V$, finishing the proof that $Z$ acts faithfully on $V$.
Since $[Z,[G,V]]=1$ by definition and $[Z,G]=1$, we have $[G,[Z,V]]=1$ by the three-subgroup lemma (see \cite[Theorem 2.2.3]{Gorenstein}). Hence $[Z,V]\le V_0$, so $Z$ acts trivially on $V/V_0$.
If $V_1<V_2$, then $G$ acts trivially on $V_2/V_1$ and on $V/V_2$, and hence acts trivially on $V/V_1$ (recall $G$ is generated by $p'$-elements). So $V/V_1=(V_2/V_1)\times(V'/V_1)$ for some $\Z/p^kG$-submodule $V'<V$ containing $V_1$ with $V'/V_1\cong V/V_2$. Also, $Z$ acts faithfully on $V'$ since it acts faithfully on $V=V'+V_2$ and trivially on $V_2$, so $G$ acts faithfully on $V'$ since $G/Z$ acts faithfully on $[G,V_2]\le V_1=V'\cap V_2$. \end{proof}
For example, when $p=2$, $G=2M_{12}$, $Z=Z(G)\cong C_2$, and $H\cong M_{11}$, then by Example \ref{ex:pG-ext}, there is a $12$-dimensional faithful $\F_2G$-module $V$ with submodules $V_0<V_1<V$, where $\dim(V_0)=1$, $\dim(V_1)=11$, $Z$ acts trivially on $V_1$ and on $V/V_0$, and where $V_1$ has index two in the $12$-dimensional permutation module for $G/Z\cong M_{12}$.
There are much more general ways to construct faithful $\Z/p^kG$-modules $V$ with $V_0<V_1<V$ as in Lemma \ref{l:pG-rep}, starting with a given $\Z/p^k\4G$-module $V_1$ ($\4G=G/Z$). But the ones we have found all seem to require certain conditions on $H^2(\4G;V_1)$ to hold.
We end the section with the following, more technical lemma needed in
Section \ref{s:A<|F}.
\begin{Lem} \label{l:C(A/A0)G} Let $A$ be a finite abelian group, and fix $\alpha\in\Aut(A)$. Let $A_0\le A$ be such that $\alpha(A_0)=A_0$. Then
$|C_{A/A_0}(\alpha)|\le|C_A(\alpha)|$. \end{Lem}
\begin{proof} Set $G=\gen{\alpha}\le\Aut(A)$. The short exact sequence $0\to A_0\longrightarrow A\longrightarrow A/A_0\to0$ induces an exact sequence in cohomology
\[ 0 \Right2{} C_{A_0}(G) \Right3{} C_A(G) \Right3{} C_{A/A_0}(G)
\Right3{} H^1(G;A_0) \Right3{} \dots, \] and hence
\[ |C_A(G)| \ge |C_{A/A_0}(G)|\cdot
|C_{A_0}(G)| \big/ |H^1(G;A_0)|. \] Since $G=\gen{\alpha}$ and $A_0$ is finite, we have
$|H^1(G;A_0)|=|H^2(G;A_0)|$ where $H^2(G;A_0)$ is a quotient group of $C_{A_0}(G)$ (see \cite[Theorem 6.2.2]{Weibel}). So
$|C_A(G)|\ge|C_{A/A_0}(G)|$. \end{proof}
\section{The Golay modules for \texorpdfstring{$M_{22}$ and $M_{23}$} {M22 and M23}} \label{s:Todd-F2}
We now apply results in Section 2 to prove that the Golay modules (i.e., dual Todd modules) for $M_{22}$ and $M_{23}$ are not fusion realizable in the sense of Definition \ref{d:realize}. We do this by showing that $\RR[+]{T}{A}=\emptyset$ (see Definition \ref{d:K&R}) whenever $T\in\syl2{M_n}$ ($n=22$ or $23$) and $A$ is the Golay module of $M_n$.
We first set up our notation for handling these groups and modules. The notation used here for doing that is based mostly on that used by Griess \cite[Chapter 4--5]{Griess}.
For a finite set $I$ and a field $K$, let $K^I$ be the vector space of maps $I\longrightarrow K$, with canonical basis
$\{e_i\,|\,i\in I\}$. Let
\[ \Perm_I(K) \le \Mon^*_I(K) \le \Aut^*(K^I) \] be the groups of permutation automorphisms, semilinear monomial automorphisms, and all semilinear automorphisms, respectively (i.e., linear
with respect to some field automorphism of $K$). Thus if $|I|=n$, then $\Perm_I(K)\cong\Sigma_n$ and $\Mon^*_I(K)\cong (K^\times)^n\rtimes(\Sigma_n\times\Aut(K))$. Let
\[ \pi = \pi_{I,K} \colon \Mon^*_I(K) \Right4{} \Perm_I(K) \] be the canonical projection that sends a monomial automorphism to the corresponding permutation automorphism; thus $\Ker(\pi_{I,K})$ is the group of semilinear automorphisms that send each $Ke_i$ to itself.
More concretely, set
\[ I=\{1,2,3,4,5,6\} \qquad\textup{and}\qquad \Omega=\F_4\times I.
\] Thus $\F_2^\Omega$ and $\F_4^I$ are the vector spaces of functions $\Omega\longrightarrow\F_2$ and $I\longrightarrow\F_4$, respectively. We also identify $\F_4^I$ with the space of 6-tuples in $\F_4$. Fix $\omega\in\F_4\smallsetminus\F_2$, and let $(x\mapsto\4x)$ be the field automorphism of $\F_4$ of order $2$. Thus $\F_4=\{0,1,\omega,\4\omega\}$, and $\4x=x^2$ for $x\in\F_4$.
Let $\scrh\subseteq\F_4^I$ be the hexacode subgroup:
\beqq \scrh=\Gen{(\omega,\4\omega,\omega,\4\omega,\omega,\4\omega),
(\4\omega,\omega,\4\omega,\omega,\omega,\4\omega),
(\4\omega,\omega,\omega,\4\omega,\4\omega,\omega),
(\omega,\4\omega,\4\omega,\omega,\4\omega,\omega) }_{\F_4} .
\label{e:H-gens} \eeqq Thus $\scrh$ is a 3-dimensional $\F_4$-linear subspace of $\F_4^I$. When making computations, we will frequently refer to the following elements in $\scrh$:
\beqq h_1=(1,1,1,1,0,0), \qquad h_2=(1,1,0,0,1,1), \qquad
h_3 = (\omega,\4\omega,1,0,1,0). \label{e:h1-h3} \eeqq
\begin{Not} \label{not:Gamma-action} Let the group $\Gamma\defeq\F_4^I\rtimes\Mon_I^*(\F_4)$ act on $\Omega=\F_4\times I$ in the usual way: $\F_4^I$ acts via translation, $(\F_4^\times)^I$ acts via multiplication in each coordinate, $\Perm_I(\F_4)$ permutes the coordinates, and $\phi\in\Aut(\F_4)$ sends $(c,i)$ to $(\4c,i)$. This in turn induces an action on $\F_2^\Omega$, where $g\in\Gamma$ sends an element $e_{(c,i)}$ to $e_{g(c,i)}$. Equivalently, for $\xi\in\F_2^\Omega$ and $(c,i)\in\Omega$, define $g(\xi)$ by $(g(\xi))(c,i)=\xi(g^{-1}(c,i))$.
As special cases, $\trs\eta\in\Aut(\F_2^\Omega)$ will denote translation by $\eta\in\F_4^I$, and $\mbf{\tau}(\alpha)\in\Aut(\F_2^\Omega)$ will be the automorphism induced by $\alpha\in\Mon_I^*(\F_4)$. Thus
\[ \trs\eta(\xi)(c,i)=\xi(c-\eta(i),i) \qquad\textup{and}\qquad
\mbf{\tau}(\alpha)(\xi)(c,i)=\xi(\alpha^{-1}(c,i)). \] \end{Not}
Now set
\[ \Aut^*(\scrh) \defeq
\bigl\{\alpha\in\Mon^*_I(\F_4) \,\big|\,
\alpha(\scrh)=\scrh\bigr\}. \] By \cite[Proposition 4.5.ii]{Griess}, $\Aut^*(\scrh)\cong3\Sigma_6$. In other words, each permutation of $I$ is the image of some automorphism of $\scrh$, unique up to multiplication by $u\cdot\Id$ for some $u\in\F_4^\times$. More explicitly, $\Aut^*(\scrh)$ is generated by the subgroup
\[ \Aut^*_0(\scrh) = \Gen{(1\,2)(3\,4),(1\,2)(5\,6),(1\,3\,5)(2\,4\,6),
(1\,3)(2\,4), (1\,2)(3\,4)(5\,6)\phi} \cong \Sigma_4\times C_2, \] where $\phi$ is the field automorphism $\phi(x_1,\dots,x_6)=(\4{x_1},\dots,\4{x_6})$, together with the elements
\[ \omega\cdot\Id \qquad\textup{and}\qquad
\alpha=(1\,2\,3)\cdot\diag(1,1,1,1,\4\omega,\omega). \]
We refer to \cite[Definition 5.15]{Griess} for a definition of the Golay code $\scrg\le\F_2^\Omega$. Here, rather than repeat that definition, we give a set of generators. Define $\mathfrak{Gr}\colon\F_4^I\longrightarrow\F_2^\Omega$ by setting
\[ \mathfrak{Gr}(\xi) = \sum\nolimits_{i\in I} e_{(\xi(i),i)} \] (the ``graph'' of $\xi$). Define elements in $\F_2^\Omega$:
\[ C_i = \sum_{c\in\F_4} e_{(c,i)} \quad \textup{(for $i\in I$)}
\qquad\textup{and}\qquad
\grf{h} = \mathfrak{Gr}(h)+\mathfrak{Gr}(0) \quad \textup{(for $h\in\F_4^I$),} \] and also $C_{ij}=C_i+C_j$ for distinct $i,j\in I$ and $C_{1234}=C_{12}+C_{34}$. Then $C_i+\mathfrak{Gr}(0)$ and $\grf{h}$ are in $\scrg$ for all $i\in I$ and all $h\in\scrh$. From the ``standard basis'' for $\scrg$ given in \cite[5.35]{Griess}, we see that
\[ \scrg = \Gen{C_i+\mathfrak{Gr}(h) \,\big|\, i\in I,~ h\in\scrh} =
\Gen{C_i+\mathfrak{Gr}(0),\grf{h}, \,\big|\, i\in I,~ h\in\scrh} . \] This is a 12-dimensional subspace of $\F_2^\Omega$, with basis consisting of the six elements $C_i+\mathfrak{Gr}(0)$ for $i\in I$, together with six elements $\grf{h}$ for $h$ in any given $\F_2$-basis of $\scrh$. By \cite[Theorem 5.8]{Griess}, the weight of each element in $\scrg$ is $0$, $8$, $12$, $16$, or $24$.
Define $\Mat24$ to be the group of permutations of $\Omega$ that preserve $\scrg$, and set $\dTodd24=\scrg/\gen{e_\Omega}$: its Golay module. Also, define
\[ \Delta_1=\{(0,6)\} \qquad\textup{and}\qquad
\Delta_2=\{(0,6),(1,6)\}, \] and for $i=1,2$ set
\[ \Mat{24}{-i} = C_{\Mat24}(\Delta_i) \qquad\textup{and}\qquad
\dTodd{24}{-i} = \bigl\{ \xi\in\scrg \,\big|\,
\supp(\xi)\cap\Delta_i=\emptyset \bigr\}. \] Thus $\dim(\dTodd24)=\dim(\dTodd23)=11$, while $\dim(\dTodd22)=10$.
Define permutations $\mbf{\tau}_{ij},\trs{h}\in\Sigma_\Omega$ for $i\ne j$ in $I$ and $h\in\F_4^I$ by letting $\mbf{\tau}_{ij}$ exchange the $i$-th and $j$-th columns and letting $\trs{h}$ be translation by $h$. More precisely,
\[ \mbf{\tau}_{ij}(c,k) = (c,\sigma(k))~ \textup{where}~
\sigma=(i\,j)\in\Sigma_6
\qquad\textup{and}\qquad
\trs{h}(c,i)=(c+h(i),i). \] Then $\trs{h}\in\Mat24$ for all $h\in\scrh$. By the above description of $\Aut^*_0(\scrh)\le\Aut^*(\scrh)$, the elements $\mbf{\tau}_{12}\mbf{\tau}_{34}$, $\mbf{\tau}_{12}\mbf{\tau}_{56}$, and $\mbf{\tau}_{13}\mbf{\tau}_{24}$ all lie in $\Mat24$.
\begin{Not} \label{n:M22-23} Fix $n=22$ or $23$. Set $\Gamma=\Mat{}n$, and define subgroups
\begin{align*}
T &= \Gen{\trsh1, \trsh[\omega]1, \trsh3, \trsh[\omega]3,
\mbf{\tau}_{12}\mbf{\tau}_{34}, \mbf{\tau}_{13}\mbf{\tau}_{24},
\mbf{\tau}_{12}\phi} \in \syl2\Gamma \\
H_1 &= \gen{\trsh1,\trsh[\omega]1,\mbf{\tau}_{12}\mbf{\tau}_{34},
\mbf{\tau}_{13}\mbf{\tau}_{24}} \\
H_2 &= \gen{\trsh1,\trsh[\omega]1, \trsh3, \trsh[\omega]3 } .
\end{align*} \end{Not}
In the next lemma, we list the basic properties of these subgroups that will be needed.
\begin{Lem} \label{l:hexad.grp} Assume Notation \ref{n:M22-23}, with $n=22$ or $23$. Then $H_1$ and $H_2$ are the only subgroups of $T$ isomorphic to $E_{16}$. If we set $A=\dTodd{}n$, then
\begin{align*}
[\trsh1,A]&=\gen{C_{12},C_{13},C_{14},\grfh1}\cong E_{16}, \\
C_A(H_1)&=C_A(T)=\gen{C_{1234}}, \\
C_A(H_2)&=\gen{C_{12},C_{13},C_{14},C_{15}}\cong E_{16}.
\end{align*} \end{Lem}
\begin{proof} The first statement is well known and easily checked. Note, for example, that $T/H_1\cong D_8$, and that $C_{H_1}(x)$ has rank $2$ for $x\in T\smallsetminus H_1$. So if $E_{16}\cong H\le T$ and $H\ne H_1$, then $HH_1=H_1\gen{\trsh3,\trsh[\omega]3}$ or $H_1\gen{\trsh3,\mbf{\tau}_{12}\phi}$, and from this one easily reduces to the case $H=H_2$. (Note that all elements of order $2$ in $H_1H_2$ lie in $H_1\cup H_2$.)
The statements about commutators and centralizers follow from Tables \ref{tbl:[x,a]} and \ref{tbl:[ch1,x]}. \end{proof}
\begin{Table}[ht] \[ \renewcommand{1.5}{1.2} \addtolength{2mm}{1mm}
\begin{array}{r|ccccccc} x & C_{1234} & C_{12} & C_{13} & C_{15} & \grfh1 & \grfh2+C_{56} & \grf{h_3+\omega h_2}+C_{56} \\\hline [\trsh1,x] & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ {}[\trsh[\omega]1,x] & 0 & 0 & 0 & 0 & C_{1234} & C_{12} & C_{23} \\ {}[\trsh3,x] & 0 & 0 & 0 & 0 & C_{12} & C_{12} & C_{25} \\ {}[\trsh[\omega]3,x] & 0 & 0 & 0 & 0 & C_{13} & C_{15} & C_{35} \\ {}[\mbf{\tau}_{12}\mbf{\tau}_{34},x] & 0 & 0 & C_{1234} & C_{12} & 0 & 0 & \grfh1 \\ {}[\mbf{\tau}_{13}\mbf{\tau}_{24},x] & 0 & C_{1234} & 0 & C_{13} & 0 & \grfh1 & \grfh1 \\ {}[\mbf{\tau}_{12}\phi,x] & 0 & 0 & C_{12} & C_{12} & 0 & 0 & \grfh2+C_{56} \end{array} \] \caption{Table of commutators $[g,x]=g(x)-x$ for $g\in T$ and $x\in\dTodd23$. The first six elements in the top row form a basis for $C_{\dTodd22}(\trsh1)$, and together with the seventh they form a basis for $C_{\dTodd23}(\trsh1)$.} \label{tbl:[x,a]} \end{Table}
\begin{Table}[ht] \[ \renewcommand{1.5}{1.5}
\begin{array}{r|cccc} x & \grfh[\omega]1 & \grfh3 & \grfh[\omega]3 & \mathfrak{Gr}(\omega h_2)+C_1 \\\hline [\trsh1,x] & C_{1234} & C_{12} & C_{13} & \grfh1+C_{12} \end{array} \] \caption{The classes of these four elements $x$ form a basis for $\dTodd{}n/C_{\dTodd{}n}(\trsh1)$.} \label{tbl:[ch1,x]} \end{Table}
\iffalse We are now ready to look at the sets $\RR[+]{T}{A}$ (see Definition \ref{d:K&R}), when $\Gamma\cong M_{22}$ or $M_{23}$ and $A$ is its Golay module.
\begin{Prop} \label{p:dTodd} Assume Notation \ref{n:M22-23}, with $n=22$ or $23$, $\Gamma=\Mat{}n$, and $T\in\syl2\Gamma$. Let $A$ be the Golay module for $\Gamma$. Then $\RR[+]{T}{A}=\emptyset$. \end{Prop}
\begin{proof} Assume the proposition is not true, and fix a triple $(\tau,B,A_*)\in\RR[+]{T}{A}$. Thus $\tau\in T$ has order $2$, $B\le T$ is an elementary abelian $2$-subgroup, and $A_*\le
C_A(\gen{B,\tau})$ is such that $|B|\ge|C_{A/A_*}(\tau)|$. By Proposition \ref{p:not.str.cl.3b} and since $[\tau,A]\le C_A(\tau)$, we have
\beqq
|B| \ge \bigl|[\tau,A]\bigr| \cdot \bigl[C_A(\tau) : A_*[\tau,A] \bigr]
\ge |[\tau,A]|. \label{e:Td*1} \eeqq
Since $\Gamma\cong M_{22}$ or $M_{23}$ has only one conjugacy class of
involution, we have $|[\tau,A]|=|[\trsh1,A]|=2^4$ by Lemma
\ref{l:hexad.grp}. Thus $|B|\ge2^4$, with equality since $\rk_2(\Gamma)=4$. Since $H_1$ and $H_2$ are the only subgroups of $T\in\syl2{\Gamma_0}$ isomorphic to $E_{16}$ (Lemma \ref{l:hexad.grp}), $B$ must be equal to one of these subgroups. Also, $C_A(\tau)=A_*[\tau,A]$ by
\eqref{e:Td*1} and since $|B|=|[\tau,A]|$, so
\[ \rk(C_A(B))\ge \rk(A_*)\ge\rk(C_A(\tau)/[\tau,A]) \ge 2. \] By Lemma \ref{l:hexad.grp} again, $C_A(H_1)=\gen{C_{1234}}$ has rank $1$ and $C_A(H_2)=\gen{C_{12},C_{13},C_{14},C_{15}}$ has rank $4$, so $B=H_2$.
By condition ($*$) in Definition \ref{d:K&R}, each element of $B^\#$ can appear as the first component in an element of $\RR[+]{T}{A}$. So we can assume that $(\tau,B,A_*)$ was chosen such that $\tau=\trsh1$ (and still $B=H_2$). Hence by Tables \ref{tbl:[x,a]} and \ref{tbl:[ch1,x]},
\[ \grfh2+C_{56}\in C_A(\trsh1) = A_*[\trsh1,A]
\le C_A(H_2)[\trsh1,A] = \gen{C_{12},C_{13},C_{14},C_{15},\grfh1}, \] a contradiction. We conclude that $\RR[+]{T}{A}=\emptyset$. \end{proof} \fi
\section{The 6-dimensional module for \texorpdfstring{$3M_{22}$}{3M22}} \label{s:3M22}
\newcommand{\mathfrak{h}}{\mathfrak{h}} \renewcommand{\textbf{\underline3}}{\textbf{\underline3}}
We again fix an element $\omega\in\F_4\smallsetminus\F_2$, and let $(a\mapsto\4a)$ denote the field automorphism of $\F_4$. Thus $\F_4=\{0,1,\omega,\4\omega\}$. We also use the bar over matrices to denote the field automorphism applied to the entries; i.e., $\4{(a_{ij})}=(\4{a_{ij}})$. Let $\Tr\colon\F_4\longrightarrow\F_2$ be the trace: $\Tr(a)=a+\4a$.
Set $V=\F_4^3$ and $A=\F_4^6$, where elements of $V$ are written as column matrices $\colthree{a}bc$ for $a,b,c\in\F_4$, and elements of $A$ are written as column matrices $\coltwo{u}v$ for $u,v\in V$. Let $\gen{-,-}$ be the hermitian form on $A$ defined by
\[ \left\langle \Coltwo{u}v, \Coltwo{x}y \right\rangle
= \Tr(u^t\4y+v^t\4x). \]
The description here of the action of $\Gamma=3M_{22}$ on $A$ is based on that in \cite[Chapter 2]{bensonthesis} and in \cite[p. 39]{atlas}, originally due to Benson and others. An element denoted \fbox{$\begin{smallmatrix}r&s&t\\x&y&z\end{smallmatrix}$} in \cite{bensonthesis} or \texttt{(rx~sy~tz)} in \cite{atlas} is written here $\Coltwo{u}v$ where $u=\colthree{r}st$ and $v=\colthree{r+x}{s+y}{t+z}$.
For $i,k=1,2,3$ and $j=1,2$, define
\[ b_{ijk} = \begin{cases}
\omega^j & \textup{if $i=k$} \\
1 & \textup{if $i\ne k$,}
\end{cases} \qquad\textup{and}\qquad
b_{ij} = \Colthree{b_{ij1}}{b_{ij2}}{b_{ij3}}\in V; \]
and set $\scrb=\{\gen{b_{ij}}\,|\,i=1,2,3,~j=1,2\}$. The following lemma is easily checked.
\begin{Lem} \label{l:B&U} Consider the hermitian form $\mathfrak{h}\:V\times V\longrightarrow\F_4$ defined by $\mathfrak{h}(v,w)=\4v^tw$. Define elements $u_1,\dots,u_6\in V$ by setting
\[ u_1=\colthree100, \quad u_2=\colthree010,\quad
u_3=\colthree001,\quad u_4=\colthree111,\quad
u_5=\colthree1\omega{\4\omega},\quad
u_6=\colthree1{\4\omega}\omega,
\]
and set $\scru=\{\gen{u_i}\,|\,1\le i\le6\}$. Then the members of $\scru$ are the only $1$-dimensional subspaces of $V$ not orthogonal to any member of $\scrb$, and the members of $\scrb$ are the only $1$-dimensional subspaces of $V$ not orthogonal to any member of $\scru$. Hence for $D\in\GL_3(4)$, the action of $D$ on $V$ permutes the members of $\scru$ if and only if the action of $\4D^t$ on $V$ permutes the members of $\scrb$. \end{Lem}
Define matrices
\[ M_{10} = \mxthree000010001, \quad
M_{20} = \mxthree100000001, \quad
M_{01} = \mxthree011101110, \quad
M_{02} = \mxthree0\omega{\4\omega}{\4\omega}0\omega\omega{\4\omega}0,\] and set $M_{00}=0$, $M_{03}=M_{01}+M_{02}$, $M_{30}=M_{10}+M_{20}$, and $M_{ij}=M_{i0}+M_{0j}$ for $i,j=1,2,3$. In other words, if we set $\textbf{\underline3}=\{0,1,2,3\}$ and regard it as an elementary abelian $2$-group via bitwise sum, then $((i,j)\mapsto M_{ij})$ is a homomorphism from $\textbf{\underline3}\times\textbf{\underline3}$ to $M_3(\F_4)$.
Finally, set
\[ N_{ij}=I+M_{ij} \qquad ((i,j)\in\textbf{\underline3}\times\textbf{\underline3}). \] Note that
\beqq N_{i0}=\4{u_i}u_i^t \qquad\textup{and}\qquad
N_{0i}=\4{u_{i+3}}u_{i+3}^t \qquad\textup{for all $i=1,2,3$.}
\label{e:Ai0} \eeqq
\begin{Not} \label{n:3M22} Define maximal isotropic subspaces $X_{ij}\le A$ (for $i,j=0,1,2,3$) and $Y_{ij}\le A$ (for $i=1,2,3$ and $j=1,2$) as follows:
\[ X_{ij} = \left\{ \left. \Coltwo{N_{ij}v}v
\,\right|\,
v\in V \right\} \quad\textup{and}\quad
Y_{ij} = \left\{\left. \Coltwo{u}{b_{ij}\4{b_{ij}}^tu} \,\right|\, u\in
V \right\} . \]
Set $\scrx=\{X_{ij}\,|\,i,j=0,1,2,3\}$ and
$\scry=\{Y_{ij}\,|\,i=1,2,3,~j=1,2\}$. Let $\Gamma\le\Aut(A)$ be the group of unitary automorphisms of $A$ that permute the members of $\scrx\cup\scry$. \end{Not}
The members of $\scrx\cup\scry$ are all totally isotropic since the matrices $N_{ij}$ and $b_{ij}\4{b_{ij}}^t$ are hermitian for all $i,j$. Following \cite{bensonthesis} and \cite{atlas}, we arrange them diagrammatically as follows:
\beqq \renewcommand{1.5}{1.5}
\renewcommand{2mm}{2mm}
\begin{array}{|cc|cc|cc|} \hline
& & X_{00} & X_{01} & X_{02} & X_{03} \\
Y_{12} & Y_{11} & X_{10} & X_{11} & X_{12} & X_{13} \\\hline
Y_{22} & Y_{21} & X_{20} & X_{21} & X_{22} & X_{23} \\
Y_{32} & Y_{31} & X_{30} & X_{31} & X_{32} & X_{33} \\\hline
\end{array} \label{e:MOG} \eeqq
\begin{Not} \label{n:3M22b} For $M\in M_3(\F_4)$ and $D\in\GL_3(\F_4)$, define $\varphi_M,\psi_D\in\Aut(A)$ by setting
\[ \varphi_M\left(\Coltwo{u}v\right) = \Mxtwo{I}C0I\Coltwo{u}v
\qquad\textup{and}\qquad
\psi_D\left(\Coltwo{u}v\right) = \Mxtwo{D}00{\4D^{-t}}\Coltwo{u}v
\] where $(-)^{-t}$ means transpose inverse. Set
\[ D_0 = \mxthree100001010, \qquad D_1 = \mxthree100101110,\qquad
D_2 = \mxthree011101001,\qquad D_3 =
\mxthree1000{\omega}000{\4\omega}; \] and set
\beq \mu_{ij}=\varphi_{M_{ij}} \qquad\textup{and}\qquad
\delta_i=\psi_{D_i} \tag{for $i,j=0,1,2,3$}. \eeq Also, define the following subgroups of $\Aut_{\F_4}(A)$ (in fact, of $\Gamma$):
\begin{align*}
H &= N_\Gamma(\scry) = N_\Gamma(\scrx)
& P_1 &= \{\mu_{ij}\,|\,i,j=0,1,2,3\} \\
H_0 &= C_\Gamma(\scry)
& P_2 &= \gen{\mu_{10},\mu_{01},\delta_0,\delta_1} \\
\Gamma_0 &= C_\Gamma(\scrx\cup\scry)
& T &= P_1P_2\gen{\delta_2} = P_1\gen{\delta_0,\delta_1,\delta_2} .
\end{align*} \end{Not}
Note that $\varphi_M$ is unitary whenever $\4M^t=M$, and $\psi_D$ is unitary for all $D\in\GL_3(4)$. In particular, the $\mu_{ij}$ and the $\delta_i$ are all unitary.
Most of the information about $\Gamma$ and its action on $A$ in the following lemma is well known and implicit in Chapter 2 of \cite{bensonthesis}, but we try here to make more explicit some of the details in the proofs.
\begin{Lem} \label{l:3M22c}
Set $A_0=\left\{\left.\coltwo{w}0\,\right|\,w\in V\right\}$. Set $\Delta=\gen{D_0,D_1,D_2,D_3}\le\GL_3(4)$, and set $\psi_\Delta=\gen{\delta_0,\delta_1,\delta_2,\delta_3}
=\{\psi_D\,|\,D\in\Delta\}\le\Aut(A)$. Then \begin{enuma}
\item $\Gamma\cong3M_{22}$ and $T\in\syl2\Gamma$;
\item $\Delta\cong\psi_\Delta\cong3A_6$;
\item $H_0=P_1\times\Gamma_0$ where
$P_1=\bigl\{\varphi\in\Gamma\,\big|\,\varphi|_{A_0}=\Id\bigr\} \cong E_{16}$ and $\Gamma_0=\gen{\omega\cdot\Id_A}$; and
\item $H=\{\varphi\in\Gamma\,|\,\varphi(A_0)=A_0\} = P_1\psi_\Delta$.
\end{enuma} \end{Lem}
\begin{proof} For each $i=1,2,3$ and $j=1,2$,
\beqq Y_{ij}\cap A_0 = \{\coltwo{u}0 \,|\, u\in V,~ \4{b_{ij}}^tu=0
\} = \{\coltwo{u}0 \,|\, u\in b_{ij}^\perp \} \label{e:Yij.A0}
\eeqq in the notation of Lemma \ref{l:B&U}. Thus $\dim_{\F_4}(Y\cap A_0)=2$ for $Y\in\scry$, and distinct members of $\scry$ have distinct intersections with $A_0$. So for each pair $Y\ne Y'$ in $\scry$, we have $Y\cap Y'\le A_0$ where $\dim(Y\cap Y')=1$, and the set of all such intersections generates $A_0$.
Thus each $\varphi\in H$ sends $A_0$ to itself. If $\varphi\in H_0$, then $\varphi$ sends each of the 1-dimensional subspaces $Y\cap Y'$ to itself (for $Y\ne Y'$ in $\scry$), and hence
$\varphi|_{A_0}\in\gen{\omega\cdot\Id_{A_0}}$.
By definition, $X\cap A_0=0$ for each $X\in\scrx$. So if $\varphi\in\Gamma$ is such that $\varphi(A_0)=A_0$, then $\varphi$ permutes the members of
$\scrx$ and those of $\scry$, and hence lies in $H$. If $\varphi|_{A_0}\in \gen{\omega\cdot\Id_{A_0}}$, then since the intersections $Y\cap A_0$ for $Y\in\scry$ are all distinct, $\varphi$ sends each member of $\scry$ to itself and hence lies in $H_0$. To summarize, we have now shown that
\beqq H = \bigl\{\varphi\in\Gamma\,\big|\,\varphi(A_0)=A_0 \bigr\}
\qquad\textup{and}\qquad
H_0 = \bigl\{\varphi\in\Gamma\,\big|\,
\varphi|_{A_0}\in\gen{\omega\cdot\Id_{A_0}} \bigr\}. \label{e:H,H0}
\eeqq
\noindent\textbf{(b) } Each of the matrices $D_i$ for $i=0,1,2,3$ permutes the
members of $\scru=\{u_i\,|\,1\le i\le6\}$, and does so via the permutations
\beqq D_0:~(2\,3)(5\,6), \qquad D_1:~(1\,4)(2\,3), \qquad
D_2:~(1\,2)(3\,4), \qquad D_3:~(4\,5\,6). \label{e:Di-perm} \eeqq These generate the group of all even permutations of the set $\scru$. In particular, there is a matrix $D_4\in\Delta$ that induces the permutation $(1\,2\,3)$, and by considering its action on the $u_i$ for $1\le i\le4$, we see that $D_4=\mxthree00rr000r0$ for some $r\in\F_4^\times$.
We claim that
\beqq \Delta = \{D\in\GL_3(4) \,|\, D(\scru)=\scru \}.
\label{e:Delta(U)} \eeqq To see this, assume $D\in\GL_3(4)$ permutes the $\gen{u_i}$. Since all even permutations of $\scru$ are realized by elements in $\Delta$, there is $D'\equiv D$ (mod $\Delta$) that sends each of the subspaces $\gen{u_1}$, $\gen{u_2}$, $\gen{u_3}$, $\gen{u_4}$ to itself. But then $D'$ must have the form $s\cdot I$ for $s\in\F_4^\times$. Since
\[ \mxthree{\omega}000\omega000\omega =
\left[ \mxthree1000{\omega}000{\4\omega},\mxthree00rr000r0\right]
\in [\Delta,\Delta], \] this proves \eqref{e:Delta(U)}, and also shows that $\Delta\cong3A_6$.
The isomorphism $\psi_\Delta\cong\Delta$ follows directly from the definitions.
\noindent\textbf{(c) } We first check, for each $i,j=0,1,2,3$, $k=1,2,3$, and $\ell=1,2$, that $\mu_{ij}(Y_{k\ell})=Y_{k\ell}$. This means showing, for $u\in V$, that
\[ b_{k\ell}\4{b_{k\ell}}^tu =
b_{k\ell}\4{b_{k\ell}}^t(u+M_{ij}b_{k\ell}\4{b_{k\ell}}^tu); \] i.e., that $\4{b_{k\ell}}^tM_{ij}b_{k\ell}=0$. It suffices to do this when $ij=0$ and $(i,j)\ne(0,0)$. In all such cases, by \eqref{e:Ai0}, there is $\gen{c_{ij}}\in\scru$ such that $c_{ij}\4{c_{ij}}^t=I+M_{ij}$. So it suffices to show that
\[ (\4{b_{k\ell}}^tc_{ij})\cdot\4{(\4{b_{k\ell}}^tc_{ij})} =
\4{b_{k\ell}}^tb_{k\ell} = 1; \] equivalently, that $b_{k\ell}\not\perp c_{ij}$ --- which follows from Lemma \ref{l:B&U}.
For the same automorphism $\mu_{ij}$ with matrix $\mxtwo{I}{M_{ij}}0I$, an element $\coltwo{N_{k\ell}u}u\in X_{k\ell}$ is sent to $\coltwo{N_{k\ell}u+M_{ij}u}u$. Since $N_{k\ell}+M_{ij}=N_{k+i,\ell+j}$ where sums of indices are taken bitwise, this shows that $\mu_{ij}(X_{k\ell})=X_{k+i,\ell+j}$. So $\mu_{ij}$ permutes the members of $\scrx$, finishing the proof that $\mu_{ij}\in H_0\le\Gamma$.
Conversely, for each $\varphi\in\Gamma$ such that $\varphi|_{A_0}=\Id$, $\varphi$ induces the identity on $A/A_0$ since it is unitary and $A_0$ is a maximal isotropic subgroup, so $\varphi$ has matrix $\mxtwo{I}M0I$ for some $M\in M_3(\F_4)$. Thus $\varphi=\varphi_M$ (see Notation \ref{n:3M22b}). Let $(i,j)$ be such that $\varphi(X_{00})=X_{ij}$; then $N_{00}+M=I+M=N_{ij}$, so $M=M_{ij}$, and $\varphi=\mu_{ij}\in P_1$. We now conclude that
\[ P_1 = \bigl\{\varphi\in\Gamma\,\big|\, \varphi|_{A_0}=\Id
\bigr\}. \]
By \eqref{e:H,H0}, $\varphi\in H_0$ implies that
$\varphi|_{A_0}\in\gen{\omega\cdot\Id_{A_0}}$, and hence that $\varphi\in P_1\times\gen{\omega\cdot\Id_A}$. Thus $H_0\le P_1\times\gen{\omega\cdot\Id_A}$, and we already proved the opposite inclusion. Also, $\gen{\omega\cdot\Id_A}\le\Gamma_0\le H_0$, and $\Gamma_0\cap P_1=1$ since $P_1$ acts faithfully on $\scrx$. So $\Gamma_0=\gen{\omega\cdot\Id_A}$.
\noindent\textbf{(d) } Fix $D\in\Delta$; we will show that $\psi_D\in H$. Let $\rho_D\:M_3(\F_4)\longrightarrow M_3(\F_4)$ be the homomorphism $\rho_D(M)=DM\4D^t$. Since $D$ permutes the members of $\scru$ by \eqref{e:Di-perm}, $\rho_D$ permutes the set
\[ \{u\4u^t\,|\,\gen{u}\in\scru\} =
\{N_{10},N_{20},N_{30},N_{01},N_{02},N_{03}\} \] (see \eqref{e:Ai0}). This, together with the relations $N_{ij}+N_{k\ell}+N_{mn}=N_{i+k+m,j+\ell+n}$ (where indices are added bitwise) shows that $\rho_D$ permutes the set of all $N_{ij}$ for $i,j=0,1,2,3$. (Note, for example, that $N_{00}=N_{10}+N_{20}+N_{30}$.) If $i,j,k,\ell$ are such that $\rho_D(N_{ij})=N_{k\ell}$, then
\[ \psi_D(X_{ij})
= \left\{ \left. \Coltwo{DN_{ij}u}{\4D^{-t}u}
\,\right|\, u\in V \right\}
= \left\{ \left. \Coltwo{DN_{ij}\4D^tv}v \,\right|\, v\in V
\right\} = X_{k\ell}, \] and thus $\psi_D$ permutes the members of $\scrx$.
By Lemma \ref{l:B&U} and since $D$ permutes the members of $\scru$, the matrix $\4D^t$ permutes the members of $\scrb$. So for each $i,j$ there is $k,\ell$ such that $\4D^tb_{k\ell}\in\gen{b_{ij}}$, and hence
\begin{multline*}
\psi_D(Y_{ij})
= \left\{ \left. \Coltwo{Du}{\4D^{-t}b_{ij}\4{b_{ij}}^tu}
\,\right|\, u\in V \right\}
= \left\{ \left. \Coltwo{v}{\4D^{-t}b_{ij}\4{b_{ij}}^tD^{-1}v}
\,\right|\, v\in V \right\} \\[1mm]
= \left\{ \left. \Coltwo{v}{b_{k\ell}\4{b_{k\ell}}^tv}
\,\right|\, v\in V \right\} = Y_{k\ell}.
\end{multline*} Thus $\psi_D$ also permutes the members of $\scry$, and it follows that $\psi_D\in H$.
Conversely, for each $\eta\in H$, $\eta(A_0)=A_0$ by \eqref{e:H,H0}, and
$\eta|_{A_0}$ permutes the subspaces $Y\cap A_0$ for $Y\in\scry$. So $\eta$ has matrix of the form $\mxtwo{D}X0{\4D^t}$, where $D$ permutes the subspaces $b^\perp\le V$ for all $\gen{b}\in\scrb$ by \eqref{e:Yij.A0}, and hence permutes the members of $\scru$ by Lemma \ref{l:B&U}. So by \eqref{e:Delta(U)}, there is $\delta\in\psi_\Delta$ such that
$\eta|_{A_0}=\delta|_{A_0}$. Then $\delta^{-1}\eta\in P_1$ by (c), and $\eta\in P_1\psi_\Delta$. This finishes the proof that $H=P_1\psi_\Delta$.
\noindent\textbf{(a) } Set $\Gamma^*=\Gamma/\Gamma_0$, regarded as a group of permutations of the set $\scrx\cup\scry$. By \cite[Theorem 2.3]{bensonthesis}, $\Gamma^*$ acts 3-transitively on the set $\scrx\cup\scry$. It is well known (see, e.g., \cite[p. 235]{Pogorelov}) that the only finite groups that act $2$-transitively on a set of order $22$ are $M_{22}$, $A_{22}$, and their automorphism groups. So once we have
shown that $T\in\syl2\Gamma$ and $|T|=2^7$, it will then follow that $\Gamma^*\cong M_{22}$, and that $\Gamma$ is a central extension of $\Gamma_0\cong C_3$ by $\Gamma^*$.
Recall that $T=P_1\gen{\delta_0,\delta_1,\delta_2}$, where by \eqref{e:Di-perm}, the action of the $\delta_i$ on $\scry$ generates a subgroup of $\Sigma_6$ isomorphic to $D_8$. Hence $T/P_1\cong D_8$, and
$|T|=2^7$. Alternatively, one can describe $T$ by looking at the subgroup of $\Aut(A_0)$ generated by restrictions of its elements.
Under the action of $\Gamma^*$, the stabilizer of a subspace $X\in\scrx\cup\scry$ acts $\F_4$-linearly on $X$. If $\varphi\in\Gamma$ is
such that $\varphi|_X=\Id_X$, then $\varphi$ sends each member of $\scrx\cup\scry$ to itself since their intersections with $X$ are distinct, and hence $\varphi\in\Gamma_0$. The point stabilizer for the action of $\Gamma^*$ on $\scrx\cup\scry$ is thus isomorphic to a subgroup of $\PGL_3(4)$, and hence the order of $\Gamma^*$ divides
$22\cdot|\PGL_3(4)|=2^7\cdot3^3\cdot5\cdot7\cdot11=3\cdot|M_{22}|$. So $T\in\syl2\Gamma$ and $\Gamma^*\cong M_{22}$. Finally, $\Gamma$ is the nonsplit central extension of $\Gamma_0\cong C_3$ by $\Gamma^*$ since it contains $\psi_D\cong3A_6$ by (b,d). \end{proof}
Thus $P_1=O_2(H)$, where $H\cong E_{16}\rtimes3A_6$ is a hexad subgroup of $\Gamma\cong3M_{22}$. One can also show that $P_2=O_2(K)$ where $K=N_\Gamma(\{Y_{11},Y_{12}\})$ is a duad subgroup of $\Gamma$. Equivalently, $K\cong C_3\times(E_{16}\rtimes\Sigma_5)$ is the group of elements of $\Gamma$ that permute the five $2\times2$ blocks in diagram \eqref{e:MOG}; i.e., send the four members of each such block to those in another block.
The next lemma collects some technical properties of the action of $\Gamma$ on $A$.
\begin{Lem} \label{l:3M22b} Let $\{e_1,e_2,\dots,e_6\}$ be the canonical basis for $A=\F_4^6$. Then for $P_1,P_2,T\le\Gamma$ and $\mu_{10}\in P_1\cap P_2$ as defined in Notation \ref{n:3M22b}, \begin{enuma} \item $P_1$ and $P_2$ are the only subgroups of $T$ isomorphic to $E_{16}$; and \item $C_A(\mu_{10})=\gen{e_1,e_2,e_3,e_4}$, $[\mu_{10},A]=\gen{e_2,e_3}$, $C_A(P_1)=\gen{e_1,e_2,e_3}$, $C_A(P_2)=\gen{e_2+e_3}$. \end{enuma} \end{Lem}
\begin{proof} For point (a), see Lemma \ref{l:hexad.grp}. Point (b) follows easily from the above descriptions of the actions. \end{proof}
\iffalse We are now ready to look at the sets $\RR[+]{T}{V}$ for this action.
\begin{Prop} \label{p:3M22} Assume $p=2$ and $\Gamma=3M_{22}$, and let $V\cong\F_4^6$ be the $6$-dimensional $\F_4\Gamma$-module. Then $\RR[+]{T}{V}=\emptyset$ for each $T\in\syl2\Gamma$. \end{Prop}
\begin{proof} We can assume $T<\Gamma$ and $V$ are as in Notation \ref{n:3M22} and \ref{n:3M22b}. Assume $\RR[+]{T}{V}\ne\emptyset$, and fix an element $(\tau,B,A_*)\in\RR[+]{T}{V}$. Thus $\tau\in T$ has order $2$, $B\le T$ is an elementary abelian $2$-group, and $A_*\le C_{V}(B\gen{\tau})$ is such that $\rk(B)\ge\rk([\tau,V])$. By Proposition \ref{p:not.str.cl.3b} and since $[\tau,V]\le C_V(\tau)$,
\beqq |B| \ge |[\tau,V]|\cdot \bigl[C_{V}(\tau) : A_*[\tau,V] \bigr]
\ge |[\tau,V]|. \label{e:3M22c} \eeqq Since $\Gamma$ has only one conjugacy class of involutions, $\tau$ is $\Gamma$-conjugate to $\mu_{10}$, and hence $\rk([\tau,V])=2\cdot\dim_{\F_4}([\mu_{10},V])=4$ (Lemma \ref{l:3M22b}(b)). Since $\rk(B)\le\rk_2(\Gamma)=4$, we have $\rk(B)=4$ and $C_{V}(\tau)=A_*[\tau,V]$ by \eqref{e:3M22c}. So
\[ \rk(C_V(B)) \ge \rk(A_*)\ge \rk(C_V(\tau))-\rk([\tau,V]) =
\rk(C_V(\mu_{10}))-\rk([\mu_{10},V]) = 4. \] where the last equality holds by Lemma \ref{l:3M22b}(b).
By Lemma \ref{l:3M22b}(a), $P_1$ and $P_2$ are the only subgroups of $T$ isomorphic to $E_{16}$. Since $\rk(C_V(P_2))=2$, we have $B=P_1$. By condition ($*$) in Definition \ref{d:K&R}, we can assume that the triple $(\tau,P_1,A_*)$ was chosen so that $\tau=\mu_{10}$. But then
\[ \gen{e_1,e_2,e_3,e_4} = C_V(\mu_{10}) =
A_*[\mu_{10},V]\le C_V(P_1)[\mu_{10},V]
= \gen{e_1,e_2,e_3} \] by Lemma \ref{l:3M22b}(b), a contradiction. \end{proof} \fi
\end{document} | arXiv |
Do electrons collapse into nucleus, if electrons in the atom are constantly excited?
From the Bohr's atomic model, it is clear that electron can have only certain definite energy levels. When the electron is present as close to the nucleus as possible, the atom has the minimum possible energy and is said to be in the ground state. When energy from some outside source is supplied to it, it can absorb a definite amount of energy and jumps to higher energy state. Such a state of an atom in which the atom possesses more energy than possessed in the ground state is called excited state. These excited states are unstable and the electron tends to come back to lower energy level. This transition (change) from upper to lower energy level occurs with a jump and energy is emitted in the form of a quantum equal to difference in energies between the two levels.
I have a doubt here, if electron absorbs energy from the outside source and jumps to the higher energy state, it will be now storing absorbed energy, as potential energy at the excited state. If electron emits all the absorbed energy (quanta-difference in energies between the two levels). By virtue of what energy does electron come down? I mean, in common situations, we say that an object at any height comes down converting it's potential energy into kinetic energy. Here in case of electron, it has already emitted absorbed energy as quanta. So, is it that electron losses some energy other than the energy absorbed from the source, to come down to ground state. I thought, if it was a possibility, then electron would constantly need to lose energy, whenever excited, at last, it would collapse into the nucleus. But, this not we really observe. I think there might be some misunderstanding by me or there might be any of the existing model like quantum mechanical model, which could account for this. If any were the case, please explain.
quantum-mechanics energy atoms atomic-physics
Immortal Player
Immortal PlayerImmortal Player
$\begingroup$ When anything at a higher potential comes down to a lower potential, it either converts its energy into KE or releases this energy. That's what happens here; it reduces its potential by emitting photons of a certain wavelength, and hence comes down to a lower state. $\endgroup$ – mikhailcazi Dec 10 '13 at 12:04
$\begingroup$ Thank you for the comment. Suppose, I have a cup of hot coffee on the table. It will be continuously losing energy in the form of heat, but it stays on the table, though there was a energy loss. Now, all of a sudden, I take off the table, the cup of coffee converts it potential energy into kinetic energy to come down. Similarly, electron losses energy in the form of quanta, it doesn't mean it should come down, it must convert some of its energy into kinetic, to come down. This what I thought, if anything wrong, please explain. $\endgroup$ – Immortal Player Dec 10 '13 at 12:39
$\begingroup$ Related: physics.stackexchange.com/q/20003/2451 , physics.stackexchange.com/q/9415/2451 and links therein. $\endgroup$ – Qmechanic♦ Dec 10 '13 at 13:24
$\begingroup$ Let me write an answer. $\endgroup$ – mikhailcazi Dec 10 '13 at 13:29
Based on some of your comments, I think what might be tripping you up is the first statement you started with:
From the Bohr's atomic model, it is clear that electron can have only certain definite energy levels.
...If suppose, we assume electron losses total energy, electron can't stay in any particular shell, as it would not have that particular value of energy.
That may be true for Bohr's atomic model, but Bohr's atomic model is wrong. And electron does not have to be in a particular, definite shell or energy level. Rather, any electron state is a superposition of states of definite energy level (energy eigenstates).
That means the expectation value of a hydrogen electron state is going to look like $$\langle E\rangle = \sum_n |a_n|^2 E_n\text{,}$$ where $\{a_n\}$ are arbitrary complex values with $\sum_{n>0}|a_n|^2 = 1$ and $E_n$ are energy levels in increasing order. Because of the sum-to-$1$ condition, taking any portion along the other energy eigenstates will increase energy compared to the ground state.
In other words, even if the electron state does not have a definite energy, you still can't go lower than the ground state.
Suppose, I have a cup of hot coffee on the table. It will be continuously losing energy in the form of heat, but it stays on the table, though there was a energy loss. Now, all of a sudden, I take off the table, the cup of coffee converts it potential energy into kinetic energy to come down.
If you don't shake the table, the coffee cup will sit there, forever. Similarly, nothing perturbs the electron in an excited energy eigenstate, then it simply will never decay. It cannot: energy eigenstates are stationary; they do not evolve into anything other than themselves.
However, being completely without external perturbation is actually impossible. The uncertainty principle provides the electromagnetic field with vacuum fluctuations, which will perturb the electron even if nothing else in the environment does. In your analogy, this (or something else) provides the "shaking of the table" for the electron. Once the electron state gains even a tiny component in some other energy eigenstate, the state can evolve in time.
In other words, one can think of spontaneous emission as a particular type of stimulated emission where it's the vacuum that does the perturbing.
Stan LiouStan Liou
Here in case of electron, it has already emitted absorbed energy as quanta. So, is it that electron losses some energy other than the energy absorbed from the source, to come down to ground state. I thought, if it was a possibility, then electron would constantly need to lose energy, whenever excited, at last, it would collapse into the nucleus. But, this not we really observe. I think there might be some misunderstanding by me or there might be any of the existing model like quantum mechanical model, which could account for this. If any were the case, please explain.
As others said the Bohr model has been superseded by the quantum mechanical formulation of particle interactions in the microcosm. There are definite energy levels that can be occupied by electrons and there is a ground state from which the electrons cannot fall lower generally.
Please study what these orbitals mean, which are the result of solving for the Schrodinger equation for the atom . They show the probability distribution in space for the electron around a nucleus:
The shapes of the first five atomic orbitals: 1s, 2s, 2px, 2py, and 2pz. The colors show the wave function phase. These are graphs of ψ(x, y, z) functions which depend on the coordinates of one electron. To see the elongated shape of ψ(x, y, z)2 functions that show probability density more directly, see the graphs of d-orbitals below.
Note that the angular momentum = 0 (S states ) have a probability for the electron to be at the center where the positive charge of the nucleus is. Because the nucleus is orders of magnitude smaller than the space covered by the orbital the probability of the electron being captured by the nucleus is infinitesimally small (together with other phase space considerations), but it happens for proton rich nuclei where the fields are strong. It is called electron capture. In these rare processes when an electron is captured it will emit a photon/gamma characteristic of the ground state it lost. There is no continuous loss because the system is quantized.
anna vanna v
I thought, if it was a possibility, then electron would constantly need to lose energy, whenever excited, at last, it would collapse into the nucleus.
You seem to have forgotten that when the electron is excited, it gets energy, which is then released when it emits it. So it wouldn't collapse because energy absorption and emission are balanced.
RuslanRuslan
Reading your comment in reply to mine, I understood what you wanted to ask.
This is where so many people are confused. Since we start middle school chemistry, we are taught about electron orbits which are like concentric circles. Everyone innately assumes that the 1st orbit is inside the 2nd orbit, which is inside the 3rd and so on.
It isn't like that!
Electron orbits are not comparable to the solar system. A higher energy shell doesn't mean one with a larger radius.
When an electron loses energy to come into a lower state, it doesn't mean it actually reduces its radius - just its energy state. Bohr's model was simply a theory, which explained a lot of occurrences, but once you start learning about the quantum mechanics of an atom, you'll see how we believe it to be right now.
If you have started learning about the s, p, d and f subshells, I can explain a bit further.
Most people who haven't pictured the atom correctly - thanks to the middle school analogy of the solar system - assume a lot of wrong stuff about the atom.
For instance, all the atomic orbitals exist centred at the nucleus. Many people picture it like this:
$$\text{This is WRONG.}\uparrow$$
Actually, every one of $p_x, p_y, p_z, s, d_{xy} $, etc are placed centred at the nucleus.
mikhailcazimikhailcazi
$\begingroup$ The answer looks reasonable, but looking at it one may not read the last sentence - but see the wrong picture. Could this maybe be marked as "(wrong)" near the picture? $\endgroup$ – Ruslan Dec 10 '13 at 14:59
$\begingroup$ Thank you for the answer. Consider the following situation, which would make more sense. Let a cup of hot coffee be displaced from one position to another position along the table using j joules of energy, similarly electron could be assumed to be displaced to higher energy shell due to absorption of energy. Assume that coffee loses j joules of energy in the form of heat to the atmosphere, but it remains at the same position, in the same way electron loses all the absorbed energy, but remains in the same shell. For the coffee cup to come back to it's old position, it needs to convert some... $\endgroup$ – Immortal Player Dec 10 '13 at 18:03
$\begingroup$ ....of its energy (other than lost energy) into kinetic to come back. In the same way, electron needs to convert some of it's energy into kinetic to return to the initial shell. Thus, electron needs to lose some of it's energy other than absorbed, to come back to initial shell. I understood, what you meant to say with respect previous analogy. If any such complexity, even with respect to this analogy, please mention. $\endgroup$ – Immortal Player Dec 10 '13 at 18:10
$\begingroup$ @Ruslan Thanks, I'll edit the answer to make the important points stand out. $\endgroup$ – mikhailcazi Dec 12 '13 at 7:33
$\begingroup$ @Vinay, talking about moving from one place to another is different than changing orbit sizes. How does anything orbit? If it has a speed just enough to let it overcome the attractive force, it will start orbiting. It's basically an orbit when it is moving so fast that it keeps evading the pull. The larger the speed, the bigger the orbit. I guess this works for electrons and the nucleus too, but the radius difference is negligible. $\endgroup$ – mikhailcazi Dec 13 '13 at 13:40
As a side note, remember the Pauli exclusion principle, that no two (fermionic) particles can share the same state. So two electrons cannot share the same exact state e.g. at the ground state there are only two electrons, each with different spin. Also, there is the uncertainty principle, that you cannot precisely the position or momentum of a particle (and the more you know of one of those factors, the less you know of the others), and that each particle has a wavelength that roughly guides the particle's "orbit" in the classical sense.
Even if an electron gives up energy (and the quantum rules require certain energy states in a bound system), the nuclear repulsive forces (not electromagnetic) would keep it from getting "near" to the nucleus itself. Take a look at the kind of speeds particles are accelerated to in CERN or LINAC to allow them to collide with a nucleus.
Carl WitthoftCarl Witthoft
$\begingroup$ If nucleus ${}^m_n\mathrm{X}$ has sufficiently more energy than ${}^m_{n-1}\mathrm{X}'$, then radioctive decay by electron capture can occur. So it doesn't seem to be accurate to blame lack of "getting near" on the presence of nuclear forces, since nuclear (weak) forces are responsible for electron capture, while the absence of such is the fault of energy conservation. $\endgroup$ – Stan Liou Dec 10 '13 at 12:50
$\begingroup$ @Carl Witthoft.Thank you for the answer, Sir. When schrodinger wave equation was solved for hydrogen atom, it gave expression for the energy of electron in any particular shell. If suppose, we assume electron losses total energy, electron can't stay in any particular shell, as it would not have that particular value of energy. Thus, it would be attracted more towards the nucleus because of the unlike charge nature of electron and proton. So, I thought electron would collapse into the nucleus. I am not an expert in quantum mechanics, please explain how CERN or LINAC deals with this concept. $\endgroup$ – Immortal Player Dec 10 '13 at 13:15
$\begingroup$ @StanLiou fair enough. I wasn't going that deep :-) $\endgroup$ – Carl Witthoft Dec 10 '13 at 13:56
$\begingroup$ "I am not an expert in quantum mechanics" lol, best physics quote ever! :P $\endgroup$ – shortstheory Dec 10 '13 at 14:02
$\begingroup$ @shortstheory or "How Can You Be in Two Places at Once When You're Not Anywhere at All" $\endgroup$ – Carl Witthoft Dec 10 '13 at 14:56
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Why don't electrons crash into the nuclei they "orbit"?
Why do electrons occupy the space around nuclei, and not collide with them?
How can the nucleus of an atom be in an excited state?
classical understanding of an atom
Some questions regarding the behaviour of electrons
Does the energy of ground and/or excited states have uncertainty?
How does an electron manage to constantly revolve around the nucleus of an atom forever?
Can an excited atom have multiple electrons in excited states?
Can a light element with excited nucleus undergo internal conversion
What force keeps electrons in their orbitals and not collapse into the positively charged nucleus?
Excited states in Bohr's model of an atom
Do electrons in an atom gain kinetic energy when a photon hits it? | CommonCrawl |
Subsets and Splits